Hörmander Operators 9811261687, 9789811261688

Table of contents :
Dedication
Contents
Foreword
Introduction
1. Basic geometry of vector fields
2. Function spaces defined by systems of vector fields
3. Homogeneous groups in ℝᴺ
4. Hypoellipticity of sublaplacians on Carnot groups
5. Hypoellipticity of general Hörmander operators
6. Fundamental solutions of Hörmander operators
7. Real analysis and singular integrals in locally doubling metric spaces
8. Sobolev and Hölder estimates for Hörmander operators on groups
9. More geometry of vector fields: metric balls and equivalent distances
10. Lifting and approximation
11. Sobolev and Hölder estimates for general Hörmander operators
12. Nonvariational operators constructed with Hörmander vector fields
Appendix: A Short summary of distribution theory
Bibliography
Index

Citation preview

Hörmander Operators

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Hörmander Operators

Marco Bramanti Politecnico di Milano, Italy

Luca Brandolini

Università degli Studi di Bergamo, Italy

World Scientific NEW JERSEY

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BEIJING

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HÖRMANDER OPERATORS Copyright © 2023 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

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Contents

Foreword

xi

Introduction

xiii

0.1 0.2 0.3

xiii xviii xxii

1.

Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scope and structure of the book . . . . . . . . . . . . . . . . . . . Why study H¨ ormander operators? . . . . . . . . . . . . . . . . .

Basic geometry of vector fields

1

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

2.

3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exponentials and commutators of vector fields . . . . . . . . . . . Lie algebras, H¨ ormander’s condition, H¨ormander operators . . . . The control distance . . . . . . . . . . . . . . . . . . . . . . . . . The weighted control distance . . . . . . . . . . . . . . . . . . . . Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other properties related to connectivity . . . . . . . . . . . . . . Maximum principles for degenerate elliptic operators . . . . . . . Propagation of maxima and strong maximum principle for sum of squares operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Propagation of maxima for operators with drift . . . . . . . . . . 1.11 Some examples of explicit computations with the control distance 1.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 48 56 65

Function spaces defined by systems of vector fields

67

2.1 2.2 2.3

67 80 90

Sobolev spaces induced by vector fields . . . . . . . . . . . . . . . H¨ older spaces induced by H¨ ormander vector fields . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 8 17 21 24 35 37

Homogeneous groups in RN

93

3.1

94

Homogeneous groups . . . . . . . . . . . . . . . . . . . . . . . . . vii

viii

H¨ ormander operators

3.2

Homogeneous Lie algebras of invariant vector fields on a homogeneous group . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Exponential maps on a homogeneous group . . . . . . . . . . . . 3.4 Convolution and mollifiers on a homogeneous group . . . . . . . . 3.5 Homogeneous stratified Lie groups and Lie algebras, and their control distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Connectivity matters and Poincar´e inequality on stratified groups 3.7 Weak solutions to Dirichlet problems for divergence form equations structured on vector fields . . . . . . . . . . . . . . . . 3.8 Homogeneous stratified Lie algebras and Lie groups of type II . . 3.9 Distributions on homogeneous groups . . . . . . . . . . . . . . . . 3.10 Examples of homogeneous groups and homogeneous H¨ormander operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.

Hypoellipticity of sublaplacians on Carnot groups 4.1 4.2 4.3 4.4 4.5 4.6

5.

6.

6.2

143 150

153

. . . . .

156 163 174 184 190 191

. . on . . . . . . . . . . . . . . . . . .

. . .

191

. . . . . . . . .

192 195 199 217 229 232 238 242 245

. . . . . . . . .

. . . . . . . . .

Fundamental solutions of H¨ ormander operators 6.1

132 135 141

.

Hypoellipticity of general H¨ ormander operators Introduction . . . . . . . . . . . . . . . . . . . . . . . . . The Fourier transform on the Schwartz space S(Rn ) and tempered distributions . . . . . . . . . . . . . . . . . . . 5.3 Fractional order Sobolev spaces . . . . . . . . . . . . . . 5.4 Some classes of operators on S(Rn ) . . . . . . . . . . . . 5.5 Subelliptic estimates . . . . . . . . . . . . . . . . . . . . 5.6 Localized subelliptic estimate . . . . . . . . . . . . . . . 5.7 Hypoellipticity of H¨ ormander operators . . . . . . . . . 5.8 Uniform subelliptic estimates . . . . . . . . . . . . . . . 5.9 Some applications of the subelliptic estimates . . . . . . 5.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 128

153

Introduction, statement of the main results and strategy of the proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation and preliminary facts about Sobolev spaces and finite differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regularity estimates for the canonical sublaplacian . . . . . . . Hypoellipticity of the canonical sublaplacian . . . . . . . . . . . General sublaplacians and uniform estimates . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1 5.2

106 116 118

Fundamental solutions and solvability of general H¨ormander operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Homogeneous H¨ ormander operators . . . . . . . . . . . . . . . . .

247 248 254

Contents

6.3 6.4 6.5 6.6 7.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Locally doubling metric spaces . . . . . . . . . . . . . . . . . Localized kernels of singular and fractional integrals . . . . . Singular and fractional integrals on H¨older spaces . . . . . . . L2 continuity of singular integrals via continuity on C α . . . Local maximal function and fractional integrals on Lp spaces Calder´ on-Zygmund theory in locally doubling metric spaces . Integral characterization of H¨ older continuity . . . . . . . . . Some geometric results . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.3 8.4 8.5 8.6 8.7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Homogeneous kernels on G, fractional integrals and Sobolev embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . Singular integrals associated to homogeneous kernels of type Global Sobolev estimates . . . . . . . . . . . . . . . . . . . . Local Sobolev estimates . . . . . . . . . . . . . . . . . . . . H¨ older estimates for solutions of Lu = f . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

263 268 285 288 291

. . . . . . . . . .

. . . . . . . . . .

Sobolev and H¨ older estimates for H¨ ormander operators on groups 8.1 8.2

9.

Existence of a global homogeneous fundamental solution and uniform estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of the global fundamental solution . . . . . . . . . . . Some explicit examples of fundamental solutions on homogeneous groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Real analysis and singular integrals in locally doubling metric spaces 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10

8.

ix

291 295 298 301 309 312 318 327 331 335 337

. . .

337

. 0 . . . .

345 354 360 370 378 397

. . . . . .

. . . . . .

More geometry of vector fields: metric balls and equivalent distances

399

9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9

399 406 408 420 437 458 462 466 477

Introduction and statement of the main results . . . . . . . . . . Dependence of the constants . . . . . . . . . . . . . . . . . . . . . The Baker-Campbell-Hausdorff formula . . . . . . . . . . . . . . Suboptimal bases and their properties . . . . . . . . . . . . . . . Structure of metric balls . . . . . . . . . . . . . . . . . . . . . . . Local equivalence of the distances d, d∗ . . . . . . . . . . . . . . . Segment properties and the global doubling condition . . . . . . Proof of the BCH formula for formal series and other consequences Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10. Lifting and approximation 10.1 Motivation and statement of the main results . . . . . . . . . . .

479 479

x

H¨ ormander operators

10.2 Lifting of H¨ ormander vector fields . . . . . . . . . . . 10.3 Approximation of free vector fields with left invariant homogeneous vector fields . . . . . . . . . . . . . . . 10.4 Some geometry of free lifted vector fields . . . . . . . 10.5 Abstract free Lie algebras and Lie groups . . . . . . 10.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

486

. . . .

498 513 523 533

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

11. Sobolev and H¨ older estimates for general H¨ormander operators 11.1 11.2 11.3 11.4 11.5

Introduction and general overview . . . . . . . . . . . . . . . . . Operators of type λ . . . . . . . . . . . . . . . . . . . . . . . . . . Parametrix and representation formulas . . . . . . . . . . . . . . Continuity of operators of type λ . . . . . . . . . . . . . . . . . . A priori estimates in Sobolev or H¨ older spaces for solutions to Lu = f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Smoothing of distributional solutions and solvability in H¨older or Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12. Nonvariational operators constructed with H¨ormander vector fields 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operators of type λ and representation formulas . . . . . . . . . H¨ older estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . Lp estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lp continuity of variable operators of type 0 and their commutators Regularization of solutions . . . . . . . . . . . . . . . . . . . . . . Proof of the estimates on spherical harmonics . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Appendix A

Short summary of distribution theory

535 535 542 560 566 577 605 611 613 613 619 627 631 635 646 654 664 667

Bibliography

681

Index

691

Foreword

In the theory of linear second order partial differential equations with nonnegative characteristic form, the study of H¨ ormander operators occupies a preeminent position, both because they arise in many different theoretical and applied areas, and because of the richness of the results obtained up to the present time. Several monographs and survey memoirs concerning H¨ ormander operators are today available in the literature. However, most of them only consider and illustrate particular, even though important, operators such as sub-Laplacians or heat-type operators. Moreover, the valuable monograph by Olejnik and Radkevic, the only one concerning general linear second order PDEs with nonnegative characteristic form, dates back to many years ago and does not take into account the remarkable geometric structures underlying the relevant operators which today provide many important tools to develop the broad and profound theory of H¨ormander’s PDEs. To cover this gap in the literature is no easy feat. On the one hand, the theory of H¨ ormander operators today involves at a very deep level geometries of nonRiemannian type, Lie groups and Lie algebras of vector fields, harmonic and Fourier analysis, distribution theory, and methodologies and techniques that arise from the classical theory of partial differential equations. All these prerequisites can be found in specialized papers and textbooks, but without a self-contained, ad hoc presentation of this big amount of background material, any book on H¨ormander operators would not be easily readable, both by graduate students, and by young and even senior researchers without strong skills in all the above topics. On the other hand, the core of the theory consists in ideas, techniques and results which are often difficult to read in the original works, both for the depth of the original mathematical ideas, and for the highly nontrivial details whose proofs are often left to the readers. The authors have taken up the major challenge of writing a monograph presenting a comprehensive theory of H¨ ormander operators, together with a self-contained exposition of all the algebraic, geometric, analytical prerequisites and tools, an exposition which can be readable and usable even by not experienced people in each subject. A monograph capable to lead the young researchers, and to help the senior ones, to do active research in the area. xi

xii

H¨ ormander operators

With their necessarily vast monograph, I can say that the authors have completely achieved their aims, by composing a very well written and extremely clear book. The treatise starts with a general Introduction, split into three parts. The first one is an historical Prologue on the notion of hypoellipticity. This is a very appropriate start: the milestone of the whole theory, indeed, is the celebrated H¨ ormander’s Theorem giving a sufficient condition — the so-called H¨ormander’s rank condition — for the hypoellipticity of a linear second order partial differential operator “sum of squares of vector fields, plus a drift”. The second part of the Introduction illustrates and comments on the general aims and contents of the book; this is a useful guide for the reader. The third part describes several important motivations for studying H¨ ormander operators, and then discusses the need to investigate the geometry of systems of vector fields, to understand the deep meaning of H¨ ormander’s rank condition. The introductions and the final notes to each chapter of the book are another extremely useful guide for the reader to understand motivations, history, ideas and methodologies of each topic presented and analyzed in the treatise. While the monograph is very valuable in its entirety, some of its parts deserve explicit mentions. The Chapter “Basic geometry of vector fields” has an interest in its own right to familiarize readers with some crucial concepts in the general theory of second order PDEs with nonnegative characteristic form: the notions of connectivity, control (or Carnot Carath´eodory) distance, propagation of maxima. The proof of the hypoellipticity of sub-Laplacians — Chapter 4 — is an original procedure due to the authors. It does not use pseudodifferential operator theory as in the celebrated J. J. Kohn’s proof of H¨ ormander’s Theorem. The one presented here strongly relies on algebraic devices, and enlightens the deep role of the Lie groups structures in studying regularity properties of solutions to left invariant homogeneous PDEs. The a priori estimates in Sobolev-type and H¨older-type spaces (Chapters 8, 11, 12), a crucial target for the authors to study the regularity properties of H¨ ormander operators, are sharp original versions of some analogous ones already present in the literature. Finally, Chapter 10 is devoted to the Rothschild and Stein lifting and approximation procedure, a set of algebraic-geometric tools allowing to connect general H¨ ormander operators with the ones on homogeneous Lie groups. These are powerful and sophisticated instruments, requiring much labor to be understood. The clear, detailed and complete description of this procedure is another merit of the authors of this book. Ermanno Lanconelli University of Bologna, Italy

Introduction

This introduction consists in three sections: Prologue. A soft introduction to the main themes of the book, where the subject matter is put into its historical perspective. Scope and structure of the book. A more detailed presentation of the aims, style and structure of the book. It should help the reader to realize what he/she can or cannot find in this book, and how the book can be used. Why study H¨ ormander operators? An informal discussion of some problems, coming from both theoretical and applied mathematics, which lead to the study of H¨ ormander operators, and geometry of systems of H¨ormander vector fields. 0.1

Prologue

The beginning of our story is the following result known as Weyl’s Lemma1 : Lemma 0.1 Let Ω be an open subset of Rn and let u be a locally integrable function in Ω. If for every smooth function ϕ with compact support in Ω we have Z u (x) ∆ϕ (x) dx = 0, Ω

P

∂x2i xi

where ∆ = is the ordinary Laplacian, then u is smooth in Ω and satisfies pointwise ∆u (x) = 0. The significant part of the above result is the assertion that u is smooth. Indeed, if u has some regularity, say u ∈ C 2 (Ω), integrating by parts we obtain that for every smooth ϕ Z ∆u (x) ϕ (x) dx = 0. Ω

This shows that u is harmonic in Ω and therefore smooth. 1 The result is usually credited to Hermann Weyl [158], 1940. Actually, already in 1934 Renato Caccioppoli [44] proved it in dimension 2, with a proof which actually works in any dimension.

xiii

xiv

H¨ ormander operators

This peculiar regularizing property of the Laplacian should be compared with the fact that in general the solutions of a partial differential equation need not be regular. For example, the solution of the vibrating string equation u = ux1 x1 − ux2 x2 = 0, can be expressed using d’Alembert formula as u (x1 , x2 ) = F (x1 + x2 ) + G (x1 − x2 ) where F and G are arbitrary C 2 functions. This means that there exist solutions of u = 0 that are only C 2 (or even less regular if we admit derivatives in weak sense). In his treatise on distributions Laurent Schwartz [149] coined the term hypoelliptic operator to identify those differential operators that have a strong regularizing property. More precisely he gave the following: Definition 0.2 We say that a linear differential operator L with smooth coefficients is hypoelliptic if for every distribution u such that Lu is smooth in an open subset of Rn then also u is smooth in the same subset. Actually in the first edition of his treatise (1950-1951) Schwartz used the word elliptic to identify this property, but when he revised the book for reprinting he realized (as himself noted, see [149, p. 143]) that this word generates too much confusion and he switched to the term hypoelliptic. In view of the above discussion, the wave operator  is not hypoelliptic, while a simple modification of Weyl’s Lemma that takes into account distributions allows to show that the ordinary Laplace operator ∆ is actually hypoelliptic. In 1955, in his Ph.D. Thesis [105], Lars H¨ ormander characterized constant coefficient differential operators that admit a fundamental solution which is smooth outside the origin. Since this fact is actually equivalent to hypoellipticity2 , H¨ormander unwittingly characterized hypoelliptic constant coefficient differential operators. A few years later, Bernard Malgrange [123] and H¨ormander [106] independently gave a sufficient condition for a differential operator with variable coefficients to be hypoelliptic: an operator L is hypoelliptic if, for every point x0 , the constant coefficient operator L (x0 ) obtained from L “freezing” its coefficient at x0 is hypoelliptic, and operators L (x0 ) frozen at different points are equivalent, in a suitable sense. As we shall see in a moment this condition is far from being necessary. Even though this result is technically complicated, the operators considered are essentially perturbations of hypoelliptic constant coefficient operators: the variable coefficients do not play any role in granting the hypoellipticity of these operators. By contrast, let us consider for example the second order differential operator 2

2

L = (∂x + 2y∂t ) + (∂y − 2x∂t ) .

(0.1)

2 The fact that an operator L possessing a fundamental solution smooth outside the pole is hypoelliptic was proved by Schwartz [149, p. 143, Th. XII].

Introduction

xv

defined in R3 . Freezing its coefficients at the origin we obtain the operator 2 2 L0 = ∂xx + ∂yy

that does not contain any derivative with respect to the variable t. This operator cannot be hypoelliptic since L0 u = 0 for every function u which is independent of x and y. Nevertheless, Gerald Folland [84] explicitly constructed a fundamental solution, which is smooth outside the pole. Therefore L is hypoelliptic. Namely, the fundamental solution of L with pole at the origin is c Γ (x, y, t) = q . 2 (x2 + y 2 ) + t2 So, for this operator the variable coefficients are essential for the hypoellipticity. A turning point in the study of this kind of operators came in 1967 when H¨ ormander, in his celebrated paper [107], gave an almost complete description of second order hypoelliptic differential operators with real smooth coefficients. First of all he noted that if the second order differential operator X X L= aij (x) ∂x2i xj + bi (x) ∂xi + c (x) (0.2) i,j

P is hypoelliptic, then at any point x the quadratic form i,j aij (x) ξi ξj is semidefinite. This condition is not sufficient to guarantee hypoellipticity, however it implies a special form for L. More precisely, in any open set Ω ⊂ Rn where the rank of the above quadratic form is constant we can always write ±L =

q X

Xj2 + X0 + c

j=1

for suitable real smooth vector fields X0 , X1 , . . . , Xq defined in Ω and a smooth function c. To ensure the hypoellipticity for operators in this form, H¨ormander introduced an algebraic condition on the iterated commutators of the vector fields X0 , X1 , . . . , Xq . To explain this condition, let us return to the above example 2

2

L = (∂x + 2y∂t ) + (∂y − 2x∂t ) that we rewrite in the form L = X12 + X22 with X1 = ∂x + 2y∂t ,

X2 = ∂y − 2x∂t .

Observe that [X1 , X2 ] = X1 X2 − X2 X1 = −4∂t . In other words the derivative with respect to the variable t that is missing when the coefficients of the operator L are frozen at the origin can be recovered using the commutator [X1 , X2 ]. The fact that this simple geometrical property implies the hypoellipticity for L is hard to be proved, and it is the content of H¨ormander’s celebrated result:

xvi

H¨ ormander operators

Theorem 0.3 (H¨ ormander’s hypoellipticity theorem) Let X0 , X1 , . . . , Xq be real smooth vector fields in a domain Ω of Rn . Let L=

q X

Xj2 + X0 + c

(0.3)

j=1

and assume that at any given point of Ω, among the iterated commutators Xj1 , [Xj1 , Xj2 ] , [Xj1 , [Xj2 , Xj3 ]] , . . . , where ji = 0, . . . , q, there exist n which are linearly independent. Then L is hypoelliptic in Ω. The above condition can be reformulated in a more algebraic fashion saying that the Lie algebra spanned by the vector fields X0 , X1 , . . . , Xq at any point has dimension n. This condition is actually “almost necessary” for hypoellipticity. Namely, if the Lie algebra spanned by Xj1 , [Xj1 , Xj2 ], [Xj1 , [Xj2 , Xj3 ]] , . . . , has constant rank k < n in some open set, by Frobenius theorem (see Chapter 1, Theorem 1.76) it is possible to find a change of variables so that in the new coordinates y1 , y2 , . . . , yn the operator L only acts on the variables y1 , . . . , yk . If, moreover, the operator does not contain a zero order term, any function u independent of such variables satisfies Lu = 0, so that L cannot be hypoelliptic. It is worth noting that in H¨ ormander’s approach the presence of variable coefficients is essential to ensure the hypoellipticity of the operator. If the coefficients were constant, all the commutators would vanish and the rank of the Lie algebra spanned by X0 , X1 , . . . , Xq would be less than or equal to q + 1. Therefore, when q + 1 < n a constant coefficient operator of type (0.3) cannot be hypoelliptic. A system of smooth vector fields defined in an open subset Ω in Rn and satisfying there the assumption of H¨ ormander’s theorem (which is now labelled “H¨ormander’s condition”) is called a system of H¨ ormander vector fields, and, under this assumption, an operator L of the kind (0.3) is called a H¨ ormander operator. This book is mainly devoted to the study of the properties of H¨ormander operators. From the point of view of the classification of PDEs, H¨ormander operators can be seen as possibly degenerate elliptic-parabolic operators. However, this “degeneracy” label does not do justice to their properties, which present differences but also deep similarities with those of uniformly elliptic or parabolic operators. Some of these similarities and differences are well illustrated by the regularizing properties of these classes of operators. A standard proof of H¨ormander’s theorem (although not the original one), due to Joseph Kohn [117] (see also [116]) and Olga Ole˘ınik and Evgeniy Radkeviˇc [137] consists in proving the so-called subelliptic estimates: there exists ε ∈ (0, 1) such that if u is a distribution satisfying the equation Lu = f and f is locally in the fractional Sobolev space H k for some k, then u is locally in H k+ε . Although this fact expresses a rather poor degree of regularization (for uniformly elliptic operators we would have 2 instead of ε), it is clearly enough

Introduction

xvii

to ensure the hypoellipticity of L. On the other hand, the small ε appearing in subelliptic estimates suggests that the properties of H¨ormander operators are not fully revealed until we make use of Cartesian derivatives and “isotropic” Sobolev spaces. A series of papers published in 1974-1976 by Folland and Elias Stein [91], Folland [85], and Linda Preiss Rothschild and Stein [142], shows that the regularizing effect of a H¨ ormander operator can be analyzed in a scale of Sobolev spaces defined replacing the Cartesian derivatives ∂xi with derivatives with respect to vector fields Pq Xi : if u is a solution to the equation Lu = f where L = j=1 Xj2 is a sum of squares of H¨ ormander vector fields, then u has two derivatives more than f , with respect to the vector fields X1 , X2 , . . . , Xq . One of the main features of this theory is an intimate connection between the properties of H¨ ormander operators and the geometry of vector fields. A cornerstone result in this area was proved independently by Wei-Liang Chow [61] in 1939 and Petr Rashevski˘ı [140] in 1938: even though the vector fields Xi are fewer than the dimension of the space, it is possible to connect any couple of points in the space following integral lines of the vector fields, that is lines that at every point are tangent to some of the vector fields. This connectivity property allows to define a distance, known as the control distance, which is topologically equivalent to the Euclidean one, but usually not metrically equivalent to it. One of the deepest classical results about the geometry of H¨ ormander vector fields, proved by Alexander Nagel, Stein and Stephen Wainger [131] in 1985, is a sharp estimate for the volume of the control balls which implies the validity of a local doubling property. The study of this connection between properties of H¨ ormander operators and the geometry of H¨ ormander vector fields is the second theme of this book. Another technically relevant issue is the following. In some important instances, the vector fields Xi are defined on the whole Rn and possess some algebraic extraproperties: they are left invariant with respect to a (nonabelian) group of translations in Rn , and they are homogeneous with respect to a suitable family of (nonisotropic) dilations so that the operator Lu =

q X

Xi2 u + X0 u

i=1

turns out to be left invariant and homogeneous of degree two, like the classical Laplacian is in Rn with respect to Euclidean translations and dilations. For H¨ ormander operators that are left invariant and homogeneous of degree two, Folland [85] proved the existence of a global homogeneous fundamental solution. The existence of a fundamental solution with good properties (one more deep analogy between H¨ ormander operators and uniformly elliptic operators) is a starting point to prove, for these operators on homogeneous groups, sharp regularizing

xviii

H¨ ormander operators

estimates in a scale of Sobolev and H¨ older spaces adapted to the vector fields and the control distance. These left invariant homogeneous H¨ ormander operators which, historically, have been the first ones to be investigated in depth, are not only interesting in themselves: in some sense these play the role that the operators with constant coefficients play in the study of nonvariational elliptic operators with continuous coefficients. Indeed, the technique developed by Rothschild and Stein [142], in order to prove Sobolev estimates for H¨ ormander operators, consists in reducing, locally, the study of a general H¨ ormander operator to that of left invariant operators, homogeneous of degree two on a suitable homogeneous group. In order to make this idea work, Rothschild and Stein developed new techniques in which the geometric and analytic properties of the vector fields are deeply interconnected with the algebraic structure of the Lie algebra that these vector fields generate. In particular they introduced the lifting of H¨ormander vector fields to new H¨ ormander vector fields which are defined on a higher dimensional space, project on the original ones, and are algebraically free. A third underlying theme of this book is the development of the algebraic tools that are needed to study H¨ ormander operators. 0.2

Scope and structure of the book

The aim of this book is to give a systematic exposition of a relevant part of the theory of H¨ ormander operators and vector fields, together with the necessary background and prerequisites. The book is intended for self-study, or as a reference book, and we hope that it could be useful to both younger and senior researchers, already working in this area or aiming to approach it. Most of the major results described in this book are originally contained in a relatively small group of outstanding papers, dating back to several decades ago. This material contains true cornerstones in the field of H¨ormander operators, that is ideas, results and techniques which every active researcher in this area has to master. However, the original papers are often difficult to read, both for the depth of the original mathematical ideas contained, and for the variety of different subjects which form the background which the papers rely on. Therefore, we think that time has come to give to this material the structured form of a book. We do not aim to give the state of the art; instead, we aim to give an in-depth presentation of the fundamental results, with the level of detail which is necessary to handle arguments and proofs designed in the classical papers and solve new problems. A remark about the level of generality of our assumptions. Since, as already declared, we are mainly interested in the study of H¨ormander operators, we have systematically assumed that the vector fields we consider are C ∞ and that our second order operators can be written as sum of squares of vector fields (plus lower

Introduction

xix

order terms). Throughout the book we will not discuss the possible validity of our results under weaker assumptions, with the only exception of some maximum principles discussed in Chapter 1. Style We have tried to make a detailed, self-contained and, hopefully, pedagogical exposition of the theory, together with its necessary prerequisites, presented in a language which anybody working in Mathematical Analysis and PDEs should find comfortable. Let us say some more words of explanation of these features. “Detailed” exposition means that also some extra-value has been added to the original sources. Some examples are the following. 1. Whenever possible, we have keeped track of the dependence of the constants involved in the main results. This allows to establish some uniform estimates for families of operators, which quantitatively improve the classical results: uniform subelliptic estimates (Chapter 5), uniform estimates for fundamental solutions of families of left invariant homogeneous H¨ ormander operators on groups (Chapter 6), uniform a priori estimates, in the scale of Sobolev or H¨older spaces, for H¨ormander operators (Chapters 8, 11). The theory of nonvariational operators structured on H¨ ormander vector fields, presented in Chapter 12, is an example of application of these uniform quantitative estimates. The estimates on the volume of metric balls (Chapter 9) are proved with a detailed analysis of the dependence of the constants which allows to prove a uniformity result for such estimates, which has shown to be useful in the application to the theory of nonsmooth H¨ormander operators (see Chapter 9 for details). 2. The possible presence of the drift term X0 in an operator (0.3) can make much more complicated the proof of important results. On the other hand, in the classical literature the drift case is sometimes considered only quickly, sometimes with no proof at all. Instead, we have always written complete proofs explicitly covering the drift case: propagation of maxima (Chapter 1), subelliptic estimates (Chapter 5), a priori estimates for H¨ ormander operators (Chapters 8, 11, 12), lifting and approximation (Chapter 10). 3. Regularity results for H¨ ormander operators, in the scale of Sobolev spaces induced by the vector fields, are classically proved without writing down explicitly a priori estimates. We have instead worked out as independent results a priori estimates, solvability results, regularization results, in the proper functional framework of both Sobolev and H¨ older spaces, for distributional solutions to the equations (Chapters 8, 11, 12). “Self-contained” exposition means that the required background from Lie groups and Lie algebras, real and Fourier analysis, differential geometry, distribution theory, is given within this book, almost from scratch, avoiding the use of too specialized jergons: our “ideal reader” has a background from analysis and/or partial differential equations, but not necessarily from other fields. We think that, for such

xx

H¨ ormander operators

a reader, studying the same amount of background on specialized textbooks could consume considerably more time, spent in “acquiring the prerequisites necessary to understand the prerequisites”. “Pedagogical exposition” means that we have also tried to make use of some graduality, even though this implies some repetition: for instance, H¨ormander’s theorem is proved first for sublaplacians on Carnot groups (Chapter 4), and then in the general case (Chapter 5); a priori estimates in the scales of Sobolev or H¨older spaces are proved first on homogeneous groups and then in the general case. For the sake of graduality we have also split the material about the geometry of vector fields in two separate (and distant) chapters: Chapter 1, where the basic ideas which are necessary from the very beginning are introduced, and Chapter 9, where much more advanced material is presented, starting with the Baker-Campbell-Hausdorff formula and proceeding with Nagel-Stein-Wainger analysis of the volume of control balls. An analogous style has been followed for Lie groups and Lie algebras, which are presented first, in Chapter 3, in the concrete form of a structure of homogeneous group G in Rn , with the Lie algebra of left invariant vector fields on G (which is enough for most of the book), and later, in Chapter 10, with the abstract construction of the free nilpotent Lie algebra of some step on a fixed number of generators, and the abstract construction of the corresponding Lie group: these abstract constructions are necessary for the lifting and approximation technique developed in Chapter 10 and applied in Chapters 11 and 12, but not before. Structure Let us now describe the structure of the volume. A significant part of the book is devoted to giving the necessary tools and background. This material is included in the first three chapters, Chapter 7 and the Appendix. More specifically, in Chapter 1 we introduce the basic terminology and properties about vector fields, such as exponential map, commutators, control distance, definition of system of H¨ ormander vector fields and H¨ormander operator. Chapter 2 contains definitions and basic properties of Sobolev and H¨older spaces induced by a system of H¨ ormander vector fields. In Chapter 3 we introduce the concept of homogeneous Lie group and Lie algebra, and develop the tools which serve to the study of left invariant homogeneous H¨ormander operators. We also establish Poincar´e’s inequality in this setting. In Chapter 7 we present some real analysis tools in the framework of locally doubling metric spaces: maximal function, singular and fractional integrals, and related material. To make our exposition self-contained, we have also collected in an Appendix of the book those elements of distribution theory that are necessary in a few key points. The theme of geometry of vector fields is dealt in Chapters 1 and 9. In Chapter 1 we prove the connectivity theorem for a system of H¨ormander vector fields, establish the first properties of the control distance, in particular its relation with

Introduction

xxi

the Euclidean distance, and study the phenomenon of the propagation of maxima, for a second order differential operator, along the integral lines of suitable vector fields. We also discuss weak and strong maximum principles for these operators. Chapter 9 is almost entirely devoted to the proof of the deep theorem about the volume of control balls, which implies a local doubling condition. H¨ ormander’s theorem is proved in Chapters 4 and 5. First, in Chapter 4, we restrict our study to sum of squares H¨ ormander operators that are left invariant and homogeneous on a suitable homogeneous group. This context allows to use ad hoc tools which make the proof easier. In Chapter 5 we establish H¨ormander’s theorem in full generality by means of the subelliptic estimates, which are a result of independent interest. This chapter also contains the necessary background in Fourier analysis. Chapter 6 contains some general facts about the fundamental solution of H¨ ormander operators, in particular the construction and properties of the global homogeneous fundamental solution of left invariant homogeneous H¨ormander operators on homogeneous groups. This is a fundamental tool for the next subject: a priori estimates and regularity results in the scales of Sobolev and H¨ older spaces. This is the content of Chapter 8 (for H¨ ormander operators on groups), Chapter 11 (for general H¨ormander operators) and Chapter 12 (for nonvariational operators structured on H¨ormander’s vector fields). Chapter 10 deals with lifting and approximation, a set of algebraic-geometric tools which allow to connect the study of general H¨ormander operators to that of H¨ ormander operators on homogeneous groups. This is an essential tool employed in Chapters 11 and 12 for establishing a priori estimates. The reader can also use the book crossing it with partial routes, for instance: • for the reader who is only interested in H¨ormander operators on groups, the book ends with Chapter 8; • the reader who, on the contrary, is mainly interested in general H¨ormander operators can skip Chapters 4 and 8; • the reader who is interested in the geometry of vector fields can read Chapters 1 and 9. In order to make this book more user-friendly, we have started every chapter with a descriptive introduction which puts into context the subject of the chapter and in some cases contains the precise statements of the main results which will be proved throughout the chapter, while in other cases just presents a discursive description of the content. Again, doing so forces some unavoidable repetition which, however, is aimed to simplify the reader’s hard job. Credits and bibliographical references are collected in the final Notes of each chapter. These notes just want to declare the sources of the material that we have actually put in the book. With a few exceptions, we have not made any attempt to describe and give references for the further developments which are not explicitly discussed here.

xxii

0.3

H¨ ormander operators

Why study H¨ ormander operators?

H¨ ormander operators are not just a convenient way of rewriting particular degenerate elliptic-parabolic operators: there are important problems in analysis and applied mathematics which produce operators already written in the form (0.3). These operators therefore appear as natural objects. Also, there are problems in both theoretical and applied mathematics which lead to the study of systems of vector fields, their integral lines and their control distance, sometimes without any relation with second order differential operators. In this section we will sketch some ideas related to both these circles of ideas: H¨ ormander operators, and geometry of H¨ ormander vector fields. This should give some more motivation for the study of this subject. Let us start with H¨ ormander operators. Many physical systems with a finite number of degrees of freedom are governed by deterministic laws with, however, smaller or larger forms of stochastic perturbation, which are often modeled as Gaussian white noise. In this situation the mathematical model takes the form of a system of stochastic ordinary differential equations, which in the formalism of Ito’s stochastic calculus3 can be written as dx (τ ) = b (x (τ ) , τ ) dτ + B (x (τ ) , τ ) dw (τ ) ; x (t) = x

(0.4)

where the n-dimensional stochastic process x (t) is the unknown, x is the initial condition, b and B are, respectively, an assigned vector and a matrix of deterministic functions, and dw (τ ) is a Gaussian n-dimensional white noise. The interesting analytic object to be determined in this situation is the transition probability density p (t, x, s, y) (s > t), defined by the relation Z P (x (s) ∈ A | x (t) = x) = p (t, x, s, y) dy. A

This function can be proved to satisfy the forward Kolmogorov equation (also called Fokker-Planck equation) in the variables (s, y): ∂s p + ∇y · (bp) −

n 1 X 2 ∂ (aij p) = 0 2 i,j=1 yi yj

(0.5)

while p (t, x, s, y) satisfies the backward Kolmogorov equation in the variables (t, x): ∂t p + b · ∇x p +

n 1 X aij ∂x2i xj p = 0. 2 i,j=1

(0.6)

3 The interested reader can found a self-contained introduction to this subject in the books [82] or [148].

Introduction

xxiii

In (0.5) and (0.6) the matrix A = (aij ) is defined by: A = BB T with B as in (0.4). Note that A is always symmetric and nonnegative, but needs not be positive. Equation (0.6) can be rewritten in the form n X Xi2 p + X0 p = 0 i=1

where, letting B = {bij (x, t)}i,j , n

1 X Xi = √ bij ∂xj , i = 1, 2, . . . , n, 2 j=1 X X0 = ∂t + b · ∇x − (bik ∂xi bjk ) ∂xj . i,k,j

In many interesting examples the matrix B is actually lower dimensional, hence the vector fields Xi are actually fewer than n, and the operator is degenerate. However, already in 1934, Andrej Kolmogorov [118] exhibited an example of operator of this type, namely Lu = uxx − xuy − ut in R3 which, despite its degeneracy, possesses a fundamental solution Γ smooth outside the pole, this fact implying the hypoellipticity of L. Actually, Γ ((x, y, t) , (x0 , y 0 , t0 ))  2  (x−x0 )  2√3 exp − 2 0 4(t−t ) − π(t−t0 ) =   0

2 !  0 0 3 y−y 0 − x+x 2 (t−t ) (t−t0 )3

for t > t0 for t < t0

This phenomenon is well understood in the framework of the theory of H¨ormander operators; actually, this operator can be written as Lu = X12 u + X0 u with X1 = ∂x ; X0 = − (x∂y + ∂t ) and since [X1 , X0 ] = −∂y we see that X1 , X0 , [X1 , X0 ] span R3 at every point of the space, hence H¨ ormander’s condition is satisfied. This operator is explicitly quoted as a motivating example in the introduction of H¨ormander’s paper [107] and, as we have shown, is part of a large class of operators of type (0.3) which represent interesting physical models. A second main motivation to study H¨ ormander operators comes from the theory of several complex variables. Here we will just sketch a few ideas, since the highly technical matter prevents us to give a short rigorous explanation accessible to the nonexpert. The theory of holomorphic functions of several complex variables leads to the study of some differential operators, one of which is the Kohn-Laplacian on the boundary of a domain of Cn (n > 1). Without defining it in general4 , let us 4 The interested reader is referred, for instance, to the book [58]. For a survey in a few pages, see [24, pp. 26-34.].

xxiv

H¨ ormander operators

just explain which form it assumes in the special model case when this domain is the generalized upper half-plane 5   n   X 2 U n = ζ ∈ Cn+1 : |ζj | < Im ζ0 .   j=1

Its boundary ∂U n =

 

ζ ∈ Cn+1 :



n X

2

|ζj | = Im ζ0

  

j=1

is a real hypersurface in Cn+1 , which can be identified with R2n+1 as follows. Letting zj = xj + iyj ; z = (z1 , z2 , . . . , zn ), we identify the point (x1 , x2 , . . . , xn , y1 , y2 , . . . , yn , t) ∈ R2n+1   2 with z, t + i |z| ∈ ∂U n . We can endow R2n+1 with the following group law: (z, t) ◦ (z 0 , t0 ) = (z + z 0 , t + t0 + 2 Im

n X

zj z 0j ).

j=1 2n+1

The set R with this group law is called the Heisenberg group Hn . The identification of ∂U n with Hn has a geometric meaning, because Hn allows to define a family of translations in Cn+1 which preserve both U n and its boundary. The Kohn Laplacian b on ∂U n then takes the following form. Let us first say that this operator by definition acts on (0, q)-forms (with q = 0, 1, 2, . . . , n) of the kind X φJ dz J |J|=q

(where dz J = dz j1 ∧ dz j2 ∧ . . . ∧ dz jq ). Define the vector fields: Xj = ∂xj + 2yj ∂t ; Yj = ∂yj − 2xj ∂t ; T = ∂t which are left invariant on ∂U n . Then for any q = 0, 1, 2, . . . , n, we have   X X b  (Lα φJ ) dz J with α = n − 2q φJ dz J  = |J|=q

|J|=q

where n

Lα = −


1X Xk2 + Yk2 + iαT. 2 k=1

5 This set is biholomorphically equivalent to the unit ball in Cn+1 , and is a model example of strongly pseudoconvex domain in Cn+1 , a central notion in the theory.

Introduction

xxv

The study of b on q-forms is therefore reduced to the study of the scalar operators Lα for α = n, n − 2, n − 4, . . . , −n. In particular, for α = 0, Lα is a H¨ormander operator, because [Xk , Yk ] = −4∂t , hence X1 , . . . Xn , Y1 , . . . , Yn , [Xk , Yk ] span R2n+1 . Therefore n
1X L0 = − Xk2 + Yk2 2 k=1

is hypoelliptic. It is usually called the sublaplacian on the Heisenberg group, and is probably the most studied among H¨ ormander operators. We end this example noting that the operators Lα have been studied for any α ∈ C. For α nonzero integer, note that Lα is not a H¨ ormander operator because has not real coefficients. However, it has been proved that for any α different from a discrete set of forbidden values, which are ±n, ± (n + 2) , ± (n + 4) , . . . the operator Lα is hypoelliptic. Let us now discuss some motivations for the study of the geometry of systems of H¨ ormander vector fields, with no reference to second order differential operators. A first motivation arises, for instance, from the study of geometric control theory and nonholonomic mechanics or, in a more abstract language, from the study of the so-called subriemannian geometry 6 . In the language of geometric control theory, the connectivity property of a system of H¨ ormander vector fields means that some physical system is controllable, that is we can lead the system from some fixed initial state to any desired final state, provided we act in a suitable way on the controls at our disposal. This is especially interesting when the total number of controls is less than the number of degrees of freedom of the physical system, that is when the vector fields are fewer than the dimension of the space. Suppose that the dynamics of a physical system is governed by a systems of ODEs of the form: m X fj (x) uj . (0.7) x0 = j=1

Here x is a point in the configuration space of the system that we suppose to be n dimensional and u1 , . . . um are the control variables. As already noted, the interesting case is m < n. A control function that drives the system (0.7) from a configuration x0 to a configuration xf in time T is a function u : [0, T ] → Rm such that the actual path of the system, in the configuration space, is a curve x = φ (t) which solves: Pm  0  φ (t) = j=1 fj (φ (t)) uj (t) x (0) = x0  x (T ) = xf 6 The reader who is interested in this circle of ideas can look into the monographs [11], [112], [127], [48].

xxvi

H¨ ormander operators

and this means that φ is an integral curve of the family of vector fields f1 , . . . , fm that connects x0 to xf . In other words the concept of controllability of the system (0.7) coincides with that of connectivity of the configuration space by means of integral curves of the vector fields f1 , . . . , fm . In particular, if H¨ormander’s condition is satisfied, then the system is controllable. Let us explain this idea on a concrete example (see [130, pp. 703-4], [135, pp. 33-36]). Let us consider a car which can move in a plane. Let (x, y) be the middle point of the rear axle; l the distance between the front and the rear axles; ϑ the angle formed by the car with the x-axis; φ the angle formed by the front wheels with respect to the car. y

φ

ϑ (x, y) x We can control the linear velocity u1 of the car, and the the angle velocity of steering φ0 = u2 . Let us express the dynamics of the system. First of all, the point (x, y) moves according to  0 x = u1 cos ϑ, y 0 = u1 sin ϑ. We also know that φ0 = u2 . We do not have a direct control on the angle ϑ, but there is a nonholonomic constraint relating ϑ to φ, which allows us to write an equation for ϑ0 . First, the quantity lϑ0 represents the linear velocity of the middle point of the front axle in the direction normal to the longitudinal axis of the car; the actual instantaneous velocity of this point is a vector v which forms an angle φ with the longitudinal axis of the car; with respect to the car, its components are lϑ0 and u1 , hence lϑ0 = u1 tan φ. We have therefore the control system  0 x = u1 cos ϑ    0 y = u1 sin ϑ  φ0 = u2   0 ϑ = ul1 tan φ This can be rewritten, letting r = (x, y, φ, ϑ) as r0 = u1 X1 + u2 X2 with X1 = cos ϑ∂x + sin ϑ∂y +

1 tan φ∂ϑ ; X2 = ∂φ . l

Introduction

xxvii

Since  1 1 + tan2 φ ∂ϑ l  1 [[X2 , X1 ] , X1 ] = 1 + tan2 φ {− sin ϑ∂x + cos ϑ∂y } l [X2 , X1 ] =

the four vectors X1 , X2 , [X2 , X1 ] , [[X2 , X1 ] , X1 ] are always independent, provided φ 6= ± π2 . Hence the system is controllable. As already noted, the interesting feature of examples like the previous one consists in the fact that the system is controllable even though the number of controls at our disposal (in this case, 2) is less than the number of degrees of freedom of the system (in this case, 4). This possibility depends on the nonholonomy of the constraints In the axiomatic foundations of thermodynamics proposed by Carath´eodory in 1909 ([55], see the book [114, Chap. 12] for an English translation, see also [22, Chap. V and Appendix 6, 7]), the second law of thermodynamics is derived from an axiom which states the existence, in any neighborhood of any state of a thermodynamical system, of other states which are inaccessible moving along adiabatic paths (which correspond to integral curves of suitable vector fields). In this context the negation of the connectivity property is used to deduce the negation of H¨ormander’s condition, a fact which in this particular case assures the existence of an integrating factor for the differential form dQ representing the heat exchanged by the system. The integrating factor is interpreted as the absolute temperature T and the relation T dQ = dφ then gives the existence of the entropy function φ, that is the second law of thermodynamics. The name of Carnot-Carath´eodory distance, to denote the control distance induced by a system of vector fields, which we will introduce in Chapter 1, was given as a credit to the pioneering works which connected thermodynamics to geometry of vector fields. The idea of connectivity is also at the core of a more recent discipline called neurogeometry. In 1981 the neurophysiologists David H. Hubel and Torsten N. Wiesel received the Nobel prize for their study on the structure and function of the visual cortex. In particular, they found that some cortical cells respond to contours of specific orientation. This raises the question of how the brain reconstructs a complete contour from a large number of neurons. Since then, many mathematical models of a functional architecture of the visual cortex have been proposed, starting from the 1989 seminal paper of William Hoffman [104]. This problem is strictly connected to the problem of perceptual completion: how the brain reconstructs missing information such as contours that are not visible. In 2006 Giovanna Citti and Alessandro Sarti [66] proposed a mathematical model of contour completion based on a system of vector fields that satisfies H¨ ormander’s condition. In a very simplified form their model is the following. The image on the retina can be represented by a function

xxviii

H¨ ormander operators

I : D → R. For every (x, y) ∈ D, let θ = θ (x, y) such that ∇I (x, y) = |∇I (x, y)| (− sin θ, cos θ) and observe that (− sin θ, cos θ) is normal to the level line in (x, y). This means that when the point (x, y) is part of a contour, then θ is the direction of the contour. Now we define a surface Σ in the three dimensional space (x, y, θ) as the graph θ = θ (x, y) and we lift the image I to the surface Σ letting: u (x, y, θ) = I (x, y) for (x, y, θ) ∈ Σ. This surface is a good model of the way the visual cortex actually represents an image: our brain does not only store an image as a set of points, but as a system of points and directions. It is not difficult to check that at every point in Σ the tangents to the level lines of u lie in the plane spanned by the vector fields X1 = (cos θ) ∂x + (sin θ) ∂y ; X2 = ∂θ . Since [X2 , X1 ] = − (sin θ) ∂x + (cos θ) ∂y the vector fields X1 , X2 and [X2 , X1 ] are linearly independent. The connectivity theorem of Chow and Rashevski˘ı assures that two points (x1 , y1 , θ1 ) (x2 , y2 , θ2 ) can be always connected following integral lines of the vector fields X1 and X2 . Citti and Sarti noted that the neural activity develops and propagate itself along these directions and proposed of simple model of this activity propagation: a linear diffusion model with respect to the H¨ ormander operator ∂t u = Lu where L = 2 2 X1 + X2 . The reader interested in this topic is referred to the monograph [67].

Chapter 1

Basic geometry of vector fields

1.1

Introduction

The aim of this chapter is twofold. On the one hand, we will introduce several basic concepts and terminology that will be used throughout the book, such as the exponential map, the commutator of two vector fields, the distance induced by a family of vector fields (the “control distance”), H¨ormander’s condition, H¨ormander operators and vector fields, Lie algebras (sections 1.2-1.5). We think that also the reader who is already acquainted with this material should give a glance at these preliminary sections, to check notation and terminology. On the other hand, we will develop a first portion of the so-called geometry of vector fields. Although so far we have not even defined the concepts which are involved, let us briefly sketch the topics and results that will be dealt with. After discussing the basic properties of the control distance induced by a general system of vector fields (sections 1.4, 1.5) we will prove that, for a system of H¨ ormander vector fields, the connectivity theorem holds: any two points of the domain can be joined by a curve composed by arcs of integral lines of the vector fields (section 1.6, Theorem 1.45). We will also derive some quantitative results related to this fact, in particular a local comparison between the control distance and the Euclidean one (section 1.7, Theorem 1.53). We will then pass to consider second order differential operators expressed in terms of smooth vector fields Xi in the form q q X X L= Xi2 + X0 or, more simply, L = Xi2 i=1

i=1

(“operators with drift” or “sum of squares operators”, see section 1.3). Depending on which is the second order operator under study, we will consider, respectively, the system of vector fields X0 , X1 , . . . , Xq where X0 has “weight” two while the others Xi have weight one, or, for sum of squares operators, simply the system X1 , . . . , Xq . After a discussion of the weak maximum principle (section 1.8, Theorem 1.57), we will apply the previous geometrical results to the study of the phenomenon of propagation of maxima for H¨ ormander operators (sections 1.9-1.10, Theorems 1.62 and 1.73), which will imply a strong maximum principle for sum of squares operators (Theorem 1.64). 1

2

H¨ ormander operators

The most technical results in this chapter are the connectivity theorem (Theorems 1.45 and 1.48) and the propagation of maxima (Theorems 1.62 and 1.73). This last result is proved in two steps, first for sum of squares operators and then for operators with drift. In this chapter we have also carried out detailed computations about exponential maps and control distances for some concrete systems of H¨ormander vector fields and operators, in order to offer to the reader, before going into the general abstract results that will be discussed throughout the book, the possibility of gaining some experience of the concepts involved. We now take the story from the beginning, and start introducing the basic definitions. 1.2

Exponentials and commutators of vector fields

Let X be a real smooth vector field, defined in a domain (that is a connected open set) Ω ⊂ Rn . As usual, X can be seen either as a differential operator, X=

n X

bj (x) ∂xj

j=1

with bj ∈ C ∞ (Ω) or as a smooth function X : Ω → Rn . We will write Xf (x) =

n X j=1

bj (x)

∂f (x) ∂xj

to denote the differential operator X acting on a function f , and Xx = (b1 (x) , b2 (x) , . . . , bn (x)) to denote the vector field X evaluated at the point x. Let us recall the standard Definition 1.1 (Exponential of a vector field) For X as above, x0 ∈ Ω and t ∈ R (small enough), we set exp (tX) (x0 ) = ϕ (t) where ϕ is the solution to the Cauchy problem  0 ϕ (τ ) = Xϕ(τ ) , ϕ (0) = x0 .

(1.1)

Equivalently, we can also write exp (tX) (x0 ) = φ (1) where φ is the solution to the Cauchy problem  0 φ (τ ) = tXφ(τ ) φ (0) = x0 .

Basic geometry of vector fields

3

The proof of the equivalence is straightforward. The differential equation in (1.1) can be written also as d (exp (tX) (x0 )) = Xexp(tX)(x0 ) , (1.2) dt which means that ϕ (t) = exp (tX) (x0 ) is an integral curve of X. The exponential of a vector field actually defines a smooth map in the joint variables (t, x0 ): Proposition 1.2 (Regularity of the exponential map) For Ω0 b Ω there exists ε > 0 such that for every (t, x0 ) ∈ (−ε, ε) × Ω0 the exponential exp (tX) (x0 ) is uniquely defined. Moreover the function F (t, x) = exp (tX) (x) 0

is smooth in (−ε, ε) × Ω . Proof. The existence of ε such that exp (tX) (x0 ) is uniquely defined in (−ε, ε)×Ω0 follows from the classical result about existence and uniqueness for solutions to Cauchy problems. Moreover, by the classical result about regular dependence of the solution of a Cauchy problem on initial condition (see e.g. [138, §21 Chap. 3 and α §29 in Chap. 4]), we read that the derivatives ∂∂xFα (t, x) exist and are continuous in a neighborhood of (0, x0 ), for any fixed α. To complete the proof, we have to ∂ α+β F check the existence and continuity of mixed derivatives ∂x α ∂tβ (t, x). If α = 0, the required regularity can be read from the equation. The general case requires an inductive reasoning. To fix ideas, let us consider the case |α| + |β| = 2. Then we have:
∂ 2 F (t, x) ∂F (t, x) ∂ = XF (t,x) = JX (F (t, x)) · ∂xi ∂t ∂xi ∂xi where JX denotes the Jacobian matrix of the map x 7→ Xx . Since we already know 2 F (t,x) (t,x) and JX are continuous, the continuity of ∂ ∂x follows. The general that ∂F∂x i i ∂t case is analogous. Notation 1.3 We will often compose two (or several) exponentials. We will simply write exp (tX) exp (sY ) (x0 ) to denote exp (tX) (exp (sY ) (x0 )) . An easy consequence of the uniqueness of solutions of the Cauchy problem is that exp ((t + s) X) (x0 ) = exp (tX) exp (sX) (x0 )

(1.3)

for s, t small enough; this also implies −1

[exp (tX)] −1

where the symbol [exp (tX)]

= exp (−tX)

(1.4)

denotes the inverse mapping of x 7→ exp (tX) (x).

4

H¨ ormander operators

If f : Ω → R is a differentiable function, using (1.1) it is immediate to check that d (f (exp (tX) (x0 ))) = (Xf ) (exp (tX) (x0 )) . dt

(1.5)

Let now X1 , . . . , Xq be a system of real smooth vector fields, defined in a domain Ω ⊂ Rn (usually q < n). Starting from a fixed point x0 ∈ Ω we want to move along the integral lines of the vector fields Xi , trying to reach another point of Ω. If the number q of vector fields is strictly less than the dimension n, we cannot in general expect that every point will be accessible in this way. However, an interesting result can occur moving alternatively along the integral lines of two different vector fields. Let us first give the following Definition 1.4 (Commutator) The commutator of two smooth vector fields X=

n X

aj (x) ∂xj ; Y =

n X

bj (x) ∂xj

j=1

j=1

is defined as: [X, Y ] = XY − Y X =

n X


aj ∂xj bi − bj ∂xj ai ∂xi .

(1.6)

i,j=1

We say that two vector fields X, Y commute when [X, Y ] ≡ 0. Note that in the previous computation second order derivatives cancel thanks to the equality of mixed partials, so that [X, Y ] is actually a vector field. Observe that if JX is the Jacobian of the map1 x 7→ Xx we can rewrite (1.6) as [X, Y ]x = JY (x) Xx − JX (x) Yx .

(1.7)

Example 1.5 Let X = ∂x ; Y = x∂y , then [X, Y ] = ∂y Let X = ∂x + y∂t ; Y = ∂y − x∂t , then [X, Y ] = −2∂t Let X = ∂x + y∂t , Y = ∂y + x∂t , then [X, Y ] = 0. With the above notation, a standard result about ODEs reads as follows: Theorem 1.6 Let X, Y be a couple of smooth vector fields in a domain Ω. Then, for every x0 ∈ Ω, as t → 0+ , exp (−tX) exp (−tY ) exp (tX) exp (tY ) (x0 )


= exp t2 [Y, X] (x0 ) + o t2 = x0 + t2 [Y, X]x0 + o t2 . 1 Here X is seen as a column vector, and accordingly the Jacobian J x X is written with rows ∇a1 , . . . , ∇an .

Basic geometry of vector fields

tY )

exp

exp

(tX )

(− tX )

exp(−

5

t2 [Y, X]x0 + o(t2 ) x0 Fig. 1.1

exp(tY

)

Commutator of two exponentials.

We skip for the moment the proof of this theorem because we will actually prove a more general result later in this chapter (see Theorem 1.49). The statement means that moving repeatedly along the integral curves of the vector fields tY, tX, −tY, −tX causes as a net result a smaller displacement in the approximate direction of the commutator t2 [Y, X]. In this sense a control in the directions of two vector fields can give a control also in the direction of their commutator. Also, this fact suggests that in some sense the commutator [Y, X] , although actually is still a first order differential operator, should be regarded as an operator of weight two, compared with the vector fields X, Y of weight one. Example 1.7 Let us check the property expressed in the above theorem in the simplest case: X = ∂x ; Y = x∂y . At the origin we have X(0,0) = ∂x , Y(0,0) = 0, hence following the integral curves of these vector fields it is impossible to move directly in the y direction. But let us move along the integral curves of X and Y as in the above theorem, that is let us compute: exp (−tX) exp (−tY ) exp (tX) exp (tY ) (0, 0) . It is more convenient here to write X = (1, 0) and Y = (0, x). Then, applying Definition 1.1 we can compute exp (tY ) (0, 0) = (0, 0) because  0    x (τ ) = 0 x (τ ) = 0 x (τ ) = 0 x (t) = 0 =⇒ =⇒ =⇒ y 0 (τ ) = x y 0 (τ ) = 0 y (τ ) = 0 y (t) = 0. Also, exp (tX) exp (tY ) (0, 0) = exp (tX) (0, 0) = (t, 0), because  0   x (τ ) = τ x (t) = t x (τ ) = 1 =⇒ =⇒ y 0 (τ ) = 0 y (τ ) = 0 y (t) = 0.

6

H¨ ormander operators

Then exp (−tY ) exp (tX) exp (tY ) (0, 0) = exp (−tY ) (t, 0) = t, −t2




because 

x0 (τ ) = 0 =⇒ y 0 (τ ) = x



x (τ ) = t =⇒ y 0 (τ ) = t



x (τ ) = t =⇒ y (τ ) = tτ



x (−t) = t y (−t) = −t2

and finally,

exp (−tX) exp (−tY ) exp (tX) exp (tY ) (0, 0) = exp (−tX) t, −t2 = 0, −t2 because 

x0 (τ ) = 1 =⇒ y 0 (τ ) = 0



x (τ ) = t + τ =⇒ y (τ ) = −t2



x (−t) = 0 y (−t) = −t2 .


On the
other hand, since [X, Y ] = ∂y = (0, 1), we have exp t2 [Y, X] (0, 0) =
0, −t2 which proves the assertion. In this case the remainder o t2 vanishes, and we have an exact equality. Example 1.8 A less elementary example is given in R3 3 (x, y, θ) by the two vector fields (related to the rotations in the plane) X = ∂θ ; Y = (cos θ) ∂x + (sin θ) ∂y . We have [X, Y ] = (− sin θ) ∂x + (cos θ) ∂y . Rewriting X = (0, 0, 1) ; Y = (cos θ, sin θ, 0) ; [Y, X] = (sin θ, − cos θ, 0) we can subsequently compute, reasoning like in the previous example: exp (tY ) (0, 0, 0) = (t, 0, 0) exp (tX) (t, 0, 0) = (t, 0, t) exp (−tY ) (t, 0, t) = (−t cos t + t, −t sin t, t) exp (−tX) (−t cos t + t, −t sin t, t) = (−t cos t + t, −t sin t, 0) . Hence exp (−tX) exp (−tY ) exp (tX) exp (tY ) (0, 0, 0) = (−t cos t + t, −t sin t, 0) .

On the other hand, exp t2 [Y, X] (0, 0, 0) = 0, −t2 , 0 . Note that
exp (−tX) exp (−tY ) exp (tX) exp (tY ) (0, 0, 0) − exp t2 [Y, X] (0, 0, 0)  3 
t4
t 3 4 = (t (1 − cos t) +, t (t − sin t) , 0) = + o t , + o t , 0 as t → 0 2 6
2 so the difference is actually o t , although does not vanish identically like in the previous example. We now want to point out that, in contrast with the property expressed by Theorem 1.6, if two vector fields X, Y commute, following consecutively their integral lines we have no chance of moving in an independent direction. Namely:

Basic geometry of vector fields

7

Theorem 1.9 Let X and Y be smooth vector fields in a domain Ω such that [X, Y ] ≡ 0. Let x0 ∈ Ω. There exist ε > 0 such that for every |t| < ε exp (tX) exp (tY ) (x0 ) = exp (t (X + Y )) (x0 ) ; exp (−tX) exp (−tY ) exp (tX) exp (tY ) (x0 ) = x0 . Proof. The second identity easily follows from the first one. To prove the first identity, let ϕ (t) = exp (tX) exp (tY ) (x0 ). Clearly ϕ (0) = x0 . We will show that ϕ0 (t) = Xϕ(t) + Yϕ(t)

(1.8)

and therefore ϕ (t) = exp (t (X + Y )) (x0 ). Let now F (u, v) = exp (uX) exp (vY ) (x0 ) , it follows that ϕ0 (t) =

∂F ∂u

(t, t) +

∂F ∂v

(t, t). For the first term we have

∂F (t, t) = Xexp(tX) exp(tY )(x0 ) = Xϕ(t) . ∂u In order to evaluate ∂F ∂v (t, t) we set Gv (u) = Yexp(vY )(x0 ) and that

∂F ∂v

(u, v), we observe that Gv (0) =


∂2F ∂ (u, v) = Xexp(uX) exp(vY )(x0 ) ∂v∂u ∂v ∂ (exp (uX) exp (vY ) (x0 )) = JX (exp (uX) exp (vY ) (x0 )) ∂v = JX (exp (uX) exp (vY ) (x0 )) Gv (u) .

G0v (u) =

Let now Hv (u) = Yexp(uX) exp(vY )(x0 ) , observe that Hv (0) = Yexp(vY )(x0 ) and that ∂ (exp (uX) exp (vY ) x0 ) ∂u = JY (exp (uX) exp (vY ) x0 ) Xexp(uX) exp(vY )x0 .

Hv0 (u) = JY (exp (uX) exp (vY ) x0 )

Since the vector fields X and Y commute, by (1.7), JX (x) Yx = JY (x) Xx and therefore Hv0 (u) = JX (exp (uX) exp (vY ) x0 ) Yexp(uX) exp(vY )x0 = JX (exp (uX) exp (vY ) x0 ) Hv (u) . It follows that Hv and Gv satisfies the same Cauchy problem so that ∂F (u, v) = Gv (u) = Hv (u) = Yexp(uX) exp(vY )(x0 ) . ∂v Hence

∂F ∂v

(t, t) = Yϕ(t) and (1.8) follows.

8

1.3

H¨ ormander operators

Lie algebras, H¨ ormander’s condition, H¨ ormander operators

At this point it can be worthwhile to recall a standard concept which will be used throughout the book. Definition 1.10 (Lie algebra) A Lie algebra (over R) is a real vector space (g, +, ·) endowed with another internal operation, called Lie bracket, [·, ·] : g × g → g enjoying the following properties. For every X, Y, Z ∈ g and λ, µ ∈ R [λX + µY, Z] = λ [X, Z] + µ [Y, Z] [X, Y ] = − [Y, X] [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0

(bilinearity) (anticommutativity) (Jacobi identity)

If g is a Lie algebra and h ⊂ g is still a Lie algebra with respect to the same operations, we also say that h is a Lie subalgebra of g. Then we have the following Proposition 1.11 Let Ω ⊆ Rn be an open set, and let X (Ω) be the family of all real smooth vector fields in Ω. Then the set X (Ω), endowed with the sum of vector fields, the product with real scalars, and the commutator [X, Y ] = XY − Y X is a Lie algebra over R. As a vector space, it is infinite dimensional. The proof is an easy exercise left to the reader, although checking Jacobi identity is a bit tedious. Definition 1.12 Given a family of real smooth vector fields X1 , . . . , Xq defined in an open set Ω ⊆ Rn , the Lie algebra generated by this family is the smallest Lie subalgebra of X (Ω) containing X1 , . . . , Xq , and will be denoted by the symbol L (X1 , . . . , Xq ) . With the symbol L (X1 , . . . , Xq )x we will denote the vector space obtained evaluating at some point x ∈ Ω the elements of L (X1 , . . . , Xq ). Definition 1.13 (Rank) For any Lie subalgebra g of X (Ω) and any x ∈ Ω, the rank of g at x is the dimension of the vector space {Xx : X ∈ g}. We can now define the main objects which will be studied in the book. Definition 1.14 We say that a family of real smooth vector fields X1 , . . . , Xq defined in an open set Ω ⊆ Rn , satisfies H¨ ormander’s condition in Ω if the Lie algebra generated by this family has rank n at any point of Ω. In other words, at every point of Ω, among the vector fields X1 , . . . , Xq and their iterated commutators we can find n independent vectors. We also say that X1 , . . . , Xq form a system of H¨ormander vector fields in Ω.

Basic geometry of vector fields

9

Definition 1.15 (H¨ ormander operators) If X0 , X1 , . . . , Xq is a system of H¨ ormander vector fields in Ω, then L=

q X

Xi2 + X0

(1.9)

i=1

is called a H¨ ormander operator in Ω. If X0 ≡ 0, that is if L=

q X

Xi2

(1.10)

i=1

where X1 , . . . , Xq is a system of H¨ ormander vector fields in Ω, then L is also called a sum of squares of H¨ ormander vector fields, or a sublaplacian. Several examples of H¨ ormander operators will be given at the end of this section. Remark 1.16 (The role of X0 ) The vector field X0 in (1.9) is often called “drift”, from the physical interpretation of L as a transport-diffusion operator, hence operators like (1.9) can also be called “ operators with drift”, when we want to stress the effective presence of this term. This class is the general framework which allows to study, for instance, operators of Kolmogorov-Fokker-Planck type (which we have briefly described in the Introduction of the book), which constitute one of the important motivations for the development of the theory. The fact that the vector field X0 is required, together with the Xi (i = 1, 2, . . . , q), to fulfill H¨ ormander’s condition, means that the first order operator X0 cannot be thought as a lower order term, but must be considered as belonging to the principal part of L. In some sense, X0 “weights” as a second order derivative, analogously to the time derivative in the 2 heat operator ∂xx − ∂t . As we will see throughout the book, the presence of the drift often poses substantially new problems which do not appear for “sum of squares” operators. If, on the other hand, L is a sum of squares of H¨ ormander vector fields like (1.10) and Y is any vector field, then the operator q X

Xi2 + Y

i=1

is a sublaplacian with a lower order term Y , not an operator with drift: in this case, Y is not required to fulfill H¨ ormander’s condition. To focus on the main problems, we will usually disregard the possible presence of lower order terms. In the following we will pervasively make computations on commutators of vector fields. To do this efficiently, and to keep into account the possible presence of a drift term, we have both to fix some notation and convention which will be used consistently throughout the book, and to establish some basic properties.

10

H¨ ormander operators

Definition 1.17 (Weighted vector fields, standard commutators) For a system X0 , X1 , . . . , Xq of smooth real vector fields, defined in a domain Ω ⊂ Rn , let us assign to each Xi a weight pi , saying that p0 = 2 and pi = 1 for i = 1, 2, . . . q. For any multiindex I = (i1 , i2 , . . . , ik ) with ij ∈ {0, 1, 2, . . . , q}, we define the weight of I as |I| =

k X

pij ,

j=1

while the length of I is simply ` (I) = k. So, for I = (1, 2) we have ` (I) = |I| = 2, while for I = (0, 1) we have ` (I) = 2 and |I| = 3. Note that if in our system of vector fields the drift X0 is lacking, weight and length of any multiindex coincide. We set XI = Xi1 Xi2 . . . Xik and      X[I] = Xi1 , Xi2 , . . . Xik−1 , Xik . . . . Note that XI are differential operators of order k, while X[I] are vector fields, that we will call standard commutators. If I = (i1 ) , then X[I] = Xi1 = XI . According to the convention already stated for vector fields, we will write
X[I] f to denote the differential operator X[I] acting on a function f , and X[I] x to denote the vector field X[I] evaluated at the point x. Note that not every iterated commutator of the vector fields Xi is a standard commutator, for instance [[X1 , X2 ] , [X3 , X4 ]] is not written in the form of a standard commutator [Xi1 , [Xi2 , [Xi3 , Xi4 ]]]. However, by simple algebraic computation we can actually rewrite it as a linear combination of standard commutators: Example 1.18 Let us expand [[X1 , X2 ] , [X3 , X4 ]] = [X1 X2 , [X3 , X4 ]] − [X2 X1 , [X3 , X4 ]] However, [X1 X2 , [X3 , X4 ]] = X1 X2 [X3 , X4 ] − X1 [X3 , X4 ] X2 +X1 [X3 , X4 ] X2 − [X3 , X4 ] X1 X2 = X1 [X2 , [X3 , X4 ]] + [X1 , [X3 , X4 ]] X2 and similarly [X2 X1 , [X3 , X4 ]] = X2 [X1 , [X3 , X4 ]] + [X2 , [X3 , X4 ]] X1

Basic geometry of vector fields

11

so that [X1 , X2 ] , [X3 , X4 ] = X1 [X2 , [X3 , X4 ]] + [X1 , [X3 , X4 ]] X2 − X2 [X1 , [X3 , X4 ]] − [X2 , [X3 , X4 ]] X1 = [X1 , [X2 , [X3 , X4 ]]] − [X2 , [X1 , [X3 , X4 ]]] = X[(1,2,3,4)] − X[(2,1,3,4)] . So we have rewritten a nonstandard iterated commutator as a linear combination of standard commutators. Proving that this computations can be carried out in general will be the content of Lemma 1.21. To prove it, it is now convenient to introduce the standard notion of adjoint map: Definition 1.19 Let X (Ω) be the Lie algebra of real smooth vector fields in Ω. Given X ∈ X (Ω), the adjoint of X is the linear map ad X : X (Ω) → X (Ω) such that (ad X) Z = [X, Z]. Let us consider two vector fields X and Y and the commutator of their adjoint map [ad X, ad Y ], that is the map [ad X, ad Y ] Z = (ad X ad Y ) Z − (ad Y ad X) Z = [X, [Y, Z]] − [Y, [X, Z]] . By the Jacobi identity we can write [X, [Y, Z]] − [Y, [X, Z]] = [[X, Y ] , Z] = (ad [X, Y ]) Z so that ad [X, Y ] = [ad X, ad Y ] .

(1.11)

Let us consider now a family of vector fields X1 , . . . , Xq ∈ X (Ω) and the associated adjoint maps ad X1 , ad X2 , . . . ad Xq . For every multiindex I = (i1 , i2 , . . . , ik ) we set      (ad X)[I] = ad Xi1 , ad Xi2 , . . . ad Xik−1 , ad Xik . . . . (1.12)
Lemma 1.20 For every multiindex I we have ad X[I] = (ad X)[I] . Proof. Let I = (i1 , i2 , . . . , ik ). The proof goes by induction on k. If k = 1 there is nothing to prove. If k = 2 we already observed in (1.11) that ad [Xi1 , Xi2 ] = [ad Xi1 , ad Xi2 ] . Let now k > 2 and assume that the property holds for every   multiindex of length k − 1. If we set I 0 = (i2 , . . . , ik ) we have X[I] = Xi1 , X[I 0 ] so that i     h ad X[I] = ad Xi1 , X[I 0 ] = ad Xi1 , ad X[I 0 ] = ad Xi1 , (ad X)[I 0 ] = (ad X)[I] .

12

H¨ ormander operators

Lemma 1.21 Given any two commutators X[I] , X[J] there exist absolute constants dK I,J such that X   X[I] , X[J] = dK (1.13) I,J X[K] . |K|=|I|+|J|

As a consequence, every iterated commutator is a linear combination of standard commutators. Proof. The last assertion follows by iteration of (1.13), so let us prove this identity. By Lemma 1.20 we have  
X[I] , X[J] = ad X[I] X[J] = (ad X)[I] X[J] . By definition (1.12), we can write (ad X)[I] as linear combination of products of Pk adjoint maps, that is maps of the kind ad X`1 ad X`2 · · · ad X`k with i=1 p`i = |I|. It follows that (ad X)[I] X[J] is a linear combination of vector fields of the kind      (ad X`1 ad X`2 · · · ad X`k ) X[J] = X`1 , X`2 , · · · , X`k , X[J] · · · which is clearly a commutator of the kind X[K] with |K| = |I| + |J|. Assume now that X0 , X1 , . . . , Xq form a system of H¨ormander vector fields in Ω ⊂ Rn , then at any point x ∈ Ω we can find n linearly independent iterated commutators; by Lemma 1.21, each of these iterated commutators is a linear combination of standard commutators X[I] , which means that we have a finite set of generators of Rn consisting in standard commutators; we can then also form a basis of Rn consisting in n standard commutators X[I1 ] , X[I2 ] , . . . , X[In ] . This shows that we can equivalently give the definition of H¨ ormander’s condition as follows: Definition 1.22 (H¨ ormander’s condition, equivalent definition) We say that a system X0 , X1 , . . . , Xq of real smooth vector fields defined in a domain Ω ⊂ Rn satisfies H¨ ormander’s condition in Ω if at every point x ∈ Ω, there exist multiindices I1 , I2 , . . . , In such that the vectors


X[I1 ] x , X[I2 ] x , . . . , X[In ] x span Rn . Observe that in the above definition the maximum weight of the commutators that are needed to span Rn is not necessarily bounded, as the next example shows: Example 1.23 Let us define (

2

x − 1−x 2

x2 < 1 x2 > 1 Q+∞ k+1 and note that ϕ ∈ C ∞ (R). Let also define a (x) = k=0 ϕ (x − 2k) . Clearly ∞ a ∈ C (R), also observe that a (x) = 0 if and only if x = 2k for some k ∈ N ϕ (x) =

1−e 1

Basic geometry of vector fields

13

and that a (x) vanishes of order 2k + 2 at 2k. Let us consider the following smooth vector fields in R2 X = ∂x ; Y = a (x) ∂y . Since a (x) vanishes only for x = 2k, X and Y span R2 at every point (x, y) ∈ R2 with x 6= 2k. Moreover since [X, Y ] = a0 (x) ∂y , a commutator of the kind [X, [X, · · · [X, Y ] · · · ]] of order 2k +2 and X span R2 at the points (2k, y). Therefore H¨ ormander’s condition holds in R2 , but the weight of commutators required to fulfil H¨ ormander’s condition is unbounded in R2 . The same phenomenon can occur on a bounded domain: it is enough to define X = ∂x ; Y = a (tan (x)) ∂y in Ω = − π2 ,


π 2

× (0, 1), with a (·) as above.

In view of the above example we introduce the following definition. Definition 1.24 (H¨ ormander’s condition of step s) We say that a system X0 , X1 , . . . , Xq of real smooth vector fields defined in a domain Ω ⊂ Rn satisfies H¨ ormander’s condition of (weighted) step s in Ω if at every point x ∈ Ω the vectors
 X[I] x |I|6s (built evaluating at x all the standard commutators of weight 6 s) span Rn . We will also say that X0 , X1 , . . . , Xq is a system of H¨ormander vector fields of (weighted) step s. In the following we will simply say “H¨ ormander’s condition of step s”, dropping the word “weighted”, when no confusion arises. A simple compactness argument implies the following Proposition 1.25 If X0 , X1 , . . . , Xq satisfy H¨ ormander’s condition in a domain 0 Ω, then for every Ω b Ω there exists a positive integer s such that X0 , X1 , . . . , Xq satisfy H¨ ormander’s condition at step s in Ω0 . We wish to remark that Example 1.23 is quite pathological: in all interesting examples of systems of H¨ ormander vector fields, the step of H¨ormander’s condition remains bounded on the whole set where the vector fields are defined, and we do not need to apply the previous proposition. Anyhow, in view of the above proposition and since most of our results will have a local nature, the following is not restrictive: Convention 1.26 Whenever we will write that X0 , X1 , . . . , Xq is a system of H¨ ormander vector fields in Ω we will mean that: i) H¨ ormander’s condition of some step s holds in Ω, and ii) the coefficients of the vector fields are smooth up to the boundary of Ω, so that they have finite C k (Ω) norms for every k, whenever Ω is bounded.

14

H¨ ormander operators

These assumptions will be useful when we will specify the quantitative dependence of the constants in our estimates. It is time to see some concrete examples of all the notions that we have introduced in this section. Example 1.27 Here we list some examples of H¨ ormander operators. Note that, once we have a system of H¨ ormander vector fields X0 , X1 , . . . , Xq in some domain Ω, both the “operator with drift” q X Xi2 + X0 i=1

and the “sum of squares” operator q X

Xi2

i=0

are H¨ ormander operators, according to Definition 1.15. Depending on the context and the problem it can be more natural to study the first or the second. Note that, however, the step of H¨ ormander’s condition will be different for the two operators, depending on the different weight given to X0 in the two cases. (a) Sublaplacian on the Heisenberg group H1 in R3 . We have already quoted this example in the Introduction, and we will present its analog in R2n+1 in Chapter 3. In R3 3 (x, y, t), let X = ∂x + 2y∂t ; Y = ∂y − 2x∂t and L = X 2 + Y 2. Since [X, Y ] = −4∂t and the three vectors X, Y, [X, Y ] are independent at every point of R3 , H¨ ormander’s condition at step 2 holds, and L is a sum of squares of H¨ ormander vector fields. According to the above remark, we could also study the H¨ ormander operator with drift X 2 + Y, which satisfies H¨ ormander’s condition at step 3 (since in this case Y has weight 2). In this specific case, the operator which is commonly studied is the sum of squares. (b) In R2 3 (x, y), let X = ∂x ; Y = xk ∂y for some positive integer k. The two vector fields are not independent at all points x = 0, however [X, Y ] = kxk−1 ∂y [X, [. . . [X, [X, Y ]]]] = k!∂y , | {z }

commutator of length k

Basic geometry of vector fields

15

hence H¨ ormander’s condition holds, and the sum of squares 2 2 ∂xx + x2k ∂yy ,

as well as the operator with drift 2 ∂xx + xk ∂y

are H¨ ormander operators. Note that H¨ ormander’s condition holds at step k for the sum of squares operator, and at step k + 1 for the operator with drift, since in this case Y has weight 2. (c) A Kolmogorov-type operator. In Rn+1 3 (x1 , x2 , . . . , xn , t) let X1 = ∂x1 ; X0 = x1 ∂x2 + x2 ∂x3 + . . . + xxn−1 ∂xn − ∂t . Then [X1 , X0 ] = ∂x2 [[X1 , X0 ] , X0 ] = ∂x3 ··· [[[[X1 , X0 ] , X0 ] , . . .] , X0 ] = ∂xn | {z } commutator of length n



hence the vector fields X[Ii ]

n i=0

with

I0 = (0) ; I1 = (1) ; I2 = (1, 0) ; I3 = (1, 0, 0) ; . . . In = (1, 0, . . . , 0) of length n span Rn+1 , and L = X12 + X0 = ∂x21 x1 + x1 ∂x2 + x2 ∂x3 + . . . + xxn−1 ∂xn − ∂t is a H¨ ormander operator with drift. H¨ ormander’s condition holds at step 2n + 1. This is an operator of Kolmogorov type, and we will say something more about it in Chapter 3 and Chapter 6. Note that L is a highly degenerate ultraparabolic operator, since its principal part contains only 1 variable out of n + 1. (d) A somewhat similar example is the following “Bony-type” sublaplacian (quoted in [21, Remarque 3.1]). In Rn+1 3 (x1 , x2 , . . . , xn , t) let X1 = ∂t ; X2 = t∂x1 + t2 ∂x2 + . . . + tn ∂xn . Then [X1 , X2 ] = ∂x1 + 2t∂x2 + 3t2 ∂x3 + . . . + ntn−1 ∂xn [X1 , [X1 , X2 ]] = 2∂x2 + 6t∂x3 + . . . + n (n − 1) tn−2 ∂xn ··· [X, [. . . , [X1 , [X1 , X2 ]]]] = n!∂xn | {z } commutator of length n+1

16

H¨ ormander operators

 n+1 hence the vector fields X[Ii ] i=1 with I1 = (1) ; I2 = (1, 2) ; I3 = (1, 1, 2) ; . . . In+1 = (1, 1, . . . , 1, 2) of length n + 1 span Rn+1 , and 2 L = X12 + X22 = ∂tt + t∂x1 + t2 ∂x2 + . . . + tn ∂xn


2

is a “sum of squares” H¨ ormander operator. H¨ ormander’s condition holds at step n + 1. (e) Mumford operator (arising in the study of the “process of random direction”, see [129]). In R4 3 (x, y, θ, t) let σ X1 = √ ∂θ ; X0 = (cos θ) ∂x + (sin θ) ∂y + ∂t 2 for some fixed parameter σ > 0. Then σ [X1 , X0 ] = √ (− (sin θ) ∂x + (cos θ) ∂y ) 2 σ2 [X1 , [X1 , X0 ]] = ((− cos θ) ∂x − (sin θ) ∂y ) 2 hence the vector fields X1 , X0 , [X1 , X0 ] , [X1 , [X1 , X0 ]] span R4 at any point, and σ2 2 ∂ + (cos θ) ∂x + (sin θ) ∂y + ∂t 2 θθ is a H¨ ormander operator with drift. H¨ ormander’s condition holds at step 4. (f ) A similar but more complicated operator is the following, introduced by August-Zucker in [4] to study the “process of random curvature”. Let (x, y, θ, κ, t) ∈ R5 and let σ X1 = √ ∂κ ; X0 = (cos θ) ∂x + (sin θ) ∂y + κ∂θ + ∂t 2 L = X12 + X0 =

for some fixed parameter σ > 0. Then it is easy to check that a basis of R5 consisting in standard commutators is given at any point by: σ X[1] = √ ∂κ 2 X[0] = (cos θ) ∂x + (sin θ) ∂y + κ∂θ + ∂t σ X[(1,0)] = √ ∂θ 2 σ X[(0,1,0)] = √ ((sin θ) ∂x − (cos θ) ∂y ) 2 2 σ ((cos θ) ∂x + (sin θ) ∂y ) . X[(1,0,0,1,0)] = 2

Basic geometry of vector fields

17

Hence σ2 2 ∂ + (cos θ) ∂x + (sin θ) ∂y + κ∂θ + ∂t 2 κκ is a H¨ ormander operator with drift. Here H¨ ormander’s condition holds at (weighted) step 8. L = X12 + X0 =

We end this section pointing out an easy property of systems of H¨ormander vector fields which should give one more reason to expect that information along X1 , . . . , Xq be enough to get a control in any direction, in a suitable sense: Proposition 1.28 Let X1 , . . . , Xq be a system of H¨ ormander vector fields in a domain Ω of Rn , and let f : Ω → R be a function such that Xi f ≡ 0 in Ω for i = 1, 2, . . . , q. Then f is constant in Ω. Proof. Since H¨ ormander’s condition holds in Ω, for every x ∈ Ω and k = 1, 2, . . . , n we can write a finite sum X
(1.14) ∂xk = cI,k (x) X[I] x for suitable numbers cI,k (x) depending on x, I, k. On the other hand, our assumption on f implies that for any multiindex I, XI f ≡ 0 in Ω, hence also X[I] f ≡ 0 in Ω, which by (1.14) implies that ∂xk f (x) = 0 for x ∈ Ω and k = 1, 2, . . . , n so that f is constant. This result is just a first instance of getting a control on a function f by a control on its “gradient” Xf = (X1 f, . . . , Xq f ). We will meet a more general result in this direction with Theorem 1.47. 1.4

The control distance

Let us now introduce the basic notion of control distance induced by a system of vector fields. This will be a key object throughout the book, related both to the geometry of vector fields and to the properties of second order differential operators built with them. Definition 1.29 Given a system of smooth vector fields X1 , . . . , Xq in a domain Ω ⊂ Rn , for any δ > 0, let Cx,y (δ) be the class of absolutely continuous mappings ϕ : [0, 1] −→ Ω which satisfy 0

ϕ (t) =

q X

ai (t) (Xi )ϕ(t) a.e.

i=1

ϕ (0) = x, ϕ (1) = y with suitable measurable functions ai : [0, 1] → R satisfying |ai (t)| 6 δ a.e. Then define dX,Ω (x, y) = inf {δ > 0 : ∃ϕ ∈ Cx,y (δ)} ,

18

H¨ ormander operators

with the convention inf ∅ = +∞. We also define the d-balls BX (x, r) = {y ∈ Ω : d (x, y) < r} . Note that, in contrast with the usual definition of distance, we explicitly allow dX,Ω (x, y) = +∞ meaning that x and y are not connected by any curve ϕ ∈ Cxy (δ). This little misuse of the definition of distance allows to introduce dX,Ω for every Ω and every system of vector fields. We will address later the problem of finding conditions on the system of vector fields that ensure that dX,Ω (x, y) is always finite. Some explicit computations of control distances will be carried out in section 1.11. Remark 1.30 (Dependence of the distance on the domain) In the definition of dX,Ω we have used absolutely continuous mappings ϕ : [0, 1] → Ω. This means that the distance dX,Ω actually depends on the domain Ω. More precisely, let Ω0 b Ω be another domain, to compute dX,Ω0 (x, y) we can use only mappings ϕ : [0, 1] → Ω0 and therefore dX,Ω0 (x, y) > dX,Ω (x, y) for every x, y ∈ Ω0 . Notation 1.31 When there is no risk of confusion, we will drop the X and/or the Ω from the symbols dX,Ω (x, y), BX (x, r), writing d (x, y), B (x, r). We will also write Br (x) for B (x, r). Remark 1.32 (Euclidean distance) If we take the system of vector fields Xi = ∂xi for i = 1, 2, . . . , n, then dX,Ω (x, y) is (comparable to2 ) the minimal length (in Euclidean sense) of the paths inside Ω joining x to y. Note that if Ω is not convex this distance is usually greater than the restriction to Ω of the Euclidean distance in Rn . This remark again stresses the dependence of the control distance on the domain. Remark 1.33 (Distance of intermediate points on a curve) Let x, y ∈ Ω and let γ ∈ Cx,y (δ). Observe that every point of this path has distance less than δ from the endpoints. Indeed, let z = γ (t0 ) for some t0 ∈ (0, 1) and let γ e (t) = γ (t0 t). A simple computation shows that γ e ∈ Cx,z (t0 δ) so that d (x, z) 6 t0 δ < δ. Analogously, if γ ∗ (t) = γ (t0 + (1 − t0 ) t), one can check that γ ∗ ∈ Cz,y ((1 − t0 ) δ) so that d (z, y) 6 (1 − t0 ) δ < δ. Note that we have actually proved a stronger statement, namely: for every γ ∈ Cxy (δ) and every point z along this curve we have d (x, z) + d (z, y) 6 δ. Remark 1.34 (Exponential and control distance) The following inequality will be often useful: for any x0 ∈ Ω, j = 1, 2, . . . , q and t small enough, d (exp (tXj ) (x0 ) , x0 ) 6 |t| . qP in Definition 1.29 we wrote a2i ≤ δ instead of |ai | ≤ δ for every i, then for Xi = ∂xi , the control distance dX,Ω would be exactly the Euclidean distance. 2 If

Basic geometry of vector fields

19

Namely, letting x = exp (tXj ) (x0 ), the curve ϕ (s) = exp (stXj ) (x0 ) belongs to the family Cx0 ,x (|t|), hence d (x, x0 ) 6 |t|. Given a couple of points x, y, the existence of any curve ϕ ∈ Cx,y (δ) is not obvious, so that the finiteness of d (x, y) is not a fact but a problem, or better a possible property of the system of vector fields: Definition 1.35 We say that the system {X1 , . . . , Xq } satisfies the connectivity property in Ω if d (x, y) < +∞ for any couple of points x, y ∈ Ω. Before going into the connectivity problem, however, let us fix a basic fact: Proposition 1.36 The function d introduced in Definition 1.29 is a distance 3 in Ω. Proof. From the definition we read that d (x, y) > 0 and that d (x, x) = 0. Let now x, y ∈ Ω with x 6= y; we want to show that d (x, y) > 0. Let BE (x, r) ⊂ Ω be a small Euclidean ball of radius r and center x such that y ∈ / BE (x, r). Let ϕ ∈ Cx,y (δ) for some δ > 0 (if no such curve exists, then d (x, y) = +∞ and we are done) and let t0 = inf {t ∈ [0, 1] : ϕ (t) ∈ ∂BE (x, r)}. Then Z t0 X Z t0 q 0 ϕ (t) dt 6 r = |ϕ (t0 ) − x| = ai (t) (Xi )ϕ(t) dt 6 0

6δ

q X

sup

i=1 z∈BE (x,r)

0

i=1

|(Xi )z | < +∞

(1.15)

and therefore d (x, y) > Pq

i=1

r sup z∈BE (x,r)

|(Xi )z |

> 0.

(1.16)

It follows that d (x, y) = 0 if and only if x = y. The symmetry of d follows from the fact that if ϕ ∈ Cx,y (δ) then ϕ e (t) = ϕ (1 − t) belongs to Cy,x (δ). To prove d (x, z) 6 d (x, y) + d (y, z) ,

(1.17)

assume d (x, y) and d (y, z) are both finite (otherwise, there is nothing to prove), hence there exist δ1 , δ2 > 0 such that Cx,y (δ1 ), Cy,z (δ2 ) are nonempty; let ϕ1 ∈ Cx,y (δ1 ) and ϕ2 ∈ Cy,z (δ2 ). Defining    h i δ1 + δ2  1  t for t ∈ 0, δ1δ+δ  ϕ1 2   δ1 ϕ (t) = h i δ δ + δ  1 1 2 1  t− for t ∈ δ1δ+δ , 1 ,  ϕ2 2 δ2 δ2 it is easy to check that ϕ ∈ Cx,z (δ1 + δ2 ). Hence d (x, z) 6 δ1 + δ2 and passing to the infimum of δ1 > 0 such that Cx,y (δ1 ) 6= ∅ and δ2 > 0 such that Cy,z (δ2 ) 6= ∅ we get (1.17). 3 As

remarked after the definition, we are allowing d (x, y) to be infinite.

20

H¨ ormander operators

The distance d (x, y) is essentially the minimal time required to go from x to y along paths such that at every point the velocity is in the span of the vector fields X1 , . . . , Xq (with coefficients 6 1). Since, as already noted, the Euclidean distance can be defined similarly declaring admissible any unitary velocity vector, it is reasonable to expect that the control distance will be generally larger than (a constant times) the Euclidean one. This is the content of the following Proposition 1.37 (Lower bound on the control distance) Let Ω0 b Ω00 b Ω. Then: (a) there exists c > 0 such that for every x, y ∈ Ω0 , |x − y| 6 cd (x, y) . Namely,  c = max 1,

diam (Ω0 ) dist (Ω0 , ∂Ω00 )

 X q · max |(Xi )x | . i=1

x∈Ω00

(where both diam (Ω0 ) and dist (Ω0 , ∂Ω00 ) are defined by the Euclidean distance). (b) If q X

sup |(Xi )x | = K < +∞,

(1.18)

i=1 x∈Ω

then |x − y| 6 Kd (x, y) ∀x, y ∈ Ω. We will apply this proposition in two typical contexts: either when Ω = RN and the vector fields have unbounded (typically, polynomial) coefficients on the whole space, hence point (a) applies while point (b) does not; or when the vector fields are defined (and bounded) only on a bounded domain Ω ⊂ RN , hence point (b) applies, and gives a global bound in Ω, instead of a local one. Proof. To prove (a), let x, y ∈ Ω0 , δ > 0 and ϕ ∈ Cx,y (δ) (if no such ϕ exists, then d (x, y) = ∞ and there is nothing to prove). If ϕ (t) ∈ Ω00 for every t ∈ [0, 1], then Z 1 Z 1 q q X X 0 |y − x| = ϕ (t) dt = max |(Xi )x | . ai (t) (Xi )ϕ(t) dt 6 δ 0 x∈Ω00 0 i=1

i=1

00

Assume now that ϕ (t) ∈ / Ω for some t and let t0 = {inf t : ϕ (t) 6∈ Ω00 }. Since ϕ (t0 ) ∈ ∂Ω00 and x ∈ Ω0 , if r0 = dist (Ω0 , ∂Ω00 ) (Euclidean distance), we have diam (Ω0 ) diam (Ω0 ) r0 6 |x − ϕ (t0 )| |x − y| 6 diam (Ω0 ) = r0 r0 Z q diam (Ω0 ) X diam (Ω0 ) t0 0 6 = ϕ (t) dt δ max |(Xi )x | . r0 r0 x∈Ω00 0 i=1

Basic geometry of vector fields

21

In both cases we have |x − y| 6 cδ so that |x − y| 6 cd (x, y) with c as in the statement of the Proposition. (b) Under the assumption (1.18) for any x, y ∈ Ω and ϕ ∈ Cxy (δ) we can write Z 1 X Z 1 q X sup |(Xi )x | . ai (t) (Xi )ϕ(t) dt 6 δ ϕ0 (t) dt = |y − x| = x∈Ω 0

0

i=1

and the proof is complete. 1.5

The weighted control distance

Although the definition of control distance given in the previous section works for any family of vector fields, from the point of view of the theory of second order differential operators that definition is well shaped only to study “sums of squares” operators L=

q X

Xi2 .

i=1

If we are interested in the more general family of “operators with drift” L=

q X

Xi2 + X0 ,

i=1

the fact that the drift term X0 weights as a second order derivative (see Remark 1.16) is a point that affects all the subsequent theory. In particular, if we want to develop a geometry of the system of vector fields shaped on the operator with drift we have to define the control distance in a modified way with respect to what we have done in section 1.4, in order to assign weight two to X0 and weight one to the other vector fields Xi , and to extend to this more general case the basic properties of the control distance that we have proved so far. This will be done in this section. Definition 1.38 (Weighted control distance) Let X0 , X1 , . . . , Xq be a system of “weighted” vector fields in a domain Ω ⊂ Rn , as in Definition 1.17. For any δ > 0, let Cx,y (δ) be the class of absolutely continuous mappings ϕ : [0, 1] −→ Ω which satisfy 0

ϕ (t) =

q X

ai (t) (Xi )ϕ(t) a.e.

i=0

ϕ (0) = x, ϕ (1) = y with suitable measurable functions ai : [0, 1] → R, satisfying |a0 (t)| 6 δ 2 ; |ai (t)| 6 δ a.e. for i = 1, 2, . . . , q. Then define dX,Ω (x, y) = inf {δ > 0 : ∃ϕ ∈ Cx,y (δ)} , with the convention inf ∅ = +∞.

22

H¨ ormander operators

The same general remarks made in section 1.4 about the function d in the unweighted case are in order. Here we are going to point out only those facts which require extra care or need to be modified. Remark 1.39 (Distance of intermediate points along a curve) Let x, y ∈ Ω and let γ ∈ Cxy (δ). It is still true that every point of this path has distance less than δ from the endpoints. However, it is worthwhile to point out something different from the unweighted case. If z = γ (t0 ) for some t0 ∈ (0, 1)
and we √ let γ e (t) = γ (t0 t), then a simple computation shows that γ e ∈ Cxz t0 δ so that √ ∗ letting γ (t) = γ (t0 + (1 − t0 ) t), one can check d (x, z) 6 t0 δ < δ. Analogously
√ that γ ∗ ∈ Czy 1 − t0 δ so that √ d (z, y) 6 1 − t0 δ < δ. Note that, although also in this case we have proved that d (x, z) 6 δ and d (y, z) 6 δ, in contrast with what we have proved in Remark 1.33 when the drift is lacking, this time we have √ √ d (x, z) + d (z, y) 6 t0 δ + 1 − t0 δ, a quantity which is no longer 6 δ. In Chapter 9 we will come back to this issue, showing that the control distance satisfies a kind of “segment property” provided the drift is lacking. Remark 1.40 (Exponential and control distance) As we have seen in Remark 1.34, for any x0 ∈ Ω, j = 1, 2, . . . , q and t small enough, d (exp (tXj ) (x0 ) , x0 ) 6 |t| for j = 1, 2, . . . , q. On the other hand, by definition of weighted control distance, p d (exp (tX0 ) (x0 ) , x0 ) 6 |t|. The two relations can be summarized by d (exp (tpj Xj ) (x0 ) , x0 ) 6 |t| for j = 0, 1, 2, . . . , q.

(1.19)

Let us prove that also in this case d is a distance. Proposition 1.41 The function d introduced in Definition 1.38 is a distance in Ω. Proof. With reference to the proof of Proposition 1.36, let us just discuss the points that require some small modification. For x, y ∈ Ω with x 6= y; we want to show that d (x, y) > 0. Let BE (x, r) ⊂ Ω be an Euclidean ball such that y ∈ / BE (x, r). Let ϕ ∈ Cx,y (δ) for some δ > 0 (if no such curve exists, then d (x, y) = +∞ and we are done) and let t0 = inf {t ∈ [0, 1] : ϕ (t) ∈ ∂BE (x, r)}. Then Z t0 Z t0 q X r = |ϕ (t0 ) − x| = ϕ0 (t) dt 6 ai (t) (Xi )ϕ(t) dt 6 0

6δ

q X

sup

i=1 z∈BE (x,r)

|(Xi )z | + δ 2

0

i=0

sup z∈BE (x,r)

|(X0 )z | 6 max δ, δ 2

q
X

sup

i=0 z∈BE (x,r)

|(Xi )z |

Basic geometry of vector fields

23

and therefore
max d (x, y) , d2 (x, y) > Pq

i=0

r sup z∈BE (x,r)

|(Xi )z |

> 0.

(1.20)

It follows that d (x, y) = 0 if and only if x = y. The symmetry of d and the triangle inequality follows as in the unweighted case. In analogy with the unweighted case we can prove the following Proposition 1.42 (Lower bound on the control distance) (a) Let Ω0 b Ω00 b Ω. Then there exists c > 0 such that for every x, y ∈ Ω0 , |x − y| 6 cd (x, y) max (1, d (x, y)) . Namely, diam (Ω0 ) c = max 1, dist (Ω0 , ∂Ω00 ) 

 X q · max |(Xi )x | . i=0

x∈Ω00

(where both diam (Ω0 ) and dist (Ω0 , ∂Ω00 ) are defined by the Euclidean distance). (b) If sup d (x, y) = H < +∞

(1.21)

x,y∈Ω

and q X

sup |(Xi )x | = K < +∞,

(1.22)

i=0 x∈Ω

then |x − y| 6 K (1 + H) d (x, y) ∀x, y ∈ Ω. Note that in (b) we have assumed that the diameter of Ω (with respect to the control distance d) is bounded, while in the analogous Proposition proved in the unweighted case this was unnecessary. Proof. To prove (a), one can reason exactly like in the proof of Proposition 1.37, getting this time, for x, y ∈ Ω0 , δ > 0 and ϕ ∈ Cxy (δ),
|x − y| 6 c1 max δ, δ 2 with  c = max 1,

diam (Ω0 ) dist (Ω0 , ∂Ω00 )

 X q · max |(Xi )x | i=0

x∈Ω00

so that   2 |x − y| 6 c max d (x, y) , d (x, y) = cd (x, y) max (1, d (x, y)) . with c as in the statement of the Proposition.

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H¨ ormander operators

(b) Under the assumption (1.22) for any x, y ∈ Ω and ϕ ∈ Cxy (δ) we can write Z 1 X Z 1 q ai (t) (Xi )ϕ(t) dt ϕ0 (t) dt = |y − x| = 0 i=0 0 ! q X
6 δ max |(Xi )x | + δ 2 max |(X0 )x | 6 K δ + δ 2 i=1

x∈Ω00

x∈Ω00

hence |y − x| 6 Kd (x, y) (1 + d (x, y)) 6 K (1 + H) d (x, y) .

1.6

Connectivity

Let us now come to the problem of the finiteness of d. In Definition 1.29 we can choose, as a particular case, piecewise constant coefficients ai (t) such that, for suitable numbers 0 = t0 < t1 < t2 < . . . < tN = 1, on each subinterval [tj−1 , tj ] only one of the ai (t) is nonzero; in this case the curve γ can be seen as a chain of integral lines of the Xi ’s. Vice versa, any chain of integral lines of the Xi ’s can be reparametrized in this way. In conclusion: if two points x, y can be joined by a chain of integral lines of the vector fields Xi , then d (x, y) is finite. Although more general admissible curves exist, it is a convenient intuitive reference to think admissible curves in this way. Example 1.43 (1) The system X = ∂x ; Y = ∂y + ∂z does not satisfy the connectivity property in R3 . Actually, starting for instance from the point (0, 0, 0), only the points on the plane z = y are reachable by admissible curves. Hence, for instance, d ((0, 0, 0) , (0, 1, 0)) = +∞. (2) The system X = ∂x ; Y = ∂y + x∂z satisfies the connectivity property in R3 . This will follow from Theorem 1.45 but one can get the main idea from the following example: Starting from (0, 0, 0), we reach any given (a, b, c) ∈ R3 moving repeatedly along integral lines of X, Y in the following way: If b 6= 0, c   c  X exp (bY ) exp X (0, 0, 0) . (a, b, c) = exp a − b b If b = 0 and a 6= 0, c  c  (a, 0, c) = exp Y exp (aX) exp Y (0, 0, 0) . a a If a = 0 and b = 0, (0, 0, c) = exp (−cY ) exp (−X) exp (cY ) exp (X) (0, 0, 0) .

Basic geometry of vector fields

25

Note, comparing the previous two examples, how a slight difference in the coefficients of the systems {X, Y } can forbid or allow connectivity. The first example is trivial: any system of constant vector fields does not have the connectivity property unless the vectors already span Rn ; in the second example, we see how properly alternating integral lines of X and Y we can reach any point. Note that in this case we have X = ∂x , Y = ∂y + x∂t and [X, Y ] = ∂z , where X, Y, [X, Y ] span R3 . This is consistent with Theorem 1.6: moving properly along the integral lines of X, Y can produce a displacement in the approximate direction of [X, Y ], hence if we know that X, Y, [X, Y ] span R3 , we can expect connectivity. One can imagine now more involved situations where also iterated commutators are needed to give new directions. Example 1.44 Does the system X = ∂x ; Y = ∂y + x2 ∂z satisfy the connectivity property in R3 ? Note that in this case X, Y and [X, Y ] = 2x∂z do not span R3 , but X, Y and [X, [X, Y ]] = 2∂z do. So perhaps the algorithm needed to move from (0, 0, 0) to (a, b, c) will become more involved. Let us try, for instance, to join (0, 0, 0) to (1, 1, −1). One can check that: (1, 1, −1) = exp (X) exp (2Y ) exp (−X) exp (−Y ) exp (X) (0, 0, 0) so we need this time 5 arcs, instead of 3 as in the case of Example 1.43 (2), where Y = ∂y + x∂z . The previous discussion may suggest the following conjecture: If the vector fields X1 , . . . , Xq satisfy H¨ ormander condition in a domain Ω ⊂ Rn , then they satisfy the connectivity property. The conjecture is actually true, and is made precise by the following result, which constitutes the main goal of this section: Theorem 1.45 Let X0 , X1 , . . . , Xq be a system of H¨ ormander vector fields in a domain Ω ⊂ Rn . Then: (1) (Local connectivity). For any x0 ∈ Ω and any neighborhood V of x0 , (V ⊂ Ω) there exists another neighborhood of x0 , U ⊂ V such that any two points of U can be connected by a curve contained in V, which is composed by a finite number of arcs, integral curves of the vector fields Xi for i = 0, 1, . . . , q.

26

H¨ ormander operators

(2) (Global connectivity). For any couple of points x, y ∈ Ω there exists a curve joining x to y and contained in Ω, which is composed by a finite number of arcs, integral curves of the vector fields Xi for i = 0, 1, . . . , q. In particular, d (x, y) < ∞ for every x, y ∈ Ω. The geometric idea behind the proof of this theorem is to generalize the statement of Theorem 1.6, in order to show that one can move in the approximate direction of any higher order commutator of the vector fields Xi just exploiting in a suitable way their integral lines. This idea is made precise in Theorem 1.49. Although the statement of this theorem is purely topological, the way we will prove it will give us some precise quantitative information, expressed in terms of “quasiexponential maps” and the control distance (see Theorem 1.48), which has independent interest and will be the object of further development in section 1.7. Definition 1.46 (Quasiexponential maps) For a multiindex I = (i1 , . . . , i` ), with i1 , . . . , i` ∈ {0, 1, . . . , q}, and t > 0 small enough, we define the quasiexponential map C` (t, XI ), in the following iterative way: C1 (t, Xi1 ) = exp (tpi1 Xi1 ) ; C2 (t, Xi1 Xi2 ) = exp (−tpi2 Xi2 ) exp (−tpi1 Xi1 ) exp (tpi2 Xi2 ) exp (tpi1 Xi1 ) ; ···

(1.23)

C` (t, Xi1 Xi2 · · · Xi` ) −1

= C`−1 (t, Xi2 · · · Xi` )

exp (−tpi1 Xi1 ) C`−1 (t, Xi2 · · · Xi` ) exp (tpi1 Xi1 )

(recall that pi is the weight of Xi , that is p0 = 2 and pi = 1 for i = 1, . . . , q). We will also denote these maps by C`(I) (t, XI ) (x). (Note that ` (I) is the length of I, which is different from the weight |I|). Remark 1.47 The map x 7−→ C`(I) (t, XI ) (x) is the composition of a fixed number, depending on ` (I), of maps of the kind exp (±tpi Xi ) with i = 0, 1, 2, . . . , q. This ensures that it is well defined, smooth and invertible for t small enough; in particular, for every Ω0 b Ω there exists T > 0 such that for |t| < T the map is well defined and injective in Ω0 . We already know (see Remark 1.40) that d (exp (tpi Xi ) (x0 ) , x0 ) 6 |t|. Therefore, if x = C` (t, Xi1 Xi2 . . . Xi` ) (x0 ), this means that the points x0 , x can be joined by a curve composed by a finite number of integral curves of the Xi ’s, so that d (C` (t, Xi1 Xi2 . . . Xi` ) (x0 ) , x0 ) 6 c |t|

(1.24)

for some constant c only depending on the number `. We are going to show that every point x in a small neighborhood of x0 can be obtained in this way: this will imply the connectivity property.

Basic geometry of vector fields

27

Since the vector fields Xi satisfy H¨ ormander’s condition at step s in Ω, for any fixed x0 ∈ Ω, we can choose a set B of n multiindices I with |I| 6 s such that n
o X[I] x 0 I∈B

is a basis of Rn . Since the vectors X[I] x vary smoothly with x, X[I] x I∈B will be actually a basis of Rn for any x in a suitable neighborhood of x0 . 0 More precisely ifn Ω0 b Ω o and x0 ∈ Ω there exists δ > 0, depending on
dist (Ω0 , ∂Ω) , det X[I] x and the C s (Ω) norms of the coefficients of the vec0

I∈B

tor fields Xi (with s the step of H¨ ormander condition), such that for |x − x0 | < δ 
det X[I] x I∈B 6= 0. Let us now define the maps ( EI (t) =


C`(I) t1/|I| , XI if t > 0  −1 1/|I| C`(I) |t| , XI if t < 0

(1.25)

for any I ∈ B, and the map EBX (x, h) EBX (x, ·) : (h1 , h2 , . . . , hn ) 7→ EI1 (h1 ) EI2 (h2 ) . . . EIn (hn ) (x) ,

(1.26)

where B = {I1 , . . . , In }. We will prove the following Theorem 1.48 (Quasiexponential maps diffeomorphism) Let Ω0 b Ω00 b Ω, 0 x0 ∈ Ω and let X[Ij ] I ∈B be any family of n commutators (with |Ij | 6 s) which j are a basis of Rn at x0 . Then there exist constants δ1 , δ2 > 0 such that the map h 7→ EBX (x0 , h) is a C 1 diffeomorphism of a neighborhood of the origin {h : |h| < δ1 } onto a neighborhood U (x0 ) of x0 containing {x : |x − x0 | < δ2 } . It is also a C 1 map in the joint variables x, h for x ∈ Ω0 and |h| < δ1 . Finally, it satisfies n X
1/|I | d EBX (x, h) , x 6 c |hj | j . (1.27) j=1

The constant c depends only on n, s; the numbers δ1 , δ2 and the C 1 norm of the diffeomorphism depend on the C k (Ω00 ) norms of the coefficients of the vector fields X0 , X1 , . . . , Xq for some k only depending on n, δ1 , δ2 also depend o n s; the
numbers 0 00 . on dist (Ω , ∂Ω ) and on a lower bound for det X[Ij ] x 0

Ij ∈B

The proof of Theorem 1.48 will proceed in several steps. We start studying the maps C`(I) (t, XI ) defined above. The key result about them is the following: Theorem 1.49 (Approximation of commutators) For any Ω0 b Ω00 b Ω, any multiindex I with |I| 6 s, and x ∈ Ω0 we have  
C`(I) (t, XI ) (x) = x + t|I| X[I] x + o t|I| as t → 0 (1.28)  
−1 C`(I) (t, XI ) (x) = x − t|I| X[I] x + o t|I| as t → 0

28

H¨ ormander operators


where the remainder o t|I| is actually a map x 7→ f (t) (x) such that |f (t) (x)| 6 ct|I|+1 for every x ∈ Ω0 , 0 6 t 6 δ, with c depending on the C k (Ω00 ) norms of the coefficients of X0 , . . . , Xq for some k only depending on ` (I), and δ also depending on dist (Ω0 , ∂Ω00 ). Note that Theorem 1.6 is a particular case of this theorem for |I| = 2. The above theorem says that moving in a suitable way along a chain of integral lines of the Xi ’s can give, as a net result, a displacement approximately in the direction of any commutator of the vector fields. Since the commutators span, this will imply that we can reach any point in this way, locally and therefore, by connectedness of Ω, globally. In turn, the proof of the above theorem is organized in several steps. Let us first state an abstract result: Proposition 1.50 Let Ω0 b Ω. For some ε > 0 and a positive integer ` let F : ` (−ε, ε) ×Ω0 → Rn be a smooth function such that F (t1 , t2 , . . . , t` ) (x) = x for every 0 x ∈ Ω as soon as at least one of the tj is equal to 0. Then, for (t1 , t2 , . . . , t` ) → 0 we have: F (t1 , . . . , t` ) (x) = x + t1 · · · t`

∂`F (0, . . . , 0) (x) + o (t1 · · · t` ) ∂t1 · · · ∂t`

∂F ∂`F (t1 , . . . , t` ) (x) = t2 · · · t` (0, . . . , 0) (x) + o (t2 · · · t` ) ∂t1 ∂t1 · · · ∂t`

(1.29)

(1.30)

where the term o (t1 · · · t` ) in (1.29) is a map x 7→ f (t, x) such that |f (t, x)| 6 |t1 · · · t` |

q t21 + · · · + t2` kF kC `+1 ((−ε,ε)` ×Ω0 )

for every x ∈ Ω0 , and an analogous bound holds for the function o (t2 · · · t` ) in (1.30). Proof. Since F (t1 , . . . , t` ) (x) = x if ti = 0 for some i, then ∂F (t1 , . . . , t` ) (x) = 0 if ti = 0 for some i 6= j. ∂tj

Basic geometry of vector fields

29

It follows that Z t1 ∂F F (t1 , . . . , t` ) (x) = x + (u1 , t2 , . . . , t` ) (x) du1 ∂t 1 0  Z t1  ∂F ∂F =x+ (u1 , t2 , . . . , t` ) (x) − (u1 , 0, t3 , . . . , t` ) (x) du1 ∂t1 ∂t1 0 Z t1 Z t2 2 ∂ F (u1 , u2 , t3 . . . , t` ) (x) du2 du1 = · · · (1.31) =x+ ∂t 1 ∂t2 0 0 Z t1 Z t` ∂`F (u1 , . . . , u` ) (x) du` · · · du1 =x+ ··· 0 0 ∂t1 · · · ∂t`  Z t1 Z t`  ∂`F =x+ ··· (0, . . . , 0) (x) + o (1) du` · · · du1 ∂t1 · · · ∂t` 0 0 = x + t1 · · · t`

∂`F (0, . . . , 0) (x) + o (t1 · · · t` ) ∂t1 · · · ∂t`

where ∂`F ∂`F (u1 , . . . , u` ) (x) − (0, . . . , 0) (x) |o (1)| = ∂t1 · · · ∂t` ∂t1 · · · ∂t` q 6 t21 + · · · + t2` kF kC `+1 ((−ε,ε)` ×Ω0 ) . This proves (1.29). Also, if in (1.31) we start again with the intermediate identity Z t1 Z t` ∂`F (u1 , u2 , . . . , u` ) (x) du` · · · du1 F (t1 , t2 , . . . , t` ) (x) = x + ··· 0 0 ∂t1 ∂t2 . . . ∂t` and we take the derivative ∂t1 , we obtain Z t2 Z t` ∂F ∂`F (t1 , . . . , t` ) (x) = ··· (t1 , u2 , . . . , u` ) (x) du` · · · du2 ∂t1 0 0 ∂t1 · · · ∂t`  Z t2 Z t`  ∂`F = ··· (0, . . . , 0) (x) + o (1) du` · · · du2 ∂t1 · · · ∂t` 0 0 = t2 . . . t`

∂`F (0, . . . , 0) (x) + o (t2 · · · t` ) , ∂t1 · · · ∂t`

which is (1.30). We now come back to the quasiexponential maps. For a fixed multiindex I = (i1 , i2 , . . . , i` ) such that |I| 6 s and {i1 , i2 , . . . , i` } ⊂ {0, 1, 2, . . . , q} , let us define recursively still another family of maps. Fix Ω0 b Ω. Then, for t1 , . . . , t` > 0 and small enough, and x ∈ Ω0 , let C1 (t1 ) (x) = exp (t1 Xi1 ) (x) C2 (t1 , t2 ) (x) = exp (−t2 Xi2 ) exp (−t1 Xi1 ) exp (t2 Xi2 ) exp (t1 Xi1 ) (x) .. . (1.32) C` (t1 , . . . , t` ) (x) = −1

C`−1 (t2 , . . . , t` )

exp (−t1 X1 ) C`−1 (t2 , . . . , t` ) exp (t1 X1 ) (x)

30

H¨ ormander operators

where ` = ` (I). The notation C` (t1 , . . . t` ) (x) leaves understood the dependence of the particular choice of the multiindex I = (i1 , i2 , . . . , i` ), which we now think as fixed. Note that C` (t, XI ) = C` (tpi1 , tpi2 , . . . , tpi` ) .

(1.33)

The previous Proposition implies the following: Theorem 1.51 For any Ω0 b Ω00 b Ω, any multiindex I with |I| 6 s, and x ∈ Ω0 we have C` (t1 , . . . t` ) (x) = x + t1 · · · t`

∂ ` C` (0, . . . , 0) (x) + o (t1 · · · t` ) ∂t1 · · · ∂t`

(1.34)

as (t1 , . . . , t` ) → 0. In particular: C` (t, XI ) (x) = C` (tpi1 , . . . , tpi` ) (x) `

(1.35)

∂ C` ∂t1 · · · ∂t`
as t → 0, where the remainder o t|I| is a map x 7→ f (t, x) such that |f (t, x)| 6 ct|I|+1 for every x ∈ Ω0 , 0 6 t 6 δ, with c depending on the C k (Ω00 ) norms of the coefficients of X0 , . . . , Xq for some k only depending on `, and δ also depending on dist (Ω0 , ∂Ω00 ). = x + t|I|

  (0, . . . , 0) (x) + o t|I|

Proof. We start by noting that C` (t1 , . . . , t` ) reduces to the identity if at least one component of (t1 , . . . , t` ) vanishes. The easy proof of this fact is left to the reader. By Proposition 1.2, the function (t1 , . . . , t` , x) 7→ C` (t1 , . . . , t` ) (x) is smooth in a neighborhood of (x0 , 0) and therefore (1.34) follows from Proposition 1.50. The identity (1.35) in Theorem 1.51 will imply (1.28) as soon as we prove that
∂ ` C` (0, . . . , 0) (x) = X[I] x . ∂t1 · · · ∂t`

(1.36)

In order to prove (1.36), yet another abstract Lemma is useful: Lemma 1.52 Let O be an open subset of Rn and let A, B : (−ε, ε) × O → Rn be two smooth functions such that for every x ∈ O we have A (0, x) = x and B (0, x) = x. Then, for every x ∈ O and t sufficiently small A (t, ·), B (t, ·) are invertible. Moreover, if A−1 (t, ·) and B −1 (t, ·) denote their inverses and
F (t, s, x) = A−1 t, B −1 (s, A (t, B (s, x))) then ∂2A ∂B ∂2B ∂A ∂2F (0, 0, x) = (0, x) (0, x) − (0, x) (0, x) . ∂t∂s ∂x∂t ∂s ∂s∂x ∂t

Basic geometry of vector fields

Here we used a compact matrix notation, where for instance for the Jacobian of the map x 7→ ∂A ∂t (0, x).

31 ∂2A ∂x∂t

(0, x) stands

Proof. Since ∂A ∂x (0, x) = In (the n × n identity matrix) it follows that for every t sufficiently small A (t, ·) is invertible; the same is true for B (t, ·). Then
∂F ∂A−1 (0, s, x) = 0, B −1 (s, A (0, B (s, x))) ∂t ∂t
∂B −1 ∂A ∂A−1 0, B −1 (s, A (0, B (s, x))) (s, A (0, B (s, x))) (0, B (s, x)) . + ∂x ∂x ∂t Since A (0, x) = x, B −1 (s, A (0, B (s, x))) = B −1 (s, B (s, x)) = x and In , the above equation reduces to

∂A−1 ∂x

(0, x) =

∂F ∂A−1 ∂B −1 ∂A (0, s, x) = (0, x) + (s, B (s, x)) (0, B (s, x)) . ∂t ∂t ∂x ∂t Let us compute ∂2F (0, 0, x) ∂s∂t   ∂ 2 B −1 ∂ 2 B −1 ∂B ∂A = (0, B (0, x)) + (0, B (0, x)) (0, x) (0, B (0, x)) ∂s∂x ∂x2 ∂s ∂t ∂B −1 ∂2A ∂B + (0, B (0, x)) (0, B (0, x)) (0, x) . ∂x ∂x∂t ∂s Since

∂B −1 ∂x

(0, B (0, x)) = In and

∂ 2 B −1 ∂x2

(0, B (0, x)) = 0 we have

∂2F ∂ 2 B −1 ∂A ∂2A ∂B (0, 0, x) = (0, x) (0, x) + (0, x) (0, x) . ∂s∂t ∂s∂x ∂t ∂x∂t ∂s Finally since B −1 (s, B (s, x)) = x a simple computation shows that − ∂B ∂s (0, x) and therefore

∂B −1 ∂s

(0, x) =

∂2B ∂A ∂2A ∂B ∂2F (0, 0, x) = − (0, x) (0, x) + (0, x) (0, x) . ∂s∂t ∂s∂x ∂t ∂x∂t ∂s

Proof of Theorem 1.49. As already noted, in order to prove (1.28) we are only left to prove (1.36). This will be done by induction on `. For ` = 1 this is trivial, since by (1.2) ∂C1 ∂ (0) (x) = exp (tXi ) (x) = (Xi )x . ∂t ∂t t=0 Let ` > 2 and assume the theorem holds for ` − 1. Let I = (i1 , i2 . . . , i` ) = (i1 , I 0 ) , A (t2 , . . . , t` , x) = C`−1 (t2 , . . . , t` ) (x) , B (t1 , x) = exp (t1 Xi1 ) (x) .

32

H¨ ormander operators

Applying Lemma 1.52 to A, B, with respect to the variables t1 , t2 (regarding t3 , . . . , t` as parameters), gives: ∂ 2 C` ∂2B ∂A ∂2A ∂B (0, 0, t3 , . . . , t` ) (x) = − (0, x) (0, x) + (0, x) (0, x) ∂t1 ∂t2 ∂t1 ∂x ∂t2 ∂x∂t2 ∂t1 =−

∂ (Xi1 )x ∂C`−1 ∂ ∂C`−1 (0, t3 , . . . , t` ) (x) + (0, t3 , . . . , t` ) (x) (Xi1 )x ∂x ∂t2 ∂x ∂t2

and taking the remaining ` − 2 derivatives in 0 we obtain ∂ ` C` (0, . . . , 0) ∂t1 · · · ∂t` =−

∂ (Xi1 )x ∂ `−1 C`−1 ∂ ∂ `−1 C`−1 (0, . . . , 0) (x) + (0, . . . , 0) (x) (Xi1 )x . ∂x ∂t2 · · · ∂t` ∂x ∂t2 · · · ∂t`

Since, by inductive assumption,   
∂ `−1 C`−1 (0, . . . , 0) (x) = Xi2 , . . . Xi`−1 , Xi` x = X[I 0 ] x ∂t2 · · · ∂t` we obtain

∂ (Xi1 )x ∂ ∂ ` C` (0, . . . , 0) = − X[I 0 ] x + X[I 0 ] x (Xi1 )x ∂t1 · · · ∂t` ∂x ∂x

= Xi1 X[I 0 ] − X[I 0 ] Xi1 x = X[I] x , and (1.36) follows. As already noted, Theorem 1.51 and (1.36) give  
C`(I) (t, XI ) (x) = x + t|I| X[I] x + o t|I| . Since C` (t1 , . . . , t` )

−1

(x) −1

= exp (−t1 X1 ) C`−1 (t2 , . . . , t` )

exp (t1 X1 ) C`−1 (t2 , . . . , t` ) (x)

−1 C|I|

the analogous result for follows in a similar way switching the roles of A and B in the previous argument. To prove Theorem 1.48, we will use the maps EI (t) defined in (1.25) for I ∈ B, 
where X[I] x I∈B is a basis of Rn for every x in some fixed neighborhood of x0 ∈ Ω0 . By Theorem 1.49, we can write the following expansions: for t → 0+
EI (t) (x) = x + t X[I] x + o (t) ; (1.37) for t → 0− |I|  −1 
1/|I| 1/|I| X[I] x + o (t) EI (t) (x) = C`(I) |t| , XI (x) = x − |t|
= x + t X[I] x + o (t) ;

Basic geometry of vector fields

33

so that (1.37) holds for t → 0± . Also note that, by (1.24), the following holds: 1/|I|

d (EI (t) (x0 ) , x0 ) 6 c`(I) |t|

(1.38)

We are now ready for the: Proof of Theorem 1.48. First of all, let us check that the map (x, h) 7−→ EI1 (h1 ) EI2 (h2 ) · · · EIn (hn ) (x)

(1.39)

is of class C 1 for x ∈ Ω0 and |h| 6 δ, for some δ > 0. This will follow, by composition, if we prove that for any multiindex I, the map (t, x) 7→ EI (t) (x) is C 1 . This map is obviously smooth in x, and has continuous derivative with respect to t for t 6= 0, the problem is for t = 0 due to the fractional power t1/|I| appearing in the definition of EI (t) (x). The expansion (1.37) shows that there exists
∂EI (1.40) (0) (x) = X[I] x . ∂t It remains to prove that
∂EI (1.41) (t) (x) → X[I] x for t → 0. ∂t Let |I| = r. By (1.25) and (1.33) we have   g (t) ≡ EI (t) (x) = C`(I) tpk1 /r , . . . , tpk` /r (x) (1.42) where each pki is the weight of a vector field (pki = 1 or 2), and pk1 + pk2 + . . . + pk` = r. 0




Since g (0) = X[I] x (by (1.37 which follows by Theorem 1.49), we have to show that g 0 (t) → g 0 (0) as t → 0. Since g is continuous at t = 0, if we show that the limit of g 0 (t) as t → 0 is finite, this also implies that this limit is g 0 (0). However, as noted in the proof of Theorem 1.51, the map C` (t1 , . . . , t` ) satisfies the assumptions of Proposition 1.50, hence by (1.30) we can write: ∂ ` C` ∂C` (t1 , . . . , t` ) (x) = t2 · · · t` (0, . . . , 0) (x) + o (t2 · · · t` ) . ∂t1 ∂t1 · · · ∂t` Then, applying (1.43) to every j-derivative of C` we obtain `  X pk ∂C`  pk /r g 0 (t) = t 1 , . . . , tpk` /r (x) j tpkj /r−1 ∂tj r j=1 =

` X pkj j=1

=

` X pkj j=1

=

r

r

` X ∂C` j=1

∂tj

tpkj /r−1



P  P ∂C` (0, . . . , 0) (x) t i6=j pki /r + o t i6=j pkj /r ∂tj

tpkj /r−1



  ∂C` (0, . . . , 0) (x) t1−pkj /r + o t1−pkj /r ∂tj `

(0, . . . , 0) (x)

(1.43)





X ∂C` pkj pk [1 + o (1)] → (0, . . . , 0) (x) j as t → 0. r ∂t r j j=1

34

H¨ ormander operators

P` pkj ` So that g 0 (t) → j=1 ∂C ∂tj (0, . . . , 0) (x) r as t → 0. This allows to conclude that the map (1.39) is C 1 . To prove that it is a diffeomorphism from a neighborhood of the origin onto a neighborhood of x0 , it is enough to show that the Jacobian determinant of EBX (x0 , ·) at the origin is nonzero. Since EIk(0) is the identity,
∂EIi (0) ∂ = EI1 (h1 ) EI2 (h2 ) · · · EIp (hn ) (x0 ) (x0 ) = X[Ii ] x 0 ∂hi ∂hi h=0 Hence the Jacobian of the map h 7→ EI1 (h1 ) EI2 (h2 ) · · · EIn (hn ) (x0 )
at zero is the matrix having as rows the vectors X[Ii ] x and since these vectors 0 are a basis of Rn , the Jacobian is nonsingular. Moreover, the same Jacobian is uniformly continuous for x ∈ Ω0 , |h| 6 δ; therefore from the standard proof of the inverse mapping theorem (see e.g. [143, p. 221]) one can see that our map is a diffeomorphism of a neighborhood of the origin {h : |h| < δ1 } onto a neighborhood U (x0 ) of x0 containing {x : |x − x0 | < δ2 } , with δ1 , δ2 depending on the quantities specified in the statement of the theorem. Finally, inequality (1.27) follows by (1.38) and the triangle inequality. Thanks to the above theorem we can now prove the main result in this section, i.e. the connectivity theorem: Proof of Theorem 1.45. 1. Fix x0 ∈ Ω and a neighborhood V (x0 ). With the same notation of Theorem 1.48, let  Uδ (x0 ) = EBX (x0 , h) : |h| < δ (here δ is the number that in Theorem 1.48 is δ1 ). Then, for any x1 , x2 ∈ Uδ (x0 ) we can write xi = EBX (x0 , hi ) with |hi | < δ (i = 1, 2); this means that both x1 and x2 can be joined to x0 by a curve γ (t) , t ∈ [0, 1], composed by integral curves of the vector fields X1 , . . . , Xq . Now, we cannot assure that every intermediate point γ (t) along this curve still belongs to Uδ (x0 ), but by definition of EBX (x0 , h) and (1.27) we can write, for any t ∈ [0, 1] n X δ 1/|Ij | 6 cδ 1/s . (1.44) d (γ (t) , x0 ) 6 c j=1

Since by Proposition 1.42 |x − x0 | n 6 cd (x, x0 ), the curveo is contained in Vδ (x0 ) = x : |x − x0 | < c1 δ 1/s , which in turn is contained in V (x0 ) for δ small enough. Finally, since EBX (x0 , ·) is a diffeomorphism, Uδ (x0 ) is diffeomorphic to the Euclidean ball BE (0, δ), hence Uδ (x0 ) is actually a neighborhood of x0 , and our assertion is proved choosing U = Uδ (x0 ). Note that by construction Uδ (x0 ) ⊂ Vδ (x0 ) ⊂ V , since every point in Uδ (x0 ) is the endpoint of a curve contained in Vδ (x0 ). 2. Now we can cover any compact connected subset Ω0 of Ω with a finite number of neighborhoods U (xi ) ⊂ V (xi ) ⊂ Ω in such a way that any two points of Ω0 can be joined by a curve as above, contained in the union of the sets V (xi ), and therefore in Ω.

