Maximal Subellipticity 9783111085173, 9783111085647, 9783111085944

Table of contents :
Contents
1 Introduction
2 Ellipticity
3 Vector fields and Carnot–Carathéodory geometry
4 Pseudo-differential operators
5 Singular integrals
6 Besov and Triebel–Lizorkin spaces
7 Zygmund–Hölder spaces
8 Linear maximally subelliptic operators
9 Nonlinear maximally subelliptic equations
A Canonical coordinates
Bibliography
Symbol Index
Index

Citation preview

Brian Street Maximal Subellipticity

De Gruyter Studies in Mathematics

�

Edited by Carsten Carstensen, Berlin, Germany Gavril Farkas, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Waco, Texas, USA Niels Jacob, Swansea, United Kingdom Zenghu Li, Beijing, China Guozhen Lu, Storrs, USA Karl-Hermann Neeb, Erlangen, Germany René L. Schilling, Dresden, Germany Volkmar Welker, Marburg, Germany

Volume 93

Brian Street

Maximal Subellipticity �

Mathematics Subject Classification 2020 Primary: 35-02, 35H20, 35B65; Secondary: 35G05, 35G20, 35J70, 35R03, 42B20, 42B25, 42B35, 42B37, 53C17 Author Prof. Brian Street University of Wisconsin – Madison Department of Mathematics 480 Lincoln Drive Madison, WI 53706-1388 USA [email protected]

ISBN 978-3-11-108517-3 e-ISBN (PDF) 978-3-11-108564-7 e-ISBN (EPUB) 978-3-11-108594-4 ISSN 0179-0986 Library of Congress Control Number: 2023932713 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2023 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Contents 1

Introduction � 1 1.1 Basic definitions � 2 1.1.1 Linear maximally subelliptic operators � 5 1.1.2 Nonlinear maximally subelliptic operators � 9 1.2 Background: elliptic theory � 12 1.3 Carnot–Carathéodory geometry � 14 1.4 Singular integrals � 15 1.5 Function spaces � 17 1.6 Linear theory � 18 1.7 Fully nonlinear theory � 20 1.7.1 The inverse function theorem � 21 1.8 Multi-parameter theory � 24 1.9 The main tool: scaling � 27 1.9.1 Scaling for elliptic operators � 28 1.9.2 Scaling for maximally subelliptic operators � 28 1.10 Outline � 29

2

Ellipticity � 31 2.1 Some basic function spaces � 32 2.2 Pseudo-differential operators and the Fourier transform � 35 2.2.1 Pseudo-differential operators without the Fourier transform � 37 2.2.2 Elliptic operators and pseudo-differential operators � 42 2.3 Singular integral operators � 47 2.3.1 Local, matrix-valued operators � 58 2.4 Besov and Triebel–Lizorkin spaces � 59 2.5 Zygmund–Hölder spaces � 70 2.5.1 Decomposition into smooth functions � 76 2.6 Compositions and product Zygmund–Hölder spaces � 78 2.7 Linear elliptic operators � 87 2.8 Nonlinear elliptic equations � 90 2.8.1 Reduction I � 93 2.8.2 Reduction II � 96 2.8.3 Completion of the proof � 98 2.8.4 Weighted estimates near the boundary � 106 2.9 Further reading and references � 111

3

Vector fields and Carnot–Carathéodory geometry � 114 3.1 Manifolds � 114 3.1.1 The exponential map � 116 3.1.2 The Baker–Campbell–Hausdorff formula � 118

VI � Contents

3.2 3.3 3.4 3.5

3.6 3.7 3.8 3.9 3.10 3.11

3.12 3.13 3.14 3.15

3.16 4

3.1.3 The Frobenius theorem � 119 The unit scale � 121 Scaling and Hörmander vector fields � 123 3.3.1 Scaling and maximal subellipticity � 127 Finitely generated setting � 129 Multi-parameter Carnot–Carathéodory geometry � 135 3.5.1 An important special case: Hörmander’s condition � 137 3.5.2 Dropping parameters � 138 The quantitative coordinate system � 139 Proofs of scaling results � 142 Approximately commuting vector fields � 148 Integrating over leaves � 155 Integrals and approximately commuting vector fields � 158 Maximal functions � 166 3.11.1 The finitely generated setting � 167 3.11.1.1 A result on the unit ball � 170 3.11.1.2 Proof of Theorem 3.11.2 � 172 3.11.2 The Hörmander setting � 174 Approximately commuting partial differential operators � 177 Filtered modules of vector fields � 186 Control of vector fields � 188 3.14.1 Further comments on equivalence � 190 The main multi-parameter setting � 194 3.15.1 The multi-parameter unit scale � 194 3.15.2 Scaling � 195 3.15.3 Quantitative scaling � 198 Further reading and references � 203

Pseudo-differential operators � 205 4.1 Symbols of pseudo-differential operators � 206 4.1.1 Connection with standard pseudo-differential operators � 208 4.1.2 Littlewood–Paley decomposition of symbols � 209 4.2 The exponential map � 215 4.3 Littlewood–Paley decompositions � 223 4.3.1 Lebesgue space bounds � 231 4.3.2 Adjoints � 234 4.3.3 Proof of the Littlewood–Paley decomposition � 236 4.4 Adding parameters � 238 4.5 The sub-Laplacian � 239 4.5.1 The sub-Laplacian is subelliptic � 240 4.5.2 Homogeneity � 250 4.5.3 Nilpotent Lie groups � 254

Contents

4.6

4.5.4 Free nilpotent Lie algebras � 256 4.5.5 The calculus � 257 4.5.6 The parametrix � 262 4.5.7 Limitations of the lifting procedure � 264 Further reading and references � 265

5

Singular integrals � 267 5.1 The three settings � 267 5.1.1 The single-parameter setting � 267 5.1.2 The multi-parameter Hörmander setting � 268 5.1.3 The general multi-parameter setting � 269 5.2 The algebras of singular integrals � 270 5.2.1 The single-parameter setting � 271 5.2.2 The multi-parameter Hörmander setting � 274 5.2.3 The general multi-parameter setting � 276 5.3 Notation � 279 5.4 Pre-elementary operators � 280 5.5 Elementary operators � 289 5.6 Pseudo-differential operators � 310 5.7 Equivalence of the definitions � 316 5.8 Basic properties � 329 5.8.1 Pseudo-locality � 334 5.8.2 Standard pseudo-differential operators � 338 5.9 An important subalgebra � 340 5.9.1 Standard pseudo-differential operators � 353 5.10 Lp bounds in the single-parameter case � 355 5.10.1 A uniform result � 358 5.10.2 Beyond Hörmander’s condition � 359 5.11 Parametrices � 369 5.11.1 Parametrices via heat equations � 370 5.11.2 Parametrices and other geometries � 378 5.12 Spectral multipliers � 387 5.13 Further reading and references � 389

6

Besov and Triebel–Lizorkin spaces � 393 6.1 Informal description of the norms � 394 6.2 The single-parameter spaces � 396 6.3 The multi-parameter spaces � 400 6.4 The main estimate � 402 6.5 Some basic properties � 408 6.5.1 Smooth functions are dense � 422 6.6 The single-parameter spaces, revisited � 425

� VII

VIII � Contents

6.7 6.8 6.9 6.10 6.11

6.12 6.13 6.14

6.6.1 The classical spaces � 427 6.6.2 Comparing single-parameter spaces � 431 6.6.3 Boundedness of operators in D � 439 Trading derivatives � 445 6.7.1 Sharpness � 448 Adding parameters � 458 Sobolev spaces � 470 Distributions of finite norm � 472 An explicit choice of norm � 475 6.11.1 The unit scale � 479 6.11.2 Scaling � 479 A spectral definition � 481 Further questions � 482 Further reading and references � 483

7

Zygmund–Hölder spaces � 486 7.1 The norm � 487 7.2 Decomposition into smooth functions � 490 7.3 Bounds of some sums � 492 7.4 Algebra � 494 7.5 Compositions � 496 7.5.1 Properties of the product Zygmund–Hölder spaces � 499 7.5.2 Proof of the composition theorem � 501 7.5.3 The classical product Zygmund–Hölder spaces � 513 7.6 Adding parameters � 514 7.7 Difference characterization � 515 7.7.1 Hölder spaces � 530

8

Linear maximally subelliptic operators � 534 8.1 The main result � 534 8.2 Further regularity properties � 537 8.2.1 Single-parameter function spaces � 537 8.2.2 Standard function spaces � 541 8.2.3 Multi-parameter function spaces � 546 8.3 A priori estimates � 551 8.3.1 Some preliminary estimates � 552 8.3.2 Subelliptic estimates � 554 8.3.3 Exponential estimates � 566 8.4 Heat equations � 579 8.4.1 Step I: the unit scale � 581 8.4.1.1 On diagonal bounds � 581 8.4.1.2 Off-diagonal bounds � 582

Contents �

8.4.2

8.5 8.6 8.7

8.8 8.9 8.10

Step II: a single point and scale � 583 8.4.2.1 On-diagonal bounds � 585 8.4.2.2 Off-diagonal bounds � 587 8.4.3 Step III: all scales � 588 Proof of the main result � 598 Vector bundles � 604 Quantitative regularity estimates � 608 8.7.1 The unit scale � 608 8.7.2 A small scale � 610 Representation theory and Rockland’s condition � 613 Positive definite forms � 614 Further reading and references � 616

9

Nonlinear maximally subelliptic equations � 619 9.1 Main qualitative results � 619 9.1.1 Single-parameter results � 620 9.1.1.1 Qualitative Schauder estimates � 622 9.1.2 Multi-parameter results � 624 9.1.2.1 Qualitative Schauder estimates � 627 9.2 Main quantitative result � 627 9.2.1 Vector fields and norms � 630 9.2.2 Bump functions � 632 9.2.3 Scaled estimates � 640 9.2.4 Step I: Perturbation of a linear operator � 646 9.2.5 Step II: A simpler form � 656 9.2.6 Scaled decompositions of Zygmund–Hölder functions � 660 9.2.7 Step III: Completion of the proof � 662 9.3 Weighted estimates near the boundary � 675 9.3.1 Weighted Schauder estimates near the boundary � 684 9.4 Examples � 685 9.4.1 Second-order equations � 685 9.4.2 The Monge–Ampère equation � 687 9.4.3 Higher-order Monge–Ampère equations � 689 9.4.4 Higher-order equations � 691 9.5 Further reading and references � 693

A

Canonical coordinates � 695 A.1 Basic notation � 695 A.2 The main results � 696 A.2.1 More on the assumptions � 698 A.2.2 Densities � 701 A.3 Qualitative consequences � 703

IX

X � Contents A.4

A.5

Proofs � 707 A.4.1 An ODE � 707 A.4.1.1 Derivation of the ODE � 707 A.4.1.2 Existence, uniqueness, and regularity � 710 A.4.2 An inverse function theorem � 716 A.4.3 Proof of the main result � 719 A.4.3.1 Linearly independent � 719 A.4.3.2 Linearly dependent � 724 A.4.4 Densities � 732 A.4.5 Proof of Theorem 3.6.5 � 738 Further reading and references � 738

Bibliography � 741 Symbol Index � 753 Index � 755

1 Introduction Elliptic partial differential equations (PDEs) hold a special place among PDEs: sharp results are known for very general linear and even fully nonlinear elliptic PDEs. Many of the classical techniques from harmonic analysis were first developed to study these sharp results; the study of elliptic PDEs leans heavily on the Fourier transform and Riemannian geometry. Starting with the work of Hörmander, Kohn, Folland, Stein, and Rothschild in the 1960s and 1970s, a far-reaching generalization of ellipticity was introduced, now known as maximal subellipticity or maximal hypoellipticity.1 Maximally subelliptic operators generalize elliptic operators and can be significantly more degenerate. In the intervening years, many authors have adapted results from elliptic PDEs to various special cases of maximally subelliptic PDEs. One of the central problems in the study of PDEs is to establish regularity theorems, for both linear and nonlinear PDEs. Let X s denote a family of spaces indexed by a regularity parameter s ∈ ℝ (for example, X s could denote the Lp Sobolev space of order s). A linear regularity theorem near a point x0 for a linear partial differential operator P is a result of the form s

t

P u ∈ X near x0 󳨐⇒ u ∈ X near x0 ,

(Linear)

for some s, t ∈ ℝ. Nonlinear regularity theorems are also of interest. Fix κ ∈ {1, 2, 3, . . .}. For a function u, we write {𝜕xα u}|α|≤κ for the vector whose components are given by 𝜕xα u, where |α| ≤ κ. For a given function F(x, ζ ), a nonlinear regularity theorem near a point x0 is a result of the form F(x, {𝜕xα u(x)}|α|≤κ ) ∈ X s near x0 󳨐⇒ u ∈ X t near x0 .

(Nonlinear)

Results like (Linear) and (Nonlinear) can be very difficult to establish. One often has to work with specific properties of the partial differential operators, and abstract general results are rare. In contrast to these difficulties, when considering elliptic PDEs, both (Linear) and (Nonlinear) are well understood and sharp results are known. One of the main goals of this text is to establish sharp results of the form (Linear) and (Nonlinear) for general maximally subelliptic PDEs, thereby generalizing the sharp results for general elliptic PDEs to a much larger class of PDEs. The development of this regularity theory will take us through many generalizations of classical notions from the area of elliptic PDEs. The Fourier transform is not a

1 The PDEs studied in this text have been called both maximally subelliptic and maximally hypoelliptic. We use the more descriptive, but less common, terminology maximally subelliptic; see Remark 1.6.2 for some comments on this choice. These operators are subelliptic (see (1.18) and Corollary 8.2.5) and therefore hypoelliptic (see Definition 2.2.37 and Proposition 2.2.38). https://doi.org/10.1515/9783111085647-001

2 � 1 Introduction decisive tool, but many ideas from elliptic theory have generalizations to the maximally subelliptic setting: – Riemannian geometry is replaced by a Carnot–Carathéodory (or sub-Riemannian) geometry, – Besov and Triebel–Lizorkin spaces have generalizations which are adapted to the Carnot–Carathéodory geometry, – pseudo-differential operators are replaced by singular integral operators adapted to the Carnot–Carathéodory geometry. While most of the main results of this text are new, they are based on previous work of many authors. At the end of most chapters, we include a section “Further reading and references,” which attempts to give a history of the ideas we use and also points the reader to references on related subjects which we do not directly cover. This text focuses on generalizing several aspects of elliptic theory to the maximally subelliptic setting. However, there are many parts of elliptic theory which we do not address; given almost any result from the vast elliptic theory, one can ask whether a generalization holds for maximally subelliptic equations. Usually, the right generalization is not immediately clear: one needs to find the right function spaces, operators, and geometry to use. One also usually needs a completely new proof. Nevertheless, it is often possible to find the right generalization (see Sections 8.10 and 9.5 for many examples from the literature). It seems likely that this will be a fruitful area of study for years to come. The author was partially supported by National Science Foundation grants DMS1764265 and DMS-2153069. The opinions, findings, and conclusions expressed are those of the author and do not necessarily reflect the views of the National Science Foundation.

1.1 Basic definitions Let M be a smooth manifold2 and let Vol be any smooth, strictly positive density on M (see Definition 3.1.5); the definitions that follow do not depend on the choice of Vol, which plays the role of “Lebesgue measure” on M. Let ℕ = {0, 1, 2, . . .} and ℕ+ = {1, 2, 3, . . .}. Definition 1.1.1. Let W = {W1 , . . . , Wr } be a finite collection of smooth vector fields on M. For m ∈ ℕ+ and x ∈ M, we say W satisfies Hörmander’s condition of order m at x if

2 All manifolds in this text are manifolds without boundary. Our main results are local, so M can usually be taken to be an open subset of ℝn . Nevertheless, it is often useful to work in the abstract setting of manifolds, since many proofs exploit changing coordinates in a quantitative way. See Section 3.3 for a first example of this and Appendix A for more details.

1.1 Basic definitions

�

3

the commutators of W1 , . . . , Wr up to order m span Tx M, that is, if W1 (x), W2 (x), . . . , Wr (x), . . . , [Wi , Wj ](x), . . . , [Wi , [Wj , Wk ]](x), . . . , . . . , commutators of order m evaluated at x

span Tx M. We say W satisfies Hörmander’s condition at x if ∃m ∈ ℕ+ such that W satisfies Hörmander’s condition of order m at x. We say W satisfies Hörmander’s condition of order m on M if ∀x ∈ M, W satisfies Hörmander’s condition of order m at x. We say W satisfies Hörmander’s condition on M if ∀x ∈ M, W satisfies Hörmander’s condition at x. Example 1.1.2. (i) With M = ℝ2 , let W1 = 𝜕x and W2 = x𝜕y . Note that span{W1 (0), W2 (0)} ≠ T0 ℝ2 . However, [W1 , W2 ] = 𝜕y , so span{W1 (0), W2 (0), [W1 , W2 ](0)} = T0 ℝ2 . Thus, {W1 , W2 } satisfies Hörmander’s condition of order 2 at 0. In fact, {W1 , W2 } satisfies Hörmander’s condition of order 2 on ℝ2 . (ii) With M = ℝ2 , let W1 = 𝜕x and W2 = x 2 𝜕y . In this case [W1 , [W1 , W2 ]] = 2𝜕y , and W = {W1 , W2 } satisfies Hörmander’s condition of order 3 on ℝ2 . More generally, {𝜕x , x m 𝜕y } satisfies Hörmander’s condition of order m + 1 on ℝ2 . (iii) On ℝ3 , with coordinates (x, y, t), let W1 = 𝜕x + 2y𝜕t and W2 = 𝜕y − 2x𝜕t . In this case W1 (ζ ) and W2 (ζ ) do not span Tζ ℝ3 for any ζ . However, [W1 , W2 ] = −4𝜕t , and therefore W1 (ζ ), W2 (ζ ), [W1 , W2 ](ζ ) span Tζ ℝ3 for every ζ ∈ ℝ3 . We conclude W = {W1 , W2 } satisfies Hörmander’s condition of order 2 on ℝ3 . In this case, W1 and W2 are left invariant vector fields on the Heisenberg group; see [216, Chapters XII and XIII] for more information on the Heisenberg group. The Heisenberg group is a simple example of a graded nilpotent Lie group, and such groups play an important role in the theory of maximal subellipticity (see Sections 4.5.3 and 8.8). (iv) On ℝn , let W1 = 𝜕x1 , . . . , Wn = 𝜕xn . Then W = {W1 , . . . , Wn } span the tangent space at every point and therefore satisfy Hörmander’s condition of order 1. This example plays an important role when we see elliptic operators as a special case of maximally subelliptic operators in Example 1.1.10 (i). (v) The above examples are all homogeneous under appropriately chosen dilations; for example the vector fields in (i) are homogeneous under the dilations (x, y) 󳨃→ (δx, δ2 y). Usually, there will be no such choice of dilations. For example, with M = ℝ2 , let W1 = 𝜕x and W2 = (x + x 2 )𝜕y . Then W = {W1 , W2 } satisfies Hörmander’s condition of order 2 on ℝ2 , but the vector fields are not homogeneous with respect to dilations. To deal with general Hörmander vector fields, we work with more abstract scaling maps instead of simple dilations; see Section 1.9. (vi) On ℝ3 , let W1 = 𝜕x and W2 = x𝜕y . Then W1 and W2 do not satisfy Hörmander’s condition at any point. However, for each t0 ∈ ℝ, if we restrict our attention to the submanifold Lt0 := {(x, y, t0 ) : x, y ∈ ℝ}, then W1 and W2 are tangent to Lt0 and satisfy Hörmander’s condition of order 2 on Lt0 . This idea can be generalized using the classical Frobenius theorem (see Section 3.1.3 and Remark 3.4.11).

4 � 1 Introduction 2

(vii) On ℝ2 , let W1 = 𝜕x and W2 = e−1/x 𝜕y . Then W = {W1 , W2 } does not satisfy Hörmander’s condition at any point of the form (0, y) ∈ ℝ2 . Unlike (vi), there is not a submanifold passing through (0, y) such that W1 and W2 are tangent to the submanifold and satisfy Hörmander’s condition. Given a set of vector fields satisfying Hörmander’s condition, we will often pair each vector field with a “formal degree,” which is an element of ℕ+ . Definition 1.1.3. We call a finite set (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )}, where each Wj is a smooth vector field on M and each dsj ∈ ℕ+ , a finite set of vector fields with formal degrees. If W = {W1 , . . . , Wr } satisfies Hörmander’s condition, we call (W , ds) a set of Hörmander vector fields with formal degrees. Let (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} be a set of Hörmander vector fields with formal degrees on M. Informally, we will treat Wj as a differential operator of “degree” dsj , even though Wj is a differential operator of order 1 in the usual sense. The case ds1 = ds2 = ⋅ ⋅ ⋅ = dsr = 1 is particularly interesting, and the reader may find it enlightening to interpret the definitions in this special case. When we consider this case, we will write (W , 1) instead of (W , ds). Because the vector fields W1 , . . . , Wr do not necessarily commute, when we consider polynomials in these vector fields we use noncommutative polynomials. For this, we use the following notation. Definition 1.1.4. Let α = (α1 , . . . , αL ) ∈ {1, . . . , r}L be a list of elements of {1, . . . , r}. We set W α := Wα1 Wα2 ⋅ ⋅ ⋅ WαL , |α| := L, and degds(α) := dsα1 + dsα2 + ⋅ ⋅ ⋅ + dsαL . We call such a list α an ordered multi-index. Remark 1.1.5. We treat [Wj , Wk ] = Wj Wk − Wk Wj as a differential operator of degree dsj + dsk . For example, when (W1 , ds1 ) = (𝜕x , 1) and (W2 , ds2 ) = (x𝜕y , 1), we will treat 𝜕y = [W1 , W2 ] as a differential operator of degree 2 (see Example 1.6.1). Thus, even though 𝜕y is a differential operator of order 1 in the classical sense, in the theory which follows we treat it as an operator of degree 2. In particular, the vector fields W1 and W2 do not even approximately commute: [W1 , W2 ] = W1 W2 − W2 W1 is a partial differential operator of the same degree as W1 W2 . This is in contrast to elliptic theory, where [𝜕x , 𝜕y ] = 0. This difference underlies nearly all the difficulties when studying maximally subelliptic operators. Nearly every proof from classical elliptic theory at some point, either implicitly or explicitly, uses the commutative nature of the elliptic setting. Because of this, these proofs do not translate to the maximally subelliptic setting and new ideas are needed. In what follows, κ ∈ ℕ+ will be such that dsj divides κ, 1 ≤ j ≤ r, and nj := κ/dsj ∈ ℕ+ . We write A ⋐ B if A is a relatively compact subset of B.

1.1 Basic definitions

�

5

1.1.1 Linear maximally subelliptic operators Fix D1 , D2 ∈ ℕ+ . Given a partial differential operator, P , with coefficients in ∞ Cloc (M; 𝕄D1 ×D2 (ℂ)),3 we write P in the form4 P=

∑

degds (α)≤κ

aα (x)W α ,

∞ aα ∈ Cloc (M; 𝕄D1 ×D2 (ℂ)).

(1.1)

Recall dsj is assumed to divide κ, ∀j. By Hörmander’s condition, every partial differential operator with smooth coefficients is of the form (1.1). Remark 1.1.6. The form (1.1) is not unique. For example, if W1 = 𝜕x and W2 = 𝜕y , then we can write 0 = [W1 , W2 ] = W1 W2 − W2 W1 . We always pick a representation of the form (1.1) with the least possible κ (for a fixed choice of (W , ds)). The value of κ depends on (W , ds) and can be considered the degree of P with respect to (W , ds). Definition 1.1.7. We say that P given by (1.1) is maximally subelliptic of degree κ with respect to (W , ds) on M, if for every relatively compact, open set Ω ⋐ M, there exists CΩ ≥ 0, such that ∀f ∈ C0∞ (Ω; ℂD2 ), r

󵄩 n 󵄩 󵄩 󵄩 󵄩 󵄩 ∑󵄩󵄩󵄩Wj j f 󵄩󵄩󵄩L2 (M,Vol;ℂD2 ) ≤ CΩ (󵄩󵄩󵄩P f 󵄩󵄩󵄩L2 (M,Vol;ℂD1 ) + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (M,Vol;ℂD2 ) ). j=1

When M is clear from the context, we say P is maximally subelliptic of degree κ with respect to (W , ds). Recall, nj = κ/dsj ∈ ℕ+ . Remark 1.1.8. Given (W , ds), Definition 1.1.7 defines a class of operators which are maximally subelliptic with respect to (W , ds). If one changes (W , ds), the class of operators may change as well. Remark 1.1.9. The density Vol does not play a role in Definition 1.1.7: one obtains an equivalent definition with any choice of smooth, strictly positive density. Given a partial differential operator with smooth coefficients, P , to show P is maximally subelliptic, one needs to: – find the right choice (W , ds) of Hörmander vector fields with formal degrees, – find the minimal κ such that P can be written in the form (1.1), with this choice of (W , ds), – prove that P is maximally subelliptic of degree κ with respect to (W , ds). 3 See Sections 2.1 and 3.1 for a discussion of C0∞ (the space of smooth functions with compact support), ∞ Cloc (the space of locally smooth functions), and C ∞ (the space of bounded smooth functions, all of whose derivatives are bounded). 4 In this text we work with trivial vector bundles, though most definitions and results can be easily extended to general vector bundles – see Section 8.6. Our main results are local in nature, so general vector bundles do not give much additional generality.

6 � 1 Introduction This may not even be possible: not every partial differential operator can be seen as maximally subelliptic with respect to some (W , ds).5 Even when it is possible, the above steps are not necessarily easy (however, there have been some recent advances in recognizing maximally subelliptic operators – see Section 8.8). Once a partial differential operator is shown to be maximally subelliptic, many sharp results follow from the theory in this text. Example 1.1.10. In the examples that follow, (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} will denote any set of Hörmander vector fields with formal degrees on M. We write (W , 1) for the special case where ds1 = ds2 = ⋅ ⋅ ⋅ = dsr = 1. Recall that nj = κ/dsj ∈ ℕ+ . (i) With M = ℝn , let (𝜕, 1) := {(𝜕x1 , 1), . . . , (𝜕xn , 1)}. Then P is maximally subelliptic of degree κ with respect to (𝜕, 1) if and only if P is locally elliptic of degree κ in the classical sense (see Definition 2.7.1). In particular, the Laplacian △ = 𝜕x∗1 𝜕x1 + ⋅ ⋅ ⋅ + 𝜕x∗n 𝜕xn = −𝜕x21 − 𝜕x22 − ⋅ ⋅ ⋅ − 𝜕x2n (ii)

is maximally subelliptic of degree 2 with respect to (𝜕, 1). The “gradient” n

n

∇κ,W f := (W1 1 f , W2 2 f , . . . , Wrnr f ) is maximally subelliptic of degree κ with respect to (W , ds) on M; this follows immediately from Definition 1.1.7. (iii) The “sub-Laplacian” ∗

r

n ∗

nj

L := ∇κ,W ∇κ,W = ∑(Wj j ) Wj j=1

is maximally subelliptic of degree κ with respect to (W , ds) on M; here ∗ denotes the formal L2 (M, Vol) adjoint. The maximal subellipticity of L follows from (ii) and Theorem 8.1.1 (i) ⇒ (vi). In the special case where (W , ds) is replaced by (W , 1) and κ = 1, the maximal subellipticity of L was shown by Rothschild and Stein [200]; in that case L is called Hörmander’s sub-Laplacian. Just as the Laplacian plays an important role in the study of elliptic PDEs, the sub-Laplacian plays an important role in the study of maximally subelliptic PDEs (see Section 4.5). The sub-Laplacian can be used to give several interesting examples of maximally subelliptic operators: (a) The operator 𝜕x2 + (x𝜕y )2 = 𝜕x2 + x 2 𝜕y2

(1.2)

5 For example, all maximally subelliptic operators are subelliptic (see (1.18) and Corollary 8.2.5), while not every partial differential operator is subelliptic.

1.1 Basic definitions

�

7

is maximally subelliptic of degree 2 with respect to {(𝜕x , 1), (x𝜕y , 1)} on ℝ2 . Note that this operator is not elliptic at x = 0. See also Example 1.1.2 (i). The operator in (1.2) is known as the Grushin operator. (b) More generally, for n1 , n2 ∈ ℕ+ , the operator 2n2

(−1)n1 𝜕x2n1 + (−1)n2 (x𝜕y )

= (−1)n1 𝜕x2n1 + (−1)n2 x 2n2 𝜕y2n2

is maximally subelliptic of degree 2n1 n2 with respect to {(𝜕x , n2 ), (x𝜕y , n1 )} on ℝ2 . (c) Item (a) can be generalized in another way: 𝜕x2 + (x m 𝜕y )2 = 𝜕x2 + x 2m 𝜕y2

(1.3)

is maximally subelliptic of degree 2 with respect to {(𝜕x , 1), (x m 𝜕y , 1)} on ℝ2 . See also Example 1.1.2 (ii). The operator in (1.3) is known as a higher-order Grushin operator. (d) The operator 2

2

(𝜕x + 2y𝜕t ) + (𝜕y − 2x𝜕t )

(1.4)

is maximally subelliptic of degree 2 with respect to {(𝜕x + 2y𝜕t , 1), (𝜕y − 2x𝜕t , 1)} on ℝ3 ; see also Example 1.1.2 (iii). Note that the operator in (1.4) is nowhere elliptic. In fact, we will see that maximally subelliptic operators are usually nowhere elliptic. For example, in the special case where ds1 = ⋅ ⋅ ⋅ = dsr = 1, the following are equivalent for a fixed x0 ∈ M and an operator P written in the form (1.1): – P is elliptic of degree κ near x0 , – P is maximally subelliptic of degree κ with respect to (W , 1) near x0 and W = {W1 , . . . , Wr } satisfies Hörmander’s condition of order 1 at x0 . Moreover, the higher the order of Hörmander’s condition at x0 , the more degenerate maximally subelliptic operators are at x0 . See Remark 8.2.7. Because of this, classical elliptic theory is not useful to study general maximally subelliptic operators. (iv) On ℝn+1 with coordinates (t, x1 , x2 , . . . , xn ), the heat operator 𝜕t + 𝜕x21 + 𝜕x22 + ⋅ ⋅ ⋅ + 𝜕x2n

(1.5)

is maximally subelliptic of degree 2 with respect to {(𝜕t , 2), (𝜕x1 , 1), (𝜕x2 , 1), . . . , (𝜕xn , 1)} on ℝn+1 . More generally, locally parabolic partial differential operators are the special case of maximally subelliptic operators when we choose (W , ds) = {(𝜕t , 2), (𝜕x1 , 1), (𝜕x2 , 1), . . . , (𝜕xn , 1)}.

8 � 1 Introduction (v)

Let W = {W1 , . . . , Wr } be vector fields satisfying Hörmander’s condition on M. Generalizing (1.5), Rothschild and Stein [200, § 18] showed that W1 − W2∗ W2 − W3∗ W3 − ⋅ ⋅ ⋅ − Wr∗ Wr

(1.6)

is maximally subelliptic of degree 2 with respect to {(W1 , 2), (W2 , 1), (W3 , 1), . . . , (Wr , 1)} on M. Operators of the type (1.6) arise in the study of Kolmogorov– Fokker–Planck equations and account for one of the main original motivating examples for studying maximally subelliptic operators; see [16, Chapter 2]. (vi) Let W = {W1 , . . . , Wr } be vector fields satisfying Hörmander’s condition. Let a1 ∈ ℕ+ be odd, let a2 , . . . , ar ∈ ℕ+ be even, and let κ ∈ ℕ+ be the least common multiple of a1 , . . . , ar . Generalizing (v), Androulidakis, Moshen, and Yuncken [2, Proposition 4.1] showed that the operator r

a

W1 1 + ∑(−1)

aj /2

j=2

a

Wj j

is maximally subelliptic of degree κ with respect to the Hörmander vector fields with formal degrees (W , ds) = {(W1 , κ/a1 ), . . . , (Wr , κ/ar )}. (vii) Let ai,j (x) : M → 𝕄r×r (ℂ) be a self-adjoint matrix depending smoothly on x ∈ M which is strictly positive definite at every point. Then the operator u 󳨃→ ∑ ai,j (x)Wi Wj u(x) i,j

is maximally subelliptic of degree 2 with respect to (W , 1) on M. See Example 8.9.2. (viii) The operator in (vii) can be generalized to higher-order operators. Indeed, let ai,j (x) : M → 𝕄r×r (ℂ) be a self-adjoint matrix depending smoothly on x ∈ M which is strictly positive definite at every point. Then the operator n

n

u 󳨃→ ∑ ai,j (x)Wi i Wj j u(x) i,j

is maximally subelliptic of degree 2κ with respect to (W , ds) on M. One can also add lower-order terms. See Proposition 8.9.1 for details. For a different generalization of (vii) to higher-order operators, see Proposition 8.9.4. (ix) Consider the Laplacian on ℝ2 : △ = −𝜕x2 − 𝜕y2 . As described in (i), △ is maximally subelliptic of degree 2 with respect to {(𝜕x , 1), (𝜕y , 1)} on ℝ2 . Consider, instead, the Hörmander vector fields with formal degrees (W , ds) = {(𝜕x , 1), (x𝜕y , 1)}. Then, on any neighborhood of (0, 0), △ is a polynomial of degree 4 in 𝜕x and x𝜕y , and is not maximally subelliptic of degree 4 (or of any degree) with respect to (W , ds) on any neighborhood of (0, 0). This highlights the fact that maximal subellipticity depends on the choice of Hörmander vector fields with formal degrees. Depending on the

1.1 Basic definitions

�

9

choice of (W , ds), elliptic operators may not be maximally subelliptic with respect to (W , ds) (see also the discussion in (iii), part (d)). Thus, ellipticity is not a stronger condition than maximal subellipticity; it is a special case of maximal subellipticity. Maximally subelliptic operators in general, and the sub-Laplacian in particular, arise in many settings, including probability theory, several complex variables, and stochastic PDEs. The text [16] gives an excellent history of the subject; see also Sections 8.10 and 9.5 for a further discussion and many references.

1.1.2 Nonlinear maximally subelliptic operators Fix D1 , D2 ∈ ℕ+ and let N := D2 × #{α an ordered multi-index : degds(α) ≤ κ}.

(1.7)

Given a sufficiently smooth function u = (u1 , . . . , uD2 ) : M → ℝD2 , we define {W α u}deg

ds (α)≤κ

: M → ℝN

to be the vector-valued function whose components are given by W α uj , degds(α) ≤ κ and 1 ≤ j ≤ D2 . Let F(x, ζ ) : M × ℝN → ℝD1 be a function.6 Given a nonlinear PDE, we write it in the form F(x, {W α u(x)}deg

ds (α)≤κ

) = g(x).

(1.8)

By Hörmander’s condition, every nonlinear PDE can be written in the form (1.8). For a fixed x0 ∈ M and u : M → ℝD2 the operator α

Pu,x0 v := dζ F(x0 , {W u(x0 )}deg

ds (α)≤κ

){W α v}deg

ds (α)≤κ

(1.9)

is a partial differential operator of the form Pu,x0 v =

∑

degds (α)≤κ

aα,u,x0 W α v,

where aα,u,x0 ∈ 𝕄D1 ×D2 (ℝ) are constant matrices which depend on x0 and a finite number of Taylor coefficients of u at x0 . 6 For the purposes of this introduction, we assume that all functions are smooth enough that every expression makes sense.

10 � 1 Introduction Definition 1.1.11. Fix x0 ∈ M, u : M → ℝD2 , and F : M × ℝN → ℝD1 ; we assume both u and F are sufficiently smooth so that all objects which follow make sense. We say the nonlinear partial differential operator u 󳨃→ F(x, {W α u(x)}deg

ds (α)≤κ

(1.10)

)

is maximally subelliptic at (x0 , u) of degree κ with respect to (W , ds) if there exists an open neighborhood U ⊆ M of x0 such that the linear partial differential operator Pu,x0 given by (1.9) is maximally subelliptic of degree κ with respect to (W , ds) on U. We do not assume the function F(x, ζ ) in Definition 1.1.11 is smooth. In particular, by considering the case where F(x, ζ ) is linear in ζ but not smooth in x, we will be able to study linear maximally subelliptic operators with non-smooth coefficients as a special case of Definition 1.1.11. This will allow us to conclude results like interior Schauder estimates by using a more general fully nonlinear theory. See, for example, Corollary 9.1.10. Because the definition of a fully nonlinear equation being maximally subelliptic is just that a “frozen coefficient” version of the linearization is maximally subelliptic, once one has examples of linear maximally subelliptic operators (as in Example 1.1.10) it is simple to write down examples of nonlinear maximally subelliptic operators. Moreover, important examples of fully nonlinear elliptic operators have generalizations to the maximally subelliptic setting, as the next examples show. Example 1.1.12 (The Monge–Ampère equation). An important nonlinear PDE is the Monge–Ampère equation. For a function u : ℝn → ℝ, we consider the Hessian Hu = (𝜕xi 𝜕xj u(x))1≤i,j≤n . If Hu(x0 ) > 0, i. e., Hu(x0 ) is a strictly positive definite matrix, then the fully nonlinear operator u 󳨃→ det Hu is elliptic at (x0 , u); this is known as the Monge–Ampère equation. Note that the condition Hu(x0 ) > 0 implies u is strictly convex near x0 (provided u is at least C 2 ). (i) As first noted in [157] and [57], a natural generalization of the Monge–Ampère equation to the maximally subelliptic setting is as follows. Let (W , 1) = {(W1 , 1), . . . , (Wr , 1)} be Hörmander vector fields each paired with formal degree 1. For a function u : M → ℝ, define a symmetric r × r matrix W u(x) whose (i, j) component is given by 21 (Wi Wj + Wj Wi )u(x). If W u(x0 ) > 0 (i. e., if the matrix is strictly positive definite at x0 ), then u 󳨃→ det W u is maximally subelliptic at (x0 , u) of degree 2 with respect to (W , 1) – this is proved, and more details are given, in Section 9.4.2. By choosing W1 , . . . , Wr to satisfy Hörmander’s condition of a very high order, this fully nonlinear equation is very “degenerate;” see the discussion in Example 1.1.10 (d). (ii) The maximally subelliptic Monge–Ampère equation from (i) can be further generalized to higher-order fully nonlinear equations. Let (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} be Hörmander vector fields with formal degrees. As above, fix κ ∈ ℕ+ such that dsj divides κ, ∀j, and set nj := κ/dsj ∈ ℕ+ . For a function u : M → ℝ, define a symmetric n

n

n

n

r×r matrix W [κ]u(x) whose (i, j) component is given by 21 (Wi i Wj j +Wj j Wi i )u(x). If

1.1 Basic definitions

�

11

W [κ]u(x0 ) > 0, then u 󳨃→ det W [κ]u is maximally subelliptic at (x0 , u) of degree 2κ with respect to (W , ds) – this is proved, and more details are given, in Section 9.4.3.

(iii) There is a different generalization (i) to higher-order operators. With the same setting as in (ii), for a function u : M → ℝ define a symmetric, square matrix, Ŵ[κ]u(x), whose components are parameterized by (α, β), where degds(α) = degds(β) = κ, by 1 (Ŵ[κ]u(x))α,β = (W α W β u(x) + W β W α u(x)), 2

degds(α) = degds(β) = κ.

If Ŵ[κ]u(x0 ) > 0, then u 󳨃→ det Ŵ[κ]u is maximally subelliptic at (x0 , u) of degree 2κ with respect to (W , ds) – this is proved, and more details are given, in Section 9.4.3. Example 1.1.13. The ellipticity of the classical Monge–Ampère equation is just one example of a second-order fully nonlinear equation whose linearization is elliptic because its principal part is given by a strictly positive definite quadratic form – a phenomenon which often appears in elliptic theory. This phenomenon can be generalized to the maximally subelliptic setting as follows. Let (W , 1) = {(W1 , 1), . . . , (Wr , 1)} be Hörmander vector fields, each paired with formal degree 1. Consider a nonlinear partial differential operator of the form u 󳨃→ F(x, {W α u(x)}|α|≤2 ),

(1.11)

where F is a real-valued function. We consider F = F(x, ζ ) as a function of x ∈ M and 2 ζ ∈ ℝr +r+1 : ζ = (ζ0 , ζ1 , . . . , ζr , . . . , ζi,j , . . . , ζr,r ), where, for example, the coordinate ζi,j corresponds to Wi Wj u. For a fixed u : M → ℝ and x0 ∈ M, if the r × r matrix 𝜕ζ1,1 F(x0 , {W α u(x0 )}|α|≤2 ) .. ( . 𝜕ζr,1 F(x0 , {W α u(x0 )}|α|≤2 )

⋅⋅⋅ .. . ⋅⋅⋅

𝜕ζ1,r F(x0 , {W α u(x0 )}|α|≤2 ) .. ) . α 𝜕ζr,r F(x0 , {W u(x0 )}|α|≤2 )

is symmetric and strictly positive definite, then (1.11) is maximally subelliptic at (x0 , u) of degree 2 with respect to (W , 1). See Section 9.4.1 for details. Moreover, similar to Example 1.1.12 (ii) and (iii), this idea can be generalized to higher-order operators. See Section 9.4.4 for details. See Section 9.5 for several other settings where nonlinear maximally subelliptic and other related nonlinear PDEs arise. Remark 1.1.14. One main goal of this text is to study the regularity theory of classical solutions to general fully nonlinear maximally subelliptic equations of arbitrary order. Because we work with general equations, we do not introduce any notion of a weak solution.

12 � 1 Introduction

1.2 Background: elliptic theory All the results of this text have classical versions for elliptic equations. We describe the main results we generalize here; these topics are discussed in more detail in Chapter 2. We work in ℝn . The sharp regularity theory of elliptic operators is often described in terms of the s s classical Besov (Bp,q (ℝn )) and Triebel–Lizorkin (Fp,q (ℝn )) spaces; here s ∈ ℝ and p, q ∈ (0, ∞]. However, for the purposes of this text, we consider only the simplest setting when s p, q ∈ [1, ∞] in general and restrict our attention to p ∈ (1, ∞) for Fp,q . A unifying aspect

of these spaces is that 𝜕xj acts as a differential operator of order 1: 𝜕xj : X s → X s−1 , s s where X s is either Bp,q (ℝn ) or Fp,q (ℝn ). See Section 2.4 for a more complete discussion. While these spaces have a technical definition, many of the classical spaces from s harmonic analysis can be seen as special cases. For example, Fp,2 (ℝn ) is the Lp Sobolev s n s n space of order s ∈ ℝ. Also, for s > 0, C (ℝ ) := B∞,∞ (ℝ ) is the so-called Zygmund– Hölder space of order s. When s ∈ (0, ∞)\ℕ+ , C s (ℝn ) = C ⌊s⌋,s−⌊s⌋ (ℝn ) is the usual Hölder space of order s. However, when s ∈ ℕ+ , C s (ℝn ) is not a Hölder space: C m+1 (ℝn ) ⊊ C m,1 (ℝn ) ⊊ C m+1 (ℝn ); see Section 2.5 for a detailed description of C s (ℝn ). Some key results we generalize from the elliptic setting are: – Parametrices: Let P be an elliptic partial differential operator of order κ with smooth coefficients. Then there is an operator T which is locally a pseudo-differential operator of order −κ such that TP ≡ I

–

∞ mod Cloc .

See Theorem 2.7.2 (g) for a more detailed statement and Proposition 2.2.27 for a proof of a similar result. Linear regularity: Let P be an elliptic partial differential operator of order κ on ℝn , with smooth coefficients. Then if X s denotes any one of the scales of spaces s s Bp,q (ℝn ) or Fp,q (ℝn ), we have, for any distribution u and any point x0 ∈ ℝn , s

P u ∈ X near x0 ⇐⇒ u ∈ X

–

s+κ

near x0 .

See Theorem 2.7.2 (e) for a more precise statement. Fully nonlinear regularity: Fix s > r > 0 and let F(x, ζ ) be a function which is C s ∞ in the x variable and Cloc in the ζ variable.7 Fix x0 ∈ ℝn and suppose u ∈ C r+κ (ℝn ) α is such that F(x, {𝜕x u(x)}|α|≤κ ) is elliptic near x0 and u. That is, suppose that the constant coefficient, linear partial differential operator α

Pu,x0 v := dζ F(x0 , {𝜕x u(x0 )}|α|≤κ ){𝜕x v}|α|≤κ 7 We only require finite regularity for F(x, ζ ) in the ζ variable, though we do not make that precise here. See Theorem 2.8.3 for more details.

1.2 Background: elliptic theory

�

13

is elliptic of order κ. Then F(x, {𝜕xα u(x)}|α|≤κ ) ∈ C s (ℝn ) near x0 ⇐⇒ u ∈ C s+κ (ℝn ) near x0 . –

See Theorem 2.8.3 for a more general and precise statement. Interior Schauder estimates: Fix s > r > 0 and let P be an elliptic partial differential operator of order κ with C s coefficients. Then, for x0 ∈ ℝn and u ∈ C r+κ (ℝn ), s

n

P u ∈ C (ℝ ) near x0 ⇐⇒ u ∈ C

–

s+κ

(ℝn ) near x0 .

This is the special case of the fully nonlinear result described above where we take F(x, ζ ) to be linear in ζ . See Corollary 2.8.5 for a more precise statement. Gaussian bounds: Let P be an elliptic partial differential operator of order κ with smooth coefficients. Let L be any non-negative self-adjoint extension of the nonnegative symmetric operator P ∗ P . Then, for t > 0, the heat operator e−tL is given by integration against a smooth function, which we denote by e−tL (x, y). Moreover, we have local Gaussian bounds for this heat kernel: for every compact set 𝒦 ⋐ ℝn , ∃c = c(𝒦) > 0, ∀α, β ∈ ℕn , ∀s ∈ ℕ, ∃C = C(α, β, s, 𝒦) ≥ 0, so that ∀t ∈ (0, 1], ∀x, y ∈ 𝒦, 2κ

1 −n−|α|−|β|−2κs |x − y| 󵄨󵄨 s α β −tL 󵄨 (x, y)󵄨󵄨󵄨 ≤ C(|x − y| + t 2κ ) exp(−c( ) 󵄨󵄨𝜕t 𝜕x 𝜕y e t

1/(2κ−1)

).

See Theorem 2.7.2 (i) for more details. In fact, Theorem 2.7.2 shows that more precise restatements of the parametrices, linear regularity, and Gaussian bounds described above are actually equivalent to ellipticity. As described in the above bullet points on parametrices and local regularity, linear ∞ elliptic operators with smooth coefficients are locally left invertible modulo Cloc as maps s+κ s s X → X , where X denotes either a scale of Besov or Triebel–Lizorkin spaces. In ∞ particular, they are locally left invertible modulo Cloc as maps C s+κ (ℝn ) → C s (ℝn ). As we will see in the proof of Theorem 2.8.7, one can use this invertibility to apply ideas similar to the contraction mapping principle (along with some careful covering lemmas) to deduce the fully nonlinear regularity described above. Elliptic operators are the only partial differential operators which have this left invertibility on the classical spaces, which prevents such a simple argument from directly working for any operators which are not elliptic. To avoid this problem, when we move to studying more general maximally subelliptic operators, we introduce generalizations of the Besov and Triebel– Lizorkin spaces, adapted to the operator in question, on which it is left invertible modulo smooth functions.

14 � 1 Introduction

1.3 Carnot–Carathéodory geometry Intimately tied to a maximally subelliptic operator is a metric geometry on M induced by the Hörmander vector fields with formal degrees (W , ds), known as a Carnot– Carathéodory (or sub-Riemannian) geometry. This geometry plays the role for maximal subellipticity with respect to (W , ds) that Riemannian geometry plays for ellipticity. Let M be a connected smooth manifold of dimension ≥ 1 and let W = {W1 , . . . , Wr } be a finite set of smooth vector fields on M. We define the Carnot–Carathéodory ball of unit radius, centered at x ∈ M, by 󵄨󵄨 󵄨󵄨 BW (x) := {y ∈ M 󵄨󵄨󵄨 ∃γ : [0, 1] → M, γ(0) = x, γ(1) = y, 󵄨󵄨 󵄨 γ is absolutely continuous, r

γ′ (t) = ∑ aj (t)Wj (γ(t)) almost everywhere,

(1.12)

j=1

󵄩󵄩 r 󵄩󵄩 󵄩󵄩 󵄩 2󵄩 󵄩 aj ∈ L ([0, 1]), 󵄩󵄩󵄩∑ |aj | 󵄩󵄩󵄩󵄩 < 1}. 󵄩󵄩j=1 󵄩󵄩 ∞ 󵄩 󵄩L ([0,1]) ∞

In particular, since γ is absolutely continuous in (1.12), we have in local coordinates t γ(t1 ) = γ(t0 ) + ∫t 1 ∑rj=1 aj (t)Wj (γ(t)) dt. 0 To define balls of other radii, we scale the vector fields. We use a list of vector fields with formal degrees (W , ds) := {(W1 , ds1 ), . . . , (Wr , dsr )}, with dsj ∈ ℕ+ . For δ > 0 set δdsW := {δds1 W1 , . . . , δdsr Wr }. We define the Carnot–Carathéodory ball of radius δ > 0, centered at x ∈ M, by B(W ,ds) (x, δ) := Bδds W (x).

(1.13)

Corresponding to the balls B(W ,ds) (x, δ) is the Carnot–Carathéodory distance ρ(W ,ds) (x, y) := inf{δ > 0 : y ∈ B(W ,ds) (x, δ)}.

(1.14)

It is not hard to see that ρ(W ,ds) is an extended metric on M (it may take the value +∞, but otherwise it satisfies the usual axioms for a metric). A main setting in which we study these Carnot–Carathéodory balls is when (W , ds) are Hörmander vector fields with formal degrees (see Definition 1.1.3). In this case, a classical theorem of Chow [45] (see Lemma 3.1.7) implies that ρ(W ,ds) is a metric on M (i. e., ρ(W ,ds) (x, y) < ∞, ∀x, y ∈ M), and the metric topology on M induced by ρ(W ,ds) agrees with the topology on M (as a manifold) – see Lemma 3.1.7. In this setting, Nagel, Stein, and Wainger [189] proved the important inequality for small δ > 0 Vol(B(W ,ds) (x, 2δ)) ≲ Vol(B(W ,ds) (x, δ)),

(1.15)

1.4 Singular integrals

�

15

Table 1.1: Analogs between ellipticity and maximal subellipticity. Elliptic

Maximally subelliptic

Riemannian geometry

Carnot–Carathéodory (CC) geometry

Classical (Riemannian) function spaces

Function spaces adapted to the CC geometry

Pseudo-differential operators

Gaussian bounds for heat equations using a Riemannian metric Standard dilation maps on ℝn Fourier transform

Singular integral operators adapted to the CC geometry Gaussian bounds for heat equations using the CC metric

Abstract scaling maps adapted to the CC geometry

No single analog

uniformly as x ranges over a compact set. Formula (1.15) implies that the balls B(W ,ds) (x, δ) when paired with the smooth, strictly positive density Vol locally give M the structure of a space of homogeneous type in the sense of Coifman and Weiss [52]. This gives access to the robust Calderón–Zygmund theory, which is an essential tool of this text (see, for example, Section 1.4). See Table 1.1 for some of the many ways Carnot–Carathéodory geometry arises when studying maximally subelliptic operators.

1.4 Singular integrals Unlike the case of elliptic operators, standard pseudo-differential operators and the Fourier transform are not decisive tools when studying maximally subelliptic operators. Since the work of Folland and Stein [91], Folland [90], Rothschild and Stein [200], and Nagel, Stein, and Wainger [189], it has been clear that singular integral operators adapted to the Carnot–Carathéodory geometry can often take the place of pseudodifferential operators. The main results of this text are local, so it suffices to consider operators whose Schwartz kernels have compact support. For t ∈ ℝ, in Chapter 5 we define a class of singular integral operators A t (W , ds) whose Schwartz kernels have compact support. These satisfy the following properties: – A t (W , ds) is a filtered algebra: For T ∈ A t (W , ds) and S ∈ A s (W , ds), we have TS ∈ A t+s (W , ds). – Operators in A t (W , ds) have good boundedness properties on function spaces adapted to (W , ds). See Section 1.5. – Left parametrices for maximally subelliptic operators of degree κ are locally in A −κ (W , ds). See Section 1.6. – If ψ ∈ C0∞ (M), then Mult[ψ]Wj ∈ A dsj (W , ds), where Mult[ψ] : f 󳨃→ ψf . More generally, if P is a partial differential operator of the form (1.1), then Mult[ψ]P ∈ A κ (W , ds).

16 � 1 Introduction –

Operators in A t (W , ds) are pseudo-local: their Schwartz kernels are smooth off the diagonal.

Remark 1.4.1. If T ∈ A t (W , ds) and S ∈ A s (W , ds), then by the filtered algebra property we have [T, S] = TS − ST ∈ A t+s (W , ds); however, unlike pseudo-differential operators we do not have [T, S] ∈ A t+s−1 (W , ds). This is unavoidable: we wish to treat [Wj , Wk ] as a differential operator of degree dsj + dsk (see Remark 1.1.5). Because of this, most proofs from elliptic theory which rely on pseudo-differential operators do not directly translate to the maximally subelliptic setting. We defer the technical definition of A t (W , ds) to Section 5.2.1 (where, in fact, four equivalent definitions are given). However, T ∈ A 0 (W , ds) implies the following: – The Schwartz kernel T(x, z) of T agrees with a smooth function (which we again denote by T(x, z)) away from the diagonal x = z. Moreover, for all ordered multiindices α, β, we have 󵄨󵄨 α β 󵄨 󵄨󵄨Wx Wz T(x, z)󵄨󵄨󵄨 ≤ Cα,β

–

ρ(W ,ds) (x, z)− degds (α)−degds (β) , Vol(B(W ,ds) (x, ρ(W ,ds) (x, z))) ∧ 1

where Wx denotes the list of vector fields W1 , . . . , Wr acting as partial differential operators in the x variable, and similarly for Wz . T has good bounds when acting on certain bump functions which are adapted to the Carnot–Carathéodory geometry.

In particular, T is a Calderón–Zygmund singular integral operator with respect to the local space of homogeneous type on M induced by the Carnot–Carathéodory metric ρ(W ,ds) . The classical Calderón–Zygmund theory can then be used to show that T extends to a bounded operator on Lp (M, Vol), 1 < p < ∞. This boundedness is an important tool in our study of the regularity theory of maximally subelliptic operators. While the singular integrals above are our main replacement for pseudo-differential operators, there is a subclass of these singular integrals which is somewhat more analogous to the classical approach to pseudo-differential operators; we call this subclass (generalized) pseudo-differential operators.8 We present and study these operators in Chapter 4. Unfortunately, though they do have some important applications, these generalized pseudo-differential operators do not seem robust enough to establish all the results we need, so we instead turn to the more general singular integral operators in Chapter 5.

8 This class of operators appears in the Rothschild and Stein lifting procedure, which is very useful, but has some limitations. See Section 4.5.7.

1.5 Function spaces

�

17

1.5 Function spaces The sharp regularity theory of maximally subelliptic operators is in terms of Besov s s (Bp,q (W , ds)) and Triebel–Lizorkin (Fp,q (W , ds)) spaces which are adapted to (W , ds);9 here s ∈ ℝ. We define and study these spaces in Chapter 6. The two most important spes cial cases are Fp,2 (W , ds), which can be viewed as Lp Sobolev spaces adapted to (W , ds), s s and C (W , ds) := B∞,∞ (W , ds), s > 0, which can be viewed as a Zygmund–Hölder space adapted to (W , ds). In Chapters 6 and 7 we establish the following properties for s s X s (W , ds) ∈ {Bp,q (W , ds), Fp,q (W , ds)}: – Wj acts as a differential operator of degree dsj . More precisely, Wj : X s (W , ds) → X s−dsj (W , ds). – If T ∈ A t (W , ds), then T : X s (W , ds) → X s−t (W , ds). s – Fp,2 (W , ds) can be viewed as the Lp Sobolev space of order s ∈ ℝ with respect to (W , ds), for 1 < p < ∞. More precisely, let κ ∈ ℕ be such that dsj divides κ, 1 ≤ j ≤ r. Then, for a distribution u with compact support, κ u ∈ Fp,2 (W , ds) ⇐⇒ W α u ∈ Lp (M, Vol), ∀ degds(α) ≤ κ.

–

0 In particular, Fp,2 (W , ds) is locally equal to Lp (M, Vol).

For s ∈ (0, 1), the space C s (W , ds) is locally equal to the Hölder space (C 0,s (W , ds)) with respect to the Carnot–Carathéodory metric ρ(W ,ds) : 󵄨 󵄨 ‖f ‖C 0,s (W ,ds) := ‖f ‖C(M) + sup ρ(W ,ds) (x, y)−s 󵄨󵄨󵄨f (x) − f (y)󵄨󵄨󵄨. x,y∈M x =y̸

Moreover, in the special case where ds1 = ds2 = ⋅ ⋅ ⋅ = dsr = 1 and s ∈ (0, ∞) \ ℕ+ , C s (W , 1) is locally equal to the naturally defined Hölder space C ⌊s⌋,s−⌊s⌋ (W , 1) with norm ‖f ‖C m,t (W ,1) := ∑ ‖W α f ‖C 0,t (W ,1) , |α|≤m

–

m ∈ ℕ, t ∈ [0, 1].

See Theorem 7.7.23 for details. s s When M = ℝn , with (𝜕, 1) = {(𝜕x1 , 1), . . . , (𝜕xn , 1)}, Bp,q (𝜕, 1) and Fp,q (𝜕, 1) are locally s n s n equal to their classical counterparts Bp,q (ℝ ) and Fp,q (ℝ ); see Section 6.6.1 for details.

One interesting aspect of our theory is that we define the Besov and Triebel–Lizorkin spaces for all regularity parameters s ∈ ℝ in such a way that we can concretely under9 In defining these spaces, we focus only on distributions with compact support. This is sufficient since the main results of this text are local. Though it is likely not necessary, we also restrict our attention to s p, q ∈ [1, ∞] in general and to p ∈ (1, ∞) when considering Fp,q .

18 � 1 Introduction stand these spaces – for example we can characterize C s (W , ds) in terms of differences. In the literature there are two typical situations similar to ours: s s – Theories which define Bp,q and/or Fp,q for all s ∈ ℝ. Such theories usually use some sort of Lie group structure to help define the spaces (e. g., for the classical spaces, ℝn is used). s s – Theories which define Bp,q and/or Fp,q for |s| small (e. g., |s| < 1). With the restriction that s must be small, one can define these function spaces for much more general underlying spaces. See Section 6.14 for many references. In our setting, we define and study the Besov and Triebel–Lizorkin spaces for all regularity parameters s ∈ ℝ, even though there is no group structure on which to rely.

1.6 Linear theory Let κ ∈ ℕ+ be such that dsj divides κ, 1 ≤ j ≤ r, and let P = ∑degds (α)≤κ aα (x)W α be a partial differential operator with smooth coefficients written in the form (1.1). If P is maximally subelliptic of degree κ with respect to (W , ds) on M, then in Theorem 8.1.1 we establish the following. – Parametrices: There is an operator T, which is locally in A −κ (W , ds), such that TP ≡ I –

∞ mod Cloc (M).

s Regularity: If X s (W , ds) denotes any one of the scales of spaces Bp,q (W , ds) or s Fp,q (W , ds), then for any distribution u and any point x0 ∈ M, s

P u ∈ X (W , ds) near x0 ⇐⇒ u ∈ X

–

s+κ

(W , ds) near x0 .

(1.16)

Gaussian bounds: Let L be any non-negative self-adjoint extension of the nonnegative symmetric operator P ∗ P . Then for t > 0, the heat operator e−tL is given by integration against a smooth function, which we denote by e−tL (x, y). Moreover, we have local Gaussian bounds for this heat kernel in terms of the metric ρ(W ,ds) : for every 𝒦 ⋐ M compact, there exists c > 0, such that for all ordered multi-indices α, β and all s ∈ ℕ, there exists C ≥ 0, ∀t ∈ (0, 1], ∀x, y ∈ 𝒦, so that 1 − deg (α)−deg (β)−2κs 󵄨󵄨 s α β −tL 󵄨 ds ds (x, y)󵄨󵄨󵄨 ≤ C(ρ(W ,ds) (x, y) + t 2κ ) 󵄨󵄨𝜕t Wx Wy e 1

1 ρ(W ,ds) (x, y)2κ 2κ−1 −1 × exp(−c ( ) )(Vol(B(W ,ds) (x, ρ(W ,ds) (x, y) + t 2κ )) ∧ 1) . t

1.6 Linear theory

� 19

See Theorem 8.1.1 for more precise versions of the above. Moreover, once made precise, Theorem 8.1.1 shows that each of the above conditions are equivalent to maximal subellipticity, and moreover gives several other equivalent conditions, including: – P0 := ∑deg(α)=κ aα W α is maximally subelliptic of degree κ with respect to (W , ds) on M. – For all x0 ∈ M, there exists an open neighborhood U ⊆ M of x0 such that the frozen coefficient operator ∑degds (α)≤κ aα (x0 )W α is maximally subelliptic of degree κ with respect to (W , ds) on U. – P ∗ P is maximally subelliptic of degree 2κ with respect to (W , ds) on M. Example 1.6.1. Let P = 𝜕x2 + x 2 𝜕y2 . Then P is maximally subelliptic of degree 2 with respect to (W , ds) := {(𝜕x , 1), (x𝜕y , 1)} on ℝ2 ; see Example 1.1.10 (iii). Applying (1.16) in the 0 case X 0 (W , ds) = Fp,2 (W , ds) = Lp (ℝ2 ), 1 < p < ∞, we see that for ζ ∈ ℝ2 , p

2

2 2

p

P u ∈ L near ζ ⇐⇒ u, 𝜕x u, x𝜕y u, 𝜕x u, x 𝜕y u, 𝜕x x𝜕y u, x𝜕y 𝜕x u ∈ L near ζ .

In particular, P u ∈ Lp near ζ implies 𝜕y u = [𝜕x , x𝜕y ]u ∈ Lp near ζ , but does not imply 𝜕y2 u ∈ Lp near ζ . This is why we treat 𝜕y as a differential operator of degree 2 in this setting; see Remark 1.1.5. The regularity theory described above is clearly sharp (the conclusion (1.16) is an “if and only if”). However, it is also of interest to study the regularity theory of maximally subelliptic operators with respect to other geometries; for example, in Section 8.2.2 we present the sharp regularity theory of maximally subelliptic operators with respect s to the standard Besov and Triebel–Lizorkin spaces on M, here denoted by Bp,q,std and s s s Fp,q,std (in any local coordinate system, Bp,q,std and Fp,q,std are locally equal to their s s classical counterparts Bp,q (ℝn ) and Fp,q (ℝn )). In Corollary 8.2.3, the following is shown, s s s with Xstd equal to either Bp,q,std or Fp,q,std : Suppose P, given by (1.1), is maximally subelliptic of degree κ with respect to (W , ds) on M. Then, for any distribution u and x0 ∈ M, s

s+κ/λstd

Pu ∈ Xstd near x0 󳨐⇒ u ∈ Xstd

near x0 ,

(1.17)

where we have an explicit formula for λstd = λstd (x0 , (W , ds)) > 0. For example, in the special case ds1 = ds2 = ⋅ ⋅ ⋅ = dsr = 1, λstd = m, where W1 , . . . , Wr satisfy Hörmander’s condition of order exactly m at x0 ; see Definition 6.6.15 for the formula for λstd (x0 , (W , ds)) in the general setting. Furthermore, Corollary 8.2.3 shows that this result is sharp: there is no better regularity result in terms of the classical Besov and Triebel–Lizorkin spaces (see Corollary 8.2.3 for a precise statement of this sharpness). See Corollary 8.2.9 for some more regularity properties in terms of the standard Besov and Triebel–Lizorkin spaces.

In particular, maximally subelliptic operators are subelliptic. Indeed, for 1 < p < ∞, p s s we may take Xstd = Fp,2,std = Ls , the standard Lp Sobolev space of order s ∈ ℝ. In this special case, (1.17) becomes

20 � 1 Introduction p

p

P u ∈ Ls near x0 󳨐⇒ u ∈ Ls+κ/λ

std

near x0 ,

(1.18)

and this subelliptic gain is sharp. See Corollary 8.2.5. See Section 8.2 for additional regularity properties of maximally subelliptic operators. In particular, we study the regularity theory with respect to function spaces X s (Z, dr), where (Z, dr) is another set of Hörmander vector fields with formal degrees. Remark 1.6.2. The above discussion justifies the name maximally subelliptic. Indeed, (1.18) shows maximally subelliptic operators are subelliptic (Definition 2.2.32); informally, u is strictly more regular than P u. Moreover, (1.16) shows that u is more regular than P u by the clearly maximal possible amount, and hence maximally subelliptic. See Corollary 8.2.9 (i) for another perspective on this maximal gain using the standard Lp Sobolev spaces. Because maximally subelliptic operators are subelliptic, they are also hypoelliptic (see Definition 2.2.37 and Proposition 2.2.38); though hypoellipticity is a more qualitative concept than subellipticity, and it is perhaps less clear what “maximally hypoelliptic” should mean. This is why we have opted to call the operators in this text maximally subelliptic rather than maximally hypoelliptic.

1.7 Fully nonlinear theory The sharp regularity theory for fully nonlinear, maximally subelliptic PDEs is in terms of s the Zygmund–Hölder spaces adapted to (W , ds): C s (W , ds) = B∞,∞ (W , ds). Important for s the nonlinear theory is that C (W , ds) forms an algebra, and in fact we can understand compositions of classical Zygmund–Hölder functions with functions in C s (W , ds) – see Sections 7.4 and 7.5. Fix κ ∈ ℕ+ , and let u 󳨃→ F(x, {W α u(x)}deg

ds (α)≤κ

)

be a fully nonlinear equation of the type described in Section 1.1.2. Fully nonlinear regularity Fix s > r > 0, x0 ∈ M, and a function u : M → ℝD2 . Suppose: (a) u ∈ C r+κ (W , ds) near x0 , (b) F(x, ζ ) is C s (W , ds) in the x variable and C ∞ in the ζ variable, for x near x0 , (c) u 󳨃→ F(x, {W α u(x)}degds (α)≤κ ) is maximally subelliptic at (x0 , u) of degree κ with respect to (W , ds) in the sense of Definition 1.1.11. Then F(x, {W α u(x)}deg

ds (α)≤κ

) ∈ C s (W , ds) near x0 ⇐⇒ u ∈ C s+κ (W , ds) near x0 .

(1.19)

1.7 Fully nonlinear theory

� 21

See Theorem 9.1.2 for a more precise and more general version of this result, where F(x, ζ ) is only assumed to have finite regularity in the ζ variable. In the above result, r should be thought of as small. Thus, (a) implies that W α u is continuous near x0 , ∀ degds(α) ≤ κ, but it does not imply much more than that. The above result shows that even though u is a priori assumed to have low regularity, since it satisfies a maximally subelliptic equation it must have a higher regularity. Since the conclusion (1.19) is an “if and only if,” it is clearly sharp. Regularity in classical Zygmund–Hölder spaces Even though (1.19) gives a sharp regularity result, it is also of interest to understand the regularity theory with respect to the classical Zygmund–Hölder spaces. Fix s > r > 0, x0 ∈ M, and a function u : M → ℝD2 . s Let Cstd denote the standard (local) Zygmund–Hölder space of order s on M.10 Suppose: r – W α u ∈ Cstd , ∀ degds(α) ≤ κ, s – F(x, ζ ) is Cstd in the x variable and C ∞ in the ζ variable, for x near x0 , α – u 󳨃→ F(x, {W u(x)}degds (α)≤κ ) is maximally subelliptic at (x0 , u) of degree κ with respect to (W , ds) in the sense of Definition 1.1.11. Then, ∀ϵ > 0, F(x, {W α u(x)}deg

ds (α)≤κ

s+κ/λstd −ϵ

s ) ∈ Cstd near x0 󳨐⇒ u ∈ Cstd

near x0 ,

(1.20)

where λstd = λstd (x0 , (W , ds)) is the same constant as in (1.17).11 See Corollary 9.1.8 for a more precise and more general version of this result and for other related regularity properties. Except for the loss of an arbitrarily small ϵ > 0, (1.20) is sharp – indeed, as described in Corollary 8.2.3, it is sharp for linear equations. We do not know if this loss of ϵ is necessary. See Section 9.1.1 for a further discussion of these results, along with a statement of regularity theory in terms of the Zygmund–Hölder spaces C s (Z, dr), with respect to a different set of Hörmander vector fields with formal degrees (Z, dr). See Section 9.3 for a description of blowup near the boundary for solutions in terms of weighted Zygmund– Hölder norms. 1.7.1 The inverse function theorem Much like classical elliptic theory, a main source of intuition for our proof of the fully nonlinear regularity theory comes from a standard proof of the inverse function theos 10 In local coordinates, Cstd = C s (ℝn ).

11 In particular, in the special case where ds1 = ⋅ ⋅ ⋅ = dsr = 1, we have λstd = m, where W1 , . . . , Wr satisfy Hörmander’s condition of order exactly m at x0 . See Definition 6.6.15 for the formula for λstd (x0 , (W , ds)) in the general setting.

22 � 1 Introduction rem in Banach spaces: we use a fixed point argument to prove regularity of fully nonlinear equations. We present this proof of the inverse function theorem, and then describe how it informs our proof of (1.19). Here X and Y will both denote Banach spaces. Given a function f : U → Y , where U ⊆ X is open, we say f is C k if it is k times continuously Fréchet differentiable. Theorem 1.7.1 (Banach space inverse function theorem). Suppose f : U → Y is C k , where k ≥ 1, and U ⊆ X is open. Fix x0 ∈ U. If df (x0 ) : X → Y has a bounded left inverse, Y → X , then there exists a ball B ⊆ U, centered at x0 , and an open set V ⊆ Y such that 󵄨 f 󵄨󵄨󵄨B : B → V has a left inverse g : V → B which is C k . Remark 1.7.2. In short, Theorem 1.7.1 says that if df (x0 ) is left invertible, then f is left invertible near x0 . A similar proof shows that if df (x0 ) is invertible, then f is invertible near x0 . A similar result can also be stated when df (x0 ) is right invertible. However, it is the proof of the case where df (x0 ) is left invertible which is most informative for our applications. Lemma 1.7.3. Suppose U ⊆ X is an open set, α ∈ (0, 1), r : U → X , F(x) = x + r(x) for x ∈ U, and r satisfies ‖r(x) − r(y)‖X ≤ α‖x − y‖X ,

∀x, y ∈ U.

(1.21)

Then F(U) is open in X and F : U → F(U) is a homeomorphism. Proof. First note that from (1.21), we have ‖x − y‖X = ‖F(x) − F(y) − (r(x) − r(y))‖X

≤ ‖F(x) − F(y)‖X + ‖r(x) − r(y)‖X ≤ ‖F(x) − F(y)‖X + α‖x − y‖X ,

from which it follows that ‖x −y‖X ≤ (1−α)−1 ‖F(x)−F(y)‖X , and therefore F is injective on U. Let V := F(U) and set G := F −1 : V → U, which exists since F is injective. We next show V is open. Take z0 = F(x0 ) = x0 + r(x0 ) ∈ V ; we wish to show that for z close to z0 , there exists x ∈ U with F(x) = z, or equivalently x = z − r(x). Fix δ > 0 such that BX (x0 , δ) ⊆ U and z ∈ BX (z0 , (1 − α)δ), where BX (x, δ) denotes the ball of radius δ, centered at x, in X . Define Tz (x) := z − r(x).

We wish to find x ∈ U such that Tz (x) = x. We will do this by showing that Tz : BX (x0 , δ) → BX (x0 , δ) is a strict contraction. Using (1.21), for x ∈ BX (x0 , δ), ‖Tz (x) − x0 ‖X = ‖Tz (x) − Tz0 (x0 )‖X = ‖z − z0 − (r(x) − r(x0 ))‖X ≤ ‖z − z0 ‖X + α‖x − x0 ‖X < (1 − α)δ + αδ = δ,

1.7 Fully nonlinear theory

�

23

showing that Tz : BX (x0 , δ) → BX (x0 , δ). Also, using (1.21), ‖Tz (x) − Tz (y)‖X = ‖r(x) − r(y)‖X ≤ α‖x − y‖X ,

(1.22)

so Tz is a strict contraction. The contraction mapping principle shows that there is a unique fixed point x ∈ BX (x0 , δ) ⊆ U such that Tz (x) = x. Since x = Tz (x) = z − F(x) + x, we see that F(x) = z, i. e., G(z) = x. This also shows B(z0 , (1 − α)δ) ⊆ F(U) = V , and therefore V is open. Finally, we wish to show that G is continuous. Using that G(z) is a fixed point of the map Tz and (1.22), for z, w ∈ V we have ‖G(z) − G(w)‖X = ‖Tz (G(z)) − Tw (G(w))‖X ≤ ‖Tz (G(z)) − Tw (G(z))‖X + ‖Tw (G(z)) − Tw (G(w))‖X ≤ ‖Tz (G(z)) − Tw (G(z))‖X + α‖G(z) − G(w)‖X . We conclude that ‖G(z) − G(w)‖X ≤ (1 − α)−1 ‖Tz (G(z)) − Tw (G(z))‖X = (1 − α)−1 ‖z − w‖X . It follows that G = F −1 : V → B is continuous, completing the proof. Proof of Theorem 1.7.1. Let Lx0 be a left inverse to df (x0 ). Define F(x) := Lx0 f (x) : X → X and set r(x) := F(x) − x. Note that dF(x0 ) = I, where I denotes the identity map X → X . Thus, dr(x0 ) = 0. By choosing a small enough open ball, B ⊆ X , centered at x0 , we have supx∈B ‖dr(x)‖X →X ≤ 1/2; therefore, for x, y ∈ B, ‖r(x) − r(y)‖X ≤ 21 ‖x − y‖X . 󵄨 Lemma 1.7.3 applies to F 󵄨󵄨󵄨B : B → X to show that F(B) ⊆ X is open and F has a continuous inverse G : F(B) → B. Set V := L−1 x0 (B) (the set theoretic pre-image), which is open in Y , and set g(y) := G(Lx0 (y)) : V → B. Note that for x ∈ B, we have g(f (x)) = G(F(x)) = x. Finally, it follows from the chain rule that G is C k ; therefore, g is C k as well. A recap of the proof of Theorem 1.7.1 is given now. To find a local left inverse to f , we apply the contraction mapping principle to Lx0 f (x), where Lx0 is a left inverse for df (x0 ). When we consider a nonlinear equation written in the form F(x, {W α u(x)}deg

ds (α)≤κ

),

(1.23)

24 � 1 Introduction understanding regularity theory for u near x0 is akin to finding a left inverse, near u and x0 , modulo smooth functions, for the map v 󳨃→ F(x, {W α v(x)}deg

ds (α)≤κ

),

C

s+κ

(W , ds) → C s (W , ds).

(1.24)

The assumption that (1.23) is maximally subelliptic at (x0 , u) of degree κ with respect to (W , ds) is the same as saying Px0 ,u , given by (1.9), is maximally subelliptic of degree κ with respect to (W , ds) on a neighborhood of x0 . Px0 ,u is not exactly the derivative of the map (1.24), but near x0 is a perturbation of this derivative. As described in Section 1.6, since Px0 ,u is maximally subelliptic, it has a local left parametrix Tx0 ,u which is locally in A −κ (W , ds). This parametrix takes C s (W , ds) → C s+κ (W , ds) (locally). Following the proof of Theorem 1.7.1, we will prove the regularity of u by applying the contraction mapping principle to a localization of v 󳨃→ Tx0 ,u F(x, {W α v(x)}deg

ds (α)≤κ

),

which takes C s+κ (W , ds) → C s+κ (W , ds). Unfortunately, the proof is not quite as simple as the proof of Theorem 1.7.1. One main issue is that Tx0 ,u is not actually an inverse for Px0 ,u , and we need to introduce cut-off functions to localize all our computations. Due to these cut-off functions, the contraction mapping principle does not directly apply. However, by employing a covering argument inspired by a work of Simon [212], we are able to get around these issues. See Section 9.2 for details. Remark 1.7.4. There are other inverse function theorems which often appear in the study of PDEs. Notably, the Nash–Moser inverse function theorem [105] is a powerful tool which applies to a wide class of fully nonlinear PDEs, a much larger class of PDEs than the Banach space inverse function theorem (Theorem 1.7.1) applies to. However, when it applies, the Banach space inverse function theorem often yields sharper and stronger results than the Nash–Moser inverse function theorem does in general. A main theme of this text, especially in Chapter 9, is that ideas from the Banach space inverse function theorem apply to maximally subelliptic equations, once the right spaces are chosen, and sharp results can be deduced.

1.8 Multi-parameter theory When proving results for maximally subelliptic operators with respect to the adapted s s Besov (Bp,q (W , ds)) and Triebel–Lizorkin (Fp,q (W , ds)) spaces we employ the classical (single-parameter) Calderón–Zygmund theory to understand the parametrix and function spaces. However, when we prove results with respect to function spaces which are not adapted to the vector fields (W , ds) (for example when we prove results concerning

1.8 Multi-parameter theory

�

25

s s the standard Besov [Bp,q,std ] and Triebel–Lizorkin [Fp,q,std ] spaces), classical Calderón– Zygmund theory is no longer sufficient because more than one geometry is relevant. To deal with this issue, we turn to multi-parameter singular integrals and function spaces of the type originally studied in [220]. For example, when we study the regularity theory of fully nonlinear maximally subelliptic equations with respect to the stan∙ dard Zygmund–Hölder spaces, Cstd , we employ an algebra of singular integral operators which (locally) contains both the parametrices for maximally subelliptic operators and standard pseudo-differential operators. This algebra is filtered by ℝ2 denoting the two regularity parameters: one for the Carnot–Carathéorody geometry adapted to (W , ds) and one for the standard Riemannian geometry. Fix ν ∈ ℕ+ . We will now discuss the ν-parameter theory. We are given ν finite sets of vector fields with formal degrees: μ

μ

(W μ , dsμ ) := {(W1 , ds1 ), . . . , (Wrμμ , dsμrμ )}, μ

1 ≤ μ ≤ ν,

μ

where each dsj ∈ ℕ+ and Wj is a smooth vector field; we will assume various condiμ

tions on these sets, depending on the context. For each μ, the balls B(W μ ,dsμ ) (x, δμ ), given by (1.13), induce a geometry on M. These ν geometries might interact in complicated ways. Let e1 , . . . , eν denote the standard basis in ℝν . Define (W , ds)⃗ = {(W1 , ds1⃗ ), . . . , (Wr , dsr⃗ )} μ

μ

:= {(Wj , dsj eμ ) : 1 ≤ μ ≤ ν, 1 ≤ j ≤ rμ },

(1.25)

so that (W , ds)⃗ is a list of smooth vector fields, Wj , each paired with a degree dsj⃗ ∈ ℕν \{0}. Usually, we will consider the case where (W 1 , ds1 ) are Hörmander vector fields with formal degrees, and we will study maximally subelliptic operators with respect to (W 1 , ds1 ). Example 1.8.1. Let M = ℝn and suppose (W 1 , ds1 ) are Hörmander vector fields with formal degrees on ℝn . Given an either linear or fully nonlinear, maximally subelliptic operator with respect to (W 1 , ds1 ), if we wish to study regularity theory with respect to the classical Besov and Triebel–Lizorkin spaces, we take ν = 2 and (W 2 , ds2 ) = {(𝜕x1 , 1), . . . , (𝜕xn , 1)}. We always assume that (W 1 , ds1 ), . . . , (W ν , dsν ) “locally weakly approximately commute” as in Definition 3.8.5. Under an additional finite type condition on each of the (W μ , dsμ ) we introduce and study the following: s⃗ ⃗ and Triebel–Lizorkin (F s⃗ (W , ds)) ⃗ – In Chapter 6, we introduce Besov (Bp,q (W , ds)) p,q ν spaces indexed by a regularity parameter s⃗ ∈ ℝ . These are defined in such a way ⃗ ⃗ ⃗ ⃗ i. e., Wj acts as a differential operator of degree that Wj : X s (W , ds)⃗ → X s−dsj (W , ds); ν dsj⃗ ∈ ℕ .

26 � 1 Introduction –

In Chapter 5, we introduce two filtered algebras of singular integral operators in⃗ ̃s⃗ (W , ds). ⃗ dexed by s⃗ ∈ ℝν : A s (W , ds)⃗ and A ⃗ t⃗ s ⃗ then TS ∈ A s+⃗ t⃗(W , ds); ⃗ a similar result can – If T ∈ A (W , ds)⃗ and S ∈ A (W , ds), ∙ ∙ ̃ ⃗ ⃗ be obtained with A (W , ds) replaced by A (W , ds). ⃗ – A s (W , ds)⃗ is a generalization of the Calderón–Zygmund theory we use in the single-parameter case, and is only defined in the special case where all (W μ , dsμ ) are Hörmander vector fields with formal degrees, which pairwise “locally weakly approximately commute.” ⃗ ̃s⃗ (W , ds)⃗ and moreover A ̃s⃗ (W , ds)⃗ is defined even when (W μ , dsμ ) – A s (W , ds)⃗ ⊆ A do not satisfy Hörmander’s condition. This is important because the assumption of Hörmander’s condition for all (W μ , dsμ ) is not quantitatively preserved ̃s⃗ (W , ds)⃗ provides an algebra with good scaling under our scaling arguments. A properties. – If P is a maximally subelliptic operator of degree κ with respect to (W 1 , ds1 ) on ̃−κe1 (W , ds)⃗ and locally in M, then P has a left parametrix which is locally in A −κe1 A (W , ds)⃗ when it is defined. – When M = ℝn , if (W μ , dsμ ) = {(𝜕x1 , 1), . . . , (𝜕xn , 1)}, then standard pseudõkeμ (W , ds)⃗ differential operators of order k ∈ ℝ with compact support are in A keμ and in A (W , ds)⃗ when it is defined. See Section 5.8.2. ̃t⃗(W , ds), ⃗ then T : X s⃗ (W , ds)⃗ → X s−⃗ t⃗(W , ds), ⃗ where X s⃗ denotes either – If T ∈ A s s Bp,q or Fp,q . ⃗

⃗

The multi-parameter theory is useful when studying both linear and nonlinear equations, but its use is especially transparent in nonlinear theory. We describe here how it can be used to prove (1.20); the same ideas are used to prove more general results in Chapter 9. Let (W 1 , ds1 ) = {(W11 , ds11 ), . . . , (Wr11 , ds1r1 )} be Hörmander vector fields with (singleparameter) formal degrees on ℝn . Consider a nonlinear operator of the form α

v 󳨃→ F(x, {(W 1 ) v(x)}deg

ds1

(1.26)

(α)≤κ )

which is maximally subelliptic at (x0 , u) of degree κ with respect to (W 1 , ds1 ). For simplicity in this section we take F(x, ζ ) ∈ C ∞ , though we deal with finite smoothness when we present this result in Corollary 9.1.8. Suppose we wish to prove (1.20) using a contraction mapping argument as described in Section 1.7.1. We immediately run into a problem. Thought of as a map on the standard s+κ s Zygmund–Hölder spaces, the best we can say in general is that (1.26) maps Cstd → Cstd , near x0 . However, if we define Pu,x0 by (1.9) and let Tu,x0 be a left parametrix for Pu,x0 s+κ/λstd

s near x0 , then (by (1.17)) the best we can say is that Tu,x0 : Cstd → Cstd that α

v 󳨃→ Tu,x0 F(x, {(W 1 ) v(x)}deg

ds (α)≤κ

s+κ/λstd

s+κ ) : Cstd → Cstd

,

. We conclude

near x0 .

(1.27)

1.9 The main tool: scaling

�

27

Unless we are in the elliptic setting we have λstd > 1, so (1.27) maps from one space to a different space, and we cannot use the contraction mapping principle. Instead, we work with a multi-parameter theory. We set (W 2 , ds2 ) := {(𝜕x1 , 1), . . . , ⃗ ⃗ ⃗ (𝜕x , 1)}, ν = 2, and define (W , ds)⃗ as in (1.25). For s⃗ ∈ (0, ∞)2 , set C s (W , ds)⃗ := B s (W , ds). ∞,∞

n

⃗ ⃗ ⃗ near x0 , and Tu,x : C s⃗ (W , ds)⃗ → Then, we will show (1.26) maps C s+κe1 (W , ds)⃗ → C s (W , ds), 0 ⃗s+κe1 ⃗ In particular, we have C (W , ds). α

v 󳨃→ Tu,x0 F(x, {(W 1 ) v(x)}deg

ds (α)≤κ

⃗ ⃗ ⃗ ) : C s+κe1 (W , ds)⃗ → C s+κe1 (W , ds),

near x0 , (1.28)

and we will be able to apply a contraction mapping type argument to (1.28) in the Banach ⃗ ⃗ space C s+κe1 (W , ds). To conclude (1.20) we argue roughly as follows (see the proof of Proposition 9.1.6 for full details in a more general case): r s – We start with u ∈ Cstd such that F(x, {W α u(x)}degds (α)≤κ ) ∈ Cstd . ′ – Fix ϵ ∈ (0, r) small. We show that there exist ϵ > 0 with: ′ r – u ∈ Cstd ⊆ C (ϵ ,r−ϵ) (W , ds)⃗ near x0 , ′ – F(x, {W α u(x)}deg (α)≤κ ) ∈ C s ⊆ C (ϵ ,s−ϵ) (W , ds)⃗ near x0 . –

–

ds

std

′

Using (1.28) we apply a contraction mapping type argument in C (ϵ +κ,s−ϵ) (W , ds)⃗ to ′ show u ∈ C (ϵ +κ,s−ϵ) (W , ds)⃗ near x0 . This is the main (and most difficult) step of the proof. We then show that s+κ/λ −ϵ u ∈ C (ϵ +κ,s−ϵ) (W , ds)⃗ ⊆ C (ϵ ,s+κ/λstd −ϵ) (W , ds)⃗ ⊆ Cstd std , ′

′

near x0 . In particular, the loss of ϵ derivatives comes from the conversion between the singleparameter Zygmund–Hölder spaces and the multi-parameter Zygmund–Hölder spaces. When working directly with multi-parameter Zygmund–Hölder spaces, our regularity results are sharp and generalize (1.19); see Theorem 9.1.14.

1.9 The main tool: scaling While most tools from classical elliptic theory (e. g., the Fourier transform) do not translate to the maximally subelliptic setting, there is one tool which does: scaling. Scaling for elliptic equations is very simple: we use the standard dilation maps on ℝn . When we turn to the maximally subelliptic setting, we cannot write down such a simple formula in general, and instead satisfy ourselves by establishing that scaling maps with appropriate properties exist.

28 � 1 Introduction 1.9.1 Scaling for elliptic operators We begin by explaining how scaling works for elliptic operators, which is very simple, and a standard tool in elliptic theory. For each x ∈ ℝn and δ > 0, we define the scaling map ∼

Φx,δ (t) = x + δt : Bn (1) 󳨀 → Bn (x, δ), where Bn (x, δ) := {y ∈ ℝn : |x − y| < δ} and Bn (1) := Bn (0, 1). Let E := ∑|α|≤κ aα (x)𝜕xα be an elliptic operator on ℝn with smooth coefficients. Set ∗

κ

Ex,δ := Φx,δ δ E (Φx,δ )∗ = ∑ aα (x + δt)δ

κ−|α|

|α|≤κ

𝜕t .

It is not hard to see that Ex,δ is an elliptic operator on Bn (1), uniformly as x ranges over compact sets and δ ∈ (0, 1]. Because of this, if we want to study the elliptic operator E on a small ball Bn (x, δ), it suffices to instead study a different elliptic operator, Ex,δ , on the unit ball Bn (1). On the Fourier transform side, this means that if we want to study high frequencies for an elliptic operator, it suffices to instead study low frequencies for a different elliptic operator. It is this invariance under scaling that makes the classical Calderón–Zygmund theory on ℝn an important tool when studying elliptic operators. 1.9.2 Scaling for maximally subelliptic operators Let (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} be Hörmander vector fields with formal degrees on M. Fix a compact set 𝒦 ⋐ M. For each x ∈ 𝒦 and δ ∈ (0, 1], Nagel, Stein, and Wainger [189] introduced a scaling map, ∼

Φx,δ : Bn (1) 󳨀 → Φx,δ (Bn (1)) ≈ B(W ,ds) (x, δ), where Φx,δ (Bn (1)) is not exactly equal to B(W ,ds) (x, δ), but is comparable to it. Let P := ∑degds (α)≤κ aα (x)W α be maximally subelliptic of degree κ with respect to (W , ds) on M. We have the following: – Define Wjx,δ := Φ∗x,δ δdsj Wj and (W x,δ , ds) := {(W1x,δ , ds1 ), . . . , (Wrx,δ , dsr )}.

–

Then (W x,δ , ds) are Hörmander vector fields with formal degrees uniformly for x ∈ 𝒦 and δ ∈ (0, 1]. Set Px,δ := Φ∗x,δ δκ P (Φx,δ )∗ . Then Px,δ is maximally subelliptic of degree κ with respect to (W x,δ , ds) on Bn (1), uniformly for x ∈ 𝒦 and δ ∈ (0, 1].

1.10 Outline

� 29

See Theorem 3.3.7 for a much more detailed discussion of these maps and Section 3.3.1 for a more detailed discussion of scaling maximally subelliptic operators; the main point is that every estimate we need for our proofs holds uniformly for x ∈ 𝒦 and δ ∈ (0, 1]. Because of the above, and just as in the elliptic setting, to study a maximally subelliptic operator at a small (Carnot–Carathéodory) scale B(W ,ds) (x, δ), it suffices to instead study a different maximally subelliptic operator at the unit scale Bn (1). This is the key fact needed to begin applying the Calderón–Zygmund theory, with respect to the Carnot– Carathéodory metric, to these operators. In our proofs, we iterate these scaling ideas several times, leading to several “layers” of scaling. In elliptic theory, adding such layers adds no complications because the scaling maps are so simple. However, in our setting, this means that we need results which are more quantitative – uniformity on compact sets is not sufficient. To achieve these more quantitative results and to allow us to study the multi-parameter theory described in Section 1.8, we use recent work on these scaling maps [228], which is based on an idea of Tao and Wright [232]. This technical, quantitative result, described in Section 3.6 and Appendix A, is the central tool in this text and forms the foundation for the proof of every main result.

1.10 Outline –

–

–

– –

Chapter 2: We describe classical results from elliptic theory which we generalize in later chapters. Many of our proofs are much simpler (and well known) in the classical setting. The proofs given in Chapter 2 may help the reader better understand the logical structure of the more abstract proofs given in the maximally subelliptic setting. Chapter 3: We study vector fields and Carnot–Carathéodory geometry. This chapter is the toolbox we need when studying maximally subelliptic operators. In particular, we define and study the scaling maps described in Section 1.9. The reader interested in understanding the proofs for maximally subelliptic operators might only skim this chapter on a first pass, referring back whenever a particular tool is needed. Chapter 4: We introduce generalized pseudo-differential operators corresponding to a finite list of vector fields with multi-parameter formal degrees as discussed in Section 1.8. These pseudo-differential operators provide several useful examples of the singular integral operators studied in Chapter 5. Chapter 5: We study singular integral operators, both in the single-parameter case described in Section 1.4 and in the multi-parameter case described in Section 1.8. Chapter 6: We study the Besov and Triebel–Lizorkin spaces described in Section 1.5, both in the single-parameter case and in the multi-parameter case described in Section 1.8.

30 � 1 Introduction –

– – –

s Chapter 7: We focus on the Zygmund–Hölder spaces C s = B∞,∞ . We study compositions of functions as well as characterizations of these spaces in terms of differences. Chapter 8: We develop the linear theory of maximally subelliptic operators. Chapter 9: We develop the fully nonlinear theory of maximally subelliptic operators. Appendix A: We present the proof from [228] of the main technical theorem underlying all the scaling results in Chapter 3. The results in Appendix A form the heart of every main proof in this text. ⃗

⃗

2 Ellipticity Maximally subelliptic partial differential operators are a generalization of elliptic partial differential operators, and all the main results in this text are generalizations of classical results concerning elliptic operators. Elliptic operators may be the most wellunderstood class of partial differential operators: not only is the linear theory of elliptic PDEs well understood, but many sharp results are known for general fully nonlinear elliptic PDEs. In the classical study of elliptic PDEs, the Fourier transform and the group structure on ℝn play a central role. In the more general setting of maximally subelliptic operators, these tools are not as applicable. Thus, while we will see that many of the classical results from linear and nonlinear elliptic PDEs generalize to the case of maximally subelliptic PDEs, the classical proofs do not generalize. In this chapter, we will reframe these classical results in such a way that the group structure and the Fourier transform play a secondary role. These reframed results and methods will generalize to maximally subelliptic operators more directly. 1 1 Set Dx = (Dx1 , . . . , Dxn ) := 2πi 𝜕x = 2πi (𝜕x1 , . . . , 𝜕xn ), where 𝜕xj = 𝜕x𝜕 . Consider a linear j

partial differential operator of order m on ℝn with smooth coefficients: α

P := ∑ aα (x)Dx ,

aα : ℝn → ℂ smooth.

|α|≤m

Corresponding to P is its characteristic polynomial or symbol P(x, ξ) := ∑ aα (x)ξ α , |α|≤m

with principal symbol Pm (x, ξ) := ∑|α|=m aα (x)ξ α . Definition 2.0.1. We say P is elliptic near x0 ∈ ℝn if Pm (x0 , ξ) = 0 ⇐⇒ ξ = 0. For Ω ⊆ ℝn open, we say P is elliptic on Ω if P is elliptic near x0 , ∀x0 ∈ Ω. The key result that we will focus on, and the reason fully nonlinear elliptic PDEs are so well understood, is that elliptic operators are locally left invertible modulo smooth functions on many spaces. For example, suppose P is a partial differential operator of p order m, which is elliptic near x0 . Let u be a distribution on ℝn . Then letting Ls denote the Lp Sobolev space of order s ∈ ℝ (see Definition 2.2.11), a standard result is, for 1 < p < ∞, p

p

P u ∈ Ls near x0 ⇐⇒ u ∈ Ls+m near x0 ; https://doi.org/10.1515/9783111085647-002

32 � 2 Ellipticity see Proposition 2.2.35. It is not just Sobolev spaces which enjoy this property. Indeed, for l ∈ ℕ and r ∈ [0, 1], let C l,r denote the standard Hölder space on ℝn (see (2.44)). Then, for l ∈ ℕ and r ∈ (0, 1), Pu ∈ C

l,r

near x0 ⇐⇒ u ∈ C l+m,r near x0 .

(2.1)

This does not hold for r ∈ {0, 1}; however, one can generalize (2.1) to integer exponents by using the classical Zygmund–Hölder spaces (see Section 2.5). We denote these by C s , s > 0. We have C l,r = C l+r for r ∈ ̸ {0, 1} and s

P u ∈ C near x0 ⇐⇒ u ∈ C

s+m

near x0 ;

see Theorem 2.7.2 (e). There is a more general framework to understand the above results: the Besov and s Triebel–Lizorkin spaces (see Section 2.4). We denote the scale of Besov spaces by Bp,q , s where 1 ≤ p, q ≤ ∞, and s ∈ ℝ, and the scale of Triebel–Lizorkin spaces by Fp,q , where p

s 1 < p < ∞, 1 < q ≤ ∞, and s ∈ ℝ.1 We have Ls = Fp,2 for 1 < p < ∞ and s ∈ ℝ, and s s C = B∞,∞ and s > 0. Thus, one many generalize the above results to s

s+m

s

s+m

P u ∈ Bp,q near x0 ⇐⇒ u ∈ Bp,q near x0 , P u ∈ Fp,q near x0 ⇐⇒ u ∈ Fp,q near x0 ;

see Theorem 2.7.2 (e). It is these results which will naturally generalize to the case of maximally subelliptic operators. Example 2.0.2. The Laplacian on ℝn , △ = −𝜕x21 − 𝜕x22 − ⋅ ⋅ ⋅ − 𝜕x2n , is an elliptic operator. Its principal symbol is 4π 2 |ξ|2 .

2.1 Some basic function spaces For any topological space Ω, we let C(Ω) denote the space of bounded, continuous functions f : Ω → ℂ endowed with the norm ‖f ‖C(Ω) := sup |f (x)|. x∈Ω

With this norm, C(Ω) is a Banach space. Let Ω ⊆ ℝn be an open set. For m ∈ ℕ let C m (Ω) := {f ∈ C(Ω) : 𝜕xα f ∈ C(Ω), ∀|α| ≤ m}, ‖f ‖C m (Ω) := ∑

‖𝜕xα f ‖C(Ω) ,

|α|≤m

1 It is possible to define these spaces for all positive p, q; see Remark 2.4.6.

(2.2)

2.1 Some basic function spaces

� 33

m so that C m (Ω) is a Banach space. Let Cloc (Ω) be the space of those f : Ω → ℂ such that 󵄨 m ∀x ∈ Ω, there exists an open neighborhood V of x such that f 󵄨󵄨󵄨V ∈ C m (V ); Cloc (Ω) is a Fréchet space (see [238, Chapter 10]). Similarly, for any function space we define the local version by adding a subscript loc. We let

C ∞ (Ω) := ⋂ C m (Ω), m∈ℕ

∞ m Cloc (Ω) := ⋂ Cloc (Ω), m∈ℕ

(2.3)

and give these spaces the usual Fréchet topologies (see [238, Chapter 10]). We let C0∞ (Ω) be the space of all f ∈ C ∞ (Ω) such that supp(f ) ⋐ Ω. Furthermore, we give C0∞ (Ω) the structure of a locally convex topological vector space in the usual way (see [238, Chapter 13, Example 2]). For all of the above spaces, we define the analogous spaces of functions taking values in a Banach space X in the obvious way, and denote these spaces by C(Ω; X ), m C m (Ω; X ), Cloc (Ω; X ), C0∞ (Ω; X ), etc. We let S (ℝn ) denote the space of Schwartz functions on ℝn , that is, the space of all f ∈ C ∞ (ℝn ) such that for each α, β ∈ ℕn the following semi-norm is finite: ‖f ‖α,β := sup |x α 𝜕xβ f (x)|. x∈ℝn

We give S (ℝn ) the coarsest topology such that all such semi-norms are continuous. Often, given a locally convex topological vector space, X , we are interested in the bounded subsets of X , defined as follows. Definition 2.1.1. A subset ℬ ⊆ X of a locally convex topological vector space X is said to be a bounded subset of X if for every continuous semi-norm ‖ ⋅ ‖ on X we have sup ‖f ‖ < ∞. f ∈ℬ

Remark 2.1.2. For most of the above spaces, the bounded subsets are easy to understand directly from the definition. However, the semi-norms which induce the topology on C0∞ (Ω) are a bit more complicated. Fortunately, the bounded subsets have a simple characterization: ℬ ⊂ C0∞ (Ω) is bounded if and only if the following two conditions hold: – ⋃f ∈ℬ supp(f ) ⋐ Ω, – for every m ∈ ℕ, supf ∈ℬ ‖f ‖C m < ∞. For a proof of this fact, see [238, Proposition 14.6 and Chapter 13, Example 2]. In C0∞ (Ω), the bounded sets are precisely the pre-compact sets. Given a locally convex topological vector space X , we denote by X ′ its dual. Thus, ∞ C0∞ (Ω)′ is the usual space of distributions on Ω, Cloc (Ω)′ is the space of distributions with compact support in Ω (see [238, Theorem 24.2]), and S (ℝn )′ is the space of tempered distributions. We give these spaces the usual weak dual topology.

34 � 2 Ellipticity Given a locally integrable function f : Ω → ℂ, we obtain a distribution by ϕ 󳨃→ ∫ f (x)ϕ(x) dx.

(2.4)

Thus, given a function, we may treat it as a distribution. Given an arbitrary distribution λ ∈ C0∞ (Ω)′ we abuse notation and for ϕ ∈ C0∞ (Ω), we write λ(ϕ) = ∫ λ(x)ϕ(x) dx, whether or not λ is given by integration against a locally integrable function. If we say that λ agrees with a locally integrable function, this means that there exists f such that λ is given by (2.4). In this case, we identify λ with f , and may then treat it as a function. If X1 and X2 are two locally convex topological vector spaces and we write T : X1 → X2 , this means that T is a continuous linear map from X1 to X2 . We write Hom(X1 , X2 ) for the space of all such continuous linear maps. The Schwartz kernel theorem gives a convenient characterization of certain spaces of linear maps. Theorem 2.1.3 (The Schwartz kernel theorem). Given K(x, y) ∈ C0∞ (Ω × Ω)′ , we obtain T ∈ Hom(C0∞ (Ω), C0∞ (Ω)′ ) by, for ϕ1 , ϕ2 ∈ C0∞ (Ω), ∫ ϕ2 (x)(Tϕ1 )(x) dx = ∬ ϕ2 (x)ϕ1 (y)K(x, y) dxdy. ∞ ∞ ∞ We similarly define maps Cloc (Ω × Ω)′ → Hom(Cloc (Ω), Cloc (Ω)′ ) and S (ℝn × ℝn )′ → n n ′ Hom(S (ℝ ), S (ℝ ) ). The above maps are all bijections. The distribution K is called the Schwartz kernel of the operator T.

Henceforth, we identify operators with their Schwartz kernel. Thus, given T : C0∞ (Ω) → C0∞ (Ω)′ , we sometimes treat T as a distribution T(x, y) ∈ C0∞ (Ω×Ω)′ , and given a distribution T(x, y) ∈ C0∞ (Ω × Ω)′ , we sometimes treat T as an operator T : C0∞ (Ω) → C0∞ (Ω)′ . Moreover, given a locally integrable function F(x, y) : Ω × Ω → ℂ, we identify F with a distribution on Ω × Ω via (2.4), and therefore with a map F : C0∞ (Ω) → C0∞ (Ω)′ . For a proof of the Schwartz kernel theorem, as well as an elegant description of a more general result, see [238, Chapter 51]. Definition 2.1.4. Given two locally convex topological vector spaces X1 and X2 , the topology of bounded convergence on Hom(X1 , X2 ) can be defined as follows. Given a bounded set ℬ ⊂ X1 and a continuous semi-norm | ⋅ | on X2 , define a semi-norm on Hom(X1 , X2 ) by ‖T‖ := sup |Tf |. f ∈ℬ

The topology of bounded convergence is the coarsest topology such that all of the above semi-norms are continuous.

2.2 Pseudo-differential operators and the Fourier transform

� 35

If (M, μ) is a measure space, we define Lp (M, μ) for p ∈ [1, ∞] in the usual way. When M is an open subset of ℝn , we give M Lebesgue measure unless otherwise mentioned. So, for example, Lp (ℝn ) denotes the usual Lp space on ℝn . Definition 2.1.5. For a distribution λ ∈ C0∞ (Ω)′ , we define λ by λ(ϕ) = λ(ϕ),

ϕ ∈ C0∞ (Ω).

Definition 2.1.6. Given any operator T ∈ Hom(C0∞ (Ω), C0∞ (Ω)′ ), we define the formal L2 adjoint of T, denoted by T ∗ , where T ∗ ∈ Hom(C0∞ (Ω), C0∞ (Ω)′ ), by ∫ ψ(x)(Tϕ)(x) dx = ∫(T ∗ ψ)(x)ϕ(x) dx,

ϕ, ψ ∈ C0∞ (Ω).

∞ We use a similar definition with C0∞ (Ω) replaced with Cloc (Ω) or S (Ω) throughout.

2.2 Pseudo-differential operators and the Fourier transform A central tool in the study of elliptic PDEs is the Fourier transform. Definition 2.2.1. Given f ∈ S (ℝn ), we define the Fourier transform of f , f ̂(ξ) : ℝn → ℂ, by f ̂(ξ) := ∫ f (x)e−2πix⋅ξ dx. We have the following well-known result. Proposition 2.2.2. The Fourier transform f 󳨃→ f ̂ is an automorphism of the Fréchet space S (ℝn ) with inverse g 󳨃→ ǧ given by ̌ g(x) := ∫ g(ξ)e2πix⋅ξ dξ. The Fourier transform allows us to introduce another key tool in the study of elliptic operators: pseudo-differential operators. To a tempered distribution a ∈ S (ℝn × ℝn )′ , we may associate a continuous linear map a(x, D) : S (ℝn ) → S (ℝn )′ by ∫ g(x)(a(x, D)f )(x) dx := ∬ g(x)e2πix⋅ξ f ̂(ξ)a(x, ξ) dxdξ. An analog of the Schwartz kernel theorem (Theorem 2.1.3) states that the map a 󳨃→ a(x, D) is a bijection S (ℝn × ℝn )′ → Hom(S (ℝn ), S (ℝn )′ ). The tempered distribution a(x, ξ) is called the symbol of the operator a(x, D) and for appropriate choices of a, operators written in this way are called pseudo-differential operators. We now introduce the class of symbols which we will use. The class that follows is the mostly commonly used class of symbols, and we refer to them as “standard symbols.”

36 � 2 Ellipticity Definition 2.2.3. Fix m ∈ ℝ. The space of standard symbols of order m, S m , is the ∞ Fréchet space consisting of functions a ∈ Cloc (ℝn × ℝn ), which satisfy, for all multin indices α, β ∈ ℕ , 󵄨󵄨 α β 󵄨 m−|β| . 󵄨󵄨𝜕x 𝜕ξ a(x, ξ)󵄨󵄨󵄨 ≤ Cα,β (1 + |ξ|) We give S m the coarsest topology such that the least possible Cα,β defines a continuous semi-norm on S m (for each α, β). Operators of the form a(x, D) for a ∈ S m are called pseudo-differential operators of order m. Remark 2.2.4. It can be shown that for a ∈ S m , a(x, D) : S (ℝn ) → S (ℝn ). The first example to consider, and the one that justifies the name, is that of linear partial differential operators with smooth coefficients. Example 2.2.5. Consider a linear partial differential operator of order m ∈ ℕ+ , P(x, D), of the form P(x, D)f (x) = ∑ aα (x)Dα f (x), |α|≤m

where aα ∈ C ∞ (ℝn ) (see (2.3)). Then P(x, D) is a pseudo-differential operator of order m, with symbol P(x, ξ) = ∑|α|≤m aα (x)ξ α . We state the next lemma without proof. Lemma 2.2.6. Let a(x, ξ) ∈ S m and let a(x, D)∗ be the formal L2 adjoint of a(x, D) (see Definition 2.1.6). Then a(x, D)∗ is a pseudo-differential operator of order m. Moreover, one can obtain an asymptotic expansion for the symbol of a(x, D)∗ in terms of a; see [216, Chapter VI, Section 6.2]. Remark 2.2.7. Combining Lemma 2.2.6 and Remark 2.2.4, for a ∈ S m we may extend a(x, D) to be a map a(x, D) : S (ℝn )′ → S (ℝn )′ by duality: ∫(a(x, D)f )(x)g(x) dx = ∫ f (x)(a(x, D)∗ g)(x) dx,

f ∈ S (ℝn )′ , g ∈ S (ℝn ).

The two central results concerning pseudo-differential operators which we use are that they form a filtered algebra and that they are bounded on appropriate Lp Sobolev spaces. We turn to stating these results in detail, though we do not include all the proofs. For further discussion and proofs, see [216, Chapter VI]. Theorem 2.2.8 (The calculus of pseudo-differential operators). Suppose a ∈ S m1 and b ∈ S m2 . We consider the operator a(x, D)b(x, D) : S (ℝn ) → S (ℝn ). There exists c ∈ S m1 +m2 such that c(x, D) = a(x, D)b(x, D). Furthermore, for every N ∈ ℕ, c− ∑ |α| 0, 󵄨󵄨 β α 󵄨 −n−m−|α|−N . 󵄨󵄨𝜕x 𝜕z K(x, z)󵄨󵄨󵄨 ≤ Cα,β,N |z|

38 � 2 Ellipticity –

(Cancelation condition) For m ≥ 0, we assume for every bounded set ℬ ⊂ C0∞ (ℝn ) and every multi-index α ∈ ℕn 󵄨󵄨 󵄨󵄨 sup sup sup R−m 󵄨󵄨󵄨∫ 𝜕xα K(x, z)ϕ(Rz) dz󵄨󵄨󵄨 < ∞. 󵄨 󵄨 n x∈ℝ R≥1 ϕ∈ℬ

–

If m < 0, the growth condition implies that for ϕ ∈ C0∞ (ℝn ), the limit lim ∫ K(x, z)ϕ(z) dz ϵ↓0

|z|>ϵ

converges for each x ∈ ℝn . We assume that ∫ K(x, z)ϕ(z) dz agrees with this limit. Proposition 2.2.14 (See Chapter VI, Section 7.4 of [216]). a 󳨃→ ǎ is a bijection between S m and kernels of order m. Remark 2.2.15. In light of Proposition 2.2.14, we may write pseudo-differential operators in another way. Namely, for a ∈ S m , ̌ z) dz, a(x, D)f (x) = ∫ f (x − z)a(x,

f ∈ S (ℝn ),

̌ z) is a kernel of order m. We will use this idea to generalize some aspects of where a(x, pseudo-differential operators to settings which have no immediate analog of the Fourier transform. Example 2.2.16. A simple example to keep in mind is the Laplacian on ℝ2 , which has Green’s function K(x − y) = − 2π1 log(|x − y|). If ϕ ∈ C0∞ (ℝn ) and we consider the distribution ϕ(x)K(x − y)ϕ(y), then this is the Schwartz kernel of a pseudo-differential operator of order −2. Here, n = 2 and m = −2. This demonstrates the need for N in the growth condition in Definition 2.2.13. We will see this phenomenon several times in this text. The second way to view pseudo-differential operators without using the Fourier transform utilizes a Littlewood–Paley decomposition. To describe this, we need to introduce some more function spaces on ℝn . Definition 2.2.17. We let S0 (ℝn ) denote the space of all f ∈ S (ℝn ) such that ∫ t α f (t) dt = 0,

∀α ∈ ℕn .

S0 (ℝn ) is called the space of Schwartz functions, all of whose moments vanish. It is a closed subspace of S (ℝn ), and as such inherits a Fréchet topology from S (ℝn ).

Lemma 2.2.18. Let f ∈ S (ℝn ). Then f ∈ S0 (ℝn ) if and only if ∀α ∈ ℕn , 𝜕ξα f ̂(0) = 0. Proof. This is immediate from the definitions.

2.2 Pseudo-differential operators and the Fourier transform

� 39

Corollary 2.2.19. Fix s ∈ ℝ. The map △s : S0 (ℝn ) → S0 (ℝn ) given by △s : f 󳨃→ (|2πξ|2s f ̂(ξ))∨ is an automorphism. Proof. It follows easily from Lemma 2.2.18 that for f ∈ S0 (ℝn ), △s f ∈ S0 (ℝn ). The closed graph theorem shows that △s is continuous. The continuous inverse of △s is △−s . Corollary 2.2.19 gives a characterization of S0 (ℝn ) which will be useful later. Corollary 2.2.20. Let 𝒯 ⊆ S (ℝn ) be the largest subset such that ∀f ∈ 𝒯 n

f = ∑ 𝜕tj fα ,

fα ∈ 𝒯 .

j=1

(2.5)

Then 𝒯 = S0 (ℝn ). Proof. For f ∈ S0 (ℝn ), we have n

f = △ △−1 f = ∑ 𝜕tj (−𝜕tj △−1 f ). j=1

It follows from Corollary 2.2.19 that (−𝜕tj △−1 f ) ∈ S0 (ℝn ). We conclude S0 (ℝn ) ⊆ 𝒯 . Suppose f ∈ 𝒯 and fix a multi-index α ∈ ℕn . Applying (2.5) |α| + 1 times, we have f =

∑

β

𝜕t fβ ,

fβ ∈ 𝒯 ⊆ S (ℝn ).

|β|=|α|+1

Thus, ∫ t α f (t) dt =

∑

β

∫ t α 𝜕t fβ (t) dt = 0.

|β|=|α|+1

Since α ∈ ℕn was arbitrary, we conclude f ∈ S0 (ℝn ). This shows 𝒯 ⊆ S0 (ℝn ) and completes the proof. ̂ S (ℝn ) be the space of all ς(x, t) ∈ C ∞ (ℝn × ℝn ) such Definition 2.2.21. We let C ∞ (ℝn )⊗ n that for each α, β, γ ∈ ℕ , the following semi-norm is finite: 󵄨 β 󵄨 ‖ς‖α,β,γ := sup 󵄨󵄨󵄨t α 𝜕t 𝜕xγ ς(x, t)󵄨󵄨󵄨. n x,t∈ℝ

̂ S (ℝn ) the coarsest topology such that all such semi-norms are conWe give C ∞ (ℝn )⊗ ̂ S (ℝn ) is a Fréchet space. tinuous; with this topology C ∞ (ℝn )⊗ ̂ S (ℝn ) is justified because the space agrees with Remark 2.2.22. The notation C ∞ (ℝn )⊗ the completed tensor product of the nuclear space S (ℝn ) and the Fréchet space C ∞ (ℝn ). ̂ it denotes the completed tensor product where one of the Whenever we use the symbol ⊗ two factors is nuclear. We will not use any details of the theory of nuclear spaces, though.

40 � 2 Ellipticity For the interested reader, [238, Part III] is an excellent introduction to the theory of such tensor products. ̂ S0 (ℝn ) be the space of those ς(x, t) ∈ C ∞ (ℝn )⊗ ̂ S (ℝn ) Definition 2.2.23. We let C ∞ (ℝn )⊗ n such that for all α ∈ ℕ , ∫ t α ς(x, t) dt = 0,

∀x ∈ ℝn .

̂ S0 (ℝn ) is a closed subspace of C ∞ (ℝn )⊗ ̂ S (ℝn ) and inherits its Fréchet topolC ∞ (ℝn )⊗ ogy. ̂ S (ℝn ), let ς(x, ̂ ξ) denote the Fourier transform of ς in the t For ς(x, t) ∈ C ∞ (ℝn )⊗ ̂ S (ℝn ) with inverse ς 󳨃→ ς,̌ variable. Note that ς 󳨃→ ς̂ is an automorphism of C ∞ (ℝn )⊗ where ς̌ denotes the inverse Fourier transform in the second variable. ̂ S0 (ℝn ), which We have analogs of Lemma 2.2.18 and Corollary 2.2.19 for C ∞ (ℝn )⊗ we state without proof, since the proofs are nearly identical. ̂ S (ℝn ). Then ς ∈ C ∞ (ℝn )⊗ ̂ S0 (ℝn ) if and only if ∀α ∈ Lemma 2.2.24. Let ς ∈ C ∞ (ℝn )⊗ 󵄨󵄨 n α ̂ ℕ , 𝜕ξ ς(x, ξ)󵄨󵄨ξ=0 = 0. Corollary 2.2.25. Fix s ∈ ℝ. The map ̂ S0 (ℝn ) → C ∞ (ℝn )⊗ ̂ S0 (ℝn ) △st : C ∞ (ℝn )⊗ ̂ ξ))∨ is an automorphism. given by △st : ς(x, t) 󳨃→ (|2πξ|2s ς(x, ̂ S (ℝn ) and j ∈ ℕ, let Dil2j (ς)(x, t) := 2jn ς(x, 2j t) ∈ C ∞ (ℝn )⊗ ̂ For ς(x, t) ∈ C ∞ (ℝn )⊗ n S (ℝ ). Theorem 2.2.26. Let a(x, ξ) ∈ S (ℝn × ℝn )′ and fix m ∈ ℝ. The following are equivalent: (i) ǎ is a kernel of order m, (ii) a ∈ S m , ̂ S (ℝn ) with ςj ∈ C ∞ (ℝn )⊗ ̂ S0 (ℝn ) for (iii) there is a bounded set {ςj : j ∈ ℕ} ⊂ C ∞ (ℝn )⊗ j > 0 such that ̌ t) = ∑ 2jm Dil2j (ςj )(x, t), a(x, j∈ℕ

where the sum is taken in the sense of S (ℝn × ℝn )′ (and every such sum converges in this sense). Before we turn to the proof of Theorem 2.2.26 we need some new notation. For two real numbers a, b ∈ ℝ, we write a ∨ b = max{a, b} and a ∧ b = min{a, b}. Proof of Theorem 2.2.26. (i) ⇔ (ii): This is Proposition 2.2.14.

2.2 Pseudo-differential operators and the Fourier transform

�

41

̂ S (ℝn ) is a bounded set with ςj ∈ (iii) ⇒ (ii): Suppose {ςj : j ∈ ℕ} ⊂ C ∞ (ℝn )⊗ n ̂ n C (ℝ )⊗S0 (ℝ ) for j > 0. Because the Fourier transform of Dil2j (ςj ) is ςĵ (x, 2−j ξ), we consider the sum ∞

a(x, ξ) = ∑ 2jm ςĵ (x, 2−j ξ). j∈ℕ

We will show that this sum converges in the sense of tempered distributions (and in ∞ Cloc (ℝn × ℝn )) to a symbol a(x, ξ) ∈ S m . In particular, we wish to show 󵄨󵄨 󵄨󵄨 󵄨󵄨 α β 󵄨 󵄨󵄨𝜕 𝜕 ∑ 2jm ς̂ (x, 2−j ξ)󵄨󵄨󵄨 ≲ (1 + |ξ|)m−|β| . 󵄨󵄨 x ξ 󵄨󵄨 j 󵄨󵄨 󵄨󵄨 j∈ℕ 󵄨 󵄨

(2.6)

β β β Note that since 𝜕xα 𝜕ξ ςĵ (x, 2−j ξ) = 2−j|β| (𝜕xα 𝜕ξ ςĵ )(x, 2−j ξ) and 𝜕xα 𝜕ξ ςĵ is of the same form as ςĵ , it suffices to prove (2.6) with α = β = 0 (by replacing m with m − |β|). We will show that

∑ (1 + |ξ|)−m 2jm ςĵ (x, 2−j ξ)

j∈ℕ+

(2.7)

converges uniformly in x and ξ. It then immediately follows that the same is true with ℕ+ replaced by ℕ, and this will complete the proof of (ii). Set L = |m| + 1. We claim, for j ∈ ℕ+ , |ςĵ (x, ξ)| ≲ (1 + |ξ|)−L ∧ |ξ|L .

(2.8)

Indeed, |ςĵ (x, ξ)| ≲ (1 + |ξ|)−L , ̂ S (ℝn ) is a bounded which follows immediately from the fact that {ςĵ : j ∈ ℕ} ⊂ C ∞ (ℝn )⊗ set. To see that |ςĵ (x, ξ)| ≲ |ξ|L ,

(2.9)

−L/2 ̂ S0 (ℝn ) is a bounded note that ςj = △L/2 ςj ), where {△−L/2 ςj : j ∈ ℕ+ } ⊂ C ∞ (ℝn )⊗ t (△t t set, by Corollary 2.2.25. Since ∧ ςĵ = |2πξ|L (△L/2 t ςj ) ,

(2.9) follows.

42 � 2 Ellipticity We separate (2.7) into two parts: ∑2≤2j 0. We say T is subelliptic on V if T is L2 subelliptic on V . Subellipticity automatically implies a stronger estimate by iterating (2.12). Lemma 2.2.33. Suppose T ∈ Hom(C0∞ (Ω)′ , C0∞ (Ω)′ ), p ∈ (1, ∞), and T is Lp subelliptic of order ϵ > 0 on an open set V ⊆ Ω. Then ∀ϕ1 , ϕ2 ∈ C0∞ (V ), with ϕ1 ≺ ϕ2 , we have the following estimate ∀s ∈ ℝ, N ≥ 0: ‖ϕ1 u‖Lps+ϵ ≤ Cϕ1 ,ϕ2 ,s,N (‖ϕ2 Tu‖Lps + ‖ϕ2 u‖Lp ),

∀u ∈ C0∞ (Ω)′ ,

s−N

(2.13)

where if the right-hand side is finite, then the left-hand side is also finite. Before we begin the proof of Lemma 2.2.33 we introduce some new notation. We write A ≲ B for A ≤ CB, where C is a constant which only depends on certain parameters. It will be clear from the context what parameters C is allowed to depend on; for example, in the proof of Lemma 2.2.33, we will allow C to depend on the same things as the constant in (2.13). Proof. We prove by induction on N ∈ ℕ that ∀ϕ1 , ϕ2 ∈ C0∞ (V ), with ϕ1 ≺ ϕ2 , and all s ∈ ℝ, N ≥ 0, ‖ϕ1 u‖Lps+ϵ ≲ ‖ϕ2 Tu‖Lps + ‖ϕ2 u‖Lp , s−Nϵ

∀u ∈ C0∞ (Ω)′ ,

(2.14)

which will complete the proof by taking N large. The base case, N = 0, of (2.14) is the assumption of the lemma. We assume (2.14) for N ∈ ℕ and prove it for N + 1. Fix ϕ1 , ϕ2 ∈ C0∞ (V ) with ϕ1 ≺ ϕ2 and let ψ ∈ C0∞ (V ) with ϕ1 ≺ ψ ≺ ϕ2 . Applying the inductive hypothesis with ϕ2 replaced by ψ, we have ‖ϕ1 u‖Lps+ϵ ≲ ‖ψTu‖Lps + ‖ψu‖Lp .

(2.15)

s−Nϵ

Applying the assumption (2.12) we see ‖ψu‖Lp

s−Nϵ

≲ ‖ϕ2 Tu‖Lp

s−(N+1)ϵ

+ ‖ϕ2 u‖Lp

s−(N+1)ϵ

.

(2.16)

Using ‖ψTu‖Lps ≲ ‖ϕ2 Tu‖Lps and ‖ϕ2 Tu‖Lp ≲ ‖ϕ2 Tu‖Lps , combining (2.15) and (2.16) s−(N+1)ϵ gives (2.14) with N replaced by N + 1, completing the proof.

46 � 2 Ellipticity Proposition 2.2.34. Suppose P(x, D) is elliptic near x0 . Then there is an open neighborhood V ⊆ Ω of x0 such that P(x, D) is Lp subelliptic of order m on V , ∀p ∈ (1, ∞). Proof. Let V ⊆ Ω be the open neighborhood of x0 from Proposition 2.2.27. Fix ϕ1 , ϕ2 , ϕ3 ∈ C0∞ (V ) with ϕ1 ≺ ϕ2 ≺ ϕ3 . For N ∈ ℕ, let bN = bN,ϕ1 ,ϕ2 ∈ S −m be as in Proposition 2.2.27. We have, using Theorem 2.2.12, ∀u ∈ C0∞ (Ω)′ , ‖ϕ1 bN (x, D)ϕ2 u‖Lps+m ≲ ‖ϕ2 u‖Lps .

(2.17)

Replacing u with P(x, D)u in (2.17), we have ‖ϕ1 bN (x, D)ϕ2 P(x, D)u‖Lps+m ≲ ‖ϕ2 P(x, D)u‖Lps .

(2.18)

Proposition 2.2.27 gives ϕ1 bN (x, D)ϕ2 P(x, D)u = ϕ1 u + ϕ1 RN ϕ3 u, where RN is a pseudodifferential operator of order −N. Plugging this into (2.18) and using Theorem 2.2.12, we have ‖ϕ1 u‖Lps+m ≲ ‖ϕ2 P(x, D)u‖Lps + ‖ϕ3 u‖Lp

s+m−N

.

Since N was arbitrary and ‖ϕ2 P(x, D)u‖Lps ≲ ‖ϕ3 P(x, D)u‖Lps , this completes the proof of Lp subellipticity (with ϕ2 replaced by ϕ3 in Definition 2.2.32). A similar proof yields the following. Proposition 2.2.35. Suppose P(x, D) is elliptic on Ω. Then P(x, D) is Lp subelliptic of order m on Ω, ∀p ∈ (1, ∞). Remark 2.2.36. Proposition 2.2.35 is optimal in the sense that a differential operator of order m can never be Lp subelliptic of order ϵ for ϵ > m (as can be immediately seen from the definition). In fact, being subelliptic of order m characterizes elliptic operators (see Theorem 2.7.2 (b)). There is a third, even weaker condition: hypoellipticity. Definition 2.2.37. Let T ∈ Hom(C0∞ (Ω)′ , C0∞ (Ω)′ ). We say T is hypoelliptic on Ω if the 󵄨 ∞ following holds. Suppose V ⊆ Ω is an open set and u ∈ C0∞ (Ω)′ satisfies Tu󵄨󵄨󵄨V ∈ Cloc (V ). 󵄨 ∞ Then u󵄨󵄨󵄨V ∈ Cloc (V ).

Proposition 2.2.38. Suppose T is Lp subelliptic on Ω for some p ∈ (1, ∞). Then T is hypoelliptic on Ω. 󵄨 ∞ Proof. Let V ⊆ Ω be open, let u ∈ C0∞ (Ω)′ , and suppose Tu󵄨󵄨󵄨V ∈ Cloc (V ). Fix x0 ∈ V . We 󵄨 ∞ wish to show that there is a neighborhood U ⊆ V of x0 such that u󵄨󵄨󵄨U ∈ Cloc (U), and this will complete the proof. Take ϕ1 , ϕ2 ∈ C0∞ (V ), with ϕ1 equal to 1 on a neighborhood of x0 , and ϕ1 ≺ ϕ2 . By subellipticity and Lemma 2.2.33, we have, for some ϵ > 0 and ∀s ∈ ℝ, N ∈ ℕ,

2.3 Singular integral operators

�

47

‖ϕ1 u‖Lps+ϵ ≤ Cϕ1 ,ϕ2 ,s,N (‖ϕ2 Tu‖Lps + ‖ϕ2 u‖Lp ). s−N

Since ϕ2 Tu ∈ C0∞ (ℝn ), by hypothesis, ‖ϕ2 Tu‖Lps < ∞, ∀s ∈ ℝ. By taking N sufficiently large, ‖ϕ2 u‖Lp < ∞ (as this it true for any distribution with compact support, for some s−N

p

N). Thus, ϕ1 u ∈ ⋂s∈ℝ Ls and the Sobolev embedding theorem implies ϕ1 u ∈ C0∞ (ℝn ). Since ϕ1 equals 1 on a neighborhood of x0 , the result follows.

Propositions 2.2.35 and 2.2.38 can be succinctly restated as the following implications: ellipticity ⇒ subellipticity ⇒ hypoellipticity. None of the reverse implications hold, in general, even for partial differential operators.2 The goal of this text is to study a class of operators which are subelliptic and which generalize elliptic operators: the maximally subelliptic operators. Elliptic operators are a special case of maximally subelliptic operators (see Example 1.1.10 (i)) and maximal subellipticity ⇒ subellipticity ⇒ hypoellipticity.

2.3 Singular integral operators The theory of pseudo-differential operators is closely tied to the group structure on ℝn . In the study of maximally subelliptic operators, we need to move beyond this group structure, and to do this we turn to the Calderón–Zygmund theory of singular integrals. We introduce a filtered algebra of singular integral operators which contains pseudodifferential operators, but is less tied to the group structure of ℝn . The ideas here are amenable to the generalizations we require in later chapters. Following Theorem 2.2.26 we define this algebra in three equivalent ways. The first way will be the most familiar to readers who have studied singular integrals. However, it is the later two ways which are more important for our generalizations and applications. ∞ Definition 2.3.1. We say T ∈ Hom(S (ℝn ), Cloc (ℝn )) is a singular integral operator of order s ∈ ℝ if: ∞ – (Growth condition) T(x, y) agrees with a Cloc function on {(x, y) : x ≠ y} and for all n multi-indices α, β ∈ ℕ and all N ≥ 0 such that n + s + |α| + |β| + N > 0,

󵄨󵄨 α β 󵄨 −n−s−|α|−|β|−N . 󵄨󵄨𝜕x 𝜕y T(x, y)󵄨󵄨󵄨 ≤ Cα,β,M |x − y| 2 See Remark 8.2.7, where it is shown that maximally subelliptic operators are subelliptic but very rarely elliptic. See [143] for examples of partial differential operators which are hypoelliptic but far from subelliptic.

48 � 2 Ellipticity –

(Cancelation condition) For all bounded sets ℬ ⊂ C0∞ (ℝn ), all multi-indices α ∈ ℕn , and all N ≥ 0 such that n + s + |α| + N > 0, there exists Cℬ,α,N ≥ 0 such that the following holds. For ϕ ∈ ℬ and z ∈ ℝn , set ϕR,z (x) := Φ(R(x − z)). We assume 󵄨 󵄨 sup sup sup R−s−|α|−N 󵄨󵄨󵄨𝜕xα TϕR,z (x)󵄨󵄨󵄨 ≤ Cℬ,α,N . ϕ∈ℬ R≥1 x,z∈ℝn

We also assume that the same holds with T replaced by its formal L2 adjoint T ∗ (see Definition 2.1.6). Remark 2.3.2. A singular integral operator of order s ∈ ℝ is a priori an operator in ∞ Hom(S (ℝn ), Cloc (ℝn )). However, we will see that, in fact, T ∈ Hom(S (ℝn ), S (ℝn )) and that T extends to an operator T ∈ Hom(S (ℝn )′ , S (ℝn )′ ). See Remark 2.3.29. The equivalent characterizations of Definition 2.3.1 which we use are closely tied to a Littlewood–Paley decomposition of the operators. To discuss this, we need to introduce a new kind of operator. Definition 2.3.3. We say ℱ ⊂ C ∞ (ℝn × ℝn ) × (0, 1] is a bounded set of pre-elementary operators if ∀m ≥ 0, ∀α, β ∈ ℕn , ∃Cm,α,β ≥ 0 such that −m 󵄨 󵄨 sup 󵄨󵄨󵄨(2−j 𝜕x )α (2−j 𝜕z )β F(x, z)󵄨󵄨󵄨 ≤ Cm,α,β 2jn (1 + 2j |x − z|) . −j

(F,2 )∈ℱ

Example 2.3.4. If ℬ ⊂ S (ℝn ) is a bounded set, then {(2jn ϕ(2j (x − z)), 2−j ) | ϕ ∈ ℬ, j ∈ [0, ∞)} is a bounded set of pre-elementary operators. Definition 2.3.5. We define the set of bounded sets of elementary operators, G , to be the largest set of subsets of C ∞ (ℝn × ℝn ) × (0, 1] such that ∀ℰ ∈ G : – ℰ is a bounded set of pre-elementary operators, – ∀(E, 2−j ) ∈ ℰ , E=

∑ 2−(2−|α|−|β|)j (2−j 𝜕x )α Eα,β (2−j 𝜕x )β ,

(2.19)

|α|,|β|≤1

where we are treating E and Eα,β as operators (by identifying an operator with its Schwartz kernel), and {(Eα,β , 2−j ) : (E, 2−j ) ∈ ℰ , |α|, |β| ≤ 1} ∈ G . Remark 2.3.6. In Definition 2.3.5, G is defined to be the largest set satisfying certain axioms. There is a unique largest such set, since if {Gα } is the collection of all sets satisfying

2.3 Singular integral operators

� 49

the axioms, then ⋃α Gα also satisfies the axioms. In practice, we will show that a set ℰ is a bounded set of elementary operators by showing that: – ℰ is a bounded set of pre-elementary operators, and – for each (E, 2−j ) ∈ ℰ , E can be written in the form (2.19), where each Eα,β is “of the same form” as E. See Example 2.3.8 for a simple version of this argument. Remark 2.3.7. Note that (2.19) is trivial when j = 0 by taking E0,0 = E and Eα,β = 0 for |α| + |β| > 0. Example 2.3.8. –

If ℬ1 ⊂ S0 (ℝn ) is a bounded set, then {(2jn ϕ(2j (x − z), 2−j ) : ϕ ∈ ℬ1 , j ∈ [0, ∞)}

(2.20)

is a bounded set of elementary operators. Indeed, the proof of Corollary 2.2.20 shows that for each ϕ ∈ ℬ1 , we have ϕ(x − z) =

∑ 𝜕xα 𝜕zβ ϕα,β (x − z), |α|,|β|=1

where {ϕα,β : ϕ ∈ ℬ1 , |α|, |β| = 1} ⊆ S0 (ℝn )

–

is a bounded set. From here, it follows easily that (2.20) is a bounded set of elementary operators. See Lemma 2.3.17 for a more general version of this fact. If ℬ2 ⊂ S (ℝn ) is a bounded set, then {(ϕ(x − z), 2−0 ) : ϕ ∈ ℬ2 } is a bounded set of elementary operators (see Remark 2.3.7).

Remark 2.3.9. See [220, Section 1.2] for a closely related perspective on singular integrals, which makes the relationship between S0 and bounded sets of elementary operators even more clear. ∞ Theorem 2.3.10. Let T ∈ Hom(S (ℝn ), Cloc (ℝn )) and fix s ∈ ℝ. The following are equivalent: (i) T is a singular integral operator of order s. (ii) For every bounded set of elementary operators ℰ ,

{(2−js TE, 2−j ) : (E, 2−j ) ∈ ℰ } is a bounded set of elementary operators.

50 � 2 Ellipticity (iii) There exists a bounded set of elementary operators {(Ej , 2−j ) : j ∈ ℕ} such that T = ∑ 2js Ej , j∈ℕ

where this sum converges in the sense of tempered distributions (we will see in Lemma 2.3.20 that every such sum converges in the sense of tempered distributions). From Theorem 2.3.10 we immediately see that singular integral operators form a filtered algebra. Corollary 2.3.11. Let T be an operator of order s1 ∈ ℝ and let S be an operator of order s2 ∈ ℝ. Then TS is an operator of order s1 + s2 . Proof. This follows immediately from Theorem 2.3.10 (ii). Remark 2.3.12. It follows immediately from either Theorem 2.3.10 (i) or (ii) that 𝜕xα is a singular integral operator of order |α|. Corollary 2.3.13. Singular integral operators of order 0 extend to bounded operators on Lp (ℝn ), 1 < p < ∞. Corollary 2.3.13 can be deduced from Definition 2.3.1 and the T(1) theorem of David and Journé [60]; see the presentation in [216, Chapter VII, § 3.2]. However, we defer the proof to later in the section, where we will see it as a relatively elementary consequence of Theorem 2.3.10. The rest of this section is devoted to Theorem 2.3.10 and Corollary 2.3.13. For Theorem 2.3.10, we prove only (ii) ⇔ (iii) as that is the equivalence that is of central importance to our later generalizations. The remaining equivalence, (i) ⇔ (iii), is proved in an essentially more general setting in Theorem 5.2.12, and we refer the reader there for details. Our first result says that pseudo-differential operators of order t are singular integral operators of order t. Proposition 2.3.14. Let a(x, D) be a pseudo-differential operator of order s ∈ ℝ. Then a(x, D) satisfies the conclusions of Theorem 2.3.10 (iii). Remark 2.3.15. Proposition 2.3.14 would follow immediately from Theorem 2.2.26 (ii) ⇒ (i) combined with Theorem 2.3.10 (i) ⇒ (iii). However, this is circular: Proposition 2.3.14 is central to the proof of Theorem 2.3.10. Remark 2.3.16. Using Proposition 2.3.14 and Corollary 2.3.13, one can “move derivatives past” singular integral operators. Indeed, suppose T is a singular integral operator of order s. Then n

n

k=1

k=1

𝜕xj T = 𝜕xj T(1 + △)−1 − ∑ (𝜕xj T(I + △)−1 𝜕xk )𝜕xk = S0(s−1) + ∑ Sk(s) 𝜕xk ,

2.3 Singular integral operators

�

51

where each Sk(s) is a singular integral operator of order s and S0(s−1) is a singular integral operator of order s−1. However, the best one can say in general is that [𝜕xj , T] is a singular integral operator of order s + 1; this is in contrast to pseudo-differential operators where the commutator is an operator of order s (see Theorem 2.2.8). This lack of commutativity is a major inconvenience when studying elliptic PDEs; however it is an essential feature in our study of maximally subelliptic PDEs (see Remark 1.4.1). ̂ S (ℝn ) be a bounded set with ςj (x, t) ∈ Lemma 2.3.17. Let {ςj (x, t) : j ∈ ℕ} ⊂ C ∞ (ℝn )⊗ ∞ n ̂ n C (ℝ )⊗S0 (ℝ ) for j > 0. Then jn

j

ℰ := {(2 ςj (x, 2 (x − y)), 2 ) : j ∈ ℕ} −j

is a bounded set of elementary operators. Proof. It follows immediately from the definitions that ℰ is a bounded set of preelementary operators. Thus, it suffices to show that each 2jn ςj (x, 2j (x − y)) is an appropriate sum of derivatives of operators of the same form. Set ςj̃ (x, t) := {

△−1 t ςj (x, t), ς0 (x, t),

j ≥ 1,

j = 0.

̂ S (ℝn ) is a bounded set and ςj̃ ∈ C ∞ (ℝn )⊗ ̂ By Corollary 2.2.25, {ςj̃ : j ∈ ℕ} ⊂ C ∞ (ℝn )⊗ n S0 (ℝ ) for j ≥ 1. The same is therefore also true of {𝜕xk ςj̃ (x, t) : j ∈ ℕ}. Furthermore, we have, for j ≥ 1, n

n

k=1

k=1

2nj ςj (x, 2j (x − y)) = ∑ 2nj (2−j 𝜕xk )(2−j 𝜕yk )ςj̃ (x, 2j (x − y)) − ∑ 2nj−j (2−j 𝜕yk )(𝜕xk ςj̃ )(x, x − y). This sees 2nj ςj (x, 2j (x − y)) as an appropriate sum of derivatives of operators of the same form as in (2.19). Since (2.19) is trivial when j = 0 (Remark 2.3.7), this completes the proof. Proof of Proposition 2.3.14. Let a ∈ S s . Then by Theorem 2.2.26 (iii) we may write ̌ t) = ∑ Dil2j (ςj )(x, t), a(x, j∈ℕ

with convergence in the sense of tempered distributions, where {ςj : j ∈ ℕ} ⊂ ̂ S (ℝn ) with ςj ∈ C ∞ (ℝn )⊗ ̂ S0 (ℝn ) for j > 0. The Schwartz kernel of a(x, D) C ∞ (ℝn )⊗ ̌ x − y) and therefore we have is given by a(x, a(x, D) = ∑ Dil2j (ςj )(x, x − y), j∈ℕ

52 � 2 Ellipticity with convergence in the sense of tempered distributions. Lemma 2.3.17 shows that {(Dil2j (ςj )(x, x − y), 2−j ) : j ∈ ℕ} is a bounded set of elementary operators, completing the proof. Proposition 2.3.18. Let ℰ be a bounded set of elementary operators. Then: (a) If ψ ∈ C ∞ (ℝn ), then {(Mult[ψ]E, 2−j ), (E Mult[ψ], 2−j ) : (E, 2−j ) ∈ ℰ } is a bounded set of elementary operators. (b) {(E ∗ , 2−j ) : (E, 2−j ) ∈ ℰ } is a bounded set of elementary operators. (c) Fix a multi-index α ∈ ℕn . Then {((2−j 𝜕x )α E, 2−j ), (E(2−j 𝜕x )α , 2−j ) : (E, 2−j ) ∈ ℰ } is a bounded set of elementary operators. (d) For all N ∈ ℕ, (E, 2−j ) ∈ ℰ can be written as E = ∑ 2−j(N−|α|) (2−j 𝜕x )α Eα , |α|≤N

where {(Eα , 2−j ) : (E, 2−j ) ∈ ℰ , |α| ≤ N} is a bounded set of elementary operators. Similarly, each (E, 2−j ) ∈ sE can be written as E = ∑ 2−j(N−|α|) Ẽα (2−j 𝜕x )α , |α|≤N

where {(Ẽα , 2−j ) : (E, 2−j ) ∈ ℰ , |α| ≤ N} is a bounded set of elementary operators. Proof. This follows easily from the definitions. See Proposition 5.5.5 for a similar result in a more general context. Lemma 2.3.19. For every m > n, ∀j1 , j2 ∈ [0, ∞), ∫ 2nj1 (1 + 2j1 |x − y|)−m 2nj2 (1 + 2j2 |y − z|)−m dy ≤ Cm,n 2n(j1 ∧j2 ) (1 + 2j1 ∧j2 |x − z|)−m . Proof. By symmetry in x and z, it suffices to prove the case where j2 ≤ j1 . By a simple dilation argument, it suffices to prove the case where j2 = 0. By a translation, it suffices to prove the case x = 0. Thus, it suffices to show that, ∀j ∈ [0, ∞), ∫ 2nj (1 + 2j |y|)−m (1 + |y − z|)−m dy ≤ Cm,n (1 + |z|)−m . We leave this elementary estimate to the reader.

2.3 Singular integral operators

� 53

Lemma 2.3.20. Fix s ∈ ℝ and let {(Ej , 2−j ) : j ∈ ℕ} be a bounded set of elementary operators. Then the sum ∑ 2js Ej

j∈ℕ

converges in the topology of bounded convergence on Hom(S (ℝn ), S (ℝn )) (and therefore converges in S (ℝn × ℝn )′ ). Proof. Let ℬ ⊂ S (ℝn ) be a bounded set. We will show that for all multi-indices α ∈ ℕn , all m ∈ ℕ, and all f ∈ ℬ, ∑ 2js (1 + |x|)m 𝜕xα Ej f (x)

j∈ℕ

(2.21)

converges absolutely and uniformly for x ∈ ℝn and f ∈ ℬ, and the result will follow. Note that ∑ 2js (1 + |x|)m 𝜕xα Ej f (x) = ∑ 2j(s+|α|) (1 + |x|)m (2−j 𝜕x )α Ej f (x).

j∈ℕ

j∈ℕ

Since {((2−j 𝜕x )α Ej , 2−j ) : j ∈ ℕ} is a bounded set of elementary operators by Proposition 2.3.18 (c), we see that it suffices to prove the convergence of (2.21) in the case α = 0 (by replacing s with s + α). Fix N ∈ ℕ. By Proposition 2.3.18 (d) we have ∑ (1 + |x|)m 2js Ej f = ∑ ∑ 2j(s−N) Ej,β 𝜕xβ f .

j∈ℕ

j∈ℕ |β|≤N

β

By taking N ≥ s + 1 and replacing f with 𝜕x f , we see that it suffices to prove the convergence of (2.21) in the case α = 0 and s = −1. By taking z = 0 in Lemma 2.3.19, we see that for f ∈ ℬ and (E, 2−j ) ∈ ℰ , |Ej f (x)| ≲ (1 + |x|)−m . Thus, 󵄨 󵄨 ∑ 󵄨󵄨󵄨2−j (1 + |x|)m Ej f (x)󵄨󵄨󵄨 ≲ ∑ 2−j .

j∈ℕ

j∈ℕ

The desired convergence of (2.21) follows (in the case α = 0, s = −1), completing the proof. Corollary 2.3.21. Let I ∈ Hom(S (ℝn ), S (ℝn )) be the identity operator. Then there exists a bounded set of elementary operators {(Ej , 2−j ) : j ∈ ℕ} such that

54 � 2 Ellipticity I = ∑ Ej , j∈ℕ

where the sum is taken in the sense of the topology of bounded convergence on Hom(S (ℝn ), S (ℝn )). Proof. Since I is a pseudo-differential operator of order 0, this follows from Proposition 2.3.14 and Lemma 2.3.20. Proof of Theorem 2.3.10 (ii) ⇒ (iii). By Corollary 2.3.21, we may write the identity operator, I, as I = ∑ Ej , j∈ℕ

where {(Ej , 2−j ) : j ∈ ℕ} is a bounded set of elementary operators, and this sum converges in the topology of bounded convergence on Hom(S (ℝn ), S (ℝn )). Thus, T = TI = ∑ 2js (2−js TEj ). j∈ℕ

Since {(2−js TEj , 2−j ) : j ∈ ℕ} is a bounded set of elementary operators, by hypothesis, this completes the proof. To prove Theorem 2.3.10 (iii) ⇒ (ii), we use the next proposition, which is central to our analysis. Proposition 2.3.22. Let ℰ be a bounded set of elementary operators. Then, ∀N ∈ ℕ, the set {(2N|j1 −j2 | E1 E1 , 2−j1 ), (2N|j1 −j2 | E1 E2 , 2−j2 ) : (E1 , 2−j1 ), (E2 , 2−j2 ) ∈ ℰ } is a bounded set of elementary operators. To prove Proposition 2.3.22 we need two lemmas. Lemma 2.3.23. Let ℰ be a bounded set of pre-elementary operators. Then, ∀α, β ∈ ℕn , m ∈ ℕ, ∃C = C(m, α, β) ≥ 0, such that ∀(F1 , 2−j1 ), (F2 , 2−j2 ) ∈ ℱ , and letting k ∈ {j1 , j2 }, 󵄨 󵄨 2−(m+n+|α|∨|β|)|j1 −j2 | 󵄨󵄨󵄨(2−k 𝜕x )α (2−k 𝜕z )β [F1 F2 ](x, z)󵄨󵄨󵄨 ≤ C2kn (1 + 2k |x − z|)−m , where [F1 F2 ](x, z) denotes the Schwartz kernel of the operator F1 F2 . Proof. We prove the result for k = j2 ; the proof for k = j1 is similar and we leave it to the reader. Thus, we wish to show 󵄨 󵄨 2−(m+n+|α|)|j1 −j2 | 󵄨󵄨󵄨(2−j2 𝜕x )α (2−j2 𝜕z )β [F1 F2 ](x, z)󵄨󵄨󵄨 ≲ 2j2 n (1 + 2j2 |x − z|)−m ;

2.3 Singular integral operators

� 55

it therefore suffices to show 󵄨 󵄨 2−(m+n)|j1 −j2 | 󵄨󵄨󵄨(2−j1 𝜕x )α (2−j2 𝜕z )β [F1 F2 ](x, z)󵄨󵄨󵄨 ≲ 2j2 n (1 + 2j2 |x − z|)−m .

(2.22)

But Lemma 2.3.19 shows 󵄨 󵄨 2−(m+n)|j1 −j2 | 󵄨󵄨󵄨(2−j1 𝜕x )α (2−j2 𝜕z )β [F1 F2 ](x, z)󵄨󵄨󵄨

≲ 2−(m+n)|j1 −j2 | 2n(j1 ∧j2 ) (1 + 2j1 ∧j2 |x − z|)−m .

Formula (2.22) follows, completing the proof. Lemma 2.3.24. Let ℰ be a bounded set of elementary operators. Then, ∀N ∈ ℕ, the set {(2N|j1 −j2 | E1 E1 , 2−j1 ), (2N|j1 −j2 | E1 E2 , 2−j2 ) : (E1 , 2−j1 ), (E2 , 2−j2 ) ∈ ℰ } is a bounded set of pre-elementary operators. Proof. Fix N0 ∈ ℕ. We will show that {(2N0 |j1 −j2 | E1 E1 , 2−j1 ) : (E1 , 2−j1 ), (E2 , 2−j2 ) ∈ ℰ } is a bounded set of pre-elementary operators. The proof of the result with j1 replaced by j2 is similar and we leave it to the reader. Fix m ∈ ℕ and let α, β ∈ ℕn . We wish to show 󵄨 󵄨 2N0 |j1 −j2 | 󵄨󵄨󵄨(2−j1 𝜕x )α (2−j1 𝜕z )β [E1 E2 ](x, z)󵄨󵄨󵄨 ≲ 2nj1 (1 + 2j1 |x − z|)−m .

(2.23)

Set N := m + n + |α| ∨ |β|. We separate the proof into two cases. When j1 ≤ j2 , we apply Proposition 2.3.18 (d) to write E2 =

∑

2j2 (|γ|−N−N0 ) (2−j2 𝜕x )γ E2,γ ,

|γ|≤N+N0

where {(E2,γ , 2−j2 ) : (E2 , 2−j2 ) ∈ ℰ , |γ| ≤ N + N0 } is a bounded set of elementary operators. Thus, E1 E2 =

∑

2−j2 (N+N0 −|γ|) 2−|γ||j1 −j2 | E1 (2−j1 𝜕x )γ E2,γ

|γ|≤N+N0

=

∑

2−|j1 −j2 |(N+N0 ) E1,γ E2,γ ,

(2.24)

|γ|≤N+N0

where E1,γ = 2−j1 (N+N1 −|γ|) E1 (2−j1 𝜕x )α , and therefore by Proposition 2.3.18 (c), {(E1,γ , 2−j1 ) : (E, 2−j ) ∈ ℰ , |γ| ≤ N + N0 } is a bounded set of elementary operators. Plugging (2.24) into (2.23), we see that it suffices to prove (2.23) with N0 replaced by −m − n − |α| ∧ |β|.

56 � 2 Ellipticity With this replacement, (2.23) follows directly from Lemma 2.3.23, which completes the proof in the case j1 ≤ j2 . When j1 ≥ j2 the proof is similar, except that we instead apply Proposition 2.3.18 (d) to E1 to write E1 =

2j1 (|γ|−N−N0 ) E1,γ (2−j1 𝜕x )γ .

∑ |γ|≤N+N0

From here the proof is similar, and we leave the details to the reader. Proof of Proposition 2.3.22. We prove that ℰN := {(2

N|j1 −j2 |

E1 E2 , 2−j1 ) : (E, 2−j1 ), (E, 2−j2 )}

is a bounded set of elementary operators. The proof for the result with j1 replaced by j2 is similar, and we leave it to the reader. Lemma 2.3.24 shows that ℰN is a bounded set of pre-elementary operators. The result will follow once we show that 2N|j1 −j2 | E1 E2 is a sum of derivatives of operators of the same form as in (2.19). Using Proposition 2.3.18 (d) we have 2N|j1 −j2 | E1 E2 =

∑ 2N|j1 −j2 |−(1−|α|)j1 −(1−|β|)j2 (2−j1 𝜕x )α E1,α E2,β (2−j2 𝜕x )β |α|,|β|≤1

=

∑ 2−(2−|α|−|β|)j1 (2−j1 𝜕x )α [2N|j1 −j2 |+(j1 −j2 ) E1,α E2,α ](2−j1 𝜕x )β ,

|α|,|β|≤1

where {(E1,α , 2−j1 ), (E2,β , 2−j2 ) : (E1 , 2−j1 ), (E2 , 2−j2 ) ∈ ℰ , |α|, |β| ≤ 1} is a bounded set of elementary operators. This completes the proof since 2N|j1 −j2 |+(j1 −j2 ) E1,α E2,α is of the same form as 2N|j1 −j2 | E1 E2 (with a different choice of N). Proof of Theorem 2.3.10 (iii) ⇒ (ii). Fix s ∈ ℝ and let T = ∑j∈ℕ 2js Ej , where {(Ej , 2−j ) : j ∈ ℕ} is a bounded set of elementary operators. Let ℰ be a bounded set of elementary operators. For (E, 2−k ) ∈ ℰ and j ∈ ℕ, set Ẽj := 2(j−k)s+|j−k| Ej E, so that {(Ẽj , 2−k ) : j ∈ ℕ, (E, 2−k ) ∈ ℰ } is a bounded set of elementary operators by Proposition 2.3.22. Set Ẽ := ∑j∈ℕ 2−|j−k| Ẽj , so ̃ 2−k ) : (E, 2−k ) ∈ ℰ } is a bounded set of elementary operators. We have that {(E, ̃ 2−ks TE = ∑ 2(j−k)s Ej E = ∑ 2−|j−k| Ẽj = E. j∈ℕ

j∈ℕ

Thus, {(2−ks TE, 2−k ) : (E, 2−k ) ∈ ℰ } is a bounded set of elementary operators, completing the proof.

2.3 Singular integral operators

�

57

Finally, we turn to the proof of Corollary 2.3.13 without using the T(1) theorem. For this, we employ the Cotlar–Stein lemma. Lemma 2.3.25 (The Cotlar–Stein lemma). Let ℋ1 and ℋ2 be Hilbert spaces. For j ∈ ℕ, let Tj : ℋ1 → ℋ2 be bounded operators such that 1

2 sup ∑ ‖Tj∗ Tk ‖ℋ →ℋ ≤ A < ∞, 1

j∈ℕ k∈ℕ

1

1 2

sup ∑ ‖Tj Tk∗ ‖ℋ →ℋ ≤ B < ∞. 2

j∈ℕ k∈ℕ

2

Then the sum ∑j∈ℕ Tj converges in the strong operator topology on Hom(ℋ1 , ℋ2 ) to a bounded operator ℋ1 → ℋ2 and 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 ≤ √AB. 󵄩󵄩 ∑ Tj 󵄩󵄩󵄩 󵄩󵄩 󵄩 j∈ℕ 󵄩ℋ1 →ℋ2 For a proof of Lemma 2.3.25, see [216, Chapter VII, § 2]. The statement in that reference is slightly different from the statement here, though the proof can be adapted to prove the result as we have stated it. Lemma 2.3.26. Let ℰ be a bounded set of pre-elementary operators. Then sup

sup ‖E‖Lp →Lp < ∞.

p∈[1,∞] (E,2−j )∈ℰ

Proof. First we claim sup sup ∫ |E(x, y)| dy < ∞.

(E,2−j )∈ℰ x∈ℝn

(2.25)

Indeed, if we take m = n + 1, we have ∫ |E(x, y)| dy ≲ ∫ 2jn (1 + 2j |x − y|)−m dy = ∫(1 + |y|)−m dy ≲ 1, as desired. Formula (2.25) implies sup ‖E‖L∞ →L∞ < ∞.

(E,2−j )∈ℰ

A similar proof shows sup sup ∫ |E(x, y)| dx < ∞,

(E,2−j )∈ℰ y∈ℝn

which implies

(2.26)

58 � 2 Ellipticity sup ‖E‖L1 →L1 < ∞.

(E,2−j )∈ℰ

(2.27)

Interpolating (2.26) and (2.27) completes the proof. Lemma 2.3.27. Let T be a singular integral operator of order s ∈ ℝ. Then the formal L2 adjoint of T, T ∗ (see Definition 2.1.6) is also a singular integral operator of order s. Proof. This can be seen directly from Definition 2.3.1. Lemma 2.3.28. Let T be a singular integral operator of order 0. Then T extends to a bounded operator on L2 (ℝn ). Proof. By Theorem 2.3.10 (iii), we may write T = ∑j∈ℕ Ej , where {(Ej , 2−j ) : j ∈ ℕ} is a bounded set of elementary operators. Combining Proposition 2.3.18 (b) with Lemma 2.3.24, we see that for every N ∈ ℕ, {(2N|j−k| Ej∗ Ek , 2−j ), (2N|j−k| Ej Ek∗ , 2−j ) : j, k ∈ ℕ} is a bounded set of elementary operators. Lemma 2.3.26 implies ‖Ej∗ Ek ‖L2 →L2 , ‖Ej Ek∗ ‖L2 →L2 ≲ 2−|j−k| . The result now follows from the Cotlar–Stein lemma (Lemma 2.3.25). Proof of Corollary 2.3.13. Let T be a singular integral operator of order 0. Lemma 2.3.28 shows that T extends to a bounded operator on L2 (ℝn ). Using the growth condition from Definition 2.3.1 it is a standard result that this implies that T is weak type (1, 1); see [216, Chapter I, § 5]. The Marcinkiewicz interpolation theorem then shows that T extends a bounded operator on Lp , 1 < p ≤ 2. Lemma 2.3.27 shows that T ∗ is also a singular integral operator of order 0. Thus, the same argument shows that T ∗ extends to a bounded operator on Lp , 1 < p ≤ 2, and therefore T extends to a bounded operator on Lp , 2 ≤ p < ∞, completing the proof. Remark 2.3.29. Let T be a singular integral operator of order s ∈ ℝ. A priori, T ∈ ∞ Hom(S (ℝn ), Cloc (ℝn )). However, it follows from Theorem 2.3.10 (iii) and Lemma 2.3.20 that, in fact, T ∈ Hom(S (ℝn ), S (ℝn )). Since T is a singular integral operator of order s if and only if T ∗ is a singular integral operator of order s (by Lemma 2.3.27), we see that T extends by duality to an operator in Hom(S (ℝn )′ , S (ℝn )′ ). 2.3.1 Local, matrix-valued operators We will often be considering results where the estimates are local. We also want to consider the possibility of operators acting on functions taking values in a finite-

2.4 Besov and Triebel–Lizorkin spaces

� 59

dimensional complex vector space. To describe this, we introduce the following definitions. Definition 2.3.30. Fix D1 , D2 ∈ ℕ+ and let 𝕄D1 ×D2 (ℂ) denote the Banach space of complex D1 × D2 matrices. Fix a connected open set Ω ⊆ ℝn . ∞ – We say T ∈ Hom(C0∞ (Ω; ℂD1 ), Cloc (Ω; ℂD2 )′ ) is locally a singular integral operator of ∞ order s ∈ ℝ if ∀ϕ1 , ϕ2 ∈ C0 (Ω), Mult[ϕ1 ]T Mult[ϕ2 ] is a D2 × D1 matrix of singular integral operators of order s. ∞ – We say T ∈ Hom(C0∞ (Ω; ℂD1 ), Cloc (Ω; ℂD2 )′ ) is locally a pseudo-differential operator of order s ∈ ℝ if ∀ϕ1 , ϕ2 ∈ C0∞ (Ω), Mult[ϕ1 ]T Mult[ϕ2 ] is a D2 × D1 matrix of pseudodifferential operators of order s. ∞ – We say ℰ ⊂ Cloc (Ω × Ω; 𝕄D1 ×D2 (ℂ)) is locally a bounded set of pre-elementary operators if ∀ϕ1 , ϕ2 ∈ C0∞ (Ω), if we write (F, 2−j ) ∈ ℰ as (Fl1 ,l2 )1≤l1 ≤D1 ,1≤l2 ≤D2 , then {(Mult[ϕ1 ]Fl1 ,l2 Mult[ϕ2 ], 2−j ) : (F, 2−j ) ∈ ℰ , 1 ≤ l1 ≤ D1 , 1 ≤ l2 ≤ D2 } is a bounded set of pre-elementary operators. Remark 2.3.31. The above definitions can be generalized to operators acting on sections of a vector bundle. See Section 8.6 for details in the more general maximally subelliptic setting.

2.4 Besov and Triebel–Lizorkin spaces Besov and Triebel–Lizorkin spaces are a central tool in the classical study of elliptic PDEs: they are the function spaces on which elliptic PDEs are most well behaved. Both the classical Lp Sobolev spaces for 1 < p < ∞ (see Proposition 2.4.9) and Hölder spaces C m,r for 0 < r < 1 (see Corollary 2.5.6) are special cases, as we will see. The usual study of these spaces is intimately tied to the Fourier transform; however, when we turn to generalizations in Chapter 6 we will have to leave behind the Fourier transform altogether. We start by describing the main definitions and results of this section, and then turn to the proofs. To begin, we need to introduce vector-valued Lebesgue spaces. Let (M, μ) be a measure space and let 𝒜 be a countable or finite set. Let fj : M → ℂ be a sequence of a. e. equivalence classes of measurable functions indexed by j ∈ 𝒜. For p, q ∈ [1, ∞], we set ‖{fj }j∈𝒜 ‖Lp (M,μ;ℓq (𝒜))

1󵄩 󵄩󵄩 󵄩 , {󵄩󵄩󵄩(∑j∈𝒜 |fj (x)|q ) q 󵄩󵄩󵄩 p 󵄩 󵄩L (M,μ) := { 󵄩󵄩 󵄩󵄩 {󵄩󵄩supj∈𝒜 |fj (x)|󵄩󵄩Lp (M,μ) , 1

‖{fj }j∈𝒜 ‖ℓq (𝒜;Lp (M,μ))

q {(∑ ‖f ‖ p )q , := { j∈𝒜 j L (M,μ) {supj∈𝒜 ‖fj ‖Lp (M,μ) ,

q ∈ [1, ∞), q = ∞,

q ∈ [1, ∞), q = ∞.

(2.28) (2.29)

60 � 2 Ellipticity For p, q ∈ [1, ∞], if 𝒱 is either Lp (M, μ; ℓq (𝒜)) or ℓq (𝒜; Lp (M, μ)), we let the space 𝒱 consist of those sequences {fj (x)}j∈𝒜 such that ‖{fj }j∈𝒜 ‖𝒱 < ∞. With the norm ‖ ⋅ ‖𝒱 , 𝒱 is a Banach space. ̂ Let ϕ̂ ∈ C0∞ (ℝn ) be such that ϕ(ξ) ≡ 1 on a neighborhood of 0 ∈ ℝn . For j ≥ 0 set ̂ ϕ(ξ), j = 0, ψ̂ j (ξ) := { ̂ ̂ ϕ(ξ) − ϕ(2ξ), j ≥ 1, so that ψ̂ j = ψ̂ 1 for j ≥ 1 and ∞

∑ ψ̂ j (2−j ξ) ≡ 1.

j=0

For j ≥ 0, define Dj : S (ℝn )′ → S (ℝn )′ by (Dj f ) (ξ) := ψ̂ j (2−j ξ)f ̂(ξ),

(2.30)

∧

where ∧ denotes the Fourier transform. In other words, if ψj ∈ S (ℝn ) is the inverse Fourier transform of ψ̂ , then j

Dj f = f ∗ Dil2j (ψj ). ∞ In particular, for f ∈ S (ℝn )′ , Dj f ∈ Cloc (ℝn ) ∩ S (ℝn )′ , and we may therefore treat Dj f as a function.

Remark 2.4.1. By choosing ϕ̂ to be real-valued and even, we can ensure ψj is real-valued, ∀j. Thus, when necessary, we may assume Dj takes real-valued functions to real-valued functions. Definition 2.4.2. For s ∈ ℝ, p, q ∈ [1, ∞], we define the extended Besov norm, 󵄩 js 󵄩 s (ℝn ) := 󵄩 ‖f ‖Bp,q 󵄩󵄩{2 Dj f }j∈ℕ 󵄩󵄩󵄩ℓq (ℕ;Lp (ℝn )) ,

f ∈ S (ℝn )′ ,

and the Besov space, s

n

n ′

s (ℝn ) < ∞}. Bp,q (ℝ ) := {f ∈ S (ℝ ) : ‖f ‖Bp,q

For s ∈ ℝ, p ∈ (1, ∞), q ∈ (1, ∞], we define the extended Triebel–Lizorkin norm, 󵄩 js 󵄩 s (ℝn ) := 󵄩 ‖f ‖Fp,q 󵄩󵄩{2 Dj f }j∈ℕ 󵄩󵄩󵄩Lp (ℝn ;ℓq (ℕ)) ,

f ∈ S (ℝn )′ ,

and the Triebel–Lizorkin space, s

n

n ′

s (ℝn ) < ∞}. Fp,q (ℝ ) := {f ∈ S (ℝ ) : ‖f ‖Fp,q

2.4 Besov and Triebel–Lizorkin spaces

�

61

s s Remark 2.4.3. It is well known that Bp,q (ℝn ) and Fp,q (ℝn ) are Banach spaces with the above norms; see [244, § 2.3.3].

Remark 2.4.4. If X is a finite-dimensional Banach space, we can similarly define the s s spaces Fp,q (ℝn ; X ) and Bp,q (ℝn ; X ) of distributions taking values in X , by making the obvious modifications. s (ℝn ) and ‖⋅‖F s (ℝn ) Proposition 2.4.5. The equivalence class of the extended norms ‖⋅‖Bp,q p,q s ̂ does not depend on the choice of ϕ. In particular, the Banach spaces B (ℝn ) and F s (ℝn )

p,q

do not depend on the choice of ϕ.̂

p,q

s (ℝn ) to a well-defined Remark 2.4.6. It is possible to extend the definition of ‖ ⋅ ‖Bp,q s (ℝn ) equivalence class of quasi-norms for p, q ∈ (0, ∞] and to similarly extend ‖ ⋅ ‖Fp,q to p ∈ (0, ∞), q ∈ (0, ∞], though the proofs in these settings require an extra twist. See [244, § 2.3.1 and § 2.3.2] for details. We will not go down this path in this text, though generalizing some of our later results in this way might be interesting for other applications.

Theorem 2.4.7. If T is a singular integral operator of order s ∈ ℝ, then T is a bounded operator: s0 s0 −s T : Bp,q (ℝn ) → Bp,q (ℝn ),

s0 s0 −s T : Fp,q (ℝn ) → Fp,q (ℝn ),

s0 ∈ ℝ, p, q ∈ [1, ∞], s0 ∈ ℝ, p ∈ (1, ∞), q ∈ (1, ∞].

Recall that Tf is defined for any f ∈ S (ℝn )′ – see Remark 2.3.29. Corollary 2.4.8. If a(x, D) is a pseudo-differential operator of order s ∈ ℝ, then a(x, D) is a bounded operator s0 s0 −s a(x, D) : Bp,q (ℝn ) → Bp,q (ℝn ),

s0 s0 −s a(x, D) : Fp,q (ℝn ) → Fp,q (ℝn ),

s0 ∈ ℝ, p, q ∈ [1, ∞], s0 ∈ ℝ, p ∈ (1, ∞), q ∈ (1, ∞].

Proof. Since a(x, D) is a singular integral operator of order s by Proposition 2.3.14, this follows immediately from Theorem 2.4.7. p

s Proposition 2.4.9. For p ∈ (1, ∞) and s ∈ ℝ, Fp,2 (ℝn ) = Ls and s ‖f ‖Fp,2 (ℝn ) ≈ ‖f ‖Lps ,

∀f ∈ Lps , p

where the implicit constants depend on p and s, but not on f . Here, Ls is the Lp Sobolev space of order s (see Definition 2.2.11).

62 � 2 Ellipticity We turn to the proofs of the above results. A convenient tool is the Calderón reproducing formula. For C0 ∈ ℕ, define ̃ C0 := D j

∑ Dk . |k−j|≤C0 k∈ℕ

Lemma 2.4.10 (Calderón reproducing formula). If C0 = C0 (ϕ)̂ ∈ ℕ is sufficiently large, then ̃ C0 = D j , Dj D j

∀j ∈ ℕ.

Proof. Note that C

̃ 0 f ) (ξ) = (D j ∧

∑ ψ̂ k (2−k ξ)f ̂(ξ). |k−j|≤C0 k∈ℕ

̃ C0 = Dj if ∑|k−j|≤C ψ̂ k (2−k+j ξ) ≡ 1 on supp(ψ̂ j ). But, Thus, Dj D 0 j k∈ℕ

̂ −C0 ξ) − ϕ(2 ̂ C0 ξ), ϕ(2 ∑ ψ̂ k (2−k+j ξ) = { −C ̂ 0 ϕ(2 ξ), |k−j|≤C0

j > C0 , j ≤ C0 ,

k∈ℕ

and ̂ − ϕ(2ξ), ̂ ϕ(ξ) j ≥ 1, ψ̂ j (ξ) = { ̂ ϕ(ξ), j = 0. From here the result follows from the fact that ϕ̂ ≡ 1 on a neighborhood of 0. Henceforth, we fix C0 = C0 (ϕ)̂ ∈ ℕ so large that Lemma 2.4.10 holds. A central part of the proofs of the above results is the following technical result. For a bounded set of elementary operators ℰ , s ∈ ℝ, and p

n

q

q

p

n

V ∈ {L (ℝ ; ℓ (ℕ)) : p ∈ (1, ∞), q ∈ (1, ∞]} ∪ {ℓ (ℕ; L (ℝ )) : p, q ∈ [1, ∞]},

define a semi-norm by ‖f ‖V ,s,ℰ :=

sup

{(Ej

,2−j ):j∈ℕ}⊆ℰ

‖{2js Ej f }j∈ℕ ‖V .

Proposition 2.4.11. For p, q ∈ [1, ∞] and s ∈ ℝ, ‖ ⋅ ‖ℓq (Lp ),s,ℰ is a continuous semi-norm on s Bp,q (ℝn ) and s (ℝn ) , ‖f ‖ℓq (Lp ),s,ℰ ≤ Cp,q,s,ℰ ‖f ‖Bp,q

∀f ∈ S (ℝn )′ .

(2.31)

2.4 Besov and Triebel–Lizorkin spaces

� 63

s For p ∈ (1, ∞), q ∈ (1, ∞], and s ∈ ℝ, ‖ ⋅ ‖Lp (ℓq ),s,ℰ is a continuous semi-norm on Fp,q (ℝn ) and

∀f ∈ S (ℝn )′ .

s (ℝn ) , ‖f ‖Lp (ℓq ),s,ℰ ≤ Cp,q,s,ℰ ‖f ‖Fp,q

(2.32)

The constants in the above depend on ℰ , p, q, s, and n, though they may be chosen uniformly on compact subsets of s ∈ ℝ. Notation 2.4.12. For the rest of the section, for s ∈ ℝ, X s will denote any one of the spaces s

s

n

s

n

X ∈ {Bp,q (ℝ ) : p, q ∈ [1, ∞]} ⋃{Fp,q (ℝ ) : p ∈ (1, ∞), q ∈ (1, ∞]}.

We set V := {

ℓq (ℕ; Lp (ℝn )), Lp (ℝn ; ℓq (ℕ)),

s if X s = Bp,q (ℝn ), s if X s = Fp,q (ℝn ).

With this notation we have 󵄩 󵄩 ‖f ‖X s = ‖f ‖V ,s,{(Dj ,2−j ):j∈ℕ} = 󵄩󵄩󵄩{2js Dj f }j∈ℕ 󵄩󵄩󵄩V . To prove Proposition 2.4.11 we require the next lemma. Lemma 2.4.13. Let ℰ be a bounded set of pre-elementary operators. Let {(Ej , 2−j ) : j ∈ ℕ} ⊆ ℰ and define an operator on sequences of L1loc (ℝn ) functions {fj (x)}j∈ℕ by 𝒯 {fj }j∈ℕ = {Ej fj }j∈ℕ .

For p

n

q

q

p

n

𝒱 ∈ {L (ℝ ; ℓ (ℕ)) : p ∈ (1, ∞), q ∈ (1, ∞]} ∪ {ℓ (ℕ; L (ℝ )) : p, q ∈ [1, ∞]},

we have 𝒯 ∈ Hom(𝒱 , 𝒱 ) and ‖𝒯 ‖𝒱→𝒱 ≤ Cℰ,𝒱 . Before we prove Lemma 2.4.13 we introduce some preliminary results. For a function f ∈ L1loc (ℝn ), we use the Hardy–Littlewood maximal function defined by ℳf (x) = sup r>0

1

Vol(Bn (x, r))

∫ |f (y)| dy,

(2.33)

Bn (x,r)

where Bn (x, r) = {y ∈ ℝn : |x − y| < r} and Vol(A) denotes the Lebesgue measure of A.

64 � 2 Ellipticity Lemma 2.4.14. For p ∈ (1, ∞) and q ∈ (1, ∞], ‖{ℳfj }j∈ℕ ‖Lp (ℝn ;ℓq (ℕ)) ≤ Cp,q,n ‖{fj }j∈ℕ ‖Lp (ℝn ;ℓq (ℕ)) , where if the right-hand side is finite, then the left-hand side is finite. Proof. This is due to C. Fefferman and E. M. Stein [83]. See also [216, Chapter II, § 1.3.1]. Lemma 2.4.15. Let ℰ be a bounded set of pre-elementary operators. Then there exists C = C(ℰ ) ≥ 0 such that ∀(E, 2−j ) ∈ ℰ and all f ∈ L1loc , |Ef (x)| ≤ C ℳf (x). Proof. This follows easily from the definitions. Proof of Lemma 2.4.13. Using Lemmas 2.4.14 and 2.4.15, for p ∈ (1, ∞), q ∈ (1, ∞] we have ‖𝒯 {fj }j∈ℕ ‖Lp (ℓq ) = ‖{Ej fj }j∈ℕ ‖Lp (ℓq ) ≲ ‖{ℳfj }j∈ℕ ‖Lp (ℓq ) ≲ ‖{fj }j∈ℕ ‖Lp (ℓq ) . Using Lemma 2.3.26, for p, q ∈ [1, ∞] we have ‖𝒯 {fj }j∈ℕ ‖ℓq (Lp ) = ‖{Ej fj }j∈ℕ ‖ℓq (Lp ) ≲ ‖{fj }j∈ℕ ‖ℓq (Lp ) , completing the proof. In the remainder of this section, we will have several operators indexed by j ∈ ℕ; for example, we may have Ej for j ∈ ℕ. For such a sequence of operators, we define Ej = 0 for j ∈ ℤ \ ℕ. Lemma 2.4.16. Let ℰ be a bounded set of elementary operators and let {(Ej1 , 2−j ) : j ∈ ℕ}, {(Ej2 , 2−j ) : j ∈ ℕ} ⊆ ℰ . For k ∈ ℤ, set 1 2

𝒯k {fj }j∈ℕ := {Ej Ej+k fj }j∈ℕ .

Then, ∀N ∈ ℕ, ∀𝒱 ∈ {Lp (ℝn ; ℓq (ℕ)) : p ∈ (1, ∞), q ∈ (1, ∞]} ∪ {ℓq (ℕ; Lp (ℝn )) : p, q ∈ [1, ∞]}, we have 𝒯k ∈ Hom(𝒱 , 𝒱 ), ∀k ∈ ℤ, and there exists Cℰ,N,𝒱 ≥ 0 such that ∀k ∈ ℤ ‖𝒯k ‖𝒱→𝒱 ≤ Cℰ,N,𝒱 2−N|k| .

2.4 Besov and Triebel–Lizorkin spaces

� 65

Proof. Lemma 2.3.24 shows that, ∀N ∈ ℕ, {(2N|k| F1 F2 , 2−j ) : (F1 , 2−j ), (F2 , 2−(j+k) ) ∈ ℰ } is a bounded set of elementary operators. From here, the result follows from Lemma 2.4.13. Lemma 2.4.17. Let ℰ be a bounded set of pre-elementary operators and let f ∈ S (ℝn )′ . Then, ∀(E, 2−k ) ∈ ℰ , E ∑ Dj f = ∑ EDj f (x) = Ef (x), j∈ℕ

j∈ℕ

where this sum converges pointwise. Here, Dj are the operators from (2.30). Proof. It follows easily from the definitions that for g ∈ S (ℝn ), ∑j∈ℕ D∗j g = g, with convergence in S (ℝn ). Thus, by duality, ∑j∈ℕ Dj f = f , with convergence in S (ℝn )′ . Since, for x fixed, E(x, ⋅) ∈ S (ℝn ) and EDj f (x) = ∫ E(x, y)(Dj f )(y) dy, the result follows. Proof of Proposition 2.4.11. We prove (2.31) and (2.32) simultaneously, by using Notation 2.4.12. Let {(Ej , 2−j ) : j ∈ ℕ} ⊆ ℰ . Using Lemmas 2.4.10 and 2.4.17, we have 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 js 󵄩󵄩 js 󵄩 󵄩󵄩 󵄩 {2 E f } = { 2 E D f } ∑ 󵄩󵄩 󵄩󵄩 j j∈ℕ 󵄩 j j+k 󵄩V 󵄩󵄩󵄩 󵄩 󵄩󵄩 k∈ℤ 󵄩V j∈ℕ 󵄩 󵄩 󵄩 󵄩 ̃ C0 f } 󵄩󵄩󵄩 , ≤ ∑ 󵄩󵄩󵄩{2js Ej Dj+k f }j∈ℕ 󵄩󵄩󵄩V = ∑ 󵄩󵄩󵄩{2js Ej Dj+k D j+k j∈ℕ 󵄩V k∈ℤ

(2.34)

k∈ℤ

where the ≤ inequality uses the triangle inequality, Fatou’s lemma, and the pointwise convergence of the sum (Lemma 2.4.17). Define, for k ∈ ℤ, the vector-valued operator 𝒯k by |k|−ks

𝒯k {fj }j∈ℕ := {2

Ej Dj+k fj }j∈ℕ .

Since {(Dj , 2−j ) : j ∈ ℕ} is a bounded set of elementary operators (see Example 2.3.8), Lemma 2.4.16 shows ‖𝒯k ‖V →V ≲ 1. Thus,

66 � 2 Ellipticity 󵄩 ̃ C0 f } 󵄩󵄩󵄩 = ∑ 2−|k| 󵄩󵄩󵄩𝒯k {2(j+k)s D ̃ C0 f } 󵄩󵄩󵄩 ∑ 󵄩󵄩󵄩{2js Ej Dj+k D 󵄩 j+k j∈ℕ 󵄩V j+k j∈ℕ 󵄩V

k∈ℤ

k∈ℤ

󵄩 ̃ C0 f } 󵄩󵄩󵄩 ≤ ∑ 2−|k| 󵄩󵄩󵄩{2js D ̃ C0 f } 󵄩󵄩󵄩 ≲ ∑ 2−|k| 󵄩󵄩󵄩{2(j+k)s D 󵄩 j∈ℕ 󵄩V j j+k j∈ℕ 󵄩V k∈ℤ

k∈ℤ

≲ ∑2 k∈ℤ

(2.35)

js

󵄩 󵄩󵄩{2 Dj f }j∈ℕ 󵄩󵄩󵄩V ≲ ‖f ‖X s ,

−|k| 󵄩 󵄩

̃ C0 . Combining (2.34) and (2.35) and where the final ≲ estimate used the definition of D j taking the supremum over all {(Ej , 2−j ) : j ∈ ℕ} ⊆ ℰ shows ‖f ‖V ,s,ℰ =

sup

{(Ej ,2−j ):j∈ℕ}⊆ℰ

󵄩󵄩 js 󵄩 󵄩󵄩{2 Ej f }j∈ℕ 󵄩󵄩󵄩V ≲ ‖f ‖X s ,

completing the proof. Proposition 2.4.11 leads to the following characterization of X s . Proposition 2.4.18. For s ∈ ℝ and f ∈ S (ℝn )′ , the following are equivalent: (i) f ∈ X s , (ii) for every bounded set of elementary operators ℰ , ‖f ‖V ,s,ℰ < ∞. Proof. (i) ⇒ (ii) follows from Proposition 2.4.11. For (ii) ⇒ (i), note that {(Dj , 2−j ) : j ∈ ℕ} is a bounded set of elementary operators (see Example 2.3.8) and ‖f ‖X s = ‖f ‖V ,s,{(Dj ,2−j ):j∈ℕ} . ̂ Proof of Proposition 2.4.5. Let γ̂ ∈ C0∞ (ℝn ) satisfy γ(ξ) ≡ 1 on a neighborhood of 0. Define Dj in terms of γ,̂ just as Dj was defined in terms of ϕ.̂ The statement of the proposition is equivalent to ‖f ‖

V ,s,{(Dj ,2−j ):j∈ℕ}

≈ ‖f ‖

V ,s,{(Dj ,2−j ):j∈ℕ}

.

(2.36)

Since the assumptions of (2.36) are symmetric in Dj and Dj , it suffices to show just the ≲ part of (2.36). Since {(Dj , 2−j ) : j ∈ ℕ} is a bounded set of elementary operators (see Example 2.3.8), this follows from Proposition 2.4.11, completing the proof. Next we turn to the proof of Theorem 2.4.7. We will use the next lemma. Lemma 2.4.19. Let T be a singular integral operator of order s ∈ ℝ and let ℰ be a bounded set of elementary operators. Then {(2−js ET, 2−j ) : (E, 2−j ) ∈ ℰ } is a bounded set of elementary operators.

2.4 Besov and Triebel–Lizorkin spaces

�

67

Proof. By Lemma 2.3.27, T ∗ is a singular integral operator of order s. By Proposition 2.3.18 (b), {(E ∗ , 2−j ) : (E, 2−j ) ∈ ℰ } is a bounded set of elementary operators. From here, Theorem 2.3.10 (ii) implies that {(2−js T ∗ E ∗ , 2−j ) : (E, 2−j ) ∈ ℰ } is a bounded set of elementary operators. Proposition 2.3.18 (b) applied to this set shows {(2−js ET, 2−j ) : (E, 2−j ) ∈ ℰ } is a bounded set of elementary operators, completing the proof. Proof of Theorem 2.4.7. Using Notation 2.4.12, we wish to show that T : X s0 → X s0 −s is bounded. Set Ej := 2−js Dj T. Since {(Dj , 2−j ) : j ∈ ℕ} is a bounded set of elementary operators (see Example 2.3.8), Lemma 2.4.19 shows that {(Ej , 2−j ) : j ∈ ℕ} is a bounded set of elementary operators. We have, for f ∈ S (ℝn )′ , 󵄩 󵄩 󵄩 󵄩 ‖Tf ‖X s−s0 = 󵄩󵄩󵄩{2j(s0 −s) Dj Tf }j∈ℕ 󵄩󵄩󵄩V = 󵄩󵄩󵄩{2js0 Ej f }j∈ℕ 󵄩󵄩󵄩V = ‖f ‖V ,s,{(Ej ,2−j ):j∈ℕ} ≲ ‖f ‖X s , where the last inequality follows from Proposition 2.4.11 and completes the proof. p

We turn to the proof of Proposition 2.4.9. We begin with the case s = 0, where L0 = p L . First, we introduce the Khintchine inequality, which we state without proof. Theorem 2.4.20 (The Khintchine inequality – see [103]). Let {ϵj }∞ j=0 be a sequence of i. i. d. random variables with mean zero taking values in ±1. Let x0 , x1 , x2 , . . . be a sequence of complex numbers with ∑ |xj |2 < ∞ and fix 0 < p < ∞. There are constants Ap and Bp , depending only on p, such that 1 󵄨󵄨 ∞ 󵄨󵄨p p1 ∞ 2 󵄨󵄨 󵄨󵄨 2 󵄨 󵄨 Ap (∑ |xj | ) ≤ (𝔼 󵄨󵄨󵄨∑ ϵj xj 󵄨󵄨󵄨 ) ≤ Bp (∑ |xj | ) . 󵄨󵄨j=0 󵄨󵄨 j=0 j=0 󵄨 󵄨

∞

2

1 2

Remark 2.4.21. The use of the Khintchine inequality in this and similar settings is very common. See, for instance, [216, page 267] and [213, Chapter 4, Section 5]. Lemma 2.4.22. We have, for 1 < p < ∞, 󵄩󵄩 󵄩 󵄩󵄩{Dj f }j∈ℕ 󵄩󵄩󵄩Lp (ℓ2 ) ≲ ‖f ‖Lp (ℝn ) ,

∀f ∈ Lp (ℝn ),

(2.37)

where the implicit constant depends on p ∈ (1, ∞) and n. Proof. Fix p ∈ (1, ∞). Given any sequence ϵj ∈ {−1, 1}, set Tϵ := ∑j∈ℕ ϵj Dj . Using the

fact that {(Dj , 2−j ) : j ∈ ℕ} is a bounded set of elementary operators (see Example 2.3.8), Corollary 2.3.13 shows that for f ∈ Lp , ‖Tϵ f ‖Lp ≲ ‖f ‖Lp . Though it is not stated there, the

68 � 2 Ellipticity same proof shows that this is uniform in ϵ: sup ‖Tϵ f ‖Lp ≲ ‖f ‖Lp .

(2.38)

ϵj ∈{−1,1}

Using the Khintchine inequality (Theorem 2.4.20), we have with ϵj a sequence of i. i. d. random variables of mean zero taking values in ±1, 󵄩󵄩 󵄩 󵄩󵄩{Dj f }j∈ℕ 󵄩󵄩󵄩Lp (ℓ2 )

󵄩󵄩 󵄨󵄨 ∞ 󵄨󵄨p p1 󵄩󵄩󵄩󵄩 󵄩󵄩 󵄨󵄨 󵄨󵄨 󵄩󵄩 󵄩󵄩 ≈ 󵄩󵄩󵄩(𝔼 󵄨󵄨󵄨󵄨∑ ϵj Dj f 󵄨󵄨󵄨󵄨 ) 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄨󵄨j=0 󵄨󵄨 󵄩󵄩 p 󵄨 󵄨 󵄩󵄩󵄩 󵄩L 1 󵄩󵄩 ∞ 󵄩󵄩p p 󵄩󵄩 󵄩󵄩 = (𝔼 󵄩󵄩󵄩󵄩∑ ϵj Dj f 󵄩󵄩󵄩󵄩 ) ≤ sup ‖Tϵ f ‖Lp ≲ ‖f ‖Lp , 󵄩󵄩j=0 󵄩󵄩 p ϵj ∈{−1,1} 󵄩 󵄩L

where the last inequality follows from (2.38), completing the proof. 0 Lemma 2.4.23. For 1 < p < ∞, Lp ⊆ Fp,2 (ℝn ) and

‖f ‖F 0

p,2 (ℝ

n)

≲ ‖f ‖Lp ,

∀f ∈ Lp ,

(2.39)

where the implicit constant depends on p ∈ (1, ∞) and n. Proof. Formula (2.39) is just a restatement of (2.37). The result follows. 0 0 Lemma 2.4.24. For p ∈ (1, ∞), Fp,2 (ℝn ) ⊆ Lp and ‖f ‖Lp ≲ ‖f ‖F 0 (ℝn ) , ∀f ∈ Fp,s (ℝn ), p,2 where the implicit constant depends on p ∈ (1, ∞) and n.

Proof. We take ⟨g, f ⟩ = ∫ g(x)f (x) dx; here if f ∈ S (ℝn )′ , f is defined as in Definition 2.1.5. 0 Suppose f ∈ Fp,2 (ℝn ). Let q ∈ (1, ∞) be dual to p (i. e., p−1 + q−1 = 1). We will show that ∀g ∈ S (ℝn ), 󵄨󵄨 󵄨 󵄨󵄨⟨g, f ⟩󵄨󵄨󵄨 ≲ ‖g‖Lq ‖f ‖F 0

p,2 (ℝ

n)

,

and the result will follow. Since ∑j∈ℕ Dj g = g, with convergence in S (ℝn ), we have, using Lemma 2.4.10, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨 ̃ C0 g, f ⟩󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∑ ⟨D ̃ C0 g, D∗ f ⟩󵄨󵄨󵄨 󵄨󵄨⟨g, f ⟩󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∑ ⟨Dj g, f ⟩󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∑ ⟨Dj D j 󵄨󵄨 󵄨󵄨 󵄨󵄨 j j 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 j∈ℕ 󵄨󵄨 󵄨j∈ℕ 󵄨 󵄨 j∈ℕ 1 2

1 2

󵄨 ̃ C0 󵄨󵄨 󵄨 󵄨 ̃ C0 󵄨2 󵄨 󵄨2 ≤ ∫ ∑ 󵄨󵄨󵄨D g(x)󵄨󵄨󵄨󵄨󵄨󵄨D∗j f (x)󵄨󵄨󵄨 dx ≤ ∫( ∑ 󵄨󵄨󵄨D g(x)󵄨󵄨󵄨 ) ( ∑ 󵄨󵄨󵄨D∗j f (x)󵄨󵄨󵄨 ) dx j j j∈ℕ

j∈ℕ

j∈ℕ

󵄩 ̃ C0 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩{D g}j∈ℕ 󵄩󵄩󵄩Lq (ℓ2 ) 󵄩󵄩󵄩{D∗j f }j∈ℕ 󵄩󵄩󵄩Lp (ℓ2 ) ≲ 󵄩󵄩󵄩{Dj g}j∈ℕ 󵄩󵄩󵄩Lq (ℓ2 ) 󵄩󵄩󵄩{D∗j f }j∈ℕ 󵄩󵄩󵄩Lp (ℓ2 ) j ≲ ‖g‖Lq ‖f ‖F 0

n p,2 (ℝ )

,

2.4 Besov and Triebel–Lizorkin spaces

� 69

̃ C0 and the second used Lemma 2.4.22 where the first ≲ estimate used the definition of D j and (2.32). Proof of Proposition 2.4.9. The case s = 0 follows by combining Lemmas 2.4.24 and 2.4.23. Fix s ∈ ℝ. Let Λs be the pseudo-differential operator of order s from Definition 2.2.10. p By Definition 2.2.11, Λs is an isomorphism Ls → Lp , 1 < p < ∞. By Corollary 2.4.8, s 0 Λs : Fp,2 (ℝn ) → Fp,2 (ℝn ) = Lp is an isomorphism with continuous inverse Λ−s . We p s conclude that the identity map I = Λ−s Λs is an isomorphism Ls → Fp,2 (ℝn ), completing the proof. We close this section with the following useful proposition. Proposition 2.4.25. For f ∈ S (ℝn )′ , the following are equivalent: (i) f ∈ X s+1 , (ii) f ∈ X s and 𝜕xj f ∈ X s , 1 ≤ j ≤ n, (iii) f ∈ X s−1 and 𝜕x2j f ∈ X s−1 , 1 ≤ j ≤ n. In this case, we have n

n

j=1

j=1

‖f ‖X s+1 ≈ ‖f ‖X s + ∑ ‖𝜕xj f ‖X s ≈ ‖f ‖X s−1 + ∑ ‖𝜕x2j f ‖X s−1 .

(2.40)

Proof. (i) ⇒ (ii): Suppose f ∈ X s+1 . Clearly f ∈ X s . Since 𝜕xj is a pseudo-differential operator of order 1, Corollary 2.4.8 shows that 𝜕xj f ∈ X s and n

‖f ‖X s + ∑ ‖𝜕xj f ‖X s ≲ ‖f ‖X s+1 . j=1

(2.41)

(ii) ⇒ (iii): Suppose f ∈ X s and 𝜕xj f ∈ X s , 1 ≤ j ≤ n. Since 𝜕xj is a pseudo-differential

operator of order 1, Corollary 2.4.8 shows 𝜕x2j f ∈ X s−1 and n

n

j=1

j=1

‖f ‖X s−1 + ∑ ‖𝜕x2j f ‖X s−1 ≲ ‖f ‖X s + ∑ ‖𝜕xj f ‖X s .

(2.42)

(iii) ⇒ (i): Suppose f ∈ X s−1 and 𝜕x2j f ∈ X s−1 , 1 ≤ j ≤ n. Since (1 + △)−1 = Λ−2 is a

pseudo-differential operator of order −2, Corollary 2.4.8 shows f = Λ−2 (1 + △)f ∈ X s+1 , and n

‖f ‖X s+1 ≲ ‖(1 + △)f ‖X s−1 ≲ ‖f ‖X s−1 + ∑ ‖𝜕x2j f ‖X s−1 . j=1

Formula (2.40) follows from (2.41), (2.42), and (2.43).

(2.43)

70 � 2 Ellipticity

2.5 Zygmund–Hölder spaces We now turn to one of the most important classes of spaces we consider: the Zygmund– Hölder spaces. Recall the classical Hölder spaces on ℝn . For r ∈ [0, 1] and f : ℝn → ℂ, we set ‖f ‖C 0,r := ‖f ‖C(ℝn ) + sup |x − y|−r |f (x) − f (y)|, n x =y∈ℝ ̸

and for m ∈ ℕ, ‖f ‖C m,r := ∑ ‖𝜕xα f ‖C 0,r . |α|≤m

We set C m,r (ℝn ) := {f ∈ C m (ℝn ) : ‖f ‖C m,r < ∞}.

(2.44)

As is well known, and as we will see, Hölder spaces with non-integer exponents (i. e., when r ∈ (0, 1)) play a central role in the study of nonlinear elliptic PDEs. However, when r ∈ {0, 1}, the Hölder space is often not the right space to use. Instead, we follow Zygmund’s lead [255], and use the Zygmund–Hölder spaces. Thus, for s ∈ (0, ∞), we will define the Zygmund–Hölder space C s (ℝn ). We will see (Corollary 2.5.6) that for m ∈ ℕ and r ∈ (0, 1), C m,r (ℝn ) = C m+r (ℝn ). However, for r ∈ {0, 1}, these spaces differ. We have, for m ∈ ℕ, C m+1,0 (ℝn ) ⊊ C m,1 (ℝn ) ⊊ C m+1 (ℝn ). We will give a few equivalent characterizations of the space C s (ℝn ). We begin by defining it as follows. s Definition 2.5.1. For s > 0, we define the Zygmund–Hölder space C s (ℝn ) := B∞,∞ (ℝn ) s and ‖ ⋅ ‖C s (ℝn ) := ‖ ⋅ ‖B∞,∞ (ℝn ) . If X is a finite-dimensional Banach space, we similarly s s define C s (ℝn ; X ) := B∞,∞ (ℝn ; X ) and ‖ ⋅ ‖C s (ℝn ;X ) := ‖ ⋅ ‖B∞,∞ (ℝn ;X ) .

C s (ℝn ) can be given a more elementary characterization in terms of differences.

For f : ℝn → ℂ and v ∈ ℝn , set

Diffv f (x) := f (x + v) − f (x). For the next definition, recall that e1 , . . . , en denotes the standard basis in ℝn . Definition 2.5.2. For s ∈ (0, ∞) and l ∈ ℕ with l > s, define l

′ ′ 󵄨 󵄨 ‖f ‖C s := ∑ max sup sup δ−sl /l 󵄨󵄨󵄨Difflδek f (x)󵄨󵄨󵄨, l

l′ =0

1≤k≤n δ∈(0,1] x∈ℝn

2.5 Zygmund–Hölder spaces

� 71

where Difflv denotes the operator Diffv applied l′ times. We set ′

s

n

n

Cl (ℝ ) := {f ∈ C(ℝ ) : ‖f ‖C s < ∞}. l

The main result of this section is the next proposition. Proposition 2.5.3. For s ∈ (0, ∞) and l ∈ ℕ with l > s, we have C s (ℝn ) = Cls (ℝn ) and ∀f ∈ C s (ℝn ),

‖f ‖C s (ℝn ) ≈ ‖f ‖C s (ℝn ) , l

where the implicit constants depend on s, l, and n. In particular, Cls (ℝn ) does not depend on the choice of l, so long as l > s. Remark 2.5.4. For another, closely related, characterization of C s (ℝn ), see Remark 2.5.13. Remark 2.5.5. Another equivalent norm for C s (ℝn ) which is commonly seen is, for s ∈ (0, 2), 󵄨 󵄨 sup |f (x)| + sup sup |h|−s 󵄨󵄨󵄨Diff2h f (x)󵄨󵄨󵄨. n n

x∈ℝn

x∈ℝ 0=h∈ℝ ̸

This definition can be extended to all s > 0 by using Corollary 2.5.7. That this norm is equivalent to ‖ ⋅ ‖C s follows easily from Remark 2.5.13. Before we prove Proposition 2.5.3, we use it to see some simple corollaries. Corollary 2.5.6. For m ∈ ℕ, r ∈ (0, 1), we have C m+r (ℝn ) = C m,r (ℝn ) and ‖f ‖C m+r ≈ ‖f ‖C m,r ,

∀f ∈ C m,r (ℝn ),

where the implicit constants depend on m, r, and n. Proof. We begin with the case m = 0. Indeed, in this case, C r = C1r = C 0,r , by Proposition 2.5.3 and the definition of C1r . Furthermore, also directly from the definition, ‖⋅‖C1r ≈ ‖ ⋅ ‖C 0,r . This completes the case m = 0. m+r When m ∈ ℕ+ , we use the fact that C m+r = B∞,∞ (ℝn ) (with equality of norms) and apply Proposition 2.4.25 m times to see ‖f ‖C m+r ≈ ∑ ‖𝜕xα f ‖C r ≈ ∑ ‖𝜕xα f ‖C1r ≈ ∑ ‖𝜕xα f ‖C 0,r = ‖f ‖C m,r . |α|≤m

|α|≤m

|α|≤m

A similar proof shows that C m+r = C m,r . Corollary 2.5.7. For m ∈ ℕ and r ∈ (0, 2), C

m+r

(ℝn ) = {f ∈ C m (ℝn ) : 𝜕xα f ∈ C2r , ∀|α| ≤ m}

72 � 2 Ellipticity and ‖f ‖C m+r ≈ ∑ ‖𝜕xα f ‖C2r , |α|≤m

where the implicit constants depend on m, r, and n. m+r Proof. Using the fact that C m+r = B∞,∞ (ℝn ) (with equality of norms), the result with r r C2 replaced by C follows from m applications of Proposition 2.4.25. Since r ∈ (0, 2), the result now follows from Proposition 2.5.3.

Remark 2.5.8. The characterization of C m+r given in Corollary 2.5.7 is the one most commonly used. We now turn to the proof of Proposition 2.5.3. Lemma 2.5.9. If g ∈ C l (ℝn ), 1 ≤ k ≤ n, and δ ∈ (0, 1], 󵄨 󵄨 󵄨 󵄨 sup 󵄨󵄨󵄨Difflδek g(x)󵄨󵄨󵄨 ≲ (sup 󵄨󵄨󵄨(δ𝜕xk )l g(x)󵄨󵄨󵄨) ∧ ‖g‖L∞ . n n

x∈ℝ

x∈ℝ

Proof. It is clear that 󵄨 󵄨 sup 󵄨󵄨󵄨Difflδek g(x)󵄨󵄨󵄨 ≲ ‖g‖L∞ .

x∈ℝn

To see that 󵄨 󵄨 󵄨 󵄨 sup 󵄨󵄨󵄨Difflδek g(x)󵄨󵄨󵄨 ≲ sup 󵄨󵄨󵄨(δ𝜕xk )l g(x)󵄨󵄨󵄨, n

x∈ℝn

x∈ℝ

(2.45)

note that the fundamental theorem of calculus gives 1

Diffδek g(x) = ∫(δ𝜕xk g)(x + δtek ) dt, 0

and therefore, l

Difflδek g(x) = ∫ ((δ𝜕xk )l g) (x + δ ∑ tr ) dt. [0,1]l

r=1

The estimate (2.45) follows, completing the proof. Lemma 2.5.10. Let l ∈ ℕ+ . Then, for all s ∈ (0, l), we have C s ⊆ Cls and ‖f ‖C s ≲ ‖f ‖C s , l ∀f ∈ C s . Proof. We proceed by induction on l ∈ ℕ+ . That is, suppose we know that the result holds for all l′ ∈ ℕ+ with l′ < l. We will then prove the result for l. Because of how we

2.5 Zygmund–Hölder spaces

�

73

have set up the induction, we do not require a base case (the case l = 1 is also included in the inductive step). Fix s ∈ (0, l). Suppose f ∈ C s . Then ‖Dj f ‖L∞ ≤ 2−js ‖f ‖C s , ∀j ∈ ℕ, where Dj is as in (2.30). In particular, since f = ∑j∈ℕ Dj f in the sense of S (ℝn )′ , this sum in fact converges uniformly and therefore f ∈ C(ℝn ) and ‖f ‖C(ℝn ) ≲ ‖f ‖C s . For 0 < l′ < l we have, by the inductive hypothesis, ′ ′ 󵄨 󵄨 max sup sup δ−sl /l 󵄨󵄨󵄨Difflδek f (x)󵄨󵄨󵄨 ≲ ‖f ‖C sl′ /l ≤ ‖f ‖C s ,

1≤k≤n δ∈(0,1] x∈ℝn

where we have used l > s and therefore l′ > sl′ /l. Thus, it suffices to show that for δ ∈ (0, 1], 1 ≤ k ≤ n, 󵄨󵄨 󵄨 l s 󵄨󵄨Diffδek f (x)󵄨󵄨󵄨 ≲ δ ‖f ‖C s .

(2.46)

Without loss of generality, we assume ‖f ‖C s = 1. Set Ej := (2−j 𝜕xk )l Dj , so that {(Ej , 2−j ) : j ∈ ℕ} is a bounded set of elementary operas tors by Proposition 2.3.18 (c). By Proposition 2.4.11, ∀j ∈ ℕ, ‖Ej f ‖L∞ ≲ 2−js ‖f ‖B∞,∞ (ℝn ) ≲ 2−js . By Lemma 2.5.9 we have

󵄨󵄨 󵄨 l l 󵄨󵄨Diffδek Dj f (x)󵄨󵄨󵄨 ≲ ‖(δ𝜕xk ) Dj f ‖L∞ ∧ ‖Dj f ‖L∞

= ((δ2j )l ‖Ej f ‖L∞ ) ∧ ‖Dj f ‖L∞ ≲ 2−js ((δ2j )l ∧ 1).

Thus, 󵄨󵄨 󵄨 󵄨 󵄨 l l −js j l 󵄨󵄨Diffδek f (x)󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨Diffδek Dj f (x)󵄨󵄨󵄨 ≲ ∑ 2 ((δ2 ) ∧ 1) j∈ℕ

j∈ℕ

j(l−s) l

= ∑ 2

δ + ∑ 2

2−j ≥δ

2−j s. This proves (2.46) and completes the proof.

We now turn to the containment Cls ⊆ C s . To prove this, we are more precise about the choice of ϕ̂ in the definition of Dj . We use the next lemma. Lemma 2.5.11. Suppose f ∈ S0 (ℝn ) with supp f ̂ ⊂ Bn (1). Then, ∀l ∈ ℕ there exist gj,l ∈ S0 (ℝn ), j = 1, . . . n, such that n

f = ∑ Difflej gj,l . j=1

74 � 2 Ellipticity Proof. Fix l ∈ ℕ. By Lemma 2.2.18, we may write n

f ̂(ξ) = ∑(2πiξj )l ĥ j,l (ξ), j=1

(2.47)

where ĥ j,l (ξ) ∈ S (ℝn ), supp(ĥ j,l ) ⊂ Bn (1), and ĥ j,l (ξ) vanishes to infinite order at 0. We set ĝj,l (ξ) := (

2πiξj

e−2πiξj − 1

l

) ĥ j,l (ξ).

(2.48)

Since ĥ j,k has compact support in Bn (1) and vanishes to infinite order at 0, it follows that ĝj,l ∈ S (ℝn ) and vanishes to infinite order at 0. Therefore, its inverse Fourier transform, gj,l , is in S0 (ℝn ) (Lemma 2.2.18). Combining (2.47) and (2.48), we have n

l

f ̂(ξ) = ∑(e−2πiξj − 1) ĝj,l (ξ). j=1

Taking the inverse Fourier transform yields the result. Fix ϕ̂ ∈ C0∞ (Bn (1)) with ϕ̂ ≡ 1 on a neighborhood of 0. Define Dj as in (2.30). Recall that ∑j∈ℕ Dj = I and {(Dj , 2−j ) : j ∈ ℕ} is a bounded set of elementary operators (see Example 2.3.8). Lemma 2.5.12. Let l ∈ ℕ+ . Then, for all s > 0, we have Cls ⊆ C s and ‖f ‖C s ≲ ‖f ‖C s , l ∀f ∈ Cls . ̂ Proof. We wish to show ‖f ‖C s ≲ ‖f ‖C s , ∀f ∈ Cls . Let ψ ∈ S (ℝn ) be defined by ψ(ξ) := l n j ̂ ̂ ϕ(ξ) − ϕ(2ξ), so that ψ ∈ S (ℝ ) and D f = f ∗ Dil (2 ), for j ≥ 1. By Lemma 2.5.11, 0

j

ψ

n

ψ = ∑ Difflek γk , k=1

where γk ∈ S0 (ℝn ). We will show, for f ∈ Cls , that ‖Dj f ‖L∞ ≲ 2−js ‖f ‖C s . l

(2.49)

This implies ‖f ‖C s ≲ ‖f ‖C s , and will complete the proof. The estimate (2.49) is easy when l j = 0; indeed, ‖D0 f ‖L∞ ≤ ‖f ‖L∞ ‖ϕ‖L1 ≲ ‖f ‖L∞ ≤ ‖f ‖C s . For j ≥ 1, we have l

2.5 Zygmund–Hölder spaces

�

75

n 󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨 l 󵄨󵄨Dj f (x)󵄨󵄨󵄨 = 󵄨󵄨󵄨∫ f (x − y) Dil2j (ψ)(y) dy󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨∫ f (x − y) Dil2j (Diffek γk )(y) dy󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 k=1

n

󵄨󵄨 󵄨󵄨 = ∑ 󵄨󵄨󵄨∫ f (x − y) Diffl2−j e Dil2j (γk )(y) dy󵄨󵄨󵄨 k 󵄨 󵄨 k=1 n 󵄨

󵄨󵄨 󵄨 = ∑ 󵄨󵄨󵄨∫(Diffl2−j e f )(x − y) Dil2j (γk )(y) dy󵄨󵄨󵄨 k 󵄨 󵄨 k=1 n

≤ ∑ ‖ Diffl2−j e f ‖L∞ ‖γk ‖L1 k=1 −ls

≲2

k

‖f ‖C s . l

This proves (2.49) and completes the proof. Proof of Proposition 2.5.3. This follows by combining Lemmas 2.5.10 and 2.5.12. Remark 2.5.13. The proof of Proposition 2.5.3 gives another characterization of C s (ℝn ). Indeed, let l > s. Then C s (ℝn ) consists of those f ∈ C(ℝn ) such that the following norm is finite: 󵄨 󵄨 ‖f ‖C(ℝn ) + max sup sup δ−s 󵄨󵄨󵄨Difflδek f (x)󵄨󵄨󵄨. 1≤k≤n δ∈(0,1] x∈ℝ

Moreover, this defines an equivalent norm on C s (ℝn ). Our first clue that C s (ℝn ) will be useful in the nonlinear setting is that it forms an algebra. Lemma 2.5.14. C s (ℝn ) is an algebra. Moreover, ‖fg‖C s ≤ Cs,n ‖f ‖C s ‖g‖C s ,

∀f , g ∈ C s (ℝn ).

(2.50)

Remark 2.5.15. For applications to PDEs, one often requires a stronger estimate than (2.50), namely, the tame estimate: ‖fg‖C s ≲ ‖f ‖C s ‖g‖L∞ + ‖f ‖L∞ ‖g‖C s . We prove this tame estimate in Theorem 2.6.2 (iii) using paraproducts. For now, we content ourselves with the much easier non-tame estimate (2.50). Proof of Lemma 2.5.14. For v ∈ ℝn , let τv f (x) = f (x + v). Note that l l j Difflv (fg) = ∑ ( )(τvj Diffl−j v f )(Diffv g). j j=0

Letting f , g ∈ C s (ℝn ), l > s, δ ∈ (0, 1], 1 ≤ k ≤ n, and 0 ≤ l′ ≤ l, we have

76 � 2 Ellipticity l′

󵄨󵄨 󵄨 󵄨 r 󵄨󵄨 󵄨 l′ l′ −r r 󵄨󵄨Diffδek (fg)(x)󵄨󵄨󵄨 ≲ ∑ 󵄨󵄨󵄨τδek Diffδek f (x)󵄨󵄨󵄨󵄨󵄨󵄨Diffδek g(x)󵄨󵄨󵄨 r=0 l′

≲ ∑ ‖ Difflδe−rk f ‖L∞ ‖ Diffrδek g‖L∞ ′

r=0 l′

≲ ∑ δs(l −r)/l δsr/l ‖f ‖C s ‖g‖C s ≲ δsl /l ‖f ‖C s ‖g‖C s . ′

′

l

r=0

l

l

l

It follows that ‖fg‖C s ≲ ‖f ‖C s ‖g‖C s . Applying Proposition 2.5.3 completes the proof. l

l

l

2.5.1 Decomposition into smooth functions One of the most convenient ways to study functions in C s (ℝn ) is to decompose them into a sum of smooth functions. The next proposition gives such a decomposition. Proposition 2.5.16. Let s > 0. For f ∈ S (ℝn )′ , the following are equivalent: (i) f ∈ C s (ℝn ). (ii) There exists a sequence {fj }j∈ℕ ⊂ C ∞ (ℝn ) such that for every multi-index α, α 󵄩 󵄩 sup 2js 󵄩󵄩󵄩(2−j 𝜕x ) fj 󵄩󵄩󵄩L∞ (ℝn ) < ∞, j∈ℕ

(2.51)

such that f = ∑j∈ℕ fj , with convergence in C(ℝn ). In this case, for M > s there exists C = C(s) ≥ 0 such that α 󵄩 󵄩 ‖f ‖C s (ℝn ) ≤ C ∑ sup 2js 󵄩󵄩󵄩(2−j 𝜕x ) fj 󵄩󵄩󵄩L∞ (ℝn ) . |α|≤M j∈ℕ

(2.52)

Furthermore, fj may be chosen such that for every multi-index α, there exists Cα,s ≥ 0 such that α 󵄩 󵄩 sup 2js 󵄩󵄩󵄩(2−j 𝜕x ) fj 󵄩󵄩󵄩L∞ (ℝn ) ≤ Cα,s ‖f ‖C s (ℝn ) . j∈ℕ

(2.53)

Proof. Suppose (i) holds, i. e., f ∈ C s (ℝn ). Set fj := Dj f and define Ej,α := (2−j 𝜕x )α Dj . Since {(Dj , 2−j ) : j ∈ ℕ} is a bounded set of elementary operators (see Example 2.3.8), Proposition 2.3.18 (c) implies that {(Ej,α , 2−j ) : j ∈ ℕ} is also a bounded set of elementary s operators. Therefore, by Proposition 2.4.11, using C s (ℝn ) = B∞,∞ (ℝn ), 󵄩󵄩 −j α 󵄩󵄩 󵄩 󵄩 −js 󵄩󵄩(2 𝜕x ) fj 󵄩󵄩L∞ (ℝn ) = 󵄩󵄩󵄩Ej,α fj 󵄩󵄩󵄩L∞ (ℝn ) ≲ 2 ‖f ‖C s (ℝn ) .

2.5 Zygmund–Hölder spaces

� 77

This shows that fj ∈ C ∞ (ℝn ) and (2.53) holds (and therefore (2.51) holds). We already know that ∑j∈ℕ Dj f = f , with convergence in S (ℝn )′ ; (2.51) with |α| = 0 implies that ∑j∈ℕ fj converges in C(ℝn ). This establishes (ii). Conversely, suppose (ii) holds so that f = ∑j∈ℕ fj , where fj satisfies (2.51). Fix M > s. We wish to show α 󵄩 󵄩 󵄩 󵄩 sup 2js 󵄩󵄩󵄩Dj f 󵄩󵄩󵄩L∞ (ℝn ) ≲ ∑ sup 2js 󵄩󵄩󵄩(2−j 𝜕x ) fj 󵄩󵄩󵄩L∞ (ℝn ) . j∈ℕ

|α|≤M j∈ℕ

(2.54)

Indeed, (2.54) is equivalent to (2.52), which implies (i). Consider, 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 2js 󵄩󵄩󵄩Dj f 󵄩󵄩󵄩L∞ (ℝn ) ≤ 2js ∑ 󵄩󵄩󵄩Dj fk 󵄩󵄩󵄩L∞ (ℝn ) + 2js ∑ 󵄩󵄩󵄩Dj fk 󵄩󵄩󵄩L∞ (ℝn ) . j≥k

j 0 we let C s,t (ℝn ×ℝN ) consist of those tempered distributions F ∈ S (ℝn × ℝN )′ such that the following norm is finite: ̂ j,l F‖ ∞ n N . ‖F‖C s,t (ℝn ×ℝN ) := sup 2js+lt ‖D L (ℝ ×ℝ ) j,l

For M ∈ ℕ+ , we define the vector-valued space C s,t (ℝn × ℝN ; ℂM ) to consist of those tempered distributions F = (F1 , . . . , FM ) with each Fk ∈ C s,t (ℝn × ℝN ). We set M

‖F‖C s,t (ℝn ×ℝN ;ℂM ) := ∑ ‖Fk ‖C s,t (ℝn ×ℝN ) . k=1

To study fully nonlinear elliptic PDEs, we use the next composition theorem. Theorem 2.6.2. Fix s, t > 0 and let F ∈ C s,t (ℝn ×ℝN ), u, v ∈ C s (ℝn ; ℝN ), and w ∈ C s (ℝn ). Then: (i) If t > ⌊s⌋ + 2, then F(x, u(x) + ζ ) ∈ C s,t−⌊s⌋−2 (ℝn × ℝN ) and ‖F(x, u(x) + ζ )‖C s,t−⌊s⌋−2 ≲ ‖F‖C s,t (1 + ‖u‖C s )(1 + ‖u‖L∞ )

⌊s⌋+1

(ii)

.

If t > ⌊s⌋ + 2, then F(x, u(x)) ∈ C s (ℝn ) and ‖F(x, u(x))‖C s ≲ ‖F‖C s,t (1 + ‖u‖C s )(1 + ‖u‖L∞ )

⌊s⌋+1

.

3 Here we are considering functions taking values in the real Banach space ℝN . In most of the text, we use complex Banach spaces, but for the purposes of this section the distinction is not relevant: every element of ℝN can be viewed as an element of ℂN , and every element of ℂN can be viewed as an element of ℝ2N . See also Remark 2.4.1.

2.6 Compositions and product Zygmund–Hölder spaces

79

�

(iii) We have u ⋅ v ∈ C s (ℝn ) and ‖u ⋅ v‖C s ≲ ‖u‖C s ‖v‖L∞ + ‖u‖L∞ ‖v‖C s . (iv) If t > ⌊s⌋ + 2, we have F(x, u(x) + ζ )w(x) ∈ C s,t−⌊s⌋−2 and ‖F(x, u(x) + ζ )w(x)‖C s,t−⌊s⌋−2 ≲ ‖F‖C s,t (‖w‖C s + ‖u‖C s ‖w‖L∞ )(1 + ‖u‖L∞ ) (v)

⌊s⌋+1

.

If t > ⌊s⌋ + 3, then ‖F(x, u(x) + ζ ) − F(x, v(x) + ζ )‖C s,t−⌊s⌋−3 ≲ ‖F‖C s,t (‖u − v‖C s + (‖u‖C s + ‖v‖C s )‖u − v‖L∞ )(1 + ‖u‖L∞ + ‖v‖L∞ )

⌊s⌋+1

.

(vi) If t > ⌊s⌋ + 3, then ‖F(x, u(x)) − F(x, v(x))‖C s ≲ ‖F‖C s,t (‖u − v‖C s + (‖u‖C s + ‖v‖C s )‖u − v‖L∞ )(1 + ‖u‖L∞ + ‖v‖L∞ )

⌊s⌋+1

.

β

(vii) Fix L ∈ ℕ+ and suppose 𝜕ζ F(x, 0) ≡ 0, ∀|β| < L. If t > ⌊s⌋ + 2 + L, then ‖F(x, u(x))‖C s ≲ ‖F‖C s,t (1 + ‖u‖L∞ )

⌊s⌋+2

‖u‖L−1 L∞ ‖u‖C s .

β

(viii) Fix L ∈ ℕ, L ≥ 2 and suppose 𝜕ζ F(x, 0) ≡ 0, ∀0 < |β| ≤ L and t > ⌊s⌋ + 2 + L. Then ‖F(x, u(x)) − F(x, v(x))‖C s ≲ ‖F‖C s,t (1 + ‖u‖L∞ + ‖v‖L∞ )

⌊s⌋+2

× (‖u‖L∞ + ‖v‖L∞ )

L−2

(‖u‖C s + ‖v‖C s )‖u − v‖L∞

+ ‖F‖C s,t (‖u‖L∞ + ‖v‖L∞ )

L−1

‖u − v‖C s .

The implicit constants do not depend on F, u, v, or w, but may depend on s, t, n, and N. Remark 2.6.3. Of particular interest is Theorem 2.6.2 (vi), which, roughly, says the map u 󳨃→ F(x, u(x)) is Lipschitz in u. This will allow us to use arguments similar to the contraction mapping principle when studying fully nonlinear elliptic PDEs. Remark 2.6.4. Theorem 2.6.2 assumes more regularity than is necessary in the ζ variable (i. e., the results are not sharp in t). We proceed with these non-sharp results because the proof of Theorem 2.6.2 easily generalizes to the more general situation we consider in Section 7.5. Understanding the sharp regularity needed in the ζ variable can be somewhat complicated; see, e. g., [14]. Remark 2.6.5. All the estimates in Theorem 2.6.2 are tame in the sense that each term on the right-hand side of the inequalities only has one factor of ‖ ⋅ ‖C s (W ,ds)⃗ ; all the other

80 � 2 Ellipticity factors are ‖ ⋅ ‖L∞ . Tame estimates are an important tool in the field of nonlinear PDEs (see, for example, [105]) and are an essential tool in our study of fully nonlinear PDEs. Before we prove Theorem 2.6.2, we state (without proof) several basic properties of

C s,t (ℝn × ℝN ); the proofs are very similar to the corresponding proofs for C s (ℝn ).

Proposition 2.6.6. For s, t > 0, C s,t (ℝn × ℝN ) is a Banach space when endowed with the norm ‖ ⋅ ‖C s,t (ℝn ×ℝN ) . Neither the space C s,t (ℝn × ℝN ) nor the equivalence class of ̃l. ‖ ⋅ ‖C s,t (ℝn ×ℝN ) depends on the choices we made when defining Dj and D Proposition 2.6.7. Let s, t > 0. For F ∈ S (ℝn × ℝN )′ , the following are equivalent: (i) We have F ∈ C s,t (ℝn × ℝN ). (ii) There exists a sequence {Fj,l }j,l∈ℕ ⊂ C ∞ (ℝn × ℝN ) such that for all multi-indices α, β, α β 󵄩 󵄩 sup 2js 2lt 󵄩󵄩󵄩(2−j 𝜕x ) (2−l 𝜕x ) Fj,k 󵄩󵄩󵄩L∞ (ℝn ×ℝN ) < ∞,

j,l∈ℕ

and F = ∑j,l∈ℕ Fj,l with convergence in C(ℝn × ℝN ). In this case, there exist M = M(s, t) ∈ ℕ and Cs,t ≥ 0 such that ‖F‖C s,t (ℝn ×ℝN ) ≤ Cs,t

∑

α β 󵄩 󵄩 sup 2js 2lt 󵄩󵄩󵄩(2−j 𝜕x ) (2−l 𝜕x ) Fj,k 󵄩󵄩󵄩L∞ (ℝn ×ℝN ) .

|α|,|β|≤M j,l∈ℕ

Furthermore, Fj,l can be chosen such that ∀α, β, there exists Cs,t,α,β ≥ 0, α β 󵄩 󵄩 sup 2js 2lt 󵄩󵄩󵄩(2−j 𝜕x ) (2−l 𝜕x ) Fj,k 󵄩󵄩󵄩L∞ (ℝn ×ℝN )

j,l∈ℕ

≤ Cs,t,α,β ‖F‖C s,t (ℝn ×ℝN ) .

(2.58)

Proposition 2.6.8. Let s > 0, let β ∈ ℕN be a multi-index, and let t > |β|. Then F(x, ζ ) 󳨃→ β 𝜕ζ F(x, ζ ) is continuous, C s,t (ℝn × ℝN ) → C s,t−|β| (ℝn × ℝN ). Lemma 2.6.9. For s, t > 0, we have: (i) The map F(x, ζ ) 󳨃→ F(x, 0) is continuous, C s,t (ℝn × ℝN ) → C s (ℝn ). (ii) The map F(x, ζ1 , . . . , ζN ) 󳨃→ F(x, ζ1 , . . . , ζN−1 , 0) is continuous, C s,t (ℝn × ℝN ) → C s,t (ℝn × ℝN−1 ). (iii) For x0 ∈ ℝn , the map F(x, ζ ) 󳨃→ F(x0 , ζ ) is continuous, C s,t (ℝn × ℝN ) → C t (ℝN ). Proof. This follows easily from Propositions 2.5.16 and 2.6.7. With the above facts in hand, we turn to sketching the proof of Theorem 2.6.2. In Section 7.5 we repeat this proof in a more difficult setting and include all the details. Lemma 2.6.10. Let s, t > 0 and set M1 := ⌊s⌋ + 1 and M2 := ⌊t⌋ + 1. Suppose ∀j, k, l ∈ ℕ, we are given Hj,k,l (x, ζ ) ∈ C ∞ (ℝn × ℝN ) such that ∀|α| ≤ M1 and |β| ≤ M2 ,

2.6 Compositions and product Zygmund–Hölder spaces

�

81

󵄩󵄩 −(k∨j) α −l β 󵄩 𝜕x ) (2 𝜕ζ ) Hj,k,l 󵄩󵄩󵄩L∞ (ℝn ×ℝN ) ≤ A2−ks 2−js 2−lt , 󵄩󵄩(2 for some A ≥ 0. Set H(x, ζ ) := ∑j,k,l∈ℕ Hj,k,l (x, ζ ). Then H ∈ C s,t (ℝn × ℝN ) and ‖H‖C s,t (ℝn ×ℝN ) ≤ Cs,t,n,N A,

(2.59)

for some Cs,t,n,N ≥ 0. Proof. We will show for j, k, k ′ , l, l′ ∈ ℕ that 󵄩󵄩 ̂ 󵄩 −M ((j∨k∨k ′ )−(j∨k ′ )) −M2 ((l∨l′ )−l′ ) −js −k ′ s −l′ t 2 2 2 2 . 󵄩󵄩Dk,l Hj,k ′ ,l′ 󵄩󵄩󵄩L∞ (ℝn ×ℝN ) ≲ A2 1

(2.60)

First we see why (2.60) will complete the proof. Indeed, using (2.60), we have 󵄩󵄩 ̂ 󵄩 󵄩󵄩Dk,l H 󵄩󵄩󵄩L∞ (ℝn ×ℝN ) ≤

󵄩̂ 󵄩󵄩 ∑ 󵄩󵄩󵄩D k,l Hj,k ′ ,l′ 󵄩 󵄩 L∞

j,k ′ ,l′ ∈ℕ

≲A

2−M1 ((j∨k∨k )−(j∨k )) 2−M2 ((l∨l )−l ) 2−js 2−k s 2−l t ≲ A2−ks 2−lt , ′

∑

j,k ′ ,l′ ∈ℕ

′

′

′

′

′

where the final estimate is elementary and uses M1 > s and M2 > t – see Lemma 7.3.2. This estimate is equivalent to (2.59), which then completes the proof. We turn to the proof of (2.60), which is separated into several cases. Using Proposition 2.3.18 (d) we may write α

Dk = ∑ 2−k(M1 −|α|) Ek,α (2−k 𝜕x ) ,

(2.61)

|α|≤M1

̃ l = ∑ 2−l(M2 −|β|) El,β (2−l 𝜕ζ )β , D

(2.62)

|β|≤M1

where {(Ek,α , 2−k ) : k ∈ ℕ, |α| ≤ M1 } and {(Ẽl,β , 2−l ) : l ∈ ℕ, |β| ≤ M2 } are bounded sets of elementary operators on ℝn and ℝN , respectively. In particular, using Lemma 2.3.26, we have ̃ l ‖L∞ →L∞ , ‖Ek,α ‖L∞ →L∞ , ‖Ẽl,β ‖L∞ →L∞ ≲ 1, ‖Dk ‖L∞ →L∞ , ‖D

(2.63)

∀k, l ∈ ℕ, |α| ≤ M1 , |β| ≤ M2 . First, we consider the case where k ′ ∨ j = k ∨ k ′ ∨ j and l′ = l ∨ l′ . Thus, using (2.63) ̂ k,l = Dk ⊗ D ̃ l , we have and the fact that D ̂ k,l Hj,k ′ ,l′ ‖L∞ ≲ ‖Hj,k ′ ,l′ ‖L∞ ≤ A2−js 2−k s 2−l t , ‖D ′

establishing (2.60) in this case.

′

82 � 2 Ellipticity We next consider the case where k > k ′ ∨ j but l ≤ l′ . Using (2.61) and (2.63), we have ̂ k,l Hj,k ′ ,l′ ‖L∞ ≤ ∑ 2−(M1 −|α|)k 󵄩󵄩󵄩(Ek,α ⊗ D ̃ l )(2−k 𝜕x )α Hj,k ′ ,l′ 󵄩󵄩󵄩 ∞ ‖D 󵄩 󵄩L |α|≤M1

α 󵄩 󵄩 ≲ ∑ 2−(M1 −|α|)k 󵄩󵄩󵄩(2−k 𝜕x ) Hj,k ′ ,l′ 󵄩󵄩󵄩L∞ |α|≤M1

′ ′ α 󵄩 󵄩 ≲ ∑ 2−(M1 −|α|)k 2−|α|(k−(k ∨j)) 󵄩󵄩󵄩(2−(k ∨j) 𝜕x ) Hj,k ′ ,l′ 󵄩󵄩󵄩L∞

|α|≤M1

≲ A2−M1 (k−(k ∨j)) 2−k s 2−js 2−l t , ′

′

′

which establishes (2.60) in this case. We next consider the case where k > k ′ ∨ j and l > l′ . Using (2.61), (2.62), and (2.63), we have ̂ k,l Hj,k ′ ,l′ ‖L∞ ‖D

α β 󵄩 󵄩 ≤ ∑ 2−(M1 −|α|)k 2−l(M2 −|β|) 󵄩󵄩󵄩(Ek,α ⊗ Ẽl,β )(2−k 𝜕x ) (2−l 𝜕ζ ) Hj,k ′ ,l′ 󵄩󵄩󵄩L∞ |α|≤M1 |β|≤M2

α β 󵄩 󵄩 ≲ ∑ 2−(M1 −|α|)k 2−l(M2 −|β|) 󵄩󵄩󵄩(2−k 𝜕x ) (2−l 𝜕ζ ) Hj,k ′ ,l′ 󵄩󵄩󵄩L∞ |α|≤M1 |β|≤M2

′ ′ ′ ′ α β 󵄩 󵄩 = ∑ 2−(M1 −|α|)k 2−l(M2 −|β|) 2−|α|(k−(k ∨j)) 2−|β|(l−l ) 󵄩󵄩󵄩(2−(k ∨j) 𝜕x ) (2−l 𝜕ζ ) Hj,k ′ ,l′ 󵄩󵄩󵄩L∞

|α|≤M1 |β|≤M2

≲ A2−M1 (k−(k ∨j)) 2−M2 (l−l ) 2−k s 2−js 2−l t , ′

′

′

′

which establishes (2.60) in this case. Finally, we consider the case where k ≤ k ′ ∨ j but l > l′ . Using (2.62) and (2.63), we have ̃ k,l Hj,k ′ ,l′ ‖L∞ ≤ ∑ 2−(M2 −|β|)l 󵄩󵄩󵄩(Dk ⊗ Ẽl,β )(2−l 𝜕ζ )β Hj,k ′ ,l′ 󵄩󵄩󵄩 ∞ ‖D 󵄩 󵄩L |β|≤M2

β 󵄩 󵄩 ≲ ∑ 2−(M2 −|β|)l 󵄩󵄩󵄩(2−l 𝜕ζ ) Hj,k ′ ,l′ 󵄩󵄩󵄩L∞ |β|≤M2

′ 󵄩 ′ β 󵄩 ≲ ∑ 2−(M2 −|β|)l 2−|β|(l−l ) 󵄩󵄩󵄩(2−l 𝜕ζ ) Hj,k ′ ,l′ 󵄩󵄩󵄩L∞

|β|≤M2

≲ A2−M2 (l−l ) 2−js 2−k s 2−l t , ′

′

′

establishing (2.60) in this case, completing the proof. Proof of Theorem 2.6.2. (i): We begin with the proof of (i), which contains most of the difficulties. To prove this, we wish to decompose F(x, u(x) + ζ ) as a sum as in Lemma 2.6.10. First, we decompose F = ∑k,l∈ℕ Fk,l as in Proposition 2.6.7, where Fk,l satisfy (2.58). Next, take PJ := ∑j≤J Dj so that PJ f = f ∗Dil2J (ϕ) (see the definition of Dj in (2.30)). In particular,

2.6 Compositions and product Zygmund–Hölder spaces

� 83

for u ∈ C s (ℝn ), we have PJ u → u in C(ℝn ). The decomposition of F(x, u(x) + ζ ) we use is given by F(x, u(x) + ζ ) = lim F(x, PJ u(x) + ζ ) J→∞

∞

= F(x, D0 u(x) + ζ ) + ∑(F(x, Pj u(x) + ζ ) − F(x, Pj−1 u(x) + ζ )) j=1

= ∑ Fk,l (x, D0 u(x) + ζ ) k,l∈ℕ

∞

+ ∑ ∑(Fk,l (x, Pj u(x) + ζ ) − Fk,l (x, Pj−1 u(x) + ζ )) k,l∈ℕ j=1

=: ∑ Hj,k,l (x, ζ ), j,k,l∈ℕ

where Hj,k,l = {

Fk,l (x, D0 u(x) + ζ ),

Fk,l (x, Pj u(x) + ζ ) − Fk,l (x, Pj−1 u(x) + ζ ),

j = 0, j ≥ 1.

To prove (i), we will show 󵄩󵄩 −(k∨j) α −l β 󵄩 𝜕x ) (2 𝜕ζ ) Hj,k,l 󵄩󵄩󵄩L∞ 󵄩󵄩(2

≤ Cα,β ‖F‖C s,t (ℝn ×ℝN ) 2−ks 2−js 2−l(t−|α|−1) (1 + ‖u‖C s )(1 + ‖u‖L∞ ) . |α|

(2.64)

To see why (2.64) completes the proof of (i), set M1 := ⌊s⌋ + 1 and M2 := ⌊t⌋ − ⌊s⌋ − 1. We have by (2.64), ∀|α| ≤ M1 , ∀|β| ≤ M2 , 󵄩󵄩 −(k∨j) α −l β 󵄩 𝜕x ) (2 𝜕ζ ) Hj,k,l 󵄩󵄩󵄩L∞ 󵄩󵄩(2

≲ ‖F‖C s,t (ℝn ×ℝN ) 2−ks 2−js 2−l(t−⌊s⌋−2) (1 + ‖u‖C s )(1 + ‖u‖L∞ )

⌊s⌋+1

.

This shows that Hj,k,l satisfies the assumptions of Lemma 2.6.10 with t replaced by t − ⌊s⌋ − 2. The conclusion of Lemma 2.6.10 gives (i). We turn to proving (2.64). When j = 0, proving (2.64) is straightforward, and we leave it to the reader. When j ≥ 1, we use Hj,k,l (x, ζ ) = Fk,l (x, Pj u(x) + ζ ) − Fk,l (x, Pj−1 u(x) + ζ ) 1

= ∫ 𝜕σ (Fk,l (x, (1 − σ)Pj u(x) + σPj−1 u(x) + ζ )) dσ 0

1

= ∫ dζ Fk,l (x, (1 − σ)Pj u(x) + σPj−1 u(x) + ζ ) dσDj−1 u(x). 0

84 � 2 Ellipticity Thus, to prove (2.64) it suffices to show that α β 󵄩 󵄩 sup 󵄩󵄩󵄩(2−(k∨j) 𝜕x ) (2−l 𝜕ζ ) (dζ Fk,l )(x, (1 − σ)Pj u(x) + σPj−1 u(x) + ζ )Dj−1 u(x)󵄩󵄩󵄩L∞

σ∈[0,1]

≲ ‖F‖C s,t (ℝn ×ℝN ) 2

2 2

−ks −js −l(t−|α|−1)

|α|

(2.65)

(1 + ‖u‖C s )(1 + ‖u‖L∞ ) .

Proving the estimate (2.65) is straightforward, and we leave it to the reader (see the proof of the analogous result Theorem 7.5.2). 󵄨 (ii): Since F(x, u(x)) = F(x, u(x) + ζ )󵄨󵄨󵄨ζ =0 , (ii) follows by combining (i) and Lemma 2.6.9 (i). (iii): We first consider the case where ‖u‖L∞ = ‖v‖L∞ = 1. Fix ϕ(ζ1 , ζ2 ) ∈ C0∞ (ℝN ×ℝN ) with ϕ(ζ1 , ζ2 ) ≡ 1 if |ζ1 |, |ζ2 | ≤ 1. Set F(x, ζ ) := ϕ(ζ1 , ζ2 )ζ1 ⋅ ζ2 ∈ C0∞ (ℝ2N ). Since ϕ ∈ C0∞ (ℝ2N ), it follows that F(x, ζ ) ∈ C s,t (ℝn × ℝ2N ), ∀t > 0. Using the fact that ‖u‖L∞ = ‖v‖L∞ = 1, we have u(x) ⋅ v(x) = F(u(x), v(x)) and (ii) implies ‖u ⋅ v‖C s = ‖F(x, u(x), v(x))‖C s ≲ ‖F‖C s,⌊s⌋+3 (1 + ‖u‖C s + ‖v‖C s )(1 + ‖u‖L∞ + ‖v‖L∞ )

(2.66)

≲ 1 + ‖u‖C s + ‖v‖C s .

For general (not identically zero) u, v ∈ C s (ℝn ; ℝN ), we apply (2.66) to u/‖u‖L∞ and v/‖v‖L∞ to see ‖u ⋅ v‖C s ≲ ‖u‖L∞ ‖v‖L∞ + ‖u‖C s ‖v‖L∞ + ‖u‖L∞ ‖v‖C s ≲ ‖u‖C s ‖v‖L∞ + ‖u‖L∞ ‖v‖C s , where the final inequality uses ‖u‖L∞ ≲ ‖u‖C s . (iv): First we prove the result when ‖w‖L∞ = 1. Let ϕ(ζ ′ ) ∈ C0∞ (ℝ) satisfy ϕ ≡ 1 for |ζ ′ | ≤ 1. Set ̃ ζ , ζ ′ ) := F(x, ζ )ϕ(ζ ′ )ζ ′ . F(x, ̃ s,t n N+1 ≲ It follows easily from Proposition 2.6.7 that F̃ ∈ C s,t (ℝn × ℝN+1 ) and ‖F‖ C (ℝ ×ℝ ) ‖F‖C s,t (ℝn ×ℝN ) . ̃ u(x) + ζ , w(x) + ζ ′ )󵄨󵄨󵄨 ′ . Thus, When ‖w‖L∞ = 1, we have F(x, u(x) + ζ )w(x) = F(x, 󵄨ζ =0 applying (i) and Lemma 2.6.9 (ii), we have ‖F(x, u(x) + ζ )w(x)‖C s,t−⌊s⌋−2 (ℝn ×ℝN ) 󵄩̃ 󵄨 󵄩 = 󵄩󵄩󵄩F(x, u(x) + ζ , w(x) + ζ ′ )󵄨󵄨󵄨ζ ′ =0 󵄩󵄩󵄩C s,t−⌊s⌋−2 (ℝn ×ℝN ) 󵄩̃ 󵄩 ≲ 󵄩󵄩󵄩F(x, u(x) + ζ , w(x) + ζ ′ )󵄩󵄩󵄩C s,t−⌊s⌋−2 (ℝn ×ℝN+1 )

(2.67) ⌊s⌋+1

≲ ‖F‖C s,t (1 + ‖u‖C s + ‖w‖C s )(1 + ‖u‖L∞ + ‖w‖L∞ ) ⌊s⌋+1

≲ ‖F‖C s,t (1 + ‖u‖C s + ‖w‖C s )(1 + ‖u‖L∞ )

.

2.6 Compositions and product Zygmund–Hölder spaces

� 85

For general, not identically zero, w ∈ C s (ℝn ), we apply (2.67) with w replaced by w/‖w‖L∞ to see ‖F(x, u(x) + ζ )w(x)‖C s,t−⌊s⌋−2 (ℝn ×ℝN ) ⌊s⌋+1

≲ ‖F‖C s,t (‖w‖L∞ + ‖w‖L∞ ‖u‖C s + ‖w‖C s )(1 + ‖u‖L∞ ) ⌊s⌋+1

≲ ‖F‖C s,t (‖w‖L∞ ‖u‖C s + ‖w‖C s )(1 + ‖u‖L∞ )

,

where the final inequality used ‖w‖L∞ ≲ ‖w‖C s . (v): By Proposition 2.6.8, the components of the vector dζ F satisfy the same assumptions as F, but with t replaced by t − 1. Using (iv), we have ‖F(x, u(x) + ζ ) − F(x, v(x) + ζ )‖C s,t−⌊s⌋−3 󵄩󵄩 1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 = 󵄩󵄩∫ dζ F(x, σu(x) + (1 − σ)v(x) + ζ )(u(x) − v(x)) dσ 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 0 󵄩󵄩C s,t−⌊s⌋−3 1

≤ ∫ ‖dζ F(x, σu(x) + (1 − σ)v(x) + ζ )(u(x) − v(x))‖C s,t−⌊s⌋−3 dσ 0

≤ sup ‖dζ F(x, σu(x) + (1 − σ)v(x) + ζ )(u(x) − v(x))‖C s,t−⌊s⌋−3 σ∈[0,1]

≲ ‖dζ F‖C s,t−1 (‖u − v‖C s + ‖σu + (1 − σ)v‖C s ‖u − v‖L∞ ) × (1 + ‖σu + (1 − σ)v‖L∞ )

⌊s⌋+1

≲ ‖F‖C s,t (‖u − v‖C s + (‖u‖C s + ‖v‖C s )‖u − v‖L∞ ) × (1 + ‖u‖L∞ + ‖v‖L∞ )

⌊s⌋+1

,

where the last inequality uses Proposition 2.6.8. (vi): Item (vi) follows by combining (v) and Lemma 2.6.9 (i). (vii): We prove (vii) by induction on L and begin with the base case, L = 1. When L = 1, using the fact that F(x, 0) ≡ 0, we have F(x, u(x)) = F(x, u(x)) − F(x, 0). From here, the result when L = 1 follows from the case v = 0 of (vi). We assume (vii) for some L ∈ ℕ+ and prove it for L + 1; thus we assume the hypotheses with L replaced by L + 1. Using the fact that F(x, 0) ≡ 0 and using (iii), we have ‖F(x, u(x))‖C s = ‖F(x, u(x)) − F(x, v(x))‖C s 󵄩󵄩 1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 = 󵄩󵄩∫ dζ F(x, σu(x))u(x) dσ 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 0 󵄩󵄩C s 1

󵄩 󵄩 ≤ ∫󵄩󵄩󵄩dζ F(x, σu(x))u(x)󵄩󵄩󵄩C s dσ 0

(2.68)

86 � 2 Ellipticity 󵄩 󵄩 ≤ sup 󵄩󵄩󵄩dζ F(x, σu(x))u(x)󵄩󵄩󵄩C s σ∈[0,1]

󵄩 󵄩 ≲ sup 󵄩󵄩󵄩dζ F(x, σu(x))󵄩󵄩󵄩C s ‖u‖L∞ + sup ‖dζ F(x, σu(x))‖L∞ ‖u‖C s σ∈[0,1]

σ∈[0,1]

=: (I) + (II), where we applied (iii) to the various summands of the matrix multiplication dζ F(x, σu(x))u(x). To bound (I) we use the inductive hypothesis applied to dζ F with t replaced by t−1 > ⌊s⌋ + 2 + L and Proposition 2.6.8 to see 󵄩 󵄩 (I) = sup 󵄩󵄩󵄩dζ F(x, σu(x))󵄩󵄩󵄩C s ‖u‖L∞ σ∈[0,1]

≲ sup ‖dζ F‖C s,t−1 (1 + ‖σu‖L∞ )

⌊s⌋+2

σ∈[0,1]

≲ ‖F‖C s,t (1 + ‖u‖L∞ )

⌊s⌋+2

‖σu‖L−1 L∞ ‖σu‖C s ‖u‖L∞

(2.69)

‖u‖LL∞ ‖u‖C s .

β

For (II) we use the fact that 𝜕ζ dζ F(x, 0) ≡ 0, ∀|β| < L, the fact that t > L + 1, and Proposition 2.6.8 to see that 󵄩 󵄩 󵄩 󵄩 sup 󵄩󵄩󵄩dζ F(x, σu(x))󵄩󵄩󵄩L∞ ≲ sup ∑ 󵄩󵄩󵄩𝜕ζL dζ F(x, ζ )󵄩󵄩󵄩L∞ ‖σu(x)‖LL∞

σ∈[0,1]

σ∈[0,1] |β|=L

≲ ‖F‖C s,t ‖u‖LL∞ . We conclude that (II) ≲ ‖F‖C s,t ‖u‖LL∞ ‖u‖C s .

(2.70)

Using (2.69) and (2.70) to bound the right-hand side of (2.68) completes the inductive step. (viii): We have, using (iii), 󵄩󵄩 1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 ‖F(x, u(x)) − F(x, v(x))‖C s = 󵄩󵄩󵄩∫ dζ F(x, σu(x) + (1 − σ)v(x))(u(x) − v(x)) dσ 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 0 󵄩󵄩C s 1

󵄩 󵄩 ≤ ∫󵄩󵄩󵄩dζ F(x, σu(x) + (1 − σ)v(x))(u(x) − v(x))󵄩󵄩󵄩C s dσ 0

󵄩 󵄩 ≤ sup 󵄩󵄩󵄩dζ F(x, σu(x) + (1 − σ)v(x))(u(x) − v(x))󵄩󵄩󵄩C s σ∈[0,1]

≲ sup ‖dζ F(x, σu(x) + (1 − σ)v(x))‖C s ‖u − v‖L∞ σ∈[0,1]

+ sup ‖dζ F(x, σu(x) + (1 − σ)v(x))‖L∞ ‖u − v‖C s σ∈[0,1]

=: (III) + (IV),

(2.71)

2.7 Linear elliptic operators

�

87

where we applied (iii) to the various summands of the matrix multiplication dζ F(x, σu(x) + (1 − σ)v(x))(u(x) − v(x)). β For (III), we have 𝜕ζ dζ F(x, 0) ≡ 0, ∀|β| ≤ L − 1, and we may therefore apply (vii) with L replaced by L − 1 and t replaced by t − 1 to see 󵄩 󵄩 sup 󵄩󵄩󵄩dζ F(x, σu(x) + (1 − σ)v(x))󵄩󵄩󵄩C s

σ∈[0,1]

≲ sup ‖dζ F‖C s,t−1 (1 + ‖σu + (1 − σ)v‖L∞ )

⌊s⌋+2

σ∈[0,1]

≲ ‖F‖C s,t (1 + ‖u‖L∞ + ‖v‖L∞ )

⌊s⌋+2

‖σu + (1 − σ)v‖L−2 L∞ ‖σu + (1 − σ)v‖C s

(‖u‖L∞ + ‖v‖L∞ )

L−2

(‖u‖C s + ‖v‖C s ),

where the final estimate used Proposition 2.6.8. It follows that L−2

(‖u‖C s + ‖v‖C s )‖u − v‖L∞ . (2.72) β For (IV), we use the fact that 𝜕ζ dζ f (x, 0) ≡ 0, ∀|β| < L − 1 to see

(III) ≲ ‖F‖C s,t (1 + ‖u‖L∞ + ‖v‖L∞ )

⌊s⌋+ν+1

(‖u‖L∞ + ‖v‖L∞ )

󵄩 󵄩 sup 󵄩󵄩󵄩dζ F(x, σu(x) + (1 − σ)v(x))󵄩󵄩󵄩L∞

σ∈[0,1]

≲ sup

󵄩 β 󵄩 󵄩 󵄩L−1 ∑ 󵄩󵄩󵄩𝜕ζ dζ F(x, σu(x) + (1 − σ)v(x))󵄩󵄩󵄩L∞ 󵄩󵄩󵄩σu + (1 − σ)v󵄩󵄩󵄩L∞

σ∈[0,1] |β|=L−1

≲ ‖F‖C s,t (‖u‖L∞ + ‖v‖L∞ )

L−1

,

where in the last inequality we used t > L and Proposition 2.6.8. It follows that (IV) ≲ ‖F‖C s,t (‖u‖L∞ + ‖v‖L∞ )

L−1

‖u − v‖C s .

(2.73)

Using (2.72) and (2.73) to bound the right-hand side of (2.71) completes the proof.

2.7 Linear elliptic operators Our main result about linear maximally subelliptic operators with smooth coefficients (Theorem 8.1.1) is a theorem giving several equivalent characterizations of such operators. Here, we state the special case of this theorem in the case of elliptic operators, a setting where the theorem is well known. Fix a connected open set Ω ⊆ ℝn , D1 , D2 , κ ∈ ℕ+ , and let α

P := ∑ aα (x)𝜕x ,

∞ aα ∈ Cloc (Ω; 𝕄D1 ×D2 (ℂ)).

|α|≤κ

Definition 2.7.1. Let Pκ (x, ξ) := ∑ aα (x)(2πi)|α| ξ α . |α|=κ

(2.74)

88 � 2 Ellipticity We say P is elliptic near x0 ∈ Ω if the D1 × D2 matrix Pκ (x0 , ξ) is left invertible ∀ξ ≠ 0. Note that this requires D1 ≥ D2 . We say P is elliptic on Ω if P is elliptic near x0 , ∀x0 ∈ Ω. Letting P ∗ P , where P ∗ is the formal adjoint of P , we have α

P P = ∑ bα (x)𝜕x , ∗

∞ bα ∈ Cloc (Ω; 𝕄D2 ×D2 (ℂ)).

|α|≤2κ

It makes sense to ask whether P ∗ P is elliptic, since it is a partial differential operator of order 2κ. We sometimes consider P ∗ P as an unbounded operator on L2 (Ω), with dense domain C0∞ (Ω); then P ∗ P is a non-negative symmetric operator. For two operators T, S ∈ Hom(C0∞ (Ω; ℂD1 ), C0∞ (Ω; ℂD2 )′ ), we write ∞ mod Cloc (Ω × Ω; 𝕄D2 ×D1 (ℂ)),

T ≡S

∞ if the Schwartz kernel of T − S is given by integration against a function in Cloc (Ω × D2 ×D1 Ω; 𝕄 (ℂ)), that is,

∫ g(x) ⋅ (T − S)f (x) dx = ∬ g(x) ⋅ (T − S)(x, y)f (y) dy, ∞ ∀f ∈ C0∞ (Ω; ℂD1 ), ∀g ∈ C0∞ (Ω; ℂD2 ), where T − S ∈ Cloc (Ω × Ω; 𝕄D2 ×D1 (ℂ)). We now state a basic theorem regarding linear elliptic partial differential operators. All aspects of this theorem are well known, and some are trivial; we will later prove a generalization for maximally subelliptic operators (see Theorem 8.1.1). In this theorem, and in the rest of the text, for a matrix M, we write |M| for the usual operator norm.

Theorem 2.7.2. Let P be given by (2.74). The following are equivalent: (a) P is elliptic on Ω. (b) For every pre-compact, open set Ω1 ⋐ Ω, there exists CΩ1 ≥ 0 such that ∀f ∈ C0∞ (Ω1 ; ℂD2 ), n

󵄩 󵄩 ∑󵄩󵄩󵄩𝜕xκj f 󵄩󵄩󵄩L2 (Ω ;ℂD2 ) ≤ CΩ1 (‖P f ‖L2 (Ω1 ;ℂD1 ) + ‖f ‖L2 (Ω1 ;ℂD2 ) ). j=1

1

(c) P0 := ∑|α|=κ aα (x)𝜕xα is elliptic on Ω. (d) The frozen coefficient operator ∑|α|≤κ aα (x0 )𝜕xα is elliptic on Ω, ∀x0 ∈ Ω. (e) For all ϕ1 , ϕ2 ∈ C0∞ (Ω) with ϕ1 ≺ ϕ2 , the following holds ∀u ∈ C0∞ (Ω; ℂD2 )′ : s s+κ – for p, q ∈ [1, ∞] and s ∈ ℝ, if ϕ2 P u ∈ Bp,q (ℝn ), then ϕ1 u ∈ Bp,q (ℝn ), and for such u s+κ (ℝn ) ≤ Cs,p,q,ϕ ,ϕ ,P (‖ϕ2 P u‖Bs (ℝn ) + ‖ϕ2 u‖Bs (ℝn ) ), ‖ϕ1 u‖Bp,q 1 2 p,q p,q

–

s s+κ for p ∈ (1, ∞), q ∈ (1, ∞], and s ∈ ℝ, if ϕ2 P u ∈ Fp,q (ℝn ), then ϕ1 u ∈ Fp,q (ℝn ), and for such u

2.7 Linear elliptic operators

� 89

s+κ (ℝn ) ≤ Cs,p,q,ϕ ,ϕ ,P (‖ϕ2 P u‖F s (ℝn ) + ‖ϕ2 u‖F s (ℝn ) ). ‖ϕ1 u‖Fp,q 1 2 p,q p,q

(f) P ∗ P is elliptic on Ω. ∞ (g) There exists an operator T ∈ Hom(C0∞ (Ω; ℂD1 ), Cloc (Ω; ℂD2 )) which is locally a singular integral operator of order −κ (see Definitions 2.3.30 and 2.3.1 for this concept), such that TP ≡ I

∞ mod Cloc (Ω × Ω; 𝕄D2 ×D2 (ℂ)).

In fact, T is locally a pseudo-differential operator of order −κ (see Definitions 2.3.30 and 2.2.3). ∞ (h) There exists an operator S ∈ Hom(C0∞ (Ω; ℂD2 ), Cloc (Ω; ℂD2 )) which is locally a singular integral operator of order −2κ such that SP ∗ P , P ∗ P S ≡ I

∞ mod Cloc (Ω × Ω; 𝕄D2 ×D2 (ℂ)).

In fact, S is locally a pseudo-differential operator of order −2κ. (i) For every non-negative self-adjoint extension L of P ∗ P and ∀t > 0, e−tL is given by ∞ integration against a Cloc (Ω × Ω; 𝕄D2 ×D2 (ℂ)) function (denoted by e−tL (x, y)), which satisfies the following. For every Ω1 ⋐ Ω, ∃c = c(Ω1 ) > 0, ∀α, β ∈ ℕn , ∀s ∈ ℕ, ∃C = C(α, β, s, Ω1 ) ≥ 0, ∀t ∈ (0, 1], ∀x, y ∈ Ω1 , such that 2κ

1 −n−|α|−|β|−2κs |x − y| 󵄨󵄨 s α β −tL 󵄨 (x, y)󵄨󵄨󵄨 ≤ C(|x − y| + t 2κ ) exp(−c ( ) 󵄨󵄨𝜕t 𝜕x 𝜕y e t

1/(2κ−1)

).

(j) There exists a non-negative self-adjoint extension L of P ∗ P satisfying the conclusions of (i). (k) For every non-negative self-adjoint extension L of P ∗ P and ∀t > 0, e−tL is given by ∞ integration against a Cloc (Ω × Ω; 𝕄D2 ×D2 (ℂ)) function (denoted by e−tL (x, y)), which satisfies the following. For every Ω1 ⋐ Ω, ∃c = c(Ω1 ) > 0, ∀α, β ∈ ℕn , ∀l, s ∈ ℕ, ∃C = C(α, β, s, l) ≥ 0, ∀t ∈ (0, 1], ∀x, y ∈ Ω1 , such that 1 −n 󵄨󵄨 s α β −tL 󵄨 (x, y)󵄨󵄨󵄨 ≤ Ct −|α|/2κ−|β|/2κ−s (|x − y| + t 2κ ) (1 + t −1/2κ |x − y|)−l . 󵄨󵄨𝜕t 𝜕x 𝜕y e

(l) There exists a non-negative self-adjoint extension L of P ∗ P satisfying the conclusions of (k). The main aspects of Theorem 2.7.2 are a special case of Theorem 8.1.1 and we refer the reader there for proofs. Remark 2.7.3. Theorem 2.7.2 concerns elliptic operators acting on vector-valued functions. One can easily generalize this theorem to operators acting on sections of a finitedimensional vector bundle. See Section 8.6 for a more general version of this for maximally subelliptic operators.

90 � 2 Ellipticity

2.8 Nonlinear elliptic equations It follows from Theorem 2.7.2 that elliptic partial differential operators of order κ are locally left invertible, modulo smooth functions as maps from C s+κ (ℝn ) → C s (ℝn ). In this section, we use this fact to deduce the sharp regularity theory for fully nonlinear elliptic equations. Fix D1 , D2 , κ ∈ ℕ+ and set N := D2 × #{α ∈ ℕn : |α| ≤ κ}. Given u = (u1 , . . . , uD2 ) : ℝn → ℝD2 , we define {𝜕xα u}|α|≤κ : ℝn → ℝN to be the vector whose components are given by 𝜕xα uj , 1 ≤ j ≤ D2 , |α| ≤ κ. Let F(x, ζ ) : ℝn × ℝN → ℝD1 be a function. For a fixed x0 ∈ ℝn and u : ℝn → ℝD2 , v 󳨃→ dζ F(x0 , {𝜕xα u(x0 )}|α|≤κ ){𝜕xα v(x)}|α|≤κ is a constant coefficient partial differential operator of degree κ of the form v 󳨃→ ∑ aα,x0 ,u 𝜕xα v, |α|≤κ

where aα,x0 ,u ∈ 𝕄D1 ×D2 (ℝ) are constant matrices which only depend on the Taylor coefficients of u at x0 up to order κ. Definition 2.8.1. Let x0 ∈ ℝn , u : ℝn → ℝD2 , and F(x, ζ ) : ℝn × ℝN → ℝD1 ; we assume that both u and F are sufficiently smooth for all objects that follow to make sense. We say the nonlinear partial differential operator F(x, {𝜕xα u(x)}|α|≤κ ) is elliptic at (x0 , u) of degree κ if the constant coefficient linear partial differential operator of degree κ v 󳨃→ dζ F(x0 , {𝜕xα u(x0 )}|α|≤κ ){𝜕xα u(x)} is elliptic. Definition 2.8.2. – For r > 0, we say u ∈ C r near x0 if there exists ψ ∈ C0∞ (ℝn ) with ψ ≡ 1 on a neighborhood of x0 such that ψu ∈ C r (ℝn ). – For r, t > 0, we say F(x, ζ ) ∈ C r,t near x0 if there exists ψ ∈ C0∞ (ℝn ) with ψ ≡ 1 on a neighborhood of x0 such that ψ(x)F(x, ζ ) ∈ C r,t (ℝn × ℝN ).

2.8 Nonlinear elliptic equations

�

91

The main result of this section (Theorem 2.8.7) is a quantitative theorem. It has, as an immediate consequence, the next qualitative theorem. Theorem 2.8.3 (Main qualitative theorem). Fix s > r > 0, t > 2⌊s⌋ + 6, and x0 ∈ ℝn . Suppose that: – F ∈ C s,t near x0 . – u ∈ C r+κ near x0 . – F(x, {𝜕xα u(x)}|α|≤κ ) ∈ C s near x0 . – F(x, {𝜕xα u(x)}|α|≤κ ) is elliptic of order κ at (x0 , u). Then u ∈ C s+κ near x0 . Remark 2.8.4. Theorem 2.8.3 is not sharp with respect to the regularity of F(x, ζ ) in the ζ variable, i. e., it is not sharp in t. However, it is sharp in s, and r (the a priori regularity of u) does not play a role in the conclusion. Theorem 2.8.3 has, as an easy consequence, the classical interior Schauder estimates. Corollary 2.8.5 (Qualitative Schauder estimates). Fix s > r > 0 and let P be a linear partial differential operator of the form α

P u(x) := ∑ aα (x)𝜕x u(x), |α|≤κ

where aα : ℝn → 𝕄D1 ×D2 (ℝ). Fix x0 ∈ ℝn and suppose that: – aα ∈ C s near x0 , for all |α| ≤ κ. – u ∈ C r+κ near x0 . – P u ∈ C s near x0 . – Set Px0 := ∑|α|≤κ aα (x0 )𝜕xα ; we suppose Px0 is elliptic of order κ. Then u ∈ C s+κ near x0 . Proof. Take ψ ∈ C0∞ (ℝn ), with ψ ≡ 1 on a neighborhood of x0 and such that ψu ∈ C r+κ (ℝn ). Dividing u by ∑|α|≤κ ‖𝜕xα ψu‖L∞ , we see that we may (without loss of generality) assume that |𝜕xα u(x)| ≤ 1 for x near x0 , ∀|α| ≤ κ. Fix ϕ(ζ ) ∈ C0∞ (ℝN ) with ϕ(ζ ) ≡ 1 on a neighborhood of {ζ : |ζα | ≤ 2}. Define F(x, ζ ) := ∑ aα (x)ζα ϕ(ζ ). |α|≤κ

Clearly, F(x, ζ ) ∈ C s,2⌊s⌋+7 near x0 . Moreover, since |𝜕xα u(x)| ≤ 1 near x0 , ∀|α| ≤ κ, we have F(x, {𝜕xα u(x)}|α|≤κ ) = P u(x) ∈ C s ,

x near x0 ,

92 � 2 Ellipticity and dζ F(x0 , {𝜕xα u(x0 )}|α|≤κ ){𝜕xα }|α|≤κ = Px0 . Thus, Theorem 2.8.3 applies to show u ∈ C s+κ near x0 , completing the proof. Remark 2.8.6. The a priori assumption that u ∈ C r+κ near x0 in Corollary 2.8.5 can be weakened considerably by an easier reprise of the proof of Theorem 2.8.3, using the fact that the PDE in Corollary 2.8.5 is linear; see [212]. Thus, seeing Schauder estimates as a consequence of estimates for fully nonlinear equations (as in Theorem 2.8.3) introduces unnecessary difficulties and assumptions. A main focus of this text is to study fully nonlinear equations, so we do not discuss such improvements here. Theorem 2.8.3 follows easily from the next quantitative theorem. Theorem 2.8.7. Let s > r > 0 and σ > 0. Suppose that: – F ∈ C s,2⌊s⌋+6+σ (ℝn × ℝN ; ℝD1 ). – u ∈ C r+κ (ℝn ; ℝD2 ). – g ∈ C s (ℝn ; ℝD1 ). – F(x, {𝜕xα u(x)}|α|≤κ ) = g(x), ∀x ∈ Bn (3/4). – Let P v := dζ F(0, {𝜕xα u(0)}|α|≤κ ){𝜕xα v}|α|≤κ , so that P is a constant coefficient partial differential operator of order κ with coefficients in 𝕄D1 ×D2 (ℝ). We assume that P is elliptic, i. e., ∃A ≥ 0 such that n

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ∑󵄩󵄩󵄩𝜕xκj f 󵄩󵄩󵄩L2 ≤ A(󵄩󵄩󵄩P f 󵄩󵄩󵄩L2 + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 ), j=1

∀f ∈ C0∞ (ℝn ; ℂD2 ).

(2.75)

Let ψ0 ∈ C0∞ (Bn (2/3)) and for δ > 0 set ψδ0 (x) := ψ0 (δ−1 x). Then ∃δ ∈ (0, 1], depending only on n, A, D1 , D2 , s, r, σ, κ and upper bounds for ‖F‖C s,2⌊s⌋+6+σ , ‖g‖C s , ‖u‖C r+κ such that ψδ0 u ∈ C s+κ and ‖ψδ0 u‖C s+κ ≤ C5 , where C5 ≥ 0 can be chosen to depend only on n, A, D1 , D2 , s, r, κ, σ, ψ0 , and upper bounds for ‖F‖C s,2⌊s⌋+6+σ , ‖g‖C s , ‖u‖C r+κ . We sketch the proof of Theorem 2.8.7; a full proof with all details of a more general result is given in the maximally subelliptic setting (see Theorem 9.2.1). There are many proofs of results like Theorem 2.8.7 in the literature (see [233, Chapter 14] for several approaches). However, these proofs do not directly generalize to the setting of maximally subelliptic PDEs (see Remark 1.1.5). Instead, we adapt a proof due to Simon [212] to this setting; while the proof we present is somewhat more complicated than other proofs of Theorem 2.8.7, it has the advantage of also working in the more general setting of maximally subelliptic operators.

2.8 Nonlinear elliptic equations

� 93

We write Dκ := {𝜕xα }|α|≤κ so that Dκ u = {𝜕xα u}|α|≤κ . Throughout this section, we take s > r > 0 and σ > 0. We begin by reducing Theorem 2.8.7 to the case of a perturbation of a constant coefficient linear elliptic operator. We do this in two steps, which can be found in Section 2.8.1 and Section 2.8.2; this reduction is straightforward. Then, once we have reduced to the case of a perturbation of a constant coefficient linear elliptic operator, in Section 2.8.3 we use ideas from [212] to deduce the result.

2.8.1 Reduction I To prove Theorem 2.8.7, we reduce it to the following, simpler proposition. Proposition 2.8.8. Let C3 ≥ 0 be given. There exists ϵ2 = ϵ2 (C3 , n, D1 , D2 , κ, s, r, σ) > 0 sufficiently small such that the following holds. Suppose that: – G ∈ C ⌊s⌋+4+σ (ℝN ; ℝD1 ) with ‖G‖C ⌊s⌋+4+σ ≤ C3 . – u ∈ C r+κ (ℝn ; ℝD2 ) with ‖u‖C r+κ ≤ ϵ2 . – R3 ∈ C s,⌊s⌋+3+σ (ℝn × ℝN ; ℝD1 ) with ‖R3 ‖C s,⌊s⌋+3+σ ≤ ϵ2 . – Let P := dG(Dκ u(0))Dκ . We assume P is elliptic of degree κ, i. e., ∃A2 ≥ 0 with n

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ∑󵄩󵄩󵄩𝜕xκj f 󵄩󵄩󵄩L2 ≤ A2 (󵄩󵄩󵄩P f 󵄩󵄩󵄩L2 + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 ), j=1

–

∀f ∈ C0∞ (ℝn ; ℂD2 ).

G(Dκ u(x)) + R3 (x, Dκ u(x)) = 0, ∀x ∈ Bn (13/16).

Then, ∀ψ0 ∈ C0∞ (Bn (2/3)), ψ0 u ∈ C s+κ (ℝn ; ℝD2 ) and ‖ψ0 u‖C s+κ ≤ C4 , where C4 ≥ 0 can be chosen to depend only on C3 , n, A2 , D1 , D2 , s, κ, r, σ, and ψ0 . The rest of this section is devoted to sketching the reduction of Theorem 2.8.7 to Proposition 2.8.8; we henceforth assume the setting of Theorem 2.8.7. By replacing F(x, ζ ) with F(x, ζ )−g(x), it suffices to consider only the case g = 0; we henceforth assume g = 0. Take J ∈ ℕ+ large, to be chosen later. Set uJ := ∑ Dj u, 0≤j 0 be as in Proposition 2.8.8 with this choice of C3 and with A2 replaced by A. By (2.80) and (2.81), if we take J ≈ 1 sufficiently large, we have ‖Ĥ J ‖C s,⌊s⌋+3+σ ≤ ϵ2 and ‖v̂J ‖C r+κ ≤ ϵ2 . Using this and the above remarks, we have shown that all the hypotheses of Proposition 2.8.8 hold for equation (2.78). The conclusion of Proposition 2.8.8 in this case is ‖ψ0 v̂J ‖C s+κ ≲ 1.

(2.84)

‖ψ0 (⋅)vJ (2−J ⋅)‖C s+κ ≲ 1.

(2.85)

Since ψ0 ≺ ψ, (2.84) is equivalent to

Since J ≈ 1, (2.85) is equivalent to ‖ψ0 (2J ⋅)vJ (⋅)‖C s+κ ≲ 1.

(2.86)

Also, since J ≈ 1, we have uJ ∈ C ∞ (ℝn ) and ‖uJ ‖C L ≲ 1, ∀L ∈ ℕ. In particular, ‖uJ ‖C s+κ ≲ 1, and therefore ‖ψ0 (2J ⋅)uJ (⋅)‖C s+κ ≲ 1. Since u = uJ + 2−κJ vJ , combining (2.86) and (2.87) shows ‖ψ0 (2J ⋅)u(⋅)‖C s+κ ≲ 1. Theorem 2.8.7 follows with δ = 2−J .

(2.87)

96 � 2 Ellipticity 2.8.2 Reduction II To prove Proposition 2.8.8, we reduce it to the following, simpler proposition. Proposition 2.8.9. Fix A, B, C1 ≥ 0. There exists ϵ1 = ϵ1 (n, A, B, C1 , s, r, σ, κ, n, D1 , D2 ) > 0 such that the following holds. Let P = ∑|α|≤κ aα 𝜕xα , where aα ∈ 𝕄D1 ×D2 (ℝ) and |aα | ≤ B, ∀α. Suppose P is elliptic in the following quantitative sense: n

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ∑󵄩󵄩󵄩𝜕xκj f 󵄩󵄩󵄩L2 ≤ A(󵄩󵄩󵄩P f 󵄩󵄩󵄩L2 + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 ), j=1

∀f ∈ C0∞ (ℝn ; ℂD2 ).

Suppose R1 (x, ζ ) ∈ C s,⌊s⌋+3+σ (ℝn × ℝN ; ℝD1 ), R2 ∈ C s,⌊s⌋+4+σ (ℝn × ℝN ; ℝD1 ), and u ∈ C r+κ (ℝn ; ℝD2 ) with ‖R1 ‖C s,⌊s⌋+3+σ ≤ ϵ1 ,

‖R2 ‖C s,⌊s⌋+4+σ ≤ C1 ,

‖u‖C r+κ ≤ ϵ1 ,

β

and 𝜕ζ R2 (x, 0) ≡ 0, ∀|β| ≤ 1. Suppose κ

κ

P u(x) = R1 (x, D u(x)) + R2 (x, D u(x)),

∀x ∈ Bn (13/16).

(2.88)

Then, ∀ψ0 ∈ C0∞ (Bn (2/3)), we have ψ0 u ∈ C s+κ (ℝn ; ℝD2 ) and ‖ψ0 u‖C s+κ ≤ C2 , where C2 = C2 (n, A, B, C1 , s, r, σ, κ, n, D1 , D2 , ψ0 ) ≥ 0. The rest of this section is devoted to the reduction of Proposition 2.8.8 to Proposition 2.8.9; thus, we assume the setting of Proposition 2.8.8. Our goal is to choose ϵ2 > 0 small (as in Proposition 2.8.8) so that the assumptions of Proposition 2.8.9 hold. The hypotheses of Proposition 2.8.8 give, for f ∈ C0∞ (ℝn ; ℂD2 ), n

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ∑ 󵄩󵄩󵄩𝜕xα f 󵄩󵄩󵄩L2 ≲ ∑󵄩󵄩󵄩𝜕xκj f 󵄩󵄩󵄩L2 + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 ≲ 󵄩󵄩󵄩dG(Dκ u(0))Dκ f 󵄩󵄩󵄩L2 + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2

|α|≤κ

j=1

󵄩 󵄩 󵄩 󵄩 󵄨 󵄨 󵄩 󵄩 ≲ 󵄩󵄩󵄩dG(0)Dκ f 󵄩󵄩󵄩L2 + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 + 󵄨󵄨󵄨dG(Dκ u(0)) − dG(0)󵄨󵄨󵄨 ∑ 󵄩󵄩󵄩𝜕xα f 󵄩󵄩󵄩L2 .

(2.89)

|α|≤κ

Since ‖Dκ u‖L∞ ≲ ‖u‖C r+κ ≤ ϵ2 , (2.89) implies that there exists A3 ≈ 1 with 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ∑ 󵄩󵄩󵄩𝜕xα f 󵄩󵄩󵄩L2 ≤ A3 (󵄩󵄩󵄩dG(0)Dκ f 󵄩󵄩󵄩L2 + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 ) + A3 ϵ2 ∑ 󵄩󵄩󵄩𝜕xα f 󵄩󵄩󵄩L2 .

|α|≤κ

(2.90)

|α|≤κ

If we choose ϵ2 ≤ 1/2A3 , (2.90) implies, for f ∈ C0∞ (ℝn ; ℂD2 ), 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ∑ 󵄩󵄩󵄩𝜕xα f 󵄩󵄩󵄩L2 ≤ 2A3 (󵄩󵄩󵄩dG(0)Dκ f 󵄩󵄩󵄩L2 + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 ).

|α|≤κ

(2.91)

2.8 Nonlinear elliptic equations

�

97

Consider we have 󵄨󵄨 󵄨 κ κ 󵄨󵄨G(D u(0)) − G(0)󵄨󵄨󵄨 ≲ ‖D u‖L∞ ≲ ‖u‖C r+κ ≲ ϵ2 .

(2.92)

But we also have, by the hypotheses of Proposition 2.8.8, 󵄨󵄨 󵄨 󵄨 󵄨 κ κ κ 󵄨󵄨G(D u(0))󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨G(D u(0)) + R3 (0, D u(0))󵄨󵄨󵄨 + ‖R3 ‖L∞ = 0 + ‖R3 ‖L∞ ≲ ϵ2 .

(2.93)

Combining (2.92) and (2.93) gives |G(0)| ≲ ϵ2 .

(2.94)

Define E(ζ ) by G(ζ ) := G(0) + dζ G(0) + E(ζ ), so that E(0) = 0, dζ E(0) = 0, and ‖E‖C ⌊s⌋+4+σ ≲ 1.

(2.95)

Set P := dζ G(0)Dκ . Rewriting the assumption that G(Dκ u(x)) + R3 (x, Dκ u(x)) = 0, ∀x ∈ Bn (13/16), we have κ

κ

∀x ∈ Bn (13/16).

P u = −G(0) − R3 (x, D u(x)) − E(D u(x)),

(2.96)

Fix ψ1 ∈ C0∞ (Bn (1)) with ψ1 ≡ 1 on Bn (13/16). Set R1 (x, ζ ) := ψ1 (x)(G(0) + R3 (x, ζ )),

R2 (x, ζ ) := −ψ1 (x)E(ζ ).

β

Since E(0) = 0 and dE(0) = 0, we have 𝜕ζ R2 (x, 0) = 0, ∀|β| ≤ 1. Formula (2.96) implies κ

κ

P u = R1 (x, D u(x)) + R2 (x, D u(x)),

∀x ∈ Bn (13/16).

(2.97)

Furthermore, using (2.94), we have ‖R1 ‖C s,⌊s⌋+3+σ ≲ |G(0)| + ‖R3 ‖C s,⌊s⌋+3+σ ≲ ϵ2 , and using (2.95), we have ‖R2 ‖C s,⌊s⌋+4+σ ≲ ‖E‖C ⌊s⌋+4+σ ≲ 1. Finally, we have |dζ G(0)| ≲ ‖G‖C ⌊s⌋+4+σ ≲ 1. Thus, if P = ∑|α|≤κ aα 𝜕xα , we have |aα | ≲ 1, ∀α. Thus, by taking ϵ2 > 0 small, all the hypotheses of Proposition 2.8.9 hold (the ellipticity of P is (2.91)). The conclusion of Proposition 2.8.9, in this case, is exactly the conclusion of Proposition 2.8.8, completing the proof of Proposition 2.8.8.

98 � 2 Ellipticity 2.8.3 Completion of the proof In this section we complete the proof of Theorem 2.8.7 by proving Proposition 2.8.9. For each δ > 0, we define a norm on C s (ℝn ) given by ‖f ‖C s (δ) := ‖f (δ⋅)‖C s (ℝn ) , and more generally, for s ∈ ℝ, we set s s ‖f ‖B∞,∞ (δ) := ‖f (δ⋅)‖B∞,∞ (ℝn ) .

In particular, ‖f ‖C s = ‖f ‖C s (1) and if δ1 ≈ δ2 , then ‖f ‖C s (δ1 ) ≈ ‖f ‖C s (δ2 ) . We write Bn (x, δ) := {y ∈ ℝn : |x − y| < δ}. For the proof we require scaled bump functions near each point. Fix ϕ0,1 ∈ C0∞ (Bn (3/4)) with ϕ0,1 ≡ 1 on Bn (2/3). For x ∈ ℝn and δ > 0, set ϕx,δ (y) := ϕ(δ−1 (y − x)). Proposition 2.8.10. There exists N1 ∈ ℕ, depending only on n, such that the following hold: (i) For all s > 0, ∃Cn,s ≥ 0, ∀x ∈ ℝn , δ > 0, there exist x1 , . . . , xN1 ∈ Bn (x, δ) with Bn (xj , δ/4) ⊆ Bn (x, δ) and N1

‖ϕx,δ u‖C s (δ) ≤ Cn,s ∑ ‖ϕxj ,δ/32 u‖C s (δ) , j=1

∀u ∈ C0∞ (ℝn )′ ,

where if the right-hand side is finite, so is the left-hand side. (ii) For all s > 0, ∀ψ ∈ C0∞ (Bn (2/3)), ∃Cn,s,ψ ≥ 0, ∃x1 , . . . , xN1 ∈ Bn (2/3) with Bn (xj , 1/4) ⊆ Bn (3/4) such that N1

‖ψu‖C s ≤ Cn,s,ψ ∑ ‖ϕxj ,1/32 u‖C s , j=1

∀u ∈ C0∞ (ℝn )′ ,

where if the right-hand side is finite, so is the left-hand side. Proof. (i): By translating and scaling, it suffices to prove (i) for x = 0 and δ = 1. Cover Bn (0, 3/4) by a finite collection of balls of the form Bn (x1 , 1/64), . . . , Bn (xN1 , 1/64), with xj ∈ Bn (3/4); note that Bn (xj , 1/4) ⊆ Bn (0, 1). Let ψxj ∈ C0∞ (Bn (xj , 1/64)) be a partition of N

unity on Bn (3/4) subordinate to this cover, i. e., ∑j=11 ψxj ≡ 1 on Bn (0, 3/4). Then we have N1

N1

j=1

j=1

‖ϕ0,1 u‖C s ≤ ∑ ‖ϕ0,1 ψxj u‖C s ≲ ∑ ‖ψxj u‖C s N1

N1

j=1

j=1

= ∑ ‖ψxj ϕxj ,1/32 u‖C s ≲ ∑ ‖ϕxj ,1/32 u‖C s , completing the proof of (i).