Basic geometry of vector fields

1.7

35

Other properties related to connectivity

We are now going to draw some quantitative consequences of Theorem 1.48. These consequences, together with what we have already proved about the control distance, allow to state the following: Theorem 1.53 (Control vs. Euclidean distance) Let X1 , . . . , Xq be a system of H¨ ormander vector fields in a domain Ω ⊂ Rn . Then, for any Ω0 b Ω00 b Ω there exist constants c1 , c2 > 0 such that c1 |x − y| 6 d (x, y) 6 c2 |x − y|

1/s

for any x, y ∈ Ω0

(1.45)

where s is the step of H¨ ormander’s condition. The same is true if X0 , X1 , . . . , Xq is a system of H¨ ormander vector fields where X0 has weight 2, and d is the corresponding weighted control distance. In particular, the topology induced by d in Ω is the Euclidean topology. The constants c1 , c2 depend on dist (Ω0 , ∂Ω00 ), the C k (Ω00 ) norms of the coefficients of X0 , . . . , Xq for some k only depending on s, n, and the number 4
 ∆Ω0 ≡ inf 0 max det X[I] x I∈B , x∈Ω

B

where B ranges over all n-tuples of multiindices (I1 , . . . , In ) with |Ij | 6 s. Note that the second inequality in (1.45) can be seen also as a quantitative formulation of the connectivity theorem and relies on the validity of H¨ormander’s condition. In section 1.11 we will see examples of explicit computation of the control distance induced by some systems of vector fields, showing a behavior consistent with (1.45). Proof. To prove the second inequality, fix x0 ∈ Ω0 , and the map EBX (x0 , h)  consider defined in (1.26), for a suitable choice of the basis X[I] I∈B , namely we choose B n
o > ∆Ω0 . Since, by Theorem 1.48, h 7−→ E X (x0 , h) is such that det X[I] x B 0 I∈B a diffeomorphism, there exist positive constants k1 , k2 such that, for x = EBX (x0 , h) in a suitable neighborhood of x0 , we have: k1 |x − x0 | 6 max |hi | 6 k2 |x − x0 | . i=1,...,n

1/s

1/s

Hence, by (1.44), d (x, x0 ) 6 c |h| 6 c0 |x − x0 | for any x with |x − x0 | < r and 0 a suitable r depending on Ω , where c0 is locally uniformly bounded with respect to 1/s x0 . Then one can also say that there exists r > 0 such that d (x, y) 6 c |x − y| for any x, y ∈ Ω0 with |x − y| < r. This is enough to say that the function y 7→ d (x, y) is continuous on Ω0 (and, by the same reasoning, on some slightly larger domain) with respect to the Euclidean distance, hence is bounded on Ω0 and we can write, 4 Since

H¨ ormander’s condition at step s holds in Ω, by compactness we have ∆Ω0 > 0.

36

H¨ ormander operators

for any x, y ∈ Ω0 with |x − y| > r, K 1/s |x − y| . r1/s Therefore the second inequality (1.45) actually holds for all x, y ∈ Ω0 . The first inequality in (1.45) is contained in Proposition 1.37 and, for the drift case, in Proposition 1.42, since we have just proved that supx,y∈Ω0 d (x, y) is finite. Then inequalities (1.45) in terms of balls rewrite as d (x, y) 6 K 6

s

BE (x, (R/c2 ) ) ⊂ BX (x, R) ⊂ BE (x, R/c1 ) 0

(1.46)

0

for every x ∈ Ω and R 6 R0 (depending on Ω ). In particular, the topology induced by d is the Euclidean topology. More precisely, the two topologies coincide in Ω (although the constants c1 , c2 also depend on Ω0 ). Another fact related to the connectivity theorem is the possibility of getting a pointwise control of the increment of a function f by means of its “gradient” Xf = (X1 f, . . . , Xq f ) . The following result, although quite easy to prove, will be a powerful tool in the following. In order to state the result we have to distinguish the unweighted and weighted case. Theorem 1.54 (“Lagrange

Theorem”) Let X1 , . . . , Xq be a system of  n 1 H¨ ormander vector fields in a domain Ω ⊂ R and let f ∈ C BX (x0 , r) , with BX (x0 , r) b Ω (here we are considering the unweighted control distance). Then there exist z ∈ BX (x0 , r) and A ∈ Rq such that √ f (x) − f (x0 ) = (Xf ) (z) · A with |A| 6 q d (x0 , x) . In particular, |f (x) − f (x0 )| 6

√

q d (x, x0 ) ·

|Xf |

sup

∀x ∈ BX (x0 , r) ,

(1.47)

BX (x0 ,r)

|f (x) − f (y)| 6

√

q d (x, y) ·

sup BX (x0 ,r)

where |Xf | =

qP q

i=1

|Xf |

 r ∀x, y ∈ BX x0 , 3

2

(Xi f ) .

Proof. Let x ∈ BX (x0 , r), d (x, x0 ) = δ < r. For every fixed 0 < ε < r − δ, there exists γ ∈ Cx0 x (δ + ε). Thanks to Remark 1.33 all the points of the path γ (t) are inside the ball BX (x0 , r). Then Z 1 Z 1X q d (f (γ (t))) dt = ai (t) (Xi )γ(t) · ∇f (γ (t)) dt f (x) − f (x0 ) = 0 i=1 0 dt Z 1X q q X = ai (t) (Xi f ) (γ (t)) dt = ai (t0 ) (Xi f ) (γ (t0 )) 0

i=1

i=1

Basic geometry of vector fields

37

for some t0 ∈ [0, 1], so letting zε = γ (t0 ) and Aε = (a1 (t0 ) , . . . , aq (t0 )) we have that for every ε > 0 √ f (x) − f (x0 ) = (Xf ) (zε ) · Aε with |Aε | 6 q (δ + ε) , zε ∈ BX (x0 , δ) . By compactness, we can find A, z such that f (x) − f (x0 ) = (Xf ) (z) · A with |A| 6

√

qδ =

√

qd (x0 , x)

and z ∈ BX (x0 , δ) ⊂ BX (x0 , r) . This also implies
(1.47). The last inequality
2 follows applying the first one to the ball BX x, 3 r , which for x, y ∈ BX x0 , 3r is contained in BX (x0 , r) and contains y. Remark 1.55 Although the previous proof does not explicitly exploit the connectivity property, the information contained in (1.47) becomes relevant in view of Theorem 1.53. Since BX (x0 , R) ⊃ BE (x0 , c1 Rs ), we read from (1.47) that the gradient Xf allows to control the increment of the function f in a full Euclidean neighborhood of x0 , and not just in particular directions. In the weighted case the previous result takes the following form: Theorem 1.56 (“Lagrange Theorem” – drift case) Let X0 , X1 , . . . , Xq be a system of H¨ ormander vector fields in a domain Ω ⊂ Rn , where X0 is  a drift of weight  1 2 and d is the corresponding weighted control distance. Let f ∈ C BX (x0 , r) with BX (x0 , r) b Ω, then |f (x) − f (x0 )| 6

√

qd (x, x0 ) ·

v u q uX 2 2 t (Xi f ) + d (x, x0 ) ·

sup BX (x0 ,r)

i=1

|X0 f |

sup BX (x0 ,r)

for any x ∈ BX (x0 , r) , while |f (x) − f (y)| 6

√

qd (x, y) ·

sup BX (x0 ,r)

v u q uX 2 2 t (Xi f ) + d (x, y) · i=1

sup

|X0 f |

BX (x0 ,r)


for any x, y ∈ BX x0 , 3r . 1.8

Maximum principles for degenerate elliptic operators

In the last part of this chapter we focus on several issues related to maximum principles for second order differential operators with nonnegative characteristic form. Although we have in mind the application of these results to H¨ormander operators, in some cases these results can be formulated in the more general setting of degenerate elliptic operators, so we will state them in this form. Theorem 1.57 (Weak maximum principle) Let n n X X Lu = aij (x) uxi xj + ak (x) uxk + a (x) u i,j=1

k=1

(1.48)

38

H¨ ormander operators n

where {aij (x)}i,j=1 is a symmetric nonnegative definite matrix at any point of a bounded open set Ω ⊂ Rn , and assume that: (i) a (x) 6 0 in Ω and a is bounded on Ω; (ii) for some i0 ∈ {1, 2, . . . , n} ai0 i0 (x) > c0 > 0 in Ω;

(1.49)

(iii) for this index i0 the
function ai0 is bounded on Ω. Let u ∈ C 2 (Ω) ∩ C 0 Ω , then: (a) if Lu > 0 in Ω, then max u 6 max u+ ; Ω

∂Ω

(b) if Lu = 0 in Ω, then max |u| 6 max |u| . Ω

∂Ω

We can refer to assumption (1.49) saying that the (possibly degenerate) elliptic operator L is uniformly not completely degenerate. Note that no regularity assumption on the coefficients aij , ak , a is required. Applying the weak maximum principle on every ball BR (0) and letting R → ∞ we also have the following useful Corollary 1.58 Assume that L satisfies the assumptions (i)-(iii) of Theorem 1.57 on every bounded domain Ω ⊂ Rn . Let u ∈ C 2 (Rn ) satisfy  Lu > 0 in Rn u (x) → 0 as |x| → ∞. Then u (x) 6 0 in Rn . Remark 1.59 If the operator L has the form L=

q X

Xi2 + X0 + a

(1.50)

i=1

with Xi =

n X

bij (x) ∂xj for i = 0, 1, . . . , q

j=1

and coefficients bij at least C 1 , then we can rewrite L in the form (1.48). Namely, we have: q n n X X X Lu = Xk2 u + X0 u + au = aij uxi xj + aj uxj + au (1.51) i,j=1

k=1

j=1

with: aij =

q X k=1

bki bkj ,

aj =

q n X X i=1 k=1

bki ∂xi bkj + b0j .

Basic geometry of vector fields

39

In particular, n X

aij ξi ξj =

i,j=1

q n X X

bki ξi bkj ξj =

i,j=1 k=1

q n X X k=1

!2 bki ξi

,

(1.52)

i=1

n

hence the matrix {aij }i,j=1 is symmetric and nonnegative. To check condition Pq 2 (1.49), note that ajj (x) = following we will ofi=1 bij (x) . Throughout the Pn ten deal with operators L of type (1.50), with X1 = ∂x1 + j=2 b1j (x) ∂xj . In this special case, a11 (x) = 1 +

q X

2

bi1 (x) > 1 in Ω

i=2 n

and condition (1.49) is fulfilled. The class of operators (1.48) with {aij }i,j=1 symmetric and nonnegative is actually larger than (1.50) 5 . However, throughout the book we will focus on the latter. A simple consequence of Theorem 1.57 and the above remark is the following: Corollary 1.60 (Weak maximum principle for degenerate operators) Let Lu =

q X

Xi2 u + X0 u + au

(1.53)

i=1

where Xi are smooth real vector fields in Ω, for some bounded open set Ω ⊂ Rn , and assume that: (i) a (x) 6 0 in Ω and a is bounded on Ω; (ii) for some j ∈ {1, 2, . . . , n} the uniform positivity condition q X

2

bij (x) > c0 > 0 in Ω

i=1

Pn


holds, where Xi = j=1 bij (x) ∂xj for i = 0, 1, . . . , q. Let u ∈ C 2 (Ω) ∩ C 0 Ω , then: (a) if Lu > 0 in Ω, then max u 6 max u+ ; Ω

∂Ω

(b) if Lu = 0 in Ω, then max |u| 6 max |u| . Ω

∂Ω

To prove Theorem 1.57 we need an algebraic fact: Lemma 1.61 Let A, B be two n × n real symmetric, nonnegative definite matrices. Then Tr (A · B) > 0. 5 A (not elementary!) example of operator of type (1.48) which cannot be written in the form (1.50) can be found in [137, p. 8].

40

H¨ ormander operators

Proof. Since A is real symmetric, there exists an orthogonal matrix Q such that QAQT = ∆A = diag (λ1 , . . . , λn ) with λi > 0. Since similar matrices have the same trace,

Tr (AB) = Tr QABQT = Tr QAQT QBQT = Tr (∆A B 0 )
where B 0 = b0ij is similar to B and in particular nonnegative. Finally, since nonnegative matrices have nonnegative diagonal 0

Tr (AB) = Tr (∆A B ) =

n X

λj b0jj > 0.

j=1


Proof of Theorem 1.57. To prove (a), let u ∈ C 2 (Ω) ∩ C 0 Ω be such that Lu > 0 in Ω. Assume for instance a11 (x) > c0 > 0 in Ω and define uε,γ (x) = u (x) + εeγx1 for constants ε, γ to be chosen later. We can compute
Luε,γ (x) = Lu (x) + εeγx1 γ 2 a11 (x) + γa1 (x) + a (x) . By the uniform positivity of a11 and the boundedness of a1 and a, for γ = γ0 large enough the function γ02 a11 (x) + γ0 a1 (x) + a (x) is strictly positive in Ω, hence Luε,γ0 (x) > 0 in Ω for any ε > 0.

(1.54)

We want to prove that maxΩ uε,γ0 6 max∂Ω u+ ε,γ0 for any ε > 0. This will imply our assertion (a) since uε,γ0 → u uniformly in Ω as ε → 0+ by the boundedness of eγx1 (since Ω is bounded). Assume, by contradiction, that for some ε0 > 0 we have max uε0 ,γ0 > max u+ ε0 ,γ0 . Ω

∂Ω

This implies that uε0 ,γ0 assumes a positive maximum at some point x0 ∈ Ω and that ∇uε0 ,γ0 (x0 ) = 0. Hence (since uε0 ,γ0 (x0 ) > 0 and a (x) 6 0) Luε0 ,γ0 (x0 ) =

n X

aij (x) ∂x2i xj uε0 ,γ0 (x0 ) + a (x0 ) uε0 ,γ0 (x0 )

i,j=1

6 Tr (A · Huε0 ,γ0 ) (x0 ) where A = {aij (x0 )} and Huε0 ,γ0 is the Hessian matrix. Since A is nonnegative and at a maximum point the Hessian matrix is nonpositive, by the previous Lemma Tr (A · Huε0 ,γ0 ) (x0 ) 6 0, hence Luε0 ,γ0 (x0 ) 6 0, which contradicts (1.54), so we are done. Finally, applying point (a) to ±u, one easily gets (b).

Basic geometry of vector fields

1.9

41

Propagation of maxima and strong maximum principle for sum of squares operators

In the previous section we have met a weak maximum principle stating that, under broad assumptions on a degenerate elliptic operator L, a classical solution to Lu > 0 in Ω takes its maxima on the boundary of Ω. This fact does not exclude that u assumes a maximum also inside Ω, whence the name of “weak” maximum principle. For sum of squares operators (possibly with lower order terms) this result can be improved to a strong maximum principle, stating that a classical solution to Lu > 0 in a bounded domain Ω cannot assume a maximum inside Ω without being constant. For instance, this fact will be used in Chapter 6 to prove the strict negativity of the homogeneous global fundamental solution of sublaplacians on Carnot groups. The validity of this strong maximum principle is strictly related to the phenomenon of propagation of maxima along the integral lines of a family vector fields, a fact which is very interesting in itself because it enlightens the strict relations between the properties of a second order differential operator built on a family of vector fields and the geometry of these vector fields. This will be the object of this section. The result that we will prove is the following: Theorem 1.62 (Propagation of maxima) Let us consider the operator q X Lu = Xk2 u + X0 u + au (1.55) k=1

where X0 , X1 , . . . , Xq are real smooth vector fields in Ω (not necessarily satisfying H¨ ormander’s condition) and the function a (x) (not necessarily regular) satisfies a (x) 6 0 in Ω. Let u ∈ C 2 (Ω) be such that Lu > 0 in Ω and assume that u attains a positive maximum M at some point x0 ∈ Ω. Then u (x) = M at any point of Ω which can be reached from x0 moving repeatedly along integral curves of the vector fields Xk for k = 1, 2, . . . , q. In other words, maxima propagate along the integral curves of the vector fields Xk for k = 1, 2, . . . , q. Remark 1.63 For coherence with the general assumptions of the book, the vector fields in the previous theorem have smooth coefficients. This assumption however is far from being necessary and a careful reading of the proof shows that it suffices that the vector fields have Lipschitz coefficients. The above result and the connectivity theorem (Theorem 1.45) imply the aforementioned maximum principle: Theorem 1.64 (Strong maximum principle) Let L be as in (1.55), assume that a (x) 6 0 in an open connected set Ω and that the vector fields
Xk for k = 1, 2, . . . , q satisfy H¨ ormander’s condition in Ω. Let u ∈ C 2 (Ω) ∩ C Ω be such that Lu > 0 in Ω. Then:

42

H¨ ormander operators

(a) u cannot assume a positive maximum at a point x0 ∈ Ω without being constant on Ω. (b) If, moreover, u solves Lu = 0 in Ω, then u cannot assume neither a maximum nor a minimum at point x0 ∈ Ω without being constant in Ω. Note that we are assuming that H¨ ormander’s condition is fulfilled by X1 , . . . , Xq , hence the operator L can be regarded as a sum of squares operator, with X0 as a lower order term, not a drift of weight two. In section 1.10 we will consider the propagation of maxima for operators with drift. Proof of Theorem 1.64. The proof of (a) has been described before the statement of the theorem (here we are using also the connectedness of Ω). As to (b), assume that for some x0 ∈ Ω we have maxΩ u = u (x0 ) . If u (x0 ) > 0 then u is constant by point (a), then assume u (x0 ) 6 0. We can also assume minΩ u < maxΩ u (otherwise u is constant in Ω), therefore minΩ u < u (x0 ) 6 0. This means that −u assumes a positive maximum inside Ω and solves L (−u) = 0, hence −u (and so u) is constant in Ω by point (a). Assume now that u assumes a minimum at some point x0 ∈ Ω, then −u assumes a maximum at x0 ∈ Ω, and by the previous reasoning u is constant.

The rest of this section will be devoted to the proof of Theorem 1.62. In the following, Ω ⊂ Rn will be a connected open set and F ⊂ Ω a relatively closed subset. Definition 1.65 (Exterior normal) We say that a nonzero vector ν is an exterior normal to F at a point x ∈ F if B (x + ν, |ν|) ∩ F = ∅. In this case we will write ν ⊥x F . We will also let F ∗ = {x ∈ F : ∃ν ⊥x F } . Clearly, F ∗ ⊂ ∂F.

x1 x3 x2 F Fig. 1.2

A few examples of exterior normals.

x4

Basic geometry of vector fields

43

Example 1.66 (See Figure 1.2) At those points of ∂F where ∂F is smooth there exist exterior normals, pointing in a unique direction (see x1 , x2 in the picture). At those points where ∂F has outer angles or cusps (see x3 ) there exist exterior normals in an infinite range of directions. At those points where ∂F has inner angles or cusps (see x4 ), no exterior normal exist. If ν is an exterior normal at x, also its smaller multiples εν (with ε ∈ (0, 1)) will be exterior normals at x; on the other hand, larger multiples kν (with k > 1) may or may not be exterior normals (for instance, in points like x2 there is an upper bound for the admissible |ν|). In the example of the picture, F ∗ = ∂F \ {x4 } . Definition 1.67 (Tangency) A real smooth vector field X defined in Ω is said to be tangent at the closed set F ⊂ Ω if for every x ∈ F ∗ and every ν ⊥x F , hXx , νi = 0. Definition 1.68 (X-invariance) Let X be a real smooth vector field in Ω. We say that a closed set F ⊂ Ω is positively X-invariant if every integral curve γ of X that originates in F is entirely contained in F . Namely, if γ : [0, T ] → Ω satisfies γ 0 (t) = Xγ(t) with γ (0) ∈ F then γ (t) ∈ F for every t ∈ [0, T ]. Proposition 1.69 Let X be a real smooth vector field in Ω, let F be positively X-invariant. Then for every x ∈ F ∗ and ν ⊥x F we have hXx , νi 6 0. Proof. Let x ∈ F ∗ , ν ⊥x F , z = x + ν, r = |ν|, so that B (z, r) ∩ F = ∅. Let γ : [0, T ] → Ω be an integral curve of X starting at x, that is  0 γ (t) = Xγ(t) γ (0) = x. Since F is positively X-invariant, γ (t) ∈ F for every t ∈ [0, T ], hence 2

2

|γ (t) − z| > r2 = |γ (0) − z| , 2

which means that the function f (t) = |γ (t) − z| defined in [0, T ] has a minimum at t = 0, so that  d  2 |γ (t) − z| > 0. dt t=0 Then 0 6 2 hγ 0 (0) , γ (0) − zi = 2 hXx , x − zi = −2 hXx , νi and we are done. A relevant fact is that the previous statement can be inverted: Theorem 1.70 Let X be a real smooth vector field in Ω, let F be closed set in Ω and assume that for every x ∈ F ∗ and ν ⊥x F we have hXx , νi 6 0. Then F is positively X-invariant.

44

H¨ ormander operators

ν31

ν1

ν32

x1

ν33

x3 ν2

Xx1

Fig. 1.3

x2 Xx2 F

Xx3 x4

A positively X-invariant set.

To understand the assumption in the previous theorem, let us look again at some examples (see Figure 1.3). At points x1 , x2 , a single inequality hXx , νi 6 0 is required, implying that Xx points inside F . At point x3 , inequalities hXx , νi 6 0 must hold for all the vectors ν ranging in a certain angle; as a consequence, a more demanding condition is required on Xx3 , implying that it still points inside F . At point x4 ∈ / F ∗ nothing is required about Xx4 . Proof of Theorem 1.70. By contradiction, assume that x : [0, T ] → Ω is an integral curve of X such that x (0) ∈ F but x (t) ∈ / F for some t > 0. Since Ω \ F is open, we can find an interval [t0 , t1 ] ⊂ [0, T ] such that x (t0 ) ∈ F but x (t) ∈ /F for any t ∈ (t0 , t1 ]. Hence, possibly changing the parametrization of x (t), we can assume that x (0) ∈ F but x (t) ∈ / F ∀t ∈ (0, T ]. Let M be the Lipschitz constant of the vector field X in a neighborhood of x (0) containing x (t) for all t ∈ [0, T ]. Shrinking if necessary the interval [0, T ], we can assume 1 . (1.56) 4 Let δ (t) be the Euclidean distance between x (t) and F . The proof now proceed in two steps. Step 1. We claim that for every t ∈ (0, T ] there exists a point y (t) ∈ F such that |x (t) − y (t)| = δ (t) and MT 6

δ 2 (t) − δ 2 (t − h) 6 2δ (t) Xx(t) − Xy(t) . h↓0 h

lim

(1.57)

Step 2. We claim that (1.57) implies that, for every 0 < t1 < t2 6 T, δ 2 (t2 ) − δ 2 (t1 ) 6 2M (t2 − t1 ) sup δ 2 (t) . t∈[t1 ,t2 ]

Since δ (0) = 0, from Step 2 and (1.56) we get, for every t ∈ (0, T ], δ 2 (t) 6 2M t sup δ 2 (s) 6 2M T sup δ 2 (s) 6 s∈(0,t]

s∈(0,T ]

1 sup δ 2 (s) . 2 s∈(0,T ]

This implies that δ ≡ 0 in (0, T ], that is x (t) ∈ F for every t ∈ (0, T ], which is a contradiction.

Basic geometry of vector fields

45

Let us prove the claim in step 1. By contradiction, assume there exists s ∈ (0, T ] such that for every point y (s) ∈ F such that |x (s) − y (s)| = δ (s) we have δ 2 (s) − δ 2 (s − h) (1.58) > 2δ (s) Xx(s) − Xy(s) . lim h↓0 h Take a sequence hn ↓ 0 such that δ 2 (s) − δ 2 (s − h) δ 2 (s) − δ 2 (s − hn ) = lim n→∞ h↓0 h hn

lim

and let zn ∈ F such that δ (s − hn ) = |x (s − hn ) − zn | . Passing if necessary to a subsequence we can assume that zn tends to some point z, such that |x (s) − z| = δ (s) . Therefore for every y (s) ∈ F such that |x (s) − y (s)| = δ (s) we have, by (1.58), δ 2 (s) − δ 2 (s − hn ) 2δ (s) Xx(s) − Xy(s) < lim n→∞ hn ) ( 2 2 |x (s) − zn | − |x (s − hn ) − zn | . 6 lim n→∞ hn 2

Applying Lagrange theorem to the function s 7→ |x (s) − zn | we have, for some sn ∈ (s − hn , s) , 2δ (s) Xx(s) − Xy(s) < lim 2 hx0 (sn ) , x (sn ) − zn i n→∞



 = lim 2 Xx(sn ) , x (sn ) − zn = 2 Xx(s) , x (s) − z n→∞

 = 2 hXz , x (s) − zi + 2 Xx(s) − Xz , x (s) − z . Since |x (s) − z| = δ (s), the ball B (x (s) , |x (s) − z|) is contained in Ω\F (otherwise a point of F different from z would have distance from x (s) less than δ (s)), hence z ∈ F ∗ and hXz , x (s) − zi 6 0, so the last line above is

 6 2 Xx(s) − Xz , x (s) − z 6 2 Xx(s) − Xz δ (s) which implies Xx(s) − Xy(s) < Xx(s) − Xz , a contradiction, since by definition of y (s) we can take y (s) = z. Let us now prove the claim in step 2. By Lipschitz continuity of X, (1.57) implies δ 2 (t) − δ 2 (t − h) 6 2δ (t) M |x (t) − y (t)| = 2M δ 2 (t) . h↓0 h For every 0 < t1 < t2 6 T , let now, lim

(1.59)

g (t) = δ 2 (t) − δ 2 (t1 ) − (C + ε) (t − t1 ) for t ∈ [t1 , t2 ] , for some fixed ε > 0 and C = 2M supt∈[t1 ,t2 ] δ 2 (t) . Then for every h > 0 small enough, δ 2 (t) − δ 2 (t − h) g (t) − g (t − h) = − (C + ε) h h

46

H¨ ormander operators

so that by (1.59), for every t ∈ [t1 , t2 ] lim h↓0

g (t) − g (t − h) 6 2M δ 2 (t) − (C + ε) < 0. h

Since g (t1 ) = 0, the last inequality implies g (t2 ) 6 0, otherwise by continuity of g we could construct a sequence hn ↓ 0 such that g (t2 ) − g (t2 − hn ) > 0, leading to lim h↓0

g (t) − g (t − h) > 0, h

a contradiction. Since g (t2 ) 6 0 we have, for every 0 < t1 < t2 6 T , δ 2 (t2 ) − δ 2 (t1 ) 6 (C + ε) (t2 − t1 ) , which still holds for ε = 0, giving the claim in step 2. So we are done. Another ingredient in the present discussion of the propagation of maxima is the following result. Theorem 1.71 Let L as in (1.51) with a (x) 6 0 in Ω and let u ∈ C 2 (Ω) such that Lu > 0 in Ω. Assume that u attains a positive maximum M in Ω and let F = {x ∈ Ω : u (x) = M } . Then the vector fields Xk are tangent to F for k = 1, 2, . . . , q. Proof. Let x0 ∈ F ∗ , ν ⊥x0 F , x1 = x0 + ν, r = |ν|, and let us show that the vectors (Xk )x0 are orthogonal to ν for k = 1, 2, . . . , q. To prove this we will show that the (nonnegative) quantity n X

α=

aij (x0 ) xi1 − xi0




xj1 − xj0



i,j=1

actually vanishes. By (1.52) this will imply 0=

q n X X k=1

!2 bki (x0 )

xi1

−

xi0




i=1

that is (Xk )x0 is orthogonal to (x1 − x0 ) = ν, for k = 1, 2, . . . , q. By contradiction, 2 2 assume α > 0 and define the function v (x) = e−K|x−x1 | − e−Kr with K > 0 to be chosen later. Clearly we have v (x) > 0 in Br (x1 ) , v (x) < 0 in Ω \ Br (x1 ), v (x0 ) = 0. A simple computation gives Lv (x0 ) = e

−K|x0 −x1 |2

2

4K α − 2K

n X i=1

aii (x0 ) +

n X i=1

!! ai (x0 )

xi0

−

xi1




.

Basic geometry of vector fields

47

Br (x1 ) x1

w(x) < M u(x) 6 M − c0 since we are far off F Bρ (x0 )

w(x0 ) = M x0 Lw > 0

w(x) < M since v < 0 out of Br (x1 )

F

Fig. 1.4

Proof of Theorem 1.71.

Hence fixing K large enough, we have Lv (x0 ) > 0, and then Lv (x) > 0 in a small ball Bρ (x0 ). Next, for λ > 0 to be chosen later, define w (x) = u (x) + λv (x) . We know that Lw > 0 in Bρ (x0 ) and w (x0 ) = u (x0 ) = M since x0 ∈ F . Also, in Ω \ Br (x1 ) we have w (x) 6 M + λv (x) < M. This in particular holds (see Figure 1.4) on ∂Bρ (x0 ) \ Br (x1 ). On the other hand, ∂Bρ (x0 ) ∩ Br (x1 ) is separated from F , hence in this set u 6 M − c0 for some c0 > 0 and, choosing λ small enough, by continuity w (x) < M also on ∂Bρ (x0 ) \ Br (x1 ). For this λ we then have Lw (x) > 0 in Bρ (x0 ) w (x) < M on ∂Bρ (x0 ) w (x0 ) = M which means that w attains a local maximum > M > 0 at some point x ∈ Bρ (x0 ). But then   n n X X Lw (x) =  aij wxi xj + ai wxi + aw (x) 6 0 i,j=1

i=1

since wxi (x) = 0, a 6 0, w (x) > M > 0, and the matrix aij is symmetric and nonnegative, so that by Lemma 1.61 n X aij wxi xj (x) 6 0. i,j=1

This contradicts the relation Lw (x) > 0 in Bρ (x0 ), proving the assertion.

48

H¨ ormander operators

Theorems 1.71 and 1.70 immediately imply that F is positively invariant for ±Xk for k = 1, 2, . . . q, which proves Theorem 1.62. 1.10

Propagation of maxima for operators with drift

Assume now that H¨ ormander’s condition is satisfied by the system X0 , X1 , . . . , Xq (but not by X1 , . . . , Xq ). Then, knowing that maxima propagate along the integral lines of Xk for k = 1, 2, . . . , q is no longer enough to derive a strong maximum principle, since also the integral lines of X0 are necessary, in general, to connect each couple of points. Example 1.72 A counterexample to the strong maximum principle in this case is given already by the heat operator: the Gaussian kernel ( 2 −x 4t e√ if t > 0 h (x, t) = 2πt 0 if t 6 0 satisfies Lu = uxx − ut = 0 in Ω = R2 \ {(0, 0)} but assumes its minima in the whole half-plane t < 0, which lies in the interior of Ω. Here X1 = ∂x , X0 = −∂t , and both vector fields are required to satisfy H¨ ormander’s condition in Ω. Actually, so far we have not studied the role of X0 in the propagation of maxima. The necessary analysis is contained in the following: Theorem 1.73 Let L be as in (1.55), with a (x) 6 0 in Ω. Assume that in Ω the Lie algebra generated by X1 , . . . , Xq has constant rank p < n. Let u ∈ C 2 (Ω) such that Lu > 0 in Ω, assume that u attains a positive maximum M at some point of Ω, and let F = {x ∈ Ω : u (x) = M }. Then for every y ∈ F ∗ and ν ⊥y F, D E (X0 )y , ν 6 0. Together with Theorems 1.70 and 1.71, this Theorem completes the picture: maxima propagate positively along the integral lines of ±X1 , . . . , ±Xq , and X0 . More precisely we have the following: Theorem 1.74 (Propagation of maxima for operators with drift) Let L be as in (1.55), with a (x) 6 0 in Ω. Assume that in Ω the Lie algebra generated by X1 , . . . , Xq has constant rank p < n. Let u ∈ C 2 (Ω) such that Lu > 0 in Ω and assume that u attains a positive maximum M at some point of Ω. Then u (x) = M at any point of Ω which can be reached from x0 moving repeatedly along integral curves of the vector fields Xk for k = 1, 2, . . . , q and moving positively along the integral lines of X0 .

Basic geometry of vector fields

49

(x0 , t0 )

D(x0 , t0 )

Fig. 1.5

Example (a).

Example 1.75 Let us look at some examples of operators L = X12 + X0 , with X1 nondegenerate, to illustrate how maxima propagate. 2 − ∂t be the heat operator in a noncylindrical domain Ω ⊂ R2 . (a) Let L = ∂xx Maxima propagate along the directions ±∂x and −∂t . If a maximum is attained at (x0 , t0 ) then it propagates in the region n D(x0 ,t0 ) shownoin the Figure 1.5. 2 (b) Let L = ∂xx + x∂y in Ω =

(x, y) ∈ [−1, 1]

2

⊂ R2 . Maxima propagate

along the directions ±∂x and x∂y . Since moving along the lines of ±∂x until x reaches a positive or negative value we can then follow a line of x∂y going upward or downward, a maximum at some point (x0 , y0 ) ∈ Ω actually propagates everywhere in Ω (see Figure 1.6).

Ω

(x0 , y0 )

Fig. 1.6

Example (b).

50

H¨ ormander operators

n o 2 (c) The same operator of (b) in a domain like Ω = (x, y) ∈ [1, 2] would present a different behavior. Now x > 0 in Ω, hence the lines of x∂y always go upward, and a maximum at some point (x0 , y0 ) ∈ Ω only propagates in the upper region y > y0 (see Figure 1.7).

(x0 , y0 )

Ω

Fig. 1.7

Example (c).

The proof of Theorem 1.73 requires two preliminary steps. The first is a general result about vector fields, the second a technical lemma related to our operator. Let us start with: Theorem 1.76 (Local form of Frobenius theorem) Assume that in Ω the Lie algebra generated by X1 , . . . , Xq has constant rank p < n. Then, for every x0 ∈ Ω there exists a neighborhood U of x0 and a diffeomorphism of U onto a neighborhood V of the origin such that, in the new variables z1 , . . . , zn , the vector fields X1 , . . . , Xq in V only acts on the variables z1 , . . . , zp . Remark 1.77 Frobenius theorem actually holds in a global form: it can be shown that there exists in Ω a global diffeomorphism making the Xi into vector fields depending on p variables only. The above local version however is easier to prove and will be enough for our purposes. Proof. Let Vx = L n (X1 , . . . , X q ) , by assumption dim Vx = p for every x ∈ Ω.
o x Fix x0 ∈ Ω and let X[I] x (|A| = p) a basis of Vx0 built evaluating at 0

I∈A

x0 a suitable family of p commutators. 
For any x ranging in a sufficiently small neighborhood U (x0 ), the vectors X[I] x I∈A will still be a basis of Vx . Let us relabel X10 , . . . , Xp0 these p vector fields which at any point of U (x0 ) form a basis of Vx , and write Xi0 =

n X j=1

bij (x) ∂xj .

Basic geometry of vector fields

51

Since the n × p matrix {bij (x)} has rank p at any x ∈ U (x0 ), we can find a p × p p submatrix, say {bij }i,j=1 , which is nonsingular at x0 and therefore in a suitably p small neighborhood U1 (x0 ) ⊂ U (x0 ). Let {gij (x)}i,j=1 be the inverse matrix of p {bij (x)}i,j=1 and set Yi =

p X

gij (x) Xj0 for i = 1, 2, . . . , p.

(1.60)

j=1 p

Clearly the vectors {(Yi )x }i=1 are still a basis of Vx for every x ∈ U1 (x0 ). We can compute:     p p p p n n X X X X X X   Yi = gij (x) gij bjk  ∂xk + gij bjk  ∂xk bjk (x) ∂xk = j=1

= ∂xi +

k=1 n X

k=1

j=1

k=p+1

j=1

αik ∂xk

(1.61)

k=p+1

for i = 1, . . . , p and some smooth functions αik . We claim that the vector fields Yi possess the further property: [Yi , Yj ] = 0 in U1 (x0 ) for i, j = 1, 2, . . . , p. To prove these identities, observe that since L (X1 , . . . Xq ) has constant rank p, by the definition of Xi0 we have p  0 0 X Xi , Xj = ckij Xk0 k=1

for any i, j = 1, 2, . . . , p and some smooth functions ckij . This also implies that for suitable smooth functions dkij , ekij " p # p p p X X X X 0 0 k 0 [Yi , Yj ] = gik (x) Xk , gjh (x) Xh = dij Xk = ekij Yk . (1.62) k=1

h=1

k=1

k=1

On the other hand, the explicit form (1.61) of the Yi gives:   n n n X X X [Yi , Yj ] = ∂xi + αik ∂xk , ∂xj + αjh ∂xh  = βik ∂xk k=p+1

h=p+1

k=p+1

while p X

ekij Yk =

k=1

p X

 ekij ∂xk +

k=1

n X

 αkh ∂xh 

h=p+1

and the identity n X k=p+1

βik ∂xk =

p X k=1

 ekij ∂xk +

n X h=p+1

 αkh ∂xh 

52

H¨ ormander operators

is possible only if ekij ≡ 0, which by (1.62) implies the claim. Next, write x0 = (x00 , x000 ) with x00 ∈ Rp , x000 ∈ Rn−p . Let W1 be a neighborhood of 0 in Rp . Let V 0 be an n − p vector subspace of Rn transversal to Vx0 = L (X1 , . . . , Xq )x0 and let W2 be a neighborhood of 0 in V 0 . For (t, y) ∈ W1 × W2 define the map f (t, y) = f (t1 , . . . , tp , yp+1 , . . . , yn ) = exp (t1 Y1 ) exp (t2 Y2 ) . . . exp (tp Yp ) (x00 , x000 + y) which for W = W1 × W2 small enough takes values in U1 (x0 ). Now, observe that since the vector fields Yi commute, by Theorem 1.9 and (1.2), we have ∂f (t, y) = (Yi )f (t,y) for i = 1, 2, . . . , p. ∂ti

(1.63)

Also, since f (0, y) = (x00 , x000 + y),   ∂f (0, y) = 0, . . . , 0, 1, 0, . . . , 0 for k = p + 1, . . . , n k ∂yk so that, recalling (1.61), the Jacobian matrix of f at points (0, y) is   Ip ∗ Jf (0, y) = 0 In−p where ∗ denotes a p × (n − p) block. Hence the matrix is nonsingular at 0 and, shrinking W if necessary, f is a diffeomorphism of W onto some neighborhood U2 (x0 ) ⊂ U1 (x0 ). In the new (t, y) variables, the vectors
Yi take the form Yit,y related to the original Yix by the relation (Yix )f (t,y) = Yit,y (t,y) · Jf (t, y). For any
fixed i = 1, 2, . . . , p, we can look at this as a linear system in the unknown Yit,y (t,y) . Since by (1.63) the first p rows of the matrix Jf (t, y) are exactly (Yix )f (t,y) ,  
Yit,y (t,y) = 0, . . . , 0, 1, 0, . . . ., 0 i

is the solution to the systems. In other words Yit,y = ∂ti for i = 1, 2, . . . , p and by p p (1.60), recalling that {bij (·)}i,j=1 is the inverse matrix of {gij (·)}i,j=1 , Xi0 =

p X

bij (f (t, y)) ∂tj for i = 1, 2, . . . , p.

j=1

Pp Hence, letting z = (t, y), Xi0 = j=1 bij (f (z)) ∂zj . In particular this is the form of X1 , . . . , Xq in the new variable z, since these can be expressed as linear combinations of X10 , . . . , Xp0 in U1 (x0 ), so the assertion is proved. Lemma 1.78 Assume that in Ω the Lie algebra generated by X1 , . . . , Xq has constant rank p < n, let L=

q X k=1

Xk2 u + X0 u + au =

n X i,j=1

aij uxi xj +

n X j=1

aj uxj + au

Basic geometry of vector fields

53

with a (x) 6 0 in Ω and let u ∈ C 2 (Ω) be such that Lu > 0 in Ω. Assume that u attains a positive maximum M at some point of Ω and let F = {x ∈ Ω : u (x) = M }. Let y ∈ F ∗ and ν ⊥y F , then there exists a local diffeomorphism T : V → U mapping a neighborhood V of y onto a neighborhood U of the origin in the space of the new variables z1 , . . . , zn , such that Lu =

p X

αij (z) uzi zj +

i,j=1

n X

αi (z) uzi + a (z) u

i=1

with the p × p matrix {αij (z)} nonnegative in U , and n X

X0 u (z) =

c0i (z) uzi with c0n (z) = αn (z) .

i=1

Moreover, T (y) = 0 and letting F 0 = T (F ∩ V ) we have en ⊥0 F 0 . w(x) < M u(x) 6 M − c0 since we are far off F

E

O Bρ (0)

F w(x) < M + λv(x) < M

Fig. 1.8

Proof of Theorem 1.73.

Proof. Frobenius theorem implies the existence of a diffeomorphism x = T (z) of a neighborhood of the origin to a neighborhood of y such that, denoting by Xix , Xiz , respectively, the vector fields expressed in the old and new variables, we have Xkz

=

p X

ckj (z) ∂zj for k = 1, 2, . . . , q.

j=1

Pn As to the vector field X0 , we can simply write X0z = j=1 cj (z) ∂zj while the function a (x) turns to a0 (z) = a (T (z)) 6 0. Then ! ! p q p q X p X X X X z 2 L u= cki (z) ckj (z) ∂zi zj u + cki (z) (∂zi ckj ) (z) ∂zj u i,j=1

+

=

n X

j=1

k=1

k=1 i=1

cj (z) ∂zj u + a0 (z) u

j=1 p X i,j=1

αi,j (z) uzi zj +

n X j=1

αj (z) ∂zj u + a0 (z) u

54

where αjk =

H¨ ormander operators

Pq

is clearly a p × p nonnegative matrix, and  q p XX   cki ∂zi ckj + cj if j = 1, 2, . . . , p αj = k=1 i=1   cj if j = p + 1, . . . , n

k=1 cki ckj

in particular αn = cn . It remains to prove that en ⊥0 F 0 . By Theorem 1.62 maxima of L propagate along the integral lines of X1z , . . . , Xqz , which are contained in the space Rp of the variables z1 , . . . , zp . Since L (X1 , . . . , Xq ) has rank p, the vector fields satisfy H¨ ormander’s condition in the Rp subspace of the first variables z1 , . . . , zp , hence by the connectivity theorem F 0 contains a neighborhood of the origin of this subspace. Since B (y + ν, |ν|) ∩ F = ∅ it follows that T (B (y + ν, |ν|) ∩ V ) ∩ F 0 = ∅. Since F 0 contains a neighborhood of the origin of the subspace generated by z1 , . . . , zp this implies that the normal to ∂T (B (y + ν, |ν|) ∩ V ) at 0 is orthogonal to such subspace. An affinity in the subspace of variables zp+1 , . . . , zn maps such normal to the vector (0, 0, . . . , 0, 1), while it does not affect the vector fields X1 , . . . , Xq and preserves the property αn = c0n in the new variables. It follows that, up to shrinking V if necessary, ∂T (B (y + ν, |ν|) ∩ V ) can be represented in a neighborhood of the origin by the graph of a function zn = Φ (z 0 ) ,

z 0 = (z1 , . . . , zn−1 ) , 2

with ∇Φ (0) = 0. This implies |Φ (z 0 )| 6 c |z 0 | for a suitable constant c > 0. Taking V small enough we can also assume that if z = (z 0 , zN ) ∈ V satisfies
zn > Φ (z 0 ) 1 then z ∈ T (B (y + ν, |ν|) ∩ V ). Let B 0 be the ball of center 0, . . . , 0, 2c and radius 2 1 0 and observe that z ∈ B if and only if z > c |z| . It follows that when z ∈ B0 n 2c we have 2

2

|Φ (z 0 )| 6 c |z 0 | 6 c |z| 6 zn and therefore z ∈ T (B (y + ν, |ν|) ∩ V ), so that B 0 ⊂ T (B (y + ν, |ν|) ∩ V ). This
1 ⊥0 F 0 . A new affinity changes the means that B 0 ∩ F
0 = ∅, hence 0, . . . , 0, 2c 1 vector 0, . . . , 0, 2c to en .

Proof of Theorem 1.73. DWe will prove that, in the new variables described in E the above Lemma, we have (X0 )y , ν 6 0 that is, since in the new variables (X0 )y 7→ (. . . , . . . , . . . , αn (0)) ν 7→ (0, 0, . . . , 0, 1) , we have to prove that αn (0) 6 0. We know that the ball 2

z12 + z22 + . . . + (zn − 1) < 1

(1.64)

Basic geometry of vector fields

55

does not intersect F . Moreover, as explained in the proof of the previous lemma, maxima propagate along the directions z1 , z2 , . . . , zp . Hence the cylinder defined by 2

2 2 zp+1 + . . . + zn−1 + (zn − 1) 6 1

(1.65)

does not contain points of F in its interior. Otherwise, starting from such point and moving in the z1 , z2 , . . . , zp directions until the linear space z1 = 0, z2 = 0, . . . , zp = 0, we will get a point of F inside the ball, which is impossible. For some ε ∈ (0, 1) , let us consider the function δ defined by
2 2 2 2 δ (z) = ε z12 + . . . + zp2 + zp+1 + . . . + zn−1 + (zn − 1) . 2

The ellipsoid Eε defined by δ (z) 6 1 is contained in the cylinder (1.65) and touches 2 F only at the origin. The function v (z) = e−δ(z) − e−1 vanishes at the origin and is positive (negative) inside (outside, respectively) the ellipsoid. Let us compute  for i = 1, 2, . . . , p  εzi 2 −δ(z) vzi (z) = −2e · zi for i = p + 1, . . . , n − 1  zn − 1 for i = n.  2  vzi zj (z) = e−δ(z) 4ε2 zi zj − 2εδij for i, j = 1, 2, . . . , p  p p p X X X 2 −δ(z)  2 Lv (z) = e 4ε αij (z) zi zj − 2ε αii (z) − 2ε αi (z) zi i,j=1

−2

n−1 X

i=1

i=1

 αi (z) zi − 2αn (z) (zn − 1) + a (z) v (z)

i=p+1

and " Lv (0) = e

−1

−2ε

p X

# αii (0) + 2αn (0) .

i=1

Clearly, if αn (0) > 0 we can pick ε > 0 small enough so that Lv (0) > 0. Then we can repeat the argument used in the proof of Theorem 1.71 (exploiting the nonnegativity of the matrix αij ) arriving to a contradiction. Let us recapitulate it for the convenience of the reader. Since Lv (0) > 0, we can say that Lv (z) > 0 in a small ball Bρ (0). Let us define w (z) = u (z) + λv (z) with λ > 0 to be chosen later. We know that Lw > 0 in Bρ (0) and w (0) = u (0) = M since 0 ∈ F . Also, outside the ellipsoid Eε we have w (z) 6 M + λv (z) < M. This in particular holds on ∂Bρ (0)\Eε . On the other hand, ∂Bρ (0)∩Eε is separated from F , hence in this set u 6 M − c0 for some c0 > 0 and, choosing λ small enough, by continuity w (z) < M also on ∂Bρ (0) ∩ Eε . For this λ we then have Lw (z) > 0 in Bρ (0) w (z) < M on ∂Bρ (0) w (0) = M

56

H¨ ormander operators

which means that w attains a local maximum = M > 0 at some point z ∈ Bρ (0). But then     p p n X X X αij wzi zj + αij wzi zj + aw (z) 6 0 αi wzi + aw (z) =  Lw (z) =  i,j=1

i=1

i,j=1

since wzi (z) = 0, a 6 0, w (z) = M > 0, and the matrix αij is symmetric and Pp nonnegative, so that by Lemma 1.61 i,j=1 αij wzi zj (z) > 0. This contradicts the relation Lw (z) > 0 in Bρ (0), hence αn (0) 6 0, and the theorem is proved. 1.11

Some examples of explicit computations with the control distance

In this section we collect a few examples of concrete systems of vector fields where some explicit computations on the control distance can be easily done. These computations can be an instructive experience of some phenomena that will be studied in depth by general methods in Chapter 9. We will establish, in three interesting examples, estimates on the control distance, on the volume of control balls and relations between the control balls and suitable “boxes”. 1.11.1

The control distance in the Heisenberg group H1

Here we come back to the Example 1.27 (a). In R3 3 (x, y, t), let X = ∂x + 2y∂t ≡ (1, 0, 2y) Y = ∂y − 2x∂t ≡ (0, 1, −2x) . We have already noted that these vector fields satisfy H¨ormander’s condition in R3 and we now want to estimate their control distance. As we will see in Chapter 3, this distance enjoys a translation invariance property (with respect to a suitable group of translations, called the Heisenberg group H1 ). It is therefore enough to estimate d ((x, y, t) , (0, 0, 0)). We have the following: Proposition 1.79 Let d as above. Then (i) The control distance satisfies the inequalities     p p max |x| , |y| , |t| /2 6 d ((x, y, t) , (0, 0, 0)) 6 6 max |x| , |y| , |t| /2 . (ii) The control balls satisfy, for every r > 0, the inclusions
B (0, r) ⊂ (−r, r) × (−r, r) × −2r2 , 2r2 ⊂ B (0, 6r) In particular,

r4 81

6 |B (0, r)| 6 16r4 .

Basic geometry of vector fields

57

In this proposition, as in the others that we will prove in this section, we have made no attempt in finding sharp constants; the numerical value of the constants therefore is not important, but is left to stress the fact that the computations are completely explicit. Note that the inequalities contained in point (i) also imply 1/2

c1 |P − Q| 6 d (P, Q) 6 c2 |P − Q|

for any P, Q ranging in a bounded set. This is consistent with the general property (1.45) of H¨ ormander vector fields, since in this case H¨ormander’s condition holds at step s = 2. Proof. Let Q = (x, y, t) and, for ε > 0, let δ = d (Q, 0)+ε, then there exists a curve γ ∈ C0,Q (δ) such that γ 0 (τ ) = a (τ ) Xγ(τ ) + b (τ ) Yγ(τ ) , with |a (τ )| 6 δ, |b (τ )| 6 δ. Letting γ (τ ) = (x (τ ) , y (τ ) , t (τ )) we obtain  0  x (τ ) = a (τ ) y 0 (τ ) = b (τ )  0 t (τ ) = 2a (τ ) y (τ ) − 2b (τ ) x (τ ) so that Z

1

|xQ | = |x (1)| 6

|a (τ )| dτ 6 δ 0

Z

1

|yQ | = |y (1)| 6

|b (s)| ds 6 δ 0 Z 1

|tQ | = |t (1)| 6

Z |2a (τ ) y (τ ) − 2b (τ ) x (τ )| dτ 6 2δ

0

1

2δτ dτ = 2δ 2 .

0



 p This implies max |x| , |y| , |t| /2 6 d (Q, 0) + ε for any ε > 0, and therefore   p max |x| , |y| , |t| /2 6 d (Q, 0)
B (0, r) ⊂ (−r, r) × (−r, r) × −2r2 , 2r2 |B (0, r)| 6 16r4 . To prove the reverse inequalities, we have to compute the exponentials of the vector fields X and Y . An explicit computation gives exp (sX) (a, b, c) = (a + s, b, 2bs + c) exp (sY ) (a, b, c) = (a, b + s, −2as + c) . Therefore exp (−βY ) exp (−αX) exp (βY ) exp (αX) (0, 0, 0) = exp (−βY ) exp (−αX) exp (βY ) (α, 0, 0) = exp (−βY ) exp (−αX) (α, β, −2αβ) = exp (−βY ) (0, β, −4αβ) = (0, 0, −4αβ) .

58

H¨ ormander operators

It follows that exp (γY ) exp (δX) exp (−βY ) exp (−αX) exp (βY ) exp (αX) (0, 0, 0) = exp (γY ) exp (δX) (0, 0, −4αβ) = exp (γY ) (δ, 0, −4αβ)

(1.66)

= (δ, γ, −2δγ − 4αβ) = (xQ , yQ , tQ ) provided that δ = xQ , γ = yQ and αβ =   p Now let r = max |xQ | , |yQ | , |tQ | /2 , then

2xQ yQ − tQ . 4

|δ| = |xQ | 6 r, |γ| = |yQ | 6 r 2xQ yQ − tQ 2r2 + 2r2 6 = r2 . |αβ| = 4 4 p p 2x y −t We now choose α and β. For c = Q 4Q Q , pick α = |c|, β = sign (c) |c|, then 2x y −t αβ = Q 4Q Q with |α| 6 r, |β| 6 r. Since d (exp (sX) (P ) , P ) 6 |s| d (exp (sY ) (P ) , P ) 6 |s| , (1.66) and the triangle inequality give d (Q, 0) 6 |γ| + |δ| + |α| + |β| + |α| + |β| 6 6r. This also implies that
(−r, r) × (−r, r) × −2r2 , 2r2 ⊂ B (0, 6r)  r r   r r   r2 r2  B (0, r) ⊃ − , × − , × − , 6 6 6 6 18 18 r4 |B (0, r)| > 81 so we are done. Remark 1.80 Our inequalities about the volume of control balls can be rewritten as  r 4 4 6 |B (0, r)| 6 (2r) . 3 In particular |B (0, 2r)| 6 124 , (1.67) |B (0, r)| which is the doubling condition for balls centered at the origin. We will see in Chapter 3 that in this case the measure of the ball actually does not depend on the center. We will also see that actually |B (0, r)| = cr4 for some constant c > 0; this will imply |B (0, 2r)| = 24 , |B (0, r)| a much more precise result than (1.67).

Basic geometry of vector fields

1.11.2

59

The weighted control distance for Kolmogorov vector fields

We now consider an example of weighted control distance. Let us consider, in R3 3 (x, y, t) , the vector fields X1 = ∂x ≡ (1, 0, 0) ; X0 = x∂y − ∂t ≡ (0, x, −1) where X0 has weight 2 while X1 has weight 1, which is the natural choice if we want to study Kolmogorov’ operator6 2 ∂xx + x∂y − ∂t .

Analogously to what happens for the Heisenberg group, also in this case the control distance turns out to be invariant with respect to a suitable family of translations, that we will discuss in Chapter 3. Therefore it is not restrictive to limit our analysis to the distance with the origin. Proposition 1.81 (i) The (weighted) control distance d satisfies the inequalities:     1/3 1/2 1/3 1/2 max |x| , |2y| , |t| 6 d ((x, y, t) , (0, 0, 0)) 6 4 max |x| , |y| , |t| . (ii) The control balls satisfy, for every r > 0, the inclusions

B (0, r) ⊂ (−r, r) × −r3 , r3 × −r2 , r2 ⊂ B (0, 4r) In particular, r6 6 |B (0, r)| 6 4r6 . 512 Also in this example, it is instructive to compare the estimates of point (i) with the general result (1.45). We can rewrite the estimates on d in the form 1/3

c1 |P − Q| 6 d (P, Q) 6 c2 |P − Q|

for any P, Q ranging in some bounded set. In this case H¨ormander’s condition holds at (weighted) step s = 3. Proof. Let Q = (x, y, t) and, for ε > 0, let δ = d (Q, 0) + ε, then there exists a curve γ ∈ C0,Q (δ) such that γ 0 (τ ) = a (τ ) (X1 )γ(τ ) + b (τ ) (X0 )γ(τ ) with |a (τ )| 6 δ and |b (τ )| 6 δ 2 . Letting γ (τ ) = (x (τ ) , y (τ ) , t (τ )), we have  0  x (τ ) = a (τ ) y 0 (τ ) = b (τ ) x (τ )  0 t (τ ) = −b (τ ) 6 This operator is the simplest instance of the operators presented in Example 1.27, (c). It is also a historically important example, which we have already met in the Introduction of the book.

60

H¨ ormander operators

and Z

τ

a (s) ds, |x (τ )| 6 δτ,

x (τ ) =

|x| 6 δ,

0

Z

1

b (s) x (s) ds, |yQ | 6 δ 2

yQ = |tQ | 6

1

δτ dτ 6 0

0

Z

Z

1 3 δ , 2

1

|−b (τ )| dτ 6 δ 2 .

0

Since this holds for any ε > 0,  this implies

 1/2 , |t| 6 d (Q, 0)  3 3
r r B (0, r) ⊂ (−r, r) × − , × −r2 , r2 2 2 max |x| , |2y|

1/3

|B (0, r)| 6 4r6 . To prove the reverse inequalities, we have to compute the exponentials: exp (sX1 ) (a, b, c) = (a + s, b, c) , and exp (sX0 ) (a, b, c) = (a, b + as, c − s) . These easily give exp (b0 X0 ) exp (a1 X1 ) exp (a0 X0 ) (0, 0, 0) = (a1 , a1 b0 , −a0 − b0 ) . Hence if Q = (x, y, t) we can write exp (b0 X0 ) exp (a1 X1 ) exp (a0 X0 ) (0, 0, 0) = (x, y, t) provided a1 = x, a1 b0 = y, − a0 − b0 = t   1/3 1/2 Now let r = max |x| , |y| , |t| , then we can satisfy (1.69) letting y y a1 = r, b0 = , a0 = −t − r r which give y |y| 6 r2 , |a0 | 6 |t| + 6 2r2 . |a1 | = r, |b0 | 6 r r Next, recall that d (exp (sX1 ) (P ) , P ) 6 |s|

d exp s2 X0 (P ) , P 6 |s| hence (1.68) and the triangle inequality give p p √ d (Q, 0) 6 |b0 | + |a1 | + |a0 | 6 r + r + 2r < 4r. This also implies that

(−r, r) × −r3 , r3 × −r2 , r2 ⊂ B (0, 4r)  r r    r 3  r 3    r 2  r 2  × − , × − , B (0, r) ⊃ − , 4 4 4 4 4 4 r6 |B (0, r)| > 512 so we are done.

(1.68) (1.69)

Basic geometry of vector fields

61

Remark 1.82 Our inequalities about the volume of control balls can be rewritten as  6 √ 6 r √ 2r . 6 |Br (0, 0, 0)| 6 2 2 In particular |B (0, 2r)| 6 86 , (1.70) |B (0, r)| which is the doubling condition for balls centered at the origin. As in the case of the Heisenberg group H1 , we will see in the Chapter 3 that the measure of the ball actually does not depend on the center, and |B (0, r)| = cr6 for some constant c > 0, which gives |B (0, 2r)| = 26 , |B (0, r)| a better result than (1.70). 1.11.3

The control distance for vector fields of Franchi-Lanconelli type

In R2 3 (x, y) let us consider the vector fields X = ∂x ; Y = x∂y .

(1.71)

Despite the apparent simplicity of these vector fields, the control distance in this case exhibits interesting phenomena. For instance, the volume of the control balls does not behave like a fixed power of the radius but, depending on whether the center lies on the line x = 0 or not, and depending on the relative size of the radius with respect to the x-coordinate of the center, can behave like r2 or r3 . In particular, differently from what happens in the other two examples within this section, here the control distance is not translation invariant with respect to any group of translations. This is the simplest instance within the family of systems of H¨ormander vector fields X = ∂x ; Y = xk ∂y for k positive integer, which we have already met in Example 1.27 (b). FranchiLanconelli in [94] studied the control distance induced in Rn by a general family of “diagonal vector fields” Xi = λi (x) ∂xi (i = 1, 2, . . . , n) having as model case the couple of vector fields α

X = ∂x ; Y = |x| ∂y in the plane, for any α > 0. Note that for noninteger α these are no longer smooth and do not satisfy H¨ ormander condition on x = 0; nevertheless, one can still define the control distance. Here we will prove some explicit estimates for the control distance and the volume of control balls for the system (1.71). We start with the following:

62

H¨ ormander operators

Lemma 1.83 For every (x0 , y0 ) ∈ R2 , any r > 0,      r2 r2 Br (x0 , y0 ) ⊂ (x0 − r, x0 + r) × y0 − r |x0 | + , y0 + r |x0 | + . (1.72) 2 2 In particular, |Br (x0 , y0 )| 6 4r2 |x0 | + 2r3 . Proof. Let (x, y) ∈ Br (x0 , y0 ). Then there exists a curve γ ∈ C(x,y),(x0 ,y0 ) (δ) for some δ < r. Letting γ (t) = (x (t) , y (t)) we have  0 x (τ ) = a (τ ) y 0 (τ ) = b (τ ) x (τ ) and Z τ a (s) ds 6 δτ, |x − x0 | 6 δ < r |x (τ ) − x0 | = 0 Z 1 Z 1 δ2 r2 |y − y0 | = (|x0 | + δτ ) dτ = δ |x0 | + b (s) x (s) ds 6 δ < r |x0 | + . 2 2 0 0

The proof of the converse inclusion is more difficult and to shorten our computation we will not cover all possible cases. We will study two situations which are significantly different: balls centered at the origin, and small balls centered at (A, 0) with A 6= 0. We start with the study at the origin. Proposition 1.84 (i) For every (x, y) ∈ R2 we have    p  p max |x| , 2 |y| 6 d ((x, y) , (0, 0)) 6 4 max |x| , |y| .

(1.73)

(ii) For every r > 0,  2 2 r r B (0, r) ⊂ (−r, r) × − , 2 2
2 2 (−r, r) × −r , r ⊂ B (0, 4r) .

(1.74) (1.75)

In particular, r3 6 |B (0, r)| 6 2r3 16

(1.76)

3 Proof. Letting (x 0 , y0 ) = (0, 0)  in (1.72) gives (1.74) which implies |B (0, r)| 6 2r . p Now, let δ = max |x| , 2 |y| and let r > 0 such that (x, y) ∈ B (0, r). By (1.74) we have δ < r. This implies   p max |x| , 2 |y| 6 d ((x, y) , (0, 0))

that is the first inequality in (1.73). To prove the converse relations, we need an admissible curve joining the origin to Q = (xQ , yQ ) . Let us start computing the exponentials. We have exp (tY ) (a, b) = (a, b + at) exp (tX) (a, b) = (a + t, b) ,

Basic geometry of vector fields

63

and therefore exp (a3 X) exp (a2 Y ) exp (a1 X) (0, 0) = (a1 + a3 , a1 a2 ) = (xQ , yQ ) provided a1 + a3 = xQ , a1 a2 = yQ .  p Let r = max |xQ | , |yQ | . Then |xQ | 6 r, |yQ | 6 r2 and choosing

(1.77)



a1 = r, a3 = xQ − r, a2 =

yQ r

the system (1.77) is fulfilled, and we have |a1 | = r, |a2 | 6 r and |a3 | 6 2r. Then by the triangle inequality, since d (exp (tX) (P ) , P ) 6 |t| and d (exp (tY ) (P ) , P ) 6 |t| , we get d (Q, 0) 6 r + r + 2r = 4r, which proves the second inequality in (1.73) which in turn gives  r r   r2 r2  ⊂ B (0, r) − , × − , 4 4 16 16 so that r r2 r3 |B (0, r)| > 2 · 2 = , 4 16 16 hence also (1.76) is proved. A different behavior of the control distance, metric balls and their volume is exhibited if we take a point (A, 0) with A > 0. Let us study the balls with this center and radius small enough. Proposition 1.85 Let A > 0. Then 2 (i) For every (x, y) such that 0 6 x 6 2A, |y| 6 A2 ,     2 |y| max |x − A| , |y| 6 d ((x, y) , (A, 0)) 6 2 max |x − A| , . 3A A (ii) For every A > 0 and r 6 A,    r 1 1 r × − Ar, Ar ⊂ Br (A, 0) A − ,A + 2 2 2 2

(1.78)

(1.79)

  3 3 ⊂ (A − r, A + r) × − Ar, Ar . 2 2 In particular, Ar2 6 |Br (A, 0)| 6 6Ar2

(1.80)

64

H¨ ormander operators

Proof. By (1.72) we have r2 r2 Br (A, 0) ⊂ (A − r, A + r) × −rA − , rA + 2 2 

 .

2

Assuming r 6 A, we have rA + r2 6 rA + 21 rA = 32 rA, and the second inclusion 2 in (1.79) follows. This also implies |Br (A, 0)| 6 2r · 3Ar = 6Ar
. The inclusion 2 just proved also means that, letting δ = max |x − A| , 3A |y| we have (x, y) ∈ Br (A, 0) =⇒ δ < r, which implies the first inequality in (1.78). To prove the converse relations, let us build an admissible curve joining (A, 0) to Q = (xQ , yQ ). It is enough to compute: exp (a2 X) exp (a1 Y ) (A, 0) = exp (a2 X) (A, Aa1 ) = (A + a2 , Aa1 ) . Hence exp (a2 X) exp (a1 Y ) (A, 0) = (xQ , yQ ) provided A + a2 = xQ , Aa1 = yQ .  |y | Letting δ = max |xQ − A| , AQ , we get (1.81) satisfied choosing

(1.81)



yQ ; |a1 | 6 δ A hence recalling that d (exp (tX) (P ) , P ) 6 |t| and the same holds for Y , by the triangle inequality we get   |yQ | d ((xQ , yQ ) , (A, 0)) 6 2δ = 2 max |xQ − A| , A a2 = xQ − A; |a2 | 6 δ and a1 =

which is the second inequality in (1.78). This also means that (A − r, A + r) × (−Ar, Ar) ⊂ B2r (0, 0) which gives the first inclusion in (1.79) and therefore also |Br (0, 0)| > Ar2 so we are done. Comparing the previous two propositions we see that p the distance of a point from the origin behaves like the “parabolic distance” |x| + |y|, while the distance of a point Q from (A, 0) when A 6= 0 and Q is close enough to (A, 0) behaves like the Euclidean distance. Also, the volume of a ball of radius r centered at the origin behaves like r3 , while the volume of a ball of radius r 6 A centered at (A, 0) behaves like r2 . It is worthwhile noting that, despite the different size of balls with different centers, the quotient |B(P,2r)| |B(P,r)| is bounded (at least, in the cases we have worked out). Namely 3

|B (0, 2r)| 2 (2r) 6 r3 = 256 |B (0, r)| 16

Basic geometry of vector fields

65

and, for A > 0 and r 6 A, 2

|B ((A, 0) , 2r)| 6A (2r) = 24. 6 |B ((A, 0) , r)| Ar2 In Chapter 9 we will prove a very general result about the volume of control balls for families of H¨ ormander vector fields which, applied to this particular example, will give the following bound (see Example 9.2):

C1 r3 + r2 |x| 6 |B ((x, y) , r)| 6 C2 r3 + r2 |x| 2 for every p (x, y) ∈ R and r > 0, with C1 , C2 depending on an upper bound on r 2 2 and x + y . This in particular will imply the doubling condition
C2 8r3 + 4r2 |x| |B ((x, y) , 2r)| 8C2 6 6 3 2 |B ((x, y) , r)| C1 (r + r |x|) C1

where, however, we will not determine an explicit value for the constants C1 , C2 . 1.12

Notes

The definitions of exponential of a vector field, commutator, Lie algebra, given in sections 1.2 and 1.3 are standard. A good reference for the material that we will need in this book from differential geometry and Lie groups is the monograph by Varadarajan [156]. A definition of control distance induced by a general system of vector fields (sections 1.4 and 1.5) has been given by Nagel-Stein-Wainger [131] in 1985, where, actually, several notions of distance are introduced and compared. This general notion has some antecedents in the papers by Franchi-Lanconelli [92], [94], [93] in 1982-1984, who introduced and studied the distance induced by some special families of vector fields (see also our remarks in section 1.11.3). The notion of connectivity (section 1.6) has a much older history. As we have already pointed out in the Introduction, the connectivity theorem is known as “Rashevski-Chow’s theorem”, from [61], [140], dating back to 1938-39. At that time these geometric properties of noncommuting vector fields were studied in the context of Pfaffian systems, motivated by the study of nonholonomic systems and by Carath´eodory’s work on the foundations of thermodynamics, in 1909 ([55], see also [22, Chap. V and Appendix 6, 7]). Rashevski-Chow’s theorem is important in the context of geometric control theory; for some history and an introduction to this circle of ideas the reader can look at the monograph by Jurdjevic [112]. H¨ ormander’s condition (section 1.3) is so labeled after the paper by H¨ormander [107], but obviously it was already known in relation with Rashevski-Chow’s theorem and its applications to geometric control theory. Our proof of the connectivity theorem and other basic properties of the control distance (section 1.6) follows the ideas originally contained in [131]; however, much detailed computation, particularly in connection with the extension of these results to a system of vector fields containing a drift term X0 of weight two is taken from the

66

H¨ ormander operators

paper by Bramanti-Brandolini-Pedroni [33], where similar results are established in a nonsmooth context; see also the paper [128] by Morbidelli. The weak maximum principle proved in section 1.8 is a version of Picone’s maximum principle, originally proved in [139], 1929. The results about propagation of maxima (section 1.9) are due to Bony, [21], 1969, see also Amano, [3], 1974. On the subject of Frobenius theorem, briefly touched in Theorem 1.76, we refer the reader to [156, §1.3]. Some more motivation and history on the subject of this chapter can be found in [24, Chap. 4]. We also point out the monograph [8, Chapter 1-11] by Biagi-Bonfiglioli, where much of the material of this chapter is dealt in details.

Chapter 2

Function spaces defined by systems of vector fields

In this chapter we introduce some function spaces which are analogous to the classical Sobolev or H¨ older spaces W k,p (Ω) , C k,α (Ω) over a domain Ω ⊆ Rn , but are defined replacing the cartesian derivatives ∂xi with a system of vector fields Xi , and the Euclidean distance |x − y| with the control distance dX (x, y) induced by the system of vector fields. We will establish the basic properties of these function spaces, such as completeness, Hilbert space properties for Sobolev spaces with p = 2, relations with the corresponding “Euclidean” spaces, approximation by smooth functions, basic norm inequalities for H¨ older spaces. Some further properties of these function spaces will be proved in Chapters 3 and 8, in the special case where Ω is RN endowed with a structure of homogeneous group, and in Chapter 11, again in the general case. Throughout this chapter, we will consider a system X = {X0 , X1 , . . . , Xq } of smooth vector fields in an open subset Ω of Rn , where X0 (if present) plays the role of a drift, having weight two, as explained in Chapter 1, section 1.5. There is a basic difference between the assumptions that we will make when studying Sobolev or H¨ older spaces. While most of the properties of Sobolev spaces can be studied for a general system of smooth vector fields, not necessarily satisfying H¨ormander’s condition, in the study of H¨ older spaces the control distance is also involved, so it is natural to assume from the beginning H¨ ormander’s condition, in order to have the finiteness of the distance (by the connectivity result in Chapter 1, see Theorem 1.45). 2.1 2.1.1

Sobolev spaces induced by vector fields Definition and basic properties of Sobolev spaces

In this section we are going to define Sobolev spaces induced by a system of smooth vector fields in an open subset Ω of Rn . We start generalizing to this context the notion of weak derivative. Definition 2.1 (Weak derivatives) Let Ω ⊆ Rn be an open set and let X be a smooth vector field in Ω. We say that a given f ∈ L1loc (Ω) is differentiable in weak 67

68

H¨ ormander operators

sense with respect to X if there exists g ∈ L1loc (Ω) such that for every ϕ ∈ C0∞ (Ω) Z Z g (x) ϕ (x) dx = f (x) X ∗ ϕ (x) dx (2.1) Ω

Ω ∗

where the transpose operator X is defined as follows: if Xϕ (x) =

n X

ci (x) ∂xi ϕ (x) then X ∗ ϕ (x) = −

i=1

n X

∂xi (ci ϕ (x)) .

(2.2)

i=1

In this case we will write g = Xf . Analogously, if X1 , . . . , Xk are smooth vector fields in Ω, the higher order weak derivative X1 X2 . . . Xk f is defined by the identity Z Z X1 X2 . . . Xk f (x) ϕ (x) dx = f (x) Xk∗ . . . X2∗ X1∗ ϕ (x) dx. Ω

Ω

If g is a smooth function, integration by parts shows that its weak derivative Pn Xg coincides with the usual derivative Xg = i=1 ci ∂xi g; actually, the consistence of the two notions of derivative is the reason of the appearance of the transpose X ∗ in (2.1). Note that the transpose of a vector field is not in general a vector field. Namely, X ∗ has the form: X ∗ϕ = −

n X

∂xi (ci ϕ) = −

i=1

n X

ci ∂xi ϕ − ϕ

i=1

n X

∂xi ci = −Xi ϕ + cϕ

(2.3)

i=1

with c smooth function. We can now give the following definition of Sobolev spaces adapted to a fixed system of smooth vector fields. Throughout the following we will keep the notation introduced in definition 1.17 about multiindices and derivatives. Definition 2.2 (Sobolev spaces) Given a system X = {X0 , X1 , X2 , . . . , Xq } of smooth vector fields on an open subset Ω of Rn , where the vector field X0 (possibly k,p lacking) has weight 2, while X1 , . . . , Xq have weight 1, we say that f ∈ WX (Ω) p (1 6 p 6 ∞) if f ∈ L (Ω) and there exist, in weak sense, all the derivatives XI f ∈ Lp (Ω) for |I| 6 k. We set kf kW k,p (Ω) = kf kLp (Ω) + X

k X

j

D f p L (Ω) j=1

where X

j

D f p = kXI f kLp (Ω) . L (Ω) |I|=j

Also, we define the Sobolev spaces of functions “vanishing at the boundary” of Ω as k,p k,p follows: the space WX,0 (Ω) is the closure of C0∞ (Ω) in the norm of WX (Ω).

Function spaces defined by systems of vector fields

69

Remark 2.3 If the vector fields are smooth up the boundary of Ω we obviously have kf kW k,p (Ω) 6 c kf kW k,p (Ω) X

so that k,p W k,p (Ω) ⊆ WX (Ω)

(see also Convention 1.26). Note that we are not assuming H¨ ormander’s condition for the system X. As already advertized, several basic results about these Sobolev spaces do not require this assumption. On the other hand, this assumption will be required when we want to prove some results which assure a control on the function f in terms of its derivatives Xi f (like in Sobolev or Poincar´e’s inequality): in this case, it is clearly important that the system X be rich enough to control every direction. For the properties which require H¨ ormander’s condition, a delicate point appears k,p in the definition of WX (Ω) in the weighted case, that is when the vector field X0 1,2 (Ω) does not of weight two is present in our system. For instance, the space WX take into any account the derivative with respect to X0 , which on the other hand is essential to get a control of the function in some directions. Hence we can expect troubles in handling this space, and more generally, Sobolev spaces of odd orders, in situations when the aim is getting a fine tuned control on a function by means of its derivatives. We will encounter these problems in Chapters 8, 11 and 12, when proving a priori estimates for solutions of second order equations defined by H¨ ormander vector fields. Proposition 2.4 The integration by parts formula Z Z f Xi g = gXi∗ f Ω

Ω

1,2 (i = 1, 2, . . . , q) holds for any couple of functions f, g ∈ WX (Ω) such that one of 1,2 them belongs to WX,0 (Ω).

Proof. This follows from the definition of weak derivative and the density of C0∞ (Ω) 1,2 (Ω), keeping into account the explicit form of the operator X ∗ = −Xi + c. in WX,0

The following basic facts are easily established: k,p 1,p Proposition 2.5 The spaces WX (Ω) , WX,0 (Ω) are Banach spaces for any p ∈ n [1, ∞] and any open subset Ω of R , separable for any p ∈ [1, ∞), and Hilbert spaces for p = 2, with respect to the inner product Z X Z hf, giW k,2 (Ω) = f (x) g (x) dx + XI f (x) XI g (x) dx X

Ω

16|I|6k

Ω

(where, for simplicity, we are considering real valued functions, and vector spaces over the field R).

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H¨ ormander operators

Proof. These facts simply follow as in the Euclidean case from the definition of weak derivative and the completeness of Lp (Ω) spaces (see e.g. [1, Thm. 3.3, Thm. 3.6]), and do not depend on particular properties of the system of vector fields.



1,2 1,2 We now focus on the Hilbert spaces WX (Ω), WX,0 (Ω) and their duals 0  0  0 1,2 1,2 1,2 WX (Ω) and WX,0 (Ω) . Let us start with T ∈ WX (Ω) . By Riesz repre-

1,2 sentation theorem (see e.g. [42, Thm. 5.5]) there exists f ∈ WX (Ω) such that for 1,2 every g ∈ WX (Ω) Z q Z X T g = hg, f iW 1,2 (Ω) = g (x) f (x) dx + Xj g (x) Xj f (x) dx. X

Ω

Ω

j=1

For given f0 , f1 , . . . , fq ∈ L2 (Ω) we can now consider the (apparently more general) linear functional defined by ! Z q X gf0 + fi Xi g dx. (2.4) T : g 7→ Ω

i=1

1,2 Clearly T is continuous on WX (Ω) since

|T g| 6 kgk2 kf0 k2 +

q X

kfi k2 kXi gk2 6

i=1

q X

!1/2 2 kfi k2

kgkW 1,2 (Ω) . X

i=0

Analogously to what happens in the Euclidean case (see [1, Thm. 3.10]) it can be  0 1,2 proved that every T ∈ WX (Ω) can be represented this way, although the functions fi are not uniquely determined, and kT k(W 1,2 (Ω))0 X   !1/2 q  X  2 = min kfi kL2 (Ω) with f0 , f1 , . . . , fq such that (2.4) holds .   i=0

1,2 WX

(Ω) is a separable Hilbert space we have (see e.g. [42, Coroll. 3.30 Since and Proposition 3.5]): ∞

1,2 Proposition 2.6 (Weak compactness of WX (Ω)) If {fk }k=1 is a bounded se∞ 1,2 1,2 quence in WX (Ω), then there exist a subsequence {fhk }k=1 and f ∈ WX (Ω) such  0 1,2 that for any T ∈ WX (Ω) we have T fhk → T f . In particular ! ! Z Z q q X X fk φ + Xi fk Xi φ dx → fφ + Xi f Xi φ dx Ω

Ω

i=1

i=1

1,2 for any φ ∈ WX (Ω), and

kf kW 1,2 (Ω) 6 lim inf kfhk kW 1,2 (Ω) . X

X

Function spaces defined by systems of vector fields

71

1,2 The above characterization of linear continuous functionals on WX (Ω) still 1,2 holds for linear continuous functionals on the space WX,0 (Ω). However, the dual  0 1,2 1,2 space of WX,0 (Ω) admits another representation. Let T ∈ WX,0 (Ω) ; as al-

ready remarked there exist functions f0 , f1 , . . . , fq ∈ L2 (Ω) such that for every 1,2 g ∈ WX,0 (Ω) Z q Z X Tg = g (x) f0 (x) dx + Xj g (x) fj dx. Ω

j=1

Ω

Pq Let us consider the distribution in Ω given by f0 + j=1 Xj∗ fj . Clearly for every ϕ ∈ C0∞ (Ω) we have q D E X f0 + Xj∗ fj , ϕ = T ϕ. j=1 1,2 (Ω), every distribution of the kind Moreover, since test functions are dense in WX,0 Pq ∗ 2 f0 + j=1 Xj fj with f0 , f1 , . . . fq ∈ L (Ω) can be uniquely extended to a functional 1,2 T ∈ WX,0 (Ω) (see e.g. [1, Thm. 3.12] for the case of the classical Sobolev spaces). This representation theorem will be implicitly assumed in the definition of weak solution to the Dirichlet problem, which we will give in Chapter 3, section 3.7.

While the above properties hold for spaces defined by any system of smooth vector fields, the next property holds for Sobolev spaces defined by systems of H¨ ormander vector fields (compare with Remark 2.3): Proposition 2.7 (Regularity by means of Sobolev spaces) Given a system X = {X0 , X1 , X2 , . . . , Xq } of H¨ ormander vector fields on an open set Ω ⊂ Rn , then: 1. For any 1 6 p 6 ∞, we have ∞ T k,p WX (Ω) ⊂ C ∞ (Ω) . k=1

2. If H¨ ormander’s condition at (weighted) step s holds in Ω, then for any positive integer k and p ∈ [1, ∞] and any Ω0 b Ω ks,p WX (Ω) ⊂ W k,p (Ω0 )

(where W k,p stands for the classical Sobolev space) and there exists a constant c > 0 such that kukW k,p (Ω0 ) 6 c kukW ks,p (Ω) X

for every u ∈

ks,p WX

(Ω).

Proof. Let x0 ∈ Ω and assume H¨ ormander’s condition at step s holds at x0 . By continuity, there exist n multiindices I1 , I2 , . . . , In , with |Ij | 6 s, and a neighborhood U (x0 ) such that for any x ∈ U (x0 )


X[I1 ] x , X[I2 ] x , . . . , X[In ] x

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H¨ ormander operators


are linearly independent. This means that the n × n matrix with rows X[I1 ] x ,

X[I2 ] x , . . . , X[In ] x belongs to C ∞ (U (x0 )) and it is invertible for any x ∈ U (x0 ); therefore also its inverse matrix is in C ∞ (U (x0 )). Hence for any x ∈ U (x0 ) we can solve the linear systems n X
ei = cij (x) X[Ij ] x i = 1, 2, . . . , n j=1

(where ei are the canonical unit vectors in Rn ), finding cij ∈ C ∞ (U (x0 )) so that n X ∂ = cij (x) X[Ij ] ∂xi j=1 and more generally X ∂α = cαβ (x) XIβ ∂xα β

with |Iβ | 6 ks and coefficients cαβ bounded on U (x0 ). This means that kukW k,p (U (x0 )) 6 c kukW ks,p (U (x0 )) . X This implies ∞ ∞ T T k,p (U (x0 )) ⊂ WX W k,p (U (x0 )) ⊂ C ∞ (U (x0 )) k=1

(2.5)

k=1

where the second inclusion follows by the usual (Euclidean) Sobolev embedding theorems (see e.g. [1, Thm. 4.12, case A]), provided we have chosen U (x0 ) smooth enough (for instance, an Euclidean ball). This implies the first assertion of the Proposition. Also, under the assumption that H¨ ormander’s condition at step s holds in Ω, every 0 point x0 ∈ Ω has a neighborhood U (x0 ) ⊂ Ω such that (2.5) holds. By compactness of Ω0 this gives the second assertion. 2.1.2

Approximation by smooth functions

k,p Now we consider the problem of approximating a function in WX (Ω) with functions ∞ in C0 (Ω), in a compact subset of Ω. In order to fix notation, we start recalling the following well-known result (see for instance [42, Theorem 4.22]):

Lemma 2.8 Euclidean ball, let J ∈ C0∞ (B1 ) with R (Mollifiers) Let B1 be the unit −n J (x) > 0, J (x) dx = 1, and let Jε (x) = ε J (x/ε) for ε > 0. For every f ∈ Lp (Ω) (1 6 p < ∞) (extended to be equal to zero outside Ω), let Z fε (x) = (Jε ∗ f ) (x) = Jε (x − y) f (y) dy. Then fε ∈ C0∞ (Rn ) and

Rn

fε → f in Lp (Ω) as ε → 0+ . Moreover, if f is bounded and continuous in Ω (extended to zero outside Ω), then fε → f locally uniformly in Ω; if f is continuous and compactly supported in Ω, then fε → f uniformly in Ω.

Function spaces defined by systems of vector fields

73

Note that we can rewrite Z Z (f ∗ Jε ) (x) = f (x − εz) J (z) dz = f (x + εy) J (y) dy provided the function J is chosen such that J (x) = J (−x) (for instance, we can take J radial). Since supp J ⊂ B1 , for x ∈ Ω0 b Ω and ε small enough, the function f (x + εy) is evaluated in a set Ω00 b Ω. The main goal of this section is to prove the following approximation result. Note that we are not assuming that the system of vector fields satisfies H¨ormander’s condition. Theorem 2.9 (Approximation in Sobolev spaces) Let X = {X0 , X1 , X2 , . . . , Xq } be a system of smooth vector fields in an open set Ω ⊆ Rn , where the vector field X0 (possibly lacking) has weight 2, while X1 , . . . , Xq have weight 1. Let k,p (Ω) with 1 6 p < ∞ and some positive integer k and let f ∈ WX fm = f ∗ J1/m where Jε are the mollifiers defined in the previous Lemma and we assume f equal to zero outside Ω. Then fm → f in Lp (Ω) and for any Ω0 b Ω, k,p fm → f in WX (Ω0 ) .

This result is the analog of Friedrichs’ theorem for classical Sobolev spaces (see for instance [42, Thm. 9.2]). Our main interest in the previous result relies in the following fact, which will be useful in Chapters 8, 11 and 12. k,p Corollary 2.10 For p ∈ [1, ∞), let f ∈ WX (Ω) and φ ∈ C0∞ (Ω). Then f φ ∈ k,p WX,0 (Ω). k,p (Ω) and φ ∈ C0∞ (Ω) then clearly Proof of the Corollary. If f ∈ WX k,p 0 f φ ∈ WX (Ω) with supp (f φ) ⊂ Ω for some Ω0 b Ω. Let us apply to f φ the approximation with smooth functions defined in the previous theorem. For ε > 0 small enough, depending on Ω0 , we have

(f φ)ε ∈ C0∞ (Ω) while k,p (f φ)ε → f φ in WX (Ω0 ) , k,p that is in WX (Ω), in view of the supports of the functions.

For the proof of Theorem 2.9 we need the following fairly standard lemma:

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H¨ ormander operators

Lemma 2.11 (Kernels with vanishing integral) Let Ω0 b Ω and let {Kε (x, y)}0 0 small enough and x ∈ Ω0 , in the integral defining Tε f the point x + εy ranges in a set compactly contained in Ω, where f is defined. Proof. By the vanishing property (ii) and H¨ older inequality, for x ∈ Ω0 and ε small enough Z Z |Tε f (x)| = Kε (x, y) f (x + εy) dy = Kε (x, y) [f (x + εy) − f (x)] dy Z 1/p0 Z 1/p p 6 |Kε (x, y)| dy |Kε (x, y)| |f (x + εy) − f (x)| dy with 1/p0 + 1/p = 1 (assume for the moment p > 1). Hence, by (iii) Z Z Z p p |Tε f (x)| dx 6 c |Kε (x, y)| |f (x + εy) − f (x)| dydx 0 0 Ω Z  ZΩ Ω p 6c sup |Kε (z, y)| |f (x + εy) − f (x)| dx dy Ω z∈Ω0 Ω0 ! Z sup |Kε (z, y)|

6c Ω

z∈Ω0

p

sup kf (· + εw) − f kLp (Ω0 )

dy

|w|61 p

6 c · sup kf (· + εw) − f kLp (Ω0 ) → 0 as ε → 0+ |w|61

for the Lp -continuity of translations (see [42, Lemma 4.3]). The case p = 1 is similar.

Proof of Theorem 2.9. We will prove the theorem assuming that all the vector fields Xi have weight 1 (the drift is lacking). At the end of the proof we will briefly

Function spaces defined by systems of vector fields

75

explain why the theorem still holds in the drift case. We initially consider the case k = 1. 1,p For f ∈ WX (Ω), let fε as in Lemma 2.8. Then fε → f in Lp (Ω) and we have to show that for Ω0 b Ω, Xi fε → Xi f in Lp (Ω0 ) . Write Xi fε − Xi f = Xi (f ∗ Jε ) − (Xi f ∗ Jε ) + (Xi f ∗ Jε ) − Xi f

(2.6)

and observe that (Xi f ∗ Jε )−Xi f tends to zero in Lp (Ω) by Lemma 2.8. It remains to show that Jεi f ≡ Xi (f ∗ Jε ) − (Xi f ∗ Jε )

(2.7)

converges to zero in Lp (Ω0 ), as ε → 0+ . Let us write Xi =

n X

bik (x) ∂xk .

k=1

Using the definition of weak derivative, since the function z 7−→ Jε (z − x) belongs to C0∞ (Ω) for x ∈ Ω0 and ε small enough, we can compute Z Z X n n X i bik (x) (∂xk Jε ) (x − z) f (z) dz + Jε f (x) = ∂zk [bik (z) Jε (x − z)] f (z) dz =

k=1 n Z X

k=1

[∂zk bik (z) Jε (x − z) + (bik (x) − bik (z)) ∂xk Jε (x − z)] f (z) dz.

k=1

Since Jε (−y) = Jε (y), ∂zk Jε (z) = ε−n−1 (∂zk J) (z/ε) and (∂yk J) (−y) = −∂yk J (y) by the change of variable x + εy = z, we obtain Z X n i Jε f (x) = ε−1 ∂yk [(bik (x + εy) − bik (x)) J (y)] f (x + εy) dy k=1

Z ≡

Kεi (x, y) f (x + εy) dy

where n

Kεi (x, y) =

1X ∂yk [(bik (x + εy) − bik (x)) J (y)] . ε

(2.8)

k=1

We claim that the kernel Kεi has the following properties: (a) Kεi is smooth in the joint variables and for every x ∈ Ω0 the function y 7→ Kεi (x, y) is a sum of partial yk -derivatives of functions compactly supported in |y| < 1.

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H¨ ormander operators

(b) For every multiindices α, β there exists c > 0 such that for every ε (small enough) sup ∂xα ∂yβ Kεi (x, y) 6 c. x∈Ω0 ,|y|61

Note that properties (a)-(b) in particular imply that the kernel Kεi satisfies the assumptions of Lemma 2.11, hence Jεi f → 0 in Lp (Ω) as ε → 0. Actually, properties (a)-(b) contain much more information than what is needed to apply Lemma 2.11, but they will be useful in the iteration process that we will use to prove the theorem when k > 1. Property (a) is apparent from the definition of Kεi . As to (b), 1 ∂xα ∂yβ ∂yk [(bik (x + εy) − bik (x)) J (y)] ε 1 |β| β = ε ∂y ∂yk ∂xα bik (x + εy) J (y) + ∂yβ [(∂xα bik (x + εy) − ∂xα bik (x)) ∂yk J (y)] ε ≡ Aε (x, y) + Bε (x, y) . Now, |Aε (x, y)| 6 c by the smoothness of the functions ∂yβ ∂yk ∂xα bik and the compact support in y, while Bε (x, y) is a sum of terms of the kind   1 ∂yβ1 (∂xα bik (x + εy) − ∂xα bik (x)) ∂yβ2 J (y) . ε If β1 = 0 we have 1 α (∂x bik (x + εy) − ∂xα bik (x)) ∂yβ2 J (y) 6 c |y| ∂yβ2 J (y) 6 c, ε while for |β1 | > 1 we can write   β 1 α ∂y 1 (∂x bik (x + εy) − ∂xα bik (x)) ∂yβ2 J (y) ε h i = ε|β1 |−1 ∂yβ1 ∂xα bik (x + εy) ∂yβ2 J (y) 6 ∂yβ1 ∂xα bik (x + εy) ∂yβ2 J (y) 6 c. It is worthwhile summarizing what we have proved so far. Recalling (2.6) we can write Xi fε − Xi f = Jεi f + (Xi f )ε − Xi f where Jεi f

Z (x) =

Kεi (x, y) f (x + εy) dy,

the kernel Kεi is given by (2.8) and satisfies (a)-(b).

Function spaces defined by systems of vector fields

77

k,p Now, for any multiindex I with |I| = k > 1 and f ∈ WX (Ω) we can write

XI fε − XI f = XI (f ∗ Jε ) − (XI f ) ∗ Jε + (XI f ) ∗ Jε − XI f ≡ JεI f + (XI f ) ∗ Jε − XI f where we set, consistently with (2.7), JεI f ≡ XI (f ∗ Jε ) − (XI f ) ∗ Jε . Since XI f ∈ Lp (Ω), by Lemma 2.8 (XI f ) ∗ Jε − XI f → 0 in Lp (Ω) , and we have to prove that JεI f → 0 in Lp (Ω0 ) . In turn, if we write XI = Xj XI 0 with |I 0 | = k − 1, we have JεI f = Xj XI 0 (f ∗ Jε ) − Xj (XI 0 f ∗ Jε ) + Xj (XI 0 f ∗ Jε ) − (Xj XI 0 f ∗ Jε ) =

0 Xj JεI f

+

(2.9)

Jεj XI 0 f.

By the assertion proved in the case k = 1, we already know that Jεj XI 0 f → 0 in 1,p Lp (Ω0 ) , because XI 0 f ∈ WX (Ω), hence we are left to prove that 0

Xj JεI f → 0 in Lp (Ω0 ) . To this end, we will show by induction on k, that: (i) For any I = (j, I 0 ) , with |I| = k, there exist kernels KεI,` (x, y), ` = 1, . . . , N , satisfying properties (a)-(b) (written for the proof of the case k,p k = 1) and multiindices I` , |I` | 6 k − 1, such that, for every f ∈ WX (Ω) , 0 Xj JεI f

(x) =

N Z X

KεI,` (x, y) XI` f (x + εy) dy.

(2.10)

`=1

(ii) An integral representation similar to (2.10) holds for JεI f. In particular, k,p JεI f → 0 in Lp (Ω0 ) for every f ∈ WX (Ω) .

When k = 1, I = i and I 0 = ∅ (i) and (ii) obviously hold since 0

JεI f = (f ∗ Jε ) − f ∗ Jε ≡ 0, and we have already proved that JεI f (x) = Jεi f (x) → 0 in Lp (Ω0 ) as ε → 0+ . Assume now that (i)-(ii) hold up to k − 1 and let us prove them for k. The point is to prove (i) for k. After that, (2.9) and (2.10) will imply that JεI f has a form similar to (2.10), which by the properties of the kernels and Lemma 2.11 implies that JεI f → 0 in Lp (Ω0 ), so the assertion will be proved.

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H¨ ormander operators

Assume now |I| = k and XI = Xj XI 0 with |I 0 | = k − 1. By (ii) we can write N Z X 0 0 JεI f (x) = KεI ,` (x, y) XI` f (x + εy) dy `=1 0

with |I` | 6 k − 2. Let us compute Xj JεI f . To make the proof easier to follow, we 0 will handle a single term in the summation that defines JεI f , which we rewrite as Z 0 Aε (x) = KεI (x, y) XJ f (x + εy) dy 0

for some |J| 6 k − 2, and KεI (x, y) satisfying (a)-(b). Assuming for the moment f smooth, if n X Xj = bj` (x) ∂x` , `=1

we can write Z Xj Aε (x) =

0 Xjx KεI

Z (x, y) XJ f (x + εy) dy +

0

KεI (x, y) (Xj XJ f ) (x + εy) dy

n Z X

0 1 KεI (x, y) [bj` (x) − bj` (x + εy)] ∂y` [XJ f (x + εy)] dy ε `=1 Z Z 0 0 = Xjx KεI (x, y) XJ f (x + εy) dy + KεI (x, y) (Xj XJ f ) (x + εy) dy

+

n Z h 0 i X 1 + ∂y` KεI (x, y) (bj` (x + εy) − bj` (x)) XJ f (x + εy) dy ε `=1 Z Z 0 ≡ KεI (x, y) (Xj XJ f ) (x + εy) dy + KεI (x, y) XJ f (x + εy) dy

(2.11) with h i 0 0 1 ∂y` (bj` (x + εy) − bj` (x)) KεI (x, y) + Xjx KεI (x, y) . ε Note that the smoothness of f has been exploited to compute ∂x` XJ f and not k−1,p just Xj XJ f , which surely exists for f ∈ WX (Ω). Before showing that actually k,p the conclusion still holds for f ∈ WX (Ω), let us check that KεI (x, y) satisfies 0 properties (a)-(b). Recall that KεI (x, y) satisfies the same properties by inductive assumption. The property of smoothness and support of KεI clearly follows by the 0 0 analogous properties of KεI (x, y) . Since KεI (x, y) is a sum of partial yk -derivatives of a smooth function supported in |y| < 1 and the differential operators Xjx and ∂yk commute, the same is true for KεI . Hence (a) holds. As to (b), KεI (x, y) =

∂xα ∂yβ KεI (x, y) h i 0 0 1 = ∂xα ∂yβ ∂y` (bj` (x + εy) − bj` (x)) KεI (x, y) + ∂xα ∂yβ Xjx KεI (x, y) ε ≡ Aε (x, y) + Bε (x, y) .

Function spaces defined by systems of vector fields

79

0

Then |Bε (x, y)| 6 c because KεI (x, y) satisfies (b) by inductive assumption, while Aε (x, y) is a sum of terms of the kind   0 1 ∂yβ1 (∂xα1 bj` (x + εy) − ∂xα1 bj` (x)) ∂xα2 ∂yβ2 KεI (x, y) ε with β1 6= 0 and this function is bounded uniformly in ε, for the same reasoning used in the case k = 1. Recalling that also the term Z 0 KεI (x, y) (Xj XJ f ) (x + εy) dy appearing in (2.11) has a kernel satisfying (a)-(b) (by inductive assumption) and acts on a derivative Xj XJ f of weight 6 k − 1, we have proved that, at least for f smooth, (i) holds. k,p We are left to show that (i) still holds for f ∈ WX (Ω). By the inductive k−1,p (Ω) there exists {fk } ⊂ C0∞ (Ω) such that assumption we know that for f ∈ WX k−1,p fk → f in WX (Ω0 ) . So we can write N Z X I0 Xj Jε fk (x) = KεI,` (x, y) XI` fk (x + εy) dy (2.12) `=1

with |I` | 6 k−1. By the properties (a)-(b) satisfied by the kernels KεI,` , we conclude that the linear integral operators with these kernels are, for any fixed ε, bounded from Lp (Ω) to Lp (Ω0 ). Therefore the right hand side in (2.12) converges in Lp (Ω0 ) to N Z X KεI,` (x, y) XI` f (x + εy) dy. `=1

Next, take φ ∈ C0∞ (Ω0 ); then, for k → ∞, Z Z Z 0 0 0 Xj JεI fk (x) φ (x) dx = JεI fk (x) Xj∗ φ (x) dx → JεI f (x) Xj∗ φ (x) dx while on the other hand Z Z 0 Xj JεI fk (x) φ (x) dx →

N Z X

! KεI,`

(x, y) XI` f (x + εy) dy φ (x) dx.

`=1

This means that (2.10) holds in the sense of weak derivatives. So the proof is complete, assuming that the drift is lacking. However, in presence of a vector field X0 with weight two, the previous proof k,p still holds. Actually, for f ∈ WX (Ω), let I be a multiindex with |I| 6 k, and let ` (I) be the usual length of the multiindex (clearly, ` (I) 6 |I| 6 k). We can then repeat the same proof, by induction on the integer ` (I) (and not |I| or k), concluding the assertion. k,p Theorem 2.9 allows to approximate every f ∈ WX (Ω), with 1 6 p < ∞, by k,∞ k,∞ smooth functions. If f ∈ WX (Ω), one cannot expect that fε → f in WX (Ω) (this already fails to be true for k = 0). We can prove, however, the following upper bound:

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H¨ ormander operators

k,∞ Proposition 2.12 Let f ∈ WX (Ω) and let fε = f ∗ Jε be as in Theorem 2.9. 0 Then, for every Ω b Ω there exists c depending on the vector fields, k, Ω and Ω0 such that for every ε small enough,

kfε kW k,∞ (Ω0 ) 6 c kf kW k,∞ (Ω) . X

X

Proof. For simplicity we will prove the statement only for k = 1. Since Z fε (x) = (Jε ∗ f ) (x) = Jε (x − y) f (y) dy n R R with Jε (x) > 0 and Rn Jε (x) dx = 1, we have kfε kL∞ (Rn ) 6 kf kL∞ (Ω)

(2.13)

(where f can be thought set equal to zero outside Ω). Moreover, from the proof of Theorem 2.9 (see (2.6)-(2.7)) we read that Xi fε = (Xi f ∗ Jε ) + Jεi f so that, by (2.13)

kXi fε k ∞ 0 6 kXi f k ∞ 0 + Jεi f ∞ 0 . L

(Ω )

L

(Ω )

L

(Ω )

Again from the the Proof of Theorem 2.9 we read that, for x ∈ Ω0 and ε > 0 small enough, Z Jεi f (x) = Kεi (x, y) f (x + εy) dy where, for every x ∈ Ω0 , Kεi (x, y) is supported in |y| 6 1 and for some constant c > 0 and every ε (small enough), we have sup Kεi (x, y) 6 c x∈Ω0 ,|y|61

with c depending on the vector fields, Ω, Ω0 . Therefore Z i

i Kε (x, y) dy 6 c kf k ∞

Jε f ∞ 0 6 kf k ∞ L (Ω) L (Ω) sup L (Ω ) x∈Ω0

|y| 0 we have

c1 |x − y| 6 d (x, y) 6 c2 |x − y|

1/s

for any x, y ∈ Ω,

where s is the step of H¨ ormander’s condition in Ω0 .

(2.14)

Function spaces defined by systems of vector fields

81

k,α Definition 2.13 (Spaces CX ) For any1 α > 0, f : Ω → R, let:   |f (x) − f (y)| |f |C α (Ω) = sup : x, y ∈ Ω, x 6= y , α X d (x, y)

kf kC α (Ω) = |f |C α (Ω) + kf kL∞ (Ω) , X n o α CX (Ω) = f : Ω → R : kf kC α (Ω) < ∞ . X

For α = 1 the space

α CX

(Ω) is called Lip (Ω). Also, for any positive integer k, let n o k,α CX (Ω) = f : Ω → R : kf kC k,α (Ω) < ∞ , X

with kf kC k,α (Ω) = X

k X

j

D f α + kf kC α (Ω) C (Ω) X

X

j=1

and X

j

D f α = kXI f kC α (Ω) . C (Ω) X

X

|I|=j

Here the derivatives must be intended as classical directional derivatives, or intrinsic derivatives: Xj f (x) =

d f (γ (t))/t=0 dt

where γ (t) is an integral curve of the vector field Xj with γ (0) = x. k,α α α (Ω), (Ω), Lip0 (Ω) and CX,0 (Ω) the subspaces of CX We will denote by CX,0 k,α Lip (Ω) and CX (Ω) of functions which are compactly supported in Ω. We have the following inclusions between classical C k,α spaces and those defined by vector fields: Proposition 2.14 Under the assumptions stated at the beginning of this section, for every α > 0 we have: α C α (Ω) ⊂ CX (Ω) ⊂ C α/s (Ω)

(2.15)

and for every positive integer k we have k,α C k,α (Ω) ⊂ CX (Ω) ks,α CX

(Ω) ⊂ C k,α/s (Ω) ,

(2.16) (2.17)

with continuous inclusions. 1 In the following we will restrict to α ∈ (0, 1], but this definition is formally meaningful for any α > 0.

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H¨ ormander operators

Proof. The inequalities (2.14) immediately give (2.15). The first inclusion in (2.15) implies (2.16), while the second one implies (2.17), expressing derivatives ∂xk as commutators of weight 6 s of the vector fields. The continuity of the above inclusions also implies that a bounded sequence in α CX (Ω) is a sequence of equibounded and equicontinuous functions in the standard sense. This allows to prove the following: k,α Proposition 2.15 Under the above assumptions CX (Ω), equipped with the norm k·kC k,α (Ω) is a Banach space. X

α α Proof. Let {fk } be a Cauchy sequence in CX (Ω). Since {fk } is bounded in CX (Ω), the functions {fk } are equicontinuous and equibounded and therefore by Arzel`aAscoli theorem (see [162, Chap. III, §3]), there exists a subsequence, that we keep calling {fk }, that converges uniformly in Ω to some continuous function f . By the Cauchy condition on |fm − fk |α , for any ε > 0 there exists n0 such that for k, m > n0 and any x1 6= x2 ,

|(fm (x1 ) − fk (x1 )) − (fm (x2 ) − fk (x2 ))| 6 ε. α d (x1 , x2 ) α (Ω). Letting k → ∞ we read that fm → f in CX 1,α If {fk } is now a bounded sequence in CX (Ω), the above reasoning shows that, passing to a subsequence {fk }, we have, for k → ∞,

kfk − f kC α (Ω) → 0 X

kXi fk − gi kC α (Ω) → 0 for i = 1, 2, . . . , q X

1,α α (Ω), hence the completeness of CX (Ω) will be proved for some f, g1 , . . . , gq ∈ CX as soon as we show that actually gi = Xi f . To see this, fix x ∈ Ω and let γ (t) be an integral curve of Xi defined in Ω at least for |t| 6 T and with γ (0) = x. Then, for k → ∞,

fk (γ (t)) → f (γ (t)) d (fk (γ (t))) = (Xi fk ) (γ (t)) → gi (γ (t)) dt uniformly in t ∈ [−T, T ]. This implies that there exists d f (γ (t)) = gi (γ (t)) , dt which for t = 0 means Xi f (x) = gi (x). k,α Completeness of CX (Ω) for k > 1 can be proved analogously using an iterative argument. α The continuous inclusion C α (Ω) ⊂ CX (Ω) immediately implies the following:

Function spaces defined by systems of vector fields

83

α Proposition 2.16 The space CX,0 (Ω) is dense in Lp (Ω) for any α ∈ (0, 1] and p ∈ [1, ∞).

The above proposition is a first hint of the fact that also in this context H¨older spaces are interesting only for α 6 1. The next result makes this idea more precise: α Proposition 2.17 Let f ∈ CX (Ω). Then:

(i) If α > 1, then Xi f ≡ 0 in Ω for i = 1, 2, . . . , q. In particular, if in our system of H¨ ormander vector fields the drift X0 is lacking, then f is constant in Ω. (ii) If α > 2, then also X0 f ≡ 0, and f is constant in Ω. Proof. (i). For any x ∈ Ω and i ∈ {1, 2, . . . , q}, let γ (t) be the integral curve of Xi such that: ( γ 0 (t) = (Xi )γ(t) γ (0) = x Then: 

 d f (γ (t)) − f (x) f (γ (t)) (0) = lim . t→0 dt t On the other hand, the curve ϕ (τ ) = γ (tτ ) satisfies ( ϕ0 (τ ) = t (Xi )ϕ(τ ) ϕ (0) = x, ϕ (1) = γ (t) , Xi f (x) =

(2.18)

which means that ϕ ∈ Cx,γ(t) (t) , hence d (x, γ (t)) 6 t and we can write α |f (γ (t)) − f (x)| 6 |f |a d (γ (t) , x) 6 |f |a tα . If α > 1 by (2.18) we have Xi f (x) ≡ 0. If the drift is lacking, the conditions Xi f ≡ 0 in Ω for i = 1, 2, . . . , q imply that f is constant, by Proposition 1.28. (ii). If γ (t) is an integral curve of X0 starting at x, the previous reasoning shows (with the same notation) that p  |t| ϕ ∈ Cx,γ(t) and α

|f (γ (t)) − f (x)| 6 |f |α d (γ (t) , x) 6 |f |α |t| Hence for α > 2 we conclude X0 f (x) ≡ 0.

α/2

.

By the previous discussion, and also in view of the results about singular integrals on H¨ older spaces that we will develop in Chapter 7, henceforth we will consider α H¨ older spaces CX (Ω) only for α ∈ (0, 1). α The next Proposition collects some simple properties of CX functions which will be useful later (Chapters 11-12).

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H¨ ormander operators

Proposition 2.18 (Basic inequalities for H¨ older norms) (i) For any f ∈ C01 (B (x, R)) , |f |C α (B(x,R)) 6 cR1−α ·

q X

! sup |Xi f | + R sup |X0 f | .

(2.19)

B(x,R)

i=1 B(x,R)

α (ii) For any couple of functions f, g ∈ CX (B (x, R)), one has

|f g|C α (B(x,R)) 6 |f |C α (B(x,R)) kgkL∞ (B(x,R)) + |g|C α (B(x,R)) kf kL∞ (B(x,R)) X X X (2.20) and kf gkC α (B(x,R)) 6 kf kC α (B(x,R)) kgkC α (B(x,R)) . X

X

X

(2.21)

Moreover, if both f and g vanish at least at a point of B (x, R), then |f g|C α (B(x,R)) 6 2Rα |f |C α (B(x,R)) |g|C α (B(x,R)) . X

X

X

(2.22)

(iii) The following relations hold between H¨ older seminorms on different domains: |f |C α (Ω1 ) 6 |f |C α (Ω2 ) if Ω1 ⊂ Ω2 ; X

X

α |f |C α (B(x,r)) = |f |C α (Ω2 ) if B (x, r) ⊂ Ω2 and f ∈ CX,0 (B (x, r)) . X

X

Moreover, if B (xi , r) (i = 1, 2, · · · , k) is a finite family of balls of the same radius r, such that ∪ki=1 B (xi , r) is connected and ∪ki=1 B (xi , 2r) ⊂ Ω, then α for any f ∈ CX (Ω), kf kC α (∪k

i=1 B(xi ,r))

X

6c

k X

kf kC α (B(xi ,2r)) X

(2.23)

i=1

with c depending on the family of balls, but not on f . 2,α (iv) There exists r0 > 0 such that for any f ∈ CX,0 (B (x, R)) and 0 < r 6 r0 , we have the following interpolation inequality: 2 (2.24) kX0 f kL∞ (B(x,R)) 6 rα/2 |X0 f |C α (B(x,R)) + kf kL∞ (B(x,R)) . X r Proof. (i) By Theorem 1.56 on B (x, 3R) we can write, for f ∈ C01 (B (x, R)) v u q uX √ 2 2 |f (x) − f (y)| 6 qd (x, y) · sup t (Xi f ) + d (x, y) · sup |X0 f | . B(x,R)

B(x,R)

i=1 α

for any x, y ∈ B (x, R) . Dividing by d (x, y) and taking the sup for x, y ∈ B (x, R) we obtain (2.19). (ii) We have |(f g) (x) − (f g) (y)| 6 |f (x) − f (y)| |g (x)| + |g (x) − g (y)| |f (y)| α

6 |f |C α (B(x,R)) d (x, y) kgkL∞ (B(x,R)) X

α

+ |g|C α (B(x,R)) d (x, y) kf kL∞ (B(x,R)) , X

(2.25)

Function spaces defined by systems of vector fields

85

which gives (2.20). From this (2.21) follows easily. Assume now that f (x1 ) = g (y1 ) = 0 for some x1 , y1 ∈ B (x, R). Then (2.25) implies |(f g) (x) − (f g) (y)| 6 |f (x) − f (y)| |g (x) − g (y1 )| + |g (x) − g (y)| |f (y) − f (x1 )| α

α

6 |f |C α (B(x,R)) d (x, y) |g|C α (B(x,R)) d (x, y1 ) X

X

α

α

+ |g|C α (B(x,R)) d (x, y) |f |C α (B(x,R)) d (y, x1 ) X

X

α

α

6 (2R) |f |C α (B(x,R)) |g|C α (B(x,R)) d (x, y) X

X

which gives (2.22). (iii) Inequality |f |C α (Ω1 ) 6 |f |C α (Ω2 ) for Ω1 ⊂ Ω2 is obvious from the definition X X α of |f |C α (Ωi ) . If f ∈ CX,0 (B (x, r)) and x1 , x2 ∈ B (x, r) we clearly have X

α

|f (x1 ) − f (x2 )| 6 d (x1 , x2 ) |f |C α (B(x,r)) X

while for x1 , x2 ∈ Ω2 \ B (x, r) we have |f (x1 ) − f (x2 )| = 0. Assume now x1 ∈ B (x, r) and x2 ∈ Ω2 \B (x, r), fix ε > 0 and let γ ∈ Cx1 x2 (d (x1 , x2 ) + ε) (an almost minimizing curve joining x1 and x2
). By the continuity of t 7→ d (x, γ (t)), the curve intersects ∂B (x, r), so let x = γ t such that d (x, x) = r. Then f (x) = 0 and we can write α

|f (x1 ) − f (x2 )| = |f (x1 )| = |f (x1 ) − f (x)| 6 d (x1 , x) |f |C α (B(x,r)) X

α

6 (d (x1 , x2 ) + ε) |f |C α (B(x,r)) X

where we have exploited the fact that any intermediate point on the curve γ has distance 6 d (x1 , x2 ) + ε from the endpoints (see Remark 1.33). From this we conclude that |f |C α (B(x,r)) = |f |C α (Ω2 ) . X X i Let ζi (i = 1, 2, . . . , k) be smooth cutoff functions such that supp ζi ⊂ B2r , Pk k i ζ = 1 in ∪ B . The triangle inequality and the monotonicity with respect i=1 r i=1 i to the domain give |f |C α (∪k

j j=1 Br

)6

k X

|f ζi |C α (∪k

i=1

j j=1 Br

)6

k X

|f ζi |C α (∪k

j j=1 B2r

)=

i=1

i since supp f ζi ⊂ B2r

=

k X i=1

|f ζi |C α (B i ) 6 2 2r

k X

kf kC α (B i ) kζi kC α (B i ) 6 c 2r 2r

i=1

k X

kf kC α (B i ) 2r

i=1

where we have used (2.21). 2,α (iv). Let f ∈ CX,0 (B (x, R)). For any x ∈ B (x, R), let γ(t) be the curve such that γ 0 (t) = (X0 )γ(t) , γ(0) = x. This γ (t) will be defined at least for t ∈ [0, r0 ] where r0 > 0 is a number only depending on B (x, R) and X0 . Then, for any r ∈ (0, r0 ) and a suitable θ ∈ (0, 1) we can write d f (γ(r)) − f (γ(0)) = r [f (γ(t))]t=θr = r (X0 f ) (γ(θr)) . dt

86

H¨ ormander operators

Hence 1 [f (γ(r)) − f (γ(0))] r 1/2 and since, by definition of γ and d, d (γ (0) , γ (θr)) 6 (θr) 6 r1/2 , 2 |(X0 f ) (x)| 6 |(X0 f ) (γ(0)) − (X0 f ) (γ(θr))| + kf kL∞ r 2 α 6 d (γ (0) , γ (θr)) |X0 f |C α (B(x,R)) + kf kL∞ (B(x,R)) X r 2 6 rα/2 |X0 f |C α (B(x,R)) + kf kL∞ (B(x,R)) , X r which is (2.24). (X0 f ) (x) = (X0 f ) (γ(0)) − (X0 f ) (γ(θr)) +

2.2.2

Approximation by smooth functions

k,α (Ω) functions Next, we need some results concerning the approximation of CX with functions in C0∞ (Ω), on subsets Ω0 b Ω. However, in contrast with what happens for Sobolev spaces, even in the Euclidean case we cannot approximate H¨ older continuous functions by smooth functions in the H¨ older norm: √ Example 2.19 Let f (x) = 3 x in [−1, 1], let g ∈ C 1 ([−1, 1]) and observe that |f (x) − g (x) − (f (0) − g (0))| |g (x) − g (0)| |f (x) − f (0)| 6 + 1= 1/3 1/3 1/3 |x| |x| |x| |g (x) − g (0)| 6 kf − gkC 1/3 + . 1/3 |x|

Since limx→0

|g(x)−g(0)| |x|1/3

= 0, we have kf − gkC 1/3 > 1.

k,α Instead of proving an approximation in CX (Ω), we will prove a locally uniform approximation by smooth functions. α Let {Jε }ε>0 be a family of mollifiers like in Lemma 2.8 and for every f ∈ CX (Ω) (0 < α < 1), let fε (x) = (Jε ∗ f ) (x). By Lemma 2.8 we know that fε ∈ C0∞ (Rn ) α and that fε → f locally uniformly in Ω as ε → 0+ . Moreover, if f ∈ CX,0 (Ω) the α convergence is uniform in Ω. This holds because, by (2.15), CX (Ω) functions are uniformly continuous in the standard sense. The following result will be enough for our purposes: k,α Theorem 2.20 (Approximation in H¨ older spaces) Let f ∈ CX (Ω) with α ∈ (0, 1) and k = 1, 2, 3, . . . and let fm = f ∗ J1/m . Then

fm → f locally uniformly in Ω 0

and for any Ω b Ω, XI fm → XI f locally uniformly in Ω0 for every |I| 6 k. The above convergences are uniform (and not just locally uniform) if f is compactly supported inside Ω.

Function spaces defined by systems of vector fields

87

To prove this result, we need some more preliminaries. α First of all we need a version of Lemma 2.11 for functions in CX . Lemma 2.21 (Kernels with vanishing integral) Let Ω0 b Ω and let {Kε (x, y)}00 of group automorphisms (“dilations”) such that Dλ (x1 , x2 , . . . , xN ) = (λα1 x1 , λα2 x2 , . . . , λαN xN )

(3.1)
for suitable positive real numbers α1 , α2 , . . . , αN . We will write G = RN , ◦, Dλ to PN denote this structure. The number Q = i=1 αi will be called homogeneous dimension of G. It is not restrictive to assume 1 = α1 6 α2 6 . . . 6 αN (normalization of the dilation exponents). The last statement needs a clarification. We can always relabel the variables so that α1 6 α2 6 . . . 6 αN . Also, if {Dλ }λ>0 are automorphisms for one choice of the exponents (α1 , α2 , . . . , αN ), the same is true for the exponents (kα1 , kα2 , . . . , kαN ) for any k > 0. Therefore we can always normalize the numbers αi so that α1 = 1. Some other remarks are in order. Under our normalization assumption, we always have Q > N , and Q = N only if the dilations are the Euclidean ones. We anticipate that under the stricter assumptions that we will make from section 3.5, speaking of stratified groups or graded groups, the numbers αi (and therefore Q) will be positive integers, as in the examples below. 1 This

assumption is not restrictive, as noted in [16, p. 13].

Homogeneous groups in RN

95

Saying that {Dλ }λ>0 are automorphisms explicitly means that Dλ (x ◦ y) = Dλ (x) ◦ Dλ (y)

(3.2)

Dλ (Dµ (x)) = Dλµ (x) .

(3.3)

while from (3.1) we read

Note that under the change of coordinates x = Dλ (y) the volume element transforms according to dx = λQ dy

(3.4)

which justifies the name of homogeneous dimension for Q. Example 3.3 The simplest example
of (noncommutative) homogeneous group is the Heisenberg group H1 = R3 , ◦, Dλ , where: (x1 , y1 , t1 ) ◦ (x2 , y2 , t2 ) = (x1 + x2 , y1 + y2 , t1 + t2 + 2(x2 y1 − x1 y2 )) and
Dλ (x, y, t) = λx, λy, λ2 t . −1

In this case Q = 4. Note that (x1 , y1 , t1 )
Example 3.4 Let G = R4 , ◦, Dλ with:

= (−x1 , −y1 , −t1 ).

(x1 , x2 , x3 , x4 ) ◦ (y1 , y2 , y3 , y4 )   1 = x1 + y1 , x2 + y2 , x3 + y3 + x2 y1 , x4 + y4 + x3 y2 + x2 y22 2 and
Dλ (x1 , x2 , x3 , x4 ) = λx1 , λx2 , λ2 x3 , λ3 x4 . Here Q = 7. In this case −1

(x1 , x2 , x3 , x4 ) 3.1.1

=

  1 −x1 , −x2 , −x3 + x1 x2 , −x4 + x2 x3 − x32 . 2

Structure of the homogeneous group operation

The dilations induce a notion of homogeneity of functions: Definition 3.5 We say that a function f , smooth in RN \ {0}, is Dλ -homogeneous of degree β ∈ R (or simply “β-homogeneous”) if f (Dλ (x)) = λβ f (x)

∀λ > 0, x ∈ RN \ {0} .

In contrast with the above definition, when we say that f is a polynomial of degree β we refer to the usual notion of degree, with no reference to the dilations. While a great variety of Lie group operations in RN can be defined, the presence of dilations imposes severe restrictions on the analytical form of the operation ◦, which makes this operation easier to study:

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H¨ ormander operators

Theorem 3.6 (Analytical properties of the group law) The map P : R N × RN → RN P (x, y) = x ◦ y has the following properties. (a) P is a polynomial; more precisely, if P (x, y) = (P1 (x, y) , P2 (x, y) , . . . , PN (x, y)) then Pk is a polynomial in x, y of total degree 6 αk . In particular, P1 (x, y) = x1 + y1 . (b) P has the following structure P (x, y) = x + y + Q (x, y) where Q (x, y) satisfies Q (x, 0) = Q (0, y) = 0 ∀x, y ∈ RN .

(3.5)

In particular, Q (x, y) does not contain pure monomials in x or y; all its terms are mixed, of degree at least 2. Also, Qk (x, y) only depends on x1 , x2 , . . . , xk−1 , y1 , y2 , . . . , yk−1 , (c) The Jacobian matrices of P have the following  1  ∂P 2 (x,y) ∂P (x, y)   =  ∂x. 1  ∂x .. 

structure: 0 1 .. .

∂PN (x,y) ∂PN (x,y) ∂x1 ∂x2



1

 ∂P (x,y) 2 ∂P (x, y)   =  ∂y. 1  ∂y .. 

0 1 .. .

∂PN (x,y) ∂PN (x,y) ∂y1 ∂y2

 ... 0 . . . . ..   ..  . 0  ... 1  ... 0 . . . . ..   ..  . 0  ... 1

(d) The Lebesgue measure is left and right invariant with respect to the translations ◦; in particular, for any fixed y ∈ RN : x = x0 ◦ y =⇒ dx = dx0 x = y ◦ x0 =⇒ dx = dx0

Homogeneous groups in RN

97

The reader is invited to check that the properties stated in this theorem actually hold in the Examples 3.3-3.4. Proof. (a)-(b). By (3.1)-(3.2) we have Pk (Dλ (x) , Dλ (y)) = λαk Pk (x, y), that is Pk is αk -homogeneous. Differentiating with respect to xj gives

∂xj Pk (Dλ (x) , Dλ (y)) = λαk −αj ∂xj Pk (x, y) (3.6) and for β = (β1 , . . . , βN )  β PN ∂ (Pk (Dλ (x) , Dλ (y))) = λαk − j=1 βj αj ∂x



∂ ∂x

!

β Pk

(x, y) .

By an analogous reasoning on the y-derivatives we conclude that the function  γ  β ∂ ∂ Pk (x, y) ∂y ∂x PN is d-homogeneous, with d ≡ αk − j=1 (βj + γj ) αj . Recalling that αj > 1, we read that d < 0 as soon as |β| + |γ| > αk .
∂ ∂ β In this case, the function ∂y Pk (x, y) must vanish, since otherwise it ∂x would be unbounded at the origin, while it must be smooth. Therefore, Pk is a polynomial of (Euclidean) degree 6 αk . Since the origin is the group identity, we must have 

γ

P (x, 0) = x and P (0, y) = y for any x, y ∈ RN . Being P a polynomial, this implies that P (x, y) = x + y + Q (x, y) with Q satisfying (3.5). Moreover Q (x, y) does not contain any pure monomial in x or y. In particular, since P1 has degree 6 α1 = 1, necessarily P1 (x, y) = x1 + y1 . This ends the proof of (a)-(b), apart from the last statement in (b) which will follow from (c). (c) From (3.6) we read that the functions ∂xk Pk (x, y) , ∂yk Pk (x, y) are homogeneous of degree zero. Since these functions are smooth this implies that they are constant. On the other hand, ∂xk Pk (x, y) = 1 + ∂xk Qk (x, y) ∂yk Pk (x, y) = 1 + ∂yk Qk (x, y) which, since Qk (x, y) cannot reduce to a monomial axk or byk , implies ∂xk Qk (x, y) = ∂yk Qk (x, y) = 0 ∂xk Pk (x, y) = ∂yk Pk (x, y) = 1

∀x, y.

Also, from (3.6) we know that ∂xj Pk (x, y) is (αk − αj )-homogeneous. Since αk − αj < 0 for k < j, in order for ∂xj Pk (x, y) to be smooth we must have ∂xj Pk (x, y) ≡

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H¨ ormander operators

0 for k < j, and an analogous property holds for ∂yj Pk (x, y). It follows that the Jacobian matrices have the required structure, and the last statement in (b) holds. (d) By (c), for any fixed y, the change of coordinates x = x0 ◦y has Jacobian matrix of determinant 1, hence dx = dx0 . The same holds for the change of coordinates x = y ◦ x0 . The reader could ask if some easy characterization of the group inverse x−1 also holds in a homogeneous group. As we will see, in most concrete examples one simply has x−1 = −x. However, this is not necessarily the case. What we can say is contained in the following: Proposition 3.7 (Analytical properties of the inversion) The inversion map x 7→ x−1 on a homogeneous group has the following structure:
x−1 i = −xi + qi (x) where qi is an αi -homogeneous polynomial, only depending on the variables x1 , x2 , . . . , xi−1 . The Jacobian matrix of the inversion map is triangular with −1 on the diagonal, hence the change of variables x0 = x−1 gives dx0 = dx. Proof. From the relations: (x ◦ y)1 = x1 + y1 (x ◦ y)2 = x2 + y2 + Q2 (x1 , y1 ) (x ◦ y)3 = x3 + y3 + Q3 (x1 , x2 , y1 , y2 ) ... letting x ◦ y = 0, y = x−1 we can write, iteratively,
x−1 1 = −x1
x−1 2 = −x2 − Q2 (x1 , −x1 ) = −x2 + q2 (x1 )
x−1 3 = −x3 − Q3 (x1 , x2 , −x1 , −x2 + q2 (x1 )) = −x3 + q3 (x1 , x2 ) ... which is the desired structure for the inverse mapping. 3.1.2

Homogeneous norms

In every homogeneous group we can define a “homogeneous norm”, which is the right substitute of the Euclidean norm. Abstractly, this concept is defined as follows: Definition
3.8 A homogeneous norm or a gauge on a homogeneous group G = RN , ◦, Dλ is a continuous function k·k : RN → [0, +∞),

Homogeneous groups in RN

99

such that, for some constant c > 0, for every x, y ∈ RN , (i) (ii) (iii) (iv)

kxk = 0 ⇐⇒ x = 0 kD λ (x)k

−1

= λ kxk ∀λ > 0

x 6 c kxk kx ◦ yk 6 c (kxk + kyk) .

If a homogeneous norm has the extra property that

−1

x = kxk for any x ∈ RN , we will say that it is symmetric. If a homogeneous norm k·k is C ∞ outside the origin, we will say that it is smooth. If a homogeneous norm has the extra property that the set  x ∈ RN : kxk = 1 is a piecewise C 1 surface, we will say that it is regular.  NoteN that if the last requirement holds, then the same is true for the sets x ∈ R : kxk = r for any r > 0, and the divergence theorem can be applied  to the sets x ∈ RN : kxk 6 r . From now on, we will always use the symbol k·k to denote a homogeneous norm, and the symbol |·| to denote the Euclidean norm. Before making explicit examples of homogeneous norms, let us prove some properties that will simplify the job of checking conditions (i)-(iv) in the above definition. Proposition 3.9 Let k·k : RN → [0, +∞), be a continuous function satisfying conditions (i)-(ii) in the above definition. Then it also satisfies (iii)-(iv). Moreover, the sets  BR = x ∈ RN : kxk 6 R are compact in the Euclidean sense. Proof. Let us start proving the last assertion. The set BR is closed since k·k is continuous, so let us prove that it is also bounded in Euclidean sense. Since the set Σ1 = {|x| = 1} is compact and does not contain the origin, there exists c > 0 such that kxk > c for |x| = 1. This implies that Bc = {kxk 6 c} ⊂ {|x| = 6 1} . However, the set Bc contains the origin and is connected, since any point of x ∈ Bc can be joined to the origin by the arc γ (t) = Dt (x), t ∈ [0, 1]. Therefore {kxk 6 c} ⊂ {|x| < 1} .

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H¨ ormander operators

If R 6 c we are done, while if R > c, by dilation  {kxk 6 R} ⊂ DR/c ({|x| < 1}) ⊂

 |x|
0, otherwise there is nothing to prove. Then kx ◦ yk = λ kDλ−1 (x ◦ y)k = λ kDλ−1 (x) ◦ Dλ−1 (y)k 6 C (kxk + kyk) because kDλ−1 (x)k + kDλ−1 (y)k = λ−1 (kxk + kyk) = 1. There are several concrete ways to define a homogeneous norm on G. Some of the most common are collected in the following: Proposition 3.10 Each of the following functions k·k is a regular homogeneous norm on G: 1

(a) (b) (c)

kxk = maxk=1,2,...,N |xk | αk ; P αN 1/αN N αk kxk = |x | ; k k=1 kxk = λ ⇐⇒ D1/λ (x) = 1 (if x 6= 0), k0k = 0.