2.8 Nonlinear elliptic equations

� 99

(ii): Since ψ ≺ ϕ0,1 , we have, by (i) with x = 0 and δ = 1, N1

‖ψu‖C s = ‖ψϕ0,1 u‖C s ≲ ‖ϕ0,1 u‖C s ≲ ∑ ‖ϕxj ,1/32 u‖C s . j=1

We henceforth take the same setting and notation as in Proposition 2.8.9. Our goal is to choose ϵ1 > 0 sufficiently small so that the conclusions of Proposition 2.8.9 hold. By ̃ where ψ̃ ∈ C ∞ (Bn (1)) and ψ̃ ≡ 1 on Bn (31/32), we may henceforth replacing u with ψu, 0 assume supp(u) ⊂ Bn (1). Lemma 2.8.11. For x ∈ ℝn and δ ∈ (0, 1], s−κ (δ) + ‖ϕx,δ/16 v‖L∞ , ‖ϕx,δ/32 v‖C s+κ (δ) ≲ ‖ϕx,δ/16 δ2κ P ∗ P v‖B∞,∞

∀v ∈ C0∞ (ℝn ; ℝD2 )′ ,

(2.98)

where if the right-hand side is finite, so is the left-hand side. Proof. Write P ∗ P = ∑|α|≤2κ bα 𝜕xα , so that bα ∈ 𝕄D2 ×D2 (ℝ) and |bα | ≲ 1. For δ ∈ (0, 1], set Lδ := ∑|α|≤2κ bα δ2κ−|α| 𝜕xα . Since P is elliptic, the same is true of P ∗ P , and in fact, n

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ∑󵄩󵄩󵄩𝜕x2κj f 󵄩󵄩󵄩L2 ≲ 󵄩󵄩󵄩P ∗ P f 󵄩󵄩󵄩L2 + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 , j=1

∀f ∈ C0∞ (ℝn ; ℂD2 ).

(2.99)

Multiplying both sides of (2.99) by δ2κ , we get n

2κ 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ∑󵄩󵄩󵄩(δ𝜕xj ) f 󵄩󵄩󵄩L2 ≲ 󵄩󵄩󵄩δ2κ P ∗ P f 󵄩󵄩󵄩L2 + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 , j=1

∀f ∈ C0∞ (ℝn ; ℂD2 ).

(2.100)

Replacing f with g(δ−1 ⋅) in (2.100) and changing variables, we get n

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ∑󵄩󵄩󵄩𝜕x2κj g 󵄩󵄩󵄩L2 ≲ 󵄩󵄩󵄩Lδ g 󵄩󵄩󵄩L2 + 󵄩󵄩󵄩g 󵄩󵄩󵄩L2 , j=1

∀g ∈ C0∞ (ℝn ; ℂD2 ),

that is, Lδ is elliptic (uniformly in δ ∈ (0, 1] and any other relevant parameters). Since ϕ0,1/32 ≺ ϕ0,1/16 , standard elliptic estimates (see Theorem 2.7.2 (e)) show that s−κ (1) + ‖ϕ0,1/16 v‖L∞ , ‖ϕ0,1/32 v‖C s+κ (1) ≲ ‖ϕ0,1/16 Lδ v‖B∞,∞

∀v ∈ C0∞ (ℝn ; ℝD2 )′ .

(2.101)

Translating and dilating (2.101) gives (2.98) and completes the proof. Fix ψ1 , ψ2 , ψ3 ∈ C0∞ (Bn (13/16)) with ψ3 ≺ ψ2 ≺ ψ1 and ψ3 ≡ 1 on a neighborhood of Since P is elliptic, Theorem 2.7.2 (a) ⇒ (h) implies that there exists an operator T which is locally a singular integral operator of order −2κ such that Bn (3/4).

100 � 2 Ellipticity T P ∗P , P ∗P T ≡ I

∞ mod Cloc (ℝn × ℝn ; 𝕄D2 ×D2 (ℂ)).

(2.102)

In fact, since P is a constant coefficient operator, we can choose T to be a multiplier, but the details of T are not important for our purposes; all we need are some basic properties of T like the boundedness on Besov spaces given by Theorem 2.4.7. Using (2.88), we define V∞ := ψ2 T P ∗ ψ1 (x)[R1 (x, Dκ u(x)) + R2 (x, Dκ u(x))]

= ψ2 T P ∗ ψ1 P u = ψ2 T P ∗ P u + ψ2 T P ∗ (1 − ψ1 )P u

(2.103)

= ψ2 u + ψ2 e∞ ,

where, by the pseudo-locality of T (see the growth condition in Definition 2.3.1) and (2.102), we have e∞ ∈ C0∞ (Bn (1)). Moreover, we have ‖e∞ ‖C r+κ , ‖e∞ ‖C s+κ ≲ ‖u‖C r+κ ≲ ϵ1 .

(2.104)

Since T is locally a singular integral operator of −2κ, T P ∗ ψ1 P is locally a singular integral operator of order 0. Thus, Theorem 2.4.7 combined with the fact that we are assuming supp(u) ⊂ Bn (1) implies ‖V∞ ‖C r+κ = ‖ψ2 T P ∗ ψ1 P u‖C r+κ ≲ ‖u‖C r+κ ≲ ϵ1 , that is, ∃C3 ≈ 1 with ‖V∞ ‖C r+κ ≤ C3 ϵ1 .

(2.105)

Set H := u − V∞ , so that ‖H‖C r+κ ≤ ‖u‖C r+κ + ‖V∞ ‖C r+κ ≲ ϵ1 . Also, by (2.103) we have ψ3 H = ψ3 e∞ , and therefore using (2.104) we have ‖ψ3 H‖C s+κ ≲ ‖e∞ ‖C s+κ ≲ ϵ1 .

(2.106)

For V ∈ C r+κ , define κ

κ

T (V ) := ψ2 T P ψ1 (x)[R1 (x, D (V + H)(x)) + R2 (x, D (V + H)(x))]. ∗

It follows from the fact that H ∈ C r+κ , Theorem 2.4.7, and Theorem 2.6.2 that T : C r+κ → C r+κ . By the definition of V∞ (see (2.103)), we have T (V∞ ) = V∞ . Let D ∈ [C3 ∨1, 1/ϵ1 ] be a fixed number, to be chosen later (we will pick D ≈ 1). Define MD,ϵ1 ,r := {V ∈ C

r+κ

: ‖V ‖C r+κ ≤ Dϵ1 }.

2.8 Nonlinear elliptic equations

�

101

Note that MD,ϵ1 ,r is a closed subspace of the Banach space C r+κ , and therefore it is a complete metric space with distance induced by ‖ ⋅ ‖C r+κ . Furthermore, since D ≥ C3 , (2.105) shows V∞ ∈ MD,ϵ1 ,r . We will show that T : MD,ϵ1 ,r → MD,ϵ1 ,r and that it is a strict contraction (with appropriate choices of D and ϵ1 ); it will then follow that V∞ is the unique fixed point of T in MD,ϵ1 ,r . Using Theorem 2.4.7, Theorem 2.6.2 (ii) applied to R1 , and Theorem 2.6.2 (vii) with L = 2 applied to R2 , we have, for V ∈ MD,ϵ1 ,r , 󵄩 󵄩 ‖T (V )‖C r+κ = 󵄩󵄩󵄩ψ2 T P ∗ ψ1 (R1 (x, Dκ (V + H)(x)) + R2 (x, Dκ (V + H)(x)))󵄩󵄩󵄩C r+κ 󵄩 󵄩 󵄩 󵄩 ≲ 󵄩󵄩󵄩R1 (x, Dκ (V + H)(x))󵄩󵄩󵄩C r + 󵄩󵄩󵄩R2 (x, Dκ (V + H)(x))󵄩󵄩󵄩C r ≲ ‖R1 ‖C r,⌊r⌋+2+σ (1 + ‖Dκ (V + H)‖C r )(1 + ‖Dκ (V + H)‖L∞ )

⌊r⌋+1

+ ‖R2 ‖C r,⌊r⌋+4+σ (1 + ‖Dκ (V + H)‖L∞ )

⌊r⌋+2

‖Dκ (V + H)‖L∞ ‖Dκ (V + H)‖C r

≲ ϵ1 (1 + Dϵ1 )(1 + Dϵ1 )⌊r⌋+1 + (1 + Dϵ1 )⌊r⌋+2 (Dϵ1 )2 ≲ ϵ1 + D2 ϵ12 , that is, ∃C4 ≈ 1 with ‖T (V )‖C r+κ ≤ C4 (ϵ1 + D2 ϵ12 ),

(2.107)

∀V ∈ MD,ϵ1 ,r .

Take D := max{C3 , 2C4 , 1}, so that D ≈ 1, and take ϵ1 ∈ (0, 1/D2 ] to be chosen later. Then C4 (ϵ1 + ϵ12 D2 ) ≤ Dϵ1 and (2.107) shows that ‖T (V )‖C r+κ ≤ Dϵ1 ,

∀V ∈ MD,ϵ1 ,r .

We conclude that T : MD,ϵ1 ,r → MD,ϵ1 ,r . Now that we have chosen D ≈ 1, we have, for V ∈ MD,ϵ1 ,r , ‖V ‖C r+κ ≤ Dϵ1 ≲ ϵ1 . Using Theorem 2.6.2 (vi) applied to R1 and Theorem 2.6.2 (viii) with L = 2 applied to R2 , we have, for V1 , V2 ∈ MD,ϵ1 ,r , ‖T (V1 ) − T (V2 )‖C r+κ

= ‖ψ2 T P ∗ ψ1 (R1 (x, Dκ (V1 + H)(x)) − R1 (x, Dκ (V2 + H)(x)) + R2 (x, Dκ (V1 + H)(x)) − R2 (x, Dκ (V2 + H)(x)))‖C r+κ

≲ ‖R1 (x, Dκ (V1 + H)(x)) − R1 (x, Dκ (V2 + H)(x))‖C r

+ ‖R2 (x, Dκ (V1 + H)(x)) − R2 (x, Dκ (V2 + H)(x))‖C r

≲ ‖R1 ‖C r,⌊r⌋+3+σ (‖Dκ (V1 − V2 )‖C r

+ (‖Dκ (V1 + H)‖C r + ‖Dκ (V2 + H)‖C r )‖Dκ (V1 − V2 )‖L∞ ) × (1 + ‖Dκ (V1 + H)‖L∞ + ‖Dκ (V2 + H)‖L∞ )

⌊r⌋+1

+ ‖R2 ‖C r,⌊r⌋+4+σ (1 + ‖Dκ (V1 + H)‖L∞ + ‖Dκ (V2 + H)‖L∞ )

⌊r⌋+2

× (‖Dκ (V1 + H)‖C r + ‖Dκ (V2 + H)‖C r )‖Dκ (V1 − V2 )‖L∞

+ ‖R2 ‖C r,⌊r⌋+4+σ (‖Dκ (V1 + H)‖L∞ + ‖Dκ (V2 + H)‖L∞ )‖Dκ (V1 − V2 )‖C r

102 � 2 Ellipticity ≲ ϵ1 (‖V1 − V2 ‖C r+κ + ϵ1 ‖Dκ (V1 − V2 )‖L∞ )(1 + ϵ1 )⌊r⌋+1 + (1 + ϵ1 )⌊r⌋+2 ϵ1 ‖Dκ (V1 − V2 )‖L∞ + ϵ1 ‖Dκ (V1 − V2 )‖L∞

≲ ϵ1 ‖V1 − V2 ‖C r+κ + ϵ1 ‖Dκ (V1 − V2 )‖L∞ ≲ ϵ1 ‖V1 − V2 ‖C r+κ .

We conclude that ∃C5 ≈ 1 with ‖T (V1 ) − T (V2 )‖C r+κ ≤ C5 ϵ1 ‖V1 − V2 ‖C r+κ . In particular, we will take ϵ1 ∈ (0, min{1/2C5 , 1/2D}], so that 1 ‖T (V1 ) − T (V2 )‖C r+κ ≤ ‖V1 − V2 ‖C r+κ , 2

∀V1 , V2 ∈ MD,ϵ1 ,r .

Thus, T : MD,ϵ1 ,r → MD,ϵ1 ,r is a strict contraction with unique fixed point V∞ . Remark 2.8.12. At this point it is tempting to try to show that T is also a strict contraction on MD,ϵ′ ,s , for some small ϵ′ > 0, and then conclude that V∞ must be the fixed point of this contraction and therefore V∞ ∈ C s+κ . Unfortunately, we only know an estimate like ‖ψ3 H‖C s ≲ ϵ1 , where we would need a stronger estimate like ‖ψ1 H‖C s ≲ ϵ1 to make such an argument work. In short, the localizations get in the way of directly applying the contraction mapping principle to T on functions in C s+κ . To get around this issue, we apply a clever covering type argument adapted from a work of Simon [212]. Set V0 := 0 and for j ≥ 1 let Vj := T (Vj−1 ). We have established Vj ∈ MD,ϵ1 ,r , ∀j, and j→∞

the contraction mapping principle shows that Vj 󳨀󳨀󳨀󳨀→ V∞ in C r+κ . Fix x0 ∈ ℝn and δ ∈ (0, 1] such that Bn (x0 , δ/4) ⊆ Bn (3/4). Applying Lemma 2.8.11, we see that s−κ (δ) + ‖ϕx ,δ/16 Vj+1 ‖L∞ . ‖ϕx0 ,δ/32 Vj+1 ‖C s+κ (δ) ≲ ‖ϕx0 ,δ/16 δ2κ P ∗ P Vj+1 ‖B∞,∞ 0

(2.108)

We will estimate the right-hand side of (2.108). For the second term on the right-hand side of (2.108), we have ‖ϕx0 ,δ/16 Vj+1 ‖L∞ ≤ ‖Vj+1 ‖L∞ ≲ ‖Vj+1 ‖C r+κ ≲ ϵ1 .

(2.109)

For the first term on the right-hand side of (2.108), we use supp(ϕx0 ,δ/16 ) ⊆ Bn (x0 , δ/4) ⊆ Bn (3/4); thus by the choice of ψ2 we have ϕx0 ,δ/16 ≺ ψ2 . So, we have ϕx0 ,δ/16 P ∗ P Mult[ψ2 ]T P ∗ Mult[ψ1 ] = ϕx0 ,δ/16 P ∗ P T P ∗ Mult[ψ1 ]

= ϕx0 ,δ/16 P ∗ Mult[ψ1 ] + ϕx0 ,δ/16 E∞ Mult[ψ1 ],

where E∞ ∈ C0∞ (Bn (1) × Bn (1)) with ‖E∞ ‖C L ≲L 1, ∀L ∈ ℕ. Thus, we have

2.8 Nonlinear elliptic equations

� 103

s−κ (δ) ‖ϕx0 ,δ/16 δ2κ P ∗ P Vj+1 ‖B∞,∞

s−κ (δ) = ‖ϕx0 ,δ/16 δ2κ P ∗ PT (Vj )‖B∞,∞ 󵄩󵄩 󵄩 2κ ∗ ∗ = 󵄩󵄩ϕx0 ,δ/16 δ P P ψ2 T P ψ1 (R1 (x, Dκ (Vj + H)(x)) + R2 (x, Dκ (Vj + H)(x)))󵄩󵄩󵄩Bs−κ (δ) ∞,∞ 󵄩 󵄩 ≤ 󵄩󵄩󵄩ϕx0 ,δ/16 δ2κ P ∗ ψ1 (R1 (x, Dκ (Vj + H)(x)) + R2 (x, Dκ (Vj + H)(x)))󵄩󵄩󵄩Bs−κ (δ) (2.110) ∞,∞ 󵄩󵄩 󵄩 + 󵄩󵄩ϕx0 ,δ/16 δ2κ E∞ ψ1 (R1 (x, Dκ (Vj + H)(x)) + R2 (x, Dκ (Vj + H)(x)))󵄩󵄩󵄩Bs−κ (δ) ∞,∞

=: (I) + (II). To bound (II), we use the fact that for some L ≈ 1, (II) ≲ δ2κ ‖E∞ ‖C L ‖R1 (x, Dκ (Vj + H)(x)) + R2 (x, Dκ (Vj + H)(x))‖L∞ ≲ δ2κ (‖R1 (x, Dκ (Vj + H)(x))‖L∞ + ‖R2 (x, Dκ (Vj + H)(x))‖L∞ ).

(2.111)

We have ‖R1 (x, Dκ (Vj + H)(x))‖L∞ ≤ ‖R1 ‖L∞ ≲ ‖R1 ‖C s,⌊s⌋+3+σ ≤ ϵ1 .

(2.112)

Since R2 (x, 0) ≡ 0, we have β

‖R2 (x, Dκ (Vj + H)(x))‖L∞ ≲ ∑ ‖𝜕ζ R2 (x, ζ )‖L∞ ‖Dκ (Vj + H)‖L∞ |β|≤1

β

≲ ∑ ‖𝜕ζ R2 ‖C s,⌊s⌋+3+σ ‖Dκ (Vj + H)‖L∞

(2.113)

|β|≤1

≲ ‖R2 ‖C s,⌊s⌋+4+σ ‖Vj + H‖C r+κ ≲ ϵ1 . Using (2.112) and (2.113) to estimate the right-hand side of (2.111) shows that (II) ≲ δ2κ ϵ1 ≲ ϵ1 .

(2.114)

For (I), we use the fact that ϕx0 ,δ/16 ≺ ϕx0 ,δ/8 to see that 󵄩󵄩 󵄩 (I) = 󵄩󵄩󵄩ϕx0 ,δ/16 δ2κ P ∗ ψ1 (R1 (x, ϕx0 ,δ/8 (x)Dκ (Vj + H)(x)) 󵄩󵄩 󵄩󵄩 󵄩 + R2 (x, ϕx0 ,δ/8 (x)Dκ (Vj + H)(x)))󵄩󵄩󵄩 󵄩󵄩Bs−κ (δ) ∞,∞ 󵄩󵄩 󵄩 ≲ δκ 󵄩󵄩󵄩ϕx0 ,δ/8 (R1 (x, ϕx0 ,δ/8 (x)Dκ (Vj + H)(x)) 󵄩󵄩 󵄩󵄩 󵄩 + R2 (x, ϕx0 ,δ/8 (x)Dκ (Vj + H)(x)))󵄩󵄩󵄩 󵄩󵄩C s (δ) 󵄩 κ󵄩 κ ≤ δ 󵄩󵄩󵄩ϕx0 ,δ/8 R1 (x, ϕx0 ,δ/8 (x)D (Vj + H)(x))󵄩󵄩󵄩C s (δ) 󵄩 󵄩 + δκ 󵄩󵄩󵄩ϕx0 ,δ/8 R2 (x, ϕx0 ,δ/8 (x)Dκ (Vj + H)(x))󵄩󵄩󵄩C s (δ) =: (III) + (IV),

(2.115)

104 � 2 Ellipticity where we used ϕx0 ,δ/8 ≺ ψ1 (since supp(ϕx0 ,δ/8 ) ⊆ Bn (x0 , δ/4) ⊆ Bn (3/4)). We turn to estimating (III) and (IV); this is the key place where the tame estimates from Theorem 2.6.2 arise (see Remark 2.6.5). Indeed, important for bounding (III) and (IV) is the following estimate: α

δκ ‖ϕx0 ,δ/8 Dκ (Vj + H)‖C s (δ) ≲ ∑ ‖ϕx0 ,δ/8 (δ𝜕x ) (Vj + H)‖C s (δ) |α|≤κ

= ∑ ‖ϕx0 /δ,1/8 𝜕xα (Vj (δ⋅) + H(δ⋅))‖C s (1) |α|≤κ

(2.116)

≲ ‖ϕx0 /δ,1/4 (Vj (δ⋅) + H(δ⋅))‖C s+κ (1) = ‖ϕx0 ,δ/4 (Vj + H)‖C s+κ (δ) .

The idea in (2.116) is that we require the factor δκ on the left-hand side to achieve the needed estimate. In contrast, to estimate a similar L∞ norm we do not require the factor of δκ . Indeed, we have ‖ϕx0 ,δ/8 Dκ (Vj + H)‖L∞ ≲ ‖Dκ (Vj + H)‖L∞ ≲ ‖Vj + H‖C r+κ ≲ ϵ1 . We also require another estimate. Since supp(ϕx0 ,δ/4 ) ⊆ Bn (x0 , δ/4) ⊆ Bn (3/4), we have ϕx0 ,δ/4 ψ3 = ϕx0 ,δ/4 . Therefore, ‖ϕx0 ,δ/4 H‖C s+κ (δ) = ‖ϕx0 ,δ/4 ψ3 H‖C s+κ (δ) = ‖ϕx0 /δ,1/4 ψ3 (δ⋅)H(δ⋅)‖C s+κ (1)

≲ ‖ψ3 (δ⋅)H(δ⋅)‖C s+κ (1) = ‖ψ3 H‖C s+κ (δ) ≲ ‖ψ3 H‖C s+κ (1) ≲ ϵ1 .

Combining (2.116) and (2.117), we get δκ ‖ϕx0 ,δ/8 Dκ (Vj + H)‖C s (δ) ≲ ‖ϕx0 ,δ/4 Vj ‖C s+κ (δ) + ϵ1 . In the next estimates, we use the above inequalities freely. We have, using Theorem 2.6.2 (ii), 󵄩 󵄩 (III) = δκ 󵄩󵄩󵄩ϕx0 ,δ/8 R1 (x, ϕx0 ,δ/8 (x)Dκ (Vj + H)(x))󵄩󵄩󵄩C s (δ) 󵄩 󵄩 = δκ 󵄩󵄩󵄩ϕx0 /δ,1/8 R1 (δx, ϕx0 /δ,1/8 (x)Dκ (Vj + H)(δx))󵄩󵄩󵄩C s (1)

≲ ‖R1 (δx, ζ )‖C s,⌊s⌋+2+σ (δκ + δκ ‖ϕx0 /δ,1/8 Dκ (Vj + H)(δx)‖C s (1) ) × (1 + ‖ϕx0 /δ,1/8 Dκ (Vj + H)(δx)‖L∞ )

⌊s⌋+1

≲ ‖R1 ‖C s,⌊s⌋+2+σ (δκ + δκ ‖ϕx0 ,δ/8 Dκ (Vj + H)‖C s (δ) ) × (1 + ‖ϕx0 ,δ/8 Dκ (Vj + H)‖L∞ )

⌊s⌋+1

≲ ϵ1 (1 + ‖ϕx0 ,δ/4 Vj ‖C s+κ (δ) + ϵ1 )(1 + ϵ1 ) ≲ ϵ1 + ϵ1 ‖ϕx0 ,δ/4 Vj ‖C s+κ (δ) .

⌊s⌋+1

(2.117)

2.8 Nonlinear elliptic equations

� 105

Using a similar argument for (IV), but with Theorem 2.6.2 (vii) in place of Theorem 2.6.2 (ii), we have 󵄩 󵄩 (IV) = δκ 󵄩󵄩󵄩ϕx0 ,δ/8 R2 (x, ϕx0 ,δ/8 (x)Dκ (Vj + H)(x))󵄩󵄩󵄩C s (δ) ≲ ‖R2 ‖C s,⌊s⌋+4+σ (1 + ‖ϕx0 ,δ/8 Dκ (Vj + H)‖L∞ )

⌊s⌋+2

× ‖ϕx0 ,δ/8 Dκ (Vj + H)‖L∞ δκ ‖ϕx0 ,δ/8 Dκ (Vj + H)‖C s (δ)

≲ (1 + ϵ1 )⌊s⌋+2 ϵ1 (‖ϕx0 ,δ/4 Vj ‖C s+κ (δ) + ϵ1 ) ≲ ϵ12 + ϵ1 ‖ϕx0 ,δ/4 Vj ‖C s+κ (δ) .

Plugging the above estimates into (2.115) gives (I) ≲ ϵ1 + ϵ1 ‖ϕx0 ,δ/4 Vj ‖C s+κ (δ) .

(2.118)

Using (2.118) and (2.111) to bound the right-hand side of (2.110) gives s−κ (δ) ≲ ϵ1 + ϵ1 ‖ϕx ,δ/4 Vj ‖C s+κ (δ) . ‖ϕx0 ,δ/16 δ2κ P ∗ P Vj+1 ‖B∞,∞ 0

(2.119)

Using (2.119) and (2.109) to bound the right-hand side of (2.108) shows that for x0 ∈ ℝn , δ ∈ (0, 1] such that Bn (x0 , δ/4) ⊆ Bn (3/4), we have ‖ϕx0 ,δ/32 Vj+1 ‖C s+κ (δ) ≲ ϵ1 + ϵ1 ‖ϕx0 ,δ/4 Vj ‖C s+κ (δ)

≈ ϵ1 + ϵ1 ‖ϕx0 ,δ/4 Vj ‖C s+κ (δ/4) .

(2.120)

Set, for j ∈ ℕ, Qj :=

sup

δ∈(0,1],x∈ℝn Bn (x,δ)⊆Bn (3/4)

‖ϕx,δ Vj ‖C s+κ (δ) ∈ [0, ∞].

Since V0 = 0, we have Q0 = 0. For x0 ∈ Bn (3/4) and δ ∈ (0, 1] such that Bn (x0 , δ/4) ⊆ Bn (3/4), (2.120) implies ‖ϕx0 ,δ/32 Vj+1 ‖C s+κ (δ) ≲ ϵ1 + ϵ1 Qj .

(2.121)

Take x ∈ Bn (3/4) and δ ∈ (0, 1] such that Bn (x, δ) ⊆ Bn (3/4). Take N1 ≈ 1 and x1 , . . . , xN1 as in Proposition 2.8.10 (i). In particular, we have Bn (xj , δ/4) ⊆ Bn (x, δ) ⊆ Bn (3/4), and therefore by (2.121), we have ‖ϕxl ,δ/32 Vj+1 ‖C s+κ (δ) ≲ ϵ1 + ϵ1 Qj , Proposition 2.8.10 (i) gives

1 ≤ l ≤ N1 .

(2.122)

106 � 2 Ellipticity N1

‖ϕx,δ Vj+1 ‖C s+κ (δ) ≲ ∑ ‖ϕxl ,δ/32 Vj+1 ‖C s+κ (δ) ≲ ϵ1 + ϵ1 Qj , l=1

(2.123)

where the final estimate used (2.122) and the fact that N1 ≈ 1. Taking the supremum over all such x and δ in (2.123) gives Qj+1 ≲ ϵ1 + ϵ1 Qj , that is, ∃C6 ≈ 1 with Qj+1 ≤ C6 (ϵ1 + ϵ1 Qj ). We take ϵ1 := min{1/2C6 , 1/2C5 , 1/D2 }. Then we have Qj+1 ≤

1 Qj + . 2 2

A simple induction, using the fact that Q0 = 0, shows that Qj ≤ 1, ∀j ∈ ℕ. Take ψ0 ∈ C0∞ (Bn (2/3)) and take x1 , . . . , xN1 as in Proposition 2.8.10 (ii). We have N1

N1

l=1

l=1

‖ψ0 Vj ‖C s+κ ≲ ∑ ‖ϕxl ,1/32 Vj ‖C s+κ ≈ ∑ ‖ϕxl ,1/32 Vj ‖C s+κ (1/32) ≤ N1 Qj ≲ 1, that is, ‖Dl ψ0 Vj ‖L∞ ≲ 2−l(s+κ) ,

∀l ∈ ℕ.

j→∞

(2.124)

Since Vj 󳨀󳨀󳨀󳨀→ V∞ in C r+κ , we have Vj → V∞ in L∞ . Since Dl : L∞ → L∞ is bounded, taking the limit as j → ∞ in (2.124) gives ‖Dl ψ0 V∞ ‖L∞ ≲ 2−l(s+κ) ,

∀l ∈ ℕ.

This is equivalent to ‖ψ0 V∞ ‖C s+κ ≲ 1. We have ‖ψ0 u‖C s+κ ≤ ‖ψ0 V∞ ‖C s+κ + ‖ψ0 H‖C s+κ

≲ ‖ψ0 V∞ ‖C s+κ + ‖ψ3 H‖C s+κ ≲ 1,

where the last inequality used (2.106), completing the proof.

2.8.4 Weighted estimates near the boundary Let Ω ⊆ ℝn be an open set. If a nonlinear elliptic equation holds on Ω, F(x, {𝜕xα u(x)}|α|≤κ ) = g(x),

x ∈ Ω,

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107

then the estimates on ‖u‖C s+κ implied by Theorem 2.8.3 blow up near 𝜕Ω. To make this blowup precise we need to introduce weighted analogs of the Zygmund–Hölder norms ‖ ⋅ ‖C s defined in Definition 2.5.2. l For v ∈ ℝn and l ∈ ℕ, set Ωv,l := {x ∈ Ω : x, x + v, x + 2v, . . . , x + lv ∈ Ω}. Note that for f ∈ Cloc (Ω), Difflv f (x) is defined for x ∈ Ωv,l . Definition 2.8.13. For l ∈ ℕ+ , s ∈ (0, l), and f ∈ Cloc (Ω), we define the weighted interior norm l

‖f ‖Ĉs (Ω) := ∑ max sup l

l′ =0

sup (dist(x, 𝜕Ω) ∧ 1)

1≤k≤n δ∈(0,1] x∈Ω

sl′ /l −sl′ /l 󵄨 󵄨

δ

δek ,l′

󵄨 l 󵄨󵄨Diffδek f (x)󵄨󵄨󵄨, ′

where dist(x, 𝜕Ω) = inf{|x − y| : y ∈ 𝜕Ω}. Theorem 2.8.14. Let s > r > 0, σ > 0. Suppose that: – F(x, ζ ) ∈ C s,3⌊s⌋+8+σ (ℝn × ℝN ; ℝD1 ). – u ∈ C r+κ (ℝn ; ℝD2 ). – g ∈ C s (ℝn ; ℝD1 ). – F(x, {𝜕xα u(x)}|α|≤κ ) = g(x), ∀x ∈ Ω; since the functions involved are continuous, this equality in fact holds on Ω. – For each x0 ∈ Ω, define the constant coefficient, linear differential operator α

α

Px0 v := dζ F(x0 , {𝜕x u(x0 )}|α|≤κ ){𝜕x v}|α|≤κ .

(2.125)

We assume Px0 is elliptic, uniformly for x0 ∈ Ω, that is, ∃A ≥ 0, ∀x0 ∈ Ω, n

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ∑󵄩󵄩󵄩𝜕xκj f 󵄩󵄩󵄩L2 ≤ A(󵄩󵄩󵄩Px0 f 󵄩󵄩󵄩L2 + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 ), j=1

∀f ∈ C0∞ (ℝn ; ℝD2 ).

(2.126)

Then, ∀l ∈ ℕ+ with l > s + κ, ∃C ≥ 0 such that ‖u‖Ĉs (Ω) ≤ C. l

Here, C ≥ 0 can be chosen to depend only on n, s, l, r, κ, σ, A, D1 , D2 , and upper bounds for ‖u‖C r+κ , ‖g‖C s , and ‖F‖C s,3⌊s⌋+8+σ . Remark 2.8.15. Results like Theorem 2.8.14 are often stated in terms of weighted Hölder norms, C m,α , where α ∈ ̸ {0, 1}. By using weighted Zygmund–Hölder norms one can state the results for all s > 0, not just s ∈ (0, ∞) \ ℕ. Using results like Corollary 2.5.6, one can

108 � 2 Ellipticity easily show that Theorem 2.8.14 implies a corresponding result about weighted Hölder norms with non-integer exponents. The rest of this section is devoted to a sketch of the proof of Theorem 2.8.14. A proof with all details of a more general result in the maximally subelliptic setting is given in Section 9.3. By replacing F(x, ζ ) by F(x, ζ ) − g(x), we may henceforth assume g = 0. For J ∈ [0, ∞), set uJ := ∑ Dj u, 0≤j 0, 0 ≤ l′ ≤ l, 1 ≤ k ≤ n, and x0 ∈ Ωξ0 δek ,l′ and set 2−J0 := dist(x0 , 𝜕Ω) ∧ 1. To prove the statement of the theorem, it suffices to show that (for some ξ0 ≈ 1) ′ ′ ′ 󵄨 󵄨 2−J0 (s+κ)l /l δ−(s+κ)l /l 󵄨󵄨󵄨Difflξ0 δek u(x0 )󵄨󵄨󵄨 ≲ 1.

(2.130)

If δ ≥ 2−J0 , then we have ′ ′ ′ 󵄨 󵄨 2−J0 (s+κ)l /l δ−(s+κ)l /l 󵄨󵄨󵄨Difflξ0 δek u(x0 )󵄨󵄨󵄨 ≲ ‖u‖L∞ ≲ ‖u‖C r+κ ≲ 1,

establishing (2.130). If l′ = 0, then δ and 2−J0 do not play a role in (2.130) and the same proof establishes (2.130). We turn to proving (2.130) in the remaining case where 2−J0 > δ and 1 ≤ l′ ≤ l. Take ψ ∈ C0∞ (Bn (1)) with ψ ≡ 1 on Bn (3/4). Set v̂x0 ,J0 (y) := vJ0 (2−J0 y + x0 ),

û x0 ,J0 (y) := ψ(y)uJ0 (2−J0 y + x0 ),

F̂x0 ,J0 (y, ζ ) := FJ0 (2−J0 y + ζ ).

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It is straightforward to show ‖û x0 ,J0 ‖C s+κ ≲ ‖u‖C r+κ ≲ 1,

‖v̂x0 ,J0 ‖C r+κ ≲ ‖u‖C r+κ ≲ 1,

(2.131)

and using Theorem 2.6.2 (i), ‖F̂x0 ,J0 ‖C s,2⌊s⌋+6+σ ≲ ‖F‖C s,3⌊s⌋+8+σ .

(2.132)

Rescaling (2.129) shows F̂x0 ,J0 (y, {𝜕yα v̂x0 ,J0 (y)}|α|≤κ ) = 0,

∀y ∈ Bn (3/4).

(2.133)

Our goal is to apply Theorem 2.8.7 to (2.133). To do so, we need to show (2.133) is elliptic, uniformly in x0 ∈ Ω. That is, define α

α

̂x ,J := dζ F̂x ,J (0, {𝜕 v̂x ,J (0)} P y 0 0 |α|≤κ ){𝜕y }|α|≤κ . 0 0 0 0 ̂x ,J is elliptic of degree κ, uniformly for x0 ∈ Ω. We with to show P 0 0 Recall Px0 from (2.125). From its definition, Px0 is of the form α

Px0 = ∑ aα,x0 𝜕x ,

aα,x0 ∈ 𝕄D1 ×D2 (ℝ),

|α|≤κ

and we have dζ F(x0 , {𝜕xα u(x0 )}|α|≤κ ){ζα }|α|≤κ = ∑ aα,x0 ζα . |α|≤κ

Using (2.127) and (2.128), we have α

α

̂x ,J w = dζ F̂x ,J (0, {𝜕 v̂x ,J (y)} P y 0 0 |α|≤κ ){𝜕y w}|α|≤κ 0 0 0 0 α

= dζ FJ0 (x0 , {(2−J0 𝜕x ) vJ0 (x0 )}){𝜕yα w}|α|≤κ = dζ F(x0 , {𝜕xα u(x0 )}|α|≤κ ){2−(κ−|α|)J0 𝜕yα w}|α|≤κ = ∑ 2−(κ−|α|)J0 aα,x0 𝜕yα w. |α|≤κ

Multiplying (2.126) by 2−κJ0 gives, ∀f ∈ C0∞ (ℝn ; ℝD2 ), n

κ 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ∑󵄩󵄩󵄩(2−J0 𝜕xj ) f 󵄩󵄩󵄩L2 ≤ A(󵄩󵄩󵄩2−κJ0 Px0 f 󵄩󵄩󵄩L2 + 2−κJ0 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 ) j=1

󵄩 󵄩 󵄩 󵄩 ≤ A(󵄩󵄩󵄩2−κJ0 Px0 f 󵄩󵄩󵄩L2 + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 ).

(2.134)

110 � 2 Ellipticity Applying (2.134) to f (x) = g(2J0 x) and changing variables shows n

󵄩 󵄩 󵄩̂ 󵄩󵄩 󵄩󵄩 󵄩󵄩 ∑󵄩󵄩󵄩𝜕yκj g 󵄩󵄩󵄩L2 ≤ A(󵄩󵄩󵄩P x0 ,J0 g 󵄩 󵄩L2 + 󵄩󵄩g 󵄩󵄩L2 ),

∀g ∈ C0∞ (ℝn ; ℝD2 ).

j=1

̂x ,J is elliptic, uniformly for x0 ∈ Ω. In particular, P 0 0 In light of the above, Theorem 2.8.7 applies to (2.133). Let ψ0 ∈ C0∞ (Bn (2/3)) with δ δ ψ0 ≡ 1 on Bn (1/2) and for δ1 > 0 set ψ01 (y) := ψ0 (δ1−1 y), so that ψ01 ≡ 1 on Bn (δ1 /2). The conclusion of Theorem 2.8.7, when applied to (2.133), shows that there exists δ1 ∈ (0, 1], δ1 ≈ 1, with δ

‖ψ01 v̂x0 ,J0 ‖C s+κ ≲ 1.

(2.135)

Recall that, by (2.131), we also have ‖û x0 ,J0 ‖C s+κ ≲ 1.

(2.136)

By Proposition 2.5.3, (2.135) and (2.136) imply δ

‖ψ01 v̂x0 ,J0 ‖C s+κ , ‖û x0 ,J0 ‖C s+κ ≲ 1. l

l

In particular, ′ δ1 󵄨󵄨 󵄨󵄨Diffl J0 ̂ 󵄨 2 δξ0 ek (ψ0 vx0 ,J0 )(0)󵄨󵄨 ≲ 1, ′ −(s+κ)l′ /l 󵄨 󵄨󵄨 󵄨󵄨Diffl J0 ̂ (2J0 δ) 󵄨 2 δξ0 ek ux0 ,J0 (0)󵄨󵄨 ≲ 1.

(2J0 δ)

−(s+κ)l′ /l 󵄨

(2.137) (2.138)

By taking ξ0 := δ1 /2l ≈ 1 and using 2J0 δ ≤ 1, we see that δ

ψ01 (2J0 δξ0 l′′ ek ) = 1,

∀0 ≤ l′′ ≤ l,

(2.139)

and since ψ ≡ 1 on Bn (3/4), ψ(2J0 δξ0 l′′ ek ) = 1,

∀0 ≤ l′′ ≤ l.

(2.140)

Using (2.137) and (2.139) shows (2J0 δ)

′ ′ 󵄨 J0 −(s+κ)l /l 󵄨󵄨 l′ 󵄨󵄨Diffl 󵄨󵄨 󵄨󵄨Diff2J0 δξ0 ek v̂x0 ,J0 (0)󵄨󵄨󵄨 ≲ 1. δξ0 ek vJ0 (x0 )󵄨󵄨 = (2 δ) 󵄨

−(s+κ)l′ /l 󵄨

(2.141)

Using (2.138) and (2.140) shows (2J0 δ)

′ ′ 󵄨󵄨 󵄨 J0 −(s+κ)l /l 󵄨󵄨 l′ 󵄨󵄨Diffl 󵄨󵄨Diff2J0 δξ0 ek û x0 ,J0 (0)󵄨󵄨󵄨 ≲ 1. δξ0 ek uJ0 (x0 )󵄨󵄨 = (2 δ) 󵄨

−(s+κ)l′ /l 󵄨

(2.142)

2.9 Further reading and references �

111

Since u = uJ0 + 2−κJ0 vJ0 (see (2.127)), (2.130) follows by combining (2.141) and (2.142), completing the proof.

2.9 Further reading and references The prototypical example of a singular integral operator, the Hilbert transform, first arose in Hilbert’s work on what is now known as the Riemann–Hilbert problem. Hilbert’s proof was published by Weyl [250]. These results were later improved by Schur [208]. The results of Hilbert, Weyl, and Schur were restricted to L2 ; Marcel Riesz extended these results to Lp (1 < p < ∞) [204]. These references were focused on “complex analysis methods,” and did not generalize to higher dimensions. Besikovitch [10], Titchmarsh [237], and Marcinkiewicz [166] offered a “real-variable” analysis of the Hilbert transform. The real-variable analysis of the Hilbert transform was a main motivating example for Calderón and Zygmund when they introduced the first iteration of what are now known as Calderón–Zygmund singular integrals [27]. The proof of Corollary 2.3.13 uses their methods; the interpolation theorem used in that proof (the “Marcinkiewicz interpolation theorem”) was proved by Marcinkiewicz [167]. Also used in Corollary 2.3.13 was the Cotlar–Stein lemma (Lemma 2.3.25). The special case of the Cotlar–Stein lemma when Tj are all assumed to be self-adjoint and to pairwise commute was originally proved by Cotlar [54]. Lemma 2.3.25 in its full generality is due to E. M. Stein; see [216, Chapter VII, § 2]. Though it may not be immediately apparent, these methods form the foundation for many of the Lp estimates later in this text. The concept of a pseudo-differential operator is rooted in the work of Marcinkiewicz [167], later work by Calderón and Zygmund, and the work of Seeley in his thesis [209]. This was follows by further work of Seeley [210] and culminated in the work of Unterberger and Bokobza [247, 246], Kohn and Nirenberg [144], and Hörmander [124]. It was these last two references that first exhibited the theory of pseudo-differential operators as covered in Section 2.2, though our presentation more closely follows the one from [216, Chapter VI]. As mentioned in Section 2.2.1, the generalizations of pseudo-differential operators in Chapter 4 cannot use the Fourier transform, and are more closely related to the earlier works mentioned above. The theory of non-translation invariant, non-homogeneous Calderón–Zygmund singular integral operators (roughly speaking, singular integral operators of order 0 from Definition 2.3.1) was the work of many authors. A systematic approach, working in more general “spaces of homogeneous type” and with much less regular kernels, was developed by Coifman and Weiss [52]. Definition 2.3.1 (at least for singular integrals of order > −n) was taken from much more recent work of Nagel, Rosay, Stein, and Wainger [182] and Koenig [140], who worked in the more general setting described in Sections 5.1.1 and 5.2.1 (see Section 5.13 for further comments on their work). Results like the charac-

112 � 2 Ellipticity terization of singular integral operators given in Theorem 2.3.10 can be found in [220, Chapter I], though they are closely related to much earlier and well-known methods – for example Theorem 2.2.26. A key idea of Theorem 2.3.10 – to characterize singular integrals in terms of their action on certain special functions or operators (e. g., Theorem 2.3.10 (ii)) – was first pointed out to the author in a graduate course given by E. M. Stein at Princeton University in 2007. The concept of defining the singular integrals partially in terms of a “cancelation condition” (as in Definition 2.3.1) is closely related to the hypotheses of the T(1) theorem of David and Journé [60]; see also the presentation in [216, pages 293–294]. This idea was further championed by E. M. Stein; see, e. g., [216, page 248]. The decomposition of singular integral operators as in Theorem 2.3.10 (iii) is called a Littlewood–Paley decomposition of the operator, named so because the first place where similar decompositions appeared was in the work of Littlewood and Paley on Fourier series [149–151, 194]. These ideas were later worked on by Zygmund and Marcinkiewicz, but were moved to higher dimensions and used in greater generality by E. M. Stein (see, e. g., [214]). Such decompositions are now common in the theory of pseudo-differential and singular integral operators; see for example Theorem 2.2.26 (iii). The Hardy–Littlewood maximal function (2.33) was introduced by Hardy and Littlewood, and the vector-valued result we used (Lemma 2.4.14) is due to C. Fefferman and E. M. Stein [83]. For more history on some of the above topics, see the expository articles of E. M. Stein [215, 217] along with Stein’s book [216]. In particular, the case s = 0 of Proposition 2.4.9 is often known as a square function estimate; [215] contains a detailed history of such square functions. s The Besov spaces Bp,q (ℝn ) were introduced by Peetre [195] (which was preceded by some earlier ideas of Hörmander in the case p = q = 2 [123]). The Triebel–Lizorkin s spaces Fp,q (ℝn ) were introduced by Triebel [240] with closely related definitions given by Lizorkin [152, 153]. Besov and Triebel–Lizorkin spaces unify a number of classical p function spaces. Two of these are of particular interest to us: the Sobolev spaces Ls (Defs inition 2.2.11) and the Zygmund–Hölder spaces C (Section 2.5). The Sobolev spaces were introduced by Aronszajn and Smith [4] and Calderón [26]. The Zygmund–Hölder spaces were introduced by Zygmund [255]. For a detailed discussion of the spaces generalized by the Besov and Triebel–Lizorkin spaces and the history of these spaces, we refer the reader to [244, § 2.2.1 and § 2.3.5]. The theory of these spaces is extensive (see, for example, the books by Triebel [244, 242, 243, 241]). We have only described the very basic definitions and results in this chapter, as that is all we generalize to the maximally subelliptic setting. However, it would be interesting to generalize other aspects of these theories to the maximally subelliptic setting. The theory of linear elliptic operators described in Section 2.7 is classical. Theorem 2.7.2 is well known, in some form or another, to any expert in the field. The history of results like Theorem 2.8.3 is long, and many improvements are known. The expository article [20] contains a detailed history. There are several known

2.9 Further reading and references

� 113

proofs of Theorem 2.8.3; however, most of them do not easily generalize to the maximally subelliptic setting. This is usually because they implicitly rely on the basic fact that 𝜕xj and 𝜕xk commute. For example, one standard approach can be found on pages 139 and 140 of [233] in the proof of Theorem 4.6. There is it used that derivatives and differences commute, a property which we do not have when we move to the maximally subelliptic setting in Chapter 9.

3 Vector fields and Carnot–Carathéodory geometry Central to the study of maximally subelliptic operators is the Carnot–Carathéodory geometry induced by the vector fields with formal degrees (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )}, with balls B(W ,ds) (x, δ) defined in (1.13) and associated metric ρ(W ,ds) (x, y) defined in (1.14). In this chapter we present the main tools we need to study these geometries and vector fields. A key goal is to define and study the associated scaling maps as discussed in Section 1.9. We study these scaling maps both in the single-parameter case (Section 3.3) and in a more general multi-parameter setting (Section 3.5). These theorems about scaling (along with other, more quantitative versions we need) all follow from a single, abstract, technical theorem of the author and Stovall [228], which is presented in Section 3.6 and proved in Appendix A. Since this chapter is the toolbox for the rest of the text, the reader wishing to understand the main ideas might start by reading up through Section 3.3, and then referring back to this chapter whenever a tool or topic is needed.

3.1 Manifolds We assume the reader is familiar with the basic definitions and results concerning manifolds and vector fields; for background see [147]. In this section, we fix notation and review some concepts which, though they are standard, may not be immediately recalled by readers who only know the basic theory of manifolds. For L ∈ ℕ and an open subset Ω ⊆ ℝn , we defined the Banach space C L (Ω) of L-times differential functions whose derivatives up to order L were bounded (see (2.2)); we simL ilarly defined the Fréchet space C ∞ (Ω). We also defined the Fréchet space Cloc (Ω) to be L the space of functions which were locally in C . Let M be a connected C ∞ manifold (all manifolds in this text are assumed to be second countable). There is no intrinsic definition of a Banach space C L (M), unless M is L compact. However, there is an intrinsic definition of Cloc (M). Thus, whenever we speak L of functions on a manifold, we will always use the local spaces Cloc (M), and we will not use any norm of the form ‖f ‖C L (M) , as there is no intrinsic definition of this norm. We let C0∞ (M) denote the locally convex topological vector space consisting of those functions ∞ in Cloc (M) which have compact support. As before we denote by C0∞ (M)′ the space of ∞ distributions on M and by Cloc (M)′ the space of distributions with compact support. Definition 3.1.1. We say Φ : U → M is a smooth coordinate system if U ⊆ ℝn and ∞ Φ(U) ⊆ M are open and Φ : U → Φ(U) is a Cloc diffeomorphism. ∞ Let V denote a C ∞ vector bundle over M. We let Cloc (M; V ) denote the space of ∞ ∞ Cloc sections of V . If V is a finite-dimensional vector space, we let Cloc (M; V ) denote the ∞ ∞ space of Cloc sections of the trivial vector bundle M × V . In particular, Cloc (M; TM) den notes the space of smooth vector fields on M. Let Ω ⊆ ℝ be an open set. Then TΩ can be https://doi.org/10.1515/9783111085647-003

3.1 Manifolds

� 115

∞ ∞ identified with the trivial vector bundle Ω × ℝn and therefore Cloc (Ω; TΩ) ≅ Cloc (Ω; ℝn ). L n Thus, for a vector field Y on Ω, it makes sense to ask if Y ∈ C (Ω; ℝ ). We do this by identifying Y = ∑nj=1 aj 𝜕xj with the function a(x) = (a1 (x), . . . , an (x)). We then define

‖Y ‖C L (Ω;ℝn ) := ‖a‖C L (Ω;ℝn ) , and we say Y ∈ C L (Ω; ℝn ) if and only if a ∈ C L (Ω; ℝn ). Such a norm and space are only defined for vector fields on open subsets of ℝn and are not defined on an abstract manifold. This is an important distinction in many of our proofs, where we will be given vector fields on an abstract manifold, and we will find a coordinate system in which they are “normalized”; see Section 3.3. ∞ Suppose M and N are C ∞ manifolds and Φ : M → N is a Cloc map. We write ∗ ∞ Φ for the pullback via Φ and Φ∗ for the pushforward via Φ. If Φ : M → N is a Cloc diffeomorphism, then Φ∗ and Φ∗ are inverses to each other, whenever they are defined. ∞ Example 3.1.2. Suppose Φ : M → N is a Cloc diffeomorphism. Then: ∞ ∗ ∞ – If f ∈ Cloc (N), then Φ f = f ∘ Φ ∈ Cloc (M). ∞ ∞ – If g ∈ Cloc (M), then Φ∗ g = g ∘ Φ−1 ∈ Cloc (N). ∞ ∗ ∞ – If X ∈ Cloc (N; TN), then Φ X ∈ Cloc (M; TM) is defined by

Φ∗ X(m) = (dΦ(m))−1 (X(Φ(m))) ∈ Tm M. –

∞ ∞ If Y ∈ Cloc (M; TM), then Φ∗ Y ∈ Cloc (N; TN) is defined by

Φ∗ Y (n) = dΦ(Φ−1 (n))(Y (Φ−1 (n))) ∈ Tn N. Remark 3.1.3. If we consider vector fields as first-order partial differential operators ∞ ∞ and Φ : M → N is a Cloc diffeomorphism, then for X ∈ Cloc (N; TN) we have Φ∗ (X(Φ∗ g)) = (Φ∗ X)g,

∞ ∀g ∈ Cloc (M).

We will have occasion to use densities on a manifold. Given a real vector space V of dimension n, we let |V | denote the space of all maps σ : ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ V × V × ⋅⋅⋅ × V → ℝ n factors

such that for any linear map A : V → V we have σ(Av1 , . . . , Avn ) = |det A|σ(v1 , . . . , vn ). There is a canonical element of |ℝn |, called the Lebesgue density, given by σLeb (v1 , . . . , vn ) = |det(v1 | ⋅ ⋅ ⋅ |vn )|,

v1 , . . . , vn ∈ ℝn ,

where (v1 | ⋅ ⋅ ⋅ |vn ) denotes the n × n matrix with columns v1 , . . . , vn . Definition 3.1.4. A density on M is a map σ, which assigns to each x ∈ M an element σ(x) ∈ |Tx M|.

116 � 3 Vector fields and Carnot–Carathéodory geometry ∞ Let M and N be two manifolds. Given a Cloc map Φ : M → N and a density σ on N, ∗ we define a density Φ σ on M by

(Φ∗ σ)(m)(v1 , . . . , vn ) = σ(Φ(m))(dΦ(m)v1 , . . . , dΦ(m)vn ),

v1 , . . . , vn ∈ Tm M.

(3.1)

Definition 3.1.5. Let M be a C ∞ manifold of dimension n and let σ a density on M. We say σ is a smooth density if the following holds. For every smooth coordinate system Φ : U → Φ(U), where U ⊆ ℝn is open, we have Φ∗ σ = fσLeb , ∞ where f ∈ Cloc (U). We say σ is a strictly positive density if f > 0 for all such Φ.

Given a strictly positive, smooth density σ on M, σ induces a non-negative Borel measure on M which is strictly positive on every open set. Indeed, this may be defined locally as follows. If S ⊆ M is a Borel set such that there exists a smooth coordinate system Φ : U → Φ(U), we define σ(S) := ∫ f (x) dx, Φ−1 (S)

where Φ∗ σ = fσLeb and dx denotes the Lebesgue measure on ℝn . Henceforth, we identify σ with this measure. For more details on densities, we refer the reader to the particularly lucid presentation in [100] (see also [191], where densities are called 1-densities).

3.1.1 The exponential map ∞ Let M be a C ∞ manifold and let X ∈ Cloc (M; TM) be a smooth vector field on M. For x ∈ M and t ∈ ℝ small, we define

etX x = E(t), where E(t) is the unique solution to the ordinary differential equation (ODE) d E(t) = X(E(t)), dt

E(0) = x.

Classical results from the field of ODEs show that (t, x) 󳨃→ etX x defines a smooth map U → M, where U is an open neighborhood of {0} × M ⊂ ℝ × M. Furthermore, we have the identity e−tX etX x = e0X x = x for (t, x) in an open neighborhood of {0} × M ⊂ ℝ × M. It follows from the definition that d f (etX x) = (Xf )(etX x). dt

(3.2)

3.1 Manifolds

� 117

Let Φt (x) := etX x. By the above remarks, given a relatively compact, open set Ω ⋐ M, ∞ for t ∈ ℝ sufficiently small (depending on Ω), Φt : Ω → Φt (Ω) is a Cloc diffeomorphism d 󵄨󵄨 with inverse Φ−t . We define the Lie derivative with respect to X by LieX = dt 󵄨󵄨t=0 Φ∗t . ∞ ∞ Example 3.1.6. For f ∈ Cloc (M), LieX f = Xf . For Y ∈ Cloc (M; TM), LieX Y = [X, Y ]. For a d 󵄨󵄨 ∗ ∗ smooth density σ on M, LieX σ = dt 󵄨󵄨t=0 Φt σ, where Φt σ is defined by (3.1); in particular, if M = ℝn , then LieX (fσLeb ) = (Xf + f ∇ ⋅ X)σLeb , where ∇ ⋅ X denotes the divergence of X.

We will have many uses for the exponential map in this text. We begin with a simple example, following the work of Nagel, Stein, and Wainger [189]. ∞ Lemma 3.1.7. Let (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} ⊂ Cloc (M; TM) × ℕ+ be a set of Hörmander vector fields with formal degrees on the connected C ∞ manifold M. Then ρ(W ,d) is a metric, and the metric topology induced by ρ(W ,ds) is the same as the underlying topology on M.

Proof. ρ(W ,d) is easily seen to be symmetric and satisfy the triangle inequality. If U is an open neighborhood of x ∈ M, then the Picard–Lindelöf theorem shows that there is a δ > 0 such that B(W ,ds) (x, δ) ⊆ U. Thus, the topology induced by ρ(W ,ds) is finer than the topology on M; this also implies ρ(W ,ds) (x, y) > 0 for x ≠ y. Next, we will show that the topology on M is finer than the metric topology induced by ρ(W ,ds) . Fix x ∈ M and δ > 0. We wish to show that there is an open set U with ∞ x ∈ U ⊆ B(W ,ds) (x, δ). For smooth vector fields S1 , S2 , . . . ∈ Cloc (M; TM) and a ∈ ℝ small, set C1 (a, S1 ) := eaS1 ,

C2 (a, S1 , S2 ) := eaS1 eaS2 e−aS1 e−aS2 ,

and more generally, Cl (a, S1 , . . . , Sl ) := eaS1 Cl−1 (a, S2 , . . . , Sl−1 )e−aS1 (Cl−1 (a, S2 , . . . , Sl ))−1 . Standard results (see Section 3.1.2) show that 󵄨 𝜕al 󵄨󵄨󵄨a=0 Cl (a, S1 , . . . , Sl )x = [S1 , [S2 , . . . , [Sl−1 , Sl ] ⋅ ⋅ ⋅]](x), 󵄨 𝜕ak 󵄨󵄨󵄨a=0 Cl (a, S1 , . . . , Sl )x = 0, ∀k < l. Pick Y1 , . . . , Yn of the form j

Yj = [Wlj , [Wlj , . . . , [Wl 1

2

kj −1

, Wlk ] ⋅ ⋅ ⋅]] j

so that Y1 (x), . . . , Yn (x) form a basis for Tx M (this is always possible since W satisfies Hörmander’s condition). Set, for t ∈ ℝ with t small,

118 � 3 Vector fields and Carnot–Carathéodory geometry 1

kj { { {Ckj (t , Wl1j , Wl2j , . . . , Wlkj j ), Cj (t) := { 1 { {Ck (|t| kj , W j , W j , . . . , W j )−1 , l1 l2 lk { j j

kj odd or kj even and t ≥ 0, kj even and t < 0.

Set D(t1 , . . . , tn ) := C1 (t1 )C2 (t2 ) ⋅ ⋅ ⋅ Cn (tn )x. 1 Note that D(0) = x. It is easy to see that D is Cloc on a neighborhood of 0 ∈ ℝn . Fur󵄨󵄨 thermore, since 𝜕tj 󵄨󵄨t=0 D(t) = Yj (x), we see that dD(0) is an invertible map. The inverse function theorem applies to show that if ϵ > 0 is sufficiently small, D(Bn (ϵ)) is open. Furthermore, if ϵ > 0 is small enough, it is easy to see that D(Bn (ϵ)) ⊆ B(W ,ds) (x, δ). Since x = D(0) ∈ D(Bn (ϵ)), it follows that the topology on M is finer than the topology induced by ρ(W ,ds) . Thus, the topology induced by ρ(W ,ds) agrees with the topology on M. To see that ρ(W ,ds) is a metric, it remains to show that ρ(W ,ds) (x, y) < ∞, ∀x, y. We have already shown that for each x, {y ∈ M : ρ(W ,ds) (x, y) < ∞} is an open set, and by the same proof its complement is open. Since M is connected, {y ∈ M : ρ(W ,ds) (x, y) < ∞} = M, completing the proof.

3.1.2 The Baker–Campbell–Hausdorff formula The Baker–Campbell–Hausdorff formula is a formal series, where if one considers the algebra of formal series in two non-commuting indeterminants X, Y , then eX eY = eBCH(X,Y ) , where BCH(X, Y ) is an infinite formal (Lie) series 1 1 1 BCH(X, Y ) = X + Y + [X, Y ] + [X, [X, Y ]] + [Y , [Y , X]] 2 12 12 + commutators of order 3 + commutators of order 4 + ⋅ ⋅ ⋅ . This formal series is universal (the coefficients are universal constants). It is possible to write down an explicit formula for the coefficients (see, for example, [12]), though what is important for our applications is that it is a universal formula. ∞ ∞ Let X ∈ Cloc (M; TM) be a smooth vector field and let f ∈ Cloc (M). Define etX f (x) := f (etX x). ∞ Then (t, x) 󳨃→ etX f (x) ∈ Cloc (U), where U is an open neighborhood of {0} × M in ℝ × M. Repeatedly using (3.2), we see that the Taylor series in the t variable of etX f (x) can be computed as

(tX)j f (x). j! j=0 ∞

etX f (x) ∼ ∑

3.1 Manifolds

� 119

∞ As a consequence, if Y ∈ Cloc (M; TM), then as a formal power series in t and s, we have

f (esY etX x) = etX esY f (x) ∼ eBCH(tX,sY ) f (x),

(3.3)

where the right-hand side is only interpreted as a formal power series. If sufficiently high-order commutators of X and Y are zero, then BCH(tX, sY ) is a finite sum and (3.3) is an equality of functions, not just of formal sums. One application of (3.3) is that it allows us to compute the Taylor series in the s variable of X(f (esY x)). Indeed, Xf (esY x) =

d 󵄨󵄨󵄨󵄨 d 󵄨󵄨󵄨 sY tX BCH(tX,sY ) f (x). 󵄨󵄨 f (e e x) ∼ 󵄨󵄨󵄨 e dt 󵄨󵄨t=0 dt 󵄨󵄨t=0

3.1.3 The Frobenius theorem Closely tied to Carnot–Carathéodory geometry is a quantitative version of the classical Frobenius theorem. In this section, we describe this classical theorem in the form which is of most use to us. The presentation here is a special case of the theory developed by Sussman [231], and we refer the reader to that paper for more details. Definition 3.1.8. Let N and M be C ∞ manifolds. An immersion of N in M is a smooth map N → M whose differential is everywhere injective. Definition 3.1.9. Let M be a C ∞ manifold. An injectively immersed submanifold N of M is a smooth manifold N along with an injective immersion i : N 󳨅→ M. Remark 3.1.10. If N 󳨅→ M is an injectively immersed submanifold of M, then we identify N with its image under the immersion, and therefore think of N as a subset of M. However, N need not be a topological subspace of M. See Example 3.1.19. Remark 3.1.11. If i : N 󳨅→ M is an injectively immersed submanifold, for x ∈ N, we may identify Tx N with its image under di(x) : Tx N 󳨅→ Ti(x) M. Henceforth, we write Tx N to denote this subspace of Ti(x) M. ∞ ∞ ∞ Note that Cloc (M; TM) is a Cloc (M) module: given f , g ∈ Cloc (M) and X, Y ∈ ∞ ∞ ∞ Cloc (M; TM), fX +gY ∈ Cloc (M; TM). For the next three definitions, let 𝒟 ⊆ Cloc (M; TM) ∞ be a Cloc (M)-submodule.

Definition 3.1.12. For x ∈ M, set 𝒟(x) := span{X(x) : X ∈ 𝒟}. ∞ Definition 3.1.13. We say 𝒟 is locally finitely generated as a Cloc (M)-module if for each x ∈ M, there is a neighborhood V ⊆ M of x and a finite collection Z1 , . . . , Zq ∈ 𝒟 such q 󵄨 󵄨 ∞ that for every Y ∈ 𝒟, there exist c1 , . . . , cq ∈ Cloc (V ) with Y 󵄨󵄨󵄨V = ∑j=1 cj Zj 󵄨󵄨󵄨V .

Definition 3.1.14. We say 𝒟 is involutive if ∀X, Y ∈ 𝒟, [X, Y ] ∈ 𝒟, that is, if 𝒟 is a Lie ∞ subalgebra of Cloc (M; TM).

120 � 3 Vector fields and Carnot–Carathéodory geometry ∞ Theorem 3.1.15 (The Frobenius theorem; see Theorem 4.2 in [231]). Let 𝒟 ⊆ Cloc (M; TM) ∞ ∞ be a Cloc (M)-submodule which is involutive and locally finitely generated as a Cloc (M)module. Then, for each point x ∈ M, there is a unique, maximal, connected, injectively immersed submanifold L 󳨅→ M, with x ∈ L and ∀y ∈ L, Ty L = 𝒟(y). The set of all such L, as x ∈ M varies, forms a partition of M into disjoint injectively immersed submanifolds.

Definition 3.1.16. In the setting of Theorem 3.1.15, we say L is the leaf passing through x, and we refer to the conclusion by saying that 𝒟 foliates M into leaves. Remark 3.1.17. Under the conditions of Theorem 3.1.15, the vector fields in 𝒟 are tangent to each leaf. Thus, we may treat them as vector fields on the leaves. By the conclusion of Theorem 3.1.15, these vector fields span the tangent space to the leaf at every point. ∞ Example 3.1.18. Consider ℝ2 with the usual coordinates (x, y). Let 𝒟 = Cloc (ℝ2 ){𝜕y }. Then the leaf passing through (x0 , y0 ) is

{(x0 , y) : y ∈ ℝ}. Thus, in this case, the Frobenius theorem decomposes ℝ2 into its usual product structure ℝ × ℝ. Example 3.1.19. Let M = ℝ/ℤ × ℝ/ℤ be the torus with inherited coordinates (x, y) from ∞ ℝ × ℝ. Fix θ ∈ ℝ \ ℚ an irrational number. Let 𝒟 = Cloc (M){𝜕x + θ𝜕y }. In this case the leaves are one-dimensional, dense subsets of M and carry a topology strictly finer than the subspace topology. ∞ Example 3.1.20. Consider the case where 𝒟 = Cloc (M; TM). In this case, there is only one leaf, and that leaf equals M. While this example is trivial from the perspective of Theorem 3.1.15, in Section 3.6 we introduce a quantitative version of the Frobenius theorem where this example is not trivial. In fact, from a quantitative perspective, this example is the most important one in this text.

Suppose 𝒟 satisfies the conditions of Theorem 3.1.15 and let Lx denote the leaf passing through x ∈ M. Note dim Lx = dim Tx Lx = dim 𝒟(x). Nowhere did we assume dim 𝒟(x) is constant in x, and therefore the dimensions of the leaves may vary from point to point. To discuss this, we introduce a definition. Definition 3.1.21. We say that x ∈ M is a singular point of 𝒟 if dim 𝒟(x) is not constant on any neighborhood of x. ∞ Example 3.1.22. Let M = ℝ2 with the usual coordinates (x, y). We set 𝒟 = Cloc (ℝ2 ){x𝜕y }. Note that

3.2 The unit scale

dim 𝒟(x, y) = {

1 0

� 121

if x ≠ 0, if x = 0.

If (x0 , y0 ) ∈ ℝ2 has x0 ≠ 0, then the leaf passing through (x0 , y0 ) is {(x0 , y) : y ∈ ℝ}. On the other hand, the leaf passing through (0, y0 ) is the point {(0, y0 )}. In this case, each point of the form (0, y0 ) is a singular point of 𝒟. The map (x, y) 󳨃→ dim 𝒟(x, y) is not continuous at the singular points. Suppose 𝒟 satisfies the conditions of Theorem 3.1.15 and let Lx denote the leaf passing through x ∈ M. As we saw in Example 3.1.22, the map x 󳨃→ dim Lx = dim 𝒟(x) may not be continuous M → ℕ. However, it is lower semi-continuous, as the next result shows. ∞ ∞ Lemma 3.1.23. Let 𝒟 ⊆ Cloc (M; TM) be a Cloc (M)-submodule which is locally finitely ∞ generated as a Cloc (M)-module. Then x 󳨃→ dim 𝒟(x), M → ℕ is lower semi-continuous.

Proof. We will show {x : dim 𝒟(x) > r} ⊆ M is open, ∀r ∈ ℝ. Letting m − 1 = ⌊r⌋ ∨ −1, it suffices to show that {x : dim 𝒟(x) > m − 1} is open. The case where m − 1 = −1 is trivial, so we assume m − 1 ≥ 0. Fix a point x0 ∈ {x : dim 𝒟(x) > m − 1}. Let V ⊆ M be an open ∼ neighborhood of x0 so small that there is a smooth coordinate system Φ : U 󳨀 → V , with n ∞ U ⊆ ℝ open, and such that 𝒟 is finitely generated on V as a Cloc (V )-module. Thus, we ∞ have Z1 , . . . , Zq ∈ 𝒟 such that Z1 , . . . , Zq generate 𝒟 on V as a Cloc (V )-module. Note that for y ∈ V , 𝒟(y) = span{Z1 (y), . . . , Zq (y)}, and therefore, {u ∈ U : max{|det A| : A is an m × m submatrix of (Φ∗ Z1 (u)| ⋅ ⋅ ⋅ |Φ∗ Zq (u))} > 0} = Φ−1 ({x : dim 𝒟(x) > m − 1}), where each Φ∗ Zj (u) is treated as an n-dimensional column vector. The left-hand side of the above equation is clearly an open neighborhood of Φ−1 (x0 ) ∈ U, and the result follows. Remark 3.1.24. The leaves of a foliation are manifolds and are therefore defined via an atlas of coordinate charts. In the usual proofs of the Frobenius theorem, one does not have good quantitative control of these coordinate charts near a singular point. In this text, we require better quantitative control than the standard proofs yield; in particular, we require uniform control near singular points. See Remark 3.5.2 for a further discussion of the sort of control we use.

3.2 The unit scale In this text, scaling is the process of changing coordinates to turn a small scale into the unit scale. In this section, we describe the unit scale, which will always take place on Bn (1), the unit ball in ℝn .

122 � 3 Vector fields and Carnot–Carathéodory geometry Let (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} ⊂ C ∞ (Bn (1); ℝn ) × ℕ+ be C ∞ vector fields on Bn (1) paired with formal degrees. We assume W = {W1 , . . . , Wr } satisfies Hörmander’s condition of order m ∈ ℕ+ . Let X1 , . . . , Xq be an enumeration of all commutators of W1 , . . . , Wr up to order m, so that, by hypothesis, X1 (x), . . . , Xq (x) span Tx Bn (1), ∀x ∈ Bn (1). We assume, moreover, that τ := inf n

max

x∈B (1) j1 ,...,jn ∈{1,...,q}

󵄨󵄨 󵄨 󵄨󵄨det(Xj1 (x)| ⋅ ⋅ ⋅ |Xjn (x))󵄨󵄨󵄨 > 0.

(3.4)

For each L ∈ ℕ, fix CL ≥ 0 so that ‖Wj ‖C L (Bn (1);ℝn ) ≤ CL ,

1 ≤ j ≤ r.

Definition 3.2.1. For a parameter ι, we say C is an ι-unit-admissible constant if there exists L ∈ ℕ, depending only on ι, and upper bounds for m, r, max1≤j≤r dsj , such that C can be chosen to depend only on ι and upper bounds for n, m, r, max1≤j≤r dsj , CL , and τ −1 . Definition 3.2.2. Let ℐ be a set, and for each κ ∈ ℐ , suppose (W κ , dsκ ) = {(W1κ , dsκ1 ), . . . , (Wrκκ , dsκrκ )} are C ∞ vector fields with formal degrees on Bnκ (1). We say (W κ , dsκ ) are Hörmander vector fields with formal degrees at the unit scale, uniformly in κ, if ι-unitadmissible constants (as in Definition 3.2.1) can be chosen independent of κ, that is, if the parameters n, m, r, max1≤j≤r dsj , CL , and τ −1 described there are bounded uniformly in κ. Example 3.2.3. For δ > 0, consider the vector fields with formal degrees, on ℝn , {(δ𝜕x1 , 1), . . . , (δ𝜕xn , 1)}. The parameters CL and τ −1 are not bounded uniformly in δ; indeed, CL ≈ δ and τ ≈ δn . Thus, these vector fields with formal degrees are not Hörmander vector fields with formal degrees at the unit scale, uniformly in δ. However, for x ∈ ℝn and δ > 0, define the map Φx,δ : Bn (1) → (x + Bn (δ)) by Φx,δ (t) = x + δt. Then Φ∗x,δ δ𝜕xj = 𝜕tj . Therefore, {(Φ∗x,δ δ𝜕x1 , 1), . . . , (Φ∗x,δ δ𝜕xn , 1)} are Hörmander vector fields with formal degrees at the unit scale, uniformly in δ. Thus, we have used Φx,δ to change coordinates that turned vector fields at “scale δ” into vector fields at the “unit scale”. Theorems 3.3.1, 3.3.7, 3.5.1, and 3.5.4 are more complicated versions of this simple idea. Whenever we prove a result about Hörmander vector fields with formal degrees (W , ds), the result will be uniform in the following sense. When we apply the result to vector fields (W κ , dsκ ) which are Hörmander vector fields with formal degrees at the unit scale, uniformly in κ, the estimates proved will all hold uniformly in κ. This is an important point, because many of our later results will apply earlier results infinitely many times in their proofs, and it is important that the estimates are uniform over these applications. In particular, by proving a uniform result at the unit scale, rescaling techniques will allow us to prove a result at every scale, uniformly in the scale. For example, we have the next result at the unit scale, which we will later use to prove results at every scale (see the proof of Theorem 3.5.1 (b)).

3.3 Scaling and Hörmander vector fields

� 123

Lemma 3.2.4. Let (W , ds) be Hörmander vector fields with formal degrees at the unit scale. Then there exists a 0-unit-admissible constant η > 0 such that Bn (η) ⊆ BW (0) = B(W ,ds) (0, 1). Proof. Lemma 3.1.7 shows that such an η > 0 exists. By keeping track of the estimates in the proof of that lemma, it is easy to see that η can be taken to be a 0-unit-admissible constant.

3.3 Scaling and Hörmander vector fields ∞ Let M be a connected C ∞ manifold and let Vol be a strictly positive Cloc density on M. ∞ Let (W , ds) ⊂ Cloc (M; TM) × ℕ+ be Hörmander vector fields with formal degrees. Fix 𝒦 ⋐ M a compact set and Ω1 ⋐ M a relatively compact, open set, with 𝒦 ⋐ Ω1 ⋐ M. In what follows we write A ≲ B for A ≤ CB, where C ≥ 0 does not depend on x, y ∈ 𝒦 or δ > 0. C may depend on any of the other ingredients in the problem. We write A ≈ B for A ≲ B and B ≲ A. All of the estimates and constants below depend on the particular choice of (W , ds) and on the choice of 𝒦 and Ω1 . We state the main theorem of this section twice, beginning with a simpler version. We defer proofs of these results to Section 3.7.

Theorem 3.3.1. There exist δ0 = δ0 (𝒦, Ω1 ) ∈ (0, 1] and ξ3 = ξ3 (𝒦, Ω1 ) ∈ (0, 1] such that the following hold: (a) ∀x ∈ 𝒦, δ ∈ (0, δ0 ], B(W ,ds) (x, δ) ⊆ Ω1 and B(W ,ds) (x, δ) is open in M. (b) ∀x ∈ 𝒦, δ ∈ (0, δ0 ], Vol(B(W ,ds) (x, 2δ)) ≲ Vol(B(W ,ds) (x, δ)). ∀x ∈ 𝒦, δ > 0, Vol(B(W ,ds) (x, 2δ) ∧ 1 ≲ Vol(B(W ,ds) (x, δ)) ∧ 1. For x ∈ 𝒦, δ ∈ (0, 1], there exists a map Φx,δ : Bn (1) → M such that: (c) Φx,δ (0) = x. (d) Φx,δ is a smooth coordinate system, that is, Φx,δ (Bn (1)) ⊆ M is open and Φx,δ : ∞ Bn (1) → Φx,δ (Bn (1)) is a Cloc diffeomorphism. n (e) B(W ,ds) (x, ξ3 δ) ⊆ Φx,δ (B (1)) ⊆ B(W ,ds) (x, δ). For x ∈ 𝒦, δ ∈ (0, 1], set Wjx,δ := Φ∗x,δ δdsj Wj . Let (W x,δ , ds) = {(W1x,δ , ds1 ), . . . , (Wrx,δ , dsr )} ⊂ C ∞ (Bn (1); TBn (1)) × ℕ+ . (f) (W x,δ , ds) are Hörmander vector fields with formal degrees at the unit scale, uniformly for x ∈ 𝒦 and δ ∈ (0, 1]; see Definition 3.2.2. Remark 3.3.2. The most important aspect of Theorem 3.3.1 is (f); this sees the maps Φx,δ as scaling maps which take the case where δ is small and changes coordinates to turn it into the case δ = 1. Thus, once we prove a result for Hörmander vector fields with formal degrees at the unit scale, we obtain a result at every small scale by pushing the result

124 � 3 Vector fields and Carnot–Carathéodory geometry forward via Φx,δ . Another important aspect is (b), which is the main inequality needed to show that the balls B(W ,ds) (x, δ), when paired with Vol, locally give M the structure of a space of homogeneous type in the sense of Coifman and Weiss [52]. Example 3.3.3. To see the maps Φx,δ as scaling maps, it is useful to consider some simple special cases. – When M = ℝn and (W , ds) = {(𝜕x1 , 1), . . . , (𝜕xn , 1)}, we may take Φx,δ (t1 , . . . , tn ) = (x1 + δt1 , . . . , xn + δtn ). In this case we have the simple identity, Φ∗x,δ δ𝜕xj = 𝜕tj . Thus, in the “elliptic setting,” the maps Φx,δ are nothing more than the usual dilation maps. – With M = ℝ2 and (W , ds) = {(𝜕x1 , 1), (x1 𝜕x2 , 1)}, the map Φx,δ depends in a non-trivial way on the base point x. Indeed, when x = 0, we may take Φ0,δ (t1 , t2 ) = (δt1 , δ2 t2 ). Then we have Φ∗0,δ δ𝜕x1 = 𝜕t1 and Φ∗0,δ δx1 𝜕x2 = t1 𝜕t2 . However, when x1 = 1 and δ ≪ 1, we instead take Φ(1,0),δ (t1 , t2 ) = (δt1 , δt2 ). The map Φx,δ from Theorem 3.3.1 correctly interpolates between these two choices in a non-trivial way which depends on both δ and x1 . We require more detailed information than Theorem 3.3.1 gives, and for this we introduce some new notation. ∞ ∞ Definition 3.3.4. Let 𝒮 ⊆ Cloc (M; TM) × ℕ+ . We let Gen(𝒮 ) ⊆ Cloc (M; TM) × ℕ+ be the ∞ smallest subset of Cloc (M; TM) × ℕ+ such that: – ∀(X1 , d1 ), (X2 , d2 ) ∈ Gen(𝒮 ), ([X1 , X2 ], d1 + d2 ) ∈ Gen(𝒮 ). – 𝒮 ⊆ Gen(𝒮 ).

For the remainder of the section, fix Ω2 ⋐ M, a relatively compact, open set, with 𝒦 ⋐ Ω1 ⋐ Ω2 ⋐ M. Lemma 3.3.5. There exists a finite set (X, d ) = {(X1 , d1 ), . . . , (Xq , dq )} ⊂ Gen((W , ds)) such that (W , ds) ⊆ (X, d ) and ∀1 ≤ j, k ≤ q, [Xj , Xk ] =

∑ dl ≤dj +dk

l cj,k Xl ,

(3.5)

l ∞ where cj,k ∈ Cloc (Ω2 ).

Proof. Since W = {W1 , . . . , Wr } satisfies Hörmander’s condition on M, by the compactness of Ω2 , there exists m ∈ ℕ+ so that W satisfies Hörmander’s condition of order m on Ω2 . Set (X, d ) := {(Z, e) ∈ Gen((W , d)) : e ≤ m max dsj }. 1≤j≤r

3.3 Scaling and Hörmander vector fields

� 125

Note that (X, d ) = {(X1 , d1 ), . . . , (Xq , dq )} is a finite set and X1 (x), . . . , Xq (x) span Tx Ω2 , ∀x ∈ Ω2 (by Hörmander’s condition of order m). We claim (X, d ) satisfies the conclusions of the lemma. It is clear that (W , ds) ⊆ (X, d ), so we only need to verify (3.5). For 1 ≤ j, k ≤ q, there are two cases. If dj + dk ≤ l m max1≤l≤r dsl , then ([Xj , Xk ], dj + dk ) ∈ (X, d ) and (3.5) follows trivially (with cj,k constants in {0, 1}). If dj + dk > m max1≤l≤r dsl ≥ maxl dl , then using the fact that X1 (x), . . . , Xq (x) span Tx Ω2 , ∀x ∈ Ω2 , we have q

l [Xj , Xk ] = ∑ cj,k Xl = l=1

∑

dl ≤dj +dk

l cj,k Xl ,

l ∞ cj,k ∈ Cloc (Ω2 ),

completing the proof. Example 3.3.6. When M = ℝ2 and (W , ds) = {(𝜕x , 1), (x𝜕y , 1)}, we may take (X, d ) = {(𝜕x , 1), (x𝜕y , 1), (𝜕y , 2)}. This encodes the fact that we consider 𝜕y = [𝜕x , x𝜕y ] as a differential operator of degree 2 in this setting. See Remark 1.1.5. Henceforth, we let (X, d ) = {(X1 , d1 ), . . . , (Xq , dq )} be as in the conclusion of Lemma 3.3.5 (there are many possible choices of (X, d ), but for what follows it is irrelevant which choice is used). For x ∈ 𝒦, δ > 0, set Λ(x, δ) :=

max

j1 ,...,jn ∈{1,...,q}

Vol(x)(δdj1 Xj1 (x), . . . , δdjn Xjn (x)).

Note that Λ(x, δ) > 0 since X1 (x), . . . , Xq (x) span Tx Ω2 and Vol is a strictly positive density. Theorem 3.3.7. There exist δ0 = δ0 (𝒦, Ω1 ) ∈ (0, 1] and ξ3 = ξ3 (𝒦, Ω1 ) ∈ (0, 1] such that the following hold: (a) ∀x ∈ 𝒦, δ ∈ (0, δ0 ], B(W ,ds) (x, δ) ⊆ B(X,d) (x, δ) ⊆ Ω1 , and B(W ,ds) (x, δ) and B(X,d) (x, δ) are open in M. (b) ∀x ∈ 𝒦, δ ∈ (0, 1], B(X,d) (x, ξ3 δ) ⊆ B(W ,ds) (x, δ) ⊆ B(X,d) (x, δ). (c) ∀x ∈ 𝒦, δ ∈ (0, δ0 ], Vol(B(W ,ds) (x, δ)) ≈ Vol(B(X,d) (x, δ)) ≈ Λ(x, δ). (d) ∀x ∈ 𝒦, δ > 0, Vol(B(W ,ds) (x, δ)) ∧ 1 ≈ Vol(B(X,d) (x, δ)) ∧ 1 ≈ Λ(x, δ) ∧ 1. (e) ∀x ∈ 𝒦, δ ∈ (0, δ0 ], Vol(B(W ,ds) (x, 2δ)) ≲ Vol(B(W ,ds) (x, δ)) and Vol(B(X,d) (x, 2δ)) ≲ Vol(B(X,d) (x, δ)). (f) ∀x ∈ 𝒦, δ > 0, Vol(B(W ,ds) (x, 2δ))∧1 ≲ Vol(B(W ,ds) (x, δ))∧1 and Vol(B(X,d) (x, 2δ))∧1 ≲ Vol(B(X,d) (x, δ)) ∧ 1. For all x ∈ 𝒦, δ ∈ (0, 1], there exists Φx,δ : Bn (1) → B(X,d) (x, δ) ∩ Ω1 such that: (g) Φx,δ (0) = x. (h) Φx,δ is a smooth coordinate system, that is, Φx,δ (Bn (1)) ⊆ M is open and Φx,δ : ∞ Bn (1) → Φx,δ (Bn (1)) is a Cloc diffeomorphism. (i) ∀x ∈ 𝒦, δ ∈ (0, 1], B(W ,ds) (x, ξ3 δ) ⊆ B(X,d) (x, ξ3 δ) ⊆ Φx,δ (Bn (1/2)) ⊆ Φx,δ (Bn (1)) ⊆ B(X,d) (x, δ).

126 � 3 Vector fields and Carnot–Carathéodory geometry Set Wjx,δ := Φ∗x,δ δdsj Wj and Xjx,δ := Φ∗x,δ δdj Xj . Let (W x,δ , ds) = {(W1x,δ , ds1 ), . . . , (Wrx,δ , dsr )}, W x,δ = {W1x,δ , . . . , Wrx,δ }, and X x,δ := {X1x,δ , . . . , Xqx,δ }.

(j) Wjx,δ and Xkx,δ are C ∞ , uniformly for x ∈ 𝒦 and δ ∈ (0, 1] in the sense that ∀L ∈ ℕ, 1 ≤ j ≤ r, 1 ≤ k ≤ q, 󵄩 󵄩 sup 󵄩󵄩󵄩Wjx,δ 󵄩󵄩󵄩C L (Bn (1);ℝn ) < ∞,

x∈𝒦 δ∈(0,1]

󵄩 󵄩 sup 󵄩󵄩󵄩Xkx,δ 󵄩󵄩󵄩C L (Bn (1);ℝn ) < ∞.

x∈𝒦 δ∈(0,1]

(k) X1x,δ (u), . . . , Xqx,δ (u) span Tu Bn (1) uniformly in x ∈ 𝒦, δ ∈ (0, 1], u ∈ Bn (1) in the sense that inf

󵄨 󵄨 inf 󵄨󵄨det(Xjx,δ (u)| ⋅ ⋅ ⋅ |Xjx,δ (u))󵄨󵄨󵄨 > 0. 1 n

max

x∈𝒦 j1 ,...jn ∈{1,...,q} u∈Bn (1)󵄨 δ∈(0,1]

(l) (W x,δ , ds) are Hörmander vector fields with formal degrees at the unit scale, uniformly for x ∈ 𝒦 and δ ∈ (0, 1]. (m) ∀x ∈ 𝒦, δ ∈ (0, 1], 󵄩 󵄩 ‖f ‖C L (Bn (1)) ≈ ∑ 󵄩󵄩󵄩(X x,δ )α f 󵄩󵄩󵄩C(Bn (1)) ,

∀f ∈ C(Bn (1)),

|α|≤L

where the implicit constants depend on L ∈ ℕ, but not on x, δ, or f . Here, we are using ordered multi-index notation; see Definition 1.1.4. For x ∈ 𝒦, δ ∈ (0, 1], define hx,δ ∈ C ∞ (Bn (1)) by Φ∗x,δ Vol = Λ(x, δ)hx,δ σLeb . (n) ∀x ∈ 𝒦, δ ∈ (0, 1], u ∈ Bn (1), hx,δ (u) ≈ 1 (with implicit constants independent of u, x, δ), and ∀L ∈ ℕ, sup ‖hx,δ ‖C L (Bn (1)) < ∞.

x∈𝒦 δ∈(0,1]

Corollary 3.3.8. For each x ∈ 𝒦, δ ∈ (0, 1], there exists ϕx,δ ∈ C0∞ (Ω1 ) such that (i) supp(ϕx,δ ) ⊆ B(W ,ds) (x, δ) ∩ Ω1 . (ii) ϕx,δ ≡ 1 on a neighborhood of the closure of B(W ,ds) (x, ξ32 δ), where ξ3 > 0 is as in Theorem 3.3.7. (iii) For every N ∈ ℕ, sup ∑ ‖(δdsW )α ϕx,δ ‖C(M) < ∞.

x,∈𝒦 |α|≤N δ∈(0,1]

Corollary 3.3.9. Let Ω ⋐ M. Then there exist C = C(Ω) ≥ 0, δ1 = δ1 (Ω) > 0, and Q1 = Q1 (Ω), Q2 = Q2 (Ω), with n ≤ Q1 ≤ Q2 , such that the following hold:

3.3 Scaling and Hörmander vector fields

� 127

(i) ∀x ∈ Ω, Vol(B(W ,ds) (x, δ1 )) ≈ Vol(B(X,d) (x, δ1 )) ≈ 1. (ii) ∀x ∈ Ω, δ ∈ (0, δ1 ], ξ ∈ (0, 1], ξ −Q1 Vol(B(W ,ds) (x, ξδ)) ≲ Vol(B(W ,ds) (x, δ))

≲ ξ −Q2 Vol(B(W ,ds) (x, ξδ)),

and the same estimates hold with (W , ds) replaced by (X, d ). (iii) ∀x ∈ Ω, δ > 0, ξ ∈ (0, 1], (ξ −Q1 Vol(B(W ,ds) (x, ξδ))) ∧ 1 ≲ Vol(B(W ,ds) (x, δ)) ∧ 1

≲ (ξ −Q2 Vol(B(W ,ds) (x, ξδ))) ∧ 1,

and the same estimates hold with (W , ds) replaced by (X, d ). Remark 3.3.10. If (W , ds) ⊂ C ∞ (Bn (1); ℝn )×ℕ+ are Hörmander vector fields with formal degrees at the unit scale, as in Section 3.2, and if one fixes sets 𝒦 ⋐ Ω1 ⋐ Bn (1), then all of the constants in this section can be taken to be unit-admissible constants in the sense of Definition 3.2.1. So, for example, if we write A ≲ B, this can be taken to mean A ≤ CB, where C is a unit-admissible constant (also depending on the choice of 𝒦 and Ω1 ). 3.3.1 Scaling and maximal subellipticity We are prepared to describe one of the main ideas in this text: maximal subellipticity is quantitatively preserved under the scaling maps Φx,δ from Theorem 3.3.7. Let P=

∑

degds (α)≤κ

aα (x)W α ,

∞ aα ∈ Cloc (M; 𝕄D1 ×D2 (ℂ)),

where κ ∈ ℕ+ is such that dsj divides κ, ∀j. Suppose that P is maximally subelliptic of degree κ with respect to (W , ds) on M. We take the same setting as in Theorem 3.3.7. In particular, since Ω1 ⋐ M is relatively compact and P is maximally subelliptic, there exists A ≥ 0 such that, with nj := κ/dsj ∈ ℕ+ , r

󵄩 n 󵄩 󵄩 󵄩 󵄩 󵄩 ∑󵄩󵄩󵄩Wj j f 󵄩󵄩󵄩L2 (M,Vol;ℂD2 ) ≤ A(󵄩󵄩󵄩P f 󵄩󵄩󵄩L2 (M,Vol;ℂD1 ) + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (M,Vol;ℂD2 ) ), j=1

∀f ∈ C0∞ (Ω1 ; ℂD2 ). For x ∈ 𝒦, δ ∈ (0, 1], set

(3.6)

128 � 3 Vector fields and Carnot–Carathéodory geometry

P

x,δ

:= Φ∗x,δ δκ P (Φx,δ )∗ =

∑

degds (α)≤κ

x,δ α bx,δ ) , α (W

κ−degds (α) where bx,δ aα ∘ Φx,δ ∈ C ∞ (Bn (1); 𝕄D1 ×D2 (ℂ)). α =δ

Proposition 3.3.11. P x,δ is maximally subelliptic of degree κ with respect to (W x,δ , ds), uniformly for x ∈ 𝒦 and δ ∈ (0, 1] in the sense that: (i) We have r

n 󵄩 󵄩 ∑󵄩󵄩󵄩(Wjx,δ ) j f 󵄩󵄩󵄩L2 (Bn (1),h σ ;ℂD2 ) x,δ Leb j=1

󵄩 󵄩 ≤ A(󵄩󵄩󵄩P x,δ f 󵄩󵄩󵄩L2 (Bn (1),h

D x,δ σLeb ;ℂ 1 )

󵄩 󵄩 + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (Bn (1),h

D x,δ σLeb ;ℂ 2 )

),

∀f ∈ C0∞ (Bn (1); ℂD2 ), where this is the same constant A as in (3.6). (ii) For every L ∈ ℕ, we have sup

max ‖bα ‖C L (Bn (1);𝕄D1 ×D2 ) < ∞.

x∈𝒦 degds (α)≤κ δ∈(0,1]

Remark 3.3.12. Proposition 3.3.11 shows that P x,δ is maximally subelliptic of degree κ with respect to (W x,δ , ds) on Bn (1), in a way which is uniform in x ∈ 𝒦 and δ ∈ (0, 1]. Theorem 3.3.7 (l) shows that (W x,δ , ds) are Hörmander vector fields at the unit scale, uniformly for x ∈ 𝒦 and δ ∈ (0, 1], and Theorem 3.3.7 (n) shows that hx,δ is smooth and bounded away from 0, uniformly for x ∈ 𝒦 and δ ∈ (0, 1]. Thus, once we prove a result for maximally subelliptic operators, that result will hold for P x,δ , uniformly for x ∈ 𝒦 and δ ∈ (0, 1]. By then pushing the result forward via Φx,δ , we will obtain results for P on Φx,δ (Bn (1)), which hold uniformly for all x ∈ 𝒦 and δ ∈ (0, 1]. It follows from Theorem 3.3.7 (i) that Φx,δ (Bn (1)) is comparable to B(W ,ds) (x, δ). Thus, by proving a result about maximally subelliptic operators at the unit scale, we will be able to conclude results about maximally subelliptic operators at every small Carnot–Carathéodory scale, δ ∈ (0, 1], uniformly in the scale. Proof of Proposition 3.3.11. (i): Multiplying (3.6) by δκ , we see that ∀f ∈ C0∞ (Ω1 ; ℂD2 ), r

n 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ∑󵄩󵄩󵄩(δdsj Wj ) j f 󵄩󵄩󵄩L2 (M,Vol;ℂD2 ) ≤ A(󵄩󵄩󵄩δκ P f 󵄩󵄩󵄩L2 (M,Vol;ℂD1 ) + δκ 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (M,Vol;ℂD2 ) ) j=1

󵄩 󵄩 󵄩 󵄩 ≤ A(󵄩󵄩󵄩δκ P f 󵄩󵄩󵄩L2 (M,Vol;ℂD1 ) + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (M,Vol;ℂD2 ) ).

(3.7)

Let g ∈ C0∞ (Bn (1); ℂD2 ). Theorem 3.3.7 (h) shows f := g∘Φx,δ ∈ C0∞ (Ω1 ; ℂD2 ). Applying (3.7) with this choice of g, using Wjx,δ = Φx,δ δdsj Wj , and changing variables, we see that r

n 󵄩 󵄩 ∑󵄩󵄩󵄩(Wjx,δ ) j g 󵄩󵄩󵄩L2 (Bn (1),Φ∗ j=1

x,δ

Vol)

󵄩 󵄩 ≤ A(󵄩󵄩󵄩P x,δ g 󵄩󵄩󵄩L2 (Bn (1),Φ∗

x,δ

Vol)

󵄩 󵄩 + 󵄩󵄩󵄩g 󵄩󵄩󵄩L2 (Bn (1),Φ∗

x,δ

Vol) ).

(3.8)

3.4 Finitely generated setting

� 129

1

Finally, using that Φ∗x,δ Vol = Λ(x, δ)hx,δ σLeb , multiplying both sides of (3.8) by Λ(x, δ)− 2 gives (i). (ii): Using Theorem 3.3.7 (m), we have, for each α with degds(α) ≤ κ and each x ∈ 𝒦, δ ∈ (0, 1], 󵄩󵄩 x,δ β x,δ 󵄩󵄩 ‖bx,δ α ‖C L (Bn (1)) ≈ ∑ 󵄩 󵄩(X ) bα 󵄩󵄩C(Bn (1)) |β|≤L

β 󵄩 󵄩 󵄩 󵄩 = ∑ δκ−degds (β) 󵄩󵄩󵄩(δd X) aα 󵄩󵄩󵄩C(Φ (Bn (1))) ≤ ∑ 󵄩󵄩󵄩X β aα 󵄩󵄩󵄩C(Ω ) . x,δ 1 |β|≤L

(3.9)

|β|≤L

Since Ω1 is relatively compact and aα and X1 , . . . , Xq are smooth, we have 󵄩 󵄩 ∑ 󵄩󵄩󵄩X β aα 󵄩󵄩󵄩C(Ω ) < ∞. 1

|β|≤L

We conclude the right-hand side of (3.9) is finite and does not depend on x ∈ 𝒦 or δ ∈ (0, 1]. Item (ii) follows.

3.4 Finitely generated setting As we will see in later sections, the assumption of Hörmander’s condition is not essential to obtain many of our results concerning Carnot–Carathéodory geometry and associated scaling maps. This is important when we study two different Carnot–Carathéodory geometries simultaneously, as Hörmander’s condition will not be quantitatively preserved under the adapted rescaling (though it will be qualitatively preserved). The key requirement is the conclusion of Lemma 3.3.5. In this section, we investigate this condition. For later applications, we fix ν ∈ ℕ+ and associate to the vector fields multi-parameter formal degrees which are elements of ℕν \{0}. Throughout this section, we let M be a connected C ∞ manifold. ∞ Definition 3.4.1. Let 𝒮 ⊆ Cloc (M; TM) × (ℕν \ {0}). For Ω ⊆ M open, we say 𝒮 is finitely generated on Ω if there is a finite subset ℱ ⊆ 𝒮 such that ∀(X0 , d 0⃗ ) ∈ 𝒮 ,

X0 =

∑ (X1 ,d ⃗1 )∈ℱ d ⃗ 1 ≤d ⃗ 0

c(X ,d ⃗ ) X1 , 1

1

∞ c(X ,d ⃗ ) ∈ Cloc (Ω), 1

1

(3.10)

where for the vectors d 0⃗ and d 1⃗ we have written d 0⃗ ≤ d 1⃗ to mean the inequality holds μ μ for each component: d 0⃗ ≤ d 1⃗ , ∀μ ∈ {1, . . . , ν}. If we wish to make the choice of ℱ explicit, we say 𝒮 is finitely generated by ℱ on Ω. Remark 3.4.2. In Definition 3.4.1, and in the rest of this text, we have drawn an ⃗over d ⃗ to remind the reader that d ⃗ is a vector.

130 � 3 Vector fields and Carnot–Carathéodory geometry ∞ Definition 3.4.3. Let 𝒮 ⊆ Cloc (M; TM) × (ℕν \ {0}). For Ω ⊆ M open, we say 𝒮 is linearly finitely generated on Ω if 𝒮 is finitely generated by ℱ on Ω, where for each (X, d) ∈ ℱ , d is non-zero in only one component. If we wish to make the choice of ℱ explicit, we say 𝒮 is linearly finitely generated by ℱ on Ω.

Remark 3.4.4. When ν = 1, finitely generated and linearly finitely generated are equivalent. ∞ Definition 3.4.5. Let 𝒮 ⊆ Cloc (M; TM) × (ℕν \ {0}). We say 𝒮 is locally finitely generated (respectively, locally linearly finitely generated) if for every relatively compact open set Ω ⋐ M, 𝒮 is finitely generated (respectively, linearly finitely generated) on Ω. If we wish to make M explicit, we say 𝒮 is locally finitely generated on M (respectively, locally linearly finitely generated on M). ∞ ∞ Lemma 3.4.6. Suppose 𝒮 ⊆ Cloc (M; TM) × (ℕν \ {0}) and let 𝒟 ⊆ Cloc (M; TM) be the ∞ Cloc (M)-module generated by {X : (X, d )⃗ ∈ 𝒮 }. If 𝒮 is locally finitely generated, then 𝒟 is ∞ locally finitely generated as a Cloc (M)-module.

Proof. This follows immediately from the definitions. Next, we adapt Definition 3.3.4 to the multi-parameter setting. ∞ ∞ Definition 3.4.7. Let 𝒮 ⊆ Cloc (M; TM) × (ℕν \ {0}). We let Gen(𝒮 ) ⊆ Cloc (M; TM) × (ℕν \ ∞ ν {0}) be the smallest subset of Cloc (M; TM) × (ℕ \ {0}) such that: – ∀(X1 , d 1⃗ ), (X2 , d 2⃗ ) ∈ Gen(𝒮 ), ([X1 , X2 ], d 1⃗ + d 2⃗ ) ∈ Gen(𝒮 ). – 𝒮 ⊆ Gen(𝒮 ). ∞ Example 3.4.8. In this text we are often given a set 𝒮 ⊆ Cloc (M; TM) × (ℕν \ {0}) and we will be interested in whether Gen(𝒮 ) is locally finitely generated. It is instructive to see an example where Gen(𝒮 ) is not locally finitely generated. On M = ℝ2 , let X1 = 𝜕x , 2 X2 = e−1/x 𝜕y , and 𝒮 = {(X1 , 1), (X2 , 1)}. Let Ω be any neighborhood of 0 ∈ ℝ2 . Then Gen(𝒮 ) is not finitely generated on Ω. Indeed, any commutator of the form

[X1 , [X1 , [X1 , . . . , [X1 , X2 ] ⋅ ⋅ ⋅]]] is not spanned (with bounded coefficients) on any neighborhood of 0 by commutators with fewer terms. This example is in contrast to Proposition 3.4.15, which shows that such a phenomenon cannot occur with real analytic vector fields. Example 3.4.9. An important example where Gen(𝒮 ) is finitely generated but not linearly finitely generated comes from the Heisenberg group. The Heisenberg group, ℍ1 , has a three-dimensional Lie algebra spanned by vector fields X, Y , T, where [X, Y ] = T and T is in the center. As a manifold, ℍ1 ≅ ℝ3 and the vector fields X, Y , T form a basis for the tangent space at every point. Set 𝒮 := {(X, (1, 0)), (Y , (0, 1))}. Then, for any nonempty open set Ω ⊂ ℍ1 , it is immediate to see that Gen(𝒮 ) is finitely generated on Ω,

3.4 Finitely generated setting

� 131

but not linearly finitely generated on Ω. A main point here is that the vector fields X and Y are far from commuting. This example is in contrast to Proposition 3.8.8, which shows that if the vector fields “weakly approximately commute,” then Gen(𝒮 ) is linearly finitely generated. ∞ ∞ Lemma 3.4.10. Let 𝒮 ⊆ Cloc (M; TM) × (ℕν \ {0}) and let 𝒟 be the Cloc (M)-module generated by {X : (X, d) ∈ Gen(𝒮 )}. Then 𝒟 is involutive.

Proof. This follows immediately from the definitions. ∞ Remark 3.4.11. Let 𝒮 ⊆ Cloc (M; TM) × (ℕν \ {0}). If Gen(𝒮 ) is locally finitely generated, then Lemmas 3.4.6 and 3.4.10 show that the Frobenius theorem (Theorem 3.1.15) applies ∞ with 𝒟 equal to the Cloc (M)-module generated by {X : (X, d )⃗ ∈ Gen(𝒮 )}. Thus, M is foliated into leaves, and the tangent space to the leaf passing through y ∈ M at y is equal to span{X(y) : (X, d )⃗ ∈ Gen(𝒮 )}. In other words, {X : (X, d )⃗ ∈ 𝒮 } satisfies Hörmander’s condition on each leaf.

Remark 3.4.12. Suppose Gen(𝒮 ) is finitely generated by (X, d )⃗ = {(X1 , d 1⃗ ), . . . , (Xq , d q⃗ )} ⊆ Gen(𝒮 ) on Ω. Then, since ([Xj , Xk ], d j⃗ + d k⃗ ) ∈ Gen(𝒮 ), we have [Xj , Xk ] =

∑

l cj,k Xl ,

l ∞ cj,k ∈ Cloc (Ω).

d ⃗ l ≤d ⃗ j +d ⃗ k

In light of Remark 3.4.12, Lemma 3.3.5 can be restated as follows: if (W , ds) ⊂ ∞ Cloc (M; TM) × ℕ+ are Hörmander vector fields with formal degrees, then Gen((W , ds)) is locally finitely generated. The next definition and result generalize this to the multiparameter setting. ∞ ∞ Definition 3.4.13. Let 𝒮1 ⊆ Cloc (M; TM) × (ℕν1 \ {0}) and 𝒮2 ⊆ Cloc (M; TM) × (ℕν2 \ {0}). We let

𝒮1 ⊠ 𝒮2 := {(X1 , (d1 , 0ν2 )) : (X1 , d1 ) ∈ 𝒮1 } ∪ {(X2 , (0ν1 , d2 )) : (X2 , d2 ) ∈ 𝒮2 } ∞ ⊆ Cloc (M; TM) × (ℕn1 +ν2 \ {0}).

Proposition 3.4.14. For μ ∈ {1, . . . , ν}, let μ

μ

∞ (W μ , dsμ ) = {(W1 , ds1 ), . . . , (Wrμμ , dsμrμ )} ⊂ Cloc (M; TM) × ℕ+

be Hörmander vector fields with formal degrees. Set (W , ds)⃗ = {(W1 , ds1⃗ ), . . . , (Wr , dsr⃗ )} := (W 1 , ds1 ) ⊠ (W 2 , ds2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (W ν , dsν ) μ

μ

∞ = {(Wj , dsj eμ ) : μ ∈ {1, . . . , ν}, 1 ≤ j ≤ qμ } ⊂ Cloc (M; TM) × (ℕν \ {0}),

132 � 3 Vector fields and Carnot–Carathéodory geometry ⃗ is locally finitely genwhere e1 , . . . , eν denotes the standard basis of ℝν . Then Gen((W , ds)) erated. μ

μ

Proof. Fix Ω ⋐ M. Since W μ = {W1 , . . . , Wrμ } satisfies Hörmander’s condition on M, for

each μ ∈ {1, . . . , ν}, by the compactness of Ω there exists m ∈ ℕ+ so that W μ satisfies Hörmander’s condition of order m on Ω, for all μ ∈ {1, . . . , ν}. Set μ

⃗ : |e|∞ ≤ m max max ds }. ℱ := {(Z, e)⃗ ∈ Gen((W , ds)) j 1≤μ≤ν 1≤j≤rμ

Note that ℱ is a finite set. Enumerate ℱ = {(X1 , d 1⃗ ), . . . , (Xq , d q⃗ )}. Here, we have used |e|∞ to denote the ℓ∞ norm of the vector e. Since (W μ , dsμ ) satisfies Hörmander’s condition of order m on Ω, we have ∀μ ∈ {1, . . . , ν}, x ∈ Ω, span{Xj (x) : dj is non-zero in only the μ component} = Tx Ω.

(3.11)

⃗ is finitely generated by ℱ on Ω. Indeed, take (X0 , d 0⃗ ) ∈ We claim Gen((W , ds)) μ ⃗ ⃗ Gen((W , ds)). If |d 0 |∞ ≤ m max1≤μ≤ν max1≤j≤rμ dsj , then (X0 , d 0⃗ ) ∈ ℱ and (3.10) holds μ μ trivially. Otherwise, there exists μ with d ⃗ > m max1≤μ≤ν max1≤j≤r ds . Using (3.11), we 0

may write

X0 =

μ

∑

d ⃗l is non-zero in only the μ component

μ

cl Xl ,

j

∞ cl ∈ Cloc (Ω).

μ

Since d l⃗ ≤ |d l⃗ |∞ < d 0⃗ , for 1 ≤ l ≤ q, (3.10) follows, completing the proof. ⃗ is locally Proposition 3.4.14 used Hörmander’s condition to prove that Gen((W , ds)) finitely generated. However, Hörmander’s condition is not always necessary for such a conclusion. A particularly compelling case is when the vector fields are real analytic. ω Proposition 3.4.15. Let M be a real analytic manifold, and let 𝒮 ⊂ Cloc (M; TM)×ℕν \{0} be a finite set of real analytic vector fields paired with ν-parameter formal degrees. Then Gen(𝒮 ) is locally finitely generated.

To prove Proposition 3.4.15 we use the next lemma. Lemma 3.4.16. Let 𝒮 be as in Proposition 3.4.15. Then, for every x ∈ M, there is a finite set ℱx ⊂ Gen(𝒮 ) such that for each (X0 , d 0⃗ ) ∈ Gen(𝒮 ), there is an open neighborhood V(X ,d ⃗ ),x of x with 0

0

X0 =

∑ (X1 ,d ⃗1 )∈ℱx d ⃗ 1 ≤d ⃗ 0

c

(X1 ,d ⃗1 ) X, x,(X0 ,d ⃗0 ) 1

c

(X1 ,d ⃗1 ) x,(X0 ,d ⃗0 )

∞ ∈ Cloc (V(X

⃗

0 ,d 0 ),x

).

3.4 Finitely generated setting

�

133

Proof of Proposition 3.4.15 given Lemma 3.4.16. Let ℱx be as in Lemma 3.4.16, let Mx := max{|d 1⃗ |∞ : (X1 , d 1⃗ ) ∈ ℱx ∪ 𝒮 }, and let ℱx′ := {(X1 , d 1⃗ ) : |d 1⃗ |∞ ≤ Mx }. Note that ℱx′ is a finite set and 𝒮 ∪ ℱx ⊆ ℱx′ . Let V(X ,d ⃗ ),x be as in Lemma 3.4.16 and set 0

0

Vx :=

⋂ (X1 ,d ⃗1 ),(X2 ,d ⃗2 )∈ℱx′

V([X ,X ],d ⃗ +d ⃗ ),x . 1

2

1

2

By the definition of Vx , we have, for (X1 , d 1⃗ ), (X2 , d 2⃗ ) ∈ ℱx′ , [X1 , X2 ] =

(X3 ,d ⃗3 ) X, (X1 ,d ⃗1 ),(X2 ,d ⃗2 ) 3

c

∑ (X3 ,d ⃗3 )∈ℱx′

c

(X3 ,d ⃗3 ) (X1 ,d ⃗1 ),(X2 ,d ⃗2 )

∈ C0∞ (Vx ).

(3.12)

d ⃗ 3 ≤d ⃗ 1 +d ⃗ 2

Let 𝒯x denote the set of all (X0 , d 0⃗ ) ∈ Gen(𝒮 ) such that X0 =

c

∑ (X1 ,d ⃗1 )∈ℱx′

(X1 ,d ⃗1 ) X, (X0 ,d ⃗0 ) 1

(X1 ,d ⃗1 ) (X0 ,d ⃗0 )

c

∞ ∈ Cloc (Vx ).

(3.13)

d ⃗ 1 ≤d ⃗ 0

We claim 𝒯x = Gen(𝒮 ). Indeed, (3.12) shows that 𝒮 ⊆ ℱx′ ⊆ 𝒯x . Also, if (X1 , d 1⃗ ), (X2 , d 2⃗ ) ∈ 𝒯x , it follows from (3.12) and (3.13) that ([X1 , X2 ], d 1⃗ + d 2⃗ ) ∈ 𝒯x . Thus, 𝒯x = Gen(𝒮 ). Fix Ω open with Ω ⋐ M. {Vx : x ∈ Ω} is an open cover for the compact set Ω. We extract a finite subcover Vx1 , . . . , VxN . Let M := max{Mxj : 1 ≤ j ≤ N} and set ℱ := {(X0 , d 0⃗ ) :∈ Gen(𝒮 ) : |d 0⃗ | ≤ M}. Note that 𝒮 ⊆ ℱx ⊆ ℱ , for 1 ≤ j ≤ N, and ℱ is a finite set. j

Since 𝒯xj = Gen(𝒮 ), a simple partition of unity argument together with the definition of 𝒯x (see (3.13)) shows that ∀(X0 , d 0⃗ ) ∈ Gen(𝒮 ), j

(X1 ,d ⃗1 ) X, (X0 ,d ⃗0 ) 1

X0 = ∑ c (X1 ,d ⃗1 ) d ⃗1 ≤d ⃗0

(X1 ,d ⃗1 ) (X0 ,d ⃗0 )

c

∞ ∈ Cloc (Ω).

This completes the proof. We turn to the proof of Lemma 3.4.16. Since it is a local result, it suffices to work on a neighborhood of 0 ∈ ℝn . We write f : ℝn0 → ℝm to denote that f is a germ of a function defined near 0 ∈ ℝn . Let n

𝒜n := {f : ℝ0 → ℝ | f is real analytic}, m

n

m

𝒜n := {f : ℝ0 → ℝ | f is real analytic}. n Note that 𝒜m n can be identified with the m-fold Cartesian product of 𝒜n and 𝒜n can be identified with the set of germs of real analytic vector fields defined near 0 ∈ ℝn .

Lemma 3.4.17. The ring 𝒜n is Noetherian.

134 � 3 Vector fields and Carnot–Carathéodory geometry Comments on the proof. This is a simple consequence of the Weierstrass preparation theorem; see page 148 of [252]. The proof in [252] is for the formal power series ring; however, as mentioned on page 130 of [252], the proof also works on the ring of convergent power series, i. e., the ring of power series with some positive radius of convergence. This is exactly the ring 𝒜n . Lemma 3.4.18. The module 𝒜m n is a Noetherian 𝒜n -module. Comments on the proof. Since 𝒜n is a Noetherian 𝒜n -module by Lemma 3.4.17, 𝒜m n is a finite direct sum of Noetherian 𝒜n -modules. It is therefore a Noetherian 𝒜n -module (see [98, Corollary 1.3]). Lemma 3.4.19. Let 𝒯 ⊆ 𝒜nn × ℕν . Then there exists a finite set ℱ ⊆ 𝒯 such that for every (X0 , d 0⃗ ) ∈ 𝒯 , X0 =

c

∑ (X1 ,d ⃗1 )∈ℱ d ⃗ 1 ≤d ⃗ 0

(X1 ,d ⃗1 ) X, (X0 ,d ⃗0 ) 1

(X1 ,d ⃗1 ) (X0 ,d ⃗0 )

c

∈ 𝒜n .

Proof. Define a map ι : 𝒜nn ×ℕν → 𝒜nn+ν by ι(X1 , d 1⃗ ) = t d 1 X1 (x), where t ∈ ℝν , x ∈ ℝn . Let M be the submodule of 𝒜nn+ν generated by ι𝒯 . M is finitely generated by Lemma 3.4.18. Let ℱ ⊆ 𝒯 be a finite set so that ιℱ generates M. We will show that ℱ satisfies the conclusions of the lemma. ⃗ Let (X0 , d 0⃗ ) ∈ 𝒯 . Since t d 0 X0 ∈ M, we have ⃗

t d 0 X0 (x) = ⃗

∑ (X1 ,d ⃗1 )∈ℱ

⃗ (X1 ,d ⃗1 ) (t, x)t d 1 X1 (x) (X0 ,d ⃗0 )

ĉ

on a neighborhood of (0, 0) ∈ ℝ × ℝn . Applying ⃗ (X ,d ⃗ ) 1 d ⃗0 󵄨󵄨 𝜕 󵄨 ĉ 1 1 (t, x)t d 1 d ⃗ ! t 󵄨t=0 (X ,d ⃗ ) 0

0

0

= 0 if d 1⃗ ≰ d 0⃗ , we have

X0 (x) =

∑ (X1 ,d ⃗1 )∈ℱ d ⃗ 1 ≤d ⃗ 0

[

1 d ⃗0 󵄨󵄨 𝜕 󵄨 d ⃗0 ! t 󵄨t=0

to both sides and using that

⃗ 1 d ⃗0 󵄨󵄨󵄨󵄨 (X ,d ⃗ ) 𝜕t 󵄨󵄨 ĉ 1 1⃗ (t, x)t d 1 ]X1 (x). (X , d ) 󵄨 d 0⃗ ! 󵄨t=0 0 0

The result follows. Proof of Lemma 3.4.16. Since the result is local, it suffices to prove the result when M is an open neighborhood of 0 ∈ ℝn and x = 0. Applying Lemma 3.4.19 to 𝒯 := Gen(𝒮 ), the result follows.

3.5 Multi-parameter Carnot–Carathéodory geometry

�

135

3.5 Multi-parameter Carnot–Carathéodory geometry Fix ν ∈ ℕ+ and let M be a connected C ∞ manifold of dimension n. Let ∞ (W , ds)⃗ = {(W1 , ds1⃗ ), . . . , (Wr , dsr⃗ )} ⊂ Cloc (M; TM) × (ℕν \ {0})

(3.14)

be a finite set. Our main assumption in this section, which we assume throughout, is the following. ⃗ is locally finitely generated on M. Main Assumption: Gen((W , ds)) Note that we do not assume that W1 , . . . , Wr satisfy Hörmander’s condition. However, Remark 3.4.11 shows that the Frobenius theorem (Theorem 3.1.15) applies to foliate M into leaves: through each point x, there is a unique, maximal, connected, injectively immersed submanifold Leafx 󳨅→ M such that W1 , . . . , Wr are tangent to Leafx and satisfy Hörmander’s condition on Leafx . Set N(x) := dim Leafx ; Lemma 3.1.23 shows x 󳨃→ N(x) is lower semi-continuous. Fix Ω ⋐ M, an open, relatively compact set. Pick a finite set (X, d )⃗ = {(X1 , d 1⃗ ), . . . , ⃗ so that Gen((W , ds)) ⃗ is finitely generated by (X, d )⃗ on Ω and (Xq , d q⃗ )} ⊂ Gen((W , ds)) ⃗ Note that, ∀x, y ∈ Ω, Ty Leafx = span{X1 (y), . . . , Xq (y)}. (W , ds)⃗ ⊆ (X, d ).

For δ ∈ [0, ∞)ν , we set δdsW = {δds1 W1 , . . . , δdsr Wr }, where δdsj is defined via stan⃗ ⃗ ⃗ dard multi-index notation. We similarly set δd X = {δd 1 X1 , . . . , δd q Xq }. For x ∈ M, we define ⃗

⃗

B(W ,ds)⃗ (x, δ) := Bδds⃗ W (x),

⃗

⃗

B(X,d)⃗ (x, δ) := Bδd ⃗ X (x),

where Bδds⃗ W (x) and Bδd ⃗ X (x) are defined via (1.12). Fix sets 𝒦 ⋐ Ω1 ⋐ Ω with 𝒦 compact and Ω1 open and relatively compact in Ω. In what follows, we write A ≲ B for A ≤ CB, where C ≥ 0 does not depend on x ∈ 𝒦 or δ ∈ (0, 1]ν . Theorem 3.5.1. There exists δ0 = δ0 (𝒦, Ω1 ) ∈ (0, 1] and ξ3 = ξ3 (𝒦, Ω1 ) ∈ (0, 1] such that the following hold: (a) ∀x ∈ 𝒦, δ ∈ (0, δ0 ]ν \ {0}, we have B(W ,ds)⃗ (x, δ) ⊆ B(X,d)⃗ (x, δ) ⊆ Ω1 ∩ Leafx , and B(W ,ds)⃗ (x, δ) and B(X,d)⃗ (x, Ω1 ) are open in Leafx . (b) ∀x ∈ 𝒦, δ ∈ (0, 1]ν , we have B(X,d)⃗ (x, ξ3 δ) ⊆ B(W ,ds)⃗ (x, δ) ⊆ B(X,d)⃗ (x, δ). For all x ∈ 𝒦, δ ∈ (0, 1]ν , there exists Φx,δ : BN(x) (1) → B(X,d)⃗ (x, δ) ∩ Ω1 such that: (c) Φx,δ (0) = 0. (d) Φx,δ is a smooth coordinate system on Leafx . (e) B(W ,ds)⃗ (x, ξ3 δ) ⊆ B(X,d)⃗ (x, ξ3 δ) ⊆ Φx,δ (BN(x) (1/2)) ⊆ Φx,δ (BN(x) (1)) ⊆ B(X,d)⃗ (x, δ)∩Ω1 ⊆ Leafx ∩ Ω1 . (f) ∀δ ∈ (0, 1]ν and m ∈ ℕ, the map (x, u) 󳨃→ Φx,δ (u) taking {x ∈ 𝒦 : N(x) = m}×Bm (1) → M is Borel measurable.

136 � 3 Vector fields and Carnot–Carathéodory geometry Set Wjx,δ := Φ∗x,δ δdsj Wj and Xjx,δ := Φ∗x,δ δd j Xj . Let ⃗

⃗

W x,δ = {W1x,δ , . . . , Wrx,δ },

X x,δ := {X1x,δ , . . . , Xqx,δ },

(W x,δ , |ds|⃗ 1 ) := {(W1x,δ , |ds1⃗ |1 ), . . . , (Wrx,δ , |dsr⃗ |1 )},

where |dsj⃗ |1 denotes the ℓ1 norm of the vector dsj⃗ . (g) Wjx,δ and Xkx,δ are C ∞ , uniformly for x ∈ 𝒦 and δ ∈ (0, 1]ν in the sense that ∀L ∈ ℕ, 1 ≤ j ≤ r, 1 ≤ k ≤ q, 󵄩 󵄩 sup 󵄩󵄩󵄩Wjx,δ 󵄩󵄩󵄩C L (BN(x) (1);ℝN(x) ) < ∞,

x∈𝒦 δ∈(0,1]ν

󵄩 󵄩 sup 󵄩󵄩󵄩Xkx,δ 󵄩󵄩󵄩C L (BN(x) (1);ℝN(x) ) < ∞.

x∈𝒦 δ∈(0,1]ν

(h) X1x,δ (u), . . . , Xqx,δ (u) span Tu BN(x) (1), uniformly for x ∈ 𝒦, δ ∈ (0, 1]ν , and u ∈ BN(x) (1) in the sense that inf

max

inf

󵄨󵄨 󵄨 x,δ x,δ 󵄨det(Xj1 (u)| ⋅ ⋅ ⋅ |XjN(x) (u))󵄨󵄨󵄨 > 0,

x∈𝒦 j1 ,...,jN(x) ∈{1,...,q} u∈BN(x) (1)󵄨 δ∈(0,1]ν

where each Xjx,δ (u) is treated as a column vector in ℝN(x) . (i) (W x,δ , |ds|⃗ 1 ) are Hörmander vector fields with formal degrees at the unit scale, uniformly for x ∈ 𝒦 and δ ∈ (0, 1]ν . (j) For all L ∈ ℕ, α 󵄩 󵄩 ‖f ‖C L (BN(x) (1)) ≈ ∑ 󵄩󵄩󵄩(X x,δ ) f 󵄩󵄩󵄩C(BN(x) (1)) ,

∀x ∈ 𝒦, δ ∈ (0, 1]ν ,

|α|≤L

where the implicit constants depend on L, but not on f , x, or δ. We defer the proof of Theorem 3.5.1 to Section 3.7. Remark 3.5.2. The map Φx,δ in Theorem 3.5.1 is a coordinate chart on a neighborhood of x in Leafx . Theorem 3.5.1 shows that this coordinate chart has good quantitative estimates, uniformly for x ∈ 𝒦 and δ ∈ (0, 1]ν . In particular, these quantitative estimates remain uniform if x approaches a singular point. The main aspects of this uniformity are Theorem 3.5.1 (g), (h), and (i). Remark 3.5.3. When working with multi-parameter balls, it is often more convenient to write δ ∈ [0, ∞)ν as δ = 2−j , where j ∈ (−∞, ∞]ν . This is defined by (δ1 , . . . , δν ) = (2−j1 , . . . , 2−jν ), where 2−∞ = 0. For j, k ∈ (−∞, ∞]ν we write j ≤ k to mean jμ ≤ kμ for μ ∈ {1, . . . , ν} and we write j ∧ k = (j1 ∧ k1 , . . . , jν ∧ kν ) ∈ (−∞, ∞]ν .

3.5 Multi-parameter Carnot–Carathéodory geometry

� 137

3.5.1 An important special case: Hörmander’s condition In this section, we describe an important special case of Theorem 3.5.1. As above, we let M be a connected C ∞ manifold of dimension n. We let Vol be a smooth, strictly positive density on M. Fix ν ∈ ℕ+ and for each μ ∈ {1, . . . , ν}, let μ

μ

∞ (W μ , dsμ ) = {(W1 , ds1 ), . . . , (Wrμμ , dsμrμ )} ⊂ Cloc (M; TM) × ℕ+

be Hörmander vector fields with single-parameter formal degrees. We define (W , ds)⃗ = {(W1 , ds1⃗ ), . . . , (Wr , dsr⃗ )} := (W 1 , ds1 ) ⊠ (W 2 , ds2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (W ν , dsν ) μ

μ

∞ = {(Wj , dsj eμ ) : μ ∈ {1, . . . , ν}, 1 ≤ j ≤ rμ } ⊂ Cloc (M; TM) × (ℕν \ {0}),

⃗ is where e1 , . . . , eν denotes the standard basis of ℝν . By Proposition 3.4.14, Gen((W , ds)) locally finitely generated on M. Thus, Theorem 3.5.1 applies in this setting. In this case, N(x) = n and Leafx = M, ∀x ∈ M. We take 𝒦 ⋐ Ω1 ⋐ Ω ⋐ M and (X, d )⃗ as in Theorem 3.5.1. For x ∈ 𝒦, δ ∈ (0, 1]ν , set Λ(x, δ) :=

max

Vol(x)(δd j1 Xj1 (x), . . . , δd jn Xjn (x)). ⃗

j1 ,...,jn ∈{1,...,q}

⃗

(3.15)

Note that Λ(x, δ) > 0, ∀x ∈ 𝒦 and δ ∈ (0, 1]ν . We have the following theorem. Theorem 3.5.4. Theorem 3.5.1 holds with N(x) = n and Leafx = M, ∀x ∈ M. Moreover, we also have: (k) ∀x ∈ 𝒦, δ ∈ (0, δ0 ]ν , we have Vol(B(W ,ds)⃗ (x, δ)) ≈ Vol(B(X,d)⃗ (x, δ)) ≈ Λ(x, δ). (l) ∀x ∈ 𝒦, δ ∈ (0, ∞)ν , we have Vol(B(W ,ds)⃗ (x, δ)) ∧ 1 ≈ Vol(B(X,d)⃗ (x, δ)) ∧ 1 ≈ Λ(x, δ) ∧ 1. (m) ∀x ∈ 𝒦, δ ∈ (0, δ0 ]ν , we have Vol(B(W ,ds)⃗ (x, 2δ)) ≲ Vol(B(W ,ds)⃗ (x, δ)) and Vol(B(X,d)⃗ (x, 2δ)) ≲ Vol(B(X,d)⃗ (x, δ)). (n) ∀x ∈ 𝒦, δ ∈ (0, ∞)ν , we have Vol(B(W ,ds)⃗ (x, 2δ)) ∧ 1 ≲ Vol(B(W ,ds)⃗ (x, δ)) ∧ 1 and Vol(B(X,d)⃗ (x, 2δ)) ∧ 1 ≲ Vol(B(X,d)⃗ (x, δ)) ∧ 1. For x ∈ 𝒦 and δ ∈ (0, 1]ν , define hx,δ ∈ C ∞ (Bn (1)) by Φ∗x,δ Vol = Λ(x, δ)hx,δ σLeb . (o) ∀x ∈ 𝒦, δ ∈ (0, 1]ν , u ∈ Bn (1), hx,δ (u) ≈ 1, with implicit constants independent of x, δ, and u. (p) For all L ∈ ℕ, sup ‖hx,δ ‖C L (Bn (1)) < ∞.

x∈𝒦 δ∈(0,1]ν

We defer the proof of Theorem 3.5.4 to Section 3.7. Example 3.5.5. Consider the following very simple special case. We let M = ℝ, ν = 2, (W 1 , ds1 ) = {(W11 , ds11 )} := {(𝜕x , 1)}, and (W 2 , ds2 ) = {(W11 , ds21 )} := {(𝜕x , 1)}. Let δ = (δ1 , δ2 ) ∈

138 � 3 Vector fields and Carnot–Carathéodory geometry (0, 1]2 . This satisfies the conditions of Theorem 3.5.4. If δ1 ≥ δ2 , then the map Φx,δ in Theds1

orems 3.5.1 and 3.5.4 can be taken to be Φx,δ (t) = x + δ1 t. In this case, Φ∗x,δ δ1 1 W11 = 𝜕t and ds2

ds2

Φ∗x,δ δ2 1 W12 = (δ2 /δ1 )𝜕t . Thus, if δ2 ≪ δ1 , then while Φ∗x,δ δ2 1 W12 satisfies Hörmander’s condition (in fact, it spans the tangent space at every point), it does not do so uniformly in δ. Therefore, the conditions of Theorem 3.5.4 are not quantitatively invariant under rescaling. However, the conditions of Theorem 3.5.1 are quantitatively invariant under rescaling. Even though our main motivations come from the setting of Theorem 3.5.4, because we will often need to rescale and prove quantitative results, it is often best to work in the more general setting of Theorem 3.5.1, since it is quantitatively preserved under rescalings.

3.5.2 Dropping parameters Theorems 3.5.1 and 3.5.4 consider only δ ∈ (0, ∞)ν . In fact, similar results hold for δ ∈ [0, ∞)ν \ {0}. This is because when some of the coordinates of δ are equal to 0, the balls B(W ,ds)⃗ (x, δ) can be viewed as lower-parameter balls where none of the coordinates of δ are equal to 0. This section is devoted to explaining this. Fix 0 ≠ E ⊆ {1, . . . , ν}. Then under the assumptions of either Theorem 3.5.1 or Theorem 3.5.4, the list of sets of vector fields with formal degrees (W μ , dsμ ),

μ ∈ E,

satisfies the same assumptions as (W μ , dsμ ),

μ ∈ {1, . . . , ν},

but with ν replaced by |E|. We let ∞ (W E , dsE⃗ ) ⊂ Cloc (M; TM) × ℕ|E|

be as in (3.14), but with (W 1 , ds1 ), . . . , (W ν , dsν ) replaced by (W μ , dsμ ), μ ∈ E. Moreover, suppose δ ∈ [0, ∞)ν \ {0}. Set E := {μ ∈ {1, . . . , ν} : δμ ≠ 0} and let δE := (δμ )μ∈E ∈ (0, ∞)|E| be the non-zero components of δ. Then, B(W E ,ds⃗ E ) (x, δE ) = B(W ,ds)⃗ (x, δ). Because of this, we can obtain versions of Theorems 3.5.1 and 3.5.4 for the balls B(W ,ds)⃗ (x, δ), when some of the coordinates of δ are 0, by applying those results to (W E , dsE⃗ ).

3.6 The quantitative coordinate system

� 139

3.6 The quantitative coordinate system All of the main scaling theorems of this chapter follow from a single, abstract theorem concerning the existence of a coordinate system with good quantitative estimates. This theorem is due to the author and Stovall [228], and we present it here. An important aspect of the theory in this section is that all assumptions and estimates remain unchanged ∞ when setting is transformed under an arbitrary Cloc diffeomorphism: every estimate is “coordinate-free.” This comes up in Remark 3.6.1; see Remarks 3.6.6 and A.2.3 for a further discussion of this invariance. ∞ Let M be a C ∞ manifold of dimension n and let X = {X1 , . . . , Xq } ⊂ Cloc (M; TM) be a finite set of smooth vector fields on M such that ∀x ∈ M, Tx M = span{X1 (x), . . . , Xq (x)}. Let Vol be a smooth, strictly positive density on M. For δ > 0 and x ∈ M, set BX (x, δ) := B{δX1 ,...,δXq } (x), where the latter ball is defined by (1.12). Remark 3.6.1. When we use the results of this section to prove results like Theorem 3.3.7, we apply the theory of this section with X1 , . . . , Xq replaced by δd1 X1 , . . . , δdq Xq , where X1 , . . . , Xq satisfy certain assumptions and δ is small. We require that the results we prove are uniform in δ (and other parameters) in such applications. However, since there is no parameter like δ in this section, we instead proceed by keeping careful track of what each estimate does depend on. This will allow us to see that the estimates do not depend on the relevant parameters (like δ) in our applications. Unfortunately, this makes the statements in this section somewhat technical. Fix x0 ∈ M and ξ > 0. We write q

l [Xj , Xk ] = ∑ cj,k Xl , l=1

l and we assume cj,k ∈ C(BX (x0 , ξ)). The ball BX (x0 , ξ) is open in M (see Lemma 3.1.7) and we may therefore treat BX (x0 , ξ) as a manifold.

Definition 3.6.2. Let Z be a one-dimensional real vector space. For z1 , z2 ∈ Z, z2 ≠ 0, we set z1 /z2 := λ(z1 )/λ(z2 ) ∈ ℝ, where λ : Z → ℝ is any non-zero linear map. It is easy to see that z1 /z2 does not depend on the choice of λ. Fix ζ ∈ (0, 1]. Reorder X1 , . . . , Xq so that 󵄨󵄨 Xj (x0 ) ∧ Xj (x0 ) ∧ ⋅ ⋅ ⋅ ∧ Xj (x0 ) 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 −1 2 n 󵄨󵄨 󵄨󵄨 ≤ ζ . j1 ,...,jn ∈{1,...,q}󵄨󵄨 X1 (x0 ) ∧ X2 (x2 ) ∧ ⋅ ⋅ ⋅ ∧ Xn (x0 ) 󵄨󵄨 max

(3.16)

This quotient is defined by Definition 3.6.2, where we have used the fact that ⋀n Tx0 M is a one-dimensional vector space, since Tx0 M is an n-dimensional vector space. Note that we may always reorder X1 , . . . , Xq in this way: it is always possible to pick X1 , . . . , Xn so that the left-hand side of (3.16) equals 1, though we will require the flexibility to take ζ < 1 in our applications. See Remark 3.6.9 for a further comment on this quantity.

140 � 3 Vector fields and Carnot–Carathéodory geometry ∞ For 1 ≤ j ≤ q, define fj ∈ Cloc (M) by LieXj Vol = fj Vol.

Definition 3.6.3. For x ∈ M, η > 0, and U ⊆ M, we say X satisfies 𝒞 (x, η, U) if for every a ∈ Bq (η) the expression ea1 X1 +⋅⋅⋅+aq Xq x exists in U. More precisely, consider the differential equation 𝜕r E(r) = a1 X1 (E(r)) + ⋅ ⋅ ⋅ + aq Xq (E(r)),

E(0) = x.

We assume that a solution to this differential equation exists up to r = 1, E : [0, 1] → U. We have E(r) = era1 X1 +⋅⋅⋅+raq Xq x. We require two more technical quantities related to X, though the reader may wish to skip these on a first reading: – Let η ∈ (0, 1] be such that {X1 , . . . , Xn } satisfies 𝒞 (x0 , η, M). – Let τ0 ∈ (0, 1] be such that for τ ∈ (0, τ0 ] the following holds: if z ∈ B{X1 ,...,Xn } (x0 , ξ) is such that {X1 , . . . , Xn } satisfies 𝒞 (z, τ, B{X1 ,...,Xn } (x0 , ξ)), t ∈ Bn (τ) is such that et1 X1 +⋅⋅⋅+tn Xn z = z, and X1 (z), . . . , Xn (z) are linearly independent, then t = 0. Such an η and τ0 always exist (by possibly shrinking ξ > 0) – see Proposition 3.6.7 and Remark 3.6.8. However, our quantitative estimates will depend on η and τ0 . Definition 3.6.4. For L ∈ ℕ, we say C is an L-admissible constant if C can be chosen to depend only on upper bounds for the following quantities, which we assume to be finite: l – ∑|α|≤L ‖X α cj,k ‖C(BX (x0 ,ξ)) , 1 ≤ j, k, l ≤ q, –

L, q, η−1 , τ0−1 , ζ −1 , and ξ −1 .

We say C is an L, Vol-admissible constant if C is an L-admissible constant which is also allowed to depend on upper bounds for the following quantity, which we assume to be finite: ∑ ‖X α fj ‖C(BX (x0 ,ξ)) ,

1 ≤ j ≤ q.

|α|≤L

We write A ≲L B for A ≤ CB, where C is a positive L-admissible constant and write A ≈L B for A ≲L B and B ≲L A. We similarly define A ≲L,Vol B and A ≈L,Vol B. Because {X1 , . . . , Xn } satisfies 𝒞 (x0 , η, M), by hypothesis, we may define the map, for t ∈ Bn (η), Φ(t) := et1 X1 +⋅⋅⋅+tn Xn x0 .

(3.17)

∞ Let η0 := min{η, ξ} so that Φ : Bn (η0 ) → BX (x0 , ξ). Note that Φ ∈ Cloc since X1 , . . . , Xq ∈ ∞ Cloc (M; TM).

3.6 The quantitative coordinate system

� 141

Theorem 3.6.5. There exist 1-admissible constants η1 , ξ2 ∈ (0, 1] such that: (a) Φ(Bn (η1 )) is an open subset of BX (x0 , ξ) and is therefore a submanifold of M. ∞ (b) Φ : Bn (η1 ) → Φ(Bn (η1 )) is a Cloc diffeomorphism. n n (c) BX (x0 , ξ2 ) ⊆ Φ(B (η1 /2)) ⊆ Φ(B (η1 )) ⊆ BX (x0 , ξ). Let Yj = Φ∗ Xj and write 𝜕t1 Y1 [Y ] [𝜕 ] [ 2] [ t] [ . ] = (I + A) [ . 2 ] , [.] [ . ] [.] [ . ] [Yn ] [𝜕tn ] where A ∈ C ∞ (Bn (η1 ); 𝕄n×n ). (d) A(0) = 0 and supt∈Bn (η1 ) ‖A‖𝕄n×n ≤ 21 . (e) We have the following quantitative regularity for Yj , 1 ≤ j ≤ q: ‖Yj ‖C L (Bn (η1 );ℝn ) ≲L 1. (f) We have the following equivalence of norms for f ∈ C(Bn (1)): ‖f ‖C L (Bn (η1 )) ≈(L−1)∨0 ∑ ‖Y α f ‖C(Bn (η1 )) . |α|≤L

Define h ∈ C ∞ (Bn (η1 )) by Φ∗ Vol = (

max

j1 ,...,jn ∈{1,...,q}

Vol(x0 )(Xj1 (x0 ), . . . , Xjn (x0 )))hσLeb .

(g) h(t) ≈0,Vol 1, ∀t ∈ Bn (η1 ), with implicit constant independent of t. (h) We have the following quantitative regularity of h: for all L ∈ ℕ, ‖h‖C L (Bn (η1 )) ≲L,Vol 1. Remark 3.6.6. Note that the estimates in Theorem 3.6.5 do not depend on the norms of the coefficients of the vector fields X1 , . . . , Xq in any coordinate system. In fact, the ∞ quantitative estimates are invariant under conjugating the setting by an arbitrary Cloc diffeomorphism. More is true: in Appendix A we prove a generalization of this result 2 which is invariant under arbitrary Cloc diffeomorphisms; thus, one does not even need ∞ to assume X1 , . . . , Xq are Cloc in order to achieve the above result. In particular, one can ∞ 1 conclude that Y1 , . . . , Yq are Cloc even in some cases when X1 , . . . , Xq are merely Cloc . See Appendix A, and in particular Remark A.2.3, for more on this. Proposition 3.6.7. η > 0 and τ0 > 0 satisfying the above assumptions always exist. Indeed, let 𝒦 ⋐ M be a compact set. Then:

142 � 3 Vector fields and Carnot–Carathéodory geometry – –

∃η ∈ (0, 1] such that ∀x0 ∈ 𝒦, X satisfies 𝒞 (x, η, M). ∃τ0 ∈ (0, 1] such that ∀θ ∈ ℝq with |θ| = 1, if x ∈ 𝒦 is such that θ1 X1 (x)+⋅ ⋅ ⋅+θq Xq (x) ≠ 0, then ∀r ∈ (0, τ0 ], erθ1 X1 +⋅⋅⋅+rθq Xq x ≠ x.

Remark 3.6.8. Proposition 3.6.7 immediately implies that η ∈ (0, 1], as in the above assumptions, exists. By possibly shrinking ξ > 0, B{X1 ,...,Xn } (x0 , ξ) is pre-compact in M; the existence of τ0 ∈ (0, 1] follows by applying Proposition 3.6.7 to the closure of this set. Remark 3.6.9. Let V be an n-dimensional real vector space, let v1 , . . . , vn ∈ V be a basis for V , and let w1 , . . . , wn ∈ V . Then, as described above, Definition 3.6.2 defines the quantity w1 ∧ w2 ∧ ⋅ ⋅ ⋅ ∧ wn ∈ ℝ. v1 ∧ v2 ∧ ⋅ ⋅ ⋅ ∧ vn

(3.18)

This quantity can be described in a much more concrete way. Indeed, pick any basis for V and let V1 , . . . , Vn and W1 , . . . , Wn be the column vectors given by writing v1 , . . . , vn and w1 , . . . , wn in this basis. Then w1 ∧ w2 ∧ ⋅ ⋅ ⋅ ∧ wn det(W1 |W2 | ⋅ ⋅ ⋅ |Wn ) = . v1 ∧ v2 ∧ ⋅ ⋅ ⋅ ∧ vn det(V1 |V2 | ⋅ ⋅ ⋅ |Vn ) The reason we have chosen the more abstract notation (3.18) is that it makes it clear that this quantity does not depend on the choice of basis or coordinate system. For proofs of the results in this section, see Appendix A.

3.7 Proofs of scaling results In this section, we use Theorem 3.6.5 to prove Theorems 3.3.1, 3.3.7, 3.5.1, and 3.5.4 and Corollaries 3.3.8 and 3.3.9. Theorem 3.3.1 is a special case of Theorem 3.3.7, which, in turn, is a special case of Theorem 3.5.4. We prove Theorems 3.5.1 and 3.5.4 simultaneously. Proof of Theorems 3.5.1 and 3.5.4. Most of this proof is in the more general setting of Theorem 3.5.1, though we will occasionally work in the special setting of Theorem 3.5.4. We will apply Theorem 3.6.5 with x0 replaced by x ∈ 𝒦, M replaced by Leafx ∩Ω, and

X = {X1 , . . . , Xq } replaced by δd X = {δd 1 X1 , . . . , δd q Xq } for δ ∈ (0, 1]ν . We need to show that the hypotheses of Theorem 3.6.5 hold uniformly for x ∈ 𝒦 and δ ∈ (0, 1]ν , with these choices. More precisely, we will show that L-admissible constants and L, Vol-admissible constants, as in Definition 3.6.4, can be chosen independent of x ∈ 𝒦 and δ ∈ (0, 1]ν . ⃗

⃗

⃗

3.7 Proofs of scaling results

143

�

By the Picard–Lindelöf theorem, we may take ξ ∈ (0, 1] so small that ∀x ∈ 𝒦, BX (x, ξ) ⊆ Ω1 . We will choose δ0 ∈ (0, ξ]. Since δ0 ≤ ξ, we have, ∀δ ∈ (0, δ0 ]ν , x ∈ 𝒦, B(X,d)⃗ (x, δ) = Bδd ⃗ X (x) ⊆ BX (x, ξ) ⊆ Ω1 .

(3.19)

⃗ This, comFor any x and δ, we have B(W ,ds)⃗ (x, δ) ⊆ B(X,d)⃗ (x, δ) (since (W , ds)⃗ ⊆ (X, d )). bined with (3.19), Lemma 3.1.7, and the immediate fact that BX (x, ξ) ⊆ Leafx , proves (a). For x ∈ 𝒦 and δ ∈ (0, 1]ν , pick j1 = j1 (x, δ), . . . , jN(x) = jN(x) (x, δ) ∈ {1, . . . , q} such that 󵄨󵄨 (δd ⃗k1 X (x)) ∧ ⋅ ⋅ ⋅ ∧ (δd ⃗kN(x) X 󵄨󵄨 󵄨󵄨 k1 kN(x) (x)) 󵄨󵄨 󵄨 󵄨󵄨 > 1/2. max 󵄨󵄨 d ⃗ 󵄨󵄨 d⃗ k1 ,...,kN(x) ∈{1,...,q}󵄨󵄨 󵄨 (δ j1 Xj1 (x)) ∧ ⋅ ⋅ ⋅ ∧ (δ jN(x) XjN(x) (x)) 󵄨󵄨

(3.20)

It is always possible to pick j1 (x, δ), . . . , jN(x) (x, δ) as in (3.20); indeed, we may pick them so that the left-hand side equals 1 (see also Remark 3.6.9). We require the choice of j1 , . . . , jn to be Borel measurable in x. Indeed, fix m ∈ ℕ, and note that the set {x ∈ 𝒦 : N(x) = m} is measurable by Lemma 3.1.23 (the function N(x) is lower semi-continuous). For each x ∈ {x ∈ 𝒦 : N(x) = m}, we pick j1 (x, δ), . . . , jm (x, δ) as in (3.20) such that the map x 󳨃→ jl (x, δ), {x ∈ 𝒦 : N(x) = m} → ℕ is Borel measurable (this is possible since the left-hand side of (3.20) is continuous in x for a fixed k1 , . . . , km , j1 , . . . , jm ). We will apply Theorem 3.6.5 with X1 , . . . , Xn replaced by δd j1 (x,δ) Xj1 (x,δ) , . . . , δ ⃗

d ⃗jN(x) (x,δ)

XjN(x) (x,δ) ;

(3.21)

thus, we are taking ζ = 1/2 in that theorem. By Proposition 3.6.7, with M replaced by Ω1 , we can find η = η(𝒦, Ω1 ) ∈ (0, 1] and τ0 = τ0 (𝒦, Ω1 ) ∈ (0, 1] satisfying the conclusions of that proposition when applied to X. It follows immediately that the same η and τ0 satisfy the hypotheses of Theorem 3.6.5 with x ∈ 𝒦 and X replaced by δd X for δ ∈ (0, 1]ν (and X1 , . . . , Xn replaced by (3.21)). ⃗ is finitely generated by (X, d )⃗ on Ω, Remark 3.4.12 shows that Since Gen((W , ds)) ⃗

[Xj , Xk ] =

l cj,k Xl ,

∑

l ∞ cj,k ∈ Cloc (Ω).

(3.22)

d ⃗l ≤d ⃗j +d ⃗k

Set, for δ ∈ (0, 1]ν , l,δ cj,k := {

l δd j +d k −d l cj,k ⃗

0

⃗

⃗

if d j⃗ + d k⃗ ≥ d l⃗ , otherwise.

l,δ ∞ Note that cj,k ∈ Cloc (Ω), uniformly for δ ∈ (0, 1]ν . Multiplying both sides of (3.22) by

δd j +d k , we have ⃗

⃗

144 � 3 Vector fields and Carnot–Carathéodory geometry q

l,δ d l [δd j Xj , δd k Xk ] = ∑ cj,k δ Xl . ⃗

⃗

⃗

l=1

l,δ l In our application of Theorem 3.6.5, cj,k plays the role of cj,k . When considering the spe∞ cial case of Theorem 3.5.4, let fj ∈ Cloc (M) be defined by LieXj Vol = fj Vol (such an fj ex-

ists, since Vol is a smooth, strictly positive density). Let fjδ := δd j fj , so that Lie ⃗

fjδ

fjδ

ν

Vol. Note that ∈ uniformly for δ ∈ (0, 1] . For every L ∈ ℕ, we have sup

max

∞ Cloc (M),

d⃗

Vol =

󵄩 ⃗ α l,δ 󵄩󵄩 󵄩 ⃗ α l,δ 󵄩󵄩 max ∑ 󵄩󵄩󵄩(δd X) cj,k ∑ 󵄩󵄩󵄩(δd X) cj,k 󵄩󵄩C(B ⃗ (x,ξ)) ≤ sup 󵄩󵄩C(Ω1 ) ν j,k,l∈{1,...,q} δd X

δ∈(0,1] j,k,l∈{1,...,q} |α|≤L x∈𝒦 ν

δ j Xj

δ∈(0,1]

≤ max

|α|≤L

󵄩 l 󵄩 󵄩󵄩 ∑ 󵄩󵄩󵄩X α cj,k 󵄩C(Ω1 ) < ∞.

d ⃗l ≤d ⃗j +d ⃗k |α|≤L

Similarly in the special case of Theorem 3.5.4, we have sup

󵄩 ⃗ α 󵄩 󵄩 󵄩 max ∑ 󵄩󵄩󵄩(δd X) fjδ 󵄩󵄩󵄩C(B ⃗ (x,ξ)) ≤ max ∑ 󵄩󵄩󵄩X α fj 󵄩󵄩󵄩C(Ω ) < ∞. 1 j∈{1,...,q} δd X

δ∈(0,1] j∈{1,...,q} |α|≤L x∈𝒦 ν

|α|≤L

The above shows that L-admissible constants for any L ∈ ℕ and L, Vol-admissible constants in the special case of Theorem 3.5.4 as in Definition 3.6.4 can be chosen independent of x ∈ 𝒦 and δ ∈ (0, 1]ν . Thus, Theorem 3.6.5 applies with x0 replaced by x ∈ 𝒦 and X replaced by δd X for δ ∈ (0, 1]ν , and any constants in that theorem can be chosen uniformly for x ∈ 𝒦 and δ ∈ (0, 1]ν . Let η1 > 0 be as in Theorem 3.6.5; as described above, η1 can be chosen independent ̃x,δ : BN(x) (η1 ) → B d ⃗ (x, ξ) ⊆ B ⃗ (x, δ) ∩ Ω1 be the map of x ∈ 𝒦 and δ ∈ (0, 1]ν . Let Φ (X,d) δ X Φ from Theorem 3.6.5, that is, ⃗

̃x,δ (t) = exp(t1 δd ⃗j1 (x,δ) Xj (x,δ) + ⋅ ⋅ ⋅ + tN(x) δd ⃗jN(x) (x,δ) Xj (x,δ) )x. Φ 1 N(x) Clearly, Φx,δ (0) = x. The Borel measurability of jl (x, δ) along with standard theorems from the field of ODEs shows that for each m ∈ ℕ and δ ∈ (0, 1]ν , the map ̃x,δ (t), (x, t) 󳨃→ Φ

{x ∈ 𝒦 : N(x) = m} × Bm (η1 ) → M

̃x,δ is a smooth coordinate system on is Borel measurable. By Theorem 3.6.5 (a) and (b), Φ Leafx . ̃x,δ (η1 t) : BN(x) (1) → B ⃗ (x, δ) ∩ Ω1 . The results (c), (d), and (f) now Set Φx,δ (t) := Φ (X,d) ̃x,δ described above. follow from the corresponding results for Φ Taking ξ2 ∈ (0, 1] as in Theorem 3.6.5 and using Theorem 3.6.5 (c) and the fact that ⃗ we have, for all x ∈ 𝒦, δ ∈ (0, 1]ν , and ξ ′ ∈ (0, ξ2 ], (W , ds)⃗ ⊆ (X, d ),

3.7 Proofs of scaling results

B(W ,ds)⃗ (x, ξ ′ δ) ⊆ B(X,d)⃗ (x, ξ ′ δ) ⊆ B(X,d)⃗ (x, ξ2 δ) ⊆ Bδd ⃗ X (x, ξ2 )

̃x,δ (BN(x) (η1 /2)) = Φx,δ (BN(x) (1/2)) ⊆ Φx,δ (BN(x) (1)) ⊆Φ

�

145

(3.23)

̃x,δ (BN(x) (η1 )) ⊆ B d ⃗ (x, ξ) ⊆ B ⃗ (x, δ). =Φ (X,d) δ X

We will later pick ξ3 ∈ (0, ξ2 ], and item (e) will follow from (3.23) with ξ ′ = ξ3 . Note that for any vector field Z, ̃∗ Φ∗x,δ Z = η−1 1 Φx,δ Z.

(3.24)

̃∗ Z into estimates about Since η1 ≈ 1, (3.24) allows us to easily turn estimates about Φ x,δ ∗ Φx,δ Z. Theorem 3.6.5 (e) combined with (3.24) shows that for 1 ≤ k ≤ q and L ∈ ℕ, ‖Xkx,δ ‖C L (BN(x) (1)) ≲ 1. Since each Wjx,δ is of the form Xkx,δ for some k, item (g) follows. Theorem 3.6.5 (d) shows that the matrix I + A(t) from that theorem is invertible with det(I + A(t)) ≳ 1, and therefore using (3.24) we have inf

󵄨󵄨 󵄨 x,δ x,δ 󵄨󵄨det(Xj (x,δ) (u)| ⋅ ⋅ ⋅ |Xj (x,δ) (u))󵄨󵄨󵄨

≈

inf

󵄨󵄨 ̃∗ δ 󵄨󵄨det(Φ x,δ

inf

󵄨󵄨 󵄨 󵄨󵄨det(I + A(u))󵄨󵄨󵄨 ≈ 1.

u∈BN(x) (1)

=

1

u∈BN(x) (1) u∈BN(x) (1)

N(x)

d ⃗j

1

̃∗ δd ⃗jN(x) Xj (u))󵄨󵄨󵄨 Xj1 (u)| ⋅ ⋅ ⋅ |Φ x,δ 󵄨 N(x)

Item (h) follows. ⃗ we see that δd ⃗j Xj is a fixed commutator of the Since each (Xk , d k⃗ ) ∈ Gen((W , ds)),

vector fields δdsW . Thus, each Xjx,δ is the same commutator of the vector fields W x,δ (since the pullback of the commutator is the commutator of the pullback). From this, (i) follows from (g) and (h). Item (j) follows immediately from (3.24) and Theorem 3.6.5 (f). We turn to (b). We always have B(W ,ds)⃗ (x, δ) ⊆ B(X,d)⃗ (x, δ) for any x and δ, since ⃗ By (3.23) with ξ ′ = ξ2 , we have, for x ∈ 𝒦 and δ ∈ (0, 1]ν , (W , ds)⃗ ⊆ (X, d ). ⃗

B(ξ δ)ds⃗ W (x) = B(W ,ds)⃗ (x, ξ2 δ) ⊆ Φx,δ (BN(x) (1/2)). 2

Pulling this equation back via Φx,δ , we have B

|ds|⃗ 1

ξ2

W x,δ

(0) = BΦ∗

x,δ

(ξ2 δ)ds W ⃗

(x) ⊆ BN(x) (1/2).

By (i) and the fact that ξ2 ≈ 1, we see that (ξ2 1 W x,δ , |ds|⃗ 1 ) are Hörmander vector fields with formal degrees at the unit scale, uniformly for x ∈ 𝒦 and δ ∈ (0, 1]ν . Lemma 3.2.4 shows that there exists η2 ∈ (0, 1/2] with η2 independent of x ∈ 𝒦 and δ ∈ (0, 1] such that |ds|⃗

146 � 3 Vector fields and Carnot–Carathéodory geometry BN(x) (η2 ) ⊆ B

|ds|⃗ 1

ξ2

W x,δ

(0) = BΦ∗

x,δ

(ξ2 δ)ds W ⃗

(x) ⊆ BN(x) (1/2).

Using (g), the Picard–Lindelöf theorem shows that there exists ξ3 ∈ (0, ξ2 ], independent of x ∈ 𝒦 and δ ∈ (0, 1]ν , so small that B(X x,δ ,|d|⃗ ) (0, ξ3 ) ⊆ BN(x) (η2 ) ⊆ B 1

|ds|⃗ 1

ξ2

W x,δ

(0).

Pushing this forward via Φx,δ , we have B(X,d)⃗ (x, ξ3 δ) = B(δd ⃗ X,|d|⃗ |) (x, ξ3 ) = Φx,δ (B(X x,δ ,|d|⃗ ) (0, ξ3 )) ⊆ Φx,δ (Bn (η2 )) ⊆ Φx,δ (B

1

1

|ds|⃗ ξ2 1 W x,δ

(0)) = B(W ,ds)⃗ (x, ξ2 δ) ⊆ B(W ,ds)⃗ (x, δ),

(3.25)

where in the last containment, we have used ξ2 ≤ 1. This completes the proof of (b). We have completed the proof of Theorem 3.5.1. For the remainder of this proof, we assume the additional assumptions of Theorem 3.5.4 – note that N(x) = n, ∀x in this case. Item (o) follows from Theorem 3.6.5 (g) and item (p) follows from Theorem 3.6.5 (h). Using (3.25) and the fact that B(W ,ds)⃗ (x, ξ3 δ) ⊆ B(X,d)⃗ (x, ξ3 δ), we have, for x ∈ 𝒦 and δ ∈ (0, 1]ν , Vol(B(W ,ds)⃗ (x, ξ3 δ)) ≤ Vol(B(X,d)⃗ (x, ξ3 δ)) ≤ Vol(Φx,δ (Bn (1))) = Λ(x, δ) ∫ hx,δ (t) dt ≈ Λ(x, δ) ≈ Λ(x, ξ3 δ),

(3.26)

Bn (1)

where in the second to last estimate we have used (o) and in the last estimate we have used ξ3 ≈ 1 and the formula for Λ. Similarly, again using (3.25), η2 ≈ 1, and (o), we have Λ(x, δ) ≈ Λ(x, δ) ∫ hx,δ (t) dt = Vol(Φx,δ (Bn (η2 ))) Bn (η2 )

(3.27)

≤ Vol(B(W ,ds)⃗ (x, δ)) ≤ Vol(B(X,d)⃗ (x, δ)). Combining (3.26) and (3.27), we have, for δ ∈ (0, ξ3 ]ν , Vol(B(W ,ds)⃗ (x, δ)) ≈ Vol(B(X,d)⃗ (x, δ)) ≈ Λ(x, δ).

(3.28)

Thus, by taking δ ∈ (0, ξ3 ], (k) follows. Next, we turn to (l). When δ ∈ (0, δ0 ]ν , (l) follows from (k). Suppose δ = (δ1 , . . . , δν ) ∈ (0, ∞)ν with δμ > δ0 for some μ. Then, using (k), we have Vol(B(W ,ds)⃗ (x, δ)) ≥ Vol(B(W ,ds)⃗ (x, δ0 eμ ))

≈ Λ(x, δ0 eμ ) ≥ inf min Λ(x ′ , δ0 eμ′ ) ≳ 1, ′ x∈𝒦 1≤μ ≤ν

3.7 Proofs of scaling results

� 147

where in the last estimate, we have used the compactness of 𝒦, the formula for Λ, the μ′

μ′

fact that δ0 ≈ 1, and the fact that X1 , . . . , Xq′ span the tangent space at every point of μ

Ω ⋑ 𝒦, ∀μ′ ∈ {1, . . . , ν}. We also have

Vol(B(X,d)⃗ (x, δ)) ≥ Vol(B(W ,ds)⃗ (x, δ)) ≳ 1. Finally, using the formula for Λ, we have Λ(x, δ) ≥ inf min Λ(x ′ , δ0 eμ′ ) ≳ 1. ′ x∈𝒦 1≤μ ≤ν

Combining the above three estimates shows that Vol(B(W ,ds)⃗ (x, δ)) ∧ 1 ≈ 1,

Vol(B(X,d)⃗ (x, δ)) ∧ 1 ≈ 1,

Λ(x, δ) ∧ 1 ≈ 1,

completing the proof of (l). Taking δ0 := min{ξ, ξ3 /2}, we have by (3.28), for x ∈ 𝒦 and δ ∈ (0, δ0 ]ν , Vol(B(W ,ds)⃗ (x, 2δ)) ≈ Λ(x, 2δ) ≈ Λ(x, δ) ≈ Vol(B(W ,ds)⃗ (x, δ)), where we have used the formula for Λ to see that Λ(x, 2δ) ≈ Λ(x, δ). A similar proof gives ⃗ This proves (m). A similar proof using the same estimate with (W , ds)⃗ replaced by (X, d ). (l) in place of (3.28) proves (n). Proof of Corollary 3.3.8. Fix ϕ ∈ C0∞ (Bn (1)) with ϕ ≡ 1 on Bn (3/4). Set ϕx,δ (t) := ϕ(Φx,ξ3 δ (t)) ∈ C0∞ (Ω1 ), where Φx,δ and ξ3 are as in Theorem 3.3.7. By Theorem 3.3.7 (b) we have supp(ϕx,δ ) ⊆ Φx,ξ3 δ (Bn (1)) ⊆ B(X,d) (x, ξ3 δ) ∩ Ω1 ⊆ B(W ,ds) (x, δ) ∩ Ω1 , proving (i). By definition, ϕx,δ equals 1 on Φx,ξ3 δ (Bn (3/4)), and therefore it equals 1 on a neighborhood of the closure of Φx,ξ3 δ (Bn (1/2)) ⊇ B(W ,ds) (x, ξ32 δ), where the containment follows from Theorem 3.3.7 (i). This proves (ii). Fix N ∈ ℕ. By Theorem 3.3.7 (j), ∑ ‖(W x,ξ3 δ )α ϕ‖C(Bn (1)) ≲ 1.

|α|≤N

Pushing this equation forward via Φx,ξ3 δ , we have ∑ ‖((ξ3 δ)dsW )α ϕx,δ ‖C(Φx,ξ δ (Bn (1))) ≲ 1.

|α|≤N

3

Since ξ3 ≈ 1 and supp(ϕx,δ ) ⊆ Φx,ξ3 δ (Bn (1)), (iii) follows, completing the proof.

148 � 3 Vector fields and Carnot–Carathéodory geometry Finally, we turn to the proof of Corollary 3.3.9. This is a simple consequence of Theorem 3.3.7. Proof of Corollary 3.3.9. Set 𝒦 := Ω and fix Ω1 ⋐ M open with 𝒦 ⋐ Ω1 . We apply Theorem 3.3.7 with this choice of 𝒦 and Ω1 . Set δ1 := δ0 , where δ0 is as in Theorem 3.3.7. By Theorem 3.3.7 (c), we have, for x ∈ 𝒦, δ ∈ (0, δ1 ], Vol(B(W ,ds) (x, δ) ≈ Vol(B(X,d) (x, δ) ≈ Λ(x, δ).

(3.29)

(i): Since x 󳨃→ Λ(x, δ1 ) and Λ(x, δ1 ) is never 0 on Ω1 , we see that inf Λ(x, δ1 ) > 0.

x∈𝒦

(3.30)

Item (i) now follows from (3.29). (ii): By the formula for Λ(x, δ), there exist Q1 = Q1 (Ω), Q2 = Q2 (Ω) with n ≤ Q1 ≤ Q2 such that for all δ > 0, ξ ∈ (0, 1], ξ −Q1 Λ(x, ξδ) ≲ Λ(x, δ) ≲ ξ −Q2 Λ(x, ξδ). Item (i) now follows from (3.29). (iii): We begin with the first inequality. We prove only the result for (W , ds); the same proof works with (X, d ) in place of (W , ds). If δ ∈ (0, δ1 ], then by (ii), ξ −Q1 Vol(B(W ,ds) (x, ξδ)) ≲ Vol(B(W ,ds) (x, δ)), and it follows that (ξ −Q1 Vol(B(W ,ds) (x, ξδ))) ∧ 1 ≲ Vol(B(W ,ds) (x, δ)) ∧ 1. If δ > δ0 , then by (i), Vol(B(W ,ds) (x, δ)) ∧ 1 ≈ 1, and the first inequality in (iii) follows. We turn to the second inequality in (iii). If δ ∈ (0, δ1 ], then by (ii), Vol(B(W ,ds) (x, δ)) ≲ ξ −Q2 Vol(B(W ,ds) (B(W ,ds) (x, ξδ)), and the second inequality follows. If δ > δ1 , then the result will follow once we show ξ −Q2 Vol(B(W ,ds) (B(W ,ds) (x, ξδ)) ≳ 1. Indeed, we have, using (3.29) and possibly increasing Q2 , ξ −Q2 Vol(B(W ,ds) (B(W ,ds) (x, ξδ)) ≥ ξ −Q2 Vol(B(W ,ds) (B(W ,ds) (x, ξδ1 )) ≈ ξ −Q2 Λ(x, ξδ1 ) ≳ Λ(x, δ1 ) ≳ 1,

where ξ −Q2 Λ(x, ξδ1 ) ≳ Λ(x, δ1 ) follows from the formula for Λ (by taking Q2 large enough), and Λ(x, δ1 ) ≳ 1 follows from (3.30).

3.8 Approximately commuting vector fields ∞ ∞ Fix ν1 , ν2 ∈ ℕ+ . Let 𝒮1 ⊆ Cloc (M; TM) × (ℕν1 \ {0}) and 𝒮2 ⊆ Cloc (M; TM) × (ℕν2 \ {0}). In this text we define various objects related to 𝒮1 and 𝒮2 (for example, the Carnot– Carathéodory balls defined in Section 3.5, the singular integrals defined in Chapter 5,

3.8 Approximately commuting vector fields

� 149

or the function spaces defined in Chapter 5). A main theme is that these objects associated with 𝒮1 behave well with the same objects associated with 𝒮2 provided 𝒮1 and 𝒮2 “locally weakly approximately commute,” a notion that we make precise in this section (see Definition 3.8.5). Throughout this section, Ω ⋐ M is an open, relatively compact set. Definition 3.8.1. We say 𝒮1 and 𝒮2 strongly approximately commute on Ω if ∀(Z1 , dr1⃗ ) ∈ 𝒮1 and (Z2 , dr2⃗ ) ∈ 𝒮2 , there are finite sets ℱ1 ⊆ {(Y , d̂)⃗ ∈ 𝒮1 : d̂⃗ ≤ dr1⃗ },

ℱ2 ⊆ {(Y , d̂)⃗ ∈ 𝒮2 : d̂⃗ ≤ dr2⃗ },

such that [Z1 , Z2 ] =

∑ ⃗ (Y ,d̂)∈ℱ 1

a(Y ,d̂)⃗ Y +

∑ ⃗ (Y ,d̂)∈ℱ 2

b(Y ,d̂)⃗ Y ,

∞ a(Y ,d̂)⃗ , b(Y ,d̂)⃗ ∈ Cloc (Ω).

(3.31)

We say 𝒮1 and 𝒮2 locally strongly approximately commute if 𝒮1 and 𝒮2 strongly approximately commute on Ω, for all Ω ⋐ M open. ∞ Example 3.8.2. In the case ν2 = 1, let 𝒮1 ⊆ Cloc (ℝn ; Tℝn ) × (ℕν1 \ {0}) be any set of vector n ∞ fields with formal degrees on ℝ . Let 𝒮2 = {(𝜕x1 , 1), . . . , (𝜕xn , 1)} ⊂ Cloc (ℝn ; Tℝn ) × ℕ+ . Then 𝒮1 and 𝒮2 locally strongly approximately commute. Indeed, for (Z1 , dr1⃗ ) ∈ 𝒮1 , we have n

l [Z1 , 𝜕xk ] = ∑ a(Z 𝜕 , ,dr⃗ ),k xl l=1

1

1

l ∞ a(Z ∈ Cloc (ℝn ). ,dr⃗ ),k 1

1

We generalize Example 3.8.2 via the next lemma. ∞ Lemma 3.8.3. Let 𝒮1 ⊆ Cloc (M; TM) × (ℕν1 \ {0}) be any set of vector fields with formal n ∞ degrees on ℝ . Let 𝒮2 ⊆ Cloc (M; TM) × ℕ+ be a set of vector fields paired with singleparameter formal degrees. Suppose

Tx M = span{Z2 (x) : (Z2 , 1) ∈ 𝒮2 },

∀x ∈ M.

Then 𝒮1 and 𝒮2 locally strongly approximately commute. Proof. Let Ω ⋐ M be an open set. Since Ω is compact, there is a finite set ℱ2 ⊆ 𝒮2 such that Tx M = span{Y (x) : (Y , 1) ∈ ℱ2 },

∀x ∈ Ω.

(3.32)

Let (Z1 , dr1⃗ ) ∈ 𝒮1 and (Z2 , dr2 ) ∈ 𝒮2 . Since [Z1 , Z2 ] is a smooth vector field, (3.32) implies [Z1 , Z2 ] =

∑ (Y ,1)∈ℱ2

a(Y ,1) Y ,

∞ a(Y ,1) ∈ Cloc (M).

150 � 3 Vector fields and Carnot–Carathéodory geometry This establishes (3.31) with ℱ1 = 0, proving that 𝒮1 and 𝒮2 strongly approximately commute on Ω. Since Ω ⋐ M was an arbitrary relatively compact open set, we conclude 𝒮1 and 𝒮2 locally strongly approximately commute. Example 3.8.4. Let M be a Lie group. Suppose the vector fields in 𝒮1 are left invariant vector fields and the vector fields in 𝒮2 are right invariant vector fields. Then 𝒮1 and 𝒮2 locally strongly approximately commute. Indeed, [Z1 , Z2 ] = 0, for all (Z1 , dr1⃗ ) ∈ 𝒮1 and (Z2 , dr2⃗ ) ∈ 𝒮2 . Definition 3.8.5. We say 𝒮1 and 𝒮2 weakly approximately commute on Ω if Gen(𝒮1 ) and Gen(𝒮2 ) strongly approximately commute on Ω. We say 𝒮1 and 𝒮2 locally weakly approximately commute if 𝒮1 and 𝒮2 weakly approximately commute on Ω, for all open, relatively compact sets Ω ⋐ M. Proposition 3.8.6. (a) Suppose 𝒮1 and 𝒮2 strongly approximately commute on Ω. Then 𝒮1 and 𝒮2 weakly approximately commute on Ω. (b) Suppose 𝒮1 and 𝒮2 locally strongly approximately commute. Then 𝒮1 and 𝒮2 locally weakly approximately commute. The proof of Proposition 3.8.6 uses the next lemma. Lemma 3.8.7. Suppose 𝒮1 and 𝒮2 strongly approximately commute on Ω. Then Gen(𝒮1 ) and 𝒮2 strongly approximately commute on Ω. Proof. Let 𝒮̂1 ⊆ Gen(𝒮1 ) be the set of all (Z1 , dr1⃗ ) ∈ Gen(𝒮1 ) such that ∀(Z2 , dr2⃗ ) ∈ 𝒮2 there are finite sets ℱ1 ⊆ {(Y , d̂)⃗ ∈ Gen(𝒮1 ) : d̂⃗ ≤ dr1⃗ },

ℱ2 ⊆ {(Y , d̂)⃗ ∈ 𝒮2 : d̂⃗ ≤ dr2⃗ },

with [Z1 , Z2 ] =

∑ ⃗ (Y ,d̂)∈ℱ 1

a(Y ,d̂)⃗ Y +

∑ ⃗ (Y ,d̂)∈ℱ 2

b(Y ,d̂)⃗ Y ,

∞ a(Y ,d̂)⃗ , b(Y ,d̂)⃗ ∈ Cloc (Ω).

The assumption of the lemma implies 𝒮1 ⊆ 𝒮̂1 . Our goal is to show 𝒮̂1 = Gen(𝒮1 ), which is equivalent to the conclusion of the lemma. Since 𝒮1 ⊆ 𝒮̂1 ⊆ Gen(𝒮1 ), it suffices to show that if (Z1 , dr1⃗ ), (X1 , d 1⃗ ) ∈ 𝒮1 , then ([Z1 , X1 ], dr1⃗ + dr1⃗ ) ∈ 𝒮1 . Thus, we wish to consider, for (Z2 , dr2⃗ ) ∈ 𝒮2 , [[Z1 , X1 ], Z2 ] = [[Z1 , Z2 ], X1 ] + [Z1 , [Z2 , X1 ]].

(3.33)

The two terms on the right-hand side of (3.33) are similar, so we only show that the first is of the desired form, that is, we wish to show that there are finite sets ̂1 ⊆ {(Y , d̂)⃗ ∈ Gen(𝒮1 ) : d̂⃗ ≤ dr1⃗ + dr2⃗ }, ℱ

̂2 ⊆ {(Y , d̂)⃗ ∈ 𝒮2 : d̂⃗ ≤ dr2⃗ } ℱ

3.8 Approximately commuting vector fields

� 151

such that [[Z1 , Z2 ], X1 ] =

∑ ⃗ ℱ ̂1 (Y ,d̂)∈

a(Y ,d̂)⃗ Y +

∑ ⃗ ℱ ̂2 (Y ,d̂)∈

b(Y ,d̂)⃗ Y ,

∞ a(Y ,d̂)⃗ , b(Y ,d̂)⃗ ∈ Cloc (Ω).

(3.34)

By the hypothesis that (Z1 , dr1⃗ ) ∈ 𝒮̂1 , there exist finite sets ℱ1 ⊆ {(Y , d̂)⃗ ∈ Gen(𝒮1 ) : d̂⃗ ≤ dr1⃗ } and ℱ2 ⊆ {(Y , d̂)⃗ ∈ 𝒮2 : d̂⃗ ≤ dr2⃗ } such that [Z1 , Z2 ] =

∑ ⃗ (Y ,d̂)∈ℱ 1

a(Y ,d̂)⃗ Y +

∑ ⃗ (Y ,d̂)∈ℱ 2

b(Y ,d̂)⃗ Y ,

∞ a(Y ,d̂)⃗ , b(Y ,d̂)⃗ ∈ Cloc (Ω).

Thus, we have [[Z1 , Z2 ], X1 ] = −

∑ ⃗ (Y ,d̂)∈ℱ 1

+

∑

(X1 a(Y ,d̂)⃗ )Y − [Y , X1 ] +

⃗ (Y ,d̂)∈ℱ 1

∑ ⃗ (Y ,d̂)∈ℱ 2

∑ ⃗ (Y ,d̂)∈ℱ 2

(X1 b(Y ,d̂)⃗ )Y

b(Y ,d̂)⃗ [Y , X1 ].

(3.35)

The first two terms on the right-hand side of (3.35) are of the desired form from (3.34). That the last term on the right-hand side of (3.35) is of the desired form follows from the fact that (X1 , d 1⃗ ) ∈ 𝒮̂1 and for (Y , d̂)⃗ ∈ ℱ2 we have d 1⃗ + d̂⃗ ≤ d 1⃗ + dr2⃗ . Finally, for the third ⃗ d 1⃗ ) ∈ term on the right-hand side of (3.35), note that for (Y , d̂)⃗ ∈ ℱ1 , we have ([Y , X1 ], d̂+ ⃗ ⃗ ⃗ ⃗ Gen(𝒮1 ) with d̂+ d 1 ≤ dr1 + d 1 , and therefore the third term is of the desired form as well. We conclude ([Z1 , X1 ], dr1⃗ + d 1⃗ ) ∈ 𝒮̂1 , completing the proof. Proof of Proposition 3.8.6. (a): By Lemma 3.8.7, Gen(𝒮1 ) and 𝒮2 strongly approximately commute on Ω. Another application of Lemma 3.8.7 then shows that Gen(𝒮1 ) and Gen(𝒮2 ) strongly approximately commute on Ω, completing the proof of (a). (b): This follows immediately from (a). ∞ The way we will usually obtain a set 𝒮 ⊆ Cloc (M; TM) × (ℕν \ {0}) is by creating it out of sets of vector fields with single-parameter formal degrees as follows. Given ∞ 𝒮1 , . . . , 𝒮ν ⊆ Cloc (M; TM) × ℕ+ , we set

𝒮 := 𝒮1 ⊠ 𝒮2 ⊠ ⋅ ⋅ ⋅ ⊠ 𝒮ν

= {(Xμ , dμ eμ ) : μ ∈ {1, . . . , ν}, (Xμ , dμ ) ∈ 𝒮μ } ⊂

∞ Cloc (M; TM)

ν

(3.36)

× ℕ \ {0}.

∞ Proposition 3.8.8. Let 𝒮1 , . . . , 𝒮ν ⊆ Cloc (M; TM) × ℕ+ and define 𝒮 = 𝒮1 ⊠ 𝒮2 ⊠ ⋅ ⋅ ⋅ ⊠ 𝒮ν as in (3.36). (a) Suppose 𝒮1 , . . . , 𝒮ν pairwise weakly approximately commute on Ω. For each μ ∈ {1, . . . , ν}, suppose Gen(𝒮μ ) is finitely generated on Ω by ℱμ ⊂ Gen(𝒮μ ). Then Gen(𝒮 ) is linearly finitely generated on Ω by

152 � 3 Vector fields and Carnot–Carathéodory geometry ℱ := ℱ1 ⊠ ℱ2 ⊠ ⋅ ⋅ ⋅ ⊠ ℱν = {(Xμ , dμ eμ ) : μ ∈ {1, . . . , ν}, (Xμ , dμ ) ∈ ℱμ } ⊆ 𝒮 .

(b) Suppose 𝒮1 , . . . , 𝒮ν pairwise locally weakly approximately commute and suppose for each μ ∈ {1, . . . , ν} that Gen(𝒮μ ) is locally finitely generated. Then Gen(𝒮 ) is locally linearly finitely generated. Proof. Item (b) follows immediately from (a), so we prove only (a). Clearly ℱ is a finite set and for each (X0 , d 0⃗ ) ∈ ℱ , d 0⃗ is non-zero in only one component. We will show that Gen(𝒮 ) is finitely generated by ℱ on Ω, which will complete the proof. Let 𝒯 be the set of all those (X0 , d 0⃗ ) ∈ Gen(𝒮 ) such that X0 =

∑ (X1 ,d ⃗1 )∈ℱ d ⃗ 1 ≤d ⃗ 0

(X1 ,d ⃗1 ) X, (X0 ,d ⃗0 ) 1

c

c

(X1 ,d ⃗1 ) (X0 ,d ⃗0 )

∞ ∈ Cloc (Ω).

(3.37)

We wish to show that 𝒯 = Gen(𝒮 ). By the definition of 𝒮 and ℱ , we have 𝒮 ⊆ 𝒯 . Suppose (X1 , d 1⃗ ), (X2 , d 2⃗ ) ∈ ℱ . We claim ([X1 , X2 ], d 1⃗ + d 2⃗ ) ∈ 𝒯 . Indeed, (X1 , d 1⃗ ) = (Xμ1 , dμ1 eμ1 ) and (X2 , d 2⃗ ) = (Xμ′ 2 , dμ′ 2 eμ2 ) for some (Xμ1 , dμ1 ) ∈ Gen(𝒮μ1 ) and (Xμ′ 2 , dμ′ 2 ) ∈ Gen(𝒮μ2 ). There are two cases. If μ1 = μ2 , then since ([Xμ1 , Xμ′ 2 ], dμ1 + dμ′ 2 ) ∈ Gen(𝒮μ1 ) and Gen(𝒮μ1 ) is finitely generated by ℱμ1 on Ω, we have [Xμ1 , Xμ′ 2 ] =

∑ (Xμ′′ ,dμ′′ )∈ℱμ1

c(Xμ′′ ,dμ′′ ) Xμ′′1 , 1

1

∞ c(Xμ′′ ,dμ′′ ) ∈ Cloc (Ω). 1

1

1 1 dμ′′ ≤dμ1 +dμ′ 1 2

From here it follows immediately from the definitions that ([X1 , X2 ], d 1⃗ + d 2⃗ ) = ([Xμ1 , Xμ′ 2 ], (dμ1 + dμ′ 2 )eμ1 ) ∈ 𝒯 . If μ1 ≠ μ2 , then since Gen(𝒮μ1 ) and Gen(𝒮μ2 ) strongly approximately commute on Ω, using the fact that Gen(𝒮μ1 ) and Gen(𝒮μ2 ) are finitely generated by ℱ1 and ℱ2 (respectively) on Ω, we have [Xμ1 , Xμ′ 2 ] =

∑ (Xμ′′ ,dμ′′ )∈ℱμ1 1 1 dμ′′ ≤dμ1 1

b(Xμ′′ ,dμ′′ ) Xμ′′1 + 1

1

∑ (Xμ′′′ ,dμ′′′ )∈ℱμ2 2 2 dμ′′′ ≤dμ′ 2 2

c(Xμ′′′ ,dμ′′′ ) Xμ′′′2 , 2

2

∞ b(Xμ′′ ,dμ′′ ) , c(Xμ′′′ ,dμ′′′ ) ∈ Cloc (Ω). 1

1

2

2

From here it follows immediately from the definitions that ([X1 , X2 ], d 1⃗ + d 2⃗ ) = ([Xμ1 , Xμ′ 2 ], dμ1 eμ1 + dμ′ 2 eμ2 ) ∈ 𝒯 . Now suppose (X1 , d 1⃗ ), (X2 , d 2⃗ ) ∈ 𝒯 . We claim ([X1 , X2 ], d 1⃗ + d 2⃗ ) ∈ 𝒯 . Indeed, this follows immediately from the definition of 𝒯 , (3.37), and the fact that we just proved that if (X1′ , d ′1⃗ ), (X2 , d ′2⃗ ) ∈ ℱ , then ([X1′ , X2′ ], d ′1⃗ + d ′2⃗ ) ∈ 𝒯 . Combining this with 𝒮 ⊆ 𝒯 ⊆ Gen(𝒮 ), it follows that 𝒯 = Gen(𝒮 ).

3.8 Approximately commuting vector fields

� 153

Saying that 𝒯 = Gen(𝒮 ) is equivalent to saying that Gen(𝒮 ) is finitely generated by ℱ on Ω, which completes the proof. Example 3.8.9. This is an important example for our applications. Suppose (W 1 , ds1 ) ⊂ ∞ Cloc (Ω; TΩ) × ℕ+ are Hörmander vector fields with formal degrees on an open set Ω ⊆ n ℝ (see Definition 1.1.3). Let (W 2 , ds2 ) = {(𝜕x1 , 1), . . . , (𝜕xn , 1)}. By Example 3.8.2, (W 1 , ds1 ) and (W 2 , ds2 ) locally strongly approximately commute, and by Proposition 3.8.6, (W 1 , ds1 ) and (W 2 , ds2 ) locally weakly approximately commute. By Proposition 3.4.14 (in the case ν = 1), Gen((W 1 , ds1 )) and Gen((W 2 , ds2 )) are both locally finitely generated. Set (W , ds)⃗ := ∞ (W 1 , ds1 )⊠(W 2 , ds2 ) ⊂ Cloc (Ω; TΩ)×(ℕ2 \{0}). Combining the above with Proposition 3.8.8 ⃗ is locally linearly finitely generated. shows that Gen((W , ds)) ∞ Proposition 3.8.10. Suppose that for j = 1, 2, 3, 𝒮j ⊆ Cloc (M; TM) × (ℕνj \ {0}), where νj ∈ ℕ+ . Define

𝒮0 := 𝒮1 ⊠ 𝒮2 ⊆ Cloc (M; TM) × (ℕ ∞

ν1 +ν2

\ {0}).

Then we have the following: (i) If, for j = 1, 2, 𝒮j and 𝒮3 strongly approximately commute on Ω, then 𝒮0 and 𝒮3 strongly approximately commute on Ω. (ii) If, for j = 1, 2, 𝒮j and 𝒮3 locally strongly approximately commute, then 𝒮0 and 𝒮3 locally strongly approximately commute. (iii) If, for j = 1, 2, 𝒮j and 𝒮3 weakly approximately commute on Ω, then 𝒮0 and 𝒮3 weakly approximately commute on Ω. (iv) If, for j = 1, 2, 𝒮j and 𝒮3 locally weakly approximately commute, then 𝒮0 and 𝒮3 locally weakly approximately commute. Proof. Item (ii) follows immediately from (i) and item (iv) follows immediately from (iii), so we prove only (i) and (iii). (i): Let (Z0 , dr0⃗ ) ∈ 𝒮0 and (Z3 , dr3⃗ ) ∈ 𝒮3 . We wish to show that there are finite sets ℱ0 ⊆ {(Y , d̂)⃗ ∈ 𝒮0 : d̂⃗ ≤ dr0⃗ },

ℱ3 ⊆ {(Y , d̂)⃗ ∈ 𝒮3 : d̂⃗ ≤ dr3⃗ }

such that [Z0 , Z3 ] =

∑ ⃗ (Y ,d̂)∈ℱ 0

a(Y ,d̂)⃗ Y +

∑ ⃗ (Y ,d̂)∈ℱ 3

b(Y ,d̂)⃗ Y ,

∞ a(Y ,d̂)⃗ , b(Y ,d̂)⃗ ∈ Cloc (Ω).

(3.38)

There are two cases: either (Z0 , dr0⃗ ) = (Z1 , (dr1⃗ , 0ν2 )) for some (Z1 , dr1⃗ ) ∈ 𝒮1 or (Z0 , dr0⃗ ) = (Z2 , (0ν1 , dr2⃗ )) for some (Z2 , dr2⃗ ) ∈ 𝒮2 . We consider only the first case, where (Z0 , dr0⃗ ) = (Z1 , (dr1⃗ , 0ν2 )) for some (Z1 , dr1⃗ ) ∈ 𝒮1 ; the other case is similar. In this case, by the assumption that 𝒮1 and 𝒮3 strongly approximately commute on Ω, there exist finite sets ℱ1 ⊆ {(Y , d̂)⃗ ∈ 𝒮1 : d̂⃗ ≤ dr1⃗ },

ℱ3 ⊆ {(Y , d̂)⃗ ∈ 𝒮3 : d̂⃗ ≤ dr3⃗ }

154 � 3 Vector fields and Carnot–Carathéodory geometry with [Z1 , Z3 ] =

∑ ⃗ (Y ,d̂)∈ℱ 0

a(Y ,d̂)⃗ Y +

∑ ⃗ (Y ,d̂)∈ℱ 3

b(Y ,d̂)⃗ Y ,

∞ a(Y ,d̂)⃗ , b(Y ,d̂)⃗ ∈ Cloc (Ω).

Setting ℱ0 := {(Y , (d̂,⃗ 0ν2 )) : (Y , d̂)⃗ ∈ ℱ1 } ⊆ {(Y , d̂)⃗ ∈ 𝒮0 : d̂⃗ ≤ dr0⃗ }

establishes (3.38) and completes the proof of (i). (iii): The assumption can be restated as follows: Gen(𝒮j ) and Gen(𝒮3 ) strongly approximately commute on Ω, for j = 1, 2. Set 𝒯 := Gen(𝒮1 ) ⊠ Gen(𝒮2 ) ⊆ Gen(𝒮0 ).

By (i), 𝒯 and Gen(𝒮3 ) strongly approximately commute on Ω. By Proposition 3.8.6 (a), 𝒯 and Gen(𝒮3 ) weakly approximately commute on Ω, i. e., Gen(𝒯 ) and Gen(𝒮3 ) strongly approximately commute on Ω. Since 𝒮0 ⊆ 𝒯 ⊆ Gen(𝒮0 ), we have Gen(𝒯 ) = Gen(𝒮0 ). We conclude that Gen(𝒮0 ) and Gen(𝒮3 ) strongly approximately commute on Ω, i. e., 𝒮0 and 𝒮3 weakly approximately commute on Ω. Corollary 3.8.11. Fix ν1 , ν2 ∈ ℕ+ . Let 𝒮1 , . . . , 𝒮ν1 , 𝒯1 , . . . , 𝒯ν2 ⊆ Cloc (M; TM) × ℕ+ . ∞

Suppose that ∀μ1 ∈ {1, . . . , ν1 } and ∀μ2 ∈ {1, . . . , ν2 }, 𝒮μ1 and 𝒯μ2 locally weakly approximately commute. Set ν

𝒮0 := 𝒮1 ⊠ 𝒮2 ⊠ ⋅ ⋅ ⋅ ⊠ 𝒮ν1 ⊆ Cloc (M; TM) × (ℕ 1 \ {0}), ∞

ν

𝒯0 := 𝒯1 ⊠ 𝒯2 ⊠ ⋅ ⋅ ⋅ ⊠ 𝒯ν2 ⊆ Cloc (M; TM) × (ℕ 2 \ {0}). ∞

Then 𝒮0 and 𝒯0 locally weakly approximately commute. Proof. For μ1 ∈ {1, . . . , ν1 } and μ2 ∈ {1, . . . , ν2 }, set 𝒮

μ1

𝒯

μ2

∞ := 𝒮1 ⊠ 𝒮2 ⊠ ⋅ ⋅ ⋅ ⊠ 𝒮μ1 ⊆ Cloc (M; TM) × (ℕμ1 \ {0}), ∞ := 𝒯1 ⊠ 𝒯2 ⊠ ⋅ ⋅ ⋅ ⊠ 𝒯μ2 ⊆ Cloc (M; TM) × (ℕμ2 \ {0}).

First claim: For all μ1 ∈ {1, . . . , ν1 } and all μ2 ∈ {1, . . . , ν2 }, 𝒮 μ1 and 𝒯μ2 locally weakly approximately commute. We prove the first claim by induction on μ1 . Since 𝒮 1 = 𝒮1 , the base case (μ1 = 1) follows from the hypotheses of the corollary. We assume the first claim for some μ1 ∈ {1, . . . , ν1 − 1} and prove it for μ1 + 1. The inductive hypothesis implies that 𝒮 μ1 and

3.9 Integrating over leaves

�

155

𝒯μ2 locally weakly approximately commute for all μ2 ∈ {1, . . . , ν2 }. The hypotheses of the corollary imply that 𝒮μ1 +1 and 𝒯μ2 locally weakly approximately commute for all μ2 ∈ {1, . . . , ν2 }. Proposition 3.8.10 (iv) with 𝒮1 in that proposition replaced with 𝒮 μ1 , 𝒮2 replaced with 𝒮μ1 +1 , and 𝒮3 replaced with 𝒯μ2 implies that 𝒮 μ1 +1 and 𝒯μ2 locally weakly

approximately commute, for all μ2 ∈ {1, . . . , ν2 }. This completes the inductive step and completes the proof of the first claim.

Second claim: For all μ2 ∈ {1, . . . , ν2 }, 𝒮 ν1 and 𝒯 μ2 locally weakly approximately commute. We prove the second claim by induction on μ2 . Since 𝒯 1 = 𝒯1 , the base case (μ2 = 1) follows from the first claim. We assume the second claim for some μ2 ∈ {1, . . . , ν2 − 1} and prove it for μ2 + 1. The inductive hypothesis implies that 𝒮 ν1 and 𝒯 μ2 weakly locally approximately commute, and the first claim implies 𝒮 ν1 and 𝒯μ2 +1 weakly locally approximately commute. Proposition 3.8.10 (iv) with 𝒮1 in that proposition replaced with 𝒯 μ2 , 𝒮2 replaced with 𝒯μ2 +1 , and 𝒮3 replaced with 𝒮 ν1 implies that 𝒮 ν1 and 𝒯 μ2 +1 weakly locally approximately commute. This completes the inductive step and completes the proof of the second claim. Since 𝒮0 = 𝒮 ν1 and 𝒯0 = 𝒯 ν2 , the corollary follows from the second claim.

3.9 Integrating over leaves Let M be a connected C ∞ manifold, with smooth strictly positive density Vol. Fix ν ∈ ℕ+ and let ∞ (W , ds)⃗ = {(W1 , ds1⃗ ), . . . , (Wr , dsr⃗ )} ⊂ Cloc (M; TM) × (ℕν \ {0})

be a finite set. Our main assumption is the following. ⃗ is locally finitely generated on M. Main assumption: Gen((W , ds)) Theorem 3.5.1 applies in this setting, though that theorem does not give any relationship between the Carnot–Carathéodory geometry on each leaf and the density Vol, which is defined on the total space M. The main result of this section gives a way to estimate an integral over the total space by integrals over leaves. Fix open sets Ω3 ⋐ Ω4 ⋐ 𝒦 ⋐ Ω1 ⋐ Ω ⋐ M, with Ω, Ω1 , Ω3 , and Ω4 open and ⃗ be finitely generated by (X, d )⃗ = relatively compact and 𝒦 compact. Let Gen((W , ds)) ⃗ ⃗ {(X1 , d 1 ), . . . , (Xq , d q )} on Ω. Theorem 3.5.1 applies with this choice of 𝒦, Ω1 , and Ω. We let Φx,δ : BN(x) (1) → M be as in that theorem. The main result of this section is the following proposition. ⃗ ∈ (0, 1] such that ∀δ ∈ (0, 1]ν , Proposition 3.9.1. There exists η1 = η1 (𝒦, Ω3 , Ω4 , (X, d )) ′ η ∈ (0, η1 ], and f ∈ C(M) with supp(f ) ⊆ Ω3 and f ≥ 0, ∫ f (x) d Vol(x) ≈ ∫ Ω3

∫

Ω4 BN(x) (η′ )

f (Φx,δ (u)) du d Vol(x),

156 � 3 Vector fields and Carnot–Carathéodory geometry where the implicit constants depend on η′ . Here, we have used Theorem 3.5.1 (f) to see that the integral on the right-hand side makes sense. To prove Proposition 3.9.1, we require two lemmas. ⃗ ∈ (0, 1] such that ∀η′ ∈ (0, η2 ] and Lemma 3.9.2. There exists η2 = η2 (𝒦, Ω3 , Ω4 , (X, d )) all f ∈ C(M) with supp(f ) ⊆ Ω3 and f ≥ 0, we have ∀δ ∈ (0, 1]ν , ∫ f (x) d Vol(x) ≈ Ω3

d⃗ 1 ∫ ∫ f (et⋅δ X x) dt d Vol(x). ′ q (η )

Ω4 Bq (η′ )

⃗ ∈ (0, 1] is sufficiently small, then for t ∈ Bq (η2 ), we Proof. If η2 = η2 (𝒦, Ω3 , Ω4 , (X, d )) 󵄨 t⋅X have e Ω4 ⊇ Ω3 . Furthermore, since dx et⋅X 󵄨󵄨󵄨t=0 = I, if η2 is sufficiently small and t ∈ q B (η2 ), a change of variables shows that ∫ f (x) d Vol(x) ≈ ∫ f (et⋅X x) d Vol(x), Ω3

∀t ∈ Bq (η2 ).

Ω4

In particular, by replacing t with (δd 1 t1 , . . . , δd q tq ), we have, for δ ∈ (0, 1]ν , ⃗

⃗

∫ f (x) d Vol(x) ≈ ∫ f (et⋅δ X x) d Vol(x),

∀t ∈ Bq (η2 ).

d⃗

Ω3

Ω4

Averaging both sides of this equation over Bq (η′ ) yields the result. ⃗ ∈ (0, 1] Lemma 3.9.3. Let η2 be as in Lemma 3.9.2. There exist η1 = η1 (𝒦, Ω3 , Ω4 , (X, d )) ′ ⃗ ∈ (0, η2 ] such that ∀η ∈ (0, η1 ], there exists η4 = and η3 = η3 (𝒦, Ω3 , Ω4 , (X, d )) ⃗ ∈ (0, η3 ] such that the following holds. For all f ∈ C(M) with η4 (η′ , 𝒦, Ω3 , Ω4 , (X, d )) f ≥ 0 and for all δ ∈ (0, 1]ν , x ∈ Ω4 , ∫ f (et⋅δ X x) dt ≲ d⃗

Bq (η4 )

f ∘ Φx,δ (u) du ≲ ∫ f (et⋅δ X x) dt. d⃗

∫ BN(x) (η′ )

Bq (η3 )

Proof. Theorem 3.5.1 (g) shows that ‖Φ∗x,δ δd j Xj ‖C 2 (BN(x) (1);ℝN(x) ) ≲ 1. ⃗

(3.39)

Let Θx,δ (t) := et⋅Φx,δ δ X 0. Using (3.39), standard results from the field of ODEs imply that if η5 ≳ 1 is sufficiently small, ∗

d⃗

‖Θx,δ ‖C 2 (BN(x) (η5 );ℝN(x) ) ≲ 1. Since

(3.40)

3.9 Integrating over leaves

� 157

⃗ 󵄨 𝜕tj 󵄨󵄨󵄨t=0 Θx,δ (t) = Φ∗x,δ δd j Xj (0),

Theorem 3.5.1 (h) shows that we can decompose the t variable into (t 1 , t 2 ), where t 1 = (tj1 , tj2 , . . . , tjN(x) ), t 2 denotes the rest of the variables in t, and 󵄨󵄨 󵄨 󵄨 󵄨󵄨det dt1 󵄨󵄨󵄨t=0 Θ(t)󵄨󵄨󵄨 ≈ 1.

(3.41)

⃗ ∈ (0, η2 ] such Combining this with (3.40) shows that there exists η3 = η3 (𝒦, Ω3 , Ω4 , (X, d )) q that ∀t ∈ B (η3 ), 󵄨󵄨 󵄨 󵄨 󵄨󵄨det dt1 󵄨󵄨󵄨t=0 Θx,δ (t)󵄨󵄨󵄨 ≈ 1.

(3.42)

Using the inverse function theorem, (3.39), (3.40), and the fact that Θx,δ (0) = 0, we ⃗ ∈ (0, 1] with can show that there exists η1 = η1 (𝒦, Ω3 , Ω4 , (X, d )) BN(x) (η1 ) ⊆ Θx,δ (Bq (η3 )).

(3.43)

Given η′ ∈ (0, η1 ], the Picard–Lindelöf theorem and (3.39) show that there exists η4 = ⃗ ∈ (0, η3 ] such that η4 (η′ , 𝒦, Ω3 , Ω4 , (X, d )) Θx,δ (t) = et⋅Φx,δ δ X 0 ∈ BN(x) (η1 ), ∗

d⃗

∀t ∈ Bq (η4 ).

(3.44)

Since x = Φx,δ (0) (see Theorem 3.5.1 (c)), we have by a change of variables, using (3.42), (3.43), and (3.44), ∫ f (et⋅δ X x) dt = ∫ f (et⋅δ X Φx,δ (0)) dt = ∫ f ∘ Φx,δ (Θx,δ (t)) dt d⃗

d⃗

Bq (η4 )

Bq (η4 )

Bq (η4 )

f ∘ Φx,δ (u) du ≲ ∫ f ∘ Φx,δ (Θx,δ (t)) dt = ∫ f (et⋅δ X x) dt, d⃗

≲

∫

Bq (η3 )

BN(x) (η′ )

Bq (η3 )

completing the proof. Proof of Proposition 3.9.1. Applying Lemmas 3.9.2 and 3.9.3, we have with η1 , η3 , and η4 as in those lemmas, for f ∈ C(M) with supp(f ) ⊆ Ω3 and f ≥ 0, ∀η′ ∈ (0, η1 ], ∫ f (x) d Vol(x) ≈ ∫ ∫ f (et⋅δ X x) dt d Vol(x) d⃗

Ω3

Ω4 Bq (η4 )

f ∘ Φx,δ (u) du d Vol(x) ≲ ∫ ∫ f (et⋅δ X x) dt d Vol(x) d⃗

≲∫

∫

Ω4 BN(x) (η′ )

≈ ∫ f (x) d Vol(x), Ω3

where the implicit constants depend on η′ ∈ (0, η1 ].

Ω4 Bq (η3 )

158 � 3 Vector fields and Carnot–Carathéodory geometry

3.10 Integrals and approximately commuting vector fields Let M be a connected C ∞ manifold with smooth, strictly positive density Vol. Fix ν ∈ ℕ+ . For each μ ∈ {1, . . . , ν}, let μ

μ

∞ (W μ , dsμ ) = {(W1 , ds1 ), . . . , (Wrμμ , dsμrμ )} ⊂ Cloc (M; TM) × ℕ+

be Hörmander vector fields with formal degrees. Set (W , ds)⃗ = {(W1 , ds1⃗ ), . . . , (Wr , dsr⃗ )} := (W 1 , ds1 ) ⊠ (W 2 , ds2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (W ν , dsν ) μ

μ

∞ := {(Wj , dsj eμ ) : μ ∈ {1, . . . , ν}, j ∈ {1, . . . , qμ }} ⊂ Cloc (M; TM) × (ℕν \ {0}),

where e1 , . . . , eν denotes the standard basis of ℝν . Main assumption: In this section, we assume the lists (W μ , dsμ ) pairwise locally weakly approximately commute, μ ∈ {1, . . . , ν}. The main result of this section is the following proposition; in it we write δ ∈ [0, ∞)ν as δ = 2−j , where j ∈ (−∞, ∞]ν . See Remark 3.5.3. Proposition 3.10.1. Fix an open, relatively compact set Ω ⋐ M. Then there exists D = ⃗ Ω) ≥ 0 such that ∀j, k ∈ (−∞, ∞]ν \ {(∞, . . . , ∞)}, ∀x, z ∈ Ω, D((W , ds), ∫ Ω

χB

(W ,ds)⃗ (x,2

−j )

χB

(y)

(W ,ds)⃗ (y,2

−k )

(z)

(Vol(B(W ,ds)⃗ (x, 2−j )) ∧ 1) (Vol(B(W ,ds)⃗ (y, 2−k )) ∧ 1) ≤D

χB

(W ,ds)⃗ (x,2⋅2

−j∧k )

(z)

Vol(B(W ,ds)⃗ (x, 2−j∧k )) ∧ 1

d Vol(y) (3.45)

.

The rest of this section is devoted to the proof of Proposition 3.10.1. Fix Ω ⋐ Ω1 ⋐ Ω2 ⋐ Ω3 ⋐ M, with Ω1 , Ω2 , and Ω3 open and relatively compact. By Proposition 3.4.14 (applied with ν = 1), each Gen((W μ , dsμ )) is finitely generated μ μ μ μ on Ω3 by some (X μ , d μ ) = {(X1 , d1 ), . . . , (Xqμ , dqμ )} ⊂ Gen((W μ , dsμ )). We define (X, d )⃗ = {(X1 , d 1⃗ ), . . . , (Xq , d q⃗ )} := (X 1 , d 1 ) ⊠ (X 2 , d 2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (X ν , d ν ) μ

μ

{(Xj , dj eμ ) : μ ∈ {1, . . . , ν}, 1 ≤ j ≤ qμ }.

⃗ is linearly finitely generated by (X, d )⃗ on Ω3 . By Proposition 3.8.8, Gen((W , ds)) ν For a > 0 and j ∈ [0, ∞] , set 1

Aa2−j f (x) = ∫ f (e−t ⋅2 Bq (a)

−j1 d 1

2

X 1 −t 2 ⋅2−j2 d X 2

e

⋅ ⋅ ⋅ e−t

ν

ν

⋅2−jν d X ν

x) dt,

3.10 Integrals and approximately commuting vector fields

μ

μ

μ

159

�

μ

μ

where t = (t 1 , . . . , t ν ) ∈ ℝq1 × ⋅ ⋅ ⋅ × ℝqν = ℝq and 2−jμ d X μ = (2−jμ d1 X1 , . . . , 2−jμ d Xqμ ) with 2−∞ = 0. For a > 0 sufficiently small, we treat Aa2−j as an operator in Hom(C0∞ (Ω2 ), ∞ Cloc (Ω3 )), and we write the Schwartz kernel as Aa2−j (x, y). ⃗ > 0 such that Lemma 3.10.2. Fix ξ4 > 0. Then there exists a0 = a0 (ξ4 , Ω1 , Ω2 , (X, d )) ⃗ ∀a ∈ (0, a0 ], ∃ξ5 = ξ5 (a, Ω1 , Ω2 , (X, d )) > 0 such that the following hold: 󵄨 (i) ∀δ ∈ [0, 1]ν \ {0}, ∀x ∈ Ω1 , supp Aaδ (x, ⋅) ∈ Ω2 , and Aaδ 󵄨󵄨󵄨Ω ×Ω is a bounded measurable 1 2 function. ⃗ ≥ 0 such that ∀x ∈ Ω1 , y ∈ Ω2 , δ ∈ [0, 1]ν \ {0}, (ii) There exists C1 = C1 (a, Ω1 , Ω2 , (X, d )) χB(W ,ds)⃗ (x,ξ5 δ) (y)

Vol(B(W ,ds)⃗ (x, ξ5 δ)) ∧ 1

≤ C1 Aaδ (x, y).

⃗ ≥ 0 such that ∀x ∈ Ω1 , y ∈ Ω2 , δ ∈ [0, 1]ν \ {0}, (iii) There exists C2 = C2 (a, Ω1 , Ω2 , (X, d )) Aaδ (x, y) ≤ C2

χB(W ,ds)⃗ (x,ξ4 δ) (y)

Vol(B(W ,ds)⃗ (x, ξ4 δ)) ∧ 1

.

⃗ ≥ 0 such that ∀2−j , 2−k ∈ [0, 1]ν with 2−j∧k ≠ 0, (iv) There exists C3 = C3 (a, Ω1 , Ω2 , (X, d )) ∀x ∈ Ω1 , y ∈ Ω2 , [Aa2−j Aa2−k ](x, y) ≤ C3

χB

(W ,ds)⃗ (x,ξ4 2

−j∧k )

Vol(B(W ,ds)⃗ (x, ξ4

(y)

2−j∧k ))

∧1

.

Proof. For (i), (ii), and (iii) we claim that it suffices to consider only the case δ ∈ (0, 1]ν , i. e., δ is non-zero in every component. Indeed, if δ ∈ [0, 1]ν \ {0}, then if we set E = {μ : δμ ≠ 0}, the above results with δ are equivalent to the same results applied to (W E , dsE⃗ ) with δ replaced by δE , where (W E , dsE⃗ ) and δE ∈ (0, 1]|E| are as in Section 3.5.2. Similarly, it suffices to prove (iv) in the case where j ∧ k is not infinite in any component. We henceforth assume δ ∈ (0, 1]ν in (i), (ii), and (iii) and j ∧ k is not infinite in any component in (iv). For t ∈ ℝq with t small, set 1

1

Γt (x) = e−t ⋅X e−t

2

⋅X 2

⋅ ⋅ ⋅ e−t

ν

⋅X ν

x

so that Aaδ f (x) = ∫ f (Γδd ⃗ t (x)) dt Bq (a)

and δd t = (δd 1 t1 , . . . , δd q tq ). Fix Ω1.5 open with Ω1 ⋐ Ω1.5 ⋐ Ω2 . We apply Theorem 3.5.1 with 𝒦 replaced by Ω1 and Ω replaced by Ω1.5 . Without loss of generality (using Theorem 3.5.4 (n)) we may shrink ⃗

⃗

⃗

160 � 3 Vector fields and Carnot–Carathéodory geometry ξ4 so that ξ4 ≤ ξ3 ∧ δ0 ∧ 1 (where ξ3 and δ0 are as in Theorem 3.5.1). In particular, by Theorem 3.5.1 (a), we have B(X,d)⃗ (x, ξ4 δ) ⊆ Ω1.5 , ∀x ∈ Ω1 , δ ∈ (0, 1]ν . If 0 < a0 ≤ ξ4 /ν is small enough that Γt (x) is defined ∀t ∈ Bq (a), x ∈ Ω1 , then we have for such t and x, Γδd ⃗ t (x) ∈ B(X,d)⃗ (x, ξ4 δ) ⊆ Ω1.5 . And therefore, for x ∈ Ω1 and a ∈ (0, a0 ], supp Aaδ (x, ⋅) ⊆ Ω1.5 ⊆ Ω2 . Note that

μ

𝜕tμ Γt (x) = Xj (x). j

μ

μ

Since, for each μ ∈ {1, . . . , ν}, the vector fields X1 (x), . . . , Xqμ (x) span the tangent space at 󵄨 every point of Ω3 (because W μ satisfies Hörmander’s condition), dt 󵄨󵄨󵄨t=0 Γt (x) : ℝq → Tx M is surjective for each x ∈ Ω3 . Since Γt (x) is smooth in both variables (see Section 3.1.1), the inverse function theorem along with a simple change of variables shows that if a0 > 0 is 󵄨 sufficiently small and a ∈ (0, a0 ], then Aaδ (x, y)󵄨󵄨󵄨Ω ×Ω is a bounded measurable function. 1 2 This completes the proof of (i). Next, we prove (iv); recall that we are assuming without loss of generality that j∧k ∈ ν ℝ . Let l = j ∧ k, j ̂ = j − l, and k̂ = k − l; note that j ̂ ∧ k̂ = 0 ∈ ℝν . Let Φx,2−l be the map μ

μ

from Theorem 3.5.1. Let Xpm,x,2 := Φ∗x,δ 2−lμ dp Xp . By Theorem 3.5.1 (h), we may pick n of −l

μ,x,2−l

the vector fields from {Xp with

: μ ∈ {1, . . . , ν}, 1 ≤ p ≤ qμ } and call them Y1x,2 , . . . , Ynx,2 , −l

−l

−l −l 󵄨 󵄨 inf 󵄨󵄨󵄨det(Y1x,2 (u)| ⋅ ⋅ ⋅ |Ynx,2 (u))󵄨󵄨󵄨 ≳ 1.

(3.46)

u∈Bn (1)

Using the fact that Φx,2−l (0) = x (Theorem 3.5.1 (c)), we have [Aa2−j Aa2−k f ](x) = ∫

1

∫ f (e−t ⋅2

−k1 d 1 1 X

Bq (a) Bq (a) 1

e−s ⋅2 = ∫

−j1 d 1

2

X 1 −s2 ⋅2−j2 d X 2

e

1

∫ f ∘ Φx,2−l (e−t ⋅2

=: ∫

−j1̂ d 1

X 1,x,2

−l

2

e−s ⋅2

2

2

⋅2−k2 d X 2

⋅ ⋅ ⋅ e−s 1

Bq (a) Bq (a)

e−s ⋅2

e−t

−j2̂ d 2

−k̂1 d 1

X 2,x,2

ν

ν

⋅2−jν d X ν

X

−l

⋅ ⋅ ⋅ e−t

1,x,2−l

ν

ν

⋅2−kν d X ν

Φx,2−l (0)) ds dt

e−t

⋅ ⋅ ⋅ e−s

ν

2

2

⋅2−k2 d X 2,x,2 ̂

ν

⋅2−jν d X ν,x,2 ̂

−l

−l

⋅ ⋅ ⋅ e−t

ν

0) ds dt

∫ f ∘ Φx,2−l (Θj,̂ k,x,l ̂ (s, t)) ds dt,

Bq (a) Bq (a)

where 1

−k̂1 d 1

X 1,x,2

1

−j1̂ d 1

X 1,x,2

−t ⋅2 Θj,̂ k,x,l ̂ (s, t) = e

e−s ⋅2

2

e−t

−l

e−s ⋅2

2

2

⋅2−k2 d X 2,x,2

−l

̂

−j2̂ d 2

X 2,x,2

−l

−l

⋅ ⋅ ⋅ e−t ⋅ ⋅ ⋅ e−s

ν

ν

ν

⋅2−kν d X ν,x,2 ̂

ν

⋅2−jν d X ν,x,2 ̂

−l

−l

ν

⋅2−kν d X ν,x,2

0.

̂

−l

3.10 Integrals and approximately commuting vector fields

� 161

Note that μ

󵄨󵄨 −jμ̂ dp μ,x,2 𝜕spμ Θj,̂ k,x,l Xp (0), ̂ 󵄨󵄨s=t=0 = −2 −l

μ

󵄨󵄨 −k̂μ dp μ,x,2 𝜕tpμ Θj,̂ k,x,l Xp (0). ̂ 󵄨󵄨s=t=0 = −2 −l

−l −l Thus, since j ̂ ∧ k̂ = 0, each of the vectors Y1x,2 (0), . . . , Ynx,2 (0) from (3.46) appears as 󵄨󵄨 󵄨󵄨 1 n either −𝜕spμ Θj,̂ k,x,l ̂ 󵄨󵄨s=t=0 or −𝜕tpμ Θj,̂ k,x,l ̂ 󵄨󵄨s=t=0 for some p and μ. Let u ∈ ℝ denote those

coordinates of (s, t) corresponding to the vectors Y1x,2 (0), . . . , Ynx,2 (0) and let u2 denote the rest of the coordinates of (s, t), so that −l

−l

󵄨 −𝜕up1 Θj,̂ k,l̂ (u1 , u2 )󵄨󵄨󵄨u1 =u2 =0 = Ypx,2 (0). −l

Combining this with (3.46), we have 󵄨󵄨 󵄨󵄨 󵄨󵄨det du1 Θj,̂ k,x,l ̂ (0, 0)󵄨󵄨 ≳ 1.

(3.47)

μ,x,2−l

Theorem 3.5.1 (g) shows that ‖Xp ‖C 1 (Bn (1);ℝn ) ≲ 1; thus, if a0 ≳ 1 is sufficiently small, standard theorems from the field of ODEs show that 󵄩󵄩 󵄩󵄩 󵄩󵄩Θj,̂ k,x,l ̂ 󵄩 󵄩C 1 (Bq (a)×Bq (a);ℝn ) ≲ 1,

∀a ∈ (0, a0 ].

(3.48)

By Lemma 3.2.4, Theorem 3.5.1 (i), and the fact that ξ4 ≈ 1, we may pick η0 ≳ 1 so small that Bn (η0 ) ⊆ B

|ds|⃗ 1

ξ4

W x,2

−l

(0). = B(W x,2−l ,|ds|⃗ ) (0, ξ4 /2), 1

so that Φx,2−l (Bn (η0 )) ⊆ B(W ,ds)⃗ (x, ξ4 2−l ).

(3.49)

1 2 Applying a change of variables in the u1 variable, with v = Θj,̂ k,x,l ̂ (u , u ), (3.47) ⃗ Ω1 , Ω2 ) > 0 is sufficiently small and a ∈ (0, a0 ], and (3.48) show that if a0 = a0 (η0 , (X, d ),

then

∫ Bq (a)×Bq (a)

= ∫

f ∘ Φx,2−l (Θj,̂ k,x,l ̂ (s, t)) ds dt 2 2 ∫ f ∘ Φx,2−l (v)Ja,j,̂ k,x,l ̂ (v, u ) du dv,

Bn (η0 ) Bq (2a) 2 where |Ja,j,̂ k,x,l ̂ (v, u )| ≲ 1.

162 � 3 Vector fields and Carnot–Carathéodory geometry Let Λ(x, 2−l ) and hx,2−l be as in Theorem 3.5.4. We have

∫

2 2 ∫ f ∘ Φx,2−l (v)Ja,j,̂ k,x,l ̂ (v, u ) du dv

Bn (η0 ) Bq (2a)

2 2 −l ∫ f ∘ Φx,2−l (v)hx,2−l (v)−1 Ja,j,̂ k,x,l ̂ (v, u ) du Λ(x, 2 )hx,2−l (v) dv.

= Λ(x, 2−l )−1 ∫

Bn (η0 ) Bq (2a)

Since Φ∗x,2−l Vol = Λ(x, 2−l )hx,2−l σLeb , applying a change of variables y = Φx,2−l (v) gives Λ(x, 2−l )−1 ∫

2 2 −l ∫ f ∘ Φx,2−l (v)hx,2−l (v)−1 Ja,j,̂ k,x,l ̂ (v, u ) du Λ(x, 2 )hx,2−l (v) dv

Bn (η0 ) Bq (2a)

= Λ(x, 2−l )−1

−1 2 2 (y))−1 Ja,j,̂ k,x,l ∫ f (y)hx,2−l (Φ−1 ̂ (Φx,2−l (y), u ) du d Vol(y). x,2−l

∫

Φx,2−l (Bn (η0 )) Bq (2a)

Combining the above equations yields Aa2−j Aa2−k (x, y) = χΦ

x,2−j

−l −1 (Bn (η0 )) (y)Λ(x, 2 )

−1 2 2 × hx,2−l ∘ Φ−1 (y)−1 ∫ Ja,j,̂ k,x,l ̂ (Φx,2−l (y), u ) du . x,2−l Bq (2a)

Since |Ja,j,̂ k,x,l ̂ | ≲ 1, hx,2−l ≈ 1 (Theorem 3.5.4 (o)), using (3.49), we have 󵄨󵄨 a a 󵄨 −l −1 󵄨󵄨A2−j A2−k (x, y)󵄨󵄨󵄨 ≲ Λ(x, 2 ) χB(W ,ds)⃗ (x,ξ4 2−l ) (y) ≲ ≈

χB

(W ,ds)⃗ (x,ξ4

2−l )

(y)

Vol(B(W ,ds)⃗ (x, 2−l )) ∧ 1

≲

χB

(W ,ds)⃗ (x,ξ4 2

Λ(x, 2−l )

χB

−l )

(y)

∧1

(W ,ds)⃗ (x,ξ4 2

−l )

(y)

Vol(B(W ,ds)⃗ (x, ξ4 2−l )) ∧ 1

,

where in the second to last estimate we have used Theorem 3.5.4 (l) and in the last estimate we have used the fact that ξ4 ≤ 1. This completes the proof of (iv).

Item (iii) follows from (iv). Indeed, when j = (∞, . . . , ∞), Aa2−j f (x) = |Bq (a)|f (x). Thus, (iv) with 2−j = 0 and 2−k = δ implies (iii).

Finally, we turn to (ii), which has a very similar proof to (iv). Again, we take Φx,δ as μ,x,δ

in Theorem 3.5.4, and we let Xj rem 3.5.1 (c)), we have

d

μ

μ

:= Φ∗x,δ δμ j Xj . Using the fact that Φx,δ (0) = x (Theo-

3.10 Integrals and approximately commuting vector fields

1

d1

Aaδ f (x) = ∫ f (e−t ⋅δ1

2

X 1 −t 2 ⋅δ2d X 2

e

⋅ ⋅ ⋅ e−t

ν

ν

⋅δνd X ν

�

163

Φx,δ (0)) dt

Bq (a)

1

= ∫ f ∘ Φx,δ (e−t ⋅X

1,x,δ

e−t

2

⋅X 2,x,δ

⋅ ⋅ ⋅ e−t

ν

⋅X ν,x,δ

0) dt

Bq (a)

=: ∫ f ∘ Φx,δ (Θx,δ (t)) dt, Bq (a)

where 1

Θx,δ (t) = e−t ⋅X

1,x,δ

e−t

2

⋅X 2,x,δ

⋅ ⋅ ⋅ e−t

ν

⋅X ν,x,δ

0.

Note that μ,x,δ 󵄨 𝜕tμ Θx,δ (t)󵄨󵄨󵄨t=0 = −Xj (0), j

and therefore by Theorem 3.5.1 (h), max

l1 ,...,ln ∈{1,...,q}

󵄨󵄨 󵄨 󵄨󵄨det(𝜕tl Θx,δ (0)| ⋅ ⋅ ⋅ |𝜕tln Θx,δ (0))󵄨󵄨󵄨 ≳ 1. 1

(3.50)

μ,x,2−l

Theorem 3.5.1 (g) shows that ‖Xj ‖C 1 (Bn (1);ℝn ) ≲ 1; thus, if a0 ≳ 1 is sufficiently small, standard theorems from the field of ODEs show that 󵄩󵄩 󵄩 󵄩󵄩Θx,δ 󵄩󵄩󵄩C 1 (Bq (a);ℝn ) ≲ 1,

∀a ∈ (0, a0 ].

(3.51)

Picking l1 , . . . , ln to achieve the max in (3.50) and applying a change of variables in the (tl1 , . . . , tln ) variable, (3.50), (3.51), and the fact that Θx,δ (0) = 0 show that ∫ f ∘ Φx,δ (Θx,δ (t)) dt = ∫ f ∘ Φx,δ,a (v)Jx,δ,a (v) dv,

(3.52)

Bn (1)

Bq (a)

⃗ Ω1 , Ω2 ) > 0 such that Jx,δ,a (v) ≳ 1 on where Jx,δ,a (v) ≥ 0 and there exists η0 = η0 (a, (X, d ), n B (η0 ). Taking Λ(x, δ) and hx,δ as in Theorem 3.5.4 and using (3.52), Aaδ f (x) = Λ(x, δ)−1 ∫ f ∘ Φx,δ (v)hx,δ (v)−1 Jx,δ,a (v)hx,δ (v)Λ(x, δ) dv. Bn (1)

Since Φ∗x,δ Vol = Λ(x, δ)hx,δ σLeb , this implies Aaδ f (x) = Λ(x, δ)−1

∫ Φx,δ (Bn (1))

−1 f (y)hx,δ (Φ−1 x,δ (y))Jx,δ,a (Φx,δ (y)) d Vol(y).

164 � 3 Vector fields and Carnot–Carathéodory geometry Thus, we see −1 Aaδ (x, y) = Λ(x, δ)−1 χΦx,δ (Bn (1)) (y)hx,δ (Φ−1 x,δ (y))Jx,δ,a (Φx,δ (y)).

that

⃗ Ω1 , Ω2 ) ∈ (0, ξ4 ] > 0 so small Using Theorem 3.5.1 (g), we may pick ξ5 = ξ5 (η0 , (X, d ), B(X x,δ ,|d|⃗ ) (0, ξ5 ) ⊆ Bn (η0 /2). 1

This implies that B(W ,ds)⃗ (x, ξ5 δ) ⊆ B(X,d)⃗ (x, ξ5 δ) ⊆ Φx,δ (Bn (η0 )). Since Jx,δ (v) ≳ 1 on Bn (η0 ) and hx,δ ≈ 1 (Theorem 3.5.4 (o)), this implies Aaδ (x, y) ≳ Λ(x, δ)−1 ,

y ∈ B(W ,ds)⃗ (x, ξ5 δ).

Since Ω1 is compact, the formula for Λ gives sup

x∈Ω1 δ∈(0,1]ν

(3.53)

Λ(x, δ) ≲ 1. Thus,

Λ(x, δ) ≈ Λ(x, δ) ∧ 1 ≈ Vol(B(W ,ds)⃗ (x, δ)) ∧ 1,

(3.54)

where the last estimate uses Theorem 3.5.4 (l). Since ξ5 ≈ 1, Theorem 3.5.4 (n) shows that Vol(B(W ,ds)⃗ (x, δ)) ∧ 1 ≈ Vol(B(W ,ds)⃗ (x, ξ5 δ)) ∧ 1.

(3.55)

Plugging (3.54) and (3.55) into (3.53) yields Aaδ (x, y) ≳ (Vol(B(W ,ds)⃗ (x, ξ5 δ)) ∧ 1) ,

y ∈ B(W ,ds)⃗ (x, ξ5 δ),

−1

which completes the proof of (ii). Proof of Proposition 3.10.1. Since B(W ,ds)⃗ (x, 2−j ), B(W ,ds)⃗ (x, 2−k ) ⊆ B(W ,ds)⃗ (x, 2−j∧k ) and the

balls r 󳨃→ B(W ,ds)⃗ (x, r2−j ) are metric balls, it follows that the integral on the left-hand side of (3.45) equals zero unless z ∈ B(W ,ds)⃗ (x, 2 ⋅ 2−j∧k ). Thus, to prove (3.45) it suffices to show that for x, z ∈ Ω, ∫ Ω

χB

(W ,ds)⃗ (x,2

(Vol(B(W ,ds)⃗

−j )

χB

(y)

(x, 2−j ))

(W ,ds)⃗ (y,2

∧ 1) (Vol(B(W ,ds)⃗

1 ≲ . Vol(B(W ,ds)⃗ (x, 2−j∧k )) ∧ 1

−k )

(z)

(y, 2−k ))

∧ 1)

d Vol(y) (3.56)

Fix Ω1 , Ω2 , and Ω3 open with Ω ⋐ Ω1 ⋐ Ω2 ⋐ Ω3 ⋐ M. We apply Lemma 3.10.2 with ξ4 = 1 and take a0 > 0 and ξ5 > 0 (with a = a0 ) as in that lemma. We separate the proof of (3.56) into two cases.

3.10 Integrals and approximately commuting vector fields

�

165

The first case is when |2−j |∞ , |2−k |∞ ≤ ξ5 . In this case, set 2−j = ξ5 2−j and 2−k = ξ5 2−k , ̂

̂

with 2−j , 2−k ∈ (0, 1]ν . We have, for x, y, z ∈ Ω, by Lemma 3.10.2 (ii), ̂

̂

χB

(W ,ds)⃗ (x,2

−j )

(y)

Vol(B(W ,ds)⃗ (x, 2−j )) ∧ 1

χB

a

≲ A −0j ̂ (x, y),

−k (W ,ds)⃗ (y,2 )

(z)

Vol(B(W ,ds)⃗ (y, 2−k )) ∧ 1

2

≲A

a0

2−k ̂

(y, z).

Thus, ∫ Ω

χB

(W ,ds)⃗ (x,2

(Vol(B(W ,ds)⃗ a0

≲ [A −j ̂ A 2

a0

2−k ̂

−j )

χB

(y)

(x, 2−j ))

(W ,ds)⃗ (y,2

∧ 1) (Vol(B(W ,ds)⃗

−k )

(z)

(y, 2−k ))

](x, z) ≲ (Vol(B(W ,ds)⃗ (x, 2

̂ k̂ −j∧

∧ 1)

d Vol(y)

(3.57)

)) ∧ 1) . −1

Since ξ5 ≈ 1, Theorem 3.5.4 (n) shows that Vol(B(W ,ds)⃗ (x, 2−j∧k )) ∧ 1 ≈ Vol(B(W ,ds)⃗ (x, 2−j∧k )) ∧ 1. ̂

̂

Combining this with (3.57) proves (3.56) and completes the case where |2−j |∞ , |2−k |∞ ≤ ξ5 . We turn to the case where either |2−j |∞ > ξ5 or |2−k |∞ > ξ5 . First, we claim that for any δ ∈ [0, ∞)ν \ {0}, x, z ∈ Ω, χB(W ,ds)⃗ (x,δ) (z)

Vol(B(W ,ds)⃗ (x, δ)) ∧ 1

≈

χB(W ,ds)⃗ (z,δ) (x)

Vol(B(W ,ds)⃗ (z, δ)) ∧ 1

(3.58)

.

To prove (3.58), it suffices to consider the case where δ ∈ (0, ∞)ν by the discussion in Section 3.5.2. For z ∈ B(W ,ds)⃗ (x, δ), B(W ,ds)⃗ (x, δ) ⊆ B(W ,ds)⃗ (z, 3δ), and therefore, χB(W ,ds)⃗ (x,δ) (z)

Vol(B(W ,ds)⃗ (x, δ)) ∧ 1

≥ =

χB(W ,ds)⃗ (x,δ) (z)

Vol(B(W ,ds)⃗ (z, 3δ)) ∧ 1 χB(W ,ds)⃗ (z,δ) (x)

Vol(B(W ,ds)⃗ (z, 3δ)) ∧ 1

≈

χB(W ,ds)⃗ (z,δ) (x)

Vol(B(W ,ds)⃗ (z, δ)) ∧ 1

,

where the last estimate uses Theorem 3.5.4 (n). This proves the ≳ part of (3.58); the ≲ part follows by reversing the roles of x and z. Formula (3.58) implies that (3.45) is symmetric in x and z, so that it suffices to prove (3.56) in the case where |2−k |∞ > ξ5 . Because our assumptions are symmetric in the parameters 1, . . . , ν, it suffices to consider only the case 2−k1 > ξ5 . Since 2−k1 > ξ5 , we have Vol(B(W ,ds)⃗ (x, 2−k )) ≥ Vol(B(W 1 ,ds1 ) (x, 2−k1 )) ≥ Vol(B(W 1 ,ds1 ) (x, ξ5 )) ≳ 1, and therefore

(3.59)

166 � 3 Vector fields and Carnot–Carathéodory geometry Vol(B(W ,ds)⃗ (x, 2−j∧k )) ≥ Vol(B(W ,ds)⃗ (x, 2−k )) ≳ 1.

(3.60)

Using (3.59) and (3.60), we see that ∫ Ω

χB

(W ,ds)⃗ (x,2

−j )

χB

(y)

(W ,ds)⃗ (y,2

−k )

(z)

(Vol(B(W ,ds)⃗ (x, 2−j )) ∧ 1) (Vol(B(W ,ds)⃗ (y, 2−k )) ∧ 1) ≲∫ Ω

=

χB

(W ,ds)⃗ (x,2

−j )

(y)

Vol(B(W ,ds)⃗ (x, 2−j )) ∧ 1

Vol(B(W ,ds)⃗ (x, 2−j ) ∩ Ω) Vol(B(W ,ds)⃗

(x, 2−j ))

∧1

d Vol(y)

d Vol(y)

≲1≲

1 , Vol(B(W ,ds)⃗ (x, 2−j∧k )) ∧ 1

where we have used Vol(Ω) < ∞, since Ω is compact. This completes the proof.

3.11 Maximal functions In this section, we present maximal functions adapted to some multi-parameter Carnot– Carathéodory geometries. Let M be a connected C ∞ manifold, with smooth, strictly positive density Vol. Fix ν ∈ ℕ+ , and for each μ ∈ {1, . . . , ν}, let μ

μ

∞ (W μ , dsμ ) = {(W1 , ds1 ), . . . , (Wrμμ , dsμrμ )} ⊂ Cloc (M; TM) × ℕ+ .

Set (W , ds)⃗ = {(W1 , ds1⃗ ), . . . , (Wr , dsr⃗ )} := (W 1 , ds1 ) ⊠ (W 2 , ds2 ) ⋅ ⋅ ⋅ ⊠ (W ν , dsν ) μ

μ

∞ = {(Wj , dsj eμ ) : μ ∈ {1, . . . , ν}, 1 ≤ j ≤ rμ } ⊂ Cloc (M; TM) × (ℕν \ {0}).

(3.61)

We study maximal functions in two situations: The finitely generated setting: In this setting, we assume Gen((W μ , dsμ )) is locally finitely generated on M for each μ ∈ {1, . . . , ν}. The Hörmander setting: In this setting, we assume (W μ , dsμ ) are Hörmander vector fields with formal degrees for each μ ∈ {1, . . . , ν} and that (W 1 , ds1 ), . . . , (W ν , dsν ) pairwise locally weakly approximately commute. By Proposition 3.4.14 (with ν = 1), the Hörmander setting is a special case of the finitely generated setting. The maximal functions we present in the Hörmander setting will perhaps seems a little more familiar to the reader, though as we will see, they are essentially a special case of the maximal functions we present in the finitely generated setting.

3.11 Maximal functions

�

167

3.11.1 The finitely generated setting In this section we assume the assumptions from the finitely generated setting described above. Fix 𝒦 ⋐ Ω1 ⋐ Ω ⋐ M, with Ω1 and Ω open and relatively compact and 𝒦 compact. For each μ ∈ {1, . . . , ν}, pick μ

μ

(X μ , d μ ) = {(X1 , d1 ), . . . , (Xqμμ , dqμμ )} ⊂ Gen((W μ , dsμ )) so that Gen((W μ , dsμ )) is finitely generated by (X μ , d μ ) on Ω. Set (X, d )⃗ = {(X1 , d 1⃗ ), . . . , (Xq , d q⃗ )} := (X 1 , d 1 ) ⊠ (X 2 , d 2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (X ν , d ν ) μ

μ

∞ = {(Xj , dj eμ ) : μ ∈ {1, . . . , ν}, 1 ≤ j ≤ qμ } ⊂ Cloc (M; TM) × (ℕν \ {0}).

(3.62)

⃗ > 0 be a small number, to be chosen later. For t ∈ ℝq , we Let a = a(𝒦, Ω1 , (X, d )) 1 2 ν write t = (t , t , . . . , t ) ∈ ℝq1 × ℝq2 × ⋅ ⋅ ⋅ × ℝqν . For δ = (δ1 , . . . , δν ) ∈ (0, 1]ν , set 1

Aaδ,(X,d)⃗ f (x) := ∫ f (e−t ⋅δ

d1

2

X 1 −t 2 ⋅δd X 2

e

⋅ ⋅ ⋅ e−t

ν

ν

⋅δd X ν

x) dt,

(3.63)

Bq (a)

μ

μ

μ

μ

μ

where, as before, δμd X μ = (δμ 1 X1 , . . . , δμ μ Xqμ ). d

dq

⃗ > 0 is sufficiently small, then for δ ∈ (0, 1]ν , Aa Lemma 3.11.1. If a = a(𝒦, Ω1 , (X, d )) δ,(X,d)⃗

extends to a continuous map Aaδ,(X,d)⃗ : Lp (𝒦, Vol) → Lp (M, Vol), p ∈ [1, ∞], and ∀f ∈ Lp (𝒦, Vol), supp(Aaδ,(X,d)⃗ f ) ⊆ Ω1 . Finally, sup

󵄩 󵄩 sup 󵄩󵄩󵄩Aaδ,(X,d)⃗ 󵄩󵄩󵄩Lp →Lp < ∞. ν

p∈[1,∞] δ∈(0,1]

(3.64)

We prove Lemma 3.11.1 below. In light of Lemma 3.11.1, Aaδ,(X,d)⃗ f (x) is defined almost

everywhere, for f ∈ L1 (𝒦, Vol). For such f , we set a

̃ ⃗ ℳ (X,d),𝒦,a := sup A2−l ,(X,d)⃗ |f |(x). l∈ℕν

(3.65)

The main result of this section is the following. p ⃗ > 0 is sufficiently small, then ℳ ̃ ⃗ Theorem 3.11.2. If a = a(𝒦, Ω1 , (X, d )) (X,d),𝒦,a : L (𝒦, Vol) → ̃ ⃗ Lp (Ω1 , Vol), for p ∈ (1, ∞]. In particular, for f ∈ Lp (𝒦, Vol), supp(ℳ (X,d),𝒦,a f ) ⊆ Ω1 . Also, for p ∈ (1, ∞],

̃ ⃗ p p ‖ℳ (X,d),𝒦,a f ‖L (Ω1 ,Vol) ≲ ‖f ‖L (𝒦,Vol) ,

f ∈ Lp (𝒦, Vol),

(3.66)

168 � 3 Vector fields and Carnot–Carathéodory geometry where the implicit constant depends on p ∈ (1, ∞] but not on f . Moreover, for p ∈ (1, ∞), q ∈ (1, ∞], p

q

p

q

̃ ⃗ ℳ (X,d),𝒦,a : L (𝒦, Vol; ℓ (ℕ)) → L (Ω1 , Vol; ℓ (ℕ)), and for p ∈ (1, ∞), q ∈ (1, ∞], 󵄩󵄩 ̃ 󵄩 󵄩 󵄩 fj }j∈ℕ 󵄩󵄩󵄩Lp (Ω ,Vol;ℓq (ℕ)) ≲ 󵄩󵄩󵄩{fj }j∈ℕ 󵄩󵄩󵄩Lp (𝒦,Vol;ℓq (ℕ)) , 󵄩󵄩{ℳ(X,d),𝒦,a ⃗ 1 {fj }j∈ℕ ∈ Lp (𝒦, Vol; ℓq (ℕ)),

(3.67)

where the implicit constant depends on p and q but not on fj . The rest of this section is devoted to the proof of Theorem 3.11.2. We begin with a result which implies Lemma 3.11.1. μ

μ

∞ Lemma 3.11.3. For each μ ∈ {1, . . . , ν}, let Z1 , . . . , Zqμ ∈ Cloc (M; TM). Let Z μ denote the μ

μ

vector of vector fields Z μ = (Z1 , . . . , Zqμ ). Then there exists a = a(Z 1 , . . . , Z ν , 𝒦, Ω1 ) > 0 such that the following holds ∀ζ ∈ L1 (Bq1 +q2 +⋅⋅⋅+qν (a)). For f ∈ C(M) with supp(f ) ⊆ 𝒦, set 1

q

Sζ f (x) := ∫ f (e−t ⋅Z e−t

2

⋅Z 2

⋅ ⋅ ⋅ e−t

ν

⋅Z ν

x)ζ (t 1 , . . . t ν ) dt 1 dt 2 ⋅ ⋅ ⋅ dt ν .

Then Sζ extends to a bounded operator Sζ : Lp (𝒦, Vol) → Lp (M, Vol), for p ∈ [1, ∞]. Moreover, supp(Sδ f ) ⊆ Ω1 , ∀f ∈ Lp (𝒦, Vol). Finally, ‖Sζ ‖Lp (𝒦,Vol)→Lp (Ω1 ,Vol) ≤ C‖ζ ‖L1 ,

(3.68)

where C ≥ 0 does not depend on p ∈ [1, ∞] or ζ , though it may depend on any other ingredient. Proof. Set q = q1 + q2 + ⋅ ⋅ ⋅ + qν . Fix Ω−1 , Ω0 open with 𝒦 ⋐ Ω−1 ⋐ Ω0 ⋐ Ω1 . For t = 1 1 2 2 ν ν (t 1 , . . . , t ν ) ∈ ℝq ≅ ℝq1 × ℝq2 × ⋅ ⋅ ⋅ × ℝqν , let Γt (x) := e−t ⋅Z e−t ⋅Z ⋅ ⋅ ⋅ e−t ⋅Z x. For a > 0 sufficiently small and t ∈ Bq (a), the Picard–Lindelöf theorem shows that Γt (x) ∈ Ω0 is defined for all x ∈ Ω−1 . Standard results from the field of ODEs show that Γt (x) is C ∞ in both variables. 󵄨 Since dx Γt (x)󵄨󵄨󵄨t=0 = I, if a is sufficiently small and t ∈ Bq (a), a simple change of variables shows that for any g ∈ C(M) and t ∈ Bq (a), ∫ g(Γt (x)) d Vol(x) = ∫ g(u)J(t, u) d Vol(u), Ω−1

Ω

where supt,u |J(t, u)| < ∞. Thus, for f ∈ C(M) with supp(f ) ⊆ Ω−1 , we have supp(Sζ (f )) ⊆ Ω0 ⊆ Ω1 , and

3.11 Maximal functions

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169

∫ |Sζ f (x)| d Vol(x) ≤ ∫ ∫ |f (Γt (x))||ζ (t)| dt d Vol(x), Ω1

Ω1

∫ ∫ |g(u)||J(t, u)||ζ (t)| dt d Vol(x) ≲ ‖g‖L1 (Ω−1 ,Vol) ‖ζ ‖L1 .

Ω1

Thus, Sζ extends to a bounded operator L1 (Ω−1 , Vol) → L1 (Ω1 , Vol) and (3.68) holds for p = 1; in particular, this extended operator is well-defined on almost everywhere equivalence classes of functions in L1 (Ω−1 , Vol). For f ∈ L∞ (Ω−1 , Vol) ⊆ L1 (Ω−1 , Vol), Sζ f is, therefore, defined almost everywhere. Furthermore, we may take fk ∈ C(Ω−1 ) with fk → f in L1 and ‖fk ‖L∞ ≤ ‖f ‖L∞ , ∀k. By taking a subsequence, we may ensure fk → f almost everywhere and Sζ fk → Sζ f almost everywhere. We have, for almost every x, 󵄨 󵄨 |Sζ f (x)| = lim |Sζ fk (x)| ≤ lim inf ∫󵄨󵄨󵄨fk (Γt (x))󵄨󵄨󵄨|ζ (t)| dt k→∞

k→∞

≤ lim inf ‖fk ‖L∞ ‖ζ ‖L1 ≤ ‖f ‖L∞ ‖ζ ‖L1 . k→∞

This proves the lemma for p = ∞. The Riesz–Thorin interpolation theorem completes the proof. Proof of Lemma 3.11.1. We have 1

Aaδ,(X,d)⃗ f (x) = ∫ f (e−t ⋅δ

d1

2

X 1 −t 2 ⋅δd X 2

e

⋅ ⋅ ⋅ e−t

ν

ν

⋅δd X ν

x) dt

Bq (a)

1

1

= ∫ f (e−t ⋅X e−t

2

⋅X 2

⋅ ⋅ ⋅ e−t

ν

⋅X ν

1

−|d 1 |1 −|d 2 |1 δ2

x)δ1

⋅ ⋅ ⋅ δν−|d

ν

|1

ν

× χBq (a) (δ1−d t 1 , . . . , δν−d t ν ) dt 1

1

=: ∫ f (e−t ⋅X e−t μ

μ

μ

−d

μ

−dqμ μ tqμ ).

where δμ−d t μ = (δμ 1 t1 , . . . , δμ

2

⋅X 2

⋅ ⋅ ⋅ e−t

ν

⋅X ν

x)ζδ (t) dt,

The result now follows from Lemma 3.11.3.

Lemma 3.11.4. It suffices to prove Theorem 3.11.2 in the special case ν = 1. Proof. Suppose we know Theorem 3.11.2 in the case ν = 1. Note that the hypotheses of Theorem 3.11.2 hold with (W , ds)⃗ and (X, d )⃗ replaced by (W μ , dsμ ) and (X μ , d μ ), for each μ ∈ {1, . . . , ν}. Fix open sets ν

ν

ν−1

𝒦 ⋐ Ω1 ⋐ Ω2 ⋐ Ω1

⋐ Ων−1 ⋐ ⋅ ⋅ ⋅ ⋐ Ω11 ⋐ Ω12 ⋐ Ω1 . 2

170 � 3 Vector fields and Carnot–Carathéodory geometry μ

μ

Let aμ = aμ (Ω1 , Ω2 , (X μ , d μ )) > 0 be as in the ν = 1 case of Theorem 3.11.2 applied ⃗ > 0 is small enough, we have, for δ = to (X μ , d μ ). Note that if a = a(𝒦, Ω1 , (X, d )) ν (δ1 , . . . , δν ) ∈ (0, 1] , 1

󵄨 Aaδ,(X,d)⃗ |f |(x) = ∫ 󵄨󵄨󵄨f (e−t ⋅δ

d1 1 X

e−t

2

2

⋅δd X 2

⋅ ⋅ ⋅ e−t

ν

ν

⋅δd X ν

Bq (a)

≤

1 d1 1 2 d 󵄨 ∫ ⋅ ⋅ ⋅ ∫ 󵄨󵄨󵄨f (e−t ⋅δ X e−t ⋅δ

∫

Bq1 (a1 ) Bq2 (a2 )

=

󵄨 x)󵄨󵄨󵄨 dt 2

X2

⋅ ⋅ ⋅ e−t

ν

ν

⋅δd X ν

Bqν (aν )

󵄨 x)󵄨󵄨󵄨 dt 1 dt 2 ⋅ ⋅ ⋅ dt ν

a a a Aδ1 ,(X 1 ,d 1 ) Aδ2 ,(X 2 ,d 2 ) ⋅ ⋅ ⋅ Aδν ,(X ν ,d ν ) |f |(x). ν 1 2

Moreover, if supp(f ) ⊆ 𝒦, then a

a

a

μ−1

supp(Aδμ ,(X μ ,d μ ) Aδμ+1 ,(X μ ,d μ ) ⋅ ⋅ ⋅ Aδν ,(X ν ,d ν ) f ) ⊂ Ω1 . μ

μ+1

ν

Thus, if supp(f ) ⊆ 𝒦, j ∈ ℕν , ̃ 1 1 1 ℳ ̃ 2 2 2 ⋅⋅⋅ℳ ̃ ν ν ν f (x), Aa2−j ,(X,d)⃗ |f |(x) ≤ ℳ (X ,d ),Ω ,a (X ,d ),Ω ,a (X ,d ),Ω ,a 1

1

ν

1

2

1

and therefore, ̃ ⃗ ̃ ̃ ̃ ℳ (X,d),𝒦,a f (x) ≤ ℳ(X 1 ,d 1 ),Ω1 ,a ℳ(X 2 ,d 2 ),Ω2 ,a ⋅ ⋅ ⋅ ℳ(X ν ,d ν ),Ων ,a f (x). 1

1

1

2

1

ν

̃ ⃗ From here the bounds for ℳ (X,d),𝒦,a follow from the case ν = 1 applied to each ̃ μ μ μ . ℳ (X ,d ),Ω ,a 1

μ

In light of Lemma 3.11.4, it suffices to prove Theorem 3.11.2 in the case ν = 1. We do this in two steps. In the first, we prove a quantitative result on the unit ball in ℝN . Then we combine this quantitative result with Proposition 3.9.1 to complete the proof. 3.11.1.1 A result on the unit ball Fix N ∈ ℕ+ and let (Y , d̂) = {(Y1 , d̂1 ), . . . , (Yq , d̂q )} ⊂ C ∞ (Bn (1)) × ℕ+ be C ∞ vector fields on BN (1), paired with formal degrees. We suppose inf

max

u∈BN (1) j1 ,...,jN ∈{1,...q}

󵄨󵄨 󵄨 󵄨󵄨det(Yj1 (u)| ⋅ ⋅ ⋅ |YjN (u))󵄨󵄨󵄨 ≥ γ0 > 0.

(3.69)

̃ Fix η4 ∈ (0, 1/2] and η3 ∈ (0, η4 ). We will study the maximal function ℳ (Y ,d̂),BN (η3 ),a defined by (3.65) in the case ν = 1. What is crucial is keeping track of what our estimates depend on. For this, we introduce the following definition, for use just in this section.

3.11 Maximal functions

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171

Definition 3.11.5. For a parameter ι, we say C ∈ ℝ is an ι-admissible constant if C can be chosen to depend only on ι and upper bounds for N, q, γ0−1 , (η4 − η3 )−1 , max1≤j≤q d̂j , and max ‖Yj ‖C L (Bq (1);ℝN ) ,

j∈{1,...q}

where L can be chosen to depend only on ι and upper bounds for N and q. We write A ≲ι B for A ≤ CB, where C is an ι-admissible constant. We say C is an admissible constant if it is a 0-admissible constant. The main result of this section is the next proposition. Proposition 3.11.6. There is an admissible constant a ∈ (0, 1] such that for p ∈ (1, ∞], we have 󵄩󵄩 ̃ 󵄩 󵄩󵄩ℳ(Y ,d̂),BN (η ),a f 󵄩󵄩󵄩Lp (BN (η4 ),σLeb ) ≤ Cp ‖f ‖Lp , 3

∀f ∈ L1 (BN (η3 ); σLeb ),

where Cp ≥ 0 is a p-admissible constant. Also, for p ∈ (1, ∞), q ∈ (1, ∞], 󵄩󵄩 ̃ 󵄩 󵄩 󵄩 󵄩󵄩{ℳ(Y ,d̂),BN (η ),a fj }j∈ℕ 󵄩󵄩󵄩Lp (BN (η4 ),σLeb ;ℓq (ℕ)) ≤ Cp,q 󵄩󵄩󵄩{fj }j∈ℕ 󵄩󵄩󵄩Lp (BN (η3 ),σLeb ;ℓq (ℕ)) , 3 {fj }j∈ℕ ∈ Lp (BN (η3 ), σLeb ; ℓq (ℕ)),

where Cp,q ≥ 0 is a p, q-admissible constant. The rest of this section is devoted to the proof of Proposition 3.11.6. To do this, we introduce a more standard maximal function. For f ∈ L1 (BN (η3 ); σLeb ) and δ1 ∈ (0, 1], we set 1 Vol(B δ∈(0,δ1 ] (Y ,d̂) (x, δ))

̂(Y ,d̂),δ f (x) := sup ℳ 1

∫

|f (u)| du.

B(Y ,d̂) (x,δ)

Proposition 3.11.7. There exists an admissible constant δ1 > 0 such that for p ∈ (1, ∞], we have 󵄩󵄩 ̂ 󵄩 󵄩󵄩ℳ(Y ,d̂),δ1 f 󵄩󵄩󵄩Lp ≤ Cp ‖f ‖Lp ,

∀f ∈ L1 (BN (η3 ); σLeb ),

(3.70)

where Cp ≥ 0 is a p-admissible constant. Also, for p ∈ (1, ∞), q ∈ (1, ∞], 󵄩󵄩 ̂ 󵄩 󵄩 󵄩 󵄩󵄩{ℳ(Y ,d̂),δ1 fj }j∈ℕ 󵄩󵄩󵄩Lp (BN (η4 ),σLeb ;ℓq (ℕ)) ≤ Cp,q 󵄩󵄩󵄩{fj }j∈ℕ 󵄩󵄩󵄩Lp (BN (η3 ),σLeb ;ℓq (ℕ)) , p

N

q

{fj }j∈ℕ ∈ L (B (η3 ), σLeb ; ℓ (ℕ)),

where Cp,q ≥ 0 is a p, q-admissible constant.

(3.71) (3.72)

172 � 3 Vector fields and Carnot–Carathéodory geometry Proof. Because Y1 , . . . , Yq span the tangent space to BN (1) at every point, they satisfy Hörmander’s condition of order 1 on BN (1). In particular, (3.69) together with the definition of admissible constants implies q

l [Yj , Yk ] = ∑ cj,k Yl , l=1

l cj,k Yl ,

l cj,k ∈ C ∞ (BN (1)),

l with ‖cj,k ‖C L (BN (1)) ≲L 1, ∀L ∈ ℕ. Thus, Theorem 3.3.7 applies with (W , ds) = (X, d ) :=

(Y , d̂), 𝒦 = BN (η4 ), and Ω = BN ((1 + η4 )/2). Moreover, by tracing through the proof, it is immediate to see that all of the constants in that theorem can be taken to be admissible constants in the sense of Definition 3.11.5. In particular, by Theorem 3.3.7 (e), we have σLeb (B(Y ,d̂) (x, 2δ)) ≲ σLeb (B(Y ,d̂) (x, δ)),

x ∈ BN (x, 2δ), δ ∈ (0, δ0 ],

(3.73)

where δ0 > 0 is the admissible constant from Theorem 3.3.7. Formula (3.73) is the main inequality needed to show that the balls B(Y ,d̂) (x, δ), δ ∈ (0, 1], when paired with σLeb , locally give BN (η4 ) the structure of a space of homogeneous type in the sense of Coifman and Weiss [52]. By taking δ1 = δ0 , the result now follows from standard methods: (3.70) follows from the methods in [216, Chapter I, Section 3.1] and (3.71) follows from the methods in [83]. Proof of Proposition 3.11.6. Let δ1 ∈ (0, 1] be the admissible constant from Proposition 3.11.7. Similar to the remarks in the proof of Proposition 3.11.7, Lemma 3.10.2 applies with ξ4 = δ1 , ν = 1, (W , ds)⃗ = (X, d )⃗ := (Y , d̂), Ω1 = BN (η4 ), and Ω2 = BN ((1 + η4 )/2); moreover, by tracing through the proof it is easy to see that each of the constants in Lemma 3.10.2 can be chosen to be an admissible constant in the sense of Definition 3.11.5. We let a = a0 ∧ 1, where a0 > 0 is as in Lemma 3.10.2. By Lemma 3.10.2 (iii), we have, for x ∈ BN (η4 ) = Ω1 and f ∈ L1 (BN (η3 )), ̂(Y ,d̂),δ f (x), Aaδ,(Y ,d̂) |f |(x) ≲ ℳ 1 and therefore, for x ∈ BN (η4 ) and f ∈ L1 (BN (η3 )), ̃ ℳ (Y ,d̂),BN (η

3 ),a

̂(Y ,d̂),δ f (x). f (x) ≲ ℳ 1

From here the result follows immediately from Proposition 3.11.7. 3.11.1.2 Proof of Theorem 3.11.2 In this section, we use Propositions 3.9.1 and 3.11.6 to complete the proof of Theorem 3.11.2. As described in Lemma 3.11.4, it suffices to consider only the case ν = 1.

3.11 Maximal functions

�

173

Thus, we assume we are given ∞ (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} ⊂ Cloc (M; TM) × ℕ+ ,

such that Gen((W , ds)) is locally finitely generated on M. We pick (X, d ) = {(X1 , d1 ), . . . , (Xq , dq )} ⊂ Gen((W , ds)) such that (W , ds) is finitely generated by (X, d ) on Ω. Theorem 3.5.1 applies (with ν = 1) to these vector fields. We let Φx,δ , N(x), and X x,δ = (X1x,δ , . . . , Xqx,δ ) ⊂ C ∞ (BN(x) (1); TBN(x) (1)) × ℕ+ be as in that theorem, for x ∈ 𝒦, δ ∈ (0, 1]. For x ∈ 𝒦, δ ∈ (0, 1], define Aaδ,(X x,1 ,d) by the ν = 1 case of (3.63). Taking (Y , d̂) = (X x,δ , d ), it follows from Theorem 3.5.1 (g) and (h) that admissible constants as defined in Definition 3.11.5 can be chosen independent of x ∈ 𝒦. This leads us to the next result. Lemma 3.11.8. Let η3 , η4 ∈ (0, 1/2], η3 < η4 . There exists a = a((X, d ), 𝒦, Ω1 , η3 , η4 ) ∈ (0, 1] such that the following holds. For p ∈ (1, ∞] there exists C = C(p, (X, d ), 𝒦, Ω1 , η3 , η4 ) ≥ 0 such that ∀x ∈ 𝒦, 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 a ≤ ‖g‖Lp (BN(x) (η3 )) , 󵄩󵄩sup A2−l ,(X x,1 ,d) |g|󵄩󵄩󵄩 󵄩󵄩 l∈ℕ 󵄩󵄩 p N(x) L (B (η4 ))

∀g ∈ Lp (BN(x) (η3 )).

For p ∈ (1, ∞), q ∈ (1, ∞], there exists C = C(p, q, (X, d ), 𝒦, Ω1 , η3 , η4 ) ≥ 0 such that ∀x ∈ 𝒦, 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 a ≤ C 󵄩󵄩󵄩{gj }j∈ℕ 󵄩󵄩󵄩Lp (BN(x) (η ),σ ;ℓq (ℕ)) , 󵄩󵄩{sup A2−l ,(X x,1 ,d) |gj |} 󵄩󵄩󵄩 3 Leb 󵄩󵄩 l∈ℕ j∈ℕ 󵄩 󵄩Lp (BN(x) (η4 ),σLeb ;ℓq (ℕ))

{gj }j∈ℕ ∈ Lp (BN(x) (η3 ), σLeb ; ℓq (ℕ)). Proof. Note that for g ∈ L1 (BN(x) (η3 )), ̃ x,1 ℳ (X ,d),BN (η

3 ),a

g(u) = sup Aa2−l ,(X x,1 ,d) |g|(u). l∈ℕ

Thus, this result follows from Proposition 3.11.6 applied with (Y , d̂) = (X x,1 , d ). The main point here, as discussed above, is that admissible constants as defined in Definition 3.11.5 can be chosen independent of x ∈ 𝒦, and therefore the conclusions of Proposition 3.11.6 are uniform in x ∈ 𝒦. Proof of Theorem 3.11.2. We will pick a = a(𝒦, Ω1 , (X, d )) > 0 in the proof; we will choose it small enough that Lemma 3.11.1 holds with our choice of a. In particular, if supp(f ) ⊆ 𝒦, then supp(Aaδ,(X,d) ) ⊆ Ω1 .

174 � 3 Vector fields and Carnot–Carathéodory geometry When p = ∞, (3.66) follows immediately from the p = ∞ case of (3.64). Thus, it suffices to consider only the case p ∈ (1, ∞). By possibly increasing the size of 𝒦, it suffices to consider the case where 𝒦 = Ω0 and Ω0 is an open set with Ω0 ⋐ Ω1 . In this case, by a density argument, it suffices to prove (3.66) and (3.67) in the case where f and fj are continuous. We henceforth assume these functions are continuous. Fix a compact set 𝒦′ and open sets Ω′1 , Ω3 , and Ω4 with 𝒦 ⋐ Ω1 ⋐ Ω3 ⋐ Ω4 ⋐ 𝒦′ ⋐ ′ Ω1 ⋐ M. We will apply our previous results (e. g., Lemma 3.11.8) with 𝒦 and Ω1 replaced by 𝒦′ and Ω′1 , respectively. Let η1 = η1 (𝒦′ , Ω3 , Ω4 , (X, d )) ∈ (0, 1] be as in Proposition 3.9.1. Set η3 := η1 /4 and η4 = η1 /2. Take a = a((X, d ), 𝒦′ , Ω′1 , η3 , η4 ) ∈ (0, 1] small enough that Lemma 3.11.8 holds. Note that for f ∈ C(M) and x ∈ 𝒦, we have (Aaδ,(X,d) f ) ∘ Φx,1 (u) = ∫ f (e−t⋅δ X Φx,1 (u)) dt d

Bq (a)

= ∫ f ∘ Φx,1 (e−t⋅δ

d

X x,1

Bq (a)

u) dt = Aaδ,(X x,1 ,d) (f ∘ Φx,1 ).

Fix M ∈ ℕ+ . For p ∈ (1, ∞) and f ∈ C(M) with supp(f ) ⊆ 𝒦, using Proposition 3.9.1 and Lemma 3.11.8, we have with implicit constants independent of M ∈ ℕ 󵄩󵄩 󵄩󵄩p 󵄩󵄩 󵄩 󵄩󵄩 sup Aa2−l ,(X,d) |f |󵄩󵄩󵄩 ≈∫ 󵄩󵄩l∈{0,...,M} 󵄩󵄩 p 󵄩 󵄩L (Ω1 ,Vol) Ω 4

=∫

∫ BN(x) (η3 )

∫

Ω4 BN(x) (η3 )

≲∫

∫

Ω4 BN(x) (η4 )

󵄨 󵄨p sup 󵄨󵄨󵄨(Aa2−l ,(X,d) f )(Φx,1 (u))󵄨󵄨󵄨 du d Vol(x)

l∈{0,...M}

󵄨 󵄨p sup 󵄨󵄨󵄨Aa2−l ,(X x,1 ,d) (f ∘ Φx,1 )(u)󵄨󵄨󵄨 du d Vol(x)

l∈{0,...M}

󵄨󵄨 󵄨 󵄨󵄨f ∘ Φx,1 (u)󵄨󵄨󵄨 du d Vol(x) p

≈ ∫ |f (x)|p d Vol(x) = ‖f ‖Lp . Ω3

Taking the limit M → ∞ and applying the monotone convergence theorem, (3.66) is proved. A very similar proof gives (3.67) in the case where all but finitely many of the fj are identically zero. The monotone convergence theorem then gives (3.67) in the general case.

3.11.2 The Hörmander setting In this section, we introduce a maximal function in the “Hörmander setting” described above. Thus, for each μ ∈ {1, . . . , ν}, we are given Hörmander vector fields with formal

3.11 Maximal functions

�

175

degrees μ

μ

∞ (W μ , dsμ ) = {(W1 , ds1 ), . . . , (Wrμμ , dsμrμ )} ⊂ Cloc (M; TM) × ℕ+ .

We assume that (W 1 , ds1 ), . . . , (W ν , dsν ) pairwise locally weakly approximately commute. We define (W , ds)⃗ by (3.61). For f ∈ L1 (M, Vol), set ℳ(W ,ds)⃗ f (x) :=

sup

δ∈(0,∞)ν

1 Vol(B(W ,ds)⃗ (x, δ)) ∧ 1

|f (y)| d Vol(y).

∫ B(W ,ds)⃗ (x,δ)

Theorem 3.11.9. Fix a relatively compact, open set Ω ⋐ M. Then, for p ∈ (1, ∞], there exists Cp,Ω,(W ,ds)⃗ ≥ 0 such that 󵄩󵄩 󵄩 󵄩󵄩ℳ(W ,ds)⃗ f 󵄩󵄩󵄩Lp (Ω,Vol) ≤ Cp,Ω,(W ,ds)⃗ ‖f ‖Lp (Ω,Vol) ,

∀f ∈ Lp (Ω, Vol).

Also, for p ∈ (1, ∞), q ∈ (1, ∞], there exists Cp,q,Ω,(W ,ds)⃗ ≥ 0 such that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩{ℳ(W ,ds)⃗ fj }j∈ℕ 󵄩󵄩󵄩Lp (Ω,Vol;ℓq (ℕ)) ≤ Cp,q,Ω,(W ,ds)⃗ 󵄩󵄩󵄩{fj }j∈ℕ 󵄩󵄩󵄩Lp (Ω,Vol;ℓq (ℕ)) , ∀{fj }j∈ℕ ∈ Lp (Ω, Vol; ℓq (ℕ)).

Proof. Let ξ5 ∈ (0, 1] be a small number, to be chosen later. Define ℳ(W ,ds),ξ f (x) := sup ⃗ 5

j∈ℕν

1 Vol(B(W ,ds)⃗ (x, ξ5 2−j )) ∧ 1

|f (y)| d Vol(y).

∫ B(W ,ds)⃗ (x,ξ5

2−j )

We claim, for f ∈ L1 (Ω, Vol) and x ∈ Ω, ℳ(W ,ds)⃗ f (x) ≲ ℳ(W ,ds),ξ f (x) + χΩ (x) ∫ |f (y)| d Vol(y). ⃗ 5

(3.74)

Indeed, we will show, for such x and f and for δ ∈ (0, ∞)ν , 1

Vol(B(W ,ds)⃗ (x, δ)) ∧ 1

|f (y)| d Vol(y)

∫ B(W ,ds)⃗ (x,δ)

(3.75)

≲ ℳ(W ,ds),ξ f (x) + χΩ (x) ∫ |f (y)| d Vol(y), ⃗ 5

which immediately implies (3.74), by taking the supremum over δ. We separate the proof of (3.75) into two cases. The first is when |δ|∞ ≤ ξ5 . In this case, pick l = (l1 , . . . , lν ) ∈ ℕν such that ∀μ ∈ {1, . . . , ν}, we have ξ5 2−lμ −1 ≤ δμ ≤ ξ5 2−lμ . Then ∫ B(W ,ds)⃗ (x,δ)

|f (y)|d Vol(y) ≤

∫ B(W ,ds)⃗ (x,ξ5 2−l )

|f (y)| d Vol(y),

(3.76)

176 � 3 Vector fields and Carnot–Carathéodory geometry and using Theorem 3.5.4 (n), Vol(B(W ,ds)⃗ (x, ξ5 2−l )) ∧ 1 ≈ Vol(B(W ,ds)⃗ (x, δ)) ∧ 1.

(3.77)

Combining (3.76) and (3.77) shows 1 Vol(B(W ,ds)⃗ (x, δ)) ∧ 1 ≲

|f (y)| d Vol(y)

∫ B(W ,ds)⃗ (x,δ)

1 Vol(B(W ,ds)⃗ (x, ξ5 2−l )) ∧ 1

|f (y)| d Vol(y)

∫ B(W ,ds)⃗ (x,ξ5

2−l )

≤ ℳ(W ,ds),ξ f (x), ⃗ 5

completing the proof of (3.75) when |δ|∞ ≤ ξ5 . Suppose |δ|∞ > ξ5 . Without loss of generality, we assume δ1 > ξ5 . We apply Theorem 3.5.4 with Ω replaced by some Ω′ ⋐ M with Ω ⋐ Ω′ . Using Theorem 3.5.4 (l) and the formula for Λ from that theorem, we have Vol(B(W ,ds)⃗ (x, δ)) ∧ 1 ≈ Λ(x, δ) ∧ 1 ≥ Λ(x, (ξ5 , 0, 0, . . . , 0)) ∧ 1 ≥ inf′ Λ(y, (ξ5 , 0, 0, . . . , 0)) ∧ 1 ≳ 1. y∈Ω

Therefore, 1 Vol(B(W ,ds)⃗ (x, δ)) ∧ 1

∫

|f (y)| d Vol(y) ≲ ∫ |f (y)| dy.

B(W ,ds)⃗ (x,δ)

Since we are only considering x ∈ Ω, this proves (3.75) and completes the proof of (3.74). Thus, it suffices to estimate the two terms on the right-hand side of (3.74). We begin with the second term. Using that Vol(Ω) < ∞, since Ω is relatively compact, we have Vol(Ω) ≲ 1, since our estimates are allowed to depend on Ω. We have, for p ∈ [1, ∞] and f ∈ L1 (Ω, Vol), using Hölder’s inequality, 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 ≲ ‖f ‖L1 (Ω,Vol) ≲ ‖f ‖Lp (Ω,Vol) . 󵄩󵄩(∫ |f (y)| d Vol(y))χΩ 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 Ω 󵄩󵄩Lp (Ω,Vol) Similarly, using the triangle inequality and Hölder’s inequality, for p, q ∈ [1, ∞] and {fj }j∈ℕ ∈ L1 (Ω, Vol; ℓq (ℕ)), we have

3.12 Approximately commuting partial differential operators �

177

󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩{(∫ |fj (y)| d Vol(y))χΩ } 󵄩󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 Ω 󵄩󵄩Lp (Ω,Vol;ℓq (ℕ)) j∈ℕ 󵄩

1 1󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 q q󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩∫( ∑ |fj (y)|q ) d Vol(y)󵄩󵄩󵄩󵄩 ≈ 󵄩󵄩󵄩󵄩( ∑ |fj |q ) 󵄩󵄩󵄩󵄩 󵄩󵄩 j∈ℕ 󵄩󵄩 󵄩󵄩 j∈ℕ 󵄩󵄩 󵄩󵄩Lp (Ω,Vol) 󵄩󵄩 󵄩󵄩L1 (Ω,Vol) 󵄩󵄩Ω

≲ ‖{fj }j∈ℕ ‖Lp (Ω,Vol;ℓq (ℕ)) . The completes the proof of the desired estimates for the second term on the right-hand side of (3.74). Finally, we estimate the first term on the right-hand side of (3.74). For this we need to choose ξ5 > 0. Fix relatively compact open sets Ω1 , Ω2 , Ω′1 , and Ω′ and a compact set 𝒦 with Ω ⋐ Ω1 ⋐ Ω2 ⋐ 𝒦 ⋐ Ω′1 ⋐ Ω′ ⋐ M. By Proposition 3.4.14 in the case ν = 1, we may pick (X μ , d μ ) ⊂ Gen((W μ , dsμ )) such that Gen((W μ , dsμ )) is finitely generated by (X μ , d μ ) on Ω′ . We define (X, d )⃗ by (3.62) and ⃗ > 0 as in Theorem 3.11.2. Aaδ,(X,d)⃗ by (3.63). Take a = a(𝒦, Ω′1 , (X, d )) ⃗ > 0 be as in Lemma 3.10.2 (with ξ4 = 1 in that lemma); Let a0 = a0 (1, Ω1 , Ω2 , (X, d )) ⃗ > 0 be by replacing a with a ∧ a0 , we may assume a ∈ (0, a0 ]. Let ξ5 = ξ5 (a, Ω1 , Ω2 , (X, d )) as in Lemma 3.10.2. Using Lemma 3.10.2 (ii), we have, for l ∈ ℕν , x ∈ Ω1 , 1 Vol(B(W ,ds)⃗ (x, ξ5 2−l )) ∧ 1 ≲

Aa2−l ,(X,d)⃗ |f |(x)

|f (y)| d Vol(y)

∫ B(W ,ds)⃗ (x,ξ5

2−l )

̃ ⃗ ≤ℳ (X,d),𝒦,a f (x).

̃ ⃗ Taking the supremum over l ∈ ℕν , we have ℳ(W ,ds),ξ f (x), for x ∈ Ω. ⃗ 5 f (x) ≲ ℳ(X,d),𝒦,a The desired estimates for ℳ(W ,ds),ξ ⃗ 5 follow immediately from Theorem 3.11.2, which completes the proof.

3.12 Approximately commuting partial differential operators ∞ Let ν1 , ν2 ∈ ℕ+ and for j = 1, 2, 𝒮j ⊆ Cloc (M; TM) × (ℕνj \ {0}). If 𝒮1 and 𝒮2 locally weakly approximately commute, the main result of this section shows that partial differential operators which are polynomials in the vector fields from 𝒮1 and 𝒮2 also “approximately commute.” We turn to making this precise. Throughout this section, Ω ⋐ M will denote a relatively compact, open set. Also, ν1 , ν2 , ν will denote elements of ℕ+ and ∞ ∞ 𝒮j ⊆ Cloc (M; TM) × (ℕνj \ {0}), for j = 1, 2 and 𝒮 ⊆ Cloc (M; TM) × (ℕν \ {0}).

178 � 3 Vector fields and Carnot–Carathéodory geometry Definition 3.12.1. Given κ ∈ ℤν and k, L ∈ ℤ, we say a partial differential operator on Ω with smooth coefficients, P , is an 𝒮 partial differential operator on Ω of degree ≤ κ, length ≤ k, and order ≤ L if: – If κ ∈ ̸ ℕν , or k ≤ −1, or L ≤ −1, then P = 0. – If κ ∈ ℕν and k, L ∈ ℕ, then P can be written as a finite sum of terms of the form Mult[g]Z1 Z2 ⋅ ⋅ ⋅ ZK , ∞ where g ∈ Cloc (Ω), K ≤ L, and (Z1 , dr1⃗ ), . . . , (ZK , drk⃗ ) ∈ 𝒮 with dr1⃗ + dr2⃗ + ⋅ ⋅ ⋅ + drK⃗ ≤ κ ⃗ ⃗ and |dr1 + dr2 + ⋅ ⋅ ⋅ + drK⃗ |1 ≤ k.

We say P is an 𝒮 partial differential operator on Ω of degree ≤ κ and length ≤ k if there is an L ∈ ℤ such that P is an 𝒮 partial differential operator on Ω of degree ≤ κ, length ≤ k, and order ≤ L. We say P is an 𝒮 partial differential operator on Ω of degree ≤ κ if there is a k ∈ ℤ such that P is an 𝒮 partial differential operator of degree ≤ κ and length ≤ k. Remark 3.12.2. Note that if P is an 𝒮 partial differential operator on Ω of degree ≤ κ ∈ ℕν , then it is a partial differential operator on Ω of degree ≤ κ and length ≤ |κ|1 . Remark 3.12.3. We will sometimes speak of 𝒮 partial differential operators on Ω of degree ≤ κ, where κ ∈ ℝν . The definition is the same, and is equivalent to replacing κ = (κ1 , . . . , κν ) with (⌊κ1 ⌋, . . . , ⌊κν ⌋) ∈ ℤν . Lemma 3.12.4. Suppose, for j = 1, 2, Pj is an 𝒮 partial differential operator on Ω of degree ≤ κj , length ≤ kj , and order ≤ Lj . Then P1 P2 is an 𝒮 partial differential operator on Ω of degree ≤ κ1 + κ2 , length ≤ k1 + k2 , and order ≤ L1 + L2 . Proof. If κ1 ∈ ℤν \ ℕν , κ2 ∈ ℤν \ ℕν , k1 ≤ −1, k2 ≤ −1, L1 ≤ −1, or L2 ≤ −1, then P1 P2 = 0 and the result is trivial. Thus, we may assume κ1 , κ2 ∈ ℕν and k1 , k2 , L1 , L2 ∈ ℕ. By the definition, P1 can be written as a finite sum of terms of the form Mult[g]Z1 Z2 ⋅ ⋅ ⋅ ZK1 ,

(3.78)

∞ where g ∈ Cloc (Ω), K1 ≤ L1 , and (Z1 , dr1⃗ ), . . . , (ZK1 , drK⃗ 1 ) ∈ 𝒮 with dr1⃗ + ⋅ ⋅ ⋅ + drK⃗ 1 ≤ κ1 and ⃗ ⃗ |dr1 + ⋅ ⋅ ⋅ + drK1 |1 ≤ k1 . And P2 can be written as a finite sum of terms of the form

Mult[f ]Y1 Y2 ⋅ ⋅ ⋅ YK2 ,

(3.79)

∞ where f ∈ Cloc (Ω), K2 ≤ L2 , and (Y1 , d̂1⃗ ), . . . , (YK2 , d̂K⃗ 2 ) ∈ 𝒮 with d̂1⃗ + ⋅ ⋅ ⋅ + d̂K⃗ 2 ≤ κ2 and |d̂1⃗ + ⋅ ⋅ ⋅ + d̂K⃗ 2 |1 ≤ k2 . It suffices to prove the case where P1 is given by a single term of the form (3.78) and P2 is given by a single term of the form (3.79). In this case, P1 P2 can be written as

3.12 Approximately commuting partial differential operators

� 179

a finite sum of terms of the form Mult[h]Zl1 Zl2 ⋅ ⋅ ⋅ ZlR Y1 Y2 ⋅ ⋅ ⋅ YK2 , ∞ where 1 ≤ l1 < l2 < ⋅ ⋅ ⋅ < lR ≤ K1 and h ∈ Cloc (Ω). This is clearly an 𝒮 partial differential operator of degree ≤ κ1 + κ2 , length ≤ k1 + k2 , and order ≤ L1 + L2 , completing the proof.

Lemma 3.12.5. Suppose P is an 𝒮 partial differential operator on Ω of degree ≤ κ, length ≤ k, and order ≤ L. Then P ∗ is an 𝒮 partial differential operator on Ω of degree ≤ κ, length ≤ k, and order ≤ L, where P ∗ denotes the formal L2 adjoint of P , thought of as an operator on C0∞ (Ω). Proof. If κ ∈ ℤν \ ℕν , k ≤ −1, or L ≤ −1, then P = 0 and the result is trivial. Thus, we may assume κ ∈ ℕν and k, L ∈ ℕ. By definition, P can be written as a finite sum of terms of the form Mult[g]Z1 Z2 ⋅ ⋅ ⋅ ZK ,

(3.80)

∞ where g ∈ Cloc (Ω), K ≤ L, and (Z1 , dr1⃗ ), . . . , (ZK , drK⃗ ) ∈ 𝒮 with dr1⃗ + ⋅ ⋅ ⋅ + drK⃗ 1 ≤ κ and ⃗ ⃗ |dr1 + ⋅ ⋅ ⋅ + drK1 |1 ≤ k. It suffices to consider the case where P is given by a single term of the form (3.80). The formal adjoint of (3.80) is given by ∗ ZK∗ ZK−1 ⋅ ⋅ ⋅ Z1∗ Mult[g].

(3.81)

∞ Since Zj∗ = −Zj + Mult[fj ], for some fj ∈ Cloc (M), we see that each Zj∗ is an 𝒮 partial differential operator on Ω of degree ≤ drj⃗ , length ≤ |drj⃗ |1 , and order ≤ 1. Combining this

with the fact that Mult[g] is an 𝒮 partial differential operator on Ω of degree ≤ 0 ∈ ℕν , length ≤ 0 ∈ ℕ, and order ≤ 0 ∈ ℕ, Lemma 3.12.4 implies that the operator in (3.81) is an 𝒮 partial differential operator on Ω of degree ≤ dr1⃗ + ⋅ ⋅ ⋅ + drK⃗ ≤ κ, length ≤ |dr1⃗ |1 + ⋅ ⋅ ⋅ + |drK⃗ |1 ≤ k, and order ≤ K ≤ L, completing the proof. The main result of this section is the next theorem. Theorem 3.12.6. Suppose 𝒮1 and 𝒮2 weakly approximately commute on Ω. Let κ1 ∈ ℤν1 , κ2 ∈ ℤν2 , and k1 , k2 ∈ ℤ. For j = 1, 2, let Pj of degree ≤ κj and length ≤ kj . Then

(κj ,kj )

(κ1 ,k1 )

P1

(κ2 ,k2 )

P2

be an 𝒮j partial differential operator on Ω

(κ2 ,k2 )

= P2

(κ1 ,k1 )

P1

+ Error,

̃(κ2 ,k2 −1) P ̃(κ1 ,k1 ) and P ̃(κ2 ,k2 ) P ̃(κ1 ,k1 −1) , where Error is a finite sum of operators of the form P 1 1 2 2 ̃(κj ,l) is an 𝒮j partial differential operator on Ω of degree ≤ κj and length ≤ l. where each P j

Before we prove Theorem 3.12.6, we introduce three corollaries.

180 � 3 Vector fields and Carnot–Carathéodory geometry Corollary 3.12.7. Suppose 𝒮1 and 𝒮2 weakly approximately commute on Ω. Let κ1 ∈ ℤν1 and κ2 ∈ ℤν2 . For j = 1, 2, let Pj ≤ κj . Then

be an 𝒮j partial differential operator on Ω of degree

(κj )

(κ1 )

P1

(κ2 )

P2

(κ )

(κ1 )

= P2 2 P1

+ Error,

̃(κ2 ,|κ2 |1 −1) P ̃(κ1 ,|κ1 |1 ) and where Error is a finite sum of operators of the form P 1 2 ̃(κ2 ,|κ2 |1 ) P ̃(κ1 ,|κ1 |1 −1) , where each P ̃(κj ,l) is an 𝒮j partial differential operator on Ω of P 1 2 j degree ≤ κj and length ≤ l. Proof. Since Pj is an 𝒮j partial differential operator on Ω of degree ≤ κj if and only if it is an 𝒮j partial differential operator on Ω of degree ≤ κj and length ≤ |κj |1 (see Remark 3.12.2) this follows immediately from Theorem 3.12.6. Corollary 3.12.8. Suppose 𝒮1 and 𝒮2 weakly approximately commute on Ω. Let κ1 ∈ ℤν1 and κ2 ∈ ℤν2 . For j = 1, 2, let Pj

be an 𝒮j partial differential operator on (κ1 ) (κ2 ) ≤ κj . Then P1 P2 can be written as a finite sum of operators of the form ̃(κj ) is an 𝒮j partial differential operator on Ω of degree ≤ κj . where each P j (κj )

Ω of degree ̃(κ2 ) P ̃(κ1 ) , P 1

2

Proof. This follows immediately from Corollary 3.12.7. Corollary 3.12.9. Suppose ν1 = ν2 = 1 and suppose 𝒮1 and 𝒮2 weakly approximately commute on Ω. Let κ1 , κ2 ∈ ℤ and for j = 1, 2, let Pj Ω of degree ≤ κj . Then

(κj )

(κ1 )

P1

(κ2 )

P2

(κ )

be an 𝒮j partial differential operator on (κ1 )

= P2 2 P1

+ Error,

̃(κ2 −1) P ̃(κ1 ) and P ̃(κ2 ) P ̃(κ1 −1) , where where Error is a finite sum of operators of the form P (γ) ̃ is an 𝒮j partial differential operator on Ω of degree ≤ γ. each P j Proof. When ν = 1, P is an 𝒮 partial differential operator on Ω of degree ≤ κ and length ≤ k if and only if P is an 𝒮 partial differential operator on Ω of degree ≤ min{κ, k} (see Definition 3.12.1). Using this, the result follows immediately from Corollary 3.12.7. We now turn to the proof of Theorem 3.12.6. To prove it, we introduce some new no-

tation. For j = 1, 2, we write Pj

(κj ,kj ,Lj )

to denote a particular (fixed) 𝒮j partial differential

̃(κj ,kj ,Lj ) to denote operator on Ω of degree ≤ κj , length ≤ kj and order ≤ Lj . We write P j an 𝒮j partial differential operator on Ω of degree ≤ κj , length ≤ kj , and order ≤ Lj which ̃(κ1 ,k1 ,L1 ) or P ̃(κ2 ,k2 ,L2 ) may change from line to line. If we have a term which involves P 1

2

and we write ∑ in front of it, we mean a finite sum of terms of that form, where the ̃(κ1 ,k1 ,L1 ) and P ̃(κ2 ,k2 ,L2 ) vary in the sum: only the terms with a ∼ over them operators P 1 2 vary in the sum. For example, (κ1 ,k1 ,L1 ) ̃(κ2 ,k2 ,L2 ) P2

∑ P1

(3.82)

3.12 Approximately commuting partial differential operators

� 181

means there is a finite collection P2,12 2 2 , . . . , P2,Q2 2 2 of 𝒮2 partial differential operators on Ω of degree ≤ κ2 , length ≤ k2 , and order ≤ L2 such that the expression in (3.82) equals (κ ,k ,L )

Q

(κ ,k ,L )

(κ1 ,k1 ,L1 )

∑ P1 j=1

(κ ,k2 ,L2 )

P2,j 2

.

Using this new notation, we begin by proving Theorem 3.12.6 in the special case where 𝒮1 and 𝒮2 strongly approximately commute, as stated in the next proposition. Proposition 3.12.10. Suppose 𝒮1 and 𝒮2 strongly approximately commute on Ω. Then (κ1 ,k1 ,L1 )

P1

(κ2 ,k2 ,L2 )

P2

(κ2 ,k2 ,L2 )

= P2

(κ1 ,k1 ,L1 )

P1

(κ2 ,k2 −1,L2 −1) ̃(κ1 ,k1 ,L1 ) P1 (κ ,k ,L ) (κ ̃ 2 2 2P ̃ 1 ,k1 −1,L1 −1) . ∑P 1 2

̃ + ∑P 2 +

Proof. If κ1 ∈ ℤν1 \ ℕν1 , k1 ≤ −1, or L1 ≤ −1, then P1

(κ1 ,k1 ,L1 )

ν2

ν2

= 0 and the result is trivial.

Similarly if κ2 ∈ ℤ \ ℕ , k2 ≤ −1, or L2 ≤ = 0 and the result is trivial. We henceforth assume κ1 ∈ ℕν1 , κ2 ∈ ℕν2 , and k1 , k2 , L1 , L2 ∈ ℕ. (κ ,k ,0) We first consider the case L2 = 0. In this case, P2 2 2 = Mult[g2 ] for some g2 ∈ ∞ Cloc (Ω). By definition, P1

(κ ,k ,L ) −1, then P1 2 2 2

(3.83)

(κ1 ,k1 ,L1 )

can be written as a finite sum of terms of the form Mult[g1 ]Z1 Z2 ⋅ ⋅ ⋅ ZK ,

∞ where g1 ∈ Cloc (Ω), K ≤ L1 , (Z1 , dr1⃗ ), . . . , (ZK , drK⃗ ) ∈ 𝒮1 , with dr1⃗ + ⋅ ⋅ ⋅ + drK⃗ ≤ κ1 and ⃗ ⃗ |dr1 + ⋅ ⋅ ⋅ + drK |1 ≤ k1 . Note that

Mult[g1 ]Z1 Z2 ⋅ ⋅ ⋅ ZK P2

(κ2 ,k2 ,0)

= Mult[g1 ]Z1 Z2 ⋅ ⋅ ⋅ ZK Mult[g2 ]

= Mult[g2 ] Mult[g1 ]Z1 Z2 ⋅ ⋅ ⋅ ZK + Error (κ2 ,k2 ,0)

= P2

Mult[g1 ]Z1 Z2 ⋅ ⋅ ⋅ ZK + Error,

where Error is a finite sum of terms of the form Mult[g3 ]Zl1 Zl2 ⋅ ⋅ ⋅ ZlR ,

∞ g ∈ Cloc (Ω), R < K ≤ L1 , 1 ≤ l1 < l2 < ⋅ ⋅ ⋅ < lR ≤ K.

̃(κ1 ,k1 −1,L1 −1) . Since |drl⃗ 1 + ⋅ ⋅ ⋅ + drl⃗ R |1 < |dr1⃗ + ⋅ ⋅ ⋅ + drK⃗ |1 ≤ k1 , we see that Error is of the form P 1 We conclude that (κ1 ,k1 ,L1 )

P1

(κ2 ,k2 ,L2 )

P2

(κ1 ,k1 ,L1 )

̃(κ1 ,k1 −1,L1 −1) + ∑ Mult[1]P 1

(κ1 ,k1 ,L1 )

̃ + ∑P 2

(κ2 ,k2 ,L2 )

P1

(κ2 ,k2 ,L2 )

P1

= P2 = P2

(κ2 ,k2 ,0) ̃(κ1 ,k1 −1,L1 −1) P1 ,

182 � 3 Vector fields and Carnot–Carathéodory geometry completing the proof when L2 = 0. A similar proof gives the result when L1 = 0, and we leave the details to the interested reader. (κ ,0,L ) (κ ,0,0) If k1 = 0 and L1 > 0, then P1 1 1 is of the form P1 1 (see Definition 3.12.1) and therefore the case k1 = 0 follows from the already established case L1 = 0. Similarly, the case k2 = 0 follows from the already established case L2 = 0. Thus, we may henceforth assume L1 , L2 , k1 , k2 ∈ ℕ+ . Next, we consider the case L1 = L2 = 1 and k1 , k2 ∈ ℕ+ . In this case, R1

= ∑ Mult[gl ]Yl + Mult[g0 ],

(κ1 ,k1 ,1)

P1

l=1 R2

= ∑ Mult[fl ]Zl + Mult[f0 ],

(κ2 ,k2 ,1)

P2

∞ (Yl , d̂l⃗ ) ∈ 𝒮1 , d̂l⃗ ≤ κ1 , gl ∈ Cloc (Ω), ∞ (Zl , drl⃗ ) ∈ 𝒮2 , drl⃗ ≤ κ2 , fl ∈ Cloc (Ω).

l=1

It suffices to consider the cases where P1

(κ1 ,k1 ,1)

hand side of (3.84) and of (3.85). If P1

(κ1 ,k1 ,1)

(κ ,k ,1) P2 2 2

(κ2 ,k2 ,0)

(3.85)

is given by one of the terms on the right-

is given by one of the terms on the right-hand side

= Mult[g0 ], then it is of the form P1

(κ1 ,k1 ,0)

from the already proved case L1 = 0. Similarly, if form P2

(3.84)

(κ ,k ,1) P2 2 2

and the result follows

= Mult[f0 ], then it is of the

, and the result follows from the already proved case L2 = 0. Thus, it suf-

fices to assume that P1

and P2

(κ1 ,k1 ,1)

= Mult[g1 ]Y1 ,

(κ2 ,k2 ,1)

= Mult[f1 ]Z1 ,

(κ1 ,k1 ,1)

P1 P2

(κ2 ,k2 ,1)

have the following forms:

∞ (Y1 , d̂1⃗ ) ∈ 𝒮1 , d̂1⃗ ≤ κ1 , g1 ∈ Cloc (Ω),

(3.86)

∞ (Z1 , dr1⃗ ) ∈ 𝒮2 , dr1⃗ ≤ κ2 , f1 ∈ Cloc (Ω).

(3.87)

Using the assumption that 𝒮1 and 𝒮2 strongly approximately commute on Ω, there are finite sets ℱ1 ⊆ {(Y0 , d̂0⃗ ) ∈ 𝒮1 : d̂0⃗ ≤ d̂1⃗ },

ℱ2 ⊆ {(Z0 , dr0⃗ ) ∈ 𝒮2 : dr0⃗ ≤ dr1⃗ }

such that [Y1 , Z1 ] =

∑ (Y0 ,d̂⃗ 0 )∈ℱ1

a(Y

⃗

0 ,d̂0 )

Y0 +

∑ (Z0 ,dr⃗ 0 )∈ℱ2

b(Z

⃗

0 ,dr0 )

Z0 ,

a(Y

⃗

0 ,d̂0 )

, b(Z

⃗

0 ,dr0 )

∞ ∈ Cloc (Ω).

̃(κ1 ,k1 ,1) . SimiNote that, since d̂1⃗ ≤ κ1 and |d̂1⃗ |1 ≤ k1 , each (Y0 , d̂0⃗ ) ∈ ℱ1 is of the form P 1 (κ ,k ,1) (κ ,k ,1) ̃ 2 2 . We have, assuming P 1 1 and P (κ2 ,k2 ,1) larly, each (Z0 , dr0⃗ ) ∈ ℱ2 is of the form P are given by (3.86) and (3.87), (κ1 ,k1 ,1)

P1

(κ2 ,k2 ,1)

P2

2

= Mult[g1 ]Y1 Mult[f1 ]Z1

= Mult[f1 ]Z1 Mult[g1 ]Y1 + Mult[g1 (Y1 f1 )]Z1 − Mult[f1 (Z1 g1 )]Y1 + Mult[g1 f1 ][Y1 , Z1 ]

1

2

3.12 Approximately commuting partial differential operators �

(κ2 ,k2 ,1)

= P2 +

(κ1 ,k1 ,1)

P1

∑ (Y0 ,d̂⃗ 0 )∈ℱ1

⃗

0 ,d̂0 )

P1

(κ2 ,k2 ,1)

P1

= P2

+ Mult[g1 (Y1 f )]Z1 − Mult[f1 (Z1 g1 )]Y1

Mult[f1 g1 a(Y

(κ2 ,k2 ,1)

= P2

183

]Y0 +

∑ (Z0 ,dr⃗ 0 )∈ℱ2

(κ2 ,k2 ,1)

Mult[f1 g1 b(Z

⃗

0 ,dr0 )

]Z0

̃ Mult[1] + ∑ Mult[1]P 1

(κ1 ,k1 ,1)

̃ + ∑P 2

(κ1 ,k1 ,1)

(κ1 ,k1 ,1)

̃(κ2 ,k2 ,1) P ̃(κ1 ,k1 −1,0) + ∑ P ̃(κ2 ,k2 −1,0) P ̃(κ1 ,k1 ,1) , + ∑P 1 1 2 2

̃(κ1 ,k1 −1,0) where in the last equality, we have used the fact that Mult[1] is of the form P 1 (κ ,k −1,0) ν ν ̃ 2 2 and of the form P (which uses the fact that κ1 ∈ ℕ 1 , κ1 ∈ ℕ 2 , and k1 , k2 ≥ 1). 2

This completes the proof of the case (L1 , L2 ) = (1, 1) and k1 , k2 ∈ ℕ+ . Combining all of the above arguments, we have proved the result for all (L1 , L2 ), with L1 ≤ 1, L2 ≤ 1, and k1 , k2 ∈ ℤ. We proceed by induction on (L1 , L2 ), the base case being (L1 , L2 ) = (1, 1), which we have already shown. Thus, fix (L1 , L2 ) with L1 ≥ 1, L2 ≥ 1, and L1 + L2 > 2; we wish to prove the result for this L1 and L2 , and as described above we may assume κ1 ∈ ℕν1 , κ2 ∈ ℕν2 , and k1 , k2 ∈ ℕ+ . We suppose we know the result for all (L′1 , L′2 ) with L′1 ≤ L1 , L′2 ≤ L2 , and L′1 + L′2 < L1 + L2 (and all κ1 ∈ ℤν1 , κ2 ∈ ℤν2 , k1 , k2 ∈ ℤ). By the inductive hypothesis, for such L′1 and L′1 (and all κ1′ ∈ ℤν1 , κ2′ ∈ ℤν2 , and k1′ , k2′ ∈ ℤ), we have (κ1′ ,k1′ ,L′1 ) ̃(κ2′ ,k2′ ,L′2 ) P2

̃ P 1

(κ2′ ,k2′ ,L′2 ) ̃(κ1′ ,k1′ ,L′1 ) P1 ,

̃ = ∑P 2

(3.88)

where we recall that the terms with a ∼ over them may vary from term to term. (κ ,k ,L ) By the definitions (see Definition 3.12.1), P1 1 1 1 is a finite sum of operators of the form Mult[g]Y1 Y2 ⋅ ⋅ ⋅ YK ,

(3.89)

where K ≤ L1 , (Y1 , d̂1⃗ ), . . . , (YK , d̂K⃗ ) ∈ 𝒮1 , d̂1⃗ + ⋅ ⋅ ⋅ + d̂K⃗ ≤ κ1 , and |d̂1⃗ + ⋅ ⋅ ⋅ + d̂K⃗ |1 ≤ k1 . When K ≥ 1, we may write (3.89) as (Mult[g]Y1 Y2 ⋅ ⋅ ⋅ YK−1 )YK . When K = 0, we may write (3.89) (κ ,k ,L ) as Mult[g] Mult[1]. This shows that in either case, we may write P1 1 1 1 as a finite sum of terms of the form (κ1,1 ,k1,1 ,L1 −1)

P1

Similarly, P2

(κ2 ,k2 ,L2 )

(κ1,2 ,k1,2 ,1)

P1

,

κ1,1 + κ1,2 = κ1 , k1,1 + k1,2 = k1 .

(3.90)

can be written as a finite sum of terms of the form

(κ2,1 ,k2,1 ,L2 −1)

P2

(κ2,2 ,k2,2 ,1)

P2

,

κ2,1 + κ2,2 = κ1 , k2,1 + k2,2 = k1 .

Thus, it suffices to prove the result in the special case where P1

(κ1 ,k1 ,L1 )

of the form (3.90) and P2

(κ2 ,k2 ,L2 )

is a single term of the form (3.91).

(3.91) is a single term

184 � 3 Vector fields and Carnot–Carathéodory geometry Using the inductive hypothesis, we have (κ1 ,k1 ,L1 )

P1

(κ2 ,k2 ,L2 )

P2

(κ1,1 ,k1,1 ,L1 −1)

P1

(κ1,1 ,k1,1 ,L1 −1)

P2

= P1 = P1

(κ1,2 ,k1,2 ,1)

(κ2,1 ,k2,1 ,L2 −1)

P2

(κ2,1 ,k2,1 ,L2 −1)

(κ1,2 ,k1,2 ,1)

P1

(κ2,2 ,k2,2 ,1)

P2

(κ2,2 ,k2,2 ,1)

P2

(κ1,1 ,k1,1 ,L1 −1) ̃(κ2,1 ,k2,1 −1,L2 −2) ̃(κ1,2 ,k1,2 ,1) (κ2,2 ,k2,2 ,1) P2 P1 P2 (κ ,k ,L −1) ̃(κ2,1 ,k2,1 ,L2 −1) ̃(κ1,2 ,k1,2 −1,0) (κ2,2 ,k2,2 ,1) P1 P2 ∑ P1 1,1 1,1 1 P 2

+ ∑ P1 +

=: (I) + (II) + (III). Using repeated application of (3.88) and Lemma 3.12.4, we have (κ2,1 ,k2,1 −1,L2 −2) ̃(κ2,2 ,k2,2 ,1) ̃(κ1,1 ,k1,1 ,L1 −1) ̃(κ1,2 ,k1,2 ,1) P2 P1 P1 ̃(κ2 ,k2 −1,L2 −1) P ̃(κ1 ,k1 ,L1 ) , ∑P 1 2

̃ (II) = ∑ P 2 =

which is of the desired form for our error term in (3.83). Similarly, we have (κ2,1 ,k2,1 ,L2 −1) ̃(κ2,2 ,k2,2 ,1) ̃(κ1,1 ,k1,1 ,L1 −1) ̃(κ1,2 ,k1,2 −1,0) P2 P1 P1 (κ ,k ,L ) (κ ,k −1,L −1) ̃ 2 2 2P ̃ 1 1 1 , ∑P 1 2

̃ (III) = ∑ P 2 =

which is also of the desired form for the error term in (3.83). Turning to (I), we have by the base case (L′1 , L′2 ) = (1, 1) (κ1,1 ,k1,1 ,L1 −1)

(I) = P1

(κ2,1 ,k2,1 ,L2 −1)

P2

(κ1,1 ,k1,1 ,L1 −1)

(κ1,2 ,k1,2 ,1)

P1

(κ2,1 ,k2,1 ,L2 −1) ̃(κ2,2 ,k2,2 −1,0) ̃(κ1,2 ,k1,2 ,1) P2 P1 (κ1,1 ,k1,1 ,L1 −1) (κ2,1 ,k2,1 ,L2 −1) ̃(κ2,2 ,k2,2 ,1) ̃(κ1,2 ,k1,2 −1,0) P2 P2 P1 ∑ P1

+ ∑ P1 +

(κ2,2 ,k2,2 ,1)

P2

P2

=: (IV) + (V) + (VI). Using repeated application of (3.88) and Lemma 3.12.4, we have (κ2,1 ,k2,1 ,L2 −1) ̃(κ2,2 ,k2,2 −1,0) ̃(κ1,1 ,k1,1 ,L1 −1) ̃(κ1,2 ,k1,2 ,1) P2 P1 P1 ̃(κ2 ,k2 −1,L2 −1) P ̃(κ1 ,k1 ,L1 ) , ∑P 1 2

̃ (V) = ∑ P 2 =

which is of the desired form for our error term in (3.83). Similarly, we have ̃(κ2,1 ,k2,1 ,L2 −1) P ̃(κ2,2 ,k2,2 ,1) P ̃(κ1,1 ,k1,1 ,L1 −1) P ̃(κ1,2 ,k1,2 −1,0) (VI) = ∑ P 1 1 2 2 (κ2 ,k2 ,L2 ) ̃(κ1 ,k1 −1,L1 −1) P2 ,

̃ =P 2

which is also of the desired form for the error term in (3.83). Turning to (IV), using the inductive hypothesis, Lemma 3.12.4, and the assumption (κ ,k ,L ) (κ ,k ,L ) that P1 1 1 1 and P2 2 2 2 are given by (3.90) and (3.91), we have

3.12 Approximately commuting partial differential operators �

(κ1,1 ,k1,1 ,L1 −1) (κ2 ,k2 ,L2 )

= P2

(κ1,2 ,k1,2 ,1)

(κ2 ,k2 ,L2 )

P1

(κ1,1 ,k1,1 ,L1 −1)

P1

(IV) = P1

P2

P1

185

(κ1,2 ,k1,2 ,1)

̃(κ2 ,k2 −1,L2 −1) P ̃(κ1,1 ,k1,1 ,L1 −1) P (κ1,2 ,k1,2 ,1) + ∑P 1 1 2 (κ2 ,k2 ,L2 ) ̃(κ1,1 ,k1,1 −1,L1 −2) (κ1,2 ,k1,2 ,1) P1 P1 (κ2 ,k2 ,L2 ) (κ1 ,k1 ,L1 ) (κ ,k 2 ̃ 2 ,L2 ) P ̃(κ1 ,k1 −1,L1 −1) P2 P1 + ∑P 1 2 (κ2 ,k2 −1,L2 −1) ̃(κ1 ,k1 ,L1 ) ̃ + ∑ P2 P1 ,

̃ + ∑P 2

=

which is of the desired form (3.83) and completes the proof. Lemma 3.12.11. P is an 𝒮 partial differential operator on Ω of degree ≤ κ and length ≤ k if and only if P is a Gen(𝒮 ) partial differential operator on Ω of degree ≤ κ and length ≤ k. Proof. If κ ∈ ℤν \ ℕν or k ∈ ℤ \ ℕ, then P = 0 and the result is trivial. We henceforth assume κ ∈ ℕν and k ∈ ℕ. Since 𝒮 ⊆ Gen(𝒮 ), if P is an 𝒮 partial differential operator on Ω of degree ≤ κ and length ≤ k, then it is a Gen(𝒮 ) partial differential operator on Ω of degree ≤ κ and length ≤ k. Suppose P is a Gen(𝒮 ) partial differential operator on Ω of degree ≤ κ and length ≤ k. Then P can be written as a finite sum of terms of the form Mult[g]Z1 Z2 ⋅ ⋅ ⋅ ZK ,

(3.92)

where (Z1 , dr1⃗ ), . . . , (ZK , drK⃗ ) ∈ Gen(𝒮 ), dr1⃗ + ⋅ ⋅ ⋅ + drK⃗ ≤ κ, and |dr1⃗ + ⋅ ⋅ ⋅ + drK⃗ |1 ≤ k. By the definition of Gen(𝒮 ), each Zj is an 𝒮 partial differential operator on Ω of degree ≤ drj⃗ and length ≤ |drj⃗ |1 . Lemma 3.12.4 then implies that terms of the form (3.92) are 𝒮 partial differential operators on Ω of degree ≤ κ and length ≤ k, and therefore the same is true of P , completing the proof. Remark 3.12.12. Lemma 3.12.11 shows that the degree and length of an 𝒮 partial differential operator on Ω do not depend on whether one uses 𝒮 or Gen(𝒮 ). However, the order does depend on whether one uses 𝒮 or Gen(𝒮 ). Proof of Theorem 3.12.6. By Lemma 3.12.11, it suffices to prove the result with 𝒮1 and 𝒮2 replaced by Gen(𝒮1 ) and Gen(𝒮2 ). Since 𝒮1 and 𝒮2 weakly approximately commute on Ω, Gen(𝒮1 ) and Gen(𝒮2 ) strongly approximately commute on Ω. The result now follows from Proposition 3.12.10. Theorem 3.12.6 has a consequence which will be of use to us. Proposition 3.12.13. Suppose 𝒮1 and 𝒮2 weakly approximately commute on Ω. Set ν = ν1 + ν2 and ν

𝒮 := 𝒮1 ⊠ 𝒮2 ⊆ Cloc (M; TM) × (ℕ \ {0}). ∞

186 � 3 Vector fields and Carnot–Carathéodory geometry Let κ = (κ1 , κ2 ) ∈ ℤν1 ×ℤν2 ≅ ℤν . Suppose P (κ) is an 𝒮 partial differential operator on Ω of ̃(κ1 ) P ̃(κ2 ) , degree ≤ κ. Then P (κ) can be written as a finite sum of operators of the form P 1 2 (κj ) ̃ is an 𝒮j partial differential operator on Ω of degree ≤ κj . where each P j

Proof. If κ ∈ ℤν \ ℕν , then P (κ) = 0 and the result is trivial. We henceforth assume κ ∈ ℕν . We claim that P (κ) can be written as a finite sum of operators of the form (σ1 )

P1

(τ )

(σ2 )

P2 1 P1

(τ2 )

P2

(σL )

⋅ ⋅ ⋅ P1

(τL )

P2

(3.93)

,

for some L ≥ 1, where σ1 , . . . , σL ∈ ℕν1 , τ1 , . . . , τL ∈ ℕν2 , σ1 + ⋅ ⋅ ⋅ + σL ≤ κ1 , τ1 + ⋅ ⋅ ⋅ + τL ≤ κ2 ,

each P1 j is an 𝒮1 partial differential operator on Ω of degree ≤ σj , and each P2 j is an 𝒮2 partial differential operator on Ω of degree ≤ τj . Indeed, by Definition 3.12.1, P (κ) can be written as a finite sum of operators of the form (σ )

(τ )

Mult[g]P2 1 P1 (τ )

(σ2 )

(τ2 )

P2

(σL )

⋅ ⋅ ⋅ P1

(τL )

P2

,

∞ where g ∈ Cloc (Ω), (σ2 + ⋅ ⋅ ⋅ + σL , τ1 + ⋅ ⋅ ⋅ + τL ) = (κ1 , κ2 ) = κ, and each P1

is of the form ⃗ = Mult[1] if R = 0), where (Y1 , d̂1 ), . . . , (YR , d̂R⃗ ) ∈ (σj )

Y1 Y2 ⋅ ⋅ ⋅ YR for some R ≥ 0 (where (τ ) 𝒮1 with d̂1⃗ + ⋅ ⋅ ⋅ + d̂R⃗ ≤ σj . Similarly, each P2 j is of the form Z1 Z2 ⋅ ⋅ ⋅ ZR (for some R ≥ 0), where (Z1 , dr1⃗ ), . . . , (ZR , drR⃗ ) ∈ 𝒮2 with dr1⃗ + ⋅ ⋅ ⋅ + drR⃗ ≤ τj . (σ ) P1 j

Setting σ1 = 0 and P1

(σ1 )

= Mult[g], we see that each P1

(σj )

(τ ) σj and each P2 j (κ)

is an 𝒮1 partial differential

operator on Ω of degree ≤ is an 𝒮2 partial differential operator on Ω of degree ≤ τj . This establishes that P can be written as a sum of terms of the form (3.93). Thus, it suffices to prove the result when P (κ) is given by a single term of the form (3.93). In that case, by repeated application of Corollary 3.12.8, P (κ) can be written as a finite sum of terms of the form (σ )

(σ2 )

̃ 1P ̃ P 1 1

(σL ) ̃(τ1 ) ̃(τ2 ) ̃(τL ) . P2 P2 ⋅ ⋅ ⋅ P 2

̃ ⋅⋅⋅P 1

(3.94)

̃(κ1 ) P ̃(κ2 ) , completing the proof. Lemma 3.12.4 shows that (3.94) is of the form P 1 2

3.13 Filtered modules of vector fields Many of the definitions in this chapter can be restated in the language of filtered modules. This can be useful when understanding certain qualitative properties of the definitions, though is not as useful in the quantitative analysis needed for our applications. We describe this perspective in this section. Our modules will be filtered by the semi-group ℕν , with the partial ordering for a, b ∈ ℕν , a ≤ b if aμ ≤ bμ for all μ ∈ {1, . . . , ν}. We will sometimes use [0, ∞)ν in place of ℕν , with the same definitions.

3.13 Filtered modules of vector fields

� 187

Definition 3.13.1. Let R be a commutative ring. An ℕν -filtration F⋅ of an R-module M assigns to each D ∈ ℕν an R-submodule FD of M, satisfying D1 ≤ D2 ⇒ FD1 ⊆ FD2 . Remark 3.13.2. In Definition 3.13.1, we do not assume that a filtration satisfies ⋃D∈ℕν FD = M. Definition 3.13.3. Let F⋅ be an ℕν -filtration of an R-module M. We say F⋅ is finitely generated if there is a finite set ℱ ⊆ M such that FD ∩ ℱ generates FD as an R-module. Definition 3.13.4. A ℕν filtration F⋅ of a Lie algebra A assigns to each D ∈ ℕν a vector subspace FD of A, such that D1 ≤ D2 ⇒ FD1 ⊆ FD2 and [FD1 , FD2 ] ⊆ FD1 +D2 , ∀D1 , D2 ∈ ℕν . Definition 3.13.5. Given two ℕν -filtrations F⋅ and G⋅ of a module M or a Lie algebra A, we say F⋅ ⊆ G⋅ if FD ⊆ GD , ∀D ∈ ℕν . We say F⋅ = G⋅ if FD = GD for all D ∈ ℕν . ∞ Let M be a smooth manifold. For open sets Ω ⊆ M, the map Ω 󳨃→ Cloc (Ω) gives a ∞ sheaf of rings on M. The map Ω 󳨃→ Cloc (Ω; TΩ) gives a sheaf of modules over this sheaf of rings. This is also a sheaf of Lie algebras via the usual Lie bracket of vector fields.

Definition 3.13.6. An ℕν -filtration of vector fields on M, Γ⋅ (⋅), is for each open set Ω ⊆ M ∞ a filtration, Γ⋅ (Ω), of the module Cloc (Ω; TΩ) such that for each D ∈ ℕν , Ω 󳨃→ ΓD (Ω) is a ∞ subpresheaf of the sheaf Ω 󳨃→ Cloc (Ω; TΩ). We say Γ is a Lie algebra ℕν -filtration of vector ∞ fields on M if in addition each Γ⋅ (Ω) is a filtration of the Lie algebra Cloc (Ω; TΩ). Definition 3.13.7. Let Γ be an ℕν -filtration of vector fields on M. We say Γ is locally finitely generated if for each relatively compact, open set Ω ⋐ M, Γ⋅ (Ω) is finitely generated. ∞ Definition 3.13.8. Given a set 𝒮 ⊆ Cloc (M; TM) × ℕν , we obtain a filtration of vector ν fields on M defined by, for D ∈ ℕ ,

󵄨 ∞ ΓD (Ω, 𝒮 ) := Cloc (Ω) module generated by {X 󵄨󵄨󵄨Ω : (X, d) ∈ 𝒮 , d ≤ D}. ∞ As we will see, many of the properties we study about sets 𝒮 ⊆ Cloc (M; TM) × ℕν only depend on the associated filtration of vector fields Γ⋅ (⋅, S). For example, we have the follow lemma, whose simple proof we leave to the interested reader. ∞ Lemma 3.13.9. Let 𝒮 ⊆ Cloc (M; TM) × ℕν . Then: (i) 𝒮 is strongly locally finitely generated if and only if Γ⋅ (⋅, 𝒮 ) is locally finitely generated. 𝒮 is strongly finitely generated on Ω if and only if Γ⋅ (Ω, 𝒮 ) is finitely generated. Moreover, for Ω ⋐ M open, if 𝒮 is finitely generated on Ω by ℱ ⊆ 𝒮 , then ΓD (Ω, 𝒮 ) is 󵄨 finitely generated by {X 󵄨󵄨󵄨Ω : (X, d ) ∈ ℱ , d ≤ D}. (ii) Γ⋅ (⋅, Gen(𝒮 )) is the Lie algebra ℕν -filtration of vector fields on M generated by Γ⋅ (⋅, 𝒮 ). More precisely, Γ⋅ (⋅, Gen(𝒮 )) is the smallest Lie algebra ℕν -filtration of vector fields on M which contains Γ⋅ (⋅, 𝒮 ).

188 � 3 Vector fields and Carnot–Carathéodory geometry ∞ ∞ (iii) Let 𝒮1 ⊆ Cloc (M; TM) × ℕν1 and 𝒮2 ⊆ Cloc (M; TM) × ℕν2 . Then 𝒮1 and 𝒮2 weakly approximately commute on Ω ⋐ M if and only if the following holds. If

𝒮 := {(X1 , (d1 , 0ν2 )), (X2 , (0ν1 , d2 )) : (X1 , d1 ) ∈ Gen(𝒮1 ), (X2 , d2 ) ∈ Gen(𝒮2 )},

then Γ⋅ (Ω, 𝒮 ) is a Lie algebra filtration.

3.14 Control of vector fields ∞ Fix ν1 , ν2 ∈ ℕ+ and for j = 1, 2 let 𝒮j ⊆ Cloc (M; TM) × (ℕνj \ {0}). If (X2 , d 2⃗ ) ∈ 𝒮2 , we think of X2 as a differential operator of “degree” d j⃗ ∈ ℕν2 . The goal of this section is to describe a situation in which we can “trade” such vector fields in 𝒮2 for vector fields in 𝒮1 . Fix a ν2 ×ν1 matrix λ with components in [0, ∞]. We use the convention that 0⋅∞ = 0. ∞ Definition 3.14.1. We define λ⊺ (𝒮2 ) ⊆ Cloc (M; TM)×[0, ∞)ν to be the set of all (X2 , λ⊺ (d 2⃗ )), ⊺ ⃗ ⃗ where (X2 , d 2 ) ∈ 𝒮2 and λ (d 2 ) is not infinite in any component.

Definition 3.14.2. We say 𝒮1 strongly λ-controls 𝒮2 on Ω if ∀(Y2 , d̂2⃗ ) ∈ λ⊺ (𝒮2 ), we have ∞ Y2 ∈ Cloc (Ω) module generated by {X1 : (X1 , d 1⃗ ) ∈ 𝒮1 , d 1⃗ ≤ d̂2⃗ }.

We say 𝒮1 locally strongly λ-controls 𝒮2 on M if 𝒮1 strongly λ-controls 𝒮2 on Ω, for all Ω ⋐ M open. When ν1 = ν2 , we say 𝒮1 strongly controls 𝒮2 on Ω if 𝒮1 strongly I-controls 𝒮2 on Ω, where I is the identity matrix. We say 𝒮1 locally strongly controls 𝒮2 on M if 𝒮1 locally strongly I-controls 𝒮2 on M. Definition 3.14.3. We say 𝒮1 weakly λ-controls 𝒮2 on Ω if Gen(𝒮1 ) strongly λ-controls 𝒮2 on Ω. We say 𝒮1 locally weakly λ-controls 𝒮2 on M if 𝒮1 weakly λ-controls 𝒮2 on Ω, for all Ω ⋐ M open. When ν1 = ν2 , we say 𝒮1 weakly controls 𝒮2 on Ω if 𝒮1 weakly I-controls 𝒮2 on Ω, where I is the identity matrix. We say 𝒮1 locally weakly controls 𝒮2 on M if 𝒮1 locally weakly I-controls 𝒮2 on M. Example 3.14.4. Let (W , 1) = {(W1 , 1), . . . , (Wr , 1)} be such that W1 , . . . , Wr1 satisfy Hörmander’s condition of order m on Ω. Let (Z, 1) = {(Z1 , 1), . . . , (Zs , 1)} be such that Z1 (x), . . . , Zs (x) span Tx Ω, ∀x ∈ Ω (i. e., Z1 , . . . , Zs satisfy Hörmander’s condition of order 1 on Ω). Then (Z, 1) strongly controls (W , 1) on Ω, and (W , 1) weakly m-controls (Z, 1) on Ω. For a generalization of this example, see the discussion of λstd and Λstd (Definition 6.6.15) in Section 6.6.2. Definition 3.14.5. When ν1 = ν2 , we say 𝒮1 and 𝒮2 are weakly (respectively, strongly) equivalent on Ω if 𝒮1 weakly (respectively, strongly) controls 𝒮2 on Ω and 𝒮2 weakly (respectively, strongly) controls 𝒮1 on Ω. We say 𝒮1 and 𝒮2 are locally weakly (respectively, strongly) equivalent on M if 𝒮1 and 𝒮2 are weakly (respectively, strongly) equivalent on Ω, ∀Ω ⋐ M open.

3.14 Control of vector fields

�

189

Remark 3.14.6. Using the notation from Section 3.13 we have the following restatements of the above definitions: – 𝒮1 strongly λ-controls 𝒮2 on Ω if and only if Γ⋅ (Ω, λ⊺ (𝒮2 )) ⊆ Γ⋅ (Ω, 𝒮1 ). Here, we are using filtration indexed by [0, ∞)ν instead of ℕν , though the definitions are the same. – 𝒮1 weakly λ-controls 𝒮2 on Ω if and only if Γ⋅ (Ω, λ⊺ (𝒮2 )) ⊆ Γ⋅ (Ω, Gen(𝒮1 )). – 𝒮1 and 𝒮2 are strongly equivalent on Ω if and only if Γ⋅ (Ω, 𝒮2 ) = Γ⋅ (Ω, 𝒮1 ). – 𝒮1 and 𝒮2 are weakly equivalent on Ω if and only if Γ⋅ (Ω, Gen(𝒮2 )) = Γ⋅ (Ω, Gen(𝒮1 )), that is, if and only if the filtered Lie algebra generated by Γ⋅ (Ω, 𝒮2 ) equals the filtered Lie algebra generated by Γ⋅ (Ω, 𝒮1 ) – see Lemma 3.13.9 (ii). Remark 3.14.7. Weak and strong control on Ω each give a partial ordering on subsets of ∞ Cloc (M; TM) × (ℕν \ {0}). Weak and strong equivalence on Ω each give an equivalence relation on such subsets. This follows immediately from Remark 3.14.6. Lemma 3.14.8. 𝒮1 weakly λ-controls 𝒮2 on Ω if and only if 𝒮1 weakly λ-controls Gen(𝒮2 ) on Ω. Proof. The “if” part of the lemma is clear, so we focus on the “only if” part. Suppose 𝒮1 weakly λ-controls 𝒮2 on Ω. Then, by Remark 3.14.6, Γ⋅ (Ω, λ⊺ (𝒮2 )) ⊆ Γ⋅ (Ω, Gen(𝒮1 )). Since Γ⋅ (Ω, Gen(𝒮1 )) is a Lie algebra filtration, this shows that Γ⋅ (Ω, Gen(𝒮1 )) contains the Lie algebra filtration generated by Γ⋅ (Ω, λ⊺ (𝒮2 )). It is easy to see that this is exactly Γ⋅ (Ω, λ⊺ (Gen(𝒮2 ))), completing the proof. ∞ Lemma 3.14.9. For j = 1, 2, let 𝒮j′ , ℱj ⊆ Cloc (M; TM) × (ℕνj \ {0}) with ℱj finite. (i) If 𝒮1 and 𝒮1′ are strongly equivalent on Ω and 𝒮2 and 𝒮2′ are strongly equivalent on Ω, then 𝒮1 strongly λ-controls 𝒮2 on Ω if and only if 𝒮1′ strongly λ-controls 𝒮2′ on Ω. (ii) If 𝒮1 and 𝒮1′ are weakly equivalent on Ω and 𝒮2 and 𝒮2′ are weakly equivalent on Ω, then 𝒮1 weakly λ-controls 𝒮2 on Ω if and only if 𝒮1′ weakly λ-controls 𝒮2′ on Ω. (iii) Suppose 𝒮1 is finitely generated by ℱ1 ⊆ 𝒮1 on Ω. Then 𝒮1 strongly λ-controls 𝒮2 on Ω if and only if ℱ1 strongly λ-controls 𝒮2 on Ω. (iv) Suppose 𝒮2 is finitely generated by ℱ2 ⊆ 𝒮2 on Ω. Then 𝒮1 strongly λ-controls 𝒮2 on Ω if and only if 𝒮1 strongly λ-controls ℱ2 on Ω. (v) Suppose Gen(𝒮1 ) is finitely generated by ℱ1 ⊂ Gen(𝒮1 ) on Ω. Then 𝒮1 weakly λ-controls 𝒮2 on Ω if and only if ℱ1 strongly λ-controls 𝒮2 on Ω. (vi) Suppose Gen(𝒮2 ) is finitely generated by ℱ2 ⊂ Gen(𝒮2 ) on Ω. Then 𝒮1 weakly λ-controls 𝒮2 on Ω if and only if 𝒮1 weakly λ-controls ℱ2 on Ω.

Proof. We use Remark 3.14.6 freely in this proof. (i): That 𝒮j and 𝒮j′ are strongly equivalent on Ω is equivalent to Γ⋅ (Ω, 𝒮j ) = Γ⋅ (Ω, 𝒮j′ ). It follows that Γ⋅ (Ω, λt (𝒮2 )) = Γ⋅ (Ω, λt (𝒮2′ )). That 𝒮1 strongly λ-controls 𝒮2 is equivalent to Γ⋅ (Ω, λt (𝒮2 )) ⊆ Γ⋅ (Ω, 𝒮1 ), and similarly for 𝒮j replaced with 𝒮j′ . The result follows.

190 � 3 Vector fields and Carnot–Carathéodory geometry (ii): By Lemma 3.14.8, 𝒮1 weakly λ-controls 𝒮2 if and only if 𝒮1 weakly λ-controls Gen(𝒮2 ). The result follows by applying (i) with 𝒮1 , 𝒮2 , 𝒮1′ , 𝒮2′ replaced with Gen(𝒮1 ), Gen(𝒮2 ), Gen(𝒮1′ ), Gen(𝒮2′ ). (iii) and (iv): Since 𝒮 is finitely generated by ℱ ⊆ 𝒮 on Ω if and only if ℱ ⊆ 𝒮 is finite and 𝒮 and ℱ are strongly equivalent on Ω, these results follow immediately from (i). (v): This follows from (iii) by replacing 𝒮1 with Gen(𝒮1 ). (vi): By Lemma 3.14.8, 𝒮1 weakly λ-controls 𝒮2 if and only if 𝒮1 weakly λ-controls Gen(𝒮2 ). Using this the result follows from (iv) with 𝒮1 and 𝒮2 replaced by Gen(𝒮1 ) and Gen(𝒮2 ). The most important definitions of this text only depend on the weak equivalence class of 𝒮 . For example, we have the next result. ∞ Proposition 3.14.10. Let 𝒮1 , 𝒮2 , 𝒮1′ , 𝒮2′ ⊆ Cloc (M; TM) × (ℕν \ {0}). Suppose 𝒮1 and 𝒮1′ are ′ weakly equivalent on Ω and 𝒮2 and 𝒮2 are weakly equivalent on Ω. Then 𝒮1 and 𝒮2 weakly approximately commute on Ω if and only if 𝒮1′ and 𝒮2′ weakly approximately commute on Ω.

Proof. Set 𝒮 := Gen(𝒮1 ) ⊠ Gen(𝒮2 ),

𝒮 := Gen(𝒮1 ) ⊠ Gen(𝒮2 ). ′

′

′

Suppose 𝒮1 and 𝒮2 weakly approximately commute on Ω. We will show that 𝒮1′ and 𝒮2′ weakly approximately commute on Ω; the reverse implication will then follow by reversing the roles of 𝒮1 and 𝒮2 with 𝒮1′ and 𝒮2′ . That 𝒮1 and 𝒮2 weakly approximately commute on Ω is easily seen to be equivalent to Γ⋅ (Ω, 𝒮 ) = Γ⋅ (Ω, Gen(𝒮 )). Since 𝒮1 and 𝒮1′ are weakly equivalent on Ω, Remark 3.14.6 shows that Γ⋅ (Ω, Gen(𝒮1 )) = Γ⋅ (Ω, Gen(𝒮1′ )) and similarly Γ⋅ (Ω, Gen(𝒮2 )) = Γ⋅ (Ω, Gen(𝒮2′ )). It follows that Γ⋅ (Ω, 𝒮 ) = Γ⋅ (Ω, 𝒮 ′ ). Since Γ⋅ (Ω, 𝒮 ) = Γ⋅ (Ω, Gen(𝒮 )), we know that Γ⋅ (Ω, 𝒮 ) = Γ⋅ (Ω, 𝒮 ′ ) is a Lie algebra filtration (see Lemma 3.13.9 (ii)), and therefore Γ⋅ (Ω, 𝒮 ′ ) = Γ⋅ (Ω, Gen(𝒮 ′ )). This is equivalent to 𝒮1′ and 𝒮2′ weakly approximately commuting on Ω, completing the proof.

3.14.1 Further comments on equivalence The main setting we will be considering in this text is as follows. Fix ν ∈ ℕ+ . For each μ ∈ μ μ μ μ ∞ {1, . . . , ν}, we are given a finite set (W μ , dsμ ) = {(W1 , ds1 ), . . . , (Wrμ , dsrμ )} ⊂ Cloc (M; TM) × μ μ ℕ+ . We assume that for each μ, Gen((W , ds )) is locally finitely generated on M, and we assume that (W 1 , ds1 ), . . . , (W ν , dsν ) pairwise weakly approximately commute on M. Set (W , ds)⃗ = {(W1 , ds1⃗ ), . . . , (Wr , dsr⃗ )} := (W 1 , ds1 ) ⊠ (W 2 , ds2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (W ν , dsν ) μ

μ

∞ = {(Wj , dsj eμ ) : μ ∈ {1, . . . , ν}, 1 ≤ j ≤ rμ } ⊂ Cloc (M; TM) × (ℕν \ {0}).

(3.95)

3.14 Control of vector fields

� 191

Fix an open, relatively compact set Ω ⋐ M. For each μ ∈ {1, . . . , ν}, let (Z μ , drμ ) = μ μ μ μ ∞ {(Z1 , dr1 ), . . . , (Zsμ , drsμ )} ⊂ Cloc (M; TM) × ℕ+ . We assume (W μ , dsμ ) is weakly equivalent to (Z μ , drμ ) on Ω. Define (Z, dr)⃗ in the same way as (W , ds)⃗ (see (3.95)), with each (W μ , dsμ ) replaced by (Z μ , drμ ). The goal of this section is to show that (W , ds)⃗ can be replaced by (Z, dr)⃗ in many settings. We will often be given a set like (W , ds)⃗ and we will define various objects associated to this set; we will see that many of these objects can be ⃗ equivalently defined using (Z, dr). μ μ μ μ μ μ Let (X , d ) = {(X1 , d1 ), . . . , (Xqμ , dqμ )} ⊂ Gen((W μ , dsμ )) be such that (W μ , dsμ ) ⊆ μ μ μ μ (X , d ) and Gen((W , ds )) is finitely generated by (X μ , d μ ) on Ω. Set (X, d )⃗ = {(X1 , d 1⃗ ), . . . , (Xq , d q⃗ )} := (X 1 , d 1 ) ⊠ (X 2 , d 2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (X ν , d ν ) μ

μ

∞ = {(Xj , dj eμ ) : μ ∈ {1, . . . , ν}, 1 ≤ j ≤ qμ } ⊂ Cloc (M; TM) × (ℕν \ {0}).

⃗ is finitely generated by (X, d )⃗ on Ω. By Proposition 3.8.8, Gen((W , ds)) Lemma 3.14.11. For each μ ∈ {1, . . . , ν}, Gen((Z μ , drμ )) is finitely generated on Ω and (Z 1 , dr1 ), . . . , (Z ν , drν ) pairwise weakly approximately commute on Ω. Finally, (W , ds)⃗ and (Z, dr)⃗ are weakly equivalent on Ω. Proof. Using Lemma 3.13.9 (i), that (W μ , dsμ ) is weakly finitely generated on Ω is equivalent to Γ⋅ (Ω, Gen((W μ , dsμ ))) being finitely generated. Since Γ⋅ (Ω, Gen((W μ , dsμ ))) = Γ⋅ (Ω, Gen((Z μ , drμ ))), by hypothesis (see Remark 3.14.6), it follows that Gen((Z μ , drμ )) is finitely generated on Ω. Proposition 3.14.10 implies that (Z 1 , dr1 ), . . . , (Z ν , drν ) pairwise weakly approximately commute on Ω. Since Γ⋅ (Ω, Gen((W μ , dsμ ))) = Γ⋅ (Ω, Gen((Z μ , drμ ))) for each μ, it follows easily that ⃗ Indeed, Γ⋅ (Ω, Gen((W , ds))) ⃗ = Γ⋅ (Ω, Gen((Z, dr))). ⃗ is nothing more than Γ⋅ (Ω, Gen((W , ds))) ∞ the filtered Lie algebra generated by the filtered Cloc (Ω) module generated by μ

μ

μ

μ

{(Vj , fj eμ ) : μ ∈ {1, . . . , ν}, (Vj , fj ) ∈ Γ⋅ (Ω, Gen((W μ , dsμ )))}, ⃗ This shows that (W , ds)⃗ and (Z, dr)⃗ are weakly and similarly for (W , ds)⃗ replaced by (Z, dr). equivalent on Ω. Lemma 3.14.11 shows that (Z 1 , dr1 ), . . . , (Z ν , drν ) satisfy the same hypotheses as μ μ μ (W 1 , ds1 ), . . . , (W ν , dsν ) on Ω. Let (Y μ , d̂μ ) = {(Y1 , d̂1μ ), . . . , (Ypμ , d̂pμ )} ⊂ Gen((Z μ , drμ )) μ μ μ μ μ μ be such that (Z , dr ) ⊆ (Y , d̂ ) and Gen((Z , dr )) is finitely generated by (Y μ , d̂μ ) on Ω, that is, (Y μ , d̂μ ) is to (Z μ , drμ ) as (X μ , d μ ) is to (W μ , dsμ ). Define (Y , d̂)⃗ in terms of (Y 1 , d̂1 ), . . . , (Y ν , d̂ν ) in the same way (X, d )⃗ was defined in terms of (X 1 , d 1 ), . . . , (X ν , d ν ). By ⃗ ⃗ is finitely generated on Ω by (Y , d̂). Proposition 3.8.8 and Lemma 3.14.11, Gen((Z, dr)) Lemma 3.14.12. (X, d )⃗ and (Y , d̂)⃗ are strongly equivalent on Ω.

192 � 3 Vector fields and Carnot–Carathéodory geometry ⃗ is finitely ⃗ is finitely generated by (X, d )⃗ on Ω, Gen((Z, dr)) Proof. Since Gen((W , ds)) generated by (Y , d̂)⃗ on Ω, and (W , ds)⃗ and (Z, dr)⃗ are weakly equivalent on Ω (see Lemma 3.14.11), we have ⃗ ⃗ = Γ⋅ (Ω, (Y , d̂)), ⃗ = Γ⋅ (Ω, Gen((W , ds))) ⃗ = Γ⋅ (Ω, Gen((Z, dr))) Γ⋅ (Ω, (X, d )) completing the proof. ⃗ Proposition 3.14.13. Fix 𝒦 ⋐ Ω, a compact set. Then there exists ξ4 = ξ4 (𝒦, Ω, (W , ds), ⃗ ∈ (0, 1] such that for all x ∈ 𝒦 and δ ∈ (0, 1]ν , (Z, dr)) B(W ,ds)⃗ (x, ξ4 δ) ⊆ B(Z,dr)⃗ (x, δ)

and

B(Z,dr)⃗ (x, ξ4 δ) ⊆ B(W ,ds)⃗ (x, δ).

Proof. We will prove the containment B(Z,dr)⃗ (x, ξ4 δ) ⊆ B(W ,ds)⃗ (x, δ); because our assumptions are symmetric in (Z, dr)⃗ and (W , ds)⃗ (by Lemma 3.14.11), the other containment fol-

lows from the same proof. Fix Ω1 open with 𝒦 ⋐ Ω1 ⋐ Ω and take ξ5 ∈ (0, 1] so small that B(Y ,d̂)⃗ (x, ξ5 δ) ⊆ Ω1 for all δ ∈ (0, 1]ν and x ∈ 𝒦. We claim that there exists C ≥ 1 with B(Y ,d̂)⃗ (x, ξ5 δ) ⊆ B(X,d)⃗ (x, Cδ),

(3.96)

∀x ∈ 𝒦 and δ ∈ (0, 1]ν . Indeed, let C ≥ 1 be a large constant to be chosen later and fix y ∈ B(Y ,d̂)⃗ (x, ξ5 δ). Then there exists an absolutely continuous γ : [0, 1] → B(Y ,d̂)⃗ (x, ξ5 δ) ⊆ Ω1 , with γ(0) = x, γ(1) = y, and γ′ (t) = ∑j aj (t)(ξ5 δ)d̂j Yj (γ(t)) almost everywhere, where ∑ |aj (t)|2 < 1 for almost every t. Lemma 3.14.12 shows that (X, d )⃗ and (Y , d̂)⃗ are strongly equivalent on Ω. Thus, ⃗

Yj = ∑ bkj Xk ,

∞ bkj ∈ Cloc (Ω).

d ⃗k ≤d̂⃗ j

Therefore, γ′ (t) = ∑( ∑ aj (t)bkj (γ(t))ξ5 j δd̂j −d k )δd k Xk (γ(t)) d̂⃗

k

⃗

⃗

⃗

d̂⃗ j ≥d ⃗k

= ∑( ∑ C −|d k |1 aj (t)bkj (γ(t))ξ5 j δd̂j −d k )(Cδ) k Xk (γ(t)) ⃗

k

d̂⃗

⃗

⃗

d⃗

d̂⃗ j ≥d ⃗k

d⃗

=: ∑ ã k (t)(Cδ) k Xk (γ(t)). k

Since γ(t) ∈ Ω1 , ∀t, and Ω1 ⋐ Ω, we have |bkj (γ(t))| ≲ 1, for almost every t. Thus, by taking C sufficiently large (independent of δ, x, and y) we have ∑ |ã k (t)|2 < 1, that is, y = γ(1) ∈ B(X,d)⃗ (x, Cδ), establishing (3.96).

3.14 Control of vector fields

�

193

Letting ξ3 ∈ (0, 1] be as in Theorem 3.5.1, Theorem 3.5.1 (b) shows that for all x ∈ 𝒦 and δ ∈ (0, 1]ν , B(X,d)⃗ (x, ξ3 δ) ⊆ B(W ,ds)⃗ (x, δ). Using that (Z, dr)⃗ ⊆ (Y , d̂)⃗ and (3.96), for x ∈ 𝒦 and δ ∈ (0, 1]ν , we have B(Z,dr)⃗ (x, ξ3 ξ5 C −1 δ) ⊆ B(Y ,d̂)⃗ (x, ξ3 ξ5 C −1 δ) ⊆ B(X,d)⃗ (x, ξ3 δ) ⊆ B(W ,ds)⃗ (x, δ), which completes the proof with ξ4 := ξ3 ξ5 C −1 . Proposition 3.14.14. Suppose, in addition to the above assumptions, that for each μ ∈ {1, . . . , ν}, (W μ , dsμ ) are Hörmander vector fields with formal degrees on Ω. Then, for each μ ∈ {1, . . . , ν}, (Z μ , drμ ) are Hörmander vector fields with formal degrees on Ω. Moreover, if 𝒦 ⋐ M is compact, then ∀x ∈ 𝒦, δ ∈ (0, ∞)ν , Vol(B(W ,ds)⃗ (x, δ)) ∧ 1 ≈ Vol(B(Z,dr)⃗ (x, δ)) ∧ 1, ⃗ but not on x or δ. ⃗ and (Z, dr), where the implicit constants depend on 𝒦, Ω, (W , ds), Proof. That (W μ , dsμ ) are Hörmander vector fields with formal degrees on Ω is equivalent to the vector fields in Γ⋅ (Ω, Gen((W μ , dsμ ))) spanning the tangent space at every point of Ω. Similarly, that (Z μ , drμ ) are Hörmander vector fields with formal degrees on Ω is equivalent to the vector fields in Γ⋅ (Ω, Gen((Z μ , drμ ))) spanning the tangent space at every point of Ω. By hypothesis, Γ⋅ (Ω, Gen((W μ , dsμ ))) = Γ⋅ (Ω, Gen((Z μ , drμ ))), so we see that (Z μ , drμ ) are Hörmander vector fields with formal degrees on Ω. With ξ4 as in Proposition 3.14.13, we have for δ ∈ (0, ξ4 ]ν , using Theorem 3.5.4 (n), Vol(B(Z,dr)⃗ (x, δ)) ∧ 1 ≤ Vol(B(W ,ds)⃗ (x, ξ4−1 δ)) ∧ 1 ≲ Vol(B(W ,ds)⃗ (x, δ)) ∧ 1. Suppose δ ∈ (0, ∞)ν \ (0, ξ4 ]ν . Without loss of generality, assume δ1 > ξ4 . Then, using Theorem 3.5.4 (l), we have for x ∈ 𝒦 Vol(B(W ,ds)⃗ (x, δ)) ∧ 1 ≥ Vol(B(W ,ds)⃗ (x, ξ4 e1 )) ∧ 1 ≈ Λ(x, ξ4 e1 ) ∧ 1 ≥ inf Λ(y, ξ4 e1 ) ∧ 1 ≳ 1, y∈𝒦

where the last line uses the formula for Λ, (3.15), and the compactness of 𝒦. Similarly, we have Vol(B(Z,dr)⃗ (x, δ)) ∧ 1 ≳ 1. Thus, Vol(B(Z,dr)⃗ (x, δ)) ∧ 1 ≈ 1 ≈ Vol(B(W ,ds)⃗ (x, δ)) ∧ 1. Combining the above estimates, we conclude that for δ ∈ (0, ∞)ν and x ∈ 𝒦, Vol(B(Z,dr)⃗ (x, δ)) ∧ 1 ≲ Vol(B(W ,ds)⃗ (x, δ)) ∧ 1. The reverse inequality follows by reversing the roles of (W , ds)⃗ and (Z, dr)⃗ throughout.

194 � 3 Vector fields and Carnot–Carathéodory geometry

3.15 The main multi-parameter setting In our applications to maximally subelliptic operators, a special case of the multiparameter setting in Section 3.5 is of central interest. In this section, we describe this setting and show that it is invariant under an adapted single-parameter scaling. Let M be a connected C ∞ manifold of dimension n ∈ ℕ+ with smooth, strictly positive density Vol. Fix ν ≥ 1. For each μ ∈ {1, . . . , ν}, we suppose we are given a list of vector fields with single-parameter formal degrees: μ

μ

∞ (W μ , dsμ ) := {(W1 , ds1 ), . . . , (Wrμμ , dsμrμ )} ⊂ Cloc (M; TM) × ℕ+ .

We suppose: (I) (W 1 , ds1 ) are Hörmander vector fields with formal degrees. (II) Gen((W μ , dsμ )) is locally finitely generated for μ ∈ {2, . . . , ν}. (III) (W 1 , ds1 ), . . . , (W ν , dsν ) pairwise weakly locally approximately commute on M. 3.15.1 The multi-parameter unit scale To present the scaling results adapted to the above setting, we need to introduce a version of the above setting at the “unit scale,” and we will rescale to this unit scale. We work on Bn (1) endowed with the density Vol = hσLeb , where h ∈ C ∞ (Bn (1); ℝ), with infx∈Bn (1) h(x) > 0. Fix ν ≥ 1 and assume that for each μ ∈ {1, . . . , ν} we are given μ

μ

(W μ , dsμ ) := {(W1 , ds1 ), . . . , (Wrμμ , dsμrμ )} ⊂ C ∞ (Bn (1); TBn (1)) × ℕ+ . We wish to make assumptions (I)–(III) described above quantitative. To do this, we assume that for each μ ∈ {1, . . . , ν} we are given a finite set μ

μ

(X μ , d μ ) = {(X1 , d1 ), . . . , (Xqμμ , dqμμ )} ⊂ Gen((W μ , dsμ )) with (W μ , dsμ ) ⊆ (X μ , d μ ) such that the following holds. Set (X, d )⃗ = {(X1 , d 1⃗ ), . . . , (Xq , d q⃗ )}

:= (X 1 , d 1 ) ⊠ (X 2 , d 2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (X ν , d ν ) ⊂ C ∞ (Bn (1); TBn (1)) × (ℕν \ {0}).

We assume that: l l (i) [Xj , Xk ] = ∑d ⃗ ≤d ⃗ +d ⃗ cj,k Xl , cj,k ∈ C ∞ (Bn (1)). l

j

k

(ii) span{X11 (x), . . . , Xq11 (x)} = Tx Bn (1), ∀x ∈ Bn (1), and moreover, max

󵄨 󵄨 inf 󵄨󵄨󵄨det(Xj11 (x)| ⋅ ⋅ ⋅ |Xj1n (x))󵄨󵄨󵄨 > 0.

j1 ,...,jn ∈{1,...,q} x∈Bn (1)

Item (ii) implies (I) holds and (i) implies (II) and (III) hold.

(3.97)

3.15 The main multi-parameter setting

� 195

Definition 3.15.1. For a parameter ι, we say C is an ι-multi-parameter unit-admissible constant if there exists L ∈ ℕ, depending only on ι and upper bounds for q, ν, and max1≤j≤q |dj |1 , such that C can be chosen to depend only on ι, upper bounds for q, l ν, max1≤j≤q |dj |1 , max1≤j≤q ‖Xj ‖C L (Bn (1);ℝn ) , max1≤j,k,l≤q ‖cj,k ‖C L (Bn (1)) , and ‖h‖C L (Bn (1)) , and lower bounds > 0 for infx∈Bn (1) h(x) and the left-hand side of (3.97). If ι0 is another parameter, we say C = C(ι0 ) is an ι-multi-parameter unit-admissible constant if C is an ι-multi-parameter unit-admissible constant, which can also depend on ι0 . 3.15.2 Scaling We work in the setting at the start of this section. Thus, we are given vector fields with ∞ formal degrees (W 1 , ds1 ), . . . , (W ν , dsν ) ⊂ Cloc (M; TM)×ℕ+ satisfying hypotheses (I)–(III). In this section, we describe a single-parameter scaling adapted to these vector fields. Fix 𝒦 ⋐ Ω1 ⋐ Ω2 ⋐ M with 𝒦 compact and Ω1 and Ω2 open and relatively compact. For each μ ∈ {1, . . . , ν}, let (X μ , d μ ) ⊂ Gen((W μ , dsμ )) be such that Gen((W μ , dsμ )) is finitely generated by (X μ , d μ ) on Ω2 and (W μ , dsμ ) ⊆ (X μ , d μ ). This is always possible: for 2 ≤ μ ≤ ν this follows from hypothesis (II) and for μ = 1 this uses hypothesis (I) and the case ν = 1 of Proposition 3.4.14. Because W11 , . . . , Wr11 satisfy Hörmander’s condition, X11 , . . . , Xq11 span the tangent space at every point of Ω2 . Set λ1 := 1 and for μ ∈ {2, . . . , ν} pick λμ > 0 such that for 1 ≤ j ≤ rm u, μ

Wj =

∑

μ dl1 ≤λμ dsj

l cj,μ Xl1 ,

l ∞ cj,μ ∈ Cloc (Ω2 );

(3.98)

this is always possible since X11 , . . . , Xq11 span the tangent space at every point of Ω2 . Let λ := (λ1 , . . . , λν ) ∈ (0, ∞)ν . Remark 3.15.2. Formula (3.98) is a more explicit version of the fact that (W 1 , ds1 ) weakly λμ controls (W μ , dsμ ) on Ω2 ; see Definition 3.14.3. Theorem 3.3.7 applies with (W 1 , ds1 ) and (X 1 , d 1 ) in place of (W , ds) and (X, d ). Let Φx,δ : Bn (1) → B(X 1 ,d 1 ) (x, δ) ∩ Ω1 be as in that theorem. Then we have the following. Theorem 3.15.3. For μ ∈ {1, . . . , ν}, x ∈ 𝒦, and δ ∈ (0, 1] we set μ,x,δ

Wj

:= Φ∗x,δ δ

μ

λμ dsj

μ,x,δ

(W μ,x,δ , dsμ ) := {(W1

μ,x,δ

(X μ,x,δ , d μ ) := {(X1

μ

Wj , μ

μ,x,δ

Xk

μ

μ

:= Φ∗x,δ δλμ dk Xk ,

, ds1 ), . . . , (Wrμ,x,δ , dsμrμ )} ⊂ C ∞ (Bn (1); TBn (1)) × ℕ+ , μ μ

, d1 ), . . . , (Xqμ,x,δ , dqμμ )} ⊂ C ∞ (Bn (1); TBn (1)) × ℕ+ . μ

Let hx,δ be the function h from this application of Theorem 3.3.7 (n). Then (W 1,x,δ , ds1 ), . . . , (W ν,x,δ , dsν ), (X 1,x,δ , d 1 ), . . . , (X ν,x,δ , d ν ), and hx,δ satisfy the hypotheses of Section 3.15.1 uniformly for x ∈ 𝒦, δ ∈ (0, 1], in the sense that multi-parameter unit-admissible constants

196 � 3 Vector fields and Carnot–Carathéodory geometry (as in Definition 3.15.1) can be chosen independent of x ∈ 𝒦 and δ ∈ (0, 1]. More precisely, we have: (a) (W μ,x,δ , dsμ ) ⊆ (X μ,x,δ , d μ ) ⊂ Gen((W μ,x,δ , dsμ )), for μ ∈ {1, . . . , ν}, x ∈ 𝒦, δ ∈ (0, 1]. μ,x,δ μ,x,δ (b) Wj and Xk are C ∞ uniformly for x ∈ 𝒦 and δ ∈ (0, 1]. More precisely, ∀L ∈ ℕ, μ ∈ {1, . . . , ν}, 1 ≤ j ≤ rμ , 1 ≤ k ≤ qμ , 󵄩 μ,x,δ 󵄩 sup 󵄩󵄩󵄩Wj 󵄩󵄩󵄩C L (Bn (1);ℝn ) < ∞,

x∈𝒦 δ∈(0,1]

󵄩 μ,x,δ 󵄩 sup 󵄩󵄩󵄩Xk 󵄩󵄩󵄩C L (Bn (1);ℝn ) < ∞.

x∈𝒦 δ∈(0,1]

(c) We have X11,x,δ , . . . , Xq1,x,δ span the tangent space to every point of Bn (1) uniformly in 1 x ∈ 𝒦 and δ ∈ (0, 1] in the sense that inf

󵄨 󵄨 inf 󵄨󵄨󵄨det(Xj1,x,δ (u), . . . , Xj1,x,δ (u))󵄨󵄨󵄨 > 0. 1 n

max

x∈𝒦,δ∈(0,1] j1 ,...,jn ∈{1,...q1 } u∈Bn (1)

(d) Define (X x,δ , d )⃗ = {(X1x,δ , d 1⃗ ), . . . , (Xqx,δ , d q⃗ )}

:= (X 1,x,δ , d 1 ) ⊠ (X 2,x,δ , d 2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (X ν,x,δ , d ν ).

Then, for 1 ≤ j, k ≤ q, x ∈ 𝒦, δ ∈ (0, 1], [Xjx,δ , Xkx,δ ] =

∑ d ⃗ l ≤d ⃗ j +d ⃗ k

l,x,δ x,δ cj,k Xl ,

l,x,δ cj,k ∈ C ∞ (Bn (1)),

and for every L ∈ ℕ, 󵄩 l,x,δ 󵄩󵄩 sup 󵄩󵄩󵄩cj,k 󵄩󵄩C L (Bn (1)) < ∞.

(3.99)

x∈𝒦 δ∈(0,1]

(e) infu∈Bn (1) infx∈𝒦,δ∈(0,1] hx,δ (u) > 0. (f) ∀L ∈ ℕ, supx∈𝒦,δ∈(0,1] ‖hx,δ ‖C L (Bn (1)) < ∞. Proof. (a): Since (W μ , dsμ ) ⊆ (X μ , d μ ) ⊂ Gen((W μ , dsμ )), we have μ

μ

μ

(δλμ ds W μ , dsμ ) ⊆ (δλμ d X μ , d μ ) ⊂ Gen((δλμ ds W μ , dsμ )).

(3.100)

Pulling (3.100) back via Φx,δ gives (a). (b): When μ = 1, (b) follows from Theorem 3.3.7 (j). Let μ ∈ {2, . . . , ν}. We first prove μ μ,x,δ λ ds the result for Wj . Multiplying (3.98) by δ μ j , we have δ

μ

λμ dsj

μ

Wj =

μ

∑ (δ μ

dl1 ≤λμ dsj

1 λμ dsj −dl1 l cj,μ )δdl Xl1 .

(3.101)

3.15 The main multi-parameter setting

� 197

Pulling (3.101) back via Φx,δ we have μ,x,δ

Wj

q1

l,x,δ 1,x,δ = ∑ cj,μ Xl , l=1

where μ

λ dsj −dl1 l cj,μ

{δ μ l,x,δ cj,μ := { 0, {

μ

dl1 ≤ λμ dsj ,

∘ Φx,δ ,

otherwise.

Since we have already established the case μ = 1, to complete the proof it suffices to show that ∀L ∈ ℕ, 󵄩 l,x,δ 󵄩󵄩 sup 󵄩󵄩󵄩cj,μ 󵄩󵄩C L (Bn (1)) < ∞.

(3.102)

x∈𝒦 δ∈(0,1]

μ

μ

The estimate (3.102) is trivial when dl1 > λμ dsj . When dl1 ≤ λμ dsj , we have, using Theorem 3.3.7 (m), for x ∈ 𝒦, δ ∈ (0, 1], 󵄩󵄩 l,x,δ 󵄩󵄩 󵄩 1,x,δ α l,x,δ 󵄩󵄩 ) cj,μ 󵄩󵄩C(Bn (1)) 󵄩󵄩cj,μ 󵄩󵄩C L (Bn (1)) ≈ ∑ 󵄩󵄩󵄩(X |α|≤L

= ∑ δ

μ

1 α l 󵄩 󵄩󵄩(X ) cj,μ 󵄩󵄩󵄩C(Φx,δ (Bn (1)))

λμ dsj −dl1 +degd 1 (α) 󵄩 󵄩

|α|≤L

α l 󵄩 󵄩 󵄩󵄩 ≲ ∑ 󵄩󵄩󵄩(X 1 ) cj,μ 󵄩C(Ω1 ) ≲ 1, |α|≤L

where the final ≲ 1 uses that Ω1 ⋐ Ω2 . This proves (3.102) and completes the proof of (b) μ,x,δ μ,x,δ μ,x,δ for Wj . Since each Xk is a fixed commutator (independent of x and δ) of the Wk , μ,x,δ

the result for Xk follows as well, completing the proof of (b). (c): This is a restatement of Theorem 3.3.7 (k). (d): Set (W , ds)⃗ = {(W1 , ds1⃗ ), . . . , (Wr , ds1⃗ )}

∞ := (W 1 , ds1 ) ⊠ (W 2 , ds2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (W ν , dsν ) ⊂ Cloc (M; TM) × (ℕν \ {0}), (X, d )⃗ = {(X1 , d 1⃗ ), . . . , (Xq , d 1⃗ )} ∞ := (X 1 , d 1 ) ⊠ (X 2 , d 2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (X ν , d ν ) ⊂ Cloc (M; TM) × (ℕν \ {0}).

⃗ By Proposition 3.8.8 (a), Gen((W , ds)) ⃗ is linearly finitely generClearly, (W , ds)⃗ ⊆ (X, d ). ated by (X, d )⃗ on Ω2 . By Remark 3.4.12, we have [Xj , Xk ] =

∑ d ⃗ l ≤d ⃗ j +d ⃗ k

l cj,k Xl ,

l ∞ cj,k ∈ Cloc (Ω2 ).

(3.103)

198 � 3 Vector fields and Carnot–Carathéodory geometry Multiplying (3.103) by δλ⋅(d j +d k ) and pulling back via Φx,δ , we obtain ⃗

⃗

[Xjx,δ , Xkx,δ ] =

∑ dl ≤dj +dk

l,x,δ x,δ cj,k Xl ,

l,x,δ l where cj,k := δλ⋅(dj +dk −dl ) cj,k . It remains to show that (3.99) holds. This can be done by an almost identical argument to the proof of (3.102); we leave the details to the interested reader. (e) and (f): These follow from Theorem 3.3.7 (n).

3.15.3 Quantitative scaling After applying Theorem 3.15.3 we obtain the following objects at the unit scale: (W 1,x,δ , ds1 ), . . . , (W ν,x,δ , dsν ), (X 1,x,δ , d 1 ), . . . , (X ν,x,δ , d ν ), and hx,δ , which satisfy the hypotheses of Section 3.15.1 uniformly in x and δ. We can then apply Theorem 3.15.3 to these new vector fields to obtain another scaling map. Sometimes, we will need to do this process repeatedly, working with several levels of iterated scaling. To do this, we need a more quantitative version of Theorems 3.3.7 and 3.15.3. The idea is that we will start with vector fields at the multi-parameter unit scale (as in Section 3.15.1) and have a scaling result in which all estimates are multi-parameter unit-admissible constants (as in Definition 3.15.1). As with all the scaling results in this text, we will see this as a simple consequence of Theorem 3.6.5. Thus, in this section, we take the same setting as in Section 3.15.1, so we are given (W 1 , ds1 ), . . . , (W ν , dsν ), (X 1 , d 1 ), . . . , (X ν , d ν ), and h satisfying the assumptions of that section (in particular, satisfying (i) and (ii)). Notation 3.15.4. In this section, we write A ≲ι;ι0 B to denote A ≤ CB, where C = C(ι0 ) ≥ 0 is an ι-multi-parameter unit-admissible constant as in Definition 3.15.1. We write A ≈ι;ι0 B for A ≲ι;ι0 B and B ≲ι;ι0 A. We write A ≲ι0 B and A ≈ι0 B for A ≲0;ι0 B and A ≈0;ι0 B, respectively. Set λ1 := 1 and for μ ∈ {2, . . . , ν} take λμ > 0 such that ∀1 ≤ j ≤ qμ we may write μ

Wj =

∑

μ dl1 ≤λμ dsj

l cj,μ Xl1 ,

l cj,μ ∈ C ∞ (Bn (1)),

󵄩 l 󵄩󵄩 and such that 󵄩󵄩󵄩cj,μ 󵄩󵄩C L (Bn (1)) ≲L 1, ∀L ∈ ℕ; this is always possible by (ii). Define Λ(x, δ) := h(x) = h(x)

1 1 󵄨 󵄨 max 󵄨󵄨󵄨det(δdj1 Xj11 (x)| ⋅ ⋅ ⋅ |δdjn Xj1n (x))󵄨󵄨󵄨 1≤j ,...,j ≤q 1

n

max

1≤j1 ,...,jn ≤q1

1

1

1

σLeb (δdj1 Xj11 (x), . . . , δdjn Xj1n (x)).

(3.104)

3.15 The main multi-parameter setting

� 199

Theorem 3.15.5. Fix σ ∈ (0, 1).1 There exist 0-multi-parameter unit-admissible constants δ0 = δ0 (σ) ∈ (0, 1] and ξ3 = ξ3 (σ) ∈ (0, 1] such that the following hold: (a) ∀x ∈ Bn (σ), δ ∈ (0, 1], B(X 1 ,d 1 ) (x, ξ3 δ) ⊆ B(W 1 ,ds1 ) (x, δ) ⊆ B(X 1 ,d 1 ) (x, δ). (b) ∀x ∈ Bn (σ), δ ∈ (0, δ0 ], Vol(B(W 1 ,ds1 ) (x, δ)) ≈σ Vol(B(X 1 ,d 1 ) (x, δ)) ≈σ Λ(x, δ). (c) ∀x ∈ Bn (σ), δ ∈ (0, δ0 ], Vol(B(W 1 ,ds1 ) (x, 2δ)) ≲ Vol(B(W 1 ,ds1 ) (x, δ)) and Vol(B(X 1 ,d 1 ) (x, 2δ)) ≲ Vol(B(X 1 ,d 1 ) (x, δ)).

For all x ∈ Bn (σ), δ ∈ (0, 1], there exists a map Φx,δ : Bn (1) → B(X 1 ,d 1 ) (x, δ) ∩ Bn ((1 + σ)/2) such that: (d) Φx,δ (0) = x. (e) Φx,δ is a smooth coordinate system, that is, Φx,δ (Bn (1)) ⊆ Bn (1) is open and Φx,δ : ∞ Bn (1) → Φx,δ (Bn (1)) is a Cloc diffeomorphism. n (f) ∀x ∈ B (σ), δ ∈ (0, 1], B(W 1 ,ds1 ) (x, ξ3 δ) ⊆ B(X 1 ,d 1 ) (x, ξ3 δ) ⊆ Φx,δ (Bn (1/2)) ⊆ Φx,δ (Bn (1)) ⊆ B(X 1 ,d 1 ) (x, δ). μ,x,δ

Set Wj

:= Φ∗x,δ δ

μ

λμ dsj

μ

μ,x,δ

Wj and Xk

μ

μ

:= Φ∗x,δ δλμ dk Xk . Let μ,x,δ

(W μ,x,δ , dsμ ) := {(W1

μ,x,δ

(X μ,x,δ , d μ ) := {(X1

μ

, ds1 ), . . . , (Wrμ,x,δ , dsμrμ )}, μ μ

, d1 ), . . . , (Xqμ,x,δ , dqμμ )}. μ

Define hx,δ by Φ∗x,δ hσLeb = Λ(x, δ)hx,δ σLeb . Then (W 1,x,δ , ds1 ), . . . , (W ν,x,δ , dsν ), (X 1,x,δ , d 1 ), . . . , (X ν,x,δ , d ν ), and hx,δ satisfy the hypotheses of Section 3.15.1 quantitatively in the sense that ι-multi-parameter unit-admissible constants C with respect to these choices can be chosen to be C = C(σ), ι-multi-parameter unit-admissible constants with respect to (W 1 , ds1 ), . . . , (W ν , dsν ), (X 1 , d 1 ), . . . , (X ν , d ν ), and h. More precisely, we have: (g) (W μ,x,δ , dsμ ) ⊆ (X μ,x,δ , d μ ) ⊂ Gen((W μ,x,δ , dsμ )). (h) ∀L ∈ ℕ, x ∈ Bn (σ), δ ∈ (0, 1], 󵄩󵄩 μ,x,δ 󵄩󵄩 󵄩󵄩Wj 󵄩󵄩C L (Bn (1);ℝn ) ≲L;σ 1, 󵄩󵄩 μ,x,δ 󵄩󵄩 󵄩󵄩Xk 󵄩󵄩C L (Bn (1);ℝn ) ≲L;σ 1. 1 In many of our applications, we will take σ = 7/8.

200 � 3 Vector fields and Carnot–Carathéodory geometry (i) For all x ∈ Bn (σ), δ ∈ (0, 1], 󵄨 󵄨 inf 󵄨󵄨󵄨det(Xj1,x,δ (u)| ⋅ ⋅ ⋅ |Xj1,x,δ (u))󵄨󵄨󵄨 ≳σ 1. 1 n

max

j1 ,...,jn ∈{1,...,q1 } u∈Bn (1)

(j) ∀x ∈ Bn (σ), δ ∈ (0, 1], 󵄩󵄩 󵄩󵄩 󵄩 1,x,δ α 󵄩󵄩 n ) f 󵄩󵄩C B (1) . 󵄩󵄩f 󵄩󵄩C L (Bn (1)) ≈L;σ ∑ 󵄩󵄩󵄩(X |α|≤L

(k) For x ∈ Bn (σ) and δ ∈ (0, 1], set (X x,δ , d )⃗ = {(X1x,δ , d 1⃗ ), . . . , (Xqx,δ , d q⃗ )}

:= (X 1,x,δ , d 1 ) ⊠ (X 2,x,δ , d 2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (X ν,x,δ , d ν ).

Then [Xjx,δ , Xkx,δ ] =

∑ d ⃗ l ≤d ⃗ j +d ⃗ k

l,x,δ x,δ cj,k Xl ,

l,x,δ cj,k ∈ C ∞ (Bn (1)),

where ∀L ∈ ℕ, 󵄩󵄩 l,x,δ 󵄩󵄩 󵄩󵄩cj,k 󵄩󵄩C L (Bn (1)) ≲L;σ 1. (l) ∀L ∈ ℕ, ‖hx,δ ‖C L (Bn (1)) ≲L;σ 1. (m) hx,δ (u) ≈σ 1, ∀u ∈ Bn (1). To prove Theorem 3.15.5, we require the next lemma. Lemma 3.15.6. For each μ ∈ {2, . . . , ν} and 1 ≤ k ≤ qμ , we have μ

Xk =

∑

μ dl1 ≤λμ dk

l ck,μ Xl ,

l ck,μ ∈ C ∞ (Bn (1)),

such that ∀L ∈ ℕ, 󵄩󵄩 l 󵄩󵄩 󵄩󵄩ck,μ 󵄩󵄩C L (Bn (1)) ≲L 1. Proof. Recursively set Gen0 ((W μ , dsμ )) := (W μ , dsμ ) and for K ≥ 1, GenK ((W μ , dsμ )) := GenK−1 ((W μ , dsμ ))

∪ {([Z1 , Z2 ], dr1 + dr2 ) : (Z1 , dr1 ), (Z2 , dr2 ) ∈ GenK−1 ((W μ , dsμ ))}.

Note that

� 201

3.15 The main multi-parameter setting

(X μ , d μ ) ⊆ Genmax1≤k≤q

μ

μ

dk ((W

μ

, dsμ )).

(3.105)

We will show that ∀(Z, dr) ∈ GenK ((W μ , dsμ )) we have ∑ bl Xl1 ,

Z=

dl1 ≤λμ dr

bl ∈ C ∞ (Bn (1)),

(3.106)

with ‖bl ‖C L (Bn (1)) ≲(L,K) 1, ∀L ∈ ℕ, and the result will follow using (3.105). We prove (3.106) by induction on K. The base case, K = 0, is (3.104). We assume (3.106) for some K − 1 ≥ 0 and prove it for K. Let (Z1 , dr1 ), (Z2 , dr2 ) ∈ Gen[ ((W μ , dsμ ))K − 1]. By the inductive hypothesis we have for j = 1, 2, Zj =

∑

dl1 ≤λμ drj

blj Xl1 ,

blj ∈ C ∞ (Bn (1)),

with ‖blj ‖C L (Bn (1)) ≲L,K 1, ∀L ∈ ℕ. Thus, we have [Z1 , Z2 ] =

=

l

l

∑

b11 b22 [Xl11 , Xl22 ] +

∑

b11 b22

dl1 ≤λμ dr1 1 dl1 ≤λμ dr2 2 dl1 ≤λμ dr1 1

dl1 ≤λμ dr2

l

l

l

∑

dl1 ≤dl1 +dl1 3

1

l

l

l

l

∑ (bl1 (Xl11 b22 )Xl12 − b22 (Xl12 b11 )Xl11 )

dl1 ≤λμ dr1 1 dl1 ≤λμ dr2 2 l ,1

cl3,l cl3,l Xl13 + 1 2

1 2

2

l

l

l

l

∑ (bl1 (Xl11 b22 )Xl12 − b22 (Xl12 b11 )Xl11 ),

dl1 ≤λμ dr1 1

dl1 ≤λμ dr2

2

2

󵄩 l ,1 󵄩 where 󵄩󵄩󵄩cl3,l 󵄩󵄩󵄩C L (Bn (1)) ≲L 1; here, the final equality uses hypothesis (i) from Section 3.15.1 1 2 applied when the vectors d j⃗ and d k⃗ are non-zero in only the first component. The above sum includes only terms of the form Xl1 where dl1 ≤ λμ (dr1 + dr2 ). This establishes (3.106) for ([Z1 , Z2 ], dr1 + dr2 ), completing the proof.

Proof of Theorem 3.15.5. The heart of the proof is to apply Theorem 3.6.5 with x0 replaced 1

1

by x ∈ Bn (σ), X1 , . . . , Xq replaced by δd1 X11 , . . . , δdq1 Xq11 , Vol := hσLeb , and M = Bn ((1 + σ)/2). The main point is that L-admissible constants and L, Vol-admissible constants C ≥ 0 as in Definition 3.6.4 are C = C(σ), L-multi-parameter unit-admissible constants as in Definition 3.15.1. To see this, we must verify that all of the quantities in Definition 3.6.4 are ≲L;σ 1. Throughout this proof, x will denote an arbitrary element of Bn (σ) and δ will denote an arbitrary element of (0, 1] By the Picard–Lindelöf theorem, if η ≳σ 1 is sufficiently small (η depending only on σ q1 and ∑j=1 ‖Xj1 ‖C 1 (Bn (1);ℝn ) ), then X11 , . . . , Xq11 satisfy 𝒞 (x, η, Bn ((1+σ)/2)) (see Definition 3.6.3), 1

1

and therefore δd1 X11 , . . . , δdq1 Xq11 also satisfy 𝒞 (x, η, Bn ((1 + σ)/2)). Lemma A.2.11 shows that τ0 = τ0 (σ) > 0 as in the hypotheses of Theorem 3.6.5 can be chosen to be a 0-multiparameter unit-admissible constant.

202 � 3 Vector fields and Carnot–Carathéodory geometry In this application of Theorem 3.6.5, we take ξ = 1. Applying assumption (i) from Section 3.15.1 applied when the vectors d j⃗ and d k⃗ are non-zero in only the first component, we have [Xj1 , Xk1 ] =

∑

dl1 ≤dj1 +dk1

1,l 1 cj,k Xl ,

l,1 cj,k ∈ C ∞ (Bn (1)),

(3.107)

1 1 󵄩 1,l 󵄩󵄩 and ∀L ∈ ℕ, 󵄩󵄩󵄩cj,k 󵄩󵄩C L (Bn (1)) ≲L;1 1. Multiplying (3.107) by δdj +dk , we have

q1

1

1

1,l,δ dl 1 [δdj Xj1 , dk1 Xk1 ] = ∑ cj,k δ Xl , l=1

where l,1,δ cj,k

1

:= {

1

1

1,l δdj +dk −dl cj,k ,

dl1 ≤ dj1 + dk1 ,

0

otherwise.

We have, for every L ∈ ℕ, α 1,l,δ 󵄩 󵄩 󵄩󵄩 max ∑ 󵄩󵄩󵄩(δd X 1 ) cj,k 󵄩C(B d 1 j,k,l 1

δ

|α|≤L

X1

(x))

≤ max

dl1 ≤dj1 +dk1

≲L;1

α l 󵄩 󵄩 󵄩󵄩 n ∑ 󵄩󵄩󵄩(X 1 ) cj,k 󵄩C(B (1))

|α|≤L

max

󵄩 l 󵄩󵄩 ∑ 󵄩󵄩󵄩cj,k 󵄩󵄩C |α| (Bn (1)) ≲L;1 1.

dl1 ≤dj1 +dk1 |α|≤L

Consider Lie

d1 δ k Xk

1

1

Vol = (δdk Xk1 h)σLeb + δdk h div(Xk )σLeb =: fk,δ σLeb .

We have, for L ∈ ℕ, α 󵄩 1 󵄩 ∑ 󵄩󵄩󵄩(δd X 1 ) fj,δ 󵄩󵄩󵄩C(B 1 d

|α|≤L

δ

X

(x))

α α 󵄩 󵄩 󵄩 󵄩 ≤ ∑ 󵄩󵄩󵄩(X 1 ) Xk h󵄩󵄩󵄩C(Bn (1)) + ∑ 󵄩󵄩󵄩(X 1 ) h󵄩󵄩󵄩C(Bn (1)) ≲L;1 1. |α|≤L

|α|≤L

We use ζ = 1 in this application of Theorem 3.6.5, so that (after reordering the vector fields) the left-hand side of (3.16) equals 1; this is always possible – see the remarks following (3.16). Combining all of the above, we have shown L-admissible constants and L, Vol-admissible constants C ≥ 0 as in Definition 3.6.4 are C = C(σ), L-multi-parameter unitadmissible constants as in Definition 3.15.1. Thus, with the above choices, Theorem 3.6.5 ∼ applies. Let η1 > 0 and let Ψx,δ : Bn (η1 ) 󳨀 → Ψx,δ (Bn (η1 )) be the map Φ from that theorem. Set Φx,δ (t) := Ψx,δ (η1 t). From here (and using Lemma 3.15.6), the result follows just as in the proofs of Theorems 3.3.7 and 3.15.3; all the estimates are multi-parameter unit-admissible constants due to the above remarks. We leave the final details to the interested reader.

3.16 Further reading and references

� 203

3.16 Further reading and references The study of Carnot–Carathéodory and sub-Riemannian geometry has a long history; we only cover the parts of that history directly relevant to the results in this chapter. The general theory of Carnot–Carahtéodory geometry began with the work of Carathéodory [33] and Chow [45]. The quantitative study of Carnot–Carathéodory geometry, more suitable to applications to PDEs, began with the work of Nagel, Stein, and Wainger [189]. They proved a result very similar to Theorem 3.3.7. In particular, they proved that Carnot–Carathéodory balls in Euclidean space give rise to a space of homogeneous type in the sense of Coifman and Weiss [52], paving the way for using techniques from harmonic analysis adapted to these geometries. Around the same time, C. Fefferman and Sanchez-Calle proved some related results [84]. Nagel, Stein, and Wainger’s methods were based on Taylor series and the Baker– Campbell–Hausdorff formula, and while they can be adapted to some settings beyond the one described in Section 3.3, they have limitations when working in the multiparameter setting or when working with vector fields with low regularity.2 Tao and Wright [232] improved on Nagel, Stein, and Wainger’s methods, when they studied twoparameter balls B(x, δ1 , δ2 ) under a weakly comparable condition δ1N ≲ δ2 ≲ δ11/N , for some large N. Importantly, they replaced some of Nagel, Stein, and Wainger’s Taylor series methods with more robust methods from the field of ODEs. The author later expanded on these ideas in [219], where general multi-parameter balls were studied without the weakly comparable hypothesis. Theorem 3.6.5 was proved by the author and Stovall in [228]; it implies all of the above previous results and is stronger than they are. A main difference is the following. In previous works, all of the estimates depended on the C m norms of the coefficients of the vector fields in some given coordinate system. Theorem 3.6.5 provides a coordinate system in which the C m norms of the vector fields have good estimates, but the hypotheses of the theorem do not assume the existence of such a coordinate system. In fact, all of the estimates in Theorem 3.6.5 remain unchanged if the whole setting is pushed forward ∞ under an arbitrary Cloc diffeomorphism. See Section A.5 for a further discussion of the history of these ideas and more results. Because Nagel, Stein, and Wainger showed that single-parameter Carnot–Carathéodory balls induce a space of homogeneous type, the theory of the maximal functions in Theorem 3.11.9 in the special case ν = 1 follows from their work. When the vector fields do not necessarily satisfy Hörmander’s condition as in the ν = 1 case of Theorem 3.11.2, the corresponding maximal function was introduced in [219]. Bounding a multi-parameter maximal function by the composition of several single-parameter

2 We present a more quantitative version of Theorem 3.3.7 in Proposition 8.3.21, which is a key step in the proof of Theorem 8.1.1. It is possible that one could obtain a similar result using the methods of [189]; however, it is much more straightforward to use Theorem 3.6.5.

204 � 3 Vector fields and Carnot–Carathéodory geometry maximal functions, as we did in the proof of Lemma 3.11.4, dates back to the work of Jessen, Marcinkiewicz, and Zygmund [135]. The particular multi-parameter maximal functions studied in Theorem 3.11.2 were first studied by the author in [219]. The study of vector-valued maximal operators began with the work of C. Fefferman and Stein [83]. Stronger multi-parameter maximal functions than the ones studied here were studied by the author and Stein in [226, 225]; these stronger results require a more intricate proof and are not useful for the applications in this text. The concepts of finitely generated and linearly finitely generated, as in Definitions 3.4.1 and 3.4.3, were first introduced in this context in [221], where their relationship to developing function spaces adapted to vector fields was described. We use similar ideas in Chapter 6. The relationship between linearly finitely generated, multi-parameter singular integrals and parametrices for differential operators was first described in [220]; we will build on these ideas in Chapter 5. For more details on, and a detailed history of, the Baker–Campbell–Hausdorff formula (sometimes called the Campbell–Hausdorff formula or the Baker–Campbell– Hausdorff–Dynkin formula) we refer the reader to [12]. The concepts of control, equivalence, and sharp control described in Section 3.14 were defined in a more general setting and in a more general way in [221]. The more general approach used in [221] might be useful for some questions like the ones discussed in this text, but this comes at the cost of more technicalities and does not change the main thrust of our results. For this reason, we chose the simpler approach in Section 3.14.

4 Pseudo-differential operators Let M be a C ∞ manifold of dimension n ∈ ℕ+ , fix ν ∈ ℕ+ , and for each μ ∈ {1, . . . , ν}, let μ

μ

∞ (X μ , d μ ) = {(X1 , d1 ), . . . , (Xqμμ , dqμμ )} ⊂ Cloc (M; TM) × ℕ+

be a finite set of smooth vector fields with formal degrees. Set (X, d )⃗ = {(X1 , d 1⃗ ), . . . , (Xq , d q⃗ )} := (X 1 , d 1 ) ⊠ (X 2 , d 2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (X ν , d ν ) μ

μ

∞ = {(Xj , dj eμ ) : μ ∈ {1, . . . , ν}, 1 ≤ j ≤ qμ } ⊂ Cloc (M; TM) × (ℕν \ {0}),

(4.1)

where we recall that eμ ∈ ℝν is the vector which equals 1 in the μ-th component and 0 in all other components. Note that q = q1 + ⋅ ⋅ ⋅ + qν and each d j⃗ is non-zero in precisely one component. The goal in this chapter is to define the concept of a pseudo-differential ⃗ operator with respect to (X, d ). Main assumption: out, is

The main assumption of this chapter, which we assume through[Xj , Xk ] =

∑

l cj,k Xl ,

l ∞ cj,k ∈ Cloc (M).

(4.2)

d ⃗ l ≤d ⃗ j +d ⃗ k

We do not assume X1 , . . . , Xq span the tangent space at any point. Recall that d l⃗ ≤ d j⃗ + d k⃗ means that the inequality holds coordinatewise. Remark 4.0.1. The main way (X, d )⃗ will arise in our applications is the following. We ∞ will be given ν sets 𝒮1 , . . . , 𝒮ν ⊆ Cloc (M; TM) × ℕ+ such that each Gen(𝒮μ ) is locally finitely generated on M and such that 𝒮1 , . . . , 𝒮ν pairwise locally weakly approximately commute. Fix Ω ⋐ M open and relatively compact and let Gen(𝒮μ ) be finitely generated by (X μ , d μ ) on Ω. Then (X, d )⃗ satisfies (4.2) with M replaced by Ω; this follows from Proposition 3.8.8 combined with Remark 3.4.12. A particularly important example is Example 3.8.9. ⃗ > 0 Fix pre-compact open sets Ω1 , Ω2 ⋐ M with Ω1 ⋐ Ω2 . For a = a(Ω1 , Ω2 , (X, d )) sufficiently small and t = (t 1 , . . . , t ν ) ∈ Bq (a) (where t μ ∈ ℝqμ ), define, for x ∈ Ω1 , 1

1

Γ(t, x) = Γ((t 1 , . . . , t ν ), x) = e−t ⋅X e−t

2

⋅X 2

⋅ ⋅ ⋅ e−t

ν

⋅X ν

x.

(4.3)

Note that if we take a > 0 sufficiently small, then Γ(t, x) ∈ Ω2 , ∀x ∈ Ω1 , t ∈ Bq (a). Following Remark 2.2.15, (X, d )⃗ pseudo-differential operators will take the form f 󳨃→ ∫ f (Γ(t, x))K(x, t) dt, https://doi.org/10.1515/9783111085647-004

206 � 4 Pseudo-differential operators for a class of kernels K(x, t) made precise in Section 4.1. One of our main uses for (X, d )⃗ pseudo-differential operators is to create a parametrix for the so-called sub-Laplacian, which is a key tool in the study of maximally subelliptic operators. See Section 4.5. Example 4.0.2. Consider the case where M = ℝ2 and we are working with the Hörmander vector fields with formal degrees (W , ds) = {(𝜕x , 1), (x𝜕y , 1)}. In this case we take ν = 1 and we may take (X, d ) = {(X1 , d1 ), (X2 , d2 ), (X3 , d3 )} = {(𝜕1 , x), (x𝜕y , 1), (𝜕y , 2)}; see Example 3.3.6. Then we are considering operators of the form f 󳨃→ ∫ f (e−t1 X1 −t2 X2 −t3 X3 x)K(x, (t1 , t2 , t3 )) dt. Here, the kernel K(x, t) will be defined in a way which respects the fact that X1 = 𝜕x and X2 = x𝜕y have formal degree 1, but X3 = 𝜕y has formal degree 2. See Definitions 4.1.1 and 4.1.2.

4.1 Symbols of pseudo-differential operators ⃗ which depend on a pre-compact, For m ∈ ℝν , we define a class of symbols S m (a, Ω, d ), μ open set Ω ⋐ M, a small number a > 0 (to be chosen later), and the degrees dj associated μ

with the vector fields Xj (though they do not depend on the particular vector fields). μ

For t =

μ μ (t1 , . . . , tqμ )

∈ ℝqμ we define dilations by, for δμ > 0, μ

μ

μ

μ

δμd t μ = (δμ 1 t1 , . . . , δμ μ tqμμ ). d

dq

(4.4)

We define a homogeneous norm as ‖t μ ‖μ :=

qμ

qμ

qμ

μ

μ

󵄨 μ 󵄨(2 ∏ d )/d (∑󵄨󵄨󵄨tl 󵄨󵄨󵄨 j=1 j l l=1

μ

1/2 ∏j=1 dj

)

qμ

μ

󵄨 μ 󵄨1/d ≈ ∑󵄨󵄨󵄨tl 󵄨󵄨󵄨 l .

(4.5)

l=1

Note that μ

‖δμd t μ ‖μ = δμ ‖t μ ‖μ .

(4.6) q

μ

μ

μ For a multi-index γ = (γ1 , . . . , γqμ ) ∈ ℕqμ , set degd μ (γ) = ∑l=1 γj dj . Note that (δμd t μ )γ =

degd μ (γ) μ

δμ

t . We also define multi-parameter dilations; for δ ∈ [0, ∞)ν and t = (t 1 , . . . , t ν ) ∈ ℝq1 × ⋅ ⋅ ⋅ × ℝqν = ℝq , we set 1

ν

δd t = (δ1d t 1 , . . . , δνd t ν ). ⃗

(4.7)

4.1 Symbols of pseudo-differential operators

� 207

For γ = (γ1 , . . . , γν ) ∈ ℕq1 × ⋅ ⋅ ⋅ × ℕqν = ℕq , we set degd ⃗ (γ) = ∑ degd μ (γμ )eμ ∈ ℕν . μ

(4.8)

Note that (δd t)γ = δdegd ⃗ (γ) t γ . ⃗

Definition 4.1.1. For m = (m1 , . . . , mν ) ∈ ℝν and a > 0, we set S m (a, Ω, d )⃗ to be the set of ∞ all b(x, s, ξ) ∈ Cloc (M × Bq (a) × ℝq ) such that: (i) {(x, s) ∈ M × Bq (a) : ∃ξ ∈ ℝq with b(x, s, ξ) ≠ 0} ⋐ Ω × Bq (a). ∼ (ii) For every smooth coordinate system Φ : Bn (1) 󳨀 → Φ(Bn (1)) ⊆ M and all multi-indices α ∈ ℕn , β ∈ ℕq , γ = (γ1 , . . . , γν ) ∈ ℕq1 × ⋅ ⋅ ⋅ × ℕqν ≅ ℕq , ν

mμ −degd μ (γμ ) 󵄨󵄨 α β γ 󵄨 μ . 󵄨󵄨𝜕u 𝜕s 𝜕ξ b(Φ(u), s, ξ)󵄨󵄨󵄨 ≤ CΦ,α,β,γ ∏(1 + ‖ξ ‖μ ) μ=1

Definition 4.1.2. For pre-compact, open sets Ω1 , Ω2 ⋐ M with Ω1 ⋐ Ω2 , if a = a(Ω1 , Ω2 , ⃗ > 0 is sufficiently small (how small to be chosen later, but in particular so small (X, d )) ⃗ for some m ∈ ℝν , we call operators that Γ(Bq (a) × Ω1 ) ⋐ Ω2 ) and b(x, s, ξ) ∈ S m (a, Ω1 , d ), of the form ̌ t, t) dt, f 󳨃→ ∫ f (Γ(t, x))b(x,

C0∞ (M) → C0∞ (M)

(4.9)

̌ s, t) ⃗ a pseudo-differential operators of order m supported in Ω1 × Ω2 . Here, b(x, (X, d ), denotes the inverse Fourier transform in the ξ variable of b(x, s, ξ) (see Remark 4.1.4) and Γ is as in (4.3). ⃗ > 0 is used in the Remark 4.1.3. In Definition 4.1.2 a small number a = a(Ω1 , Ω2 , (X, d )) ⃗ definition of (X, d ), a pseudo-differential operators. Various results which follow require ⃗ > 0 be “sufficiently small.” In these results, how small a has to that a = a(Ω1 , Ω2 , (X, d )) ⃗ be can depend on Ω1 , Ω2 , and the particular vector fields with formal degrees (X, d ). ⃗ Since there are only finitely many of these results, if a = a(Ω1 , Ω2 , (X, d )) > 0 is chosen sufficiently small, all of these results hold. In our applications, we will take a that small.

̌ s, t) is a priori only defined as a tempered distribution, and this does Remark 4.1.4. b(x, ̌ t, t) a meaning. However, we define the integral in (4.9) as not immediately give b(x, ̌ t, t) dt := ∬ f (Γ(t, x))b(x, t, ξ)e2πit⋅ξ dt dξ. ∫ f (Γ(t, x))b(x, It is easy to see that this converges for f ∈ C0∞ (M) and a > 0 sufficiently small. Theorem 4.3.3 extends this definition to f ∈ C0∞ (M)′ . ⃗ a pseudo-differential operators have many similarities with the Remark 4.1.5. (X, d ), standard pseudo-differential operators described in Definition 2.2.3. However, unlike the

208 � 4 Pseudo-differential operators standard pseudo-differential operators, we do not have a robust calculus like the one in Theorem 2.2.8. In fact, a very useful fact about standard pseudo-differential operators is that if b1 (x, D) and b2 (x, D) are standard pseudo-differential operators of orders m1 and m2 , then [b1 (x, D), b2 (x, D)] is a standard pseudo-differential operator of order m1 +m2 −1. In particular, the commutator is of lower order than the composition b1 (x, D)b2 (x, D), ⃗ a pseudo-differential operators have no such gain when which is of order m1 +m2 . (X, d ), considering commutators; in fact, this is a necessary part of our theory – see Proposition 4.5.31. There are some remnants of the calculus of pseudo-differential operators, however; see Proposition 4.5.31. See also Section 4.6 for some comments on recent results for related pseudo-differential operators where a calculus can be obtained. ⃗ a pseudoExample 4.1.6. Let ψ ∈ C0∞ (Ω1 ). Then for any a > 0, Mult[ψ] is an (X, d ), ∞ q differential operator of order 0 supported in Ω1 × Ω2 . Indeed, take η ∈ C0 (B (a)), with η ≡ 1 on a neighborhood of 0. Set b(x, s, ξ) := ψ(x)η(s); it is immediate to verify that ̌ t, t) = ψ(x)η(t)δ (t) = ψ(x)δ (t), where δ (t) denotes the ⃗ Note that b(x, b ∈ S 0 (a, Ω1 , d ). 0 0 0 Dirac δ function at 0. Thus, ̌ t, t) dt = ψ(x)f (Γ(0, x)) = ψ(x)f (x). ∫ f (Γ(t, x))b(x, See Proposition 6.11.3 for a more detailed discussion of this perspective for multiplication operators.

4.1.1 Connection with standard pseudo-differential operators Formally, Definition 4.1.2 is very similar to the formula for standard pseudo-differential operators on ℝn given in Remark 2.2.15. In this section, we make precise the connection between the two. Let M = ℝn and consider the vector fields on ℝn with formal degrees (𝜕, 1) = l {(𝜕x1 , 1), . . . , (𝜕xn , 1)}. (𝜕, 1) satisfies the main assumption (4.2) (with cj,k = 0 for all j, k, l). Thus, it makes sense to talk about (𝜕, 1) pseudo-differential operators. These are not exactly the same as standard pseudo-differential operators as described in Definition 2.2.3, though they are almost the same, as the next proposition shows. Proposition 4.1.7. Fix relatively compact, open sets Ω1 ⋐ Ω2 ⋐ ℝn . (i) Let b(x, D) : S (ℝn ) → S (ℝn ) be a (standard) pseudo-differential operator of order m ∈ ℝ and let ψ ∈ C0∞ (Ω1 ). Then if a = a(Ω1 , Ω2 ) > 0 is sufficiently small, Mult[ψ]b(x, D) = T + R, where T is a (𝜕, 1), a pseudo-differential operator of order m supported in Ω1 × Ω2 and ∞ R ∈ Cloc (ℝn × ℝn ) (i. e., the Schwartz kernel of R is infinitely smooth).

4.1 Symbols of pseudo-differential operators

� 209

(ii) Let a > 0 and suppose T is a (𝜕, 1), a pseudo-differential operator of order m ∈ ℝ supported in Ω1 × Ω2 . Then T is a (standard) pseudo-differential operator of order m. Proof. (i): Take a = a(Ω1 , Ω2 ) > 0 so small that {x + t : x ∈ Ω1 , |t| < a} ⋐ Ω2 . Let η ∈ C0∞ (Bn (a)) equal 1 on a neighborhood of 0. For b ∈ S m (where S m is defined in Definition 2.2.3), we have ̌ t) dt Mult[ψ]b(x, D)f (x) = ∫ f (x − t)ψ(x)b(x, ̌ t) dt + ∫ f (x − t)ψ(x)(1 − η(t))b(x, ̌ t) dt = ∫ f (e−t⋅𝜕 x)ψ(x)η(t)b(x, =: Tf (x) + Rf (x), where we have used e−t⋅𝜕 x = x − t. Note that Tf (x) is a (𝜕, 1), a pseudo-differential operator of order m with symbol η(s)b(x, ξ) ∈ S m (a, Ω, 1). By Proposition 2.2.14, using the ̌ t) ∈ C ∞ , and fact that 1 − η(t) is zero on a neighborhood of 0, we have ψ(x)(1 − η(t))b(x, loc ∞ therefore the Schwartz kernel of R is in Cloc . (ii): Since T is a (𝜕, 1), a pseudo-differential operator of order m supported in Ω1 ×Ω2 , there is b ∈ S m (a, Ω1 , 1) so that Tf (x) = ∫ f (x − t)b(x, t, ξ)e2πit⋅ξ dt dξ = ∫ f (y)c(x, y, ξ)e2πi(y−x)⋅ξ dy dξ, where c(x, y, ξ) = b(x, y − x, ξ). For all α, β, γ ∈ ℕn , we have 󵄨󵄨 α β γ 󵄨 m−|γ| . 󵄨󵄨𝜕x 𝜕y 𝜕ξ c(x, y, ξ)󵄨󵄨󵄨 ≤ Cα,β,γ (1 + |ξ|) It follows that c is a “compound symbol of order m” in the sense of [216, Chapter VI, Section 6.1], and the results there show that T is a pseudo-differential operator of order m. 4.1.2 Littlewood–Paley decomposition of symbols Theorem 2.2.26 (ii) ⇔ (iii) provides a decomposition of standard symbols. We require a ⃗ For this, we require some definitions; similar decomposition for symbols in S m (a, Ω, d ). throughout, a > 0 and Ω ⋐ M is open and relatively compact. ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) is the space of all f (x, s, t) ∈ Definition 4.1.8. The space C0∞ (Ω)⊗ ∞ Cloc (M × Bq (a) × ℝq ) such that: (i) {(x, s) ∈ Ω × Bq (a) : ∃t ∈ ℝq , f (x, s, t) ≠ 0} ⋐ Ω × Bq (a). ∼ (ii) For every smooth coordinate system Φ : Bn (1) 󳨀 → Φ(Bn (1)) ⊆ M, ∀m ∈ ℕ, for all multi-indices α ∈ ℕn , β, γ ∈ ℕq , 󵄨󵄨 α β γ 󵄨 −m 󵄨󵄨𝜕u 𝜕s 𝜕t f (Φ(u), s, t)󵄨󵄨󵄨 ≤ CΦ,m,α,β,γ (1 + |t|) .

210 � 4 Pseudo-differential operators ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) is a locally convex topological vector space. Remark 4.1.9. C0∞ (Ω)⊗ The topology can be understood as follows. For compact sets 𝒦1 ⋐ Ω and 𝒦2 ⋐ Bq (a), ̂ C0∞ (𝒦2 )⊗ ̂ S (ℝq ) consisting of functions in C0∞ (Ω)⊗ ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) the space C0∞ (𝒦1 )⊗ whose support in the (x, s) variable is contained in 𝒦1 × 𝒦2 can be given a Fréchet topology in the usual way. Letting Ω = ⋃ 𝒦1m and Bq (a) = ⋃ 𝒦2m , where 𝒦1m and 𝒦2m are ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) the topology increasing sequences of compact sets, we give C0∞ (Ω)⊗ ̂ C0∞ (𝒦2m )⊗ ̂ S (ℝq ); it is easy to see of the inductive limit of the Fréchet spaces C0∞ (𝒦1m )⊗ that this limit does not depend on the choice of sequences of compact sets. Thus, not only ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) a locally convex topological vector space, it is an LF space is C0∞ (Ω)⊗ in the sense of [238, Chapter 13]. ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) is a bounded set if and only if: Lemma 4.1.10. ℬ ⊂ C0∞ (Ω)⊗ q q – {(x, s) ∈ Ω × B (a) : ∃f ∈ ℬ, t ∈ ℝ , f (x, s, t) ≠ 0} ⋐ Ω × Bq (a). – The constant CΦ,m,α,β,γ from Definition 4.1.8 (ii) can be chosen independent of f ∈ ℬ. Proof. This follows from [238, Proposition 14.6], combined with Remark 4.1.9. ̂ C0∞ (Bq (a))⊗ ̂ SE (ℝq ) be the subDefinition 4.1.11. For a set E ⊆ {1, . . . , ν}, we let C0∞ (Ω)⊗ ∞ ∞ q q ̂ C0 (B (a))⊗ ̂ S (ℝ ) such that space of all f ∈ C0 (Ω)⊗ ∫ (t μ )α f (x, s, t) dt μ = 0,

∀μ ∈ E, α ∈ ℕqμ .

ℝqμ

̂ C0∞ (Bq (a))⊗ ̂ SE (ℝq ) is a closed subspace of C0∞ (Ω)⊗ ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) and we C0∞ (Ω)⊗ give it the subspace topology. ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) and 2j = (2j1 , . . . , 2jν ) ∈ Definition 4.1.12. Given f (x, s, t) ∈ C0∞ (Ω)⊗ ⃗ ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) by (0, ∞)ν , we define Dild2j (f )(x, s, t) ∈ C0∞ (Ω)⊗ q1

1

q2

2

qν

ν

Dild2j (f )(x, s, t) = 2j1 (∑l=1 dl )+j2 (∑l=1 dl )+⋅⋅⋅+jν (∑l=1 dl ) f (x, s, 2jd t), ⃗

⃗

where 2jd t = (2j )d t is defined by (4.7). ⃗

⃗

Remark 4.1.13. Note that Dild2j (f ) is defined so that ∫ Dild2j (f )(x, s, t) dt = ∫ f (x, s, t) dt. ⃗

⃗

The main proposition of this section is the following. Proposition 4.1.14. Fix a > 0 and m ∈ ℝν . The following are equivalent: ⃗ (i) b ∈ S m (a, Ω, d ). (ii) There exists a bounded set ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) {ςj : j ∈ ℕν } ⊂ C0∞ (Ω)⊗ such that

4.1 Symbols of pseudo-differential operators

� 211

̂ C0∞ (Bq (a))⊗ ̂ S{μ:j =0} ςj ∈ C0∞ (Ω)⊗ (ℝq ) μ ̸ and ̌ s, t) = ∑ 2j⋅m Dildj ⃗ (ς )(x, s, t), b(x, 2 j j∈ℕν

̌ s, t) denotes the inverse Fourier transform of b(x, s, ξ) in the ξ variable, where b(x, and the above sum converges in the weak topology on ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ))′ . (C0∞ (Ω)⊗ Moreover, every such sum converges in this sense. The rest of this section is devoted to the proof of Proposition 4.1.14, which uses ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ), we write ς(x, ̂ s, ξ) ∈ standard techniques. For ς(x, s, t) ∈ C0∞ (Ω)⊗ ∞ ∞ q q ̂ C0 (B (a))⊗ ̂ S (ℝ ) for the Fourier transform of ς in the t variable. It is easy C0 (Ω)⊗ ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) (see also [238, to see that this gives an automorphism of C0∞ (Ω)⊗ ∞ ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) Proposition 13.1]). In particular, it takes bounded subsets of C0 (Ω)⊗ ∞ ∞ q q ̂ C0 (B (a))⊗ ̂ S (ℝ ). to bounded subsets of C0 (Ω)⊗ ̂ C0∞ (Bq (a))⊗ ̂ ŜE (ℝq ) be the subDefinition 4.1.15. For a set E ⊆ {1, . . . , ν}, we let C0∞ (Ω)⊗ ∞ ∞ q q ̂ C0 (B (a))⊗ ̂ S (ℝ ) such that space of all f (x, s, ξ) ∈ C0 (Ω)⊗ 󵄨 𝜕ξαμ f (x, s, ξ)󵄨󵄨󵄨ξ μ= 0 = 0,

∀μ ∈ E, α ∈ ℕqμ .

̂ C0∞ (Bq (a))⊗ ̂ ŜE (ℝq ) is a closed subspace of C0∞ (Ω)⊗ ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) and we C0∞ (Ω)⊗ give it the subspace topology. ̂ C0∞ (Bq (a))⊗ ̂ SE (ℝq ) → Lemma 4.1.16. The map ς 󳨃→ ς̂ is an isomorphism C0∞ (Ω)⊗ ∞ ∞ q q ̂ ̂ C0 (B (a))⊗ ̂ SE (ℝ ). C0 (Ω)⊗ Proof. This follows easily from the definitions. Lemma 4.1.17. Fix s ∈ ℝ and E ⊆ {1, . . . , ν}. Then, ∀μ ∈ E, the maps f (x, s, ξ) 󳨃→ ̂ C0∞ (Bq (a))⊗ ̂ |ξ μ |s f (x, s, ξ) and f (x, s, ξ) 󳨃→ ‖ξ μ ‖sμ f (x, s, ξ) are automorphisms of C0∞ (Ω)⊗ q ŜE (ℝ ). Proof. Note ‖ξ μ ‖μ is smooth away from ξ μ = 0 (see (4.5)). Thus, it follows that the map f (x, s, ξ) 󳨃→ ‖ξ μ ‖sμ f (x, s, ξ) is a bijection ̂ C0∞ (Bq (a))⊗ ̂ ŜE (ℝq ) → C0∞ (Ω)⊗ ̂ C0∞ (Bq (a))⊗ ̂ ŜE (ℝq ), C0∞ (Ω)⊗ with inverse f (x, s, ξ) 󳨃→ ‖ξ μ ‖−s μ f (x, s, ξ). It is elementary to show that this map is continuous (see [238, Proposition 13.1]), and we leave the details to the interested reader. A similar proof gives the result for f (x, s, ξ) 󳨃→ |ξ μ |s f (x, s, ξ).

212 � 4 Pseudo-differential operators Definition 4.1.18. For s ∈ ℝ, μ ∈ {1, . . . , ν}, set ̂ s, ξ))∨ , △sμ ς(x, s, t) = (|2πξ μ |2s ς(x, where ∨ denotes the inverse Fourier transform in the ξ variable. Corollary 4.1.19. Fix s ∈ ℝ and E ⊆ {1, . . . , ν}. Then ∀E such that μ ∈ E, ̂ C0∞ (Bq (a))⊗ ̂ SE (ℝq ) → C0∞ (Ω)⊗ ̂ C0∞ (Bq (a))⊗ ̂ SE (ℝq ) △sμ : C0∞ (Ω)⊗ is an automorphism. Proof. This follows immediately from Lemmas 4.1.16 and 4.1.17. Proof of Proposition 4.1.14. Applying the Fourier transform in the t variable, Lemma 4.1.16 shows that (ii) is equivalent to: (ii)’ There is a bounded set ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) {bj (x, s, ξ) : j ∈ ℕν } ⊂ C0∞ (Ω)⊗ such that ̂ C0∞ (Bq (a))⊗ ̂ Ŝ{μ:j =0} bj (x, s, ξ) ∈ C0∞ (Ω)⊗ (ℝq ) μ ̸ and such that b(x, s, ξ) = ∑ 2j⋅m bj (x, s, 2−jd ξ), ⃗

j∈ℕν

with convergence in the weak topology on ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ))′ , (C0∞ (Ω)⊗ and every such sum converges in this sense. (ii)’ ⇒ (i): Let bj (x, s, ξ) be as in (ii)’. We consider the sum b(x, s, ξ) = ∑ ∑ 2j⋅m bj (x, s, 2−jd ξ). ⃗

j∈ℕν j∈ℕν

(4.10)

We will show that this sum converges in the weak topology on ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ))′ , (C0∞ (Ω)⊗ ∞ ⃗ That such b and converges in Cloc (M × Bq (a) × ℝq ) to a symbol b(s, x, ξ) ∈ S m (a, Ω, d ). m ⃗ satisfies Definition 4.1.8 (i) is obvious. Thus, to show b ∈ S (a, Ω, d ), we wish to show ∼ that for any smooth coordinate system Φ : Bn (1) 󳨀 → Φ(Bn (1)) ⊆ M,

4.1 Symbols of pseudo-differential operators

󵄨󵄨 󵄨󵄨 ν 󵄨󵄨 α β γ 󵄨 󵄨󵄨𝜕 𝜕 𝜕 ∑ 2j⋅m bj (Φ(u), s, 2−jd ⃗ ξ)󵄨󵄨󵄨 ≲ ∏(1 + ‖ξ μ ‖μ )mμ −degμ (γμ ) . 󵄨󵄨 󵄨󵄨󵄨 u s ξ 󵄨󵄨 μ=1 j∈ℕ 󵄨

� 213

(4.11)

Note that with Φ∗ bj (u, s, ξ) := bj (Φ(u), s, ξ), γ

γ

𝜕uα 𝜕sβ 𝜕ξ bj (Φ(u), s, 2−jd ξ) = 2−j⋅degd (γ) (𝜕uα 𝜕sβ 𝜕ξ Φ∗ bj )(u, s, 2−jd ξ), it suffices to prove (4.11) in the case α = 0, β = 0, and γ = 0, by replacing m with m − degd ⃗ (γ). We will show that for each E ⊆ {1, . . . , ν}, ν

∑ [∏(1 + ‖ξ μ ‖μ )

−mμ jμ mμ

μ=1 j∈ℕν jμ >0,μ∈E jμ =0,μ∈E ̸

2

⃗ 󵄨 󵄨 ]󵄨󵄨󵄨bj (x, s, 2−j⋅d ξ)󵄨󵄨󵄨

(4.12)

converges absolutely and is bounded independent of x, s, and ξ, which, when combined with the above remarks, implies both the desired convergence of (4.10) and the desired bound (4.11). Set L ≥ max1≤μ≤ν |mμ | + 1. We claim, with E = {μ : jμ ≠ 0}, |ςj (x, s, ξ)| ≲ [∏(1 + ‖ξ μ ‖μ )−L ] × [∏(1 + ‖ξ μ ‖μ )−L ∧ ‖ξ μ ‖Lμ ]. μ∈E ̸

μ∈E

(4.13)

Indeed, take F ⊆ E and note that by Lemma 4.1.17, if bj,F (x, s, ξ) := ‖ξ μ ‖−L μ bj (x, s, ξ), then ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) {bj,F : j ∈ ℕν , jμ ≠ 0, ∀μ ∈ F} ⊂ C0∞ (Ω)⊗ is a bounded set. Thus, it follows immediately from the definitions that |bj (x, s, ξ)| = ‖ξ μ ‖Lμ |bj,F (x, s, ξ)| ≲ [∏(1 + ‖ξ μ ‖μ )−L ] × [∏(1 + ‖ξ μ ‖μ )−L ] × [∏ ‖ξ μ ‖Lμ ]. μ∈E ̸

μ∈F̸

μ∈F

Taking the minimum of the right-hand side over F ⊆ E yields (4.13). We separate the sum (4.12) into several parts, one for each subset F ⊆ E. For a fixed ξ ∈ ℝq , we set AF,E (ξ) := {j ∈ ℕν :∀μ ∈ ̸ E, jμ = 0, ∀μ ∈ E, jμ > 0,

∀μ ∈ F, ‖ξ μ ‖μ ∨ 2 < 2jμ ,

∀μ ∈ E \ F, 2 ≤ 2jμ ≤ 2 ∨ ‖ξ μ ‖μ }.

214 � 4 Pseudo-differential operators Note that ⋃ AF,E (ξ) = {j ∈ ℕν : jμ = 0 ⇔ μ ∈ E}.

(4.14)

F⊆E

Using (4.13), we have ∑ [ ∏ (1 + ‖ξ μ ‖μ )−mμ 2jμ ]|bj (x, s, 2−jd ξ)| ⃗

j∈AF,E (ξ) μ∈{1,...,ν}

≲

∑ [ ∏ (1 + ‖ξ μ ‖μ )−mμ 2jmμ ] × [∏(1 + 2−jμ ‖ξ μ ‖μ )−L ] μ∈F̸

j∈AF,E (ξ) μ∈{1,...,ν}

× [∏(2−jμ ‖ξ μ ‖μ )L ] μ∈F

≲

∑ [ ∏ 1] × [ ∏ (1 + ‖ξ μ ‖μ )−mμ 2jμ (mμ +L) ‖ξ μ ‖−L μ ]

j∈AF,E (ξ) μ∈E c

μ∈F c \E c

× [∏(1 + ‖ξ μ ‖μ )−mμ 2jμ (mμ −L) ‖ξ μ ‖Lμ ] μ∈F

m +L

≲ [ ∏ (1 + ‖ξ μ ‖μ )−mμ ‖ξ μ ‖−L ‖ξ μ ‖μ μ ] μ∈F c \E c

m −L

× [∏(1 + ‖ξ μ ‖μ )−mμ ‖ξ μ ‖μ μ ‖ξ μ ‖Lμ ] μ∈F

≲ 1, where in the second to last estimate we have used the definition of AF,E (ξ) and the fact that these are geometric sums. Combining this estimate with (4.14) proves (4.12) and completes the proof of (i). (i) ⇒ (ii)’: For each μ ∈ {1, . . . , ν}, let ϕμ ∈ C0∞ (ℝqμ ) satisfy ϕμ (ξ μ ) = 1 on a neighborμ

μ

hood of 0 ∈ ℝqμ . Set ψμ (ξ μ ) := ϕμ (ξ μ )−ϕμ (2d ξ μ ). Note that 1 = ϕμ (ξ μ )+∑jμ >0 ψμ (2−jμ d ξ μ ). Set bj (x, s, 2−jd ξ) := 2−j⋅m [ ⃗

∏

μ with jμ =0

ϕμ (ξ μ )] × [

μ

∏

μ with jμ >0

Note that b(x, s, ξ) = ∑j∈ℕν 2j⋅m bj (x, s, 2−jd ξ), ∀ξ ∈ ℝq .

ψμ (2−jμ d ξ μ )]b(x, s, ξ).

⃗

μ

Since ϕμ (ξ μ ) is supported on ‖ξ μ ‖μ ≲ 1 and ψμ (2−jμ d ξ μ ) is supported on ‖ξ μ ‖μ ≈ 2jμ , it follows easily from the definitions that ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) {bj (x, s, ξ) : j ∈ ℕν } ⊂ C0∞ (Ω)⊗

4.2 The exponential map

� 215

is a bounded set. Furthermore, since ψμ (ξ μ ) vanishes on a neighborhood of 0 ∈ ξ μ , we ̂ C0∞ (Bq (a))⊗ ̂ Ŝ{μ:j =0} see bj (x, s, ξ) ∈ C0∞ (Ω)⊗ (ℝq ). μ ̸

∞ Thus, by the proof of (ii)’ ⇒ (i), the sum ∑j∈ℕν 2j⋅m bj (x, s, 2−jd ξ) converges in Cloc (M× q q B (a) × ℝ ) and in the sense of the weak topology on ⃗

̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ))′ . (C0∞ (Ω)⊗ Since we already know that the sum converges pointwise to b(x, s, ξ), the sum converges in the above senses to b(x, s, ξ). This completes the proof.

4.2 The exponential map In this section, we revisit the exponential map defined in Section 3.1.1 and describe the ⃗ namely (4.2), and the map Γ derelationship between our main assumption on (X, d ), fined in (4.3). Recall the degree of a multi-index α ∈ ℕq , degd ⃗ (α) ∈ ℕν , defined in (4.8). ∞ Definition 4.2.1. For a function f ∈ Cloc (U), where U ⊆ ℝq ×M is an open neighborhood ν ⃗ of {0} × M, and for d 0 ∈ ℤ , we say

degd ⃗ (f ) ≥ d 0⃗ ∞ if f (t, x) = ∑Kk=1 t αk fk (t, x), where degd ⃗ (αk ) ≥ d 0⃗ for 1 ≤ k ≤ K and fk ∈ Cloc (U ′ ) for some open neighborhood U ′ ⊆ ℝq × M of {0} × M. The inequality degd ⃗ (αk ) ≥ d 0⃗ means the vector degd ⃗ (αk ) − d 0⃗ is non-negative in every component.

The main proposition of this section is the following. ∞ Proposition 4.2.2. For f ∈ Cloc (M), we have the following identities for 1 ≤ j ≤ q: q k (i) Xj (f (Γ(t, x))) = ((Xj + ∑k=1 g1,j Xk )f )(Γ(t, x)). q

k (ii) (Xj f )(Γ(t, x)) = (Xj + ∑k=1 g2,j Xk )(f (Γ(t, x))). q

k (iii) 𝜕tj f (Γ(t, x)) = ((−Xj + ∑k=1 g3,j Xk )f )(Γ(t, x)). q

k (iv) 𝜕tj f (Γ(t, x)) = (−Xj + ∑k=1 g4,j Xk )(f (Γ(t, x))). q

k (v) Xj (f (Γ(t, x))) = (−𝜕tj + ∑k=1 g5,j 𝜕tk )(f (Γ(t, x))). q

k (vi) (Xj f )(Γ(t, x)) = (−𝜕tj + ∑k=1 g6,j 𝜕tk )(f (Γ(t, x))). l ∞ k Here gl,j (t, x) ∈ Cloc (U), U ⊆ ℝq × M is an open neighborhood of {0} × M, degd ⃗ (gl,j ) ≥ k ⃗ ⃗ d k − d j , and g (0, x) = 0. l,j

To prove Proposition 4.2.2 we introduce some matrix notation, which we will use k only in this section. X will denote the column vector of vector fields [X1 , . . . , Xq ]⊤ . If gl,j is as in the conclusion of Proposition 4.2.2, we write Gl (t, x) for the q × q matrix with

216 � 4 Pseudo-differential operators k ∞ the (j, k) component equal to gl,j (t, x), so that Gl ∈ Cloc (M; 𝕄q×q ). For example, Proposition 4.2.2 (i) can be rewritten as X(f (Γ(t, x))) = ((I + G1 )Xf )(Γ(t, x)) and (ii) can be rewritten as (Xf )(Γ(t, x)) = (I + G2 )X(f (Γ(t, x))). ∞ Lemma 4.2.3. Let A, B ∈ Cloc (U; 𝕄q×q ) be q × q matrices with components Akj (t, x) and

Bjk (t, x), respectively, where U is an open neighborhood of {0} × M in ℝq × M. Suppose that degd ⃗ (Akj ), degd ⃗ (Bjk ) ≥ d k⃗ − d j⃗ . Then if (AB)kj (t, x) is the (j, k) component of AB, we have degd ⃗ ((AB)kj ) ≥ d k⃗ − d j⃗ .

(4.15)

degd ⃗ (Akj (t, Γ(t, x))) ≥ d k⃗ − d j⃗ ,

(4.16)

Also,

and similarly with Γ(t, x) replaced by Γ(−t, x). Finally, if A(0, x) ≡ 0, then there is an open ∞ neighborhood U1 of {0} × M in ℝq × M such that (I + A)−1 ∈ Cloc (U1 ; 𝕄q×q ), and degd ⃗ (((I + A)−1 − I)kj ) ≥ d k⃗ − d j⃗ ,

(4.17)

where ((I + A)−1 − I)kj denotes the (j, k) component of the matrix (I + A)−1 − I. Proof. It follows immediately from the definitions that degd ⃗ (Alj Blk ) ≥ d l⃗ − d j⃗ + d k⃗ − d l⃗ = d k⃗ − d j⃗ , and (4.15) follows. Formula (4.16) follows immediately from the definitions. l l For (4.17), since A(0, x) ≡ 0, for t sufficiently small we have (I + A)−1 − I = ∑∞ l=1 (−1) A . Using (4.15), a straightforward argument completes the proof; we leave the details to the interested reader.

The next lemma is a key step in the proof of Proposition 4.2.2. ̂ x) := (t, Γ(t, x)) ∈ ℝq × M. Then, for 1 ≤ j ≤ q, Lemma 4.2.4. Let Γ(t, q

k Γ̂∗ Xj = Xj + ∑ g1,j Xk , k=1

q

k Γ̂∗ 𝜕tj = 𝜕tj − Xj + ∑ g3,j Xk , k=1

k k where g1,j and g3,j are as in Proposition 4.2.2.

To prove Lemma 4.2.4 we require a few lemmas. For the next lemma, let Z ∈ ∞ Cloc (M; TM), and for s ∈ ℝ small, set Φs (x) = esZ x. Φs (x) is defined on a neighborhood

4.2 The exponential map

�

217

of {0} × M in ℝ × M. Since Φ0 (x) = x, given x ∈ M, for s small, there is a neighborhood V 󵄨 of x such that Φs 󵄨󵄨󵄨V is a diffeomorphism onto its image (by the inverse function theorem). Thus, if Y is another vector field, it makes sense to consider (Φs )∗ Y , which is an element ∞ of Cloc (U; TM), where U is a neighborhood of {0} × M in ℝ × M. Lemma 4.2.5. (i) (Φ−s )∗ Z = Z. (ii) 𝜕s (Φ−s )∗ Y = (Φ−s )∗ ([Z, Y ]). ∞ Proof. For (i), let f ∈ Cloc (M; TM). Then

󵄨 ((Φ−s )∗ Z)f = (Φ−s )∗ ZΦ∗−s f (x) = (Φ−s )∗ Zf (e−sZ x) = (Φ−s )∗ 𝜕u 󵄨󵄨󵄨u=0 f (e−sZ euZ x) 󵄨 = (Φ−s )∗ 𝜕u 󵄨󵄨󵄨u=0 f (euZ e−sZ x) = (Φ−s )∗ (Zf )(e−sZ x) = Zf (x), proving (i). We claim 󵄨 𝜕s 󵄨󵄨󵄨s=0 (Φ−s )∗ Y = [Z, Y ].

(4.18)

∞ Indeed, for f ∈ Cloc (M; TM),

󵄨 󵄨 (𝜕s 󵄨󵄨󵄨s=0 (Φ−s )∗ Y )f (x) = 𝜕s 󵄨󵄨󵄨s=0 (Φ−s )∗ Y Φ∗−s f (x) 󵄨 󵄨 = 𝜕s 󵄨󵄨󵄨s=0 (Φ−s )∗ Yf (x) + 𝜕s 󵄨󵄨󵄨s=0 Y Φ∗−s f (x) 󵄨 󵄨 = 𝜕s 󵄨󵄨󵄨s=0 (Yf )(esZ x) + 𝜕s 󵄨󵄨󵄨s=0 Y (f (e−sZ x)) = ZYf − YZf , proving (4.18). Consider, using (4.18), 󵄨 󵄨 𝜕s (Φ−s )∗ Y = 𝜕ϵ 󵄨󵄨󵄨ϵ=0 (Φ−s−ϵ )∗ Y = 𝜕ϵ 󵄨󵄨󵄨ϵ=0 (Φ−ϵ )∗ (Φ−s )∗ Y = [Z, (Φ−s )∗ Y ].

(4.19)

∞ For f ∈ Cloc (M; TM), by (i) we have Zf = (Φ−s )∗ ZΦ∗−s f , and therefore,

f = (Z(Φ−s )∗ Y Φ∗−s − (Φ−s )∗ Y Φ∗−s Z)f

= ((Φ−s )∗ ZY Φ∗−s − (Φ−s )∗ YZΦ∗−s )f = ((Φ−s )∗ [Z, Y ])f .

Combining (4.19) and (4.20) proves (ii). For 1 ≤ κ ≤ μ, set Γκ (t κ , x) = e−t

κ

⋅X κ

x,

and for ϵ ∈ [0, 1], set Γ̂ϵ,κ (t, x) = Γ̂ϵ,κ ((t 1 , . . . , t ν ), x) := ((t 1 , . . . , t ν ), Γκ (ϵt κ , x)).

(4.20)

218 � 4 Pseudo-differential operators 󵄨 If U is a sufficiently small neighborhood of {0} × M in ℝq × M, Γ̂ϵ,κ 󵄨󵄨󵄨U is a smooth diffeomorphism onto its image, for all ϵ ∈ [0, 1], and depends smoothly on ϵ. For the rest of this section, all functions of (t, x) will be smooth and defined on some open neighborhood of {0}×M in ℝq ×M. All functions of (ϵ, t, x) will be smooth and defined on [0, 1]×U, where U is some open neighborhood of {0} × M in ℝq × M. Given a function f (ϵ, t, x), we define degd ⃗ (f ) ≥ d 0⃗ just as in Definition 4.2.1, but the functions fk from that definition depend on (ϵ, t, x). In the next lemmas, we write t κ for an element of ℝqκ , where κ ∈ {1, . . . , ν}. However, for a multi-index α ∈ ℕq , we write t α for the usual multi-index notation. α, β, γ will all denote multi-indices in ℕq , while κ will denote an element of {1, . . . , ν}. Lemma 4.2.6. For 1 ≤ j ≤ q, q

k,1 (Γ̂ϵ,κ )∗ Xj = ∑ hj,κ (ϵ, t, x)Xk , j=1

k,1 where degd ⃗ (hj,κ ) ≥ d k⃗ − d j⃗ .

Proof. Set M0 := 2ν max1≤l≤q |d l⃗ |1 . Fix 1 ≤ k ≤ q and consider, for α ∈ ℕq with |α| ≤ M0 and degd ⃗ (α) ≥ d k⃗ − d j⃗ , using Lemma 4.2.5 (ii), we have qκ

𝜕ϵ (Γ̂ϵ,κ )∗ (t α Xk ) = (Γ̂ϵ,κ )∗ [t κ ⋅ X κ , t α Xk ] = ∑ tjκ′ t α (Γϵ,κ )∗ [Xjκ′ , Xk ] j′ =1

κ

= ∑ tjκ′ t α (Γ̂ϵ,κ )∗ j′ =1

cjl,κ ′ ,k Xl

(4.21)

j′

qκ

=∑

∑

d ⃗l ≤d κ eκ +d ⃗k

∑

j′ =1 d ⃗l ≤d κ′ eκ +d ⃗k

κ κ α ̂ cjl,κ ′ ,k (Γϵ,κ (−t , x))(Γϵ,κ )∗ tj ′ t Xl ,

j

∞ where cjl,κ ′ ,k ∈ Cloc (M) come from our main hypothesis (4.2). We consider the right-hand

side of (4.21). Note that in the sum, the term t β := tjκ′ t α , where degd ⃗ (β) = degd ⃗ (α) + djκ′ eκ ≥ d k⃗ − d j⃗ + d κ′ eκ ≥ d l⃗ − d j⃗ . Thus, Xl is multiplied by t β , where deg ⃗ (β) ≥ d l⃗ − d j⃗ . j

d

There are two cases. If |α| < M0 , then |β| ≤ M0 , in which case (4.21) shows ∞ 𝜕ϵ (Γ̂ϵ,κ )∗ (t α Xk ) is a Cloc linear combination of terms of the same form as (Γ̂ϵ,κ )∗ (t α Xk ). If |α| = M0 , then |β| = M0 + 1 and at least one component of degd ⃗ (β) must be greater than M0 /ν. Thus, there exist γ1 , γ2 ∈ ℕq , with β = γ1 + γ2 , |γ1 | ≤ M0 , and degd ⃗ (γ1 ) ≥ d l⃗ − d j⃗ . In fact, one may take γ1 = β − γ2 ∈ ℕq , where γ2 ∈ ℕq is equal to 1 in one component, zero in all other components, and such that degd ⃗ (γ2 ) is non-zero in the component where degd ⃗ (β) is greater than M0 /ν; then, in that component degd ⃗ (γ1 ) will be ≥ max1≤l≤q |d l⃗ |1 . Since degd ⃗ (γ1 ) equals degd ⃗ (β) in all other components, it follows that degd ⃗ (γ1 ) ≥ d l⃗ − d j⃗ . ∞ Thus, in this case (4.21) again shows that 𝜕ϵ (Γ̂ϵ,κ )∗ (t α Xk ) is a Cloc linear combination of α terms of the same form as (Γ̂ϵ,κ )∗ (t Xk ).

4.2 The exponential map

� 219

Hence, if we consider the vector of vector fields Vj (ϵ, t, x) = ((Γ̂ϵ,κ )∗ t α Xk ), where k ranges over 1 ≤ k ≤ q and α ∈ ℕq ranges over all |α| ≤ M0 with degd ⃗ (α) ≥ d k⃗ −d j⃗ , then Vj satisfies an ODE of the form 𝜕ϵ Vj (ϵ, t, x) = M(ϵ, t, x)Vj (ϵ, t, x), for some smooth matrix M(ϵ, t, x). Solving this ODE, we see that Vj (ϵ, t, x) = A(ϵ, t, x)Vj (0, t, x), where A(ϵ, t, x) is some smooth matrix. Since (Γ̂ϵ,κ )∗ Xj is one of the components of Vj (ϵ, t, x) and (Γ̂0,κ )∗ t α Xk = t α Xk , the result follows. q k,2 k,2 Lemma 4.2.7. (Γ̂ϵ,κ )∗ Xj = Xj + ∑k=1 hj,κ (ϵ, t, x)Xk , where degd ⃗ (hj,κ ) ≥ d k⃗ − d j⃗ and k,2 hj,κ (ϵ, 0, x) = 0.

Proof. Using Lemma 4.2.5 (ii) and (4.2), we have qκ

𝜕ϵ (Γ̂ϵ,κ )∗ Xj = (Γ̂ϵ,κ )∗ [t κ ⋅ X κ , Xj ] = ∑

∑

j′ =1 d ⃗l ≤d κ′ eκ +d ⃗j

l,κ cj,k (Γϵ,κ (t, x))(Γ̂ϵ,κ )∗ tjκ′ Xl ,

(4.22)

j

l,κ ∞ where cj,k ∈ Cloc (M). Using Lemma 4.2.6, for d l⃗ ≤ djκ′ eκ + d j⃗ , we have q

r,1 (Γ̂ϵ,κ )∗ tjκ′ Xl = ∑ tjκ′ hk,κ (ϵ, t, x)Xr , r=1

r,1 r,1 where degd ⃗ (hk,κ ) ≥ d r⃗ − d l⃗ , and therefore degd ⃗ (tjκ′ hk,κ ) ≥ d r⃗ − d l⃗ + djκ′ eκ ≥ d r⃗ − d j⃗ . Plugging this into (4.22), we see q

k,3 𝜕ϵ (Γ̂ϵ,κ )∗ Xj = ∑ hj,κ (ϵ, t, x)Xk , k=1

k,3 k,3 where hj,κ (ϵ, 0, x) = 0 and degd ⃗ (hj,κ ) ≥ d k⃗ − d j⃗ . Integrating (4.23) and using (Γ̂0,κ )∗ Xj = Xj , we see that q

ϵ

k=1

0

k,3 ′ (Γ̂ϵ,κ )∗ Xj = Xj + ∑ (∫ hj,κ (ϵ , t, x) dϵ′ ) Xk ,

and the result follows.

(4.23)

220 � 4 Pseudo-differential operators Recall that we have separated t ∈ ℝq as t = (t 1 , . . . , t ν ), where t κ ∈ ℝqκ . Thus, for a given j ∈ {1, . . . , q}, tj is a coordinate of precisely one t κ . Lemma 4.2.8. We have {𝜕t , (Γ̂ϵ,κ )∗ 𝜕tj = { j q k,4 𝜕 − ϵXj + ∑k=1 hj,κ (ϵ, t, x)Xk , { tj

tj is not a coordinate of t κ , tj is a coordinate of t κ ,

k,4 k,4 where degd ⃗ (hj,κ ) ≥ d k⃗ − d j⃗ and hj,κ (ϵ, 0, x) = 0.

Proof. We lift the vector fields Xkκ to ℝq × M, by viewing them as tangent to {t} × M, ∀t ∈ ℝq , and we use the same notation Xkκ for the lifted vector fields. Let Y κ (t, x) = κ ∞ t κ ⋅ X κ (x) ∈ Cloc (ℝq × M; T(ℝq × M)). Using this notation, Γ̂(ϵ,κ) (t, x) = eϵY (t, x). Thus, using Lemma 4.2.5 (ii), 𝜕ϵ (Γ̂ϵ,κ )∗ 𝜕tj = (Γ̂ϵ,κ )∗ [Y κ , 𝜕tj ] = {

0, tj is not a coordinate of t κ , (Γ̂ϵ,κ )∗ (−Xj ), tj is a coordinate of t κ ,

0, ={ q k,2 −Xj − ∑k=1 hj,κ Xk ,

tj is not a coordinate of t κ , tj is a coordinate of t κ ,

k,2 where hj,κ (ϵ, t, x) is as in Lemma 4.2.7. Using the fact that (Γ̂0,κ )∗ 𝜕tj = 𝜕tj and integrating we see

{𝜕t , (Γ̂ϵ,κ )∗ 𝜕tj = { j ϵ k,2 ′ q 𝜕 − ϵXj − ∑k=1 (∫0 hj,κ (ϵ , t, x) dϵ′ ) Xk , { tj

tj is not a coordinate of t κ , tj is a coordinate of t κ .

The result follows. Corollary 4.2.9. We have q

k,5 (Γ̂1,κ )∗ Xj = Xj + ∑ hj,κ (t, x)Xk , k=1

{𝜕t , (Γ̂1,κ )∗ 𝜕tj = { j q k,6 𝜕 − Xj + ∑k=1 hj,κ (t, x)Xk , { tj

tj is not a coordinate of t κ , tj is a coordinate of t κ ,

k,l k,l where hj,κ (0, x) = 0 and degd ⃗ (hj,κ ) ≥ d k⃗ − d j⃗ , for l = 5, 6.

Proof. This follows by taking ϵ = 1 in Lemmas 4.2.7 and 4.2.8. Lemma 4.2.10. For κ ∈ {1, . . . , ν}, j ∈ {1, . . . , q}, q

k,7 (Γ̂1,κ ∘ Γ̂1,κ+1 ∘ ⋅ ⋅ ⋅ ∘ Γ̂1,ν )∗ Xj = Xj + ∑ hj,κ (t, x)Xk , k=1

(4.24)

4.2 The exponential map

(Γ̂1,κ ∘ Γ̂1,κ+1 ∘ ⋅ ⋅ ⋅ ∘ Γ̂1,ν )∗ 𝜕tj ={

𝜕tj , q k,8 𝜕tj − Xj + ∑k=1 hj,κ (t, x)Xk ,

221

�

tj is not a coordinate of (t κ , t κ+1 , . . . , t ν ), tj is a coordinate of (t κ , t κ+1 , . . . , t ν ),

(4.25)

k,l k,l where hj,κ (0, x) = 0 and degd ⃗ (hj,κ ) ≥ d k⃗ − d j⃗ , for l = 7, 8. k,l k,l Proof. We write Hκ,l (t, x) for the q × q matrix whose (j, k) component is hj,κ , where hj,κ is either as in the statement of the lemma (for l = 7, 8) or as in Corollary 4.2.9 (for l = 5, 6). We freely use (4.15) from Lemma 4.2.3, so that the product of two matrices of the type Hκ,l is again of the same form. We begin with (4.24). We proceed by induction on κ; the base case, κ = ν, follows from Corollary 4.2.9. Thus, we assume the result for κ replaced by κ + 1 and prove it for κ. By the inductive hypothesis and Corollary 4.2.9, we have

(Γ̂1,κ ∘ ⋅ ⋅ ⋅ ∘ Γ̂1,ν )∗ X = (Γ̂1,κ )∗ (Γ̂1,κ+1 ∘ ⋅ ⋅ ⋅ ∘ Γ̂1,ν )∗ = (Γ̂1,κ )∗ (I + Hκ+1,7 )X = (I + Hκ+1,7 (t, Γκ (t, x)))(I + Hκ,5 (t, x))X =: (I + Hκ,7 (t, x)).

That Hκ,7 (t, x) is of the desired form follows from Lemma 4.2.3; this completes the proof of (4.24). We turn to (4.25). Let Bκ denote the q × q matrix which equals −1 in the (j, j) component if tj is a coordinate of t κ and is equal to 0 in all other components. Let 𝜕t denote the column vector 𝜕t = [𝜕t1 , . . . , 𝜕tq ]⊤ . We claim that ν

ν

κ′ =κ

κ′ =κ

(Γ̂1,κ ∘ Γ̂1,κ+1 ∘ ⋅ ⋅ ⋅ ∘ Γ̂1,ν )∗ 𝜕t = 𝜕t − ( ∑ Bκ′ ) X + ( ∑ Bκ′ H9,κ′ (t, x)) X,

(4.26)

k,9 k,0 where H9,κ (0, x) = 0 and if hj,κ is the (j, k) component of H9,κ , then degd ⃗ (hj,κ ) ≥ d k⃗ − d j⃗ . Formula (4.25) follows immediately from (4.26) and we prove (4.26) by induction on κ. The base case, κ = ν, follows from Corollary 4.2.9. For the inductive step, we assume (4.26) with κ replaced by κ + 1 and prove it for κ. We have, by the inductive hypothesis and Corollary 4.2.9,

(Γ̂1,κ ∘ ⋅ ⋅ ⋅ ∘ Γ̂1,ν )∗ 𝜕t = (Γ̂1,κ )∗ (Γ̂1,κ+1 ∘ ⋅ ⋅ ⋅ ∘ Γ̂1,ν )∗ 𝜕t ν

ν

= (Γ̂1,κ )∗ (𝜕t − ( ∑ Bκ′ ) X + ( ∑ Bκ′ H9,κ′ ) X) κ′ =κ+1

ν

κ′ =κ+1

= 𝜕t − Bκ X + Bκ Hκ,6 X − ( ∑ Bκ′ ) (X + Hκ,5 X) ν

κ′ =κ+1

+ ( ∑ Bκ′ H9,κ′ ) (I + Hκ,5 )X. κ′ =κ+1

This is of the desired form by Lemma 4.2.3, completing the proof.

222 � 4 Pseudo-differential operators ̂ x) = Γ̂1,1 ∘ Γ̂1,2 ∘ ⋅ ⋅ ⋅ ∘ Γ̂1,ν (t, x), this follows immediately Proof of Lemma 4.2.4. Since Γ(t, from Lemma 4.2.10. Proof of Proposition 4.2.2. We prove parts of the proposition with f (x) replaced by a more general function F(t, x) which is allowed to depend on t ∈ ℝq . Using Lemma 4.2.4, we have Xj F(t, Γ(t, x)) = Γ̂∗ Γ̂∗ Xj Γ̂∗ F(t, x) = Γ̂∗ ((Γ̂∗ Xj )F(t, x)) q

q

k=1

k=1

k k = Γ̂∗ ((Xj + ∑ g1,j Xk )F)(t, x) = ((Xj + ∑ g1,j Xk )F)(t, Γ(t, x)),

proving (i). For (ii), we define G2 by G2 (t, Γt (x)) = (I + G1 (t, x))−1 − I, that is, G2 (t, x) = (I + G1 )−1 (t, Γ−t (x)) − I. By Lemma 4.2.3, G2 is of the desired form, and we have, using (i), ((I + G2 )X)(F(t, Γt (x))) = ((I + G1 )−1 (I + G1 )XF)(t, Γt (x)) = (XF)(t, Γt (x)), proving (ii). For (iii), similar to the proof of (i), we have, by Lemma 4.2.4, q

k 𝜕tj F(t, Γt (x)) = Γ̂∗ Γ̂∗ 𝜕tj Γ̂∗ F(t, x) = Γ̂∗ ((𝜕tj − Xj + ∑ g3,j Xk ) F) (t, x) q

k=1

k = ((𝜕tj − Xj + ∑ g3,j Xk ) F) (t, Γt (x)), k=1

proving (iii). For (iv), we have, by using (ii) and (iii), 𝜕tj f (Γt (x)) = ((−I + G3 )Xf )(Γt (x)) = (−I + G3 (t, Γt (x)))(I + G2 (t, x))X(f (Γt (x))), which is of the desired form by Lemma 4.2.3. For (v), we have, by using (iv), X(f (Γ(t, x))) = (−I + G4 (t, x))−1 (I + G4 (t, x))X(f (Γ(t, x))) = (−I + G4 (t, x))−1 𝜕t (f (Γ(t, x))), which is of the desired form by Lemma 4.2.3. For (vi), we have, by using (iii), (Xf )(Γt (x)) = (−I + G3 (t, Γ(t, x)))−1 ((−I + G3 )Xf )(Γ(t, x)) = (−I + G3 (t, Γ(t, x)))−1 𝜕t (f (Γ(t, x))), which is of the desired form by Lemma 4.2.3.

4.3 Littlewood–Paley decompositions

� 223

4.3 Littlewood–Paley decompositions Proposition 4.1.14 gave a decomposition of symbols in S m (a, Ω, d )⃗ into Littlewood– Paley scales. This immediately gives a corresponding decomposition of (X, d )⃗ pseudodifferential operators as defined in Definition 4.1.2. In this section, we give a description of the properties of this decomposition. In this section, fix relatively compact open sets Ω1 , Ω2 ⋐ M, with Ω1 ⋐ Ω2 , and take ⃗ > 0 small, in particular so small that a = a(Ω1 , Ω2 , (X, d )) Γ(Bq (a) × Ω1 ) ⋐ Ω2 . ⃗ > 0 as needed to make Throughout this section, we may shrink a = a(Ω1 , Ω2 , (X, d )) various results hold; we will only need to shrink a a finite number of times, so we will be left with a strictly positive number. As in Section 4.1.2, for j ∈ ℤν we write 2j = (2j1 , 2j2 , . . . , 2jν ) ∈ (0, ∞)ν . We write ⃗ ⃗ ⃗ ⃗ 2j⋅d X = (2j )d X = {2j⋅d 1 X1 , . . . , 2j⋅d ν Xν }. ⃗ a pre-pseudo-differential operator scales Definition 4.3.1. (i) A bounded set of (X, d ), ∞ supported in Ω1 × Ω2 is a subset ℰ ⊂ Hom(Cloc (M), C0∞ (Ω1 )) × (0, 1]ν such that each (E, 2−j ) ∈ ℰ is of the form Ef (x) = ∫ f (Γ(t, x)) Dild2j (ς)(x, t, t) dt, ⃗

where ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) {ς : (E, 2−j ) ∈ ℰ } ⊂ C0∞ (Ω1 )⊗ is a bounded set. ̂ C0∞ (Bq (a))⊗ ̂ S{μ:j =0} (ii) If, in addition, for each (Ej , 2−j ) ∈ ℰ , we have ς ∈ C0∞ (Ω1 )⊗ (ℝq ), μ ̸ ⃗ a pseudo-differential operator scales supported we say ℰ is a bounded set of (X, d ), in Ω1 × Ω2 .

⃗ a pre-pseudo-differential operRemark 4.3.2. Note that if ℰ is a bounded set of (X, d ), −j ator scales supported in Ω1 × Ω2 , then for each (E, 2 ) ∈ ℰ , the Schwartz kernel of E is supported in Ω1 × Ω2 . ⃗ > 0 is sufficiently small, then for T ∈ Hom(C ∞ (M), Theorem 4.3.3. If a = a(Ω1 , Ω2 , (X, d )) loc C0∞ (Ω1 )) the following are equivalent for m ∈ ℝν : ⃗ a pseudo-differential operator of order m supported in Ω1 × Ω2 . (i) T is an (X, d ), ⃗ a pseudo-differential operators scales supported in (ii) There is a bounded set of (X, d ), −j ν Ω1 × Ω2 , {(Ej , 2 ) : j ∈ ℕ }, such that T = ∑j∈ℕν 2j⋅m Ej , where this sum converges in ∞ the topology of bounded convergence on Hom(Cloc (M), C0∞ (Ω1 )).

224 � 4 Pseudo-differential operators Furthermore, every sum of the type described in (ii) converges in the topology of ∞ bounded convergence on Hom(Cloc (M), C0∞ (Ω1 )) and in the weak operator topology on ∞ ∞ Hom(C0∞ (M)′ , Cloc (Ω1 )′ ), where Cloc (Ω1 )′ is given the weak topology. In particular, every ⃗ a pseudo-differential operator extends to an operator in Hom(C ∞ (M)′ , C ∞ (Ω1 )′ ). (X, d ), 0 loc We complete the proof of Theorem 4.3.3 in Section 4.3.3; we begin with some basic properties of (X, d )⃗ pseudo-differential operator scales. ⃗ > 0 is sufficiently small, the following holds. Lemma 4.3.4. If a = a(Ω1 , Ω2 , (X, d )) ⃗ (i) Let ℰ be a bounded set of (X, d ), a pre-pseudo-differential operator scales supported in Ω1 × Ω2 . Let α be an ordered multi-index. Then −j⋅d ⃗

ℰα := {((2

X)α E, 2−j ), (E(2−j⋅d X)α , 2−j ) : (E, 2−j ) ∈ ℰ } ⃗

⃗ a pre-pseudo-differential operator scales supported in Ω1 × Ω2 . is a bounded set (X, d ), ⃗ a pseudo-differential operator scales sup(ii) If, in addition, ℰ is a bounded set of (X, d ), ⃗ a pseudo-differential operator ported in Ω1 ×Ω2 , then ℰα is also a bounded set of (X, d ), scales supported in Ω1 × Ω2 . Proof. We prove only (ii); the proof of (i) follows from a simpler version of the same argument. We will show that for l ∈ {1, . . . , q}, {(2−j⋅d l Xl E, 2−j ), (E2−j⋅d l Xl , 2−j ) : (E, 2−j ) ∈ ℰ } ⃗

⃗

⃗ a pseudo-differential operator scales supported in Ω1 × Ω2 . The is a bounded set of (X, d ), result for general α then follows by repeated application of this result. Consider, using Proposition 4.2.2 (v), q

k Xl f (Γ(t, x)) = (−𝜕tl + ∑ g5,l 𝜕tk ) f (Γ(t, x)), k=1

k k where degd ⃗ g5,l ≥ d k⃗ − d l⃗ and g5,l (t, x) is defined on some open neighborhood of {0} × M q k in ℝ × M. Treating l as fixed, we write g5,l (t, x) = ∑Kr=1 t βr,k gr,k (t, x), where gr,k (t, x) ∈ ∞ q Cloc (B (a) × Ω1 ) and degd ⃗ (βr,k ) ≥ d k⃗ − d l⃗ (where we have taken a > 0 sufficiently small). For (E, 2−j ) ∈ ℰ , we have

Ef (x) = ∫ f (Γ(t, x)) Dild2j (ς)(x, t, t) dt, ⃗

k where ς is as in Definition 4.3.1. Thus, if a > 0 is chosen so small that g5,l (t, x) is defined q and smooth for x ∈ Ω1 and t ∈ B (a), we have by integration by parts

4.3 Littlewood–Paley decompositions q

� 225

k 2−j⋅d l Xl Ef (x) = ∫ 2−j⋅d l ((−𝜕tl + ∑ g5,l 𝜕tk ) f (Γ(t, x))) Dild2j (ς)(x, t, t) dt ⃗

⃗

⃗

k=1

=

⃗ ̃ t, t) ∫ f (Γ(t, x)) Dild2j (ς)(x,

dt,

where K

̃ s, t) = 𝜕tl ς(x, s, t) − [ ∑ 𝜕sk (sβr,k gr,k (s, x))] ς(x, s, t) ς(x, r=1

K

− ∑ gr,k (s, x)2−j⋅(d l −d k +degd ⃗ (βr,k )) 𝜕tk t βr,k ς(x, s, t). ⃗

⃗

r=1

Since d l⃗ − d k⃗ + degd ⃗ (βr,k ) ≥ 0, we have 2−j⋅(d l −d k +degd ⃗ (βr,k )) ∈ (0, 1]. Note that 𝜕tk , multiplication by t βr,k , and multiplication by any smooth function g(s, x) are all con̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) → C0∞ (Ω1 )⊗ ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ), which pretinuous maps C0∞ (Ω1 )⊗ ∞ ∞ q q ̂ C0 (B (a))⊗ ̂ SE (ℝ ), ∀E ⊆ {1, . . . , ν}. Thus, we have ς̃ ∈ serve the subspaces C0 (Ω1 )⊗ ∞ ∞ q q ̂ ̂ C0 (Ω1 )⊗C0 (B (a))⊗S{μ:jμ =0} ̸ (ℝ ) and ⃗

⃗

̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) {ς̃ : (E, 2−j ) ∈ ℰ } ⊂ C0∞ (Ω1 )⊗ is a bounded set, completing the proof that {(2−j⋅d l Xl E, 2−j ) : (E, 2−j ) ∈ ℰ } ⃗

⃗ a pseudo-differential operator scales supported in Ω1 × Ω2 . is a bounded set of (X, d ), A similar proof shows that {(E2−j⋅d l Xl , 2−j ) : (E, 2−j ) ∈ ℰ } ⃗

is a bounded set of (X, d )⃗ pseudo-differential operator scales supported in Ω1 ×Ω2 , where we use Proposition 4.2.2 (vi) in place of Proposition 4.2.2 (v). Along with the conclusions of Lemma 4.3.4 (ii), one of the main properties of pseudodifferential operator scales is the next proposition. ⃗ > 0 is sufficiently small, the following holds. Let Proposition 4.3.5. If a = a(Ω1 , Ω2 , (X, d )) ⃗ ℰ be a bounded set of (X, d ), a pseudo-differential operator scales supported in Ω1 × Ω2 . Then, ∀μ ∈ {1, . . . , ν}, (E, 2−j ) ∈ ℰ , μ

α

μ

β

E = ∑ 2−jμ (2−|αμ |−|βμ |) (2−jμ d X μ ) μ Eμ,αμ ,βμ (2−jμ d X μ ) μ , |αμ |≤1 |βμ |≤1

where

226 � 4 Pseudo-differential operators {(Eμ,αμ ,βμ , 2−j ) : (E, 2−j ) ∈ ℰ , μ ∈ {1, . . . , ν}, |αμ |, |βμ | ≤ 1} ⃗ a pseudo-differential scales supported in Ω1 × Ω2 . is a bounded set of (X, d ), To prove Proposition 4.3.5, we require two lemmas. ∞ Lemma 4.3.6. Fix μ0 ∈ {1, . . . , ν}, 1 ≤ l ≤ qμ0 , 1 ≤ k ≤ q. Let g(t, x) ∈ Cloc (U), where U ⊆ μ q ℝ × M is an open neighborhood of {0} × M, and suppose g satisfies degd ⃗ (g) ≥ d k⃗ − dl 0 eμ0 . ⃗ > 0 is small enough, the following holds. Then if b = b(g, Ω1 , Ω2 , (X, d )) ∞ ∞ ̂ C0 (Bq (b))⊗ ̂ S (ℝq ) be a bounded set. For ς ∈ ℬ, j ∈ ℕν , define Let ℬ ⊂ C0 (Ω1 )⊗ operators F1 = F1 [j, ς] and F2 = F2 [j, ς] by μ0

F1 f (x) = 2−jμ0 dl ∫(gXk f )(Γ(t, x)) Dild2j (ς)(x, t, t) dt, ⃗

μ0

F2 f (x) = 2−jμ0 dl ∫[gXk (f (Γ(t, x)))] Dild2j (ς)(x, t, t) dt. ⃗

̂ C0∞ (Bq (b))⊗ ̂ S{μ:j =0} Then if ς ∈ C0∞ (Ω1 )⊗ (ℝq ) ∩ ℬ, we have μ ̸ μ0

F1 = ∑ 2−jμ0 (1−|αμ0 |) Eαμ ,1 (2−jμ0 d X μ0 )

αμ0

,

(4.27)

Eαμ ,2 ,

(4.28)

0

|αμ0 |≤1

μ0

F2 = ∑ 2−jμ0 (1−|αμ0 |) (2−jμ0 d X μ0 ) |αμ0 |≤1

α μ0

0

where {(Eαμ ,1 , 2−j ), (Eαμ ,2 , 2−j ) : j ∈ [0, ∞)ν , 0

ς∈

0

̂ C0∞ (Bq (b))⊗ ̂ S{μ:j =0} C0∞ (Ω1 )⊗ (ℝq ) μ ̸

∩ ℬ, |αμ0 | ≤ 1}

⃗ b pseudo-differential operator scales supported in Ω1 × Ω2 . is a bounded set of (X, d ), Proof. By taking b sufficiently small, we may write R

g(t, x) = ∑ t γr gr (t, x), r=1

μ ∞ q where gr (t, x) ∈ Cloc (B (b) × Ω2 ) and degd ⃗ (γr ) ≥ d k⃗ − dl 0 eμ0 . This decomposes the righthand sides of (4.27) and (4.28) into R terms, and it suffices to study each of these terms ∞ q separately. Thus, we henceforth take g(t, x) = t γ h(t, x), where h(t, x) ∈ Cloc (B (b) × Ω2 ) μ 0 and degd ⃗ (γ) ≥ d k⃗ − dl eμ0 . The formal degree d k⃗ ∈ ℕν is non-zero in precisely one component, and we separate the proof into two cases. The first case is when d k⃗ is non-zero in the μ0 component. In μ μ μ μ this case, (Xk , d k⃗ ) = (Xp 0 , dp 0 eμ0 ) for some 1 ≤ p ≤ qμ0 and degd ⃗ (γ)μ0 ≥ dp 0 − dl 0 .

4.3 Littlewood–Paley decompositions

� 227

For (4.27), we have μ0

F1 f (x) = 2−jμ0 dl ∫(t γ hXpμ0 f )(Γ(t, x)) Dild2j (ς)(x, t, t) dt ⃗

μ0

= ∫(2−jμ0 dp Xpμ0 f )(Γ(t, x)) Dild2j (ς1̃ )(x, t, t) dt, ⃗

where μ0

ς1̃ (x, s, t) = 2jμ0 (dp μ0

Since 2jμ0 (dp

μ

−dl 0 )−j⋅degd ⃗ (γ)

μ

−dl 0 )−j⋅degd ⃗ (γ)

h(s, Γ(s, x))t γ ς(x, s, t).

̂ C0∞ (Bq (b))⊗ ̂ S{μ:j =0} ∈ (0, 1], we see ς1̃ ∈ C0∞ (Ω1 )⊗ (ℝq ) and μ ̸

̂ C0∞ (Bq (b))⊗ ̂ S{μ:j =0} {ς1̃ : j ∈ [0, ∞)ν , ς ∈ C0∞ (Ω1 )⊗ (ℝq ) ∩ ℬ} μ ̸ ̂ C0∞ (Bq (b))⊗ ̂ S (ℝq ) ⊂ C0∞ (Ω1 )⊗

is a bounded set. Formula (4.27) follows. For (4.28), we have μ0

F2 f (x) = 2−jμ0 dl ∫[t γ hXpμ0 (f (Γ(t, x)))] Dild2j (ς)(x, t, t) dt ⃗

= 2−jμ0 ∫ f (Γ(t, x)) Dild2j (ς2̃ )(x, t, t) dt ⃗

μ0

+ 2−jμ0 dp Xpμ0 ∫ f (Γ(t, x)) Dild2j (ς3̃ )(x, t, t) dt, ⃗

where μ0

ς2 (x, s, t) = −2−jμ0 (dl

−1) μ0 γ Xp [s h(s, x)ς(x, s, t)],

μ μ jμ0 (dp 0 −dl 0 )−j⋅degd ⃗ (γ)

ς3 (x, s, t) = 2 μ0

μ0

h(s, x)t γ ς(x, s, t).

μ0

Since 2−jμ0 (dl −1) , 2jμ0 (dp −dl )−j⋅degd ⃗ (γ) ∈ (0, 1], it is immediate to verify that ς2̃ , ς3̃ ∈ ̂ C0∞ (Bq (b))⊗ ̂ S{μ:j =0} C0∞ (Ω1 )⊗ (ℝq ) and μ ̸ ̂ C0∞ (Bq (b))⊗ ̂ S{μ:j =0} {ς2̃ , ς3̃ : j ∈ [0, ∞)ν , ς ∈ C0∞ (Ω1 )⊗ (ℝq ) ∩ ℬ} μ ̸ ̂ C0∞ (Bq (b))⊗ ̂ S (ℝq ) ⊂ C0∞ (Ω1 )⊗

is a bounded set. Formula (4.28) follows. We now consider the case where d k⃗ is non-zero is some component other than μ0 . Thus, degd ⃗ (γ) ≥ d k⃗ (since d k⃗ is non-zero in precisely one component). For (4.27), we have μ0

F1 f (x) = 2−jμ0 dl ∫(t γ hXk f )(Γ(t, x)) Dild2j (ς)(x, t, t) dt ⃗

= 2−jμ0 ∫(2−j⋅d k Xk f )(Γ(t, x)) Dild2j (ς4̃ )(x, t, t) dt, ⃗

⃗

228 � 4 Pseudo-differential operators where μ0

ς4̃ (x, s, t) = 2j⋅d k −j⋅degd ⃗ (γ) 2−jμ0 (dl ⃗

−1) γ

t h(s, x)ς(x, s, t).

̂ C0∞ (Bq (b))⊗ ̂ S{μ:j =0} Since 2j⋅d k −j⋅degd ⃗ (γ) 2−jμ0 ∈ (0, 1], it follows that ς4̃ ∈ C0∞ (Ω1 )⊗ (ℝq ) and μ ̸ ⃗

̂ C0∞ (Bq (b))⊗ ̂ S{μ:j =0} {ς4̃ : j ∈ [0, ∞)ν , ς ∈ C0∞ (Ω1 )⊗ (ℝq ) ∩ ℬ} μ ̸ ̂ C0∞ (Bq (b))⊗ ̂ S (ℝq ) ⊂ C0∞ (Ω1 )⊗

is a bounded set. Thus, we have shown that F1 = 2−jμ0 E(2−j⋅d k Xk ), ⃗

where ̂ C0∞ (Bq (b))⊗ ̂ S{μ:j =0} {(E, 2−j ) : j ∈ [0, ∞)ν , ς ∈ C0∞ (Ω1 )⊗ (ℝq ) ∩ ℬ} μ ̸ ⃗ b pseudo-differential operator scales supported in Ω1 × Ω2 . is a bounded set of (X, d ), ⃗ Applying Lemma 4.3.4 (ii) to E(2−j⋅d k Xk ) proves ̃ F1 = 2−jμ0 E, where ̃ 2−j ) : j ∈ [0, ∞)ν , ς ∈ C ∞ (Ω1 )⊗ ̂ C0∞ (Bq (b))⊗ ̂ S{μ:j =0} {(E, (ℝq ) ∩ ℬ} 0 μ ̸ ⃗ b pseudo-differential operator scales supported in Ω1 ×Ω2 ; (4.27) is a bounded set of (X, d ), follows (with only one term in that formula: the one when |αμ | = 0). For (4.28), we have μ0

F2 f (x) = 2−jμ0 dl ∫[t γ hXk (f (Γ(t, x)))] Dild2j (ς)(x, t, t) dt ⃗

= 2−jμ0 ∫ f (Γ(t, x)) Dild2j (ς5̃ )(x, t, t) dt ⃗

+ 2−jμ0 2−j⋅d k Xk ∫ f (Γ(t, x)) Dild2j (ς6̃ )(x, t, t) dt, ⃗

⃗

where μ0

ς5 (x, s, t) = −2−jμ0 (dl μ0

ς6 (x, s, t) = 2−jμ0 (dl μ0

Since 2−jμ0 (dl

−1)

−1)

Xk [sγ h(s, x)ς(x, s, t)],

−1) j⋅d ⃗k −degd ⃗ (γ)⋅j

2

, 2j⋅d k −degd ⃗ (γ)⋅j ∈ (0, 1], it follows that ⃗

h(s, x)t γ ς(x, s, t).

4.3 Littlewood–Paley decompositions

� 229

̂ C0∞ (Bq (b))⊗ ̂ S{μ:j =0} ς5̃ , ς6̃ ∈ C0∞ (Ω1 )⊗ (ℝq ) μ ̸ and ̂ C0∞ (Bq (b))⊗ ̂ S{μ:j =0} {ς5̃ , ς6̃ : j ∈ [0, ∞)ν , ς ∈ C0∞ (Ω1 )⊗ (ℝq ) ∩ ℬ} μ ̸ ̂ C0∞ (Bq (b))⊗ ̂ S (ℝq ) ⊂ C0∞ (Ω1 )⊗

is a bounded set. Thus, we have shown that F2 = 2−jμ0 E1 + 2−jμ0 2−j⋅d k Xk E2 , ⃗

where ̂ C0∞ (Bq (b))⊗ ̂ S{μ:j =0} {(E1 , 2−j ), (E2 , 2−j ) : j ∈ [0, ∞)ν , ς ∈ C0∞ (Ω1 )⊗ (ℝq ) ∩ ℬ} μ ̸ ⃗ b pseudo-differential operator scales supported in Ω1 × Ω2 . is a bounded set of (X, d ), ⃗ As in the proof of (4.27), applying Lemma 4.3.4 (ii) to 2−j⋅d k Xk E2 completes the proof of (4.28). ⃗ > 0 is sufficiently small, the following holds. Let Lemma 4.3.7. If a = a(Ω1 , Ω2 , (X, d )) ∞ ∞ q q ̂ C0 (B (a))⊗ ̂ S (ℝ ) be a bounded set. For ς ∈ ℬ, j ∈ ℕν , define an operator ℬ ⊂ C0 (Ω1 )⊗ F = F[j, ς] by Ff (x) = ∫ f (Γ(t, x)) Dild2j (ς)(x, t, t) dt. ⃗

̂ C0∞ (Bq (a))⊗ ̂ S{μ:j =0} Then if ς ∈ C0∞ (Ω1 )⊗ (ℝq ) ∩ ℬ, we have μ ̸ μ0

α

−j (1−|αμ0 |) Eαμ ,1 (2−jμ0 d X μ0 ) μ0 , {∑|α |≤1 2 μ0 0 F = { μ0 μ0 α ∑|αμ |≤1 2−jμ0 (1−|αμ0 |) (2−jμ0 d X μ0 ) μ0 Eαμ ,2 , { 0 0

where {(Eαμ ,1 , 2−j ), (Eαμ ,2 , 2−j ) : |αμ0 | ≤ 1, |βμ0 | ≤ 1, j ∈ [0, ∞)ν , 0

0

̂ C0∞ (Bq (a))⊗ ̂ S{μ:j =0} ς ∈ C0∞ (Ω1 )⊗ (ℝq ) ∩ ℬ} μ ̸

⃗ a pseudo-differential operator scales supported in Ω1 × Ω2 . is a bounded set of (X, d ), Proof. We take a so small that a is less than b(g, Ω1 , Ω2 ) from Lemma 4.3.6, where g can k k be any of the functions g3,j or g4,j from Proposition 4.2.2 (note there are only 2q2 such functions, and in particular, only finitely many). ̂ C0∞ (Bq (a))⊗ ̂ S{μ:j =0} Fix j ∈ [0, ∞)ν and ς ∈ (C0∞ (Ω1 )⊗ (ℝq )) ∩ ℬ. If jμ0 = 0, the result μ ̸ is trivial, as we can take

230 � 4 Pseudo-differential operators

E αμ

,1 0

= E αμ

,2 0

={

F,

|αμ0 | = 0,

0,

otherwise.

Thus, we may assume jμ0 > 0, so we have by Corollary 4.1.19, with △μ0 the Laplacian in the t μ0 variable, qμ0

−1 μ μ ς(x, s, t) = △μ0 △−1 μ0 = ∑ −𝜕t 0 (𝜕t 0 △μ0 ς(x, s, t)), l

l=1

l

∞ q ̂ ∞ q ̂ where 𝜕tμ0 △−1 ̸ (ℝ ) and μ0 ς ∈ C0 (Ω1 )⊗C0 (B (a))⊗S{μ:jμ =0} l

∞ q ̂ ∞ q ̂ {(𝜕tμ0 △−1 μ0 ς) : ς ∈ C0 (Ω1 )⊗C0 (B (a))⊗S{μ0 } (ℝ ) ∩ ℬ } l

̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) ⊂ C0∞ (Ω1 )⊗

is a bounded set. Thus, it suffices to prove the result with ς(x, s, t) replaced by −𝜕tμ0 ς(x, s, t) l for some l ∈ {1, . . . , qμ0 }. We henceforth do this. Using integration by parts we have Ff (x) = ∫ f (Γ(t, x)) Dild2j (−𝜕tμ0 ς)(x, t, t) dt ⃗

l

=2

−jμ0

+2 =: 2

⃗ ̃ t, t) ∫ f (Γ(t, x)) Dild2j (ς)(x,

μ −jμ0 dl 0

−jμ0

dt

∫[𝜕tμ0 (f (Γ(t, x)))] Dild2j (ς)(x, t, t) dt

(4.29)

⃗

l

G1 f (x) + G2 f (x),

where μ0

̃ s, t) = 2−jμ0 (dl ς(x, μ0

Since 2−jμ0 (dl

−1)

−1)

𝜕sμ0 ς(x, s, t). l

̂ C0∞ (Bq (a))⊗ ̂ S{μ:j =0} ∈ (0, 1], it follows easily that ς̃ ∈ C0∞ (Ω1 )⊗ (ℝq ) and μ ̸

̂ C0∞ (Bq (b))⊗ ̂ S{μ:j =0} {ς̃ : j ∈ [0, ∞)ν , ς ∈ C0∞ (Ω1 )⊗ (ℝq ) ∩ ℬ} μ ̸ ̂ C0∞ (Bq (b))⊗ ̂ S (ℝq ) ⊂ C0∞ (Ω1 )⊗

is a bounded set. Thus, ̂ C0∞ (Bq (a))⊗ ̂ S{μ:j =0} {(G1 , 2−j ) : j ∈ [0, ∞)ν , ς ∈ C0∞ (Ω1 )⊗ (ℝq ) ∩ ℬ} μ ̸ ⃗ a pseudo-differential scales supported in Ω1 ×Ω2 , and therefore is a bounded set of (X, d ), −jμ0 the term 2 G1 on the right-hand side of (4.29) is of the desired form.

4.3 Littlewood–Paley decompositions

� 231

It remains to show that operator G2 is of the desired form. Applying Proposiμ μ tion 4.2.2 (iii) and (iv), by taking l0 so that (Xl 0 , dl 0 eμ0 ) = (Xl0 , d l⃗ 0 ), μ0

q

k 2−jμ0 dl ∫((−Xl + ∑k=1 g3,l X )f )(Γ(t, x)) Dild2j (ς)(x, t, t) dt, 0 k G2 f (x) = { −j d μ0 ⃗ q k 2 μ0 l ∫(−Xl + ∑k=1 g4,l X )(f (Γ(t, x))) Dild2j (ς)(x, t, t) dt, 0 k

=: {

⃗

F1 f (x),

F2 f (x),

μ k where degd ⃗ (gr,l ) ≥ d k⃗ − dl 0 eμ0 . Note also that for the constant function g(t, x) = −1 0 μ we have degd ⃗ (g) ≥ −dl 0 eμ0 . Thus, F1 and F2 are sums of operators of the type of the same name in Lemma 4.3.6. Lemma 4.3.6 shows these operators are of the desired form, completing the proof.

Proof of Proposition 4.3.5. Lemma 4.3.7 is equivalent to saying μ0

α

−j (1−|αμ0 |) Eαμ ,1 (2−jμ0 d X μ0 ) μ0 , {∑|α |≤1 2 μ0 0 E = { μ0 μ0 α ∑|αμ |≤1 2−jμ0 (1−|αμ0 |) (2−jμ0 d X μ0 ) μ0 Eαμ ,2 , { 0 0

(4.30)

where {(Eαμ ,1 , 2−j ), (Eαμ ,2 , 2−j ) : |αμ0 | ≤ 1, |βμ0 | ≤ 1, j ∈ [0, ∞)ν , 0

ς∈

0

̂ C0∞ (Bq (a))⊗ ̂ S{μ:j =0} C0∞ (Ω1 )⊗ (ℝq ) μ ̸

∩ ℬ}

⃗ a pseudo-differential operators supported in Ω1 ×Ω2 . The result is a bounded set of (X, d ), follows by two applications of (4.30).

4.3.1 Lebesgue space bounds ⃗ a pseudo-differential operators We will discuss the boundedness properties of (X, d ), on various function spaces as part of a more general theory in Chapter 6. However, in ⃗ a pseudo-differential this section, we describe the Lp boundedness properties of (X, d ), operator scales, which will be an essential tool in later chapters. Fix a smooth, strictly positive density Vol on M. ⃗ > 0 is sufficiently small, the following holds. Let Proposition 4.3.8. If a = a(Ω1 , Ω2 , (X, d )) ⃗ ℰ be a bounded set of (X, d ), a pre-pseudo-differential operator scales supported in Ω1 ×Ω2 . Then sup

sup ‖E‖Lp (M,Vol)→Lp < ∞.

(E,2−j )∈ℰ p∈[1,∞]

(4.31)

232 � 4 Pseudo-differential operators Also, for p ∈ (1, ∞), q ∈ (1, ∞], there exists C = C(p, q, ℰ ) ≥ 0 such that the following holds. Let ℰ ′ = {(El , 2−jl ) : l ∈ ℕ} ⊆ ℰ . Then sup

sup

ℰ ′ (El ,2−jl )∈ℰ ′

󵄩󵄩 󵄩 󵄩󵄩{El fl }l∈ℕ 󵄩󵄩󵄩Lp (M,Vol;ℓq (ℕ)) ≤ C‖{fl }l∈ℕ ‖Lp (M,Vol;ℓq (ℕ)) ,

(4.32)

for all {fl }l∈ℕ ∈ Lp (M, Vol; ℓq (ℕ)), where the supremum supℰ ′ is taken over all such countable subsets of ℰ . This section is devoted to the proof of Proposition 4.3.8. ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) be a bounded set. Then, for j ∈ [0, ∞)ν Lemma 4.3.9. Let ℬ ⊂ C0∞ (Ω1 )⊗ and ς ∈ ℬ, there exists {γk : k ∈ ℕν , k ≤ j} ⊂ C0∞ (Ω1 × Bq (a) × Bq (a)) such that Dild2j (ς)(x, t, t) = ∑ Dild2k (γk )(x, t, t). ⃗

⃗

k≤j k∈ℕν

(4.33)

Furthermore, for every M ∈ ℕ, {2M|j−k| γk : j ∈ [0, ∞)ν , k ≤ j, k ∈ ℕν , ς ∈ ℬ} ⊂ C0∞ (Ω1 × Bq (a) × Bq (a))

(4.34)

is a bounded set. Proof. Fix η ∈ C0∞ (Bq (a)) so that η ≡ 1 on a neighborhood of the closure of {s ∈ Bq (a) : ∃ς ∈ ℬ, x ∈ Ω, t ∈ ℝq , ς(x, s, t) ≠ 0}. See Lemma 4.1.10 for why this is possible. Take ς ∈ ℬ and j ∈ [0, ∞)ν . For k ∈ ℕν with k ≤ j, let σk (t) :=

∑ (−1)p1 +p2 +⋅⋅⋅+pν η(2p⋅d t), ⃗

p∈{0,1}ν k+p≤j

so that η(t) = ∑0≤k≤j σk (2k⋅d t). Note that σk (t) = 0 if kμ ≤ jμ − 1 and |t μ | is sufficiently small (independent of j, k). Define, for k ∈ ℕν , k ≤ j, ⃗

γk (x, s, t) := σk (t) Dild2j−k (ς)(x, s, t). ⃗

It follows immediately from the definitions that {γk : j ∈ [0, ∞)ν , k ≤ j, k ∈ ℕν , |j − k|∞ < 1, ς ∈ ℬ} ⊂ C0∞ (Ω1 × Bq (a) × Bq (a))

4.3 Littlewood–Paley decompositions

� 233

is a bounded set, so to prove (4.34) is a bounded set it suffices to consider the case where |j − k|∞ ≥ 1. Take μ such that |j − k|∞ = jμ − kμ ≥ 1. Because σk (t) = 0 for |tμ | sufficiently

small, we see by Lemma 4.1.10 that for every smooth coordinate system Φ : Bn (1) 󳨀 → Φ(Bn (1)) ⊆ M, ∀m ∈ ℕ, for all multi-indices α ∈ ℕn , β1 , β2 ∈ ℕq , ∼

󵄨󵄨 α β1 β2 󵄨 (j −k )d μ μ −m −m|j−k|∞ . 󵄨󵄨𝜕u 𝜕s 𝜕t γk (Φ(u), s, t)󵄨󵄨󵄨 ≲ χ{|tμ |≈1} (1 + |2 μ μ t |) ≲ 2 Since γk is clearly supported in a compact set of Ω1 × Bq (a) × Bq (a) independent of j, k, it follows that for every M ∈ ℕ, {2M|j−k| γk : j ∈ [0, ∞)ν , k ≤ j, k ∈ ℕν , |j − k|∞ ≥ 1, ς ∈ ℬ} ⊂ C0∞ (Ω1 × Bq (a) × Bq (a))

is a bounded set, completing the proof that (4.34) is bounded. Since Dild2j (ς)(x, t, t) = η(t) Dild2j (ς)(x, t, t) = ∑ Dild2k (γk )(x, t, t), ⃗

⃗

⃗

k≤j

formula (4.33) follows, completing the proof. Define Aaδ,(X,d)⃗ as in (3.63). Set 𝒦 := Ω2 and let Ω3 be an open, relatively compact set such that 𝒦 ⋐ Ω3 ⋐ M. In what follows, we think of Ω3 as being chosen depending only on Ω2 . ⃗ > 0, the following holds. Let ℰ be a bounded set of Lemma 4.3.10. If a = a(Ω1 , Ω2 , (X, d )) ⃗ (X, d ), a pre-pseudo-differential scales supported in Ω1 ×Ω2 . Then there exists C = C(ℰ ) ≥ 0 such that for (E, 2−j ) ∈ ℰ , |Ef (x)| ≤ CχΩ3 ∑ 2−|j−k| Aa2−k ,(X,d)⃗ |χΩ2 f |(x),

(4.35)

̃ ⃗ |Ef (x)| ≤ CχΩ3 ℳ (X,d),𝒦,a (χΩ2 f )(x).

(4.36)

k≤j k∈ℕν

Proof. For (E, 2−j ) ∈ ℰ , Ef = EχΩ2 f , and we have Ef (x) = ∫(χΩ2 f )(Γt (x)) Dild2j (ς)(x, t, t) dt, ⃗

where ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) {ς : (E, 2−j ) ∈ ℰ } ⊂ C0∞ (Ω1 )⊗ is bounded. We decompose

234 � 4 Pseudo-differential operators Dild2j (ς)(x, t, t) = ∑ Dild2k (γk )(x, t, t), ⃗

⃗

k≤j k∈ℕν

where γk ∈ C0∞ (Ω1 × Bq (a) × Bq (a)) is as in Lemma 4.3.9. We have ⃗ 󵄨 󵄨 |Ef (x)| ≤ ∑ ∫ |(χΩ2 f )(Γ(t, x))|󵄨󵄨󵄨Dild2k (γk )(x, t, t)󵄨󵄨󵄨 dt k≤j k∈ℕν

⃗ ⃗ 󵄨 󵄨 = ∑ ∫ |(χΩ2 f )(Γ(2−j⋅d t, x))|󵄨󵄨󵄨γk (x, 2−j⋅d t, t)󵄨󵄨󵄨 dt k≤j k∈ℕν

≲ ∑ 2−|j−k| ∫ |(χΩ2 f )(Γ(2−j⋅d t, x))| dt ⃗

k≤j k∈ℕν

Bq (a)

= ∑ 2−|j−k| Aa2−k ,(X,d)⃗ (χΩ2 f )(x), k≤j k∈ℕν

where in the second to last line, we used that (4.34) is a bounded set. This implies (4.35), without the factor of χΩ3 on the right-hand side. By Lemma 3.11.1, if a > 0 is sufficiently small, then supp(Aaδ,(X,d)⃗ χΩ2 f ) ⊆ Ω3 , and we may therefore include the term χΩ3 on the right-hand side of (4.35). ̃ ⃗ ̃ ⃗ Since χΩ3 Aaδ,(X,d)⃗ |χΩ2 f |(x) ≤ χΩ3 ℳ , (X,d),𝒦,a (χΩ2 f )(x), by the definition of ℳ(X,d),𝒦,a (4.36) follows from (4.35). Proof of Proposition 4.3.8. By taking a > 0 sufficiently small, (4.31) follows from (4.35) and Lemma 3.11.1 (with Ω1 replaced by Ω3 in that lemma). Also, (4.32) follows from (4.36) and Theorem 3.11.2 (again, with Ω1 replaced by Ω3 in that theorem). 4.3.2 Adjoints ⃗ a Fix a strictly positive, smooth density Vol on M. The L2 (M, Vol) adjoint of an (X, d ), ⃗ pseudo-differential operator is not precisely an (X, d ), a pseudo-differential operator: taking adjoints reverses the order of (X 1 , d 1 ), . . . , (X ν , d ν ). However, since our assump⃗ a pseudo-differential tions are symmetric in (X 1 , d 1 ), . . . , (X ν , d ν ), the adjoint of an (X, d ), operator is “of the same form” as the original operator. To make this precise, for a permutation σ of {1, . . . , ν}, let (Z 1 , dr1 ) = (X σ(1) , d σ(1) ), . . . , (Z ν , drν ) = (X σ(ν) , d σ(ν) ), ∞ and let (Z, dr)⃗ ⊂ Cloc (M) × ℕν be given by

(Z, dr)⃗ = (Z 1 , dr1 ) ⊠ (Z 2 , dr2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (Z ν , drν ).

4.3 Littlewood–Paley decompositions

� 235

⃗ so it makes sense to talk about Then (Z, dr)⃗ satisfies all the same hypotheses as (X, d ), ⃗ a pseudo-differential operators supported on Ω1 × Ω2 . In the rest of this section, (Z, dr),

we take the particular permutation σ(1) = ν, σ(2) = ν − 1, . . . , σ(ν) = 1, and we define (Z, dr)⃗ as above. Fix a relatively compact open set Ω3 ⋐ M, with Ω2 ⋐ Ω3 . ⃗ > 0 is sufficiently small, the following hold: Lemma 4.3.11. If a = a(Ω1 , Ω2 , Ω3 , (X, d )) ⃗ a pre-pseudo-differential operator scales supported (i) Let ℰ be a bounded set of (X, d ), in Ω1 × Ω2 . Then ℰ := {(E , 2 ) : (E, 2 ) ∈ ℰ } ∗

∗

−j

−j

(4.37)

⃗ a pre-pseudo-differential scales supported in Ω2 × Ω3 . Here, is a bounded set of (Z, dr), E ∗ denotes the L2 (M, Vol) adjoint of E. ⃗ a pseudo-differential operator scales sup(ii) If, in addition, ℰ is a bounded set of (X, d ), ∗ ⃗ a pseudo-differential operator ported in Ω1 ×Ω2 , then ℰ is also a bounded set of (X, d ), scales supported in Ω1 × Ω2 . Proof. We prove only (ii); (i) follows by a simpler version of the same proof. We write Γt (x) = Γ(t, x), and for each t we think of Γt (⋅) as a map. Note that t Γ−1 t (x) = e

ν

X ν t ν−1 ⋅X ν−1

e

1

1

⋅ ⋅ ⋅ et ⋅X x,

⃗ ⃗ that is, Γ−1 −t is to (Z, dr) as Γt is to (X, d ). 󵄨 Since Γ(0, x) = x, for t sufficiently small, Γ(t, ⋅)󵄨󵄨󵄨Ω is a diffeomorphism onto its image 2 ∞ q and Γ(t, Ω1 ) ⋐ Ω2 . Let h(t, x) Vol = Γ(t, ⋅)∗ Vol, so that h ∈ Cloc (B (a) × Ω2 ), provided a is chosen sufficiently small. For each (E, 2−j ) ∈ ℰ , we have Ef (x) = ∫ f (Γ(t, x)) Dild2j (ς)(x, t, t) dt, ⃗

̂ C0∞ (Bq (a))⊗ ̂ S{μ:j =0} where ς ∈ C0∞ (Ω1 )⊗ (ℝq ), and μ ̸ ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) {ς : (E, 2−j ) ∈ ℰ } ⊂ C0∞ (Ω1 )⊗ is a bounded set. We have ∫(Ef )(x)g(x) d Vol(x) = ∬ f (Γ(t, x))g(x) Dild2j (ς)(x, t, t) d Vol(x) dt ⃗

d −1 = (−1)q ∬ f (v)g(Γ−1 −t (v)) Dil2j (ς)(Γ−t (v), −t, −t)h(−t, v) d Vol(v) dt. ⃗

We conclude that

236 � 4 Pseudo-differential operators ̃ E ∗ g(x) = ∫ g(Γ−1 −t (x)) Dil2j (ς)(x, t, t) dt, where ̃ s, t) = (−1)q h(−s, x)ς(Γ−1 ς(x, −s (x), −s, −t). ̂ C0∞ (Bq (a))⊗ ̂ S{μ:j =0} It is straightforward to verify that ς̃ ∈ C0∞ (Ω2 )⊗ (ℝq ), and μ ̸ ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) {ς̃ : (E, 2−j ) ∈ ℰ } ⊂ C0∞ (Ω2 )⊗ ⃗ a is a bounded set. Thus, if a > 0 is sufficiently small, (4.37) is a bounded set of (Z, dr), pseudo-differential operator scales supported in Ω2 × Ω3 , as desired. 4.3.3 Proof of the Littlewood–Paley decomposition In this section, we complete the proof of Theorem 4.3.3. Fix smooth vector fields ∞ Y1 , . . . , YL ∈ Cloc (M; TM) so that ∀x ∈ Ω2 , span{Y1 (x), . . . , YL (x)} = Tx Ω2 . ⃗ > 0 is sufficiently small, the following holds. Let Lemma 4.3.12. If a = a(Ω1 , Ω2 , (X, d )) ⃗ ℰ be a bounded set of (X, d ), a pseudo-differential operator scales supported in Ω1 × Ω2 . Then, for (E, 2−j ) ∈ ℰ , 1 ≤ k ≤ L, L

Yk E = E0 + ∑ El Yl , l=1

⃗ a where {(E0 , 2−j ), (E1 , 2−j ), . . . , (EL , 2−j ) : (E, 2−j ), 1 ≤ k ≤ L ∈ ℰ } is a bounded set of (X, d ), pseudo-differential operator scales supported in Ω1 × Ω2 . Proof. Let (E, 2−j ) ∈ ℰ . Then Ef (x) = ∫ f (Γ(t, x)) Dild2j (ς)(x, t, t) dt, ⃗

̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) is a bounded set and ς ∈ where {ς : (E, 2−j ) ∈ ℰ } ⊂ C0∞ (Ω1 )⊗ ̂ C0∞ (Bq (a))⊗ ̂ S{μ:j =0} C0∞ (Ω1 )⊗ (ℝq ). By the chain rule, we have μ ̸ L

Yk Ef (x) = ∫ f (Γ(t, x)) Dild2j (Yk ς)(x, t, t) dt + ∑(Yl f )(Γ(t, x))hl (t, x) Dild2j (ς)(x, t, t) dt ⃗

L

⃗

l=1

=: E0 f (x) + ∑ El Yl f (x), l=1

∞ q where hl (t, x) ∈ Cloc (B (a)×Ω2 ). Set ς0 (x, s, t) := Yk ς(x, s, t) and ςl (x, s, t) := hl (s, x)ς(x, s, t) ̂ C0∞ (Bq (a))⊗ ̂ S{μ:j =0} for l = 1, . . . , L. We have ςl ∈ C0∞ (Ω1 )⊗ (ℝq ) and μ ̸

4.3 Littlewood–Paley decompositions

� 237

̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) {ςl : (E, 2−j ) ∈ ℰ , 0 ≤ l ≤ L, 1 ≤ k ≤ L} ⊂ C0∞ (Ω1 )⊗ is a bounded set. Since El f (x) = ∫ f (Γ(t, x)) Dil2j (ςl )(x, t, t) dt, the result follows. ⃗ > 0 is sufficiently small, the following holds. Let ℰ Lemma 4.3.13. If a = a(Ω1 , Ω2 , (X, d )) ⃗ a pseudo-differential operator scales supported in Ω1 × Ω2 , and be a bounded set of (X, d ), ∞ let ℬ ⊂ Cloc (M) be a bounded set. Then, for every M ∈ ℕ and every ordered multi-index α, sup 2M|j|∞ ‖Y α Ef ‖L∞ (M,Vol) < ∞.

(E,2−j )∈ℰ f ∈ℬ

Proof. Let (E, 2−j ) ∈ ℰ . Repeated application of Lemma 4.3.12 shows that Y α Ef = ∑ Eβ Y β f , |β|≤|α|

⃗ a pseudo-differential where {(Eβ , 2−j ) : (E, 2−j ) ∈ ℰ , |β| ≤ |α|} is a bounded set of (X, d ), β ∞ scales. Since {Y f : f ∈ ℬ} ⊂ Cloc (M) is a bounded set, it suffices to prove the case |α| = 0. Fix μ ∈ {1, . . . , ν} so that jμ = |j|∞ . Applying Proposition 4.3.5 M times we see that μ

α

μ

β

Ef = ∑ 2−jμ (2M−|αμ |−|βμ |) (2−jμ d X μ ) μ Eμ,αμ ,βμ (2−jμ d X μ ) μ f |αμ |≤M |βμ |≤M

=: ∑ 2−jμ (M−|βμ |+degd μ (βμ )) 2−jμ (M−|αμ |) Ẽμ,αμ ,βμ fβμ , |αμ |≤M |βμ |≤M μ

where Ẽμ,αμ ,βμ = (2−jμ d X μ )αμ Eμ,αμ ,βμ and fβμ = (X μ )βμ f . By Proposition 4.3.5, {(Eμ,αμ ,βμ , 2−j ) : ⃗ a pseudo-differential scales supported in Ω1 × Ω2 (E, 2−j ) ∈ ℰ } is a bounded set of (X, d ), ⃗ a and therefore by Lemma 4.3.4 (ii), {(Ẽμ,α ,β , 2−j ) : (E, 2−j ) ∈ ℰ } is a bounded set of (X, d ), μ

μ

∞ pseudo-differential scales supported in Ω1 × Ω2 . Clearly, {fβμ : f ∈ ℬ} ⊂ Cloc (M) is a

bounded set. Thus, since 2−jμ (M−|βμ |+degd μ (βμ )) 2−jμ (M−|αμ |) ≤ 2−Mjμ = 2−M|j|∞ , we see that it suffices to prove the result in the case M = 0. For the case M = 0, take ϕ ∈ C0∞ (M) so that ϕ = 1 on Ω2 . We then have, by Proposition 4.3.8, sup ‖Ef ‖L∞ =

(E,2−j )∈ℰ f ∈ℬ

sup ‖Eϕf ‖L∞ ≲ sup ‖ϕf ‖L∞ < ∞,

(E,2−j )∈ℰ f ∈ℬ

f ∈ℬ

completing the proof. ⃗ a pseudoProof of Theorem 4.3.3. Let {(Ej , 2−j ) : j ∈ ℕν } be a bounded set of (X, d ), differential operator scales supported in Ω1 × Ω2 .

238 � 4 Pseudo-differential operators We begin by showing that ∑j∈ℕν 2j⋅m Ej converges in the typology of bounded con∞ ∞ vergence on Hom(Cloc (M), C0∞ (Ω1 )). Let ℬ ⊂ Cloc (M) be a bounded set. By the definition ⃗ of bounded set of (X, d ), a pseudo-differential operator data, we have ⋃ ⋃ supp(Ej f ) ⋐ Ω1 .

j∈ℕν f ∈ℬ

Thus, to prove the desired convergence, it suffices to show that for every ordered multiindex α, ∑j∈ℕν Y α Ej f (x) converges uniformly in x and f ∈ ℬ. This follows immediately from Lemma 4.3.13, completing the proof of the convergence in the typology of bounded ∞ convergence on Hom(Cloc (M), C0∞ (Ω1 )). ⃗ a pseudo-differential scales supported in By the definition of bounded sets of (X, d ), ∞ Ω1 × Ω2 , we may pick ϕ ∈ C0 (Ω1 ) so that Mult[ϕ]Ej = Ej , ∀j ∈ ℕ. Let f ∈ C0∞ (M)′ and ∞ consider that for g ∈ Cloc (Ω1 ), we have ⟨ ∑ 2m⋅j Ej f , g⟩ |j|≤M

L2 (M,Vol)

= ⟨f , ∑ 2m⋅j Ej∗ ϕg⟩ |j|≤M

L2 (M,Vol)

,

(4.38)

where ⟨⋅, ⋅⟩L2 denotes the (formal) L2 inner product. By Lemma 4.3.11 (ii), {(Ej∗ , 2−j ) : j ∈ ⃗ a pseudo-differential operator scales supported in Ω2 × ℕν } is a bounded set of (X, d ), Ω3 , for an open set Ω3 with Ω2 ⋐ Ω3 ⋐ M. Thus, ∑j∈ℕν 2m⋅j Ej∗ ϕg converges in C0∞ (Ω2 ). ∞ By (4.38), we conclude that ∑j∈ℕν 2m⋅j Ej f converges in the weak topology on Cloc (Ω1 )′ , ∞ that is, ∑j∈ℕν 2m⋅j Ej converges in the weak operator topology on Hom(C0∞ (M)′ , Cloc (Ω1 )′ ), ∞ where Cloc (Ω1 )′ is given the weak topology. Now that the desired convergence of the sums has been shown, (i) ⇔ (ii) follows immediately from Proposition 4.1.14.

4.4 Adding parameters Let E = {μ1 < μ2 < ⋅ ⋅ ⋅ < μ|E| } ⊆ {1, . . . , ν}. For a vector in v ∈ ℝ|E| , we index the coordinates of v by μ ∈ E, so it makes sense to write vμ for μ ∈ E. We consider the set of vector fields with |E|-parameter formal degrees by (X E , d E⃗ ) = {(X1E d E1⃗ ), . . . , (XqEE , d Eq⃗ E )}

∞ := (X μ1 , d μ1 ) ⊠ (X μ2 , d μ2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (X μ|E| , d μ|E| ) ⊂ Cloc (M; TM) × (ℕ|E| \ {0}).

Note that (X E , d E⃗ ) satisfies (4.2). It therefore makes sense to talk of (X E , d E⃗ ), a pseudodifferential operators. ⃗ > 0 is chosen sufficiently small, then the folProposition 4.4.1. If a = a(Ω1 , Ω2 , (X, d )) E E lowing holds. Suppose T is an (X , d ⃗ ), a pseudo-differential operator of order mE ∈ ℝ|E|

4.5 The sub-Laplacian

�

239

supported in Ω1 × Ω2 . Let m ∈ ℝν be such that mμ = mμE , for μ ∈ E and mμ = 0 for μ ∈ ̸ E. ⃗ a pseudo-differential operator of order m supported in Ω1 × Ω2 . Then T is an (X, d ), c

Proof. Let E = {μ1 < μ2 < ⋅ ⋅ ⋅ < μ|E| }. Set t E = (t μ1 , t μ2 , . . . , t μ|E| ) and define t E similarly with E replaced by E c . By hypothesis, Tf (x) = ∫ f (e−t

μ1

⋅X μ1 −t μ2 ⋅X μ2

e

⋅ ⋅ ⋅ e−t

μ|E| μ|E| ⋅X

x)b̌ E (x, t E , t E ) dt E ,

c

c

where bE (x, sE , ξ E ) ∈ S mE (a, Ω1 , d E⃗ ). Fix η(t E ) ∈ C0∞ (ℝqEc ) with η(t E ) = 1 on a neighborhood of 0, and with small support. Set c

b(x, s, ξ) := bE (x, sE , ξ E )η(sE ). ⃗ and Note that if the support of η is sufficiently small, we have b(x, s, ξ) ∈ S m (a, Ω1 , d ), c

̌ t, t) = b̌ (x, t E , t E ) ⊗ δ (t E ), b(x, E 0 where δ0 denotes the Dirac δ function at 0. We have Tf (x) = ∫ f (e−t

μ1

⋅X μ1 −t μ2 ⋅X μ2

e

⋅ ⋅ ⋅ e−t

μ|E|

⋅X μ|E|

x)b̌ E (x, t E , t E ) dt E

c

̌ t, t) dt, = ∫ f (Γ(t, x))b̌ E (x, t E , t E ) ⊗ δ0 (t E ) dt = ∫ f (Γ(t, x))b(x, completing the proof.

4.5 The sub-Laplacian ∞ Let (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} ⊂ Cloc (M; TM) × ℕ+ be such that Gen((W , ds)) is locally finitely generated. Fix κ ∈ ℕ+ such that dsj divides κ, 1 ≤ j ≤ r. Fix a smooth, strictly positive density Vol on M. We define the sub-Laplacian, associated with (W , ds), by r

κ/dsj ∗

L := ∑(Wj j=1

κ/dsj

) Wj

,

(4.39)

where ∗ denotes the formal L2 (M, Vol) adjoint. Fix relatively compact open sets Ω1 ⋐ Ω2 ⋐ M. By hypothesis, Gen((W , ds)) is finitely generated by some (X, d ) = {(X1 , d1 ), . . . , (Xq , dq )} ⊂ Gen((W , ds)) on Ω2 . In Section 4.5.5 we will pick a particular choice of (X, d ); with this choice, the main result of this section is the following. Theorem 4.5.1. For every a > 0 (sufficiently small) and every ψ ∈ C0∞ (Ω1 ), there is an (X, d ), a pseudo-differential operator, Tψ , of order −2κ supported in Ω1 × Ω2 such that

240 � 4 Pseudo-differential operators Tψ L = Mult[ψ] + E, where E is an (X, d ), a pseudo-differential operator of order −1 supported in Ω1 × Ω2 . This section is devoted to the proof of Theorem 4.5.1. Remark 4.5.2. A primary example to consider is the case where (W , ds) are Hörmander vector fields with formal degrees (see Proposition 3.4.14). In this case, L is maximally subelliptic (see Example 1.1.10 (iii)), and we can obtain much more detailed information about L (see Theorem 8.1.1). Remark 4.5.3. The proof of Theorem 4.5.1 uses a method of Rothschild and Stein [200], where we use a fundamental solution for a sub-Laplacian on a high-dimensional nilpotent Lie group to deduce the result. After Rothschild and Stein’s work, this idea was furthered by Goodman [99], and our approach more closely follows his. One thing that sets our analysis apart from these references is that we do not assume the vector fields W1 , . . . , Wr satisfy Hörmander’s condition. We only assume the weaker condition that Gen((W , ds)) is locally finitely generated. For example, when the vector fields are real analytic, this condition always holds (Proposition 3.4.15). The key to being able to work under this weaker condition is Proposition 4.2.2. Remark 4.5.4. Theorem 4.5.1 gives a left parametrix for L . A similar proof gives a right parametrix, that is, there is an (X, d ), a pseudo-differential operator, Sψ , of order −2κ supported in Ω1 × Ω2 such that L Sψ = Mult[ψ] + E1 , where E1 is an (X, d ), a pseudodifferential operator of order −1 supported in Ω1 × Ω2 . The proof of this is a straightforward modification of the proof of Theorem 4.5.1 and we leave the details to the interested reader.

4.5.1 The sub-Laplacian is subelliptic ∞ Let M be an open subset of ℝn and let (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} ⊂ Cloc (M; TM) × ℕ+ be Hörmander vector fields with formal degrees. Define L as in (4.39), where Vol is taken to be the Lebesgue measure. The main result of this section is the following.

Theorem 4.5.5. For every relatively compact, open set Ω1 ⋐ M, L is subelliptic on Ω1 . More precisely, there exists ϵ0 = ϵ0 (Ω1 ) > 0 such that ∀s ∈ ℝ, N ≥ 0, ∀ϕ1 , ϕ2 ∈ C0∞ (Ω1 ) with ϕ1 ≺ ϕ2 , ‖ϕ1 u‖L2s+ϵ ≤ Cϕ1 ,ϕ2 ,s,N (‖ϕ2 L u‖L2s + ‖ϕ2 u‖L2 ), 0

s−N

∀u ∈ C0∞ (M)′ ,

where if the right-hand side is finite, so is the left-hand side. This section is devoted to the proof of Theorem 4.5.5. The results in this section are not sharp: we do not obtain the optimal value for ϵ0 here. However, we do obtain the op-

4.5 The sub-Laplacian

�

241

timal value for ϵ0 in Section 8.2.2, where we also obtain results for more general spaces. The results here are an important step in obtaining these more general and sharper results. We require some new notation. Throughout this section we write ⟨⋅, ⋅⟩ for the usual L2 (ℝn ) inner product. We write Λs for the standard pseudo-differential operator of order s given in Definition 2.2.10. We write ‖f1 ‖L2s ≲ ‖f2 ‖L2s + ‖f3 ‖L2−∞ 1

2

to mean that ∀N ∈ ℕ, there exists CN such that ‖f1 ‖L2s ≤ CN (‖f2 ‖L2s + ‖f3 ‖L2 ). 1

2

−N

We write (l.c.) to denote “large constant” and (s.c.) to denote “small constant.” An expression of the form A ≤ (l.c.)B1 + (s.c.)B2

(4.40)

means that ∀ϵ > 0, ∃Cϵ ≥ 0 with A ≤ Cϵ B1 + ϵB2 . Note that AB ≤ (l.c.)A + (s.c.)B. Also, if A ≤ (l.c.)B + (s.c.)A, then A ≲ B. We can combine the two above notations as follows. If we write ‖f1 ‖L2s ≤ (s.c.)‖f2 ‖L2s + (l.c.)(‖f3 ‖L2s + ‖f4 ‖L2−∞ ), 1

2

3

this means that ∀ϵ > 0, N ∈ ℕ, ∃Cϵ,N ≥ 0, such that ‖f1 ‖L2s ≤ ϵ‖f2 ‖L2s + Cϵ,N (‖f3 ‖L2s + ‖f4 ‖L2 ). 1

2

3

−N

We write S (s) to denote some standard pseudo-differential operator of order s, on ℝ , which may change from line to line. In each case, the operator S (s) can be written down explicitly, but what is important is only that it is a pseudo-differential operator of order s. Thus, using Theorem 2.2.12, we have estimates like n

󵄩󵄩 (s) 󵄩󵄩 󵄩󵄩S u󵄩󵄩L2 ≲ ‖u‖L2s +s . s1

1

We use the calculus of pseudo-differential operators (Theorem 2.2.8) freely in this section. Fix relatively compact open sets Ω1 ⋐ Ω2 ⋐ Ω3 ⋐ M. Because W1 , . . . , Wr satisfy Hörmander’s condition on M, there exists m ∈ ℕ+ such that W1 , . . . , Wr satisfy Hörmander’s condition of order m on Ω2 ; we henceforth fix the smallest such m. Fix ψ ∈ C0∞ (Ω3 ) with

242 � 4 Pseudo-differential operators ψ ≡ 1 on a neighborhood of Ω2 . By replacing W1 , . . . , Wr with ψW1 , . . . ψWr , the vector fields still satisfy Hörmander’s condition of order m on Ω2 , and they can be thought of as globally defined standard pseudo-differential operators on ℝn . We henceforth make this replacement. Proposition 4.5.6. For any s < s2 ∈ ℝ, we have u ∈ C0∞ (ℝn ).

‖u‖L2s ≤ (s.c.)‖u‖L2s + (l.c.)‖u‖L2−∞ , 2

To prove Proposition 4.5.6, we prove the following intermediate lemma. Lemma 4.5.7. Let s < s2 . Then ‖u‖L2s ≤ (s.c.)‖u‖L2s + (l.c.)‖u‖L2

2s−s2

2

u ∈ C0∞ (ℝn ).

,

Proof. We have ‖u‖2L2 = ⟨Λs u, Λs u⟩ = ⟨Λs2 u, Λ2s−s2 u⟩ ≤ (s.c.)‖u‖2L2 + (l.c.)‖u‖L2 s

2s−s2

s2

,

where the last estimate follows from the Cauchy–Schwartz inequality. Proof of Proposition 4.5.6. We claim that for all j ∈ ℕ, ‖u‖L2s ≤ (s.c.)‖u‖L2s + (l.c.)‖u‖L2

2j s−(2j −1)s2

2

u ∈ C0∞ (ℝn ).

,

(4.41)

The result then follows from (4.41) by taking j large. We prove (4.41) by induction on j. The base case (j = 0) is obvious (in this case, the small constant can be taken to be 0 and the large constant can be taken to be 1). We assume we have (4.41) for j and prove it for j + 1. Lemma 4.5.7 shows ‖u‖L2

2j s−(2j −1)s2

≤ (s.c.)‖u‖L2s + (l.c.)‖u‖L2 2

2j+1 s−(2j+1 −1)s2

.

Combining this with the inductive hypothesis proves (4.41) with j replaced by j + 1, completing the proof. The next proposition is the key step where Hörmander’s condition comes into play. Proposition 4.5.8. For all s ∈ ℝ, r

‖u‖L2

s+2(1−m)

≲ ∑ ‖Wj u‖L2s + ‖u‖L2s , j=1

u ∈ C0∞ (Ω1 ).

(4.42)

Remark 4.5.9. s + 2(1−m) is not the optimal value on the left-hand side of (4.42). In fact, the optimal value is essentially s + m1 (see Corollary 8.2.5 for precise details on what the

4.5 The sub-Laplacian

�

243

optimal value is). The optimal estimate is a result of Rothschild and Stein [200, Theorem 12]. This follows, along with many other sharp results, from the theory of Chapter 8. However, we use the results from this chapter to prove the results in Chapter 8, and therefore the non-optimal result Proposition 4.5.8 plays an important role in establishing the optimal results. Let 𝒴1 := {W1 , . . . , Wr }, and for j > 1 recursively define 𝒴j := {[Wl , V ] : 1 ≤ l ≤ r, V ∈ 𝒴j−1 }.

Each 𝒴j is a finite set and our hypothesis of Hörmander’s condition implies that the vector fields in ⋃m j=1 𝒴j span the tangent space at every point of Ω2 . To prove Proposition 4.5.8, we use the following lemma. Lemma 4.5.10. Let Zj ∈ 𝒴j . Then ‖Zj u‖L2

r

s+21−j −1

≲ ∑ ‖Wk u‖L2s + ‖u‖L2s , k=1

u ∈ C0∞ (Ω1 ).

Proof. We prove the result by induction on j. The base case, j = 1, is obvious. Fix j ≥ 2 for which we wish to prove the result. We assume the result for all lesser values of j. Take Zj ∈ 𝒴j and write Zj = [Wl , Zj−1 ] for some 1 ≤ l ≤ r and some Zj−1 ∈ 𝒴j−1 . We have ‖Zj u‖2L2

s+21−j −1

= ⟨Zj u, Λ2s+2

2−j

−2

Zj u⟩

= ⟨Wl Zj−1 u, Λ2s+2

2−j

−2

Zj u⟩ − ⟨Zj−1 Wl u, Λ2s+2

2−j

−2

Zj u⟩.

(4.43)

We bound the two terms on the right-hand side of (4.43) separately. We write Wl∗ = −Wl + Mult[f ] for some f ∈ C0∞ (Ω3 ). We have, for u ∈ C0∞ (Ω1 ), 󵄨󵄨 󵄨 2s+22−j −2 Zj u⟩󵄨󵄨󵄨 󵄨󵄨⟨Wl Zj−1 u, Λ 2−j 2−j 󵄨 󵄨 = 󵄨󵄨󵄨⟨Zj−1 u, Wl Λ2s+2 −2 Zj u⟩ − ⟨Zj−1 u, f Λ2s+2 −2 Zj u⟩󵄨󵄨󵄨 2−j 2−j 󵄨 󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨⟨Zj−1 u, Λ2s+2 −2 Zj Wl u⟩󵄨󵄨󵄨 + 󵄨󵄨󵄨⟨Zj−1 u, S (2s+2 −1) u⟩󵄨󵄨󵄨,

(4.44)

where in the last inequality we have used the calculus of pseudo-differential operators. Consider 2−j 󵄨󵄨 󵄨 󵄨 (2s+22−j −1) 󵄨󵄨 u⟩󵄨󵄨 = 󵄨󵄨󵄨⟨S (s+2 −1) Zj−1 u, Λs u⟩󵄨󵄨󵄨 󵄨󵄨⟨Zj−1 u, S

≲ ‖Zj−1 u‖2L2

s+22−j −1

r

+ ‖u‖2L2 ≲ ∑ ‖Wk u‖2L2 + ‖u‖2L2 , s

k=1

s

where the last inequality follows from the inductive hypothesis. Also,

s

(4.45)

244 � 4 Pseudo-differential operators r

󵄨󵄨 󵄨 2s+22−j −2 Zj Wl u⟩󵄨󵄨󵄨 ≲ ‖Zj−1 u‖2L2 󵄨󵄨⟨Zj−1 u, Λ

s+22−j −1

+ ‖Wl u‖2L2 ≲ ∑ ‖Wk u‖2L2 + ‖u‖2L2 , s

s

k=1

s

(4.46)

where in the first inequality we have used Λs−1 Zj Wl u = S (s) Wl u (since Zj is a pseudodifferential operator of order 1) and in the second inequality we have used the inductive hypothesis. Combining (4.44), (4.45), and (4.46), we see that r

󵄨󵄨 󵄨 2s+22−j −2 Zj u⟩󵄨󵄨󵄨 ≲ ∑ ‖Wk u‖2L2 + ‖u‖2L2 , 󵄨󵄨⟨Wl Zj−1 u, Λ s

k=1

s

∀u ∈ C0∞ (Ω1 ).

(4.47)

We now turn to the second term on the right-hand side of (4.43). Recall that for j ≥ 2, 2−j 2−j 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 2s+22−j −2 Zj u⟩󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨⟨Wl u, Zj−1 Λ2s+2 −2 Zj u⟩󵄨󵄨󵄨 + 󵄨󵄨󵄨⟨Wl u, S (2s+2 −1) u⟩󵄨󵄨󵄨 󵄨󵄨⟨Zj−1 Wl u, Λ 2−j 󵄨 󵄨 ≲ 󵄨󵄨󵄨⟨Wl u, Zj−1 Λ2s+2 −2 Zj u⟩󵄨󵄨󵄨 + ‖Wl u‖2L2 + ‖u‖2L2 s s+22−j −1 2−j 2−j 󵄨 󵄨 󵄨 󵄨 ≲ 󵄨󵄨󵄨⟨Wl u, Λ2s+2 −2 Zj Zj−1 u⟩󵄨󵄨󵄨 + 󵄨󵄨󵄨⟨Wl u, S (2s+2 −1) u⟩󵄨󵄨󵄨

+ ‖Wl u‖2L2

s+22−j −1

+ ‖u‖2L2 s

2−j 󵄨 󵄨 ≲ 󵄨󵄨󵄨⟨Wl u, Λ2s+2 −2 Zj Zj−1 u⟩󵄨󵄨󵄨 + ‖Wl u‖2L2 + ‖u‖2L2 s s+22−j −1

≲ ‖Wl u‖2L2 + ‖Zj−1 u‖2L2 s

s+22−j −1

r

(4.48)

+ ‖u‖2L2 s

≲ ∑ ‖Wk u‖2L2 + ‖u‖2L2 , s

k=1

s

where the last inequality follows from the inductive hypothesis. Combining (4.43), (4.47), and (4.48) completes the proof. Proof of Proposition 4.5.8. Note that for u ∈ C0∞ (Ω1 ), n

‖u‖L2

s+21−m

≲ ∑ ‖𝜕xj u‖L2

s+21−m −1

j=1

+ ‖u‖L2s r

≲

∑

Z∈⋃m j=1 𝒴j

‖Zu‖L2

s+21−m −1

+ ‖u‖Ls2 ≲ ∑ ‖Wk u‖L2s + ‖u‖L2s , k=1

where the last inequality follows from Lemma 4.5.10. Set nj := κ/dsj and pick ϵ0 > 0 such that ∀j, j′ ∈ {1, . . . r}, (nj − (nj′ − 1))ϵ0 ≤ 2−m . Lemma 4.5.11. Let Y ∈ C0∞ (Ω3 ; TΩ3 ). Then, ∀s ∈ ℝ, k ∈ ℕ+ , ‖Y k u‖L2s ≤ (s.c.)‖Y k−1 u‖L2s+ϵ + (l.c.)(‖Y k+1 u‖L2s−ϵ + ‖u‖L2−∞ ), 0

0

u ∈ C0∞ (Ω1 ).

4.5 The sub-Laplacian

�

245

Proof. Using that Y ∗ = −Y +Mult[f ], where f ∈ C0∞ (Ω3 ), we have, using Proposition 4.5.6, 󵄨 󵄨 ‖Y k u‖L2s = 󵄨󵄨󵄨⟨Λ2s Y k u, Y k u⟩󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨⟨Λs+ϵ0 Y k−1 u, Λs−ϵ0 Y k+1 u⟩󵄨󵄨󵄨 + 󵄨󵄨󵄨⟨S (s+ϵ0 ) Y k−1 u, S (s−ϵ0 ) Y k u⟩󵄨󵄨󵄨 ≤ (s.c.)‖Y k−1 u‖2L2 ≤ (s.c.)(‖Y

k−1

s+ϵ0

+ (l.c.)(‖Y k+1 u‖2L2

u‖2L2 s+ϵ

≤ (s.c.)(‖Y k−1 u‖2L2

0

s+ϵ0

+ ‖Y

k

u‖2L2 ) s

s−ϵ0

+ ‖Y k u‖2L2 )

+ (l.c.)(‖Y

k+1

s−ϵ0

u‖2L2 s−ϵ

+ ‖Y k u‖2L2 ) + (l.c.)(‖Y k+1 u‖2L2 s

0

s−ϵ0

+ ‖Y k u‖L2−∞ ) + ‖u‖L2−∞ ),

where the second to last inequality uses Proposition 4.5.6. Subtracting a small constant times ‖Y k u‖2L2 from both sides completes the proof. s

Lemma 4.5.12. Let Y ∈ C0∞ (Ω3 ; TΩ3 ). Then, ∀s ∈ ℝ, l ∈ ℕ+ , l−1

∑ ‖Y k u‖L2

s+(l−k)ϵ0

k=1

≤ (l.c.)(‖Y l u‖L2s + ‖u‖L2−∞ ) + (s.c.)‖u‖L2

s+lϵ0

u ∈ C0∞ (Ω1 ).

,

Proof. When l = 1, the result is trivial since the left-hand side equals 0, so we assume l ≥ 2. We prove, for 1 ≤ j ≤ l − 1, l−1

∑ ‖Y k u‖L2

s+(l−k)ϵ0

k=j

≤ (l.c.)(‖Y l u‖L2s + ‖u‖L2−∞ ) + (s.c.)‖Y j−1 u‖L2

s+(l−(j−1))ϵ0

,

(4.49)

and the result will follow by taking j = 1. We prove (4.49) by induction on j. The base case, j = l − 1, follows directly from Lemma 4.5.11. We assume (4.49) with j replaced by j + 1 and prove it for j. Using Lemma 4.5.11 and then the inductive hypothesis, we have l−1

∑ ‖Y k u‖L2

s+(l−k)ϵ0

k=j

l−1

≤ ∑ ‖Y k u‖L2

s+(l−k)ϵ0

k=j+1

+ (l.c.)(‖Y j+1 u‖L2

s+(l−(j+1))ϵ0

+ ‖u‖L2−∞ )

+ (s.c.)‖Y j−1 u‖L2

s+(l−(j−1))ϵ0

≤ (l.c.)(‖Y l u‖L2s + ‖u‖L2−∞ ) + (s.c.)(‖Y j u‖L2

s+(l−j)ϵ0

Subtracting a small constant times ‖Y j u‖L2

s+(l−j)ϵ0

pletes the proof.

+ ‖Y j−1 u‖L2

s+(l−(j−1))ϵ0

).

from both sides yields (4.49) and com-

Lemma 4.5.13. For s ∈ ℝ, r nj −1

∑ ∑ ‖Wjl u‖L2 j=1 l=0

s+(nj −l)ϵ0

r

n

≲ ∑ ‖Wj j u‖L2s + ‖u‖L2−∞ , j=1

u ∈ C0∞ (Ω1 ).

246 � 4 Pseudo-differential operators Proof. By Lemma 4.5.12, we have r nj −1

r

∑ ∑ ‖Wjl u‖L2

s+(nj −l)ϵ0

j=1 l=0

r

n

≲ ∑ ‖Wj j u‖L2s + ∑ ‖u‖L2s+n ϵ . j=1

(4.50)

j 0

j=1

Using Proposition 4.5.8, the fact that ϵ0 satisfies nj ϵ0 − 21−m ≤ (nj′ − 1)ϵ0 − 2−m , ∀j, j′ , and Proposition 4.5.6, we have r

r

r

r

∑ ‖u‖L2s+n ϵ ≲ ∑ ∑ ‖Wj′ u‖L2 j=1

j 0

s+nj ϵ0 −21−m

j=1 j′ =1

+ ∑ ‖u‖L2

r

r

≲ ∑ ‖Wj′ u‖L2

s+nj ϵ0 −21−m

j=1

r

r

≤ (s.c.) ( ∑ ‖Wj′ u‖L2

s+(n ′ −1)ϵ0 j

j′ =1

(4.51)

+ ∑ ‖u‖L2

s+(n ′ −1)ϵ0 −2−m j

j′ =1

s+nj ϵ0 −21−m

j=1

+ ∑ ‖u‖L2s+n ϵ ) + (l.c.)‖u‖L2−∞ . j 0

j=1

Plugging (4.51) into (4.50), we have r nj −1

r

∑ ∑ ‖Wjl u‖L2 j=1 l=0

n

≤ (l.c.) (∑ ‖Wj j u‖L2s + ‖u‖L2−∞ )

s+(nj −l)ϵ0

j=1

r

+ (s.c.) (∑ ‖Wj u‖L2

s+(nj −1)ϵ0

j=1

Subtracting a small constant times ∑rj=1 ‖Wj u‖L2

s+(nj −1)ϵ0

of (4.52) completes the proof.

(4.52)

r

+ ∑ ‖u‖L2s+n ϵ ) . j 0

j=1

+ ∑rj=1 ‖u‖L2s+n ϵ from both sides j 0

Lemma 4.5.14. For all s ∈ ℝ, r nj −1

r

n

∑ ∑ ‖Wjl u‖L2s+ϵ ≲ ∑ ‖Wj j u‖L2s + ‖u‖L2−∞ , j=1 l=0

0

j=1

u ∈ C0∞ (Ω1 ).

Proof. This follows immediately from Lemma 4.5.13. Theorem 4.5.5 is a statement about all distributions u ∈ C0∞ (M)′ . All of our results, up to this point, have been about functions u ∈ C0∞ (Ω1 ). To make the move from smooth functions to distributions, we approximate each distribution by a smooth function. To do this, let ψ̂ ∈ C0∞ (ℝn ) equal 1 on a neighborhood of 0. For δ > 0, let ψδ (D) denote the pseudo-differential operator ∨ ̂ f 󳨃→ (ψ(δξ) f ̂(ξ)) .

(4.53)

̂ Note that ψ(δξ) ∈ S −∞ = ⋂m∈ℝ S m , where S m denotes the space of standard symbols from Definition 2.2.3; to indicate this, we say ψδ (D) is a pseudo-differential operator of

4.5 The sub-Laplacian

�

247

∞ ̂ ̂ order −∞. Because of this, ψ(δξ) : S (ℝn )′ → Cloc (ℝn ). However, ψ(δξ) ∈ S 0 , uniformly in 0 ̂ δ ∈ (0, 1]; more precisely, {ψ(δξ) : δ ∈ (0, 1]} ⊂ S is a bounded set. Also, limδ→0 ψδ (D) = I. We write S (s,δ) , perhaps with a subscript, to denote some standard pseudo-differential operator of order −∞, such that {S (s,δ) : δ ∈ (0, 1]} is a bounded set of pseudo-differential operators of order s. When we write S̃(s,δ) , perhaps with a subscript, we mean a finite 󵄩 󵄩 number of terms of that form. For example, if we write 󵄩󵄩󵄩S̃(s,δ) u󵄩󵄩󵄩L2 , it means ‖S1(s,δ) u‖L2 + (s,δ) (s,δ) (s,δ) ⋅ ⋅ ⋅ + ‖SQ u‖L2 , for some finite collection S1 , . . . , SQ . Here the finite collection does not depend on the function or distribution u under consideration.

Lemma 4.5.15. Let P be any partial differential operator with smooth coefficients. Suppose that for all s ∈ ℝ and all ϕ1 , ϕ2 ∈ C0∞ (Ω1 ), with ϕ1 ≺ ϕ2 , and ∀S (s,δ) as described above, r

nj

r nj −1

∑ ∑ ‖Wjl ϕ1 S (s,δ) u‖L2 ≲ ‖ϕ2 P u‖L2s + ∑ ∑ ‖Wjl ϕ2 S̃(s,δ) u‖L2 j=1 l=0

j=1 l=0

+ ‖u‖L2−∞ ,

(4.54)

∀u ∈ S (ℝn )′ .

Then, for all s ∈ ℝ and ϕ1 , ϕ2 ∈ C0∞ (Ω1 ) with ϕ1 ≺ ϕ2 , we have ‖ϕ1 u‖L2s+ϵ ≲ ‖ϕ2 P u‖L2s + ‖ϕ2 u‖L2−∞ , 0

∀u ∈ C0∞ (Ω1 )′ .

(4.55)

Proof. Recall that the term ‖Wjl ϕ2 S̃(s,δ) u‖L2 really denotes a finite sum of such terms. Because of the localizations ϕ1 and ϕ2 in (4.55), it suffices to prove (4.55) for u ∈ S (ℝn )′ . For the remainder of the proof, u will denote an arbitrary element of S (ℝn )′ . Fix ϕ1 , ϕ2 , ϕ3 ∈ C0∞ (Ω1 ) with ϕ1 ≺ ϕ3 ≺ ϕ2 and let ηj ∈ C0∞ (Ω1 ), j ∈ ℕ, be such that ϕ1 ≺ η0 ≺ η1 ≺ η2 ≺ ⋅ ⋅ ⋅ ≺ ϕ3 . (s−(k−1)ϵ0 ,δ) Let S0(s,δ) be given. We claim that ∀k ∈ ℕ+ , there exists S̃k such that nj

r

r nj −1

(s−(k−1)ϵ0 ,δ) 󵄩 󵄩 󵄩 󵄩 u󵄩󵄩󵄩L2 + ‖u‖L2−∞ . ∑ ∑󵄩󵄩󵄩Wjl η0 S0(s,δ) u󵄩󵄩󵄩L2 ≲ ‖ηk P u‖L2s + ∑ ∑ 󵄩󵄩󵄩Wjl ηk S̃k j=1 l=0

j=1 l=0

(4.56)

We prove (4.56) by induction on k. The base case, k = 1, follows from the hypothesis (4.54). We assume (4.56) for k ∈ ℕ+ and prove it for k + 1. Fix η̃ k,1 , η̃ k,2 ∈ C0∞ (Ω1 ) with ηk ≺ η̃ k,1 ≺ η̃ k,2 ≺ ηk+1 . By the inductive hypothesis and Lemma 4.5.14, r

nj

r nj −1

(s−(k−1)ϵ0 ,δ) 󵄩 󵄩 󵄩 󵄩 u󵄩󵄩󵄩L2 + ‖u‖L2−∞ ∑ ∑󵄩󵄩󵄩Wjl η0 S0(s,δ) u󵄩󵄩󵄩L2 ≲ ‖ηk P u‖L2s + ∑ ∑ 󵄩󵄩󵄩Wjl ηk S̃k j=1 l=0

j=1 l=0 r

nj

(s−(k−1)ϵ0 ,δ) 󵄩 󵄩 ≲ ‖ηk P u‖L2s + ∑ ∑󵄩󵄩󵄩Wjl ηk S̃k u󵄩󵄩󵄩L2 + ‖u‖L2−∞ . −ϵ0 j=1 l=0

Consider, using the calculus of pseudo-differential operators,

(4.57)

248 � 4 Pseudo-differential operators r

nj

r

(s−(k−1)ϵ0 ,δ) 󵄩 󵄩 u󵄩󵄩󵄩L2 ∑ ∑󵄩󵄩󵄩Wjl ηk S̃k

−ϵ0

j=1 l=0

nj

(s−(k−1)ϵ0 ,δ) 󵄩 󵄩 ≲ ∑ ∑󵄩󵄩󵄩η̃ k,1 Λ−ϵ0 Wjl S̃k u󵄩󵄩󵄩L2 + ‖u‖L2−∞ j=1 l=0 r

nj

(s−kϵ ,δ) 󵄩 󵄩 ≲ ∑ ∑󵄩󵄩󵄩η̃ k,1 Wjl S̃k+1 0 u󵄩󵄩󵄩L2 + ‖u‖L2−∞ j=1 l=0 r

nj

(4.58)

(s−kϵ ,δ) 󵄩 󵄩 = ∑ ∑󵄩󵄩󵄩η̃ k,1 Wjl η̃ k,2 S̃k+1 0 u󵄩󵄩󵄩L2 + ‖u‖L2−∞ j=1 l=0 r

nj

(s−kϵ ,δ) 󵄩 󵄩 ≲ ∑ ∑󵄩󵄩󵄩Wjl η̃ k,2 S̃k+1 0 u󵄩󵄩󵄩L2 + ‖u‖L2−∞ . j=1 l=0

The hypothesis (4.54) gives r

nj

(s−kϵ ,δ) 󵄩 󵄩 ∑ ∑󵄩󵄩󵄩Wjl η̃ k,2 S̃k+1 0 u󵄩󵄩󵄩L2 j=1 l=0

≲ ‖ηk+1 P u‖L2

(4.59)

r nj −1

s−kϵ0

(s−kϵ ,δ) 󵄩 󵄩 + ∑ ∑ 󵄩󵄩󵄩Wjl ηk+1 S̃k+1 0 u󵄩󵄩󵄩L2 + ‖u‖L2−∞ . j=1 l=0

Plugging (4.58) and (4.59) into (4.57), we get r

nj

r nj −1

(s−kϵ ,δ) 󵄩 󵄩 󵄩 󵄩 ∑ ∑󵄩󵄩󵄩Wjl η0 S0(s,δ) u󵄩󵄩󵄩L2 ≲ ‖ηk+1 P u‖L2s + ∑ ∑ 󵄩󵄩󵄩Wjl ηk+1 S̃k+1 0 u󵄩󵄩󵄩L2 + ‖u‖L2−∞ , j=1 l=0

j=1 l=0

which completes the inductive step and therefore the proof of (4.56). Taking k large in (4.56) implies r

nj

󵄩 󵄩 ∑ ∑󵄩󵄩󵄩Wjl η0 S0(s,δ) u󵄩󵄩󵄩L2 ≲ ‖ϕ3 P u‖L2s + ‖u‖L2−∞ . j=1 l=0

By Lemma 4.5.14, this implies 󵄩󵄩 (s,δ) 󵄩󵄩 󵄩󵄩η0 S0 u󵄩󵄩L2 ≲ ‖ϕ3 P u‖L2s + ‖u‖L2−∞ . ϵ0

(4.60)

We may now choose a particular S0(s,δ) . Namely, we take S0(s,δ) := ψδ (D)Λs , where ψδ (D) is as defined in (4.53). Consider we have ‖ϕ1 u‖L2s+ϵ ≲ sup ‖η0 Λs ψδ (D)ϕ1 u‖L2ϵ + ‖u‖L2−∞ 0

δ>0

0

󵄩 󵄩 = sup󵄩󵄩󵄩η0 S0(s,δ) u󵄩󵄩󵄩L2 + ‖u‖L2−∞ ϵ0 δ>0

(4.61)

≲ ‖ϕ3 P u‖L2s + ‖u‖L2−∞ , where the last inequality follows from (4.60). Replacing u with ϕ2 u in (4.61), we obtain

4.5 The sub-Laplacian

�

249

‖ϕ1 u‖L2s+ϵ ≲ ‖ϕ3 P u‖L2s + ‖ϕ2 u‖L2−∞ ≲ ‖ϕ2 P u‖L2s + ‖ϕ2 u‖L2−∞ , 0

completing the proof. Lemma 4.5.16. Let ϕ1 , ϕ2 , ϕ3 ∈ C0∞ (Ω1 ) with ϕ1 ≺ ϕ2 ≺ ϕ3 . Then r 2nj −1

(s,δ) [L , Mult[ϕ1 ]S (s,δ) ] = ∑ ∑ Mult[ϕ2 ]Wjl Mult[ϕ3 ]Sj,l , j=1 l=1

(s,δ) where S (s,δ) and Sj,l are pseudo-differential operators of order −∞ which are pseudodifferential operators of order s, uniformly in δ ∈ (0, 1].

Proof. This follows easily from the calculus of pseudo-differential operators (Theorem 2.2.8). Lemma 4.5.17. Let ϕ1 , ϕ2 ∈ C0∞ (Ω1 ) with ϕ1 ≺ ϕ2 . Then, for u ∈ S (ℝn )′ , 󵄨󵄨 (s,δ) (s,δ) 󵄨 󵄨󵄨⟨[Mult[ϕ1 ]S0 , L ]u, Mult[ϕ1 ]S0 u⟩󵄨󵄨󵄨 r nj −1

r

nj

󵄩 󵄩2 󵄩 󵄩2 ≤ (l.c.) ∑ ∑ 󵄩󵄩󵄩Wjl ϕ2 S̃(s,δ) u󵄩󵄩󵄩L2 + (s.c.) ∑ ∑󵄩󵄩󵄩Wjl ϕ1 S0(s,δ) u󵄩󵄩󵄩L2 . j=1 l=0

j=1 l=0

Proof. Fix ϕ1 , ϕ2 , ϕ3 ∈ C0∞ (Ω1 ) with ϕ1 ≺ ϕ2 ≺ ϕ3 . We prove the result with ϕ2 replaced by ϕ3 . Applying Lemma 4.5.16, we have 󵄨󵄨 (s,δ) (s,δ) 󵄨 󵄨󵄨⟨[Mult[ϕ1 ]S0 , L ]u, Mult[ϕ1 ]S0 u⟩󵄨󵄨󵄨 󵄨󵄨 r 2nj −1 󵄨󵄨 󵄨󵄨 󵄨 l (s,δ) (s,δ) 󵄨󵄨 󵄨 󵄨 = 󵄨󵄨∑ ∑ ⟨Mult[ϕ2 ]Wj Mult[ϕ3 ]Sj,l u, Mult[ϕ1 ]S0 u⟩󵄨󵄨󵄨 󵄨󵄨j=1 l=0 󵄨󵄨 󵄨 󵄨 󵄨󵄨 r 2nj −1 󵄨󵄨 󵄨󵄨 󵄨 󵄨 (s,δ) = 󵄨󵄨󵄨󵄨∑ ∑ ⟨Wjl Mult[ϕ3 ]Sj,l u, Mult[ϕ1 ]S0(s,δ) u⟩󵄨󵄨󵄨󵄨 . 󵄨󵄨j=1 l=0 󵄨󵄨 󵄨 󵄨

(4.62)

∞ Since Wj∗ = −Wj + Mult[fj ], where fj ∈ Cloc (ℝn ), the right-hand side of (4.62) is bounded by a sum of terms of the form r nj −1 nj

l 󵄨 l 󵄨 ∑ ∑ ∑ 󵄨󵄨󵄨⟨Wj 1 Mult[ϕ3 ]S̃(s,δ) u, Wj 2 Mult[ϕ1 ]S0(s,δ) u⟩󵄨󵄨󵄨 j=1 l1 =0 l2 =0

r nj −1

r

nj

󵄩 󵄩2 󵄩 󵄩2 ≤ (l.c.) ∑ ∑ 󵄩󵄩󵄩Wjl Mult[ϕ3 ]S̃(s,δ) u󵄩󵄩󵄩L2 + (s.c.) ∑ ∑󵄩󵄩󵄩Wjl Mult[ϕ1 ]S0(s,δ) u󵄩󵄩󵄩L2 , j=1 l=0

completing the proof.

j=1 l=0

250 � 4 Pseudo-differential operators Proof of Theorem 4.5.5. We prove the result by verifying the conditions of Lemma 4.5.15. Let u ∈ S (ℝn )′ and ϕ1 , ϕ2 ∈ C0∞ (Ω1 ) with ϕ1 ≺ ϕ2 . Then, using Lemma 4.5.17 and the Cauchy–Schwartz inequality, r

nj

󵄩 󵄩2 󵄨 󵄨 ∑ ∑󵄩󵄩󵄩Wjl ϕ1 S0(s,δ) u󵄩󵄩󵄩L2 = 󵄨󵄨󵄨⟨L ϕ1 S0(s,δ) u, ϕ1 S0(s,δ) u⟩󵄨󵄨󵄨 j=1 l=0

󵄨 󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨⟨ϕ1 S0(s,δ) L u, ϕ1 S0(s,δ) u⟩󵄨󵄨󵄨 + 󵄨󵄨󵄨⟨[L , ϕ1 S0(s,δ) ]u, ϕ1 S0(s,δ) u⟩󵄨󵄨󵄨 r nj −1

󵄩 󵄩2 󵄩 󵄩2 ≤ (l.c.) (󵄩󵄩󵄩ϕ1 S0(s,δ) L u󵄩󵄩󵄩L2 + ∑ ∑ 󵄩󵄩󵄩Wjl ϕ2 S̃(s,δ) u󵄩󵄩󵄩L2 )

(4.63)

j=1 l=0

r

nj

󵄩 󵄩2 + (s.c.) ∑ ∑󵄩󵄩󵄩Wjl ϕ1 S0(s,δ) u󵄩󵄩󵄩L2 . j=1 l=0

nj 󵄩 l 󵄩󵄩W ϕ1 S (s,δ) u󵄩󵄩󵄩2 2 from both sides of (4.63) and using Subtracting (s.c.) ∑rj=1 ∑l=0 󵄩 j 󵄩L 0

󵄩󵄩 (s,δ) 󵄩2 2 2 󵄩󵄩ϕ1 S0 L u󵄩󵄩󵄩L2 ≲ ‖ϕ2 L u‖L2 + ‖u‖L2 , s −∞ we have r

nj

r nj −1

󵄩 󵄩2 󵄩 󵄩2 ∑ ∑󵄩󵄩󵄩Wjl ϕ1 S0(s,δ) u󵄩󵄩󵄩L2 ≲ ‖ϕ2 L u‖2L2 + ∑ ∑ 󵄩󵄩󵄩Wjl ϕ2 S̃(s,δ) u󵄩󵄩󵄩L2 + ‖u‖2L2 . s −∞ j=1 l=0

j=1 l=0

The result now follows from Lemma 4.5.15. 4.5.2 Homogeneity Understanding a fundamental solution for a hypoelliptic operator is particularly accessible when the operator is homogeneous under a dilation structure. In this section, we present these ideas. Suppose we are given dilations on ℝN : d = (d1 , . . . , dN ), where dj ∈ ℕ+ , that is, for δ > 0 and t ∈ ℝN , δd t = (δd1 t1 , . . . , δdN tN ). We set Q := d1 + ⋅ ⋅ ⋅ + dN ∈ ℕ+ , the so-called homogeneous dimension of ℝN with respect to these dilations. Definition 4.5.18. We say a partial differential operator with smooth coefficients, P : ∞ ∞ Cloc (ℝN ) → Cloc (ℝN ), is homogeneous of degree s ∈ ℝ if d

s

d

P (f (δ t)) = δ (P f )(δ t),

∞ f ∈ Cloc (ℝN ).

Definition 4.5.19. For K ∈ C0∞ (ℝN )′ and δ > 0, we define K(δd t) ∈ C0∞ (ℝN )′ by ∫ K(δd t)f (t) dt := δ−Q ∫ K(t)f (δ−d t) dt,

f ∈ C0∞ (ℝN ).

4.5 The sub-Laplacian

�

251

Note that when K agrees with integration against a locally integrable function, this definition agrees with the usual, pointwise, definition of the function K(δd t). Definition 4.5.20. We say a distribution is homogeneous of degree s ∈ ℝ if K(δd t) = δs K(t), ∀δ > 0. The next result, due to Folland [90], is the key result of this section, which we state without proof. For a proof, the reader is referred to [90, Theorem 2.1] and also [99, Chapter 3, Section 3]. ∞ ∞ Theorem 4.5.21. Let P : Cloc (ℝN ) → Cloc (ℝN ) be a partial differential operator with smooth coefficients, which is homogeneous of degree s ∈ (0, Q). Suppose P and P ∗ are both hypoelliptic. Then there exists a unique distribution K ∈ C0∞ (ℝN )′ such that K is homogeneous of degree s − Q and P K = δ0 (where δ0 denotes the Dirac δ function at 0). ∞ This K(t) agrees with a Cloc (ℝn ) function on {t ≠ 0}.

Following (4.5) we define a homogeneous norm on ℝN by N

(2 ∏Nj=1 dj )/dl

‖t‖ = (∑ |tl | l=1

1/2 ∏Nj=1 dj

)

N

≈ ∑ |tl |1/dl . l=1

Note that ‖δd t‖ = δ‖t‖, for δ > 0. For a multi-index α ∈ ℕN , define degd (α) := ∑Nl=1 αl dl ∈ ℕ. Proposition 4.5.22. Let K ∈ C0∞ (ℝN )′ be homogeneous of degree s − Q, for s ∈ (−∞, Q], ∞ and let η ∈ C0∞ (ℝn ). Suppose K(t) agrees with a Cloc (ℝn ) function on {t ≠ 0}. Then, ∀α ∈ ℕN , 󵄨󵄨 α ̂ 󵄨󵄨 −s−degd (α) . 󵄨󵄨𝜕ξ ηK(ξ)󵄨󵄨 ≤ Cα (1 + ‖ξ‖)

(4.64)

Proof. We have ̂ = ∫ K(t)(−2πit)α η(t)e−2πit⋅ξ dt. 𝜕ξ α ηK(ξ) Note that {(−2πit)α η(t)e−2πit⋅ξ : ‖ξ‖ ≤ 1} ⊂ C0∞ (ℝN ) is a bounded set, and therefore 󵄨 ̂ 󵄨󵄨 sup 󵄨󵄨󵄨𝜕ξα ηK(ξ) 󵄨󵄨 < ∞.

‖ξ‖≤1

This proves (4.64) when ‖ξ‖ ≤ 1. To prove (4.64) when ‖ξ‖ ≥ 1, we will show, with ηδ (t) := η(δd t), 󵄨 󵄨󵄨 sup sup 󵄨󵄨󵄨𝜕ξα η̂ δ K(ξ)󵄨󵄨 < ∞.

‖ξ‖=1 δ∈(0,1]

(4.65)

First, we see why (4.65) implies (4.64) for ‖ξ‖ ≥ 1. Note that by the homogeneity of K, we have

252 � 4 Pseudo-differential operators ̂ −d ξ) = ∫ η(t)K(t)e−2πi(δ ηK(δ

−d

t)⋅ξ

dt = δs ∫ η(δd t)K(t)e−2πit⋅ξ dt = δs η̂ δ K(ξ).

Thus, using (4.65), we have, for δ ∈ (0, 1], 󵄨 ̂ 󵄨󵄨 󵄨 α ̂ −d 󵄨󵄨 deg (α) sup 󵄨󵄨󵄨𝜕ξα ηK(ξ) ξ))󵄨󵄨 󵄨󵄨 = δ d sup 󵄨󵄨󵄨𝜕ξ (ηK(δ −1 ‖ξ‖=1

‖ξ‖=δ

=δ

degd (α)+s

󵄨 󵄨󵄨 degd (α)+s sup 󵄨󵄨󵄨𝜕ξα η̂ . δ K(ξ)󵄨󵄨 ≲ δ

‖ξ‖=1

This immediately implies (4.64) for ‖ξ‖ ≥ 1. We turn to proving (4.65). Fix ϕ ∈ C0∞ ({t : ‖t‖ < 2}) with ϕ ≡ 1 on {t : ‖t‖ ≤ 3/2}. We have α −2πit⋅ξ 𝜕ξα η̂ dt δ K(ξ) = ∫ K(t)ηδ (t)(−2πit) e

= ∫ K(t)ηδ (t)(−2πit)α ϕ(t)e−2πit⋅ξ dt

(4.66)

+ ∫ K(t)ηδ (t)(−2πit)α (1 − ϕ(t))e−2πit⋅ξ dt. Note that {ηδ (t)ϕ(t)(−2πit)α e−2πit⋅ξ : ‖ξ‖ = 1, δ ∈ (0, 1]} ⊂ C0∞ (ℝN ) is a bounded set, and therefore, 󵄨󵄨 󵄨󵄨 sup sup 󵄨󵄨󵄨∫ K(t)ηδ (t)(−2πit)α ϕ(t)e−2πit⋅ξ dt 󵄨󵄨󵄨 < ∞. 󵄨 󵄨 ‖ξ‖=1 δ∈(0,1]

(4.67)

Thus, it suffices to estimate the second term on the right-hand side of (4.66). We claim that, ∀β ∈ ℕN , t ≠ 0, 󵄨󵄨 β 󵄨 α −s−degd (β)+degd (α) . 󵄨󵄨𝜕t ((−2πit) ηδ (t)K(t))󵄨󵄨󵄨 ≤ Cα,β ‖t‖

(4.68)

Indeed, fix D > 0 such that supp(η) ⊆ {t : ‖t‖ ≤ D}. Then 󵄨󵄨 β 󵄨 α 󵄨󵄨𝜕t ((−2πit) ηδ (t)K(t))󵄨󵄨󵄨 ≲ ≲ ≲

∑

β1 +β2 +β3 =β

β β 󵄨󵄨 β1 󵄨 α 󵄨󵄨(𝜕t ((−2πit) ))(𝜕t 2 ηδ (t))(𝜕t 3 K(t))󵄨󵄨󵄨

∑

‖t‖degd (α)−degd (β1 ) δdegd (β2 ) χ{‖t‖≤Dδ−1 } ‖t‖− degd (β3 )

∑

‖t‖degd (α)−degd (β1 )−degd (β2 )−degd (β3 ) ,

β1 +β2 +β3 β1 +β2 +β3

which proves (4.68). We claim that, ∀β ∈ ℕN , 󵄨󵄨 β 󵄨 α ′ −s−|β|+degd (α) χ{‖t‖≥1} . 󵄨󵄨𝜕t ((−2πit) (1 − ϕ(t))ηδ (t)K(t))󵄨󵄨󵄨 ≤ Cα,β ‖t‖

(4.69)

4.5 The sub-Laplacian

�

253

Indeed, 󵄨󵄨 β 󵄨 α 󵄨󵄨𝜕t ((−2πit) (1 − ϕ(t))ηδ (t)K(t))󵄨󵄨󵄨 β 󵄨 β 󵄨 ≲ ∑ 󵄨󵄨󵄨(𝜕t 1 (1 − ϕ(t)))(𝜕t 2 ((−2πit)α ηδ (t)K(t)))󵄨󵄨󵄨. β1 +β2 =β

β

When β1 ≠ 0, 𝜕t 1 (1 − ϕ(t)) is supported on {3/2 ≤ ‖t‖ ≤ 2}, and therefore, using (4.68), we have β 󵄨󵄨 β1 󵄨 α 󵄨󵄨(𝜕t (1 − ϕ(t)))(𝜕t 2 ((−2πit) ηδ (t)K(t)))󵄨󵄨󵄨

≲ χ{3/2≤‖t‖≤2} ‖t‖−s−degd (β2 )+degd (α) ≲ χ{‖t‖≥1} .

When β1 = 0, then β2 = β. Using that (1 − ϕ(t)) is supported on {‖t‖ ≥ 3/2} and (4.68), we have β 󵄨󵄨 β1 󵄨 α 󵄨󵄨(𝜕t (1 − ϕ(t)))(𝜕t 2 ((−2πit) ηδ (t)K(t)))󵄨󵄨󵄨 β 󵄨󵄨 󵄨 α 󵄨󵄨((1 − ϕ(t)))(𝜕t ((−2πit) ηδ (t)K(t)))󵄨󵄨󵄨

≲ χ{‖t‖≥3/2} ‖t‖−s−degd (β)+degd (α) ≲ χ{‖t‖≥3/2} ‖t‖−s−|β|+degd (α) ,

where in the last inequality we have used |β| ≤ degd (β). Combining the previous three equations proves (4.69). Consider, for the second term on the right-hand side of (4.66), we have for L ∈ ℕ sufficiently large 󵄨󵄨 󵄨󵄨 sup sup 󵄨󵄨󵄨∫ K(t)ηδ (t)(−2πit)α (1 − ϕ(x))e−2πit⋅ξ dt 󵄨󵄨󵄨 󵄨 󵄨 ‖ξ‖=1 δ∈(0,1] 󵄨󵄨 󵄨󵄨 = sup sup (−|2πξ|2 )−L 󵄨󵄨󵄨∫(△Lt K(t)ηδ (t)(−2πit)α (1 − ϕ(x)))e−2πit⋅ξ dt 󵄨󵄨󵄨 󵄨 󵄨 ‖ξ‖=1 δ∈(0,1]

(4.70)

≲ sup sup (−|2πξ|2 )−L ∫ ‖t‖−s−2L+degd (α) dt < ∞, ξ=1 δ∈(0,1]

‖t‖≥1

where the second to last inequality used (4.69). Combining (4.67) and (4.70) proves (4.65) and completes the proof. Remark 4.5.23. We use Proposition 4.5.22 in the following way. Suppose K ∈ C0∞ (ℝN )′ satisfies the hypotheses of Proposition 4.5.22 and fix η ∈ C0∞ (BN (a)) and ψ ∈ C0∞ (Ω), where Ω ⋐ M and M is some manifold. We wish to see the distribution ψ(x)η(t)K(t) ̌ t, t) = ψ(x)η(t)K(t) for some b(x, u, ξ) ∈ S −s (a, Ω, d). Indeed, take any as of the form b(x, ̌ u, t) := ψ(x)η(u)η′ (t)K(t). Then Proposition 4.5.22 η′ (t) ∈ C0∞ (ℝN ) with η ≺ η′ and set b(x, −s ̌ t, t) = ψ(x)η(t)K(t). shows b(x, u, ξ) ∈ S (a, Ω, d), and it is immediate to see that b(x,

254 � 4 Pseudo-differential operators 4.5.3 Nilpotent Lie groups An important model case of vector fields with formal degrees comes from graded nilpotent Lie groups. Following an idea of Rothschild and Stein [200], we will construct a parametrix for a general sub-Laplacian, by using a fundamental solution for a subLaplacian on a graded nilpotent Lie group. In this section, we introduce the basic background information regarding such groups. We refer the reader to the books [53] and [99] for more information. Let g be a finite-dimensional Lie algebra. Define g(0) = g and recursively set g(k+1) := (k) [g, g ]. Definition 4.5.24. We say g is nilpotent of step k ∈ ℕ if g(k) = {0}. We say g is nilpotent if g is nilpotent of step k for some k. Definition 4.5.25. For a Lie group G, we say G is nilpotent if its Lie algebra is nilpotent. It is well known that if g is a nilpotent Lie algebra, then there is a unique connected, simply connected, nilpotent Lie group G whose Lie algebra is g. In this case, the exponential map exp : g → G is a diffeomorphism (see [53, Theorem 1.2.1]). In particular, as a manifold G ≅ ℝdim G . Given two elements of the Lie algebra X, Y ∈ g, we can compute the group multiplication eX eY via the Baker–Campbell–Hausdorff formula, namely, eX eY = eBCH(X,Y ) , where BCH(X, Y ) is a finite sum due to the nilpotent hypothesis; see [53, Theorem 1.2.1]. Given a basis V1 , . . . , Vdim G of g, we obtain global coordinates on G ∼ by (v1 , . . . , vdim G ) 󳨃→ ev1 V1 +⋅⋅⋅+vdim G Vdim G , ℝdim G 󳨀 → G. In this coordinate system, the group multiplication can be computed explicitly via the Baker–Campbell–Hausdorff formula: the group product of (v1 , . . . , vdim G ) and (u1 , . . . , udim G ) is given by the coefficients of V1 , . . . , Vdim G in the formula BCH(v1 V1 + ⋅ ⋅ ⋅ + vdim G Vdim G , u1 V1 + ⋅ ⋅ ⋅ udim G Vdim G ); this is a polynomial in v and u. Henceforth, we identify G with g via the exponential map; therefore, g has a group structure defined by the Baker–Campbell–Hausdorff formula. With this identification, 0 is the identity element, and the group inverse corresponds with multiplication by −1. If, further, we pick a basis for g, this identifies G with ℝdim G and ℝdim G has a group structure defined by the Baker–Campbell–Hausdorff formula. The group G is unimodular and with the above identification G ≅ ℝdim G , the Lebesgue measure is a Haar measure on G; see [53, Theorem 1.2.10]. By the Schwartz kernel theorem, right invariant operators C0∞ (G) → C0∞ (G)′ are of the form f (x) 󳨃→ K ∗ f (x) = ∫ K(xy−1 )f (y) dy for some K ∈ C0∞ (G)′ . Left invariant operators are of the form f 󳨃→ f ∗ K. Definition 4.5.26. We say a finite-dimensional, nilpotent Lie algebra g is graded if g = ⊕νμ=1 vμ , where [vμ1 , vμ2 ] ⊆ vμ1 +μ2 and we have taken vμ = {0} for μ > ν.

4.5 The sub-Laplacian

�

255

Definition 4.5.27. We say a nilpotent Lie group is graded if its Lie algebra is graded. Henceforth, fix a graded nilpotent Lie algebra g = ⊕νμ=1 vμ and let G be the associated graded nilpotent Lie group. Define dilations on g by δ ⋅ X := δμ X, for δ > 0 and X ∈ vμ , and extend this by linearity. These dilations are Lie algebra automorphisms of g. We henceforth give g this dilation structure. By the Baker–Campbell–Hausdorff formula, these dilations are also group automorphisms. μ μ μ For each μ ∈ {1, . . . , ν}, fix a basis X1 , . . . , Xl of vμ , so that {Xj : μ ∈ {1, . . . , ν}, 1 ≤ μ

j ≤ lμ } is a basis for g, lμ = dim vμ , and ∑μ lμ = dim G. We write t ∈ ℝdim G as μ

μ μ

t = (tj )μ∈{1,...,ν},1≤j≤lμ and identify ℝdim G with g ≅ G via t 󳨃→ ∑ tj Xj . Under this iden-

tification, the dilation structure on g gives single-parameter dilations on ℝdim G , via μ δd t := (δμ tj ); here d ∈ ℕdim G is the vector which equals μ in the coordinate corμ

responding to tj . We henceforth make this identification and treat both g and G as ℝdim G , with this dilation structure. Thus, ℝdim G has a group structure inherited from G. Set Q = ∑μ∈{1,...,ν} μ dim vμ , the homogeneous dimension of G with this dilation structure. We identify g with T0 G, the tangent space to the identity of G. For V ∈ g, we identify V with the distribution which takes f ∈ C0∞ (ℝn )′ and computes the directional derivative at 0 given by Vf . There is a unique left invariant vector field V L on G such that V L (0) = V and a unique right invariant vector field V R with V R (0) = V . In fact, V L f = f ∗ V and μ μ,L V R f = V ∗ f . In particular, the elements Xj ∈ vμ induce left invariant vector fields Xj μ,R

and right invariant vector fields Xj , which (in the above chosen coordinates) satisfy μ,L

μ,R

Xj (0) = Xj (0) = 𝜕tμ . We have j

μ,L

μ,R

Xj f = f ∗ (𝜕tμ δ0 ),

Xj f = (𝜕tμ δ0 ) ∗ f ,

j

j

where δ0 denotes the Dirac δ function at 0. We have (V L )∗ = −V L and (V R )∗ = −V R . Lemma 4.5.28. Let V ∈ vμ . Then V L and V R are homogeneous differential operators of degree μ, in the sense of Definition 4.5.18. μ

Proof. It suffices to prove the result for V = Xj . We have μ,L

Xj (f (δd x)) = ∫ f (δd (xy−1 ))(𝜕tμ δ0 )(y) dy = ∫ f ((δd x)(δd y)−1 )(𝜕tμ δ0 )(y) dy j

=δ

−Q

d

j

∫ f ((δ x)y )(𝜕tμ δ0 )(δ y) dy = δ −1

−d

j

μ

d

= δ ∫ f (δ (x)y )(𝜕tμ δ0 )(y) dy = δ −1

μ

j

μ,R

A similar computation proves the result for Xj .

μ−Q

d

∫ f ((δ x)y−1 )𝜕tμ (δ0 (δ−d y)) dy

μ,L (Xj f )(δd x).

j

256 � 4 Pseudo-differential operators ̂ , ds) = {(W ̂1 , ds1 ), . . . , (W ̂r , dsr )} ⊂ g × ℕ+ satisfy W ̂j ∈ vds such that W ̂1 , . . . , W ̂r Let (W j ̂ R be the right invariant vector fields associated generate g as a Lie algebra. Let W j ̂j . Fix κ ∈ ℕ+ such that dsj divides κ for every j. We define a right invariant with W sub-Laplacian by

R

r

L := ∑(−1)

κ/dsj

j=1

̂ R )2κ/dsj = ∑((W ̂ R )∗ )κ/dsj (W ̂ R )κ/dsj . (W j j j j=1

(4.71)

Proposition 4.5.29. Suppose 2κ < Q. Then there is a unique distribution K ∈ C0∞ (G)′ which is homogeneous of degree 2κ − Q such that L R K = δ0 . This K(t) agrees with a ∞ Cloc (ℝdim G ) function on {t ≠ 0}. Proof. L R is subelliptic by Theorem 4.5.5 and is therefore hypoelliptic. By (4.71), (L R )∗ = L R , and therefore (L R )∗ is also hypoelliptic. Lemma 4.5.28 implies that L R is homogeneous of degree 2κ. The result follows from Theorem 4.5.21. A similar result holds for the left invariant sub-Laplacian, with the same proof.

4.5.4 Free nilpotent Lie algebras We use a special nilpotent Lie algebra: the free nilpotent Lie algebra of step K ∈ ℕ on r generators, nK ,r . Informally, we take nK ,r to be the nilpotent Lie algebra of step k, ̂1 , . . . , W ̂r , but otherwise has which is generated as a Lie algebra by r formal elements W ̂ ̂r , is the unique1 as few relations as possible. Formally, nK ,r , with generators W1 , . . . , W nilpotent Lie algebra of step K satisfying the following universal property: if g is an̂1 , . . . , W ̂r } → g is any function, other nilpotent Lie algebra of step at most K and ψ : {W then there is a unique Lie algebra homomorphism which makes the following diagram commute:

? nk,r ?? ̂ ̂ {W1 , . . . , Wr }

ψ

? ?g

∃!

It is easy to show that such a Lie algebra exists. Furthermore, this Lie algebra can be ̂j a formal degree dsj ∈ ℕ+ . Then there is given a gradation as follows. Assign to each W ν ̂j ∈ vds . a unique grading nK ,r = ⊕μ=1 vμ , as in Definition 4.5.26, such that W j

1 The universal property determines nK ,r uniquely up to isomorphisms of Lie algebras.

4.5 The sub-Laplacian

�

257

4.5.5 The calculus In this section, we present a weak analog of the calculus of pseudo-differential opera∞ tors.2 Let (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} ⊂ Cloc (M; TM) × ℕ+ be such that Gen((W , ds)) is locally finitely generated. Fix relatively compact, open sets Ω1 ⋐ Ω2 ⋐ Ω3 ⋐ M. Pick a finite set ℱ ⊂ Gen((W , ds)) so that Gen((W , ds)) is finitely generated by ℱ on Ω3 and (W , ds) ⊆ ℱ . Set M0 := max{e : (Z, e) ∈ ℱ }, let K ≥ 2M0 − 1, and let nK ,r be the ̂1 , . . . , W ̂r . We assign to each W ̂j free nilpotent Lie algebra of step K with r generators W the formal degree dsj and give nK ,r the associated grading as described in Section 4.5.4: nK ,r = ⊕νμ=1 vμ . We let NK ,r be the connected, simply connected Lie group whose Lie algebra is nK ,r . ̂1 , . . . , X ̂q , of the form Let q = dim nK ,r and pick a basis of nK ,r , X ̂j = ad(W ̂ j ) ad(W ̂ j ) ⋅ ⋅ ⋅ ad(W ̂ j )W ̂j , X l l l l 1

j

2

Lj−1

Lj

j

for some choice of l1 , . . . , lL ∈ {1, . . . , r}. Here, ad(V1 )V2 = [V1 , V2 ]. Choose this basis in j ̂1 , . . . , W ̂r are elements of the basis. Note that X ̂j ∈ vd where dj := such a way that W j dslj + ⋅ ⋅ ⋅ + dslj . We use this basis to identify NK ,r and nK ,r with ℝq as in Section 4.5.3. 1

Lj

For t, s ∈ ℝq we write t × s for the group multiplication induced by NK ,r , and we write δd t for the dilations described in Section 4.5.3. On M, we define vector fields X1 , . . . , Xq as follows: Xj := ad(Wlj ) ad(Wlj ) ⋅ ⋅ ⋅ ad(Wlj )Wlj . 1

2

Lj−1

Lj

We set (X, d) := {(X1 , d1 ), . . . , (Xq , dq )}, and we set ℱ1 := {(Xj , dj ) : dj ≤ M0 }.

(4.72)

Lemma 4.5.30. Gen((W , ds)) is finitely generated by ℱ1 and by (X, d ) on Ω3 . Also, (W , ds) ⊆ ℱ1 ⊆ (X, d ). Proof. By the above construction, (W , ds) ⊆ ℱ1 ⊆ (X, d) ⊂ Gen((W , ds)). Furthermore, since nK ,r is the free nilpotent Lie algebra of step K with r generators and M0 ≤ K , for every (Z, e) ∈ Gen((W , ds)) with e ≤ M0 , Z can be written as a linear combination, with constant coefficients, of elements of the form Xj where dj = e. In particular, this is true for every element of ℱ . Since Gen((W , ds)) is finitely generated by ℱ on Ω3 , it follows that Gen((W , ds)) is finitely generated by ℱ1 on Ω3 . Since ℱ1 ⊆ (X, d ), the same is true of (X, d ). 2 A much more powerful calculus of pseudo-differential operators is obtained in [2], though such results are not necessary for our purposes.

258 � 4 Pseudo-differential operators ̂ L, W ̂ R and X ̂L , X ̂R be the left and right invariant vector fields corresponding We let W j j j j ̂j and X ̂j , respectively. By Lemma 4.5.28, W ̂ R is homogeneous of degree dsj and X ̂R is to W j

j

homogeneous of degree dj . The main result of this section is the following proposition.

Proposition 4.5.31. If a = a(Ω1 , Ω2 , (X, d )) > 0 is sufficiently small, then the following holds. Suppose b ∈ S s (a, Ω1 , d ) and S is the (X, d ), a pseudo-differential operator of order s supported in Ω1 × Ω2 given by ̌ t, t) dt. Sf (x) = ∫ f (e−t⋅X x)b(x, ̌ u, t) and b̌ (x, u, t) = ̂ L b(x, Let bj,L (x, u, ξ) and bj,R (x, u, ξ) be defined by b̌ j,L (x, u, t) = W j,R j R ̌ L R ̂ ̂ ̂ W b(x, u, t), where W and W are acting in the t variable. Then j

j

j

Wj Sf (x) = Sj,L f (x) + Rj,L f (x),

SWj f (x) = Sj,R f (x) + Rj,R f (x),

where Sj,L f (x) = ∫ f (e−t⋅X x)b̌ j,L (x, t, t) dt,

Sj,R f (x) = ∫ f (e−t⋅X x)b̌ j,R (x, t, t) dt,

Rj,L and Rj,R are (X, d ), a pseudo-differential operators of order s supported in Ω1 × Ω2 , and Sj,L and Sj,R are (X, d ), a pseudo-differential operators of order s + dj supported in Ω1 × Ω2 . The rest of this section is devoted to the proof of Proposition 4.5.31. We prove only the result for SWj as that is the only part we use in our later results and the proof for Wj S is similar. Lemma 4.5.32. Let b(x, u, ξ) ∈ S s (a, Ω2 , d ), for some s ∈ ℝ and a > 0. Then: ∞ q (i) Suppose g(t, x) ∈ Cloc (B (a) × Ω2 ) satisfies degd (g) ≥ e for some e ∈ ℕ (see Definǐ t, t) = b̌ (x, t, t), where b (x, u, ξ) ∈ S s−e (a, Ω , d ). tion 4.2.1). Then g(t, x)b(x, g g 1 ̂R as a differential operator in the t ∈ ℝq ≅ NK ,r (ii) For each j ∈ {1, . . . , q}, treating each X j variable, ̌ t, t) = b̌ (x, t, t), ̂R b(x, X j,1 j

̌ t, t) = b̌ (x, t, t), 𝜕tj b(x, j,2

̌ u, t) + ř (x, u, t) and ̂R b(x, where bj,l (x, u, ξ) ∈ S m+dj (a, Ω1 , d ), b̌ j,1 (x, u, t) = X j,1 j m ̌ u, t) + ř (x, u, t), where r (x, u, ξ) ∈ S (a, Ω , d ). b̌ (x, u, t) = 𝜕 b(x, j,2

tj

j,2

j,l

1

Proof. By Proposition 4.1.14, we may write ̌ u, t) = ∑ 2js Dildj (ς )(x, u, t), b(x, 2 j j∈ℕ

(4.73)

̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) is a bounded set and ςj ∈ C0∞ (Ω1 )⊗ ̂ where {ςj : j ∈ ℕ} ⊂ C0∞ (Ω1 )⊗ ∞ q q ̂ S0 (ℝ ) for j > 0. C0 (B (a))⊗

4.5 The sub-Laplacian

�

259

∞ q For (i), write g(t, x) = ∑Ll=1 t αl gl (t, x), where gl (t, x) ∈ Cloc (B (a) × Ω2 ) and degd (αl ) ≥ e. Then we take L

̌ u, t). b̌ g (x, u, t) := ∑ gl (u, x)t αl b(x, l=1

̌ u, t) satisfies c (x, u, ξ) ∈ Thus, to prove (i), it suffices to show cľ (x, u, t) := gl (u, x)t αl b(x, l s−e S (a, Ω1 , d ). We have, by (4.73), cľ (x, u, t) = ∑ 2js g(u, x)t αl Dild2j (ςj )(x, u, t) = ∑ 2js−degd (αl ) Dild2j (ςj̃ )(x, u, t), j∈ℕ

j∈ℕ

where ςj̃ (x, u, t) := g(u, x)t αl ςj (x, u, t). It follows immediately from the hypotheses on ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) is a bounded set and ςj̃ ∈ C0∞ (Ω1 )⊗ ̂ ςj that {ςj̃ : j ∈ ℕ} ⊂ C0∞ (Ω1 )⊗ ∞ q q s−degd (αl ) ̂ C0 (B (a))⊗S0 (ℝ ) for j > 0. Proposition 4.1.14 implies cl (x, u, ξ) ∈ S (a, Ω1 , d ) ⊆ S s−e (a, Ω1 , d ), completing the proof of (i). ̂R and 𝜕t simultaneously. Indeed, if we let Turning to (ii), we prove the results for X j j ̂R , 𝜕t }, then Vj is a vector field with polynomial coefficients which is homogeneous Vj ∈ {X j

j

̌ t, t) = of degree dj in the sense of Definition 4.5.18 (see Lemma 4.5.28). Note that V b(x, ̌ u, t)), r (x, u, ξ) = V b(x, u, ξ), and V and b̌ V (x, t, t)+rV (x, t, t), where b̌ V (x, u, t) = Vt (b(x, V u t Vu denote V acting in the t and u variable, respectively. It is clear that rV ∈ S m (a, Ω1 , d). Thus, (ii) will follow if we show that bV ∈ S m+dj (a, Ω1 , d). Using (4.73), we have b̌ V (x, u, t) = ∑ 2js Vt Dild2j (ςj )(x, u, t) = ∑ 2js+dj Dild2j (ςj̃ )(x, u, t), j∈ℕ

j∈ℕ

where ςj̃ (x, u, t) := Vt ςj (x, u, t) and we have used the homogeneity of V . Since V is a veĉ C0∞ (Bq (a))⊗ ̂ tor field with polynomial coefficients, it follows that {ςj̃ : j ∈ ℕ} ⊂ C0∞ (Ω1 )⊗ q ∞ ∞ q q ̂ ̂ ̃ S (ℝ ) is a bounded set and ςj ∈ C0 (Ω1 )⊗C0 (B (a))⊗S0 (ℝ ) for j > 0. Proposition 4.1.14 implies bV (x, u, ξ) ∈ S m+dj (a, Ω1 , d), completing the proof. Lemma 4.5.33. If a = a(Ω1 , Ω2 , (X, d )) > 0 is sufficiently small, the following holds. Let S be an (X, d ), a pseudo-differential operator of order s ∈ ℝ supported in Ω1 × Ω2 . Then, for j ∈ {1, . . . , q}, SXj is an (X, d ), a pseudo-differential operator of order s + dj . Proof. This is a straightforward consequence of Theorem 4.3.3 and Lemma 4.3.4 (ii). Indeed, by Theorem 4.3.3, S = ∑k∈ℕ 2ks Ek , where {(Ek , 2−k ) : k ∈ ℕ} is a bounded set of (X, d ), a pseudo-differential operator scales supported in Ω1 × Ω2 . Thus, SXk = ∑k∈ℕ 2k(s+dj ) Ek 2−kdj Xj =: ∑k∈ℕ 2k(s+dj ) Ẽk . By Lemma 4.3.4 (ii), {(Ẽk , 2−k ) : k ∈ ℕ} is a bounded set of (X, d ), a pseudo-differential operator scales supported in Ω1 × Ω2 , and therefore, by another application of Theorem 4.3.3, SXk is an (X, d ), a pseudo-differential operator of order s + dj .