Moreover, if the group inverse is just x−1 = −x, then the previous homogeneous norms are symmetric. Finally, the homogeneous norm in (c) is also smooth.

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Remark 3.11 We will show in Proposition 3.38 that the number αN appearing in (b), which is also the maximum of α1 , . . . , αN , coincides with the step s of nilpotency of the Lie algebra g of the group G (see Definition 3.27 below), a number which will play an important role in the following. Another commonly used homogeneous norm will be introduced in sections 3.5 and 3.8 exploiting the control distance. Also, a quite different type of homogeneous norm will be introduced for stratified groups in Chapter 6, section 6.4, by means of the homogeneous fundamental solution of a sublaplacian. Proof. The functions (a) and (b) clearly satisfy (i) and (ii) and are continuous. Hence, by Proposition 3.9, (a) and (b) are homogeneous norms. Let us come to (c). First of all, it is well defined because for any x 6= 0, the function f (λ) = D1/λ (x) is continuous, strictly decreasing, f (0+ ) = +∞, f (+∞) = 0, hence there exists a unique λ such that |Dλ (x)| = 1. Moreover, since the function Fλ (x) = D1/λ (x) − 1 is smooth for x 6= 0 and ∇Fλ (x) 6= 0 for x 6= 0, by the implicit function theorem the function k·k is smooth outside the origin. Property (i) is obvious. Moreover, since D1/kxk (x) = 1, by (3.3) D1/λkxk (Dλ (x)) = D1/kxk (x) = 1 hence (ii) holds. Hence also (c) is a homogeneous norm. −1 Now, = −x, then from the analytic form of (a) and (b) one reads

−1if x

= kxk in these cases. For case (c), we note that from that x Dλ (−x) = Dkxk (−x) = −D (x) (which follows from the explicit form of dilations) we infer

λ Dkxk (x) = 1 hence x−1 = kxk. Finally, each of the three homogeneous norms is obviously regular, in the sense of Definition 3.8. Now we state some other important properties that are satisfied by any homogeneous norm.
Theorem 3.12 (Properties of homogeneous norms) Let G = RN , ◦, Dλ be a homogeneous group and k·k a homogeneous norm on G. Then: (a) For any R > 0 there exist two positive constants c1 , c2 (depending on R) such that 1/αN

c1 |x| 6 kxk 6 c2 |x|

if |x| 6 R (or kxk 6 R).

(3.7)

(b) Any two homogeneous norms k·k1 , k·k2 on G are equivalent: there exist positive constants k1 , k2 such that k1 kxk1 6 kxk2 6 k2 kxk1 ∀x ∈ RN .

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See Remark 3.11 for the number αN appearing in (a). Proof. Let us first prove (b). The compactness of {kxk1 = 1} (by Proposition 3.9) and the continuity of k·k2 immediately give 0 < k1 6 kxk2 6 k2 < ∞ whenever kxk1 = 1.

Then, for any x ∈ RN , k1 6 D1/kxk1 (x) 2 6 k2 that is k1 kxk1 6 kxk2 6 k2 kxk1 which is (b). Now, since all homogeneous norms are equivalent, it is enough to prove (a) for a particular homogeneous norm. Let kxk =

1

max

k=1,2,...,N

|xk | αk

(as in point (a) in the previous Proposition). Then for |x| 6 R, since 1 6 αi 6 αN , we have c01 |x| 6 c1

max

k=1,2,...,N

|xk | 6

1

max

k=1,2,...,N

|xk | αk 6 c2

max

k=1,2,...,N

1

|xk | αN 6 c02 |x|

1/αN

.

Finally, since by Proposition 3.9 the set {kxk 6 R} is compact, kxk 6 R =⇒ |x| 6 CR for some constant CR . Therefore inequalities (3.7) hold, for some constants c1 , c2 depending on R, also under the condition kxk 6 R. Remark 3.13 Inequalities (a) in the previous theorem cannot hold with uniform constants k1 , k2 when x ranges in an unbounded set, since the inequality k1 |x| 6 k2 |x|

1/αN

breaks down as |x| → ∞. This also means that without knowing the boundedness of the sets {kxk 6 R} by Proposition 3.9, we could not infer this property from the inequalities in (a). Also note that each of the two intermediate inequalities in (a) cannot hold for arbitrarily large x. This can be seen as follows. Let x = Dλ (ei ) = (0, 0, . . . , λαi , . . . , 0) . Then for any λ > 0 kxk = kDλ (ei )k = λ kei k = ci λ |x| = λαi 1/αN

so that the inequalities c1 |x| 6 kxk 6 c2 |x|

rewrite as

c1 λαi 6 ci λ 6 c2 λαi /αN , which cannot hold for arbitrarily large λ, since αi < αN and αi > 1 sometimes.

Homogeneous groups in RN

3.1.3

103

Quasidistances and gauge balls

We start with the following abstract Definition 3.14 (Quasidistance) Let S be a set. A function d : S × S → [0, +∞) is called a quasidistance on S if for some constant c > 1 and every x, y, z ∈ S: (i) d (x, y) = 0 ⇔ x = y (ii) d (y, x) = d (x, y) (iii) d (x, y) 6 c [d (x, z) + d (z, y)] . The function d is called a quasisymmetric quasidistance if the above properties hold, with (ii) replaced by the weaker requirement (iv) d (y, x) 6 cd (x, y) . Property (iii) is sometimes called quasitriangle inequality. A homogeneous norm allows to define in a natural way a (quasisymmetric) quasidistance on a homogeneous group, which in turn induces a family of balls and a topology:
Definition 3.15 (The gauge quasidistance) Let G = RN , ◦, Dλ be a homogeneous group and k·k a homogeneous norm on G. The gauge quasidistance is given by:

ρ (x, y) = y −1 ◦ x . Proposition 3.16 (Properties of the gauge quasidistance) The function ρ is a quasisymmetric quasidistance on RN . Moreover ρ is homogeneous and translation invariant, that is, for any λ > 0, x, y, z ∈ RN : ρ (Dλ (x) , Dλ (y)) = λρ (x, y) ρ (x ◦ z, y ◦ z) = ρ (x, y) . If the group inverse is just x symmetric).

−1

= −x, then ρ is a quasidistance (that is, it is

The proof of this proposition is immediate from the properties of the homogeneous norm. Although the Euclidean distance |y − x| is not equivalent

−1

to the quasidistance

y ◦ x , it is locally equivalent, near the pole, to y −1 ◦ x , as expressed by the following proposition, which will be sometimes useful: Proposition 3.17 There exist positive constants c1 , c2 and a neighborhood of the origin V such that when x, y ∈ V we have c1 |x − y| 6 x−1 ◦ y 6 c2 |x − y| .

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H¨ ormander operators


Proof. Let P (x, y) = x ◦ y. Then, expanding the function y 7→ P x−1 , y near x, gives
x−1 ◦ y = ∂y P x−1 , x (y − x) + o (y − x) .

By Theorem 3.6 we know that ∂y P x−1 , y = I + ∂y Q x−1 , y with Q (0, y) ≡ 0, which also implies ∂y Q (0, y) ≡ 0 so that
x−1 ◦ y = y − x + ∂y Q x−1 , x (y − x) + o (y − x) with
∂y Q x−1 , x 6 c |x| for |x| < 1. It follows that there are constants c1 and c2 and −1 a neighborhood of the origin V such that if x, y ∈ V we have c1 |x − y| 6 x ◦ y 6 c2 |x − y| . Definition 3.18 (Gauge balls) For any r > 0 and x ∈ RN we define:  Br (x) = B (x, r) = y ∈ RN : ρ (y, x) < r . Whenever we have a family of “balls”, as above, we can use them to define a topology, saying that a set A is open whenever for each x ∈ A there exists a ball Br (x) ⊂ A. (The axioms of topology are then easily checked). Note, however, that the balls Br (x) themselves need not to be open with respect to this topology. Namely, the possible presence of a constant c > 1 in the quasitriangle inequality prevents us from writing the “reverse” triangle inequality d (x, y) > |d (x, z) − d (y, z)| . As a consequence, there is no way to prove (in this generality) the following fact (which holds whenever d is a distance): ∀x ∈ BR (x0 ) ∃Br (x) ⊂ BR (x0 ) . With this in mind, we give the following: Definition 3.19 (Space of homogeneous type) Let (S, d) be a set endowed with a quasidistance, such that the d-balls are open with respect to the topology they induce. Let µ be a Borel measure on S (endowed with this topology) that satisfies the doubling condition: there exists c > 0 such that µ (B2r (x)) 6 cµ (Br (x)) < +∞ ∀x ∈ S, r > 0.

(3.8)

Then we say that (S, d, µ) is a space of homogeneous type. Property (3.8) easily implies that if for some ball one has µ (Br (x)) = 0, then this is true for any ball, hence µ is identically zero. Therefore, apart from the trivial case µ ≡ 0 one always has µ (Br (x)) > 0. Coming back to our more concrete context, we can now collect the properties related to the gauge balls in homogeneous groups, in the following

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105


Theorem 3.20 (Properties of the gauge balls) Let G = RN , ◦, Dλ be a homogeneous group, k·k a homogeneous norm on G, ρ the quasidistance induced by k·k. Then: (a) The gauge balls induce the Euclidean topology; moreover, they are open. (b) A set is bounded in Euclidean sense if and only if it is bounded with respect to the quasidistance ρ. (c) The measure of balls equals: |Br (x)| = rQ |B1 (0)| ∀x ∈ RN , r > 0, where Q is the homogeneous dimension. In particular, the doubling property holds, in the form: |B2r (x)| = 2Q |Br (x)| .

(3.8)

(d) If x−1 = kxk (this happens if the group inverse is just x−1 = −x and the homogeneous
norm is one of those appearing in Proposition 3.10) then RN , ρ, dx is a space of homogeneous type. Otherwise the definition is fulfilled apart from the fact that ρ is only quasisymmetric. Proof. Point (b) is contained in Theorem 3.12. As to (a), by (3.7) and Proposition 3.17, for x, y in a neighborhood of the origin we can write c1 |y − x| 6 ρ (x, y) 6 c2 |y − x|

1/Q

.

This implies that every ρ-ball B (x, r) contains an Euclidean ball BE (x, r1 ) (even though the number r1 does not depend in an easy way on r) and viceversa,

hence the two topologies coincide. Since, moreover, the function y 7→ ρ (x, y) = y −1 ◦ x is continuous, the ρ-balls are open. Then (a) holds. Next, the translation invariance of the Lebesgue measure (see Theorem 3.6) and (3.4) gives Z Z Q |Br (x)| = dy = r dw = rQ |B1 (0)| , kx−1 ◦ykR kxk for every R > 0, α > 0, with the constant c depending on G and on the homogeneous norm chosen.

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H¨ ormander operators

Proof. Let us prove the assertion about the origin, the other being analogous. A dilation x = DR (y) in the integral gives Z Z dx dy α = R , Q−α Q−α kxk6R kxk kyk61 kyk provided the last integral converges, so let us discuss this point. Let α > 0, then Z Z ∞ Z ∞ X X
dy dy k+1 Q−α dy = 6 2 Q−α Q−α 1 1 kyk6 1k kyk61 kyk k=0 2k+1

∞ X k=0


k Q−α

Z

dy = c2Q−α

2

kyk6

1 2k

∞ X 1 = +∞. kα 2

k=0

Homogeneous Lie algebras of invariant vector fields on a homogeneous group

3.2.1

Translation invariant and homogeneous differential operators on homogeneous groups

We are now interested in studying differential operators, and in particular vector fields, on a homogeneous group. Some interesting relations which may hold between the properties of a differential operator and the structure of homogeneous group are expressed in the following:
Definition 3.22 Let G = RN , ◦, Dλ be a homogeneous group and let  α X ∂ P = cα (x) ∂x α be any differential operator with smooth coefficients on RN . We say that: (a) P is left invariant if P (Ly f ) = Ly (P f ) for every smooth function f , where Ly f (x) = f (y ◦ x) . (b) P is right invariant if P (Ry f ) = Ry (P f ) for every smooth function f , where Ry f (x) = f (x ◦ y) . (c) P is β-homogeneous (for some β ∈ R) if P (f (Dλ (x))) = λβ (P f ) (Dλ (x)) for every smooth function f , λ > 0 and x ∈ RN \ {0}.

Homogeneous groups in RN

107


∂ For instance, on any homogeneous group G = RN , ◦, Dλ the operator ∂x is i ∂ αi -homogeneous, just by the definition of dilations. Also, the operator P = xj ∂x i is (αi − αj )-homogeneous, because   ∂f ∂ f (Dλ (x)) = xj λαi (Dλ (x)) = λαi −αj P f (Dλ (x)) . xj ∂xi ∂xi More in general, an operator of the form  β ∂ ∂ βN ∂ β1 ∂ β2 γ P =x = xγ11 xγ22 . . . xγNN β1 β2 . . . βN ∂x ∂x1 ∂x2 ∂xN Pn is d-homogeneous with d = i=1 αi (βi − γi ). If f is a γ-homogeneous function, smooth outside the origin, and P is a βhomogeneous differential operator, then P f is a (γ − β)-homogeneous function, since: P [f (Dλ (x))] = λβ (P f ) (Dλ (x)) , P [f (Dλ (x))] = P [λγ f (x)] = λγ (P f ) (x) hence (P f ) (Dλ (x)) = λγ−β (P f ) (x) . Analogously, it is immediate to check that the composition of two differential operators Pi (i = 1, 2) which are βi -homogeneous, respectively, is a (β1 + β2 )homogeneous differential operator. For vector fields we have the following simple characterization of homogeneity. PN ∂ Proposition 3.23 The vector field P = j=1 bj (x) ∂x is β-homogeneous if and j only if the coefficients bj (x) are (αj − β)-homogeneous. Proof. The if part is a simple consequence of the definition of homogeneous function and homogeneous operator. As far as the only if part, assume that P is β-homogeneous, then λβ (P f ) (Dλ (x)) = P [f (Dλ (x))] =

N X

N

bk (x)

X ∂f ∂ (f (Dλ (x))) = bk (x) λαk (Dλ (x)) . ∂xk ∂xk k=1

k=1 β

αj

Taking f (x) = xj , gives λ bj (Dλ (x)) = bj (x) λ , so that bj (x) is (αj − β)homogeneous. Given a differential operator P, recall that its transpose operator P ∗ is defined by the identity Z Z P f · g = f · P ∗g for any f, g ∈ C0∞ (G). The following holds:

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H¨ ormander operators

Proposition 3.24 Let P be a differential operator on G and P ∗ its transpose operator. Then: (a) If P is left (right) invariant, then P ∗ is left (right) invariant. (b) If P is β-homogeneous for some β ∈ R, then P ∗ is β-homogeneous. Proof. (a) We have to show that for every f ∈ C0∞ (G) and y ∈ G, P ∗ (Ly f ) = Ly (P ∗ f ) . For any g ∈ C0∞ (G) we have Z Z Z ∗ g · P (Ly f ) = P g · Ly f = Ly−1 (P g) · f Z Z Z
= P Ly−1 g · f = Ly−1 g · P ∗ f = g · Ly (P ∗ f ) which gives the assertion. (b) We have to show that for every f ∈ C0∞ (G) and x ∈ G, P ∗ [f (Dλ (x))] = λβ (P ∗ f ) (Dλ (x)) . For any g ∈ C0∞ (G) we have Z Z (P ∗ f ) (Dλ (x)) g (x) dx = (P ∗ f ) (y) g (Dλ−1 (y)) λ−Q dy G ZG = f (y) λ−β (P g) (Dλ−1 (y)) λ−Q dy G Z = P ∗ [f (Dλ (x))] λ−β g (x) dx G

which gives the assertion. Let us now illustrate how the notion of left or right invariant differential operator specializes to vector fields. There is a natural way to construct a left (or right) invariant vector field X. It is enough to fix a vector v ∈ RN and then define X as the unique left (right) invariant vector field which agrees with v at the origin. In turn, this can be done explicitly computing d [f (x ◦ tv)]|t=0 (left invariant), or (3.9) dt d [f (tv ◦ x)]|t=0 (right invariant). Xf (x) = dt It is immediate to check that the right hand side in (3.9) represents the action on f of a vector field which is left invariant and takes the value v at x = 0. Hence, it is exactly the only left invariant vector field which agrees with v at the origin. In particular, if v = ei is the unit vector pointing in the xi -direction, then Xf (x) =

Xf (x) = ∂yi [f (x ◦ y)]|y=0 (left invariant), or Xf (x) = ∂yi [f (y ◦ x)]|y=0 (right invariant).

(3.10)

Homogeneous groups in RN

109


Example 3.25 Let G = H1 = R3 , ◦, Dλ be the Heisenberg group, defined in Example 3.3. Let us compute the left invariant vector fields X1 , X2 , X3 which agree ∂ , ∂ , ∂ , respectively. By (3.10) we have at the origin with ∂x 1 ∂x2 ∂x3 ∂ [f (x1 + y1 , x2 + y2 , x3 + y3 + 2(y1 x2 − y2 x1 ))]|y=0 ∂y1 ∂f ∂f (x) + 2x2 (x) , = ∂x1 ∂x3

X1 f (x) =

∂ [f (x1 + y1 , x2 + y2 , x3 + y3 + 2(y1 x2 − y2 x1 ))]|y=0 ∂y2 ∂f ∂f = (x) − 2x1 (x) , ∂x2 ∂x3

X2 f (x) =

and ∂ [f (x1 + y1 , x2 + y2 , x3 + y3 + 2(y1 x2 − y2 x1 ))]|y=0 ∂y3 ∂f = (x) . ∂x3

X3 f (x) =

∂ ∂ ∂ ∂ ∂ + 2x2 ∂x , X2 = ∂x − 2x1 ∂x and X3 = ∂x . Analogously we can Hence X1 = ∂x 1 3 2 3 3 ∂ at the origin, for compute the right invariant vector fields XiR which agree with ∂x i instance: ∂ X1R f (x) = [f (x1 + y1 , x2 + y2 , x3 + y3 + 2(x1 y2 − x2 y1 ))]|y=0 ∂y1 ∂f ∂f = (x) − 2x2 (x) . ∂x1 ∂x3

We will see that this method of generating translation invariant vector fields is particularly useful in the development of the general theory. 3.2.2

The Lie algebra of left invariant vector fields

In Chapter 1, section 1.5 we recalled the definition of Lie algebra and introduced, for a domain Ω ⊂ RN , the Lie algebra X (Ω) of all
smooth vector fields X : Ω → RN . In the present situation, the Lie algebra X RN contains an interesting finite dimensional Lie subalgebra.
Proposition 3.26 Let G = RN , ◦, Dλ be a homogeneous group, and let g be the family of all left (respectively, right) invariant real smooth vector fields on RN . Then
N g is a Lie subalgebra of X R . As a vector space g is isomorphic to RN . In view of the above proposition we can give the following:
Definition 3.27 Let G = RN , ◦, Dλ be a homogeneous group. The Lie algebra g of G is the Lie algebra of real smooth vector fields on RN that are left invariant with respect to G.

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H¨ ormander operators


Proof of Proposition 3.26. The fact that g is a Lie subalgebra of X RN is straightforward. Let us prove that g is isomorphic to RN . Toward this aim let us consider the mapping (see (3.9)): M : RN → g v 7→ X with Xf (x) =

d dt

[f (x ◦ tv)]|t=0 . This is linear because

(M (λ1 v1 + λ2 v2 ))0 = λ1 v1 + λ2 v2 = (λ1 M v1 + λ2 M v2 )0 , where X0 is the vector field X evaluated at x = 0. Then, since M (λ1 v1 + λ2 v2 ) and λ1 M v1 + λ2 M v2 are left invariant vector fields which agree at the origin, they are actually the same vector field. The map M is also bijective, with inverse M −1 : g → RN X 7→ X0 . It follows that g is isomorphic to RN as a vector space. Reasoning as in the last part of the above proof, we see that a basis of g as a vector space is given by the vector fields Yi (i = 1, 2, . . . , N ) which are uniquely defined by the requirement: ∂ at the origin. Yi is left invariant and coincides with ∂xi We will call the set {Y1 , Y2 , . . . , YN } the canonical basis of g. Notation 3.28 Throughout the book, we will usually denote vector fields with the symbols X, Xi , . . . and we will reserve the symbol Yi to denote the vector fields of the canonical basis on a homogeneous group G. In the following theorem we study in detail the structure of these vector fields. Theorem 3.29 (Structure of the canonical basis) The left invariant vector ∂ field Yi which coincides with ∂x at the origin is αi -homogeneous and has the foli lowing structure: X ∂ ∂ Yi = + qik (x) (i = 1, 2, . . . , N ) (3.11) ∂xi ∂xk i 0. Choosing f (x) = xk we get ck = ck λαk −β for any λ > 0 which P implies that ck 6= 0 only for αk = β. This means that X = j:αj =β cj Yj so that X Dλ (X) = cj λαj Yj = λβ X. j:αj =β

Observe that if X is different from zero, then β = αk for some k. Assume now that Dλ (X) = λβ X. Since ! N N X X Dλ (X) = Dλ ci Yi = ci λαi Yi i=1

i=1

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H¨ ormander operators

PN PN αi β we can write i=1 ci λ Yi = i=1 ci λ Yi . Using the fact that the vector fields Y1 , . . . , YN form a basis we have ci λαi = ci λβ so that ci 6= 0 only if αi = β. P Hence X = i:αi =β ci Yi . Since the vector fields Yi are αi -homogeneous X is βhomogeneous. Now we show that the family of dilations Dλ allows to decompose g as direct sum of vector spaces of homogeneous vector fields and that this decomposition is consistent with the Lie algebra structure. We start rewriting the dilation exponents in increasing order without repetition: let β1 < β2 < · · · < βM , so that for every j ∈ {1, . . . , N }, αj = βk for some k ∈ {1, . . . , M }. Then, for k ∈ {1, . . . , M } we set  Wβk = X ∈ g : Dλ (X) = λβk X . Proposition 3.35 The Lie algebra g of left invariant vector fields has the following decomposition as direct sum g =Wβ1 ⊕ Wβ2 ⊕ · · · ⊕ WβM .

(3.20)

Moreover, if we fix two exponents, βi and βj , then either βi +βj = βk for some k    and Wβi , Wβj ⊆ Wβk or Wβi , Wβj = {0} (here Wβi , Wβj denotes the vector space of vector fields of the form [X1 , X2 ] with X1 ∈ Wβi and X2 ∈ Wβj ). Proof. The decomposition (3.20) is a simple consequence of the fact that the elements of the canonical basis are homogeneous. Fix X ∈ g and write X=

N X

ci Yi .

(3.21)

i=1

For every j ∈ {1, . . . , M } let Xj =

P

i:αi =βj

X=

M X

ci Yi . Then Xj

(3.22)

j=1

and clearly Xj ∈ Wβj . Now, let us show that the decomposition (3.22) is unique. PM Assume that X = j=1 Xj0 for some Xj0 ∈ Wβj . Arguing as in the proof of Lemma P 3.34 there exist coefficients di such that Xj0 = i:αi =βj di Yi . Hence X=

M X X j=1 i:αi =βj

di Yi =

N X

di Yi .

i=1

Since the vector fields Yi form a basis, comparing with (3.21) gives di = ci and therefore Xj0 = Xj . Now, let X1 ∈ Wβi and X2 ∈ Wβj and let X = [X1 , X2 ]. By Lemma 3.34 X1 and X2 are respectively βi -homogeneous and βj -homogeneous; it follows that X is βi + βj -homogeneous. Again by Lemma 3.34, then, either X = 0 or βi + βj = βk for some k, so that X ∈ Wβk .

Homogeneous groups in RN

115

Definition 3.36 (Nilpotent Lie algebra) A Lie algebra h is said to be nilpotent of step s, if the chain of vector spaces h1 = h h2 = [h1 , h] h3 = [h2 , h] ··· is such that hs+1 = {0} while hs 6= {0}. Definition 3.37 (Homogeneous Lie algebra) Let h be a Lie algebra endowed with a family of Lie algebra automorphisms {Dλ }λ>0 such that Dλ Dµ = Dλµ for every λ, µ > 0. We will say that h is a homogeneous Lie algebra with dilations {Dλ }λ>0 . Proposition 3.38 (Homogeneity and nilpotency of g) Let G be a homogeneous group. Its Lie algebra g, equipped with the dilations {Dλ }λ>0 induced by the group, is a homogeneous Lie algebra that is also nilpotent of a suitable step s. Namely, s = αN . Proof. By definition the maps Dλ on g preserve the linear structure and are bijective. Let us show that they also preserve the Lie bracket. Since the elements Yi of the basis of g are αi -homogeneous given Yi , Yj their commutator [Yi , Yj ] is clearly (αi + αj )-homogeneous. Hence, Dλ ([Yi , Yj ]) = λαi +αj [Yi , Yj ] = [λαi Yi , λαj Yj ] = [Dλ (Yi ) , Dλ (Yj )] . By linearity, Dλ preserves the Lie bracket on g and is a Lie algebra automorphism. Moreover, for any Yi in the basis of g we have Dλ Dµ Yi = λαi µαi Yi = Dλµ Yi . By linearity, this implies that for any X ∈ g Dλ Dµ X = Dλµ X for every λ, µ > 0 hence g is a homogeneous Lie algebra. Next, let us consider the vector spaces g1 = g and gk+1 = [gk , g] . Every element of g1 is contained in the span of the vector fields Y1 , Y2 , . . . , YN , which are homogeneous of degree α1 = 1, α2 , . . . , αN , respectively. Therefore every element of g2 is contained in the span of the vector fields [Yi , Yj ] which are homogeneous of degree at least 2. Iterating the reasoning, we see that gk is contained in the span of a set of vector fields which are homogeneous of degree at least k. However there are no left invariant vector fields homogeneous of any degree larger than αN , hence for k > αN we must have gk = {0}, and we are done.

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H¨ ormander operators

Summarizing: the Lie algebra
g of left (or right) invariant vector fields over a homogeneous group RN , ◦, Dλ is isomorphic (as a vector space) to RN . It has a (vector space) basis Y1 , Y2 , . . . , YN such that Yi coincides at the origin with ∂xi ; the Yi ’s have polynomial coefficients and can be expressed in terms of the ∂xk by a triangular matrix. Each Yi is αi -homogeneous. Then the dilations can be extended to the Lie algebra g, which becomes a homogeneous Lie algebra and, as a consequence, is nilpotent of some step s, that is any iterated commutator of more than s left invariant vector fields vanishes. We end this section pointing out a relation between left and right invariant operators which will be very useful in the following. Proposition 3.39 Let L, R be any two differential operators with smooth coefficients, left and right invariant, respectively. Then L and R commute: for any smooth function f LRf = RLf. Proof. Since L is left invariant we can write Ly [f (x ◦ y)] = Lf (x ◦ y), so that Lf (x) = Ly [f (x ◦ y)]|y=0 = Ly0 [f (x ◦ y)]|y=0 where L0 is the constant coefficients operator obtained evaluating L at the origin. Similarly we obtain Rf (x) = Rz0 [f (z ◦ x)]|z=0 and therefore LRf (x) = Ly0 [Rf (x ◦ y)]|y=0 = Ly0 [ Rz0 [f (z ◦ x ◦ y)]|z=0 ]|y=0 = Ly0 Rz0 [f (z ◦ x ◦ y)]|(z,y)=(0,0) = Rz0 Ly0 [f (z ◦ x ◦ y)]|(z,y)=(0,0) = RLf (x) since

Rz0 ,

Ly0

obviously commute.

Example 3.40 As we have seen in Example 3.25, on the Heisenberg group H1 the left invariant vector fields which agree at the origin with ∂x1 , ∂x2 are, respectively ∂ ∂ ∂ ∂ + 2x2 ∂x and Y2 = ∂x − 2x1 ∂x while the right invariant vector field Y1 = ∂x 1 3 2 3 ∂ ∂ R which agrees  at the  origin with R∂x1 is Y1 = ∂x1 − 2x2 ∂x3 . The reader is invited to R check that Y1 , Y1 = 0, Y2 , Y1 = 0, while [Y1 , Y2 ] 6= 0. Also, the reader is invited to try to repeat the proof of Proposition 3.39 to prove the (false!) assertion that any two left invariant differential operators commute, to realize what goes wrong. 3.3

Exponential maps on a homogeneous group

In Chapter 1, section 1.2, we have introduced the notion of exponential of a smooth vector field. For a left invariant vector field on a homogeneous group, the exponential has some special properties which is worthwhile to point out here. Namely, it is defined globally in time and it is left invariant and homogeneous. The precise statements of these properties are contained in the following two propositions.

Homogeneous groups in RN

117

Proposition 3.41 Let G be a homogeneous group and let X be a left invariant vector field on G. Then, for any x, y ∈ G and t small enough exp (tX) (x ◦ y) = x ◦ exp (tX) (y) .

(3.23)

Proof. Let ϕ (t) = x ◦ exp (tX) (y) and P (x, y) = x ◦ y. Then d ∂P (x, exp (tX) (y)) (exp (tX) (y)) ∂y dt ∂P = (x, exp (tX) (y)) Xexp(tX)(y) . ∂y

ϕ0 (t) =

By (3.15), ∂P (x, exp (tX) (y)) Xexp(tX)(y) = Xx◦exp(tX)(y) ∂y hence ϕ0 (t) = Xx◦exp(tX)(y) = Xϕ(t) . Since ϕ (0) = x ◦ y we obtain ϕ (t) = exp (tX) (x ◦ y). Since (3.23) with y = 0 gives exp (tX) (x) = x ◦ exp (tX) (0) it is natural to define Exp (tX) = exp (tX) (0)

(3.24)

exp (tX) (x) = x ◦ Exp (tX) .

(3.25)

so that

In the next proposition we summarize some of the properties of Exp that we will use in the following. Proposition 3.42 (Properties of Exp on G) Let G be a homogeneous group and let X be a left invariant vector field on G. Then: (a) Exp (tX) is defined for every t ∈ R and (3.23) holds for every t ∈ R. (b) If X is β-homogeneous for some β > 1, for every λ > 0 and every t ∈ R we have
Exp λβ tX = Dλ (Exp (tX)) . (3.26) Proof. (a) We know there exists ε > 0 such that Exp (tX) = exp (tX) (0) is defined for t ∈ (−ε, ε). Then iterating the identity (3.25) we can write exp (2tX) (x) = exp (tX) exp (tX) (x) = (exp (tX) (x)) ◦ Exp (tX) = x ◦ Exp (tX) ◦ Exp (tX) k

exp (ktX) (x) = x ◦ (Exp (tX))

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H¨ ormander operators

for any positive integer k, and t small enough. This in particular shows that for any left invariant vector field X and any x ∈ G, the exponential exp (tX) (x) is well defined for any t ∈ R and (3.23) holds for every t ∈ R. (b) Let ϕλ (t) = Dλ (Exp (tX)) for some λ > 0 and observe that  
d ϕ0λ (t) = Dλ Exp (tX) = Dλ XExp(tX) . dt Since X is β-homogeneous by Proposition 3.23 we have Dλ (Xx ) = (λα1 b1 (x) , λα2 b2 (x) , . . . , λαN bN (x)) = λβ λα1 −β b1 (x) , λα2 −β b2 (x) , . . . , λαN −β bN (x)




= λβ X(Dλ (x)) . and therefore ϕ0λ (t) = λβ XDλ (Exp(tX)) = λβ Xϕλ (t) . Since ϕλ (0) = 0 we obtain that

ϕλ (t) = exp tλβ X (0) = Exp tλβ X .

3.4

Convolution and mollifiers on a homogeneous group

In Chapter 2 we have shown how the standard mollification technique can be used to prove approximation results for functions in Sobolev or H¨older spaces defined with respect to a family of general H¨ ormander vector fields. The fact that the usual convolution does not fit well with vector fields made the proofs of those results (see Chapter 2, sections 2.1.2 and 2.2.2) quite long and involved. In this section we will define the convolution between functions on a homogeneous group. Since it is defined using the group structure, left and right invariant differential operators act in a very natural way on this convolution product, which will therefore be a useful tool for the analysis of homogeneous H¨ormander operators on groups. In particular, the group convolution will allow an easy proof of a global density result (Theorem 3.49). Throughout the book we will work sometimes in the context of general H¨ ormander vector fields, sometimes with homogeneous left invariant H¨ormander vector fields on groups. Depending on the context, convolution, although always denoted with the same symbol ∗, will have two different meanings. Namely, we adopt the following: Convention 3.43 Whenever a homogeneous group structure is present, the symbol ∗ will always denote the convolution with respect such a group structure, defined here below, while in the other cases ∗ will just denote the Euclidean convolution.
Definition 3.44 Let G = RN , ◦, Dλ be a homogeneous group. Then the convolution f ∗ g is defined as Z
(f ∗ g) (x) = f (y) g y −1 ◦ x dy RN

for any couple of functions f, g : R

N

→ R for which the integral makes sense.

Homogeneous groups in RN

119

Note that with the change of variables y −1 ◦ x = w, y = x ◦ w−1 , by Theorem 3.6 and Proposition 3.7 we have dy = dw, so that Z Z

−1 (f ∗ g) (x) = f (y) g y ◦ x dy = f x ◦ w−1 g (w) dw. (3.27) RN

RN

Even though we can write the convolution f ∗ g in two different ways, the convolution on a homogeneous group is not commutative unless the group itself is commutative. On the other hand, it is still associative and bilinear. Thanks to the invariance of the Lebesgue measure with respect to translation and inversion, the usual proof of Young’s inequality can be repeated, giving the following

Proposition 3.45 (Young’s inequality) Let f ∈
Lp RN and g ∈ Lq RN with p, q ∈ [1, +∞] , p1 + 1q − 1 > 0. Then f ∗ g ∈ Lr RN with 1r = p1 + 1q − 1, and kf ∗ gkLr (RN ) 6 kf kLp (RN ) kgkLq (RN ) . Proof. Let us write Z
|f ∗ g (x)| 6 |f (y)| g y −1 ◦ x dy N ZR
q/r
1−q/r p/r 1−p/r = |f (y)| g y −1 ◦ x |f (y)| g y −1 ◦ x dy RN Z ≡ A (x, y) B (y) C (x, y) dy. RN

with p/r

A (x, y) = |f (y)|


g y −1 ◦ x q/r ,

1−p/r

B (y) = |f (y)| ,
1−q/r −1 C (x, y) = g y ◦ x . pr qr Since the exponents r, r−p and r−q satisfy (with three exponents) we obtain

1 r

+

|f ∗ g (x)| 6 kA (x, ·)kr kBk

r−p pr

pr r−p

+

r−q qr

= 1 by H¨older inequality

kC (x, ·)k

r−p

qr r−q

. r−q

A simple computation shows that kBk pr = kf kp r and kC (x, ·)k qr = kgkq r so r−p r−q that Z Z r r−p r−q r |f ∗ g (x)| dx 6 kf kp kgkq kA (x, ·)kr dx RN RN Z Z
q r−p r−q p = kf kp kgkq |f (y)| g y −1 ◦ x dydx RN

=

r kf kp

r kgkq

.

RN

120

H¨ ormander operators

The reader could ask if, in view of the Rnoncommutativity
of the convolution, the choice
of defining (f ∗ g) (x) as RN f (y) g y −1 ◦ x dy instead of R g (y) f y −1 ◦ x dy has any reason or is just arbitrary. The answer appears RN when one applies an invariant differential operator to the convolution:
Proposition 3.46 (Derivative of a convolution) Let f ∈ Lp RN , let g ∈
Lq RN with p1 + 1q > 1 and let 1r = p1 + 1q − 1. If X is a left invariant vector
q N field such that the weak derivative Xg ∈ L R , then f ∗ g has weak derivative in
r N L R and X (f ∗ g) = f ∗ Xg.
Similarly, if Y is a right invariant vector field such that Y f ∈ Lp RN then f ∗ g has weak derivative in Lr and Y (f ∗ g) = Y f ∗ g. Hence we see that a left (right) invariant vector field is also invariant with respect to left (right) convolution, a justification of the way we have given the definition of convolution. Proof. Let ϕ be a test function on RN . Since X ∗ = −X we have to show that Z Z (f ∗ g) (x) Xϕ (x) dx = − (f ∗ Xg) (x) ϕ (x) dx. RN

RN

Since, by H¨ older inequality and Proposition 3.45, Z Z
f (y) g y −1 ◦ x Xϕ (x) dydx RN

RN

6 kXϕkr0 k|f | ∗ |g|kr 6 kXϕkr0 kf kp kgkq with

1 r

+ Z

1 r0

= 1, by Fubini’s theorem we have Z Z
(f ∗ g) (x) Xϕ (x) dx = f (y) g y −1 ◦ x Xϕ (x) dx dy RN RN RN Z Z Z
=− f (y) Xg y −1 ◦ x ϕ (x) dxdy = − (f ∗ Xg) (x) ϕ (x) dx. RN

RN

RN r

Note that the function f ∗ Xg belongs to L by Proposition 3.45. The proof for the right invariant vector field is similar.

A well known property of the Euclidean convolution is the fact that ∂xi f ∗ g = f ∗ ∂xi g.

(3.28)

Due to the lack of commutativity however, a similar property does not hold on homogeneous groups, even replacing the partial derivatives with left invariant vector fields. To obtain an analogous of (3.28) we need both, a left invariant and a right invariant differential operator. Namely, the following holds:

Homogeneous groups in RN

121

Proposition 3.47 Let L, R be two differential operators, left and right invariant, respectively, which agree at the origin. Then, for any couple of functions f, g for which the integrals defining the convolutions converge absolutely, we have: Lf ∗ g = f ∗ Rg. In particular we have Yi f ∗ g = f ∗ YiR g for i = 1, 2, . . . , N. Proof. Let D0 the constant coefficient operator which agrees at the origin with both L and R. Then, reasoning like in the proof of Proposition 3.39 we can write: Z Z

−1 (Lf ∗ g) (x) = Lf (y) g y ◦ x dy = D0w [f (y ◦ w)]|w=0 g y −1 ◦ x dy RN

RN

(3.29) Z (f ∗ Rg) (x) =


f (z) Rg z −1 ◦ x dz =

RN

Z RN


 f (z) D0w g w ◦ z −1 ◦ x w=0 dz (3.30)

On the other hand the change of variables y ◦ w = z, dy = dz shows that Z Z

f (y ◦ w) g y −1 ◦ x dy = f (z) g w ◦ z −1 ◦ x dz ∀x, w ∈ RN . RN

RN

D0w

Taking the derivative at the origin of both sides of the last identity gives Z  Z 

 w −1 w −1 D0 [f (y ◦ w)] g y ◦ x dy = f (z) D0 g w ◦ z ◦ x dz RN

RN

w=0

w=0

which by comparison with (3.29)-(3.30) gives the desired identity. Let now show how the properties of the group convolution allow to define efficiently a family of mollifiers adapted to the group structure. ∞ Proposition 3.48 (Mollifiers adapted R to G) Pick a function φ ∈ C0 (G) such that φ > 0, φ (x) = 0 for kxk > 1 and G φ (x) dx = 1 and define, for every ε > 0, φε (x) = ε−Q φ (Dε−1 (x)). Let f ∈ Lp (G) with p ∈ [1, +∞) and let Z
fε (x) = (φε ∗ f ) (x) = φε (y) f y −1 ◦ x dy. G

Then: (i) fε → f in Lp (G) as ε → 0+ . (ii) If for some left invariant vector field Z we have Zf ∈ Lp (G) then Zfε = φε ∗ Zf → Zf in Lp (G) as ε → 0+ . (iii) If ZIR is a right invariant differential operator of any order, then ZIR fε = ZIR φε ∗ f. In particular if f has compact support then fε ∈ C0∞ (G).

122

H¨ ormander operators

Proof. The proof of (i) is standard, we include it for the sake of completeness. Assume first that f ∈ C0 (G). Then Z Z 
 fε (x) − f (x) = φε (y) f y −1 ◦ x dy − φε (y) f (x) dy G G Z 
 = ε−Q φ (Dε−1 (y)) f y −1 ◦ x − f (x) dy ZG h   i −1 = φ (z) f (Dε (z)) ◦ x − f (x) dz {kzk 0 and let g ∈ C0∞ (G) such that kf − gkp 6 η. Then kfε − f kp 6 kfε − gε kp + kgε − gkp + kg − f kp . Since, by Proposition 3.45, kfε − gε kp = kφε ∗ (f − g)kp 6 kφε k1 kf − gkp = kf − gkp 6 η and kgε − gkp 6 c kgε − gk∞ with c depending on the support of g, for sufficiently small ε we obtain kfε − f kp 6 3η and this gives (i). Property (ii) now follows from Proposition 3.46 and (i). Similarly (iii) follows from Proposition 3.46 and the fact that it is possible to write partial derivative of any order using right invariant vector fields as explained in Remark 3.32. Note that the support of fε is compact since both f and φε are compactly supported. Let now Z = {Z1 , Z2 , . . . , Zn } be a system of left invariant vector fields on G and let WZk,p (G) be the Sobolev space defined using the vector fields contained in Z as described in Chapter 2 (see Definition 2.2). Using the mollifiers introduced in the above proposition we can now easily prove the following result. Theorem 3.49 (A density result for Sobolev spaces on G) For any system Z of left invariant vector fields on G, the space C0∞ (G) is dense in WZk,p (G) for any p ∈ [1, ∞), k = 1, 2, 3 . . . In symbols, k,p WZk,p (G) = WZ,0 (G) . R The same is true for WZk,p is a system of right invariant vector fields. R (G) where Z

Homogeneous groups in RN

123

Proof. Let f ∈ WZk,p (G) for p ∈ [1, ∞), some k = 1, 2, 3, . . . , and assume first f compactly supported. By the previous Proposition, fε ∈ C0∞ (G) and fε → f in WZk,p (G). To remove the restriction on the support on f take a function ψ ∈ C0∞ (G) such that ψ (x) = 1 for kxk 6 1 and define ψR (x) = ψ (DR−1 (x)). If f ∈ WZk,p (G) a routine computation shows that f ψR has compact support, f ψR ∈ WZk,p (G) and that kf ψR − f kW k,p → 0 as R → +∞, so the density of C0∞ (G) in WZk,p (G) is Z

proved. The reasoning for the spaces WZk,p R (G) is analogous; in this case one should exploit the mollificators: fεR (x) = f ∗ φε

3.5

(3.31)

Homogeneous stratified Lie groups and Lie algebras, and their control distance

Starting with this section we will make a stronger assumption on the structure of the Lie algebra of left invariant vector fields g associated to a homogeneous group G. Recall that by Proposition 3.35 there exist vector spaces Wβ1 , Wβ2 , . . . , WβM such that g =Wβ1 ⊕ Wβ2 ⊕ · · · ⊕ WβM . where each Wβi comprises the vector fields that are homogeneous  of degree βi and Wβi , Wβj ⊆ Wβk whenever βi + βj = βk for some βk , otherwise Wβi , Wβj = {0}. Also, recall that by Definition 3.2 β1 = 1; let Y1 , Y2 , . . . , Yq be the elements of the canonical base that span W1 . Under the additional assumption that Y1 , Y2 , . . . , Yq satisfy H¨ ormander’s condition we will show that all the dilation exponents βj are integral and every Wβj is an iterated commutator of Wi . We start with the following: Definition 3.50 (Stratified Lie algebra) We say that a Lie algebra h is stratified if it admits, as a vector space, a decomposition h = V1 ⊕ V2 ⊕ . . . ⊕ Vs where: Vk = [Vk−1 , V1 ] for k = 2, 3, . . . , s [Vs , V1 ] = {0} where the symbol [Vk−1 , V1 ] denotes the vector space of all possible elements [X, Y ] for X ∈ Vk−1 , Y ∈ V1 . In this case s is called the step of the Lie algebra, and the Vi ’s are called layers.

124

H¨ ormander operators

Definition 3.51 (Homogeneous stratified Lie algebra) Let h be a homogeneous Lie algebra which is also a stratified Lie algebra. If the first layer V1 is spanned by 1-homogeneous vector fields, i.e. V1 = {X ∈ h : Dλ (X) = λX} we will say that h is a homogeneous stratified Lie algebra.
Proposition 3.52 Let G = RN , ◦, Dλ be a homogeneous group and let Y1 , . . . , YN be the canonical basis of its Lie algebra g. Assume that for some integer q, 1 < q 6 N the vector fields Y1 , Y2 , . . . Yq are 1-homogeneous and satisfy H¨ ormander’s condition at step s (at some point, and therefore at every point of RN , by left invariance). Then g is a homogeneous stratified Lie algebra of step s, with V1 = span (Y1 , Y2 , . . . , Yq ) . The vector fields Y1 , . . . , Yq are called generators of the Lie algebra g. Also, in this case the exponents αi of the dilations of G are just the integers 1, 2, . . . , s and Y1 , Y2 , . . . Yq have the following structure: X ∂ ∂ Yi = + qik (x) (i = 1, 2, . . . , q) , where ∂xi ∂xk q 0. iii) the function kxk = d (x, 0) is a homogeneous norm. More precisely, it also satisfies the stronger properties

−1

x = kxk and kx ◦ yk 6 kxk + kyk . Also,

d (x, y) = y −1 ◦ x

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H¨ ormander operators 0

This in particular implies that for any homogeneous norm kxk on a stratified group we have

−1

y ◦ x 0 globally equivalent to d (x, y) .
Finally, RN , d, dx is a space of homogeneous type (see Definition 3.19). Summarizing some of the facts contained in this proposition and Proposition 3.16 we can say that: • any homogeneous norm induces a gauge quasidistance; • the control distance induces a particular homogeneous norm, such that the corresponding gauge quasidistance is the control distance; • since all the homogeneous norms are equivalent, all gauge quasidistances are equivalent, and they are also equivalent to the control distance.

Proof. Let us start by noting that (3.34) follows from (3.33). Namely, by (3.33) and the properties of distance
d x−1 , 0 = d (0, x) = d (x, 0) (3.35) and

d (x1 ◦ x2 , 0) = d x1 , x−1 6 d (x1 , 0) + d x−1 2 2 , 0 = d (x1 , 0) + d (x2 , 0) . Iteration gives (3.34). Now, by definition of control distance, (3.33) and (ii) will follow as soon as we show that for any absolutely continuous curve ϕ in RN , any δ, λ > 0, x, y, z ∈ RN , ϕ ∈ Cx,y (δ) ⇒ z ◦ ϕ ∈ Cz◦x,z◦y (δ) .

(3.36)

ϕ ∈ Cx,y (δ) ⇒ Dλ (ϕ) ∈ CDλ (x),Dλ (y) (λδ)

(3.37)

Namely, let ϕz (t) = z ◦ ϕ (t) with ϕ0 (t) =

q X

ai (t) (Yi )ϕ(t) , |ai (t)| 6 δ

i=1

then ϕz (0) = z ◦ ϕ (0) = z ◦ x and ϕz (1) = z ◦ ϕ (1) = z ◦ y, while by (3.15) q

ϕ0z

X ∂P ∂P (t) = (z, ϕ (t)) ϕ0 (t) = ai (t) (z, ϕ (t)) (Yi )ϕ(t) ∂y ∂y i=1 =

q X i=1

ai (t) (Yi )z◦ϕ(t) =

q X

ai (t) (Yi )ϕz (t)

i=1

and (3.36) follows. To show (3.37), with the above notation let ϕλ (t) = Dλ (ϕ (t)), then ϕλ (0) = Dλ (ϕ (0)) = Dλ (x) and ϕλ (1) = Dλ (ϕ (1)) = Dλ (y) while ϕ0λ (t) = (λα1 ϕ01 (t) , λα2 ϕ02 (t) , . . . , λαN ϕ0N (t))

Homogeneous groups in RN

127

with ϕ (t) = (ϕ1 (t) , ϕ2 (t) , . . . , ϕN (t)) and λαk ϕ0k (t) = λαk

q X

k

ai (t) (Yi )ϕ(t) .

i=1

If X is 1-homogeneous vector field, bj (x) = (Xx )j is a (αj − 1) homogeneous polyk

k

nomial, hence (Yi )Dλ (x) = λαk −1 (Yi )x , so that λ

αk

ϕ0k

(t) = λ

q X

k

ai (t) (Yi )Dλ (ϕ(t))

i=1

and ϕ0λ (t) = λ

q X

ai (t) (Yi )ϕλ (t) with |λai (t)| 6 λδ,

i=1

so (3.37) is proved, and (i)-(ii) follow. As to (iii), the gauge has the positivity and homogeneity properties because the

−1

= kxk. Also, by (3.34), distance has them. Moreover, by (3.35) we have x kx ◦ yk = d (x ◦ y, 0) 6 d (x, 0) + d (0, y) = kxk + kyk . Hence k·k has the required properties. Finally, note that, by translation invariance of d,

−1


y ◦ x = d y −1 ◦ x, 0 = d (x, y) . Since, by Theorem 3.12, any two homogeneous norms on a homogeneous group are

−1 0

y ◦ x 0 is globally equivalent, for any other homogeneous norm k·k we get that

globally equivalent to ( y −1 ◦ x and then
to) d (x, y). Finally, since d is a distance and |B (x, r)| = |B (0, 1)| rQ , RN , d, dx is a space of homogeneous type. In the following, it will be useful also the following property of the control distance on stratified groups: Lemma 3.55 Let h = Exp (tYj ) for some j = 1, 2, . . . , q and t ∈ R. Then d (h, 0) = |t|. Proof. By Remark 1.34 we already know that (for the control distance of any system of vector fields, not necessarily on groups) d (exp (tYj ) (x0 ) , x0 ) 6 |t|, hence d (h, 0) 6 |t|. To prove the converse, let Exp (tYj ) = φ (1) with ( φ0 (s) = (tYj )φ(s) (3.38) φ (0) = 0.

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H¨ ormander operators

Recalling that the vector fields of the first layer have the structure (see Proposition 3.52) X ∂ ∂ Yj = + qjk (x) , ∂xj ∂xk k>q

projecting (3.38) on the j-th variable for j = 1, 2, . . . , q gives (φ0 )j (s) = t so that Z 1 hj = (φ0 )j (s) ds = t for j = 1, 2, . . . , q. (3.39) 0

Let now ψ be a curve ψ : [0, 1] → G such that, for some δ > 0, ψ (0) = 0 , ψ 0 (s) =

q X

ψ (1) = Exp (tYj ) = h, ak (s) (Yk )ψ(s)

with |ak (s)| 6 δ.

k=1

Since we are considering only vector fields in the first layer a derivative with respect to xi (for i = 1, 2, . . . , q) only appears in the vector field Yj . Then projecting the differential equation in ψ on the i-variable gives (ψ 0 )i (s) = ai (s) , i = 1, 2, . . . , q and therefore by (3.39) Z |t| = |hj | =

0

1

Z (ψ 0 )j (s) ds =

0

1

aj (s) ds 6 sup |aj (s)| 6 δ

and therefore, taking the inf on δ > 0, d (0, h) > |t|. 3.6

Connectivity matters and Poincar´ e inequality on stratified groups

We now want to reinterpret the connectivity results proved in Chapter 1, section 1.6 in the context of stratified groups. The relevant result is the following:
Proposition 3.56 Let G = RN , ◦, Dλ be a stratified group with generators Y1 , . . . , Yq . Then, there exist constants M, c > 0 and for every x ∈ RN there exist numbers t1 , t2 , . . . , tM such that x = Exp (tM YkM ) ◦ · · · ◦ Exp (t2 Yk2 ) ◦ Exp (t1 Yk1 ) with k1 , . . . , kM ∈ {1, 2, . . . , q} and |ti | 6 cd (x, 0) . Proof. Let us consider again the canonical basis Y1 , Y2 , . . . , YN ; all these vector fields can also be expressed in terms of commutators of Y1 , Y2 , . . . , Yq of some step 6 s. With the language of Theorem 1.48 in Chapter 1, let us fix once and for all a basis Y[Ii ] I ∈B of such commutators. By that theorem it is well defined a i diffeomorphism h 7→ y = EBY (0, h)

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129

from a neighborhood U of the origin in RN to a neighborhood V of the origin in RN which is given by the composition of N quasiexponential maps; in turn, each quasiexponential fixed number of exponentials   map EIi (hi ) is the composition of a  1/|Ii | exp ±hi Yi for i = 1, 2, . . . , q; moreover, the box h ∈ RN : |h| < δ1 is diffeomorphically mapped on a neighborhood of the origin containing an Euclidean ball BE (0, δ2 ), and therefore also a metric ball B (0, cδ2 ) (by Theorem 1.53). We can summarize the previous information saying that: for every z ∈ B (0, cδ2 ) we can write z = exp (t1 Yk1 ) exp (t2 Yk2 ) · · · exp (tM YkM ) (0) (3.40) 1/|Ik | for suitable numbers ti = ±hki i , i = 1, . . . M where M is an absolute constant 1/s

depending on G. Since |hki | < δ1 , we have |ti | 6 δ1 rewrite (3.40) as

≡ δ3 . Using (3.25) we can

z = Exp (tM YkM ) ◦ · · · ◦ Exp (t2 Yk2 ) ◦ Exp (t1 Yk1 ) . Let now x ∈ R write

N

and let λ = cδ2 /2d (x, 0), then z = Dλ (x) ∈ B (0, cδ2 ) and we can

Dλ (x) = Exp (tM YkM ) ◦ · · · ◦ Exp (t2 Yk2 ) ◦ Exp (t1 Yk1 ) with |ti | 6 δ3 . Using the fact that Dλ is a group automorphism and (3.26) we obtain x = (Dλ−1 (Exp (tM YkM ))) ◦ · · · ◦ (Dλ−1 (Exp (t2 Yk2 ))) ◦ (Dλ−1 (Exp (t1 Yk1 )))       t1 t2 tM Yk Yk ◦ Exp Yk ◦ · · · ◦ Exp = Exp λ M λ 2 λ 1 = Exp (t0M YkM ) ◦ · · · ◦ Exp (t02 Yk2 ) ◦ Exp (t01 Yk1 ) with |t0i | =

|ti | λ

6

2δ3 cδ2 d (x, 0),

so we are done.

In contrast with the local properties proved in Chapter 1, the previous proposition contains a global quantitative connectivity property of RN in terms of integral lines of the vector fields Y1 , Y2 , . . . , Yq . In the next chapter we will use this fact to prove the subelliptic estimates. Here we show a first important consequence of the previous proposition: Theorem 3.57 (Poincar´ e inequality on stratified groups) Let G = (RN , ◦, Dλ ) be a stratified group with generators Y1 , . . . , Yq . For every p ∈ [1, ∞) there exist positive constants c, λ
(depending on G and p) such that for any ball B = B (x0 , r) and any u ∈ C 1 λB (with λB = B (x0 , λr)) we have: Z 1/p  1/p Z p p |u (x) − u (y)| dxdy 6 cr |B| |Y u (x)| dx (3.41) B×B

λB

P

q i=1

2

1/2

where |Y u| = |Yi u| . Also:  1/p  1/p Z Z 1 1 p p |u (x) − uB | dx 6 cr |Y u (x)| dx . |B| B |λB| λB

(3.42)

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H¨ ormander operators

By the density result contained in Theorem 3.49, this theorem extends to any u ∈ WY1,p (Ω) with Ω bounded domain containing B. Note that the constants c, λ in the previous Poincar´e inequality are independent of r and x0 . Remark 3.58 (Changing the set of generators of G) The above result is stated with reference to the generators Y1 , . . . , Yq . If X1 , . . . , Xq is a different set of left invariant, 1-homogeneous vector fields satisfying H¨ ormander’s condition, then X1 , . . . , Xq are linear combinations (with constants coefficients) of Y1 , . . . , Yq . We will show in Chapter 4 that this Poincar´e’s inequality still holds for the system X1 , . . . , Xq and the corresponding control distance. Proof. By Proposition 3.56, any h ∈ G can be written as
h = Exp (t1 Yk1 ) ◦ · · · ◦ Exp tM −1 YkM −1 ◦ Exp (tM YkM ) = h1 ◦ h2 ◦ . . . ◦ hM
with hj = Exp tj Ykj and |ti | 6 cd (h, 0). Setting x0 = x and xj = xj−1 ◦ hj we have u (x ◦ h) − u (x) =

M X

u (xj ) − u (xj−1 ) =

j=1

=

u (xj−1 ◦ hj ) − u (xj−1 )

j=1

M Z X

tj

0

j=1

=

M X

M Z X



 d  u xj−1 ◦ Exp sYkj ds ds

tj

Ykj u xj−1 ◦ Exp sYkj





ds

0

j=1

where in the last equality we have exploited (3.25) and (1.5). Since |ti | 6 cd (h, 0) 6 cr, for any x ∈ B = Br (x0 ) and any fixed h ∈ B2r (0), we can write (with a small abuse of notation) M Z tj X

|u (x ◦ h) − u (x)| 6 Ykj u x ◦ h1 ◦ h2 ◦ . . . ◦ hj−1 ◦ Exp sYkj ds 6

j=1

0

M Z X

cr

j=1



Ykj u x ◦ h1 ◦ h2 ◦ . . . ◦ hj−1 ◦ Exp sYkj ds.

0

(3.43) Applying Minkowski integral inequality we obtain Z 1/p p |u (x ◦ h) − u (x)| dx B

6

M Z X j=1

0

cr

Z B



Yk u x ◦ h1 ◦ h2 ◦ . . . ◦ hj−1 ◦ Exp sYk p dx j j

1/p ds

(3.44)

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131

By (3.34) and Remark 1.34 we have j−1

X d h1 ◦ h2 ◦ . . . ◦ hj−1 ◦ Exp sYkj , 0 6 |ti | + |s| 6 cr

(3.45)

i=1


and the change of variables y = x ◦ h1 ◦ h2 ◦ . . . ◦ hj ◦ Exp sYkj gives for a suitable λ > 0, 1/p 1/p Z Z

Ykj u (y) p dy Ykj u x ◦ h1 ◦ h2 ◦ . . . ◦ hj−1 ◦ Exp sYkj p dx . 6 λB

B

Therefore, for every h ∈ B2r (0), Z Z 1/p p 6 Cr |u (x ◦ h) − u (x)| dx

p

1/p

|Y u (x)| dx

.

(3.46)

λB

B

Now, by the change of variables y = x ◦ h Z p |u (x) − u (y)| dydx B×B

Z

!

Z

p

|u (x ◦ h) − u (x)| dh dx

= {h:x◦h∈Br (x0 )}

B

Z

Z

 |u (x ◦ h) − u (x)| dx dh p

= B2r (0)

B



Z 6

C p rp

B2r (0)

 Z p |Y u (x)| dx dh = crp |B|

Z λB

p

|Y u (x)| dx

λB

since |B2r (0)| = 2Q |Br (x0 )|, which is (3.41). Also, by H¨older inequality, 

1 |B|

p 1/p 1/p  Z Z 1 1 (3.47) u (x) − |u (x) − uB | dx = u (y) dy dx |B| B |B| B B p 1/p  Z Z 1 1 dx = [u (x) − u (y)] dy |B| B |B| B !1/p Z Z 1 p 6 |u (x) − u (y)| dydx 1+p−p/q B B |B|

Z

p

and applying (3.41) 6 = cr

1

p p

1+p−p/q

|B| 

1 |B|

which is (3.42).

Z

p

c r |B|

|Y u (x)| dx λB

p

|Y u (x)| dx λB

!1/p

Z 1/p

= cλQ/p r



1 |λB|

Z

p

|Y u (x)| dx λB

1/p

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H¨ ormander operators


1,p Corollary 3.59 (Equivalent norms on WY,0 (Ω)) Let G = RN , ◦, Dλ be a stratified group with generators Y1 , . . . , Yq . For every p ∈ [1, ∞) there exists a positive constant c, depending on G, p, such that for every ball B = B (x0 , r) (with 1,p respect to the control distance) and every u ∈ WY,0 (B) , Z 1/p Z 1/p p p . 6 cr |Y u (x)| dx |u (x)| dx B

B

1,p Also, for every bounded open set Ω ⊂ RN and every u ∈ WY,0 (Ω), Z 1/p Z 1/p p p . 6 c · diam (Ω) |Y u (x)| dx |u (x)| dx Ω

Ω

This means that, like for classical Sobolev spaces, on bounded open sets Ω the Sobolev 1,p norm of WY,0 (Ω) is equivalent to kY ukLp (Ω) . Proof. It is enough to prove the two results for u ∈ C0∞ (B), u ∈ C0∞ (Ω), respectively; the general cases then will follow by density (Theorem 3.49). The second inequality follows from the first one choosing a ball B (x0 , r) for some x0 ∈ Ω and r = diam (Ω) and applying the inequality to u ∈ C0∞ (Ω). To prove the first inequality, apply (3.46) to u ∈ C0∞ (B) with d (h, 0) = 2r, then for any x ∈ B we have, since d is a distance,

d (x ◦ h, x0 ) = d h, x−1 ◦ x0 > d (h, 0) − d 0, x−1 ◦ x0 = d (h, 0) − d (x, x0 ) > r, hence u (x ◦ h) = 0 and we get Z Z p |u (x)| dx 6 Cr1+p/q B

Z

λB

Z

p

|Y u (x)| dx B

1/p Z 1/p p p 1/p |u (x)| dx 6C r |Y u (x)| dx .

B

3.7

p

|Y u (x)| dx = Cr1+p/q

B

Weak solutions to Dirichlet problems for divergence form equations structured on vector fields

Poincar´e inequality (or, more precisely, Corollary 3.59), together with elementary properties of Sobolev spaces induced by vector fields (see Chapter 2), allows to develop easily the first steps of a theory of weak solutions for divergence form equations
structured on H¨ ormander vector fields. Let G = RN , ◦, Dλ be a stratified group q with generators Y1 , . . . , Yq , Ω a bounded open subset of RN , and {aij (x)}i,j=1 a symmetric matrix of real functions belonging to L∞ (Ω). Assume this matrix satisfies in Ω a uniform ellipticity condition, expressed in the form 2

ν |ξ| 6

q X i,j=1

2

aij (x) ξi ξj 6 ν −1 |ξ|

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133

for some constant ν > 0, every ξ ∈ Rq , a.e. x ∈ Ω. We can consider, analogously to what is done in the theory of linear elliptic equations, the second order operator in divergence form: Lu =

q X

Yi∗ (aij Yj u) = −

i,j=1

q X

Yi (aij Yj u) .

i,j=1

Recall that in our context Yi∗ = −Yi , so that integration by parts takes now the familiar form Z Z f Yi g = − (Yi f ) g Ω

for any f ∈

WY1,2

(Ω) , g ∈

Ω

1,2 WY,0

(Ω). Also (see Chapter 2, section 2.1), recall that  0 1,2 an element T of the dual space WY,0 (Ω) can be represented as T = f0 −

q X

Yi fi with fi ∈ L2 (Ω) for i = 0, 1, 2, . . . , q.

i=1 1,2 The basic theory of weak solutions in the Hilbert Sobolev space WY,0 (Ω) can then be developed along the usual lines.2

Definition 3.60 With the previous assumptions and notation, we say that u is a weak solution to the Dirichlet problem  Lu = T in Ω (3.48) u = 0 on ∂Ω 0  Pq 1,2 (Ω) , T = f0 − i=1 Yi fi , fi ∈ L2 (Ω) for i = 0, 1, 2, . . . , q if with T ∈ WY,0 1,2 (Ω) and u ∈ WY,0

 Z

q X

 Ω

 Z aij Yj uYi φ dx =

f0 φ + Ω

i,j=1

q X

! fi Yi φ dx

(3.49)

i=1

1,2 for every φ ∈ WY,0 (Ω). 1,2 If, on the Hilbert space WY,0 (Ω), we define the bilinear form 1,2 1,2 a (·, ·) : WY,0 (Ω) × WY,0 (Ω) → R   Z q X  a (u, v) = aij Yj uYi v  dx Ω

i,j=1

a reference, see e.g. [42, §5.3] for the basics about the abstract Hilbert space framework, and [42, §9.5] for the concrete application to the weak formulation of boundary value problems for uniformly elliptic equations. 2 As

134

H¨ ormander operators

then we can write |a (u, v)| 6 ν

−1

Z Ω

|Y u| |Y v| dx 6 ν −1 kY ukL2 (Ω) kY vkL2 (Ω)

6 ν −1 kukW 1,2 (Ω) kvkW 1,2 (Ω) , Y,0

Y,0

1,2 WY,0

that is the bilinear form is continuous on (Ω); also, by Corollary 3.59 for p = 2 we have Z 2 2 a (u, u) > ν |Y u| dx > νc (Ω) kukW 1,2 (Ω) , Y,0

Ω

1,2 that is the bilinear form is coercive on WY,0 (Ω). We can then apply Lax-Milgram theorem (see e.g. [42, Corollary 5.8 p. 140]), concluding the following:

Theorem 3.61 Under the previous assumptions, the Dirichlet problem (3.48) is 1,2 (Ω) well posed: for every f0 , f1 , . . . , fq ∈ L2 (Ω) there exists a unique u ∈ WY,0 satisfying (3.49). Moreover, there exists a constant c, only depending on G, ν, Ω such that kukW 1,2 (Ω) 6 c

q X

Y

kf kL2 (Ω)

i=0

or also kukW 1,2 (Ω) 6 c kT k(W 1,2 (Ω))0 . Y

Y,0

As in the standard theory, the weak formulation (and the corresponding existence theorem) of a a boundary value problem with nonzero boundary datum is a straightforward adaptation of the previous case, at least in the case, which will be enough for our purposes, that the boundary datum is actually a function globally defined in Ω and belonging to WY1,2 (Ω): Definition 3.62 With the previous assumptions and notation, we say that u is a weak solution to the Dirichlet problem  Lu = T in Ω (3.50) u = g on ∂Ω  0 1,2 1,2 with T ∈ WY,0 (Ω) , g ∈ WY1,2 (Ω) assigned if: u ∈ WY1,2 (Ω), u−g ∈ and WY,0 (Ω) and v = u − g is a weak solution (in the sense of the previous definition) to the problem  Lv = T − Lg in Ω v = 0 on ∂Ω.  0 1,2 Actually, note that g ∈ WY1,2 (Ω) =⇒ Lg ∈ WY,0 (Ω) . Hence by the previous theorem we can immediately conclude the following.

Homogeneous groups in RN

135

Theorem 3.63 Under the previous assumptions, the Dirichlet problem (3.50) is  0 1,2 well posed: for every T ∈ WY,0 (Ω) , g ∈ WY1,2 (Ω) there exists a unique u ∈ WY1,2 (Ω) satisfying (3.50) in the sense of the previous definition. Moreover, there exists a constant c, only depending on G, ν, Ω such that n o kukW 1,2 (Ω) 6 c kT k(W 1,2 (Ω))0 + kgkW 1,2 (Ω) . Y

3.8

Y,0

Y

Homogeneous stratified Lie algebras and Lie groups of type II

In this section we want to revise the theory previously developed about homogeneous groups and Lie algebras having in mind the possible application to H¨ ormander operators with drift q X L= Yi2 + Y0 . i=1

In order for the operator L to be left invariant and 2-homogeneous, Y0 has to be 2-homogeneous, Y1 , Y2 , . . . , Yq have to be 1-homogeneous, and all of them must be left invariant. 3.8.1

Basic definitions

To begin with, note that all the theory developed in this chapter about homogeneous groups in RN , homogeneous Lie algebras of left invariant vector fields on homogeneous groups, exponentials of left invariant vector fields, and convolution on homogeneous groups, does not depend in any way on the homogeneity degrees of the vector fields which are required to satisfy H¨ormander’s condition. The notion of homogeneous stratified Lie algebra, on the other hand, describes the situation when the Lie algebra of left invariant vector fields is generated by a system of H¨ormander vector fields Y1 , . . . , Yq which are homogeneous of degree 1. In order to study operators with drift this notion is no longer the useful one, and we need the following variation: Definition 3.64 (Homogeneous stratified Lie algebra of type II) Let G be a homogeneous group and let Y1 , . . . , YN be the canonical basis of its Lie algebra g. Assume that for some integer q, 1 6 q 6 N the vector fields Y1 , Y2 , . . . Yq are 1homogeneous, Yq+1 (which from now on we relabel as Y0 ) is homogeneous of degree 2 and Y0 , Y1 , . . . , Yq satisfy H¨ ormander’s condition at weighted3 step s (at some point, and therefore at every point of RN , by left invariance). Then the Lie algebra g is called homogeneous stratified of type II at step s, and G is called homogeneous stratified group of type II. The operator q X L= Yi2 + Y0 i=1

will be called canonical homogeneous H¨ ormander operator with drift on G. 3 See

Definition 1.24

136

H¨ ormander operators

In Proposition 3.35 we proved that the Lie algebra g of a homogeneous group always admits the decomposition g = Wβ1 ⊕ Wβ2 ⊕ . . . ⊕ WβM where Wβ is the vector space of left invariant vector fields that are homogeneous of degree β. If g is a homogeneous stratified Lie algebra of type II, since the iterated commutators (up to the weighted step s) of vector fields Y0 , Y1 , . . . , Yq span the whole Lie algebra, then β1 = 1, β2 = 2 and the other exponents must be integral with βM = s. Hence g = W1 ⊕ W2 ⊕ . . . ⊕ Ws . In view of this we give the following: Definition 3.65 (Graded Lie algebra) We say that a Lie algebra g is graded if it admits, as a vector space, the decomposition g = W1 ⊕ W2 ⊕ . . . ⊕ Ws where: [Wi , Wj ] ⊆ Wi+j for i + j 6 s [Wi , Wj ] = {0} otherwise. Now, it is easy to prove the following:
Proposition 3.66 Let G = RN , ◦, Dλ be a homogeneous stratified group of type II and let g be its Lie algebra. Then: (i) the exponents αi of the dilations of G are the integers 1, 2, . . . , s and the vector fields Y0 , Y1 , . . . Yq have the following structure: Yi =

N X

bij (x) ∂xj for i = 0, 1, . . . , q,

j=1

where bij is an (αj − pi )-homogeneous polynomial and pi is the weight of Yi ; (ii) g is graded, with Wi the set of left invariant vector fields which are homogeneous of degree i; moreover, W1 = span (Y1 , Y2 , . . . , Yq ) and W2 = span ([W1 , W1 ] , Y0 ) . Remark 3.67 Our definition of homogeneous stratified Lie algebra of type II requires that the vectors fields Y0 , Y1 , . . . , Yq satisfy H¨ ormander’s condition at weighted step s without any assumptions on the Lie algebra generated by Y1 , Y2 , . . . , Yq alone. In view of the above proposition we can therefore distinguish between two different possibilities: either [W1 , W1 ] $ W2 which means that Y0 is linearly independent of

Homogeneous groups in RN

137

the commutators [Yi , Yj ] with i, j = 1, . . . , q and therefore it is necessary to satisfy H¨ ormander’s condition, or [W1 , W1 ] = W2 . In this latter case the vector fields Y1 , Y2 , . . . , Yq already satisfy H¨ ormander’s condition and therefore g is also a homogeneous stratified Lie algebra and G a Carnot group. These two notions however address two different circumstances. In both cases q X L= Yi2 + Y0 i=1

is a left invariant, 2-homogeneous H¨ ormander operator, but while in the second case Pq also the operator L = i=1 Yi2 is a H¨ ormander operator, this is no longer true in the first case. In the second case the term Y0 is actually a linear combination of commutators of the kind [Yi , Yj ] for i, j ∈ {1, 2, . . . , q} .
Example 3.68 (Kolmogorov-type operators) Let G = R3 , ◦, Dλ with (x1 , x2 , x3 ) ◦ (y1 , y2 , y3 ) = (x1 + y1 , x2 + y2 , x3 + y3 − x1 y2 )
Dλ (x1 , x2 , x3 ) = λx1 , λ2 x2 , λ3 x3 and let Y1 , Y2 , Y3 the canonical base of the Lie algebra. ∂yi [f (x ◦ y)]|y=0 we easily obtain

Since Yi f

=

Y1 = ∂x1 Y2 = ∂x2 − x1 ∂x3 Y3 = ∂x3 . Observe that [Y1 , Y2 ] = −Y3 so that the vector field Y1 which is 1-homogeneous and the vector field Y2 which is 2-homogeneous satisfy H¨ ormander’s condition. If we relabel the vector field Y2 as Y0 , according to Definition 3.64, G is a homogeneous stratified group of type II. In this case the canonical homogeneouos H¨ ormander operator is L = Y12 + Y0 = ∂x21 x1 + ∂x2 − x1 ∂x3 , known as Kolmogorov operator. If we set W1 = span (Y1 ), W2 = span (Y0 ) and W3 = span (Y3 ) we see that [W1 , W1 ] = {0} ⊂ W2 [W1 , W2 ] = W3 g = W1 ⊕ W2 ⊕ W3 , which shows that g is graded.
Example 3.69 Let G = R5 , ◦, Dλ with (x1 , x2 , x3 , x4 , x5 ) ◦ (y1 , y2 , y3 , y4 , y5 )  = x1 + y1 , x2 + y2 , x3 + y3 , x4 + y4 + x1 y2 , x5 + y5 − x1 y1 (x2 + y2 )  1 2 − y1 x2 + x1 y4 + x1 y3 , 2

138

H¨ ormander operators

Dλ (x1 , x2 , x3 , x4 , x5 ) = λx1 , λx2 , λ2 x3 , λ2 x4 , λ3 x5




and let Y1 , . . . , Y5 the canonical base of its Lie algebra g. Then Y1 = ∂x1 − x1 x2 ∂x5 , Y2 = ∂x2 + x1 ∂x4 , Y3 = ∂x3 + x1 ∂x5 , Y4 = ∂x4 + x1 ∂x5 , Y5 = ∂x5 . Moreover [Y1 , Y2 ] = ∂x4 + x1 ∂x5 [Y1 , Y3 ] = ∂x5 . It follows that Y1 , Y2 , Y3 , [Y1 , Y2 ] , [Y1 , Y3 ] span g. Note that Y1 and Y2 are 1homogeneous while Y3 is 2-homogeneous. Relabeling Y3 as Y0 we obtain that g is generated by Y0 , Y1 , Y2 so that it is a homogeneous stratified Lie algebra of type II. The canonical homogeneous H¨ ormander operator is 2

2

L = Y12 + Y22 + Y0 = (∂x1 − x1 x2 ∂x5 ) + (∂x2 + x1 ∂x4 ) + ∂x3 + x1 ∂x5 , Letting W1 = span (Y1 , Y2 ) W2 = span (Y0 , [Y1 , Y2 ]) W3 = span ([Y1 , Y0 ]) we obtain the decomposition g =W1 ⊕ W2 ⊕ W3 . Also, since [Y2 , Y0 ] = 0, [Y1 , [Y1 , Y2 ]] = ∂x4 and [Y2 , [Y1 , Y2 ]] = 0 we have [W1 , W1 ] = span ([Y1 , Y2 ]) ⊂ W2 [W1 , W2 ] = span ([Y1 , Y0 ]) = W3 [W1 , W3 ] = [W2 , W2 ] = [W2 , W3 ] = {0} hence g is graded. 3.8.2

Some remarks on the notions of stratified and graded Lie algebra

In this chapter we have introduced the notions of stratified Lie algebra and graded Lie algebra. These two concepts involve a decomposition of the Lie algebra as direct sum of vector spaces in a way that is compatible with the Lie bracket. Since in the following we will be mainly interested in Lie algebras of vector fields, the above notions of stratified and graded Lie algebras given in an abstract setting will not

Homogeneous groups in RN

139

play an important role. However, in order to clarify the general picture it can be useful to compare these concepts. Recall that a Lie algebra g is stratified if it admits, as a vector space, a decomposition g = V1 ⊕ V2 ⊕ . . . ⊕ Vs where: Vk = [Vk−1 , V1 ] for k = 2, 3, . . . , s [Vs , V1 ] = {0} . Also, a Lie algebra g is graded if it admits, as a vector space, a decomposition g = W1 ⊕ W2 ⊕ . . . ⊕ Ws where: [Wi , Wj ] ⊆ Wi+j for i + j 6 s [Wi , Wj ] = {0} otherwise. Then: • If g is any stratified Lie algebra, then g is graded. Actually, if Vi are the layers of a stratified Lie algebra, it is easy to prove (by induction on j) that [Vi , Vj ] ⊆ Vi+j for any i, j with i + j 6 s, hence g is graded with Wi = Vi . • When g is the Lie algebra of left invariant vector fields on a homogeneous group and g is homogeneous stratified, with the first layer V1 consisting in span (Y1 , Y2 , . . . , Yq ), with Y1 , Y2 , . . . , Yq 1-homogeneous, then we can define a natural grading of g letting Wi be the span of i-homogeneous left invariant vector fields. Note that by definition Vi ⊆ Wi for i = 1, 2, . . . , s; these inclusions, together with the identity W1 ⊕ W2 ⊕ . . . ⊕ Ws = V1 ⊕ V2 ⊕ . . . ⊕ Vs imply Vi = Wi for i = 1, 2, . . . , s. • If g is graded, then g may or may not be stratified; if g is stratified, its decomposition g = V1 ⊕ V2 ⊕ . . . ⊕ Vs0 can differ from its natural decomposition as graded algebra g = W1 ⊕ W2 ⊕ . . . ⊕ Ws , where Wi is the span of i-homogeneous left invariant vector fields; also, s0 can be smaller than s. The next examples will justify these last statements. Example 3.70 Let G be as in Example 3.68 and let g be its Lie algebra. If V1 = span (Y1 , Y0 ) and V2 = span (Y3 ), then V2 = [V1 , V1 ] [V1 , V2 ] = {0} g = V1 ⊕ V2 therefore g is stratified of step 2. Since V1 is not spanned by 1-homogeneous vector fields g is both homogeneous and stratified, but not “homogeneous stratified”. If we define the natural grading letting Wi = span (Yi ) for i = 1, 2, 3, then g is both graded and stratified, but the natural grading differs from the stratification.

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Example 3.71 Let G = R5 , ◦, Dλ as in Example 3.69 and let g its Lie algebra. We have already shown that g is graded. Let us show that it is not stratified. Since in a stratified Lie algebra the elements of V1 generate the whole Lie algebra we are forced to set V1 = span (Y1 , Y2 , Y0 ). Then V2 = [V1 , V1 ] = span ([Y1 , Y2 ] , [Y1 , Y0 ]) = span (∂x4 + x1 ∂x5 , ∂x5 ) and g = V1 ⊕ V2 . However, [V1 , V2 ] = span ([Y1 , ∂x4 + x1 ∂x5 ]) = {∂x5 } = 6 {0} , hence g is not stratified. 3.8.3

Weighted control distance and Poincar´ e inequality on homogeneous stratified groups of type II

For a system Y0 , Y1 , . . . , Yq of H¨ ormander vector fields which are generators of a stratified Lie algebra of type II, we can define as in Chapter 1, section 1.5, a weighted control distance, assigning weight 2 to Y0 and weight 1 to the other Yi ’s. The natural relations between this distance and the group structure, established in section 3.5 in the stratified case, are still in force. All the properties stated in Theorem 3.54 still hold for the weighted control distance induced by Y0 , Y1 , . . . , Yq in RN , essentially with the same proof. Let us end this section with a few words about the Poincar´e inequality in this case. Proposition 3.56 assumes in the weighted case the following form:
Proposition 3.72 Let G = RN , ◦, Dλ a stratified group of type II with generators Y0 , Y1 , . . . , Yq . Then, there exist constants M, c > 0 and for every x ∈ RN there exist numbers t1 , t2 , . . . , tM such that p 

p p k x = Exp tMM YkM ◦ · · · ◦ Exp t2k2 Yk2 ◦ Exp t1k1 Yk1 with k1 , . . . , kM ∈ {0, 1, 2, . . . , q} , pi the weight of Yi , and |ti | 6 cd (x, 0). The proof is a straightforward adaptation of the analogous unweighted result, applying Theorem 1.48 in Chapter 1 in the weighted case and Proposition 3.42 (b) with β = 1 or 2. We then have: Theorem 3.73
(Poincar´ e inequality on stratified groups of type II) Let N G = R , ◦, Dλ a stratified group of type II with generators Y0 , Y1 , . . . , Yq . For every p ∈ [1, ∞) there exist positive constants c, λ (depending on G and p,
independent of r and x0 ) such that for any ball B = B (x0 , r) , any u ∈ C 1 λB (with

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141

λB = B (x0 , λr)) we have: Z 1/p p |u (x) − u (y)| dxdy

(3.51)

B×B



 1/p Z 2 + cr |B| |Y u (x)| dx

Z

6 cr |B| λB

where |Y u| = 

1 |B|

P

q i=1

p

λB 2

|Yi u|

1/2

. Also

1/p |u (x) − uB | dx

Z

p

(3.52)

B

 6 cr

1/p |Y0 u (x)| dx

p

1/p  Z 2 + cr |B| |Y u (x)| dx

Z

1 |λB|

p

λB

p

1/p

|Y0 u (x)| dx

.

λB

Proof. We can repeat the the argument in the proof of Theorem 3.57. This time, in (3.43) we will also have some terms Z 2 tj Y0 u (x ◦ h1 ◦ h2 ◦ . . . ◦ hj−1 ◦ Exp (sY0 )) ds 0 Z cr2 6 |Y0 u (x ◦ h1 ◦ h2 ◦ . . . ◦ hj−1 ◦ Exp (sY0 ))| ds. 0

Since, by (1.19), d (Exp (sY0 ) , 0) 6

√

s 6 cr,

inequality (3.45) still holds, and one can conclude 1/p Z cr2 Z p |Y0 u (x ◦ h1 ◦ h2 ◦ . . . ◦ hj−1 ◦ Exp (sY0 ))| dx ds 0

6 Cr

B 2

Z

p

|Y0 u (x)| dx

1/p .

λB

The other terms in (3.43) are handled as in the unweighted case. The previous theorem is a less useful tool than the corresponding result in the stratified case. The point is that the derivative Y0 u is, morally speaking, a second order derivative, and its appearance in a bound on the oscillation of u is unsatisfactory. On the other hand, if Y0 is required together with Y1 , . . . , Yq to fulfil H¨ ormander’s condition this means that without Y0 we cannot control the increment of a function in all directions. 3.9

Distributions on homogeneous groups

Throughout the following we will use a bit of the classical theory of distributions (in the sense of L. Schwartz). The reader is referred to the Appendix at the end

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of this book for a summary of the properties of distributions which will be relevant for us. In this section we discuss a couple of notions which involve distributions in connection with the structure of homogeneous group and therefore are not completely standard (although they are actually straightforward adaptations of the corresponding notions in the Euclidean context). The first notion that we will sometimes need is the convolution (with respect to the group structure) between a test function and a distribution (while the general definition of convolution of distributions will not be useful for us). Definition 3.74 (Convolution with a distribution) Let φD∈ Cb0∞ (G) E and T ∈ 0 D (G). Then φ ∗ Tbis the distribution defined by hφ ∗ T, ϕi = T, φ ∗ ϕ for bevery E D
ϕ ∈ C0∞ (G). Here φ (x) = φ x−1 . Analogously, we set hT ∗ φ, ϕi = T, ϕ ∗ φ . Note that, since the convolution of functions on G is not commutative, two different definitions are needed for φ ∗ T and T ∗ φ. The definition of φ ∗ T is justified by the following computation, if T is a locally integrable function: Z Z
hφ ∗ T, ϕi = ϕ (x) φ (y) T y −1 ◦ x dydx ZG ZG
= ϕ (x) φ x ◦ w−1 T (w) dwdx G Z  ZG E D b

= T (w) φ y −1 ϕ y −1 ◦ w dy dw = T, φ ∗ ϕ . G

G

An analogous computation justifies the definition of T ∗ φ. The following useful property extends Proposition 3.46 to distributions: Proposition 3.75 (Properties of convolution) 1. Let φ ∈ C0∞ (G), T ∈ D0 (G) and let P be a left invariant differential operator. Then P (φ ∗ T ) = φ ∗ P T . 2. Let δ be the Dirac distribution and φ ∈ C0∞ (G), then φ ∗ δ = δ ∗ φ = φ. Proof. Let P ∗ be the transpose operator of P, and recall that P ∗ is still left invariant, by Proposition 3.24 (a). For every ϕb∈ C0∞ (G) we can write: D E hP (φ ∗ T ) , ϕi = hφ ∗ T, P ∗ ϕi = T, φ ∗ P ∗ ϕ b D b E D E = T, P ∗ φ ∗ ϕ = P T, φ ∗ ϕ = hφ ∗ P T, ϕi which gives point 1. As for pointb 2, Z D b E  

hφ ∗ δ, ϕi = δ, φ ∗ ϕ = φ ∗ ϕ (0) = φ x−1 ϕ x−1 dx = hφ, ϕi . G

The fact that δ ∗ φ = φ is similar. A second notion which we need is contained in the next

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143

Definition 3.76 Let T ∈ D0 (G). We will say that T is a homogeneous distribution of degree α ∈ R (on G) if for every test function ϕ ∈ C0∞ (G) we have hT, ϕ (Dλ (·))i = λ−Q−α hT, ϕi ∀λ > 0. The above definition is motivated by the following fact. Assume that T is a locally integrable function which is homogeneous of degree α, i.e. T (Dλ (x)) = λα T (x) . Then Z

Z T (x) ϕ (Dλ (x)) dx = T (Dλ−1 (y)) ϕ (y) λ−Q dy G G Z −α−Q =λ T (y) ϕ (y) dy = λ−α−Q hT, ϕi .

hT, ϕ (Dλ ) (·)i =

G

Remark 3.77 (Homogeneity of the Dirac δ) According with the above definition, the Dirac delta function δ is homogeneous of degree −Q. Indeed, hδ, ϕ (Dλ ) (·)i = ϕ ◦ Dλ (0) = ϕ (0) = hδ, ϕi . Proposition 3.78 (Derivative of a homogeneous distribution) If T ∈ D0 (G) is a homogeneous distribution of degree α and P is a differential operator on G with smooth coefficients, homogeneous of degree β, then P T is a homogeneous distribution of degree α − β. Proof. Let P ∗ be the transpose operator of P, and recall that P ∗ is still βhomogeneous, by Proposition 3.24 (b). Then we can write hP T, ϕ (Dλ (·))i = hT, P ∗ (ϕ (Dλ (·)))i = λβ hT, (P ∗ ϕ) (Dλ (·))i = λ−Q−α λβ hT, P ∗ ϕi = λ−Q−(α−β) hP T, ϕi .

3.10

Examples of homogeneous groups and homogeneous H¨ ormander operators

In this section we collect some more examples of homogeneous stratified (or stratified of type II) groups, together with the left invariant 2-homogeneous H¨ormander operators canonically attached to them. 3.10.1

Sublaplacians on the Heisenberg group

This example generalizes Examples 3.3 and 3.25. The Heisenberg group Hn is the
2n+1 group R , ◦, Dλ where, denoting the points of R2n+1 with (x, y, t), x = (x1 , . . . , xn ) , y = (y1 , . . . , yn ) ∈ Rn , t ∈ R,

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H¨ ormander operators

we let:  (x, y, t) ◦ (x0 , y 0 , t0 ) = x + x0 , y + y 0 , t + t0 + 2

n X


0

x0j yj − xj yj 

j=1

Dλ (x, y, t) = λx, λy, λt

2




.

It is an exercise to check that Hn is a homogeneous group, with homogeneous −1 dimension Q = 2n + 2. Note that (x1 , y1 , t1 ) = (−x1 , −y1 , −t1 ) . It is also customary to introduce the more compact complex notation
(z, t) ◦ (z 0 , t0 ) = z + z 0 , t + t0 + 2 Im z · z 0
Dλ (z, t) = λz, λ2 t where z = (z1 , . . . , zn ) ∈ Cn , z·z 0 is the scalar product in Cn and we identify Cn with R2n . The computation carried out in Example 3.25 shows that the left invariant vector fields4 Xj , Yj , T which agree at the origin with ∂xj , ∂yj , T , respectively, are given by: Xj = ∂xj + 2yj ∂t Yj = ∂yj − 2xj ∂t T = ∂t and it is immediate to check that Xj , Yj are 1-homogeneous while T is 2homogeneous. Moreover, [Xj , Yj ] = −4∂t = −4T. Hence X1 , . . . , Xn , Y1 , . . . , Yn are 1-homogeneous and satisfy H¨ormander’s condition at step 2, which by Proposition 3.52 implies that Hn is a Carnot group and its canonical sublaplacian is L=

n X

Xj2 + Yj2




j=1

3.10.2

Sublaplacians on homogeneous groups of Heisenberg type (or H-groups)

These groups are a generalization of the Heisenberg groups. Consider the Lie group in Rm+n 3 (x, t):   1 (x, t) ◦ (ξ, τ ) = x + ξ, t + τ + hBx, ξi 2 where hBx, ξi =

D

E D E B (1) x, ξ , . . . , B (n) x, ξ ,

4 Here for the names of the vector fields we adopt the notation which is standard in the study of the Heisenberg group.

Homogeneous groups in RN

145

and B (1) , . . . , B (n) are constant m × m skew symmetric orthogonal matrices (this forces m to be even), also satisfying B (i) B (j) = −B (j) B (i) for every i 6= j.

(3.53)

The validity of these assumptions is possible only under some relation5 between n, m, in particular we will assume n < m. With this translation ◦ and the dilations
D (λ) (x, t) = λx, λ2 t , Rm+n becomes a homogeneous group of step two, of homogeneous dimension Q = m + 2n, called group of Heisenberg type, or H-group. In particular, from the skew symmetry of the matrices B (i) it follows that (x, t)

−1

= (−x, −t) .

The m left invariant vector fields which agree at the origin with ∂xi are ! n m 1 X X (k) bi,l xl ∂tk ; i = 1, 2, . . . , m Yi = ∂xi + 2 k=1



(k)

where B (k) = bi,l

m

l=1

. They are 1-homogeneous, and satisfy the following com-

i,l=1

mutation relations: " [Yi , Yj ] = ∂xi

n m 1 X X (k) + bi,l xl 2 k=1

=

! ∂tk , ∂xj

l=1

n m 1 X X (h) + bj,r xr 2 r=1

!

# ∂th

(3.54)

h=1

n

n

k=1

k=1

 X (k) 1 X  (k) (k) bj,i − bi,j ∂tk = bj,i ∂tk 2

by the skew symmetry of B (k) . Let us show that H¨ormander’s condition holds. First of all let us show that the matrices B (1) , B (2) , . . . , B (n) are linearly independent. P Suppose there are coefficients α1 , α2 , . . . , an such that αi B (i) = 0. Since the matrices B (i) are orthogonal, skew symmetric and satisfy (3.53) we obtain  ! n n T X X  0= αi B (i)  αj B (j)  i=1

=

n X

j=1

 

αi2 B (i) B (i)

i=1

=

n X i=1

T

+

 X



αi αj B (i) B (j)

T 

i6=j

αi2 −

X i6=j

αi αj B (i) B (j) =

n X

αi2

i=1

5 For the exact relation, see [113, Corollary 1]. Note that in that statement the symbols n, m are switched with respect to our use.

146

H¨ ormander operators

(note that in the second-last sum the terms pairwise cancel). Therefore αi = 0. Now, let us consider the m2 × n matrix   (1) (2) (n) b11 b11 · · · b11    b(1) b(2) · · · b(n)    12 12 12 Q=    (1) (2) (n) bmm bmm · · · bmm where the k-th column is obtained writing the m2 entries of the matrix B (k) in some fixed order. Since the matrices B (i) are linearly independent also the columns of Q are linearly independent and therefore rank (Q) = n. This implies that there are n rows of this matrix that are linearly independent, and by (3.54) this implies that span {[Yi , Yj ] ; i, j = 1, 2, . . . , m} = span {∂ti ; i = 1, . . . , n} . It follows that the vector fields Y1 , Y2 , . . . , Ym and their commutators [Yi , Yj ] span Rm+n so that G is stratified with canonical sublaplacian m X L= Yi2 . (3.55) i=1

Note that for n = 1, m = 2k, and B (1) the 2k × 2k matrix    0 1 0 ... 0   −1 0       0 1  0  0   B (1) = 4  −1 0     ...  ...     0 1  0 ... −1 0 we recover the Heisenberg group Hk . Let us give an explicit example of H-group which does not reduce to a group Hk . Another example of this kind can be found in [16, §3.6.]. Example 3.79 Let n = 2, m = 4,    0 0 −1 0 0    1 0 0 0  (2)  0 B (1) =  0 0 0 −1 ; B = −1 0 0 1 0 0

 01 0 0 0 −1  00 0 10 0

then in R6 = R4+2 3 (x, t) we have the group operation   1 (x, t) ◦ (ξ, τ ) = x + ξ, t + τ + hBx, ξi 2  = x1 + ξ1 , x2 + ξ2 , x3 + ξ3 , x4 + ξ4 , 1 1 t1 + τ1 + (x1 ξ2 − x2 ξ1 + x3 ξ4 − x4 ξ3 ) , t2 + τ2 + (x3 ξ1 − x4 ξ2 + x2 ξ4 − x1 ξ3 ) 2 2



Homogeneous groups in RN

147


with dilations D (λ) (x, t) = λx, λ2 t . The generators of the Lie algebra are 1 (−x2 ∂t1 + x3 ∂t2 ) 2 1 + (x1 ∂t1 − x4 ∂t2 ) 2 1 + (−x4 ∂t1 − x1 ∂t2 ) 2 1 + (x3 ∂t1 + x2 ∂t2 ) , 2

Y1 = ∂x1 + Y2 = ∂x2 Y3 = ∂x3 Y4 = ∂x4 which satisfy the relations

[Y1 , Y2 ] = [Y3 , Y4 ] = ∂t1 [Y3 , Y1 ] = [Y2 , Y4 ] = ∂t2 , P4 the group is stratified, and the canonical sublaplacian is L = i=1 Yi2 . Note that in this case we could also regard G as a stratified group of type II, letting Y0 = ∂t1 (which is 2-homogeneous). The canonical homogeneous H¨ ormander operator would P4 2 be L = i=1 Yi + ∂t1 .This is an instance of the somewhat hybrid situation described in Remark 3.67: the drift term ∂t1 is not necessary to fulfil H¨ ormander’s condition, but cannot be regarded as a lower order term, since it is a commutator of the generators; the operator L is however a left invariant, 2-homogeneous H¨ ormander operator, of ultraparabolic type. 3.10.3

Kolmogorov-Fokker-Planck type operators

This example generalizes Example 3.68. In Rn+1 3 (x, t) let us define the composition law: (x, t) ◦ (x0 , t0 ) = (x0 + E (t0 ) x, t + t0 )
where E (t) is the exponential matrix E (t) = exp −tB T , B T is the transpose matrix of B, which is an n × n triangular matrix with the following structure:   0 B1 ... 0 0  0 0 B2 0 0     B=  0 ... ... ... ...  ... ... ... 0 Br  0

0

... 0

0

where Bj is a pj−1 × pj block with rank equal to pj , for every j = 1, 2, . . . , r. Moreover p0 > p1 > . . . > pr and p0 + p1 + . . . + pr = n. −1 One can check that (x, t) = (−E (−t) x, −t). Splitting   RN = Rp0 × Rp1 × . . . × Rpr 3 x(0) , x(1) , . . . , x(r) , one sees that this is a homogeneous group with respect to the dilations:   D (λ) (x, t) = λx(0) , λ3 x(1) , λ5 x(2) , . . . , λ2r+1 x(r) , λ2 t

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H¨ ormander operators

and homogeneous dimension Q = p0 + 3p1 + . . . + (2r + 1) pr + 2. The vector fields Yj = ∂xj for j = 1, 2, . . . , p0 are (obviously) 1-homogeneous and left invariant. Much less obviously, it can be proved (see [122]) that, under the above assumption on the matrix B, the vector field Y0 = hx, B∇x i − ∂t =

n X

xi bij ∂xj − ∂t

i,j=1

is left invariant, 2-homogeneous, and the system Y0 , Y1 , . . . , Yp0 satisfies H¨ ormander’s condition. Hence the group is homogeneous stratified of type II, and the canonical homogeneous H¨ ormander operator is6 L=

p0 X

Yj + Y0 .

j=1

Since p0 < n these are ultraparabolic operators. Example 3.80 The simplest (ultraparabolic) operator in this class is obtained letting n = 2, p0 = p1 = 1   01 B= 00  
1 0 so that B 2 = 0 and E (s) = exp −sB T = I − sB T = . Hence −s 1 (x1 , x2 , t) ◦ (x01 , x02 , t0 ) = (x1 + x01 , x2 + x02 − t0 x1 , t + t0 )
D (λ) (x1 , x2 , t) = λx1 , λ3 x2 , λ2 t and Q = 6. Also Y1 = ∂x1 Y0 = x1 ∂x2 − ∂t and the canonical homogeneous H¨ ormander operator is L = ∂x21 x1 + x1 ∂x2 − ∂t which is the operator of Example 3.68. This, apart from a different sign of the term x1 ∂x2 , is the operator met in 1934 by Kolmogorov as an instance of degenerate parabolic operator possessing a fundamental solution smooth outside the pole, as discussed in the introduction of the book. A slightly more involved example is the following:

6 Strictly

speaking, to fit the definition of canonical basis we should consider −X0 , which at the origin coincides with +∂t . The difference however is immaterial; our choice of the sign makes the operator L a forward Kolmogorov operator.

Homogeneous groups in RN

149

Example 3.81 Let n = 3, p0 = p1 = p2 = 1, and   010 B = 0 0 1 . 000 A simple computation shows that 

 1 00
  exp sB T =  s 1 0 . 2 s 2 s 1 Hence   t02 0 0 0 0 0 0 x1 , t + t (x1 , x2 , x3 , t) ◦ = x1 + x1 , x2 + x2 + t x1 , x3 + x3 + t x2 + 2
D (λ) (x1 , x2 , x3 , t) = λx1 , λ2 x2 , λ3 x3 , λ2 t . (x01 , x02 , x03 , t0 )

The vector fields Y1 = ∂x1 Y0 = x1 ∂x2 + x2 ∂x3 − ∂t are left invariant, 1 and 2-homogeneous, respectively, and coincide at the origin with ∂x1 , −∂t . Moreover, [Y1 , Y0 ] = ∂x2 [[Y1 , Y0 ] , Y0 ] = [∂x2 , Y0 ] = ∂x3 hence Y1 , Y0 satisfy H¨ ormander’s condition at (weighted) step 5. The homogeneous group is stratified of type II and its canonical H¨ ormander operator is L = ∂x21 x1 + x1 ∂x2 + x2 ∂x3 − ∂t We end with some general remarks about the “parabolic” counterpart of sublaplacians: 3.10.4

Heat type operators

Let G = (Rn , ◦, D (λ)) be a homogeneous stratified group of homogeneous dimension Q and let L=

q X

Yi2

i=1

be the corresponding canonical sublaplacian. Let us define the homogeneous group
H = Rn+1 , ∗, D0 (λ) such that: (x, t) ∗ (y, s) = (x ◦ y, t + s)
D0 (λ) (x, t) = D (λ) x, λ2 t .

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H¨ ormander operators

The above structure actually defines a homogeneous group stratified of type II, of homogeneous dimension Q0 = Q + 2. The vector fields Y1 , . . . , Yq and Y0 = ∂t are left invariant and, respectively, 1 and 2-homogeneous; they satisfy H¨ormander’s condition. The corresponding canonical H¨ ormander operator is H+ = L − ∂t or also H− = L + ∂t and can be seen as “forward” and “backward” heat-type operators, respectively. 3.11

Notes

As already pointed out, the use of the general notion of homogeneous group in the study of H¨ ormander operators starts with the paper by Folland in [85], which is the main source for this chapter. This general theory had been somewhat anticipated by the study of Heisenberg groups in connection with the Kohn Laplacian and the study of the Neumann problem for the Cauchy-Riemann complex, see the monograph by Folland-Kohn [90], and the paper [91] by Folland-Stein. Besides the paper [85], some useful references on the basic properties of homogeneous groups collected in sections 3.1 to 3.5 are the monograph by Bonfiglioli-Lanconelli-Uguzzoni [16, Chaps. 1-4] and some sections of the monograph by Stein [150, Chap. XIII, §5]. A general reference for Lie groups and Lie algebras is the monograph by Varadarajan [156]. The concept of space of homogeneous type introduced in section 3.1 is originally due to Coifman-Weiss [70], and the definition is subject to small variations in the literature. For instance, sometimes d it is allowed to be only a quasisymmetric quasidistance. Proposition 3.56 in section 3.6 is taken from [16, Thm. 19.2.1.]; its use to give a quick proof of Poincar´e’s inequality on stratified groups seems to be new. The Hilbert space approach for dealing with the Dirichlet problem for H¨ ormander operators (that we adapted in section 3.7 to study operators in divergence form) appears in the 1971 paper [75, Ch. 3] by Derridj. The term “stratified group of type II” has been introduced by Rothschild-Stein in [142]. Let us now make some comments on the examples contained in section 3.10. The Heisenberg group Hn (section 3.10.1) was originally introduced by Hermann Weyl in his 1931 book [159], in the context of quantum mechanics. A more modern explanation of the emergence of the Heisenberg group in quantum mechanics can be found in Folland’s book [87, Chap. 1] or, more briefly, in Stein’s book [150, Chap. XII, §3]. The relevance of the Heisenberg group in the theory of functions of several complex variables and the study of the related sublaplacian appeared in the 1970s starting with the papers by Folland [84], Folland-Stein [91]. The groups of Heisenberg type (section 3.10.2) were introduced by Kaplan in 1980 [113] as a class of Carnot groups of step 2 for which an explicit form of the

Homogeneous groups in RN

151

fundamental solution can be found for the sublaplacian. We will come back to this topic in Chapter 6, when dealing with fundamental solutions. For these groups we refer the reader to the book by Bonfiglioli-Lanconelli-Uguzzoni [16, §3.6 and Chap. 18]. We also refer to [16, Chaps. 3, 4, 18] for more examples of stratified groups and their sublaplacians. The class of Kolmogorov-Fokker-Planck operators discussed in section 3.10.3, and the underlying Lie group structure, has been introduced and studied by Lanconelli-Polidoro [122], 1993 (see also the references quoted in that paper for previous works on that subject). This class generalizes Kolmogorov’ 1934 example (see [118]) that we have discussed in the introduction of the book. This class of operators was also quoted by H¨ ormander in the introduction of his 1967 paper [107], in the same spirit of Kolmogorov’ remark: an example of degenerate operator possessing a fundamental solution smooth outside the pole. Actually in [122] the authors also study a more general class of operators, which are left invariant with respect to a Lie group of translations but are not homogeneous with respect to any family of dilations. This framework constitutes an interesting intermediate situation between the setting of left invariant homogeneous H¨ormander operators and that of completely general H¨ ormander operators. However, throughout the book we will not pursue this line of research. Heat-type operators related to sublaplacians on stratified groups (3.10.4) were explicitly considered by Folland in [85], 1975. Another source for the study of heat-type operators related to sublaplacians is the book by Varopoulos, SaloffCoste, Coulhon [157], and the references therein; see also the more recent work by Bonfiglioli-Lanconelli-Uguzzoni [14].

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Chapter 4

Hypoellipticity of sublaplacians on Carnot groups

4.1

Introduction, statement of the main results and strategy of the proofs

In this chapter we will prove the celebrated H¨ormander’s hypoellipticity theorem, which we have briefly discussed in the Introduction, in the special case of sublaplacians on Carnot groups. Let us begin with a precise Definition 4.1 A linear differential operator L, with coefficients in C ∞ (Ω) for some open set Ω ⊂ RN , is said to be hypoelliptic in Ω if, whenever a distribution u ∈ D0 (Ω) solves the equation Lu = f in Ω, with f ∈ D0 (Ω), then for every open set Ω0 ⊂ Ω we have f ∈ C ∞ (Ω0 ) =⇒ u ∈ C ∞ (Ω0 ) . Pq Here we will prove that if L = i=1 Xi2 is any sublaplacian on a Carnot group, that is X1 , . . . , Xq are left invariant, independent, 1-homogeneous vector fields on a Carnot group G and satisfy H¨ ormander’s condition (see Chapter 3, section 3.5), N then L is hypoelliptic in R (Theorem 4.26). More precisely, we will prove this fact Pq for the canonical sublaplacian L = i=1 Yi2 and then we will extend this result to any sublaplacian on G. This result is a special case of H¨ ormander’s theorem in two senses. First, it does not cover operators with drift (see Chapter 3, section 3.8); second, it does not apply to general H¨ ormander vector fields, but only to 1-homogeneous left invariant H¨ ormander vector fields on a Carnot group. The general version of H¨ormander’s theorem will be proved in Chapter 5, with a completely different proof, independent from the arguments in this chapter. So the reader may ask what is the real scope of this chapter which, on a purely logical basis, is somewhat superfluous. To answer, let us first say something more about H¨ ormander’s theorem. The original proof, in [107], is hard. A detailed exposition of the complete argument, especially in the case of operators with drift, would be very long. It makes use of heavy computation involving exponentials of vector fields and the Baker-Campbell-Hausdorff formula (which we will introduce in Chapter 9), Sobolev 153

154

H¨ ormander operators

norms of fractional order, special H¨ older norms which measure the differentiability of functions along vector fields, and a number of technical ad hoc tools. A somewhat easier proof of H¨ ormander’s theorem was given by Kohn [117], and independently by Oleinik-Radkevic1 in the book [137, Part II, §5], deeply exploiting techniques of pseudodifferential operators. This approach emphasizes an intermediate result which has great independent interest, namely the so-called subelliptic estimates, allowing to control a fractional Sobolev norm H s+ε of u, for some small ε > 0, in terms of a norm H s of Lu and some other less regular norm of u. These fractional Sobolev spaces are the classical ones, defined via Fourier transform2 ; in particular, they are isotropic in nature, that is they do not incorporate in any way the geometry of the vector fields. Although throughout the years several authors have written other proofs of H¨ ormander’s theorem, it is remarkable that, apart from the probabilistic approach due to Malliavin [124], all the analytic proofs of this result substantially go along the line of Kohn and Oleinik-Radkevic, based on subelliptic estimates and pseudodifferential operators. The general proof that we will present in Chapter 5 will also follow these ideas, although we will not make any explicit use of the general notion of pseudodifferential operator. In this chapter, instead, we present a simplified proof adapted to the special situation we consider here. The resulting proof is not only shorter and easier but, we hope, more transparent. Moreover, it could teach to the reader some techniques, typical of Carnot groups, which can be useful when working in this context, also for other problems. The plan of the chapter is the following. In section 4.2 we develop some introductory material about Sobolev spaces and certain function spaces defined by finite differences. In section 4.3, we will prove a priori estimates, stating that if 1,2 f ∈ WY,loc is a local weak solution to Lf = F in a domain, then the L2 norm of any fixed number of YiR -derivatives of f can be controlled in terms of the L2 norm of f and those of a (much larger) number of derivatives of F . The precise result is the following. In the next statement s is the step of the Lie algebra of G; WYk,2 and WYk,2 R are the Sobolev spaces induced by the systems of 1-homogeneous left invariant vector fields Y1 , . . . , Yq and the corresponding right invariant vector fields Y1R , . . . , YqR . 1,2 Theorem 4.2 (Regularity estimates) Let f ∈ WY,loc (G) and assume that Lf ∈ L2loc (G) (see Definition 4.12). For some open set Ω ⊂ G, let ζ, ζ1 ∈ C0∞ (Ω) be such that ζ1 = 1 on supp ζ. Then: 2 −1,2 (i) For any k = 1, 2, . . . , there exists c > 0 such that if Lf ∈ WYk+s (Ω) then R ,loc

1 Actually,

these authors prove a somewhat more general result than H¨ ormander’s theorem, since they allow H¨ ormander’s condition to be violated on a suitably small set. 2 We will define them precisely in Chapter 5.

Hypoellipticity of sublaplacians on Carnot groups

155

f ∈ WYk,2 R ,loc (Ω) and n o kζf kW k,2 (G) 6 c kζ1 Lf kW k+s−1,2 (G) + kζ1 f kL2 (G) . YR

(4.1)

YR

(ii) In particular, if Lf ∈ C ∞ (Ω) then f ∈ C ∞ (Ω). Although the estimate (4.1) is not sharp, it has the same independent interest as Kohn’s subelliptic estimates. Not only it implies that if Lf is smooth also f is smooth, but also expresses a quantitative a priori control on any finite number of derivatives, which is useful in several situations. Our proof will make use of both the Sobolev spaces induced by the vector fields Yi (as well as their right-invariant counterparts YiR ) and seminorms defined in terms of finite differences that measure the H¨ older continuity in L2 sense along the vector fields of the first layer. Therefore our regularity results are expressed exploiting the geometric and algebraic properties of the operator. 1,2 In section 4.4 we will weaken the assumption on f , from WY,loc to distribution, therefore getting the hypoellipticity of L. Finally, in section 4.5, we will extend Theorem 4.2 replacing L with a general sublaplacian L and showing that the regularity estimates are uniform when this sublaplacian is chosen in a suitably fixed class Lν (see Definition 4.37). Let us also note that, besides its intrinsic interest, the hypoellipticity of sublaplacians on Carnot groups will be a key ingredient in the proof of the existence of a global homogeneous fundamental solution for L, which will be performed in Chapter 6 and, in turn, will be a basic tool to prove sharp estimates in Sobolev and H¨ older spaces induced by the vector fields, both for operators on groups, in Chapter 8, and in more general contexts, in Chapters 11 and 12. We now want to say some words about the strategy of the proof of Theorem 4.2, also in order to motivate the reader for the preliminary analysis of section 4.2. Our starting point is the following elementary computation. Let f ∈ C0∞ (G), then, recalling that Yi∗ = −Yi (see (3.12)), Z X Z q 2 2 k∇Y f k2 ≡ (Yi f (x)) dx = − Lf (x) f (x) dx G i=1

G

 1 2 2 6 kLf k2 kf k2 6 kLf k2 + kf k2 2

(4.2)

which gives the estimate kf kW 1,2 6 c (kLf k2 + kf k2 ) ∀f ∈ C0∞ (G) .

(4.3)

Y

Now, assume for a moment that we knew, instead of (4.3), the following apparently similar estimate kf kW 1,2 6 c (kLf k2 + kf k2 ) ∀f ∈ C0∞ YR

(4.4)

156

H¨ ormander operators

where the Sobolev norm in the left hand side is computed using the right invariant vector fields. Then, exploiting the fact that the left invariant operator L commutes with the right invariant operators YiR (see Proposition 3.39), we could apply (4.4) . . . YiR f , obtaining YiR to the function YiR 2 1 k

R R






Yi Yi . . . YiR f 1,2 6 c LYiR YiR . . . YiR f + YiR YiR . . . YiR f 1 2 1 2 1 2 k k k W R 2 2 Y

R R

R R


R R = c Yi Yi . . . Yi Lf + Yi Yi . . . Yi f 1

2

k

2

1

2

k

2

hence   kf kW k+1,2 6 c kLf kW k,2 + kf kW k,2 YR

YR

YR

and iteratively   kf kW k+1,2 6 c kLf kW k,2 + kf k2 , YR

YR

an estimate that allows to control the derivatives of f in terms of the derivatives of Lf . In other words, the idea of measuring the degree of smoothness of a solution to Lf = F (with L left invariant) using right invariant derivatives apparently trivializes the problem, as if we were handling an elliptic operator with constant coefficients. Unfortunately, we do not have the bound (4.4). Nevertheless, we will see that it is possible to control the regularity expressed using right invariant vector fields in terms of the left invariant ones, but this implies a loss of regularity: if s is the step of the Lie algebra of G, using one derivative with respect to the left invariant vector fields, we can only control a right invariant regularity “of order 1/s”. More precisely we will show that for h ∈ G the following bound holds Z 1/2 2 1/s 1/s |f (h ◦ x) − f (x)| dx 6 c khk k∇Y f k2 6 c khk (kLf k2 + kf k2 ) . G

(4.5) Iterating the above estimate it is possible to control a full derivative with respect to the right invariant vector fields using s − 1 derivatives of Lf . More generally we will show that it is possible to control k derivatives YiR of f using k + s − 1 derivatives 1,2 of Lf and from this we will deduce that for a function f ∈ WY,loc , whenever Lf is smooth also f is smooth. The strategy that we will use in section 4.4 to extend the regularity result from functions in WY1,2 to distributions will be explained at the beginning of that section. 4.2

Notation and preliminary facts about Sobolev spaces and finite differences

Throughout this chapter G will be a Carnot group and in accordance with the notation introduced3 in Chapter 3 we will denote by ◦ the group operation, by {Dλ }λ>0 the family of dilations on G, by g the Lie algebra of left invariant vector 3 In

this chapter we will use much material from Chapter 3, sections 3.1 to 3.5.

Hypoellipticity of sublaplacians on Carnot groups

157

fields over G and by s the step of g. We will denote by Y1 , . . . , YN the canonical base of g and we will assume that the first layer of g is spanned by Y1 , . . . , Yq . Also Y1R , . . . , YNR will denote the right invariant vectors fields that agree with ∂x1 , . . . , ∂xN (and hence with Y1 , . . . , YN ) at the origin. Definition 4.3 (Sublaplacians on a Carnot group) Under the previous asPq sumptions, the operator L = j=1 Yj2 is called the canonical sublaplacian on G. If X1 , . . . , Xq is any other system of left invariant independent vector fields on G which Pq are 1-homogeneous and satisfy H¨ ormander’s condition, the operator L = j=1 Xj2 is said to be a sublaplacian on G. Analogous definitions can be given for right invariant sublaplacians. Our emphasis on the distinction between a sublaplacian and the canonical one is due to the fact that the special structure of the vector fields Y1 , . . . , Yq , studied in Chapter 3, makes very convenient doing computations on the canonical sublaplacian. On the other hand, we would like to know that the results we establish hold for any sublaplacian, but clearly this is not automatic. To study the regularity of the operator L we will make use of the Sobolev spaces k,p WY (G), WYk,p ormander vector fields R (G) induced by the systems of H¨  R R Y = {Y1 , . . . , Yq } , Y = Y1 , . . . , YqR , respectively. It will be sometimes useful the compact notation |∇Y f | =

q X j=1

|Yj f | ,

q X R Yj f . |∇Y R f | = j=1

Also, k·k = dY (x, 0) will denote the homogeneous norm on G constructed using the distance induced by Y1 , . . . , Yq (see Theorem 3.54 for the relevant properties of this homogeneous norm). In particular we will use that and that

−1

h = khk for everyh ∈ G, kh ◦ kk 6 khk + kkk for every h, k ∈ G. Actually this choice of homogeneous norm is relevant only within the proofs, while the statements of all our results will still hold, with possibly different constants, if we replace this homogeneous norm with a different one. Definition 4.4 (Finite difference operators) For every h ∈ G let us define the operators ∆h f (x) = f (x ◦ h) − f (x) , e h f (x) = f (h ◦ x) − f (x) . ∆ Note that the operator ∆h , which acts on functions computing the increment of f corresponding to an increment of its variable on the right (x 7→ x ◦ h) is e h computes actually a left invariant operator (Ly ∆h f = ∆h Ly f ); analogously, ∆

158

H¨ ormander operators

the increment of f corresponding to an increment of its variable on the left and is a right invariant operator. This “duality” is a central point in the techniques which will be used throughout this chapter. Proposition 4.5 There exists a constant c = c (G) such that for any h ∈ G and any f ∈ WY1,2 (G) we have k∆h f k2 6 c khk k|∇Y f |k2 and for any f ∈ WY1,2 R (G) we have

e

∆h f 6 c khk k|∇Y R f |k2 . 2

This means that k|∇Y f |k2 allows to control the increment of f in any direction, and an analogous property holds for right invariant vector fields using the operators e h . This property expresses once again the fact that a system of H¨ormander vector ∆ fields Y1 , . . . , Yq is enough to control N > q directions. Proof. It’s enough to prove the assertion for f ∈ C0∞ (G) , the case f ∈ WY1,2 (G) then follows by density (Theorem 3.49). Also, it is enough to prove the first assertion, since the second one is analogous. By Proposition 3.56, reasoning like in the proof of Poincar´e’s inequality in Carnot groups (Theorem 3.57), for any h ∈ G we can write M Z tj X

f (x ◦ h) − f (x) = Ykj f x ◦ h1 ◦ · · · ◦ hj−1 ◦ Exp sYkj ds j=1

0

(we implicitly assume h1 ◦ · · · ◦ hj−1 = 0 when j = 1) where M is
an absolute constant depending on G, k1 , . . . , kM ∈ {1, . . . , q}, hj = Exp tj Ykj and |tj | 6 cd (h, 0) = c khk. Hence, by Minkowsky integral inequality M Z |tj | X





Ykj f x ◦ h1 ◦ · · · ◦ hj−1 ◦ Exp sYkj ds. k∆h f k2 6 2 j=1

0

Since





Ykj f x ◦ h1 ◦ · · · ◦ hj−1 ◦ Exp sYkj 2 = 2

Z

Ykj f (x) 2 dx

G

we have k∆h f k2 6

max |tj | ·

j=1,...,M

M X

Ykj f 6 c khk k|∇Y f |k . 2 2 j=1

In the above proposition we have obtained a control

on k∆h f k2 in terms of

e k|∇Y f |k2 . Now we need to turn this into a control on ∆ h f . The next lemma shows that this is possible, with a loss in the exponent of khk.

2

Hypoellipticity of sublaplacians on Carnot groups

159

Lemma 4.6 Let f ∈ L2 (G) , U b G, supp f ⊂ U . There exists c > 0, depending on U , such that

e

∆h f k∆h f k2 2 6 c sup sup 1/s khk 0 0 such that   m−1 X R R |ζ0 f |m,m/s 6 c  |ζLf |j + kζf k2  , (4.15) j=0

whenever the right hand side is finite. Proof. For any two ζ0 , ζ ∈ C0∞ (G) with ζ0 ≺ ζ we can always choose a fixed number of cutoff functions ζ1 ≺ . . . ≺ ζn such that ζ0 ≺ ζ1 and ζn = ζ. Also, for fixed n, m there exists ε > 0 such that if khk < ε, then ∆m−1 ζi ≺ ζi+1 for h i = 0, 1, . . . , n. The number n will vary in different steps of the proof. We will prove the theorem by induction on m. Let m = 1. Applying (4.12) to ζ0 f and

166

H¨ ormander operators

using Lemma 4.17 we obtain R

|ζ0 f |1,1/s 6 c (kL (ζ0 f )k2 + kζ0 f k2 )

q !

X

Yi ζ0 Yi f + kf Lζ 0 k2 + kζ0 f k2 6 c kζ0 Lf k2 +

i=1 2 ! q X 6 c kζ0 Lf k2 + kYi (ζ1 f )k2 + kζ1 f k2 .

(4.16)

i=1

Let us bound Z q Z q X X 2 Yi (ζ1 f ) Yi (ζ1 f ) dx = − L (ζ1 f ) (ζ1 f ) dx kYi (ζ1 f )k2 = i=1

i=1

Z =−

Z ζ1 Lf · ζ1 f dx −

(Lζ1 ) f 2 ζ1 dx − 2

q Z X

(Yi ζ1 ) (Yi f ) ζ1 f dx

i=1 2

6 kζ1 Lf k2 kζ1 f k2 + c kζ2 f k2 − 2

q Z X

(Yi ζ1 ) (Yi (ζ1 f )) f dx

i=1

+2

q Z X

2

(Yi ζ1 ) f 2 dx

i=1 2

6 kζ1 Lf k2 kζ1 f k2 + c kζ2 f k2 + 2

q X

kYi (ζ1 f )k2 kζ2 f k2

i=1

6c



2 kζ1 Lf k2

+

2 kζ2 f k2



+ε

q X

2

kYi (ζ1 f )k2 +

i=1

c 2 kζ2 f k2 ε

therefore q X

kYi (ζ1 f )k2 6 c (kζ2 Lf k2 + kζ2 f k2 )

(4.17)

i=1

which inserted into (4.16) gives R

|ζ0 f |1,1/s 6 c (kζ2 Lf k2 + kζ2 f k2 ) ,

(4.18)

which is (4.15) for m = 1 (with ζ2 = ζ). Assume now that (4.15) holds for every m0 < m and let h = Exp (tYj ) for some i = 1, . . . , q such that 0 < khk 6 ε. We start writing





em

e e m−1

e

∆ (ζ f ) = ∆ (ζ g )

∆h (ζ0 f ) = ∆

h h 0 h 1 h 2

2

2

e m−1 (ζ0 f ) ≺ ζ1 . Then where gh = ∆ h

em

1/s R 1/s

∆h (ζ0 f ) 6 khk |ζ1 gh |1,1/s 6 c khk (kζ2 Lgh k2 + kζ2 gh k2 ) , 2

where we have applied our theorem for m = 1. Hence,



 

em

e m−1

1/s e m−1 (ζ f )

∆h (ζ0 f ) 6 c khk

L∆h (ζ0 f ) + ∆ 0 h 2

2

2

Hypoellipticity of sublaplacians on Carnot groups

167

e h commute (Lemma 4.18) since L and ∆

 

e m−1

1/s e m−1 = c khk (ζ0 f )

∆h L (ζ0 f ) + ∆ h 2 2    



X



1/s  e m−1  m−1 e  = c khk ζ0 Lf + 2 Yj ζ0 · Yj f + f Lζ0 

∆h

+ ∆h (ζ0 f ) 2

j 2 



X

e m−1

e m−1

1/s 6 c khk  ∆ (ζ0 Lf ) +

∆h (Yj ζ0 Yj f ) h 2

2

j





e m−1

e m−1

+ ∆ (f Lζ0 ) + ∆ (ζ0 f ) . h h 2

(4.19)

2

By Lemma 4.19 we have m−1

X

e m−1

k e m−k−1 ζ0

∆h (ζ0 Lf ) 6 c

Lh ∆h 2

Since, by (4.9)

k e m−k−1 ζ0

Lh ∆h

∞

∞

k=0



e m−k−1 ζ0 = ∆ h



ek

∆h (ζ1 Lf ) . 2

m−k−1

∞

m−k−1

6 kζ0 kW m−k−1,∞ khk

6 c khk

Y

we obtain m−1 m−1



X X

e m−1

m−k−1 e k R m−1 khk |ζ1 Lf |k .

∆h (ζ1 Lf ) 6 c khk

∆h (ζ0 Lf ) 6 c 2

2

k=0

k=0

(4.20) As for the second term in (4.19) we have m−1

X

ek

e m−1 e m−1−k f ∆ (Y ζ Y f ) 6 c

∆h (Yj ζ0 ) Lkh Yj ∆

h j 0 j h 2

6c

(4.21)

2

k=0 m−1 X



ek

∆h (Yj ζ0 )

∞

k=0

 

e m−1−k f

Yj ζ1 ∆

, h 2

e m−1−k commute and ∆ e k (Yj ζ0 ) ≺ ζ1 . By where we used the fact that Yj and ∆ h h m−1−k e (4.17) and the fact that L and ∆h commute we have



  

e m−1−k

e m−1−k e m−1−k f 6 c ∆ Lf + ∆ f

Yj ζ1 ∆

ζ

ζ

2 2 h h h 2 2

2 

e m−1−k

e m−1−k

6 c ∆ (ζ Lf ) + ∆ (ζ f )

h 3 3 h 6 c khk

m−1−k

2 R |ζ3 Lf |m−1−k

2 (m−1−k)/s

+ c khk

R

|ζ3 f |m−1−k,(m−1−k)/s .

Using the inductive assumption we obtain R |ζ3 f |m−1−k,(m−1−k)/s

6c

m−k−2 X i=0

! R |ζ4 Lf |i

+ kζ4 f k2

168

H¨ ormander operators

(we implicitly assume that for k = m − 1 the summation is empty). Therefore

 

e m−k−1 f

Yj ζ1 ∆

h 2 ! m−k−2 X m−1−k R (m−1−k)/s R 6 c khk |ζ4 Lf |m−1−k + c khk |ζ4 Lf |i + kζ4 f k2 . i=0

Again by (4.9),

ek

∆h ζj

k

∞

k

6 c kζj kW k,∞ khk 6 c khk Y

then using (4.21) we obtain

e m−1

∆h (Yj ζ Yj f ) 2

m−1

6 c khk

m−1 X

R |ζ4 Lf |k

+c

k=0 m−1

6 c khk

m−1 X

m−1 X

k

(m−1−k)/s

khk khk

m−2 X

k=0 R |ζ4 Lf |k

6 c khk

+ kζ4 f k2

k=0 m−2 X

(m−1)/s

+ c khk

k=0 (m−1)/s

! R |ζ4 Lf |k

! R |ζ4 Lf |k

+ kζ4 f k2

(4.22)

k=0 m−1 X

! R |ζ4 Lf |k

+ kζ4 f k2

.

k=0

As to the last two terms in (4.19) we have



e m−1

e m−1 (ζ f )

∆h (f Lζ0 ) + ∆ 0 h 2 2   (m−1)/s R R 6 khk |f Lζ0 |m−1,(m−1)/s + |ζ0 f |m−1,(m−1)/s

(4.23)

Inserting (4.20), (4.22) and (4.23) in (4.19) we get

em

∆h (ζ0 f ) 6 c khk

2 m−1 X

m/s

! R |ζ4 Lf |k

+ kζ4 f k2 +

R |f Lζ0 |m−1,(m−1)/s

+

R |ζ0 f |m−1,(m−1)/s

.

k=0

By the inductive assumption, we obtain 



|f Lζ0 |m−1,(m−1)/s + |ζ0 f |m−1,(m−1)/s 6 c 

|ζ2 Lf |j + kζ2 f k2 

R

m−2 X

R

R

j=1

so that

em

m/s

∆h (ζ0 f ) 6 c khk 2

m−1 X

! |ζ4 Lf |i + kζ4 f k2

.

i=0

Recall now that we chose h = Exp (tYi ) with 0 < khk 6 ε. To obtain (4.15) it is enough to observe that when khk > ε one has the trivial estimate

em

∆h (ζ0 f ) kζ0 f k 2 6 c m/s 2 6 c kζ4 f k2 . m/s ε khk

Hypoellipticity of sublaplacians on Carnot groups

169

The previous theorem allows to control the regularity of a function f , measured using difference operators, by means of the regularity of Lf . Unfortunately we cannot apply directly Proposition 4.5 to bound YiR f since it requires an estimate for the first order difference of f while (4.16) contains higher order differences. A result of M. A. Marchaud allows to bound the first order difference operator using higher order one (see e.g. [76, Chapter 2, Theorem 8.1]). In the following proposition we adapt this classical result to our setting. Lemma 4.21 (Marchaud inequality on Carnot groups) Let f ∈ L2 (G) and assume that for some positive constant A, integer m > 1 and 1 < α < 2 we have

em α (4.24)

∆h f 6 A khk . 2

Then, there exists c > 0, independent of f such that for khk 6 1

e

∆h f 6 c (A + kf k2 ) khk . 2

Proof. For every integer k > 1 let Qk (x) =

1 − 2−k (x + 1) x−1

k

k+1

and observe that multiplying both sides by (x − 1) k

(x − 1) = (x − 1)

k+1

Qk (x) + 2−k

we obtain
k x2 − 1 .