260 � 4 Pseudo-differential operators Lemma 4.5.34. If a = a(Ω1 , Ω2 , (X, d )) > 0 is sufficiently small, the following holds. Let ∞ c ∈ Cloc (Ω3 ) and let S be an (X, d ), a pseudo-differential operator of order s ∈ ℝ supported in Ω1 × Ω2 . Then S Mult[c] is an (X, d ), a pseudo-differential operator of order s supported in Ω1 × Ω2 . ̌ t, t), then S Mult[c]f (x) = ∫ f (e−t⋅X )b̌ (x, t, t), where Proof. If Sf (x) = ∫ f (e−t⋅X x)b(x, 1 −u⋅X b1 (x, u, ξ) = c(e x)b(x, u, ξ). It follows easily from the definitions that if b ∈ S s (a, Ω1 , d ), then b1 ∈ S s (a, Ω1 , d ). This completes the proof. Lemma 4.5.35. If a = a(Ω1 , Ω2 , (X, d )) > 0 is sufficiently small, the following holds. Suppose (Z, e) ∈ Gen((W , ds)) and S is an (X, d ), a pseudo-differential operator of order s ∈ ℝ supported in Ω1 × Ω2 . Then SZ is an (X, d ), a pseudo-differential operator of order s + M0 supported in Ω1 × Ω2 . Proof. By Lemma 4.5.30, Gen((W , ds)) is finitely generated by ℱ1 on Ω3 . Thus, since ℱ1 is given by (4.72), we may write Z = ∑dj ≤M0 cj Xj , so SZ = ∑dj ≤M0 S Mult[cj ]Xj . By Lemma 4.5.34, S Mult[cj ] is an (X, d ), a pseudo-differential operator of order s supported on Ω1 × Ω2 , and therefore by Lemma 4.5.33, S Mult[cj ]Xj is an (X, d ), a pseudo-differential operator of order s+ dj supported on Ω1 ×Ω2 . It follows that SZ = ∑dj ≤M0 S Mult[cj ]Xj is an (X, d ), a pseudo-differential operator of order s + M0 supported on Ω1 × Ω2 , completing the proof. Lemma 4.5.36. If a = a(Ω1 , Ω2 , (X, d )) > 0 is sufficiently small, then for 1 ≤ j ≤ q, f ∈ ∞ Cloc (Ω3 ), ̂R (f (e−t⋅X x)) + (Xj f )(e−t⋅X x) = −X j

q

∑

k (t, x)t α (Xk f )(e−t⋅X x), ∑ hj,α

degd (α)≥K +1−dj k=1

k where the sum ∑degd (α)≥K +1−dj denotes a finite sum of such terms and hj,α (t, x) ∈ ∞ q Cloc (B (a) × Ω2 ).

Proof. By Proposition 4.2.2 (iii) we know that q

̂R f (e−t⋅X x) = ∑ pk (t, x)(Xk f )(e−t⋅X x), X j j k=1

∞ q for some pkj (t, x) ∈ Cloc (B (a) × Ω2 ). Thus, the result will follow if we can show that

̂R f (e−t⋅X x) − (Xj f )(e−t⋅X x) = O(|t|K ). X j

(4.74)

We have, by the Baker–Campbell–Hausdorff formula and the definition of multiplication on ℝq ≅ NK ,r ,

4.5 The sub-Laplacian

�

261

󵄨 󵄨 (Xj f )(e−t⋅X x) = 𝜕sj 󵄨󵄨󵄨s=0 f (es⋅X e−t⋅X x) = 𝜕sj 󵄨󵄨󵄨s=0 (e−t⋅X es⋅X f )(x) 󵄨 = 𝜕sj 󵄨󵄨󵄨s=0 (e((−s)×t)

−1

⋅X

󵄨 f )(x) + 𝜕sj 󵄨󵄨󵄨s=0 ((e−t⋅X es⋅X f )(x) − (e((−s)×t)

󵄨 󵄨 = 𝜕sj 󵄨󵄨󵄨s=0 (e−((−s)×t)⋅X f )(x) + 𝜕sj 󵄨󵄨󵄨s=0 O((‖t‖ + ‖s‖) =

̂R e−t⋅X f (x) −X j

+ O(‖t‖

K +1−dj

)=

K +1

̂R f (e−t⋅X x) −X j

−1

⋅X

f )(x))

)

+ O(‖t‖K +1−dj ),

where we have used the fact that s−1 = −s in the group NK ,r ≅ ℝq . This establishes (4.74) and completes the proof. Lemma 4.5.37. If a = a(Ω1 , Ω2 , (X, d )) > 0 is sufficiently small, the following holds. Suppose b(x, u, ξ) ∈ S s (a, Ω1 , d ) and S is the (X, d ), a pseudo-differential operator of order s supported in Ω1 × Ω2 given by ̌ t, t) dt. Sf (x) = ∫ f (e−t⋅X x)b(x, Then, for (Xj , dj ) ∈ ℱ1 , SXj f (x) = ∫ f (e−t⋅X x)b̌ j (x, t, t) dt + Rj f (x), ̌ u, t) (where X ̂R b(x, ̂R acts in the t variable) and R is an (X, d ), a pseudowhere b̌ j (x, u, t) = X j j differential operator of order s supported in Ω1 × Ω2 . Finally, bj ∈ S s+dj (a, Ω, d ). Proof. Lemma 4.5.36 shows that ̌ t, t) dt SXj f (x) = ∫(Xj f )(e−t⋅X x)b(x, q

̌ t, t) dt + ∑ ∫(X f )(e−t⋅X x)g k (t, x)b(x, ̌ t, t) dt ̂R f (e−t⋅X x))b(x, = − ∫(X k j j k=1

= (I) + (II), k k ∞ q where gjk (t, x) is a finite sum of the form ∑degd (α)≥K +1−dj hj,α (t, x)t α and hj,α ∈ Cloc (B (a)×

Ω2 ). Note that degd (gjk ) ≥ K + 1 − dj . ̌ t, t) = b̌ (x, t, t), where By Lemma 4.5.32 (i), gjk (t, x)b(x, j,k

bj,k (x, u, ξ) ∈ S s−K −1+dj (a, Ω1 , d ). Thus, q

(II) = ∑ Sk Xk f (x), k=1

where each Sk is an (X, d ), a pseudo-differential operator of order s− K −1+ dj supported in Ω1 ×Ω2 . Lemma 4.5.35 shows that each Sk Xk is an (X, d ), a pseudo-differential operator

262 � 4 Pseudo-differential operators of order s − K − 1 + dj + M0 ≤ s supported in Ω1 × Ω2 ; here we have used dj ≤ M0 since (Xj , dj ) ∈ ℱ1 and we have used the fact that K was chosen to be ≥ 2M0 −1. This shows that (II) = Rj,1 f (x), where Rj,1 is an (X, d ), a pseudo-differential operator of order s supported in Ω1 × Ω2 . For (I), integrating by parts shows that ̌ t, t) dt. ̂R b(x, (I) = ∫ f (e−t⋅X x)X j ̌ t, t) = b̌ (x, t, t) + ř (x, t, t), where r ∈ S s (a, Ω , d ). ̂R b(x, Lemma 4.5.32 (ii) shows that X j j,2 j,2 1 j This proves that (I) is of the desired form. Finally, it follows from Lemma 4.5.32 (ii) that bj ∈ S s+dj (a, Ω, d ). Proof of Proposition 4.5.31. The result for SWj follows immediately from Lemma 4.5.37, as (Wj , dsj ) ∈ ℱ1 (see Lemma 4.5.30). The proof for Wj S is similar, and we leave it to the reader. 4.5.6 The parametrix In this section, we complete the proof of Theorem 4.5.1; we take all the same assumptions and notation as in that theorem. In particular, we have vector fields (W , ds) = ∞ {(W1 , ds1 ), . . . , (Wr , dsr )} ⊂ Cloc (M; TM) × ℕ+ such that Gen((W , ds)) is locally finitely generated, and we are given κ ∈ ℕ+ such that dsj divides κ for every j. ̂ R be We take nK ,r as in Section 4.5.5, where we pick K so that K ≥ 2κ + 1. We let W j as in Section 4.5.5, and we define L R by (4.71). We define (X, d ) = {(X1 , d1 ), . . . , (Xq , dq )} as in Section 4.5.5. In particular, NK ,r ≅ ℝq . With Q the homogeneous dimension of NK ,r ≅ ℝq , we have Q ≥ K > 2κ. Thus, Proposition 4.5.29 shows that there is a unique distribution K ∈ C0∞ (ℝq )′ which is homo∞ geneous of degree 2κ − Q such that L R K = δ0 . This K(t) agrees with a Cloc (ℝq ) function on {t ≠ 0}. Take a = a(Ω1 , Ω2 , (X, d )) > 0 small and η ∈ C0∞ (Bq (a)) with η = 1 on a neighborhood of 0 ∈ ℝq . For ψ ∈ C0∞ (Ω1 ) we set Tψ f (x) := ∫ f (e−t⋅X x)ψ(x)η(t)K(t) dt. Remark 4.5.23 shows that Tψ is an (X, d ), a pseudo-differential operator of order −2κ supported in Ω1 ×Ω2 . We will show that this Tψ satisfies the conclusions of Theorem 4.5.1. Note that we may take a > 0 as small as we like in the definition of Tψ . In what follows, we take all the same notation as in Section 4.5.5. Lemma 4.5.38. If a = a(Ω1 , Ω2 , (X, d )) > 0 is sufficiently small, the following holds. Suppose b(x, u, ξ) ∈ S s (a, Ω1 , d ) and let S be the (X, d ), a pseudo-differential operator of order s supported in Ω1 × Ω2 given by

4.5 The sub-Laplacian

�

263

̌ t, t) dt. Sf (x) = ∫ f (e−t⋅X x)b(x, Then, for every l ∈ ℕ, j ∈ {1, . . . , r}, SWjl f (x) = Sj,l f (x) + Rj,l f (x), where Rj,l is an (X, d ), a pseudo-differential operator of order s + dsj (l − 1) supported in Ω1 × Ω2 and Sj,l is the (X, d ), a pseudo-differential operator of order s + dsj supported in Ω1 × Ω2 given by Sj,l f (x) = ∫ f (et⋅X x)b̌ j,l (x, t, t) dt, ̌ u, t), where W ̂ R )l b(x, ̂ R is where bj,l (x, u, ξ) ∈ S s+dsj (a, Ω1 , d ) is defined by b̌ j,l (x, u, t) = (W j j q acting in the t ∈ ℝ ≅ NK ,r variable. Proof. We proceed by induction on l. The base case, l = 0, is trivial (one can take Rj,0 = 0 and Sj,0 = S). We suppose we have the result for l and prove it for l replaced by l + 1. By the inductive hypothesis, we have SWjl f (x) = Sj,l Wj f (x) + Rj,l Wj f (x), where Sj,l and Rj,l are as in the statement of the lemma. Proposition 4.5.31 applied to Sj,l Wj shows that Sj,l Wj is of the desired form. Lemma 4.5.33 (and the fact that (Wj , dsj ) ∈ (X, d )) applied to Rj,l Wj shows that Rj,l Wj is an (X, d ), a pseudo-differential operator of order s + dsj (l − 1) + dsj = s + dsj l supported in Ω1 × Ω2 and therefore of the desired form. This completes the proof. Proof of Theorem 4.5.1. A simple computation shows that r

L = ∑(−1) j=1

κ/dsj

2κ/dsj

Wj

r 2κ/dsj −1

+ ∑ ∑ cj,l Wjl , j=1

l=κ

∞ where cj,k ∈ Cloc (M). Applying Lemma 4.5.34 once and Lemma 4.5.33 l times shows that

Tψ cj,l Wjl is an (X, d ), a pseudo-differential operator of order −2κ + dsj l supported in Ω1 × Ω2 . Thus, r 2κ/dsj −1

Tψ ∑ ∑ cj,l Wjl j=1

l=κ

is an (X, d ), a pseudo-differential operator of order −1 supported in Ω1 × Ω2 .

264 � 4 Pseudo-differential operators Lemma 4.5.38 shows that r

2κ/dsj

Tψ ∑(−1)κ/dsj Wj j=1

= S + R,

where R is an (X, d ), a pseudo-differential operator of order −1 supported on Ω1 × Ω2 , and Sf (x) = ∫ f (e−t⋅X x)ψ(x)η(t)(L R K(t)) dt = ∫ f (e−t⋅X x)ψ(x)η(t)δ0 (t) dt = ψ(x)f (x). Combining the above, we conclude Tψ L ≡ Mult[ψ], modulo (X, d ), a pseudodifferential operators of order −1 supported on Ω1 × Ω2 , completing the proof.

4.5.7 Limitations of the lifting procedure The proof of Theorem 4.5.1, sometimes called the Rothschild–Stein lifting3 procedure, is a powerful tool in the study of parametrices for partial differential operators based on vector fields. When it works, it provides detailed information which can be difficult or impossible to obtain by other methods. For example, in Theorem 4.5.1 we did not even require the underlying vector fields to satisfy Hörmander’s condition. Unfortunately, it is not possible to obtain a parametrix for every maximally subelliptic operator in this way: there are maximally subelliptic operators which have no corresponding lifted maximally subelliptic operator on a nilpotent Lie group. Since the goal of this text is to study maximally subelliptic operators in full generality, we are forced to move beyond the Rothschild–Stein lifting procedure; this is taken up in Chapter 8. However, Theorem 4.5.1 is a key tool in some of our methods for proving more general results. Thus, there are several situations we consider where the Rothschild–Stein lifting procedure is not directly applicable, but still plays an important role in the proof. To see an example of an operator which is maximally subelliptic but does not have a corresponding lifted maximally subelliptic operator, we consider operators of the following form, where W1 , W2 are Hörmander vector fields and λ ∈ ℂ: 2

2

Lλ = W1 + W2 + iλ[W1 , W2 ].

3 Unlike the proof due to Rothschild and Stein [200], we did not lift the operator to a nilpotent Lie group. Instead, following Goodman [99], we took a fundamental solution on a nilpotent Lie group and pushed it forward to a lower-dimensional space to obtain a parametrix. Thus, with this perspective, “lifting” is not the most appropriate word to use.

4.6 Further reading and references �

265

It is a result of Rothschild and Stein [200] that when |λ| < 1, Lλ is maximally subelliptic of degree 2 with respect to (W , 1) = {(W1 , 1), (W2 , 1)}. However, the situation for |λ| ≥ 1 is more subtle. When W1 and W2 are the left invariant vector fields on the Heisenberg group described in Example 1.1.2 (iii), then it is a result of Folland and Stein [91] that Lλ is maximally subelliptic for all but a discrete set of λ ∈ ℂ; see also the presentation in [216, Chapter XIII]. On the other hand, Helffer [119] (building on work of Rothschild and Stein [200]) showed that ℝ2 and the Heisenberg group are the only stratified Lie groups with this property: if W1 and W2 are generating left invariant vector fields on a stratified nilpotent Lie group of step higher than 2, then Lλ is only hypoelliptic for |λ| < 1. Unlike the special case of stratified nilpotent Lie groups, there are choices of Hörmander vector fields which satisfy Hörmander’s condition of arbitrarily high order, where Lλ is maximally subelliptic for all but a discrete set of λ. Indeed, on ℝ2 we take W1 = 𝜕x and W2 = x k 𝜕y , where k ∈ ℕ is fixed, so that W1 and W2 satisfy Hörmander’s condition of order k + 1 on ℝ2 . Then, except for a discrete set of λ ∈ ℂ, Lλ is maximally subelliptic of degree 2 with respect to (W , 1) on ℝ2 . When k is odd this follows from methods of Gilioli and Trèves [97] and when k is even this follows from methods of Menikoff [170]. Thus, when λ ∈ ℂ is outside a discrete set, Lλ is maximally subelliptic of degree 2 with respect to (W , 1), but when |λ| ≥ 1 and k ≥ 2 it cannot be lifted to a maximally subelliptic operator on a nilpotent Lie group. Hence, the Rothschild–Stein lifting procedure destroys the maximally subellipticity of Lλ in this case. A recent, elegant pseudo-differential calculus of Androulidakis, Moshen, and Yuncken [2] salvages many aspects of the Rothschild–Stein lifting procedure to the general setting. See Sections 4.6 and 8.8.

4.6 Further reading and references In the single-parameter (ν = 1) case, (X, d ) pseudo-differential operators were first studied by Folland and Stein [91] and Rothschild and Stein [200]. In particular, Rothschild and Stein used them to obtain results similar to Theorem 4.5.1. Since these original works, similar operators have been used in many different contexts. For example, they were clarified by Goodman [99] and used by Rothschild [198], and a related theory with a calculus was developed by Christ, Geller, Głowacki, and Polin [49]. In the multi-parameter setting (ν > 1), similar operators were used by the author in [220] and [221]. When κ = 1 and dsj = 1 for every j, the sub-Laplacian (4.39) was originally studied by Hörmander [125]; it is sometimes called Hörmander’s sub-Laplacian. In this context, Hörmander proved Theorem 4.5.5. Kohn [142] gave a simpler proof of Hörmander’s result, and it is this simpler proof that we use to prove Theorem 4.5.5. As mentioned earlier, the choice of ϵ0 in Theorem 4.5.5 is not optimal. When κ = 2 and dsj = 1 for every j, it is a result of Rothschild and Stein [200] that one can take ϵ0 = 2/m, where W1 , . . . , Wr satisfy Hörmander’s condition of order m on Ω1 , and this is optimal. See Corollary 8.2.5

266 � 4 Pseudo-differential operators for a further discussion of the optimal ϵ0 . For general κ and dsj , we presented the theory for L given by (4.39) using the same methods that were originally developed for κ = 1, dsj = 1. One of the early settings where this more general operator was explicitly used was in the work of Helffer and Nourrigat [121]. In the single-parameter setting, Androulidakis, Moshen, and Yuncken [2] recently introduced a pseudo-differential calculus which contains the parametrices for maximally subelliptic operators, based on the ideas of [3]. This is a powerful and general approach, which yields parametrices for all maximally subelliptic operators; however, it does not seem to directly apply to the multi-parameter theory we use in this text or to directly imply regularity theory for maximally subelliptic operators outside the setting of adapted L2 Sobolev spaces. It is likely that these pseudo-differential operators could be combined with some methods of this text to give a pseudo-differential approach to some of the regularity theory of linear maximally subelliptic operators, though we do not pursue that here.

5 Singular integrals Parametrices for elliptic partial differential operators are often constructed using the Fourier transform; see Proposition 2.2.27. Unfortunately, the Fourier transform is not a decisive tool when studying maximally subelliptic partial differential operators. Instead, we construct parametrices for maximally subelliptic operators by using a filtered algebra of singular integral operators. We present singular integral operators in three different settings, each of which is useful in a different way when studying maximally subelliptic operators. Throughout this chapter, M is a connected C ∞ manifold, endowed with a smooth, strictly positive density Vol. Fix Ω ⋐ M a relatively compact, open set.

5.1 The three settings In this section, we describe the three settings under which we introduce singular integral operators. We call these the single-parameter setting, the multi-parameter Hörmander setting, and the general multi-parameter setting.

5.1.1 The single-parameter setting In this setting, we are given Hörmander vector fields with formal degrees (W , ds) = ∞ {(W1 , ds1 ), . . . , (Wr , dsr )} ⊂ Cloc (M; TM) × ℕ+ . For t ∈ ℝ, we will define a filtered alget bra A (Ω, (W , ds)) consisting of operators in Hom(C0∞ (M), C0∞ (M)′ ), whose Schwartz kernels are supported in Ω × Ω. These operators have the following properties: – A t (Ω, (W , ds)) is a filtered algebra: for T ∈ A t (Ω, (W , ds)) and S ∈ A s (Ω, (W , ds)), we have TS ∈ A t+s (Ω, (W , ds)). See Proposition 5.8.6. – If t ≤ s, then A t (Ω, (W , ds)) ⊆ A s (Ω, (W , ds)). See Proposition 5.8.2. – Operators in A 0 (Ω, (W , ds)) extend to bounded operators on Lp (M, Vol). See Proposition 5.10.1. – More generally, operators in A t (Ω, (W , ds)) have good boundedness properties on function spaces adapted to (W , ds). See Theorem 6.2.10. – Left parametrices for maximally subelliptic operators of degree κ are “locally” in A −κ (⋅, (W , ds)). See Theorem 8.1.1 (vii). – If ψ ∈ C0∞ (Ω), then Mult[ψ] ∈ A 0 (Ω, (W , ds)) and Mult[ψ]Wj ∈ A dsj (Ω, (W , ds)). – More generally, if ψ ∈ C0∞ (Ω) and α is an ordered multi-index, then Mult[ψ]W α ∈ A degds (α) (Ω, (W , ds)). See Proposition 5.8.3. – A t (Ω, (W , ds)) depends on (W , ds) only though the weak equivalence class of (W , ds). See Proposition 5.8.12. https://doi.org/10.1515/9783111085647-005

268 � 5 Singular integrals In fact, we will give four definitions of the filtered algebra A t (Ω, (W , ds)). We will define sets A1t (Ω, (W , ds)), A2t (Ω, (W , ds)), A4t (Ω, (W , ds)), and for K ∈ ℕ, A3t (Ω, (W , ds), K). A main theorem (Theorem 5.2.12) is that these four definitions all give the same set: for all K ∈ ℕ, t

t

t

t

A1 (Ω, (W , ds)) = A2 (Ω, (W , ds)) = A3 (Ω, (W , ds), K) = A4 (Ω, (W , ds)).

We let A t (Ω, (W , ds)) be the common value. These single-parameter singular integrals are one of our main tools when proving properties of both linear and nonlinear maximally subelliptic PDEs in terms of function spaces adapted to (W , ds). 5.1.2 The multi-parameter Hörmander setting In this setting, we are given ν ∈ ℕ+ and for each μ ∈ {1, . . . , ν} Hörmander vector fields μ μ μ μ ∞ with formal degrees (W μ , dsμ ) = {(W1 , ds1 ), . . . , (Wrμ , dsrμ )} ⊂ Cloc (M; TM)×ℕ+ . We make the following key assumption. Key assumption: We assume (W 1 , ds1 ), . . . , (W ν , dsν ) pairwise locally weakly approximately commute. Set (W , ds)⃗ = {(W1 , ds1⃗ ), . . . , (Wr , dsr⃗ )} := (W 1 , ds1 ) ⊠ (W 2 , ds2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (W ν , dsν ) μ

μ

∞ = {(Wj , dsj eμ ) : μ ∈ {1, . . . , ν}, 1 ≤ j ≤ rμ } ⊂ Cloc (M; TM) × (ℕν \ {0}).

(5.1)

⃗ consisting of operators in For t ∈ ℝν , we will define a filtered algebra A t (Ω, (W , ds)) ∞ ∞ ′ Hom(C0 (M), C0 (M) ), whose Schwartz kernels are supported in Ω×Ω. These operators have the following properties: ⃗ coincides with the single-parameter algebra of the same – When ν = 1, A t (Ω, (W , ds)) – – – – – – – –

name in Section 5.1.1. ⃗ is a filtered algebra: for T ∈ A t (Ω, (W , ds)) ⃗ and S ∈ A s (Ω, (W , ds)), ⃗ we A t (Ω, (W , ds)) t+s ⃗ See Proposition 5.8.6. have TS ∈ A (Ω, (W , ds)). ⃗ ⊆ A s (Ω, (W , ds)). ⃗ See Proposition 5.8.2. If t ≤ s (coordinatewise), then A t (Ω, (W , ds)) 0 ⃗ Operators in A (Ω, (W , ds)) extend to bounded operators on Lp (M, Vol). See Theorem 6.2.10 and Proposition 6.2.13. ⃗ have good boundedness properties on More generally, operators in A t (Ω, (W , ds)) ⃗ See Theorem 6.2.10. function spaces adapted to (W , ds). ⃗ are pseudo-local, i. e., their Schwartz kernels are smooth Operators in A t (Ω, (W , ds)) off the diagonal. See Theorem 5.8.18. ⃗ Left parametrices for maximally subelliptic operators are locally in A (⋅, (W , ds)). ∞ 0 ds⃗ j ⃗ ⃗ If ψ ∈ C (Ω), then Mult[ψ] ∈ A (Ω, (W , ds)) and Mult[ψ]Wj ∈ A (Ω, (W , ds)). 0

More generally, if ψ ∈ C0∞ (Ω) and α is an ordered multi-index, then Mult[ψ]W α ∈ ⃗ See Proposition 5.8.3. A degds⃗ (α) (Ω, (W , ds)).

5.1 The three settings

–

�

269

⃗ depends on (W μ , dsμ ) only though the weak equivalence class of A t (Ω, (W , ds)) μ μ (W , ds ). See Proposition 5.8.12.

Example 5.1.1. A main example to consider is the following. Suppose (W 1 , ds1 ) are Hörmander vector fields with formal degrees. Suppose (W 2 , ds2 ) = {(W12 , 1), . . . , (Wr22 , 1)} are such that W12 (x), . . . , Wr22 (x) span Tx M, ∀x ∈ M, i. e., they satisfy Hörmander’s condi-

tion of order 1. In this case, (W 1 , ds1 ) and (W 2 , ds2 ) locally weakly approximately commute – in fact, they locally strongly approximately commute; see Lemma 3.8.3. We will see that parametrices for partial differential operators which are maximally subelliptic ⃗ (see Section 5.11.2) of degree κ with respect to (W 1 , ds1 ) are locally in A (−κ,0) (⋅, (W , ds)) ⃗ and standard pseudo-differential operators of order m are locally in A (0,m) (⋅, (W , ds))

(see Section 5.8.2). Thus, we have a pseudo-local algebra of singular integral operators containing both the parametrices for maximally subelliptic operators and the standard pseudo-differential operators. ⃗ we generalize two of the definitions from the singleTo define A t (Ω, (W , ds)), ⃗ and parameter setting in Section 5.1.1. Thus, we define two sets A1t (Ω, (W , ds)) t ⃗ A2 (Ω, (W , ds)). A main theorem (see Theorem 5.2.18) is that these two sets are equal, ⃗ and we denote their common value by A t (Ω, (W , ds)). These pseudo-local, multi-parameter singular integrals generalize the singular integrals from Section 5.1.1 to a multi-parameter setting. This will allow us to derive certain regularity properties of maximally subelliptic operators with respect to function spaces which are not adapted to the same vector fields, for example in the setting described in Example 3.5.5. Unfortunately, the setting described in this section is not quantitatively invariant under rescalings (see Example 3.5.5). Because of this, the singular integrals operators described here are not general enough to prove the results we require for nonlinear maximally subelliptic PDEs. For this, we introduce a more general setting for our singular integrals, described in Section 5.1.3.

5.1.3 The general multi-parameter setting In this setting, we are given ν ∈ ℕ+ and for each μ ∈ {1, . . . , ν} vector fields with singleμ μ μ μ ∞ parameter formal degrees (W μ , dsμ ) = {(W1 , ds1 ), . . . , (Wrμ , dsrμ )} ⊂ Cloc (M; TM) × ℕ+ . We make the following key assumption. Key assumption: We assume (W 1 , ds1 ), . . . , (W ν , dsν ) pairwise locally weakly approximately commute and for each μ ∈ {1, . . . , ν}, Gen((W μ , dsμ )) is locally finitely generated on M. Remark 5.1.2. By Proposition 3.4.14 (in the case ν = 1 applied to each (W μ , dsμ )), the setting in Section 5.1.2 is a special case of the setting described here.

270 � 5 Singular integrals As in (5.1), set (W , ds)⃗ = {(W1 , ds1⃗ ), . . . , (Wr , dsr⃗ )} := (W 1 , ds1 ) ⊠ (W 2 , ds2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (W ν , dsν ) μ

μ

∞ = {(Wj , dsj eμ ) : μ ∈ {1, . . . , ν}, 1 ≤ j ≤ rμ } ⊂ Cloc (M; TM) × (ℕν \ {0}).

̃t (Ω, (W , ds)) ⃗ consisting of operators For t ∈ ℝν , we will define a filtered algebra A whose Schwartz kernels are supported in Ω × Ω. These operators have the following properties: ̃t (Ω, (W , ds)) ̃t (Ω, (W , ds)) ̃s (Ω, (W , ds)), ⃗ is a filtered algebra: for T ∈ A ⃗ and S ∈ A ⃗ we – A t+s ̃ ⃗ have TS ∈ A (Ω, (W , ds)). See Proposition 5.8.6. ̃t (Ω, (W , ds)) ̃s (Ω, (W , ds)). ⃗ ⊆A ⃗ See Proposition 5.8.2. – If t ≤ s (coordinatewise), then A 0 ̃ (Ω, (W , ds)) ⃗ extend to bounded operators on Lp (M, Vol). See Theo– Operators in A rem 6.2.10 and Proposition 6.2.13. ̃t (Ω, (W , ds)) ⃗ have good boundedness properties on – More generally, operators in A ⃗ See Theorem 6.2.10. function spaces adapted to (W , ds). ̃(⋅, (W , ds)). ⃗ – Left parametrices for maximally subelliptic operators are locally in A ⃗j ∞ 0 d s ̃ (Ω, (W , ds)) ̃ (Ω, (W , ds)). ⃗ and Mult[ψ]Wj ∈ A ⃗ – If ψ ∈ C (Ω), then Mult[ψ] ∈ A – – –

0

More generally, if ψ ∈ C0∞ (Ω) and α is an ordered multi-index, then Mult[ψ]W α ∈ ̃degds⃗ (α) (Ω, (W , ds)). See Proposition 5.8.3. A ⃗ ⊆ If (W , ds)⃗ satisfies the stronger hypotheses of Section 5.1.2, then A t (Ω, (W , ds)) ̃t (Ω, (W , ds)). ⃗ A ̃t (Ω, (W , ds)) ⃗ depends on (W μ , dsμ ) only though the weak equivalence class of A (W μ , dsμ ). See Proposition 5.8.4.

̃t (Ω, (W , ds)) ⃗ we modify the definition from the multi-parameter HörmanTo define A ̃t (Ω, (W , ds)) ⃗ and der setting described in Section 5.1.2. Thus, we define two sets A 1 t ̃ (Ω, (W , ds)). ⃗ A main theorem (see Theorem 5.2.30) is that these two sets are equal, A 2 ̃t (Ω, (W , ds)). ⃗ and we denote their common value by A

These generalized multi-parameter singular integrals are one of our main tools when proving regularity properties of nonlinear maximally subelliptic PDEs with respect to function spaces which are not adapted to the same vector fields (for example, when proving regularity properties of nonlinear maximally subelliptic PDEs with respect to the classical Zygmund–Hölder spaces).

5.2 The algebras of singular integrals In this section, we define the filtered algebras of singular integrals in the three settings described in Section 5.1. The definitions in this section are somewhat complicated, and the reader may wish to understand them first in the elliptic setting; this is contained in Section 2.3.

5.2 The algebras of singular integrals �

271

In what follows, using the Schwartz kernel theorem, we identify Hom(C0∞ (M), ≅ C0∞ (M × M)′ . Furthermore, we abuse notation and for a distribution T(x, y) ∈ C0∞ (M × M), we write the pairing between T(x, y) and f (x, y) ∈ C0∞ (M × M) as integration: C0∞ (M)′ )

∫ T(x, y)f (x, y) d Vol(x) d Vol(y). 󵄨 If, for some open set U ⊆ M × M, T 󵄨󵄨󵄨U is given by integration against an L1loc (U, Vol ⊗ Vol) function, then we identify T with this function on that open set. Thus, given an operator T ∈ Hom(C0∞ (M), C0∞ (M)′ ), we treat T as a distribution T(x, y), and conversely. In particular, given F(x, y) ∈ L1loc (M × M, Vol ⊗ Vol), we identify F with a distribution and therefore with an operator. In this way, we may treat functions of two variables as operators. For an operator T ∈ Hom(C0∞ (M), C0∞ (M)′ ), we write T ∗ ∈ Hom(C0∞ (M), C0∞ (M)′ ) for the formal L2 (M, Vol) adjoint of T, which satisfies ∫ f (x)(Tg)(x) d Vol(x) = ∫(T ∗ f )(x)g(x) d Vol(x),

∀f , g ∈ C0∞ (M).

5.2.1 The single-parameter setting We take the setting described in Section 5.1.1; thus, we are given Hörmander vector fields with formal degrees (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )}. As described in Section 5.1.1, we will introduce four sets of operators, which all turn out to be the same set: A1t (Ω, (W , ds)), A2t (Ω, (W , ds)), A4t (Ω, (W , ds)), and for K ∈ ℕ, A3t (Ω, (W , ds), K). The definitions of A3t and A4t are the most similar to definitions which are usually seen in the literature, and we begin with those. For j ∈ ℝ, we write 2−jdsW for {2−jds1 W1 , . . . , 2−jdsr Wr }. Definition 5.2.1. We say a set ℬ ⊂ C0∞ (M) × M × (0, 1] is a bounded set of (W , ds) bump functions in Ω if: – ∀(ϕ, x, 2−j ) ∈ ℬ, we have x ∈ Ω. – ∀(ϕ, x, 2−j ) ∈ ℬ, we have supp(ϕ) ⊆ B(W ,ds) (x, 2−j ) ∩ Ω. – For all ordered multi-indices α, there exists Cα ≥ 0 such that ∀(ϕ, x, 2−j ) ∈ ℬ, −1 󵄨 󵄨 sup 󵄨󵄨󵄨(2−jdsW )α ϕ(z)󵄨󵄨󵄨 ≤ Cα (Vol(B(W ,ds) (x, 2−j )) ∧ 1) .

z∈M

Definition 5.2.2. For t ∈ ℝ, K ≥ 0, we let A3t (Ω, (W , ds), K) ⊆ Hom(C0∞ (M), C0∞ (M)′ ) be the set of all T ∈ Hom(C0∞ (M), C0∞ (M)′ ) such that the following hold: – supp(T) ⊆ Ω × Ω. ∞ – (Growth condition) The Schwartz kernel of T agrees with a Cloc function on {(x, z) ∈ M × M : x ≠ z} and satisfies, for all ordered multi-indices α and β,

272 � 5 Singular integrals ρ(W ,ds) (x, z)−((t+degds (α)+degds (β))∨K) 󵄨󵄨 α β 󵄨 , 󵄨󵄨Wx Wz T(x, z)󵄨󵄨󵄨 ≤ Cα,β Vol(B (x, ρ (x, z))) ∧ 1 (W ,ds)

–

(W ,ds)

where Wx denotes the list of vector fields W1 , . . . , Wr acting as partial differential operators in the x variable, and similarly for Wz . (Cancelation condition) For every bounded set of (W , ds) bump functions in Ω, ℬ ⊂ C0∞ (M) × M × (0, 1], and each ordered multi-index α, there exists Cℬ,α ≥ 0 with sup −j

(ϕ,z,2 )∈ℬ x∈M

󵄨 󵄨 2−j((t+degds (α))∨K) (Vol(B(W ,ds) (x, 2−j )) ∧ 1)󵄨󵄨󵄨W α Tϕ(x)󵄨󵄨󵄨 ≤ Cℬ,α

with the same estimate for T ∗ in place of T. Remark 5.2.3. A priori, A3t (Ω, (W , ds), K) ⊆ A3t (Ω, (W , ds), K + 1). As a consequence of Theorem 5.2.12, we will see that A3t (Ω, (W , ds), K) does not depend on K. The estimates given in the growth condition of A3t (Ω, (W , ds), K) are only sharp when t + degds(α) + degds(β) ≥ K: the definition actually implies better estimates than are assumed when t +degds(α)+degds(β) < K. A similar remark holds for the cancelation condition. The next definition makes precise the estimates for lower-order derivatives. For the remainder of this chapter, fix Q1 = Q1 (Ω) ≥ n as in Corollary 3.3.9. Definition 5.2.4. For t ∈ ℝ, we let A4t (Ω, (W , ds)) ⊆ Hom(C0∞ (M), C0∞ (M)′ ) be the set of all T ∈ Hom(C0∞ (M), C0∞ (M)′ ) such that the following hold: – supp(T) ⊆ Ω × Ω. ∞ – (Growth condition) The Schwartz kernel of T agrees with a Cloc function on {(x, z) ∈ M × M : x ≠ z} and satisfies, for all ordered multi-indices α and β and all M ≥ 0 such that t + degds(α) + degds(β) + M > −Q1 , ρ(W ,ds) (x, z) 󵄨󵄨 α β 󵄨 󵄨󵄨Wx Wz T(x, z)󵄨󵄨󵄨 ≤ Cα,β Vol(B (x, ρ

−(t+degds (α)+degds (β)+M)

(W ,ds)

–

(W ,ds) (x, z)))

∧1

.

(Cancelation condition) For every bounded set of (W , ds) bump functions in Ω, ℬ ⊂ C0∞ (M) × M × (0, 1], each ordered multi-index α, and each M ≥ 0 such that t + degds(α) + M > −Q1 , there exists Cℬ,α,M ≥ 0 with sup −j

(ϕ,z,2 )∈ℬ x∈M

󵄨 󵄨 2−j(t+degds (α)+M) (Vol(B(W ,ds) (x, 2−j )) ∧ 1)󵄨󵄨󵄨W α Tϕ(x)󵄨󵄨󵄨 ≤ Cℬ,α,M

with the same estimate for T ∗ in place of T. Remark 5.2.5. In the definition of A4t (Ω, (W , ds)), because Ω is relatively compact, the conditions become stronger the smaller M is. For example, in the growth condition, if t + degds(α) + degds(β) > −Q1 , then one may simply take M = 0. In particular, if t > −Q1 ,

5.2 The algebras of singular integrals

� 273

one does not need the parameter M at all. A similar remark holds for the cancelation condition. For the definitions of A1t and A2t , we require some additional definitions. Definition 5.2.6. We say ℰ ⊂ C0∞ (Ω×Ω)×(0, 1] is a bounded set of (W , ds) pre-elementary operators supported in Ω if the following hold: – ⋃(E,2−j )∈ℰ supp(E) ⋐ Ω × Ω. – For all ordered multi-indices α, β, for all m ∈ ℕ, there exists C = C(ℰ , m, α, β) ≥ 0 such that for all (E, 2−j ) ∈ ℰ , (1 + 2j ρ(W ,ds) (x, z))−m α −jds β 󵄨󵄨 −jds 󵄨 . 󵄨󵄨(2 Wx ) (2 Wz ) E(x, z)󵄨󵄨󵄨 ≤ C Vol(B(W ,ds) (x, 2−j + ρ(W ,ds) (x, z))) ∧ 1 Remark 5.2.7. Definition 5.2.6 uses the quantity (1 + 2j ρ(W ,ds) (x, z))−m

Vol(B(W ,ds) (x, 2−j + ρ(W ,ds) (x, z))) ∧ 1

.

It is important for what follows that this is essentially symmetric in x and z. Namely, for x, z ∈ Ω, (1 + 2j ρ(W ,ds) (x, z))−m

Vol(B(W ,ds) (x, 2−j + ρ(W ,ds) (x, z))) ∧ 1

≈

(1 + 2j ρ(W ,ds) (z, x))−m

Vol(B(W ,ds) (z, 2−j + ρ(W ,ds) (z, x))) ∧ 1

.

See Proposition 5.4.3. Definition 5.2.8. We define the set of bounded sets of (W , ds) elementary operators supported in Ω, GΩ , to be the largest set of subsets of C0∞ (Ω × Ω) × (0, 1] such that for all ℰ ∈ GΩ : – ℰ is a bounded set of (W , ds) pre-elementary operators supported in Ω. – ∀(E, 2−j ) ∈ ℰ , E=

α

β

∑ 2−(2−|α|−|β|)j (2−jdsW ) Eα,β (2−jdsW ) ,

(5.2)

|α|,|β|≤1

where {(Eα,β , 2−j ) : (E, 2−j ) ∈ ℰ , |α|, |β| ≤ 1} ∈ GΩ . For ℰ ∈ GΩ , we say ℰ is a bounded set of (W , ds) elementary operators supported in Ω. We say ℰ is a bounded set of (W , ds) elementary operators if there exists an Ω ⋐ M such that ℰ is a bounded set of (W , ds) elementary operators supported in Ω. Remark 5.2.9. In Definition 5.2.8 (and in all similar definitions), we have defined GΩ to be the largest set satisfying certain axioms. That there is a largest such set follows from the fact that if {GΩα } is the collection of all sets satisfying the axioms, then ⋃α GΩα also satisfies the axioms. See Remark 2.3.6 for further comments on this definition.

274 � 5 Singular integrals Definition 5.2.10. For t ∈ ℝ, we let A1t (Ω, (W , ds)) ⊆ Hom(C0∞ (M), C0∞ (M)′ ) be the set of all T ∈ Hom(C0∞ (M), C0∞ (M)′ ) such that supp(T) ⊆ Ω × Ω and for every bounded set of (W , ds) elementary operators supported in Ω, ℰ , {(2−jt TE, 2−j ) : (E, 2−j ) ∈ ℰ } is a bounded set of (W , ds) elementary operators supported in Ω. Definition 5.2.11. For t ∈ ℝ, we let A2t (Ω, (W , ds)) ⊆ Hom(C0∞ (M), C0∞ (M)′ ) be the set of all T ∈ Hom(C0∞ (M), C0∞ (M)′ ) such that there exists a bounded set of (W , ds) elementary operators supported in Ω, {(Ej , 2−j ) : j ∈ ℕ}, with T = ∑j∈ℕ 2jt Ej . See Proposition 5.5.10 for a description of the convergence of this sum. The next theorem is our main theorem concerning the above definitions. It is proved in Section 5.7. Theorem 5.2.12. For every K ∈ ℕ, t ∈ ℝ, t

t

t

t

A1 (Ω, (W , ds)) = A2 (Ω, (W , ds)) = A3 (Ω, (W , ds), K) = A4 (Ω, (W , ds)).

We denote their common value by A t (Ω, (W , ds)). Remark 5.2.13. The definition of A4t (Ω, (W , ds)) depends on Q1 from Corollary 3.3.9. If Q1 satisfies the conclusions of Corollary 3.3.9, then any Q1′ ∈ [n, Q1 ] also satisfies the conclusions in place of Q1 . However, the algebra defined by A4t (Ω, (W , ds)) does not depend on which such Q1 is used: this follows from Theorem 5.2.12, as none of the other characterizations of A t (Ω, (W , ds)) depend on the choice of Q1 . 5.2.2 The multi-parameter Hörmander setting We take the setting described in Section 5.1.2. Thus, we are given ν families of Hörmander vector fields with formal degrees (W 1 , ds1 ), . . . , (W ν , dsν ) which pairwise locally weakly approximately commute, and we define (W , ds)⃗ := (W 1 , ds1 ) ⊠ (W 2 , ds2 ) ⊠ ⋅ ⋅ ⋅ ⊠ ⃗ for t ∈ ℝν . To do this, we (W ν , dsν ). Our goal is to define the algebra A t (Ω, (W , ds)), generalize Definitions 5.2.10 and 5.2.11 to this setting. For j ∈ ℝν , we write 2−jdsW for ⃗ ⃗ {2−j⋅ds1 W1 , . . . , 2−j⋅dsr Wr }. For x ∈ M, δ > 0, j ∈ ℝν , set ⃗

B2−j (x, δ) := B(W ,ds)⃗ (x, δ2−j ). B2−j (x, δ) are metric balls, and we let ρ2−j (x, z) be the corresponding metric. Note that ρ2−j (x, z) = ρ(2−jds⃗ W ,|ds|⃗ ) (x, z), where |ds|⃗ 1 denotes the list of single-parameter formal de1 grees |ds1⃗ |1 , . . . , |dsq⃗ |1 .

5.2 The algebras of singular integrals

� 275

Definition 5.2.14. We say ℰ ⊂ C0∞ (Ω × Ω) × (0, 1]ν is a bounded set of (W , ds)⃗ preelementary operators supported in Ω if the following hold: – ⋃(E,2−j )∈ℰ supp(E) ⋐ Ω × Ω. – For all ordered multi-indices α, β, for all m ∈ ℕ, there exists C = C(ℰ , α, β, m) ≥ 0 such that ∀(E, 2−j ) ∈ ℰ , (1 + ρ2−j (x, z)) α −jds⃗ β 󵄨󵄨 −jds⃗ 󵄨 . 󵄨󵄨(2 Wx ) (2 Wz ) E(x, z)󵄨󵄨󵄨 ≤ C Vol(B −j (x, 1 + ρ −j (x, z))) ∧ 1 −m

2

2

Definition 5.2.15. We define the set of bounded sets of (W , ds)⃗ elementary operators supported in Ω, GΩ , to be the largest set of subsets of C0∞ (Ω × Ω) × (0, 1]ν such that ∀ℰ ∈ GΩ : – ℰ is a bounded set of (W , ds)⃗ pre-elementary operators supported in Ω. – For all (E, 2−j ) ∈ ℰ , μ ∈ {1, . . . , ν}, E=

μ

α

μ

β

2−(2−|αμ |−|βμ |)jμ (2−jμ ds W μ ) μ Eμ,αμ ,βμ (2−jμ ds W μ ) μ ,

∑

(5.3)

|αμ |,|βμ |≤1

where {(Eμ,αμ ,βμ , 2−j ) : (E, 2−j ) ∈ ℰ , μ ∈ {1, . . . , ν}, |αμ |, |βμ | ≤ 1} ∈ GΩ . For ℰ ∈ GΩ , we say ℰ is a bounded set of (W , ds)⃗ elementary operators supported in Ω. We say ℰ is a bounded set of (W , ds)⃗ elementary operators if there exists Ω ⋐ M such that ℰ is a bounded set of (W , ds)⃗ elementary operators supported in Ω. ⃗ ⊆ Hom(C ∞ (M), C ∞ (M)′ ) be the set Definition 5.2.16. For t ∈ ℝν , we let A1t (Ω, (W , ds)) 0 0 ∞ ∞ ′ of all T ∈ Hom(C0 (M), C0 (M) ) such that supp(T) ⊆ Ω × Ω and for every bounded set of (W , ds)⃗ elementary operators supported in Ω, ℰ , {(2−jt TE, 2−j ) : (E, 2−j ) ∈ ℰ } is a bounded set of (W , ds)⃗ elementary operators supported in Ω. ⃗ ⊆ Hom(C ∞ (M), C ∞ (M)′ ) be the Definition 5.2.17. For t ∈ ℝν , we let A2t (Ω, (W , ds)) 0 0 ∞ ∞ ′ set of all T ∈ Hom(C0 (M), C0 (M) ) such that there exists a bounded set of (W , ds)⃗ elementary operators supported in Ω, {(Ej , 2−j ) : j ∈ ℕν }, with T = ∑j∈ℕν 2j⋅t Ej . See Proposition 5.5.10 for a description of the convergence of this sum. The next theorem is our main theorem concerning the above definitions. It is proved in Section 5.7. Theorem 5.2.18. For every t ∈ ℝν , t

t

⃗ = A (Ω, (W , ds)). ⃗ A1 (Ω, (W , ds)) 2 ⃗ We denote their common value by A t (Ω, (W , ds)).

276 � 5 Singular integrals Remark 5.2.19. It is straightforward to see that when ν = 1, all of the above definitions coincide with the similar definitions in Section 5.2.1. This uses the fact that in the case ν = 1, 2j ρ(W ,ds) (x, z) = ρ2−j (x, z) and B(W ,ds) (x, 2−j δ) = B2−j (x, δ); these follow directly from the definitions. In particular,

(1+2j ρ(W ,ds) (x,z))−m Vol(B(W ,ds) (x,2−j +ρ(W ,ds) (x,z)))∧1

=

(1+ρ2−j (x,z))−m . Vol(B2−j (x,1+ρ2−j (x,z)))∧1

Thus, the definitions here generalize the classical Calderón–Zygmund paradigm to this multi-parameter setting.

Remark 5.2.20. Definitions 5.2.16 and 5.2.17 generalize the definitions of A1t and A2t from Section 5.2.1 to the multi-parameter setting. However, we did not present conditions analogous to the more familiar A3t and A4t from Definitions 5.2.2 and 5.2.4. It is possible to do this to a certain extent; however, the definitions are a bit complicated and not useful in our applications. For a presentation of this in a more general context, see [220, Chapter 5].

5.2.3 The general multi-parameter setting We take the setting described in Section 5.1.3. Thus, we are given ν sets of vector fields with single-parameter formal degrees, (W 1 , ds1 ), . . . , (W ν , dsν ), which pairwise locally weakly approximately commute and such that Gen((W μ , dsμ )) is locally finitely generated for each μ ∈ {1, . . . , ν}. We define (W , ds)⃗ := (W 1 , ds1 ) ⊠ (W 2 , ds2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (W ν , dsν ). ̃t (Ω, (W , ds)). ⃗ To do this, we present analogs of DefOur goal is to define the algebra A ⃗ initions 5.2.16 and 5.2.17 in this more general setting. For j ∈ ℝν , we write 2−jdsW for ⃗ ⃗ {2−j⋅ds1 W1 , . . . , 2−j⋅dsr Wr }. Definition 5.2.21. For L ∈ ℕ and U ⊆ M open, define the vector space L CW (U) := {f ∈ C(U) : W α f ∈ C(U), ∀|α| ≤ L},

with norm 󵄨 󵄨 ‖f ‖C L (U) := sup ∑ 󵄨󵄨󵄨W α f (x)󵄨󵄨󵄨. W x∈U |α|≤L

L With this norm, CW (U) is a Banach space.

Definition 5.2.22. We set 󵄨󵄨 ∞ L CW ,loc (M) := {f : M → ℂ | f 󵄨󵄨U ∈ CW (U), ∀U ⋐ M open, ∀L ∈ ℕ}. ∞ We give CW ,loc (M) the usual projective limit topology (similar to [238, Chapter 10, Exam∞ ple I]), making CW ,loc (M) a Fréchet space.

5.2 The algebras of singular integrals �

277

Definition 5.2.23. We set ∞ ∞ CW ,0 (M) := {f ∈ CW ,loc (M) : supp(f ) is compact}. ∞ We give CW ,0 (M) the usual inductive limit topology (similar to [238, Chapter 13, Exam∞ ple II]), making CW ,0 (M) an LF space. ∞ ∞ ∞ ∞ Remark 5.2.24. We have Cloc (M) 󳨅→ CW ,loc (M) and C0 (M) 󳨅→ CW ,0 (M), where the in∞ clusions are continuous. If W1 , . . . , Wr satisfy Hörmander’s condition, then CW ,loc (M) = ∞ ∞ ∞ Cloc (M) and CW ,0 (M) = C0 (M) with equality of topologies. ∞ ∞ ′ ∞ ∞ ′ Remark 5.2.25. By Remark 5.2.24, Hom(CW ,0 (M), CW ,0 (M) ) ⊆ Hom(C0 (M), C0 (M) ), ∞ ∞ ′ and therefore we identify elements of Hom(CW ,0 (M), CW ,0 (M) ) with their Schwartz kernels. ∞ ∞ ν Definition 5.2.26. We say ℰ ⊂ Hom(CW ,0 (M), CW ,loc (M)) × (0, 1] is a bounded set of generalized (W , ds)⃗ pre-elementary operators supported in Ω if the following hold: – ⋃(E,2−j )∈ℰ supp(E) ⋐ Ω × Ω. – For all ordered multi-indices α, β the operator α

(2−jdsW ) E(2−jdsW ) ⃗

⃗

β

(5.4)

extends to a bounded operator L1 (M, Vol) → L1 (M, Vol), and ⃗ ⃗ α β󵄩 󵄩 sup 󵄩󵄩󵄩(2−jdsW ) E(2−jdsW ) 󵄩󵄩󵄩L1 (M,Vol)→L1 (M,Vol) < ∞. −j

(5.5)

(E,2 )∈ℰ

–

We henceforth identify (5.4) with this extended operator. Note that since supp(E) ⋐ ⃗ ⃗ M × M, it follows that (2−jdsW )α E(2−jdsW )β f is defined for all f ∈ L1loc (M, Vol). For all ordered multi-indices α, β, ⃗ ⃗ α β󵄩 󵄩 sup 󵄩󵄩󵄩(2−jdsW ) E(2−jdsW ) 󵄩󵄩󵄩L∞ (M,Vol)→L∞ (M,Vol) < ∞. −j

(5.6)

(E,2 )∈ℰ

–

For all ordered multi-indices α, β and for every countable set ℰ ′ := {(Ek , δk ) : k ∈ ℕ} ⊆ ℰ , set α

β

𝒯ℰ ′ ,α,β {fk }k∈ℕ := {(δk W ) Ek (δk W ) }k∈ℕ . ds⃗

ds⃗

Then, for all p ∈ (1, ∞), q ∈ (1, ∞], sup ‖𝒯ℰ ′ ,α,β ‖Lp (M,Vol;ℓq (ℕ))→Lp (M,Vol;ℓq (ℕ)) < ∞, ℰ′

where the supremum is taken over all such countable subsets ℰ ′ ⊆ ℰ .

(5.7)

278 � 5 Singular integrals –

∞ ′ For (E, 2−j ) ∈ ℰ , we assume E ∗ initially defined on CW ,loc (M) restricts to an operator ∞ ∞ in Hom(CW ,0 (M), CW ,loc (M)). Furthermore, set

ℰ := {(E , 2 ) : (E, 2 ) ∈ ℰ } ∗

∗

−j

−j

∞ ∞ ν ⊆ Hom(CW ,0 (M), CW ,loc (M)) × (0, 1] .

We assume that all of the above also holds with ℰ replaced by ℰ ∗ . Definition 5.2.27. We define the set of bounded sets of generalized (W , ds)⃗ elementary ∞ ∞ operators supported in Ω, G̃Ω , to be the largest set of subsets of Hom(CW ,0 (M), CW ,loc (M)) such that ∀ℰ ∈ G̃Ω : – ℰ is a bounded set of generalized (W , ds)⃗ pre-elementary operators supported in Ω. – For all (E, 2−j ) ∈ ℰ , μ ∈ {1, . . . , ν}, E=

μ

α

μ

β

2−(2−|αμ |−|βμ |)jμ (2−jμ ds W μ ) μ Eμ,αμ ,βμ (2−jμ ds W μ ) μ ,

∑ |αμ |,|βμ |≤1

where {(Eμ,αμ ,βμ , 2−j ) : (E, 2−j ) ∈ ℰ , μ ∈ {1, . . . , ν}, |αμ |, |βμ | ≤ 1} ∈ G̃Ω . For ℰ ∈ G̃Ω , we say ℰ is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω. We say ℰ is a bounded set of generalized (W , ds)⃗ elementary operators if there exists Ω ⋐ M such that ℰ is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω. ̃t (Ω, (W , ds)) ⃗ ⊆ Hom(C ∞ (M), C ∞ (M)′ ) be the Definition 5.2.28. For t ∈ ℝν , we let A 1 W ,0 W ,0 ∞ ∞ ′ set of all T ∈ Hom(CW ,0 (M), CW ,0 (M) ) such that supp(T) ⊆ Ω×Ω and for every bounded set of generalized (W , ds)⃗ elementary operators supported in Ω, ℰ , {(2−jt TE, 2−j ) : (E, 2−j ) ∈ ℰ } is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω. ̃t (Ω, (W , ds)) ⃗ ⊆ Hom(C ∞ (M), C ∞ (M)′ ) be the Definition 5.2.29. For t ∈ ℝν , we let A 2 W ,0 W ,0 ∞ ∞ ′ set of all T ∈ Hom(CW ,0 (M), CW ,0 (M) ) such that there exists a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω, {(Ej , 2−j ) : j ∈ ℕν }, with T = ∑j∈ℕν 2j⋅t Ej . See Proposition 5.5.10 for a description of the convergence of this sum. The next theorem is our main theorem concerning the above definitions. It is proved in Section 5.7. Theorem 5.2.30. For every t ∈ ℝν , t

t

̃ (Ω, (W , ds)) ̃ (Ω, (W , ds)). ⃗ =A ⃗ A 1 2 ̃t (Ω, (W , ds)). ⃗ We denote their common value by A

5.3 Notation

� 279

Remark 5.2.31. The main difference between the definitions in this section and those in Section 5.2.2 is the definition of bounded sets of (W , ds)⃗ pre-elementary operators (Definition 5.2.14) versus the definition of bounded sets of generalized (W , ds)⃗ pre-elementary operators (Definition 5.2.26). Other than this one difference, the rest of the definitions are formally the same. Because of this, many of our proofs in the multi-parameter Hörmander setting will be formally the same as proofs of corresponding results in the general multi-parameter setting.

5.3 Notation We will prove similar results in all three settings described in this chapter: the singleparameter Hörmander setting described in Section 5.1.1, the multi-parameter Hörmander setting described in Section 5.1.2, and the general multi-parameter setting described in Section 5.1.3. Most of our results in the single-parameter Hörmander setting are a special case of the results in the multi-parameter Hörmander setting. The results we prove in the multi-parameter Hörmander setting and the general multi-parameter setting are similar and have similar proofs, so we study these two cases simultaneously. Thus, in this chapter, we will refer to two settings: the Hörmander setting (short for the multi-parameter Hörmander setting of Section 5.1.2) and the general setting (short for the multi-parameter general setting of Section 5.1.3). In either setting, we are given ν ∈ ℕ+ and ν lists of vector fields with singleparameter formal degrees: μ

μ

∞ (W μ , dsμ ) = {(W1 , ds1 ), . . . , (Wrμμ , dsμrμ )} ⊂ Cloc (M; TM) × ℕ+ .

The assumptions in the two settings are the following. Hörmander setting: We assume that each (W μ , dsμ ) are Hörmander vector fields with formal degrees and that (W 1 , ds1 ), . . . , (W ν , dsν ) pairwise locally weakly approximately commute. General setting: We assume that for each μ ∈ {1, . . . , ν}, Gen((W μ , dsμ )) is locally finitely generated on M and that (W 1 , ds1 ), . . . , (W ν , dsν ) pairwise locally weakly approximately commute. Remark 5.1.2 shows that the Hörmander setting is a special case of the general setting. In either setting, we define (W , ds)⃗ = {(W1 , ds1⃗ ), . . . , (Wr , dsr⃗ )} := (W 1 , ds1 ) ⊠ (W 2 , ds2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (W ν , dsν ) μ

μ

∞ = {(Wj , dsj eμ ) : μ ∈ {1, . . . , ν}, 1 ≤ j ≤ rμ } ⊂ Cloc (M; TM) × (ℕν \ {0}).

When we consider the Hörmander setting, we will consider objects like bounded sets of (W , ds)⃗ pre-elementary operators supported in Ω (Definition 5.2.14), bounded sets

280 � 5 Singular integrals ⃗ of (W , ds)⃗ elementary operators supported in Ω (Definition 5.2.15), and A t (Ω, (W , ds)). In the general setting, we will consider objects like bounded sets of generalized (W , ds)⃗ pre-elementary operators supported in Ω (Definition 5.2.26), bounded sets of generalized ̃t (Ω, (W , ds)). ⃗ (W , ds)⃗ elementary operators supported in Ω (Definition 5.2.27), and A

5.4 Pre-elementary operators In this section we work in the Hörmander setting (see Section 5.3) and prove some results we need concerning bounded sets of (W , ds)⃗ pre-elementary operators (Definition 5.2.14). Bounded sets of (W , ds)⃗ pre-elementary operators are based on the quantity (1 + ρ2−j (x, z))−m , Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1

(5.8)

and many of the results of this section focus on a better understanding of this quantity. We will sometimes allow some of the components of j in (5.8) to be equal to ∞, i. e., we will allow some of the components of 2−j to be equal to 0. More precisely, we will consider j ∈ (−∞, ∞]ν \ {(∞, . . . , ∞)}. The definition of (5.8) in this more general setting is the same when some of the components of 2−j equal 0. This corresponds to considering a setting with a lower number of parameters, where ν denotes the number of parameters, as the next lemma shows. Lemma 5.4.1. Let j ∈ (−∞, ∞]ν \ {(∞, . . . , ∞)} and let E = {μ ∈ {1, . . . , ν} : jμ ≠ ∞}. Let jE ∈ ℝ|E| be the vector consisting of just the finite components of j. Then ̃ −jE (x, δ), B2−j (x, δ) = B 2 ρ2−j (x, y) −m

(5.9)

= ρ̃ 2−jE (x, y),

(5.10)

(1 + ρ2−j (x, z)) (1 + ρ̃ 2−jE (x, z)) = , ̃ −jE (x, 1 + ρ̃ −jE (x, z))) ∧ 1 Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1 Vol(B 2 2 −m

(5.11)

̃ −jE are obtained by replacing (W , ds)⃗ with (W E , dsE⃗ ) in the definition of where ρ̃ 2−jE and B 2 ρ2−j and B2−j , where (W E , dsE⃗ ) is as described in Section 3.5.2. Proof. Indeed, directly from the definitions, since (δ2−j )μ = 0 for all μ ∈ ̸ E, we have ̃ −jE (x, δ), B2−j (x, δ) = B(W ,ds)⃗ (x, δ2−j ) = B(W E ,ds⃗ E ) (x, δ2−jE ) = B 2 proving (5.9). Since ρ2−j is the metric associated with the balls B2−j and ρ̃ 2−jE is the metric associated with the balls B2−jE , (5.10) follows from (5.9). Finally, (5.11) follows by combining (5.9) and (5.10).

5.4 Pre-elementary operators

� 281

Remark 5.4.2. For 0 ≠ E ⊆ {1, . . . , ν}, (W E , dsE⃗ ) satisfies all the same hypotheses as ⃗ Thus, in light of Lemma 5.4.1, many of the results in this section in the general (W , ds). case j ∈ (−∞, ∞]ν \ {(∞, . . . , ∞)} follow from the special case j ∈ ℝν . Proposition 5.4.3. For x, z ∈ Ω, j ∈ (−∞, ∞]ν \ {(∞, . . . , ∞)}, m ∈ ℕ, (1 + ρ2−j (x, z))−m (1 + ρ2−j (z, x))−m ≈ , Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1 Vol(B2−j (z, 1 + ρ2−j (z, x))) ∧ 1

(5.12)

⃗ where the implicit constants depend only on Ω and (W , ds). Proof. By Remark 5.4.2, it suffices to prove the result when j ∈ ℝν . Since ρ2−j (x, z) = ρ2−j (z, x), we only need to consider the denominator of both sides of (5.12). Using Theorem 3.5.4 (n), we have Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1 ≤ Vol(B2−j (z, 2(1 + ρ2−j (x, z)))) ∧ 1 = Vol(B(W ,ds)⃗ (x, 2(1 + ρ2−j (x, z))2−j )) ∧ 1 ≲ Vol(B(W ,ds)⃗ (x, (1 + ρ2−j (x, z))2−j )) ∧ 1 = Vol(B2−j (z, 1 + ρ2−j (z, x))) ∧ 1. By reversing the roles of x and z, we also have Vol(B2−j (z, 1 + ρ2−j (z, x))) ∧ 1 ≲ Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1, completing the proof. ⃗ ∈ ℕ and C = Proposition 5.4.4. For all m ∈ ℕ, there exist m′ = m′ (m, Ω, (W , ds)) ν ⃗ C(m, Ω, (W , ds)) ≥ 0 such that ∀x, z ∈ Ω, j, k ∈ (−∞, ∞] \ {(∞, . . . , ∞)}, (1 + ρ2−k (y, z))−m (1 + ρ2−j (x, y))−m )( ) d Vol(y) ∫( Vol(B2−j (x, 1 + ρ2−j (x, y))) ∧ 1 Vol(B2−k (y, 1 + ρ2−k (y, z))) ∧ 1 ′

Ω

≤C

′

(1 + ρ2−j∧k (x, z))−m . Vol(B2−j∧k (x, 1 + ρ2−j∧k (x, z))) ∧ 1

The proof of Proposition 5.4.4 requires several preliminary results, which will also be useful for some other purposes. Lemma 5.4.5. For all x, z ∈ M and j, k ∈ ℝν , we have 2−|k−j| ρ2−j (x, z) ≤ ρ2−k (x, z). Proof. Directly from the definition, we have 2−|k−j| ρ2−j (x, z) = ρ2|k−j| 2−j (x, z) ≤ ρ2−k (x, z), where the inequality follows from the fact that 2|k−j| 2−j ≥ 2−k , coordinatewise.

282 � 5 Singular integrals ⃗ ≥ 0 such that for all x, z ∈ Ω and j, k ∈ ℝν , Lemma 5.4.6. There exists C = C(Ω, (W , ds)) we have 2−C|k−j| (Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1) ≤ Vol(B2−k (x, 1 + ρ2−k (x, z))) ∧ 1. ⃗ ≥ 0 such that for all Proof. By Theorem 3.5.4 (n), there is a constant D = D(Ω, (W , ds)) ν x ∈ Ω and r ∈ (0, ∞) , Vol(B(W ,ds)⃗ (x, 2r)) ∧ 1 ≤ 2D (Vol(B(W ,ds)⃗ (x, r)) ∧ 1).

(5.13)

We have, using Lemma 5.4.5 and (5.13), Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1 = Vol(B(W ,ds)⃗ (x, (1 + ρ2−j (x, z))2−j )) ∧ 1

≤ Vol(B(W ,ds)⃗ (x, (1 + 2|k−j| ρ2−k (x, z))2|k−j| 2−k )) ∧ 1 ≤ Vol(B(W ,ds)⃗ (x, (1 + ρ2−k (x, z))22|k−j| 2−k )) ∧ 1

≤ 22D|k−j| (Vol(B(W ,ds)⃗ (x, (1 + ρ2−k (x, z))2−k )) ∧ 1) = 22D|k−j| (Vol(B2−k (x, 1 + ρ2−k (x, z))) ∧ 1). Taking C = 2D completes the proof. ⃗ Ω) ≥ 0 such that for all Lemma 5.4.7. For all m ∈ ℕ, there exists M = M(m, (W , ds), ν x, z ∈ Ω and j, k ∈ ℝ , we have 2−M|j−k|

(1 + ρ2−k (x, z))−m (1 + ρ2−j (x, z))−m ≤ . Vol(B2−k (x, 1 + ρ2−k (x, z))) ∧ 1 Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1

⃗ Ω) > 0 sufficiently Proof. We use Lemmas 5.4.5 and 5.4.6 to see that for M = M(m, (W , ds), large, 2−M|j−k| ≤ ≤

(1 + ρ2−k (x, z))−m Vol(B2−k (x, 1 + ρ2−k (x, z))) ∧ 1

(1 + 2|k−j| ρ2−k (x, z))−m 2−k (x, 1 + ρ2−k (x, z))) ∧ 1)

2(M−m)|j−k| (Vol(B

(1 + ρ2−j (x, z))−m . Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1

⃗ Ω) ≥ 0 such that ∀j ∈ ℝν , Lemma 5.4.8. For all m ∈ ℕ, there exists M = M(m, (W , ds), ∑ 2−M|j−k|

k∈ℤν

(1 + ρ2−k (x, z))−m (1 + ρ2−j (x, z))−m ≤ ν2ν . Vol(B2−k (x, 1 + ρ2−k (x, z))) ∧ 1 Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1

Proof. We prove the result with M replaced by M + 1, where M is as in Lemma 5.4.7. Using Lemma 5.4.7, we have

5.4 Pre-elementary operators

∑ 2−(M+1)|j−k|

k∈ℤν

≤

�

283

(1 + ρ2−k (x, z))−m Vol(B2−k (x, 1 + ρ2−k (x, z))) ∧ 1

(1 + ρ2−j (x, z))−m ∑ 2−|j−k| . Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1 k∈ℤν

The result follows. In the rest of this section, for j, k ∈ (−∞, ∞]ν , when we compute |j − k| we use the convention ∞ − ∞ = 0. Whenever we compute |j − k|, j and k will be infinite in the same components, and because of our convention, |j − k| will always be a finite number. Lemma 5.4.9. Fix 0 ≠ E ⊆ {1, . . . , ν}. Set ℐE := {k ∈ (ℤ ∪ {∞})ν : kμ ≠ ∞ iff μ ∈ E}. For ⃗ Ω) ≥ 0, such that for all D > 0, there exists all m ∈ ℕ, there exists M = M(m, (W , ds), ⃗ Ω) ≥ 0, such that ∀x, z ∈ Ω, ∀j ∈ (−∞, ∞]ν with jμ ≠ ∞ iff μ ∈ E, C1 = C1 (D, m, (W , ds), ∑ 2−M|k−j|

k∈ℐE

χ{ρ −k (x,z) 0, there exists C2 = C2 (m, D, (W , ds), ν ∀x, z ∈ Ω, ∀j ∈ ℝ , χ{ρ −k (x,z) 1, it is somewhat involved to find the optimal estimate (see, for example, [229, Section 9.1]). Moreover, even if the optimal estimates were obtained, we would not know of any use for these estimates; in particular, they would not simplify any of our definitions or proofs.

338 � 5 Singular integrals 5.8.2 Standard pseudo-differential operators To understand the regularity properties of maximally subelliptic operators with respect ̃ to the standard Besov and Triebel–Lizorkin spaces, an important property of A and A is that we may set them up in such a way to contain the standard pseudo-differential operators on M.2 ∞ For ν ≥ 1, we let (W 1 , ds1 ), . . . , (W ν−1 , dsν−1 ) ⊂ Cloc (M; TM) × ℕ+ satisfy the assumptions of this chapter. That is, we consider two settings. Hörmander setting: We assume (W 1 , ds1 ), . . . , (W ν−1 , dsν−1 ) pairwise locally weakly approximately commute and each (W μ , dsμ ) is a set of Hörmander vector fields with formal degrees on M. General setting: We assume (W 1 , ds1 ), . . . , (W ν−1 , dsν−1 ) pairwise locally weakly approximately commute and each Gen((W μ , dsμ )) is locally finitely generated on M. ∞ Let (W ν , dsν ) = {(W1ν , 1), . . . , (Wrνν , 1)} ⊂ Cloc (M; TM) × ℕ+ be such that

span{W1ν (x), . . . , Wrνν (x)} = Tx M,

∀x ∈ M.

Lemma 5.8.22. (W 1 , ds1 ), . . . , (W ν , dsν ) satisfy the hypotheses of this chapter. More precisely, we have the following. Hörmander setting: (W 1 , ds1 ), . . . , (W ν , dsν ) pairwise locally weakly approximately commute and each (W μ , dsμ ) is a set of Hörmander vector fields with formal degrees on M. General setting: (W 1 , ds1 ), . . . , (W ν , dsν ) pairwise locally weakly approximately commute and each Gen((W μ , dsμ )) is locally finitely generated on M. Proof. Lemma 3.8.3 shows us that (W ν , dsν ) and (W μ , dsμ ) locally strongly approximately commute, ∀μ ∈ {1, . . . , ν}. Therefore, by Proposition 3.8.6 (b), (W ν , dsν ) and (W μ , dsμ ) locally weakly approximately commute. Since (W ν , dsν ) are Hörmander vector fields with formal degrees (W1ν , . . . , Wrνν satisfy Hörmander’s condition of order 1 on M), the result follows (see also the case ν = 1 of Proposition 3.4.14). We define (W , ds)⃗ := (W 1 , ds1 ) ⊠ (W 2 , ds2 ) ⊠ ⋅ ⋅ ⋅ (W ν , dsν ). Theorem 5.8.23. Let b(x, D) be a standard pseudo-differential operator of order m ∈ ℝ on M with supp(b(x, D)) ⊆ Ω × Ω. Then we have the following. ⃗ Hörmander setting: We have b(x, D) ∈ A meν (Ω, (W , ds)). meν ̃ ⃗ General setting: We have b(x, D) ∈ A (Ω, (W , ds)). The rest of this section is devoted to the proof of Theorem 5.8.23. 2 A pseudo-differential operator of order m ∈ ℝ on a manifold is a pseudo-local operator which when restricted to any local coordinate system equals a standard pseudo-differential operator of order m on ℝn . See [239, Chapter 1, Section 5] for a detailed definition.

5.8 Basic properties �

339

Lemma 5.8.24. Suppose b(x, D) is a standard pseudo-differential operator of order m ∈ ∞ ℝ. Let U ⋐ ℝn and V ⋐ M be relatively compact open sets such that there is a Cloc diffeomorphism Φ : U 󳨀 → V . Suppose supp(b(x, D)) ⋐ V × V . Then we have the following. ⃗ Hörmander setting: We have b(x, D) ∈ A meν (V , (W , ds)). ̃meν (V , (W , ds)). ⃗ General setting: We have b(x, D) ∈ A ∼

Proof. We prove the result in the Hörmander setting; the proof in the general setting is nearly identical. For each μ ∈ {1, . . . , ν}, set μ

μ

(Z μ , drμ ) = {(Z1 , dr1 ), . . . , (Zrμμ , drμrμ )} μ

μ

:= {(Φ∗ W1 , ds1 ), . . . , (Φ∗ Wrμμ , dsμrμ )} ∞ ⊂ Cloc (U; TU) × ℕ+ .

Set (𝜕, 1) := {(𝜕x1 , 1), . . . , (𝜕xn , 1)}. Note that (Z ν , drν ) and (𝜕, 1) are locally strongly equivalent on U (and are therefore locally weakly equivalent on U). We define two lists of vector fields with ν-parameter formal degrees: (Z ′ , dr′⃗ ) := (Z 1 , dr1 ) ⊠ (Z 2 , dr2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (Z ν−1 , drν−1 ) ⊠ (𝜕, 1), (Z, dr)⃗ := (Z 1 , dr1 ) ⊠ (Z 2 , dr2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (Z ν , drν ).

By our assumption, Φ∗ b(x, D)Φ∗ is a standard pseudo-differential operator of order m supported in U ⋐ ℝn . Take U1 ⋐ U2 ⋐ U3 ⋐ U open with supp(Φ∗ b(x, D)Φ∗ ) ⋐ U1 × U1 . Proposition 4.1.7 implies that for any a > 0 sufficiently small, we may write Φ∗ b(x, D)Φ∗ = Ta + Ra , where Ta is a (𝜕, 1), a pseudo-differential operator of order m supported in U1 × U2 and Ra ∈ C0∞ (U2 × U2 ). By Corollary 5.8.14, if a > 0 is chosen sufficiently small, we have T ∈ A meν (U , (Z ′ , dr′⃗ )). Since R ∈ C ∞ (U × U ), Proposition 5.8.11 a

3

a

0

2

2

implies that Ra ∈ A meν (U3 , (Z ′ , dr′⃗ )). Thus, Φ∗ b(x, D)Φ∗ = Ta + Ra ∈ A meν (U3 , (Z ′ , dr′⃗ )). Since (Z ν , drν ) and (𝜕, 1) are locally, weakly equivalent on U, Proposition 5.8.12 ⃗ We conclude that Φ∗ b(x, D)Φ ∈ implies that A meν (U , (Z ′ , dr′⃗ )) = A meν (U , (Z, dr)). 3

3

∗

⃗ By Proposition 5.8.17, this is equivalent to b(x, D) ∈ A meν (Φ(U3 ), (W , ds)). ⃗ A meν (U3 , (Z, dr)). meν ⃗ Since Φ(U3 ) ⊆ V , Proposition 5.8.7 shows that b(x, D) ∈ A (V , (W , ds)), completing the proof.

Proof of Theorem 5.8.23. We prove the result in the Hörmander setting; the proof in the general setting is nearly identical. Let ϕ1 , . . . , ϕL ∈ C0∞ (M) be such that L

∑ ϕl ≡ 1 l=1

on Ω

340 � 5 Singular integrals and such that if supp(ϕj ) ∩ supp(ϕk ) ≠ 0, then there exists Vj,k ⋐ M open and relatively

∞ compact with supp(ϕj ) ∪ supp(ϕk ) ⋐ Vj,k and a Cloc diffeomorphism Φ : Uj,k 󳨀 → Vj,k , n where Uj,k ⋐ ℝ is open. Since supp(b(x, D)) ⊆ Ω × Ω, we have b(x, D) = ∑j,k Mult[ϕj ]b(x, D) Mult[ϕk ]. Thus, ⃗ it suffices to show that for j, k ∈ {1, . . . L}, Mult[ϕj ]b(x, D) Mult[ϕk ] ∈ A meν (Ω, (W , ds)). If supp(ϕj ) ∩ supp(ϕk ) ≠ 0, then by Lemma 5.8.24, Mult[ϕj ]b(x, D) Mult[ϕk ] ∈ ⃗ Since supp(Mult[ϕj ]b(x, D) Mult[ϕk ]) ⊆ Ω ∩ Vj,k , Proposition 5.8.7 A meν (Vj,k , (W , ds)). ⃗ ⊆ A meν (Ω, (W , ds)), ⃗ as shows that Mult[ϕj ]b(x, D) Mult[ϕk ] ∈ A meν (Vj,k ∩ Ω, (W , ds)) desired. If supp(ϕj ) ∩ supp(ϕk ) = 0, then since standard pseudo-differential operators are pseudo-local, we have Mult[ϕj ]b(x, D) Mult[ϕk ] ∈ C0∞ (Ω × Ω). Proposition 5.8.11 then ⃗ completing the proof. implies that Mult[ϕj ]b(x, D) Mult[ϕk ] ∈ A meν (Ω, (W , ds)), ∼

5.9 An important subalgebra We again consider the two settings described in Section 5.3. Fix ν1 ∈ {1, . . . , ν − 1}. Set (Y , d̂)⃗ = {(Y1 , d̂1⃗ ), . . . , (Ys1 , d̂s⃗ 1 )}

∞ := (W 1 , ds1 ) ⊠ (W 2 , ds2 ) ⊠ ⋅ ⋅ ⋅ (W ν1 , dsν1 ) ⊂ Cloc (M; TM) × (ℕν1 \ {0}).

In either the Hörmander setting or the general setting, (Y , d̂)⃗ satisfies the same assump⃗ with ν replaced by ν1 . Set ν2 = ν − ν1 and similarly define tions as (W , ds), (Z, dr)⃗ = {(Z1 , dr1⃗ ), . . . , (Zs2 , drs⃗ 2 )}

∞ := (W ν1 +1 , dsν1 +1 ) ⊠ (W ν1 +2 , dsν1 +2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (W ν , dsν ) ⊂ Cloc (M; TM) × (ℕν2 \ {0}).

We describe the thrust of this section in the Hörmander setting; though as we will see similar results hold in the general setting as well. For t ∈ ℝν1 , there are two very ⃗ and A (t,0ν2 ) (Ω, (W , ds)). ⃗ These similar filtered algebras of singular integrals: A t (Ω, (Y , d̂)) algebras are not the same. In fact, in general we have t

⃗ ⊊A A (Ω, (Y , d̂))

(t,0ν2 )

⃗ (Ω, (W , ds)),

A

(t,0ν2 )

⃗ ⃗ ⊊ A t (Ω, (Y , d̂)). (Ω, (W , ds))

However, many important operators lie in the intersection: t

⃗ ⋂A A (Ω, (Y , d̂))

(t,0ν2 )

⃗ (Ω, (W , ds)).

An example of such operators can be obtained by using the theory described in Section 4.4 (see, for example, Corollary 5.8.14). We used this to show that parametrices for the sub-Laplacian can be viewed as elements of higher parameter algebras (Corollary 5.8.15). In fact, we will see in Section 5.11.2 that a similar result holds for all maximally subelliptic operators.

5.9 An important subalgebra

�

341

⃗ ⋂ A (t,0ν2 ) (Ω, In this section, we introduce a filtered subalgebra of A t (Ω, (Y , d̂)) ⃗ which contains many important examples and has convenient properties. (W , ds)) ⃗ which “approximately Roughly speaking, this is the algebra of operators in A t (Ω, (Y , d̂)) ⃗ commute” with (Z, dr). ⃗ (Z, dr)) ⃗ ⃗ ⊆ A t (Ω, (Y , d̂)) Definition 5.9.1. Hörmander setting: For t ∈ ℝν1 , let D t (Ω, (Y , d̂), (t) t ν2 ⃗ be the set of those T ∈ A (Ω, (Y , d̂)) such that for every κ ∈ ℕ and every (Z, dr)⃗ partial differential operator on Ω of degree ≤ κ, P (κ) , we have: (1) T (t) P (κ) can be written as a finite sum of operators of the form ̃ T̃ , P (κ) (t)

̃(κ) is a (Z, dr)⃗ partial differential operator on Ω of degree ≤ κ and T̃ (t) ∈ where P t ⃗ A (Ω, (Y , d̂)). (κ) (t) (2) P T can be written as a finite sum of operators of the form ̃(κ) , T̃ (t) P ̃(κ) is a (Z, dr)⃗ partial differential operator on Ω of degree ≤ κ and T̃ (t) ∈ where P t ⃗ A (Ω, (Y , d̂)). ⃗ (Z, dr)) ⃗ as above by replacing A with A ̃t (Ω, (Y , d̂), ̃ General setting: We define D throughout. ⃗ It follows immediately from Proposition 5.8.7 that if Ω1 ⊆ Ω2 , then D t (Ω1 , (Y , d̂), ⃗ (Z, dr)) ⃗ (Z, dr)) ⃗ (Z, dr)). ⃗ ⊆ D t (Ω2 , (Y , d̂), ⃗ and D ⃗ ⊆D ⃗ This jus̃t (Ω1 , (Y , d̂), ̃t (Ω2 , (Y , d̂), (Z, dr)) tifies the next definition.

Definition 5.9.2. Hörmander setting: We set t

t

⃗ (Z, dr)) ⃗ (Z, dr)). ⃗ := ⋃ D (Ω, (Y , d̂), ⃗ D ((Y , d̂), Ω⋐M

General setting: We set ⃗ (Z, dr)) ⃗ (Z, dr)). ⃗ := ⋃ D ⃗ ̃t ((Y , d̂), ̃t (Ω, (Y , d̂), D Ω⋐M

We begin by proving some very simple properties of these operators. ⃗ (Z, dr)) ⃗ is a filtered subalgebra of Proposition 5.9.3. Hörmander setting: D t (Ω, (Y , d̂), t t ⃗ In other words, if T ∈ D (Ω, (Y , d̂), ⃗ (Z, dr)) ⃗ (Z, dr)), ⃗ and S ∈ D s (Ω, (Y , d̂), ⃗ A (Ω, (Y , d̂)). t+s ⃗ ⃗ then TS ∈ D (Ω, (Y , d̂), (Z, dr)). ̃ and A replaced by A ̃ General setting: The same result holds with D replaced by D throughout. Proof. We prove only the result in the Hörmander setting; the proof in the general set⃗ (Z, dr)) ⃗ (Z, dr)). ⃗ and S (s) ∈ D s (Ω, (Y , d̂), ⃗ First, ting is similar. Suppose T (t) ∈ D t (Ω, (Y , d̂),

342 � 5 Singular integrals ⃗ Let P (κ) be a (Z, dr)⃗ parwe note by Proposition 5.8.6 that T (t) S (s) ∈ A (t+s) (Ω, (Y , d̂)). tial differential operator on Ω of degree ≤ κ. Applying the definition of D twice, we ̃(κ) , where P ̃(κ) may write P (κ) T (t) S (s) as a finite sum of operators of the form T̃ (t) 𝒮 (s) P (t) t ⃗ ⃗ ̃ is a (Z, dr) partial differential operator on Ω of degree ≤ κ, T ∈ A (Ω, (Y , d̂)), and ⃗ By Proposition 5.8.6, T̃ (t) 𝒮 (s) ∈ A t+s (Ω, (Y , d̂)). ⃗ This shows that S̃(s) ∈ A s (Ω, (Y , d̂)). (κ) (t) (s) P T S is a finite sum of operators of the correct form. A similar proof shows that T (t) S (s) P (κ) is also a finite sum of operators of the correct form, which completes the ⃗ (Z, dr)). ⃗ proof that TS ∈ D t+s (Ω, (Y , d̂), Proposition 5.9.4. Let β ∈ ℤν1 and let Q be a (Y , d̂)⃗ partial differential operator on Ω of degree ≤ β. ⃗ (Z, dr)). ⃗ Then Q T, T Q ∈ D t+β . Hörmander setting: Let T ∈ D t (Ω, (Y , d̂), ⃗ (Z, dr)). ⃗ Then Q T, T Q ∈ D ̃t (Ω, (Y , d̂), ̃t+β . General setting: Let T ∈ D Proof. We prove only the result in the Hörmander setting and only the result for Q T; the proofs for the remaining parts are similar. First, we note that by Corollary 5.8.10, Q T ∈ ⃗ Let P (κ) be a (Z, dr)⃗ partial differential operator on Ω of degree ≤ κ. By A t+β (Ω, (Y , d̂)). ̃(β) P ̃(κ) , Corollary 3.12.8, P (κ) Q can be written as a finite sum of operators of the form Q (β) (κ) ⃗ ̃ ̃ where Q is a (Y , d̂) partial differential operator on Ω of degree ≤ β and P is a (Z, dr)⃗ ⃗ (Z, dr)), ⃗ partial differential operator on Ω of degree ≤ κ. By the definition of D t (Ω, (Y , d̂), (κ) (t) ̂(κ) (t) ̃ ̃ ̃ P T can be written as a finite sum of operators of the form T P , where T ∈ ⃗ and P ̂(κ) is a (Z, dr)⃗ partial differential operator on Ω of degree ≤ κ. Putting A t (Ω, (Y , d̂)) these together, we see that P (κ) Q T can be written as a finite sum of operators of the form ⃗ which completes the ̃(β) T̃ (t) P ̂(κ) . Corollary 5.8.10 shows that Q ̃(β) T̃ (t) ∈ A t+β (Ω, (Y , d̂)), Q (κ) proof that P Q T can be written as a sum of terms of the desired form. ̂(κ) Q ̃(β) T̃ (t) , Similarly, Q T P (κ) can be written as a finite sum of terms of the form P (β) ̃ (t) t+β (κ) ⃗ ̃ and by Corollary 5.8.10, Q T ∈ A (Ω, (Y , d̂)), which shows that Q T P can be written as a sum of terms of the desired form. ⃗ (Z, dr)), ⃗ completing the proof. It follows that Q T ∈ D t+β (Ω, (Y , d̂), Proposition 5.9.5. Hörmander setting: Let T ∈ D t . Then T ∗ ∈ D t . ̃t . Then T ∗ ∈ D ̃t . General setting: Let T ∈ D Proof. We prove only the result in the Hörmander setting; the proof in the general set⃗ (Z, dr)) ⃗ By Proposi⃗ ⊆ A t (Ω, (Y , d̂)). ting is nearly identical. Suppose T ∈ D t (Ω, (Y , d̂), ∗ t ⃗ tion 5.8.1, T ∈ A (Ω, (Y , d̂)). Let P be a (Z, dr)⃗ partial differential operator on Ω of degree ≤ κ. By Lemma 3.12.5, ∗ P is also a (Z, dr)⃗ partial differential operator on Ω of degree ≤ κ. Thus, P T ∗ = [T P ∗ ] ̃ ∗ = T̃ ∗ P ̃T] ̃∗ , where P ̃ is a (Z, dr)⃗ can be written as a finite sum of terms of the form [P t ⃗ By Lemma 3.12.5, partial differential operator on Ω of degree ≤ κ and T̃ ∈ A (Ω, (Y , d̂)). ∗ ⃗ ̃ P is a (Z, dr) partial differential operator on Ω of degree ≤ κ, and by Proposition 5.8.1, ⃗ This shows that P T ∗ is of the desired form. T̃ ∗ ∈ A t (Ω, (Y , d̂)).

5.9 An important subalgebra

� 343

A similar argument proves that T ∗ P is of the desired form, which completes the proof. ̃t . We now present a main theorem regarding the algebras D t and D ⃗ (Z, dr)) ⃗ ⊆ A (t,0ν2 ) (Ω, Theorem 5.9.6. Hörmander setting: For all t ∈ ℝν1 , D t (Ω, (Y , d̂), ⃗ (W , ds)). ⃗ (Z, dr)) ⃗ ⊆A ̃t (Ω, (Y , d̂), ̃(t,0ν2 ) (Ω, (W , ds)). ⃗ General setting: For all t ∈ ℝν1 , D The rest of this section is devoted to the proof of Theorem 5.9.6. To prove Theorem 5.9.6 in the Hörmander setting, we will need to study the interaction between (Y , d̂)⃗ pre-elementary operators supported in Ω and (W , ds)⃗ pre-elementary operators supported in Ω. (W , ds)⃗ pre-elementary operators are defined in terms of the quantity (1 + ρ2−j (x, z))−m , Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1 where j ∈ [0, ∞)ν and ρ2−j and B2−j are defined in terms of (W , ds)⃗ as in Section 5.1.2. Anal̃ −k in the same way, but with (W , ds)⃗ replaced ogously, if for k ∈ ℝν1 we define ρ̃ 2−k and B 2 ⃗ ⃗ by (Y , d̂), then (Y , d̂) pre-elementary operators are defined in terms of the quantity (1 + ρ̃ 2−k (x, z))−m . ̃ −k (x, 1 + ρ̃ −k (x, z))) ∧ 1 Vol(B 2 2 Lemma 5.4.1 shows that (1 + ρ̃ 2−k (x, z))−m (1 + ρ2−(k,∞) (x, z))−m = . ̃ −k (x, 1 + ρ̃ −k (x, z))) ∧ 1 Vol(B2−(k,∞) (x, 1 + ρ2−(k,∞) (x, z))) ∧ 1 Vol(B 2 2

(5.89)

For j ∈ ℝν ≅ ℝν1 × ℝν2 , we write j = (j1 , j2 ), where j1 ∈ ℝν1 and j2 ∈ ℝν2 . Lemma 5.9.7. In the Hörmander setting, for every m ∈ ℕ, there exist m′ = m′ (m, Ω, ⃗ ∈ ℕ and C = C(m, Ω, (W , ds)) ⃗ ≥ 0 such that ∀x, z ∈ Ω, j ∈ ℝν , k ∈ ℝν1 , we have (W , ds)) (1 + ρ̃ 2−k (y, z))−m (1 + ρ2−j (x, y))−m )( ) d Vol(y) ∫( ̃ −k (y, 1 + ρ̃ −k (y, z))) ∧ 1 Vol(B2−j (x, 1 + ρ2−j (x, y))) ∧ 1 Vol(B 2 2 ′

′

Ω

≤C

(1 + ρ2−(j1 ∧k,j2 ) (x, z))−m

Vol(B2−(j1 ∧k,j2 ) (x, 1 + ρ2−(j1 ∧k,j2 ) (x, z))) ∧ 1

,

(1 + ρ̃ 2−k (y, z))−m (1 + ρ2−j (x, y))−m )( ) d Vol(y) ∫( ̃ −k (y, 1 + ρ̃ −k (y, z))) ∧ 1 Vol(B2−j (x, 1 + ρ2−j (x, y))) ∧ 1 Vol(B 2 2 ′

Ω

≤C

(1 + ρ2−(j1 ∧k,j2 ) (x, z))−m

Vol(B2−(j1 ∧k,j2 ) (x, 1 + ρ2−(j1 ∧k,j2 ) (x, z))) ∧ 1

(5.90)

′

.

(5.91)

344 � 5 Singular integrals Proof. The inequality (5.90) follows from Proposition 5.4.4 by replacing k in that proposition with (k, ∞) and applying (5.89). Similarly, (5.91) follows from Proposition 5.4.4 by taking j in that proposition to be (k, ∞), taking k in that proposition to be j, and applying (5.89). Lemma 5.9.8. Hörmander setting: Let ℰ1 be a bounded set of (Y , d̂)⃗ elementary operators supported in Ω and let ℰ2 be a bounded set of (W , ds)⃗ elementary operators supported in Ω. Then, ∀N ∈ ℕ, m ∈ ℕ, there exists C = C(N, m, ℰ1 , ℰ2 ) ≥ 0 such that for all (E1 , 2−k ) ∈ ℰ1 and (E2 , 2−j ) ∈ ℰ2 and all x, y ∈ Ω, 1 󵄨 󵄨 2N|j −k| 󵄨󵄨󵄨[E1 E2 ](x, y)󵄨󵄨󵄨 ≤ C 1 󵄨 󵄨 2N|j −k| 󵄨󵄨󵄨[E2 E1 ](x, y)󵄨󵄨󵄨 ≤ C

(1 + ρ2−j (x, y))−m , Vol(B2−j (x, 1 + ρ2−j (x, y))) ∧ 1 (1 + ρ2−j (x, y))−m . Vol(B2−j (x, 1 + ρ2−j (x, y))) ∧ 1

(5.92) (5.93)

General setting: Let ℰ1 be a bounded set of generalized (Y , d̂)⃗ elementary operators supported in Ω and let ℰ2 be a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω. Then, for all N ∈ ℕ, the following hold: – For p ∈ {1, ∞}, 1

sup

(E1 ,2−k )∈ℰ1 (E2 ,2−j )∈ℰ2

–

2N|j −k| (‖E1 E2 ‖Lp →Lp + ‖E2 E1 ‖Lp →Lp ) < ∞.

(5.94)

For all countable sets ℰ̂1 = {(E1,l , 2−kl ) : l ∈ ℕ} ⊆ ℰ1 and ℰ̂2 = {(E2,l , 2−jl ) : l ∈ ℕ} ⊆ ℰ2 , if we set 1

N|jl1 −kl |

E1,l E2,l fl }l∈ℕ ,

N|jl1 −kl |

E2,l E1,l fl }l∈ℕ ,

𝒯ℰ̂ ,ℰ̂ ,N {fl }l∈ℕ := {2 1

2

2

𝒯ℰ̂ ,ℰ̂ ,N {fl }l∈ℕ := {2 1

2

then, for all p ∈ (1, ∞), q ∈ (1, ∞], sup (‖𝒯ℰ̂1 ,ℰ̂ ,N ‖Lp (ℓq )→Lp (ℓq ) + ‖𝒯ℰ̂2 ,ℰ̂ ,N ‖Lp (ℓq )→Lp (ℓq ) ) < ∞.

ℰ̂1 ,ℰ̂2

1

2

1

2

Here, the supremum is taking over all such choices of ℰ̂1 and ℰ̂2 . Proof. We begin with the Hörmander setting and prove only (5.92); the proof for (5.93) ⃗ Ω) large, to be chosen later. We first claim that it suffices is similar. Fix M = M(m, (W , ds), to prove (5.92) with N replaced by −M. For (E1 , 2−k ) ∈ ℰ1 and (E2 , 2−j ) ∈ ℰ2 , we separate the proof of this claim into two cases.

� 345

5.9 An important subalgebra

The first case is when there is a μ ∈ {1, . . . , ν1 } such that kμ − jμ = |j1 − k|∞ . We apply Proposition 5.5.5 (h) with N replaced by N1 := ⌈√ν1 N + √ν1 M⌉ to E1 to see 1

1

μ

2N|j −k| E1 E2 = 2N|j −k| ∑ 2(|αμ |−N1 )kμ E1,αμ (2−kμ ds W μ )αμ E2 |αμ |≤N1

=2

N|j1 −k|

μ

α

∑ 2(|αμ |−N1 )kμ −degdsμ (αμ )(kμ −jμ ) E1,α−μ [(2−jμ ds W μ ) μ E2 ] |αμ |≤N1 1

1

1

α

μ

= ∑ 2−N1 |j −k|∞ +N|j −k| E1,αμ [2N1 |j −k|∞ −(|αμ |−N1 )kμ −degdsμ (kμ −jμ ) (2−jμ ds W μ ) μ E2 ] |αμ |≤N1 1

1

=: 2−N1 |j −k|∞ +N|j −k| E1,αμ E2,αμ , where {(E1,αμ , 2−k ) : (E1 , 2−k ) ∈ ℰ1 , |αμ | ≤ N1 } is a bounded set of (Y , d̂)⃗ elementary

operators supported in Ω, and by Proposition 5.5.5 (a) and (e), {(E2,αμ , 2−j ) : (E2 , 2−j ) ∈ ℰ2 , kμ − jμ = |j1 − k|∞ , |αμ | ≤ N1 } is a bounded set of (W , ds)⃗ elementary operators sup1

ported in Ω (where we have used the fact that 2N1 |j −k|∞ −(|αμ |−N1 )kμ −degdsμ (kμ −jμ ) ≤ 1 in the 1 1 1 sum). Since 2−N1 |j −k|∞ +N|j −k| ≤ 2−M|j −k| , we see that it suffices to prove (5.92) with N replaced by −M in this case. The second case is when there is a μ ∈ {1, . . . , ν1 } such that jμ − kμ = |j′ 1 − k|∞ ; the proof is similar to the first case. Indeed, we apply Proposition 5.5.5 (h) with N replaced by N1 := ⌈√ν1 N + √ν1 M⌉ to E2 to see that 1

1

μ

α

2N|j −k| E1 E2 = 2N|j −k| ∑ 2(|αμ |−N1 )jμ E1 (2−jμ ds W μ ) μ E2,αμ |αμ |≤N1 N|j1 −k|

=2

μ

α

∑ 2(|αμ |−N1 )jμ −degdsμ (αμ )(jμ −kμ ) E1 (2−kμ ds W μ ) μ E2,αμ |αμ |≤N1 1

1

μ

α

= ∑ 2−N1 |j −k|+N|j −k| [2N1 |j−k|∞ −(|αμ |−N1 )jμ −degdsμ (αμ )(jμ −kμ ) E1 (2−kμ ds W μ ) μ ]E2,αμ |αμ |≤N1 1

1

=: ∑ 2−N1 |j −k|+N|j −k| E1,αμ E2,αμ , |αμ |≤N1

where {(E2,αμ , 2−j ) : (E2 , 2−j ) ∈ ℰ2 , |αμ | ≤ N1 } is a bounded set of (W , ds)⃗ elementary

operators supported in Ω, and by Proposition 5.5.5 (a) and (e), {(E1,αμ , 2−k ) : (E1 , 2−k ) ∈ ℰ1 , jμ − kμ = |j1 − k|∞ , |αμ | ≤ N1 } is a bounded set of (Y , d̂)⃗ elementary operators sup-

ported in Ω (where we have used the fact that 2N1 |j−k|∞ −(|αμ |−N1 )jμ −degdsμ (αμ )(jμ −kμ ) ≤ 1 in the 1 1 1 sum). Since 2−N1 |j −k|∞ +N|j −k| ≤ 2−M|j −k| , we see that it suffices to prove (5.92) with N replaced by −M. ⃗ Ω) can We turn to proving (5.92) with N replaced by −M, where M = M(m, (W , ds), ⃗ Ω) be as large as we like. By Lemmas 5.4.7 and 5.9.7, we have, by taking M = M(m, (W , ds), sufficiently large, for all x, y ∈ Ω,

346 � 5 Singular integrals

1

(1 + ρ2−(j1 ∧k,j2 ) (x, y))−m

1

󵄨 󵄨 2−M|j −k| 󵄨󵄨󵄨[E1 E2 ](x, y)󵄨󵄨󵄨 ≲ 2−M|j −k| 1

Vol(B2−(j1 ∧k,j2 ) (x, 1 + ρ2−(j1 ∧k,j2 ) (x, y))) ∧ 1

≤ 2−M|j−(j ∧k,j ≲

2

)|

(1 + ρ2−(j1 ∧k,j2 ) (x, y))−m

Vol(B2−(j1 ∧k,j2 ) (x, 1 + ρ2−(j1 ∧k,j2 ) (x, y))) ∧ 1

(1 + ρ2−j (x, y))−m , Vol(B2−j (x, 1 + ρ2−j (x, y))) ∧ 1

completing the proof in the Hörmander setting. The proof in the general setting is similar and easier. Indeed, by the same argument as above, it suffices to prove the result with N replaced with 0. Once N is replaced with 0, the result follows immediately from the definition of generalized pre-elementary operators supported in Ω (Definition 5.2.26). Lemma 5.9.9. Let κ1 ∈ ℕν1 and let P1 be a (Y , d̂)⃗ partial differential operator on Ω of degree ≤ κ1 . Let t ∈ ℝν1 . ⃗ and let ℰ be a bounded set of (W , ds)⃗ Hörmander setting: Suppose T ∈ A t (Ω, (Y , d̂)) elementary operators supported in Ω. Then, for every m ∈ ℕ, there exists C ≥ 0 such that for all (E, 2−j ) ∈ ℰ and all (x, z) ∈ Ω, 1 1 󵄨 󵄨 2−(j ⋅t+j ⋅κ1 ) 󵄨󵄨󵄨(P1 TE)(x, z)󵄨󵄨󵄨 ≤ C 1 1 󵄨 󵄨 2−(j ⋅t+j ⋅κ1 ) 󵄨󵄨󵄨(ET P1 )(x, z)󵄨󵄨󵄨 ≤ C

(1 + ρ2−j (x, z))−m , Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1

(1 + ρ2−j (x, z))−m . Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1

(5.95) (5.96)

⃗ and let ℰ be a bounded set of (W , ds)⃗ elemeñt (Ω, (Y , d̂)) General setting: Suppose T ∈ A tary operators supported in Ω. Then: (i) For p ∈ {1, ∞}, 1

1

sup 2−(j ⋅t+j ⋅κ1 ) (‖P1 TE‖Lp →Lp + ‖ET P1 ‖Lp →Lp ) < ∞.

(E,2−j )∈ℰ

(ii) For each countable set ℰ̃ = {(El , 2−jl ) : l ∈ ℕ} ⊆ ℰ , we set 1

−(t⋅jl1 +κ1 ⋅jl1 )

2

−(t⋅jl1 +κ1 ⋅jl1 )

𝒯ℰ̃{fl }l∈ℕ := {2 𝒯ℰ̃{fl }l∈ℕ := {2

P1 TEl fl }l∈ℕ ,

El T P1 fl }l∈ℕ .

Then, for p ∈ (1, ∞), q ∈ (1, ∞], we have sup(‖𝒯ℰ̃1 ‖Lp (ℓq )→Lp (ℓq ) + ‖𝒯ℰ̃2 ‖Lp (ℓq )→Lp (ℓq ) ) < ∞, ℰ̃

where the supremum is taken over all countable subsets of ℰ .

(5.97)

5.9 An important subalgebra

�

347

Proof. We begin with the Hörmander setting. First, we note that (5.96) follows from (5.95). Indeed, [ET P1 ](x, z) = [(ET P1 ) ](z, x) = [P1∗ T ∗ E ∗ ](z, x). ∗

By Lemma 3.12.5, P1∗ is a (Y , d̂)⃗ partial differential operator on Ω of degree ≤ κ1 , by ⃗ and by Proposition 5.5.5 (d), {(E ∗ , 2−j ) : (E, 2−j ) ∈ Proposition 5.8.1, T ∗ ∈ A t (Ω, (Y , d̂)), ℰ } is a bounded set of (W , ds)⃗ elementary operators supported in Ω. Thus, once we prove (5.95), we will have 1

(1 + ρ2−j (z, x))−m Vol(B2−j (z, 1 + ρ2−j (z, x))) ∧ 1

1

󵄨 󵄨 2−(j ⋅t+j ⋅κ1 ) 󵄨󵄨󵄨(ET P1 )(x, z)󵄨󵄨󵄨 ≲

(1 + ρ2−j (x, z))−m , Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1

≈

where the last estimate follows from Proposition 5.4.3. ⃗ Thus, by reWe turn to proving (5.95). By Corollary 5.8.10, P1 T ∈ A t+κ1 (Ω, (Y , d̂)). placing T with P1 T and t with t + κ1 , it suffices to prove that for p ∈ {1, ∞}, 1

󵄨 󵄨 2−(j ⋅t) 󵄨󵄨󵄨(TE)(x, z)󵄨󵄨󵄨 ≤ C

(1 + ρ2−j (x, z))−m . Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1

(5.98)

⃗ = A t (Ω, (Y , d̂)), ⃗ there is a bounded set of (Y , d̂)⃗ elementary Since T ∈ A t (Ω, (Y , d̂)) 2 −k operators supported in Ω, {(Ek , 2 ) : k ∈ ℕν1 }, such that T = ∑k∈ℕν1 2k⋅t Ek . Let x, z ∈ Ω. Using Lemma 5.9.8, we have, for every m ∈ ℕ, 󵄨󵄨 −j1 ⋅t 󵄨 󵄨 (k−j1 )⋅t 󵄨 Ek E(x, z)󵄨󵄨󵄨 󵄨󵄨2 TE(x, z)󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨2 k∈ℕν1

1

≲ ∑ 2−|k−j | k∈ℕν1

≲

(1 + ρ2−j (x, z))−m Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1

(1 + ρ2−j (x, z))−m , Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1

proving (5.98) and completing the proof in the Hörmander setting. We now turn to the general setting. Using Corollary 5.8.10, we have P1 T, T P1 ∈ t+κ1 ⃗ Thus, by replacing T with either P1 T or T P1 , it suffices to prove the ̃ A (Ω, (Y , d̂)). ⃗ =A ⃗ there is a ̃t (Ω, (Y , d̂)) ̃t (Ω, (Y , d̂)), result when κ1 = 0 and P1 = Mult[1]. Since T ∈ A 2 bounded set of generalized (Y , d̂)⃗ elementary operators supported in Ω, {(Ek , 2−k ) : k ∈ ℕν1 }, with T = ∑k∈ℕν1 2k⋅t Ek . We take the convention that Ek := 0 for k ∈ ℤν1 \ ℕν1 . We begin with (i). We have, for (E, 2−j ) ∈ ℰ and p ∈ {1, ∞}, using (5.94), 1

1

1

2−j ⋅t ‖TE‖Lp →Lp ≤ ∑ 2(k−j )⋅t ‖Ek E‖Lp →Lp ≲ ∑ 2−|k−j | ≲ 1. k∈ℕν1

k∈ℕν1

348 � 5 Singular integrals 1

A similar proof shows that 2−j ⋅t ‖ET‖Lp →Lp ≲ 1, completing the proof of (i). We next turn to (ii) and prove only the result for 𝒯ℰ̃1 ; the proof for 𝒯ℰ̃2 is similar and we leave it to the reader. Let ℰ̃ = {(Ẽl , 2−jl ) : l ∈ ℕ} ⊆ ℰ and consider for p ∈ (1, ∞), q ∈ (1, ∞],

󵄩󵄩 󵄩󵄩 1 󵄩󵄩 󵄩󵄩 󵄩 󵄩 sup󵄩󵄩󵄩𝒯ℰ̃1 {fl }l∈ℕ 󵄩󵄩󵄩Lp (ℓq ) = sup 󵄩󵄩󵄩{ ∑ 2t⋅(k−jl ) Ek Ẽjl fl } 󵄩󵄩󵄩 󵄩 󵄩 p q ̃ ̃ 󵄩 ℰ⊆ℰ ℰ⊆ℰ 󵄩 k∈ℕν1 󵄩L (ℓ ) l∈ℕ 󵄩 󵄩󵄩 󵄩󵄩 ′ 1 1 1 󵄩󵄩 󵄩󵄩 = sup 󵄩󵄩󵄩{ ∑ 2−|k−jl | (2t⋅(k−jl )+|k−jl | 2t⋅(k−jl ) Ek Ẽjl fl )} 󵄩󵄩󵄩 󵄩 󵄩 p q ̃ 󵄩 ℰ⊆ℰ 󵄩 k∈ℕν1 󵄩L (ℓ ) l∈ℕ 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 = sup 󵄩󵄩󵄩{ ∑ 2−|r| (2t⋅r+|r| 2t⋅r Ej1 +r Ẽjl fl )} 󵄩󵄩󵄩 l 󵄩 󵄩 p q ̃ 󵄩 ℰ⊆ℰ 󵄩 r∈ℤν1 󵄩L (ℓ ) l∈ℕ 󵄩 󵄩󵄩 t⋅r+|r| t⋅r 󵄩 −|r| ≤ ∑ 2 sup󵄩󵄩{2 2 Ej1 +r Ẽjl fl }l∈ℕ 󵄩󵄩󵄩Lp (ℓq ) l r∈ℤν1

̃ ℰ⊆ℰ

≲ ∑ 2

−|r|

r∈ℤν1

sup ‖{fl }l∈ℕ ‖Lp (ℓq ) ̃ ℰ⊆ℰ

≲ ‖{fl }l∈ℕ ‖Lp (ℓq ) , where the second to last inequality follows from (5.97). This completes the proof of (ii). Lemma 5.9.10. In both the Hörmander setting and the general setting, (Y , d̂)⃗ and (Z, dr)⃗ locally weakly approximately commute. Proof. This follows from Corollary 3.8.11. Indeed, for μ1 ∈ {1, . . . , ν1 }, we take 𝒮μ1 := (W μ1 , dsμ1 ), and for μ2 ∈ {1, . . . , ν2 }, we take 𝒯μ2 := (W ν1 +μ2 , dsν1 +μ2 ). Then Corollary 3.8.11 applies to 𝒮1 , . . . , 𝒮ν1 , 𝒯1 , . . . , 𝒯ν2 . From that corollary, 𝒮0 equals (Y , d̂)⃗ and 𝒯0 equals ⃗ Thus, (Y , d̂)⃗ and (Z, dr)⃗ locally weakly approximately commute, completing he (Z, dr). proof. ⃗ (Z, dr)) ⃗ ⊆ Hom(C ∞ (M), C ∞ (M)). ̃t (Ω, (Y , d̂), Lemma 5.9.11. In the general setting, D W ,0 W ,loc Proof. First we claim t

⃗ ⊆ Hom(C ̃ (Ω, (Y , d̂)) A W ,loc (M), CY ,0 (M)). ∞

∞

(5.99)

⃗ we have T̃ ∈ Hom(C ∞ (M), ̃t (Ω, (Y , d̂)), Indeed, Proposition 5.8.5 shows that for T̃ ∈ A Y ,loc ∞ ∞ CY∞,0 (M)). Since Y ⊆ W , CW (M) embeds continuously into C (M) and therefore ,loc Y ,loc ∞ ∞ T̃ ∈ Hom(CW ,loc (M), CY ,0 (M)), establishing (5.99). As a consequence of (5.99), for any (Y , d̂)⃗ partial differential operator on Ω of degree (κ ) ⃗ we have ̃t (Ω, (Y , d̂)), ≤ κ1 ∈ ℤ, P1 1 , and any T̃ ∈ A ∞ T ∈ Hom(CW ,loc (Ω), C(Ω)).

(κ1 ) ̃

P1

(5.100)

5.9 An important subalgebra

� 349

⃗ (Z, dr)). ⃗ We wish to show ̃t (Ω, (Y , d̂), Let T ∈ D ∞ ∞ T ∈ Hom(CW ,loc (M), CW ,0 (M)).

(5.101)

⃗ we have supp(T) ⋐ Ω × Ω, and therefore (5.101) is equivalent to ̃t (Ω, (Y , d̂)), Since T ∈ A ∞ ∞ T ∈ Hom(CW ,0 (Ω), CW ,loc (Ω)).

(5.102)

To prove (5.102), we will show that for every ordered multi-index α, ∞ W α T ∈ Hom(CW ,0 (Ω), C(Ω)).

(5.103)

We turn to proving (5.103). Let κ := degds⃗ (α) ∈ ℕν and write κ = (κ1 , κ2 ) ∈ ℕν1 × ℕ ≅ ℕν . Note that W α is a (W , ds)⃗ partial differential operator on Ω of degree ≤ κ. Lemma 5.9.10 implies that (Y , d̂)⃗ and (Z, dr)⃗ weakly approximately commute on Ω. Proposition 3.12.13 then implies that W α can be written as a finite sum of terms of the form (κ ) (κ ) (κ ) P1 1 P2 2 , where P1 1 is a (Y , d̂)⃗ partial differential operator on Ω of degree ≤ κ1 and (κ2 ) P is a (Z, dr)⃗ partial differential operator on Ω of degree ≤ κ2 . Thus, to prove (5.103) ν2

2

it suffices to show that for such P1

(κ1 )

(κ1 )

P1

(κ2 )

P2

and P2 2 , we have (κ )

∞ T ∈ Hom(CW ,0 (Ω), C(Ω)).

(5.104)

⃗ (Z, dr)), ⃗ P (κ1 ) P (κ2 ) T can be written as a finite sum ̃t (Ω, (Y , d̂), By the definition of D 1 2 (κ1 ) ̃ ̃(κ2 ) ⃗ and P ̃t (Ω, (Y , d̂)) ̃(κ2 ) is a (Z, dr)⃗ partial of terms of the form P T P , where T̃ ∈ A 1

2

2

differential operator on Ω of degree ≤ κ2 . Thus, to prove (5.104) it suffices to show that ̃(κ2 ) , for such T̃ and P 2 (κ1 ) ̃ ̃(κ2 ) TP2

P1

∞ ∈ Hom(CW ,0 (Ω), C(Ω)).

(5.105)

̃(κ2 ) is a (Z, dr)⃗ partial differential operator on Ω of degree ≤ κ2 , it is also a (W , ds)⃗ Since P 2 partial differential operator on Ω of degree ≤ (0ν1 , κ2 ). Thus, (κ2 )

̃ P 2

∞ ∞ ∈ Hom(CW ,0 (Ω), CW ,0 (Ω)).

(5.106)

Combining (5.100) and (5.106) gives (5.105) and completes the proof. Lemma 5.9.12. Fix t ∈ ℝν1 . ⃗ (Z, dr)) ⃗ and suppose ℰ is a bounded set of Hörmander setting: Let T (t) ∈ D t (Ω, (Y , d̂), ⃗ (W , ds) elementary operators supported in Ω. Then 1

1

{(2−j ⋅t T (t) E, 2−j ), (2−j ⋅t ET (t) , 2−j ) : (E, 2−j ) ∈ ℰ } is a bounded set of (W , ds)⃗ pre-elementary operators supported in Ω.

350 � 5 Singular integrals ⃗ (Z, dr)) ⃗ and suppose ℰ is a bounded set of gener̃t (Ω, (Y , d̂), General setting: Let T (t) ∈ D alized (W , ds)⃗ elementary operators supported in Ω. Then 1

1

{(2−j ⋅t T (t) E, 2−j ), (2−j ⋅t ET (t) , 2−j ) : (E, 2−j ) ∈ ℰ }

(5.107)

is a bounded set of generalized (W , ds)⃗ pre-elementary operators supported in Ω. Proof. In either setting, we have supp(T (t) ) ⋐ Ω × Ω and ⋃(E,2−j )∈ℰ supp(E) ⋐ Ω × Ω. Therefore, 1

supp(2−j ⋅t T (t) E) ⋐ Ω × Ω.

⋃

(5.108)

(E,2−j )∈ℰ

We turn to the Hörmander setting. We prove only that 1

{(2−j ⋅t T (t) E, 2−j ) : (E, 2−j ) ∈ ℰ } is a bounded set of (W , ds)⃗ pre-elementary operators supported in Ω. The proof with T (t) E replaced by ET (t) is similar, and we leave it to the reader. In light of (5.108), it suffices to show that for all ordered multi-indices α and β, α −jds⃗ β (t) 󵄨󵄨 −j1 ⋅t −jds⃗ 󵄨 󵄨󵄨2 (2 Wx ) (2 Wz ) [T E](x, z)󵄨󵄨󵄨 (1 + ρ2−j (x, z))−m ≤ Cα,β , Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1

(5.109)

for all (E, 2−j ) ∈ ℰ and x, z ∈ Ω. Note that 1

α

β

2−j ⋅t (2−jdsWx ) (2−jdsWz ) [T (t) E](x, z) ⃗

1

⃗

α

β ∗

= 2−j ⋅t [(2−jdsW ) T (t) E((2−jdsW ) ) ](x, z). ⃗

⃗

(5.110)

⃗ Since ((2−jdsW )β )∗ = 2−j⋅degds⃗ (β) (W β )∗ and W β is a (W , ds)⃗ partial differential operator on Ω of degree ≤ degds⃗ (β), Proposition 5.5.5 (g) and Lemma 3.12.5 imply that β ∗

{(E((2−jdsW ) ) , 2−j ) : (E, 2−j ) ∈ ℰ } ⃗

is a bounded set of (W , ds)⃗ elementary operators supported in Ω. Combining this with (5.110) shows that it suffices to prove (5.109) in the special case where |β| = 0, i. e., to show that for all x, z ∈ Ω and (E, 2−j ) ∈ ℰ , (1 + ρ2−j (x, z))−m 󵄨󵄨 −j1 ⋅t −jds⃗ α (t) 󵄨 . 󵄨󵄨2 [(2 W ) T E](x, z)󵄨󵄨󵄨 ≲ Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1

(5.111)

5.9 An important subalgebra

�

351

⃗ Since (2−jdsW )α = 2−j⋅degds⃗ (α) W α and W α is a (W , ds)⃗ partial differential operator on Ω of degree ≤ degds⃗ (α), to prove (5.111), it suffices to show that for every P (κ) a (W , ds)⃗ partial differential operator on Ω of degree ≤ κ ∈ ℕν , we have for all x, z ∈ Ω, for all (E, 2−j ) ∈ ℰ ,

(1 + ρ2−j (x, z)) 󵄨󵄨 −j1 ⋅t−j⋅κ (κ) (t) 󵄨 [P T E](x, z)󵄨󵄨󵄨 ≲ . 󵄨󵄨2 Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1 −m

(5.112)

Let κ = (κ1 , κ2 ) ∈ ℕν1 × ℕν2 ≅ ℕν . Lemma 5.9.10 implies that (Y , d̂)⃗ and (Z, dr)⃗ weakly approximately commute on Ω. Proposition 3.12.13 then implies that P (κ) can be written (κ ) (κ ) (κ ) as a finite sum of terms of the form P1 1 P2 2 , where P1 1 is a (Y , d̂)⃗ partial differential (κ ) operator on Ω of degree ≤ κ1 and P 2 is a (Z, dr)⃗ partial differential operator on Ω of 2

degree ≤ κ2 . Thus, to prove (5.112), it suffices to show that for such P1 have for all x, z ∈ Ω, for all (E, 2−j ) ∈ ℰ ,

(κ1 )

and P2 2 , we

(κ ) (κ ) 󵄨󵄨 −j1 ⋅t −j1 ⋅κ1 −j2 ⋅κ2 󵄨 2 [P1 1 P2 2 T (t) E](x, z)󵄨󵄨󵄨 󵄨󵄨2 2 (1 + ρ2−j (x, z))−m ≲ . Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1

(κ )

(5.113)

⃗ (Z, dr)), ⃗ P (κ2 ) T can be written as a finite sum ̃t (Ω, (Y , d̂), By the assumption that T (t) ∈ D 2 ⃗ and P ̃(κ2 ) , where T̃ (t) ∈ A t (Ω, (Y , d̂)) ̃(κ2 ) is a (Z, dr)⃗ partial of terms of the form T̃ (t) P 2

2

differential operator on Ω of degree ≤ κ2 . Thus, to prove (5.113), it suffices to show that ̃(κ2 ) , for such T̃ (t) and P 2

2 (κ ) 󵄨󵄨 −j1 ⋅t −j1 ⋅κ1 ̃(κ2 ) E](x, z)󵄨󵄨󵄨 [P1 1 T̃ (t) 2−j ⋅κ2 P 󵄨󵄨2 2 󵄨 2 (1 + ρ2−j (x, z))−m ≲ . Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1

(5.114)

̃(κ2 ) is a (Z, dr)⃗ partial differential operator on Ω of degree ≤ κ2 , it is a (W , ds)⃗ Since P 2 partial differential operator on Ω of degree ≤ (0ν1 , κ2 ). Combining this with the fact that j2 ⋅ κ2 = j ⋅ (0ν1 , κ2 ), Proposition 5.5.5 (g) implies that {(2−j

2

⋅κ2 ̃(κ2 ) −j P 2 E, 2 )

: (E, 2−j ) ∈ ℰ }

is a bounded set of (W , ds)⃗ elementary operators supported in Ω. Thus, by replacing E 2 ̃(κ2 ) E, we see that to prove (5.114) it suffices to show that for all x, z ∈ Ω, for with 2−j ⋅κ2 P 2 all (E, 2−j ) ∈ ℰ , (κ ) 󵄨󵄨 −j1 ⋅t −j1 ⋅κ1 󵄨 [P1 1 T̃ (t) E](x, z)󵄨󵄨󵄨 ≲ 󵄨󵄨2 2

(1 + ρ2−j (x, z))−m . Vol(B2−j (x, 1 + ρ2−j (x, z))) ∧ 1

(5.115)

The estimate (5.115) follows from (5.95), completing the proof in the Hörmander setting. We now turn to the general setting. In light of (5.108), it suffices to prove the following.

352 � 5 Singular integrals (A) For all (E, 2−j ) ∈ ℰ , ∞ ∞ T (t) E, ET (t) , (T (t) E) , (ET (t) ) ∈ Hom(CW ,0 (M), CW ,loc (M)). ∗

∗

(B) For p ∈ {1, ∞} and for SE ∈ {T (t) E, ET (t) , (T (t) E)∗ , (ET (t) )∗ }, for all ordered multiindices α and β, 1 ⃗ ⃗ α β󵄩 󵄩 sup 2−j ⋅t 󵄩󵄩󵄩(2−jdsW ) SE (2−jdsW ) 󵄩󵄩󵄩Lp →Lp < ∞.

(E,2−j )∈ℰ

(C) For p ∈ (1, ∞) and q ∈ (1, ∞], the following holds. Let ℰ̃ = {(El , 2−jl ) : l ∈ ℕ} ⊆ ℰ be a 1 1 1 countable set. For r = 1, . . . , 4, let Sr,l be 2−jl ⋅t TEl if r = 1, 2−jl ⋅t El T if r = 2, (2−jl ⋅t TEl )∗ 1

if r = 3, and (2−jl ⋅t El T)∗ if r = 4. Set, for ordered multi-indices α and β, r

𝒯ℰ,α,β {fl }l∈ℕ := {(2

α

β

W ) Sr,l (2−jl dsW ) fl }l∈ℕ .

−jl ds⃗

⃗

Then sup ‖𝒯ℰ,α,β ‖Lp (ℓq )→Lp (ℓq ) < ∞, ̃ ℰ⊆ℰ

where the supremum is taken over all such countable subsets of ℰ . We turn to proving the above results. By Proposition 5.5.5 (d), {(E ∗ , 2−j ) : (E, 2−j ) ∈ ℰ } is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω. By Proposi⃗ (Z, dr)). ⃗ Thus, (T (t) E)∗ is of the same form as ET (t) and ̃t (Ω, (Y , d̂), tion 5.9.5, (T (t) )∗ ∈ D (t) ∗ (t) (ET ) is of the same form as T E. Because of this, it suffices to prove (A) for T (t) E and ET (t) , (B) when SE ∈ {T (t) E, ET (t) }, and (C) for r = 1, 2. ∞ ∞ ν We turn to (A). Since ℰ ⊆ Hom(CW ,0 (M), CW ,loc (M)) × (0, 1] (see Definition 5.2.26) ∞ ∞ ν and ⋃(E,2−j )∈ℰ supp(E) ⋐ Ω × Ω, we have ℰ ⊆ Hom(CW ,loc (M), CW ,0 (M)) × (0, 1] . Com∞ ∞ bining this with Lemma 5.9.11 implies that T (t) E, ET (t) ∈ Hom(CW ,0 (M), CW ,loc (M)), for −j (E, 2 ) ∈ ℰ , as desired. For (B), the proof when SE = ET (t) is similar to the proof when SE = T (t) E, so we only prove (B) when SE = T (t) E. Similarly, we only prove (C) for r = 1 (the proof for r = 2 being similar). ⃗ By Proposition 5.5.5 (e), {(E(2−jdsW )β , 2−j ) : (E, 2−j ) ∈ ℰ } is a bounded set of general⃗ ized (W , ds)⃗ elementary operators supported in Ω. Thus, by replacing E with E(2−jdsW )β it suffices to prove (B) and (C) in the special case |β| = 0. As in the proof in the Hörmander ⃗ setting, we may write (2−jdsW )α T (t) as a finite sum of terms of the form 1

(κ1 ) ̃ (t) −j2 ⋅κ2 ̃(κ2 ) T 2 P2 ,

2−j ⋅κ1 P1

5.9 An important subalgebra

� 353

(κ ) where κ = (κ1 , κ2 ) := degds⃗ (α) ∈ ℕν , P1 1 is a (Y , d̂)⃗ partial differential operator on Ω (κ ) ̃ 2 is a (Z, dr)⃗ partial differential operator on Ω of degree ≤ κ2 , and of degree ≤ κ1 , P 2

⃗ Thus, it suffices to prove (B) and (C) with (2−jds⃗ W )α T (t) E replaced by ̃t (Ω, (Y , d̂)). T̃ (t) ∈ A 1

(κ1 ) ̃ (t) −j2 ⋅κ2 ̃(κ2 ) T 2 P 2 E,

2−j ⋅κ1 P1 where P1

̃(κ2 ) are as described above. , T̃ (t) , and P 2 ̃(κ2 ) is a (Z, dr)⃗ partial differential operator on Ω of degree ≤ κ2 , it is a (W , ds)⃗ Since P (κ1 )

2

partial differential operator on Ω of degree ≤ (0ν1 , κ2 ). Combining this with the fact that j2 ⋅ κ2 = j ⋅ (0ν1 , κ2 ), Proposition 5.5.5 (g) implies that {(2−j

2

⋅κ2 ̃(κ2 ) −j P 2 E, 2 )

: (E, 2−j ) ∈ ℰ }

is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω. Thus, by re2 ̃(κ2 ) E, we see that it suffices to prove (B) and (C) with (2−jds⃗ W )α T (t) E placing E with 2−j ⋅κ2 P 2 replaced by 1

2−j ⋅κ1 P1

T E.

(κ1 ) ̃ (t)

From here, the result follows from Lemma 5.9.9, completing the proof. ⃗ Proof of Theorem 5.9.6. We begin with the Hörmander setting. Let T ∈ D t (Ω, (Y , d̂), t ∞ ∞ ⃗ By Proposition 5.8.5, T ∈ Hom(C (Ω), C (Ω)), and by the defini⃗ ⊆ A (Ω, (Y , d̂)). (Z, dr)) 0 0 t ⃗ ⃗ tion of A (Ω, (Y , d̂)) we have supp(T) ⊆ Ω × Ω. From here, that T ∈ A (t,0ν2 ) (Ω, (W , ds))

follows by combining Proposition 5.7.1 and Lemma 5.9.12. ⃗ (Z, dr)) ⃗ ⃗ ⊆A ̃t (Ω, (Y , d̂), ̃t (Ω, (Y , d̂)). The proof in the general setting is similar. Let T ∈ D ∞ ∞ t ⃗ we have ̃ (Ω, (Y , d̂)), By Lemma 5.9.11, T ∈ Hom(CW ,0 (Ω), CW ,0 (Ω)). By the definition of A ̃(t,0ν2 ) (Ω, (W , ds)) ⃗ follows by combining Proposisupp(T) ⊆ Ω × Ω. From here, that T ∈ A tion 5.7.1 and Lemma 5.9.12.

5.9.1 Standard pseudo-differential operators ∞ Let (Y , 1) = {(Y1 , 1), . . . , (Yv , 1)} ⊂ Cloc (M; TM) × ℕ+ be such that

span{Y1 (x), . . . , Yv (x)} = Tx M,

∀x ∈ M.

Fix a relatively compact open set Ω ⋐ M and m ∈ ℝ. Suppose b(x, D) is a standard pseudo-differential operator of order m on M. Theorem 5.8.23 implies b(x, D) ∈ A m (Ω, (Y , 1)). In this section we show that more is true: b(x, D) lies in the algebras D m ̃m . We turn to making this precise. and D

354 � 5 Singular integrals We consider the two settings of Section 5.3, so we are given a list (W 1 , ds1 ), . . . , (W , dsν ) as in that section and set ν

(W , ds)⃗ = {(W1 , ds1⃗ ), . . . , (Wr , dsr⃗ )}

∞ := (W 1 , ds1 ) ⊠ (W 2 , ds2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (W ν , dsν ) ⊂ Cloc (M; TM) × (ℕν \ {0}).

By Lemma 3.8.3, for each μ ∈ {1, . . . , ν}, (W μ , dsμ ) and (Y , 1) strongly locally approximately commute, and therefore by Proposition 3.8.6 (b) they weakly locally approximately commute. It is evident that (Y , 1) are Hörmander vector fields with formal degrees (indeed, Y1 , . . . , Yv satisfy Hörmander’s condition of order 1 at every point). Thus, in either the Hörmander setting or the general setting, (W 1 , ds1 ), . . . , (W ν , dsν ), (Y , 1) satisfy the same hypotheses as (W 1 , ds1 ), . . . , (W ν , dsν ) with ν replaced by ν + 1. In particular, ⃗ and in the in the Hörmander setting, it makes sense to talk about D t (Ω, (Y , 1), (W , ds)), t ̃ (Ω, (Y , 1), (W , ds)). ⃗ general setting, it makes sense to talk about D The main result of this setting is the next proposition.

Proposition 5.9.13. Suppose b(x, D) is a standard pseudo-differential operator of order m ∈ ℝ with supp(b(x, D)) ⊆ Ω × Ω. Then we have the following. ⃗ Hörmander setting: We have b(x, D) ∈ D m (Ω, (Y , 1), (W , ds)). m ̃ (Ω, (Y , 1), (W , ds)). ⃗ General setting: We have b(x, D) ∈ D Proof. We prove the result in the Hörmander setting; the proof in the general setting ̃ and D ̃ throughout. follows by replacing A and D with A (κ) Let P be a (W , ds)⃗ partial differential operator on Ω of degree ≤ κ ∈ ℤν . We will show that: (i) b(x, D)P (κ) can be written as a finite sum of operators of the form ̃ D), ̃ b(x, P (κ)

̃ D) is ̃(κ) is a (W , ds)⃗ partial differential operator on Ω of degree ≤ κ and b(x, where P ̃ a standard pseudo-differential operator of order m ∈ ℝ with supp(b(x, D)) ⊆ Ω × Ω. (ii) P (κ) b(x, D) can be written as a finite sum of operators of the form ̃ D)P ̃(κ) , b(x,

(5.116)

̃ D) is ̃(κ) is a (W , ds)⃗ partial differential operator on Ω of degree ≤ κ and b(x, where P ̃ a standard pseudo-differential operator of order m ∈ ℝ with supp(b(x, D)) ⊆ Ω × Ω. To see why this will complete the proof, note that Theorem 5.8.23 implies that b(x, D) ∈ ̃ D) ∈ A m (Ω, (Y , 1)) for each of the above b.̃ Thus, if (i) and (ii) hold, A m (Ω, (Y , 1)) and b(x, ⃗ are verified. the conditions for b(x, D) ∈ D m (Ω, (Y , 1), (W , ds)) The proofs for (i) and (ii) are similar, so we prove only (ii). If κ ∈ ℤν \ ℕν , then (κ) P = 0 and (ii) is trivial. We henceforth assume κ ∈ ℕν .

5.10 Lp bounds in the single-parameter case

� 355

By definition P (κ) (see Definition 3.12.1) can be written as a finite sum of terms of the form Mult[g]Wl1 Wl2 ⋅ ⋅ ⋅ WlK ,

(5.117)

∞ where g ∈ Cloc (Ω) and dsl⃗ 1 + ⋅ ⋅ ⋅ + dsl⃗ K ≤ κ. It suffices to consider the case where P (κ) is given by just one term of the form (5.117); we henceforth assume this. Since supp(b(x, D)) ⊆ Ω × Ω, supp(b(x, D)) is closed, and Ω ⋐ M, we have b(x, D) ⋐ Ω × Ω. Take ϕ ∈ C0∞ (Ω) such that ϕ ⊗ ϕ ≡ 1 on a neighborhood of supp(b(x, D)). It suffices to prove the result with P (κ) replaced by Mult[ϕ]P (κ) . That is, we may replace P (κ) with

Mult[gϕ]Wl1 Wl2 ⋅ ⋅ ⋅ WlK . From here, the calculus of pseudo-differential operators shows that Mult[ϕ]P (κ) b(x, D) can be written as a finite sum of terms of the form ̃ D) Mult[g]Z ̃ l Zl ⋅ ⋅ ⋅ Zl , b(x, k k k 1

2

R

̃ D) is a standard pseudowhere g̃ ∈ C0∞ (Ω), 1 ≤ k1 < k2 < ⋅ ⋅ ⋅ < kR ≤ K, and b(x, differential operator of order m. This is of the desired form (5.116), which completes the proof.

5.10 Lp bounds in the single-parameter case ⃗ and, more generally, operators In the multi-parameter case, operators in A 0 (Ω, (W , ds)) 0 p ̃ (Ω, (W , ds)) ⃗ are bounded on L (M, Vol), 1 < p < ∞. We will see this as part of our in A more general theory of Triebel–Lizorkin spaces in Chapter 6.3 However, in the singleparameter case (ν = 1), the operators in A 0 (Ω, (W , ds)) fall under the general Calderón– Zygmund paradigm, and we can prove the Lp boundedness using standard Calderón– Zygmund theory. We present this here. The setting is the single-parameter setting of Section 5.2.1. Thus, we are given Hörmander vector fields with formal degrees on M: (W , ds) = {(W1 , ds1 ), . . . , (Wr , ds)} ⊂ ∞ Cloc (M; TM) × ℕ+ . As before, Ω ⋐ M denotes a relatively compact, open set. Proposition 5.10.1. If T ∈ A 0 (Ω, (W , ds)), then T extends to a bounded operator T : Lp (M, Vol) → Lp (M, Vol), 1 < p < ∞. ̃0 (Ω, (W , ds)) ⃗ and A ⃗ is easy to see from elementary 3 The L2 boundedness of operators in A 0 (Ω, (W , ds)) considerations. See Remark 5.10.3.

356 � 5 Singular integrals By Theorem 3.3.1 (b), the balls B(W ,ds) (x, δ) paired with the measure Vol locally give M the structure of a space of homogeneous type. Since operators in A 0 (Ω, (W , ds)) have Schwartz kernels with compact support, the classical Calderón–Zygmund theory can be applied to them. In particular, Proposition 5.10.1 follows from the T(1) theorem on spaces of homogeneous type due to David, Journé, and Semmes [59]; here we use A 0 (Ω, (W , ds)) = A40 (Ω, (W , ds)). However, in later sections, we will apply Proposition 5.10.1 an infinite number of times, and we will require that the Lp bounds of the operator are uniform over this infinite number of applications. While it is possible to use the proof of the T(1) theorem to deduce the required uniformity, it is easier to see from a more elementary proof, which we present. A uniform result is presented in Section 5.10.1. Lemma 5.10.2. If T ∈ A 0 (Ω, (W , ds)), then T extends to a bounded operator T : L2 (M, Vol) → L2 (M, Vol). Proof. Since T ∈ A 0 (Ω, (W , ds)) = A2t (Ω, (W , ds)), there is a bounded set of (W , ds) elementary operators supported in Ω, {(Ej , 2−j ) : j ∈ ℕ}, such that T = ∑j∈ℕ Ej . The result will follow from the Cotlar–Stein lemma (Lemma 2.3.25) once we show that, ∀j, k ∈ ℕ, ‖Ej Ek∗ ‖L2 →L2 , ‖Ek∗ Ej ‖L2 →L2 ≲ 2−|j−k| .

(5.118)

By Proposition 5.5.5 (d), {(Ek∗ , 2−k ) : k ∈ ℕ} is a bounded set of (W , ds) elementary operators supported in Ω. Therefore, by Proposition 5.5.11, {(2|j−k| Ej Ek∗ , 2−j ), (2|j−k| Ek∗ Ej , 2−j ) : j, k ∈ ℕ} is a bounded set of (W , ds) elementary operators supported in Ω. Corollary 5.4.11 implies that sup ‖2|j−k| Ej Ek∗ ‖L2 →L2 , sup ‖2|j−k| Ek∗ Ej ‖L2 →L2 < ∞.

j,k∈ℕ

j,k∈ℕ

This is equivalent to (5.118), completing the proof. Remark 5.10.3. In the multi-parameter setting, ν > 1, it is straightforward to modify the ̃0 (Ω, (W , ds)) ⃗ are bounded on L2 , and proof of Lemma 5.10.2 to prove that operators in A 0 ⃗ as a special case that operators in A (Ω, (W , ds)) are bounded in L2 . However, we will see this as part of more general results in Chapter 6. Proof of Proposition 5.10.1. We will show that T extends to a bounded operator T : Lp (M, Vol) → Lp (M, Vol), 1 < p ≤ 2. Since T ∗ ∈ A 0 (Ω, (W , ds)), by Proposition 5.8.1, the result for 2 ≤ p < ∞ will follow by duality. We use the proof method of [216, Chapter 1, Section 5]. For the remainder of the proof, we use T ∈ A 0 (Ω, (W , ds)) = A40 (Ω, (W , ds)). Since we already know that T extends

5.10 Lp bounds in the single-parameter case

�

357

to a bounded operator on L2 , by Lemma 5.10.2, by the methods of [216, Chapter 1, Section 5] it suffices to show that there exists C ≥ 0 such that ∀δ > 0, y ∈ Ω, y ∈ B(W ,ds) (y, δ), ∫ B(W ,ds) (y,2δ)c ∩Ω

󵄨󵄨 󵄨 󵄨󵄨T(x, y) − T(x, y)󵄨󵄨󵄨 d Vol(x) ≤ C.

(5.119)

For such y, y, let γ : [0, 1] → B(W ,ds) (y, δ) be an absolutely continuous path with γ(0) = y, γ(1) = y, and γ′ (t) = ∑ δdsl al (t)Wl (γ(t)), with ‖ ∑ |al |2 ‖L∞ < 1. Note that for x ∈ B(W ,ds) (y, 2δ)c , ∀t ∈ [0, 1], ρ(W ,ds) (x, γ(t)) ≈ ρ(W ,ds) (x, y), since γ(t) ∈ B(W ,ds) (y, δ). Thus, for x ∈ B(W ,ds) (y, 2δ)c we have, using Theorem 3.3.1 (b), 1

r

1

󵄨󵄨 󵄨 󵄨 󵄨 󵄨 ds 󵄨 󵄨󵄨T(x, y) − T(x, y)󵄨󵄨󵄨 ≤ ∫󵄨󵄨󵄨𝜕t T(x, γ(t))󵄨󵄨󵄨 dt ≤ ∑ ∫󵄨󵄨󵄨δ l (Wl,y T)(x, γ(t))󵄨󵄨󵄨 dt 0

r

1

l=1

0

≲ ∑ δdsl ∫ r

≈ ∑ δdsl l=1

l=1 0

ρ(W ,ds) (x, γ(t))−dsl Vol(B(W ,ds) (x, ρ(W ,ds) (x, γ(t)))) ∧ 1

(5.120)

ρ(W ,ds) (x, y)−dsl . Vol(B(W ,ds) (x, ρ(W ,ds) (x, y))) ∧ 1

Using (5.120) and Theorem 3.3.1 (b), we have ∫ B(W ,ds) (y,2δ)c ∩Ω

󵄨󵄨 󵄨 󵄨󵄨T(x, y) − T(x, y)󵄨󵄨󵄨 d Vol(x)

∞

=∑ j=1

∫ (B(W ,ds) (y,2j+1 δ)\B(W ,ds) (y,2j δ))∩Ω

󵄨󵄨 󵄨 󵄨󵄨T(x, y) − T(x, y)󵄨󵄨󵄨 d Vol(x)

∞ r

≲ ∑∑ j=1 l=1

∫ (B(W ,ds) (y,2j+1 δ)\B(W ,ds) (y,2j δ))∩Ω

∞ r

≈ ∑∑ j=1 l=1 ∞ r

∫ (B(W ,ds) (y,2j+1 δ)\B(W ,ds) (y,2j δ))∩Ω

≲ ∑ ∑ 2−jdsl j=1 l=1

δdsl ρ(W ,ds) (x, y)−dsl d Vol(x) Vol(B(W ,ds) (y, ρ(W ,ds) (x, y))) ∧ 1 2−jdsl d Vol(x) Vol(B(W ,ds) (y, 2j δ)) ∧ 1

Vol(B(W ,ds) (y, 2j+1 δ) ∩ Ω) Vol(B(W ,ds) (y, 2j δ)) ∧ 1

∞ r

≈ ∑ ∑ 2−jdsl ≈ 1, j=1 l=1

where we have used Vol(Ω) < ∞, since Ω is relatively compact. This completes the proof of (5.119).

358 � 5 Singular integrals 5.10.1 A uniform result Fix n ∈ ℕ and let (Y , d̂) = {(Y1 , d̂1 ), . . . , (Yq , d̂q )} ⊂ C ∞ (Bn (1); ℝn ) × ℕ+ be C ∞ vector fields with formal degrees on Bn (1). We suppose: – span{Y1 (u), . . . , Yq (u)} = Tu Bn (1), ∀u ∈ Bn (1), uniformly in u in the sense that inf

max

u∈Bn (1) j1 ,...,jn ∈{1,...,q}

–

󵄨󵄨 󵄨 󵄨󵄨det(Yj1 (u)| ⋅ ⋅ ⋅ |Yjn (u))󵄨󵄨󵄨 > 0.

(5.121)

l l [Yj , Yk ] = ∑d̂l ≤d̂j +d̂k cj,k Yl , where cj,k ∈ C ∞ (Bn (1)).

Fix η ∈ (0, 1). If a = a(η, (Y , d̂)) > 0 is sufficiently small, the following holds. Let T be a (Y , d̂), a pseudo-differential operator of order 0 supported in Bn (η/2) × Bn (η). Then, by Proposition 5.8.13, T ∈ A 0 (Bn (1), (Y , d̂)), and therefore by Proposition 5.10.1, T extends to a bounded operator T : Lp (Bn (1), σLeb ) → Lp (Bn (1), σLeb ), 1 < p < ∞, where σLeb denotes the usual Lebesgue density on ℝn . In other words, for 1 < p < ∞, ‖T‖Lp (Bn (1),σLeb )→Lp (Bn (1),σLeb ) ≤ Cp ,

(5.122)

for some constant Cp ≥ 0. By keeping track of constants in the abovementioned proofs, we can be more explicit about what Cp ≥ 0 and a > 0 can be chosen to depend on. Here we describe this dependence. The constant a: The constant a > 0 can be chosen to depend only on upper bounds for η−1 , (1 − η)−1 , n, q, and max1≤j≤q ‖Yj ‖C 1 (Bn (1);ℝn ) . Since T is a (Y , d̂), a pseudo-differential operator of order 0 supported in Bn (η/2) × n B (η), T can be written as T = ∑ Ej , j∈ℕ

where Ej f (x) = ∫ f (e−t⋅Y x) Dild2j (ςj )(x, t, t) dt, ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) is a bounded set and ςj ∈ where {ςj : j ∈ ℕ} ⊂ C0∞ (Bn (η/2))⊗ ∞ n ∞ q q ̂ C0 (B (a))⊗ ̂ S0 (ℝ ) for j > 0. C0 (B (η/2))⊗ The constant Cp : There is a constant L ∈ ℕ, depending only on upper bounds for n and q, such that the constant Cp ≥ 0 from (5.122) can be chosen to depend only on: – upper bounds for n, q, η−1 , (1 − η)−1 , p, (1 − p)−1 , and max{d̂j : 1 ≤ j ≤ q}, – an upper bound for max ‖Yj ‖C L (Bn (1);ℝn ) , 1≤j≤q

5.10 Lp bounds in the single-parameter case

–

an upper bound for sup

∑ |α|,|β|,|γ|≤L

–

� 359

x∈Bn (η/2)

L󵄨 γ 󵄨 sup sup sup(1 + |t|) 󵄨󵄨󵄨𝜕xα 𝜕sβ 𝜕t ςj (x, s, t)󵄨󵄨󵄨, q q

s∈B (a) t∈ℝ j∈ℕ

a lower bound, > 0, for the left-hand side of (5.121).

5.10.2 Beyond Hörmander’s condition Proposition 5.10.1 concerned the case where (W , ds) were Hörmander vector fields with formal degrees, which is the setting to which the classical Calderón–Zygmund theory most directly applies. However, the uniformity described in Section 5.10.1 allows us to deduce the Lp boundedness of some operators outside of the setting of Hörmander’s condition. At the end of this section, we use this to develop a Littlewood–Paley square function, which will be a crucial tool in our study of Sobolev spaces in Section 6.9. ∞ ∞ Let (X, d ) = {(X1 , d1 ), . . . , (Xq , dq )} ⊂ Cloc (M; TM) × ℕ+ be Cloc vector fields with formal degrees satisfying [Xj , Xk ] =

∑ dl ≤dj +dk

l cj,k Xl ,

l ∞ cj,k ∈ Cloc (M).

(5.123)

Note that we are not assuming that X1 , . . . , Xq span the tangent space at any point, and we can therefore not directly use Proposition 5.10.1 to deduce the Lp boundedness of (X, d ), a pseudo-differential operators of order 0. Fix relatively compact, open sets Ω3 ⋐ Ω4 ⋐ M. The main result of this section is the following. Proposition 5.10.4. If a = a(Ω3 , Ω4 , (X, d )) > 0 is sufficiently small, the following holds. Let ℰ be a bounded set of (X, d ), a pseudo-differential operator scales supported in Ω3 ×Ω4 . For ℰ ′ = {(Ej , 2−j ) : j ∈ ℕ} ⊆ ℰ , set T = T[ℰ ′ ] := ∑j∈ℕ Ej . Then, for p ∈ (1, ∞), T : Lp (M, Vol) → Lp (M, Vol) and sup

{(Ej ,2−j ):j∈ℕ} ⊆ℰ

‖T‖Lp (M,Vol)→Lp (M,Vol) < ∞.

We turn to the proof of Proposition 5.10.4. The idea is the following: because of (5.123) the classical Frobenius theorem (Theorem 3.1.15) foliates M into leaves, and X1 , . . . , Xq span the tangent space at every point on the leaves. The idea is to apply Proposition 5.10.1 on each leaf (using the uniform version described in Section 5.10.1) to deduce the Lp boundedness of T. There is a difficulty with this proof outline, though: the foliation may have singular points (see Definition 3.1.21). In this case, the usual proofs of the Frobenius theorem do not give uniform control over the coordinate charts defining the foliation near a singular point, and we do not immediately obtain enough uniformity to deduce

360 � 5 Singular integrals the Lp boundedness of T. Fortunately, Theorem 3.5.1 (in the case ν = 1) does give the required uniformity for these coordinate charts. Pick Ω1 , Ω5 open and relatively compact and 𝒦 compact with Ω3 ⋐ Ω4 ⋐ Ω5 ⋐ 𝒦 ⋐ Ω1 ⋐ M. We apply Theorem 3.5.1 (in the case ν = 1) with this choice of 𝒦 and Ω1 and with (X, d ) = (W , ds). For x ∈ 𝒦 and δ ∈ (0, 1], let Φx,δ and N(x) = dim span{X1 (x), . . . , Xq (x)} be as in that theorem. We will use Proposition 3.9.1, with Ω3 in that proposition replaced by Ω4 and Ω4 replaced by Ω5 and with 𝒦 playing the role of the set with the same name. We let η1 > 0 be as in that proposition. Lemma 5.10.5. If a = a(Ω3 , Ω4 , (X, d )) > 0 is sufficiently small, the following holds. Let ℰ be a bounded set of (X, d ), a pseudo-differential operator scales supported in Ω3 × Ω4 . For each p ∈ (1, ∞), there exists Dp ≥ 0 such that for all x ∈ 𝒦 and f ∈ C0∞ (Ω1 ), sup −j

∫

{(Ej ,2 ):j∈ℕ} N(x) B (η1 /4) ⊆ℰ

󵄨󵄨 󵄨p 󵄨󵄨Tf (Φx,1 (u))󵄨󵄨󵄨 du ≤ Dp

∫ BN(x) (η1 )

󵄨󵄨 󵄨p 󵄨󵄨f (Φx,1 (u))󵄨󵄨󵄨 du,

(5.124)

where T = T[{(Ej , 2−j ) : j ∈ ℕ}] = ∑j∈ℕ Ej , as in Proposition 5.10.4. Proof. Since ℰ is a bounded set of (X, d ), a pseudo-differential operator scales, by Definition 4.3.1, for (Ej , 2−j ) ∈ ℰ , we can write Ej f (x) = ∫ f (e−t⋅X x) Dild2j (ςj )(x, t, t) dt,

(5.125)

̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) is a bounded set and ςj ∈ where {ςj : (Ej , 2−j ) ∈ ℰ } ⊂ C0∞ (Ω3 )⊗ ∞ ∞ q q ̂ C0 (B (a))⊗ ̂ S0 (ℝ ) for j > 0. C0 (Ω3 )⊗ We begin with the case where N(x) = 0, i. e., X1 (x) = ⋅ ⋅ ⋅ = Xq (x) = 0, Φx,1 (u) ≡ x, BN(x) (η) = {0}, and du denotes the point mass at 0 ∈ ℝ0 . In this (degenerate) case, we will show that 󵄨󵄨 󵄨 −j 󵄨󵄨Ej f (x)󵄨󵄨󵄨 ≲ 2 |f (x)|.

(5.126)

First we see why (5.126) yields (5.124) in this case. Indeed, by (5.126), we have |Tf (x)| ≲ ∑j∈ℕ 2−j |f (x)| ≲ |f (x)|. Therefore, we have sup −j

∫

{(Ej ,2 ):j∈ℕ} N(x) B (η1 /4) ⊆ℰ

󵄨󵄨 󵄨p 󵄨󵄨Tf (Φx,1 (u))󵄨󵄨󵄨 du =

sup −j

{(Ej ,2 ):j∈ℕ} ⊆ℰ

󵄨󵄨 󵄨p 󵄨󵄨Tf (x)󵄨󵄨󵄨 du

≲ |f (x)|p =

∫ BN(x) (η1 )

󵄨󵄨 󵄨p 󵄨󵄨f (Φx,1 (u))󵄨󵄨󵄨 du,

which yields (5.124) in this case. We turn to proving (5.126). Since X1 (x) = ⋅ ⋅ ⋅ = Xq (x) = 0, we have e−t⋅X x = x. Thus, (5.125) can be rewritten as

5.10 Lp bounds in the single-parameter case

�

361

Ej f (x) = f (x) ∫ Dild2j (ςj )(x, t, t) dt, so to prove (5.126), it suffices to show that 󵄨󵄨 󵄨 󵄨󵄨∫ Dildj (ςj )(x, t, t) dt 󵄨󵄨󵄨 ≲ 2−j . 󵄨󵄨 󵄨󵄨 2

(5.127)

When j = 0, we have, by Definition 4.1.8 (ii), 󵄨󵄨 󵄨󵄨 d 󵄨 󵄨󵄨 −q−1 dt ≲ 1, 󵄨󵄨Dil20 (ςj )(x, t, t) dt 󵄨󵄨󵄨 = 󵄨󵄨󵄨∫ ς0 (x, t, t) dt 󵄨󵄨󵄨 ≲ ∫(1 + |t|) 󵄨 󵄨 ̂ C0∞ (Bq (a))⊗ ̂ proving (5.127) when j = 0. We now assume j > 0, and therefore ςj ∈ C0∞ (Ω3 )⊗ q S0 (ℝ ). By Corollary 4.1.19, we can write q

q

k=1

k=1

−1 ̃ ςj (x, s, t) = △t △−1 t ςj (x, s, t) = ∑ 𝜕tk (−𝜕tk △t ςj (x, s, t)) =: ∑ 𝜕tk ςj,k (x, s, t),

̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) is a bounded ̃ : (Ej , 2−j ) ∈ ℰ , j > 0, k ∈ {1, . . . , q}} ⊂ C0∞ (Ω3 )⊗ where {ςj,k set. Thus, we have, integrating by parts and using Definition 4.1.8 (ii), q 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨∫ Dildj (ςj )(x, t, t) dt 󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨∫ Dildj (𝜕t ςj,k ̃ )(x, s, t) dt 󵄨 󵄨 󵄨󵄨 󵄨 󵄨 󵄨 2 2 k 󵄨 󵄨 󵄨󵄨s=t 󵄨󵄨󵄨 k=1 󵄨 q 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 ̃ )(x, s, t))󵄨󵄨󵄨󵄨 dt 󵄨󵄨󵄨󵄨 = ∑ 2−jdk 󵄨󵄨󵄨∫(𝜕tk Dild2j (ςj,k 󵄨󵄨 󵄨󵄨s=t 󵄨󵄨 k=1 q 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 ̃ )(x, s, t))󵄨󵄨󵄨󵄨 dt 󵄨󵄨󵄨󵄨 = ∑ 2−jdk 󵄨󵄨󵄨∫(𝜕sk Dild2j (ςj,k 󵄨󵄨 󵄨󵄨s=t 󵄨󵄨 k=1 q 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 ̃ )(x, s, t))󵄨󵄨󵄨󵄨 dt 󵄨󵄨󵄨󵄨 = ∑ 2−jdk 󵄨󵄨󵄨∫(Dild2j (𝜕sk ςj,k 󵄨󵄨 󵄨󵄨s=t 󵄨󵄨 k=1 q 󵄨󵄨 󵄨 ̃ )(x, 2−jd t, t) dt 󵄨󵄨󵄨󵄨 = ∑ 2−jdk 󵄨󵄨󵄨∫(𝜕sk ςj,k 󵄨 󵄨 k=1 q

≲ ∑ 2−jdk ∫(1 + |t|)

−q−1

k=1

q

dt ≲ ∑ 2−jdk ≲ 2−j . k=1

This establishes (5.127) and completes the proof in the case N(x) = 0. It remains to prove the result when N(x) ∈ {1, . . . , n = dim M}. For each m ∈ {1, . . . , n}, let ϕm ∈ C0∞ (Bm (η1 /2)) satisfy ϕm ≡ 1 on Bm (η1 /4); note that we have only made n = dim M ≈ 1 such choices of ϕm . We will show that sup −j

∫

{(Ej ,2 ):j∈ℕ} N(x) B (η1 /4) ⊆ℰ

󵄨󵄨 󵄨p 󵄨󵄨ϕN(x) (u)Tf (Φx,1 (u))󵄨󵄨󵄨 du ≤ Dp

∫ BN(x) (η1 )

󵄨󵄨 󵄨p 󵄨󵄨f (Φx,1 (u))󵄨󵄨󵄨 du,

(5.128)

362 � 5 Singular integrals and the result will follow. Let Xjx,1 := Φ∗x,1 Xj , as in Theorem 3.5.1. We will apply the results in Section 5.10.1 with x

(Y , d̂) = (X x,1 , d ) = {(X1x,1 , d1 ), . . . , (X1 1 , dq )} to obtain (5.128). Thus, we need to show that the assumptions in Section 5.10.1 hold uniformly for x ∈ 𝒦. Pulling back (5.123) via Φx,1 , we have [Xjx,1 , Xkx,1 ] =

∑ dl ≤dj +dk

l cj,k ∘ Φx,1 Xl .

By Theorem 3.5.1 (j), using the fact that Φx,1 (BN(x) (1)) ⊆ Ω1 (see Theorem 3.5.1), we have, for every L ∈ ℕ, α l 󵄩 󵄩 l sup ‖cj,k ∘ Φx,1 ‖C L (BN(x) (1)) ≈ sup ∑ 󵄩󵄩󵄩(X x,1 ) cj,k ∘ Φx,1 󵄩󵄩󵄩C(BN(x) (1)) x∈𝒦

x∈𝒦 |α|≤L

󵄩 󵄩 󵄩 α l 󵄩 l 󵄩 = sup ∑ 󵄩󵄩󵄩X α cj,k 󵄩󵄩C(Φx,1 (BN(x) (1))) ≤ sup ∑ 󵄩󵄩󵄩X cj,k 󵄩󵄩󵄩C(Ω1 ) < ∞, x∈𝒦 |α|≤L

x∈𝒦 |α|≤L

l ∞ where the < ∞ follows from the fact that cj,k ∈ Cloc (M) and Ω1 ⋐ M. Combining this

with Theorem 3.5.1 (g) and (h), we see that the vector fields (Y , d̂) := (X x,1 , d ) satisfy the hypotheses of Section 5.10.1, uniformly for x ∈ 𝒦. With ςj as in (5.125), for x ∈ 𝒦, set ςj,x (u, s, t) := ϕN(x) ςj (Φx,1 (u), s, t).

(5.129)

̂ C0∞ (BN(x) (η1 /2))⊗ ̂ S (ℝq ) and ςx,j ∈ C0∞ (Ω3 )⊗ ̂ C0∞ (BN(x) (η1 /2))⊗ ̂ Note that ςx,j ∈ C0∞ (Ω3 )⊗ q S0 (ℝ ) for j > 0. We claim, for all L ∈ ℕ, sup

∑

L󵄨 γ 󵄨 sup (1 + |t|) 󵄨󵄨󵄨𝜕uα 𝜕sβ 𝜕t ςj,x (u, s, t)󵄨󵄨󵄨 < ∞, q

sup

(1) s∈B (a) |α|,|β|,|γ|≤L x∈𝒦 u∈B q N(x)=0 ̸ (Ej ,2−j )∈ℰ t∈ℝ N(x)

(5.130)

where ςj = ςj (Ej , 2−j ) is as in (5.125) and ςj,x = ςj,x (ςj ) is defined in (5.129). Indeed, by Theorem 3.5.1 (g) and (h), using the fact that Φx,1 (BN(x) (1)) ⊆ Ω1 (see Theorem 3.5.1), we have ∑

sup

sup

L󵄨 γ 󵄨 sup (1 + |t|) 󵄨󵄨󵄨𝜕uα 𝜕sβ 𝜕t ςj,x (u, s, t)󵄨󵄨󵄨

q |α|,|β|,|γ|≤L x∈𝒦 u∈BN(x) (1) s∈B (a) q N(x)=0 ̸ (Ej ,2−j )∈ℰ t∈ℝ

≲

∑

sup

sup

L󵄨 α γ 󵄨 sup (1 + |t|) 󵄨󵄨󵄨(Xux,1 ) 𝜕sβ 𝜕t ςj,x (u, s, t)󵄨󵄨󵄨 q

(1) s∈B (a) |α|,|β|,|γ|≤L x∈𝒦 u∈B q N(x)=0 ̸ (Ej ,2−j )∈ℰ t∈ℝ N(x)

5.10 Lp bounds in the single-parameter case

=

L󵄨 γ 󵄨 sup sup (1 + |t|) 󵄨󵄨󵄨Xyα 𝜕sβ 𝜕t ςj (y, s, t)󵄨󵄨󵄨 q x∈𝒦 N(x) s∈B (a) |α|,|β|,|γ|≤L (1)) q N(x)=0 ̸ y∈Φx,1 (B

sup

∑

(Ej ,2−j )∈ℰ

≤

� 363

sup

∑

t∈ℝ

L󵄨 γ 󵄨 sup (1 + |t|) 󵄨󵄨󵄨Xyα 𝜕sβ 𝜕t ςj (y, s, t)󵄨󵄨󵄨 < ∞,

q |α|,|β|,|γ|≤L y∈Ω1 s∈B (a) q (Ej ,2−j )∈ℰ t∈ℝ

̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) where the < ∞ follows from the fact that {ςj : (Ej , 2−j ) ∈ ℰ } ⊂ C0∞ (Ω3 )⊗ is a bounded set. This establishes (5.130). For (Ej , 2−j ) ∈ ℰ , we have x,1

ϕN(x) (u)Ef (Φx,1 (u)) = ∫ f ∘ Φx,1 (e−t⋅X x)ϕN(x) (u) Dild2j (ςj,x )(u, t, t) =: Ej,x (f ∘ Φx,1 )(u), where by (5.130), ℰx := {(Ex , 2−j ) : (E, 2−j ) ∈ ℰ } is a bounded set of (X x,1 , d ), a pseudodifferential operator scales supported in BN(x) (η1 /2) × BN(x) (η1 ), uniformly for x ∈ 𝒦, provided a > 0 is sufficiently small as described in Section 5.10.1. If {(Ej,x , 2−j ) : j ∈ ℕ} ⊆ ℰx , then by the above remarks, the results in Section 5.10.1 show that if a > 0 is sufficiently small (as described in that section, and therefore it can be chosen independent of x ∈ 𝒦) and Tx := ∑j∈ℕ Ej,x , then sup

sup

x∈𝒦 {(E ,2−j ):j∈ℕ} j,x ⊆ℰ

‖Tx ‖Lp (BN(x) ,σLeb )→Lp (BN(x) ,σLeb ) < ∞.

(5.131)

Furthermore, since ℰx is a bounded set of (X x,1 , d ), a pseudo-differential operator scales supported in BN(x) (η1 /2) × BN(x) (η1 ), we have supp(Tx ) ⊆ BN(x) (η1 /2) × BN(x) (η1 ). Combining this with (5.131), we have, for f ∈ C0∞ (Ω1 ), 󵄨 󵄨p 󵄨 󵄨p ∫󵄨󵄨󵄨ϕN(x) (u)Tf (ϕx,1 (u))󵄨󵄨󵄨 du = ∫󵄨󵄨󵄨Tx (f ∘ Φx,1 )(u)󵄨󵄨󵄨 du ≲

󵄨 󵄨p ∫ 󵄨󵄨󵄨f ∘ Φx,1 (u)󵄨󵄨󵄨 du.

BN(x) (1)

This establishes (5.128) and completes the proof. Proof of Proposition 5.10.4. In what follows, all implicit constants will be independent of the choice of {(Ej , 2−j ) : j ∈ ℕ} ⊆ ℰ with T = ∑j∈ℕ Ej . Since T is an (X, d ), a pseudo-differential operator supported in Ω3 × Ω4 , we have supp(T) ⋐ Ω3 × Ω4 . We will prove that, for 1 < p < ∞, ‖Tf ‖Lp (Ω3 ,Vol) ≲ ‖f ‖Lp (M,Vol) ,

∀f ∈ C0∞ (Ω4 ),

(5.132)

and the result will follow by the density of C0∞ (Ω4 ) in Lp (Ω4 , Vol). We have, for 1 < p < ∞,

364 � 5 Singular integrals 󵄨 󵄨p ∫ 󵄨󵄨󵄨Tf (x)󵄨󵄨󵄨 d Vol(x)

Ω3

≈∫

∫

Ω5 BN(x) (η1 /4)

≲∫

∫

Ω5 BN(x) (η1 )

󵄨󵄨 󵄨p 󵄨󵄨Tf (Φx,1 (u))󵄨󵄨󵄨 du d Vol(x)

󵄨󵄨 󵄨 󵄨󵄨f ∘ Φx,1 (u)󵄨󵄨󵄨 du d Vol(x)

󵄨 󵄨p ≈ ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 d Vol(x)

Proposition 3.9.1 Lemma 5.10.5 Proposition 3.9.1.

Ω4

Formula (5.132) follows, completing the proof. A square function: We now use Proposition 5.10.4 to develop a Littlewood–Paley square function. Fix a compact set 𝒦0 ⋐ Ω3 . Fix ψ ∈ C0∞ (Ω3 ) with ψ ≡ 1 on a neighborhood of 𝒦0 . Let a > 0 be a small number; in particular small enough that Proposition 5.10.4 applies. Example 4.1.6 shows that Mult[ψ] is an (X, d ), a pseudo-differential operator of order 0 supported in Ω3 × Ω4 , and therefore Theorem 4.3.3 shows that there is a bounded set of (X, d ), a pseudo-differential operators supported in Ω3 × Ω4 , {(Dj , 2−j ) : j ∈ ℕ}, with Mult[ψ] = ∑j∈ℕ Dj . Proposition 5.10.6. Suppose p ∈ (1, ∞). Then 1󵄩 󵄩󵄩 2󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 2 󵄨 󵄨 ≤ Cp ‖f ‖Lp (M,Vol) , 󵄩󵄩󵄩( ∑ 󵄨󵄨󵄨Dj f 󵄨󵄨󵄨 ) 󵄩󵄩󵄩 󵄩󵄩 j∈ℕ 󵄩󵄩 󵄩󵄩 󵄩󵄩Lp (M,Vol)

∀f ∈ Lp (M, Vol).

(5.133)

Moreover, if supp(f ) ⊆ 𝒦0 , we also have the reverse inequality,

‖f ‖Lp (M,Vol)

1󵄩 󵄩󵄩 2󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 2 󵄨 󵄨 ≤ Cp 󵄩󵄩󵄩( ∑ 󵄨󵄨󵄨Dj f 󵄨󵄨󵄨 ) 󵄩󵄩󵄩󵄩 , 󵄩󵄩 j∈ℕ 󵄩󵄩 󵄩󵄩 󵄩󵄩Lp (M,Vol)

∀f ∈ Lp (𝒦0 , Vol).

(5.134)

Here, Cp ≥ 0 does not depend on f , but may depend on any of the other ingredients in the proposition. Before we prove Proposition 5.10.6, we present a corollary which is how we will use the proposition in our applications. Corollary 5.10.7. Let {ϵj }j∈ℕ be a sequence of i. i. d. random variables of mean 0 taking values in ±1 and fix p ∈ (1, ∞). We have, for all f ∈ Lp (𝒦0 , Vol), 1 1󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩p p 2󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 2 󵄨 󵄨 󵄩 󵄩 (𝔼 󵄩󵄩󵄩 ∑ ϵj Dj f 󵄩󵄩󵄩 ) ≈ 󵄩󵄩󵄩( ∑ 󵄨󵄨󵄨Dj f 󵄨󵄨󵄨 ) 󵄩󵄩󵄩󵄩 ≈ ‖f ‖Lp (M,Vol) . 󵄩󵄩j∈ℕ 󵄩󵄩 p 󵄩󵄩 j∈ℕ 󵄩󵄩 󵄩 󵄩L (M,Vol) 󵄩󵄩 󵄩󵄩Lp (M,Vol)

5.10 Lp bounds in the single-parameter case

� 365

Proof. It follows from the Khintchine inequality (Theorem 2.4.20) that 1󵄩 1 󵄩󵄩 1󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩p p 󵄩󵄩 󵄨󵄨 󵄨󵄨p p 󵄩󵄩󵄩 2󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄨 󵄨 󵄩󵄩 2 󵄩 󵄨 󵄨 󵄩 󵄨 󵄨 󵄩 󵄩 󵄩 (𝔼 󵄩󵄩󵄩 ∑ ϵj Dj f 󵄩󵄩󵄩 ) = 󵄩󵄩󵄩(𝔼󵄨󵄨󵄨 ∑ ϵj Dj f 󵄨󵄨󵄨 ) 󵄩󵄩󵄩 ≈ 󵄩󵄩󵄩( ∑ 󵄨󵄨󵄨Dj f 󵄨󵄨󵄨 ) 󵄩󵄩󵄩 . 󵄩󵄩 󵄨󵄨 󵄨󵄨 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩j∈ℕ 󵄩󵄩 p 󵄩󵄩 󵄨 j∈ℕ 󵄨 󵄩󵄩 p 󵄩󵄩 j∈ℕ 󵄩󵄩Lp (M,Vol) 󵄩 󵄩L (M,Vol) 󵄩 󵄩L (M,Vol)

It follows from Proposition 5.10.6 that 1󵄩 󵄩󵄩 2󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 2 󵄨 󵄨 󵄩󵄩( ∑ 󵄨󵄨󵄨Dj f 󵄨󵄨󵄨 ) 󵄩󵄩󵄩 ≈ ‖f ‖Lp (M,Vol) . 󵄩󵄩 󵄩󵄩 󵄩󵄩 p 󵄩󵄩󵄩 j∈ℕ 󵄩L (M,Vol)

We now turn to the proof of Proposition 5.10.6. A difficulty we must overcome is that Dj do not satisfy a nice version of the Calderón reproducing formula. However, we are able get around this by using the fact that Dj are “almost orthogonal.” For j ∈ ℤ \ ℕ, set Dj := 0. For M > 0, set UM := ∑ Dj Dj+l , j∈ℕ |l|≤M

RM := ∑ Dj Dj+l . j∈ℕ |l|>M

Note that UM + RM = ∑j,k∈ℕ Dj Dk = Mult[ψ2 ]. Proposition 5.10.8 (Calderón reproducing type formula). Fix p ∈ (1, ∞) and ψ1 ∈ C0∞ (Ω3 ) with ψ ≡ 1 on a neighborhood of supp(ψ1 ). There exist M = M(p) and VM : Lp (M, Vol) → Lp (M, Vol) such that Mult[ψ1 ]UM VM = VM UM Mult[ψ1 ] = Mult[ψ1 ]. To prove Proposition 5.10.8, we use the next lemma. Lemma 5.10.9. Fix p ∈ (1, ∞). Then lim ‖RM ‖Lp (M,Vol)→Lp (M,Vol) = 0.

M→∞

Before we prove Lemma 5.10.9, we see how it implies Proposition 5.10.8. Proof of Proposition 5.10.8. Recall that UM = Mult[ψ2 ]−RM ; Lemma 5.10.9 shows that RM is bounded on Lp , and therefore the same is true of UM . Take ψ2 ∈ C0∞ (Ω3 ) with ψ1 ≺ ψ2 ≺ ψ. By Lemma 5.10.9, we may take M = M(p) so large that ‖RM Mult[ψ2 ]‖Lp →Lp < 1. Define ∞ l l VM := ∑∞ l=0 Mult[ψ2 ](RM Mult[ψ2 ]) = ∑l=0 (Mult[ψ2 ]RM ) Mult[ψ2 ], with convergence in p p the uniform operator topology on Hom(L , L ). Using the fact that Mult[ψ1 ] Mult[ψ2 ] = Mult[ψ1 ] and Mult[ψ2 ] Mult[ψ2 ] = Mult[ψ2 ], we have ∞

l

Mult[ψ1 ]UM VM = Mult[ψ1 ](Mult[ψ2 ] − RM ) (∑ Mult[ψ2 ](RM Mult[ψ2 ]) ) l=0

∞

l

∞

l

= Mult[ψ1 ](∑ Mult[ψ2 ](RM Mult[ψ2 ]) − ∑ RM Mult[ψ2 ](RM Mult[ψ2 ]) ) l=0

l=0

366 � 5 Singular integrals ∞

l

∞

l

= Mult[ψ1 ] (∑ Mult[ψ2 ](RM Mult[ψ2 ]) − ∑ Mult[ψ2 ](RM Mult[ψ2 ]) ) l=0

l=1

= Mult[ψ1 ] Mult[ψ2 ] = Mult[ψ1 ]. A similar proof gives VM UM Mult[ψ1 ] = Mult[ψ1 ], which completes the proof. Lemma 5.10.9 follows by interpolating the next two lemmas. Lemma 5.10.10. For p ∈ (1, ∞), M ≥ 1, ‖RM ‖Lp (M,Vol)→Lp (M,Vol) ≤ Cp M. Lemma 5.10.11. For every N ≥ 0, ‖RM ‖L2 (M,Vol)→L2 (M,Vol) ≤ CM 2−NM . To prove Lemma 5.10.10, we require two more lemmas, the first of which implies (5.133). Lemma 5.10.12. Fix p ∈ (1, ∞). Then 1󵄩 󵄩󵄩 2󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄨󵄨 󵄨󵄨2 󵄩󵄩󵄩 ≤ Cp ‖f ‖Lp (M,Vol) , 󵄩󵄩( ∑ 󵄨󵄨Dj f 󵄨󵄨 ) 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 j∈ℕ 󵄩󵄩Lp (M,Vol)

∀f ∈ Lp (M, Vol).

The same result holds with Dj replaced by D∗j throughout. Proof. In the proof that follows, we only use the fact that {(Dj , 2−j ) : j ∈ ℕ} is a bounded set of (X, d ), a pseudo-differential operator scales supported in Ω3 × Ω4 . By taking a > 0 small enough, Lemma 4.3.11 (ii) implies that {(D∗j , 2−j ) : j ∈ ℕ} is a bounded set of (X, d ), a pseudo-differential operator scales supported in Ω4 × Ω5 . Thus, the same proof applies with D∗j in place of Dj , so it suffices to consider only the result for Dj . For any sequence {ϵj }j∈ℕ ⊆ {±1}, let T = T[{ϵj }j∈ℕ ] = ∑j∈ℕ ϵj Dj . Since {(ϵj Dj , 2−j ) : ϵj ∈ {±1}, j ∈ ℕ} is a bounded set of (X, d ), a pseudo-differential operator scales supported in Ω3 × Ω4 , Proposition 5.10.4 implies that sup {ϵj }j∈ℕ ⊆{±1}

‖T[{ϵj }]‖Lp →Lp < ∞.

(5.135)

Now let {ϵj }j∈ℕ be a sequence of i. i. d. random variables of mean 0 taking values ±1. Using the Khintchine inequality (Theorem 2.4.20) and (5.135), we have 1 1󵄩 󵄩󵄩 󵄩󵄩󵄩 󵄨󵄨 󵄩󵄩 󵄩󵄩p p1 󵄨󵄨p p 󵄩󵄩󵄩󵄩 2󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄨 󵄨 󵄩 󵄩 󵄩 2 󵄩󵄩 󵄨 󵄨 󵄩 󵄨 󵄨 󵄩󵄩( ∑ 󵄨󵄨󵄨Dj f 󵄨󵄨󵄨 ) 󵄩󵄩󵄩 ≈ 󵄩󵄩󵄩󵄩(𝔼󵄨󵄨󵄨 ∑ ϵj Dj 󵄨󵄨󵄨 ) 󵄩󵄩󵄩󵄩 = (𝔼 󵄩󵄩󵄩󵄩 ∑ ϵj Dj f 󵄩󵄩󵄩󵄩 ) 󵄩󵄩 󵄩󵄩 󵄨 󵄨 󵄩󵄩 󵄩󵄩j∈ℕ 󵄩󵄩 p 󵄨󵄨 󵄩󵄩Lp 󵄩󵄩󵄩󵄩 󵄨󵄨 j∈ℕ 󵄩󵄩Lp 󵄩 󵄩L 󵄩󵄩 j∈ℕ 1 1 p p 󵄩 󵄩 󵄩 󵄩 = (𝔼󵄩󵄩󵄩T[{ϵj }]f 󵄩󵄩󵄩Lp ) p ≲ (𝔼󵄩󵄩󵄩f 󵄩󵄩󵄩Lp ) p = ‖f ‖Lp ,

completing the proof.

5.10 Lp bounds in the single-parameter case

�

367

Lemma 5.10.13. For p ∈ (1, ∞) and M ≥ 1, the sum UM = ∑|l|≤M ∑j∈ℕ Dj Dj+l converges in weak operator topology on Hom(Lp , Lp ) and ‖UM ‖Lp →Lp ≤ Cp M. Proof. Let q ∈ (1, ∞) be dual to p (i. e., p1 + q1 = 1). We will show that for f ∈ Lp and g ∈ Lq with ‖g‖Lq = 1, we have 󵄨 󵄨 ∑ ∑ 󵄨󵄨󵄨⟨g, Dj Dj+l f ⟩󵄨󵄨󵄨 ≲ M‖f ‖Lp ,

|l|≤M j∈ℕ

which will complete the proof. Indeed, we have, using Lemma 5.10.12, 󵄨 󵄨 󵄨 󵄨 ∑ ∑ 󵄨󵄨󵄨⟨g, Dj Dj+l f ⟩󵄨󵄨󵄨 = ∑ ∑ 󵄨󵄨󵄨⟨D∗j g, Dj+l f ⟩󵄨󵄨󵄨

|l|≤M j∈ℕ

|l|≤M j∈ℕ

≤ ∑ ∫(∑ |l|≤M

j∈ℕ

≤ ∑ ∫(∑ |l|≤M

j∈ℕ

|D∗j g|2 ) |D∗j g|2 )

1 2

1 2

2

( ∑ |Dj+l f | ) d Vol j∈ℕ

1 2

2

1 2

( ∑ |Dj f | ) d Vol j∈ℕ

1󵄩 1󵄩 󵄩󵄩 󵄩 2󵄩 2󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 ∗ 2 2 ≤ 2M 󵄩󵄩( ∑ |Dj g| ) 󵄩󵄩 󵄩󵄩( ∑ |Dj f | ) 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 j∈ℕ j∈ℕ 󵄩󵄩Lq 󵄩󵄩 󵄩󵄩Lp 󵄩󵄩

≲ M‖g‖Lq ‖f ‖Lp = M‖f ‖Lp , as desired, completing the proof.

Proof of Lemma 5.10.10. Since RM = Mult[ψ2 ] − UM , it suffices to prove the result with UM in place of RM , and this follows from Lemma 5.10.13. Proof of Lemma 5.10.11. Since {(Dj , 2−j ) : j ∈ ℕ} is a bounded set of (X, d ), a pseudodifferential operator scales supported in Ω3 ×Ω4 , if a > 0 is small enough, Lemma 4.3.11 (ii) implies that {(D∗j , 2−j ) : j ∈ ℕ} is a bounded set of (X, d ), a pseudo-differential operator scales supported in Ω4 × Ω5 . Thus, if a > 0 is small enough, Proposition 5.6.3 (with (W , ds) = (X, d )) implies that {(Dj , 2−j ), (D∗j , 2−j ) : j ∈ ℕ} is a bounded set of generalized (X, d ) elementary operators supported in Ω1 . Thus, Proposition 5.5.11 implies that for every N ∈ ℕ, {(2N|l| Dj Dj+l , 2−j ), (2N|j1 −j2 | Dj1 D∗j2 , 2−j1 ), (2−N|l| D∗j+l D∗j , 2−j ), (2N|j1 −j2 | D∗j1 Dj2 , 2−j1 ) : j, j1 ∈ ℕ, j2 , l ∈ ℤ}

is a bounded set of generalized (X, d ) elementary operators supported in Ω1 . Thus, we have (see Definition 5.2.26), for every N ∈ ℕ, j, j1 ∈ ℕ, and j2 , l ∈ ℤ, ‖Dj Dj1 +l ‖L2 →L2 ≲ 2−N|l| ,

‖Dj1 D∗j2 ‖L2 →L2 ≲ 2−N|j1 −j2 | ,

368 � 5 Singular integrals ‖D∗j+l D∗j ‖L2 →L2 ≲ 2−N|l| , ‖D∗j1 Dj2 ‖L2 →L2 ≲ 2−N|j1 −j2 | . It follows that for every N ∈ ℕ, there exists CN ≥ 0 such that for all j1 , j2 ∈ ℕ and l1 , l2 ∈ ℤ, ‖Dj Dj1 +l1 D∗j2 +l2 D∗j2 ‖L2 →L2 , ‖D∗j1 +l1 D∗j1 Dj2 Dj2 +l2 ‖L2 →L2 ≤ CN 2−N diam{j1 ,j1 +l1 ,j2 +l2 ,j2 } , where diam denotes the diameter of the set. The Cotlar–Stein lemma (Lemma 2.3.25) applies to show that ∀N ∈ ℕ, ‖RM ‖L2 →L2 ≲ sup ∑ 2−N diam{j1 ,j1 +l1 ,j2 +l2 ,j2 }/2 ≲ 2−NM/2 . j1 ∈ℕ j2 ∈ℕ |l1 |>M |l2 |>M

Replacing N with 2N completes the proof. Proof of Proposition 5.10.6. Formula (5.133) follows from Lemma 5.10.12, so we prove only (5.134). Fix ψ1 ∈ C0∞ (Ω3 ), with ψ1 ≡ 1 on a neighborhood of 𝒦0 , and ψ1 ≺ ψ. Fix p ∈ (1, ∞) and take M = M(p) and VM as in Proposition 5.10.8, so that Mult[ψ1 ]UM VM = VM UM Mult[ψ1 ] = Mult[ψ1 ]. Let q ∈ (1, ∞) be dual to p (i. e., p1 + q1 = 1), so that VM∗ : Lq → Lq is bounded. Let g ∈ Lq (M, Vol) with ‖g‖Lq = 1. Let f ∈ Lp (𝒦0 , Vol). We have 󵄨󵄨 󵄨 󵄨󵄨⟨g, f ⟩󵄨󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨⟨g, Mult[ψ1 ]f ⟩󵄨󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨⟨g, VM UM Mult[ψ1 ]f ⟩󵄨󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨⟨VM∗ g, UM f ⟩󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 = 󵄨󵄨󵄨 ∑ ∑ ⟨VM∗ g, Dj Dj+l f ⟩󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 |l|≤M j∈ℕ 󵄨 󵄨󵄨 ∗ ∗ 󵄨 ≤ ∑ ∑ 󵄨󵄨⟨Dj VM g, Dj+l f ⟩󵄨󵄨󵄨

using Mult[ψ1 ]f = f using VM UM Mult[ψ1 ] = Mult[ψ1 ] using Mult[ψ1 ]f = f using Lemma 5.10.13

|l|≤M j∈ℕ

󵄨 󵄨󵄨 󵄨 ≤ ∑ ∫ ∑ 󵄨󵄨󵄨D∗j VM∗ g 󵄨󵄨󵄨󵄨󵄨󵄨Dj+l f 󵄨󵄨󵄨 d Vol |l|≤M

j∈ℕ

1 2

1 2

󵄨 󵄨2 󵄨 󵄨2 ≤ ∑ ∫ ( ∑ 󵄨󵄨󵄨D∗j VM∗ g 󵄨󵄨󵄨 ) ( ∑ 󵄨󵄨󵄨Dj+l f 󵄨󵄨󵄨 ) d Vol

using Cauchy–Schwartz

1󵄩 1󵄩 󵄩󵄩 󵄩 2󵄩 2󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄨󵄨 ∗ ∗ 󵄨󵄨2 󵄩󵄩󵄩 󵄩󵄩󵄩 󵄨󵄨 󵄨󵄨2 󵄩󵄩󵄩 ≤ ∑ 󵄩󵄩( ∑ 󵄨󵄨Dj VM g 󵄨󵄨 ) 󵄩󵄩 󵄩󵄩( ∑ 󵄨󵄨Dj f 󵄨󵄨 ) 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 |l|≤M 󵄩 󵄩󵄩Lq 󵄩󵄩 j∈ℕ 󵄩󵄩Lp 󵄩󵄩 j∈ℕ

using Hölder’s inequality

|l|≤M

j∈ℕ

j∈ℕ

5.11 Parametrices � 369 1󵄩 󵄩󵄩 2󵄩 󵄩 󵄩 󵄩󵄩 ∗ 󵄩󵄩 󵄩󵄩󵄩 󵄨󵄨 󵄨󵄨2 󵄩󵄩󵄩 ≲ M 󵄩󵄩VM g 󵄩󵄩Lq 󵄩󵄩( ∑ 󵄨󵄨Dj f 󵄨󵄨 ) 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 j∈ℕ 󵄩󵄩Lp 1󵄩 󵄩󵄩 2󵄩 󵄩 󵄩󵄩󵄩 󵄨 󵄨2 󵄩󵄩 ≲ 󵄩󵄩󵄩( ∑ 󵄨󵄨󵄨Dj f 󵄨󵄨󵄨 ) 󵄩󵄩󵄩 󵄩󵄩󵄩 j∈ℕ 󵄩󵄩󵄩 p 󵄩 󵄩L

using Lemma 5.10.12 using VM : Lq → Lq , M ≈ 1, ‖g‖Lq = 1.

Taking the supremum over all such g establishes (5.134) and completes the proof.

5.11 Parametrices ̃t (Ω, (W , ds)) ̃t (Ω, (W , ds)) ⃗ and A ⃗ are useful to us is that The main reason the algebras A parametrices for maximally subelliptic PDEs are locally in these algebras. In this section we present the main results we will use to deduce that the parametrices lie in these algebras. As in the previous sections, we consider the Hörmander setting of Section 5.1.2 and ⃗ the general setting of Section 5.1.3. The Schwartz kernels of operators in A t (Ω, (W , ds)) t ̃ ⃗ and A (Ω, (W , ds)) have compact support, which will usually not be the case for the parametrices we consider. To get around this issue we introduce the next definition; for this, ̃t (W , ds)⃗ = ⋃Ω⋐M A ̃t (Ω, (W , ds)) ⃗ and A ⃗ (see recall that A t (W , ds)⃗ = ⋃Ω⋐M A t (Ω, (W , ds)) Remark 5.8.8). Definition 5.11.1. Let t ∈ ℝν . t Hörmander setting: We let Aloc (W , ds)⃗ be the set of all T ∈ Hom(C0∞ (M), C0∞ (M)′ ) ⃗ such that ∀ψ1 , ψ2 ∈ C0∞ (M), Mult[ψ1 ]T Mult[ψ2 ] ∈ A t (W , ds). t ̃ (W , ds)⃗ be the set of all T ∈ Hom(C ∞ (M), C ∞ (M)′ ) such General setting: We let A W ,0 W ,0 loc ∞ ̃t (W , ds). ⃗ that ∀ψ1 , ψ2 ∈ C (M), Mult[ψ1 ]T Mult[ψ2 ] ∈ A 0

⃗ (Z, dr), ⃗ and ν1 be as in Section 5.9 and let t ∈ ℝν1 . Definition 5.11.2. Let (Y , d̂), t ⃗ (Z, dr)) ⃗ be the set of all T ∈ Hom(C ∞ (M), Hörmander setting: We let Dloc ((Y , d̂), 0 ∞ ′ ∞ ⃗ (Z, dr)) ⃗ (see C0 (M) ) such that ∀ψ1 , ψ2 ∈ C0 (M), Mult[ψ1 ]T Mult[ψ2 ] ∈ D t ((Y , d̂), Definition 5.9.2). ⃗ (Z, dr)) ⃗ be the set of all T ∈ Hom(C ∞ (M), C ∞ (M)′ ) ̃t ((Y , d̂), General setting: We let D Y ,0 Y ,0 loc ∞ ⃗ (Z, dr)) ⃗ (see Definĩt ((Y , d̂), such that ∀ψ1 , ψ2 ∈ CY ,0 (M), Mult[ψ1 ]T Mult[ψ2 ] ∈ D tion 5.9.2). Definition 5.11.3. Fix t ∈ ℝν and D1 , D2 ∈ ℕ+ . ⃗ ℂD1 , ℂD2 ) be the set of all T ∈ Hom(C ∞ (M; ℂD1 ), Hörmander setting: We let A t ((W , ds); 0 ∞ D2 ′ ⃗ Similarly, we let C0 (M; ℂ ) ) which are D2 × D1 matrices of operators in A t (W , ds). t ⃗ ℂD1 , ℂD2 ) be the set of all T ∈ Hom(C ∞ (M; ℂD1 ), C ∞ (M; ℂD2 )′ ) which Aloc ((W , ds); 0 0 t ⃗ Also, for Ω ⋐ M open and relatively are D2 × D1 matrices of operators in Aloc (W , ds).

370 � 5 Singular integrals ⃗ ℂD1 , ℂD2 ) be the set of all T ∈ A t ((W , ds); ⃗ ℂD1 , ℂD2 ) compact, we let A t (Ω, (W , ds); with supp(T) ⊆ Ω × Ω. ̃t ((W , ds); ⃗ ℂD1 , ℂD2 ) be the set of all T ∈ Hom(C ∞ (M; ℂD1 ), General setting: We let A W ,0 ∞ D2 ′ ̃t ⃗ CW ,0 (M; ℂ ) ) which are D2 × D1 matrices of operators in A (W , ds). Similarly, we ̃t ((W , ds); ⃗ ℂD1 , ℂD2 ) be the set of all T ∈ Hom(C ∞ (M; ℂD1 ), C ∞ (M; ℂD2 )′ ) let A W ,0 W ,0 loc ̃t (W , ds). ⃗ Also, for Ω ⋐ M open and which are D2 × D1 matrices of operators in A loc ̃t (Ω, (W , ds); ̃t ((W , ds); ⃗ ℂD1 , ℂD2 ) be the set of all T ∈ A ⃗ relatively compact, we let A ℂD1 , ℂD2 ) with supp(T) ⊆ Ω × Ω.

̃. We have a definition similar to Definition 5.11.3 for the algebras D and D ⃗ (Z, dr), ⃗ and ν1 be as in Section 5.9, let t ∈ ℝν1 , and let Definition 5.11.4. Let (Y , d̂), D1 , D2 ∈ ℕ+ . ⃗ (Z, dr); ⃗ ℂD1 , ℂD2 ) be the set of all Hörmander setting: We let D t ((Y , d̂), T ∈ Hom(C0∞ (M; ℂD1 ), C0∞ (M; ℂD2 )′ ) ⃗ (Z, dr)). ⃗ Similarly, we let which are D2 × D1 matrices of operators in D t ((Y , d̂), t D1 D2 ∞ ⃗ ⃗ Dloc ((Y , d̂), (Z, dr); ℂ , ℂ ) be the set of all T ∈ Hom(C0 (M; ℂD1 ), C0∞ (M; ℂD2 )′ ) t ⃗ (Z, dr)). ⃗ Also, for Ω ⋐ M which are D2 × D1 matrices of operators in Dloc ((Y , d̂), t ⃗ (Z, dr); ⃗ ℂD1 , ℂD2 ) be the set of all open and relatively compact, we let D (Ω, (Y , d̂), ⃗ (Z, dr); ⃗ ℂD1 , ℂD2 ) with supp(T) ⊆ Ω × Ω. T ∈ D t ((Y , d̂), ⃗ (Z, dr); ⃗ ℂD1 , ℂD2 ) be the set of all ̃t ((Y , d̂), General setting: We let D T ∈ Hom(CY∞,0 (M; ℂD1 ), CY∞,0 (M; ℂD2 )′ ) ⃗ (Z, dr)). ⃗ Similarly, we let ̃t ((Y , d̂), which are D2 × D1 matrices of operators in D t D D ∞ ⃗ (Z, dr); ⃗ ℂ 1 , ℂ 2 ) be the set of all T ∈ Hom(C (M; ℂD1 ), C ∞ (M; ℂD2 )′ ) ̃ ((Y , d̂), D Y ,0 Y ,0 loc ⃗ (Z, dr)). ⃗ Also, for Ω ⋐ M ̃t ((Y , d̂), which are D2 × D1 matrices of operators in D loc ⃗ (Z, dr); ⃗ ℂD1 , ℂD2 ) be the set of all ̃t (Ω, (Y , d̂), open and relatively compact, we let D t D1 D2 ⃗ ⃗ ̃ T ∈ D ((Y , d̂), (Z, dr); ℂ , ℂ ) with supp(T) ⊆ Ω × Ω. 5.11.1 Parametrices via heat equations The main way we will construct parametrices for general maximally subelliptic PDEs is via heat equations, and in this section, we describe how this works. The setting is the single-parameter setting of Section 5.1.1; thus, we are given Hörmander vector fields ∞ with formal degrees on M: (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} ⊂ Cloc (M; TM) × ℕ+ . α ∞ Fix κ, D ∈ ℕ+ and let L := ∑degds (α)≤2κ aα (x)W , where aα ∈ Cloc (M; 𝕄D×D (ℂ)). We think of L as a densely defined operator on L2 (M, Vol; ℂD ) with dense domain C0∞ (M; ℂD ). We assume L is non-negative and symmetric, and we let L be a nonnegative, self-adjoint extension of L . The goal of this section is to give conditions on the

5.11 Parametrices

� 371

−2κ heat operator e−tL which ensure L has a two-sided parametrix in Aloc ((W , ds); ℂD , ℂD ); 2κ D D because L ∈ Aloc ((W , ds); ℂ , ℂ ) (see Propositions 5.8.3 and 5.8.2), this is the best possible situation. ∞ Definition 5.11.5. We say ℰ ⊂ Cloc (M × M) × (0, 1] is a bounded set of locally (W , ds) preelementary operators if for every relatively compact, open set Ω ⋐ M, for all ordered multi-indices α and β, and for all m ∈ ℕ, there exists C = C(ℰ , Ω, α, β, m) ≥ 0 such that, ∀(F, 2−j ) ∈ ℰ , ∀x, z ∈ Ω,

(1 + 2j ρ(W ,ds) (x, z))−m α −jds β 󵄨󵄨 −jds 󵄨 . 󵄨󵄨(2 Wx ) (2 Wz ) F(x, z)󵄨󵄨󵄨 ≤ C Vol(B(W ,ds) (x, 2−j + ρ(W ,ds) (x, z))) ∧ 1 Remark 5.11.6. We define bounded sets of (W , ds) pre-elementary operators supported in Ω, bounded sets of (W , ds) elementary operators supported in Ω, and bounded sets of locally (W , ds) pre-elementary operators taking values in 𝕄D×D (instead of ℂ) in the obvious way. In this section, all such operators take values in 𝕄D×D , and for A ∈ 𝕄D×D we write |A| for the usual operator norm as an operator on ℂD . The main result of this section is the next theorem. Theorem 5.11.7. Suppose {(e−tL , t 1/2κ ) : t ∈ (0, 1]} is a bounded set of locally (W , ds) pre-elementary operators. Then there exists S ∈ −2κ Aloc ((W , ds); ℂD , ℂD ) such that L S, S L ≡ I

∞ mod Cloc (M × M; 𝕄D×D (ℂ)).

There are many constructions of parametrices in the literature where the conclusions are similar to Theorem 5.11.7; see for example [200, 182, 140]. However, these references all assume that 2κ < Q1 , where Q1 is as in Corollary 3.3.9. This allows those references to use a definition like A4−2κ (Definition 5.2.4) with M = 0. Here, we use our more general definitions to construct a parametrix no matter how large κ is. To prove Theorem 5.11.7 we require several preliminary results. Throughout this section, we let Ω ⋐ M be a relatively compact, open set. Lemma 5.11.8. Let ℰ1 be a bounded set of (W , ds) pre-elementary operators supported in ∞ Ω and let ℰ2 be a bounded set of locally (W , ds) pre-elementary operators. Let ψ ∈ Cloc (M) and let α be an ordered multi-index. Then: (i) {(Mult[ψ](2−jdsW )α F, 2−j ), (F Mult[ψ](2−jdsW )α , 2−j ) : (F, 2−j ) ∈ ℰ1 } is a bounded set of (W , ds) pre-elementary operators supported in Ω. (ii) {(Mult[ψ](2−jdsW )α F, 2−j ), (F Mult[ψ](2−jdsW )α , 2−j ) : (F, 2−j ) ∈ ℰ2 } is a bounded set of locally (W , ds) pre-elementary operators.

372 � 5 Singular integrals Proof. This follows immediately from the definitions. Lemma 5.11.9. For all ordered multi-indices α and β and all m ∈ ℕ, there exists N ∈ ℕ such that the following holds. Let ℰ1 be a bounded set of locally (W , ds) pre-elementary operators and let ℰ2 be a bounded set of (W , ds) pre-elementary operators supported in Ω. Then there exists C = C(ℰ1 , ℰ2 , m, α, β, Ω, (W , ds)) ≥ 0 such that for all x ∈ Ω, z ∈ M, (F1 , 2−j1 ) ∈ ℰ1 , (F2 , 2−j2 ) ∈ ℰ2 , k ∈ {j1 , j2 }, we have α β 󵄨 󵄨 2−N|j1 −j2 | 󵄨󵄨󵄨(2−kdsWx ) (2−kdsWz ) [F1 F2 ](x, z)󵄨󵄨󵄨

≤C

(1 + 2k ρ(W ,ds) (x, z))−m

Vol(B(W ,ds) (x, 2−k + ρ(W ,ds) (x, z))) ∧ 1

.

(5.136)

Proof. Fix Ω1 ⋐ M with Ω ⋐ Ω1 and let ψ ∈ C0∞ (Ω1 ) satisfy ψ ≡ 1 on Ω. Note that for (F1 , 2−j1 ) ∈ ℰ1 , (F2 , 2−j2 ) ∈ ℰ2 , [F1 F2 ](x, z) = [F̃1 F2 ](x, z),

x ∈ Ω, z ∈ M,

(5.137)

where F̃1 = Mult[ψ]F1 Mult[ψ]. Also, directly from the definitions, {(F̃1 , 2−j1 ) : (F1 , 2−j1 ) ∈ ℰ1 } ⋃ ℰ2 is a bounded set of (W , ds) pre-elementary operators supported in Ω1 . Lemma 5.5.12 implies (5.136) with F1 replaced by F̃1 (here, we are using Remark 5.2.19), and the result follows by (5.137). Lemma 5.11.10. Let ℰ1 be a bounded set of locally (W , ds) pre-elementary operators, let ℰ2 be a bounded set of (W , ds) elementary operators supported in Ω, and let ψ ∈ C0∞ (Ω). Then, for all N ∈ ℕ, the set {(2Nj Mult[ψ]FE, 2−j ) : (F, 1) ∈ ℰ1 , (E, 2−j ) ∈ ℰ2 } is a bounded set of (W , ds) elementary operators supported in Ω. Proof. Since ⋃(E,2−j )∈ℰ2 supp(E) ⋐ Ω, we may pick ψ1 ∈ C0∞ (Ω) which is equal to 1 on

a large enough set that Mult[ψ1 ]E = E, ∀(E, 2−j ) ∈ ℰ2 . It follows immediately from the definitions that {(Mult[ψ]F Mult[ψ1 ], 1) : (F, 1) ∈ ℰ1 }

(5.138)

is a bounded set of (W , ds) pre-elementary operators supported in Ω. Since we are only considering those (F, 2−j ) with j = 0, it is in fact a bounded set of (W , ds) elementary operators supported in Ω: in (5.2), when j = 0, one may take E0,0 = E and Eα,β = 0 for |α|, |β| > 0.

5.11 Parametrices

� 373

Because (5.138) is a bounded set of (W , ds) elementary operators supported in Ω, Proposition 5.5.11 shows that for every N ∈ ℕ, {(2Nj Mult[ψ]FE, 2−j ) : (F, 1) ∈ ℰ1 , (E, 2−j ) ∈ ℰ2 }

= {(2Nj Mult[ψ]F Mult[ψ1 ]E, 2−j ) : (F, 1) ∈ ℰ1 , (E, 2−j ) ∈ ℰ2 }

is a bounded set of (W , ds) elementary operators supported in Ω, completing the proof. Lemma 5.11.11. For all ordered multi-indices α and β and all m ∈ ℕ, there exists M = M(α, β, m, (W , ds), Ω) ∈ ℕ such that the following holds. Let ℰ be a bounded set of (W , ds) pre-elementary operators supported in Ω. Let Ft (x, z) = F(t, x, z) : (0, 1]×M×M → 𝕄D×D be a measurable function such that {(Ft , t 1/2κ ) : t ∈ (0, 1]} is a bounded set of locally (W , ds) pre-elementary operators. Then there exists C ≥ 0 such that, ∀x ∈ Ω, ∀z ∈ M, ∀(G, 2−j ) ∈ ℰ , ∀M1 ≥ M, 󵄨󵄨 󵄨󵄨 1 M1 󵄨󵄨 󵄨󵄨 t 2−2κj α −jds β 󵄨󵄨 −jds 󵄨 ) [Ft G](x, z)󵄨󵄨󵄨 󵄨󵄨(2 Wx ) (2 Wz ) ∫ ( −2κj ∧ 󵄨󵄨 󵄨󵄨 t 2 󵄨󵄨 󵄨󵄨 0 ≤ C2−2κj

(1 + 2j ρ(W ,ds) (x, z))−m

Vol(B(W ,ds) (x, 2−j + ρ(W ,ds) (x, z))) ∧ 1

(5.139)

.

Proof. We may replace M1 with M as the left-hand side of (5.139) is decreasing in M1 . By taking M sufficiently large and applying Lemma 5.11.9 (with M − 2 playing the role of N in that lemma), we have, for x ∈ Ω and z ∈ M, 󵄨󵄨 󵄨󵄨 1 M 󵄨󵄨 󵄨󵄨 t 2−2κj α −jds β 󵄨󵄨 −jds 󵄨 ) [Ft G](x, z)󵄨󵄨󵄨 󵄨󵄨(2 Wx ) (2 Wz ) ∫ ( −2κj ∧ 󵄨󵄨 󵄨󵄨 t 2 󵄨󵄨 󵄨󵄨 0 (1 + 2j ρ(W ,ds) (x, z))−m

≲

Vol(B(W ,ds) (x, 2−j + ρ(W ,ds) (x, z))) ∧ 1

1

∫( 0

t

2−2κj

∧

2

2−2κj ) dt. t

But, 1

∫( 0

t

2−2κj

2

2−2κj

1

0

0

2−2κj t2 2−4κj ∧ ) dt = ∫ −4κj dt + ∫ 2 dt ≲ 2−2κj . t t 2

This completes the proof. For the remainder of this section, we assume the hypotheses of Theorem 5.11.7. Lemma 5.11.12. For all ordered multi-indices α and β and all m ∈ ℕ, there exists M = M(α, β, m, (W , ds), Ω) ∈ ℕ such that the following holds. Let ℰ be a bounded set of (W , ds)

374 � 5 Singular integrals pre-elementary operators supported in Ω. Then, ∀M1 ≥ M, ∀M2 ≥ 4κM1 , ∀|γ| ≤ M2 , γ an ordered multi-index, there exists C ≥ 0 such that ∀(F, 2−j ) ∈ ℰ , ∀x ∈ Ω, ∀z ∈ M, 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 t M1 −tL −jds γ −j(M2 −|γ|) α −jds β 󵄨󵄨 −jds 󵄨 (2 W ) 2 F](x, z) dt 󵄨󵄨󵄨 󵄨󵄨󵄨(2 Wx ) (2 Wz ) ∫ ( 2−2κj ) [e 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 0 ≤ C2−2κj

(1 + 2j ρ(W ,ds) (x, z))−m

Vol(B(W ,ds) (x, 2−j + ρ(W ,ds) (x, z))) ∧ 1 1

2−2κj

.

1

Proof. We separate the integral into ∫0 = ∫0 + ∫2−2κj . We begin with the first integral. It follows from Lemma 5.11.8 (i) that (for any M2 and |γ| ≤ M2 ), γ

{((2−jdsW ) 2−j(M2 −|γ|) F, 2−j ) : (F, 2−j ) ∈ ℰ } is a bounded set of (W , ds) pre-elementary operators supported in Ω. Applying Lemma 5.11.11 with G replaced by (2−jdsW )γ 2−j(M2 −|γ|) F and Ft replaced by e−tL , it follows that for every m ∈ ℕ, if M is sufficiently large, we have, with γ, M1 , M2 , and F as in the statement of the lemma, 󵄨󵄨 󵄨󵄨 2−2κj 󵄨󵄨 󵄨󵄨 t M1 −tL −jds γ −j(M2 −|γ|) 󵄨󵄨 −jds 󵄨󵄨 α −jds β 󵄨󵄨(2 Wx ) (2 Wz ) ∫ ( 󵄨󵄨 ) [e (2 W ) 2 F](x, z) dt −2κj 󵄨󵄨 󵄨󵄨 2 󵄨󵄨 󵄨󵄨 0 󵄨 󵄨 −2κj 󵄨󵄨 󵄨󵄨 2 M 󵄨󵄨 󵄨󵄨 1 t 2−2κj 󵄨󵄨 −jds 󵄨 α −jds β γ = 󵄨󵄨󵄨(2 Wx ) (2 Wz ) ∫ ( −2κj ∧ ) [e−tL (2−jdsW ) 2−j(M2 −|γ|) F](x, z) dt 󵄨󵄨󵄨󵄨 t 󵄨󵄨 󵄨󵄨 2 󵄨󵄨 0 󵄨󵄨 ≲ 2−2κj

(1 + 2j ρ(W ,ds) (x, z))−m

Vol(B(W ,ds) (x, 2−j + ρ(W ,ds) (x, z))) ∧ 1

,

as desired. 1 We now turn to the term ∫2−2κj . With M1 ≥ M and M2 ≥ 4κM1 , we have, for some measurable functions B1,j (t), B2,j (t) with |B1,j (t)|, |B2,j (t)| ≤ 1, ∀t, 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 t M1 −tL −jds γ −j(M2 −|γ|) 󵄨󵄨 −jds 󵄨󵄨 α −jds β 󵄨󵄨(2 Wx ) (2 Wz ) ∫ ( 󵄨󵄨 ) [e (2 W ) 2 F](x, z) dt −2κj 󵄨󵄨 󵄨󵄨 2 󵄨󵄨 󵄨󵄨󵄨 −2κj 󵄨 2 1 󵄨󵄨 󵄨󵄨 t M1 α β = 󵄨󵄨󵄨(2−jdsWx ) (2−jdsWz ) ∫ ( −2κj ) 2−j(degds (γ)+M2 −|γ|) t − degds (γ)/2κ 󵄨󵄨 2 󵄨 2−2κj 󵄨󵄨󵄨 γ 󵄨 × [2−tL (t ds/2κ W ) F](x, z) dt 󵄨󵄨󵄨 󵄨󵄨 󵄨

5.11 Parametrices

� 375

1 󵄨󵄨 󵄨󵄨 󵄨󵄨 −jds 󵄨󵄨 t M1 −jM2 −M2 /2κ α −jds β γ −tL ds/2κ 󵄨 = 󵄨󵄨(2 Wx ) (2 Wz ) ∫ ( −2κj ) 2 t B1,j (t)[2 (t W ) F](x, z) dt 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 2 󵄨 󵄨 2−2κj 1 M1 󵄨󵄨 󵄨󵄨 󵄨󵄨 −jds 󵄨󵄨 t 2−2κj α −jds β γ −tL ds/2κ 󵄨 = 󵄨󵄨(2 Wx ) (2 Wz ) ∫ ( −2κj ∧ ) B2,j (t)[2 (t W ) F](x, z) dt 󵄨󵄨󵄨. 󵄨󵄨 󵄨󵄨 t 2 󵄨 󵄨 2−2κj

It follows from the hypotheses and Lemma 5.11.8 (ii) that γ

{(B2,j (t)e−tL (t ds/2κ W ) , t 1/2κ ) : t ∈ (0, 1], j ∈ [0, ∞)} is a bounded set of locally (W , ds) pre-elementary operators. From here, the desired estimate follows from Lemma 5.11.11, completing the proof. Lemma 5.11.13. For f ∈ C0∞ (M; ℂD ), taken in the sense of distributions.

d −tL e f dt

= −L e−tL f = −e−tL L f , where L is

Proof. Because L agrees with L on C0∞ (M; ℂD ), we have

d −tL e f dt

= −e−tL L f =

−e−tL L f . Also, on the domain of L , L agrees with L in the sense of distributions, d −tL so dt e f = −L e−tL f = −L e−tL f . Lemma 5.11.14. For all ψ ∈ C0∞ (Ω), for every ℰ a bounded set of (W , ds) elementary operators supported in Ω, ∀a ∈ ℕ, 1

{ 2κj } t a −tL E dt, 2−j ) : (E, 2−j ) ∈ ℰ } {(2 Mult[ψ] ∫ ( 2−2κj ) e 0 { } is a bounded set of (W , ds) pre-elementary operators supported in Ω. Proof. It is clear that 1

supp (Mult[ψ] ∫ (

⋃ (E,2−j )∈ℰ

0

t

2−2κj

a

) e−tL E dt) ⋐ Ω × Ω.

Fix ordered multi-indices α and β and m ∈ ℕ. The result will follow once we show that there exists C ≥ 0 such that ∀(E, 2−j ) ∈ ℰ , ∀x, z ∈ Ω, 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 t a −tL α −jds β 󵄨󵄨 −jds 󵄨 E](x, z) dt 󵄨󵄨󵄨 󵄨󵄨(2 Wx ) (2 Wz ) ∫ ( −2κj ) [e 󵄨󵄨 󵄨󵄨 2 󵄨󵄨 󵄨󵄨 0 ≤ C2−2κj

(1 + 2j ρ(W ,ds) (x, z))−m

Vol(B(W ,ds) (x, 2−j + ρ(W ,ds) (x, z))) ∧ 1

.

(5.140)

376 � 5 Singular integrals First we claim that if (5.140) holds with a replaced by a + 1 (for every ℰ with C depending on ℰ ), then it holds for a as well. Indeed, for (E, 2−j ) ∈ ℰ , Lemma 5.11.13 shows d −tL that dt e E = −e−tL L E. Thus, we have 1

∫( 0

t

a

2−2κj

) e

1

−tL

2−2κj d t a+1 E dt = ∫ ( ( −2κj ) ) e−tL E dt a+1 dt 2

2κja

=

0

1

2 1 t e−L E + ∫ ( −2κj ) a+1 a+1 2

a+1

(5.141)

e−tL 2−2κj L E dt.

0

2κja

It follows from Lemma 5.11.10 that 2a+1 e−L E satisfies the desired estimates in (5.140). For the second term on the right-hand side of (5.141), note that Proposition 5.5.5 (a), (c), and (e) implies that {(2−2κj L E, 2−j ) : (E, 2−j ) ∈ ℰ } is a bounded set of (W , ds) elementary operators supported in Ω. Thus, the second term on the right-hand side of (5.141) is of the same form as (5.140), but with a replaced by a + 1. The above discussion shows that it suffices to prove (5.140) for a sufficiently large (depending on α, β, m, (W , ds), and Ω). Let M = M(α, β, m, (W , ds), Ω) ∈ ℕ be as in Lemma 5.11.12 and take a ≥ M. We take M2 ≥ 4κa. Applying Proposition 5.5.5 (h) with N = M2 , we see that 1

∫( 0

t

a

) e 2−2κj

1 −tL

E dt = ∑ ∫ ( |γ|≤M2 0

t

a

γ

) e−tL 2−j(M2 −|γ|) (2−jdsW ) Eγ dt, 2−2κj

where {(Eγ , 2−j ) : (E, 2−j ) ∈ ℰ , |γ| ≤ M2 } is a bounded set of (W , ds) elementary operators supported in Ω. From here, (5.140) follows from Lemma 5.11.12. Lemma 5.11.15. For all ψ ∈ C0∞ (Ω), for all bounded sets of (W , ds) elementary operators supported in Ω, ℰ , ∀a, N ∈ ℕ, ∀c1 , c2 ∈ ℂ, Nj

{(c1 2 Mult[ψ]e

−L

2κj

E1 + c2 2

1

Mult[ψ] ∫ ( 0

2

t

a

) e−tL E2 dt, 2−j ) −2κj

(5.142)

: (E1 , 2−j ), (E2 , 2−j ) ∈ ℰ } is a bounded set of (W , ds) elementary operators supported in Ω. Proof. Lemmas 5.11.10 and 5.11.14 show that (5.142) is a bounded set of (W , ds) preelementary operators supported in Ω. To show that it is a bounded set of (W , ds) elementary operators supported in Ω, we use the characterization given in Corollary 5.5.7. We do this by proving (5.40) (in the case ν = 1) for the operators in (5.142), where Eμ,l1 = El1 and Ẽμ,l2 = Ẽl2 are a linear combination of operators of the same form as the ones in (5.142)

5.11 Parametrices

� 377

(perhaps with different choices of c1 , c2 , N, a, and ψ). It then follows that the set in (5.142) is an element of GΩ′′ as defined in Corollary 5.5.7, which will complete the proof. We begin with the El1 operators from (5.40). Let ψ2 ∈ C0∞ (Ω) satisfy ψ ≺ ψ2 . Note that

d −tL for (E, 2−j ) ∈ ℰ , E ∈ C0∞ (M×M; ℂD ), and Lemma 5.11.13 shows that dt e E = −L e−tL E. Using this, we have, for (E1 , 2−j ), (E2 , 2−j ) ∈ ℰ , 1

c1 2Nj Mult[ψ]e−L E1 + c2 22κj Mult[ψ] ∫ ( 0 Nj

−tL

Nj

−L

= c1 2 Mult[ψ]e

a

t

2−2κj

) e−tL E2 dt

1

c d t a+1 E1 + 2 Mult[ψ] ∫ ( ( −2κj ) ) e−tL E dt a+1 dt 2 0

= c1 2 Mult[ψ]e

c E1 + 2 22κj(a+1) Mult[ψ]e−L E2 a+1

(5.143)

1

+ 2−2κj Mult[ψ]L

c2 t a+1 ∫ ( −2κj ) e−tL E2 dt a+1 2 0

= 2 (c1 2 −j

(N+1)j

Mult[ψ]e

−L

c2 j(2κ(a+1)+1) 2 Mult[ψ]e−L E2 ) a+1

E1 +

1

+2

−2κj

c t a+1 Mult[ψ]L 2 ∫ ( −2κj ) e−tL E2 dt. a+1 2 0

The first term on the right-hand side of (5.143) is 2−j times a linear combination of terms of the same form as the ones in (5.142) (with different choices of N) and is therefore of the desired form. For the second term, we use the fact that Mult[ψ]L is a D × D matrix of (W , ds) partial differential operator in Ω of degree ≤ 2κ, and therefore the second term on the right-hand side of (5.143) is of the desired form as well. We now turn to the Ẽl2 terms from (5.40). For this, we apply Proposition 5.5.5 (h) with N = 1 to E2 to see that Nj

c1 2 Mult[ψ]e

−L

E1 + c2 2

2κj

1

Mult[ψ] ∫ ( 0

2

a

t

) e−tL E2 dt −2κj (5.144)

= 2−j (c1 2(N+1)j Mult[ψ]e−L E1 ) 1

+ ∑ 2j(|β|−1) (c2 22κj Mult[ψ] ∫ ( |β|≤1

0

t

2−2κj

a

β

) e−tL Eβ dt) (2−jdsW ) ,

where {(Eβ , 2−j ) : (E, 2−j ) ∈ ℰ , |β| ≤ 1} is a bounded set of (W , ds) elementary operators supported in Ω. The first term on the right-hand side of (5.144) is 2−j times a term of the same form as the one in (5.142) (with different choices of N) and is therefore of the desired form. Using the fact that (2−jdsW )β = 2−j degds (β) W β and W β is a (W , ds) partial

378 � 5 Singular integrals differential operator of degree ≤ degds(β), we see that the second term on the right-hand side of (5.144) is also of the desired form, completing the proof. 1

−2κ Corollary 5.11.16. We have ∫0 e−tL dt ∈ Aloc ((W , ds); ℂD , ℂD ).

Proof. Let ψ1 , ψ2 ∈ C0∞ (M) and pick Ω ⋐ M open and relatively compact such that 1

ψ1 , ψ2 ∈ C0∞ (Ω). We wish to show Mult[ψ1 ](∫0 e−tL dt) Mult[ψ2 ] ∈ A −2κ (Ω, (W , ds); ℂD , ℂD ). Let ℰ be a bounded set of (W , ds) elementary operators supported in Ω. By Proposition 5.5.5 (c), {(Mult[ψ2 ]E, 2−j ) : (E, 2−j ) ∈ ℰ } is a bounded set of (W , ds) elementary operators supported in Ω. Thus, by Lemma 5.11.15 with a = 0, c1 = 0, and c2 = 1, 1

{ 2κj } −tL dt) Mult[ψ2 ]E, 2−j ) : (E, 2−j ) ∈ ℰ } {(2 Mult[ψ1 ] (∫ e 0 { } is a bounded set of (W , ds) elementary operators supported in Ω. It follows that 1 ⃗ ℂD , ℂD ) = A −2κ (Ω, (W , ds); ⃗ ℂD , ℂD ), Mult[ψ1 ](∫0 e−tL dt) Mult[ψ2 ] ∈ A2−2κ (Ω, (W , ds); completing the proof. 1

−2κ Proof of Theorem 5.11.7. Set S = ∫0 e−tL dt, so that S ∈ Aloc ((W , ds); ℂD , ℂD ) by Corol∞ D lary 5.11.16. Using Lemma 5.11.13, we have, for f ∈ C0 (M; ℂ ), 1

1

(∫ e−tL dt) L f = − ∫ 0

0

d −tL e f dt = f − e−L f , dt

∞ that is, S L = I − e−L ≡ I mod Cloc (M × M; 𝕄D×D ). Also, for f , g ∈ C0∞ (M; ℂD ), we have (thinking of L as being taken in the sense of distributions),

⟨f , S L g⟩L2 (M,Vol;ℂD ) = ⟨L Sf , g⟩L2 (M,Vol;ℂD ) , ∞ and it follows that L S ≡ I mod Cloc (M × M; 𝕄D×D ) as well.

5.11.2 Parametrices and other geometries In this section, we work in either the Hörmander setting or the general setting described in Section 5.3. However, in the general setting, we impose the additional hypothesis that (W 1 , ds1 ) are Hörmander vector fields with formal degrees. Thus, in either setting, (W 1 , ds1 ) are Hörmander vector fields with formal degrees. We assume ν ≥ 2.

5.11 Parametrices

� 379

Fix κ1 , D ∈ ℕ+ and suppose L is an operator of the form L :=

∑

α

degds1 (α)≤κ1

aα (x)(W 1 ) ,

∞ aα ∈ Cloc (M; 𝕄D×D (ℂ)).

Suppose that there exists a parametrix for L : L −1 ∈ Aloc 1 ((W 1 , ds1 ); ℂD , ℂD ), that is, L −1 satisfies −κ

LL

−1

, L −1 L ≡ I

∞ mod Cloc (M × M; 𝕄D×D ).

Note that despite our notation, L −1 is not assumed to be an exact inverse of L , but just an inverse modulo infinitely smoothing operators. Set (Z, dr)⃗ = {(Z1 , dr1⃗ ), . . . , (Zs , drs⃗ )} := (W 2 , ds2 ) ⊠ (W 3 , ds3 ) ⊠ ⋅ ⋅ ⋅ ⊠ (W ν , dsν ) ∞ ⊂ Cloc (M; TM) × (ℕν−1 \ {0}).

The main theorem of this section is the following. Theorem 5.11.17. Under the above hypotheses, we have the following. −κ ⃗ ℂD , ℂD ). Hörmander setting: We have L −1 ∈ Dloc1 ((W 1 , ds1 ), (Z, dr); −κ −1 1 1 D 1 ⃗ ℂ , ℂD ). ̃ ((W , ds ), (Z, dr); General setting: We have L ∈ D loc Corollary 5.11.18. Under the above hypotheses, we have the following. (−κ ,0 ) ⃗ ℂD , ℂD ). Hörmander setting: We have L −1 ∈ Aloc 1 ν−1 ((W , ds); ̃(−κ1 ,0ν−1 ) ((W , ds); ⃗ ℂD , ℂD ). General setting: We have L −1 ∈ A loc

Proof. We prove only the result in the Hörmander setting; the proof in the general setting is similar. Let ψ1 , ψ2 ∈ C0∞ (M). By Theorem 5.11.17, Mult[ψ1 ]L −1 Mult[ψ2 ] ∈ ⃗ ℂD , ℂD ). Therefore, by Theorem 5.9.6, Mult[ψ1 ]L −1 Mult[ψ2 ] ∈ D −κ ((W 1 , ds1 ), (Z, dr); (−κ1 ,0ν2 ) ⃗ ℂD , ℂD ). Since ψ1 , ψ2 ∈ C ∞ (M) were arbitrary, we have L −1 ∈ A ((W , ds); 0 (−κ1 ,0ν−1 ) ⃗ ℂD , ℂD ), as desired, completing the proof. Aloc ((W , ds); The rest of this section is devoted to the proof of Theorem 5.11.17. Most of the proof will be the same in the Hörmander setting and the general setting, so we often do not specify which setting we are in (when unspecified, the reader may assume we are in the general setting, since the Hörmander setting is a special case of the general setting). We begin by introducing some notation to help with the proof. ̃(κ) denote an Fix a relatively compact, open set Ω ⋐ M. For κ ∈ ℕ, P1(κ) and P 1 arbitrary operator of the form ∑

degds1 (α)≤κ

bα (x)(W 1 )α ,

∞ bα ∈ Cloc (Ω; 𝕄D×D (ℂ)).

380 � 5 Singular integrals ̃(κ) . In other words, P (κ) or P ̃(κ) are D × D matrices of When κ ∈ ℤ \ ℕ, P1(κ) = 0 = P 1 1 1 1 1 (W , ds ) partial differential operators on Ω of degree ≤ κ. (κ ,k) ̃(κ2 ,k) denote arbitrary (scalarFor κ2 ∈ ℤν−1 and k ∈ ℤ, we let P2 2 and P 2 valued) (Z, dr)⃗ partial differential operators on Ω of order ≤ κ2 and length ≤ k (see (κ ) ̃(κ2 ) will denote arbitrary (Z, dr)⃗ partial differential Definition 3.12.1). Also, P2 2 and P 2 operators on Ω of order ≤ κ2 . For t ∈ ℝ, T (t) and T̃ (t) denote arbitrary operators in A t (Ω, (W 1 , ds1 ); ℂD , ℂD ). Also, (−∞) T and T̃ (−∞) denote an arbitrary element of ⋂t∈ℝ A t (Ω, (W 1 , ds1 ); ℂD , ℂD ). Remark 5.11.19. If R ∈ C0∞ (Ω × Ω; 𝕄D×D ), then Proposition 5.8.11 implies that R is of the form T (−∞) . ̃(κ) , P (κ2 ,k) , P ̃(κ2 ,k) , T (t) , or T̃ (t) denotes may The particular operator which P1(κ) , P 1 2 2 change from line to line. All that is important is that it is of the correct form. For the remainder of the proof, fix ψ1 , ψ2 ∈ C0∞ (Ω). The basic idea of the proof of Theorem 5.11.17 is the following. If two operators A and B commute and A is invertible, then A−1 and B commute. Corollary 3.12.7 implies (κ ) (κ ) that P2 2 and L “approximately commute” and we will use this to show that P2 2 and Mult[ψ1 ]L −1 Mult[ψ2 ] “approximately commute,” which will complete the proof. We turn to making this precise. Lemma 5.11.20. Suppose ϕ1 , ϕ2 ∈ C0∞ (Ω) with ϕ1 ≺ ϕ2 . Then for κ2 ∈ ℤν−1 and (κ ,k) ̃(κ2 ,k) Mult[ϕ2 ] and P (κ2 ,k) Mult[ϕ1 ] is of the k ∈ ℤ, Mult[ϕ1 ]P2 2 is of the form P 2 2 ̃(κ2 ,k) . form Mult[ϕ2 ]P 2

Proof. Since

(κ ,k) P2 2

is a partial differential operator, we have Mult[ϕ1 ]P2

(κ2 ,k)

= Mult[ϕ1 ]P2

(κ2 ,k)

Mult[ϕ2 ].

Since Mult[ϕ1 ] is a (Z, dr)⃗ partial differential operator of degree ≤ 0ν−1 and length ≤ (κ ,k) ̃(κ2 ,k) . This completes the proof for 0, Lemma 3.12.4 shows that Mult[ϕ1 ]P2 2 = P 2 Mult[ϕ1 ]P2 reader.

(κ2 ,k)

. A similar proof works for P2

(κ2 ,k)

Mult[ϕ1 ]; we leave the details to the

Lemma 5.11.21. There is a K = K(Ω) ∈ ℕ such that every operator of the form P2 a (W 1 , ds1 ) partial differential operator on Ω of degree ≤ Kk.

(κ2 ,k)

is

Proof. When k ≤ −1, P2 2 = 0 is trivial, so we assume k ∈ ℕ. The hypothesis that W11 , . . . , Wr11 are Hörmander vector fields and the compactness (κ ,k)

of Ω imply that W11 , . . . , Wr11 satisfy Hörmander’s condition of order m on Ω for some

∞ m ∈ ℕ. Thus, every vector field V ∈ Cloc (Ω; TΩ) is a (W 1 , ds1 ) partial differential operator 1 on Ω of degree ≤ K := m max1≤j≤r1 {dsj }.

5.11 Parametrices

By definition, P2

(κ2 ,k)

� 381

can be written as a finite sum of operators of the form Mult[g]V1 V2 ⋅ ⋅ ⋅ VR ,

∞ where g ∈ Cloc (Ω) and each (Vj , drj⃗ ) ∈ (Z, dr)⃗ with |dr1⃗ |1 + ⋅ ⋅ ⋅ + |drR⃗ |1 ≤ k. In particular, R ≤ k. Thus,

Mult[g]V1 V2 ⋅ ⋅ ⋅ VR is a (W 1 , ds1 ) partial differential operator on Ω of degree ≤ RK ≤ kK, completing the proof. Lemma 5.11.22. Let κ2 ∈ ℕν−1 . Any operator of the form T (−κ) where κ ≥ κ1 is of the form ̃(κ2 ) T̃ (−κ1 ) P 2

(5.145)

̃ 2 T̃ P 2

(5.146)

and of the form (κ ) (−κ1 )

.

In particular, any operator of the form T (−∞) is of the forms (5.145) and (5.146). Proof. Since supp(T (−κ) ) ⋐ Ω × Ω, we may take ψ ∈ C0∞ (Ω) with ψ ≡ 1 on a large enough set that T (−κ) Mult[ψ] = T (−κ) = Mult[ψ]T (−κ) . Since κ2 ∈ ℕν−1 , Mult[ψ] is of the form ̃(κ2 ) , and therefore P 2 ̃(κ2 ) T̃ (−κ) = T̃ (−κ) Mult[ψ] = T̃ (−κ) P 2 and ̃(κ2 ) T̃ (−κ) . T̃ (−κ) = Mult[ψ]T̃ (−κ) = P 2 Since κ ≥ κ1 , Proposition 5.8.2 shows that T̃ (−κ) is of the form T̃ (−κ1 ) . Lemma 5.11.23. Fix ϕ0 , ϕ1 , ϕ2 , . . . ∈ C0∞ (Ω) with ψ1 , ψ2 ≺ ϕ0 ≺ ϕ1 ≺ ϕ2 ≺ ⋅ ⋅ ⋅ . (A) For every N ∈ ℕ, any operator of the form P2 ten as a finite sum of terms of the form

(κ2 )

Mult[ψ1 ]L −1 Mult[ψ2 ] can be writ-

̃(κ2 ) T̃ (−κ1 ) P 2

(5.147)

and ̃(κ2 ,|κ2 |1 −N+j) Mult[ϕN ]L −1 Mult[ψ2 ], T̃ (−j) P 2

j = 0, 1, . . . , N.

(5.148)

382 � 5 Singular integrals (B) For every N ∈ ℕ, any operator of the form Mult[ψ1 ]L −1 Mult[ψ2 ]P2 as a finite sum of terms of the form

(κ2 )

can be written

̃ 2 T̃ P 2

(κ ) (−κ1 )

and ̃ Mult[ψ1 ]L −1 Mult[ϕN ]P 2

T

(κ2 ,|κ2 |1 −N+j) ̃ (−j)

,

j = 0, 1, . . . , N.

Proof. The proofs of (A) and (B) are similar (they just involve reversing the roles of the left and right sides of the operators in question), so we prove only (A). When κ2 ∈ ℤν−1 \ (κ ) ℕν−1 , then P2 2 = 0 and the result is trivial. Thus, we henceforth assume κ2 ∈ ℕν−1 . We prove (A) by induction on N. We begin with the base case, N = 0. Proposition 5.8.3 shows that Mult[ϕ0 ] is of the form T (0) . Thus, since ψ1 ≺ ϕ0 , using Lemma 5.11.20, we have (κ2 )

P2

Mult[ψ1 ]L −1 Mult[ψ2 ] = P2

(κ2 )

̃ = Mult[ϕ0 ]P 2

(κ2 )

Mult[ψ1 ] Mult[ϕ0 ]L −1 Mult[ψ2 ]

̃ Mult[ϕ0 ]L −1 Mult[ψ2 ] = T (0) P 2

(κ2 )

Mult[ϕ0 ]L −1 Mult[ψ2 ],

thereby proving that P2 2 Mult[ψ1 ]L −1 Mult[ψ2 ] is of the form (5.148) with j = 0, N = 0. We assume (A) for some N ∈ ℕ and prove it for N + 1. Thus, we wish to show that any term of the form (5.148) can be written as a finite sum of terms of the form (5.147) and (5.148) with N replaced by N + 1. We consider an arbitrary term of the form (5.147). For j = 0, 1, . . . , N, we have, by Lemma 5.11.20, (κ )

T (−j) P2

(κ2 ,|κ2 |1 −N+j)

Mult[ϕN ]L −1 Mult[ψ2 ]

̃(κ2 ,|κ2 |1 −N+j) L −1 Mult[ψ2 ] = T (−j) Mult[ϕN+1 ]P ̃(κ2 ,|κ2 |1 −N+j) Mult[ψ2 ] = T (−j) Mult[ϕN+1 ]L −1 P ̃ + T (−j) Mult[ϕN+1 ][P 2

(κ2 ,|κ2 |1 −N+j)

, L −1 ] Mult[ψ2 ]

=: (I) + (II). We show that the terms (I) and (II) above are of the desired form separately. We begin with (I). −κ By the hypothesis that L −1 Aloc 1 (W 1 , ds1 ), we have Mult[ϕN+1 ]L −1 Mult[ϕ0 ] = T̃ (−κ1 ) . ̃(κ2 ,|κ2 |1 −N+j) possibly changing Thus, using Lemma 5.11.20, we have (with the operator P from line to line) ̃(κ2 ,|κ2 |1 −N+j) Mult[ψ2 ] (I) = T (−j) Mult[ϕN+1 ]L −1 P ̃(κ2 ,|κ2 |1 −N+j) = T (−j) Mult[ϕN+1 ]L −1 Mult[ϕ0 ]P ̃(κ2 ,|κ2 |1 −N+j) = T (−j) T̃ (−κ1 ) P ̃(κ2 ) , = T̃ (−κ1 −j) P

5.11 Parametrices � 383

where the last equality uses Proposition 5.8.6. Since j ≥ 0, Proposition 5.8.2 shows T̃ (−κ1 −j) is of the form T̃ (−κ1 ) . Thus, (I) is of the form (5.147), as desired. ∞ Turning to (II), we use the fact that L L −1 , L −1 L ∈ Cloc (Ω × Ω; 𝕄D×D (ℂ)) and that L −1 is pseudo-local (see the growth condition in Definition 5.2.4). We see that for some ∞ R1 , R2 ∈ Cloc (M × M; 𝕄D×D ), Mult[ϕN+1 ]L −1 Mult[ϕN+2 ]L = Mult[ϕN+1 ] + Mult[ϕN+3 ]R1 Mult[ϕN+3 ], L Mult[ϕ0 ]L

−1

(5.149)

Mult[ψ2 ] = Mult[ψ2 ] + Mult[ϕN+3 ]R2 Mult[ϕN+3 ].

(5.150)

Using (5.149) and (5.150), we have ̃ (II) = T (−j) Mult[ϕN+1 ](P 2

(κ2 ,|κ2 |1 −N+j)

L

−1

(κ2 ,|κ2 |1 −N+j)

̃ − L −1 P 2

) Mult[ψ2 ]

̃(κ2 ,|κ2 |1 −N+j) L −1 − L −1 P ̃(κ2 ,|κ2 |1 −N+j) ) = T (−j) Mult[ϕN+1 ]L −1 Mult[ϕN+2 ]L (P 2 2 × L Mult[ϕ0 ]L −1 Mult[ψ2 ]

̃(κ2 ,|κ2 |1 −N+j) L −1 − L −1 P ̃(κ2 ,|κ2 |1 −N+j) ) − T (−j) Mult[ϕN+1 ]L −1 Mult[ϕN+2 ]L (P 2 2 × Mult[ϕN+3 ]R2 Mult[ϕN+3 ]

̃ − T (−j) Mult[ϕN+3 ]R1 Mult[ϕN+3 ](P 2

(κ2 ,|κ2 |1 −N+j)

L

−1

× Mult[ψ2 ]

(κ2 ,|κ2 |1 −N+j)

̃ − L −1 P 2

)

=: (III) − (IV) − (V). We study the terms (III) and (IV) + (V) separately. We begin with (IV) + (V). Note that if ξ1 , ξ2 ∈ C0∞ (Ω), we have Mult[ξ1 ]L −1 Mult[ξ2 ] = T (−κ1 ) ,

(5.151)

Mult[ξ1 ]L = T (κ1 ) .

(5.152)

and by Proposition 5.8.3,

∞ Using Remark 5.11.19, if R ∈ Cloc (Ω × Ω; 𝕄D×D (ℂ)), then

Mult[ξ1 ]R Mult[ξ2 ] = T (−∞) .

(5.153)

̃(κ2 ) = P ̃(κ2 ,|κ2 |1 ) Also, if K = K(Ω) ∈ ℕ is as in Lemma 5.11.21, then, using Remark 3.12.2, P 2 2 1 1 is a (W , ds ) partial differential operator on Ω of degree ≤ |κ2 |1 K, and therefore by Proposition 5.8.3, ̃(κ2 ) = T (|κ2 |1 K) . Mult[ξ1 ]P 2 Using (5.151), (5.152), (5.153), and (5.154), we have

(5.154)

384 � 5 Singular integrals ̃(κ2 ) Mult[ϕN+4 ] (IV) + (V) = T (−j) Mult[ϕN+1 ]L −1 Mult[ϕN+2 ]L Mult[ϕN+3 ]P 2 × L −1 Mult[ϕN+3 ]R2 Mult[ϕN+3 ]

− T (−j) Mult[ϕN+1 ]L −1 Mult[ϕN+2 ]L Mult[ϕN+3 ]L −1 Mult[ϕN+4 ] ̃(κ2 ) Mult[ϕN+3 ]R2 Mult[ϕN+3 ] ×P 2 ̃ + T (−j) Mult[ϕN+3 ]R1 Mult[ϕN+3 ]P 2

(κ2 )

Mult[ϕN+4 ]L −1 Mult[ψ2 ]

̃(κ2 ) Mult[ψ2 ] − T (−j) Mult[ϕN+3 ]R1 Mult[ϕN+3 ]L −1 Mult[ϕ0 ]P 2

= T (−j) T (−κ1 ) T (κ1 ) T (|κ2 |1 K) T (−κ1 ) T (−∞) + T (−j) T (−κ1 ) T (κ1 ) T (−κ1 ) T (|κ2 |1 K) T (−∞) + T (−j) T (−∞) T (|κ2 |1 K) T (−κ1 ) + T (−j) T (−∞) T (−κ1 ) T (|κ2 |1 K) = T̃ (−∞) , where the last equality follows from Proposition 5.8.6. Lemma 5.11.22 now shows that (IV) + (V) is of the form (5.147). We turn to (III). Using the fact that L L −1 = R3 and L −1 L = R4 , where R3 , R4 ∈ ∞ Cloc (M × M; 𝕄D×D (ℂ)), ̃(κ2 ,|κ2 |1 −N+j) − P ̃(κ2 ,|κ2 |1 −N+j) L ) (III) = T (−j) Mult[ϕN+1 ]L −1 Mult[ϕN+2 ](L P 2 2 × Mult[ϕ0 ]L −1 Mult[ψ2 ] ̃(κ2 ,|κ2 |1 −N+j) R4 Mult[ϕ0 ]L −1 Mult[ψ2 ] + T (−j) Mult[ϕN+1 ]L −1 Mult[ϕN+2 ]L P 2 ̃(κ2 ,|κ2 |1 −N+j) L Mult[ϕ0 ]L −1 Mult[ψ2 ] − T (−j) Mult[ϕN+1 ]L −1 Mult[ϕN+2 ]R3 P 2 = (VI) + (VII) − (VIII). Using (5.151), (5.152), (5.153), and (5.154), we have ̃(κ2 ) Mult[ΦN+4 ]R4 Mult[ϕ0 ] (VII) = T (−j) Mult[ϕN+1 ]L −1 Mult[ϕN+2 ]L Mult[ϕN+3 ]P 2 × L −1 Mult[ψ2 ] = T (−j) T (−κ1 ) T (κ1 ) T (|κ2 |1 K) T (−∞) T (−κ1 ) = T̃ (−∞) , where the last equality follows from Proposition 5.8.6. Lemma 5.11.22 now shows that (VII) is of the form (5.147). A similar proof shows that (VIII) = T̃ (−∞) and is therefore of the form (5.147) (by Lemma 5.11.22). We turn to (VI). Corollary 3.8.11 with ν1 = 1, ν2 = ν − 1, 𝒮1 = (W 1 , ds1 ), and 𝒯μ = (W μ−1 , dsμ−1 ), 2 ≤ μ ≤ ν, shows that (W 1 , ds1 ) and (Z, dr)⃗ locally weakly approximately ̃(κ2 ,|κ2 |1 −N+j) is a scalar-valued (Z, dr)⃗ partial differential operator on Ω commute. Since P 2 of degree ≤ κ2 and length ≤ |κ2 |1 − N + j and L is a matrix of (W 1 , ds1 ) partial differential operators on Ω of degree ≤ κ1 and length ≤ κ1 (see Remark 3.12.2), Theorem 3.12.6 implies ̃(κ2 ,|κ2 |1 −N+j) − P ̃(κ2 ,|κ2 |1 −N+j) L can be written as a finite sum of terms of the form that L P 2 2

5.11 Parametrices � 385

(κ1 ,κ1 )

P1

(κ2 ,|κ2 |1 −N+j−1)

P2

and

(κ1 ,κ1 −1)

P1

(κ2 ,|κ2 |1 −N+j−1)

P2

,

where each P1(κ,k) is a D × D matrix of (W 1 , ds1 ) partial differential operators on Ω of degree ≤ κ and length ≤ k. Since (W 1 , ds1 ) is a single-parameter list, each P1(κ,k) is in fact a D × D matrix of (W 1 , ds1 ) partial differential operators on Ω of degree ≤ min{κ, k}. Thus, ̃(κ2 ,|κ2 |1 −N+j) − P ̃(κ2 ,|κ2 |1 −N+j) L can be written as a finite sum of terms of the form LP 2 2 (κ1 )

P1

(κ2 ,|κ2 |1 −N+j−1)

P2

and

(κ1 −1)

P1

(κ2 ,|κ2 |1 −N+j−1)

P2

.

Thus (VI) can be written as a finite sum of terms of the form (IX) = T (−j) Mult[ϕN+1 ]L −1 Mult[ϕN+2 ]P1

(κ1 )

(κ2 ,|κ2 |1 −N+j−1)

Mult[ϕ0 ]L −1 Mult[ψ2 ],

(κ2 ,|κ2 |1 −N+j)

Mult[ϕ0 ]L −1 Mult[ψ2 ].

P2

(X) = T (−j) Mult[ϕN+1 ]L −1 Mult[ϕN+2 ]P1

(κ1 −1)

P2

By Proposition 5.8.3 we have, for ξ ∈ C0∞ (Ω), Mult[ξ]P1(k) = T (k) .

(5.155)

Using (5.151), (5.155), and Lemma 5.11.20, we have (IX) = T (−j) Mult[ϕN+1 ]L −1 Mult[ϕN+2 ]P1

(κ1 )

̃ = T (−j) T (−κ1 ) T (κ1 ) P 2

(κ2 ,|κ2 |1 −N+j−1)

̃(κ2 ,|κ2 |1 −N+j−1) L −1 Mult[ψ2 ] Mult[ϕ1 ]P 2

Mult[ϕN+1 ]L −1 Mult[ψ2 ]

̃(κ2 ,|κ2 |1 −N+j−1) Mult[ϕN+1 ]L −1 Mult[ψ2 ], = T̃ (−j) P 2 where the last equality uses Proposition 5.8.6. This is of the form (5.148) with N replaced by N + 1, as desired. Similarly, using (5.151), (5.155), and Lemma 5.11.20, we have (X) = T (−j) Mult[ϕN+1 ]L −1 Mult[ϕN+2 ]P1

(κ1 −1)

= =

Mult[ϕ1 ]

̃(κ2 ,|κ2 |1 −N+j) Mult[ϕN+1 ]L −1 Mult[ψ2 ] ×P 2 (−j) (−κ1 ) (κ1 −1) ̃(κ2 ,|κ2 |1 −N+j) T T T P2 Mult[ϕN+1 ]L −1 Mult[ψ2 ] ̃(κ2 ,|κ2 |1 −N+j) Mult[ϕN+1 ]L −1 Mult[ψ2 ], T̃ (−(j+1)) P 2

where the last equality uses Proposition 5.8.6. This is of the form (5.148) with N replaced by N + 1 and j replaced by j + 1, as desired, completing the proof. Lemma 5.11.24. (A) P2 of the form

(κ2 )

Mult[ψ1 ]L −1 Mult[ψ2 ] can be written as a finite sum of terms ̃(κ2 ) . T̃ (−κ1 ) P 2

(5.156)

386 � 5 Singular integrals (B) Mult[ψ1 ]L −1 Mult[ψ2 ]P2

(κ2 )

can be written as a finite sum of terms of the form ̃ 2 T̃ P 2

(κ ) (−κ1 )

(5.157)

.

Proof. The proofs for (A) and (B) are similar (only the roles of the left- and right-hand sides of the operators are switched), so we prove only (A). We use Lemma 5.11.23. Since (5.147) is already of the form (5.156), it suffices to show that if N is sufficiently large, any term of the form (5.148) is of the form (5.156). Take N := (K + 1)|κ2 |1 , where K = K(Ω) is as in Lemma 5.11.21. We claim − j + K(|κ2 |1 − N + j) − κ1 ≤ −κ1 ,

j = 0, 1, 2, . . . , N.

(5.158)

Indeed, if N − j ≥ |κ2 |1 , then K(|κ2 |1 − N + j) ≤ 0, and (5.158) holds in this case. On the other hand, if N − j ≤ |κ2 |1 , then j ≥ N − |κ2 |1 = K|κ2 |1 ≥ K(|κ2 |1 − N + j), proving (5.158). ̃(κ2 ,|κ2 |1 −N+j) is a (W 1 , ds1 ) We consider a term of the form (5.148). By Lemma 5.11.21, P 2 partial differential operator on Ω of degree ≤ K(|κ2 |1 − N + j). Combining this with ̃(κ2 ,|κ2 |1 −N+j) = T̃ (−j+K(|κ2 |1 −N+j)) . By hypothesis, Corollary 5.8.10 shows that T̃ (−j) P 2 −1 −κ1 Mult[ψ1 ]L Mult[ψ2 ] ∈ A (Ω, (W 1 , ds1 ); ℂD , ℂD ), and therefore by Proposition 5.8.6, T̃ (−j+K(|κ2 |1 −N+j)) Mult[ψ1 ]L −1 Mult[ψ2 ] = T̃ (−j+(|κ2 |1 −N+j)−κ1 ) . Combining this with (5.158) and Proposition 5.8.2 shows that (5.148) is of the form T̃ (−κ1 ) . Lemma 5.11.22 shows that this is of the form (5.156), completing the proof. Proof of Theorem 5.11.17. Since Ω ⋐ M was an arbitrary relatively compact, open set and ψ1 , ψ2 ∈ C0∞ (Ω) were arbitrary, it suffices to show that T := Mult[ψ1 ]L −1 Mult[ψ2 ] ∈{

⃗ ℂD , ℂD ), D −κ1 (Ω, (W 1 , ds1 ), (Z, dr);

in the Hörmander setting, −κ1 1 1 D D ⃗ ̃ D (Ω, (W , ds ), (Z, dr); ℂ , ℂ ), in the general setting.

(5.159)

We think of T as a D × D matrix of operators. Hörmander setting: By hypothesis, we have T ∈ A −κ1 (Ω, (W 1 , ds1 ); ℂD , ℂD ). This along ⃗ ℂD , ℂD ), with Lemma 5.11.24 is exactly the statement that T ∈ D −κ1 (Ω, (W 1 , ds1 ), (Z, dr); which proves (5.159) and completes the proof in this case. General setting: By hypothesis, we have T ∈ A −κ1 (Ω, (W 1 , ds1 ); ℂD , ℂD ). Thus, Proposĩ−κ1 (Ω, (W 1 , ds1 ); ℂD , ℂD ). Lemma 5.11.24 shows that P (κ2 ) T tion 5.8.4 implies that T ∈ A 2

can be written as a finite sum of terms of the form (5.156), while T P2 2 can be written as a finite sum of terms of the form (5.157), where each T̃ (−κ1 ) ∈ A −κ1 (Ω, (W 1 , ds1 ); ℂD , ℂD ). ̃−κ1 (Ω, (W 1 , ds1 ); ℂD , ℂD ). This combining these reBy Proposition 5.8.4, each T̃ (−κ1 ) ∈ A ⃗ ℂD , ℂD ), which proves (5.159) and com̃−κ1 (Ω, (W 1 , ds1 ), (Z, dr); marks verifies that T ∈ D pletes the proof. (κ )

5.12 Spectral multipliers

�

387

5.12 Spectral multipliers In this section, we briefly describe that spectral multipliers of Hörmander’s subLaplacian are singular integrals in the sense of Section 5.2.1. This is taken from [220, Section 2.6], and we refer the reader there for further details. In this section, we consider M a smooth, compact manifold with smooth, strictly positive density Vol. Let W1 , . . . , Wr be Hörmander vector fields on M and assign to each Wj the formal degree 1: (W , 1) = {(W1 , 1), . . . , (Wr , 1)}. Consider Hörmander’s sub-Laplacian: L = W1 W1 + ⋅ ⋅ ⋅ + Wr Wr , ∗

∗

where Wj∗ denotes the formal L2 (M, Vol) adjoint of Wj . L , with dense domain C ∞ (M), is clearly a non-negative symmetric operator on L2 (M, Vol). By [220, Lemma 2.6.3] it is essentially self-adjoint: there is a unique selfadjoint extension. Henceforth, we identify L with this self-adjoint extension. Let E be the spectral decomposition of L , so that for a Borel measurable function m : [0, ∞) → ℂ, we have m(L ) = ∫ m(λ)dE(λ). [0,∞)

The eigenspace corresponding to eigenvalue 0 of L consists of only the constant functions (see [220, Lemma 2.6.2]). In other words, E({0}) is equal to the orthogonal projection onto the constant functions. The main theorem of this section is the following. Theorem 5.12.1 (Theorem 2.6.6 of [220]). Fix t ∈ ℝ and let m : [0, ∞) → ℂ be a Borel 󵄨 ∞ measurable function with m󵄨󵄨󵄨(0,∞) ∈ Cloc ((0, ∞)) which satisfies a 󵄩 󵄩 sup λ−t 󵄩󵄩󵄩(λ𝜕λ ) m(λ)󵄩󵄩󵄩 < ∞, λ>0

Then m(L ) ∈ A 2t (M, (W , 1)). Corollary 5.12.2. For s ∈ ℝ define s

s

L = ∫ λ dE(λ). (0,∞)

Then L s ∈ A 2s (M, (W , 1)).

∀a ∈ ℕ.

388 � 5 Singular integrals Proof. The function m(λ) = {

λs 0

if λ > 0,

if λ = 0

satisfies the hypotheses of Theorem 5.12.1 with t = s. Corollary 5.12.3. For s ∈ ℝ, (I + L )s ∈ A 2s (M, (W , 1)). Proof. The function m(λ) = (1 + λ)s satisfies the hypotheses of Theorem 5.12.1 with t = s. We devote the rest of this section to a brief discussion of the proof of Theorem 5.12.1. j We refer the reader to [220, Section 2.6] for a full proof. Let Ŝ0 (ℝ) = {f ∈ S : 𝜕ξ f (0) = 0, ∀j ∈ ℕ}: the space of Schwartz functions which vanish to infinite order at 0. Ŝ0 (ℝ) is

a closed subspace of S (ℝ), and we give it the subspace topology. Note that the Fourier transform is an isomorphism S0 (ℝ) → Ŝ0 (ℝ), justifying the notation. The key to Theorem 5.12.1 is the next proposition, which we state without proof. The proof can be found in [220, Section 2.6]. Proposition 5.12.4 (Proposition 2.6.11 of [220]). Let ℬ ⊂ Ŝ0 (ℝ) be a bounded set. Then {(m(s2 L ), s) : s ∈ (0, 1], m ∈ ℬ} is a bounded set of (W , 1) elementary operators supported in M. Proof of Theorem 5.12.1. By [220, Lemma 2.6.4], the spectrum of L is discrete. Let λ0 > 0 be the least non-zero eigenvalue of L . Let ψ ∈ C0∞ (ℝ) be a non-negative function which equals 1 on (−3λ0 /4, 3λ0 /4) and which equals 0 outside (−7λ0 /8, 7λ0 /8). Define ϕ(λ) = −2j ψ(λ) − ψ(4λ). Note that 1 = ψ(λ) + ∑∞ j=1 ϕ(2 λ) and therefore ∞

m(L ) = m(0)ψ(L ) + ∑ ϕ(2−2j L )m(L ). j=1

(5.160)

Note that m(0)ψ(L ) = m(0)E({0}), where E({0}) is projection onto the constant functions. It is straightforward to verify that the singleton set {(m(0)E({0}), 1)} is a bounded set of (W , 1) elementary operators supported in M. From the assumptions on m and ϕ, {2−2jt m(22j λ)ϕ(λ) : j ∈ ℕ} ⊂ Ŝ0 (ℝ) is a bounded set. Proposition 5.12.4 implies that {(2−2jt ϕ(2−2j L )m(L ), 2−j )} is a bounded set of (W , 1) elementary operators supported in M.

5.13 Further reading and references

� 389

Combining the above, (5.160) shows that m(L ) = ∑ 22jt Ej , j∈ℕ

where {(Ej , 2−j ) : j ∈ ℕ} is a bounded set of (W , 1) elementary operators supported in M. We conclude that m(L ) ∈ A22t (M, (W , 1)) = A 2t (M, (W , 1)), completing the proof. Remark 5.12.5. The above proof in the special case m(λ) ≡ 1 gives a decomposition of the identity operator: ∞

I = ψ(L ) + ∑ ϕ(2−2j L ) =: ∑ Dj , j=1

j∈ℕ

where {(Dj , 2−j ) : j ∈ ℕ} is a bounded set of (W , 1) elementary operators supported in M. Remark 5.12.6. In this section, we focused on the case where the formal degrees are all equal to 1: ds1 = ds2 = ⋅ ⋅ ⋅ = dsr = 1. Recently, using the theory from this text, Lingxiao Zhang [254] obtained similar results for the more general sub-Laplacian (4.39) and other maximally subelliptic operators. To achieve this, Zhang utilized the short-time Gaussian type bounds for the heat equation established in Theorem 8.1.1 (ix). The proof of Proposition 5.12.4 given in [220, Section 2.6] does not extend to this more general setting, because that proof relies on the finite propagation speed (in terms of ρ(W ,1) ) of the wave equation for L (which is a result of Melrose [169]) and hence cannot work for higher-order operators whose wave equations do not have finite speed of propagation.

5.13 Further reading and references The single-parameter singular integrals in this chapter have a long history. They take their roots in the work on nilpotent Lie groups initiated by Folland and Stein [91] and Folland [90]. The general, quantitative study of Carnot-Carathéodory geometry due to Nagel, Stein, and Wainger [189] (see Section 3.3) shows that single-parameter Carnot– Carathéodory balls give rise to a space of homogeneous type in the sense of Coifman and Weiss [52, 59], and therefore it makes sense to pursue a theory of singular integrals on these spaces. Singular integrals, in the single-parameter setting, of the type A t (Ω, (W , ds)) (see Section 5.2.1) were first introduced under the name NIS operators by Nagel, Rosay, Stein, and Wainger [182], though an important precursor was introduced by Nagel and Stein [186]. These operators were later used by Chang, Nagel, and Stein [34] and Koenig [140]. These references all used a definition similar to A4t (Ω, (W , ds)) (see Definition 5.2.4) and restricted attention to t > −Q1 so that M in Definition 5.2.4 could be taken equal to 0 (see Remark 5.2.5); thus, our definitions extend these ideas to all t ∈ ℝ in a way that maintains that the operators form a filtered algebra (Proposition 5.8.6). In these references,

390 � 5 Singular integrals an extra assumption was placed on the singular integrals: in [182, 34] it was assumed that there is a sequence Tj ∈ C0∞ (M × M) satisfying the assumptions of Definition 5.2.4 uniformly in j and such that Tj → T in distribution. In [140], this was replaced with an a priori estimate. What we have used, instead of these extra assumptions, is the seemingly weaker, qualitative hypothesis that T ∈ Hom(C0∞ (M), C0∞ (M)′ ), without which we would not even be able to talk about the Schwartz kernel of T. By Theorem 5.2.12, the a priori stronger assumptions in [182, 34, 140] actually give rise to the same class of singular integrals. For example, the equality A2t (Ω, (W , ds)) = A4t (Ω, (W , ds)) gives such a sequence of smooth approximations (namely the partial sums of T = ∑j∈ℕ 2jt Ej ). Elementary operators and the definitions of A1t (Ω, (W , ds)) and A2t (Ω, (W , ds)), on a compact manifold, were introduced by the author in [220, Chapter 2]. The single-parameter results in this text improve on the theory in [220, Chapter 2] in three ways: we do not assume the manifold is compact, we extend the definition of A4t to t ≤ −Q1 (see Definition 5.2.4), and we introduced the fourth equivalent definition A3t (see Definition 5.2.2). The general relationship between single-parameter singular integrals and the heat equation, described in Section 5.11.1, is classical. In the specific setting we are considert ing, regarding the singular integrals on Carnot–Carathéodory spaces, Aloc , it was used by Nagel and Stein in [188]. In this particular setting, though, parametrices for maximally subelliptic operators are not usually constructed via the heat equation. This is the case, for example, in [182, 34, 140], and more generally in [220, Theorem 2.4.8]. In those references, one starts with an a priori parametrix (guaranteed by the hypoellipticity of the operator and [238, Theorem 52.3]), and then one proves estimates on this parametrix via an a priori subelliptic estimate. See [220, Theorem 2.4.8] for an exposition of this method. The methods from Section 5.11.1 give an easier way to obtain good quantitative estimates on the operator, which we require for the study of nonlinear equations in Chapter 9. We turn to describing some of the history of the singular integrals in the multiparameter Hörmander setting, described in Section 5.2.2. This provides a theory of pseudo-local (see Section 5.8.1) singular integrals which form a filtered algebra (Proposition 5.8.6) and can be set up in such a way to contain both the parametrices for maximally subelliptic operators (see Section 5.11.2 and Theorem 8.1.1 (i) ⇒ (vii)) and the standard pseudo-differential operators (see Section 5.8.2). Traditionally, the phrase “multi-parameter singular integrals” has often referred to the product theory of singular integrals, which are not pseudo-local in general. The product theory of singular integrals began with the work of R. Fefferman and Stein [86]. Another important early work is the work of Journé [136]. After these early works, many works followed. See, for example, [85, 37, 201, 180] for some works which have some relationship to the ideas in this chapter – this is just a small sampling of the many papers on the product theory of singular integrals. The main idea of the product theory of singular integrals is that the ambient space is a product of spaces M = M1 × M2 × ⋅ ⋅ ⋅ × Mν , and the corresponding balls are of a product type B((x1 , . . . , xν ), (δ1 , . . . , δν )) = B1 (x1 , δ1 ) × ⋅ ⋅ ⋅ × Bν (xν , δν ). Informally, we might say that the ν different geometries do not “overlap.” This is the opposite situation

5.13 Further reading and references �

391

to the one studied in Section 5.2.2, where the geometries all are on the same space, and completely “overlap.” The first papers which dealt with multi-parameter geometries which at least partially overlapped were on flag type singular integral operators. This began with the work of Müller, Ricci, and Stein [180, 181] on certain spectral multipliers on the Heisenberg group. The study of these flag kernels was later taken up by Nagel, Ricci, and Stein [184] and Nagel, Ricci, Stein, and Wainger [183]. Since these original works, several authors have studied these flag kernels; see, for example, [111], where end point Hardy space estimates were shown for the operators introduced in [180, 181]. In [220], the author put forth a theory which unified the above settings, at least when the underlying balls were assumed to be multi-parameter Carnot–Carathéodory balls. Special cases of this theory reproduced the product theory and the theory of flag kernels, which are inherently not pseudo-local. However, it was also shown (see [219, Section 5.1.4]) that in the multi-parameter Hörmander setting of Section 5.2.2, where the geometries completely overlap, the singular integrals described there are pseudo-local. In defining these algebras, [220] introduced the notion of elementary operators described in Definition 5.2.15. The algebra defined in Section 5.2.2 is essentially a special case of the general theory in [220]; however, by working in this special case we are able to make the theory considerably less technical and the main results have a cleaner statement. Beyond [220] there have been a few other theories which introduce an algebra of pseudo-local, multi-parameter singular integrals which contain both the standard pseudo-differential operators and parametrices for certain subelliptic operators; see, for example, [185] and [227]. Neither of these theories seems general enough to attack general maximally subelliptic operators. We do not know the relationship between these theories and the general theory put forth in [220] or the theory from Section 5.2.2. A precursor to the general theory in [220], due to the author, developed a theory of multi-parameter convolution operators on a nilpotent Lie group which contained both the left and right invariant operators on the group [229]. This is essentially a special case of the setting in Section 5.2.2. Even though the operators in Section 5.2.2 are pseudo-local, we did not define algebra in terms of pointwise estimates for the Schwartz kernel. This is in contrast to many of the previous theories for multi-parameter singular integrals. In fact, even when we proved the operators were pseudo-local (in Section 5.8.1), the estimates we obtained were far from optimal. In the special case studied in [229], a similar proof was used to obtain much sharper pointwise estimates for the Schwartz kernel. Even there, though, these estimates do not seem strong enough to define the algebra. In [220, Chapter 5] another definition of the algebra is given in terms of certain kinds of “test functions.” This is perhaps closer to the idea of pointwise estimates for the Schwartz kernel, though it is still somewhat different. The multi-parameter singular integrals in the general setting from Section 5.2.3 (which are not necessarily pseudo-local) seem to be new. The idea here is to axiomatize what we need from the theory in the Hörmander setting to be able to derive the

392 � 5 Singular integrals function space estimates we require. The reason we must introduce these singular integrals is that the hypotheses are quantitatively invariant under the rescalings described in Section 3.5. We need this quantitative scale invariance when we move to studying nonlinear PDEs in Chapter 9. Alternatively, we could have introduced a theory similar to the one in [220, Chapter 5] which generalized the singular integrals from Section 5.2.2 and is appropriately quantitatively scale invariant, though this is somewhat technical, because one can no longer use the underlying compact support of the operator to deduce any quantitative estimates. Proceeding as in Section 5.2.3, which is sufficient for our purposes, is more straightforward. Special cases of the theory in Section 5.2.3 include the flag kernels and product theory described above. See [220, Sections 4.1 and 4.2] for more details. Spectral multipliers, like the ones described in Section 5.12, have been studied by many authors. In the group setting, see [127, 179, 1, 48]. A very general result of Stein [214] deals with the case where m is of Laplace transform type. The proof of Theorem 5.12.1, which can be found in [220, Section 2.6], uses the finite propagation speed of the wave equation for Hörmander’s sub-Laplacian, which is a result of Melrose [169]. The proof also uses ideas of Sikora [211] and Coulhon and Duong [55].

6 Besov and Triebel–Lizorkin spaces As described in Section 1.2 (see also Section 1.7.1), one of the main useful properties of elliptic operators is that they are locally left invertible modulo smooth functions on the classical Besov and Triebel–Lizorkin spaces; among these important spaces are the Lp Sobolev spaces (1 < p < ∞) and the Zygmund–Hölder spaces. In this chapter, we generalize these spaces to achieve the same sort of left invertibility for maximally subelliptic operators. We present the results in two settings. Throughout this chapter, M will be a connected C ∞ manifold of dimension n ∈ ℕ+ , endowed with a smooth, strictly positive density Vol. The first setting is the single-parameter setting, where we are given (W , ds) = ∞ {(W1 , ds1 ), . . . , (Wr , dsr )} ⊂ Cloc (M; TM) × ℕ+ , which are Hörmander vector fields with formal degrees. For 𝒦 ⋐ M a compact set, we define two scales of spaces of distributions supported in 𝒦: s – The Besov spaces: for s ∈ ℝ, p, q ∈ [1, ∞], Bp,q (𝒦, (W , ds)). s – The Triebel–Lizorkin spaces: for s ∈ ℝ, p ∈ (1, ∞), q ∈ (1, ∞], Fp,q (𝒦, (W , ds)). The spaces are defined in such a way that each Wj acts as a differential operator of s “degree” dsj . Of particular interest to us are the spaces Fp,2 (𝒦, (W , ds)), which we see as p s a non-isotropic L Sobolev space, and B∞,∞ (𝒦, (W , ds)), which we see as a non-isotropic Zygmund–Hölder space. These Zygmund–Hölder spaces play a key role in the study of nonlinear maximally subelliptic equations. The single-parameter Besov and Triebel– Lizorkin spaces are the right spaces in which to state the optimal regularity properties of maximally subelliptic operators. Unfortunately the single-parameter theory described above is not powerful enough to deduce regularity properties of nonlinear maximally subelliptic equations with respect to other geometries. For example, when considering the classical (Euclidean) Besov and Triebel–Lizorkin spaces from Section 2.4, the single-parameter theory is not strong enough to deduce the regularity theory of fully nonlinear maximally subelliptic operators with respect to these classical spaces (as we will see, it is enough to deduce much of the regularity theory for linear operators). Instead, we introduce multi-parameter Besov and Triebel–Lizorkin spaces in the general setting of Section 5.1.3. These spaces generalize the single-parameter spaces described above. They give scales of spaces which can involve various geometries (for example the sub-Riemannian geometry associated with (W , ds) combined with the classical Euclidean geometry) and on which maximally subelliptic operators are essentially isomorphisms. A main difficulty we face is that we do not have a simple Calderón reproducing formula in the generality in which we work.1 We make do by repeatedly leveraging various

1 However, in some more specialized settings, we can recreate a Calderón reproducing formula; see the proof of Proposition 6.12.1. https://doi.org/10.1515/9783111085647-006

394 � 6 Besov and Triebel–Lizorkin spaces types of almost orthogonality, though this makes the proofs somewhat more technical than in the classical setting. Also, for this same reason, we define the spaces by generalizing Proposition 2.4.18 (ii). First, we describe some main ideas in a simple special case in Section 6.1. Then we state some of the main definitions and results of this chapter in Sections 6.2 and 6.3, before turning to the proofs in later sections.

6.1 Informal description of the norms There are several subtleties involved in defining the Besov and Triebel–Lizorkin spaces. We defer those subtleties to later sections, and in this section informally describe the norms we use in a special case while ignoring some difficulties; the reader should consult Section 6.2 for rigorous and more general versions of the concepts described here. We begin by reminding the reader of the definition of the classical Besov and Triebel–Lizorkin norms on ℝn , described in Section 2.4. Let ϕ̂ ∈ C0∞ (ℝn ) be such that ̂ ϕ(ξ) ≡ 1 on a neighborhood of 0 ∈ ℝn . For j ≥ 0 set ςĵ (ξ) := {

̂ ϕ(ξ), ̂ − ϕ(2ξ), ̂ ϕ(ξ)

j = 0, j ≥ 1.

For j ≥ 0, define D1j : S (ℝn )′ → S (ℝn )′ by (D1j f ) (ξ) := ςĵ (2−j ξ)f ̂(ξ).

(6.1)

∧

We have 󵄩󵄩 js 1 󵄩󵄩 s (ℝn ) = 󵄩 ‖f ‖Bp,q 󵄩{2 Dj f }j∈ℕ 󵄩󵄩ℓq (ℕ;Lp (ℝn )) , 󵄩 js 1 󵄩 s (ℝn ) = 󵄩 ‖f ‖Fp,q 󵄩󵄩{2 Dj f }j∈ℕ 󵄩󵄩󵄩Lp (ℝn ;ℓq (ℕ)) ,

f ∈ S (ℝn )′ , f ∈ S (ℝn )′ .

Formula (6.1) can be equivalently written as D1j f (x) = ∫ f (x − t) Dil2j (ςj )(t) dt, where Dil2j (ςj )(t) = 2nj ςj (2j t). Let η ∈ C0∞ (ℝn ) satisfy η ≡ 1 on a neighborhood of 0. Set D2j f (x) := ∫ f (x − t)η(t) Dil2j (ςj )(t) dt. It is not hard to see that we have 󵄩󵄩 js 2 󵄩󵄩 s (ℝn ) ≈ 󵄩 ‖f ‖Bp,q 󵄩{2 Dj f }j∈ℕ 󵄩󵄩ℓq (ℕ;Lp (ℝn )) , 󵄩 js 2 󵄩 s (ℝn ) ≈ 󵄩 ‖f ‖Fp,q 󵄩󵄩{2 Dj f }j∈ℕ 󵄩󵄩󵄩Lp (ℝn ;ℓq (ℕ)) ,

f ∈ S (ℝn )′ , f ∈ S (ℝn )′ .

6.1 Informal description of the norms

� 395

Finally, fix a compact set 𝒦 ⋐ ℝn and let ψ ∈ C0∞ (ℝn ) have ψ ≡ 1 on a neighborhood of 𝒦. Set D3j f (x) := ψ(x) ∫ f (x − t)η(t) Dil2j (ςj )(t) dt

(6.2)

= ψ(x) ∫ f (e−t⋅𝜕 x)η(t) Dil2j (ςj )(t) dt. We have 󵄩 js 3 󵄩 s (ℝn ) ≈ 󵄩 ‖f ‖Bp,q 󵄩󵄩{2 Dj f }j∈ℕ 󵄩󵄩󵄩ℓq (ℕ;Lp (ℝn )) , 󵄩󵄩 js 3 󵄩󵄩 s (ℝn ) ≈ 󵄩 ‖f ‖Fp,q 󵄩{2 Dj f }j∈ℕ 󵄩󵄩Lp (ℝn ;ℓq (ℕ)) ,

f ∈ S (ℝn )′ with supp(f ) ⊆ 𝒦, f ∈ S (ℝn )′ with supp(f ) ⊆ 𝒦.

Let us now turn to the simple case where M = ℝ2 and we are given the list of Hörmander vector fields with formal degrees (W , ds) = {(𝜕x , 1), (x𝜕y , 1)}, and fix a compact set s (W ,ds) and ‖f ‖F s (W ,ds) when supp(f ) ⊆ 𝒦. 𝒦 ⋐ ℝ2 . We wish to define the norms ‖f ‖Bp,q p,q s s−1 We will define these spaces so that 𝜕x , x𝜕y : Bp,q (𝒦, (W , ds)) → Bp,q (𝒦, (W , ds)), and s s similarly for Fp,q (𝒦, (W , ds)). In particular, we have 𝜕y = [𝜕x , x𝜕y ] : Bp,q (𝒦, (W , ds)) → s−2 Bp,q (𝒦, (W , ds)).

Let

(X, d ) = {(X1 , d1 ), (X2 , d2 ), (X3 , d3 )} := {(𝜕x , 1), (x𝜕y , 1), (𝜕y , 2)}; see Example 3.3.6. Take ϕ̂ ∈ C0∞ (ℝ3 ) as above, so that ϕ̂ ≡ 1 on a neighborhood of 0. Define ςĵ (ξ1 , ξ2 , ξ3 ) := {

̂ , ξ , ξ ), ϕ(ξ 1 2 3 2 ̂ , ξ , ξ ) − ϕ(2ξ ̂ ̂ ̂ d ϕ(ξ 1 2 3 1 , 2ξ2 , 2 ξ3 ) = ϕ(ξ) − ϕ(2 ξ),

j = 0, j ≥ 1.

Fix η ∈ C0∞ (ℝ3 ) with η ≡ 1 on a neighborhood of 0 and η with small support. Fix ψ ∈ C0∞ ℝ2 with ψ ≡ 1 on a neighborhood of 𝒦. As an analog to (6.2), we set Dj f (x) := ψ(x) ∫ f (e−t⋅X x)η(t) Dild2j (ςj )(t) dt, where Dild2j (ςj )(t1 , t2 , t3 ) = 24j ςj (2j t1 , 2j t2 , 22j t3 ); see Definition 4.1.12. Note that because ψ has compact support and η has small support, the expression e−t⋅X x, x ∈ supp(ψ) and t ∈ supp(η) will always make sense, as long as the support of η is chosen sufficiently small – how small will, in general, depend on the vector fields X.

396 � 6 Besov and Triebel–Lizorkin spaces For f with supp(f ) ⋐ 𝒦, we set 󵄩󵄩 js 󵄩󵄩 s (W ,ds) = 󵄩 ‖f ‖Bp,q 󵄩{2 Dj f }j∈ℕ 󵄩󵄩ℓq (ℕ;Lp (ℝ2 )) , 󵄩 js 󵄩 s (W ,ds) = 󵄩 ‖f ‖Fp,q 󵄩󵄩{2 Dj f }j∈ℕ 󵄩󵄩󵄩Lp (ℝ2 ;ℓq (ℕ)) ,

supp(f ) ⊆ 𝒦, supp(f ) ⊆ 𝒦.

We have made many choices, but we will show that the equivalence class of the above norms does not depend on these choices. Away from x = 0, 𝜕x and x𝜕y span the tangent space to ℝ2 . Because of this (and the fact that 𝜕x and x𝜕y have formal degree 1), Theorem 6.6.7 shows that the norms s (W ,ds) and ‖ ⋅ ‖F s (W ,ds) are locally equivalent to the standard norms ‖ ⋅ ‖ s ‖ ⋅ ‖Bp,q Bp,q (ℝ2 ) p,q and ‖ ⋅ ‖Fp,q s (ℝ2 ) , away from x = 0. However, as x → 0, these norms are not comparable: near x = 0 and when s > 0, ‖ ⋅ ‖Bp,q s (ℝ2 ) and ‖ ⋅ ‖F s (ℝ2 ) are larger (stronger) norms than p,q s (W ,ds) and ‖ ⋅ ‖F s (W ,ds) , respectively. Theorem 6.6.13 and Remark 6.6.14 describe ‖ ⋅ ‖Bp,q p,q how to compare these norms. For example, in Corollary 6.2.14, we see that for κ ∈ ℕ and 1 < p < ∞, α 󵄩 󵄩 κ (W ,ds) ≈ ∑ 󵄩 ‖f ‖Fp,2 󵄩󵄩(𝜕x , x𝜕y ) f 󵄩󵄩󵄩Lp (ℝ2 ) ,

supp(f ) ⊆ 𝒦.

(6.3)

|α|≤κ

From (6.3) it can be seen that, near x = 0, the best one can say is ‖f ‖F κ

p,2 (ℝ

2)

≲ ‖f ‖F 2κ (W ,ds) ≲ ‖f ‖F 2κ (ℝ2 ) . p,2

p,2

(6.4)

One can think of (6.4) as reflecting the fact that 𝜕y = [𝜕x , x𝜕y ] is treated as a differential s operator of degree 2 with respect to the norms ‖ ⋅ ‖Fp,2 (W ,ds) .

6.2 The single-parameter spaces ∞ Let (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} ⊂ Cloc (M; TM) × ℕ+ be Hörmander vector fields with formal degrees on M. Fix a compact set 𝒦 ⋐ M. Our goal is to define the following spaces consisting of distributions in C0∞ (M)′ with support in 𝒦: s – The Besov spaces: for s ∈ ℝ, p, q ∈ [1, ∞], Bp,q (𝒦, (W , ds)). s – The Triebel–Lizorkin spaces: for s ∈ ℝ, p ∈ (1, ∞), q ∈ (1, ∞], Fp,q (𝒦, (W , ds)).

Notation 6.2.1. V will denote one of the spaces: {ℓq (ℕ; Lp (M, Vol)) : p, q ∈ [1, ∞]} ⋃{Lp (M, Vol; ℓq (ℕ)) : p ∈ (1, ∞), q ∈ (1, ∞]}. See (2.28) and (2.29) for further details.

6.2 The single-parameter spaces

�

397

Notation 6.2.2. For s ∈ ℝ, X s (𝒦, (W , ds)) will denote any one of the spaces (to be defined) s s {Bp,q (𝒦, (W , ds)) : p, q ∈ [1, ∞]} ⋃{Fp,q (𝒦, (W , ds)) : p ∈ (1, ∞), q ∈ (1, ∞]}.

We take V := {

ℓq (ℕ; Lp (M, Vol)), p

q

L (M, Vol; ℓ (ℕ)),

s when X s (𝒦, (W , ds)) = Bp,q (𝒦, (W , ds)), s when X s (𝒦, (W , ds)) = Fp,q (𝒦, (W , ds)).

Definition 6.2.3. Let ℰ be a bounded set of (W , ds) elementary operators (see Definition 5.2.8). We set, for f ∈ C0∞ (M)′ and s ∈ ℝ, ‖f ‖V ,s,ℰ :=

sup {(Ej

,2−j ):j∈ℕ}⊆ℰ

󵄩󵄩 js 󵄩 󵄩󵄩{2 Ej f }j∈ℕ 󵄩󵄩󵄩V .

This defines an extended2 semi-norm on C0∞ (M)′ . Motivated by Proposition 2.4.18 we make the next definition. Definition 6.2.4. For s ∈ ℝ, we define X s (𝒦, (W , ds)) to be the space of all those f ∈ C0∞ (M)′ such that supp(f ) ⊆ 𝒦 and for every bounded set of (W , ds) elementary operators, ℰ , we have ‖f ‖V ,s,ℰ < ∞. It is clear from the definition that X s (𝒦, (W , ds)) can be given the structure of a locally convex topological vector space. In fact, X s (𝒦, (W , ds)) is a Banach space. We turn to introducing a norm on X s (𝒦, (W , ds)) which achieves this. Fix ψ ∈ C0∞ (M) with ψ ≡ 1 on a neighborhood of 𝒦. Let Ω ⋐ M be an open set with ψ ∈ C0∞ (Ω). By Proposition 5.8.3, Mult[ψ] ∈ A 0 (Ω, (W , ds)) = A20 (Ω, (W , ds)). Thus, there exists a bounded set of (W , ds) elementary operators, 𝒟0 = {(Dj , 2−j ) : j ∈ ℕ} with Mult[ψ] = ∑j∈ℕ Dj (see Proposition 5.5.10 for the convergence of this sum). Proposition 6.2.5. ‖⋅‖V ,s,𝒟0 defines a norm on X s (𝒦, (W , ds)). With respect to this norm, X s (𝒦, (W , ds)) is a Banach space. Proposition 6.2.6. The equivalence class of the norm ‖ ⋅ ‖V ,s,𝒟0 does not depend on any of the choices we made above. More precisely, if ψ̃ ∈ C0∞ (M) is such that ψ̃ ≡ 1 on a ̃ j , 2−j ) : j ∈ ℕ} is a bounded set of (W , ds) elementary ̃ = {(D neighborhood of 𝒦 and 𝒟 ̃ ̃ operators with Mult[ψ] = ∑j∈ℕ Dj , then ‖f ‖V ,s,𝒟0 ≈ ‖f ‖V ,s,𝒟 ̃,

∀f ∈ X s (𝒦, (W , ds)).

2 It is an extended semi-norm in that it satisfies all the axioms of a semi-norm, except that it may take the value ∞.

398 � 6 Besov and Triebel–Lizorkin spaces Here, the implicit constants do not depend on f , but may depend on any of the other choices. In light of Proposition 6.2.6, the next definition makes sense. Definition 6.2.7. For f ∈ X s (𝒦, (W , ds)), we write 󵄩 󵄩 ‖f ‖X s (W ,ds) := ‖f ‖V ,s,𝒟0 = 󵄩󵄩󵄩{2js Dj f }j∈ℕ 󵄩󵄩󵄩V . Remark 6.2.8. Only the equivalence class of the norm ‖ ⋅ ‖X s (W ,ds) is well-defined on X s (𝒦, (W , ds)), though that is all we require. The notation ‖ ⋅ ‖X s (W ,ds) does not involve the compact set 𝒦, since given two compact sets 𝒦 and 𝒦′ , we may pick ψ so that ψ ≡ 1 on a neighborhood of 𝒦 ∪ 𝒦′ , and we may therefore use the same norm on X s (𝒦, (W , ds)) and X s (𝒦′ , (W , ds)). Remark 6.2.9. Proposition 6.2.6 only shows that the equivalence class of the norm ‖f ‖X s (W ,ds) is well-defined for f ∈ X s (𝒦, (W , ds)), not for all f ∈ C0∞ (M)′ with supp f ⊆ 𝒦. In particular, we do not claim here that if ‖f ‖X s (W ,ds) < ∞, then f ∈ X s (𝒦, (W , ds)). This problem can be fixed by modifying the norm; see Remark 6.4.6 and also Section 6.10 for further comments. Theorem 6.2.10. Let Ω be an open set with Ω ⊆ 𝒦. Then operators in A t (Ω, (W , ds)) are bounded X s (𝒦, (W , ds)) → X s−t (𝒦, (W , ds)). The next two results show that the vector field Wj acts like a differential operator of degree dsj on these spaces. Proposition 6.2.11. For any ordered multi-index α, W α defines a bounded operator X s (𝒦, (W , ds)) → X s−degds (α) (𝒦, (W , ds)). Theorem 6.2.12. Fix κ ∈ ℕ+ such that dsj divides κ for 1 ≤ j ≤ r. Set nj := κ/dsj ∈ ℕ+ n

n

and L := ∑rj=1 (Wj j )∗ Wj j , the sub-Laplacian (see Section 4.5). Then, for f ∈ C0∞ (M)′ , the following are equivalent: (i) f ∈ X s (𝒦, (W , ds)). (ii) For every ordered multi-index α with degds(α) ≤ κ, we have W α f ∈ X s−κ (𝒦, (W , ds)). n (iii) f ∈ X s−κ (𝒦, (W , ds)) and Wj j f ∈ X s−κ (𝒦, (W , ds)) for 1 ≤ j ≤ r. (iv) f , L f ∈ X s−2κ (𝒦, (W , ds)).

Furthermore, in this case we have r

‖f ‖X s (W ,ds) ≈

∑

degds (α)≤κ

n

‖W α f ‖X s−κ (W ,ds) ≈ ‖f ‖X s−κ (W ,ds) + ∑ ‖Wj j f ‖X s−κ (W ,ds)

≈ ‖f ‖X s−2κ (W ,ds) + ‖L f ‖X s−2κ (W ,ds) .

j=1

6.2 The single-parameter spaces

� 399

s One space of particular importance is Fp,2 (for 1 < p < ∞), which can be thought p of as a non-isotropic L Sobolev space. This interpretation is justified by the next two results. 0 Proposition 6.2.13. For 1 < p < ∞, we have Fp,2 (𝒦, (W , ds)) = Lp (𝒦, Vol) and

∀f ∈ Lp (𝒦, Vol).

‖f ‖F 0 ≈ ‖f ‖Lp (𝒦,Vol) , p,2

Corollary 6.2.14. Fix κ ∈ ℕ+ such that dsj divides κ for 1 ≤ j ≤ r; set nj := κ/dsj ∈ ℕ+ . Then, for 1 < p < ∞ and f ∈ C0∞ (M)′ , the following are equivalent: κ (i) f ∈ Fp,2 (𝒦, (W , ds)). (ii) For every ordered multi-index α with degds(α) ≤ κ, we have W α f ∈ Lp (𝒦, Vol). n (iii) f ∈ Lp (𝒦, Vol) and Wj j f ∈ Lp (𝒦, Vol) for 1 ≤ j ≤ r. Furthermore, in this case we have r

κ (W ,ds) ≈ ‖f ‖Fp,2

n

∑

degds (α)≤κ

n

‖W α f ‖Lp ≈ ‖f ‖Lp + ∑ ‖Wj j f ‖Lp . j=1

(6.5)

n

Also, if L := ∑rj=1 (Wj j )∗ Wj j is the sub-Laplacian, then the following are equivalent: 2κ (a) f ∈ Fp,2 (𝒦, (W , ds)). (b) f , L f ∈ Lp (𝒦, Vol).

Furthermore, in this case we have ‖f ‖F 2κ (W ,ds) ≈ ‖f ‖Lp + ‖L f ‖Lp . p,2

Proof. This follows by combining Proposition 6.2.13 and Theorem 6.2.12. All of the above results follow from more general results in the multi-parameter setting. See Section 6.3. Finally, there is one last space which will be important for our applications, namely, the Zygmund–Hölder space, for s > 0, s

s

C (𝒦, (W , ds)) := B∞,∞ (𝒦, (W , ds)).

This space plays the same role in the study of nonlinear maximally subelliptic equations that the classical Zygmund–Hölder spaces play for nonlinear elliptic equations. We defer discussion of these spaces to Chapter 7. The classical Besov and Triebel–Lizorkin spaces are a special case of the above spaces: with the right choice of (W , ds), these spaces exactly correspond to the classical spaces with the restriction that the distributions are supported in 𝒦. See Section 6.6.1 for a further discussion.

400 � 6 Besov and Triebel–Lizorkin spaces See Section 6.12 for another approach to these function spaces using spectral multipliers.

6.3 The multi-parameter spaces We work in the general setting described in Section 5.1.3. Thus, for some ν ∈ ℕ+ we are given ν lists of vector fields with formal degrees: μ

μ

∞ (W μ , dsμ ) = {(W1 , ds1 ), . . . , (Wrμμ , dsμrμ )} ⊂ Cloc (M; TM) × ℕ+ .

We assume that (W 1 , ds1 ), . . . , (W ν , dsν ) pairwise locally weakly approximately commute and that for each μ ∈ {1, . . . , ν}, Gen((W μ , dsμ )) is locally finitely generated on M. We define (W , ds)⃗ = {(W1 , ds1⃗ ), . . . , (Wr , dsr⃗ )}

∞ := (W 1 , ds1 ) ⊠ (W 2 , ds2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (W ν , dsν ) ⊂ Cloc (M; TM) × (ℕν \ {0}).

Fix 𝒦 ⋐ M. The goal of this section is to define the following spaces consisting of ∞ ′ distributions in CW ,0 (M) with support in 𝒦: s ⃗ – The Besov spaces: for s ∈ ℝν , p, q, ∈ [1, ∞], Bp,q (𝒦, (W , ds)). ν ⃗ – The Triebel–Lizorkin spaces: for s ∈ ℝ , p ∈ (1, ∞), q ∈ (1, ∞], F s (𝒦, (W , ds)). p,q

Similar to Notations 6.2.1 and 6.2.2, we introduce the following. Notation 6.3.1. V will denote one of the spaces: {ℓq (ℕν ; Lp (M, Vol)) : p, q ∈ [1, ∞]} ⋃{Lp (M, Vol; ℓq (ℕν )) : p ∈ (1, ∞), q ∈ (1, ∞]}. For a sequence of distributions {fj }j∈ℕ ⊂ C0∞ (M)′ , when we consider ‖{fj }j∈ℕ ‖V we set this equal to ∞ if any of the fj does not agree with integration against an L1loc (M, Vol) function. If all fj agree with integration against an L1loc (M, Vol) function, ‖{fj }j∈ℕ ‖V has the usual definition (see (2.28) and (2.29)). ⃗ will denote any one of the spaces (to be deNotation 6.3.2. For s ∈ ℝ, X s (𝒦, (W , ds)) fined) s ⃗ : p, q ∈ [1, ∞]} ⋃{F s (𝒦, (W , ds)) ⃗ : p ∈ (1, ∞), q ∈ (1, ∞]}. {Bp,q (𝒦, (W , ds)) p,q

⃗ = B s (𝒦, (W , ds)) ⃗ we take V to be ℓq (ℕν ; Lp (M, Vol)), and when When X s (𝒦, (W , ds)) p,q ⃗ = F s (𝒦, (W , ds)) ⃗ we take V to be Lp (M, Vol; ℓq (ℕν )). X s (𝒦, (W , ds)) p,q Definition 6.3.3. Let ℰ be a bounded set of generalized (W , ds)⃗ elementary operators ∞ ′ ν (see Definition 5.2.27). We set, for f ∈ CW ,0 (M) and s ∈ ℝ ,

6.3 The multi-parameter spaces

‖f ‖V ,s,ℰ :=

sup

{(Ej ,2−j ):j∈ℕν }⊆ℰ

� 401

󵄩󵄩 j⋅s 󵄩 󵄩󵄩{2 Ej f }󵄩󵄩󵄩V .

∞ ′ This defines an extended semi-norm on CW ,0 (M) .

Remark 6.3.4. In Definition 6.3.3, we applied Ej to f , where (Ej , 2−j ) ∈ ℰ and f ∈ ∞ ′ ⃗ CW ,0 (M) . If one looks at the definition of generalized (W , ds) pre-elementary operators ∞ ∞ (Definition 5.2.26), Ej is only given as an element of Hom(CW ,0 (M), CW ,loc (M)). How∗ ∞ ∞ ever, as part of Definition 5.2.26 it is also assumed that Ej ∈ Hom(CW ,0 (M), CW ,loc (M)). ∞ ′ ∞ ′ Thus, by duality, Ej extends to an operator in Hom(CW ,loc (M) , CW ,0 (M) ). Combining this with the assumed compact support of Ej (also in Definition 5.2.26), we see that ∞ ′ ∞ ′ ∞ ′ Ej ∈ Hom(CW ,0 (M) , CW ,loc (M) ). In particular, Ej f makes sense for f ∈ CW ,0 (M) . ⃗ to be the space of all those f ∈ Definition 6.3.5. For s ∈ ℝν , we define X s (𝒦, (W , ds)) ∞ ′ CW (M) such that supp(f ) ⊆ 𝒦 and for every bounded set of generalized (W , ds)⃗ ele,0 mentary operators, ℰ , we have ‖f ‖V ,s,ℰ < ∞. ⃗ into a Banach space. Fix ψ ∈ We next present a norm which turns X s (𝒦, (W , ds)) with ψ ≡ 1 on a neighborhood of 𝒦. Let Ω ⋐ M be an open set with ψ ∈ C0∞ (Ω). ̃0 (Ω, (W , ds)) = A ̃0 (Ω, (W , ds)). Thus, there exists a By Proposition 5.8.3, Mult[ψ] ∈ A 2 bounded set of generalized (W , ds)⃗ elementary operators, 𝒟0 = {(Dj , 2−j ) : j ∈ ℕ} with Mult[ψ] = ∑j∈ℕ Dj (see Proposition 5.5.10 for the convergence of this sum). C0∞ (M)

⃗ With respect to this norm, Proposition 6.3.6. ‖⋅‖V ,s,𝒟0 defines a norm on X s (𝒦, (W , ds)). s ⃗ X (𝒦, (W , ds)) is a Banach space. Proposition 6.3.7. The equivalence class of the norm ‖ ⋅ ‖V ,s,𝒟0 does not depend on any of the choices we made above. More precisely, if ψ̃ ∈ C0∞ (M) is such that ψ̃ ≡ 1 on a ̃ j , 2−j ) : j ∈ ℕ} is a bounded set of generalized (W , ds)⃗ ̃ = {(D neighborhood of 𝒦 and 𝒟 ̃ j , then elementary operators with Mult[ψ]̃ = ∑j∈ℕ D ‖f ‖V ,s,𝒟0 ≈ ‖f ‖V ,s,𝒟 ̃,

⃗ ∀f ∈ X s (𝒦, (W , ds)).

Here, the implicit constants do not depend on f , but may depend on any of the other choices. In light of Proposition 6.3.7, the next definition makes sense. ⃗ we write Definition 6.3.8. For f ∈ X s (𝒦, (W , ds)), 󵄩 󵄩 ‖f ‖X s (W ,ds)⃗ := ‖f ‖V ,s,𝒟0 = 󵄩󵄩󵄩{2j⋅s Dj }j∈ℕ 󵄩󵄩󵄩V . Remarks 6.2.8 and 6.2.9 also apply in this more general setting. In particular, we have the following.

402 � 6 Besov and Triebel–Lizorkin spaces Remark 6.3.9. The notation ‖ ⋅ ‖X s (W ,ds)⃗ does not involve the compact set 𝒦, since given two compact sets 𝒦 and 𝒦′ , we may pick ψ so that ψ ≡ 1 on a neighborhood of 𝒦 ∪ 𝒦′ , and we may therefore use the same norm on X s (𝒦, (W , ds)) and X s (𝒦′ , (W , ds)). ̃t (Ω, (W , ds)) are Theorem 6.3.10. Let Ω be an open set with Ω ⊆ 𝒦. Then operators in A s s−t bounded X (𝒦, (W , ds)) → X (𝒦, (W , ds)). The above results are proved in Section 6.5. Analogs of all the results in Section 6.2 hold in this more general setting as well, and will be presented throughout the chapter. In particular, similar to Proposition 6.2.11 and Theorem 6.2.12, Wj acts like a differ⃗ (see Corollary 6.5.11, ential operator of degree dsj⃗ ∈ ℕν on the spaces X s (𝒦, (W , ds)) Proposition 6.5.12, and Corollary 6.5.13). Similar to Proposition 6.2.13 and Corollary 6.2.14, s ⃗ can be viewed as non-isotropic Lp Sobolev spaces for 1 < p < ∞ (see SecFp,2 (𝒦, (W , ds)) tion 6.9). s ⃗ ℂN )) Definition 6.3.11. For N ∈ ℕ+ , we define the vector-valued Besov (Bp,q (𝒦, (W , ds); s N s N ⃗ ℂ )), by letting X (𝒦, (W , ds); ⃗ ℂ ) consist of and Triebel–Lizorkin (Fp,q (𝒦, (W , ds); s ⃗ those distributions u = (u1 , . . . , uN ) with each uj ∈ X (𝒦, (W , ds)). We set N

‖u‖X s ((W ,ds);ℂ ⃗ N ) := ∑ ‖uj ‖X s (W ,ds)⃗ . j=1

6.4 The main estimate This section is devoted to the main technical estimate which is at the heart of many of the proofs in this chapter. Recall that in Section 6.3, we fixed ψ ∈ C0∞ (M) with ψ ≡ 1 on 𝒦 and a bounded set of generalized (W , ds)⃗ elementary operators, 𝒟0 = {(Dj , 2−j ) : j ∈ ℕν }, with Mult[ψ] = ∑j∈ℕν Dj . Notation 6.4.1. For any sequence of operators indexed by j ∈ ℕν , Ej , we define Ej := 0 for j ∈ ℤν \ ℕν . For example, using this notation, we have Mult[ψ] = ∑j∈ℤν Dj . For j, k, l ∈ ℤν and N ≥ 1, set FN,j,k,l := 2N|k|+|l| Dj+k Dj+k+l . Note that using Notation 6.4.1, FN,j,k,l = 0 if j + k ∈ ℤν \ ℕν or j + k + l ∈ ℤν \ ℕν . Set 𝒟N := {(FN,j,k,l , 2

−(j+k)

) : j, k, l ∈ ℤν , j + k ∈ ℕν , |l| > |k|}.

(6.6)

Lemma 6.4.2. For N ≥ 1, 𝒟N is a bounded set of generalized (W , ds)⃗ elementary operators. Moreover, if 𝒟0 is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω, then so is 𝒟N .

6.4 The main estimate

�

403

Proof. Let Ω ⋐ M be open and relatively compact such that 𝒟0 is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω. Since |l| > |k| in the definition of 𝒟N , by Proposition 5.5.5 (a), it suffices to show that {(2(N+2)|l| Dj+k Dj+k+l , 2−(j+k) ) : j, k, l ∈ ℤν , j + k ∈ ℕν } is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω. This follows directly from Proposition 5.5.11, completing the proof. The next result is the main technical estimate which lies at the heart of many of the proofs in this chapter. Theorem 6.4.3. Fix K0 ⋐ ℝν compact and set N := sup{|s| : s ∈ K0 } + 1. Suppose s ∈ ∞ ′ K0 , f ∈ CW ,0 (M) with supp f ⊆ 𝒦, and ‖f ‖V ,s,𝒟N < ∞. Then, for every bounded set of generalized (W , ds)⃗ elementary operators ℰ , there exists C = C(ℰ , K0 , V ) ≥ 0 such that ‖f ‖V ,s,ℰ ≤ C‖f ‖V ,s,𝒟0 , where if the right-hand side is finite, so is the left-hand side. Here, C does not depend on f , ‖f ‖V ,s,𝒟N , or the particular s ∈ K0 . Before we prove Theorem 6.4.3 we present three simple corollaries. Corollary 6.4.4. Let s ∈ ℝν and let ℰ be a bounded set of generalized (W , ds)⃗ elementary ⃗ In particular, operators. Then ‖ ⋅ ‖V ,s,ℰ defines a continuous semi-norm on X s (𝒦, (W , ds)). there is a constant C ≥ 0 with ‖f ‖V ,s,ℰ ≤ C‖f ‖V ,s,𝒟0 = ‖f ‖X s (W ,ds)⃗ ,

⃗ ∀f ∈ X s (𝒦, (W , ds)).

Furthermore, if K0 ⋐ ℝν is a compact set, then the constant C ≥ 0 can be chosen independent of s ∈ K0 . ⃗ Set N := sup{|s| : s ∈ K0 } + 1. By Lemma 6.4.2, 𝒟N Proof. Suppose f ∈ X s (𝒦, (W , ds)). is a bounded set of generalized (W , ds)⃗ elementary operators. Thus, by the definition of ⃗ (Definition 6.3.5), we have ‖f ‖V ,s,𝒟 < ∞. Applying Theorem 6.4.3 we see X s (𝒦, (W , ds)) N that ‖f ‖V ,s,ℰ ≲ ‖f ‖V ,s,𝒟0 = ‖f ‖X s (W ,ds)⃗ , as desired, completing the proof. ∞ ′ s ⃗ Corollary 6.4.5. For f ∈ CW ,0 (M) , we have f ∈ X (𝒦, (W , ds)) if and only if ‖f ‖V ,s,𝒟0 + ‖f ‖V ,s,𝒟|s|+1 < ∞ and supp f ⊆ 𝒦. Moreover, the norm ‖ ⋅ ‖V ,s,𝒟0 + ‖ ⋅ ‖V ,s,𝒟|s|+1 is equivalent s ⃗ to the norm ‖ ⋅ ‖ s ⃗ on X (𝒦, (W , ds)). X (W ,ds)

404 � 6 Besov and Triebel–Lizorkin spaces ⃗ Then by the definition of X s (𝒦, (W , ds)) ⃗ (DefiniProof. Suppose f ∈ X s (𝒦, (W , ds)). tion 6.3.5) we have supp(f ) ⊆ 𝒦 and for every bounded set of generalized (W , ds)⃗ elementary operators, ℰ , we have ‖f ‖V ,s,ℰ < ∞. Since 𝒟0 is a bounded set of generalized (W , ds)⃗ elementary operators by construction and 𝒟|s|+1 is a bounded set of generalized (W , ds)⃗ elementary operators by Lemma 6.4.2, we have ‖f ‖V ,s,𝒟0 + ‖f ‖V ,s,𝒟|s|+1 < ∞. Conversely, suppose supp f ⊆ 𝒦 and ‖f ‖V ,s,𝒟0 + ‖f ‖V ,s,𝒟|s|+1 < ∞. Let ℰ be a bounded set of generalized (W , ds)⃗ elementary operators. Applying Theorem 6.4.3 with K0 = {s}, we have

‖f ‖V ,s,ℰ ≲ ‖f ‖V ,s,𝒟0 < ∞. ⃗ Since ℰ was arbitrary, we see that f ∈ X s (𝒦, (W , ds)). ⃗ Finally, we show that the two norms are equivalent. Clearly, for f ∈ X s (𝒦, (W , ds)), we have ‖f ‖X s (W ,ds)⃗ = ‖f ‖V ,s,𝒟0 ≤ ‖f ‖V ,s,𝒟0 + ‖f ‖V ,s,𝒟|s|+1 . ⃗ Conversely, Corollary 6.4.4 shows that for f ∈ X s (𝒦, (W , ds)), ‖f ‖V ,s,𝒟0 + ‖f ‖V ,s,𝒟|s|+1 ≲ ‖f ‖X s (W ,ds)⃗ , completing the proof. Remark 6.4.6. Proposition 6.3.7 states that the equivalence class of the norm ‖f ‖X s (W ,ds)⃗ ⃗ However, it does not make any claim for f ∈ ̸ is well-defined for f ∈ X s (𝒦, (W , ds)). s ⃗ In particular, it is not claimed that if ‖f ‖ s X (𝒦, (W , ds)). ⃗ < ∞ and supp(f ) ⊆ 𝒦, X (W ,ds)

⃗ However, if one replaces ‖f ‖ s then f ∈ X s (𝒦, (W , ds)). X (W ,ds)⃗ with the equivalent norm ‖f ‖V ,s,𝒟0 + ‖f ‖V ,s,𝒟|s|+1 , then Corollary 6.4.5 shows that if supp(f ) ⊆ 𝒦 and the norm is ⃗ See also Section 6.10 for some more comments on this finite, then f ∈ X s (𝒦, (W , ds)). issue.

Corollary 6.4.7. Let Ω ⋐ M be any relatively compact, open set with 𝒦 ⋐ Ω. Then for ∞ ′ f ∈ CW ,0 (M) with supp(f ) ⊆ 𝒦, the following are equivalent: ⃗ (i) f ∈ X s (𝒦, (W , ds)). (ii) For every bounded set of generalized (W , ds)⃗ elementary operators supported in Ω, ℰ , we have ‖f ‖V ,s,ℰ < ∞. Proof. (i) ⇒ (ii): This follows immediately from the definitions. (ii) ⇒ (i): By construction, we may take 𝒟0 to be a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω. By Lemma 6.4.2, 𝒟N is then also a bounded set of

6.4 The main estimate

�

405

generalized (W , ds)⃗ elementary operators supported in Ω, for every N ≥ 1. Thus, if (ii) holds, then ‖f ‖V ,s,𝒟0 + ‖f ‖V ,s,𝒟|s|+1 < ∞ and Corollary 6.4.5 implies that (i) holds. ⃗ only depend on Remark 6.4.8. Corollary 6.4.7 shows that the spaces X s (𝒦, (W , ds)) ⃗ the choice of (W , ds) on an arbitrarily small neighborhood of 𝒦. This justifies the fact ⃗ does not mention the ambient manifold M, as M that the notation X s (𝒦, (W , ds)) can be replaced by any open set containing 𝒦, and we would obtain the same space ⃗ X s (𝒦, (W , ds)). We turn to the proof of Theorem 6.4.3. We begin with two lemmas. Lemma 6.4.9. Let ℰ be a bounded set of generalized (W , ds)⃗ elementary operators and let 1

ν

ℰ1 := {(Ej , 2 ) : j ∈ ℕ }, −j

2

ν

ℰ2 := {(Ej , 2 ) : j ∈ ℕ } ⊆ ℰ . −j

For k ∈ ℤν , define 𝒯k = 𝒯k (ℰ1 , ℰ2 ) by 1 2

𝒯k {fj }j∈ℕν := {Ej Ej+k fj }j∈ℕν .

Then 𝒯k ∈ Hom(V , V ) and there exists C = C(ℰ , N, 𝒱 ) ≥ 0 such that for all k ∈ ℤν , sup ‖𝒯k ‖V →V ≤ C2−N|k| ,

ℰ1 ,ℰ2 ⊆ℰ

where the supremum is taken over all such ℰ1 , ℰ2 ⊆ ℰ . Proof. Proposition 5.5.11 shows that for every N ∈ ℕ, ℰN := {(2

N|k|

E1 E2 , 2−j ) : (E1 , 2−j ), (E2 , 2−(j+k) ) ∈ ℰ }

is a bounded set of generalized (W , ds)⃗ elementary operators. For ℱN := {(Fj , 2−j ) : j ∈ ℕν } ⊆ ℰN , define 𝒮 = 𝒮 (ℱN ) by 𝒮 {fj }j∈ℕν := {Fj fj }j∈ℕν . Thus, we have 2N|k| sup ‖𝒯k ‖V →V ≲ sup ‖𝒮 ‖V →V ≲ 1, ℰ1 ,ℰ2 ⊆ℰ

ℱN ⊆ℰN

where the ≲ 1 follows from the fact that ℰN is a bounded set of generalized (W , ds)⃗ elementary operators supported and we have used the definition of bounded sets of generalized (W , ds)⃗ pre-elementary operators (Definition 5.2.26). This completes the proof. Lemma 6.4.10 (A triangle inequality). Suppose that for j ∈ ℕν , k ∈ ℕ, fj,k ∈ C0∞ (M)′ is such that 󵄩 󵄩 ∑ 󵄩󵄩󵄩{fj,k }j∈ℕν 󵄩󵄩󵄩V < ∞.

k∈ℕ

406 � 6 Besov and Triebel–Lizorkin spaces Then ∑k∈ℕ fj,k converges in C0∞ (M)′ for every j and 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩{ ∑ f } 󵄩󵄩 ≤ ∑ 󵄩󵄩󵄩{f } ν 󵄩󵄩󵄩 . 󵄩󵄩 󵄩󵄩 j,k 󵄩 j,k j∈ℕ 󵄩V 󵄩󵄩 k∈ℕ 󵄩 ν 󵄩 󵄩V k∈ℕ j∈ℕ 󵄩

(6.7)

Proof. Recall Notation 6.3.1, so that V is either Lp (M, Vol; ℓq (ℕν )) or ℓq (ℕν ; Lp (M, Vol)). We present the proof when p, q ≠ ∞; the case where either p or q is infinite is similar and easier. 󵄩 󵄩 Since ∑k∈ℕ 󵄩󵄩󵄩{fj,k }j∈ℕν 󵄩󵄩󵄩V < ∞, we have, for every j ∈ ℕ, ∑k∈ℕ ‖fj,k ‖Lp < ∞, which implies ∑k∈ℕ fj,k converges in Lp (M, Vol), and therefore in C0∞ (M)′ . It remains to prove (6.7). Since ∑k∈ℕ fj,k converges in Lp (M, Vol) we may choose l→∞

N

l a sequence Nl → ∞ such that ∑k=0 fj,k 󳨀󳨀󳨀󳨀→ ∑k∈ℕ fj,k pointwise almost everywhere. Applying Fatou’s lemma and the triangle inequality, we see that

󵄩󵄩 󵄩󵄩 󵄩󵄩 Nl 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩{ ∑ f } 󵄩󵄩 ≤ lim inf 󵄩󵄩󵄩{ ∑ f } 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 j,k j,k 󵄩 󵄩 l→∞ 󵄩󵄩 k∈ℕ 󵄩 󵄩 󵄩 󵄩 󵄩 ν j∈ℕ 󵄩 󵄩 󵄩 k=0 󵄩V j∈ℕν 󵄩V Nl

󵄩 󵄩 󵄩 󵄩 ≤ lim inf ∑ 󵄩󵄩󵄩{fj,k }j∈ℕν 󵄩󵄩󵄩V = ∑ 󵄩󵄩󵄩{fj,k }j∈ℕν 󵄩󵄩󵄩V , l→∞ k=0

k∈ℕ

completing the proof. Proof of Theorem 6.4.3. Set ℰ ′ := ℰ ∪𝒟N . It follows from Lemma 6.4.2 that ℰ ′ is a bounded set of generalized (W , ds)⃗ elementary operators. Note that max{‖f ‖V ,s,ℰ , ‖f ‖V ,s,𝒟N } ≤ ‖f ‖V ,s,ℰ ′ .

(6.8)

Thus, it suffices to prove the result with ℰ replaced by ℰ ′ . Let {(Ej , 2−j ) : j ∈ ℕν } ⊆ ℰ ′ . Since supp(f ) ⊆ 𝒦, we have Ej f = Ej Mult[ψ] Mult[ψ]f = Ej ( ∑ Dj+k ) ( ∑ Dj+k+l ) f k∈ℕν

l∈ℤν

= ∑ Ej Dj+k Dj+k+l f , k,l∈ℤν

∞ ′ where the above sums all converge in the sense of CW ,0 (M) (see Proposition 5.5.10) and ∞ ′ ∞ we are using the fact that each of Ej , Dj ∈ Hom(CW ,0 (M) , CW ,loc (M)′ ) (see Remark 6.3.4). Thus, using Lemma 6.4.10, we have

󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 j⋅s 󵄩󵄩 󵄩󵄩 󵄩󵄩 j⋅s 󵄩󵄩{2 Ej }j∈ℕν 󵄩󵄩V = 󵄩󵄩{ ∑ 2 Ej Dj+k Dj+k+l f } 󵄩󵄩 󵄩󵄩 󵄩󵄩 ν 󵄩󵄩 k,l∈ℤ j∈ℕν 󵄩 󵄩V 󵄩 󵄩 ≤ ∑ 󵄩󵄩󵄩{2j⋅s Ej Dj+k Dj+k+l }j∈ℕν 󵄩󵄩󵄩V , k,l∈ℤν

(6.9)

6.4 The main estimate

407

�

where if the right-hand side of (6.9) is finite, then the left-hand side is as well. Fix M ∈ ℕ large to be chosen later. We wish to bound the right-hand side of (6.9). To do so, we separate it into three terms: (I) := ∑ ⋅, k,l∈ℤν |l|≤M

(II) :=

∑

k,l∈ℤν |l|>M,|k|≤|l|

⋅,

(III) :=

∑

k,l∈ℤν |l|>M,|l|>|k|

⋅.

Note that the right-hand side of (6.9) equals (I) + (II) + (III). In what follows A ≲ B means A ≤ CB, where C may depend on V , K0 , ℰ , 𝒟0 , and 𝒟N , but not on k, l, M, f , or the particular choice of {(Ej , 2−j ) : j ∈ ℕν } ⊆ ℰ ′ . We begin by estimating (I). We define a vector-valued operator, for k ∈ ℤν , by 1

|k|−k⋅s

𝒯k {fj }j∈ℕν := {2

Ej Dj+k fj }j∈ℕν .

Lemma 6.4.9 shows that ‖𝒯k1 ‖V →V ≲ 1. Thus, we have 󵄩 󵄩 (I) = ∑ 2−l⋅s−|k| 󵄩󵄩󵄩{(2|k|−k⋅s Ej Dj+k )(2(j+k+l)⋅s Dj+k+l )f }j∈ℕν 󵄩󵄩󵄩V k,l∈ℤν |l|≤M

󵄩 󵄩 = ∑ 2−l⋅s−|k| 󵄩󵄩󵄩𝒯k1 {2(j+k+l)⋅s Dj+k+l f }j∈ℕν 󵄩󵄩󵄩V k,l∈ℤν |l|≤M

󵄩 󵄩 ≲ ∑ 2−l⋅s−|k| 󵄩󵄩󵄩{2(j+k+l)⋅s Dj+k+l f }j∈ℕν 󵄩󵄩󵄩V

(6.10)

k,l∈ℤν |l|≤M

≤ ∑ 2−l⋅s−|k| ‖f ‖V ,s,𝒟0 ≲ 2M|s| ‖f ‖V ,s,𝒟0 . k,l∈ℤν |l|≤M

We turn to (II). For k, l ∈ ℤν with |k| ≥ |l|, we define the vector-valued operator 2

|k|+|l|−(k+l)⋅s

𝒯k,l {fj }j∈ℕν := {2

Ej Dj+k }j∈ℕν .

2 Using the fact that |k| ≥ |l|, Lemma 6.4.9 shows that ‖𝒯k,l ‖V →V ≲ 1. Thus, we have

󵄩 󵄩 (II) = ∑ 2−|k|−|l| 󵄩󵄩󵄩{(2|k|+|l|−(k+l)⋅s Ej Dj+k )(2(j+k+l)⋅s Dj+k+l f )}j∈ℕν 󵄩󵄩󵄩V k,l∈ℤν |l|>M,|k|≥|l|

󵄩 2 (j+k+l)⋅s 󵄩 = ∑ 2−|k|−|l| 󵄩󵄩󵄩𝒯k,l {2 Dj+k+l }j∈ℕν 󵄩󵄩󵄩V

k,l∈ℤν |l|>M,|k|≥|l|

󵄩 󵄩 ≲ ∑ 2−|k|−|l| 󵄩󵄩󵄩{2(j+k+l)⋅s Dj+k+l }j∈ℕν 󵄩󵄩󵄩V

k,l∈ℤν |l|>M,|k|≥|l|

≤ ∑ 2−|k|−|l| ‖f ‖V ,s,𝒟0 ≲ 2−M ‖f ‖V ,s,𝒟0 .

k,l∈ℤν |l|>M,|k|≥|l|

(6.11)

408 � 6 Besov and Triebel–Lizorkin spaces Finally, we turn to (III). Define a vector-valued operator 3

𝒯 {fj }j∈ℕν := {Ej fj }j∈ℕν .

Since ℰ ′ is a bounded set of generalized (W , ds)⃗ elementary operators, it follows from the definition (see Definition 5.2.26) that ‖𝒯 3 ‖V →V ≲ 1. Thus, we have, using 2−N|k|−k⋅s+|k| ≤ 1, by the definition of N, 󵄩 󵄩 (III) = ∑ 2−|k|−|l| 󵄩󵄩󵄩{Ej (2(j+k)⋅s 2−k⋅s+|k|+|l| Dj+k Dj+k+l f )}j∈ℕν 󵄩󵄩󵄩V k,l∈ℤν |l|>M,|l|>|k|

󵄩 󵄩 = ∑ 2−|k|−|l| 󵄩󵄩󵄩𝒯 3 {2−N|k|−k⋅s+|k| 2(j+k)⋅s FN,j,k,l f }j∈ℕν 󵄩󵄩󵄩V

k,l∈ℤν |l|>M,|l|>|k|

󵄩 󵄩 ≲ ∑ 2−|k|−|l| 󵄩󵄩󵄩{2−N|k|−k⋅s+|k| 2(j+k)⋅s FN,j,k,l f }j∈ℕν 󵄩󵄩󵄩V

k,l∈ℤν |l|>M,|l|>|k|

(6.12)

󵄩 󵄩 ≤ ∑ 2−|k|−|l| 󵄩󵄩󵄩{2(j+k)⋅s FN,j,k,l f }j∈ℕν 󵄩󵄩󵄩V

k,l∈ℤν |l|>M,|l|>|k|

≤ ∑ 2−|k|−|l| ‖f ‖V ,s,𝒟N ≲ 2−M ‖f ‖V ,s,𝒟N .

k,l∈ℤν |l|>M,|l|>|k|

Plugging (6.10), (6.11), and (6.12) into (6.9) we see that there exists C ≲ 1 with ‖f ‖V ,s,ℰ ′ ≤ C(2M|s| ‖f ‖V ,s,𝒟0 + 2−M ‖f ‖V ,s,𝒟N ). Taking M so large that C2−M ≤ ing (6.8), we see that

1 , 2

using the hypothesis that ‖f ‖V ,s,𝒟N < ∞, and us-

‖f ‖V ,s,ℰ ′ ≤ 2C2M|s| ‖f ‖V ,s,𝒟0 , completing the proof.

6.5 Some basic properties In this section we prove some basic properties of the Besov and Triebel–Lizorkin spaces. Among other things, we prove the results stated in Section 6.3. We begin with Proposition 6.3.7, that the equivalence class of the norm ‖ ⋅ ‖X s (W ,ds)⃗ is well-defined. ̃, Proof of Proposition 6.3.7. The statement of the proposition is symmetric in 𝒟0 and 𝒟 so it suffices to show that

6.5 Some basic properties

‖f ‖V ,s,𝒟 ̃ ≲ ‖f ‖V ,s,𝒟0 ,

⃗ ∀f ∈ X s (𝒦, (W , ds)).

�

409 (6.13)

̃ is a bounded set of generalized (W , ds)⃗ elementary operators, (6.13) follows diSince 𝒟 rectly from Corollary 6.4.4, completing the proof. ̃t (Ω, (W , ds)). ⃗ Next, we turn to Theorem 6.3.10: the boundedness of operators in A ̃t (Ω, (W , ds)) ⃗ and f ∈ Proof of Theorem 6.3.10. By assumption, Ω ⋐ 𝒦 is open. Fix T ∈ A s ⃗ X (𝒦, (W , ds)). ⃗ Clearly, supp(Tf ) ⊆ Ω ⊆ 𝒦. Let ℰ be a We wish to show Tf ∈ X s−t (𝒦, (W , ds)). ⃗ bounded set of generalized (W , ds) elementary operators. Thus, there exists some Ω′ ⋐ M open such that ℰ is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω′ . By replacing Ω′ with Ω′ ∪ Ω, we may assume Ω ⊆ Ω′ . Set ℰ ′ := {(2−j⋅t ET, 2−j ) : (E, 2−j ) ∈ ℰ }. We claim ℰ ′ is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω′ . Indeed, by Proposition 5.5.5 (d), {(E ∗ , 2−j ) : (E, 2−j ) ∈ ℰ } is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω′ . By Proposition 5.8.1 and Proposĩt (Ω, (W , ds)) ̃t (Ω′ , (W , ds)) ̃t (Ω′ , (W , ds)). ⃗ ⊆A ⃗ =A ⃗ Thus, {(2−j⋅t T ∗ E ∗ , 2−j ) : tion 5.8.7, T ∗ ∈ A 1 −j (E, 2 ) ∈ ℰ } is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω′ . Therefore, by Proposition 5.5.5 (d), ℰ ′ is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω′ as desired. Thus, we have ‖Tf ‖V ,s−t,ℰ = ≤

{(Ej

sup

󵄩󵄩 (s−t)⋅j 󵄩 Ej Tf }j∈ℕν 󵄩󵄩󵄩V 󵄩󵄩{2

sup

󵄩󵄩 s⋅j ′ 󵄩 󵄩󵄩{2 Ej f }j∈ℕν 󵄩󵄩󵄩V = ‖f ‖V ,s,ℰ ′ ≲ ‖f ‖X s (W ,ds)⃗ < ∞, ′

,2−j ):j∈ℕν }⊆ℰ

{(Ej′ ,2−j ):j∈ℕν }⊆ℰ

(6.14)

⃗ where the last ≲ uses Corollary 6.4.4 and the fact that f ∈ X s (𝒦, (W , ds)). ⃗ Since ℰ was an arbitrary bounded set of generalized (W , ds) elementary operators, ⃗ Moreover, applying (6.14) in the special case ℰ = 𝒟0 , we conclude Tf ∈ X s−t (𝒦, (W , ds)). we see that ‖Tf ‖X s−t (W ,ds)⃗ = ‖Tf ‖V ,s−t,𝒟0 ≲ ‖f ‖X s (W ,ds)⃗ , ⃗ → X s−t (𝒦, (W , ds)) ⃗ is bounded. proving that T : X s (𝒦, (W , ds)) ⃗ is a Banach space with Next we turn to proving Proportion 6.3.6: that X s (𝒦, (W , ds)) norm ‖ ⋅ ‖V ,s,𝒟0 = ‖ ⋅ ‖X s (W ,ds)⃗ . To do this, we introduce some new notation just for this section and several preliminary lemmas. For μ ∈ {1, . . . , ν}, set ℐμ := {j ∈ ℕν : jμ = |j|∞ }. ∞ ⃗ For g ∈ CW ,0 (M), a bounded set of generalized (W , ds) elementary operators ℰ , N ∈ ∞ ′ ℝ, and μ ∈ {1, . . . , ν}, set, for f ∈ CW ,0 (M) ,

410 � 6 Besov and Triebel–Lizorkin spaces

‖f ‖g,N,ℰ := ‖f ‖g,N,ℰ,μ :=

sup

󵄨󵄨 󵄨󵄨 ∑ 2N|j| 󵄨󵄨󵄨∫ g(x)Ej f (x) d Vol(x)󵄨󵄨󵄨 , 󵄨 󵄨 ν

sup

󵄨󵄨 󵄨󵄨 ∑ 2N|j|∞ 󵄨󵄨󵄨∫ g(x)Ej f (x) d Vol(x)󵄨󵄨󵄨 . 󵄨 󵄨

{(Ej ,2−j ):j∈ℕν }⊆ℰ j∈ℕ

{(Ej ,2−j ):j∈ℐμ }⊆ℰ j∈ℐμ

Lemma 6.5.1. Let ℰ be a bounded set of generalized (W , ds)⃗ elementary operators, g ∈ ∞ s ⃗ CW ,0 (M), and N ∈ ℝ. Then, ‖ ⋅ ‖g,N,ℰ defines a continuous semi-norm on X (𝒦, (W , ds)). In other words, ‖f ‖g,N,ℰ ≤ Cg,N,ℰ ‖f ‖V ,s,𝒟0 = Cg,N,ℰ ‖f ‖X s (W ,ds)⃗ . To prove Lemma 6.5.1 we begin with two easier lemmas. Lemma 6.5.2. Let ℰ be a bounded set of generalized (W , ds)⃗ elementary operators, g ∈ ∞ CW ,0 (M), and N ≤ −|s|√ν − 1. Then for μ ∈ {1, . . . , ν}, ‖ ⋅ ‖g,N,ℰ,μ defines a continuous ⃗ semi-norm on X s (𝒦, (W , ds)). Proof. Under the hypotheses on N we claim (N + 1)|j|∞ ≤ s ⋅ j.

(6.15)

Indeed, if s ⋅ j ≥ 0, (6.15) is clear. If s ⋅ j < 0, then (6.15) is equivalent to |s ⋅ j| ≤ |N + 1||j|∞ = (−N − 1)|j|∞ . By Cauchy–Schwartz we have |s ⋅ j| ≤ |s||j| ≤ √ν|s||j|∞ ≤ (−N − 1)|j|∞ as desired, completing the proof of (6.15). Without loss of generality, we may replace ℰ with ν

ℰ ∪ {(0, 2 ) : j ∈ ℕ }. −j

(6.16)

⃗ = B s (𝒦, (W , ds)) ⃗ for some p, q ∈ We first prove the result when X s (𝒦, (W , ds)) p,q [1, ∞]. We prove the result only in the case q ∈ (1, ∞); the case where q ∈ {1, ∞} is similar and easier, so we leave it to the reader. Let p′ be dual to p ( p1′ + p1 = 1) and let q′ ⃗ using (6.15), be dual to q. We have, for {(Ej , 2−j ) : j ∈ ℐμ } ⊆ ℰ and f ∈ B s (𝒦, (W , ds)), p,q

󵄨󵄨 󵄨󵄨 ∑ 2N|j|∞ 󵄨󵄨󵄨∫ g(x)Ej f (x) d Vol(x)󵄨󵄨󵄨 󵄨 󵄨

j∈ℐμ

1 q′

q

≤ ( ∑ 2−|j|∞ q ) ( ∑ (2(N+1)|j|∞ ‖g‖Lp′ ‖Ej f ‖Lp ) ) ′

j∈ℐμ

1 q

j∈ℐμ

󵄩 󵄩 󵄩 󵄩 ≲ 󵄩󵄩󵄩{2(N+1)|j|∞ Ej f }j∈ℐ 󵄩󵄩󵄩ℓq (ℐ ;Lp (M,Vol)) ≲ 󵄩󵄩󵄩{2s⋅j Ej f }j∈ℕν 󵄩󵄩󵄩ℓq (ℐ ;Lp (M,Vol)) μ μ μ ≤ ‖f ‖V ,s,ℰ ≲ ‖f ‖Bs

⃗

p,q (W ,ds)

,

6.5 Some basic properties

� 411

s ⃗ and Corollary 6.4.4 and the second to where the last estimate uses f ∈ Bp,q (𝒦, (W , ds)) last estimate used the fact that we replaced ℰ with (6.16). Taking the supremum over all {(Ej , 2−j ) : j ∈ ℐμ } ⊆ ℰ , we see that

‖f ‖g,N,ℰ,μ ≲ ‖f ‖Bs

⃗

p,q (W ,ds)

,

s ⃗ ∀f ∈ Bp,q (𝒦, (W , ds)),

⃗ = B s (𝒦, (W , ds)). ⃗ completing the proof in the case X s (𝒦, (W , ds)) p,q s s ⃗ ⃗ for some p ∈ We now turn to the case where X (𝒦, (W , ds)) = Fp,q (𝒦, (W , ds)) (1, ∞), q ∈ (1, ∞]. We prove only the case where q ∈ (1, ∞); the case where q = ∞ follows by an obvious modification, which we leave to the reader. Let p′ be dual to p and s ⃗ using (6.15), let q′ be dual to q. We have, for {(Ej , 2−j ) : j ∈ ℐμ } ⊆ ℰ and f ∈ Fp,q (𝒦, (W , ds)), 󵄨󵄨 󵄨󵄨 ∑ 2N|j|∞ 󵄨󵄨󵄨∫ g(x)Ej f (x) d Vol(x)󵄨󵄨󵄨 󵄨 󵄨

j∈ℐμ

󵄨 󵄨 ≤ ∫ |g(x)| ∑ 󵄨󵄨󵄨Ej f (x)󵄨󵄨󵄨 d Vol(x) j∈ℐμ

−|j|∞ q′

≤ ∫ |g(x)|( ∑ 2 j∈ℐμ

1 q′

) ( ∑ (2 j∈ℐμ

1 q

󵄨q 󵄨󵄨Ej f (x)󵄨󵄨󵄨) ) d Vol(x)

(N+1)|j|∞ 󵄨󵄨

󵄩 󵄩 ≲ ‖g‖Lp′ 󵄩󵄩󵄩{2(N+1)|j|∞ Ej f }j∈ℕν 󵄩󵄩󵄩Lp (M,Vol;ℓq (ℐ )) μ 󵄩 󵄩 ≲ 󵄩󵄩󵄩{2s⋅j Ej f }j∈ℕν 󵄩󵄩󵄩Lp (M,Vol;ℓq (ℐ )) ≤ ‖f ‖V ,s,ℰ ≲ ‖f ‖F s (W ,ds)⃗ , μ p,q s ⃗ and Corollary 6.4.4 and the second to where the last estimate uses f ∈ Fp,q (𝒦, (W , ds)) last estimate used the fact that we replaced ℰ with (6.16). Taking the supremum over all {(Ej , 2−j ) : j ∈ ℐμ } ⊆ ℰ , we see that

‖f ‖g,N,ℰ,μ ≲ ‖f ‖F s

⃗

p,q (W ,ds)

,

s ⃗ ∀f ∈ Fp,q (𝒦, (W , ds)),

completing the proof. Lemma 6.5.3. Let ℰ be a bounded set of generalized (W , ds)⃗ elementary operators, g ∈ ∞ CW ,0 (M), and N ∈ ℝ. Then for μ ∈ {1, . . . , ν}, ‖ ⋅ ‖g,N,ℰ,μ defines a continuous semi-norm on ⃗ X s (𝒦, (W , ds)). Proof. Take M ∈ ℕ with M ≥ |s|√ν + 1 + N. Applying Proposition 5.5.5 (h), we have, for (Ej , 2−j ) ∈ ℰ , μ

α

Ej = ∑ 2−(M−|αμ |)jμ (2−jμ ds W μ ) μ Ej,μ,αμ , |αμ |≤M

where ℰ̃ := {(Ej,μ,αμ , 2−j ) : (E, 2−j ) ∈ ℰ , |αμ | ≤ M} is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω.

412 � 6 Besov and Triebel–Lizorkin spaces ⃗ using the fact that jμ = |j|∞ for j ∈ ℐμ , We have, for f ∈ X s (𝒦, (W , ds)), ‖f ‖g,N,ℰ,μ =

sup {(Ej ,2−j ):j∈ℐμ }⊆ℰ

≤ ∑

|αμ |≤M {(Ej

󵄨󵄨 󵄨󵄨 ∑ 2N|j|∞ 󵄨󵄨󵄨∫ g(x)Ej f (x) d Vol(x)󵄨󵄨󵄨 󵄨 󵄨

j∈ℐμ

sup ,2−j ):j∈ℐ

μ }⊆ℰ

∑ 2(N−M)|j|∞ +(|αμ |−degdsμ (αμ ))|j|∞

j∈ℐμ

󵄨󵄨 󵄨󵄨 α × 󵄨󵄨󵄨∫ g(x)(W μ ) μ Ej,μ,αμ f (x) d Vol(X)󵄨󵄨󵄨 󵄨 󵄨 ≤ ∑

|αμ |≤M {(Ej

sup

,2−j ):j∈ℐ

μ }⊆ℰ

∑ 2−(|s|√ν+1)|j|∞

j∈ℐμ

󵄨󵄨 󵄨󵄨 α ∗ × 󵄨󵄨󵄨∫(((W μ ) μ ) g(x))Ej,μ,αμ f (x) d Vol(X)󵄨󵄨󵄨 󵄨 󵄨 ≤ ∑ ‖f ‖((W μ )αμ )∗ g,−(|s|√ν+1),ℰ,μ ̃ ≲ ‖f ‖X s (W ,ds)⃗ , |αμ |≤M

where the last estimate used Lemma 6.5.2. This completes the proof. Proof of Lemma 6.5.1. It follows directly from the definitions that ‖f ‖g,N,ℰ ≤

∑

μ∈{1,...,ν}

‖f ‖g,(√νN)∨0,ℰ,μ .

Thus, the result follows immediately from Lemma 6.5.3. ∞ ∞ ′ Lemma 6.5.4. For g ∈ CW ,0 (M), the map λg : CW ,0 (M) → ℂ given by λg (f ) = ∫ fg s ⃗ restricts to a bounded linear functional on X (𝒦, (W , ds)).

⃗ Since f = Mult[ψ]f = ∑j∈ℕν Dj f (with convergence in Proof. Let f ∈ X s (𝒦, (W , ds)). ∞ ′ CW ,loc (M) by Proposition 5.5.10), we see that 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨λg (f )󵄨󵄨 = 󵄨󵄨 ∑ ∫ g(x)Dj f (x) d Vol(x)󵄨󵄨󵄨󵄨 󵄨󵄨j∈ℕν 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ ∑ 󵄨󵄨∫ g(x)Dj f (x)󵄨󵄨 = ‖f ‖g,0,𝒟0 ≲ ‖f ‖X s (W ,ds)⃗ , 󵄨 󵄨󵄨 ν󵄨 j∈ℕ

where the last estimate follows from Lemma 6.5.1, completing the proof. ⃗ (as Proof of Proportion 6.3.6. First we show that ‖ ⋅ ‖V ,s,𝒟0 is a norm on X s (𝒦, (W , ds)) s ⃗ opposed to merely a semi-norm). Indeed, for f ∈ X (𝒦, (W , ds)), we have f = Mult[ψ]f = ∞ ′ ∑j∈ℕν Dj f , with convergence in CW ,loc (M) by Proposition 5.5.10. If ‖f ‖V ,s,𝒟0 = 0, then Dj f = 0 for every j ∈ ℕν , and therefore f = 0. We conclude that ‖ ⋅ ‖V ,s,𝒟0 is a norm on ⃗ X s (𝒦, (W , ds)).

6.5 Some basic properties

�

413

⃗ is complete with respect to this norm. Let It remains to show that X s (𝒦, (W , ds)) s ⃗ be a Cauchy sequence. Our goal is to show that fl converges in {fl }l∈ℕ ⊂ X (𝒦, (W , ds)) ⃗ X s (𝒦, (W , ds)).

Let Ω ⋐ M be open with 𝒦 ⋐ Ω. Set 𝒦1 := Ω, so that 𝒦1 is compact. This ∞ ∞ makes CW ,0 (𝒦1 ) (the space of CW ,0 (M) functions supported in 𝒦1 ) a Fréchet space. ∞ ′ ∞ For f ∈ CW ,0 (M) and g ∈ CW ,0 (𝒦1 ), set μf (g) = ∫ gf . It follows from Lemma 6.5.4 ∞ that {μfl (g)}l∈ℕ ⊂ ℂ, for every g ∈ CW ,0 (𝒦1 ). Set μ(g) := liml→∞ μfl (g). By the Banach– ∞ Steinhaus theorem (see [206, Theorem 2.8]), μ : CW ,0 (𝒦1 ) → ℂ is continuous. In partic∞ ∞ ′ ular, μ : CW ,0 (Ω) → ℂ is continuous, and therefore μ(g) = ∫ fg for some f ∈ CW ,0 (Ω) . ∞ ∞ ′ Since μfl (g) → μ(g), ∀g ∈ CW ,0 (Ω), we have fl → f in CW ,0 (Ω) . Since supp fl ⊆ 𝒦 for every l, we have supp f ⊆ 𝒦 ⋐ Ω. Therefore, f extends to an ∞ ′ ∞ ′ ∞ ′ element of CW ,loc (M) ⊆ CW ,0 (M) and fl → f in CW ,0 (M) . Take N := |s| + 1 and define 𝒟N as in (6.6). Let ℰ := 𝒟N ∪ 𝒟0 . For each (E, 2−j ) ∈ ℰ , by ∞ ′ ∞ ′ ∞ ′ Remark 6.3.4, E ∈ Hom(CW ,0 (M) , CW ,0 (M) ), and therefore Efl → f in CW ,0 (M) . ⃗ and fl 󳨀l→∞ The proof will be complete once we show that f ∈ X s (𝒦, (W , ds)) 󳨀󳨀󳨀→ f in ⃗ X s (𝒦, (W , ds)). ⃗ equals B s (𝒦, (W , ds)) ⃗ or F s (𝒦, (W , ds)), ⃗ V Depending on whether X s (𝒦, (W , ds)) p,q p,q q ν p p q ν is equal to either ℓ (ℕ ; L (M, Vol)) or L (M, Vol; ℓ (ℕ )) (see Notation 6.3.2). In either case, let p ∈ [1, ∞] be the corresponding p. We claim, for every (E, 2−j ) ∈ ℰ , that Efl is Cauchy in Lp (M, Vol). Indeed, using Corollary 6.4.4, we have 2−j⋅s ‖E(fa − fb )‖Lp ≤ ‖fa − fb ‖V ,s,𝒟0 + ‖fa − fb ‖V ,s,𝒟N ≲ ‖fa − fb ‖V ,a,𝒟0 . Since {fl }l∈ℕ is Cauchy in ‖ ⋅ ‖V ,s,𝒟0 , it follows that {Efl }l∈ℕ is Cauchy in Lp (M, Vol). Since ∞ ′ p we already have Efl → Ef in CW ,0 (M) , it follows that Ef ∈ L (M, Vol) and Efl → Ef in p L (M, Vol). Since Efl → Ef in Lp for every (E, 2−j ) ∈ ℰ and ℰ is countable, there exists a subsequence flk such that Eflk → Ef pointwise almost everywhere for every (E, 2−j ) ∈ ℰ . We have, for any ℰ ′ := {(Ej , 2−j ) : j ∈ ℕν } ⊆ ℰ (with constants independent of the choice of ℰ ′ ), using Fatou’s lemma, 󵄩󵄩 j⋅s 󵄩 󵄩 j⋅s 󵄩 󵄩󵄩{2 Ej f }j∈ℕν 󵄩󵄩󵄩V ≤ lim inf󵄩󵄩󵄩{2 Ej flk }j∈ℕν 󵄩󵄩󵄩V k→∞ ≤ lim inf ‖flk ‖V ,s,ℰ ≲ lim inf ‖flk ‖V ,s,𝒟0 ≲ 1, k→∞

k→∞

where the second to last estimate follows from Corollary 6.4.4 and the final estimate follows from the fact that flk is Cauchy in ‖ ⋅ ‖V ,s,𝒟0 . Taking the supremum over all such ℰ ′ ⊆ ℰ , we see that ‖f ‖V ,s,𝒟0 + ‖f ‖V ,s,𝒟N ≤ 2‖f ‖V ,s,ℰ < ∞. ⃗ By Corollary 6.4.5, we have f ∈ X s (𝒦, (W , ds)).

414 � 6 Besov and Triebel–Lizorkin spaces Similarly, using the fact that 𝒟0 ⊆ ℰ and Fatou’s lemma, we have 󵄩 󵄩 lim sup ‖f − fm ‖V ,s,𝒟0 = lim sup󵄩󵄩󵄩{2j⋅s Dj (f − fm )}j∈ℕν 󵄩󵄩󵄩V m→∞

m→∞

󵄩 󵄩 ≤ lim sup lim inf󵄩󵄩󵄩{2j⋅s Dj (flk − fm )}j∈ℕν 󵄩󵄩󵄩V m→∞

k→∞

= lim sup lim inf ‖flk − fm ‖V ,s,𝒟0 = 0. m→∞

k→∞

⃗ completing the proof. Thus, fl → f in X s (𝒦, (W , ds)), ∞ ∞ Proposition 6.5.5. Let CW ,0 (𝒦) be the space of those CW ,0 (M) functions supported in 𝒦. ∞ s ν ⃗ Then CW ,0 (𝒦) ⊆ X (𝒦, (W , ds)) (for every s ∈ ℝ ). ∞ ⃗ Proof. Let f ∈ CW ,0 (𝒦) and let ℰ be a bounded set of generalized (W , ds) elementary operators supported in Ω for some Ω ⋐ M. We will show that

(6.17)

‖f ‖V ,s,ℰ < ∞, which will complete the proof. Set ℐμ := {j ∈ ℕν : jμ = |j|∞ } and set Vμ := {

ℓq (ℐμ ; Lp (M, Vol))

if V = ℓq (ℕν ; Lp (M, Vol)),

Lp (M, Vol; ℓq (ℐμ ))

if V = Lp (M, Vol; ℓq (ℕν )).

(6.18)

We have ‖f ‖V ,s,ℰ = ≤

sup

{(Ej ,2−j ):j∈ℕν }⊆ℰ

∑

󵄩󵄩 j⋅s 󵄩 󵄩󵄩{2 Ej f }j∈ℕν 󵄩󵄩󵄩V

sup

μ∈{1,...,ν} {(Ej ,2−j ):j∈ℐμ }⊆ℰ

󵄩󵄩 j⋅s 󵄩 󵄩󵄩{2 Ej f }j∈ℐμ 󵄩󵄩󵄩Vμ .

(6.19)

Let N := ⌈|s|√ν⌉+1. In light of (6.19), to show (6.17) it suffices to show that for μ ∈ {1, . . . , ν}, sup {(Ej ,2−j ):j∈ℐμ }⊆ℰ

󵄩󵄩 (N−1)jμ 󵄩 Ej f }j∈ℐμ 󵄩󵄩󵄩V < ∞, 󵄩󵄩{2 μ

(6.20)

where we have used the fact that jμ = |j|∞ for j ∈ ℐμ . Fix μ ∈ {1, . . . , ν}. For (Ej , 2−j ) ∈ ℰ , we apply Proposition 5.5.5 (h) to see that μ

αμ

Ej = ∑ 2−(N−|αμ |)jμ Ej,μ,αμ (2−jμ ds W μ ) |αμ |≤N

= ∑ 2−Njμ 2−(degdsμ (αμ )−|αμ |)jμ Ej,μ,αμ (W μ )αμ ,

(6.21)

|αμ |≤N

where ℰ ′ := {(Ej,μ,αμ , 2−j ) : (Ej , 2−j ) ∈ ℰ , |αμ | ≤ N} is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω.

6.5 Some basic properties

�

415

Since degdsμ (αμ ) − |αμ | ≥ 0 for all αμ , using (6.21), we see that sup {(Ej ,2−j ):j∈ℐμ }⊆ℰ

󵄩󵄩 (N−1)jμ 󵄩 Ej f }j∈ℐμ 󵄩󵄩󵄩V 󵄩󵄩{2 μ sup

≤ ∑

|αμ |≤N {(Ej

≲

,2−j ):j∈ℐ

sup {(Ej′ ,2−j ):j∈ℐμ }⊆ℰ ′

μ }⊆ℰ

󵄩󵄩 −jμ 󵄩 μ αμ 󵄩󵄩{2 Ej,μ,αμ (W ) f }j∈ℐμ 󵄩󵄩󵄩Vμ

(6.22)

󵄩󵄩 −jμ ′ μ αμ 󵄩 󵄩󵄩{2 Ej (W ) f }j∈ℐμ 󵄩󵄩󵄩Vμ .

Let p ∈ [1, ∞] be as in (6.18). Using the definition of bounded sets of generalized (W , ds)⃗ pre-elementary operators (Definition 5.2.26), we have sup {(Ej′ ,2−j ):j∈ℐμ }⊆ℰ ′

󵄩󵄩 −jμ ′ μ αμ 󵄩 󵄩󵄩{2 Ej (W ) f }j∈ℐμ 󵄩󵄩󵄩Vμ (6.23)

α 󵄩 󵄩 ≲ 󵄩󵄩󵄩{2−jμ (W μ ) μ f }j∈ℐμ 󵄩󵄩󵄩V μ

α α 󵄩 󵄩 󵄩 󵄩 = 󵄩󵄩󵄩{2−|j|∞ (W μ ) μ f }j∈ℐμ 󵄩󵄩󵄩V ≲ 󵄩󵄩󵄩(W μ ) μ f 󵄩󵄩󵄩Lp (M,Vol) < ∞. μ

Combining (6.22) and (6.23) gives (6.20) and completes the proof. Proposition 6.5.6. Suppose 𝒦1 , 𝒦2 ⋐ M are two compact sets with 𝒦1 ⊆ 𝒦2 . Then ⃗ ⊆ X s (𝒦2 , (W , ds)) ⃗ is a closed subspace (and the subspace topology on X s (𝒦1 , (W , ds)) s ⃗ ⃗ consists X (𝒦1 , (W , ds)) agrees with its original topology). Furthermore, X s (𝒦1 , (W , ds)) s ⃗ such that supp(f ) ⊆ 𝒦1 . of precisely those f ∈ X (𝒦2 , (W , ds)) ∞ ′ Proof. It follows directly from Definition 6.3.5 that for f ∈ CW ,0 (M) , the following are equivalent: ⃗ and supp(f ) ⊆ 𝒦1 . – f ∈ X s (𝒦2 , (W , ds)) s ⃗ – f ∈ X (𝒦1 , (W , ds)).

⃗ ⊆ X s (𝒦2 , (W , ds)) ⃗ and X s (𝒦1 , (W , ds)) ⃗ consists of preThis proves that X s (𝒦1 , (W , ds)) s ⃗ cisely those f ∈ X (𝒦2 , (W , ds)) such that supp(f ) ⊆ 𝒦1 . ⃗ and By Remark 6.3.9 we may choose the same norm on X s (𝒦2 , (W , ds)) s s ⃗ ⃗ X (𝒦1 , (W , ds)) so that the subspace topology on X (𝒦1 , (W , ds)) agrees with its original ⃗ is complete (Proposition 6.3.6), we see that topology. In particular, since X s (𝒦1 , (W , ds)) s s ⃗ is a closed subspace of X (𝒦2 , (W , ds)). ⃗ X (𝒦1 , (W , ds)) In light of Proposition 6.5.6, the next definition makes sense. Definition 6.5.7. For s ∈ ℝν , we set s

Xcpt (W , ds)⃗ :=

⋃

𝒦⋐M 𝒦 compact

s

⃗ X (𝒦, (W , ds)).

416 � 6 Besov and Triebel–Lizorkin spaces Remark 6.5.8. Proposition 6.5.6 can be restated as saying that the following are equivalent: ⃗ – u ∈ X s (𝒦, (W , ds)). s ⃗ – u ∈ Xcpt (W , ds) and supp(u) ⊆ 𝒦. The next results show that each vector field Wj acts as a differential operator of degree dsj⃗ ∈ ℕν . Proposition 6.5.9. Let Ω ⋐ M be an open set with 𝒦 ⋐ Ω. Let κ ∈ ℤν and let P be a (W , ds)⃗ partial differential operator on Ω of degree ≤ κ. Then, for every s ∈ ℝν , P : ⃗ → X s−κ (𝒦, (W , ds)) ⃗ and is a bounded operator. X s (𝒦, (W , ds)) Proof. If κ ∈ ℤν \ℕν , then P = 0 and the result is trivial. We henceforth assume κ ∈ ℕν . ⃗ we have Take ϕ ∈ C0∞ (Ω) with ϕ ≡ 1 on a neighborhood of 𝒦. For f ∈ X s (𝒦, (W , ds)) supp(f ) ⊆ 𝒦 and therefore P f = Mult[ϕ]P f . Thus, it suffices to prove the result with P replaced by Mult[ϕ]P . By definition (see Definition 3.12.1), Mult[ϕ]P can be written as a finite sum of op∞ erators of the form Mult[ϕgα ]W α , where gα ∈ Cloc (Ω) and degds⃗ (α) ≤ κ. By Proposiα deg (α) ⃗ ̃ ds (Ω, (W , ds)) ̃κ (Ω, (W , ds)). ⃗ ⊆A ⃗ We conclude tions 5.8.3 and 5.8.2, Mult[ϕgα ]W ∈ A κ ̃ (Ω, (W , ds)). ⃗ that Mult[ϕ]P ∈ A ̃κ (Ω, (W , ds)), ⃗ by Theorem 6.3.10, Mult[ϕ]P : Set 𝒦1 := Ω ⋐ M. Since Mult[ϕ]P ∈ A s s−κ ⃗ ⃗ is a bounded operator. By Proposition 6.5.6, X (𝒦1 , (W , ds)) → X (𝒦1 , (W , ds)) s ⃗ ⃗ and therefore Mult[ϕ]P : X (𝒦, (W , ds)) is a closed subspace of X s (𝒦1 , (W , ds)), s s−κ ⃗ ⃗ X (𝒦, (W , ds)) → X (𝒦1 , (W , ds)) is a bounded operator. ⃗ agrees with its topology as By Proposition 6.5.6, the topology on X s−κ (𝒦, (W , ds)) s−κ ⃗ a subspace of X (𝒦1 , (W , ds)). Thus, to complete the proof, it suffices to show that ⃗ → X s−κ (𝒦, (W , ds)). ⃗ Take f ∈ X s (𝒦, (W , ds)). ⃗ We have alMult[ϕ]P : X s (𝒦, (W , ds)) s−κ ⃗ Since supp(f ) ⊆ 𝒦 and P is a parready shown that Mult[ϕ]P f ∈ X (𝒦1 , (W , ds)). tial differential operator, we have supp(Mult[ϕ]P f ) ⊆ 𝒦. Proposition 6.5.6 implies that ⃗ completing the proof. Mult[ϕ]P f ∈ X s−κ (𝒦, (W , ds)),

∞ ⃗ → Corollary 6.5.10. Let ϕ ∈ Cloc (M). Then Mult[ϕ] is a bounded operator X s (𝒦, (W , ds)) s ⃗ X (𝒦, (W , ds)).

Proof. For any Ω ⋐ M, Mult[ϕ] is a (W , ds)⃗ partial differential operator on Ω of degree ≤ 0. Thus, the result follows immediately from Proposition 6.5.9. Corollary 6.5.11. For any ordered multi-index α, ⃗ → X s−degds⃗ (α) (𝒦, (W , ds)) ⃗ W α : X s (𝒦, (W , ds)) ⃗ → X s−ds⃗ j (𝒦, (W , ds)) ⃗ is a is a bounded operator. In particular, Wj : X s (𝒦, (W , ds)) bounded operator.

6.5 Some basic properties

� 417

Proof. Let Ω ⋐ M be open with 𝒦 ⋐ Ω. Then W α is a (W , ds)⃗ partial differential operator on Ω of degree ≤ degds⃗ (α). Thus, the claimed boundedness of W α follows directly from Proposition 6.5.9. The claim about Wj is the special case |α| = 1 of the claim about W α . μ

Proposition 6.5.12. Fix μ ∈ {1, . . . , ν} and κμ ∈ ℕ+ such that dsj divides κμ for 1 ≤ j ≤ rμ . μ

μ

Set nj := κμ /dsj ∈ ℕ+ and let

rμ

μ n

μ

∗

μ n

μ

Lμ := ∑((Wj ) j ) (Wj ) j . j=1

(6.24)

∞ ′ For s ∈ ℝν and f ∈ CW ,0 (M) , the following are equivalent: ⃗ (i) f ∈ X s (𝒦, (W , ds)). (ii) For all ordered multi-indices αμ with degdsμ (αμ ) ≤ κμ , we have α

⃗ (W μ ) μ f ∈ X s−κμ eμ (𝒦, (W , ds)). μ nμ

⃗ and (W ) j f ∈ X s−κμ eμ (𝒦, (W , ds)) ⃗ for 1 ≤ j ≤ rμ . (iii) f ∈ X s−κμ eμ (𝒦, (W , ds)) j s−2κμ eμ ⃗ (iv) f , Lμ f ∈ X (𝒦, (W , ds)). ⃗ Furthermore, we have, for all f ∈ X s (𝒦, (W , ds)), ‖f ‖X s (W ,ds)⃗ ≈

∑

degdsμ (αμ )≤κμ

󵄩󵄩 μ αμ 󵄩󵄩 󵄩󵄩(W ) f 󵄩󵄩X s−κμ (W ,ds)⃗ rμ

μ

󵄩 μ n 󵄩 ≈ ‖f ‖X s−κμ eμ (W ,ds)⃗ + ∑󵄩󵄩󵄩(Wj ) j f 󵄩󵄩󵄩X s−κμ eμ (W ,ds)⃗

(6.25)

j=1

≈ ‖f ‖X s−2κμ eμ (W ,ds)⃗ + ‖Lμ f ‖X s−2κμ eμ (W ,ds)⃗ , where the implicit constants do not depend on f (but may depend on any of the other ingredients). ⃗ and degdsμ (αμ ) ≤ κμ . Since (W μ )αμ is a (W , ds)⃗ Proof. (i) ⇒ (ii): Suppose f ∈ X s (𝒦, (W , ds)) partial differential operator of degree ≤ degdsμ (αμ )eμ ≤ κμ eμ on Ω (for any Ω open with ⃗ and 𝒦 ⋐ Ω), Proposition 6.5.9 shows that (W μ )αμ f ∈ X s−κμ eμ (𝒦, (W , ds)) ∑

degdsμ (αμ )≤κμ

󵄩󵄩 μ αμ 󵄩󵄩 󵄩󵄩(W ) f 󵄩󵄩X s−κμ (W ,ds)⃗ ≲ ‖f ‖X s (W ,ds)⃗ . μ njμ

(ii) ⇒ (iii): Suppose (ii) holds. Since (Wj ) holds, and moreover,

(6.26)

= (W μ )αμ , for some degdsμ (αμ ) = κμ , (iii)

418 � 6 Besov and Triebel–Lizorkin spaces rμ

μ

󵄩 μ n 󵄩 ‖f ‖X s−κμ eμ (W ,ds)⃗ + ∑󵄩󵄩󵄩(Wj ) j f 󵄩󵄩󵄩X s−κμ eμ (W ,ds)⃗ j=1

≲

∑

degdsμ (αμ )≤κμ

(6.27)

󵄩󵄩 μ αμ 󵄩󵄩 󵄩󵄩(W ) f 󵄩󵄩X s−κμ (W ,ds)⃗ . μ nμ

(iii) ⇒ (iv): Suppose (iii) holds. Fix Ω ⋐ M open with 𝒦 ⋐ Ω. (Wj ) j is a (W , ds)⃗ partial μ nμ

differential operator on Ω of degree ≤ κμ eμ , and therefore by Lemma 3.12.5, ((Wj ) j )∗ is a (W , ds)⃗ partial differential operator on Ω of degree ≤ κμ eμ . Thus, since (by assumpμ nμ

μ nμ

μ nμ

⃗ Proposition 6.5.9 shows that ((W ) j )∗ (W ) j f ∈ tion) (Wj ) j f ∈ X s−κμ eμ (𝒦, (W , ds)), j j ⃗ and X s−2κμ eμ (𝒦, (W , ds)) μ

μ

μ

μ n ∗ μ n 󵄩 󵄩󵄩 󵄩 μ n 󵄩 󵄩󵄩((Wj ) j ) (Wj ) j f 󵄩󵄩󵄩X s−2κμ eμ (W ,ds)⃗ ≲ 󵄩󵄩󵄩(Wj ) j f 󵄩󵄩󵄩X s−κμ eμ (W ,ds)⃗ .

Using the formula for Lμ , we conclude that Lμ f ∈ X s−2κμ eμ and μ

󵄩󵄩 󵄩 󵄩 μ n 󵄩 󵄩󵄩Lμ f 󵄩󵄩󵄩X s−2κμ eμ (W ,ds)⃗ ≲ 󵄩󵄩󵄩(Wj ) j f 󵄩󵄩󵄩X s−κμ eμ (W ,ds)⃗ .

(6.28)

⃗ ⊆ X s−2κμ eμ (𝒦, (W , ds)), ⃗ which comBy assumption, we have f ∈ X s−κμ eμ (𝒦, (W , ds)) pletes the proof of (iv). Moreover, ‖f ‖X s−2κμ eμ (W ,ds)⃗ ≤ ‖f ‖X s−κμ eμ (W ,ds)⃗ .

(6.29)

By combining (6.28) and (6.29), we obtain ‖f ‖X s−2κμ eμ (W ,ds)⃗ + ‖Lμ f ‖X s−2κμ eμ (W ,ds)⃗ rμ

μ

󵄩 μ n 󵄩 ≲ ‖f ‖X s−κμ eμ (W ,ds)⃗ + ∑󵄩󵄩󵄩(Wj ) j f 󵄩󵄩󵄩X s−κμ eμ (W ,ds)⃗ .

(6.30)

j=1

(iv) ⇒ (i): We claim the following. Claim: Under the assumptions of (iv), for L ∈ {0, 1, . . . , 2κμ }, we have ⃗ f ∈ X s−2κμ eμ +Leμ (𝒦, (W , ds)) and ‖f ‖X s−2κμ eμ +Leμ (W ,ds)⃗ ≲ ‖f ‖X s−2κμ eμ (W ,ds)⃗ + ‖Lμ f ‖X s−2κμ eμ (W ,ds)⃗ .

(6.31)

We prove the claim by induction on L. The base case, L = 0, follows immediately from the assumption (iv). Thus, we assume the claim for L ∈ {0, 1, . . . , 2κμ − 1} and prove it for L + 1.

6.5 Some basic properties

�

419

Fix Ω ⋐ M open with 𝒦 ⋐ Ω. Let 𝒦1 := Ω, so that 𝒦1 is compact with 𝒦 ⋐ 𝒦1 . Fix ⃗ with ϕ ∈ C0∞ (Ω) with ϕ ≡ 1 on 𝒦. By Corollary 5.8.15 there exists T ∈ A −2κμ eμ (Ω, (W , ds)) T Lμ = Mult[ϕ] + R, ̃−eμ (Ω, (W , ds)). ⃗ where R ∈ A By Proposition 6.5.6 and assumption (iv), Lμ f ∈ X

s−2κμ eμ

⃗ ⊆ X s−2κμ eμ (𝒦1 , (W , ds)). ⃗ (𝒦, (W , ds))

⃗ and supp(f ) ⊆ 𝒦, we Thus, by Theorem 6.3.10, using the fact that T ∈ A −2κμ eμ (Ω, (W , ds)) have ⃗ f + Rf = Mult[ϕ]f + Rf = T Lμ f ∈ X s (𝒦1 , (W , ds)), with ‖f + Rf ‖X s (W ,ds)⃗ ≲ ‖Lμ f ‖X s−2κμ eμ (W ,ds)⃗ .

(6.32)

Using the inductive hypothesis and Proposition 6.5.6, we have ⃗ ⊆ X s−2κμ eμ +Leμ (𝒦1 , (W , ds)). ⃗ f ∈ X s−2κμ eμ +Leμ (𝒦, (W , ds)) ̃−eμ (Ω, (W , ds)), ⃗ applying Theorem 6.3.10, we have Thus, since R ∈ A ⃗ Rf ∈ X s−2κμ eμ +(L+1)eμ (𝒦1 , (W , ds)) with ‖Rf ‖X s−2κμ eμ +(L+1)eμ (W ,ds)⃗ ≲ ‖f ‖X s−2κμ eμ +Leμ (W ,ds)⃗

≲ ‖f ‖X s−2κμ eμ (W ,ds)⃗ + ‖Lμ f ‖X s−2κμ eμ (W ,ds)⃗ ,

(6.33)

where the last estimate uses the inductive hypothesis. ⃗ Combining the above, we see that f = (f + Rf ) − Rf ∈ X s−2κμ eμ +(L+1)eμ (𝒦1 , (W , ds)). ⃗ Finally, Since supp(f ) ⊆ 𝒦, Proposition 6.5.6 implies that f ∈ X s−2κμ eμ +(L+1)eμ (𝒦, (W , ds)). combining (6.32) and (6.33), we obtain (6.31) with L replaced by L + 1. This completes the proof of the claim. Applying the claim with L = 2κμ shows that (i) holds, and moreover, ‖f ‖X s (W ,ds)⃗ ≲ ‖f ‖X s−2κμ eμ (W ,ds)⃗ + ‖Lμ f ‖X s−2κμ eμ (W ,ds)⃗ .

(6.34)

Thus, we have completed the proof of the equivalence of the conditions. Moreover, (6.25) follows by combining (6.26), (6.27), (6.30), and (6.34).

420 � 6 Besov and Triebel–Lizorkin spaces μ

Corollary 6.5.13. For each μ ∈ {1, . . . , ν}, let κμ ∈ ℕ be such that dsj divides κμ for 1 ≤ j ≤ rμ (we allow the possibility κμ = 0). Set κ := (κ1 , . . . , κν ) ∈ ℕν . Then, for s ∈ ℝν and ∞ ′ f ∈ CW ,0 (M) , the following are equivalent: s ⃗ (i) f ∈ X (𝒦, (W , ds)). ⃗ (ii) For every ordered multi-index α with degds⃗ (α) ≤ κ, we have W α f ∈ X s−κ (𝒦, (W , ds)). ⃗ we have Furthermore, for f ∈ X s (𝒦, (W , ds)), ‖f ‖X s (W ,ds)⃗ ≈

∑

degds⃗ (α)≤κ

‖W α f ‖X s−κ (W ,ds)⃗ ,

(6.35)

where the implicit constants do not depend on f (but may depend on any of the other ingredients). Proof. (i) ⇒ (ii): By Corollary 6.5.11, for degds⃗ (α) ≤ κ, ⃗ → X s−degds⃗ (α) (𝒦, (W , ds)) ⃗ ⊆ X s−κ (𝒦, (W , ds)) ⃗ W α : X s (𝒦, (W , ds)) is bounded, and therefore (i) ⇒ (ii) and moreover ∑

degds⃗ (α)≤κ

‖W α f ‖X s−κ (W ,ds)⃗ ≲ ‖f ‖X s (W ,ds)⃗ .

(6.36)

(ii) ⇒ (i): Suppose (ii) holds. For each μ ∈ {1, . . . , ν+1} set κμ = (0, 0, . . . , 0, κμ , κμ+1 , . . . , κν ) ∈ ℕν , with κν+1 = 0 ∈ ℕν . Claim: For μ ∈ {1, . . . , ν + 1} and for every ordered multi-index α with degds⃗ (α) ≤ κμ , we μ ⃗ and have W α f ∈ X s−κ (𝒦, (W , ds)) ∑

degds⃗

(α)≤κμ

‖W α f ‖X s−κμ (W ,ds)⃗ ≲

∑

degds⃗ (α)≤κ

‖W α f ‖X s−κ (W ,ds)⃗ .

(6.37)

We prove the claim by induction on μ. The base case, μ = 1, follows immediately from hypothesis (ii) ((6.37) being trivial in the case μ = 1). We assume the result for some μ ∈ {1, . . . , ν} and prove it for μ + 1. If κμ = 0, then κμ = κμ+1 and there is nothing to prove. Thus, we assume κμ ≠ 0. Fix β with degds⃗ (β) ≤ κμ+1 . For any αμ with degdsμ (αμ ) ≤ κμ , (W μ )αμ W β is of the form W γ for some γ with degds⃗ (γ) ≤ κμ . Thus, by the inductive hypothesis, we have (W μ )αμ W β f ∈ μ ⃗ and X s−κ (𝒦, (W , ds)) ∑

degdsμ (αμ )≤κμ

󵄩󵄩 μ αμ β 󵄩󵄩 󵄩󵄩(W ) W f 󵄩󵄩X s−κμ (W ,ds)⃗ ≲

∑

degds⃗ (α)≤κ μ+1

‖W α f ‖X s−κ (W ,ds)⃗ .

⃗ Moreover, (6.25) imProposition 6.5.12 (ii) ⇒ (i) implies that W β f ∈ X s−κ (𝒦, (W , ds)). plies that

6.5 Some basic properties

‖W β f ‖X s−κμ+1 (W ,ds)⃗ ≲

∑

degdsμ (αμ )≤κμ

≲

∑

degds⃗ (α)≤κ

� 421

󵄩󵄩 μ αμ β 󵄩󵄩 󵄩󵄩(W ) W f 󵄩󵄩X s−κμ (W ,ds)⃗

‖W α f ‖X s−κ (W ,ds)⃗ .

(6.38)

Since β was arbitrary with degds⃗ (β) ≤ κμ+1 , summing (6.38) over such β completes the proof of the claim. Applying the claim with μ = ν + 1 shows that (i) holds, and moreover, ‖f ‖X s (W ,ds)⃗ ≲

∑

degds⃗ (α)≤κ

‖W α f ‖X s−κ (W ,ds)⃗ .

(6.39)

Finally, (6.35) follows by combining (6.36) and (6.39). Proposition 6.5.14. For each μ ∈ {1, . . . , ν}, let μ

μ

∞ (Z μ , drμ ) = {(Z1 , dr1 ), . . . , (Zsμμ , drμsμ )} ⊂ Cloc (M; TM) × ℕ+

be such that (Z μ , drμ ) and (W μ , dsμ ) are locally weakly equivalent on M. Define (Z, dr)⃗ = (Z 1 , dr1 ) ⊠ (Z 2 , dr2 ) ⊠ ⋅ ⋅ ⋅ (Z ν , drν ), i. e., (Z, dr)⃗ is to (Z μ , drμ ), 1 ≤ μ ≤ ν, as (W , ds)⃗ is to (W μ , dsμ ), ⃗ with equivalent norms. ⃗ = X s (𝒦, (Z, dr)), 1 ≤ μ ≤ ν. Then X s (𝒦, (W , ds)) Proof. By Proposition 5.5.14, ℰ is a bounded set of generalized (W , ds)⃗ elementary operators if and only if ℰ is a bounded set of generalized (Z, dr)⃗ elementary operators. Using this, the result follows directly from the definitions. ⃗ are invariant under diffeomorphisms. Proposition 6.5.15. The spaces X s (𝒦, (W , ds)) ∼ ∞ ∞ More precisely, suppose N is another C manifold and Ψ : M 󳨀 → N is a Cloc diffeomorphism. Set ∞ (Ψ∗ W , ds)⃗ := {(Ψ∗ W1 , ds1⃗ ), . . . , (Ψ∗ Wr , dsr⃗ )} ⊂ Cloc (N; TN) × (ℕν \ {0}).

Then s

s

⃗ = X (Ψ(𝒦), (Ψ∗ W , ds)) ⃗ X (𝒦, (W , ds)) and ‖f ‖X s (W ,ds)⃗ ≈ ‖Ψ∗ f ‖

X s (Ψ∗ W ,ds⃗ )

,

⃗ ∀f ∈ X s (𝒦, (W , ds)).

(6.40)

Proof. It is easy to see that all of our assumptions and definitions are invariant under diffeomorphisms, so this result follows just by tracing through the definitions. s s Proposition 6.5.16. Fix p ∈ [1, ∞] such that X s = Bp,q or X s = Fp,q (see Notation 6.3.2). ν Then, ∀s ∈ (0, ∞) , we have

422 � 6 Besov and Triebel–Lizorkin spaces s

p

⃗ ⊆ L (𝒦, Vol) ⊆ X X (𝒦, (W , ds))

−s

(6.41)

⃗ (𝒦, (W , ds)),

and the inclusions are continuous. Proof. We begin with the second inclusion in (6.41). Let f ∈ Lp (𝒦, Vol) and let ℰ be a bounded set of generalized (W , ds)⃗ elementary operators. Interpolation of (5.5) and (5.6) (with |α| = |β| = 0) shows that 󵄩 󵄩 sup 󵄩󵄩󵄩E 󵄩󵄩󵄩Lp (M,Vol)→Lp (M,Vol) < ∞. −j

(E,2 )∈ℰ

Let {(Ej , 2−j ) : j ∈ ℕν } ⊆ ℰ . We have 󵄩󵄩 −j⋅s 󵄩 󵄩 −j⋅s 󵄩 󵄩 −j⋅s 󵄩 󵄩󵄩{2 Ej f }j∈ℕν 󵄩󵄩󵄩V ≤ 󵄩󵄩󵄩{2 Ej f }j∈ℕν 󵄩󵄩󵄩ℓ1 (ℕν ;Lp (M,Vol)) = ∑ 2 󵄩󵄩󵄩Ej f 󵄩󵄩󵄩Lp (M,Vol) j∈ℕν

󵄩 󵄩 󵄩 󵄩 ≲ ∑ 2−j⋅s 󵄩󵄩󵄩f 󵄩󵄩󵄩Lp (M,Vol) ≲ 󵄩󵄩󵄩f 󵄩󵄩󵄩Lp (M,Vol) .

(6.42)

j∈ℕν

Taking the supremum of the left-hand side of (6.42) over {(Ej , 2−j ) : j ∈ ℕν } ⊆ ℰ shows that (6.43)

‖f ‖V ,s,ℰ ≲ ‖f ‖Lp (M,Vol) < ∞.

⃗ proving the second containment in (6.41). Moreover, This shows that f ∈ X −s (𝒦, (W , ds)), setting ℰ = 𝒟0 in (6.43) shows that ‖f ‖X −s (W ,ds)⃗ ≲ ‖f ‖Lp , proving that the inclusion is continuous. ⃗ Since supp(f ) ⊆ 𝒦, it We turn to the first inclusion in (6.41). Let f ∈ X s (𝒦, (W , ds)). p suffices to show that f ∈ L (M, Vol) and ‖f ‖Lp (M,Vol) ≲ ‖f ‖X s (W ,ds)⃗ . Since supp(f ) ⊆ 𝒦, we ∞ ′ have f = Mult[ψ]f = ∑j∈ℕν Dj f with convergence in CW ,0 (M) (see Proposition 5.5.10). s p ⃗ we have Dj f ∈ L (M, Vol) and ‖Dj f ‖Lp ≲ 2−j⋅s ‖f ‖ s Since f ∈ X (𝒦, (W , ds)), ⃗ . This implies that f = ∑j∈ℕν Dj f converges in Lp (M, Vol), and moreover,

X (W ,ds)

‖f ‖Lp ≤ ∑ ‖Dj f ‖Lp ≲ ∑ 2−j⋅s ‖f ‖X s (W ,ds)⃗ ≲ ‖f ‖X s (W ,ds)⃗ , j∈ℕν

j∈ℕν

completing the proof.

6.5.1 Smooth functions are dense s Recall Notation 6.2.2, which says that X s is either of the form Bp,q , for some p, q ∈ [1, ∞], s or of the form Fp,q , for some p ∈ (1, ∞), q ∈ (1, ∞]; for the remainder of this section, fix such p, q. The main result of this section shows that if q ≠ ∞, then “smooth” functions ⃗ are dense in X s (W , ds). cpt

6.5 Some basic properties

�

423

Most of the proofs of this section are stated in the general setting of this chapter, but we do obtain a better result in the special case of the Hörmander setting (see Section 5.3 for the distinction between the general setting and the Hörmander setting). The main result of this section is the next proposition. Proposition 6.5.17. Suppose q ≠ ∞. Fix s ∈ ℝν . Let 𝒦 ⋐ M be compact and let Ω ⋐ M be an open, relatively compact set with 𝒦 ⋐ Ω. ⃗ there exists a sequence {fL }L∈ℕ ⊂ C ∞ (Ω) such Hörmander setting: ∀f ∈ X s (𝒦, (W , ds)), 0 L→∞ s ⃗ that fL 󳨀󳨀󳨀󳨀󳨀→ f in X (Ω, (W , ds)). ⃗ there exists a sequence General setting: ∀f ∈ X s (𝒦, (W , ds)), ⃗ {fL }L∈ℕ ⊂ ⋂ X t (Ω, (W , ds)) t∈ℝν

L→∞ ⃗ such that fL 󳨀󳨀󳨀󳨀󳨀→ f in X s (Ω, (W , ds)).

The rest of this section is devoted to the proof of Proposition 6.5.17. We take 𝒦 ⋐ Ω ⋐ M as in that proposition and fix s ∈ ℝν . Throughout this section, p ∈ [1, ∞] and q ∈ [1, ∞) are as above and we insist q ≠ ∞. We define V as in Notation 6.3.2. Lemma 6.5.18. Suppose ℰ is a bounded set of generalized (W , ds)⃗ elementary operators ⃗ and let {(Ej,k , 2−j ) : j, l ∈ ℕν } ⊆ ℰ . Then, ∀f ∈ X s (𝒦, (W , ds)), 󵄩 󵄩 lim ∑ 2−|l| 󵄩󵄩󵄩{2j⋅s χ{|j|∞ >L} Ej,l f }j∈ℕν 󵄩󵄩󵄩V = 0,

L→∞

l∈ℤν

(6.44)

where 1, χ{|j|∞ >L} = { 0,

|j|∞ > L,

otherwise.

Proof. We prove the result in the case where V = Lp (M, Vol; ℓq (ℕν )), p ∈ (1, ∞); the proof in the case where V = ℓq (ℕν ; Lp (M, Vol)), p ∈ [1, ∞], is similar and easier. Note that 󵄨󵄨 j⋅s 󵄨 󵄨 j⋅s 󵄨 󵄨󵄨2 χ{|j|∞ >L} Ej,l f 󵄨󵄨󵄨 ≤ 2 󵄨󵄨󵄨Ej,l f 󵄨󵄨󵄨, and by Corollary 6.4.4, 󵄩 󵄨 󵄨 󵄩 ∑ 2−|l| 󵄩󵄩󵄩{2j⋅s 󵄨󵄨󵄨Ej,l f 󵄨󵄨󵄨}j∈ℕν 󵄩󵄩󵄩V ≤ ∑ 2−|l| ‖f ‖V ,s,ℰ

j∈ℤν

j∈ℤν

≲ ∑ 2−|l| ‖f ‖X s (W ,ds)⃗ < ∞. l∈ℤν

424 � 6 Besov and Triebel–Lizorkin spaces Thus, Lebesgue’s dominated convergence theorem can be applied to the left-hand side of (6.44) to show that (using p < ∞) 󵄩 󵄩 lim ∑ 2−|l| 󵄩󵄩󵄩{2j⋅s χ{|j|∞ >L} Ej,l f }j∈ℕν 󵄩󵄩󵄩V

L→∞

l∈ℤν

1󵄩 󵄩󵄩 q󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 q 󵄨 󵄨 = lim ∑ 2−|l| 󵄩󵄩󵄩󵄩( ∑ 󵄨󵄨󵄨2j⋅s Ej,l f 󵄨󵄨󵄨 ) 󵄩󵄩󵄩󵄩 L→∞ 󵄩󵄩 |j| >L 󵄩󵄩 l∈ℤν 󵄩󵄩 ∞ 󵄩󵄩Lp (M,Vol) 1󵄩 󵄩󵄩 q󵄩 󵄩󵄩 󵄩 󵄩 󵄨 󵄨q 󵄩󵄩 = ∑ 2−|l| 󵄩󵄩󵄩󵄩( lim ∑ 󵄨󵄨󵄨2j⋅s Ej,l f 󵄨󵄨󵄨 ) 󵄩󵄩󵄩󵄩 = 0, 󵄩󵄩 L→∞ |j| >L 󵄩󵄩 l∈ℤν 󵄩󵄩 󵄩󵄩Lp (M,Vol) ∞

where we have used the fact that if ∑j∈ℕν |aj | < ∞, for a sequence aj ∈ ℂ, then limL→∞ ∑|j|∞ >L |aj | = 0. As before, we take ψ ∈ C0∞ (Ω) with ψ ≡ 1 on 𝒦. We let 𝒟0 = {(Dj , 2−j ) : j ∈ ℕν } be a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω with Mult[ψ] = ∑j∈ℕν Dj . For L ∈ ℕ, set PL := ∑|j|∞ ≤L Dj . Using Definition 5.2.29, we have ̃t (Ω, (W , ds)) ̃t (Ω, (W , ds)). ⃗ = ⋂t∈ℝν A ⃗ PL ∈ ⋂t∈ℝν A 2 L→∞

⃗ PL f 󳨀󳨀󳨀󳨀󳨀→ f in X s (Ω, (W , ds)). ⃗ Lemma 6.5.19. We have, ∀f ∈ X s (𝒦, (W , ds)), ⃗ In particular, since supp(f ) ⊆ 𝒦, we have f = Mult[ψ]f Proof. Let f ∈ X s (𝒦, (W , ds)). and therefore (I − PL )f = (Mult[ψ] − PL )f = ∑ f . |j|∞ >L

̃0 (Ω, (W , ds)), ⃗ ⊆ X s (Ω, (W , ds)). ⃗ Since PL ∈ A ⃗ By Proposition 6.5.6, f ∈ X s (𝒦, (W , ds)) s ⃗ Theorem 6.3.10 shows that PL f ∈ X (Ω, (W , ds)), ∀L ∈ ℕ. Thus, to prove the result it suffices to show that 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 L→∞ 󵄩󵄩 ∑ D f 󵄩󵄩󵄩 󳨀󳨀󳨀󳨀󳨀→ 0. 󵄩󵄩 k 󵄩 󵄩󵄩 󵄩󵄩|k| >L 󵄩󵄩X s (W ,ds)⃗ 󵄩 ∞

(6.45)

̃ j , 2−j ) : j ∈ ℕν } be a Fix ψ̃ ∈ C0∞ (M) with ψ̃ ≡ 1 on a neighborhood of Ω. Let {(D ̃ We ̃ j = Mult[ψ]. bounded set of generalized (W , ds)⃗ elementary operators with ∑j∈ℕν D ν ν ν |l| −l⋅s ̃ j for j ∈ ℤ \ ℕ . Set, for j, l ∈ ℤ , Ej+l,l := 2 2 D ̃ j Dj+l . take the convention Dj = 0 = D Proposition 5.5.11 implies that

{(Ej,l , 2−j ) : j ∈ ℕν , l ∈ ℤν } is a bounded set of generalized (W , ds)⃗ elementary operators.

6.6 The single-parameter spaces, revisited

� 425

Using the definition of ‖ ⋅ ‖X s (W ,ds)⃗ (see Definition 6.3.8), we have 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ∑ D f 󵄩󵄩󵄩 󵄩󵄩{ ∑ 2j⋅s D 󵄩󵄩 ̃ ≈ D f } 󵄩󵄩 󵄩 󵄩󵄩 k 󵄩 j k 󵄩󵄩 󵄩󵄩 󵄩󵄩|k| >L ν󵄩 󵄩 󵄩 󵄩󵄩V |k| >L s j∈ℕ ⃗ 󵄩 ∞ 󵄩X (W ,ds) 󵄩 ∞ 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 −|l| (j+l)⋅s 󵄩 󵄩󵄩 = 󵄩󵄩{ ∑ 2 2 χ{|j+l|∞ >L} Ej+l,l f } 󵄩󵄩 󵄩󵄩 󵄩 l∈ℤν 󵄩V j∈ℕν 󵄩 L→∞ 󵄩 󵄩 ≤ ∑ 2−|l| 󵄩󵄩󵄩{2j⋅s χ{|j|∞ >L} Ej,l f }j∈ℕν 󵄩󵄩󵄩V 󳨀󳨀󳨀󳨀󳨀→ 0, l∈ℤν

where the final limit uses Lemma 6.5.18. This establishes (6.45) and completes the proof. Proof of Proposition 6.5.17. We take fL := PL f . Lemma 6.5.19 shows that fL → f in ⃗ X s (Ω, (W , ds)). ⃗ ⊆ In the general setting, we use Proposition 6.5.6 to see that f ∈ X s (𝒦, (W , ds)) s t ̃ ⃗ ⃗ X (Ω, (W , ds)). Since PL ∈ ⋂t∈ℝν A (Ω, (W , ds)), Theorem 6.3.10 shows that fL = PL f ∈ ⃗ ∀L ∈ ℕ, completing the proof in the general setting. ⋂t∈ℝν X t (Ω, (W , ds)), In the Hörmander setting, we note that we may take {(Dj , 2−j ) : j ∈ ℕν } to be a bounded set of (W , ds)⃗ elementary operators supported in Ω (see Corollary 5.6.2). In particular, Dj ∈ C0∞ (Ω × Ω) and therefore PL ∈ C0∞ (Ω × Ω). It follows that fL = PL f ∈ C0∞ (Ω), completing the proof.

6.6 The single-parameter spaces, revisited We return to the single-parameter setting of Section 6.2; thus, we are given Hörmander vector fields with single-parameter formal degrees: (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} ⊂ ∞ Cloc (M; TM) × ℕ+ . This is a special case of the general multi-parameter setting covered in Section 6.3. The next result shows that the spaces defined in this single-parameter setting are a special case of the spaces defined in the general multi-parameter setting covered in Section 6.3. Proposition 6.6.1. For s ∈ ℝ, X s (𝒦, (W , ds)) as defined in Definition 6.2.4 is the same as X s (𝒦, (W , ds)) as defined in Definition 6.3.5. More precisely, in the setting of this section, ∞ ′ ∞ ′ C0∞ (M)′ = CW ,0 (M) , and for f ∈ C0 (M) the following are equivalent: (i) Fix Ω ⋐ M relatively compact and open with 𝒦 ⋐ Ω. Then, for every bounded set of (W , ds) elementary operators supported in Ω, ℰ , we have ‖f ‖V ,s,ℰ < ∞. (ii) For every bounded set of (W , ds) elementary operators, ℰ , ‖f ‖V ,s,ℰ < ∞. (iii) For every bounded set of generalized (W , ds) elementary operators, ℰ , ‖f ‖V ,s,ℰ < ∞. Furthermore, in the definition of the norm ‖ ⋅ ‖X s (W ,ds) = ‖ ⋅ ‖V ,s,𝒟0 , we may take the set 𝒟0 to be a bounded set of (W , ds) elementary operators.

426 � 6 Besov and Triebel–Lizorkin spaces ∞ ′ Proof. It follows from Remark 5.2.24 that C0∞ (M)′ = CW ,0 (M) . Since every bounded set of (W , ds) elementary operators is a bounded set of generalized (W , ds) elementary operators, by Proposition 5.5.8, we have (ii) ⇒ (iii). Also, (ii) ⇒ (i) is trivial. Fix Ω ⋐ M as in (i). Let ψ ∈ C0∞ (Ω) satisfy ψ ≡ 1 on a neighborhood of 𝒦. Proposition 5.8.3 shows that Mult[ψ] ∈ A 0 (Ω, (W , ds)) = A20 (Ω, (W , ds)). In particular, there exists a bounded set of (W , ds) elementary operators supported in Ω, 𝒟0 = {(Dj , 2−j ) : j ∈ ℕ} such that Mult[ψ] = ∑j∈ℕ Dj . This shows that we may take 𝒟0 in the definition of the norm ‖ ⋅ ‖X s (W ,ds) = ‖ ⋅ ‖V ,s,𝒟0 to be a bounded set of (W , ds) elementary operators. Finally, we turn to showing (i) ⇒ (iii). Suppose f ∈ C0∞ (M)′ is such that (i) holds. To show that (iii) holds, Corollary 6.4.5 shows that it suffices to show ‖f ‖V ,s,𝒟0 < ∞ and ‖f ‖V ,s,𝒟N < ∞ for all N ≥ 1, where 𝒟N is defined in (6.6). Since 𝒟0 is a bounded set of (W , ds) elementary operators supported in Ω, (i) implies ‖f ‖V ,s,𝒟0 < ∞. Similarly, ‖f ‖V ,s,𝒟N < ∞ will follow once we show that 𝒟N is a bounded set of (W , ds) elementary operators supported in Ω, for every N ≥ 1. Since |l| > |k| in the definition of 𝒟N (see (6.6)), by Proposition 5.5.5 (a), it suffices to show that

{(2(N+2)|l| Dj+k Dj+k+l , 2−(j+k) ) : j, k, l ∈ ℤ, j + k ∈ ℕ}

(6.46)

is a bounded set of (W , ds) elementary operators supported in Ω (recall that Dj+k+l = 0 if j + k + l < 0; see Notation 6.4.1). It follows directly from Proposition 5.5.11 that the set in (6.46) is a bounded set of (W , ds) elementary operators supported in Ω, completing the proof. Remark 6.6.2. By a proof very similar to the one of Proposition 6.6.1, one can see that in the multi-parameter Hörmander setting of Section 5.1.2 (which is a special case of the ⃗ general multi-parameter setting described in this chapter), the spaces X s (𝒦, (W , ds)) can be defined using bounded sets of (W , ds)⃗ elementary operators, instead of bounded sets of generalized (W , ds)⃗ elementary operators. In light of Proposition 6.6.1, many of our main results concerning the singleparameter spaces follow from more general results concerning multi-parameter spaces. For example, we can now prove many of the results from Section 6.2. Proof of Proposition 6.2.5. In light of Proposition 6.6.1, this is a special case of Proposition 6.3.6. Proof of Proposition 6.2.6. Since every bounded set of (W , ds) elementary operators is a bounded set of generalized (W , ds) elementary operators, by Proposition 5.5.8, this is a special case of Proposition 6.3.7. ̃t (Ω, (W , ds)). ⃗ Thus, in Proof of Theorem 6.2.10. By Proposition 5.8.4, A t (Ω, (W , ds)) ⊆ A light of Proposition 6.6.1, this result is a special case of Theorem 6.3.10. Proof of Proposition 6.2.11. In light of Proposition 6.6.1, this is a special case of Corollary 6.5.11.

6.6 The single-parameter spaces, revisited

� 427

Proof of Theorem 6.2.12. In light of Proposition 6.6.1, this is a special case of Proposition 6.5.12. We defer the proof of Proposition 6.2.13 to Section 6.9.

6.6.1 The classical spaces The Besov and Triebel–Lizorkin spaces defined in this chapter generalize the classical Besov and Triebel–Lizorkin spaces on ℝn as described in Section 2.4. For s ∈ ℝ, we write s s X s to denote any one of Bp,q , p, q ∈ [1, ∞], or Fp,q , p ∈ (1, ∞), q ∈ (1, ∞]. Thus, X s (ℝn ) s s denotes one of the spaces Bp,q (ℝn ) or Fp,q (ℝn ) from Section 2.4. We begin by stating and proving a result on ℝn , after which we will turn to a more general result on manifolds. Proposition 6.6.3. Let M = ℝn (endowed with Vol = Lebesgue measure) and let 𝒦 ⋐ ℝn ∞ be a compact subset. Set (𝜕, 1) := {(𝜕x1 , 1), (𝜕xn , 1)} ⊂ Cloc (ℝn ; Tℝn ) × ℕ+ . Then, for s ∈ ℝ, s

s

n

X (𝒦, (𝜕, 1)) = {f ∈ X (ℝ ) : supp(f ) ⊆ 𝒦}.

(6.47)

∀f ∈ X s (𝒦, (𝜕, 1)),

(6.48)

Moreover, ‖f ‖X s (𝜕,1) ≈ ‖f ‖X s (ℝn ) ,

where the implicit constants do not depend on f , but may depend on any of the other ingredients. Proof. First we claim ρ(𝜕,1) (x, y) ≈ |x − y|,

∀x, y ∈ ℝn .

(6.49)

Indeed, ρ(𝜕,1) (x, y) and |x − y| are both translation invariant, so it suffices to prove (6.49) in the case x = 0. It also follows easily from the definitions that for δ > 0, ρ(𝜕,1) (δx, δy) = δρ(𝜕,1) (x, y) and of course |δx − δy| = δ|x − y|. Thus, to show (6.49) it suffices to show that ρ(𝜕,1) (0, y) ≈ 1,

∀y ∈ ℝn with |y| = 1.

(6.50)

Since y 󳨃→ ρ(𝜕,1) (0, y) is continuous (see Lemma 3.1.7) and {y ∈ ℝn : |y| = 1} is compact, (6.50) follows, completing the proof of (6.49). Let Ω ⋐ ℝn be open and relatively compact. In light of (6.49), it follows easily from the definitions that if ℰ is a bounded set of (𝜕, 1) elementary operators supported on Ω, then ℰ is a bounded set of elementary operators in the sense of Definition 2.3.5. Thus, if f ∈ X s (ℝn ), it follows from Proposition 2.4.18 that for every bounded set of (𝜕, 1) elementary operators supported in Ω, ℰ , we have

428 � 6 Besov and Triebel–Lizorkin spaces

‖f ‖V ,s,ℰ < ∞. Since Ω ⋐ ℝn was arbitrary, we see that if we also have supp(f ) ⊆ 𝒦, then f ∈ X s (𝒦, (𝜕, 1)). This proves the ⊇ part of (6.47). Moreover, if ψ ∈ C0∞ (ℝn ) satisfies ψ ≡ 1 on a neighborhood of 𝒦 and 𝒟0 = {(Dj , 2−j ) : j ∈ ℕ} is a bounded set of (𝜕, 1) elementary operators with Mult[ψ] = ∑j∈ℕ Dj , then Proposition 2.4.11 shows that ‖f ‖X s (𝜕,1) = ‖f ‖V ,s,𝒟0 ≲ ‖f ‖X s (ℝn ) .

(6.51)

Suppose f ∈ X s (𝒦, (𝜕, 1)). The same proof as in Proposition 6.4.3 and Corollary 6.4.4 shows that ℰ in those results can be taken to instead be a bounded set of elementary operators as in Definition 2.3.5 (see also the proof of Proposition 2.4.11). Thus, we have, for any such bounded set of elementary operators, ℰ , ‖f ‖V ,s,ℰ ≲ ‖f ‖X s (𝜕,1) < ∞.

(6.52)

We conclude that f ∈ X s (ℝn ) (see Proposition 2.4.18). This proves the ⊆ part of (6.47). ̃ j , 2−j ) : j ∈ ℕ}, where D ̃ j is the operator Dj from ̃0 = {(D In particular, if we let 𝒟 ̃0 is a bounded set of elementary operators and (6.52) with ℰ = 𝒟 ̃0 Section 2.4, then 𝒟 implies ‖f ‖X s (ℝn ) = ‖f ‖V ,s,𝒟 ̃ ≲ ‖f ‖X s (𝜕,1) . 0

(6.53)

Combining (6.51) and (6.53) yields (6.48) and completes the proof. There is a more general version of Proposition 6.6.3 on an arbitrary connected, smooth manifold, which we now present. To begin, we need to define the classical Besov and Triebel–Lizorkin spaces on an arbitrary manifold, and we restrict our attention to distributions with compact support. We denote these spaces by, for 𝒦 ⋐ M, s

Bp,q,std (𝒦),

s

Fp,q,std (𝒦).

s s s As before, we let Xstd (𝒦) denote any one of Bp,q,std (𝒦), p, q ∈ [1, ∞], or Fp,q,std (𝒦), p ∈ (1, ∞), q ∈ (1, ∞]. s Definition 6.6.4. Xstd (𝒦) consists of those f ∈ C0∞ (M)′ with supp(f ) ⊆ 𝒦 such that for

∞ each x ∈ 𝒦, there is a Cloc diffeomorphism Φx : Bn (1) 󳨀 → Φx (Bn (1)), where Φx (Bn (1)) is ∞ n an open neighborhood of x, and ψx ∈ C0 (Φx (B (1))) with ψx ≡ 1 on a neighborhood of x such that Φ∗x ψx f ∈ X s (ℝn ). We set ∼

s

Xstd,cpt (M) :=

⋃

𝒦⋐M 𝒦 compact

s

Xstd (𝒦).

(6.54)

6.6 The single-parameter spaces, revisited

� 429

s We wish to endow Xstd (𝒦) with a norm. For each x ∈ M, let Φx : Bn (1) 󳨀 → Φx (Bn (1)) ∞ n be a Cloc diffeomorphism such that Φx (0) = x and Φx (B (1)) ⋐ M is open and relatively compact. Cover 𝒦 by a finite collection of open sets of the form Φx1 (Bn (1)), . . . , ΦxL (Bn (1)). Take ψj ∈ C0∞ (Φxj (Bn (1))) such that ∑Lj=1 ψj ≡ 1 on a neighborhood of 𝒦. ∼

s Definition 6.6.5. For f ∈ Xstd (𝒦) we define the norm L

‖f ‖X s

(𝒦) std

:= ∑ ‖Φ∗xj ψj f ‖X s (ℝn ) . j=1

s With this norm, Xstd (𝒦) is a Banach space. We state the next simple lemma without proof.

Lemma 6.6.6. For f ∈ C0∞ (M)′ , the following are equivalent: s – f ∈ Xstd (𝒦). – supp(f ) ⊆ 𝒦 and for each 1 ≤ j ≤ L, Φ∗xj ψj f ∈ X s (ℝn ). s Moreover, on Xstd (𝒦), the equivalence class of the norm ‖ ⋅ ‖X s (𝒦) does not depend on std any of the choices we made.

We can now present the main result of this section, which is a generalization of Proposition 6.6.3. ∞ Theorem 6.6.7. Let (Y , 1) = {(Y1 , 1), . . . , (Ys , 1)} ⊂ Cloc (M; TM) × ℕ+ be such that

span{Y1 (x), . . . , Ys (x)} = Tx M,

∀x ∈ M.

(6.55)

Then s

s

Xstd (𝒦) = X (𝒦, (Y , 1)),

with equality of topologies. To prove Theorem 6.6.7, we require a lemma. Lemma 6.6.8. For f ∈ C0∞ (M)′ with supp(f ) ⊆ 𝒦, the following are equivalent: (i) f ∈ X s (𝒦, (Y , 1)). (ii) ψj f ∈ X s (𝒦, (Y , 1)), for 1 ≤ j ≤ L. Moreover, for f ∈ X s (𝒦, (Y , 1)), L

‖f ‖X s (Y ,1) ≈ ∑ ‖ψj f ‖X s (Y ,1) . j=1

(6.56)

Proof. (i) ⇒ (ii): Suppose f ∈ X s (𝒦, (Y , 1)). By Corollary 6.5.10, Mult[ψj ] : X s (𝒦, (Y , 1)) → X s (𝒦, (Y , 1)) is a bounded operator. Thus, ψj f ∈ X s (𝒦, (Y , 1)) and

430 � 6 Besov and Triebel–Lizorkin spaces L

∑ ‖ψj f ‖X s (Y ,1) ≲ ‖f ‖X s (Y ,1) .

(6.57)

j=1

(ii) ⇒ (i): Suppose (ii) holds. Then, since ∑Lj=1 ψj ≡ 1 on 𝒦, we have f = ∑Lj=1 ψj f ∈ X (𝒦, (Y , 1)). Moreover, by the triangle inequality, s

L

‖f ‖X s (Y ,1) ≤ ∑ ‖ψj f ‖X s (Y ,1) .

(6.58)

j=1

Combining (6.57) and (6.58) yields (6.56) and completes the proof. Proof of Theorem 6.6.7. In light of Lemmas 6.6.6 and 6.6.8, to prove the result it suffices to show that for 1 ≤ j ≤ L, ψj f ∈ X s (𝒦, (Y , 1)) ⇐⇒ Φ∗xj ψj f ∈ X s (ℝn ),

(6.59)

‖ψj f ‖X s (Y ,1) ≈ ‖Φxj ψj f ‖X s (ℝn ) .

(6.60)

Set 𝒦j := supp(ψj f ) ⋐ Φxj (Bn (1)). By Proposition 6.5.6, ψj f ∈ X s (𝒦, (Y , 1)) ⇐⇒ ψj f ∈ X s (𝒦j , (Y , 1)).

(6.61)

By Proposition 6.5.15 (see also Remark 6.4.8), we have ∗ ψj f ∈ X s (𝒦j , (Y , 1)) ⇐⇒ Φ∗xj ψj f ∈ X s (Φ−1 xj (𝒦j ), (Φxj Y , 1)),

(6.62)

‖ψj f ‖X s (Y ,1) ≈ ‖Φ∗xj ψj f ‖X s (Φ∗x Y ,1) .

(6.63)

j

The hypothesis (6.55) implies span{Φ∗xj Y1 (u), . . . Φx ∗ Ys (u)} = Tu Bn (1), j

∀u ∈ Bn (1).

It follows that (Φ∗xj Y , 1) and (𝜕, 1) are locally strongly equivalent on Bn (1), and therefore they are locally weakly equivalent on Bn (1). Thus, by Proposition 6.5.14,

∗ ∗ s −1 Φ∗xj ψj f ∈ X s (Φ−1 xj (𝒦j ), (Φxj Y , 1)) ⇐⇒ Φxj ψj f ∈ X (Φxj (𝒦j ), (𝜕, 1)),

(6.64)

‖Φ∗xj ψj f ‖X s (Φ∗x Y ,1) j

(6.65)

≈

‖Φ∗xj ψj f ‖X s (𝜕,1) .

n Since supp(Φ∗xj ψj f ) ⊆ Φ−1 xj (𝒦j ) ⋐ B (1), Proposition 6.6.3 (see also Remark 6.4.8) shows ∗ s n Φ∗xj ψj f ∈ X s (Φ−1 xj (𝒦j ), (𝜕, 1)) ⇐⇒ Φxj ψj f ∈ X (ℝ ),

(6.66)

‖Φ∗xj ψj f ‖X s (𝜕,1)

(6.67)

≈

‖Φ∗xj ψj f ‖X s (ℝn ) .

Combining (6.61), (6.62), (6.63), (6.64), (6.65), (6.66), and (6.67) proves (6.59) and (6.60) and completes the proof.

6.6 The single-parameter spaces, revisited

� 431

6.6.2 Comparing single-parameter spaces Let (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} and (Z, dr) = {(Z1 , dr1 ), . . . , (Zv , drv )} be two lists of Hörmander vector fields with formal degrees on M. In this section, we compare the spaces X s1 (𝒦, (W , ds)) and X s2 (𝒦, (Z, dr)). Definition 6.6.9. For x ∈ M and 1 ≤ j ≤ v, define 󵄨 λ(x, (W , ds), (Zj , drj )) := inf{λ′ > 0 :∃ a neighborhood U of x such that Zj 󵄨󵄨󵄨U ∞ is in the Cloc (U) module generated by

(6.68)

{X1 : (X1 , d1 ) ∈ Gen((W , ds)), d1 ≤ λ drj }}. ′

Set λ(x, (W , ds), (Z, dr)) := max λ(x, (W , ds), (Z, dr)).

(6.69)

1≤j≤v

Lemma 6.6.10. We have λ(x, (W , ds), (Z, dr)) < ∞ and the infimum in (6.68) is achieved (i. e., inf can be replaced with min). Proof. Because W1 , . . . , Wr satisfy Hörmander’s condition on M, given a relatively compact neighborhood U of x, there exists M ∈ ℕ such that span{X1 (y) : (X1 , d1 ) ∈ Gen((W , ds)), d1 ≤ M} = Ty M,

∀y ∈ U.

Thus, λ(x, (W , ds), (Z, dr)) ≤ M < ∞. To see that the infimum is achieved, it suffices to show that the set over which we are taking the infimum is closed, that is, we wish to show that the following set is open: 󵄨 ∞ Λ := {λ′ > 0 :∃ a neighborhood U of x such that Zj 󵄨󵄨󵄨U is not in the Cloc (U) module generated by {X1 : (X1 , d1 ) ∈ Gen((W , ds)), d1 ≤ λ′ drj }}.

Indeed, since each d1 ∈ ℕ+ , for (X1 , d1 ) ∈ Gen((W , ds)), if λ′ ∈ Λ, then λ′ + ϵ ∈ Λ for all ϵ sufficiently small. This shows Λ is open and completes the proof. Corollary 6.6.11. There is an open neighborhood, Ω ⋐ M, of x such that (W , ds) weakly λ(x, (W , ds), (Z, dr))-controls (Z, dr) on Ω. Proof. Since the infimum in (6.68) is achieved, by Lemma 6.6.10, and since (Z, dr) is a fi󵄨 ∞ nite set, there is a neighborhood Ω of x such that for 1 ≤ j ≤ v, Zj 󵄨󵄨󵄨Ω is in the Cloc (Ω) module generated by {X1 : (X1 , d1 ) ∈ Gen((W , ds)), d1 ≤ λdrj }, where λ := λ(x, (W , ds), (Z, dr)). In other words, (W , ds) weakly λ-controls (Z, dr) on Ω (see Definition 3.14.3). This completes the proof.

432 � 6 Besov and Triebel–Lizorkin spaces Remark 6.6.12. In fact, λ(x, (W , ds), (Z, dr)) is chosen to be the least number such that Corollary 6.6.11 holds. In many important cases, this implies our results are sharp. See Section 6.7.1. The main result of this section is the next theorem. Informally, it allows us to trade derivatives with respect to (Z, dr) for derivatives with respect to (W , ds), with the cost of a factor of λ(x, (W , ds), (Z, dr)). Theorem 6.6.13. For every x ∈ M, there exists a relatively compact neighborhood Ω0 ⋐ M of x such that if 𝒦 := Ω0 and λ := λ(x, (W , ds), (Z, dr)), then: (A) For all s0 , ϵ > 0, X

λs0 +ϵ

(𝒦, (W , ds)) ⊆ X s0 ((Z, dr), 𝒦),

(6.70)

and the inclusion is continuous. (B) For all s0 , ϵ > 0, X

−(s0 /λ)+ϵ

(𝒦, (Z, dr)) ⊆ X −s0 (𝒦, (W , ds)),

(6.71)

and the inclusion is continuous. Remark 6.6.14. Theorem 6.6.13 involves a “loss” of ϵ derivatives, for an arbitrarily small ϵ > 0. It is not clear if this loss is necessary or merely an artifact of the proof, and in some situations we have similar results without such a loss (see, for example, Proposition 6.6.21 and Section 6.7). It seems especially likely that the loss is not needed in the s special case X s = Fp,2 , and it seems likely that methods similar to Proposition 6.8.8 could be used to prove such a result; however, we do not pursue that here. When (W , ds) and (Z, dr) weakly locally approximately commute, then we do have for s0 > 0 ‖f ‖F s0 (Z,dr) ≲ ‖f ‖F λs0 (W ,ds) , p,2

p,2

‖f ‖F −s0 (Z,dr) ≲ ‖f ‖F −(s0 /λ) (W ,ds) , p,2

p,2

for all f ∈ C0∞ (M) with supp(f ) ⊆ 𝒦. This can be seen by combining Proposition 6.8.8 and Theorem 6.7.2. We do not use this result, though, so we do not present the full proof. Before we prove Theorem 6.6.13 we introduce an important corollary: the comparison between the Besov and Triebel–Lizorkin spaces associated with (W , ds) and the standard Besov and Triebel–Lizorkin spaces (see Definition 6.6.4). Definition 6.6.15. For x ∈ M, set λstd (x, (W , ds)) := min{ max{d1 , . . . , dn } : (X1 , d1 ), . . . , (Xn , dn ) ∈ Gen((W , ds)), satisfy span{X1 (x), . . . , Xn (x)} = Tx M}.

6.6 The single-parameter spaces, revisited

� 433

Also, set Λstd (x, (W , ds))−1 := max {dsj :Wj is not identically zero on any neighborhood of x}.

Example 6.6.16. If ds1 = ⋅ ⋅ ⋅ = dsr = 1, then λstd (x, (W , ds)) = m, where W1 , . . . , Wr satisfy Hörmander’s condition of order m at x (see Definition 1.1.1). In this special case, Λstd (x, (W , ds)) = 1. ∞ Lemma 6.6.17. If (Y , 1) = {(Y1 , 1), . . . , (Yv , 1)} ⊂ Cloc (M; TM)×ℕ+ are such that Y1 , . . . , Yv span the tangent space at every point of M, then λstd (x, (W , ds)) = λ(x, (W , ds), (Y , 1)) and Λstd (x, (W , ds)) = λ(x, (Y , 1), (W , ds)).

Proof. This follows easily from the definitions. Corollary 6.6.18. For every x ∈ M, there exists a relatively compact neighborhood Ω0 ⋐ M of x such that if 𝒦 = Ω0 , λstd := λstd (x, (W , ds)), and Λstd := Λstd (x, (W , ds)), then we have, for all s0 , ϵ > 0, X

λstd s0 +ϵ

s

(𝒦, (W , ds)) ⊆ Xstd0 (𝒦), Λ s0 +ϵ

Xstdstd

(𝒦) ⊆ X s0 (𝒦, (W , ds)),

−(s0 /λstd )+ϵ

(𝒦) ⊆ X −s0 (𝒦, (W , ds)),

Xstd X

−(s0 /Λstd )+ϵ

−s

(𝒦, (W , ds)) ⊆ Xstd 0 (𝒦),

and the inclusions are all continuous. ∞ Proof. Pick (Y , 1) = {(Y1 , 1), . . . , (Yn , 1)} ⊂ Cloc (M; TM) × ℕ+ such that Y1 (x), . . . , Yn (x) form a basis for Tx M. Thus, there is an open neighborhood N ⊆ M of x such that Y1 (y), . . . , Yn (y) form a basis for Ty N, ∀y ∈ N. By Remark 6.4.8, it suffices to prove the result with M replaced by N; in particular, we may assume Y1 , . . . , Yn span the tangent space at every point. Combining Lemma 6.6.17 and Theorem 6.6.13 shows that the result holds with s Xstd (𝒦) replaced by X s (𝒦, (Y , 1)) throughout. Theorem 6.6.7 shows that X s (𝒦, (Y , 1)) = s Xstd (𝒦), with equality of topologies, completing the proof.

The rest of this section is devoted to the proof of Theorem 6.6.13. By Corollary 6.6.11, we may take Ω ⋐ M an open, relatively compact neighborhood of x, such that (W , ds) weakly λ-controls (Z, dr) on Ω. Let Ω0 ⋐ Ω be an open neighborhood of x and set 𝒦 := Ω0 ⋐ Ω. For the remainder of the section, we write A ≲ B to mean A ≤ Cϵ B, where Cϵ ≥ 0 may depend on ϵ > 0. Lemma 6.6.19. Let ℰ1 be a bounded set of (W , ds) elementary operators supported in Ω and let ℰ2 be a bounded set of (Z, dr) elementary operators supported in Ω. Then, for every N ∈ ℕ, there exists L = L(N) ∈ ℕ, ℰ1′ = ℰ1′ (N) a bounded set of (W , ds) elementary

434 � 6 Besov and Triebel–Lizorkin spaces operators supported in Ω, and ℰ2′ = ℰ2′ (N) a bounded set of (Z, dr) elementary operators supported in Ω such that the following holds ∀(E1 , 2−j1 ) ∈ ℰ1 , (E2 , 2−j2 ) ∈ ℰ2 : L

2N|j2 ∨(λj1 )−λj1 | E1 E2 = ∑ E1K E2K ,

(6.72)

K=1

where (E1K , 2−j1 ) ∈ ℰ1′ and (E2K , 2−j2 ) ∈ ℰ2′ . Similarly, L

2N|j2 ∨(λj1 )−λj1 | E2 E1 = ∑ F2K F1K ,

(6.73)

K=1

where (F1K , 2−j1 ) ∈ ℰ1′ and (F2K , 2−j2 ) ∈ ℰ2′ . Proof. The proofs of (6.72) and (6.73) are similar, so we prove only (6.72). It suffices to prove (6.72) in the case N = 1, as the result for general N follows from repeated application of the result for N = 1. By picking ℰ1′ and ℰ2′ such that ′

′

ℰ1 ⊆ ℰ1 ,

ℰ2 ⊆ ℰ2 ,

{(0, 2−j ) : j ≥ 0} ⊆ ℰ1′ ∩ ℰ2′ ,

(6.72) is trivial when λj1 ≥ j2 . Thus, we turn to proving (6.72) in the case where N = 1 and j2 > λj1 . By Proposition 5.5.5 (h) (with N = 1), we can write v

2j2 −λj1 E1 E2 = 2j2 −λj1 (2−j2 E1 E2,0 + ∑ E1 2−j2 drl Zl E2,l ) =2

−λj1

v

l=1

E1 E2,0 + ∑ 2 l=1

(j2 −λj1 )(1−drl )

E1 2

−λj1 drl

Zl E2,l ,

(6.74)

where {(E2,l , 2−j2 ) : l ∈ {0, . . . , v}, (E2 , 2−j2 ) ∈ ℰ2 } is a bounded set of (Z, dr) elementary operators supported in Ω. We set Ẽ2,0 := 2−λj1 E2,0 and Ẽ2,l := 2(j2 −λj1 )(1−drl ) E2,l for l = 1, . . . , v. Then, since 2−λj1 , 2(j2 −λj1 )(1−drl ) ≤ 1, Proposition 5.5.5 (a) shows that {(Ẽ2,l , 2−j2 ) : (E2 , 2−j2 ) ∈ ℰ ,2 > λj1 } is a bounded set of (Z, dr) elementary operators supported in Ω. Since (W , ds) weakly λ-controls (Z, dr) on Ω (by the choice of Ω), it follows from the definitions that Zl is a (W , ds) partial differential operator on Ω of degree ≤ λdrl . Thus, if we set Ẽ1,l := E1 2−λdrl Zl , it follows from Proposition 5.5.5 (f) that {(Ẽ1,l , 2−j1 ) : (E2 , 2−j ) ∈ ℰ1 , l ∈ {1, . . . , v}} is a bounded set of (W , ds) elementary operators supported in Ω; see also Remark 3.12.3.

6.6 The single-parameter spaces, revisited

� 435

Combining the above discussion with (6.74) shows that v

2j2 −λj1 E1 E2 = E1 Ẽ2,0 + ∑ Ẽ1,l Ẽ2,l , l=1

completing the proof. Recall that by Notation 6.2.2, V is either Lp (M, Vol; ℓq (ℕ)) or ℓq (ℕ; Lp (M, Vol)). Set Ṽ := {

Lp (M, Vol; ℓq (ℕ2 )), ℓq (ℕ2 ; Lp (M, Vol)),

if V = Lp (M, Vol; ℓq (ℕ)), if V = ℓq (ℕ; Lp (M, Vol)).

(6.75)

We fix these p, q ∈ [1, ∞] for the remainder of the section. Lemma 6.6.20. For every ϵ > 0 and every sequence of measurable functions {fj1 ,j2 }j1 ,j2 ∈ℕ , 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩{ ∑ f } 󵄩󵄩 ≲ 󵄩󵄩󵄩{2ϵj1 f } 󵄩󵄩 󵄩󵄩 j1 ,j2 j1 ,j2 (j1 ,j2 )∈ℕ2 󵄩 󵄩 󵄩Ṽ, 󵄩󵄩 j ∈ℕ 󵄩 j2 ∈ℕ 󵄩 󵄩 1 󵄩V 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩{fj1 ,j2 }(j1 ,j2 )∈ℕ2 󵄩󵄩󵄩Ṽ ≤ ∑ 󵄩󵄩󵄩{fj1 ,j2 }j2 ∈ℕ 󵄩󵄩󵄩V .

(6.76) (6.77)

j1 ∈ℕ

Proof. For (6.76), we use ∑ 2

−ϵj1

j1 ∈ℕ

q

1 q

|bj1 | ≲ sup |bj1 | ≤ ( ∑ |bj1 | ) . j1 ∈ℕ

(6.78)

j1 ∈ℕ

Thus, if V = ℓq (ℕ; Lp (M, Vol)), we have 1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 q 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 q 󵄨 󵄨 ϵj 󵄩󵄩{ ∑ f } 󵄩󵄩 ≲ 󵄩󵄩{( ∑ 󵄨󵄨2 1 f 󵄨󵄨 ) } 󵄩󵄩 = 󵄩󵄩󵄩󵄩{2ϵj1 fj ,j }(j ,j )∈ℕ2 󵄩󵄩󵄩󵄩Ṽ, 󵄩󵄩 󵄩󵄩 󵄩 j1 ,j2 j1 ,j2 󵄨 󵄨 1 2 󵄩 󵄩󵄩 1 2 󵄩󵄩 j ∈ℕ 󵄩 󵄩 j2 ∈ℕ 󵄩 j2 ∈ℕ 󵄩 󵄩 1 󵄩V 󵄩󵄩󵄩 j1 ∈ℕ 󵄩󵄩V

as desired. Similarly, if V = Lp (M, Vol; ℓq (ℕ)), we have, using the triangle inequality and (6.78), 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩{ ∑ f } 󵄩󵄩 ≤ ∑ 󵄩󵄩󵄩{f } 󵄩󵄩 󵄩󵄩 j1 ,j2 󵄩 j1 ,j2 j2 ∈ℕ 󵄩󵄩V 󵄩󵄩 j ∈ℕ 󵄩 󵄩 j1 ∈ℕ j2 ∈ℕ 󵄩V 󵄩 1 ≲(∑

j1 ∈ℕ

completing the proof of (6.76).

󵄩󵄩 ϵj1 󵄩q 󵄩󵄩{2 fj1 ,j2 }j2 ∈ℕ 󵄩󵄩󵄩V

1 q

󵄩 󵄩 ) = 󵄩󵄩󵄩{2ϵj1 fj1 ,j2 }(j ,j )∈ℕ2 󵄩󵄩󵄩Ṽ, 1 2

436 � 6 Besov and Triebel–Lizorkin spaces When V = ℓq (ℕ; Lp (M, Vol)), (6.77) follows immediately from the inequality ‖ ⋅ ‖ℓq (ℕ) ≤ ‖ ⋅ ‖ℓ1 (ℕ) applied in the j1 variable. When V = Lp (M, Vol; ℓq (ℕ)), again using ‖ ⋅ ‖ℓq (ℕ) ≤ ‖ ⋅ ‖ℓ1 (ℕ) and the triangle inequality, we have 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 ≤ ∑ 󵄩󵄩󵄩{f } 󵄩󵄩{fj1 ,j2 }(j1 ,j2 )∈ℕ2 󵄩󵄩󵄩Ṽ ≤ 󵄩󵄩󵄩󵄩{ ∑ |fj1 ,j2 |} 󵄩󵄩 󵄩 j1 ,j2 j2 ∈ℕ 󵄩󵄩V , 󵄩󵄩 j ∈ℕ 󵄩 j2 ∈ℕ 󵄩 󵄩 1 󵄩V j1 ∈ℕ completing the proof. Proof of Theorem 6.6.13 (A). Set ϵ1 := ϵ/(1 + λ) > 0. Let s̃ := s0 + ϵ1 > 0. Set s1 := ϵ1 + λs̃ = ϵ + λs0 and s2 := s0 − s̃ = −ϵ1 . Let s⃗ := (s1 , s2 ). Note that s⃗ + (−λs,̃ s)̃ = (ϵ1 , s0 ).

(6.79)

Let ℰ2 be a bounded set of (Z, dr) elementary operators supported in Ω. Let {(Ej2 , 2−j2 ) : j2 ∈ ℕ} ⊆ ℰ2 . In the estimates which follow, implicit constants will not depend on the particular choice of {(Ej2 , 2−j2 ) : j2 ∈ ℕ} ⊆ ℰ2 . Let {(Dj1 , 2−j1 ) : j1 ∈ ℕ} be a bounded set of (W , ds) elementary operators supported in Ω such that ∑j1 ∈ℕ Dj1 = Mult[ψ], where ψ ∈ C0∞ (Ω) satisfies ψ ≡ 1 on 𝒦. We have, for f ∈ X λs0 +ϵ ((W , ds), 𝒦), using (6.76) and (6.79),

󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 j2 s0 󵄩 js 󵄩󵄩 󵄩󵄩{2 Ej2 f }j2 ∈ℕ 󵄩󵄩󵄩V = 󵄩󵄩󵄩󵄩{ ∑ 2 2 0 Ej2 Dj1 f } 󵄩󵄩 󵄩󵄩 j ∈ℕ j2 ∈ℕ 󵄩 󵄩 1 󵄩V ⃗ ̃ ̃ 󵄩 󵄩 󵄩 󵄩 ≲ 󵄩󵄩󵄩{2j⋅(ϵ1 ,s0 ) Ej2 Dj1 f }j∈ℕ2 󵄩󵄩󵄩Ṽ = 󵄩󵄩󵄩{2j⋅s 2j⋅(−λs,s) Ej2 Dj1 f }j∈ℕ2 󵄩󵄩󵄩Ṽ ⃗ ̃ 󵄩 󵄩 = 󵄩󵄩󵄩{2j⋅s 2s(j2 −λj1 ) Ej2 Dj1 f }j∈ℕ2 󵄩󵄩󵄩Ṽ ⃗ ̃ 󵄩 󵄩 ≤ 󵄩󵄩󵄩{2j⋅s 2⌈s⌉|(j2 ∨(λj1 ))−λj1 | Ej2 Dj1 f }j∈ℕ2 󵄩󵄩󵄩Ṽ.

(6.80)

By Lemma 6.6.19, L

̃ 2⌈s⌉|(j2 ∨(λj1 ))−λj1 | Ej2 Dj1 = ∑ ẼjK2 F̃jK1 , K=1

(6.81)

where {(ẼjK2 , 2−j2 ) : (Ej2 , 2−j2 ) ∈ ℰ2 , j1 ∈ ℕ, 1 ≤ L ≤ K} is a bounded set of (Z, dr) elementary operators supported in Ω and {(F̃ K , 2−j1 ) : (Ej , 2−j2 ) ∈ ℰ2 , j1 ∈ ℕ, 1 ≤ L ≤ K} is a bounded j1

2

set of (W , ds) elementary operators supported in Ω, where L ≈ 1. Here, ẼjK2 depends on j1 and F̃ K depends on Ẽ K , though we suppress this dependence. j1

j2

Plugging (6.81) into (6.80), we see that

6.6 The single-parameter spaces, revisited

� 437

L

󵄩󵄩 j2 s0 󵄩 󵄩 j⋅s⃗ K K 󵄩 󵄩󵄩{2 Ej2 f }j2 ∈ℕ 󵄩󵄩󵄩V ≲ ∑ 󵄩󵄩󵄩{2 Ẽj2 F̃j1 f }j∈ℕ2 󵄩󵄩󵄩Ṽ K=1 L

󵄩 󵄩 = ∑ 󵄩󵄩󵄩{2−ϵ1 j2 2j1 s1 ẼjK2 F̃jK1 f }j∈ℕ2 󵄩󵄩󵄩Ṽ K=1

(6.82)

L

󵄩 󵄩 ≤ ∑ ∑ 2−ϵ1 j2 󵄩󵄩󵄩{2j1 s1 ẼjK2 F̃jK1 f }j ∈ℕ 󵄩󵄩󵄩V , 1 K=1 j2 ∈ℕ

where the last inequality uses (6.77). Since {(ẼjK2 , 2−j2 ) : (Ej2 , 2−j2 ) ∈ ℰ2 , j1 ∈ ℕ, 1 ≤ L ≤ K} is a bounded set of (Z, dr) elementary operators supported in Ω, it follows from Corollary 5.4.11 and Lemma 5.4.12 that 󵄩󵄩 j1 s1 ̃ K ̃ K 󵄩 󵄩 js K 󵄩 󵄩󵄩{2 Ej2 Fj1 f }j1 ∈ℕ 󵄩󵄩󵄩V ≲ 󵄩󵄩󵄩{2 1 1 F̃j1 f }j1 ∈ℕ 󵄩󵄩󵄩V .

(6.83)

Since {(F̃jK1 , 2−j1 ) : (Ej2 , 2−j2 ) ∈ ℰ2 , j1 ∈ ℕ, 1 ≤ L ≤ K} is a bounded set of (W , ds) elementary operators supported in Ω, Theorem 6.4.3 shows that 󵄩󵄩 j1 s1 ̃ K 󵄩 󵄩󵄩{2 Fj1 f }j1 ∈ℕ 󵄩󵄩󵄩V ≲ ‖f ‖X s1 (W ,ds) = ‖f ‖X ϵ+λs0 (W ,ds) .

(6.84)

Plugging (6.83) and (6.84) into (6.82) and using the fact that L ≈ 1, we obtain L

󵄩󵄩 j2 s0 󵄩 −ϵ j 󵄩󵄩{2 Ej2 f }j2 ∈ℕ 󵄩󵄩󵄩V ≲ ∑ ∑ 2 1 2 ‖f ‖X ϵ+λs0 (W ,ds) ≈ ‖f ‖X ϵ+λs0 (W ,ds) . K=1 j2 ∈ℕ

(6.85)

Taking the supremum of (6.85) over {(Ej2 , 2−j2 ) : j2 ∈ ℕ} ⊆ ℰ2 shows that ‖f ‖V ,s0 ,ℰ2 ≲ ‖f ‖X ϵ+λs0 (W ,ds) < ∞.

(6.86)

Since ℰ2 was an arbitrary bounded set of (W , ds) elementary operators supported in Ω, it follows from Proposition 6.6.1 (i) ⇒ (ii) that f ∈ X s0 (𝒦, (Z, dr)). This completes the proof of the containment (6.70). ̃ j , 2−j2 ) : j2 ∈ ℕ} be a bounded set ̃ = {(D To see that the inclusion is continuous, let 𝒟 2 ̃ j = Mult[ψ]. Applying (6.86) of (Z, dr) elementary operators supported in Ω with ∑j2 ∈ℕ D 2 ̃ with ℰ2 = 𝒟 shows that ‖f ‖X s0 (Z,dr) ≈ ‖f ‖V ,s

̃

0 ,𝒟

≲ ‖f ‖X ϵ+λs0 (W ,ds) .

Thus, the inclusion map is bounded, completing the proof. Proof of Theorem 6.6.13 (B). Take ϵ1 > 0 so small that ϵ1 (1+ λ1 ) ≤ ϵ and s̃ := (s0 −ϵ1 )/λ > 0. Set s2 := ϵ1 − s̃ = (1 + 1/λ)ϵ1 − s0 /λ ≤ ϵ − s0 /λ. Set s1 := −ϵ1 = λs̃ − s0 , and let s⃗ := (s1 , s2 ). Note that s⃗ + (−λs,̃ s)̃ = (−s0 , ϵ1 ).

(6.87)

438 � 6 Besov and Triebel–Lizorkin spaces Let ℰ1 be a bounded set of (W , ds) elementary operators supported in Ω. Let {(Ej1 , 2−j1 ) : j1 ∈ ℕ} ⊆ ℰ1 . In the estimates which follow, implicit constants will not depend on the particular choice of {(Ej1 , 2−j1 ) : j1 ∈ ℕ} ⊆ ℰ1 . Let {(Dj2 , 2−j2 ) : j2 ∈ ℕ} be a bounded set of (Z, dr) elementary operators supported in Ω such that ∑j2 ∈ℕ Dj2 = Mult[ψ], where ψ ∈ C0∞ (Ω) satisfies ψ ≡ 1 on 𝒦. We have, for f ∈ X −(s0 /λ)+ϵ (𝒦, (Z, dr)), using (6.76) and (6.87),

󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 −j1 s0 󵄩󵄩 −j1 s0 󵄩 󵄩󵄩 󵄩 Ej1 f }j ∈ℕ 󵄩󵄩V = 󵄩󵄩{ ∑ 2 Ej1 Dj2 f } 󵄩󵄩{2 󵄩󵄩 1 󵄩󵄩 j ∈ℕ 󵄩󵄩 j ∈ℕ 󵄩 2 󵄩V 1 ⃗ ̃ ̃ 󵄩󵄩 j⋅(−s0 ,ϵ1 ) 󵄩󵄩 󵄩 󵄩 ≲ 󵄩󵄩{2 Ej1 Dj2 f }j∈ℕ2 󵄩󵄩Ṽ = 󵄩󵄩󵄩{2j⋅s+j⋅(−λs,s) Ej1 Dj2 f }j∈ℕ2 󵄩󵄩󵄩Ṽ ⃗ ̃ 󵄩 󵄩 = 󵄩󵄩󵄩{2j⋅s+s(j2 −λj1 ) Ej1 Dj2 f }j∈ℕ2 󵄩󵄩󵄩Ṽ ⃗ ̃ 󵄩 󵄩 ≤ 󵄩󵄩󵄩{2j⋅s+⌈s⌉|(j2 ∨(λj1 ))−λj1 | Ej1 Dj2 f }j∈ℕ2 󵄩󵄩󵄩Ṽ.

(6.88)

By Lemma 6.6.19, L

2⌈s⌉|(j2 ∨(λj1 ))−λj1 | Ej1 Dj2 = ∑ ẼjK1 F̃jK2 , ̃

K=1

(6.89)

where {(ẼjK1 , 2−j1 ) : (Ej1 , 2−j1 ) ∈ ℰ1 , j2 ∈ ℕ, 1 ≤ L ≤ K} is a bounded set of (W , ds) elementary operators supported in Ω and {(F̃ K , 2−j2 ) : (Ej , 2−j1 ) ∈ ℰ1 , j2 ∈ ℕ, 1 ≤ L ≤ K} is a bounded j2

1

set of (Z, dr) elementary operators supported in Ω, where L ≈ 1. Here, ẼjK1 depends on j2 and F̃ K depends on Ẽ K , though we suppress this dependence. j2

j1

Plugging (6.89) into (6.88), we obtain L

⃗ 󵄩󵄩 −j1 s0 󵄩 󵄩 󵄩 Ej1 f }j ∈ℕ 󵄩󵄩󵄩V ≲ ∑ 󵄩󵄩󵄩{2j⋅s ẼjK1 F̃jK2 f }j∈ℕ2 󵄩󵄩󵄩Ṽ 󵄩󵄩{2 1 K=1 L

󵄩 󵄩 = ∑ 󵄩󵄩󵄩{2−ϵ1 j1 2j2 s2 ẼjK1 F̃jK2 f }j∈ℕ2 󵄩󵄩󵄩Ṽ K=1

(6.90)

L

󵄩 󵄩 ≤ ∑ ∑ 2−ϵ1 j1 󵄩󵄩󵄩{2j2 s2 ẼjK1 F̃jK2 f }j ∈ℕ 󵄩󵄩󵄩V , 2 K=1 j1 ∈ℕ

where the last inequality uses (6.77). Since {(ẼjK1 , 2−j1 ) : (Ej1 , 2−j1 ) ∈ ℰ1 , j2 ∈ ℕ, 1 ≤ L ≤ K} is a bounded set of (W , ds) elementary operators supported in Ω, it follows from Corollary 5.4.11 and Lemma 5.4.12 that 󵄩󵄩 j2 s2 ̃ K ̃ K 󵄩 󵄩 js K 󵄩 󵄩󵄩{2 Ej1 Fj2 f }j2 ∈ℕ 󵄩󵄩󵄩V ≲ 󵄩󵄩󵄩{2 2 2 F̃j2 f }j2 ∈ℕ 󵄩󵄩󵄩V .

(6.91)

Since {(F̃jK2 , 2−j2 ) : (Ej1 , 2−j1 ) ∈ ℰ1 , j1 ∈ ℕ, 1 ≤ L ≤ K} is a bounded set of (W , ds) elementary operators supported in Ω, Theorem 6.4.3 shows that

6.6 The single-parameter spaces, revisited

� 439

󵄩󵄩 j2 s2 ̃ K 󵄩 󵄩󵄩{2 Fj2 f }j2 ∈ℕ 󵄩󵄩󵄩V ≲ ‖f ‖X s2 (Z,dr) = ‖f ‖X −(s0 /λ)+ϵ (Z,dr) .

(6.92)

Plugging (6.91) and (6.92) into (6.90) and using the fact that L ≈ 1, we obtain L

󵄩󵄩 −j1 s0 󵄩 Ej1 f }j ∈ℕ 󵄩󵄩󵄩V ≲ ∑ ∑ 2−ϵ1 j1 ‖f ‖X −(s0 /λ)+ϵ (Z,dr) ≈ ‖f ‖X −(s0 /λ)+ϵ (Z,dr) . 󵄩󵄩{2 1 K=1 j1 ∈ℕ

(6.93)

Taking the supremum of (6.93) over {(Ej1 , 2−j1 ) : j1 ∈ ℕ} ⊆ ℰ1 shows that ‖f ‖V ,−s0 ,ℰ1 ≲ ‖f ‖X −(s0 /λ)+ϵ (Z,dr) < ∞.

(6.94)

Since ℰ1 was an arbitrary bounded set of (Z, dr) elementary operators supported in Ω, it follows from Proposition 6.6.1 (i) ⇒ (ii) that f ∈ X −s0 (𝒦, (W , ds)). This completes the proof of the containment (6.71). ̃ j , 2−j1 ) : j1 ∈ ℕ} be a bounded set ̃ = {(D To see that the inclusion is continuous, let 𝒟 1 ̃ j = Mult[ψ]. Applying (6.94) of (W , ds) elementary operators supported in Ω with ∑j1 ∈ℕ D 1 ̃ shows that with ℰ1 = 𝒟 ‖f ‖X −s0 (W ,ds) ≈ ‖f ‖V ,−s

̃

0 ,𝒟

≲ ‖f ‖X −(s0 /λ)+ϵ (Z,dr) .

Thus, the inclusion map is bounded, completing the proof.

6.6.3 Boundedness of operators in D Let (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} and (Z, dr) = {(Z1 , dr1 ), . . . , (Zv , drv )} be two lists of Hörmander vector fields with formal degrees on M. We assume (W , ds) and (Z, dr) locally weakly approximately commute. Because (W , ds) and (Z, dr) locally weakly approximately commute, it makes sense to talk of the filtered algebra D t (Ω, (W , ds), (Z, dr)) ⊆ A t (Ω, (W , ds)) as defined in Section 5.9. Informally speaking, D t (Ω, (W , ds), (Z, dr)) consists of those operators in A t (Ω, (W , ds)) which “approximately commute” with the vector fields in (Z, dr). This is enough to show that the operators in D κ (Ω, (W , ds), (Z, dr)), for κ > 0, have good mapping properties on the spaces X s (𝒦, (Z, dr)), as the next result shows. Proposition 6.6.21. For every x ∈ M, there exists a relatively compact, open neighborhood Ω0 ⋐ M of x such that if λ := λ(x, (W , ds), (Z, dr)) (see Definition 6.6.9) and 𝒦 := Ω0 , then for every κ > 0 and T ∈ D κ (Ω0 , (W , ds), (Z, dr)), T : X s−κ/λ (𝒦, (Z, dr)) → X s (𝒦, (Z, dr)) is bounded. Proposition 6.6.21 in particular shows that standard pseudo-differential operators have the expected mapping properties on the spaces X s (𝒦, (Z, dr)).

440 � 6 Besov and Triebel–Lizorkin spaces Corollary 6.6.22. For every x ∈ M, there exists a relatively compact, open neighborhood Ω0 ⋐ M of x such that if Λstd := Λstd (x, (Z, dr)) (see Definition 6.6.15) and 𝒦 := Ω0 , then for every κ > 0 and b(x, D) a standard pseudo-differential operator of order κ with supp(b(x, D)) ⊆ Ω × Ω, b(x, D) : X s−κ/Λstd (𝒦, (Z, dr)) → X s (𝒦, (Z, dr)) is bounded. ∞ Proof. Pick Y1 , . . . , Yn ∈ Cloc (M; TM) such that Y1 (x), . . . , Yn (x) form a basis for Tx M. By continuity, there is a neighborhood N ⊆ M of x such that Y1 (y), . . . , Yn (y) form a basis for Ty N, for all y ∈ N. By Remark 6.4.8 it suffices to prove the result with M replaced by N. We henceforth make this replacement. Let (Y , 1) = {(Y1 , 1), . . . , (Yn , 1)}. By Lemma 6.6.17, Λstd (x, (Z, dr)) = λ(x, (Y , 1), (Z, dr)). Let Ω0 and 𝒦 be as in Proposition 6.6.21 with (W , ds) replaced by (Y , 1). By Proposition 5.9.13 we have b(x, D) ∈ D κ (Ω0 , (Y , 1), (Z, dr)). From here, Proposition 6.6.21 completes the proof.

An important application of Proposition 6.6.21 is to give regularity properties of maximally subelliptic operators with respect to Besov and Triebel–Lizorkin spaces corresponding to other geometries. In particular, it gives sharp regularity of maximally subelliptic operators with respect the standard Besov and Triebel–Lizorkin spaces. See Section 8.2.2. The rest of this section is devoted to the proof of Proposition 6.6.21. Fix x ∈ M, for which we wish to prove Proposition 6.6.21. By Corollary 6.6.11 there is a neighborhood Ω ⋐ M such that (W , ds) weakly λ = λ(x, (W , ds), (Z, dr))-controls (Z, dr) on Ω. Let Ω0 ⋐ Ω be an open neighborhood of x and let 𝒦 := Ω0 ⋐ Ω. These are the Ω0 and 𝒦 for which we prove Proposition 6.6.21. We recall Notation 6.4.1, where for any operator, Ej , indexed by j ∈ ℕ, we set Ej := 0 for j < 0. Lemma 6.6.23. Let ℱ be a bounded set of (W , ds) elementary operators supported in Ω and let ℰ be a bounded set of (Z, dr) elementary operators supported in Ω. Let 1

ℰ1 = {(Ej , 2 ) : j ∈ ℕ}, ′

−j

2

ℰ2 = {(Ej , 2 ) : j ∈ ℕ} ⊆ ℰ , ′

(6.95)

−j

ℱ = {(Fk , 2 ) : k ∈ ℕ} ⊆ ℱ . ′

(6.96)

−k

Then, for every M ∈ ℕ and s ∈ ℝ, 󵄩 󵄩 2 sup sup 2M|k| 󵄩󵄩󵄩{Ej1 F⌈ j+l ⌉+k 2s(j+l) Ej+l f }j∈ℕ 󵄩󵄩󵄩V ≲ ‖f ‖X s (Z,dr) ,

ℰ1′ ,ℰ2′ ′

k 0 and T ∈ A −κ (Ω, (W , ds)). Let ℰ be a bounded set of (Z, dr) elementary operators supported in Ω. Let 1

ℰ1 = {(Ej , 2 ) : j ∈ ℕ}, ′

−j

2

ℰ2 = {(Ej , 2 ) : j ∈ ℕ} ⊆ ℰ . ′

−j

(6.101)

Then, for every s ∈ ℝ, 󵄩 󵄩 2 sup sup 󵄩󵄩󵄩{Ej1 T2(j+l)s Ej+l f }j∈ℕ 󵄩󵄩󵄩V ≲ ‖f ‖X s−κ/λ . l∈ℤ ℰ1′ ,ℰ2′

(6.102)

442 � 6 Besov and Triebel–Lizorkin spaces Proof. Using T ∈ A −κ (Ω, (W , ds)) = A2−κ (Ω, (W , ds)), we may write T = ∑k∈ℕ 2−kκ Fk , where {(Fk , 2−k ) : k ∈ ℕ} is a bounded set of (W , ds) elementary operators supported in Ω. By Proposition 5.5.10, this sum converges in the topology of bounded convergence on ∞ 2 Hom(Cloc (M), C0∞ (M)). In particular, with ℰ1′ and ℰ2′ as in (6.101), since Ej1 , Ej+l ∈ C0∞ (Ω × ∞ ′ Ω), we have, for any f ∈ C0 (M) , 2 Ej1 T2(j+l)s Ej+l f = ∑ Ej1 2−(k+⌈ k∈ℤ

j+l ⌉)κ λ

2 F⌈ j+l ⌉+k 2(j+l)s Ej+l f.

(6.103)

λ

In what follows, the implicit constants will be independent of the particular choice

of and ℰ2′ in (6.101) and independent of l ∈ ℤ. Note that 2⌈ for f ∈ X s−κ/λ (𝒦, (Z, dr)), using (6.103), ℰ1′

j+l ⌉κ λ

j+l

≈ 2κ λ . Thus, we have,

󵄩󵄩 1 (j+l)s 2 󵄩 Ej+l f }j∈ℕ 󵄩󵄩󵄩V 󵄩󵄩{Ej T2 j+l 󵄩 󵄩 2 ≤ ∑ 󵄩󵄩󵄩{Ej1 2−(k+⌈ λ ⌉)κ F⌈ j+l ⌉+k 2(j+l)s Ej+l f }j∈ℕ 󵄩󵄩󵄩V λ k∈ℤ

(6.104)

󵄩 󵄩 2 ≈ ∑ 2−kκ 󵄩󵄩󵄩{Ej1 F⌈ j+l ⌉+k 2(j+l)(s−κ/λ) Ej+l f }j∈ℕ 󵄩󵄩󵄩V . λ

k∈ℤ

We separate the sum on the right-hand side of (6.104) into two parts: k ≥ 0 and k < 0. For k ≥ 0, using Corollary 5.4.11 and Lemma 5.4.12 we have 󵄩󵄩 1 󵄩 󵄩 󵄩 (j+l)(s−κ/λ) 2 2 Ej+l f }j∈ℕ 󵄩󵄩󵄩V ≲ 󵄩󵄩󵄩{F⌈ j+l ⌉+k 2(j+l)(s−κ/λ) Ej+l f }j∈ℕ 󵄩󵄩󵄩V 󵄩󵄩{Ej F⌈ j+l ⌉+k 2 λ λ 󵄩 󵄩 2 ≲ 󵄩󵄩󵄩{2(j+l)(s−κ/λ) Ej+l f }j∈ℕ 󵄩󵄩󵄩V ≲ ‖f ‖X s−κ/λ (Z,dr) , where the last estimate follows from Corollary 6.4.4. Thus, 󵄩 󵄩 2 f }j∈ℕ 󵄩󵄩󵄩V ∑ 2−kκ 󵄩󵄩󵄩{Ej1 F⌈ j+l ⌉+k 2(j+l)(s−κ/λ) Ej+l λ

k≥0

≲ ∑ 2−kκ ‖f ‖X s−κ/λ (Z,dr) ≲ ‖f ‖X s−κ/λ (Z,dr) ,

(6.105)

k≥0

where we have used κ > 0. For k < 0, we apply Lemma 6.6.23 with M = κ + 1 and s replaced by s − κ/λ to see that 󵄩 󵄩 2 f }j∈ℕ 󵄩󵄩󵄩V ∑ 2−kκ 󵄩󵄩󵄩{Ej1 F⌈ j+l ⌉+k 2(j+l)(s−κ/λ) Ej+l λ

k 0, ϵ1 , ϵ2 ≥ 0, a space X0s of the form s

s

s

X0 ∈ {Bp,q : p, q ∈ [1, ∞]} ⋃{Fp,q : p ∈ (1, ∞), q ∈ (1, ∞]},

(6.120)

a neighborhood U of x, and C ≥ 0 such that ̃ ‖f ‖X s+(−λ(s),̃ s)+(ϵ 1 ,ϵ2 ) (W ,ds) ≤ C‖f ‖X s (W ,ds) , 0 0

∀f ∈ C0∞ (U).

(6.121)

Then ϵ1 = ϵ2 = 0. The bound (6.119) is a special case of inclusion map in Theorem 6.7.2 being continuous; thus, we need only to prove the sharpness part of Proposition 6.7.7. The rest of this section is devoted to the proof of this sharpness. Fix x ∈ M, s ∈ ℝ2 , s̃ > 0, ϵ1 , ϵ2 ≥ 0, X0s , and U ⋐ M as in Proposition 6.7.7, such that (6.121) holds. We assume λ(x, (Y , d̂), (Z, dr)) is sharp. Our goal is to show ϵ1 = ϵ2 = 0. Recall that X0s is of the form (6.120). Fix p ∈ [1, ∞] s s so that X0s is equal to either Bp,q or Fp,q , for some q ∈ [1, ∞]. By Corollary 6.6.11, there is a neighborhood Ω ⋐ U of x such that (Y , d̂) weakly λ := λ(x, (Y , d̂), (Z, dr))-controls (Z, dr) on Ω. Fix Ω0 ⋐ Ω1 ⋐ Ω2 ⋐ Ω open with x ∈ Ω0 . Using Proposition 3.4.14, in the case ν = 1, Gen((Y , d̂)) is finitely generated on Ω by some (X 1 , d 1 ) = {(X11 , d11 ), . . . , (Xq11 , dq11 )} ⊆ Gen((Y , d̂)) and Gen((Z, dr)) is finitely generated on Ω by some (X 2 , d 2 ) = {(X12 , d12 ), . . . , (Xq22 , dq22 )} ⊆ Gen((Z, dr)). Since λ(x, (Y , d̂), (Z, dr))

is sharp, by possibly adding an element to (X 2 , d 2 ), we may assume (X12 , d12 ) = (V , dr), where (V , dr) ∈ Gen((Z, dr)) is as in Definition 6.7.4 with U = Ω0 . We will be considering pseudo-differential operators corresponding to (X 1 , d 1 ), (X 2 , d 2 ) as in Chapter 4. To this end, define (X, d )⃗ as in (4.1), that is, (X, d )⃗ = {(X1 , d 1⃗ ), . . . , (Xq , d q⃗ )} := (X 1 , d 1 ) ⊠ (X 2 , d 2 ) ∞ ⊂ Cloc (M; TM) × (ℕ2 \ {0}).

Note that q = q1 + q2 . For t ∈ ℝq , we write t = (t 1 , t 2 ) ∈ ℝq1 × ℝq2 = ℝq . Fix a > 0 small and N ∈ ℕ large, to be chosen later. Let ψ ∈ C0∞ (Ω1 ) and ς ∈ ∞ q C0 (B (a)) be such that α

∫(t 1 ) 1 ς(t 1 , t 2 ) dt 1 ≡ 0,

α

∫(t 2 ) 2 ς(t 1 , t 2 ) dt 2 ≡ 0,

∀|α1 |, |α2 | ≤ N.

(6.122)

450 � 6 Besov and Triebel–Lizorkin spaces Define, for j ≥ 0, with λ := λ(x, (Y , d̂), (Z, dr)), 1

1

Tj f (x) := ∫ f (e−t ⋅X e−t

2

⋅X 2

x)ψ(x) Dild2(j,λj) (ς)(t) dt,

(6.123)

⃗

where Dild2(j,λj) (ς)(t) is defined as in Definition 4.1.12 (here we are taking f (x, s, t) = ς(t) in Definition 4.1.12). The proof of Proposition 6.7.7 follows by combining the next two lemmas. ⃗

Lemma 6.7.8. Suppose ϵ1 + ϵ2 > 0. Then if a > 0 is sufficiently small and N ∈ ℕ is sufficiently large, we have lim ‖Tj Tj ‖Lp (M,Vol)→Lp (M,Vol) = 0.

j→∞

Lemma 6.7.9. For every a > 0 and N ∈ ℕ, there exist a choice of ψ and ς as above with lim sup ‖Tj Tj ‖Lp (M,Vol)→Lp (M,Vol) > 0. j→∞

Proof of Proposition 6.7.7. In light of Lemmas 6.7.8 and 6.7.9, we must have ϵ1 + ϵ2 ≤ 0. Since ϵ1 , ϵ2 ≥ 0, this implies ϵ1 = ϵ2 = 0, completing the proof. We begin by proving Lemma 6.7.8; this lemma does not require that λ(x, (Y , d̂), (Z, dr)) be sharp. We begin with two auxiliary lemmas. ⃗ ∈ ℕ is sufficiently large and a = Lemma 6.7.10. Fix v ∈ ℝ2 . If N = N(v, (X, d )) (j,λj)⋅v ⃗ ⃗ a pseudo-differential a(Ω1 , Ω2 , (X, d )) > 0 is sufficiently small, then 2 Tj is an (X, d ), operator of order v supported in Ω1 × Ω2 , uniformly for j ≥ 0. More precisely, if Tj is associated with the kernel b̌ j (x, s, t) as in Definition 4.1.2, then {bj : j ≥ 0} ⊂ S m (a, Ω1 , d )⃗ is a bounded set, where S m (a, Ω1 , d )⃗ is defined in Definition 4.1.1 and is given the obvious locally convex topology. Here, b (x, s, ξ) is the Fourier transform of b̌ (x, s, t) in the t variable.

j

j

Proof. Take ς0 ∈ C0∞ (Bq (a)) such that ς ≺ ς0 . Thus, if we set ⃗ b̌ j (x, s, t) := ψ(x)ς0 (s)(2(j,λj)⋅v Dild2(j,λj) (ς)(t)),

then 1

1

Tj f (x) = ∫ f (e−t ⋅X e−t

2

⋅X 2

x)b̌ j (x, t, t) dt.

Thus, if bj (x, s, ξ) is the Fourier transform of b̌ j (x, s, t) in the t variable, we wish to show that {bj : j ≥ 0} ⊂ S m (a, Ω1 , d )⃗ is a bounded set. In other words, we wish to show that the conditions of Definition 4.1.1 hold uniformly in j ≥ 0.

6.7 Trading derivatives

�

451

Note that ⋃{(x, s) ∈ M × Bq (a) : ∃ξ ∈ ℝq with bj (x, s, ξ) ≠ 0} ⊆ supp(ψ(x)ς0 (s)) ⋐ Ω1 × Bq (a). j≥0

Thus, Definition 4.1.1 (i) holds uniformly in j ≥ 0. All that remains to show is that Definition 4.1.1 (ii) holds uniformly in j ≥ 0. Since ̂ bj (x, s, ξ) = ψ(x)ς0 (s)2(j,λj)⋅v ς(ξ), where ς̂ is the Fourier transform of ς, to show Definition 4.1.1 (ii) holds uniformly in j ≥ 0, it suffices to show that, ∀α, β, 󵄨󵄨 α β (j,λj)⋅v ̂ −jd 1 1 −jλd 2 2 󵄨󵄨 ς(2 ξ , 2 ξ )󵄨󵄨 󵄨󵄨𝜕ξ1 𝜕ξ 2 2 ≤ Cα,β (1 + ‖ξ 1 ‖1 )

v1 −degd 1 (α)

v2 −degd 2 (β)

(1 + ‖ξ 2 ‖2 )

(6.124) ,

where Cα,β does not depend on j ≥ 0. To prove (6.124), we use the following elementary inequality. For w1 ∈ ℝ, L ≥ 0 with L ≥ |w1 |, we have w1 ∨0

2j1 w1 (2−j1 ‖ξ 1 ‖1 )

(1 + 2−j1 ‖ξ 1 ‖1 )

−L

w

≲ (1 + ‖ξ 1 ‖1 ) 1 ,

(6.125)

where the implicit constant does not depend on j1 ≥ 0 or ξ 1 , but may depend on any of the other ingredients. Indeed, to see (6.125), we consider three cases. When w1 ≥ 0, it is w immediate that the left-hand side of (6.125) is ≲ ‖ξ 1 ‖1 1 ≲ (1 + ‖ξ 1 ‖1 )w1 , establishing (6.125) in this case. When w1 < 0 and 2−j1 ‖ξ 1 ‖1 ≤ 1, we have, using the fact that j ≥ 0, w1 ∨0

2j1 w1 (2−j1 ‖ξ 1 ‖1 )

(1 + 2−j1 ‖ξ 1 ‖1 )

−L

≲ 2j1 w1 = 2−j1 |w1 | ≲ ‖ξ 1 ‖1

−|w1 |

w

w

∧ 1 = ‖ξ 1 ‖1 1 ∧ 1 ≈ (1 + ‖ξ 1 ‖1 ) 1 ,

establishing (6.125) in this case. Finally, if w1 < 0 and 2−j1 ‖ξ 1 ‖1 > 1, we have, using the fact that L ≥ |w1 |, 2jw1 (2−j1 ‖ξ 1 ‖1 )

w1 ∨0

(1 + 2−j1 ‖ξ 1 ‖1 )

−L

≲ 2j1 w1 2Lj1 ‖ξ 1 ‖−L 1 w

= ‖ξ 1 ‖1 1 (‖ξ 1 ‖1 w

−w1 −j1 w1

2

w

)(2Lj1 ‖ξ 1 ‖−L 1 )

≤ ‖ξ 1 ‖1 1 ≲ (1 + ‖ξ 1 ‖1 1 ),

where the final estimate uses the fact that ‖ξ1 ‖1 ≥ 2−j1 ≥ 1 in this case. This completes the proof of (6.125). ̂ 1 , ξ 2 ) vanishes to as high an order as we like at By (6.122), by taking N ∈ ℕ large, ς(ξ 1 2 q ξ = 0 and at ξ = 0. Since ς̂ ∈ S (ℝ ) as well, we see that by taking N = N(v, d )⃗ ∈ ℕ large, we have, ∀α, β, L, −L −L 󵄨󵄨 α β ̂ 1 󵄨 1 (v1 −degd 1 (α))∨0 2 (v2 −degd 2 (β))∨0 ‖ξ ‖2 (1 + ‖ξ 1 ‖1 ) (1 + ‖ξ 2 ‖2 ) . 󵄨󵄨𝜕ξ 1 𝜕ξ 2 ς(ξ , ξ2 )󵄨󵄨󵄨 ≲ ‖ξ ‖1

(6.126)

452 � 6 Besov and Triebel–Lizorkin spaces Thus, using (4.6), we have, for every α, β, L, 󵄨󵄨 α β (j,λj)⋅v ̂ −jd 1 1 −jλd 2 2 󵄨󵄨 ς(2 ξ , 2 ξ )󵄨󵄨 󵄨󵄨𝜕ξ1 𝜕ξ 2 2 1 󵄩 󵄩(v −degd 1 (α))∨0 ≲ 2j(v1 −degd 1 (α)) 2λj(v2 −degd 2 (β)) 󵄩󵄩󵄩2−jd ξ 1 󵄩󵄩󵄩1 1

2 1 2 󵄩 󵄩(v −degd 2 (β))∨0 󵄩 󵄩 −L 󵄩 󵄩 −L × 󵄩󵄩󵄩2−λjd ξ 2 󵄩󵄩󵄩2 2 (1 + 󵄩󵄩󵄩2−jd ξ 1 󵄩󵄩󵄩1 ) (1 + 󵄩󵄩󵄩2−λjd ξ 2 󵄩󵄩󵄩2 ) 󵄩 󵄩 (v −degd 1 (α))∨0 󵄩 󵄩 −L = 2j(v1 −degd 1 (α)) (2−j 󵄩󵄩󵄩ξ 1 󵄩󵄩󵄩1 ) 1 (1 + 2−j 󵄩󵄩󵄩ξ 1 󵄩󵄩󵄩1 ) 󵄩 󵄩 (v −degd 2 (β))∨0 󵄩 󵄩 −L × 2λj(v2 −degd 2 (β)) (2−λj 󵄩󵄩󵄩ξ 2 󵄩󵄩󵄩2 ) 2 (1 + 2−λj 󵄩󵄩󵄩ξ 2 󵄩󵄩󵄩2 ) 󵄩 󵄩 v −degd 1 (α) 󵄩 󵄩 v −degd 2 (β) ≲ (1 + 󵄩󵄩󵄩ξ 1 󵄩󵄩󵄩1 ) 1 (1 + 󵄩󵄩󵄩ξ 2 󵄩󵄩󵄩2 ) 2 ,

where the final estimate follows by taking L = L(v1 , v2 , α, β) large and applying (6.125) twice (once with j1 = j and ‖ξ 1 ‖1 = ‖ξ 1 ‖1 and once with j1 = λj and ‖ξ 1 ‖1 replaced by ‖ξ 2 ‖2 ). This establishes (6.124) and completes the proof. Lemma 6.7.11. If a > 0 is sufficiently small, the following holds. For v ∈ ℝ2 , if N = N(v) ∈ ℕ is sufficiently large, then we have, ∀w ∈ ℝ2 , 󵄩󵄩 v⋅(j,λj) 󵄩󵄩 Tj f 󵄩󵄩X w (W ,ds)⃗ ≲ ‖f ‖X w+v (W ,ds)⃗ , 󵄩󵄩2 0 0

∀f ∈ C0∞ (Ω),

(6.127)

where the implicit constant depends on neither j ≥ 0 nor f . ⃗ a pseudo-differential operator of order Proof. By Lemma 6.7.10, 2v⋅(j,λj) Tj is an (X, d ), v supported in Ω1 × Ω2 , uniformly for j ≥ 0. Thus, by Proposition 5.8.13, 2v⋅(j,λj) Tj ∈ ̃v (Ω, (W , ds)), ⃗ uniformly in j ≥ 0, and therefore by Proposition 6.3.10, A 󵄩󵄩 v⋅(j,λj) 󵄩󵄩 Tj 󵄩󵄩X w+v (Ω,(W ,ds))→X 󵄩󵄩2 w ⃗ ⃗ ≲ 1. 0 0 (Ω,(W ,ds))

(6.128)

∞ w+v ⃗ Thus, (6.127) follows from By Proposition 6.5.5, C0∞ (Ω) ⊆ CW (Ω, (W , ds)). ,0 (Ω) ⊆ X0 (6.128), completing the proof.

Proof of Lemma 6.7.8. The assumption of the lemma is that (6.121) holds for some ϵ1 , ϵ2 ≥ 0 with ϵ1 + ϵ2 > 0. We first claim that we may assume ϵ1 , ϵ2 > 0. Indeed, if ϵ1 = 0 and ϵ2 > 0, then by (6.119) (which we have already seen as a consequence of Theorem 6.7.2), (6.121) also holds with s̃ replaced by s̃ + ϵ′ , ϵ1 replaced by ϵ1 + λϵ′ , and ϵ2 replaced by ϵ2 − ϵ′ . Taking ϵ′ ∈ (0, ϵ2 ) shows that we may assume ϵ1 , ϵ2 > 0. Similarly, if ϵ2 = 0 and ϵ1 > 0, then by (6.119), (6.121) also holds with s̃ replaced by s̃ − ϵ′ , ϵ1 replaced by ϵ1 − λϵ′ , and ϵ2 replaced by ϵ2 + ϵ′ . Thus, taking ϵ′ > 0 so small that ϵ1 − λϵ′ > 0 and s̃ − ϵ′ > 0 shows that we may assume ϵ1 , ϵ2 > 0. Set ϵ1′ := ϵ1 /2 > 0 and ϵ2′ := ϵ2 /2 > 0. We henceforth assume (6.121) holds for some s̃ > 0 and ϵ1 , ϵ2 > 0. Fix ψ0 ∈ C0∞ (M) with ψ0 ≡ 1 on a neighborhood of Ω. Let {(Dj , 2−j ) : j ∈ ℕ2 } be a bounded set of (W , ds)⃗ elementary operators satisfying ∑j∈ℕ2 Dj = Mult[ψ0 ].

6.7 Trading derivatives

� 453

s We let V0 be to X0s as V is to X s (see Notation 6.3.2), that is, if X0s = Bp,q , then V0

q

s is ℓ (ℕ2 ; Lp (M, Vol)), and if X0s = Fp,q , then V0 is Lq (M, Vol; ℓp (ℕ2 )). By a proof very similar to Lemma 6.6.20, we have the following elementary inequalities:

󵄩󵄩 󵄩 󵄩 (ϵ′ ,ϵ′ )⋅j 󵄩 󵄩󵄩{fj }j∈ℕ2 󵄩󵄩󵄩Lp (M,Vol;ℓ1 (ℕ2 )) ≲ 󵄩󵄩󵄩{2 1 2 fj }󵄩󵄩󵄩V0 , 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩{fj }j∈ℕ2 󵄩󵄩V0 ≤ 󵄩󵄩{fj }j∈ℕ2 󵄩󵄩ℓ1 (ℕ2 ;Lp (M,Vol)) .

(6.129) (6.130)

By the formula for Tj , if a > 0 is sufficiently small, we have Tj : C0∞ (M) → C0∞ (Ω1 ). In particular, for f ∈ C0∞ (M), by Proposition 5.5.10, Tj Tj f = Mult[ψ0 ]Tj Tj f = ∑k∈ℕ Dk Tj Tj f , with convergence in C0∞ (M). By taking N = N(s, s,̃ λ, ϵ1 , ϵ2 ) ∈ ℕ sufficiently large and applying Lemma 6.7.11 and (6.121), we have, for f ∈ C0∞ (Ω2 ), 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ‖Tj Tj f ‖Lp = 󵄩󵄩󵄩 ∑ Dk Tj Tj f 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 p 󵄩k∈ℕ 󵄩L ′ ′ 󵄩󵄩 k⋅(ϵ1 ,ϵ2 ) 󵄩 ≲ 󵄩󵄩{2 Dk Tj Tj f }󵄩󵄩󵄩V 0 󵄩󵄩 󵄩󵄩 ′ ′ ≈ 󵄩󵄩Tj Tj f 󵄩󵄩 (ϵ1 ,ϵ2 ) X (W ,ds)⃗

by (6.129)

0

′ ′ ̃ −(s+(ϵ1′ ,ϵ2′ ))⋅(j,λj) 󵄩 󵄩 (s+(ϵ1 ,ϵ2 )+(−λ(s),̃ s))⋅(j,λj)

=2

󵄩󵄩2

󵄩 Tj Tj f 󵄩󵄩󵄩

(ϵ′ ,ϵ′ ) 1 2 (W ,ds) ⃗

X0

′ ′ 󵄩 󵄩 ≲ 2−(s+(ϵ1 ,ϵ2 ))⋅(j,λj) 󵄩󵄩󵄩Tj f 󵄩󵄩󵄩X s+(ϵ1 ,ϵ2 )+(−λ(s),̃ s)̃ (W ,ds)⃗

by Lemma 6.7.11

≲2

󵄩󵄩 󵄩󵄩 󵄩󵄩Tj f 󵄩󵄩X s (W ,ds)⃗ 0 ′ ′ 󵄩 󵄩 ≲ 2−(ϵ1 /2,ϵ2 /2)⋅(j,λj) 󵄩󵄩󵄩f 󵄩󵄩󵄩 (−ϵ1′ /2,−ϵ2′ /2) X (W ,ds)⃗ ′ ′ ′ ′ 󵄩 󵄩 ≈ 2−(ϵ1 /2,ϵ2 /2)⋅(j,λj) 󵄩󵄩󵄩{2−(ϵ1 /2,ϵ2 /2)⋅k Dk f }k∈ℕ2 󵄩󵄩󵄩V

by (6.121)

≤2

by (6.130)

0

−(s+(ϵ1′ ,ϵ2′ ))⋅(j,λj)

by Lemma 6.7.11

0

−(ϵ1′ /2,ϵ2′ /2)⋅(j,λj)

∑ 2

−(ϵ1′ /2,ϵ2′ /2)⋅k

k∈ℕ2

‖Dk f ‖Lp

≲ 2−(ϵ1 /2,ϵ2 /2)⋅(j,λj) ∑ 2−(ϵ1 /2,ϵ2 /2)⋅k ‖f ‖Lp ′

′

′

by Corollary 5.4.11

′

k∈ℕ2

≲ 2−(ϵ1 /2,ϵ2 /2)⋅(j,λj) ‖f ‖Lp . ′

′

⃗ a elementary operator supported in Ω1 × Ω2 and the above equaSince Tj is an (X, d ),

tion holds for all f ∈ C0∞ (Ω2 ), we conclude that ‖Tj Tj ‖Lp →Lp ≲ 2−(ϵ1 /2,ϵ2 /2)⋅(j,λj) . The result follows. ′

′

We now turn to the proof of Lemma 6.7.9, where the sharpness of λ = λ(x, (Y , d̂), (Z, dr)) is essential. Recall that we have chosen (X12 , d12 ) = (V , dr), where (V , dr) ∈ Gen((Z, dr)) is as in Definition 6.7.4 with U = Ω0 . Thus, there is a point x0 ∈ Ω0 such that X12 (x0 ) ∈ ̸ span{Xk1 (x0 ) : dk1 < λd12 }.

(6.131)

454 � 6 Besov and Triebel–Lizorkin spaces For δ ∈ (0, 1], let Φx,δ : Bn (1) → Ω be the map from Theorem 3.3.7 when applied with 𝒦 = {x0 }, Ω1 = Ω1 , (W , ds) = (Y , d̂), and (X, d ) = (X 1 , d 1 ). Lemma 6.7.12. For every L ∈ ℕ, we have 1 󵄩 󵄩 sup 󵄩󵄩󵄩Φ∗x0 ,δ δdj Xj1 󵄩󵄩󵄩C L (Bn (1);ℝn ) < ∞,

1 ≤ j ≤ q1 ,

(6.132)

2 󵄩 󵄩 sup 󵄩󵄩󵄩Φ∗x0 ,δ δλdj Xj2 󵄩󵄩󵄩C L (Bn (1);ℝn ) < ∞,

1 ≤ j ≤ q2 .

(6.133)

δ∈(0,1] δ∈(0,1]

Furthermore, there exists a sequence δl ↓ 0 such that d1

l→∞

Φ∗x0 ,δl δl j Xj1 󳨀󳨀󳨀󳨀→ Yj1 ∈ C ∞ (Bn (1); ℝn ),

1 ≤ j ≤ q1 ,

(6.134)

Yj2 ∈ C ∞ (Bn (1); ℝn ),

1 ≤ j ≤ q2 ,

(6.135)

λd 2 l→∞ Φ∗x0 ,δl δl j Xj2 󳨀󳨀󳨀󳨀→

with convergence in C ∞ (Bn (1); ℝn ). Finally, we have Y12 (0) ≠ 0, and by possibly reordering (X 1 , d 1 ), we also have Y11 (0) ≠ 0. Proof. The estimate (6.132) is Theorem 3.3.7 (j). Next, we establish (6.133). By the choice of Ω, (Y , d̂) weakly λ-controls (Z, dr) on Ω. Thus, Lemma 3.14.9 (v) and (vi) implies (X 1 , d 1 ) strongly λ-controls (X 2 , d 2 ) on Ω. In other words, for 1 ≤ j ≤ q2 , we may write Xj2 = ∑ cjk Xk1 ,

∞ cjk ∈ Cloc (Ω).

dk1 ≤λdj2

Thus, for δ ∈ (0, 1] and 1 ≤ j ≤ q2 , we have 2

2

1

1

Φ∗x0 ,δ δλdj Xj2 = ∑ δλdj −dj (cjk ∘ Φx0 ,δ )Φ∗x0 ,δ δdk Xk1 . dk1 ≤λdj2

(6.136)

In light of (6.132) and (6.136), to prove (6.133) it suffices to show that for every L ∈ ℕ, 󵄩 󵄩 sup 󵄩󵄩󵄩cjk ∘ Φx0 ,δ 󵄩󵄩󵄩C L (Bn (1);ℝn ) < ∞,

δ∈(0,1]

1 ≤ j ≤ q2 , 1 ≤ k ≤ q1 .

(6.137)

By Theorem 3.3.7 (m), (6.137) is equivalent to α 󵄩 󵄩 sup ∑ 󵄩󵄩󵄩(Φ∗x0 ,δ δd1 X 1 ) cjk ∘ Φx0 ,δ 󵄩󵄩󵄩C(Bn (1);ℝn ) < ∞.

δ∈(0,1] |α|≤L

(6.138)

But (Φ∗x0 ,δ δd1 X 1 )α cjk ∘ Φx0 ,δ = ((δd1 X 1 )α cjk ) ∘ Φx0 ,δ . Thus, (6.138) is equivalent to α 󵄩 󵄩 1 sup ∑ 󵄩󵄩󵄩(δd X 1 ) cjk 󵄩󵄩󵄩C(Φ (Bn (1));ℝn ) < ∞. x0 ,δ

δ∈(0,1] |α|≤L

(6.139)

6.7 Trading derivatives

� 455

Theorem 3.3.7 shows that Φx0 ,δ (Bn (1)) ⊂ Ω1 , and therefore, using δ ∈ (0, 1], α 󵄩 α 󵄩 󵄩 1 󵄩 sup ∑ 󵄩󵄩󵄩(δd X 1 ) cjk 󵄩󵄩󵄩C(Φ (Bn (1));ℝn ) ≤ ∑ 󵄩󵄩󵄩(X 1 ) cjk 󵄩󵄩󵄩C(Ω ;ℝn ) < ∞, x0 ,δ 1

δ∈(0,1] |α|≤L

|α|≤L

∞ where the < ∞ follows from the fact that cjk ∈ Cloc (Ω) and Ω1 ⋐ Ω. This establishes (6.139) and completes the proof of (6.133). By (6.132) and (6.133), 1

2

{Φ∗x0 ,δ δdk Xk1 , Φ∗x0 ,δ δλdj Xj2 : δ ∈ (0, 1], 1 ≤ k ≤ q1 , 1 ≤ j ≤ q2 } ⊂ C ∞ (Bn (1); ℝn ) is a bounded set and therefore relatively compact. Thus, there exists a sequence δl ↓ 0 such that (6.134) and (6.135) hold. λd 2

l→∞

We claim Y12 (0) ≠ 0. For contradiction, suppose Y12 (0) = 0. Thus, Φ∗x0 ,δ δl 1 X12 (0) 󳨀󳨀󳨀󳨀→ 0. By Theorem 3.3.7 (k) and (j), for every ϵ > 0, there exists L ∈ ℕ such that l ≥ L implies λd12

(Φ∗x0 ,δl δl

q1

d1

X12 )(0) = ∑ alϵ,k (Φ∗x0 ,δ δl k Xk1 )(0), k=1

|alk,ϵ | < ϵ.

(6.140)

Since Φx0 ,δl (0) = x0 , (6.140) is equivalent to λd12

δl

q1

d1

X12 (x0 ) = ∑ alϵ,k δl k Xk1 (x0 ). k=1

We therefore have d 1 −λd12

X12 (x0 ) − ∑ alϵ,k δl k λd12 ≤dk1

Xk1 (x0 ) ∈ span{Xk (x0 ) : dk1 < λd12 }.

Taking ϵ ↓ 0 and l → ∞ and using the fact that |alϵ,k | < ϵ and that finite-dimensional vector spaces are closed, we see that X12 (x0 ) ∈ span{Xk1 (x0 ) : dk1 < λd12 }. This contradicts (6.131) and completes the proof that Y12 (0) ≠ 0. Finally, we claim that, by possibly reordering (X 1 , d 1 ), we have Y11 (0) ≠ 0. Indeed, by taking liml→∞ of Theorem 3.3.7 (k), we see max

k1 ,...,kn ∈{1,...,q1 }

󵄨󵄨 󵄨 1 1 󵄨󵄨det(Yk1 (0)| ⋅ ⋅ ⋅ |Ykn (0))󵄨󵄨󵄨 > 0.

In particular, Yk1 (0) ≠ 0 for some k ∈ {1, . . . , q1 }. Thus, by possibly reordering (X 1 , d 1 ) to switch (Xk1 , dk1 ) and (X11 , d11 ), we have Y11 (0) ≠ 0.

456 � 6 Besov and Triebel–Lizorkin spaces For the rest of this section, we take δl ↓ 0 and Y 1 = (Y11 , . . . , Yq11 ) and Y 2 = (Y12 , . . . , Yq22 )

as in Lemma 6.7.12. In particular, Y11 (0) ≠ 0 and Y12 (0) ≠ 0. Given a > 0 small and N ∈ ℕ, pick f0 ∈ C0∞ (Bn (1/2)) and ς ∈ C0∞ (Bq (a)) satisfying (6.122) such that 1

1

∫ f0 (e−t ⋅Y e−t

2

⋅Y 2 −t 3 ⋅Y 1 −t 4 ⋅Y 2

e

e

0)ς(t 1 , t 2 )ς(t 3 , t 4 ) dt 2 dt 2 dt 3 dt 4 ≠ 0.

Since Y11 (0) ≠ 0 and Y12 (0) ≠ 0, this is always possible. Note that by continuity, we have, for u in some open neighborhood of 0 ∈ ℝn , 1

1

∫ f0 (e−t ⋅Y e−t

2

⋅Y 2 −t 3 ⋅Y 1 −t 4 ⋅Y 2

e

e

u)ς(t 1 , t 2 )ς(t 3 , t 4 ) dt 2 dt 2 dt 3 dt 4 ≠ 0.

(6.141)

Take ψ ∈ C0∞ (Ω1 ) with ψ ≡ 1 on a neighborhood of x0 . Define Tj as in (6.123) with this choice of ψ and ς. We will show that lim supj→∞ ‖Tj Tj ‖Lp →Lp > 0, which will prove Lemma 6.7.9, since a > 0 and N ∈ ℕ were arbitrary. Define jl ↑ ∞ by 2−jl = δl . Set Fl := Φ∗x0 ,δl Tjl Tjl (Φx0 ,δl )∗ . Lemma 6.7.13. For every l ∈ ℕ, ‖Fl ‖Lp (Bn (1),σLeb )→Lp (Bn (1),σLeb ) ≲ ‖Tjl Tjl ‖Lp (M,Vol)→Lp (M,Vol) , where σLeb denotes the usual Lebesgue density on ℝn . Proof. Write Sl := Tjl Tjl . Let hx0 ,δl ∈ C ∞ (Bn (1)) be as in Theorem 3.3.7 (n), so that Φ∗x0 ,δl Vol = Λ(x0 , δl )hx0 ,δl σLeb , where Λ(x0 , δl ) is as in Theorem 3.3.7 (n). In particular, by Theorem 3.3.7 (n), we have hx0 ,δ ≈ 1. Note that for g ∈ C0∞ (Bn (1)), we have supp((Φx0 ,δ )∗ g) ⊆ Φx0 ,δl (Bn (1)). Thus, we have, for g ∈ C0∞ (Bn (1)), 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩Fl g 󵄩󵄩Lp (Bn (1),σLeb ) ≈ 󵄩󵄩󵄩Fl g 󵄩󵄩󵄩Lp (Bn (1),hx ,δ σLeb ) 0 󵄩 󵄩 = Λ(x0 , δl )−1/p 󵄩󵄩󵄩Fl g 󵄩󵄩󵄩Lp (Bn (1),Φ∗ Vol) x0 ,δl 󵄩 ∗ 󵄩 −1/p 󵄩 = Λ(x0 , δl ) 󵄩󵄩Φx0 ,δl Sl (Φx0 ,δl )∗ g 󵄩󵄩󵄩Lp (Bn (1),Φ∗ Vol) x0 ,δl 󵄩 󵄩 −1/p 󵄩 󵄩 = Λ(x0 , δl ) 󵄩󵄩Sl (Φx0 ,δl )∗ g 󵄩󵄩Lp (Φ (Bn (1)),Vol) x0 ,δ 󵄩 󵄩 −1/p p p ≤ Λ(x0 , δl ) ‖Sl ‖L (M,Vol)→L (M,Vol) 󵄩󵄩󵄩(Φx0 ,δl )∗ g 󵄩󵄩󵄩Lp (M,Vol) 󵄩 󵄩 = Λ(x0 , δl )−1/p ‖Sl ‖Lp (M,Vol)→Lp (M,Vol) 󵄩󵄩󵄩(Φx0 ,δl )∗ g 󵄩󵄩󵄩Lp (Φ (Bn (1)),Vol) x0 ,δ 󵄩󵄩 󵄩󵄩 −1/p p p = Λ(x0 , δl ) ‖Sl ‖L (M,Vol)→L (M,Vol) 󵄩󵄩g 󵄩󵄩Lp (Bn (1),Λ(x ,δ )h σ ) 0 l x0 ,δl Leb 󵄩 󵄩 = ‖Sl ‖Lp (M,Vol)→Lp (M,Vol) 󵄩󵄩󵄩g 󵄩󵄩󵄩Lp (Bn (1),h σ ) x0 ,δl Leb 󵄩 󵄩 󵄩 󵄩 ≈ ‖Sl ‖Lp (M,Vol)→Lp (M,Vol) 󵄩󵄩g 󵄩󵄩Lp (Bn (1),σ ) . Leb The result follows.

6.7 Trading derivatives

�

457

Lemma 6.7.14. The limit liml→∞ Fl f0 (u) exists (pointwise) for all u ∈ Bn (1) and is nonzero for u in a neighborhood of 0. Here, f0 ∈ C0∞ (Bn (1/2)) is the particular function chosen so that (6.141) holds. Proof. Consider, for u ∈ Bn (1), 1

1

Fl f0 (u) = ∫f0 ∘ Φx0 ,δ∗ (e−t ⋅X e−t l

ψ(Φx0 ,δl (u))ψ(e−t

3

2

⋅X 2 −t 3 ⋅X 1 −t 4 ⋅X 2

e

e

⋅X 1 −t 4 ⋅X 2

e

Φx0 ,δl (u))

Φx0 ,δl (u)) Dild2(jl ,λjl ) (ς)(t 1 , t 2 ) ⃗

(6.142)

Dild2(jl ,λjl ) (ς)(t 3 , t 4 ) dt 1 dt 2 dt 3 dt 4 . ⃗

By Theorem 3.3.7, Φx0 ,δl (Bn (1)) ⊆ B(X,d) (x0 , δl ). Therefore, for any neighborhood V of x0 , since δl ↓ 0, if l is sufficiently large, Φx0 ,δl (Bn (1)) ⊆ V ; this uses Lemma 3.1.7. Since ψ ≡ 1 on a neighborhood of x0 , this shows that for l sufficiently large, ψ(Φx0 ,δl (u)) ≡ 1. Similarly, since jl ↑ ∞, for l large the integral in (6.142) is only over t 3 and t 4 small. 3 1 4 2 Thus, for l sufficiently large, we also have ψ(e−t ⋅X e−t ⋅X Φx0 ,δl (u)) ≡ 1 on the domain of integration of (6.142). Thus, for l large, we have (using δl = 2−jl ) 1

1

−t ⋅X −t Fl f0 (u) = ∫ f0 ∘ Φ−1 e x0 ,δ (e

2

⋅X 2 −t 3 ⋅X 1 −t 4 ⋅X 2

e

e

Φx0 ,δl (u))

Dild2(jl ,λjl ) (ς)(t 1 , t 2 ) Dild2(jl ,λjl ) (ς)(t 3 , t 4 ) dt 1 dt 2 dt 3 dt 4 ⃗

⃗

1

−t ⋅2 = ∫ f0 ∘ Φ−1 x0 ,δ (e

−d 1 jl

2

1

2

X 1 −t 2 ⋅2−λd jl X 2 −t 3 ⋅2−d jl X 1 −t 4 ⋅2−λd jl X 2

e

e

e

Φx0 ,δl (u))

ς(t 1 , t 2 )ς(t 3 , t 4 ) dt 1 dt 2 dt 3 dt 4 = ∫ f0 (e

1

2

1

2

−t 1 ⋅Φ∗x,δ δld X 1 −t 2 ⋅Φ∗x,δ δlλd X 2 −t 3 ⋅Φ∗x,δ δld X 1 −t 4 ⋅Φ∗x,δ δlλd X 2 l

e

l

e

l

e

l

u)

ς(t 1 , t 2 )ς(t 3 , t 4 ) dt 1 dt 2 dt 3 dt 4 l→∞

1

1

󳨀󳨀󳨀󳨀→ ∫ f0 (e−t ⋅Y e−t

2

⋅Y 2 −t 3 ⋅Y 1 −t 4 ⋅Y 2

e

e

u)ς(t 1 , t 2 )ς(t 3 , t 4 ) dt 1 dt 2 dt 3 dt 4 ,

where the last line follows from Lemma 6.7.12. Thus the limit exists pointwise, and by (6.141) is non-zero for u in a neighborhood of 0. Proof of Lemma 6.7.9. Suppose, for contradiction, that limj→∞ ‖Tj Tj ‖Lp (M,Vol)→Lp (M,Vol) = 0. Then by Lemma 6.7.13, liml→∞ ‖Fl ‖Lp (Bn (1),σLeb )→Lp (Bn (1),σLeb ) = 0. In particular, k→∞

liml→∞ ‖Fl f0 ‖Lp (Bn (1),σLeb ) = 0. Thus, there is a subsequence lk → ∞ such that Flk f0 󳨀󳨀󳨀󳨀󳨀→ 0, pointwise almost everywhere. This contradicts Lemma 6.7.14 and completes the proof.

458 � 6 Besov and Triebel–Lizorkin spaces

6.8 Adding parameters Suppose ν ≥ 2. Set (Y μ , d̂μ ) := (W μ , dsμ ),

1 ≤ μ ≤ ν − 1,

(Y , d̂)⃗ = {(Y1 , d̂1⃗ ), . . . , (YvY , d̂v⃗ Y )} := (Y 1 , d̂1 ) ⊠ (Y 2 , d̂2 ) ⊠ ⋅ ⋅ ⋅ (Y ν−1 , d̂ν−1 ) ∞ ⊂ Cloc (M; TM) × (ℕν−1 \ {0}).

⃗ but with ν replaced by ν−1. For s ∈ ℝν , (Y , d̂)⃗ satisfies all the same assumptions as (W , ds), s ⃗ and X (s,0) (𝒦, (W , ds)). ⃗ These spaces we have two different scales of spaces: X (𝒦, (Y , d̂)) ∙ ∙ are similar, but unless X = Fp,2 , they are not the same (for a discussion of the case ∙ X ∙ = Fp,2 , see Proposition 6.8.2). Fix a compact set 𝒦 ⋐ M and s ∈ ℝν−1 . The main results of this section are the following. Proposition 6.8.1. For every ϵ > 0, we have X

(s,ϵ)

⃗ ⊆ X (s,−ϵ) (𝒦, (W , ds)), ⃗ ⊆ X s (𝒦, (Y , d̂)) ⃗ (𝒦, (W , ds))

and the inclusions are continuous. Proposition 6.8.2. For 1 < p < ∞, (s,0)

s

⃗ ⃗ = F (𝒦, (Y , d̂)), Fp,2 (𝒦, (W , ds)) p,2 with equality of topologies. We begin with the proof of Proposition 6.8.1, for which we need to introduce some notation and preliminary lemmas. ′ Fix ψ′ , ψ ∈ C0∞ (M), with ψ′ , ψ ≡ 1 on a neighborhood of 𝒦. Let 𝒟0′ = {(D′j′ , 2−j ) : j′ ∈ ℕν−1 } be a bounded set of generalized (Y , d̂)⃗ elementary operators with ∑ ′ ν−1 D′′ = j ∈ℕ

j

Mult[ψ′ ], and let 𝒟0 = {(Dj , 2−j ) : j ∈ ℕν } be a bounded set of generalized (W , ds)⃗ elementary operators with ∑j∈ℕν Dj = Mult[ψ]. Given j ∈ ℕν , we write j = (j′ , jν ) ∈ ∙ ℕν−1 × ℕ. As in Notation 6.3.2, if X ∙ = Bp,q , we set V = ℓq (ℕν ; Lp (M, Vol)) and

∙ V ′ = ℓq (ℕν−1 ; Lp (M, Vol)). If X ∙ = Fp,q , we set V = Lp (M, Vol; ℓq (ℕν )) and V ′ =

Lp (M, Vol; ℓq (ℕν−1 )). Throughout this section, we fix p, q ∈ [1, ∞] as above. Lemma 6.8.3. Let ϵ > 0. (I) For {fj }j∈ℕν ∈ V , 󵄩 󵄩 󵄩 󵄩 ∑ 2−ϵjν 󵄩󵄩󵄩{fj }j′ ∈ℕν−1 󵄩󵄩󵄩V ′ ≲ϵ 󵄩󵄩󵄩{fj }j∈ℕν 󵄩󵄩󵄩V .

jν ∈ℕ

6.8 Adding parameters

� 459

(II) For {fj′ }j′ ∈ℕν−1 ∈ V ′ and ϵ > 0, 󵄩󵄩 −ϵjν ′ 󵄩 󵄩 󵄩 󵄩󵄩{2 fj }j∈ℕν 󵄩󵄩󵄩V ≲ϵ 󵄩󵄩󵄩{fj′ }j′ ∈ℕν−1 󵄩󵄩󵄩V ′ . Proof. (I): When V = ℓq (Lp ), (I) follows immediately. When V = Lp (ℓq ), we have 󵄩 󵄩 󵄩 󵄩 ∑ 2−ϵjν 󵄩󵄩󵄩{fj }j′ ∈ℕν−1 󵄩󵄩󵄩V ′ ≲ sup 󵄩󵄩󵄩{fj }j′ ∈ℕν−1 󵄩󵄩󵄩V ′ jν ∈ℕ

j∈ℕν

󵄩󵄩 󵄩󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩{sup |fj |} ′ ν−1 󵄩󵄩󵄩 ′ ≤ 󵄩󵄩󵄩{fj }j∈ℕν 󵄩󵄩󵄩V , 󵄩 j ∈ℕ j ∈ℕ 󵄩V ν

as desired. (II): When V = Lp (ℓq ), (II) follows immediately. When V = ℓq (Lp ), we have 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 −ϵjν ′ 󵄩 −ϵj 󵄩󵄩 󵄩󵄩{2 fj }j∈ℕν 󵄩󵄩󵄩V ≤ 󵄩󵄩󵄩󵄩{ ∑ 2 ν |fj′ |} 󵄩󵄩 󵄩󵄩 j ∈ℕ ′ ν−1 j ∈ℕ 󵄩 󵄩 ν 󵄩V ′ 󵄩 󵄩 󵄩 −ϵjν 󵄩 󵄩 󵄩 ≤ ∑ 2 󵄩󵄩{fj′ }j′ ∈ℕν−1 󵄩󵄩V ′ ≲ 󵄩󵄩󵄩{fj′ }j′ ∈ℕν−1 󵄩󵄩󵄩V ′ , jν ∈ℕ

as desired. Lemma 6.8.4. Let ϵ > 0. (I) Let ℰ ′ be a bounded set of generalized (W , ds)⃗ pre-elementary operators. Then

{(E ′′ ,2 j

−j′

󵄩 󵄩 󵄩 󵄩 ∑ 󵄩󵄩󵄩{2−jν ϵ Ej′′ fj }j′ ∈ℕν−1 󵄩󵄩󵄩V ′ ≲ϵ 󵄩󵄩󵄩{fj }j∈ℕν 󵄩󵄩󵄩V ,

sup

):j′ ∈ℕν−1 }⊆ℰ ′ jν ∈ℕ

for all {fj }j∈ℕν ∈ V . (II) Let ℰ be a bounded set of generalized (W , ds)⃗ pre-elementary operators. Then sup

{(Ej

,2−j ):j∈ℕν }⊆ℰ

󵄩󵄩 −jν ϵ ′ 󵄩 󵄩 󵄩 󵄩󵄩{2 Ej fj }j∈ℕν 󵄩󵄩󵄩V ≲ϵ 󵄩󵄩󵄩{fj′ }j′ ∈ℕν−1 󵄩󵄩󵄩V ′ ,

for all {fj′ }j′ ∈ℕν−1 ∈ V ′ . Proof. (I): Using Definition 5.2.26 and Lemma 6.8.3 (I), we have

{(E ′′ ,2 j

−j′

sup

󵄩 󵄩 ∑ 󵄩󵄩󵄩{2−jν ϵ Ej′′ fj }j′ ∈ℕν−1 󵄩󵄩󵄩V ′

):j′ ∈ℕν−1 }⊆ℰ ′ jν ∈ℕ

≤ ∑

jν ∈ℕ

sup

′ {(E ′′ ,2−j ):j′ ∈ℕν−1 }⊆ℰ ′ j

󵄩󵄩 −jν ϵ ′ 󵄩 󵄩󵄩{2 Ej′ fj }j′ ∈ℕν−1 󵄩󵄩󵄩V ′

󵄩 󵄩 󵄩 󵄩 ≲ ∑ 󵄩󵄩󵄩{2−jν ϵ fj }j′ ∈ℕν−1 󵄩󵄩󵄩V ′ ≲ 󵄩󵄩󵄩{fj }j∈ℕν 󵄩󵄩󵄩V . jν ∈ℕ

460 � 6 Besov and Triebel–Lizorkin spaces (II): Set gj := 2−jν ϵ fj′ . Then by Lemma 6.8.3 (II), {gj }j∈ℕν ∈ V and ‖{gj }j∈ℕν ‖V ≲ ‖{fj′ }j′ ∈ℕν−1 ‖V ′ . Using this and Definition 5.2.26, we have sup

{(Ej ,2−j ):j∈ℕν }⊆ℰ

󵄩󵄩 −jν ϵ ′ 󵄩 󵄩󵄩{2 Ej fj }j∈ℕν 󵄩󵄩󵄩V

sup

{(Ej ,2−j ):j∈ℕν }⊆ℰ

󵄩󵄩 󵄩 󵄩󵄩{Ej gj }j∈ℕν 󵄩󵄩󵄩V

󵄩 󵄩 ≲ 󵄩󵄩󵄩{gj }j∈ℕν 󵄩󵄩󵄩V ≲ ‖{fj′ }j′ ∈ℕν−1 ‖V ′ . ⃗ ⊆ C ∞ (M)′ and ∀f ∈ X (s,ϵ) (𝒦, (W , ds)), ⃗ Lemma 6.8.5. For all ϵ > 0, X (s,ϵ) (𝒦, (W , ds)) Y ,0 ∞ ′ ∑j∈ℕν Dj f = f with convergence in CY ,0 (M) . ⃗ ⊆ C ∞ (M)′ and Proposition 5.5.10 Proof. A priori, Definition 6.3.5 gives X (s,ϵ) (𝒦, (W , ds)) W ,0 shows f = Mult[ψ]f = ∑ Dj f , j∈ℕν

⃗ ∀f ∈ X (s,ϵ) (𝒦, (W , ds)),

(6.143)

∞ ′ with convergence in CW ,0 (M) . We will use the fact that ϵ > 0 to show that the sum ∞ in (6.143) converges in CY ,0 (M)′ , which will complete the proof. Fix g ∈ CY∞,0 (M) and let N = N(s) ∈ ℕ be so large that |j′ |∞ N ≥ |j′ ⋅ s| for every ′ j ∈ ℕν−1 . We will show that, for all j ∈ ℕν ,

󵄨󵄨 󵄨 α −j′ ⋅s−|j′ |∞ N−jν ϵ ‖f ‖X (s,ϵ) (W ,ds)⃗ . 󵄨󵄨⟨g, Dj f ⟩󵄨󵄨󵄨 ≲ ∑ ‖Y g‖Lq 2

(6.144)

|α|≤N

Once we show (6.144), it will immediately follow that the sum in (6.143) converges in CY∞,0 (M)′ , completing the proof. For j ∈ ℕν , let μ ∈ {1, . . . , ν − 1} be such that |j′ |∞ = jμ . Using Proposition 5.5.5 (h), we can write μ

α

Dj = ∑ 2(|αμ |−N)jμ (2−jμ ds W μ ) μ Ej,μ,αμ |αμ |≤N μ

α

= ∑ 2(|αμ |−N)jμ (2−jμ d̂ Y μ ) μ Ej,μ,αμ , ′

|αμ |≤N

where {(Ej,μ,αμ , 2−j ) : j ∈ ℕν , μ ∈ {1, . . . , ν − 1}, |αμ | ≤ N} is a bounded set of generalized ⃗ it follows from Corollary 6.4.4 (W , ds)⃗ elementary operators. Since f ∈ X (s,ϵ) (𝒦, (W , ds)), 󵄩󵄩 󵄩󵄩 −j⋅(s,ϵ) that 󵄩󵄩Ejμ ,μ,αμ f 󵄩󵄩Lp (M,Vol) ≲ 2 ‖f ‖X (s,ϵ) (W ,ds)⃗ . Thus, 󵄨󵄨 󵄨 󵄨 (|α |−N)jμ 󵄨󵄨 −j′ d̂μ μ αμ 󵄨󵄨⟨g, Dj f ⟩󵄨󵄨󵄨 ≤ ∑ 2 μ 󵄨󵄨⟨g, (2 μ Y ) Ej,μ,αμ f ⟩󵄨󵄨󵄨 |αμ |≤N

′ μ α ∗ 󵄨 󵄨 = ∑ 2(|αμ |−N−degd̂μ (αμ ))jμ 󵄨󵄨󵄨⟨((2−jμ d̂ Y μ ) μ ) g, Ej,μ,αμ f ⟩󵄨󵄨󵄨

|αμ |≤N

6.8 Adding parameters

�

461

β 󵄩 󵄩 󵄩 󵄩 ≲ 2−Njμ ( ∑ 󵄩󵄩󵄩(Y μ ) μ g 󵄩󵄩󵄩Lq )( max 󵄩󵄩󵄩Ej,μ,αμ f 󵄩󵄩󵄩Lp ) |α |≤N μ

|βμ |≤N

β 󵄩 󵄩 ≲ 2−N|j |∞ 2−j ⋅s−jν ϵ ∑ 󵄩󵄩󵄩(Y μ ) μ g 󵄩󵄩󵄩Lq ‖f ‖X (s,ϵ) (W ,ds)⃗ , ′

′

|βμ |≤N

establishing (6.144) and completing the proof. Lemma 6.8.6. Let ℰ be a bounded set of generalized (W , ds)⃗ elementary operators and let ℰ ′ be a bounded set of generalized (Y , d̂)⃗ elementary operators. Then, for every N ∈ ℕ, ′ there exists L = L(N) ∈ ℕ such that ∀(Ej , 2−j ) ∈ ℰ , (Ek′ ′ , 2−k ) ∈ ℰ ′ , L

Ej Ek′ ′ = ∑ 2−N|k −j | Fj,K Fk′ ′ ,K ′

(6.145)

′

K=1

and L

Ek′ ′ Ej = ∑ 2−N|k −j | F̃k′ ′ ,K F̃j,K , ′

(6.146)

′

K=1

where {(Fj,K , 2−j ), (F̃j,K , 2−j ) : (Ej , 2−j ) ∈ ℰ , (Ek′ ′ , 2−k ) ∈ ℰ ′ , 1 ≤ K ≤ L} ′

is a bounded set of generalized (W , ds)⃗ elementary operators and {(Fk′ ′ ,K , 2−k ), (F̃k′ ′ ,K , 2−k ) : (Ej , 2−j ) ∈ ℰ , (Ek′ ′ , 2−k ) ∈ ℰ ′ , 1 ≤ K ≤ L} ′

′

′

is a bounded set of generalized (Y , d̂)⃗ elementary operators. Proof. We prove only (6.145); the proof of (6.146) is similar and we leave it to the reader. ′ ′ ′ ′ We will prove (6.145) with 2−N|k −j | replaced by 2−N|k −j |∞ ; the result will then follow by increasing N and using Proposition 5.5.5 (a). Pick μ ∈ {1, . . . , ν − 1} such that |jμ′ − kμ′ | = |j′ − k ′ |∞ . When kμ′ ≥ jμ′ , we apply Proposition 5.5.5 (h) to Ek′ ′ to see that μ

α

Ek′ ′ = ∑ 2(|αμ |−N)kμ (2−kμ d̂ Y μ ) μ Fk′ ′ ,αμ ′

′

|αμ |≤N

μ

α

= ∑ 2(|αμ |−N)kμ (2−kμ ds W μ ) μ Fk′ ′ ,αμ ′

′

|αμ |≤N

where {(Fk′ ′ ,α , 2−k ) : (Ek′ ′ , 2−k ) ∈ ℰ ′ } is a bounded set of (Y , d̂)⃗ elementary operators ′

μ

′

and we have used (W μ , dsμ ) = (Y μ , d̂μ ) for 1 ≤ μ ≤ ν − 1. Thus, we have

462 � 6 Besov and Triebel–Lizorkin spaces μ

α

Ej Ek′ ′ = 2−N(kμ −jμ ) ∑ [2−Njμ +(|αμ |−degdsμ (αμ ))(kμ −jμ ) Ej (2−jμ ds W μ ) μ ]Fk′ ′ ,αμ ′

′

′

′

′

′

|αμ |≤N

=: 2

−N|k ′ −j′ |∞

∑ Fj,αμ Fk′ ′ ,αμ ,

|αμ |≤N

where by Proposition 5.5.5 (e) and (a), {(Fj,αμ , 2−j ) : (Ej , 2−j ) ∈ ℰ , (Ek′ ′ , 2−k ) ∈ ℰ ′ , |αμ | ≤ N} is a bounded set of generalized (W , ds)⃗ elementary operators. This shows that Ej E ′ ′ is of ′

k

the form (6.145). If kμ′ < jμ′ , then a similar proof works where we apply Proposition 5.5.5 (h) to Ej and Proposition 5.5.5 (e) and (a) to Ek′ ′ ; we leave the details to the interested reader.

Proof of Proposition 6.8.1. In the proof which follows, for an operator indexed by j ∈ ℕν , Ej (for example, Ej = Dj ), we take the convention Ej = 0 for j ∈ ℤν \ {0}. We use a similar convention for operators indexed by j′ ∈ ℕν−1 . ⃗ Let ℰ ′ be a bounded ⃗ 󳨅→ X s (𝒦, (Y , d̂)). We begin with the inclusion X (s,ϵ) (𝒦, (W , ds)) (s,ϵ) set of generalized (Y , d̂)⃗ elementary operators and fix f ∈ X . We will show that ‖f ‖V ′ ,s,ℰ ′ ≲ ‖f ‖X (s,ϵ) (W ,ds)⃗ .

(6.147)

Lemma 6.8.5 shows that f ∈ CY∞,0 (M)′ , and therefore the left-hand side of (6.147) is de⃗ Morefined. Since ℰ ′ was arbitrary and supp(f ) ⊆ 𝒦, (6.147) implies f ∈ X s (𝒦, (Y , d̂)). ′ ′ over, by taking ℰ = 𝒟0 , (6.147) implies ‖f ‖X s (Y ,d̂)⃗ ≲ ‖f ‖X (s,ϵ) (W ,ds)⃗ , proving that the ⃗ is continuous. ⃗ 󳨅→ X s (𝒦, (Y , d̂)) inclusion X (s,ϵ) (𝒦, (W , ds)) (s,ϵ) ⃗ it suffices to ⃗ 󳨅→ X s (𝒦, (Y , d̂)), Thus, to prove the inclusion X (𝒦, (W , ds)) prove (6.147). Fix {(Ej′ , 2−j ) : j′ ∈ ℕν−1 } ⊆ ℰ . We will show that ′

󵄩󵄩 −j′ ⋅s ′ 󵄩 󵄩󵄩{2 Ej′ f }j′ ∈ℕν−1 󵄩󵄩󵄩V ′ ≲ ‖f ‖X (s,ϵ) (W ,ds)⃗ .

(6.148)

Taking the supremum of (6.148) over {(Ej′ , 2−j ) : j′ ∈ ℕν−1 } ⊆ ℰ establishes (6.147). Thus, we turn to establishing (6.148). Lemma 6.8.5 shows that f = ∑j∈ℕν Dj f with convergence in CY∞,0 (M)′ , so ′

󵄩󵄩 󵄩󵄩 󵄩󵄩 −j′ ⋅s ′ 󵄩󵄩 󵄩󵄩 −j′ ⋅s ′ 󵄩 󵄩󵄩 󵄩󵄩{2 Ej′ f }j′ ∈ℕν−1 󵄩󵄩󵄩V ′ = 󵄩󵄩󵄩󵄩{2 Ej′ ∑ ∑ D(j′ +l,jν ) f } 󵄩󵄩 󵄩󵄩 󵄩 ′ ν−1 ′ ν−1 jν ∈ℕ l∈ℤ j ∈ℕ 󵄩 󵄩 󵄩V ′ 󵄩󵄩 −j ⋅s ′ ′ 󵄩󵄩 ≤ ∑ ∑ 󵄩󵄩{2 Ej′ D(j +l,jν ) f }j′ ∈ℕν−1 󵄩󵄩V ′ .

(6.149)

l∈ℤν−1 jν ∈ℕ

Take N > |s|. By Lemma 6.8.6, we may write (with K ≈ 1) K

Ej′′ D(j′ +l,jν ) = ∑ 2−N|l| Fj′′ ,L F(j′ +l,jν ),L , L=1

(6.150)

6.8 Adding parameters

� 463

where {(Fj′′ ,L , 2−j ) : (Ej′′ , 2−j ) ∈ ℰ ′ , jν ∈ ℕ, l ∈ ℤν−1 , 1 ≤ L ≤ K} is a bounded set of ′ generalized (Y , d̂)⃗ elementary operators and {(Fk,L , 2−k ) : (E ′′ , 2−j ) ∈ ℰ ′ , k ∈ ℕν , 1 ≤ L ≤ ′

′

j

K} is a bounded set of generalized (W , ds)⃗ elementary operators. Plugging (6.150) into (6.149), we see that 󵄩󵄩 −j′ ⋅s ′ 󵄩 󵄩󵄩{2 Ej′ f }j′ ∈ℕν−1 󵄩󵄩󵄩V ′ K

′ ′ 󵄩 󵄩 ≤ ∑ ∑ ∑ 2−N|l|−s⋅l 󵄩󵄩󵄩{2−jν ϵ Fj′′ ,L 2(s,ϵ)⋅(j +l ,jν ) F(j′ +l′ ,jν ),L f }j′ ∈ℕν−1 󵄩󵄩󵄩V ′

l∈ℤν−1 jν ∈ℕ L=1 K

′ ′ 󵄩 󵄩 ≲ ∑ ∑ 2−N|l|−s⋅l 󵄩󵄩󵄩{2(s,ϵ)⋅(j +l ,jν ) F(j′ +l′ ,jν ),L f }j∈ℕν 󵄩󵄩󵄩V

l∈ℤν−1 L=1 K

≲ ∑ ∑ 2−N|l|−s⋅l ‖f ‖X (s,ϵ) (W ,ds)⃗

Lemma 6.8.4 (I) Corollary 6.4.4

l∈ℤν−1 L=1

≲ ‖f ‖X (s,ϵ) (W ,ds)⃗ , ⃗ 󳨅→ establishing (6.148) and completing the proof of the inclusion X (s,ϵ) (𝒦, (W , ds)) s ⃗ X (𝒦, (Y , d̂)). ⃗ 󳨅→ X (s,−ϵ) (𝒦, (W , ds)). ⃗ Let ℰ be a bounded We turn to the inclusion X s (𝒦, (Y , d̂)) ⃗ We will show set of generalized (W , ds)⃗ elementary operators and let f ∈ X s (𝒦, (Y , d̂)). that ‖f ‖V ,(s,−ϵ),ℰ ≲ ‖f ‖X s (Y ,d̂)⃗ .

(6.151)

⃗ ⊆ C ∞ (M)′ ⊆ C ∞ (M)′ , the left-hand side of (6.151) makes Since f ∈ X s (𝒦, (Y , d̂)) Y ,0 W ,0 ⃗ sense. Since ℰ was arbitrary and supp(f ) ⊆ 𝒦, (6.151) implies f ∈ X (s,−ϵ) (𝒦, (W , ds)). Moreover, by taking ℰ = 𝒟0 , (6.151) gives ‖f ‖X (s,−ϵ) (W ,ds)⃗ ≲ ‖f ‖X s (Y ,d̂)⃗ , establishing the ⃗ 󳨅→ X (s,−ϵ) (𝒦, (W , ds)). ⃗ We turn to proving (6.151). continuous inclusion X s (𝒦, (Y , d̂)) Let {(Ej , 2−j ) : j ∈ ℕν } ⊆ ℰ . We will show that

󵄩󵄩 −j⋅(s,−ϵ) 󵄩 Ej f }j∈ℕν 󵄩󵄩󵄩V ≲ ‖f ‖X s (Y ,d̂)⃗ . 󵄩󵄩{2

(6.152)

Taking the supremum of (6.152) over {(Ej , 2−j ) : j ∈ ℕν } ⊆ ℰ establishes (6.151). Thus, we turn to establishing (6.152). Proposition 5.5.10 shows f = Mult[ψ′ ]f = ∑j′ ∈ℕν−1 D′j′ f with convergence in CY∞,0 (M)′ ; ∞ ′ therefore, this sum also converges in the weaker sense of CW ,0 (M) . Thus,

󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 −j⋅(s,−ϵ) 󵄩󵄩 󵄩󵄩󵄩 −j⋅(s,−ϵ) ′ 󵄩󵄩 {2 E f } = {2 E D f } ∑ 󵄩󵄩 j j∈ℕν 󵄩 j j′ +l 󵄩V 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩V l∈ℤν−1 j∈ℕν 󵄩 󵄩󵄩 −j⋅(s,−ϵ) 󵄩󵄩 ′ ≤ ∑ 󵄩󵄩{2 Ej Dj′ +l f }j∈ℕν 󵄩󵄩V . l∈ℤν−1

(6.153)

464 � 6 Besov and Triebel–Lizorkin spaces Take N > |s|. By Lemma 6.8.6, we may write (with K ≈ 1) K

Ej D′j′ +l = ∑ 2−N|l| Fj,L Fj′′ +l,L ,

(6.154)

L=1

where {(Fj,L , 2−j ) : (Ej , 2−j ) ∈ ℰ , l ∈ ℤν−1 , 1 ≤ L ≤ K} is a bounded set of generalized ′ (W , ds)⃗ elementary operators and {(Fk′ ′ ,L , 2−k ) : (Ej , 2−j ) ∈ ℰ , k ′ ∈ ℕν−1 , 1 ≤ L ≤ K} is a bounded set of generalized (Y , d̂)⃗ elementary operators. Plugging (6.154) into (6.153), we see that 󵄩󵄩 −j⋅(s,−ϵ) 󵄩 Ej f }j∈ℕν 󵄩󵄩󵄩V 󵄩󵄩{2 K

′ ′ 󵄩 󵄩 ≤ ∑ ∑ 2−N|l|+l⋅s 󵄩󵄩󵄩{2−jν ϵ Fj,L 2−(j +l )⋅s Fj′′ +l,L f }j∈ℕν 󵄩󵄩󵄩V

l∈ℤν−1 L=1 K

′ 󵄩 󵄩 ≲ ∑ ∑ 2−|l|(N−|s|) 󵄩󵄩󵄩{2s⋅(j +l) Fj′′ +l′ ,L f }j′ ∈ℕν−1 󵄩󵄩󵄩V ′

Lemma 6.8.4(II)

l∈ℤν−1 L=1 K

≲ ∑ ∑ 2−|l|(N−|s|) ‖f ‖X s (Y ,d̂)⃗

Corollary 6.4.4

l∈ℤν−1 L=1

≲ ‖f ‖X s (Y ,d̂)⃗ , establishing (6.152) and completing the proof. The remainder of this section is devoted to the proof of Proposition 6.8.2. This is achieved by combining the next two results. Lemma 6.8.7. Fix ϵ0 > 0 and s = (s1 , . . . , sν ) ∈ ℝν . Suppose ⃗ f ∈ ⋂ X (s1 −ϵ,s2 −ϵ,...,sν −ϵ) (𝒦, (W , ds)) ϵ∈(0,ϵ0 ]

⃗ This result holds even in the case ν = 1. and ‖f ‖V ,s,𝒟0 < ∞. Then f ∈ X s (𝒦, (W , ds)). Proposition 6.8.8. One may choose 𝒟0 and 𝒟0′ such that the following holds. Fix s ∈ ℝν−1 and p ∈ (1, ∞). For f ∈ C0∞ (M)′ such ‖f ‖Lp (M,Vol;ℓ2 (ℕν−1 )),s,𝒟′ < ∞. Then 0

‖f ‖Lp (M,Vol;ℓ2 (ℕν−1 )),s,𝒟′ ≈ ‖f ‖Lp (M,Vol;ℓ2 (ℕν )),(s,0),𝒟0 , 0

(6.155)

where the implicit constants depend on p ∈ (1, ∞) and the choices of 𝒟0 and 𝒟0′ . Here, the left-hand side of (6.155) is finite if and only if the right-hand side is. First we see why the above two results give Proposition 6.8.2.

6.8 Adding parameters

� 465

Proof of Proposition 6.8.2. Since (s,0) ⃗ ∀f ∈ Fp,2 (W , ds),

‖f ‖F (s,0) (W ,ds)⃗ ≈ ‖f ‖Lp (M,Vol;ℓ2 (ℕν )),(s,0),𝒟0 , p,2

‖f ‖F s

⃗

p,2 (Y ,d̂)

s ⃗ ∀f ∈ Fp,2 (Y , d̂),

≈ ‖f ‖Lp (M,Vol;ℓ2 (ℕν−1 )),s,𝒟′ , 0

(s,0) ⃗ it follows immediately from (6.155) ⃗ = F s (𝒦, (Y , d̂)), once we establish Fp,2 (𝒦, (W , ds)) p,2 that the topologies are equal. (s,0) ⃗ and fix Ω ⋐ M open and relatively compact such that 𝒦 ⋐ Ω Let f ∈ Fp,2 (𝒦, (W , ds))

and ψ and ψ′ are equal to 1 on a neighborhood of Ω. Fix ϵ > 0. By Proposition 6.5.17 there L→∞ ⃗ Ω) such that fL 󳨀󳨀 ⃗ By is a sequence {fL }L∈ℕ ⊂ F (s,ϵ) ((W , ds), 󳨀󳨀󳨀→ f in F (s,0) (Ω, (W , ds)). p,2

p,2

s ⃗ ∀L ∈ ℕ. Therefore, using (6.155), we have Proposition 6.8.1, we have fL ∈ Fp,2 (Ω, (Y , d̂)),

‖fL − fL′ ‖F s

⃗

p,2 (Y ,d̂)

≈ ‖fL − fL′ ‖Lp (M,Vol;ℓ2 (ℕν−1 )),s,𝒟′

0

≈ ‖fL − fL′ ‖Lp (M,Vol;ℓ2 (ℕν )),(s,0),𝒟0 ≈ ‖fL − fL′ ‖F (s,0) (W ,ds)⃗ . p,2

s ⃗ Since F s (Ω, (Y , d̂)) ⃗ is comWe conclude that {fL }L∈ℕ is Cauchy in Fp,2 (Ω, (Y , d̂)). p,2 s ⃗ plete (Proposition 6.3.6), we conclude that f ∈ F (Ω, (Y , d̂)). Since supp(f ) ⊆ 𝒦, p,2

s ⃗ This establishes the containment Proposition 6.5.6 shows that f ∈ Fp,2 (𝒦, (Y , d̂)). (s,0) s ⃗ ⃗ ⊆ F (𝒦, (Y , d̂)). F (𝒦, (W , ds)) p,2

p,2

s ⃗ Then by (6.155) we have Suppose f ∈ Fp,2 (𝒦, (Y , d̂)).

‖f ‖Lp (M,Vol;ℓ2 (ℕν )),(s,0),𝒟0 ≈ ‖f ‖Lp (M,Vol;ℓ2 (ℕν−1 )),s,𝒟′ ≈ ‖f ‖F s 0

⃗

p,2 (Y ,d̂)

< ∞.

Proposition 6.8.1 shows that (s,−ϵ) ⃗ ⊆ ⋂ F (s,0)−(ϵ,ϵ,...,ϵ) (𝒦, (W , ds)). ⃗ f ∈ ⋂ Fp,2 (𝒦, (W , ds)) p,2 ϵ>0

ϵ>0

Using the above two equations combined with Lemma 6.8.7 shows that (s,0) ⃗ f ∈ Fp,2 (𝒦, (W , ds)). s ⃗ ⊆ F (s,0) (𝒦, (W , ds)) ⃗ and completes the This establishes the containment Fp,2 (𝒦, (Y , d̂)) p,2 proof.

Proof of Lemma 6.8.7. Let ℰ be a bounded set of generalized (W , ds)⃗ elementary oper⃗ ators. We wish to show ‖f ‖V ,s,ℰ < ∞. Since f ∈ ⋂ϵ∈(0,ϵ0 ] X (s1 −ϵ,s2 −ϵ,...,sν −ϵ) (𝒦, (W , ds)), Corollary 6.4.4 with K0 = [s1 − ϵ0 , s1 ] × ⋅ ⋅ ⋅ × [sν − ϵ0 , sν ] shows that there exists C ≥ 0 such that for every ϵ ∈ (0, ϵ0 ], ‖f ‖V ,s−(ϵ,ϵ,...,ϵ),ℰ ≤ C‖f ‖V ,s−(ϵ,ϵ,...,ϵ),𝒟0 ≤ C‖f ‖V ,s,𝒟0 .

466 � 6 Besov and Triebel–Lizorkin spaces Thus, by the monotone convergence theorem,3 we have, for {(Ej , 2−j ) ∈ ℰ : j ∈ ℕν } ⊆ ℰ , 󵄩󵄩 j⋅s 󵄩 󵄩 j⋅(s−(ϵ,...,ϵ)) 󵄩 Ej f }j∈ℕν 󵄩󵄩󵄩V 󵄩󵄩{2 Ej f }j∈ℕν 󵄩󵄩󵄩V = lim󵄩󵄩󵄩{2 ϵ↓0 ≤ lim ‖f ‖V ,s−(ϵ,ϵ,...,ϵ),ℰ ≤ C‖f ‖V ,s,𝒟0 . ϵ↓0

Therefore, taking the supremum over all {(Ej , 2−j ) ∈ ℰ : j ∈ ℕν } ⊆ ℰ , we see that ‖f ‖V ,s,ℰ ≤ C‖f ‖V ,s,𝒟0 < ∞.

(6.156)

⃗ we have supp(f ) ⊆ 𝒦. Combining this Since f ∈ X (s1 −ϵ0 ,s2 −ϵ0 ,...,sν −ϵ0 ) (𝒦, (W , ds)), with (6.156) and using the fact that ℰ was an arbitrary bounded set of generalized (W , ds)⃗ ⃗ completing the proof. elementary operators, we obtain f ∈ X s (𝒦, (W , ds)), We close this section by proving Proposition 6.8.8. For this, we must introduce the choices of 𝒟0 and 𝒟0′ we use. Fix open sets 1

1

2

2

ν

ν

𝒦 ⋐ Ω1 ⋐ Ω2 ⋐ Ω1 ⋐ Ω2 ⋐ ⋅ ⋅ ⋅ ⋐ Ω1 ⋐ Ω2 ⋐ Ω1 ⋐ Ω2 ⋐ Ω ⋐ M. μ

μ−1

For each μ ∈ {1, . . . , ν}, let ψμ ∈ C0∞ (Ω1 ) satisfy ψμ ≡ 1 on a neighborhood of Ω2 if μ ≥ 2 and ψ1 ≡ 1 on a neighborhood of 𝒦. By hypothesis, for each μ ∈ {1, . . . , ν}, Gen((W μ , dsμ )) is locally finitely generated. Pick μ

μ

(X μ , d μ ) := {(X1 , d1 ), . . . , (Xqμμ , dqμμ )} ⊂ Gen((W μ , dsμ )) such that Gen((W μ , dsμ )) is finitely generated by (X μ , d μ ) on Ω. Set ∞ (X, d )⃗ := (X 1 , d 1 ) ⊠ (X 2 , d 2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (X ν , d ν ) ⊂ Cloc (M; TM) × (ℕν \ {0}),

and similarly, ∞ (X ′ , d ′⃗ ) := (X 1 , d 1 ) ⊠ (X 2 , d 2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (X ν−1 , d ν−1 ) ⊂ Cloc (M; TM) × (ℕν−1 \ {0}).

⃗ Note that (X, d )⃗ is to (W , ds)⃗ as (X ′ , d ′⃗ ) is to (Y , d̂). Let aμ > 0 be a small number to be chosen later. Example 4.1.6 shows that Mult[ψμ ] μ μ is an (X μ , d μ ), aμ partial differential operator of order 0 supported in Ω1 × Ω2 . Therefore, μ μ −jμ by Theorem 4.3.3, we can write Mult[ψμ ] = ∑jμ ∈ℕ Dj , where {(Dj , 2 ) : jμ ∈ ℕ} is a μ

μ

μ

μ

bounded set of (X μ , d μ ), a pseudo-differential operator scales supported in Ω1 × Ω2 .

3 We use the monotone convergence theorem when V = Lp (ℓq ) or V = ℓq (Lp ), where p, q ∈ [1, ∞). If either p or q is infinite, one can combine an elementary argument with the monotone convergence theorem to obtain the same result.

6.8 Adding parameters

�

467

Define ν−2 1 D′j′ := Dν−1 jν−1 Djν−2 ⋅ ⋅ ⋅ Dj1 ,

1 ν ′ Dj = Dνjν Dν−1 jν−1 ⋅ ⋅ ⋅ Dj1 = Djν Dj′ .

Note that 1 ∑ Dj = ( ∑ Dνjν )( ∑ Dν−1 jν−1 ) ⋅ ⋅ ⋅ ( ∑ Dj1 )

j∈ℕν

jν ∈ℕ

jν−1 ∈ℕ

j1 ∈ℕ

= Mult[ψν ] Mult[ψν−1 ] ⋅ ⋅ ⋅ Mult[ψ1 ] = Mult[ψ1 ]. Similarly, ∑ D′j′ = Mult[ψν−1 ] Mult[ψν−2 ] ⋅ ⋅ ⋅ Mult[ψ1 ] = Mult[ψ1 ].

j′ ∈ℕν−1

Set ν

𝒟0 := {(Dj , 2 ) : j ∈ ℕ }, −j

𝒟0 := {(Dj′ , 2 ′

′

−j′

) : j′ ∈ ℕν−1 }.

Remark 6.8.9. Note that 𝒟0′ is of the same form as 𝒟0 , but with ν replaced by ν − 1. ⃗ Ω1 , Ω1 , . . . , Ω1 , Ω2 , Lemma 6.8.10. Let a > 0 be a given small number. If aμ = aμ (a, (X, d ), 1 2 ν ν ⃗ a Ω1 , Ω2 ) > 0 is sufficiently small, for μ ∈ {1, . . . , ν}, then 𝒟0 is a bounded set of (X, d ), pseudo-differential operator scales supported in Ω1 × Ω2 and 𝒟0′ is a bounded set of (X ′ , d ′⃗ ), a pseudo-differential operator scales supported in Ω1 × Ω2 . Proof. The result for 𝒟0′ is the same as the result for 𝒟0 , with ν replaced by ν − 1, so it suffices to prove only the result for 𝒟0 . μ By construction, for each μ ∈ {1, . . . , ν}, {(Dj , 2−jμ ) : jμ ∈ ℕ} is a bounded set of μ

(X μ , d μ ), aμ pseudo-differential operator scales supported in Ω1μ × Ω2μ . Thus, by Definition 4.3.1, we may write μ

Dj f (x) = ∫ f (e−t

μ

⋅X μ

μ

μ

μ

μ

x) Dild2jμ (ςj )(x, t μ , t μ ), μ

μ

̂ C0∞ (Bqμ (aμ ))⊗ ̂ S0 (ℝqμ ) if jμ ≠ 0 and where ςj ∈ C0∞ (Ω1 )⊗ μ

μ

μ

̂ C0∞ (Bqμ (aμ ))⊗ ̂ S (ℝqμ ) {ςj : jμ ∈ ℕ} ⊂ C0∞ (Ω1 )⊗ μ

is a bounded set. For j ∈ ℕν , set 2

2

3

3

ςj (x, (s1 , . . . , sν ), (t 1 , . . . , t ν )) = ςj11 (e−s ⋅X e−s ⋅X ⋅ ⋅ ⋅ e−s 3

3

× ςj22 (e−s ⋅X e−s

4

⋅X 4

× ⋅ ⋅ ⋅ ςjνν (x, sν , t ν ).

ν

⋅X ν

⋅ ⋅ ⋅ e−s

x, s1 , t 1 )

ν

⋅X ν

x, s2 , t 2 )

468 � 6 Besov and Triebel–Lizorkin spaces We only consider sμ ∈ Bqμ (aμ ), where we may take aμ > 0 small. Thus, the above expression makes sense. Moreover, since Ω1 μ ⋐ Ω1 , we see that if aμ > 0 is sufficiently small for each μ ∈ {1, . . . , ν}, then ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) {ςj : j ∈ ℕν } ⊂ C0∞ (Ω1 )⊗ μ

μ

̂ C0∞ (Bqμ (aμ ))⊗ ̂ S0 (ℝqμ ) if jμ ≠ 0, it is a bounded set. Furthermore, since ςj ∈ C0∞ (Ω1 )⊗ μ

̂ C0∞ (Bq (a))⊗ ̂ S{μ:j =0} follows from the definitions that ςj ∈ C0∞ (Ω1 )⊗ (ℝq ). μ ̸ Finally, we have 1 Dj f (x) = Dνjν Dν−1 jν−1 ⋅ ⋅ ⋅ Dj1 f (x) 1

1

= ∫ f (e−t ⋅X e−t

2

⋅X 2

⋅ ⋅ ⋅ e−t

ν

⋅X ν

x) Dild2j (ς)(x, t, t) dt. ⃗

⃗ a pseudo-differential Combining the above, we see that 𝒟0 is a bounded set of (X, d ), operator scales supported in Ω1 × Ω2 , completing the proof. ⃗ Ω1 , Ω1 , . . . , Ω1 , Ω2 , Ω1 , Ω2 , Ω) > 0 is sufficiently small, Corollary 6.8.11. If aμ = aμ (a, (X, d ), 1 2 ν ν for μ ∈ {1, . . . , ν}, then 𝒟0 is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω and 𝒟0′ is a bounded set of generalized (Y , d̂)⃗ elementary operators supported in Ω. Proof. Given any a > 0, by choosing aμ > 0 small enough, μ ∈ {1, . . . , ν}, Lemma 6.8.10 ⃗ a pseudo-differential operator scales supported in Ω1 × Ω2 and shows that 𝒟0 is a (X, d ), ′ 𝒟0 is a bounded set of (X ′ , d ′⃗ ), a pseudo-differential operator scales supported in Ω1 ×Ω2 . ⃗ Ω1 , Ω2 ) > 0 small enough, the result follows from From here, by taking a = a((X, d ), Proposition 5.6.3. To prove Proposition 6.8.8, we need a well-known multi-parameter version of the Khintchine inequality. μ

Theorem 6.8.12 (Multi-parameter Khintchine inequality). Let {ϵj }j∈ℕ,μ∈{1,...,ν} be i. i. d. random variables of mean zero taking values in ±1. For j = (j1 , . . . , jν ) ∈ ℕν , set ϵj := ϵj11 ϵj22 ⋅ ⋅ ⋅ ϵjνν . Fix p ∈ (1, ∞). Then for any sequence of complex numbers {aj }j∈ℕν with ∑ |aj |2 < ∞, we have

1 2

1

󵄨󵄨 󵄨󵄨p p 󵄨󵄨 󵄨󵄨 2 󵄨 ( ∑ |aj | ) ≈ (𝔼󵄨󵄨 ∑ ϵj aj 󵄨󵄨󵄨 ) , 󵄨 󵄨󵄨 󵄨󵄨 j∈ℕν j∈ℕμ 󵄨 where the implicit constants depend on p ∈ (0, ∞). See [213, Appendix D] for a proof of Theorem 6.8.12.

6.8 Adding parameters

� 469

Proof of Proposition 6.8.8. We use Corollary 5.10.7 with 𝒦0 ⋐ Ω3 ⋐ Ω4 replaced by Ων−1 ⋐ 2 Ων1 ⋐ Ων2 . The results of that corollary hold with {(Dj , 2−j ) : j ∈ ℕ} replaced by {(Dνjν , 2−jν ) : jν ∈ ℕ}. μ Let {ϵj }j∈ℕ,μ∈{1,...,ν} be i. i. d. random variables of mean zero taking values in ±1. For j = (j1 , . . . , jν ) ∈ ℕν , set ϵj := ϵj11 ϵj22 ⋅ ⋅ ⋅ ϵjνν and ϵj′ := ϵj11 ϵj22 ⋅ ⋅ ⋅ ϵjν−1 . By Corollary 5.10.7, we ν−1 have, for every g ∈ Lp (Ων−1 2 , Vol),

󵄩󵄩 󵄩󵄩p 󵄩󵄩 󵄩 󵄩 󵄩p ν ν 󵄩 󵄩 󵄩 𝔼 󵄩󵄩 ∑ ϵjν Djν g 󵄩󵄩󵄩󵄩 ≈ 󵄩󵄩󵄩g 󵄩󵄩󵄩Lp . 󵄩󵄩j ∈ℕ 󵄩󵄩 p 󵄩ν 󵄩L

(6.157)

Let f ∈ C0∞ (M)′ be such that ‖f ‖Lp (M,Vol;ℓ2 (ℕν−1 )),s,𝒟′ < ∞. Using Theorem 6.8.12, we 0 have 1 1 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩 󵄨󵄨 󵄨󵄨p p 󵄩󵄩󵄩󵄩 p 󵄩󵄩 󵄩 󵄩 ′ ′ 󵄨 󵄨 󵄩 󵄩 󵄩 󵄨 󵄨 (𝔼 󵄩󵄩󵄩󵄩 ∑ ϵj′ 2j ⋅s D′j′ f 󵄩󵄩󵄩󵄩 ) = 󵄩󵄩󵄩󵄩(𝔼󵄨󵄨󵄨 ∑ ϵj′ 2j ⋅s D′j′ f 󵄨󵄨󵄨 ) 󵄩󵄩󵄩󵄩 󵄨󵄨 󵄩󵄩j′ ∈ℕν−1 󵄩󵄩 󵄩󵄩 󵄨󵄨󵄨 j′ ∈ℕν−1 󵄩󵄩 󵄨 󵄩 󵄩 Lp 󵄩󵄩 󵄩󵄩Lp ≈ ‖f ‖Lp (M,Vol;ℓ2 (ℕν−1 )),s,𝒟′ < ∞. 0

In particular, ∑j′ ∈ℕν−1 ϵj′ 2j ⋅s D′j′ f ∈ Lp (M, Vol), 𝔼-almost surely. We also know that ′

ν−1 ν−1 ′ ν−1 ν−2 1 supp(Dν−1 jν−1 ) ⊆ Ω1 × Ω2 , and since Dj′ = Djν−1 Djν−2 ⋅ ⋅ ⋅ Dj1 , we have

supp( ∑ ϵj′ 2j ⋅s D′j′ f ) ⊆ Ων−1 2 . ′

j′ ∈ℕν−1

Thus, we may apply (6.157) with g = ∑j′ ∈ℕν−1 ϵj′ 2j ⋅s D′j′ f . We have ′

‖f ‖Lp (M,Vol;ℓ2 (ℕν−1 )),s,𝒟′ 0 󵄩󵄩 󵄩󵄩󵄩 ′ 󵄩 󵄨 󵄨2 󵄩󵄩 = 󵄩󵄩󵄩󵄩( ∑ 󵄨󵄨󵄨2j ⋅s D′j′ f 󵄨󵄨󵄨 )󵄩󵄩󵄩󵄩 󵄩󵄩 j′ ∈ℕν−1 󵄩󵄩 󵄩 󵄩Lp 1 󵄩󵄩 󵄨 󵄨󵄨p p 󵄩󵄩󵄩󵄩 󵄩󵄩 󵄨󵄨 󵄩󵄩 󵄨󵄨 󵄩 j′ ⋅s ′ 󵄨󵄨󵄨 ≈ 󵄩󵄩󵄩(𝔼󵄨󵄨 ∑ ϵj′ 2 Dj′ f 󵄨󵄨 ) 󵄩󵄩󵄩󵄩 󵄨󵄨 󵄩󵄩 󵄨󵄨󵄨 j′ ∈ℕν−1 󵄩󵄩 󵄨 󵄩󵄩 󵄩󵄩Lp 1 󵄩󵄩 󵄩󵄩p p 󵄩󵄩 󵄩 j′ ⋅s ′ 󵄩 󵄩 󵄩 = (𝔼 󵄩󵄩 ∑ ϵj′ 2 Dj′ f 󵄩󵄩󵄩󵄩 ) 󵄩󵄩j′ ∈ℕν−1 󵄩󵄩 󵄩 󵄩 Lp

󵄩󵄩 󵄩󵄩p p1 󵄩󵄩 󵄩󵄩 ′ ≈ (𝔼 󵄩󵄩󵄩󵄩 ∑ ∑ ϵjνν ϵj′ 2j ⋅s Dνjν D′j′ f 󵄩󵄩󵄩󵄩 ) 󵄩󵄩j ∈ℕ j′ ∈ℕν−1 󵄩󵄩 󵄩ν 󵄩 Lp

by Theorem 6.8.12

by (6.157)

470 � 6 Besov and Triebel–Lizorkin spaces 󵄩󵄩 󵄩󵄩p p1 󵄩󵄩 󵄩󵄩 j′ ⋅s 󵄩 = (𝔼 󵄩󵄩󵄩 ∑ ϵj 2 Dj f 󵄩󵄩󵄩󵄩 ) 󵄩󵄩j∈ℕν 󵄩󵄩 p 󵄩 󵄩L 1 󵄩󵄩 󵄨 󵄨󵄨p p 󵄩󵄩󵄩󵄩 󵄩󵄩 󵄨󵄨 ′ 󵄨 󵄩󵄩 󵄨󵄨 󵄩 󵄨 = 󵄩󵄩󵄩(𝔼󵄨󵄨 ∑ ϵj 2j ⋅s Dj f 󵄨󵄨󵄨 ) 󵄩󵄩󵄩󵄩 󵄨 󵄨 󵄩󵄩 󵄨󵄨 j∈ℕν 󵄩󵄩 󵄨󵄨 󵄩󵄩 󵄩󵄩Lp 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄨 ′ 󵄨2 󵄩󵄩 ≈ 󵄩󵄩󵄩󵄩( ∑ 󵄨󵄨󵄨2j ⋅s Dj f 󵄨󵄨󵄨 )󵄩󵄩󵄩󵄩 󵄩󵄩 j∈ℕν 󵄩󵄩 p 󵄩 󵄩L

by Theorem 6.8.12

= ‖f ‖Lp (M,Vol;ℓ2 (ℕν−1 )),(s,0),𝒟0 , completing the proof. Along with Proposition 6.8.8, the same proof ideas give the following useful corollary. Corollary 6.8.13. For a1 , . . . , aν > 0 sufficiently small, we have the following. For f ∈ Lp (M, Vol) with supp(f ) ⊆ 𝒦, we have ‖f ‖Lp (M,Vol;ℓ2 (ℕν )),0,𝒟0 ≈ ‖f ‖Lp (M,Vol) . This corollary holds even when ν = 1. Proof. We prove the result by induction on ν. For the base case, ν = 1, we use Corollary 5.10.7 with 𝒦0 ⋐ Ω3 ⋐ Ω4 replaced by 𝒦 ⋐ Ω11 ⋐ Ω12 , with D1j1 playing the role of Dj in that result, to see that, for f ∈ Lp (𝒦, Vol), 1󵄩 󵄩󵄩 2󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 2 󵄨 󵄨 1 󵄩󵄩( ∑ 󵄨󵄨󵄨Dj f 󵄨󵄨󵄨 ) 󵄩󵄩󵄩 ≈ ‖f ‖Lp , 󵄩󵄩 󵄩󵄩 1 󵄩󵄩 p 󵄩󵄩󵄩 j1 ∈ℕ 󵄩L

which is exactly the claimed result when ν = 1. The inductive step is the case s = 0 of Proposition 6.8.8 (see also Remark 6.8.9).

6.9 Sobolev spaces s ⃗ An important special case of the Triebel–Lizorkin spaces are the spaces Fp,2 (𝒦, (W , ds)), p ⃗ consisting of 1 < p < ∞, which we think of as L Sobolev spaces adapted to (W , ds), distributions supported in 𝒦. The key to understanding this is the next proposition and its corollary. 0 ⃗ = Lp (𝒦, Vol), and the corresponding Proposition 6.9.1. For 1 < p < ∞, Fp,2 (𝒦, (W , ds)) norms are equivalent.

6.9 Sobolev spaces

�

471

Proof. Let 𝒟0 be as in Corollary 6.8.13. Since (see Proposition 6.3.7) ‖ ⋅ ‖F 0

⃗

p,2 (W ,ds)

≈ ‖ ⋅ ‖Lp (M,Vol;ℓ2 (ℕν )),0,𝒟0 ,

Corollary 6.8.13 shows that the norms ‖⋅‖Lp and ‖⋅‖F 0

⃗

p,2 (W ,ds)

are equivalent on Lp (𝒦, Vol) ⋂

0 ⃗ Thus, once we have the equality of the spaces, the equivalence of the Fp,2 (𝒦, (W , ds)).

norms follows from Corollary 6.8.13. First we prove the result in the special case where 𝒦 = Ω1 , for some open set Ω1 ⋐ M. For f ∈ Lp (𝒦, Vol), by Corollary 6.8.13 we have ‖f ‖Lp (𝒦,Vol) ≈ ‖f ‖Lp (M,Vol;ℓ2 (ℕν )),s,𝒟0 . 0 ⃗ and therefore for f ∈ C ∞ (Ω1 ), we have By Proposition 6.5.5, C0∞ (Ω1 ) ⊆ Fp,2 (𝒦, (W , ds)), 0

‖f ‖Lp ≈ ‖f ‖Lp (M,Vol;ℓ2 (ℕν )),0,𝒟0 ≈ ‖f ‖F 0

⃗

p,2 (W ,ds)

(6.158)

.

0 ⃗ is complete Using the fact that C0∞ (Ω1 ) is dense in Lp (𝒦, Vol) and that Fp,2 (𝒦, (W , ds)) (Proposition 6.3.6), we have 0 ⃗ Lp (𝒦, Vol) = C0∞ (Ω1 ) ⊆ Fp,2 (𝒦, (W , ds)),

where C0∞ (Ω1 ) denotes the completion of C0∞ (Ω1 ) with respect to any of the equivalent norms (6.158). 0 ⃗ ⊆ Lp (𝒦, Vol). Fix f ∈ F 0 (𝒦, (W , ds)). ⃗ Conversely, we wish to show Fp,2 (𝒦, (W , ds)) 2,p Let Ω ⋐ M be an open, relatively compact set and ϵ > 0. By Proposition 6.5.17, there L→∞ ⃗ such that fL 󳨀󳨀 ⃗ exists a sequence {fL }L∈ℕ ⊂ F (ϵ,ϵ,...,ϵ) (Ω, (W , ds)) 󳨀󳨀󳨀→ f in F 0 (Ω, (W , ds)). s,2

p,2

Proposition 6.5.16 shows that fL ∈ Lp (Ω, Vol) for every L. Thus, since the sequence fL is Cauchy in the norm ‖ ⋅ ‖F 0 , it is also Cauchy in the equivalent norm ‖ ⋅ ‖Lp . We conclude p,2

that fL converges in Lp (Ω, Vol), and therefore f ∈ Lp (M, Vol). Since supp(f ) ⊆ 𝒦, we conclude that f ∈ Lp (𝒦, Vol), as desired. This completes the proof in the case where 𝒦 = Ω1 . For general 𝒦, pick Ω1 ⋐ M open with 𝒦 ⋐ Ω1 and set 𝒦1 := Ω1 . Proposition 6.5.6 shows that 0 ⃗ ⇐⇒ f ∈ F 0 (𝒦1 , (W , ds)) ⃗ and supp(f ) ⊆ 𝒦, f ∈ Fp,2 (𝒦, (W , ds)) p,2

and it is a standard fact that f ∈ Lp (𝒦, Vol) ⇐⇒ f ∈ Lp (𝒦1 , Vol) and supp(f ) ⊆ 𝒦. 0 ⃗ follows from the already proved equality Thus, the equality Lp (𝒦, Vol) = Fp,2 (𝒦, (W , ds)) p 0 ⃗ completing the proof. L (𝒦1 , Vol) = F (𝒦1 , (W , ds)), p,2

472 � 6 Besov and Triebel–Lizorkin spaces Proof of Proposition 6.2.13. In light of Proposition 6.6.1, this is a special case of Proposition 6.9.1. μ

Corollary 6.9.2. Fix p ∈ (1, ∞). For each μ ∈ {1, . . . , ν}, let κμ ∈ ℕ be such that dsj divides κμ for 1 ≤ j ≤ rμ (we allow the possibility κμ = 0). Set κ := (κ1 , . . . , κν ) ∈ ℕν . Then, for ∞ ′ s ∈ ℝν and f ∈ CW ,0 (M) , the following are equivalent: κ ⃗ (i) f ∈ F (𝒦, (W , ds)). p,2

(ii) For every ordered multi-index α with degds⃗ (α) ≤ κ, we have W α f ∈ Lp (𝒦, Vol). κ ⃗ we have Furthermore, for f ∈ Fp,2 (𝒦, (W , ds)),

‖f ‖F κ

⃗

p,2 (W ,ds)

≈

∑

degds⃗ (α)≤κ

‖W α f ‖Lp ,

where the implicit constants do not depend on f (but may depend on any of the other ingredients). Proof. In light of Proposition 6.9.1, this follows immediately from Corollary 6.5.13 with s = κ.

6.10 Distributions of finite norm In Definition 6.3.8, we defined the norm ‖f ‖X s (W ,ds)⃗ := ‖f ‖V ,s,𝒟0 . However, if f ∈ ∞ ′ CW ,0 (M) satisfies supp(f ) ⊆ 𝒦 and ‖f ‖V ,s,𝒟0 < ∞, we have not shown that f ∈ s ⃗ Remark 6.4.6 shows that we can replace ‖ ⋅ ‖V ,s,𝒟 with a different X (𝒦, (W , ds)). 0 ⃗ and whose finiteness does imply belonging norm which is equivalent on X s (𝒦, (W , ds)) ⃗ to the space X s (𝒦, (W , ds)). For some of our applications, the norm ‖ ⋅ ‖V ,s,𝒟0 is the most convenient choice. The main result of this section shows that if s ∈ (0, ∞)ν , then finiteness of this norm does ⃗ imply belonging to the space X s (𝒦, (W , ds)). ∞ ′ Proposition 6.10.1. Fix s ∈ (0, ∞)ν . Let f ∈ CW ,0 (M) with supp(f ) ⊆ 𝒦. Then the following are equivalent: ⃗ (i) f ∈ X s (𝒦, (W , ds)). (ii) ‖f ‖V ,s,𝒟0 < ∞.

Proof. (i) ⇒ (ii) is trivial, so we prove only the reverse implication. Thus, suppose f ∈ ∞ ′ −j ν CW ,0 (M) with supp(f ) ⊆ 𝒦 and (ii) holds. Recall that 𝒟0 = {(Dj , 2 ) : j ∈ ℕ } is a bounded set of generalized (W , ds)⃗ elementary operators with ∑j∈ℕν Dj = Mult[ψ], where ψ ∈ C0∞ (M) satisfies ψ ≡ 1 on a neighborhood of 𝒦. V is equal to either Lp (M, Vol; ℓq (ℕν )) or ℓq (ℕν ; Lp (M, Vol)) (see Notation 6.3.1). In either case, fix this p, q ∈ [1, ∞].

6.10 Distributions of finite norm

�

473

We claim f ∈ Lp (M, Vol). Indeed, since supp(f ) ⊆ 𝒦, we have ∑j∈ℕν Dj f = ∞ ′ Mult[ψ]f = f , with convergence in CW ,loc (M) , by Proposition 5.5.10. We also have 󵄩󵄩 j⋅s 󵄩 󵄩󵄩{2 Dj f }j∈ℕν 󵄩󵄩󵄩V = ‖f ‖V ,s,𝒟0 < ∞. In particular, this implies ‖Dj f ‖Lp (M,Vol) ≲ 2−j⋅s .

(6.159)

Thus, ∑j∈ℕν ‖Dj f ‖Lp < ∞, and therefore f = ∑j∈ℕν Dj f converges in the norm topology on Lp (M, Vol), showing that f ∈ Lp (M, Vol). Set minμ∈{1,...,ν} sμ

ϵ0 :=

2

> 0.

⃗ and (i) will then follow from We will show ∀ϵ ∈ (0, ϵ0 ] that f ∈ X s−ϵ(1,1,...,1) (𝒦, (W , ds)), Lemma 6.8.7. For the remainder of the proof, ϵ will denote an arbitrary element of (0, ϵ0 ]. Fix a bounded set of (W , ds)⃗ of generalized (W , ds)⃗ elementary operators, ℰ . In what follows, (Ej , 2−j ) will denote an arbitrary element of ℰ . We write A ≲ B for A ≤ CB, where C ≥ 0 does not depend on (Ej , 2−j ) ∈ ℰ or on the variable k ∈ ℕν . However, C may depend on f . We claim that, ∀N ∈ ℕ, ‖Ej Dk ‖Lp (M,Vol)→Lp (M,Vol) ≲ 2−N|j1 −k1 | 2−N|j2 −k2 | ⋅ ⋅ ⋅ 2−N|jν −kν | .

(6.160)

Indeed, by Proposition 5.5.11, for every N0 ∈ ℕ, {(2N0 |j−k| Ej Dk , 2−j ) : (Ej , 2−j ) ∈ ℰ , k ∈ ℕν } is a bounded set of generalized (W , ds) elementary operators and therefore by Definition 5.2.26, ‖Ej Dk ‖Lp (M,Vol)→Lp (M,Vol) ≲ 2−N0 |j−k| . The estimate (6.160) follows by taking N0 ≥ Nν. Fix δ = δ(ϵ) ∈ (0, 1) large enough that if we set cμ := is always possible since

sμ −ϵ sμ

sμ −ϵ , δsμ

we have cμ ∈ (0, 1); this

∈ (0, 1) by the choice of ϵ. By (6.160) with N replaced by

N/(1 − δ) and the already proved fact that f ∈ Lp , we have N

N

N

‖Ej Dk f ‖Lp ≲ 2− 1−δ |j1 −k1 | 2− 1−δ |j2 −k2 | ⋅ ⋅ ⋅ 2− 1−δ |jν −kν | .

(6.161)

By (6.159) and the fact that sup(Ej ,2−j )∈ℰ ‖Ej ‖Lp →Lp ≲ 1 (see Definition 5.2.26), we have ‖Ej Dk f ‖Lp ≲ 2−k⋅s . Combining (6.161) and (6.162), we have

(6.162)

474 � 6 Besov and Triebel–Lizorkin spaces ‖Ej Dk f ‖Lp ≲ 2−k1 δs1 2−k2 δs2 ⋅ ⋅ ⋅ 2−kν δsν 2−N|j1 −k1 | 2−N|j2 −k2 | ⋅ ⋅ ⋅ 2−N|jν −kν | .

(6.163)

We claim that, ∀ϵ ∈ (0, ϵ0 ], ‖Ej f ‖Lp (M,Vol) ≤ Cϵ 2−j⋅(s−ϵ(1,1,...,1)) .

(6.164)

Indeed, using (6.163) with N ≥ maxμ∈{1,...,ν} (sμ − ϵ)/(1 − cμ ), we have 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ‖Ej f ‖Lp = 󵄩󵄩󵄩Ej ∑ Dk f 󵄩󵄩󵄩 ≤ ∑ ‖Ej Dk f ‖Lp 󵄩󵄩 󵄩󵄩 p k∈ℕν 󵄩 k∈ℕν 󵄩L =

≲

=

≲

∑

∑

‖Ej Dk f ‖Lp

∑

∑

(∏ 2−kμ δμ sμ )(∏ 2−N(jμ −kμ ) )

F⊆{1,...,ν} kμ ≥cμ jμ ,μ∈F kμ 0 is sufficiently small, then Γ(t, x) is defined for x ∈ Ω1 and t ∈ Bq (a) and satisfies x ∈ Ω1 , t ∈ Bq (a) ⇒ Γ(t, x) ∈ Ω2 , q

Γ(t, x) ∈ 𝒦, t ∈ B (a) ⇒ x ∈ Ω0 .

(6.168) (6.169)

For j ∈ ℕν , set Dj f (x) := ψ(x) ∫ f (Γ(t, x)) Dild2j (ςj )(x, t, t) dt ⃗

= ψ(x) ∫ f (Γ(t, x))η(t) Dild2j (τ{μ:jμ =0} ̸ )(t) dt, ⃗

(6.170)

Pj := ∑ Dj . k∈ℕν k≤j

⃗ 𝒦, Ω0 , Ω1 , Ω2 ) > 0 is suffiProposition 6.11.1. With Dj and Pj as above, if a = a((X, d ), ciently small, we have: (i) {(Dj , 2−j ) : j ∈ ℕν } is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω.

6.11 An explicit choice of norm

� 477

(ii) {(Pj , 2−j ) : j ∈ ℕν } is a bounded set of generalized (W , ds)⃗ pre-elementary operators supported in Ω. (iii) ∑j∈ℕν Dj = Mult[ψ]; see Proposition 5.5.10 for the convergence of this sum. (iv) If supp(f ) ⊆ 𝒦, then Dj f (x) = ∫ f (Γ(t, x))η(t) Dild2j (τ{μ:jμ =0} ̸ )(t) dt; ⃗

in particular, Dj f does not depend on the choice of ψ. In light of Proposition 6.11.1, the next definition makes sense and yields an explicit choice of norm in the equivalence class of ‖ ⋅ ‖X s (W ,ds)⃗ . Definition 6.11.2. For f ∈ C0∞ (M)′ with supp(f ) ⊆ 𝒦 and s ∈ ℝν , we set ‖f ‖X s (W ,ds),(X, ⃗ ⃗ d),a,Ω

0 ,Ω1 ,Ω2 ,Vol

:= ‖f ‖V ,s,{(Dj ,2−j ):j∈ℕν } .

The rest of this section is devoted to the proof of Proposition 6.11.1. To prove it, we introduce the following more precise version. Proposition 6.11.3. If a = a(Ω1 , Ω2 , 𝒦) > 0 is sufficiently small, then: (i) ∑j∈ℕν Dj = Mult[ψ] with convergence in the topology of bounded convergence on ∞ ∞ Hom(Cloc (M), C0∞ (Ω1 )) and in the weak operator topology on Hom(C0∞ (M)′ , Cloc (Ω1 )′ ), ∞ ′ where Cloc (Ω1 ) is given the weak topology. ⃗ a pseudo-differential operator scales (ii) {(Dj , 2−j ) : j ∈ ℕν } is a bounded set of (X, d ), supported in Ω1 × Ω2 . ⃗ a pre-pseudo-differential operator scales (iii) {(Pj , 2−j ) : j ∈ ℕν } is a bounded set of (X, d ), supported in Ω1 × Ω2 . ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) Proof. (ii): Since {ςj : j ∈ ℕν } = {ψ ⊗ η ⊗ τF : F ⊆ {1, . . . , ν}} ⊂ C0∞ (Ω1 )⊗ is a finite set, it is a bounded set. It follows from (6.166) that ςj = ψ ⊗ η ⊗ τ{μ:jμ =0} ∈ ̸ q ̂ C0∞ (Bq (a))⊗ ̂ S{μ:j =0} C0∞ (Ω1 )⊗ (ℝ ). From here, (ii) follows immediately from the defini̸ μ

tions (see Definition 4.3.1 (ii)). ⃗ a pseudo-differential operator (i): By (ii), {(Dj , 2−j ) : j ∈ ℕν } is a bounded set of (X, d ), scales supported in Ω1 × Ω2 . Thus, by Theorem 4.3.3, if a > 0 is sufficiently small, ∑j∈ℕν Dj ∞ converges in the topology of bounded convergence on Hom(Cloc (M), C0∞ (Ω1 )) and in the ∞ ∞ weak operator topology on Hom(C0∞ (M)′ , Cloc (Ω1 )′ ), where Cloc (Ω1 )′ is given the weak topology. Thus, to prove (i) it suffices to show that ∑ Dj f (x) = ψ(x)f (x),

j∈ℕν

∀f ∈ C0∞ (M).

(6.171)

478 � 6 Besov and Triebel–Lizorkin spaces By Remark 4.1.4, ∑ Dj f (x) = ∫ f (Γ(t, x))( ∑ Dild2j (ςj )(x, t, t)) dt ⃗

j∈ℕν

j∈ℕν

(6.172)

= ∬ f (Γ(t, x))( ∑ ςĵ (x, t, 2

−jd ⃗

j∈ℕν

ξ))e

2πit⋅ξ

dt dξ.

Because of this, to prove (6.171), it suffices to show that ∑ ςĵ (x, s, ξ) = ψ(x)η(s).

(6.173)

j∈ℕν

Indeed, once we have (6.173), then (6.172) together with the fact that η(0) = 1 yields ∑ Dj f (x) = ∬ f (Γ(t, x))ψ(x)η(t)e2πit⋅ξ dt dξ

j∈ℕν

= ∬ f (Γ(t, x))ψ(x)η(t)δ0 (t) dt = f (Γ(0, x))ψ(x)η(0) = f (x)ψ(x). We complete the proof of (i) by proving (6.173). In fact, we have, for j ∈ ℕν , ∑ ςk̂ (x, s, 2−k d ξ) = ψ(x)η(s) ⃗

k≤j k∈ℕν

μ

μ

∏ (ρ̂ μ (ξ μ ) + ∑ (ρ̂ μ (2−kμ d ξ μ ) − ρ̂ μ (2−(kμ −1)d ξ μ )))

μ∈{1,...,ν}

1≤kμ ≤jμ

1

ν

= ψ(x)η(s)ρ̂ 1 (2−j1 d ξ 1 ) ⋅ ⋅ ⋅ ρ̂ ν (2−jν d ξ ν ).

(6.174)

Since each ρ̂ μ (ξ μ ) = 1 for ξ μ small, the right-hand side of (6.174) clearly tends to ψ(x)η(s) as j1 , . . . , jν → ∞, proving (6.173) and completing the proof of (i). (iii): By (6.174), we have 1

ν

∑ ςk̂ (x, s, 2−k d ξ) = ψ(x)η(s)ρ̂ 1 (2−j1 d ξ 1 ) ⋅ ⋅ ⋅ ρ̂ ν (2−jν d ξ ν ) = τ0̂ (x, s, 2−jd ξ). ⃗

⃗

k≤j k∈ℕν

Thus, Pj f (x) = ∫ f (Γ(t, x)) Dild2j (τ0 )(x, t, t) dt. ⃗

̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ), if a > 0 is sufficiently small, (iii) follows immeSince τ0 ∈ C0∞ (Ω1 )⊗ ⃗ a pre-pseudo-differential operator diately from the definition of bounded sets of (X, d ), scales supported in Ω1 × Ω2 (see Definition 4.3.1 (i)). Proof of Proposition 6.11.1. (i): In light of Proposition 6.11.3 (ii), this follows from Proposition 5.6.3. (ii): In light of Proposition 6.11.3 (iii), this follows from Lemma 5.6.6.

6.11 An explicit choice of norm

�

479

(iii): This follows from Proposition 6.11.3 (i). (iv): By (6.169), if supp(f ) ⊆ 𝒦, then the integral ∫ f (Γ(t, x))η(t) Dild2j (τ{μ:jμ =0} ̸ )(t) dt ⃗

is supported on x ∈ Ω0 . For such x, ψ(x) = 1, and therefore, Dj f (x) = ψ(x) ∫ f (Γ(t, x))η(t) Dild2j (τ{μ:jμ =0} ̸ )(t) dt ⃗

= ∫ f (Γ(t, x))η(t) Dild2j (τ{μ:jμ =0} ̸ )(t) dt, ⃗

completing the proof. 6.11.1 The unit scale In many of our main results, we require quantitative estimates at the multi-parameter unit scale described in Section 3.15.1. Here, M = Bn (1) and Vol = hσLeb , where h ∈ C ∞ (Bn (1); (0, ∞)) with infu∈Bn (1) h(u) > 0. Letting (W 1 , ds1 ), . . . , (W ν , dsν ) and (X 1 , d 1 ), . . . , (X ν , d ν ) be as in Section 3.15.1, we set (W , ds)⃗ := (W 1 , ds1 ) ⊠ ⋅ ⋅ ⋅ ⊠ (W ν , dsν ) and (X, d )⃗ := (X 1 , d 1 ) ⊠ ⋅ ⋅ ⋅ ⊠ (X ν , d ν ). For σ ∈ (0, 1) and f ∈ C0∞ (Bn (1))′ with supp(f ) ⊆ Bn (σ), we define ‖f ‖X s (W ,ds),(X, := ‖f ‖X s (W ,ds),(X, n (σ+(1−σ)/5),Bn (σ+2(1−σ)/5),Bn (σ+3(1−σ)/5),Vol . ⃗ ⃗ ⃗ ⃗ d),a,σ,h d),a,B In our applications, we will usually take σ = 7/8. The main idea is that when we prove results in this setting, the estimates will only ⃗ (X, d ), ⃗ and h through multi-parameter unit-admissible constants as depend on (W , ds), described in Definition 3.15.1. 6.11.2 Scaling The norm at the unit scale in Section 6.11.1 plays a particular role in some scaling estimates. Fix compact sets 𝒦0 , 𝒦 ⋐ M and relatively compact, open sets U, Ω1 , Ω2 ⋐ M, with 𝒦0 ⋐ U ⋐ 𝒦 ⋐ Ω1 ⋐ Ω2 ⋐ M. Let (W 1 , ds1 ), . . . , (W ν , dsν ), (X 1 , d 1 ), . . . , (X ν , d ν ) ⊂ ∞ Cloc (M; TM), and λ ∈ (0, ∞)ν be as in Section 3.15.2, with these choices of 𝒦, Ω1 , and Ω2 . As usual, set (W , ds)⃗ := (W 1 , ds1 ) ⊠ ⋅ ⋅ ⋅ ⊠ (W ν , dsν ) and (X, d )⃗ := (X 1 , d 1 ) ⊠ ⋅ ⋅ ⋅ ⊠ (X ν , d ν ). Consider the following norm from Section 6.11: ‖f ‖X s (W ,ds),(X, ⃗ ⃗ d),a,Ω

0 ,Ω1 ,Ω2 ,Vol

,

⃗ f ∈ X s (𝒦, (W , ds)),

defined in Definition 6.11.2. For δ ∈ (0, 1], we also have the norm

480 � 6 Besov and Triebel–Lizorkin spaces

‖f ‖X s (δλds⃗ W ,ds),(δ λd ⃗ X,d),a,Ω ⃗ ⃗

0 ,Ω1 ,Ω2 ,Vol

,

⃗ f ∈ X s (𝒦, (W , ds)).

These norms are both in the equivalence class of ‖f ‖X s (W ,ds)⃗ and therefore, for f ∈ ⃗ X s (𝒦, (W , ds)), ‖f ‖X s (W ,ds),(X, ⃗ ⃗ d),a,Ω

0 ,Ω1 ,Ω2 ,Vol

≈δ ‖f ‖X s (δλds⃗ W ,ds),(δ λd ⃗ X,d),a,Ω ⃗ ⃗

0 ,Ω1 ,Ω2 ,Vol

,

where the implicit constants depend on δ. When we turn to use scaling, as in Theorems 3.15.3 and 3.15.5 to prove quantitative results, the estimates are often in terms of ‖f ‖X s (δλds⃗ W ,ds),(δ instead of ‖f ‖X s (W ,ds),(X, . This is due to the λd ⃗ X,d),a,Ω ⃗ ⃗ ⃗ ⃗ d),a,Ω 0 ,Ω1 ,Ω2 ,Vol 0 ,Ω1 ,Ω2 ,Vol next proposition. ⃗ Ω0 , Ω1 , Ω2 , σ) > 0 is sufficiently small, Proposition 6.11.4. Fix σ ∈ (0, 1). If a = a(𝒦, (X, d ), the following holds. Let Φx,δ be as in Theorem 3.15.3, with 𝒦 replaced by 𝒦0 and Ω1 replaced by U, and let (W μ,x,δ , dsμ ), (X μ,x,δ , d μ ) ⊂ C ∞ (Bn (1); TBn (1)) × ℕ+ and hx,δ ∈ C ∞ (Bn (1)) be as in that theorem. Set (W x,δ , ds)⃗ := (W 1,x,δ , ds1 ) ⊠ ⋅ ⋅ ⋅ ⊠ (W ν,x,δ , dsν ) and (X x,δ , d )⃗ := (X 1,x,δ , ds1 ) ⊠ ⋅ ⋅ ⋅ ⊠ (X ν,x,δ , dsν ). Fix ϕ ∈ C0∞ (Bn (σ)). Then, for any u ∈ C0∞ (M)′ , x ∈ 𝒦0 , δ ∈ (0, 1], and s ∈ ℝν , we have4 ‖ϕΦ∗x,δ u‖X s (W x,δ ,ds),(X x,δ ,d),a,σ,h ⃗ ⃗

x,δ

󵄩 󵄩󵄩 = Λ(x, δ)−1/p 󵄩󵄩󵄩(ϕ ∘ Φ−1 λd ⃗ X,d),a,Ω ⃗ ⃗ x,δ )u󵄩 󵄩X s (δλds⃗ W ,ds),(δ

0 ,Ω1 ,Ω2 ,Vol

,

where Λ(x, δ) is given by Λ(x, δ) :=

max

1

j1 ,...,jn ∈{1,...,q1 }

1

Vol(x)(δdj1 Xj11 (x), . . . , δdjn Xj1n (x))

(6.175)

and we use the convention Λ(x, δ)1/∞ = 1. ̂ x,δ be the operator Dj from (6.170), with (X, d )⃗ replaced by (X x,δ , d ). ⃗ Similarly, Proof. Let D j ⃗ Recall that τF is let Dδ be the operator from (6.170) with (X, d )⃗ replaced by (δλd X, d ). j

chosen to depend only on q1 , . . . , qν , and η is chosen to depend only on a and q. Thus, we ̂ x,δ and Dδ . choose the same τF and η in the definitions of D j j By Proposition 6.11.1 (iv), we have ̂ x,δ (ϕΦ∗ u)(v) D x,δ j 1

= ∫(ϕΦ∗x,δ u)(e−t ⋅X

1,x,δ

1

e−t

−t ⋅δ = ∫((ϕ ∘ Φ−1 x,δ )u)(e

d1

2

⋅X 2,x,δ

⋅ ⋅ ⋅ e−t d2

ν

⋅X ν,x,δ

X 1 −t 2 ⋅(λ2 δ) X 2

e

v)η(t) Dild2j (τ{μ:jμ =0} ̸ )(t) dt

⋅ ⋅ ⋅ e−t

(6.176)

⃗

ν

dν

⋅(λν δ) X ν

= [Dδj ((ϕ ∘ Φx,δ )u)](Φx,δ (v)) = Φ∗x,δ Dδj ((ϕ ∘ Φ−1 x,δ )u)(v),

s s 4 Here, p ∈ [1, ∞] is the p for which X s = Fp,q or X s = Bp,q .