(4.25)

For every h ∈ G let Lh be the left translation operator (4.14) and I the identity e h f (x) = (Lh − I) f . Also note that Lh Lh = L2h , where form operator. Then ∆ now on we will write nh = h · · ◦ h}. | ◦ ·{z n times

Replacing in (4.25) the variable x with the translation operator Lh we obtain k

k

k+1

(Lh − I) = 2−k (L2h − I) + Qk (Lh ) (Lh − I)

(note that all the operators involved commute). Since kLh f k2 = kf k2 we have kQk (Lh ) gk2 6 Mk kgk2 where Mk only depends on the coefficients of the polynomial Qk (x). It follows that





ek

ek

k e k+1

∆h f = (Lh − I) f 6 2−k ∆ 2h f + Qk (Lh ) ∆h f 2 2

2

2

e k+1 −k e k 6 2 ∆2h f + Mk ∆h f 2 2

ek Applying n times the above inequality to the term ∆ 2h f we get 2

n



X

ek

e k+1

−k(n+1) e k ∆ f 6 M 2−jk ∆

h

∆2n+1 h f . k 2j h f + 2 2

j=0

2

2

(4.26)

170

H¨ ormander operators

We will now prove that from (4.24) it follows that ( α

ck (A + kf k2 ) khk for 1 < k 6 m

ek .

∆h f 6 2 ck (A + kf k2 ) khk for k = 1

(4.27)

Clearly (4.24) gives the above bound when k = m, while the desired assertion is (4.27) with k = 1. So, we now assume that (4.27) holds for k + 1 and we will prove it for k (assuming k > 1). For khk 6 1, using (4.26) we obtain n



X

α

ek

ek f 2−jk (A + kf k2 ) 2j h + 2−k(n+1) ∆

∆h f 6 Mk n+1 2 h 2

2

j=0

6 Mk (A + kf k2 )

n X

α

2−jk 2jα khk + 2−kn kf k2

j=0

where in the last inequality we used knhk = kh ◦ · · · ◦ hk 6 n khk | {z } n times

and

ek

∆h f 6 2k kf k2 . 2

Choosing n ∈ N in such a way that 2

−n

6 khk 6 2−n+1 we obtain

n

X

ek k α 2j(α−k) + ck khk kf k2 .

∆h f 6 Mk (A + kf k2 ) khk 2

j=0

Since α < 2, if k > 2 the sum is bounded independently of n so that

ek α

∆ f 6 ck (A + kf k ) khk . h

2

2

If k = 1 we obtain, since α > 1, n

X

e α 2j(α−1) + c1 khk kf k2

∆h f 6 M1 (A + kf k2 ) khk 2

j=0 α

6 M1 (A + kf k2 ) khk 2(α−1)(n+1) + c1 khk kf k2 α

6 M1 (A + kf k2 ) khk c khk

1−α

+ c1 khk kf k2 6 c (A + kf k2 ) khk .

Corollary 4.22 Let f ∈ L2 (G) and assume that for ε ∈ (0, 1) and some integer R m > 1 the seminorm |f |m,1+ε is finite. Then n o R R |f |1 6 c |f |m,1+ε + kf k2 .

Hypoellipticity of sublaplacians on Carnot groups

171



em R 1+ε R . By the Proof. Since |f |m,1+ε is finite we can write ∆ h f 6 |f |m,1+ε khk 2

R

previous lemma with α = 1 + ε and A = |f |m,1+ε , we obtain

 

e R

∆h f 6 c |f |m,1+ε + kf k2 khk 2

and the proof is complete. The proof of Theorem 4.14, which is the main result of this section, requires three steps. First we will establish the regularity estimate of point (i) in the case k = 1; next, we will prove a technical result necessary to regularize a distributional solution from L2loc to WY1,2 R ,loc ; then we will be in position to prove the full statement of Theorem 4.14. 1,2 Proposition 4.23 Let f ∈ WY,loc (G) be such that Lf ∈ L2loc (G). Fix ζ, ζ1 ∈ 2

C0∞ (G) , with ζ ≺ ζ1 . Then there exists c > 0 such that whenever Lf ∈ WYs R,2,loc (G) then f ∈ WY1,2 R ,loc (G) and n o kζf kW 1,2 (G) 6 c kζ1 Lf kW s,2 (G) + kζ1 f kL2 (G) . YR

YR

2

1,2 Proof. Let f ∈ WY,loc (G) be a local weak solution to Lf = F ∈ WYs R,2,loc (Ω). By Corollary 4.22 and Theorem 4.20 with m = s + 1 and ε = 1/s (actually, we just exploit F ∈ WYs,2 R ,loc (Ω)) we have   s  X n o R R R |ζ1 Lf |j + kζ1 f k2 . |ζf |1 6 c |ζf |s+1,1+1/s + kζf k2 6 c   j=0

By Proposition 4.11 and Theorem 4.9 we obtain   s X  n o R kζf kW 1,2 (G) 6 c |ζ1 Lf |j + kζ1 f k2 6 c kζ1 Lf kW s,2 (G) + kζ1 f k2 .   YR YR j=0

Proposition 4.24 Let f ∈ L2loc (G) be a distributional solution to Lf = F ∈ L2loc (G), that is Z Z f Lφ = F φ for every φ ∈ C0∞ (G) . G

G

2

∞ If F ∈ WYs R,2,loc (G) then f ∈ WY1,2 R ,loc (G) and for every ζ, ζ1 ∈ C0 (G) with ζ ≺ ζ1 we have o n kζf kW 1,2 (G) 6 c kζ1 Lf kW s,2 (G) + kζ1 f kL2 (G) . (4.28) YR

YR

172

H¨ ormander operators

Proof. Let us define the ε-mollified fε of f like in Proposition 3.48. We can apply to fε the estimate proved in Proposition 4.23: n o kζfε kW 1,2 (G) 6 c kζ1 L (fε )kW s,2 (G) + kζ1 fε kL2 (G) . (4.29) YR

YR

By Proposition 3.48 we know that kζ1 fε kL2 (G) → kζ1 f kL2 (G) . We also claim that L (fε ) = Fε . This is not entirely trivial since Lf exists in distributional sense and we cannot simply write L (fε ) = (Lf )ε . However, for every ϕ ∈ C0∞ (Ω) we can write: Z Z L (fε ) (x) ϕ (x) dx = fε (x) Lϕ (x) dx Z  Z
φε (y) f y −1 ◦ x dy dx = Lϕ (x) Z  Z = φε (y) Lϕ (y ◦ z) f (z) dz dy letting ψy (z) = ϕ (y ◦ z) and exploiting the fact that ψy ∈ C0∞ (Ω) for any fixed ϕ and ε small enough, and f is a distributional solution to Lf = F Z  Z  Z Z = φε (y) Lψy (z) f (z) dz dy = φε (y) ψy (z) F (z) dz dy Z  Z
= φε (y) ϕ (x) F y −1 ◦ x dx dy Z  Z Z
−1 = ϕ (x) φε (y) F y ◦ x dy dx = ϕ (x) Fε (x) dx and therefore L (fε ) = Fε . Hence by Proposition 3.48, ζ1 L (fε ) = ζ1 Fε → ζ1 F in WYk,2 (G) k,2 as soon as F ∈ WY,loc (G) . Since we need instead convergence in WYs,2 R ,loc (G) we are forced to make the following very rough estimates:

kζ1 L (fε ) − ζ1 F kW s,2 (G) 6 c kζ1 (Lf )ε − ζ1 F kW s,2 (G) YR

6 c kζ1 Fε − ζ1 F kW s2 ,2 (G) → 0 Y

s2 ,2 WY,loc

since F ∈ (G). Here we have bounded the Sobolev norm WYs,2 R (on a compact set containing the support of ζ1 ) with the Euclidean Sobolev norm on the same domain and then we have exploited Proposition 2.7 (2). Therefore the right hand side of (4.29) is bounded, hence the sequence ζfε is bounded in WY1,2 R (Ω), and by Proposition 2.6 there exists a subsequence of ζfε weakly converging in WY1,2 R (Ω) to 2 some g and in particular weakly converging in L (Ω) to ζf . This is enough to say that ζf ∈ WY1,2 R (Ω). Moreover, again by Proposition 2.6, o n kζf kW 1,2 (G) 6 lim inf kζfε kW 1,2 (G) 6 c kζ1 Lf kW s,2 (G) + kζ1 f kL2 (G) YR

hence (4.28) holds.

YR

YR

Hypoellipticity of sublaplacians on Carnot groups

173

Proof of Theorem 4.14. First of all we observe that point (ii) of the theorem folq lows applying point (i) to any Ω0 b Ω, since since YiR i=1 is a system of H¨ormander vector fields and by Proposition 2.7 we have ∞ \

0 ∞ WYk,2 (Ω0 ) . R (Ω ) ⊂ C

k=1

We will prove (i) by induction on k. The case k = 1 is just Proposition 4.23. 1,2 Assume that (i) holds up to the integer k. Let f ∈ WY,loc (G) be a local weak 2

,2 solution to Lf = F ∈ WYk+s Let YIR be any right invariant differential R ,loc (Ω).

operator with |I| 6 k. By the inductive assumption we know that f ∈ WYk,2 R ,loc (G). R 2 Then YI f ∈ Lloc (Ω) and we claim that it is a distributional solution in Ω to
2 L YIR f = YIR F ∈ WYs R,2,loc (Ω) .
∗ Namely, let YIR be the transpose operator (which by Proposition 3.24 is also right invariant hence commutes with L) and let φ ∈ C0∞ (Ω). Then, since f is a distributional solution to Lf = F in Ω, Z Z Z

∗
∗ YIR f Lφ = f YIR Lφ = f L YIR φ Z Z

R ∗ = F YI φ= YIR F φ, so the claim is proved. We can now apply Proposition 4.24 to YIR f and conclude that YIR f ∈ WY1,2 R ,loc (Ω) and satisfies the estimate  

R



ζYI f 1,2

ζ1 YIR F s,2

ζ1 YIR f 2 6 c + . W (G) W (G) L (G) YR

YR

This means that f ∈ WYk+1,2 R ,loc (Ω) and, introducing other cutoff functions such that ζ0 ≺ ζ ≺ ζ1 ≺ ζ2 ≺ ζ3 and exploiting the inductive assumption, X

YIR (ζ0 f ) 1,2 kζ0 f kW k+1,2 (G) = kζ0 f kW k,2 (G) + W (G) YR

YR

|I|=k

YR

    X

ζ1 YIR f 1,2 6 c kζ1 f kW k,2 (G) + W R (G)   YR Y |I|=k     X 



ζ2 YIR F s,2 + ζ2 YIR f L2 (G) 6 c kζ2 F kW s+k−1,2 (G) + W R (G)   YR Y |I|=k n o 6 c kζ3 F kW s+k,2 (G) + kζ3 f kL2 (G) YR

so we are done.

174

4.4

H¨ ormander operators

Hypoellipticity of the canonical sublaplacian

We are now left to extend Theorem 4.14 from WY1,2 functions to generic distributions, obtaining the hypoellipticity result. Definition 4.25 For an open set Ω ⊂ G and f ∈ D0 (Ω), we say that Lf ∈ C ∞ (Ω) if there exists a function g ∈ C ∞ (Ω) such that for every test function ϕ ∈ C0∞ (Ω) Z hf, Lϕi = g (x) ϕ (x) dx. Theorem 4.26 (Hypoellipticity of the sublaplacian) For an open set Ω ⊂ G, let f ∈ D0 (Ω) such that Lf ∈ C ∞ (Ω), then f ∈ C ∞ (Ω). That is, L is hypoelliptic in G. The strategy of the proof is the following. We are going to consider the second order differential operator ER =

N X

YjR


2

j=1

built using the whole canonical base of right invariant vector fields. This operator is right invariant, although no longer homogeneous, and uniformly elliptic in a neighborhood of the origin, since it coincides with the classical Laplacian at the origin. Using the fundamental solution of the Laplacian we can construct a parametrix for E R (that is an approximate fundamental solution) that we will name γ e (Proposition 4.27). We will then study the operator T f = γ e ∗ f , and we will show that if f is a distribution, then for K large enough T K f ∈ WY1,2 (Proposition 4.30 and Corollary 4.31). Since T and L commute (Proposition 4.32), then if Lf is smooth also LT K f = T K Lf is smooth and therefore by Theorem 4.14 (since T K f ∈ WY1,2 ) we see that T K f is smooth. Now, if γ e were the fundamental solution of E R then R K just applying K times E to T f we would obtain that f is smooth. The fact that γ e is only an approximate fundamental solution introduces a minor difficulty that is addressed in Lemma 4.34. Let us start with the construction of γ e. Proposition 4.27 (Parametrix of a right invariant elliptic operator) Let V ⊂ G be a neighborhood of the origin. There exist γ e ∈ C ∞ (G \ {0}) and ω ∈ ∞ C (G \ {0}), both supported in V , satisfying c |e γ (x)| 6 (4.30) N −2 |x| c |∂xi γ e (x)| 6 i = 1, 2, . . . , N (4.31) N −1 |x| c (4.32) |ω (x)| 6 N −2 |x|

Hypoellipticity of sublaplacians on Carnot groups

175

and such that, in the sense of distributions, ERγ e = −δ + ω. Explicitly, this means that for every ϕ ∈ C0∞ (G) and every y ∈ G we have Z Z

γ e x ◦ y −1 E R ϕ (x) dx = −ϕ (y) + ω x ◦ y −1 ϕ (x) dx. G

G

We will need the following elementary Lemma 4.28 For any N > 2 there exists cN such that for any x ∈ RN \ {0} Z 1−N 1−N 2−N |x − w| |w| dw = cN |x| . RN

w = |x| y gives 1−N x 1−N |y| dy − y RN |x|

Proof of the Lemma. The change of variables Z Z 1−N 1−N 2−N |x − w| |w| dw = |x| RN

by rotation invariance of the integral, letting e1 = (1, 0, . . . , 0), Z 1−N 1−N 2−N 2−N |e1 − y| |y| dy = cN |x| = |x| RN

since the integral, independent of x, is convergent for N > 2. Proof of Proposition 4.27. By Theorem 3.29 we now that YiR =

X ∂ ∂ + qik (x) ∂xi ∂xk i n/2, H s (Rn ) ⊂ L∞ (Rn ) with continuous embedding. (v) For every positive integer k, H s (Rn ) ⊂ C k (Rn ), with continuous embedding, whenever s > m for some integer m > k + n/2. Another useful property, which is less immediate to prove, is the following Proposition 5.17 For every dimension n and any m > n2 there exists a constant c (n, m) such that kf kH −m 6 c kf kL1 (Rn ) for any f ∈ L1 (Rn ). Proof. It is not difficult to check that when f ∈ L1 (Rn ) then the distribution fb is actually a continuous function given by Z b f (ξ) = f (x) e−2πix·ξ dξ,



so that fb ∞ n 6 kf kL1 (Rn ) . Hence L (R ) Z  Z 

2 −m −m b 2

2 2 2 kf kH −m = 1 + |ξ| 1 + |ξ| dξ f (ξ) dξ 6 fb ∞ n L

6

(R )

2 c (n, m) kf kL1 (Rn )

as soon as m > n/2, which makes

R

1 + |ξ|

2

−m

dξ convergent.

For every ϕ ∈ S (Rn ) letbus define the translation operator τx ϕ (y) = ϕ (y − x) and the reflection operator ϕ (y) = ϕ (−y).

Hypoellipticity of general H¨ ormander operators

197

Definition 5.18 (Convolution between S and S 0 ) For u ∈ S 0 (Rn ) and ϕ ∈ S (Rn ), the convolution u ∗ ϕ is defined by b u ∗ ϕ (x) = hu, τx ϕi . The convolution u ∗ ϕ is clearly a function, but it is also a tempered distribution: for every ψ ∈ S (Rn ) we can write  Z b  Z b b hu ∗ ϕ, ψi = hu, τx ϕi ψ (x) dx = u, ϕ (· − x) ψ (x) dx = hu, ϕ ∗ ψi (5.6) b with ϕ ∗ ψ ∈ S (Rn ). Moreover, u ∗ ϕ ∈ C ∞ (Rn ), since b 

b


∂xj (u ∗ ϕ) (x) = u, ∂xj (τx ϕ) = u, −τx ∂xj ϕ , an identity which can be iterated at any order. Finally, the usual identity on the Fourier transform of a convolution (Proposition 5.7) extends to this convolution: u[ ∗ϕ=u bϕ b

(5.7)

where both sides can be seen as a tempered distribution. To see this, let ψ ∈ S (Rn ), then, by (5.6), E E D b

 D u[ ∗ ϕ, ψ = u ∗ ϕ, ψb = u, ϕ ∗ ψb and also D E D b E c = u, ϕ ∗ ψb . hb uϕ, b ψi = hb u, ϕψi b = u, ϕψ b In Chapter 2, Lemma 2.8, we have already introduced the standard mollifiers. We are now interested in seeing them as operators acting on S 0 (Rn ). Definition 5.19 (Mollifiers) R Let J be a smooth, positive function supported on n the unit ball of R such that J (x) dx = 1, and let x . Jδ (x) = δ −n J b δ Let us also assume that J is even, so that J = J. For every tempered distribution u define Sδ u = u ∗ Jδ . We have the following: Lemma 5.20 For every ψ ∈ S (Rn ) , Sδ ψ converges to ψ in S (Rn ) as δ → 0. Moreover, for every u ∈ S 0 (Rn ) and ψ ∈ S (Rn ) we have lim hSδ u, ψi = hu, ψi .

δ→0

198

H¨ ormander operators

Proof. To prove the first assertion, let us show that, for any fixed multiindices α β α β α, β, supx x D (Sδ ψ) − x D ψ → 0 as δ → 0. Now, Z
  Dβ (Sδ ψ) − ψ (x) = Jδ (y) Dβ ψ (x − y) − Dβ ψ (x) dy and, for |y| < δ, β D ψ (x − y) − Dβ ψ (x) 6 δ sup ∇Dβ ψ , Bδ (x)

hence α β α β α β sup x D (Sδ ψ) − x D ψ 6 δ sup x sup ∇D ψ → 0 as δ → 0. x x Bδ (x) Let us show the second assertion. By (5.6) we have, since J is even, lim hSδ u, ϕi = lim hu ∗ Jδ , ϕi = lim hu, Jδ ∗ ψi = hu, ψi .

δ→0

δ→0

δ→0

We will also need mollifiers to regularize tempered distributions in H s (Rn ), with a control on the H s norm: Lemma 5.21 Let s ∈ R. If u ∈ H s (Rn ), then Sδ u ∈ C ∞ (Rn ) ∩ H s (Rn ) and kSδ ukH s 6 kukH s . Conversely, assume that u ∈ S 0 (Rn ) and there exists C > 0 such that for suitably small δ > 0, kSδ ukH s 6 C, then u ∈ H s (Rn ) and kukH s 6 C. d Proof. Let u ∈ H s (Rn ), then u b is a function and by (5.7) S bJbδ , so that δu = u Z 2  s 2 2 2 u (ξ)| Jbδ (ξ) 1 + |ξ| kSδ ukH s = |b dξ Z  s 2 2 2 6 |b u (ξ)| 1 + |ξ| dξ = kukH s since Z Z b Jδ (ξ) = Jδ (x) e−2πix·ξ dx 6 |Jδ (x)| dx = 1. Moreover, Sδ u is smooth as noted after Definition 5.18. Conversely, assume now thatkSδ ukH s6 Cfor small δ. Since the set of functions n o s 2 d Sδ u is bounded in L2 1 + |ξ| dξ , by the Banach-Alaoglu theorem (see δ>0 s   2 [42, Thm. 3.16]), there exists a function g ∈ L2 1 + |ξ| dξ and a sequence [ δn → 0 such that S δn u converges weakly to g. Moreover kgkL2 ((1+|ξ|2 )s dξ ) 6 C.

Hypoellipticity of general H¨ ormander operators

199

 −s 2 Let us prove that u b = g. Fix ψ ∈ S (Rn ) and let ψs (ξ) = ψ (ξ) 1 + |ξ| . By [ Lemma 5.20 and the weak convergence of S δn u we have D E D D E E [ hb u, ψi = u, ψb = lim Sδn u, ψb = lim S δn u, ψ n→+∞ n→+∞ Z  s 2 [ = lim S dξ δn u (ξ) ψs (ξ) 1 + |ξ| n→+∞ Z = g (ξ) ψ (ξ) dξ = hg, ψi so that u b = g. Finally, Z  s 2 2 2 2 kukH s = |b u (ξ)| 1 + |ξ| dξ = kgkL2 ((1+|ξ|2 )s dξ) 6 C 2 .

5.4

Some classes of operators on S(Rn )

Notation 5.22 Throughout this and the following two sections, the symbol h·, ·i always stands for h·, ·iL2 . In what follows we will introduce some classes of pseudodifferential operators that will be useful for the proof of H¨ ormander’s theorem. As already explained in the introduction of this chapter, we actually do not define any standard class of general pseudodifferential operators. Instead, we present in a self-contained way the specific classes of operators which will be useful throughout this chapter. We start with the following: Definition 5.23 (Fractional derivative) For every σ ∈ R we define the linear operator Λσ : S (Rn ) → S (Rn ) by requiring that  σ/2 σ ϕ (ξ) = 1 + |ξ|2 Λd ϕ b (ξ) , that is Λσ ϕ (x) =

Z 

1 + |ξ|

2

σ/2

ϕ b (ξ) e2πiξ·x dξ.

Remark 5.24 Although the definition of Λσ ϕ involves the Fourier transform ϕ, b note that if ϕ is real valued then Λσ ϕ is real valued, too. Indeed, one can easily check that Λσ ϕ (x) = Λσ ϕ (x). The next proposition collects some of the properties that we will use throughout the following. Proposition 5.25 (Properties of Λσ ) For every σ, τ ∈ R and every ϕ, ψ ∈ S(Rn ) we have

200

H¨ ormander operators

(i) Λσ Λτ = Λσ+τ (ii) kϕkH σ (Rn ) = kΛσ ϕkL2 (Rn ) , (iii) kΛσ ϕkH τ (Rn ) = kϕkH σ+τ (Rn ) , (iv) hϕ, ψi = hΛσ ϕ, Λ−σ ψi (v) hϕ, Λσ ψi = hΛσ ϕ, ψi ,

 2 (vi) kϕkH σ (Rn ) = ϕ, Λ2σ ϕ , (vii) |hϕ, ψi| 6 kϕkH σ kψkH −σ , (viii) The map Λσ : S(Rn ) → S(Rn ) is a bijection. Proof. These properties are simple consequences of the definition of the operator Λσ . We omit the details. Definition 5.26 (Operator of order m) Let T : S(Rn ) → S(Rn ) be a linear operator. We will say that T is of order m if for every σ ∈ R there exists a constant cσ > 0 such that for every ϕ ∈ S(Rn ) kT ϕkH σ 6 cσ kϕkH σ+m .

(5.8)

Remark 5.27 Note that, with this definition, if T is an operator of order m then it is also an operator of order m0 for every m0 > m. Let us collect some of the elementary properties connected with the order of an operator in the following: Proposition 5.28 (Properties of operators of order m) (i) For every σ ∈ R the operator Λσ is of order σ. (ii) Let T1 , T2 operators of order m1 and m2 respectively. Then the composition T1 T2 is an operator of order m1 + m2 . m (iii) Let α be a multiindex with |α| = m and let T ϕ = ∂∂xαϕ . Then T is of order m. (iv) An operator T is of order m if and only for every σ ∈ R the operator Λσ T Λ−σ−m is bounded on L2 (Rn ). Proof. (i) is a consequence of Proposition 5.25 point (iii), while (ii) is trivial. ∂ϕ . Then, by (vi) To prove (iii) it is enough to check the case m = 1. Let T ϕ = ∂x k in Proposition 5.5 Z  σ 2 2 2 2 kT ϕkH σ = 1 + |ξ| |2πiξk | |ϕ b (ξ)| dξ Z  σ+1 2 2 2 6c 1 + |ξ| |ϕ b (ξ)| dξ = c kϕkH σ+1 . To prove (iv) assume first that Λσ T Λ−σ−m ϕ is bounded on L2 . Then



kT ϕkH σ = Λσ T Λ−σ−m Λσ+m ϕ 2 6 c Λσ+m ϕ 2 = c kϕkH σ+m .

Hypoellipticity of general H¨ ormander operators

201

Conversely, assume that T is of order m, then



σ −σ−m

Λ T Λ ϕ 2 = T Λ−σ−m ϕ H σ 6 c Λ−σ−m ϕ H σ+m = c kϕk2 .

Proposition 5.29 (Multiplication operator) Let g ∈ S (Rn ). The operator T ϕ = gϕ is of order 0. Before proving this proposition we introduce a few conventions. Notation 5.30 Throughout this chapter we will handle compositions of several linear operators and, to ease notation, we will avoid the use of parentheses, writing T1 T2 f

for

T1 (T2 f ) .

Also, with a small abuse of notation we will simply denote by f the multiplication operator φ 7→ f φ. This means that, for instance, the expression Λσ f Λτ g

means

Λσ (f · Λτ g)

and not

(Λσ f ) · (Λτ g) .

This convention will be used consistently throughout this chapter. To improve the readability of the formulas in the following proofs we also introduce the symbol  1/2 2 hξi = 1 + |ξ| . s u (ξ) = hξis u d For example we will write Λ b (ξ) .

Lemma 5.31 Let σ ∈ R. For every x, y ∈ Rn we have σ hxi |σ| |σ|/2 hx − yi σ 62 hyi

(5.9)

Proof. Clearly, it is enough to prove (5.9) when σ > 0. From   2 2 2 |x| 6 2 |x − y| + |y| one easily deduce    2 2 2 1 + |x| 6 2 1 + |x − y| 1 + |y| and therefore σ/2 2  σ/2 1 + |x| 2 σ/2 . 1 + |x − y|  σ/2 6 2 2 1 + |y|



Proof of Proposition 5.29. By Proposition 5.28 (iv) it is enough to show that for every σ ∈ R we have kΛσ gΛ−σ ϕk2 6 c kϕk2 . By (5.4) we have Z Z 2 2

σ −σ 2 σ −σ ϕ (ξ) dξ = −σ ϕ (ξ) dξ \ \

Λ gΛ ϕ = hξiσ gΛ g b ∗ Λ hξi 2 Z 2 = |P ϕ b (ξ)| dξ

202

H¨ ormander operators

where Z Pϕ b (ξ) =

σ

hξi b (ξ − η) ϕ b (η) dη. σg hηi

By Lemma 5.31 the kernel K (ξ, η) of this operator satisfies |K (ξ, η)| 6 c hξ − ηi n

1

|σ|

|b g (ξ − η)| ≡ k (ξ − η) .

n

Since gb ∈ S (R ), k ∈ L (R ) so that P is bounded on L2 . It follows that

σ −σ 2 2 2

Λ gΛ ϕ 6 c kϕk b 2 = c kϕk2 . 2

We will use several times the following consequence of the fact that the multiplication by a function in S (Rn ) is an operator of order 0. Corollary 5.32 Let η1 , η2 ∈ C0∞ (Rn ), such that η1 ≺ η2 (that is, η2 = 1 on supp η1 ). For every σ ∈ R there exists c > 0 such that for every u ∈ S (Rn ) kη1 ukH σ 6 c kη2 ukH σ . Proof. Since the multiplication by η1 is an operator of degree 0, we have kη1 ukH σ = kη1 η2 ukH σ 6 kη2 ukH σ .

Definition 5.33 (Commutator) If T1 , T2 are two operators on S (Rn ), we define their commutator as [T1 , T2 ] = T1 T2 − T2 T1 where Ti Tj is the composition. According to Notation 5.30, for a function g and an operator T we will write, for instance, [T, g] u = T (gu) − gT u = T gu − gT u. We will often need to compute the order of the commutator of two operators. Morally speaking, if T1 , T2 are two operators of orders m1 , m2 , we expect [T1 , T2 ] to have order m1 + m2 − 1. However, our definition of “operator of order m” is too general (in some sense, too vague) for proving such a general result. Instead, we will prove this fact in a number of specific cases which will be enough for our purposes. We start with the following easy: Proposition 5.34 (Differential operators and their commutators) P (i) A differential operator of order m, Dm = |α|6m aα (x) ∂xα with coefficients aα ∈ S (Rn ) is an operator of order m. (ii) If g ∈ S (Rn ) and Dm is as above, then [Dm , g] is a differential operator of order m − 1 with coefficients in S (Rn ), so it is an operator of order m − 1.

Hypoellipticity of general H¨ ormander operators

203

(iii) In particular, the commutator of a vector field with coefficients in S (Rn ) with a multiplication operator is a multiplication operator, hence of order 0. (iv) More generally, if Dn , Dm are differential operators with coefficients in S (Rn ) of order n, m respectively, then [Dn , Dm ] is a differential operator of order n + m − 1 with coefficients in S (Rn ); in particular, it is an operator of order n + m − 1. Proof. Point (i) follows by Propositions 5.29 and 5.28 (iii). Point (ii) follows by point (i) and a direct computation, and (iii) follows from (ii); (iv) is then still a straightforward computation. The above kind of result extends to commutators with the fractional differential operators Λs . This, however, requires some work. Proposition 5.35 (Commutators of Λσ with multiplication) Let g, h ∈ S (Rn ), then for every σ, τ ∈ R we have (i) the operator [Λσ , g] is of order σ − 1, (ii) the operator [Λσ , [Λτ , g]] is of order σ + τ − 2 (iii) the operator [[Λσ , g] , h] is of order σ − 2. Proof. (i) By Proposition 5.28 (iv) it is enough to show that for every τ ∈ R, Λτ [Λσ , g] Λ−τ −σ+1 is bounded on L2 . For every ϕ ∈ S (Rn ) let T1 ϕ = Λτ [Λσ , g] Λ−τ −σ+1 ϕ = Λτ +σ gΛ−τ −σ+1 ϕ − Λτ gΛ−τ +1 ϕ, hence, by (5.4),
∧
∧ τ +σ τ Td gb ∗ Λ−τ −σ+1 ϕ (ξ) − hξi gb ∗ Λ−τ +1 ϕ (ξ) 1 ϕ (ξ) = hξi Z τ hξi σ σ = b (ξ − η) ϕ b (η) dη τ +σ−1 (hξi − hηi ) g hηi Z ≡ K1 (ξ, η) ϕ b (η) dη. By the mean value theorem we obtain     σ−1  σ−1 2 2 2 σ σ 2 + 1 + |η| |hξi − hηi | 6 c |ξ − η| 1 + |ξ|   σ−1 σ−1 = c |ξ − η| hξi + hηi so that, using Lemma 5.31, τ +σ−1

|K1 (ξ, η)| 6 c

hξi

τ +σ−1

hηi

τ

hξi + τ hηi

! |ξ − η| gb (ξ − η)

  τ +σ−1 τ 6 c hξ − ηi + hξ − ηi |ξ − η| |b g (ξ − η)| ≡ k1 (ξ − η) .

204

H¨ ormander operators



b 2 . It follows Since gb ∈ S (Rn ) the kernel k1 ∈ L1 (R) and therefore Td 1 ϕ 6 c kϕk 2

that [Λσ , g] is of order σ − 1. (ii) We consider now the operator [Λσ , [Λτ , g]]. For every ϕ ∈ S (Rn ) let T2 ϕ = [Λσ , [Λτ , g]] ϕ = Λσ [Λτ , g] ϕ − [Λτ , g] Λσ ϕ = Λσ Λτ gϕ − Λσ gΛτ ϕ − Λτ gΛσ ϕ + gΛτ Λσ ϕ. Then, Z h i σ+τ σ τ τ σ τ +σ d T2 ϕ (ξ) = hξi − hξi hηi − hξi hηi + hηi gb (ξ − η) ϕ b (η) dη Z τ τ σ σ = (hξi − hηi ) (hξi − hηi ) gb (ξ − η) ϕ b (η) dη Z ≡ K2 (ξ, η) ϕ b (η) dη. Arguing as in the previous case we have    2 τ −1 τ −1 σ−1 σ−1 |K2 (ξ, η)| 6 c |ξ − η| hξi + hηi hξi + hηi |b g (ξ − η)| . Using Proposition 5.28 (iv) we will show that T2 is an operator of order σ + τ − 2 checking that for every ρ ∈ R the operator Λρ T2 Λ−ρ−σ−τ +2 is bounded on L2 . By (5.2) this is equivalent to show that the operator with kernel e 2 (ξ, η) = hξiρ K2 (ξ, η) hηi−ρ−σ−τ +2 K is bounded on L2 . Again, by Lemma 5.31, we have ! ! ρ+τ −1 ρ σ−1 hξi hξi hξi e 2 b (ξ − η) K2 (ξ, η) 6 c ρ ρ+τ −1 + σ−1 + 1 |ξ − η| g hηi hηi hηi    |ρ+τ −1| |ρ| |σ−1| 2 6 c hξ − ηi + (ξ − η) hξ − ηi + 1 |ξ − η| |b g (ξ − η)| ≡e k2 (ξ − η) . Since gb ∈ S (Rn ) we have e k2 ∈ L1 (Rn ) and therefore the operator with kernel 2 e K2 (ξ, η) is bounded on L . (iii) Now, let us consider the operator [[Λσ , g] , h]. If ϕ ∈ S (Rn ), arguing as in point (i) we have Z ∧ σ σ ([Λσ , g] ϕ) (ξ) = (hξi − hηi ) gb (ξ − η) ϕ b (η) dη. Using Taylor expansion we can write σ

σ

hξi − hηi =

n X ∂ σ (hξi ) (ξk − ηk ) + E (ξ, η) ∂ξk

k=1

where 2

|E (ξ, η)| 6 c |ξ − η|



hξi

σ−2

σ−2

+ hηi



.

(5.10)

Hypoellipticity of general H¨ ormander operators

It follows that ∧

205

Z X n ∂ σ (hξi ) (ξk − ηk ) gb (ξ − η) ϕ b (η) dη ∂ξk k=1 Z 0 00 d + E (ξ, η) gb (ξ − η) ϕ b (η) dη ≡ Td 3 ϕ (ξ) + T3 ϕ (ξ) .

([Λσ , g] ϕ) (ξ) =

Now, T300 is an operator of order σ − 2, since by (5.10) and Lemma 5.31 Z   2 s+σ−2 s σ−2 s 00 |ξ − η| hξi + hξi hηi |b g (ξ − η) ϕ b (η)| dη hξi Td 3 ϕ (ξ) 6 ! Z s+σ−2 s hξi hξi s+σ−2 2 hηi |b g (ξ − η) ϕ b (η)| dη = |ξ − η| + s s+σ−2 hηi hηi Z   2 |s+σ−2| |s| s+σ−2 6 c |ξ − η| hξ − ηi + hξ − ηi |b g (ξ − η)| hηi |ϕ b (η)| dη   s+σ−2 = k300 ∗ hηi ϕ b (ξ)

00 1 n with k3 ∈ L (R ). Hence kΛs T300 ϕk 6 c Λs+σ−2 ϕ so that [T300 , h] = T300 h − hT300 2

2

is trivially an operator of order σ − 2 since the multiplication by h is an operator of order 0. It remains to show that [T30 , h] is also an operator of order σ − 2. Let n n X X ∂ σ−2 σ K30 (ξ, η) = hξi ξk (ξk − ηk ) gb (ξ − η) (hξi ) (ξk − ηk ) gb (ξ − η) = σ ∂ξk k=1

k=1

so that

Z

∧ 0 0 0 [ ([T30 , h] ϕ) (ξ) = T[ K30 (ξ, η) b h∗ϕ b (η) dη − b h ∗ Td 3 hϕ (ξ) − hT3 ϕ (ξ) = 3 ϕ (ξ) Z Z Z Z 0 b b = K3 (ξ, η) h (η − ζ) ϕ b (ζ) dζdη − h (ξ − η) K30 (η, ζ) ϕ b (ζ) dζdη Z Z = (K30 (ξ, ξ − η + ζ) − K30 (η, ζ)) b h (ξ − η) dη ϕ b (ζ) dζ Z ≡ K4 (ξ, ζ) ϕ b (ζ) dζ

A simple computation shows that n  X  σ−2 σ−2 0 0 |K3 (ξ, ξ − η + ζ) − K3 (η, ζ)| = σ hξi ξk − hηi ηk (ηk − ζk ) gb (η − ζ) k=1   σ−2 σ−2 6 c |ξ − η| |η − ζ| |b g (η − ζ)| hξi + hηi , so that

Z

  σ−2 σ−2 b c |ξ − η| |η − ζ| |b g (η − ζ)| hξi + hηi h (ξ − η) dη ! Z σ−2 hηi b σ−2 h (ξ − η) = c hξi |ξ − η| |η − ζ| |b g (η − ζ)| 1 + dη σ−2 hξi Z o n σ−2 |σ−2| b 6 c hξi {|η − ζ| |b g (η − ζ)|} · |ξ − η| hξ − ηi h (ξ − η) dη Z σ−2 σ−2 ≡ c hξi f1 (η − ζ) · f2 (ξ − η) dη = c hξi (f1 ∗ f2 ) (ξ − ζ) |K4 (ξ, ζ)| 6

206

H¨ ormander operators

Since gb, b h ∈ S (Rn ), the functions f1 , f2 inside the last integral have fast decay at infinity, in the sense that for every γ > 0 γ

sup hξi |fi (ξ)| < +∞. ξ∈Rn

Using Lemma 5.31 it is easy to see that k4 = f1 ∗f2 has the same property. Therefore σ−2

σ−2

|K4 (ξ, ζ)| 6 c hξi

k4 (ξ − ζ) = c |σ−2|

6 c hξ − ζi

hξi

hζi

σ−2 k4 σ−2

k4 (ξ − ζ) hζi

σ−2

(ξ − ζ) hζi

≡ k5 (ξ − ζ) hζi

σ−2

with k5 having fast decay at infinity. Hence Z σ−2 0 \ ϕ b (ζ) dζ [T3 , h] ϕ (ξ) = k5 (ξ − ζ) hζi and this easily implies that [T1 , h] is of order σ − 2. Indeed, using Lemma 5.31 we obtain Z ρ 0 \ ρ −ρ |k5 (ξ − ζ)| hζi |ϕ b (ζ)| dζ Λ [T3 , h] Λ−ρ−σ+2 ϕ (ξ) 6 hξi Z Z |ρ| 6 2|ρ|/2 k5 (ξ − ζ) hξ − ζi |ϕ b (ζ)| dζ ≡ k6 (ξ − ζ) |ϕ b (ζ)| dζ with k6 ∈ L1 (Rn ), so that Λρ [T30 , h] Λ−ρ−σ+2 is bounded on L2 (Rn ). By Proposition 5.28 (iv) the operator [T30 , h] has order σ − 2. Proposition 5.36 (Commutators of Λs with differential operators) Let P and Q differential operators with coefficients in S (Rn ) of orders m and n respectively. For every σ, τ ∈ R we have: (i) the operator [Λσ , P ] is of order m + σ − 1 (ii) the operator [Λσ , [Λτ , P ]] is of order σ + τ + m − 2 (iii) the operator [[Λσ , P ] , Q] is of order σ + m + n − 2. Proof. Let us start by noting that, by the definition of Λσ and Proposition 5.5 (vi), derivatives commute with Λσ : ∂α σ ∂αf = Λ f, ∂xα ∂xα as one can check taking the Fourier transform of both sides. Then, letting Λσ

P =

X |α|6m

aα (x)

∂ |α| ∂xα

we have [Λσ , P ] =

X  |α|6m

Λσ , aα

 X ∂ |α| ∂ |α| = [Λσ , aα ] α . α ∂x ∂x |α|6m

(5.11)

Hypoellipticity of general H¨ ormander operators

207

By Proposition 5.35 (i) and Proposition 5.28 (ii)-(iii) [Λσ , P ] has order m + σ − 1. This shows (i). As to (ii), again by (5.11) we can write for any operator T   ∂ |α| ∂ |α| σ Λ , T α = [Λσ , T ] α ∂x ∂x τ hence, for T = [Λ , aα ],  X  X ∂ |α| ∂ |α| σ τ σ τ [Λ , [Λ , P ]] = Λ , [Λ , aα ] α = [Λσ , [Λτ , aα ]] α ∂x ∂x |α|6m

|α|6m

and the assertion follows applying now Proposition 5.35 (ii). Finally,  X X  ∂ |β| ∂ |α| [[Λσ , P ] , Q] = [Λσ , aα ] α , aβ β . ∂x ∂x |α|6m |β|6n

Now we apply an identity that can be checked directly expanding the commutators involved. For every choice of operators A, B, C, D the following holds: [AB, CD] = [ABD, C] − C [D, A] B + AB [C, D] . Hence,   ∂ |α| ∂ |β| [Λσ , aα ] α , aβ β ∂x ∂x   |α|     |β| |α| |β| ∂ ∂ ∂ ∂ |α| ∂ |β| ∂ σ σ = [Λσ , aα ] α β , aβ − aβ , [Λ , a ] + [Λ , a ] a , α α β ∂x ∂x ∂xβ ∂xα ∂xα ∂xβ ≡ I − II + III. i h |β| Now, since aβ , ∂∂xβ is a differential operator of order |β| − 1, by Proposition 5.35 (i) III has order (σ − 1) + |α| + (|β| − 1) 6 σ + n + m − 2. As to II, exploiting the |β| fact that ∂∂xβ and Λσ commute,      |β| ∂ |β| ∂ σ σ , [Λ , a ] = − Λ , a , α α ∂xβ ∂xβ has order σ + |β| − 2, by point (i). Hence II has order 0 + (σ + |β| − 2) + |α| 6 σ + n + m − 2. Finally,   ∂ |α| ∂ |β| ∂ |α| ∂ |β| σ σ I = [[Λ , aα ] , aβ ] α β − [Λ , aα ] aβ , α β = I1 − I2 ∂x ∂x ∂x ∂x 5.35 (iii) I1 has order (σ − 2) + |α| + |β| 6 σ + n + m − 2, while i hwhere by Proposition |α|

|β|

aβ , ∂∂xα ∂∂xβ

is a differential operator of order |α| + |β| − 1, hence by Proposition

5.35 (iii) I2 has order (σ − 1) + (|α| + |β| − 1) 6 σ + n + m − 2 so we are done. Proposition 5.37 (Commutator of Sδ with vector fields) Let Sδ be the mollification operator as in Definition 5.19 and let X be a vector field with real coefficients in S (Rn ). Let ζ1 , ζ2 ∈ C0∞ (Rn ), with ζ1 ≺ ζ2 . Then there exists C > 0 such that for suitably small δ > 0 and for every u ∈ S (Rn ), k[Sδ ζ1 , X] ukH σ = k[Sδ ζ1 , X] ζ2 ukH σ 6 C kζ2 ukH σ ,

(5.12)

k[[Sδ ζ1 , X, ] , X] ukH σ = k[[Sδ ζ1 , X, ] , X] ζ2 ukH σ 6 C kζ2 ukH σ .

(5.13)

208

H¨ ormander operators

Moreover, if T σ = ζ1 Λσ ζ1 then k[Sδ T σ , X] uk2 6 C kζ2 ukH σ , σ

k[[Sδ T , X] , X] uk2 6 C kζ2 ukH σ .

(5.14) (5.15)

The constant C in (5.12)-(5.15) is independent of δ. Proof. Let us prove (5.12). The fact that [Sδ ζ1 , X] u = [Sδ ζ1 , X] ζ2 u is a straightforward computation. It remains to check that [Sδ ζ1 , X] is an operator of order 0. Since [Sδ ζ1 , X] u = [Sδ , X] ζ1 u − Sδ (uXζ1 ) . and by Lemma 5.21, kSδ (uXζ1 )kH σ 6 C kuXζ1 kH σ 6 C kukH σ , it remains to consider [Sδ , X]. By Proposition 5.28 (iv) it is enough to show that the operator Λσ [Sδ , X] Λ−σ is bounded on L2 for every σ ∈ R with constant indepen∂ dent of δ and using the linearity we can assume X = a (x) ∂x for some a ∈ S (Rn ) j and some j. We have [Sδ , X] u (x) = Jδ ∗ Xu (x) − X (Jδ ∗ u) (x) Z Z ∂ = Jδ (x − y) a (y) uj (y) dy − a (x) Jδ (x − y) u (y) dy ∂xj Z Z = ∂j Jδ (x − y) a (y) u (y) dy − Jδ (x − y) ∂j a (y) u (y) dy Z − a (x) ∂j Jδ (x − y) u (y) dy = [∂j Jδ ∗ (au) (x) − a (x) ∂j Jδ ∗ u (x)] − [Jδ ∗ (u∂j a) (x)] ≡ T1 u − T2 u, Since kT2 ukH σ = kJδ ∗ (u∂j a)kH σ 6 ku∂j akH σ 6 c kukH σ it remains to bound T1 . We have   d d T a∗u b (ξ)) − b a ∗ ∂d b (ξ) 1 u (ξ) = ∂j Jδ (ξ) (b j Jδ u Z   d = ∂d a (ξ − ζ) u b (ζ) dζ j Jδ (ξ) − ∂j Jδ (ζ) b Z   = 2πi ξj Jb (δξ) − ζj Jb (δζ) b a (ξ − ζ) u b (ζ) dζ Z = 2πiδ −1 (ψ (δξ) − ψ (δζ)) b a (ξ − ζ) u b (ζ) dζ with ψ (ξ) = ξj Jb (ξ) ∈ S (Rn ), and therefore Z d a (ξ − ζ)| |b u (ζ)| dζ T1 u (ξ) 6 c |ξ − ζ| |b

Hypoellipticity of general H¨ ormander operators

209

Thus, by Lemma 5.31, Z σ\ −σ σ |ξ − ζ| |b a (ξ − ζ)| hζi |b u (ζ)| dζ Λ T1 Λ−σ u (ξ) 6 hξi Z |σ| 6 c |ξ − ζ| |b a (ξ − ζ)| hξ − ζi |b u (ζ)| dζ ≡ k ∗ u b (ξ) . |σ|

Since the function k (ξ) = c |ξ| |b a (ξ)| hξi



σ\

Λ T1 Λ−σ u 2

is in L1 (Rn ), we obtain

L (Rn )

6 c kb ukL2 (Rn ) ,

so that kT1 ukH σ 6 c kukH σ . Observe that the constant c is independent of δ. Let us come to the proof of (5.13). Since [Sδ ζ1 , X] u = [Sδ , X] ζ1 u − Sδ (Xζ1 ) u, and [Sδ , X] (Xζ1 ) u = Sδ (XXζ1 ) u + Sδ (Xζ1 ) Xu − XSδ (Xζ1 ) u we have [[Sδ ζ1 , X, ] , X] u = [Sδ , X] ζ1 Xu − Sδ (Xζ1 ) Xu − X [Sδ , X] ζ1 u + XSδ (Xζ1 ) u = [Sδ , X] X (ζ1 u) − [Sδ , X] (Xζ1 ) u − Sδ (Xζ1 ) Xu − X [Sδ , X] ζ1 u + XSδ (Xζ1 ) u = [[Sδ , X] , X] ζ1 u − [Sδ , X] (Xζ1 ) u − Sδ (Xζ1 ) Xu + XSδ (Xζ1 ) u = [[Sδ , X] , X] ζ1 u − 2 [Sδ , X] (Xζ1 ) u + Sδ (XXζ1 ) u Now, by (5.12), k[Sδ , X] (Xζ1 ) ukH σ 6 c kukH σ kSδ (XXζ1 ) ukH σ 6 c kukH σ . To bound [[Sδ , X] , X] ζ1 u we can use the computation in the estimate of [Sδ , X] writing, with the same notation [[Sδ , X] , X] ζ1 u = [T1 , X] ζ1 u − [T2 , X] ζ1 u. Now, by (5.12), k[T2 , X] ζ1 ukH σ = k[Sδ ∂j a, X] ζ1 ukH σ = k[Sδ ζ2 ∂j a, X] ζ1 ukH σ 6 c kukH σ and we are left to bound [T1 , X] ζ1 u that is, reasoning as above, we need to prove that

σ \ −σ ukL2 (Rn ) .

Λ [T1 , X] Λ u 2 n 6 c kb L (R )

Again by linearity, it is enough to consider X = b (x) ∂xk , so that recalling that T1 u = ∂j Jδ ∗ (au) − a (∂j Jδ ∗ u) we are interested in [T1 , b∂k ] u = T1 (b∂k u) − b∂k (T1 u) = ∂j Jδ ∗ (ab∂k u) − a (∂j Jδ ∗ (b∂k u)) − (b (∂k ∂j Jδ ) ∗ (au) − ab (∂k ∂j Jδ ) ∗ u) + b∂k a (∂j Jδ ∗ u) .

(5.16)

210

H¨ ormander operators

We have ∂j Jδ ∗ (ab∂k u) = ∂j Jδ ∗ (∂k (abu) − (∂k a) bu − a (∂k b) u) = ∂j ∂k Jδ ∗ (abu) − ∂j Jδ ∗ ((∂k a) bu) − ∂j Jδ ∗ (a (∂k b) u) .

(5.17)

Moreover ∂j Jδ ∗ (b∂k u) = ∂j Jδ ∗ (∂k (bu) − (∂k b) u) = ∂j ∂k Jδ ∗ (bu) − ∂j Jδ ∗ ((∂k b) u) .

(5.18)

Substituting (5.17) and (5.18) into (5.16) we obtain [T1 , b∂k ] = (∂j ∂k Jδ ∗ (abu) − ∂j Jδ ∗ ((∂k a) bu) − ∂j Jδ ∗ (a (∂k b) u)) − a (∂j ∂k Jδ ∗ (bu) − ∂j Jδ ∗ ((∂k b) u)) − (b (∂j ∂k Jδ ) ∗ (au) − ab (∂j ∂k Jδ ) ∗ u) + b∂k a (∂j Jδ ∗ u) = {∂j ∂k Jδ ∗ (abu) − a (∂j ∂k Jδ ) ∗ (bu) − b (∂j ∂k Jδ ) ∗ (au) + ab (∂j ∂k Jδ ) ∗ u} + {b∂k a (∂j Jδ ∗ u) − ∂j Jδ ∗ ((∂k a) bu)} + {a (∂j Jδ ) ∗ ((∂k b) u) − ∂j Jδ ∗ (a (∂k b) u)} ≡ {T3 u} + {T4 u} + {T5 u} The terms T4 u and T5 u are similar to T1 u so that



σ\

T5 Λ−σ u 2

Λ T4 Λ−σ u 2 n + Λσ\ L (R )

L (Rn )

6 c kb ukL2 (Rn ) .

As for T3 we have Z Z
2 d b T3 u (ξ) = −4π ξj ξk J (δξ) b a (ξ − η) bb (η − z) u b (z) dzdη Z Z
− b a (ξ − η) −4π 2 ηj ηk Jb (δη) bb (η − z) u b (z) dzdη Z Z
− bb (ξ − η) −4π 2 ηj ηk Jb (δη) b a (η − z) u b (z) dzdη Z Z  
b (z) dzdη + bb (ξ − η) b a (η − z) −4π 2 zj zk Jb (δz) u Z Z n = − 4π 2 b a (ξ − η) bb (η − z) ξj ξk Jb (δξ) − b a (ξ − η) ηj ηk Jb (δη) bb (η − z) o −bb (ξ − η) ηj ηk Jb (δη) b a (η − z) + bb (ξ − η) b a (η − z) zj zk Jb (δz) dηb u (z) dz Letting Φ (z) = zj zk Jb (δz), then Z Z −4π 2 d T b a (ξ − η) bb (η − z) (Φ (δξ) − Φ (δη)) dηb u (z) dz 3 u (ξ) = δ2 Z Z 4π 2 b a (w − z) bb (ξ − w) (Φ (δw) − Φ (δz)) dwb u (z) dz. + 2 δ

Hypoellipticity of general H¨ ormander operators

211

Also, the change of variable w = ξ − η + z, dw = dη in the last integral gives Z Z −4π 2 d T3 u (ξ) = b a (ξ − η) bb (η − z) δ2 × ([Φ (δξ) − Φ (δη)] − [Φ (δ (ξ − η + z)) − Φ (δz)]) dηb u (z) dz Z ≡ k3 (ξ, z) u b (z) dz. Now, |[Φ (δξ) − Φ (δη)] − [Φ (δ (ξ − η + z)) − Φ (δz)]| Z 1 d = (Φ (δ (ξ + t (z − η))) − Φ (δ (η + t (z − η)))) dt 0 dt Z 1 Z 1 d d Φ (δ (η + s (ξ − η) + t (z − η))) dtds 6 Cδ 2 |ξ − η| |z − η| = 0 ds dt 0 so that Z C b a (ξ − η) b (η − z) ([Φ (δξ) − Φ (δη)] − [Φ (δ (ξ − η + z)) − Φ (δz)]) dη |k3 (ξ, z)| 6 2 b δ Z 6 C |b a (ξ − η)| |ξ − η| bb (η − z) |z − η| dη Z = C |b a (ξ − z − w)| |ξ − z − w| bb (w) |w| dz ≡ Cγ (ξ − z) where γ has fast decay at infinity (see the proof of Proposition 5.35). By Lemma 5.31 we obtain Z σ\ σ −σ γ (ξ − z) hζi |b u (ζ)| dζ Λ T3 Λ−σ u (ξ) 6 C hξi Z |σ| 6 C γ (ξ − z) hξ − ζi |b u (ζ)| dζ 0

≡ k3 ∗ u b (ξ) .

0

with k3 (ξ) in L1 (Rn ) so that Λσ\ T3 Λ−σ u

L2 (Rn )

6 c kb ukL2 (Rn ) .

To prove (5.14) we use the identity [AB, C] = A [B, C] + [A, C] B.

(5.19)

that holds for generic operators A, B and C. Then, [Sδ T σ , X] = [Sδ ζ1 Λσ ζ1 , X] = Sδ ζ1 [Λσ ζ1 , X] + [Sδ ζ1 , X] Λσ ζ1 so that, by Lemma 5.21 and (5.12), k[Sδ T σ , X] uk2 6 kSδ ζ1 [Λσ ζ1 , X] uk2 + k[Sδ ζ1 , X] Λσ ζ1 uk2 6 c k[Λσ ζ1 , X] ζ2 uk2 + c kζ2 Λσ ζ1 uk2 6 c kζ2 ukH σ where in the last inequality we used the fact that ζ2 Λσ ζ1 and [Λσ ζ1 , X] are operators of order σ. Similarly to prove (5.15) we use again (5.19) to write [[Sδ T σ , X, ] , X] = [Sδ ζ1 [Λσ ζ1 , X] , X] + [[Sδ ζ1 , X] Λσ ζ1 , X] = Sδ ζ1 [[Λσ ζ1 , X] , X] + [Sδ ζ1 , X] [Λσ ζ1 , X] + [Sδ ζ1 , X] [Λσ ζ1 , X] + [[Sδ ζ1 , X] , X] Λσ ζ1 . Using (5.12) and (5.13) we easily obtain k[[Sδ T σ , X, ] , X] uk2 6 c kζ2 ukH σ .

212

H¨ ormander operators

Definition 5.38 (Simple operators) We will say that an operator T is a simple operator if T is the composition of finitely many operators of the kind Λσ , Sδ , vector fields with real coefficients in S (Rn ), and multiplication operators with a real function in S (Rn ). Remark 5.39 Keeping into account Remark 5.24, if T is a simple operator, then T ϕ = T ϕ. In particular, T ϕ is real whenever ϕ is real. Proposition 5.40 (Commutators with “simple” operators) Let P and Q be differential operators with coefficients in S (Rn ) of orders m and n respectively and let T be a simple operator of order σ which does not contain operators Sδ . Then the operators [P, T ] (5.20) and [Q, [P, T ]] (5.21) have respectively orders m + σ − 1 and m + n + σ − 2. If T contains some operator Sδ , the previous assertions hold for m = n = 1 (that is, P and Q vector fields); in this case the constant cσ in (5.8) is independent of δ. Proof. By the definition of simple operator we can write T = T1 T2 · · · TM where Tk are operators of order σk of the kind Λσk , Sδ , vector fields or multiplication operators. We will prove the proposition by induction on M . Assume M = 1. If T1 = Λσ1 the order of (5.20) and (5.21) is given by Proposition 5.36 and if T = Sδ (and n = m = 1) it is given by Proposition 5.37 (with constants independent of δ). If T1 is a vector field or a multiplication operator then the order of (5.20) and (5.21) can be computed straightforwardly. Let now M > 1. Observe that for generic operators A, B, C we have [A, BC] = [A, B] C + B [A, C] . (5.22) Using this identity we obtain [P, T1 T2 · · · TM −1 TM ] = [P, T1 T2 · · · TM −1 ] TM + T1 T2 · · · TM −1 [P, TM ] . By the induction assumption [P, T1 T2 · · · TM −1 ] is of order m + σ1 · · · + σM −1 − 1 and [P, TM ] is of order m + σM − 1 so that [P, T1 T2 · · · TM −1 TM ] is of order m + σ1 · · · + σM − 1. We now consider the operator in (5.21). Using again (5.22) we obtain [Q, [P, T1 T2 · · · TM ]] = [Q, [P, T1 T2 · · · TM −1 ] TM ] + [Q, T1 T2 · · · TM −1 [P, TM ]] = [Q, [P, T1 T2 · · · TM −1 ]] TM + [P, T1 T2 · · · TM −1 ] [Q, TM ] + [Q, T1 T2 · · · TM −1 ] [P, TM ] + T1 T2 · · · TM −1 [Q, [P, TM ]] . By the induction assumption the order of these operators is m + n + σ1 · · · + σM − 2.

Now, we need to define the transpose of an operator.

Hypoellipticity of general H¨ ormander operators

213

Definition 5.41 Let T : S (Rn ) → S (Rn ) be a linear operator. Its transpose is a linear operator T ∗ : S (Rn ) → S (Rn ) such that for every ϕ, ψ ∈ S (Rn ) hT ϕ, ψi = hϕ, T ∗ ψi . When such operator T ∗ exists we say that T can be transposed. Remark 5.42 It T is a simple operator, by Remark 5.39 the above condition can also be written as Z Z (T ϕ) ψ = ϕ (T ∗ ψ) . Example 5.43 Let X be a vector field with real coefficients in S (Rn ), then by (2.3) X can be transposed and X ∗ = −X + g with a real valued g ∈ S (Rn ). 2

Example 5.44 Let T : S (R) → S (R) such that T f (x) = f (0) e−x . Then T is a well defined linear operator on S (R) but it cannot be transposed. Indeed, let ϕ, ψ ∈ S (R),then Z 2 hT ϕ, ψi = ϕ (0) e−x ψ (x) dx and for a fixed ψ ∈ S (R) there is not any g = T ∗ ψ ∈ S (R) such that Z Z 2 ϕ (x) g (x) dx = ϕ (0) e−x ψ (x) dx for every ϕ ∈ S (R) . Actually, in this case the object T ∗ ψ is not a function but the distribution Z  ∗ −x2 T ψ= e ψ (x) dx δ with δ the Dirac mass. The next proposition shows that all the operators we are interested in can be actually transposed, with a very simple transpose operator. Proposition 5.45 The transpose of: (i) a multiplication operator T u = au (with a ∈ S (Rn ) real valued) is still T ; (ii) the fractional derivative operator Λσ is still Λσ ; (iii) the mollification operator Sδ is still Sδ (assuming the function J even, as in Definition 5.19); (iv) a vector field X (with real coefficients in S (Rn )) is the usual transpose vector field X ∗ = −X + c with c ∈ S (Rn ), c real valued; (v) a differential operator of order m (with real coefficients in S (Rn )) is a differential operator of order m with real coefficients in S (Rn ); (vi) the commutator [A, B] of any two linear operators that can be transposed ∗ A, B : S (Rn ) → S (Rn ) is [A, B] = [B ∗ , A∗ ]; (vii) the composition AB of any two linear operators that can be transposed A, B : ∗ S (Rn ) → S (Rn ) is (AB) = B ∗ A∗ .

214

H¨ ormander operators

(viii) a simple operator T = T1 T2 · · · TM (with Tk operators of the kind Λσ , vector fields with real coefficients in S (Rn ) and multiplication by a real function ∗ in S (Rn )) is T ∗ = TM · · · T2∗ T1∗ . In particular: every simple operator can be transposed, and its transpose is still a simple operator. The proof of this proposition is straightforward. We can now compute the order of the transpose. Proposition 5.46 Let T : S (Rn ) → S (Rn ) be a linear operator that can be transposed (in particular, T can be a simple operator). (i) If for some σ ∈ R there exists cσ > 0 such that kT ϕkH σ 6 cσ kϕkH σ+m , for every ϕ ∈ S (Rn ), then kT ∗ ψkH −σ−m 6 cσ kψkH −σ . (ii) If T is an operator of order m, then also T ∗ is of order m, with the same constant cσ of T. Proof. (i). Let ϕ, ψ ∈ S (Rn ). Then

kT ∗ ψkH −σ−m = Λ−σ−m T ∗ ψ 2 =

sup ϕ∈S(Rn ), kϕk2 =1

−σ−m ∗  Λ T ψ, ϕ .

Using Proposition 5.25 we have −σ−m ∗  



Λ T ψ, ϕ = ψ, T Λ−σ−m ϕ = Λ−σ ψ, Λσ T Λ−σ−m ϕ

6 kψkH −σ T Λ−σ−m ϕ H σ

6 cσ kψkH −σ Λ−σ−m ϕ H σ+m = cσ kψkH −σ kϕk2 . So that kT ∗ ψkH −σ−m 6 cσ kψkH −σ . Point (ii) immediately follows from (i). In general, Λσ is not a local operator: fix ψ ∈ C0∞ (Rn ), then supp (Λσ ψu) can be larger than supp (ψ). The next proposition essentially shows that away from supp (ψ), Λσ ψu is smooth. Proposition 5.47 Let ϕ, ψ ∈ C0∞ (Rn ) such that ψ ≺ ϕ and let σ ∈ R. Then the operator u 7→ (1 − ϕ) Λσ ψu is of order −τ for every τ ∈ R. Proof. It is enough to prove that the operator is of order −τ for every τ > 0. More precisely, we should prove that for every ρ ∈ R, τ > 0, σ ∈ R, there exists cρ,σ,τ such that k(1 − ϕ) Λσ ψukH ρ 6 cρ,σ,τ kukH ρ−τ for every u ∈ S (Rn ). However, by the genericity of ρ, τ it is enough to show that for every ρ ∈ R and every τ > 0 there exists cρ,τ such that k(1 − ϕ) Λσ ψukH ρ 6 cρ,τ kukH −τ . We consider first the case ρ 6 0. Let T = (1 − ϕ) Λσ ψΛτ and assume to know that T is bounded on L2 (Rn ). Then

k(1 − ϕ) Λσ ψukH ρ 6 k(1 − ϕ) Λσ ψuk2 = (1 − ϕ) Λσ ψΛτ Λ−τ u 2

6 c Λ−τ u = c kuk −τ . 2

H

Hypoellipticity of general H¨ ormander operators

To prove that T is bounded on L2 observe that Z σ \ τ u (ξ) e2πiξ·x dξ T u (x) = (1 − ϕ (x)) hξi ψΛ  Z Z σ τ = (1 − ϕ (x)) hξi ψb (ξ − ζ) hζi e2πiξ·x dξ u b (ζ) dζ Z ≡ K (x, ζ) u b (ζ) dζ

215

(5.23)

where τ

Z

σ

ψb (ξ − ζ) e2πiξ·x dξ Z τ σ 2πiζ·x = (1 − ϕ (x)) hζi e hζ + ηi ψb (η) e2πiη·x dη

K (x, ζ) = (1 − ϕ (x)) hζi

hξi

σ

Let Ψζ (η) = hζ + ηi , then, for an integer m > 0 to be chosen later, σ

hζ + ηi =

X 1 ∂ |α| Ψζ (0) η α + α! ∂η α

|α|6m

X |α|=m+1

1 ∂ |α| Ψζ (ζ + tη) η α α! ∂η α

for a suitable t ∈ (0, 1). Observe now that Z Z \ 1 ∂ |α| ψ αb 2πiη·x (1 − ϕ (x)) η ψ (η) e dη = (1 − ϕ (x)) (η) e2πiη·x dη |α| ∂xα (2πi) =

1 |α|

(1 − ϕ (x))

(2πi)

∂ |α| ψ (x) = 0 ∂xα

since (1 − ϕ) and ψ have disjoint support. If follows that X Z 1 ∂ |α| Ψζ τ 2πiζ·x (ζ + tη) η α ψb (η) e2πiη·x dη K (x, ζ) = e (1 − ϕ (x)) hζi α! ∂η α |α|=m+1

|α| ∂ Ψ σ−|α| we obtain and since ∂ηα ζ (ζ + tη) 6 c hζ + tηi Z σ−m−1 m+1 b τ hζ + tηi |η| |K (x, ζ)| 6 c |1 − ϕ (x)| hζi ψ (η) dη. σ

|σ|

σ

By Lemma 5.31, (in the form hx + yi 6 c hxi hyi ) we obtain Z τ |σ−m−1| σ−m−1 m+1 b |K (x, ζ)| 6 c |1 − ϕ (x)| htηi hζi |η| ψ (η) hζi dη Z τ +σ−m−1 |σ−m−1| m+1 b 6 c hζi hηi |η| ψ (η) dη τ +σ−m−1

6 c hζi

.

Moreover since 
 2 (I − ∆η ) e2πiη·x = 1 + 4π 2 |x| e2πiη·x

216

H¨ ormander operators

(here I stands for the identity operator and ∆η for the standard Laplacian), for an integer k > 0 to be chosen later, we have K (x, ζ) τ

(1 − ϕ (x)) hζi = e2πiζ·x  k 2 1 + 4π 2 |x|

|α|=m+1

τ

=e

2πiζ·x

(1 − ϕ (x)) hζi  k 2 1 + 4π 2 |x|


∂ |α| Ψζ k (ζ + tη) η α ψb (η) (I − ∆η ) e2πiη·x dη α ∂η

Z

X

Z

X

(I − ∆η )

|α|=m+1

k



 ∂ |α| Ψζ αb (ζ + tη) η ψ (η) e2πiη·x dη. ∂η α k

Observe now that expanding the power (I − ∆η ) and taking the derivatives with respect to η we obtain a sum of terms of the type ∂ |δ| ψb ∂ |β| Ψζ (ζ + tη) η γ (η) β ∂η ∂η δ with |β| 6 m + 1 + 2k. Arguing as before we have Z ∂ |β| Ψ |δ| b ζ σ−m−1−|β| σ−m−1 γ∂ ψ 2πiη·x 6 c hζi . (ζ + tη) η (η) e dη 6 c hζi ∂η β ∂η δ τ +σ−m−1

Hence |K (x, ζ)| 6 c hζi Z

2

Z Z

−2k

hxi

and therefore

τ +σ−m−1

−2k

2

|T u (x)| dx 6 c hζi hxi |b u (ζ)| dζ dx Z Z  −2k  τ +σ−m−1  2 2 2 6 c 1 + |x| kuk2 1 + |ζ| dζ dx 6 c kuk2 . provided that m and k are large enough. Let us now consider the case ρ > 0 and let k be an integer larger than ρ. Since the norm in H k is equivalent to the norm in W k,2 we have

X

∂ |α|

−τ

−τ

σ −τ



k(1 − ϕ) Λ ψukH ρ = T Λ u H ρ 6 T Λ u H k 6 c

∂xα T Λ u . 2 |α|6k

By (5.23) the kernel of the operator Z

|α1 |

∂ |α| ∂xα T

is a finite sum of terms of the kind

∂ |α1 | (1 − ϕ) σ τ α (x) hξi ψb (ξ − ζ) hζi (2πiξ) 2 e2πiξ·x dξ. ∂xα1

Since also ∂∂xα1ϕ and ψ have disjoint support, the only relevant difference with α respect to the previous computation is the presence of the term (2πiξ) 2 . However, |α| |α | this term can be easily controlled by hξi 2 . It follows that also ∂∂xα T is bounded 2 σ −τ on L and therefore kϕΛ ψukH ρ 6 c kΛ uk2 = c kukH −τ .

Hypoellipticity of general H¨ ormander operators

5.5

217

Subelliptic estimates

From now on X0 , X1 , . . . , Xq will be a system of H¨ormander vector fields of step s in a bounded domain Ω ⊂ Rn (see Definition 1.24). Shrinking a bit the domain Ω if necessary, and multiplying their coefficients for a suitable cutoff function, we can assume that the vector fields are defined on the whole Rn , have coefficients in S (Rn ), and satisfy H¨ ormander’s condition in Ω. Hence the vector fields Xi and their commutators X[I] are differential operators of order 1 to which we can apply the results of the previous section. We explicitly remark that, differently from the rest of the book, in this chapter the weight of vector fields or commutators does not play any role. Actually, a key point of the techniques used in this chapter is that differential operators are just considered for their order : every vector field (in particular, every iterated commutator of vector fields) is simply regarded as a differential operator of order 1. We will consider the differential operator L=

q X

Xj2 + X0 + b

(5.24)

j=1

where b ∈ S (Rn ) is a fixed real valued function (also this function b can be obtained extending to the whole Rn some original function b ∈ C ∞ (Ω)). Remark 5.48 (Use of real valued functions) The goal of this and the following section is to prove Theorem 5.1, that is the so-called the subelliptic estimates. Since the operator L has real coefficients, by linearity it is clearly enough to prove subelliptic estimates for real valued functions or distributions. Moreover, since all the operators that we will handle in proving these estimates are simple operators (in the sense of Definition 5.38), which transform real valued functions into real valued functions (see Remark 5.39), from now on within this and the next section we will assume that the elements of S (Rn ) and S 0 (Rn ) are real valued. This will assure that all the functions and distributions that we handle are real valued, which will be important in one key point (see Remark 5.52). Our first result is the following: Theorem 5.49 Let ε = 21−s . There exists a positive constant c such that for every u ∈ C0∞ (Ω)   q X 2 2 2 kukH ε 6 c  kXj uk2 + kuk2  . (5.25) j=0

Note that this estimate gives a control on “fractional derivatives of u in any direction” in terms of derivatives along the vector fields Xi alone. Clearly, this is possible because the vector fields satisfy H¨ ormander’s condition.

218

H¨ ormander operators

Proof. Let Cε > 0 be such that   ε  ε−1  2 2 2 1 + |ξ| 6 Cε 1 + |ξ| 1 + |ξ| . Then 2 kukH ε

Z



6 Cε

 ε−1  2 2 1 + |ξ| 1 + |ξ| |b u (ξ)| dξ 2

Rn

Z

2

= Cε kuk2 + Cε  6

2 Cε kuk2

+

n  X

1 + |ξ|

2

ε−1

Rn j=1

n X

2

|ξj u b (ξ)| dξ

(5.26)

 2 k∂j ukH ε−1  .

j=1

From the proof of Proposition 2.7 we know that x0 ∈ Ω there exist a   for every neighborhood U (x0 ) and coefficients cj,I ∈ C ∞ U (x0 ) such that X ∂ = cI,j X[I] . ∂xj

(5.27)

|I|6s

We can now cover Ω with a finite number of such neighborhoods and choose a P partition of unity {ϕk } adapted to such covering so that k ϕk = 1 in Ω and on (k) the support of every ϕk there exist coefficients cI,j such that (5.27) holds. Therefore, using Proposition 5.29 we obtain

2

X X X

(k) 2 2

k∂j ukH ε−1 6 c kϕk ∂j ukH ε−1 6 c ϕk cI,j X[I] u

ε−1 k k |I|6s H

2 X X X

2

(k)

X[I] u ε−1 . (5.28) 6c

ϕk cI,j X[I] u ε−1 6 c H |I|6s k

H

|I|6s

2

Now, we will prove that every term X[I] u H ε−1 can be bounded by the right handside in (5.25). By (5.26) and (5.28), this will give our assertion. Let us fix a multiindex I such that |I| 6 s. If |I| = 1 then X[I] = Xj for some 0 6 j 6 q and since ε < 1 we have

X[I] u ε−1 6 kXj uk 2 H so that there is nothing to prove. Assume now |I| > 1, let 0 6 j 6 q and I 0 such that X[I] = Xj X[I 0 ] − X[I 0 ] Xj . Then, by Proposition 5.25 (vi),



X[I] u 2 ε−1 = X[I] u, Λ2ε−2 X[I] u H



 = Xj X[I 0 ] u, Λ2ε−2 X[I] u − X[I 0 ] Xj u, Λ2ε−2 X[I] u (5.29) ≡ A − B.

Hypoellipticity of general H¨ ormander operators

219

We start estimating A. By (2.3) we have, for some g ∈ S (Rn ),





 Xj X[I 0 ] u, Λ2ε−2 X[I] u = − X[I 0 ] u, Xj Λ2ε−2 X[I] u + X[I 0 ] u, gΛ2ε−2 X[I] u ≡ −A1 + A2 . Let us write     Xj Λ2ε−2 X[I] u = Λ2ε−2 X[I] Xj u + Λ2ε−2 Xj , X[I] u + Xj , Λ2ε−2 X[I] u, so that



   A1 = X[I 0 ] u, Λ2ε−2 X[I] Xj u + X[I 0 ] u, Λ2ε−2 Xj , X[I] u 

  + X[I 0 ] u, Xj , Λ2ε−2 X[I] u ≡A1,1 + A1,2 + A1,3 . Since Λ−1 X[I] is an operator of order 0, by Proposition 5.25 (iv) we have

 |A1,1 | = Λ2ε−1 X[I 0 ] u, Λ−1 X[I] Xj u  

2 2 6 c X[I 0 ] u H 2ε−1 kXj uk2 6 c X[I 0 ] u H 2ε−1 + kXj uk2 . Similarly  

2

   2 |A1,2 | = Λ2ε−1 X[I 0 ] u, Λ−1 Xj , X[I] u 6 c X[I 0 ] u H 2ε−1 + kuk2 .   2ε−2 Since by Proposition 5.36 X , Λ is an operator of order 2ε − 2, the operator j  −2ε+1 2ε−2 Λ Xj , Λ X[I] is of order 0 and, again by Proposition 5.25 (iv), we have  

2 2ε−1    2 X[I 0 ] u, Λ−2ε+1 Xj , Λ2ε−2 X[I] u 6 c X[I 0 ] u H 2ε−1 + kuk2 . |A1,3 | = Λ Moreover, by Proposition 5.29 we have

 |A2 | = Λ2ε−1 X[I 0 ] u, Λ−2ε+1 gΛ2ε−2 X[I] u  

2

2 6 c X[I 0 ] u H 2ε−1 kuk2 6 c X[I 0 ] u H 2ε−1 + kuk2 . Hence  

2 2 2 |A| 6 c kXj uk2 + X[I 0 ] u H 2ε−1 + kuk2

(5.30)

Let us consider the term B. We have, for a suitable function h ∈ S (Rn ),

 B = X[I 0 ] Xj u, Λ2ε−2 X[I] u



 = − Xj u, X[I 0 ] Λ2ε−2 X[I] u + Xj u, hΛ2ε−2 X[I] u . It follows that

1

X[I 0 ] Λ2ε−2 X[I] u 2 + 1 hΛ2ε−2 X[I] u 2 2 2 2 2

1 2 2 2 6 kXj uk2 + X[I 0 ] Λ2ε−2 X[I] u 2 + c kukH 2ε−1 . 2 2

|B| 6 kXj uk2 +

Since   X[I 0 ] Λ2ε−2 X[I] u = Λ2ε−2 X[I 0 ] X[I] u + X[I 0 ] , Λ2ε−2 X[I] u     = Λ2ε−2 X[I] X[I 0 ] u + Λ2ε−2 X[I 0 ] , X[I] u + X[I 0 ] , Λ2ε−2 X[I] u

220

H¨ ormander operators

  and X[I 0 ] , Λ2ε−2 is an operator of degree 2ε − 2, we have






X[I 0 ] Λ2ε−2 X[I] u 6 c X[I 0 ] u 2ε−1 + kuk 2ε−1 . H H 2  

2 2 2 Therefore, using 2ε − 1 6 0 we obtain |B| 6 c kXj uk2 + X[I 0 ] u H 2ε−1 + kuk2 and by (5.29) and (5.30)  



X[I] u 2 ε−1 6 c kXj uk2 + X[I 0 ] u 2 2ε−1 + kuk2 2 2 H H   q X

2 2 2 6 c kXj uk2 + X[I 0 ] u H 2ε−1 + kuk2  .

(5.31) (5.32)

j=0

Let ` = ` (I) be the length of the multindex I. Recursively applying (5.32) gives   q q X X

2 2 2 2

X[I] u ε−1 6 c  kXj ukH 2`−1 ε−1 + kuk2  , kXj uk2 + H j=0

j=0 `−1

ε − 1 6 0 and therefore   q X

2 2 2

X[I] u ε−1 6 c  kXj uk2 + kuk2  . H

and since ` 6 s, we have 2

j=0

Remark 5.50 (Dependence of the constant) Note that in (5.27) we have exploited H¨ ormander’s condition. Actually, this is the only point where we exploit this assumption throughout the proof of subelliptic estimates, in this section. This is important in connection with the analysis of the dependence of the final constant appearing in the subelliptic estimates on our vector fields. Whereas in all the other estimates we will simply bound (from above) the supremum, on some compact set, of the coefficients of the Xi (and the derivatives of these coefficients, up to some fixed order), in the identity (5.27) we are inverting the matrix having as rows some basis of Rn built selecting n suitable commutators the Xi , of step up to s. Therefore the constant appearing in the estimate proved as a consequence of (5.27) will also depend on a positive constant c0 such that the following bound holds:



det X[I ] , X[I ] , . . . , X[I ] > c0 . max inf x∈Ω |I1 |,|I2 |,...,|In |6s

1

x

2

x

n

x

We now want to improve the estimate of the previous theorem getting, on the right hand side, L2 norms of Lu and u, instead of of Xj u. The following result is easy but crucial (note the similarity with the computation (4.2) in the previous chapter): Lemma 5.51 There exists a positive constant C such that for every (real valued) ϕ ∈ S (Rn ) we have q   X 2 2 kXj ϕk2 6 C |hLϕ, ϕi| + kϕk2 . j=1

Hypoellipticity of general H¨ ormander operators

221

Note that the norm of X0 ϕ does not appear on the left-hand side. Also, recall that, as noted at the beginning of this section, the vector fields Xj and the operator L have been extended to the whole space, with coefficients in S (Rn ), although H¨ ormander’s condition holds in some domain Ω. In the following we will need to apply this Lemma also to functions in S (Rn ), which are not compactly supported in Ω. Proof. Let Xj∗ = −Xj + gj be the transpose of the vector fields Xj and let ϕ ∈ C0∞ (Ω). Then, recalling that L is given by (5.24), * q + q q q X X X X 2 2 hXj ϕ, Xj ϕi = − kXj ϕk2 = Xj ϕ, ϕ + hgj ϕ, Xj ϕi j=1

j=1

j=1

j=1

= − hLϕ, ϕi + hX0 ϕ, ϕi + hbϕ, ϕi +

q X

hgj ϕ, Xj ϕi .

j=1

Since hX0 ϕ, ϕi = hϕ, X0∗ ϕi = − hϕ, X0 ϕi + hϕ, g0 ϕi

(5.33)

we have 2

|hX0 ϕ, ϕi| 6 c kϕk2 so that

q X

2

2

kXj ϕk2 6 |hLϕ, ϕi| + c kϕk2 +

j=1

q X

(5.34)

kgj ϕk2 kXj ϕk2

j=1 q

q

1X 1X 2 2 6 |hLϕ, ϕi| + + kgj ϕk2 + kXj ϕk2 2 j=1 2 j=1   Pq 2 2 and therefore j=1 kXj ϕk2 6 c |hLϕ, ϕi| + kϕk2 . 2 c kϕk2

Remark 5.52 In the above proof we have exploited the assumption of handling real valued functions ϕ (see Remark 5.48). Otherwise (5.33) would give 2 Re hX0 ϕ, ϕi = hϕ, g0 ϕi and we could not conclude (5.34). Sometimes in the following we will exchange the operator L with another operator T using the commutator [L, T ]. The next lemma will be useful to handle this term. Lemma 5.53 Let T be a simple operator of order σ. There exist simple operators Tk and Tek , k = 0, . . . q of order σ such that q X [L, T ] = Tk Xk + T0 k=1

and [L, T ] =

q X k=1

Xk Tek + Te0 .

222

H¨ ormander operators

Proof. We have [L, T ] =

=

=

=

q X  2  Xk , T + [X0 , T ] + [b, T ] k=1 q X k=1 q X k=1 q X

([Xk , T ] Xk + Xk [Xk , T ]) + [X0 , T ] + [b, T ] 2 [Xk , T ] Xk +

q X

[Xk , [Xk , T ]] + [X0 , T ] + [b, T ]

k=1

Tk Xk + T0 .

k=1

The second identity can be proved similarly. By Proposition 5.40 all the operators Tk have order σ or better ([b, T ] has order σ − 1). Moreover, by construction they are simple operators. The next result contains the first kind of subelliptic estimate: a derivative of some small order of u is controlled by the L2 norms of Lu and u: Theorem 5.54 (The basic subelliptic estimate) There exist positive constants ε and C such that for every u ∈ C0∞ (Ω) kukH ε 6 C (kLuk2 + kuk2 ) . Actually, ε =

2 4s

where s is the step of H¨ ormander’s condition in Ω.

Remark 5.55 If the operator L had no drift term X0 , combining Theorem 5.49 and Lemma 5.51 we would immediately get   q   X 2 2 2 2 kukH ε 6 c  kXj uk2 + kuk2  6 c |hLu, ui| + kuk2 j=1

that is Theorem 5.54 or, more precisely, an even stronger result, since |hLu, ui| can 2 2 be bounded also by δ kLuk2 + δc kuk2 for some small δ > 0, and also since in this case we would get the subelliptic estimate with ε = 21−s , which is a more precise value than the one that will be given in the following proof. This means that the following long proof of Theorem 5.54 is entirely due to the presence of the drift term X0 . This also means that the value of ε given in the statement of the theorem is not optimal. Proof. Arguing as in the proof of Theorem 5.49 (see (5.26)-(5.28)), it is enough to show there exist ε and C such that for every multiindex I, with ` (|I|) 6 s we have  

X[I] u 2 ε−1 6 c kLuk2 + kuk2 . (5.35) 2 2 H We split the proof in several steps. The value of ε will be taken smaller and smaller throughout the steps of the proof. We start assuming 0 < ε < 1.

Hypoellipticity of general H¨ ormander operators

223

Step 1: the case ` (I) = 1. Since ε < 1 we have



X[I] u 2 ε−1 6 X[I] u 2 H

2

and the estimate (5.35) follows from Lemma 5.51 when X[I] = Xj and j > 0. Pq When j = 0 we use the identity X0 = L− j=1 Xj2 − b, so that

 2 kX0 ukH ε−1 = X0 u, Λ2ε−2 X0 u (5.36)

2ε−2

= Lu, Λ

q  X

2 

 X0 u − Xj u, Λ2ε−2 X0 u − bu, Λ2ε−2 X0 u . j=1

For the first term, assuming now 2ε − 1 < 0, that is 0 < ε < 21 , we have  

 Lu, Λ2ε−2 X0 u 6 c kLuk kuk 2ε−1 6 c kLuk2 + kuk2 . 2 H 2 2

(5.37)

Also,

 bu, Λ2ε−2 X0 u 6 c kuk kuk 2ε−1 6 c kuk2 . 2 H 2 As to the second term in (5.36), by Lemma 5.51 (letting Xj∗ = −Xj + gj with gj ∈ S (Rn )), we have 2  



Xj u, Λ2ε−2 X0 u 6 Xj u, Xj Λ2ε−2 X0 u + Xj u, gj Λ2ε−2 X0 u

2

2 2 6 2 kXj uk2 + Xj Λ2ε−2 X0 u 2 + gj Λ2ε−2 X0 u 2 (5.38)

2 2 2 6 2 kXj uk2 + Xj Λ2ε−2 X0 u 2 + kukH 2ε−1  

2 2 6 c |hLu, ui| + kuk2 + Xj Λ2ε−2 X0 u 2 . Again, by Lemma 5.51, setting T 2ε−1 = Λ2ε−2 X0 , we obtain 



 

Xj Λ2ε−2 X0 u 2 6 c LT 2ε−1 u, T 2ε−1 u + T 2ε−1 u 2 2 2  

 2 6 c LT 2ε−1 u, T 2ε−1 u + kuk2 . Since

(5.39)



    LT 2ε−1 u, T 2ε−1 u = T 2ε−1 Lu, T 2ε−1 u + L, T 2ε−1 u, T 2ε−1 u D E D  E
∗ ∗ = Lu, T 2ε−1 T 2ε−1 u + u, L, T 2ε−1 T 2ε−1 u
∗  ∗ and by Propositions 5.40, 5.46, T 2ε−1 T 2ε−1 and L, T 2ε−1 T 2ε−1 are respectively operators of order 4ε − 2 and 4ε − 1, by (5.38)-(5.39) we obtain, assuming now ε < 1/4,   2  Xj u, Λ2ε−2 X0 u 6 c kLuk2 + kuk2 + kuk2 4ε−2 + kuk2 4ε−1 2 2 H H   2 2 6 c kLuk2 + kuk2 which by (5.36) and (5.37) gives   2 2 2 kX0 ukH ε−1 6 c kLuk2 + kuk2 , completing the proof in the case ` (I) = 1.

224

H¨ ormander operators

Step 2: iterative estimate for the case ` (I) > 1. Let now ` (I) > 1 and write X[I] = Xj X[I 0 ] − X[I 0 ] Xj for suitable 0 6 j 6 q and ` (I 0 ) = ` (I) − 1. If 1 6 j 6 q, by (5.31) and Lemma 5.51 we obtain  



X[I] u 2 ε−1 6 c kXj uk2 + X[I 0 ] u 2 2ε−1 + kuk2 2 2 H H  

2 2 2

6 c kLuk2 + X[I 0 ] u H 2ε−1 + kuk2 . (5.40) For j = 0, we will show that the previous estimate holds in the modified form  



X[I] u 2 ε−1 6 c kLuk2 + X[I 0 ] u 2 4ε−1 + kuk2 . (5.41) 2 2 H H To this aim, let T 2ε−1 = Λ2ε−2 X[I] , so that T 2ε−1 is a simple operator of order 2ε − 1, then





X[I] u 2 ε−1 = X[I] u, Λ2ε−2 X[I] u H



 = X0 X[I 0 ] u, T 2ε−1 u − X[I 0 ] X0 u, T 2ε−1 u = A − B.

(5.42)

We start estimating A. It is immediate to check that for suitable functions ck and d in S (Rn ) we can write L∗ =

q X

Xk2 − X0 +

k=1

q X

ck Xk + d

k=1

so that ∗

X0 = −L +

q X

Xk2

k=1

+

q X

ck Xk + d.

k=1

Therefore q

 X

2  A = − L∗ X[I 0 ] u, T 2ε−1 u + Xk X[I 0 ] u, T 2ε−1 u k=1

+

q X



 ck Xk X[I 0 ] u, T 2ε−1 u + dX[I 0 ] u, T 2ε−1 u

k=1

= A1 + A2 + A3 + A4 . As to A3 , since



 ck Xk X[I 0 ] u, T 2ε−1 u = X[I 0 ] u, Xk∗ ck T 2ε−1 u arguing as in the proof of Theorem 5.49 (estimate of term A) we obtain  

 ck Xk X[I 0 ] u, T 2ε−1 u 6 c kXk uk2 + X[I 0 ] u 2 2ε−1 + kuk2 2 2 H

(5.43)

Hypoellipticity of general H¨ ormander operators

225

so that by Lemma 5.51  

2 2 2 |A3 | 6 c kLuk2 + X[I 0 ] u H 3ε−1 + kuk2 . Moreover, by Proposition 5.25 (iv),



 |A4 | = X[I 0 ] u, dT 2ε−1 u 6 X[I 0 ] u H 2ε−1 dT 2ε−1 u H −2ε+1  

2

2 6 c X[I 0 ] u H 2ε−1 kuk2 6 c X[I 0 ] u H 2ε−1 + kuk2 . As to A1 , by Lemma 5.53 q   X 2ε−1 L, T = Tk2ε−1 Xk + T02ε−1 k=1

where that

Tk2ε−1 , k

= 0, . . . q are suitable simple operators of order 2ε − 1. It follows



 |A1 | = X[I 0 ] u, LT 2ε−1 u q



 X 

 X[I 0 ] u, T 2ε−1 Xk u + X[I 0 ] u, T 2ε−1 u 6 X[I 0 ] u, T 2ε−1 Lu + 0 k k=1 q X





X[I 0 ] u 2ε−1 kXk uk + c X[I 0 ] u 2ε−1 kuk 6 X[I 0 ] u H 2ε−1 kLuk2 + c 2 2 H H k=1

6c



2 kLuk2



2 2 + X[I 0 ] u H 2ε−1 + kuk2 .

Next, we have, for some gk ∈ S (Rn ) 





A2 = Xk2 X[I 0 ] u, T 2ε−1 u = − Xk X[I 0 ] u, Xk T 2ε−1 u + Xk X[I 0 ] u, gk T 2ε−1 u





 = − Xk X[I 0 ] u, Xk T 2ε−1 u − X[I 0 ] u, Xk gk T 2ε−1 u + X[I 0 ] u, gk gk T 2ε−1 u = −A2,1 − A2,2 + A2,3 . Now,

 |A2,2 | = X[I 0 ] u, Xk gk T 2ε−1 u



   6 X[I 0 ] u, gk T 2ε−1 Xk u + X[I 0 ] u, Xk , gk T 2ε−1 u   By Proposition 5.40 the operator Xk , gk T 2ε−1 is of order 2ε − 1, hence by Lemma 5.51 (recall k = 1, 2, . . . , q) we have, since 2ε − 1 < 0,



|A2,2 | 6 c kXk uk2 X[I 0 ] u H 2ε−1 + c X[I 0 ] u H 2ε−1 kuk2  

2 2 2 6 c kuk2 + X[I 0 ] u H 2ε−1 + kLuk2 . Also  

2  2 |A2,3 | = X[I 0 ] u, gk gk T 2ε−1 u 6 c X[I 0 ] u H 2ε−1 + kuk2 .

226

H¨ ormander operators

Next, we have





   A2,1 = Xk X[I 0 ] u, Xk T 2ε−1 u = Xk X[I 0 ] u, T 2ε−1 Xk u + Xk X[I 0 ] u, Xk , T 2ε−1 u D E


∗    = T 2ε−1 Xk X[I 0 ] u, Xk u − X[I 0 ] u, Xk Xk , T 2ε−1 u

   + X[I 0 ] u, gk Xk , T 2ε−1 u = C1 − C2 + C3 . Now

  

    |C2 | 6 X[I 0 ] u, Xk , T 2ε−1 Xk u + X[I 0 ] u, Xk , Xk , T 2ε−1 u



6 c X[I 0 ] u H 2ε−1 kXk uk2 + c X[I 0 ] u H 2ε−1 kuk2  

2 2 2 6 c kuk2 + X[I 0 ] u H 2ε−1 + kLuk2 and  

2

2 |C3 | 6 c kuk2 + X[I 0 ] u H 2ε−1 . Let us consider D E Dh i E
∗
∗ |C1 | 6 Xk T 2ε−1 X[I 0 ] u, Xk u + T 2ε−1 , Xk X[I 0 ] u, Xk u

2

2
∗

2 6 Xk T 2ε−1 X[I 0 ] u + kXk uk2 + X[I 0 ] u H 2ε−1 . 2

By Lemma 5.51 we have

2
∗

Xk T 2ε−1 X[I 0 ] u 2  D

2  E

∗ 2ε−1
∗

2ε−1 2ε−1 ∗ X[I 0 ] u, T X[I 0 ] u + T X[I 0 ] u 6c L T 2  E  D

2

2ε−1 ∗ 2ε−1 ∗

X[I 0 ] u, T X[I 0 ] u + X[I 0 ] u H 2ε−1 . 6c L T Let us write E D E D
∗
∗
∗
∗ L T 2ε−1 X[I 0 ] u, T 2ε−1 X[I 0 ] u = T 2ε−1 X[I 0 ] Lu, T 2ε−1 X[I 0 ] u Dh i E
∗
∗ + L, T 2ε−1 X[I 0 ] u, T 2ε−1 X[I 0 ] u . The first term is bounded by

c kLuk2 X[I 0 ] u 4ε−1 while, for the second term, using again Lemma 5.53 we can write q h i X
∗ L, T 2ε−1 X[I 0 ] = Tk2ε Xk + T02ε k=1

Hypoellipticity of general H¨ ormander operators

227

(where Tk2ε , k = 0, . . . q are suitable simple operators of order 2ε) so that Dh E i
∗
∗ L, T 2ε−1 X[I 0 ] u, T 2ε−1 X[I 0 ] u 6

q D E E D X
∗
∗ 2ε Tk Xk u, T 2ε−1 X[I 0 ] u + T02ε u, T 2ε−1 X[I 0 ] u k=1 q X

6c



kXk uk2 X[I 0 ] u H 4ε−1 + c X[I 0 ] u H 4ε−1 kuk2

k=1

 

2 2 2 6 c X[I 0 ] u H 4ε−1 + kLuk2 + kuk2 . Collecting our inequalities we obtain (see (5.42)-(5.43))  

2 2 2 |A| 6 c X[I 0 ] u H 4ε−1 + kLuk2 + kuk2 . Let us consider now the term B in equation (5.42). Using X0 = L − we can write

 B = X[I 0 ] X0 u, T 2ε−1 u

Pq

j=1

Xj2 − b

q

 X



 = X[I 0 ] Lu, T 2ε−1 u − X[I 0 ] Xj2 u, T 2ε−1 u − X[I 0 ] bu, T 2ε−1 u j=1

≡ B1 −

q X

B2,k − B3 .

k=1

For the first term we have, for a suitable real function g ∈ S (Rn ),



 B1 = − Lu, X[I 0 ] T 2ε−1 u + Lu, gT 2ε−1 u



   = − Lu, T 2ε−1 X[I 0 ] u − Lu, X[I 0 ] , T 2ε−1 u

 + Lu, gT 2ε−1 u . Hence

|B1 | 6 c kLuk2 X[I 0 ] u H 2ε−1 + c kLuk2 kukH 2ε−1 + c kLuk2 kukH 2ε−1 . Since ε
0 as in Theorem 5.54 and let η1 , η2 ∈ C0∞ (Ω) such that η1 ≺ η2 . For every σ ∈ R there exists a constant c, such that for every u ∈ C0∞ (Ω) kη1 ukH σ+ε 6 C (kη2 LukH σ + kη2 ukH σ ) .

(5.48)

Proof. Let η 0 ∈ C0∞ (Ω) such that η1 ≺ η 0 ≺ η2 . Then kη1 ukH σ+ε = kΛσ η1 ukH ε 6 kη 0 Λσ η1 ukH ε + k(1 − η 0 ) Λσ η1 ukH ε .

(5.49)

By Proposition 5.47 we have k(1 − η 0 ) Λσ η1 ukH ε = k(1 − η 0 ) Λσ η1 η2 ukH ε 6 c kη2 ukH σ .

(5.50)

230

H¨ ormander operators

Applying Theorem 5.54 to η 0 Λσ η1 u (and then Corollary 5.32) we get kη 0 Λσ η1 ukH ε 6 c (kLη 0 Λσ η1 uk2 + kη 0 Λσ η1 uk2 ) 6 c (kη 0 Λσ η1 Luk2 + k[L, η 0 Λσ η1 ] uk2 + kη2 ukH σ ) 0

(5.51)

σ

6 c (kη2 LukH σ + k[L, η Λ η1 ] uk2 + kη2 ukH σ ) . Observe now that [L, η 0 Λσ η1 ] =

=

q X  2 0 σ  Xk , η Λ η1 + [X0 , η 0 Λσ η1 ] + [b, η 0 Λσ η1 ] k=1 q X

(5.52)

(2Xk [Xk , η 0 Λσ η1 ] + [[Xk , η 0 Λσ η1 ] , Xk ]) + [X0 , η 0 Λσ η1 ] + [b, η 0 Λσ η1 ] .

k=1

Since [[Xk , η 0 Λσ η1 ] , Xk ] u = [[Xk , η 0 Λσ η1 ] , Xk ] η2 u, [X0 , η 0 Λσ η1 ] u = [X0 , η 0 Λσ η1 ] η2 u, and [b, η 0 Λσ η1 ] u = [b, η 0 Λσ η1 ] η2 u, we easily obtain k[L, η 0 Λσ η1 ] uk2 6 2

q X

kXk [Xk , η 0 Λσ η1 ] uk2 + c kη2 ukH σ

(5.53)

k=1

Let now Tkσ = [Xk , η 0 Λσ η1 ], by Lemma 5.51 we obtain   2 2 kXk Tkσ uk2 6 c |hLTkσ u, Tkσ ui| + kTkσ uk2 .

(5.54)

To bound the first term in the right hand side of (5.54) we write LTkσ u = Tkσ Lu + [L, Tkσ ] u. Since Tkσ u = Tkσ η2 u we obtain |hLTkσ u, Tkσ ui| 6 |hTkσ Lu, Tkσ ui| + |h[L, Tkσ ] u, Tkσ ui| 6 c kη2 LukH σ kη2 ukH σ + |h[L, Tkσ ] u, Tkσ ui| . (5.55) 0 σ σ To estimate the last term in (5.55) we use (5.52) with η Λ η1 replaced by Tk . Then |h[L, Tkσ ] u, Tkσ ui| 62

q X j=1

6 6 6

|hXj [Xj , Tkσ ] u, Tkσ ui| +

q X

|h[[Xj , Tkσ ] , Xj ] u, Tkσ ui|

j=1

+ |h[X0 , Tkσ ] u, Tkσ ui| + |h[b, Tkσ ] u, Tkσ ui| q X

 [Xj , Tkσ ] u, Xj∗ Tkσ u + c kη2 uk2 σ 2 H j=1 q X 2 (|h[Xj , Tkσ ] u, Xj Tkσ ui| + |h[Xj , Tkσ ] u, cj Tkσ ui|) j=1 q X 2 2 kη2 ukH σ kXj Tkσ uk2 + c kη2 ukH σ . j=1

(5.56) 2

+ c kη2 ukH σ

Hypoellipticity of general H¨ ormander operators

Collecting (5.54), (5.55) and (5.56) we obtain  2

kXk Tkσ uk2 6 c kη2 LukH σ kη2 ukH σ + kη2 ukH σ

q X

231

 2

kXj Tkσ uk2 + kη2 ukH σ 

j=1

 2

2

6 c kη2 LukH σ + kη2 ukH σ

 1 2 2 +δ kXj Tkσ uk2 + kη2 ukH σ  . δ j=1 q X

For suitably small δ, this gives q   X 2 2 2 kXk Tkσ uk2 6 c kη2 LukH σ + kη2 ukH σ k=1

and by (5.49), (5.50), (5.51) and (5.53) we finally obtain   2 2 kη1 ukH σ+ε 6 c kη2 LukH σ + kη2 ukH σ .

We can now finally come to: Theorem 5.57 (Localized higher order subelliptic estimates) Let ε > 0 as in Theorem 5.54 and let η, η 0 ∈ C0∞ (Ω) satisfying η ≺ η 0 . For every σ, τ > 0, there exists a constant C > 0, such that for every u ∈ C0∞ (Ω) (5.57) kηukH σ+ε 6 C (kη 0 LukH σ + kη 0 ukH −τ ) . Remark 5.58 Let us note that the term kηk ukH −τ appearing in the right hand side of the above subelliptic estimate can be obviously bounded by kηk ukL2 or also, by Lemma 5.17, by kηk ukL1 , with a proper choice of τ . Although the estimate contained in this theorem is exactly the subelliptic estimate stated in the introduction, this theorem does not contain yet the full strength of the desired result, since for the moment (5.57) is proved only for u ∈ C0∞ (Ω). In the next section we will extend this to any distribution u such that the right hand side is finite. Proof. Let ηk ∈ C0∞ (Ω) be a sequence of cut-off functions such that η ≺ η1 ≺ η2 ≺ · · · ≺ η 0 . By Theorem 5.56 we have kηukH σ+ε 6 C (kη1 LukH σ + kη1 ukH σ ) . Applying once again Theorem 5.56 to the term kη1 ukH σ gives kη1 ukH σ 6 C (kη2 LukH σ−ε + kη2 ukH σ−ε ) 6 C (kη2 LukH σ + kη2 ukH σ−ε ) so that kηukH σ+ε 6 C (kη2 LukH σ + kη2 ukH σ−ε ) . Iterating the above reasoning gives kηukH σ+ε 6 Ck (kηk LukH σ + kηk ukH σ−(k−1)ε ) so that, for k large enough, kηukH σ+ε 6 C (kηk LukH σ + kηk ukH −τ ) .

232

H¨ ormander operators

Remark 5.59 Note that if we knew an priori estimate like kηukH σ 6 C (kη 0 LukH σ + kη 0 ukH −τ )

(5.58)

(i.e. the subelliptic estimate of any order σ with ε = 0), this will be enough to assure that u is smooth as soon as Lu is smooth, so one can wander what is the true importance of the ε-regularization contained in (5.57). However, the iterative reasoning of the above proof shows how the presence of this ε in the previous steps of the proof (Theorem 5.54 and Theorem 5.56) is crucial in order to obtain an estimate like (5.57), or even like (5.58). 5.7

Hypoellipticity of H¨ ormander operators

So far, we have proved the validity of subelliptic estimates only for smooth functions since in Theorem 5.57 we assume u ∈ C0∞ (Ω). Our aim now is to show that the estimates (5.57) actually hold for any distribution u ∈ D0 (Ω) such that the right hand side is finite. This not only immediately implies H¨ormander’s theorem, but makes these estimates a powerful tool for the subsequent development of the theory. However, to apply the previous estimates to distributions instead of smooth functions, we have preliminarily to extend the operators considered on S (Rn ) to distributions in S 0 (Rn ). Let T : S (Rn ) → S (Rn ) be a simple operator. Then we know that it can be transposed to another simple operator T ∗ . We can then define a linear operator T 0 on S 0 (Rn ) requiring that hT 0 u, ϕiS 0 = hu, T ∗ ϕiS 0 for every u ∈ S 0 (Rn ) , ϕ ∈ S (Rn ) .

(5.59)

From now on we remove the restriction consisting in assuming the elements of S (Rn ) and S 0 (Rn ) real valued. By Remark 5.42 if T is a simple operator we have, for ψ, ϕ ∈ S (Rn ) , Z Z hT ψ, ϕiS 0 = (T ψ) ϕ = ψT ∗ ϕ = hψ, T ∗ ϕiS 0 = hT 0 ψ, ϕiS 0 . Hence the new operator T 0 coincides with T on S (Rn ) and we can conclude the following Proposition 5.60 If T : S (Rn ) → S (Rn ) is a simple operator, then it can be extended as an operator T 0 : S 0 (Rn ) → S 0 (Rn ) by (5.59). We will say that T 0 is an extension of T to distributions and from now on we will use the same symbol T to denote the original operator on S (Rn ) and its extension to S 0 (Rn ). Revising the terminology introduced in section 5.4, we define the concept of operator of order m acting on tempered distributions: Definition 5.61 (Operators of order m on S 0 (Rn )) We say that a linear operator T : S 0 (Rn ) → S 0 (Rn ) is of order m if, for every σ ∈ R, it can be restricted to a linear continuous operator T : H σ+m (Rn ) → H σ (Rn ).

Hypoellipticity of general H¨ ormander operators

233

In the next proposition we show that if the operator T is of order m on S (Rn ) then it is also of order m on S 0 (Rn ). Proposition 5.62 Let T : S (Rn ) → S (Rn ) be a simple operator of order m. Then, for every σ ∈ R, its extension T : S 0 (Rn ) → S 0 (Rn ), can be restricted to a linear continuous operator T : H σ+m (Rn ) → H σ (Rn ) such that for every u ∈ H σ (Rn ), kT ukH σ 6 cσ kukH σ+m . Moreover, the constants cσ are the same for the original operator and its extension. Proof. Let us fix σ ∈ R and let v ∈ H σ+m . Note that T v and Tcv are in general tempered distributions. We have to show that T v ∈ H σ (Rn ) and kT vkH σ 6 c kvkH σ+m . More precisely, we have to show that Tcv is actually a function satisfying Z Z 2 c 2σ 2 2(σ+m) v (ξ)| hξi dξ. T v (ξ) hξi dξ 6 c |b To see this let us fix ϕ ∈ S (Rn ) and let ψ ∈ S (Rn ) such that ψb = T ∗ ϕ. b Then E D E D Tcv, ϕ 0 = hT v, ϕi b S 0 = hv, T ∗ ϕi b S 0 = v, ψb 0 = hb v , ψiS 0 . (5.60) S

σ+m

S

n

Now, since v ∈ H (R ) we can write Z σ+m −σ−m ψ (ξ) hξi dξ |hb v , ψiS 0 | = vb (ξ) hξi Z 1/2 Z 1/2 2 2(σ+m) 2 −2(σ+m) 6 |b v (ξ)| hξi dξ |ψ (ξ)| hξi dξ Z = kvkH σ+m

2

−2(σ+m)

|ψ (ξ)| hξi

1/2 dξ

.

Since, by Proposition 5.46, T ∗ has order m we obtain Z Z b 2 −2σ 2 −2(σ+m) 2 ∗ |ψ (ξ)| hξi dξ = kT ϕk b H −σ−m 6 c ϕ b (ξ) hξi Z 2 −2σ = c |ϕ (ξ)| hξi dξ. Using (5.60) and (5.61) gives, for every ϕ ∈ S (Rn ) Z 1/2 D E c 2 −2σ |ϕ (ξ)| hξi dξ . T v, ϕ 6 c kvkH σ+m S0

2σ

In particular for any g ∈ S (Rn ), letting ϕ (ξ) = hξi g (ξ) we obtain Z 1/2 D E c 2σ 2 4σ −2σ |g (ξ)| hξi hξi dξ T v, h·i g 0 6 c kvkH σ+m S

Z = c kvkH σ+m

2

2σ

|g (ξ)| hξi

1/2 dξ

.

(5.61)

234

H¨ ormander operators

D E 2σ This means that the map g → Tcv, h·i g 0 is a linear continuous functional on S    2σ 2σ L2 hξi dξ , which implies that Tcv (ξ) ∈ L2 hξi dξ , Z 2 c 2σ (5.62) T v (ξ) hξi dξ 6 c kvkH σ+m and that D

2σ Tcv, h·i g

E S0

Z =

2σ Tcv (ξ) g (ξ) hξi dξ

D E 2σ for every g ∈ S (Rn ). In particular for ϕ = h·i g we have Tcv, ϕ 0 = S R Tcv (ξ) ϕ (ξ) dξ. This means that the distribution Tcv is actually a function and by (5.62) we have kT vkH σ 6 c kvkH σ+m . Lemma 5.63 Let u be a distribution in Ω and let ζ ∈ C0∞ (Ω) then uζ ∈ H −m for some m > 0. Proof. By Theorem A.29 (see the Appendix to the book) every distribution is locally the derivative of a compactly supported continuous function. Therefore we can write uζ = Dα g where g ∈ C0 (Ω) and α is a suitable multiindex α. If follows c = (2πi)|α| gb (ξ) ξ α . From this one can easily obtain uζ ∈ H −m for some that uζ m > 0. Theorem 5.64 (Hypoellipticity of H¨ ormander operators) Let u, f be distributions in Ω such that Lu = f . Assume that f is smooth in an open subset V of Ω. Then u is smooth in V . Also, the subelliptic estimates of Theorem 5.57 hold whenever u ∈ D0 (Ω) and the right hand side of (5.57) is finite. Remark 5.65 The smoothness of distributional solutions to Lu = f (with f smooth) also implies that these solutions are actually classical solutions. Indeed, if u, f are smooth in V and u is a distributional solution to the equation Lu = f in V , then for every φ ∈ C0∞ (V ) Z Z Z (Lu) φ = uL∗ φ = f φ, V

V

V

which implies that Lu (x) = f (x) for every x ∈ V . To prove the above theorem we need the following lemma in which we establish a basic regularization estimate that we will then iterate to obtain the final result. Lemma 5.66 Let η1 , η2 ∈ C0∞ (Ω) such that η1 ≺ η2 and let σ ∈ R. Then if u is a distribution such that η2 u ∈ H σ and η2 Lu ∈ H σ then η1 u ∈ H σ+ε and kη1 ukH σ+ε 6 C (kη2 LukH σ + kη2 ukH σ ) , where ε is the same constants appearing in the subelliptic estimate (5.48) and C is independent of u.

Hypoellipticity of general H¨ ormander operators

235

Proof. Let Sδ be the mollification operator (see Definition 5.19). Since for suitably small δ > 0, Sδ η1 u ∈ C0∞ (Ω), by Theorem 5.56 we have kSδ η1 ukH σ+ε 6 C (kLSδ η1 ukH σ + kSδ η1 ukH σ ) . Observe now that kLSδ η1 ukH σ 6 kSδ η1 LukH σ + k[L, Sδ η1 ] ukH σ 6 kSδ η1 LukH σ

(5.63)

q X

 2 

Xj , Sδ η1 u +

Hσ

+ k[X0 , Sδ η1 ] ukH σ + k[b, Sδ η1 ] ukH σ

k=1

By Lemma 5.21 we have kSδ η1 LukH σ 6 kη1 LukH σ . Also, using Proposition 5.37 (which by Proposition 5.62 still holds for u ∈ H σ ), we obtain k[X0 , Sδ η1 ] ukH σ 6 C kη2 ukH σ with C independent of δ. As for the second term we can write

 2 

Xj , Sδ η1 u σ 6 k[Xj , Sδ η1 ] Xj uk σ + k[Xj , [Xj , Sδ η1 ]] uk σ H H H and applying again Proposition 5.37 we obtain k[Xj , [Xj , Sδ η1 ]] ukH σ 6 C kη2 ukH σ . Also

2 k[Xj , Sδ η1 ] Xj ukH σ 6 C Xj (η2 ) u

Hσ

2

(note that since η1 ≺ η2 we also have η1 ≺ (η2 ) ) and



2 2 = Λσ Xj (η2 ) u 6 kXj η2 Λσ η2 uk2 + k[Λσ , Xj η2 ] η2 uk2

Xj (η2 ) u Hσ

2

6 c kXj η2 Λσ η2 uk2 + c kη2 ukH σ . As for the fourth term in (5.63) by Lemma 5.21 we have k[b, Sδ η1 ] ukH σ 6 kbSδ η1 ukH σ + kSδ η1 bukH σ 6 c kη1 ukH σ . Collecting all the above computation we obtain   q X kSδ η1 ukH σ+ε 6 C kη1 LukH σ + kXj η2 Λσ η2 uk2 + kη2 ukH σ  .

(5.64)

j=1

Let T σ u = η2 Λσ η2 u. We are going to show that kXj T σ uk2 6 c (kη2 LukH σ + kη2 ukH σ ) .

(5.65)

In view of Lemma 5.21 it is enough to show that for suitable small δ > 0 kSδ Xj T σ uk2 6 c (kη2 LukH σ + kη2 ukH σ ) .

(5.66)

Let ζ ∈ C0∞ (Ω) be such that η2 ≺ ζ. Since Xj T σ u = ζXj T σ u we can write Sδ Xj T σ u = Sδ ζXj T σ u = Xj Sδ ζT σ u + [Sδ ζ, Xj ] T σ u.

(5.67)

236

H¨ ormander operators

The second term can by bounded using (5.12), k[Sδ ζ, Xj ] T σ uk2 6 c kT σ uk2 6 c kη2 ukH σ .

(5.68)

To bound the first term we are going to repeat the computation made in the proof of Theorem 5.56 to control the term Xk Tkσ u, replacing Tkσ u with Sδ T σ u (which is a smooth function). Applying Lemma 5.51 we have   2 2 (5.69) kXj Sδ T σ uk2 6 c |hLSδ T σ u, Sδ T σ ui| + kSδ T σ uk2 . Since LSδ T σ u = Sδ T σ Lu + [L, Sδ T σ ] u, we have |hLSδ T σ u, Sδ T σ ui| 6 kSδ T σ Luk2 kSδ T σ uk2 + |h[L, Sδ T σ ] u, Sδ T σ ui| 6 c kη2 LukH σ kη2 ukH σ + |h[L, Sδ T σ ] u, Sδ T σ ui| .

(5.70)

Also, σ

[L, Sδ T ] =

q X

(2Xj [Xj , Sδ T σ ] + [[Xj , Sδ T σ ] , Xj ]) + [X0 , Sδ T σ ] + [b, Sδ T σ ]

j=1

and therefore |h[L, Sδ T σ ] u, Sδ T σ ui| 62

q X

|hXj [Xj , Sδ T σ ] u, Sδ T σ ui| +

j=1

q X

|h[[Xj , Sδ T σ ] , Xj ] u, Sδ T σ ui|

j=1 σ

+ |h[X0 , Sδ T ] u, Sδ T ui| + |h[b, Sδ T σ ] u, Sδ T σ ui| 62

q X

σ

(5.71)



 [Xj , Sδ T σ ] u, Xj∗ Sδ T σ u + c k[[Xj , Sδ T σ ] , Xj ] uk2 2

j=1 2

2

2

+ k[X0 , Sδ T σ ] uk2 + k[b, Sδ T σ ] uk2 + c kη2 ukH σ 2

The term k[b, Sδ T σ ] uk2 can be trivially bounded k[b, Sδ T σ ] uk2 = kbSδ T σ uk2 + kSδ T σ buk2 6 c kη2 ukH σ . The terms k[[Xj , Sδ T σ ] , Xj ] uk2 and k[X0 , Sδ T σ ] uk2 can also be bounded by c kη2 ukH σ by (5.14) and (5.15) (together with Proposition 5.62). Hence |h[L, Sδ T σ ] u, Sδ T σ ui| 62

q X

2

(|h[Xj , Sδ T σ ] u, Xj Sδ T σ ui| + |h[Xj , Sδ T σ ] u, cj Sδ T σ ui|) + c kη2 ukH σ

j=1

6c

q X j=1

(5.72) σ

σ

σ




k[Xj , Sδ T ] uk2 kXj Sδ T uk2 + kSδ T uk2 +

2 c kη2 ukH σ

.

Hypoellipticity of general H¨ ormander operators

237

Since by (5.14), k[Xj , Sδ T σ ] uk2 6 c kη2 ukH σ using (5.69) and (5.72) we conclude that 2

kXj Sδ T σ uk2 6 c kη2 LukH σ kη2 ukH σ q X

+c


2 kη2 ukH σ kXj Sδ T σ uk2 + kSδ T σ uk2 + c kη2 ukH σ

j=1

6c



2 kη2 LukH σ

+

2 kη2 ukH σ



+ε

q X

2

2

kXj Sδ T σ uk2 + c (ε) kη2 ukH σ

j=1

so that q X

  2 2 2 kXj Sδ T σ uk2 6 c kη2 LukH σ + kη2 ukH σ

j=1

which together with (5.67) and (5.68) gives (5.66) and then (5.65). Inserted in (5.64), this gives kSδ η1 ukH σ+ε 6 C (kη2 LukH σ + kη2 ukH σ ) which by Lemma 5.21 implies that η1 u ∈ H σ+ε and kη1 ukH σ+ε 6 C (kη2 LukH σ + kη2 ukH σ ) .

Proof of Theorem 5.64. We start establishing the subelliptic estimates in D0 (Ω). Let ζ, ζ 0 ∈ C0∞ (Ω) such that ζ ≺ ζ 0 and let σ, m > 0. We want to show that if u is a distribution such that ζ 0 Lu ∈ H σ and ζ 0 u ∈ H −m then ζu ∈ H σ+ε and kζukH σ+ε 6 C (kζ 0 LukH σ + kζ 0 ukH −m ) with C independent of u. Let ε0 6 ε such that k ≡ (m + σ) /ε0 + 1 is an integer (the reason for this choice will be apparent later) and let ζ1 , ζ2 , . . . , ζk+1 ∈ C0∞ (Ω) be cutoff functions such that ζ 0 = ζ1  ζ2  · · ·  ζk+1 = ζ. Since ζ1 Lu ∈ H σ ⊂ H −m and ζ1 u ∈ H −m by 0 Lemma 5.66 we have ζ2 u ∈ H −m+ε (observe that the lemma clearly works with ε replaced by ε0 6 ε) and kζ2 ukH −m+ε0 6 C (kζ1 LukH −m + kζ1 ukH −m ) . 0

Iterating the above argument k − 1 times we obtain ζk u ∈ H −m+ε (k−1) and kζk ukH −m+ε0 (k−1) 6 C (kζk−1 LukH −m+ε(k−2) + kζk−1 ukH −m+ε(k−2) ) 6 C (kζk−1 LukH −m+ε(k−2) + kζ1 ukH −m ) . Letting σ = −m + ε0 (k − 1) we can now apply once again Lemma 5.66, so that ζu = ζk+1 u ∈ H σ+ε and kζukH σ+ε 6 C (kζk LukH σ + kζ1 ukH −m ) 6 C (kζ 0 LukH σ + kζ 0 ukH −m ) .

238

H¨ ormander operators

We have therefore established the validity of subelliptic estimates for every distribution u such that the right hand side is finite. In particular, if ζ 0 Lu is smooth we can take σ as large as we want, and ζu is smooth. If u ∈ D0 (Ω) and Lu = f with f smooth in an open set V ⊂ Ω, for every x0 ∈ V we can take two neighborhoods of x0 , U1 (x0 ) b U2 (x0 ) b V and two cutoff functions ζ, ζ 0 ∈ C0∞ (Ω) such that ζ ≺ ζ 0 , ζ = 1 in U1 (x0 ) and supp ζ 0 ⊂ U2 (x0 ). Then ζ 0 Lu is smooth, so ζu is smooth, that is u is smooth in U (x0 ), and by the genericity of x0 and U (x0 ), u is smooth in V . This shows that L is hypoelliptic. 5.8

Uniform subelliptic estimates

The subelliptic estimates proved for general H¨ormander operators in this chapter (and for sublaplacians on Carnot groups in Chapter 4) are a useful tool in several matters, since they offer a quantitative control on any desired degree of regularity on an assigned function u, in terms of Lu and some very weak norm of u itself. The constant that appears in estimate (5.57) will depend on the domain Ω, the order of the derivative that we want to control, and obviously on the operator L. Here we want to investigate more explicitly the dependence of the constant on L. Actually, in some applications of the general theory, a family of operators Lα is involved, and a crucial information is the validity of some uniform (with respect to α) regularity estimate, a fact that can be assured only knowing how the constant depends on the operator. We start with the following definition. Definition 5.67 Let us fix a domain Ω ⊆ Rn , a positive integer q and a step s. Let K ∈ N and κ0 , κ1 > 0. We will say that a system of H¨ ormander vector fields ∞ X0 , X1 , . . . , Xq in Ω and a function b ∈ C (Ω) are in the class Λ (K, κ0 , κ1 ) = Λ (Ω, K, κ0 , κ1 ) if



det X[I ] , X[I ] , . . . , X[I ] > κ0 , inf max (5.73) 1 2 n x x x x∈Ω |I1 |,|I2 |,...,|In |6s

and, for i = 0, 1, . . . , q, j = 1, . . . , n, sup sup |Dα bij (x)| + sup sup |Dα b (x)| 6 κ1 , |α|6K x∈Ω

(5.74)

|α|6K x∈Ω

where bij are the coefficients of the vector fields. With a small abuse of notation we Pq will also say that the operator L = j=1 Xj2 + X0 + b is in the class Λ (Ω, K, κ0 , κ1 ). Theorem 5.68 Let us fix a domain Ω ⊂ Rn , a positive integer q, two cut-off functions η, η 0 ∈ C0∞ (Ω) such that η ≺ η 0 and a step s. There exist ε > 0 and K ∈ N such that for every choice of positive constants κ0 , κ1 , σ, τ there exists C > 0 such that for every operator L in the class Λ (K, κ0 , κ1 ) and every u ∈ D0 (Ω) such that η 0 Lu ∈ H σ and η 0 u ∈ H −τ we have kηukH σ+ε 6 C (kη 0 LukH σ + kη 0 ukH −τ ) .

(5.75)

Hypoellipticity of general H¨ ormander operators

239

Proof. First of all observe that arguing as in §5.5 we can assume that the operators in the class Λ (K, κ0 , κ1 ) are constructed using vector fields that are defined on the whole Rn , have coefficients in S (Rn ) and that satisfy H¨ormander’s condition in Ω. For a single operator L the estimate (5.75) has been proved in Theorem 5.56. The fact that the constant C can be uniformly bounded in the class Λ (K, κ0 , κ1 ) comes from a careful reading of the proofs of the previous sections. In particular, as already remarked (see Remark 5.50) the bound (5.73) guarantees that Theorem 5.49 holds uniformly. Note that the proof of Theorem 5.49 is the only one that depends explicitly on H¨ ormander’s condition. All the other constants involved in the results of the previous sections depends on a bound for a finite number of derivatives of the coefficients of the vector fields and of b. These bounds come from (5.74). We will now extend Theorem 5.68 to a slightly more general class of operators. This will be particularly useful when dealing with families of H¨ormander operators on groups, as will appear in Chapter 6. Proposition 5.69 Let us consider a system of H¨ ormander vector fields X0 , X1 , . . . , Xq in Ω and a function b ∈ C0∞ (Ω) in the class Λ (K, κ0 , κ1 ), and let s be q the step of H¨ ormander’s condition. If {aij }i,j=1 , {cij }16i r − 1, by inductive assumption, while H Z 

2   2 −m

2 α0 1 + |ξ| Dα0 fk ) (ξ) dξ

∂xj φ2 D fk −m = 2πiξj (φ2\ H Z  2 −m+1 \ 2 0 6c 1 + |ξ| (φ2 Dα fk ) (ξ) dξ

2 0

= c φ2 Dα fk −(m−1) → 0 H

by the inductive assumption, because m − 1 > r − 1 since m > r.

244

H¨ ormander operators

This shows that for some m > 0 we have kφ2 (uk − u)kH −m → 0, so we are done.

A classical theorem by Montel states that a family of holomorphic functions that is uniformly bounded on compact sets is a normal family: every sequence of functions in such family contains a subsequence that converges uniformly on compact sets. The following theorem generalizes this kind of result to a family of solutions of Lu = 0. ∞

Theorem 5.73 Let L be a H¨ ormander operator in Ω and let {uk }k=1 ⊂ D0 (Ω) be a sequence of (distributional) solutions to the equation Lv = 0, such that for every compact set K ⊂ Ω Z 2 |uk (x)| dx < +∞. (5.78) sup k

K

Then there exists a subsequence ukj and a classical solution u to Lv = 0 such that for any multiindex α and compact K ⊂ Ω, Dα ukj → Dα u uniformly in K. Proof. Let us fix a sequence of bounded open domains Ωn such that Ωn ⊂ Ωn+1 +∞ [ Ωn = Ω. We will construct the subsequence ukj by means of an iterative and n=1

process. At the first step we consider φ1 ≺ φ01 ∈ C0∞ (Ω) such that φ1 ≡ 1 on Ω1 . By Theorem 5.57, for positive integers h, m and suitable numbers ε, c > 0, we can write kφ1 uk kH h+ε 6 c kφ01 uk kH −m 6 c kφ01 uk k2 (recall that Luk = 0). Since the last term is uniformly bounded by (5.78) and since for h large enough the Sobolev
space H h+ε is compactly embedded into the space of continuous function C Ω1 (see e.g. [1, Theorem 6.3]), we can extract n o∞ (1) ∞ from {uk }k=1 a subsequence uk that converges uniformly to a continuous k=1
v1 ∈ C Ω1 . Iterating this argument, for every n, we can fix a couple of cutoff again Theorem 5.57. functions φn ≺ φ0n ∈ C0∞ (Ω)nsuch that o φn ≡ 1 on Ωn and apply n o This allows to extract from

(n−1)

∞

(n)

∞

a subsequence uk that converges k=1 k=1
uniformly on Ωn to a continuous function vn ∈ C Ωn . Observe that clearly vn coincides with vn−1 on Ωn−1 . In this way we can define a function w ∈ C (Ω) that (k) when restricted to Ωn coincides with vn . For n every ok = 1, . . . let also set wk = uk . ∞

uk

(n)

By construction {wk }k=1 is a subsequence of uk ∞ {uk }k=1 .

∞ {wk }k=1

∞

for every n and therefore a

k=1

subsequence of It follows that converges uniformly in every Ωn to w and therefore on every compact subset K ⊂ Ω. The thesis follows applying ∞ Theorem 5.71 to {wk }k=1 .

Hypoellipticity of general H¨ ormander operators

5.10

245

Notes

A basic reference for the material of section 5.2 is the book of Stein and Weiss [151, Chap. 1]. Fractional Sobolev spaces, defined via Fourier transform as in section 5.3, are dealt for instance in the book by F. Demengel, G. Demengel [74, Chap. 4]. See also the book by Rauch [141, Chap. 2], which deals with the material in both these sections. As already noted in Chapter 4, the original proof of the hypoellipticity theorem for H¨ ormander operator is due to H¨ ormander, [107]. The most commonly reported analytical proof of H¨ ormander’s theorem (here presented in sections 5.5-5.7) is the one based on subelliptic estimates and pseudodifferential operators. It is due to Kohn [117], and independently to Oleinik and Radkevic [137]. Our presentation, avoiding the explicit use of the general notion of pseudodifferential operator, goes along the line followed in the book by Chen-Shaw [58, §8.1-8.2]. The book by Oleinik and Radkevic [137] contains also some extension of H¨ ormander’s theorem, consisting in allowing H¨ormander’s condition to be violated in some lower dimensional subset of Ω, without losing the hypoellipticity of the operator. For the precise assumptions we refer to the book [137, Chap. II, §5]. A completely different proof of H¨ ormander’s theorem existing in the literature is the probabilistic one, based on Malliavin’s calculus, due to Malliavin [124] (see also the exposition on Malliavin’s calculus and the proof of H¨ormander’s theorem contained in the book by Nualart [136, §2.3]). The conclusion of Theorem 5.71 in section 5.9 actually holds whenever both L and L∗ are hypoelliptic (not necessarily H¨ ormander operators). See the discussion in the book by Tr`eves in [155, Chap. 52]. For a comprehensive treatment of Montel theorem and normal families see the book [147] by Schiff.

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Chapter 6

Fundamental solutions of H¨ ormander operators

Fundamental solutions are a key tool in the study of H¨ormander operators, especially in presence of an underlying group structure which makes the operator left invariant and 2-homogeneous. Such operators admit a unique global, left invariant and homogeneous fundamental solution, enjoying properties which are strictly analogous to those of the fundamental solution for the classical Laplacian, although an explicit form of this fundamental solution is generally unknown. This function will be the key tool to prove sharp a priori estimates for solutions to Lu = f (Chapter 8), and is a powerful tool for many other purposes (see for instance the applications described in [16, Chap. 5]). The construction of this fundamental solution and the study of its properties, famous results due to Folland [85], are the main theme of this chapter. Before attacking this problem, however, we will consider (section 6.1) a general H¨ ormander operator (i.e. without an underlying group structure) and will prove that, under the quite weak assumption of not total degeneration, on any bounded domain it admits a (one sided) fundamental solution (Theorem 6.3). This fact will be established as a consequence of the weak maximum principle for linear second order operators with nonnegative characteristic form, proved in Chapter 1, and the hypoellipticity of H¨ ormander operators, proved in Chapter 5. A consequence of this existence result is that every H¨ ormander operator is solvable on any bounded domain Ω, that is the equation Lu = f admits a classical, smooth solution in Ω for any f ∈ C0∞ (Ω) (Theorem 6.4). Then we will turn to the study of homogeneous H¨ormander operators on homogeneous groups. After defining precisely, in section 6.2, the class of operators that we will study, in section 6.3 we will prove in this context the existence of a global homogeneous fundamental solution (Theorem 6.18). The argument of the proof consists in considering the local fundamental solution whose existence has been previously granted for any H¨ ormander operator, and using properly the dilations on the group, extending it to a global homogeneous fundamental solution. This proof also exploits once more the hypoellipticity of the operator. In section 6.4 we will prove the main properties of this fundamental solution (Theorem 6.20) and the representation formula for second order derivatives of solutions to the equation 247

248

H¨ ormander operators

Lu = f (Theorem 6.32). These results, and others related to them and contained in section 6.4, will be fundamental tools in the proof of a priori estimates that we will carry out in Chapter 8 for operators on homogeneous groups, and in Chapter 11 for general H¨ ormander operators. Finally, in section 6.5 we will present a few examples of homogeneous groups where an explicit simple form of the homogeneous fundamental solution is known. We remark that if one is only interested in the results about fundamental solutions of sublaplacians on Carnot groups, then the theory presented in this chapter can be considered independent from the subelliptic estimates of Chapter 5, and founded instead on the hypoellipticity result proved in Chapter 4. A relevant feature of the existence result for homogeneous fundamental solutions that we prove in section 6.3 is that we keep under control some constants, getting uniform bounds on the derivatives of any order of the fundamental solutions relative to suitable families of H¨ ormander operators. This is a useful and remarkable fact, in view of the lacking of an explicit formula assigning these fundamental solutions. We will exploit this result in Chapter 12, when proving a priori estimates for nonvariational operators with nonsmooth coefficients, structured on H¨ormander vector fields. These uniform estimates on the fundamental solution rely on the uniform subelliptic estimates established in Chapter 5. 6.1

Fundamental solutions and solvability of general H¨ ormander operators

Let Ω be a bounded domain in Rn and let L=

q X

Xj2 + X0 =

j=1

n X

ajk (x) ∂x2j xk +

j,k=1

n X

bk (x) ∂xk

k=1

be a H¨ ormander operator in Ω. We will apply to L the weak maximum principle proved in Chapter 1 (Theorem 1.57), hence we will assume that L is uniformly not totally degenerate, that is, for some j ∈ {1, . . . , n} and some constant c0 > 0, ajj (x) > c0 in Ω.

(6.1)

We recall that, in terms of the vector fields Xi =

n X

bij (x) ∂xj ,

(6.2)

j=1

condition (6.1) rewrites as q X

2

bij (x) > c0 > 0 in Ω.

i=1

Under this assumption we can establish the following a priori estimate:

(6.3)

Fundamental solutions of H¨ ormander operators

249

Pq Proposition 6.1 Let L = j=1 Xj2 + X0 satisfying (6.1) (or (6.3)) in a bounded domain Ω ⊂ Rn with Xi as in (6.2) and bij smooth in Ω. There exists a constant C = CΩ,L such that for every f ∈ C02 (Ω) sup |f | 6 CΩ,L sup |Lf | . Ω

Ω

Proof. Without loss of generality we can assume that a11 (x) =

q X

2

bi1 (x) > c0 > 0

i=1

and Ω b {x ∈ Rn : −d < x1 < d}. Let g (x) = e2γd − eγ(x1 +d) with γ > 0 to be fixed later (note that g (x) > 0 in Ω). Let us compute Xi g (x) = −γeγ(x1 +d) bi1 (x) n   X Xi2 g (x) = bij (x) ∂xj −γeγ(x1 +d) bi1 (x) j=1 n X

2

= −γ 2 eγ(x1 +d) bi1 (x) +

  bij (x) −γeγ(x1 +d) ∂xj bi1 (x)

j=1

 Lg (x) = −γeγ(x1 +d) γ

q X

2

bi1 (x) +

q X n X

 bij (x) ∂xj bi1 (x) + b01 (x) .

i=1 j=1

i=1

Now, let X n q X bij (x) ∂xj bi1 (x) + b01 (x) . β ≡ max x∈Ω i=1 j=1 Pq 2 Since i=1 bi1 (x) > c0 and eγ(x1 +d) > 1, for γ > 1+β c0 we have Lg 6 −γ (γc0 − β), and choosing γ = max (1, (1 + β) /c0 ) we conclude Lg 6 −1 in Ω. Let now f ∈ C02 (Ω) and h (x) = f (x) − g (x) max |Lf | , Ω

then Lh (x) = Lf (x) − Lg (x) max |Lf | > Lf (x) + max |Lf (x)| > 0. Ω

Ω

Since on ∂Ω we have h (x) = −g (x) maxΩ |Lf | 6 0, by the weak maximum principle (Theorem 1.57) we have h (x) 6 0 in Ω, that is f (x) 6 g (x) max |Lf | 6 sup g · max |Lf | . Ω

Ω

Ω

The same argument applied to −f gives −f (x) 6 sup g · max |Lf | Ω

Ω

250

H¨ ormander operators

and therefore sup |f | 6 CΩ,L sup |Lf | Ω

with CΩ,L = e

2γd

Ω

and d, γ as above.

The previous proposition is useful in view of the following abstract result: Proposition 6.2 Let L be any linear differential operator with smooth coefficients in Ω ⊂ Rn . Assume there exists a positive constant C such that for every ϕ ∈ C0∞ (Ω) the transpose operator L∗ satisfies sup |ϕ| 6 C sup |L∗ ϕ| . Ω

(6.4)

Ω

Then, for every Radon measure ν on Ω there exists a Radon measure µ with |µ| (Ω) 6 C |ν| (Ω) such that Lµ = ν in the distributional sense, that is Z Z L∗ φdµ = φdν for every φ ∈ C0∞ (Ω) . Ω

Ω

Proof. Let V = L∗ (C0∞ (Ω)) = {L∗ ϕ : ϕ ∈ C0∞ (Ω)}. By (6.4) L∗ is injective on C0∞ (Ω) and therefore there is a one-to-one correspondence between V and C0∞ (Ω). Using this fact we can define on V a linear functional Λ, setting for every g = L∗ ϕ, Z Λg = ϕdν. Ω

Since Z |Λg| = ϕdν 6 sup |ϕ| |ν| (Ω) 6 C sup |L∗ ϕ| |ν| (Ω) = C |ν| (Ω) sup |g| , Ω Ω

Ω

Ω

Λ is linear continuous on V, which we regard as a subspace of C∗ (Ω) (continu
ous functions on Ω “vanishing at infinity”, which just coincides with C Ω if Ω is bounded). Using Hahn-Banach theorem (see e.g. [42, Thm. 1.1, p. 1]) we can extend Λ to a linear continuous functional on C∗ (Ω) and by the Riesz representation theorem (see e.g. [89, Thm. (7.17)]) there exists a Radon measure µ, satisfying |µ| (Ω) 6 C |ν| (Ω), such that for every g ∈ C∗ (Ω) we have Z Λg = gdµ. Ω

C0∞

In particular, for every ϕ ∈ (Ω) , letting g = L∗ ϕ we have Z Z Z hν, ϕi = ϕdν = Λg = gdµ = L∗ ϕdµ = hµ, L∗ ϕi = hLµ, ϕi . Ω

Ω

Ω

Fundamental solutions of H¨ ormander operators

251

Theorem 6.3 (Fundamental solution on a bounded domain) Let Ω be a bounded domain in Rn and let q n n X X X 2 2 L= Xj + X0 = ajk (x) ∂xj xk + bk (x) ∂xk j=1

j,k=1

k=1

be a H¨ ormander operator satisfying (6.1) (or (6.3)) in Ω, with vector fields Xi smooth up to the boundary of Ω. Then for every x ∈ Ω there exists a smooth L1 (Ω) function γ (x, ·) : Ω \ {x} → R, γ (x, ·) ∈ L1 (Ω), such that Lγ (x, ·) = δx in Ω in the distributional sense, that is Z φ (x) = γ (x, y) L∗ φ (y) dy for every φ ∈ C0∞ (Ω) . Ω

Moreover Z |γ (x, y)| dy 6 CΩ,L

sup x∈Ω

Ω

with CΩ,L as in Proposition 6.1. Proof. Note that L and L∗ , when written in the form n X ajk (x) ∂x2j xk + . . . , j,k=1

have the same coefficients ajk , hence if L satisfies (6.1) in Ω, the same is true for L∗ . By Propositions 6.1 and 6.2, for any fixed x ∈ Ω, letting ν = δx , there exists a Radon measure µx with |µx | (Ω) 6 CΩ,L such that Lµx = δx . Since Lµx = 0 in Ω \ {x} and L is hypoelliptic (by H¨ ormander’s theorem, see Theorem 5.64), the measure µx coincides in Ω\{x} (in the sense of distributions) with a smooth function γ (x, ·) : Ω \ {x} → R. This means that for every test function ϕ ∈ C0∞ (Ω \ {x}) we have Z Z ϕ (y) dµx (y) = ϕ (y) γ (x, y) dy. Moreover kγ (x, ·)kL1 (Ω) 6 |µx | (Ω \ {x}) 6 CΩ,L .

(6.5)

Since the measure µx − γ (x, ·) dy is supported on {x} it is a multiple of a delta measure, that is, µx = γ (x, ·) dy + α (x) δx for a suitable number α (x). Since Lµx = δx , for every ϕ ∈ C0∞ (Ω) we have Z Z ∗ ϕ (x) = L ϕ (y) dµ (y) = L∗ ϕ (y) γ (x, y) dy + α (x) L∗ ϕ (x) . Ω

Ω

To conclude the proof it is enough to show that α (x) ≡ 0. Recall that L∗ =

n X j,k=1

ajk (x) ∂x2j xk +

n X k=1

ck (x) ∂xk + d (x)

(6.6)

252

H¨ ormander operators

with the same coefficients ajk as L, coefficients ck possibly different from the bk of L, and a zero order term d. We now fix a point x ∈ Ω, a ball Br (x) ⊂ Ω and a test function ϕ ∈ C0∞ (Br (x)) such that n X ajk (x) ∂x2j xk ϕ (x) 6= 0. (6.7) j,k=1

This can be done as follows. Let ( 2 2 e1/(|x−y| −r ) ϕ (y) = 0

if |x − y| < r if |x − y| > r.

Then ϕ ∈ C0∞ (Br (x)) and ∂x2j xk ϕ (x) = cr δjk so that n X

ajk (x) ∂x2j xk ϕ (x) = cr

n X

ajj (x) 6= 0

j=1

j,k=1

since {ajk } is nonnegative and satisfies (6.1). Applying (6.6) to   y−x ϕε (y) = ϕ x + ε (with ε ∈ (0, 1)), evaluating the identity at x, we obtain  Z X n   y−x γ (x, y) dy ε2 ϕ (x) = ajk (y) ∂x2j xk ϕ x + ε j,k=1   Z X n y−x +ε ck (y) (∂xk ϕ) x + γ (x, y) dy ε k=1   Z y−x 2 +ε d (y) ϕ x + γ (x, y) dy ε   n n X    X + α (x) ajk (x) ∂x2j xk ϕ (x) + ε ck (x) (∂xk ϕ) (x) + ε2 d (x) ϕ (x) .   j,k=1

k=1

(6.8) Now, for ε → 0, by (6.5), Z n   X y−x γ (x, y) dy 6 ε sup |ck | sup |∂xk ϕ| CΩ,L → 0, ck (y) (∂xk ϕ) x + ε ε Ω Ω Z k=1   2 y−x ε γ (x, y) dy 6 ε2 sup |d| sup |ϕ| CΩ,L → 0, d (y) ϕ x + ε Ω Ω and Z X  n   y−x 2 ϕ x + γ (x, y) dy a (y) ∂ jk xj xk ε j,k=1 Z n X 6 sup |ajk | sup ∂x2j xk ϕ |γ (x, y)| dy → 0, j,k=1

Ω

Ω

Bεr (x)

Fundamental solutions of H¨ ormander operators

  where the last conclusion holds since ∂x2j xk ϕ x +

y−x ε




253

is supported in Bεr (x).

Then letting ε → 0 in (6.8) we get 0 = α (x)

n X

  ajk (x) ∂x2j xk ϕ (x) ,

j,k=1

hence α (x) = 0, by (6.7). So far, we have shown the existence of a fundamental solution for L on any bounded domain Ω. Clearly, the same result can be applied to L∗ , hence there exists also a fundamental solution γ ∗ (x, y) for L∗ , such that Z γ ∗ (x, y) Lφ (y) dy for every φ ∈ C0∞ (Ω) . φ (x) = Ω

Note, however, that we have no information about the regularity of γ or γ ∗ with respect to the first variable, and we do not know any relation between γ and γ ∗ ; both these unpleasant facts are a consequence of the obvious nonuniqueness of the objects that we have built. The same argument used to prove the above existence theorem can be used to establish another interesting result: Theorem 6.4 (Solvability on bounded domains) Let L be a H¨ ormander operator satisfying (6.1) (or (6.3)) in a bounded domain Ω ⊂ Rn . Then for any ψ ∈ C0∞ (Ω) there exists u ∈ C ∞ (Ω) ∩ L1 (Ω) satisfying (in the distributional and then in the classical sense) the equation Lu = ψ and such that kukL1 (Ω) 6 C kψkL1 (Ω)

(6.9)

with C = CΩ,L as in Proposition 6.1. The same property holds for the transpose operator L∗ . We can rephrase the previous theorem saying that an operator like L is globally solvable on any bounded domain. Proof. Applying Proposition 6.2 to the measure dν = ψ (x) dx we find a Radon measure µ satisfying in the distributional sense Lµ = ψ and |µ| (Ω) 6 C kψkL1 (Ω) . However, since L is hypoelliptic (Theorem 5.64) actually dµ = u (x) dx for some u ∈ C ∞ (Ω), and then u solves the equation Lu = ψ in the classical sense. Moreover, |µ| (Ω) = kukL1 (Ω) hence (6.9) holds.

254

6.2

H¨ ormander operators

Homogeneous H¨ ormander operators

We now specialize our study of fundamental solutions to left invariant 2-homogeneous H¨ ormander operators on homogeneous groups. As we will see, within this class of operators the theory is much more satisfactory. In order to define precisely the class of operators that we will consider, let us first recapitulate, for the reader’s convenience, some concepts introduced in Chapter 3. Let G be a homogeneous group on RN and let Y1 , . . . , YN be the canonical basis of the Lie algebra g of left invariant vector fields on G, that is Yi (for i = 1, . . . , N ) is the only left invariant vector field in G which agrees with ∂xi at the origin. We assume that for some positive integer q < N one of the following two assumptions holds. 1. Y1 , . . . , Yq are 1-homogeneous and satisfy H¨ormander’s condition. In this case the operator q X L= Yj2 (6.10) j=1

has been called the canonical sublaplacian on G (see Definition 3.53); it is left invariant, 2-homogeneous and hypoelliptic by H¨ormander’s theorem (see Chapter 5, or the simplified version proved in Chapter 4). Moreover, since Yi∗ = −Yi (see (3.14)), we have L∗ = L. 2. Y1 , . . . , Yq are 1-homogeneous, Yq+1 , which we relabel Y0 , is 2-homogeneous, Y0 , Y1 , . . . , Yq satisfy H¨ ormander’s condition. In this case the operator q X L= Yj2 + Y0 , (6.11) j=1

has been called canonical homogeneous H¨ ormander operator with drift on G (see Definition 3.64); it is left invariant, 2-homogeneous, and hypoelliptic, by H¨ ormander’s theorem (see Chapter 5). Since Yi∗ = −Yi we have q X ∗ L = Yi2 − Y0 6= L. i=1

Note that also L∗ is hypoelliptic, since Y1 , . . . , Yq , −Y0 are a system of H¨ormander vector fields. In both cases the operator L is left invariant, 2-homogeneous, and hypoelliptic (and the same is true for L∗ ). Let us summarize the previous discussion in the following: Definition 6.5 (Canonical homogeneous H¨ ormander operator) We say that L is a canonical homogeneous H¨ ormander operator (on G) if either G is a Carnot group and L is its canonical sublaplacian or G is a homogeneous stratified group of type II and L is its canonical homogeneous H¨ ormander operator with drift. It is obviously possible to consider a more general family of (non-canonical) homogeneous H¨ ormander operators with drift, analogously to what we have done in Chapter 4 for sublaplacians:

Fundamental solutions of H¨ ormander operators

255

Definition 6.6 (Homogeneous H¨ ormander operator) Let G be a homogeneous stratified group of type II. We say that q X L= Xj2 + X0 j=1

is a homogeneous H¨ ormander operator on G with drift if X0 , X1 , . . . , Xq are a system of left invariant H¨ ormander vector fields on G, X0 is 2-homogeneous and Xi is 1-homogeneous for i = 1, . . . , q. We will simply say that L is a homogeneous H¨ ormander operator to say that either L is a sublaplacian on a Carnot group G (see Definition 4.3), or that L is a homogeneous H¨ ormander operator with drift on a homogeneous stratified group of type II G. We can prove the following: Proposition 6.7 The class of homogeneous H¨ ormander operators on a homogePq neous stratified group G of type II, L = j=1 Xj2 + X0 , coincides with the class of operators q X X L= ahk Yi Yj + a0 Y0 + cij [Yi , Yj ] (6.12) 16i ν

q X

q X

q X

j,h=1

i=1

! mij mih

βj1 βh1

2 βj1 =ν

j=1

j,h=1

since βj1 = δj1 for j = 1, . . . , q (see the structure of the canonical vector fields, Theorem 3.29). Hence L is uniformly not totally degenerate, with c0 = ν. Next, let us compute, in our situation, the constant X n q X β ≡ max bij (x) ∂xj bi1 (x) + b01 (x) x∈Ω i=1 j=1 appearing in the proof of Proposition 6.1. Since bi1 =

q X

mij βj1 = mi1 = constant,

j=1

we have ∂xj bi1 (x) ≡ 0. Also, since Y0 is 2-homogeneous and ∂x1 is 1-homogeneous, b01 (x) ≡ 0. Hence β = 0. The constant computed in the proof of Proposition 6.1 is CΩ,L = e2γd with     1 1 1+β γ = max 1, = max 1, = , c0 ν ν hence CΩ,L = e2d/ν and we are done. Remark 6.10 We explicitly note that the above proposition also applies to sublaplacians on Carnot groups, where the class Lν is defined as in Definition 4.37. Another useful property of this class of operators is the possibility of proving uniform subelliptic estimates for them:

258

H¨ ormander operators

Theorem 6.11 Let Ω be a fixed bounded domain in G. Let L be a homogeneous H¨ ormander operator on G, and assume that L belongs to the class Lν for some ν > 0. Then, for every positive integer K, the operator L belongs to the class Λ (Ω, K, κ0 , κ1 ) (see Definition 5.67) for positive constants κ0 , κ1 depending on K, G and ν. In particular, the operators of the class Lν satisfy uniform subelliptic estimates in the sense of Theorem 5.68. Proof. Let us first consider the canonical homogeneous H¨ormander operator L on G. For any fixed bounded domain Ω ⊂ G, let us show that for any positive integer K the operator L belongs to the class Λ (Ω, K, κ0 , κ1 ) for positive constants κ0 , κ1 depending on K, G. The upper bound (5.74) on the derivatives of coefficients of the vector fields is obvious. As for the lower bound (5.73) on determinants, given the canonical vector fields Y0 , Y1 , . . . , Yq on G we can select n commutators Y[I1 ] , Y[I2 ] , . . . , Y[In ] with |Ij | 6 s such that these vectors span Rn at the origin, and therefore at any point x ∈ G (due to left invariance). Since the function



f (x) = det Y[I ] , Y[I ] , . . . , Y[I ] 1

x

2

x

n

x

is continuous and positive everywhere, its minimum on every compact set is positive, hence for any bounded domain Ω condition (5.73) holds. This proves that the canonical operator belongs to Λ (Ω, K, κ0 , κ1 ) for every K, with constants κ0 , κ1 depending on K. Let now L be any operator in the class Lν , in particular assume that (6.13) is in force. Then by Proposition 6.7 and Proposition 5.69, L belongs to the class Λ (Ω, K, c0 · c (ν, n, s) , c1 · c (q, ν)) and satisfies uniform subelliptic estimates, in the sense of Theorem 5.68. If the drift X0 in L is not required to fulfill H¨ormander’s condition, then condition ν 1/2 6 |a0 | 6 ν −1/2 in the definition of the class Lν is weakened to |a0 | 6 ν −1/2 (see Definition 6.8). On the other hand, since X0 is 2homogeneous, it can be written as a linear combination of commutators of step 2. Hence, by Remark 5.70, Proposition 5.69 can be still applied. We can also show that the Sobolev spaces generated using the canonical generators Yi or another system Xi as above coincide, with equivalent norms within a fixed class Lν , and that the control distances induced by Yi and Xi are equivalent, with uniform constants within Lν . This fact will also imply an analogous equivalence of H¨ older norms with respect to Yi and Xi . Proposition 6.12 Let 0 < ν < 1, let q X Xj = mjk Yk for j = 1, 2, . . . , q k=1

X0 = a0 Y0 +

X

cij [Yi , Yj ] 16i 1 and let us prove it for k + 1. Let us write X

2

(XI f ) =

q X X i=1

|I|=k+1

2

(XI 0 Xi f ) +

|I 0 |=k

X

2

(XJ X0 f ) ≡ A + B.

|J|=k−1 2

(If k = 1 the term B reduces to (X0 f ) but the proof does not need any change). To handle A, let us recall that, as proved in (4.46), for every differential operator D with smooth coefficients, ν

q X

2

(DYk f ) 6

q X

2

(DXj f ) 6 ν −1

j=1

k=1

q X

2

(DYk f ) .

(6.17)

k=1

To shorten notation, throughout this proof we will write A ' B to mean that c1 (k, ν) B 6 A 6 c2 (k, ν) B. Applying (6.17) to D = XI 0 and then summing up for |I 0 | = k we get A'

q X X i=1 |I 0 |=k

2

(XI 0 Yi f ) .

260

H¨ ormander operators

Then, applying the inductive assumption to the function Yi f and then summing up for i = 1, . . . , q we get q q X X X X X 2 2 2 0 (XI Yi f ) ' (YI 0 Yi f ) 6 (YI f ) . i=1 |I 0 |=k

i=1 |I 0 |=k

|I|=k+1

To handle B, we first apply the inductive assumption to the function X0 f and then summing up for |J| = k − 1 we get X 2 B' (YJ X0 f ) |J|=k−1

and recalling that X0 = a0 Y0 + 

P

16i (1 − ε) (a0 YJ Y0 f ) + 1 − cij YJ [Yi , Yj ] f  ε 16i (1 − ε) ν (YJ Y0 f ) + 1 − ε 16i c1 (k, ν) (1 − ε)

X

2

(YJ Y0 f )

|J|=k−1

 +

1−

1 ε



X q X 2 c2 (k, ν) + c3 (k, ν) (YI 0 Yi f ) i=1 |I 0 |=k

with c1
, c2 , c3 > 0 and any ε ∈ (0, 1). Now, choosing ε close enough to 1 so that 1 − 1ε c2 + c3 > 0 we conclude A + B > c4

X

2

(YJ Y0 f ) + c5

|J|=k−1

which completes the proof.

q X X i=1 |I 0 |=k

2

(YI 0 Yi f ) > c

X |I|=k+1

(YI f )

2

Fundamental solutions of H¨ ormander operators

261

Proposition 6.13 Under the same assumptions of the previous proposition, if dX , dY are the control distances induced (as in Definition 1.38) by the systems q q {Xi }i=0 , {Yi }i=0 , then there exist constants c1 , c2 depending on G, ν such that c1 dY (x, y) 6 dX (x, y) 6 c2 dY (x, y) for every x, y ∈ G. In particular, if we define the homogeneous norms on G kxkX = dX (x, 0) kxkY = dY (x, 0) , then all these homogeneous norms are equivalent, with uniform constants as X ranges in the class Lν . A complete proof of this proposition will be given in Chapter 9 (see Proposition 9.58), for the following reasons. Recall that if G is a Carnot group this proposition has been proved in Chapter 4, see Proposition 4.42. A strictly analogous proof holds if a drift X0 is present, but the cij in (6.14) vanish, that is X0 = a0 Y0 . Let us present this proof, also to explain the problem which arises in the most general case: X (δ) , that is Proof of Proposition 6.13 in the case X0 = a0 Y0 . Let γ ∈ Cx,y

γ 0 (t) =

q X

αi (t) (Xi )γ(t)

i=0

with γ (0) = x, γ (1) = y, |α0 (t)| 6 δ 2 ,|αi (t)| 6 δ for i = 1, . . . , q. Then γ 0 (t) =

q X

αi (t)

i=1

=

≡

mij (Yj )γ(t) + α0 (t) a0 (Y0 )γ(t)

j=1

q q X X j=1 q X

q X

! αi (t) mij

(Yj )γ(t) + α0 (t) a0 (Y0 )γ(t)

i=1

bj (t) (Yj )γ(t) + b0 (t) (Y0 )γ(t)

j=1

where (see Remark 4.43) |bj (t)| 6

q X

|αi (t)| |mij | 6 qδν −1/2

i=1

|b0 (t)| 6 δ 2 ν −1/2 Y which implies γ ∈ Cx,y (cδ). Exploiting the inverse relations

Yi =

q X i,j=1

m0ij Xj ,

Y0 =

1 X0 a0

262

H¨ ormander operators

and the corresponding bounds m0ij 6 ν −1/2 and |a0 | > ν one can also show that Y X γ ∈ Cx,y (δ) =⇒ γ ∈ Cx,y (c (ν) δ) ,

and the two distances are equivalent. q

For a general system {Xi }i=0 in Lν , with X X0 = a0 Y0 + cij [Yi , Yj ] 16i 2 is harmless, since it only rules out uniformly elliptic operators in two variables. Also, note that if L is an operator with drift, the theorem applies also to L∗ . Proof. Let us start proving the uniqueness. Let γ1 , γ2 be two global fundamental solutions, homogeneous of degree 2 − Q and observe that L
(γ1 − γ2 ) = 0 in RN . Since L is hypoelliptic this implies that γ1 − γ2 ∈ C ∞ RN . On the other hand, γ1 − γ2 is also homogeneous of degree 2 − Q < 0, and therefore unbounded near the origin, unless it is identically zero. Hence γ1 = γ2 . To prove the existence result, let Ω be a neighborhood of the origin and let γ be the local fundamental solution in Ω constructed in Theorem 6.15. We choose {x : |x| < 1} ⊂ Ω b {x : |x1 | < 2} ,

Fundamental solutions of H¨ ormander operators

265

so that kγkL1 (Ω) 6 e4/ν .

(6.21)

To extend this function to the whole group G, we fix two cut-off functions η1 , η2 ∈ C0∞ (Ω), such that η2 is identically 1 in a neighborhood of the origin and η1 is identically 1 on the support of η2 and we set Γ1 = γη1 . We also set Γs (x) = s2−Q Γ1 (Ds−1 (x)) , for s > 0. We will show that the required homogeneous fundamental solution Γ is given by Γ (x) = lim Γs (x) . s→+∞

As a first step we will show that LΓs = δ0 + Gs

(6.22)

where Gs (x) = s−Q G (Ds−1 (x)) and G ∈ C0∞ (G). Namely, for every ϕ ∈ C0∞ (G) we have Z ∗ hLΓs , ϕi = hΓs , L ϕi = s2−Q Γ1 (Ds−1 (x)) L∗ ϕ (x) dx Z = s2 Γ1 (x) L∗ ϕ (Ds (x)) dx Z = Γ1 (x) L∗ [ϕ (Ds (x))] dx = hΓ1 , L∗ ϕs i where ϕs (y) = ϕ (Ds (y)). Hence hLΓs , ϕi = hΓ1 , L∗ (η2 ϕs )i + hΓ1 , L∗ [(1 − η2 ) ϕs ]i .

(6.23)

Now, ∗

s

Z

γ (x) η1 (x) L∗ (η2 ϕs ) (x) dx

Z

γ (x) L∗ (η2 ϕs ) (x) dx = η2 (0) ϕs (0) = ϕ (0)

hΓ1 , L (η2 ϕ )i = =

(6.24)

since Lγ = δ0 in Ω and η2 ϕs is supported where η1 is identically 1. Moreover, since (1 − η2 ) is supported away from the origin we have Z hΓ1 , L∗ [(1 − η2 ) ϕs ]i = γ (x) η1 (x) L∗ [(1 − η2 ) ϕs ] (x) dx Z = L (γη1 ) (x) (1 − η2 (x)) ϕs (x) dx Z Z = G (x) ϕs (x) dx = Gs (y) ϕ (y) dx, (6.25) where G = L (γη1 ) (1 − η2 ) .

(6.26)

266

H¨ ormander operators

From (6.23)-(6.24)-(6.26) we obtain (6.22). Let us consider ω = Γ2 −Γ1 . Note that ω is a compactly supported integrable function on Ω. By (6.22), Lω = L (Γ2 − Γ1 ) = G2 − G1 and by the hypoellipticity of L it follows that ω ∈ C0∞ (G). Since Γ2j+1 (x) − Γ2j (x) = 2j(2−Q) ω (D2−j (x)) , for every x ∈ G, +∞ X

|Γ2j+1 (x) − Γ2j (x)| 6

+∞ X

j=0

2j(2−Q) |ω (D2−j (x))|

j=0

6 kωkL∞ (G)

+∞ X

2j(2−Q)

(6.27)

j=0

(here we have exploited the assumption Q > 2). It follows that for every x 6= 0, Γ2n (x) = Γ1 (x) +

n−1 X

(Γ2j+1 (x) − Γ2j (x))

(6.28)

j=0

converges uniformly to a function Γ ∈ L1loc (G). Let us show that the same is true for Γs as s → +∞. Let 2n 6 s < 2n+1 and r = 2sn , then Γs (x) − Γ2n (x) = s2−Q Γ1 (Ds−1 (x)) − 2n(2−Q) Γ1 (D2−n (x))
= 2n(2−Q) r2−Q Γ1 (Dr−1 (D2−n (x))) − Γ1 (D2−n x) = 2n(2−Q) [Γr (D2−n x) − Γ1 (D2−n x)] . Let us consider the function Γr (x) − Γ1 (x); the same argument used for ω shows that this function is smooth. Also, since 1 6 r < 2 this function belongs to C0∞ (2Ω), and by Proposition 6.1 kΓr − Γ1 kL∞ (2Ω) 6 C kL (Γr − Γ1 )kL∞ (2Ω) 6 C kGr − G1 kL∞ (2Ω) . Since kGr − G1 kL∞ (2Ω) is uniformly bounded for 1 6 |Γs (x) − Γ2n (x)| 6 C 2n(2−Q) so that uniformly in x 6= 0,

r

6

2 we have

lim Γs (x) = Γ (x) .

s→+∞

To show that Γ is (2 − Q)-homogeneous observe that t

Q−2

Q−2

Γ (Dt (x)) = lim t s→+∞

 Q−2 t Γs (Dt (x)) = lim Γ1 (Dts−1 (x)) s→+∞ s

= lim s2−Q Γ1 (Ds−1 (x)) = Γ (x) . s→+∞

C0∞

Finally, let ϕ ∈ Z

(G), then by (6.22)

Γ (x) L∗ ϕ (x) dx = lim

Z

Γs (x) L∗ ϕ (x) dx Z = ϕ (0) + lim Gs (x) ϕ (x) dx. s→+∞

s→+∞

Fundamental solutions of H¨ ormander operators

267

Since by (6.26) G vanishes in a neighborhood of the origin, for s large enough Gs in the above integral vanishes and therefore Z Γ (x) L∗ ϕ (x) dx = ϕ (0) , that is LΓ = δ0 in G. To prove the uniform bounds (6.20), let us now look again into the previous computation. From (6.27)-(6.28) we read |Γ2n (x)| 6 |Γ1 (x)| + cQ kωkL∞ (G) and since Γ2n converges uniformly to Γ we have Z Z |Γ (x)| dx 6 |Γ1 (x)| dx + cQ kωkL∞ (G) . kxk62

kxk62

Recalling that Γ1 = γη1 and assuming supp η1 ⊂ B2 (0) we obtain Z |Γ (x)| dx 6 kγkL1 (B2 (0)) + cQ kωkL∞ (G) .

(6.29)

kxk62

Since ω ∈ C0∞ (Ω), by the classical Sobolev embedding theorem and the uniform subelliptic estimates (Theorem 6.11) there exist k, ε such that kωkL∞ 6 c (n) kωkH k+ε 6 c (n, m, G, ν) {kLωkH k + kωkH −m } for any m > 0. By Proposition 5.17 we can choose m >

n 2

so that

kωkH −m 6 c (n, m) kωkL1 Recalling that Lω = G2 − G1 we get kωkL∞ 6 c (G, ν) {kG2 − G1 kH k + kωkL1 } .

(6.30)

Next we bound Z

2−Q
2 kωkL1 (G) = kΓ2 − Γ1 kL1 (G) = (γη1 ) D1/2 (x) − (γη1 ) (x) dx G Z 6 2 |(γη1 ) (x)| dx = 2 kγkL1 (Ω) . (6.31) G


On the other hand, G1 = G = L (γη1 ) (1 − η2 ), G2 (x) = 2−Q G D1/2 (x) and their derivatives of any order can be bounded in terms of derivatives of γ in a set of the kind 12 6 kxk 6 2. Again by the uniform subelliptic estimates, for two cutoff functions φ1 ≺ φ2 vanishing outside a neighborhood of the shell 12 6 kxk 6 2 we have kG2 − G1 kH k 6 c (G) kφ1 γkH k+2 6 c (G, ν, m) {kφ2 LγkH k+2 + kφ2 γkH −m } = c (G, ν, m) kφ2 γkH −m 6 c (G, ν) kγkL1 (Ω)

(6.32)

where in the last inequality we have applied again Proposition 5.17, hence we have finally chosen an m only depending on the dimension n. Putting (6.32) and (6.31) in (6.30) we get kωkL∞ 6 c (G, ν) kγkL1 (Ω) which in (6.29) gives, by (6.21), Z |Γ (x)| dx 6 c (G, ν) kγkL1 (Ω) 6 c (G, ν) e4/ν . (6.33) kxk62

268

H¨ ormander operators

If we now want to bound the derivatives of any order of Γ we can still apply subelliptic estimates, choosing two cutoff functions φ1 ≺ φ2 supported in the shell 1 2 6 kxk 6 2 and equal to 1 in a neighborhood of the sphere kxk = 1; applying again Proposition 5.17 and the bound (6.33) we have kφ1 ΓkH k+ε 6 c (k, G, ν, m) {kφ2 LΓkH k + kφ2 ΓkH −m } = c (k, G, ν, m) kφ2 ΓkH −m 6 c (k, G, ν) kΓkL1 (B2 ) 6 c (k, G, ν) . Finally, by the classical Sobolev embedding theorems, for any positive integer r there exists k such that sup sup |Dxα Γ (x)| 6 c (r, n) kφ1 ΓkH k 6 c (r, G, ν)

kxk=1 |α|6r

and we are done. 6.4

Properties of the global fundamental solution

In this section we are going to study the main properties of the homogeneous fundamental solution built in Theorem 6.18, particularly in connection with representation formulas for solutions to Lu = f and their derivatives Xi Xj u. These properties will be crucial to apply the theory of singular integrals (which we will develop in Chapter 7), to get sharp a priori estimates (Chapter 8). The next theorem collects the basic properties of Γ. We will then pass to study in detail some further properties of the singular kernel Xi Xj Γ. Notation 6.19 From now on we will make use of the homogeneous norm in G defined by kxk = dY (x, 0), where dY is the control distance of the canonical basis in G. This is a homogeneous norm by Theorem 3.54 (and its extension to the drift case, as noted in section 3.8.3), satisfies the triangle inequality, and is uniquely determined by the group G. If we replace this norm with kxk = dX (x, 0) where q the system {Xi }i=0 belongs to the class Lν , then the two norms are equivalent with uniform constants depending on ν. This fact will be implicitly used in the following. Theorem 6.20 (Properties of the fundamental solution) Let L be a homogeneous H¨ ormander operator on G and assume that L belongs to the class Lν for some ν > 0. Let Γ be the homogeneous fundamental solution to L, as constructed in Theorem 6.18 (we are assuming Q > 2) and let Γ∗ be the analogous fundamental solution for L∗ . Then: 1. Upper bounds. There exists c = c (G, ν) > 0 such that for every x ∈ G \ {0} c |Γ (x)| 6 ; Q−2 kxk for any differential monomial XI homogeneous of degree k there exists ck = c (k, G, ν) such that for every x ∈ G \ {0} ck |XI Γ (x)| 6 . Q−2+k kxk

Fundamental solutions of H¨ ormander operators

269

In particular, the functions Γ and Xi Γ (i = 1, 2, . . . , q) are locally integrable. More generally, the above bound holds for Z k Γ where Z k is any k-homogeneous differential operator (even not translation invariant), with a constant also depending on the coefficients of Z k . 2. Representation formula. For every φ ∈ C0∞ (G), x ∈ G Z
φ (x) = Γ x−1 ◦ y L∗ φ (y) dy, G −1




that is LΓ x ◦ · = δx 3. Solvability of L. For every φ ∈ C0∞ (G), the function Z
u (x) = Γ y −1 ◦ x φ (y) dy G

belongs to C ∞ (G) and vanishes at infinity, with the decay estimates |u (x)| 6 |Xi u (x)| 6

c (G, ν, φ) Q−2

kxk c (G, ν, φ) Q−1

kxk

i = 1, . . . , q.

Moreover, u solves the equation Lu = φ in G.
4. Symmetry property of Γ. For every x ∈ G \ {0}, Γ∗ (x) = Γ
x−1 . In particular, if L is a sublaplacian on a Carnot group, then Γ (x) = Γ x−1 . 5. Nonpositivity of Γ. For every x ∈ G \ {0}, Γ (x) 6 0. Moreover, if L is a sublaplacian on a Carnot group, then Γ (x) is strictly negative. Remark 6.21 (Γ is a two-sided fundamental solution) By point 3 in the above theorem, we can write Z
φ (x) = L Γ y −1 ◦ x φ (y) dy. G

Also, applying point 2 to L∗ and recalling point 4, Z
φ (x) = Γ y −1 ◦ x Lφ (y) dy. G

The previous relations can be rewritten in terms of convolutions (see (3.27)) as follows: φ = Lφ ∗ Γ = L (φ ∗ Γ) C0∞

(6.34)

for every φ ∈ (G). The two identities (6.34) are summarized saying that Γ is a two-sided global fundamental solution for Γ. Compare with what we have established in Theorem 6.3 for a general (uniformly not totally degenerate) H¨ ormander operator: in that case we have proved the existence of a one-sided fundamental solution, on any bounded domain. Proving in that generality the existence of a two-sided fundamental solution is actually possible, under suitable assumptions, but would require a longer path. We will say something more on this topic in the Notes at the end of the chapter.

270

H¨ ormander operators

Remark 6.22 (Fundamental solution of a sublaplacian) If
the operator L is a sublaplacian, then keeping in mind that L∗ = L and that Γ x−1 = Γ (x) we can rewrite the identities (6.34) in different useful ways: for every φ ∈ C0∞ (G), Z Z

−1 φ (x) = Γ x ◦ y Lφ (y) dy = Γ y −1 ◦ x Lφ (y) dy G G Z Z

−1 = L Γ y ◦ x φ (y) dy = L Γ x−1 ◦ y φ (y) dy. G

G

Proof of Theorem 6.20. 1. The assertion follows since Γ and XI Γ are smooth outside the origin and, respectively, (2 − Q) and (2 − Q − k)-homogeneous. The form of the constants depends on the uniform bounds (6.20). 2. By the definition of Γ we have Z φ (0) = Γ (y) L∗ φ (y) dy ∀φ ∈ C0∞ (G) . G

Applying this identity to the function φx (y) = φ (x ◦ y) we have, by the left invariance of L∗ , Z φ (x) = φx (0) = Γ (y) L∗ φx (y) dy G Z Z
= Γ (y) (L∗ φ) (x ◦ y) dy = Γ x−1 ◦ y L∗ φ (y) dy. G

G

3. Let φ ∈ C0∞ (G) and let Z Z

−1 u (x) = Γ y ◦ x φ (y) dy = Γ (y) φ x ◦ y −1 dy. G

G

L1loc

Since Γ ∈ and φ is smooth and compactly supported, the above integral is absolutely convergent. We can also compute, for any right invariant differential operator XIR , Z
XIR u (x) = Γ (y) XIR φ x ◦ y −1 dy. G k,p Hence u belongs to the Sobolev spaces WX R (Ω) for any integer k, p ∈ [1, ∞] and bounded domain Ω, therefore the function u is smooth (see Chapter 2, Proposition 2.7). Since our homogeneous norm satisfies the triangle inequality (see Notation 6.19), so that we can write

−1

y ◦ x > kxk if kyk 6 R and kxk > 2R.

Then, if φ ∈ C0∞ (G) is supported in {kyk 6 R} and kxk > 2R we have Z c |u (x)| 6 cφ dy Q−2 kyk6R ky −1 ◦ xk Z c c 6 cφ dy = cφ cRQ Q−2 Q−2 kxk kxk kyk6R

Fundamental solutions of H¨ ormander operators

Since we already know that u is smooth, the inequality |u (x)| 6 every x ∈ G. We can also write Z
Xi u (x) = Xi Γ y −1 ◦ x φ (y) dy

271 cφ kxkQ−2

holds for

G

and exploiting the decay |Xi Γ (x)| 6 cφ . kxkQ−1

|Xi u (x)| 6 from the identities

c kxkQ−1

we can prove analogously that

The fact that u satisfies the equation Lu = φ can be read L (φ ∗ Γ) = φ ∗ LΓ = φ ∗ δ0 = φ,

(see Proposition 3.75). 4. Let φ ∈ C0∞ (G) and Z


Γ y −1 ◦ x Lφ (y) dy.

u (x) = G

By point 3, Lu = Lφ and u (x) → 0 as kxk → ∞. Since φ is compactly supported, the function w = u − φ satisfies  Lw = 0 in G w (x) → 0 for kxk → ∞. and by Corollary 1.58 we conclude that w ≡ 0. Therefore Z
Γ y −1 ◦ x Lφ (y) dy = φ (x) G


which means that the function x 7→ Γ x−1 is a fundamental solution of L∗ in RN , and by Theorem 6.18 it is actually the unique fundamental solution of L∗ ,
global −1 ∗ homogeneous of degree 2 − Q, that is Γ x = Γ (x). 5. For every φ ∈ C0∞ (G) , φ > 0, the function Z
u (x) = Γ y −1 ◦ x φ (y) dy G

satisfies Lu = φ > 0 in G and vanishes at infinity. By Corollary 1.58, u (x) 6 0 in G. Knowing that for any φ ∈ C0∞ (G) , φ > 0 one has Z
Γ y −1 φ (y) dy 6 0 G

Pq implies Γ 6 0. Moreover, if L= i=1 Xi2 is a sublaplacian on a Carnot group, then the function Γ is strictly negative in G \ {0}. Actually, this follows from the strong maximum principle in Chapter 1 (Theorem 1.64). In any domain Ω ⊂ G \ {0} we know that LΓ = 0 and Γ 6 0. Then if at some point x0 ∈ Ω we had Γ (x0 ) = 0, we could conclude Γ ≡ 0 in Ω (and therefore in G \ {0}) which, however, contradicts the representation formula Z
φ (x) = Γ x−1 ◦ y Lφ (y) dy G

which must hold for every φ ∈ C0∞ (G) and x ∈ G. Note that the last proof requires knowing in advance the weaker result Γ 6 0.

272

H¨ ormander operators

Remark 6.23 Observe that the strict negativity result is false for H¨ ormander opn erators with drift: the heat operator in R has a Gaussian fundamental solution which is nonnegative but vanishes in a halfspace. The fact that in a Carnot group the fundamental solution Γ is strictly negative allows to define a homogeneous norm adapted to Γ. This is similar to what happens with the fundamental solution of the classical Laplacian in Rn (for n > 3) 1/(2−n) n and |Γ (x)| is a multiple of the Euclidean norm (letting where Γ (x) = |x|cn−2 |Γ (0)|

1/(2−n)

= 0 by continuity). We have the following:

Proposition 6.24 Let G be a Carnot group of homogeneous dimension Q > 2, L a sublaplacian on G and Γ the homogeneous global fundamental solution of L. Then ( 1/(2−Q) |Γ (x)| if x 6= 0 kxk = 0 if x = 0 is a symmetric, smooth homogeneous norm in G (in the sense of Definition 3.8). Proof. By definition we have kxk > 0 and k0k = 0. By Theorem 6.20, points 4 and 5, we know that |Γ (x)| is symmetric and strictly positive. Moreover, |Γ (x)| is smooth outside the origin. Hence k·k is smooth and strictly positive outside the origin, in particular kxk = 0 implies x = 0, and k·k is symmetric. Since Γ is (2 − Q)homogeneous, k·k is 1-homogeneous. By Proposition 3.9, then, k·k is a (symmetric, smooth) homogeneous norm. In the following we will use the homogeneous fundamental solution to prove a representation formula for second order derivatives Xi Xj u in terms of Lu and u, which will be the starting point to derive a priori estimates on Lp or H¨older norms of Xi Xj u, exploiting a suitable abstract theory of singular integrals. In view of this goal, it is also necessary to prove some key properties of the kernel Xi Xj Γ, which are strictly analogous to those enjoyed by classical Calder´on-Zygmund kernels. Since we do not know the explicit form of Γ, all these properties must be derived from general considerations about homogeneity, distributions, and so on, which makes this study even more instructive. We start with the following: Proposition 6.25 (Mean value inequality for homogeneous kernels) Let f ∈ C 1 (Rn \ {0}) be homogeneous of degree α < 1. Then there exist constants M > 1, c > 0 such that α−1

|f (x ◦ y) − f (x)| + |f (y ◦ x) − f (x)| 6 c kyk kxk

for every x, y ∈ G with kxk > M kyk. Moreover M only depends on G while c = c (G, α) sup |∇f (z)| . kzk=1

In particular this implies −Q−1

|Xi Xj Γ (x ◦ y) − Xi Xj Γ (x)| 6 c kyk kxk

(6.35)

Fundamental solutions of H¨ ormander operators

273

−Q−1

|Xi Xj Γ (y ◦ x) − Xi Xj Γ (x)| 6 c kyk kxk for kxk > M kyk , i, j = 1, 2, . . . , q. The constant c has the form c (G, ν) for L ∈ Lν . Proof. Throughout this proof we use the homogeneous norm kxk = dX (x, 0), recalling that it satisfies the triangle inequality and keeping in mind its equivalence with dY (x, 0) (see Notation 6.19). We denote by B (x, r) the dX -balls. Let y 6= 0, otherwise there is nothing to prove. Then x ◦ y ∈ B (x, 2 kyk). If kxk > 4 kyk then for every z ∈ B (x, 2 kyk) we have

kxk kzk > kxk − x−1 ◦ z > kxk − 2 kyk > . 2 By Theorem 1.56, for every f ∈ C 1 (BX (x0 , r)) v u q uX √ 2 2 |f (x) − f (x0 )| 6 qd (x, x0 ) · sup t (Xi f ) + d (x, x0 ) · sup |X0 f | BX (x0 ,r)

BX (x0 ,r)

i=1

for any x ∈ BX (x0 , r), hence in our setting v u q uX √ 2 2 (Xi f ) + kyk |f (x ◦ y) − f (x)| 6 q kyk sup t B(x,2kyk)

6

√

q kyk

sup

v u q uX 2 2 t (Xi f ) (z) + kyk

kzk>kxk/2

|X0 f |

sup B(x,2kyk)

i=1

sup

|X0 f (z)|

kzk>kxk/2

i=1

since Xi f is (α − 1)-homogeneous and X0 f is (α − 2)-homogeneous,   α−1 2 α−2 α−1 6 c sup |Xf (z)| · kyk kxk + kyk kxk 6 c sup |∇f (z)| · kyk kxk kzk=1

kzk=1

(6.36) where the last inequality exploits the fact that kxk > 4 kyk. To obtain a similar
estimate for |f (y ◦ x) − f (x)| we apply (6.36) to g (x) = f x−1 (note that also g is homogeneous of degree α) writing





g x−1 ◦ y −1 − g x−1 6 c sup |Xg (z)| · y −1 x−1 α−1 kzk=1



when x−1 > M y −1 . Now, by the properties of homogeneous norm we can write

c1 kzk 6 z −1 6 c2 kzk for suitable constants c1 , cq only depending on G. Then α−1 |f (y ◦ x) − f (x)| 6 c sup |Xg (z)| · kyk kxk kzk=1

when kxk > M 0 kyk for a suitable M 0 . Finally, using the homogeneity of Xg (z) we obtain sup |Xg (z)| 6 sup |Xg (z)| 6 c sup |Xg (z)| kzk=1

c1 6kz −1 k6c2

kz −1 k=1


= c sup Xg z −1 6 c sup |∇f (z)| kzk=1

that gives (6.35) with the required constant.

kzk=1

274

H¨ ormander operators

Proposition 6.26 (Integral of singular kernels over spherical shells) Let f ∈ L1loc (G \ {0}) homogeneous of degree −Q. Then: 1. For every c2 > c1 > 0, r > 0, Z Z f (x) dx = f (x) dx. (6.37) c1 r6kxk6c2 r

c1 6kxk6c2

2. There exist a finite value M (f ) (“mean value of f ”) and a constant c = c (G) such that for every R > r > 0   Z R f (x) dx = cM (f ) log . (6.38) r r6kxk6R Proof. Point 1 immediately follows by the change of variables x = Dr (y) since f is −Q-homogeneous. To prove 2, let Z L (r) = f (x) dx for r > 1. 16kxk6r

We claim that L (rs) = L (r) + L (s)

(6.39)

for every r, s > 1. Note that since L (1) = 0, if r or s equals 1 there is nothing to prove. For r, s > 1, using (6.37) Z Z L (rs) = f (x) dx + f (x) dx = L (r) + L (s) . 16kxk6r

r6kxk6rs


From this we easily obtain that for positive integers p and q, L sp/q = pq L (s) for every s > 1 and using the continuity of L that for every α > 0 and s > 1, L (sα ) = αL (s). Hence, for r > 1
L (r) = L elog r = L (e) log r. Therefore Z



Z f (x) dx = L

f (x) dx = r6kxk6R

16kxk6 R r

R r



 = L (e) log

R r



and (6.38) is proved, with Z M (f ) =

f (x) dx. 16kxk6e

Note that since f ∈ L1loc (G \ {0}) the quantity M (f ) is finite. Remark 6.27 (Vanishing vs. boundedness) The above proposition implies in particular that for a −Q-homogeneous function f the boundedness property Z f (x) dx 6 c for every R > r > 0 (6.40) r6kxk6R

Fundamental solutions of H¨ ormander operators

275

holds if and only if the vanishing property holds: Z f (x) dx = 0 for every R > r > 0. r6kxk6R

Namely, for R/r → ∞ in (6.38), the only way to have boundedness is that M (f ) = 0. This fact is even strengthened by the following remark. The definition of M (f ) clearly also depends on the particular homogeneous norm chosen, so one could ask if a kernel f which enjoys the vanishing property with respect to a homogeneous norm 0 k·k still satisfies it with respect to a different (equivalent) homogeneous norm k·k . This is actually the case, in view of the following: 0

Proposition 6.28 Let f ∈ L1loc (G \ {0}) homogeneous of degree −Q. Let k·k , k·k be two different (equivalent 1 ) homogeneous norms. Then the boundedness property 0 (6.40) holds with respect to k·k if and only if it holds with respect to k·k ; equivalently (by the above remark), Mk·k (f ) = 0 if and only if Mk·k0 (f ) = 0. Proof. We assume that Z f (x) dx 6 c for every R > r > 0 r6kxk0 6R 0

and we prove that the same holds with k·k replaced by k·k. Assuming 0

c1 kxk 6 kxk 6 c2 kxk for every x, we can write  0 {x : r 6 kxk 6 R} ⊂ x : c1 r 6 kxk 6 c2 R and  0 x : c1 r 6 kxk 6 c2 R \ {x : r 6 kxk 6 R}     c1 c2 ⊂ x : r 6 kxk 6 r ∪ x : R 6 kxk 6 R . c2 c1 Hence Z f (x) dx r6kxk6R

Z

Z

= f (x) dx − f (x) dx c r6kxk0 6c2 R {x:c1 r6kxk0 6c2 R}\{x:r6kxk6R} Z 1 f (x) dx r6kxk6R Z Z Z |f (x)| dx + |f (x)| dx 6 f (x) dx + c c1 c1 r6kxk0 6c2 R r6kxk6r R6kxk6 2 R c2

1 We

c1

have proved in Theorem 3.12 that actually any two homogeneous norms on G are equivalent.

276

H¨ ormander operators

by (6.37) Z Z Z f (x) dx + = |f (x)| dx + |f (x)| dx c1 c c1 r6kxk0 6c2 R 6kxk61 16kxk6 2 c2

c1

6 K + c (f ) where c (f ) does not depend on r, R. Let now f ∈ C (G \ {0}) be homogeneous of degree −Q. Then f is not integrable in a neighborhood of the origin, so it does not identify with a distribution in the standard way. However, if its mean value M (f ) vanishes, then it actually defines a distribution as follows: Proposition 6.29 (Principal value distribution) Let f ∈ C (G \ {0}) be homogeneous of degree −Q and let M (f ) = 0. Then: 1. It is possible to define the distribution PV f (“principal value of f ”) as follows: Z hPV f, φi = lim f (x) φ (x) dx ε→0 kxk>ε Z Z = f (x) [φ (x) − φ (0)] dx + f (x) φ (x) dx kxk1

for every φ ∈ C0∞ (G). The distribution PV f is −Q-homogeneous. 2. For every ψ ∈ C0∞ (G), the limit Z
lim f y −1 ◦ x ψ (y) dy ε→0

ky −1 ◦xk>ε

defines a locally integrable function, that we will denote by PV (ψ ∗ f ) (x). Moreover, for every φ ∈ C0∞ (G), Z hψ ∗ PV f, φi = PV (ψ ∗ f ) (x) φ (x) dx. G

In other words, the distribution defined by the L1loc function PV (ψ ∗ f ) coincides with the distributional convolution ψ ∗ PV f . (See Definition 3.74). Proof. 1. Since M (f ) = 0 we can write Z Z Z f (x) φ (x) dx = f (x) φ (x) dx + f (x) φ (x) dx kxk>ε εε

hence PV f is −Q-homogeneous as a distribution. 2. By the above argument, for every x, Z
lim f y −1 ◦ x ψ (y) dx ε→0 ky −1 ◦xk>ε Z Z
= f y −1 ◦ x [ψ (y) − ψ (x)] dy + ky −1 ◦xk1

By Proposition 1.37 and Theorem 3.54, we have

|ψ (y) − ψ (x)| 6 c1 |x − y| 6 c2 dY (x, y) 6 c3 y −1 ◦ x . It follows that the first integral can be uniformly bounded in x, by Z c dy = c < ∞ Q−1 −1 −1 ky ◦ xk ky ◦xk1

kxk 0 the last integral is less than Z  Z |f (w)| |ψ (x)| dx dw 6 c. 1ε

!

Z φ (w)

G

f (x) ψ w ◦ x

φ (w) G

Z =

!

Z

lim

ε→0

f y

−1




◦ w ψ (y) dy dw

ky −1 ◦wk>ε

Z φ (w) PV (ψ ∗ f ) (w) dw.

= G

We have seen that to any kernel f ∈ C (G \ {0}) which is −Q-homogeneous and satisfies the vanishing property M (f ) = 0 it is naturally associated a principal value distribution. A very interesting fact is that this assertion can be somewhat inverted: Theorem 6.30 (−Q-homogeneous distribution) Let T be a distribution, homogeneous of degree −Q, which agrees outside the origin with a function f ∈ C (G \ {0}) . Then M (f ) = 0 and there exists a constant c such that T = PV f + cδ0 . Moreover the function f is −Q-homogeneous. Proof. Let us define the distribution F given by: Z Z hF, φi = f (x) [φ (x) − φ (0)] dx + kxk1

The computation in the proof of Proposition 6.29 shows that even without assuming M (f ) = 0 this is a well defined object, which is easily checked to be a distribution. Moreover, outside the origin it coincides with T , since for any φ ∈ C0∞ (G) with 0∈ / supp φ we have Z Z Z hF, φi = f (x) φ (x) dx + f (x) φ (x) dx = f (x) φ (x) dx = hT, φi . kxk1

G

Hence F −T is a distribution supported at the origin, which implies that it is a linear combination of the Dirac δ0 and a finite number of its derivatives (see Theorem A.27 in the Appendix) F − T = c0 δ 0 +

N X i=1

ci Dβi δ0 .

(6.41)

Fundamental solutions of H¨ ormander operators

279

Since T is −Q homogeneous, for every λ > 0 we have hT, φ ◦ Dλ i = hT, φi, hence hF, φ ◦ Dλ i − hF, φi = hF − T, φ ◦ Dλ i − hF − T, φi * + N X βi = c0 δ0 + ci D δ 0 , φ ◦ D λ − φ i=1

=

N X

D

|βi |

ci δ0 , (−1)

E Dβi (φ ◦ Dλ − φ) .

i=1

Now, since
Dβi (φ (Dλ (x))) = λγi Dβi φ (Dλ (x)) for some exponent γi > 0 depending on the multiindex βi , hF, φ ◦ Dλ i − hF, φi =

N X

ci (−1)

|βi |

 γ

 λ βi Dβi φ (0) − Dβi φ (0) ,

(6.42)

i=1

a quantity that remains bounded as λ → 0+ . On the other hand, by definition of F, for any λ ∈ (0, 1) hF, φ ◦ Dλ i − hF, φi Z Z = f (x) [φ (Dλ (x)) − φ (0)] dx + f (x) φ (Dλ (x)) dx kxk1 Z Z − f (x) [φ (x) − φ (0)] dx − f (x) φ (x) dx kxk1 Z Z = f (y) [φ (y) − φ (0)] dy + f (y) φ (y) dy kykλ Z Z − f (x) [φ (x) − φ (0)] dx − f (x) φ (x) dx kxk1 Z Z =− f (x) [φ (x) − φ (0)] dx + f (x) φ (x) dx λ r > 0. r6kxk6R

Also, if φ : G → R is a radial function, φ (x) = Φ (kxk) with Φ : (0, ∞) → R continuous, then Z f (x) φ (x) dx = 0 for every R > r > 0. r 1 and let ψε (x) = ψ D1/ε (x) . We claim that Z Z

Xi f y −1 ◦ x φ (y) dy = lim Xi (f ψε ) y −1 ◦ x φ (y) dy (6.52) ε→0 G G
R at least in the distributional sense. Namely: since h (x) = G f y −1 ◦ x φ (y) dy is an L1loc function, Xi h exists as a distribution. Moreover, Z
(f ψε ) y −1 ◦ x φ (y) dy → h (x) G

in L1loc . Hence (recall that by Remark 3.30, Xi∗ = −Xi ), for every ϕ ∈ C0∞ (G), Z 
−1 hXi h, ϕi = hh, −Xi ϕi = lim (f ψε ) y ◦ x φ (y) dy, −Xi ϕ ε→0 G  Z 
−1 = lim Xi (f ψε ) y ◦ x φ (y) dy, ϕ ε→0

G

and (6.52) holds. Note that the existence of the limit in the last expression also implies the fact that the right hand side of (6.52) is actually a distribution, by Theorem A.30 in the Appendix. Since Xi (f ψε ) ∈ L1loc we can then compute Z Z

Xi (f ψε ) y −1 ◦ x φ (y) dy = (Xi (f ψε )) y −1 ◦ x φ (y) dy G ZG Z

−1 = (Xi f ) y ◦ x φ (y) dy + (Xi (f ψε )) y −1 ◦ x φ (y) dy ky −1 ◦xk>ε

≡ I1 (ε, x) + I2 (ε, x) .

ky −1 ◦xk6ε

284

H¨ ormander operators

Let us show that for ε → 0, the function I1 (ε, x) converges uniformly to some I1 (x). Namely, for ε2 > ε1 > 0 we have Z
I1 (ε1 , x) − I1 (ε2 , x) = (Xi f ) y −1 ◦ x φ (y) dy ε1 R0 where R0 is a number such that d0 (x, y) < R0 =⇒ d (x, y) < R. Moreover, assume that K can be written as K (x, y) = K0 (x, y) + K1 (x, y) where

326

H¨ ormander operators

(i) K0 is a singular integral kernel of exponent β0 in B (x, R), and moreover satisfies the strong vanishing property: Z K0 (x, y) dµ (y) = 0 r1 3κ we have n o α |u∗ (x) − u∗ (y)| 6 c Mα,Ω0 ,Ω1 u + kukL1 (Ω1 ) d (x, y) .

(7.30)

Remark 7.39 The possibility of extending the bound (7.29) to any x, y ∈ Ω0 (suppressing the term kukL1 (Ω1 ) in (7.30)) relies on some global property of the quasidistance. For instance, if for every couple of points x, y ∈ Ω0 and any positive integer N we can find a chain of points x0 = x, x1 , x2 , . . . , xN = y such that d (xj , xj−1 ) is comparable to d (x, y) /N , this is the case. In our abstract setting, this cannot be generally assured.

328

H¨ ormander operators

To shorten notation, henceforth we will write Mα u in place of Mα,Ω0 ,Ω1 u. For u ∈ Lα (Ω0 , Ω1 ) , x0 ∈ Ω0 , r 6 3κ, let c (x0 , r, u) be the constant such that Z Z |u (y) − c (x0 , r, u)| dµ (y) = inf |u (y) − c| dµ (y) . c∈R

B(x0 ,r)

B(x0 ,r)

R

This is possible because the function c 7−→ B(x0 ,r) |u (y) − c| dµ (y) is continuous, nonnegative, and goes to +∞ for c → ±∞, hence attains a minimum. Then: Lemma 7.40 There exists c1 depending on Ω, α such that for every u ∈ Lα (Ω0 , Ω1 )  r  (7.31) c (x0 , r, u) − c x0 , k , u 6 c1 rα Mα u 2 for every x0 ∈ Ω0 , r 6 3κ, positive integer k.
r Proof. For any x0 ∈ Ω0 , r 6 3κ, positive integer h and y ∈ B x0 , 2h+1 we can write    r  r c x0 , h , u − c x0 , h+1 , u 2 2    r  r 6 c x0 , h , u − u (y) + u (y) − c x0 , h+1 , u 2 2
r , and integrating over B x0 , 2h+1      r r  r B x0 , h+1 c x0 , h , u − c x0 , h+1 , u 2 2 2 Z  r  6 c x0 , h , u − u (y) dµ (y) + r 2 B (x0 , h ) Z 2   r + u (y) − c x0 , h+1 , u dµ (y) r 2 B (x0 , h+1 ) n 2  r   r α o r   r α  + B x0 , h+1 h+1 , 6 Mα u B x0 , h h 2 2 2 2 hence by the local doubling property        1 r  c r 1 α = cr M u + . c x0 , h , u − c x0 , h+1 , u 6 rα Mα u α 2 2 2hα 2hα 2(h+1)α P∞ 1 we have for any positive integer k Setting H = h=0 2hα ∞    r  X  r  r c (x0 , r, u) − c x0 , k , u 6 c x0 , h , u − c x0 , h+1 , u 6 cHrα Mα u. 2 2 2 h=0 (7.32)

Lemma 7.41 For any u ∈ Lα (Ω0 , Ω1 ) we have supx∈Ω0 |c (x, 3κ, u)| < ∞. Namely, n o |c (x, 3κ, u)| 6 c1 Mα u + kukL1 (Ω1 ) for some constant c1 depending on Ω and α.

Real analysis and singular integrals in locally doubling metric spaces

329

Proof. For any fixed x ∈ Ω0 and a.e. ξ ∈ B (x, r), |c (x, 3κ, u)| 6 |c (x, 3κ, u) − u (ξ)| + |u (ξ)| hence averaging in ξ over B (x, 3κ) we get Z n o 1 α |c (x, 3κ, u)| 6 (3κ) Mα u + |u (ξ)| dµ (ξ) 6 c Mα u + kukL1 (Ω1 ) |B (x, 3κ)| Ω1 since inf x∈Ω0 |B (x, 3κ)| > 0, by Lemma 7.47. Lemma 7.42 For any u ∈ Lα (Ω0 , Ω1 ) and any x ∈ Ω1 lim c (x, r, u) ≡ u∗ (x) .

r→0+

is finite. Moreover, |c (x, r, u) − u∗ (x)| 6 c1 rα Mα u for every r 6 3κ for some constant c1 depending on Ω, α. Proof. For any x ∈ Ω1 and r 6 3κ, let k > h be two positive integers, then  r   r   r  k−1   r X c x, i , u − c x, i+1 , u c x, k , u − c x, h , u 6 2 2 2 2 i=h

and since by (7.32) this is the remainder of a convergent series,  r  lim c x, k , u (7.33) k→∞ 2 exists and is finite. Let us
show that it does not depend on r. Let 0 < r1 < r2 6 3κ, then for any ξ ∈ B x, 2rh1  r  r   r   r   1 1 2 2 c x, h , u − c x, h , u 6 c x, h , u − u (ξ) + c x, h , u − u (ξ) 2 2 2 2
and integrating in ξ over B x, 2rh1 we get  r   r   r  1 1 2 B x, h c x, h , u − c x, h , u 2 2 2  r α  r   r α  r  1 1 2 2 6 h B x, h Mα u + h B x, h Mα u 2 2 2 2 so that, by Lemma 7.4 and for some β > 0, (
) r2  r   r  M u 1 2 α α α B x, 2h
c x, h , u − c x, h , u 6 αh r1 + r2 2 2 2 B x, 2rh1 (  β ) Mα u r2 α α 6 αh r1 + r2 c →0 2 r1 for h → ∞ and r1 , r2 fixed. Hence the limit (7.33) does not depend on r. Let, x ∈ Ω1 and r 6 3κ,  r  u∗ (x) = lim c x, h , u . h→∞ 2 Passing to the limh→∞ in (7.31) we get |c (x, r, u) − u∗ (x)| 6 c1 rα Mα u which implies the existence of limr→0 c (x, r, u) = u∗ (x), and the Lemma is proved.

330

H¨ ormander operators

Proposition 7.43 For any u ∈ Lα (Ω0 , Ω1 ), u∗ is bounded: n o |u∗ (x)| 6 c Mα u + kukL1 (Ω1 ) . Moreover, u∗ (x) = u (x) a.e. in Ω0 . Proof. By the previous Lemma we get, for any x ∈ Ωn α

|c (x, 3κ, u) − u∗ (x)| 6 c (3κ) Mα u. Hence by Lemma 7.41 n o |u∗ (x)| 6 |c (x, 3κ, u)| + |c (x, 3κ, u) − u∗ (x)| 6 c1 Mα u + kukL1 (Ω1 ) which shows the boundedness of u∗ . Moreover, for any x ∈ Ω0 and a.e. ξ ∈ B (x, r) we have |c (x, r, u) − u (x)| 6 |c (x, r, u) − u (ξ)| + |u (ξ) − u (x)| and averaging in ξ over B (x, r) , Z 1 |c (x, r, u) − u (ξ)| dµ (ξ) + |B (x, r)| B(x,r) Z 1 + |u (ξ) − u (x)| dµ (ξ) |B (x, r)| B(x,r) Z 1 |u (ξ) − u (x)| dµ (ξ) . 6 r α Mα u + |B (x, r)| B(x,r)

|c (x, r, u) − u (x)| 6

On the other hand, since u ∈ L1 (Ω1 ), by Lebesgue’s theorem (Theorem 7.27), for a.e. x ∈ Ω0 , Z 1 |u (ξ) − u (x)| dµ (ξ) = 0 lim r→0 |B (x, r)| B(x,r) hence for a.e. x ∈ Ω0 we have |c (x, r, u) − u (x)| → 0 for r → 0, that is u∗ (x) = u (x) a.e. in Ω0 . Lemma 7.44 For any u ∈ Lα (Ω0 , Ω1 ), any x, y ∈ Ω0 with 2d (x, y) < 3κ, we have α

|c (x, 2d (x, y) , u) − c (y, 2d (x, y) , u)| 6 cMα ud (x, y) . Proof. Let r = d (x, y) < 3κ/2 and I0 = B (x, 2r) ∩ B (y, 2r). For any ξ ∈ I0 , |c (x, 2r, u) − c (y, 2r, u)| 6 |c (x, 2r, u) − u (ξ)| + |c (y, 2r, u) − u (ξ)| and integrating over I0 we get |I0 | |c (x, 2r, u) − c (y, 2r, u)| Z Z 6 |c (x, 2r, u) − u (ξ)| dµ (ξ) + B(x,2r) α

6 (2r) Mα u {|B (x, 2r)| + |B (y, 2r)|} .

B(y,2r)

|c (y, 2r, u) − u (ξ)| dµ (ξ)

Real analysis and singular integrals in locally doubling metric spaces

331

Note that B (x, r) ∪ B (y, r) ⊂ I0 so that 2r 6 3κ implies, by the local doubling condition |I0 | > c {|B (x, 2r)| + |B (y, 2r)|}, hence |c (x, 2r, u) − c (y, 2r, u)| 6 crα Mα u and we are done. We can finally come to the Proof of Theorem 7.38. Let x, y ∈ Ω0 with d (x, y) = r and 2r 6 3κ, then |u∗ (x) − u∗ (y)| 6 |u∗ (x) − c (x, 2r, u)| + |u∗ (y) − c (y, 2r, u)| + |c (x, 2r, u) − c (y, 2r, u)| ≡ A + B + C. Now, by Lemma 7.42, α

A + B 6 c (2r) Mα u, while by Lemma 7.44 C 6 crα Mα u so (7.29) follows. If now 2d (x, y) > 3κ, by Proposition 7.43 o n |u∗ (x) − u∗ (y)| 6 |u∗ (x)| + |u∗ (y)| 6 c Mα u + kukL1 (Ω1 ) o n α 6 c Mα u + kukL1 (Ω1 ) d (x, y) , since d (x, y) > 3κ/2. So we are done. 7.9

Some geometric results

In this section we have collected three geometric results which we have used previously. In the situations we will be interested in, we will always know that for small radii µ (B (x, ρ)) behaves like ρQ for some Q > 0, which makes these facts quite obvious; however, here we will prove them in the general framework of locally doubling metric spaces. We also prove here a version of the classical Marcinkievic interpolation inequality. ∞

Lemma 7.45 For a sequence of pairwise disjoint balls {B (xi , ρi )}i=1 contained in Ω1 we necessarily have lim ρi = 0. i→∞

Proof. Since ∞ X i=1

µ (B (xi , ρi )) = µ

∞ [

! B (xi , ρi )

6 µ (Ω1 ) < ∞,

i=1

we have limi→∞ µ (B (xi , ρi )) = 0. Let us show that this implies that the radii also tend to zero. By contradiction, assume ρi > δ > 0 for infinitely many i. So ∞ there exists a (new) sequence of pairwise disjoint balls {B (xi , δ)}i=1 contained in

332

H¨ ormander operators

Ω1 and with µ (B (xi , δ)) → 0. It is not restrictive to assume δ 6 κ. Also, by compactness, we can cover Ω1 with a finite number of balls B (yj , δ) , yj ∈ Ω1 and one of these balls, say B (y, δ) , must intersect infinitely many balls B (xi , δ) for which µ (B (xi , δ)) → 0. Then d (y, xi ) 6 2δ and B (y, δ) ⊂ B (xi , 3δ). Hence the local doubling condition implies µ (B (y, δ)) 6 c (Ω, δ) µ (B (xi , δ)) → 0 which means µ (B (y, δ)) = 0 and this is impossible. Proposition 7.46 For any fixed Ω1 and δ ∈ (0, 1) there exists an integer N , only depending on Ω and δ, such that every ball B (x, R) ⊂ Ω1 does not contain more than N pairwise disjoint balls of radii δR. Proof. Let {B (xi , δR)}i be a family of pairwise disjoint balls contained in B (x, R), and assume first that R 6 κ. Then B (x, R) ⊂ B (xi , 2R) and the local doubling condition implies that µ (B (x, R)) 6 µ (B (xi , 2R)) 6 c (Ω, δ) µ (B (xi , δR)) . Assume there exist at least N such disjoint balls B (xi , δR), then adding up these inequalities for i = 1, . . . , N we get N µ (B (x, R)) 6 c (Ω, δ)

N X

µ (B (xi , δR)) 6 c (Ω, δ) µ (B (x, R)) ,

i=1

hence N 6 c (Ω, δ). Note that the constant c (Ω, δ) does not increase if we increase δ. Let us now deal with the case R > κ. Again, let {B (xi , δR)}i be a family of pairwise disjoint balls contained in B (x, R). Let η ∈ (0, 1) be a constant to be fixed later. Since [ Ω1 ⊂ B (y, ηκ) , y∈Ω1

by compactness there exist y1 , . . . , yJ ∈ Ω1 , with J only depending on Ω, η, such that J [ B (yj , ηκ) . Ω1 ⊂ j=1

For ρ = min (δR, ηκ), let us consider the family of pairwise disjoint balls {B (xi , ρ)}i and the subcollection of those balls which intersect a fixed B (yj , ηκ). For each of these balls we can write d (xi , yj ) 6 ρ + ηκ 6 2ηκ B (xi , ρ) ⊂ B (yj , 3ηκ) . Choosing η = 1/3 we have B (xi , ρ) ⊂ B (yj , κ) ⊂ Ω2 for each ball B (xi , ρ) intersecting B (yj , ηκ). We now want to apply the first part of the proof to the ball B (yj , κ). There are two cases:

Real analysis and singular integrals in locally doubling metric spaces

333

i) δR < ηκ. Then δ = κ/3, a constant only depending on Ω, hence for some integer N (Ω) there are at most N (Ω) balls B (xi , ρ) intersecting B (yj , ηκ) (and therefore contained in B (yj , κ)), and there are at most N (Ω) J balls B (xi , ρ) at all, with J only depending on Ω. ii) δR > ηκ. Then ρ = δR and (since we are assuming R > κ) κρ = δR κ > δ. We know that for some integer N (Ω, δ) there are at most N (Ω, δ) disjoint balls of radius δκ contained in B (yj , κ); a fortiori, there are at most N (Ω, δ) disjoint balls of radius ρ > δκ contained in B (yj , κ). Again, the total number of balls B (xi , ρ) is bounded by N (Ω, δ) · J.

Lemma 7.47 We have inf µ (B (x, κ)) ≡ c > 0.

x∈Ω1

∞

Proof. By contradiction, assume there exists a sequence of points {xk }k=1 ⊂ Ω1 such that µ (B (xk , κ)) → 0 as k → ∞. By compactness, there exists a subsequence, ∞ which we still relabel {xk }k=1
, converging
to some point x ∈ Ω1 . Now for k large enough, we have xk ∈ B x, κ2 and B x, κ2 ⊂ B (xk , κ). Hence   κ  µ B x, 6 µ (B (xk , κ)) → 0 as k → ∞ 2

and µ B x, κ2 = 0, which is impossible. We also recall the well-known Theorem 7.48 (Marcinkievicz interpolation theorem) Let  X = f = f1 + f2 : f1 ∈ L1 (Ω1 ) , f2 ∈ L2 (Ω1 ) and let T be a subadditive mapping defined on X taking values in the space of measurable functions on Ω0 , that is |T (f1 + f2 ) (x)| 6 |T (f1 ) (x)| + |T (f2 ) (x)| . Assume that for some p0 ∈ (1, +∞] there exist constants c1 , c2 > 0 such that kf kL1 (Ω1 ) for any f ∈ L1 (Ω1 ) , α > 0 µ ({x ∈ Ω0 : |T f (x)| > α}) 6 c1 α and, if p0 < ∞ !p0 kf kLp0 (Ω1 ) µ ({x ∈ Ω0 : |T f (x)| > α}) 6 c2 for any f ∈ Lp0 (Ω1 ) α while if p0 = ∞ kT f kL∞ (Ω0 ) 6 c2 kf kL∞ (Ω1 ) . Then for every p ∈ (1, p0 ) the operator T extends to Lp (Ω1 ) and there exists cp,p0 > 0 such that for every f ∈ Lp (Ω1 ) kT f kLp (Ω0 ) 6 cp,p0 kf kLp (Ω1 ) .

334

H¨ ormander operators

Proof. Let us first assume p0 < ∞. For any fixed α > 0 and f ∈ Lp (Ω1 ) with 1 < p < p0 , let us split f = f χ{|f |>α} + f χ{|f |6α} ≡ f1 + f2 . It is easy to see that f1 ∈ L1 (Ω1 ) while f2 ∈ Lp0 (Ω1 ). Therefore by our assumptions on T µ ({x ∈ Ω0 : |T f (x)| > α}) 6 µ ({x ∈ Ω0 : |T f1 (x)| > α/2}) + µ ({x ∈ Ω0 : |T f2 (x)| > α/2}) 2c1 2p0 c2 p kf1 kL1 (Ω1 ) + p0 kf2 kL0p0 (Ω1 ) α Z α Z 2c1 2p0 c2 p = |f | 0 dµ. |f | dµ + p0 α |f |>α α |f |6α

6

Using this bound on the distribution function of T f we can compute, by (7.22), Z +∞ p kT f kLp (Ω0 ) = pαp−1 µ ({x ∈ Ω0 : |T f (x)| > α}) dα 0

(

+∞

2p0 c2 6 pαp−1 |f (x)| dµ (x) + p0 α 0 Ω1 ,|f |>α ! Z |f (x)| Z p−2 |f (x)| = 2pc1 α dα dµ (x) Z

2c1 α

Ω1

p0

Z

α

p−1−p0

dα dµ (x)

p−1

Z |f (x)| · Ω1

=p

dµ (x) dα

Ω1 ,|f |6α

|f (x)|

Ω1



|f (x)|

p0

!

+∞

|f (x)|

+ 2 pc2

= 2pc1

)

Z

0

Z

p0

Z

|f (x)| p−1 Z

2c1 2p0 c2 + p − 1 p0 − p

dµ (x) + 2p0 pc2

Z |f (x)| Ω1

p0

p−p0

·

|f (x)| p0 − p

dµ (x)

p

|f (x)| dµ (x) . Ω1

Assume now p0 = ∞; for any fixed α > 0 and f ∈ Lp (Ω1 ) with 1 < p < ∞, let us split f = f χn|f |>

α 2c2

o

+ f χn|f |6

α 2c2

o

≡ f1 + f2 .

Then µ ({x ∈ Ω0 : |T f (x)| > α}) 6 µ ({x ∈ Ω0 : |T f1 (x)| > α/2}) + µ ({x ∈ Ω0 : |T f2 (x)| > α/2}) . However, since kT f2 kL∞ (Ω0 ) 6 c2 kf2 kL∞ (Ω1 ) 6 c2

α α = 2c2 2

(7.34)

Real analysis and singular integrals in locally doubling metric spaces

335

the second set in (7.34) is empty and we have Z 2c1 2c1 kf1 kL1 (Ω1 ) 6 |f (x)| dµ (x) α α Ω1 ,|f |> 2cα 2 ! Z Z +∞ 2c 1 p−1 |f (x)| dµ (x) dα 6 pα α α Ω1 ,|f |> 2c 0 2 ! Z Z 2c2 |f (x)| = 2pc1 |f (x)| αp−2 dα dµ (x)

µ ({x ∈ Ω0 : |T f (x)| > α}) 6 p

kT f kLp (Ω0 )

Ω1

0

(2c2 |f (x)|) p−1

Z |f (x)|

= 2pc1 Ω1

p−1

= 2pc1

(2c2 ) p−1

Z

p−1

dµ (x)

p

|f (x)| dµ (x) Ω1

and kT f kLp (Ω0 ) 6 cp kf kLp (Ω1 ) . 7.10

Notes

In section 7.1.1 we have just given some fundamental references for the classical theory of singular integrals and some of its less classical developments. Here we want to give the references for the path that we have followed in this chapter. Our notion of locally doubling metric space is a particular case of the more general notion of locally homogeneous space, introduced by Bramanti-Zhu in [40], and much of the material in the present chapter is an adaptation to this context of results previously developed in the literature, in (globally) homogeneous spaces. The estimates of fractional and singular integrals in H¨older spaces proved in section 7.4 are adapted from the paper by Bramanti-Brandolini [28], see also Wittman [160]. The general strategy consisting in proving L2 estimates for singular integrals as a consequence of estimates in H¨ older spaces as an application of Krein’s theorem (see [119]) is contained for the first time in the paper by Wittman [160]. That paper deals with singular integrals in spaces of homogeneous type, although the author’s assumptions are not immediately comparable to ours. The same technique has been applied later by Fabes-Mitrea-Mitrea [83] to give a new, simplified proof (in the Euclidean setting) of the T (1) theorem by David-Journ´ee [72], while Bramanti [23] has applied this idea in the context of nondoubling spaces. Our exposition of the proof of Krein’s theorem in section 7.5 follows [83], while the proof of L2 continuity is adapted from [23]. The Calder´ on-Zygmund theory developed in section 7.7 is adapted to the locally doubling context from the doubling theory by CoifmanWeiss [70]. The integral characterization of H¨ older spaces is due to Campanato [50] for the classical Euclidean case. The extension to the locally doubling context presented in section 7.8 is taken from [40].

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Chapter 8

Sobolev and H¨ older estimates for H¨ ormander operators on groups

8.1 8.1.1

Introduction Statement of the problem, notation and main results

The subelliptic estimates proved in Chapter 5 describe the regularizing properties of a H¨ ormander operator L in terms of classical Sobolev spaces. Roughly speaking, these estimates show that there exists ε > 0 such that when a distribution u is such that Lu is in the fractional Sobolev space H α then u is in H α+ε . This gain in regularity ε depends on the step s of the Lie algebra and the higher is the step the smaller is ε. This fact should be compared with the regularity properties of an elliptic operator (see e.g. [100, Chapter 9]) where a similar property holds with ε = 2 (or, in the Hilbert space theory of weak solutions ε = 1). There is a simple explanation for this phenomenon: the fractional Sobolev classes H α are isotropic, so that the measure of the regularity is the same on all directions. However, H¨ormander operators can be strongly anisotropic and we cannot expect a gain in regularity equal to 2 in all directions. Example 8.1 Let us consider, in the Heisenberg group H1 = R3 (see section 3.10.1), the vector fields X=

∂ ∂ ∂ ∂ + 2y ; Y = − 2x ∂x ∂t ∂y ∂t

and the associated sublaplacian L = X 2 + Y 2 . Recall that X, Y are 1-homogeneous with respect to the dilations defined in H1 , so that L is 2-homogeneous. Also, if k·k is any homogeneous norm on H1 , the function α

u (x) = kxk

is α-homogeneous and therefore Lu is α − 2 homogeneous. A simple computation 0 shows that, since H1 has homogeneous dimension Q = 4, then Lu ∈ Hloc = L2loc ∂ for every α > 0 (see Proposition 3.21). Now, let T = [X, Y ] = −4 ∂t and observe that since T is 2-homogeneous, then T 2 u is (α − 4)-homogeneous, hence T 2 u ∈ L2loc 2 only for α > 2. This means that when 0 < α < 2, we cannot have u ∈ Hloc even if 0 Lu ∈ Hloc . 337

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H¨ ormander operators

However, observe that the second order derivatives X 2 u and Y 2 u are (α − 2)homogeneous and therefore we have X 2 u, Y 2 u ∈ L2loc for every α > 0. In other words, the anisotropic nature of the operator L prevents a gain in regularity equal in all directions. Nevertheless, restricting our analysis to the derivatives along the vector fields X and Y used to construct the operator L, a gain in regularity of order two is possible. In this chapter we will consider a group G which is either a Carnot group or a homogeneous stratified group of type II and we will show that when L is a homogeneous H¨ ormander operator on G (see Definition 6.6) and we measure the regularity k,p using the Sobolev spaces WX that are adapted to the vector fields used to construct L (see Chapter 2), it is possible to recover estimates that are similar to those k,p k+2,p of elliptic operators, namely: if Lu ∈ WX then u ∈ WX . A similar result also k,α holds for the H¨ older classes CX . To state precisely our main results and fix some notation that we will use in the following, let us recall some of the basic facts about homogeneous groups. From now on (G, ◦, Dλ ) will be either a Carnot group or a homogeneous stratified group of type II on RN , of homogeneous dimension Q (see Chapter 3, section 3.1). We will consider a homogeneous H¨ ormander operator, as defined in section 6.2, that is q X L= Xi2 i=1

on Carnot groups or L=

q X

Xi2 + X0

i=1

on stratified groups of type II. As we have shown (see section 4.5 for sublaplacians, and section 6.2 for operators with drift), the operator L can be rewritten in terms of the canonical generators Y0 , Y1 , . . . , Yq , and the dependence of the system {Xi } on the system {Yi } can be quantitatively expressed saying that L belongs to some class Lν , for ν ∈ (0, 1) (see Definitions 4.37 and 6.8). We will denote with d (x, y) the control distance associated to the vector fields Xi (see section 1.29). Using the distance d we can also define a homogeneous norm on G (see Theorem 3.54 and section 3.8) setting kxk = d (x, 0). Recall that this homogeneous norm satisfies the triangle inequality kx ◦ yk 6 kxk + kyk . k,p Let us also recall the definition of the Sobolev spaces WX (Ω) and H¨older spaces α,k CX (Ω) that have been introduced in Chapter 2 (sections 2.1-2.2). If u ∈ Lp (Ω) k,p we will say that u ∈ WX (Ω) (1 6 p 6 ∞) if there exist, in weak sense, all the p derivatives XI u ∈ L (Ω) for |I| 6 k. In this case we will set X kukW k,p (Ω) = kukLp (Ω) + kXI ukLp (Ω) . X

|I|6k

Sobolev and H¨ older estimates for H¨ ormander operators on groups

339

k,p Moreover, the Sobolev space WX,0 (Ω) of functions “vanishing at the boundary” of k,p ∞ Ω is the closure of C0 (Ω) in the norm of WX (Ω). For 1 6 p 6 +∞ we will also set X

k

D u p = kXI uk p . L (Ω)

L (Ω)

|I|=k

Note that since kDukLp (Ω) =

q X

kXj ukLp (Ω)

i=1 1,p the Sobolev space WX (Ω) does not take into account derivatives with respect to the vector field X0 even when this vector field is necessary to generate the Lie algebra of G. On the other hand the second order derivatives term includes

one

derivative k

along the vector field X0 . In some cases when Ω = G instead of D u Lp (RN ) we

will use the simpler notation Dk u . p

k,α The spaces CX (Ω) are defined as follows. For a function u defined in Ω let   |u (x) − u (y)| |u|C α (Ω) = sup : x, y ∈ Ω, x 6= y , α X kx−1 ◦ yk

kukC α (Ω) = kukL∞ (Ω) + |u|C α (Ω) . X

X

k,α We say that u ∈ CX (Ω) (0 < α 6 1) if there exist (as classical directional derivatives) all the derivatives XI f for |I| 6 k and the quantity X kukC k,α (Ω) = kukC α (Ω) + kXI ukC α (Ω) X

X

X

|I|6k

is finite. We will also use the shorthand notation X

k

D u α = kXI ukC α (Ω) . C (Ω) X

X

|I|=k

For operators in the class Lν , in this chapter we obtain three kinds of regularity estimates for distributional solutions to the equation Lu = f : global Sobolev estimates, local Sobolev estimates and local H¨older estimates. We stress that all our a priori estimates are uniform in the class Lν , since the constants involved depends on the “ellipticity constant” ν but not on the specific operator L in the class Lν . We start with the global result: Theorem 8.2 (Global Sobolev regularity) Let L be in Lν for some ν > 0, let 1 < p < +∞ and k ∈ N (assume k even if G is a stratified homogeneous group of type II). Moreover, we assume the homogeneous dimension Q > 2 if G is a Carnot group and Q > 4 if G is type II.
Then, there exists c = c (p, k, G, ν)
stratified of k,p such that if u ∈ Lp RN and Lu ∈ WX RN (which means that the distribution

k,p k+2,p Lu can be identified with a function in WX RN ), then u ∈ WX RN and   kukW k+2,p (RN ) 6 c kLukW k,p (RN ) + kukLp (RN ) . (8.1) X

X

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H¨ ormander operators

Theorem 8.2 will be proved through Theorems 8.27, 8.30, 8.35. Remark 8.3 (Assumption on k) The restriction on the index k when G is a stratified homogeneous group of type II is due to the presence of the drift term X0 . Namely, the technique that we will use in section 8.4.2 to prove representation formulas for derivatives of order k + 2 of u in terms of derivatives of order k of Lu does not work in presence of the drift (see Remark 8.33 for an explanation of the exact nature of this technical problem). In the drift case we will prove, with a different technique, a representation formula for the derivatives of even order of u (see section 8.4.3). Remark 8.4 (Assumption on Q) The different assumptions made on the homogeneous dimension of the group are due to the techniques used to prove higher order estimates for operators with or without drift term. While the condition Q > 2 is necessary for the existence of a global homogeneous fundamental solution of L (see Theorem 6.18), which is the basic tool in the proof of the a priori estimates, the assumption Q > 4 in the case of a stratified group of type II is more a technical requirement, related to the possibility of handling the convolution of two fundamental solutions (see Proposition 8.16 and the proof of Lemma 8.36). It is important to realize that both these assumptions on Q are harmless, as far as we are interested in developing a theory dealing with degenerate equations. Actually, if G is a Carnot group, then Q = 2 happens only if we are considering R2 with the Euclidean structure, which means that L is a constant coefficient elliptic operator in 2 variables. On the other hand, if G is homogeneous stratified of type II with Q 6 4, then the operator L (with drift) is necessarily parabolic. Actually, in order for L to be an ultraparabolic degenerate operator, we must have at least three variables x0 , x1 , x2 , with L = X12 + X0 ; then the variables x1 , x0 corresponding to X1 and X0 must have homogeneities 1, 2, respectively, while the variable x3 must correspond to a commutator, hence must have homogeneity at least 3, which forces Q > 6. The next theorem is a local version of the previous global result, useful when Lu possesses a certain amount of regularity only on a bounded domain Ω ⊂ RN . Moreover, it also contain an existence result. Theorem 8.5 (Local Sobolev regularity and solvability) Let L be in the class Lν for some ν > 0, let Ω be a bounded domain in G, let Ω0 b Ω00 b Ω and let k ∈ N (assume k even if G is a stratified group of type II). Moreover, we assume the homogeneous dimension Q > 2 if G is a Carnot group, and Q > 4 if G is stratified k,p of type II. If u is a distribution in Ω such that Lu ∈ WX (Ω) (1 < p < ∞) then k+2,p 00 u ∈ WX (Ω ) and n o kukW k+2,p (Ω0 ) 6 c kLukW k,p (Ω) + kukLp (Ω00 ) X

X

where c = c(p, k, G, Ω, Ω0 , ν). Moreover, for every f ∈ Lp (Ω) there exists u ∈ 2,p WX,loc (Ω) satisfying the equation Lu = f in Ω (and therefore satisfying the above a priori estimate for k = 0).

Sobolev and H¨ older estimates for H¨ ormander operators on groups

341

Theorem 8.5 will be proved through Theorem 8.44, Proposition 8.45 and section 8.5.3. As it should be apparent, the above theorem greatly sharpens the regularity results proved by means of the subelliptic estimates in Chapters 4 and 5 (for sublaplacians on homogeneous groups and for general H¨ormander operators, respectively). While those results have been precious to prove hypoellipticity of H¨ ormander operators (which corresponds to the regularity estimates for k = ∞), it is when Lu has only a partial regularity that the present result shows the power of its quantitative information: the idea is simply that, if we measure the regularity of u in terms of derivatives with respect to the vector fields, then the operator L regularizes u with a gain of exactly 2 derivatives, as uniformly elliptic operators do with respect to Cartesian derivatives. Remark 8.6 (Local vs. global Sobolev regularity) Comparing the statements of Theorems 8.2 and 8.5 one
can ask: if u is a distributional global solution to Lu
= k,p RN (but we do not know in advance that u ∈ Lp RN ), f in RN and f ∈ WX what can we say about u? Applying Theorem 8.5 to any Ω0 b Ω00 b RN one can

k+2,p k+2,p conclude that u ∈ WX,loc RN . However, we cannot assure that u ∈ WX RN
p N without knowing that u ∈ L R , as shown by the trivial example u ≡ 1, which is
k+2,p a distributional solution to Lu = 0 but does not belong to any space WX RN . Finally we can also prove similar local results in H¨older spaces: Theorem 8.7 (Local H¨ older regularity and solvability) Let L be in the class Lν for some ν > 0, let Ω be a bounded domain in G, let Ω0 b Ω00 b Ω and let 0 < α < 1 and k ∈ N (assume k even if G is a stratified group of type II). Moreover, we assume the homogeneous dimension Q > 2 if G is a Carnot group, and Q > 4 k,α if G is stratified of type II. If u is a distribution such that Lu ∈ CX (Ω) for some k+2,α (Ω00 ) and integer k = 0, 1, 2, . . . then u ∈ CX n o kukC k+2,α (Ω0 ) 6 c kLukC k,α (Ω00 ) + kukL∞ (Ω00 ) X

X

α (Ω) there exists u ∈ where c = c(α, k, G, Ω0 , Ω00 , ν). Moreover, for every f ∈ CX,0 2,α CX,loc (Ω) satisfying the equation Lu = f in Ω (and therefore satisfying the above a priori estimate for k = 0).

Theorem 8.7 will be proved through Theorems 8.57, 8.60, 8.65 and Proposition 8.58. H¨ older spaces are not well suited to global estimates, so we confine ourselves to the above local result. However, we point out the following global estimate on the H¨ older seminorms of second order derivatives (which will be proved as Corollary 8.51):

342

H¨ ormander operators

α Proposition 8.8 (Global estimate for CX seminorm) Let L be in the class 2,α Lν for some ν > 0 and let 0 < α < 1. For every u ∈ CX,0 (G) we have 2 D u α 6 c (G, ν, α) |Lu|C α (G) . C (G) k,p Remark 8.9 (Equivalent norms) In view of Proposition 6.12, the WX (Ω) norms are equivalent as the system {Xi } ranges in a fixed class Lν ; in particular, they are equivalent to the norm WYk,p (Ω) induced by the canonical generators {Yi } , with constants depending on G, ν, k, p. This gives to all the estimates stated above a further uniformity. A completely analogous statement holds for H¨ older norms k,α CX (Ω), in view of Proposition 6.14. Also, recall that the homogeneous norms q kxk = dX (x, 0) are all equivalent with uniform constants, as the system {Xi }i=0 ranges in the class Lν (see Proposition 6.13). This equivalence will be tacitly used in the following.

In Chapter 11 we will address the regularity problem analogous to that studied in this chapter, in the more general context of H¨ormander vector fields, without an underlying group structure. As the reader will see, the results will be quite similar to those of this chapter, although the proofs will be more difficult and considerably longer. In particular, some results of this chapter are strictly contained in the corresponding results of Chapter 11, so that the reader could ask the reason of this double procedure. A couple of remarks are in order on this subject. First, in this chapter we have emphasized also some results which do not have a corresponding result in the more general context: the possibility of proving global estimates in Sobolev spaces (Theorem 8.2) is peculiar of the group case, and does not have an analog for any system of H¨ ormander vector fields. Also, for operators with drift we prove in this chapter higher order Sobolev and H¨older regularity estimates (although with the restriction of even order), while in the case of general H¨ormander vector fields, in Chapter 11, we shall content of a priori estimates on the second order derivatives, for major technical difficulties in extending the result, in the drift case, to higher order derivatives. A second remark is the following. Also for the case of local a priori estimates for sublaplacians, a case which is strictly covered by the results that we will prove in Chapter 11 for general H¨ormander vector fields, we believe that a presentation of the shorter and more transparent proof which is possible on groups is a precious guide to orient the reader in the study of the more involved proof which is necessary in the general case. 8.1.2

Strategy of the proof and plan of the chapter

As already said, one of the main tools used for the proof of the above results is the homogeneous fundamental solution for operators in the class Lν that has been constructed in Chapter 6, and the associated representation formulas. We will always denote this fundamental solution with Γ. In the next theorem we collect some of the properties of Γ proved in Chapter 6, that will be useful in the following.

Sobolev and H¨ older estimates for H¨ ormander operators on groups

343

Note that all the constants depend on the operator L only through the “ellipticity constant” ν. Theorem 8.10 Let L ∈ Lν . There exists a fundamental solution Γ for L which is (2 − Q)-homogeneous and satisfies the following bounds: (i) There exists c = c (G, ν) such that for every x ∈ G\ {0} c |Γ (x)| 6 . Q−2 kxk (ii) For every multiindex I and differential operator XI there exists c = c (I, G, ν) such that for every x ∈ G\ {0} c . |XI Γ (x)| 6 Q+|I|−2 kxk More generally, for every k-homogeneous differential operator Z k (even not translation invariant) and x ∈ G\ {0}
k k Z Γ (x) 6 c Z , G . Q+k−2 kxk (iii) For every ϕ ∈ C0∞ (G), x ∈ G Z
ϕ (x) = Γ y −1 ◦ x Lϕ (y) dy G Z
L Γ y −1 ◦ x ϕ (y) dy = Lϕ (x)

(8.2) (8.3)

G

(iv) For every ϕ ∈ C0∞ (G), i = 1, . . . , q, x ∈ G Z
Xj ϕ (x) = Xj Γ y −1 ◦ x Lϕ (y) dy.

(8.4)

G

(v) For every i, j = 1, . . . , q there exist constants cij such that for every ϕ ∈ C0∞ (G), x ∈ G, Z
Xi Xj ϕ (x) = lim Xi Xj Γ y −1 ◦ x Lϕ (y) dy + cij Lϕ (x) (8.5) ε→0

ky −1 ◦xk>ε

To have an idea of the tools necessary to prove Theorem 8.2 let us consider the case k = 0. The estimate in (8.1) reduces to  

2

D u p N + kDuk p N + kuk p N 6 c kLuk p N + kuk p N (8.6) L (R ) L (R ) L (R ) L (R ) L (R ) If u is a test function we can apply the theory of singular integrals developed in Chapter 7 to the representation formula for second order derivatives (8.5) to get

2

D u p N 6 c kLuk p N . L (R ) L (R ) Also, an interpolation inequality of the kind  

kXi ukLp (RN ) 6 c Xi2 u Lp (RN ) + kukLp (RN )

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H¨ ormander operators

allows to write    

kDukLp (RN ) 6 c D2 u Lp (RN ) + kukLp (RN ) 6 c kLukLp (RN ) + kukLp (RN ) that gives (8.6). To control higher order derivatives of u in terms of higher order derivatives of Lu, the basic idea is that of transferring the derivative Xj from Γ to Lu in the representation formula (8.4). Doing this is not trivial, but requires expressing left invariant vector fields in terms of right invariant vector fields, which serves the scope thanks to the identity Xi f ∗ g = f ∗ XiR g (see Proposition 3.47). This idea is useful on Carnot groups, while in presence of a drift X0 of weight 2 does not work, and must be replaced by a different technique which, however, only works for derivatives of even order. Finally, localization of the global estimates is a procedure which can be carried out imitating what is usually done in the study of linear elliptic equations: here the standard tools are suitable cutoff functions and interpolation inequalities for Lp or C α norms of the derivatives of intermediate order. Producing these tools in the context of Sobolev or H¨older spaces induced by H¨ ormander vector fields is therefore another necessary step to achieve the desired results. The structure of this chapter is the following. In sections 8.2 and 8.3 we present some preliminary material which is related to fractional and singular integrals in homogeneous groups. The results are partly local and partly global. While the local results are just applications to our setting of the theory developed in Chapter 7 in abstract locally doubling spaces, the global results are essentially based on the structure of homogeneous group. In section 8.2.2 we have also inserted some material about Sobolev embeddings in Carnot groups that, although will not be used in the following, are strictly related to the results presented here and have an independent interest. The main results of the chapter are contained in the subsequent three sections. Section 8.4 contains global a priori estimates and regularity results in Sobolev spaces (Theorem 8.2). First, in section 8.4.1 we deal with second order estimates. Higher order estimates are proved, separately for sublaplacians on Carnot groups and for H¨ ormander operators with drift on stratified groups of type II, in sections 8.4.2 and 8.4.3. In section 8.5 we prove analogous results in local form (Theorem 8.5). After introducing, in section 8.5.1, some properties of cutoff functions and interpolation inequalities for Sobolev norms, we deal in section 8.5.2 with the case of second order derivatives and in section 8.5.3 with higher order estimates. Finally, in section 8.6 we deal with a priori local estimates and k,α regularity results in the H¨ older spaces CX (Ω) (Theorem 8.7). Sections 8.6.1 and 8.6.2 contain some auxiliary results. Section 8.6.3 and 8.6.4 contain the proof of Theorems 8.57 and 8.7, respectively. This chapter relies on the results of Chapter 3 on homogeneous groups, of Chapter 6 on the global homogeneous fundamental solution of L in homogeneous groups, and on some general properties of function spaces studied in Chapter 2.

Sobolev and H¨ older estimates for H¨ ormander operators on groups

8.2

345

Homogeneous kernels on G, fractional integrals and Sobolev embeddings

8.2.1

Homogeneous kernels and fractional integrals on G

As already explained, in this chapter we will apply the real variable theory in locally doubling spaces developed in Chapter 7 to deduce the continuity, on Lp or C α spaces, of suitable singular or fractional integral operators. Sometimes, however, we will also need to handle integral operators defined on the whole space, which do not fit the framework of that local theory, but on the other hand are easily analyzed by similar techniques, exploiting the fact that on the homogeneous group G the doubling condition holds globally, i.e. with no restriction on the center or radius of the balls, and there is a notion of convolution, analogous to the Euclidean one but induced by the group structure. In this section we will discuss some results which complement the real analysis material presented in Chapter 7 with some global results which are typical of homogeneous groups. We start with the definition of (global) Hardy-Littlewood maximal function on G, with the related Lp inequality. Definition 8.11 The Hardy-Littlewood maximal function on G is defined as follows. For f ∈ L1loc (G), Z 1 Mf (x) = sup |f (z)| dz. r>0 |B (x, r)| B(x,r) Then: Theorem 8.12 For every p ∈ (1, ∞) there exists c = c (p, G) such that for every f ∈ Lp (G) , kMf kLp (G) 6 c kf kLp (G) . Proof. The result is very similar to that proved in Chapter 7, section 7.6, so we will limit ourselves to some brief remarks. The general idea is that the present result is easier than the one we have already proved (Theorem 7.25) because we can use a stronger assumption: the global doubling condition. Therefore, Vitali covering lemma (Lemma 7.23) can be proved in the same way and, with this in hand, the (1, 1)-weak-type estimate (Theorem 7.25 (b)), which the Lp continuity is based on, can be proved analogously. The only point which is worthwhile to be noted is the following. In the present definition of Mf , there is not, a priori, an upper bound for the radii of the balls involved; on the other hand, Vitali covering Lemma requires that the initial covering consists in a family of balls with bounded radii. However
1 N (see the proof of Theorem 7.25 (b)), if f ∈ L R , then for every x ∈ At ≡  x ∈ RN : Mf (x) > t there exists B (xx , rx ) 3 x such that xx ∈ RN , and Z 1 |f (y)| dy > t. |B (xx , rx )| B(xx ,rx ) But this implies crxQ 6 kf kL1 (RN ) /t, which gives an upper bound for rx .

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H¨ ormander operators

In this chapter we will use extensively the convolution with the fundamental solution Γ and its derivatives with respect to the vector fields. Since these functions are homogeneous it is natural to introduce the concept of homogeneous kernel and to study its convolution properties. We start with the following definition. Definition 8.13 (Kernel of
type α > 0 on G) A kernel of type α > 0 on G is a function T ∈ C ∞ RN \ {0} which is homogeneous of degree α − Q. In this case, for every k ∈ N we set ΛT,k = sup sup ∂ β T (w) . |β|6k kwk=1

Note that a kernel of type α > 0 is locally integrable, because by homogeneity and the continuity of f outside the origin, α−Q

|T (x)| 6 max |T (w)| kxk kwk=1

α−Q

= ΛT,0 kxk

(8.7)

which is locally integrable for every α > 0, by Proposition 3.21. The next proposition shows that the convolution of a kernel of type α > 0 with a function in Lp is well defined, for a certain range of p and α. Moreover the convolution is Lp improving, in the sense that it maps functions in Lp to functions in Lq for a certain q > p. Proposition 8.14 Let T be a kernel of type α, 0 < α < Q, 1 < p < Q/α and let
1/q = 1/p − α/Q. If f ∈ Lp RN then T ∗ f and f ∗ T are defined a.e. and kf ∗ T kq 6 c kf kp ,

(8.8)

kT ∗ f kq 6 c kf kp .

(8.9)

Moreover c = c (G, ΛT,0 , α, p). Proof. We will prove (8.8), the proof
of (8.9) being analogous. By (8.7) we have α−Q p N |T (x)| 6 c kxk . Let f ∈ L R and let R > 0, then Z Z

f x ◦ y −1 kykα−Q dy + c f x ◦ y −1 kykα−Q dy |f ∗ T (x)| 6 c kyk6R

kyk>R

≡ I1 (x) + I2 (x) . As for I1 we have I1 (x) = c

6

+∞ Z X


f x ◦ y −1 kykα−Q dy

−k−1 R

Z 6 kf kp

R

αp0 −Qp0

αp0 −Qp0

kwk

!1/p0 dwR

= c (α, p) kf kp Rα−Q/p .

Q

kwk>1

Note that since p
1, Z

q

RN

(Mf (x))

q−p

|f ∗ T (x)| dx 6 c kf kp

Z RN

p

q

[Mf (x)] dx 6 c kf kp .

An immediate consequence of the previous proposition is that the convolution of a kernel with a function in Lp can be iterated. The next proposition establishes when this is possible and shows that it is associative. Proposition 8.15 Let T be a kernel of type α, with 0 <
α < Q, let 1 6 p 6
+∞, 1 1 1 α p 1 0. If f ∈ L
f ∗ (g ∗ T ) and (f ∗ g) ∗ T are well defined as elements of Lr RN and they are equal. Proof. We start by proving that the
operators (f, (f ∗
g) ∗ T and (f, g) 7→
g) 7→ α p N q N r , f ∗ (g ∗ T ) are bounded from L R ×L R to L RN . Let 1s = 1q − Q then by Proposition 8.14 we have kg ∗ T ks 6 c kgkq and by Young’s inequality (see α Proposition 3.45) since 1r = p1 + 1s − 1 = p1 + 1q − Q − 1 > 0 we obtain kf ∗ (g ∗ T )kLr (RN ) 6 kf kp kg ∗ T ks 6 c kf kp kgkq . Similarly let s10 = p1 + 1q − 1 > 0, then kf ∗ gks0 6 kf kp kgkq and since 1 < s0 < we can apply Proposition 8.14 to obtain k(f ∗ g) ∗ T kr 6 c kf ∗ gks0 6 c kf kp kgkq .

Q α

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H¨ ormander operators


Assume that f, g have compact support. Since T ∈ L1loc RN by Fubini’s theorem  Z Z

−1 (f ∗ g) ∗ T (x) = f (z) g z ◦ y dz T y −1 ◦ x dy N RN ZR Z

= f (z) g z −1 ◦ y T y −1 ◦ x dydz N N ZR ZR
= f (z) g (w) T w−1 ◦ z −1 ◦ x dwdz N RN ZR
= f (z) (g ∗ T ) z −1 ◦ x dz = f ∗ (g ∗ T ) . RN


Exploiting the continuity of the operators (f ∗ g) ∗ T and f ∗ (g
∗ T ) on Lp RN ×

Lq RN we see that (f ∗ g)∗T = f ∗(g ∗ T ) for every f ∈ Lp RN and g ∈ Lq RN .

In the next proposition we consider the convolution of two homogeneous kernels. Proposition 8.16 Let T1 be a kernel of type α1 > 0 and T2 be kernel of type α2 > 0. Suppose α1 + α2 < Q. Then Z
T1 ∗ T2 (x) = T1 x ◦ y −1 T2 (y) dy RN


defines a kernel of type α1 + α2 and for every ϕ ∈ C0∞ RN we have (ϕ ∗ T1 ) ∗ T2 = ϕ ∗ (T1 ∗ T2 ) .

(8.10)

Moreover for every k ∈ N there exists K ∈ N such that ΛT1 ∗T2 ,k 6 c (G) ΛT1 ,K ΛT2 ,K .

(8.11)

Proof. We will show that the function T1 ∗ T2 (x) is well defined and smooth in a neighborhood of every fixed x0 ∈ RN , x0 6= 0. Let ε = 13 kx0 k, pick φ1 ∈ C0∞ (Rn ) such that φ
1 (y) = 1 if kyk < ε/2 and φ1 (y) = 0 if kyk > ε and set φ2 (y) = N φ1 x0 ◦ y −1

. Note

that φ1 and φ2 have disjoint supports. For every x ∈ R −1

satisfying x0 ◦ x < ε/4 write Z
T1 ∗ T2 (x) = T1 x ◦ y −1 T2 (y) dy RN Z Z Z = {· · · } φ1 (y) dy + {· · · } φ2 (y) dy + {· · · } (1 − φ1 (y) − φ2 (y)) dy ≡ I1 (x) + I2 (x) + I3 (x) . Writing x0 = x0 ◦ x−1 ◦ x ◦ y −1 ◦ y we see that the choices of ε and x imply

ε 3ε = kx0 k 6 + x ◦ y −1 + kyk , 4 so that

11 ε 6 x ◦ y −1 + kyk . (8.12) 4

Sobolev and H¨ older estimates for H¨ ormander operators on groups

349



Similarly, 3ε 6 x0 ◦ y −1 +  kyk. By (8.12), if kyk < ε we have x ◦ y −1 > 74 ε so  α −Q that the integrand in I1 is O kyk 2 and I1 is absolutely convergent. Similarly  



x0 ◦ y −1 < ε implies kyk > 2ε, in I2 the integrand is O x ◦ y −1 α1 −Q so that I2 is absolutely convergent. Finally in I3 we necessarily have kyk > ε/2

−1

and x0 ◦ y > ε/2 so that





ε

ε/2 < x0 ◦ y −1 < x0 ◦ x−1 + x ◦ y −1 < + x ◦ y −1 4

−1

implies x ◦ y > ε/4. It follows   that in I3 when kyk is large we have
α1 +α2 −2Q −1 T1 x ◦ y T2 (y) = O kyk . Since α1 + α2 < Q, also I3 is absolutely convergent (see Proposition 3.21). It follows that T1 ∗ T2 (x) is defined for every x 6= 0. Moreover if P is a right invariant differential operator of arbitrary order we have Z
P I1 (x) = P T1 x ◦ y −1 T2 (y) φ1 (y) dy (8.13) N R

so that I1 is smooth in x0 ◦ x−1 < ε. Similarly if P is a left invariant differential operator of arbitrary order we have Z 
P I2 (x) = P T1 x ◦ y −1 T2 (y) φ2 (y) dy Z 

−1 −1 =P T1 (z) T2 z ◦ x φ2 z ◦ x dz (8.14) Z
= T1 (z) P (T2 φ2 ) z −1 ◦ x dz

so that I2 is smooth in x0 ◦ x−1 < ε. As for I3 observe that integrand is now smooth. If P is a β-homogeneous right invariant operator then Z
P I3 (x) = P T1 x ◦ y −1 T2 (y) (1 − φ1 (y) − φ2 (y)) dy. (8.15)
P T1 x ◦ y −1 T2 (y) = Indeed, since P T (y) is α − Q − β homogeneous we have 1 1   α +α −2Q−β O kyk 1 2 so that integral is absolutely convergent. The above computation shows that T1 ∗ T2 (x) is smooth away from the origin. Observe now that for every λ > 0, Z
T1 ∗ T2 (Dλ (x)) = T1 Dλ (x) ◦ y −1 T2 (y) dy N ZR   −1 T2 (Dλ (z)) λQ dz = T1 Dλ (x) ◦ Dλ (z) N R Z

= λQ T1 Dλ x ◦ z −1 T2 (Dλ (z)) dz RN

= λα1 +α2 −Q T1 ∗ T2 (x) , so that T1 ∗ T2 is a kernel of type α1 + α2 . The bound (8.11) can be obtained from (8.13), (8.14) and (8.15) writing any derivative as a linear combination of left

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H¨ ormander operators


invariant or right invariant differential operators. To prove (8.10) let ϕ ∈ C0∞ RN
q and let 1 < p < α1 Q RN with 1q = p1 − αQ1 . It +α2 . By Proposition 8.14, ϕ ∗ T1 ∈ L
follows that (ϕ ∗ T1 ) ∗ T2 is well defined and belongs to Lr RN with 1r = 1q − αQ2 =
α1 +α2 1 r RN . It p − Q . Applying Proposition 8.14 we see that also ϕ ∗ (T1 ∗ T2 ) ∈ L remains to show that these two functions are actually the same. To see this let us write T1 = S0 + S∞ where S0 (x) = T1 (x) when kxk
6 1 and S0 (x) = 0 otherwise.
Q . Then for ε ∈ (0, q), S0 ∈ Lq−ε RN and S∞ ∈ Lq+ε RN . Pick Let q = Q−α 1 ε > 0 so that 1 < q − ε < q + ε < αQ2 (note that this is possible since α1 + α1 < Q), then by
Proposition 8.15 (ϕ ∗ S0 ) ∗ T2 and ϕ ∗ (S0 ∗ T2 ) are equal as elements of 1 − αQ2 − 1. Similarly (ϕ ∗ S∞ ) ∗ T2 and ϕ ∗ (S∞ ∗ T2 ) Lr0 RN , with r10 = p1 + q−ε
1 are equal as elements of Lr1 RN , with r11 = p1 + q+ε − αQ2 − 1. It follows that (ϕ ∗ T1 )∗T2 and ϕ∗(T1 ∗ T2 ) are equal almost everywhere, and therefore everywhere, because both sides are smooth functions, since ϕ is smooth. We also need the following result: Proposition 8.17 (Kernel of type α > 0 and H¨ older spaces) Let T be a kernel of type α, with 0 < α < Q and let Br be a ball of radius r in G. For every δ (Br ) we δ ∈ (0, α), there exists c = c (G, ΛT,1 , r, δ, α) > 0 such that every u ∈ CX,0 have ku ∗ T kC δ (Br ) 6 c kukL∞ (Br ) . Proof. To prove the above inequality we will use the abstract result of Chapter 7 for fractional integrals (Theorem 7.14). To apply this theory we need a suitable setting of locally doubling space that satisfies condition H1-H4 of section 7.2. We make the following choice: Ω0 = B2r , Ω1 = B20r , Ω2 = B40r and κ = 18r. Let us show that the kernel of the convolution operator satisfies the required estimate of fractional integral. We can write Z Z
u ∗ T (x) = T y −1 ◦ x u (y) dy ≡ K (x, y) u (y) dy. G

G

Q and since1 B (x; y) = c (G) y −1 ◦ x α

α−Q d (x, y) . |K (x, y)| 6 y −1 ◦ x sup |T (w)| 6 c (G) sup |T (w)| B (x; y) kwk=1 kwk=1

Also, by Proposition 6.25 when d (x0 , y) > M (G) d (x0 , x) we have

α c (ΛT,1 , G) x−1 ◦ x0 d (x0 , y) d (x0 , x) |K (x0 , y) − K (x, y)| 6 = c (Λ , G) . T,1 Q−α+1 B (x; y) d (x0 , y) ky −1 ◦ x0 k Rephrasing these estimates with the language of Chapter 7, section 7.3, we can say that K (x, y) is a fractional integral kernel of exponent β = 1 and ν = α. We now 1 Recall

the notation B (x; y) = |B (x, d (x, y))| .

Sobolev and H¨ older estimates for H¨ ormander operators on groups

351

define a localized kernel as in Definition 7.9. Let R = 2r, let a be cutoff function e (x, y) = a (x) K (x, y) a (y). By Proposition 7.11, such that Br ≺ a ≺ B 32 r and set K e (x, y) is a fractional integral kernel with the same exponents as K (x, y). Also K e (x, y) = 0. By Theorem 7.14 and Remark 7.16 observe that if d (x, y) > 4r then K for every δ < ν = α we have ku ∗ T kC δ (B4r ) 6 c kukL∞ (B4r ) . Since u is supported in Br we obtain ku ∗ T kC δ (Br ) 6 c kukL∞ (Br ) . 8.2.2

Sobolev embedding in Carnot groups

An interesting consequence of Proposition 8.14, coupled with the properties of the homogeneous fundamental solution Γ, is a version of Sobolev embedding which can be easily proved in Carnot groups. Although we will not make use explicitly of this property, it is worthwhile to be stated for its independent interest. Theorem 8.18 (Sobolev embedding in Carnot groups) Let G be a Carnot 1,p group with homogeneous dimension Q and let WX (G) be the Sobolev space defined Pq by a system X1 , . . . , Xq of generators such that L = i=1 Xi2 belongs to Lν . Then ∗ 1,p (G) ⊂ Lp (G) with 1/p∗ = 1/p − 1/Q and for every p ∈ (1, Q), WX q X kφkp∗ 6 c (p, G, ν) kXi φkp . (8.16) i=1 ∗

1,p Also, for any domain Ω we have WX,0 (Ω) ⊂ Lp (Ω) with 1/p∗ = 1/p − 1/Q and the same estimate.

Proof. Let Γ be the homogeneous fundamental solution L. By Remark 6.22, for every φ ∈ C0∞ (G), Z q q Z X
X
2 −1 φ (x) = Γ x ◦y Xi φ (y) dy = − Xi Γ x−1 ◦ y Xi φ (y) dy G q  X

i=1

i=1

G

d

=− Xi φ ∗ Xi Γ (x) d i=1
where Xi Γ (x) = (Xi Γ) x−1 is clearly a kernel of type 1. Then, by Proposition Pq 8.14, for every p ∈ (1, Q) and 1/q = 1/p − 1/Q, we have kφkq 6 c i=1 kXi φkp with c depending on G, p and
ΛXi Γ,0 = sup Xi Γ w−1 6 c (G, ν) kwk=1 1,p by Theorem 6.20, (1). By Theorem 3.49 we know that C0∞ (G) is dense in WX (G), 1,p hence the estimate (8.16) holds for any φ ∈ WX (G). On the other hand, if we apply 1,p (Ω) . (8.16) to every φ ∈ C0∞ (Ω) by density we conclude (8.16) for every φ ∈ WX,0

With a little more effort we can also prove the embedding of Sobolev spaces 1,p WX,0 (Ω) for Ω bounded and p > Q in suitable H¨older spaces. We first need a result on the regularizing property of fractional integrals in a bounded domain.

352

H¨ ormander operators

Proposition 8.19 Let Z I1 f (x) =

f (y)

Ω ky −1

◦ xk

Q−1

dy

α for x ∈ Ω ⊂ B (0, R) and f ∈ Lp (Ω) for some p > Q. Then I1 f ∈ CX (Ω) with α = 1 − Q/p and

kI1 f kL∞ (Ω) 6 c (G, p) R1−Q/p kf kLp (Ω) |I1 f |C α (Ω) 6 c (G, p) kf kLp (Ω) . X

Proof. By H¨ older’s inequality, with 1/p + 1/q = 1, Z |I1 f (x)| 6 kf kLp (Ω)

Ω

!1/q

dy ky −1 ◦ xk

(Q−1)q

.

For x, y ∈ Ω we have y −1 ◦ x 6 2R; moreover, p > Q implies (Q − 1) q < Q, hence, by Proposition 3.21 Z Z dy dy 6 = cRQ−(Q−1)q (Q−1)q (Q−1)q −1 kyk2kx−1 2 ◦x1 k !1/q Z

−1

dy 6 c x2 ◦ x1 kf kLp (Ω) . Qq ky −1 ◦ x1 k Ω,ky −1 ◦x1 k>2kx−1 2 ◦x1 k Again by Proposition 3.21, since Qq > Q, Z Z dy dy c (G) 6 =

Qq−Q Qq Qq −1 −1 −1 −1 −1

◦ x1 k Ω,ky ◦x1 k>2kx2 ◦x1 k ky kyk>2kx2 ◦x1 k kyk x2 ◦ x1 hence

1−(Qq−Q)/q

1−Q/p

A 6 c x−1 kf kLp (Ω) = c x−1 kf kLp (Ω) . 2 ◦ x1 2 ◦ x1

Sobolev and H¨ older estimates for H¨ ormander operators on groups

353

Also, Z B6

|f (y)|

|f (y)|

Ω,ky −1 ◦x

1 kε

Then, for every 1 < p < +∞, there exists a constant c = c (p, G, ΛT,1 ) > 0 such
∞ N that for every u ∈ C0 R we have kT ukp 6 c kukp . Remark 8.23 Observe that the limit in (8.19) exists by Proposition 6.29 and is independent of the particular choice of homogeneous norm used to define T . We defer the proof of the theorem for a lemma that shows how to truncate the kernel inside (8.19).
Lemma 8.24 Let f ∈ C ∞ RN \ {0} be a −Q-homogeneous function with vanishing mean, i.e. M (f ) = 0 (see Proposition 6.26 for the definition of M (f )). Let R0 > 0 and let φ : [0, ∞) → [0, 1] be a smooth function such that φ (t) = 0 for t > 2R0 and φ (t) = 1 for t < R0 . Consider the truncated kernel



K (x, y) = f y −1 ◦ x φ y −1 ◦ x ,

Sobolev and H¨ older estimates for H¨ ormander operators on groups

355

then, there exist positive constants A, B depending on G, R0 and Λf,1 such that A , B (x; y) B d (x0 , x) |K (x0 , y) − K (x, y)| 6 B (x; y) d (x0 , y) |K (x, y)| 6

(8.20) if d (x0 , y) > 2d (x0 , x) .

(8.21)

Moreover, for 0 < r1 < r2 < R0 we have Z K (x, y) dy = 0.

(8.22)

r1 M x−1 ◦ x0 , then for suitable constants c1 and c2 , we have



c1 y −1 ◦ x0 6 y −1 ◦ x 6 c2 y −1 ◦ x0 .

Let e c = 2R0 max (1, c2 ) and assume

y −1 ◦ x

> e c. Since e c > 2R0 we see that

K (x, y) = 0. Also, since y −1 ◦ x0 > c12 y −1 ◦ x > cec2 > 2R0 we have K (x0 , y) = 0 and therefore

this case there is nothing to−1prove. We are re −1 in

c/c1 and c. From this we obtain y ◦ x0 6 e duced to considering y ◦ x 6 e

x−1 ◦ x0 6 e c/ (M c1 ). In this case we have, again by Proposition 6.25,







|K (x0 , y) − K (x, y)| = f y −1 ◦ x0 φ y −1 ◦ x0 − f y −1 ◦ x φ y −1 ◦ x




6 φ y −1 ◦ x0 f y −1 ◦ x0 − f y −1 ◦ x






+ f y −1 ◦ x φ y −1 ◦ x0 − φ y −1 ◦ x




c (ΛT,1 , G)

x−1 ◦ x0 φ y −1 ◦ x0 Q+1 −1 ky ◦ xk





c (ΛT,1 , G) φ y −1 ◦ x0 − φ y −1 ◦ x . + Q −1 ky ◦ xk 6

Since φ is Lipschitz we obtain −1





φ y ◦ x0 − φ y −1 ◦ x





6 c (R0 ) y −1 ◦ x0 − y −1 ◦ x 6 c (R0 ) x−1 ◦ x 0

356

H¨ ormander operators

where we have exploited our choice of the homogeneous norm kxk = dX (x, 0), which satisfies the triangle inequality. It follows that c d (x0 , x) c + d (x0 , x) B (x; y) ky −1 ◦ x0 k B (x; y)

−1

y ◦ x0 d (x0 , x) c c + d (x0 , x) 6 B (x; y) d (x0 , y) B (x; y) d (x0 , y)

|K (x0 , y) − K (x, y)| 6

and since y −1 ◦ x0 is bounded we obtain |K (x0 , y) − K (x, y)| 6

c d (x0 , x) B (x; y) d (x0 , y)



for y −1 ◦ x0 > M x−1 ◦ x0 . It remains to consider the case d (x0 , y) > 2d (x0 , x)

−1



y ◦ x0 6 M x−1 ◦ x0 . In this case we have |K (x0 , y) − K (x, y)| 6

c ky −1

◦ x0 k

Q

+

c

c ky −1

Q

◦ xk

!

c

−1

x ◦ x0 M −1 ky ◦ x0 k

+ Q Q ky −1 ◦ x0 k ky −1 ◦ xk   1 1 d (x0 , x) 6c + . B (x0 , y) B (x; y) d (x0 , y) 6

Since d (x0 , y) > 2d (x0 , x) the measures of the two balls are equivalent and therefore |K (x0 , y) − K (x, y)| 6 c

d (x0 , x) 1 . B (x; y) d (x0 , y)

To prove (8.22) observe that, by assumption Z Z



f y −1 ◦ x φ y −1 ◦ x dy K (x, y) dy = −1 −1 r1 3 and choose Ω0 ⊂ Ω1 ⊂ Ω2 ⊂ RN in such a way that 0 ∈ Ω0 and that conditions H1–H4 of section 7.2 are satisfied. Let R0 < κ and let φ as in Lemma 8.24. We rewrite (8.19) as Z



T u (x) = lim f y −1 ◦ x φ y −1 ◦ x u (y) dy ε→0 ky −1 ◦xk>ε Z



 + f y −1 ◦ x 1 − φ y −1 ◦ x u (y) dy ≡ T1 u (x) + T2 u (x) .