Φx,δ (v))η(t) Dild2j (τ{μ:jμ =0} ̸ )(t) dt ⃗

6.12 A spectral definition

�

481

where the final equality uses Proposition 6.11.1 (iv) again. s We complete the proof in the case where X s = Fp,q . A similar proof works in the s s case where X = Bp,q ; we leave the details to the reader. Using (6.176) and Φ∗x,δ Vol = Λ(x, δ)hx,δ σLeb , we have ‖ϕΦ∗x,δ u‖X s (W x,δ ,ds),(X x,δ ,d),a,σ,h ⃗ ⃗

x,δ

󵄩 ̂ x,δ 󵄩󵄩 ∗ = 󵄩󵄩󵄩{D 󵄩Lp (Bn (1),hx,δ σLeb ;ℓq (ℕν )) j (ϕΦx,δ u)}j∈ℕν 󵄩 −1/p 󵄩 x,δ ̂ (ϕΦ∗ u)} ν 󵄩󵄩󵄩 p n = Λ(x, δ) 󵄩󵄩󵄩{D x,δ j∈ℕ 󵄩L (B (1),Φ∗x,δ Vol;ℓq (ℕν )) j 󵄩 󵄩󵄩 = Λ(x, δ)−1/p 󵄩󵄩󵄩{Φ∗x,δ Dδj ((ϕ ∘ Φ−1 x,δ )u)}j∈ℕν 󵄩 󵄩Lp (Bn (1),Φ∗x,δ Vol;ℓq (ℕν )) 󵄩 −1/p 󵄩 δ −1 = Λ(x, δ) 󵄩󵄩󵄩{Dj ((ϕ ∘ Φx,δ )u)}j∈ℕν 󵄩󵄩󵄩Lp (M,Vol;ℓq (ℕν )) 󵄩 󵄩󵄩 = Λ(x, δ)−1/p 󵄩󵄩󵄩(ϕ ∘ Φ−1 . λd ⃗ X,d),a,Ω ⃗ ⃗ x,δ )u󵄩 󵄩X s (δλds⃗ W ,ds),(δ 0 ,Ω1 ,Ω2 ,Vol

6.12 A spectral definition In the single-parameter setting, the Besov and Triebel–Lizorkin spaces can sometimes be equivalently defined in terms of spectral multipliers for Hörmander’s sub-Laplacian. We outline this in the simplest setting. In this section, we assume M is a smooth, compact manifold with smooth strictly positive density Vol. On M we are given Hörmander vector fields W1 , . . . , Wr . We assign to each Wj the formal degree 1: (W , 1) = {(W1 , 1), . . . , (Wr , 1)}. Let L = W1∗ W1 + ⋅ ⋅ ⋅ + Wr∗ Wr : Hörmander’s sub-Laplacian. As discussed in Section 5.12, L is essentially self-adjoint, and we identify L with its self-adjoint extension. In Remark 5.12.5 we present a decomposition of the identity operator I = ∑ Dj , j∈ℕ

where 𝒟0 := {(Dj , 2−j ) : j ∈ ℕ} is a bounded set of (W , 1) elementary operators supported in M. Moreover, there is ψ ∈ C0∞ (ℝ) with ψ ≡ 1 on a neighborhood of 0 such that if ϕ(λ) = ψ(λ) − ψ(4λ), then Dj = {

ψ(L )

ϕ(2−2j L )

if j = 0, if j ≥ 1.

s s The Besov, Bp,q (M, (W , 1)), and Triebel–Lizorkin, Fp,q (M, (W , 1)), spaces can be equivalently defined using these spectral multipliers, as the next result shows. We use Notations 6.2.1 and 6.2.2 and Definition 6.2.3.

482 � 6 Besov and Triebel–Lizorkin spaces Proposition 6.12.1. Fix s ∈ ℝ. For f ∈ C0∞ (M)′ the following are equivalent: (i) f ∈ X s (M, (W , 1)), (ii) ‖f ‖V ,s,𝒟0 < ∞. Moreover, in this case we have ‖f ‖X s (W ,1) ≈ ‖f ‖V ,s,𝒟0 . Proof. (i) ⇔ (ii): This follows from a simple reprise of the proof of Proposition 2.4.18. The key is that due to the spectral theorem we have an analog of the Calderón reproducing formula (Lemma 2.4.10). Indeed, for C0 ∈ ℕ define ̃ C0 := D j

∑ Dk . |k−j|≤C0 k∈ℕ

̃ C0 , ∀j. Using this and replacThen if C0 = C0 (ψ) is sufficiently large, we have Dj = Dj D j ing Lemma 2.4.16 with Lemma 6.4.9, the proof of Proposition 2.4.18 goes through nearly unchanged to establish (i) ⇔ (ii). That ‖f ‖X s (W ,1) ≈ ‖f ‖V ,s,𝒟0 is a special case of Proposition 6.2.6. Remark 6.12.2. We presented the above in the special case where ds1 = ⋅ ⋅ ⋅ = dsr = 1. Similar results for more general formal degrees and more general operators L have recently been obtained by Lingxiao Zhang [254] using the theory from this text. See also Remark 5.12.6.

6.13 Further questions In this chapter, we have restricted attention to those results which are most directly useful for our study of maximally subelliptic PDEs. There are a number of very natural questions which we did not pursue. We mention a few of those here. s – We restricted our attention to p, q ∈ [1, ∞] for the Besov spaces Bp,q and to p ∈ s (1, ∞), q ∈ (1, ∞] for the Triebel–Lizorkin spaces Fp,q . In particular, we did not 0 study the local Hardy spaces Fp,2 where p ∈ (0, 1] or bounded mean oscillation

–

0 5 (bmo) F∞,2 , though the local Hardy and bmo spaces have been studied by other authors on more general spaces of homogeneous type (see the references in Section 6.14). It would be interesting to extend the values of p and q for which our spaces are defined. s s In the definitions of Bp,q and Fp,q , we used the space ℓq (ℕν ). When, for example,

ν = 2, one could replace ℓq (ℕ2 ) in the definitions with ℓq1 (ℕ; ℓq1 (ℕ)) and obtain

s 5 In the classical setting, when studying F∞,q one needs a modification of the definition; see [244, Section 2.3.4]. To study such spaces in the generality of this text would likely require a similar modification.

6.14 Further reading and references

–

–

– –

� 483

s s a more general definition. One could even mix the definitions of Bp,q and Fp,q by q1 p q2 using the space ℓ (ℕ; L (M, Vol; ℓ (ℕ))). In this way one could create many more scales of multi-parameter function spaces which are naturally associated with our multi-parameter geometries. It would be interesting to have a general theory of such function spaces. There is an extensive interpolation and embedding theory for the classical Besov and Triebel–Lizorkin spaces. There are many interesting questions relating to what extent these results generalize to the spaces studied here. The classical Besov and Triebel–Lizorkin spaces can be defined using the heat equation; see [244, Section 2.12.2]. In light of Theorem 8.1.1 (i) ⇒ (ix), it seems likely that in the single-parameter case ν = 1, the same methods of this chapter could be used s (W ,ds) and ‖ ⋅ ‖F s (W ,ds) can be equivalently defined to prove that the norms ‖ ⋅ ‖Bp,q p,s using the heat equation for any non-negative self-adjoint extension of P ∗ P , where P is any maximally subelliptic operator. To what extent is the loss of ϵ derivatives in Theorem 6.6.13 necessary? See Remark 6.6.14. In Section 6.10 it was shown that if s ∈ (0, ∞)ν , the Besov and Triebel–Lizorkin spaces consisted of precisely those distributions of finite norm (see Proposition 6.10.1). If s ∈ ℝν , then by Remark 6.4.6 one has a similar result with a more complicated norm, but it is unclear if one can use the simpler norm used in Proposition 6.10.1. Can Proposition 6.10.1 be extended to all s ∈ ℝν ? By being more precise about the norm, this can be done in some cases; see Proposition 6.12.1.

6.14 Further reading and references s s There is a vast body of literature6 on Besov (Bp,q ) and Triebel–Lizorkin (Fp,q ) spaces, mostly in the single-parameter case ν = 1, where s ∈ ℝ. We are interested in settings where we can develop concrete characterizations of these spaces. For the most part, this literature can be separated into two nearly distinct categories: s s (i) Theories which define Bp,q and/or Fp,q for all s ∈ ℝ. These are usually based off n of ℝ as in the classical theory (see [244] for a detailed history) or more generally based on some sort of Lie groups (see [92, 90, 207] for some early examples of this).7 s s (ii) Theories which define Bp,q and/or Fp,q for |s| small (e. g., |s| < 1). With the restriction that |s| be small, one can define these function spaces for much more general underlying spaces. There has been much work on this topic when the underlying

6 Because the literature on function spaces is so extensive, we make no attempt to cite every important work. Rather, we hope to give the reader a starting point for further reading in any topic of interest.

0 7 Much work has been done on the Hardy spaces Fp,2 , p ∈ (0, 1]. Even though we did not address Hardy spaces in this chapter, many of the methods developed for Hardy spaces are important for studying the Besov and Triebel–Lizorkin spaces in generality.

484 � 6 Besov and Triebel–Lizorkin spaces space is a space of homogeneous type. This began with the work of Han and Sawyer [106], under some additional hypotheses on the space of homogeneous type. Some other early important works are [114, 115, 112, 113]. A breakthrough by Auscher and Hytönen [5, 6] allowed the creation of an appropriate Littlewood–Paley theory on general spaces of homogeneous type using wavelets, leading to a theory of Besov and Triebel–Lizorkin spaces (for |s| small) on general spaces of homogeneous type. See the recent papers [110, 107, 108, 116, 138, 23, 24, 22, 21, 95, 96, 117] for various approaches to this theory. The distinction between (i) and (ii) comes from the following observation. On ℝn (or more generally a Lie group) one can make sense of what it means to have high-order moments vanish (or, similarly, to make sense of high-order differences). On a general space of homogeneous type, it is only clear how to define first moments vanishing (similarly, first-order differences). The setting of this chapter lies between the two categories described above. We do not have any sort of group structure with which to work, though (at least when ν = 1) we are working on a (local) space of homogeneous type (see Remark 3.3.2). At this intermediate level of generality, we are able to develop the Besov and Triebel–Lizorkin spaces for all s. The idea is that, though there is no underlying group structure (even locally), there is an intrinsic and quantitative notion of high levels of smoothness adapted to our geometry (see Definition 5.2.21). This is in contrast to a general space of homogeneous type, where one can only define smoothness up to Lipschitz regularity. Elementary operators are appropriately scaled high-order derivatives of elementary operators up to controllable error terms (see Proposition 5.5.5 (h)): this is the analog we use of having “high-order moments vanish.” See Example 2.3.8 for a description of why this can be viewed as an analog of having high-order moments vanish. See Section 7.7 for the analog we use of high-order differences. s ⃗ can be viewed as Lp Sobolev As described in Section 6.9, the spaces Fp,2 (𝒦, (W , ds)) spaces. From this perspective, these generalize the non-isotropic Sobolev spaces which were first used outside of the group setting by Rothschild and Stein [200, Section 16], where the definition was restricted to the special case s ∈ ℕ. These Sobolev spaces were s later used in [182, 140]. A special case of the spaces Fp,2 (𝒦, (W , ds)) described in this chapter was defined in [220, Section 2.10], where the connection with the sub-Laplacian was made more explicit. In the multi-parameter setting, ν ≥ 2, there has been considerable work studying Besov and Triebel–Lizorkin spaces when the underlying space is a product of singleparameter spaces. Early important work on this in ℝn1 × ℝn2 was due to Chang and R. Fefferman [35–37], Journé [136], and Pipher [197]; see also [160, 68, 70, 71, 41]. For some results beyond the Euclidean setting see, for example, [109, 43, 42, 148, 159]. Beyond the product setting, less is known. For the flag setting, see for example [111, 72]. For spaces on ℝn with two different underlying homogeneities, see for example [69, 67, 66]. Lu, Shen, and Zhang [158] studied the boundedness of the multi-parameter singular integrals from

6.14 Further reading and references

� 485

[220] on Hardy spaces and local Hardy spaces, results which have many similarities to Theorem 6.3.10. s ⃗ in the multi-parameter setting, were defined The Sobolev spaces Fp,2 (𝒦, (W , ds)), and studied in [220, Section 5.1.1], and were defined in an even more general setting in [221].

7 Zygmund–Hölder spaces s As we saw in Section 2.8, the classical Zygmund–Hölder spaces C s (ℝn ) = B∞,∞ (ℝn ), where s > 0, play an important role in the study of nonlinear elliptic PDEs. For maxs ⃗ where imally subelliptic operators, an analogous role is played by B∞,∞ (𝒦, (W , ds)), ν s ∈ (0, ∞) . In this chapter, we develop the theory of this space, which will be useful when we turn to studying nonlinear equations in Chapter 9. Throughout this chapter, M is a connected C ∞ manifold of dimension n ∈ ℕ+ .1 We work in the general setting described in Section 5.1.3. Thus, for some ν ∈ ℕ+ we are given ν lists of vector fields with formal degrees: μ

μ

∞ (W μ , dsμ ) = {(W1 , ds1 ), . . . , (Wrμμ , dsμrμ )} ⊂ Cloc (M; TM) × ℕ+ .

We assume that (W 1 , ds1 ), . . . , (W ν , dsν ) pairwise locally weakly approximately commute and that for each μ ∈ {1, . . . , ν}, Gen((W μ , dsμ )) is locally finitely generated on M. We define (W , ds)⃗ = {(W1 , ds1⃗ ), . . . , (Wr , dsr⃗ )}

∞ := (W 1 , ds1 ) ⊠ (W 2 , ds2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (W ν , dsν ) ⊂ Cloc (M; TM) × (ℕν \ {0}).

In addition, we make the additional assumption that W1 , . . . , Wr satisfy Hörmander’s condition on M. Throughout this chapter, 𝒦 will denote a compact set and Ω will denote an open, relatively compact set, with 𝒦 ⋐ Ω ⋐ M. Definition 7.0.1. For s ∈ (0, ∞)ν we define the Zygmund–Hölder spaces with respect to ⃗ := B s (𝒦, (W , ds)) ⃗ and C s (𝒦, (W , ds); ⃗ ℂN ) := B s (𝒦, (W , ds); ⃗ (W , ds)⃗ by C s (𝒦, (W , ds)) ∞,∞ ∞,∞ N ℂ ) with equality of norms: ‖ ⋅ ‖C s (W ,ds)⃗ := ‖ ⋅ ‖Bs

⃗

∞,∞ (W ,ds)

,

‖ ⋅ ‖C s ((W ,ds);ℂ ⃗ N ) := ‖ ⋅ ‖Bs

⃗ N ∞,∞ ((W ,ds);ℂ )

.

As in Definition 6.5.7, we set s

Ccpt (W , ds)⃗ :=

⋃

𝒦⋐M 𝒦 compact

s

⃗ C (𝒦, (W , ds)).

We define the classical Zygmund–Hölder spaces consisting of functions with compact s s support by Cstd (𝒦) := B∞,∞,std (𝒦) and 1 In this chapter, the only Lp space we consider is L∞ . Because of this, it does not matter what strictly positive, smooth density Vol we use on M: they all give rise to the same norm ‖ ⋅ ‖L∞ (M) . https://doi.org/10.1515/9783111085647-007

7.1 The norm

s

Cstd,cpt (M) :=

⋃

� 487

s

𝒦⋐M 𝒦 compact

Cstd (𝒦).

⃗ The main purpose of this section is to develop the parts of the theory of C s (𝒦, (W , ds)) which are useful when studying nonlinear PDEs. The key results are: ⃗ forms an algebra (Proposition 7.4.1). – C s (𝒦, (W , ds)) – For appropriately smooth functions F(x, ζ ), where x ∈ M and ζ ∈ ℝN , we have ⃗ F(x, u(x)) ∈ C s (𝒦, (W , ds)), –

⃗ ℝN ). ∀u ∈ C s (𝒦, (W , ds);

See Section 7.5. In the single-parameter case, ν = 1, we present a difference characterization of C s (𝒦, (W , ds)) and describe the connection with the classical Hölder spaces with respect to the metric ρ(W ,ds) (x, y). See Sections 7.7 and 7.7.1.

7.1 The norm ⃗ the norm ‖f ‖ s For s ∈ (0, ∞)ν and f ∈ C s (𝒦, (W , ds)), C (W ,ds)⃗ is defined as follows. Let ∞ ψ ∈ C0 (Ω) satisfy ψ ≡ 1 on a neighborhood of 𝒦. By Proposition 5.8.3 we may decompose Mult[ψ] = ∑j∈ℕν Dj , where {(Dj , 2−j ) : j ∈ ℕν } is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω. Then ‖f ‖C s (W ,ds)⃗ = ‖f ‖Bs

⃗

∞,∞ (W ,ds)

= sup 2j⋅s ‖Dj f ‖L∞ . j∈ℕν

Proposition 6.3.7 shows that the equivalence class of this norm does not depend on any of the choices we have made, and Proposition 6.10.1 shows that s

⃗ = {f ∈ C (M) : supp(f ) ⊆ 𝒦, ‖f ‖ s C (𝒦, (W , ds)) W ,0 C (W ,ds)⃗ < ∞}. ∞

′

(7.1)

We use the flexibility that we may choose any such {(Dj , 2−j ) : j ∈ ℕν }, and we will use the particular choice from Section 6.11, that is, we choose Ω0 , Ω1 , and Ω2 as in that section, we let a > 0 be small, and we define Dj as in (6.170). The main result of this section is that with this choice, we have the following proposition. Proposition 7.1.1. Fix ψ ∈ C0∞ (Ω) with ψ ≡ 1 on a neighborhood of 𝒦. There exists a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω, {(Dj , 2−j ) : j ∈ ℕν } such that the following holds. Set Pj := ∑k≤j Dj . Then: (i) ∑j∈ℕν Dj = Mult[ψ]; see Proposition 5.5.10 for the convergence of this sum. (ii) ∀j ∈ ℕν , we have Pj , Dj ∈ C0∞ (Ω × Ω). ⃗ (iii) ∀s ∈ (0, ∞)ν , ∀f ∈ C s (𝒦, (W , ds)), ∑ Dj f = ψf ,

j∈ℕν

488 � 7 Zygmund–Hölder spaces with convergence in C(M), that is, Pj f → f in C(M) as j1 , . . . , jν → ∞. (iv) {(Pj , 2−j ) : j ∈ ℕν } is a bounded set of generalized (W , ds)⃗ pre-elementary operators supported in Ω. Remark 7.1.2. Proposition 7.1.1 (ii) is the reason we assumed the additional assumption in this chapter that W1 , . . . , Wr satisfy Hörmander’s condition on M. ⃗ Remark 7.1.3. We take, for f ∈ C s (𝒦, (W , ds)), ‖f ‖C s (W ,ds)⃗ := ‖f ‖ℓ∞ (ℕν ;L∞ (M)),s,𝒟0 = sup 2j⋅s ‖Dj f ‖L∞ (M) j∈ℕν

= ‖f ‖Bs

⃗

⃗

∞,∞ (W ,ds),(X,d),a,Ω0 ,Ω1 ,Ω2 ,Vol

.

⃗ ℂN ), we take See Definition 6.11.2. More generally, for f ∈ C s (𝒦, (W , ds); j⋅s ‖f ‖C s ((W ,ds);ℂ ⃗ N ) := ‖f ‖ℓ∞ (ℕν ;L∞ (M;ℂN )),s,𝒟0 = sup 2 ‖Dj f ‖L∞ (M;ℂN ) . j∈ℕν

Before we prove Proposition 7.1.1, we state a corollary which is the main way Proposition 7.1.1 will be used. Corollary 7.1.4. Let Dj and Pj be as in Proposition 7.1.1. Then: (i) For every ordered multi-index α, ⃗ α 󵄩 󵄩 sup 󵄩󵄩󵄩(2−jdsW ) Pj 󵄩󵄩󵄩L∞ (M)→L∞ (M) < ∞,

j∈ℕν

⃗ α 󵄩 󵄩 sup 󵄩󵄩󵄩(2−jdsW ) Dj 󵄩󵄩󵄩L∞ (M)→L∞ (M) < ∞.

j∈ℕν

(ii) For each μ ∈ {1, . . . , ν} and M ∈ ℕ, we can write μ

Dj = ∑ 2−(M−|αμ |)jμ Ej,αμ ,μ,M (2−jμ ds W μ )αμ ,

(7.2)

󵄩 󵄩 sup sup 󵄩󵄩󵄩Ej,αμ ,μ,M 󵄩󵄩󵄩L∞ (M)→L∞ (M) < ∞.

(7.3)

|αμ |≤M

where j∈ℕν |αμ |≤M

Proof. (i): Since {(Dj , 2−j ) : j ∈ ℕν } is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω, it is a bounded set of generalized (W , ds)⃗ pre-elementary operators supported in Ω (see Definition 5.2.27). Also, by Proposition 7.1.1 (iv), {(Pj , 2−j ) : j ∈ ℕν } is a bounded set of generalized (W , ds)⃗ pre-elementary operators supported in Ω. From here, (i) follows from the definition of bounded sets of generalized (W , ds)⃗ preelementary operators (see Definition 5.2.26). (ii): Because {(Dj , 2−j ) : j ∈ ℕν } is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω, Proposition 5.5.5 (h) shows that we may write Dj as in (7.2),

7.1 The norm

� 489

where ℰμ,M := {(Ej,αμ ,μ,M , 2−j ) : j ∈ ℕν , |αμ | ≤ M} is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω. In particular, ℰμ,M is a bounded set of generalized (W , ds)⃗ pre-elementary operators supported in Ω (see Definition 5.2.27). From here, (7.3) follows from the definition of bounded sets of generalized (W , ds)⃗ pre-elementary operators (see Definition 5.2.26). Proof of Proposition 7.1.1. By Proposition 6.11.1 (i), {(Dj , 2−j ) : j ∈ ℕν } is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω. Item (iv) follows from Proposition 6.11.1 (ii). Item (i) follows from Proposition 6.11.1 (iii). (ii): Since each Pj is given by the finite sum Pj = ∑k≤j Dj , it suffices to show that Dj ∈ C0∞ (Ω1 × Ω2 ). We use the formula for Dj given in (6.170). By Proposition 6.11.3 (ii) ⃗ a pseudo-differential we already know that {(Dj , 2−j ) : j ∈ ℕν } is a bounded set of (X, d ), operator scales supported in Ω1 × Ω2 . In particular, supp(Dj ) ⋐ Ω1 × Ω2 , for each j ∈ ℕν . ∞ ⃗ Thus, it suffices to show that Dj ∈ Cloc (M × M). By Proposition 3.8.8 (a), Gen((W , ds)) is finitely generated by (X, d )⃗ on Ω. In particular, since W1 , . . . , Wr satisfy Hörmander’s condition, X1 (x), . . . , Xq (x) must span Tx M, ∀x ∈ Ω. By the formula for Γ(t, x) (see (6.167)), 󵄨 we have 𝜕tj 󵄨󵄨󵄨t=0 Γ(t, x) = −Xj (x), for 1 ≤ j ≤ q. In particular, for x ∈ Ω, the q × n matrix ⃗ 󵄨 dt Γ(t, x)󵄨󵄨󵄨t=0 has full rank. The function Dild2j (ςj )(x, t, t) is supported in x ∈ Ω1 and t ∈ Bq (a). Since Ω1 ⋐ Ω, if a > 0 is sufficiently small, we may apply the change of variables u = Γ(t, x) in the t variable to see Dj f (x) = ∫ f (Γ(t, x)) Dild2j (ςj )(x, t, t) dt = ∫ f (u)Kj (x, u) du, ⃗

∞ where Kj (x, u) ∈ Cloc (M × M), completing the proof of (ii). ⃗ It follows from (i) that ∑j∈ℕν Dj f = ψf with convergence (iii): Let f ∈ C s (𝒦, (W , ds)). ∞ ′ in Cloc (M) . Thus, to complete the proof of (iii) it suffices to show that ∑j∈ℕν Dj f converges in C(M). It follows from (ii) that Dj f ∈ C0∞ (M) ⊆ C(M) for every j ∈ ℕν , and therefore it suffices to show that ∑j∈ℕν Dj f converges in L∞ (M). Since {(Dj , 2−j ) : j ∈ ℕν } is a bounded set of generalized (W , ds)⃗ elementary operators, we have, by Corollary 6.4.4 s (with V = ℓ∞ (ℕν ; L∞ (M)), X s = B∞,∞ , and ℰ = {(Dj , 2−j ) : j ∈ ℕν }),

sup 2j⋅s ‖Dj f ‖L∞ ≲ ‖f ‖C s (W ,ds)⃗ .

j∈ℕν

Since s ∈ (0, ∞)ν , it follows that ∑j∈ℕν Dj f converges in L∞ (M), completing the proof. Remark 7.1.5. In Section 7.7, we will use the above choice of Dj in the single-parameter setting (ν = 1). Thus, we are given Hörmander vector fields with formal degrees (W , ds), and we pick (X, d ) = {(X1 , d1 ), . . . , (Xq , dq )} ⊂ Gen((W , ds)) such that Gen((W , ds)) is finitely generated by (X, d ) on Ω. The construction is: – Pick any ψ ∈ C0∞ (Ω) with ψ ≡ 1 on a neighborhood of 𝒦.

490 � 7 Zygmund–Hölder spaces – – –

̂ Pick any ρ ∈ S (ℝq ) with ρ(ξ) ≡ 1 on a neighborhood of 0. For a > 0 sufficiently small (how small depends on supp(ψ)), pick any η ∈ C0∞ (Bq (a)) with η ≡ 1 on a neighborhood of 0. For j ∈ ℕ, we define ςj ∈ S (ℝq ) by ςĵ (ξ) := {

–

̂ ρ(ξ),

̂ − ρ(2 ̂ ξ), ρ(ξ) d

j=0

j ≥ 1.

Note that ςj ∈ S0 (ℝq ) for j ≥ 1 and ςj = ς1 for all j ≥ 1. Set Dj f (x) = ψ(x) ∫ f (e−t⋅X x)η(t) Dild2j (ςj )(t) dt.

7.2 Decomposition into smooth functions Similar to the classical setting on ℝn (see Section 2.5.1), one of the most convenient ways ⃗ is to decompose them into a sum of smooth functions. to study functions in C s (𝒦, (W , ds)) The next proposition gives such a decomposition. Proposition 7.2.1. Let s ∈ (0, ∞)ν . For f ∈ C0∞ (M)′ , the following are equivalent: ⃗ (i) f ∈ C s (𝒦, (W , ds)). (ii) supp f ⊆ 𝒦 and there exists a sequence {fj }j∈ℕν ⊂ C0∞ (M) such that for every ordered multi-index α ⃗ α 󵄩 󵄩 sup 2j⋅s 󵄩󵄩󵄩(2−jdsW ) fj 󵄩󵄩󵄩L∞ (M) < ∞

j∈ℕν

and f = ∑j∈ℕν fj with convergence in C(M). In this case, there exists M = M(s, ν) ∈ ℕ such that ⃗ α 󵄩 󵄩 ‖f ‖C s (W ,ds)⃗ ≤ C ∑ sup 2j⋅s 󵄩󵄩󵄩(2−jdsW ) fj 󵄩󵄩󵄩L∞ (M) . ν |α|≤M j∈ℕ

(7.4)

Furthermore, fj may be chosen such that fj ∈ C0∞ (Ω), for every j ∈ ℕν , and for every ordered multi-index α, ⃗ α 󵄩 󵄩 sup 2j⋅s 󵄩󵄩󵄩(2−jdsW ) fj 󵄩󵄩󵄩L∞ (M) ≤ Cα ‖f ‖C s (W ,ds)⃗ .

j∈ℕν

(7.5)

Here, C and Cα do not depend on f , but may depend on any other ingredient in the proposition.

7.2 Decomposition into smooth functions

� 491

⃗ Then by definition supp(f ) ⊆ 𝒦. Let Dj Proof. Suppose (i) holds, i. e., f ∈ C s (𝒦, (W , ds)). be as in Proposition 7.1.1 and set fj := Dj f . By Proposition 7.1.1 (ii), Dj ∈ C0∞ (Ω × Ω) and therefore fj ∈ C0∞ (Ω). By Proposition 7.1.1 (iii), ∑j∈ℕν fj = f , with convergence in C(M). Fi-

nally, let α be an ordered multi-index and set Ej,α := (2−jdsW )α Dj . Since {(Dj , 2−j ) : j ∈ ℕν } is a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω, Proposition 5.5.5 (e) implies that ℰα := {(Ej,α , 2−j ) : j ∈ ℕν } is also a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω. Thus, by Corollary 6.4.4, we have ⃗

⃗ α 󵄩 󵄩 󵄩 󵄩 sup 2j⋅s 󵄩󵄩󵄩(2−jdsW ) fj 󵄩󵄩󵄩L∞ (M) = sup 2j⋅s 󵄩󵄩󵄩Ej,α f 󵄩󵄩󵄩L∞ (M)

j∈ℕν

j∈ℕν

= ‖f ‖ℓ∞ (ℕν ;L∞ (M)),s,ℰα ≲ ‖f ‖Bs

⃗

∞,∞ (W ,ds)

= ‖f ‖C s (W ,ds)⃗ .

Thus, (ii) holds, and moreover we have fj ∈ C0∞ (Ω), ∀j ∈ ℕν , and (7.5) holds. Suppose (ii) holds. Fix M ≥ ν(|s|∞ + 1). We will show that for all j, k ∈ ℕν , ⃗ α 󵄩 󵄩󵄩 󵄩 󵄩 −M|(j∨k)−k|∞ ∑ 󵄩󵄩󵄩(2−k dsW ) fk 󵄩󵄩󵄩L∞ . 󵄩󵄩Dj fk 󵄩󵄩󵄩L∞ ≲ 2

(7.6)

|α|≤M

First, we see why (7.6) will complete the proof. Indeed, we then have 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩Dj f 󵄩󵄩L∞ ≤ ∑ 󵄩󵄩󵄩Dj fk 󵄩󵄩󵄩L∞ k∈ℕν

′ ′ ⃗ α 󵄩 󵄩 ≲ ( sup 2k ⋅s ∑ 󵄩󵄩󵄩(2−k dsW ) fk ′ 󵄩󵄩󵄩L∞ ) ∑ 2−M|(j∨k)−k|∞ 2−k⋅s . ′ ν

k ∈ℕ

(7.7)

k∈ℕν

|α|≤M

Also, ∑ 2−M|(j∨k)−k|∞ 2−k⋅s ≤ ∑ 2−|(j∨k)−k|1 2−(j∨k)⋅s

k∈ℕν

k∈ℕν

=

≤ ≲

∏

∑ 2−((jμ ∨kμ )−kμ ) 2−(jμ ∨kμ )sμ

μ∈{1,...,ν} kμ ∈ℕ

∏ ( ∑ 2−(jμ −kμ ) 2−jμ sμ + ∑ 2−kμ sμ )

μ∈{1,...,ν} kμ ≤jμ

∏

μ∈{1,...,ν}

2

−jμ sμ

(7.8)

kμ ≥jμ

=2

−j⋅s

.

Combining (7.7) and (7.8) establishes (7.4). In particular, since ‖f ‖C s (W ,ds)⃗ < ∞ and ⃗ and therefore (i) holds. supp f ⊆ 𝒦, (7.1) shows that f ∈ C s (𝒦, (W , ds)) Thus, we will complete the proof by proving (7.6). If |(j ∨ k) − k|∞ = 0, then since 󵄩 󵄩 supj∈ℕν ‖Dj ‖L∞ →L∞ < ∞ (Corollary 7.1.4 (i)), we have 󵄩󵄩󵄩Dj fk 󵄩󵄩󵄩L∞ ≲ ‖fk ‖L∞ , proving (7.6) in this case.

492 � 7 Zygmund–Hölder spaces We turn to proving (7.6) when |(j∨k)−k|∞ > 0. Fix μ ∈ {1, . . . , ν} such that jμ −kμ = |(j∨ k) − j|∞ . Letting Ej,αμ ,μ,M be as in Corollary 7.1.4 (ii), so that ‖Ej,αμ ,μ,M ‖L∞ →L∞ ≲ 1, we have μ 󵄩󵄩 󵄩 −j (M−|αμ |) 󵄩 󵄩󵄩Ej,α ,μ,M (2−jμ ds W μ )αμ fk 󵄩󵄩󵄩 ∞ 󵄩󵄩Dj fk 󵄩󵄩󵄩L∞ ≤ ∑ 2 μ 󵄩 μ 󵄩L

|αμ |≤M

μ α 󵄩 󵄩 ≲ ∑ 2−jμ (M−|αμ |) 2−(jμ −kμ ) degdsμ (αμ ) 󵄩󵄩󵄩(2−kμ ds W μ ) μ fk 󵄩󵄩󵄩L∞

|αμ |≤M

μ α 󵄩 󵄩 ≤ ∑ 2−M(jμ −kμ ) 󵄩󵄩󵄩(2−kμ ds W μ ) μ fk 󵄩󵄩󵄩L∞

|αμ |≤M

⃗ α 󵄩 󵄩 ≤ 2−M|(j∨k)−k|∞ ∑ 󵄩󵄩󵄩(2−k dsW ) fk 󵄩󵄩󵄩L∞ , |α|≤M

establishing (7.6) and completing the proof. ⃗ ⊆ C(M) and the inclusion is continuous. Corollary 7.2.2. For s ∈ (0, ∞)ν , C s (𝒦, (W , ds)) ⃗ Proposition 7.2.1 shows that f = ∑j∈ℕν fj , where fj ∈ Proof. For f ∈ C s (𝒦, (W , ds)), ∞ C0 (M) ⊆ C(M), with convergence in C(M). This proves that f ∈ C(M). Moreover, by choosing fj so that (7.5) holds, we have ‖f ‖C(M) ≤ ∑ ‖fj ‖L∞ ≲ ∑ 2−j⋅s ‖f ‖C s (W ,ds)⃗ ≲ ‖f ‖C s (W ,ds)⃗ , j∈ℕν

j∈ℕν

proving that the inclusion map is continuous.

7.3 Bounds of some sums In this chapter, we require some elementary estimates for sums, which we present in this section. The two main estimates we require are the next two lemmas. Lemma 7.3.1. Let s ∈ (0, ∞)ν and M > |s|1 . Then, ∀k ∈ ℕν , ∑ 2−M|(k∨k ∨j)−(k ∨j)|∞ 2−k ⋅s 2−j⋅s ≤ Cs,M,ν 2−k⋅s . ′

′

′

j,k ′ ∈ℕν

Lemma 7.3.2. Let s ∈ (0, ∞)ν , t > 0, M1 > |s|1 , and M2 > t. Then, for all k ∈ ℕν and l ∈ ℕ, ∑ 2−M1 |(k∨k ∨j)−(k ∨j)|∞ 2−M2 |(l∨l )−l | 2−k ⋅s 2−j⋅s 2−l t ≤ Cs,t,M1 ,M2 ,ν 2−k⋅s 2−lt . ′

′

′

′

′

′

j,k ′ ∈ℕν l′ ∈ℕ

We begin by proving two easier lemmas in the single-parameter case.

7.3 Bounds of some sums

� 493

Lemma 7.3.3. Let s > 0 and M > s. Then, ∀k ∈ ℕ, ∑ 2−M|(k∨k )−k | 2−k s ≤ Cs,M 2−ks . ′

′

′

k ′ ∈ℕ

Proof. We have ∑ 2−M|(k∨k )−k | 2−k s = ∑ 2−k s ≲ 2−ks ′

′

′

′

k ′ ≥k

k ′ ≥k

(7.9)

and ∑ 2−M|(k∨k )−k | 2−k s = ∑ 2−M(k−k ) 2−k s ′

′

′

′

k ′ ≤k

′

k ′ ≤k

≤ 2−ks ∑ 2−(M−s)|k−k | ≲ 2−ks . ′

(7.10)

k ′ ∈ℕ

Combining (7.9) and (7.10) completes the proof. Lemma 7.3.4. Let s > 0 and M > s. Then, ∀k ∈ ℕ, ∑ 2−M|(k∨k ∨j)−(k ∨j)| 2−k s 2−js ≤ Cs,M 2−ks . ′

′

′

j,k ′ ∈ℕ

Proof. The sum is symmetric in j and k ′ , so it suffices to estimate just the sum ∑j≤k ′ . We have ∑ 2−M|(k∨k ∨j)−(k ∨j)| 2−k s 2−js ′

′

′

j≤k ′

≤ ( ∑ 2−js )( ∑ 2−M|(k∨k )−k | 2−ks ) ≲ 2−ks , ′

j∈ℕ

′

k ′ ∈ℕ

where the final estimate follows from Lemma 7.3.3, completing the proof. Proof of Lemma 7.3.1. Pick M1 , . . . , Mν with M = ∑νμ=1 Mμ and Mμ > sμ . We have, using Lemma 7.3.4, ∑ 2−M|(k∨k ∨j)−(k ∨j)|∞ 2−k ⋅s 2−j⋅s ′

′

′

j,k ′ ∈ℕν

≤ ≲ completing the proof.

∏ ( ∑ 2−Mμ |(kμ ∨kμ ∨jμ )−(kμ ∨jμ )| 2−kμ sμ 2−jμ sμ ) ′

μ∈{1,...,ν} jμ ,kμ′ ∈ℕ

∏

μ∈{1,...,ν}

2−kμ sμ = 2−k⋅s ,

′

′

494 � 7 Zygmund–Hölder spaces Proof of Lemma 7.3.2. Applying Lemmas 7.3.3 and 7.3.1, we have ∑ 2−M1 |(k∨k ∨j)−(k ∨j)|∞ 2−M2 |(l∨l )−l | 2−k ⋅s 2−j⋅s 2−l t ′

′

′

′

′

′

j,k ′ ∈ℕν l′ ∈ℕ

= ( ∑ 2−M1 |(k∨k ∨j)−(k ∨j)|∞ 2−k ⋅s 2−j⋅s )( ∑ 2−M2 |(l∨l )−l | 2−l t ) ′

′

j,k ′ ∈ℕν

′

′

′

′

l′ ∈ℕ

≲ 2−k⋅s 2−lt , completing the proof.

7.4 Algebra ⃗ is When considering nonlinear PDEs, an important property of the space C s (𝒦, (W , ds)) that it forms an algebra. Indeed, we have the following. ⃗ we have fg ∈ C s (𝒦, (W , ds)), ⃗ Proposition 7.4.1. Fix s ∈ (0, ∞)ν . For f , g ∈ C s (𝒦, (W , ds)), and moreover, ‖fg‖C s (W ,ds)⃗ ≤ C‖f ‖C s (W ,ds)⃗ ‖g‖L∞ + ‖f ‖L∞ ‖g‖C s (W ,ds)⃗ ,

(7.11)

⃗ ≥ 0. where C = C(s, 𝒦, (W , ds)) The estimate (7.11) is called a tame estimate and such estimates are very important in the study of nonlinear PDEs (see Remark 7.5.4). It is somewhat easier to prove the weaker non-tame estimate: (7.12)

‖fg‖C s (W ,ds)⃗ ≲ ‖f ‖C s (W ,ds)⃗ ‖g‖C s (W ,ds)⃗ .

We defer the proof of the tame estimate (7.11) to Theorem 7.5.2 (iii), and in this section prove only the easier non-tame estimate (7.12), as a warm-up for the more difficult proofs which follow. The proof of (7.11) uses a more complicated para-paraproduct decomposition which we introduce to study more general compositions. Before we prove (7.12), we need some new notation. Notation 7.4.2 (The product rule). We often consider derivatives of the form (2−jdsW )α , where α is an ordered multi-index. Recall that this means that α = (α1 , . . . , αL ) is a list with αj ∈ {1, . . . , r}. We then have ⃗

α

(2−jdsW ) = (2−j⋅dsα1 Wα1 ) ⋅ ⋅ ⋅ (2−j⋅dsαL WαL ). ⃗

⃗

⃗

We write ∑β1 +β̇ 2 =α to denote the sum over non-overlapping sublists β1 , β2 of α which combine to make the full list α. For example, ∑β1 +β̇ 2 =(1,2) sums over the choices

7.4 Algebra

� 495

{β1 = (), β2 = (1, 2)}, {β1 = (1), β2 = (2)}, {β1 = (2), β2 = (1)}, {β1 = (1, 2), β2 = ()}. We take this sum with repetitions, so for example ∑β1 +β̇ 2 =(1,1) sums over {β1 = (), β2 = (1, 1)}, {β1 = (1), β2 = (1)}, {β1 = (1), β2 = (1)}, {β1 = (1, 1), β2 = ()}. With this notation, we have the product rule: α

(2−jdsW ) (fg) = ⃗

β

β

∑ ((2−jdsW ) 1 f )((2−jdsW ) 2 g). ⃗

⃗

̇ 2 =α β1 +β

We similarly define sums with more terms, for example ∑β1 +β̇ 2 +⋅⋅⋅ ̇ +β ̇ R =α , and obtain a similar product rule. ⃗ ⊆ C(M), so the Proof of the algebra property and (7.12). By Corollary 7.2.2, C s (𝒦, (W , ds)) product fg is defined pointwise. Without loss of generality, we assume ‖f ‖C s (W ,ds)⃗ = ‖g‖C s (W ,ds)⃗ = 1. By Proposition 7.2.1, we may decompose f = ∑k∈ℕν fk and g = ∑l∈ℕν gl , where for every ordered multi-index α, ⃗ ⃗ α 󵄩 α 󵄩 󵄩 󵄩 2k⋅s 󵄩󵄩󵄩(2−k dsW ) fk 󵄩󵄩󵄩L∞ , 2l⋅s 󵄩󵄩󵄩(2−ldsW ) gl 󵄩󵄩󵄩L∞ ≲ 1.

Fix M > |s|1 . We will show that ‖Dj (fk gl )‖L∞ ≲ 2−M|(j∨k∨l)−(k∨l)| 2−k⋅s 2−l⋅s .

(7.13)

First, we see why (7.13) completes the proof. Indeed, we have, using (7.13) and Lemma 7.3.1, 󵄩󵄩 󵄩 󵄩 󵄩 −M|(j∨k∨l)−(k∨l)| −k⋅s −l⋅s 2 2 ≲ 2−j⋅s . 󵄩󵄩Dj fg 󵄩󵄩󵄩L∞ ≤ ∑ 󵄩󵄩󵄩Dj fk gl 󵄩󵄩󵄩L∞ ≲ ∑ 2 k,l∈ℕν

k,l∈ℕν

This establishes (7.12). Moreover, since supp(f ), supp(g) ⊆ 𝒦, we have supp(fg) ⊆ 𝒦. ⃗ Since (7.12) implies ‖fg‖C s (W ,ds)⃗ < ∞, it follows from (7.1) that fg ∈ C s (𝒦, (W , ds)). We turn to proving (7.13). If k ∨ l = j ∨ k ∨ l, then using ‖Dj ‖L∞ →L∞ ≲ 1 (see Corollary 7.1.4 (i)), we have 󵄩󵄩 󵄩 −k⋅s −l⋅s 󵄩󵄩Dj (fk gl )󵄩󵄩󵄩L∞ ≲ ‖fk gl ‖L∞ ≤ ‖fk ‖L∞ ‖gl ‖L∞ ≲ 2 2 , proving (7.13) in this case. Suppose |(j ∨ k ∨ l) − (k ∨ l)|∞ > 0. Pick μ ∈ {1, . . . , ν} such that jμ − (kμ ∨ lμ ) = |(j ∨ k ∨ l) − (k ∨ l)|∞ . Letting Ej,αμ ,μ,M be as in Corollary 7.1.4 (ii), we have ‖Ej,αμ ,μ,M ‖L∞ →L∞ ≲ 1 and (using Notation 7.4.2 and Corollary 7.1.4 (ii))

496 � 7 Zygmund–Hölder spaces μ 󵄩󵄩 󵄩 −j (M−|αμ |) 󵄩 󵄩󵄩Ej,α ,μ,M (2−jμ ds W μ )αμ (fk gl )󵄩󵄩󵄩 ∞ 󵄩󵄩Dj (fk gl )󵄩󵄩󵄩L∞ ≤ ∑ 2 μ 󵄩 μ 󵄩L

|αμ |≤M

μ α 󵄩 󵄩 ≲ ∑ 2−jμ (M−|αμ |) 󵄩󵄩󵄩(2−jμ ds W μ ) μ (fk gl )󵄩󵄩󵄩L∞

|αμ |≤M

μ

≤ ∑

∑

= ∑

∑

|αμ |≤M βμ +β ̇ 2μ =αμ 1

μ

μ μ β β 󵄩 󵄩 󵄩 󵄩 2−jμ (M−|αμ |) 󵄩󵄩󵄩(2−jμ ds W μ ) 1 fk 󵄩󵄩󵄩L∞ 󵄩󵄩󵄩(2−jμ ds W μ ) 2 gl 󵄩󵄩󵄩L∞ 1

|αμ |≤M βμ +β ̇ 2μ =αμ 1

2

2−jμ (M−|αμ |) 2−(jμ −kμ ) degdsμ (βμ ) 2−(jμ −lμ ) degdsμ (βμ ) μ

μ

μ μ β β 󵄩 󵄩 󵄩 󵄩 × 󵄩󵄩󵄩(2−kμ ds W μ ) 1 fk 󵄩󵄩󵄩L∞ 󵄩󵄩󵄩(2−lμ ds W μ ) 2 gl 󵄩󵄩󵄩L∞

≲ ∑

1

∑

|αμ |≤M βμ +β ̇ 2μ =αμ 1

2

2−jμ (M−|αμ |) 2−(jμ −kμ ) degdsμ (βμ ) 2−(jμ −lμ ) degdsμ (βμ )

≤ 2−(jμ −(kμ ∨lμ ))M = 2−|(j∨k∨l)−(k∨l)|∞ M , where in the final ≤ we have used degdsμ (βμ1 ) + degdsμ (βμ2 ) = degdsμ (αμ ) ≥ |αμ |. This proves (7.13) and completes the proof.

7.5 Compositions ⃗ ℝN ), we wish to consider compositions of Fix N ∈ ℕ+ . For functions u ∈ C s (𝒦, (W , ds); N the form F(x, u(x)), where F(x, ζ ) : M × ℝ → ℂ is a sufficiently smooth function.2 ⃗ in the x variInformally, the class of functions F(x, ζ ) we consider are C s (𝒦, (W , ds)) t N able and C (ℝ ) in the ζ variable (see Definition 2.5.1). To rigorously define this product Zygmund–Hölder space, let ψ ∈ C0∞ (Ω) satisfy ψ ≡ 1 on a neighborhood of 𝒦 and let {(Dj , 2−j ) : j ∈ ℕν } be a bounded set of generalized (W , ds)⃗ elementary operators supported in Ω with ∑j∈ℕν Dj = Mult[ψ] (we will usually take Dj from Proposition 7.1.1). Let ̃ l , 2−l ) : l ∈ ℕν } be given by (2.30) with n replaced by N; in particular, it is a bounded {(D ̃ l = I. Set set of elementary operators on ℝN in the sense of Definition 2.3.5 and ∑l∈ℕ D ̂ ̃ Dj,l := Dj ⊗ Dl . Definition 7.5.1. For s ∈ (0, ∞)ν and t > 0, we let C s,t (𝒦 × ℝN , (W , ds)⃗ ⊠ ∇ℝN ) denote the space of all F ∈ C0∞ (M × ℝN )′ such that the following norm is finite: ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

)

󵄩 ̂ 󵄩󵄩 := sup 2j⋅s 2lt 󵄩󵄩󵄩D j,l F 󵄩 󵄩L∞ (M×ℝN ) < ∞. j∈ℕν l∈ℕ

2 Here, we are considering functions u(x) taking values in ℝN instead of ℂN (as was used in most of the rest of the text). There is no real difference for our purposes: functions taking values in ℝN can be viewed as functions taking values in ℂN , while functions taking values in ℂN can be viewed as functions taking values in ℝ2N . The use of ℝN in this section makes the notation a little simpler.

7.5 Compositions

� 497

For M ∈ ℕ+ , we define the vector-valued space C s,t (𝒦 × ℝN , (W , ds)⃗ ⊠ ∇ℝN ; ℂM ) to consist of those distributions F = (F1 , . . . , FM ) with each Fk ∈ C s,t (𝒦 × ℝN , (W , ds)⃗ ⊠ ∇ℝN ). We set M

‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

;ℂM )

:= ∑ ‖Fk ‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

k=1

)

and s,t

Ccpt ((W , ds)⃗ ⊠ ∇ℝN ) :=

C

⋃

s,t

𝒦⋐M 𝒦 compact

(𝒦 × ℝN , (W , ds)⃗ ⊠ ∇ℝN ).

(7.14)

The main result of this section is the next theorem. Theorem 7.5.2. Fix s ∈ (0, ∞)ν and t > 0. Let F ∈ C s,t (𝒦 × ℝN , (W , ds)⃗ ⊠ ∇ℝN ), u, v ∈ ⃗ ℝN ), and w ∈ C s (𝒦, (W , ds)). ⃗ Then: C s (𝒦, (W , ds); (i) If t > ⌊|s|1 ⌋ + 1 + ν, then F(x, u(x) + ζ ) ∈ C s,t−⌊|s|1 ⌋−1−ν (𝒦 × ℝN , (W , ds)⃗ ⊠ ∇ℝN ) and ‖F(x, u(x) + ζ )‖C s,t−⌊|s|1 ⌋−1−ν ((W ,ds)⊠∇ ⃗

ℝN

≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

(ii)

) (1

) ⌊|s|1 ⌋+ν

+ ‖u‖C s (W ,ds)⃗ )(1 + ‖u‖L∞ )

.

If t > ⌊|s|1 ⌋ + 1 + ν, then ⃗ F(x, u(x)) ∈ C s (𝒦, (W , ds)) and ‖F(x, u(x))‖C s (W ,ds)⃗ ≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

) (1

⌊|s|1 ⌋+ν

+ ‖u‖C s (W ,ds)⃗ )(1 + ‖u‖L∞ )

.

⃗ and (iii) We have u ⋅ v ∈ C s (𝒦, (W , ds)) ‖u ⋅ v‖C s (W ,ds)⃗ ≲ ‖u‖C s (W ,ds)⃗ ‖v‖L∞ + ‖u‖L∞ ‖v‖C s (W ,ds)⃗ . (iv) If t > ⌊|s|1 ⌋ + 1 + ν we have F(x, u(x) + ζ )w(x) ∈ C s,t−⌊|s|1 ⌋−1−ν (𝒦 × ℝN , (W , ds)⃗ ⊠ ∇ℝN ) and ‖F(x, u(x) + ζ )w(z)‖C s,t−⌊|s|1 ⌋−1−ν ((W ,ds)⊠∇ ⃗

ℝN

≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

) (‖w‖C s (W ,ds)⃗

) ⌊|s|1 ⌋+ν

+ ‖u‖C s (W ,ds)⃗ ‖w‖L∞ )(1 + ‖u‖L∞ )

.

498 � 7 Zygmund–Hölder spaces (v)

If t > ⌊|s|1 ⌋ + 2 + ν, ‖F(x, u(x) + ζ ) − F(x, v(x) + ζ )‖C s,t−⌊|s|1 ⌋−2−ν ((W ,ds)⊠∇ ⃗

ℝN

≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

) (‖u

)

− v‖C s (W ,ds)⃗ + (‖u‖C s (W ,ds)⃗ + ‖v‖C s (W ,ds)⃗ )‖u − v‖L∞ ) ⌊|s|1 ⌋+ν

× (1 + ‖u‖L∞ + ‖v‖L∞ )

.

(vi) If t > ⌊|s|1 ⌋ + 2 + ν, then ‖F(x, u(x)) − F(x, v(x))‖C s (W ,ds)⃗ ≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

) (‖u

− v‖C s (W ,ds)⃗ + (‖u‖C s (W ,ds)⃗ + ‖v‖C s (W ,ds)⃗ )‖u − v‖L∞ ) ⌊|s|1 ⌋+ν

× (1 + ‖u‖L∞ + ‖v‖L∞ )

.

β

(vii) Fix L ∈ ℕ+ and suppose 𝜕ζ F(x, 0) ≡ 0, ∀|β| < L and t > ⌊|s|1 ⌋ + 1 + ν + L. Then ‖F(x, u(x))‖C s (W ,ds)⃗ ≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

) (1

⌊|s|1 ⌋+ν+1

+ ‖u‖L∞ )

‖u‖L−1 L∞ ‖u‖C s (W ,ds)⃗ .

β

(viii) Fix L ∈ ℕ, L ≥ 2 and suppose 𝜕ζ F(x, 0) ≡ 0, ∀0 < |β| < L and t > ⌊|s|1 ⌋ + 1 + ν + L. Then ‖F(x, u(x)) − F(x, v(x))‖C s (W ,ds)⃗ ≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

) (1

+ ‖u‖L∞ + ‖v‖L∞ )

⌊|s|1 ⌋+ν+1

L−2

(‖u‖L∞ + ‖v‖L∞ )

× (‖u‖C s (W ,ds)⃗ + ‖v‖C s (W ,ds)⃗ )‖u − v‖L∞ + ‖F‖C s,t ((W ,ds)⊠∇ ⃗

) ℝN

L−1

(‖u‖L∞ + ‖v‖L∞ )

‖u − v‖C s (W ,ds)⃗ .

The implicit constants do not depend on F, u, v, or w, but may depend on any of the other ingredients in the result. Remark 7.5.3. Theorem 7.5.2 (vi) says, roughly, that the map u 󳨃→ F(x, u(x)) is Lipschitz in u. This will allow us to use arguments similar to the contraction mapping principle when studying fully nonlinear maximally subelliptic PDEs. Remark 7.5.4. All the estimates in Theorem 7.5.2 are tame in the sense that each term on the right-hand side of the inequalities only has one factor of ‖ ⋅ ‖C s (W ,ds)⃗ ; all the other factors are ‖ ⋅ ‖L∞ . See also Remark 2.6.5. Tame estimates play an essential role in the proof in Section 9.2.4. Remark 7.5.5. It is likely that Theorem 7.5.2 assumes too much regularity in the ζ variable (i. e., the result is probably not sharp in t). However, finding the sharp result seems to be difficult. See also Remark 2.6.4. Remark 7.5.6. In Definition 7.5.1, we have used the fact that W1 , . . . , Wr satisfy Hörman∞ ∞ der’s condition, and therefore CW ,0 (M) = C0 (M) (see Remark 5.2.24). Thus, in the proof

7.5 Compositions

� 499

̂ j,l F = (Dj ⊗ D ̃ l )F, for any F ∈ C ∞ (M × ℝN )′ ≅ which follows, it makes sense to consider D 0 ∞ ′̂ ∞ N ′ C0 (M) ⊗C0 (ℝ ) . 7.5.1 Properties of the product Zygmund–Hölder spaces ⃗ The space C s,t (𝒦 ×ℝN , (W , ds)⊠∇ ℝN ) is closely related to the (ν+1)-parameter Zygmund– Hölder spaces C (s,t) associated with the vector fields with (ν + 1)-parameter formal degrees: (W , ds)⃗ ⊠ {(𝜕x1 , 1), . . . , (𝜕xn , 1)} The difference is that in C s,t (𝒦 × ℝN , (W , ds)⃗ ⊠ ∇ℝN ) we do not require that the functions F(x, ζ ) have compact support in the ζ variable. This is possible since the vector fields 𝜕x1 , . . . , 𝜕xn are easy to study at both large and small scales (since they are homogeneous under the usual global family of dilations on ℝn ), while general smooth vector fields can only be easily studied as small scales. ⃗ can be easily modified to prove The proofs that we have exhibited for C s (𝒦, (W , ds)) s,t N ⃗ similar results for C (𝒦×ℝ , (W , ds)⊠∇ℝN ). We state some of these results without proof here. Proposition 7.5.7. For s ∈ (0, ∞)ν and t > 0, the following are equivalent: (i) F ∈ C s,t (𝒦 × ℝN , (W , ds)⃗ ⊠ ∇ℝN ). (ii) supp(F) ⊆ 𝒦 × ℝN and for every bounded set of generalized (W , ds)⃗ elementary operators, ℰ1 , and every bounded set of elementary operators on ℝN , ℰ2 (see Definition 2.3.5), we have sup

{(Ej ,2−j ):j∈ℕν }⊆ℰ1

sup

󵄩 󵄩 sup 2j⋅s 2lt 󵄩󵄩󵄩(Ej ⊗ Ẽl )F 󵄩󵄩󵄩L∞ (M×ℝN ) < ∞. ν

̃l ,2−l ):l∈ℕ}⊆ℰ2 j∈ℕ {(E l∈ℕ

⃗ Proposition 7.5.8. For s ∈ (0, ∞)ν and t > 0, C s,t (𝒦 ×ℝN , (W , ds)⊠∇ ℝN ) is a Banach space when endowed with the norm ‖ ⋅ ‖C s,t ((W ,ds)⊠∇ . ⃗ N) ℝ

ν

Proposition 7.5.9. Let s ∈ (0, ∞) and t > 0. For F ∈ C0∞ (M × ℝN )′ , the following are equivalent: (i) F ∈ C s,t (𝒦 × ℝN , (W , ds)⃗ ⊠ ∇ℝN ). ∞ (ii) supp(F) ⊆ 𝒦 × ℝN and there exists a sequence {Fj,l }j∈ℕν ⊂ Cloc (M × ℝN ) such that for l∈ℕ

every ordered multi-index α and every multi-index β,

⃗ α β 󵄩 󵄩 sup 2j⋅s 2lt 󵄩󵄩󵄩(2−jdsWx ) (2−l 𝜕ζ ) Fj,l (x, ζ )󵄩󵄩󵄩L∞ (M×ℝN ) < ∞,

j∈ℕν l∈ℕ

with F = ∑j,l Fj,l , where the sum converges in C(M × ℝN ).

500 � 7 Zygmund–Hölder spaces In this case, there exists M = M(s, t, ν) ∈ ℕ such that ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

)

≤C

∑

⃗ α β 󵄩 󵄩 sup 2j⋅s 2lt 󵄩󵄩󵄩(2−jdsWx ) (2−l 𝜕ζ ) Fj,l (x, ζ )󵄩󵄩󵄩L∞ (M×ℝN ) .

ν |α|,|β|≤M j∈ℕ l∈ℕ

Furthermore, Fj,l can be chosen such that supp(Fj,l ) ⊆ Ω × ℝN , ∀j, l, and for each ordered multi-index α and multi-index β, ⃗ α β 󵄩 󵄩 sup 2j⋅s 2lt 󵄩󵄩󵄩(2−jdsWx ) (2−l 𝜕ζ ) Fj,l (x, ζ )󵄩󵄩󵄩L∞ (M×ℝN ) ≤ Cα,β ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

j∈ℕν l∈ℕ

).

(7.15)

Here, C and Cα,β do not depend on F or Fj,l , but may depend on any of the other ingredients in the result. Proposition 7.5.10. For s ∈ (0, ∞)ν and t > 0, C s,t (𝒦 × ℝN , (W , ds)⃗ ⊠ ∇ℝN ) ⊆ C(M × ℝN ) and the inclusion is continuous. Proposition 7.5.11. Let s ∈ (0, ∞)ν , β ∈ ℕN a multi-index, and t > |β|. Then F(x, ζ ) 󳨃→ β s,t−|β| ⃗ ⃗ 𝜕ζ F(x, ζ ) is a continuous map C s,t (𝒦 ×ℝN , (W , ds)⊠∇ (𝒦 ×ℝN , (W , ds)⊠∇ ℝN ) 󳨃→ C ℝN ). We include one final result in this section, which is not an analog of a previous result, but instead a simple consequence of Proposition 7.5.9. Corollary 7.5.12. For s ∈ (0, ∞)ν and t > 0, we have: s ⃗ ⃗ (i) The map F(x, ζ ) 󳨃→ F(x, 0) is continuous C s,t (𝒦 ×ℝN , (W , ds)⊠∇ ℝN ) → C (𝒦, (W , ds)). s,t N ⃗ (ii) The map F(x, ζ1 , . . . , ζN ) 󳨃→ F(x, ζ1 , . . . , ζN−1 , 0) is continuous C (𝒦 × ℝ , (W , ds) ⊠ ∇ℝN ) → C s,t (𝒦 × ℝN−1 , (W , ds)⃗ ⊠ ∇ℝN−1 ). (iii) For x0 ∈ M, the map F(x, ζ ) 󳨃→ F(x0 , ζ ) is continuous C s,t (𝒦 × ℝN , (W , ds)⃗ ⊠ ∇ℝN ) → C t (ℝN ). Proof. Let F ∈ C s,t (𝒦 × ℝN , (W , ds)⃗ ⊠ ∇ℝN ). By Proposition 7.5.10, F ∈ C(M × ℝN ), so the functions F(x, 0) ∈ C(M) and F(x, ζ1 , . . . , ζN−1 , 0) are defined pointwise. Proposition 7.5.9 ∞ shows that we may write F = ∑j∈ℕν Fj,l , where Fj,l ∈ Cloc (M × ℝN ) and for every ordered l∈ℕ

multi-index α and every multi-index β ∈ ℕN we have α −l β 󵄩󵄩 −jds⃗ 󵄩 −j⋅s −lt 󵄩󵄩(2 Wx ) (2 𝜕ζ ) Fj,l (x, ζ )󵄩󵄩󵄩L∞ (M×ℝN ) ≲ 2 2 ‖F‖C s,t ((W ,ds)⊠∇ ⃗ ). ℝN

(7.16)

(i): Set Gj (x) := ∑l∈ℕ Fj,l (x, 0) so that F(x, 0) = ∑j∈ℕν Gj (x). Using (7.16) with β = 0, we have, for every ordered multi-index α, α 󵄩󵄩 −jds⃗ α 󵄩󵄩 󵄩 −jds⃗ 󵄩 󵄩󵄩(2 W ) Gj 󵄩󵄩L∞ (M) ≤ ∑ 󵄩󵄩󵄩(2 Wx ) Fj,l (x, ζ )󵄩󵄩󵄩L∞ (M×ℝN ) l∈ℕ

≲ ∑ 2−j⋅s 2−lt ‖F‖C s,t ((W ,ds)⊠∇ ⃗ l∈ℕ

) ℝN

≲ 2−j⋅s ‖F‖C s,t ((W ,ds)⊠∇ ⃗

) ℝN

.

(7.17)

� 501

7.5 Compositions

Combining (7.17) with the fact that supp(F(⋅, 0)) ⊆ 𝒦 (since supp(F) ⊆ 𝒦 × ℝN ), Propo⃗ with ‖F(⋅, 0)‖ s sition 7.2.1 shows that F(⋅, 0) ∈ C s (𝒦, (W , ds)) ⃗ C (W ,ds)⃗ ≲ ‖F‖C s,t ((W ,ds)⊠∇ ), ℝN completing the proof. (ii): Let ζ ′ = (ζ1 , . . . , ζN−1 ). It follows directly from (7.16) that for every ordered multiindex α and every multi-index β′ ∈ ℕN−1 , we have α −l β′ 󵄩󵄩 −jds⃗ 󵄩 ′ −j⋅s −lt 󵄩󵄩(2 Wx ) (2 𝜕ζ ′ ) Fj,l (x, ζ , 0)󵄩󵄩󵄩L∞ (M×ℝN ) ≲ 2 2 ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

).

Since F(x, ζ ′ , 0) = ∑j∈ℕν Fj,l (x, ζ ′ , 0) and supp(F(x, ζ ′ , 0)) ⊆ 𝒦 × ℝN−1 , since supp(F) ⊆ l∈ℕ

𝒦 × ℝN , Proposition 7.5.9 implies F(x, ζ ′ , 0) ∈ C s,t (𝒦 × ℝN−1 , (W , ds)⃗ ⊠ ∇ℝN−1 ) and

‖F(x, ζ ′ , 0)‖C s,t ((W ,ds)⊠∇ ⃗

ℝN−1

)

≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

),

completing the proof.

(iii): Set Gl (ζ ) := ∑j∈ℕν Fj,l (x0 , ζ ) so that F(x0 , ζ ) = ∑l∈ℕ Gl (ζ ). Using (7.16) with |α| = 0, we have, for every multi-index β, α 󵄩󵄩 −lt β 󵄩󵄩 󵄩 −jds⃗ 󵄩 󵄩󵄩(2 𝜕ζ ) Gl 󵄩󵄩L∞ (ℝN ) ≤ ∑ 󵄩󵄩󵄩(2 Wx ) Fj,l (x, ζ )󵄩󵄩󵄩L∞ (M×ℝN ) j∈ℕν

≲ ∑ 2−j⋅s 2−lt ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

j∈ℕν

)

≲ 2−lt ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝ

. N)

(7.18)

Combining (7.18) with Proposition 2.5.16 shows that ‖F(x0 , ζ )‖C t (ℝN ) ≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗ ), ℝN completing the proof.

7.5.2 Proof of the composition theorem In this section, we prove Theorem 7.5.2. As we will see, the main difficulties are contained in Theorem 7.5.2 (i); the rest of Theorem 7.5.2 follows as an easy consequence. To prove Theorem 7.5.2 (i), we use the next lemma. Lemma 7.5.13. Fix s ∈ (0, ∞)ν and t > 0. Set M1 := ⌊|s|1 ⌋ + 1 and M2 := ⌊t⌋ + 1. Suppose ∞ that for each j, k ∈ ℕν and l ∈ ℕ, Hj,k,l (x, ζ ) ∈ Cloc (M × ℝN ) are such that ∀μ ∈ {1, . . . , ν}, |αμ | ≤ M1 , and |β| ≤ M2 , 󵄩󵄩 −(kμ ∨jμ )dsμ μ αμ −l β 󵄩 W ) (2 𝜕ζ ) Hj,k,l 󵄩󵄩󵄩L∞ (M×ℝN ) ≤ A2−k⋅s 2−j⋅s 2−lt , 󵄩󵄩(2 for some constant A ≥ 0. Set H(x, ζ ) := ∑j,k∈ℕν ∑l∈ℕ Hj,k,l (x, ζ ) and suppose supp(H) ⊆ 𝒦 × ℝN . Then H ∈ C s,t (𝒦 × ℝN , (W , ds)⃗ ⊠ ∇ℝN ) and ‖H‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

)

≤ Cs,t,(W ,ds),𝒦,N A. ⃗

502 � 7 Zygmund–Hölder spaces Proof. We will show that for j, k, k ′ ∈ ℕν and l, l′ ∈ ℕ, 󵄩󵄩 ̂ 󵄩 −M |(j∨k∨k ′ )−(j∨k ′ )|∞ −M2 |(l∨l′ )−l′ | −j⋅s −k ′ ⋅s −l′ t 2 2 2 2 . 󵄩󵄩Dk,l Hj,k ′ ,l′ 󵄩󵄩󵄩L∞ (M×ℝN ) ≲ A2 1

(7.19)

First, we see why (7.19) will complete the proof. Indeed, using (7.19) we have 󵄩󵄩 ̂ 󵄩 󵄩̂ 󵄩󵄩 󵄩󵄩Dk,l H 󵄩󵄩󵄩L∞ (M×ℝN ) ≤ ∑ 󵄩󵄩󵄩D k,l Hj,k ′ ,l′ 󵄩 󵄩L∞ (M×ℝN ) j,k ′ ∈ℕν l′ ∈ℕ

≲ A ∑ 2−M1 |(j∨k∨k )−(j∨k )|∞ 2−M2 |(l∨l )−l | 2−j⋅s 2−k ⋅s 2−l t ′

′

′

′

′

′

j,k ′ ∈ℕν l′ ∈ℕ −k⋅s −lt

≲ A2

(7.20)

2 ,

where in the final estimate we have used M1 > |s|1 , M2 > t and applied Lemma 7.3.2. By hypothesis, we have supp(H) ⊆ 𝒦 × ℝN , and (7.20) implies ‖H‖C s,t ((W ,ds)⊠∇ ⃗ ) ≲ A, which ℝN completes the proof. We turn to proving (7.19), which we separate into several cases. Suppose |(k ∨ k ′ ∨ j) − (k ′ ∨ j)|∞ = 0 and |(l ∨ l′ ) − l′ | = 0. By Corollary 7.1.4 (i) and Lemma 2.3.26, we have ̂ k,l ‖ ∞ ̃ ‖D L (M×ℝN )→L∞ (M×ℝN ) = ‖Dk ‖L∞ (M)→L∞ (M) ‖Dl ‖L∞ (ℝN )→L∞ (ℝN ) ≲ 1. Thus, 󵄩󵄩 ̂ 󵄩 󵄩 󵄩 −j⋅s −k ′ ⋅s −l′ t 2 , 󵄩󵄩Dk,l Hj,k ′ ,l′ 󵄩󵄩󵄩L∞ ≲ 󵄩󵄩󵄩Hj,k ′ ,l′ 󵄩󵄩󵄩L∞ ≤ A2 2 completing the proof of (7.19) in this case. Next we consider the case where |(k ∨ k ′ ∨ j) − (k ′ ∨ j)|∞ > 0. Pick μ ∈ {1, . . . , ν} such that kμ − (kμ′ ∨ jμ ) = |(k ∨ k ′ ∨ j) − (k ′ ∨ j)|∞ . Let Ek,αμ ,μ,M1 be as in Corollary 7.1.4 (ii). We consider two subcases: |(l ∨ l′ ) − l′ | = 0 and |(l ∨ l′ ) − l′ | > 0. If |(l ∨ l′ ) − l′ | = 0, we use ̃ l ‖ ∞ N ∞ N ≲ 1 (Lemma 2.3.26) to see that ‖D L (ℝ )→L (ℝ ) μ 󵄩󵄩 ̂ 󵄩 −(M −|α |)k 󵄩 ̃ l )(2−kμ ds W μ )αμ Hj,k ′ ,l′ 󵄩󵄩󵄩 ∞ 󵄩󵄩Dk,l Hj,k ′ ,l′ 󵄩󵄩󵄩L∞ ≤ ∑ 2 1 μ μ 󵄩󵄩󵄩(Ek,αμ ,μ,M1 ⊗ D 󵄩L

|αμ |≤M1

μ α 󵄩 󵄩 ≲ ∑ 2−(M1 −|αμ |)kμ 󵄩󵄩󵄩(2−kμ ds W μ ) μ Hj,k ′ ,l′ 󵄩󵄩󵄩L∞

|αμ |≤M1

′ ′ μ α 󵄩 󵄩 ≲ ∑ 2−(M1 −|αμ |)kμ 2− degdsμ (αμ )(kμ −(kμ ∨jμ )) 󵄩󵄩󵄩(2−(kμ ∨jμ )ds W μ ) μ Hj,k ′ ,l′ 󵄩󵄩󵄩L∞

|αμ |≤M1

≲ 2−M1 (kμ −(kμ ∨jμ )) A2−k ⋅s 2−j⋅s 2−l t ′

′

′

= A2−M1 |(j∨k∨k )−(j∨k )|∞ 2−k ⋅s 2−j⋅s 2−l t , ′

′

′

′

where we have used degdsμ (αμ ) ≥ |αμ |, establishing (7.19). If |(l ∨ l′ ) − l′ | > 0, we apply Proposition 2.3.18 (d) to write

� 503

7.5 Compositions

̃ l = ∑ 2−l(M2 −|β|) Ẽl,β (2−l 𝜕ζ )β , D

(7.21)

|β|≤M2

where {(Ẽl,β , 2−l ) : l ∈ ℕ, |β| ≤ M2 } is a bounded set of elementary operators on ℝN , and therefore, by Lemma 2.3.26, ‖Ẽl,β ‖L∞ (ℝN )→L∞ (ℝN ) ≲ 1. Thus, we have 󵄩󵄩 ̂ 󵄩 󵄩󵄩Dk,l Hj,k ′ ,l′ 󵄩󵄩󵄩L∞ ≤ ∑

μ α β 󵄩 󵄩 ∑ 2−(M1 −|αμ |)kμ 2−l(M2 −|β|) 󵄩󵄩󵄩(Ek,αμ ,μ,M1 ⊗ Ẽl,β )(2−kμ ds W μ ) μ (2−l 𝜕ζ ) Hj,k ′ ,l′ 󵄩󵄩󵄩L∞

|αμ |≤M1 |β|≤M2

≲ ∑

μ α β 󵄩 󵄩 ∑ 2−(M1 −|αμ |)kμ 2−l(M2 −|β|) 󵄩󵄩󵄩(2−kμ ds W μ ) μ (2−l 𝜕ζ ) Hj,k ′ ,l′ 󵄩󵄩󵄩L∞

|αμ |≤M1 |β|≤M2

≲ ∑

∑ 2−(M1 −|αμ |)kμ 2− degdsμ (αμ )(kμ −(kμ ∨jμ )) 2−l(M2 −|β|) 2−|β|(l−l ) ′

′

|αμ |≤M1 |β|≤M2 ′ μ ′ α β 󵄩 󵄩 × 󵄩󵄩󵄩(2−(kμ ∨jμ )ds W μ ) μ (2−l 𝜕ζ ) Hj,k ′ ,l′ 󵄩󵄩󵄩L∞

≲ A2−M1 (kμ −(kμ ∨jμ )) 2−M2 (l−l ) A2−k ⋅s 2−j⋅s 2−l t ′

′

′

′

= A2−M1 |(j∨k∨k )−(j∨k )|∞ 2−M2 |(l∨l )−l | 2−j⋅s 2−k ⋅s 2−l t , ′

′

′

′

′

′

where we have used degdsμ (αμ ) ≥ |αμ |, establishing (7.19). Finally, we consider the case where |(k ∨ k ′ ∨ j) − (k ′ ∨ j)|∞ = 0 and |(l ∨ l′ ) − l′ | > 0. As in the previous case, we use (7.21), where ‖Ẽl,β ‖L∞ →L∞ ≲ 1. Using ‖Dk ‖L∞ →L∞ ≲ 1 (Corollary 7.1.4 (i)), we have 󵄩󵄩 ̂ 󵄩 −l(M2 −|β|) 󵄩 󵄩󵄩(Dk ⊗ Ẽl,β )(2−l 𝜕ζ )β Hj,k ′ ,l′ 󵄩󵄩󵄩 ∞ 󵄩󵄩Dk,l Hj,k ′ ,l′ 󵄩󵄩󵄩L∞ ≤ ∑ 2 󵄩 󵄩L |β|≤M2

′ 󵄩 ′ β 󵄩 ≲ ∑ 2−l(M2 −|β|) 2−|β|(l−l ) 󵄩󵄩󵄩(2−l 𝜕ζ ) Hj,k ′ ,l′ 󵄩󵄩󵄩L∞

|β|≤M2

≲ 2−M2 (l−l ) A2−j⋅s 2−k ⋅s 2−l t = A2−M2 |(l∨l )−l | 2−j⋅s 2−k ⋅s 2−l t , ′

′

′

′

′

′

′

establishing (7.19) and completing the proof. To prove Theorem 7.5.2 (i), we wish to decompose F(x, u(x) + ζ ) as in Lemma 7.5.13. ⃗ For E ⊆ {1, . . . , ν}, let eE = Let F ∈ C s,t (𝒦 × ℝN , (W , ds)⃗ ⊠ ∇ℝN ) and u ∈ C s (𝒦, (W , ds)). ∑μ∈E eμ , where e1 , . . . , eν is the standard basis for ℝν . Decompose F = ∑ Fk,l k∈ℕν l∈ℕ

(7.22)

∞ as in Proposition 7.5.9, with Fk,l ∈ Cloc (M × ℝN ) satisfying (7.15). The decomposition of F(x, u(x) + ζ ) we use is the following:

504 � 7 Zygmund–Hölder spaces F(x, u(x) + ζ ) = lim F(x, P(J,J,...,J) u(x) + ζ ) J→∞

= ∑

∑

j∈ℕν E⊆{μ:jμ =0} ̸

= ∑

∑

(−1)|E| F(x, Pj−eE u(x) + ζ )

̸ j,k∈ℕν E⊆{μ:jμ =0} l∈ℕ

(7.23)

(−1)|E| Fk,l (x, Pj−eE u(x) + ζ )

=: ∑ Hj,k,l (x, ζ ). j,k∈ℕν l∈ℕ

∞ Note that Hj,k,l ∈ Cloc (M × ℝN ). The decomposition (7.23) is the decomposition to which we wish to apply Lemma 7.5.13. Thus, we wish to estimate the functions Hj,k,l (x, ζ ). Before we enter into details, we describe how we will study Hj,k,l in the special case ν = 2 to help give the reader an overview of how we will proceed. Suppose g(σ1 , σ2 ) is a function of two variables. Then

g(1, 1) − g(1, 0) − g(0, 1) + g(0, 0) = ∬ 𝜕σ1 𝜕σ2 g(σ1 , σ2 ) dσ. [0,1]2

In the case ν = 2, when j ∈ ℕ2+ , we write Hj,k,l (x, ζ ) as Hj,k,l (x, ζ ) = Fk,l (x, P(j1 ,j2 ) u(x) + ζ ) − Fk,l (x, P(j1 −1,j2 ) u(x) + ζ )

− Fk,l (x, P(j1 ,j2 −1) u(x) + ζ ) + Fk,l (x, P(j1 −1,j2 −1) u(x) + ζ )

= ∬ 𝜕σ1 𝜕σ2 Fk,l (x, Pj1 −1,j2 −1 u(x) [0,1]2

(7.24)

+ σ1 Dj1 ,j2 −1 u(x) + σ2 Dj1 −1,j2 u(x) + σ1 σ2 Dj1 ,j2 u(x) + ζ ) dσ.

Thus, to estimate Hj,k,l , it suffices to instead estimate the above integrand. By the chain ⃗ rule, this integrand involves the factors Dj1 −1,j2 u, Dj1 ,j2 −1 u, or Dj1 ,j2 u. If u ∈ C s (𝒦, (W , ds)), −j⋅s then these factors give a gain of 2 , which is the key to our estimates. If j1 ≥ 1, but j2 = 0, we instead use Hj,k,l (x, ζ ) = Fk,l (x, P(j1 ,0) u(x) + ζ ) − Fk,l (x, P(j1 −1,0) u(x) + ζ ) = ∫ 𝜕σ1 Fk,l (x, Pj1 −1,0 u(x) + σ1 Dj1 ,0 u(x) + ζ ) dσ1 . [0,1]

Again the chain rule gives a gain of 2−j⋅s . A similar formula works when j2 ≥ 1 and j1 = 0. Finally, when j = (j1 , j2 ) = (0, 0), 2−j⋅s = 1, so we can directly use the formula Hj,k,l (x, ζ ) = Fk,l (x, P0 u(x) + ζ ).

7.5 Compositions

� 505

∞ Notation 7.5.14. For a function G(x, ζ ) ∈ Cloc (M × ℝN ) and L ∈ ℕ+ , we write DLζ G(x, ζ ) for the L-th derivative of G(x, ζ ) in the ζ variable. Thus, DLζ G(x, ζ ) is an L-linear form 󵄩 󵄩 on ℝN . We write 󵄩󵄩󵄩DLζ G(x, ζ )󵄩󵄩󵄩L∞ for the L∞ (M × ℝN ) norm of DLζ G(x, ζ ) thought of as taking values in the space of L-linear forms on ℝN with the usual norm. In other words, 󵄩󵄩 L 󵄩󵄩 N 󵄩󵄩Dζ G(x, ζ )󵄩󵄩L∞ is the least constant such that for all v1 , . . . , vL ∈ ℝ

󵄩󵄩 L 󵄩 󵄩 L 󵄩 󵄩󵄩(Dζ G(x, ζ ))[v1 , . . . , vL ]󵄩󵄩󵄩L∞ (M×ℝN ) ≤ 󵄩󵄩󵄩Dζ G(x, ζ )󵄩󵄩󵄩L∞ |v1 | ⋅ ⋅ ⋅ |vL |. When L = 1, we may identify Dζ G(x, ζ ) with a vector in ℝN . Thus, it makes sense to consider ‖Dζ G(x, ζ )‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

)

:= ‖Dζ G(x, ζ )‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

;ℝN ) .

Lemma 7.5.15. Fix s ∈ (0, ∞)ν , t ∈ ℝ, and A ≥ 0. Fix k ∈ ℕν and l ∈ ℕ and suppose we ∞ are given Gk,l (x, ζ ) ∈ Cloc (M × ℝN ) satisfying, for all α and β, 󵄩󵄩 −k ds⃗ α −l 󵄩 −k⋅s −lt 󵄩󵄩(2 W ) (2 𝜕ζ )Gk,l 󵄩󵄩󵄩L∞ (M×ℝN ) ≤ Cα,β A0 2 2 .

(7.25)

Let u ∈ L∞ (M; ℝN ). For E ⊆ {1, . . . , ν}, let aE ∈ [0, 1]. For j ∈ ℕν , set {Pj−e{μ:j =0}̸ u + ∑E⊊{μ:jμ =0} ̸ aE Dj−eE u, j ≠ 0, μ uj := { j = 0. {P0 u, Then, for all μ ∈ {1, . . . , ν}, αμ , and β, 󵄩󵄩 −(jμ ∨kμ )dsμ μ αμ −l β 󵄩 W ) (2 𝜕ζ ) Gk,l (x, uj (x) + ζ )󵄩󵄩󵄩L∞ (M×ℝN ) 󵄩󵄩(2 |α | ≤ C̃μ,α ,β A0 2−k⋅s 2−l(t−|αμ |) (1 + ‖u‖L∞ ) μ ,

(7.26)

μ

where C̃μ,αμ ,β does not depend on A0 , j, k ∈ ℕν , l ∈ ℕ, aE ∈ [0, 1], or u ∈ L∞ (M; ℝN ). Proof. By Corollary 7.1.4 (i), we have, for all α, 󵄩󵄩 −jds⃗ α 󵄩󵄩 󵄩󵄩(2 W ) uj 󵄩󵄩L∞ ≲ 1.

(7.27)

Consider, using Notations 7.4.2 and 7.5.14, μ

α

β

(2−(jμ ∨kμ )ds W μ ) μ (2−l 𝜕ζ ) Gk,l (x, uj (x) + ζ ) α

μ

β

= ((2−(jμ ∨kμ )ds Wxμ ) μ (2−l 𝜕ζ ) Gk,l )(x, uj (x) + ζ ) +

∑

∑

̇ 2 +⋅⋅⋅ ̇ +γ ̇ L ̇ μ2 γ1 +γ αμ1 +α =αμ2 1≤L≤|αμ2 |

μ

α1

μ

γ

β

(((2−(jμ ∨kμ )ds Wxμ ) μ (2−l 𝜕ζ ) DLζ Gk,l )(x, uj (x) + ζ )) μ

γ

[(2−(jμ ∨kμ )ds W μ ) 1 uj (x), . . . , (2−(jμ ∨kμ )ds W μ ) L uj (x)].

(7.28)

506 � 7 Zygmund–Hölder spaces For the first term on the right-hand side of (7.28), we have, using (7.25), 󵄩󵄩 −(jμ ∨kμ )dsμ μ αμ −l β 󵄩 Wx ) (2 𝜕ζ ) Gk,l )(x, uj (x) + ζ )󵄩󵄩󵄩L∞ 󵄩󵄩((2 μ α β 󵄩 󵄩 ≤ 󵄩󵄩󵄩(2−kμ ds W μ ) μ (2−l 𝜕ζ ) Gk,l 󵄩󵄩󵄩L∞ ≲ A0 2−k⋅s 2−lt ,

which is even better than the desired bound (7.26). We turn to bounding the sum on the right-hand side of (7.28); we bound each term of the sum separately. We have, using L ≤ |αμ2 | ≤ |αμ |, (7.25), and (7.27), 1 μ 󵄩󵄩 󵄩󵄩(((2−(jμ ∨kμ )ds W μ )αμ (2−l 𝜕ζ )β DL Gk,l )(x, uj (x) + ζ )) x ζ 󵄩󵄩

μ μ 󵄩󵄩 γ γ [(2−(jμ ∨kμ )ds W μ ) 1 uj (x), . . . , (2−(jμ ∨kμ )ds W μ ) L uj (x)]󵄩󵄩󵄩 ∞ 󵄩L

L

μ μ α1 β L γ 󵄩 󵄩 󵄩 󵄩 ≤ 2lL 󵄩󵄩󵄩(2−kμ ds W μ ) μ (2−l 𝜕ζ ) (2−l Dζ ) Gk,l 󵄩󵄩󵄩L∞ ∏󵄩󵄩󵄩(2−jμ ds W μ ) s uj 󵄩󵄩󵄩L∞

s=1

lL

≲ 2 A0 2

2

−k⋅s −lt

‖u‖LL∞

≤ A0 2

2

−k⋅s −l(t−|αμ |)

(1 + ‖u‖L∞ )

|αμ |

,

completing the proof. ⃗ ℝN ). Then, for all μ ∈ Lemma 7.5.16. Let Hj,k,l be as in (7.23), where u ∈ C s (𝒦, (W , ds); {1, . . . , ν}, αμ , and β, 󵄩󵄩 −(kμ ∨jμ )dsμ μ αμ −l β 󵄩 W ) (2 𝜕ζ ) Hj,k,l (x, ζ )󵄩󵄩󵄩L∞ 󵄩󵄩(2 ≤ Cμ,αμ ,β ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

−k⋅s −j⋅s −l(t−|αμ |−ν) 2 2 )2

× (1 + ‖u‖C s ((W ,ds);ℝ ⃗ N ) )(1 + ‖u‖L∞ )

|αμ |+ν−1

(7.29)

,

where Cμ,αμ ,β ≥ 0 does not depend on j, k ∈ ℕν , l ∈ ℕ, F, or u. Proof. Before delving into details, we encourage the reader to review (7.24), which contains the main formula we use in a simple special case. By (7.23), we have Hj,k,l (x, ζ ) =

∑

E⊆{μ:jμ =0} ̸

Fk,l (x, Pj−eE u(x) + ζ ),

where Fk,l satisfy (7.15). Note that this implies that Fk,l satisfies the assumptions of Gk,l in Lemma 7.5.15 with A0 = ‖F‖C s,t ((W ,ds)⊠∇ ⃗ ). ℝN First we consider the case j = 0, so that H0,k,l (x, ζ ) = Fk,l (x, P0 u(x) + ζ ). Applying Lemma 7.5.15 with j = 0 gives

7.5 Compositions

󵄩󵄩 −kμ dsμ μ αμ −l 󵄩 W ) (2 𝜕ζ )Fk,l (x, P0 u(x) + ζ )󵄩󵄩󵄩L∞ 󵄩󵄩(2 ≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

−k⋅s −l(t+|αμ |) 2 (1 )2

|αμ |

+ ‖u‖L∞ )

� 507

,

which is even better than the desired estimate (7.29). Let j ∈ ℕν \ {0}. Without loss of generality, we assume {μ : jμ ≠ 0} = {1, . . . , ν0 } for some 1 ≤ ν0 ≤ ν. We write σ = (σ1 , . . . , σν0 ) ∈ ℝν0 . For E ⊆ {1, . . . , ν0 }, we write σE := ∏μ∈E σμ and E c := {1, . . . , ν0 } \ E. We have Hj,k,l (x, ζ ) =

∑

E⊆{1,...,ν0 }

(−1)|E| Fk,l (x, Pj−eE u(x) + ζ )

= ∫ 𝜕σ1 𝜕σ2 ⋅ ⋅ ⋅ 𝜕σν (Fk,l (x, Pj−e{1,...,ν } u(x) + 0

[0,1]ν0

0

∑

E⊊{1,...,ν0 }

σEc Dj−eE u(x) + ζ )) dσ.

Thus, to prove (7.29) it suffices to show that for all σ ∈ [0, 1]ν0 , 󵄩󵄩 󵄩󵄩 󵄩󵄩 −(jμ ∨kμ )dsμ μ αμ −l β 󵄩󵄩 󵄩󵄩(2 W ) (2 𝜕ζ ) 𝜕σ1 ⋅ ⋅ ⋅ 𝜕σν Fk,l (x, Pj−e{1,...,ν } u(x) + ∑ σEc Dj−eE u(x) + ζ )󵄩󵄩󵄩 󵄩󵄩 0 0 󵄩󵄩 ∞ 󵄩󵄩 E⊊{1,...,ν0 } 󵄩L ≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

−k⋅s −j⋅s −l(t−|αμ |−ν0 ) 2 2 (1 )2

+ ‖u‖C s ((W ,ds);ℝ ⃗ N ) )(1 + ‖u‖L∞ )

|αμ |+ν−1

.

(7.30)

Let 1 ≤ L ≤ ν0 and E1 , . . . , EL , Ẽ1 , . . . , ẼL ⊆ {1, . . . , μ0 }. Then, since ν0 ≥ 1, 𝜕σ1 ⋅ ⋅ ⋅ 𝜕σν Fk,l (x, Pj−e{1,...,ν } u(x) + 0

0

∑

E⊊{1,...,ν0 }

σEc Dj−eE u(x) + ζ )

can be written as a finite sum of terms of the form (DLζ Fk,l )(x, Pj−e{1,...,ν } u(x) + 0

∑

E⊊{1,...,ν0 }

σEc Dj−eE u(x) + ζ )[σE1 Dj−eẼ u, . . . , σEL Dj−eẼ u]. 1

L

Since each σEs ∈ [0, 1], to prove (7.30) it suffices to show that 󵄩󵄩 󵄩󵄩 −(jμ ∨kμ )dsμ μ αμ −l β 󵄩󵄩(2 W ) (2 𝜕ζ ) [(DLζ Fk,l )(x, Pj−e{1,...,ν } u(x) + ∑ σEc Dj−eE u(x) + ζ ) 󵄩󵄩 0 󵄩󵄩 E⊊{1,...,ν0 } 󵄩󵄩 󵄩󵄩 [Dj−eẼ u, . . . , Dj−eẼ u]]󵄩󵄩󵄩 (7.31) 󵄩󵄩 ∞ L 1 󵄩L ≲ ‖F‖C s,t ((W ,ds)⊠∇ 2−k⋅s 2−j⋅s 2−l(t−|αμ |−ν0 ) ⃗ N) ℝ

× (1 + ‖u‖C s ((W ,ds);ℝ ⃗ N ) )(1 + ‖u‖L∞ )

|αμ |+ν−1

.

508 � 7 Zygmund–Hölder spaces Using Notations 7.4.2 and 7.5.14, we have 󵄩󵄩 󵄩󵄩 −(jμ ∨kμ )dsμ μ αμ −l β 󵄩󵄩(2 W ) (2 𝜕ζ ) [(DLζ Fk,l )(x, Pj−e{1,...,ν } u(x) + ∑ σEc Dj−eE u(x) + ζ ) 󵄩󵄩 0 󵄩󵄩 E⊊{1,...,ν0 } 󵄩󵄩 󵄩󵄩 [Dj−eẼ u, . . . , Dj−eẼ u]]󵄩󵄩󵄩 󵄩󵄩 ∞ L 1 󵄩L 󵄩󵄩 μ 󵄩󵄩󵄩 −(jμ ∨kμ )ds μ γ0 −l β ≤ W ) (2 𝜕ζ ) (7.32) ∑ 󵄩󵄩(2 󵄩 ̇ 1 +⋅⋅⋅ ̇ +γ ̇ L󵄩 󵄩 γ0 +γ =αμ

(DLζ Fk,l )(x, Pj−e{1,...,ν } u(x) + 0

L

󵄩󵄩 󵄩󵄩 σEc Dj−eE u(x) + ζ )󵄩󵄩󵄩 󵄩󵄩 ∞ E⊊{1,...,ν0 } 󵄩L ∑

μ γ 󵄩 󵄩 × ∏󵄩󵄩󵄩(2−(jμ ∨kμ )ds W μ ) s Dj−eẼ u󵄩󵄩󵄩L∞ . s

s=1

We bound each term in the sum in (7.32) separately. By (7.15), the components of the tensor (2−l Dζ )L Fk,l satisfy the assumptions of Gk,l in Lemma 7.5.15 with A0 = ‖F‖C s,t ((W ,ds)⊠∇ . Thus, Lemma 7.5.15 implies ⃗ N) ℝ

󵄩󵄩 󵄩󵄩 󵄩󵄩 −(jμ ∨kμ )dsμ μ αμ −l β L 󵄩󵄩 󵄩󵄩(2 󵄩󵄩 c W ) (2 𝜕 ) (D F )(x, P u(x) + σ D u(x) + ζ ) ∑ ζ j−e{1,...,ν0 } E j−eE ζ k,l 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩L∞ E⊊{1,...,ν0 } ≲ 2Ll ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

≤ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

)2

2

−k⋅s −l(t−|γ0 |)

|γ0 |

(1 + ‖u‖L∞ )

−k⋅s −l(t−|αμ |−ν) 2 (1 )2

+ ‖u‖L∞ )

|αμ |

(7.33)

,

where in the last estimate we have used L ≤ ν0 ≤ ν and |γ0 | ≤ |αμ |. Since |j − (j − eẼ )|∞ ≤ 1, Proposition 5.5.5 (a) and (e) implies that 1

μ

γ

{((2−jμ ds W μ ) 1 Dj−eẼ , 2−j ) : j ∈ ℕν } 1

is a bounded set of generalized (W , ds)⃗ elementary operators, and therefore by Corollary 6.4.4, μ γ 󵄩󵄩 −(jμ ∨kμ )dsμ μ γ1 󵄩 󵄩 󵄩 W ) Dj−eẼ u󵄩󵄩󵄩L∞ ≤ 󵄩󵄩󵄩(2−jμ ds W μ ) 1 Dj−eẼ u󵄩󵄩󵄩L∞ 󵄩󵄩(2 1 1

≲ 2−j⋅s ‖u‖C s ((W ,ds);ℝ ⃗ N ).

(7.34)

7.5 Compositions

� 509

We use Corollary 7.1.4 (i) to see that L

μ γ 󵄩 󵄩 ∏󵄩󵄩󵄩(2−(jμ ∨kμ )ds W μ ) s Dj−eẼ u󵄩󵄩󵄩L∞ s

s=2

L

μ γ 󵄩 󵄩 ≤ ∏󵄩󵄩󵄩(2−jμ ds W μ ) s Dj−eẼ u󵄩󵄩󵄩L∞ s

(7.35)

s=2 L

ν−1 󵄩 󵄩 ≲ ∏󵄩󵄩󵄩u󵄩󵄩󵄩L∞ ≤ (1 + ‖u‖L∞ ) , s=2

where in the last estimate we have used L ≤ ν0 ≤ ν. Plugging (7.33), (7.34), and (7.35) into (7.32) yields (7.31) and completes the proof. Proof of Theorem 7.5.2. (i): Take Hj,k,l as in (7.23), so that F(x, u(x)+ζ ) = ∑j,k∈ℕν Hj,k,l (x, ζ ). l∈ℕ

Set M1 := ⌊|s|1 ⌋ + 1 and M2 := ⌊t⌋ − ⌊|s|1 ⌋ − ν. For μ ∈ {1, . . . , ν}, |αμ | ≤ M1 , and |β| ≤ M2 , Lemma 7.5.16 implies 󵄩󵄩 −(kμ ∨jμ )dsμ μ αμ −l β 󵄩 W ) (2 𝜕ζ ) Hj,k,l 󵄩󵄩󵄩L∞ 󵄩󵄩(2 ≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

−k⋅s −j⋅s −l(t−|αμ |−ν) 2 2 )2

× (1 + ‖u‖C s ((W ,ds);ℝ ⃗ N ) )(1 + ‖u‖L∞ ) ≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

|αμ |+ν−1

−k⋅s −j⋅s −l(t−⌊|s|1 ⌋−1−ν) 2 2 )2

× (1 + ‖u‖C s ((W ,ds);ℝ ⃗ N ) )(1 + ‖u‖L∞ )

⌊|s|1 ⌋+ν

.

This shows that Hj,k,l satisfies the assumptions of Lemma 7.5.13 with t replaced by t − ⌊|s|1 ⌋+ν ⌊|s|1 ⌋−1−ν and A = ‖F‖C s,t ((W ,ds)⊠∇ . The conclusion ⃗ ⃗ N ) )(1+‖u‖L∞ ) ) (1+‖u‖C s ((W ,ds);ℝ ℝN of Lemma 7.5.13, using F(x, u(x) + ζ ) = ∑j,k∈ℕν Hj,k,l (x, ζ ), is exactly the conclusion (i). l∈ℕ

󵄨 (ii): Since F(x, u(x)) = F(x, u(x) + ζ )󵄨󵄨󵄨ζ =0 , (ii) follows by combining (i) and Corollary 7.5.12 (i). (iii): We first consider the case where ‖u‖L∞ = ‖v‖L∞ = 1. Fix ψ0 ∈ C0∞ (M) with ψ0 ≡ 1 on 𝒦 and ϕ(ζ1 , ζ2 ) ∈ C0∞ (ℝN × ℝN ) with ϕ(ζ1 , ζ2 ) ≡ 1 if |ζ1 |, |ζ2 | ≤ 1. Set F(x, ζ ) := ψ0 (x)ϕ(ζ1 , ζ2 )ζ1 ⋅ ζ2 . It follows immediately from Proposition 7.5.9 that F ∈ C s,t (supp(ϕ) × ℝ2N , (W , ds)⃗ ⊠ ∇ℝ2N ), ∀t > 0. If ‖u‖L∞ = ‖v‖L∞ = 1, then u(x) ⋅ v(x) = F(x, u(x), v(x)) and (ii) implies ‖u ⋅ v‖C s (W ,ds)⃗ = ‖F(x, u(x), v(x))‖C s (W ,ds)⃗ ≲ ‖F‖C s,⌊|s|1 ⌋+2+ν ((W ,ds)⊠∇ ⃗

ℝN

× (1 + ‖u‖L∞ + ‖v‖L∞ )

) (1

+ ‖u‖C s (W ,ds)⃗ + ‖v‖C s (W ,ds)⃗ )

≲ 1 + ‖u‖C s (W ,ds)⃗ + ‖v‖C s (W ,ds)⃗ .

(7.36)

510 � 7 Zygmund–Hölder spaces ⃗ ℝN ), we apply (7.36) to u/‖u‖L∞ For general (not identically zero) u, v ∈ C s (𝒦, (W , ds); and v/‖v‖L∞ to see that ‖u ⋅ v‖C s (W ,ds)⃗ ≲ ‖u‖L∞ ‖v‖L∞ + ‖u‖C s (W ,ds)⃗ ‖v‖L∞ + ‖u‖L∞ ‖v‖C s (W ,ds)⃗ ≲ ‖u‖C s (W ,ds)⃗ ‖v‖L∞ + ‖u‖L∞ ‖v‖C s (W ,ds)⃗ , where the final inequality uses Corollary 7.2.2. (iv): First we prove the result with ‖w‖L∞ = 1. Let ϕ(ζ ′ ) ∈ C0∞ (ℝ) satisfy ϕ ≡ 1 for |ζ ′ | ≤ 1. Set ̃ ζ , ζ ′ ) := F(x, ζ )ϕ(ζ ′ )ζ ′ . F(x, It follows easily from Proposition 7.5.9 that F̃ ∈ C s,t (𝒦 × ℝN+1 , (W , ds)⃗ ⊠ ∇ℝN+1 ) and ̃ s,t ‖F‖ ⃗ ⃗ C ((W ,ds)⊠∇ ) ≲ ‖F‖C s,t ((W ,ds)⊠∇ ). ℝN ℝN+1 ̃ u(x) + ζ , w(x) + ζ ′ )󵄨󵄨󵄨 ′ . Thus, When ‖w‖L∞ = 1, we have F(x, u(x) + ζ )w(x) = F(x, 󵄨ζ =0 applying (i) and Corollary 7.5.12 (ii), we have ‖F(x, u(x) + ζ )w(x)‖C s,t−⌊|s|1 ⌋−1−ν ((W ,ds)⊠∇ ⃗

ℝN

)

󵄩̃ 󵄨 󵄩 = 󵄩󵄩󵄩F(x, u(x) + ζ , w(x) + ζ ′ )󵄨󵄨󵄨ζ ′ =0 󵄩󵄩󵄩C s,t−⌊|s|1 ⌋−1−ν ((W ,ds)⊠∇ ⃗ ) ℝN 󵄩󵄩 ̃ ′ 󵄩 󵄩 ≲ 󵄩󵄩F(x, u(x) + ζ , w(x) + ζ )󵄩󵄩C s,t−⌊|s|1 ⌋−1−ν ((W ,ds)⊠∇ ⃗ ) ℝN+1 ̃ s,t ≲ ‖F‖ ⃗ C ((W ,ds)⊠∇

ℝN+1

≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

) (1

) (1

ℝN

(7.37)

+ ‖u‖C s (W ,ds)⃗ + ‖w‖C s (W ,ds)⃗ )(1 + ‖u‖L∞ + ‖w‖L∞ )

+ ‖u‖C s (W ,ds)⃗ + ‖w‖C s (W ,ds)⃗ )(1 + ‖u‖L∞ )

⌊|s|1 ⌋+ν

⌊|s|1 ⌋+ν

.

⃗ we apply (7.37) to w/‖w‖L∞ to For general (not identically zero) w ∈ C s (𝒦, (W , ds)), see that ‖F(x, u(x) + ζ )w(x)‖C s,t−⌊|s|1 ⌋−1−ν ((W ,ds)⊠∇ ⃗

ℝN

≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

) (‖w‖L

∞

≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

) (‖w‖L

∞

ℝN ℝN

) ⌊|s|1 ⌋+ν

+ ‖w‖L∞ ‖u‖C s (W ,ds)⃗ + ‖w‖C s (W ,ds)⃗ )(1 + ‖u‖L∞ ) ⌊|s|1 ⌋+ν

‖u‖C s (W ,ds)⃗ + ‖w‖C s (W ,ds)⃗ )(1 + ‖u‖L∞ )

,

where the final inequality uses Corollary 7.2.2. (v): Using (iv) and Proposition 7.5.11, we have 󵄩󵄩 󵄩 󵄩󵄩F(x, u(x) + ζ ) − F(x, v(x) + ζ )󵄩󵄩󵄩C s,t−⌊|s|1 ⌋−2−ν ((W ,ds)⊠∇ ⃗

ℝN

)

󵄩󵄩 1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 = 󵄩󵄩∫ Dζ F(x, σu(x) + (1 − σ)v(x) + ζ )(u(x) − v(x)) dσ 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 0 󵄩󵄩C s,t−⌊|s|1 ⌋−2−ν ((W ,ds)⊠∇ ⃗

ℝN

1

󵄩 󵄩 ≤ ∫󵄩󵄩󵄩Dζ F(x, σu(x) + (1 − σ)v(x) + ζ )(u(x) − v(x))󵄩󵄩󵄩C s,t−⌊|s|1 ⌋−2−ν ((W ,ds)⊠∇ ⃗ 0

ℝN

)

)

dσ

7.5 Compositions

� 511

󵄩 󵄩 ≤ sup 󵄩󵄩󵄩Dζ F(x, σu(x) + (1 − σ)v(x) + ζ )(u(x) − v(x))󵄩󵄩󵄩C s,t−⌊|s|1 ⌋−2−ν ((W ,ds)⊠∇ ⃗ ) ℝN σ∈[0,1]

≲ ‖Dζ F‖C s,t−1 ((W ,ds)⊠∇ ⃗

ℝN

) (‖u

− v‖C s (W ,ds)⃗ + ‖σu + (1 − σ)v‖C s (W ,ds)⃗ ‖u − v‖L∞ )

× (1 + ‖σu + (1 − σ)v‖L∞ )

⌊|s|1 ⌋+ν

≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

) (‖u

− v‖C s (W ,ds)⃗ + (‖u‖C s (W ,ds)⃗ + ‖v‖C s (W ,ds)⃗ )‖u − v‖L∞ ) ⌊|s|1 ⌋+ν

× (1 + ‖u‖L∞ + ‖v‖L∞ )

.

󵄨 (vi): Since F(x, u(x)) − F(x, v(x)) = F(x, u(x) + ζ ) − F(x, v(x) + ζ )󵄨󵄨󵄨ζ =0 , this follows from (v) and Corollary 7.5.12 (i). (vii): We prove (vii) by induction on L. We begin with the base case, L = 1. When L = 1, using the fact that F(x, 0) ≡ 0, we have F(x, u(x)) = F(x, u(x)) − F(x, 0). From here, the result when L = 1 follows from (vi) with v = 0. We assume (vi) for some L ∈ ℕ+ and prove it for L + 1; thus, we assume the hypotheses with L replaced by L + 1. Using the fact that F(x, 0) ≡ 0 and using (iii), we have ‖F(x, u(x))‖C s (W ,ds)⃗ = ‖F(x, u(x)) − F(x, v(x))‖C s (W ,ds)⃗ 󵄩󵄩 1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 = 󵄩󵄩∫ Dζ F(x, σu(x))u(x) dσ 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 0 󵄩󵄩C s (W ,ds)⃗ 1

󵄩 󵄩 ≤ ∫󵄩󵄩󵄩Dζ F(x, σu(x))u(x)󵄩󵄩󵄩C s (W ,ds)⃗ dσ 0

󵄩 󵄩 ≤ sup 󵄩󵄩󵄩Dζ F(x, σu(x))u(x)󵄩󵄩󵄩C s (W ,ds)⃗

(7.38)

σ∈[0,1]

󵄩 󵄩 ≲ sup 󵄩󵄩󵄩Dζ F(x, σu(x))󵄩󵄩󵄩C s (W ,ds)⃗ ‖u‖L∞ σ∈[0,1]

+ sup ‖Dζ F(x, σu(x))‖L∞ ‖u‖C s (W ,ds)⃗ σ∈[0,1]

=: (I) + (II), where when we applied (iii) to the various summands of the matrix multiplication Dζ F(x, σu(x))u(x). To bound (I) we use the inductive hypothesis applied to Dζ F with t replaced by t − 1 > ⌊|s|1 ⌋ + 1 + ν + L and Proposition 7.5.11 to see 󵄩 󵄩 (I) = sup 󵄩󵄩󵄩Dζ F(x, σu(x))󵄩󵄩󵄩C s (W ,ds)⃗ ‖u‖L∞ σ∈[0,1]

≲ sup ‖Dζ F‖C s,t−1 ((W ,ds)⊠∇ ⃗

ℝN

σ∈[0,1]

≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

) (1

) (1

+ ‖u‖L∞ )

⌊|s|1 ⌋+ν+1

+ ‖σu‖L∞ )

⌊|s|1 ⌋+ν+1

‖σu‖L−1 L∞ ‖σu‖C s (W ,ds)⃗ ‖u‖L∞

‖u‖LL∞ ‖u‖C s (W ,ds)⃗ .

(7.39)

512 � 7 Zygmund–Hölder spaces β

For (II) we use the fact that 𝜕ζ Dζ F(x, 0) ≡ 0, ∀|β| < L, the fact that t > L + 1, and Propositions 7.5.11 and 7.5.10 to see that 󵄩 󵄩 󵄩 󵄩 sup 󵄩󵄩󵄩Dζ F(x, σu(x))󵄩󵄩󵄩L∞ ≲ sup ∑ 󵄩󵄩󵄩𝜕ζL Dζ F(x, ζ )󵄩󵄩󵄩L∞ ‖σu(x)‖LL∞

σ∈[0,1]

σ∈[0,1] |β|=L

≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

L ) ‖u‖L∞ .

We conclude (II) ≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

L ) ‖u‖L∞ ‖u‖C s (W ,ds)⃗ .

(7.40)

Using (7.39) and (7.40) to bound the right-hand side of (7.38) completes the inductive step. (viii): We have, using (iii), ‖F(x, u(x)) − F(x, v(x))‖C s (W ,ds)⃗ 󵄩󵄩 1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 = 󵄩󵄩∫ Dζ F(x, σu(x) + (1 − σ)v(x))(u(x) − v(x)) dσ 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 0 󵄩󵄩C s (W ,ds)⃗ 1

󵄩 󵄩 ≤ ∫󵄩󵄩󵄩Dζ F(x, σu(x) + (1 − σ)v(x))(u(x) − v(x))󵄩󵄩󵄩C s (W ,ds)⃗ dσ 0

󵄩 󵄩 ≤ sup 󵄩󵄩󵄩Dζ F(x, σu(x) + (1 − σ)v(x))(u(x) − v(x))󵄩󵄩󵄩C s (W ,ds)⃗

(7.41)

σ∈[0,1]

≲ sup ‖Dζ F(x, σu(x) + (1 − σ)v(x))‖C s (W ,ds)⃗ ‖u − v‖L∞ σ∈[0,1]

+ sup ‖Dζ F(x, σu(x) + (1 − σ)v(x))‖L∞ ‖u − v‖C s (W ,ds)⃗ σ∈[0,1]

=: (III) + (IV), where we applied (iii) to the various summands of the matrix multiplication Dζ F(x, σu(x) + (1 − σ)v(x))(u(x) − v(x)). β

For (III), we have 𝜕ζ Dζ F(x, 0) ≡ 0, ∀|β| ≤ L − 1, and we may therefore apply (vii) with L replaced by L − 1 and t replaced by t − 1 to see that 󵄩 󵄩 sup 󵄩󵄩󵄩Dζ F(x, σu(x) + (1 − σ)v(x))󵄩󵄩󵄩C s (W ,ds)⃗

σ∈[0,1]

≲ sup ‖Dζ F‖C s,t−1 ((W ,ds)⊠∇ ⃗

ℝN

σ∈[0,1]

) (1

+ ‖σu + (1 − σ)v‖L∞ )

× ‖σu + (1 − σ)v‖L−2 L∞ ‖σu + (1 − σ)v‖C s (W ,ds)⃗

≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

) (1

× (‖u‖L∞ + ‖v‖L∞ )

+ ‖u‖L∞ + ‖v‖L∞ )

L−2

⌊|s|1 ⌋+ν+1

(‖u‖C s (W ,ds)⃗ + ‖v‖C s (W ,ds)⃗ ),

⌊|s|1 ⌋+ν+1

7.5 Compositions

� 513

where the final estimate used Proposition 7.5.11. It follows that (III) ≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

) (1

⌊|s|1 ⌋+ν+1

+ ‖u‖L∞ + ‖v‖L∞ )

L−2

× (‖u‖L∞ + ‖v‖L∞ )

(‖u‖C s (W ,ds)⃗ + ‖v‖C s (W ,ds)⃗ )‖u − v‖L∞ .

(7.42)

β

For (IV), we use the fact that 𝜕ζ Dζ f (x, 0) ≡ 0, ∀|β| < L − 1, to see that 󵄩 󵄩 sup 󵄩󵄩󵄩Dζ F(x, σu(x) + (1 − σ)v(x))󵄩󵄩󵄩L∞

σ∈[0,1]

≲ sup

󵄩 β 󵄩 󵄩 󵄩L−1 ∑ 󵄩󵄩󵄩𝜕ζ Dζ F(x, σu(x) + (1 − σ)v(x))󵄩󵄩󵄩L∞ 󵄩󵄩󵄩σu + (1 − σ)v󵄩󵄩󵄩L∞

σ∈[0,1] |β|=L−1

≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

) (‖u‖L

∞

L−1

+ ‖v‖L∞ )

,

where in the last inequality we used t > L and Propositions 7.5.11 and 7.5.10. It follows that (IV) ≲ ‖F‖C s,t ((W ,ds)⊠∇ ⃗

ℝN

) (‖u‖L

∞

L−1

+ ‖v‖L∞ )

‖u − v‖C s (W ,ds)⃗ .

(7.43)

Using (7.42) and (7.43) to bound the right-hand side of (7.41) completes the proof.

7.5.3 The classical product Zygmund–Hölder spaces Just as in Section 6.6.1, we can understand classical product Zygmund–Hölder spaces in terms of the framework of this section. s,t Definition 7.5.17. Fix 𝒦 ⋐ M compact. For s, t > 0, we let Cstd (𝒦 × ℝN ) be the space consisting of those F ∈ C0∞ (M × ℝN )′ with supp(F) ⊆ 𝒦 × ℝN such that for each x ∈ 𝒦, ∞ there exists a Cloc diffeomorphism Φx :󳨀 → Φx (Bn (1)), where Φx (Bn (1)) is an open neighborhood of x and ψx ∈ C0∞ (Φx (Bn (1))) with ψx ≡ 1 on a neighborhood of x such that Φ∗x ψx F ∈ C s,t (ℝn × ℝN ). ∼

Remark 7.5.18. In Definition 7.5.17, and in the rest of this section, we used Φ∗x to denote the pullback in the first variable. So, for example, Φ∗x ψx F could be more explicitly written as (Φx ⊗ IdℝN )∗ ψx F, where IdℝN : ℝN → ℝN denotes the identity map. s,t Similar to Section 6.6.1, Cstd (𝒦 × ℝN ) can be endowed with a norm which turns it

∞ into a Banach space. Indeed, for each x ∈ M, let Φx : Bn (1) 󳨀 → Φx (Bn (1)) be a Cloc n diffeomorphism such that Φx (B (1)) ⊆ M is open and x = Φx (0). Cover 𝒦 by a finite collection of open sets of the form Φx1 (Bn (1)), . . . , ΦxL (Bn (1)). Take ψj ∈ C0∞ (Φxj (Bn (1))) ∼

s,t such that ∑Lj=1 ψj ≡ 1 on a neighborhood of 𝒦. For F ∈ Cstd (𝒦 × ℝN ), we define the norm

514 � 7 Zygmund–Hölder spaces L

‖F‖C s,t (𝒦×ℝN ) := ∑ ‖Φ∗xj ψj F‖C s,t (ℝn ×ℝN ) . std

j=1

s,t With this norm, Cstd (𝒦 × ℝN ) is a Banach space, and the equivalence class of the norm does not depend on any of the choices we made. We set s,t

N

Cstd,cpt (M × ℝ ) :=

s,t

⋃

𝒦⋐M𝒦 compact

N

Cstd (𝒦 × ℝ ).

We have the following analog of Theorem 6.6.7, which has a similar proof, which we omit. ∞ Theorem 7.5.19. Let (Y , 1) = {(Y1 , 1), . . . , (Ys , 1)} ⊂ Cloc (M; TM) × ℕ+ be such that

span{Y1 (x), . . . , Ys (x)} = Tx M,

∀x ∈ M.

Then s,t

N

Cstd (𝒦 × ℝ ) = C

s,t

(𝒦 × ℝN , (Y , 1) ⊠ ∇ℝN ),

with equality of topologies.

7.6 Adding parameters Using the results from Section 6.8, we can relate single-parameter Zygmund–Hölder spaces with higher-parameter Zygmund–Hölder spaces, as the next result shows. ∞ Proposition 7.6.1. Let (Y , d̂), (Z, dr) ⊂ Cloc (M; TM)×ℕ+ be two lists of Hörmander vector fields with formal degrees on M. Suppose (Y , d̂) and (Z, dr) locally weakly approximately ∞ commute on M and set (W , ds)⃗ := (Y , d̂) ⊠ (Z, dr) ⊂ Cloc (M; TM) × (ℕ2 \ {0}). Then, ∀ϵ > 0, ′ ∃ϵ > 0, ∀s > ϵ, ∀t > 0, s

C (𝒦, (Z, dr)) ⊆ C C

s,t

N

(𝒦 × ℝ , (Z, dr) ⊠ ∇ℝN ) ⊆ C

(ϵ′ ,s−ϵ)

⃗ (𝒦, (W , ds)),

′

(ϵ ,s−ϵ),t

N

(𝒦 × ℝ , (W , ds)⃗ ⊠ ∇ℝN ),

(7.44) (7.45)

and ∀ϵ > 0, ∀s, t > 0, ⃗ ⊆ C s (𝒦, (Z, dr)), (𝒦, (W , ds))

(7.46)

(𝒦 × ℝ , (W , ds)⃗ ⊠ ∇ℝN ) ⊆ C (𝒦 × ℝ , (Z, dr) ⊠ ∇ℝN ),

(7.47)

C C

(ϵ,s),t

N

(ϵ,s)

s,t

N

where the inclusions are continuous. Moreover, if Ω ⋐ M is open and relatively compact with 𝒦 ⋐ Ω and (Z, dr) weakly λ-controls (Y , d̂) on Ω, then we may take ϵ′ = ϵ/2λ.

7.7 Difference characterization

515

�

Proof. We prove (7.44) and (7.46) by applying results from Chapter 6. The proofs for (7.45) and (7.47) are similar and follow by a straightforward adaptation of these previous results to the similar spaces C ∙,∙ (𝒦 × ℝN , (W , ds)⃗ ⊠ ∇ℝN ); we leave the details of this adaptation to the reader. In this proof, all inclusions of spaces are continuous. For (7.46), we have, using Proposition 6.8.1, C

(ϵ,s)

⃗ = B (ϵ,s) (𝒦, (W , ds)) ⃗ ⊆ B s (𝒦, (Z, dr)) = C s (𝒦, (Z, dr)). (𝒦, (W , ds)) ∞,∞ ∞,∞

For (7.44) we take λ > 0 as in the statement of the proposition (such a λ always exists by Hörmander’s condition and the relative compactness of Ω). Set ϵ′ := ϵ/2λ. Proposition 6.8.1 gives s

s

(−ϵ′ ,s)

⃗ C (𝒦, (Z, dr)) = B∞,∞ (𝒦, (Z, dr)) ⊆ B∞,∞ (𝒦, (W , ds)).

(7.48)

Applying Theorem 6.7.2 with s̃ = ϵ/λ, we have (−ϵ′ ,s)

(−ϵ′ ,s)+(ϵ/λ,−ϵ)

⃗ ⊆B B∞,∞ (𝒦, (W , ds)) ∞,∞ ′

⃗ (𝒦, (W , ds))

′

(ϵ ,s−ϵ) ⃗ = C (ϵ ,s−ϵ) (𝒦, (W , ds)). ⃗ = B∞,∞ (𝒦, (W , ds))

(7.49)

Combining (7.48) and (7.49) yields (7.44) and completes the proof.

7.7 Difference characterization Proposition 2.5.3 gives an elementary difference characterization of the classical spaces C s (ℝn ). In this section, we present a similar difference characterization of the spaces C s (𝒦, (W , ds)) in the single-parameter3 setting ν = 1, where (W , ds) are Hörmander vector fields with (single-parameter) formal degrees. In Section 7.7.1 we use this character⃗ and more elemenization to describe the connection between the spaces C s (𝒦, (W , ds)) tary Hölder type spaces. The Hölder spaces, C 0,r for r ∈ [0, 1], can be defined on any metric space, because they are defined in terms of first-order differences. However, similar to Proposition 2.5.3, to characterize C s (𝒦, (W , ds)) we need to use higher-order differences. This is a difficulty, since higher-order differences generally require some kind of group structure, and we have no such structure in our setting. Instead, we proceed by considering higher-order differences of functions restricted to sufficiently regular paths, as we make precise below.

3 The restriction to the single-parameter case is probably not necessary, and is only for simplicity.

516 � 7 Zygmund–Hölder spaces Definition 7.7.1. For k ∈ ℕ and l > 0 we let L∞ k ([0, l]) be defined as L∞ ([0, l]), k = 0, L∞ ([0, l]) = { k k−1,1 C ([0, l]), k ≥ 1. Fix a finite collection ∞ (X, d ) = {(X1 , d1 ), . . . , (Xq , dq )} ⊂ Cloc (M; TM) × ℕ+ .

Definition 7.7.2. For l ∈ ℕ+ and δ > 0, we let 𝒫δl (M, (X, d )) denote the set of those absolutely continuous paths γ : [0, l] → M such that q

d γ(t) = ∑ aj (t)Xj (γ(t)), dt j=1

a. e.,

where aj ∈ L∞ l−1 ([0, l]) and ‖𝜕tl aj ‖L∞ ([0,l]) < δdj ∨(l +1) , ′

′

0 ≤ l′ ≤ l − 1.

(7.50)

For l = 0, we let 𝒫s0 (M, (X, d )) consist of all paths γ : {0} → M, i. e., 𝒫s0 (M, (X, d )) ≅ M. Definition 7.7.3. For K ∈ ℕ+ and g : [0, K] → ℂ, we set Diff(g)(t) := g(t + 1) − g(t) : [0, K − 1] → ℂ. Remark 7.7.4. If g : [0, K] → ℂ, then DiffK (g) : {0} → ℂ. In particular, it makes sense to consider DiffK (g)(0). Definition 7.7.5. For l ∈ ℕ, γ : [0, l] → M, and f : M → ℂ, we set △lγ f := Diffl (f ∘ γ)(0) ∈ ℂ. Definition 7.7.6. For l ∈ ℕ+ , s ∈ (0, l), and f ∈ C(M), we set l

‖f ‖C s (X,d) := ∑ sup l

sup

l′ =0 δ∈(0,∞) γ∈𝒫δl (M,(X,d)) ′

′ 󵄨 ′ 󵄨 δ−sl /l 󵄨󵄨󵄨△lγ f 󵄨󵄨󵄨

and s

Cl (M, (X, d )) := {f ∈ C(M) : ‖f ‖C s (X,d) < ∞}. l

Proposition 7.7.7. For l ∈ ℕ+ and s ∈ (0, l), Cls (M, (X, d )) is an algebra and ∀f , g ∈ Cls (M, (X, d )) ‖fg‖C s (X,d) ≤ Cl ‖f ‖C s (X,d) ‖g‖C s (X,d) , l

where Cl is a constant depending only on l.

l

l

7.7 Difference characterization

� 517

Proof. Fix 0 ≤ l′ ≤ l and γ ∈ 𝒫δl (M, (X, d )). We wish to show ′

󵄨󵄨 l′ 󵄨 sl′ /l 󵄨󵄨△γ (fg)󵄨󵄨󵄨 ≲ δ ‖f ‖Cls (X,d) ‖g‖Cls (X,d) ,

(7.51)

and this will complete the proof. We have △lγ (fg) = Diffl ((f ∘ γ)(g ∘ γ))(0) ′

′

l ′ l′ = ∑ ( )(Diffl −k (f ∘ γ))(k)(Diffk (g ∘ γ))(0). k k=0 ′

(7.52)

For 0 ≤ k ≤ l′ , we set γk (t) := γ(t + k). Then γk ∈ 𝒫δl −k (M, (X, d )), and therefore ′

󵄨󵄨 󵄨 󵄨 l′ −k 󵄨 l′ −k s(l′ −k)/l ‖f ‖C s (X,d) . 󵄨󵄨Diff (f ∘ γ)(k)󵄨󵄨󵄨 = 󵄨󵄨󵄨△γk f 󵄨󵄨󵄨 ≤ δ l

(7.53)

Also, γ ∈ 𝒫δl (M, (X, d )) ⊆ 𝒫δk (M, (X, d )) and Diffk (g ∘ γ)(0) = △kγ g. Therefore, ′

󵄨󵄨 󵄨 󵄨 k 󵄨 k sk/l 󵄨󵄨Diff (g ∘ γ)(0)󵄨󵄨󵄨 = 󵄨󵄨󵄨△γ g 󵄨󵄨󵄨 ≤ δ ‖g‖Cls (X,d) .

(7.54)

Plugging (7.53) and (7.54) into (7.52) establishes (7.51) and completes the proof. Now let ∞ (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} ⊂ Cloc (M; TM) × ℕ+

be Hörmander vector fields with (single-parameter) formal degrees. Recall that 𝒦 ⋐ M denotes a compact set and Ω ⋐ M denotes a relatively compact, open set with 𝒦 ⋐ Ω. ̃ ⋐ M with Ω ⋐ Ω. ̃ By Proposition 3.4.14 (in the Fix a new relatively compact, open set Ω case ν = 1), we may pick a finite collection (X, d ) = {(X1 , d1 ), . . . , (Xq , dq )} ⊂ Gen((W , ds)) ̃ such that Gen((W , ds)) is finitely generated by (X, d ) on Ω. ̃ Theorem 7.7.8. Suppose Gen((W , ds)) is finitely generated by (X, d ) ⊂ Gen((W , ds)) on Ω. Then, for all l ∈ ℕ+ and s ∈ (0, l), s

s

C (𝒦, (W , ds)) = {f ∈ Cl (M, (X, d )) : supp(f ) ⊆ 𝒦}.

Moreover, these two spaces have equivalent norms: for f ∈ C s (𝒦, (W , ds)), ‖f ‖C s (W ,ds) ≈ ‖f ‖C s (X,d) , l

where the implicit constants do not depend on f , but may depend on any of the other ingredients.

518 � 7 Zygmund–Hölder spaces The rest of this section is devoted to the proof of Theorem 7.7.8. We henceforth as̃ We begin with the sume Gen((W , ds)) is finitely generated by (X, d ) ⊂ Gen((W , ds)) on Ω. easier containment C s (𝒦, (W , ds)) ⊆ Cls (M, (X, d )). For this, the next two lemmas are useful. Lemma 7.7.9. Let l ∈ ℕ+ . Then, for all δ ∈ (0, 1], h ∈ C0∞ (M), γ ∈ 𝒫δl (M, (X, d )), and 0 ≤ l′ ≤ l, 𝜕tl (h ∘ γ)(t) = ∑ aβl (t)(X β h)(γ(t)), ′

(7.55)

′

|β|≤l′

where for all 0 ≤ k ≤ l − l′ , 󵄩󵄩 k l′ 󵄩󵄩 deg (β)∨(k+l′ ) . 󵄩󵄩𝜕t aβ 󵄩󵄩L∞ ([0,l]) ≤ Ck,q,l δ d Proof. We prove the result by induction on l′ . The base case, l′ = 0, follows by taking a00 ≡ 1. Let l ≥ l′ ≥ 1. We assume the result for l′ − 1 and prove it for l′ . We have, letting aj (t) be as in Definition 7.7.2, 𝜕tl h ∘ γ(t) = 𝜕t 𝜕tl −1 h ∘ γ(t) = 𝜕t ∑ aβl −1 (t)(X β h)(γ(t)) ′

′

′

|β|≤l′ −1

=

q

∑ (𝜕t aβl −1 )(t)(X β h)(γ(t)) + ∑ ∑ aβl −1 aj (t)(X β Xj h)(γ(t)) ′

′

|β|≤l′ −1 j=1

|β|≤l′ −1

:= (I) + (II). We wish to show that (I) and (II) are of the desired form (7.55). For (I), for 0 ≤ k ≤ l − l′ we have k + 1 ≤ l − l′ + 1 = l − (l′ − 1), and therefore by the inductive hypothesis, ′ 󵄩󵄩 k l′ −1 󵄩󵄩 󵄩 k+1 l′ −1 󵄩 deg (β)∨(k+1+(l′ −1)) = δdegd (β)∨(k+l ) , 󵄩󵄩𝜕t 𝜕t aβ 󵄩󵄩L∞ = 󵄩󵄩󵄩𝜕t aβ 󵄩󵄩󵄩L∞ ≲ δ d

establishing that (I) is of the form (7.55). ̃ For (II), we have X β Xj = X β , with degd (β)̃ = degd (β) + dj , and we therefore wish to show that for all 0 ≤ k ≤ l − l′ , 󵄩󵄩 k l′ −1 󵄩 (deg (β)+dj )∨(k+l′ ) . 󵄩󵄩𝜕t (aβ (t)aj (t))󵄩󵄩󵄩L∞ ≲ δ d

(7.56)

Using (7.50) and the inductive hypothesis, we have, for 0 ≤ k ≤ l − l′ , k

󵄩󵄩 k l′ −1 󵄩󵄩 󵄩 k l′ −1 󵄩 󵄩 k−k 󵄩 󵄩󵄩𝜕t (aβ aj )󵄩󵄩L∞ ≲ ∑ 󵄩󵄩󵄩𝜕t 1 aβ 󵄩󵄩󵄩L∞ 󵄩󵄩󵄩𝜕t 1 aj 󵄩󵄩󵄩L∞ k1 =0 k

≲ ∑ δdegd (β)∨(k1 +l −1) δdj ∨(k−k1 +1) ≲ δ(degd (β)+dj )∨(k+l ) . ′

′

k1 =0

This establishes (7.56) and proves (II) is of the desired form, completing the proof.

7.7 Difference characterization

�

519

Lemma 7.7.10. Fix l ∈ ℕ. Then for all h ∈ C0∞ (M), δ ∈ (0, ∞), and γ ∈ 𝒫δl (M, (X, d )), 󵄨󵄨 l 󵄨󵄨 deg (β)∨l ‖X β h‖L∞ ) ∧ ‖h‖L∞ . 󵄨󵄨△γ h󵄨󵄨 ≲ ( ∑ δ d |β|≤l

Proof. Clearly, | △lγ h| ≲ ‖h‖L∞ . Also, for δ > 1, ∑ δdegd (β)∨l ‖X β h‖L∞ ≥ ‖h‖L∞ . |β|≤l

Thus, it suffices to assume that δ ∈ (0, 1] and to prove that 󵄨󵄨 l 󵄨󵄨 deg (β)∨l ‖X β h‖L∞ . 󵄨󵄨△γ h󵄨󵄨 ≲ ∑ δ d

(7.57)

|β|≤l

Consider 1

Diff(h ∘ γ)(t) = ∫ 𝜕t (h ∘ γ)(t + u) du. 0

Therefore, △lγ h = Diffl (h ∘ γ)(0) = ∫ (𝜕tl (h ∘ γ))(u1 + u2 + ⋅ ⋅ ⋅ + ul ) du, [0,1]l

so with aβl as in Lemma 7.7.9, we have 󵄨󵄨 l 󵄨󵄨 󵄩󵄩 l 󵄩 󵄩 l󵄩 󵄩 β 󵄩 deg (β)∨l 󵄩 󵄩󵄩X β h󵄩󵄩󵄩 ∞ . 󵄨󵄨△γ h󵄨󵄨 ≤ 󵄩󵄩𝜕t (h ∘ γ)󵄩󵄩󵄩L∞ ([0,l]) ≤ ∑ 󵄩󵄩󵄩aβ 󵄩󵄩󵄩L∞ 󵄩󵄩󵄩X h󵄩󵄩󵄩L∞ ≲ ∑ δ d 󵄩 󵄩L |β|≤l

|β|≤l

This establishes (7.57) and completes the proof. We are prepared to prove the easy half of Theorem 7.7.8. Proposition 7.7.11. Let l ∈ ℕ+ and s ∈ (0, l). Then ‖f ‖C s (X,d) ≲ ‖f ‖C s (W ,ds) , l

∀f ∈ C s (𝒦, (W , ds)),

(7.58)

and C s (𝒦, (W , ds)) ⊆ Cls (M, (X, d )). Proof. We proceed by induction on l ∈ ℕ+ . Suppose we know the result holds ∀l′ ∈ ℕ+ with l′ < l; we will prove the result for l. Because of how we have set up this induction, we do not require a base case (the base case l = 1 is included in the inductive step). Suppose f ∈ C s (𝒦, (W , ds)). By Corollary 7.2.2, f ∈ C(M) and ‖f ‖C(M) ≲ ‖f ‖C s (W ,ds) . For 0 < l′ < l, we have, by the inductive hypothesis,

520 � 7 Zygmund–Hölder spaces ′ 󵄨 ′ 󵄨 δ−sl /l 󵄨󵄨󵄨△lγ f 󵄨󵄨󵄨 ≤ ‖f ‖

sup

δ∈(0,∞)

′ γ∈𝒫δl (M,(X,d))

C sl′ /l (X,d) ′

l

≲ ‖f ‖C sl′ /l (W ,ds) ≤ ‖f ‖C s (W ,ds) .

Thus, to prove (7.58) it suffices to show that for δ > 0, γ ∈ 𝒫δl (M, (X, d )), 󵄨󵄨 l 󵄨󵄨 s 󵄨󵄨△γ f 󵄨󵄨 ≲ δ ‖f ‖C s (W ,ds) .

(7.59)

By Proposition 7.2.1 we may write f = ∑j∈ℕ fj , where fj ∈ C0∞ (Ω) satisfy, for all α, 󵄩󵄩 −jds α 󵄩󵄩 −js 󵄩󵄩(2 W ) fj 󵄩󵄩L∞ ≲ 2 ‖f ‖C s (W ,ds) . Since (X, d ) ⊂ Gen((W , ds)), we therefore have, for all β, 󵄩󵄩 −jd β 󵄩󵄩 −js 󵄩󵄩(2 X) fj 󵄩󵄩L∞ ≲ 2 ‖f ‖C s (W ,ds) .

(7.60)

By Lemma 7.7.10 and (7.60) we have 󵄨󵄨 l 󵄨󵄨 deg (β)∨l 󵄩 󵄩󵄩X β fj 󵄩󵄩󵄩 ∞ ) ∧ ‖fj ‖L∞ 󵄨󵄨△γ fj 󵄨󵄨 ≲ ( ∑ δ d 󵄩 󵄩L |β|≤l

≲ ((2−js ∑ δdegd (β)∨l 2j degd (β) ) ∧ 2−js )‖f ‖C s (W ,ds) |β|≤l j

l

≲ 2 ((2 δ) ∧ 1)‖f ‖C s (W ,ds) . −js

Hence, using l > s, we have 󵄨󵄨 l 󵄨󵄨 󵄨 l 󵄨 −js j l 󵄨󵄨△γ f 󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨△γ fj 󵄨󵄨󵄨 ≲ ‖f ‖C s (W ,ds) ∑ 2 ((δ2 ) ∧ 1) j∈ℕ

j∈ℕ

≤ ‖f ‖C s (W ,ds) ( ∑ 2j(l−s) δl + ∑ 2−js ) 2−j ≥δ s s

2−j ≤δ

≲ ‖f ‖C s (W ,ds) (δ + δ ) ≲ ‖f ‖C s (W ,ds) δs . This establishes (7.59) and completes the proof of (7.58). The containment C s (𝒦, (W , ds)) ⊆ Cls (M, (X, d )) is an immediate consequence of (7.58). We turn to the other half of Theorem 7.7.8: the containment {f ∈ Cls (M, (X, d )) : supp(f ) ⊆ 𝒦} ⊆ C s (𝒦, (W , ds)). The basic proof idea is the same as in Lemma 2.5.12, but here the operators Dj are more complicated: see Section 7.1. To make the same proof outline work requires a number of technical estimates. To do this, we introduce several lemmas.

7.7 Difference characterization

� 521

̃ is an open neighborhood of {0} × Ω ̃ and g(t, x) ∈ C ∞ (U). Then, Suppose U ⊆ ℝq × Ω loc ̃ it makes sense to ask if deg (g) ≥ d0 for by Definition 4.2.1 (with M replaced by Ω), d d0 ∈ ℤ. Since this is the single-parameter case (ν = 1), we can define degd (g) ∈ ℕ to be the least number e ∈ ℕ such that degd (g) ≥ d0 ⇔ e ≥ d0 , ∀d0 ∈ ℤ. ̃ be an open neighborhood of {0} × Ω ̃ and let 𝒦0 ⋐ Ω ̃ Lemma 7.7.12. Let U ⊆ ℝq × Ω be compact. Fix l ∈ ℕ+ . Then if a0 = a0 (𝒦0 , U, (X, d ), l) > 0 is sufficiently small, the ∞ following holds. Suppose γ : [0, l] → 𝒦0 satisfies, for some t ∈ Bq (a0 ), g1 , . . . , gq ∈ Cloc (U), −j ′ k ∈ {1, . . . , q}, j ∈ (0, ∞) with 2 ≤ a0 , j ∈ [0, j], q

′ d p γ(s) = ∑ 2−jdk gk (2−j d t + 2−jdk sek , γ(s))Xp (γ(s)), ds p=1

(7.61)

p

where degd (gk ) ≥ dp − dk . Then there exists A = A(𝒦0 , (X, d ), l, g1 , . . . , gq ) ≥ 0, not depending on t, j, or j′ , such that γ ∈ 𝒫 l −j′ (M, (X, d )). A2

Proof. Note that a0 > 0 must be sufficiently small for the right-hand side of (7.61) to even p make sense because gk is only defined on U. In particular, g(t, x) is defined for all x ∈ 𝒦0 for t sufficiently small (independent of x ∈ 𝒦0 ). l The conclusion that γ ∈ 𝒫A2 −j (M, (X, d )), for some A ≲ 1, is equivalent to stating that ′ for 0 ≤ l ≤ l − 1, ′ ′ ′ p 󵄨 ′ 󵄨 sup sup 󵄨󵄨󵄨𝜕sl 2−jdk gk (2−j d t + 2−jdk sek , γ(s))󵄨󵄨󵄨 ≲ 2−j (dp ∨(l +1)) .

t∈Bq (a0 ) 0≤s≤l

(7.62)

∞ To prove (7.62), we prove a more general estimate. Let g ∈ Cloc (U) satisfy degd (g) ≥ d0 − dk for some d0 ∈ ℕ. We will show for all l1 , l2 ∈ ℕ with 0 ≤ l1 + l2 ≤ l,

sup

′ ′ 󵄨 󵄨 sup 󵄨󵄨󵄨𝜕sl11 𝜕sl22 2−jdk g(2−j d t + 2−jdk s1 ek , γ(s2 ))󵄨󵄨󵄨 ≲ 2−j (d0 ∨(l1 +l2 +1)) .

t∈Bq (a0 ) 0≤s1 ,s2 ≤l

(7.63)

Indeed, setting s1 = s2 = s in (7.63) and summing over all l1 + l2 = l′ proves (7.62). We prove (7.63) by induction on l2 . We begin with the base case, l2 = 0. By the definition of degd (g) ≥ d0 − dk , we may write g(t, x) = ∑Pp=1 t αp hp (t, x), where P ≥ 1, ∞ hp ∈ Cloc (U), and degd (αp ) ≥ d0 − dk . It suffices to prove (7.63) when g is given by one such term, so we assume g(t, x) = t α h(t, x), ∞ where h ∈ Cloc (U) and degd (α) ≥ d0 − dk .

522 � 7 Zygmund–Hölder spaces l

We have, for some constants cl′1 , 1

′ 𝜕sl11 2−jdk g(2−j d t

=

+2

−jdk

′ 𝜕sl11 [2−jdk (2−j d t

l1 ∧αk

s1 ek , γ(s2 )) α

+ 2−jdk sek ) h(2−j d t + 2−jdk s1 ek , γ(s2 ))]

l

′

= ∑ cl′1 2−j(1+l1 )dk (2−jd t + 2−jdk s1 ek ) l1′ =0

α−l1′ ek

1

l −l′ (𝜕tk1 1 h)(2−jd t

(7.64)

+ 2−jdk s1 ek , γ(s2 )).

In the above expression, we are only considering |t| < a0 ≲ 1, |s| ≤ l ≲ 1, and 2−j ≤ 2−j . Therefore, for any multi-index β, ′

β󵄨 󵄨󵄨 −j′ d −jd −j′ degd (β) . 󵄨󵄨(2 t + 2 k sek ) 󵄨󵄨󵄨 ≲ 2

(7.65)

∞ Since γ(s2 ) ∈ 𝒦0 , |2−j d t + 2−jdk sek | ≤ (l + 1)a0 , and h ∈ Cloc (U), if a0 > 0 is sufficiently small, we have by compactness ′

󵄨󵄨 l1 −l1′ 󵄨 −j′ d −jd 󵄨󵄨(𝜕t h)(2 t + 2 k s1 ek , γ(s2 ))󵄨󵄨󵄨 ≲ 1.

(7.66)

Plugging (7.65) and (7.66) into (7.64) and using degd (α) ≥ d0 − dk and 2−j ≤ 2−j , we have ′

l1 ∧αk

′ ′ ′ 󵄨󵄨 l1 −jdk 󵄨 g(2−j d t + 2−jdk s1 ek , γ(s2 ))󵄨󵄨󵄨 ≲ ∑ 2−j(1+l1 )dk 2−j degds (α−l1 ek ) 󵄨󵄨𝜕s1 2

l1′ =0

l1

≲ 2−j(1+l1 )dk ∧ ( ∑ 2−j(1+l1 )dk 2−j (d0 −dk −l1 dk ) ) ′

′

l1′ =0

≤ 2−j (1+l1 )dk ∧ 2−j d0 ≤ 2−j (d0 ∨(1+l1 )) . ′

′

′

This proves (7.63) in the base case l2 = 0. Let l2 ≥ 1. We assume that (7.63) holds for all lesser values of l2 . Applying the inductive hypothesis, we have, for all 0 ≤ l2′ < l2 , ′ 󵄨 󵄨 sup 󵄨󵄨󵄨𝜕s2 2−jdk g(2−j d t + 2−jdk s2 ek , γ(s2 ))󵄨󵄨󵄨

0≤s2 ≤l

′ 󵄨 l l −jdk 󵄨 2 g(2−j d t + 2−jdk s1 ek , γ(s2 ))󵄨󵄨󵄨 ∑ 󵄨󵄨󵄨𝜕s1,11 𝜕s2,1 2

≲ sup

0≤s1 ,s2 ≤l l +l =l′ 1,1 2,1 2

(7.67)

≲ 2−j (d0 ∨(l2 +1)) . ′

′

Consider, by (7.61), 𝜕s2 2−jdk g(2−j d t + 2−jdk s1 ek , γ(s2 )) ′

q

p

= ∑ 2−2jdk gk (2−j d t + 2−jdk s2 ek , γ(s2 ))(Xp g)(2−j d t + 2−jdk s1 ek , γ(s2 )). p=1

′

′

(7.68)

7.7 Difference characterization

�

523

p

We have degd (gk ) ≥ dp − dk by hypothesis and degd (Xp g) ≥ degd (g) ≥ d0 − dk . Thus, p

we can apply (7.67) to 2−jdk gk (2−j d t + 2−jdk s2 ek , γ(s2 )) (with d0 replaced by dp ) and the ′

inductive hypothesis to 2−jdk (Xp g)(2−j d t+2−jdk s1 ek , γ(s2 )). Applying these to (7.68) shows that ′

′ 󵄨󵄨 l1 l2 −jdk 󵄨 g(2−j d t + 2−jdk s1 ek , γ(s2 ))󵄨󵄨󵄨 󵄨󵄨𝜕s1 𝜕s2 2 ′ l p 󵄨 󵄨 ≲ gk (2−j d t + 2−jdk s2 ek , γ(s2 ))󵄨󵄨󵄨 ∑ 󵄨󵄨󵄨2−jdk 𝜕s2,1 2

l2,1 +l2,2 =l2 −1

′ l 󵄨 󵄨 × 󵄨󵄨󵄨2−jdk 𝜕sl11 𝜕s2,2 (Xp g)(2−j d t + 2−jdk s1 ek , γ(s2 ))󵄨󵄨󵄨 2

≲

∑

2−j (dp ∨(l2,1 +1)) 2−j (d0 ∨(l1 +l2,2 +1)) ′

l2,1 +l2,2 =l2 −1

′

≲ 2−j (d0 ∨(l1 +l2 +1)) . ′

This establishes (7.63) and completes the proof. ̃ Ω, (X, d ), l) > 0 is sufficiently small, the following Lemma 7.7.13. Fix l ∈ ℕ+ . If a0 = a0 (Ω, ̃ holds. There exists A = A(Ω, Ω, (X, d ), l) ≥ 0 such that for x ∈ Ω, t ∈ Bq (a0 ), j ∈ (0, ∞) with 2−j ≤ a0 , j′ ∈ [0, j], and k ∈ {1, . . . , q}, if we set γx,t,j,j′ (s) := exp(−(2−j d t + 2−jdk sek ) ⋅ X)x, ′

then γx,t,j,j′ ∈ 𝒫 l

A2−j

′

(M, (X, d )).

Proof. By Remark 3.4.12, we have [Xj , Xk ] =

∑ dl ≤dj +dk

l cj,k Xl ,

l ∞ cj,k ∈ Cloc (Ω).

̃ and In other words, (X, d ) satisfies the hypotheses of Chapter 4 with M replaced by Ω ν = 1. Thus, we can apply Proposition 4.2.2 to understand the exponential map Γ(t, x) = ̃ If a0 > 0 is sufficiently small, then γx,t,j,j′ (s) ∈ 𝒦0 , e−t⋅X x. Fix 𝒦0 compact with Ω ⋐ 𝒦0 ⋐ Ω. ∀s ∈ [0, l]. Proposition 4.2.2 (iv) implies q

′ d p γx,t,j,j′ (s) = −2−jdk Xk (γx,t,j,j′ ) + ∑ 2−jdk g4,k (2−j d t + 2−jdk sek , γx,t,j,j′ (s))Xp (γx,t,j,j′ ), ds p=1

p

p

where g4,k (t, x) is as in that result. In particular, degd (g4,k ) ≥ dp − dk . From here, the result follows from Lemma 7.7.12. For a function F : ℝq → ℂ and v ∈ ℝq , we define Diffv F : ℝq → ℂ by Diffv F(t) := F(t + v) − F(t).

(7.69)

524 � 7 Zygmund–Hölder spaces If F is only defined on a subset of ℝq , then Diffv F(t) is only defined for t where (7.69) makes sense. ̃ Ω, (X, d ), l) > 0 such that Lemma 7.7.14. Fix l ∈ ℕ+ and s ∈ (0, l). There exists a0 = a0 (Ω, the following holds. There exists C ≥ 0 such that for all j ∈ (0, ∞) with 2−j ≤ a0 , j′ ∈ [0, j], k ∈ {1, . . . , q}, 0 ≤ l′ ≤ l, f ∈ Cls (M, (X, d )), ′

′ −j d ′ ′ 󵄨 󵄨 sup 󵄨󵄨󵄨Diffl (j′ −j)dk f (e−2 t⋅X x)󵄨󵄨󵄨 ≤ C2−j sl /l ‖f ‖C s (X,d) . l 2 ek q

t∈B (a0 ) x∈Ω

−j′ d

Here, Diff2(j′ −j)dk e is acting on f (e−2 k

t⋅X

x) in the t ∈ ℝq variable.

Proof. For t ∈ Bq (a0 ) and x ∈ Ω, we set γ(s) := exp(−(2−j d t + 2−jdk sek )x). Lemma 7.7.13 ′ shows that there exists A ≲ 1 with γ ∈ 𝒫 l −j′ (M, (X, d )) ⊆ 𝒫 l −j′ (M, (X, d )). Thus, ′

A2

󵄨󵄨 l′ −2 󵄨󵄨Diff2(j′ −j)dk e f (e k

−j′ d

t⋅X

A2

′ 󵄨 󵄨 󵄨 󵄨 ′ 󵄨 x)󵄨󵄨󵄨 = 󵄨󵄨󵄨(Diffl (f ∘ γ))(0)󵄨󵄨󵄨 = 󵄨󵄨󵄨△lγ f 󵄨󵄨󵄨 ′

sl′ /l

≤ (A2−j )

‖f ‖C s (X,d) ≲ 2−j sl /l ‖f ‖C s (X,d) . ′

′

l

l

̃ Ω, (X, d ), l) > 0 is sufficiently Lemma 7.7.15. Fix l ∈ ℕ+ and s ∈ (0, l). If a0 = a0 (Ω, ∞ q small, the following holds. Let ℬ ⊂ C0 (B (a0 )) be a bounded set. Then there exists C ≥ 0 such that for all j ∈ (0, ∞) with 2−j ≤ a0 , j′ ∈ [0, j], 0 ≤ l′ ≤ l, k ∈ {1, . . . , q}, and f ∈ Cls (M, (W , ds)), ′ ′ ′ 󵄨󵄨 󵄨󵄨 sup sup 󵄨󵄨󵄨∫ f (e−t⋅X x) Diffl−2−jdk e (η(t) Dild−j′ (ς)(t)) dt 󵄨󵄨󵄨 ≤ C2−j sl /l ‖f ‖C s (X,d) . 2 l k 󵄨 󵄨 x∈Ω η,ς∈ℬ

Proof. Let a0 > 0 be as in Lemma 7.7.14. Applying that lemma, we have, for x ∈ Ω and η, ς ∈ ℬ, ′ 󵄨󵄨 󵄨 󵄨󵄨∫ f (e−t⋅X x) Diffl −jd (η(t) Dild−j′ (ς)(t)) dt 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 −2 k ek 2 ′ 󵄨󵄨 󵄨󵄨 −j d ′ ′ = 󵄨󵄨󵄨󵄨∫ f (e−2 t⋅X x) Diffl (j′ −j)dk (η(2−j d t)ς(t)) dt 󵄨󵄨󵄨󵄨 −2 ek 󵄨 󵄨 󵄨󵄨 󵄨 ′ −j′ d ′ 󵄨 = 󵄨󵄨󵄨󵄨∫(Diffl (j′ −j)dk f (e−2 t⋅X x))η(2−j d t)ς(t) dt 󵄨󵄨󵄨󵄨 2 ek 󵄨 󵄨 ′ ′ 󵄨 󵄨 −j sl /l −j′ d 󵄨 󵄨 ≲2 ‖f ‖C s (X,d) ∫󵄨󵄨η(2 t)ς(t)󵄨󵄨 dt l

−j′ sl′ /l

≲2

‖f ‖C s (X,d) . l

Lemma 7.7.15 required ς ∈ C0∞ (Bq (a0 )); however, we require a similar result for ς ∈ S (ℝq ). This can be achieved by decomposing the case of ς ∈ S (ℝq ) into a sum of the cases where ς ∈ C0∞ (Bq (a0 )). This decomposition is described in the next lemma.

7.7 Difference characterization

�

525

Lemma 7.7.16. Fix a0 > 0. Let ℬ1 ⊂ C0∞ (Bq (a0 )) and ℬ2 ⊂ S (ℝq ) be bounded sets. Then, for all η ∈ ℬ1 , ς ∈ ℬ2 , j ∈ [0, ∞), there exists {γj : j′ ∈ ℕ, j′ ≤ j} ⊂ C0∞ (Bq (a0 )) such that η(t) Dild2j (ς) = η ∑ Dildj′ (γj′ )(t). 2

j′ ∈ℕ j′ ≤j

(7.70)

Furthermore, for every M ∈ ℕ, {2M|j−j | γj′ : j ∈ [0, ∞), j′ ≤ j, ς ∈ ℬ2 , η ∈ ℬ1 } ⊂ C0∞ (Bq (a0 )) ′

(7.71)

is a bounded set. Proof. Fix η̃ ∈ C0∞ (Bq (a0 )) such that η̃ ≡ 1 on ⋃η∈ℬ1 supp(η) and such that η̃ ≡ 1 on a neighborhood of 0. Fix j ∈ ℕ. For j′ ∈ ℕ with j′ ≤ j, let σj′ (t) := {

̃ − η(2 ̃ d t), η(t)

j′ + 1 ≤ j,

̃ η(t),

j′ + 1 > j.

Note that j

̃ = ∑ σj′ (2j d t) η(t) ′

j′ =0

(7.72)

and σj′ (t) = 0 if j′ ≤ j − 1 and |t| is sufficiently small (independent of j, j′ ). For ς ∈ ℬ2 , set γj′ (t) := σj′ (t) Dildj−j′ (ς)(t). 2

It follows immediately from the definitions that {γj′ : j ∈ [0, ∞), j′ ≤ j, (j − j′ ) ≤ 1, ς ∈ ℬ2 } is a bounded set. So, to prove (7.71) is a bounded set, it suffices to consider the case where j − j′ > 1. For j − j′ > 1, since σj′ (t) = 0 for |t| small, using the Schwartz bounds for ς we have, for all m ∈ ℕ and multi-indices α ∈ ℕq , ′ 󵄨󵄨 α ′ 󵄨󵄨 󵄨 (j−j′ )d 󵄨󵄨 −m−|α| max dk (j−j′ )|α| max dk t 󵄨󵄨) 2 ≲ 2−m|j−j | . 󵄨󵄨𝜕t γj (t)󵄨󵄨 ≲ χ{|t|≈1} (1 + 󵄨󵄨󵄨2

(7.73)

By the definition of γj′ , there is a compact set 𝒦0 ⋐ Bq (a0 ) independent of j, j′ , and ς such that supp(γj′ ) ⊆ 𝒦0 . Combining this with (7.73) shows that ∀M ∈ ℕ,

526 � 7 Zygmund–Hölder spaces {2M|j−j | γj′ : j ∈ [0, ∞), j′ ≤ j, j′ ∈ ℕ, j − j′ ≥ 1, ς ∈ ℬ} ⊂ C0∞ (Bq (a0 )) ′

is a bounded set. This completes the proof that (7.71) is a bounded set. Using (7.72), we have ̃ Dild2j (ς)(t) = η(t) ∑ Dildj′ (γj′ )(t), η(t) Dild2j (ς)(t) = η(t)η(t) j′ ∈ℕ j′ ≤j

2

establishing (7.70) and completing the proof. ̃ Ω, (X, d ), l) > 0 is sufficiently small, Lemma 7.7.17. Fix l ∈ ℕ+ and s ∈ (0, l). If a0 = a0 (Ω, ∞ q the following holds. Let ℬ1 ⊂ C0 (B (a0 )) and ℬ2 ⊂ S (ℝq ) be bounded sets. Then there exists C ≥ 0 such that for all j ∈ [0, ∞) with 2−j ≤ a0 , η ∈ ℬ1 , ς ∈ ℬ2 , 0 ≤ l′ ≤ l, k ∈ {1, . . . , q}, and f ∈ Cls (M, (X, d )), ′ ′ 󵄨󵄨 󵄨󵄨 sup sup 󵄨󵄨󵄨∫ f (2−t⋅X x) Diffl−2−jdk e (η(t) Dild2j (ς)(t)) dt 󵄨󵄨󵄨 ≤ C2−jsl /l ‖f ‖C s (X,d) . l k 󵄨 󵄨 x∈Ω η∈ℬ1

ς∈ℬ2

Proof. By Lemma 7.7.16, for η ∈ ℬ1 and ς ∈ ℬ2 , we have η(t) Dild2j (ς)(t) = η(t) ∑ Dildj′ (γj′ )(t), j′ ∈ℕ j′ ≤j

2

where ∀M ∈ ℕ, {2M|j−j | γj′ : j ∈ [0, ∞), j′ ≤ j, j′ ∈ ℕ, ς ∈ ℬ2 , η ∈ ℬ1 } ⊂ C0∞ (Bq (a0 )) ′

is a bounded set. By taking M > s and applying Lemma 7.7.15, we have, for x ∈ Ω, ′ 󵄨󵄨 󵄨 󵄨󵄨∫ f (e−t⋅X x) Diffl −jd (η(t) Dildj (ς)(t)) dt 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 2 −2 k ek ′ 󵄨󵄨 󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨∫ f (e−t⋅X x) Diffl−2−jdk e (η(t) Dildj′ (γj′ )(t)) dt 󵄨󵄨󵄨 2 k 󵄨 󵄨 ′

j ∈ℕ j′ ≤j

≲ ∑ 2−M|j−j | 2−j sl /l ‖f ‖C s (X,d) ≲ 2−jsl /l ‖f ‖C s (X,d) . ′

j′ ∈ℕ j′ ≤j

′

′

′

l

l

̃ Ω, (X, d ), l) > 0 is sufficiently small, Lemma 7.7.18. Fix l ∈ ℕ+ and s ∈ (0, l). If a1 = a1 (Ω, ∞ q the following holds. Let ℬ1 ⊂ C0 (B (a1 )) and ℬ2 ⊂ S (ℝq ) be bounded sets. Then there

7.7 Difference characterization

� 527

exists C ≥ 0 such that for all j ∈ [0, ∞), η ∈ ℬ1 , ς ∈ ℬ2 , k ∈ {1, . . . , q}, 󵄨󵄨 󵄨󵄨 sup sup 󵄨󵄨󵄨∫ f (2−t⋅X x)η(t) Dild2j [Diffl−ek ς](t) dt 󵄨󵄨󵄨 ≤ C2−js ‖f ‖C s (X,d) . l 󵄨 󵄨 x∈Ω η∈ℬ1 ς∈ℬ2

Proof. We take a1 = a0 /2, where a0 is as in Lemma 7.7.17. If 2−j ≥ a1 /l, then 2−j ≈ 1 and we have 󵄨󵄨 󵄨󵄨 sup sup 󵄨󵄨󵄨∫ f (2−t⋅X x)η(t) Dild2j [Diffl−ek ς](t) dt 󵄨󵄨󵄨 󵄨 x∈Ω η∈ℬ1 󵄨 ς∈ℬ2

󵄨 󵄨 ≲ ‖f ‖C(M) sup ∫󵄨󵄨󵄨Dild2j [Diffl−ek ς](t)󵄨󵄨󵄨 dt ς∈ℬ2

󵄨 󵄨 = ‖f ‖C(M) sup ∫󵄨󵄨󵄨Diffl−ek ς(t)󵄨󵄨󵄨 dt ς∈ℬ2

≲ ‖f ‖C(M) ≤ ‖f ‖C s (X,d) ≈ 2−js ‖f ‖C s (X,d) . l

l

Thus we may henceforth assume 2−j < a1 /l. For v ∈ ℝq and F : ℝq → ℂ, we set τv F(t) := F(t + v). We have Diffv (FG) = (τv F)(Diffv G) + (Diffv F)G. In other words, F Diffv G = Diffv ((τ−v F)G) − (Diffv (τ−v F))G. Therefore, for l ∈ ℕ+ , l ′ ′ ′ l l F Difflv G = ∑ ( ′ )(−1)l−l Difflv ((Diffl−l v τ−v F)G). l l′ =0

(7.74)

Using (7.74) with F = η and G = Dildς (2j ) we have ∫ f (e−t⋅X x)η(t) Dild2j [Diffl−ek ς](t) dt = ∫ f (e−t⋅X x)η(t) Diffl−2−jdk e (Dild2j (ς)(t)) dt k

(7.75)

l

′ ′ l = ∑ ( ′ )(−1)l−l ∫ f (2−t⋅X x) Diffl−2−jdk e [ηj,l,l′ (t) Dild2j (ς)(t)] dt, k l l′ =0

where ηj,l,l′ := Diffl−l τl η. −2−jdk e 2−jdk e ′

k

k

(7.76)

528 � 7 Zygmund–Hölder spaces In light of (7.75), it suffices to show that for 0 ≤ l′ ≤ l, x ∈ Ω, ′ 󵄨󵄨 󵄨 󵄨󵄨∫ f (e−t⋅X x) Diffl −jd [ηj,l,l′ (t) Dildj (ς)(t)] dt 󵄨󵄨󵄨 ≲ 2−js ‖f ‖C s (X,d) . 󵄨󵄨 󵄨󵄨 2 −2 k ek l

(7.77)

We claim that {2(l−l )jdk ηj,l,l′ : 2−j ≤ a1 /l, η ∈ ℬ1 , 0 ≤ l′ ≤ l} ⊂ C0∞ (Bq (a0 )) ′

(7.78)

is a bounded set. It is clear from (7.76) (using 2−j < a1 /l) that supp(ηj,l,l′ ) ⊆ supp(η) + Bq (a1 ). Since a1 = a0 /2 and η ∈ ℬ1 ⊂ C0∞ (Bq (a1 )) is a bounded set, this shows ⋃ supp(ηj,l,l′ ) ⋐ Bq (a0 ).

⋃

(7.79)

η∈ℬ1 2−j 0 We follow the construction in Section 7.1; see Remark 7.1.5. Let a1 = a1 (Ω, be a small number, in particular so small that Lemma 7.7.18 holds. Let η ∈ C0∞ (Bq (a1 )) ̂ equal 1 on a neighborhood of 0. Take ρ ∈ S (ℝq ) with ρ(ξ) ≡ 1 on a neighborhood of q q ξ = 0 and such that supp(ρ)̂ ⊂ B (1). Define ς0 , ς1 ∈ S (ℝ ) by ̂ ς0̂ (ξ) := ρ(ξ),

̂ − ρ(2 ̂ d ξ). ς1̂ (ξ) = ρ(ξ)

Note that supp(ς1̂ ) ⊂ Bq (1) and ς1̂ (ξ) ≡ 1 for ξ on a neighborhood of 0. Define Dj f (x) := {

ψ(x) ∫ f (e−t⋅X x)η(t)ς0 (t) dt, ψ(x) ∫ f (e

−t⋅X

x)η(t) Dild2j (ς1 )(t)

dt,

j = 0, j ≥ 1.

Proposition 6.11.3 (see also Remark 7.1.5) implies that if a1 > 0 is sufficiently small in the above construction, then the norm ‖f ‖C s (W ,ds)⃗ can be defined by (7.80). Lemma 7.7.19. Fix l ∈ ℕ+ and s ∈ (0, l). With the above choice of Dj , ‖Dj f ‖L∞ ≲ 2−js ‖f ‖C s (X,d) , l

∀f ∈ Cls (X, d ).

(7.81)

Proof. For j = 0, we use ‖D0 ‖L∞ →L∞ ≲ 1 (see Corollary 7.1.4 (i)) to see that ‖D0 f ‖L∞ ≲ ‖f ‖L∞ ≤ ‖f ‖C s (X,d) , l

establishing (7.81) in the case j = 0. For j > 0, we apply Lemma 2.5.11 to write4 q

ς1 = ∑ Diffl−ek ς1,k , k=1

where ς1,k ∈ S (ℝq ). Applying Lemma 7.7.18 and using supp(Dj f ) ⊆ supp(ψ) ⊆ Ω, we have 󵄨 󵄨 ‖Dj f ‖L∞ = sup󵄨󵄨󵄨Dj f (x)󵄨󵄨󵄨 x∈Ω q

󵄨󵄨 󵄨󵄨 ≤ ∑ sup 󵄨󵄨󵄨∫ f (e−t⋅X x)η(t) Dild2j [Diffl−ek ς1,k ](t) dt 󵄨󵄨󵄨 󵄨 󵄨 x∈Ω k=1 −js

≲ 2 ‖f ‖C s (X,d) , l

completing the proof. 4 Lemma 2.5.11 was stated using Difflek but the same proof yields the result with Diffl−ek .

530 � 7 Zygmund–Hölder spaces Proposition 7.7.20. Fix l ∈ ℕ+ and s ∈ (0, l). Then {f ∈ Cls (X, d ) : supp(f ) ⊆ 𝒦} ⊆ C s (𝒦, (W , ds)),

(7.82)

and for f ∈ Cls (X, d ) with supp(f ) ⊆ 𝒦, we have ‖f ‖C s (W ,ds) ≲ ‖f ‖C s (X,d) . l

(7.83)

Proof. Inequality (7.83) follows directly from Lemma 7.7.19 (see (7.80)). The containment (7.82) follows from inequality (7.83) (see (7.1)). Proof of Theorem 7.7.8. This follows by combining Propositions 7.7.11 and 7.7.20. 7.7.1 Hölder spaces Because the spaces C1s (M, (X, d )) involve only first-order differences, they are equal to more standard Hölder spaces, as we will make precise in this section. We can use this observation and Theorem 7.7.8 to connect the Zygmund–Hölder spaces C s (𝒦, (W , ds)) with standard Hölder spaces in some cases. We begin by defining the standard Hölder spaces. For this, let (X, d ) = {(X1 , d1 ), . . . , ∞ (Xq , dq )} ⊂ Cloc (M; TM) × ℕ+ be any finite collection of smooth vector fields with singleparameter formal degrees. Definition 7.7.21. For s ∈ [0, 1] we set 󵄨 󵄨 ‖f ‖C 0,s (X,d) := ‖f ‖C(M) + sup ρ(X,d) (x, y)−s 󵄨󵄨󵄨f (x) − f (y)󵄨󵄨󵄨 x,y∈M x =y̸

and C 0,s (M, (X, d )) := {f ∈ C(M) : ‖f ‖C 0,r (X,d) < ∞}. Here, ρ(X,d) is the extended metric defined in (1.14). We will sometimes be interested in the special case where d1 = d2 = ⋅ ⋅ ⋅ dq = 1. We denote this case by (X, 1) = {(X1 , 1), . . . , (Xq , 1)}. In this special case, it is possible to extend Definition 7.7.21 to the spaces C m,s (M, (X, 1)) for m ∈ ℕ, as the next definition shows. Definition 7.7.22. For m ∈ ℕ and s ∈ [0, 1], we set C m,s (M, (X, 1)) := {f ∈ C(M) : X α f ∈ C 0,s (M, (X, 1)), ∀|α| ≤ m}

7.7 Difference characterization

�

531

and ‖f ‖C m,s (X,1) := ∑ ‖X α ‖C 0,s (X,1) . |α|≤m

For the remainder of this section, let ∞ (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} ⊂ Cloc (M; TM) × ℕ+

be Hörmander vector fields with (single-parameter) formal degrees. We let (W , 1) denote the special case where ds1 = ⋅ ⋅ ⋅ = dsr = 1. The main result of this section is the next theorem. Theorem 7.7.23. Fix s ∈ (0, 1). Then: (i) {f ∈ C 0,s (M, (W , ds)) : supp(f ) ⊆ 𝒦} = C s (𝒦, (W , ds)), with equivalent norms. (ii) For m ∈ ℕ, {f ∈ C m,s (M, (W , 1)) : supp(f ) ⊆ 𝒦} = C m+s (𝒦, (W , 1)), with equivalent norms. The rest of this section is devoted to the proof of Theorem 7.7.23. Lemma 7.7.24. For s ∈ (0, 1), C 0,s (M, (X, d )) = C1s (M, (X, d )), with equivalent norms. Proof. We will show that ‖f ‖C 0,s (X,d) ≈ ‖f ‖C1s (X,d) ,

∀f ∈ C(M),

where if one side is infinite, the other side is as well. The result will then follow. By unraveling the definitions we have sup

sup

δ∈(0,∞) γ∈𝒫 0 (M,(X,d)) δ

󵄨󵄨 0 󵄨󵄨 󵄨󵄨△γ f 󵄨󵄨 = sup

sup

δ∈(0,∞) γ∈𝒫 0 (M,(X,d)) δ

󵄨󵄨 󵄨 󵄨󵄨f (γ(0))󵄨󵄨󵄨

= sup |f (x)| = ‖f ‖C(M) . x∈M

Thus, it suffices to show that sup

sup

δ∈(0,∞) γ∈𝒫δ1 (M,(X,d))

󵄨 󵄨 󵄨 󵄨 δ−s 󵄨󵄨󵄨△1γ f 󵄨󵄨󵄨 ≈ sup ρ(X,d) (x, y)−s 󵄨󵄨󵄨f (x) − f (y)󵄨󵄨󵄨. x,y∈M x =y̸

(7.84)

We begin with the ≲ part of (7.84). Let δ ∈ (0, ∞) and γ ∈ 𝒫δ1 (M, (X, d )). If γ(1) = γ(0), then the left-hand side of (7.84) equals 0 and there is nothing to show. Otherwise, by the definition of ρ(X,d) (see (1.14)), we have ρ(X,d) (γ(0), γ(1)) < √qδ. Therefore, 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 δ−s 󵄨󵄨󵄨△1γ f 󵄨󵄨󵄨 = δ−s 󵄨󵄨󵄨f (γ(1)) − f (γ(0))󵄨󵄨󵄨 ≲ sup ρ(X,d) (x, y)−s 󵄨󵄨󵄨f (x) − f (y)󵄨󵄨󵄨, x,y∈M x =y̸

532 � 7 Zygmund–Hölder spaces taking the supremum over such δ and γ proves the ≲ part of (7.84). We turn to the ≳ part of (7.84). Let x, y ∈ M with x ≠ y. If ρ(X,d) (x, y) = ∞, then the right-hand side of (7.84) equals 0 and there is nothing to show. Otherwise, take δ > 0 such that ρ(X,d) (x, y) < δ. By the definition of ρ(X,d) (see (1.14)), there is an absolutely q continuous path γ : [0, 1] → M such that γ(0) = x, γ(1) = y, and γ′ (t) = ∑j=1 aj (t)Xj (γ(t)), with 󵄩󵄩 q 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩∑ |a |2 󵄩󵄩󵄩 < 1. 󵄩󵄩 j 󵄩 󵄩󵄩 󵄩󵄩j=1 󵄩󵄩L∞ ([0,1]) 󵄩 This implies γ ∈ 𝒫δ1 (M, (X, d )), so 󵄨 󵄨 δ−s 󵄨󵄨󵄨f (x) − f (y)󵄨󵄨󵄨 ≤ sup

δ∈(0,∞)

sup

γ∈𝒫δ1 (M,(X,d))

󵄨 󵄨 δ−s 󵄨󵄨󵄨△1γ f 󵄨󵄨󵄨.

Taking the supremum over all δ with ρ(X,d) (x, y) < δ proves the ≳ part of (7.84) (in fact proves ≥). Remark 7.7.25. In the definition of 𝒫δ1 (M, (X, d )) (Definition 7.7.2), if we replace (7.50) with the slight modification 󵄩󵄩 q 󵄩 󵄩󵄩 󵄨 −(d ∨(l′ +1)) l′ 󵄨2 󵄩󵄩󵄩 󵄩󵄩∑󵄨󵄨δ j 󵄨 𝜕t aj 󵄨󵄨 󵄩󵄩󵄩󵄩 < 1, 󵄩󵄩 󵄨 󵄩󵄩j=1 󵄩󵄩 ∞ 󵄩 󵄩L then one obtains the same spaces Cls (M, (X, d )), with an equivalent (though not equal) norm. With this change, one has equality of norms in Lemma 7.7.24, instead of merely equivalence of norms. ̃ ⋐ M with 𝒦 ⋐ Ω ⋐ Ω ̃ ⋐ M. Fix a relatively compact open set Ω ̃ and Lemma 7.7.26. Fix s ∈ (0, 1). Suppose Gen((W , ds)) is finitely generated by (X, d ) on Ω (W , ds) ⊆ (X, d ). Then {f ∈ C 0,s (M, (W , ds)) : supp(f ) ⊆ 𝒦} = {f ∈ C 0,s (M, (X, d )) : supp(f ) ⊆ 𝒦}, with equivalence of norms. Proof. Let f ∈ C(M) with supp(f ) ⊆ 𝒦 and let x, y ∈ M with x ≠ y. We will show that ‖f ‖C(M) + ρ(W ,ds) (x, y)−s |f (x) − f (y)|

≈ ‖f ‖C(M) + ρ(X,d) (x, y)−s |f (x) − f (y)|,

(7.85)

with constants independent of f , x, and y. Taking the supremum over such x and y completes the proof.

7.7 Difference characterization

� 533

If x ∈ ̸ 𝒦 and y ∈ ̸ 𝒦, then both sides of (7.85) equal ‖f ‖C(M) . Thus, it suffices to consider the case where at least one of x or y is in 𝒦. Without loss of generality, we assume x ∈ 𝒦. ̃ We apply Theorem 3.3.7 with 𝒦 Set 𝒦0 := Ω and pick Ω1 open with 𝒦0 ⋐ Ω1 ⋐ Ω. replaced by 𝒦0 and this choice of Ω0 to (W , ds) and (X, d ). Let ξ3 ∈ (0, 1] be as in that theorem. Suppose ρ(X,d) (x, y) < ξ3 . Then, by Theorem 3.3.7 (b) for δ ∈ (0, 1], ρ(X,d) (x, y) < ξ3 δ ⇒ ρ(W ,ds) (x, y) < δ ⇒ ρ(X,d) (x, y) < δ. We conclude that ρ(X,d) (x, y) ≈ ρ(W ,ds) (x, y), and (7.85) follows in this case. Suppose ρ(X,d) (x, y) ≥ ξ3 ; in particular, ρ(X,d) (x, y) ≳ 1. Since (W , ds) ⊆ (X, d ), we also have ρ(W ,ds) (x, y) ≥ ρ(X,d) (x, y) ≳ 1. Thus, ρ(W ,ds) (x, y)−s |f (x) − f (y)| ≲ ‖f ‖C(M) ,

ρ(X,d) (x, y)−s |f (x) − f (y)| ≲ ‖f ‖C(M) .

Formula (7.85) follows, completing the proof. Proof of Theorem 7.7.23. Item (i) follows by combining Theorem 7.7.8 and Lemmas 7.7.24 and 7.7.26. We turn to (ii), where ds1 = ⋅ ⋅ ⋅ = dsr = 1. In this case, for an ordered multi-index α, we have degds(α) = |α|. Let f ∈ C0∞ (M)′ with supp(f ) ⊆ 𝒦. Applying (i) and Theorem 6.2.12 s (with s replaced by s + m, X s = B∞,∞ = C s , and κ = m), we see that f ∈ C m+s (𝒦, (W , 1)) if and only if for all |α| ≤ m, W α f ∈ C s (𝒦, (W , 1)) = C 0,s (M, (W , 1)) if and only if f ∈ C m,s (M, (W , 1)). Moreover, in this case, ‖f ‖C m+s (W ,1) ≈ ∑ ‖W α f ‖C s (W ,1) ≈ ∑ ‖W α f ‖C 0,s (W ,1) = ‖f ‖C m,s (W ,1) , |α|≤m

completing the proof.

|α|≤m

8 Linear maximally subelliptic operators In this chapter, we study linear maximally subelliptic operators with smooth coefficients. Similar to the case of elliptic operators, linear maximally subelliptic operators with constant coefficients are a decisive tool for studying fully nonlinear maximally subelliptic operators and for studying linear maximally subelliptic operators with nonsmooth coefficients. See Chapter 9 for more details. Throughout this chapter, M will be a connected C ∞ manifold endowed with a smooth, strictly positive density Vol. Let ∞ (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} ⊂ Cloc (M; TM) × ℕ+

be Hörmander vector fields with (single-parameter) formal degrees on M. Fix κ ∈ ℕ+ such that dsj divides κ for 1 ≤ j ≤ r and set nj := κ/dsj ∈ ℕ+ . Fix D1 , D2 ∈ ℕ+ . We consider partial differential operators of the form P :=

∑

degds (α)≤κ

aα (x)W α ,

∞ aα ∈ Cloc (M; 𝕄D1 ×D2 (ℂ)).

(8.1)

We repeat here the definition of maximal subellipticity (Definition 1.1.7), as it is the central concept of this chapter (and even of this text). Definition 8.0.1. We say that P given by (8.1) is maximally subelliptic of degree κ with respect to (W , ds) on M if for every relatively compact, open set Ω ⋐ M, there exists CΩ ≥ 0 such that for all f ∈ C0∞ (Ω; ℂD2 ), r

󵄩 n 󵄩 󵄩 󵄩 󵄩 󵄩 ∑󵄩󵄩󵄩Wj j f 󵄩󵄩󵄩L2 (M,Vol;ℂD2 ) ≤ CΩ (󵄩󵄩󵄩P f 󵄩󵄩󵄩L2 (M,Vol;ℂD1 ) + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (M,Vol;ℂD2 ) ). j=1

(8.2)

When M is clear from the context, we just say P is maximally subelliptic of degree κ with respect to (W , ds). Remark 8.0.2. The density Vol does not play a role in Definition 8.0.1: one obtains an equivalent definition with any choice of smooth, strictly positive density.

8.1 The main result The main result of this chapter is a theorem which includes several equivalent characterizations of maximally subelliptic operators (in terms of regularity properties, heat equations, and parametrices). Using the L2 structure on (M, Vol), we may take the formal L2 (M, Vol) adjoint of Wj : Wj∗ = −Wj + Mult[fj ], https://doi.org/10.1515/9783111085647-008

∞ fj ∈ Cloc (M).

(8.3)

535

8.1 The main result �

Letting P be as in (8.1), we can also take its formal adjoint. Using (8.3), we have ∗

P P=

∑ degds (α),degds (β)≤κ

bα,β (x)W α W β ,

∞ bα,β ∈ Cloc (M; 𝕄D2 ×D2 (ℂ)).

(8.4)

In light of (8.4), it makes sense to ask whether P ∗ P is maximally subelliptic of degree 2κ with respect to (W , ds). We also treat P ∗ P as a densely defined operator on L2 (M, Vol; ℂD2 ), with dense domain C0∞ (M; ℂD2 ). Thought of in this way, P ∗ P is a nonnegative, symmetric, densely defined operator. We are now prepared to state the main result of this chapter. ∞ Theorem 8.1.1. Let (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} ⊂ Cloc (M; TM)×ℕ+ be Hörmander vector fields with formal degrees on M and let P be of the form (8.1). The following are equivalent: (i) P is maximally subelliptic of degree κ with respect to (W , ds) on M. (ii) P0 := ∑degds (α)=κ aα (x)W α is maximally subelliptic of degree κ with respect to (W , ds) on M. Recall that P = ∑degds (α)≤κ aα (x)W α , so that P0 consists of just those terms of degree precisely κ. (iii) ∀x0 ∈ M, there exists an open neighborhood U ⊆ M of x0 such that P is maximally subelliptic of degree κ with respect to (W , ds) on U. (iv) ∀x0 ∈ M, there exists an open neighborhood U ⊆ M of x0 such that the frozen coefficient operator ∑degds (α)≤κ aα (x0 )W α is maximally subelliptic of degree κ with respect to (W , ds) on U. (v) For any scale of spaces X s of the form s

s

s

X ∈ {Bp,q : p, q ∈ [1, ∞]} ⋃{Fp,q : p ∈ (1, ∞), q ∈ (1, ∞]},

we have the following. Let ϕ1 , ϕ2 , ϕ3 ∈ C0∞ (M) with ϕ1 ≺ ϕ2 ≺ ϕ3 . Then, ∀s ∈ ℝ, s s+κ ϕ3 P u ∈ Xcpt ((W , ds); ℂD1 ) ⇒ ϕ1 u ∈ Xcpt ((W , ds); ℂD2 ),

∀u ∈ C0∞ (M)′ .

(8.5)

Moreover, for every N ≥ 0 there exists C = C(s, N, X s , (W , ds), ϕ1 , ϕ2 , ϕ3 ) ≥ 0 such that ‖ϕ1 u‖X s+κ ((W ,ds);ℂD2 ) ≤ C(‖ϕ3 P u‖X s ((W ,ds);ℂD1 ) + ‖ϕ2 u‖X s−N ((W ,ds);ℂD2 ) ), s ∀u ∈ C0∞ (M; ℂD2 )′ with ϕ3 P u ∈ Xcpt (W , ds).

(8.6)

(vi) P ∗ P is maximally subelliptic of degree 2κ with respect to (W , ds) on M (see (8.4)). −κ (vii) There exists T ∈ Aloc ((W , ds); ℂD2 , ℂD1 ) (see Definition 5.11.3) such that TP ≡ I

∞ mod Cloc (M × M; 𝕄D2 ×D2 (ℂ)).

536 � 8 Linear maximally subelliptic operators −2κ (viii) There exists S ∈ Aloc ((W , ds); ℂD2 , ℂD2 ) such that ∞ mod Cloc (M × M; 𝕄D2 ×D2 (ℂ)).

SP ∗ P , P ∗ P S ≡ I

(ix) For every non-negative self-adjoint extension L of P ∗ P and all t > 0, the Schwartz ∞ kernel of e−tL is a Cloc (M × M; 𝕄D2 ×D2 (ℂ)) function (denoted by e−tL (x, y)), which satisfies the following. For every 𝒦 ⋐ M compact, there exists c > 0 such that for all ordered multi-indices α, β, for all s ∈ ℕ, there exists C ≥ 0, ∀t ∈ (0, 1], ∀x, y ∈ 𝒦, 1 − deg (α)−deg (β)−2κs 󵄨󵄨 s α β −tL 󵄨 ds ds (x, y)󵄨󵄨󵄨 ≤ C(ρ(W ,ds) (x, y) + t 2κ ) 󵄨󵄨𝜕t Wx Wy e 1

1 ρ(W ,ds) (x, y)2κ 2κ−1 −1 × exp(−c ( ) )(Vol(B(W ,ds) (x, ρ(W ,ds) (x, y) + t 2κ )) ∧ 1) . t

(x)

There exists a non-negative self-adjoint extension L of P ∗ P satisfying the conclusions of (ix). (xi) For every non-negative self-adjoint extension L of P ∗ P , {(e−tL , t 1/2κ ) : t ∈ (0, 1]} is a bounded set of locally (W , ds) pre-elementary operators. (xii) There exists a non-negative self-adjoint extension L of P ∗ P satisfying the conclusions of (xi). This chapter is devoted to proving Theorem 8.1.1 and understanding some of its consequences. The proof of Theorem 8.1.1 is completed in Section 8.5. Remark 8.1.2. Item (v) is a priori stronger than (i) in several ways. Indeed, fix a relatively compact, open set Ω ⋐ M and choose ϕ1 , ϕ2 , ϕ3 as in (v) with ϕ1 ≡ 1 on Ω. Then, s applying (v) with X s = Fp,2 for p ∈ (1, ∞) and N = s = 0 and using Corollary 6.2.14 shows that ∀f ∈ C0∞ (Ω; ℂD2 ), ∑

degds (α)≤κ

󵄩󵄩 α 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩W f 󵄩󵄩Lp (M,Vol;ℂD2 ) ≲ 󵄩󵄩󵄩P f 󵄩󵄩󵄩Lp (M,Vol;ℂD1 ) + 󵄩󵄩󵄩f 󵄩󵄩󵄩Lp (M,Vol;ℂD2 ) .

(8.7)

Specializing (8.7) to p = 2, we obtain an a priori stronger inequality than (8.2). Thus, (v) improves the left-hand side of (8.2), generalizes (8.2) from p = 2 to all p ∈ (1, ∞), s extends (8.2) beyond just the L2 Sobolev spaces F2,2 to all Besov and Triebel–Lizorkin spaces, and localizes (8.2) using the cut-off functions ϕ1 , ϕ2 , and ϕ3 . Remark 8.1.3. The estimate defining maximal subellipticity, (8.2), is in some sense the strongest possible local estimate one can assume. Indeed, due to the form (8.1) of P , it follows immediately that the reverse inequality in (8.7) holds. Thus, if P is maximally

8.2 Further regularity properties �

537

subelliptic of degree κ with respect to (W , ds) on M, Ω ⋐ M is open and relatively compact, and p ∈ (1, ∞), we have ∀f ∈ C0∞ (Ω; ℂD2 ), ∑

degds (α)≤κ

󵄩󵄩 α 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩W f 󵄩󵄩Lp (M,Vol;ℂD2 ) ≈ 󵄩󵄩󵄩P f 󵄩󵄩󵄩Lp (M,Vol;ℂD1 ) + 󵄩󵄩󵄩f 󵄩󵄩󵄩Lp (M,Vol;ℂD2 ) .

Remark 8.1.4. Recently, another condition equivalent to maximal subellipticity, in terms of the representation theory of certain nilpotent Lie groups, was established by Androulidakis, Moshen, and Yuncken [2]. See Section 8.8. Remark 8.1.5. Though Theorem 8.1.1 is not a local theorem, the main arguments used to prove it are. Because of this, there is a more general result, with the same proof, for vector bundles over M, whereas Theorem 8.1.1 is just stated for trivial vector bundles. See Section 8.6 for more details.

8.2 Further regularity properties Combining Theorem 8.1.1 with the results from Chapters 5 and 6, we can deduce sharp regularity properties of maximally subelliptic operators with respect to the Besov and Triebel–Lizorkin spaces corresponding to other geometries. We separate this into three parts: – Regularity properties of maximally subelliptic operators with respect to the function spaces corresponding to different lists of Hörmander vector fields with formal degrees. See Section 8.2.1. – Sharp regularity properties of maximally subelliptic operators with respect to the standard Besov and Triebel–Lizorkin spaces on M. In particular, we see that maximally subelliptic operators are subelliptic and prove the sharp subelliptic gain. This is a special case of the previous point. See Section 8.2.2. – Sharp regularity properties of maximally subelliptic operators with respect to multi-parameter function spaces. See Section 8.2.3. Even if one is only interested in single-parameter function spaces, the results on multi-parameter function spaces are essential for understanding the regularity properties of fully nonlinear maximally subelliptic equations in Section 9.1.1 (see the discussion in Section 1.8). They are also useful for obtaining some single-parameter results about linear operators (for example, Corollary 8.2.9).

8.2.1 Single-parameter function spaces Proposition 8.2.1. Let P be given by (8.1) and suppose P is maximally subelliptic of de∞ gree κ with respect to (W , ds) on M. Let (Z, dr) = {(Z1 , dr1 ), . . . , (Zv , drv )} ⊂ Cloc (M; TM)×ℕ+ be another set of Hörmander vector fields with single-parameter formal degrees on M.

538 � 8 Linear maximally subelliptic operators Suppose (W , ds) and (Z, dr) locally weakly approximately commute on M. Fix x0 ∈ M and let λ := λ(x0 , (W , ds), (Z, dr)) > 0 be as in Definition 6.6.9. Then there exists a relatively compact, open neighborhood Ω ⋐ M of x0 such that the following holds. For every family of spaces X s of the form s

s

s

X ∈ {Bp,q : p, q ∈ [1, ∞]} ⋃{Fp,q : p ∈ (1, ∞), q ∈ (1, ∞]}

(8.8)

and for every ϕ1 , ϕ2 , ϕ3 ∈ C0∞ (Ω) with ϕ1 ≺ ϕ2 ≺ ϕ3 , we have s s+κ/λ ϕ3 P u ∈ Xcpt (Z, dr) ⇒ ϕ1 u ∈ Xcpt (Z, dr),

∀u ∈ C0∞ (M; ℂD2 )′ .

(8.9)

Moreover, for every N ≥ 0 there exists C = C(s, N, P , X s , (W , ds), (Z, dr), ϕ1 , ϕ2 , ϕ3 ) ≥ 0 such that 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩ϕ1 u󵄩󵄩X s+κ/λ (Z,dr) ≤ C(󵄩󵄩󵄩ϕ3 P u󵄩󵄩󵄩X s (Z,dr) + 󵄩󵄩󵄩ϕ2 u󵄩󵄩󵄩X s−N (Z,dr) ), s ∀u ∈ C0∞ (M; ℂD2 )′ with ϕ3 P u ∈ Xcpt (Z, dr).

(8.10)

Furthermore, if λ(x0 , (W , ds), (Z, dr)) is sharp (see Definition 6.7.4), this result is optimal in the following sense. Suppose P is any partial differential operator of the form (8.1) (not necessarily maximally subelliptic) and suppose that there exists a neighborhood Ω′ of x0 , a scale of spaces X0s of the form (8.8), s ∈ ℝ, C ≥ 0, and ϵ ≥ 0 such that ∀u ∈ C0∞ (Ω′ ; ℂD2 ) we have ‖u‖X s+κ/λ+ϵ (Z,dr) ≤ C(‖P u‖X0s (Z,dr) + ‖u‖X0s (Z,dr) ). 0

(8.11)

Then ϵ = 0. −2κ Proof. By Theorem 8.1.1 (i) ⇒ (viii), there exists S ∈ Aloc ((W , ds); ℂD2 , ℂD2 ) such that

SP ∗ P , P ∗ P S ≡ I

∞ mod Cloc (M × M; 𝕄D2 ×D2 (ℂ)).

−2κ By Theorem 5.11.17, S ∈ Dloc ((W , ds), (Z, dr); ℂD2 , ℂD2 ). Set T := S P ∗ so that T satisfies

TP ≡ I

∞ mod Cloc (M × M; 𝕄D2 ×D2 (ℂ)).

(8.12)

−κ By Proposition 5.9.4, T ∈ Dloc ((W , ds), (Z, dr); ℂD2 , ℂD1 ). In particular, since −κ −κ T ∈ Dloc ((W , ds), (Z, dr); ℂD2 , ℂD1 ) ⊆ Aloc ((W , ds); ℂD2 , ℂD1 ), ∞ T is pseudo-local, i. e., its Schwartz kernel, T(x, y), is Cloc on x ≠ y (see, for example, the growth condition in Definition 5.2.2).

8.2 Further regularity properties

� 539

Let Ω0 ⋐ M be an open neighborhood of x0 as guaranteed by Proposition 6.6.21 (with x replaced by x0 in that proposition). Set 𝒦1 := Ω0 . By Proposition 6.6.21, we have, ∀γ1 , γ2 ∈ C0∞ (Ω0 ), Mult[γ1 ]T Mult[γ2 ] : X s (𝒦1 , (Z, dr)) → X s+κ/λ (𝒦1 , (Z, dr)) is bounded.

(8.13)

Fix Ω ⋐ Ω0 an open neighborhood of x0 and let 𝒦 := Ω ⋐ Ω0 . This is the Ω for which we prove the result. Let Ψ1 , Ψ2 , Ψ3 ∈ C0∞ (Ω0 ), with Ψ1 ≺ Ψ2 ≺ Ψ3 , and Ψ1 ≡ 1 on 𝒦. Note that ϕ1 ≺ ϕ2 ≺ ϕ3 ≺ Ψ1 , and therefore (8.9) and (8.10) are equivalent with u replaced by Ψ1 u. We henceforth make this replacement, so that Ψ2 u = u. ∞ Let R1 , R2 denote elements of Cloc (M × M; 𝕄D2 ×D2 ) (which depend on ϕ3 ). We have, using the pseudo-locality of T and (8.12), ϕ1 u = ϕ1 Ψ2 u = ϕ1 T P Ψ2 u + ϕ1 R1 Ψ2 u = ϕ1 Tϕ3 P Ψ2 u + ϕ1 R2 Ψ2 u

= ϕ1 Tϕ3 P u + ϕ1 R2 Ψ2 u = (Mult[ϕ1 ]T Mult[Ψ2 ])ϕ3 P u + ϕ1 R2 Ψ2 u.

(8.14)

s Using (8.14), we can prove (8.9). Indeed, suppose ϕ3 P u ∈ Xcpt (Z, dr); since supp(ϕ3 ) ⊆ s 𝒦1 , Remark 6.5.8 shows that ϕ3 P u ∈ X (𝒦1 , (Z, dr)). Therefore, (8.13) implies ∞ (Mult[ϕ1 ]T Mult[Ψ2 ])ϕ3 P u ∈ X s+κ/λ (𝒦1 , (Z, dr)). Also, since R2 ∈ Cloc (M × M; 𝕄D2 ×D2 ), ∞ s+κ/λ we have ϕ1 R2 Ψ2 u ∈ Cloc (Ω) ⊆ X (𝒦1 , (Z, dr)), where the ⊆ follows from Proposition 6.5.5. Combining the above with (8.14), we conclude ϕ1 u ∈ X s+κ/λ (𝒦1 , (Z, dr)) ⊆ s+κ/λ Xcpt (Z, dr), as desired. We turn to (8.10). Let ψ ∈ C0∞ (Ω) with ϕ2 ≺ ψ ≺ ϕ3 . Applying (8.14) with ϕ3 replaced ∞ by ψ we obtain (with a possibly different choice of R2 ∈ Cloc (M × M; 𝕄D2 ×D2 )), for u ∈ s X (𝒦, (Z, dr)),

‖ϕ1 u‖X s+κ/λ (Z,dr) ≤ ‖ Mult[ϕ1 ]T Mult[Ψ2 ]ψP u‖X s+κ/λ (Z,dr) + ‖ϕ1 R2 Ψ2 u‖X s+κ/λ (Z,dr) ≲ ‖ψP u‖X s (Z,dr) + ‖Ψ2 u‖X s−N (Z,dr) ,

(8.15)

where the final estimate used (8.13) for the first term, and for the second term we used the fact that Mult[ϕ1 ]R2 Mult[Ψ2 ] ∈ C0∞ (Ω0 × Ω0 ; 𝕄D2 ×D2 (ℂ)) ⊆ A −κ/λ−N (Ω0 , (Z, dr); ℂD2 , ℂD2 ) (Proposition 5.8.11) and Theorem 6.2.10. Replacing u with ϕ2 u in (8.15)1 shows that ‖ϕ1 u‖X s+κ/λ (Z,dr) ≲ ‖ψP u‖X s (Z,dr) + ‖ϕ2 u‖X s−N (Z,dr)

= ‖ψϕ3 P u‖X s (Z,dr) + ‖ϕ2 u‖X s−N (Z,dr) ≲ ‖ϕ3 P u‖X s (Z,dr) + ‖ϕ2 u‖X s−N (Z,dr) ,

where the last estimate used Corollary 6.5.10. This completes the proof of (8.10). Finally we turn to the sharpness. Suppose λ(x0 , (W , ds), (Z, dr)) is sharp. For contradiction, suppose (8.11) holds for some ϵ > 0 and some choice of X0s , C ≥ 0, and Ω′ . By s 1 Since ϕ3 Pu ∈ Xcpt (Z, dr), the first part of the proof with ϕ1 replaced with ϕ2 implies ϕ2 u ∈ s+κ/λ s Xcpt (Z, dr) ⊆ Xcpt (Z, dr).

540 � 8 Linear maximally subelliptic operators Corollary 6.6.11, we may shrink Ω′ so that (W , ds) weakly λ = λ(x0 , (W , ds), (Z, dr))-controls (Z, dr) on Ω′ . Let (Y , d̂)⃗ be the set of vector fields with two parameter degrees given by (Y , d̂)⃗ := {(W1 , (ds1 , 0)), . . . , (Wr , (dsr , 0))} ⋃{(Z1 , (0, dr1 )), . . . , (Zv , (0, drv ))} ∞ ⊂ Cloc (M; TM) × (ℕ2 \ {0}).

Using Proposition 6.8.1, using our assumption (8.11), and using ϵ > 0, we have, for u ∈ C0∞ (Ω′ ; ℂD2 ), ‖u‖X (κ+λϵ/4,s)+(−λ(κ/λ+ϵ/2),κ/λ+ϵ/2)+(λϵ/8,ϵ/2) (Y ,d̂)⃗ 0

= ‖u‖X (−λϵ/8,s+κ/λ+ϵ) (Y ,d̂)⃗ 0

≲ ‖u‖X s+κ/λ+ϵ (Z,dr) ≲ ‖P u‖X0s (Z,dr) + ‖u‖X0s (Z,dr)

(8.16)

0

≲ ‖P u‖X (λϵ/4,s) (Y ,d̂) + ‖u‖X (λϵ/4,s) ≲ ‖u‖X (λϵ/4+κ,s) (Y ,d̂) , 0

0

0

where in the last estimate we used Proposition 6.5.9. Combining (8.16) with Proposition 6.7.7 shows ϵ = 0, a contradiction, completing the proof. Proposition 8.2.1 assumed that (W , ds) and (Z, dr) locally weakly approximately commute, which covers the most important examples. Without this assumption, we can still obtain a weaker result. Proposition 8.2.2. Let P be given by (8.1) and suppose P is maximally subelliptic of de∞ gree κ with respect to (W , ds) on M. Let (Z, dr) = {(Z1 , dr1 ), . . . , (Zv , drv )} ⊂ Cloc (M; TM)×ℕ+ be another set of Hörmander vector fields with single-parameter formal degrees on M. Fix x0 ∈ M and let λ := λ(x0 , (W , ds), (Z, dr)) > 0 be as in Definition 6.6.9. Then there exists a relatively compact, open neighborhood Ω ⋐ M of x0 such that the following holds. For every family of spaces X s of the form s

s

s

X ∈ {Bp,q : p, q ∈ [1, ∞]} ⋃{Fp,q : p ∈ (1, ∞), q ∈ (1, ∞]}

and for every ϕ1 , ϕ2 , ϕ3 ∈ C0∞ (Ω) with ϕ1 ≺ ϕ2 ≺ ϕ3 , we have, ∀ϵ > 0, 0 −ϵ+κ/λ ϕ3 P u ∈ Xcpt (Z, dr) ⇒ ϕ1 u ∈ Xcpt (Z, dr),

∀u ∈ C0∞ (M; ℂD2 )′ .

(8.17)

Moreover, for every N ≥ 0 there exists C = C(ϵ, N, P , X s , (W , ds), (Z, dr), ϕ1 , ϕ2 , ϕ3 ) ≥ 0 such that 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩ϕ1 u󵄩󵄩X −ϵ+κ/λ (Z,dr) ≤ C(󵄩󵄩󵄩ϕ3 P u󵄩󵄩󵄩X 0 (Z,dr) + 󵄩󵄩󵄩ϕ2 u󵄩󵄩󵄩X −N (Z,dr) ), 0 ∀u ∈ C0∞ (M; ℂD2 )′ with ϕ3 P u ∈ Xcpt (Z, dr).

(8.18)

8.2 Further regularity properties �

541

Proof. Let Ω ⋐ M be the relatively compact, open neighborhood of x0 called Ω0 in Theorem 6.6.13 (with x0 playing the role of x in that theorem). Set 𝒦 := Ω. Fix ϵ > 0 and 0 suppose u ∈ C0∞ (M; ℂD2 )′ with ϕ3 u ∈ Xcpt (Z, dr). Then, since supp(ϕ3 ) ⊆ 𝒦, Remark 6.5.8 shows that ϕ3 u ∈ X 0 (Z, dr)𝒦. The result is stronger the smaller ϵ is, so we may assume ϵ ∈ (0, κ/λ). By Theorem 6.6.13 (B) with s0 = λϵ/2 and ϵ replaced by ϵ/2, we have ϕ3 P u ∈ X −λϵ/2 (𝒦, (W , ds)),

‖ϕ3 P u‖X −λϵ/2 (W ,ds) ≲ ‖ϕ3 P u‖X 0 (Z,dr) .

(8.19)

Applying Theorem 8.1.1 (i) ⇒ (v) we see that ϕ1 u ∈ X κ−λϵ/2 (𝒦, (W , ds)),

‖ϕ1 u‖X κ−λϵ/2 (W ,ds) ≲ ‖ϕ3 P u‖X −λϵ/2 (W ,ds) + ‖ϕ2 u‖X −(N+ϵ)λ (W ,ds) .

(8.20)

Using κ−λϵ/2 = λ(κ/λ−ϵ)+λϵ/2, Theorem 6.6.13 (A) with s0 = −ϵ+κ/λ > 0 and ϵ replaced by λϵ/2 shows that κ/λ−ϵ ϕ1 u ∈ X κ/λ−ϵ (𝒦, (Z, dr)) ⊆ Xcpt (Z, dr),

‖ϕ1 u‖X κ/λ−ϵ (Z,dr) ≲ ‖ϕ1 u‖X κ−λϵ/2 (W ,ds) .

(8.21)

This establishes (8.17). For (8.18), combining (8.19), (8.20), and (8.21), we have ‖ϕ1 u‖X κ/λ−ϵ (Z,dr) ≲ ‖ϕ3 P u‖X 0 (Z,dr) + ‖ϕ2 u‖X −(N+ϵ)λ (W ,ds) .

(8.22)

Applying Theorem 6.6.13 (B) with s0 = (N + ϵ)λ, we have ‖ϕ2 u‖X −(N+ϵ)λ (W ,ds) ≲ ‖ϕ2 u‖X −N (Z,dr) .

(8.23)

Combining (8.22) and (8.23) yields (8.18) and completes the proof.

8.2.2 Standard function spaces As a simple consequence of Proposition 8.2.1, in this section we will obtain the sharp regularity properties of maximally subelliptic operators with respect to the standard Besov and Triebel–Lizorkin spaces on M. For this we recall the spaces, for 𝒦 ⋐ M compact, s

Bp,q,std (𝒦),

s

Fp,q,std (𝒦),

defined in Section 6.6.1 (see Definition 6.6.4). These are nothing more than the usual Besov and Triebel–Lizorkin spaces consisting of distributions supported in 𝒦. We also

542 � 8 Linear maximally subelliptic operators recall the positive numbers λstd (x, (W , ds)) and Λstd (x, (W , ds)) from Definition 6.6.15. In the important special case where ds1 = ds2 = ⋅ ⋅ ⋅ = dsr = 1, then λstd (x, (W , ds)) = m, where W1 , . . . , Wr satisfy Hörmander’s condition of order m at x, and Λstd (x, (W , ds)) = 1. s s s See Example 6.6.16. Finally, if Xstd is one of Bp,q,std or Fp,q,std , recall the definition of s s Xstd,cpt (M) given in (6.54) – the space of those distributions in Xstd (𝒦) for some 𝒦 ⋐ M. Corollary 8.2.3. Let P be given by (8.1) and suppose P is maximally subelliptic of degree κ with respect to (W , ds) on M. Fix x0 ∈ M. There exists a relatively compact, open neighborhood Ω ⋐ M of x0 such that the following holds with λ := λstd (x0 , (W , ds)). For every s family of spaces Xstd of the form s

s

s

Xstd ∈ {Bp,q,std : p, q ∈ [1, ∞]} ⋃{Fp,q,std : p ∈ (1, ∞), q ∈ (1, ∞]}

(8.24)

and for every ϕ1 , ϕ2 , ϕ3 ∈ C0∞ (Ω) with ϕ1 ≺ ϕ2 ≺ ϕ3 , we have s s+κ/λ ϕ3 P u ∈ Xstd,cpt (M) ⇒ ϕ1 u ∈ Xstd,cpt (M),

∀u ∈ C0∞ (M; ℂD2 )′ .

s Moreover, for every N ≥ 0 there exists C = C(s, N, P , Xstd , (W , ds), ϕ1 , ϕ2 , ϕ3 ) ≥ 0 such that

󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩ϕ1 u󵄩󵄩X s+κ/λ ≤ C(󵄩󵄩󵄩ϕ3 P u󵄩󵄩󵄩X s + 󵄩󵄩󵄩ϕ2 u󵄩󵄩󵄩X s−N ), std std std

s ∀u ∈ C0∞ (M; ℂD2 )′ with ϕ3 P u ∈ Xstd,cpt (M).

Furthermore, this result is optimal in the following sense. Suppose P is any partial differential operator of the form (8.1) (not necessarily maximally subelliptic) and suppose that s there exist a neighborhood Ω′ of x0 , a scale of spaces X0,std of the form (8.24), s ∈ ℝ, C ≥ 0, ∞ ′ D2 and ϵ ≥ 0 such that ∀u ∈ C0 (Ω ; ℂ ) we have ‖u‖X s+κ/λ+ϵ ≤ C(‖P u‖X s + ‖u‖X s ). 0,std

0,std

0,std

Then ϵ = 0. To prove Corollary 8.2.3 (and Corollary 8.2.8 below), we use the next lemma. Lemma 8.2.4. Fix x0 ∈ M. There exist an open neighborhood N ⊆ M of x0 and vector ∞ fields Z1 , . . . , Zn ∈ Cloc (N; TN) such that if (Z, 1) = {(Z1 , 1), . . . , (Zn , 1)}, we have: (i) span{Z1 (y), . . . , Zn (y)} = Ty N, ∀y ∈ N. (ii) λstd (x0 , (W , ds)) = λ(x0 , (W , ds), (Z, 1)) and Λstd (x0 , (W , ds)) = λ(x0 , (Z, 1), (W , ds)). (iii) λ(x0 , (W , ds), (Z, 1)) and λ(x0 , (Z, 1), (W , ds)) are sharp. (iv) (W , ds) and (Z, 1) locally weakly approximately commute on N. s (v) For every compact set 𝒦 ⋐ N and every space Xstd of the form (8.24), we have s

s

Xstd (𝒦) = X (𝒦, (Z, 1)), s s with equivalent norms. In particular, Xstd,cpt (N) = Xcpt (N, (Z, 1)).

8.2 Further regularity properties

� 543

∞ Proof. Pick Z1 , . . . , Zn ∈ Cloc (M; TM) such that Z1 (x0 ), . . . , Zn (x0 ) form a basis for Tx0 M. By continuity, there is a neighborhood N ⊆ M of x0 such that (i) holds. Lemma 6.6.17 with M replaced by N implies (ii). Item (iii) follows from the definitions (see also Example 6.7.6). Lemma 3.8.3 implies that (W , ds) and (Z, 1) locally strongly approximately commute on N, and therefore Proposition 3.8.6 (b) implies they locally weakly approximately commute on N; item (iv) follows. Item (v) is an immediate consequence of Theorem 6.6.7.

Proof of Corollary 8.2.3. Let N and (Z, 1) be as in Lemma 8.2.4. In light of Lemma 8.2.4, the result is an immediate consequence of Proposition 8.2.1 with (Z, dr) replaced by (Z, 1). Corollary 8.2.3 shows that maximally subelliptic operators are subelliptic and gives p the maximal possible subelliptic gain. Indeed, letting Ls denote the standard Lp Sobolev space of order s ∈ ℝ, we have the following. Corollary 8.2.5. Let P be given by (8.1) and suppose P is maximally subelliptic of degree κ with respect to (W , ds) on M. Fix x0 ∈ M. There exists a relatively compact, open neighborhood Ω ⋐ M of x0 such that the following holds with λ := λstd (x0 , (W , ds)). For every p ∈ (1, ∞), s ∈ ℝ, and ϕ1 , ϕ2 , ϕ3 ∈ C0∞ (Ω) with ϕ1 ≺ ϕ2 ≺ ϕ3 , we have p

ϕ3 P u ∈ Lps ⇒ ϕ1 u ∈ Ls+κ/λ ,

∀u ∈ C0∞ (M; ℂD2 )′ .

Moreover, for every N ≥ 0, there exists C = C(p, s, N, P , (W , ds), ϕ1 , ϕ2 , ϕ3 ) ≥ 0 such that 󵄩󵄩 󵄩󵄩 p 󵄩 󵄩 󵄩 󵄩 ≤ C(󵄩󵄩󵄩ϕ3 P u󵄩󵄩󵄩Lp + 󵄩󵄩󵄩ϕ2 u󵄩󵄩󵄩Lp ), 󵄩󵄩ϕ1 u󵄩󵄩L s s−N s+κ/λ

∀u ∈ C0∞ (M; ℂD2 )′ ,

where if the right-hand side is finite, so is the left-hand side. Furthermore, this result is optimal in the following sense. Suppose P is any partial differential operator of the form (8.1) (not necessarily maximally subelliptic) and suppose that there exist a neighborhood Ω′ of x0 , p ∈ (1, ∞), s ∈ ℝ, C ≥ 0, and ϵ ≥ 0 such that ∀u ∈ C0∞ (Ω′ ; ℂD2 ) we have ‖u‖Lp

s+κ/λ+ϵ

≤ C(‖P u‖Lps + ‖u‖Lps ).

Then ϵ = 0. s s Proof. This is the special case of Corollary 8.2.3 with Xstd = Fp,2,std , since when rep s stricted to distributions with fixed compact support, the spaces Fp,2,std and Ls are equal and have equivalent norms.

Remark 8.2.6. Corollary 8.2.5 shows that Corollary 8.2.3 implies maximally subelliptic operators are subelliptic and gives the sharp subelliptic gain. Corollary 8.2.3 does more than this: it gives the sharp gain on Besov and Triebel–Lizorkin spaces other than just the standard Sobolev spaces. For example, Corollary 8.2.3 gives the sharp gain on the s classical Zygmund–Hölder spaces B∞,∞,std .

544 � 8 Linear maximally subelliptic operators Remark 8.2.7. If P is given by (8.1), then it follows from Corollary 8.2.5 that the following are equivalent: – P is elliptic of degree κ near x0 . – P is maximally subelliptic of degree κ with respect to (W , ds) and span{Wj (x0 ) : dsj = 1} = Tx0 M. In particular, even in the special case where ds1 = ds2 = ⋅ ⋅ ⋅ = dsr = 1, the only way an operator of the form (8.1) can be elliptic of degree κ is if W1 , . . . , Wr satisfy Hörmander’s condition of order 1. Moreover, Corollary 8.2.5 shows that the higher the order of Hörmander’s condition, the more degenerate the operator will be. ̂ be a partial differential operator of order κ on M, with smooth Corollary 8.2.8. Let P ̂ is of order κ elliptic at x0 . Let Λ := Λstd (x0 , (W , ds)). coefficients. Fix x0 ∈ M and suppose P Then there exists a relatively compact, open neighborhood Ω ⋐ M of x0 such that the following holds. For every family of spaces X s of the form s

s

s

X ∈ {Bp,q : p, q ∈ [1, ∞]} ⋃{Fp,q : p ∈ (1, ∞), q ∈ (1, ∞]}

(8.25)

and for every ϕ1 , ϕ2 , ϕ3 ∈ C0∞ (Ω) with ϕ1 ≺ ϕ2 ≺ ϕ3 , we have ̂u ∈ X s (W , ds) ⇒ ϕ1 u ∈ X s+κ/Λ (W , ds), ϕ3 P cpt cpt ∀u ∈ C0∞ (M; ℂD2 )′ .

̂, X s , (W , ds), ϕ1 , ϕ2 , ϕ3 ) ≥ 0 such that Moreover, for every N ≥ 0 there exists C = C(s, N, P 󵄩󵄩 󵄩󵄩 󵄩 ̂ 󵄩󵄩 󵄩 󵄩 u󵄩󵄩X s (W ,ds) + 󵄩󵄩󵄩ϕ2 u󵄩󵄩󵄩X s−N (W ,ds) ), 󵄩󵄩ϕ1 u󵄩󵄩X s+κ/Λ (W ,ds) ≤ C(󵄩󵄩󵄩ϕ3 P ̂u ∈ X s (W , ds). ∀u ∈ C ∞ (M; ℂD2 )′ with ϕ3 P 0

cpt

̂ be any partial differential Furthermore, this result is optimal in the following sense. Let P operator of order κ on M, with smooth coefficients (not necessarily elliptic). Suppose that there exist a neighborhood Ω′ of x0 , a scale of spaces X0s of the form (8.25), s ∈ ℝ, C ≥ 0, and ϵ ≥ 0 such that ∀u ∈ C0∞ (Ω′ ; ℂD2 ) we have ̂u‖X s (W ,ds) + ‖u‖X s (W ,ds) ). ‖u‖X s+κ/Λ+ϵ (W ,ds) ≤ C(‖P 0 0 0

Then ϵ = 0. ̂ Proof. Let N and (Z, 1) be as in Lemma 8.2.4. By possibly shrinking N, we may assume P ̂ is maximally subelliptic of degree κ with is elliptic on N. In particular, this implies that P respect to (Z, 1) on N. By Remark 6.4.8, it suffices to prove the result with M replaced ̂, (W , ds) by N. Lemma 8.2.4 shows that Proposition 8.2.1 applies with P replaced by P replaced by (Z, 1), (Z, dr) replaced by (W , ds), M replaced by N, and λ replaced by Λ. The

8.2 Further regularity properties

� 545

conclusions of Proposition 8.2.1 with these replacements are exactly the conclusions of the corollary. As described in those results, Corollaries 8.2.3 and 8.2.5 are sharp. However, we can give a more precise version of Corollary 8.2.5, and if we allow a loss of ϵ derivatives, a more precise version of Corollary 8.2.3 can also be given. This is contained in the next result. Corollary 8.2.9. Let P be given by (8.1) and suppose P is maximally subelliptic of degree κ with respect to (W , ds) on M. Fix ϕ1 , ϕ2 , ϕ3 ∈ C0∞ (M) with ϕ1 ≺ ϕ2 ≺ ϕ3 . Then, ∀s ∈ ℝ, we have the following: (i) For every p ∈ (1, ∞), we have ϕ3 P u ∈ Lps ⇒ ϕ1 W α u ∈ Lps , ∀u ∈ C0∞ (M)′ ,

∀ degds(α) ≤ κ.

Moreover, for every N ≥ 0 there exists C = C(s, N, p, P , (W , ds), ϕ1 , ϕ2 , ϕ3 ) ≥ 0 with ∑

degds (α)≤κ

󵄩󵄩 󵄩 󵄩 󵄩 󵄩 α 󵄩 󵄩󵄩ϕ1 W u󵄩󵄩󵄩Lps ≤ C(󵄩󵄩󵄩ϕ3 P u󵄩󵄩󵄩Lps + 󵄩󵄩󵄩ϕ2 u󵄩󵄩󵄩Lp ), s−N

∀u ∈ C0∞ (M)′ ,

where if the right-hand side is finite, so is the left-hand side. s (ii) Let Xstd be any space of the form (8.24). Then, ∀ϵ > 0, s s−ϵ ϕ3 P u ∈ Xstd,cpt ⇒ ϕ1 W α u ∈ Xstd,cpt ,

∀u ∈ C0∞ (M)′ ,

∀ degds(α) ≤ κ.

s Moreover, for every N ≥ 0 there exists C = C(s, N, ϵ, Xstd , P , (W , ds), ϕ1 , ϕ2 , ϕ3 ) ≥ 0 with

∑

degds (α)≤κ

󵄩󵄩 󵄩 󵄩 󵄩 󵄩 α 󵄩 󵄩󵄩ϕ1 W u󵄩󵄩󵄩X s ≤ C(󵄩󵄩󵄩ϕ3 P u󵄩󵄩󵄩X s+ϵ + 󵄩󵄩󵄩ϕ2 u󵄩󵄩󵄩X s−N ), std std std

∀u ∈ C0∞ (M)′ ,

where if the right-hand side is finite, so is the left-hand side. We defer the proof of Corollary 8.2.9 to Section 8.2.3, where we see it as a consequence of more general multi-parameter results. Remark 8.2.10. It would be interesting to establish Corollary 8.2.9 (ii) with ϵ = 0. It s s seems likely that this is possible: in the special case where Xstd = Fp,2,std , this is exactly Corollary 8.2.9 (i). Also, Corollary 8.2.3 makes it seem likely that Corollary 8.2.9 (ii) holds with ϵ = 0.

546 � 8 Linear maximally subelliptic operators 8.2.3 Multi-parameter function spaces Essential to our study of the regularity properties of fully nonlinear maximally subelliptic equations (with respect to, for example, the standard Zygmund–Hölder spaces) are the regularity properties of linear maximally subelliptic operators with respect to multiparameter Besov and Triebel–Lizorkin spaces. We present the relevant multi-parameter results in this section. μ μ μ μ ∞ For each μ ∈ {1, . . . , ν}, let (W μ , dsμ ) = {(W1 , ds1 ), . . . , (Wrμ , dsrμ )} ⊂ Cloc (M; TM) × ℕ+ ∞ be Cloc vector fields with formal degrees. We assume that: – (W 1 , ds1 ) are Hörmander vector fields with formal degrees on M. – For 2 ≤ μ ≤ ν, Gen((W μ , dsμ )) is locally finitely generated on M. – (W 1 , ds1 ), . . . , (W ν , dsν ) pairwise locally weakly approximately commute on M. Remark 8.2.11. The above assumptions are exactly the assumptions of the main multiparameter setting described in Section 3.15. Define (W , ds)⃗ := (W 1 , ds1 ) ⊠ (W 2 , ds2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (W ν , dsν ). Since (W 1 , ds1 ) are Hörmander vector fields with formal degrees, Proposition 3.4.14 (in the case ν = 1) implies Gen((W 1 , ds1 )) is locally finitely generated on M. In particular, (W , ds)⃗ satisfies all the hypotheses of Section 5.1.3, and it therefore makes sense to talk about singular integrals ̃s (W , ds)⃗ (for s ∈ ℝν ) and the ν-parameter Besov and Triebel–Lizorkin like those in A loc spaces with respect to (W , ds)⃗ as defined in Section 6.3. Fix κ ∈ ℕ+ such that ds1j divides κ for 1 ≤ j ≤ r1 . We consider partial differential operators of the form P :=

∑

degds1 (α)≤κ

α

aα (x)(W 1 ) ,

∞ aα ∈ Cloc (M; 𝕄D1 ×D2 (ℂ)).

(8.26)

Proposition 8.2.12. Let P be given by (8.26) and suppose P is maximally subelliptic of degree κ with respect to (W 1 , ds1 ) on M. Then for any scale of spaces X s , s ∈ ℝν , of the form s

s

s

X ∈ {Bp,q : p, q ∈ [1, ∞]} ⋃{Fp,q : p ∈ (1, ∞), q ∈ (1, ∞]},

the following holds. Let ϕ1 , ϕ2 , ϕ3 ∈ C0∞ (M) with ϕ1 ≺ ϕ2 ≺ ϕ3 . Then, ∀s ∈ ℝν , s+κe1

s ϕ3 P u ∈ Xcpt (W , ds)⃗ ⇒ ϕ1 u ∈ Xcpt

⃗ (W , ds),

∀u ∈ C0∞ (M; ℂD2 )′ .

(8.27)

⃗ ϕ1 , ϕ2 , ϕ3 ) ≥ 0 such that Moreover, ∀N⃗ ∈ [0, ∞)ν , there exists C = C(s, N,⃗ P , X s , (W , ds), 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩ϕ1 u󵄩󵄩X s+κe1 (W ,ds)⃗ ≤ C(󵄩󵄩󵄩ϕ3 P u󵄩󵄩󵄩X s (W ,ds)⃗ + 󵄩󵄩󵄩ϕ2 u󵄩󵄩󵄩X s (W ,ds)⃗ ), ⃗ ∀u ∈ C ∞ (M; ℂD2 )′ with ϕ3 P u ∈ X s (W , ds). 0

cpt

(8.28)

8.2 Further regularity properties �

547

−2κ Proof. By Theorem 8.1.1 (i) ⇒ (viii), there exists S ∈ Aloc ((W 1 , ds1 ); ℂD2 , ℂD2 ) such that

SP ∗ P , P ∗ P S ≡ I

∞ mod Cloc (M × M; 𝕄D2 ×D2 (ℂ)).

̃ 1 ((W , ds); ⃗ ℂD2 , ℂD2 ). Set T := S P ∗ so that T satisBy Corollary 5.11.18, we have S ∈ A loc fies −2κe

TP ≡ I

∞ mod Cloc (M × M; 𝕄D2 ×D2 (ℂ)).

(8.29)

−κ ̃−κe1 ((W , ds); ⃗ ℂD2 , It follows from Corollary 5.8.10 that T ∈ Aloc ((W 1 , ds1 ); ℂD2 , ℂD1 ) ⋂ A loc D1 −κ 1 1 D2 D1 ℂ ). In particular, since T ∈ Aloc ((W , ds ); ℂ , ℂ ), T is pseudo-local, i. e., its Schwartz ∞ kernel, T(x, y), is Cloc on x ≠ y (see, for example, the growth condition in Definition 5.2.2). Set 𝒦 := supp(ϕ3 ). Let Ψ1 , Ψ2 , Ψ3 ∈ C0∞ (M), with Ψ1 ≺ Ψ2 ≺ Ψ3 , and Ψ1 ≡ 1 on 𝒦. Take Ω0 ⋐ M open and relatively compact with Ψ3 ∈ C0∞ (Ω0 ) and set 𝒦1 := Ω0 . Note that ϕ1 ≺ ϕ2 ≺ ϕ3 ≺ Ψ1 , and therefore (8.27) and (8.28) are equivalent with u replaced by Ψ1 u. We henceforth make this replacement, so that Ψ2 u = u. ∞ Let R1 , R2 denote elements of Cloc (M × M; 𝕄D2 ×D2 ) (which depend on ϕ3 ). We have, using the pseudo-locality of T and (8.29),

ϕ1 u = ϕ1 Ψ2 u = ϕ1 T P Ψ2 u + ϕ1 R1 Ψ2 u = ϕ1 Tϕ3 P Ψ2 u + ϕ1 R2 Ψ2 u

= ϕ1 Tϕ3 P u + ϕ1 R2 Ψ2 u = (Mult[ϕ1 ]T Mult[Ψ2 ])ϕ3 P u + ϕ1 R2 Ψ2 u.

(8.30)

s ⃗ Then, since Using (8.30), we can prove (8.27). Indeed, suppose ϕ3 P u ∈ Xcpt (W , ds). s ⃗ supp(ϕ3 ) ⊆ 𝒦1 , Remark 6.5.8 implies ϕ3 P u ∈ X (𝒦1 , (W , ds)). Since Mult[ϕ1 ]T Mult[Ψ2 ] ∈ ̃−κe1 (Ω0 , (W , ds); ⃗ ℂD2 , ℂD1 ), Theorem 6.3.10 implies that (Mult[ϕ1 ]T Mult[Ψ2 ])ϕ3 P u ∈ A s+κe1 ⃗ Also, since R2 ∈ C ∞ (M × M; 𝕄D2 ×D2 ), we have ϕ1 R2 Ψ2 u ∈ C ∞ (Ω) ⊆ X (𝒦1 , (W , ds)). loc loc s+κe1 ⃗ where the ⊆ follows from Proposition 6.5.5. Combining the above X (𝒦1 , (W , ds)), ⃗ ⊆ X s+κe1 (W , ds), ⃗ as desired. with (8.14), we conclude that ϕ1 u ∈ X s+κe1 (𝒦1 , (W , ds)) cpt ∞ We turn to (8.28). Let ψ ∈ C0 (Ω) with ϕ2 ≺ ψ ≺ ϕ3 . Applying (8.30) with ϕ3 replaced ∞ by ψ we obtain (with a possibly different choice of R2 ∈ Cloc (M × M; 𝕄D2 ×D2 )), for u ∈ s ⃗ X (𝒦, (W , ds)),

‖ϕ1 u‖X s+κe1 (W ,ds)⃗ ≤ ‖ Mult[ϕ1 ]T Mult[Ψ2 ]ψP u‖X s+κe1 (W ,ds)⃗ + ‖ϕ1 R2 Ψ2 u‖X s+κe1 (W ,ds)⃗

(8.31)

≲ ‖ψP u‖X s (W ,ds)⃗ + ‖Ψ2 u‖X s−N⃗ (W ,ds)⃗ , ̃−κe1 (Ω0 , (W , ds); ⃗ ℂD2 , ℂD1 ) and where the final estimate used Mult[ϕ1 ]T Mult[Ψ2 ] ∈ A Theorem 6.3.10 for the first term, and for the second term we used the fact that ̃−κe1 −N⃗ (Ω0 , (W , ds); ⃗ ℂD2 , ℂD2 ) (PropoMult[ϕ1 ]R2 Mult[Ψ2 ] ∈ C0∞ (Ω0 × Ω0 ; 𝕄D2 ×D2 (ℂ)) ⊆ A sition 5.8.11) and Theorem 6.3.10. Replacing u with ϕ2 u in (8.31)2 shows that s ⃗ the first part of the proof with ϕ1 replaced by ϕ2 shows that ϕ2 u ∈ 2 Since ϕ3 Pu ∈ Xcpt (W , ds), s+κe1 ⃗ ⊆ X s (𝒦, (W , ds)), ⃗ and therefore (8.31) applies. X (𝒦, (W , ds))

548 � 8 Linear maximally subelliptic operators ‖ϕ1 u‖X s+κe1 (W ,ds)⃗ ≲ ‖ψP u‖X s (W ,ds)⃗ + ‖ϕ2 u‖X s−N⃗ (W ,ds)⃗

= ‖ψϕ3 P u‖X s (W ,ds)⃗ + ‖ϕ2 u‖X s−N⃗ (W ,ds)⃗ ≲ ‖ϕ3 P u‖X s (W ,ds)⃗ + ‖ϕ2 u‖X s−N⃗ (W ,ds)⃗ ,

where the last estimate used Corollary 6.5.10. This completes the proof. Using results from Chapter 6, we can deduce new regularity estimates for maximally subelliptic operators directly from Proposition 8.2.12. We present two such estimates in the next two corollaries. Corollary 8.2.13. Let P be given by (8.26) and suppose P is maximally subelliptic of degree κ with respect to (W 1 , ds1 ) on M. Let (Y , d̂)⃗ := (W 2 , ds2 ) ⊠ (W 3 , ds3 ) ⊠ ⋅ ⋅ ⋅ (W ν , dsν ). Fix ϕ1 , ϕ2 , ϕ3 ∈ C0∞ (M) with ϕ1 ≺ ϕ2 ≺ ϕ3 . Then, for all s ∈ ℝν−1 , p ∈ (1, ∞), α s s ⃗ ϕ3 P u ∈ Fp,2,cpt (Y , d̂)⃗ ⇒ ϕ1 (W 1 ) u ∈ Fp,2,cpt (Y , d̂),

∀u ∈ C0∞ (M)′ ,

(8.32)

∀ degds1 (α) ≤ κ.

⃗ ϕ1 , ϕ2 ) ≥ 0 such that Moreover, for every N⃗ ∈ [0, ∞)ν−1 , there exists C = C(s, P , p, (W , ds), ∑

degds1 (α)≤κ

󵄩󵄩 󵄩 󵄩 󵄩 󵄩 1 α 󵄩 󵄩󵄩ϕ1 (W ) u󵄩󵄩󵄩F s (Y ,d̂)⃗ ≤ C(󵄩󵄩󵄩ϕ3 P u󵄩󵄩󵄩F s (Y ,d̂)⃗ + 󵄩󵄩󵄩ϕ2 u󵄩󵄩󵄩F s−N⃗ (Y ,d̂)⃗ ), p,2 p,2 p,2

(8.33)

s ⃗ for all u ∈ C0∞ (M)′ with ϕ3 P u ∈ Fp,2,cpt (Y , d̂).

Proof. For 𝒦 ⋐ M compact, Proposition 6.8.2 shows that s

⃗ =F ⃗ Fp,2 (𝒦, (Y , d̂)) p,2 (𝒦, (W , ds)), (0,s)

(0,s) s ⃗ with equivalent norms. In particular, this shows that Fp,2,cpt (Y , d̂)⃗ = Fp,2,cpt (W , ds). ⃗ Fix ϕ1.5 ∈ C ∞ (M) with ϕ1 ≺ ϕ1.5 ≺ ϕ2 . Suppose ϕ3 P u ∈ F s (Y , d̂)⃗ = F (0,s) (W , ds). 0

p,2,cpt

p,2,cpt

(κ,s) Proposition 8.2.12 implies ϕ1.5 u ∈ Fp,2,cpt (W , ds)⃗ and moreover we also have ϕ3 u ∈ (0,s) ⃗ so Proposition 8.2.12 also implies that Fs (Y , d̂)⃗ = F (W , ds), p,2,cpt

p,2,cpt

‖ϕ1.5 u‖F (κ,s) (W ,ds)⃗ ≲ ‖ϕ3 P u‖F (0,s) (W ,ds)⃗ + ‖ϕ2 u‖F (0,s)−(0,N)⃗ (W ,ds)⃗ p,2

p,2

≈ ‖ϕ3 P u‖F s

⃗ p,2 (Y ,d̂)

p,2

+ ‖ϕ2 u‖F s−N⃗ (Y ,d̂)⃗ .

(8.34)

p,2

(κ,s) ⃗ Corollaries 6.5.10 and 6.5.11 imply, for deg 1 (α) ≤ κ, Since ϕ1.5 u ∈ Fp,2,cpt (W , ds), ds α

α

ϕ1 (W 1 ) u = ϕ1 (W 1 ) ϕ1.5 u ∈ Fp,2,cpt

(κ−degds1 (α),s)

(W , ds)⃗

(0,s) s ⃗ ⊆ Fp,2,cpt (W , ds)⃗ = Fp,2,cpt (Y , d̂),

8.2 Further regularity properties

� 549

establishing (8.32), and moreover ∑

degds1 (α)≤κ

󵄩󵄩 1 α 󵄩 󵄩󵄩ϕ1 (W ) u󵄩󵄩󵄩F s (Y ,d̂)⃗ ≈ p,2

∑

degds1 (α)≤κ

󵄩󵄩 1 α 󵄩 󵄩󵄩ϕ1 (W ) u󵄩󵄩󵄩F (0,s) (W ,ds)⃗ p,2

󵄩 󵄩 ≲ 󵄩󵄩󵄩ϕ1.5 u󵄩󵄩󵄩F (κ,s) (W ,ds)⃗ .

(8.35)

p,2

Combining (8.34) and (8.35) yields (8.33) and completes the proof. s Corollary 8.2.13 only concerned the Sobolev spaces Fp,2 . If one is willing to “lose ϵ derivatives,” then it is sometimes possible to prove a similar result for more general spaces, as the next result shows.

Corollary 8.2.14. Consider the special case where ν = 2 and suppose (W 2 , ds2 ) are Hörmander vector fields with formal degrees. Let P be given by (8.26) and suppose P is maximally subelliptic of degree κ with respect to (W 1 , ds1 ) on M. Then, for any scale of spaces X s , s ∈ ℝ, of the form s

s

s

X ∈ {Bp,q : p, q ∈ [1, ∞]} ⋃{Fp,q : p ∈ (1, ∞), q ∈ (1, ∞]},

the following holds. Let ϕ1 , ϕ2 , ϕ3 ∈ C0∞ (M) with ϕ1 ≺ ϕ2 ≺ ϕ3 and ϵ > 0. We have α

s s−ϵ ϕ3 P u ∈ Xcpt (W 2 , ds2 ) ⇒ ϕ1 (W 1 ) u ∈ Xcpt (W 2 , ds2 ),

∀u ∈ C0∞ (M)′ ,

∀ degds1 (α) ≤ κ.

(8.36)

Moreover, ∀N ≥ 0, there exists C = C(s, N, ϵ, P , X s , (W 1 , ds1 ), (W 2 , ds2 ), ϕ1 , ϕ2 ) ≥ 0 such that ∑

degds1 (α)≤κ

󵄩󵄩 󵄩 󵄩 󵄩 󵄩 1 α 󵄩 󵄩󵄩ϕ1 (W ) u󵄩󵄩󵄩X s−ϵ (W 2 ,ds2 ) ≤ C(󵄩󵄩󵄩ϕ3 P u󵄩󵄩󵄩X s (W 2 ,ds2 ) + 󵄩󵄩󵄩ϕ2 u󵄩󵄩󵄩X s−N (W 2 ,ds2 ) ),

(8.37)

s ∀u ∈ C0∞ (M)′ with ϕ3 P u ∈ Xcpt (W 2 , ds2 ).

Proof. Set 𝒦 := supp(ϕ3 ) and let Ω ⋐ M be an open, relatively compact set with 𝒦 ⊆ Ω. Since W12 , . . . , Wr22 satisfy Hörmander’s condition on M, it follows directly from the definitions that (W 2 , ds2 ) weakly λ-controls (W 1 , ds1 ) on Ω for some λ > 0. Set ϵ1 := ϵ/2λ > 0. Fix ϕ1.5 ∈ C0∞ (M) with ϕ1 ≺ ϕ1.5 ≺ ϕ2 . Suppose ϕ3 P u ∈ X s (𝒦, (W 2 , ds2 )). Then, ⃗ Proposiby Proposition 6.8.1, we have ϕ3 P u ∈ X s (𝒦, (W 2 , ds2 )) ⊆ X (−ϵ1 ,s) (𝒦, (W , ds)). (−ϵ1 +κ,s) ⃗ and moreover, Propositions 8.2.12 tion 8.2.12 implies that ϕ1.5 u ∈ X (𝒦, (W , ds)) and 6.8.1 imply that ⃗ ‖ϕ1.5 u‖X (−ϵ1 +κ,s) (W ,ds)⃗ ≲ ‖ϕ3 P u‖X (−ϵ1 ,s) (W ,ds)⃗ + ‖ϕ2 u‖X [ (−ϵ1 , s) − (0, N)][(W , ds)] ≲ ‖ϕ3 P u‖X s (W 2 ,ds2 ) + ‖ϕ2 u‖X s−N (W 2 ,ds2 ) .

(8.38)

550 � 8 Linear maximally subelliptic operators ⃗ Corollaries 6.5.10 and 6.5.11 imply, for deg 1 (α) ≤ κ, Since ϕ1.5 u ∈ X (−ϵ1 +κ,s) (𝒦, (W , ds)), ds α α ⃗ ϕ1 (W 1 ) u = ϕ1 (W 1 ) ϕ1.5 u ∈ X (−ϵ1 +κ−degds1 (α),s) (𝒦, (W , ds))

⃗ ⊆ X (−ϵ1 ,s) (𝒦, (W , ds)) and ∑

degds1 (α)≤κ

󵄩󵄩 󵄩 󵄩 1 α 󵄩 󵄩󵄩ϕ1 (W ) u󵄩󵄩󵄩X (−ϵ1 ,s) (W ,ds)⃗ ≲ 󵄩󵄩󵄩ϕ1.5 u󵄩󵄩󵄩X (−ϵ1 +κ,s) (W ,ds)⃗ .

(8.39)

By Theorem 6.7.2, we have, for degds1 (α) ≤ κ, α ⃗ ⊆ X (−ϵ1 ,s)+(ϵ/λ,−ϵ) (𝒦, (W , ds)) ⃗ = X (ϵ1 ,s−ϵ) (𝒦, (W , ds)) ⃗ ϕ1 (W 1 ) u ∈ X (−ϵ1 ,s) (𝒦, (W , ds))

and ∑

degds1 (α)≤κ

󵄩󵄩 1 α 󵄩 󵄩󵄩ϕ1 (W ) u󵄩󵄩󵄩X (ϵ1 ,s−ϵ) (W ,ds)⃗ ≲

∑

degds1 (α)≤κ

󵄩󵄩 1 α 󵄩 󵄩󵄩ϕ1 (W ) u󵄩󵄩󵄩X (−ϵ1 ,s) (W ,ds)⃗ .

(8.40)

Finally, by Proposition 6.8.1 we have, for degds1 (α) ≤ κ, α ⃗ ⊆ X s−ϵ (𝒦, (W 2 , ds2 )), ϕ1 (W 1 ) u ∈ X (ϵ1 ,s−ϵ) (𝒦, (W , ds))

establishing (8.36), and moreover ∑

degds1 (α)≤κ

󵄩󵄩 1 α 󵄩 󵄩󵄩ϕ1 (W ) u󵄩󵄩󵄩X s−ϵ (W 2 ,ds2 ) ≲

∑

degds1 (α)≤κ

󵄩󵄩 1 α 󵄩 󵄩󵄩ϕ1 (W ) u󵄩󵄩󵄩X (ϵ1 ,s−ϵ) (W ,ds)⃗ .

(8.41)

Combining (8.38), (8.39), (8.40), and (8.41) gives (8.37) and completes the proof. Proof of Corollary 8.2.9. Set 𝒦 := supp(ϕ2 ) ⋐ M and let N ⋐ M be an open and relatively ∞ compact subset of M with 𝒦 ⋐ N. Let (Z, 1) := {(Z1 , 1), . . . , (Zv , 1)} ⊂ Cloc (N; TN) × ℕ+ be such that span{Z1 (x), . . . , Zv (x)} = Tx N, ∀x ∈ N. Lemma 3.8.3 shows (Z, 1) and (W , ds) locally strongly approximately commute on N, and therefore Proposition 3.8.6 (b) implies they locally weakly approximately commute on N. s Theorem 6.6.7 shows that Xstd (𝒦) = X s (𝒦, (Z, 1)), with equivalent norms. We take 1 1 ν = 2 and replace (W , ds ) with (W , ds) and (W 2 , ds2 ) with (Z, 1). Using the above remarks, (i) follows from Corollary 8.2.13 and (ii) follows from Corollary 8.2.14. Remark 8.2.15. When (W 1 , ds1 ) = (W 2 , ds2 ), the conclusion of Corollary 8.2.14 with ϵ = 0 follows immediately from Theorem 8.1.1 (i) ⇒ (v).

8.3 A priori estimates

�

551

8.3 A priori estimates The key to proving Theorem 8.1.1 is combining scaling techniques with a priori estimates which take place on the unit ball. Because we combine these results with scaling techniques, the results in this section will be applied an infinite number of times in the proof of Theorem 8.1.1. Thus, it is important to be explicit about what our estimates depend on, to later show that they remain uniform over this infinite number of applications. We suppose we are given Hörmander vector fields with formal degrees on Bn (1), (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} ⊂ C ∞ (Bn (1); TBn (1)), and we assume W1 , . . . , Wr satisfy Hörmander’s condition of order m ∈ ℕ+ on Bn (1). We make all the same assumptions as in Section 3.2. Fix a strictly positive smooth function h(u) ∈ C ∞ (Bn (1)) with infu∈Bn (1) h(u) > 0; we will sometimes endow Bn (1) with the measure h(u)σLeb , where σLeb denotes the usual Lebesgue density on Bn (1). We will also sometimes endow Bn (1) with the usual Lebesgue density σLeb . Fix κ ∈ ℕ+ such that dsj divides κ, for 1 ≤ j ≤ r. Set nj := κ/dsj ∈ ℕ+ . Fix D ∈ ℕ+ . The basic object of study in this section is a partial differential operator of the form L :=

∑ degds (α),degds (β)≤κ

bα,β (x)W α W β ,

bα,β ∈ C ∞ (Bn (1); 𝕄D×D (ℂ)).

(8.42)

Definition 8.3.1. We say an operator L of the form (8.42) is maximally subelliptic of type 2 of degree 2κ at the unit scale with respect to (W , ds) and hσLeb if there exists A ≥ 0 such that ∀f ∈ C0∞ (Bn (1); ℂD ), r

󵄩 n 󵄩2 ∑󵄩󵄩󵄩Wj j f 󵄩󵄩󵄩L2 (Bn (1),hσ j=1

Leb ;ℂ

D)

󵄨 󵄨 󵄩 󵄩2 ≤ A(󵄨󵄨󵄨⟨L f , f ⟩L2 (Bn (1),hσ ;ℂD ) 󵄨󵄨󵄨 + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (Bn (1),hσ ;ℂD ) ). Leb Leb

(8.43)

See Remark 8.5.3 and Corollary 8.5.4 for some comments on the relationship between maximal subellipticity of type 2 and maximal subellipticity. The main result of this section (Proposition 8.3.6) shows that maximally subelliptic operators of type 2 at the unit scale are subelliptic; we use this result to conclude some important corollaries. As mentioned before, it is important to be explicit about what our estimates depend on. For this, we introduce the next definition. Definition 8.3.2. For a parameter ι, we say C is an ι-L -unit-admissible constant if C can be chosen to depend only on: – Anything an (ι, κ)-unit-admissible constant can depend on in Definition 3.2.1. – An upper bound for D and the constant A ≥ 0 from (8.43).

552 � 8 Linear maximally subelliptic operators –

An upper bound for ‖h‖C L (Bn (1)) +

–

max

degds (α),degds (β)≤κ

‖bα,β ‖C L (Bn (1);𝕄D×D ) ,

where L ∈ ℕ+ can be chosen to depend only on ι and upper bounds for m, κ, n, r, and max{ds1 , . . . , dsr }. A lower bound > 0 for infu∈Bn (1) h(u).

We say C is an L -unit-admissible constant if it is a 0-L -unit-admissible constant. For another parameter ι0 , we say C = C(ι0 ) is an ι-L -unit-admissible constant if C is an ι-L -unit-admissible constant which may also depend on ι0 . In this section, we write A ≲ B to mean A ≤ CB, where C = C(ι0 ) ≥ 0 is an ι-L -unitadmissible constant, for appropriate choices of ι and ι0 , which will usually be clear from the context. When the choices of ι and ι0 are not clear, we will be explicit. All that matters for our applications, though, is that the estimates do not depend on the particular choice of L , (W , ds), or h, except through the quantities discussed in Definition 8.3.2. 8.3.1 Some preliminary estimates Set ϵ0 := (m max{ds1 , . . . , dsr })

−1

> 0;

(8.44)

recall that W1 , . . . , Wr satisfy Hörmander’s condition of order m ∈ ℕ+ on Bn (1). For s ∈ ℝ, we let ‖ ⋅ ‖L2s denote the standard L2 Sobolev norm of order s on ℝn . Lemma 8.3.3. Fix σ ∈ (0, 1) and κ ∈ ℕ+ such that dsj divides κ for 1 ≤ j ≤ r. Then: (i) ∀f ∈ C0∞ (Bn (1 − σ); ℂD ), r

∑

degds (α)≤κ

󵄩󵄩 α 󵄩󵄩 󵄩 nj 󵄩 󵄩 󵄩 󵄩󵄩W f 󵄩󵄩L2 (Bn (1),σLeb ;ℂD ) ≈ ∑󵄩󵄩󵄩Wj f 󵄩󵄩󵄩L2 (Bn (1),σLeb ;ℂD ) + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (Bn (1),σLeb ;ℂD ) . j=1

(ii) ∀f ∈ C0∞ (Bn (1 − σ); ℂD ), ∑

degds (α)≤κ−1

󵄩󵄩 α 󵄩󵄩 󵄩󵄩W f 󵄩󵄩L2 (Bn (1),σLeb ;ℂD ) ≲ α

∑

degds (α)≤κ−1

‖W f ‖L2ϵ ≲ 0

∑

󵄩󵄩 α 󵄩󵄩 󵄩󵄩W f 󵄩󵄩L2 , −ϵ0

∑

󵄩󵄩 α 󵄩󵄩 󵄩󵄩W f 󵄩󵄩L2 (Bn (1),σLeb ;ℂD ) .

degds (α)≤κ degds (α)≤κ

(iii) ∀f ∈ C0∞ (Bn (1 − σ); ℂD ), ‖f ‖L2κϵ ≲ 0

∑

degds (α)≤κ

󵄩󵄩 α 󵄩󵄩 󵄩󵄩W f 󵄩󵄩L2 (Bn (1),σLeb ;ℂD ) .

8.3 A priori estimates

�

553

Here, the implicit constants are κ-unit-admissible constants as in Definition 3.2.1 that can also depend on a lower bound > 0 for σ. Remark 8.3.4. The choice of ϵ0 in Lemma 8.3.3 (ii) and (iii) is not optimal. However, finding the optimal choice is not important for our purposes: all we use at this point is the fact that ϵ0 > 0 depends only on m and ds1 , . . . , dsr . When we combine this with scaling and techniques from harmonic analysis, we will obtain optimal results like those discussed in Section 8.2.2. Thus, the non-optimal Lemma 8.3.3 is an important tool in later proving optimal results. Remark 8.3.5. In the important special case ds1 = ⋅ ⋅ ⋅ = dsr = 1, Lemma 8.3.3 (ii) (with a different choice of ϵ0 > 0) follows from Proposition 4.5.8, which has a much simpler and much more elementary proof. This is one of the main ways in which working with general ds1 , . . . , dsr is harder than the case where ds1 = ⋅ ⋅ ⋅ = dsr = 1. Even in the general case, one can easily prove Lemma 8.3.3 (iii) (with a different choice of ϵ0 ) using Proposition 4.5.8, though we do not know of a direct, elementary proof of Lemma 8.3.3 (ii). Proof of Lemma 8.3.3. We prove the result without keeping track of the dependence of the constants. The dependence follows by keeping track of constants in the results we use, which we leave to the reader. The estimate (i) follows from Corollary 6.2.14 (in particular, (6.5)), with M = Bn (1), 𝒦 := Bn (1 − σ), and Vol = σLeb . The estimates (ii) and (iii) follow from a more general claim. Namely, for s ∈ ℝ and μ ∈ {0, . . . , κ}, ∑

degds (α)≤κ−μ

‖W α f ‖L2s ≲

∑

degds (α)≤κ

‖W α f ‖L2s−μϵ , 0

∀f ∈ C0∞ (Bn (1 − σ)).

(8.45)

Indeed, (ii) follows from the cases s = 0, μ = 1 and s = ϵ0 , μ = 1 of (8.45), while (iii) is the special case s = κϵ0 , μ = κ. We turn to proving (8.45). Throughout this proof, f will denote an arbitrary element of C0∞ (Bn (1 − σ)). We consider the Hörmander vector fields with formal degrees on Bn (1) given by (𝜕, 1) = {(𝜕x1 , 1), . . . , (𝜕xn , 1)}. Lemma 3.8.3 implies (W , ds) and (𝜕, 1) locally strongly approximately commute on Bn (1), and therefore Proposition 3.8.6 (b) implies they locally weakly approximately commute on Bn (1). We define the list of vector fields with two⃗ by parameter formal degrees, (Y , d̂), (Y , d̂)⃗ = {(W1 , (ds1 , 0)), . . . , (Wr , (dsr , 0))} ⋃{(𝜕x1 , (0, 1)), . . . , (𝜕xn , (0, 1))}. (Y , d̂)⃗ satisfies all the assumptions of Section 5.1.2 and therefore satisfies the assumptions of Section 5.1.3 (see Remark 5.1.2). In particular, all of the results concerning the function ⃗ spaces from Chapter 6 apply with (W , ds)⃗ replaced by (Y , d̂).

554 � 8 Linear maximally subelliptic operators By Theorem 6.6.7 (see also Definition 6.6.4), we have, ∀s ∈ ℝ, ‖f ‖L2s ≈ ‖f ‖F s

(8.46)

≈ ‖f ‖F2,2s (𝜕,1) .

2,2,std

Applying Proposition 6.8.8, we have, ∀s ∈ ℝ, (8.47)

‖f ‖F2,2s (𝜕,1) ≈ ‖f ‖F (0,s) (Y ,d̂)⃗ . 2,2

Thus, using (8.46), (8.47), and Corollary 6.5.11, we have ∑

degds (α)≤κ−μ

‖W α f ‖L2s ≈

∑

degds (α)≤κ−μ

‖W α f ‖F (0,s) (Y ,d̂)⃗ 2,2

(8.48)

≲ ‖f ‖F (κ−μ,s) (Y ,d̂)⃗ . 2,2

Because W1 , . . . , Wr satisfy Hörmander’s condition of order m, for each j = 1, . . . , n we have 𝜕xj = ∑ cjk Xk ,

∞ n (Xk , dk ) ∈ Gen((W , ds)), dk ≤ ϵ0−1 , cjk ∈ Cloc (B (1)).

k

This shows that (W , ds) weakly ϵ0−1 -controls (𝜕, 1) on Bn (1) (see Definition 3.14.3). By Theorem 6.7.2, we have ‖f ‖F (κ−μ,s) (Y ,d̂)⃗ = ‖f ‖F (κ,s−μϵ0 )+(−μ,μϵ0 ) (Y ,d̂)⃗ ≲ ‖f ‖F (κ,s−μϵ0 ) (Y ,d̂)⃗ . 2,2

2,2

2,2

(8.49)

By Proposition 6.5.12 (in particular, (6.25)), we have, ∀s ∈ ℝ, ‖f ‖F (κ,s−μϵ0 ) (Y ,d̂)⃗ ≈ 2,2

∑

degds (α)≤κ

‖W α f ‖F (0,s−μϵ0 ) (Y ,d̂)⃗ . 2,2

(8.50)

Applying (8.46) and (8.47), we have, ∀s ∈ ℝ, ∑

degds (α)≤κ

‖W α f ‖F (0,s−μϵ0 ) (Y ,d̂)⃗ ≈ 2,2

≈

∑

‖W α f ‖F s−μϵ0 (𝜕,1)

∑

‖W α f ‖L2s−μϵ .

degds (α)≤κ degds (α)≤κ

2,2

(8.51)

0

Combining (8.48), (8.49), (8.50), and (8.51) establishes (8.45) and completes the proof. 8.3.2 Subelliptic estimates In this section we present the subelliptic estimate which lies at the heart of the proof of Theorem 8.1.1. Throughout this section, we assume that L is a partial differential operator of the form (8.42) and we assume that L is maximally subelliptic of type 2 of

8.3 A priori estimates

�

555

degree 2κ at the unit scale with respect to (W , ds) and hσLeb (see Definition 8.3.1). Let ϵ0 > 0 be as in (8.44). The main result of this section is the next proposition. Proposition 8.3.6. L is subelliptic on Bn (1) in the following sense. For every s ∈ ℝ and ϕ1 , ϕ2 ∈ C0∞ (Bn (1)) with ϕ1 ≺ ϕ2 , we have ‖ϕ1 f ‖L2s+κϵ ≲ ‖ϕ2 L f ‖L2s + ‖ϕ2 f ‖L2−∞ , 0

∀f ∈ C0∞ (Bn (1); ℂD )′ ,

(8.52)

where if the right-hand side is finite, so is the left-hand side. More precisely, for every s ∈ ℝ, N ∈ ℕ, there exists an (s, N)-L -unit-admissible constant C = C(ϕ1 , ϕ2 ) ≥ 0 (see Definition 8.3.2) such that ∀f ∈ C0∞ (Bn (1); ℂD )′ , ‖ϕ1 f ‖L2s+κϵ ≤ C(‖ϕ2 L f ‖L2s + ‖ϕ2 f ‖L2 ). 0

−N

Remark 8.3.7. The gain of κϵ0 on the left-hand side of (8.52) is not optimal. Similar to Remark 8.3.4, finding the optimal gain is not important for our applications: any subelliptic gain will allow us to deduce optimal estimates for maximally subelliptic operators as in Section 8.2.2. In particular, if L is, in addition, known to be symmetric and non-negative, then Corollary 8.5.4 shows that L is maximally subelliptic. The optimal subelliptic gain then follows from Corollary 8.2.5. it.

Before we prove Proposition 8.3.6, we state two important results which follow from

Corollary 8.3.8. For all s > 0 and ϕ1 , ϕ2 ∈ C0∞ (Bn (1)) with ϕ1 ≺ ϕ2 , there exists an s-L unit-admissible constant C = C(ϕ1 , ϕ2 ) ≥ 0 with ⌈ κϵs ⌉ 0

‖ϕ1 f ‖L2s ≤ C ∑ ‖ϕ2 L j f ‖L2 (Bn (1),σLeb ;ℂD ) , j=0

∀f ∈ C0∞ (Bn (1); ℂD )′ ,

where if the right-hand side is finite, so is the left-hand side. Proposition 8.3.9. Think of L as a densely defined operator on L2 (Bn (1), hσLeb ; ℂD ) with dense domain C0∞ (Bn (1); ℂD ). Suppose L is non-negative and symmetric. Then there is an L -unit-admissible constant C ≥ 0 such that the following holds. For every ϵ ∈ (0, 1/2], ∞ ∀u(t, x) ∈ Cloc ((−1, 1) × Bn (1); ℂD )) such that 𝜕t u(t, x) = −ϵ−2κ L u(t, x), ∀k ∈ [1, 1/2ϵ], ‖𝜕t u‖L2 ((−1+kϵ,1−kϵ)×Bn (1/2),dt×d(hσLeb );ℂD )

≤ Cϵ−2κ ‖u‖L2 ((−1+(k−1)ϵ,1−(k−1)ϵ)×Bn (1),dt×d(hσLeb );ℂD ) .

Remark 8.3.10. The measure hσLeb plays a secondary role in the above results. Indeed, because h is smooth, it is bounded below, and all its derivatives are bounded, in most estimates it is not important whether one uses σLeb or hσLeb . For example, since h is bounded above and below by L -unit-admissible constants, we have

556 � 8 Linear maximally subelliptic operators

‖f ‖L2 (Bn (1),hσLeb ) ≈ ‖f ‖L2 (Bn (1),σLeb ) ,

∀f ∈ L2 (Bn (1), σLeb ).

(8.53)

One important place h plays a role is in Proposition 8.3.9: L being symmetric and nonnegative on L2 (Bn (1), hσLeb ; ℂD ) is not the same thing as L being symmetric and nonnegative on L2 (Bn (1), σLeb ; ℂD ). As described at the start of Section 8.3, the above results will be applied an infinite number of times, and h will vary over these applications, though we will be able to estimate h uniformly across these applications. In this section, every L2 inner product will be taken with respect to the measure hσLeb . We turn to proving Propositions 8.3.6 and 8.3.9 and Corollary 8.3.8. We begin with the proof of Proposition 8.3.6. Note that the conclusions of Proposition 8.3.6 remain unchanged if we replace W1 , . . . , Wr with ϕ3 W1 , . . . , ϕ3 Wr , where ϕ3 ∈ C0∞ (Bn (1)) with ϕ2 ≺ ϕ3 . With this replacement, W1 , . . . , Wr are standard pseudodifferential operators of order 1. Because of this, we may treat W1 , . . . , Wr as standard pseudo-differential operators of order 1 in the proofs which follow. Similarly, we can view the coefficients bα,β in (8.42) as elements of C0∞ (Bn (1); 𝕄D×D (ℂ)) rather than as elements of the larger space C ∞ (Bn (1); 𝕄D×D (ℂ)). As in Section 4.5.1, we write S (s,δ) , perhaps with a subscript to denote a standard pseudo-differential operator of order −∞ such that {S (s,δ) : δ ∈ (0, 1]} is a bounded set of standard pseudo-differential operators of order s. When we write S̃(s,δ) we mean a 󵄩 󵄩 finite sum of such terms. For example, if we write 󵄩󵄩󵄩S̃(s,δ) u󵄩󵄩󵄩L2 , it means ‖S1(s,δ) u‖L2 + ⋅ ⋅ ⋅ + (s,δ) (s,δ) (s,δ) ‖SQ u‖L2 , for some finite collection S1 , . . . , SQ . Here the finite collection does not depend on the function or distribution u under consideration. Lemma 8.3.11. Suppose Q is a partial differential operator on Bn (1) with smooth coefficients taking values in 𝕄D×D (ℂ). Suppose that for all ϕ1 , ϕ2 ∈ C0∞ (Bn (1)) with ϕ1 ≺ ϕ2 , ∀s ∈ ℝ, ∑

degds (α)≤κ

‖W α ϕ1 S (s,δ) f ‖L2 (Bn (1),σLeb ;ℂD )

≲ ‖ϕ2 Q f ‖L2s +

∑

degds (α)≤κ

󵄩󵄩 α ̃(s,δ) 󵄩󵄩 󵄩󵄩W ϕ2 S f 󵄩󵄩L2 + ‖f ‖L2−∞ , −ϵ0

(8.54)

∀f ∈ S (ℝn ; ℂD )′ . Then, ∀ϕ1 , ϕ2 ∈ C0∞ (Bn (1)) with ϕ1 ≺ ϕ2 , ∀s ∈ ℝ, ‖ϕ1 f ‖L2s+κϵ ≲ ‖ϕ2 Q f ‖L2s + ‖ϕ2 f ‖L2−∞ , 0

(8.55)

∀f ∈ C0∞ (Bn (1); ℂD )′ . 󵄩 󵄩 Proof. Recall that 󵄩󵄩󵄩W α ϕ2 S̃(s,δ) f 󵄩󵄩󵄩L2 denotes a finite sum of such terms. In this proof we −ϵ0

write L2 for the space L2 (Bn (1), σLeb ; ℂD ). Because of the localizations ϕ1 and ϕ2 in (8.55), it suffices to prove (8.55) for f ∈ S (ℝn ; ℂD )′ ; henceforth, f denotes an arbitrary element of S (ℝn ; ℂD )′ .

8.3 A priori estimates

�

557

Fix ϕ1 , ϕ2 , ϕ1.5 ∈ C0∞ (Bn (1)) with ϕ1 ≺ ϕ1.5 ≺ ϕ2 . For j ∈ ℕ, let ηj ∈ C0∞ (Bn (1)) satisfy ϕ1 ≺ η0 ≺ η1 ≺ η2 ≺ ⋅ ⋅ ⋅ ≺ ϕ1.5 . (s−kϵ ,δ) Let S0(s,δ) be given. We claim that, ∀k ∈ ℕ, there exist S̃k 0 such that ∑

degds (α)≤κ

󵄩󵄩 α (s,δ) 󵄩 󵄩󵄩W η0 S0 f 󵄩󵄩󵄩L2

≲ ‖ηk Q f ‖L2s +

∑

degds (α)≤κ

󵄩󵄩 α ̃(s−kϵ0 ) 󵄩󵄩 f 󵄩󵄩L2 + ‖f ‖L2−∞ . 󵄩󵄩W ηk Sk

(8.56)

We prove (8.56) by induction on k. The base case, k = 0, is trivial. We assume (8.56) for some k ∈ ℕ and prove it for k + 1. Fix η̃ k,1 , η̃ k,2 ∈ C0∞ (Bn (1)) with ηk ≺ η̃ k,1 ≺ η̃ k,2 ≺ ηk+1 . By the inductive hypothesis and (8.54) (with ϕ1 replaced by ηk and ϕ2 replaced by η̃ k,1 ), we have ∑

degds (α)≤κ

‖W α η0 S0(s,δ) f ‖L2

≲ ‖ηk Q f ‖L2s +

∑

degds (α)≤κ

≲ ‖η̃ k,1 Q f ‖L2s +

󵄩󵄩 α ̃(s−kϵ0 ,δ) 󵄩󵄩 f 󵄩󵄩L2 + ‖f ‖L2−∞ 󵄩󵄩W ηk S

∑

󵄩󵄩 α ̃ ̃(s−kϵ0 ,δ) 󵄩󵄩 f 󵄩󵄩L2 + ‖f ‖L2−∞ 󵄩󵄩W ηk,1 S −ϵ0

∑

󵄩󵄩 α ̃ ̃(s−kϵ0 ,δ) 󵄩󵄩 f 󵄩󵄩L2 + ‖f ‖L2−∞ . 󵄩󵄩W ηk,1 S −ϵ0

degds (α)≤κ

≲ ‖ηk+1 Q f ‖L2s +

degds (α)≤κ

Thus, to prove the inductive step, it suffices to show that ∑

degds (α)≤κ

󵄩󵄩 α ̃ ̃(s−kϵ0 ,δ) 󵄩󵄩 f 󵄩󵄩L2 ≲ 󵄩󵄩W ηk,1 S −ϵ0

∑

degds (α)≤κ

󵄩󵄩 α (s−(k+1)ϵ0 ,δ) 󵄩 f 󵄩󵄩󵄩L2 . 󵄩󵄩W ηk+1 S̃

(8.57)

Using the calculus of pseudo-differential operators (Theorem 2.2.8), we have, with Λs as in Definition 2.2.10, ∑

degds (α)≤κ

≲ ≲

󵄩󵄩 α ̃ ̃(s−kϵ0 ,δ) 󵄩󵄩 f 󵄩󵄩L2 ≲ 󵄩󵄩W ηk,1 S −ϵ0

∑

degds (α)≤κ

∑

󵄩󵄩 ̃ α (s−(k+1)ϵ0 ,δ) 󵄩 f 󵄩󵄩󵄩L2 = 󵄩󵄩ηk,2 W S̃

∑

󵄩󵄩 α (s−(k+1)ϵ0 ,δ) 󵄩 f 󵄩󵄩󵄩L2 , 󵄩󵄩W ηk+1 S̃

degds (α)≤κ degds (α)≤κ

󵄩󵄩 ̃ −ϵ α (s−kϵ0 ,δ) 󵄩 f 󵄩󵄩󵄩L2 󵄩󵄩ηk,2 Λ 0 W η̃ k,1 S̃ ∑

degds (α)≤κ

󵄩󵄩 ̃ α (s−(k+1)ϵ0 ,δ) 󵄩 f 󵄩󵄩󵄩L2 󵄩󵄩ηk,2 W ηk+1 S̃

establishing (8.57) and completing the proof of (8.56). Taking k large in (8.56) and using ηk ≺ ϕ1.5 shows that ∑

degds (α)≤κ

󵄩󵄩 α (s,δ) 󵄩 󵄩󵄩W η0 S0 f 󵄩󵄩󵄩L2 ≲ ‖ϕ1.5 Q f ‖L2s + ‖f ‖L2−∞ .

558 � 8 Linear maximally subelliptic operators By Lemma 8.3.3 (iii), this implies ‖η0 S0(s,δ) f ‖L2ϵ κ ≲ ‖ϕ1.5 Q f ‖L2s + ‖f ‖L2−∞ .

(8.58)

0

We now choose a particular S0(s,δ) . Namely, we take S0(s,δ) = Λs ψδ (D) Mult[ϕ1 ], where ψ̂ ∈ C0∞ (ℝn ) equals 1 on a neighborhood of 0 and ψδ (D) is given by (4.53). We have, using (8.58), ‖ϕ1 f ‖L2s+ϵ κ ≲ sup ‖η0 Λs ψδ (D)ϕ1 f ‖L2κϵ + ‖f ‖L2−∞ 0

0

δ>0

=

sup ‖η0 S0(s,δ) f ‖L2κϵ 0 δ>0

(8.59)

+ ‖f ‖L2−∞ ≲ ‖ϕ1.5 Q f ‖L2s + ‖f ‖L2−∞ .

Replacing f with ϕ2 f in (8.59) and using ϕ1 ≺ ϕ1.5 ≺ ϕ2 , we have ‖ϕ1 f ‖L2s+ϵ κ ≲ ‖ϕ1.5 Q f ‖L2s + ‖ϕ2 f ‖L2−∞ ≲ ‖ϕ2 Q f ‖L2s + ‖ϕ2 f ‖L2−∞ , 0

completing the proof. Lemma 8.3.12. Let ϕ1 , ϕ2 , ϕ3 ∈ C0∞ (Bn (1)) satisfy ϕ1 ≺ ϕ2 ≺ ϕ3 . Then [L , Mult[ϕ1 ]S (s,δ) ] =

∑ degds (α),degds (β)≤κ degds (α)+degds (β)≤2κ−1

(s,δ) Mult[ϕ2 ]W α W β Mult[ϕ3 ]Sα,β ,

(s,δ) where S (s,δ) and Sj,l are pseudo-differential operators of order −∞ which are pseudodifferential operators of order s, uniformly in δ ∈ (0, 1].

Proof. This follows easily from the calculus of pseudo-differential operators (Theorem 2.2.8). We recall the large constant (l.c.) and small constant (s.c.) notation defined in Section 4.5.1; see the discussion surrounding (4.40). Lemma 8.3.13. For ϕ1 , ϕ2 ∈ C0∞ (Bn (1)) with ϕ1 ≺ ϕ2 , we have 󵄨󵄨 󵄨 (s,δ) (s,δ) 󵄨󵄨⟨[Mult[ϕ1 ]S0 , L ]u, ϕ1 S0 u⟩L2 (Bn (1),hσLeb ;ℂD ) 󵄨󵄨󵄨 󵄩 󵄩2 ≤ (l.c.) ∑ 󵄩󵄩󵄩W α ϕ2 S̃(s,δ) u󵄩󵄩󵄩L2 + (s.c.) ∑ degds (α)≤κ

−ϵ0

degds (α)≤κ

󵄩󵄩 α (s,δ) 󵄩2 󵄩󵄩W ϕ1 S0 u󵄩󵄩󵄩L2 (Bn (1),σ

Leb ;ℂ

D)

,

for all u ∈ S (ℝn ; ℂD )′ . Furthermore, if u ∈ S (ℝn ; ℂD ), we may replace S0(s,δ) with S (s) and S̃(s,δ) with S̃(s) . Proof. We prove only the result for u ∈ S (ℝn ; ℂD )′ . The same proof applies for u ∈ S (ℝn ; ℂD ) (by replacing S0(s,δ) with S (s) and S̃(s,δ) with S̃(s) throughout the proof). When

8.3 A priori estimates

559

�

u ∈ S (ℝn ; ℂD )′ we require that the pseudo-differential operators be (qualitatively) infinitely smoothing to guarantee that we are working with smooth functions; this is not necessary when u ∈ S (ℝn ; ℂD ). Fix ϕ1 , ϕ2 , ϕ3 ∈ C0∞ (Bn (1)) with ϕ1 ≺ ϕ2 ≺ ϕ3 . We prove the result with ϕ2 replaced by ϕ3 . Applying Lemma 8.3.12, we have 󵄨󵄨 󵄨 (s,δ) (s,δ) 󵄨󵄨⟨[Mult[ϕ1 ]S0 , L ]u, ϕ1 S0 u⟩L2 (Bn (1),hσLeb ;ℂD ) 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 (s,δ) = 󵄨󵄨󵄨 ⟨ϕ2 W α W β ϕ3 Sα,β u, ϕ1 S0(s,δ) u⟩L2 (Bn (1),hσ ;ℂD ) 󵄨󵄨󵄨. ∑ Leb 󵄨󵄨 󵄨󵄨 󵄨 degds (α),degds (β)≤κ 󵄨

(8.60)

degds (α)+degds (β)≤2κ−1

By the calculus of pseudo-differential operators, we see that the right-hand side of (8.60) is bounded by a finite sum of terms of the form ∑ degds (α),degds (β)≤κ degds (α)+degds (β)≤2κ−1

󵄨󵄨 󵄨󵄨 ≤ 󵄨󵄨󵄨 󵄨󵄨 󵄨

∑

degds (α)≤κ degds (β)≤κ−1

󵄨󵄨 β ̃(s,δ) 󵄨 α (s,δ) 󵄨󵄨⟨W ϕ3 Sα,β u, W ϕ1 S0 u⟩L2 (Bn (1),hσLeb ;ℂD ) 󵄨󵄨󵄨

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ⋅󵄨󵄨󵄨 + 󵄨󵄨󵄨 ⋅󵄨󵄨󵄨 =: (I) + (II). ∑ 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨degds (α)≤κ−1 󵄨󵄨 degds (β)≤κ

We begin with (I). Using the Cauchy–Schwartz inequality, (8.53), and Lemma 8.3.3 (ii), we have (I) ≤ (l.c.)

∑ degds (β)≤κ−1

≤ (l.c.)

∑ degds (β)≤κ

󵄩󵄩 β ̃(s,δ) 󵄩󵄩2 󵄩󵄩W ϕ3 Sα,β u󵄩󵄩L2 (Bn (1),σLeb ) + (s.c.)

󵄩󵄩 β ̃(s,δ) 󵄩󵄩 󵄩󵄩W ϕ3 Sα,β u󵄩󵄩L2 + (s.c.) −ϵ0

∑

degds (α)≤κ

∑

degds (α)≤κ

󵄩󵄩 α (s,δ) 󵄩2 󵄩󵄩W ϕ1 S0 u󵄩󵄩󵄩L2 (Bn (1),σLeb )

󵄩󵄩 α (s,δ) 󵄩2 󵄩󵄩W ϕ1 S0 u󵄩󵄩󵄩L2 (Bn (1),σLeb ) ,

which is the desired bound. We turn to (II). Let Λs be as in Definition 2.2.10 and let S (−ϵ0 ) denote a standard pseudo-differential operator of order −ϵ0 . Using the Cauchy–Schwartz inequality, the calculus of pseudo-differential operators, (8.53), and Lemma 8.3.3 (ii), we have (II) ≤

∑

degds (α)≤κ−1 degds (β)≤κ

≤ (l.c.)

󵄨󵄨 (−ϵ0 ) β ̃(s,δ) W ϕ3 Sα,β u, Λϵ0 W α ϕ1 S0(s,δ) u⟩L2 (Bn (1),hσ 󵄨󵄨⟨S

∑ degds (β)≤κ

≤ (l.c.)

∑ degds (β)≤κ

󵄩󵄩 β ̃(s,δ) 󵄩󵄩2 󵄩󵄩W ϕ3 Sα,β u󵄩󵄩L2 + (s.c.) −ϵ0 󵄩󵄩 β ̃(s,δ) 󵄩󵄩2 󵄩󵄩W ϕ3 Sα,β u󵄩󵄩L2 + (s.c.) −ϵ0

∑

degds (α)≤κ−1

∑

degds (α)≤κ

which is the desired bound, completing the proof.

Leb ;ℂ

D)

󵄨󵄨 󵄨󵄨

󵄩󵄩 α (s,δ) 󵄩2 󵄩󵄩W ϕ1 S0 u󵄩󵄩󵄩L2

ϵ0

󵄩󵄩 α (s,δ) 󵄩2 󵄩󵄩W ϕ1 S0 u󵄩󵄩󵄩L2 (Bn (1),σLeb ) ,

560 � 8 Linear maximally subelliptic operators Proof of Proposition 8.3.6. We prove the proposition by verifying the conditions of Lemma 8.3.11. Let ϕ1 , ϕ2 ∈ C0∞ (Bn (1)) with ϕ1 ≺ ϕ2 . Using (8.53), Lemma 8.3.3 (i), the Cauchy–Schwartz inequality, Lemma 8.3.13, and Lemma 8.3.3 (ii), we have, for u ∈ S (ℝn ; ℂD )′ , ∑

degds (α)≤κ

󵄩󵄩 α (s,δ) 󵄩2 󵄩󵄩W ϕ1 S0 u󵄩󵄩󵄩L2 (Bn (1),σLeb )

r

󵄩 n 󵄩2 ≈ ∑󵄩󵄩󵄩Wj j ϕ1 S0(s,δ) u󵄩󵄩󵄩L2 (Bn (1),σ j=1

Leb )

󵄩 󵄩2 + 󵄩󵄩󵄩ϕ1 S0(s,δ) u󵄩󵄩󵄩L2 (Bn (1),σ

r

󵄩 n 󵄩2 ≲ ∑󵄩󵄩󵄩Wj j ϕ1 S0(s,δ) u󵄩󵄩󵄩L2 (Bn (1),hσ j=1

Leb )

Leb )

󵄩 󵄩2 + 󵄩󵄩󵄩ϕ2 S0(s,δ) u󵄩󵄩󵄩L2 (Bn (1),σ

Leb )

󵄨 󵄨 󵄩 󵄩2 ≲ 󵄨󵄨󵄨⟨L ϕ1 S0(s,δ) u, ϕ1 S0(s,δ) u⟩L2 (Bn (1),hσ ;ℂD ) 󵄨󵄨󵄨 + 󵄩󵄩󵄩ϕ2 S0(s,δ) u󵄩󵄩󵄩L2 (Bn (1),σ ) Leb Leb 󵄨󵄨 󵄨󵄨 (s,δ) (s,δ) ≲ 󵄨󵄨⟨ϕ1 S0 L u, ϕ1 S0 u⟩L2 (Bn (1),hσ ;ℂD ) 󵄨󵄨 Leb

󵄨 + 󵄨󵄨󵄨⟨[L , Mult[ϕ1 ]S0(s,δ) ]u, ϕ1 S0(s,δ) u⟩L2 (Bn (1),hσ

󵄩 󵄩2 ≤ (l.c.)(󵄩󵄩󵄩ϕ1 S0(s,δ) L u󵄩󵄩󵄩L2 (Bn (1),σ + (s.c.)(

∑

degds (α)≤κ

Leb )

+

∑

degds (α)≤κ

Leb ;ℂ

D)

󵄨󵄨 󵄩󵄩 (s,δ) 󵄩󵄩2 󵄨󵄨 + 󵄩󵄩ϕ2 S0 u󵄩󵄩L2 (Bn (1),σLeb )

‖W α ϕ2 S̃(s,δ) u‖2L2 ) −ϵ0

󵄩󵄩 α (s,δ) 󵄩2 󵄩󵄩W ϕ1 S0 u󵄩󵄩󵄩L2 (Bn (1),σLeb ) ).

󵄩 󵄩2 Subtracting (s.c.)(∑degds (α)≤κ 󵄩󵄩󵄩W α ϕ1 S0(s,δ) u󵄩󵄩󵄩L2 (Bn (1),σ ) ) from both sides of the above equaLeb tion yields ∑

degds (α)≤κ

󵄩󵄩 α (s,δ) 󵄩2 󵄩󵄩W ϕ1 S0 u󵄩󵄩󵄩L2 (Bn (1),σ

󵄩 󵄩2 ≲ 󵄩󵄩󵄩ϕ1 S0(s,δ) L u󵄩󵄩󵄩L2 (Bn (1),σ

Leb

Leb )

)+

∑

degds (α)≤κ

‖W α ϕ2 S̃(s,δ) u‖2L2 . −ϵ0

Combining 󵄩󵄩 (s,δ) 󵄩2 2 󵄩󵄩ϕ1 S0 L u󵄩󵄩󵄩L2 (Bn (1),σLeb ) ≲ ‖ϕ2 L u‖L2 + ‖u‖L2−∞ s with (8.61) shows that ∑

degds (α)≤κ

󵄩󵄩 α (s,δ) 󵄩2 󵄩󵄩W ϕ1 S0 u󵄩󵄩󵄩L2 (Bn (1),σLeb )

≲ ‖ϕ2 L u‖2L2 + s

∑

degds (α)≤κ

‖W α ϕ2 S̃(s,δ) u‖2L2 + ‖u‖2L2 .

Lemma 8.3.11 applies to complete the proof.

−ϵ0

−∞

(8.61)

8.3 A priori estimates

�

561

Proof of Corollary 8.3.8. Let L2 denote L2 (Bn (1), σLeb ; ℂD ). Let ηj ∈ C0∞ (Bn (1)), j ∈ ℕ, be a sequence with ϕ1 = η0 ≺ η1 ≺ η2 ≺ ⋅ ⋅ ⋅ ≺ ϕ2 . Then, using Proposition 8.3.6, we have, for N ∈ ℕ, ‖ϕ1 f ‖L2s = ‖η0 f ‖L2s ≲ ‖η1 L f ‖L2s−κϵ + ‖η1 f ‖L2 0

≲ ‖η2 L 2 f ‖L2

s−2κϵ0

+ ‖η2 L f ‖L2 + ‖η2 f ‖L2

≲ ⋅⋅⋅ ≲ ‖ηN L N f ‖L2

s−Nκϵ0

+ ‖ηN L N−1 f ‖L2 + ‖ηN L N−2 f ‖L2 + ⋅ ⋅ ⋅ + ‖ηN f ‖L2 .