Since, when x, y ∈ B (0, R0 ), we have x−1 ◦ y < 2R0 , the L1 norm of the kernel f (·) [1 − φ (k·k)] can actually be computed over B (0, 2R0 ) (but not on the whole

358

H¨ ormander operators

space). Also, due to the presence of the cutoff function, the above kernel is regular at the origin. Hence, kT2 ukLp (B(0,R0 )) 6 kukLp (B(0,R0 )) kf [1 − φ (k·k)]kL1 (B(0,2R0 )) .

(8.24)

We are left to bound T1 u in Lp (B (0, R0 )). By Lemma 8.24 the kernel



f y −1 ◦ x φ y −1 ◦ x satisfies the assumptions of Theorem 7.35. Let a, b cutoff functions satisfying B (0, R0 /2) ≺ a ≺ B (0, R0 ) , B (0, R0 /2) ≺ b ≺ B (0, R0 ) . Then the operator Z Te1 f (x) = lim

ε→0

ky −1 ◦xk>ε



a (x) f y −1 ◦ x φ y −1 ◦ x b (y) f (y) dy

can be extended to a continuous operator on Lp (B (0, R0 )) and

e 6 c1 kf kLp (B(0,R0 ))

T1 f p

(8.25)

L (B(0,R0 ))

with c1 depending on p, R0 , the cutoff a and b and the constants A, B that appear in Lemma 8.24. In turn A, B depend only on R0 , G and ΛT,1 . Let now r0 = R0 /2 and let u ∈ C0∞ (B (0, r0 )) , then kT ukLp (B(0,r0 )) 6 kT1 ukLp (B(0,r0 )) + kT2 ukLp (B(0,r0 )) . By (8.24) we have kT2 ukLp (B(0,r0 )) 6 kT2 ukLp (B(0,R0 )) 6 c2 kukLp (B(0,R0 )) = c2 kukLp (B(0,r0 )) , with c2 = c2 (φ, R0 , ΛT,1 ) and by (8.25) we have





6 Te1 u kT1 ukLp (B(0,r0 )) = Te1 u p L (B(0,r0 ))

Lp (B(0,R0 ))

6 c kukLp (B(0,R0 )) = c kukLp (B(0,r0 )) . Since R0 and the cutoff a and b can be chosen once and for we see that kT ukLp (B(0,r0 )) 6 c kukLp (B(0,r0 )) with c = c (p, G, ΛT,1 ). The following is the analog of Theorem 8.22 for H¨older spaces:
Theorem 8.25 Let T be a kernel of type 0 that agrees with f ∈ C ∞ RN \ {0} away from the origin. Fix 0 < α < 1, R0 > 0, and let BR0 be a ball of radius R0 . α For every u ∈ CX,0 (BR0 ) define Z
T u (x) = lim f y −1 ◦ x u (y) dy. (8.26) ε→0

ky −1 ◦xk>ε

There exists a constant c = c (α, G, R0 , ΛT,1 ) > 0 such that for every u ∈ C0α (BR0 ) we have kT ukC α (BR ) 6 c kukC α (BR ) . 0 0

(8.27)

Sobolev and H¨ older estimates for H¨ ormander operators on groups

359

Proof. Since the operator T commutes with left translation it is clearly enough to prove (8.27) when BR0 = B (0, R0 ). Using the notation introduced in Chapter 7 let Ω0 = BR0 , fix κ > max (3, R0 ) and choose Ω0 ⊂ Ω1 ⊂ Ω2 ⊂ RN in such a way that the conditions H1–H4 of section 7.2 are satisfied. As in the proof of Theorem 8.22 we split the integral in (8.26) as Z



T u (x) = lim f y −1 ◦ x φ y −1 ◦ x u (y) dy ε→0 ky −1 ◦xk>ε Z



 + f y −1 ◦ x 1 − φ y −1 ◦ x u (y) dy ≡ T1 u (x) + T2 u (x) . Let g (x) = f (x) [1 − φ (kxk)]. Due to the presence of the cutoff function g ∈ L1loc , for every u is supported B (0, R0 ) we have kT2 ukL∞ (B(0,R0 )) 6 kgkL1 (B(0,2R0 )) kukL∞ (B(0,R0 )) . Also, by Theorem 1.54 or Theorem 1.56 we have Z Z

−1 −1 |T2 u (x1 ) − T2 u (x2 )| = g y ◦ x1 u (y) dy − g y ◦ x2 u (y) dy Z

6 kukL∞ (B(0,R0 )) g y −1 ◦ x1 − g y −1 ◦ x2 dy 6 kukL∞ (B(0,R0 )) |B (0, 3R0 )|

sup

|Xg| · d (x1 , x2 )

B(0,3R0 )

= c (G, ΛT,1 , R0 ) d (x1 , x2 ) kukL∞ (B(0,R0 )) . So that kT2 ukC α (B(0,R)) 6 c kukL∞ (B(0,R0 )) . Let us consider T1 . By Lemma 8.24



the kernel f y −1 ◦ x φ y −1 ◦ x is a singular kernel of exponent 1 and by Proposition 7.17 we have T1 (1) ∈ C γ (B (0, R0 )). By Corollary 7.19 for every 0 < α < 1 there exists c > 0 such that kT ukC α (B(0,R)) 6 c kukC α (B(0,R)) . Unlike what happens for the Lp estimates, dilations do not allow to extend (8.27) to the whole RN since the two terms that appear in the definition of kukC α rescale differently. The best we can obtain is a control for the H¨older seminorm: Corollary 8.26 Let T as in the previous theorem and
let 0 < α < 1. There exists c = c (α, G, ΛT,1 ) > 0 such that for every u ∈ C0∞ RN |T u|C α (RN ) 6 c |u|C α (RN ) . Proof. Let u ∈ C0∞ (B (0, R)) and let v (x) = u (DR (x)), then v ∈ C0∞ (B1 ). Observe that |u (DR (x)) − u (DR (y))| |v|C α (B1 ) = sup α kx−1 ◦ yk x6=y |u (x0 ) − u (y 0 )|   α = Rα |v|C α (BR ) . = sup

−1 0 0 x0 6=y 0 −1 (x ) ◦ y

DR

360

H¨ ormander operators

Also, kvk∞ = kuk∞ and T v (x) = T u (DR (x)). Applying the previous theorem to v we have   kT vkL∞ (B1 ) + |T v|C α (B1 ) 6 c kvkL∞ (B1 ) + |v|C α (B1 ) , hence   kT ukL∞ (BR ) + Rα |T u|C α (BR ) 6 c kvkL∞ (BR ) + Rα |v|C α (BR ) . Letting R → +∞ gives |T u|C α (RN ) 6 c |v|C α (RN ) . 8.4

Global Sobolev estimates

In this section we prove the global Sobolev regularity estimates contained in Theorem 8.2. We start with the result for second order derivatives and then we will consider separately the case of Carnot groups and the case of homogeneous stratified groups of type II. 8.4.1

2,p -regularity Global WX

We now prove the regularity result for second order derivatives for both Carnot groups and homogeneous stratified groups of type II:
Theorem 8.27 Let L be in L
ν for some ν > 0, let 1 < p < +∞, let u ∈ Lp RN and assume that Lu ∈ Lp RN (which means that
the distribution Lu can be iden2,p tified with a function in Lp ). Then u ∈ WX RN and   kukW 2,p (RN ) 6 c kLukp + kukp X

with c = c (p, G, ν). To prove the theorem we need a few intermediate results. We start with the following: Proposition 8.28 Let ν > 0 and let 1 < p < +∞. There exists a constant c =
2,p c (p, G, ν) such that if L ∈ Lv and u ∈ WX RN then, for every i, j = 1, . . . q, kXi Xj ukp 6 c kLukp .

(8.28)

Moreover if G is a homogeneous stratified group of type II we also have kX0 ukp 6 c kLukp .

(8.29)

Proof. Let Γ be the fundamental solution of L constructed in
Theorem 6.18 and observe that ΛT,1 only depends on G and ν. For u ∈ C0∞ RN , by Corollary 6.34 we have Z
Xi Xj u (x) = lim Xi Xj Γ y −1 ◦ x Lu (y) dy + dij Lu (x) (8.30) ε→0

ky −1 ◦xk>ε

Sobolev and H¨ older estimates for H¨ ormander operators on groups

361

with |dij | 6 c (G, ν) . Clearly Xi Xj Γ is homogenous of degree −Q, satisfies M (Xi Xj Γ) = 0 by Theorem 6.30 and, by Proposition 6.29, defines a principal value distribution which is a kernel of type 0 on G. The estimate (8.28) is an immediate consequence of Theorem 8.22. To prove (8.29) it is enough to write



q q X X

2 2

Xi u 6 c kLuk . Xi u 6 kLukp + kX0 ukp = Lu − p p

i,j=1 i,j=1 p


2,p is dense in WX RN (Theorem 3.49) the above estimate Finally, since C0∞ R
2,p extends to functions in WX RN .
N

In the next proposition we show how to control the Lp norm of first order derivatives interpolating between the Lp norm of the function and the Lp norm of second order derivatives.
2,p Proposition 8.29 Let 1 6 p < +∞ and let u ∈ WX RN . For every ε > 0 and j = 1, . . . , q we have

2 kXj ukp 6 ε Xj2 u p + kukp . (8.31) ε Proof. Assume first u smooth, the general case will follow by density (Theorem 3.49). Let F (t) = u (exp (tXj ) (x)) and observe that F 0 (t) = Xj u (exp (tXj ) (x)) and F 00 (t) = Xj2 u (exp (tXj ) (x)). Applying Taylor expansion to F we obtain Z 1 u (exp (Xj ) (x)) = u (x) + Xj u (x) + (1 − t) Xj2 u (exp (tXj ) (x)) dt. 0

Since exp (tXj ) (x) = x ◦ Exp (tXj ), using the translation invariance of the measure, we easily obtain

(8.32) kXj ukp 6 2 kukp + Xj2 u p , which is the assertion for ε = 1. Applying (8.32) to the function v (x) = u (Dε (x)) we get


ε k(Xj u) (Dε (·))kp 6 2 ku (Dε (·))kp + ε2 Xj2 u (Dε (·)) p

ε1+Q/p kXj uk 6 2εQ/p kuk + ε2+Q/p Xj2 u p

p

p

which gives (8.31). Proof of Theorem 8.27. Let φε be a mollifier defined as in Proposition 3.48. Let εn → 0, define un = φεn ∗ u and note that un converges to u in Lp norm. By Proposition 3.75
we have Lun = φεn ∗ Lu so that also Lun converges to Lu in the norm of Lp RN . Applying Proposition 8.28 we obtain kXi Xj un kp 6 c kLun kp

(8.33)

362

H¨ ormander operators

Also, by Lemma 8.29 we have

kXi un kp 6 Xi2 un p + 2 kun kp 6 c kLun kp + 2 kun kp .

(8.34)

2,p From (8.33)-(8.34) it follows that un is a Cauchy sequence in WX R . Therefore

2,p 2,p N N u ∈ WX R and un → u in WX R . Finally, by (8.33) and (8.34) we have   kukW 2,p (RN ) = lim kun kW 2,p (RN ) 6 lim c kLun kp + kun kp X X n→+∞ n→+∞   = c kLukp + kukp .


N

8.4.2

Higher order estimates: the case of Carnot groups

We will now prove higher order estimates for the derivatives of solutions to the equation Lu = f in the case of Carnot groups. Theorem 8.30 (Global Sobolev regularity in Carnot groups) Let G be a Carnot group, let L be in the class Lν for some ν > 0, let 1 < p
< +∞ and k ∈ N. Then, there exists c = c (p, k, G, ν) such that if u ∈ Lp RN and Lu ∈
k,p WX RN (which means that the distribution Lu can be identified with a function

k+2,p k,p RN and RN ), then u ∈ WX in WX   (8.35) kukW k+2,p (RN ) 6 c kLukW k,p (RN ) + kukLp (RN ) . X

X

We start fixing a useful Notation 8.31 We will say that P k is a differential monomial of degree k if P k = XI with |I| = k. In particular, P k is left invariant and homogeneous of degree k. Let us begin with the following estimate for higher order derivatives. Proposition 8.32 Let G be a Carnot group and let L be in the class Lν for some ν > 0. For every 1 < p < +∞ and
every k ∈ N there exists a constant c = ∞ N c (p, k, G, ν) such that if u ∈ C0 R then

k+2

D

u 6 c Dk Lu . p

p

Proof. We will prove the desired estimate first for derivatives with respect to the canonical vector fields Yi , that is with X

k

D u = kYI ukp . p |I|=k

k

The general case then will follow since, by Proposition 6.12, DX u p and DYk u p are equivalent with uniform constants depending on ν, as L ranges in the class Lν . We start with the representation formula Z
u (x) = Γ y −1 ◦ x Lu (y) dy RN

Sobolev and H¨ older estimates for H¨ ormander operators on groups

(with L =

P

363

Xi2 ) so that for every 1 6 i 6 q we have Z
Yi u (x) = Yi Γ y −1 ◦ x Lu (y) dy. RN

To begin with, our goal is to control third order derivatives of u with first order derivatives of Lu. In order to do that we need to move the derivative with respect to Yi to Lu before computing the remaining two derivatives of the integral. This can be done using Proposition 3.47 but we need a right invariant vector field applied to Γ to obtain a left invariant vector field applied to Lu. By Remark 3.32 we can write Yi =

N X

rij (x) YjR

j=i ∂ where YjR denotes the right invariant vector field that agrees with ∂x at the origin j and rij (x) is an (αj − αi )-homogeneous polynomial (denoting by αi the exponents appearing in the dilations on G). Note that since αj − αi < αj the coefficients rij (x) do not contain the variable xk for any k > j and therefore rij (x) and YjR commute, by the structure of YjR (see Theorem 3.29). It follows that N X

Z Yi u (x) =

rij (·) YjR Γ

RN j=i N X

Z =

RN j=i





y −1 ◦ x Lu (y) dy


YjR (rij Γ) y −1 ◦ x Lu (y) dy.

Since the vector fields Y1R , Y2R , . . . , YqR generate the Lie algebra of right invariant vector, fields for any q < j 6 N we can write YjR (which is homogeneous of degree αj ) as linear combination of commutators of length αj of the vector fields of the first layer. It follows that there exists constants θij1 ...iα such that j

YjR

q X

=

YiR · · · YiR . θij1 ...iα YiR 1 2 α j

j

i1 ,...,iαj =1

Hence Z Yi u (x) =

q X

RN j=i

Z +


YjR (rij Γ) y −1 ◦ x Lu (y) dy

N X

q X

RN j=q+1 i ,...,i =1 1 αj


θij1 ...iα YiR YiR · · · YiR (rij Γ) y −1 ◦ x Lu (y) dy 1 2 α j

j

(8.36)

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H¨ ormander operators

Applying Proposition 3.47 we obtain Z X q
Yi u (x) = (rij Γ) y −1 ◦ x Yj Lu (y) dy RN j=i q X

N X

Z +

RN j=q+1 i ,...i =1 1 αj


θij1 ...iα YiR · · · YiR (rij Γ) y −1 ◦ x Yi1 Lu (y) dy. 2 α j

j

(8.37) · · · YiR (rij Γ) in the above integrals Observe now that the kernels rij Γ and YiR αj 2 are homogenous of degree 2 − Q. Indeed if 1 6 j 6 q, since αj = 1, then rij is homogeneous of degree αj − αi = 0. When j > q, then rij Γ is homogeneous of degree 2−Q+αj −1. Since we apply to rij Γ, αj −1 operators that are homogeneous · · · YiR (rij Γ) is homogeneous of degree 2 − Q. It follows of degree 1 we see that YiR αj 2 that we can write q Z X
Yi u (x) = Γk y −1 ◦ x Yk Lu (y) dy k=1

RN

where Γk are homogeneous functions of degree 2 − Q that are smooth outside the origin. The above argument can be easily iterated. If P k = Yi1 Yi2 · · · Yik , 1 6 i` 6 q, is a differential monomial of degree k we have Z q X
P k u (x) = Γj1 ,...,jk y −1 ◦ x Yj1 Yj2 · · · Yjk Lu (y) dy (8.38) j1 ,j2 ,··· ,jk =1

RN

where the functions Γj1 ,...,jk are homogeneous of degree 2 − Q and smooth outside the origin. Now we want to take two more derivatives. Let 1 6 h, j 6 q. Then Z q X
Yh P k u (x) = Yh Γj1 ,...,jk y −1 ◦ x Yj1 Yj2 · · · Yjk Lu (y) dy. j1 ,j2 ,··· ,jk =1

RN

Also, by Theorem 6.33 (see also the proof of Corollary 6.34) we obtain Yj Yh P k u (x) =

q X

Z lim

j1 ,j2 ,··· ,jk =1

ε→0

ky −1 ◦xk>ε


Yj Yh Γj1 ,...,jk y −1 ◦ x Yj1 Yj2 · · · Yjk Lu (y) dy

+ cjhj1 ,...,jk Yj1 Yj2 · · · Yjk Lu (x) .

(8.39)

Applying Theorem 8.22 we obtain

Yj Yh P k u 6 c p

q X

kYj1 Yj2 · · · Yjk Lu (y)kp

j1 ,j2 ,··· ,jk =1

and therefore

k+2

D u p 6 c Dk Lu p .

(8.40)

Sobolev and H¨ older estimates for H¨ ormander operators on groups

365


Finally observe that the kernels Yj Yh Γj1 ,...,jk y −1 ◦ x have been obtained multiplying Γ by the homogeneous polynomial rij and applying the right invariant vector fields. Since by Theorem 6.18 |α| ∂ Γ (x) 6 c (j, G, ν) sup sup ∂xα kxk=1 |α|6j we see that the constant that appears in Theorem 8.22 depends only on k, G and ν. Remark 8.33 The reason why the above result is proved in Carnot groups (and not on homogeneous stratified groups of type II) appears in the passage from (8.36) to (8.37), which is troublesome in presence of the drift. Actually, if the vector field YiR 1 appearing in (8.36)-(8.37) is Y0R , then (8.37) expresses Yi u as a fractional integral of Y0 Lu, and taking two more derivatives we would get a representation of a third order derivative of u in terms of Y0 Lu, that is in terms of a second order derivative of Lu, instead of a first order derivative as expected. The conclusion is that the presence of the drift prevents us from getting a useful representation formula of the derivatives of odd orders of the solution. Next, we extend the previous result to complete Sobolev norms, getting the desired a priori estimates, for the moment only for smooth functions. Theorem 8.34 Let G be a Carnot group and let L be in Lν for some ν > 0. For every 1 < p 0, by Theorem 8.34 we have   kun − um kW k+2,p (RN ) 6 c kLun − Lum kW k,p (RN ) + kun − um kp . X

X

366

H¨ ormander operators


k,p Since Lun = φεn ∗ Lu, the sequence Lun converges to Lu ∈ WX RN . Simi
larly un converges to u in Lp RN . It follows that kun − um kW k+2,p (RN ) → 0 as X
k+2,p n, m → +∞ which means that un is a Cauchy sequence in WX RN . Hence
k+2,p u ∈ WX RN and satisfies (8.35). 8.4.3

Higher order estimates: the case of homogeneous stratified groups of type II

We will now prove higher order estimates for the derivatives of solutions to the equation Lu = f in the case of homogeneous stratified groups of type II of homogeneous dimension Q > 4 (we alert the reader that the assumption Q > 4 will be explicitly used in the proofs of Lemma 8.36 and Proposition 8.37). The techniques used in the previous section no longer work in this case, due to the presence of the vector field X0 that has weight 2, for the reason explained in Remark 8.33. To overcome this difficulty we are forced to assume k even. The result reads as follows: Theorem 8.35 (Global Sobolev regularity in the drift case) Let G be a stratified homogeneous group of type II of homogeneous dimension Q > 4, let L be in Lν for some ν > 0 and let 1 < p < N. Then, there exists
+∞ and k ∈2k,p
c = c (p, k, G, ν) such that if u ∈ Lp RN and Lu ∈ WX RN (which means
k,p that the distribution Lu can be identified with a function in WX RN ), then u ∈
2k+2,p RN and WX   kukW 2k+2,p (RN ) 6 c kLukW 2k,p (RN ) + kukLp (RN ) . X

X

The next Lemma will be useful for several purposes. Lemma 8.36 Let G be as above and let L be in the class Lν for some ν > 0. For every integer k > 2 and any couple of differential monomials P 2k−1 and P 2k−2 of degrees 2k − 1 and 2k − 2, respectively, we can determine two kernels K1 , K2 (depending only on these monomials) homogeneous of degrees 1 and 2, respectively,
such that for u ∈ C0∞ RN P 2k−1 u = Lk u ∗ K1 , P

2k−2

k

u(x) = L u ∗ K2 .

(8.43) (8.44)

Moreover for every k ∈ N there exists k0 such that ΛKj ,k 6 c (k, G, ν) ΛΓ,k0 . Proof. The proof is by induction on k and we will apply several times Proposition 8.16. We start with k = 2. Then u = Lu ∗ Γ = (LLu ∗ Γ) ∗ Γ = LLu ∗ K4 , where, by Proposition 8.16, K4 = Γ ∗ Γ is a kernel of type 4 since Q > 4. Hence P 3 u = LLu ∗ P 3 K4 = LLu ∗ K1 , P 2 u = LLu ∗ P 2 K4 = LLu ∗ K2 ,

Sobolev and H¨ older estimates for H¨ ormander operators on groups

367

with K1 and K2 kernels of type 1 and 2 respectively. Now, assume (8.43) and (8.44) hold for k − 1, that is P 2k−3 u = Lk−1 u ∗ K1 , P

2k−4

u=L

k−1

u ∗ K2

(8.45) (8.46)

with K1 and K2 kernels of type 1 and 2 respectively, as required, and let us prove (8.45) for k. Any differential monomial P 2k−1 can be written either as P 2 P 2k−3 (with P 2 = X0 or P 2 = Xi Xj for some i, j = 1, . . . , q) or as P 3 P 2k−4 (with P 3 = Xi X0 for some i = 1, . . . , q). In the first case, we can write, with the obvious meaning of symbols, by (8.45),

P 2k−1 u = P 2 P 2k−3 u = P 2 Lk−1 u ∗ K1

= P 2 Lk u ∗ (Γ ∗ K1 ) = P 2 Lk u ∗ K3 = Lk u ∗ P 2 K3 = Lk u ∗ K10 In the second case, by (8.46)

P 2k−1 u = P 3 P 2k−4 u = P 3 Lk−1 u ∗ K2

= P 3 Lk u ∗ (Γ ∗ K2 ) = P 3 Lk u ∗ K4 = Lk u ∗ P 3 K4 = Lk u ∗ K1 so (8.45) is proved. To prove (8.46) for k, any differential monomial P 2k−2 can be written either as Xi P 2k−3 (for some i = 1, . . . , q) or as X0 P 2k−4 . In the first case, we can write, by (8.45),

P 2k−2 u = Xi Lk−1 u ∗ K1 = Xi Lk u ∗ (Γ ∗ K1 ) = Lk u ∗ Xi K3 = Lk u ∗ K20 . In the second case,

P 2k−2 u = X0 Lk−1 u ∗ K2 = X0 Lk u ∗ (Γ ∗ K2 ) = Lk u ∗ X0 K4 = Lk u ∗ K200 , so (8.46) is proved, too. Next, we prove the following. Proposition 8.37 Let L be in the class Lν for some ν > 0. Let 1 < p < +∞, let k be a positive integer and let P 2k be a differential monomial of
degree 2k. Then, there exists c = c (p, k, G, ν) > 0 such that for every u ∈ C0∞ RN we have

2k

P u 6 c Lk u (8.47) p p Proof. The case k = 1 has been proved in Theorem 8.28. For k > 2, let us consider a differential monomial P 2k . Then it can be written either as Xj P 2k−1 (for some j = 1, 2, . . . , q) or as X0 P 2k−2 , for suitable differential monomials P 2k−1 , P 2k−2 . Applying Lemma 8.36 we can write either

P 2k u = Xj Lk u ∗ K1 or P 2k u = X0 Lk u ∗ K2
for u ∈ C0∞ RN and suitable kernels K1 , K2 (depending only on the monomials P 2k−1 , P 2k−2 ) homogeneous of degrees 1 and 2, respectively. In the first case, by Proposition 3.75 we have, in the distributional sense, P 2k u = Lk u ∗ Xj K1 and by

368

H¨ ormander operators

Remark 3.78 Xj K1 is a distribution that is homogeneous of degree −Q. This distribution, away from the origin, agrees with a smooth function that is homogeneous of degree −Q. Let us denote this function by f . By Proposition 6.30 we have Xj K1 = PV f1 + c1 δ with M (f1 ) = 0 and by Proposition 6.29 we obtain Z
P 2k u (x) = lim f1 y −1 ◦ x Lk u (y) dy + c1 Lk u (x) . ε→0

ky −1 ◦xk>ε

We can now apply Theorem 8.22 so that

2k



P u 6 c Lk u 6 c D2k u . p p p
2k k Analogously, if P u = X0 L u ∗ K2 we can rewrite it as P 2k u = Lk u ∗ X0 K2 = Lk u ∗ (PV f2 + c2 δ) for some constant c2 and smooth function f2 that

is homogeneous

of degree

−Q and satisfies M (f2 ) = 0. Again, we infer P 2k u p 6 c Lk u p 6 c D2k u p that gives the assertion. In the next lemma we prove “interpolation inequalities” adapted to the context of homogeneous stratified groups of type II. Lemma 8.38 For every integer k > 2 there exists a constant c = c(G, k, ν) such
∞ N that for every u ∈ C0 R , we have  

2k−1

2k 1

D

(8.48) u p 6 c ε D u p + 2k−1 kukp ε  

2k−2 1

D u p 6 c ε2 D2k u p + 2k−2 kukp (8.49) ε Proof. Let P 2k−1 be a differential monomial of degree 2k − 1, and let K1 be as in Lemma 8.36. We split the kernel K1 as K1 = ϕK1 + (1 − ϕ) K1 ≡ K1,0 + K1,∞ , where ϕ is a cutoff function, B1 (0) ≺ ϕ ≺ B2 (0), so that P 2k−1 u = Lk u ∗ K1,0 + Lk u ∗ K1,∞ . Clearly K1,0 is homogeneous of degree (1 − Q) near the origin and has compact support, hence it is integrable, while K1,∞ is homogeneous of degree (1 − Q) near e 1,∞ (x) = K1,∞ (x−1 ), then infinity and vanishes near the origin. Let K Z Z

e 1,∞ x−1 ◦ y Lk u(y) dy Lk u ∗ K1,∞ (x) = K1,∞ y −1 ◦ x Lk u(y) dy = K Z
0 k e −1 = (L∗ ) K ◦ y u(y)dy = u ∗ K1 (x) , 1,∞ x

Sobolev and H¨ older estimates for H¨ ormander operators on groups

369


k e −1 where K10 (x) = (L∗ ) K and L∗ is the transpose operator of L. Therefore 1,∞ x 0 P 2k−1 u = Lk u ∗ K1,0 + u ∗ K1 where K10 is homogeneous of degree (1 − Q − 2k) at infinity and vanishes near the origin, hence it is integrable. The integrability of K1,0 and K10 gives n o

2k−1

D u p 6 c (G, k, ν) D2k u p + kukp , (8.50) which is the assertion for ε = 1. Let now uε (x) = u (Dε (x)). Then if P k is a homogeneous monomial of degree k we have 1/p Z

k

k P u (Dε (x)) p dx

P uε = εk = εk−Q/p P k u p p RN

so that applying (8.50) to uε we obtain 

2k−1

D u p 6 c(G, k, ν) ε D2k u p +

1 ε2k−1

 kukp ,

that is (8.48). The proof of (8.49) can be carried out analogously, exploiting (8.44) instead of (8.43). We can now prove the following a priori estimate in Sobolev spaces of even degree: Theorem 8.39 Let L be in the class Lν for some ν > 0. Let 1 < p < +∞ and let
k ∈ N. Then, there exists c = c (k, p, G, ν) > 0 such that for every u ∈ C0∞ RN we have   kukW 2k+2,p (RN ) 6 c kLukW 2k,p (RN ) + kukp . (8.51) X

X

Proof. The proof is by induction on k. The case k = 0 is contained in Theorem 8.27. Now assume to know that (8.51) holds for k and let us prove it for k + 1. Let P h be any differential monomial of degree h 6 2k + 4. When h 6 2k + 2 by (8.51) we obtain  

h

P u 6 kuk 2k+2,p N 6 c kLuk 2k,p N + kuk . p W (R ) W (R ) p X

X

It remains to consider the cases h = 2k + 3 and h = 2k + 4. Assume first h = 2k + 4, then by Proposition 8.37 we have

2k+4

P u p 6 c Lk+2 u p 6 c kLukW 2(k+1),p (RN ) . (8.52) X

Finally let h = 2k + 3. By (8.48) (with ε = 1) and (8.52) we have  

2k+3



P u p 6 D2k+4−1 u p 6 c D2k+4 u p + kukp   6 c kLukW 2(k+1),p (RN ) + kukp . X

Proof of Theorem 8.35. The Theorem follows from Theorem 8.39 by a density argument as in the Proof of Theorem 8.30. We omit the details.

370

H¨ ormander operators

8.5

Local Sobolev estimates

8.5.1

Cutoff functions and interpolation inequalities

This section is devoted to the proof of Theorem 8.5. We start constructing a family of cutoff functions adapted to the homogeneous balls in G. These functions will be used to make the solution of Lu = f compactly supported in order to apply the results of the previous section. Lemma 8.40 For every σ ∈

1 2, 1





, r > 0 there exists ϕ ∈ C0∞ RN ,

B (0, σr) ≺ ϕ ≺ B (0, σ 0 r)

with σ 0 = (1 + σ) /2,

such that if P k is a differential monomial of degree k (see Notation 8.31), then

k ck

P ϕ ∞ N 6 , k L (R ) (1 − σ) rk with ck depending on k, G, ν but independent of σ and r. Proof. Let ψ : R → [0, 1] be a smooth function such that ψ (t) = 1 for t ∈ (−∞, 1], f (t) = 0 for t ∈ [2, ∞) and let   2t 1 − 3σ f (t) = ψ + . (1 − σ) r 1−σ Observe that f (t) ≡ 1 for t ∈ (−∞, σr] and f (t) ≡ 0 for t ∈ [σ 0 r, +∞). Moreover for suitable constants cj , cj (j) , j ∈ N. f (t) 6 j (1 − σ) rj Finally set ϕ (x) = f (kxk). If P k is a differential monomial of degree k, a simple iterative argument shows that P k ϕ is a finite sum of terms of the kind f (j) (kxk) P1 (kxk) P2 (kxk) · · · Pj (kxk) with 1 6 j 6 k, where P1 , P2 , . . . , Pj are differential monomials of degrees α1 , α2 , . . . , αj with α1 + α2 + . . . + αj = k. Since f (j) (kxk) is different from zero only for σr 6 kxk 6 σ 0 r, j 6 α and the norm is homogeneous of degree 1 we have (j) f (kxk) P1 (kxk) P2 (kxk) · Pj (kxk) 6

cj

1−α1

j

(1 − σ)

rj

(σr)

(σr)

1−α2

· · · (σr)

1−αj

6

cj σ j−k (1 − σ)

j

rk

6

cj k

(1 − σ) rk

.

The next theorem provides a localized interpolation inequality. We start with a definition.

Sobolev and H¨ older estimates for H¨ ormander operators on groups

371

Definition 8.41 Let Br be a ball of radius r > 0 and let k = 0, 1, 2, . . . For u ∈ k,p WX (Br ) we set h i

k Φk,r (u) = sup (1 − σ) rk Dk u Lp (B ) . 1 2 0. Hence (

c(G, ν) kDukLp (Bσ0 r ) + kDukLp (Bσr ) 6 ε D2 u Lp (B 0 ) + σ r (1 − σ) r ) c(G, ν) 2 + + kukLp (Bσ0 r ) . 2 2 kukLp (Bσ0 r ) ε (1 − σ) r

Multiplying both sides by (1 − σ) r, choosing ε = δ (1 − σ) r and noting that (1 − σ) = 2(1 − σ 0 ) we find

2 (1 − σ) r kDukLp (Bσr ) 6 δ (1 − σ) r2 D2 u Lp (B 0 ) σ r   2 kukLp (Bσ0 r ) + cδ (1 − σ) r kDukLp (Bσ0 r ) + cδ + δ   2 Φ0 . 6 4δΦ2 + cδΦ1 + cδ + δ (cδ+ 2 ) 4δ Therefore, for δ small enough, Φ1 6 1−cδ Φ2 + 1−cδδ Φ0 which, for δ small enough, is equivalent to Φ1 6 δΦ2 + δc Φ0 . The interpolation inequality in the previous theorem will be enough to prove estimates for high order derivatives in the case of Carnot groups. For homogeneous stratified groups of type II we need the higher order interpolation inequality contained in the following Theorem. Theorem 8.43 (Interpolation inequality for Φj ) Let G be a homogeneous 2k,p stratified group of type II. Let Br be a ball of radius r > 0 and let u ∈ WX (Br ). For some k > 1, p ∈ [1, ∞). Then Φj 6 εΦ2k + c (k, G, ν, ε) Φ0 for every integer j with 1 6 j 6 2k − 1 and every ε > 0.

372

H¨ ormander operators

Proof. Let ϕ be a cutoff function as in Lemma 8.40, then we easily obtain h X

h

D (uϕ) 6 c p j=0

1 h−j

(1 − σ)

rh−j

j

D u p L (B

σ0 r )

,

h > 0.

(8.54)

2k,p Since, by Corollary 2.10, uϕ ∈ WX,0 (Br ), (8.48) is applicable to uϕ and gives, also exploiting (8.54),  

2k−1



1

D u Lp (Bσr ) 6 D2k−1 (uϕ) p 6 c(G, k, ν) ε D2k (uϕ) p + 2k−1 k(uϕ)kp ε   2k   X

1 1

Dj u p kuk 6 c(G, k, ν) ε + p (B 0 ) L 2k−j 2k−j L (Bσ0 r ) σ r   ε2k−1 (1 − σ) r j=0

and therefore

2k−1 2k−1

D2k−1 u p (1 − σ) r L (Bσr )   2k 2k−1 2k−1  X 

(1 − σ) r j−1 j−1 j D u Lp (B 0 ) + 6c ε (1 − σ) r kuk p L (Bσ0 r ) σ r   ε2k−1 j=0   2k 2k−1 2k−1  X  1 (1 − σ) r 6c ε Φj + Φ . 0   (1 − σ) r ε2k−1 j=0 Choosing ε = δ (1 − σ) r gives

2k−1 2k−1

D2k−1 u p (1 − σ) r L (B

σr )

  2k  X  1 6c δ Φj + 2k−1 Φ0   δ j=0

and therefore Φ2k−1

  2k   X 1 6c δ Φj + 2k−1 Φ0 .   δ j=0

(8.55)

Similarly applying (8.49) we obtain   2k  X  1 Φ2k−2 6 c δ 2 Φj + 2k−2 Φ0 ,   δ j=0 which we rewrite as Φ2k−2

  2k   X 1 6c δ Φj + k−1 Φ0   δ j=0

(8.56)

Sobolev and H¨ older estimates for H¨ ormander operators on groups

373

Inequalities (8.55)-(8.56) clearly hold if k is replaced by any integer i with 1 6 i 6 k. Adding up these inequalities and also recalling (8.53) we get 2k−1 k X X Φj = Φ1 + (Φ2i−1 + Φ2i−2 ) j=1

i=2

    k  X k  X 2i 2i   X X c 1 1 6 δΦ2 + Φ0 + c δ δ Φj + i−1 Φ0 Φj + 2i−1 Φ0 + c     δ δ δ i=2 i=2 j=0 j=0 6 ck δ

2k X

Φj + c (δ, k) Φ0 .

j=0

It follows that 2k−1 X

Φj − ck δ

2k−1 X

j=1

Let δ
0. 8.5.2

Local Lp estimates for second order derivatives

In this section we will prove the local regularity result of Theorem 8.5 for second order derivatives: Theorem 8.44 Let G be either a Carnot group or a homogeneous stratified group of type II, with homogeneous dimension Q > 2. Let L be in the class Lν for some ν > 0, let Ω be a bounded domain in G, let Ω0 b Ω00 b Ω and let 1 < p < +∞. If u 2,p is a distribution in Ω such that Lu ∈ Lp (Ω), then u ∈ WX (Ω00 ) and o n kukW 2,p (Ω0 ) 6 c kLukLp (Ω00 ) + kukLp (Ω00 ) X

where c = c(p, G, Ω0 , Ω00 , ν). To prove the above theorem we need the following solvability result. 2,p Proposition 8.45 (Solvability in WX (Ω)) Let L be in the class Lν for some
N ν > 0, let Ω be a bounded domain of R and let 1 < p < +∞. Let f ∈ Lp RN 2,p with support in Ω, then v = f ∗ Γ defines a function v ∈ WX (Ω) such that Lv = f . Moreover for i = 1, . . . q Xi v = f ∗ Xi Γ (8.57) and for i, j = 1, . . . , q Z
Xi Xj v = lim Xi Xj Γ y −1 ◦ x f (y) dy + cij f (x) (8.58) ε→0

ky −1 ◦xk>ε

for suitable constants cij . Finally, an expression for X0 v similar to (8.58) holds in the case of homogeneous stratified groups of type II.

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H¨ ormander operators


Proof. Let ϕn be a sequence of test functions that converges to f in Lp RN and set vn = ϕn ∗ Γ. Note that we can always assume that the functions ϕn are supported in a large ball BR that contains Ω. Let us show that vn converges in 2 2 Lp (Ω). Fix a number p0 satisfying p1 6 p10 6 p1 + Q and Q < p10 < 1. Such number i h 2 and exists since under the conditions p ∈ (1, ∞) , Q > 2 the intervals p1 , p1 + Q   1 1 2 2 Q , 1 always overlap. Letting q 0 = p0 − Q , by Proposition 8.14 we have kvn − vm kLq0 (Ω) 6 k(ϕn − ϕm ) ∗ ΓkLq0 (RN ) 6 c1 kϕn − ϕm kLp0 (BR ) . Exploiting the fact that p0 6 p and

1 q0

=

1 p0

−

2 Q

6

1 p

we obtain

kvn − vm kLp (Ω) 6 c2 kvn − vm kLq0 (Ω) 6 c1 c2 kϕn − ϕm kLp0 (BR ) 6 c3 kϕn − ϕm kLp (RN ). Similarly, since Xj vn = ϕn ∗ Xj Γ

(8.59)

and Xj Γ is a kernel of type 1, we obtain kXj vn − Xj vm kLp (Ω) 6 c4 kϕn − ϕm kLp (RN ). Now, arguing as in the proof of Theorem 6.32 we can write Xi Xj vn (x) = PV (ϕn ∗ Xi Xj Γ) + cij ϕn Z
= lim Xi Xj Γ y −1 ◦ x ϕn (y) dy + cij ϕn (x) ε→0

(8.60)

ky −1 ◦xk>ε

(with an analogous formula for X0 vn ) and by Theorem 8.22 we obtain kXi Xj vn − Xi Xj vm kLp (Ω) 6 c5 kϕn − ϕm kLp (RN ). 2,p (Ω) and therefore converges in It follows that {vn } is a Cauchy sequence in WX 2,p WX (Ω) to a function v. Finally, since, by (8.3), Lvn = ϕn , taking the limit in Lp (Ω) we obtain Lv = f . The representation formulas (8.57) and (8.58) follow respectively from (8.59) and (8.60) exploiting the continuity in Lp (Ω) of the associated operators.

Proof of Theorem 8.44. Let Ω000 such that Ω00 b Ω000 b Ω
and let ψ ∈ C0∞ (Ω) such that Ω000 ≺ ψ ≺ Ω. Since Lu ∈ Lp (Ω), ψLu ∈ Lp RN and by Proposition 8.45 there exists v ∈ W 2,p (Ω) such that Lv = ψLu. Now, observe that L (u − v) = (1 − ψ) Lu and since (1 − ψ) Lu vanishes in Ω000 by H¨ ormander’s theorem (Theorem 5.64) u − v is smooth in Ω000 . It follows that 2,p u ∈ WX (Ω00 ). Let us fix r < dist (Ω0 , ∂Ω00 ) and let
Br be a ball with
center in Ω0 and radius r, then Br ⊂ Ω00 . For every σ ∈ 21 , 1 let ϕ ∈ C0∞ RN as in Lemma 8.40, so

Sobolev and H¨ older estimates for H¨ ormander operators on groups

375

2,p that Bσr ≺ ϕ ≺
Bσ0 r , with σ 0 = (1 + σ) /2. Since u ∈ WX (Ω00 ), by Corollary 2.10 2,p N uϕ ∈ WX R and by Theorem 8.27   kXi Xj (uϕ)kp 6 c kL (uϕ)kp + kuϕkp (8.61)

with c = c (p, G, ν). Since L (ϕu) = ϕLu + uLϕ +

q X

Xi uXi ϕ,

(8.62)

i=1

from (8.61) and Lemma 8.40 we get

2

D u p L (Bσr ) ( 1 6 c kLukLp (Bσ0 r ) + kDukLp (Bσ0 r ) + (1 − σ) r

!

1 2

(1 − σ) r2

)

+ 1 kukLp (Bσ0 r )

2

Now, multiplying both sides by (1 − σ) r2 , adding (1 − σ) r kDukLp (Bσr ) , using (1 − σ) = 2(1 − σ 0 ) and taking the supremum over 12 < σ < 1 gives o n Φ1 + Φ2 6 c r2 kLukLp (Br ) + Φ1 + kukLp (Br ) . By Theorem 8.42 we obtain o n c Φ1 + Φ2 6 c r2 kLukLp (Br ) + δΦ2 + Φ0 + kukLp (Br ) δ Fixing a suitably small δ we obtain n o Φ1 + Φ2 6 c r2 kLukLp (Br ) + kukLp (Br ) and therefore

r2 D2 u Lp (B

r/2

n o 2 6 c r kLuk , + r kDuk p (B ) + kukLp (B ) p B L L r r ( r/2 ) )

that is n o kukW 2,p (Br/2 ) 6 c kLukLp (Br ) + kukLp (Br ) . X

We can now cover Ω0 with a finite number of balls Bj of radius r/2 so that o X Xn kukW 2,p (Ω0 ) 6 kukW 2,p (Bj ) 6 c kLukLp (2Bj ) + kukLp (2Bj ) X X n o 6 c kLukLp (Ω00 ) + kukLp (Ω00 ) . with c = c(p, G, Ω0 , Ω00 , ν).

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H¨ ormander operators

8.5.3

Local Lp estimates for higher order derivatives

In this section we complete the proof of Theorem 8.5 for derivatives of order k > 2. The proof requires two lemmas: the first one addresses the case of Carnot groups and the second one the case of homogeneous stratified groups of type II. Lemma 8.46 Let G be a Carnot group. Then Theorem 8.5 holds under the addik+1,p tional requirement that u ∈ WX (Ω00 ). Proof. We proceed by induction on k. For k = 0 the result is contained in Theorem 8.44. Assume the result holds up to k − 1, and let us prove it for k. So, k+1,p k,p for some k > 1, let u ∈ WX (Ω00 ) and Lu ∈ WX (Ω). Let r > 0 and let Br 0 be a ball of radius r and center in Ω such that B2k−1 r b Ω00 . Let σ = 12 and let
∞ N ϕ ∈ C0 R as in Lemma 8.40, so that Br/2 ≺ ϕ ≺ Br . Using (8.62), we see

k,p RN . Since ϕu ∈ Lp RN , applying Theorem 8.2 we obtain that L (ϕu) ∈ WX
k+2,p uϕ ∈ WX RN and   kϕukW k+2,p (Br ) 6 c kL (ϕu)kW k,p (Br ) + kϕukLp (Br ) X

X

so that   kukW k+2,p (Br/2 ) 6 c kLukW k,p (Br ) + kukW k+1,p (Br ) . X

X

X

Applying the inductive assumption we can write   kukW k+1,p (Br ) 6 c kLukW k−1,p (B2r ) + kukLp (B2r ) X

X

hence   kukW k+2,p (Br/2 ) 6 c kLukW k,p (B2r ) + kukLp (B2r ) . X

X

The same covering argument used in the proof of Theorem 8.44 gives the desired result. In the next lemma we consider homogeneous stratified groups of type II. Unfortunately we cannot use the previous argument since in this case the estimate for kukW k+2,p (RN ) only works for k even (see Theorem 8.39). X

Lemma 8.47 Let G be a homogeneous stratified group of type II. Then Theorem k+1,p 8.5 holds under the additional requirement that u ∈ WX (Ω00 ). Proof. Fix r ε

2,α Proof. Since u ∈ CX,0 (Ω) the left hand sides of the above equations are continuous functions. By Proposition 8.17 and Theorem 8.25 also the right hand sides defines continuous functions. Hence, it is enough to show that (8.64), (8.65) and (8.66) 2,p hold for almost every x ∈ Ω. Fix 1 < p < Q 2 . Since u ∈ WX,0 (Ω), by Theorem 2,p 2.9 we can pick a sequence un ∈ C0∞ (Ω) that converges to u in WX,0 (BR ). By Proposition 8.10 (iii) we have un = Lun ∗ Γ. Let us show that Lun ∗ Γ converges to Lu ∗ Γ in Lp (Ω). Since Γ is a kernel of type 2, we can apply Theorem 8.14 with 1/q = 1/p − 2/Q. Note that q > p, so that

kLun ∗ Γ − Lu ∗ ΓkLp (Ω) 6 c k(Lun − Lu) ∗ ΓkLq (Ω) 6 c kLun − LukLp (Ω) . Since Lun and un converge in Lp (Ω) to Lu and u respectively we obtain u = Lu ∗ Γ in the sense of Lp (Ω) functions and therefore almost everywhere. A similar argument shows that also (8.65) holds for almost every x. The argument for (8.66) is also similar, but in this case to ensure that Z
lim Xi Xj Γ y −1 ◦ x Lun (y) dy ε→0

ky −1 ◦xk>ε

converges to Z lim

ε→0


Xi Xj Γ y −1 ◦ x Lu (y) dy

ky −1 ◦xk>ε

in Lp (Ω) we appeal to Theorem 8.22. The first consequence of the above representation formulas is the following: α Theorem 8.50 (CX estimate for second derivatives of test functions) Let L be in the class Lν for some ν > 0, let Br be a ball of radius r in G and let

380

H¨ ormander operators

0 < α < 1. There exists a positive constant c = c (G, ν, α, r) such that for every 2,α u ∈ CX,0 (Br ) we have

2

D u α 6 c kLukC α (Br ) . C (Br ) X

X

Proof. Applying Theorem 8.25 with T = Xi Xj Γ to (8.66) we easily obtain kXi Xj ukC α (Br ) 6 c kLukC α (Br ) . X

X

Note that the constant c in Theorem 8.25 depends on ΛT,1 and therefore on ΛΓ,3 . By Theorem 8.10 this in turn depends on ν. When X0 is present we can control kX0 ukC α (Br ) as in the proof of Theorem 8.28. X

α Corollary 8.51 (Global estimate for CX seminorm) Let L be in the class Lν 2,α for some ν > 0 and let 0 < α < 1. For every u ∈ CX,0 (G) we have 2 D u α 6 c |Lu| α , C (G)

C (G)

where c is the constant appearing in Theorem 8.50. Proof. Let R large enough so that supp u ⊂ BR (0) and let v (x) = u (DR (x)). 2,α Clearly v ∈ CX,0 (B1 ) and by Theorem 8.50 we have

2

D v α 6 c kLvkC α (B1 ). (8.67) C (B1 ) X

X

If P is a differential operator homogeneous of degree 2, we claim that kP vkC α (B1 ) = R2+α |P u|C α (BR ) + R2 kP ukL∞ (BR ) . X

Indeed, kP vkC α (B1 ) = |P v|C α (B1 ) + sup |P v| X

B1

= R2

sup x,y∈B1 ,x6=y

= R2

|P u (DR (x)) − P u (DR (y))| + R2 sup |P u (DR (x))| α kx−1 ◦ yk B1

sup x0 ,y 0 ∈BR ,x0 6=y 0

=R

2+α

|P u (x0 ) − P u (y 0 )| + R2 sup |P u (x0 )| 1 0−1 ◦ y 0 kα kx BR α R

|P u|C α (BR ) + R2 kP ukL∞ (BR ) .

Using this in (8.67) we obtain

R2+α D2 u C α (B ) + R2 D2 u L∞ (B R

and therefore 2 D u α

R)

  6 c R2+α |Lu|C α (BR ) + R2 kLvkC α (BR )

  −α 6 c |Lu| + R kLvk α α C (BR ) C (BR ) C (BR R) 2 with c independent of R. If we let R → +∞ obtain D v C α (RN ) 6 c |Lv|C α (RN ) .

+ R−α D2 u L∞ (B )

k,α To prove regularity results in the scale of H¨older spaces CX , we will need to use the mollifiers adapted to G, as already done for Sobolev spaces. However, as

Sobolev and H¨ older estimates for H¨ ormander operators on groups

381

k,α noted in Chapter 2, section 2.2, test functions are not dense in the spaces CX (Ω), so we cannot hope to prove the convergence of the mollified functions fε to f in k,α CX -norm. What we have proved in Chapter 2, in the context of general vector fields and using the usual mollifiers, is a local uniform convergence of fε (and its derivatives) to f (see Theorem 2.20). Here we will see that on homogeneous groups, using the mollifiers adapted to G and exploiting the translation invariance of the k,α control distance, we can improve that result with a convergence of the CX norms, at least for compactly supported functions:

Theorem 8.52 (Bounded convergence of H¨ older norms on G) Let Ω be a k,α bounded open domain in G and let f ∈ CX,0 (Ω), for some integer k = 0, 1, 2, . . . and α ∈ (0, 1). Let fε = φε ∗ f where φε is a mollifier defined as in Proposition 3.48. Then: (i) fε ∈ C0∞ (Ω) for ε small enough; (ii) as ε → 0, fε → f uniformly in Ω and XI fε → XI f uniformly in Ω for |I| 6 k; (iii) for every ε > 0 small enough we have kfε kC k,α (Ω) 6 kf kC k,α (Ω) and X X kfε kC k,α (Ω) → kf kC k,α (Ω) as ε → 0. X

X

Point (iii) in the above theorem is the extra result that holds on homogeneous groups, while we could not establish it in the general context of Theorem 2.20. Proof. Points (i)-(ii) are actually contained in the proof of Proposition 3.48. To prove (iii), let ε > 0 be small enough so that supp fε ⊂ Ω. Then for every x ∈ Ω, Z Z
−1 |fε (x)| 6 φε (y) f y ◦ x dy 6 kf kL∞ (Ω) φε (y) dy = kf kL∞ (Ω) , G

G

and if XI = Xi1 Xi2 . . . Xij is any left invariant differential operator with |I| 6 k, Z
XI fε (x) = φε (y) XI f y −1 ◦ x dy G

which gives kfε kC k,0 (Ω) 6 kf kC k,0 (Ω) . Also, by the translation invariance of the X X control distance d, Z

|fε (x1 ) − fε (x2 )| 6 φε (y) f y −1 ◦ x1 − f y −1 ◦ x2 dy ZG α α 6 φε (y) |f |α d (x1 , x2 ) dy = |f |α d (x1 , x2 ) G

so that |fε |α 6 |f |α and an analogous inequality holds for the derivatives up to order k, so that (iii) holds. To show that |fε |α → |f |α , fix δ > 0. By definition of |f |α , there exist x1 , x2 ∈ Ω, x1 6= x2 , such that |f (x1 ) − f (x2 )| > |f |α − δ. α d (x1 , x2 ) Then for ε > 0 small enough, since fε → f uniformly, we have |f (x1 ) − f (x2 )| |fε (x1 ) − fε (x2 )| > − δ > |f |α − 2δ α α d (x1 , x2 ) d (x1 , x2 )

382

H¨ ormander operators

which gives |f |α − 2δ 6 |fε |α for ε small enough. Since, on the other hand, |fε |α 6 |f |α for every ε > 0, we conclude |fε |α → |f |α . The same argument works for the derivatives XI fε . 8.6.2

Interpolation inequalities in H¨ older spaces

In analogy with the technique used to prove the estimates for second order deriva1,α tives in Sobolev spaces, we need suitable interpolation formulas for CX norm and a family of cutoff functions. We start with the following interpolation formula for compactly supported functions. Proposition 8.53 Let Ω be a bounded open domain in G, then there exist positive 2,α constants c = c (G, ν, Ω, α) and γ = γ (G) such that for every u ∈ CX,0 (Ω) and every ε ∈ (0, 1) we have kukC 1,α (Ω) 6 ε kLf kL∞ (Ω) + cε−γ kukL∞ (Ω) . X

Proof. Let φ ∈ C0∞ (R) such that φ (t) = 0
for |t| > 1 and φ
(t) = 1 for |t| < 21 and let φ1 (x) = φ (kxk) and φε (x) = φ ε−1 kxk = φ1 D1/ε (x) . By Proposition 8.49 for every x ∈ Ω and i = 1, . . . , q we can write Xi u (x) = Lu ∗ (φε Xi Γ) (x) + Lu ∗ ((1 − φε ) Xi Γ) (x) ≡ Aε (x) + Bε (x) Let us estimate kAε kC α and kBε kC α separately. For the first term we have Z





Aε (x) = φ ε−1 y −1 ◦ x Xi Γ y −1 ◦ x Lu (y) dy N ZR ≡ Kε (x, y) Lu (y) dy. RN


ε Now, fix δ ∈ (0, 1). Since φ ε−1 y −1 ◦ x 6= 0 only if d(x,y) > 1, we can write
1−Q 1−δ−Q |Kε (x, y)| 6 cφ ε−1 d (x, y) d (x, y) 6 cεδ d (x, y) . Moreover for d (x1 , y) > 2d (x, x1 ) |Kε (x, y) − Kε (x1 , y)| 6 cεδ

d (x, x1 ) d (x1 , y)

1−δ−Q

Q+δ

= cεδ d (x1 , y)



d (x, x1 ) d (x1 , y)

 .

It follows that Kε is a fractional integral kernel of exponents β = 1, ν = 1 − δ. By Theorem 7.14 and Remark 7.16, this implies kAε kC α (Ω) 6 cεδ kLf kL∞ (Ω) . Let us consider the second term Bε . Set H (x) = (1 − φε (x)) Xi Γ (x) so that Bε (x) = Lu ∗ H. The idea now is to use the smoothness of H (we have removed the singularity using the cutoff) to move the derivatives from u to H. By Proposition 3.47 this can be done using the right invariant differential operator LR =

q X i=1

XiR


2

+ X0R .

Sobolev and H¨ older estimates for H¨ ormander operators on groups

383

We can write (at least formally), Bε = Lu ∗ H = u ∗ LR H. In order to estimate Bε (and check that the integral defining this last convolution converges absolutely) we need an estimate for LR H. Since



LR H (x) = − Xi Γ (x) ε−2 LR φ1 ε−1 kxk + 1 − φ ε−1 kxk LR Xi Γ (x) − ε−1

q X

XjR φ1





ε−1 kxk XjR Xi Γ (x) ,

j=1

exploiting the homogeneity of left and right invariant vector fields, when kxk > ε/2 we obtain   R L H (x) 6 c ε−2 kxk1−Q + kxk−Q−1 + ε−1 kxk−Q 6 cε−1−Q ,

while for kxk < ε/2 we simply have H (x) = 0. It follows that LR H ∞ 6 cε−1−Q and therefore Z R
L H y −1 ◦ x |u (y)| dy |Bε (x)| 6 N

R

6 LR H ∞ kukL1 (Ω) 6 cε−1−Q |Ω| kukL∞ (Ω) . With a similar computation we can also obtain the bounds



Xj LR H 6 cε−2−Q and X0 LR H 6 cε−3−Q . ∞ ∞ Therefore, using Lagrange’s theorem (see Theorem 1.56), we have Z R

L H y −1 ◦ x1 − LR H y −1 ◦ x2 |u (y)| dy |Bε (x1 ) − Bε (x2 )| = RN Z X





Xi LR H + d (x1 , x2 ) X0 LR H 6 d (x1 , x2 ) |u (y)| dy ∞ ∞ RN
6 d (x1 , x2 ) cε−2−Q + d (x1 , x2 ) cε−3−Q kukL∞ (Ω) . 6 d (x1 , x2 ) ε−3−Q kukL∞ (Ω) so that kBε kC α 6 ε−3−Q kukL∞ (Ω) . It follows that kXi u (x)kC α (Ω) 6 kAε (x)kC α (Ω) + kBε (x)kC α (Ω) 6 cεδ kLf kL∞ (Ω) + ε−3−Q kukL∞ (Ω) . Setting ε1 = cεδ gives kDukC α (Ω) 6 ε1 kLf kL∞ (Ω) + cε−γ 1 kukL∞ (Ω) X

for γ = (3 + Q) /δ. A similar argument applied to (8.64) gives kukC α (Ω) 6 ε kLf kL∞ (Ω) + cε−γ kukL∞ (Ω) X

and we are done. To extend the previous interpolation inequality to functions that do not have compact support we need the following:

384

H¨ ormander operators

Lemma 8.54 (Cutoff functions) For any 0 < 2s < t < s and every x ∈ G, there
∞ N exists ϕ ∈ C0 R with the following properties: B (x, t) ≺ ϕ ≺ B (x, s) ,

(8.68)

and for every k = 0, 1, 2, . . . there exists a constant ck = c (k, G, ν, α) (independent of s, t, ϕ) such that

k ck

D ϕ ∞ ; (8.69) 6 k L (G) (s − t)

k ck

D ϕ α 6 . (8.70) k+α CX (G) (s − t) α Moreover, for any f ∈ CX (G) c (8.71) kf Xi ϕkC α (G) 6 α (G) 2 kf kCX X (s − t) c (8.72) kf X0 ϕkC α (G) , kf Xi Xj ϕkC α (G) 6 α (G) . 3 kf kCX X X (s − t)



Proof. Using the translation invariance of Dk ϕ L∞ (G) and Dk ϕ C α (G) we can X clearly assume x = 0. Let us apply Lemma 8.40 with the balls B (0, σr) , B (0, σ 0 r) replaced by the balls B (0, t) , B (0, s) respectively. Then the corresponding cutoff function ϕ satisfies (8.68), c c |Xi ϕ| 6 and |X0 ϕ| + |Xi Xj ϕ| 6 , (s − t) (s − t)2 hence an obvious iteration gives (8.69). Let us show that this implies (8.70). Applying Lagrange theorem to XI ϕ for |I| = k we can write, for every x1 , x2 ∈ G, |XI ϕ (x1 ) − XI ϕ (x2 )| q X

6 cd (x1 , x2 ) ·

! kXi XI ϕkL∞ (G) + d (x1 , x2 ) kX0 XI ϕkL∞ (G)

i=1

6c

d (x1 , x2 ) k+1

  d (x1 , x2 ) · 1+ . (s − t)

(s − t) by (8.69). Next, let us distinguish two cases. If d (x1 , x2 ) 6 s − t we have α d (x1 , x2 ) d (x1 , x2 ) 6 c |XI ϕ (x1 ) − XI ϕ (x2 )| 6 c k+1 k+α (s − t) (s − t) If d (x1 , x2 ) > s − t then |XI ϕ (x1 ) − XI ϕ (x2 )| 6 |XI ϕ (x1 )| + |XI ϕ (x2 )| 6 2 kXI ϕkL∞ (G) α

α

d (x1 , x2 ) c d (x1 , x2 ) =c . α · k k+α (s − t) (s − t) (s − t) In any case, |XI ϕ|C α (G) 6 (r−t)c k+α , which gives (8.70). Finally, (8.71) and (8.72) X follow from the inequality (2.18): 62

kf gkC α (B(x,R)) 6 2 kf kC α (B(x,R)) kgkC α (B(x,R)) . X

X

Next, we need the following technical lemma:

X

Sobolev and H¨ older estimates for H¨ ormander operators on groups

385

Lemma 8.55 Let ψ (t) be a bounded nonnegative function defined on the interval [T0 , T1 ], where T1 > T0 > 0. Suppose that for any T0 6 t < s 6 T1 , the function ψ satisfies ψ (t) 6 ϑψ (s) +

A β

(s − t)

+ B,

(8.73)

where ϑ, A, B, β are nonnegative constants, and ϑ < 31 . Then, there exists a constant cβ (only depending on β) such that for T0 6 ρ < R 6 T1 " # A ψ (ρ) 6 cβ +B . (8.74) β (R − ρ) Proof. Let t0 = ρ, ti+1 = ti + (1 − τ ) τ i (R − ρ) (i = 0, 1, 2, . . .), where τ ∈ (0, 1) is to be determined. Note that ti ↑ R, in particular ti ∈ [T0 , T1 ]. From (8.73) ψ (ti ) 6 ϑψ (ti+1 ) +

A β

[(1 − τ ) τ i (R − ρ)]

+ B (i = 0, 1, 2, . . .) .

By iteration, ψ (t0 ) 6 ϑk ψ (tk ) +

k−1 X

A β

[(1 − τ ) (R − ρ)]

ϑi τ −iβ + B

i=0

k−1 X

ϑi .

i=0

Since ϑ < 31 , we can choose τ such that ϑτ −β = 12 ; then ψ (t0 ) 6 ϑk ψ (tk ) + h

3 A 2 + B. iβ · β
1/β 2 (R − ρ) 1 − 32

For k → +∞, we get (8.74). Finally we have the following: Proposition 8.56 (Interpolation inequality) Let R > 0, There exist positive constants c = c (G, ν, α) and γ = γ (G) such that for any u ∈ C 2,α (B (x, R)), 0 < δ < 1/3, R/2 < r < R, kDukC α (B(x,r)) 6 δ kLukL∞ (B(x,R)) +

δγ

c γ kukL∞ (B(x,R)). (R − r)

The constant γ is as in Proposition 8.53, the constant c depends on α, G, R and γ. Proof. Let u ∈ C 2,α (B (x, R)) , let R/2 < t < s 6 R and let ϕ as Lemma 8.54. Applying Proposition 8.53 to uϕ, we get: kDukC α (B(x,t)) 6 kD (ϕu)kC α (B(x,s)) 6 ε kL (ϕu)kL∞ (B(x,s)) +

c kukL∞ (B(x,s)) εγ (8.75)

386

H¨ ormander operators

where, by Lemma 8.54, kL (ϕu)kL∞ (B(x,s)) 6 kϕLukL∞ (B(x,s)) + 2

q X

kXj ϕXj ukL∞ (B(x,s)) + kuLϕkL∞ (B(x,s))

j=1

6 kLukL∞ (B(x,s)) +

c c kDukL∞ (B(x,s)) + 2 kukL∞ (B(x,s)) . s−t (s − t)

Next, we insert the last inequality in (8.75). Choosing ε = δ (s − t) /c we get (s − t) kLukL∞ (B(x,s)) kDukC α (B(x,t)) 6 δ kDukL∞ (B(x,s)) + δ X c   δ 1 kukL∞ (B(x,s)) +c + γ (s − t) δ (s − t)γ Now, let ψ (t) = kDukC α (B(x,t)) and fix ϑ < 1/3. Since γ > 1, for all δ < ϑ and for X any 0 < t < s < R we have ψ (t) 6 ϑψ (s) +

cR γ kukL∞ (B(x,R)) + cR δ kLukL∞ (B(x,R)) δ γ (s − t)

By Lemma 8.55 we get ψ (r) 6

δγ

c γ kukL∞ (B(x,R)) + cR δ kLukL∞ (B(x,R)) (R − r)

for any 0 < r < R, which gives the assertion. 8.6.3

Local H¨ older estimates for second order derivatives

In this section we prove the following result: Theorem 8.57 (Local H¨ older regularity for second derivatives) Let L be in the class Lν for some ν > 0, let Ω be a bounded domain in G, with the homogeneous dimension Q > 2, let Ω0 b Ω00 b Ω and let 0 < α < 1. If u is a distribution such 2,α α that Lu ∈ CX (Ω), then u ∈ CX (Ω00 ) and n o kukC 2,α (Ω0 ) 6 c kLukC α (Ω00 ) + kukL∞ (Ω00 ) X

X

where c = c(p, G, Ω0 , Ω00 , ν). We start with the following solvability result in H¨older spaces. 2,α Proposition 8.58 (Solvability in CX (Ω)) Let L be in the class Lν for some α ν > 0, let Ω be a bounded domain in G and let 0 < α < 1. For every f ∈ CX,0 (Ω) 2,α there exists v ∈ CX (Ω) such that Lv = f . Moreover,

kvkC 2,α (Ω) 6 c (G, ν, α) kf kC α (Ω) . X

X

Sobolev and H¨ older estimates for H¨ ormander operators on groups

387

Proof. Let fε be the mollified of f as defined in Theorem 8.52 and let vε = fε ∗ Γ, with ε > 0 small enough so that fε ∈ C0∞ (Ω). Note that also vε ∈ C ∞ (Ω), by the smoothness of fε . On the other hand, by our results on kernels of type 2, 1 and 0 (see Proposition 8.17 and Theorem 8.25), we can write kvε kC 2,α (Ω) 6 c (G, ν, α) kfε kC α (Ω) 6 c kf kC α (Ω) X

X

X

where in the last bound we exploited Theorem 8.52 (iii). In particular, the sequence 2,α vε is bounded in CX (Ω), which implies that the functions XI vε with 0 6 |I| 6 2 are equicontinuous and equibounded. By Ascoli-Arzel`a’s theorem, there exists a sequence εn → 0 such that XI vεn converges locally uniformly in Ω, and then α uniformly because of the compact support. Also, the uniform limit of a CX (Ω) 2,α α function is still a CX (Ω) function, hence the limit of vε , say u, belongs to CX (Ω). On the other hand, if we let v = f ∗ Γ, then a similar argument shows that v is at least a weak solution to Lv = f , while the identity vε − v = (fε − f ) ∗ Γ implies, by Proposition 8.17, kvε − vkC α (Ω) 6 c kfε − f kL∞ (Ω) → 0 X

(the uniform convergence of fε is contained in Theorem 8.52). Then v = u, so that 2,α v ∈ CX (Ω) and kvkC 2,α (Ω) 6 c (G, ν, α) kf kC α (Ω) . X

X

We can now establish the following local a priori estimate: 2,α Proposition 8.59 (Local CX estimates) Let R > 0 and 0 < α < 1. There exist positive constants c = c (R, G, ν, α) and β = β (G, α) such that, if x ∈ G and 2,α u ∈ CX (B(x, R)) and R/2 < t < R, then o n c kukC 2,α (B(x,t)) 6 . kLuk α (B(x,R)) + kukL∞ (B(x,R)) C X X (R − t)β 2,α Proof. Let u ∈ CX (B (x, R)) and let s = R+t so that t < s < R. Applying 2 Lemma 8.54 we construct a cutoff function ϕ satisfying B (x, t) ≺ ϕ ≺ B (x, s). We can now apply Proposition 8.50 and Proposition 8.53 (with ε = 1) to uϕ getting,

kukC 2,α (B(x,t)) 6 D2 (ϕu) C α (B(x,s)) + kϕukC 1,α (B(x,s)) X X X   6 c kL (ϕu)kC α (B(x,s)) + kϕukL∞ (B(x,s)) X ( q X 6 c kϕLukC α (B(x,s)) + kXi ϕXi ukC α (B(x,s)) X

X

i=1

) + kuLϕkC α (B(x,s)) + kϕukL∞ (B(x,s)) X

( 6c

1 1 kLukC α (B(x,s)) + α (B(x,s)) 2 kDukCX X s−t (s − t) )

+ kuLϕkC α (B(x,s)) + kukL∞ (B(x,s)) X

388

H¨ ormander operators

We handle the term kuLϕkC α (B(x,s)) separately. Applying (2.19), Lemma 8.54 and X (2.24) we can write, for some small η to be chosen later:   |uLϕ|C α (B(x,s)) 6 R1−α kD (uLϕ)kL∞ (B(x,s)) + R kX0 (uLϕ)kL∞ (B(x,s)) X c c 6 (8.76) 2 kDukL∞ (B(x,s)) + 3 kukL∞ (B(x,s)) (s − t) (s − t) c + cη α/2 |X0 (uLϕ)|C α (B(x,s)) + kuLϕkL∞ (B(x,s)) X η where we are allowing c to depend on R. Now, again by Lemma 8.54 and (2.19), |X0 (uLϕ)|C α (B(x,s)) 6 |(X0 u) Lϕ|C α (B(x,s)) + |uX0 Lϕ|C α (B(x,s)) X X X c 6 α (B(x,s)) 3 kX0 ukCX (s − t)   + R1−α kD (uX0 Lϕ)kL∞ (B(x,s)) + R kX0 (uX0 Lϕ)kL∞ (B(x,s)) c c c 6 α (B(x,s)) + 3 kX0 ukCX 4 kDukL∞ (B(x,s)) + 5 kukL∞ (B(x,s)) (s − t) (s − t) (s − t) c c + 4 kX0 ukL∞ (B(x,s)) + 6 kukL∞ (B(x,s)) (s − t) (s − t) c c c 6 α (B(x,s)) + 4 kX0 ukCX 4 kDukL∞ (B(x,s)) + 6 kukL∞ (B(x,s)) . (s − t) (s − t) (s − t) Inserting this last in (8.76) we obtain kuLϕkC α (B(x,s)) = kuLϕkL∞ (B(x,s)) + |uLϕ|C α (B(x,s)) X X c c 6 3 kukL∞ (B(x,s)) + 2 kDukL∞ (B(x,s)) (s − t) (s − t) ( 1 1 α/2 + cη α (B(x,s)) + 4 kX0 ukCX 4 kDukL∞ (B(x,s)) (s − t) (s − t) ) c 1 + + 6 kukL∞ (B(x,s)) 2 kukL∞ (B(x,s)) (s − t) η (s − t) 4

Let now cη α/2 / (s − t) = ε (with ε to be chosen later), then c c kukC 2,α (B(x,t)) 6 kLukC α (B(x,s)) + (8.77) α (B(x,s)) 2 kDukCX X X s−t (s − t) c kukL∞ (B(x,s)) . + ε kX0 ukC α (B(x,s)) + 2+8/α X 2/α ε (s − t) Next, we apply the interpolation inequality of Proposition 8.56 with δ = (s − t), and recalling that by our choice of t, s, R, we have s − t = R − s = R−t 2 , we obtain kukC 2,α (B(x,t)) 6 X

c c kLukC α (B(x,R)) + ε kukC 2,α (B(x,s)) + 0 kukL∞ (B(x,R)) . β0 X X R−t εα (s − t)

Let now ψ (t) = kukC 2,α (B(x,t)) and fix ε < 1/3. Then X   c ψ (t) 6 εψ (R) + kLuk + kuk α ∞ CX (B(x,R)) L (B(x,R)) , β0 (R − t)

Sobolev and H¨ older estimates for H¨ ormander operators on groups

389

and by Lemma 8.55 we get kukC 2,α (B(x,t)) 6 X

c

 β0

(R − t)

 kLukC 2,α (B(x,R)) + kukL∞ (B(x,R)) . X

Proof of Theorem 8.57. The argument is similar to that of Theorem 8.44. Let Ω000 such that Ω00 b Ω000 b Ω and let ψ ∈ C0∞ (Ω) such that Ω000 ≺ ψ ≺ Ω. Since 2,α α ψLu ∈ CX,0 (Ω) by Proposition 8.58 there exists v ∈ CX (Ω) such that Lv = ψLu. Now, observe that L (u − v) = (1 − ψ) Lu and since (1 − ψ) Lu vanishes in Ω000 by H¨ ormander’s theorem (Theorem 5.64) u − v is smooth in Ω000 . It follows that 2,α u ∈ CX (Ω00 ). Let us now fix r < dist (Ω0 , ∂Ω00 ) and let Br be a ball with center in Ω0 and radius r, then Br ⊂ Ω00 . By Proposition 8.59 we have o c n kukC 2,α (Br/2 ) 6 β kLukC α (Br ) + kukL∞ (Br ) . X X r We can now cover Ω0 with a finite number of balls Bj of radius r/2 so that (recalling (2.23)) o Xn X kLukC α (2Bj ) + kukL∞ (2Bj ) kukC 2,α (Bj ) 6 c kukC 2,α (Ω0 ) 6 X X X o n 6 c kLukC α (Ω00 ) + kukL∞ (Ω00 ) . X

with c = c(α, G, Ω0 , Ω00 , ν). 8.6.4

Local H¨ older estimates for higher order derivatives in Carnot groups

We are now going to prove the following higher order regularity result in Carnot groups, analogous to Theorem 8.34. Theorem 8.60 Let G be a Carnot group and let L be in the class Lν for some ν > 0. Let Ω be a bounded domain in G, Ω0 b Ω00 b Ω and 0 < α < 1. If k,α (Ω) for some integer k = 0, 1, 2, . . . then u is a distribution such that Lu ∈ CX k+2,α 00 u ∈ CX (Ω ) and o n kukC k+2,α (Ω0 ) 6 c kLukC k,α (Ω00 ) + kukL∞ (Ω00 ) X

X

where c = c(α, G, Ω0 , Ω00 , ν). We start with the following, which is the analog, for the H¨older case, of Proposition 8.32. Proposition 8.61 Let G be a Carnot group and let L be in the class Lν for some ν > 0. For every α ∈ (0, 1) and every k ∈ N there exists a constant c = c (α, k, G, ν) k+2,α such that if u ∈ CX,0 (BR ) then

k+2

D u α 6 c Dk Lu α . (8.78) CX (BR )

CX (BR )

Also, we have   kukC k+2,α (BR ) 6 c kLukC k,α (BR ) + kukL∞ (BR ) . X

X

390

H¨ ormander operators

Proof. As in the proof of Proposition 8.32, we can establish (8.78) first for derivatives with respect to the canonical generators Yi , and then get the general case, by Proposition 6.12, with the constant c also depending on ν. In the proof of Proposition 8.32 we have established the following representation formula (8.39): Yj Yh P k u (x) =

q X

Z lim

j1 ,j2 ,··· ,jk =1

ε→0

ky −1 ◦xk>ε


Yj Yh Γj1 ,...,jk y −1 ◦ x Yj1 Yj2 · · · Yjk Lu (y) dy

+ cjhj1 ,...,jk Yj1 Yj2 · · · Yjk Lu (x)
for any u ∈ C0∞ RN and differential monomial P k of degree k (see Notation 8.31), for suitable kernels Γj1 ,...,jk of type 2. By Theorem 8.25 we deduce

k+2

D u C α (B ) 6 c Dk Lu C α (B ) X

R

X

R

C0∞

for every u ∈ (BR ). Moreover, reasoning like in the proof of Proposition 8.49, the previous representation formula and estimate can be extended to any k+2,α u ∈ CX,0 (BR ). Summing up inequalities (8.78) for j = 0, 1, . . . , k and recalling that u is compactly supported we get, using also Theorem 8.57,     kukC k+2,α (BR ) 6 c kLukC k,α (BR ) + kukC 1,α (BR ) 6 c kLukC k,α (BR ) + kukL∞ (BR ) . X

X

X

X

Next, we can prove the following regularity result: Proposition 8.62 Let G be a Carnot group and let L be in the class Lν for some ν > 0. For every α ∈ (0, 1) and every k ∈ N there exists a constant c = c (p, k, G, ν) k,α k+2,α such that if u ∈ C0α (BR ) and Lu ∈ CX,0 (BR ), then u ∈ CX (BR ) and   kukC k+2,α (BR ) 6 c kLukC k,α (BR ) + kukL∞ (BR ) . X

X

Proof. Let uε be the mollified of u (defined according to Proposition 3.48). Then L (uε ) = (Lu)ε . This identity is not trivial because a priori u does not belong to 2,α CX , however it can be checked reasoning on derivatives in weak sense. For ε 2,α small enough uε ∈ CX,0 (BR ), hence applying Proposition 8.61 to uε and recalling Theorem 8.52 we can write   kuε kC k+2,α (BR ) 6 c kLuε kC k,α (BR ) + kuε kL∞ (BR ) X X   6 c kLukC k,α (BR ) + kukL∞ (BR ) . X

k+2,α In particular, the sequence uε is bounded in CX (BR ), and up to passing to a k+2,0 k+2,α subsequence it converges in CX (BR ) to a function v ∈ CX (BR ) which must k+2,α coincide with u. Therefore u ∈ CX (BR ) and the a priori estimate holds.

Sobolev and H¨ older estimates for H¨ ormander operators on groups

391

We now want to remove the assumption of compact support on the function u. The first step is the following. Proposition 8.63 Let G be a Carnot group. Then Theorem 8.60 holds under the k+1,α additional requirement that u ∈ CX (Ω00 ). Proof. We proceed by induction on k. For k = 0 the result is contained in Theorem 8.57. Assume the result holds up to k − 1, and let us prove it for k. So, let k+1,α k,α u ∈ CX (Ω00 ) and Lu ∈ CX (Ω). Let r > 0 and let Br be a ball
of radius r and 0 center in Ω such that B2r b Ω00 . Let σ = 21 and let ϕ ∈ C0∞ RN as in Lemma 8.40, so that Br/2 ≺ ϕ ≺ Br . Since L (ϕu) = ϕLu + uLϕ +

q X

Xi uXi ϕ

i=1


k,α we see that L (ϕu) ∈ CX,0 RN . Applying Proposition 8.62 we obtain uϕ ∈ k+2,α CX,0 (Br ) and   kϕukC k+2,α (Br ) 6 c kL (ϕu)kC k,α (Br ) + kϕukL∞ (Br ) X

X

so that   kukC k+2,α (Br/2 ) 6 c kLukC k,α (Br ) + kukC k+1,α (Br ) . X

X

X

Applying the inductive assumption we can write   kukC k+1,α (Br ) 6 c kLukC k−1,α (B2r ) + kukL∞ (B2r ) X

X

hence   kukC k+2,α (Br/2 ) 6 c kLukC k,α (B2r ) + kukL∞ (B2r ) . X

X

By Proposition 2.18 (iii), a covering argument gives the desired result. k+1,α To remove the assumption u ∈ CX (Ω00 ) we need the following:

Lemma 8.64 Let G be a Carnot group. Let u satisfy the same assumptions of k+1,α Theorem 8.60, let ψ ∈ C0∞ (Ω) and let v = (ψLu) ∗ Γ. Then v ∈ CX (Ω00 ). Proof. As in the proof of Proposition 8.32, we will work with derivatives with respect to the canonical generators Yi , and will exploit the fact that, by Proposition k,α 6.12, CX (Ω) = CYk,α (Ω), with equivalent norms. Fix 0 6 ` 6 k and let P ` = Yi1 Yi2 · · · Yi` with 1 6 ij 6 q. As in the proof of Proposition 8.32 we can write q X P `v = Yj1 Yj2 · · · Yj` (ψLu) ∗ Γj1 ,j2 ,...,j` Yj P k v =

j1 ,j2 ,...,j` =1 q X j1 ,j2 ,...,jk =1

Yj1 Yj2 · · · Yjk (ψLu) ∗ Xj Γj1 ,j2 ,...,jk

392

H¨ ormander operators

where Γj1 ,j2 ,...,j` are suitable kernels of type 2 and Yj Γj1 ,j2 ,...,jk are kernels of type k,α α 1. Since Lu ∈ CX (Ω) and ψ ∈ C0∞ (Ω), Yj1 Yj2 · · · Yj` (ψLu) ∈ CX,0 (Ω). Then, by k+1,α α Proposition 8.17, v, P ` v, Yj P k v belong to CX (Ω), that is v ∈ CX (Ω). Proof of Theorem 8.60. Choose Ω000 such that Ω00 b Ω000 b Ω, let ψ ∈ C0∞ (Ω) k+1,α satisfying Ω000 ≺ ψ ≺ Ω and let v = (ψLu) ∗ Γ, then by Lemma 8.64 v ∈ CX (Ω). In the sense of distributions we have L (u − v) = (1 − ψ) Lu. Hence L (u − v) vanishes on Ω000 and since L is hypoelliptic u − v is smooth in Ω000 . k+1,α k+2,α If follows that u ∈ CX (Ω00 ). By Proposition 8.63 we obtain u ∈ CX (Ω00 ) and n o kukC k+2,α (Ω0 ) 6 c kLukC k,α (Ω) + kukL∞ (Ω00 ) . X

8.6.5

X

Local H¨ older estimates for higher order derivatives in homogeneous groups stratified of type II

In this section we will prove the following: Theorem 8.65 Let G be a homogeneous group stratified of type II with homogeneous dimension Q > 4 and let L be in the class Lν for some ν > 0. Let Ω be a bounded domain in G, let Ω0 b Ω00 b Ω and let 0 < α < 1. If u is a distribution 2k+2,α 2k,α (Ω00 ) (Ω) for some integer k = 0, 1, 2, . . . , then u ∈ CX such that Lu ∈ CX and o n kukC 2k+2,α (Ω0 ) 6 c kLukC 2k,α (Ω00 ) + kukL∞ (Ω00 ) X

X

0

00

where c = c(α, k, G, Ω , Ω , ν). The line of the proof of this result is a mix of that of higher order estimates in Sobolev spaces on homogeneous groups stratified of type II (section 8.5.3) and that of higher order estimates in H¨ older spaces in Carnot groups (section 8.6.4). The first step is the following. Proposition 8.66 Let L be in the class Lν for some ν > 0. Let α ∈ (0, 1), let R > 0, k be a positive integer and let P 2k be a left invariant differential operator homogeneous of degree 2k. Then, there exists c = c (α, k, G, ν, R) > 0 such that for every u ∈ C0∞ (BR ) we have

2k

P u α 6 c Lk u α . CX (BR )

CX (BR )

Proof. This can be proved analogously to Proposition 8.37, applying Theorem 8.25 instead of Theorem 8.22 to the same representation formulas.
2k,α Lemma 8.36 still holds taking u ∈ CX,0 (BR ) instead of u ∈ C0∞ RN .

Sobolev and H¨ older estimates for H¨ ormander operators on groups

393

Lemma 8.67 For every integer k > 2 and α ∈ (0, 1), there exist constants 2k,α c(G, ν, k, α), γ(G, k) such that for every u ∈ CX,0 (BR ), we have  

2k−1

2k 1

D

u C α (B ) 6 c ε D u L∞ (B ) + γ kukL∞ (BR ) . R R X ε Proof. Let P 2k−1 be a differential monomial, homogeneous of degree 2k − 1, and let K1 be as in Lemma 8.36. Let φ ∈ C0∞ (R)
such that φ (t)
= 0 for |t| > 1 and φ (t) = 1 for |t| < 21 and let φε (x) = φ ε−1 kxk = φ1 D1/ε (x) . ε ε We split the kernel K1 as K1 = φε K1 + (1 − φε ) K1 ≡ K1,0 + K1,∞ so that ε ε P 2k−1 u = Lk u ∗ K1,0 + Lk u ∗ K1,∞ . ε Clearly K1,0 is homogeneous of degree (1 − Q) near the origin and has compact ε support, hence it is integrable, while K1,∞ is homogeneous of degree (1 − Q) at ε ε e (x) = K1,∞ (x−1 ), then, as in the infinity and vanishes near the origin. Let K1,∞ proof of Lemma 8.38, Z
k eε −1 ε ◦ y u(y)dy = u ∗ Kε (x) Lk u ∗ K1,∞ (x) = (L∗ ) K 1,∞ x  
k eε where Kε (x) = (L∗ ) K x−1 and L∗ is the transpose operator of L. There1,∞ fore ε P 2k−1 u = Lk u ∗ K1,0 + u ∗ Kε (x)

(8.79)

where Kε is homogeneous of degree (1 − Q − 2k) at infinity and vanishes near the origin, hence it is integrable. Now, fixing δ ∈ (0, 1), for the first term in (8.79) we have that (as in the proof of ε Proposition 8.53) K1,0 is a fractional integral kernel of exponents β = 1, ν = 1 − δ, and operator norm 6 cεδ . By Theorem 7.14 and Remark 7.16, this implies

k



ε

L u ∗ K1,0 6 cεδ Lk u ∞ 6 cεδ D2k u ∞ . α L

CX (BR )

(BR )

L

(BR )

For the second term in (8.79), |Kε (x)| 6 c/εγ for some γ > 0, hence c c ku ∗ Kε k∞ 6 γ kukL1 (BR ) 6 γ kukL∞ (BR ) . ε ε In order to bound |u ∗ Kε |C α , using Lagrange’s theorem we can write X

|u ∗ Kε (x1 ) − u ∗ Kε (x2 )| Z

Kε y −1 ◦ x1 − Kε y −1 ◦ x2 |u (y)| dy = BR Z X  6 d (x1 , x2 ) kXi Kε k∞ + d (x1 , x2 ) kX0 Kε k∞ |u (y)| dy BR −γ1

6 d (x1 , x2 ) cε


+ d (x1 , x2 ) cε−γ2 kukL∞ (BR )

6 cd (x1 , x2 ) ε−γ3 kukL∞ (BR ) hence, for some γ 0 > 0 0

ku ∗ Kε kC α (BR ) 6 cε−γ kukL∞ (BR ) X

and we are done.

394

H¨ ormander operators

Theorem 8.68 (A priori estimates for compactly supported functions) Let L be in the class Lν for some ν > 0. Let α ∈ (0, 1) , R > 0 and let k ∈ N. Then, k,α there exists c = c (k, α, G, ν, R) > 0 such that if u ∈ C0α (BR ) and Lu ∈ CX,0 (BR ), k+2,α then u ∈ CX (BR ) and   kukC 2k+2,α (BR ) 6 c kLukC 2k,α (BR ) + kukL∞ (BR ) . (8.80) X

X

2k+2,α Proof. It is enough to prove (8.80) assuming u ∈ CX,0 (BR ); the regularization result then follows as in Proposition 8.62. The proof of (8.80) is by induction on k. The case k = 0 is contained in Theorem 8.57. Now assume to know that (8.80) holds for k and let us prove it for k + 1. Let P h be any left invariant differential operator that is homogeneous of degree h 6 2k + 4. When h 6 2k + 2 by (8.80) we obtain  

h

P u α kLuk + kuk 6 kuk 6 c 2k,α 2k+2,α ∞ L (BR ) . C (BR ) C (BR ) C (B ) X

R

X

X

It remains to consider the cases h = 2k + 3 and h = 2k + 4. Assume first h = 2k + 4, then by Proposition 8.66 we have

2k+4

P u C α (B ) 6 c Lk+2 u C α (B ) 6 c kLukC 2(k+1),α (BR ) . X

R

X

R

X

Finally let h = 2k + 3. By Lemma 8.67 (with ε = 1) and Proposition 8.66 we have  



2k+3

P u C α (B ) 6 D2k+4−1 u C α (B ) 6 c D2k+4 u L∞ (B ) + kukL∞ (BR ) R R R X X   6 c kLukC 2(k+1),α (BR ) + kukL∞ (BR ) . X

To remove the assumption on the compact support of u, we need the following interpolation inequality: Proposition 8.69 (Interpolation inequality) Let R > 0. There exist positive 2k,α constants c, γ such that for any u ∈ CX (B (x, R)), δ > 0 small enough, R/2 < r < R,

2k−1

D u C α (B(x,r)) 6 δ D2k u L∞ (B(x,R))   2k X

c

D2k−j u ∞  + kukL∞ (B(x,R))  . + γ γ L (B(x,R)) δ (R − r) j=2 2k,α Proof. Let u ∈ CX (B (x, R)) , let R/2 < t < s 6 R and let ϕ as Lemma 8.54. Applying Lemma 8.67 to uϕ, we get:

2k−1

D u C α (B(x,t)) 6 D2k−1 (ϕu) C α (B(x,s))

c 6 ε D2k (ϕu) L∞ (B(x,s)) + γ kukL∞ (B(x,s)) ε

Sobolev and H¨ older estimates for H¨ ormander operators on groups

395

where, by Lemma 8.54,

2k

D (ϕu) ∞ L (B(x,s)) 2k X

2k−j

D 6 D2k u L∞ (B(x,s)) + u L∞ (B(x,s)) Dj ϕ L∞ (B(x,s)) j=1 2k X

2k−j

D u L∞ (B(x,s)) 6 D2k u L∞ (B(x,s)) + j=1

c j

(s − t)

.

2k−1

c

D2k−1 u ∞

D u C α (B(x,t)) 6 ε D2k u L∞ (B(x,s)) + ε L (B(x,s)) s−t 2k X

2k−j c c

D + γ kukL∞ (B(x,s)) . +ε u L∞ (B(x,s)) j ε (s − t) j=2 Next, we set ε = (s − t) δ/c, with δ > 0, getting



2k−1

D u C α (B(x,t)) 6 c1 δ D2k u L∞ (B(x,s)) + δ D2k−1 u C α (B(x,s)) +

2k X

2k−j

D u L∞ (B(x,s))

c j

(s − t) j=2

2k−1 Let ψ (t) = D u C α (B(x,t)) hence for a fixed ϑ < for some γ 0 > 0 and every

+ 1 3

δγ

c γ kukL∞ (B(x,s)) . (s − t)

and every δ 6 ϑ we have,

R 2

< t < s < R,

2k ψ (t) 6 ϑψ (s) + c1 δ D u L∞ (B(x,s))   2k X

2k−j c

D  u L∞ (B(x,s)) + kukL∞ (B(x,s))  , + 0 γ0 δ γ (s − t) j=2

which by Lemma 8.55 implies

2k−1

D u C α (B(x,t)) 6 c1 δ D2k u L∞ (B(x,s))   2k X

2k−j c

D  + 0 u L∞ (B(x,s)) + kukL∞ (B(x,s))  . γ0 δ γ (s − t) j=2

As in the case of Carnot groups, the proof of local estimates is firstly done under an additional regularity assumption on u, which is removed later. Lemma 8.70 Let G be a homogeneous stratified group of type II. Then Theorem 2k+1,α 8.65 holds under the additional requirement that u ∈ CX (Ω00 ). Proof. Fix r < dist (Ω0 , ∂Ω00 ) and let Br
be a ball of radius r and center in Ω0 . For t < s and s0 = (t + s) /2, let ϕ ∈ C0∞ RN as in Lemma 8.40, so that Bt ≺ ϕ ≺ Bs0 .

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H¨ ormander operators

2k,α From (8.62) we read that L (ϕu) ∈ CX,0 (BR ). Applying Theorem 8.68 to uϕ and using the estimates in Lemma 8.54 after some computation we obtain   kukC 2k+2,α (Bt ) 6 c kL (ϕu)kC 2k,α (B 0 ) + kϕukL∞ (Bs0 ) X

s

X

kϕLukC 2k,α (B

6c +

X

q X

s0 )

+ kuLϕkC 2k,α (B X

s0 )

! kXi ϕXi ukC 2k,α (B

s0 )

X

+ kϕukL∞ (Bs0 )

i=1

6

c 2k+1

(s0 − t) +c

kLukC 2k,α (B

2k X h=0

X

1 (s0

− t)

2k+3−h

s0 )

+

c 2k+2

(s0 − t)

h

D u α C (B X

s0 )

2k+1

D u C α (B X

s0 )

.

By Proposition 8.69,

2k+1

D u C α (B 0 ) 6 δ D2k+2 u L∞ (B(x,s)) + s X   2k+2 X

c

D2k+2−j u ∞  + kukL∞ (B(x,s))  . γ L (B(x,s)) δ γ (s − s0 ) j=2 2k+2

Pick δ =

(s0 −t)

ε

, hence

2k+2 u L∞ (B(x,s)) kukC 2k+2,α (Bt ) 6 ε D X   2k+2 X

c

D2k+2−j u ∞  + + kukL∞ (B(x,s))  3γ+2kγ L (B(x,s)) γ ε (s − t) j=2 c

+

c 2k+1

(s − t)

kLukC 2k,α (Bs ) + c

2k X

1 2k+3−h

X

h=0

(s − t)

h

D u α . C (B ) X

Let ψ (t) = kukC 2k+2,α (Bt ) , by Lemma 8.55 X

kukC 2k+2,α (Bt ) 6 X

c (s − t)

2k+1

kLukC 2k,α (Bs ) + X

c β

(s − t)

2k X

h

D u α . C (Bs ) h=0

X

Now we can make an iterative reasoning, letting s = 2t and writing

c c kukC 2k+2,α (Bt ) 6 β1 kLukC 2k,α (B k ) + β2 D2 u C α (B ) X X 2 t X 2k t t t which together with Theorem 8.57 gives c c kukC 2k+2,α (Bt ) 6 β1 kLukC 2k,α (B k+1 ) + β2 kukL∞ (B k+1 ) . X X 2 t 2 t t t This by a covering argument gives the assertion.

s

Sobolev and H¨ older estimates for H¨ ormander operators on groups

397

Lemma 8.71 Let G be a homogeneous stratified group of type II. Let u satisfy the assumptions of Theorem 8.65, let ψ ∈ C0∞ (Ω) and let v = (ψLu) ∗ Γ. Then 2k+1,α v ∈ CX (Ω00 ). 2k,α Proof. Let Lu ∈ CX (Ω). Arguing as in Lemma 8.36, if 2 6 2` 6 k and P 2`+1 and P 2` are left invariant differential monomials of degree 2`+1 and 2` respectively, we can determine kernels K1 and K2 of type 1 and 2 such that P 2`+1 v = L` (ψLu) ∗ K1 and P 2` v = L` (ψLu) ∗ K2 . α Since ψLu, L` (ψLu) ∈ CX,0 (Ω), the same argument used in the proof of Lemma 2k+1,α 2`+1 α 8.64 shows that P v and P 2` v are in CX (Ω) and therefore v ∈ CX (Ω).

Finally, from the above results Theorem 8.65 follows with the same proof as Theorem 8.60. 8.7

Notes

The study of the regularity of solutions of the equation Lu = f in the scale of Sobolev spaces defined by vector fields was initiated by Folland and Stein in [91], in which the Heisenberg group was a model for a much more complex setting: the ∂ b complex on the boundary of a complex domain with nondegenerate Levi form. The Lp and H¨ older estimates proved by the above authors on the Heisenberg group are a key tool for the study of the regularity in this more general setting. A much more complete analysis of the regularity of sublaplacians on nilpotent Lie groups with dilations was carried out by Folland in [85]. In his paper the author defines Sobolev spaces, Lipschitz and H¨ older spaces adapted to the context of homogeneous groups. In particular, if L is a sublaplacian on a Carnot group, using the diffusion semigroup generated by −L he is able to define fractional powers of L and then Sobolev spaces of fractional order. When the order is an integer his Sobolev spaces k,p (G) defined in Chapter 2. The paper [85] is the same coincide with the spaces WX where the existence and properties of the homogeneous fundamental solution for L on G was proved, hence that paper is the backbone of our Chapters 3, 6 and 8. Clearly, much of the material in this chapter is presented here with a language and a point of view different from that of [85]. In particular, we have systematically adopted the language (completely standard in the context of elliptic PDEs) of a priori estimates involving Banach space norms, taking on the left hand side of the estimate the derivatives of intermediate order. This requires the use of density results, interpolation inequalities, cutoff functions and some techniques of Sobolev and H¨ older spaces which were not present in [85]. Interpolation inequalities for Sobolev norms in homogeneous groups have been firstly proved, in the context of H¨ ormander vector fields, by Bramanti-Brandolini in [26]. As we have written in the introduction (section 8.1.1), in Chapter 11 we will show that regularity results similar to the local ones proved in this chapter can be proved for general H¨ ormander vector fields, with more refined techniques.

398

H¨ ormander operators

Between the case of left invariant vector fields on homogeneous groups and that of general H¨ ormander vector fields there is an interesting intermediate case, given by left invariant H¨ ormander vector fields on a nonhomogeneous group, i.e. without an underlying family of adapted dilations. This is the case of a class of operators of Kolmogorov-Fokker-Planck type with linear drift, which has been introduced by Lanconelli-Polidoro in [122] and has been studied in a number of subsequent papers. The literature on this subject is by now very large, we just point out the survey paper by Lanconelli-Pascucci-Polidoro [121] and the paper by BramantiCupini-Lanconelli-Priola [36], with the references therein. There is also a line of research, started with the paper by Bonfiglioli-Lanconelli [13], consisting in giving sufficient conditions for the existence of a Lie group in Rn which makes an assigned H¨ ormander operator left invariant. Another interesting intermediate case is that of sums of squares of H¨ormander vector fields which are 1-homogeneous on Rn with respect to a family of nonisotropic dilations, without being left invariant with respect to any Lie group. This situation has been studied in [9], where global Sobolev estimates have been established in this context, extending (at least for sums of squares) the result of Theorem 8.2.

Chapter 9

More geometry of vector fields: metric balls and equivalent distances

9.1

Introduction and statement of the main results

In Chapter 1 we have introduced the control distance induced by a family of H¨ ormander vector fields X1 , . . . , Xq , or its “weighted” version where among the vector fields there is also a drift term X0 with weight 2, and we have proved the connectivity theorem, together with some related results. In this chapter we address the problem of estimating the volume of the metric balls. The explicit examples worked out in Chapter 1, section 1.11 already suggest that, in general, we can expect that, at any fixed point of the space, a metric ball of radius r should look like a “box” with sidelengths of different sizes: namely, those pointing in the directions of the basic vector fields X1 , . . . , Xq will have the size of r, while those pointing in the directions spanned by some commutators of these vector fields will have the size of some higher power of r, corresponding to the weight of the commutator. The word “box” is used here just as a suggestion of the size of metric balls, with no aim in describing their actual shape. We will see in this chapter that a very precise result can be proved for the volume of the control balls of a general system of (weighted) H¨ ormander vector fields, stating that at every point of the domain the volume of an r-ball behaves (for small r) like a polynomial of r, with the dominant term crα possibly changing from point to point. In particular, the volume of metric balls is not generally equivalent to a fixed power of the radius, as happens in stratified groups (see Chapter 3). Nevertheless, the control on volumes is enough to imply the validity of a local doubling condition, which is a powerful result. To prove this, it turns out that the control distance d as defined in Chapter 1 is not the most convenient one for actual computations. Instead, it is more natural to use another distance d∗ induced by the vector fields, defined by allowing as admissible curves the integral lines of both the basic vector fields and their commutators (but penalizing the integral lines of commutators). Most of the proof of the main theorem about the volume of metric balls is carried out using d∗ ; this distance will be eventually proved to be equivalent to d, which will complete the proof of the estimate.

399

400

H¨ ormander operators

To state precisely the main results that we will prove in this chapter, we are going to introduce some notation. Let X0 , X1 , . . . , Xq be a system of H¨ ormander vector fields in an open bounded domain Ω ⊂Rn . According to Convention 1.26 this means that there exists a step
s such that X[I] x |I|6s generates Rn at every point x ∈ Ω, where      X[I] = Xi1 , Xi2 , . . . Xik−1 , Xik . . . for I = (i1 , . . . , ik ) and that the coefficients of the vector fields are smooth up to the boundary of Ω, so that they have finite C k (Ω) norms for every k. It is clear that, in general,  for
a fixed x ∈ Ω, many different bases can be extracted from the collection X[I] x |I|6s . Also, changing the point x, a different choice of the vector fields could be necessary, to get a base. For an n-tuple of multiindices B = (I1 , I2 , . . . , In ) with |Ij | 6 s, we will set |B| =

n X

|Ii |

i=1

 XB = X[I] I∈B λB (x) = det

X[I1 ]


x

, X[I2 ]


x

, . . . , X[In ]



x

ΩB = {x ∈ Ω : λB (x) 6= 0} . Note that x ∈ ΩB if and only if XB gives a base at x. Now, for any δ > 0, x ∈ Ω, we define X Λ (x, δ) = |λB (x)| δ |B| (9.1) B

where in the sum B ranges over all the possible n-tuples of multiindices of length 6 s. By H¨ ormander’s condition the function Λ (x, δ) is strictly positive for any δ > 0 and x ∈ Ω. One of the main results in this chapter is the local equivalence of Λ (x, δ) with the volume of the balls B (x, δ) associated to the control distance d which we have defined in Chapter 1 and that we will recall later. We can now state the first main result of this chapter: Theorem 9.1 (Volume of control balls and doubling property) For every Ω0 b Ω there exist positive constants δ0 , c, c1 , c2 such that for every δ 6 δ0 , x ∈ Ω0 c1 Λ (x, δ) 6 |B (x, δ)| 6 c2 Λ (x, δ)

(9.2)

|B (x, 2δ)| 6 c |B (x, δ)| .

(9.3)

and

Note that the doubling property (9.3) is an easy consequence of the estimate (9.2). Namely, since B is an n-tuple of multiindices of length 6 s, X X |B| Λ (x, 2δ) = |λB (x)| (2δ) 6 2ns |λB (x)| δ |B| = 2ns Λ (x, δ) B

hence |B (x, 2δ)| 6

c2 2ns c1

B

|B (x, δ)|.

More geometry of vector fields: metric balls and equivalent distances

401

The proof of (9.2) will be completed in section 9.5, according to the strategy that will be explained in the following of this introduction. Let us first exemplify (9.2) in a concrete case. Example 9.2 Let us consider the vector fields: X1 = ∂x ; X2 = x∂y in R2 that we have already met in Chapter 1, section 1.11.3. There are just two interesting choices of B: B1 = ((1) , (2)) (i.e. X1 , X2 ) which gives 
|B1 | = 2 and det X[I] I∈B = |x| 1 B2 = ((1) , (1, 2)) (i.e. X1 and [X1 , X2 ] ) which gives 
|B2 | = 3 and det X[I] I∈B = 1. 2

The first choice is possible only at points x 6= 0. Hence, at any point (0, y0 ) we will have to choose B2 , getting n o
3 |B2 | δ det X[I] (0,y0 ) I∈B2 = δ while as soon as we move to (x0 , y0 ) with x0 6= 0, choosing B1 (which has smaller weight than B2 ) we get n o
= δ 2 |x0 | δ |B1 | det X[I] (x ,y ) 0 0 I∈B1 Now, Theorem 9.1 states that

C1 δ 3 + δ 2 |x| 6 |B ((x, y) , δ)| 6 C2 δ 3 + δ 2 |x| p with C1 , C2 depending on an upper bound on δ and x2 + y 2 . In particular, the balls of center (0, y0 ) have volume comparable to δ 3 , while the balls of center (x0 , y0 ) with large x0 and small radius δ have volume comparable to δ 2 . This result is consistent with the estimate on the distance d that we got for this system of vector fields, by direct computation, in Chapter 1, section 1.11.3. As already said, another theme of this chapter is to introduce other metrics associated to the vector fields and to show their equivalence to the control distance. Let us start recalling the definition of control distance given in Chapter 1. Let X0 , X1 , . . . , Xq be a family of smooth real vector fields in an open connected subset Ω ⊂ Rn .

402

H¨ ormander operators

Definition 9.3 (Distance d) For any δ > 0, x, y ∈ Ω, let Cx,y (δ) be the class of absolutely continuous mappings ϕ : [0, 1] −→ Ω which satisfy ϕ0 (t) =

q X

ai (t) (Xi )ϕ(t) a.e.

i=0

ϕ (0) = x, ϕ (1) = y with ai : [0, 1] → R measurable functions, |ai (t)| 6 δ a.e., i = 1, . . . , q, and |a0 (t)| 6 δ 2 a.e. Then define d (x, y) = inf {δ > 0 : ∃ϕ ∈ Cx,y (δ)} and the related d-balls B (x, r) = {y ∈ Ω : d (x, y) < r}. We have proved in Chapter 1 that d (x, y) is finite for any couple of points in Ω (connectivity property, Theorem 1.45). The other important distance that we need is defined enlarging  the class of admissible curves allowing to move along all the commutators X[I] |6s| instead of the vector fields Xi alone: ∗ Definition 9.4 (Distance d∗ ) For any δ > 0, x, y ∈ Ω, let Cx,y (δ) be the class of absolutely continuous mappings ϕ : [0, 1] −→ Ω which satisfy X
ϕ0 (t) = aI (t) X[I] ϕ(t) a.e. |I|6s

ϕ (0) = x, ϕ (1) = y with aI : [0, 1] → R measurable functions such that |aI (t)| 6 δ |I| a.e. Then define  ∗ d∗ (x, y) = inf δ > 0 : ∃ϕ ∈ Cx,y (δ) and the related balls B ∗ (x, r) = {y : d∗ (x, y) < r}. Remark 9.5 Comparing the above definition with that of control distance d, one ∗ (δ) contains Cx,y (δ), hence can see that the class of admissible curves Cx,y d∗ (x, y) 6 d (x, y) ,

(9.4)

∗

which in particularly implies the finiteness of d for any couple of points. This fact 1 could also be established directly 
as follows. Let nϕ : [0, 1] −→ Ω be any C0 curve joining x to y; since the X[I] x |I|6s span R at any point x ∈ Ω, ϕ (t) can always be expressed in the form X
aI (t) X[I] ϕ(t) , |I|6s

for suitable bounded functions aI (t) ; then the curve ϕ will belong to the class ∗ Cx,y (δ) , for δ > 0 large enough, and d∗ (x, y) will be finite and not exceeding this δ.

More geometry of vector fields: metric balls and equivalent distances

403

We will prove in section 9.6 a converse inequality of (9.4), namely: Theorem 9.6 (Equivalence of distances) For every Ω0 b Ω there exist positive constants δ0 and C such that for every x, y ∈ Ω0 with d∗ (x, y) < δ0 we have d (x, y) 6 Cd∗ (x, y) . The above theorem is one of the main results in this chapter. Let us start establishing the following easy: Proposition 9.7 The function d∗ : Ω × Ω → R is a distance. Moreover, for any Ω0 b Ω there exist positive constants c1 , c2 such that 1/s

c1 |x − y| 6 d∗ (x, y) 6 c2 |x − y|

for any x, y ∈ Ω0 .

(9.5)

Proof. The fact that d∗ is a distance can be proved following step by step the proof of the analogous property of the control distance d, given in Chapter 1, see Proposition 1.36 and Proposition 1.41. Since d∗ (x, y) 6 d (x, y), the upper bound in (9.5) follows from the analogous bound for d, proved in Theorem 1.53. The lower bound in (9.5) is easier and can be proved as for d, see Propositions 1.37 if the vector field X0 is lacking and Proposition 1.42 for the general case (observe that since Ω0 is bounded the previous estimate immediately gives supx,y∈Ω0 d∗ (x, y) < +∞). We will also need the following modification of d∗ which, however, will turn out to be just a quasidistance. # Definition 9.8 For any δ > 0, x, y ∈ Ω, let Cx,y (δ) be the class of absolutely continuous mappings ϕ : [0, 1] −→ Ω which satisfy X
ϕ0 (t) = aI X[I] ϕ(t) a.e. |I|6s

ϕ (0) = x, ϕ (1) = y with aI constants such that |aI | 6 δ |I| . Then define  # ρ# (x, y) = inf δ > 0 : ∃ϕ ∈ Cx,y (δ)  and the related balls B # (x, r) = y : ρ# (x, y) < r . Remark 9.9 Comparing the above definition with that of d∗ , one can see that the ∗ # class of admissible curves Cx,y (δ) contains Cx,y (δ), hence d∗ (x, y) 6 ρ# (x, y) . # Note, however, that the nonemptiness of Cx,y (δ) (and therefore the finiteness of # ρ (x, y)) is not obvious. We will prove later that an inequality of the kind

ρ# (x, y) 6 cd∗ (x, y) also holds, locally, which actually implies that ρ# (x, y) is finite for every couple of points (close enough) and is at least a quasidistance (although in general not a distance). The basic disadvantage of ρ# with respect to d and d∗ is that joining two admissible curves we do not get, in general, an admissible curve.

404

H¨ ormander operators

Remark 9.10 It is easy to see that the balls B # (x, r) can be alternatively represented, in terms of exponential maps, as follow:       X
B # (x, r) = y = exp  aI X[I]  (x) : |aI | < r|I| .   |I|6s

To give a first idea of the strategy used to prove Theorem 9.1, and also to state an intermediate result which has a great independent interest, let us fix some more notation which will be used throughout the chapter. Let Ω0 b Ω and let x ∈ Ω0 . We fix a base  XB = X[I1 ] , X[I2 ] , . . . , X[In ] in a neighborhood of x and, for u ∈ Rn sufficiently small, we define ! n X ui X[Ii ] (x) . Φx,B (u) = exp

(9.6)

i=1

 Let Qδ = u ∈ Rn : |uj | < δ |Ij | and consider the family of balls that are images under the map Φx,B of the these boxes, BB (x, δ) ≡ Φx,B (Qδ ) . By extension, also these sets BB (x, δ) are usually called “boxes”. By Remarks 9.10 and 9.9, we have the inclusions: BB (x, δ) ⊆ B # (x, δ) ⊆ B ∗ (x, δ) . The next theorem collects important quantitative information about the maps Φx,B and states the equivalence between d∗ -balls and the “boxes” Φx,B (Qδ ): Theorem 9.11 (Structure of balls) For a fixed Ω0 b Ω there exist constants c, c1 , c2 , δ0 ∈ (0, 1) such that if x ∈ Ω0 , 0 < δ 6 δ0 , and XB is a base at x satisfying the suboptimality condition 1 |λB (x)| δ |B| > max |λC (x)| δ |C| 2 C (where the max is taken over all n-tuples C = (J1 , . . . , Jn ) with |Jj | 6 s), then: (1) if JΦx,B (u) is the Jacobian matrix of Φx,B then for every u ∈ Qc1 δ we have
1 |λB (x)| 6 det JΦx,B (u) 6 4 |λB (x)| , 4 (2) B ∗ (x, cδ) ⊂ BB (x, δ) ⊂ B # (x, δ) ⊂ B ∗ (x, δ) . (3) Φx,B is one-to-one on the box Qc2 δ . One of the qualifying features of this theorem is the uniformity of its quantitative conclusions with respect to x ∈ Ω0 and δ ∈ (0, δ0 ). Point (2) is commonly known as “the ball-box Theorem”. Let us show how this theorem implies the following:

More geometry of vector fields: metric balls and equivalent distances

405

Theorem 9.12 (Volume of d∗ -balls) The statement of Theorem 9.1 holds if d-balls are replaced with d∗ -balls. Proof. Let B be as in Theorem 9.11. For every x ∈ Ω0 , 0 < δ 6 δ0 , assuming, as we can that c1 > c2 , we have Z Z Z ∗ |B (x, δ)| > |BB (x, δ)| = dy > dy > dy Φx,B (Qδ ) Φx,B (Qc1 δ ) Φx,B (Qc2 δ ) Z Z
det JΦ (u) du > 1 |λB (x)| = du = c |λB (x)| δ |B| . x,B 4 Q c2 δ Qc2 δ By the assumption on XB and recalling the definition (9.1) we obtain X c c |λB (x)| δ |B| > max |λC (x)| δ |C| > C1 |λC (x)| δ |C| = C1 Λ (x, δ) . 2 C C

Analogously, for 0 < δ 6 cδ0 ,   Z δ ∗ 6 4 |λB (x)| |B (x, δ)| 6 BB x, du = c0 |λB (x)| δ |B| 6 C2 Λ (x, δ) . c Qδ c

Since, by Theorem 9.6, d and d∗ are equivalent, Theorem 9.1 will follow by Theorem 9.12. So, the main task of this chapter is proving Theorem 9.11 and Theorem 9.6. This will be a hard job, requiring several tools. Let us briefly explain them, also to describe the global structure of the chapter. (1) The Baker-Campbell-Hausdorff formula. This is a well-known result from noncommutative algebra which in particular gives a useful way of computing the composition of the exponentials of two vector fields X, Y as the exponential of a single vector field Z (X, Y ) obtained from X, Y with a suitable infinite procedure involving commutators. This powerful and delicate tool is by now standard in the context of H¨ormander vector fields. We have collected in section 9.3 the statement of this purely algebraic result and the derivation of all the consequences that we will need. The last section of this chapter (section 9.8) contains a detailed proof of the algebraic result. The use of suitable consequences of the Baker-Campbell-Hausdorff formula, throughout this chapter, is pervasive. (2) Some preliminary algebraic computation which is necessary in particular to prove the estimate on the determinants involved in point (1) of Theorem 9.11. These preliminaries will be the content of section 9.4, culminating in the proof of Theorem 9.28. We will say some more words of explanation about this in section 9.4, after the statement of that theorem. (3) Some delicate topological analysis related to the problem of assuring the global invertibility of a locally invertible map. This will enter the proof of points (2) and (3) in Theorem 9.11, since we are interested in establishing

406

H¨ ormander operators

that a certain map is invertible on a neighborhood of size independent of the point. All these tools will allow us to prove, in section 9.5, Theorem 9.11. The proof of Theorem 9.6, given in section 9.6, will be quite brief, although it actually relies on Theorem 9.11, some results proved in Chapter 1, and some nontrivial consequences of the Baker-Campbell-Hausdorff formula proved in section 9.3. Finally, in section 9.7 we will discuss the possible validity of a global doubling condition on subdomains Ω0 b Ω. 9.2

Dependence of the constants

Since the main results in this chapter have a long and difficult proof, in view of their possible applications it is quite important to state precisely how the constants which are involved in the final estimates depend on the vector fields. To this aim, let us introduce an important quantity related to our system of H¨ormander vector fields, which our constants will depend on: Definition 9.13 For any domain Ω0 b Ω, let ∆Ω0 ≡ inf 0 max |λB0 (y)| , 0 y∈Ω

B

(9.7)

where B 0 ranges on all possible n-tuples (I1 , . . . , In ) with |Ij | 6 s. By H¨ ormander’s condition, maxB0 |λB0 (y)| > 0 for every y ∈ Ω and since this is the maximum of a finite number of continuous functions, it is a continuous function of y; then ∆Ω0 is positive by the compactness of Ω0 . In order to give precise statements avoiding tedious repetitions it can be useful also the following convention. Convention 9.14 (Local regularity) A lot of theorems within this chapter have the following logical structure: “For every Ω0 b Ω there exist constants δ0 > 0 (and possibly other ones), such that for every x ∈ Ω0 , δ ∈ (0, δ0 ) we have (some consequence)”. This sentence has always the following meaning: For every Ω0 b Ω there exists a number δ0 > 0, depending on dist (Ω0 , ∂Ω) and on the C h (Ω) norms of the coefficients of the vector fields X0 , . . . , Xq such that for every x ∈ Ω0 , δ ∈ (0, δ0 ) we have the stated consequences. The integer h only depends on q, the dimension n and the step s of H¨ ormander’s condition in Ω. For the seak of brevity, we will simply write that δ0 , and possibly other constants, depend on the local regularity of the vector fields to indicate the previous dependence. Any other kind of dependence will be specified. We can now make some first precise statements:

More geometry of vector fields: metric balls and equivalent distances

407

Claim 9.15 In Theorem 9.11 (structure of balls) the constants c, c1 , δ0 depend on the local regularity of the vector fields; c, c1 also depend on ∆Ω0 . In Theorem 9.6 (equivalent distances) the constants δ0 and C depend on ∆Ω0 and the local regularity of the vector fields. The constant c2 in point (3) of Theorem 9.11, and consequently the constants in Theorem 9.1 (volume of balls), depend on the vector fields in a way which seems not easy to bound in terms of a few explicit constants. Nevertheless, we can prove that these constants remain uniformly controlled if our system of H¨ormander vector fields depends smoothly on a parameter varying in a compact set. This information is crucial, for instance, in the application of these results to nonsmooth H¨ormander vector fields, as we will explain better at the end of this section. Assume we have a system of smooth vector fields X0σ , . . . , Xqσ in Ω, depending on some parameter σ ∈ Σ, where Σ is a compact subset of RM for some M , and satisfying, for every σ ∈ Σ, H¨ ormander’s condition in Ω. We assume that the dependence on σ is such that the commutators   (σ) ≡ aI (x, σ) , X[I] x

for |I| 6 s, together with their partial derivatives of any order with respect to the variables xj , are continuous in the joint variables (x, σ) ∈ Ω × Σ. Remark 9.16 Note that any quantity which for every fixed σ depends on the local regularity of the vector fields X0σ , . . . , Xqσ in some Ω0 b Ω is uniformly bounded as σ varies in the compact set Σ. Also, the quantity ∆Ω0 in (9.7) remains uniformly positive as σ ranges in the compact Σ, because it is positive for every σ ∈ Σ and depends continuously on σ. Therefore, the constants which depend on the local regularity of the vector fields and the number ∆Ω0 still remain uniformly controlled if the system of H¨ormander vector fields depends on a parameter σ as specified above. We will prove that, moreover: Claim 9.17 The constant c2 in point (3) of Theorem 9.11, and consequently the constants in Theorem 9.1 (volume of balls), are uniformly controlled if the system of H¨ ormander vector fields depends on a parameter σ as specified above. We end this section saying some more words about the possible use of the information contained in the above analysis of the dependence of the constants appearing in the main results of this chapter. Pn Let us consider a family of (nonsmooth) vector fields Xi = j=1 bij (x) ∂xj , i = 0, 1, . . . , q, satisfying H¨ ormander’s condition of step s > 2 in Ω, where bij ∈ C s−1 (Ω) for i = 1, . . . , q and b0j ∈ C s−2 (Ω) . For any x0 ∈ Ω we can consider the smooth vector fields Six0 obtained taking the Taylor expansion of degree s − pi of the coefficients of Xi near x0 :

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H¨ ormander operators

Six0

S0x0



 α D b (x ) ij 0 α  = (x − x0 )  ∂xj for i = 1, . . . , q α! j=1 |α|6s−1   n X X Dα b0j (x0 ) α  = (x − x0 )  ∂xj . α! j=1 n X

X

|α|6s−2

o n x0 Then the commutators S[I]

|I|6s

are smooth vector fields (for x0 fixed); their coef-

ficients, and the xj derivatives of any order of their coefficients, depend continuously on (x, x0 ). Moreover,    n
o x0 S[I] = X[I] x x0

0

|I|6s

|I|6s

q

ormander’s condition at x0 , and therefore, hence the vector fields {Six0 }i=0 satisfy H¨ by continuity, in some small compact neighborhood Σ of any fixed x0 . We can then q apply the results of this chapter to the family of H¨ormander vector fields {Six0 }i=0 for x0 ∈ Σ, getting estimates with constants uniformly controlled as x0 ∈ Σ. This is one of the key tools which has been used in the papers [31], [33], [32] to build a theory for nonsmooth H¨ ormander vector fields and operators. 9.3 9.3.1

The Baker-Campbell-Hausdorff formula The BCH formula for formal series

Let x, y be two noncommuting variables. For every positive integer N , let us write the formal expression (ex )N =

N X xk k=0

k!

(ex · ey )N =

, (ey )N =

N X yk k=0

k!

N X k X xj y k−j . j! (k − j)! j=0

k=0 x

y

Note that (e · e )N can be viewed as the N -th partial sum of the series obtained as the Cauchy product of the formal exponential series of ex , ey . Theorem 9.18 (BCH formula for formal series) There exists a sequence ∞

{Ck (x, y)}k=2 of homogeneous polynomials of degree k in x, y, where each Ck (x, y) is expressible as a linear combination of repeated commutators of length k of the noncommuting variables x, y such that if, for every N = 2, 3, 4, . . . , we set zN = x + y +

N X k=2

Ck (x, y)

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and (ezN )N =

N X zk

N

k=0

k!

then we have (ezN )N = (ex · ey )N + RN +1 (x, y) where RN +1 (x, y) is a polynomial in x, y containing only monomials of degree > N + 1. In particular, 1 C2 (x, y) = [x, y] . 2 The above theorem can be rephrased in a more concise way saying that, in the P∞ sense of formal series, ex · ey = ez with z = x + y + k=2 Ck (x, y). The meaning of the expression “in the sense of formal series” is exactly the content of the previous theorem. For the curious reader, the first polynomials after C2 are the following: 1 C3 (x, y) = {[x, [x, y]] − [y, [x, y]]} 12 1 C4 (x, y) = − {[y, [x, [x, y]]] + [x, [y, [x, y]]]} . 48 However, we will never need the explicit form of Ck (x, y) for k > 2. For a proof of this abstract result, and some more details about the notion of formal series, we refer the reader to section 9.8. 9.3.2

The BCH formula for vector fields and some consequences

The aim of this section is to derive from the abstract Baker-Campbell-Hausdorff formula the following more “concrete” version involving vector fields. We will prove the following: Theorem 9.19 (BCH formula for vector fields) Let X, Y be two smooth vector fields defined in some domain Ω ⊂ Rn . For any N = 2, 3, 4, . . . , and σ, τ ∈ R, let N X Ck (σY, τ X) (9.8) ZN (σY, τ X) = σY + τ X + k=2

where Ck (x, y) are the polynomials appearing in Theorem 9.18. Then, for any N = 2, 3, 4 . . . and Ω0 b Ω there exist positive constants C, δ depending on Ω0 , Ω end the C 2N −1 (Ω) norms of the coefficients of X, Y , such that for any x0 ∈ Ω0 and |σ| , |τ | < δ we have    N2−1  2 2 exp (τ X) exp (σY ) (x0 ) = exp (ZN (σY, τ X)) (x0 ) + O |σ| |τ | |σ| + |τ | where    N −1    N −1 O |σ| |τ | |σ|2 + |τ |2 2 6 C |σ| |τ | |σ|2 + |τ |2 2 and all the exponential maps appearing in the assertion are well defined.

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H¨ ormander operators

Remark 9.20 A relevant point of the above result, as well as the others that will be proved in this section 9.3 for vector fields, is the fact that the constants depend on the regularity of the coefficients of the vector fields only through a finite number of derivatives (obviously depending on the integer N which specifies the smallness of the error term). To prove this theorem, it is convenient to introduce some terminology. Definition 9.21 We say that a smooth function f : (−δ, δ) → R has the formal Taylor expansion f (x) ∼

∞ X

ak xk

k=0 (k)

if ak = f k!(0) , for every k = 0, 1, 2, . . . , regardless to the fact that the radius of convergence of the power series could be zero. Analogously, we say that a smooth 2 function f : (−δ, δ) → R has the formal Taylor expansion f (x, y) ∼

∞ X k X

akj xj y k−j

k=0 j=0

if akj =

∂k f 1 j!(k−j)! ∂xj ∂y k−j

(0, 0) for every k = 0, 1, 2 . . . , j = 0, 1, . . . , k.

Knowing the formal Taylor expansion of a function allows to write Taylor formula at any fixed order. For instance, if f (x, y) ∼

∞ X k X

akj xj y k−j

k=0 j=0

then, for every N = 1, 2, . . . and (x, y) is a suitable neighborhood of the origin f (x, y) =

N X k X

akj xj y k−j + O x2 + y 2


N2+1

.

k=0 j=0

As we have seen in Chapter 1, section 1, if f : Ω → R is a differentiable function, then d (f (exp (τ X) (x0 ))) = (Xf ) (exp (τ X) (x0 )) . dτ If we then define the action of the exponential map on a smooth function f letting exp (τ X) f (x0 ) = f (exp (τ X) (x0 )) then we can write d exp (τ X) f (x0 ) = (Xf ) (exp (τ X) (x0 )) = exp (τ X) Xf (x0 ) dτ

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411

and, by iteration, dk exp (τ X) f (x0 ) = exp (τ X) X k f (x0 ) = X k f (exp (τ X) (x0 )) dτ k dk = X k f (x0 ) exp (τ X) f (x0 ) dτ k τ =0

(9.9)

hence the function τ 7→ exp (τ X) f (x0 ) has the formal Taylor series ∞ X τ kXk k=0

k!

f (x0 )

which, passing to the Taylor formula at any fixed finite order, gives the expansion exp (τ X) f (x0 ) =

N X τ kXk k=0

k!

f (x0 ) + O τ N +1




which holds for τ small enough. Proof of Theorem 9.19. Let us consider the composition g (τ, σ) = f (exp (τ X) exp (σY ) (x0 )) and let us compute its partial derivatives at the origin. Since we know a priori that the composed function g is smooth in the joint variables, we can compute mixed derivatives in the most convenient order. Then, by (9.9) ∂ j+h g (0, 0) = Y h X j f (x0 ) ∂σ h ∂τ j hence, in the sense of formal Taylor series, g (τ, σ) ∼

∞ X k X Y k−j X j f (x0 ) k=0 j=0

j! (k − j)!

τ j σ k−j

which is the formal Cauchy product of the power series ∞ k ∞ j X (σY ) X (τ X) k=0

k!

j=0

j!

f (x0 ) = exp (σY ) exp (τ X) f (x0 )

(note that this expression is consistent with the definition of exp (τ X) f given above) and therefore, by the abstract BCH formula (Theorem 9.18), equals, in the sense of P∞ k formal series, k=0 Zk! f (x0 ) with Z (σY, τ X) = σY + τ X +

∞ X

Ck (σY, τ X) .

k=2

If we interpret this identity in terms of exact identities between polynomials in τ X, σY of degree N for any fixed N (as in the statement of the abstract BCH formula) and we regard these polynomials as the Taylor polynomials at the origin of functions of (τ, σ), we conclude that ! N k X ∂ j+h g ∂ j+h ZN (σY, τ X) (0, 0) = f (x0 ) (0, 0) for j + h 6 N (9.10) ∂σ h ∂τ j ∂σ h ∂τ j k! k=0

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H¨ ormander operators

where ZN (σY, τ X) = σY + τ X +

N X

Ck (σY, τ X) .

k=2

Since the two functions in (9.10) coincide up to order N at (0, 0), we can also write   N k X
N2+1 ZN (σY, τ X) 2 2 g (τ, σ) = f (x0 ) + O σ + τ k! k=0

for (τ, σ) is a suitable neighborhood of the origin. We claim that this can also be rewritten as  
N2+1 2 2 g (τ, σ) = exp (ZN (σY, τ X)) f (x0 ) + O σ + τ . (9.11) To see this, let us consider the function ϕ (u) = exp (uZN (σY, τ X)) f (x0 ) and observe that if (σ, τ ) is small enough ϕ (u) is defined in [0, 1]. Hence Z u (N +1) N X ϕ (τ ) ϕ(k) (0) k N u + (u − τ ) dτ. ϕ (u) = k! N ! 0 k=0

Taking u = 1 and using (9.9) we obtain exp (ZN (σY, τ X)) f (x0 ) =

N k X ZN (σY, τ X)

k!

k=0

f (x0 ) +

1 N!

Z

1

ZN (σY, τ X)

N +1

N

f (x0 ) (1 − τ ) dτ.

0

Observe now that Z 1 N +1 N +1 N 6 C σ 2 + τ 2 2 Z (σY, τ X) f (x ) (1 − τ ) dτ N 0 0

with C depending on the derivatives of the coefficients of X, Y up to the order 2N − 1. Hence   N k X
N +1 ZN (σY, τ X) f (x0 ) = exp (ZN (σY, τ X)) f (x0 ) + O σ 2 + τ 2 2 , k! k=0

and (9.11) holds. The form of the error term in (9.11) can be made more precise noting that ZN (σY, 0) = σY and ZN (0, τ X) = τ X. Hence, letting h (τ, σ) = exp (τ X) exp (σY ) f (x0 ) − exp (ZN (σY, τ X)) f (x0 ) ∂h we have h (τ, 0) = h (0, σ) ≡ 0 which also implies ∂σ (0, σ) ≡ 0 and therefore Z σZ τ 2 Z σ ∂ h ∂h (τ, σ 0 ) dσ 0 = (τ 0 , σ 0 ) dτ 0 dσ 0 . h (τ, σ) = 0 0 ∂τ ∂s 0 ∂σ  
N2+1 2 2 Since h is smooth, from h (τ, σ) = O σ + τ we obtain that

∂ j+h h (0, 0) = 0 ∂σ h ∂τ j

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413

2

∂ h for j + h 6 N and from the Taylor expansion of ∂τ ∂σ (τ, σ) we obtain  
N2−1 ∂2h 2 2 . (τ, σ) = O σ + τ ∂τ ∂σ

Hence 2 ∂ h
N −1 0 0 (τ , σ ) 6 C |στ | σ 2 + τ 2 2 . |h (τ, σ)| 6 |στ | 0 max0 |σ |6σ,|τ |6τ ∂τ ∂s We conclude that  
N −1 exp (τ X) exp (σY ) f (x0 ) = exp (ZN (σY, τ X)) f (x0 ) + O |στ | σ 2 + τ 2 2 for any N = 1, 2, 3 . . . and (τ, σ) in a suitable neighborhood of the origin. By the genericity of the function f , we can also write  
N −1 exp (τ X) exp (σY ) (x0 ) = exp (ZN (σY, τ X)) (x0 ) + O |στ | σ 2 + τ 2 2 for (τ, σ) is a suitable neighborhood of the origin. We will also need some generalization of the previous result to a family of more than two vector fields. We start with the following Lemma 9.22 Let X1 , . . . , Xp be smooth vector fields defined in a domain Ω ⊂ Rn , x0 ∈ Ω and let f be smooth function defined in a neighborhood of x0 . (a) Let us consider the function     p X F (u1 , . . . , up ) = f exp  uj Xj  (x0 ) , j=1

defined for u = (u1 , . . . , up ) in a neighborhood of the origin in Rp . Then the formal Taylor series of F at 0 ∈ Rp is  k p ∞ X X 1  uj Xj  f (x0 ) . F (u1 , . . . , up ) ∼ k! j=1 k=0

(b) Also, if 

     p p X X G (σ1 , . . . , σp , τ1 , . . . , τp ) = f exp  σj Xj  exp  τj Xj  (x0 ) j=1

j=1

is defined for (σ, τ ) in a neighborhood of the origin in R2p , then the formal Taylor series of G at 0 ∈ R2p is  m  k p p ∞ X X 1 X G (σ1 , . . . , σp , τ1 , . . . , τp ) ∼ τj Xj   σj Xj  f (x0 ) . m!k! j=1 j=1 k,m=0

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H¨ ormander operators

Proof. (a) Let H (τ, u1 , . . . , up ) = F (τ u1 , . . . , τ up ). Then     p X H (τ, u1 , . . . , up ) = f exp τ uj Xj  (x0 ) j=1

hence by (9.9)  k p X ∂kH (0, u1 , . . . , up ) =  uj Xj  f (x0 ) . ∂τ k j=1 On the other hand, X k! ∂ α F ∂kH (0, u , . . . , u ) = (0) uα 1 p ∂τ k α! ∂uα |α|=k

hence  k p X X 1 ∂αF 1  (0) uα = uj Xj  f (x0 ) α! ∂uα k! j=1

|α|=k

and the first assertion is proved. (b) Applying (a) to σ 7→ G (σ, τ ), for τ small enough we can write  k     p p X 1 ∂ |α| G X X 1  (0, τ ) σ α = σj Xj  f exp  τj Xj  (x0 ) . α! ∂σ α k! j=1 j=1 |α|=k

Applying again (a), with f replaced with X |α|=k |β|=m

1 k!

P

p j=1

σj Xj

k

f , we get

 m  k p p X X G 1  1  1 ∂ (0, 0) σ α τ β = τj Xj  σj Xj  f (x0 ) β!α! ∂σ α ∂τ β m! j=1 k! j=1 |α|+|β|

and we are done. Theorem 9.23 (Extension of BCH formula to several vector fields) For X1 , . . . , Xp smooth vector fields defined in a domain Ω ⊂ Rn , any N = 2, 3, . . . and σ, τ ∈ Rp , let σ·X =

p X

σj Xj , τ · X =

j=1

ZN (σ · X, τ · X) = σ · X + τ · X +

p X

τj Xj

j=1 N X

Ck (σ · X, τ · X)

k=2

where Ck (x, y) are the polynomials appearing in Theorem 9.18. Then, for any N = 2, 3, . . . and Ω0 b Ω there exist positive constants C, δ depending on Ω0 , Ω and

More geometry of vector fields: metric balls and equivalent distances

415

the C 2N −1 (Ω) norms of the coefficients of X1 , . . . , Xp , such that for any x0 ∈ Ω0 and |σ| , |τ | < δ we have exp (τ · X) exp (σ · X) (x0 )    N2−1  2 2 = exp (ZN (σ · X, τ · X)) (x0 ) + O |σ| |τ | |σ| + |τ | where    N −1    N −1   O |σ| |τ | |σ|2 + |τ |2 2 6 C |σ| |τ | |σ|2 + |τ |2 2 6 C |σ|N +1 + |τ |N +1 and all the exponential maps appearing in the assertion are well defined. Proof. Let us write (τ1 , . . . , τp ) = τ τ10 , . . . , τp0





0

(σ1 , . . . , σp ) = σ σ10 , . . . , σp

with with

p X

2

(τi0 ) = 1

i=1 p X

2

(σi0 ) = 1

i=1

Xτ =

p X j=1

τj0 Xj , Xσ =

p X

σj0 Xj

j=1

so that τ · X = τ Xτ and σ · X = σXσ . To apply Theorem 9.19 observe that the C 2N −1 (Ω) norms of the coefficients of the vector fields Xσ and Xτ can be bounded independently of σ and τ . It follows that exp (τ · X) exp (σ · X) (x0 ) = exp (τ Xτ ) exp (σXσ ) (x0 )  
N2−1 2 2 = exp (ZN (σXσ , τ Xτ )) (x0 ) + O |στ | σ + τ    N2−1  2 2 = exp (ZN (σ · Xσ , τ · Xτ )) (x0 ) + O |σ| |τ | |σ| + |τ | where    N −1   N −1  O |σ| |τ | |σ|2 + |τ |2 2 6 C |σ| |τ | |σ|2 + |τ |2 2 with C independent of σ and τ . We end this section showing a consequence of Theorem 9.19 that we will be useful in the next chapter in the context of the Lie algebra of a nilpotent homogeneous group: Theorem 9.24 (BCH formula in the nilpotent case) Let X, Y be two smooth vector fields defined in some domain Ω ⊂ Rn . Assume moreover that: (1) for some positive integer s, every commutator of X, Y of length larger than s vanishes;

416

H¨ ormander operators

(2) X and Y have analytic coefficients. Then, for any Ω0 b Ω there exists a positive constant δ depending on the C s (Ω) norms of the coefficients of X, Y , such that for any x0 ∈ Ω0 and |σ| , |τ | < δ we have exp (τ X) exp (σY ) (x0 ) = exp (Z (σY, τ X)) (x0 ) Ps where Z (σY, τ X) = σY + τ X + k=2 Ck (σY, τ X) and all the exponential maps appearing in the assertion are well defined. Proof. For any Ω0 b Ω there exists a positive constant δ depending on the C 1 (Ω) norms of the coefficients of X,Y ,Z (σY, τ X), that is on the C s (Ω) norms of the coefficients of X, Y , such that for any x0 ∈ Ω0 and |σ| , |τ | < δ the functions (σ, τ ) 7−→ exp (τ X) exp (σY ) (x0 ) (σ, τ ) 7−→ exp (Z (σY, τ X)) (x0 ) are well defined. Since X, Y have analytic coefficients the same is true for Z (σY, τ X). It follows that the above exponential maps are solutions of analytic differential equations in which τ and σ are analytic parameters. It is a standard fact (see e.g. [109, Chapter 1]) that such solutions have analytic dependence from parameters and therefore the above functions are also analytic for |σ| , |τ | < δ. By Theorem 9.19 for every N > s, we can write  
N −1 exp (τ X) exp (σY ) (x0 ) = exp (Z (σY, τ X)) (x0 ) + O |στ | σ 2 + τ 2 2 , which means that the two analytic functions coincide at any finite order at the origin, therefore they are identical for |σ| , |τ | < δ. 9.3.3

The BCH formula and H¨ ormander vector fields

The results of the previous section are applicable to generic smooth vector fields. We now turn to consider a family of H¨ ormander vector fields, X0 , X1 , . . . , Xq defined in n an open bounded domain Ω ⊂ R and according to Convention 1.26 we assume that
there exists a step s such that X[I] x |I|6s generates Rn at every point x ∈ Ω and that the coefficients of the vector fields are smooth up to the boundary of Ω, so that they have finite C k (Ω) norms for every k. We will need a way to approximate the exponential of a linear combination of commutators X[I] with |I| 6 s by the repeated composition of exponentials of vector fields X0 , X1 , . . . , Xq . Namely, the following result will be useful: Theorem 9.25 For every Ω0 b Ω there exist a positive integer M and constants δ0 ∈ (0, 1) , C > 0 such that for every δ ∈ (0, δ0 ) , x ∈ Ω0 and any set of constants M {aI }|I|6s with |aI | 6 δ |I| there exists a set of constants {cj }j=1 with |cj | 6 Cδ pkj such that     M X Y
exp  6 Cδ s+1    a X (x) − exp c X (x) (9.12) I [I] j kj j=1 |I|6s

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417

where kj ∈ {0, 1, 2, . . . , q} . The constants M, δ0 , C depend on the local regularity of the vector fields (see Convention 9.14). Note that |cj | 6 Cδ if kj ∈ {1, 2, . . . , q} , |cj | 6 Cδ 2 if kj = 0. To prove the proposition we need the following Lemma, related to the quasiexponential maps introduced in Chapter 1, section 1.6. Let us recall this concept. For any multiindex I = (i1 , i2 , . . . , i` ), with i1 , i2 , . . . ∈ {0, 1, . . . , q}, and τ > 0 we define the quasiexponential map C` (τ, XI ), in the following iterative way: C1 (τ, Xi1 ) = exp (τ pi1 Xi1 ) ; C` (τ, Xi1 Xi2 . . . Xi` ) −1

= C`−1 (τ, Xi2 . . . Xi` )

exp (−τ pi1 Xi1 ) C`−1 (τ, Xi2 . . . Xi` ) exp (τ pi1 Xi1 )

Then we have: Lemma 9.26 For I = (i1 , . . . , i` ), ij ∈ {0, 1, . . . , q}, |I| 6 s, we have   s X
C` (τ, Xi1 . . . Xi` ) = exp τ |I| X[I] + τ k Ck0  + O τ s+1

(9.13)

k=|I|+1

and  −1

= exp −τ |I| X[I] −

C` (τ, Xi1 . . . Xi` )



s X

τ k Ck0  + O τ s+1




(9.14)

k=|I|+1

Ck0

where denote linear combinations of commutators of weight
k of the vector fields X0 , X1 , . . . , Xq . The implicit constant in the terms O τ s+1 depends on the C s+`−3 (Ω) norms of the coefficients of X0 , . . . , Xq . Note that this result is somewhat similar to Theorem 1.49 in Chapter 1, however it is sharper with respect to the quantification of the error term, which here is small of order s + 1 even for |I| < s. Proof of the Lemma. First of all we observe that if (9.13) holds for a certain ` then also (9.14) holds. Indeed, let s X

Z = τ |I| X[I] +

τ k Ck0 .

k=|I|+1

Then, since exp (Z)

−1

= exp (−Z) we have

C` (τ, Xi1 . . . Xi` )

−1

= exp (Z) + O τ s+1



−1



−1 = I + exp (−Z) O τ s+1 (exp (−Z))


s+1 = I +O τ exp (−Z) = exp (−Z) + O τ s+1 and therefore  −1

C` (τ, Xi1 . . . Xi` )

= exp −τ |I| X[I] −

s X k=|I|+1


τ k Ck0  + O τ s+1 .

(9.15)

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H¨ ormander operators

We will prove (9.13) by induction on `. For ` = 1 this is obvious. Assume we have proved (9.13) (and therefore (9.14)) up to ` − 1, and let us prove it for `. Let I = (i1 , i2 , . . . , i` ) and I 0 = (i2 , . . . , i` ); then, C`−1 (τ, Xi2 . . . Xi` ) exp (τ pi1 Xi1 )     s X
0 = exp τ |I | X[I 0 ] + τ k Ck0  + O τ s+1  exp (τ pi1 Xi1 ) k=|I 0 |+1





s X

0 = exp τ |I | X[I 0 ] +


τ k Ck0  exp (τ pi1 Xi1 ) + O τ s+1 .

k=|I 0 |+1

Now, we apply Theorem 9.19 with N = s to the vector fields s X 0 τ |I | X[I 0 ] + τ k Ck0 and τ pi1 Xi1 . k=|I 0 |+1

Note that the terms that appear in (9.8) may also include commutators of the vector fields X0 , X1 , . . . , Xq with weigth larger than s. Clearly these commutators give a
s+1 contribution of magnitude O τ and therefore can be omitted. Hence
C`−1 (τ, Xi2 . . . Xi` ) exp (τ pi1 Xi1 ) = exp (W ) + O τ s+1 , where s X

0 W = τ |I | X[I 0 ] +τ pi1 Xi1 +

τ k Ck0 +

k=|I 0 |+1

i 1 h |I 0 | τ X[I 0 ] , τ pi1 Xi1 + 2

s X

τ k Ck00 .

k=|I 0 |+1+pi1

Similarly, using (9.15) and again Theorem 9.19 we have −1

exp (−τ pi1 Xi1 )  s X
0 = exp −τ |I | X[I 0 ] + −τ k Ck0  exp (−τ pi1 Xi1 ) + O τ s+1

C`−1 (τ, Xi2 . . . Xi` ) 

k=|I 0 |+1


= exp (W 0 ) + O τ s+1 , where 0 W 0 = −τ |I | X[I 0 ] −τ pi1 Xi1 −

s X

τ k Ck0 +

k=|I 0 |+1

i 1 h |I 0 | τ X[I 0 ] , τ pi1 Xi1 + 2

s X 0

k=|I| +1+pi1

Using again Theorem 9.19 we obtain
C` (τ, Xi1 Xi2 . . . Xi` ) = exp (W 0 ) exp (W ) + O τ s+1   s h 0 i X
τ k (Ck000 + Ck00 ) + O τ s+1 = exp  τ |I | X[I 0 ] , τ pi1 Xi1 + k=|I|0 +1+pi1

 = exp τ |I| X[I] +

s X k=|I|+1


τ k Ck0000  + O τ s+1 .

τ k Ck000 .

More geometry of vector fields: metric balls and equivalent distances

419


which is (9.13) for `. Here the implicit constant in the term O τ s+1 depends on the C s−1 (Ω) norms of the coefficients of X[I 0 ] , X0 , . . . , Xq , hence (since I 0 has length 6 ` − 1 and the coefficients of X[I 0 ] involve derivatives up to order ` − 2 of Xi ) on the C s+`−3 (Ω) norms of the coefficients of X0 , . . . , Xq .

Proof of Proposition 9.25. Our task is to approximate a given exponential of the kind X  exp aI X[I] (x) (9.16) |I|6s

with a composition of the kind

M Q

exp cj Xkj




 (x), where the last expression

j=1

involves only vector fields chosen among X0 , X1 , . . . , Xq , while the first one involves commutators any weightn6 s.oMore precisely we will find s groups of coefficients n on1 n of o ns n2 (s) (2) (1) that roughly speaking correspond to terms , . . . , cj , cj cj j=1

j=1

j=1

appearing in (9.16) with the same weight, so that nk s Y   Y (k) exp cj Xkj (x) k=1

j=1

satisfies (9.12). Toward this aim we write X

aI X[I] =

|I|6s

s X X

aI X[I]

j=1 |I|=j

and we start working on the group of terms corresponding to j = 1. In this case X

aI X[I] =

|I|=1

q X

ak Xk

k=1

(1)

and we set cj

= aj , j = 1 . . . , q. A repeated application of Theorem 9.19, gives   q s   Y X X
(1) exp cj Xj = exp  aI X[I] + Ak  + O δ s+1 (9.17)

j=1

|I|=1

k=2

where Ak denotes a linear combination of commutators of weight k of the
vector k fields a1 X1 , . . . , aq Xq , in particular since ak = O
(δ) we have Ak = O δ . hNote s+1 will now depend on the C (Ω) that the implicit constant in the term O δ norms of the coefficients of X1 , . . . , Xq for some (large) h only depending on s and nj (which in turn depends on s and q). Next we consider the sum of vector fields of weight 2 given by X X aI X[I] − A2 ≡ a0I X[I] |I|=2

|I|=2

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H¨ ormander operators


where A2 is as in (9.17). Since A2 = O δ 2 we have |a0I | 6 cδ 2 . Applying Lemma nI 9.26 for ` = 2 to a0I X[I] we can find constants cIj j=1 and indices iI,j ∈ {0, 1, . . . , q} such that cIj = O (δ piI,j ) nI Y

exp cIj XiI,j = exp a0I X[I] +


j=1

s X

! A0k

+ O δ s+1




k=3

A0k

where denotes a linear combination of commutators of
weight k of the vector fields X0 , X1 , . . . , Xq , with coefficients such that A0k = O δ k . Composing the above identity for I ranging among all multiindices of weight two and applying Theorem (2) (2) 9.19 we get that for suitable indices ij and constants cj satisfying cj = O (δ pij )   n2 s   Y X X
(2) exp cj Xij = exp  a0I X[I] + A00k  + O δ s+1 j=1

|I|=2

k=3

with an analogous meaning of the symbol A00k . Hence     q n2 s X X Y


Y (2) (1) s+1   Ak + O δ exp cj Xij = exp aI X[I] + exp cj Xj j=1

j=1



|I|=1

 exp 

X

a0I X[I] +

|I|=2

s X

 A00k  + O δ s+1




k=2



k=3

and by Theorem 9.19 and the definition of a0I   s X X X
 + O δ s+1 . = exp  aI X[I] + a0I X[I] + A000 k |I|=1

|I|=2

k=3

An iterative reasoning then gives the desired result. 9.4

Suboptimal bases and their properties

In this section we will use the notation introduced or recalled at the beginning of section 9.1. We start with the following: Definition 9.27 For fixed numbers t, δ ∈ (0, 1) and x0 ∈ Ω we say that XB =  X[I] I∈B , with B = (I1 , . . . , In ) is a t-suboptimal base, at the point x0 and radius δ if 0 |λB (x0 )| δ |B| > t max |λB0 (x0 )| δ |B | , (9.18) 0 B

where the maximum is taken over all n-tuples B 0 = (J1 , . . . , Jn ) with |Jj | 6 s. Note that there are finitely many n-tuples B 0 = (J1 , . . . , Jn ) with |Jj | 6 s since |B 0 | 6 ns. Also observe that x0 ∈ ΩB , that is XB is a base of Rn at x0 . One of the main results in this section, which will be useful in the proof of point (1) of Theorem 9.11, is the following:

More geometry of vector fields: metric balls and equivalent distances

421

Theorem 9.28 Let Ω0 b Ω. There exists δ0 ∈ (0, 1), such that if XB is a tsuboptimal base at some point x0 ∈ Ω0 with radius δ ∈ (0, δ0 ), then: (a) There exists ε > 0 such that for every y ∈ B # (x0 , εδ) we have 1 |λB (x0 ) − λB (y)| 6 |λB (x0 )| 2 and therefore 3 1 |λB (x0 )| 6 |λB (y)| 6 |λB (x0 )| . 2 2 In particular λB (y) 6= 0, hence B # (x0 , εδ) ⊂ ΩB . (b) If B 0 = (J1 , . . . , Jn ) is any other n-tuple, for every y ∈ B # (x0 , εδ), we have 0

δ |B|−|B | |λ (y)| 6 C |λB (x0 )| tn for a constant C independent of x0 , δ, t. The above constants δ0 , ε, C depend on the local regularity of the vector fields (see Convention 9.14); the constant ε also depends on t, ∆Ω0 (but is independent of x0 and δ). B0

Point (a) will be useful in the proof of Theorem 9.11 point (1). Note that both these statements involve the determinants λB (x) formed with suitable bases. Point (b) will be useful in the proof of Corollary 9.40, another tool which will be used in the proof of Theorem 9.11. Let us say a few words about the strategy of the proof of Theorem 9.28. To estimate the increment |λB (x0 ) − λB (y)| we will need information on the derivatives of the function λB ; in turn, these derivatives will involve complicated expressions involving derivatives of the commutators X[I] . To get the desired quantitative estimates we need: - a precise way expressing any complicated commutator in terms of the com of mutators X[I] I∈B of a fixed base; this will lead to introduce explicit classes of variable coefficients, involved in these representation formulas; - quantitative estimate on these coefficients and their derivatives; - general tools for handling the derivative of a determinant, and in particular a way to re-express any derivative of the determinant of a matrix of the kind  X[I] I∈B in terms of the determinant itself. So, the first step is the introduction of the aforementioned coefficients, which will be the basic objects involved in all the subsequent estimates. Let us consider an n-tuple B = (I1 , I2 , . . ., In ), |I
i | 6 s such that ΩB 6= ∅, which n means that there are points x ∈ Ω such that X[Ii ] x i=1 is a base of Rn . We have the following: Proposition 9.29 For every multiindex J with |J| 6 s we can write, in ΩB , X[J] =

n X i=1

aiJ (x) X[Ii ]

(9.19)

422

H¨ ormander operators

with aiJ ∈ C ∞ (ΩB ). Actually, aiJ (x) =

λBi,J (x) λB (x)

where Bi,J is the n-tuple of multiindices obtained from B replacing Ii with J. Proof. For any fixed x ∈ ΩB we can consider the system of n linear equations X[J]


x

=

n X

aiJ (x) X[Ii ]


x

i=1

 n in the unknowns aiJ (x) i=1 . The determinant of the system is λB (x), which is different from zero for x ∈ ΩB ; hence we can solve the system by Cramer’s rule, getting exactly aiJ (x) =

λBi,J (x) λB (x)


n  where λBi,J (x) is the determinant of the matrix obtained from X[Ij ] x j=1 re

placing the row X[Ii ] x with X[J] x , which also justifies the symbol used: Bi,J denotes the n-tuple of multiindces obtained from B replacing Ii with J, and λBi,J (x) is the determinant of the corresponding matrix. From the expression of the aiJ (x) we also read that these functions belong to C ∞ (ΩB ). Let us recall the following fact that proved in Chapter 1 (see Lemma 1.21): Lemma 9.30 Given any two commutators X[I] , X[J] there exist absolute constants dK I,J such that X   X[I] , X[J] = dK I,J X[K] . |K|=|I|+|J|

If |I|+|J| 6 s we can combine theabove lemma with Proposition to control
9.29   n X[Ij ] x j=1 . The next a commutator of the form X[I] , X[J] in terms of a base lemma allows a similar control when |I| + |J| > s. Lemma 9.31 For every Ω0 b Ω and multiindex K (of any weight) we can write in Ω0 X X[K] = cJK (x) X[J] |J|6s

with cJK ∈ C ∞ (Ω0 ) and, for every k = 0, 1, 2, . . . , cJK C k (Ω0 ) can be bounded in terms of ∆Ω0 and the C h (Ω) norms of the coefficients of the vector fields X0 , . . . , Xq , for some h only depending on s, n, k and |K|.

More geometry of vector fields: metric balls and equivalent distances

423


Proof. Let x ∈ Ω0 and XB = X[I1 ] , . . . , X[In ] be a base such that |λB (x)| > ∆Ω0 . There exists a number r, depending on the local regularity of the vector fields (and independent of x), such that in the spherical neighborhood Ur (x) ⊂ Ω we have inf y∈Ur (x)

|λB (y)| >

1 ∆Ω0 . 2

(9.20)

Then, as in Proposition 9.29, we can write (even if |K| > s) X[K] =

n X

ci,K X[Ii ] with ci,K =

i=1

λBi ,K . λB

Note that ci,K ∈ C ∞ (Ur (x)), and for every k = 1, 2, . . . , kci,K kC k (Ur (x)) can be bounded in terms of ∆Ω0 and the C h (Ur (x)) norms of the coefficients of the vector fields X0 , . . . , Xq , for some h only depending on s, k and |K|. Let us cover Ω0 with N a finite number of smaller neighborhoods Ur/2 (xj ) and let {φj }j=1 be a partition of unity such that X φj = 1 on Ω0 and supp φj ⊂ Ur (xj ) .  N It is now possible to construct a finite covering Ur/2 (xj ) j=1 possessing a finite PN overlapping property, with j=1 χUr (xj ) (x) controlled in terms of the dimension n (this follows from the standard Vitali covering lemma, see e.g. Lemma 7.23). Also, the functions φj can be constructed in such a way that kφj kC k (Ur (xj )) is controlled in terms of r, n, k. Then, for every j, let XBj be a base satisfying (9.20), so that X[K] =

n X

cji,K X[I j ] in Ur (xj ) i

i=1

and X[K] =

N X n X

φj cji,K X[I j ] =

X

i

j=1 i=1

cJK X[J]

|J|6s





with cJK ∈ C ∞ (Ω0 ) and cJK C k (Ω0 ) is bounded in terms of cji,K

C k (Ur (xj ))

, that

is, by the above remarks on the function φj and their finite overlapping property, in terms of ∆Ω0 and the C h (Ω) norms of the coefficients of the vector fields X0 , . . . , Xq , for some h only depending on s, n, k and |K|. We can now state a general result which specifies how to express general iterated commutators in term of the elements of a fixed base, with coefficients explicitly expressed in terms of the basic coefficients aiJ introduced in (9.19): Proposition 9.32 Let B = (I1 , . . . , In ) be such that ΩB 6= ∅ and let Ω0 b Ω.

424

H¨ ormander operators

(a) For every choice of multiindices I, J with |I| , |J| 6 s we can write, in Ω B ∩ Ω0 : n   X X[I] , X[J] = aiI,J X[Ii ] i=1

with

aiI,J

X

=

i cL I,J aL ,

(9.21)

|L|6min(|I|+|J|,s)

 where aiL ∈ C ∞ (ΩB ) are as in Proposition 9.29, cL I,J (x) |I|,|J|,|L|6s is a

finite set of C ∞ (Ω0 ) functions and for every k = 0, 1, 2, . . . , cL I,J C k (Ω0 ) is bounded in terms of ∆Ω0 and the C h (Ω) norms of the coefficients of the vector fields X0 , . . . , Xq , for some h only depending on k, n and s. (b) For every choice of multiindices K1 , K2 , . . . , Kr with |Ki | 6 s (and r > 3) we can write: n   X   XK1 , . . . , X[Kr−1 ] , X[Kr ] = aiK1 ,...,Kr X[Ii ] i=1

with n X

aiK1 ,...,Kr =

i

i

aiK1 Ii1 aiK1 2 Ii . . . aKr−3 r−2 Ii 2

r−2

aKr−2 . r−1 Kr

(9.22)

i1 ,...,ir−2 =1

Proof. (a) Using the notation introduced in Lemma 9.30 and Proposition 9.29, for |I| + |J| 6 s, we can write X

  X[I] , X[J] =

|K|=|I|+|J|

=

n X

X

dK I,J X[K] =

dK I,J

n X

aiK (x) X[Ii ]

i=1

|K|=|I|+|J|

aiI,J (x) X[Ii ] ,

i=1

where aiI,J (x) =

X

i dK I,J aK (x)

|K|=|I|+|J|

and ∈ C ∞ (ΩB ). Also, when |I| + |J| > s, by Lemma 9.31 we have   X X X   L   X[I] , X[J] = dK dK I,J X[K] = I,J cK (x) X[L] |K|=|I|+|J|

≡

X

|L|6s

cL I,J (x) X[L] =

|L|6s

=

n X

cL I,J (x)

|L|6s



 X

 i=1

X

|L|6s

|K|=|I|+|J|

i  cL X[Ii ] . I,J aL

n X i=1

aiL (x) X[Ii ]

More geometry of vector fields: metric balls and equivalent distances

Hence for any choice of I, J with |I| , |J| 6 s we can write   n X   X i   X[I] , X[J] = cL X[Ii ] I,J aL i=1

425

(9.23)

|L|6min(|I|+|J|,s)

 ∞ where cL (Ω0 ) functions with the properties I,J (x) |I|,|J|,|L|6s is a finite set of C required in the statement of (a). A careful iteration of (a) gives (b). The second step in the proof of Theorem 9.28 consists in proving quantitative estimates on the functions obtained as linear combinations of products of the coefficients aiJ , or differentiating these functions. This is done by introducing suitable classes of functions structured on these coefficients:
Definition 9.33 For a fixed base XB = X[I1 ] , . . . , X[In ] , every integer p and positive integer r, we will call generators of type Gpr (B) the functions, defined in ΩB , of the form alJ11 · alJ22 · · · alJkk (with alJii as in (9.19)) where k 6 r and p 6 |Il1 | + |Il2 | + · · · + |Ilk | − (|J1 | + |J2 | + · · · + |Jk |) . We will write alJ11 · alJ22 · · · alJkk ∈ Gpr (B) . Note that the definition of Gpr (B) depends on the fixed base XB . Also, note that for any fixed p and r the set Gpr (B) is finite since Ilj 6 s for every j, |Ji | 6 s and k 6 r. The following proposition collects some properties of Gpr (B) which follow immediately from the definition: Proposition 9.34 For every choice of multiindices J, J1 , . . . , Jk and l, l1 , . . . , lk ∈ {1, . . . , n} we have |I |−|J|

alJ ∈ G1 l

(B)

(|Il |+···+|Ilk |)−(|J1 |+···+|Jk |) alJ11 · · · alJkk ∈ Gk 1 (B) . The following inclusions hold: Gpr 1 (B) ⊇ Gpr 2 (B) for p1 6 p2 Gpr1 (B) ⊆ Gpr2 (B) for r1 6 r2 .

(9.24)

Also, +p2 if f1 ∈ Gpr11 (B) and f2 ∈ Gpr22 (B) then f1 f2 ∈ Gpr11+r (B) . 2

(9.25)

The role of Gpr (B) in the proof of Theorem 9.28 is shown in the following

426

H¨ ormander operators

Proposition 9.35 Assume that for some t, δ ∈ (0, 1) the base XB is t-suboptimal at some x0 ∈ Ω with radius δ. Then, if f ∈ Gpr (B), we have |f (x0 )| 6

δp . tr

Proof. By Proposition 9.29 and (9.18) we know that i λBi,J (x0 ) 1 1 aJ (x0 ) = 6 δ |B|−|Bi,J | = δ |Ii |−|J| . |λB (x0 )| t t

(9.26)

Then, saying that a function f ∈ Gpr (B) means that f = alJ11 · · · alJkk with k 6 r and p 6 |Il1 | + · · · + |Ilk | − (|J1 | + · · · + |Jk |) . By (9.26), we then have, since t, δ ∈ (0, 1),   1 δp l1 l aJ1 · · · aJkk (x0 ) 6 k δ |Il1 |+...+|Ilk |−(|J1 |+...+|Jk |) 6 r . t t

Let us now study how differentiation acts on functions in Gpr (B): Proposition 9.36 Let f ∈ Gpr (B), let L1 , . . . , Lm be multiindices satisfying |Li | 6 s and let Ω0 b Ω. (a) In ΩB ∩ Ω0 we have X[L1 ] . . . X[Lm ] f =

nm X

cj fj

j=1 p−(|L |+···+|Lm |)

for suitable generators fj ∈ Gr+m 1 satisfying, for every k = 0, 1, 2, . . . , nm X

(B) and smooth functions cj

kcj kC k (Ω0 ∩ΩB ) 6 Ck ,

(9.27)

j=1

with Ck depending on ∆Ω0 and the C h (Ω) norms of the coefficients of the vector fields X0 , . . . , Xq , for some h only depending on k, m, n and s. (b) If the base XB is t-suboptimal at some x0 ∈ Ω0 with radius δ, then p−(|L1 |+···+|Lm |) X[L ] . . . X[L ] f (x0 ) 6 C0 δ 1 m tr+m

for C0 as in point (a) (for k = 0). In particular, |Ii |−(|L1 |+···+|Lm |+|J|) X[L ] . . . X[L ] aiJ (x0 ) 6 C0 δ 1 m tm+1

More geometry of vector fields: metric balls and equivalent distances

427

Proof. Point (b) follows from (a) by Proposition 9.35. Observe that it is enough Pn to prove (a) for m = 1. Actually, once we know that X[L1 ] f = j11=1 cj1 fj1 with p−|L1 |

fj1 ∈ Gr+1

(B) and smooth functions cj1 satisfying n1 X

kcj1 kC k (Ω0 ∩ΩB ) 6 Ck ,

j1 =1

we can apply the same result to each fj1 , getting X[L2 ] X[L1 ] f =

n1 X

n1 X
X[L2 ] cj1 fj + cj1 X[L2 ] fj

j=1

=

n1 X




X[L2 ] cj1 fj +

j=1

j=1 n1 X

cj 1

j1 =1

p−(|L |+|L |)

p−|L |

n2 X

cj1 ,j2 fj1 ,j2

j2 =1 p−(|L |+|L |)

2 2 with fj1 ,j2 ∈ Gr+2 1 (B) , fj ∈ Gr+1 1 (B) ⊂ Gr+2 1 (B) . Iteration then gives the general case. So let us concentrate on computing X[L1 ] f, for some f = alJ11 · · · alJkk with k 6 r and

p 6 |Il1 | + · · · + |Ilk | − (|J1 | + · · · + |Jk |) . By Leibniz rule, X[L1 ] f is the sum of k terms of the kind   alJ11 · alJ22 · · · X[L1 ] alJii · · · alJkk |Il |−|Ji | Then it is enough to prove that for alJii ∈ G1 i (B) we have X[L1 ] alJii =

n1 X

cj fj

j=1

|Il |−|Ji |−|L1 | with fj ∈ G2 i (B) and cj smooth functions. Actually, once this fact is established, we can write n1   X alJ11 · · · X[L1 ] alJii · · · alJkk = cj alJ11 · · · fj · · · alJkk j=1

where, by (9.25) and (9.24) (|Il |+···+|Ilk |)−(|J1 |+···+|Jk |+|L1 |) alJ11 · · · fj · · · alJkk ∈ Gk+11 (B) ⊂ Gpr+1 (B) |I |−|J|

so we are done. Let us prove that (simplifying notation) for alJ ∈ G1 l have n1 X l X[L] aJ = cj fj j=1

with fj ∈

|I |−|J|−|L| G2 l

(B). By (9.23) we can write   n X   X i   X[L] , X[J] = cK X[Ii ] . L,J aK i=1

|K|6min(|L|+|J|,s)

(B) we

428

H¨ ormander operators

On the other hand " # n n n X X     X
i X[L] , X[J] = X[L] , aJ X[Ii ] = ajJ X[L] , X[Ij ] + X[L] aiJ X[Ii ] i=1

=

n X n X i=1

j=1

ajJ

j=1

i=1 n  X



X

i cK L,Ij aK X[Ii ] +

 X[L] ai[J] X[Ii ] .

i=1

|K|6min(|L|+|Ij |,s)

Hence n  X i=1

=

 i cK a L,J K X[Ii ]

X |K|6min(|L|+|J|,s)

n X n X i=1

j=1





X

 n   X i i cK a X + X a [Ii ] [L] [J] X[Ii ] L,Ij K

|K|6min(|L|+|Ij |,s)

i=1

ajJ

and since X[Ii ] is a base in ΩB we obtain X

X[L] ai[J] =

i cK L,J aK −

n X j=1

|K|6min(|L|+|J|,s)

X

ajJ

i cK L,Ij aK

|K|6min(|L|+|Ij |,s)

where (using the properties in Proposition 9.34) in the first sum |I |−|K|

aiK ∈ G1 i

|I |−|L|−|J|

(B) ⊆ G1 i

|I |−|L|−|J|

(B) ⊂ G2 i

(B)

while in the second sum |I |−|J|+|Ii |−|K|

ajJ aiK ∈ G2 j

|I |−|L|−|J|

(B) ⊆ G2 i

(B) .

K k 0 Moreover, cK L,J , cL,Ij have C (Ω ∩ ΩB ) norms controlled in terms of ∆Ω0 and the h 00 C (Ω ) norms of the coefficients of the vector fields X0 , . . . , Xq , for some h only depending on k, n and s. This finishes the proof of (a).

The third step in the proof of Theorem 9.28 deals with the computation of the derivatives of the determinant λB (x). We start with a general algebraic fact related to the differentiation of a determinant: Lemma 9.37 Let us consider the C 1 (Ω) vector fields Zj =

n X

αjk (x) ∂xk , j = 1, . . . , n

k=1

and

T =

n X

βl (x) ∂xl .

l=1

Then T (det (Z1 , Z2 , . . . , Zn )) = (∇ · T ) det (Z1 , Z2 , . . . , Zn ) n X + det (Z1 , . . . , Zj−1 , [T, Zj ] , Zj+1 , . . . , Zn ) . j=1

(9.28)

More geometry of vector fields: metric balls and equivalent distances

429

Proof. The standard formula for differentiating a determinant (which can be easily proved by induction on n) reads: T (det (Z1 , Z2 , . . . , Zn ))   n n X X = det Z1 , . . . , Zj−1 , (T (αjk )) ∂xk , Zj+1 , . . . , Zn j=1

(9.29)

k=1

On the other hand,   n X n n X n X X [T, Zj ] = βl (∂xl αjk ) ∂xk − αjk (∂xk βl ) ∂xl =

k=1 n X

l=1

l=1

(T (αjk )) ∂xk −

k=1

n X n X l=1

k=1

 αjk (∂xk βl ) ∂xl .

k=1

Hence det (Z1 , . . . , Zj−1 , [T, Zj ] , Zj+1 , . . . , Zn ) = det Z1 , . . . , Zj−1 ,  − det Z1 , . . . , Zj−1 ,

n X

! (T (αjk )) ∂xk , Zj+1 , . . . , Zn

k=1 n  X l=1



X



αjk (∂xk βl ) ∂xl , Zj+1 , . . . , Zn

k=1,...,n k6=l

  n X (αjl (∂xl βl )) ∂xl , Zj+1 , . . . , Zn − det Z1 , . . . , Zj−1 , l=1

≡ Aj − Bj − Cj . Note that by (9.29), n X

Aj = T (det (Z1 , Z2 , . . . , Zn )) .

j=1

Also, computing the determinant in Cj by Laplace theorem we have Cj =

n X

(−1)

j+l

αjl (∂xl βl ) Mj,l

l=1

where Mj,k is the (j, k)-minor of the matrix (Z1 , . . . , Zn ). Hence n X

Cj =

j=1

n X n X

(−1)

j+l

αjl (∂xl βl ) Mj,l =

j=1 l=1

n X l=1

(∂xl βl )

n X

(−1)

j=1

= (∇ · T ) det (Z1 , . . . , Zn ) . Similarly, Bj =

n X l=1

(−1)

j+l



X k=1,...,n k6=l

 αjk (∂xk βl ) Mj,l

j+l

αjl Mj,l

430

H¨ ormander operators

and n X

Bj =

j=1

=

n X

X

(∂xk βl )

l=1 k=1,...,n k6=l n X X

n X

(−1)

j+l

αjk Mj,l

j=1

(∂xk βl ) det (Z1 , . . . , Zk−1 , Zl , Zk+1 , . . . Zn ) = 0

l=1 k=1,...,n k6=l

since in every determinant there is a repeated line, because k 6= l. Hence from n X

det (Z1 , . . . , Zj−1 , [T, Zj ] , Zj+1 , . . . , Zn ) =

j=1

n X

(Aj − Bj − Cj )

j=1

we conclude (9.28). We can now get
a formula

the X[I] derivative of a determinant
for computing λB (x) = det X[I1 ] x , X[I2 ] x , . . . , X[In ] x : Proposition 9.38 Let Ω0 b Ω. Then (a) For every choice of multiindices I1 , . . . , Ik , there exists a function FI1 ,...,Ik such that in ΩB ∩ Ω0 X[I1 ] X[I2 ] . . . X[Ik ] λB = FI1 ,...,Ik λB with nI1 ,...,Ik

FI1 ,...,Ik =

g0I1 ,...,Ik

X

+

gjI1 ,...,Ik fjI1 ,...,Ik

j=1 −(|I1 |+...+|Ik |)

for suitable generators fjI1 ,...,Ik ∈ Gk functions gjI1 ,...,Ik such that

(B) and suitable smooth

m X I1 ,...,Ik (x) 6 C for x ∈ Ω0 ∩ ΩB gj j=0

for a constant C depending on ∆Ω0 and the C h (Ω) norms of the coefficients of the vector fields X0 , . . . , Xq , for some h depending on k, n, q, s. (b) In particular, if the base XB it t-suboptimal at some x0 ∈ Ω0 with radius δ ∈ (0, 1), then −(|I1 |+...+|Ik |) X[I ] X[I ] . . . X[I ] λB (x0 ) 6 C δ |λB (x0 )| 1 2 k tk

for a constant C depending on the aforementioned quantities, but independent of x0 , δ, t.

More geometry of vector fields: metric balls and equivalent distances

431

Proof. Point (b) follows from (a) by Proposition 9.35, so let us prove (a). We start with the case k = 1. By Lemma 9.37 and (9.23) we have, in ΩB ∩ Ω0 :
X[I] λB = X[I] det X[I1 ] , . . . , X[In ] n X
 
= ∇ · X[I] λB + det X[I1 ] , . . . , X[Ij−1 ] , X[I] , X[Ij ] , X[Ij+1 ] , . . . , X[In ] j=1




= ∇ · X[I] λB n X X +

  i det X[I1 ] , . . . , X[Ij−1 ] , cL a X , X , . . . , X [Ij+1 ] [In ] . I,Ij L [Ii ]

i,j=1 |L|6min(s,|I|+|Ij |)

On the other hand n X

  i det X[I1 ] , . . . , X[Ij−1 ] , cL I,Ij aL X[Ii ] , X[Ij+1 ] , . . . , X[In ]

X

i=1 |L|6min(s,|I|+|Ij |)

=

n X

X

i cL I,Ij aL det X[I1 ] , . . . , X[Ij−1 ] , X[Ii ] , X[Ij+1 ] , . . . , X[In ]




i=1 |L|6min(s,|I|+|Ij |)

X

=

j cL I,Ij aL det X[I1 ] , . . . , X[Ij−1 ] , X[Ij ] , X[Ij+1 ] , . . . , X[In ]




|L|6min(s,|I|+|Ij |)

 =

X

j cL I,Ij aL

 λB

|L|6min(s,|I|+|Ij |)

since the determinant of a matrix with repeated rows is zero. We conclude X  n X
j L cI,Ij aL λB X[I] λB = ∇ · X[I] λB + j=1 |L|6min(s,|I|+|Ij |) |I |−|L|

−|I|

where ajL ∈ G1 j (B) ⊂ G1 (B) and cL I,Ij are smooth functions as in Proposition 9.32. So, point (a) is proved for k = 1. Let us now prove (a) for k = 2, the general case follows analogously by iteration. Applying the result for k = 1 we get:  X[I2 ] X[I1 ] λB = X[I2 ]  =

gI1 +

nI 1 X

  cj1 fj1 λB

j1 =1 nI 1

 X[I2 ] gI1 +

X

cIj11 fjI11



nI 1   X λB + gI1 + cIj11 fjI11 X[I2 ] λB

j1 =1

j1 =1

nI 1 nI 1     X X = X[I2 ] gI1 + cIj11 X[I2 ] fjI11 + X[I2 ] cIj11 fjI11 λB j1 =1

j1 =1

nI 1 nI 2    X X + gI1 + cjI11 fjI11 gI2 + cIj12 fjI12 λB j1 =1

j2 =1

432

H¨ ormander operators −|I1 |

with fjI11 ∈ G1

−|I2 |

(B) , fjI12 ∈ G1

(B) . Next, by Proposition 9.36 we can rewrite nI1 ,I2

X[I2 ] fjI11 =

X

cIj1 ,I2 fjI1 ,I2

j=1 −|I1 |−|I2 |

with fjI1 ,I2 ∈ G2 conclude

(B) . Recalling also the properties in Proposition 9.34 we 



nI1 ,I2

X[I2 ] X[I1 ] λB = g0 +

X

gj fjI1 ,I2  λB

j=1

PnI1 ,I2 −|I |−|I2 | |gj (x)| 6 C for with fjI1 ,I2 ∈ G2 1 (B) and gj smooth functions with j=0 x ∈ Ω0 ∩ ΩB , with C depending on the quantities specified in the statement of the proposition. We will also need a formula for expressing in terms of λB (x) the derivative of the determinant λB0 (x) corresponding to a different n-tuple
XB0 = X[J1 ] , X[J2 ] , . . . , X[Jn ] , which might be a base or not. Proposition 9.39 Let Ω0 b Ω. (a) For every choice of multiindices I1 , . . . , Ik there exists a function FI1 ,...,Ik ,B0 such that, in Ω0 ∩ ΩB , X[I1 ] X[I2 ] . . . X[Ik ] λB0 = FI1 ,...,Ik ,B0 λB where nI1 ,...,Ik

FI1 ,...,Ik ,B0 =

X

gj fB0 ,I1 ,...,Ik ,j

j=1

for suitables generators fB0 ,I1 ,...,Ik ,j smooth functions gj such that

|B|−|B0 |−(|I1 |+...+|Ik |)

∈ Gn+k

(B) and

nI1 ,...,Ik

X

|gj (x)| 6 C

j=1

in Ω0 , with C depending on ∆Ω0 and the C h (Ω) norms of the coefficients of the vector fields X0 , . . . , Xq , for some h depending on k, n, q, s. (b) In particular, if XB is a t-suboptimal base at some x0 ∈ Ω0 with radius δ ∈ (0, 1), then 0

|B|−(|B |+|I1 |+...+|Ik |) X[I ] X[I ] . . . X[I ] λB0 (x0 ) 6 C δ |λB (x0 )| 1 2 k tn+k for a constant C depending on the aforementioned quantities, but independent of x0 , δ, t.

More geometry of vector fields: metric balls and equivalent distances

433

Proof. Again, point (b) follows from (a) by Proposition 9.35, so let us prove (a). By (9.19) we can write, in ΩB , ! n n X X
l1 ln λB0 = det X[J1 ] , X[J2 ] , . . . , X[Jn ] = det aJ1 X[Il ] , . . . , aJn X[Iln ] 1 l1 =1

=

n X

ln =1

  alJ11 · · · alJnn det X[Il ] , . . . , X[Iln ] . 1

l1 ,...,ln =1

Now, in the last sum the determinant is nonzero if and only if (l1 , . . . , ln ) is a permutation of (1, 2, . . . , n). Denoting by σ (l1 , . . . , ln ) the sign of this permutation, the last expression equals λB ·

X

σ (l1 , . . . , ln ) alJ11 · · · alJnn = λB

n! X

fB0 ,j ≡ λB fB0

j=1 (|I1 |+···+|In |)−(|J1 |+···+|Jn |)

with fB0 ,j ∈ Gn 9.38 and 9.36

|B|−|B0 |

(B) = Gn

(B). Then by Propositions

X[I] λB0 = X[I] (λB fB0 ) = fB0 X[I] λB + λB X[I] fB0 X  nI I = fB0 FI λB + cj fB0 ,j λB ≡ FI,B0 λB j=1

where FI,B0 =

X n!

 fB0 ,j

gI +

j=1

nI X

 cj fj

j=1

+

nI X

cj fBI 0 ,j

j=1

|B|−|B0 |−|I| |B|−|B | −|I| with fB0 ,j ∈ Gn (B) , fj ∈ G1 (B) , fBI 0 ,j ∈ Gn+1 (B) and gI , cj , c0j are k 0 smooth functions with every C (Ω ) norm bounded by a constant depending on ∆Ω0 and the C h (Ω) norms of the coefficients of the vector fields X0 , . . . , Xq , for some h depending on k. By Proposition 9.34 we can rewrite 0

FI,B0 =

nI X

gj fB0 ,I,j

j=1

PnI |B|−|B0 |−|I| (B) and with fB0 ,I,j ∈ Gn+1 j=1 |gj (x)| 6 C with C depending on the quantities specified in the statement. This proves (a) for k = 1. Let us prove it for k = 2, the general case then will follow by iteration. By the case k = 1,

X[I1 ] X[I2 ] λB0 = X[I1 ] (FI2 ,B0 λB ) = X[I1 ] FI2 ,B0 λB + FI2 ,B0 X[I1 ] λB
= X[I1 ] FI2 ,B0 + FI2 ,B0 FI1 ,B0 λB . Keeping into account the structure of FI1 ,B0 , FI1 ,B0 , Proposition 9.36 and Proposition 9.34 we get the desired result.

434

H¨ ormander operators

We can now come to the Proof of Theorem 9.28. Recall that, by Remark 9.10,  X  
B # (x0 , r) = y = exp uI X[I] (x0 ) : |uI | < r|I| |I|6s

We can also relabel the “coordinates” (uI )|I|6s as (u1 , . . . , uN ) for some N = N (q, s). For u = (uI )|I|6s = (u1 , . . . , uN ), let  X  
Fx (u) = λB exp uI X[I] (x) , |I|6s 0

then, for any fixed Ω b Ω, there exists δ0 ∈ (0, 1) such that  is P (u, x) 7→ Fx
(u)  N 0 (x) smooth on u ∈ R : |u| < δ0 × Ω and the point y = exp |I|6s uI X[I] belongs to Ω whenever x ∈ Ω0 and |u| < δ0 . Hence δ0 depends on dist (Ω0 , ∂Ω) and the coefficients of X[I] in Ω. In particular, for any positive integer m, there is a constant Am such that X 1 ∂ α Fx m+1 α Fx (u) − (0) u (9.30) 6 Am |u| α α! ∂u |α|6m for every x ∈ Ω0 and |u| < δ0 . The constant Am depends on the C m+1 (Ω) norms of the coefficients of the vector fields X[I] for |I| 6 s. By (9.18), 0

|λB (x0 )| δ |B| > t max |λB0 (x0 )| δ |B | > tδ ns max |λB0 (x0 )| > tδ ns ∆Ω0 , 0 0 B

B

and therefore, |λB (x0 )| > tδ ns−|B| ∆Ω0 > tδ ns ∆Ω0 .

(9.31)

|I|

Choosing m = ns − 1 in (9.30) for any |uI | < (εδ) , we have m+1

Am |u|

6 Am N (m+1)/2 (εδ)

ns

6 c (n, s, q)

εns 1 |λB (x0 )| 6 |λB (x0 )| ∆Ω0 t 4

for a suitable choice of ε, depending on t, ∆Ω0 , n, s, q and the C ns (Ω) norms of the coefficients of the vectorfields X[I] for |I| 6 s. This means that for any y ∈
P |I| B # (x0 , εδ) , that is y = exp (x0 ) with |uI | < (εδ) , we have |I|6s uI X[I] X 1 ∂ α Fx 1 0 α Fx0 (u) − (0) u 6 |λB (x0 )| . α α! ∂u 4 |α|6m

Since |λB (y) − λB (x0 )| = |Fx0 (u) − Fx0 (0)| X 1 ∂ α Fx X 1 ∂ α Fx 0 0 α α (0) u + (0) u , 6 Fx0 (u) − α α α! ∂u 16|α|6m α! ∂u |α|6m

More geometry of vector fields: metric balls and equivalent distances

point (a) will follow as soon as we prove that X 1 ∂ α Fx0 1 α (0) u 6 4 |λB (x0 )| . α 16|α|6m α! ∂u

435

(9.32)

But by Lemma 9.22,  k X 1 ∂ α Fx X
1  0 (0) uα = uI X[I]  λB (x0 ) . α! ∂uα k!

|α|=k

|I|6s

Now, the right-hand side is the sum of a controlled number of terms of the kind uI1 uI2 . . . uIk X[I1 ] . . . X[Ik ] λB (x0 ) , |I | and by (9.18), Proposition 9.38 and our assumption uIj < (εδ) j we have uI uI . . . uI X[I ] . . . X[I ] λB (x0 ) 1 2 k 1 k δ −(|I1 |+|I2 |+...|Ik |) |λB (x0 )| tk   ε k ε(|I1 |+|I2 |+...|Ik |) |λB (x0 )| 6 C |λB (x0 )| =C k t t for a constant C depending on ∆Ω0 and the C h (Ω) norms of the coefficients of the vector fields X0 , . . . , Xq , for some h depending on k (but recall that k 6 m = ns−1). Taking if necessary a smaller ε, (9.32) holds. Let us come to the proof of (b). By (9.18) we have 0 δ |B|−|B | |λB0 (x0 )| 6 |λB (x0 )| t so let us show that for y ∈ B # (x0 , εδ) 0 |λ 0 (y) − λ 0 (x )| 6 Ct−n δ |B|−|B | |λ (x )| , 6 (εδ)

B

(|I1 |+|I2 |+...|Ik |)

B

C

B

0

0

which will imply the desired estimate. Let us define     X
Gx (u) = λB0 exp  uI X[I]  (x) . |I|6s

|I | Arguing as above we have, for a certain m0 to be fixed later and uIj < (εδ) j , X 1 ∂ α Gx m0 +1 m0 +1 0 α Gx0 (u) − (0) u 6 Bm0 |u| 6 Bm0 (εδ) α α! ∂u |α|6m0 0

m +1 0 with B (Ω) norms of the coefficients of the vector fields  m depending on the C X[I] |I|6s . By (9.18) we know that |λB (x0 )| > t∆Ω0 δ ns hence choosing now m0 = 2ns − 1 we get, since |B| − |B 0 | 6 |B| 6 ns, X 1 ∂ α Gx |λB (x0 )| 0 α Gx0 (u) − (9.33) (0) u 6 Bm0 ε2ns δ ns α α! ∂u t∆Ω0 |α|6m0 0

Bm0 ε2ns |B|−|B0 | δ |B|−|B | 6 δ |λB (x0 )| 6 C |λB (x0 )| ∆Ω0 t t

436

H¨ ormander operators

choosing ε 6 ∆Ω0 . On the other hand |λB0 (y) − λB0 (x0 )| = |Gx0 (u) − Gx0 (0)| α α X X 1 1 ∂ G ∂ G x x 0 0 α α 6 Gx0 (u) − (0) u + (0) u , α α α! ∂u α! ∂u 16|α|62ns−1 |α|62ns−1 and the last quantity can be bounded by the sum of a controlled number of terms of the kind uI1 uI2 . . . uIk X[I1 ] . . . X[Ik ] λB0 (x0 ) which by (9.18), Proposition 9.39 |I | (b) and our assumption uIj < (εδ) j are bounded by 0

δ |B|−|B |−(|I1 |+...+|Ik |) |λB (x0 )| (εδ) C tn+k 0 0  ε k δ |B|−|B | δ |B|−|B | =C |λ (x )| 6 C |λB (x0 )| B 0 t tn tn (|I1 |+|I2 |+...|Ik |)

choosing also ε 6 t, and the proof is completed. We also have the following important consequence, which will be used several times in the computation of the next section: Corollary 9.40 With the same assumptions and notation of Theorem 9.28, and with the same ε > 0, for every positive integer n0 , there exists a constant C depending on n0 and the local regularity of the vector fields (but independent of x0 , δ and t) such that if y ∈ B # (x0 , εδ) and K1 , . . . , Kr are multiindices with r 6 n0 and |Ki | 6 s, then l aK

1 ,...,Kr

δ |Il |−(|K1 |+...+|Kr |) (y) 6 C tnn0

where alK1 ,...,Kr are the functions defined in Proposition 9.32. Proof. By Proposition 9.29, for y ∈ B # (x0 , εδ) ⊂ ΩB we know that alK (y) = λBl,K (y) λB (y)

where Bl,K is the n-tuple obtained from B replacing Il with K. Hence by Theorem 9.28 (b) we have λB

l,K

and |λB (y)| >

1 2

δ |B|−|Bl,K | δ |Il |−|K| |λ (x )| = C |λB (x0 )| (y) 6 C B 0 tn tn

|λB (x0 )|, therefore |Il |−|K| l aK (y) 6 C δ . tn

Next, by (9.21), we can write i aI,J 6

X |L|6min(|I|+|J|,s)

L δ |Ii |−|L| δ |Ii |−(|I|+|J|) cI,J C 6c . n t tn

(9.34)

More geometry of vector fields: metric balls and equivalent distances

Finally, by (9.22) we have, for r > 3, n X i i i aK ,...,K 6 aK1 Ii1 aiK1 2 Ii2 · · · aKr−3 1 r r−2 Ii

r−2

437

i aKr−2 r−1 Kr

i1 ,...,ir−2 =1

6c

n X i1 ,...,ir−2

δ |Ii |−(|K1 |+|I11 |) δ |Ii1 |−(|K2 |+|I12 |) δ |Iir−2 |−(|Kr−1 |+|Kr |) · · · tn tn tn =1

δ |Ii |−(|K1 |+|K2 |+...+|Kr |) tn(r−1) so we are done. =c

9.5

Structure of metric balls

We now come to the proof of the first main result in this chapter, i.e. Theorem 9.11, which will be based on the results of the last two sections. Actually, in view of the application of this theorem to another important result which will be established in the next chapter (see Theorem 10.40), we will prove a slightly more general result, as will be clear in a moment. We start by introducing some new notation and definitions  recalling some of the given in section 9.1. For a fixed n-tuple XB = X[I1 ] , X[I2 ] , . . . , X[In ] we consider the box n o Qδ = u ∈ Rn : |uj | < δ |Ij | . Denoting by J1 , J2 , . . . , Jm any enumeration of all the multiindices of weight 6 s we also write n o Q0δ = v ∈ Rm : |vj | < δ |Jj | and v·X =

m X

vi X[Ji ] .

i=1

For x ∈ Ω we define the map ! n m X X Φx,B,v (u) = exp ui X[Ii ] + vi X[Ji ] (x) = exp (u · XB + v · X) (x) (9.35) i=1

i=1

with u ∈ Rn and v ∈ Rm sufficiently small and consider the balls that are images under the map Φx,B,v of the boxes Qδ BB (x, δ, v) = Φx,B,v (Qδ ) . Note that for v = 0 the balls BB (x, δ, v) are exactly the balls BB (x, δ) defined in section 9.1. Let now v ∈ Q0δ and observe that since the multiindices I1 , I2 , . . . , In are repeated in J1 , J2 , . . . , Jm (which is the set of all multiindices of weight 6 s) in the expression n m X X ui X[Ii ] + vi X[Ji ] i=1

i=1

438

H¨ ormander operators

the same multiindex may appear two times. By Remarks 9.9 and 9.10 we have the inclusions: BB (x0 , δ, v) ⊆ B # (x0 , 2δ) ⊆ B ∗ (x0 , 2δ)

(9.36)

(see section 9.1 for the definitions of the balls B # and B ∗ ). Notation 9.41 Throughout this section, we fix Ω0 b Ω. Note that there exists δ0 ∈ (0, 1) such that for |u| < δ0 , |v| < δ0 and x ∈ Ω0 , Φx,B,v (u) is well defined and belongs to Ω. This number δ0 depends on the local regularity of the vector fields (see Convention 9.14). Our goal is to prove the following: Theorem 9.42 (Structure of balls) With the above Notation 9.41, for any t ∈ (0, 1) there exist constants c, c1 , c2 ∈ (0, 1) such that if x0 ∈ Ω0 , 0 < δ 6 δ0 , and XB is a t-suboptimal base at x0 with radius δ, then: (1) if we denote by JΦx0 ,B,v (u) the Jacobian matrix of Φx0 ,B,v then for every u ∈ Qc1 δ and v ∈ Q0c1 δ we have
1 |λB (x0 )| 6 det JΦx0 ,B,v (u) 6 4 |λB (x0 )| , 4 (2) B ∗ (x0 , cδ) ⊂ BB (x0 , δ, v) ⊂ B # (x0 , 2δ) ⊂ B ∗ (x0 , 2δ) , (3) for every v ∈ Q0c1 δ the map Φx0 ,B,v is one-to-one on the box Qc2 δ . The constants c, c1 depend on t, ∆Ω0 and the local regularity of the vector fields (see Claim 9.15). The constant c2 has the dependence specified in Claim 9.17. This theorem contains the one stated in the introduction (Theorem 9.11), just letting v ≡ 0 and t = 21 . We split the proof of this theorem in three separate propositions. We also need some preliminary results. 9.5.1

Some basic facts related to global invertibility of maps

Point (1) in Theorem 9.42 shows that the map Φx0 ,B,v is a local diffeomorphism. To prove point (2) and point (3) however we need to invert globally the map Φx0 ,B,v . In this section we forget for a moment the map Φx0 ,B,v and we develop some basic material related to the global invertibility of maps. Let Ω1 , Ω2 ⊆ Rn . A function f : Ω1 → Ω2 is called a local homeomorphism if for every x0 ∈ Ω1 there exists a neighborhood U of x0 and a neighborhood V of f (x0 ) such that f : U → V is a homeomorphism. We start recalling the classical definition of path and homotopy in a topological space. Definition 9.43 (Path) A path in Ω ⊆ Rn is a continuous function p : I → Ω where I ⊆ R is an interval. If a is the left endpoint of I and a ∈ I we will say that p (a) is the origin of the path.

More geometry of vector fields: metric balls and equivalent distances

439

Definition 9.44 (Homotopy) A homotopy in Ω is a continuous map H : [0, 1] × [0, 1] → Ω. The map H0 : [0, 1] → Ω defined by H0 (s) = H (s, 0) is called the base of the homotopy.

H(0, t)

H(s, 1) H(s, t)

H(1, t)

H0 (s) = H(s, 0)

Fig. 9.1

Ω

Homotopy.

Definition 9.45 (Lifting) Let f : Ω1 → Ω2 be a local homeomorphism and let p : I → Ω2 a path in Ω2 . A lifting of p by f is a path q : I → Ω1 such that f (q (s)) = p (s)

for every s ∈ I.

f

p(s) = f (q(s))

q(s)

Ω2 Ω1

Fig. 9.2

Lifting.

A remarkable property of local homeomorphisms is the following uniqueness of lifting.

440

H¨ ormander operators

Proposition 9.46 Let f : Ω1 → Ω2 be a local homeomorphism, let p : I → Ω2 a path in Ω2 and let q1 and q2 be two lifted paths of p by f . Then, either q1 (s) = q2 (s) for every s ∈ I or q1 (s) 6= q2 (s) for every s ∈ I. Proof. Let A = {s ∈ I : q1 (s) = q2 (s)}. We have to show that either A = ∅ or A = I. Assume A 6= ∅. Since q1 and q2 are continuous, A is closed (in I). We will show that A is also relatively open in I so that, since I is connected, A = I. Let s0 ∈ A and set x0 = q1 (s0 ) = q2 (s0 ). Since f is a local homeomorphism there exists a neighborhood U of x0 and a neighborhood V of f (x0 ) such that f : U → V is a homeomorphism. Using the continuity of q1 and q2 we can find a neighborhood J of s0 such that for every s ∈ J, q1 (s) , q2 (s) ∈ U . Since f on U is one-to-one, from f (q1 (s)) = f (q2 (s)) = p (s) we obtain that q1 (s) = q2 (s) for every s ∈ J. Hence J ⊆ A and therefore A is open. In the next proposition we consider the lifting of a homotopy and we show that when a map lifts every path in a homotopy then it lifts the whole homotopy. Theorem 9.47 (Homotopy lifting) Let f : Ω1 → Ω2 be a local homeomorphism, let H be a homotopy in Ω2 with base H0 and let K0 be a lifting of H0 . Assume that for every s ∈ [0, 1] the paths H (s, ·) can be lifted with origin K0 (s) and let K (s, ·) be such a lifting. Then, K : [0, 1] × [0, 1] → Ω1 is a homotopy with base K0 satisfying f (K (s, t)) = H (s, t) for every

(s, t) ∈ [0, 1] × [0, 1] .

(9.37)

In particular K is continuous.

f

H(s, ·)

H(s, 1)

K(s, ·) H0 (s)

K0 (s) Ω1

Fig. 9.3

Homotopy lifting.

Ω2

More geometry of vector fields: metric balls and equivalent distances

441

Proof. The property (9.37) immediately follows from the definition of K. We have only to prove that K is continuous. If f were a global homeomorphism this would be obvious. In our setting we will exploit the continuity of H, the continuity of K0 and the uniqueness of lifting to ensure that K = f −1 (H) holds locally and from this we will obtain the continuity of K. Fix s0 ∈ [0, 1] and let Σs0 = {t ∈ [0, 1] : K in not continuous at (s0 , t)} . Assume by contradiction that Σs0 is not empty for some s0 and let t0 = inf Σs0 . Since f is a local homeomorphism there exist a neighborhood U of K (s0 , t0 ) and a neighborhood V of f (K (s0 , t0 )) = H (s0 , y0 ) such that f : U → V is a homeomorphism. By construction the map t 7→ K (s0 , t) is continuous and therefore there exists a neighborhood J of t0 such that K (s0 , t) ∈ U for every t ∈ J. Also, since H is continuous we can fix a neighborhood W of s0 and restrict J if necessary so that H (W × J) ⊆ V . Now pick t ∈ J such that t
< t0 when t0 > 0, or t = 0 if t0 = 0. Observe that in both cases the map s 7→ K s, t is continuous at s0 . Indeed, when t0 > 0, such a map is continuous at s0 by the minimality of t0 , while for t0 =
t = 0, we have K (s, 0) = K0 (s) which is continuous by assumption. Since K s0 , t ∈ U , using this continuity we can restrict (if necessary) the neighborhood
W of s0 to ensure that K s, t ∈ U for every s ∈ W . Let f −1 the inverse of the map f : U → V . We claim that for every (s, t) ∈ W × J K (s, t) = f −1 (H (s, t)) .

(9.38)

Indeed observe that for every fixed s ∈ W , both K (s, t) and f −1 (H (s, t)) are lifting
of the path
t 7→ H (s, t) defined for J. However
t ∈−1

for t = t we have K s, t ∈ U and H s, t ∈ V so that K s, t = f H s, t . By Proposition 9.46 the two paths coincide for every t ∈ J. Since f −1 is continuous from (9.38) we obtain that K is continuous at (s0 , t) for every t ∈ J and this contradicts the minimality of t0 .

9.5.2

Proof of Theorem 9.42 (1): estimate on the Jacobian

We start with a general result which gives an estimate for the partial derivatives of the exponential map. Lemma 9.48 Let W1 , . . . , Wm be vector fields in Ω and let Ω0 b Ω. For every x ∈ Ω0 and suitably small z ∈ Rm define z · W = z1 W1 + · · · + zm Wm and Ψx (z) = exp (z · W ) (x) . Then, for any integer N there exist constants δ, c > 0 and α2 , . . . , αN such that for every x ∈ Ω0 and |z| < δ we have ( ) N ∂Ψ X x N (z) − (Wi )Ψx (z) + αk [z · W, [· · · , [z · W, Wi ]]]Ψx (z) 6 c |z| , ∂zi k=2

442

H¨ ormander operators

where in the term αk [z · W, [· · · , [z · W, Wi ]]] , the commutator has length k. The constants c, δ depend on N , d (Ω0 , ∂Ω) and the C h (Ω) norms of the coefficients of W1 , . . . , Wm , for some h depending on N . The constants α2 , . . . , αN only depend on N . Proof. Let us chose δ sufficiently small so that for x ∈ Ω and |z| < δ y = exp (z · W ) (x) 00

0

(9.39)

00

is in an open set Ω , with Ω b Ω b Ω. To compute d ∂Ψx (z) = (exp (z · W + hWi ) (x)) ∂zi dh h=0 we apply Theorem 9.23 to exp (z · W + hWi ) exp (−z · W ) (y) with y as in (9.39) and Xj = Wj for j = 1, . . . , m,

Xm+1 = Wi ,

(τ1 , . . . , τm+1 ) = (z1 , z2 , . . . , zm , h) , (σ1 , . . . , σm+1 ) = (−z1 , −z2 , . . . , −zm , 0) . Then, reducing the value of δ if necessary, for |z| 6 δ and |h| 6 δ, we have   N +1 N +1 |exp (z · W + hWi ) exp (−z · W ) (y) − exp (ZN ) (y)| 6 c |z| + |h| . (9.40) Also, since [z · W + hWi , z · W ] = [hWi , z · W ], we have ZN = hWi +

nk N X X

αkj Ckj (z · W, . . . , hWi )

k=2 j=1

Ckj

where every (. . .) is a commutator of length k containing at least one term z · W and one term hWi . If, in the previous sum we select only the commutators containing exactly one term hWi , we get the expression hWi + h

N X

αk Ck (z · W, . . . , z · W, Wi ) ≡ hRN

(9.41)

k=2

where every Ck (. . .) is a commutator of length k containing exactly (k − 1) terms z · W and one Wi . Note that the difference ZN − hRN is a sum of commutators containing at least two terms hWi , so that |ZN − hRN | 6 ch2 .

(9.42)

Now, let ψ (z, h) = exp (z · W + hWi ) exp (−z · W ) (y) − exp (ZN ) (y) and observe that ψ (z, 0) = 0. Also, by (9.40), all the partial  of ψ (z, h)  derivatives N N so that up to the order N vanish at (0, 0). Hence |∂u ψ (z, u)| 6 c |z| + |u| Z h   N N |ψ (z, h)| 6 |∂u ψ (z, u)| du 6 c |h| |z| + |u| . 0

More geometry of vector fields: metric balls and equivalent distances

443

Note also that by (9.42), |exp (ZN ) (y) − exp (hRN ) (y)| 6 c |ZN − hRN | 6 ch2 therefore |exp (z · W + hWi ) exp (−z · W ) (y) − exp (hRN ) (y)| 6 |exp (z · W + hWi ) exp (−z · W ) (y) − exp (ZN ) (y)| + ch2     N N N 6 c |h| |z| + |h| + ch2 6 c |h| |z| + h2 . Recalling that y = exp (z · W ) (x) we obtain   N |exp (z · W + hWi ) (x) − exp (hRN ) exp (z · W ) (x)| 6 c |h| |z| + h2 . Let now f (h) = exp (z · W + hWi ) (x) − exp (hRN ) exp (z · W ) (x) . N

Since f (h) = 0, we easily obtain |f 0 (0)| 6 c |z| . Since ∂Ψx d (exp (z · W + hWi ) (x)) (z) = ∂zi dh h=0 d 0 = f (0) + (exp (hRN ) exp (z · W ) (x)) = f 0 (0) + (RN )Ψx (z) , dh h=0 using the definition of RN (see (9.41)) we finally obtain ( ) N ∂Ψ X x N (z) − (Wi )Ψx (z) + αk [z · W, [· · · , [z · W, Wi ]]]Ψx (z) 6 c |z| . ∂zi k=2

Next, we apply the previous lemma to get an expansion for the Jacobian of the map ΦB,x,v defined in (9.35) and appearing in Theorem 9.42. Here we keep the notation introduced at the beginning of section 9.5. In particular recall that we have fixed Ω0 b Ω and δ0 ∈ (0, 1) so that ΦB,x,v (u) is well defined for u ∈ Qδ and v ∈ Q0δ . Lemma 9.49 For every 0 such that if XB =  κ, t ∈ (0, 1) there exists ε > 0 X[I1 ] , X[I2 ] , . . . , X[In ] is a t-suboptimal base at x ∈ Ω with radius δ ∈ (0, δ0 ), then for u ∈ Qεδ and v ∈ Q0εδ we have n X

∂ΦB,x,v (u) = X[Ii ] Φ + bij (u, v) X[Ij ] Φ (u) B,x,v B,x,v (u) ∂ui j=1

for 1 6 i 6 n, with |bij (u, v)| 6 κδ |Ij |−|Ii | . The constant ε depends on κ, t, ∆Ω0 and the local regularity of the vector fields.

444

H¨ ormander operators

Proof. Let J1 , . . . , Jm be any enumeration of the multiindices of weight 6 s. Reducing the value of δ0 defined in Notation 9.41 if necessary, we can apply Lemma 9.48 with Wi = X[Ii ] , Wn+i = X[Ji ] ,

zi = ui , zi+n = vi ,

i = 1, . . . , n, i = 1, . . . , m.

Then, for |u| 6 δ0 and |v| 6 δ0
∂ΦB,x,v (u) − X[Ii ] Φ B,x,v (u) ∂ui N X     = αk u · XB + v · X, · · · , u · XB + v · X, X[Ii ] Φ

(9.43)

B,x,v

  N N + O |u| + |v| . (u)

k=2

We will show that for a suitable ε, if u ∈ Qεδ and v ∈ Q0εδ then both the sum and the error term satisfy the required estimate. We start with the error term which is a vector that, for any i = 1, . . . , n, we can represent in the base XB as fi (u, v) =

n X

cij (u, v) X[Ij ]


ΦB,x,v (u)

.

j=1

  N N Since fi (u, v) = O |u| + |v| , solving in terms of cij by Cramer’s rule we find N

|cij (u, v)| 6 c

N

|u| + |v| |λB (ΦB,x,v (u))|

with c independent of t, δ. Let N = ns+s and let ε be half of the corresponding value of Theorem 9.28, then since u ∈ Qεδ and v ∈ Q0εδ , by (9.36), we have BB (x, εδ, v) ⊆ B # (x, 2εδ) and by Theorem 9.28 (a) and (9.31), the above quantity is bounded by   N N |u| + |v| c cεns+s s N 6 (εδ) = δ 6 κδ |Ij |−|Ii | c |λB (x)| ∆Ω0 tδ ns ∆Ω0 t for ε small enough, depending on κ, t, ∆Ω0 and c, which in turn also depends on the C h (Ω) norms of the coefficients of X0 , . . . , Xq for some fixed h. We now estimate the sum in (9.43) for N = ns + s. Observe that the iterated commutators can be expressed as a finite sum of terms of the kind     z1 X[K1 ] , · · · , zk−1 X[Kk−1 ] , X[Ii ] where zi is either some uji or vji and Ki are suitable multiindices. In both cases |K | |zi | 6 (εδ) i . By Proposition 9.32, we can write n     X z1 X[K1 ] , · · · , zk−1 X[Kk−1 ] , X[Ii ] = z1 · · · zk−1 alK1 ,...Kk−1 ,Ii X[Il ] . l=1

Shrinking ε if necessary, we can apply Corollary 9.40 (recall that k 6 N which is now fixed) so that for y = ΦB,x,v (u) ∈ BB (x, εδ, v) ⊂ B # (x, 2εδ) ,

More geometry of vector fields: metric balls and equivalent distances

445

we have l aK1 ,...Kk−1 ,Ii (y) 6 Ct−nN δ |Il |−(|K1 |+···+|Kk−1 |+|Ii |) so that |K |+···+|Kk−1 | Ct−nN δ |Il |−(|K1 |+···+|Kk−1 |+|Ii |) z1 · · · zk−1 alK1 ,...Kk−1 ,Ii (y) 6 (εδ) 1 6 ε|K1 |+···+|Kk−1 | Ct−nN δ |Il |−|Ii | . Choosing ε sufficiently small (depending on κ and t) we obtain |bij (u, v)| 6 κδ |Ij |−|Ii | . The next proposition proves point (1) of Theorem 9.42. Proposition 9.50 For every t ∈ (0, 1) there exists ε > 0 such that if XB is a tsuboptimal base at x ∈ Ω0 with radius δ ∈ (0, δ0 ) then for u ∈ Qεδ and v ∈ Q0εδ we have 1 |λB (x)| 6 det JΦB,x,v (u) 6 4 |λB (x)| 4 where JΦB,x,v stands for the Jacobian of the map ΦB,x,v . The constant ε depends on t, ∆Ω0 and the local regularity of the vector fields. Proof. By Lemma 9.49 we have n X

∂ΦB,x,v bij (u, v) X[Ij ] Φ (u) = X[Ii ] Φ + (u) B,x,v B,x,v (u) ∂ui j=1 =

n X

(δij + bij (u, v)) X[Ij ]


ΦB,x,v (u)

j=1

where |bij (u, v)| 6 κδ |Ij |−|Ii | if u ∈ Qεδ and v ∈ Q0εδ , with κ to be chosen later and ε depending on κ. Then det JΦB,x,v (u) = det [δij + bij (u, v)] · λB (ΦB,x,v (u)) . Now, in order to evaluate det [δij + bij (u, v)], observe that the determinant does not change if we multiply the j-th column by δ −|Ij | (for j = 1, 2, . . . , n) and the i-th row by δ |Ii | (i = 1, . . . , n). Therefore h i det [δij + bij (u, v)] = det δij + δ −|Ij |+|Ii | bij (u, v) and since δ −|Ij |+|Ii | |bij (u)| 6 κ, it is immediate to recognize that det ((δij + bij (u))) = 1 + O (κ) ,

(9.44)

with the implicit constant only depending on n. Choosing κ small enough we obtain 1 |λB (ΦB,x,v (u))| 6 det JΦB,x,v (u) 6 2 |λB (ΦB,x,v (u))| 2 provided that u ∈ Qεδ and v ∈ Q0εδ with ε small enough also depending on κ. The proposition follows since by Theorem 9.28 (a) 1 |λB (x)| 6 |λB (ΦB,x,v (u))| 6 2 |λB (x)| . 2

446

9.5.3

H¨ ormander operators

Proof of Theorem 9.42 (2): the ball-box theorem

Throughout this section we keep Notation 9.41. By Proposition 9.50 the Jacobian JΦB,x,v (u) is nonsingular for every u ∈ Qεδ (provided that v ∈ Q0εδ ). It follows that ΦB,x,v is locally invertible at any point u ∈ Qεδ . In the next lemma we estimate the derivatives of this local inverse. Lemma 9.51 For every t ∈ (0, 1) there exist ε, C > 0 such that if XB is a tsuboptimal base at x ∈ Ω0 of radius δ ∈ (0, δ0 ), then for u ∈ Qεδ , v ∈ Q0εδ if ψ = (ψ1 , ψ2 , . . . ψn ) is the local inverse of ΦB,x,v defined in a neighborhood of y = ΦB,x,v (u), we have X[I ] ψk (y) 6 Cδ |Ik |−|Il | l for k, l = 1, . . . , n. The constants ε, C depend on t, ∆Ω0 and the local regularity of the vector fields. We want to stress that the map ψ may depend on the point y: for every point y in a fixed neighborhood we bound a (possible different) local inverse ψ, which is defined on some neighborhood of that y. The relevant fact is that, nevertheless, the constants C, ε can be bounded in terms of the specified quantities. Proof. Let κ > 0 to be chosen later. By Lemma 9.49 there exists ε (depending on κ, t) such that if u ∈ Qεδ and v ∈ Q0εδ then n

X
∂ΦB,x,v (u) = (δij + bij (u, v)) X[Ij ] Φ B,x,v (u) ∂ui j=1 with |bij (u, v)| 6 κδ |Ij |−|Ii | . We now want to solve this system of equations in terms of X[Ij ] . By (9.44), we can choose κ is small enough so that det [δij + bij ] >

1 . 2

(9.45)

Then, solving this system we obtain X[Ii ]


ΦB,x,v (u)

=

n X j=1

i+j

with cij (u, v) = (−1)

det Mji det[δij +bij ] ,

cij (u, v)

∂ΦB,x,v (u) ∂uj

(9.46)

where Mji denotes the minor obtained from the

matrix [δlk + blk ] removing the j-th row and the i-th column. To evaluate det Mji we argue as in Proposition 9.50 and we multiply the k-th column  of the matrix [δlk+ blk ] by δ −|Ik | and the l-th row by δ |Il | obtaining the matrix δlk + δ −|Ik | δ |Il | blk . Let 0 Mji the minor obtained removing the i-th row and the j-th column from this last matrix. Clearly 0 det Mji = δ |Ij | δ −|Ii | det Mji .

More geometry of vector fields: metric balls and equivalent distances

447

0 6 C so that |det Mji | 6 Since δ −|Ij | δ |Ii | |bij | 6 κ we easily obtain det Mji Cδ |Ij | δ −|Ii | and by (9.45) |cij (u, v)| 6 2Cδ |Ij |−|Ii | .

(9.47)

Finally by (9.46) we have X[Il ] ψk (y) = X[Il ] =

n X




· ∇ψk (y) = X[Il ] y

clj (u, v)

j=1


ΦB,x0 (u)

· (∇ψk ) (ΦB,x,v (u))

∂ΦB,x,v (u) · (∇ψk ) (ΦB,x,v (u)) ∂uj

however, since ψk (ΦB,x,v (u)) = uk we have δjk =

∂ΦB,x,v ∂ (ψk (ΦB,x,v (u))) = (∇ψk ) (ΦB,x,v (u)) · ∂uj ∂uj

and therefore X[Il ] ψk (y) = clk (u, v) which by (9.47) implies the assertion. The following result, together with (9.36) implies point (2) of Theorem 9.42: Proposition 9.52 For t ∈ (0, 1) let ε be as in Proposition 9.50. There exists a positive constant κ > 0 such that if XB is a t-suboptimal base at x ∈ Ω0 with radius δ ∈ (0, δ0 ), y ∈ B ∗ (x0 , κεδ) and z = ΦB,x,v (0) with v ∈ Q0εδ then the map ΦB,x,v : Qεδ → Ω lifts every path belonging to

∗ Cz,y

(κεδ) (see Definition 9.45). In particular

B ∗ (z, κεδ) ⊆ BB (x, εδ, v)

(9.48)

and for v ∈ Q0κ εδ 3

 κ  (9.49) B ∗ x, εδ ⊆ BB (x, εδ, v) . 3 The constant κ depends on t, ∆Ω0 and the local regularity of the vector fields (but not on x and δ). ∗ (κεδ) (with κ to be chosen later). Recall Proof. Let y ∈ B ∗ (z, κεδ) and ϕ ∈ Cz,y that this means that

ϕ (0) = z, ϕ (1) = y X
ϕ0 (t) = aI (t) X[I] ϕ(t) |I|6s

with |I|

|aI (t)| 6 (κεδ)

.

Let v ∈ Q0εδ , we want to construct a lifted path ϑ : [0, 1] → Q 12 εδ such that, ϕ (s) = ΦB,x,v (ϑ (s)) = exp (ϑ (s) · XB + v · X) (x) for every s ∈ [0, 1]

(9.50)

448

H¨ ormander operators

and ϑ (0) = 0. This in particular implies that y = ϕ (1) = exp (ϑ (1) · XB + v · X) (x) ∈ BB (x, εδ, v) , that is (9.48). Observe that by Proposition 9.46 the lifted path ϑ is unique. By Proposition 9.50, det JΦB,x,v (0) > 41 |λB (x)| > 0 so that ΦB,x,v is invertible in a −1 neighborhood of 0 and for small values of s, we can define ϑ (s) = (ΦB,x,v ) (ϕ (s)). We will show that it is possible to extend ϑ to the whole interval [0, 1]. We start by showing that for κ > 0 small enough (hence, for ϕ close enough to z), whenever ϑ : [0, s0 ] → Rn is an absolutely continuous map satisfying ϕ (s) = ΦB,x,v (ϑ (s)) in [0, s0 ] , then for every s ∈ [0, s0 ], ϑ (s) ∈ Q 21 εδ . Indeed, assume by contradiction that this is not true and let s1 ∈ (0, s0 ] be smallest value such that for some j0 = 1, . . . , n  |Ij0 | 1 |ϑj0 (s1 )| = εδ . 2 Let now s < s1 . Since ϑ (s) ∈ Q 1 εδ , by Proposition 9.50 det JΦ (ϑ (s)) 6= 0 so B,x,v

2

that ΦB,x,v is locally invertible in a neighborhood of ϑ (s). Denoting with Ψs such inverse, we have ϑ (τ ) = Ψs (ϕ (τ )) for every τ in a neighborhood of s. It follows that ϑ0 (τ ) = JΨs (ϕ (τ )) · ϕ0 (τ ) and in particular ϑ0 (s) = JΨs (ϕ (s)) · ϕ0 (s) . Using the fact that n X X X

aI (t) alI (ϕ (t)) X[Il ] ϕ(t) ϕ0 (s) = aI (t) X[I] ϕ(t) = |I|6s

|I|6s

l=1




with XB = X[I1 ] , . . . , X[In ] , we obtain 0

ϑ (s) = J

Ψs

0

(ϕ (s)) · ϕ (s) =

X |I|6s

aI (t)

n X


alI (ϕ (t)) X[I` ] Ψs (ϕ (t))

l=1

so that 

|Ij0 | Z s1 1 εδ = ϑj0 (s1 ) = ϑ0j0 (s) ds 2 0 Z s1 X n X
= aI (s) alI (ϕ (s)) X[Il ] Ψsj0 (ϕ (s)) ds 0

|I|6s

l=1

denotes the j0 -th component of the map Ψs . Since, for every s ∈ (0, s1 ),   1 ϕ (s) ∈ BB x, εδ, v ⊆ B # (x, εδ) , 2 l by Corollary 9.40 we have aI (ϕ (s)) 6 Ct−n δ |Il |−|I| . Also, by Lemma 9.51,
X[I ] Ψsj (ϕ (s)) 6 Cδ |Ij0 |−|Il | l 0

where

Ψsj0

(9.51)

More geometry of vector fields: metric balls and equivalent distances

449

so that, using (9.50), by (9.51) we obtain  |Ij0 | n Z s1 XX 1 |I| εδ 6 (κεδ) Ct−n δ |Il |−|I| Cδ |Ij0 |−|Il | ds 2 0 |I|6s l=1

= Ct−n

X

Z

|I|6s

s1

|I|

(κε)

n

δ |Ij0 | ds 6 Ct−n κεδ |Ij0 |
max |λB0 (x; σ)| δ |B | , 0

(9.53)

B

∀ (y, σ) ∈ U (x) × V (σ)

|λB (y; σ)| δ

∀ (y, v, σ) ∈ U (x) × Q0δ × V (σ) ,

453

|B |

> t max |λB0 (y; σ)| δ 0

|B0 |

B

,

ΦB,y,v,σ (u) is one-to-one on QB . δ

(9.54) (9.55)

B 0 = (J1 , J2 , . . . , Jn ) with Let B = B (x; σ) be a n-tuple chosen among all n-tuples |Ji | 6 s and λB0 (x; σ) 6= 0 in such a way that B is minimal and (9.52) holds. We claim that there exists δ such that (9.53) hold. Indeed, if |B 0 | = B , then 0

|λB (x; σ)| δ |B| > |λB0 (x; σ)| δ |B | holds for every δ > 0. If, on the other hand, |B 0 | > B , we have 0

0

δ |B | |λB0 (x; σ)| = δ |B| |λB (x; σ)| δ |B |−|B|

|λB0 (x; σ)| |λB (x; σ)|

6 δ |B| |λB (x; σ)| provided that |λB (x; σ)| |B0 |−|B| ≡δ . max |λB0 (x; σ)| 0 |B |>|B|   In order to show (9.55), let F σ (u, v, y) = ΦB,y,v,σ (u) , v, y . Then, 0

δ |B |−|B| 6



∂ ∂u ΦB,y,v,σ

(u) 0 0



 ∂  JF σ (u, v, y) =   ∂v ΦB,y,v,σ (u) I 0 ∂ ∂y ΦB,y,v,σ (u) 0 I and therefore det JF σ (u, v, y) = det JΦB,y,v,σ (u). Since ΦB,x,0,σ (0) = x we have      σ σ , . . . , X[I 6 0. det JΦB,x,0,σ (0) = det X[I = |λB (x; σ)| = 1] n] x

x

Moreover, by Corollary 9.54, the function JF σ (u, v, y) is continuous in the joint variables (u, v, y, σ). Therefore, for any assigned ε > 0 there exists a neighborhood of (0, 0, x, σ) which, possibly reducing the value of δ, we can assume of the form QB × Q0δ × U (x) × V (σ), such that for every (u, v, y, σ) ∈ QB × Qδ0 × U (x) × V (σ) δ δ we have kJF σ (u, v, y) − JF σ (0, 0, x)k < ε. By the standard proof of the inverse function theorem (see e.g. [143, pp. 221-222]) × Q0δ × this implies that for every σ ∈ V (σ) the map F σ (u, v, y) is invertible in QB δ , for U (x). In turn, this implies that the map u 7→ ΦB,y,v,σ (u) is one-to-one on QB δ

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H¨ ormander operators

every (y, σ) ∈ U (x) × V (σ), and v ∈ Qδ0 which is (9.55). Finally, since for every n-tuple B 0 |λB (x; σ)| δ

|B |

> |λB0 (x; σ)| δ

|B0 |

,

shrinking U (x) × V (σ) enough we can also assume that (9.54) holds for every y ∈ U (x). Using the compactness of K × Σ we can now select a finite number of points (x1 , σ 1 ) , . . . , (xN , σ N ), positive numbers δ 1 , . . . , δ N , n-tuples B 1 , . . . , B N and neighborhoods U (x1 ) × V (σ 1 ) , . . . , U (xN ) × V (σ N ) that cover K × Σ, where (9.52), (9.53), (9.54) and (9.55) hold. For the time being, let us fix δ ∗ = mink δ k . In the following we will reduce this value if necessary. Let (x, σ) ∈ K × Σ, 0 < δ < δ ∗ and consider a t-suboptimal base B at x with radius δ. We have to prove that for a suitable constant κ independent B ∗ 0 of B and x, the map ΦB,x,v,σ
is one-to-one on Qκδ for 0 < δ < δ and v ∈ Qκδ . Assume that (x, σ) ∈ U xk × V (σ k ) for some k, and let B0 = B k and δ0 = δ k . For a given n-tuple B 0 the set n o 0 ∆B0 ,x,σ = µ > 0 : |λB0 (x; σ)| µ|B | > t max |λC (x; σ)| µ|C| C o \n 0 B = µ > 0 : |λB0 (x; σ)| µ| | > t |λC (x; σ)| µ|C| C

is a finite intersection of intervals, closed in (0, +∞) and therefore it is an interval, closed in (0, +∞). Note that [ ∆B0 ,x,σ = (0, +∞) (9.56) B0

since for any fixed µ, if we choose B 0 such that 0

|λB0 (x; σ)| µ|B | = max |λC (x; σ)| µ|C| C

∗

then µ ∈ ∆B0 ,x . Since δ < δ = min δ k we have in particular δ < δ0 . Since δ ∈ ∆B,x,σ and δ0 ∈ ∆B0 ,x,σ , using (9.56) and the fact that the intervals ∆B0 ,x are closed we can construct a finite sequence of intervals ∆B0 ,x,σ , ∆B1 ,x,σ , . . . , ∆BM ,x,σ , with ∆BM ,x.σ = ∆B,x,σ , in such a way that for k = 0, . . . , M − 1 the interval ∆Bk+1 ,x,σ is more to the left of ∆Bk ,x,σ and ∆Bk+1 ,x,σ ∩ ∆Bk ,x,σ 6= ∅. In particular denoting with δk the left end point of ∆Bk ,x,σ we have δM 6 · · · 6 δ1 6 δ0 so that [δk+1 , δk ] ⊂ ∆Bk ,x,σ , hence |λBk (x; σ)| µ|Bk | > t max |λC (x; σ)| µ|C| C

for µ ∈ [δk+1 , δk ] .

(9.57)

0 0 Now, we know that ΦB0 ,x,v,σ is one-to-one on QB δ0 , for every v ∈ Qδ0 and we want B 0 to show that ΦB,x,v,σ is one-to-one on Qκδ for every v ∈ Qκδ for a suitable κ that is independent of B and (x, σ) (but depends on the compact set K × Σ that contains

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(x, σ)). The idea is to transfer this property from ΦB0 ,x,v,σ to ΦB1 ,x,v,σ , then from ΦB1 ,x,v,σ to ΦB2 ,x,v,σ , . . . , until we reach ΦB,x,v,σ . At every step we will have to reduce the value of the constant κ, but this is not a problem in view of the finite number of steps. From now on we can also avoid pointing out the dependence on the parameter σ, since this dependence will be automatically uniform, for the reason remarked before the statement of this Proposition. 0 1 Let us start by showing that ΦB1 ,x,v is one-to-one on QB c1 δ1 for every v ∈ Qc1 δ1 for some c1 only depending on the compact set K × Σ. By (9.57) with k = 0 and k = 1 we have |B0 |

|λB0 (x)| δ1

|C|

> t max |λC (x)| δ1 C

and |B1 |

|λB1 (x)| δ1

|C|

> t max |λC (x)| δ1 . C

Now, since B1 is t-suboptimal at x with radius δ1 , by Proposition 9.52 there exist ε1 , κ1 > 0 such that for v ∈ Q01 κ1 ε1 δ1 and every σ we have 3   1 ∗ BB1 (x, ε1 δ1 , v) ⊇ B x, κ1 ε1 δ1 . 3 Using (9.36) we obtain BB1 (x, ε1 δ1 , v) ⊇ B

∗



1 x, κ1 ε1 δ1 3



  1 ⊇ BB0 x, κ1 ε1 δ1 , 0 6

(9.58)

Let now c1 = ε1 and c01 6 16 κ1 ε1 . Since B0 is t-suboptimal at x with radius δ1 it is also t0 -suboptimal at x with radius c01 δ1 for some smaller t0 . Namely: 1 1 1−|B0 | |B | |C| |B | |λB0 (x)| (c01 δ1 ) 0 max |λC (x)| (c01 δ1 ) 6 c01 |λB0 (x)| δ1 0 = (c01 ) C t t |B |−1

hence t0 = t (c01 ) 0 . In the following we shall use both Lemma 9.49 and Propo|B |−1 sition 9.52 (applied to the map ΦB0 ,x,v ) with t0 = t (c01 ) 0 and κ = 1. Let ε2 be the smallest of the corresponding numbers appearing in these two results. Then, by (9.58), BB1 (x, c1 δ1 , v) ⊇ BB0 (x, c01 δ1 , 0) ⊇ BB0 (x, ε2 c01 δ1 , 0) . By Proposition 9.52 there exists κ2 > 0 such that, again by (9.36),   1 BB0 (x, ε2 c01 δ1 , 0) ⊇ B ∗ (x, κ2 ε2 c01 δ1 ) ⊇ BB1 x, κ2 ε2 c01 δ1 , v . 3 So we have proved the inclusions BB1 (x, c001 δ1 , v) ⊆ BB0 (x, ε2 c01 δ1 , 0) ⊆ BB1 (x, c1 δ1 , v) c001

1 0 3 κ2 ε2 c1

c01

(9.59)

can be arbitrary small. Let us show that these imply where = and 0 1 that ΦB1 ,x,v is one-to-one on QB c00 δ1 for every v ∈ Qc00 δ1 . Suppose, there exist 1

00 1 u, u0 ∈ QB c00 δ1 and y ∈ BB1 (x, c1 δ1 ) such that 1

y = ΦB1 ,x,v (u) = ΦB1 ,x,v (u0 )

1

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H¨ ormander operators

and consider the segment r (s) = (1 − s) u + su0 . Its image γ (s) = ΦB1 ,x,v (r (s)) is a closed path in BB1 (x, c001 δ1 , v) ⊆ BB0 (x, ε2 c01 δ1 , 0). Since by (9.55) 0 0 ΦB0 ,x,0 : QB ε2 c0 δ1 → BB0 (x, ε2 c1 δ1 , 0) 1

is invertible, we can deform γ (s) to the point y using the homotopy   −1 H (s, µ) = ΦB0 ,x,0 µΦ−1 B0 ,x,0 (y) + (1 − µ) ΦB0 ,x,0 (γ (s)) , for s, µ ∈ [0, 1]. Indeed,   H (s, 0) = ΦB0 ,x,0 Φ−1 B0 ,x,0 (γ (s)) = γ (s) while   H (s, 1) = ΦB0 ,x,0 Φ−1 (y) = y. B0 ,x,0

(9.60)

B0 −1 Since Φ−1 B0 ,x,0 (y) and ΦB0 ,x,0 (γ (s)) are in Qε2 c01 δ1 which is convex, by (9.59),   0 0 H (s, µ) ∈ ΦB0 ,x,0 QB ε2 c0 δ1 = BB0 (x, ε2 c1 δ1 , 0) ⊆ BB1 (x, c1 δ1 , v) 1

for every (s, µ) ∈ [0, 1] × [0, 1]. We are going to show that for every fixed s ∈ [0, 1] 1 and every v ∈ Q0c00 δ1 we can construct a unique path K (s, µ) in QB c1 δ1 such that 1

K (s, 0) = r (s) ΦB1 ,x,v (K (s, µ)) = H (s, µ)

(9.61) for every µ ∈ [0, 1] .

(9.62)

The following argument is similar to that in the proof of Proposition 9.52. For a fixed s, let ψB1 the local inverse of ΦB1 ,x,v defined in a neighborhood of r (s) . The path K (s, µ) can be defined for small values of µ by K (s, µ) = ψB1 (H (s, µ)) .

(9.63)

Let n o 1 Σ = µ1 ∈ [0, 1] : ∃K (s, ·) : [0, µ1 ] → QB c1 δ1 satisfying (9.61) and (9.62) . Let µ = sup Σ. First of all we show that, up to reducing the value of c01 , K (s, µ) ∈ QB1 1c δ for every µ ∈ [0, µ]. Assume by contradiction that this is not true and let µ1 2 1 1
| Ij | the smallest value of µ such that for some j0 = 1, . . . , n, Kj0 (s, µ) = 12 c1 δ1 0 . Then, |Ij0 | Z µ1  ∂Kj0 1 c1 δ 1 (s, µ) dµ + K (s, 0)j0 . = 2 ∂µ 0 Let B0 = (J1 , J2 , . . . , Jn ) and B1 = (I1 , I2 , . . . , In ). Also, to simplify notation, let −1 h = µΦ−1 B0 ,x,0 (y) + (1 − µ) ΦB0 ,x,0 (γ (s))

and h=

∂h −1 = Φ−1 B0 ,x,0 (y) − ΦB0 ,x,0 (γ (s)) . ∂µ

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457

B0 −1 Since both Φ−1 B0 ,x,0 (y) and ΦB0 ,x,0 (γ (s)) are in Qε2 c01 δ1 , we have hi 6 2 (ε2 c01 δ1 )|Ji | 6 2 (c01 δ1 )|Ji | . 0 Since B0 is t0 -suboptimal of radius c01 δ1 and h ∈ QB ε2 c0 δ1 , by Lemma 9.49 1

n

X
∂ΦB0 ,x,0 (h) = (δij + bij ) X[Jj ] Φ B0 ,x,0 (h) ∂ui j=1 with |bij | 6 (c01 δ1 )

|Jj |−|Ji |

, hence

n n X X
∂H (s, µ) = hi (δij + bij ) X[Jj ] Φ . B0 ,x,0 (h) ∂µ i=1 j=1

On the other hand, if ψ = (ψ1 , ψ2 , . . . , ψn ) is any local inverse of ΦB1 ,x,v such that (9.63) holds, then ∂K ∂H (s, µ) = Jψ (H (s, µ)) (s, µ) ∂µ ∂µ n X n X
hi (δij + bij ) Jψ (H (s, µ)) X[Jj ] Φ = =

i=1 j=1 n X n X

B0 ,x,0 (h)

hi (δij + bij ) X[Jj ] ψ (H (s, µ)) .

i=1 j=1

Since ΦB0 ,x,0 (h) ∈ BB0 (x, c01 δ1 , 0) ⊆ B # (x, 2c01 δ1 ) ⊆ B # (x, δ1 ) , by Corollary 9.40 we have X[Jj ]


ΦB0 ,x,0 (h)

=

n X

alJj X[Il ]

l=1


ΦB0 ,x,0 (h)

|Il |−|Jj | |I |−|Jj | 6 C2 δ1 l . Hence with alJj 6 C1 (δ1 ) tn n

n

n

XXX ∂Kj0 (s, µ) = hi (δij + bij ) alJj X[Il ] ψj0 (H (s, µ)) . ∂µ i=1 j=1 l=1

Since H (s, µ) ∈ BB1 (x, c1 δ1 , v) and v ∈ Q0c00 δ1 , by Lemma 9.51 we have 1

X[I ] ψj0 (H (s, µ)) 6 C3 δ |Ij0 |−|Il | . 1 l It follows that hi (δij + bij ) alJj X[Il ] ψj0 (H (s, µ))   |Ij |−|Il | |J | |J |−|Ji | |I |−|Jj | 6 2 (c01 δ1 ) i δij + (c01 δ1 ) j C2 δ1 l C3 δ1 0 |Ji |

6 2 (c01 δ1 )

(c01 δ1 )

|Jj |−|Ji |

|I |−|Jj |

(δij + 1) C2 δ1 l

|Ij |−|Il | |J | |Ij | C3 δ1 0 6 C (c01 ) j δ1 0

458

H¨ ormander operators

and therefore  |Ij0 | Z µ1 n X 1 ∂Kj0 |J | I c1 δ1 (s, µ) dµ + r (0)j0 6 µ1 C (c01 ) i (δ1 )| j0 | + uj0 = 2 ∂µ 0 i=1 ! n X |Ij | I |J | 6 δ 0 (c00 )| j0 | + C (c0 ) i . 1

1

1

i=1 c001 6 c01 )

Taking c01 small enough (recall that gives a contradiction. This shows that K (s, µ) ∈ QB1 1c δ for every µ ∈ [0, µ]. It remains to show that µ = 1. This can 2 1 1 be done exactly as in the second part of the proof of Proposition 9.52. Applying Theorem 9.47, we conclude that K (s, µ) is actually continuous on [0, 1] × [0, 1]. Since   H (0, µ) = ΦB0 ,x,0 Φ−1 B0 ,x,0 (y) = y

we have ΦB1 ,x,v (K (0, µ)) = H (0, µ) = y. Since ΦB1 ,x,v is a local diffeomorphism this implies that K (0, µ) is constant and therefore K (0, µ) = K (0, 0) = r (0) = u. Similarly H (1, µ) = y so that ΦB1 ,x,v (K (1, µ)) = H (1, µ) = y and therefore K (1, µ) is constant. It follows that K (1, µ) = K (1, 0) = r (1) = u0 . Finally by (9.60) ΦB1 ,x.v (K (s, 1)) = H (s, 1) = y so that K (s, 1) is constant in s. Since K (0, 1) = u and K (1, 1) = u0 we obtain 1 for every v ∈ Q0c0 δ1 . We u = u0 . This proves that ΦB1 ,x,v is one-to-one on QB c00 1 1 δ1 can now conclude the proof by the iterative procedure already described. Since |B2 |

|λB2 (x)| δ2

|C|

> t max |λC (x)| δ2 C

and |B2 |

|λB1 (x)| δ2

|C|

> t max |λC (x)| δ2 , C

2 arguing as in the previous step we can prove that ΦB2 ,x,v is one-to-one on QB c2 δ2 for 0 every v ∈ Qc2 δ2 . Iterating M times (observe that M is bounded by the number of n-tuples available), and recalling that BM = B, we can show that ΦB,x,v is one-toB 0 one on QB cM δM ⊇ QcM δ for every v ∈ QcM δ , for a constant cM only depending on the compact set K.

9.6

Local equivalence of the distances d, d∗

We now come to the second main result of this chapter, that is the local equivalence of the distance d∗ with the control distance d. By Proposition 9.7, we know that for any Ω0 b Ω there exist positive constants c1 , c2 such that c1 |x − y| 6 d∗ (x, y) 6 c2 |x − y|

1/s

for any x, y ∈ Ω0

while by the results proved in Chapter 1, we also know that 1/s

c3 |x − y| 6 d (x, y) 6 c4 |x − y|

for any x, y ∈ Ω0 .

Note that the last inequality expresses in a quantitative way the connectivity property with respect to the integral lines of a family of H¨ormander vector fields. Since we already know that d∗ (x, y) 6 d (x, y), our main task is to prove the following

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459

Theorem 9.56 For every Ω0 b Ω0 b Ω there exist constants δ0 , C > 0 such that for every x, y ∈ Ω0 with d∗ (x, y) < δ0 we have d (x, y) 6 Cd∗ (x, y) . The constants δ0 , C depend on ∆Ω0 and the local regularity of the vector fields. The proof mainly relies on the following Lemma 9.57 There exist constants δ0 , C > 0 such that for 0 < ε < δ0 , if z1 , z2 ∈ Ω0 and d∗ (z1 , z2 ) 6 ε, then there exists z3 ∈ Ω with 1

d (z1 , z3 ) < Cε and d∗ (z3 , z2 ) < Cε1+ s . The constants δ0 , C depend on ∆Ω0 and the local regularity of the vector fields. Proof. By Theorem 9.42, point (2) there exist constants C, δ0 > 0 such that for every ε < δ0 and z1 , z2 ∈ Ω0 , d∗ (z1 , z2 ) 6 ε implies ρ# (z1 , z2 ) < Cε which means |I| there exist constants {aI }|I|6s such that |aI | 6 (Cε) and X  z2 = exp aI X[I] (z1 ) . |I|6s M

Applying Theorem 9.25 to these constants {aI }|I|6s we find constants {cj }j=1 , M
Q satisfying |cj | 6 cεpkj , such that, letting z3 = exp cj Xkj (z1 ), we have j=1

|z2 − z3 | 6 cεs+1 .

(9.64)

From the definition of z3 and the bound of the cj we read d (z1 , z3 ) 6 cε. On the 1/s other hand, (9.64) implies d∗ (z2 , z3 ) 6 c2 |z2 − z3 | 6 cε1+1/s , and we are done.

Proof of Theorem 9.56. Let x, y be any two points in Ω0 with d∗ (x, y) < δ0 . Applying Lemma 9.57 with ε = d∗ (x, y), z1 = x and z2 = y we find x2 ∈ Ω such that 1 1+ 1 d (z1 , x2 ) < Cd∗ (x, y) and d∗ (x2 , z2 ) < Cd∗ (x, y) s 6 d∗ (x, y) 2 provided d∗ (x, y) 6

1 s. (2C)

(9.65)

Since y ∈ Ω0 and d∗ (x2 , y) 6 δ20 , shrinking if necessary the number δ0 (depending on dist (Ω0 , ∂Ω0 )) we can assure that x2 ∈ Ω0 . Applying again the lemma with ε = 12 d∗ (x, y), z1 = x2 and z2 = y we find x3 ∈ Ω with  1+ 1s 1 ∗ 1 ∗ 1 ∗ d (x2 , x3 ) < C d (x, y) and d (x3 , z2 ) < C d (x, y) < d∗ (x, y) 2 2 4

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H¨ ormander operators

since, by (9.65),  1+ 1s  1+ 1s 1 1 1 1 1 1 ∗ s C d (x, y) 6 C = 2+ 1 < . 2 2 2C 4 s 2 Again, since d∗ (x3 , y) < (j = 2, 3, 4, . . .) such that

1 4 δ0 ,

x3 ∈ Ω0 . Proceeding this way we find xj ∈ Ω0

1 ∗ 1 d (x, y) and d∗ (xj , y) < j−1 d∗ (x, y) . 2j−2 2 Since d satisfies the triangle inequality, d (xj−1 , xj ) < C

d (x1 , xj ) 6

j X

d (xk−1 , xk ) 6

k=2

j X

C

k=2

1 ∗ d (x, y) < 2Cd∗ (x, y) . 2k−2

However, we know that c1 |xj − y| 6 d∗ (xj , y)
0 and x ∈ Ω0

(9.72)

which is somewhat different from what we actually know. If d were the Euclidean distance, condition (9.72) would hold provided Ω0 satisfies an interior cone condition; for instance, asking Ω0 to have Lipschitz boundary would be enough. In our more general context, the kind of regularity asked to Ω0 by (9.72) is less transparent. To discuss this issue, let us fix first of all some terminology.

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463

Definition 9.59 A domain Ω0 b Ω is said to be d-regular if for some constants c, r0 > 0 we have |B (x, r) ∩ Ω0 | > c |B (x, r)| for every x ∈ Ω0 , 0 < r 6 r0 . We have the following: Proposition 9.60 (a) If a domain Ω0 ⊂ Ω is d-regular, then condition (9.72) holds, hence (Ω0 , d, dx) is a space of homogeneous type. Also, (b) The union of a finite number of d-regular domains is d-regular. Proof. (a) By d-regularity and the doubling condition in Theorem 9.1 we have, for r 6 r ≡ min (δ0 , r0 ), |B (x, 2r) ∩ Ω0 | 6 |B (x, 2r)| 6 c |B (x, r)| 6 c0 |B (x, r) ∩ Ω0 | . On the other hand, for r > r, |B (x, 2r) ∩ Ω0 | |Ω0 | |Ω0 | 6 6 6 c00 |B (x, r) ∩ Ω0 | |B (x, r) ∩ Ω0 | c |B (x, r)| because, by Theorem 9.1 and Definition 9.13 (recalling that in the next sum B ranges over all the possible n-tuples of multiindices of weight 6 s), X |λB (x)| r|B| > c1 rns ∆Ω0 . inf |B (x, r)| > c1 inf x∈Ω0

x∈Ω0

B

(b) Let Ω1 , Ω2 be two d-regular domains, and pick x ∈ Ω1 ∪ Ω2 . Then, assuming for instance that x ∈ Ω1 , |Br (x) ∩ (Ω1 ∪ Ω2 )| > |Br (x) ∩ Ω1 | > c |Br (x)| , because Ω1 is d-regular. The same reasoning holds for n sets, with c the minimum of the constants corresponding to the sets Ωi . (Hence it is important the finiteness of the number of sets). Hence, the problem is to find some d-regular domain in Ω. This problem is related to the geometry of the integral lines of the vector fields Xi . To explain this, let us introduce the following notion: Definition 9.61 (Strong segment property) We say that (Ω, d) enjoys the strong segment property if, for any x, y ∈ Ω, if L = d (x, y), for every λ ∈ (0, L) there exists x ∈ Ω such that d (x, x) = λ and d (x, y) = L − λ. Definition 9.62 (Weak segment property) We say that (Ω, d) enjoys the weak segment property if, for any x, y ∈ Ω, if L = d (x, y), for every λ ∈ (0, L) and ε > 0 there exists x ∈ Ω such that λ − ε 6 d (x, x) 6 λ L − λ 6 d (x, y) 6 L − λ + ε.

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H¨ ormander operators

Although in the following we will need only the weak segment property, we have defined also the strong property because it has a more transparent geometric meaning. On the other hand, note that the validity of the strong segment property does not just require good local properties of the distance d, but also some global property of the set Ω: Example 9.63 Let Ω = R2 \ {(0, 0)} and d the Euclidean distance restricted to Ω, i.e. d (x, y) is the infimum of the length of curves contained in Ω and joining x, y. Then clearly d ((1, 0) , (−1, 0)) = 2 but there is not any point of Ω at distance 1 from both (1, 0) and (−1, 0). Hence (Ω, d) satisfies the weak but not the strong segment property. Theorem 9.64 (Regularity of control balls) If (Ω, d) enjoys the weak segment property, then any control ball B (x0 , R) b Ω is d-regular. Proof. Let B (x0 , R) be a metric ball, x ∈ B (x0 , R), d (x, x0 ) = ρ < R. Assume r < 3ρ and, applying the weak segment property with L = ρ, λ = ρ − 3r , ε = 3r , let x1 ∈ Ω such that 2 r ρ − r 6 d (x0 , x1 ) 6 ρ − 3 3 2 r 6 d (x1 , x) 6 r. 3 3 Then  r B x1 , ⊂ B (x0 , R) ∩ B (x, r) . 3
Namely, for z ∈ B x1 , 3r we have: r r d (z, x0 ) 6 d (x1 , x0 ) + d (x1 , z) < ρ − + = ρ < R 3 3 r 2 d (z, x) 6 d (z, x1 ) + d (x1 , x) < + r = r. 3 3 But then, since B (x0 , R) b Ω, by the doubling property in Theorem 9.1, for r 6 δ0 ,  r  |B (x0 , R) ∩ B (x, r)| > B x1 , > c |B (x1 , 2r)| > c |B (x, r)| 3 since d (x1 , x) 6 23 r ⇒ B (x, r) ⊂ B (x1 , 2r). If r > 3ρ the previous argument works taking x1 = x0 , that is, assuming r 6 R,  r ⊂ B (x0 , R) ∩ B (x, r) . B x0 , 3
Namely, for z ∈ B x0 , 3r we have: d (z, x) 6 d (z, x0 ) + d (x0 , x)
B x0 , > c |B (x0 , 2r)| > c |B (x, r)| 3 r since d (x0 , x) = ρ 6 3 ⇒ B (x, r) ⊂ B (x0 , 2r). Summarizing we have proved that |B (x, r) ∩ B (x0 , R)| > c |B (x, r)| for every x ∈ B (x0 , R) , 0 < r 6 min (R, δ0 ) , where δ0 is the constant appearing in Theorem 9.1 and depending on Ω and the set Ω0 b Ω with Ω0 ⊇ B (x0 , R). So B (x0 , R) is d-regular.

We are left to prove that the weak segment property actually holds in (Ω, d). This fact, however, cannot be easily proved in full generality: Theorem 9.65 If, in the set X1 , . . . , Xq of H¨ ormander vector fields, the drift term X0 of weight two is lacking, then (Ω, d) satisfies the weak segment property. Therefore, if the drift is lacking, every control ball B (x0 , R) b Ω is d-regular. Proof. Let x, y ∈ Ω with L = d (x, y) and let ε > 0, then there exists γ ∈ Cxy (L + ε). For λ ∈ (0, L) and t0 ∈ (0, 1), letting z = γ (t0 ) and γ e (t) = γ (t0 t), a simple computation shows that γ e ∈ Cxz (t0 (L + ε)) (see Remark 1.33) so that d (x, z) 6 t0 (L + ε) . ∗

Analogously, letting γ (t) = γ (t0 + (1 − t0 ) t), one can check that γ ∗ ∈ Czy ((1 − t0 ) (L + ε)) so that d (z, y) 6 (1 − t0 ) (L + ε) . Then the desired upper bounds d (x, z) 6 λ and d (y, z) 6 L − λ + ε are satisfied provided we set t0 (L + ε) = λ that is t0 =

λ L+ε

so that 

 λ (L + ε) = L − λ + ε. L+ε This also implies that the desired lower bounds hold: d (z, y) 6 (1 − t0 ) (L + ε) =

1−

λ − ε 6 d (x, z) and L − λ 6 d (z, y) because d (x, z) > d (x, y) − d (y, z) > L − (L − λ + ε) = λ − ε and d (z, y) > d (x, y) − d (x, z) > L − λ, so the weak segment property holds and, by Theorem 9.64, every control ball B (x0 , R) b Ω is d-regular.

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H¨ ormander operators

Remark 9.66 (Possible breakdown of the weak segment property) If the drift term X0 is present, the weak segment property may actually fail. This already happens for the parabolic distance in the plane. Let X1 = ∂x ; X0 = ∂y in R2 , with X0 having weight two. The control distance is just the parabolic distance  p  d ((x, y) , (0, 0)) = max |x| , |y| . Consider the points (0, 0) , (0, 1), at distance 1, and for some small λ > 0, take a point (x, y) at distance (of the order of ) λ from (0, 0). Then |y| 6 λ2 , hence p 1 d ((x, y) , (0, 1)) > 1 − λ2 ' 1 − λ2 , 2 which for small λ is much larger than 1 − λ. Remark 9.67 (Can d-regularity of control balls hold if the weak segment property does not?) The breaking down of the weak segment property does not automatically imply the breaking down of the d-regularity of control balls. For instance, in the above example of the parabolic distance in the plane, one can directly check that parabolic balls are d-regular. More generally, it can be proved that if X1 , . . . , Xq is a system of H¨ ormander vector fields in Ω ⊂ Rn and X0 = ±∂t , where t is an extra variable, as happens in the study of heat type H¨ ormander operators q X

Xi2 − ∂t

i=1

then, with respect to the weighted control distance d0 ((x, t) , (y, s)), which is equivalent to q 2 d (x, y) + |t − s| where d is the (unweighted) control distance of X1 , . . . , Xq , the control balls are actually d0 -regular. This has been proved in [28, Lemma 3.3]. Instead, proving in full generality the d-regularity of control balls in presence of a drift seems to be an open problem. This is the reason why, to prove a priori estimates for second orded operators with drift, we will use the theory of singular integrals in locally doubling spaces, while in absence of the drift the theory of doubling spaces would be fit. 9.8

Proof of the BCH formula for formal series and other consequences

The main goal of this section is to give a complete proof of the Baker-CampbellHausdorff formula stated in section 9.3.1 as Theorem 9.18. In order to do this, it is better to introduce now some more formal definition.

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467

Our basic setting is a noncommutative ring Rm of polynomials (with real coefficients) in m (possibly non commuting) indeterminates x1 , . . . , xm . If ∞

{pk (x1 , . . . , xm )}k=0 is a sequence of homogeneous polynomials of degree k in Rm , then, by definition, the formal power series or simply formal series ∞ X

pk (x1 , . . . , xm )

k=0

is the sequence in Rm of the partial sums: ( n )∞ X . pk (x1 , . . . , xm ) k=0

n=0

We do not introduce any notion of convergence for sequences and formal series in Rm . A particular case of formal series is the exponential of a single indeterminate: ( n )∞ ∞ X X xk xk x e = = . k! k! k=0

k=0

n=0

We need to introduce two kinds of operations on formal series: product and composition. Given two formal series ∞ X

pk (x1 , . . . , xm ) and

k=0

∞ X

qk (x1 , . . . , xm )

k=0

we define their product as follows: ∞ X

pk (x1 , . . . , xm ) ·

∞ X

qj (x1 , . . . , xm ) =

j=0

k=0

∞ X

ri (x1 , . . . , xm )

i=0

with ri (x1 , . . . , xm ) =

X

pk (x1 , . . . , xm ) · qj (x1 , . . . , xm ) .

k+j=i

For instance ex · ey =

   ! ∞ ∞ X xk y j X yj X  = . · k! j! k! j! n=0 j=0

∞ X xk k=0

k+j=n

It is useful to note the following fact:   ∞ ∞ X X xk y j X zl  · (ex · ey ) · ez = k! j! l! n=0 k+j=n l=0      ∞ ∞ X X X xk y j z l X X    =  = k! j! l! s=0 s=0 n+l=s

k+j=n

k+j+l=s

 xk y j z l  k! j! l!

(9.73)

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H¨ ormander operators

where we have used the properties which hold in Rm , like associativity of the product and distributivity. Since also ex · (ey · ez ) can be written in the same way we obtain the associativity of the product of exponentials (ex · ey ) · ez = ex · (ey · ez ) .

(9.74)

It is also worthwhile noting that, whenever we compute ex · ey for commuting variables x, y (for instance, when both x and y are multiples of the same indeterminate), then we simply get ex+y . Actually, n   n X xk y j X 1 X n k n−k xk y n−k 1 n = = x y (x + y) = k! j! k! (n − k)! n! k n! k+j=n

k=0

k=0

if x and y commute, hence in this case ex · ey =

∞ X 1 n (x + y) = ex+y . n! n=0

Let us now define (just in a particular case that we will need) the composition of formal series. Given two formal series ∞ ∞ X X pk (x1 , . . . , xm ) , qj (x1 , . . . , xm ) , j=0

k=1

assume that the first one does not contain the term of degree 0 (i.e. p0 = 0). The composition ! ∞ ∞ X X qj pk (x1 , . . . , xm ) , x2 , . . . , xm j=0

is the formal series

k=1

P∞

i=0 ri

(x1 , . . . , xm ) defined by the following partial sums:  ! n n n X X X ri (x1 , . . . , xm ) ≡  qj pk (x1 , . . . , xm ) , x2 , . . . , xm  (9.75) i=0

j=0

k=1

n

where [. . .]n denotes the truncation operator : [p (x1 , . . . , xm )]n is the polynomial obtained from p (x1 , . . . , xm ) suppressing all the monomials of degree greater than Pn n. Since p0 = 0, the term qj ( k=1 pk (x1 , . . . , xm ) , x2 , . . . , xm ) is a polynomial (no longer homogeneous) which does not contain any monomial of degree less than j. For this reason, the expression on the right hand side of (9.75) contains all the possible monomials of degree 6 n which would appear taking  ! N N X X  qj pk (x1 , . . . , xm ) , x2 , . . . , xm  j=0

k=1

n

for some N > n; this means that (9.75) is a consistent definition of partial sum for the composed formal series. In particular, the exponential of a formal series (without term of degree zero) is a well defined formal series.

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469

A commutator in Rm can be defined extending by bilinearity the commutator [x, y] = xy − yx of two indeterminates. This commutator obviously enjoys the properties of a Lie bracket. We can also define the standard repeated commutators as the polynomials [xi2 , xi1 ] , [xi3 , [xi2 , xi1 ]] , [xi4 , [xi3 , [xi2 , xi1 ]]] , . . . and the (general) repeated commutators as the polynomial obtained from a standard repeated commutator replacing some indeterminates with some standard repeated commutators or other indeterminates, like [[xi4 , xi1 ] , [xi2 , [xi1 , xi3 ]]] . A polynomial p (x1 , x2 , . . . , xm ) which is entirely expressible as a linear combination of repeated commutators is called a Lie polynomial. We are now in position to state and prove the following version of the BCH formula: Theorem 9.68 (Abstract BCH formula) There exists a unique formal series in two indeterminates z=

∞ X

Ck (x, y)

k=1

with C1 (x, y) = x + y, C2 (x, y) = 21 [x, y] and (for k > 2) Ck (x, y) homogeneous Lie polynomial of degree k in x, y, such that ex · ey = ez . Note that ez is well defined by the above discussion, since the formal series z does not contain the constant term. Note also that this statement implies the one PN stated in section 9.3.1. To see this, let zN = k=1 Ck (x, y) for N = 1, 2, 3, . . . By our definitions, ex · ey is the formal series with N -th partial sum N X k N X X X xi y j xj y k−j = , i! j! j! (k − j)! j=0

k=0 i+j=k

k=0

which is the expression previously denoted by (ex · ey )N , while the composition ez is the formal series having for N -th partial sum    !j  N N N j X X X z 1 N  Ck (x, y)  =  . j! j! j=0 j=0 k=1

N

N

Then the present statement means that for every N = 1, 2, . . . we have   N j X z N (ex · ey )N =  , j! j=0 N

470

H¨ ormander operators

which means N X zj

N

j=0

j!

= (ex · ey )N + RN +1 (x, y)

where RN +1 (x, y) is a polynomial containing only monomials of degree > N + 1. This is exactly the identity previously stated in Theorem 9.18. Throughout this section the symbol RN (x, y) will always denote a “remainder” polynomial (which can vary from line to line) which is sum of monomials of degree > N. We start with some lemmas. Lemma 9.69 The statement of the above theorem holds in the following weaker form: Ck (x, y) are homogeneous polynomials of degree k in x, y (but for the moment we do not pretend they are Lie polynomials). In particular, C1 (x, y) = x + y and C2 (x, y) = 12 [x, y]. Proof. By induction on N . For N = 2, (ex · ey )2 =

2 X k X
1 2 xj y k−j = 1 + (x + y) + x + 2xy + y 2 . j! (k − j)! 2 j=0

k=0

Let z2 = x + y + (ez2 )2 =

1 2

2 X k=0

[x, y], then z2k 1 1 = 1 + x + y + (xy − yx) + k! 2 2

 x+y+

2 1 (xy − yx) 2


1 2 x + 2xy + y 2 + R3 (x, y) 2 = (ex · ey )2 + R3 (x, y) .

=1+x+y+

Assume the assertion holds up to N − 1, and let us prove it for N (N > 3). This means that we know zN −1 so that we can write zN = zN −1 +CN (x, y) with CN (x, y) to be determined. Then N X 1 k (zN −1 + CN (x, y)) . (ezN )N = k! k=0

Since CN (x, y) is a sum of monomials of degree N , for any k > 2, k

k (zN −1 + CN (x, y)) = zN −1 + RN +1 (x + 1) .

Hence, by the inductive assumption, (ezN )N = 1 + zN −1 + CN (x, y) +

N k X zN −1 k=2

k!

+ RN +1 (x, y)

N zN −1 + CN (x, y) + RN +1 (x, y) N! zN = (ex · ey )N −1 + RN (x, y) + N −1 + CN (x, y) . N!

= eZN −1




+ N −1

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471

So that CN (x, y) = (ex · ey )N − (ex · ey )N −1 − RN (x, y) −

N zN −1 + RN +1 (x, y) . N!