Taking N = ⌈ κϵs ⌉ and using ηN ≺ ϕ2 completes the proof. 0

All that remains in this section is to prove Proposition 8.3.9. We henceforth assume that L is symmetric and non-negative on L2 (Bn (1), hσLeb ; ℂD ) with dense domain C0∞ (Bn (1); ℂD ). We continue to assume that L is maximally subelliptic of type 2 of degree 2κ at the unit scale with respect to (W , ds) and hσLeb (see Definition 8.3.1). In particular, since L is non-negative, this implies there exists A ≥ 0 with r

󵄩 n 󵄩2 ∑󵄩󵄩󵄩Wj j f 󵄩󵄩󵄩L2 (Bn (1),hσ j=1

Leb ;ℂ

D)

≤ A(⟨L f , f ⟩L2 (Bn (1),hσ

Leb ;ℂ

D)

+ ‖f ‖2L2 (Bn (1),hσ

Leb ;ℂ

D)

),

(8.62)

∀f ∈ C0∞ (Bn (1); ℂD ). Lemma 8.3.14. For all s ∈ ℝ and ϕ1 , ϕ2 ∈ C0∞ (Bn (1)) with ϕ1 ≺ ϕ2 , there exist s, ϕ1 , ϕ2 -L unit-admissible constants C1 , D1 ≥ 0 such that ∀f ∈ C0∞ (Bn (1); ℂD ), ∑

degds (α)≤κ

‖W α ϕ1 f ‖2L2 ≤ C1 Re⟨ϕ2 Λs ϕ1 L f , ϕ2 Λs ϕ1 f ⟩L2 (Bn (1),hσ s

+ D1

∑

degds (α)≤κ

Leb ;ℂ

D)

‖W α ϕ2 f ‖2L2 . s−ϵ0

Proof. In this proof, we use the (l.c.) and (s.c.) notation, but with an added twist. For real numbers A1 , A2 ∈ ℝ, B1 , B2 ≥ 0, we write A1 ≤ A2 + (l.c.)B1 + (s.c.)B2 to mean that for all ϵ > 0, there exists Cϵ > 0 with A1 ≤ A2 + Cϵ B1 + ϵB2 . Throughout this proof, f will denote an arbitrary element of C0∞ (Bn (1); ℂD ). Using the calculus of pseudo-differential operators (Theorem 2.2.8), we have, with ϕ3 ∈ C0∞ (Bn (1)), with ϕ2 ≺ ϕ3 , and using (8.53) and Lemma 8.3.3 (i) and (ii), where S (s) and S̃(s) , perhaps with a subscript, are standard pseudo-differential operators of order s which may change from line to line, ∑

degds (α)≤κ

‖W α ϕ1 f ‖L2s ≈ ≲

∑

‖ϕ2 Λs W α ϕ1 f ‖L2 (σLeb )

∑

‖W α ϕ2 Λs ϕ1 f ‖L2 (σLeb ) +

degds (α)≤κ degds (α)≤κ

∑

degds (α)≤κ−1

‖W α ϕ3 Sα(s) ϕ1 f ‖L2 (σLeb )

562 � 8 Linear maximally subelliptic operators r

󵄩 n 󵄩 ≲ ∑󵄩󵄩󵄩Wj j ϕ2 Λs ϕ1 f 󵄩󵄩󵄩L2 (σ j=1

Leb )

+

r

󵄩 n 󵄩 ≲ ∑󵄩󵄩󵄩Wj j ϕ2 Λs ϕ1 f 󵄩󵄩󵄩L2 (σ j=1

Leb )

+

r

󵄩 n 󵄩 ≲ ∑󵄩󵄩󵄩Wj j ϕ2 Λs ϕ1 f 󵄩󵄩󵄩L2 (σ j=1

Leb )

+

r

󵄩 n 󵄩 ≲ ∑󵄩󵄩󵄩Wj j ϕ2 Λs ϕ1 f 󵄩󵄩󵄩L2 (hσ j=1

Leb )

∑

degds (α)≤κ−1

∑

󵄩󵄩 α ̃(s) 󵄩󵄩 󵄩󵄩W ϕ2 Sα ϕ1 f 󵄩󵄩L2 −ϵ0

∑

󵄩󵄩̃(s) α 󵄩 󵄩󵄩Sα W ϕ2 f 󵄩󵄩󵄩L2 −ϵ0

degds (α)≤κ

degds (α)≤κ

+

󵄩󵄩 α ̃(s) 󵄩󵄩 󵄩󵄩W ϕ2 Sα ϕ1 f 󵄩󵄩L2 (σ

∑

degds (α)≤κ

Leb )

(8.63)

󵄩󵄩 α 󵄩 󵄩󵄩W ϕ2 f 󵄩󵄩󵄩L2 . s−ϵ0

Using Lemma 8.3.13 (with ϕ1 in that lemma replaced by ϕ2 ) and the calculus of pseudo-differential operators, we have, for ϕ3 ∈ C0∞ (Bn (1)) with ϕ3 ≺ ϕ2 , 󵄨󵄨 󵄨 s s 󵄨󵄨⟨[L , ϕ2 Λ ϕ1 ]f , ϕ2 Λ ϕ1 f ⟩L2 (hσLeb ) 󵄨󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨⟨[L , ϕ2 (Λs ϕ1 )]ϕ2 f , ϕ2 (Λs ϕ1 )ϕ2 f ⟩L2 (hσ ) 󵄨󵄨󵄨 Leb 󵄩 󵄩2 ≤ (l.c.) ∑ 󵄩󵄩󵄩W α ϕ3 S̃(s,δ) ϕ2 f 󵄩󵄩󵄩L2 + (s.c.) ∑ −ϵ0 degds (α)≤κ

≤ (l.c.)

∑

degds (α)≤κ

󵄩󵄩 α 󵄩2 󵄩󵄩W ϕ2 f 󵄩󵄩󵄩L2 + (s.c.) s−ϵ0

degds (α)≤κ

∑

degds (α)≤κ

󵄩󵄩 α 󵄩2 s 󵄩󵄩W ϕ2 (Λ ϕ1 )ϕ2 f 󵄩󵄩󵄩L2

(8.64)

󵄩󵄩 α 󵄩2 s 󵄩󵄩W ϕ2 Λ ϕ1 f 󵄩󵄩󵄩L2 .

Turning to the first term on the right-hand side of (8.63), using (8.62), (8.53), Lemma 8.3.3 (iii), and (8.64), we have r

󵄩 n 󵄩2 ∑󵄩󵄩󵄩Wj j ϕ2 Λs ϕ1 f 󵄩󵄩󵄩L2 (hσ

Leb )

j=1

≲ Re⟨L ϕ2 Λs ϕ1 f , ϕ2 Λs ϕ1 f ⟩L2 (hσ s

s

≲ Re⟨L ϕ2 Λ ϕ1 f , ϕ2 Λ ϕ1 f ⟩L2 (hσ

Leb ) Leb )

≲ Re⟨L ϕ2 Λs ϕ1 f , ϕ2 Λs ϕ1 f ⟩L2 (hσ

Leb )

= Re⟨ϕ2 Λs ϕ1 L f , ϕ2 Λs ϕ1 f ⟩L2 (hσ

Leb )

+

∑

degds (α)≤κ

‖W α ϕ2 f ‖2L2

∑

degds (α)≤κ

+ +

α

Leb )

∑

‖W ϕ2 Λs ϕ1 f ‖2L2

∑

‖W α ϕ2 f ‖2L2

degds (α)≤κ degds (α)≤κ

−κϵ0

s−κϵ0

+ Re⟨[L , ϕ2 Λs ϕ1 ]f , ϕ2 Λs ϕ1 f ⟩L2 (hσ

s−κϵ0

≤ Re⟨ϕ2 Λs ϕ1 L f , ϕ2 Λs ϕ1 f ⟩L2 (hσ + (s.c.)

󵄩 󵄩2 + 󵄩󵄩󵄩ϕ2 Λs ϕ1 f 󵄩󵄩󵄩L2 (hσ

‖W α ϕ1 f ‖2L2 . s

Leb )

+ (l.c.)

∑

degds (α)≤κ

‖W α ϕ2 f ‖2L2

s−ϵ0

(8.65) Leb )

8.3 A priori estimates

563

�

Combining (8.63) and (8.65) shows that there exists a constant C ≥ 0 such that ∑

degds (α)≤κ

‖W α ϕ1 f ‖L2s ≤ C Re⟨ϕ2 Λs ϕ1 L f , ϕ2 Λs ϕ1 f ⟩L2 (hσ

Leb )

+ (l.c.)

∑

degds (α)≤κ

‖W α ϕ2 f ‖2L2

s−ϵ0

+ (s.c.)

∑

degds (α)≤κ

‖W α ϕ1 f ‖2L2 .

(8.66)

s

Subtracting (s.c.) ∑degds (α)≤κ ‖W α ϕ1 f ‖2L2 from both sides of (8.66) completes the proof. s

For the next lemmas, we consider functions u(t, x), where t ∈ ℝ and x ∈ ℝn . We let Λsx denote the pseudo-differential operator in the x variable with symbol (1 + |ξ|2 )s/2 , where ξ is dual to x. We use the space L2 (ℝ, dt; L2s ) to denote the Hilbert space with norm ‖u‖2L2 (ℝ,dt;L2 ) s

∞

:= ∫ ‖u(t, ⋅)‖2L2 (ℝn ) dt. s

−∞ ∞ Lemma 8.3.15. Let u(t, x) ∈ Cloc ((−1, 1) × Bn (1); ℂD ) and let ϕ1 , ϕ2 ∈ C0∞ (Bn (1)) and ∞ ψ1 , ψ2 ∈ C0 ((−1, 1)) satisfy ϕ1 ≺ ϕ2 and ψ1 ≺ ψ2 . Then, for s ∈ ℝ,

󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 ∫ 󵄨󵄨ψ (t)󵄨󵄨󵄨2 Re⟨ϕ Λs ϕ 𝜕 u(t, ⋅), ϕ Λs ϕ u(t, ⋅)⟩ 2 n 󵄨󵄨 ≲ ‖ψ ϕ u‖2 2 dt D 󵄨󵄨 2 x 1 t 2 x 1 2 1 L (ℝ,dt;L2s ) . L (B (1),hσLeb ;ℂ ) 󵄨󵄨󵄨 󵄨 1 󵄨 󵄨󵄨 󵄨󵄨−∞ 󵄨 Proof. We have 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 ∫ 󵄨󵄨ψ (t)󵄨󵄨󵄨2 Re⟨ϕ Λs ϕ 𝜕 u(t, ⋅), ϕ Λs ϕ u(t, ⋅)⟩ 2 󵄨󵄨 󵄨󵄨 󵄨 1 󵄨 2 x 1 t 2 x 1 L (hσLeb ) dt 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨∞ 󵄨 󵄨󵄨 ∞ 󵄨󵄨 󵄨 󵄨 1󵄨 󵄨 󵄨 󵄨2 󵄩 󵄩2 = 󵄨󵄨󵄨󵄨 ∫ 󵄨󵄨󵄨ψ1 (t)󵄨󵄨󵄨 𝜕t 󵄩󵄩󵄩ϕ2 Λsx ϕ1 u(t, ⋅)󵄩󵄩󵄩L2 (hσ ) dt 󵄨󵄨󵄨󵄨 Leb 2 󵄨󵄨 󵄨󵄨 󵄨−∞ 󵄨 󵄨󵄨 ∞ 󵄨󵄨 󵄨 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 󵄨󵄨2 󵄩󵄩 󵄩󵄩2 s 󵄨 = 󵄨󵄨 ∫ (𝜕t 󵄨󵄨ψ1 (t)󵄨󵄨 )󵄩󵄩ϕ2 Λx ϕ1 u(t, ⋅)󵄩󵄩L2 (hσ ) dt 󵄨󵄨󵄨󵄨 Leb 2 󵄨󵄨 󵄨󵄨 󵄨−∞ 󵄨 ∞

󵄨 󵄨2 ≲ ∫ 󵄨󵄨󵄨ψ2 (t)󵄨󵄨󵄨 ‖ϕ1 (⋅)u(t, ⋅)‖2L2 dt = ‖ψ2 ϕ1 u‖2L2 (ℝ,dt;L2 ) . s

s

−∞ ∞ Lemma 8.3.16. Let u(t, x) ∈ Cloc ((−1, 1) × Bn (1); ℂD ) satisfy 𝜕t u = −L u. Let ϕ1 , ϕ2 ∈ ∞ n ∞ C0 (B (1)) and ψ1 , ψ2 ∈ C0 ((−1, 1)) satisfy ϕ1 ≺ ϕ2 and ψ1 ≺ ψ2 . Then, for s ∈ ℝ,

∑

degds (α)≤κ

󵄩󵄩 󵄩 α 󵄩󵄩ψ1 W ϕ1 u󵄩󵄩󵄩L2 (ℝ,dt;L2 ) ≲ s

∑

degds (α)≤κ

󵄩󵄩 󵄩 α 󵄩󵄩ψ2 W ϕ2 u󵄩󵄩󵄩L2 (ℝ,dt;L2 ) . s−ϵ0

564 � 8 Linear maximally subelliptic operators Proof. We have, using Lemma 8.3.14, ∞

∑

degds (α)≤κ

󵄩󵄩 󵄩2 󵄨 󵄨2 α 󵄩󵄩ψ1 W ϕ1 u󵄩󵄩󵄩L2 (ℝ,dt;L2 ) = ∫ 󵄨󵄨󵄨ψ1 (t)󵄨󵄨󵄨 s −∞

∑

degds (α)≤κ

󵄩󵄩 α 󵄩2 󵄩󵄩W ϕ1 u(t, ⋅)󵄩󵄩󵄩L2 dt s

∞

󵄨 󵄨2 ≤ C1 ∫ 󵄨󵄨󵄨ψ1 (t)󵄨󵄨󵄨 Re⟨ϕ2 Λsx ϕ1 L u(t, ⋅), ϕ2 Λsx ϕ1 u(t, ⋅)⟩L2 (hσ ) dt Leb −∞ ∞

󵄨 󵄨2 + D1 ∫ 󵄨󵄨󵄨ψ1 (t)󵄨󵄨󵄨 −∞ ∞

∑

degds (α)≤κ

󵄩󵄩 α 󵄩2 󵄩󵄩W ϕ2 u(t, ⋅)󵄩󵄩󵄩L2 dt s−ϵ0

(8.67)

󵄨 󵄨2 = −C1 ∫ 󵄨󵄨󵄨ψ1 (t)󵄨󵄨󵄨 Re⟨ϕ2 Λsx ϕ1 𝜕t u(t, ⋅), ϕ2 Λsx ϕ1 u(t, ⋅)⟩L2 (hσ ) dt Leb −∞

+ D1

∑

degds (α)≤κ

󵄩󵄩 󵄩2 α 󵄩󵄩ψ1 W ϕ2 u󵄩󵄩󵄩L2 (ℝ,dt;L2 ) . s−ϵ0

󵄩 󵄩2 ≲ 󵄩󵄩󵄩ψ2 ϕ1 u󵄩󵄩󵄩L2 (ℝ,dt;L2 ) + s

∑

degds (α)≤κ

󵄩󵄩 󵄩2 α 󵄩󵄩ψ1 W ϕ2 u󵄩󵄩󵄩L2 (ℝ,dt;L2 ) , s−ϵ0

where the last inequality follows from Lemma 8.3.15. Since ψ1 ≺ ψ2 , we have ∑

degds (α)≤κ

󵄩󵄩 󵄩 α 󵄩󵄩ψ1 W ϕ2 u󵄩󵄩󵄩L2 (ℝ,dt;L2 ) ≲ s−ϵ0

∑

degds (α)≤κ

󵄩󵄩 󵄩2 α 󵄩󵄩ψ2 W ϕ2 u󵄩󵄩󵄩L2 (ℝ,dt;L2 ) , s−ϵ0

and therefore the second term on the right-hand side of (8.67) has the desired bound. We finish the proof by estimating the first term on the right-hand side of (8.67). Using Lemma 8.3.3 (iii) and the calculus of pseudo-differential operators, we have, for ∞ n f ∈ Cloc (B (1)), 󵄩 󵄩 ‖ϕ1 f ‖L2s ≈ 󵄩󵄩󵄩ϕ2 Λs ϕ1 f 󵄩󵄩󵄩L2 (σ ) ≲ Leb ≲ ≤

∑

degds (α)≤κ

󵄩󵄩 α 󵄩 s 󵄩󵄩W ϕ2 Λ ϕ1 f 󵄩󵄩󵄩L2 −κϵ0

∑

‖S̃(s) W α ϕ2 f ‖L2−κϵ ≲

∑

‖W α ϕ2 f ‖L2s−ϵ .

degds (α)≤κ degds (α)≤κ

0

∑

degds (α)≤κ

‖W α ϕ2 f ‖L2s−κϵ

0

0

Thus, we have 󵄩󵄩 󵄩 󵄩󵄩ψ2 ϕ1 u󵄩󵄩󵄩L2 (ℝ,dt;L2 ) ≲ s

∑

degds (α)≤κ

󵄩󵄩 󵄩2 α 󵄩󵄩ψ2 W ϕ2 u󵄩󵄩󵄩L2 (ℝ,dt;L2 ) , s−ϵ0

which shows that the first term on the right-hand side of (8.67) also has the desired bound, completing the proof. ∞ Lemma 8.3.17. Let u(t, x) ∈ Cloc ((−1, 1) × Bn (1); ℂD ) satisfy 𝜕t u = −L u. Let ϕ1 , ϕ2 ∈ ∞ n ∞ C0 (B (1)) and ψ1 , ψ2 ∈ C0 ((−1, 1)) satisfy ϕ1 ≺ ϕ2 and ψ1 ≺ ψ2 . Then, for s ∈ ℝ,

󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩ψ1 ϕ1 u󵄩󵄩󵄩L2 (ℝ,dt;L2 ) ≲ 󵄩󵄩󵄩ψ2 ϕ2 u󵄩󵄩󵄩L2 (ℝ×Bn (1),dt×(hσLeb )) . s

8.3 A priori estimates

565

�

Proof. Pick sequences {ηj }j∈ℕ ⊂ C0∞ (Bn (1)) and {γj }j∈ℕ ⊂ C0∞ ((−1, 1)) with ϕ1 = η0 ≺ η1 ≺ η2 ≺ ⋅ ⋅ ⋅ ≺ ϕ2 and ψ1 = γ0 ≺ γ1 ≺ γ2 ≺ ⋅ ⋅ ⋅ ≺ ψ2 . By repeated application of Lemma 8.3.16, we have, for every N ∈ ℕ, 󵄩󵄩 󵄩 󵄩󵄩ψ1 ϕ1 u󵄩󵄩󵄩L2 (ℝ,dt;L2 ) ≤ s ≲

∑

󵄩󵄩 󵄩 α 󵄩󵄩γ0 W η0 u󵄩󵄩󵄩L2 (ℝ,dt;L2 ) s

∑

󵄩󵄩 󵄩 α 󵄩󵄩γ1 W η1 u󵄩󵄩󵄩L2 (ℝ,dt;L2 ) ≲ s−ϵ0

degds (α)≤κ degds (α)≤κ

≲ ⋅⋅⋅ ≲

∑

degds (α)≤κ

∑

degds (α)≤κ

󵄩󵄩 󵄩 α 󵄩󵄩γ2 W η2 u󵄩󵄩󵄩L2 (ℝ,dt;L2 ) s−2ϵ0

󵄩󵄩 󵄩 α 󵄩󵄩γN W ηN u󵄩󵄩󵄩L2 (ℝ,dt;L2 ) s−Nϵ0

󵄩 󵄩 ≲ 󵄩󵄩󵄩γN ηN u󵄩󵄩󵄩L2 (ℝ,dt;L2

s+κ−Nϵ0 )

󵄩 󵄩 ≲ 󵄩󵄩󵄩ψ2 ϕ2 u󵄩󵄩󵄩L2 (ℝ,dt;L2

s+κ−Nϵ0 )

.

Taking N so large that Nϵ0 ≥ s + κ and using (8.53) completes the proof. ∞ Lemma 8.3.18. Let u(t, x) ∈ Cloc ((−1, 1) × Bn (1); ℂD ) satisfy 𝜕t u = −L u. Then

󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩𝜕t u󵄩󵄩L2 ((−1/2,1/2)×Bn (1/2),dt×d(hσLeb );ℂD ) ≲ 󵄩󵄩󵄩u󵄩󵄩󵄩L2 ((−1,1)×Bn (1),dt×d(hσLeb );ℂD ) . Proof. Let ϕ1 , ϕ2 ∈ C0∞ (Bn (1)) and ψ1 , ψ2 ∈ C0∞ ((−1, 1)) satisfy ϕ1 ≺ ϕ2 , ψ1 ≺ ψ2 , ϕ1 ≡ 1 on Bn (1/2), and ψ1 ≡ 1 on Bn (−1/2, 1/2). Using (8.53) and Lemma 8.3.17, we have 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩𝜕t u󵄩󵄩L2 ((−1/2,1/2)×Bn (1/2),dt×(hσLeb )) = 󵄩󵄩󵄩L u󵄩󵄩󵄩L2 ((−1/2,1/2)×Bn (1/2),dt×(hσLeb )) 󵄩 󵄩 ≈ 󵄩󵄩󵄩L u󵄩󵄩󵄩L2 ((−1/2,1/2)×Bn (1/2),dt×σ ) Leb

≲ ‖ψ1 ϕ1 u‖L2 (ℝ,dt;L2 ) ≲ ‖ψ2 ϕ2 u‖L2 (ℝ×Bn (1),dt×d(hσLeb )) 2κ

≲ ‖u‖L2 ((−1,1)×Bn (1),dt×d(σLeb )) . Lemma 8.3.19. There exists an L -unit-admissible constant C ≥ 0 such that ∀ϵ ∈ (0, 1/2], ∞ ∀u ∈ Cloc ((−ϵ−2κ , ϵ−2κ ) × Bn (1); ℂD ) satisfying 𝜕t u = −L u, ∀k ∈ [1, (2ϵ)−1 ], we have ‖𝜕t u‖L2 ((−ϵ−2κ +(kϵ)ϵ−2κ ,ϵ−2κ −(kϵ)ϵ−2κ )×Bn (1/2),dt×d(hσLeb );ℂD )

≤ C‖u‖L2 ((−ϵ−2κ +((k−1)ϵ)ϵ−2κ ,ϵ−2κ −((k−1)ϵ)ϵ−2κ )×Bn (1),dt×d(hσLeb );ℂD ) .

(8.68)

Proof. Note that (−ϵ−2κ + ((k − 1)ϵ)ϵ−2κ , ϵ−2κ − ((k − 1)ϵ)ϵ−2κ ) \ (−ϵ−2κ + (kϵ)ϵ−2κ , ϵ−2κ − (kϵ)ϵ−2κ ) = (−ϵ−2κ + ((k − 1)ϵ)ϵ−2κ , −ϵ−2κ + (kϵ)ϵ−2κ ] ⋃[ϵ−2κ − (kϵ)ϵ−2κ , ϵ−2κ − ((k − 1)ϵ)ϵ−2κ ), which are two intervals of length ϵ−2κ+1 ≥ ϵ−1 ≥ 2. Thus, we may cover (ϵ−2κ + (kϵ)ϵ−2κ , ϵ−2κ − (kϵ)ϵ−2κ ) by a finite collection of intervals of the form (ti − 1/2, ti + 1/2), i = 1, . . . , L, where ti ∈ (ϵ−2κ +(kϵ)ϵ−2κ , ϵ−2κ −(kϵ)ϵ−2κ ), (ti −1, ti +1) ⊆ (−ϵ−2κ +((k −1)ϵ)ϵ−2κ , ϵ−2κ −((k −

566 � 8 Linear maximally subelliptic operators 1)ϵ)ϵ−2κ ), and no point lies in more than four of the intervals (ti − 1, ti + 1). Lemma 8.3.18 shows that ‖𝜕t u‖L2 ((ti −1/2,ti +1/2)×Bn (1/2),dt×d(hσLeb )) ≲ ‖u‖L2 ((ti −1,ti +1)×Bn (1),dt×d(hσLeb )) . We conclude that ‖𝜕t u‖2L2 ((−ϵ−2κ +(kϵ)ϵ−2κ ,ϵ−2κ −(kϵ)ϵ−2κ )×Bn (1/2),dt×d(hσ L

≤ ∑ ‖𝜕t u‖2L2 ((t −1/2,t +1/2)×Bn (1/2),dt×d(hσ i

i=1

i

Leb ))

Leb ))

L

≲ ∑ ‖u‖2L2 ((t −1,t +1)×Bn (1),dt×d(hσ i=1

i

i

Leb ))

≲ ‖u‖2L2 ((−ϵ−2κ +((k−1)ϵ)ϵ−2κ ,ϵ−2κ −((k−1)ϵ)ϵ−2κ )×Bn (1),dt×d(hσ

Leb ))

.

Proof of Proposition 8.3.9. Set uϵ (t, x) := u(ϵ2κ t, x), so that 𝜕t uϵ = −L u. Lemma 8.3.19 implies that (8.68) holds with u replaced by uϵ . The result follows. 8.3.3 Exponential estimates To prove Theorem 8.1.1 (ix) we need to conclude exponential estimates. Remarkably, Proposition 8.3.9 combined with some technical scaling techniques implies the required exponential estimates. In fact, in the proof of Proposition 8.3.9, we used simple scaling techniques in the t variable applied to Lemma 8.3.18. Thus, we will obtain the required exponential estimates by combining scaling techniques (in both variables (t, x)) with the result at the unit scale Lemma 8.3.18. The main result of this section is the following. Proposition 8.3.20. Let L be given by (8.42). Suppose that L , thought of as a densely defined operator on L2 (Bn (1), hσLeb ; ℂD ) with dense domain C0∞ (Bn (1); ℂD ), is symmetric and non-negative and suppose that L is maximally subelliptic of type 2 of degree 2κ at the unit scale with respect to (W , ds) and hσLeb . Then there exists an L -unit-admissible constant c > 0 such that for every ordered multi-index α, there exists an α-L -unit-admissible ∞ constant Cα ≥ 0 such that the following holds. If u(t, x) ∈ Cloc ((−1, 1) × Bn (1); ℂD ) satisfies 𝜕t u(t, x) = −L u(t, x) and u(t, x) ≡ 0 for t ∈ (−1, 0], then ∀t ∈ (−1/4, 1/4), 1 󵄨 󵄨 sup 󵄨󵄨󵄨W α u(t, x)󵄨󵄨󵄨 ≤ Cα exp(−ct − 2κ−1 )‖u‖L2 ((−1,1)×Bn (1);ℂD ) . n

x∈B (1/4)

We prove Proposition 8.3.20 by proving u(t, x) has a certain Gevrey regularity in the t variable. This idea originates in the work of Jerison and Sánchez-Calle [134], who use it to study the heat kernel associated with Hörmander’s sub-Laplacian – see Section 8.10 for more details on the history of this proof.

8.3 A priori estimates

�

567

The rest of this section is devoted to the proof of Proposition 8.3.20; we henceforth assume L is as given in that proposition. The heart of the proof is a technical, quantitative scaling result. Fix 0 < ϵ1 < ϵ2 ≤ 1 and set ϵ := ϵ2 − ϵ1 ; assume ϵ ∈ (0, 1/2]. It is important for the proof that all estimates in this section are independent of the particular choice of ϵ1 , ϵ2 . Informally, we wish to apply Theorem 3.3.7 with 𝒦 = Bn (ϵ1 ) and Ω1 = Bn (ϵ2 ); however, if one proceeds in this way without care, then the estimates so obtained depend on ϵ1 and ϵ2 . Instead, we prove an analog of Theorem 3.3.7 where all of the estimates are L -unitadmissible and, in particular, do not depend on ϵ1 or ϵ2 . Like Theorem 3.3.7, this result follows easily from Theorem 3.6.5. First set (X, d ) = {(X1 , d1 ), . . . , (Xq , dq )} := {(Z, e) ∈ Gen((W , ds)) : e ≤ m max dsj }. 1≤j≤r

In particular, since W1 , . . . , Wr satisfy Hörmander’s condition of order m on Bn (1), we have span{X1 (x), . . . , Xq (x)} = Tx Bn (1),

∀x ∈ Bn (1).

Moreover, by the definition of unit-admissible constants (in particular, see (3.4)), we have inf

max

x∈Bn (1) j1 ,...,jn ∈{1,...,q}

󵄨󵄨 󵄨 󵄨󵄨det(Xj1 (x)| ⋅ ⋅ ⋅ |Xjn (x))󵄨󵄨󵄨 ≳ 1.

(8.69)

Set Vol := hσLeb and Λ(x, δ) := =

max

Vol(x)(δdj1 Xj1 (x), . . . , δdjn Xjn (x))

max

󵄨 󵄨 h(x)󵄨󵄨󵄨det(Xj1 (x)| ⋅ ⋅ ⋅ |Xjn (x))󵄨󵄨󵄨.

j1 ,...,jn ∈{1,...,q} j1 ,...,jn ∈{1,...,q}

(8.70)

The main scaling result we use is the next proposition. Proposition 8.3.21. There exist L -unit-admissible constants c1 , c2 ∈ (0, 1], not depending on ϵ1 or ϵ2 , such that the following hold: (a) B(X,d) (x, 2c1 ϵ) ⊆ Bn (ϵ2 ), ∀x ∈ Bn (ϵ1 ). (b) ∀δ ∈ (0, 6c1 ϵ], ∀x ∈ Bn (ϵ1 ), Vol(B(X,d) (x, 2δ)) ≲ Vol(B(X,d) (x, δ)). For every x ∈ Bn (ϵ1 ), δ ∈ (0, c1 ϵ], there exists Φx,δ : Bn (1) → B(X,d) (x, δ) such that: (c) Φx,δ (0) = x. (d) Φx,δ is a smooth coordinate system, that is, Φx,δ (Bn (1)) ⊆ Bn (1) is open and Φx,δ : ∞ Bn (1) → Φx,δ (Bn (1)) is a Cloc diffeomorphism. n (e) B(X,d) (x, c2 δ) ⊆ Φx,δ (B (1/2)) ⊆ Φx,δ (Bn (1)) ⊆ B(X,d) (x, δ).

568 � 8 Linear maximally subelliptic operators Set Wjx,δ := Φ∗x,δ δdsj Wj and Xkx,δ := δdk Xk . Let (W x,δ , ds) = {(W1x,δ , ds1 ), . . . , (Wrx,δ , dsr )} and (X x,δ , d ) = {(X1x,δ , d1 ), . . . , (Xqx,δ , dq )}.

(f) Wjx,δ and Xkx,δ are in C ∞ (Bn (1); ℝn ) in the quantitative sense that ∀L ∈ ℕ, there exists an L-L -unit-admissible constant CL such that for 1 ≤ j ≤ r, 1 ≤ k ≤ q, 󵄩 󵄩 sup 󵄩󵄩󵄩Wjx,δ 󵄩󵄩󵄩C L (Bn (1);ℝn ) ≤ CL , n

x∈B (ϵ1 ) δ∈(0,c1 ϵ]

󵄩 󵄩 sup 󵄩󵄩󵄩Xkx,δ 󵄩󵄩󵄩C L (Bn (1);ℝn ) ≤ CL . n

x∈B (ϵ1 ) δ∈(0,c1 ϵ]

(g) X1x,δ (u), . . . , Xqx,δ (u) span Tu Bn (1) uniformly in the sense that inf n

󵄨 󵄨 inf 󵄨󵄨󵄨det(Xjx,δ (u)| ⋅ ⋅ ⋅ |Xjx,δ (u))󵄨󵄨󵄨 ≳ 1. 1 n

max

x∈B (ϵ1 ) j1 ,...,jn ∈{1,...,q} u∈Bn (1) δ∈(0,c1 ϵ]

(h) (W x,δ , ds) are Hörmander vector fields with formal degrees at the unit scale and all of the estimates defining L-unit-admissible constants with respect to (W x,δ , ds) as in Definition 3.2.1 can be chosen to be L-L -unit-admissible constants. (i) ∀x ∈ Bn (ϵ1 ), δ ∈ (0, c1 ϵ], L ∈ ℕ, α 󵄩 󵄩 ‖f ‖C L (Bn (1)) ≈ ∑ 󵄩󵄩󵄩(X x,δ ) f 󵄩󵄩󵄩C(Bn (1)) ,

∀f ∈ C(Bn (1)).

|α|≤L

For x ∈ Bn (ϵ1 ) and δ ∈ (0, c1 ϵ] define hx,δ ∈ C ∞ (Bn (1)) by Φ∗x,δ Vol = Λ(x, δ)hx,δ σLeb . Then: (j) ∀x ∈ Bn (ϵ1 ), δ ∈ (0, c1 ϵ], u ∈ Bn (1), hx,δ (u) ≈ 1. Furthermore, for every L ∈ ℕ, 󵄩 󵄩 sup 󵄩󵄩󵄩hx,δ 󵄩󵄩󵄩C L (Bn (1)) ≲ 1,

x∈Bn (ϵ1 ) δ∈(0,c1 ϵ]

where the implicit constant is an L-L -unit-admissible constant. Proof. We prove this result by applying Theorem 3.6.5 with x0 replaced by x ∈ Bn (ϵ1 ) and X1 , . . . , Xq replaced by δd1 X1 , . . . , δdq Xq for δ ∈ (0, c3 ϵ], where c3 ∈ (0, 1] is a small L -unit-admissible constant to be chosen later. The central aspect of this proof is to show that L-admissible constants and L, Vol-admissible constants (as in Definition 3.6.4) in this application of Theorem 3.6.5 are L-L unit-admissible constants. Thus, we must verify that each of the quantities in Definition 3.6.4 can be bounded by L-L -unit-admissible constants. By the Picard–Lindelöf theorem, if c3 ∈ (0, 1] is sufficiently small (depending only on m, q, and max1≤j≤q ‖Xj ‖C(Bn (1);ℝn ) ), then for δ ∈ (0, c3 ϵ], δd1 X1 , . . . , δdq Xq satisfy 𝒞 (x, 1, Bn (1)), ∀x ∈ Bn (ϵ1 ) (see Definition 3.6.3). Indeed, δdj ≤ (c3 ϵ)dj ≤ c3 ϵ. Therefore, by taking c3 > 0 small, each δdk Xk is a small linear combination of ϵ𝜕x1 , . . . , ϵ𝜕xn . Since ϵ1 + ϵ = ϵ2 ≤ 1, it follows that δd1 X1 , . . . , δdq Xq satisfy 𝒞 (x, 1, Bn (1)). In particular, we

8.3 A priori estimates

�

569

make take η = 1 in our application of Theorem 3.6.5. Moreover, by the same argument, by taking c3 > 0 small, we have, for x ∈ Bn (ϵ1 ) and δ ∈ (0, c3 ϵ], Bδd X (x) = B(X,d) (x, δ) ⊆ Bn (ϵ2 ). We will choose c1 ≤ c3 /2, and therefore (a) holds. Lemma A.2.11 shows that τ0 > 0 can be chosen to be an L -unit-admissible constant. In the application of Theorem 3.6.5, we take ξ = 1. We claim that [Xj , Xk ] =

∑ dl ≤dj +dk

l cj,k Xl ,

l cj,k ∈ C ∞ (Bn (1)),

(8.71)

l and ∀L ∈ ℕ, ‖cj,k ‖C L (Bn (1)) ≲ 1. Indeed, if dj + dk ≤ m max1≤p≤r dsp , then ([Xj , Xk ], dj + dk ) ∈ l (X, d ), and therefore (8.71) holds with one of the cj,k equal to 1 and the rest equal to 0. If dj + dk > m max1≤p≤r dsp , then we use ‖Xp ‖C L (Bn (1)) ≲ 1 and (8.69) to write q

l [Xj , Xk ] = ∑ cj,k Xl ,

l cj,k ∈ C ∞ (Bn (1)),

l=1

l where ∀L ∈ ℕ, ‖cj,k ‖C L (Bn (1)) ≲ 1. Since dj + dk > m max1≤p≤r dsp ≥ max{dl : 1 ≤ l ≤ q}, (8.71) follows. Multiplying (8.71) by δdj +dk , we see that q

l,δ dl [δdj Xj , δdk Xk ] = ∑ cj,k δ Xl , l=1

where l δdl +dk −dl cj,k , l,δ cj,k ={ 0,

dl ≤ dj + dk , dl > dj + dk .

Consider, ∀δ ∈ (0, c3 ϵ] ⊆ (0, 1] and x ∈ Bn (1), we have, for any ordered multi-index α, α l,δ 󵄩 󵄩 󵄩󵄩 d α l,δ 󵄩󵄩 󵄩󵄩 max󵄩󵄩󵄩(δd X) cj,k 󵄩󵄩(δ X) cj,k 󵄩󵄩C(B(X,d) (x,δ)) 󵄩C(Bδd X (x)) = max j,k,l j,k,l

󵄩 l 󵄩 󵄩󵄩 n ≲ 1, ≤ max 󵄩󵄩󵄩X α cj,k 󵄩C(B (1)) d ≤d +d l

j

k

where the implicit constant is an α-L -unit-admissible constant. Consider Lieδdk Xk Vol = (δdk Xk h)σLeb + δdk h div(Xk )σLeb =: fk,δ σLeb . We have, for L ∈ ℕ,

570 � 8 Linear maximally subelliptic operators α 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ∑ 󵄩󵄩󵄩(δd X) fj,δ 󵄩󵄩󵄩C(B d (x)) ≲ ∑ 󵄩󵄩󵄩X α Xj h󵄩󵄩󵄩C(Bn (1)) + ∑ 󵄩󵄩󵄩X α (h div(Xk ))󵄩󵄩󵄩C(Bn (1)) ≲ 1. δ X

|α|≤L

|α|≤L

|α|≤L

We use ζ = 1 in Theorem 3.6.5, so that (after reordering the vector fields) the lefthand side of (3.16) equals 1. This is always possible: see the remarks following (3.16). In Theorem 3.6.5, the vector fields are reordered so that the first n vector fields satisfy (3.16). This reordering depends on x and δ. We proceed without reordering the vector fields, so that X1 , . . . , Xn in that theorem are really of the form δdj1 (x,δ) Xj1 (x,δ) , . . . , δdjn (x,δ) Xjn (x,δ) for some j1 (x, δ), . . . , jn (x, δ) ∈ {1, . . . , q}. It follows from the above that L-admissible and L, Vol-admissible constants (as in Definition 3.6.4) are L-L -unit-admissible constants in our application of Theorem 3.6.5. We let Ψx,δ be the map Φ from Theorem 3.6.5 when applied with x0 = x ∈ Bn (ϵ1 ) and X1 , . . . , Xq replaced by δd1 X1 , . . . , δdq Xq , where δ ∈ (0, c3 ϵ], and we let η1 ∈ (0, 1]

be the 1-admissible constant from that theorem. We set Φx,δ (t) := Ψx,δ (η1 t) : Bn (1) 󳨀 → Φx,δ (Bn (1)). Item (c) follows from the definition of Φx,δ (t) (see (3.17)). Item (d) follows from Theorem 3.6.5, (a), and (b). For (e), we let c2 = ξ2 , where ξ2 is as in Theorem 3.6.5. Then, by Theorem 3.6.5 (c), we have ∼

B(X,d) (x, c2 δ) = B(c2 δ)d X (x) ⊆ Bξ2 δd X (x) ⊆ Φx,δ (Bn (1/2))

= Ψx,δ (Bn (1/2)) ⊆ Φx,δ (Bn (1)) = Ψx,δ (Bn (1)) ⊆ Bδd X (x)

(8.72)

= B(X,d) (x, δ).

Thus, (e) holds (by picking c1 ≤ c3 ). ∗ dk −1 Since Φ∗x,δ δdk Xk (u) = η−1 1 Ψx,δ δ Xk (η1 u) = η1 Yk (η1 u), where Yk is as in Theo󵄩 󵄩 rem 3.6.5 and η1 ≈ 1, it follows from Theorem 3.6.5 (e) that 󵄩󵄩󵄩Xkx,δ 󵄩󵄩󵄩C L (Bn (1);ℝn ) ≲ 1. Since each Wjx,δ is of the form Xkx,δ for some k, (f) follows. By Theorem 3.6.5 (d), we have inf

max

u∈Bn (η) j1 ,...,jn ∈{1,...,q}

󵄨󵄨 󵄨 󵄨󵄨det(Yj1 (u)| ⋅ ⋅ ⋅ |Yjn (u))󵄨󵄨󵄨 ≥ σn ,

where σn > 0 depends only on n. Since Xkx,δ (u) = η−1 1 Yk (η1 u) and η1 ≈ 1, (g) follows. (h): Since (X, d ) = {(Z, e) ∈ Gen((W , ds)) : e ≤ m max dsj }, it follows that (X x,δ , d ) = {(Z x,δ , e) ∈ Gen((W x,δ , ds)) : e ≤ m max dsj }. Thus, (h) follows from (f) and (g). Again using the fact that Xkx,δ (u) = η−1 1 Yk (η1 u) and η1 ≈ 1, (i) follows from Theorem 3.6.5 (f). (j): Let h̃ be the function h from Theorem 3.6.5. Since Φx,δ (t) = Ψx,δ (η1 t), we see that ̃ hx,δ (t) = η−n 1 h(η1 t). Since η1 ≈ 1, (j) follows from Theorem 3.6.5 (g) and (h).

8.3 A priori estimates

571

�

All that remains to show is (b). This will follow once we show that if c1 ∈ (0, 1] is a sufficiently small L -unit-admissible constant, then for δ ∈ (0, 12c1 ϵ], we have Vol(B(X,d) (x, δ)) ≈ Λ(x, δ).

(8.73)

It is immediately clear that Λ(x, 2δ) ≲ Λ(x, δ) (see (8.70)), and therefore (b) follows from (8.73). We turn to proving (8.73). By (e) we have, for δ ∈ (0, c3 ϵ] and x ∈ Bn (ϵ1 ), Vol(B(X,d) (x, c2 δ)) ≤ Vol(Φx,δ (Bn (1))) ≤ Vol(B(X,d) (x, δ)).

(8.74)

Vol(Φx,δ (Bn (1))) = Λ(x, δ) ∫ hx,δ (t) dt ≈ Λ(x, δ).

(8.75)

By (j),

Bn (1)

Combining (8.74) and (8.75) shows that Λ(x, δ) ≲ Vol(B(X,d) (x, δ)),

δ ∈ (0, c3 ϵ],

Vol(B(X,d) (x, δ)) ≲ Λ(x, δ/c2 ) ≈ Λ(x, δ), Thus, so long as c1 ≤

c2 c3 , 12

δ ∈ (0, c2 c3 ϵ].

(8.73) follows, completing the proof.

Let c1 , c2 ∈ (0, 1] be the unit-L -admissible constants in Proposition 8.3.21. Henceforth, we fix δ := c1 ϵ ≈ ϵ and we apply Proposition 8.3.21 with this choice of δ. Let Φx,δ , (W x,δ , ds), (X x,δ , d ), and Volx,δ := Φ∗x,δ (hσLeb ) = Λ(x, δ)hx,δ σLeb be as in that proposition. Set, for x ∈ Bn (ϵ1 ), L

x,ϵ

:= Φ∗x,δ ϵ2κ L (Φx,δ )∗ .

Note that L x,ϵ is a partial differential operator with smooth coefficients on Bn (1). Lemma 8.3.22. With L = L x,ϵ , h = hx,δ , and (W , ds) = (W x,δ , ds), the hypotheses of Proposition 8.3.9 hold, and L x,ϵ -unit-admissible constants can be taken to be L -unitadmissible constants. In particular, the conclusion of Proposition 8.3.9 holds for L x,ϵ and hx,δ , where C ≥ 0 is an L -unit-admissible constant. Proof. That the vector fields with formal degree (W x,δ , ds) are of the correct form is Proposition 8.3.21 (h). That hx,δ satisfies the correct estimates is Proposition 8.3.21 (j). We next show that L x,ϵ is of the correct form. Multiplying (8.42) by ϵ2κ and using δ = c1 ϵ, we have ϵ2κ L =

∑ degds (α),degds (β)≤κ

degds (α)+degds (β)

ϵ2κ−degds (α)−degds (β) c1

α

β

bα,β (x)(δdsW ) (δdsW ) .

572 � 8 Linear maximally subelliptic operators Therefore, L

x,ϵ

=

α

∑ degds (α),degds (β)≤κ

β

x,δ bx,ϵ ) (W x,δ ) , α,β (W

where degds (α)+degds (β)

2κ−degds (α)−degds (β) bx,ϵ c1 α,β = ϵ

bα,β ∘ Φx,δ .

We wish to show that, for every L ∈ ℕ, 󵄩󵄩 x,ϵ 󵄩󵄩 󵄩󵄩bα,β 󵄩󵄩C L (Bn (1)) ≲ 1,

(8.76)

where the implicit constant is an L-L -unit-admissible constant. By Proposition 8.3.21 (i), we have, using 0 < ϵ, c1 , δ ≤ 1, 󵄩󵄩 x,ϵ 󵄩󵄩 󵄩 x,δ α x,ϵ 󵄩 󵄩󵄩bα,β 󵄩󵄩C L (Bn (1)) ≈ ∑ 󵄩󵄩󵄩(X ) bα,β 󵄩󵄩󵄩C(Bn (1)) |α|≤L

α 󵄩󵄩 d 󵄩󵄩(δ X) bα,β 󵄩󵄩C(Φx,δ (Bn (1)))

degds (α)+degds (β) 󵄩 󵄩

= ∑ ϵ2κ−degds (α)−degds (β) c1 |α|≤L

󵄩 󵄩 ≤ ∑ 󵄩󵄩󵄩X α bα,β 󵄩󵄩󵄩C(Bn (1)) ≲ 1, |α|≤L

establishing (8.76). We require that L x,ϵ is symmetric and non-negative on L2 (Bn (1), hx,δ σLeb ; ℂD ) when thought of as an operator with dense domain C0∞ (Bn (1); ℂD ). By assumption, L is symmetric and non-negative, and therefore, ∀f1 , f2 ∈ C0∞ (Bn (1); ℂD ), we have ⟨ϵ2κ L f1 , f2 ⟩L2 (hσ 2κ

⟨ϵ L f1 , f1 ⟩L2 (hσ

Leb )

Leb )

= ⟨f1 , ϵ2κ L f2 ⟩L2 (hσ

≥ 0.

Leb )

(8.77)

,

(8.78)

Let g1 , g2 ∈ C0∞ (Bn (1); ℂD ) and set f1 := g1 ∘Φx,δ and f2 := g2 ∘Φx,δ . By Proposition 8.3.21 (d), f1 , f2 ∈ C0∞ (Bn (1); ℂD ). Plugging this choice of f1 , f2 into (8.77) and (8.78), we obtain ⟨L x,ϵ g1 , g2 ⟩L2 (Vol ⟨L x,ϵ g1 , g1 ⟩L2 (Vol

x,δ )

x,δ )

= ⟨g1 , L x,ϵ g2 ⟩L2 (Vol

x,δ )

(8.79)

,

≥ 0.

(8.80)

Dividing (8.79) and (8.80) by Λ(x, δ) shows that ⟨L x,ϵ g1 , g2 ⟩L2 (h ⟨L x,ϵ g1 , g1 ⟩L2 (h

x,δ σLeb )

x,δ σLeb )

= ⟨g1 , L x,ϵ g2 ⟩L2 (h ≥ 0.

Therefore, L x,ϵ is symmetric and non-negative, as desired.

x,δ σLeb )

,

8.3 A priori estimates

�

573

Finally, we wish to show that L x,ϵ is maximally subelliptic of type 2 of degree 2κ at the unit scale with respect to (W x,δ , ds) and hx,δ σLeb and where the constant in the definition of maximally subelliptic of type 2 (see Definition 8.3.1) can be taken to be an L -unit-admissible constant. By assumption, L is symmetric, non-negative, and maximally subelliptic of type 2 of degree 2κ at the unit scale with respect to (W , ds) and hσLeb . In other words, for f ∈ C0∞ (Bn (1); ℂD ), we have r

󵄩 n 󵄩2 󵄩 󵄩2 ∑󵄩󵄩󵄩Wj j f 󵄩󵄩󵄩L2 (hσ ) ≤ A(⟨L f , f ⟩L2 (hσ ) + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (hσ ) ), Leb Leb Leb

(8.81)

j=1

where, by definition, A ≥ 0 is an L -unit-admissible constant. Multiplying both sides of (8.81) by δ2κ and using δ ≈ ϵ and 0 < ϵ ≤ 1, we see that r

n 󵄩2 󵄩 󵄩 󵄩2 ∑󵄩󵄩󵄩(δdsj Wj ) j f 󵄩󵄩󵄩L2 (hσ ) ≲ (⟨ϵ2κ L f , f ⟩L2 (hσ ) + ϵ2κ 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (hσ ) ) Leb Leb Leb j=1

2κ

≤ (⟨ϵ L f , f ⟩L2 (hσ

Leb

󵄩󵄩 󵄩󵄩2 󵄩f 󵄩󵄩L2 (hσ )+󵄩

Leb )

(8.82)

).

Let g ∈ C0∞ (Bn (1); ℂD ) and let f := g ∘ Φx,δ . By Proposition 8.3.21 (d), f ∈ C0∞ (Bn (1); ℂD ). Plugging this choice of f into (8.82), we obtain r

n 󵄩2 󵄩 ∑󵄩󵄩󵄩(Wjx,δ ) j g 󵄩󵄩󵄩L2 (Vol j=1

x,δ )

≤ (⟨L x,ϵ g, g⟩L2 (Vol

x,δ )

󵄩 󵄩2 + 󵄩󵄩󵄩g 󵄩󵄩󵄩L2 (Vol

x,δ )

).

(8.83)

Dividing (8.83) by Λ(x, δ) gives r

n 󵄩2 󵄩 󵄩 󵄩2 ∑󵄩󵄩󵄩(Wjx,δ ) j g 󵄩󵄩󵄩L2 (h σ ) ≤ (⟨L x,ϵ g, g⟩L2 (h σ ) + 󵄩󵄩󵄩g 󵄩󵄩󵄩L2 (h σ ) ), x,δ Leb x,δ Leb x,δ Leb j=1

completing the proof. ∞ n For the rest of this section, let u(t, y) ∈ Cloc (B (1) × (−1, 1); ℂD ) satisfy 𝜕t u = −L u and u(t, y) ≡ 0 for t ∈ (−1, 0].

Lemma 8.3.23. For k ∈ [1, 1/2ϵ] and x ∈ Bn (ϵ1 ), 󵄩󵄩 󵄩󵄩 󵄩󵄩𝜕t u󵄩󵄩L2 ((−1+kϵ,1−kϵ)×B(X,d) (x,ϵc2 c1 ),dt×d(hσLeb );ℂD ) 󵄩 󵄩 ≲ ϵ−2κ 󵄩󵄩󵄩u󵄩󵄩󵄩L2 ((−1+(k−1)ϵ,1−(k−1)ϵ)×B (x,ϵc ),dt×d(hσ );ℂD ) . (X,d) 1 Leb

(8.84)

Proof. Fix x ∈ Bn (ϵ1 ) and set ux,ϵ (t, y) := u(t, Φx,δ (y)). Note that ux,ϵ satisfies 𝜕t ux,ϵ (t, y) = −ϵ−2κ L x,ϵ ux,ϵ (t, y), where L x,ϵ is acting in the y ∈ Bn (1) variable. Also, by Proposi∞ tion 8.3.21 (d), ux,ϵ ∈ Cloc ((−1, 1) × Bn (1); ℂD ).

574 � 8 Linear maximally subelliptic operators Lemma 8.3.22 shows that Proposition 8.3.9 applies to ux,ϵ and the constant C ≥ 0 from Proposition 8.3.9 is an L -unit-admissible constant. In other words, for k ∈ [1, 1/2ϵ], 󵄩󵄩 󵄩 󵄩󵄩𝜕t ux,ϵ 󵄩󵄩󵄩L2 ((−1+kϵ,1−kϵ)×Bn (1/2),dt×d(hx,δ σLeb )) 󵄩 󵄩 ≲ ϵ−2κ 󵄩󵄩󵄩ux,ϵ 󵄩󵄩󵄩L2 ((−1+(k−1)ϵ,1−(k−1)ϵ)×Bn (1),dt×d(h σ )) . x,δ Leb

(8.85)

1

Multiplying both sides of (8.85) by Λ(x, δ) 2 , we obtain 󵄩󵄩 󵄩 󵄩󵄩𝜕t ux,ϵ 󵄩󵄩󵄩L2 ((−1+kϵ,1−kϵ)×Bn (1/2),dt×d Volx,δ ) 󵄩 󵄩 ≲ ϵ−2κ 󵄩󵄩󵄩ux,ϵ 󵄩󵄩󵄩L2 ((−1+(k−1)ϵ,1−(k−1)ϵ)×Bn (1),dt×d Vol ) . x,δ

(8.86)

Changing variables, (8.86) is equivalent to 󵄩󵄩 󵄩󵄩 󵄩󵄩𝜕t u󵄩󵄩L2 ((−1+kϵ,1−kϵ)×Φx,δ (Bn (1/2)),dt×d(hσLeb )) 󵄩 󵄩 ≲ ϵ−2κ 󵄩󵄩󵄩u󵄩󵄩󵄩L2 ((−1+(k−1)ϵ,1−(k−1)ϵ)×Φ (Bn (1)),dt×d(hσ )) . x,δ Leb

(8.87)

By Proposition 8.3.21 (e) and the definition δ = c1 ϵ, we have B(X,d) (x, c1 c2 ϵ) ⊆ Φx,δ (Bn (1/2)) ⊆ Φx,δ (Bn (1)) ⊆ B(X,d) (x, c1 ϵ). Thus, (8.87) implies (8.84), completing the proof. The next lemma has a standard proof (see, for example, [216, page 32]), though we include the proof to make clear how all our estimates involve only L -unit-admissible constants. Lemma 8.3.24. There is a finite collection x1 , . . . , xL ∈ Bn (ϵ1 ) and an L -unit-admissible constant M ∈ ℕ+ such that {B(X,d) (xi , ϵc1 c2 ) : i = 1, . . . , L} is a cover for Bn (ϵ1 ), no point lies in more than M of the balls B(X,d) (xi , ϵc1 ), i = 1, . . . L, and for each i, B(X,d) (xi , ϵc1 ) ⊆ Bn (ϵ2 ). Proof. The keys to this proof are the doubling estimate Proposition 8.3.21 (b) and the fact that the balls B(X,d) (x, δ) are metric balls. Let B(X,d) (xi , (ϵc1 c2 )/2), i = 1, . . . , L, be a maximal disjoint subcollection of {B(X,d) (x, (ϵc1 c2 )/2) : x ∈ Bn (ϵ1 )}. Note that ⋃Li=1 B(X,d) (xi , ϵc1 c2 ) ⊇ Bn (ϵ1 ), as if there were y ∈ Bn (ϵ1 ) \ ⋃Li=1 B(X,d) (x0 , ϵc1 c2 ), then B(X,d) (y, (ϵc1 c2 )/2) would be disjoint from B(X,d) (xi , (ϵc1 c2 )/2), i = 1, . . . L, contradicting maximality. Now suppose y ∈ B(X,d) (xi1 , ϵc1 ) ∩ ⋅ ⋅ ⋅ ∩ B(X,d) (xiM , ϵc1 ); we wish to show that M ≲ 1. Since y ∈ B(X,d) (xil , ϵc1 ), we have B(X,d) (xil , (ϵc1 c2 )/2) ⊆ B(X,d) (xil , ϵc1 ) ⊆ B(X,d) (y, 2ϵc1 ). Because the balls B(X,d) (xil , (ϵc1 c2 /2)) are disjoint, (8.88) implies that

(8.88)

8.3 A priori estimates M

Vol(B(X,d) (y, 2ϵc1 )) ≥ ∑ Vol(B(X,d) (xil , (ϵc1 c2 )/2)).

�

575

(8.89)

l=1

Since B(X,d) (y, ϵc1 ) ⊆ B(X,d) (xil , 2ϵc1 ), repeated application of Proposition 8.3.21 (b) shows that Vol(B(X,d) (xil , (ϵc1 c2 )/2)) ≈ Vol(B(X,d) (xil , 2ϵc1 ))

≥ Vol(B(X,d) (y, ϵc1 )) ≈ Vol(B(X,d) (y, 2ϵc1 )).

(8.90)

Plugging (8.90) into (8.89) shows that Vol(B(X,d) (y, 2ϵc1 )) ≳ M Vol(B(X,d) (y, 2ϵc1 )), which implies M ≲ 1, as desired. Finally, it follows from the fact that xi ∈ Bn (ϵ1 ) and Proposition 8.3.21 (a) that B(X,d) (xi , ϵc1 ) ⊆ Bn (ϵ2 ). Lemma 8.3.25. For k ∈ [1, 1/2ϵ], 󵄩󵄩 󵄩󵄩 󵄩󵄩𝜕t u󵄩󵄩L2 ((−1+kϵ,1−kϵ)×Bn (ϵ1 ),dt×d(hσLeb );ℂD ) 󵄩 󵄩 ≲ ϵ−2κ 󵄩󵄩󵄩u󵄩󵄩󵄩L2 ((−1+(k−1)ϵ,1−(k−1)ϵ)×Bn (ϵ ),dt×d(hσ );ℂD ) . 2 Leb Proof. Take x1 , . . . , xL ∈ Bn (ϵ1 ) as in Lemma 8.3.24. Using the fact that B(X,d) (xi , ϵc1 c2 ) is a cover for Bn (ϵ1 ), we have 󵄩󵄩 󵄩󵄩2 󵄩󵄩𝜕t u󵄩󵄩L2 ((−1+kϵ,1−kϵ)×Bn (ϵ1 ),dt×d(hσLeb )) L

󵄩 󵄩2 ≤ ∑󵄩󵄩󵄩𝜕t u󵄩󵄩󵄩L2 ((−1+kϵ,1−kϵ)×B i=1

(X,d) (xi ,ϵc1 c2 ),dt×d(hσLeb ))

(8.91)

L

󵄩 󵄩2 ≲ ∑ ϵ−4κ 󵄩󵄩󵄩u󵄩󵄩󵄩L2 ((−1+(k−1)ϵ,1−(k−1)ϵ)×B i=1

(X,d) (xi ,ϵc1 ),dt×d(hσLeb ))

,

where the last estimate uses Lemma 8.3.23. Using the fact that no point lies in more than M ≲ 1 of the balls B(X,d) (xi , c1 ϵ) ⊆ Bn (ϵ2 ), we have L

󵄩 󵄩2 ∑ ϵ−4κ 󵄩󵄩󵄩u󵄩󵄩󵄩L2 ((−1+(k−1)ϵ,1−(k−1)ϵ)×B i=1

≲ϵ

(X,d) (xi ,ϵc1 ),dt×d(hσLeb ))

󵄩2 󵄩󵄩u󵄩󵄩󵄩L2 ((−1+(k−1)ϵ,1−(k−1)ϵ)×Bn (ϵ2 ),dt×d(hσLeb )) .

−4κ 󵄩 󵄩

Combining (8.91) and (8.92) and taking square roots completes the proof. Lemma 8.3.26. For l ∈ ℕ+ and k ∈ {1, . . . , l}, we have 󵄩󵄩 󵄩󵄩 󵄩󵄩𝜕t u󵄩󵄩L2 ((−1+ k ,1− k )×Bn (1− k ),dt×(hσ );ℂD ) Leb 2l 2l 2l 󵄩 󵄩󵄩 2κ 󵄩 ≲ l 󵄩󵄩u󵄩󵄩L2 ((−1+ k−1 ,1− k−1 )×Bn (1− k−1 ),dt×d(hσ );ℂD ) . Leb 2l 2l 2l

(8.92)

576 � 8 Linear maximally subelliptic operators Proof. Apply Lemma 8.3.25 with ϵ1 = 1 − result follows.

k 2l

and ϵ2 = 1 −

k−1 , 2l

so that ϵ = ϵ2 − ϵ1 =

1 . 2l

The

Lemma 8.3.27. There exists an L -unit-admissible constant R1 ≥ 0 such that ∀l ∈ ℕ, 󵄩󵄩 l 󵄩󵄩 󵄩󵄩𝜕t u󵄩󵄩L2 ((−1/2,1/2)×Bn (1/2),dt×d(hσLeb );ℂD ) 󵄩 󵄩 ≤ Rl1 (l!)2κ 󵄩󵄩󵄩u󵄩󵄩󵄩L2 ((−1,1)×Bn (1),dt×d(hσ );ℂD ) . Leb Proof. The case l = 0 is trivial, so we assume l ≥ 1. We claim that there exists an L -unitadmissible constant C1 ≥ 0 such that ∀k ∈ {1, . . . , l}, 󵄩󵄩 k 󵄩󵄩 󵄩󵄩𝜕t u󵄩󵄩L2 ((−1+ k ,1− k )×Bn (1− k ),dt×d(hσ )) Leb 2l 2l 2l 󵄩 k−1 󵄩󵄩 2κ 󵄩 ≤ C1 l 󵄩󵄩𝜕t u󵄩󵄩L2 ((−1+ k−1 ,1− k−1 )×Bn (1− k−1 ),dt×d(hσ )) . Leb 2l 2l 2l

(8.93)

Indeed, (8.93) follows by applying Lemma 8.3.26 to 𝜕tk−1 u, which satisfies the same hypotheses as u (in particular, 𝜕t 𝜕tk−1 u = −L 𝜕tk−1 u). Applying (8.93) l times gives 󵄩󵄩 l 󵄩󵄩 l 2κl 󵄩 󵄩 󵄩󵄩𝜕t u󵄩󵄩L2 ((−1/2,1/2)×Bn (1/2),dt×d(hσLeb )) ≤ C1 l 󵄩󵄩󵄩u󵄩󵄩󵄩L2 ((−1,1)×Bn (1),dt×d(hσLeb )) . The result now follows by Stirling’s approximation. ∞ Lemma 8.3.28. Let v(t, x) ∈ Cloc ((0, 1) × Bn (1); ℂD ). Then, for every ordered multi-index α, there exist an L -unit-admissible constant N = N(α) ∈ ℕ and an α-L -unit-admissible constant Cα ≥ 0 such that

sup

N(α) N(α)

t∈(−1/4,1/4) x∈Bn (1/4)

󵄨󵄨 α 󵄨 󵄩 r r 󵄩 󵄨󵄨Wx v(t, x)󵄨󵄨󵄨 ≤ Cα ∑ ∑ 󵄩󵄩󵄩𝜕t 1 L 2 v󵄩󵄩󵄩L2 ((−1/2,1/2)×Bn (1/2),dt×d(hσ r1 =0 r2 =0

Leb );ℂ

D)

.

Proof. Pick ϕ1 , ϕ2 ∈ C0∞ (Bn (3/8)) with ϕ1 ≺ ϕ2 and ϕ1 ≡ 1 on Bn (1/4) and pick ψ1 , ψ2 ∈ C0∞ ((−3/8, 3/8)) with ψ1 ≺ ψ2 and ψ1 ≡ 1 on (−1/4, 1/4). By the Sobolev embedding theorem there exists s = s(|α|, n) > 0 (s depending only on |α| and n) so that the following holds: sup

t∈(−1/4,1/4) x∈Bn (1/4)

󵄨󵄨 α 󵄨 󵄨 α 󵄨 󵄨󵄨Wx v(t, x)󵄨󵄨󵄨 ≤ sup󵄨󵄨󵄨Wx ϕ1 (x)ψ1 (t)v(t, x)󵄨󵄨󵄨 t,x

⌈s⌉

r 󵄩 󵄩 󵄩 󵄩 ≲ 󵄩󵄩󵄩ϕ1 ψ1 v󵄩󵄩󵄩L2 (ℝ×ℝn ) ≲ ∑ 󵄩󵄩󵄩ψ2 ϕ1 𝜕t 1 v(t, ⋅)󵄩󵄩󵄩L2 (ℝ,dt;L2 (ℝn )) . s

Corollary 8.3.8 implies that

r1 =0

s

(8.94)

8.3 A priori estimates

�

⌈ κϵs ⌉ 0

r r 󵄩󵄩 󵄩 󵄩 r 󵄩 󵄩󵄩ψ2 ϕ1 𝜕t 1 v(t, ⋅)󵄩󵄩󵄩L2 (ℝ,dt;L2 (ℝn )) ≲ ∑ 󵄩󵄩󵄩ψ2 ϕ1 𝜕t 1 L 2 v󵄩󵄩󵄩L2 (ℝ×ℝn ,dt×dσLeb ) . s r2 =0

577

(8.95)

Combining (8.94) and (8.95), using the support of ϕ2 and ψ2 , and using (8.53), we have s ⌈s⌉ ⌈ κϵ0 ⌉

sup

t∈(−1/4,1/4) x∈Bn (1/4)

r 󵄨󵄨 α 󵄨 󵄩 r 󵄩 󵄨󵄨Wx v(t, x)󵄨󵄨󵄨 ≲ ∑ ∑ 󵄩󵄩󵄩ψ2 ϕ1 𝜕t 1 L 2 v󵄩󵄩󵄩L2 (ℝ×ℝn ,dt×dσLeb ) r1 =0 r2 =0

s ⌈s⌉ ⌈ κϵ0 ⌉

󵄩 r 󵄩 ≲ ∑ ∑ 󵄩󵄩󵄩𝜕t 1 L r2 v󵄩󵄩󵄩L2 ((−1/2,1/2)×Bn (1/2),dt×d(hσ )) . Leb r1 =0 r2 =0

Taking N(α) := max{⌈s⌉, ⌈ κϵs ⌉} completes the proof. 0

Lemma 8.3.29. There exists an L -unit-admissible constant R2 > 0 such that for all ordered multi-indices α, there exists an α-L -unit-admissible constant Cα ≥ 0 such that for all l ∈ ℕ, sup

t∈(−1/4,1/4) x∈Bn (1/4)

2κ 󵄩 󵄩 󵄨󵄨 l α 󵄨 l 󵄨󵄨𝜕t W u(t, x)󵄨󵄨󵄨 ≤ Cα R2 (l!) 󵄩󵄩󵄩u󵄩󵄩󵄩L2 ((−1,1)×Bn (1),dt×d(hσLeb );ℂD ) .

(8.96)

Proof. By Lemma 8.3.28, there exists an L -unit-admissible constant N(α) ∈ ℕ such that sup

N(α) N(α)

t∈(−1/4,1/4) x∈Bn (1/4)

󵄨󵄨 l α 󵄨 󵄩 l+r r 󵄩 󵄨󵄨𝜕t W u(t, x)󵄨󵄨󵄨 ≲ ∑ ∑ 󵄩󵄩󵄩𝜕t 1 L 2 u󵄩󵄩󵄩L2 ((−1/2,1/2)×Bn (1/2),dt×d(hσLeb )) r1 =0 r2 =0

(8.97)

N(α) N(α)

󵄩 l+r +r 󵄩 = ∑ ∑ 󵄩󵄩󵄩𝜕t 1 2 u󵄩󵄩󵄩L2 ((−1/2,1/2)×Bn (1/2),dt×d(hσ )) , Leb r1 =0 r2 =0

where in the equality we have used 𝜕t u = −L u. Applying Lemma 8.3.27, we have N(α) N(α)

󵄩 l+r +r 󵄩 ∑ ∑ 󵄩󵄩󵄩𝜕t 1 2 u󵄩󵄩󵄩L2 ((−1/2,1/2)×Bn (1/2),dt×d(hσ

r1 =0 r2 =0

N(α) N(α)

󵄩 󵄩 ≲ 󵄩󵄩󵄩u󵄩󵄩󵄩L2 ((−1,1)×Bn (1),dt×d(hσ

Leb ))

󵄩 󵄩 ≲ 󵄩󵄩󵄩u󵄩󵄩󵄩L2 ((−1,1)×Bn (1),dt×d(hσ

Rl Leb )) 1

Leb ))

2κ l+r1 +r2

∑ ∑ ((l + r1 + r2 )!) R1

r1 =0 r2 =0

N(α) N(α)

2κ

∑ ∑ ((l + r1 + r2 )!) .

r1 =0 r2 =0

We use the fact that for l ∈ ℕ+ and p ∈ ℕ, p

l (l + p)! ≤ (l + p) (l!) ≤ Cp,1 lp (l!) ≤ Cp,2 2l (l!) ≤ Cp,3 (l!),

(8.98)

578 � 8 Linear maximally subelliptic operators where each Cp,j ≥ 0 depends only on p. Thus, there exists an L -unit-admissible constant R2 > 0 such that N(α) N(α)

2κ

Rl1 ∑ ∑ ((l + r1 + r2 )!) r1 =0 r2 =0

2κ

≲ Rl2 (l!) .

(8.99)

Combining (8.97), (8.98), and (8.99) completes the proof. The next lemma is [134, Lemma 2] (when the parameter γ = 2) and [118, Lemma 6] (for general γ > 1). We include the proof for completeness. ∞ Lemma 8.3.30. Let f ∈ Cloc ((−1/4, 1/4)) satisfy f (t) = 0 for t ∈ (−1/4, 0]. If there exist γ > 1, C ≥ 0, and R > 0 such that γ 󵄨󵄨 l 󵄨 l 󵄨󵄨𝜕t f (t)󵄨󵄨󵄨 ≤ CR (l!) ,

∀t ∈ (0, 1/4), l ∈ ℕ,

(8.100)

then − 1 󵄨󵄨 󵄨 γ−1 −1 󵄨󵄨f (t)󵄨󵄨󵄨 ≤ Ce exp{−(γ − 1)e (Rt) γ−1 },

t ∈ (−1/4, 1/4).

Proof. The result for C = 0 is trivial. For C > 0, by replacing f with f /C, we see that it suffices to prove the result for C = 1. We henceforth assume C = 1. Thus, we wish to show that − 1 󵄨󵄨 󵄨 γ−1 −1 󵄨󵄨f (t)󵄨󵄨󵄨 ≤ e exp{−(γ − 1)e (Rt) γ−1 },

t ∈ (−1/4, 1/4).

(8.101)

The inequality (8.101) is trivial for t ∈ (−1/4, 0] (because f (t) = 0 for t ≤ 0), so we assume t > 0. If 0 ≤ e−1 (Rt)−1/(γ−1) < 1, then the right-hand side of (8.101) ≥ 1, so (8.101) follows from the case l = 0 of (8.100) (recall that we are assuming C = 1). 1 Otherwise, pick l ∈ ℕ such that l +1 ≤ e−1 (Rt)−1/(γ−1) < l +2, so that (Rt)1/(γ−1) ≤ e(l+1) . We have t

l+1 (t − s)l 󵄨󵄨 (l+1) 󵄨󵄨 γ t 󵄨󵄨 󵄨 (s)󵄨󵄨 ds ≤ Rl+1 ((l + 1)!) 󵄨󵄨f (t)󵄨󵄨󵄨 ≤ ∫ 󵄨󵄨f l! (l + 1)! 0

= Rl+1 ((l + 1)!)

γ−1 l+1

t

1

≤ [(Rt) γ−1 (l + 1)]

≤ exp{−(γ − 1)(e−1 (Rt)

−1/(γ−1)

(γ−1)(l+1)

≤ e−(γ−1)((l+2)−1)

− 1)},

which gives (8.101), completing the proof. Proof of Proposition 8.3.20. Lemma 8.3.29 shows that (8.96) holds. Thus, Lemma 8.3.30 applies to the function t 󳨃→ W α u(t, x) to show that

8.4 Heat equations

�

579

1 − 1 󵄨 󵄨 󵄩 󵄩 sup 󵄨󵄨󵄨W α u(t, x)󵄨󵄨󵄨 ≤ Cα e2κ−1 exp{−(2κ − 1)e−1 R2 2κ−1 t − 2κ−1 }󵄩󵄩󵄩u󵄩󵄩󵄩L2 ((−1,1)×Bn (1),dt×d(hσ )) , Leb n

x∈B (1/4)

for t ∈ (−1/4, 1/4), where Cα and R2 are the constants from Lemma 8.3.29. The result now 1

follows with c := (2κ − 1)e−1 R2 2κ−1 and Cα replaced by Cα e2κ−1 . −

Remark 8.3.31. There are many proofs in harmonic analysis that work for more general quasi-metric balls instead of metric balls. However, the proof of Proposition 8.3.20 used the fact that the balls B(X,d) (x, δ) are metric balls in an essential way. Quasi-metric balls arise if we allow dsj to be elements of (0, 1). Since we take dsj ∈ ℕ+ , we always have dsj ≥ 1 and dk ≥ 1, and the balls B(W ,ds) (x, δ) and B(X,d) (x, δ) are metric balls. The part of the proof of Proposition 8.3.20 that used the fact that dk ≥ 1, k = 1, . . . , q, is when we chose c1 > 0 so that B(X,d) (x, 2c1 ϵ) ⊆ Bn (ϵ2 ), for all x ∈ Bn (ϵ1 ); recall that ϵ = ϵ2 − ϵ1 . If some of the dk were less than 1, then choosing such a c1 > 0 might not be possible.

8.4 Heat equations In this section, we present the desired heat kernel bounds. We work on a connected C ∞ manifold M with smooth, strictly positive density Vol. Let ∞ (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} ⊂ Cloc (M; TM) × ℕ+

be Hörmander vector fields with (single-parameter) formal degrees on M. Fix D ∈ ℕ+ and, as in the beginning of this section, let κ ∈ ℕ+ be such that dsj divides κ for 1 ≤ j ≤ r. Set nj := dκs ∈ ℕ+ . In this section, we consider partial differential operators of the form L =

∑ degds (α),degds (β)≤κ

j

bα,β (x)W α W β ,

∞ bα,β ∈ Cloc (M; 𝕄D×D (ℂ)).

(8.102)

Definition 8.4.1. We say that L given by (8.102) is maximally subelliptic of type 2 of degree 2κ with respect to (W , ds) on (M, Vol) if for every relatively compact, open set Ω ⋐ M, there exists CΩ ≥ 0 such that for all f ∈ C0∞ (Ω; ℂD2 ), r

󵄩 n 󵄩2 󵄨 󵄨 󵄩 󵄩2 ∑󵄩󵄩󵄩Wj j f 󵄩󵄩󵄩L2 (M,Vol;ℂD2 ) ≤ CΩ (󵄨󵄨󵄨⟨L f , f ⟩L2 (M,Vol;ℂD ) 󵄨󵄨󵄨 + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (M,Vol;ℂD2 ) ). j=1

When (M, Vol) is also clear from the context, we say P is maximally subelliptic of type 2 of degree κ with respect to (W , ds). Assumption. Throughout this section, we assume that L , when thought of as a densely defined operator on L2 (M, Vol; ℂD ) with dense domain C0∞ (M; ℂD ), is symmetric and non-negative. Let L be any non-negative, self-adjoint extension of L .

580 � 8 Linear maximally subelliptic operators Theorem 8.4.2. Suppose L given by (8.102) is maximally subelliptic of type 2 of degree 2κ with respect to (W , ds) on (M, Vol) (and is symmetric and non-negative as in the above assumption). Then, for t > 0, the Schwartz kernel of e−tL is a smooth function, which ∞ we denote by e−tL (x, y) ∈ Cloc (M × M; 𝕄D×D (ℂ)). Furthermore, for every 𝒦 ⋐ M, there exists c = c(𝒦) > 0 such that for all ordered multi-indices α and β and all s ∈ ℕ, there exists C = C(𝒦, s, α, β) such that ∀t > 0, ∀x, y ∈ 𝒦, 1 − degds (α)−degds (β)−2κs 󵄨󵄨 s α β −tL 󵄨 (x, y)󵄨󵄨󵄨 ≤ C((ρ(W ,ds) (x, y) + t 2κ ) ∧ 1) 󵄨󵄨𝜕t Wx Wy e 1

ρ(W ,ds) (x, y)2κ 2κ−1 × exp(−c ( ) ) t 1

(8.103)

× (Vol(B(W ,ds) (x, ρ(W ,ds) (x, y) + t 2κ )) ∧ 1) . −1

The rest of this section is devoted to the proof of Theorem 8.4.2. Fix 𝒦 ⋐ Ω1 ⋐ Ω2 ⋐ M, with 𝒦 compact and Ω1 and Ω2 open and relatively compact. Theorem 3.3.7 applies to give for each x ∈ 𝒦 and δ ∈ (0, 1] a map Φx,δ as in that theorem. Set, for x ∈ 𝒦 and δ ∈ (0, 1], L

x,δ

:= Φ∗x,δ δ2κ L (Φx,δ )∗ .

(8.104)

L x,δ is a partial differential operator on Bn (1). Let (X, d ) = {(X1 , d1 ), . . . , (Xq , dq )} ⊂ Gen((W , ds)) be as in Theorem 3.3.7, i. e., (W , ds) ⊆ (X, d ) and Gen((W , ds)) is finitely generated by (X, d ) on Ω2 .

We prove Theorem 8.4.2 in three steps. In the first step, in Section 8.4.1, we prove results “at the unit scale” in the case where L is a partial differential operator on Bn (1); this step uses the a priori estimates from Section 8.3. In the second step, we prove abstract results taking place on B(X,d) (x, δ) for some fixed x ∈ 𝒦 and δ ∈ (0, 1]. We do this by changing variables via Theorem 3.3.7 to translate the result to Bn (1) and apply the result from the first step to L x,δ ; importantly, the estimates in this step will not depend on x ∈ 𝒦 or δ ∈ (0, 1]. Finally, in the third step we use the results from the second step to conclude Theorem 8.4.2. Remark 8.4.3. Note that for f in the domain of L , L f = L f , where L f is taken in the sense of distributions. Indeed, for such f and for g ∈ C0∞ (M; ℂD ), we have ⟨L f , g⟩L2 (M,Vol) = ⟨f , L g⟩L2 (M,Vol) = ⟨f , L g⟩L2 (M,Vol) . Since L is symmetric, this shows that L f = L f . Remark 8.4.4. For some comments on the relationship between maximal subellipticity and maximal subellipticity of type 2, see Remark 8.5.3 and Corollary 8.5.4.

8.4 Heat equations

�

581

8.4.1 Step I: the unit scale In this section, we work on Bn (1) and use the same setting as in Section 8.3. Thus, we are given (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} ⊂ C ∞ (Bn (1); TBn (1)), a strictly positive smooth function h ∈ C ∞ (Bn (1)) with infu∈Bn (1) h(u) > 0, and an operator L of the form (8.42) which is maximally subelliptic of type 2 of degree 2κ at the unit scale with respect to (W , ds) and hσLeb , in the sense that Definition 8.3.1 holds with these choices. We assume all the same assumptions as in Section 8.3 and define ι-L -unitadmissible constants as in Definition 8.3.2. Fix a constant Λ > 0; none of the constants in our estimates will depend on the choice of Λ. We will often endow Bn (1) with the measure Vol0 := ΛhσLeb . For U ⊆ Bn (1) open, 󵄨 let RU : L2 (Bn (1), Vol0 ) → L2 (U, Vol0 ) denote the restriction operator RU : f 󳨃→ f 󵄨󵄨󵄨U . 8.4.1.1 On diagonal bounds Fix D ∈ ℕ+ and suppose (M1 , μ1 ) is a σ-finite measure space. Let T : L2 (M, μ1 ; ℂD0 ) → L2 (Bn (1), Vol0 ; ℂD ) be a bounded linear map such that ∀L ∈ ℕ+ , ∃CL ≥ 0 with L

󵄩 󵄩 ∑󵄩󵄩󵄩L j T 󵄩󵄩󵄩L2 (M ,μ )→L2 (Bn (1),Vol ) ≤ CL . 1

j=0

1

0

Remark 8.4.5. When Λ = 1, we have in mind T = e−tL , where t ≈ 1 and L is any non-negative self-adjoint extension of L . Proposition 8.4.6. Under the above conditions, RBn (1/2) T : L2 (M1 , μ1 ; ℂD ) → L2 (Bn (1/2), Vol0 ; ℂD ) is given by integration against a measurable function T(x, y) : Bn (1/2) × M1 → 𝕄D×D0 (ℂ). More precisely, for all f ∈ L2 (Bn (1/2), Vol0 ; ℂD ) and g ∈ L2 (M1 , μ1 ; ℂD0 ), we have ∫ f (x) ⋅ (Tg)(x) d Vol0 (x) = ∬ f (x) ⋅ (T(x, y)g(y)) d Vol0 (x) dμ1 (y). Furthermore, for every M ∈ ℕ, there exists L ∈ ℕ depending only on upper bounds for M, n, m, and max{dsj : 1 ≤ j ≤ r} and an M-L -unit-admissible constant EM = EM (CL ) ≥ 0 such that sup

∑

x∈Bn (1/2) deg (α)≤M ds

󵄩󵄩 α 󵄩 −1 󵄩󵄩Wx T(x, ⋅)󵄩󵄩󵄩L2 (M1 ,μ1 ;𝕄D×D0 ) ≤ EM Λ 2 .

582 � 8 Linear maximally subelliptic operators Proof. Fix f ∈ L2 (M1 , μ1 ; ℂD ). Since Vol0 = ΛhσLeb and Λ > 0 is a constant, we have, ∀j ∈ ℕ, L j Tf ∈ L2 (Bn (1), Vol1 ; ℂD ) = L2 (Bn (1), hσLeb ; ℂD ). Corollary 8.3.8 applies to show ∞ n that Tf ∈ Cloc (B (1); ℂD ). Furthermore, fix ϕ1 , ϕ2 ∈ C0∞ (Bn (1)) with ϕ1 ≡ 1 on Bn (1/2) and ϕ1 ≺ ϕ2 . Then we have, for s = s(n) ∈ (0, ∞) sufficiently large, by the Sobolev embedding theorem and Corollary 8.3.8, with L(s, M) := ⌈ s+M ⌉, κϵ 0

sup

x∈Bn (1/2)

∑

degds (α)≤M

󵄨󵄨 α 󵄨 󵄨󵄨W Tf (x)󵄨󵄨󵄨 ≲ ‖ϕ1 Tf ‖L2s+M

L(s,M)

L(s,M)

1 󵄩 󵄩 󵄩 󵄩 ≲ ∑ 󵄩󵄩󵄩ϕ2 L j Tf 󵄩󵄩󵄩L2 (Bn (1),hσ ) = Λ− 2 ∑ 󵄩󵄩󵄩ϕ2 L j Tf 󵄩󵄩󵄩L2 (Bn (1),Vol ) Leb 0

j=0

j=0

− 21 󵄩 󵄩

󵄩 ≤ CL(s,M) Λ 󵄩󵄩f 󵄩󵄩󵄩L2 (M ,μ ) . 1 1 Taking the supremum over all such f with ‖f ‖L2 (M1 ,μ1 ) = 1 yields the result. 8.4.1.2 Off-diagonal bounds Let (M1 , μ1 ) be a σ-finite measure space and fix D0 ∈ ℕ. Let σ > 0 and for t ∈ [0, σ) suppose that Tt : L2 (M1 , μ1 ; ℂD0 ) → L2 (Bn (1), Vol0 ; ℂD ) is a bounded linear map and there is a dense subspace D ⊆ L2 (M1 , μ1 ; ℂD0 ) such that the following hold: ∞ – For all f ∈ D , Tt f ∈ Cloc ([0, σ)×Bn (1); ℂD ), where we are treating Tt f (x) as a function n of (t, x) ∈ [0, σ) × B (1). – For all f ∈ D , Tt f (x) vanishes to infinite order in t as t ↓ 0. – For all f ∈ D , (𝜕t + σ −1 L )Tt f = 0. 󵄩 󵄩 – 󵄩󵄩󵄩Tt 󵄩󵄩󵄩L2 (M ,μ ;ℂD0 )→L2 (Bn (1),Vol ;ℂD ) ≤ 1, ∀t ∈ [0, σ). 1

1

0

Remark 8.4.7. When Λ = σ = 1, informally an example of Tt can be given as follows. Let U and V be disjoint open sets and let L be any non-negative self-adjoint extension of L . 󵄨 Then Tt can be taken to be the operator f 󳨃→ e−tL f 󵄨󵄨󵄨U , where we restrict our attention to 󵄨 supp(f ) ⊆ V . Since U ∩ V = 0, if f ∈ C0∞ (V ), e−tL f 󵄨󵄨󵄨U vanishes to infinite order as t ↓ 0. Proposition 8.4.8. Suppose L , when thought of as a densely defined operator on L2 (Bn (1), Vol0 ; ℂD ) with dense domain C0∞ (Bn (1); ℂD ), is symmetric and non-negative (and maximally subelliptic of type 2 of degree 2κ at the unit scale with respect to (W , ds) and hσLeb , as assumed above). Then, for every t ∈ [0, σ/4), RBn (1/4) Tt is given by integration against a measurable function Tt (x, y) : Bn (1/4) × M1 → 𝕄D×D0 (ℂ). More precisely, for all f ∈ L2 (Bn (1/4), Vol0 ; ℂD ) and g ∈ L2 (M1 , μ1 ; ℂD0 ), for t ∈ [0, σ/4) we have ∫ f (x) ⋅ (Tt g)(x) d Vol0 (x) = ∬ f (x) ⋅ (Tt (x, y)g(y)) d Vol(x)dμ1 (y).

8.4 Heat equations

�

583

Furthermore, there exists an L -unit-admissible constant c > 0 such that for all s ∈ ℕ and all ordered multi-indices α and β, there exists an (s, α)-L -unit-admissible constant Ds,α ≥ 0 such that for t ∈ [0, σ/4), 󵄩 󵄩 sup 󵄩󵄩󵄩𝜕ts Wxα Tt (x, ⋅)󵄩󵄩󵄩L2 (M ,μ ;𝕄D×D0 ) 1 1 n

x∈B (1/4)

1

1

(8.105)

1

≤ Ds,α σ −s Λ− 2 exp(−cσ 2κ−1 t − 2κ−1 ).

Proof. Since Vol0 = ΛhσLeb and Λ > 0 is constant, L , when thought of as a densely defined operator on L2 (Bn (1), hσLeb ; ℂD ) with dense domain C0∞ (Bn (1); ℂD ), is symmetric and non-negative. Because 𝜕ts Tt = (−1)s σ −s L s Tt and due to the form of L , it suffices to prove (8.105) for s = 0. Fix f ∈ D and set u(t, x) := {

Tσt f (x), 0,

t ∈ [0, 1), t ∈ (−1, 0).

∞ Note that u ∈ Cloc ((−1, 1)×Bn (1); ℂD ) and 𝜕t u(t, x) = −L u(t, x). Proposition 8.3.20 implies that there exists an L -unit-admissible constant c > 0 such that for t ∈ [0, 1/4), 1 󵄨 󵄨 󵄩 󵄩 sup 󵄨󵄨󵄨Wxα u(t, x)󵄨󵄨󵄨 ≲ exp(−ct − 2κ−1 )󵄩󵄩󵄩u󵄩󵄩󵄩L2 ((−1,1)×Bn (1),dt×hσ ) Leb n

x∈B (1/4)

1 1 󵄩 󵄩 = Λ− 2 exp(−ct − 2κ−1 )󵄩󵄩󵄩Tσt f 󵄩󵄩󵄩L2 ((−1,1)×Bn (1),dt×d Vol

≤Λ

− 21

exp(−ct

1 − 2κ−1

0)

󵄩 󵄩 )󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (M ,μ ) . 1 1

Taking the supremum over all f ∈ D with ‖f ‖L2 (M1 ,μ1 ) yields, for t ∈ [0, 1/4), 1 1 󵄩 󵄩 sup 󵄩󵄩󵄩Wxα Tσt (x, ⋅)󵄩󵄩󵄩L2 (M ,μ ;𝕄D×D0 ) ≲ Λ− 2 exp(−ct − 2κ−1 ). 1 1 n

x∈B (1/4)

From here the result follows. 8.4.2 Step II: a single point and scale We return to the main setting of this section on an abstract manifold M with smooth, strictly positive density Vol and Hörmander vector fields with formal degrees (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )}. We let L be given by (8.42) and assume that L is symmetric, non-negative, and maximally subelliptic of type 2 of degree 2κ with respect to (W , ds) on (M, Vol). The main thrust of this section is that Propositions 8.4.6 and 8.4.8 apply to L x,δ , uniformly for x ∈ 𝒦 and δ ∈ (0, 1], where L x,δ is defined in (8.104). Indeed, we have the following proposition.

584 � 8 Linear maximally subelliptic operators Proposition 8.4.9. L x,δ satisfies the hypotheses of Section 8.4.1 uniformly for x ∈ 𝒦 and δ ∈ (0, 1], and ι-L x,δ -unit-admissible constants (as in Definition 8.3.2) can be chosen independent of x ∈ 𝒦 and δ ∈ (0, 1]. More precisely, we have the following: (a) Let Wjx,δ : Φ∗x,δ δdsj Wj and set (W x,δ , ds) := {(W1x,δ , ds1 ), . . . , (Wrx,δ , dsr )}. Then (W x,δ , ds) are Hörmander vector fields at the unit scale, uniformly for x ∈ 𝒦, δ ∈ (0, 1]. (b) We give Bn (1) the density Φ∗x,δ Vol = Λ(x, δ)hx,δ σLeb , where Λ(x, δ) > 0 and hx,δ ∈ C ∞ (Bn (1)) are as in Theorem 3.3.7. Then inf

inf hx,δ (u) > 0

x∈Bn (1) u∈Bn (1) δ∈(0,1]

and for every L ∈ ℕ sup ‖hx,δ ‖C L (Bn (1)) < ∞.

x∈Bn (1) δ∈(0,1]

(c) L x,δ can be written in the form L

x,δ

=

α

∑ degds (α),degds (β)≤κ

β

x,δ bx,δ ) (W x,δ ) , α,β (W

where bx,δ ∈ C ∞ (Bn (1); 𝕄D×D (ℂ)) satisfy, for every L ∈ ℕ, α,β 󵄩 󵄩󵄩 sup 󵄩󵄩󵄩bx,δ 󵄩C L (Bn (1)) < ∞. α,β 󵄩

x∈Bn (1) δ∈(0,1]

(d) L x,δ is symmetric and non-negative when thought of as a densely defined operator on L2 (Bn (1), Φ∗x,δ Vol; ℂD ) with dense domain C0∞ (Bn (1); ℂD ) is symmetric and nonnegative. (e) L x,δ is maximally subelliptic of type 2 of order 2κ at the unit scale with respect to (W x,δ , ds) and hx,δ σLeb , uniformly in x ∈ 𝒦 and δ ∈ (0, 1] in the sense that there exists A ≥ 0 such that ∀x ∈ 𝒦, ∀δ ∈ (0, 1], ∀f ∈ C0∞ (Bn (1); ℂD ), r

n 󵄩2 󵄩 ∑󵄩󵄩󵄩(Wjx,δ ) j f 󵄩󵄩󵄩L2 (Bn (1),h σ ;ℂD ) x,δ Leb j=1

󵄨 ≤ A(󵄨󵄨󵄨⟨L x,δ f , f ⟩L2 (Bn (1),h

x,δ σLeb ;ℂ

D)

󵄨󵄨 󵄩󵄩 󵄩󵄩2 󵄨󵄨 + 󵄩󵄩f 󵄩󵄩L2 (Bn (1),hx,δ σLeb ;ℂD ) ).

Proof. (a): This follows from Theorem 3.3.7 (l). (b): This follows from Theorem 3.3.7 (n). (c): Let f , g, ∈ C0∞ (Bn (1); ℂD ). By Theorem 3.3.7 (h), f ∘ Φx,δ , g ∘ Φx,δ ∈ C0∞ (M; ℂD ). Thus, by changing variables and using the fact that L is symmetric, we have

8.4 Heat equations

⟨L x,δ f , g⟩L2 (Bn (1),Φ∗

x,δ

Vol)

= δ2κ ⟨Φ∗x,δ L (Φx,δ )∗ f , g⟩L2 (Bn (1),Φ∗

x,δ

2κ

= δ2κ ⟨f ∘ Φx,δ , L (g ∘ Φx,δ )⟩L2 (Bn (1),Vol) x,δ

585

Vol)

= δ ⟨L (f ∘ Φx,δ ), g ∘ Φx,δ ⟩L2 (Bn (1),Vol) = ⟨f , L x,δ g⟩L2 (Bn (1),Φ∗

�

(8.106)

Vol) .

We conclude that L x,δ is symmetric. Plugging f = g into (8.106) and using the fact that L is non-negative, we see that ⟨L x,δ f , f ⟩L2 (Bn (1),Φ∗

x,δ

Vol)

= δ2κ ⟨L (f ∘ Φx,δ ), f ∘ Φx,δ ⟩L2 (Bn (1),Vol) ≥ 0,

and we conclude that L x,δ is non-negative. (e): Let CΩ1 ≥ 0 be the constant from Definition 8.4.1, when applied to L . Let f ∈ C0∞ (Bn (1); ℂD ), x ∈ 𝒦, and δ ∈ (0, 1]. Set u := f ∘ Φx,δ . By Theorem 3.3.7 (h) and the fact that Φx,δ (Bn (1)) ⊆ Ω1 (as stated in Theorem 3.3.7), we see that u ∈ C0∞ (Ω1 ; ℂD ). Thus, we have r

n 󵄩2 󵄩 ∑󵄩󵄩󵄩(Wjx,δ ) j f 󵄩󵄩󵄩L2 (Bn (1),hσ j=1

r

Leb )

n 󵄩 󵄩2 = Λ−1 ∑󵄩󵄩󵄩(Φ∗x,δ δdsj Wj ) j Φ∗x,δ u󵄩󵄩󵄩L2 (Bn (1),Φ∗

x,δ

j=1

Vol)

r

󵄩 n 󵄩2 = Λ−1 δ2κ ∑󵄩󵄩󵄩Wj j u󵄩󵄩󵄩L2 (M,Vol) j=1

󵄨 󵄨 󵄩 󵄩2 ≤ CΩ1 Λ δ (󵄨󵄨󵄨⟨L u, u⟩L2 (M,Vol) 󵄨󵄨󵄨 + 󵄩󵄩󵄩u󵄩󵄩󵄩L2 (M,Vol) ) 󵄨 󵄨 󵄩 󵄩2 ≤ CΩ1 Λ−1 (󵄨󵄨󵄨⟨δ2κ L u, u⟩L2 (M,Vol) 󵄨󵄨󵄨 + 󵄩󵄩󵄩u󵄩󵄩󵄩L2 (M,Vol) ) 󵄨 󵄨 󵄩 󵄩2 = CΩ1 Λ−1 (󵄨󵄨󵄨⟨L x,δ f , f ⟩L2 (Bn (1),Φ∗ Vol) 󵄨󵄨󵄨 + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (Bn (1),Φ∗ Vol) ) x,δ x,δ 󵄨󵄨 x,δ 󵄨󵄨 󵄩󵄩 󵄩󵄩2 = CΩ1 (󵄨󵄨⟨L f , f ⟩L2 (Bn (1),h σ ) 󵄨󵄨 + 󵄩󵄩f 󵄩󵄩L2 (Bn (1),h σ ) ), x,δ Leb x,δ Leb −1 2κ

completing the proof. For the remainder of this section, fix x0 ∈ 𝒦 and δ0 ∈ (0, 1]. Importantly, the estimates which follow are all uniform in x0 ∈ 𝒦 and δ0 ∈ (0, 1]. For U ⊆ B(X,d) (x0 , δ0 ) open, let RU : L2 (B(X,d) (x0 , δ0 ), Vol; ℂD ) → L2 (U, Vol; ℂD ) denote the restriction map 󵄨 RU : f 󳨃→ f 󵄨󵄨󵄨U . 8.4.2.1 On-diagonal bounds Fix D0 ∈ ℕ+ and suppose that (M1 , μ1 ) is a σ-finite measure space and Tx0 ,δ0 : L2 (M1 , μ1 ; ℂD0 ) → L2 (B(X,d) (x0 , δ0 ), Vol; ℂD ) is a bounded linear map such that ∀L ∈ ℕ, ∃CL ≥ 0 with

586 � 8 Linear maximally subelliptic operators L

j 󵄩 󵄩 ∑󵄩󵄩󵄩(δ02κ L ) Tx0 ,δ0 󵄩󵄩󵄩L2 (M ,μ ;ℂD0 )→L2 (B (x ,δ ),Vol;ℂD ) ≤ CL . 1 1 (X,d) 0 0

j=0

Remark 8.4.10. One should think of Tx0 ,δ0 as e−tL , where t ≈ δ2κ and L is any nonnegative self-adjoint extension of L . Let ξ3 > 0 be the constant from Theorem 3.3.7; note that ξ3 does not depend on the choice of x0 ∈ 𝒦 or δ0 ∈ (0, 1]. Proposition 8.4.11. Under the above conditions, RB(X,d) (x0 ,ξ3 δ0 ) Tx0 ,δ0 : L2 (M1 , μ1 ; ℂD0 ) → L2 (B(X,d) (x0 , δ0 ξ3 ), Vol; ℂD ) is given by integration against a measurable function Tx0 ,δ0 (x, y) : B(X,d) (x0 , ξ3 δ0 ) × M1 → 𝕄D×D0 (ℂ) in the sense that for all f ∈ L2 (B(X,d) (x0 , ξ3 δ0 ), Vol; ℂD ) and g ∈ L2 (M1 , μ1 ; ℂD0 ), we have ∫ f (x) ⋅ (Tx0 ,δ0 g)(x) d Vol(x)

(8.107)

= ∬ f (x) ⋅ (Tx0 ,δ0 (x, y)g(y)) d Vol(x)dμ1 (y).

Furthermore, for every M ∈ ℕ, there exists L = L(M) ∈ ℕ and EM = EM (CL ) ≥ 0, not depending on x0 ∈ 𝒦 or δ0 ∈ (0, 1], such that sup

∑

x∈B(W ,ds) (x0 ,ξ3 δ0 ) deg (α)≤M ds

α 󵄩󵄩 ds 󵄩 󵄩󵄩(δ0 Wx ) Tx0 ,δ0 (x, ⋅)󵄩󵄩󵄩L2 (M1 ,μ1 ;𝕄D×D0 ) − 21

(8.108)

≤ EM (Vol(B(W ,ds) (x0 , δ0 )) ∧ 1) . Proof. Proposition 8.4.9 shows that Proposition 8.4.6 applies with (W , ds) replaced by (W x0 ,δ0 , ds), L replaced by L x0 ,δ0 , T replaced by Φ∗x0 ,δ Tx0 ,δ0 , and Vol0 replaced by Volx0 ,δ0 := Φ∗x0 ,δ0 Vol = Λ(x0 , δ0 )hx0 ,δ0 σLeb . Moreover, all of the estimates obtained from Proposition 8.4.6 are uniform in x0 ∈ 𝒦 and δ ∈ (0, 1]. Thus, there is a measurable function S(u, y) : Bn (1/2) × M1 → 𝕄D×D0 (ℂ) such that for all f ∈ L2 (Bn (1/2), Volx0 ,δ0 ) and g ∈ L2 (M1 , μ1 ; ℂD0 ), we have ∫ f (u) ⋅ (Φ∗x0 ,δ0 Tx0 ,δ0 g)(u) d Volx0 ,δ0 (u)

(8.109)

= ∬ f (u) ⋅ (S(u, y)g(y)) d Volx0 ,δ0 (u)dμ1 (y), which satisfies sup

u∈Bn (1/2)

∑

degds (α)≤M 1

󵄩󵄩 x0 ,δ0 α 󵄩 󵄩󵄩(Wu ) S(u, ⋅)󵄩󵄩󵄩L2 (M1 ,μ1 )

≲ Λ(x0 , δ0 )− 2 ≤ (Λ(x0 , δ0 ) ∧ 1)

− 21

−1

≈ (Vol(B(W ,ds) (x0 , δ0 )) ∧ 1) 2 ,

(8.110)

8.4 Heat equations

�

587

where the final estimate follows from Theorem 3.3.7 (d). Setting Tx0 ,δ0 (x, y) := S(Φ−1 x0 ,δ0 (x), y), changing variables x = Φx0 ,δ0 (u) in (8.109), and using B(X,d) (x0 , ξ3 δ0 ) ⊆ Φx0 ,δ0 (Bn (1/2)) (by Theorem 3.3.7 (i)) shows that (8.107) holds. Similarly, again using B(X,d) (x0 , ξ3 δ0 ) ⊆ Φx0 ,δ0 (Bn (1/2)), (8.108) follows from (8.110). 8.4.2.2 Off-diagonal bounds Suppose (M1 , μ1 ) is a σ-finite measure space. Fix D0 ∈ ℕ+ and σ > 0. For t ∈ [0, σ), suppose that Tt,x0 ,δ0 : L2 (M1 , μ1 ; ℂD0 ) → L2 (B(X,d) (x0 , δ0 ), Vol; ℂD ) is a bounded linear map and there is a dense subspace D ⊆ L2 (M, μ1 ) such that the following hold: ∞ – For all f ∈ D , Tt,x0 ,δ0 f ∈ Cloc ([0, σ) × B(X,d) (x0 , δ0 ); ℂD ), where we are treating Tt,x0 ,δ0 f (x) as a function of (t, x) ∈ [0, σ) × B(X,d) (x0 , δ0 ). – For all f ∈ D , Tt,x0 ,δ0 f (x) vanishes to infinite order in t as t ↓ 0. – For all f ∈ D , (𝜕t + σ −1 δ02κ L )Tt,x0 ,δ0 f = 0. 󵄩 󵄩 – We have 󵄩󵄩󵄩Tt,x0 ,δ0 󵄩󵄩󵄩L2 (M ,μ ;ℂD0 )→L2 (B (x ,δ ),Vol;ℂD ) ≤ 1, ∀t ∈ [0, σ). 1

1

(X,d)

0

0

Proposition 8.4.12. For every t ∈ [0, σ/4), RΦx

0 ,δ0

(Bn (1/4)) Tt,x0 ,δ0 n

is given by integration

against a measurable function Tt,x0 ,δ0 (x, y) : Φx0 ,δ (B (1/4)) × M1 → 𝕄D×D0 (ℂ). More precisely, for all f ∈ L2 (Φx0 ,δ0 (Bn (1/4)), Vol; ℂD ) and g ∈ L2 (M1 , μ1 ), for t ∈ [0, σ/4), we have ∫ f (x) ⋅ (Tt,x0 ,δ0 g)(x) d Vol(x)

(8.111)

= ∬ f (x) ⋅ (Tt,x0 ,δ0 (x, y)g(y)) d Vol(x)dμ1 (y).

Furthermore, there exists a constant c = c(𝒦, (W , ds)) > 0 such that for all s ∈ ℕ and all ordered multi-indices α and β, there exists Ds,α ≥ 0 such that for t ∈ [0, σ/4), sup

x∈Φx0 ,δ0

(Bn (1/4))

α 󵄩󵄩 s ds 󵄩 󵄩󵄩𝜕t (δ0 Wx ) Tt,x0 ,δ0 (x, ⋅)󵄩󵄩󵄩L2 (M ,μ ;𝕄D×D0 )

≤ Ds,α σ (Vol(B(W ,ds) (x0 , δ)) ∧ 1) −s

1

− 21

1

exp(−cσ

1 2κ−1

t

1 − 2κ−1

(8.112) ).

Here c > 0 and Ds,α ≥ 0 do not depend on x0 ∈ 𝒦, δ0 ∈ (0, 1], or Tt,x0 ,δ0 . Proof. Let St := Φ∗x0 ,δ Tt,x0 ,δ0 : L2 (M1 , μ1 ; ℂD0 ) → L2 (Bn (1), Φ∗x0 ,δ0 Vol). Proposition 8.4.9

shows that Proposition 8.4.8 applies with (W , ds) replaced by (W x0 ,δ0 , ds), L replaced by L x0 ,δ0 , Tt replaced by St , and Vol0 replaced by Volx0 ,δ0 := Φ∗x0 ,δ0 Vol = Λ(x0 , δ0 )hx0 ,δ0 σLeb . Moreover, all of the estimates obtained from Proposition 8.4.8 are uniform in x0 ∈ 𝒦 and δ ∈ (0, 1].

588 � 8 Linear maximally subelliptic operators Proposition 8.4.8 implies that for t ∈ [0, σ/4) there is a measurable function St : Bn (1/4) × M1 → 𝕄D×D0 (ℂ) such that for all f ∈ L2 (Bn (1/4), Vol0 ; ℂD ) and g ∈ L2 (M1 , μ1 ; ℂD0 ), we have ∫ f (u) ⋅ (St g)(u) d Volx0 ,δ0 (u)

(8.113)

= ∬ f (u) ⋅ (St (u, y)g(y)) d Volx0 ,δ0 (u)dμ1 (y).

Furthermore, Proposition 8.4.8 shows that there exists c ≳ 1 such that for all s ∈ ℕ, for all ordered multi-indices α and β, and for t ∈ [0, σ/4), α 󵄩 󵄩 sup 󵄩󵄩󵄩𝜕ts (Wux0 ,δ0 ) St (u, ⋅)󵄩󵄩󵄩L2 (M ,μ ;𝕄D×D0 ) 1 1 n

u∈B (1/4)

1

1

1

≲ σ −s Λ(x0 , δ0 )− 2 exp(−cσ 2κ−1 t − 2κ−1 ) ≤ σ −s (Λ(x0 , δ0 ) ∧ 1)

− 21

1

(8.114)

1

exp(−cσ 2κ−1 t − 2κ−1 ) − 21

≲ σ −s (Vol(B(W ,ds) (x0 , δ0 )) ∧ 1)

1

1

exp(−cσ 2κ−1 t − 2κ−1 ),

where the final estimate used Theorem 3.3.7 (d). Setting Tt,x0 ,δ0 (x, y) := St (Φ−1 x0 ,δ0 (x), y) and changing variables x = Φx0 ,δ0 (u) in (8.113) establishes (8.111). Similarly, (8.112) follows from (8.114).

8.4.3 Step III: all scales In this section, we use the results from Section 8.4.2 to complete the proof of Theorem 8.4.2. We separate the proof into three parts: ∞ – Large time: ∀t > 0, the Schwartz kernel of e−tL is a Cloc (M × M; 𝕄D×D (ℂ)) function, which we denote by e−tL (x, y). Furthermore, ∀δ2 > 0, ∀s ∈ ℕ, ∀α, β, ∃C𝒦,δ2 ,s,α,β ≥ 0 such that ∀t ≥ δ22κ , x, y ∈ 𝒦, we have 󵄨󵄨 s α β −tL 󵄨 (x, y)󵄨󵄨󵄨 ≤ C𝒦,δ2 ,s,α,β . 󵄨󵄨𝜕t Wx Wy e –

(8.115)

On-diagonal bounds: ∀s ∈ ℕ, ∀α, β, ∃C𝒦,s,α,β ≥ 0 such that ∀x, y ∈ 𝒦, t > 0 with 1

ρ(X,d) (x, y) ≤ 16t 2κ ≤ 1, we have ds ds t 󵄨󵄨 s α β −tL 󵄨 (x, y)󵄨󵄨󵄨 ≤ C𝒦,s,α,β . 󵄨󵄨𝜕t Wx Wy e 1 Vol(B(W ,ds) (x, t 2κ )) ∧ 1

−(deg (α)+deg (β)+2κs)/2κ

–

(8.116)

Off-diagonal bounds: There exists c1 = c1 (𝒦) > 0 such that ∀s ∈ ℕ, ∀α, β, ∃C𝒦,s,α,β ≥ 1

0 such that ∀x, y ∈ 𝒦, t > 0 with 16t 2κ ≤ ρ(X,d) (x, y) ≤ 1, we have

8.4 Heat equations

�

589

󵄨󵄨 s α β −tL 󵄨 (x, y)󵄨󵄨󵄨 ≤ C𝒦,s,α,β ρ(X,d) (x, y)− degds (α)−degds (β)−2κs 󵄨󵄨𝜕t Wx Wy e 1

ρ(X,d) (x, y)2κ 2κ−1 × exp(−c1 ( ) ) t

(8.117)

× (Vol(B(W ,ds) (x, ρ(X,d) (x, y))) ∧ 1) . −1

Before we prove the above estimates we see how they imply Theorem 8.4.2. For this, we need the next lemma. Lemma 8.4.13. For all x, y ∈ 𝒦, ρ(W ,ds) (x, y) ≈ ρ(X,d) (x, y). Proof. This lemma follows easily from the case j = 0, ν = 1 of Lemma 5.4.14,3 but we include the proof in this simple special case. Since (W , ds) ⊆ (X, d ), we have ρ(X,d) (x, y) ≤ ρ(W ,ds) (x, y), ∀x, y ∈ M. Let x, y ∈ 𝒦. Theorem 3.3.7 (b) shows that if δ ∈ (0, 1] and ρ(X,d) (x, y) < ξ3 δ, then ρ(W ,ds) (x, y) < δ. Thus, if ρ(X,d) (x, y) < ξ3 , we have ρ(W ,ds) (x, y) < ξ3−1 ρ(X,d) (x, y). Finally, if ρ(X,d) (x, y) ≥ ξ3 , since 𝒦 is compact with respect to ρ(W ,ds) (see Lemma 3.1.7), we have ρ(W ,ds) (x, y) ≲ 1 ≲ ρ(X,d) (x, y), completing the proof. ∞ Proof of Theorem 8.4.2. By the large time estimates, we have e−tL (x, y) ∈ Cloc (M × D×D M; 𝕄 (ℂ)), for t > 0. Thus, it suffices to prove (8.103). Fix 𝒦 ⋐ M, x, y ∈ 𝒦, and t > 0. 1 We separate the proof of (8.103) into the following three cases: 16t 2κ ≤ 1 ∧ ρ(X,d) (x, y), 1

1

ρ(X,d) (x, y) ≤ 16t 2κ , and 16t 2κ ≥ 1. In all cases, we use Lemma 8.4.13, so that ρ(X,d) (x, y) ≈ ρ(W ,ds) (x, y). 1

16t 2κ ≤ ρ(X,d) (x, y) ∧ 1: In this case, we have 1

1

ρ(X,d) (x, y) ≈ ρ(X,d) (x, y) + t 2κ ≈ ρ(W ,ds) (x, y) + t 2κ .

(8.118)

Since 𝒦 is compact with respect to ρ(W ,ds) (Lemma 3.1.7), we have ρ(W ,ds) (x, y) ≲ 1, and therefore since we also have t ≤ 1, 1

1

ρ(W ,ds) (x, y) + t 2κ ≈ (ρ(W ,ds) (x, y) + t 2κ ) ∧ 1.

(8.119)

Combining (8.118) with Theorem 3.3.7 (f), we see that Vol(B(W ,ds) (x, ρ(X,d) (x, y))) ∧ 1

1

≈ Vol(B(W ,ds) (x, ρ(W ,ds) (x, y) + t 2κ )) ∧ 1.

(8.120)

Let c1 = c1 (𝒦) > 0 be as in (8.117) and take c = c(𝒦) > 0 such that 2κ

2κ

cρ(W ,ds) (x, y) 2κ−1 ≤ c1 ρ(X,d) (x, y) 2κ−1 .

(8.121)

3 In fact, Lemma 5.4.14 implies that ρ(W ,ds) and ρ(X,d) are equivalent on any compact subset of M.

590 � 8 Linear maximally subelliptic operators Using (8.118), (8.119), (8.120), and (8.121) to bound (8.117) yields (8.103), completing the proof in this case. We use this choice of c = c(𝒦) > 0 in the rest of the proof. 1 ρ(X,d) (x, y) ≤ 16t 2κ ≤ 1: In this case, we have 1

1

1

t 2κ ≈ ρ(X,d) (x, y) + t 2κ ≈ ρ(W ,ds) (x, y) + t 2κ , t

1 2κ

(8.122)

1 2κ

1 2κ

≈ (ρ(X,d) (x, y) + t ) ∧ 1 ≈ (ρ(W ,ds) (x, y) + t ) ∧ 1.

(8.123)

Combining (8.122) with Theorem 3.3.7 (f), we see that 1

1

Vol(B(W ,ds) (x, t 2κ )) ∧ 1 ≈ Vol(B(W ,ds) (x, ρ(W ,ds) (x, y) + t 2κ )) ∧ 1.

(8.124)

1

Finally, in this case we have ρ(W ,ds) (x, y) ≈ ρ(X,d) (x, y) ≤ 16t 2κ , and therefore 1

ρ(W ,ds) (x, y)2κ 2κ−1 exp(−c ( ) ) ≳ 1. t

(8.125)

Using (8.123), (8.124), and (8.125) to bound (8.116) yields (8.103), completing the proof in this case. 1 16t 2κ ≥ 1: In this case, we have 1

(ρ(W ,ds) (x, y) + t 2κ ) ∧ 1 ≈ 1.

(8.126)

Clearly, 1

(Vol(B(W ,ds) (x, ρ(W ,ds) (x, y) + t 2κ )) ∧ 1)

−1

≥ 1.

(8.127)

Since 𝒦 is compact with respect to ρ(W ,ds) (see Lemma 3.1.7), we have ρ(W ,ds) (x, y) ≲ 1. Since t ≳ 1, we see that 1

ρ(W ,ds) (x, y)2κ 2κ−1 exp(−c ( ) ) ≈ 1. t

(8.128)

Combining (8.126), (8.127), and (8.128) shows that the right-hand side of (8.103) is ≳ 1. Therefore, (8.103) follows from (8.115) with δ2 = 161 . We turn to proving the estimates (8.115), (8.116), and (8.117). Lemma 8.4.14. For t > 0, the Schwartz kernel of e−tL is an L1loc (M × M, Vol ⊗ Vol; 𝕄D×D (ℂ)) function e−tL (x, y) which satisfies ∀α, ∃C𝒦,α ≥ 0 such that ∀x ∈ 𝒦, ∀t > 0, we have 󵄩󵄩 α −(t/2)L 󵄩 󵄩 󵄩 (x, ⋅)󵄩󵄩󵄩L2 (M,Vol;𝕄D×D ) , 󵄩󵄩󵄩Wxα e−(t/2)L (⋅, x)󵄩󵄩󵄩L2 (M,Vol;𝕄D×D ) 󵄩󵄩Wx e 1

− degds (α)

≤ C𝒦,α (t 2κ ∧ 1)

1

−1

(Vol(B(W ,ds) (x, t 2κ ∧ 1)) ∧ 1) 2 .

8.4 Heat equations

�

591

Proof. Set t1 := t ∧ 1. Note that, ∀L ∈ ℕ, ∃CL ≥ 0 (CL depending only on L), such that L

󵄩 󵄩 ∑󵄩󵄩󵄩(t1 L )j e−(t/2)L 󵄩󵄩󵄩L2 (M,Vol)→L2 (M,Vol)

j=0

(8.129)

L

󵄩 󵄩 = ∑󵄩󵄩󵄩(t1 L )j e−(t/2)L 󵄩󵄩󵄩L2 (M,Vol)→L2 (M,Vol) ≤ CL , j=0

where we have used the fact that L agrees with L on the domain of L , where L is taken in the sense of distributions (see Remark 8.4.3). 1

Fix x0 ∈ 𝒦. By restricting (8.129) to B(X,d) (x0 , t12κ ), we see that L

󵄩 󵄩 ∑󵄩󵄩󵄩(t1 L )j e−(t/2)L 󵄩󵄩󵄩

j=0

1

L2 (M,Vol)→L2 (B(X,d) (x0 ,t12κ ),Vol)

≤ CL .

(8.130)

1 󵄨 Letting RU : f 󳨃→ f 󵄨󵄨󵄨U denote the restriction map, Proposition 8.4.11 (with δ0 = t12κ

and Tx0 ,δ0 = e−(t/2)L ) shows that R

1

B(X,d) (x0 ,ξ3 t12κ )

e−(t/2)L is given by integration against a

measurable function e−(t/2)L (x, y) satisfying, for every α, sup

1

x∈B(X,d) (x0 ,ξ3 t12κ )

α −(t/2)L 󵄩󵄩 2κds 󵄩 (x, ⋅)󵄩󵄩󵄩L2 (M,Vol) 󵄩󵄩(t1 Wx ) e 1 2κ

(8.131)

− 21

≲ (Vol(B(W ,ds) (x0 , t1 )) ∧ 1) , 1

where ξ3 = ξ3 (𝒦) > 0. In particular, e−(t/2)L (x, y) ∈ L1loc (B(X,d) (x0 , ξ3 t12κ ) × M, Vol ⊗ 1

Vol; 𝕄D×D ) and ∀ϕ1 ∈ C0∞ (B(X,d) (x0 , ξ3 t12κ ); ℂD ), ∀ϕ2 ∈ C0∞ (M; ℂD ) ∫ ϕ1 (x) ⋅ (e−(t/2)L ϕ2 )(x) d Vol(x)

= ∬ ϕ1 (x) ⋅ (e−(t/2)L (y)ϕ2 (y)) d Vol(x)d Vol(y). 1

Since B(X,d) (x0 , ξ3 t 2κ ) is an open neighborhood of x0 , x0 ∈ 𝒦 is arbitrary, and 𝒦 ⋐ M is ar-

bitrary, it follows that the Schwartz kernel of e−(t/2)L is an L1loc (M×M, Vol ⊗ Vol; 𝕄D×D (ℂ)) function. Moreover, by setting x = x0 in (8.131), we have 󵄩󵄩 α −(t/2)L 󵄩 (x, ⋅)󵄩󵄩󵄩L2 (M,Vol;𝕄D×D ) 󵄩󵄩Wx e 1

− degds (α)

≲ (t 2κ ∧ 1)

1

−1

(Vol(B(W ,ds) (x, t 2κ ∧ 1)) ∧ 1) 2 .

󵄩 󵄩 Since e−(t/2)L is self-adjoint, the same estimate holds for 󵄩󵄩󵄩Wxα e−(t/2)L (⋅, x)󵄩󵄩󵄩L2 (M,Vol;𝕄D×D ) , completing the proof.

592 � 8 Linear maximally subelliptic operators Using the fact that e−tL = e−(t/2)L ∘ e−(t/2)L , we have, for t > 0 and all α, β, using Lemma 8.4.14 and the Cauchy–Schwartz inequality, 󵄨󵄨 α β −tL 󵄨 (x, y)󵄨󵄨󵄨 󵄨󵄨Wx Wy e 󵄨󵄨 󵄨 = 󵄨󵄨󵄨∫(Wxα e−(t/2)L (x, z)) (Wyβ e−(t/2)L (z, y)) d Vol(z)󵄨󵄨󵄨 󵄨 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩Wxα e−(t/2)L (x, ⋅)󵄩󵄩󵄩L2 (M,Vol) 󵄩󵄩󵄩Wyβ e−(t/2)L (⋅, y)󵄩󵄩󵄩L2 (M,Vol) .

(8.132)

s

Proof of (8.115). Because 𝜕ts e−tL = (−1)s L e−tL = (−1)s L e−tL , where L is taken in the sense of distributions (see Remark 8.4.3), and due to the form of L , it suffices to prove (8.115) in the case s = 0. Let δ2 > 0; the estimates which follow depend on δ2 . For t ≥ δ22κ and x, y ∈ 𝒦, we have by Lemma 8.4.14 and (8.132) 󵄨󵄨 α β −tL 󵄨 (x, y)󵄨󵄨󵄨 󵄨󵄨Wx Wy e 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩Wxα e−(t/2)L (x, ⋅)󵄩󵄩󵄩L2 (M,Vol) 󵄩󵄩󵄩Wyβ e−(t/2)L (⋅, y)󵄩󵄩󵄩L2 (M,Vol) − degds (α)−degds (β)

1

≲ (t 2κ ∧ 1)

1

(Vol(B(W ,ds) (x, t 2κ ∧ 1)) ∧ 1)

− 21

(8.133)

−1

1

× (Vol(B(W ,ds) (y, t 2κ ∧ 1)) ∧ 1) 2 . 1

Since t ≥ δ22κ , we have t 2κ ∧ 1 ≈ 1. Also, letting δ1 = δ1 (Ω1 ) > 0 be as in Corollary 3.3.9 (recall that 𝒦 ⋐ Ω1 ⋐ M and Ω1 is open), using repeated application of Theorem 3.3.7 (f), we have, for z ∈ 𝒦, 1

Vol(B(W ,ds) (z, t 2κ ∧ 1)) ∧ 1 ≥ Vol(B(W ,ds) (z, δ2 ∧ 1)) ∧ 1

≈ Vol(B(W ,ds) (z, δ1 )) ∧ 1 ≈ 1,

where the final ≈ used Corollary 3.3.9. Thus, the right-hand side of (8.133) is ≲ 1, completing the proof. ∞ Corollary 8.4.15. We have e−tL (x, y) ∈ Cloc (M × M; 𝕄D×D ). β

Proof. The estimate (8.115) shows that for every α and β, Wxα Wy e−tL (x, y) is a locally bounded function on M × M. The result follows, since W1 , . . . , Wr satisfy Hörmander’s condition. s

Proof of (8.116). Because 𝜕ts e−tL = (−1)s L e−tL = (−1)s L e−tL , where L is taken in the sense of distributions (see Remark 8.4.3), and due to the form of L , it suffices to prove (8.116) in the case s = 0. 1 Suppose x, y ∈ 𝒦 with ρ(X,d) (x, y) ≤ 16t 2κ ≤ 1. By Lemma 8.4.14 and (8.132), using 1

t 2κ < 1, we have

8.4 Heat equations

󵄨󵄨 α β −tL 󵄨 (x, y)󵄨󵄨󵄨 󵄨󵄨Wx Wy e 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩Wxα e−(t/2)L (x, ⋅)󵄩󵄩󵄩L2 (M,Vol) 󵄩󵄩󵄩Wyβ e−(t/2)L (⋅, y)󵄩󵄩󵄩L2 (M,Vol) 1

≲ t −(degds (α)+degds (β))/2κ (Vol(B(W ,ds) (x, t 2κ )) ∧ 1)

− 21

�

593

(8.134)

−1

1

× (Vol(B(W ,ds) (y, t 2κ )) ∧ 1) 2 . 1

1

1

Since t 2κ ≥ ρ(X,d) (x, y)/16, we have B(X,d) (x, t 2κ ) ⊆ B(X,d) (y, 48t 2κ ). Thus, by Theorem 3.3.7 (d) and repeated application of Theorem 3.3.7 (f), we have 1

1

Vol(B(W ,ds) (x, t 2κ )) ∧ 1 ≈ Vol(B(X,d) (x, t 2κ )) ∧ 1 1

1

≤ Vol(B(X,d) (y, 48t 2κ )) ∧ 1 ≈ Vol(B(X,d) (y, t 2κ )) ∧ 1

(8.135)

1 2κ

≈ Vol(B(W ,ds) (y, t )) ∧ 1. Using (8.135) to estimate the right-hand side of (8.134) gives the case s = 0 of (8.116) and completes the proof. Lemma 8.4.16. Fix δ > 0 and x0 ∈ M. Fix f ∈ C0∞ (M \ B(X,d) (x0 , δ); ℂD ). For t ∈ ℝ and x ∈ B(X,d) (x0 , δ), set u(t, x) = {

e−tL f (x),

t ≥ 0,

0,

t < 0.

∞ Then u ∈ Cloc (ℝ × B(X,d) (x0 , δ); ℂD ) and u(t, x) vanishes to infinite order in t as t → 0.

Proof. B(X,d) (x0 , δ) ⊆ M is an open set (see Lemma 3.1.7) and therefore it is a submanifold. Fix y0 ∈ B(X,d) (x0 , δ) and let U ⊆ B(X,d) (x0 , δ) be an open neighborhood of y0 such ∞ that there exists a Cloc diffeomorphism Φ : Bn (2) 󳨀 → U with Φ(0) = y0 . We give Bn (2) the ∞ n density hσLeb := Φ∗ Vol. Note that h ∈ Cloc (B (2)) and h(v) > 0, ∀v ∈ Bn (2). ∞ We will show that u(t, Φ(v)) ∈ Cloc (ℝ × Bn (1/2); ℂD ) and vanishes to infinite order as t → 0. This will complete the proof since y0 ∈ B(X,d) (x0 , δ) was arbitrary and Φ(Bn (1/2)) is an open neighborhood of y0 . For t > 0, we have 𝜕t u(t, x) = −L u(t, x) = −L u(t, x) (see Remark 8.4.3). Thus, 𝜕t u(t, Φ(v)) = −Φ∗ L Φ∗ u(t, Φ(v)), where Φ∗ L Φ∗ is a partial differential operator on Bn (2), acting in the v variable. Since L is maximally subelliptic of type 2 of degree 2κ with respect to (W , ds) on (M, Vol), it follows immediately from the definitions that Φ∗ L Φ∗ is maximally subelliptic of type 2 of degree 2κ with respect to (Φ∗ W , ds) := {(Φ∗ W1 , ds1 ), . . . , (Φ∗ Wr , dsr )} on (Bn (2), Φ∗ Vol). In particular, by restricting our attention to Bn (1), we see that Φ∗ L Φ∗ is maximally subelliptic of type 2 of degree 2κ at the unit scale with respect to (Φ∗ W , ds) and hσLeb . ∼

594 � 8 Linear maximally subelliptic operators Fix M ∈ ℕ. We begin by showing u(t, Φ(v)) ∈ C ∞ ((0, M) × Bn (1/2); ℂD ). Fix ϕ1 , ϕ2 ∈ with ϕ1 ≺ ϕ2 and ϕ1 ≡ 1 on Bn (3/4). Corollary 8.3.8 applied in the v variable implies that ∀s > 0, ∀k ∈ ℕ, ∃N(s) ∈ ℕ, Cs ≥ 0 such that C0∞ (Bn (1))

󵄩󵄩 󵄩 k 󵄩󵄩ϕ1 (⋅)𝜕t u(t, Φ(⋅))󵄩󵄩󵄩L2 ((0,M);L2 (ℝn )) s k 󵄩 󵄩 = 󵄩󵄩󵄩ϕ1 (⋅)(Φ∗ L Φ∗ ) u(t, Φ(⋅))󵄩󵄩󵄩L2 ((0,M);L2 (ℝn )) s N(s)+k

(8.136)

j 󵄩 󵄩 ≤ Cs ∑ 󵄩󵄩󵄩ϕ(⋅)(Φ∗ L Φ∗ ) u(t, Φ(⋅))󵄩󵄩󵄩L2 ((0,M)×ℝn ,dt×dhσ ) . Leb j=k

But, using Remark 8.4.3, j

(Φ∗ L Φ∗ ) u(t, Φ(v)) = (L j u)(t, Φ(v)) = (L j e−tL f )(Φ(v)) j

j

= (L e−tL f )(Φ(v)) = (e−tL L f )(Φ(v)) = (e−tL L j f )(Φ(v)). Thus, changing variables, we have j 󵄩󵄩 󵄩 ∗ 󵄩󵄩ϕ(⋅)(Φ L Φ∗ ) u(t, Φ(⋅))󵄩󵄩󵄩L2 ((0,M)×ℝn ,dt×dhσLeb ) 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩e−tL L j f 󵄩󵄩󵄩L2 ((0,M)×M,dt×d Vol) ≤ 󵄩󵄩󵄩L j f 󵄩󵄩󵄩L2 ((0,M)×M,dt×d Vol) 1 2

M

(8.137)

󵄩 󵄩2 = (∫󵄩󵄩󵄩L j f 󵄩󵄩󵄩L2 (M,Vol) ) < ∞, 0

where we have used L j f ∈ C0∞ (M; ℂD ) ⊆ L2 (M, Vol; ℂD ). Combining (8.136) and (8.137), we see that ∀M, k ∈ ℕ, ∀s > 0, 󵄩󵄩 󵄩 k 󵄩󵄩ϕ1 (⋅)𝜕t u(t, Φ(⋅))󵄩󵄩󵄩L2 ((0,M);L2 (ℝn )) < ∞. s Using the fact that ϕ1 ≡ 1 on Bn (3/4) ⊇ Bn (1/2), taking k and s large, and applying the Sobolev embedding theorem, we see that u(t, Φ(v)) ∈ C ∞ ((0, M) × Bn (1/2); ℂD ). In par∞ ticular, since M was arbitrary, we obtain u(t, Φ(v)) ∈ Cloc ((0, ∞) × Bn (1/2); ℂD ).

Note that for x ∈ B(X,d) (x0 , δ), u(0, x) = e−0L f (x) = f (x) = 0. Therefore, u(t, Φ(v)) ≡ 0 for t ≤ 0. Thus, to complete the proof, it suffices to show that for every k ∈ ℕ, lim 𝜕tk u(t, Φ(⋅)) = 0, t↓0

where the limit is taken in the Fréchet space C ∞ (Bn (1/2)). To prove this, fix k ∈ ℕ and let {tj }j∈ℕ ⊂ (0, 1) be any sequence with tj → 0. We will show that there is a subsequence tjl such that lim (𝜕tk u)(tjl , Φ(⋅)) = 0,

l→∞

8.4 Heat equations

�

595

where the limit is taken in C ∞ (Bn (1/2)); this will complete the proof. Since u(t, Φ(v)) ∈ C ∞ ((0, 1) × Bn (1/2)), {u(tj , Φ(⋅)) : j ∈ ℕ} ⊂ C ∞ (Bn (1/2)) is a bounded set, so there is a subsequence tjl such that lim (𝜕tk u)(tjl , Φ(⋅)) = u0 (⋅),

l→∞

where u0 ∈ C ∞ (Bn (1/2)), with convergence in C ∞ (Bn (1/2)). Thus, to complete the proof we wish to show that u0 = 0. This will follow if we show that lim 𝜕tk u(t, Φ(⋅)) = 0,

(8.138)

t↓0

with convergence in the sense of C0∞ (Bn (1/2); ℂD )′ . Let ϕ ∈ C0∞ (Bn (1/2); ℂD ). We have lim ∫ ϕ(v)𝜕tk u(t, Φ(v))h(v) dv = lim ∫ ϕ(Φ−1 (x))𝜕tk u(t, x) d Vol(x) t↓0

t↓0

= lim ∫ ϕ(Φ−1 (x))𝜕tk (e−tL f )(x) d Vol(x) t↓0

= lim ∫ ϕ(Φ−1 (x))(−1)k (e−tL L k f )(x) d Vol(x) t↓0

= ∫ ϕ(Φ−1 (x))(−1)k (L k f )(x) d Vol(x) = 0, where in the final equality we have used the fact that ϕ(Φ−1 (⋅)) is supported in B(X,d) (x0 , δ) and f is supported in M \ B(X,d) (x0 , δ). This establishes (8.138) and completes the proof. Lemma 8.4.17. There exists c2 = c2 (𝒦) > 0 such that ∀α, ∃C𝒦,α ≥ 0 such that for all δ ∈ (0, 1], x ∈ 𝒦, and t ∈ [0, δ2κ /4), 󵄩󵄩 α −tL 󵄩 (x, ⋅)󵄩󵄩󵄩L2 (M\B 󵄩󵄩Wx e

(X,d) (x,δ),Vol;𝕄

≤ C𝒦,α δ− degds (α) exp(−c2 δ

2κ 2κ−1

t

D×D )

1 − 2κ−1

󵄩 󵄩 , 󵄩󵄩󵄩Wxα e−tL (⋅, x)󵄩󵄩󵄩L2 (M\B

(X,d) (x,δ),Vol;𝕄

D×D )

− 21

)(Vol(B(W ,ds) (x, δ)) ∧ 1) .

󵄨 Proof. Fix δ0 ∈ (0, 1] and x0 ∈ 𝒦. Let RB(X,d) (x0 ,δ0 ) : f 󳨃→ f 󵄨󵄨󵄨B (x ,δ ) denote the restriction (X,d) 0 0 map. Define, for t ≥ 0, Tt,x0 ,δ0 := RB(X,d) (x0 ,δ0 ) e−tL : L2 (M \ B(X,d) (x0 , δ0 ), Vol; ℂD ) → L2 (B(X,d) (x0 , δ0 ); ℂD ). We will apply Proposition 8.4.12 with this choice of Tt,x0 ,δ0 , (M1 , μ1 ) = (M \ B(X,d) (x0 , δ0 ), Vol), σ = δ02κ , D = C0∞ (M \ B(X,d) (x0 , δ0 )), and D0 = D. We begin by verifying the hypotheses of Proposition 8.4.12. Clearly, 󵄩󵄩 󵄩 󵄩󵄩Tt,x0 ,δ0 󵄩󵄩󵄩L2 (M\B (x ,δ ),Vol)→L2 (B (x ,δ ),Vol) ≤ 1. (X,d) 0 0 (X,d) 0 0

596 � 8 Linear maximally subelliptic operators ∞ For f ∈ D , it follows immediately from Lemma 8.4.16 that Tt,x0 ,δ0 f ∈ Cloc ([0, σ) × D B(X,d) (x0 , δ0 ); ℂ ) and vanishes to infinite order as t ↓ 0. Finally, for f ∈ D , using Remark 8.4.3 and the fact that B(X,d) (x0 , δ0 ) ⊆ M is open (Lemma 3.1.7), we have

𝜕t Tt,x0 ,δ0 f = −RB(X,d) (x0 ,δ0 ) L e−tL f = −RB(X,d) (x0 ,δ0 ) L e−tL f = −L RB(X,d) (x0 ,δ0 ) e−tL f = −L Tt,x0 ,δ0 f = −σ −1 δ02κ L Tt,x0 ,δ0 . Thus, all of the hypotheses of Proposition 8.4.12 are satisfied, so the proposition implies that there exists c2 = c2 (𝒦) > 0 such that the following holds: sup

x∈Φx0 ,δ0

=

(Bn (1/4))

α −tL 󵄩󵄩 ds 󵄩 (x, ⋅)󵄩󵄩󵄩L2 (M\B (x ,δ ),Vol) 󵄩󵄩(δ0 Wx ) e (X,d) 0 0

sup

x∈Φx0 ,δ0 (Bn (1/4))

α 󵄩󵄩 ds 󵄩 󵄩󵄩(δ0 Wx ) Tt,x0 ,δ0 (x, ⋅)󵄩󵄩󵄩L2 (M\B (x ,δ ),Vol) (X,d) 0 0 − 21

≲ (Vol(B(W ,ds) (x0 , δ0 )) ∧ 1)

2κ

1

exp(−c2 δ02κ−1 t − 2κ−1 ).

Setting x = Φx0 ,δ0 (0) = x0 in the above equation and using the fact that (δ0dsW )α = deg (α) 󵄩 󵄩 δ0 ds W α proves the desired bound for 󵄩󵄩󵄩Wxα e−tL (x, ⋅)󵄩󵄩󵄩L2 (M\B (x,δ),Vol;𝕄D×D ) (with x (X,d) replaced by x0 and δ replaced by δ0 ). Since e−tL is self-adjoint, the desired bound for 󵄩󵄩 α −tL 󵄩 (⋅, x)󵄩󵄩󵄩L2 (M\B (x,δ),Vol;𝕄D×D ) also follows. 󵄩󵄩Wx e (X,d) Lemma 8.4.18. Suppose a, b, c > 0 are given. Then 1

δ a (Vol(B(W ,ds) (x, δ)) ∧ 1) 2 δ b sup sup ( ′ ) exp (−c ( ) ) < ∞. 1 δ′ x∈𝒦 δ′ ,δ∈(0,∞) δ (Vol(B(W ,ds) (x, δ′ )) ∧ 1) 2 δ′ ≤δ

Proof. It follows from repeated application of Theorem 3.3.7 (f) that there exists p = p(𝒦) > 0 and C𝒦 ≥ 0 such that ∀x ∈ 𝒦, δ, δ′ ∈ (0, ∞) with δ′ ≤ δ, Vol(B(W ,ds) (x, δ)) ∧ 1 = Vol(B(W ,ds) (x, (δ/δ′ )δ′ )) ∧ 1

≤ C𝒦 (δ/δ′ )p (Vol(B(W ,ds) (x, δ)) ∧ 1).

Thus, we have 1

δ a (Vol(B(W ,ds) (x, δ)) ∧ 1) 2 δ b ( ′) exp (−c ( ) ) 1 δ δ′ (Vol(B(W ,ds) (x, δ′ )) ∧ 1) 2 1

≤ C𝒦2 ( From here, the result follows.

δ a+p/2 δ b ) exp (−c ( ) ). δ′ δ′

8.4 Heat equations

�

597

s

Proof of (8.117). Because 𝜕ts e−tL = (−1)s L e−tL = (−1)s L e−tL , where L is taken in the sense of distributions (see Remark 8.4.3), and due to the form of L , it suffices to prove (8.117) in the case s = 0. 1 Fix x, y ∈ 𝒦 and t > 0 with 16t 2κ ≤ ρ(X,d) (x, y) ≤ 1. Set δ := ρ(X,d) (x, y)/3. Note that t < δ2κ /8 and B(X,d) (x, δ) ∩ B(X,d) (y, δ) = 0. Using (8.132), we have 󵄨󵄨 α β −tL 󵄨 󵄨󵄨 󵄨 (x, y)󵄨󵄨󵄨 = 󵄨󵄨󵄨∫(Wxα e−(t/2)L (x, z)) (Wyβ e−(t/2)L (z, y)) d Vol(z)󵄨󵄨󵄨 󵄨󵄨Wx Wy e 󵄨 󵄨󵄨 α −(t/2)L 󵄨 ≤ (x, z))(Wyβ e−(t/2)L (z, y))󵄨󵄨󵄨 d Vol(z) ∫ 󵄨󵄨(Wx e z∈M\B(X,d) (x,δ)

+

󵄨󵄨 α −(t/2)L 󵄨 (x, z))(Wyβ e−(t/2)L (z, y))󵄨󵄨󵄨 d Vol(z) 󵄨󵄨(Wx e

∫ z∈M\B(X,d) (y,δ)

=: (I) + (II). We estimate only (I); the estimate for (II) follows by a similar proof by reversing the roles of x and y. In the next equation we use t < δ2κ /4 < 1. We also freely use Theorem 3.3.7 (f). Applying the Cauchy–Schwartz inequality and Lemmas 8.4.17 and 8.4.14, we have, with c2 > 0 as in Lemmas 8.4.17 and c3 := c2 /9 > 0, 󵄩 󵄩 (I) ≤ 󵄩󵄩󵄩Wxα e−(t/2)L (x, ⋅)󵄩󵄩󵄩L2 (M\B

(X,d) (x,δ))

2κ

󵄩󵄩 β −(t/2)L 󵄩 (⋅, y)󵄩󵄩󵄩L2 (M,Vol) 󵄩󵄩Wy e − 21

1

≲ δ− degds (α) exp(−c2 δ 2κ−1 (t/2)− 2κ−1 )(Vol(B(W ,ds) (x, δ)) ∧ 1) × (t 1/2κ )

− degds (β)

≈ ρ(X,d) (x, y)

1

(Vol(B(W ,ds) (y, t 2κ )) ∧ 1)

− degds (α) − degds (β)/2κ

t

− 21

1

ρ(X,d) (x, y)2κ 2κ−1 exp(−c3 ( ) ) t −1

1

× (Vol(B(W ,ds) (x, ρ(X,d) (x, y))) ∧ 1) 2 (Vol(B(X,d) (y, t 2κ )) ∧ 1) = ρ(X,d) (x, y)− degds (α)−degds (β) (Vol(B(W ,ds) (x, ρ(X,d) (x, y))) ∧ 1) × (Vol(B(W ,ds) (y, ρ(X,d) (x, y))) ∧ 1)

1

(Vol(B(W ,ds) (y, ρ(X,d) (x, y))) ∧ 1) 2 1 2κ

(Vol(B(W ,ds) (y, t )) ∧ 1)

exp(−(c3 /2) (

1 2

ρ(X,d) (x, y)2κ ) t

− 21 1

ρ(X,d) (x, y)2κ 2κ−1 exp(−(c3 /2) ( ) ) t

− 21

×{

− 21

(

ρ(X,d) (x, y) t

1 2κ

degds (β)

)

1 2κ−1

)}.

By Lemma 8.4.18, the term inside {⋅} is ≲ 1. Thus, we have, with c1 = c3 /2,

598 � 8 Linear maximally subelliptic operators (I) ≲ ρ(X,d) (x, y)− degds (α)−degds (β) (Vol(B(W ,ds) (x, ρ(X,d) (x, y))) ∧ 1) − 21

× (Vol(B(W ,ds) (y, ρ(X,d) (x, y))) ∧ 1)

− 21 1

ρ(X,d) (x, y)2κ 2κ−1 exp(−c1 ( ) ). t

(8.139)

Finally, using Theorem 3.3.7 (d) and (f), we have Vol(B(W ,ds) (x, ρ(X,d) (x, y))) ∧ 1 ≈ Vol(B(X,d) (x, ρ(X,d) (x, y))) ∧ 1

≤ Vol(B(X,d) (y, 3ρ(X,d) (x, y))) ∧ 1 ≲ Vol(B(X,d) (y, ρ(X,d) (x, y))) ∧ 1

(8.140)

≈ Vol(B(W ,ds) (y, ρ(X,d) (x, y))) ∧ 1.

Using (8.140) to bound the right-hand side of (8.139) gives the desired bound in the case s = 0 of (8.117) and completes the proof.

8.5 Proof of the main result In this section, we use the theory developed up to this point to complete the proof of Theorem 8.1.1. Proof of Theorem 8.1.1 except for (ii) and (iv). Here we prove the equivalence of all parts of Theorem 8.1.1 except for (ii) and (iv). We do this by proving the following implications: (i)

(viii)

(ix)

(x)

(xi)

(xii) (vi)

(vii)

(viii)

(vii) (i)

(v) (iii)

(i) (i).

(i) ⇒ (ix): It follows immediately from the definitions that if P is maximally subelliptic of degree κ with respect to (W , ds) on M (see Definition 8.0.1), then L := P ∗ P is maximally subelliptic of type 2 of degree 2κ with respect to (W , ds) on (M, Vol) (see Definition 8.4.1), where the formal adjoint P ∗ is taken in the sense of L2 (M, Vol). Clearly, L is symmetric and non-negative when treated as a densely defined operator on L2 (M, Vol; ℂD2 ) with dense domain C0∞ (M; ℂD2 ). Item (ix) now follows from Theorem 8.4.2. We make one comment. In the estimate in Theorem 8.4.2, it is written 1 that (ρ(W ,ds) (x, y) + t 2κ ) ∧ 1, while in (ix) there is no ∧1. Because (ix) is restricted to t ∈ (0, 1] and x, y ∈ 𝒦 and 𝒦 is compact with respect to ρ(W ,ds) (Lemma 3.1.7), we have 1

ρ(W ,ds) (x, y) + t 2κ ≲ 1. Thus, one may remove this ∧1 in the statement of Theorem 8.4.2 in the case we are considering. Item (ix) follows. (ix) ⇒ (x) and (xi) ⇒ (xii): Since P ∗ P is a symmetric, non-negative operator, it has at least one non-negative, self-adjoint extension (for example, the Friedrichs extension). Thus, both (ix) ⇒ (x) and (xi) ⇒ (xii) are immediate. (ix) ⇒ (xi) and (x) ⇒ (xii): This follows immediately from the definitions.

8.5 Proof of the main result

� 599

(xii) ⇒ (viii): This follows from Theorem 5.11.7. −2κ (viii) ⇒ (vii): Let S ∈ Aloc ((W , ds); ℂD2 , ℂD2 ) be the operator from (viii) and set T := ∗ ∞ S L . Clearly, T P ≡ I mod Cloc (M × M; 𝕄D2 ×D2 (ℂ)). It follows from Corollary 5.8.10 −κ that T ∈ Aloc ((W , ds); ℂD2 , ℂD1 ), which establishes (vii). −κ (vii) ⇒ (v): Let T ∈ Aloc ((W , ds); ℂD2 , ℂD1 ) be the operator from (vii). Let 𝒦 := supp(ϕ2 ) and fix Ψ1 , Ψ2 , Ψ3 ∈ C0∞ (M) with Ψ1 ≺ Ψ2 ≺ Ψ3 and Ψ1 ≡ 1 on 𝒦. Fix Ω1 ⋐ M open and relatively compact with Ψ3 ∈ C0∞ (Ω1 ) and set 𝒦1 := Ω1 ⋐ M. Formulas (8.5) and (8.6) remain unchanged if u is replaced by Ψ1 u; we henceforth make this replacement, so that we may assume Ψ2 u = u. ∞ Let R1 , R2 denote elements of Cloc (M × M; 𝕄D2 ×D2 ). Using the fact that T P ≡ I ∞ mod Cloc (M × M; 𝕄D2 ×D2 ) and the pseudo-locality of T (see, for example, the growth condition in Definition 5.2.2), we have ϕ1 u = ϕ1 Ψ2 u = ϕ1 T P Ψ2 u + ϕ1 R1 Ψ2 u = ϕ1 Tϕ3 P Ψ2 u + ϕ1 R2 Ψ2 u = ϕ1 Tϕ3 P u + ϕ1 R2 Ψ2 u = ϕ1 TΨ2 ϕ3 P u + ϕ1 R2 Ψ2 u.

(8.141)

s We now prove (8.5). If ϕ3 P u ∈ Xcpt (W , ds), then by Remark 6.5.8 (since supp(ϕ2 ) ⊆ s 𝒦1 ), we have ϕ3 P u ∈ X (𝒦1 , (W , ds)). By assumption, we have Mult[ϕ1 ]T Mult[Ψ2 ] ∈ A −κ (Ω1 , (W , ds)), and therefore by Theorem 6.2.10, we have

ϕ1 TΨ2 ϕ3 P u = (Mult[ϕ1 ]T Mult[Ψ2 ])ϕ3 P u ∈ X s+κ1 (𝒦1 , (W , ds)).

(8.142)

For any distribution u ∈ C0∞ (M; ℂD2 )′ , we have ϕ1 R2 Ψ2 u ∈ C0∞ (M) and supp(ϕ1 R2 Ψ2 u) ⊆ 𝒦1 . Thus, by Proposition 6.5.5 (see also Remark 5.2.24), we have ϕ1 R2 Ψ2 u ∈ X s+κ (𝒦1 , (W , ds)).

(8.143)

Combining (8.141), (8.142), and (8.143), we conclude that ϕ1 u ∈ X s+κ (𝒦1 , (W , ds)) ⊆ s+κ Xcpt (W , ds), completing the proof of (8.5). We turn to (8.6). Let ψ ∈ C0∞ (M) satisfy ϕ2 ≺ ψ ≺ ϕ3 . We begin by assuming u ∈ X s (𝒦1 , (W , ds)) – we will later replace u with ϕ2 u. Applying (8.141) with ϕ3 replaced by ψ we have (with a different choice of R2 ) 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩ϕ1 u󵄩󵄩X s+κ (W ,ds) ≤ 󵄩󵄩󵄩ϕ1 TΨ2 ψP u󵄩󵄩󵄩X s+κ (W ,ds) + 󵄩󵄩󵄩ϕ1 R2 Ψ2 u󵄩󵄩󵄩X s+κ (W ,ds) 󵄩 󵄩 = 󵄩󵄩󵄩(Mult[ϕ1 ]T Mult[Ψ2 ])(ψP u)󵄩󵄩󵄩X s+κ (W ,ds) 󵄩 󵄩 + 󵄩󵄩󵄩(Mult[ϕ1 ]R2 Mult[Ψ3 ])Ψ2 u󵄩󵄩󵄩X s+κ (W ,ds) .

(8.144)

By assumption, Mult[ϕ1 ]T Mult[Ψ2 ] ∈ A −κ (Ω1 , (W , ds)). Since Mult[ϕ1 ]R2 Mult[Ψ3 ] ∈ C0∞ (Ω1 × Ω1 ), Proposition 5.8.11 shows that Mult[ϕ1 ]R2 Mult[Ψ3 ] ∈ A −κ−N (Ω1 , (W , ds)). Applying Theorem 6.2.10 to (8.144), we conclude that 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩ϕ1 u󵄩󵄩X s+κ (W ,ds) ≲ 󵄩󵄩󵄩ψP u󵄩󵄩󵄩X s (W ,ds) + 󵄩󵄩󵄩Ψ2 u󵄩󵄩󵄩X s−N (W ,ds) .

(8.145)

600 � 8 Linear maximally subelliptic operators s Now suppose u ∈ C0∞ (M; ℂD2 )′ with ϕ3 P u ∈ Xcpt (W , ds); by (8.5) with ϕ1 replaced s+κ s by ϕ2 , we have ϕ2 u ∈ Xcpt (W , ds) ⊆ Xcpt (W , ds). Applying (8.145) to ϕ2 u and using ϕ1 ≺ ϕ2 ≺ ψ ≺ ϕ3 ≺ Ψ2 , we have

󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩ϕ1 u󵄩󵄩X s+κ (W ,ds) ≲ 󵄩󵄩󵄩ψP u󵄩󵄩󵄩X s (W ,ds) + 󵄩󵄩󵄩ϕ2 u󵄩󵄩󵄩X s−N (W ,ds) 󵄩 󵄩 󵄩 󵄩 = 󵄩󵄩󵄩ψϕ3 P u󵄩󵄩󵄩X s (W ,ds) + 󵄩󵄩󵄩ϕ2 u󵄩󵄩󵄩X s−N (W ,ds) 󵄩 󵄩 󵄩 󵄩 ≲ 󵄩󵄩󵄩ϕ3 P u󵄩󵄩󵄩X s (W ,ds) + 󵄩󵄩󵄩ϕ2 u󵄩󵄩󵄩X s−N (W ,ds) , where the final estimate used Corollary 6.5.10. This completes the proof of (v). (v) ⇒ (i): Fix a relatively compact, open set Ω ⋐ M and ϕ1 , ϕ2 , ϕ3 ∈ C0∞ (M) with ϕ1 ≺ ϕ2 ≺ ϕ3 and ϕ1 ≡ 1 on Ω. Let 𝒦 := supp(ϕ2 ). By Proposition 6.5.5 (see also Remark 5.2.24), s we have C0∞ (Ω) ⊆ F2,2 (𝒦, (W , ds)), ∀s ∈ ℝ. Applying (8.6) with s = N = 0 we see that ∞ ∀u ∈ C0 (Ω), ‖u‖F2,2κ (W ,ds) = ‖ϕ1 u‖F2,2κ (W ,ds) ≲ ‖ϕ3 P u‖F 0 (W ,ds) + ‖ϕ2 u‖F 0 (W ,ds) 2,2 2,2 󵄩 󵄩 󵄩 󵄩 = ‖P u‖F 0 (W ,ds) + ‖u‖F 0 (W ,ds) ≈ 󵄩󵄩󵄩P u󵄩󵄩󵄩L2 (M,Vol) + 󵄩󵄩󵄩u󵄩󵄩󵄩L2 (M,Vol) , 2,2 2,2

(8.146)

where the last ≈ step uses Proposition 6.2.13. By Corollary 6.2.14 (in particular, (6.5)), we have r

󵄩 n 󵄩 ∑󵄩󵄩󵄩Wj j u󵄩󵄩󵄩L2 (M,Vol) ≲ ‖u‖F2,2κ (W ,ds) . j=1

(8.147)

Combining (8.146) and (8.147) establishes (i). We have so far established the equivalence (i) ⇔ (ix) ⇔ (x) ⇔ (xi) ⇔ (xii) ⇔ (viii) ⇔ (vii) ⇔ (v). (viii) ⇒ (vi): This follows from the already proved implication (vii) ⇒ (i) applied to the operator P ∗ P . (vi) ⇒ (vii): Suppose (vi) holds. Using the already proved implication (i) ⇒ (vii) ap−2κ plied to the operator P ∗ P shows that there is S ∈ Aloc ((W , ds); ℂD2 , ℂD2 ) such that ∗ ∞ D2 ×D2 ∗ ∞ S P P ≡ I mod Cloc (M×M; 𝕄 (ℂ)). Set T := S L . Clearly, T P ≡ I mod Cloc (M× D2 ×D2 −κ D2 D1 M; 𝕄 (ℂ)). It follows from Corollary 5.8.10 that T ∈ Aloc ((W , ds); ℂ , ℂ ), which establishes (vii). (i) ⇒ (iii): This is obvious. (iii) ⇒ (i): Suppose (iii) holds and fix a relatively compact, open set Ω ⋐ M. For each x ∈ M, by hypothesis, there is an open neighborhood Ux of x such that P is maximally subelliptic of degree κ with respect to (W , ds) on Ux . The collection {Ux : x ∈ M} is an open cover for the compact set Ω; let Ux1 , . . . , UxL be a finite subcover. For j = 1, . . . , L, pick ϕ1,j ∈ C0∞ (Uxj ) such that ∑Lj=1 ϕ1,j ≡ 1 on Ω. Let ϕ2,j ∈ C0∞ (Uxj ) be such that ϕ1,j ≺ ϕ2,j . Let f ∈ C0∞ (Ω; ℂD2 ). We use the already proved implication (i) ⇒ (v) with M replaced by Uxj and Proposition 6.5.5 (see also Remark 5.2.24) to see that

8.5 Proof of the main result

� 601

‖ϕ1,j f ‖F2,2κ (W ,ds) ≲ ‖ϕ2,j P f ‖F 0 (W ,ds) + ‖ϕ2,j f ‖F 0 (W ,ds) 2,2 2,2 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≈ 󵄩󵄩ϕ2,j P f 󵄩󵄩L2 (M,Vol) + 󵄩󵄩ϕ2,j f 󵄩󵄩L2 (M,Vol) ≲ 󵄩󵄩󵄩P f 󵄩󵄩󵄩L2 (M,Vol) + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (M,Vol) , where use have also used Proposition 6.2.13. Thus, using Corollary 6.2.14 (in particular, (6.5)), we have r

L

k=1

j=1

󵄩 n 󵄩 ∑ 󵄩󵄩󵄩Wk k f 󵄩󵄩󵄩L2 ≲ ‖f ‖F2,2κ (W ,ds) ≤ ∑ ‖ϕ1,j f ‖F2,2κ (W ,ds) ≲ ‖P f ‖L2 + ‖f ‖L2 .

This establishes (i). To complete the proof of Theorem 8.1.1 it remains to show that (ii) and (iv) are equivalent to the other conditions. Lemma 8.5.1. Suppose we have two partial differential operators of the form (8.1): P1 :=

∑

degds (α)≤κ

bα (x)W α ,

P2 :=

∑

degds (α)≤κ

cα (x)W α ,

∞ where bα , cα ∈ Cloc (M; 𝕄D1 ×D2 (ℂ)). Suppose bα = cα when degds(α) = κ. Then P1 is maximally subelliptic of degree κ with respect to (W , ds) on M if and only if P2 is maximally subelliptic of degree κ with respect to (W , ds) on M.

Proof. By symmetry, we only need to show that if P1 is maximally subelliptic of degree κ with respect to (W , ds) on M, then P2 is maximally subelliptic of degree κ with respect to (W , ds) on M. Thus, we suppose P1 is maximally subelliptic of degree κ with respect to (W , ds) on M. Fix a relatively compact open set Ω ⋐ M and let ϕ1 , ϕ2 ∈ C0∞ (M) satisfy ϕ1 ≺ ϕ2 and ϕ1 ≡ 1 on Ω. In what follows, f will denote an element of C0∞ (Ω), and we will freely use s the fact that C0∞ (Ω) ⊆ F2,2 (Ω, (W , ds)), ∀s ∈ ℝ (see Proposition 6.5.5 and Remark 5.2.24). For any s ∈ ℝ, using the already proved implication Theorem 8.1.1 (i) ⇒ (v), we have, ∀f ∈ C0∞ (Ω), ‖f ‖F2,2s+κ (W ,ds) = ‖ϕ1 f ‖F2,2s+κ (W ,ds) ≲ ‖ϕ2 P1 f ‖F2,2s (W ,ds) + ‖ϕ2 f ‖F2,2s (W ,ds) 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩ϕ2 P2 f 󵄩󵄩󵄩F s (W ,ds) + 󵄩󵄩󵄩ϕ2 (P1 − P2 )f 󵄩󵄩󵄩F s (W ,ds) + 󵄩󵄩󵄩ϕ2 f 󵄩󵄩󵄩F s (W ,ds) 2,2 2,2 2,2 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ≲ 󵄩󵄩P2 f 󵄩󵄩F s (W ,ds) + 󵄩󵄩f 󵄩󵄩F s+κ−1 (W ,ds) , 2,2 2,2

(8.148)

where the last estimate used Proposition 6.5.9 and Corollary 6.5.10. We claim, for j ∈ {0, . . . , κ}, that ‖f ‖F2,2κ (W ,ds) ≲ ‖P2 f ‖F 0 (W ,ds) + ‖f ‖F κ−j (W ,ds) . 2,2

2,2

(8.149)

602 � 8 Linear maximally subelliptic operators We prove (8.149) by induction on j. The base case, j = 0, is trivial. Suppose we have (8.149) for some j ∈ {0, . . . , κ − 1}. We wish to prove it for j + 1. Applying the inductive hypothesis and (8.148) with s = −j, we have ‖f ‖F2,2κ (W ,ds) ≲ ‖P2 f ‖F 0 (W ,ds) + ‖f ‖F κ−j (W ,ds) 2,2

2,2

≲ ‖P2 f ‖F 0 (W ,ds) + ‖P2 f ‖F −j (W ,ds) + ‖f ‖F κ−j−1 (W ,ds) 2,2

2,2

2,2

≲ ‖P2 f ‖F 0 (W ,ds) + ‖f ‖F κ−j−1 (W ,ds) , 2,2

2,2

completing the proof of (8.149). Applying (8.149) and using Corollary 6.2.14 (in particular, (6.5)) and Proposition 6.2.13, we have r

󵄩 n 󵄩 ∑ 󵄩󵄩󵄩Wk k f 󵄩󵄩󵄩L2 ≲ ‖f ‖F2,2κ (W ,ds) ≲ ‖P2 f ‖F 0 + ‖f ‖F 0 ≈ ‖P2 f ‖L2 + ‖f ‖L2 . 2,2

k=1

2,2

We conclude that P2 is maximally subelliptic of degree κ with respect to (W , ds) on M, completing the proof. Proof the equivalence of Theorem 8.1.1 (ii). (i) ⇔ (ii): This is a special case of Lemma 8.5.1. Lemma 8.5.2. Suppose we have two partial differential operators of the form (8.1): P1 :=

∑

degds (α)≤κ

bα (x)W α ,

P2 :=

∑

degds (α)≤κ

cα (x)W α ,

∞ where bα , cα ∈ Cloc (M; 𝕄D1 ×D2 (ℂ)). Fix x0 ∈ M and suppose bα (x0 ) = cα (x0 ), ∀α. Then the following are equivalent: (a) There exists a neighborhood U of x0 such that P1 is maximally subelliptic of degree κ with respect to (W , ds) on U. (b) There exists a neighborhood U of x0 such that P2 is maximally subelliptic of degree κ with respect to (W , ds) on U.

Proof. By symmetry, it suffices to prove (a) ⇒ (b). Suppose P1 is maximally subelliptic of degree κ with respect to (W , ds) on U, where U is a neighborhood of x0 . Let U1 ⋐ U be a neighborhood of x0 and let ϕ1 , ϕ2 ∈ C0∞ (U), with ϕ1 ≺ ϕ2 and ϕ1 ≡ 1 on U1 . s We will freely use the fact that C0∞ (U) ⊆ F2,2 (U, (W , ds)), ∀s ∈ ℝ (see Proposition 6.5.5 and Remark 5.2.24). Using Corollary 6.2.14 (in particular, (6.5)) and the already proved implication Theorem 8.1.1 (i) ⇒ (v), we have, for f ∈ C0∞ (U1 ; ℂD2 ),

8.5 Proof of the main result

∑

degds (α)≤κ

�

603

󵄩󵄩 α 󵄩󵄩 󵄩󵄩W f 󵄩󵄩L2 ≈ ‖f ‖F2,2κ (W ,ds) = ‖ϕ1 f ‖F2,2κ (W ,ds) ≲ ‖ϕ2 P1 f ‖F 0 (W ,ds) + ‖ϕ2 f ‖F 0 (W ,ds) = ‖P1 f ‖F 0 (W ,ds) + ‖f ‖F 0 (W ,ds) 2,2

2,2

2,2

2,2

≈ ‖P1 f ‖L2 + ‖f ‖L2 . Thus, there exists C > 0 with ∑

degds (α)≤κ

󵄩󵄩 α 󵄩󵄩 󵄩󵄩W f 󵄩󵄩L2 ≤ C(‖P1 f ‖L2 + ‖f ‖L2 ).

Take V ⊆ U1 a neighborhood of x0 so small that 1 󵄩 󵄩 sup max 󵄩󵄩󵄩bα(x) − cα (x)󵄩󵄩󵄩𝕄D1 ×D2 ≤ . 2C x∈V degds (α)≤κ It follows that for f ∈ C0∞ (V ; ℂD2 ), 1 󵄩󵄩 α 󵄩󵄩 󵄩 󵄩 ∑ 󵄩󵄩󵄩W α f 󵄩󵄩󵄩L2 + C(‖P2 f ‖L2 + ‖f ‖L2 ). 󵄩󵄩W f 󵄩󵄩L2 ≤ 2 (α)≤κ deg (α)≤κ

∑

degds

ds

We conclude that ∀f ∈

C0∞ (V ; ℂD2 ),

r

󵄩 n 󵄩 ∑󵄩󵄩󵄩Wj j f 󵄩󵄩󵄩L2 ≤ j=1

∑

degds (α)≤κ

󵄩󵄩 α 󵄩󵄩 󵄩󵄩W f 󵄩󵄩L2 ≤ 2C(‖P2 f ‖L2 + ‖f ‖L2 ).

We have shown that P2 is maximally subelliptic of degree κ with respect to (W , ds) on V , completing the proof. Proof the equivalence of Theorem 8.1.1 (iv). (iii) ⇔ (iv) This is a special case of Lemma 8.5.2. Remark 8.5.3. The exact relationship between maximal subellipticity and maximal subellipticity of type 2 is not immediately clear. It is obvious that P is maximally subelliptic of degree κ with respect to (W , ds) on M if and only if P ∗ P is maximally subelliptic of type 2 of degree 2κ with respect to (W , ds) on (M, Vol), where the formal adjoint P ∗ is taken with respect to L2 (M, Vol). It seems likely that the methods of [2] can be used to show that every operator which is maximally subelliptic of type 2 is maximally subelliptic; however, we do not pursue this here. We content ourselves with the next result, which follows easily from the results we have already shown. Corollary 8.5.4. Suppose L is a partial differential operator of the form (8.102) and satisfies: – L is maximally subelliptic of type 2 of degree 2κ with respect to (W , ds) on (M, Vol). – L is symmetric and non-negative, when thought of as a densely defined operator on L2 (M, Vol; ℂD ) with dense domain C0∞ (M; ℂD ). Then L is maximally subelliptic of degree 2κ with respect to (W , ds) on M.

604 � 8 Linear maximally subelliptic operators Proof. Let L be any non-negative, self-adjoint extension of L ; for example, we may take L to be the Friedrichs extension. Fix a compact set 𝒦 ⋐ M. Since 𝒦 is compact with respect to ρ(W ,ds) (see Lemma 3.1.7), we have ρ(W ,ds) (x, y) ≲ 1, ∀x, y ∈ 𝒦. In particular, for x, y ∈ 𝒦 and t ∈ (0, 1], we have 1

1

ρ(W ,ds) (x, y) + t 2κ ≈ (ρ(W ,ds) (x, y) + t 2κ ) ∧ 1.

(8.150)

∞ By Theorem 8.4.2 and (8.150), we have e−tL (x, y) ∈ Cloc (M × M; 𝕄D×D (ℂ)) and there exists c > 0 such that for all ordered multi-indices α, β, there exists Cα,β such that ∀x, y ∈ 𝒦 and t ∈ (0, 1], 1 − deg (α)−deg (β) 󵄨󵄨 α β −tL 󵄨 ds ds (x, y)󵄨󵄨󵄨 ≤ Cα,β (ρ(W ,ds) (x, y) + t 2κ ) 󵄨󵄨Wx Wy e 1

ρ(W ,ds) (x, y)2κ 2κ−1 × exp(−c ( ) ) t 1

× (Vol(B(W ,ds) (x, ρ(W ,ds) (x, y) + t 2κ )) ∧ 1) . −1

It follows that {(e−tL , t 1/2κ ) : t ∈ (0, 1]} is a bounded set of locally (W , ds) pre-elementary −2κ operators. By Theorem 5.11.7, there exists S ∈ Aloc ((W , ds); ℂD , ℂD ) such that L S, S L ≡ I

∞ mod Cloc (M × M; 𝕄D×D (ℂ)).

Theorem 8.1.1 (vii) ⇒ (i) now implies that L is maximally subelliptic of degree 2κ with respect to (W , ds) on M.

8.6 Vector bundles Theorem 8.1.1 (along most other results in this text) is stated for trivial vector bundles over M. However, there is an analogous result for arbitrary vector bundles over M. The idea is that while Theorem 8.1.1 is not (strictly speaking) a local theorem, the main methods to prove it are local. Thus, a similar result holds for arbitrary vector bundles with the same proof. In this section, we formulate this version on vector bundles. Elliptic operators on vector bundles have many applications to geometry; we state the below results for maximally subelliptic operators on vector bundles in hopes that they may find similar applications. Let M be a connected C ∞ manifold of dimension n ≥ 1 and let (W , ds) = {(W1 , ds1 ), . . . , ∞ (Wr , dsr )} ⊂ Cloc (M; TM) × ℕ+ be Hörmander vector fields with formal degrees on M. Fix κ ∈ ℕ+ such that dsj divides κ for j = 1, . . . , r and set nj := κ/dsj ∈ ℕ+ . Let π1 : 𝒱1 → M and π2 : 𝒱2 → M be smooth, complex vector bundles over M of finite ranks D1 and D2 , respectively.

8.6 Vector bundles

�

605

Fix a smooth connection ∇ on π2 : 𝒱2 → M and smooth inner products ⟨⋅, ⋅⟩1,x and ⟨⋅, ⋅⟩2,x on 𝒱1 and 𝒱2 , respectively. Most of the results and definitions which follow do not depend on the choice of connection or the choices of inner products (see Remarks 8.6.1, 8.6.3, 8.6.5, and 8.6.6). For an ordered multi-index α = (α1 , α2 , . . . , αL ), we write α ∇W = ∇Wα ∇Wα ⋅ ⋅ ⋅ ∇Wα . 1

L

2

Since 𝒱1 and 𝒱2 have inner products, and therefore norms, it makes sense to consider Lp (M, Vol; 𝒱1 ) and Lp (M, Vol; 𝒱2 ): the Banach spaces of Lp sections of 𝒱1 and 𝒱2 . We write Hom(𝒱2 , 𝒱1 ) for the smooth vector bundle over M whose fiber at x ∈ M is given by Hom(𝒱2 , 𝒱1 )x = Hom(𝒱2,x , 𝒱1,x ). We write Hom(𝒱2 , 𝒱1 ) for the smooth vector bundle over M × M whose fiber at (x, y) ∈ M × M is given by Hom(𝒱2 , 𝒱1 )(x,y) = Hom(𝒱2,y , 𝒱1,x ). We consider partial differential operators of the form P=

∑

degds (α)≤κ

α aα ∇W ,

∞ aα ∈ Cloc (M; Hom(𝒱2 , 𝒱1 )).

(8.151)

Remark 8.6.1. While (8.151) uses the connection, ∇, the class of operators covered by (8.151) does not depend on ∇. Indeed, it is straightforward to verify that P is of the form (8.151) if and only if the following holds. For every x ∈ M, let U ⊆ M be an open ∼ ∼ neighborhood of x with local trivializations ϕ1 : π1−1 (U) 󳨀 → U × ℂD1 , ϕ2 : π2−1 (U) 󳨀 → ϕ ,ϕ ϕ ,ϕ ∞ U × ℂD2 . We assume (ϕ1 )∗ P ϕ∗2 = ∑degds (α)≤κ aα 1 2 W α , where aα 1 2 ∈ Cloc (U; 𝕄D1 ×D2 (ℂ)). Definition 8.6.2. We say P given by (8.151) is maximally subelliptic of degree κ with respect to (W , ds) on M if the following holds. For every relatively compact, open set Ω ⋐ M, there exists CΩ ≥ 0 such that ∀f ∈ C0∞ (Ω; 𝒱2 ), r

󵄩 n 󵄩 󵄩 󵄩 󵄩 󵄩 ∑󵄩󵄩󵄩∇Wj f 󵄩󵄩󵄩L2 (M,Vol;𝒱 ) ≤ CΩ (󵄩󵄩󵄩P f 󵄩󵄩󵄩L2 (M,Vol;𝒱 ) + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (M,Vol;𝒱 ) ). j=1

j

2

1

2

(8.152)

Remark 8.6.3. For a fixed operator P , whether P is maximally subelliptic does not depend on the choice of connection, ∇, even though the expression (8.152) involves the connection. To see this, fix a relatively compact, open set Ω ⋐ M; we consider f ∈ C0∞ (M, Vol; 𝒱2 ). A simple integration by parts shows that for l ∈ ℕ+ , 󵄩󵄩 l 󵄩󵄩2 󵄩 l+1 󵄩2 󵄩 l−1 󵄩2 󵄩󵄩∇Wj f 󵄩󵄩L2 (M,Vol;𝒱2 ) ≤ (l.c.)󵄩󵄩󵄩∇Wj f 󵄩󵄩󵄩L2 (M,Vol;𝒱2 ) + (s.c.)󵄩󵄩󵄩∇Wj f 󵄩󵄩󵄩L2 (M,Vol;𝒱2 ) . Using this, a straightforward induction argument shows that nj

󵄩󵄩 nj 󵄩󵄩 󵄩 󵄩 󵄩 l 󵄩 󵄩󵄩∇Wj f 󵄩󵄩L2 (M,Vol;𝒱2 ) + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (M,Vol;𝒱2 ) ≈ ∑󵄩󵄩󵄩∇Wj f 󵄩󵄩󵄩L2 (M,Vol;𝒱2 ) , l=0

and therefore (8.152) is equivalent to

606 � 8 Linear maximally subelliptic operators r

nj

󵄩 l 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 f󵄩 2 ≲ 󵄩󵄩󵄩P f 󵄩󵄩󵄩L2 (M,Vol;𝒱 ) + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (M,Vol;𝒱 ) . ∑ ∑󵄩󵄩󵄩∇W j 󵄩L (M,Vol;𝒱2 ) 1 2 j=1 l=0

(8.153)

̃ is another smooth connection on π2 : 𝒱2 → M, A simple computation shows that if ∇ then r

nj

r

nj

󵄩 l 󵄩󵄩 󵄩 ̃ l 󵄩󵄩 f󵄩 2 ≈ ∑ ∑󵄩󵄩󵄩∇ ∑ ∑󵄩󵄩󵄩∇W Wj f 󵄩 󵄩L2 (M,Vol;𝒱 ) , j 󵄩L (M,Vol;𝒱 ) j=1 l=0

2

2

j=1 l=0

(8.154)

where the implicit constants do not depend on f , but do depend on Ω ⋐ M and the various other choices we have made. Combining (8.154) and (8.153) shows that (8.152) does not depend on the choice of connection. Using their inner products, we may identify 𝒱1 and 𝒱2 with their duals, and we ∞ do so henceforth. We will be considering operators T : C0∞ (M; 𝒱1 ) → Cloc (M; 𝒱2 ). We say such an operator is given by integration against a function T(x, y) ∈ L1loc (M × M, Vol × Vol; Hom(𝒱1 , 𝒱2 )) if ∀f ∈ C0∞ (M; 𝒱1 ) and g ∈ C0∞ (M; 𝒱2 ) we have ∫⟨g(x), Tf (x)⟩2,x d Vol(x) = ∬⟨g(x), T(x, y)f (y)⟩2,x d Vol(x) d Vol(y). ∞ We write T ≡ S mod Cloc (M × M; Hom(𝒱1 , 𝒱2 )) if T − S is given by integration against ∞ an element of Cloc (M × M; Hom(𝒱1 , 𝒱2 )). ∞ For an element E ∈ Cloc (M × M; Hom(𝒱1 , 𝒱2 )) it makes sense to consider |E(x, y)|, the operator norm of E(x, y) : 𝒱1,y → 𝒱2,x , where the norms on 𝒱1,y and 𝒱2,x are given by their inner products. Using this, one can generalize all of the definitions concerning A t (Ω, (W , ds)) to define objects like A t (Ω, (W , ds); Hom(𝒱1 , 𝒱2 )); wherever we have α used W α , one instead uses ∇W . It is perhaps more simple to proceed locally to obtain an equivalent definition: t Definition 8.6.4. For t ∈ ℝ, we let Aloc ((W , ds); Hom(𝒱1 , 𝒱2 )) be the space of those T : ∞ ∞ C0 (M; 𝒱1 ) → Cloc (M; 𝒱2 ) such that the following holds. For every x ∈ M, let U ⊆ M be

a neighborhood of x with local trivializations ϕ1 : π1−1 (U) 󳨀 → U × ℂD1 , ϕ2 : π2−1 (U) 󳨀 → D2 ∗ t D1 D2 U × ℂ . We assume (ϕ2 )∗ Tϕ1 ∈ Aloc ((W , ds); ℂ , ℂ ), where we have replaced M with t U in the definition of Aloc ((W , ds); ℂD1 , ℂD2 ). ∼

∼

Remark 8.6.5. Definition 8.6.4 does not depend on the choices of connection or inner products. Similarly, one can define the Besov and Triebel–Lizorkin spaces of sections in 𝒱 for s s 𝒦 ⋐ M compact Bp,q (𝒦, (W , ds); 𝒱 ) and Fp,q (𝒦, (W , ds); 𝒱 ), either by a simple reprise of our definitions or by using a partition of unity and reducing to local trivializations. One obtains the same spaces and equivalent norms either way.

8.6 Vector bundles

�

607

Remark 8.6.6. Using the inner products on 𝒱1 and 𝒱2 , given P of the form (8.151), it makes sense to consider the formal adjoint P ∗ , in the sense of operators L2 (M, Vol; 𝒱2 ) → L2 (M, Vol; 𝒱1 ). This adjoint depends on the choices of inner products on 𝒱1 and 𝒱2 ; however, Theorem 8.6.7 holds for any choice of smooth inner products. We are prepared to state the vector bundle analog of Theorem 8.1.1. We present analogs of all parts of Theorem 8.1.1 except for Theorem 8.1.1 (iv) as the process of freezing coefficients cannot be done intrinsically on a vector bundle. Theorem 8.6.7. In the setting above, let P be of the form (8.151). The following are equivalent: (a) P is maximally subelliptic of degree κ with respect to (W , ds) on M. α (b) P0 := ∑degds (α)=κ aα ∇W is maximally subelliptic of degree κ with respect to (W , ds) on M. (c) ∀x0 ∈ M, there exists an open neighborhood U ⊆ M of x0 such that P is maximally subelliptic of degree κ with respect to (W , ds) on U. (d) For any scale of spaces X s of the form s

s

s

X ∈ {Bp,q : p, q ∈ [1, ∞]} ⋃{Fp,q : p ∈ (1, ∞), q ∈ (1, ∞]},

we have the following. Let ϕ1 , ϕ2 , ϕ3 ∈ C0∞ (M) with ϕ1 ≺ ϕ2 ≺ ϕ3 . Then ∀s ∈ ℝ, s s+κ ϕ3 P u ∈ Xcpt ((W , ds); 𝒱1 ) ⇒ ϕ1 u ∈ Xcpt ((W , ds); 𝒱2 ),

∀u ∈ C0∞ (M)′ . Moreover, for every N ≥ 0, there exists C = C(s, N, X s , (W , ds), ϕ1 , ϕ2 , ϕ3 ) ≥ 0 such that ‖ϕ1 u‖X s+κ ((W ,ds);𝒱2 ) ≤ C(‖ϕ3 P u‖X s ((W ,ds);𝒱1 ) + ‖ϕ2 u‖X s−N ((W ,ds);𝒱2 ) ), s ∀u ∈ C0∞ (M; 𝒱2 )′ with ϕ3 P u ∈ Xcpt (W , ds).

(e) P ∗ P is maximally subelliptic of degree 2κ with respect to (W , ds) on M. −κ (f) There exists T ∈ Aloc ((W , ds); Hom(𝒱1 , 𝒱2 )) such that TP ≡ I

∞ mod Cloc (M × M; Hom(𝒱2 , 𝒱2 )).

−2κ (g) There exists S ∈ Aloc ((W , ds); Hom(𝒱2 , 𝒱2 )) such that

SP ∗ P , P ∗ P S ≡ I

∞ mod Cloc (M × M; Hom(𝒱2 , 𝒱2 )).

(h) For every non-negative self-adjoint extension L of P ∗ P and all t > 0, e−tL is given ∞ by integration against a Cloc (M × M; Hom(𝒱2 , 𝒱2 )) function (denoted by e−tL (x, y)) which satisfies the following. For every 𝒦 ⋐ M compact, there exists c > 0 such that

608 � 8 Linear maximally subelliptic operators for all ordered multi-indices α, β, for all s ∈ ℕ, there exists C ≥ 0 such that ∀t ∈ (0, 1], ∀x, y ∈ 𝒦, 1 − deg (α)−deg (β)−2κs 󵄨󵄨 s α β −tL 󵄨 ds ds (x, y)󵄨󵄨󵄨 ≤ C(ρ(W ,ds) (x, y) + t 2κ ) 󵄨󵄨𝜕t Wx Wy e 1

1 ρ(W ,ds) (x, y)2κ 2κ−1 −1 × exp(−c ( ) )(Vol(B(W ,ds) (x, ρ(W ,ds) (x, y) + t 2κ )) ∧ 1) . t

(i) There exists a non-negative self-adjoint extension L of P ∗ P satisfying the conclusions of (h). (j) For every non-negative self-adjoint extension L of P ∗ P , e−tL is given by integration ∞ against a Cloc (M × M; Hom(𝒱2 , 𝒱2 )) function (denoted by e−tL (x, y)), which satisfies the following. For every 𝒦 ⋐ M, ∀α, β ordered multi-indices, ∀l ∈ ℕ, ∃C ≥ 0 such that ∀t ∈ (0, 1], ∀x, y ∈ 𝒦, 1 −l 󵄨󵄨 α β −tL 󵄨 (x, y)󵄨󵄨󵄨 ≤ Ct −(degds (α)−degds (β))/2κ (1 + t − 2κ ρ(W ,ds) (x, y)) 󵄨󵄨∇Wx ∇Wy e 1

× (Vol(B(W ,ds) (x, ρ(W ,ds) (x, y) + t 2κ )) ∧ 1) . −1

(k) There exists a non-negative self-adjoint extension L of P ∗ P satisfying the conclusions of (j). We also have analogs of the regularity properties from Section 8.2, though those results are local in nature, so the results on arbitrary vector bundles easily follow from the corresponding results on trivial vector bundles.

8.7 Quantitative regularity estimates The regularity results in Theorem 8.1.1 (v) and more generally in Section 8.2 are qualitative in the sense that we have not been precise about what quantities the various constants depend on. For example, in (8.6), if ϕ1 and ϕ2 have small support, we wish to understand what the right estimate is to incorporate this dependence in a quantitative way. This is important when we turn to applications to nonlinear equations in Chapter 9. In this section, we describe how to use the methods of this chapter combined with the scaling ideas in Section 3.15.2 to deduce the correct quantitative results.

8.7.1 The unit scale We begin by describing a quantitative version of Proposition 8.2.12 at the unit scale. We work in the setting of Section 3.15.1. In particular, we assume M = Bn (1) and Vol = hσLeb , where h ∈ C ∞ (Bn (1); (0, ∞)) with infu h(u) > 0. We assume we are

8.7 Quantitative regularity estimates

� 609

given (W 1 , ds1 ), . . . , (W ν , dsν ) and (X 1 , d 1 ), . . . , (X ν , d ν ) satisfying the hypotheses of Section 3.15.1. These choices satisfy the hypotheses of Section 8.2.3 in a quantitative way. As described in Section 3.15.1, (W 1 , ds1 ) are Hörmander vector fields with formal degrees, and therefore it makes sense to ask whether a partial differential operator is maximally subelliptic with respect to (W 1 , ds1 ). The quantitative estimates are terms of the norm ‖f ‖X s (W ,ds),(X, described ⃗ ⃗ d),a,7/8,h in Section 6.11.1; here a ≳ 1 can be chosen to be a 0-multi-parameter unit-admissible constant as in Definition 3.15.1. Fix κ ∈ ℕ+ such that ds1j divides κ, 1 ≤ j ≤ r1 . Set nj := κ/ds1j . Fix D1 , D2 ∈ ℕ+ . We consider a partial differential operator of the form P=

∑

degds1 (α)≤κ

α

(W 1 ) ,

aα ∈ C ∞ (Bn (1); 𝕄D1 ×D2 (ℂ)).

(8.155)

We assume that P is maximally subelliptic of degree κ with respect to (W 1 , ds1 ) in the sense that ∃A ≥ 0 such that ∀f ∈ C0∞ (Bn (1)), r1

n 󵄩 󵄩 ∑󵄩󵄩󵄩(Wj1 ) j f 󵄩󵄩󵄩L2 (Bn (1),hσ ) ≤ A(‖P f ‖L2 (Bn (1),hσLeb ) + ‖f ‖L2 (Bn (1),hσLeb ) ). Leb j=1

Proposition 8.7.1. For any scale of spaces X s , s ∈ ℝν , of the form s

s

s

X ∈ {Bp,q : p, q ∈ [1, ∞]} ⋃{Fp,q : p ∈ (1, ∞), q ∈ (1, ∞]},

the following holds. Let ℬ ⊂ C0∞ (Bn (7/8)) be a bounded set. Then, ∀s ∈ ℝν , ∀N⃗ ∈ [0, ∞)ν , there exists C ≥ 0 such that for all ϕ1 , ϕ2 , ϕ3 ∈ ℬ with ϕ1 ≺ ϕ2 ≺ ϕ3 , 󵄩󵄩 󵄩󵄩 󵄩 󵄩 ≤ C(󵄩󵄩󵄩ϕ3 P u󵄩󵄩󵄩X s (W ,ds),(X, 󵄩󵄩ϕ1 u󵄩󵄩X s+κe1 (W ,ds),(X, ⃗ ⃗ ⃗ ⃗ d),a,7/8,h d),a,7/8,h 󵄩󵄩 󵄩󵄩 + 󵄩󵄩ϕ2 u󵄩󵄩X s−N⃗ (W ,ds),(X, ), ⃗ ⃗ d),a,7/8,h

(8.156)

s ⃗ Moreover, there exists L ∈ ℕ, depending ∀u ∈ C0∞ (Bn (1); ℂD2 )′ with ϕ3 P u ∈ Xcpt (W , ds). s only on max1≤j≤q |d j⃗ |1 , κ, s, N,⃗ and X , such that C = C(A, maxα ‖aα ‖C L (Bn (1)) , ℬ, D1 , D2 ) ≥ 0 is a (κ, s, N,⃗ X s )-multi-parameter unit-admissible constant as in Definition 3.15.1.

Except for the dependence of C, Proposition 8.7.1 is a special case of Proposition 8.2.12. One can prove Proposition 8.7.1 by keeping careful track of the estimates in the proof of Proposition 8.2.12; we leave the details to the reader (see, for example, Theorem 3.15.5). We state a simple corollary of Proposition 8.7.1 which is important for our applications. Corollary 8.7.2. For any scale of spaces X s , s ∈ ℝν , of the form s

s

s

X ∈ {Bp,q : p, q ∈ [1, ∞]} ⋃{Fp,q : p ∈ (1, ∞), q ∈ (1, ∞]},

610 � 8 Linear maximally subelliptic operators s s the following holds. Let p ∈ [1, ∞] be such that X s = Bp,q or X s = Fp,q and let ℬ ⊂ ∞ n ν C0 (B (7/8)) be a bounded set. Then, ∀s ∈ ℝ , there exists C ≥ 0 such that for all ϕ1 , ϕ2 ∈ ℬ with ϕ1 ≺ ϕ2 ,

󵄩󵄩 󵄩󵄩 󵄩 󵄩 ≤ C(󵄩󵄩󵄩ϕ2 P u󵄩󵄩󵄩X s (W ,ds),(X, 󵄩󵄩ϕ1 u󵄩󵄩X s+κe1 (W ,ds),(X, ⃗ ⃗ ⃗ ⃗ d),a,7/8,h d),a,7/8,h + ‖ϕ2 u‖Lp (Bn (1),hσLeb ) ),

(8.157)

s ∀u ∈ C0∞ (Bn (1); ℂD2 )′ with ϕ2 P u ∈ Xcpt (W , ds)⃗ and ϕ2 u ∈ Lp (Bn (1), hσLeb ). Moreover, there exists L ∈ ℕ, depending only on max1≤j≤q |d j⃗ |1 , κ, s, and X s , such that C = C(A, maxα ‖aα ‖C L (Bn (1)) , ℬ, D1 , D2 ) ≥ 0 is a (κ, s, X s )-multi-parameter unit-admissible constant as in Definition 3.15.1.

Proof. For notational simplicity, in this proof we write ‖ ⋅ ‖X s (W ,ds)⃗ to mean 󵄩󵄩 󵄩󵄩 . 󵄩󵄩⋅󵄩󵄩X s (W ,ds),(X, ⃗ ⃗ d),a,7/8,h ⃗ Define N ∈ [0, ∞)ν given by N⃗ μ := (sμ + 1) ∨ 0. For ϕ1 , ϕ2 ∈ ℬ with ϕ1 ≺ ϕ2 , let ϕ1.5 ∈ C0∞ (Bn (7/8)) be such that ϕ1 ≺ ϕ1.5 ≺ ϕ2 and {ϕ1.5 : ϕ1 , ϕ2 ∈ ℬ} ⊂ C0∞ (Bn (7/8)) is a bounded set. Proposition 8.7.1 shows that ‖ϕ1 u‖X s+κe1 (W ,ds)⃗ ≲ ‖ϕ2 P u‖X s (W ,ds)⃗ + ‖ϕ1.5 u‖X s−N⃗ (W ,ds)⃗ .

(8.158)

Since sμ − N⃗ μ ≤ −1, ∀μ, Proposition 6.5.16 shows that ‖ϕ1.5 u‖X s−N⃗ (W ,ds)⃗ ≤ ‖ϕ1.5 u‖X (−1,−1,...,−1) (W ,ds)⃗ ≲ ‖ϕ1.5 u‖Lp ≲ ‖ϕ2 u‖Lp ,

(8.159)

where keeping track of estimates in the proof of Proposition 6.5.16 shows that the implicit constant is of the desired form. Combining (8.158) and (8.159) completes the proof.

8.7.2 A small scale We now turn to a quantitative version of Proposition 8.2.12 which takes place on a small scale. We work in the same setting as Proposition 8.2.12 (i. e., the main multi-parameter setting of Section 3.15). Thus, we are given vector fields with single-parameter formal ∞ degrees (W 1 , ds1 ), . . . , (W ν , dsν ) ⊂ Cloc (M; TM) satisfying the hypotheses given in Sec1 1 tion 3.15. In particular, (W , ds ) = {(W11 , ds11 ), . . . , (Wr11 , ds1r1 )} are Hörmander vector fields with formal degrees on M. Fix 𝒦 ⋐ Ω0 ⋐ Ω1 ⋐ Ω2 ⋐ M, with 𝒦 compact and Ω0 , Ω1 , and Ω2 open and relatively ∞ compact. Pick (X 1 , d 1 ), . . . , (X ν , d ν ) ⊂ Cloc (M; TM) and λ = (1, λ2 , . . . , λν ) ∈ (0, ∞)ν as in Section 3.15.2 (with the above choices of 𝒦, Ω1 , and Ω2 ). Let Φx,δ be as in Theorem 3.15.3, with these choices. As usual, we set (W , ds)⃗ := (W 1 , ds1 ) ⊠ ⋅ ⋅ ⋅ ⊠ (W ν , dsν ) and (X, d )⃗ := (X 1 , d 1 ) ⊠ ⋅ ⋅ ⋅ ⊠ (X ν , d ν ).

8.7 Quantitative regularity estimates

� 611

Proposition 8.7.3. There exists a > 0 such that the following holds. Fix κ ∈ ℕ+ such that ds1j divides κ for 1 ≤ j ≤ r1 . Let P be given by (8.26) and suppose P is maximally subelliptic of degree κ with respect to (W 1 , ds1 ) on M. Then, for any scale of spaces X s , s ∈ ℝν , of the form s

s

s

X ∈ {Bp,q : p, q ∈ [1, ∞]} ⋃{Fp,q : p ∈ (1, ∞), q ∈ (1, ∞]},

we have the following. Let Φx,δ be as in Theorem 3.15.3. Fix ϕ1 , ϕ2 , ϕ3 ∈ C0∞ (Bn (7/8)) with ϕ1 ≺ ϕ2 ≺ ϕ3 . Then, for every s ∈ ℝν and N⃗ ∈ [0, ∞)ν , there exists C ≥ 0 such that the following holds ∀x ∈ 𝒦, δ ∈ (0, 1]: 󵄩󵄩 󵄩 −1 󵄩󵄩(ϕ1 ∘ Φx,δ )u󵄩󵄩󵄩X s+κe1 (δλds⃗ W ,ds),(δ λd ⃗ X,d),a,Ω ⃗ ⃗ 0 ,Ω1 ,Ω2 ,Vol 󵄩 κ󵄩 −1 ≤ C(δ 󵄩󵄩󵄩(ϕ3 ∘ Φx,δ )P u󵄩󵄩󵄩X s (δλds⃗ W ,ds),(δ λd ⃗ X,d),a,Ω ⃗ ⃗ 0 ,Ω1 ,Ω2 ,Vol 󵄩 󵄩 −1 󵄩 󵄩 + 󵄩󵄩(ϕ2 ∘ Φx,δ )u󵄩󵄩X s−N⃗ (δλds⃗ W ,ds),(δ ), λd ⃗ X,d),a,Ω ⃗ ⃗ 0 ,Ω1 ,Ω2 ,Vol

(8.160)

s ⃗ ∀u ∈ C0∞ (M; ℂD2 )′ with (ϕ3 ∘ Φ−1 x,δ )P u ∈ X (𝒦, (W , ds)). Here, C ≥ 0 does not depend on x ∈ 𝒦 or δ ∈ (0, 1], but may depend on any of the other ingredients.

Remark 8.7.4. In light of Theorem 3.3.7 (i) one should think of Φx,δ (Bn (7/8)) as being comparable to B(W 1 ,ds1 ) (x, δ) and B(X 1 ,d 1 ) (x, δ). Thus, the estimate (8.160) is taking place at scale ≈ δ with respect to ρ(W 1 ,ds1 ) (alternatively, ρ(X 1 ,d 1 ) ). Proof. Set P x,δ := Φ∗x,δ δκ P (Φx,δ )∗ . Using Proposition 6.11.4 and the fact that Φ∗x,δ δκ P u = P x,δ Φ∗x,δ u, (8.160) is equivalent to 󵄩󵄩 ∗ 󵄩 󵄩󵄩ϕ1 Φx,δ u󵄩󵄩󵄩X s+κe1 (W x,δ ,ds),(X x,δ ,d),a,7/8,h ⃗ ⃗ x,δ 󵄩󵄩 x,δ ∗ 󵄩 ≤ C(󵄩󵄩ϕ3 P Φx,δ u󵄩󵄩󵄩X s (W x,δ ,ds),(X x,δ ,d),a,7/8,h ⃗ ⃗ x,δ 󵄩󵄩 ∗ 󵄩 + 󵄩󵄩ϕ2 Φx,δ u󵄩󵄩󵄩X s−N⃗ (W x,δ ,ds),(X ), x,δ ,d),a,7/8,h ⃗ ⃗ x,δ

(8.161)

where (W x,δ , ds)⃗ and (X x,δ , d )⃗ are as in Proposition 6.11.4. We will prove (8.161) by apply⃗ (X, d ), ⃗ and h replaced by (W x,δ , ds), ⃗ (X x,δ , d ), ⃗ and hx,δ , ing Proposition 8.7.1 with (W , ds), respectively. Here, x ∈ 𝒦 and δ ∈ (0, 1]. We use the same scaling ideas as in Section 3.3.1. ⃗ (X x,δ , d ), ⃗ and hx,δ satisfy the hypotheses of Theorem 3.15.3 shows that (W x,δ , ds), Proposition 8.7.1 uniformly for x ∈ 𝒦 and δ ∈ (0, 1]. Thus, it remains to show that P x,δ also satisfies the hypotheses uniformly in x ∈ 𝒦 and δ ∈ (0, 1]. First we must show that P x,δ is of the form (8.155). We have P

x,δ

= Φ∗x,δ δκ P (Φx,δ )∗ = =:

∑

degds1 (α)≤κ

∑

degds1 (α)≤κ α

aαx,δ (W 1,x,δ ) .

α

δκ−degds1 (α) (aα ∘ Φx,δ )(W 1,x,δ )

612 � 8 Linear maximally subelliptic operators We claim, for every L ∈ ℕ and degds1 (α) ≤ κ, sup ‖aαx,δ ‖C L (Bn (1);𝕄D1 ×D2 ) < ∞.

x∈𝒦 δ∈(0,1]

(8.162)

Indeed, Theorem 3.3.7 (m) gives ‖aαx,δ ‖C L (Bn (1);𝕄D1 ×D2 ) ≈

∑ degd 1 (β)≤L

=

∑ degd 1 (β)≤L

=

∑ degd 1 (β)≤L

≤

∑ degd 1 (β)≤L

󵄩󵄩 1,x,δ β x,δ 󵄩󵄩 ) aα 󵄩󵄩C(Bn (1)) 󵄩󵄩(X β 󵄩 󵄩 δκ−degds1 (α) 󵄩󵄩󵄩(X 1,x,δ ) aα ∘ Φx,δ 󵄩󵄩󵄩C(Bn (1)) β 󵄩 󵄩 δκ−degds1 (α)+degd 1 (β) 󵄩󵄩󵄩(X 1 ) aα 󵄩󵄩󵄩C(Φ (Bn (1))) x,δ

󵄩󵄩 1 β 󵄩󵄩 󵄩󵄩(X ) aα 󵄩󵄩C(Ω ) ≲ 1, 1

where the ≲ 1 uses the relative compactness of Ω1 . This proves (8.162) and shows P x,δ is of the form (8.155) uniformly for x ∈ 𝒦 and δ ∈ (0, 1]. Finally, we must show that P x,δ is maximally subelliptic of degree κ with respect to (W 1,x,δ , ds1 ), uniformly in x ∈ 𝒦 and δ ∈ (0, 1]. By the assumption that P is maximally subelliptic of degree κ with respect to (W 1 , ds1 ) on M, there exists A ≥ 0 such that ∀f ∈ C0∞ (Ω; ℂD2 ), we have, with nj := κ/ds1j ∈ ℕ+ , r

n 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ∑󵄩󵄩󵄩(Wj1 ) j f 󵄩󵄩󵄩L2 (M,Vol;ℂD2 ) ≤ A(󵄩󵄩󵄩P f 󵄩󵄩󵄩L2 (M,Vol;ℂD1 ) + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (M,Vol;ℂD2 ) ). j=1

(8.163)

Multiplying (8.163) by δκ , we have r

n 󵄩 󵄩 1 󵄩 󵄩 󵄩 󵄩 ∑󵄩󵄩󵄩(δdsj Wj1 ) j f 󵄩󵄩󵄩L2 (M,Vol;ℂD2 ) ≤ A(󵄩󵄩󵄩δκ P f 󵄩󵄩󵄩L2 (M,Vol;ℂD1 ) + δκ 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (M,Vol;ℂD2 ) ) j=1

󵄩 󵄩 󵄩 󵄩 ≤ A(󵄩󵄩󵄩δκ P f 󵄩󵄩󵄩L2 (M,Vol;ℂD1 ) + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (M,Vol;ℂD2 ) ).

(8.164)

∞ D2 For g ∈ C0∞ (Bn (1); ℂD2 ), we have f := g ∘ Φ−1 x,δ ∈ C0 (Ω1 ; ℂ ). Plugging this choice of ∗ f into (8.164) and using Φx,δ Vol = Λ(x, δ)hx,δ σLeb (where Λ(x, δ) is given by (6.175)), we have r

n 󵄩 󵄩 ∑󵄩󵄩󵄩(Wj1,x,δ ) j g 󵄩󵄩󵄩L2 (Bn (1),Λ(x,δ)h σ ;ℂD2 ) x,δ Leb j=1

󵄩 󵄩 󵄩 󵄩 ≤ A(󵄩󵄩󵄩P x,δ g 󵄩󵄩󵄩L2 (Bn (1),Λ(x,δ)h σ ;ℂD1 ) + 󵄩󵄩󵄩g 󵄩󵄩󵄩L2 (Bn (1),Λ(x,δ)h σ ;ℂD2 ) ). x,δ Leb x,δ Leb 1

Multiplying both sides of (8.165) by Λ(x, δ)− 2 gives

(8.165)

8.8 Representation theory and Rockland’s condition

�

613

r

n 󵄩 󵄩 ∑󵄩󵄩󵄩(Wj1,x,δ ) j g 󵄩󵄩󵄩L2 (Bn (1),h σ ;ℂD2 ) x,δ Leb j=1

󵄩 󵄩 󵄩 󵄩 ≤ A(󵄩󵄩󵄩P x,δ g 󵄩󵄩󵄩L2 (Bn (1),h σ ;ℂD1 ) + 󵄩󵄩󵄩g 󵄩󵄩󵄩L2 (Bn (1),h σ ;ℂD2 ) ). x,δ Leb x,δ Leb

(8.166)

Since A does not depend on x ∈ 𝒦 or δ ∈ (0, 1], (8.166) shows that P x,δ is maximally subelliptic of degree κ with respect to (W 1,x,δ , ds1 ), uniformly in x ∈ 𝒦 and δ ∈ (0, 1]. This completes the proof that the assumptions of Proposition 8.7.1 hold uniformly for x ∈ 𝒦 and δ ∈ (0, 1]. Thus, (8.161) follows from Proposition 8.7.1, which completes the proof.

8.8 Representation theory and Rockland’s condition Probably the most familiar characterization of elliptic partial differential operators comes via the Fourier transform. Let E be a constant coefficient partial differential operator on ℝn which is homogeneous of degree κ: α

E = ∑ aα 𝜕x ,

aα ∈ ℂ.

|α|=κ

Since E is translation invariant on ℝn , conjugating it with the Fourier transform gives a one-dimensional operator for each ξ ∈ ℝn : κ

α

Ê(ξ) = (−2πi) ∑ aα ξ . |α|=κ

Then E is elliptic if and only if Ê(ξ) ≠ 0, ∀ξ ≠ 0. Thinking of the Fourier transform as the unitary representation theory of ℝn , this characterization of ellipticity has a particularly elegant generalization to nilpotent Lie groups. Let G be a connected, simply connected, graded, nilpotent Lie group (see Section 4.5.3), with Lie algebra g with grading g = ⊕νμ=1 vμ (see Definition 4.5.26). As in Section 4.5.3, the grading induces dilations on the Lie group. Fix (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} ⊂ g × ℕ+ such that W1 , . . . , Wr generate g and Wj ∈ vdsj . Fix κ ∈ ℕ+ such that dsj divides κ for each 1 ≤ j ≤ r. We identify W1 , . . . , Wr with the left invariant vector fields on G. Let P be a left invariant partial differential operator which is homogeneous of degree κ with respect to the dilations. Note that P is of the form P=

∑

degds (α)=κ

aα W α ,

aα ∈ ℂ.

Theorem 8.8.1. Let P be as above. Then the following are equivalent: – P is hypoelliptic. – P is maximally subelliptic of degree κ with respect to (W , ds) on G. – For every non-trivial irreducible unitary representation π of G, the operator π(P ) is injective in the space of C ∞ vectors of the representation.

614 � 8 Linear maximally subelliptic operators Theorem 8.8.1 was first proved in the case of the Heisenberg group by Rockland [205], who conjectured the general result; see also [171, 78]. Theorem 8.8.1 was proved in general by Helffer and Nourrigat [121]; see also [120, 9]. Helffer and Nourrigat, in turn, conjectured an extension of Theorem 8.8.1 to general partial differential operators on manifolds like the ones studied in this chapter [122]. This more general conjecture was recently proved by Androulidakis, Moshen, and Yuncken [2], who construct parametrices for maximally subelliptic operators using a pseudo-differential calculus based on the ideas of [3].

8.9 Positive definite forms Let (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} be Hörmander vector fields with formal degrees on M. Fix κ ∈ ℕ+ such that dsj divides κ for every j; set nj := κ/dsj ∈ ℕ+ . In this section, we describe the following result. ∞ Proposition 8.9.1. Let a(x) = (ai,j (x)) ∈ Cloc (M; 𝕄r×r (ℂ)) be a smooth, self-adjoint matrix which is strictly positive definite at every point. For degds(α) < κ, let bα (x) ∈ ∞ Cloc (M; ℂ). Then the operator n

n

La u := ∑ ai,j (x)Wi i Wj j u(x) + 1≤i,j≤r

∑

degds (α) 0 (i. e., the a priori regularity of u) does not play a role in the conclusion of Theorem 9.1.2; however, the assumption u ∈ C r+κ (W , ds) near x0 does ensure that all aspects of (9.3) are defined in the classical sense. Theorem 9.1.2 is clearly sharp with respect to all the other ingredients. Remark 9.1.4. If (9.3) holds on a relatively compact open set Ω, then the estimates for u blow up as one approaches the boundary of Ω. This blowup is made precise in Section 9.3. Corollary 9.1.5. Let r > 0. Suppose that: ∞ – F(x, ζ ) ∈ Cloc (M × ℝN ) for x near x0 ,

9.1 Main qualitative results

– –

� 621

∞ g ∈ Cloc near x0 , u ∈ C r+κ near x0 .

∞ Then u ∈ Cloc near x0 .

Proof. By the maximal subellipticity hypothesis, the linear partial differential operator v 󳨃→ dζ F(x0 , {W α u(x0 )}deg

ds (α)≤κ

){W α v}deg

ds (α)≤κ

is maximally subelliptic of degree κ with respect to (W , ds) on a neighborhood of x0 . By continuity, it follows that there is a neighborhood V of x0 such that for all x1 ∈ V , v 󳨃→ dζ F(x1 , {W α u(x1 )}deg

ds (α)≤κ

){W α v}deg

ds (α)≤κ

is maximally subelliptic at of degree κ with respect to (W , ds) on a neighborhood of x1 ; see Lemma 8.5.2 for a similar continuity argument. That is, F(x, {W α u(x)}deg

ds (α)≤κ

)

is maximally subelliptic at (x1 , u) of degree κ with respect to (W , ds), ∀x1 ∈ V . Fix U ⋐ V with x0 ∈ U and ϕ ∈ C0∞ (V ) with ϕ ≡ 1 on a neighborhood of x0 . By possibly shrinking U, we may apply Theorem 9.1.2 to each x1 ∈ U, and using the compactness of U and a partition of unity we see that ϕu ∈ C s+κ (U, (W , ds)), ∀s > 0. The result follows. It is also of interest to understand the regularity theory with respect to the Zygmund– Hölder spaces adapted to different vector fields with formal degrees. The next theorem gives such a result, which is sharp up to a “loss of ϵ derivatives.” Proposition 9.1.6. Fix s > r > 0, t > 2⌊s⌋ + 8. Let (Z, dr) = {(Z1 , dr1 ), . . . , (Zv , drv )} ⊂ ∞ Cloc (M; TM)×ℕ+ be another set of Hörmander vector fields with single-parameter formal degrees on M. Let λ := λ(x0 , (W , ds), (Z, dr)) > 0 be as in Definition 6.6.9. Suppose that: – (W , ds) and (Z, dr) locally weakly approximately commute on M, – F ∈ C s,t ((Z, dr) ⊠ ∇ℝN ) near x0 , – g ∈ C s (Z, dr) near x0 , – W α u ∈ C r (Z, dr) near x0 , ∀ degds(α) ≤ κ. Then, ∀ϵ > 0, we have: (I) u ∈ C s+κ/λ−ϵ (Z, dr) near x0 , and (II) W α u ∈ C s−ϵ (Z, dr) near x0 , ∀ degds(α) ≤ κ. We see Proposition 9.1.6 as a corollary of a multi-parameter result in Section 9.1.2. Proposition 8.2.1 gives the same gain of κ/λ derivatives for linear maximally subelliptic PDEs (without the loss of ϵ), and it is shown there that this gain is sharp. Thus,

622 � 9 Nonlinear maximally subelliptic equations Proposition 9.1.6 is sharp up to the endpoint ϵ = 0. It would be interesting to determine whether Proposition 9.1.6 holds with ϵ = 0. Remark 9.1.7. In Proposition 9.1.6, (I) is a consequence of (II) by Proposition 8.2.1; however, we will prove (I) directly. A special case of Proposition 9.1.6 gives the sharp (up to a loss of ϵ > 0 derivatives) for fully nonlinear maximally subelliptic PDEs with respect to the classical Zygmund– Hölder spaces. Corollary 9.1.8. Fix s > r > 0 and t > 2⌊s⌋ + 8. Let λ := λstd (x0 , (W , ds)) (see Definition 6.6.15). Suppose that: s,t – F ∈ Cstd near x0 , s – g ∈ Cstd near x0 , r – W α u ∈ Cstd near x0 , ∀ degds(α) ≤ κ. Then, ∀ϵ > 0, we have: s+κ/λ−ϵ (I) u ∈ Cstd near x0 , and α s−ϵ (II) W u ∈ Cstd near x0 , ∀ degds(α) ≤ κ. Proof. Let N and (Z, 1) be as in Lemma 8.2.4. In light of Lemma 8.2.4, the result is an immediate consequence of Proposition 9.1.6 with (Z, dr) replaced by (Z, 1). 9.1.1.1 Qualitative Schauder estimates The above results immediately imply corresponding qualitative Schauder estimate type results. We describe these qualitative results here and discuss more quantitative versions (which describe the blowup near the boundary) in Section 9.3.1. Let P u(x) = ∑degds (α)≤κ aα (x)W α u(x), where aα : M → 𝕄D1 ×D2 (ℝ). Fix x0 ∈ M and define the frozen coefficient operator Px0 :=

∑

degds (α)≤κ

aα (x0 )W α .

We make the following assumption. Main assumption: There exists a neighborhood Ω of x0 such that Px0 is maximally subelliptic of degree κ with respect to (W , ds) on Ω. Furthermore, we suppose we are given functions u : M → ℝD2 and g : M → ℝD1 satisfying P u = g.

Example 9.1.9. Suppose (W , ds) = (W , 1) = {(W1 , 1), . . . , (Wr , 1)}, i. e., each dsj = 1. Consider the operator

9.1 Main qualitative results

�

623

u 󳨃→ ∑ ai,j (x)Wi Wj u(x). i,j

If the r × r matrix (ai,j (x0 )) is strictly positive definite, then the main assumption above is satisfied for this operator. See Section 8.9 for this and generalizations. Corollary 9.1.10. Let s > r > 0. Suppose that: – aα ∈ C s (W , ds) near x0 , ∀ degds(α) ≤ κ, – g ∈ C s (W , ds) near x0 , – u ∈ C r+κ (W , ds) near x0 . Then u ∈ C s+κ (W , ds) near x0 . ∞ Corollary 9.1.11. Fix s > r > 0. Let (Z, dr) = {(Z1 , dr1 ), . . . , (Zv , drv )} ⊂ Cloc (M; TM) × ℕ+ be another set of Hörmander vector fields with single-parameter formal degrees on M. Let λ := λ(x0 , (W , ds), (Z, dr)) > 0 be as in Definition 6.6.9. Suppose that: – (W , ds) and (Z, dr) locally weakly approximately commute on M, – aα ∈ C s (Z, dr) near x0 , ∀ degds(α) ≤ κ, – g ∈ C s (Z, dr) near x0 , – W α u ∈ C r (Z, dr) near x0 , ∀ degds(α) ≤ κ.

Then, ∀ϵ > 0, we have: – u ∈ C s+κ/λ−ϵ (Z, dr) near x0 , and – W α u ∈ C s−ϵ (Z, dr) near x0 , ∀ degds(α) ≤ κ. Corollary 9.1.12. Fix s > r > 0. Let λ := λstd (x0 , (W , ds)) (see Definition 6.6.15). Suppose that: s – aα ∈ Cstd near x0 , ∀ degds(α) ≤ κ, s – g ∈ Cstd near x0 , r – W α u ∈ Cstd near x0 , ∀ degds(α) ≤ κ. Then, ∀ϵ > 0, we have: s+κ/λ−ϵ – u ∈ Cstd near x0 , and α s−ϵ – W u ∈ Cstd near x0 , ∀ degds(α) ≤ κ. Each of the above corollaries is a nearly immediate consequence of the corresponding fully nonlinear result above. We include the proof reducing Corollary 9.1.10 to Theorem 9.1.2; the other proofs are similar, and we leave them to the interested reader. Proof of Corollary 9.1.10. Fix a small, compact neighborhood 𝒦 ⋐ M of x0 and let ψ ∈ r+κ C0∞ (M) be such that ψ ≡ 1 on 𝒦 with small enough support that ψu ∈ Ccpt (W , ds). Using Corollaries 7.2.2 and 6.5.11, we have, for degds(α) ≤ κ, ‖W α u‖L∞ (𝒦) ≲ ‖W α ψu‖C r (W ,ds) ≲ ‖ψu‖C r+κ (W ,ds) < ∞.

624 � 9 Nonlinear maximally subelliptic equations Dividing u and g by ∑degds (α)≤κ ‖W α u‖L∞ (𝒦) we see that we may assume ‖W α u‖L∞ (𝒦) ≤ 1, ∀ degds(α) ≤ κ. Fix ϕ(ζ ) ∈ C0∞ (ℝN ) with ϕ(ζ ) ≡ 1 on a neighborhood of {ζ ∈ ℝN : |ζα | ≤ 2, ∀ degds(α) ≤ κ}. Define F(x, ζ ) :=

∑

degds (α)≤κ

aα (x)ζα ϕ(ζ ).

Clearly, F ∈ C s,2⌊s⌋+7 ((W , ds) ⊠ ∇ℝN ) near x0 . Note that F(x, {W α u(x)}deg

ds (α)≤κ

) = P u(x),

x near x0 .

(9.4)

Also, dζ F(x0 , {W α u(x0 )}deg

ds (α)≤κ

){W α }deg

ds (α)≤κ

= Px0 ,

so the assumption of the maximal subellipticity of Px0 gives that F(x, {W α u(x)}degds (α)≤κ ) is maximally subelliptic at (x0 , u) of degree κ with respect to (W , ds). We have shown that all the hypotheses of Theorem 9.1.2 hold for (9.4). The conclusion of Theorem 9.1.2 in this case is u ∈ C s+κ (W , ds) near x0 , completing the proof. Remark 9.1.13. The proof above of Corollary 9.1.10 does not fully exploit the fact that P is a linear partial differential operator. By exploiting this, following a proof very similar to Theorem 9.1.14, one can weaken the a priori assumption that u ∈ C r+κ (W , ds) near x0 . To push this idea to its logical conclusion would involve introducing considerable notation, and we do not do so here since the main focus of this chapter is to study fully nonlinear equations. See Remark 2.8.6.

9.1.2 Multi-parameter results ∞ Fix ν ∈ ℕ+ and let (W 1 , ds1 ), . . . , (W ν , dsν ) ⊂ Cloc (M; TM) × ℕ+ be finite lists of vector fields with formal degrees satisfying the hypotheses of the main multi-parameter setting described in Section 3.15. Namely, we assume that: – (W 1 , ds1 ) are Hörmander vector fields with formal degrees on M. – Gen((W μ , dsμ )) is locally finitely generated for μ ∈ {2, . . . , ν}. – (W 1 , ds1 ), . . . , (W ν , dsν ) pairwise weakly locally approximately commute on M. ∞ Set (W , ds)⃗ := (W 1 , ds1 ) ⊠ (W 2 , ds2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (W ν , dsν ) ⊂ Cloc (M; TM) × (ℕν \ {0}). Fix κ ∈ ℕ+ 1 such that dsj divides κ, ∀j. Let D1 , D2 ∈ ℕ+ and set

N := D2 × #{α an ordered multi-index : degds1 (α) ≤ κ}.

(9.5)

9.1 Main qualitative results

�

625

We consider the following PDE, where F(x, ζ ) : M × ℝN → ℝD1 , g(x) : M → ℝD1 , and u(x) : M → ℝD2 are given: α

F(x, {(W 1 ) u(x)}deg

ds1

(α)≤κ )

= g(x).

(9.6)

Fix x0 ∈ M. Main assumptions: We assume that α

F(x, {(W 1 ) u(x)}deg

ds1

(α)≤κ )

is maximally subelliptic at (x0 , u) of degree κ with respect to (W 1 , ds1 ), and we assume (9.3) holds for x in a neighborhood of x0 . Theorem 9.1.14. Fix r,⃗ s⃗ ∈ (0, ∞)ν and t > 2⌊|s|⃗ 1 ⌋ + 4 + 2ν. Suppose that: ⃗ – F ∈ C s,t ((W , ds)⃗ ⊠ ∇ℝN ) near x0 , ⃗ – g ∈ C s (W , ds)⃗ near x0 , ⃗ r+κe 1 – u∈C (W , ds)⃗ near x0 . ⃗ Then u ∈ C s+κe1 (W , ds)⃗ near x0 .

Proof. This is a qualitative version of the (stronger) quantitative result Theorem 9.2.1. Indeed, the hypotheses of this theorem are invariant under diffeomorphisms. By picking ∼ a coordinate chart Φ : Bn (2) 󳨀 → Φ(Bn (2)) ⊆ M, where Φ(0) = x0 and Φ(Bn (2)) is a small neighborhood of x0 ∈ M, writing the above assumptions in this coordinate system, and restricting attention to Bn (1), we see that all the hypotheses of Theorem 9.2.1 hold. The conclusion of Theorem 9.2.1 implies the conclusion of this theorem. We next present the reduction of Proposition 9.1.6 to Theorem 9.1.14. This combines several ideas from previous chapters. Proof of Proposition 9.1.6. We rename (W , ds) to be (W 1 , ds1 ), set (W 2 , ds2 ) := (Z, dr), and let (W , ds)⃗ := (W 1 , ds1 ) ⊠ (W 2 , ds2 ). By the assumptions of Proposition 9.1.6, we may pick ψ1 , ψ2 ∈ C0∞ (M), with ψ1 ≺ ψ2 , with ψ1 ≡ 1 on a neighborhood of x0 , and with small enough support that if 𝒦 := supp(ψ2 ), we have ψ1 (x)F(x, ζ ) ∈ C s,t (𝒦 × ℝN , (W 2 , ds2 ) ⊠ ∇ℝN ), s

2

(9.7)

2

ψ1 g ∈ C (𝒦, (W , ds )),

1 α

ψ2 (W ) u ∈ C r (𝒦, (W 2 , ds2 )),

(9.8)

∀ degds1 (α) ≤ κ.

Note that for degds1 (α) ≤ κ, we have α

α

(W 1 ) ψ1 u = ψ2 (W 1 ) ψ1 u =

∑

degds1 (β)≤κ

β

gαβ ψ2 (W 1 ) u,

(9.9)

626 � 9 Nonlinear maximally subelliptic equations β

where gα ∈ C0∞ (M). Therefore, Corollary 6.5.10 combined with (9.9) implies that α

(W 1 ) ψ1 u ∈ C r (𝒦, (W 2 , ds2 )),

∀ degds1 (α) ≤ κ.

(9.10)

The result is stronger the smaller ϵ > 0 is, so we may assume ϵ ∈ (0, r). Proposition 7.6.1 applied to (9.7), (9.8), and (9.10) shows that there exists ϵ′ > 0 with ψ1 (x)F(x, ζ ) ∈ C (ϵ ,s−ϵ),t (𝒦 × ℝN , (W , ds)⃗ ⊠ ∇ℝN ), ′ ′

α

⃗ ψ1 g ∈ C (ϵ ,s−ϵ) (𝒦, (W , ds)),

⃗ (W 1 ) ψ1 u ∈ C (ϵ ,r−ϵ) (𝒦, (W , ds)), ′

∀ degds1 (α) ≤ κ.

′

⃗ Proposition 6.5.12 shows that ψ1 u ∈ C (ϵ +κ,r−ϵ) (𝒦, (W , ds)). Since ψ1 ≡ 1 on a neighborhood of x0 , the above shows that Theorem 9.1.14 applies to give that there exists ϕ̃ ∈ C0∞ (M), with ϕ̃ ≡ 1 on a neighborhood of x0 , such that ′

̃ ∈ C (ϵ +κ,s−ϵ) (W , ds). ⃗ ϕu cpt

(9.11)

(II): Take ϕ ∈ C0∞ (M) with ϕ ≺ ϕ̃ and ϕ ≡ 1 on a neighborhood of x0 . Using (9.11), Corollaries 6.5.10 and 6.5.11 show that for all degds1 (α) ≤ κ, (ϵ +κ−degds1 (α),s−ϵ) α α ̃ (ϵ ,s−ϵ) ⃗ ϕ(W 1 ) u = ϕ(W 1 ) ϕu ∈ Ccpt (W , ds)⃗ ⊆ Ccpt (W , ds). ′

′

From here, Proposition 7.6.1 implies that for all degds1 (α) ≤ κ, α

(ϵ ,s−ϵ) s−ϵ ϕ(W 1 ) u ∈ Ccpt (W , ds)⃗ ⊆ Ccpt (W 2 , ds2 ), ′

completing the proof of (II). (I): Letting λ := λ(x0 , (W 1 , ds1 ), (W 2 , ds2 )) > 0 be as in Definition 6.6.9, Corollary 6.6.11 shows that (W 1 , ds1 ) weakly λ-controls (W 2 , ds2 ) on Ω′ for some relatively compact open neighborhood Ω′ ⋐ M of x0 . Take ϕ ∈ C0∞ (Ω′ ) with ϕ ≺ ϕ̃ and ϕ ≡ 1 on a neighborhood of x0 and set 𝒦1 := supp(ϕ) ⋐ Ω′ . Corollary 6.5.10 and (9.11) combine to show that ′

̃ ∈ C (ϵ +κ,s−ϵ) (W , ds), ⃗ ϕu = ϕϕu cpt ′

⃗ By Theorem 6.7.2 with and therefore, since supp(ϕ) ⊆ 𝒦1 , ϕu ∈ C (ϵ +κ,s−ϵ) (𝒦1 , (W , ds)). s̃ = κ/λ, we have ′

′

⃗ ⊆ C (ϵ +κ,s−ϵ)+(−κ,κ/λ) (𝒦1 , (W , ds)) ⃗ ϕu ∈ C (ϵ +κ,s−ϵ) (𝒦1 , (W , ds)) ′

⃗ = C (ϵ ,s+κ/λ−ϵ) (𝒦1 , (W , ds)). From here, Proposition 7.6.1 implies

9.2 Main quantitative result

� 627

⃗ ⊆ C s+κ/λ−ϵ (𝒦1 , (W 2 , ds2 )), ϕu ∈ C (ϵ ,s+κ/λ−ϵ) (𝒦1 , (W , ds)) ′

completing the proof. 9.1.2.1 Qualitative Schauder estimates Similar to Corollary 9.1.10, Theorem 9.1.14 immediately gives a qualitative Schauder type estimate. We state this result here; the proof is similar to the proof of Corollary 9.1.10, and we leave it to the interested reader. Let P u(x) = ∑deg 1 (α)≤κ aα (x)(W 1 )α u(x), where aα : M → 𝕄D1 ×D2 (ℝ). Fix x0 ∈ M ds and define the frozen coefficient operator Px0 :=

∑

degds1 (α)≤κ

aα (x0 )(W 1 )α .

We make the following assumption. Main assumption: There exists a neighborhood Ω of x0 such that Px0 is maximally subelliptic of degree κ with respect to (W , ds) on Ω. Furthermore, we suppose that we are given functions u : M → ℝD2 and g : M → D1 ℝ satisfying P u = g.

Corollary 9.1.15. Let s,⃗ r⃗ ∈ (0, ∞)ν . Suppose that: ⃗ – aα ∈ C s (W , ds)⃗ near x0 , ∀ degds1 (α) ≤ κ, s⃗ – g ∈ C (W , ds)⃗ near x0 , ⃗ – u ∈ C r+κ (W , ds)⃗ near x0 . Then u ∈ C s+κ (W , ds)⃗ near x0 . ⃗

Proof. This is an immediate consequence of Theorem 9.1.14; see the proof of Corollary 9.1.10 for a similar reduction.

9.2 Main quantitative result The main results of this chapter follow from a quantitative version of Theorem 9.1.14, which we state and prove in this section. We work in the multi-parameter unit scale setting described in Section 3.15.1. Thus, we work on Bn (1) endowed with the density Vol = hσLeb , where h ∈ C ∞ (Bn (1); ℝ), with infx∈Bn (1) h(x) > 0. Fix ν ≥ 1; we assume that for each μ ∈ {1, . . . , ν} we are given μ

μ

(W μ , dsμ ) := {(W1 , ds1 ), . . . , (Wrμμ , dsμrμ )} ⊂ C ∞ (Bn (1); TBn (1)) × ℕ+ , μ

μ

(X μ , d μ ) = {(X1 , d1 ), . . . , (Xqμμ , dqμμ )} ⊂ Gen((W μ , dsμ ))

628 � 9 Nonlinear maximally subelliptic equations with (W μ , dsμ ) ⊆ (X μ , d μ ) and such that the assumptions (i) and (ii) from Section 3.15.1 hold. We will speak of ι-multi-parameter unit-admissible constants as in Definition 3.15.1. Fix D1 , D2 , κ ∈ ℕ+ such that ds1j divides κ, ∀j; define N as in (9.5). Fix r,⃗ s⃗ ∈ (0, ∞)ν and σ > 0. Suppose that: ⃗ ⃗ F(x, ζ ) ∈ C s,2⌊|s|1 ⌋+4+2ν+σ (Bn (7/8) × ℝN , (W , ds)⃗ ⊠ ∇ℝN ; ℝD1 ),

⃗ ℝD2 ), u ∈ C r+κe1 (Bn (7/8), (W , ds); ⃗

⃗ ⃗ ℝD1 ). g ∈ C s (Bn (7/8), (W , ds);

Suppose also that we have the following: α

F(x, {(W 1 ) u(x)}deg

ds1

(α)≤κ )

= g(x),

∀x ∈ Bn (3/4).

(9.12)

Because the results in this section are quantitative, it is important to use an explicit choice of norm for ‖⋅‖C s⃗ (W ,ds)⃗ and ‖⋅‖C s,t⃗ ((W ,ds)⊠∇ ⃗ ) , rather than just work with an equivℝN alence class of norms as we usually do. The choice we use is described in Section 9.2.1; for the rest of this section, we use this choice. Fix C0 ≥ 0 such that ‖F‖C s,2⌊| ⃗ s|⃗ 1 ⌋+4+2ν+σ ⃗ ((W ,ds)⊠∇

ℝN

⃗ ⃗ , ‖g‖C s⃗ (W ,ds)⃗ 1 (W ,ds) ) , ‖u‖C r+κe

≤ C0 .

Set nj := κ/ds1j ∈ ℕ+ and define P by 1 α

P v := dζ F(0, {(W ) u(0)}deg

ds1

(α)≤κ ){(W

1 α

) v}deg

ds1

(α)≤κ .

(9.13)

We assume P is maximally subelliptic in the quantitative sense that there exists A ≥ 0 with r1

n 󵄩 󵄩 ∑󵄩󵄩󵄩(Wj1 ) j f 󵄩󵄩󵄩L2 (Bn (1),hσ ;ℂD2 ) Leb

(9.14)

j=1

󵄩 󵄩 󵄩 󵄩 ≤ A(󵄩󵄩󵄩P f 󵄩󵄩󵄩L2 (Bn (1),hσ ;ℂD1 ) + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (Bn (1),hσ ;ℂD2 ) ), Leb Leb for all f ∈ C0∞ (Bn (1); ℂD2 ).

We apply Theorem 3.15.5 with σ = 7/8; let Φx,δ : Bn (1) 󳨀 → Φx,δ (Bn (1)) ⊆ B(X 1 ,d 1 ) (x, δ) ⊆ Bn (1) be as in that theorem.2 Our main nonlinear regularity theorem is the following ∼

2 Since the vector fields X11 , . . . , Xq11 are only defined on Bn (1), we have B(X 1 ,d 1 ) (x, δ) ⊆ Bn (1), ∀δ > 0, by the definition of B(X 1 ,d 1 ) (x, δ) – see (1.12) and (1.13).

9.2 Main quantitative result

�

629

Theorem 9.2.1. Suppose all of the above assumptions hold; in particular, u satisfies (9.12). There exists an (s,⃗ r,⃗ κ, σ)-multi-parameter unit-admissible constant δ = δ(A, C0 , D1 , D2 ) ∈ (0, 1) such that the following holds. Let ψ0 ∈ C0∞ (Bn (2/3)). Then (ψ0 ∘ Φ0,δ )u ∈ ⃗ ⃗ ℂD2 ) and C s+κe1 (Bn (7/8), (W , ds); 󵄩󵄩 󵄩 −1 󵄩󵄩(ψ0 ∘ Φ0,δ )u󵄩󵄩󵄩C s+κe ⃗ ⃗ ≤ C5 , 1 (W ,ds) where C5 = C5 (A, C0 , D1 , D2 , ψ0 ) ≥ 0 is an (s,⃗ r,⃗ κ, σ)-multi-parameter unit-admissible constant. Remark 9.2.2. Theorem 9.2.1 is a generalization of the corresponding result for elliptic equations: Theorem 2.8.7. The reader interested in learning the proof of Theorem 9.2.1 might find it useful to first understand the proof of Theorem 2.8.7, which has the same outline, though many of the technicalities become much simpler. To ease notation, in the proof of Theorem 9.2.1, we write Wκ1 u(x) := {(W 1 )α u(x)}deg

ds1

(α)≤κ .

We prove Theorem 9.2.1 in three steps of increasing generality: – Step I: We first prove Theorem 9.2.1 in the case of a perturbation of a linear operator, which is the heart of the proof. More precisely, when (9.12) is of the form κ

κ

P u(x) = R1 (x, W1 u(x)) + R2 (x, W1 u(x)),

(9.15)

where P is maximally subelliptic as above, ‖R1 ‖C s,⌊|⃗ s|⃗ 1 ⌋+2+ν+σ ((W ,ds)⊠∇ ⃗

)

is small, R2 ∈

((W , ds)⃗ ⊠ ∇ℝN ) with R2 (x, 0) ≡ 0 and dζ R2 (x, 0) ≡ 0, and ‖u‖C r+κe ⃗ ⃗ 1 (W ,ds) is small. Because R2 vanishes to second order at ζ = 0 and ‖u‖C r+κe ⃗ ⃗ is small, 1 (W ,ds) the right-hand side of (9.15) is small. We may therefore study the equation as a perturbation of a linear maximally subelliptic equation, and contraction mapping type arguments can be used to prove the desired regularity. Step II: We next prove Theorem 9.2.1 in the case where (9.12) is of the form C

–

ℝN

⃗ s|⃗ 1 ⌋+3+ν+σ s,⌊|

G(Wκ1 u(x)) + R3 (x, Wκ1 u(x)) = 0,

(9.16)

⌊|s|1 ⌋+3+ν+σ where ‖R3 ‖C s,⌊|⃗ s|⃗ 1 ⌋+2+ν+σ ((W ,ds)⊠∇ (ℝN ), and ‖u‖C r+κe ⃗ ⃗ ⃗ is 1 (W ,ds) ) is small, G ∈ C ℝN small. We assume that

v 󳨃→ dζ G(Wκ1 u(0))Wκ1 v

–

is maximally subelliptic, as described above. We will see that it is a straightforward calculus exercise to reduce this case to the previous one. Step III: Finally, we prove Theorem 9.2.1 by reducing it to Step II. We do this by showing that the hypotheses of Step II hold if we “zoom in” around x = 0. This zooming

630 � 9 Nonlinear maximally subelliptic equations is achieved by rescaling via Theorem 3.15.5, which retains all of the hypotheses in a quantitative way. Remark 9.2.3. In Step III of the above outline, we require more regularity in the ζ variable than in the previous steps. This is because the reduction from Step III to Step II loses some regularity in the ζ variable. This loss is probably not necessary, though we do not pursue that here. Remark 9.2.4. In this section, we will apply several results from previous chapters. Not all of these results are stated in a quantitative way. However, by keeping track of the proofs of these results, one can prove that all estimates can be taken to be in terms of multi-parameter unit-admissible constants (when applied in the setting of this section). We will freely use this fact without mentioning it in the proofs which follow. We will write A ≲ B, where A ≤ CB and C is an appropriate multi-parameter unit-admissible constant and A ≈ B for A ≲ B and B ≲ A. It will be clear from the context exactly what C depends on; in particular, we will make the dependence explicit in the statement of results and in the proofs the dependence will usually be the same as in the statement of the result. Throughout this section, r,⃗ s⃗ ∈ (0, ∞)ν , σ > 0, D1 , D2 ∈ ℕ+ , κ ∈ ℕ+ such that ds1j divides κ, ∀1 ≤ j1 ≤ r1 and nj := κ/ds1 ∈ ℕ+ , N ∈ ℕ+ , and (W , ds)⃗ are as described above. j

9.2.1 Vector fields and norms Let λ = (1, λ1 , . . . , λν ) ∈ (0, ∞)ν be as in Section 3.15.3. For δ ∈ (0, 1] and x ∈ Bn (7/8), we define μ,x,δ

(W μ,x,δ , dsμ ) := {(W1

μ

, ds1 ), . . . , (Wrμ,x,δ , dsμrμ )} μ

⊂ C ∞ (Bn (1); TBn (1)) × ℕ+ , μ,x,δ

(X μ,x,δ , d μ ) := {(X1

μ

, d1 ), . . . , (Xqμ,x,δ , dqμμ )} μ

⊂ C ∞ (Bn (1); TBn (1)) × ℕ+ ,

and hx,δ ∈ C ∞ (Bn (1)) as in Theorem 3.15.5 with σ = 7/8. We set (W x,δ , ds)⃗ := (W 1,x,δ , ds1 ) ⊠ (W 2,x,δ , ds2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (W ν,x,δ , dsν ) ⊂ C ∞ (Bn (1); TBn (1)) × (ℕν \ {0}),

(X x,δ , d )⃗ := (X 1,x,δ , d 1 ) ⊠ (X 2,x,δ , d 2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (X ν,x,δ , d ν ) ⊂ C ∞ (Bn (1); TBn (1)) × (ℕν \ {0}).

9.2 Main quantitative result

�

631

⃗ (X x,δ , d ), ⃗ and hx,δ can Remark 9.2.5. For each x ∈ Bn (7/8) and δ ∈ (0, 1], (W x,δ , ds), be viewed as data for the multi-parameter unit scale setting of Section 3.15.1. Importantly, Theorem 3.15.5 shows that multi-parameter unit-admissible constants (as in Def⃗ (X x,δ , d ), ⃗ and hx,δ can be chosen to be multiinition 3.15.1) with respect to (W x,δ , ds), ⃗ (X, d ), ⃗ and h. In short, if parameter unit-admissible constants with respect to (W , ds), ⃗ (X, d ), ⃗ and h, we may apply that result to (W x,δ , ds), ⃗ we prove a result in terms of (W , ds), x,δ ⃗ (X , d ), and hx,δ , and multi-parameter unit-admissible constants will be preserved. In this section, all results will be in terms of multi-parameter unit-admissible constants, and this allows us to freely scale those results. For each δ ∈ (0, 1]ν , we also consider the vector fields with formal degrees: 1

⃗ (δλdsW , ds)⃗ := (δλ1 ds W 1 , ds1 ) ⊠ (δλ2 ds2 W 2 , ds2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (δλν ν W ν , dsν )

⊂ C ∞ (Bn (1); TBn (1)) × (ℕν \ {0}), 1

(δλd X, d )⃗ := (δλ1 d W 1 , d 1 ) ⊠ (δλ2 d2 X 2 , d 2 ) ⊠ ⋅ ⋅ ⋅ ⊠ (δλν ν X ν , d ν ) ⃗

⊂ C ∞ (Bn (1); TBn (1)) × (ℕν \ {0}),

⃗ ⃗ ⃗ in particular, (W x,δ , ds)⃗ = (Φ∗x,δ δλdsW , ds)⃗ and (X x,δ , d )⃗ = (Φ∗x,δ δλd X, d ). ν We define the operator Dj , j ∈ ℕ , as in Section 6.11, with 𝒦 = Bn (7/8), Ω0 = n B (15/16), and a > 0 a small, 0-multi-parameter unit-admissible constant. Note that Dj ⃗ For x ∈ Bn (7/8) and δ ∈ (0, 1] we define Dx,δ in the same way with depends on (X, d ). j

⃗ and we define Dδ with (X, d )⃗ replaced by (δλd ⃗ X, d ). ⃗ (X, d )⃗ replaced by (X x,δ , d ), j ∞ n We define the following norms: for a function f ∈ L (B (1)), ‖f ‖C s⃗ (W ,ds)⃗ := sup 2j⋅s ‖Dj f ‖L∞ = ‖f ‖Bs⃗ ⃗

⃗

⃗

∞,∞ (W ,ds),(X,d),a,σ,h

j∈ℕν

‖f ‖C s⃗ (δλds⃗ W ,ds)⃗ := sup 2j⋅s ‖Dδj f ‖L∞ = ‖f ‖Bs⃗ ⃗

∞,∞ (δ

j∈ℕν

‖f ‖C s⃗ (W x,δ ,ds)⃗ := sup 2j⋅s ‖Dx,δ j f ‖L∞ = ‖f ‖Bs⃗

λds⃗ W ,ds),(δ λd ⃗ X,d),a,σ,h ⃗ ⃗

⃗

∞,∞ (W

j∈ℕν

,

x,δ ,ds),(X x,δ ,d),a,σ,h ⃗ ⃗

,

.

̃ l as in Section 7.5, For a function F(x, ζ ) ∈ L∞ (Bn (1) × ℝN ), with D ‖F‖C s,t⃗ ((W ,ds)⊠∇ ⃗

)

̃ l )F‖L∞ , := sup sup 2j⋅s+lt ‖(Dj ⊗ D

‖F‖C s,t⃗ ((δλds⃗ W ,ds)⊠∇ ⃗

)

⃗ ̃ l )F‖L∞ , := sup sup 2j⋅s+lt ‖(Dδj ⊗ D

‖F‖C s,t⃗ ((W x,δ ,ds)⊠∇ ⃗

)

̃ := sup sup 2j⋅s+lt ‖(Dx,δ j ⊗ Dl )F‖L∞ .

ℝN

ℝN

ℝN

⃗

j∈ℕν

l∈ℕ

j∈ℕν l∈ℕ

⃗

j∈ℕν l∈ℕ

s⃗ The above definitions all extend to all s⃗ ∈ ℝν , though we use the notation B∞,∞ instead

of C s to make it clear that we are not working with a Zygmund–Hölder space. That is, ⃗

632 � 9 Nonlinear maximally subelliptic equations we set, for s⃗ ∈ ℝν , ‖f ‖Bs⃗

⃗

∞,∞ (W ,ds)

‖f ‖Bs⃗

∞,∞ (δ

λds⃗ W ,ds) ⃗

‖f ‖Bs⃗

∞,∞ (W

x,δ ,ds) ⃗

:= sup 2j⋅s ‖Dj f ‖L∞ = ‖f ‖Bs⃗ ⃗

⃗

⃗

∞,∞ (W ,ds),(X,d),a,σ,h

j∈ℕν

:= sup 2j⋅s ‖Dδj f ‖L∞ = ‖f ‖Bs⃗ ⃗

∞,∞ (δ

j∈ℕν

:= sup 2j⋅s ‖Dx,δ j f ‖L∞ = ‖f ‖Bs⃗

λds⃗ W ,ds),(δ λd ⃗ X,d),a,σ,h ⃗ ⃗

⃗

j∈ℕν

,

∞,∞ (W

x,δ ,ds),(X x,δ ,d),a,σ,h ⃗ ⃗

,

.

The norms ‖ ⋅ ‖C s⃗ (W ,ds)⃗ and ‖ ⋅ ‖C s⃗ (W x,δ ,ds)⃗ take place at the “unit scale,” and because of this we will be able to directly prove quantitative estimates in terms of these norms. When δ > 0 is small, we cannot directly prove quantitative results about ‖f ‖C s⃗ (δλ W ,ds)⃗ . However, the next two results allow us to indirectly conclude quantitative estimates about these norms. ⃗ Lemma 9.2.6. Fix c0 ∈ (0, 1] and s⃗ ∈ (0, ∞)ν . There exists an s-multi-parameter units⃗ n ⃗ admissible constant C1 = C1 (c0 ) ≥ 1 such that ∀δ ∈ [c0 , 1], ∀f ∈ C (B (7/8), (W , ds)), C1−1 ‖f ‖C s⃗ (W ,ds)⃗ ≤ ‖f ‖C s⃗ (δλ W ,ds)⃗ ≤ C1 ‖f ‖C s⃗ (W ,ds)⃗ . Proof. This follows from a simple reprise of the proof of Proposition 6.3.7, keeping track of constants in that proof. Lemma 9.2.7. For x ∈ Bn (7/8), δ ∈ (0, 1], s⃗ ∈ (0, ∞)ν , and ϕ ∈ C0∞ (Bn (7/8)), we have 󵄩 󵄩󵄩 ‖ϕΦ∗x,δ f ‖C s⃗ (W x,δ ,ds)⃗ = 󵄩󵄩󵄩(ϕ ∘ Φ−1 x,δ )f 󵄩 󵄩C s⃗ (δλ W ,ds)⃗ , ∀f ∈ C0∞ (Bn (1))′ . Proof. This is the case p = ∞ of Proposition 6.11.4. 9.2.2 Bump functions A key element of Step I is the existence of bump functions at every sub-Riemannian scale, which is due to Nagel and Stein [187]. We apply Theorem 3.15.5 (with σ = 7/8); in this section we use all the same notation as in that theorem. In particular, we let ξ3 ∈ (0, 1] be the 0-multi-parameter unitadmissible constant from that theorem. Theorem 9.2.8. There exist 0-multi-parameter unit-admissible constants ξ4 , ξ5 ∈ (0, 1] with ξ5 < ξ4 < ξ3 such that ∀x ∈ Bn (7/8) and δ ∈ (0, 1], there exists ϕx,δ ∈ C0∞ (Bn (1)) and, setting ξ6 := ξ54 /ξ43 ∈ (0, ξ5 ), we have ∀x ∈ Bn (7/8), δ ∈ (0, 1]: (i) supp(ϕx,δ ) ⊆ B(W 1 ,ds1 ) (x, ξ4 δ). (ii) 0 ≤ ϕx,δ ≤ 1.

9.2 Main quantitative result

� 633

(iii) ϕx,δ ≡ 1 on a neighborhood of B(W 1 ,ds1 ) (x, ξ5 δ). (iv) ϕx, ξ5 δ ≺ ϕx,δ . (v)

ξ4

For ξ0 ∈ (0, 1], supp(ϕx,ξ0 δ ) ⊆ Φx,δ (Bn (1/2)). In particular, supp(ϕx, ξ4 ξ δ ), supp(ϕx,( ξ4 )2 ξ δ ), supp(ϕx,( ξ4 )3 ξ δ ) ⊆ Φx,δ (Bn (1/2)). ξ5 6

ξ5

6

ξ5

6

(vi) For every α, ⃗ α 󵄩 󵄩 sup 󵄩󵄩󵄩(δλd X) ϕx,δ 󵄩󵄩󵄩C(Bn (1)) ≤ Cα , n

x∈B (7/8) δ∈(0,1]

where Cα is an α-multi-parameter unit-admissible constant. (vii) For ξ0 ∈ (0, 1] and L ∈ ℕ, 1

−L max1≤j≤q1 dj 󵄩󵄩 󵄩 , 󵄩󵄩ϕx,ξ0 δ ∘ Φx,δ 󵄩󵄩󵄩C L (Bn (1)) ≤ CL ξ0

where CL ≥ 0 is an L-multi-parameter unit-admissible constant. (viii) There exists a 0-multi-parameter unit-admissible constant N1 ∈ ℕ+ such that ∀s⃗ ∈ ⃗ (0, ∞)ν , there exists Cs⃗ ≥ 0 an s-multi-parameter unit-admissible constant such that n ∀x ∈ B (3/4) and δ ∈ (0, 1] with B(W 1 ,ds1 ) (x, δ) ⊆ Bn (3/4), there exist x1 , . . . , xN1 ∈ Bn (3/4) with B(W 1 ,ds1 ) (xj ,

ξ43 ξ6 ξ53

δ) ⊆ Bn (3/4),

(9.17)

N1

‖ϕx,δ u‖C s⃗ (δλds⃗ W ,ds)⃗ ≤ Cs⃗ ∑ ‖ϕxj ,ξ6 δ u‖C s⃗ (δλds⃗ W ,ds)⃗ , j=1

(9.18)

for all u ∈ C0∞ (Bn (1))′ . (ix) There exist 0-multi-parameter unit-admissible constants N2 ∈ ℕ+ and δ1 ∈ (0, 1] and x1 , . . . , xN2 ∈ Bn (3/4) with B(W 1 ,ds1 ) (xj , δ1 ) ⊆ Bn (3/4), 1 ≤ j ≤ N2 , such that ⃗ ∀s⃗ ∈ (0, ∞)ν , ∀ψ ∈ C0∞ (Bn (2/3)), there exists an s-multi-parameter unit-admissible constant Cs⃗ = Cs⃗ (ψ) ≥ 0 such that N2

‖ψu‖C s⃗ (W ,ds)⃗ ≤ Cs⃗ ∑ ‖ϕxj ,δ1 u‖C s⃗ (δλds⃗ W ,ds)⃗ . j=1

(x)

1

(9.19)

Fix ξ0 ∈ (0, 1] and s⃗ ∈ (0, ∞)ν . Then 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩ϕx,ξ0 δ u󵄩󵄩󵄩C s⃗ (δλds⃗ W ,ds)⃗ ≈ 󵄩󵄩󵄩ϕx,ξ0 δ u󵄩󵄩󵄩C s⃗ ((ξ δ)λds⃗ W ,ds)⃗ , 0 ⃗ ∀u ∈ C0∞ (Bn (1))′ , where the implicit constants are s-multi-parameter unit-admissible constants which depend on ξ0 .

634 � 9 Nonlinear maximally subelliptic equations Proof. With ξ3 , δ0 ∈ (0, 1] as in Theorem 3.15.5, we set ξ4 := (ξ3 /2)∧(δ0 /2). By Lemma 3.2.4 and Theorem 3.15.5, there exists a 0-multi-parameter unit-admissible constant η2 ∈ (0, 1/2] so small that ∀x ∈ Bn (7/8) and δ ∈ (0, 1], Bn (η2 ) ⊆ B(ξ ds1 W 1,x,δ ,ds1 ) (0, 1) = B(W 1,x,δ ,ds1 ) (0, ξ4 ).

(9.20)

4

1

Using the fact that Φx,δ (B(W 1,x,δ ,ds1 ) (0, ξ3 )) = B(W 1 ,ds1 ) (x, ξ3 δ) (since Φ∗x,δ δds W 1 =

W 1,x,δ , by definition), (9.20) and Theorem 3.15.5 (f) show that

Bn (η2 ) ⊆ B(W 1,x,δ ,ds1 ) (0, ξ4 ) ⊆ B(W 1,x,δ ,ds1 ) (0, ξ3 ) ⊆ Bn (1/2).

(9.21)

Take ϕ0 ∈ C0∞ (Bn (η2 )) such that ϕ0 ≡ 1 on Bn (η2 /2) and 0 ≤ ϕ0 ≤ 1; note that ϕ0 depends only on η2 and n – it does not depend on x or δ. By the Picard–Lindelöf theorem and Theorem 3.15.5 (h) (with L = 1), we make take ξ5 ∈ (0, ξ4 ) a multi-parameter unit-admissible constant so small that B(W 1,x,δ ,ds1 ) (0, ξ5 ) ⊆ Bn (η2 /4), ∀x ∈ Bn (7/8), δ ∈ (0, 1]. In particular, we have ϕ0 ≡ 1 on a neighborhood of B(W 1,x,δ ,ds1 ) (0, ξ5 ) ⊆ Bn (η4 /2),

(9.22)

∀x ∈ Bn (7/8), δ ∈ (0, 1]. n Set ϕx,δ := ϕ0 ∘ Φ−1 x,δ . Since ϕ0 is supported on B (η2 ) ⊆ B(W 1,x,δ ,ds1 ) (x, ξ4 ) and Φx,δ (B(W 1,x,δ,ds1 ) (x, ξ4 )) = B(δds1 W 1 ,ds1 ) (x, ξ4 ) = B(W 1 ,ds1 ) (x, ξ4 δ), (i) follows. Since 0 ≤ ϕ0 ≤ 1, (ii) follows immediately from the definition. Applying Φx,δ to (9.22) shows that ϕx,δ ≡ 1 on a neighborhood of B(W 1 ,ds1 ) (x, ξ5 δ), establishing (iii). (iv): Using (i) we have supp(ϕx, ξ5 δ) ⊆ B(W 1 ,ds1 ) (x, ξ5 δ). ξ4

Applying (iii) shows that ϕx,δ ≡ 1 on a neighborhood of supp(ϕx, ξ5 δ), establishing (iv). (v): For ξ0 ∈ (0, 1], we have, using (i) and Theorem 3.15.5 (f),

ξ4

supp(ϕx,ξ0 δ ) ⊆ B(W 1 ,ds1 ) (x, ξ4 ξ0 δ) ⊆ B(W 1 ,ds1 ) (x, ξ3 δ) ⊆ Φx,δ (Bn (1/2)), establishing (v). 1 (vi): We have, for x ∈ Bn (7/8) and δ ∈ (0, 1], using the fact that Φ∗x,δ δd X 1 = X 1,x,δ , 󵄩󵄩 λd ⃗ α 󵄩 󵄩 x,δ α −1 󵄩 󵄩󵄩(δ X) ϕx,δ 󵄩󵄩󵄩C(Bn (1)) = 󵄩󵄩󵄩((X ) ϕ0 ) ∘ Φx,δ 󵄩󵄩󵄩C(Bn (1)) α 󵄩 󵄩 = 󵄩󵄩󵄩(X x,δ ) ϕ0 󵄩󵄩󵄩C(Φ (Bn (1))) ≲ 1, x,δ

9.2 Main quantitative result

� 635

where in the last estimate, we have used the fact that ϕ0 ∈ C0∞ (Bn (η2 )) is a fixed function and we have used Theorem 3.15.5 (h). (vii): Using Theorem 3.15.5 (j), we have 󵄩󵄩 󵄩 󵄩 1,x,δ α 󵄩 ) ϕx,ξ0 δ ∘ Φx,δ 󵄩󵄩󵄩C(Bn (1)) 󵄩󵄩ϕx,ξ0 δ ∘ Φx,δ 󵄩󵄩󵄩C L (Bn (1)) ≈ ∑ 󵄩󵄩󵄩(X |α|≤L

− degd 1 (α) 󵄩

1 󵄩󵄩((ξ0 δ)d X 1 )α ϕx,ξ δ 󵄩󵄩󵄩 n ≲ ∑ ξ − degd 1 (α) , 󵄩 0 0 󵄩C(B (1))

= ∑ ξ0 |α|≤L

|α|≤L

where the final estimate used (vi). Item (vii) follows. (viii): Item (viii) is trivial when the right-hand side of (9.18) is infinite, so we assume the right-hand side is finite. The statement of (viii) has N1 independent of x and δ, but by allowing repetitions in x1 , . . . , xN1 , it suffices to prove (viii) with N1 ≤ C1 , where C1 is a 0-multi-parameter unit-admissible constant (that is, N1 may depend on x and δ, so long as it is bounded by a 0-multi-parameter unit-admissible constant). Let δ ∈ (0, 1] and x ∈ Bn (3/4) be such that B(W 1 ,ds1 ) (x, δ) ⊆ Bn (3/4). We will pick x1 , . . . , xN1 ∈ B(W 1 ,ds1 ) (x, ξ4 δ). Note that we will then have, by the choices of ξ6 and ξ4 , for 1 ≤ j ≤ N1 , B(W 1 ,ds1 ) (xj ,

ξ43 ξ6 ξ53

δ) = B(W 1 ,ds1 ) (xj , ξ5 δ) ⊆ B(W 1 ,ds1 ) (x, 2ξ4 δ) n

(9.23)

⊆ B(W 1 ,ds1 ) (x, δ) ⊆ B (3/4),

and therefore (9.17) follows. In particular, xj ∈ Bn (3/4). We turn to choosing x1 , . . . , xN1 . Pick from the collection {B(W 1 ,ds1 ) (y,

ξ5 ξ6 δ) : y ∈ B(W 1 ,ds1 ) (x, ξ4 δ)} 8

a maximal disjoint subcollection: B(W 1 ,ds1 ) (x1 ,

ξ5 ξ6 ξξ δ) , . . . , B(W 1 ,ds1 ) (xN1 , 5 6 ) . 8 8

B(W 1 ,ds1 ) (x1 ,

ξ5 ξ6 ξξ δ) , . . . , B(W 1 ,ds1 ) (xN1 , 5 6 ) 2 2

(9.24)

By maximality,

is a cover for B(W 1 ,ds1 ) (x, ξ4 δ). Indeed, suppose that there were a y ∈ B(W 1 ,ds1 ) (x, ξ4 δ) \ ξξ

ξξ

⋃ B(W 1 ,ds1 ) (xj , 52 6 δ). Then B(W 1 ,ds1 ) (y, 58 6 δ) would be disjoint from the collection (9.24), contradicting maximality. We claim N1 ≤ C0 , where C0 ≥ 0 is a 0-multi-parameter unit-admissible constant. Indeed, (9.23) implies each xj ∈ Bn (3/4), and therefore since 2ξ4 ≤ δ0 , repeated application of Theorem 3.15.5 (c) gives

636 � 9 Nonlinear maximally subelliptic equations Vol (B(W 1 ,ds1 ) (xj ,

ξ5 ξ6 δ)) ≳ Vol(B(W 1 ,ds1 ) (xj , 2ξ4 δ)) 8 ≥ Vol(B(W 1 ,ds1 ) (x, ξ4 δ)),

(9.25)

where in the final estimate we used B(W 1 ,ds1 ) (x, ξ4 δ) ⊆ B(W 1 ,ds1 ) (xj , 2ξ4 δ). ξξ

Since ξ5 ξ6 /8 ≤ ξ4 , we have B(W 1 ,ds1 ) (xj , 58 6 δ) ⊆ B(W 1 ,ds1 ) (x, 2ξ4 ). Combining this with the disjointness of the balls in (9.24) and Theorem 3.15.5 (c), we have Vol(B(W 1 ,ds1 ) (x, ξ4 δ)) ≳ Vol(B(W 1 ,ds1 ) (x, 2ξ4 δ)) N1

≥ ∑ Vol (B(W 1 ,ds1 ) (xj , j=1

ξ5 ξ6 δ)) ≳ N1 Vol(B(W 1 ,ds1 ) (x, ξ4 δ)), 8

where the last estimate used (9.25). We conclude that N1 ≲ 1, as desired. Using (ii), we have N1

0 ≤ ∑ ϕxj ,ξ6 δ ≤ N1 . j=1

Moreover, by (iii) we have ϕxj ,ξ6 δ ≡ 1 on B(W 1 ,ds1 ) (xj , ξ5 ξ6 δ), and since B(W 1 ,ds1 ) (xj , ξ5 ξ6 δ/2), 1 ≤ j ≤ N1 , is a cover for B(W 1 ,ds1 ) (x, ξ4 δ), we have N1

1 ≤ ∑ ϕxj ,ξ6 δ on B(W 1 ,ds1 ) (x, ξ4 δ). j=1

(9.26)

Set gx,δ :=

ϕx,δ . N1 ∑j=1 ϕxj ,ξ6 δ

(9.27)

By (9.26), (i), and (ii), we have 0 ≤ gx,δ ≤ 1. Moreover, using (vi) we also have, ∀L ∈ ℕ, α 󵄩 1 󵄩 ∑ 󵄩󵄩󵄩(δd X 1 ) gx,δ 󵄩󵄩󵄩C(Bn (1)) ≤ CL ,

|α|≤L

where CL ≥ 0 is an L-multi-parameter unit-admissible constant. In particular, we have α α 󵄩 󵄩 󵄩 1 󵄩 ∑ 󵄩󵄩󵄩(X 1,x,δ ) (gx,δ ∘ Φx,δ )󵄩󵄩󵄩C(Bn (1)) ≤ ∑ 󵄩󵄩󵄩(δd X 1 ) gx,δ 󵄩󵄩󵄩C(Bn (1)) ≤ CL .

|α|≤L

|α|≤L

By Theorem 3.15.5 (j), this gives 󵄩󵄩 −1 󵄩 󵄩󵄩gx,δ ∘ Φx,δ 󵄩󵄩󵄩C L ≲L 1, where the implicit constant is an L-multi-parameter unit-admissible constant.

(9.28)

9.2 Main quantitative result

�

637

We claim supp(ϕxj ,ξ6 δ ∘ Φx,δ ) ⊆ Bn (1/2). Indeed, we have, by (i) and the fact that xj ∈ B(W 1 ,ds1 ) (x, ξ4 δ), supp(ϕxj ,ξ6 δ ) ⊆ B(W 1 ,ds1 ) (xj , ξ6 ξ4 δ)

⊆ B(W 1 ,ds1 ) (xj , ξ4 δ) ⊆ B(W 1 ,ds1 ) (x, 2ξ4 δ) ⊆ B(W 1 ,ds1 ) (x, ξ3 δ).

Therefore, supp(ϕxj ,ξ6 δ ∘ Φx,δ ) ⊆ B(W 1,x,δ ,ds1 ) (x, ξ3 ) ⊆ Bn (1/2),

(9.29)

where the final containment uses Theorem 3.15.5 (f). We have 󵄩󵄩 N1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩ϕx,δ u󵄩󵄩C s⃗ (δλds⃗ W ,ds)⃗ = 󵄩󵄩󵄩(∑ ϕxj ,ξ6 δ ) gx,δ u󵄩󵄩󵄩󵄩 󵄩󵄩 j=1 󵄩󵄩 s⃗ λds⃗ 󵄩 󵄩C (δ W ,ds)⃗ N1

(9.30)

󵄩 󵄩 ∑󵄩󵄩󵄩gx,δ ϕxj ,ξ6 δ u󵄩󵄩󵄩C s⃗ (δλds⃗ W ,ds)⃗ j=1

N1

󵄩 󵄩 = ∑󵄩󵄩󵄩(gx,δ ∘ Φx,δ )(ϕxj ,ξ6 δ ∘ Φx,δ )u ∘ Φx,δ 󵄩󵄩󵄩C s⃗ (W x,δ ,ds)⃗ , j=1

where the last line uses Lemma 9.2.7 and the fact that ϕx,ξ6 δ ∘ Φx,δ ∈ C0∞ (Bn (1/2)) (which uses (9.29)). By (9.28), the C L norm of gx,δ ∘ Φx,δ is bounded by an L-multi-parameter unitadmissible constant. Thus, it follows from Corollary 6.5.10 that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩(gx,δ ∘ Φx,δ )w󵄩󵄩󵄩C s⃗ (W x,δ ,ds)⃗ ≤ Cs⃗ 󵄩󵄩󵄩w󵄩󵄩󵄩C s⃗ (W x,δ ,ds)⃗ ,

(9.31)

⃗ ⃗ where Cs⃗ ≥ 0 is an s-multi-parameter ⃗ ∀w ∈ C s (Bn (7/8), (W x,δ , ds)), unit-admissible constant. At the start of the proof to this part, we reduced to the case where the right-hand side of (9.18) is finite. Therefore, by Lemma 9.2.7 (and using (9.29)), we have

󵄩󵄩 󵄩 󵄩󵄩(ϕxj ,ξ6 δ ∘ Φx,δ )u ∘ Φx,δ 󵄩󵄩󵄩C s⃗ (W x,δ ,ds)⃗ < ∞. Combining this with (9.29), Proposition 6.10.1 shows that ⃗ (ϕxj ,ξ6 δ ∘ Φx,δ )u ∘ Φx,δ ∈ C s (Bn (7/8), (W x,δ , ds)). ⃗

Thus, (9.31) applies to show that 󵄩󵄩 󵄩 󵄩󵄩(gx,δ ∘ Φx,δ )(ϕxj ,ξ6 δ ∘ Φx,δ )u ∘ Φx,δ 󵄩󵄩󵄩C s⃗ (W x,δ ,ds)⃗ 󵄩 󵄩 ≤ Cs⃗ 󵄩󵄩󵄩(ϕxj ,ξ6 δ ∘ Φx,δ )u ∘ Φx,δ 󵄩󵄩󵄩C s⃗ (W x,δ ,ds)⃗ .

(9.32)

638 � 9 Nonlinear maximally subelliptic equations Plugging (9.32) into (9.30) gives N1

󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩ϕx,δ u󵄩󵄩󵄩C s⃗ (δλds⃗ W ,ds)⃗ ≤ Cs⃗ ∑󵄩󵄩󵄩(ϕxj ,ξ6 δ ∘ Φx,δ )u ∘ Φx,δ 󵄩󵄩󵄩C s⃗ (W x,δ ,ds)⃗ . j=1

(9.33)

Lemma 9.2.7 (also using (9.29)) shows that (9.33) is equivalent to (9.18), which completes the proof of (viii). (ix): This is a simple reprise of some of the ideas in the proof of (viii). As in the proof of (viii), by allowing repetitions in x1 , . . . , xN2 , it suffices to prove the result with N2 ≤ C0 , where C0 ≥ 0 is a 0-multi-parameter unit-admissible constant. Also, the result is trivial when the right-hand side of (9.19) is infinite, so we assume that the right-hand side is finite. By the Picard–Lindelöf theorem and the case L = 1 of Theorem 3.15.5 (h), we may take δ0 ∈ (0, 1] to be a 0-multi-parameter unit-admissible constant so small that B(W 1 ,ds1 ) (x, δ1 ) ⊆ Bn (3/4),

∀x ∈ Bn (2/3).

(9.34)

Pick from the collection {B(W 1 ,ds1 ) (y, ξ5 δ1 /8) : y ∈ Bn (2/3)} a maximal disjoint subcollection B(W 1 ,ds1 ) (x1 , ξ5 δ1 /8), . . . , B(W 1 ,ds1 ) (xN2 , ξ5 δ1 /8).

(9.35)

As in the proof of (viii), maximality implies that B(W 1 ,ds1 ) (x1 , ξ5 δ1 /2), . . . , B(W 1 ,ds1 ) (xN2 , ξ5 δ1 /2) is a cover for Bn (2/3). Since ξ5 ≤ ξ4 ≤ δ0 /2, we have ξ5 δ1 /2 ∈ (0, δ0 ]. In particular, it follows from Theorem 3.15.5 (b) that Vol(B(W 1 ,ds1 ) (xj , ξ5 δ1 /8)) ≈ Λ(xj , ξ5 δ1 /8) ≈ 1, where the final ≈ used the formula for Λ and the fact that ξ5 δ2 /2 ≈ 1. Thus, using the disjointness of the balls in (9.35) and using (9.34), we have N2

1 ≈ Vol(Bn (3/4)) ≥ ∑ Vol(B(W 1 ,ds1 ) (xj , ξ5 δ1 /8)) ≳ N2 . j=1

We conclude N2 ≲ 1, i. e., N2 ≤ C0 , where C0 ≥ 0 is a 0-multi-parameter unit-admissible constant.

9.2 Main quantitative result

� 639

N

By (ii), 0 ≤ ∑j=12 ϕxj ,δ1 ≤ N2 . Since ϕxj ,δ1 ≡ 1 on B(W 1 ,ds1 ) (xj , ξ5 δ1 ) (by (iii)) and N

B(W 1 ,ds1 ) (xj , ξ5 δ1 /2) form a cover for Bn (2/3), we have 1 ≤ ∑j=12 ϕxj ,δ1 on Bn (2/3). Since δ1 ≈ 1, (vii) shows that ‖ϕxj ,δ1 ‖C L (Bn (1)) ≲L 1, for every L ∈ ℕ, where the implicit constant is an L-multi-parameter unit-admissible constant. For ψ ∈ C0∞ (Bn (2/3)), we set ψ

gψ :=

N ∑j=12 ϕxj ,δ

.

By the above remarks, we have gψ ∈ C0∞ (Bn (2/3)) and ‖gψ ‖C L (Bn (1)) ≤ CL , where CL = CL (ψ) ≥ 0 is an L-multi-parameter unit-admissible constant. Thus, we have 󵄩󵄩 N2 󵄩󵄩 N2 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 ≤ ∑󵄩󵄩󵄩gψ ϕxj ,δ1 u󵄩󵄩󵄩C s⃗ (W ,ds)⃗ . 󵄩󵄩ψu󵄩󵄩C s⃗ (W ,ds)⃗ = 󵄩󵄩󵄩󵄩∑ gψ ϕxj ,δ1 u󵄩󵄩󵄩󵄩 󵄩󵄩j=1 󵄩󵄩 s⃗ j=1 󵄩 󵄩C (W ,ds)⃗

(9.36)

Since supp(ϕxj ,δ1 ) ⊆ B(W 1 ,ds1 ) (xj , ξ4 δ1 ) ⊆ Bn (3/4), by (9.34), as we are assuming 󵄩󵄩 󵄩 ⃗ ⃗ Thus, 󵄩󵄩ϕxj ,δ1 u󵄩󵄩󵄩C s⃗ (W ,ds)⃗ < ∞, Proposition 6.10.1 implies that ϕxj ,δ1 u ∈ C s (Bn (3/4), (W , ds)). using Corollary 6.5.10, we have 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩gψ ϕxj ,δ1 u󵄩󵄩󵄩C s⃗ (W ,ds)⃗ ≤ Cs⃗ 󵄩󵄩󵄩ϕxj ,δ1 u󵄩󵄩󵄩C s⃗ (W ,ds)⃗ ,

(9.37)

where Cs⃗ = Cs⃗ (ψ) ≥ 0 is an L-multi-parameter unit-admissible constant. Using Lemma 9.2.6 and the fact that δ1 ≈ 1, we have 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩ϕxj ,δ1 u󵄩󵄩󵄩C s⃗ (W ,ds)⃗ ≈s⃗ 󵄩󵄩󵄩ϕxj ,δ1 u󵄩󵄩󵄩C s⃗ (δλds⃗ W ,ds)⃗ . 1

(9.38)

Combining (9.36), (9.37), and (9.38) yields (9.19) and completes the proof. (x): Fix γ1 , γ2 ∈ [ξ0 , 1]. We will show that ‖ϕx,ξ0 δ u‖C s⃗ ((γ δ)λds⃗ W ,ds)⃗ ≲ ‖ϕx,ξ0 δ u‖C s⃗ ((γ δ)λds⃗ W ,ds)⃗ . 1

2

(9.39)

The result will then follow by applying (9.39) with γ1 = ξ0 and γ2 = 1 and with γ1 = 1 and γ2 = ξ0 . Suppose that ‖ϕx,ξ0 δ u‖C s⃗ ((γ δ)λds⃗ W ,ds)⃗ ≤ B < ∞. 2

(9.40)

We will show that ‖ϕx,ξ0 δ u‖C s⃗ ((γ δ)λds⃗ W ,ds)⃗ ≲ B, which will establish (9.39) and complete 1 the proof. By (i), we have supp(ϕx,ξ0 δ ) ⊆ B(W 1 ,ds1 ) (x, ξ0 ξ4 δ) = B(W 1 ,ds1 ) (x, γ2 δξ4 ξ0 /γ2 ). Therefore, supp(ϕx,ξ0 δ ∘ Φx,γ2 δ ) ⊆ B(W 1,x,γ2 δ ,ds1 ) (0, ξ0 ξ4 /γ2 ) ⊆ B(W 1,x,γ2 δ ,ds1 ) (0, ξ4 ) ⊆ Bn (7/8).

640 � 9 Nonlinear maximally subelliptic equations Using this and Lemma 9.2.7, we see that (9.40) is equivalent to 󵄩󵄩 󵄩 󵄩󵄩(ϕx,ξ0 δ ∘ Φx,γ2 δ )u ∘ Φx,γ2 δ 󵄩󵄩󵄩C s⃗ (W 1,x,γ2 δ ,ds)⃗ ≤ B.

(9.41)

⃗ ⃗ By Proposition 6.10.1, this implies that (ϕx,ξ0 δ ∘Φx,γ2 δ )u∘Φx,γ2 δ ∈ C s (Bn (7/8), (W 1,x,γ2 δ , ds)). ∞ n Proposition 7.2.1 shows that (9.41) implies that there exists a sequence v̂j ∈ C (B (1)), j ∈ ℕν , such that

∑ v̂j = (ϕx,ξ0 δ ∘ Φx,γ2 δ )u ∘ Φx,γ2 δ

j∈ℕν

and 󵄩󵄩 −jds⃗ x,γ2 δ α ̂ 󵄩󵄩 ) vj 󵄩󵄩L∞ (Bn (1)) ≲α 2−j⋅s B, 󵄩󵄩(2 W

∀α.

∞ Letting vj := v̂j ∘ Φ−1 x,γ2 δ , we have vj ∈ C0 (M), ∑j∈ℕν vj = ϕx,ξ0 δ u, and λds⃗ −jds⃗ α 󵄩 󵄩󵄩 −j⋅s 󵄩󵄩((γ2 δ) 2 W ) vj 󵄩󵄩󵄩L∞ ≲α 2 B.

Since γ2 ≈ γ1 , this implies that λds⃗ −jds⃗ α 󵄩 󵄩󵄩 −j⋅s 󵄩󵄩((γ1 δ) 2 W ) vj 󵄩󵄩󵄩L∞ ≲α 2 B.

Letting ṽj := vj ∘ Φx,γ1 ,δ , we see that ṽj ∈ C ∞ (Bn (1)) with 󵄩󵄩 −jds⃗ x,γ1 δ α ̃ 󵄩󵄩 ) vj 󵄩󵄩L∞ (Bn (1)) ≲α 2−j⋅s B, 󵄩󵄩(2 W

∀α,

and ∑j∈ℕν ṽj = (ϕx,ξ0 δ ∘ Φx,γ1 δ )u ∘ Φx,γ1 δ . By Proposition 7.2.1, this implies that 󵄩󵄩 󵄩 󵄩󵄩(ϕx,ξ0 δ ∘ Φx,γ1 δ )u ∘ Φx,γ1 δ 󵄩󵄩󵄩C s⃗ (W 1,x,γ1 δ ,ds)⃗ ≲ B. As before, Lemma 9.2.7 implies that this is equivalent to ‖ϕx,ξ0 δ u‖C s⃗ ((γ δ)λds⃗ W ,ds)⃗ ≲ B, 1

completing the proof.

9.2.3 Scaled estimates As described in Section 9.2.1, we may apply the previous results in this text to the norms ‖ ⋅ ‖C s⃗ (W x,δ ,ds)⃗ and all estimates will be multi-parameter unit-admissible constants. This is not true of the norms ‖ ⋅ ‖C s⃗ (δλds⃗ W ,ds)⃗ ; when previous proofs are applied to these norms,

9.2 Main quantitative result

�

641

the estimates blow up as δ ↓ 0. However, by using Lemma 9.2.7 we can conclude results about the norms ‖⋅‖C s⃗ (δλds⃗ W ,ds)⃗ . In this section, we use this idea to rephrase several earlier results in the text in terms of the norms ‖ ⋅ ‖C s⃗ (δλds⃗ W ,ds)⃗ . We let ϕx,δ and ξ4 , ξ5 , and ξ6 be as in Theorem 9.2.8. Recall that κ ∈ ℕ+ is such that ds1j divides κ, ∀1 ≤ j ≤ r1 , and nj = κ/ds1j ∈ ℕ+ . Let P be a partial differential operator of the form P=

∑

α

degds1 (α)≤κ

aα (W 1 ) ,

where aα ∈ 𝕄D1 ×D2 (ℝ) are constant matrices. Fix B ≥ 0 such that maxα |aα | ≤ B. Proposition 9.2.9. Suppose P is maximally subelliptic of degree κ with respect to (W 1 , ds1 ), in the sense that ∃A ≥ 0 with r1

n 󵄩 󵄩 󵄩 󵄩 ∑󵄩󵄩󵄩(Wj1 ) j f 󵄩󵄩󵄩L2 (Bn (1),hσ ) ≤ A(󵄩󵄩󵄩P f 󵄩󵄩󵄩L2 (Bn (1),hσ ) + ‖f ‖L2 (Bn (1),hσLeb ) ), Leb Leb j=1

∀f ∈ C0∞ (Bn (31/32); ℂD2 ). Fix s⃗ ∈ ℝν . For every x ∈ Bn (7/8) and δ ∈ (0, 1], we have ‖ϕx,ξ6 δ u‖Bs+κe ⃗ 1 (δ λds⃗ W ,ds) ⃗ ∞,∞

󵄩󵄩 󵄩󵄩 ≤ C(‖ϕx, ξ4 ξ6 δ δ2κ P ∗ P u‖Bs−κe ⃗ 1 (δ λds⃗ W ,ds) ⃗ +󵄩 󵄩ϕx, ξ4 ξ6 δ u󵄩󵄩L∞ ), ∞,∞

ξ5

(9.42)

ξ5

⃗ s−κe 1 ⃗ where C = C(A, B, D1 , D2 ) ≥ ∀u ∈ C0∞ (Bn (1); ℂD2 )′ with P ∗ P u ∈ B∞,∞ (Bn (7/8), (W , ds)), 0 is an (s,⃗ κ)-multi-parameter unit-admissible constant.

Proof. By Theorem 8.1.1 (i) ⇒ (iv), P ∗ P is maximally subelliptic of degree 2κ with respect to (W , ds)⃗ in the sense that r1

2n 󵄩 󵄩 ∑󵄩󵄩󵄩(Wj1 ) j f 󵄩󵄩󵄩L2 (Bn (1),hσ

Leb )

j=1

󵄩 󵄩 ≲ 󵄩󵄩󵄩P ∗ P f 󵄩󵄩󵄩L2 (Bn (1),hσ

Leb )

+ ‖f ‖L2 (Bn (1),hσLeb ) ,

(9.43)

∀f ∈ C0∞ (Bn (15/16)). For ξ0 ∈ {ξ6 , ξ4 ξ6 /ξ5 }, set ψx,ξ0 δ := ϕx,ξ0 δ ∘ Φx,δ . Theorem 9.2.8 (v) and (vii) shows that ψx,ξ0 δ ∈ C0∞ (Bn (1)) and for every L ∈ ℕ, ‖ψx,ξ0 δ ‖C L ≲L 1, where the implicit constant is an L-multi-parameter unit-admissible constant. Theorem 9.2.8 (iv) shows that ψx,ξ6 δ ≺ ψx, ξ4 ξ6 δ . ξ5

From here, the proof is very similar to the proof of Proposition 8.7.3, using Corollary 8.7.2; we describe the main ideas. Set L x,δ := Φ∗x,δ δ2κ P ∗ P (Φx,δ )∗ . By Lemma 9.2.7, (9.42) follows from x,δ ‖ψx,ξ6 δ v‖Bs+κe v‖Bs−κe ⃗ ⃗ 1 (W x,δ ,ds) 1 (W x,δ ,ds) ⃗ ≲ ‖ψx, ξ4 ξ6 δ L ⃗ + ‖ψx, ξ4 ξ6 δ v‖L∞ , ∞,∞

ξ5

∞,∞

ξ5

(9.44)

642 � 9 Nonlinear maximally subelliptic equations for all v ∈ C0∞ (Bn (1))′ with ψx, ξ4 ξ6 δ v ∈ L∞ (Bn (1)). ξ5

To prove (9.44), we will apply Corollary 8.7.2. Just as in Section 3.3.1 and the proof of Proposition 8.7.3, we have L

x,δ

=

α

∑

degds1 (α)≤2κ

1,x,δ bx,δ ) , α (W

∞ n D2 ×D2 where bx,δ ) and for every L ∈ ℕ, ‖bx,δ α ∈ C (B (1); 𝕄 α ‖C L ≲L 1. Also, following the proof in Section 3.3.1 (or, equivalently, the proof of Proposition 8.7.3), (9.43) implies that r1

2n 󵄩 󵄩 ∑󵄩󵄩󵄩(Wj1,x,δ ) j f 󵄩󵄩󵄩L2 (Bn (1),h j=1

󵄩 󵄩 ≲ 󵄩󵄩󵄩L x,δ f 󵄩󵄩󵄩L2 (Bn (1),h

x,δ σLeb )

x,δ σLeb )

+ ‖f ‖L2 (Bn (1),hx,δ σLeb ) ,

∀f ∈ C0∞ (Bn (1)), where hx,δ is as in Theorem 3.15.5. Thus, we have verified that all the hypotheses of Corollary 8.7.2 hold (when applied to L x,δ ) and that result (with p = ∞) implies (9.44) and completes the proof. Proposition 9.2.10. Let E∞ ∈ C ∞ (Bn (1) × Bn (1)) and fix ξ0 ∈ (0, 1]. Then, ∀s⃗ ∈ ℝν , there exists L ∈ ℕ, Cs⃗ ≥ 0, such that ∀x ∈ Bn (7/8), δ ∈ (0, 1], ‖ϕx,ξ0 δ E∞ u‖Bs⃗

∞,∞ (δ

λds⃗ W ,ds) ⃗

≤ Cs⃗ ‖E∞ ‖C L (Bn (1)×Bn (1)) ‖u‖L∞ ,

(9.45)

∀u ∈ L∞ (Bn (1)), where L = L(s)⃗ ∈ ℕ is a 0-multi-parameter unit-admissible constant and ⃗ Cs⃗ = Cs⃗ (ξ0 ) ≥ 0 is an s-multi-parameter unit-admissible constant. Proof. Since the left-hand side of (9.45) is increasing in s,⃗ it suffices to prove the result for s⃗ ∈ (0, ∞)ν , when ‖ ⋅ ‖Bs⃗ (δλds⃗ W ,ds)⃗ = ‖ ⋅ ‖C s⃗ (δλds⃗ W ,ds)⃗ . ∞,∞

⃗ that, ∀L ∈ ℕ, It follows from Theorem 9.2.8 (vi) (and using (W , ds)⃗ ⊆ (X, d )) 󵄩󵄩 λds⃗ α 󵄩 󵄩󵄩(δ W ) ϕx,ξ0 δ E∞ u󵄩󵄩󵄩L∞ (Bn (1)) ≲L ‖E∞ ‖C L ‖u‖L∞ .

∑

degds⃗ (α)≤L

Thus, ∀L ∈ ℕ, ∑

degds⃗ (α)≤L

󵄩󵄩 x,δ α ∗ 󵄩 󵄩󵄩(W ) Φx,δ (ϕx,ξ0 δ E∞ u)󵄩󵄩󵄩L∞ (Bn (1)) ≲L ‖E∞ ‖C L ‖u‖L∞ .

By Proposition 7.2.1 (see (7.4)), it follows that for some multi-parameter unit-admissible constant L = L(s)⃗ ∈ ℕ, we have 󵄩󵄩 ∗ 󵄩 󵄩󵄩Φx,δ (ϕx,ξ0 δ E∞ u)󵄩󵄩󵄩C s⃗ (W x,δ ,ds)⃗ ≲ ‖E∞ ‖C L ‖u‖L∞ . Lemma 9.2.7 shows that (9.46) is equivalent to (9.45), completing the proof.

(9.46)

9.2 Main quantitative result

� 643

Proposition 9.2.11. Fix ξ0 ∈ (0, 1]. Then, ∀s⃗ ∈ ℝν , ∃Cs⃗ ≥ 0 such that ∀x ∈ Bn (7/8), δ ∈ (0, 1], ‖ϕx,ξ

ξ5 0ξ 4

δ

δκ P u‖Bs⃗

∞,∞ (δ

λds⃗ W ,ds) ⃗

≤ Cs⃗ B‖ϕx,ξ0 δ u‖Bs+κe ⃗ 1 (δ λds⃗ W ,ds) ⃗ , ∞,∞

⃗ s+κe 1 ⃗ Similarly, ∀u ∈ C0∞ (Bn (1); ℂD2 )′ with ϕx,ξ0 δ u ∈ B∞,∞ (W , ds).

‖ϕx,ξ

ξ5 0ξ 4

δ

δκ P ∗ v‖Bs⃗

∞,∞ (δ

λds⃗ W ,ds) ⃗

≤ Cs⃗ B‖ϕx,ξ0 δ v‖Bs+κe ⃗ 1 (δ λds⃗ W ,ds) ⃗ , ∞,∞

⃗ s+κe 1 ⃗ Here, Cs⃗ = Cs⃗ (ξ0 , D1 , D2 ) ≥ 0 is an ∀u ∈ C0∞ (Bn (1); ℂD1 )′ with ϕx,ξ0 δ u ∈ B∞,∞ (W , ds). (s,⃗ κ)-multi-parameter unit-admissible constant.

Proof. Without loss of generality, we take B = 1 and it suffices to consider only the case D1 = D2 = 1. Both P and P ∗ are of the form ̃= P

∑

degds1 (α)≤κ

α

bα (W 1 ) ,

where bα ∈ C ∞ (Bn (1)) and for every L, ‖bα ‖C L (Bn (1)) ≲L 1, where the implicit constant is an L-multi-parameter unit-admissible constant. For x ∈ Bn (7/8), δ ∈ (0, 1], set x,δ

̃ P

̃(Φx,δ ) = := Φ∗x,δ δκ P ∗

∑

degds1 (α)≤κ

1,x,δ bx,δ ), α (W

κ−degds1 (α) where bx,δ bα ∘ Φx,δ . Using Theorem 3.15.5 (j), we have, ∀L ∈ ℕ, α =δ

󵄩󵄩 1,x,δ α x,δ 󵄩󵄩 ‖bx,δ ) bα 󵄩󵄩C(Bn (1)) α ‖C L (Bn (1)) ≈L ∑ 󵄩 󵄩(X |β|≤L

α 󵄩 󵄩 = ∑ δβ+κ−degds1 (α) 󵄩󵄩󵄩(X 1 ) bα 󵄩󵄩󵄩C(Φ |β|≤L

We also have, by Theorem 9.2.8 (v) and (vii), ψx,ξ for every L, ‖ψx,ξ

ξ5 0ξ 4

‖ L δ C

≲L 1.

ξ5 0ξ 4

δ

Combining the above and using the fact that ψx,ξ

it follows from Proposition 6.5.9 that ‖ψx,ξ

ξ5 0ξ 4

δ

x,δ

̃ v‖ s⃗ P B

∞,∞ (W

x,δ ,ds) ⃗

= ‖ψx,ξ

ξ5 0ξ 4

δ

x,δ (B

:= ϕx,ξ ξ5 0ξ 4

δ

ξ5 0ξ 4

n (1)))

δ

∘ Φx,δ ∈ C0∞ (Bn (1)), and

≺ ψx,ξ0 (by Theorem 9.2.8 (iv)),

x,δ

̃ ψx,ξ δ v‖ s⃗ P B 0

∞,∞ (W

≲s,κ ⃗ ⃗ ‖ψx,ξ0 δ v‖Bs+κe 1 (W x,δ ,ds) ⃗ , ∞,∞

s+κe1 ∀v ∈ C0∞ (Bn (1))′ with ψx,ξ0 δ v ∈ B∞,∞ (W x,δ ). ⃗

≲L 1.

x,δ ,ds) ⃗

(9.47)

644 � 9 Nonlinear maximally subelliptic equations We have, using Lemma 9.2.7 and (9.47), ‖ϕx,ξ

ξ5 0ξ 4

δ

̃u‖ s⃗ δκ P B

∞,∞ (δ

λds⃗ W ,ds) ⃗

= ‖ϕx,ξ

ξ5 0ξ 4

δ

x,δ

̃ Φ u‖ s⃗ P x,δ B ∗

∞,∞ (W

x,δ ,ds) ⃗

∗ ≲s,κ ⃗ ⃗ ⃗ ‖ψx,ξ0 δ Φx,δ u‖Bs+κe 1 (W x,δ ,ds) 1 (δ λds⃗ W ,ds) ⃗ = ‖ϕx,ξ0 δ u‖Bs+κe ⃗ , ∞,∞

∞,∞

completing the proof. ⃗ ⃗ and ψ ∈ Lemma 9.2.12. Let s⃗ ∈ (0, ∞)ν and t > 0. Then, for u ∈ C s (Bn (7/8), (W , ds)) ∞ n C0 (B (7/8)),

‖ψΦ∗x,δ u‖C s⃗ (W x,δ ,ds)⃗ ≤ Cs⃗ ‖ψ‖C L ‖u‖C s⃗ (W ,ds)⃗ ,

(9.48)

⃗ where L ∈ ℕ and Cs⃗ ≥ 0 are s-multi-parameter unit-admissible constants. Similarly, for ⃗ F ∈ C s,t ((W , ds)⃗ ⊠ ∇ℝN ) and ψ ∈ C0∞ (Bn (7/8)), ‖ψ(⋅)F(Φx,δ (⋅), ζ )‖C s,t⃗ ((W x,δ ,ds)⊠∇ ⃗

ℝN

)

≤ Cs,t⃗ ‖ψ‖C L ‖F‖C s,t⃗ ((W ,ds)⊠∇ ⃗

ℝN

),

(9.49)

where L ∈ ℕ and Cs,t⃗ = Cs,t⃗ (N) ≥ 0 are (s,⃗ t)-multi-parameter unit-admissible constants. ⃗ ⃗ By Proposition 7.2.1 we may write u = ∑j∈ℕν uj , Proof. Let u ∈ C s (Bn (7/8), (W , ds)). ∞ n where uj ∈ C0 (B (1)) and for every α, ⃗ α 󵄩 󵄩 sup 2j⋅s 󵄩󵄩󵄩(2−jdsW ) uj 󵄩󵄩󵄩L∞ (Bn (1)) ≲s,α ⃗ ‖u‖C s (W ,ds)⃗ .

j∈ℕν

Thus, we have ⃗ α 󵄩 󵄩 sup 2j⋅s 󵄩󵄩󵄩(2−jdsW x,δ ) Φ∗x,δ uj 󵄩󵄩󵄩L∞ (Bn (1))

j∈ℕν

⃗ α 󵄩 󵄩 ≤ sup 2j⋅s δdegds1 (α) 󵄩󵄩󵄩(2−jdsW ) uj 󵄩󵄩󵄩L∞ (Bn (1)) ≲s,α ⃗ ‖u‖C s (W ,ds)⃗ .

(9.50)

j∈ℕν

It follows that for every L ∈ ℕ, ⃗ α 󵄩 󵄩 ∑ sup 2j⋅s 󵄩󵄩󵄩(2−jdsW x,δ ) ψΦ∗x,δ uj 󵄩󵄩󵄩L∞ (Bn (1)) ≲s,L ⃗ ‖u‖C s (W ,ds)⃗ . ν

|α|≤L j∈ℕ

Since ψΦ∗x,δ u = ∑j∈ℕν ψΦ∗x,δ uj , (9.48) follows from Proposition 7.2.1 (in particular, (7.4)). The same proof gives (9.49), by replacing Proposition 7.2.1 with Proposition 7.5.9. Corollary 9.2.13. Fix ξ0 ∈ (0, 1] and s ∈ (0, ∞)ν . Then, ∀x ∈ Bn (7/8), δ ∈ (0, 1], u ∈ ⃗ ⃗ C s (Bn (7/8), (W , ds)), ‖ϕx,ξ0 δ u‖C s⃗ (δλds⃗ W ,ds)⃗ ≤ Cs⃗ ‖u‖C s⃗ (W ,ds)⃗ , ⃗ where Cs⃗ = Cs⃗ (ξ0 ) ≥ 0 is an s-multi-parameter unit-admissible constant.

(9.51)

9.2 Main quantitative result

� 645

Proof. By Lemma 9.2.7, (9.51) is equivalent to ‖(ϕx,ξ0 δ ∘ Φx,δ )Φ∗x,δ u‖C s⃗ (W x,δ ,ds)⃗ ≤ Cs⃗ ‖u‖C s⃗ (W ,ds)⃗ .

(9.52)

Formula (9.52) follows from (9.48) with ψ = ϕx,ξ0 δ ∘ Φx,δ , where we use Theorem 9.2.8 (v) and (vii) to see ϕx,ξ0 δ ∘ Φx,δ ∈ C0∞ (Bn (1/2)) and ‖ϕx,ξ0 δ ∘ Φx,δ ‖C L ≲L 1, ∀L ∈ ℕ. ⃗ Proposition 9.2.14. Fix s⃗ ∈ (0, ∞)ν , t > 0, ξ0 ∈ (0, 1], N ∈ ℕ+ . Let F(x, ζ ) ∈ C s,t (BN (7/8)× ⃗ N s N n ⃗ ℝ ). Then, ∀x ∈ B (7/8) and δ ∈ (0, 1], the ℝ , (W , ds)⃗ ⊠ ∇ℝN ) and u ∈ C (Bn (7/8), (W , ds); following hold: (i) If t > ⌊|s|⃗ 1 ⌋ + 1 + ν, then

󵄩󵄩 󵄩 󵄩󵄩ϕx,ξ0 δ (⋅)F(⋅, ϕx,ξ0 δ (⋅)u(⋅))󵄩󵄩󵄩C s⃗ (δλds⃗ W ,ds)⃗ ≤ Cs,t⃗ ‖F‖C s,t⃗ ((W ,ds)⊠∇ ⃗

ℝN

) (1

+ ‖ϕx,ξ0 δ u‖C s⃗ (δλds⃗ W ,ds)⃗ )(1 + ‖ϕx,ξ0 δ u‖L∞ )

⌊|s|1 ⌋+ν

,

where Cs,t⃗ = Cs,t⃗ (ξ0 , N) ≥ 0 is an (s,⃗ t)-multi-parameter unit-admissible constant. β (ii) Fix L ∈ ℕ+ and suppose that 𝜕ζ F(x, 0) ≡ 0, ∀|β| < L, and t > ⌊|s|⃗ 1 ⌋ + 1 + ν + L. Then 󵄩󵄩 󵄩 󵄩󵄩ϕx,ξ0 δ (⋅)F(⋅, ϕx,ξ0 δ (⋅)u(⋅))󵄩󵄩󵄩C s⃗ (δλds⃗ W ,ds)⃗ ≤ Cs,t,L ⃗ ⃗ ‖F‖C s,t⃗ ((W ,ds)⊠∇

ℝN

) (1

+ ‖ϕx,ξ0 δ u‖L∞ )

⌊|s|1 ⌋+ν+1

‖ϕx,ξ0 δ u‖L−1 L∞ ‖ϕx,ξ0 δ u‖C s⃗ (δλds⃗ W ,ds)⃗ ,

where Cs,t,L = Cs,t,L ⃗ ⃗ (ξ0 , N) ≥ 0 is an (s,⃗ t, L)-multi-parameter unit-admissible constant. Proof. The proofs of (i) and (ii) are nearly identical: we pull back the setting via Φx,δ and then apply the corresponding part of Theorem 7.5.2. Indeed, for (i) we apply Lemma 9.2.7 and Theorem 7.5.2 (ii) to see that, with ψx,ξ0 δ := ϕx,ξ0 δ ∘ Φx,δ , 󵄩󵄩 󵄩 󵄩󵄩ϕx,ξ0 δ (⋅)F(⋅, ϕx,ξ0 δ (⋅)u(⋅))󵄩󵄩󵄩C s⃗ (δλds⃗ W ,ds)⃗ 󵄩 󵄩 = 󵄩󵄩󵄩ψx,ξ0 δ (⋅)F(Φx,δ (⋅), ψx,ξ0 δ (⋅)Φ∗x,δ u(⋅))󵄩󵄩󵄩C s⃗ (W x,δ ,ds)⃗ 󵄩 󵄩 ∗ ≲s,t⃗ 󵄩󵄩󵄩ψx,ξ0 δ (⋅)F(Φx,δ (⋅), ζ )󵄩󵄩󵄩C s,t⃗ ((W x,δ ,ds)⊠∇ ⃗ ) (1 + ‖ψx,ξ0 δ Φx,δ u‖C s⃗ (W x,δ ,ds)⃗ ) ℝN 󵄩 󵄩 ⌊|s| ⌋+ν × (1 + 󵄩󵄩󵄩ψx,ξ0 δ Φ∗x,δ u󵄩󵄩󵄩L∞ ) 1 . Note that 󵄩󵄩 󵄩 󵄩 ∗ 󵄩 󵄩󵄩ψx,ξ0 δ Φx,δ u󵄩󵄩󵄩L∞ = 󵄩󵄩󵄩ϕx,ξ0 δ u󵄩󵄩󵄩L∞ , and by Lemma 9.2.7 we have ‖ψx,ξ0 δ Φ∗x,δ u‖C s⃗ (W x,δ ,ds)⃗ = ‖ϕx,ξ0 δ u‖C s⃗ (δλds⃗ W ,ds)⃗ .

646 � 9 Nonlinear maximally subelliptic equations Thus, to complete the proof of (i) it suffices to show that 󵄩󵄩 󵄩 󵄩󵄩ψx,ξ0 δ (⋅)F(Φx,δ (⋅), ζ )󵄩󵄩󵄩C s,t⃗ ((W x,δ ,ds)⊠∇ ⃗ ⃗ ⃗ ‖F‖C s,t⃗ ((W ,ds)⊠∇ ) ≲s,t ). ℝN ℝN

(9.53)

The estimate (9.53) follows from (9.49) with ψ = ψx,ξ0 δ = ϕx,ξ0 δ ∘ Φx,δ , where we use Theorem 9.2.8 (v) and (vii) to see that ϕx,ξ0 δ ∘Φx,δ ∈ C0∞ (Bn (1/2)) and ‖ϕx,ξ0 δ ∘Φx,δ ‖C L ≲L 1, ∀L ∈ ℕ. The proof for (ii) is the same, but with Theorem 7.5.2 (ii) replaced by Theorem 7.5.2 (vii).

9.2.4 Step I: Perturbation of a linear operator Let P=

α

∑

degds1 (α)≤κ

aα (W 1 ) ,

aα ∈ 𝕄D1 ×D2 (ℝ)

be maximally subelliptic of degree κ with respect to (W 1 , ds1 ) in the sense that r1

n 󵄩 󵄩 ∑󵄩󵄩󵄩(Wj1 ) j u󵄩󵄩󵄩L2 (Bn (1),hσ j=1

Leb )

󵄩 󵄩 ≤ A(󵄩󵄩󵄩P u󵄩󵄩󵄩L2 (Bn (1),hσ

Leb )

󵄩 󵄩 + 󵄩󵄩󵄩u󵄩󵄩󵄩L2 (Bn (1),hσ

Leb )

),

∀u ∈ C0∞ (Bn (31/32); ℂD2 ). Here aα ∈ 𝕄D1 ×D2 (ℝ) are constant matrices. Fix B ≥ 0 with |aα | ≤ B, ∀α. Let ϵ1 ∈ (0, 1] be a small number to be chosen later and fix C1 ≥ 0. Suppose that ⃗ ⃗ ⃗ ⃗ R1 ∈ C s,⌊|s|1 ⌋+2+ν+σ (Bn (7/8) × ℝN , (W , ds)⃗ ⊠ ∇ℝN ; ℝD1 ) and R2 ∈ C s,⌊|s|1 ⌋+3+ν+σ (Bn (7/8) × N D ℝ , (W , ds)⃗ ⊠ ∇ℝN ; ℝ 1 ) with ‖R1 ‖C s,⌊|⃗ s|⃗ 1 ⌋+2+ν+σ ((W ,ds)⊠∇ ⃗

ℝN

)

≤ ϵ1 ,

‖R2 ‖C s,⌊|⃗ s|⃗ 1 ⌋+3+ν+σ ((W ,ds)⊠∇ ⃗

ℝN

)

≤ C1 .

We suppose that β

𝜕ζ R2 (x, 0) ≡ 0,

∀|β| ≤ 1.

⃗ ℝD2 ) with Let u ∈ C r+κe1 (Bn (7/8), (W , ds); ⃗

‖u‖C r+κe ⃗ ⃗ ≤ ϵ1 1 (W ,ds) such that u satisfies the following nonlinear PDE: κ

κ

P u(x) = R1 (x, W1 u(x)) + R2 (x, W1 u(x)),

∀x ∈ Bn (13/16).

9.2 Main quantitative result

�

647

Proposition 9.2.15. There exists ϵ1 ∈ (0, 1] sufficiently small such that if the above holds, then ∀ψ0 ∈ C0∞ (Bn (2/3)), we have ⃗ ℝD2 ) ψ0 u ∈ C s+κe1 (Bn (7/8), (W , ds); ⃗

and ‖ψ0 u‖C s+κe ⃗ ⃗ ≤ C2 . 1 (W ,ds) Here, ϵ1 = ϵ1 (C1 , A, B, D1 , D2 ) ∈ (0, 1] and C2 = C2 (C1 , A, B, D1 , D2 , ψ0 ) ≥ 0 are (κ, s,⃗ r,⃗ σ)multi-parameter unit-admissible constants. We use Remark 9.2.4 freely throughout this section. That is, we will apply results from previous chapters to the setting of this section; all the estimates from those results are in terms of appropriate multi-parameter unit-admissible constants, as can be seen by keeping track of the estimates in the proofs. Throughout the proof, we shrink ϵ1 > 0 a finite number of times (while still keeping it an appropriate admissible constant). Without loss of generality, we assume that s⃗ ≥ r,⃗ as we may replace r⃗ with r⃗ ∧ s.⃗ Fix ψ1 , ψ2 , ψ3 ∈ C0∞ (Bn (13/16)) with ψ3 ≺ ψ2 ≺ ψ1 and ψ3 ≡ 1 on a neighborhood of Bn (3/4). Since P is maximally subelliptic, Theorem 8.1.1 (i) ⇒ (viii) implies that there exists −2κ T ∈ Aloc ((W 1 , ds1 ); ℂD2 , ℂD2 ) (where we are using the ambient manifold Bn (31/32)) such that T P ∗P , P ∗P T ≡ I

∞ n mod Cloc (B (31/32) × Bn (31/32); 𝕄D2 ×D2 (ℂ)).

(9.54)

̃−2κ ((W 1 , ds1 ); ℂD2 , ℂD2 ). Corollary 5.11.18 shows that Proposition 5.8.4 shows that T ∈ A loc ̃−2κe1 ((W , ds); ⃗ ℂD2 , ℂD2 ). T ∈A loc We define V∞ := ψ2 T P ∗ ψ1 (R1 (x, Wκ1 u(x)) + R2 (x, Wκ1 u(x))) = ψ2 T P ∗ ψ1 P u

(9.55)

= ψ2 T P P u + ψ2 T P (1 − ψ1 )P u = ψ2 u + ψ2 e∞ , ∗

∗

where by the pseudo-locality of T (see the growth condition in Definition 5.2.4) and (9.54), we have e∞ ∈ C0∞ (Bn (7/8)). Moreover (using supp(u) ⊆ Bn (7/8), Proposition 5.8.11, and Theorem 6.3.10), ‖e∞ ‖C r+κe ⃗ ⃗ ⃗ ⃗ , ‖e∞ ‖C s+κe ⃗ ≲ ‖u‖C r+κe ⃗ ≲ ϵ1 . 1 (W ,ds) 1 (W ,ds) 1 (W ,ds) ̃0ν ((W , ds); ⃗ ℂD2 , ℂD2 ) (see Corollary 5.8.10) and using the fact Since T P ∗ ψ1 P ∈ A loc n that supp(u) ⊆ B (7/8), Theorem 6.3.10 shows that ∗ ‖V∞ ‖C r+κe ⃗ ⃗ ⃗ ⃗ = ‖ψ2 T P P u‖C r+κe ⃗ ≲ ‖u‖C r+κe ⃗ ≲ ϵ1 , 1 (W ,ds) 1 (W ,ds) 1 (W ,ds)

648 � 9 Nonlinear maximally subelliptic equations so there exists C3 ≈ 1 with ‖V∞ ‖C r+κe ⃗ ⃗ ≤ C3 ϵ1 . 1 (W ,ds)

(9.56)

⃗ and Set H := u − V∞ . We have H ∈ C r+κe1 (Bn (7/8), (W , ds)) ⃗

‖H‖C r+κe ⃗ ⃗ ⃗ ⃗ ≤ ‖u‖C r+κe ⃗ + ‖V∞ ‖C r+κe ⃗ ≲ ϵ1 . 1 (W ,ds) 1 (W ,ds) 1 (W ,ds)

(9.57)

Also, since ψ3 H = ψ3 e∞ , using Corollary 6.5.10, we have ‖ψ3 H‖C s+κe ⃗ ⃗ ⃗ ≲ ‖e∞ ‖C s+κe ⃗ ≲ ϵ1 . 1 (W ,ds) 1 (W ,ds)

(9.58)

⃗ set For V ∈ C r+κe1 (Bn (7/8), (W , ds)), ⃗

κ

κ

T (V ) := Mult[ψ2 ]T P ψ1 (⋅)(R1 (⋅, W1 (V + H)(⋅)) + R2 (⋅, W1 (V + H)(⋅))). ∗

⃗ Theorem 6.3.10, and TheoIt follows from the fact that H ∈ C r+κe1 (Bn (7/8), (W , ds)), ⃗ ⃗ r+κe r+κe n 1 1 ⃗ ⃗ We also have rem 7.5.2 (ii) that T : C (B (7/8), (W , ds)) → C (Bn (7/8), (W , ds)). T (V∞ ) = V∞ , by (9.55). Let D ∈ [C3 ∨ 1, 1/ϵ1 ], to be chosen later; we will choose D ≈ 1. Define ⃗

MD,ϵ1 ,r⃗ := {V ∈ C

⃗ r+κe 1

⃗ : ‖V ‖ r+κe (Bn (7/8), (W , ds)) C ⃗ 1 (W ,ds)⃗ ≤ Dϵ1 }.

⃗ ⃗ (see Note that MD,ϵ1 ,r⃗ is a closed subspace of the Banach space C r+κe1 (Bn (7/8), (W , ds)) Proposition 6.3.6), and therefore it is a complete metric space with distance induced by ‖ ⋅ ‖C r+κe ⃗ ⃗ . Furthermore, since D ≥ C3 , (9.56) shows that V∞ ∈ MD,ϵ1 ,r⃗ . We will show 1 (W ,ds) that T : MD,ϵ1 ,r⃗ → MD,ϵ1 ,r⃗ and that it is a strict contraction (with appropriate choices of D and ϵ1 ); it will then follow that V∞ is the unique fixed point of T in MD,ϵ1 ,r⃗ . For V ∈ MD,ϵ1 ,r⃗ , we have, using Corollary 6.5.11, (9.57), and D ≥ 1,

‖Wκ1 (V + H)‖C r⃗ (W ,ds)⃗ ≲ ‖V + H‖C r+κe ⃗ ⃗ 1 (W ,ds)

≤ ‖V ‖C r+κe ⃗ ⃗ (W ,ds)⃗ + ‖H‖C r+κe (W ,ds)⃗ ≲ Dϵ1 + ϵ1 ≲ Dϵ1 .

(9.59)

In particular, using Corollary 7.2.2, we have ‖Wκ1 (V + H)‖L∞ ≲ ‖Wκ1 (V + H)‖C r⃗ (W ,ds)⃗ ≲ Dϵ1 .

(9.60)

Using Theorem 6.3.10, Theorem 7.5.2 (ii) applied to R1 , and Theorem 7.5.2 (vii) with L = 2 applied to R2 , for V ∈ MD,ϵ1 ,r⃗ , we have ‖T (V )‖C r+κe ⃗ ⃗ 1 (W ,ds) 󵄩 󵄩 ∗ 󵄩 = 󵄩󵄩ψ2 T P ψ1 (R1 (x, Wκ1 (V + H)(x)) + R2 (x, Wκ1 (V + H)(x)))󵄩󵄩󵄩C r+κe ⃗ ⃗ 1 (W ,ds) 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 κ κ ≲ 󵄩󵄩R1 (x, W1 (V + H)(x))󵄩󵄩C r⃗ (W ,ds)⃗ + 󵄩󵄩R2 (x, W1 (V + H)(x))󵄩󵄩C r⃗ (W ,ds)⃗

� 649

9.2 Main quantitative result

≲ ‖R1 ‖C r,⌊| ⃗ r|⃗ 1 ⌋+1+ν+σ ⃗ ((W ,ds)⊠∇

ℝN

) (1

+ ‖R2 ‖C r,⌊| ⃗ r|⃗ 1 ⌋+3+ν+σ ⃗ ((W ,ds)⊠∇

ℝN

+ ‖Wκ1 (V + H)‖C r⃗ (W ,ds)⃗ )(1 + ‖Wκ1 (V + H)‖L∞ )

) (1

+ ‖Wκ1 (V + H)‖L∞ )

⌊|r|⃗ 1 ⌋+ν

⌊|r|⃗ 1 ⌋+ν+1

× ‖Wκ1 (V + H)‖L∞ ‖Wκ1 (V + H)‖C r⃗ (W ,ds)⃗

≲ ϵ1 (1 + Dϵ1 )(1 + Dϵ1 )⌊|r|1 ⌋+1 + (1 + Dϵ1 )⌊|r|1 ⌋+ν+1 (Dϵ1 )2 ≲ ϵ1 + D2 ϵ12 , ⃗

⃗

where in the second to last inequality we used (9.59) and (9.60), and in the last we used D ≤ 1/ϵ1 . That is, there exists C4 ≈ 1 with 2 2 ‖T (V )‖C r+κe ⃗ ⃗ ≤ C4 (ϵ1 + D ϵ1 ), 1 (W ,ds)

(9.61)

∀V ∈ MD,ϵ1 ,r⃗ .

We take D := max{C3 , 2C4 , 1}, so that D ≈ 1, and we take ϵ1 ∈ (0, 1/D2 ] to be chosen later. Then C4 (ϵ1 + ϵ12 D2 ) ≤ Dϵ1 and (9.61) implies ‖T (V )‖C r+κe ⃗ ⃗ ≤ Dϵ1 , 1 (W ,ds)

∀V ∈ MD,ϵ1 ,r⃗ .

That is, T : MD,ϵ1 ,r⃗ → MD,ϵ1 ,r⃗ . Now that we have chosen D ≈ 1, (9.59) and (9.60) become ‖Wκ1 (V + H)‖C r⃗ (W ,ds)⃗ , ‖Wκ1 (V + H)‖L∞ ≲ ϵ1 ,

(9.62)

∀V ∈ MD,ϵ1 ,r⃗ .

Using (9.62), Theorem 6.3.10, Theorem 7.5.2 (vi) applied to R1 , and Theorem 7.5.2 (viii) with L = 2 applied to R2 , we have ‖T (V1 ) − T (V2 )‖C r+κe ⃗ ⃗ 1 (W ,ds)

= ‖ψ2 T P ∗ ψ1 (R1 (x, Wκ1 (V1 + H)(x)) − R1 (x, Wκ1 (V2 + H)(x))

+ R2 (x, Wκ1 (V1 + H)(x)) − R2 (x, Wκ1 (V2 + H)(x)))‖C r+κe ⃗ ⃗ 1 (W ,ds)

≲ ‖R1 (x, Wκ1 (V1 + H)(x)) − R1 (x, Wκ1 (V2 + H)(x))‖C r⃗ (W ,ds)⃗

+ ‖R2 (x, Wκ1 (V1 + H)(x)) − R2 (x, Wκ1 (V2 + H)(x))‖C r⃗ (W ,ds)⃗

≲ ‖R1 ‖C r,⌊| ⃗ r|⃗ 1 ⌋+2+ν+σ ⃗ ((W ,ds)⊠∇

ℝN

κ ) (‖W1 (V1

− V2 )‖C r⃗ (W ,ds)⃗

+ (‖Wκ1 (V1 + H)‖C r⃗ (W ,ds)⃗ + ‖Wκ1 (V2 + H)‖C r⃗ (W ,ds)⃗ )‖Wκ1 (V1 − V2 )‖L∞ ) ⌊|r|⃗ 1 ⌋+ν

× (1 + ‖Wκ1 (V1 + H)‖L∞ + ‖Wκ1 (V2 + H)‖L∞ ) ) (1

+ ‖R2 ‖C r,⌊| ⃗ r|⃗ 1 ⌋+3+ν+σ ⃗ ((W ,ds)⊠∇

κ ) (‖W1 (V1

ℝN

⌊|r|⃗ 1 ⌋+ν+1

+ ‖Wκ1 (V1 + H)‖L∞ + ‖Wκ1 (V2 + H)‖L∞ )

+ ‖R2 ‖C r,⌊| ⃗ r|⃗ 1 ⌋+3+ν+σ ⃗ ((W ,ds)⊠∇

× (‖Wκ1 (V1 + H)‖C r⃗ (W ,ds)⃗ + ‖Wκ1 (V2 + H)‖C r⃗ (W ,ds)⃗ )‖Wκ1 (V1 − V2 )‖L∞ ℝN

+ H)‖L∞

+ ‖Wκ1 (V2 + H)‖L∞ )‖Wκ1 (V1 − V2 )‖C r⃗ (W ,ds)⃗

650 � 9 Nonlinear maximally subelliptic equations κ ⌊|r|1 ⌋+ν ≲ ϵ1 (‖V1 − V2 ‖C r+κe ⃗ ⃗ + ϵ1 ‖W1 (V1 − V2 )‖L∞ )(1 + ϵ1 ) 1 (W ,ds) ⃗

+ (1 + ϵ1 )⌊|r|1 ⌋+ν+1 ϵ1 ‖Wκ1 (V1 − V2 )‖L∞ ⃗

+ ϵ1 ‖Wκ1 (V1 − V2 )‖L∞

κ ≲ ϵ1 ‖V1 − V2 ‖C r+κe ⃗ ⃗ + ϵ1 ‖W1 (V1 − V2 )‖L∞ 1 (W ,ds)

≲ ϵ1 ‖V1 − V2 ‖C r+κe ⃗ ⃗ , 1 (W ,ds)

where we have used Corollaries 7.2.2 and 6.5.11 to see that ‖Wκ1 (V1 − V2 )‖L∞ ≲ ‖Wκ1 (V1 − V2 )‖C r⃗ (W ,ds)⃗ ≲ ‖V1 − V2 ‖C r+κe ⃗ ⃗ . 1 (W ,ds) We conclude that ∃C5 ≈ 1 with ‖T (V1 ) − T (V2 )‖C r+κe ⃗ ⃗ ⃗ ≤ C5 ϵ1 ‖V1 − V2 ‖C r+κe ⃗ . 1 (W ,ds) 1 (W ,ds) In particular, if ϵ1 ∈ (0, min{1/2C5 , 1/D2 }], we have 1 ‖T (V1 ) − T (V2 )‖C r+κe ⃗ ⃗ ⃗ ≤ ‖V1 − V2 ‖C r+κe ⃗ , 1 (W ,ds) 1 (W ,ds) 2 and therefore T : MD,ϵ1 ,r⃗ → MD,ϵ1 ,r⃗ is a strict contraction with unique fixed point V∞ . Set V0 := 0 and for j ≥ 0 set Vj+1 := T (Vj ). We have established Vj ∈ MD,ϵ1 ,r⃗ , ∀j, and j→∞

⃗ the contraction mapping principle shows that Vj 󳨀󳨀󳨀󳨀→ V∞ in C r+κe1 (Bn (7/8), (W , ds)). Since Vj ∈ MD,ϵ1 ,r⃗ , we have ⃗

‖Vj ‖C r+κe ⃗ ⃗ ≲ ϵ1 , 1 (W ,ds) and therefore just as in (9.62), we have ‖Wκ1 (Vj + H)‖C r⃗ (W ,ds)⃗ , ‖Wκ1 (Vj + H)‖L∞ ≲ ‖Vj ‖C r+κe ⃗ ⃗ + ϵ1 ≲ ϵ1 , 1 (W ,ds)

(9.63)

∀j ∈ ℕ. Similarly, ‖Wκ1 Vj ‖C r⃗ (W ,ds)⃗ , ‖Wκ1 Vj ‖L∞ ≲ ‖Vj ‖C r+κe ⃗ ⃗ ≲ ϵ1 , 1 (W ,ds)

(9.64)

∀j ∈ ℕ. Lemma 9.2.16. For ψ′ ∈ C0∞ (Bn (13/16)) with ψ′ ≺ ψ3 , we have ⃗ ⃗ ψ′ Vj ∈ C s+κe1 (Bn (7/8), (W , ds)),

∀j ∈ ℕ.

Proof. We prove the lemma by induction on j. Since V0 = 0, the base case, j = 0, is trivial. We assume the result for j and prove it for j + 1.

9.2 Main quantitative result

�

651

Take ψ′′ , ψ′′′ ∈ C0∞ (Bn (13/16)) with ψ′ ≺ ψ′′ ≺ ψ′′′ ≺ ψ3 . By the inductive ⃗ ⃗ and we wish to show ψ′ Vj ∈ hypothesis, we have ψ′′′ Vj ∈ C s+κe1 (Bn (7/8), (W , ds)) ⃗ ⃗ C s+κe1 (Bn (7/8), (W , ds)). We have

ψ′ Vj+1 = ψ′ T (Vj )

= ψ′ T P ∗ ψ1 (R1 (x, Wκ1 (Vj + H)(x)) + R2 (x, Wκ1 (Vj + H)(x)))

= ψ′ T P ∗ (1 − ψ′′ )ψ1 (R1 (x, Wκ1 (Vj + H)(x)) + R2 (x, Wκ1 (Vj + H)(x))) + ψ′ T P ∗ ψ′′ (R1 (x, Wκ1 (Vj + H)(x)) + R2 (x, Wκ1 (Vj + H)(x)))

=: (I) + (II). −2κ Since ψ′ ≺ ψ′′ and T ∈ Aloc ((W 1 , ds1 ); ℂD2 , ℂD2 ) is pseudo-local (see the growth condition in Definition 5.2.4), we have Mult[ψ′ ]T P ∗ Mult[(1 − ψ′′ )ψ1 ] ∈ C0∞ (Bn (13/16) × Bn (13/16)). It follows that

(I) = ψ′ T P ∗ (1 − ψ′′ )ψ1 (R1 (x, Wκ1 (Vj + H)(x)) + R2 (x, Wκ1 (Vj + H)(x))) ⃗ ⃗ ∈ C0∞ (Bn (13/16)) ⊆ C s+κe1 (Bn (7/8), (W , ds)),

where the ⊆ uses Proposition 6.5.5. So to complete the proof, it suffices to show that ⃗ ⃗ (II) ∈ C s+κe1 (Bn (7/8), (W , ds)). We have (II) = ψ′ T P ∗ ψ′′ (R1 (x, Wκ1 ψ′′′ (Vj + H)(x)) + R2 (x, Wκ1 ψ′′′ (Vj + H)(x))). ⃗ ⃗ and since ψ′′′ H = ψ′′′ ψ3 H = By the inductive hypothesis, ψ′′′ Vj ∈ C s+κe1 (Bn (7/8), (W , ds)), ⃗ ′′′ ∞ n ′′′ s+κe 1 ⃗ Corollary 6.5.11 ψ ψ3 e∞ ∈ C0 (B (13/16)), we also have ψ H ∈ C (Bn (7/8), (W , ds)). κ ′′′ s⃗ n ⃗ shows that W ψ (Vj + H) ∈ C (B (7/8), (W , ds)) and Theorem 7.5.2 (ii) implies that 1

⃗ Since R1 (x, Wκ1 ψ′′′ (Vj + H)(x)) + R2 (x, Wκ1 ψ′′′ (Vj + H)(x)) ∈ C s (Bn (7/8), (W , ds)). ′ ∗ ′′ −κe1 n ̃ ⃗ Mult[ψ ]T P Mult[ψ ] ∈ A (B (7/8), (W , ds)), Theorem 6.3.10 implies that (II) ∈ ⃗ ⃗ as desired, completing the proof. C s+κe1 (Bn (7/8), (W , ds)), ⃗

We use the functions ϕx,δ and the constants ξ4 , ξ5 , and ξ6 from Theorem 9.2.8. Fix x0 ∈ Bn (3/4) and δ ∈ (0, 1] with B(W 1 ,ds1 ) (x0 , (ξ43 ξ6 /ξ53 )δ) ⊆ Bn (3/4). We have, by Proposition 9.2.9 (and using Lemma 9.2.16), ‖ϕx0 ,ξ6 δ Vj+1 ‖C s+κe ⃗ ⃗ ⃗ ≲ ‖ϕx 1 (δ λds W ,ds)

ξ4 ξ6 0, ξ 5

δ

δ2κ P ∗ P Vj+1 ‖Bs−κe ⃗ 1 (δ λds⃗ W ,ds) ⃗ + ‖ϕx ∞,∞

ξ4 ξ6 0, ξ 5

V ‖ ∞. δ j+1 L (9.65)

We estimate the two terms on the right-hand side of (9.65). By (9.64) we have ‖ϕx

ξ4 ξ6 0, ξ 5

V ‖ ∞ δ j+1 L

≤ ‖Vj+1 ‖L∞ ≲ ϵ1 .

(9.66)

652 � 9 Nonlinear maximally subelliptic equations To bound the first term on the right-hand side of (9.65), we use the fact that supp(ϕx , ξ4 ξ6 δ ) ⊆ B(W 1 ,ds1 ) (x0 , (ξ43 ξ6 /ξ53 )δ) ⊆ Bn (3/4) and therefore ϕx , ξ4 ξ6 δ ≺ ψ2 to see 0

that

0

ξ5

ϕx

ξ4 ξ6 0, ξ 5

δ

= ϕx = ϕx

ξ5

P P Mult[ψ2 ]T P Mult[ψ1 ] ∗

∗

ξ4 ξ6 0, ξ 5

δ

P P T P Mult[ψ1 ]

ξ4 ξ6 0, ξ 5

δ

P Mult[ψ1 ] + ϕx

∗

∗

∗

ξ4 ξ6 0, ξ 5

E Mult[ψ1 ], δ ∞

where E∞ ∈ C0∞ (Bn (7/8) × Bn (7/8)) with ‖E∞ ‖C L ≲L 1, ∀L ∈ ℕ. Thus, we have ‖ϕx

ξ4 ξ6 0, ξ 5

= ‖ϕx = ‖ϕx ≤ ‖ϕx

δ

δ2κ P ∗ P Vj+1 ‖Bs−κe ⃗ 1 (δ λds⃗ W ,ds) ⃗

ξ4 ξ6 0, ξ 5 ξ4 ξ6 0, ξ 5 ξ4 ξ6 0, ξ 5

+ ‖ϕx

∞,∞

δ δ δ

2κ

∗

2κ

∗

2κ

∗

δ P PT (Vj )‖Bs−κe ⃗ 1 (δ λds⃗ W ,ds) ⃗ ∞,∞

δ P P ψ2 T P ψ1 (R1 (x, Wκ1 (Vj +H)(x))+R2 (x, Wκ1 (Vj +H)(x)))‖Bs−κe ⃗ 1 (δ λds⃗ W ,ds) ⃗ ∗

∞,∞

δ P

ξ4 ξ6 0, ξ 5

δ

ψ1 (R1 (x, Wκ1 (Vj

+ H)(x)) +

R2 (x, Wκ1 (Vj

+ H)(x)))‖Bs−κe ⃗ 1 (δ λds⃗ W ,ds) ⃗

(9.67)

∞,∞

δ2κ E∞ ψ1 (R1 (x, Wκ1 (Vj + H)(x)) + R2 (x, Wκ1 (Vj + H)(x)))‖Bs−κe ⃗ 1 (δ λds⃗ W ,ds) ⃗ ∞,∞

=: (III) + (IV). To bound (IV), we use Proposition 9.2.10 to see that, for some L ≈ 1, (IV) ≲ δ2κ ‖E∞ ‖C L ‖R1 (x, Wκ1 (Vj + H)(x)) + R2 (x, Wκ1 (Vj + H)(x))‖L∞ ≲ δ2κ (‖R1 (x, Wκ1 (Vj + H)(x))‖L∞ + ‖R2 (x, Wκ1 (Vj + H)(x))‖L∞ ).

(9.68)

We have, using Proposition 7.5.10, ‖R1 (x, Wκ1 (Vj + H)(x))‖L∞ ≤ ‖R1 ‖L∞ ≲ ‖R1 ‖C s,⌊|⃗ s|⃗ 1 ⌋+2+ν+σ ((W ,ds)⊠∇ ⃗ Since R2 (x, 0) ≡ 0, we have, using Propositions 7.5.10 and 7.5.11,

ℝN

)

≤ ϵ1 .

(9.69)

β

‖R2 (x, Wκ1 (Vj + H)(x))‖L∞ ≲ ∑ ‖𝜕ζ R2 (x, ζ )‖L∞ ‖Wκ1 (Vj + H)‖L∞ |β|≤1

β

≲ ∑ ‖𝜕ζ R2 ‖C s,⌊|⃗ s|⃗ 1 ⌋+2+ν+σ ((W ,ds)⊠∇ ⃗ |β|≤1

≲

ℝN

κ ) ‖W1 (Vj

κ ‖R2 ‖C s,⌊|⃗ s|⃗ 1 ⌋+3+ν+σ ((W ,ds)⊠∇ ⃗ ) ‖W1 (Vj ℝN

+ H)‖L∞

(9.70)

+ H)‖L∞

≲ ‖Wκ1 (Vj + H)‖L∞ ≲ ϵ1 ,

where the final estimate used (9.63). Using (9.69) and (9.70) to estimate the right-hand side of (9.68) shows that

9.2 Main quantitative result

(IV) ≲ δ2κ ϵ1 ≲ ϵ1 . For (III), using the fact that ϕx

ξ4 ξ6 0, ξ 5

δ

≺ϕ

x0 ,

ξ 2 ξ6 4 ξ2 5

δ

� 653

(9.71)

(see Theorem 9.2.8 (iv)), we have

󵄩󵄩 (III) = 󵄩󵄩󵄩ϕx , ξ4 ξ6 δ δ2κ P ∗ ψ1 (R1 (x, ϕ ξ42 ξ6 (x)Wκ1 (Vj + H)(x)) 󵄩 0 ξ5 x0 , 2 δ ξ

+ R2 (x, ϕ

ξ2 ξ x0 , 4 2 6 ξ 5

δ

(x)Wκ1 (Vj

5

󵄩󵄩 + H)(x)))󵄩󵄩󵄩 s−κe ⃗ 1 (δ λds⃗ W ,ds) ⃗ 󵄩B∞,∞

󵄩󵄩 󵄩 ≲ δκ 󵄩󵄩󵄩ϕ ξ42 ξ6 (R1 (x, ϕ ξ42 ξ6 (x)Wκ1 (Vj + H)(x)) 󵄩󵄩 x0 , 2 δ x0 , 2 δ ξ ξ 5 5 󵄩󵄩 󵄩 κ + R2 (x, ϕ ξ42 ξ6 (x)W1 (Vj + H)(x)))󵄩󵄩󵄩 󵄩󵄩C s⃗ (δλds⃗ W ,ds)⃗ x0 , 2 δ ξ 5 󵄩󵄩 󵄩󵄩 ≤ δκ 󵄩󵄩󵄩ϕ ξ42 ξ6 R1 (x, ϕ ξ42 ξ6 (x)Wκ1 (Vj + H)(x))󵄩󵄩󵄩 s⃗ λds⃗ 󵄩 x0 , 2 δ 󵄩C (δ W ,ds)⃗ x0 , 2 δ ξ

󵄩󵄩 κ󵄩

ξ

5

(9.72)

5

󵄩󵄩 + δ 󵄩󵄩ϕ ξ42 ξ6 R2 (x, ϕ ξ42 ξ6 (x)Wκ1 (Vj + H)(x))󵄩󵄩󵄩 s⃗ λds⃗ 󵄩 x0 , 2 δ 󵄩C (δ W ,ds)⃗ x0 , 2 δ ξ

ξ

5

5

=: (V) + (VI), where we used Proposition 9.2.11 and ϕ

x0 ,

ξ 2 ξ6 4 ξ2 5

δ

≺ ψ1 (since supp(ϕ

x0 ,

ξ 2 ξ6 4 ξ2 5

δ

) ⊆ B(W 1 ,ds1 ) (x0 ,

(ξ43 ξ6 /ξ53 )δ) ⊆ Bn (3/4)) (see Theorem 9.2.8 (i)). Before we estimate (V) and (VI), we need some preliminary estimates. First, using Corollary 9.2.13 and the fact that ϕ ξ43 ξ6 ≺ ψ3 (since supp(ϕ ξ43 ξ6 ) ⊆ B(W 1 ,ds1 ) (x0 , x0 ,

(ξ43 ξ6 /ξ53 )δ)

n

ξ3 5

δ

x0 ,

⊆ B (3/4)) (see Theorem 9.2.8 (i)), we have

ξ3 5

δ

󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩ϕ ξ 3 ξ H 󵄩󵄩󵄩 = 󵄩󵄩󵄩ϕ ξ43 ξ6 ψ3 H 󵄩󵄩󵄩 s+κe ⃗ ⃗ 󵄩󵄩 x , 4 6 δ 󵄩󵄩C s+κe ⃗ 1 (δ λds W ,ds) 󵄩 󵄩C ⃗ 1 (δλds⃗ W ,ds)⃗ x , δ 0 0 3 3 ξ

ξ

5

5

󵄩 󵄩 ≲ 󵄩󵄩󵄩ψ3 H 󵄩󵄩󵄩C s+κe ⃗ ⃗ ≲ ϵ1 , 1 (W ,ds)

(9.73)

where the final estimate used (9.58). Next, using Proposition 9.2.11 and (9.73), we have 󵄩󵄩 󵄩󵄩 δκ 󵄩󵄩󵄩ϕ ξ42 ξ6 Wκ1 (Vj + H)󵄩󵄩󵄩 s⃗ λds⃗ 󵄩 x0 , 2 δ 󵄩C (δ W ,ds)⃗ ξ

5

󵄩󵄩 󵄩󵄩 ≲ 󵄩󵄩󵄩ϕ ξ43 ξ6 (Vj + H)󵄩󵄩󵄩 s+κe 󵄩 x0 , 3 δ 󵄩C ⃗ 1 (δλds⃗ W ,ds)⃗ ξ

5

󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ≤ 󵄩󵄩󵄩ϕ ξ43 ξ6 Vj 󵄩󵄩󵄩 s+κe + 󵄩󵄩ϕ 3 H 󵄩󵄩 ⃗ ⃗ ⃗ 1 (δ λds W ,ds) 󵄩 x0 , 3 δ 󵄩C ⃗ 1 (δλds⃗ W ,ds)⃗ 󵄩󵄩 x0 , ξ4 3ξ6 δ 󵄩󵄩C s+κe ξ

5

󵄩󵄩 󵄩󵄩 ≲ 󵄩󵄩󵄩ϕ ξ43 ξ6 Vj 󵄩󵄩󵄩 s+κe + ϵ1 . 󵄩 x0 , 3 δ 󵄩C ⃗ 1 (δλds⃗ W ,ds)⃗ ξ

5

ξ

5

(9.74)

654 � 9 Nonlinear maximally subelliptic equations Finally, since 0 ≤ ϕ

x0 ,

ξ 2 ξ6 4 ξ2 5

≤ 1 (Theorem 9.2.8 (ii)), we have, using (9.63),

δ

󵄩󵄩 󵄩 󵄩󵄩ϕ ξ 2 ξ Wκ (Vj + H)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩Wκ (Vj + H)󵄩󵄩󵄩 ∞ ≲ ϵ1 . 󵄩󵄩 x , 4 6 δ 1 󵄩󵄩L∞ 󵄩 1 󵄩L 0 2 ξ

(9.75)

5

For the following estimates of (V) and (VI), we freely use (9.73), (9.74), and (9.75). For (V) we have, by Proposition 9.2.14 (i), 󵄩󵄩 󵄩󵄩 (V) = δκ 󵄩󵄩󵄩ϕ ξ42 ξ6 R1 (x, ϕ ξ42 ξ6 (x)Wκ1 (Vj + H)(x))󵄩󵄩󵄩 s⃗ λds⃗ 󵄩 x0 , 2 δ 󵄩C (δ W ,ds)⃗ x0 , 2 δ ξ

ξ

5

5

≲ ‖R1 ‖C s,⌊|s|1 ⌋+1+ν+σ ((W ,ds)⊠∇ ⃗

ℝN

) (δ

κ

󵄩󵄩 󵄩󵄩 + δκ 󵄩󵄩󵄩ϕ ξ42 ξ6 Wκ1 (Vj + H)󵄩󵄩󵄩 s⃗ λds⃗ ) 󵄩 x0 , 2 δ 󵄩C (δ W ,ds)⃗ ξ

󵄩󵄩 󵄩󵄩 × (1 + 󵄩󵄩󵄩ϕ ξ42 ξ6 Wκ1 (Vj + H)󵄩󵄩󵄩 ∞ ) 󵄩 x0 , 2 δ 󵄩L

5

⌊|s|1 ⌋+ν

(9.76)

ξ 5

󵄩󵄩 󵄩󵄩 ⌊|s| ⌋+ν ≲ ϵ1 (1 + 󵄩󵄩󵄩ϕ ξ43 ξ6 Vj 󵄩󵄩󵄩 s+κe + ϵ1 )(1 + ϵ1 ) 1 󵄩 x0 , 3 δ 󵄩C ⃗ 1 (δλds⃗ W ,ds)⃗ ξ

5

󵄩󵄩 󵄩󵄩 ≲ ϵ1 + ϵ1 󵄩󵄩󵄩ϕ ξ43 ξ6 Vj 󵄩󵄩󵄩 s+κe . 󵄩 x0 , 3 δ 󵄩C ⃗ 1 (δλds⃗ W ,ds)⃗ ξ

5

For (VI) we have, by Proposition 9.2.14 (ii) with L = 2, 󵄩󵄩 󵄩󵄩 δκ 󵄩󵄩󵄩ϕ ξ42 ξ6 R2 (x, ϕ ξ42 ξ6 (x)Wκ1 (Vj + H)(x))󵄩󵄩󵄩 s⃗ λds⃗ 󵄩 x0 , 2 δ 󵄩C (δ W ,ds)⃗ x0 , 2 δ ξ

ξ

5

5

≲ ‖R2 ‖C s,⌊|⃗ s|⃗ 1 ⌋+3+ν+σ ((W ,ds)⊠∇ ⃗

) (1 ℝN

⌊|s|1 ⌋+ν+1 󵄩󵄩 󵄩󵄩 + 󵄩󵄩󵄩ϕ ξ42 ξ6 (x)Wκ1 (Vj + H)󵄩󵄩󵄩 ∞ ) 󵄩 x0 , 2 δ 󵄩L ξ

5

󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 × 󵄩󵄩󵄩ϕ ξ42 ξ6 Wκ1 (Vj + H)󵄩󵄩󵄩 ∞ δκ 󵄩󵄩󵄩ϕ ξ42 ξ6 Wκ1 (Vj + H)󵄩󵄩󵄩 s⃗ λds⃗ 󵄩 x0 , 2 δ 󵄩L 󵄩 x0 , 2 δ 󵄩C (δ W ,ds)⃗ ξ

≲ (1 + ϵ1 ) ≲

ϵ12

ξ

5

⌊|s|1 ⌋+ν+1

(9.77)

5

󵄩󵄩 󵄩󵄩 ϵ1 (󵄩󵄩󵄩ϕ ξ43 ξ6 Vj 󵄩󵄩󵄩 s+κe + ϵ1 ) 󵄩 x0 , 3 δ 󵄩C ⃗ 1 (δλds⃗ W ,ds)⃗ ξ

5

󵄩󵄩 󵄩󵄩 + ϵ1 󵄩󵄩󵄩ϕ ξ43 ξ6 Vj 󵄩󵄩󵄩 s+κe . 󵄩 x0 , 3 δ 󵄩C ⃗ 1 (δλds⃗ W ,ds)⃗ ξ

5

Using (9.76) and (9.77) to bound the right-hand side of (9.72) shows that 󵄩󵄩 󵄩󵄩 (III) ≲ ϵ1 + ϵ1 󵄩󵄩󵄩ϕ ξ43 ξ6 Vj 󵄩󵄩󵄩 s+κe . 󵄩 x0 , 3 δ 󵄩C ⃗ 1 (δλds⃗ W ,ds)⃗ ξ

(9.78)

5

Using (9.71) and (9.78) to bound the right-hand side of (9.67) shows that ‖ϕx

ξ4 ξ6 0, ξ 5

δ

󵄩󵄩 󵄩 󵄩ϕ ξ 3 ξ Vj 󵄩󵄩󵄩 δ2κ P ∗ P Vj+1 ‖Bs−κe . ⃗ ⃗ 1 λds⃗ ⃗ ≲ ϵ1 + ϵ1 󵄩 ⃗ ⃗ 1 (δ λds W ,ds) 󵄩󵄩 x0 , 4 36 δ 󵄩󵄩C s+κe ∞,∞ (δ W ,ds) ξ

5

(9.79)

9.2 Main quantitative result

� 655

Using (9.79) and (9.66) to bound the right-hand side of (9.65) and using Theorem 9.2.8 (x) shows that 󵄩󵄩 󵄩 󵄩ϕ ξ 3 ξ Vj 󵄩󵄩󵄩 ‖ϕx0 ,ξ6 δ Vj+1 ‖C s+κe ⃗ ⃗ ⃗ ⃗ ≲ ϵ1 + ϵ1 󵄩 ⃗ 1 (δ λds W ,ds) ⃗ 1 (δ λds W ,ds) 󵄩󵄩 x0 , 4 36 δ 󵄩󵄩C s+κe ξ

5

󵄩󵄩 󵄩󵄩 ≈ ϵ1 + ϵ1 󵄩󵄩󵄩ϕ ξ43 ξ6 Vj 󵄩󵄩󵄩 . λds⃗ ⃗ 󵄩 x0 , 3 δ 󵄩C s+κe ⃗ 1 (((ξ 3 ξ6 /ξ 3 )δ ) W ,ds) ξ

5

4

(9.80)

5

Set Qj :=

sup

δ∈(0,1] B(W 1 ,ds1 ) (x,δ)⊆Bn (3/4)

‖ϕx,δ Vj ‖C s+κe ⃗ ⃗ ⃗ ∈ [0, ∞], 1 (δ λds W ,ds)

where the supremum is taken over all such x and δ. Since V0 = 0, we have Q0 = 0. For x0 ∈ Bn (3/4) and δ ∈ (0, 1] such that B(W 1 ,ds1 ) (x0 , (ξ43 ξ6 /ξ53 )δ) ⊆ Bn (3/4), (9.80) implies that ‖ϕx0 ,ξ6 δ Vj+1 ‖C s+κe ⃗ ⃗ ⃗ ≲ ϵ1 + ϵ1 Q j . 1 (δ λds W ,ds)

(9.81)

Take x ∈ Bn (3/4), δ ∈ (0, 1] such that B(W 1 ,ds1 ) (x, δ) ⊆ Bn (3/4). Take N1 ≈ 1 and x1 , . . . , xN1 as in Theorem 9.2.8 (viii). Because (9.17) holds, (9.81) implies ‖ϕxj ,ξ6 δ Vj+1 ‖C s+κe ⃗ ⃗ ⃗ ≲ ϵ1 + ϵ1 Q j , 1 (δ λds W ,ds)

1 ≤ j ≤ N1 .

(9.82)

Plugging (9.82) into (9.18) shows that N1

‖ϕx,δ Vj+1 ‖C s+κe ⃗ ⃗ ⃗ ⃗ ⃗ ≲ ∑ ‖ϕxj ,ξ6 δ Vj+1 ‖C s+κe ⃗ ≲ ϵ1 + ϵ1 Qj . 1 (δ λds W ,ds) 1 (δ λds W ,ds) j=1

Taking the supremum over such x and δ shows that Qj+1 ≲ ϵ1 + ϵ1 Qj . That is, ∃C6 ≈ 1 with Qj+1 ≤ C6 (ϵ1 + ϵ1 Qj ). We take ϵ1 := min{1/2C6 , 1/2C5 , 1/D2 }. Then we have Qj+1 ≤

1 Qj + . 2 2

Since Q0 = 0, a simple induction using (9.83) shows that Qj ≤ 1, ∀j ∈ ℕ.

(9.83)

656 � 9 Nonlinear maximally subelliptic equations Finally, we take ψ0 ∈ C0∞ (Bn (2/3)). Take x1 , . . . , xN2 and δ1 ∈ (0, 1] as in Theorem 9.2.8 (ix). In particular, we have B(W 1 ,ds1 ) (xj , δ1 ) ⊆ Bn (3/4), δ1 ≈ 1, and N2 ≈ 1. By (9.19), we have N2

‖ψ0 Vj ‖C s+κe ⃗ ⃗ ⃗ ⃗ ≲ ∑ ‖ϕxj ,δ1 Vj ‖C s+κe ⃗ ≲ Qj ≤ 1. 1 (W ,ds) 1 (δ λds W ,ds) 1

j=1

That is, for l ∈ N ν , ‖Dl ψ0 Vj ‖L∞ ≲ 2−l⋅(s+κe1 ) . ⃗

(9.84)

⃗ Corollary 7.2.2 shows that Vj → V∞ in L∞ . Since Vj → V∞ in C r+κe1 (Bn (7/8), (W , ds)), Since each Dl is bounded on L∞ (see (5.6)), taking the limit as j → ∞ in (9.84) shows that ⃗

‖Dl ψ0 V∞ ‖L∞ ≲ 2−l⋅(s+κe1 ) , ⃗

∀l ∈ ℕν .

(9.85)

⃗ ⃗ with Since supp(ψ0 Vj ) ⊆ Bn (7/8), (9.85) implies that ψ0 V∞ ∈ C s+κe1 (Bn (7/8), (W , ds))

‖ψ0 V∞ ‖C s+κe ⃗ ⃗ ≲ 1. 1 (W ,ds)

(9.86)

Since ψ0 ≺ ψ3 , we have ψ0 H = ψ0 ψ3 H = ψ0 ψ3 e∞ ∈ C0∞ (Bn (7/8)). Using this, Propo⃗ ⃗ Moreover, using Corollary 6.5.10 sition 6.5.5 shows that ψ0 H ∈ C s+κe1 (Bn (7/8), (W , ds)). and (9.58), we have ‖ψ0 H‖C s+κe ⃗ ⃗ ⃗ ⃗ = ‖ψ0 ψ3 H‖C s+κe ⃗ ≲ ‖ψ3 H‖C s+κe ⃗ ≲ ϵ1 . 1 (W ,ds) 1 (W ,ds) 1 (W ,ds)

(9.87)

⃗ ⃗ and using (9.86) and (9.87) Since u = H + V∞ , we have ψ0 u ∈ C s+κe1 (Bn (7/8), (W , ds)), we see that

‖ψ0 u‖C s+κe ⃗ ⃗ ⃗ ⃗ ≤ ‖ψ0 H‖C s+κe ⃗ + ‖ψ0 V∞ ‖C s+κe ⃗ ≲ 1, 1 (W ,ds) 1 (W ,ds) 1 (W ,ds) completing the proof.

9.2.5 Step II: A simpler form In this section, we present a special case of Proposition 9.2.15 whose hypotheses will prove easier to verify in Step III (Section 9.2.7). Let ϵ2 ∈ (0, 1] be a small number, to be chosen later. Suppose R3 (x, ζ ) ∈ C s,⌊|s|1 ⌋+2+ν+σ (Bn (7/8) × ℝN , (W , ds)⃗ ⊠ ∇ℝN ; ℝD1 ) ⃗

with

⃗

9.2 Main quantitative result

‖R3 ‖C s,⌊|s|⃗ 1 ⌋+2+ν+σ ((W ,ds)⊠∇ ⃗

ℝN

)

≤ ϵ2 .

�

657 (9.88)

Let G(ζ ) ∈ C ⌊|s|1 ⌋+3+ν+σ (ℝN ; ℝD1 ) and take C3 ≥ 0 with ‖G‖C ⌊|s|1 ⌋+3+ν+σ (ℝN ) ≤ C3 ⃗ ⃗ ℝD2 ) with and u ∈ C r+κe1 (Bn (7/8), (W , ds);

‖u‖C r+κe ⃗ ⃗ ≤ ϵ2 . 1 (W ,ds) We assume dG(Wκ1 u(0))Wκ1 is maximally subelliptic of degree κ with respect to (W , ds)⃗ in the sense that there exists A2 ≥ 0 such that r1

n 󵄩 󵄩 ∑󵄩󵄩󵄩(Wj1 ) j v󵄩󵄩󵄩L2 (Bn (1),hσ ) Leb j=1

󵄩 󵄩 󵄩 󵄩 ≤ A2 (󵄩󵄩󵄩dG(Wκ1 u(0))Wκ1 v󵄩󵄩󵄩L2 (Bn (1),hσ ) + 󵄩󵄩󵄩v󵄩󵄩󵄩L2 (Bn (1),hσ ) ), Leb Leb

(9.89)

for all v ∈ C0∞ (Bn (1); ℂD2 ). Finally, we suppose u satisfies the following nonlinear PDE: G(Wκ1 u(x)) + R3 (x, Wκ1 u(x)) = 0,

∀x ∈ Bn (13/16).

(9.90)

Proposition 9.2.17. There exists ϵ2 ∈ (0, 1] sufficiently small such that if the above holds, then ∀ψ0 ∈ C0∞ (Bn (2/3)) we have ⃗ ℝD2 ) ψ0 u ∈ C s+κe1 ((W , ds); ⃗

and ∃C4 ≥ 0 with ‖ψ0 u‖C s+κe ⃗ ⃗ ≤ C4 . 1 (W ,ds) Here, ϵ2 = ϵ2 (C3 , A2 , D1 , D2 ) ∈ (0, 1] and C4 = C4 (C3 , A2 , D1 , D2 , ψ0 ) ≥ 0 are (κ, s,⃗ r,⃗ σ)multi-parameter unit-admissible constants. Proof. Lemma 8.3.3 (i) combined with the fact that h ≈ 1 shows that ∑

degds1 (α)≤κ

󵄩󵄩 1 α 󵄩󵄩 󵄩󵄩(W ) v󵄩󵄩L2 (Bn (1),hσ

Leb )

r1

󵄩 󵄩 ≈ ∑󵄩󵄩󵄩(Wj1 )nj v󵄩󵄩󵄩L2 (Bn (1),hσ j=1

Leb )

+ ‖v‖L2 (Bn (1),hσLeb ) ,

∀v ∈ C0∞ (Bn (31/32); ℂD2 ). Combining this with (9.89) shows that there exists A3 ≈ 1 with

658 � 9 Nonlinear maximally subelliptic equations

∑

degds1 (α)≤κ

󵄩󵄩 1 α 󵄩󵄩 󵄩󵄩(W ) v󵄩󵄩L2 (Bn (1),hσLeb )

󵄩 󵄩 󵄩 󵄩 ≤ A3 (󵄩󵄩󵄩dG(Wκ1 u(0))Wκ1 v󵄩󵄩󵄩L2 (Bn (1),hσ ) + 󵄩󵄩󵄩v󵄩󵄩󵄩L2 (Bn (1),hσ ) ), Leb Leb

(9.91)

∀v ∈ C0∞ (Bn (31/32); ℂD2 ). We claim that if ϵ2 > 0 is sufficiently small, then ∑

degds1 (α)≤κ

󵄩󵄩 1 α 󵄩󵄩 󵄩󵄩(W ) v󵄩󵄩L2 (Bn (1),hσLeb )

󵄩 󵄩 ≤ 2A3 (󵄩󵄩󵄩dG(0)Wκ1 v󵄩󵄩󵄩L2 (Bn (1),hσ

Leb

󵄩󵄩 󵄩󵄩 󵄩v󵄩󵄩L2 (Bn (1),hσ )+󵄩

(9.92) Leb

) ),

∀v ∈ C0∞ (Bn (31/32); ℂD2 ). To see (9.92), we first use Corollaries 7.2.2 and 6.5.11 to see that 󵄩󵄩 κ 󵄩󵄩 κ 󵄩󵄩W1 u󵄩󵄩L∞ ≲ ‖W1 u‖C r⃗ (W ,ds)⃗ ≲ ‖u‖C r+κe ⃗ ⃗ ≲ ϵ2 . 1 (W ,ds)

(9.93)

Thus, since ⌊|s|1 ⌋ + 3 + ν + σ > 2, we have β 󵄨󵄨 󵄨 󵄨 κ 󵄨 κ 󵄨󵄨dG(W1 u(0)) − dG(0)󵄨󵄨󵄨 ≲ ∑ ‖𝜕ζ G‖L∞ (ℝN ) 󵄨󵄨󵄨W1 u(0)󵄨󵄨󵄨 |β|≤2

≲ ‖G‖C ⌊|s|1 ⌋+3+ν+σ (ℝN ) ‖Wκ1 u‖L∞ ≲ ϵ2 .

(9.94)

Plugging (9.94) into (9.91) we conclude that for some C5 ≈ 1, we have 󵄩󵄩 κ 󵄩󵄩 󵄩󵄩W1 v󵄩󵄩L2 (Bn (1),hσ

Leb )

󵄩 󵄩 󵄩 󵄩 ≤ A3 (󵄩󵄩󵄩dG(0)Wκ1 v󵄩󵄩󵄩L2 (Bn (1),hσ ) + 󵄩󵄩󵄩v󵄩󵄩󵄩L2 (Bn (1),hσ ) ) Leb Leb 󵄩 󵄩 + A3 C5 ϵ2 󵄩󵄩󵄩Wκ1 v󵄩󵄩󵄩L2 (Bn (1),hσ ) , Leb

∀v ∈ C0∞ (Bn (31/32); ℂD2 ). Taking ϵ2 ∈ (0, 1 ∧ 1/2A3 C5 ] gives (9.92). Next, using ⌊|s|1 ⌋ + 3 + ν + σ > 1 and (9.93), we have β 󵄨󵄨 󵄨 󵄨 κ 󵄨 κ 󵄨󵄨G(W1 u(0)) − G(0)󵄨󵄨󵄨 ≲ ∑ ‖𝜕ζ G‖L∞ 󵄨󵄨󵄨W1 u(0)󵄨󵄨󵄨 |β|≤1

≲ ‖G‖C ⌊|s|1 ⌋+3+ν+σ (ℝN ) ‖Wκ1 u‖L∞ ≲ ϵ2 .

(9.95)

Also, using (9.90) and Proposition 7.5.10 gives 󵄨󵄨 󵄨 󵄨 󵄨 κ κ κ 󵄨󵄨G(W1 u(0))󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨G(W1 u(0)) + R3 (0, W1 u(0))󵄨󵄨󵄨 + ‖R3 ‖L∞ = 0 + ‖R3 ‖L∞ ≲ ‖R3 ‖C s,⌊|s|1 ⌋+2+ν+σ ((W ,ds)⊠∇ ≲ ϵ2 . ⃗ N)

(9.96)

ℝ

Combining (9.95) and (9.96) gives |G(0)| ≲ ϵ2 .

(9.97)

Define E(ζ ) by G(ζ ) = G(0) + dζ G(0)ζ + E(ζ ), so that E(0) = 0, dζ E(0) = 0, E ∈

C ⌊|s|1 ⌋+3+ν+σ (ℝN ), and

9.2 Main quantitative result

‖E‖C ⌊|s|1 ⌋+3+ν+σ (ℝN ) ≲ 1.

� 659

(9.98)

Set P := dζ G(0)Wκ1 . Rewriting (9.90), we have κ

κ

∀x ∈ Bn (13/16).

P u(x) = −G(0) − R3 (x, W1 u(x)) − E(W1 u(x)),

(9.99)

Fix ψ1 ∈ C0∞ (Bn (7/8)) with ψ1 ≡ 1 on Bn (13/16). Set R1 (x, ζ ) := −ψ1 (x)(G(0) + R3 (x, ζ )),

R2 (x, ζ ) := −ψ1 (x)E(ζ ).

Then (9.99) implies κ

κ

P u(x) = R1 (x, W1 u(x)) + R2 (x, W1 u(x)),

∀x ∈ Bn (13/16).

(9.100)

Furthermore, using (9.97) and (9.88), we have ‖R1 ‖C s,⌊|s|1 ⌋+2+ν+σ ((W ,ds)⊠∇ ⃗

ℝN

)

≲ |G(0)| + ‖R3 ‖C s,⌊|s|1 ⌋+2+ν+σ ((W ,ds)⊠∇ ⃗

)

≲ ϵ2 ,

)

≤ C6 .

ℝN

and using (9.98), ‖R2 ‖C s,⌊|s|1 ⌋+3+ν+σ ((W ,ds)⊠∇ ⃗

ℝN

)

≲ ‖E‖C ⌊|s|1 ⌋+3+ν+σ (ℝN ) ≲ 1.

That is, there is C6 ≈ 1 with ‖R1 ‖C s,⌊|s|1 ⌋+2+ν+σ ((W ,ds)⊠∇ ⃗

ℝN

)

≤ C6 ϵ2 ,

‖R2 ‖C s,⌊|s|1 ⌋+3+ν+σ ((W ,ds)⊠∇ ⃗

ℝN

Also note that |dζ G(0)| ≤ ‖G‖C ⌊|s|1 ⌋+3+ν+σ (ℝN ) ≲ 1, i. e., there exists C7 ≈ 1 with |dζ G(0)| ≤ C7 . β

Since E(0) = 0 and dζ E(0) = 0, we have 𝜕ζ R2 (x, 0) ≡ 0, ∀|β| ≤ 1. Let ϵ1 = ϵ1 (C6 , 2A3 , C7 , D1 , D2 ) ∈ (0, 1] be the (κ, s,⃗ r,⃗ σ)-multi-parameter unitadmissible constant of the same name from Proposition 9.2.15. Taking ϵ2 := min{ϵ1 /C6 , 1}, P , R1 , and R2 satisfy all the hypotheses of Proposition 9.2.15 (the maximal subellipticity of P is (9.92)). The conclusions of Proposition 9.2.15, in this case, are exactly the conclusions of this proposition.

660 � 9 Nonlinear maximally subelliptic equations 9.2.6 Scaled decompositions of Zygmund–Hölder functions To complete the proof of Theorem 9.2.1, we require scaled analogs of Propositions 7.2.1 and 7.5.9. In this section, we use only the scaling maps Φ0,δ ; even though the results can be generalized to work with all Φx,δ , we only require them with x = 0 and we restrict our attention to that case for simplicity. Let λ = (1, λ2 , . . . , λν ) ∈ (0, ∞)ν be as in Section 3.15.3. Proposition 9.2.18. Fix J ≥ 0, s⃗ ∈ (0, ∞)ν , t > 0, and ψ ∈ C0∞ (Bn (7/8)). ∞ n (i) Suppose that for each j ∈ ℕν , uj ∈ Cloc (B (1)) satisfies, ∀L ∈ ℕ, ⃗ α 󵄩 ⃗ 󵄩 ∑ 󵄩󵄩󵄩(2−(j+λJ)dsW ) uj 󵄩󵄩󵄩L∞ (B 1 1 (0,2−J )) ≤ CL 2−j⋅s E, (X ,d )

|α|≤L

for some constants CL , E ≥ 0. Set ̂ := ψ(y) ∑ uj (Φ0,2−J (y)). u(y) j∈ℕν

⃗ and Then û ∈ C s (Bn (7/8), (W 0,2 , ds)) −J

‖u‖̂ C s⃗ (W 0,2−J ,ds)⃗ ≤ Cs⃗ E, where there exists a multi-parameter unit-admissible constant L = L(s)⃗ ∈ ℕ such ⃗ that Cs⃗ = Cs⃗ (CL , ψ) is an s-multi-parameter unit-admissible constant. ∞ n (ii) Suppose that for j ∈ ℕν and k ∈ ℕ, Hj,k ∈ Cloc (B (1) × ℝN ) satisfies, ∀L ∈ ℕ, ⃗ α β ⃗ 󵄩 󵄩 ∑ 󵄩󵄩󵄩(2−(j+λJ)dsW ) (𝜕ζ ) Hj,k (x, ζ )󵄩󵄩󵄩L∞ (B 1 1 (0,2−J )×ℝN ) ≤ CL E2−j⋅s−kt , (X ,d )

(9.101)

|α|,|β|≤L

for some constants CL , E ≥ 0. Set ̂ ζ ) := ψ(y) ∑ Hj,k (Φ −J (y), ζ ). H(y, 0,2 j∈ℕν k∈ℕ

⃗ Then Ĥ ∈ C s,t (Bn (7/8) × ℝN , (W 0,2 , ds)⃗ ⊠ ∇ℝN ) and −J

̂ s,t⃗ ‖H‖ −J ⃗ C ((W 0,2 ,ds)⊠∇

ℝN

)

≤ Cs,t⃗ E,

where there exists a multi-parameter unit-admissible constant L = L(s,⃗ t) ∈ ℕ such that Cs,t⃗ = Cs,t⃗ (CL , N, ψ) is an s,⃗ t-multi-parameter unit-admissible constant. Proof. The proofs for (i) and (ii) are similar; we begin with the proof of (ii). Set Ĥ j,k (y, ζ ) := ψ(y)Hj,k (Φ0,2−J (y, ζ )). Since ψ ∈ C0∞ (Bn (7/8)) is a fixed function, μ,0,2−J

Φ∗0,2−J 2−Jλ⋅dsW = W 0,2 (Theorem 3.15.5), and ‖Wl ⃗

−J

‖C L ≲ 1, ∀l, μ ∈ {1, . . . , ν} (see Theo-

9.2 Main quantitative result

�

661

rem 3.15.5 (h)), we have −J α ⃗ β 󵄩 󵄩 ∑ 󵄩󵄩󵄩(2−jdsW 0,2 ) (2−k 𝜕ζ ) Ĥ j,k (y, ζ )󵄩󵄩󵄩L∞ (Bn (1)×ℝN )

|α|,|β|≤L

≲L

−J α ⃗ β 󵄩 󵄩 ∑ 󵄩󵄩󵄩(2−jdsW 0,2 ) (2−k 𝜕ζ ) Hj,k (Φ0,2−J (y), ζ )󵄩󵄩󵄩L∞ (Bn (1)×ℝN )

|α|,|β|≤L

=

⃗ α β 󵄩 󵄩 ∑ 󵄩󵄩󵄩(2−(j+λJ)dsW ) (2−k 𝜕ζ ) Hj,k (x, ζ )󵄩󵄩󵄩L∞ (Φ −J (Bn (1))×ℝN ) 0,2

|α|,|β|≤L

≤ ≤

⃗ α β 󵄩 󵄩 ∑ 󵄩󵄩󵄩(2−(j+λJ)dsW ) (2−k 𝜕ζ ) Hj,k (x, ζ )󵄩󵄩󵄩L∞ (B 1 1 (0,2−J )×ℝN ) (X ,d )

|α|,|β|≤L ⃗ CL 2−j⋅s−k⋅t E,

where in the second to last inequality we used Φ0,2−J (Bn (1)) ⊆ B(X 1 ,d 1 ) (0, 2−J ) (see Theorem 3.15.5) and in the last inequality we used the assumption (9.101). Since Ĥ = ∑j,k Ĥ j,k , the result follows from Proposition 7.5.9. The proof for (i) is similar, where there is no ζ variable and we use Proposition 7.2.1 in place of Proposition 7.5.9. ∞ n Lemma 9.2.19. Fix s⃗ ∈ (0, ∞)ν . Let u ∈ Cloc (B (1)) and J ∈ ℕ. Suppose that, ∀L ∈ ℕ, 1 β 󵄩 󵄩 ∑ 󵄩󵄩󵄩(2−Jds W 1 ) u󵄩󵄩󵄩L∞ (B 1 1 (0,2−J )) ≤ CL E, (X ,d )

(9.102)

|β|≤L

for some constants CL , E ≥ 0. Fix ψ ∈ C0∞ (Bn (7/8)) and set ̂ := ψ(y)u(Φ0,2−J (y)). u(y) ⃗ and Then u ∈ C s (Bn (7/8), (W 0,2 , ds)) −J

⃗

‖u‖̂ C s⃗ (W 0,2−J ,ds)⃗ ≤ Cs⃗ E,

(9.103)

where there exists a multi-parameter unit-admissible constant L = L(s)⃗ ∈ ℕ such that ⃗ Cs⃗ = Cs⃗ (CL , ψ) is an s-multi-parameter unit-admissible constant. 1

Proof. Using the fact that ψ ∈ C0∞ (Bn (7/8)) is a fixed function, Φ∗0,2−J 2−J⋅ds W 1 = W 1,0,2 (Theorem 3.15.5), and ‖Wl1,0,2 ‖C L ≲ 1, ∀l (see Theorem 3.15.5 (h)), we have −J

−J β −J β 󵄩 󵄩 󵄩 󵄩 ∑ 󵄩󵄩󵄩(W 1,0,2 ) û 󵄩󵄩󵄩L∞ (Bn (1)) ≲L ∑ 󵄩󵄩󵄩(W 1,0,2 ) u ∘ Φ0,2−J 󵄩󵄩󵄩L∞ (Bn (1))

|β|≤L

|β|≤L

1 β 󵄩 󵄩 = ∑ 󵄩󵄩󵄩(2−Jds W 1 ) u󵄩󵄩󵄩L∞ (Φ −J (Bn (1))) 0,2

|β|≤L

1 β 󵄩 󵄩 ≤ ∑ 󵄩󵄩󵄩(2−Jds W 1 ) u󵄩󵄩󵄩L∞ (B 1 1 (0,2−J )) (X ,d )

|β|≤L

≤ CL E,

−J

662 � 9 Nonlinear maximally subelliptic equations where the second to last inequality uses Φ0,2−J (Bn (1)) ⊆ B(X 1 ,d 1 ) (0, 2−J ) (see Theorem 3.15.5) and in the last inequality we used the assumption (9.102). −J −J Using the fact that each Xl1,0,2 is a commutator of the Wl1,0,2 vector fields (see Theorem 3.15.5), it follows that −J β 󵄩 󵄩 ∑ 󵄩󵄩󵄩(X 1,0,2 ) û 󵄩󵄩󵄩L∞ (Bn (1)) ≲L E.

(9.104)

|β|≤L

By Theorem 3.15.5 (j), (9.104) implies that ‖u‖̂ C L (Bn (1)) ≲L E.

(9.105)

Using Theorem 3.15.5 (h), (9.105) implies that −J α 󵄩 󵄩 ∑ 󵄩󵄩󵄩(W 0,2 ) û 󵄩󵄩󵄩L∞ (Bn (1)) ≲L E.

|α|≤L

Using Proposition 7.2.1 (with only one term) and the fact that supp(u)̂ ⊆ Bn (7/8), it fol−J ⃗ ⃗ (for every s⃗ ∈ (0, ∞)ν ) and (9.103) holds. lows that û ∈ C s (Bn (7/8), (W 0,2 , ds)) 9.2.7 Step III: Completion of the proof To prove Theorem 9.2.1, we “zoom in” near 0 ∈ ℝn by using the scaling maps Φ0,δ . We will see that if we zoom in sufficiently close to zero, then (9.12) is of the form covered by Proposition 9.2.17. First, note that it suffices to prove Theorem 9.2.1 with g = 0 by replacing F(x, ζ ) with F(x, ζ ) − g(x); we henceforth assume that g = 0. Take J ∈ ℕ large, to be chosen later. Set uJ :=

∑

0≤j1 0, and κ ∈ ℕ+ . ⃗ For J ∈ [0, ∞) and Lemma 9.2.20. Fix ψ ∈ C0∞ (Bn (7/8)) and u ∈ C r+κe1 (Bn (7/8), (W , ds)). degds1 (α) ≤ κ, set ⃗

uJ,α (y) := ũ J,α (y) :=

α

α

∑ ((W 1 ) Dj u)(Φ0,2−J (y)) = ((W 1 ) uJ )(Φ0,2−J (y)),

0≤j1 δ, we have

̂ γ([0, l′ ]) ⊆ B

(X x0 ,2

−J0

,d)

(0, qlξ0 2J0 δ) ⊆ B

(X x0 ,2

−J0

,d)

(0, ξ3̂ δ1 ).

(9.181)

Combining (9.180) and (9.181) shows that ̂−1 )v̂x ,J = △l ̂ v̂x ,J . △lγ̂ (ψ0 ∘ Φ 0,δ1 γ 0 0 0 0 ′

′

(9.182)

̂ Since ψ ≡ 1 on Bn (3/4) and we have already established that γ([0, l′ ]) ⊆ Bn (1/2), using (9.158), we have △lγ̂ v̂x0 ,J0 = △lγ̂ vJ0 ∘ Φx0 ,2−J0 . ′

′

(9.183)

Plugging (9.183) and (9.182) into (9.179) establishes (9.174) and completes the proof.

684 � 9 Nonlinear maximally subelliptic equations 9.3.1 Weighted Schauder estimates near the boundary We take the same setting as above, but we now assume that our partial differential operator is linear and has the form P u(x) =

∑

degds (α)≤κ

aα (x)W α u(x),

where aα ∈ C s (𝒦, (W , ds); 𝕄D1 ×D2 (ℝ)). Maximally subelliptic assumption: For each x0 ∈ Ω, define the frozen coefficient operator Px0 v :=

∑

degds (α)≤κ

aα (x0 )W α u.

We assume that there exists A ≥ 0 such that ∀f ∈ C0∞ (Ω; ℂD2 ), ∀x0 ∈ Ω, r

󵄩 n 󵄩 󵄩 󵄩 󵄩 󵄩 ∑󵄩󵄩󵄩Wj j f 󵄩󵄩󵄩L2 (M,Vol;ℂD2 ) ≤ A(󵄩󵄩󵄩Px0 f 󵄩󵄩󵄩L2 (M,Vol;ℂD1 ) + 󵄩󵄩󵄩f 󵄩󵄩󵄩L2 (M,Vol;ℂD2 ) ). j=1

Theorem 9.3.5. Under the above hypotheses, suppose that u ∈ C r+κ (𝒦, (W , ds); ℝD2 ) and g ∈ C s (𝒦, (W , ds); ℝD1 ) such that P u(x) = g(x),

∀x ∈ Ω.

Then, ∀l ∈ ℕ+ with l > s, we have ‖u‖Ĉs+κ (Ω,(W ,ds),(X,d)) ≤ C(‖u‖C r+κ (W ,ds) + ‖g‖C s (W ,ds) ). l

Here, C ≥ 0 does not depend on u or g, but may depend on any of the other ingredients in the result. Remark 9.3.6. Theorem 9.3.5 can likely be improved in several ways by using the methods of this text, though we do not do so here. See Remark 9.1.13. Proof of Theorem 9.3.5. By replacing u and g with u/(‖u‖C r+κ (W ,ds) + ‖g‖C s (W ,ds) ) and g/(‖u‖C r+κ (W ,ds) + ‖g‖C s (W ,ds) ), respectively, we see that we may assume ‖u‖C r+κ (W ,ds) , ‖g‖C s (W ,ds) ≤ 1 and in this case it suffices to prove ‖u‖Ĉs+κ (Ω,(W ,ds),(X,d)) ≲ 1. l

Using Corollaries 7.2.2 and 6.5.11, we have, ∀ degds(α) ≤ κ, 󵄩󵄩 α 󵄩󵄩 α 󵄩󵄩W u󵄩󵄩L∞ ≲ ‖W u‖C r (W ,ds) ≲ ‖u‖C r+κ (W ,ds) ≤ 1.

(9.184)

� 685

9.4 Examples

󵄩 󵄩 We conclude that there exists C ≈ 1 with 󵄩󵄩󵄩W α u󵄩󵄩󵄩L∞ ≤ C, ∀ degds(α) ≤ κ. Fix ϕ(ζ ) ∈ ∞ N C0 (ℝ ; ℝ) with ϕ(ζ ) ≡ 1 on a neighborhood of {ζ : |ζα | ≤ 2C, ∀ degds(α) ≤ κ}. Define F(x, ζ ) :=

∑

degds (α)≤κ

aα (x)ζα ϕ(ζ ).

Note that F(x, ζ ) ∈ C s,3⌊s⌋+9 (𝒦 × ℝN , (W , ds) ⊠ ∇ℝN ; ℝD1 ) with ‖F‖C s,3⌊s⌋+9 ((W ,ds)⊠∇

ℝN

Since |W α u(x)| ≤ C, ∀ degds(α) ≤ κ, ∀x, we have α

P u(x) = F(x, {W u(x)}deg

ds (α)≤κ

)

≲ 1.

),

and therefore, F(x, {W α u(x)}deg

ds (α)≤κ

) = g(x),

∀x ∈ Ω.

(9.185)

Again using the fact that |W α u(x)| ≤ C, ∀ degds(α) ≤ κ, ∀x, we have dζ F(x, {W α u(x)}deg

ds (α)≤κ

){ζα }deg

ds (α)≤κ

=

∑

degds (α)≤κ

aα (x)ζα ,

∀x ∈ Ω.

Therefore, α

Px0 = dζ F(x0 , {W u(x0 )}deg

ds (α)≤κ

){W α }deg

ds (α)≤κ

.

We have established that all the hypotheses of Theorem 9.3.2 hold for (9.185). The conclusion of Theorem 9.3.2 in this case is exactly (9.184), completing the proof.

9.4 Examples In this section, we present some examples of fully nonlinear maximally subelliptic equations, which generalize ideas from the classical elliptic theory. 9.4.1 Second-order equations Many classical examples of fully nonlinear elliptic equations are second-order equations whose linearization is given by a positive definite form. These equations often have generalizations to the maximally subelliptic setting. On ℝn , consider a nonlinear partial differential operator given by u 󳨃→ F(x, {𝜕xα u(x)}|α|≤2 ),

(9.186)

where F is a real-valued function. If we consider F as a function of x ∈ ℝn and ζ , then ζ 2 is a vector in ℝn +n+1 ,

686 � 9 Nonlinear maximally subelliptic equations ζ = (ζ0 , ζ1 , . . . , ζn , ζ1,1 , . . . , ζi,j , . . . , ζn,n ), where, for example, the coordinate ζi,j corresponds to 𝜕xi 𝜕xj u. For a fixed u and x0 , if the n × n matrix 𝜕ζ1,1 F(x0 , {𝜕xα u(x0 )}|α|≤2 ) .. ( . 𝜕ζn,1 F(x0 , {𝜕xα u(x0 )}|α|≤2 )

⋅⋅⋅ .. . ⋅⋅⋅

𝜕ζ1,n F(x0 , {𝜕xα u(x0 )}|α|≤2 ) .. ) . α 𝜕ζn,n F(x0 , {𝜕x u(x0 )}|α|≤2 )

is symmetric and strictly positive definite, then (9.186) is elliptic at (x0 , u). A similar phenomenon occurs in the maximally subelliptic setting. Let (W , 1) = {(W1 , 1), . . . , (Wr , 1)} be Hörmander vector fields on a smooth, connected manifold M, each paired with formal degree 1. We consider a nonlinear partial differential operator given by u 󳨃→ F(x, {W α u(x)}|α|≤2 ),

(9.187)

where F is a real-valued function. Similar to the above, we consider F = F(x, ζ ) as a 2 function of x ∈ M and ζ ∈ ℝr +r+1 , ζ = (ζ0 , ζ1 , . . . , ζr , ζ1,1 , . . . , ζi,j , . . . , ζr,r ), where, for example, the coordinate ζi,j corresponds to Wi Wj u. Proposition 9.4.1. Fix x0 ∈ M and u : M → ℝ. If the r × r matrix 𝜕ζ1,1 F(x0 , {W α u(x0 )}|α|≤2 ) .. ( . 𝜕ζr,1 F(x0 , {W α u(x0 )}|α|≤2 )

⋅⋅⋅ .. . ⋅⋅⋅

𝜕ζ1,r F(x0 , {W α u(x0 )}|α|≤2 ) .. ) . α 𝜕ζr,r F(x0 , {W u(x0 )}|α|≤2 )

(9.188)

is symmetric and strictly positive definite, then (9.187) is maximally subelliptic at (x0 , u) of degree 2 with respect to (W , 1) in the sense of Definition 1.1.11. Proof. By the definition (Definition 1.1.11) the maximal subellipticity of (9.187) at (x0 , u) is equivalent to the maximal subellipticity of the linear operator: v 󳨃→ ∇ζ F(x0 , {W α u(x0 )}|α|≤2 ) ⋅ (v, W1 v, . . . , Wr v, W1 W1 v, . . . , Wi Wj v, . . . , Wr Wr v) r

= 𝜕ζ0 F(x0 , {W α u(x0 )}|α|≤2 )v + ∑ 𝜕ζj F(x0 , {W α u(x0 )}|α|≤2 )Wj v r

j=1

+ ∑ 𝜕ζj,k F(x0 , {W α u(x0 )}|α|≤2 )Wj Wk v. j,k=1

The maximal subellipticity of (9.189) follows from Lemma 8.9.3.

(9.189)

9.4 Examples

� 687

Corollary 9.4.2. Let F, u, and g satisfy F(x, {W α u(x)}|α|≤2 ) = g(x) and fix x0 ∈ M and s > r > 0, t > 2⌊s⌋ + 6. Suppose that: – F ∈ C s,t ((W , 1) ⊠ ∇ℝN ) near x0 , – g ∈ C s (W , 1) near x0 , – u ∈ C r+2 (W , 1) near x0 , – the matrix given in (9.188) is symmetric and strictly positive definite. Then u ∈ C s+2 (W , 1) near x0 . Proof. This follows by combining Theorem 9.1.2 and Proposition 9.4.1. Corollary 9.4.3. Let F, u, and g satisfy F(x, {W α u(x)}|α|≤2 ) = g(x) and fix x0 ∈ M and r > 0. Suppose that: ∞ – F ∈ Cloc (M × ℝn ), ∞ – g ∈ Cloc (M) near x0 , – u ∈ C r+2 (W , 1) near x0 , – the matrix given in (9.188) is symmetric and strictly positive definite. ∞ Then u ∈ Cloc (M) near x0 .

Proof. This follows by combining Corollary 9.1.5 and Proposition 9.4.1. Remark 9.4.4. Similarly, we can apply the many other regularity results in this chapter to immediately obtain other regularity results about solutions to these second-order, maximally subelliptic equations. For example, Corollary 9.1.8 gives regularity properties in terms of the classical Zygmund–Hölder spaces. Remark 9.4.5. In this text we only address classical solutions to fully nonlinear equations; this is because our main theorems deal with general fully nonlinear equations (of arbitrary order), so we cannot talk about weak solutions. In various settings, the theory of second-order elliptic equations has been extended to many types of weak solutions; see, for example, [25]. There has been some work on weak solutions for maximally subelliptic equations; see, for example, [168]. See also the references given in Section 9.5. It seems likely that much more can be done in parallel with elliptic theory, though we do not pursue that here.

9.4.2 The Monge–Ampère equation An important classical example of a second-order fully nonlinear elliptic equation is the Monge–Ampère equation; see, e. g., [87]. For a function u : ℝn → ℝ, let (𝜕xi 𝜕xj u) denote the n × n matrix with the (i, j) component equal to 𝜕xi 𝜕xj u. In the Monge–Ampère equation, one considers the Hessian: u 󳨃→ det(𝜕xi 𝜕xj u).

(9.190)

688 � 9 Nonlinear maximally subelliptic equations When u is convex, this equation is elliptic. When u is at least C 2 , one can rephrase this as saying that if (𝜕xi 𝜕xj u(x0 )) > 0, i. e., if the matrix is positive definite, then (9.190) is elliptic at (x0 , u). In [157] and [57] a generalization of convexity and the Monge–Ampère equation were introduced (and their relationship discussed) in the setting of stratified nilpotent Lie groups. Several papers followed, extending this notion to other settings; see [56, 58, 101, 102, 137, 245, 248, 32, 7, 168]. In this section, we describe this generalized Monge– Ampère equation in the general setting of Hörmander vector fields on a manifold and show that when the function is “convex” (in the sense of positive “Hessian”), the Monge– Ampère equation is maximally subelliptic. This is a special case of the more general second-order results described above. As in Section 9.4.1, let (W , 1) = {(W1 , 1), . . . , (Wr , 1)} be Hörmander vector fields, each paired with formal degree 1, on a connected smooth manifold M. For a real-valued function u : M → ℝ, let Wi,j u(x) := 21 (Wi Wj + Wj Wi )u(x) and let W u := (Wi,j u(x)) denote the r × r matrix with the (i, j) component equal to Wi,j u(x). The Monge–Ampére equation in [57] takes the form u 󳨃→ det W u.

(9.191)

Proposition 9.4.6. Fix x0 ∈ M and suppose that W u(x0 ) > 0, i. e., W u(x0 ) is a strictly positive definite matrix. Then (9.191) is maximally subelliptic at (x0 , u) of degree 2 with respect to (W , 1) in the sense of Definition 1.1.11. Proof. Let W i,j u denote the (i, j) component of the inverse matrix (W u)−1 . Using the matrix identity d 󵄨󵄨󵄨󵄨 −1 󵄨 det(A + ϵB) = det(A)tr(A B), dϵ 󵄨󵄨󵄨ϵ=0

(9.192)

we see that the linearization of (9.191) with respect to u at the point x0 is the operator v 󳨃→ ∑ (det W u(x0 ))W i,j u(x0 )Wi,j v. 1≤i,j≤r

(9.193)

The matrix with (i, j) component equal to (det W u(x0 ))W i,j u(x0 ) is strictly positive definite, and Lemma 8.9.3 implies that (9.193) is maximally subelliptic of degree 2 with respect to (W , 1) on M, which completes the proof. In light of Proposition 9.4.6, the conclusions of Corollaries 9.4.2 and 9.4.3 hold when det W u(x0 ) > 0. See Corollaries 9.4.11 and 9.4.12 for a precise statement of this in a more general setting. Remark 9.4.7. As discussed in Remark 9.4.5, we only address classical solutions to the generalized Monge–Ampère equation (9.191). This is because we deduce results about (9.191) by applying the general fully nonlinear theory, where we cannot talk

9.4 Examples

� 689

about weak solutions in general. For the classical Monge–Ampère equation (9.190) one can use the special form to talk about weak solutions; see the discussion of Alexandrov solutions in Chapter 2 of [87]. Similar ideas have been developed in the setting of Carnot groups; see, for example, [157]. It would be interesting to generalize these ideas to the general maximally subelliptic setting.

9.4.3 Higher-order Monge–Ampère equations The generalized Monge–Ampère equation in Section 9.4.2 can be further generalized to higher-order equations in at least two different ways, which we present in this section. Let (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} be Hörmander vector fields with formal degrees on M. Fix κ ∈ ℕ+ such that dsj divides κ for every j and set nj := κ/dsj ∈ ℕ+ . The first generalization: For a real-valued function u : M → ℝ, let W [κ]u(x) denote the symmetric r × r real matrix with the (i, j) component equal to W [κ]i,j u(x) :=

1 n n n n (W i W j u(x) + Wj j Wi i u(x)). 2 i j

We consider a higher-order generalized Monge–Ampère equation given by u 󳨃→ det W [κ]u(x).

(9.194)

Remark 9.4.8. When dsj = 1, ∀1 ≤ j ≤ r, and κ = 1, then (9.194) is the same as (9.191). However, by taking κ > 1, (9.194) gives new higher-order operators, even in the special case where dsj = 1, ∀1 ≤ j ≤ r. Proposition 9.4.9. Fix x0 ∈ M and suppose that W [κ]u(x0 ) > 0, i. e., W [κ]u(x0 ) is a strictly positive definite matrix. Then (9.191) is maximally subelliptic at (x0 , u) of degree 2κ with respect to (W , ds) in the sense of Definition 1.1.11. Proof. Let W [κ]i,j u denote the (i, j) component of the inverse matrix (W [κ]u)−1 . Using (9.192) we see that the linearization of (9.194) with respect to u at the point x0 is the operator v 󳨃→ ∑ (det W [κ]u(x0 ))W [κ]i,j u(x0 )v. 1≤i,j≤r

(9.195)

The matrix with the (i, j) component equal to (det W [κ]u(x0 ))W [κ]i,j (x0 ) is strictly positive definite, and Lemma 8.9.3 implies that (9.195) is maximally subelliptic of degree 2κ with respect to (W , ds) on M, which completes the proof.

690 � 9 Nonlinear maximally subelliptic equations The second generalization: Set D := #{α : degds(α) = κ}.

(9.196)

For a real-valued function u : M → ℝ, let Ŵ[κ]u(x) denote the symmetric D × D real matrix with the (α, β) component equal to Ŵ[κ]α,β u(x) :=

1 (W α W β u(x) + W β W α u(x)), 2

degds(α) = degds(β) = κ.

We consider a higher-order generalized Monge–Ampère equation given by u 󳨃→ det Ŵ[κ]u(x).

(9.197)

Proposition 9.4.10. Fix x0 ∈ M and suppose Ŵ[κ]u(x0 ) > 0, i. e., Ŵ[κ]u(x0 ) is a strictly positive definite matrix. Then (9.197) is maximally subelliptic at (x0 , u) of degree 2κ with respect to (W , ds) in the sense of Definition 1.1.11. Proof. This follows just as in the proof of Proposition 9.4.9 by replacing Lemma 8.9.3 with Proposition 8.9.4. Corollary 9.4.11. Fix s > r > 0 and x0 ∈ M. Suppose that: – W [κ]u(x0 ) > 0, – u ∈ C r+2κ (W , ds) near x0 , – det W [κ]u ∈ C s (W , ds) near x0 . Then u ∈ C s+2κ (W , ds) near x0 . The same result holds with W replaced by Ŵ throughout. Proof. This follows by combining Theorem 9.1.2 and Propositions 9.4.9 and 9.4.10. Corollary 9.4.12. Fix r > 0 and x0 ∈ M. Suppose that: – W [κ]u(x0 ) > 0, – u ∈ C r+2κ (W , ds) near x0 , – det W [κ]u ∈ C ∞ near x0 . Then u ∈ C ∞ near x0 . The same result holds with W replaced by Ŵ throughout. Proof. This follows by combining Corollary 9.1.5 and Propositions 9.4.9 and 9.4.10. Remark 9.4.13. Similarly, we can apply the other regularity theorems in this chapter to immediately deduce results about (9.194) when either W [κ]u(x0 ) > 0 or Ŵ[κ]u(x0 ) > 0. For example, Corollary 9.1.8 gives regularity properties in terms of the classical Zygmund–Hölder spaces.

9.4 Examples

� 691

9.4.4 Higher-order equations Just as the generalized Monge–Ampère equation from Section 9.4.2 can be seen as a special case of the more general second-order equations from Section 9.4.1, the higherorder generalized Monge–Ampère equations from Section 9.4.3 can be seen as a special case of higher-order operators whose maximal subellipticity is nearly immediate. The main idea is that the definition of a nonlinear partial differential operator being maximally subelliptic (Definition 1.1.11) is just that a frozen coefficient version of its linearization is maximally subelliptic. When this linearization is given by a positive definite quadratic form, the maximal subellipticity can often be deduced from results like Propositions 8.9.1 and 8.9.4. In this section we present two such results, which generalize the two types of higher-order Monge–Ampère equations from Section 9.4.3. Let (W , ds) = {(W1 , ds1 ), . . . , (Wr , dsr )} be Hörmander vector fields with formal degrees on M. Fix κ ∈ ℕ+ such that dsj divides κ, ∀1 ≤ j ≤ r, and set nj := κ/dsj ∈ ℕ+ . Consider a nonlinear partial differential operator of the form u 󳨃→ F(x, {W β u(x)}deg

ds (β)≤2κ

(9.198)

),

where F is a real-valued function. We consider F = F(x, ζ ) as a function of x ∈ M and ζ = {ζα }degds (α)≤κ , where ζα ∈ ℝ. For each 1 ≤ i, j ≤ r define the ordered multi-index αi,j := (i, i, . . . , i, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ j, j, . . . , j). ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ni terms n

nj terms

n

Note that degds(αi,j ) = 2κ and W αi,j = Wi i Wj j . Our first result generalizes the higher-order Monge–Ampère operators from Proposition 9.4.9. Proposition 9.4.14. Fix x0 ∈ M and u : M → ℝ. Suppose that: (i) For each degds(α) = 2κ such that α ≠ αi,j , ∀i, j, we have 𝜕ζα F(x0 , {W β u(x0 )}degds (β)≤2κ ) = 0. (ii) The r × r matrix 𝜕ζα F(x0 , {W β u(x0 )}deg (β)≤2κ ) ds 1,1 .. ( . 𝜕ζα F(x0 , {W β u(x0 )}deg (β)≤2κ ) r,1

ds

⋅⋅⋅ ..

. ⋅⋅⋅

𝜕ζα F(x0 , {W β u(x0 )}deg (β)≤2κ ) ds 1,r .. ) . β 𝜕ζα F(x0 , {W u(x0 )}deg (β)≤2κ ) r,r

ds

is symmetric and strictly positive definite. Then (9.198) is maximally subelliptic at (x0 , u) of degree 2κ with respect to (W , ds) in the sense of Definition 1.1.11.

692 � 9 Nonlinear maximally subelliptic equations Remark 9.4.15. The assumption (i) in Proposition 9.4.14 is somewhat artificial, and only serves to make it easy to recognize the maximal subellipticity of (9.198). Note that (i) automatically holds if F(x, ζ ) does not depend on any of the coordinates ζα , where degds(α) = 2κ and α ≠ αi,j for all 1 ≤ i, j ≤ r. For example, this is the case in the higher-order generalized Monge–Ampère equation (9.194). Proof of Proposition 9.4.14. By the definition (Definition 1.1.11) the maximal subellipticity of (9.198) at (x0 , u) is equivalent to the maximal subellipticity of the linear operator: v 󳨃→ ∇ζ F(x0 , {W β u(x0 )}deg β

ds (β)≤2κ

) ⋅ {W α v}deg

= ∑ 𝜕ζα F(x0 , {W u(x0 )}deg i,j

1≤i,j≤r

+

∑

degds (α) r > 0, t > 2⌊s⌋ + 6. Suppose that: – F ∈ C s,t ((W , ds) ⊠ ∇ℝN ) near x0 , – g ∈ C s (W , ds) near x0 ,

9.5 Further reading and references

– –

� 693

u ∈ C r+2κ (W , ds) near x0 , the assumptions of either Proposition 9.4.14 or Proposition 9.4.16 hold.

Then u ∈ C s+2κ (W , ds) near x0 . Proof. This follows by combining Theorem 9.1.2 and Propositions 9.4.14 and 9.4.16 Corollary 9.4.19. Let F, u, and g satisfy F(x, {W α u(x)}degds (α)≤2κ ) = g(x) and fix x0 ∈ M and r > 0. Suppose that: ∞ – F ∈ Cloc (M × ℝn ), ∞ – g ∈ Cloc (M) near x0 , – u ∈ C r+2κ (W , 1) near x0 , – the assumptions of either Proposition 9.4.14 or Proposition 9.4.16 hold. ∞ Then u ∈ Cloc (M) near x0 .

Proof. This follows by combining Corollary 9.1.5 and Propositions 9.4.14 and 9.4.16 Remark 9.4.20. Similarly, we can apply the many other regularity results in this chapter to immediately obtain other regularity results about solutions to these higher-order, maximally subelliptic equations. For example, Corollary 9.1.8 gives regularity properties in terms of the classical Zygmund–Hölder spaces.

9.5 Further reading and references There has been much less work on nonlinear maximally subelliptic equations than there has been for linear maximally subelliptic equations. Nearly all the work has been on operators of order 2, whose linearizations are similar to the sub-Laplacian (and where the Rothschild–Stein lifting theorem applies; cf. Section 4.5.7). In contrast, we have studied general fully nonlinear equations. Because previous studies work with equations of a more special form, they are often able to introduce notions of weak solutions and are therefore able to deal with less regularity than we do. However, it seems likely that the methods of this chapter can, when applied to operators of a special form, be used to study weak solutions (for example, see Remark 9.1.13). Our results on weighted estimates near the boundary (in Section 9.3) also seem to be new. Much of the previous work has been on quasi-linear maximally subelliptic equations, usually of the special form r

∑ Wj∗ Aj (x, u, W1 u, . . . , Wr u) = f (x, u, W1 u, . . . , Wr u), j=1

(9.200)

where W1 , . . . , Wr are Hörmander vector fields; this is a second-order quasi-linear equation. Except for a few exceptions, this is usually done in the context of Lie groups. The

694 � 9 Nonlinear maximally subelliptic equations earliest work on quasi-linear maximally subelliptic equations of the form (9.200) was due to Xu [251], Capogna, Danielli, and Garofalo [30], and Capogna [28]. More recent results in the same vein include [31, 73, 76, 77, 74, 75, 64, 65, 89, 253, 164, 163, 165, 176–178, 29, 162, 202, 203, 19, 249]. For some other quasi-linear maximally subelliptic equations see [156]. The methods of these papers are unrelated to the methods of this text. Beyond the quasi-linear setting, there has been some work on second-order fully nonlinear equations where weak solutions can be considered [168, 8, 11] and on linear operators with non-smooth coefficients [155, 18, 15, 17]. There is also a large body of literature on degenerate elliptic equations, some of which is closely related to maximally subellipticity. A recent paper which shares some themes of this chapter is [139]. Nonlinear maximally subelliptic equations appear in the geometry of several complex variables. See, for example, [145, 126, 173], where the Levi curvature equations are studied, related to domains of holomorphy, and where a real hypersurface evolves through a motion involving the Levi curvature (here, the vector fields themselves are taken to be nonlinear), and [130], where an analog of the Yamabe problem for CR manifolds is studied. Nonlinear maximally subelliptic equations have also had applications to finance [146] and have been connected to the study of human vision and image processing [196, 51, 13, 50]. As mentioned earlier, the proof due to Simon [212] was an inspiration for the proof of Theorem 9.2.1.

A Canonical coordinates In this appendix, we discuss and prove the results from Section 3.6, which form the basis of the proofs of every main theorem of this text. The central result (Theorem A.2.4) is a slight generalization of Theorem 3.6.5 where 2 ∞ we work with Cloc vector fields instead of Cloc vector fields. The results in this section are a simplified special case of the results in [228]; see Section A.5 for details on the history of these results as well as references to some other, stronger results.

A.1 Basic notation Throughout this appendix, M will denote a C 2 manifold of dimension n ∈ ℕ+ . Given 1 1 X = {X1 , . . . , Xq } ⊂ Cloc (M; TM), a finite set of Cloc vector fields on M, we set BX (x, δ) := BδX1 ,...,δXq (x),

x ∈ M, δ > 0,

where BδX1 ,...,δXq (x) is defined by (1.12).

1 1 Given a Cloc vector field Y ∈ Cloc (M; TM) and a function f : M → ℂ, we define

Yf (x) :=

d 󵄨󵄨󵄨󵄨 tY 󵄨 f (e x). dt 󵄨󵄨󵄨t=0

(A.1)

If we say Yf (x) exists, this means that this derivative exists in the classical sense, ∀x. Note that this allows us to define Yf (x) for x such that etY x is defined for t small; in particular, f (x) need not be defined on an open set containing x. For α = (α1 , . . . , αL ) ∈ {1, . . . , q}L , we define X α f = Xα1 (Xα2 (⋅ ⋅ ⋅ (XαL f ))), and we say X α f exists if at each stage the derivatives in the sense of (A.1) exist. Since M is merely a C 2 manifold, it does not make sense to talk about functions 2 of regularity any higher than Cloc . However, it does make sense to talk about functions which are of higher regularity with respect to the vector fields X. To describe this, we 1 generalize the Banach spaces CXL (M) from Definition 5.2.21 to the setting of Cloc vector fields. Definition A.1.1. For L ∈ ℕ and U ⊆ M open, define the vector space CXL (U) := {f ∈ C(U) : X α f exists and X α f ∈ C(U), ∀|α| ≤ L}, with norm 󵄨 󵄨 ‖f ‖C L (U) := sup ∑ 󵄨󵄨󵄨X α f (x)󵄨󵄨󵄨. X x∈U |α|≤L

https://doi.org/10.1515/9783111085647-010

696 � A Canonical coordinates With this norm, CXL (U) is a Banach space. We make the same definition for any set U ⊆ M such that etXj x ∈ U for all x ∈ U, 1 ≤ j ≤ q, and t ∈ ℝ sufficiently small (how small may depend on x and j). We set CX∞ (U) := ⋂L CXL (U). ∞ Likewise, we define CX,loc (M) just as in Definition 5.2.22; in particular, even though 2 the manifold is only C , it makes sense to talk about functions which are infinitely smooth with respect to the vector fields X. Let n

ℐ (n, q) := {(i1 , . . . , in ) : ij ∈ {1, . . . q}} = {1, . . . , q} ,

ℐ0 (n, q) := {(i1 , . . . , in ) ∈ ℐ (n, q) : i1 < i2 < ⋅ ⋅ ⋅ < in }.

Given J = (j1 , . . . , jn ) ∈ ℐ (n, q), we write XJ = (Xj1 , . . . , Xjn ) and ⋀ XJ := Xj1 ∧ Xj2 ∧ ⋅ ⋅ ⋅ ∧ Xjn . If we also have K ∈ ℐ (n, q) and ⋀ XJ (x0 ) ≠ 0, then Definition 3.6.2 defines the function ⋀ XK (x) ⋀ XJ (x) for x near x0 .

A.2 The main results 1 Let M be a C 2 manifold of dimension n ∈ ℕ+ and let X = {X1 , . . . , Xq } ⊂ Cloc (M; TM) be 1 a finite set of Cloc vector fields on M such that ∀x ∈ M, Tx M = span{X1 (x), . . . , Xq (x)}. An important aspect of all the assumptions and estimates which follow is that they are 2 quantitatively invariant under arbitrary Cloc diffeomorphisms. See Remark A.2.3. Fix x0 ∈ M and ξ ∈ (0, 1]. We write q

l [Xj , Xk ] = ∑ cj,k Xl , l=1

(A.2)

l and we assume cj,k ∈ C(BX (x0 , ξ)); since X1 , . . . , Xq span the tangent space at every point, it is easy to see that BX (x0 , ξ) is open in M – see Lemma A.2.14. Fix ζ ∈ (0, 1]. Let J0 ∈ ℐ (n, q) be such that ⋀ XJ0 (x0 ) ≠ 0 and moreover

󵄨󵄨 ⋀ X (x ) 󵄨󵄨 󵄨 J 0 󵄨󵄨 󵄨󵄨 ≤ ζ −1 ; max 󵄨󵄨󵄨󵄨 J∈ℐ(n,q) 󵄨󵄨 ⋀ XJ (x0 ) 󵄨󵄨󵄨 0

(A.3)

see Definition 3.6.2 for the definition of this quotient. Note that this is always possible: we may always choose J0 so that the left-hand side of (A.3) equals 1, though it is sometimes convenient to have the flexibility to allow it to be greater than 1. Without loss of generality, we reorder X1 , . . . , Xq so that J0 = (1, . . . , n).

A.2 The main results

– –

�

697

As in Section 3.6, we require two technical quantities associated with X: Let η ∈ (0, 1] be such that XJ0 satisfies 𝒞 (x0 , η, M). See Definition 3.6.3. Let τ0 ∈ (0, 1] be such that for τ ∈ (0, τ0 ] the following holds: if z ∈ BXJ (x0 , ξ) 0

is such that XJ0 satisfies 𝒞 (z, τ, BXJ (x0 , ξ)), t ∈ Bn (τ) is such that et⋅XJ0 z = z, and 0 X1 (z), . . . , Xn (z) are linearly independent, then t = 0.

Such an η and τ0 always exist (by possibly shrinking ξ > 0 – see Proposition A.2.10); however, our quantitative estimates will depend on η and τ0 . See Section A.2.1 for a further discussion of these quantities. Definition A.2.1. We say C is a 0-admissible constant if C can be chosen to depend only l on upper bounds for q, ζ −1 , ξ −1 , and ‖cj,k ‖C(BX (x0 ,ξ)) , 1 ≤ j, k, l ≤ q. J0

Definition A.2.2. For m ∈ ℕ+ , if we say C is an m-admissible constant, this means that l we assume cj,k ∈ CXmJ (BXJ (x0 , ξ)) for 1 ≤ j, k, l ≤ q. C is allowed to depend on any0

0

thing a 0-admissible constant can depend on and upper bounds for η−1 , τ0−1 , m, and l ‖cj,k ‖CXm (BX (x ,ξ) ) . J0

0

For m ∈ ℕ, we write A ≲m B for A ≤ CB, where C ≥ 0 is an m-admissible constant. We write A ≈m B for A ≲m B and B ≲m A. Remark A.2.3. An important aspect of the admissible constants in Definitions A.2.1 and A.2.2 is that they are quantitative invariant under arbitrary C 2 diffeomorphisms. 1 For example, because each Xj is Cloc , in local coordinates we may write Xj = ∑k ajk 𝜕xk , where ajk ∈ C 1 . However, our admissible constants do not depend on any sort of upper bound of ‖ajk ‖C 1 , because this norm depends on the choice of coordinate system. Similarly, since span{X1 (x0 ), . . . , Xq (x0 )} = Tx0 M, we may choose j1 , . . . , jn such that |det(ajkl )l,k=1,...,n | > 0; however, admissible constants do not depend on a lower bound of this quantity since it depends on the choice of coordinate system.

Because XJ0 satisfies 𝒞 (x0 , η, M), by hypothesis, we may define the map, for t ∈ Bn (η), Φ(t) := et⋅XJ0 x0 = et1 X1 +⋅⋅⋅+tn Xn x0 .

(A.4)

1 Let η0 := min{η, ξ} so that Φ : Bn (η0 ) → BXJ (x0 , ξ). Note that, a priori, Φ is Cloc since 1 X1 , . . . , Xn are Cloc .

0

Theorem A.2.4. There exists a 0-admissible constant χ ∈ (0, ξ] such that: (a) ∀y ∈ BXJ (x0 , χ), ⋀ XJ0 (y) ≠ 0. 0 (b) ∀y ∈ BXJ (x0 , χ), 0

󵄨󵄨 ⋀ X (y) 󵄨󵄨 󵄨 󵄨󵄨 J 󵄨󵄨 ≈0 1. max 󵄨󵄨󵄨󵄨 J∈ℐ(n,q) 󵄨󵄨 ⋀ XJ (y) 󵄨󵄨󵄨 0

(A.5)

698 � A Canonical coordinates (c) ∀χ ′ ∈ (0, χ], BXJ (x0 , χ ′ ) is an open subset of M. 0

l For the rest of the theorem, we assume cj,k ∈ CX1 J (BXJ (x0 , ξ)), for all 1 ≤ j, k, l ≤ q. There 0 0 exist 1-admissible constants η1 , ξ1 , ξ2 ∈ (0, 1] such that: (d) Φ(Bn (η1 )) is an open subset of BXJ (x0 , χ) and is therefore an open subset of M. 0

2 (e) Φ : Bn (η1 ) → Φ(Bn (η1 )) is a Cloc diffeomorphism. (f) BX (x0 , ξ2 ) ⊆ BXJ (x0 , ξ1 ) ⊆ Φ(Bn (η1 /2)) ⊆ Φ(Bn (η1 )) ⊆ BXJ (x0 , χ) ⊆ BX (x0 , ξ). 0

0

Let Yj := Φ∗ Xj and write YJ0 = (I + A)∇, where YJ0 denotes the column vector of vector fields YJ0 = [Y1 , Y2 , . . . , Yn ]⊤ , ∇ denotes the gradient in ℝn thought of as a column vector, and A ∈ Cloc (Bn (η1 ); 𝕄n×n ). (g) A(0) = 0 and supt∈Bn (η1 ) ‖A(t)‖𝕄n×n ≤ 21 . (h) For 1 ≤ j ≤ q and m ∈ ℕ+ , ‖Yj ‖C m (Bn (η1 );ℝn ) ≲m 1. (i) We have the following equivalence of norms for f ∈ C(Bn (η1 )) and m ∈ ℕ+ : ‖f ‖C m (Bn (η1 )) ≈(m−1)∨0 ‖f ‖CYm

J0

(Bn (η1 ))

≈(m−1)∨0 ‖f ‖CYm (Bn (η1 )) .

Remark A.2.5. In the context of Lie groups, the coordinates given by Φ in (A.4) are sometimes called canonical coordinates of the first kind. Remark A.2.6. The regularity given in Theorem A.2.4 (h) is not sharp – indeed it “loses one derivative.” Sharp results are obtained in terms of Zygmund–Hölder spaces in [224] and [230]. In [230] it is shown that to obtain these sharp results one cannot use the map Φ given by (A.4). Instead, a more complicated argument involving elliptic PDEs is used. Remark A.2.7. In Theorem A.2.4 (and in the rest of the appendix), if we write an inequality which implies ‖f ‖C m (U) < ∞, we take this to mean that f ∈ C m (U) and that the inequality holds, and similarly for any other norm. See, for example, Theorem A.2.4 (h) and (i). A.2.1 More on the assumptions We further consider the constants η > 0 and τ0 > 0 used in Theorem A.2.4. First, we present two examples which show why these constants cannot be dispensed with in our definition of m-admissible constants (Definition A.2.2), and then we state a result which shows that such constants always exist. Example A.2.8. This example demonstrates the importance of η. Let M = ℝ, q = 1, x0 > 0, and X1 = x 2 𝜕x . In this case, η > 0 can be taken no larger than 1/x0 , i. e., X1

A.2 The main results �

699

satisfies 𝒞 (x0 , x0−1 , ℝ) but does not satisfy 𝒞 (x0 , η′ , ℝ) for any η′ > x0−1 (because the ODE ̇ γ(t) = γ(t)2 , γ(0) = x0 exists only for t < x0−1 ). If Theorem A.2.4 held with constants independent of η (and therefore independent of x0 ), then we could conclude X1 satisfied 𝒞 (x0 , η′ , ℝ) for some η′ independent of x0 . This is because the condition 𝒞 is invariant under a change of coordinates, and we can therefore check it in the coordinate system given by Φ in Theorem A.2.4. This is a contradiction, showing that η must play a role in the quantitative estimates in Theorem A.2.4. Example A.2.9. This example demonstrates the importance of τ0 and also shows its topological nature. The point of τ0 is to ensure the map Φ in Theorem A.2.4 is injective. Let M = S 1 , q = 1, x0 ∈ S 1 , and X1 = K𝜕θ for some large constant K ≥ 1. For this example, we must take τ0 ≤ 2π/K. If the constants in Theorem A.2.4 did not depend on τ0 , they would also not depend on K. We could then conclude that τ0 could be taken independent of K – this is because τ0 is invariant under a chance of coordinates, and we can check it in the coordinate system given by Φ in Theorem A.2.4. This shows that τ0 must play a role in the quantitative estimates in Theorem A.2.4. The next result is a generalization of Proposition 3.6.7 and shows that such an η > 0 and τ0 > 0 always exist. 1 1 Proposition A.2.10. Let X1 , . . . , Xq ∈ Cloc (M; TM) be Cloc vector fields on the C 2 manifold M. Fix an open set U ⊆ M and a compact set 𝒦 ⋐ U. Then: (i) ∃η > 0 such that ∀x0 ∈ 𝒦, X satisfies 𝒞 (x0 , η, U). (ii) ∃τ0 > 0 such that ∀θ ∈ S q−1 , if x ∈ 𝒦 is such that θ1 X1 (x) + ⋅ ⋅ ⋅ + θq Xq (x) ≠ 0, then ∀r ∈ (0, τ0 ],

erθ1 X1 +⋅⋅⋅+rθq Xq x ≠ x. Proposition A.2.10 is a local result, so it suffices to instead prove the following local version. Lemma A.2.11. Fix U ⊆ ℝn open and fix 𝒦 ⋐ U compact. Let X1 , . . . , Xq ∈ C 1 (U; TU) be C 1 vector fields on U. Then: (i) ∃η > 0 such that ∀x0 ∈ 𝒦, X satisfies 𝒞 (x0 , η, U). (ii) ∃τ0 > 0 such that ∀θ ∈ S q−1 , if x ∈ 𝒦 is such that θ1 X1 (x) + ⋅ ⋅ ⋅ + θq Xq (x) ≠ 0, then ∀r ∈ (0, τ0 ], erθ1 X1 +⋅⋅⋅+rθq Xq x ≠ x. Moreover, η > 0 and τ0 > 0 can be chosen to depend only on n, the distance from 𝒦 to 𝜕U, and upper bounds for q and max1≤j≤q ‖Xj ‖C 1 (U;ℝn ) . Before we prove Lemma A.2.11, we need another standard lemma.

700 � A Canonical coordinates Lemma A.2.12. Suppose Z ∈ C 1 (V ; TV ) is a C 1 vector field on an open set V ⊆ ℝn . Then there exists δ > 0, depending only on n, such that if ‖Z‖C 1 (V ;ℝn ) ≤ δ, then there does not exist x ∈ V with: – etZ x ∈ V , ∀t ∈ [0, 1]. – eZ x = x. – Z(x) ≠ 0. Proof. Suppose the lemma does not hold and we have an x ∈ V and Z satisfying the above conditions. In the proof of this lemma, we use big-O notation – the implicit constants will only depend on n. Differentiating the identity d tZ e x = Z(etZ x), dt we obtain 󵄨󵄨 d 󵄨󵄨 d 2 tZ e x = O (δ 󵄨󵄨󵄨󵄨 etZ x 󵄨󵄨󵄨󵄨) . 2 󵄨 dt 󵄨 dt

(A.6)

Thus, by Grönwall’s inequality, 󵄨󵄨 d 󵄨󵄨 󵄨󵄨 d tZ 󵄨 󵄨 󵄨 󵄨 󵄨 e x = O (󵄨󵄨󵄨 󵄨󵄨󵄨 etZ x 󵄨󵄨󵄨) = O(󵄨󵄨󵄨Z(x)󵄨󵄨󵄨), 󵄨󵄨 dt 󵄨󵄨t=0 󵄨󵄨 dt

t ≤ 1.

(A.7)

Plugging (A.7) into (A.6), we see that d 2 tZ 󵄨 󵄨 e x = O(δ󵄨󵄨󵄨Z(x)󵄨󵄨󵄨). dt 2

(A.8)

d tZ e x = Z(x) + O(δ|t||Z(x)|). dt

(A.9)

Integrating (A.8), we obtain

Integrating (A.9), we have x = eZ x = x + Z(x) + O(δ|Z(x)|), which is impossible if δ is sufficiently small (since Z(x) ≠ 0). This completes the proof. Proof of Lemma A.2.11. Item (i) is a standard consequence of the Picard–Lindelöf theorem, so we only prove (ii). Take δ = δ(n) > 0 as in Lemma A.2.12 and take δ1 > 0 so small that ∀y ∈ 𝒦, t ∈ Bq (δ1 ), we have et1 X1 +t2 X2 +⋅⋅⋅+tq Xq y ∈ U; by the Picard–Lindelöf theorem τ0 , can be chosen to depend only on n, the distance from 𝒦 to 𝜕U, and upper bounds for q and max1≤j≤q ‖Xj ‖C 1 (U;ℝn ) . Set C := max1≤j≤q ‖Xj ‖C 1 (U;ℝn ) and let τ0 := min{δ1 , δ/qC}. Now, (ii) follows from Lemma A.2.12.

A.2 The main results

�

701

Remark A.2.13. Proposition A.2.10 shows that η > 0 and τ0 > 0 as in Theorem A.2.4 always exist. However, the proof of Lemma A.2.11 relies on the C 1 norms of the vector fields X1 , . . . , Xq in some fixed coordinate system. Theorem A.2.4 provides a coordinate system in which we have good bounds on the C 1 norms of these vector fields, but does not a priori assume the existence of such a coordinate system. By working with η and τ0 in Definition A.2.2 we are able to present a theorem which does not assume any sort of quantitative bounds on the C 1 norms of X1 , . . . , Xq . This allows us to apply Theorem A.2.4 to some vector fields which may be a priori rough or large in the coordinate system in which they are given, and Theorem A.2.4 provides a coordinate system where they are not rough or large. We close this section with a lemma which shows that BX (x0 , ξ) is open in M. Lemma A.2.14. Let M be an n-dimensional C 2 manifold and fix x ∈ M and δ > 0. Suppose 1 1 V1 , . . . , Vq ∈ Cloc (M; TM) are Cloc vector fields which span the tangent space at every point of BV (x, δ). Then BV (x, δ) is open. Proof. Fix x0 ∈ BV (x, δ) and pick j1 , . . . , jn ∈ {1, . . . , q} such that Vj1 (x0 ), . . . , Vjn (x0 ) form a basis for Tx0 M. Define Ψ(t) = et1 Vj1 +⋅⋅⋅+tn Vjn x0 so that dΨ(0) = (Vj1 (x0 )| ⋅ ⋅ ⋅ |Vjn (x0 )) and therefore dΨ(0) is non-singular. Standard theo1 rems show that Ψ is Cloc . It is clear that for δ′ > 0 sufficiently small, Ψ(Bn (δ′ )) ⊆ BV (x, δ), and the inverse function theorem shows that for δ′ > 0 sufficiently small, Ψ(Bn (δ′ )) ⊆ BV (x, δ) is open. Since x0 = Ψ(0), Ψ(Bn (δ′ )) is an open neighborhood of x0 in BV (x, δ). Since x0 ∈ BV (x, δ) was arbitrary, we conclude BV (x, δ) is open. A.2.2 Densities Under the same setting as Theorem A.2.4, let χ ∈ (0, ξ] be as in that theorem. In most of the applications in this text, we are given a density,1 Vol, on BXJ (x0 , χ), and we need to 0 measure certain sets with respect to this density. 1 Let Vol be a Cloc density on the manifold BXJ (x0 , χ). Suppose 0

LieXj Vol = fj Vol,

1 ≤ j ≤ n, fj ∈ C(BXJ (x0 , χ)). 0

(A.10)

Our goal is to understand Φ∗ Vol and Vol(BX (x0 , ξ2 )), where Φ and ξ2 are as in Theorem A.2.4. 1 In the rest of this text, Vol is assumed to be strictly positive, though we make no such assumption in this appendix.

702 � A Canonical coordinates Definition A.2.15. We say C is a {0; Vol}-admissible constant if C is a 0-admissible constant (as in Definition A.2.1), which is also allowed to depend on upper bounds for ‖fj ‖C(BX (x0 ,χ)) , 1 ≤ j ≤ n. J0

Definition A.2.16. For m ∈ ℕ+ , if we say C is an {m; Vol}-admissible constant, this means that we assume fj ∈ CXmJ (BXJ (x0 , χ)) and C is an m-admissible constant (as in Defini0 0 tion A.2.2) which is also allowed to depend on upper bounds for ‖fj ‖CXm (BX (x0 ,χ)) , 1 ≤ j ≤ n. J0

J0

We write A ≲{m;Vol} B for A ≤ CB, where C ≥ 0 is an {m; Vol}-admissible constant. We write A ≈{m;Vol}B for A ≲{m;Vol} B and B ≲{m;Vol} A. To help understand Vol, we introduce a distinguished density, Vol0 , on BXJ (x0 , χ): 0

󵄨󵄨 Z ∧ Z ∧ ⋅ ⋅ ⋅ ∧ Z 󵄨󵄨 󵄨 2 n 󵄨󵄨 Vol0 (Z1 , . . . , Zn ) := 󵄨󵄨󵄨 1 󵄨. 󵄨󵄨 X1 ∧ X2 ∧ ⋅ ⋅ ⋅ ∧ Xn 󵄨󵄨󵄨

(A.11)

Note that Vol0 is defined since X1 ∧ X2 ∧ ⋅ ⋅ ⋅ ∧ Xn = ⋀ XJ0 is never zero on BXJ (x0 , χ) by 0 Theorem A.2.4 (a); Vol0 is clearly a density. Theorem A.2.17. There exists g ∈ C(BXJ (x0 , χ)) such that Vol = g Vol0 and: 0 (i) We have g(x) ≈{0;Vol} g(x0 ) = Vol(X1 , . . . , Xn )(x0 ), ∀x ∈ BXJ (x0 , χ). In particular, g 0 always has the same sign, and is either never zero or always zero. (ii) We have the following regularity on g: 󵄨 󵄨 ‖g‖CXm ≲{(m−1)∨0;Vol} 󵄨󵄨󵄨Vol(X1 , . . . , Xn )(x0 )󵄨󵄨󵄨. J 0

Define h ∈ C 1 (Bn (η1 )) by Φ∗ Vol = hσLeb . (iii) We have h(t) ≈{0;Vol} Vol(X1 , . . . , Xn )(x0 ), ∀t ∈ Bn (η1 ). In particular, h always has the same sign and is either never zero or always zero. (iv) We have the following regularity on h: 󵄨 󵄨 ‖h‖C m (Bn (η1 )) ≲{m;Vol} 󵄨󵄨󵄨Vol(X1 , . . . , Xn )(x0 )󵄨󵄨󵄨. Corollary A.2.18. Let ξ2 ∈ (0, 1] be as in Theorem A.2.4. Then Vol(BXJ (x0 , ξ2 )) ≈{1;Vol} Vol(BX (x0 , ξ2 )) ≈{1;Vol} Vol(X1 , . . . , Xn )(x0 ), 0

(A.12)

and therefore 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨Vol(BXJ (x0 , ξ2 )󵄨󵄨󵄨 ≈{1;Vol} 󵄨󵄨󵄨Vol(BX (x0 , ξ2 ))󵄨󵄨󵄨 0 󵄨 󵄨 ≈{1;Vol} 󵄨󵄨󵄨Vol(X1 , . . . , Xn )(x0 )󵄨󵄨󵄨 ≈0

󵄨󵄨 󵄨 (A.13) 󵄨󵄨Vol(Xj1 , . . . , Xjn )(x0 )󵄨󵄨󵄨. ∈{1,...,q}

max

j1 ,...,jn

A.3 Qualitative consequences

� 703

A.3 Qualitative consequences For the main applications of this text, we require the quantitative nature of Theorem A.2.4 (see, e. g., Remark 3.6.1). However, Theorem A.2.4 also has interesting qualita1 tive consequences: it provides a coordinate system in which the given Cloc vector fields may have a higher level of regularity. 1 1 Let X1 , . . . , Xq ∈ Cloc (M; TM) be Cloc vector fields on the C 2 manifold, M, of dimension n, which span the tangent space at every point. Theorem A.3.1 (The local theorem). Fix x0 ∈ M. The following are equivalent: (i) There are an open neighborhood V ⊆ M of x0 and a C 2 diffeomorphism Φ : U → V , where U ⊆ ℝn is open, such that Φ∗ X1 , . . . , Φ∗ Xq ∈ C ∞ (U; ℝn ). (ii) Reorder the vector fields so that X1 (x0 ), . . . , Xn (x0 ) are linearly independent. There is an open neighborhood V ⊆ M of x0 such that: ̂l Xl , 1 ≤ j, k ≤ n, where cj,k ̂l ∈ CX∞ (V ). – [Xj , Xk ] = ∑nk=1 cj,k –

For n + 1 ≤ j ≤ q, Xj = ∑nk=1 bkj Xk , where bkj ∈ CX∞ (V ).

q

l (iii) There exists an open neighborhood V ⊆ M of x0 such that [Xj , Xk ] = ∑l=1 cj,k Xl , l 1 ≤ j, k ≤ q, where cj,k ∈ CX∞ (V ).

Remark A.3.2. Theorem A.3.1 (ii) and (iii) are similar, though have different advantages. ̂l In (ii), because X1 , . . . , Xn are a basis for the tangent space of M near x0 , the functions cj,k

and bkj are uniquely determined (so long as V is chosen sufficiently small). Moreover, one can directly check to see if (ii) holds by computing these functions and this computation l can be done in any coordinate system. If q > n, X1 , . . . , Xq are linearly dependent and cj,k l in (iii) are not uniquely determined; (iii) only asks that there exists a choice of cj,k satisfying the conditions in (iii). Despite this lack of uniqueness, it is often more convenient to use the setting of (iii) since it is symmetric in X1 , . . . , Xq .

Proof of Theorem A.3.1. (i) ⇒ (ii): Let U, V , x0 , and Φ be as in (i). Without loss of generality, assume 0 ∈ U and Φ(0) = x0 . Reorder X1 , . . . , Xq so that X1 (x0 ), . . . , Xn (x0 ) are linearly independent and let Yj := Φ∗ Xj so that Yj ∈ C ∞ (U; ℝn ), 1 ≤ j ≤ q. Note that Y1 (0), . . . , Yn (0) form a basis for the tangent space T0 U. Let η > 0 be so small that Bn (2η) ⊆ U and Y1 , . . . , Yn form a basis for the tangent space at every point of Bn (2η). Since Y1 , . . . , Yq ∈ C ∞ (Bn (η)), it follows immediately from the definitions that CY∞ (Bn (η)) ⊆ C ∞ (Bn (η)). Furthermore, since Y1 , . . . , Yn form a basis for the tangent space at every point of Bn (2η), it is easy to see that C ∞ (Bn (η)) ⊆ CY∞ (Bn (η)). We conclude that CY∞ (Bn (η)) = C ∞ (Bn (η)).

(A.14)

Using the fact that Y1 , . . . , Yn ∈ C ∞ (Bn (2η)) form a basis for the tangent space at every point of Bn (2η), we have, for 1 ≤ j, k ≤ q,

704 � A Canonical coordinates n

̃ l Yl , [Yj , Yk ] = ∑ cj,k

̃l ∈ C ∞ (Bn (η)). cj,k

l=1

(A.15)

̃l ∈ CY∞ (Bn (η)). Pushing (A.15) forward via Φ shows that [Xj , Xk ] = By (A.14), we have cj,k n l̂ l̂ ̃l ∘ Φ−1 ∈ CΦ∞ Y (Φ(Bn (η))) = CX∞ (Φ(Bn (η))). ∑l=1 cj,k Xl , with cj,k = cj,k ∗ Again using the fact that Y1 , . . . , Yn ∈ C ∞ (Bn (2η)) form a basis for the tangent space at every point of Bn (2η), we have, for n + 1 ≤ j ≤ q, n

Yj = ∑ b̃ kj Yk , k=1

b̃ kj ∈ C ∞ (Bn (η)).

(A.16)

By (A.14), we have b̃ kj ∈ CY∞ (Bn (η)). Pushing (A.16) forward via Φ shows that Xj = ∑nk=1 bkj Xk , where bkj = b̃ kj ∘ Φ−1 ∈ CΦ∞∗ Y (Φ(Bn (η))) = CX∞ (Φ(Bn (η))). This completes the proof of (ii) with V = Φ(Bn (η)). (ii) ⇒ (iii): Suppose (ii) holds. We wish to show that for 1 ≤ j, k ≤ q, q

l [Xj , Xk ] = ∑ cj,k ,

l cj,k ∈ CX∞ (V ),

l=1

(A.17)

where V is as in (ii). For 1 ≤ j, k ≤ n, (A.17) is contained in (ii). We prove (A.17) for n + 1 ≤ j, k ≤ q; the remaining cases when 1 ≤ j ≤ n and n + 1 ≤ k ≤ q or n + 1 ≤ j ≤ q and 1 ≤ k ≤ n are similar and easier. Consider n

n

l

l

[Xj , Xk ] = [ ∑ bj1 Xl1 , ∑ bk2 Xl2 ] l1 =1 n

l2 =1

l

l

l

n

l

l

l

= ∑ (bj1 (Xl1 bk2 )Xl2 − bk2 (Xl2 bj1 )Xl1 + ∑ bj1 bk2 cl̂i1 ,l2 Xi ) . i=1

l1 ,l2 =1

We are given bkj ∈ CX∞ (V ) and ck̂l 1 ,k2 ∈ CX∞ (V ); it follows that Xl bkj ∈ CX∞ (V ). From here, (A.17) follows immediately from the easily shown fact that CX∞ (V ) is an algebra, completing the proof of (iii). (iii) ⇒ (i): This is a consequence of Theorem A.2.4. We make a few comments to this end. As discussed in Section A.2.1, η > 0 and τ0 > 0 as in the hypotheses of Theorem A.2.4 always exist (see also Remark A.2.13). Take ξ > 0 so small that BX (x0 , ξ) ⊆ V and take J0 ∈ ℐ (n, q) so that (A.3) holds with ζ = 1. We have, directly from the definitions, l cj,k ∈ CX∞ (V ) ⊆ CX∞ (BX (x0 , ξ)) ⊆ CX∞J (BXJ (x0 , ξ)). 0

0

In particular, all the hypotheses of Theorem A.2.4 hold and m-admissible constants as in Definition A.2.2 are finite for every m ∈ ℕ+ . This yields a map Φ as in Theorem A.2.4. This map satisfies the conclusions of (i) with U = Bn (η1 ); in particular, it follows from Theorem A.2.4 (h) that Φ∗ X1 , . . . , Φ∗ Xq ∈ C ∞ (Bn (η1 ); ℝn ).

A.3 Qualitative consequences

� 705

Theorem A.3.1 has the following global consequence. Theorem A.3.3 (The global theorem). The following are equivalent: (i) There exists a C ∞ atlas on M, compatible with its C 2 structure, such that X1 , . . . , Xq ∞ are Cloc with respect to this atlas. (ii) For each x0 ∈ M, any of the three equivalent conditions from Theorem A.3.1 hold for this choice of x0 . q l l ∞ (iii) [Xj , Xk ] = ∑l=1 cj,k Xl , 1 ≤ j, k ≤ q, where cj,k ∈ CX,loc (M). Furthermore, under these conditions, the C ∞ structure on M induced by the atlas in (i) is unique, in the sense that if there is another C ∞ atlas on M, compatible with its C 2 structure and such that X1 , . . . , Xq are also C ∞ with respect to this second atlas, then the identity ∞ map M → M is a Cloc diffeomorphism between these two C ∞ manifold structures on M. Remark A.3.4. Recall that for m ∈ ℕ ∪ {∞}, a C m atlas of dimension n ∈ ℕ on a topological space M is a collection {(ψα , Vα ) : α ∈ ℐ }, where ℐ is an index set, {Vα : α ∈ ℐ } is an open cover of M, ψα : Vα → ψ(Vα ) is a homeomorphism where ψα (Vα ) ⊆ ℝn is open, and the transition functions ψβ ∘ ψ−1 α : ψα (Vα ∩ Vβ ) → ψβ (Vα ∩ Vβ ) m are Cloc .

The uniqueness of the C ∞ structure described in Theorem A.3.3 follows from the next lemma. Lemma A.3.5. Let M and N be two n-dimensional C ∞ manifolds, suppose X1 , . . . , Xq are ∞ Cloc vector fields on M which span the tangent space at every point, and let Z1 , . . . , Zq be ∞ 2 Cloc vector fields on N. Let Ψ : M → N be a Cloc diffeomorphism such that Ψ∗ Xj = Zj . Then ∞ Ψ is a Cloc diffeomorphism. Proof. We begin with the case where M and N are open subsets of ℝn ; we use x to denote points in M and y to denote points in N. ∞ Fix a point x0 ∈ M. We wish to show that Ψ is Cloc on a neighborhood of x0 . Reorder X1 , . . . , Xq so that X1 (x0 ), . . . , Xn (x0 ) are linearly independent and reorder Z1 , . . . , Zq in the same way so that we still have Ψ∗ Xj = Zj . Since X1 (x0 ), . . . , Xn (x0 ) form a basis of Tx0 M, we may pick an open neighborhood U of x0 so that X1 (x), . . . , Xn (x) form a basis for Tx M space at every x ∈ U. Define two n × n matrices X (x) := (X1 (x)| ⋅ ⋅ ⋅ |Xn (x)) and Z (y) = (Z1 (y)| ⋅ ⋅ ⋅ |Zn (y)). ∞ ∞ By hypothesis, we have X ∈ Cloc (M; 𝕄n×n ) and Z ∈ Cloc (N; 𝕄n×n ). Since Ψ∗ Xj = Zj , we have the matrix equation dΨ(x)X (x) = Z (Ψ(x)).

(A.18)

706 � A Canonical coordinates Since X1 , . . . , Xn span the tangent space at every point of U, the matrix X (x) is invertible, ∞ ∀x ∈ U, and we have X (x)−1 ∈ Cloc (U; 𝕄n×n ). From (A.18), we obtain dΨ(x) = Z (Ψ(x))X (x)−1 .

(A.19)

m We prove, for all m ∈ ℕ, m ≥ 2, that Ψ ∈ Cloc (U) by induction on m. The base case, 2 m m = 2, follows from the hypothesis that Ψ is a Cloc diffeomorphism. Suppose Ψ ∈ Cloc (U). m m+1 Then by (A.19) it follows that dΨ ∈ Cloc (U), and therefore Ψ ∈ Cloc (U). This achieves the inductive step and completes the proof in the case where both M and N are open subsets of ℝn . ∞ We now turn to the general case where M and N are Cloc manifolds. Since M and ∞ ∞ N are C manifolds, they have associated C atlases, {(ϕα , Vα )} and {(ψβ , Wβ )}; see Re∞ mark A.3.4. Saying that Ψ is Cloc is equivalent to the statement that ∀α, β, −1 Ψα,β := ψβ ∘ Ψ ∘ ϕ−1 α : ϕα (Vα ∩ Ψ (Wβ )) → ψβ (Ψ(Vα ) ∩ Wβ ) ∞ is Cloc . By hypothesis, we have

(Ψα,β )∗ (ϕα )∗ Xj = (ψβ )∗ Zj . ∞ Since (ϕα )∗ X1 , . . . , (ϕα )∗ Xq and (ψβ )∗ Z1 , . . . , (ψβ )∗ Zq are Cloc by hypothesis and (ϕα )∗ X1 , . . . , (ϕα )∗ Xq span the tangent space at every point of ϕα (Vα ) by hypothesis, it follows from ∞ the above case where M and N are open subsets of ℝn that Ψα,β is a Cloc diffeomorphism, completing the proof.

Proof of Theorem A.3.3. (ii) ⇒ (i): Under the condition (ii), for each x ∈ M, there exist 2 open sets Ux ⊆ ℝn , Vx ⊆ M with x ∈ Vx , and a Cloc diffeomorphism Φx : Ux → Vx such x ∗ x ∞ n that if Yj := Φx Xj , then Yj ∈ Cloc (Ux ; ℝ ); note that Y1x , . . . , Yqx span the tangent space to Ux at every point. ∞ We wish to show that the collection {(Φ−1 x , Vx ) : x ∈ M} forms a C atlas on M in the ∞ sense of Remark A.3.4. Once that is shown, (i) will follow since Xj will be Cloc with respect 2 to this atlas by definition and the atlas is clearly compatible with the C structure on M. ∞ All that needs to be shown is that the transition functions are Cloc . Take x1 , x2 ∈ M and −1 −1 consider the transition function Ψx1 ,x2 := Φx2 ∘ Φx1 : Ux1 ∩ Φx1 (Vx2 ) → Ux2 ∩ Φ−1 x2 (Vx1 ); we

∞ 2 wish to show that Ψx1 ,x2 is Cloc . We already know that Ψx1 ,x2 is a Cloc diffeomorphism and x1 x2 ∞ (Ψx1 ,x2 )∗ Yj = Yj . It now follows from Lemma A.3.5 that Ψx1 ,x2 is a Cloc diffeomorphism, completing the proof of (i). (i) ⇒ (iii): Suppose (i) holds, so that we may regard M as a smooth manifold and X1 , . . . , Xq as smooth vector fields on M. Since X1 , . . . , Xq span the tangent space at every point, a simple partition of unity argument shows that we may write [Xj , Xk ] = q l l ∞ ∞ Xl , where cj,k ∈ Cloc (M). Since X1 , . . . , Xq ∈ Cloc (M; TM), it follows easily from ∑l=1 cj,k l ∞ the definitions that cj,k ∈ CX,loc (M), completing the proof of (iii).

A.4 Proofs

� 707

(iii) ⇒ (ii): This is obvious. ∞ Finally, as mentioned before, the uniqueness of the Cloc manifold structure as described in the theorem is an immediate consequence of Lemma A.3.5. ∞ Remark A.3.6. Theorems A.3.1 and A.3.3 deal with Cloc functions. Similar results hold for a finite level of smoothness provided one uses appropriate Zygmund–Hölder spaces; see [224, 230]. There are also analogous results for real analyticity; see [222].

Remark A.3.7. Though we do not use results like Theorems A.3.1 and A.3.3 in the main part of this text, it is possible they could be useful when studying PDEs. For example, one might be given vector fields whose coefficients come from the solution to a PDE which are not a priori known to be smooth. In such a situation, it is possible that Theorems A.3.1 and A.3.3 could give a manifold structure in which the vector fields are smooth, giving access to the theory of this text.

A.4 Proofs We turn to the proofs of the main results of this appendix: Theorems A.2.4 and A.2.17 and Corollary A.2.18. The heart of the proof is the study of a certain ODE which arises in canonical coordinates; this is presented in Section A.4.1. We also require a quantitative special case of the inverse function theorem, described in Section A.4.2. With those preparations in hand, we prove Theorem A.2.4 in Section A.4.3. Finally, we prove the result about densities (Theorem A.2.17 and Corollary A.2.18) in Section A.4.4.

A.4.1 An ODE The proof of Theorem A.2.4 rests on the study of the following ODE, defined for an n × n matrix A(u), depending on u ∈ Bn (η) for some η > 0. Write u = rθ, r > 0, θ ∈ S n−1 . The ODE is 𝜕 rA(rθ) = −A(rθ)2 − C(rθ)A(rθ) − C(rθ), 𝜕r

(A.20)

where C(u) ∈ C(Bn (η); 𝕄n×n (ℝ)) is a given function. A.4.1.1 Derivation of the ODE 1 1 Let X1 , . . . , Xn ∈ Cloc (M; TM) be Cloc vector fields on a C 2 manifold M of dimension n ∈ ℕ+ . Fix x ∈ M and ϵ > 0 and suppose that: – X1 , . . . , Xn span the tangent space at every point of M. – Φ(u) := eu1 X1 +u2 X2 +⋅⋅⋅+un Xn x exists for u ∈ Bn (ϵ).

708 � A Canonical coordinates l Write [Xj , Xk ] = ∑nl=1 cj,k Xl . Since X1 , . . . , Xn form a basis for the tangent space of M at l every point, the functions cj,k ∈ Cloc (M) are uniquely determined. Classical theorems

1 from the field of ODEs show that Φ is Cloc since X1 , . . . , Xn are. 󵄨 n 1 Let U ⊆ M and V ⊆ B (ϵ) be open sets such that Φ󵄨󵄨󵄨V : V → U is a Cloc diffeomor∗ 󵄨󵄨 0 phism. Let Yj := Φ󵄨󵄨V Xj so that Yj is a Cloc vector field on V . Write n

Yj = 𝜕uj + ∑ ajk (u)𝜕uk ,

(A.21)

k=1

where ajk ∈ Cloc (V ). Let A(u) denote the n × n matrix with (j, k) component ajk (u) and let

k C(u) denote the n × n matrix with (j, k) component ∑l ul cj,l ∘ Φ(u). We write u in polar coordinates as u = rθ, r ≥ 0.

Proposition A.4.1. In the above setting, A(u) satisfies the differential equation 𝜕 rA(rθ) = −A(rθ)2 − C(rθ)A(rθ) − C(rθ). 𝜕r In particular, the derivative

𝜕 rA(rθ) 𝜕r

(A.22)

exists in the classical sense.

We begin by proving Proposition A.4.1 in the smooth case. Lemma A.4.2. Proposition A.4.1 holds in the special case where M is a C ∞ manifold and ∞ X1 , . . . , Xn are Cloc vector fields on M. 󵄨 ∞ ∞ ∞ Proof. When X1 , . . . , Xn are Cloc , then Φ is Cloc and Φ󵄨󵄨󵄨V : V → U is a Cloc diffeomorphism. ∞ It follows that Y1 , . . . , Yn are Cloc vector fields on V . Furthermore, ̃ l Yl , [Yj , Yk ] = ∑ cj,k l

l ̃l := cj,k where cj,k ∘ Φ.

(A.23)

Note that dΦ(rθ)r𝜕r = rdΦ(rθ)𝜕r = rθ ⋅ X(Φ(rθ)), since Φ(rθ) = er(θ⋅X) x, and we are identifying the X with the vector of vector fields X = (X1 , . . . , Xn ). Writing this in Cartesian coordinates, we have n

n

j=1

j=1

∑ uj 𝜕uj = ∑ uj Yj .

(A.24)

Taking the Lie bracket of Yi with (A.24) and using (A.23), we obtain n

n

j=1

j=1

∑((Yi uj )𝜕uj + uj [Yi , 𝜕uj ]) = ∑((Yi uj )Yj + uj [Yi , Yj ]) n

= ∑ ((Yi uj )Yj + j=1

n

̃l (u)Yl ) . uj ∑ ci,j l=1

(A.25)

A.4 Proofs

� 709

We rewrite (A.25) as n

(∑ uj [𝜕uj , Yi − 𝜕ui ]) + Yi − 𝜕ui j=1

n

n

= − (∑((Yi − 𝜕ui )(uj ))(Yj − 𝜕uj )) − j=1

n

̃l (u)Yl . ∑ ∑ uj ci,j j=1 l=1

(A.26)

Plugging (A.21) into (A.26), we have n

n

n

∑ ∑ uj (𝜕uj aik )𝜕uk + ∑ aik 𝜕uk j=1 k=1

k=1

n

=

n

j − ∑ ∑ ai ajk 𝜕uk k=1 j=1

n

−

n

̃k 𝜕uk ∑ ∑ uj ci,j k=1 j=1

n

−

n

n

̃l alk 𝜕uk . ∑ ∑ ∑ uj ci,j l=1 k=1 j=1

(A.27)

Taking the 𝜕uk component of (A.27) and writing 1 + ∑nj=1 uj 𝜕uj = 𝜕r r, we have n

j

n

n

n

j=1

l=1

j=1

̃k − ∑ (∑ uj ci,j ̃l ) alk . 𝜕r raik = − ∑ ai ajk − ∑ uj ci,j j=1

This is exactly (A.22) and completes the proof. Proof of Proposition A.4.1. By a classical theorem of Whitney, there is a C ∞ manifold structure on M, compatible with its C 2 manifold structure, so we may assume M is a 󵄨 C ∞ manifold. Pick relatively compact open sets Ṽ ⋐ V and Ũ ⋐ U with Φ󵄨󵄨󵄨Ṽ : Ṽ → Ũ a 1 Cloc diffeomorphism. Fix u0 ∈ Ṽ . We will prove the result with V replaced by Bn (u0 , δ0 ) for some δ0 > 0, and the result will follow since the conclusion is local. Fix ϵ′ ∈ (0, ϵ) so large that Ṽ ⊆ Bn (ϵ′ ). Let Xjσ be smooth vector fields on M such 1 that Xjσ → Xj in Cloc as σ ↓ 0. Define

σ

σ

Φσ (u) := eu1 X1 +⋅⋅⋅+un Xn x. Then, for σ sufficiently small, Φσ (u) is defined for u ∈ Bn (ϵ′ ) and X1σ , . . . , Xnσ form a basis for the tangent space at every point of a neighborhood of the closure of Φσ (Bn (ϵ′ )). Thus, k,σ σ k,σ k 0 we may write [Xiσ , Xjσ ] = ∑k ci,k Xk , with ci,j → ci,j in Cloc as σ ↓ 0. Also, Φσ → Φ in

1 Cloc (Bn (ϵ′ )) as σ ↓ 0, by standard theorems. For σ sufficiently small, |det dΦσ (u0 )| ≥ 21 |det dΦ(u0 )| > 0. The inverse function 󵄨 theorem shows that there is a δ0 > 0, independent of σ, such that for σ small, Φσ 󵄨󵄨󵄨Bn (u ,δ ) 0 0 is a diffeomorphism onto its image. Define Aσ and Cσ in the obvious way on Bn (u0 , δ0 ), by using the vector fields 0 X1σ , . . . , Xnσ in place of X1 , . . . , Xn . We have Aσ → A and Cσ → C in Cloc (Bn (u0 , δ0 )). 2 n Furthermore, by Lemma A.4.2, 𝜕r rAσ = −Aσ − Cσ Aσ − Cσ , i. e., for r0 θ ∈ B (u0 , δ0 ) and r near r0 , we have

710 � A Canonical coordinates r

rAσ (rθ) = r0 Aσ (r0 θ) + ∫ −Aσ (sθ)2 − Cσ (sθ)Aσ (sθ) − Cσ (sθ) ds.

(A.28)

r0

Taking the limit as σ ↓ 0 in (A.28) shows that r

rA(rθ) = r0 A(r0 θ) + ∫ −A(sθ)2 − C(sθ)A(sθ) − C(sθ) ds. r0

It follows that 𝜕r rA exists in the classical sense and 𝜕r rA = −A2 − CA − C, completing the proof. A.4.1.2 Existence, uniqueness, and regularity In this section, we describe the existence, uniqueness, and regularity of solutions to (A.20) satisfying A(0) = 0. If C is a matrix, we write |C| for the usual matrix norm. Proposition A.4.3. Let C ∈ C(Bn (η); 𝕄n×n ) be given with C(0) = 0. Suppose that |C(x)| ≤ 0 D|x|, for x ∈ Bn (η). Then if η ≤ (10D)−1 , there exists a unique A ∈ Cloc (Bn (η); 𝕄n×n ) with A(0) = 0 satisfying (A.20). This unique solution satisfies 5 1 D|x| and |A(x)| ≤ , 8 16

∀x ∈ Bn (η).

(A.29)

C ∈ C m (Bn (η); 𝕄n×n ) 󳨐⇒ A ∈ C m (Bn (η); 𝕄n×n )

(A.30)

‖A‖C m (Bn (η);𝕄n×n ) ≤ Kn,m ,

(A.31)

|A(x)| ≤

Furthermore, for this solution A and m ∈ ℕ,

and

where Kn,m can be chosen to depend only on n, m, and an upper bound for ‖C‖C m (Bn (η);𝕄n×n ) . The rest of this section is devoted to the proof of Proposition A.4.3. Define T : Cloc (Bn (η); M n×n ) → Cloc (Bn (η); 𝕄n×n ) by 1

2

T (A)(x) := ∫ −A(sx) − C(sx)A(sx) − C(sx) ds. 0

The relevance of T is the following lemma.

A.4 Proofs

� 711

Lemma A.4.4. A ∈ Cloc (Bn (η); 𝕄n×n ) is a solution to (A.20) if and only if T (A) = A. Also, writing x = rθ, we have the following formula for T when r > 0: r

1 2 T (A)(rθ) = ∫ −A(sθ) − C(sθ)A(sθ) − C(sθ) ds. r

(A.32)

0

Proof. Formula (A.32) follows from a straightforward change of variables in the definition of T . It follows from (A.32) that A ∈ Cloc (Bn (η); 𝕄n×n ) is a solution to (A.20) if and only if T (A) = A. We begin with the uniqueness claim in Proposition A.4.3. Suppose A1 , A2 ∈ Cloc (Bn (η); 𝕄n×n ) are two solutions to (A.20) with A1 (0) = A2 (0) = 0. By Lemma A.4.4 we have T (A1 ) = A1 and T (A2 ) = A2 . We first claim that |Aj (x)| = O(|x|) as |x| → 0, for j = 1, 2; we prove this for A1 and the same will hold for A2 by symmetry. Set F(r) := sup|x|≤r |A1 (x)|, note that F : [0, η) → ℝ is continuous increasing, and consider F(0) = 0. Since T (A1 ) = A1 and |C(sx)| ≤ Ds|x| by assumption, we have 1

|A1 (x)| ≤ ∫ F(s|x|)2 + Ds|x|F(s|x|) + Ds|x| ds 0

1 1 ≤ F(|x|)2 + D|x|F(|x|) + D|x|, 2 2 so F(r) ≤ F(r)2 + 21 DrF(r) + 21 Dr, and therefore F(r)(1 − F(r)) ≤ 21 DrF(r) + 21 Dr. Taking r so small that F(r) ≤ 21 , we have F(r) ≤ 32 Dr. Thus, |A1 (x)| = O(|x|) as x → 0, as desired. Writing x in polar coordinates x = rθ and using (A.32), we have, for r > 0, r

󵄨󵄨 󵄨 󵄨 󵄨 −1 󵄨 󵄨 −1 −1 󵄨󵄨r(A1 (rθ) − A2 (rθ))󵄨󵄨󵄨 ≤ ∫ 󵄨󵄨󵄨s(A1 (sθ) − A2 (sθ))󵄨󵄨󵄨(s 󵄨󵄨󵄨A1 (sθ)󵄨󵄨󵄨 + s |A2 (sθ)| + s |C(sθ)|) ds. 0

Using the fact that |A1 (sθ)|, |A2 (sθ)|, |C(sθ)| = O(s), the integral form of Grönwall’s inequality shows that A1 (rθ) = A2 (rθ) for r > 0 and therefore A1 = A2 . This completes the proof of uniqueness. We now turn to the existence part of Proposition A.4.3, which we prove using the contraction mapping principle. Let n

M := {A ∈ C(B (η); 𝕄

sup |A(x)| ≤

x∈Bn (η)

We give M the metric

n×n

) : A(0) = 0,

1 }. 10

sup

n (η) 0=x∈B ̸

1 |A(x)| < ∞, |x|

712 � A Canonical coordinates d(A, B) :=

sup

n (η) 0=x∈B ̸

1 󵄨󵄨 󵄨 󵄨A(x) − B(x)󵄨󵄨󵄨. |x| 󵄨

With this metric, M is a complete metric space. Lemma A.4.5. T : M → M and ∀A, B ∈ M , d(T (A), T (B)) ≤ d(T (0), 0) ≤ D/2.

1 d(A, B). 5

Also,

Proof. Let A ∈ M . It follows easily from the definition of T that T (A) is continuous. For x ∈ Bn (η), using η < (10D)−1 , we have 1

󵄨󵄨 󵄨 2 󵄨󵄨T (A)(x)󵄨󵄨󵄨 ≤ ∫ ‖A‖C 0 + Ds|x|‖A‖C 0 + Ds|x| ds 0

1 D 1 D + η + η 100 2 10 2 1 1 1 1 ≤ + + ≤ . 100 200 20 10 ≤

(A.33)

Also, for 0 ≠ x ∈ Bn (η), 1

1 󵄨󵄨 1 1 󵄨 ∫ Ds|x| ds ≤ D, 󵄨T (0)(x)󵄨󵄨󵄨 ≤ |x| 󵄨 |x| 2

(A.34)

0

so T (0) ∈ M with d(T (0), 0) ≤ D/2. Finally, for A, B ∈ M , 0 ≠ x ∈ Bn (η), we have 1 󵄨󵄨 󵄨 󵄨T (A)(x) − T (B)(x)󵄨󵄨󵄨 |x| 󵄨 1

≤

1 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∫󵄨A(sx) − B(sx)󵄨󵄨󵄨(󵄨󵄨󵄨A(sx)󵄨󵄨󵄨 + 󵄨󵄨󵄨B(sx)󵄨󵄨󵄨 + 󵄨󵄨󵄨C(sx)󵄨󵄨󵄨) ds |x| 󵄨 0

≤

1

1

0

0

1 1 1 s ∫ s|x|d(A, B) ( + Ds|x|) ds ≤ ∫ sd(A, B) ( + ) ds |x| 5 5 10

1 1 1 ≤ d(A, B) ( + ) ≤ d(A, B). 10 30 5

(A.35)

1 Putting 0 = B in (A.35) and using (A.34) shows sup0=x∈B |T (A)(x)| < ∞. Combining n (η) ̸ |x| this with (A.33) shows that T : M → M . Further, (A.35) with arbitrary A, B ∈ M shows that d(T (A), T (B)) ≤ 51 d(A, B), and this completes the proof.

By Lemma A.4.5, T : M → M is a strict contraction, and the contraction mapping principle applies to show that if A0 = 0, Aa = T (Aa−1 ), a ≥ 1, then Aa → A∞ in M , where T (A∞ ) = A∞ . A∞ is the desired solution to (A.20) (see Lemma A.4.4).

A.4 Proofs

� 713

Also, for a ∈ ℕ ∪ {∞}, we have, using Lemma A.4.5, for 0 ≠ x ∈ Bn (η), a−1 1 󵄨󵄨 󵄨 b+1 b 󵄨󵄨Aa (x)󵄨󵄨󵄨 ≤ d(Aa , 0) ≤ ∑ d(T (0), T (0)) |x| b=0 a−1

5 ≤ ∑ 5 d(T (0), 0) ≤ D. 8 b=0

(A.36)

−b

In particular, for x ∈ Bn (η), we have |A∞ (x)| ≤ 85 D|x|. Also, since η ≤ (10D)−1 , it follows that |A∞ (x)| ≤ 161 ; this establishes (A.29). It remains to prove the regularity properties of A∞ in terms of the regularity of C, i. e., to establish (A.30) and (A.31) for A∞ . To do this, we use a modification of the contraction mapping principle due to Izzo. Lemma A.4.6 (Izzo’s contraction mapping principle [128]). Suppose (M, d) is a metric space and {Qa }a∈ℕ is a sequence of contractions on M for which there exists c < 1 with d(Qa (x), Qa (y)) ≤ cd(x, y),

∀x, y ∈ M, a ∈ ℕ.

Suppose ∃x∞ ∈ M with lima→∞ Qa (x∞ ) = x∞ . Let x0 ∈ M be arbitrary and define xa recursively by xa+1 = Qa (xa ). Then lima→∞ xa = x∞ . Proof. We include a slightly modified version of the proof in [128]. For each a ∈ ℕ, d(xa+1 , x∞ ) = d(Qa (xa ), x∞ )

≤ d(Qa (xa ), Qa (x∞ )) + d(Qa (x∞ ), x∞ )

(A.37)

≤ cd(xa , x∞ ) + d(Qa (x∞ ), x∞ ).

First we claim that the sequence d(xa , x∞ ) is bounded. Since Qa (x∞ ) → x∞ , ∃N, a ≥ N ⇒ d(Qa (x∞ ), x∞ ) < 1 − c. Suppose d(xa , x∞ ) is not bounded; then ∃a ≥ N with max{d(xa , x∞ ), 1} ≤ d(xa+1 , x∞ ). Applying this to (A.37), we have d(xa+1 , x∞ ) ≤ cd(xa , x∞ ) + d(Qa (x∞ ), x∞ ) < cd(xa+1 , x∞ ) + 1 − c, so d(xa+1 , x∞ ) < 1 ≤ d(xa+1 , x∞ ), a contradiction. Thus, the sequence d(xa , x∞ ) is bounded. Since Qa (x∞ ) → x∞ , (A.37) implies lim sup d(xa , x∞ ) ≤ c lim sup d(xa , x∞ ). a→∞

a→∞

Since lim supa→∞ d(xa , x∞ ) < ∞, this gives lim supa→∞ d(xa , x∞ ) = 0, completing the proof. To establish (A.30) and (A.31) we prove the following when C ∈ C m (Bn (η); 𝕄n×n ): (I) Aa → A∞ in C m (Bn (η); 𝕄n×n ),

714 � A Canonical coordinates (II) ‖A∞ ‖C m (Bn (η);𝕄n×n ) ≲ 1, where the implicit constant can be chosen to depend only on n, m, and an upper bound for ‖C‖C m (Bn (η);𝕄n×n ) . We prove (I) and (II) by induction on m. The base case, m = 0, was established above: (I) was proved by the contraction mapping principle and (II) follows from (A.29). Let m ≥ 1, assume (I) and (II) hold with m replaced by m − 1, and let C ∈ C m (Bn (η); 𝕄n×n ). Fix a multi-index β with |β| = m. We will show (under our inductive hypothesis) that: (a) Aa ∈ C m (Bn (η); 𝕄n×n ), ∀a ∈ ℕ. a→∞

β

(b) ∃B∞ ∈ C(Bn (η); 𝕄n×n ) such that 𝜕x Aa 󳨀 󳨀󳨀󳨀󳨀→ B∞ in C(Bn (η); 𝕄n×n ). (c) ‖B∞ ‖C(Bn (η);𝕄n×n ) ≲ 1, where the implicit constant can be chosen to depend only on n, m, and an upper bound for ‖C‖C m (Bn (η);𝕄n×n ) .

To see why the above will complete the proof, note that (b) along with the fact that β Aa → A∞ in C(Bn (η); 𝕄n×n ) gives 𝜕x A∞ = B∞ . Since β was an arbitrary multi-index with |β| = m, (b) along with the inductive hypothesis gives (I) and (c) along with the inductive hypothesis gives (II). Having shown them to be sufficient, we turn to proving (a), (b), and (c). Since Aa = T a (0), (a) follows from the easily established fact that when C ∈ m n C (B (η); 𝕄n×n ), T : C m (Bn (η); 𝕄n×n ) → C m (Bn (η); 𝕄n×n ). To establish (b) and (c), we use Lemma A.4.6. For M1 , M2 ∈ C m (Bn (η); 𝕄n×n ), define Rβ (M1 , M2 ) by 𝜕xβ (M1 M2 )(x) = (𝜕xβ M1 (x))M2 (x) + M1 (x)(𝜕xβ M2 (x)) + Rβ (M1 , M2 )(x), that is, Rβ (M1 , M2 ) =

β ∑ ( 1 )(𝜕xβ1 M1 )(𝜕xβ2 M2 ). β β +β =β 1

2

β1 ,β2 =0 ̸

By the inductive hypothesis, we have the following limits in C(Bn (η); 𝕄n×n ): a→∞

Rβ (Aa , Aa ) 󳨀 󳨀󳨀󳨀󳨀 → Rβ (A∞ , A∞ ),

a→∞

Rβ (C, Aa ) 󳨀󳨀󳨀󳨀󳨀→ Rβ (C, A∞ ),

(A.38)

and the following estimates for a ∈ ℕ ∪ {∞}: ‖Rβ (Aa , Aa )‖C(Bn (η);𝕄n×n ) , ‖Rβ (C, Aa )‖C(Bn (η);𝕄n×n ) ≲ 1.

(A.39)

A.4 Proofs

� 715

For a ∈ ℕ ∪ {∞}, B ∈ C(Bn (η); 𝕄n×n ), set 1

Qa (B)(x) := − ∫ s [B(sx)Aa (sx) + Aa (sx)B(sx) + C(sx)B(sx) |β|

0

+ (𝜕xβ C)(sx)Aa (sx) + (𝜕xβ C)(sx) + Rβ (Aa , Aa )(sx) + Rβ (C, Aa )(sx)] ds. Lemma A.4.7. For a ∈ ℕ ∪ {∞}, Qa : C(Bn (η); 𝕄n×n ) → C(Bn (η); 𝕄n×n ) and satisfies ‖Qa (B1 ) − Qa (B2 )‖C(Bn (η);𝕄n×n ) ≤

1 ‖B − B2 ‖C(Bn (η);𝕄n×n ) , 8 1

(A.40)

a→∞

∀B1 , B2 ∈ C(Bn (η); 𝕄n×n ). Furthermore, ∀B ∈ C(Bn (η); 𝕄n×n ), Qa (B) 󳨀󳨀󳨀󳨀󳨀→ Q∞ (B), with convergence in C(Bn (η); 𝕄n×n ). Finally, ‖Qa (0)‖C(Bn (η);𝕄n×n ) ≲ 1. Proof. It follows immediately from the definitions, the inductive hypothesis, and the assumption that C ∈ C m (Bn (η); 𝕄n×n ) that Qa : C(Bn (η); 𝕄n×n ) → C(Bn (η); 𝕄n×n ). a→∞

It follows from (A.38) and the fact that Aa → A∞ in C(Bn (η); 𝕄n×n ) that Qa (B) 󳨀󳨀󳨀󳨀󳨀→ Q∞ (B). It follows from (A.39), ‖C‖C Bn (η) (𝕄n×n ) ≲ 1, and ‖Aa ‖C(Bn (η);𝕄n×n ) ≲ 1 that ‖Qa (0)‖C(Bn (η);𝕄n×n ) ≲ 1. We turn to (A.40). Using (A.36) and the assumption |C(x)| ≤ D|x|, we have, for x ∈ n B (η), 1

󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨 |β| 󵄨 󵄨󵄨Qa (B1 )(x) − Qa (B2 )(x)󵄨󵄨󵄨 ≤ ∫ s (2󵄨󵄨󵄨Aa (sx)󵄨󵄨󵄨 + 󵄨󵄨󵄨C(sx)󵄨󵄨󵄨)󵄨󵄨󵄨B1 (sx) − B2 (sx)󵄨󵄨󵄨 ds 0

1

9 ≤ ‖B1 − B2 ‖C(Bn (η);𝕄n×n ) ∫ D|x|s ds 4 0

9 9 ≤ Dη‖B1 − B2 ‖C(Bn (η);𝕄n×n ) ≤ ‖B − B2 ‖C(Bn (η);𝕄n×n ) , 8 80 1 establishing (A.40) and completing the proof. β

For a ∈ ℕ, set Ba := 𝜕x Aa ; note that Ba ∈ C(Bn (η); 𝕄n×n ) since we have already β β shown that Aa ∈ C m (Bn (η); 𝕄n×n ). Also, Ba+1 = 𝜕x Aa+1 = 𝜕x T (Aa ) = Qa (Ba ). Lemma A.4.7 shows that Q∞ is a strict contraction on C(Bn (η); 𝕄n×n ), and the contraction mapping principle guarantees that it has a unique fixed point B∞ ∈ a→∞

C(Bn (η); 𝕄n×n ). Lemma A.4.7 shows that Qa (B∞ ) 󳨀 󳨀󳨀󳨀󳨀→ Q∞ (B∞ ) = B∞ . This, along with a→∞

Lemma A.4.7, shows that Lemma A.4.6 applies to give Ba 󳨀󳨀󳨀󳨀󳨀→ B∞ in C(Bn (η); 𝕄n×n ). β Since Ba = 𝜕x Aa , this establishes (b). Finally, to prove (c), note that B∞ is the fixed point of the strict contraction Q∞ , so a Q∞ (0) → B∞ . Hence, using (A.40),

716 � A Canonical coordinates ∞

a+1 a ‖B∞ ‖C(Bn (η);𝕄n×n ) ≤ ∑ ‖Q∞ (0) − Q∞ (0)‖C(Bn (η);𝕄n×n ) a=0 ∞

≤ ∑ 8−a ‖Q∞ (0) − 0‖C(Bn (η);𝕄n×n ) = a=0

8 ‖Q (0)‖C(Bn (η);𝕄n×n ) ≲ 1, 7 ∞

where the last inequality uses Lemma A.4.7.

A.4.2 An inverse function theorem We require a quantitative version of a special case of the inverse function theorem that does not follow from the standard statement of that theorem, though we achieve it by keeping track of constants in a standard proof. We present it here. Fix η > 0, let Y1 , . . . , Yn ∈ C 1 (Bn (η); Tℝn ) be vector fields on Bn (η), and suppose they satisfy 󵄨 󵄨 inf 󵄨󵄨󵄨det(Y1 (u)| ⋅ ⋅ ⋅ |Yn (u))󵄨󵄨󵄨 ≥ c0 > 0.

u∈Bn (η)

Take C0 > 0 such that ‖Yj ‖C 1 (Bn (η);ℝn ) ≤ C0 , ∀1 ≤ j ≤ n. Define Ψu (v) := ev1 Y1 +⋅⋅⋅+vn Yn u. Proposition A.4.8. There exist κ = κ(C0 , c0 , n) > 0 and Δ0 = Δ0 (C0 , c0 , n, η) > 0 such that for all δ ∈ (0, Δ0 ] and u ∈ Bn (κδ), v 󳨃→ Ψu (v) is defined and injective on v ∈ Bn (δ). Furthermore, Bn (κδ) ⊆ Ψu (Bn (δ)). The rest of this section is devoted to the proof of Proposition A.4.8. Lemma A.4.9. Let δ0 > 0, let F ∈ C 1 (Bn (δ0 ); ℝn ), and suppose dF(0) is non-singular and supx∈Bn (δ0 ) ‖dF(0)−1 dF(x) − I‖𝕄n×n ≤ 21 . Then F(Bn (δ0 )) ⊆ ℝn is open and F : Bn (δ0 ) → 1 F(Bn (δ0 )) is a Cloc diffeomorphism. Furthermore, F(Bn (δ0 )) ⊇ Bn (F(0), κ0 δ0 ), where 󵄨󵄨 󵄨󵄨 −(n−1) κ0 := ‖d(F −1 )‖−1 C(F(Bn (δ0 ));𝕄n×n ) ≥ cn 󵄨󵄨det dF(0)󵄨󵄨‖F‖C 1 (Bn (δ

0 );ℝ

n)

,

(A.41)

where cn > 0 can be chosen to depend only on n. Proof. We first show that F is injective. Fix y ∈ ℝn and set ϕ(x) = x + dF(0)−1 (y − F(x)). Note that F(x) = y ⇔ ϕ(x) = x. Also, ∀x ∈ Bn (δ0 ), ‖dϕ(x)‖𝕄n×n = ‖I −dF(0)−1 dF(x)‖𝕄n×n ≤ 1 . Hence, |ϕ(x1 ) − ϕ(x2 )| ≤ 21 |x1 − x2 |, and there is at most one solution to ϕ(x) = x. In 2 other words, there is at most one solution to F(x) = y, proving that F is injective. Since ‖dF(0)−1 dF(x) − I‖𝕄n×n ≤ 21 , ∀x ∈ Bn (δ0 ), it follows that dF(x) is invertible, ∀x ∈ Bn (δ0 ). Combining this with the fact that F is injective, the inverse function theorem 1 shows that F(Bn (δ0 )) is open and F : Bn (δ0 ) → F(Bn (δ0 )) is a Cloc diffeomorphism.

A.4 Proofs

� 717

Next we prove the lower bound for κ0 given in (A.41). In what follows, we use A ≲ B to denote A ≤ Cn B, where Cn can be chosen to depend only on n. Since ‖dF(0)−1 dF(x) − I‖𝕄n×n ≤ 21 by assumption, we have 󵄨 󵄨 󵄨 󵄨 inf 󵄨󵄨󵄨det dF(x)󵄨󵄨󵄨 ≳ 󵄨󵄨󵄨det dF(0)󵄨󵄨󵄨.

x∈Bn (δ0 )

(A.42)

Also, ∀x ∈ Bn (δ0 ), −1 󵄩 󵄩󵄩 󵄨 󵄨−1 n−1 󵄩󵄩(dF(x)) 󵄩󵄩󵄩𝕄n×n ≲ 󵄨󵄨󵄨det dF(x)󵄨󵄨󵄨 ‖dF‖C(Bn (δ0 );𝕄n×n ) ,

as can be seen by the cofactor representation of dF(x)−1 . Hence, −1 󵄩 󵄩 󵄨󵄨 󵄨󵄨 −1 n−1 sup 󵄩󵄩󵄩(dF(x)) 󵄩󵄩󵄩𝕄n×n ≲ ( inf det dF(y) 󵄨 󵄨󵄨) ‖dF‖C(Bn (δ0 );𝕄n×n ) , 󵄨 n y∈Bn (δ )

x∈B (δ0 )

0

and therefore 󵄨󵄨 󵄨 −1 n−1 ‖d(F −1 )‖C(F(Bn (δ0 ));𝕄n×n ) ≲ ( inf 󵄨󵄨det dF(x)󵄨󵄨󵄨) ‖dF‖C(Bn (δ0 );𝕄n×n ) n x∈B (δ ) 0

󵄨󵄨 󵄨 −1 n−1 ≲ ( inf 󵄨󵄨det dF(x)󵄨󵄨󵄨) ‖F‖C 1 (Bn (δ0 );𝕄n×n ) . x∈Bn (δ )

(A.43)

0

Combining (A.42) and (A.43) yields the inequality in (A.41). Finally, we prove that F(Bn (δ0 )) ⊇ Bn (F(0), κ0 δ0 ). Take ϵ > 0 to be the largest ϵ so that Bn (F(0), ϵ) ⊆ F(Bn (δ0 )); note that ϵ > 0 by the inverse function theorem. The proof will be complete once we show ϵ ≥ δ0 κ0 . Suppose, for contradiction, ϵ < δ0 κ0 . We have, by the mean value theorem, F −1 (Bn (F(0), ϵ)) ⊆ Bn (0, ϵ‖dF −1 ‖C(F(Bn (δ0 ));𝕄n×n ) ). Thus, if ϵ < κ0 δ0 , then F −1 (Bn (F(0), ϵ)) ⋐ Bn (0, δ0 ), which contradicts the maximality of ϵ and completes the proof. Lemma A.4.10. Let Yj , C0 , n, η, and Ψ be as in Proposition A.4.8. There exists δ1 = δ1 (C0 , n, η) > 0 such that ∀u ∈ Bn (η/2), Ψu is defined on Bn (δ1 ) and satisfies ‖Ψu ‖C 1 (Bn (δ1 );ℝn ) ≤ C(C0 , n)

(A.44)

‖dv Ψu (v) − dv Ψu (0)‖𝕄n×n ≤ C(C0 , n)|v|,

(A.45)

and ∀u ∈ Bn (η/2), v ∈ Bn (δ1 ),

where C(C0 , n) ≥ 0 can be chosen to depend only on C0 and n.

718 � A Canonical coordinates Proof. The existence of δ1 > 0 so that ∀u ∈ Bn (η/2), Ψu (v) is defined for v ∈ Bn (δ1 ) and (A.44) are classical theorems from the field of ODEs. Thus, we prove only (A.45). We write A ≲ B for A ≤ CB, where C can be chosen to depend only on C0 and n. Using the equation 𝜕r Ψu (rv) = v ⋅ Y (Ψu (r, v)), we have 1

Ψu (v) = ∫ v ⋅ Y (Ψu (sv)) ds. 0

Since dv Ψu (0) = (Y1 (u)| ⋅ ⋅ ⋅ |Yn (u)), we have, ∀u ∈ Bn (η/2), v ∈ Bn (δ1 ), 1

Ψu (v) − (dv Ψu (0))v = ∫ v ⋅ (Y (Ψu (sv)) − Y (Ψu (0))) ds.

(A.46)

0

Applying dv to (A.46) and using the chain rule, we have, ∀u ∈ Bn (η/2), v ∈ Bn (δ1 ), ‖dv Ψu (v) − dv Ψu (0)‖𝕄n×n

󵄩󵄩 1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 ⊤ = 󵄩󵄩∫ Y (Ψu (sv)) − Y (Ψu (0)) + sv dY (Ψu (sv))dΨu (sv) ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 0 󵄩󵄩𝕄n×n

≲ |v|‖Y ∘ Ψu ‖C 1 (Bn (δ1 );𝕄n×n ) + |v|‖Y ‖C 1 (Bn (η);𝕄n×n ) ‖Ψu ‖C 1 (Bn (δ1 );ℝn ) ≲ |v|‖Y ‖C 1 (Bn (η);𝕄n×n ) ‖Ψu ‖C 1 (Bn (δ1 );ℝn ) ≲ |v|, where we have written Y (u) for the 𝕄n×n -valued function (Y1 (u)| ⋅ ⋅ ⋅ |Yn (u)) and used (A.44). Proof of Proposition A.4.8. In what follows, we write A ≲ B for A ≤ CB, where C can be chosen to depend only on n, C0 , and c0 , and we write A ≲η B if C can also depend on η. By taking δ1 ≳η 1 as in Lemma A.4.10, Ψu (v) is defined for all u ∈ Bn (η/2) and v ∈ Bn (δ1 ). For such u, since dΨu (0) = (Y1 (u)| ⋅ ⋅ ⋅ |Yn (u)) and |det(Y1 (u)| ⋅ ⋅ ⋅ |Yn (0))| ≳ 1, we have ‖dv Ψu (0)−1 ‖𝕄n×n ≲ 1. Hence, using (A.45), for u ∈ Bn (η/2) v ∈ Bn (δ1 ), we have 󵄩󵄩 󵄩 󵄩 󵄩 −1 󵄩󵄩dv Ψu (0) dv Ψu (v) − I 󵄩󵄩󵄩𝕄n×n ≲ 󵄩󵄩󵄩dv Ψu (v) − dv Ψu (0)󵄩󵄩󵄩𝕄n×n ≲ |v|. Thus, if δ2 ≳η 1 is sufficiently small, we have, for all u ∈ Bn (η/2) and v ∈ Bn (δ2 ), 1 󵄩󵄩 󵄩 −1 󵄩󵄩dv Ψu (0) dv Ψu (v) − I 󵄩󵄩󵄩𝕄n×n ≤ . 2

(A.47)

Using (A.47), Lemma A.4.9 implies that for u ∈ Bn (η/2), Ψu : Bn (δ2 ) → Ψu (Bn (δ2 )) is a 1 1 −1 Cloc diffeomorphism. Set κ := 21 ‖d(Ψ−1 u )‖C(Ψu (Bn (δ2 ));𝕄n×n ) ; note the extra factor of 2 in κ as related to the definition of κ0 in (A.41). The estimate (A.41) implies κ ≳ 1. Take Δ0 < δ2 , Δ0 ≳η 1 sufficiently small so that κΔ0 < η/2. Then, for δ ∈ (0, Δ0 ] and u ∈ Bn (κδ), Lemma A.4.9 shows that

A.4 Proofs

� 719

Ψu (Bn (δ)) ⊇ Bn (Ψu (0), 2κδ) = Bn (u, 2κδ) ⊇ Bn (0, κδ), which completes the proof.

A.4.3 Proof of the main result We turn to the proof of Theorem A.2.4, which we separate into two parts: when X1 , . . . , Xq are linearly independent (i. e., when n = q) and more generally when X1 , . . . , Xq may be linearly dependent (i. e., when q ≥ n). A.4.3.1 Linearly independent We start by proving Theorem A.2.4 in the special case where n = q, i. e., when X1 , . . . , Xq form a basis for the tangent space at every point of M. Note that in this case, XJ0 = X, so we may replace XJ0 with X throughout the statement of Theorem A.2.4. We take χ = ξ. By assumption, X1 , . . . , Xn form a basis for the tangent space at every point of M, in particular at every point of BX (x0 , ξ), and (a) follows. Item (b) is trivial when n = q since the left-hand side of (A.5) equals 1 in this case. Since X1 , . . . , Xn span the tangent space at every point, Lemma A.2.14 shows that all balls of the form BX (x0 , χ ′ ) are open, which establishes (c). With (a), (b), and (c) proved, we henceforth assume that l cj,k ∈ CX1 J (BXJ (x0 , ξ)) = CX1 (BX (x0 , ξ)), 1 ≤ j, k, l ≤ n. 0

0

Consider the map Φ : Bn (η0 ) → BX (x0 , ξ) defined in (A.4), which we a priori know 1 to be Cloc . Clearly, dΦ(0)𝜕tj = Xj (x0 ). Since X1 (x0 ), . . . , Xn (x0 ) form a basis of the tangent space Tx0 BX (x0 , ξ), the inverse function theorem shows that there exists a (non1 ̂j := admissible) δ > 0 such that Φ : Bn (δ) → Φ(Bn (δ)) is a Cloc diffeomorphism. Let Y 󵄨󵄨∗ n 0 n k ̂j is a C vector field on B (δ). Write Y ̂j = 𝜕t + ∑ â (t)𝜕t . Let Φ󵄨󵄨Bn (δ) Xj , so that Y k=1 j loc j k n n×n ̂ A(t) ∈ Cloc (B (δ); 𝕄 ) denote the n×n matrix with (j, k) component â k (t) and let C(t) ∈ j

k C(Bn (η0 ); 𝕄n×n ) denote the n × n matrix with (j, k) component equal to ∑nl=1 tl cj,l ∘ Φ(t).

Proposition A.4.11. Write t in polar coordinates, t = rθ, and consider the differential equation 𝜕 rA(rθ) = −A(rθ)2 − C(rθ)A(rθ) − C(rθ), 𝜕r

(A.48)

defined for A : Bn (η0 ) → 𝕄n×n . There exists a constant η′ ∈ (0, η0 ], which is a 0-admissible constant which can also depend on a lower bound for η > 0, such that there exists a unique continuous solution A ∈ Cloc (Bn (η′ ); 𝕄n×n ) to (A.48) with A(0) = 0. Moreover, this solution lies in C 1 (Bn (η′ ); 𝕄n×n ) and satisfies ‖A(t)‖𝕄n×n ≲0 |t|

1 and ‖A(t)‖𝕄n×n ≤ , 2

∀t ∈ Bn (η′ ).

(A.49)

720 � A Canonical coordinates l l For m ∈ ℕ, if cj,k ∘ Φ ∈ C m (Bn (η′ )) with ‖cj,k ∘ Φ‖C m (Bn (η′ )) ≤ Dm , ∀j, k, l, then A ∈ m n ′ n×n C (B (η ); 𝕄 ) and there exists a constant Cm ≥ 0, which depends only on n, m, and 󵄨󵄨 ̂ 󵄨󵄨󵄨 n Dm , such that ‖A‖C m (Bn (η′ );𝕄n×n ) ≤ Cm . Finally, A 󵄨B (min{η′ ,δ}) = A󵄨󵄨Bn (min{η′ ,δ}) .

Proof. Note that by the definition of C(t) and 0-admissible constants, we have ̂ satisfies (A.48) on Bn (δ) by Proposition A.4.1. Since dΦ(0)𝜕t = ‖C(t)‖𝕄n×n ≲ |t|. Also, A j ̂ Xj (x0 ), we have A(0) = 0. With these remarks in hand, the proposition (except for the

claim A ∈ C 1 (Bn (η); 𝕄n×n )) follows directly from Proposition A.4.3. The claim that A ∈ C 1 (Bn (η′ ); 𝕄n×n ) can be seen as follows. First note that we may l assume η′ < η0 , as if η′ = η0 we can replace η′ with η0 /2. Since cj,k ∈ CX1 J (BXJ (x0 , ξ)) = 0

0

CX1 (BX (x0 , ξ)) and X1 , . . . , Xn span the tangent space at every point of BX (x0 , ξ), it follows l 1 1 that cj,k ∈ Cloc (BX (x0 , ξ)). Since Φ : Bn (η0 ) → BX (x0 , ξ) is a priori known to be Cloc , we

l 1 l have cj,k ∘ Φ ∈ Cloc (Bn (η0 )), and therefore cj,k ∘ Φ ∈ C 1 (Bn (η′ )). We conclude that C ∈

C 1 (Bn (η′ ); 𝕄n×n ), and it follows from Proposition A.4.3 that A ∈ C 1 (Bn (η′ ); 𝕄n×n ).

Fix η′ > 0 and A as in Proposition A.4.11. Write ajk (t) for the (j, k) component of A(t)

and set Yj := 𝜕tj + ∑nk=1 ajk (t)𝜕tk . Note that Y1 , . . . , Yn are C 1 vector fields on Bn (η′ ). By 󵄨 ̂j 󵄨󵄨󵄨 n Proposition A.4.11, Yj 󵄨󵄨󵄨Bn (min{η′ ,δ}) = Y 󵄨B (min{η′ ,δ}) . Since δ is not admissible, we think of δ ̂j . as being much smaller than η′ , so Yj should be thought of as extending Y Proposition A.4.12. We have, ∀t ∈ Bn (η′ ), dΦ(t)Yj (t) = Xj (Φ(t)), 1 ≤ j ≤ n. Proof. Fix θ ∈ S n−1 and set r1 := sup{r ∈ [0, η′ ) : dΦ(r ′ θ)Yj (r ′ θ) = Xj (Φ(r ′ θ)), 0 ≤ r ′ ≤ r, 1 ≤ j ≤ n}. We wish to show that r1 = η′ , and this will complete the proof since θ ∈ S n−1 was 󵄨 ̂j 󵄨󵄨󵄨 n arbitrary. Suppose, for contradiction, r1 < η′ . Since Yj 󵄨󵄨󵄨Bn (min{η′ ,δ}) = Y 󵄨B (min{η′ ,δ}) and ̂ dΦ(u)Yj (u) = Xj (Φ(u)), we know r1 > 0. By continuity, we have dΦ(r1 θ)Yj (r1 θ) = Xj (Φ(r1 θ)).

(A.50)

By assumption, X1 (Φ(r1 θ)), . . . , Xn (Φ(r1 θ)) span TΦ(r1 θ) M; combining this with (A.50) shows that dΦ(r1 θ) is non-singular and therefore the inverse function theorem applies to Φ at the point r1 θ. Thus, there exists a neighborhood V of r1 θ such that Φ : V → Φ(V ) 1 is a Cloc diffeomorphism. Pick 0 < r2 < r3 < r1 < r4 < η′ such that {r ′ θ : r2 ≤ r ′ ≤ r4 } ⊂ V . ̃j := Φ󵄨󵄨󵄨∗ Xj . By the choice of r1 , we have Let Y 󵄨V ̃j (r ′ θ) = Yj (r ′ θ), Y

r2 ≤ r ′ ≤ r3 , 1 ≤ j ≤ n.

(A.51)

̃j (t) = 𝜕t + ∑n ã k (t)𝜕t and let A(t) ̃ denote the matrix with (j, k) component Write Y k=1 j j k ̃ ′ θ) = A(r ′ θ) for r2 ≤ r ′ ≤ r3 . Proposition A.4.1 shows that ã k (t). By (A.51), we have A(r j

̃ satisfies (A.48). Away from r = 0, (A.48) is a standard ODE that both A and A ̃ satisfy. A

A.4 Proofs

� 721

Thus, by standard uniqueness theorems (using, for example, Grönwall’s inequality) we ̃ ′ θ) = A(r ′ θ) for r2 ≤ r ′ ≤ r4 , that is, Yj (r ′ θ) = Y ̃j (r ′ θ), r2 ≤ r ′ ≤ r4 , 1 ≤ j ≤ n. have A(r ′ ′ ′ ̃ ′ We therefore have dΦ(r θ)Yj (r θ) = dΦ(r θ)Yj (r θ) = Xj (Φ(r ′ θ)) for r2 ≤ r ′ ≤ r4 . Since r2 < r1 < r4 , this contradicts our choice of r1 , completing the proof. 2 Lemma A.4.13. Φ : Bn (η′ ) → BX (x0 , ξ) is Cloc . 1 Proof. Since we already know Φ : Bn (η′ ) → M is Cloc , it suffices to show that the map 1 u 󳨃→ dΦ(u), u ∈ Bn (η′ ), is Cloc . We have already remarked that Y1 , . . . , Yn are C 1 (see Proposition A.4.11). Since Y = (I + A)∇, with ‖A(t)‖𝕄n×n ≤ 21 , ∀t (see (A.49)), we conclude that Y1 , . . . , Yn are a basis for the tangent space at every point of Bn (η′ ). Also, using Propo1 1 1 sition A.4.12, dΦ(u)Yj (u) = Xj (Φ(u)) ∈ Cloc , since Xj ∈ Cloc and Φ ∈ Cloc . Since Y1 , . . . , Yn 1 are C and a basis for the tangent space at every point, we conclude that u 󳨃→ dΦ(u) is 1 2 Cloc , and therefore Φ is Cloc , completing the proof.

Proposition A.4.14. For m ∈ ℕ and η′′ ∈ (0, η′ ], we have, for any function f , ‖f ‖C m (Bn (η′′ )) ≈(m−1)∨0 ‖f ‖CYm (Bn (η′′ ))

(A.52)

‖Yj ‖C m (Bn (η′′ );ℝn ) ≲m 1.

(A.53)

and

Furthermore, for m ∈ ℕ, 1 ≤ j, k, l ≤ n, we have l ‖cj,k ∘ Φ‖C m (Bn (η′ )) ≲m 1.

(A.54)

Proof. Rewrite (A.53) as the equivalent statement ‖A‖C m (Bn (η′′ );𝕄n×n ) ≲m 1.

(A.55)

We prove (A.52), (A.55), and (A.54) by induction on m. Consider the base case, m = 0. Since ‖f ‖C 0 (Bn (η′′ )) = ‖f ‖C 0 (Bn (η′′ )) , (A.52) is trivial when Y

m = 0. The estimate (A.49) gives ‖A(t)‖𝕄n×n ≤ 21 , ∀t ∈ Bn (η′ ), and the case m = 0 of (A.55) follows. Finally, using the definition of 0-admissible constants (Definition A.2.1), we have l l ‖cj,k ∘ Φ‖C 0 (Bn (η′ )) = ‖cj,k ∘ Φ‖C(Bn (η′ ))

l l = ‖cj,k ‖C(Φ(Bn (η′ ))) ≤ ‖cj,k ‖C(BX (x0 ,ξ)) ≲0 1,

establishing the case m = 0 of (A.54). Let m ≥ 1. We assume (A.52), (A.55), and (A.54) for m − 1 and prove them for m. In particular, the inductive hypothesis for (A.55) is equivalent to ‖Yj ‖C m−1 (Bn (η′′ );ℝn ) ≲m−1 1,

1 ≤ j ≤ n.

(A.56)

722 � A Canonical coordinates (A.52): Using (A.56) it follows immediately from the definitions that ‖f ‖CYm (Bn (η′′ )) ≲m−1 ‖f ‖C m (Bn (η′′ )) . For the reverse inequality, note that Y = (I + A)∇ and (A.49) gives supt∈Bn (η′ ) ‖A(t)‖𝕄n×n ≤ 1 , so I + A is invertible and ∇ = (I + A)−1 Y . In particular, since (A.56) holds, we may 2 β

write any 𝜕x , with |β| ≤ m, as a linear combination of Y α , |α| ≤ m, with coefficients in C(Bn (η′′ )), which are ≲m−1 1. It follows that ‖f ‖C m (Bn (η′′ )) ≲m−1 ‖f ‖CYm (Bn (η′′ )) , establishing (A.52). (A.54): Since Φ∗ Xj = Yj , we have Y α (g ∘ Φ) = (X α g) ∘ Φ.

(A.57)

Using (A.52) and (A.57), we have l l ‖cj,k ∘ Φ‖C m (Bn (η′ )) ≈m−1 ‖cj,k ∘ Φ‖CYm (Bn (η′ )) 󵄩󵄩 α l 󵄩 󵄩 l 󵄩 󵄩󵄩 = ∑ 󵄩󵄩Y (cj,k ∘ Φ)󵄩󵄩󵄩C(Bn (η′ )) = ∑ 󵄩󵄩󵄩X α cj,k 󵄩C(Φ(Bn (η′ ))) |α|≤m

|α|≤m

󵄩 l 󵄩 l 󵄩󵄩 ≤ ∑ 󵄩󵄩󵄩X α cj,k 󵄩C(BX (x0 ,ξ)) = ‖cj,k ‖CXm (BX (x0 ,ξ)) ≲m 1, |α|≤m

where we have used the fact that Φ(Bn (η′ )) ⊆ BX (x0 , ξ). This establishes (A.54). (A.55): The estimate (A.54) gives ‖C‖C m (Bn (η′ );𝕄n×n ) ≲m 1. Proposition A.4.11 gives ‖A‖C m (Bn (η′ );𝕄n×n ) ≲m 1, which implies (A.55). 󵄨 Proposition A.4.15. There exists a 1-admissible constant η1 ∈ (0, η′ ] such that Φ󵄨󵄨󵄨Bn (η ) is 1

2 injective. Furthermore, Φ(Bn (η1 )) ⊆ BX (x0 , ξ) is open and Φ : Bn (η1 ) → Φ(Bn (η1 )) is a Cloc diffeomorphism.

Proof. Consider the maps, defined for u, v ∈ ℝn sufficiently small, given by Ψu (v) = ev1 Y1 +⋅⋅⋅+vn Yn u. Since Y = (I + A)∇ and supt∈Bn (η′ ) ‖A(t)‖𝕄n×n ≤

1 2

(see (A.49)), we have

󵄨󵄨 󵄨 󵄨󵄨det(Y1 (t)| ⋅ ⋅ ⋅ |Yn (t))󵄨󵄨󵄨 ≥ cn > 0,

∀t ∈ Bn (η′ ),

where cn > 0 can be chosen to depend only on n. Furthermore, Proposition A.4.14 gives ‖Yj ‖C 1 (Bn (η′ );ℝn ) ≲ 1.

A.4 Proofs

� 723

Take Δ0 , κ > 0 as in Proposition A.4.8 (with η′ playing the role of η in that proposition). In light of the above comments, Δ0 and κ can be taken to be 1-admissible constants. Set δ1 := min{Δ0 , τ0 , 1} so that δ1 > 0 is a 1-admissible constant; see Section A.2 for the definition of τ0 > 0. Let η1 := min{δ1 κ, η′ } > 0, so that η1 is also a 1-admissible constant. 󵄨 We claim Φ󵄨󵄨󵄨Bn (η ) is injective. Let u1 , u2 ∈ Bn (η1 ) be such that Φ(u1 ) = Φ(u2 ); we wish 1 to show u1 = u2 . By Proposition A.4.8 there exists v ∈ Bn (δ1 ) such that u2 = Ψu1 (v), i. e., u2 = ev⋅Y u1 . Since dΦ(u)Yj (u) = Xj (Φ(u)) (Proposition A.4.12), it follows that Φ(u1 ) = Φ(u2 ) = Φ(ev⋅Y u1 ) = ev⋅X Φ(u1 ). We have assumed that X1 , . . . , Xn are linearly independent everywhere. Also, X satisfies 𝒞 (Φ(u1 ), δ1 , BX (x0 , ξ)) because Y satisfies 𝒞 (u1 , δ1 , Bn (η′ )) (by Proposition A.4.8). Hence, by the definition of τ0 , we have v = 0. We conclude that u2 = ev⋅Y u1 = u1 , and therefore Φ is injective. Proposition A.4.12 shows that dΦ(u)Yj (u) = Xj (Φ(u)), and we have assumed that X1 , . . . , Xn form a basis for the tangent space at every point of M. It follows that dΦ(u) is non-singular, ∀u ∈ Bn (η′ ). The inverse function theorem applies to show that Φ : 1 Bn (η′ ) → BX (x0 , ξ) is an open map and is locally a Cloc diffeomorphism. In particular, 󵄨󵄨 n 1 2 Φ(B (η1 )) is open. Hence, since Φ󵄨󵄨Bn (η ) is injective, locally a Cloc diffeomorphism, and Cloc 1 2 (Lemma A.4.13), we conclude that Φ : Bn (η1 ) → Φ(Bn (η1 )) is a Cloc diffeomorphism.

Henceforth, we let η1 be the 1-admissible constant from Proposition A.4.15. Lemma A.4.16. There exists a 1-admissible constant ξ1 ∈ (0, ξ] such that BX (x0 , ξ1 ) ⊆ Φ(Bn (η1 /2)). Proof. Fix ξ1 ∈ (0, ξ] to be chosen later and suppose that y ∈ BX (x0 , ξ1 ). Thus, there exists an absolutely continuous γ : [0, 1] → BX (x0 , ξ) with γ(0) = x0 , γ(1) = y, γ′ (t) = ∑nj=1 bj (t)ξ1 Xj (γ(t)), with ‖ ∑ |bj (t)|2 ‖L∞ ([0,1]) < 1. Define t0 := sup{t ∈ [0, 1] : γ(t ′ ) ∈ Φ(Bn (η1 /2)), ∀0 ≤ t ′ ≤ t}. We want to show that by taking ξ1 > 0 to be a sufficiently small 1-admissible constant, we have t0 = 1 and y = γ(1) ∈ Φ(Bn (η1 /2)). Note that t0 ≥ 0 since γ(0) = x0 = Φ(0). Suppose that t0 < 1 or t0 = 1 and γ(1) ∈ ̸ Φ(Bn (η1 /2)). In either case, we have −1 |Φ (γ(t0 ))| = η1 /2. Using the fact that ‖Yj ‖C 0 (Bn (η1 );ℝn ) ≲0 1 and Φ(0) = x0 , we have 󵄨󵄨 t0 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 −1 󵄨󵄨 󵄨󵄨󵄨 d −1 󵄨 η1 /2 = 󵄨󵄨Φ (γ(t1 ))󵄨󵄨 = 󵄨󵄨∫ Φ ∘ γ(t) dt 󵄨󵄨󵄨 󵄨󵄨 dt 󵄨󵄨 󵄨󵄨 0 󵄨󵄨 󵄨󵄨 t0 n 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨∫ ∑ bj (t)ξ1 Yj (Φ−1 ∘ γ(t)) dt 󵄨󵄨󵄨 ≲0 ξ1 . 󵄨󵄨 󵄨󵄨 󵄨󵄨 0 j=1 󵄨󵄨

724 � A Canonical coordinates This is a contradiction if ξ1 is a sufficiently small 1-admissible constant, completing the proof. Taking χ = ξ and ξ1 = ξ2 and using the fact that X = XJ0 and Y = YJ0 , the above results combine to complete the proof of Theorem A.2.4 in the case n = q. A.4.3.2 Linearly dependent We prove Theorem A.2.4 in the general case q ≥ n. We take the same setting and notation as in Theorem A.2.4. We start with some general comments about Lie derivatives. Let Z, W1 , . . . , Wn ∈ 1 1 Cloc (M; TM) be Cloc vector fields on M. The Lie derivative of W1 ∧ W2 ∧ ⋅ ⋅ ⋅ ∧ Wn with respect to Z is given by LieZ (W1 ∧ W2 ∧ ⋅ ⋅ ⋅ ∧ Wn ) = [Z, W1 ] ∧ W2 ∧ W3 ∧ ⋅ ⋅ ⋅ ∧ Wn

+ W1 ∧ [Z, W2 ] ∧ W3 ∧ ⋅ ⋅ ⋅ ∧ Wn + ⋅ ⋅ ⋅

(A.58)

+ W1 ∧ W2 ∧ ⋅ ⋅ ⋅ ∧ Wn−1 ∧ [Z, Wn ].

1 1 Let V1 , . . . , Vn ∈ Cloc (M; TM) be Cloc vector fields which span the tangent space near a point x0 ∈ M. Near x0 , using Definition 3.6.2, we may define a real-valued function by

x 󳨃→

W1 (x) ∧ W2 (x) ∧ ⋅ ⋅ ⋅ ∧ Wn (x) . V1 (x) ∧ V2 (x) ∧ ⋅ ⋅ ⋅ ∧ Vn (x)

The derivative of this function with respect to Z is exactly what one would expect, as the next lemma shows. Lemma A.4.17. We have Z

W1 ∧ W2 ∧ ⋅ ⋅ ⋅ ∧ Wn LieZ (W1 ∧ W2 ∧ ⋅ ⋅ ⋅ ∧ Wn ) = V1 ∧ V2 ∧ ⋅ ⋅ ⋅ ∧ Vn V1 ∧ V2 ∧ ⋅ ⋅ ⋅ ∧ Vn −

W1 ∧ W2 ∧ ⋅ ⋅ ⋅ ∧ Wn LieZ (V1 ∧ V2 ∧ ⋅ ⋅ ⋅ ∧ Vn ) . V1 ∧ V2 ∧ ⋅ ⋅ ⋅ ∧ Vn V1 ∧ V2 ∧ ⋅ ⋅ ⋅ ∧ Vn

1 Proof. Let V := V1 ∧ V2 ∧ ⋅ ⋅ ⋅ ∧ Vn and W := W1 ∧ W2 ∧ ⋅ ⋅ ⋅ ∧ Wn . Let ν be any Cloc n-form which is non-zero near x0 , so that by Definition 3.6.2,

W ν(W) = . V ν(V) Because ν is non-zero near x0 (and the space of n-forms is one-dimensional at each point), we may write LieZ ν = fν for some continuous function f near x0 . Using [147, Proposition 18.9], we have Zν(W) = (LieZ ν)(W) + ν(LieZ W) = fν(W) + ν(LieZ W),

A.4 Proofs

� 725

and similarly with W replaced by V. We conclude that near x0 , Z

W ν(W) Zν(W) ν(W) Zν(V) =Z = − V ν(V) ν(V) ν(V) ν(V) fν(W) + ν(LieZ W) ν(W) fν(V) + ν(LieZ V) = − ν(V) ν(V) ν(V) ν(LieZ W) ν(W) ν(LieZ V) LieZ W W LieZ V = − = − , ν(V) ν(V) ν(V) V V V

completing the proof. Lemma A.4.18. For J ∈ ℐ (n, q), 1 ≤ j ≤ n, LieXj ⋀ XJ =

∑

K∈ℐ0 (n,q)

K gj,J ⋀ XK ,

on BXJ (x0 , ξ), 0

where, ∀m ∈ ℕ, K ‖gj,J ‖CXm

J0

(BXJ (x0 ,ξ)) 0

≲m 1.

(A.59)

Proof. Let J = (j1 , . . . , jn ). We have, using (A.58) and (A.2), LieXj ⋀ XJ = LieXj (Xj1 ∧ Xj2 ∧ ⋅ ⋅ ⋅ ∧ Xjn ) n

= ∑ Xj1 ∧ Xj2 ∧ ⋅ ⋅ ⋅ ∧ Xjl−1 ∧ [Xj , Xjl ] ∧ Xjl+1 ∧ ⋅ ⋅ ⋅ ∧ Xjn l=1 n

q

k = ∑ ∑ cj,j X ∧ Xj2 ∧ ⋅ ⋅ ⋅ ∧ Xjl−1 ∧ Xk ∧ Xjl+1 ∧ ⋅ ⋅ ⋅ ∧ Xjn . l j1 l=1 k=1

The result follows from the anti-commutativity of ∧ and the definition of m-admissible constants (see Definitions A.2.1 and A.2.2). Lemma A.4.19. Let χ ′ ∈ (0, ξ]. Suppose that for all y ∈ BXJ (x0 , χ ′ ), ⋀ XJ0 (y) ≠ 0. Then, for 0 J ∈ ℐ (n, q), 1 ≤ j ≤ n, Xj

⋀ XJ

⋀ X J0

=

∑

K∈ℐ0 (n,q)

K gj,J

⋀ XJ ⋀ XK ⋀ XK − ∑ gK ⋀ XJ0 K∈ℐ (n,q) j,J0 ⋀ XJ0 ⋀ XJ0

on BXJ (x0 , χ ′ ),

0

K where gj,J are the functions from Lemma A.4.18.

Proof. This follows by combining Lemmas A.4.17 and A.4.18. Lemma A.4.20. Fix C, u0 > 0 and let uu0 ,C (t) be the unique solution to d u (t) = C(uu0 ,C (t) + uu0 ,C (t)2 ), dt u0 ,C

uu0 ,C (0) = u0 ,

0

726 � A Canonical coordinates defined on some maximum interval [0, Ru0 ,C ). Let F(t) be a non-negative, absolutely continuous function defined on [0, R′ ) with R′ ≤ Ru0 ,C satisfying d F(t) ≤ C(F(t) + F(t)2 ), dt

F(0) ≤ u0 ,

where the above inequality holds almost everywhere. Then, for t ∈ [0, R′ ), F(t) ≤ uu0 ,C (t). Proof. This is a special case of the Bihari–LaSalle inequality. Lemma A.4.21. There exists a 0-admissible constant χ ∈ (0, ξ] such that the following holds. Suppose γ : [0, χ] → BXJ (x0 , ξ) is an absolutely continuous curve satisfying γ(0) = 0

x0 , γ′ (t) = ∑nj=1 Xj (γ(t)) almost everywhere and ‖ ∑ |aj (t)|2 ‖L∞ ([0,χ]) < 1. Suppose further that for some χ ′ ∈ (0, χ], ⋀ XJ0 (γ(t)) ≠ 0 for t ∈ [0, χ ′ ]. Then 󵄨󵄨 ⋀ X (γ(t)) 󵄨󵄨 󵄨 󵄨󵄨 J 󵄨󵄨 ≲0 1. sup 󵄨󵄨󵄨󵄨 󵄨󵄨 X (γ(t)) ⋀ 󵄨 J∈ℐ(n,q) 󵄨 J0 󵄨

(A.60)

t∈[0,χ ′ ]

Here, the implicit constant depends on neither χ ′ nor γ. Proof. Let χ ∈ (0, ξ] be a 0-admissible constant to be chosen later and let γ and χ ′ ∈ (0, χ] be as in the statement of the lemma. We wish to show that if χ is chosen to be a sufficiently small 0-admissible constant (which forces χ ′ to be small), then (A.60) holds. Set F(t) :=

󵄨󵄨 ⋀ X (γ(t)) 󵄨󵄨2 󵄨󵄨 󵄨󵄨 J 󵄨󵄨 󵄨 󵄨󵄨 ⋀ XJ (γ(t)) 󵄨󵄨󵄨 . 󵄨 J∈ℐ0 (n,q) 󵄨 0 ∑

We will show that if χ is a sufficiently small 0-admissible constant, then F(t) ≲0 1, ∀t ∈ [0, χ ′ ], and this will complete the proof.2 Using Lemma A.4.19, we have, for almost every t, ⋀ XJ (γ(t)) n ⋀ XJ d F(t) = ∑ 2 ) (γ(t)) ∑ aj (t) (Xj dt X (γ(t)) X J0 ⋀ ⋀ J0 J∈ℐ (n,q) j=1 0

=

∑

n

∑ 2aj (t)

J∈ℐ0 (n,q) j=1 K∈ℐ0 (n,q)

K × (gj,J (γ(t))

⋀ XJ (γ(t))

⋀ XJ0 (γ(t))

(A.61)

⋀ XJ (γ(t)) ⋀ XK (γ(t)) ⋀ XK (γ(t)) K − gj,J (γ(t)) ) 0 ⋀ XJ0 (γ(t)) ⋀ XJ0 (γ(t)) ⋀ XJ0 (γ(t))

≲0 F(t) + F(t)3/2 ≲0 F(t) + F(t)2 .

2 Here we are using the fact that ∀K ∈ ℐ(n, q), either ⋀ XK ≡ 0 or ∃J ∈ ℐ0 (n, q) with ⋀ XK = ± ⋀ XJ , by the basic properties of wedge products.

A.4 Proofs

� 727

Also, we have, using (A.3), F(0) =

󵄨󵄨 ⋀ X (x ) 󵄨󵄨2 󵄨󵄨 J 0 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ⋀ XJ (x0 ) 󵄨󵄨󵄨 ≲0 1. 󵄨 J∈ℐ0 (n,q) 󵄨 0 ∑

(A.62)

Using (A.61) and (A.62) we see that there exist 0-admissible constants C, u0 > 0 such that d F(t) ≤ C(F(t) + F(t)2 ), dt

F(0) ≤ u0 ,

where the inequality holds almost everywhere. Standard theorems from the field of ODEs show that if χ0 = χ0 (C, u0 ) > 0 is chosen sufficiently small, then the unique solution u(t) to d u(t) = C(u(t) + u(t)2 ), dt

u(0) = u0 ,

exists for t ∈ [0, χ0 ] and satisfies u(t) ≤ 2u0 , ∀t ∈ [0, χ0 ]. We take χ := min{χ0 , ξ} ∈ (0, χ]. For this choice of χ, which is 0-admissible since C and u0 are 0-admissible, Lemma A.4.20 shows that F(t) ≤ 2u0 ≲0 1, ∀t ∈ [0, χ ′ ], completing the proof. Proposition A.4.22. There exists a 0-admissible constant χ ∈ (0, ξ] such that ∀y ∈ BXJ (x0 , χ), ⋀ XJ0 (y) ≠ 0 and 0

sup

J∈ℐ(n,q) y∈BXJ (x0 ,χ)

󵄨󵄨 ⋀ X (y) 󵄨󵄨 󵄨󵄨 󵄨󵄨 J 󵄨󵄨 󵄨 󵄨󵄨 ⋀ XJ (y) 󵄨󵄨󵄨 ≲0 1. 󵄨 󵄨 0

(A.63)

0

Proof. Take χ as in Lemma A.4.21. First we claim that ∀y ∈ BXJ (x0 , χ), ⋀ XJ0 (y) ≠ 0 0. Fix y ∈ BXJ (x0 , χ), so that there exists an absolutely continuous γ : [0, χ] → 0

BXJ (x0 , χ) with γ(0) = x0 , γ(χ) = y, γ′ (t) = ∑nj=1 aj (t)Xj (γ(t)) almost everywhere, and 0

‖ ∑ |aj (t)|2 ‖L∞ ([0,χ]) < 1. We will show that ∀t ∈ [0, χ] we have ⋀ XJ0 (γ(t)) ≠ 0, and it will then follow that ⋀ XJ0 (y) = ⋀ XJ0 (γ(χ)) ≠ 0. Suppose not, so that ⋀ XJ0 (γ(t)) = 0 for some t ∈ [0, χ], and let t0 := inf{t ∈ [0, χ] : ⋀ XJ0 (γ(t)) = 0}, so that ⋀ XJ0 (γ(t0 )) = 0 but ⋀ XJ0 (γ(t)) ≠ 0, ∀t ∈ [0, t0 ). Note that t0 > 0 since ⋀ XJ0 (γ(t)) = ⋀ XJ0 (x0 ) ≠ 0. 1 Let ν be a Cloc n-form, defined on a neighborhood of γ(t0 ) and which is non-zero at γ(t0 ). We have lim ν(⋀ XJ0 )(γ(t)) = 0, t↑t0

lim ν(⋀ XJ (γ(t))) ≠ 0, t↑t0

by continuity, the fact that X1 , . . . , Xq span the tangent space at every point (and in particular at γ(t0 )), and the fact that ν is non-zero at γ(t0 ). We conclude, using Definition 3.6.2, that

728 � A Canonical coordinates 󵄨󵄨 ⋀ X (γ(t)) 󵄨󵄨 󵄨󵄨 ν(⋀ X )(γ(t)) 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨 J J 󵄨󵄨 = lim max 󵄨󵄨󵄨 󵄨󵄨 = ∞. lim max 󵄨󵄨󵄨󵄨 t↑t0 J∈ℐ(n,q) 󵄨󵄨 ⋀ XJ (γ(t)) 󵄨󵄨󵄨 t↑t0 J∈ℐ(n,q) 󵄨󵄨󵄨 ν(⋀ XJ )(γ(t)) 󵄨󵄨󵄨 0 0 that

(A.64)

Take any χ ′ ∈ (0, t0 ). We know that ∀t ∈ [0, χ ′ ], ⋀ XJ0 (γ(t)) ≠ 0. Lemma A.4.21 implies 󵄨󵄨 ⋀ X (γ(t)) 󵄨󵄨 󵄨 󵄨󵄨 J 󵄨󵄨 ≲0 1. sup 󵄨󵄨󵄨󵄨 󵄨 J∈ℐ(n,q) 󵄨󵄨 ⋀ XJ0 (γ(t)) 󵄨󵄨 t∈[0,χ ′ ]

Since χ ′ ∈ (0, t0 ) was arbitrary, we have 󵄨󵄨 ⋀ X (γ(t)) 󵄨󵄨 󵄨 󵄨󵄨 J 󵄨󵄨 ≲0 1. sup 󵄨󵄨󵄨󵄨 󵄨 J∈ℐ(n,q) 󵄨󵄨 ⋀ XJ0 (γ(t)) 󵄨󵄨 t∈[0,t0 )

This contradicts (A.64) and completes the proof that ⋀ XJ0 (y) ≠ 0, ∀y ∈ BXJ (x0 , χ). 0 To prove (A.63), take y ∈ BXJ (x0 , χ). Then there exists an absolutely continuous γ : 0

[0, χ] → BXJ (x0 , χ) ⊆ BXJ (x0 , ξ) with γ(0) = x0 , γ(χ) = y, γ′ (t) = ∑nj=1 aj (t)Xj (γ(t)) almost 0

0

everywhere, and ‖ ∑ |aj (t)|2 ‖L∞ ([0,χ]) < 1. We have already shown that ⋀ XJ0 (γ(t)) ≠ 0, ∀t ∈ [0, χ]. Lemma A.4.21 gives 󵄨󵄨 ⋀ X (y) 󵄨󵄨 󵄨󵄨󵄨 ⋀ XJ (γ(χ)) 󵄨󵄨󵄨 󵄨 󵄨󵄨 J 󵄨󵄨 ≲ 1. 󵄨󵄨 = max 󵄨󵄨󵄨 max 󵄨󵄨󵄨󵄨 󵄨 0 J∈ℐ(n,q) 󵄨󵄨 ⋀ XJ (y) 󵄨󵄨󵄨 J∈ℐ(n,q) 󵄨󵄨󵄨 ⋀ XJ (γ(χ)) 󵄨󵄨󵄨 0 0 Since y ∈ BXJ (x0 , χ) was arbitrary, (A.63) follows. 0

For the remainder of this section, we fix χ ∈ (0, ξ] as in Proposition A.4.22. Lemma A.4.23. BXJ (x0 , χ) ⊆ M is open. 0

Proof. By Proposition A.4.22, X1 , . . . , Xn form a basis for the tangent space at every point of BXJ (x0 , χ). The result now follows from Lemma A.2.14. 0

Lemma A.4.24. For m ∈ ℕ, J ∈ ℐ (n, q), 󵄩󵄩 ⋀ X 󵄩󵄩 󵄩󵄩 J 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ⋀ XJ 󵄩󵄩󵄩 m 󵄩 0 󵄩C

XJ

≲(m−1)∨0 1. 0

(A.65)

(BXJ (x0 ,χ)) 0

Proof. We prove (A.65) by induction on m. The base case, m = 0, follows from Proposition A.4.22. Let m ≥ 1; we assume (A.65) for m − 1 and prove it for m. We have, directly from Definition A.1.1, 󵄩󵄩 ⋀ X 󵄩󵄩 󵄩󵄩 J 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ⋀ XJ 󵄩󵄩󵄩 m 󵄩 0 󵄩C X

J0

(BXJ (x0 ,χ)) 0

󵄩󵄩 ⋀ X 󵄩󵄩 󵄩 J 󵄩 󵄩󵄩 = 󵄩󵄩󵄩󵄩 󵄩 󵄩󵄩 ⋀ XJ0 󵄩󵄩󵄩C m−1 (B XJ X J0

0

n 󵄩 󵄩󵄩 ⋀ XJ 󵄩󵄩󵄩 󵄩󵄩 + ∑ 󵄩󵄩󵄩󵄩Xj 󵄩󵄩 X ⋀ 󵄩 󵄩 J 0 󵄩C m−1 (BX j=1 󵄩 (x0 ,χ)) X J J0

. 0

(x0 ,χ))

(A.66)

� 729

A.4 Proofs

The first term on the right-hand side of (A.66) is ≲(m−2)∨0 1 by the inductive hypothesis, so we focus only on the second term. We have, using Lemma A.4.19 and letting Cm,n ≥ 0 be a constant which depends only on m and n, 󵄩󵄩 ⋀ X 󵄩󵄩 󵄩󵄩 J 󵄩 󵄩󵄩 󵄩󵄩Xj 󵄩󵄩 ⋀ XJ 󵄩󵄩󵄩 m−1 󵄩 0 󵄩C X (BXJ J0

≤ Cm,n

0

∑

(x0 ,χ))

K∈ℐ0 (n,q)

+ Cm,n

∑

K ‖gj,J ‖C m−1 (BX

K∈ℐ0 (n,q)

XJ

0

J0

󵄩󵄩 󵄩 󵄩󵄩 ⋀ XK 󵄩󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 ⋀ XJ0 󵄩󵄩󵄩C m−1 (B XJ X

(x0 ,ξ)) 󵄩 󵄩

K ‖gj,J ‖ m−1 (BX 0 C XJ

0

J0

0

(x0 ,χ))

󵄩󵄩 ⋀ X 󵄩󵄩 󵄩󵄩 J 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 ⋀ XJ0 󵄩󵄩󵄩C m−1 (B XJ X

(x0 ,ξ)) 󵄩 󵄩

J0

J0

≲m−1 1,

0

󵄩󵄩 󵄩 󵄩󵄩 ⋀ XK 󵄩󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ⋀ XJ 󵄩󵄩󵄩 m−1 0 󵄩C (x0 ,χ)) 󵄩 X (BXJ J0

0

(x0 ,χ))

where the last inequality follows from the bounds described in (A.59) and the inductive hypothesis. This completes the inductive step and the proof. Lemma A.4.25. For 1 ≤ k ≤ q, 1 ≤ l ≤ n, there exist b̃ lk ∈ C(BXJ (x0 , χ)) such that 0

n

Xk = ∑ b̃ lk Xl ,

(A.67)

l=1

where for m ∈ ℕ, ‖b̃ lk ‖CXm

J0

(BXJ (x0 ,χ)) 0

≲(m−1)∨0 1.

(A.68)

Proof. For 1 ≤ k ≤ n this is trivial (take b̃ lk = 1 if k = l and b̃ lk = 0 if k ≠ l); however, the proof which follows deals with all 1 ≤ k ≤ q simultaneously. For 1 ≤ k ≤ q, 1 ≤ l ≤ n, let J(l, k) = (1, 2, . . . , l − 1, k, l + 1, . . . , n) ∈ ℐ (n, q). We have, by Cramer’s rule, n

Xk = ∑

⋀ XJ(l,k) ⋀ X J0

l=1

Xl .

Now, the result follows from Lemma A.4.24. ̂l ∈ C(BXJ (x0 , χ)) such that [Xi , Xj ] = ∑nl=1 ci,j ̂ l Xl , Lemma A.4.26. For 1 ≤ i, j, l ≤ n, ∃ci,j 0 where for m ∈ ℕ ̂ l ‖C m ‖ci,j X

J0

(BXJ (x0 ,χ)) 0

≲m 1.

730 � A Canonical coordinates Proof. For 1 ≤ i, j ≤ n, using (A.2) and Lemma A.4.25, we have q

n

q

k=1

l=1

k=1

k k ̃l [Xi , Xj ] = ∑ ci,j Xk = ∑ ( ∑ ci,j bk ) Xl . k ̃l ̂l := ∑qk=1 ci,j Setting ci,j bk , the result follows from Lemma A.4.25 and the definition of m-admissible constants (Definitions A.2.1 and A.2.2).

Proposition A.4.22 and Lemmas A.4.23 and A.4.26 show that the (already proved) case n = q of Theorem A.2.4 applies to X1 , . . . , Xn with ξ replaced by χ and M replaced by BXJ (x0 , χ).3 In light of Lemma A.4.26, any constants which are m-admissible in the sense 0 of this application of the case of n = q of Theorem A.2.4 are m-admissible in the sense of this proof. Thus, from the case n = q of Theorem A.2.4 we obtain 1-admissible constants ξ1 , η1 > 0 and a map Φ : Bn (η1 ) → BXJ (x0 , χ) as in that theorem. Most of the case q ≥ n of 0 Theorem A.2.4 has either already been shown above or follows immediately from this application of the case q = n. All that remains to show are (b), (c), there exists ξ2 > 0 as in (f), (h) for n + 1 ≤ j ≤ q, and the second equivalence in (i). Proof of (b). That 󵄨󵄨 ⋀ X (y) 󵄨󵄨 󵄨 󵄨󵄨 J 󵄨󵄨 ≥ 1, max 󵄨󵄨󵄨󵄨 J∈ℐ(n,q) 󵄨󵄨 ⋀ XJ (y) 󵄨󵄨󵄨 0

∀y ∈ BXJ (x0 , χ) 0

is clear (by taking J = J0 ). That 󵄨󵄨 ⋀ X (y) 󵄨󵄨 󵄨 󵄨󵄨 J 󵄨󵄨 ≲0 1, max 󵄨󵄨󵄨󵄨 J∈ℐ(n,q) 󵄨󵄨 ⋀ XJ (y) 󵄨󵄨󵄨 0

∀y ∈ BXJ (x0 , χ) 0

is Proposition A.4.22. Proof of (c). It follows from Proposition A.4.22 that X1 , . . . , Xn form a basis for the tangent space at every point of BXJ (x0 , χ). From this, (c) follows immediately from Lemma A.2.14. 0

That there exists a 1-admissible constant ξ2 > 0 as in (f) follows by applying the next lemma with ζ1 = ξ1 . Lemma A.4.27. Fix ζ1 ∈ (0, χ]. Then there is a 0-admissible constant ζ2 > 0 (which also depends on ζ1 ) such that BX (x0 , ζ2 ) ⊆ BXJ (x0 , ζ1 ). 0

Proof. Let ζ2 ∈ (0, ζ1 ]. We will pick ζ2 at the end of the proof. Suppose y ∈ BX (x0 , ζ2 ), so that there is an absolutely continuous γ : [0, 1] → BX (x0 , ζ2 ) with γ(0) = x0 , γ(1) = y, q γ′ (t) = ∑j=1 aj (t)ζ2 Xj (γ(t)), and ‖ ∑ |aj (t)|2 ‖L∞ ([0,1]) < 1. Let

3 When we proved Theorem A.2.4 in the case n = q, we took ξ = χ.

A.4 Proofs

� 731

t0 := sup{t ∈ [0, 1] : γ(t ′ ) ∈ BXJ (x0 , ζ1 /2), ∀t ′ ∈ [0, t]}. 0

We wish to show that if ζ2 = ζ2 (ζ1 ) > 0 is taken to be a sufficiently small 0-admissible constant, then we have t0 = 1 and y = γ(1) ∈ BXJ (x0 , ζ1 ). 0 In fact, we will prove γ(t0 ) ∈ BXJ (x0 , ζ1 /2). The result will then follow as if t0 < 1, 0 the fact that BXJ (x0 , ζ1 /2) is open (see (c)) and the fact that γ is continuous show that 0

γ(t ′ ) ∈ BXJ (x0 , ζ1 /2) for t ′ ∈ [0, t0 + ϵ) for some ϵ > 0, which contradicts the choice of t0 . 0 We turn to proving γ(t0 ) ∈ BXJ (x0 , ζ2 /2). We have, using Lemma A.4.25, 0

q

n

q

k=1 n

l=1

k=1

γ′ (t) = ∑ ak (t)ζ2 Xl (γ(t)) = ∑ ( ∑ ak (t)ζ2 b̃ lk (t)) Xl (γ(t)) ζ =: ∑ ã l (t) 1 Xl (γ(t)), 2 l=1 where by Lemma A.4.25, ‖ ∑ |ã l (t)|2 ‖L∞ ([0,1]) ≲0

ζ2 . ζ1

Thus, by taking ζ2 = ζ2 (ζ1 ) > 0 to be a

sufficiently small 0-admissible constant, we have ‖ ∑ |ã l (t)|2 ‖L∞ ([0,1]) < 1. It follows that for this choice of ζ2 > 0, we have γ(t0 ) ∈ BXJ (x0 , ζ1 /2), completing the proof. 0

Proof of (h) for n + 1 ≤ j ≤ q. The application of Theorem A.2.4 in the case n = q already established (h) for 1 ≤ j ≤ n. Set blk := b̃ lk ∘ Φ, where b̃ lk is as in Lemma A.4.25. Since Yj = Φ∗ Xj , we have, for any ordered multi-index α, Y α bkj = (X α b̃ kj ) ∘ Φ. It follows that for every m, ‖bkj ‖CYm

J0

(Bn (η1 ))

󵄩 󵄩 = ∑ 󵄩󵄩󵄩YJα0 bkj 󵄩󵄩󵄩C(Bn (η )) 1 |α|≤m

󵄩 󵄩 󵄩 󵄩 = ∑ 󵄩󵄩󵄩(XJα0 b̃ kj ) ∘ Φ󵄩󵄩󵄩C(Bn (η )) = ∑ 󵄩󵄩󵄩XJα0 b̃ kj 󵄩󵄩󵄩C(Φ(Bn (η ))) 1 1 |α|≤m

󵄩 󵄩 ≤ ∑ 󵄩󵄩󵄩XJα0 b̃ kj 󵄩󵄩󵄩C(B XJ |α|≤m

|α|≤m

0

(x0 ,χ))

= ‖b̃ kj ‖CXm

J0

(BXJ (x0 ,χ)) 0

(A.69)

≲(m−1)∨0 1,

where the last inequality uses (A.68). The first equivalence in (i) (which we have already established in the case n = q of Theorem A.2.4) shows that ‖bkj ‖C m (Bn (η1 )) ≈(m−1)∨0 ‖bkj ‖CYm

J0

(Bn (η1 )) .

(A.70)

Combining (A.69) and (A.70) shows that ‖bkj ‖C m (Bn (η1 )) ≲(m−1)∨0 1.

(A.71)

732 � A Canonical coordinates Pulling back (A.67) via Φ shows that n

Yj = ∑ blk Yl l=1

on Bn (η1 ).

(A.72)

The application of Theorem A.2.4 in the case n = q already established (h) for 1 ≤ j ≤ n. Combining this with (A.71) and (A.72) establishes (h) for n + 1 ≤ j ≤ q, completing the proof. Proof of (i). The case n = q of Theorem A.2.4 already established the first equivalence, so it suffices to show that, ∀m ∈ ℕ, ‖f ‖CYm

J0

(Bn (η1 ))

≤ ‖f ‖CYm (Bn (η1 )) ,

‖f ‖CYm (Bn (η1 )) ≲(m−1)∨0 ‖f ‖C m (Bn (η1 )) .

(A.73) (A.74)

Since YJ0 is a sublist of Y , (A.73) follows immediately from the definitions. The estimate (A.74) follows from (h), completing the proof.

A.4.4 Densities In this section, we prove the results from Section A.2.2. Recall the density Vol0 from (A.11): 󵄨󵄨 Z (x) ∧ Z (x) ∧ ⋅ ⋅ ⋅ ∧ Z (x) 󵄨󵄨 󵄨 󵄨󵄨 2 n Vol0 (x)(Z1 (x), . . . , Zn (x)) := 󵄨󵄨󵄨 1 󵄨. 󵄨󵄨 X1 (x) ∧ X2 (x) ∧ ⋅ ⋅ ⋅ ∧ Xn (x) 󵄨󵄨󵄨

(A.75)

Throughout this section, we use the same notation as in Theorem A.2.4. Lemma A.4.28. We have Vol0 (X1 , . . . , Xn ) ≡ 1 and for j1 , . . . , jn ∈ {1, . . . , q}, Vol0 (Xj1 , . . . , Xjn ) ≲0 1. Proof. It follows directly from the definition (A.75) that Vol0 (X1 , . . . , Xn ) ≡ 1. It follows from Theorem A.2.4 (b) that Vol0 (Xj1 , . . . , Xjn ) ≲0 1. Lemma A.4.29. Let V and W be n-dimensional real vector spaces and let A : W → V be an invertible linear transformation. Let v1 , . . . , vn be a basis for V and let w1 , . . . , wn ∈ W . Then Aw1 ∧ Aw2 ∧ ⋅ ⋅ ⋅ ∧ Awn w ∧ w ∧ ⋅ ⋅ ⋅ ∧ wn = −1 1 −12 . v1 ∧ v2 ∧ ⋅ ⋅ ⋅ ∧ vn A v1 ∧ A v2 ∧ ⋅ ⋅ ⋅ ∧ A−1 vn Proof. Let Z1 , Z2 be one-dimensional real vector spaces and let B : Z1 → Z2 be an invertible linear transformation. Let z1 ∈ Z1 and 0 ≠ z2 ∈ Z2 . We claim that Bz1 z = −11 . z2 B z2

(A.76)

A.4 Proofs

� 733

Indeed, let λ2 : Z2 → ℝ be any non-zero linear functional and set λ1 := λ2 ∘ B : Z1 → ℝ so that λ1 is a non-zero linear functional. We have Bz1 λ2 (Bz1 ) λ1 (z1 ) z1 = = = . z2 λ2 (z2 ) λ1 (B−1 z2 ) B−1 z2 Applying (A.76) in the case Z1 = ⋀n W , Z2 = ⋀n V , and B : Z1 → Z2 given by B(w1 ∧ w2 ∧ ⋅ ⋅ ⋅ ∧ wn ) = Aw1 ∧ Aw2 ∧ ⋅ ⋅ ⋅ ∧ Awn completes the proof. Lemma A.4.30. For 1 ≤ j ≤ n, LieXj Vol0 = fj0 Vol0 , where fj0 ∈ C(BXJ (x0 , χ)). Further0 more, for m ∈ ℕ, ‖fj0 ‖CXm

J0

(BXJ (x0 ,χ)) 0

≲m 1.

(A.77)

󵄨 Proof. Set ϕt (x) := etXj x so that LieXj Vol0 = 𝜕𝜕 󵄨󵄨󵄨t=0 ϕ∗t Vol0 . We write dϕt (x) to denote the t differential of ϕt (x) in the x variable. Using Lemma A.4.29, we have for t small (ϕ∗t Vol0 )(x)(Z1 , . . . , Zn ) = Vol0 (Φt (x))(dϕt (x)Z1 (x), . . . , dϕt (x)Zn (x)) 󵄨󵄨 dϕ (x)Z (x) ∧ ⋅ ⋅ ⋅ ∧ dϕ (x)Z (x) 󵄨󵄨 󵄨 1 t n 󵄨󵄨󵄨 = 󵄨󵄨󵄨 t 󵄨󵄨 X1 (ϕt (x)) ∧ ⋅ ⋅ ⋅ ∧ Xn (ϕt (x)) 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 Z1 (x) ∧ ⋅ ⋅ ⋅ ∧ Zn (x) 󵄨 󵄨󵄨 = 󵄨󵄨󵄨 󵄨 󵄨󵄨 dϕt (x)−1 X1 (ϕt (x)) ∧ ⋅ ⋅ ⋅ ∧ dϕt (x)−1 Xn (ϕt (x)) 󵄨󵄨󵄨 󵄨󵄨 Z (x) ∧ ⋅ ⋅ ⋅ ∧ Z (x) 󵄨󵄨 󵄨 󵄨󵄨 n = 󵄨󵄨󵄨 ∗ 1 󵄨󵄨 . ∗ 󵄨󵄨 ϕt X1 (x) ∧ ⋅ ⋅ ⋅ ∧ ϕt Xn (x) 󵄨󵄨

(A.78)

Fix x ∈ BXJ (x0 , χ). We claim that the sign of 0

Z1 (x) ∗ ϕt X1 (x)

∧ ⋅ ⋅ ⋅ ∧ Zn (x) ∧ ⋅ ⋅ ⋅ ∧ ϕ∗t Xn (x)

1 does not change for t small. To this end, let θ be a Cloc n-form which is non-zero near x. Since X1 (x)∧X( x)∧⋅ ⋅ ⋅∧Xn (x) ≠ 0 (Theorem A.2.4 (a)), we have θ(x)(X1 (x)∧⋅ ⋅ ⋅∧Xn (x)) ≠ 0, so by continuity, for t small, θ(x)(ϕ∗t X1 (x) ∧ ⋅ ⋅ ⋅ ∧ ϕ∗t Xn (x)) ≠ 0. We conclude that for t sufficiently small,

θ(x)(Z1 (x) ∧ ⋅ ⋅ ⋅ ∧ Zn (x)) Z1 (x) ∧ ⋅ ⋅ ⋅ ∧ Zn (x) = ϕ∗t X1 (x) ∧ ⋅ ⋅ ⋅ ∧ ϕ∗t Xn (x) θ(x)(ϕ∗t X1 (x) ∧ ⋅ ⋅ ⋅ ∧ ϕ∗t Xn (x)) does not change sign, and is either never zero or always zero. Set, for t small, ϵ := sgn

Z1 (x) ∗ ϕt X1 (x)

∧ ⋅ ⋅ ⋅ ∧ Zn (x) , ∧ ⋅ ⋅ ⋅ ∧ ϕ∗t Xn (x)

734 � A Canonical coordinates and in the case that the quantity inside sgn equals zero, the choice of ϵ does not matter. By the above discussion, ϵ does not depend on t (for t small). By (A.78), using the functions K gj,J from Lemmas A.4.18 and A.4.19, we have 𝜕 󵄨󵄨󵄨󵄨 ∗ 󵄨 (ϕ Vol0 )(x)(Z1 (x), . . . , Zn (x)) 𝜕t 󵄨󵄨󵄨t=0 t 𝜕 󵄨󵄨󵄨 󵄨󵄨󵄨 Z (x) ∧ ⋅ ⋅ ⋅ ∧ Zn (x) 󵄨󵄨󵄨󵄨 = 󵄨󵄨󵄨 󵄨󵄨󵄨 ∗ 1 󵄨 𝜕t 󵄨󵄨t=0 󵄨󵄨 ϕt X1 (x) ∧ ⋅ ⋅ ⋅ ∧ ϕ∗t Xn (x) 󵄨󵄨󵄨 Z (x) ∧ ⋅ ⋅ ⋅ ∧ Zn (x) 𝜕 󵄨󵄨󵄨 = 󵄨󵄨󵄨 ϵ ∗ 1 󵄨 𝜕t 󵄨t=0 ϕt X1 (x) ∧ ⋅ ⋅ ⋅ ∧ ϕ∗t Xn (x) θ(x)(Z1 (x) ∧ ⋅ ⋅ ⋅ ∧ Zn (x)) 𝜕 󵄨󵄨󵄨 = 󵄨󵄨󵄨 ϵ 𝜕t 󵄨󵄨t=0 θ(x)(ϕ∗ X1 (x) ∧ ⋅ ⋅ ⋅ ∧ ϕ∗ Xn (x)) t

t

θ(x)(Z1 (x) ∧ ⋅ ⋅ ⋅ ∧ Zn (x)) 𝜕 󵄨󵄨󵄨󵄨 ∗ ∗ = −ϵ 󵄨󵄨 θ(x)(ϕt X1 (x) ∧ ⋅ ⋅ ⋅ ∧ ϕt (x)Xn (x)) 2 θ(x)(X1 (x) ∧ ⋅ ⋅ ⋅ ∧ Xn (x)) 𝜕t 󵄨󵄨t=0 = −ϵ = −ϵ

θ(x)(Z1 (x) ∧ ⋅ ⋅ ⋅ ∧ Zn (x)) θ(x)(LieXj (X1 ∧ ⋅ ⋅ ⋅ ∧ Xn ))

θ(x)(X1 (x) ∧ ⋅ ⋅ ⋅ ∧ Xn (x)) θ(x)(X1 (x) ∧ ⋅ ⋅ ⋅ ∧ Xn (x)) Z1 (x) ∧ ⋅ ⋅ ⋅ ∧ Zn (x) LieXj (X1 ∧ ⋅ ⋅ ⋅ ∧ Xn ) X1 (x) ∧ ⋅ ⋅ ⋅ ∧ Xn (x) X1 (x) ∧ ⋅ ⋅ ⋅ ∧ Xn (x)

LieXj (X1 ∧ ⋅ ⋅ ⋅ ∧ Xn )

Vol0 (x)(Z1 (x), . . . , Zn (x)) X1 (x) ∧ ⋅ ⋅ ⋅ ∧ Xn (x) ⋀ XK (x) K = − ∑ gj,J (x) Vol0 (x)(Z1 (X), . . . , Zn (x)). 0 XJ0 (x) ⋀ K∈ℐ (n,q) =−

0

We conclude that fj0 = −

∑

K∈ℐ0 (n,q)

K gj,J 0

⋀ XK . ⋀ X J0

The estimate (A.77) now follows from Lemma A.4.24 and (A.59). Let σ0 := Φ∗ Vol0 , so that σ0 is a density on Bn (η1 ). Define a function h0 by σ0 = h0 σLeb , so that h0 ∈ Cloc (Bn (η1 )). Lemma A.4.31. We have h0 (t) = det(I + A(t))−1 , where A(t) is the matrix from Theorem A.2.4. In particular, h0 (t) ≈0 1, ∀t ∈ Bn (η1 ), and for m ∈ ℕ, ‖h0 ‖C m (Bn (η1 )) ≲m 1.

(A.79)

Proof. Because ‖A(t)‖𝕄n×n ≤ 21 , ∀t ∈ Bn (η1 ) (Theorem A.2.4 (g)), I + A(t) is invertible, and moreover, 󵄨󵄨 −1 󵄨 −1 󵄨󵄨det(I + A(t)) 󵄨󵄨󵄨 = det(I + A(t)) ≈0 1,

∀t ∈ Bn (η1 ).

(A.80)

A.4 Proofs

� 735

We have h0 (t) = σ0 (t)(𝜕t1 , . . . , 𝜕tn ) = σ0 (t)((I + A(t)) Y1 (t), . . . , (I + A(t)) Yn (t)) −1 󵄨 󵄨 = 󵄨󵄨󵄨det(I + A(t)) 󵄨󵄨󵄨σ0 (t)(Y1 (t), . . . , Yn (t)) −1

−1

= det(I + A(t)) Vol0 (Φ(t))(X1 (Φ(t)), . . . , Xn (Φ(t))) −1

= det(I + A(t)) . −1

It now follows from (A.80) that h0 (t) ≈0 1. Recall that YJ0 = (I + A)∇, so Theorem A.2.4 (h) implies that, ∀m ∈ ℕ, ‖A‖C m (Bn (η1 );𝕄n×n ) ≲m 1.

(A.81)

Combining (A.81) with the fact that ‖A(t)‖𝕄n×n ≤ 21 , ∀t ∈ Bn (η1 ) (Theorem A.2.4 (g)), shows that, ∀m ∈ ℕ, ‖(I + A)−1 ‖C m (Bn (η1 );𝕄n×n ) ≲m 1, and therefore ‖h0 ‖C m (Bn (η1 )) = ‖ det(I + A)−1 ‖C m (Bn (η1 )) ≲m 1. We turn to studying the density Vol from Section A.2.2; thus, we use the functions fj from (A.10). Because Vol0 is a non-zero density on BXJ (x0 , χ), there is a unique g ∈ 0 Cloc (BXJ (x0 , χ)) such that Vol = g Vol0 . 0

Lemma A.4.32. For 1 ≤ j ≤ n, Xj g = (fj − fj0 )g. Proof. Using Lemma A.4.30, we have fj g Vol0 = fj Vol = LieXj Vol = LieXj (g Vol0 )

= (Xj g) Vol0 +g LieXj Vol0 = (Xj g) Vol0 +gfj0 Vol0 .

The result follows. Proof of Theorem A.2.17 (i). Note that g(x0 ) = g(x0 ) Vol0 (x0 )(X1 (x0 ), . . . , Xn (x0 )) = Vol(x0 )(X1 (x0 ), . . . , Xn (x0 )), by definition, so it suffices to show that g(x) ≈{0;Vol} g(x0 ), for x ∈ BXJ (x0 , χ). 0 Fix x ∈ BXJ (x0 , χ), so that there exists an absolutely continuous γ : [0, 1] → 0

BXJ (x0 , χ) with γ(0) = x0 , γ(1) = x, γ′ (t) = ∑nj=1 aj (t)χXj (γ(t)), and ‖ ∑ |aj (t)|2 ‖L∞ ([0,1]) < 1. 0 We have, using Lemma A.4.32,

736 � A Canonical coordinates n n d g(γ(t)) = ∑ χaj (t)Xj g(γ(t)) = ∑ χaj (t)(fj (γ(t)) − fj0 (γ(t)))g(γ(t)). dt j=1 j=1

That is, g(γ(t)) satisfies a linear ODE. Solving this ODE we have 1

n

0

g(x) = g(γ(t)) = e∫0 ∑j=1 χaj (t)(fj (γ(s))−fj The estimate (A.77) implies ‖fj0 ‖C(BX

J0

(x0 ,χ))

(γ(s))) ds

g(x0 ).

(A.82)

≲0 1. Using this and the definition of {0; Vol}-

admissible constants (Definition A.2.15), (A.82) gives g(x) ≈{0;Vol} g(x0 ), completing the proof.

Proof of Theorem A.2.17 (ii). We prove the result by induction on m ∈ ℕ. The base case, m = 0, follows from Theorem A.2.17 (i). Let m ∈ ℕ+ . We assume the result for m − 1 and prove it for m. We have n

‖g‖CXm

J0

(BXJ (x0 ,χ)) 0

= ‖g‖C m−1 (BX XJ

0

J0

(x0 ,χ))

+ ∑ ‖Xj g‖C m−1 (BX j=1

XJ

0

J0

(x0 ,χ)) .

(A.83)

The first term on the right-hand side of (A.83) is ≲{m−1;Vol} |Vol(X1 , . . . , Xn )(x0 )| by the inductive hypothesis. For the second term, using Lemma A.4.32, we have, for a constant Cm,n ≥ 0 depending only on m and n, for 1 ≤ j ≤ n, ‖Xj g‖C m−1 (BX XJ

0

J0

(x0 ,χ))

= ‖(fj − fj0 )g‖C m−1 (BX XJ

0

J0

≤ Cm,n ‖fj − fj0 ‖C m−1 (BX XJ

0

(x0 ,χ)) J0

(x0 ,χ)) ‖g‖CXm−1 (BXJ (x0 ,χ))

󵄨 󵄨 ≲{m−1;Vol} 󵄨󵄨󵄨Vol(X1 , . . . , Xn )(x0 )󵄨󵄨󵄨,

J0

0

(A.84)

where the final inequality follows from the inductive hypothesis, (A.77), and the definition of {m − 1; Vol}-admissible constants (Definitions A.2.15 and A.2.16). Using (A.84) to bound the second term on the right-hand side of (A.83) completes the proof. Lemma A.4.33. Let h(t) be as in Theorem A.2.17. Then h(t) = h0 (t)g ∘ Φ(t). Proof. We have hσLeb = Φ∗ Vol = Φ∗ g Vol0 = (g ∘ Φ)Φ∗ Vol0 = (g ∘ Φ)h0 σLeb , completing the proof. Proof of Theorem A.2.17 (iii). Since h(t) = h0 (t)g ∘ Φ(t) (Lemma A.4.33), this follows from Theorem A.2.17 (i) and Lemma A.4.31. Proof of Theorem A.2.17 (iv). Using Theorem A.2.4 (i) and the fact that Yj = Φ∗ Xj , we have, for m ∈ ℕ,

A.4 Proofs

‖g ∘ Φ‖C m (Bn (η1 )) ≈(m−1)∨0 ‖g ∘ Φ‖CYm

J0

(Bn (η1 ))

󵄩 󵄩 󵄩 󵄩 = ∑ 󵄩󵄩󵄩YJα0 (g ∘ Φ)󵄩󵄩󵄩C(Bn (η )) = ∑ 󵄩󵄩󵄩XJα0 g 󵄩󵄩󵄩C(Φ(Bn (η ))) 1 1 |α|≤m

󵄩 󵄩 ≤ ∑ 󵄩󵄩󵄩XJα0 g 󵄩󵄩󵄩C(B |α|≤m

|α|≤m

XJ

(x0 ,χ))

0

� 737

= ‖g‖CXm

J0

(A.85)

(BXJ (x0 ,χ)) 0

󵄨 󵄨 ≲{(m−1)∨0;Vol} 󵄨󵄨󵄨Vol(X1 , . . . , Xn )(x0 )󵄨󵄨󵄨, where the final inequality uses Theorem A.2.17 (ii). Since h(t) = h0 (t)g ∘ Φ(t) (Lemma A.4.33), combining (A.85) and (A.79) completes the proof. Having completed the proof of Theorem A.2.17, we turn to Corollary A.2.18. To facilitate this, we introduce a corollary of Theorem A.2.4. Corollary A.4.34. Let η1 , ξ1 , ξ2 > 0 be as in Theorem A.2.4. There exist 1-admissible constants 0 < η2 ≤ η1 and 0 < ξ4 ≤ ξ3 ≤ ξ2 such that BX (x0 , ξ4 ) ⊆ BXJ (x0 , ξ3 ) ⊆ Φ(Bn (η2 /2)) ⊆ Φ(Bn (η2 )) 0

⊆ BXJ (x0 , ξ2 ) ⊆ BX (x0 , ξ2 ) ⊆ BXJ (x0 , ξ1 ) ⊆ Φ(Bn (η1 /2)) 0

0

⊆ Φ(Bn (η1 )) ⊆ BXJ (x0 , χ) ⊆ BX (x0 , ξ). 0

Proof. After obtaining η1 , ξ1 , and ξ2 from Theorem A.2.4, we apply Theorem A.2.4 again with ξ replaced by ξ2 to obtain η2 , ξ3 , and ξ4 as in the statement of the corollary. Proof of Corollary A.2.18. We have Vol(BXJ (x0 , ξ2 )) = 0

d Vol

∫ BXJ (x0 ,ξ2 ) 0

=

∫

dΦ∗ Vol =

Φ−1 (BXJ (x0 ,ξ2 ))

∫

h(t) dt

(A.86)

Φ−1 (BXJ (x0 ,ξ2 ))

0

0

≈{0;Vol} σLeb (Φ−1 (BXJ (x0 , ξ2 ))) Vol(X1 , . . . , Xn )(x0 ), 0

where the last estimate uses Theorem A.2.17 (iii). By Corollary A.4.34 and the fact that η1 , η2 > 0 are 1-admissible constants, we have 1 ≈1 σLeb (Bn (η2 )) ≤ σLeb (Φ−1 (BXJ (x0 , ξ2 ))) ≤ σLeb (Bn (η1 )) ≈1 1. 0

(A.87)

Combining (A.86) and (A.87) shows that Vol(BXJ (x0 , ξ2 )) ≈{1;Vol} Vol(X1 , . . . , Xn )(x0 ). The 0 same proof works with BXJ (x0 , ξ2 ) replaced by BX (x0 , ξ2 ), which completes the proof 0 of (A.12). The first two estimates in (A.13) follow from (A.12). All that remains to prove (A.13) is to show that

738 � A Canonical coordinates 󵄨󵄨 󵄨 󵄨󵄨Vol(X1 , . . . , Xn )(x0 )󵄨󵄨󵄨 ≈0

max

j1 ,...,jn ∈{1,...,q}

󵄨󵄨 󵄨 󵄨󵄨Vol(Xj1 , . . . , Xjn )(x0 )󵄨󵄨󵄨.

Using Lemma A.4.28, we have 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨Vol(X1 , . . . , Xn )(x0 )󵄨󵄨󵄨 = 󵄨󵄨󵄨g(x0 ) Vol0 (X1 , . . . , Xn )(x0 )󵄨󵄨󵄨 = |g(x0 )| 󵄨 󵄨 ≈0 |g(x0 )| max 󵄨󵄨󵄨Vol0 (Xj1 , . . . , Xjn )(x0 )󵄨󵄨󵄨 j1 ,...,jn ∈{1,...q}

= =

max

󵄨󵄨 󵄨 󵄨󵄨g(x0 ) Vol0 (Xj1 , . . . , Xjn )(x0 )󵄨󵄨󵄨

max

󵄨󵄨 󵄨 󵄨󵄨Vol(Xj1 , . . . , Xjn )(x0 )󵄨󵄨󵄨,

j1 ,...,jn ∈{1,...q} j1 ,...,jn ∈{1,...q}

completing the proof. A.4.5 Proof of Theorem 3.6.5 Theorem 3.6.5 follows easily from Theorems A.2.4 and A.2.17. We make a few comments on this. ∞ In Theorem 3.6.5 (b), Φ is claimed to be a Cloc diffeomorphism, while in Theo2 rem A.2.4 (e), Φ is only shown to be a Cloc diffeomorphism. However, in the setting of ∞ Theorem 3.6.5, since X1 , . . . , Xn are Cloc vector fields, classical theorems from the field of ∞ ODEs show that the map Φ given by (3.17) is Cloc . The function h in Theorem 3.6.5 is obtained by taking the function h from Theorem A.2.17 and dividing it by max

j1 ,...,jn ∈{1,...,q}

Vol(Xj1 , . . . , Xjn )(x0 ).

In the setting of Theorem 3.6.5, Vol is assumed to be strictly positive and therefore Corollary A.2.18 gives max

j1 ,...,jn ∈{1,...,q}

󵄨 󵄨 Vol(Xj1 , . . . , Xjn )(x0 ) ≈0 󵄨󵄨󵄨Vol(X1 , . . . , Xn )(x0 )󵄨󵄨󵄨.

(A.88)

Using (A.88), Theorem 3.6.5 (g) follows from Theorem A.2.17 (iii) and Theorem 3.6.5 (h) follows from Theorem A.2.17 (iv). The rest of Theorem 3.6.5 follows directly from Theorem A.2.4.

A.5 Further reading and references The first result similar to Theorem A.2.4 was due to Nagel, Stein, and Wainger [189] and was essentially the same as Theorem 3.3.7; around the same time similar results were proved by C. Fefferman and Sánchez-Calle [84]. The methods of [189] are based on studying the Taylor series in t of the map (A.4). This leads to several issues: the methods are

A.5 Further reading and references �

739

not sufficient for studying multi-parameter balls as in Section 3.5, they require that the vector fields be smooth in some given coordinate system, and they are not quantitatively invariant under a change of coordinates. Motivated by questions involving the smoothing properties of Radon transforms, Tao and Wright [232] studied two-parameter Carnot–Carathéodory balls, B(x, (δ1 , δ2 )), where δ1 , δ2 ∈ (0, 1] were assumed to be “weakly comparable,” i. e., δ1N ≲ δ2 and δ2N ≲ δ1 for some N. Even though, as was noted in that paper, the methods of [189] were sufficient to obtain the results they needed, Tao and Wright replaced a key step in the proof of [189] with the study of the ODE (A.20). That this ODE appears in canonical coordinates is classical (see [44, p. 155] for a similar ODE), though [232] seems to be the first reference which used it for the quantitative study of Carnot–Carathéodory balls, and we have followed their derivation. In [219], building on this idea of Tao and Wright, the remaining Taylor series methods from [189] were replaced with techniques from the field of ODEs. This allowed for the study of multi-parameter balls (without any weak-comparability hypothesis) as in Section 3.5, though these results still required the vector fields to be given in some coordinate system with good a priori estimates on their coefficients. Theorems A.2.4 and A.2.17 were proved in [228], and we have closely followed the proofs in that reference. [228] is the first paper on these coordinate systems where all 2 the main results are quantitatively invariant under arbitrary Cloc diffeomorphisms: in the previously mentioned references, estimates were given in terms of the C m norms of the coefficients of the vector fields in some fixed coordinate system. Thus, the vector fields had to be a priori “smooth” and “not large” in some fixed coordinate system; the concepts of vector fields being “smooth” and “not large” are not invariant under a C 2 change of coordinates. Theorems A.2.4 and A.2.17, on the other hand, remain completely 2 unchanged if one pushes the entire setting forward under a Cloc diffeomorphism. This is because the notions of admissible constants (in Definitions A.2.1, A.2.2, A.2.15, and A.2.16) do not depend on any choice of coordinate system. The results in [228] are a bit more involved than Theorems A.2.4 and A.2.17; they work with some more general function spaces which are useful for proving sharp results (see the next paragraph), but are not useful for the applications in this text. We refer the reader to [228] for the more general results proved there. Theorem A.2.4 “loses one derivative” and hence is not sharp. Indeed, one expects k the vector fields Y1 , . . . , Yq to be one derivative smoother than the functions ci,j , i. e., one expects Theorem A.2.4 (h) with ≲m replaced by ≲(m−1)∨0 . Unfortunately, as shown in [230], this is not true. However, if one chooses a different map Φ and works with Zygmund– Hölder spaces instead of C m spaces, then such an improved result is obtained in [224, 230], where methods from the area of elliptic PDEs are used to obtain sharp results. A result similar to Theorem A.2.4 in the real analytic category is obtained in [222]. 1 ∞ Unlike [189, 232], Theorem A.2.4 works with Cloc vector fields, instead of Cloc vector s fields; see [230] for similar results for vector fields which are C for some s > 0. Results

740 � A Canonical coordinates in the single-parameter case, similar to Theorem 3.3.7 but working with non-smooth vector fields, were obtained by Montanari and Morbidelli [174]. A result similar to Proposition A.4.1 in the case where ϵ was assumed to be small was obtained in [175, Appendix A]. A result very similar to Proposition A.4.8 can be found in [175, Theorem 4.5]. Theorems A.3.1 and A.3.3 are strengthened to a finite level of smoothness (in terms of Zygmund–Hölder spaces) in [224, 230] and an analogous result for real analyticity was obtained in [222]. These results are perhaps reminiscent of the celebrated results of DeTurck and Kazdan [63] regarding a coordinate system in which a Riemannian metric tensor has optimal regularity; for further comments on this, see [230]. Results like Theorem 3.3.7 have been used many times in the theory of several complex variables (see, e. g., [182, 140, 34, 188, 218, 40, 39, 38]). Because of this, it is often of interest to obtain a result like Theorem A.2.4, but where the map Φ is holomorphic, because if Φ is not holomorphic, rescalings will destroy the complex nature of the problem. Such general holomorphic coordinate systems were studied in [223].

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Symbol Index A t (W , ds) 15 A t (W , ds)⃗ 331 ⃗ ℂD1 , ℂD2 ) 369 – A t ((W , ds); t – Aloc (W , ds)⃗ 369 t ⃗ ℂD1 , ℂD2 ) 369 – Aloc ((W , ds); t A (Ω, (W , ds)) 267 – A1t (Ω, (W , ds)) 274 – A2t (Ω, (W , ds)) 274 – A3t (Ω, (W , ds), K ) 271 – A4t (Ω, (W , ds)) 272 ⃗ 268 A t (Ω, (W , ds)) ⃗ ℂD1 , ℂD2 ) 369 – A t (Ω, (W , ds); t ⃗ 275 – A1 (Ω, (W , ds)) t ⃗ 275 – A2 (Ω, (W , ds)) ̃t (W , ds)⃗ 331 A ̃t ((W , ds); ⃗ ℂD1 , ℂD2 ) 369 –A t ̃ ⃗ ℂD1 , ℂD2 ) 369 – A loc ((W , ds); t ̃ ⃗ – A loc (W , ds) 369 ̃t (Ω, (W , ds)) ⃗ 270 A ̃t (Ω, (W , ds); ⃗ ℂD1 , ℂD2 ) 369 –A t ̃ ⃗ 278 – A 1 (Ω, (W , ds)) t ̃ ⃗ 278 – A 2 (Ω, (W , ds)) BCH 118 s Bp,q 32 s – Bp,q (ℝn ) 60 s – Bp,q,std 428 s – Bp,q (𝒦, (W , ds)) 396 s ⃗ 400 – Bp,q (𝒦, (W , ds)) ⃗ ℂN ) 402 – B s (𝒦, (W , ds); p,q

B(W ,ds) (x, δ) 14 – B(W ,ds)⃗ (x, δ) 135 BW (x) 14

C(Ω) 32 ǧ 35 C ∞ 33 ∞ Cloc 33, 114 ∞ Cloc (M; V ) 114 ̂ S (ℝn ) 39 C ∞ (ℝn )⊗ ∞ n ̂ C (ℝ )⊗S0 (ℝn ) 40 ∞ Cloc (Ω)′ 33, 114 m C 32 m Cloc 33 C m,r – C m,r (ℝn ) 70 – C 0,s (M, (X, d)) 530 https://doi.org/10.1515/9783111085647-012

– C m,r (M, (X, d)) 530 C s 32 – C s,t (ℝn × ℝN ) 78 – C s,t (ℝn × ℝN ; ℂM ) 78 – C s (ℝn ) 70 – C s (ℝn ; X ) 70 s,t – Cstd,cpt (M × ℝN ) 514

s,t – Cstd (𝒦 × ℝN ) 513 s – Cstd,cpt 486 s – Cstd (𝒦) 486 s,t – Ccpt ((W , ds)⃗ ⊠ ∇ℝN ) 497 s,t – C (𝒦 × ℝN , (W , ds)⃗ ⊠ ∇ℝN ) 496 – C s,t (𝒦 × ℝN , (W , ds)⃗ ⊠ ∇ℝN ; ℂM ) 496 – C s (W , ds)⃗ 486 cpt

⃗ 486 – C s (𝒦, (W , ds)) s ⃗ ℂN ) 486 – C (𝒦, (W , ds); Cl s – Cls (ℝn ) 71 – Ĉls (Ω) 107 – Cls (M, (W , ds)) 516 – Ĉls (Ω, (W , ds), (X, d)) 675 CWL 276, 695 CW∞,loc (M) 276 CW∞,0 (M) 276 C0∞ 33, 114 ̂ C0∞ (Bq (a))⊗ ̂ S (ℝq ) 209 C0∞ (Ω)⊗ ∞ ∞ q ̂ ̂ ŜE (ℝq ) 211 C0 (Ω)⊗C0 (B (a))⊗ ̂ C0∞ (Bq (a))⊗ ̂ SE (ℝq ) 210 C0∞ (Ω)⊗ ∞ ′ C0 (Ω) 33, 114 Dx 31 deg – degds 4 – degd ⃗ 207, 215 – degd μ 206 △ 32 △lγ 516 Diff 70, 516 Dil 40 ⃗ δ d t 206 dμ μ δμ t 206

Dildδ (f ) 210 dist(W ,ds) (x, 𝜕Ω) 675 ⃗ 341 ⃗ (Z, dr)) D t (Ω, (Y , d̂), ⃗ ℂD1 , ℂD2 ) 370 ⃗ (Z, dr); – D t (Ω, (Y , d̂), ⃗

754 � Symbol Index

⃗ 341 ⃗ (Z, dr)) D t ((Y , d̂), t ⃗ ℂD1 , ℂD2 ) 370 ⃗ – D ((Y , d̂), (Z, dr); t ⃗ 369 ⃗ (Z, dr)) – Dloc ((Y , d̂), t ⃗ ℂD1 , ℂD2 ) 370 ⃗ (Z, dr); – Dloc ((Y , d̂), t ⃗ ̃ ⃗ D (Ω, (Y , d̂), (Z, dr)) 341 ⃗ ℂD1 , ℂD2 ) 370 ̃t (Ω, (Y , d̂), ⃗ (Z, dr); –D ⃗ 341 ̃t ((Y , d̂), ⃗ (Z, dr)) D ⃗ ℂD1 , ℂD2 ) 370 ̃t ((Y , d̂), ⃗ (Z, dr); –D t ⃗ 369 ̃ ⃗ – D loc ((Y , d̂), (Z, dr)) t ⃗ ℂD1 , ℂD2 ) 370 ̃ ⃗ – D ((Y , d̂), (Z, dr); loc

ej 25 etX x 116 s Fp,q 32 s – Fp,q (ℝn ) 60 s – Fp,q,std 428 s – Fp,q (𝒦, (W , ds)) 396 s ⃗ 400 – Fp,q (𝒦, (W , ds)) ⃗ ℂN ) 402 – F s (𝒦, (W , ds); p,q

ΓD (Ω, S) 187 Gen(𝒮) 124, 130 f ̂ 35 Hom 34

s

−∞

ℕ 2 ℕ+ 2 ‖t μ ‖μ 206 ̂ 39 ⊗ 𝒫δl (M, (X, d)) 516 Φ∗ 115 Φ∗ 115 +̇ 494 ≺ 43 ρ(W ,ds) (x, y) 14 S 33 S m 36, 207 Ŝ0 388

(s.c.) 241 S0 38 S (ℝn )′ 33 ⋐ 4 ⊠ 131 T ∗ 35

ℐ(n, q) 696 ℐ0 (n, q) 696 ‖ ⋅ ‖ L2

̃ ⃗ ℳ (X,d),𝒦,a 167 Mult[ϕ] 43

241

Λ 37 λstd 432 Λstd 432 (l.c.) 241 ≲ 45 Lie 117 L∞ k ([0, l]) 515 Lp 35 Lp (M, μ; ℓq (𝒜)) 60 p Ls 31, 37 ℓq (𝒜; Lp (M, μ)) 60 𝕄D1 ×D2 59 ℳ 63 ℳ(W ,ds)⃗ 175

V 396, 400 ∨ 40 ‖ ⋅ ‖V ,s,ℰ 397, 400 Wκ1 629 ∧ 40 Xs s – Xstd,cpt 428 s – Xstd (𝒦) 428, 433 s – Xcpt (W , ds)⃗ 415 – X s (𝒦, (W , ds)) 397 ⃗ 400 – X s (𝒦, (W , ds)) s ⃗ ℂN ) 402 – X (𝒦, (W , ds); ‖ ⋅ ‖X s – ‖ ⋅ ‖X s (W ,ds) 398 – ‖ ⋅ ‖X s (W ,ds)⃗ 401 – ‖ ⋅ ‖X s (W ,ds),(X, 479 ⃗ ⃗ d),a,σ,h – ‖ ⋅ ‖X s (W ,ds),(X, ⃗ ⃗ d),a,Ω ,Ω ,Ω ,Vol 477 0

1

2

Index admissible constant – L, Vol-admissible constant 140 – L-admissible constant 140 – multi-parameter unit-admissible constant 195 – L -unit-admissible constant 551 – unit-admissible constant 122 approximately commute – strongly 149 – weakly 150 Baker–Campbell–Hausdorff formula 118 Besov space 32, 60, 396, 400 bounded set – (W , ds) bump functions 271 – (W , ds) elementary operators 273 – (W , ds) pre-elementary operators 273 – (W , ds)⃗ elementary operators 275 – (W , ds)⃗ pre-elementary operators 275 – elementary operators 48 – generalized (W , ds)⃗ elementary operators 278 – generalized (W , ds)⃗ pre-elementary operators 277 – locally (W , ds) pre-elementary operator 371 – locally convex topological vector space 33 – pre-elementary operators 48 – pre-pseudo-differential operator scales 223 – pseudo-differential operator scales 223 calculus of pseudo-differential operators 36 Calderón Reproducing Formula 62, 482 canonical coordinates 698 Carnot–Carathéodory – ball 14 – distance 14 characteristic polynomial 31 control – locally strong control 188 – locally strong λ-control 188 – locally weak control 188 – locally weak λ-control 188 – strong control 188 – strong λ-control 188 – weak control 188 – weak λ-control 188 coordinate system 114 Cotlar–Stein lemma 57 density 115 – Lebesgue 115 https://doi.org/10.1515/9783111085647-013

– smooth 116 – strictly positive 116 distribution – on ℝn 33 – on a manifold 114 – tempered 33 – with compact support 33, 114 elementary operator 48 – (W , ds) elementary operator 273 – (W , ds)⃗ elementary operator 275 – generalized (W , ds)⃗ elementary operator 278 elliptic 31, 87 – nonlinear 90 equivalent sets of vector fields – locally strong 188 – locally weak 188 – strong 188 – weak 188 exponential map 116 filtration – finitely generated 187 – Lie algebra 187 – Lie algebra filtration of vector fields 187 – locally finitely generated 187 – module 187 – vector fields 187 finitely generated 129 – locally 130 foliation 120 formal adjoint 35 formal degree 4 Fourier transform 35 free nilpotent Lie algebra 256 Frobenius Theorem 120 graded – Lie algebra 254 – Lie group 255 Hölder space 32, 70, 530 homogeneous – differential operator 250 – distribution 251 homogeneous dimension 250 homogeneous norm 206 Hörmander vector fields with formal degrees 4

756 � Index

Hörmander’s condition 2 hypoelliptic 46 immersion 119 injectively immersed submanifold 119 inverse function theorem – Banach Space 22 – Nash–Moser 24 involutive 119 kernels of order m 37 Khintchine Inequality 67 – Multi-parameter 468 λ is sharp 448 Laplacian 32 leaf 120 Lie derivative 117 linearly finitely generated 130 – locally 130 locally a pseudo-differential operator 59 locally a singular integral operator 59 locally approximately commute – strongly 149 – weakly 150 locally finitely generated 119 maximal function 63, 167, 175 maximally subelliptic 5, 534, 605 – nonlinear 10, 619 – of type 2 551, 579 Monge–Ampère equation 10, 687, 689 nilpotent – Lie algebra 254 – Lie group 254 ordered multi-index notation 4 pre-elementary operator 48 – (W , ds) pre-elementary operator 273 – (W , ds)⃗ pre-elementary operator 275 – generalized (W , ds)⃗ pre-elementary operator 277 – locally (W , ds) pre-elementary operator 371

pre-pseudo-differential operator scales 223 principal symbol 31 pseudo-differential operator – on ℝn 35 – on a manifold 338, 354 – with respect to Carnot–Carathéodory geometries 207 pseudo-differential operator scales 223 pullback 115 pushforward 115 𝒮 partial differential operator 177 Schauder Estimates 622, 627, 684 Schwartz kernel 34 Schwartz kernel theorem 34 Schwartz space 33 – all of whose moments vanish 38 singular integral operator 47 singular integrals – general multi-parameter 269 – multi-parameter Hörmander 268 – single-parameter 267 singular point 120 Sobolev space 31, 37, 470 Square Function 364 sub-Laplacian 6, 239 subelliptic 45 symbol – of a differential operator 31 – of a pseudo-differential operator 35 – of an (X, d)⃗ pseudo-differential operator 207 – standard 36 tame estimate 75, 79, 498 topology of bounded convergence 34 Triebel–Lizorkin space 32, 60, 396, 400 vector bundle 604 weighted interior norm 107, 675 Zygmund–Hölder space 32, 70, 486, 516 – product Zygmund Hölder Space 78, 496

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