(9.76)

Since by definition (ex · ey )N − (ex · ey )N −1 =

N X xj y N −j j! (k − j)! j=0

is a homogeneous polynomial of degree N and by the inductive assumption zN −1 is a polynomial of degree N − 1 without constant term, the expression on the right hand side of (9.76) has the form RN (x, y) and choosing CN (x, y) equal to the sum of terms of degree exactly N in RN (x, y) we are done. Lemma 9.70 (Properties of Lie polynomials) (a) If F (x1 , . . . , xm ) is a Lie polynomial of degree n, and we write F = Pn k=1 Fk with Fk homogeneous polynomial of degree k, then each Fk is a Lie polynomial. (b) If F (x1 , x2 ) and and G (x1 , . . . , xm ) are Lie polynomials, then also the composition F (G ((x1 , . . . , xm )) , x2 ) is a Lie polynomial. Proof. Point (a) is obvious since every repeated commutator of length k is a homogeneous polynomial of degree k, hence if we expand a Lie polynomial as a sum of monomials and then group the monomials of the same degree, all the monomials coming from a single commutator belong to the same group. Point (b) is also easy because if we write F, G as linear combination of commutators, their composition F (G ((x1 , . . . , xm )) , x2 ) is a linear combination of commutators of commutators. Lemma 9.71 (Properties of Ck (x, y)) The polynomials Ck (x, y) constructed in Lemma 9.69 have the following properties for every k > 2: (a) Ck (x, y) does not contain pure monomials in x or y. Formally Ck (x, 0) = Ck (0, y) = 0. (b) Ck (nx, mx) = 0 for any n, m ∈ Z. Proof. Point (a) follows from (b). To prove (b) observe that since the variables nx, mx commute, we have, in the sense of formal series, enx emx = ez with z = nx + mx, hence Ck (nx, mx) = 0 for every k. We are now in position for the

472

H¨ ormander operators

Proof of Theorem 9.68. By Lemma 9.69 we know there exists a sequence ∞ {Ck (x, y)}k=2 of homogeneous polynomials of degree k in x, y, such that the assertion of the theorem holds, and we have to prove that each Ck is actually a Lie polynomial. We proceed by induction on k. For k = 2 this has been proved in Lemma 9.69, since C2 (x, y) = 21 [x, y]. Assume this is true for every k < n (for some n > 2) and let us prove it for n. By Lemma 9.69 we have, in the sense of formal series, ea eb = ez with z =

∞ X

Cj (a, b)

j=1

ea e


b

ec = ez ec = ew with w =

∞ X

Ci (z, c) =

i=1

∞ X

  ∞ X Ci  Cj (a, b) , c

i=1

j=1

and by (9.74) we can write, again in the sense of formal series,     n n ∞ ∞ X X X X Cj (b, c) . Ci a, Cj (a, b) , c = Ci  i=1

i=1

j=1

(9.77)

j=1

This means in particular that the homogeneous polynomial of degree n coming from the expansion on the left hand side, say Pn (a, b, c), equals the analogous of the right hand side, say Qn (a, b, c). Now, Pn (a, b, c) is the sum of terms of order n extracted from   n n X X Ci a, Cj (b, c) . (9.78) i=1

j=1

By inductive assumption we know that Cj is a Lie polynomial for every j 6 n − 1, hence by Lemma 9.70, also   n−1 n−1 X X Ci  Cj (a, b) , c (9.79) i=1

j=1

is a Lie polynomial. The only terms of degree 6 n coming from (9.78) and not present in (9.79) are C1 (Cn (a, b) , c) = Cn (a, b) + c and Cn (C1 (a, b) , c) = Cn (a + b, c) . In turn, this implies that Pn (a, b, c) = Cn (a + b, c) + Cn (a, b) + homogeneous Lie polynomial of degree n. An analogous reasoning for the right hand side of (9.77) gives Qn (a, b, c) = Cn (a, b + c) + Cn (b, c) + homogeneous Lie polynomial of degree n.

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473

Therefore we can write Cn (a + b, c) + Cn (a, b) ∼ Cn (a, b + c) + Cn (b, c)

(9.80)

where P ∼ Q means that P − Q is a Lie polynomial. Our final goal is to prove that Cn (a, b) ∼ 0. Inserting c = −b in (9.80) and using Lemma 9.71, we easily obtain Cn (a, b) ∼ −Cn (a + b, −b) .

(9.81)

Inserting a = −b in (9.80) we get, analogously, Cn (b, c) ∼ −Cn (−b, b + c) which we rewrite, changing the names of the variables, as Cn (a, b) ∼ −Cn (−a, a + b) .

(9.82)

By (9.82), exchanging a and b and applying (9.81) with a replaced by −b and b replaced by a + b n

Cn (b, a) ∼ −Cn (−b, a + b) ∼ Cn (a, − (a + b)) = (−1) Cn (−a, a + b) (recall that Cn is homogeneous of degree n). By (9.82) we obtain Cn (b, a) ∼ (−1)

n+1

Cn (a, b) .

(9.83)

Next, we insert c = − 12 b in (9.80), getting         1 1 1 1 Cn a + b, − b + Cn (a, b) ∼ Cn a, b + Cn b, − b = Cn a, b , 2 2 2 2     1 1 (9.84) Cn (a, b) ∼ Cn a, b − Cn a + b, − b . 2 2 Similarly inserting a = − 12 b in (9.80) we get     1 1 b, c − Cn − b, b + c , Cn (b, c) ∼ Cn 2 2 which we rewrite, changing the names of variables, as     1 1 a, b − Cn − a, a + b Cn (a, b) ∼ Cn 2 2

(9.85)

Next we apply (9.84) to both terms of the right hand side of (9.85), getting     1 1 1 1 Cn (a, b) ∼ Cn a, b − Cn a + b, − b 2 2 2 2     1 a+b 1 a+b + Cn a + b, − . − Cn − a, 2 2 2 2 We now apply (9.82) to the third term on the right, and (9.81) to the second and fourth on the right:         1 1 a+b 1 1 1 1 a+b Cn (a, b) ∼ Cn a, b + Cn , b + Cn a, b − Cn b, 2 2 2 2 2 2 2 2 since Cn is homogeneous of degree n this is equal to 2−n Cn (a, b) + 2−n Cn (a + b, b) + 2−n Cn (a, b) − 2−n Cn (b, a + b)

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H¨ ormander operators

by (9.83) n

∼ 21−n Cn (a, b) + 2−n (1 + (−1) ) Cn (a + b, b) so we have proved
n 1 − 21−n Cn (a, b) ∼ 2−n (1 + (−1) ) Cn (a + b, b) .

(9.86)

For odd n > 1 this gives Cn (a, b) ∼ 0 which is our desired result. For even n we insert a − b in place of a in (9.86)
1 − 21−n Cn (a − b, b) ∼ 21−n Cn (a, b) and apply (9.81) to the left hand side of this, getting
− 1 − 21−n Cn (a, −b) ∼ 21−n Cn (a, b) hence −Cn (a, −b) ∼

21−n Cn (a, b) (1 − 21−n )

which also can be written (replacing b with −b) as Cn (a, b) ∼ −

21−n Cn (a, −b) (1 − 21−n )

and inserting the up to last relation in the last one we get  2 21−n Cn (a, b) ∼ Cn (a, b) . (1 − 21−n )  1−n 2 2 Since n > 2, we have 1−2 6= 1, so the last relation means Cn (a, b) ∼ 0, and 1−n we have finished. Let us now prove some more properties of the formal series S (x, y) = x + y +

∞ X

Ck (x, y) ,

k=2

which will be useful in the next chapter, in the context of the abstract construction of the homogeneous group corresponding to the nilpotent Lie algebra which locally approximates a given system of H¨ ormander’s free vector fields. Keeping the notation and assumptions of this section, we have: Proposition 9.72 (Group properties of S (x, y)) The following identities of formal series hold: (i) S (x, S (y, z)) = S (S (x, y) , z) (ii) S (x, 0) = x (iii) S (x, −x) = 0.

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475

This proposition says that the mapping (x, y) 7→ S (x, y) enjoys the properties of a group operation. In the next chapter we will apply this result in the particular context of a stratified Lie algebra (or “stratified of type II”, see Chapter 3), in order to define a group operation in this setting. Proof. Points (ii) and (iii) are already known by Lemma 9.71, and we have collected them in this proposition just for convenience of future reference. So, let us prove (i). Since S (x, y) is a formal series without term of degree zero, the compositions S (x, S (y, z)) , S (S (x, y) , z) are two well defined formal series, and we have to prove that they actually coincide. Note that both the composed formal series do not contain terms of degree zero. We already know that the product of exponentials is associative (ex · ey ) · ez = ex · (ey · ez ) . On the other hand, we know that ex · ey = eS(x,y) so we have eS(x,y) · ez = ex · eS(y,z) eS(S(x,y),z) = eS(x,S(y,z)) . To prove (i) we have to show that, if S1 , S2 are two formal series with no term of degree zero such that eS1 = eS2 then S1 = S2 . Multiplying both sides of eS1 = eS2 for e−S2 we have eS1 · e−S2 = eS2 · e−S2 = eS(S2 ,−S2 ) = 1, since S (S2 , −S2 ) = 0 by Lemma 9.71 (ii). On the other hand eS1 ·e−S2 = eS(S1 ,−S2 ) hence eS(S1 ,−S2 ) = 1. We now claim that: (a) for a given formal series S without term of degree zero if eS = 1 then S = 0, which will imply S (S1 , −S2 ) = 0; (b) for any two given formal series S1 , S2 without terms of degree zero if S (S1 , −S2 ) = 0 then S1 = S2 , which will conclude our proof. P∞ To prove (a), let us write S = k=1 pk with pk homogeneous polynomial of degree k. Then, by definition of eS , we have  !j  n n X X 1  pk  = 1 for n = 0, 1, 2, . . . j! j=0 k=1

n

For n = 1 this means 1 + p1 = 1, hence p1 = 0. Assume pk = 0 for every k 6 n − 1 and let us prove that pn = 0, which will imply that S = 0. We have    !j    n n n X X X 1 1 pnn j     pk = (pn ) = 1 + pn 1= = 1 + pn + . . . + j! j! n! n j=0 j=0 k=1

n

n

since pn is a homogeneous polynomial of degree n, therefore But then pn = 0, and (a) is proved.

h

p2n 2

+ ... +

pn n n!

i

= 0. n

476

H¨ ormander operators

To prove (b), let us write S1 =

∞ X

pk , S2 =

∞ X

qj

j=1

k=1

0 = S (S1 , −S2 ) = S1 − S2 +

∞ X

Ci (S1 , −S2 )

i=2

=

∞ X

pk −

∞ X

qj +

j=1

k=1

∞ X

 Ci 

i=2

∞ X

pk , −

∞ X

 qj 

j=1

k=1

and by definition of composition of formal series this means that for every n = 1, 2, 3 . . . we have    n n n n n X X X X X  pk − qj + Ci  pk , − qj  = 0. k=1

j=1

i=2

j=1

k=1

n

For n = 1 this means p1 − q1 = 0, hence p1 = q1 . Assume that pk = qk for every k 6 n − 1 and let us prove that pn = qn , which will imply S1 = S2 , that is (b). Then, by inductive assumption    n n n n n X X X X X qj  Ci  pk , − qj + pk − 0= k=1

i=2

j=1

 = pn − qn +

n X

j=1

k=1



n−1 X

Ci 

i=2

pk + pn , −

n−1 X

n

 pj − qn  .

j=1

k=1

n

Let us consider the expression   n−1 n−1 X X Ci  pk + pn , − pj − qn  k=1

j=1

Pn−1 for some i > 2. Since the polynomial Pn−1 = k=1 pk contains monomials of degrees from 1 to n − 1, pn , qn are homogeneous of degree n, and Ci (x, y) is homogeneous of degree 2 and does not contain pure monomials in x or y, in the expansion of Ci every term containing some monomial of pn or qn is actually a monomial of degree > n + 1, which is canceled by the operator [·]n . Therefore " # n X 0 = pn − qn + Ci (Pn−1 + pn , −Pn−1 − qn ) i=2

" = pn − qn +

n X i=2

n

# Ci (Pn−1 , −Pn−1 )

= [pn − qn ]n = pn − qn n

by Lemma 9.71 (ii). Hence pn = qn and we are done.

More geometry of vector fields: metric balls and equivalent distances

9.9

477

Notes

The core of this chapter, namely the sharp estimate on the volume of metric balls, the ball-box theorem, and the equivalence between several metrics induced by a family of vector fields, consists in results contained in the paper by Nagel-SteinWainger [131], a detailed exposition of which forms our sections 9.4, 9.5, 9.6. In the exposition of section 9.5 we have also relied on some arguments contained in the paper by Morbidelli [128]. Our proof of the equivalence of metrics (section 9.6) has taken advantage of the previous work done in Chapter 1. The Baker-Campbell-Hausdorff formula is an old result of noncommutative algebra, actually due to the contribution of several authors: Campbell [51], Baker [5], Hausdorff [103], Dynkin [79]. A standard reference is the book by Varadarajan [156]. We also point out to the reader the monograph by Bonfiglioli-Fulci [12], entirely devoted to this theorem. Here we have followed the approach of Eichler [80], and the proof given in section 9.8 is a detailed exposition of that very short paper. The rest of the material contained in section 9.3 (that is, consequences of the abstract result in the context of vector fields) is again an expanded account of results contained in the paper by Nagel-Stein-Wainger [131]. The d-regularity of control balls and the connection between this property and the segment property of the distance (section 9.7) have been pointed out for the first time in the paper by Franchi-Lanconelli [93, Prop. 2.10], in the particular case of a suitable family of diagonal vector fields, that is Xi = λi (x) ∂xi (i = 1, 2, . . . , n) with possibly degenerate λi . For general H¨ ormander vector fields this property has been proved, more or less explicitly, in several papers, see Capogna-Garofalo [54, Proposition 2], Bramanti-Brandolini [27, Lemma 4.2].

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Chapter 10

Lifting and approximation

10.1

Motivation and statement of the main results

In Chapter 8 we have seen how to use the global homogeneous fundamental solution constructed for homogeneous left invariant H¨ormander operators on groups to prove precise a priori estimates, exploiting representation formulas for second order derivatives along the vector fields and suitable abstract theories of singular integrals. We now turn to the task of proving analogous a priori estimates in the general situation where such underlying group structure is lacking. The techniques necessary to settle this problem have been developed by Rothschild and Stein in their celebrated paper [142]. In this chapter we will introduce their lifting and approximation procedure that allows to construct left invariant vector fields on a homogeneous group which, in a suitable sense, approximate the original ones. In view of the elegant theory developed in Chapter 8 for operators on groups this is a natural strategy to attack the problem. As we shall see, however, to carry out this strategy introduces several technical difficulties due to the interplay between the analytic properties of vector fields and the algebraic structure of their Lie algebra. In the next chapter we will apply these techniques to complete the task of deriving a priori estimates for general H¨ ormander operators, analogous to those proved in Chapter 8 on groups. Let X0 , X1 , . . . , Xq be vector fields satisfying H¨ormander’s condition at step s in Ω ⊆ Rn . As usual we assign weight 2 to the vector field X0 and weight 1 to the remaining vector fields. We anticipate that the constructions of this chapter will work equally well if we assign weight 1 to all vector fields X0 , X1 , . . . , Xq . In this sense also the case of “missing X0 ” will be included in our construction. Let us start describing informally the lifting and approximation method, to give a motivation for the several steps which will be performed. The idea is to approximate an operator of the form L=

q X

Xi2 + X0

i=1

479

480

H¨ ormander operators

with an operator of the same form L=

q X

Yi2 + Y0

i=1

constructed using vector fields Y0 , Y1 , . . . , Yq that are left invariant on a suitable homogeneous group G and satisfy H¨ ormander’s condition at the same step s. Example 10.1 Consider the vector fields X1 = ∂x + 2y∂t ; X2 = ∂y − 2 (ex − 1) ∂t that satisfy H¨ ormander’s condition at step 2. The coefficients of X2 are not polynomials, hence they cannot be left invariant with respect to any structure of homogeneous group, but if we take the first order expansion near x = 0, ex − 1 ∼ x, we get the vector fields X1 = ∂x + 2y∂t ; X20 = ∂y − 2x∂t which are 1-homogeneous and left invariant on the Heisenberg group H1 . We note that: (1) X1 , X20 , [X1 , X20 ] are three independent vectors at every point, exactly like X1 , X2 , [X1 , X2 ]. In other words, X2 and X20 are indistinguishable from the point of view of the Lie algebra structure, up to the step which is required to check H¨ ormander’s condition. (2) Moreover, near x = 0 we have X2 − X20 = −2 (ex − 1 − x) ∂t . The interesting property of this vector field is that, with respect to the homogeneities of the Heisenberg group (x has weight 1, t has weight 2), X2 − X20 is “approximately homogeneous of order 0 near x = 0”, since −2 (ex − 1 − x) ∼ −x2 and −x2 ∂t is homogeneous of degree 0, while X20 is homogeneous of degree 1. This implies that if Γ is the (2 − Q)-homogeneous fundamental solution of the approximating operator  0 2 L0 = X12 + X2 , then X20 Γ will be (1 − Q)-homogeneous, that is more singular than Γ, but (X2 − X20 ) Γ will be approximately (2 − Q)-homogeneous, that is as singular as Γ. As we will see later in more detail, this is the key point which makes this approximation useful. Generalizing the above example, the idea is to introduce, in a neighborhood of any fixed point x0 , suitable “canonical coordinates” (already used in the previous chapter) and then, with respect to these coordinates, approximate the coefficients of the vector fields Xi with suitable polynomials (a kind of Taylor expansion), getting an approximation of Xi with a vector field Yi with polynomial coefficients. More precisely, if the original vector fields satisfy H¨ormander’s condition at step

Lifting and approximation

481

s, this means that not more than s − 1 derivatives of the coefficients need to be computed when checking this condition, hence the vector fields obtained replacing each coefficient with its Taylor polynomial of degree s − 1 will satisfy the same relevant commutator relations. Actually, the Lie algebras generated by the two sets of vector fields will be indistinguishable up to step s. Observe now that the Lie algebra generated by a system of homogeneous left invariant vector field always possesses some special property: for instance, its structure is the same at any point. By this expression we mean that for any choice of N vector fields among X0 , X1 , . . . , Xq and their commutators up to step s, if these vectors are independent at some point then they are independent at every point. Therefore we cannot expect to approximate our original system of vector fields with a “good” one unless our original system already satisfies some additional algebraic condition which, in particular, makes its Lie algebra “of constant structure”. Example 10.2 The Franchi-Lanconelli vector fields already considered in section 1.11.3 provide the simplest example of vector fields that do not fulfill this basic requirement. Indeed, let us consider in R2 the vector fields X1 = ∂x , X2 = x∂y . At any point (x, y) with x 6= 0 these vector fields are independent and span R2 . When x = 0 however X2 vanishes and to span R2 we are forced to consider the commutator [X1 , X2 ] = ∂y . To solve this problem Rothschild and Stein introduced in their construction a preliminary step: the lifting of a system of vector fields. The key idea is to consider new vector fields defined in a higher dimensional space obtained adding extra variables to the original vector fields in order to obtain the degrees of freedom that are necessary to avoid undesired commutation relations, and obtain a Lie algebra with constant structure from point to point. Let us describe how this idea works in the previous example: Example 10.3 Let X1 , X2 as in the above example and define the new vector fields in R3 : f1 = X1 ; X f2 = X2 + ∂t = x∂y + ∂t X where t his the inew added variable. A straightforward computation shows that e1 , X e2 are independent at any point of R3 . Their Lie algebra is the same e1 , X e2 , X X as that of the Heisenberg group H1 , and actually a smooth change of variables in R3 can turn these vector fields into the “canonical form” X10 = ∂x0 + 2y 0 ∂t0 , X20 = e1 , X e2 satisfy the desired condition, moreover ∂y0 − 2x0 ∂t0 of H1 . The vector fields X they project onto X1 , X2 , in the sense that for any function f (x, y) which does not depend on the added t variable, we have e1 f, X2 f = X e2 f. X1 f = X As we shall see this property allows to get the desired a priori estimates for Xi Xj u ei X ej u in a higher dimensional space. once we have proved analogous estimates for X

482

H¨ ormander operators

e1 , X e2 . In the above We say that the vector fields X1 , X2 have been lifted to X simple example, the lifted vector fields are already left invariant and homogeneous. More generally one expects to build up a two-step process: Example 10.4 Let us consider the operator L = X12 + X22 with X1 = ∂x , X2 = (ex − 1) ∂y in R2 . These vector fields have the same structure as those in Example 10.2, but nonpolynomial coefficients. Then: x 3 e e First step: h we lift i X1 , X2 to X1 = ∂x , X2 = (e − 1) ∂y + ∂t in R . Note that e1 , X e2 , X e1 , X e2 are independent at any point of R3 . X e1 , X e2 with Y1 = ∂x , Y2 = x∂y + ∂t which are left Second step: we approximate X invariant and 1-homogeneous in H1 (up to a smooth change of coordinates). As we have already remarked, the purpose of the lifting procedure is to construct new vector fields that generate a Lie algebra of constant structure. We are now going to introduce a key notion which will make precise this requirement. Definition 10.5 (Free vector fields, first definition) We say that a system of smooth vector fields Z0 , Z1 , . . . , Zq is free up to (weighted) step s in a domain Ω of RN if the vector fields Zi and their commutators up to step s, evaluated at any point of Ω, do not satisfy any linear relation other than those which hold automatically as a consequence of antisymmetry of the Lie bracket and Jacobi identity. To put it into another way, Z0 , Z1 , . . . , Zq are free up to weight s if and only if the only relations of linear dependence which we can write among them and their commutators up to step s (at any point of Ω), are those which can be established without knowing the components of the Zj ’s. If the vector fields satisfy H¨ormander’s condition at step s and are free up to step s, then in particular their Lie algebra has “constant structure”. Starting with a system of vector fields X0 , X1 , . . . , Xq that satisfy H¨ormander’s condition at step s in a neighborhood of x0 ∈ Rn , we will build a new family e0 , X e1 , . . . , X eq , which are free up to step s and satisfy of “lifted” vector fields X H¨ ormander’s condition at step s in a neighborhood of (x0 , 0) ∈ Rn+m for some m > 1. The second step of the theory consists in approximating these free vector fields with homogeneous left invariant vector fields on a suitable homogeneous group. To approach this problem, we start with an algebraic remark. Disregarding ek in Cartesian coordinates, the structure of the explicit form of the vector fields X their Lie algebra up to step s (that is, the number and type of independent objects ek ’s and their commutators up to step s) is completely determined by the among the X requirement of being free up to step s. This also means that the dimension N = n + m of the lifted space only depends on the numbers q and s. Now the idea is that the Lie algebra of homogeneous left invariant vector fields Yk which approximate locally ek ’s can be defined abstractly as the free Lie algebra of step s on q generators the X

Lifting and approximation

483

(a precise definition will be given later), which turns out to be a homogeneous Lie algebra; this means that the vector fields Yk and their commutators up to step s ek ’s, but moreover all their commutators satisfy exactly the same relations as the X of step > s vanish. The structure of homogeneous group G in RN is, in turn, determined by that of the corresponding Lie algebra, as we will explain later. The ek are defined in a neighborhood of ξ0 = (x0 , 0) ∈ RN ; the vector fields vector fields X Yk are defined in the whole RN , and their behavior near the origin will approximate ek near ξ0 . This means that the approximation between X ek the behavior of the X and Yk is realized in a suitable system of coordinates. We can now summarize the previous discussion giving a precise statement which collects all the main instrumental results which will be proved throughout this chapter, and will be used in Chapter 11 in order to prove a priori estimates for general H¨ ormander operators. All the concepts involved in this statement will be precisely defined throughout this chapter: Theorem 10.6 (Lifting and approximation) Let X0 , X1 , . . . , Xq be vector fields in a domain Ω ⊂ Rn satisfying H¨ ormander’s condition of (weighted) step s at x0 ∈ Ω. Then: ek (k = 0, 1, . . . , q) defined in a (1) There exist an integer m and vector fields X neighborhood of (x0 , 0) ∈ Rn+m , of the form m X ek = Xk + X ukj (x, t1 , t2 , . . . , tj−1 ) ∂tj , j=1

ek ’s are free up to step s and where ukj are polynomials, such that the X satisfy H¨ ormander’s condition at step s in this neighborhood. (Meaning ek ’s and their commutators up to weighted step s span Rn+m ). that the X (2) In RN = Rn+m there exists a structure of stratified homogeneous group G of type II, where u−1 = −u, with canonical generators Y0 , Y1 , . . . , Yq , and for any η in a neighborhood of (x0 , 0) there exists a smooth diffeomorphism ξ 7→ Θη (ξ) from a neighborhood of η onto a neighborhood of the origin in G, smoothly depending on η, such that for any smooth function f : G → R, ei (f (Θη (·))) (ξ) = (Yi f + Rη,i f ) (Θη (ξ)) X (10.1) (i = 0, 1, . . . , q), where the “remainder” Rη,i is a smooth vector field of weight1 > 0 if i = 1, . . . , q and > −1 if i = 0, that depends smoothly on η. (3) The function Θ (η, ξ) = Θη (ξ) satisfies −1

Θ (η, ξ)

= −Θ (ξ, η) .

Moreover, the change of coordinates in RN given by ξ 7→ u = Θ (η, ξ) has a Jacobian determinant given by dξ = c (η) (1 + O (|u|)) du where c (η) is a smooth function, bounded and bounded away from zero. 1 See

Definition 10.24 for an explanation of this term.

484

H¨ ormander operators

(4) The function ρ (ξ, η) = kΘ (ξ, η)k , where k·k is any homogeneous norm on G, is a quasidistance, locally equivaek . If Q is the homogeneous lent to the distance induced by the vector fields X e (ξ, R) indimension of the group G, then the measure of the metric balls B Q e duced by the vector fields Xk is locally equivalent to R . This theorem will be proved throughout Theorems 10.19 and 10.30, Propositions 10.29, 10.33, 10.38 and 10.55. In order to better understand in which sense the vector field Rη,i in (10.1) can be ei on a function f (Θη (ξ)) seen as a “small remainder”, let us consider the action of X when f is homogeneous of some negative degree −α on G and smooth outside the origin. In this case we have Yi f is (−α − 1)-homogeneous, hence |Yi f (Θη (ξ))| 6

c α+1

kΘη (ξ)k

=

c

α+1 ,

ρ (ξ, η)

while Rη,i f is not a homogeneous function, but nevertheless, saying that Rη,i has local degree 6 0 implies, as we will see later, that c c |Rη,i f (Θη (ξ))| 6 α = α kΘη (ξ)k ρ (ξ, η) for ξ near η, that is: this term is less singular than Yi f . The results collected in the previous theorem (lifting; approximation; properties of the map Θ) will be proved throughout sections 10.2, 10.3, 10.4. We point out that these tools have further applications than the ones for which they were originally designed (that is, the proof of a priori estimates that we will carry out in the next chapter). The general idea is that, thanks to this technique, the study of local properties of a general H¨ ormander operator can sometimes be reduced to the study of a homogeneous left invariant H¨ormander operator, of the kind studied by Folland in [85]. Also, sometimes it is useful the first part of the result (lifting theorem), to reduce the study of a general system of H¨ormander vector fields to a system of free (although not homogeneous) H¨ormander vector fields, which has some advantages. In several circumstances, in order to apply successfully the technique of lifting and approximation, we need a way to come back from the lifted to the original variables, transferring quantitative geometric information. There are a few results e d, , de the control balls and distances induced in this direction. Let us denote by B, B, n oq q ei respectively by {Xi }i=0 in Rn and by X in RN . First of all, it is easy to see i=1

e ((x, t) , r) on Rn is B (x, r), which means that the projection of the control ball B that de((x, t) , (y, s)) > d (x, y). e In section 10.4 we A deeper result relates the volumes of the control balls B, B. will prove the following:

Lifting and approximation

485

Theorem 10.7 Let Ω0 b Ω and x0 ∈ Ω0 . There exist constants δ0 , κ ∈ (0, 1) and c1 , c2 > 0 such that in V = B (x0 , δ0 ) we can lift the vector fields X0 , X1 , . . . , Xq to e0 , X e1 , . . . , X eq , defined in a neighborhood U of (x0 , 0) in RN for some N = n + m, X and free at step s in U , so that for every δ ∈ (0, δ0 ) and every y ∈ Ω e n o B ((x , 0) , δ) 0 m e ((x0 , 0) , δ) 6 c1 , s ∈ R : (y, s) ∈ B |B (x0 , δ)| and for every δ ∈ (0, δ0 ) and every y ∈ B (x0 , κδ) e n o B ((x , 0) , δ) 0 m e ((x0 , 0) , δ) > c2 . s ∈ R : (y, s) ∈ B |B (x0 , δ)| Here |A| denotes the Lebesgue measure of the set A in Rk for the suitable dimension k (which is m, N, n for the three sets involved in the inequalities). This theorem will be proved by Theorem 10.40 and Remark 10.41. A geometric picture of the previous theorem is the following: the lifted ball e B ((x0 , 0) , δ) has volume comparable to that of the “cylinder” n o e ((x0 , 0) , δ) . B (x0 , δ) × s ∈ Rm : (y, s) ∈ B Note, however, that we will not prove simple set inclusions between these sets, but just the comparability of their volumes. Remark 10.8 (Dependence of the constants) The lifting result contained in Theorem 10.6, point (1), is mainly a qualitative result. Quantitative results are the smooth dependence of the remainder vector fields Rη,i in point (2), the smoothness of the diffeomorphism Θη (·), and the estimates implicitly contained in the properties stated in points (3)-(4) of Theorem 10.6. Now, the proof of Theorem 10.6 that we have chosen to reproduce in this book does not allow an easy control of these q quantities in terms of the original vector fields {Xk }k=0 . Actually, as we will see from the proof, the lifting procedure does not uniquely determine the polynomials ek . This implies that, ukj (x, t) appearing in the expression of the lifted vector fields X although all the quantities which are related to the lifted vector fields in principle only q depend on the local properties of the original vector fields {Xk }k=0 in a neighborhood of the point where the lifting procedure is performed (in particular, they depend on a finite number of derivatives of the components of the vector fields), this dependence is not easily expressed in a quantitative way in terms of a few parameters related q to the original system {Xk }k=0 . A more technical explanation of this fact will be given in Remark 10.18, after the proof of a key step of Theorem 10.6. Also, the constants in Theorem 10.7 depend on the constants appearing in Theq orem 9.42 (structure n oq of metric balls), applied both to the system {Xk }k=0 and to ek the system X . Therefore, this dependence inherits from Theorem 10.6 the k=0

q

complicated relation with the original system {Xk }k=0 .

486

H¨ ormander operators

For these reasons, differently from the previous chapters, throughout this chapter we will not make the extra effort of describing the quantitative dependence of the constants appearing in Theorems 10.6 and 10.7. In the application of this theory that we will make in Chapters 11 and 12, the relevant fact is that a bounded domain Ω0 where local a priori estimates are established will be covered by a finite number of neighborhoods, each supporting a lifting and approximation procedure applied to the q same system {Xk }k=0 at different points. Therefore, the final constants will depend q on the system {Xk }k=0 . 10.2

Lifting of H¨ ormander vector fields

We start recalling once more some notation which has been used several times. Let X0 , X1 , . . . , Xq be a system of real smooth vector fields, defined in a domain Ω ⊆ Rn . Let us assign to each Xi a weight pi , saying that p0 = 2 and pi = 1 for i = 1, 2, . . . , q. For any multiindex I = (i1 , i2 , . . . , ik ) we define the weight of I as Pk |I| = j=1 pij and the length of I, ` (I) = k. For I = (i1 , i2 , . . . , ik ) we set: XI = Xi1 Xi2 . . . Xik and      X[I] = Xi1 , Xi2 , . . . Xik−1 , Xik . . . . If I = (i1 ) , then X[I] = Xi1 = XI . As usual, we will write
X[I] f to denote the differential operator X[I] acting on a function f , and X[I] x to denote the vector field X[I] evaluated at the point x. We are now going to define the concept of free system of vector fields. Actually, this notion has been already introduced in section 10.1; the following definition is formulated in a more technical way, which actually turns out to be more useful in the proofs. We will show later that this definition agrees with the more intuitive one previously given. Let us start with the following remark. Any vector field X[I] with |I| 6 s can be rewritten explicitly as a linear combination of operators of the kind XJ for |J| = |I|: X X[I] = AIJ XJ J

where [AIJ ]|I|,|J|6s is a matrix of universal constants, built exploiting only those relations between X[I] and XJ which hold automatically, as a consequence of the definition of X[I] , regardless of the specific properties of the vector fields X0 , X1 , . . . , Xq . In particular, we see that AIJ = 0 if |J| = 6 |I| and AIJ = δIJ if |J| = |I| = 1.

(10.2)

Lifting and approximation

487

Example 10.9 For the system {X1 , X2 } and s = 2 we have 6 possible multiindices: 1, 2, (1, 1) (1, 2) , (2, 1) , (2, 2) . The only nonzero elements of the matrix {AIJ }|I|,|J|62 are: A1,1 = A2,2 = 1 A(1,2),(1,2) = 1 = A(2,1),(2,1) A(1,2),(2,1) = −1 = A(2,1),(1,2) . Actually, X[(1,2)] = [X1 , X2 ] = X1 X2 − X2 X1 = X(1,2) − X(2,1) . Note that if {aI }I∈B is any finite set of constants such that X X aI AIJ = 0 ∀J, then aI X[I] ≡ 0 I∈B

(10.3)

I∈B

for arbitrary vector fields X0 , X1 , . . . , Xq , since in this case ! X

aI X[I] =

I∈B

X

aI

X

I∈B

AIJ XJ =

J

X X J

aI AIJ

XJ = 0.

I∈B

Reversing this property we get an alternative definition of free vector fields: Definition 10.10 (Free vector fields, second definition) For any positive integer σ, we say that the vector fields X0 , X1 , . . . , Xq are free up to (weighted) step σ at x, if, for any family of constants {aI }|I|6σ , X |I|6σ

aI X[I]


x

= 0 =⇒

X

aI AIJ = 0 ∀J.

|I|6σ

As for the step of H¨ ormander’s condition, in the following we will simply say “free up to step σ” dropping the word “weighted”. This definition is consistent with Definition 10.5. Actually, assume that the vector fields X0 , X1 , . . . , Xq are
free up to step σ at x (in the sense of Definition P P 10.10) and that |I|6σ aI X[I] x = 0; then |I|6σ aI AIJ = 0 ∀J which, in view of
P discussion before the definition, implies that |I|6σ aI Y[I] x = 0 for any system of vector fields Y0 , Y1 , . . . , Yq , regardless of the coefficients of the Yi ’s. This means
P that the relation |I|6σ aI X[I] x = 0 is one of those which automatically follow from anticommutativity of the Lie bracket and Jacobi identity; in other words: if the vector fields X0 , X1 , . . . , Xq are free up to step σ at x (in the sense  of Definition
10.10), then the only linear relations which hold between the vectors X[I] x |I|6σ are those which follow axiomatically from the properties of commutators. One can also prove the converse, however we will never need this equivalence, so we do not give further details.

488

H¨ ormander operators

Example 10.11 (a) In R3 , X1 = ∂x1 + 2x2 ∂x3 ; X2 = ∂x2 − 2x1 ∂x3 (both weighting 1) are free up to step 2 at 0 (Actually, they are free at any point, but for simplicity we check this fact at the origin). Namely, if 0 = a1 (X1 )0 + a2 (X2 )0 + a12 ([X1 , X2 ])0 = a1 ∂x1 + a2 ∂x2 + a12 ∂x3 then a1 = a2 = a12 = 0. If we assign weight 1 to X1 and 2 to X2 , then they are free up to weight 3 at 0. (b) Instead, in R5 , X1 = ∂x1 + 2x2 ∂x5 ;

X2 = ∂x2 − 2x1 ∂x5

X3 = ∂x3 + 2x4 ∂x5 ;

X4 = ∂x4 − 2x3 ∂x5

(all weighting 1) are not free up to step 2 at 0. Namely, [X1 , X2 ] − [X3 , X4 ] = 0, which should imply A(1,2),J − A(3,4),J = 0 ∀J, while for J = (1, 2) we have A(1,2),(1,2) − A(3,4),(1,2) = 1 − 0 = 1. (c) In R3 , X1 = x3 ∂x1 + 2x2 ∂x3 ; X2 = ∂x2 − 2x1 ∂x3 (both weighting 1) are free up to step 2 at (0, 0, 1), for in this case the situation is like in example (a), but they are not free up to step 2 at (0, 0, 0), since (X1 )0 = 0 but A11 = 1. Remark 10.12 Let X0 , X1 , . . . , Xq be vector fields free up to step
σ at x, and P assume that for some constants {aI }|I|6σ we have a X [I] x = 0. Then |I|6σ I P is, the same linear relations hold at every point of the |I|6σ aI X[I] ≡ 0 (that P domain). Indeed, since |I|6σ aI AIJ = 0 for every J, we can write  X |J|6σ

aJ X[J] =

X |J|6σ

aJ

X

AJK XK =

K

X

 X

 K

aJ AJK  XK = 0.

|J|6σ

We also note the following basic fact: Proposition 10.13 If the vector fields X0 , X1 , . . . , Xq are free up to step σ at some point x0 , then there exists a neighborhood U (x0 ) such that they are free up to step σ at any point x ∈ U (x0 ).

Lifting and approximation

489

n
o Proof. Let B a set of multiindices such that X[I] x is a base for the span 0 I∈B n
o of X[I] x . Then there exists a neighborhood U of x0 such that for every 0 
|I|6σ / B. For suitable x ∈ U , X[I] x I∈B are still linearly independent. Let |J| 6 σ, J ∈

P coefficients cIJ we have X[J] x = c X[I] x . Let x ∈ U ; by Remark I∈B 0 0

IJ P 10.12 we have X[J] x = I∈B cIJ X[I] x . Assume now that for some constants {aI }|I|6σ we have X
aI X[I] x = 0. |I|6σ

Then 0=

X

aI X[I]




+ x

I∈B

X

aJ X[J]


x

=

X

aI X[I]




+ x

I∈B

J ∈B /

X J ∈B /

aJ

X

cIJ X[I]


x

I∈B

! =

X I∈B

aI +

X

aJ cIJ

X[I]


x

.

J ∈B /


 P Since X[I] x I∈B are linearly independent, we obtain aI + J ∈B / aJ cIJ = 0 for everyI ∈ B and and therefore ! X X X

aI X[I] x . 0= aI + aJ cIJ X[I] x = 0

0

I∈B

|I|6σ

J ∈B /

Since the vector fields are free at x0 , this implies which means that the vector fields are free at x.

P

|I|6σ

aI AIJ = 0 for every J

In the next proposition we show that if a system of H¨ormander vector fields at some step s is free up to the same step then the dimension of the space is uniquely determinated by the step s and the number of vector fields. Proposition 10.14 Let X0 , X1 , . . . , Xq be vector fields defined in a neighborhood of x ∈ Rn . 
(i) If X[I] x |I|6s span Rn , then n 6 rank [AIJ ]|I|,|J|6s . (ii) If the vector fields X0 , X1 , . . . , Xq are free up to step s at x then n > rank [AIJ ]|I|,|J|6s . 
Proof. (i) Assume that X[I] x |I|6s span Rn . Then there exist multiindices

I1 , I2 , . . . , In , with |Ij | 6 s, such that X[I1 ] x , . . . , X[In ] x are linearly indepenPn dent. Suppose there are constants a1 , a2 , . . . , an such that i=1 ai AIi J = 0 for every J, with |J| 6 s. Then n n X X X
ai X[Ii ] x = ai AIi J (XJ )x = 0. i=1

|J|6s i=1

490

H¨ ormander operators



Since the vectors X[I1 ] x , . . . , X[In ] x are linearly independent we have ai = 0 for i = 1, . . . , n. This means that the rows (AIi J )|J|6s are independent, so that rank [AIJ ]|I|,|J|6s > n. (ii) Let (AI1 J )|J|6s , (AI2 J )|J|6s , . . . , (AIk J )|J|6s
be independent rows of [AIJ ]|I|,|J|6s . Let us show that the vectors X[I1 ] x ,


Pk X[I2 ] x , . . . , X[Ik ] x are linearly independent. Let i=1 ai X[Ii ] x = 0 for some constants a1 , a2 , . . . , ak . Since the vector fields X0 , X1 , . . . , Xq are free up to the Pk step s we have i=1 ai AIi J = 0 for every J and since these rows are independent
we obtain ai = 0 for i = 1, . . . , k. This means that the vector fields X[Ii ] x are linearly independent and therefore k 6 n. Hence rank [AIJ ]|I|,|J|6s 6 n. Remark 10.15 We already noted that [AIJ ]|I|,|J|6s is a matrix of universal constants. More precisely, since this matrix only exploits those relations between X[I] and XJ that hold automatically as a consequence of the definition of X[I] , it only depends on the step s, the number of vector fields q and their weights. Hence, for vector fields X0 , X1 , . . . , Xq in Rn that are free at step s and satisfy H¨ ormander’s condition at the same step the dimension n of the space is uniquely determinated by s, q and the weights. The next proposition contains a deeper property of free vector fields: Proposition 10.16 Let X0 , X1 , . . . , X be vector fields defined in a neighborhood of the origin in Rn that are free up to the step σ at 0. Then for any family of constants {cI }|I|6σ ⊂ R there exists a polynomial u in Rn such that XI u(0) = cI when |I| 6 σ. This proposition is the technical core of this section, a key step towards the proof of the lifting theorem, and its proof is delicate. In order to help the reader getting first a general picture of the strategy used to prove the lifting theorem, we prefer to postpone the proof of this proposition at the end of this section. Proposition 10.17 Let X0 , X1 , . . . , Xq be free of step σ − 1 but not of step σ at 0. ej in Rn+1 of the form Then one can find vector fields X ej = Xj + uj (x) ∂ (j = 0, 1, . . . , q) X (10.4) ∂t with uj polynomial such that ej remain free up to the step σ − 1; (1) the vector fields X (2) for every s > σ, D  E


 e[I] dim X = dim X[I] 0 |I|6s + 1 0 |I|6s

where the symbol hYα iα∈B denotes the vector space spanned by the vectors Yα for α ∈ B.

Lifting and approximation

491

Proof. Let us show that condition 1 in the above statement holds for any choice of the functions uj (x) in (10.4). To see this, we first claim that (10.4) implies e[I] = X[I] + uI (x) ∂ X (10.5) ∂t for any multiindex I and some uI ∈ C ∞ (Rn ) . Namely, we can proceed by induction on ` (I) . For ` (I) = 1, this is just (10.4); assume (10.5) holds for ` (I) = j − 1. For ` (I) = j, let I = (i, J) for some i = 0, 1, . . . , q and ` (J) = j − 1. Then, by inductive assumption,  h i  e[I] = X e[i,J] = X ei , X e[J] = Xi + ui (x) ∂ , X[J] + uJ (x) ∂ = X ∂t ∂t  
∂ ∂ = Xi , X[J] + Xi uJ − X[J] ui = X[I] + uI (x) . ∂t ∂t ei are free of step σ − 1 at Next, we show that (10.5) implies that the vector fields X   P e[I] = 0 for constants aI , then by (10.5) we have 0. If aI X |I|6σ−1

0=

X |I|6σ−1

Since

∂ ∂t

0

 aI

∂ X[I] + uI (x) ∂t



 = 0

X

aI X[I]


0

+

 X

|I|6σ−1

|I|6σ−1

aI uI (0)

∂ . ∂t




is independent from the vectors X[I] 0 , this implies that X X
aI uI (0) = 0 and aI X[I] 0 = 0. |I|6σ−1

|I|6σ−1

But the vector fields Xi are free of step σ − 1 at 0, hence X aI AIJ = 0 for any J with |J| 6 σ − 1. |I|6σ−1

ei are free of step σ − 1 at 0. Therefore also the vector fields X We now show that it is possible to choose polynomial functions uj in (10.4) such that condition 2 in the statement of this proposition holds. To show this, we will prove that there exist polynomials uj and constants {aI }|I|6σ such that: X
aI X[I] 0 = 0 (10.6) |I|6σ

and X |I|6σ

  e[I] 6= 0 aI X 0

From (10.6)-(10.7), condition 2 will follow; namely,     X X
∂ e[I] = 0 6= aI X aI X[I] 0 + uI (0) ∂t 0 |I|6σ |I|6σ   X ∂ ∂ = aI uI (0) =b ∂t ∂t |I|6σ

(10.7)

492

H¨ ormander operators

with b 6= 0, hence that

∂ ∂t

=

D

P

aI |I|6σ b

e[I] X



 E

e[I] X

=



0 |I|6s



and this, keeping in mind (10.5), shows 0

X[I]


 0

 ⊕ |I|6s

∂ ∂t

 ,

which implies condition 2. To prove (10.6)-(10.7), we use our assumption on the vector fields Xi : since they are not free of step σ, there exist coefficients {aI }|I|6σ such that (10.6) holds but X aI AIJ 6= 0 for some J with |J| 6 σ. (10.8) |I|6σ

It remains to prove that there exist polynomials uj such that (10.7) holds for these uj ’s and aI ’s. To determine these uj ’s, let us examine the action of the vector field X X X e[I] = eJ aI aI X AIJ X |I|6σ

|I|6σ

|J|6σ

on the function t. For any J with |J| 6 σ, let us write J = (J 0 j) for some j = 0, 1, . . . , q. Then    ∂ eJ 0 uj = XJ 0 uj eJ t = X eJ 0 X ej t = X eJ 0 t =X X Xj + uj ∂t since uj does not depend on t. We then have:   X X X e[I] (t) (0) =  aI X aI AIJ (XJ 0 uj ) (0) . |I|6σ

|I|6σ

|J|6σ

where in the inner summation, J = (J 0 j). Note that  |J| − 1 for j = 1, 2, . . . , q |J 0 | = |J| − 2 for j = 0 hence, in any case, |J| 6 σ implies |J 0 | 6 σ − 1. Since the vector fields Xi are free of step σ − 1 at 0, by Proposition 10.16 for any choice of constants {cJ 0 }|J 0 |6σ−1 there exists a polynomial u ∈ C ∞ (Rn ) such that (XJ 0 u) (0) = cJ 0 . On the other hand, by (10.8), there exists a set of constants {cJ }|J|6σ such that X X aI AIJ cJ 6= 0. |I|6σ |J|6σ

Setting

cjJ 0

0

= cJ ifn J = o (J j) and applying q + 1 times Proposition 10.16 to the q + 1 j sets of constants cJ 0 , j = 0, 1, 2, . . . , q, we find polynomials u0 , u1 , . . . , uq |J 0 |6σ−1

such that 

 X

 |I|6σ

e[I] (t) (0) = aI X

X X

aI AIJ cJ 6= 0.

|I|6σ |J|6σ

Hence (10.7) holds. This completes the proof of the proposition.

Lifting and approximation

493

Remark 10.18 (Nonuniqueness of the lifting) The above proposition is the core of the constructive part of the lifting procedure, and from its proof we can realize the nonuniqueness of the polynomial function uj (x): the constants {aI }|I|6σ satisfying conditions (10.6) and (10.8) are by no reason uniquely determined. Also, once a particular set {aI }|I|6σ is chosen, any set {λaI }|I|6σ with λ 6= 0 would be good as well. The constants {cJ }|J|6σ are chosen (again, in a nonunique way) depending on {aI }|I|6σ ; finally, the polynomials uj (x) can be determined (by Proposition 10.16) in terms of the set {cJ }|J|6σ . In the end, we could, for instance, decide to normalize the constants {aI }|I|6σ and {cJ }|J|6σ in order to keep under control the C k norms of the polynomials uj (x), for some fixed k on some fixed neighborhood of the origin. This, n however, o would not allow to control the complicated algebraic structure of the e bases X[I] built by commutators of the lifted vector fields getting suitable lower I∈B

q

bounds (in terms of the system {Xi }i=0 ) on the determinants of the corresponding matrices. In turn, the quantitative properties of the diffeomorphism Θη (·) clearly depend also on this lower bound. This is the technical reason why we will not prove a statement expressing the dependence of the constants appearing in Theorem 10.6 q in terms of the system {Xi }i=0 . (See Remark 10.8). Theorem 10.19 (Lifting) Let X0 , X1 , . . . , Xq be vector fields defined in a neighborhood of the origin in Rn satisfying H¨ ormander’s condition of step s at x = 0. ek in Rn+m , of the form Then there exist an integer m and vector fields X m X ∂ ek = Xk + X ukj (x, t1 , t2 , . . . , tj−1 ) ∂t j j=1 ek ’s are free of step (k = 0,n 1, . . . ,q)o , where the ukj ’s are polynomials, such that the X n+m e[I] X span R . s and 0

|I|6s

This theorem has an obvious reformulation at any point x0 ∈ Rn , with the lifted vector fields defined in a neighborhood of (x0 , 0) ∈ Rn+m . Moreover, in view of Proposition 10.13, both the conclusions of the theorem (freeness and H¨ormander’s condition at step s for the lifted vector fields) will hold in a suitable neighborhood of this point. 
Proof. Let X[I] 0 I∈B be a basis of Rn , for some set B of n multiindices of weight 6 s. Recall that by Proposition 10.14 we have n 6 rank [AIJ ]|I|,|J|6s ≡ c (s, q) ,

(10.9)

an absolute constant only depending on s, q. Now, let σ 6 s be such that X0 , X1 , . . . , Xq are free of step σ − 1 but not of step σ, at 0. (If the vector fields Xi were already free of step s, there would be nothing to prove. We also agree to say that the vector fields Xi are free of step 0 if they are not free of step 1). We can then apply Proposition 10.17 and build vector fields ej = Xj + uj (x) ∂ (j = 0, 1, . . . , q) X ∂t

494

H¨ ormander operators

in Rn+1 , free of step σ − 1 and such that D  E


 e[I] dim X = dim X[I] 0 |I|6s + 1 = n + 1 0 |I|6s

n  o n e[I] span R ). Hence the X still 0 |I|6s 0 |I|6s n  o e[I] span the whole space Rn+1 . Now: either the vector fields X are free of (because by assumption the



X[I]




0

|I|6s

step s, and we are done, or the assumptions of Proposition 10.17 are still satisfied, and we can iterate our argument; in this case, by (10.9), condition n + 1 6 c (s, q) must hold. After a suitable finite number m of iterations, condition n + m 6 c (s, q) ej must be free of step cannot hold anymore, and this means that the vector fields X s. The iterative argument also shows that the ukj ’s are polynomials only depending on the variables x, t1 , t2 , . . . , tj−1 . We are left to prove Proposition 10.16. We want to give the reader an informal idea of the problem involved in its proof. We want to find a smooth function u which solves the system XI u(0) = cI for |I| 6 σ. For simplicity, think that the vector field X0 with weight two is not present, so that weight and length of a multiindex coincide. Writing explicitly the differential operators (XI )0 in terms of cartesian derivatives, we would find a linear system of equations X aiα Dα u (0) = ci |α|6σ

in the unknowns {Dα u (0)}|α|6σ . (The following reasoning is not rigorous). We can expect the number of equations to be less than the number of unknowns, since the differential operators XI are built on q vector fields, with q generally less than n. (Recall that I is a multiindex with entries in {1, 2, . . . , q} while α is a multiindex with entries in {1, 2, . . . , n}). Assume we can prove that this system is solvable (the assumption that the vector fields are free of step σ should imply the necessary independence among the equations), then we could easily define a function u possessing the assigned derivatives up to order σ at the origin, just taking u to Pσ be a suitable Taylor polynomial of degree σ. Also, writing u = k=1 uk with uk homogeneous polynomial of degree k, we can think that for each k = 1, 2, . . . , σ, uk is determined by the equations involving operators XI with |I| = k. This would decouple the system into groups of equations, making easier an iterative proof of its solvability. Trying to implement this na¨ıf idea, two main problems arise. Let us explain them working a concrete example. The vector fields of the Heisenberg group H1 are enough to appreciate the problem. Let X1 = ∂x1 + 2x2 ∂x3 ; X2 = ∂x2 − 2x1 ∂x3 .

Lifting and approximation

Then the system we have to solve is:   X1 u (0) = c1 ∂x1 u (0) = c1           X u (0) = c ∂ 2 2 x2 u (0) = c2        X 2 u (0) = c  ∂ 2 u (0) = c 11 11 x1 x1 1 that is 2   ∂x1 x2 u (0) − 2∂x3 u (0) = c12 X1 X2 u (0) = c12        X X u (0) = c    ∂x2 x u (0) + 2∂x3 u (0) = c21 2 1 21      2  21 2 X2 u (0) = c22 ∂x2 x2 u (0) = c22

495

(10.10)

Now, if we wanted to look for a solution u = u1 +u2 with ui homogeneous polynomial of degree i (in the usual sense), u1 solving the system of the first two equations and u2 solving the system of the last four equations, we would find (due to the vanishing of ∂x3 u2 (0)) the generally incompatible conditions ∂x21 x2 u2 (0) = c12 ∂x21 x2 u2 (0) = c21 . We can see that the system (10.10) is actually solvable, but in order to solve it we need to choose properly also the value ∂x3 u (0) which is not determined by the group of the first two equations. A solution u is the polynomial:  1 c21 − c12 x3 + c11 x21 + (c12 + c21 ) x1 x2 + c22 x22 . u (x1 , x2 , x3 ) = c1 x1 + c2 x2 + 4 2 The reader can appreciate that a symmetry issue arises here: cartesian mixed derivative commute while mixed derivatives with respect to the vector fields do not; this seems to threaten the solvability of the system. A second issue is the following. Trying to prove, in the abstract context, the solvability of the system, we need to exploit the only assumption we have, namely the fact that the vector fields are free of step σ. However, this assumption is formulated in terms of the commutators X[I] , while the system itself is written in terms of the differential monomials XI . Hence in order to exploit our assumption we have to reformulate the problem in a way involving commutators. This will be done seeing both the XI ’s and the X[I] ’s as particular polynomials in the vector fields. As the reader will see, the proof of Proposition 10.16 will exploit the language of polynomials in noncommuting variables, and a key step in the proof will consist in proving the symmetry of a suitable j-linear form. Proof of Proposition 10.16. Let us introduce some notations which will be used throughout this proof. We consider polynomials in the noncommuting variables ξ0 , ξ1 , . . . , ξq and we assign to ξ0 the weight p0 = 2, and to ξi , for i = 1, . . . , q the weight pi = 1. As we did with the vector fields X0 , X1 , . . . , Xq , for any multiindex I we define ξ[I] inductively letting ξ[(i)] = ξi if 0 6 i 6 q and ξ[I] = ξi ξ[I 0 ] − ξ[I 0 ] ξi if I = (i1 , i2 , . . . , ik ) and I 0 = (i2 , . . . , ik ). Finally, we let V be the vector space spanned by the monomials ξI , with |I| 6 σ, and V 0 be its dual space. Every function u ∈ C ∞ (Rn ) gives rise to the linear map Λu ∈ V 0 defined for p ∈ V by Λu (p) = p(X0 , . . . , Xq )u(0),

496

H¨ ormander operators

where we agree that if p = ξI , then p(X0 , . . . , Xq )u(0) = (XI u) (0) , and the definition is extended to general polynomials by linearity. Thus we have a mapping u 7→ Λu from C ∞ (Rn ) to V 0 and our aim is to show that it is surjective. More precisely, if L ∈ V 0 is defined letting L(ξI ) = cI (with L defined on the whole V by linearity), we have to find u ∈ C ∞ (Rn ) such that Λu = L. Let us denote with Vj the subspace of V spanned by the products ξ[I1 ] · · · ξ[Iν ] , with ν 6 j (and |I1 | + · · · + |Iν | 6 σ). Note that Vσ = V . We will show by induction on j, with 1 6 j 6 σ, that there exists a polynomial u such that Λu = L on Vj , that is X[I1 ] · · · X[Iν ] u(0) = L(ξ[I1 ] · · · ξ[Iν ] )

(10.11)

if ν 6 j and |I1 | + · · · + |Iν | 6 σ. If j = 1, then ν = 1 and (10.11) can be written as
X[I] u(0) = L ξ[I] for any |I| 6 σ. Since the vector fields Xi are free of step σ at 0, we have that X X
=⇒ aI ξ[I] = 0. aI X[I] 0 = 0 Namely,

P

|I|6σ

(10.12)

|I|6σ

|I|6σ

aI X[I]




= 0 implies that

P

aI AIJ = 0 for any J, hence  X X X X  = aI AIJ ξJ = aI AIJ  ξJ = 0. 0

|I|6σ



X |I|6σ

aI ξ[I]

|I|6σ

J

J

|I|6σ

By (10.12), there is a
(unique) linear
form defined on the span of the vectors
{ X[I] 0 }|I|6σ by X[I] 0 7→ L ξ[I] . We can extend this form to Rn and then find a first degree homogeneous polynomial, such that its (first
order) differential at 0, u(1) (0), coincides with such an extension. Since u(1) (0) X[I] 0 = X[I] u(0), the case j = 1 is done. Assume now that for any L ∈ V 0 there exists u0 ∈ C ∞ (Rn ) such that Λu0 = L on Vj−1 . This means that X[I1 ] · · · X[Iν ] u0 (0) = L(ξ[I1 ] · · · ξ[Iν ] )

(10.13)

when ν 6 j − 1 and |I1 | + · · · + |Iν | 6 σ. If u = u0 + v, with v vanishing of order j at 0 (in the usual sense), then Λu = L on Vj−1 , meaning that we must find v in such a way that (10.11) is solved for ν = j. In this case, the equation takes the form 

 v (j) (0) X[I1 ] 0 , . . . , X[Ij ] 0 = L(ξ[I1 ] · · · ξ[Ij ] ) − X[I1 ] · · · X[Ij ] u0 (0), (10.14) where v (j) (0) is the j-th differential of v at 0, seen as a j-linear form on Rn . Namely, X[I1 ] · · · X[Ij ] u(0) = X[I1 ] · · · X[Ij ] u0 (0) + X[I1 ] · · · X[Ij ] v(0) = L(ξ[I1 ] · · · ξ[Iν ] ) 

 but X[I1 ] · · · X[Ij ] v(0) simply equals v (j) (0) X[I1 ] 0 , . . . , X[Ij ] 0 , because all the derivatives of v of intermediate order (which appears expanding the differential operator X[I1 ] · · · X[Ij ] ) actually vanish because v vanishes of order j at 0.

Lifting and approximation

497

Thus, if we show that the right-hand side of (10.14) defines a symmetric j
linear form on the span of the tangent vectors X[I] 0 , where |I| 6 σ, then we are done, because we can then extend this form to Rn and therefore find a function v ∈ C ∞ (Rn ) vanishing of order j at 0, e.g. a j-th degree homogeneous polynomial, such that its j-th differential v (j) (0) at 0 coincides with the extended j-linear form. Note that each of the two terms at the right hand side of (10.14) is surely not symmetric, however we are claiming that its difference actually is. So, let H be the form defined by 

 H : X[I1 ] 0 , . . . , X[Ij ] 0 7→ L(ξ[I1 ] · · · ξ[Ij ] ) − X[I1 ] · · · X[Ij ] u0 (0). The check that this j-linear form is actually well defined amounts to show that for any choice of constants n o n o n o (1) (2) (j) aI1 , aI2 , . . . , aIj |I1 |6σ

such that X

(1)

aI1

X[I1 ]


0

= 0,

X

|I2 |6σ

(2)

X[I2 ]

a I2


0

|Ij |6σ

= 0, . . . ,

(j)

aIj

X[Ij ]


0

=0

|Ij |6σ

|I2 |6σ

|I1 |6σ

X

we have (1) (2)

X

(j)

aI1 aI2 . . . aIj



L(ξ[I1 ] · · · ξ[Ij ] ) − X[I1 ] · · · X[Ij ] u0 (0) = 0

|I1 |6σ,|I2 |6σ,...,|Ij |6σ

This is almost the same as in the
j = 1 case. Indeed, the implication (10.12) still P (i) P (i) aIi ξ[Ii ] = 0 for i = 1, 2, . . . , q; hence holds, and therefore aIi X[Ii ] 0 = 0 =⇒ X (1) (2) (j)  aI1 aI2 . . . aIj L(ξ[I1 ] · · · ξ[Ij ] ) − X[I1 ] · · · X[Ij ] u0 (0)  X X (2) X (j) (1) aI1 ξ[I1 ] aI2 ξ[I2 ] . . . aIj ξ[Ij] + =L X (1) X (2) X (j) − aI1 X[I1 ] aI2 X[I2 ] . . . aIj X[Ij ] u0 (0) = 0. To show the symmetry of H let us introduce dI1 ,...,Ij = L(ξ[I1 ] · · · ξ[Ij ] ) − X[I1 ] · · · X[Ij ] u0 (0), and let us prove that they are symmetric in the (multi)indices. By Lemma 9.30, we can write X [ξ[I] , ξ[J] ] = bK ξ[K] , |K|=|I|+|J|

where the bK ’s are absolute constants, only depending on the multiindices I, J, K. Let us show now the desired symmetry result. It is clearly sufficient to show it for consecutive indices. We limit ourselves to verify the symmetry with respect to the first two indices, the other cases being a straightforward generalization of it. Indeed, we have that dI1 ,I2 ,...,Ij − dI2 ,I1 ,...,Ij = L([ξ[I1 ] , ξ[I2 ] ] · · · ξ[Ij ] ) − [X[I1 ] , X[I2 ] ] · · · X[Ij ] u0 (0) X
= bK L(ξ[K] ξ[I3 ] · · · ξ[Ij ] ) − X[K] X[I3 ] · · · X[Ij ] u0 (0) = 0 |K|=|I|+|J|

498

H¨ ormander operators

by the induction hypothesis (10.13). Now we are (almost) done, because we have shown that the
right-hand side of (10.14) defines a symmetric j-linear form on the span of { X[I] 0 )}|I|6σ . As we did for j = 1, we can extend this form to Rn and then find a function v ∈ C ∞ (Rn ) vanishing of order j at 0, e.g. a j-th degree homogeneous polynomial, such that its j-th differential v (j) (0) at 0 coincides with the extended j-linear form. This completes the proof. Example 10.20 Let us illustrate the above proof on the example of H1
. The linear form L satisfies L (ξ1 ξ2 ) = c12 , L (ξ2 ξ1 ) = c21 and, by linearity, L ξ[1,2] = c12 −c21 . The function u0 has been defined solving the equations X1 u0 (0) = L (ξ1 ) = c1 , X2 u0 (0) = L (ξ2 ) = c2 ,
[X1 , X2 ] u0 (0) = L ξ[1,2] = c12 − c21 that is (keeping the notation used before the proof ) ∂x1 u0 (0) = c1 , ∂x2 u0 (0) = c1 ,

− 4∂x3 u0 (0) = c12 − c21

hence 1 (c12 − c21 ) x3 . 4 Now we seek u = u0 + v, with v vanishing of order 2 in the usual sense, satisfying u0 (x1 , x2 , x3 ) = c1 x1 + c2 x2 −

X12 u (0) = c11 , X1 X2 u (0) = c12 X2 X1 u (0) = c21 , X22 u (0) = c22 that is, recalling that (X1 X2 )0 = ∂x21 x2 − 2∂x3 and (X2 X1 )0 = ∂x21 x2 + 2∂x3 , X12 v (0) = ∂x21 x1 v (0) = c11 − X12 u0 (0) = c11 1 (c12 − c21 ) = 2 1 X2 X1 v (0) = ∂x22 x1 v (0) = c21 − X2 X1 u0 (0) = c21 + (c12 − c21 ) = 2 X22 v (0) = ∂x22 x2 v (0) = c22 − X22 u0 (0) = c22 .

X1 X2 v (0) = ∂x21 x2 v (0) = c12 − X1 X2 u0 (0) = c12 −

1 (c12 + c21 ) 2 1 (c12 + c21 ) 2

As we can see, the second and third equations are now the same, that is the symmetry problem has been overcome, and we can pick  1 c11 x21 + (c12 + c21 ) x1 x2 + c22 x22 v (x1 , x2 , x3 ) = 2 which, added to u0 gives the desired solution u. 10.3

Approximation of free vector fields with left invariant homogeneous vector fields

In this section we carry out the second part of the procedure which has been described in section 10.1, that is the approximation of free vector fields by left invariant vector fields on a homogeneous group. By the lifting theorem (Theorem 10.19),

Lifting and approximation

499

starting from any system of vector fields satisfying H¨ormander’s condition at step s e0 , . . . , X eq in some neighborhood of the origin in Rn we can define new vector fields X N n+m in a neighborhood  U
of 0 ∈ R ≡ R N that are free up to step s at any point of U e[I] spans R for any ξ ∈ U . We start with the following and such that X ξ |I|6s e0 , X e1 , . . . , X eq is a system of vector fields in a bounded domain Remark 10.21 If X N U ⊂ R , free up to the step s and satisfying H¨ ormander’s condition of step s in U , then it is possible to choose a set B of N multiindices I with |I|
6 s, such that  N e[I] e[I] is a basis is a basis of R at every point ξ ∈ U. Indeed, if X X ξ I∈B I∈B 
e[I] are of RN for some ξ ∈ U , then for any other η ∈ U the N vectors X η I∈B

linearly independent (hence a basis of RN ); otherwise by Remark 10.12 we should have, for some coefficients cI with at least one nonzero cI ,  X  X e[I] = 0 =⇒ e[I] ≡ 0, cI X cI X I∈B

in particular



e[I] X


ξ I∈B

η

I∈B

would not be a basis.

We assume this set B fixed once and for all. 10.3.1

Canonical coordinates and weights of vector fields

For any ξ ∈ U, let us introduce in a neighborhood of ξ the set of local (“canonical”) coordinates (already used in the previous chapter, see (9.6)) X

e[I] ξ , RN  u 7−→ Φξ,B (u) = exp (10.15) uI X I∈B

defined for u in a suitable neighborhood of 0. Note that since card (B) = N we can represent points of RN as (uI )I∈B . Recall that    
 ∂Φξ,B d  e[J] , e[J] ξ = X (0) = exp uJ X ∂uJ duJ ξ /uJ =0 hence the Jacobian of the map Φξ,B at u = 0, equals the matrix of the vector fields
 e[I] X and therefore it is nonsingular. ξ I∈B
This allows to define canonical coordinates in a suitable neighborhood U ξ of ξ. 
e[I] depends continuously on the point ξ, the Moreover, since the basis X ξ I∈B radius of this neighborhood can be taken uniformly bounded away from zero for ξ ranging in a compact set. Henceforth in this section, all the computations will be made with respect to this system of coordinates defined in a neighborhood of the point ξ (which has canonical coordinates u = 0). First of all, we need to check that expressing the vector fields in terms of these new variables preserve their important properties:

500

H¨ ormander operators

Proposition 10.22 Let ξ = T (u) be
a local diffeomorphism from a neighborhood U (0) 3 u onto a neighborhood V ξ 3 ξ such that ξ = T (0). Assume that eξ, X eξ, . . . , X eqξ (the superscript ξ recalls that the vector fields are expressed in terms X 0 1
of the ξ-variables) satisfy H¨ ormander’s condition at step s in V ξ and are free at e u, X e u, . . . , X equ through the diffeostep s at ξ. Then the transformed vector fields X 0 1 morphism satisfy H¨ ormander’s condition at step s in U (0) and are free at step s at 0 (and therefore, in a neighborhood of 0). Proof. Denote by JT −1 the Jacobian matrix of the inverse of T . Then: eξ = X

N X

eu = bj (ξ) ∂ξj =⇒ X

j=1

N X

ebj (u) ∂u j

j=1

e (u) = (JT −1 b) (T (u)) or, explicitly, with b ebj (u) =

N X

∂ T −1 (·)




∂ξh

h=1

! j

bh

(T (u)) .

e ξ , Y ξ one has: Using this relation one can check that for any two vector fields X h iu h i eξ, Y ξ = X e u, Y u X u  eξ e u . Since a diffeoeξ, X eξ, . . . , X eqξ , X = X and, iteratively, for our system X 0 1 [I] [I] morphism turns independent vectors into independent vectors, this implies that if
ξ e u ’s satisfy e the Xi ’s satisfy H¨ ormander’s condition at step s in V ξ , then the X i H¨ ormander’s condition at step s in U (0). Next, assume    X eu = 0. aI X [I]

|I|6σ

0

Then u   e[I]  . aI X

 0=

X |I|6σ

aI



e[I] X

u  0

=

X

|I|6σ

ξ

Since T is a diffeomorphism, a vector Y ξ vanishes if and only if Y u vanishes, hence   X e[I] aI X 0= |I|6σ

ξ

e ξ ’s are free up to step σ at ξ, implies P which, since the X i |I|6σ aI AIJ = 0 for every u e J, so that the X ’s are free up to step σ at 0. i

e u , recalling that all our vector From now on we will skip the superscript u from X i fields will be expressed in canonical coordinates and all functions will be defined in a neighborhood of the origin. We start with the following:

Lifting and approximation

Lemma 10.23 In the canonical coordinates we have X X e[I] . uI ∂uI = uI X I∈B

501

(10.16)

I∈B

P P ∂ ∂ Proof. We start by noting that, if Y = I∈B yI (u) ∂u and Z = I∈B zI (u) ∂u I I are two vector fields such that Z (uJ ) = Y (uJ ) for any J ∈ B, (that is, the vector fields act at the same way on the functions u 7→ uJ ) then yI (u) = zI (u) for any I ∈ B, hence Y = Z. Therefore, it will be enough to show that ! ! X X ∂ e[I] (uJ ) = uI (uJ ) uI X ∂uI I∈B I∈B P  e[I] (uJ ) = uJ . Now, for any vector field Y, that is u X I I∈B



d f exp (tY ) ξ where ξ = exp (t0 Y ) ξ . Y f (ξ) = dt t=t0 P e Hence, if Y = I∈B uI X[I] , by definition of the coordinates uI ! !! ! X X
d e[I] (uJ ) = e[I] ξ uI X uJ exp t uI X dt I∈B I∈B t=t0 d = = uJ . (tuJ ) dt t=t0 Recall that the aim of this section is to show that a system of free H¨ormander vector fields can be locally approximated, in a suitable coordinate system, by a family of left invariant homogeneous vector fields on a homogeneous group. In Chapter 3 we have seen that if Y0 , Y1 , . . . , Yq is such a system of homogeneous vector fields, then a commutator Y[I] is homogeneous of degree |I|, which in particular means that if f (u) is a function which is smooth outside the origin and homogeneous of degree α ∈ R, then Y[I] f (u) is homogeneous of degree α − |I|. In particular, if α < 0, that is f has a singularity at the origin, the action of Y[I] makes its −|I| singularity worse by a factor kuk . For our system of free H¨ormander vector fields, we cannot speak of homogeneity, but we still want to keep under control the amount of singularity that the action of the vector field on a function having a singularity at some point introduces. This control is expressed by the notion of weight of a vector field, which we now introduce. This notion will be also used to measure the error we make approximating a free vector field with a suitable left invariant homogeneous one. Definition 10.24 (Weights)2 We assign the weight |I| to the coordinate uI and the weight − |I| to ∂uI . In the following we will say that a C ∞ function f has 2 We

alert the reader that the above convention about weights of functions and differential operators is the one made in [108], and is different from that made in [85] and [142]: in the last two papers, the authors assign positive weight also to derivatives.

502

H¨ ormander operators

weight > σ if the Taylor expansion of f at the origin does not include terms of the kind auI1 uI2 · · · uIk with a 6= 0 and |I1 | + |I2 | + . . . + |Ik | < σ. A vector field P Y = I∈B fI ∂uI has weight > σ if fI has weight > σ + |I| for every I ∈ B. Note that the weight of a smooth function is always > 0, while the weight of a vector field is > −s, since |I| 6 s for every I ∈ B. We want to stress that the definition of weight relies on the canonical coordinates, therefore it depends on the choice of a particular basis B of RN . In order to quantify the properties of functions and vector fields as far as weights are concerned, we need to introduce some notation and terminology: Definition 10.25 We denote with Fσp the set of (smooth) functions such that in their Taylor expansion at the origin of degree 6 p (in the standard sense), all terms have weight > σ. Also, Vσp will denote the set of the vector fields with a similar property. More e = P cI (u) ∂u is a vector field, we take the Taylor expansion (at explicitly, if X I I the origin) of degree 6 p of each cI (u) and require that for each term cIK (u) of this expansion, the weight of the differential operator cIK (u) ∂uI be > σ, which happens if cIK (u) has weight > σ + |I|. Finally, the subsets of Fσp and Vσp of elements that vanish at u = 0 will be ˚σp . ˚σp and V denoted by F Lemma 10.26 The following inclusions hold p Fσp Fτp ⊂ Fσ+τ

˚p ˚σp Fτp−1 ⊂ F F σ+τ

p Fσp Vτp ⊂ Vσ+τ

˚p V p−1 ⊂ V ˚p F σ τ σ+τ

p−1 Vσp−1 (Fτp ) ⊂ Fσ+τ

˚p ˚σp (Fτp ) ⊂ F V σ+τ i h p−1 p ˚σ , Vτ ⊂ V p−1 V

p−1 [Vσp , Vτp ] ⊂ Vσ+τ

˚p ⊂ V ˚p Fσp−1 V τ σ+τ

σ+τ

with the obvious meaning of the symbols. Proof. If f ∈ Fσp and g ∈ Fτp , then all terms of the product of their Taylor expansion of degree 6 p have weight > σ + τ . Therefore, the same is true for the p Taylor expansion of degree 6 p of f g, so that f g ∈ Fσ+τ . This shows that the ˚p and g ∈ F p−1 the first inclusion holds. As to the second one, note that if f ∈ F σ τ expansion of f g of degree p contains the terms of the expansion of g only up to degree p − 1, hence the conclusion follows. The other inclusions can be proved by means of similar arguments. The next theorem contains a fundamental piece of information about the vector e[I] expressed in canonical coordinates, which parallels the properties of left fields X invariant homogeneous vector fields on a homogeneous group:

Lifting and approximation

503

Theorem 10.27 (Weight of a vector field) For every multiindex I, the vector e[I] has weight > − |I|. field X Proof. We shall prove by induction on p > 0 that for every multiindex I we have e[I] ∈ V p when e[I] ∈ V p . First of all we observe that it is enough to show that X X −|I| −|I| e[I] ∈ V p for I ∈ B. Indeed let us fix a certain p > 0 and assume to know that X −|I|

e every I ∈ B. Let J be a multiindex, J ∈ / B. n Foroany vector field X[J] with |J| 6 σ, e[J] in terms of the basis X e[I] we can express X , at any point, writing I∈B

e[J] = X

X

e[I] cJI X

I∈B

for suitable constants cJI (see the proof of Proposition 10.13). Moreover, since the e0 , X e1 , . . . , X eq are free up to step s, we can assume that in the last vector fields X sum cJI is nonzero only if |I| = |J|. Hence the last identity shows that it is enough to prove i) for I ∈ B. Now, to begin our inductive proof, let p = 0. Observe that composing the operator [∂uI , ·] with (10.16) we get i h X e[I] + e[K] (10.17) ∂uI = X uK ∂uI , X K∈B

This implies 

e[I] X

 0

= ∂uI

(10.18) 



e[I] ∈ V 0 . (By the way, note that the property X e[I] = ∂u and therefore that X I −|I| 0 is shared with the system of vector fields forming the canonical basis of a stratified Lie algebra, see Chapter 3). e[I] ∈ V p for every I, and let us prove that Assume now that, for a certain p, X −|I| the same is true with p replaced by p + 1. We claim that it is enough to show that for every J ∈ B h i e[J] , ∂u ∈ V p W = X I −|I|−|J| h i e[K] ∈ V p+1 by Lemma 10.26, and since Indeed, if this is true then uK ∂uI , X −|I| p+1 e[I] ∈ V p+1 . ∂uI ∈ V−|I| by (10.17) we have X −|I| h i p e[J] , · with (10.17). This yields To show that W ∈ V−|I|−|J| we compose X h i X h h ii h i X e[J] , X e[I] + e[J] , ∂u , X e[K] + + e[J] (uK ) ∂u , X e[K] . W = X uK X X I I K∈B

K∈B

(10.19) Also, from (10.17) we get e[J] (uK ) = δJK − X

X L∈B

h i e[L] (uK ) . uL ∂uJ , X

(10.20)

504

H¨ ormander operators

e[L] ∈ V p , hence by Lemma 10.26 By inductive assumption, X −|L| h i e[L] ∈ V p−1 ∂uJ , X −|J|−|L| h i e[L] ∈ V ˚p uL ∂uJ , X −|J| h i e[L] (uK ) ∈ F ˚p uL ∂ u , X

(10.21)

−|J|+|K|

J

and by (10.20) e[J] (uK ) − δJK ∈ F ˚p X −|J|+|K| which by (10.21) and Lemma 10.26 implies h i X e[J] (uK ) − δJK ∂u , X e[K] ∈ V ˚p X I −|J|−|I| . K∈B

On the other hand, h i X e[J] (uK ) − δJK ∂u , X e[K] X I K∈B

=

X

i i h i h h X e[J] (uK ) ∂u , X e[K] + W e[J] = e[J] (uK ) ∂u , X e[K] − ∂u , X X X I I I K∈B

K∈B

which can be rephrased writing: h i X e[J] (uK ) ∂u , X e[K] ≡ −W mod V ˚p X I −|J|−|I| .

(10.22)

K∈B

By Lemma 9.30, h

i e[J] , X e[I] = X

X

e[L] cL X

|L|=|J|+|I|

i h e[J] , X e[I] ∈ for suitable coefficients cL . This implies, by the inductive assumption, X p V−|J|−|I| . Hence by (10.22) and (10.19) ii h h X p e[J] , ∂u , X e[K] mod V−|J|−|I| W ≡ −W + uK X . I

(10.23)

K∈B

By the Jacobi identity h h ii h h ii h h ii e[J] , ∂u , X e[K] = −uK X e[K] , X e[J] , ∂u e[K] , X e[J] . uK X − u ∂ , X K u I I I ˚p We claim that the last term in the previous identity belongs to V −|J|−|I| . Namely, h i e[K] , X e[J] ∈ V p X −|J|−|K| by the inductive assumption while h h ii e[K] , X e[J] ∈ V p−1 ∂uI , X −|J|−|K|−|I|

Lifting and approximation

505

and h h ii e[K] , X e[J] ∈ V ˚p uK ∂uI , X −|J|−|I| by Lemma 10.26. Hence h h ii h h ii p e[J] , ∂u , X e[K] ≡ −uK X e[K] , X e[J] , ∂u uK X mod V−|I|−|J| I I which inserted in (10.23) gives h i X e[K] , W mod V p 2W ≡ − uK X −|I|−|J| .

(10.24)

K

e[K] by ∂u . Indeed we have We now use (10.16) to replace X K " # h i X X X e[K] , W = e[K] , W + e[K] uK X uK X W (uK ) X K

K

K

" = =

# X K

K

X

X

uK [∂uK , W ] +

K

h

i

e[J] , ∂u ∈ Since W = X I e ˚p (10.18) X[K] − ∂u ∈ V

X

uK ∂uK , W +

e[K] W (uK ) X

  e[K] − ∂u . W (uK ) X K

K

p−1 V−|I|−|J|

p−1 we have W (uK ) ∈ F|K|−|I|−|J| . Also, since by

we have   p e[K] − ∂u W (uK ) X ∈ V−|I|−|J| K P p and, by (10.24), 2W ≡ − K uK [∂uK , W ] mod V−|I|−|J| . Now, let us define the follows operator that acts on vector fields: X T : X 7→ 2X + uK [∂uK , X] . (10.25) K

−|K| ,

K p p We know that T W ∈ V−|I|−|J| , and we claim that this implies that W ∈ V−|I|−|J| . P Actually, for a vector field X = L fL ∂uL , we have: X X TX = 2 fL ∂uL + uK [∂uK , fL ∂uL ] =

=2

L

L,K

X

X

L

fL ∂uL +

L,K

X X ∂fL ∂fL uK ∂uL = 2fL + uK ∂uK ∂uK L

! ∂uL .

K

Let g be a homogeneous function of degree µ, then, by Euler’s theorem on homogeneous functions, X ∂g = µg uK ∂uK K P ∂f which shows that the operator f 7→ 2f + K uK ∂u acts on the Taylor expansion K of a function multiplying a term of degree µ by (2 + µ). This implies that p p T X ∈ V−|I|−|J| ⇔ X ∈ V−|I|−|J| . p p Hence W ∈ V−|I|−|J| , since T W ∈ V−|I|−|J| , and we are done.

506

10.3.2

H¨ ormander operators

Pointwise approximation

Summarizing the assumptions and results of the previous subsection, we assume e0 , X e1 , . . . , X eq are free up to step s in a bounded domain that the vector fields X    e[I] U ⊂ RN and for some fixed point ξ ∈ U , a basis X for RN is chosen ξ

I∈B

once and for all; this choice induces a system of canonical coordinates u, mapping a neighborhood of ξ onto a neighborhood of the origin in the space of u variables, such that expressing the vector fields in canonical coordinates the following relation holds: X X e[I] uI ∂uI = uI X (10.26) I∈B

I∈B

e[I] has weight > − |I|, in the sense of and for every multiindex I the vector field X Definition 10.24. We can now prove the approximation theorem for free weighted vector fields: Theorem 10.28 (Approximation, pointwise version) Let Y0 , Y1 , . . . , Yq be another system of vector fields defined in a neighborhood of the origin using the e[I] and satisfying canonical coordinates {uI }I∈B induced by X X X uI ∂uI = uI Y[I] . (10.27) I∈B

I∈B

Then e[I] − Y[I] has weight > 1 − |I| X ei − Yi for any multiindex I with |I| 6 s. In particular, for I = (i) we have that X e0 − Y0 has weight > −1. has weight > 0 for i = 1, 2, . . . , q while X Proof. The proof exploits the same techniques as in the proof of Theorem 10.27. We shall prove by induction on p that e[I] − Y[I] ∈ V p . X 1−|I|

(10.28)

Observe that this is obvious when |I| > s. Indeed, every vector field Z = P J∈B fJ ∂uJ has a weight > −s since if J ∈ B then |J| 6 s. Next we prove that it is enough to show (10.28) n when  o I ∈ B. Indeed, let I be any multiindex e[J] of weight 6 s. Then since X spans RN there exist coefficients cIJ 0 J∈B     e[I] = P e such that X . Let aJ = δIJ − cIJ where we assume that J∈B cIJ X[J] 0  0  P e[J] = 0. Since the vector fields are free of cIJ = 0 if J ∈ / B. Then |J|6s aJ X 0P step s at 0, this implies that for every K, |J|6s aJ AJK = 0. Therefore, for any family of (weighted) vector fields Z0 , Z1 , . . . , Zq we have X X X X X 0= aJ AJK ZK = aJ AJK ZK = aJ Z[J] K, |K|=|I| J

J, |J|=|I| K

|J|=|I|

Lifting and approximation

and finally Z[I] =

P

J∈B,|J|=|I| cIJ Z[J] .

X

e[I] = X

e[J] cIJ X

507

In particular we have X and Y[I] = cIJ Y[J] .

J∈B,|J|=|I|

J∈B,|J|=|I|

These identities show that if (10.28) holds for I ∈ B, then it holds for every I. Now we prove that (10.28) holds for I ∈ B. Let p = 0. We have seen in the proof of Theorem 10.27 that (10.27) implies (see (10.17)) h i X e[I] + e[K] = ∂u X uK ∂uI , X (10.29) I K∈B

and Y[I] +

X

  uK ∂uI , Y[K] = ∂uI .

(10.30)

K∈B

 
e[I] = ∂u and Y[I] = ∂u , hence (10.28) holds for p = 0. In particular, X I I 0 0

We now assume that (10.28) holds for a certain p and we prove that the same is true with p replaced by p + 1. We claim that it is enough to show that h i e[J] − Y[J] , ∂u ∈ V p Z= X (10.31) I 1−|I|−|J| . Indeed, by (10.29), (10.30) we have e[I] − Y[I] = X

X

h i e[K] − Y[K] , ∂u uK X I

(10.32)

K∈B

h i e[K] − Y[K] , ∂u ∈ V p+1 . Let us show and by Lemma 10.26, (10.31) implies uK X I 1−|I| h i h i p e[J] − Y[J] , ∂u ∈ V e[J] , · , (10.30) that Z = X . Composing (10.29) with X I 1−|I|−|J|   with Y[J] , · and taking the difference gives h i   e[J] , X e[I] − Y[J] , Y[I] Z= X h ii X  Xh   e[J] , uK ∂u , X e[K] − + X Y[J] , uK ∂uI , Y[K] I K∈B

h

e[J] , X e[I] = X

K∈B

i

ii h   Xh e[J] − Y[J] , uK ∂u , X e[K] − Y[J] , Y[I] + X I

(10.33)

K∈B

h ii Xh e[K] − Y[K] + Y[J] , uK ∂uI , X K∈B

By Lemma 9.30 h i   e[J] , X e[I] − Y[J] , Y[I] = X

X

  e[K] − Y[K] cK X

|K|=|J|+|I|

with cK universal constants, hence by inductive hypothesis h i   e[J] , X e[I] − Y[J] , Y[I] ∈ V p X 1−|I|−|J| .

(10.34)

508

H¨ ormander operators

e[K] ∈ V p+1 , by Lemma 10.26 we have Also, since by Theorem 10.27 X −|K|i h i h p e e ˚p+1 . Since X e[J] − Y[J] ∈ ∂uI , X[K] ∈ V−|I|−|K| and therefore uK ∂uI , X[K] ∈ V −|I| p we have V1−|J|

h

h ii e[J] − Y[J] , uK ∂u , X e[K] ∈ V p X I 1−|I|−|J| .

(10.35)

p we have By (10.33), (10.34), (10.35) we read that modulo V1−|I|−|J| h h ii X e[K] − Y[K] Z≡ Y[J] , uK ∂uI , X K∈B

=

Xh

h ii X h h ii e[K] − Y[K] + e[K] − Y[K] Y[J] − ∂uJ , uK ∂uI , X ∂uJ , uK ∂uI , X K∈B

K∈B

≡

Xh

h

e[K] − Y[K] ∂uJ , uK ∂uI , X

ii

K∈B

h i ˚p+1 and uK ∂u , X e[K] − Y[K] ∈ V p . So, we have since by (10.30) Y[J] − ∂uJ ∈ V I −|J| 1−|I| proved that h ii Xh p e[K] − Y[K] Z≡ ∂uJ , uK ∂uI , X mod V1−|I|−|J| (10.36) K∈B

and since h  i h i   e[K] − Y[K] = uK ∂u , X e[K] − Y[K] + δIK X e[K] − Y[K] ∂uI , uK X I we have Z≡

X nh

h  ii h  io e[K] − Y[K] e[K] − Y[K] ∂uJ , ∂uI , uK X − ∂uJ , δIK X

K∈B

"

"

= ∂uJ , ∂uI ,

X



e[K] − Y[K] uK X



##

h i e[I] − Y[I] − ∂uJ , X

K∈B

h

i h i e[I] − Y[I] = X e[I] − Y[I] , ∂u = − ∂uJ , X J   P e since = 0 by (10.26) and (10.27). This shows that for K∈B uK X[K] − Y[K] every multiindex I and J h i h i p e[J] − Y[J] , ∂u ≡ X e[I] − Y[I] , ∂u X mod V1−|I|−|J| . I J Using this in (10.36) yields h ii h ii Xh Xh e[K] − Y[K] ≡ e[I] − Y[I] Z≡ ∂uJ , uK ∂uI , X ∂uJ , uK ∂uK , X K∈B

K∈B

h i X h h ii e[I] − Y[I] + e[I] − Y[I] ≡ ∂uJ , X uK ∂uJ , ∂uK , X K∈B

h i X h h ii e[I] − Y[I] + e[I] − Y[I] = ∂uJ , X uK ∂uK , ∂uJ , X K∈B

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509

where the last identity can be checked exploiting the fact that ∂uK and ∂uJ comP mutes. Hence Z ≡ −Z − K∈B uK [∂uK , Z] so that X p . 2Z + uK [∂uK , Z] ≡ 0 mod V1−|I|−|J| K∈B

We can now reason as in the proof of Thm. 10.27 (the argument for the operator T p . defined in (10.25)) and conclude from the last relation that Z ∈ V1−|I|−|J| 10.3.3

From pointwise to local. The map Θ

In order to turn Theorem 10.28 into a really useful statement, some work has still to be done. First, we have to pass from the pointwise statement of Theorem 10.28 to an analogous local statement. This involves the introduction of a suitable diffeomorphism, the “map Θ”, and the study of some of its properties. Second, we have to apply this theorem to the case where the vector fields Yi are homogeneous left invariant with respect to a structure of homogeneous group, and deduce some information on the “remainders” in this approximation procedure. These tasks will be performed in this and the next subsection, respectively. We now revise the construction of local coordinates uI made in section 10.3.1. Let U be as at the beginning of section 10.3; for any U 0 b U there exists a neighborhood U (0) ⊂ RN such that for every η ∈ U 0 the map X
e[I] (η) (10.37) Φη,B : u ≡ (uI )I∈B 7−→ ξ ≡ exp uI X I∈B

is well defined and smooth; moreover, the map (u, η) 7→ Φη,B (u) is smooth in the joint variables (u, η) ∈ U (0) × U 0 . Next, we define F (u, η, ξ) = Φη,B (u) − ξ on U (0) × U 0 × RN . Noting that F (0, η, η) = 0 and that the Jacobian

of F with e respect to the u variables, at (0, η, η) , has determinant det X[I] η , which does  e[I] not vanish since X span RN , by the implicit function theorem we can I∈B

define a function u = Θ (η, ξ), smooth in some neighborhood W of (η, η), such that Φη,B (Θ (η, ξ)) = ξ. Summarizing the above discussion we can state the following: Proposition 10.29 (The map Θ) Up to possibly shrinking the neighborhood U ⊂ RN previously defined, the following hold. i) There exist a neighborhood W of {(η, η) : η ∈ U } in R2N , a neighborhood V of 0 in RN and a smooth map Θ (·, ·) : W → V such that: X
e[I] (η) for u = Θ (η, ξ) ; ξ = exp uI X (10.38) I∈B

ii) The map Θ satisfies Θ (ξ, η) = −Θ (η, ξ) ;

(10.39)

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H¨ ormander operators

iii) for any fixed η ∈ U, the map u = Θ (η, ξ) is a diffeomorphism from a neighborhood of η onto a neighborhood of 0, in RN ; iv) analogously, for any fixed ξ ∈ U, the map u = Θ (η, ξ) is a diffeomorphism from a neighborhood of ξ onto a neighborhood of 0. Proof. We have already proved (i) and (iii); (ii) follows from the fact that, for e by the definition of exponential, if ξ = exp X e (η) then η = any vector field X,   e (ξ); (iv) is then a consequence of (ii) and (iii). exp −X Recall that a vector field Z has weight k at some fixed point η if Z, expressed in terms of the local coordinates u = Θ (η, ξ) , has weight k at u = 0, in the sense of Definition 10.24. Also, recall that if we denote by Z ξ and Z u , the vector field Z written as a differential operator which acts on the variables ξ or u, respectively, then Z ξ [f (Θ (η, ξ))] = (Z u f ) (Θ (η, ξ)) .

(10.40)

for any smooth function f (u). We can then restate Theorem 10.28 as follows: Theorem 10.30 (Approximation, local version) For every multiindex I the e[I] has weight > − |I| at any point of U . If Y0 , Y1 , . . . , Yq is any other vector field X system of vector fields defined in a neighborhood of the origin using the coordinates {uI }I∈B and satisfying X X uI ∂uI = uI Y[I] (10.41) I∈B

I∈B

e[I] − Y[I] has weight > 1 − |I| at any point of U . Moreover, for any point then X η ∈ U there exists a system of vector fields Rη,[I] , of weight > 1 − |I| at η (when expressed in the coordinates u) and smoothly depending on the point η, such that

e[I] [f (Θ (η, ·))] (ξ) = Y[I] f (Θ (η, ξ)) + Rη,[I] f (Θ (η, ξ)) . X (10.42) Finally, the vector fields Rη,[I] vanish at u = 0. Proof. The first part of the theorem is exactly Theorem 10.27 and Theorem 10.28, e[I] − Y[I] has weight > 1 − |I| at η, just by stated at any point η. Saying that X e u − Y[I] has weight > 1 − |I| at definition means that the vector field Rη,[I] = X [I] e u emphasizes that this vector field is expressed u = 0. Here the superscript u in X [I]

in terms of the coordinates u. By (10.40), we can rewrite it in terms of coordinates ξ, getting (10.42). It remains to check that Rη,[I] depends smoothly on η. Let X Rη,[I] = bIJ (η, u) ∂uJ ; J

then, applying (10.42) to the function f (u) = uJ we get
e[I] [(Θ (η, ·)) ] (ξ) − Y[I] uJ (Θ (η, ξ)) . bIJ (η, Θ (η, ξ)) = X J

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The right-hand side of this equation is a smooth function of (η , ξ) , since Θ is smooth (see Proposition 10.29); hence the functions (η , ξ) 7−→ bIJ (η, Θ (η, ξ)) are smooth; fixing ξ and composing with the diffeomorphism u = Θ (η, ξ) we read that bIJ (η, u) are smooth functions. The fact that the vector fields Rη,[I] vanish at u = 0 follows from the factthat,as we have seen in the proof of Theorem 10.28, in

e[I] = ∂u = Y[I] , hence Rη,[I] = 0. the coordinates u we have X I 0 0 0

10.3.4

Approximation by left invariant vector fields

Although Theorem 10.30 contains a general result, allowing to approximate the ei , in a suitable coordinate system, by any other system of system of vector fields X vector fields Yi satisfying (10.41), what makes this fact really useful is the possibility of choosing as approximating vector fields a family of homogeneous left invariant vector fields on a homogeneous group. This, however, requires an abstract construction. We will state here some results, and show their consequences about our approximation problem, postponing to the last section in this chapter the proofs which require some algebraic background. We consider two different cases. In the first one we have q vector fields e1 , . . . , X eq of weight 1. These vector fields are free up to step s and satisfy X H¨ ormander’s condition at step s in some domain U of RN . In view of Remark 10.15 the dimension of the space only depends on s, q and the weights. We will write N = N1 (q, s) . e0 of weight 2 and q vector fields X e1 , . . . , X eq In the second case we have a vector field X of weight 1 that are free up to step s and satisfy H¨ormander’s condition at step s in some domain U of RN and we will write N = N2 (q, s) . Observe that even if formally the lifting procedure described in the previous sections has been developed for the second case one can always reduce to this case assigning e0 the weight 1. also to X The following theorem addresses the first case. Theorem 10.31 Let N = N1 (q, s). There exist in RN a system of smooth vector fields Y1 , . . . , Yq and a structure of Carnot group G (see Chapter 3, Definition 3.53) such that: (i) the . . , Yq are free up to step s in RN and the vectors  vector
fields Y1 , . N Y[I] u |I|6s span R at any point u of the space; (ii) the Y[I] ’s are left invariant and homogeneous of degree |I| with respect to the dilations in G;

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H¨ ormander operators

(iii) for I ∈ B the vector field Y[I] at u =
0 coincide with the local basis associated ∂ to the coordinates uI , that is, Y[I] 0 = ∂u ; I (iv) the vector fields Y[I] satisfy (10.41); (v) in the group G, the inverse u−1 of an element is just its (Euclidean) opposite −u. For the second case we have a strictly analogous result: Theorem 10.32 Let N = N2 (q, s). There exist in RN a system of smooth vector fields Y0 , Y1 , . . . , Yq and a structure of homogeneous stratified group G of type II (see Chapter 3, Definition 3.64) such that: N (i) the  vector
fields Y0 , Y1N, . . . , Yq are free up to step s in R and the vectors Y[I] u |I|6s span R at any point u of the space; points (ii)-(v) in the above theorem still hold. The proof of the above theorems will be given in section 10.5. Theorem 10.30 can now be applied choosing the left invariant vector fields Y[I] as the approximating system. The map u = Θ (η, ξ) can now be regarded as a diffeomorphism from a neighborhood of η onto a neighborhood of 0 in the group G. In other words, Θ (η, ξ) is an element of the group G, and one has: −1

Θ (ξ, η) = −Θ (η, ξ) = Θ (η, ξ)

.

For this choice of the vector fields Yi we also have the following useful Proposition 10.33 The change of coordinates in RN given by ξ 7→ u = Θ (η, ξ) ≡ Θη (ξ) has a Jacobian determinant given by dξ = c (η) (1 + ω (η, u)) du where c (η) is a smooth function, bounded and bounded away from zero, ω (η, u) is smooth and O (|u|), that is |ω (η, u)| 6 c |u| for small u and any η in the fixed neighborhood. Proof. We will compute the Jacobian determinant of the inverse mapping ξ = PN ∂ e Θ−1 η (u). To do this, set X[I] = k=1 cIk (ξ) ∂ξk for every I ∈ B and rewrite the left hand side of (10.42) as X X ∂f  ∂  cIk (ξ) (Θη (ξ)) (Θη (ξ))J . ∂uJ ∂ξk k

J∈B

so that (10.42), evaluated at ξ = η, becomes: N X k=1

cIk (η)

X ∂f 

∂  (0) (Θη (ξ))J ξ=η = Y[I] f + Rη,[I] f (0) = Y[I] f (0) ∂uJ ∂ξk

J∈B

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where the last identity follows from the fact that the vector fields Rη,[I] vanish at 0 by Theorem 10.30. Choosing f (u) = uK for K ∈ B, we have N X k=1

cIk (η)

 ∂  (Θη (ξ))K ξ=η = YI [uK ] (0) = δIK ∂ξk

where the last equality follows by the structure of the vector fields Yl (see Chapter 3). Let us define the square matrix C(η) = {cIk (η)}I∈B,k=1,...N (recall that card B = N ) and let J(η) be the Jacobian determinant of the mapping u = Θη (ξ) at ξ = η. Then Det [C(η)] · J(η) = 1. It follows that the determinant of the Jacobian of the mapping ξ = Θ−1 η (u) at u = 0 equals Det [C(η)] ≡ c(η). Since the Jacobian determinant of the map ξ = Θ−1 η (u) is a smooth function in u, it equals c(η) · (1 + O (|u|)) . This also implies the analogous result for the change of coordinates η 7→ u = Θη (ξ), by the smoothness of the map u 7→ u−1 in G. 10.4

Some geometry of free lifted vector fields

The aim of this section is twofold. On the one hand, we want to point out some properties of free systems of H¨ ormander vector fields which are somewhat simpler than the corresponding properties of general systems of H¨ormander vector fields; on the other hand we need to study some relations existing between the geometry of free lifted vector fields and that of the corresponding vector fields in the space of original variables. First of all, we are going to apply to the context of free vector fields the results proved in the previous chapter about the volume of metric balls. The setting is the e0 , . . . , X eq same as in the previous section: we consider a family of vector fields X N defined in some open subset U of R , free up to step s at any point of U and ei recall that these satisfying H¨ ormander’s condition at step s in U . The symbols X vector fields can be thought as obtained lifting, in a neighborhood of some point x0 ∈ Rn (with n < N ) another system of vector fields X0 , . . . , Xq satisfying H¨ormander’s condition at step s. Notation 10.34 From now on we will adapt to lifted vector fields the notation introduced in section 1 of the previous chapter for the different distances and the e de∗ to denote balls and distances, e B e∗, B eB and d, corresponding balls. We will write B, ei instead of the original Xi . defined using the lifted vector fields X  ei are free, if X e[I] We already know (see Remark 10.21) that, since the X is I∈B a basis of RN at some ξ ∈ U (for some family B of N multiindices I with |I| 6 s), then it is a basis at every point of U . We can also prove the following:

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H¨ ormander operators

Proposition 10.35 (Bases of free systems of vector fields) With the previous notation and  we have:  assumptions e[I] e[I] are two different bases of RN (at every point , X (a) If X I∈B2 I∈B1 of U ) obtained by two different families B1 , B2 of N multiindices I with |I| 6 s, then |B1 | = |B2 | , that is X X |I| . |I| = I∈B2

I∈B1 0

(b) Moreover, for n anyo fixed U b U , there exists a number t ∈ (0, 1) such that e[I] every basis X is t-suboptimal at every point ξ ∈ U 0 and for every I∈B

radius δ ∈ (0, 1) in the sense of Definition 9.27, that is: 0

|λB (ξ)| δ |B| > t max |λB0 (ξ)| δ |B | , 0 B

where the max is taken over all the N -tuples B 0 = (J1 , . . . , Jn ) with |Jj | 6 s. Proof. First of all we observe that if, for some coefficients {cI }|I|6s one   P e has = 0, then for every k = 1, 2, . . . , s one also has |I|6s cI X[I] ξ   P e = 0. Actually, by definition of free vector fields |I|=k cI X[I] ξ

X

  X e[I] = 0 =⇒ cI X cI AIJ = 0 ∀J. ξ

|I|6s

|I|6s

Then for every k = 1, 2, . . . , s if |J| = k, recalling that AIJ = 0 unless |I| = |J| (see (10.2)) we have X cI AIJ = 0 |I|=k

and therefore  X |I|=k



e[I] cI X



= ξ

X |I|=k

cI

X



eJ AIJ X



J

= ξ

X

 X

 J

  eJ = 0. cI AIJ  X

|I|=k

ξ

e[J] can be uniquely written in terms of the Now, let J ∈ B2 . The vector field X basis B1 as     X e[J] = e[I] X cIJ X ξ

I∈B1

and by the previous claim this implies that   X e[J] = X ξ

I∈B1 ,|I|=|J|

ξ

  e[I] . cIJ X ξ

In other words: every vector field of the basis B2 having weight k is a linear combination of elements of the basis B1 having weight k. Then the spaces spanned by the

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elements of B1 and B2 having the same weight k have the same dimension, which implies X X |I| = |J| for k = 1, 2, . . . , s. I∈B1 ,|I|=k

J∈B2 ,|J|=k

Summing for k = 1, 2, . . . , s we get |B1 | = |B2 |, hence (a) is proved. To prove (b) 0 let us look at the quantity |λB0 (ξ)| δ |B | . We already know that λB0 (ξ) 6= 0 at some ξ ∈ U if and only if λB0 (ξ) 6= 0 at every ξ ∈ U . In this case, by point (a), |B 0 | = Q for some integer Q independent from the particular basis. This means that for every ξ∈U 0

max |λB0 (ξ)| δ |B | = δ Q · max |λB0 (ξ)| . 0 0 B

B

Then, for ξ ranging in a fixed U 0 b U and every B 0 such that λB0 (ξ) 6= 0, by continuity we can write 0 < c1 6 |λB0 (ξ)| 6 c2 and this means that if B is any basis we can write |λB (ξ)| δ |B| = |λB (ξ)| δ Q > c1 δ Q >

0 c1 Q c1 δ max |λB0 (ξ)| = max |λB0 (ξ)| δ |B | 0 0 B c2 c2 B

and (b) holds for t = c1 /c2 . n o e[I] in U by Q = Definition 10.36 The number Q, defined for every basis X I∈B P I∈B |I|, is called homogeneous dimension of U with respect to the free vector fields e Xi . By point (a) in the previous proposition, the function X Λ (x, δ) = |λB (x)| δ |B| B

introduced in the previous chapter to control the volume of balls takes in this case the simple form Λ (ξ, δ) = Λ (ξ) δ Q with 0 < c1 6 Λ (ξ) 6 c2 for every ξ ∈ U 0 b U. Therefore by Theorem 9.1 we have the Corollary 10.37 For every ξ ∈ U 0 b U there exist constants δ0 , c1 , c2 > 0 such that for every δ 6 δ0 , ξ ∈ U 0 e c1 δ Q 6 B (ξ, δ) 6 c2 δ Q e denotes the control ball with respect to the vector fields X ei . where B

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H¨ ormander operators

Up to shrinking if necessary the neighborhood U, we can assume that the previous result holds for any ξ ∈ U . By the equivalence between the distances de and e ∗ (ξ, δ) . de∗ (see Theorem 9.6) the same result holds for the balls B Next, we want to introduce in the context of lifted variables a convenient quasidistance, which will turn out to be equivalent to the control distance. First of all, adapting the notations introduced in the previous chapter to the lifted context, let ( ! )  X  |I| e e BB (ξ, r) = η = exp uI X[I] (ξ) : |uI | < r . I∈B

Recall that the canonical variables u ∈ RN can be seen as varying in a homogeneous group G. Let us define: 1/|I|

kuk = max |uI | I∈B

.

It is easy to see that  k·k isNa homogeneous norm on G (see Chapter 3, Definition 3.8). eB (ξ, r) = Φξ,B (Qr ). Recalling also Letting also Qr = u ∈ R : kuk < r we have B that Φη,B (Θ (η, ξ)) = ξ we have eB (ξ, r) = {η : kΘ (η, ξ)k < r} . B It is therefore natural to introduce the function ρ (ξ, η) = kΘ (η, ξ)k, so that the eB (ξ, r) are balls with respect to the distance-like function ρ. Up to shrinking boxes B U if necessary, we can assume that ρ (ξ, η) is defined for ξ, η ∈ U . Actually, ρ turns out to be a quasidistance; we have the following: Proposition 10.38 (Properties of ρ) There exist constants c, c1 , c2 > 0 such that for any ξ, η ∈ U, ρ (ξ, η) > 0 and ρ (ξ, η) = 0 ⇐⇒ ξ = η ρ (ξ, η) = ρ (η, ξ) ρ (ξ, η) 6 c {ρ (ξ, ζ) + ρ (ζ, η)} e∗

e∗

c1 d (ξ, η) 6 ρ (ξ, η) 6 c2 d (ξ, η) .

(10.43) (10.44)

Proof. The first three properties follow by definition and by Proposition 10.29, while (10.43) follows by (10.44), since de∗ satisfies the triangle inequality. So let eB (ξ, r), Theorem 9.11 us prove (10.44). Since the ρ-ball Bρ (ξ, r) coincides with B (structure of balls) applied in the lifted context says that the following inclusions hold: e ∗ (ξ, cδ) ⊂ B eB (ξ, δ) ⊂ B e ∗ (ξ, δ) B for any ξ ∈ U , δ 6 δ0 , and some positive constant c. But this implies (10.44), so we are done. We now come to the results which describe the relations between distances and balls in the space of the original variables with those in the space of lifted variables. To begin with, it is easy to see that:

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517

e ((x, t) , r) on Rn is B (x, r). Proposition 10.39 The projection of the metric ball B In particular de((x, t) , (y, s)) > d (x, y) . ei project onto the vector fields Xi , Proof. This holds just because the vector fields X hence any admissible curve in the space of lifted variables project onto an admissible in the original space. More in detail, let π1 be the projection operator of Rn+m to Rn defined by π1 (u, v) = u. Let y ∈ B (x, r), γ ∈ Cx,y (δ) with δ < r, that is  0 Pq   γ (τ ) = i=0 ai (τ ) (Xi )γ(τ ) γ (0) = x, γ (1) = y   |a0 (τ )| 6 δ 2 , |ai (τ )| 6 δ for i = 1, 2, . . . , q. Define, for the same functions ai (τ ),   ( Pq ei γ e0 (τ ) = i=0 ai (τ ) X γ e (0) = (x, t) .

γ e(τ )

We claim that γ e (1) = (y, t∗ ) for some t∗ ∈ Rm , which implies that γ e ∈  e ((x, t) , r) and y ∈ π1 B e ((x, t) , r) . To compute C(x,t),(y,t∗ ) (δ), hence (y, t∗ ) ∈ B γ e (1) let us recall the structure of the lifted vector fields: ei = Xi + X

m X

bij (x, t) ∂tj ,

j=1

hence if we write only the first n equations of the differential system defining γ e (τ ) = (α (τ ) , β (τ )) ∈ Rn+m we find α0 (τ ) =

q X

ai (τ ) (Xi )α(τ )

i=0

which coupled with γ e (0) = (x, 0) , that is α (0) = x, gives (by uniqueness) α (t) e (1) = (γ (1) , β (1)) = (y, t∗ ). This shows that B (x, r) ⊂  = γ (t) and  γ e ((x, t) , r) . π1 B   e ((x, t) , r) that is there exists γ Conversely, let y ∈ π1 B e ∈ C(x,t),(y,t∗ ) (δ) with δ < r. Then:    Pq 0 ei  γ e (τ ) = a (τ ) X i  i=0  γ e(τ ) ∗

γ e (0) = (x, t) , γ e (1) = (y, t )    2 |a0 (τ )| 6 δ , |ai (τ )| 6 δ for i = 1, 2, . . . , q. If we write the first n equations of this system, letting γ e (τ ) = (α (τ ) , β (τ )) ∈ Rn+m , we find ( Pq α0 (τ ) = i=0 ai (τ ) (Xi )α(τ ) α (0) = x, α (1) = y

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H¨ ormander operators

  e ((x, t) , r) ⊂ B (x, r) and we are hence α ∈ Cx,y (δ) , and y ∈ B (x, r). Hence π1 B done. It is not possible, however, to prove a control between d and de in the reverse sense. For instance, the validity of a “reasonable” inequality like de((x, 0) , (y, 0)) 6 cd (x, y) is unclear. What one can prove is a control (in both senses) at the level of volumes of balls. This is a deeper result, relying on the analysis of the structure of metric balls given in the previous chapter, and reads as follows: Theorem 10.40 Let X0 , X1 , . . . , Xq be a system of H¨ ormander vector fields of step n 0 0 s in Ω ⊂ R , let Ω b Ω and x0 ∈ Ω . There exist constants δ0 , κ ∈ (0, 1) , c1 , c2 > 0 such that in V = B ∗ (x0 , δ0 ) we can lift the vector fields X0 , X1 , . . . , Xq to vector e0 , X e1 , . . . , X eq , defined in a neighborhood U of (x0 , 0) in RN for some N = fields X n + m, and free up to step s in U and, for every δ ∈ (0, δ0 ), for every y ∈ Ω e∗ n o B ((x , 0) , δ) 0 m ∗ e , s ∈ R : (y, s) ∈ B ((x0 , 0) , δ) 6 c1 |B ∗ (x0 , δ)| and for every y ∈ B ∗ (x0 , κδ0 ), e∗ n o B ((x , 0) , δ) 0 m ∗ e . s ∈ R : (y, s) ∈ B ((x0 , 0) , δ) > c2 |B ∗ (x0 , δ)| Here |A| denotes the Lebesgue measure of the set A in Rk for the suitable dimension k (which is m, N, n for the three sets involved in the inequalities). ∗ Remark 10.41 Due to the local equivalence d and the local doubling between d and e ∗ ((x0 , 0) , δ) , an analogous statement condition which holds for |B ∗ (x0 , δ)| and B e ∗ with B, B, e respectively. In other words, the above theorem, holds replacing B ∗ , B

which we will prove in this section, implies Theorem 10.7 stated in the introduction. Before proving the theorem For a fixed point x0 ∈  let us fix some notation. Ω b Ω, a fixed base  XB m= X[I1 ] , X[I2 ] , . . . , X[In ] (in a neighborhood of x0 ) and any enumeration X[Ji ] i=1 of the other commutators (with |Ji | 6 s) which do not belong to the family XB , let us consider the map ! n m X X Φx,B,v (u) = exp ui X[Ii ] + vi X[Ji ] (x) 0

i=1

i=1

with u ∈ Rn , v ∈ Rm sufficiently small. We set n o Qδ = u ∈ Rn : |uj | < δ |Ij | , n o Q0δ = v ∈ Rm : |vj | < δ |Jj | .

Lifting and approximation

519

For a fixed v small enough, we define the balls that are images under the map Φx,B,v of the boxes Qδ : BB (x, δ, v) = Φx,B,v (Qδ ). Proof of Theorem 10.40. For a fixed Ω0 b Ω, let us consider a point x0 ∈ Ω0 and a 21 -suboptimal base XB with radius δ at x0 . We can lift the vector fields e0 , X e1 , . . . , X eq , defined in a neighborhood U of (x0 , 0) in RN for X0 , X1 , . . . , Xq to X  e[I] project some N > n, and free up to step s in U . Since the vector fields X I∈B  n onto X[I] I∈B , which are a base of R at every point of some neighborhood V  e[I] will be linearly independent at every point of U . of x0 , the vector fields X I∈B

ei ’s are free up to step s, we can obtain a base of RN at every point Since the X  e0 , X e1 , . . . , X eq up to e[I] all the other commutators of X of U just adding to X I∈B step s (simply disregarding the commutators which, by antisymmetry and Jacobi identities, depend on the others). So, all the essentially independent commutators ei ’s up to step s can be grouped in two sets: of the X om on n n e[J ] e[I ] X , X i

i=1

i

i=1

where: n on n  e[I ] the vector fields X project onto X[Ii ] i=1 which are a base of Rn at every i i=1 point of V ; n on n om e[I ] e[J ] the vector fields X , X form one base of RN (N = n + m) at every i

i=1

i

i=1

point of U . By Proposition 10.35 (b), this is a t-suboptimal base for some “absolute” t depending on the neighborhood U . Shrinking if necessary the neighborhood V (and therefore U ) we can assume that Theorem 9.42 is applicable to the Xi in V ei in U . and to the X More preciselynthere are constants κ, κ1 ,oδ0 ∈ (0, 1) such that for 0 < δ < δ0 and |Jj |

every v ∈ Q0κ1 δ = v ∈ Rm : |vj | < (κ1 δ) Φx0 ,B,v (u) = exp

n X

the map

ui X[Ii ] +

i=1

m X

! vi X[Ji ]

(x0 )

i=1

is one to one from the cube n o |I | Qκ1 δ = u ∈ Rn : |uj | < (κ1 δ) j onto BB (x0 , δ, v) and B ∗ (x0 , κδ) ⊂ BB (x0 , δ, v) ⊂ B ∗ (x0 , 2δ) ⊂ V. Similarly, for every 0 < δ < δ0 the map e x ,B (u, v) = exp Φ 0

n X i=1

e[I ] + ui X i

m X

! e[J ] vi X i

(x0 , 0)

i=1

eB ((x0 , 0) , δ) and is one to one from the cube Qκ1 δ × Q0κ1 δ ⊂ Rn+m onto B e ∗ ((x0 , 0) , κδ) ⊂ B eB ((x0 , 0) , δ) ⊂ B e ∗ ((x0 , 0) , 2δ) ⊂ U. B

520

H¨ ormander operators

e x ,B (u, v): in the Note the different roles of v in the maps Φx0 ,B,v (u) and Φ 0 second map, both u and v multiply elements of the basis, while in the first map only u multiplies elements of the basis, and v plays the role of a “parameter”: it is for this reason that here we need the full statement of Theorem 9.42 proved in the previous chapter and not the simpler one stated in the introduction as Theorem 9.11. Let us define the two projection operators: π1 : Rn+m → Rn π2 : Rn+m → Rm that project the  first  n coordinates and the last m coordinates respectively. ei Since π1 ◦ X = (Xi )x , we also have (x,h)

e x ,B (u, v) = Φx ,B,v (u) . π1 ◦ Φ 0 0

(10.45)

Now, fix y ∈ B ∗ (x0 , κδ) and consider  Σy = (u, v) ∈ Qc1 δ × Q0c1 δ : Φx0 ,B,v (u) = y . Since for every v ∈ Q0κ1 δ the map ΦB,x0 ,v is bijective from Qκ1 δ onto BB (x0 , δ, v) ⊃ B ∗ (x0 , κδ), if follows that for every v ∈ Q0κ1 δ we can define a unique u = θ (y, v) such that Φx0 ,B,v (θ (y, v)) = y.

(10.46)

Therefore  Σy = (θ (y, v) , v) : v ∈ Q0κ1 δ . Observe now that by Theorem 9.42 (1) we have 1 det JΦ (θ (y, v)) > |λB (x0 )| . x0 ,B,v 4 and since the map (u, v, y) 7→ Φx0 ,B,v (u) − y is C 1 , by the implicit function theorem θ : B ∗ (x0 , κδ) × Q0κ1 δ → Qκ1 δ is C 1 . Differentiating (10.46) with respect to y gives JΦx0 ,B,v (θ (y, v)) ·

∂θ (y, v) = I ∂y

so that det

 −1 ∂θ (y, v) = det JΦx0 ,B,v (θ (y, v)) . ∂y

(10.47)

Hence det ∂θ (y, v) 6 4 |λB (x0 )|−1 ∂y for every (y, v) ∈ B ∗ (x0 , κδ) × Q0κ1 δ .

(10.48)

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521

Let us now consider the map: σ : B ∗ (x0 , κδ) × Q0κ1 δ → Rm e x ,B (θ (y, v) , v) . σ (y, v) = π2 ◦ Φ 0 Clearly by (10.45) for (y, v) ∈ B ∗ (x0 , κδ) × Q0κ1 δ we have   e x ,B (θ (y, v) , v) = π1 ◦ Φ e x ,B (θ (y, v) , v) , π2 ◦ Φ e x ,B (θ (y, v) , v) Φ 0 0 0 = (Φx0 ,B,v (θ (y, v)) , σ (y, v)) = (y, σ (y, v)) . Computing the Jacobian of both sides (with respect to the variables y and v) we obtain ! ! ∂θ ∂θ (y, v) ∂v (y, v) I 0 ∂y JΦ = ∂σ e x ,B (θ (y, v) , v) ∂σ 0 0 I ∂y (y, v) ∂v (y, v) so that det JΦ ex

0 ,B

(θ (y, v) , v) det

∂σ ∂θ (y, v) = det (y, v) . ∂y ∂v

(10.49)

e x ,B and (10.48) we obtain Using the smoothness of Φ 0 ∂θ ∂σ −1 det (y, v) 6 c det (y, v) 6 c |λB (x0 )| ∂v ∂y for some constant c only depending on the vector fields Xi and Ω0 . Let now now κ2 = κ1 κ and consider the set n o e ∗ ((x0 , 0) , κ2 δ) . Sy (κ2 δ) ≡ s ∈ Rm : (y, s) ∈ B e ∗ ((x0 , 0) , κ2 δ) ⊂ B eB ((x0 , 0) , κ1 δ) , if s ∈ Sy (κ2 δ) there exists (u, v) ∈ Since B 0 e x ,B (u, v). By (10.46) u = θ (y, v) so that s = Qκ1 δ × Qκ1 δ such that (y, s) = Φ 0
σ (y, v). In other words, Sy (κ2 δ) ⊂ σ Q0κ1 δ . Since e x ,B (θ (y, v) , v) = (y, σ (y, v)) Φ 0 e x ,B is one-to-one on Qκ δ × Q0 , for every fixed y also v 7→ σ (y, v) is and Φ 0 1 κ1 δ one-to-one and we can change variables in the following integral: Z Z det ∂σ dv |Sy (κ2 δ)| = ds 6 ∂v Sy (κ2 δ) Q0κ δ 1 Pm c |J | 6 · (κ1 δ) i=1 i |λB (x0 )| Pm Pn c i=1 |Ji | . i=1 |Ii |+ Pn = · δ |λB (x0 )| δ i=1 |Ii | Now recall that since XB is 21 -suboptimal of radius δ we have: X Pn 1 |λB (x0 )| δ i=1 |Ii | > max |λC (x0 )| δ |C| > c |λC (x0 )| δ |C| 2 C C

522

H¨ ormander operators

so that

Pn Pm c Pn · δ i=1 |Ii |+ i=1 |Ji | |I | i |λB (x0 )| δ i=1 e∗ B ((x0 , 0) , δ) δQ 6c· ∗ 6c |B (x0 , δ)| |B ∗ (x0 , δ)| where we have applied Theorem 9.1 in the space of the original variables in the denominator, while in the numerator we exploited the estimate n m X X e∗ |Ii | + |Ji | B ((x0 , 0) , δ) ' δ Q with Q =

|Sy (κ2 δ)| 6

i=1

i=1

(see Corollary 10.37). We have therefore proved that e∗ n o B ((x , 0) , δ) 0 m ∗ e s ∈ R : (y, s) ∈ B ((x0 , 0) , κ2 δ) 6 c ∗ |B (x0 , δ)| for every y ∈ B ∗ (x0 , κδ) and δ < δ0 . Shrinking δ0 and replacing κδ and κ2 δ with δ, in view of the local doubling condition this means that e∗ n o B ((x , 0) , δ) 0 m ∗ e ((x0 , 0) , δ) 6 c s ∈ R : (y, s) ∈ B ∗ |B (x0 , δ)| for every y ∈ B ∗ (x0 , δ). On the other hand, / B ∗ (x0 , δ) then for every s ∈ Rm  if y ∈ e ∗ ((x0 , 0) , δ) (since π2 B e ∗ ((x0 , 0) , δ) = B ∗ (x0 , δ)) and the set we have (y, s) ∈ /B n o e ∗ ((x0 , 0) , δ) is empty, so that the above inequality trivially s ∈ Rm : (y, s) ∈ B holds. Therefore, this bound holds for every y. To prove a converse inequality, we need to exploit the other inclusion in Theorem 9.42 (2), eB ((x0 , 0) , δ) ⊂ B e ∗ ((x0 , 0) , 2δ) . B Let y ∈ B ∗ (x0 , κδ) and let s ∈ σ (y, v) for some v ∈ Q0κ1 δ . Since e B,x (θ (y, v) , v) ∈ B eB ((x0 , 0) , δ) ⊂ B e ∗ ((x0 , 0) , 2δ) (y, s) = Φ 0

then (replacing δ with δ/2) n o e ∗ ((x0 , 0) , δ) s ∈ Sy (δ/2) = s ∈ Rm : (y, s) ∈ B so that n o Z e ∗ ((x0 , 0) , δ) > s ∈ Rm : (y, s) ∈ B

Sy (δ/2)

Z ds >

Q0κ

1δ

det ∂σ dv. ∂v

Using (10.49) and (10.47) we easily obtain det ∂σ ∂v > c3 so that n o Pm e ∗ ((x0 , 0) , δ) > c (κ1 δ) i=1 |Ji | s ∈ Rm : (y, s) ∈ B Pn Pm c Pn · δ i=1 |Ii |+ i=1 |Ji | . = |I | |λB (x0 )| δ i=1 i By Theorem 9.1, e∗ n o B ((x , 0) , δ) 0 m ∗ e . s ∈ R : (y, s) ∈ B ((x0 , 0) , δ) > c |B ∗ (x0 , δ)| and the theorem is completely proved.

Lifting and approximation

10.5

523

Abstract free Lie algebras and Lie groups

In this section we will prove Theorem 10.31. In order to do this, we will first introduce the notion of free nilpotent Lie algebra of step s with q generators. We will then show that this Lie algebra can be seen as the Lie algebra of a suitable group which is a homogeneous stratified group of type II (see Definition 3.64). These two steps will be accomplished in the following two subsections. 10.5.1

The free nilpotent Lie algebra of step s with q generators

Let us fix a set of q elements, Σ = {σ1 , σ2 , . . . , σq }. The elements of this set are just placeholders and it will be clear from our construction that different choices for Σ give isomorphic constructions. A word w in Σ is a finite sequence of elements of Σ. We will write w = σi1 σi2 · · · σik . The words in Σ can also be represented using multiindices. If I = (i1 , . . . , ik ) with 1 6 ij 6 q we will denote by σI the word σI = σi1 σi2 · · · σik . Given two words σI = σi1 σi2 · · · σik and σJ = σj1 σj2 · · · σj` we can define the product of this two words by concatenation: σI σJ = σi1 σi2 · · · σik σj1 σj2 · · · σj` = σIJ where IJ = (i1 , . . . , ik , j1 , . . . , j` ). This product is clearly associative. The free associative algebra fq (over R) with q generators is the associative algebra spanned by the words σI . Every element in fq can be written uniquely as k X

uIi σIi .

i=1

for suitable multiindices Ii and real numbers uIi . With a small abuse of notation we will denote by σj either the elements of the set Σ or the corresponding elements of the algebra fq . Suppose now that to every symbol σi ∈ Σ is assigned a weight pi (positive integer). This induces in fq a structure of graded algebra, in the sense of the following: Proposition 10.42 For every word σI = σi1 σi2 · · · σik let us define its weight by Pk |σI | = j=1 pij and let Pk be the subspace of fq spanned by words of weight k, then fq =

+∞ M

Pk .

k=0

Moreover if a ∈ Pi and b ∈ Pj then ab ∈ Pi+j .

(10.50)

524

H¨ ormander operators

The proof is immediate. We can now define the concept of free Lie algebra with q generators. First of all observe that we can endow fq with a structure of Lie algebra setting for every g1 , g2 ∈ fq [g1 , g2 ] = g1 g2 − g2 g1 . Following the usual notation, for every multiindex I = (i1 , i2 , . . . , ik ), we define the iterated commutators     σ[I] = σi1 , σi2 , . . . , σik−1 , σik . Also, for I = (i), i = 1, 2, . . . , q, we set σ[I] = σi . Definition 10.43 The free Lie algebra with q generator gq is the subspace of fq which is spanned by the iterated commutators σ[I] . Observe that gq is a proper subspace of fq . For example σ1 σ2 ∈ fq but σ1 σ2 ∈ / gq . The fact that gq is actually a Lie algebra (that is [gq , gq ] ⊆ gq ) is a consequence of Lemma 1.21 (note that even if this Lemma has been proved for a Lie algebra of vector fields the proof is the same for fq ): X   dK σ[I] , σ[J] = I,J σ[K] ∈ gq . |K|=|I|+|J|

Clearly gq inherits from fq the structure of graded algebra. Indeed, if we set Vk = gq ∩ Pk we obtain gq =

+∞ M

Vk

(10.51)

k=1

and [Vk , Vj ] ⊆ Vk+j . Let us fix a step s > 2, assume that pi 6 s for 1 6 i 6 q and set M hs = Vk . k>s

It is easy to check that hs is an ideal in gq , that is [gq , hs ] ⊆ hs . This allows to define a structure of Lie algebra over the quotient gq,s = gq /hs . Consider two cosets g1 + hs , g2 + hs ∈ gq /hs and define [g1 + hs , g2 + hs ] = [g1 , g2 ] + hs .

(10.52)

Since hq is an ideal this definition is independent of the particular choice of the g1 and g2 used to represent the cosets. Proposition 10.44 The quotient gq,s = gq /hs is a finite dimensional graded nilpotent Lie algebra of step s. More precisely, if we set Wk = {g + hs : g ∈ Vk } for k 6 s, then s M gq,s = Wk . k=1

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525

Moreover, if k + ` 6 s, then [Wk , W` ] ⊆ Wk+` while, for k + ` > s, [Wk , W` ] = 0. Proof. Let g + hs ∈ gq,s . Using (10.51) we can write g = and h ∈ hs . Hence g + hs =

s X

Ps

k=1 gk + h

with gk ∈ Vk

(gk + hs ) .

k=1

Such a decomposition is clearly unique, since it is unique the decomposition in gq . Finally if g1 + hs ∈ Wk and g2 + hs ∈ W` , since [g1 , g2 ] ∈ Vk+` when k + ` 6 s we have [g1 + hs , g2 + hs ] = [g1 , g2 ] + hs ∈ Wk+` . If k + ` > s then [g1 , g2 ] ∈ hs so that [g1 + hs , g2 + hs ] = [g1 , g2 ] + hs = 0. Since every Wk has finite dimension, the Lie algebra gq,s is clearly finite dimensional.

Definition 10.45 The Lie algebra gq,s = gq /hs is called the free nilpotent Lie algebra of step s with q generators. Remark 10.46 Since Vk is spanned by the iterated commutators of weight k of the elements σ1 , σ2 , . . . , σk , using (10.52) we see that Wk is spanned by iterated commutators of weight k of the elements σ1 + hs , σ2 + hs , . . . , σk + hs . It follows that gq,s is spanned by iterated commutators up to the step s of the same elements. Ps Definition 10.47 Let a ∈ gq,s and write a = k=1 wk with wk ∈ Wk . For every λ > 0 let us define the dilation operators Dλ Dλ : gq,s → gq,s s X Dλ (a) = λk wk . k=1

Proposition 10.48 The dilations Dλ are automorphisms of the Lie algebra gq,s : for every a1 , a2 ∈ gq,s and c1 , c2 ∈ R, we have Dλ (c1 a1 + c2 a2 ) = c1 Dλ (a1 ) + c2 Dλ (a2 )

(10.53)

[Dλ (a1 ) , Dλ (a2 )] = Dλ ([a1 , a2 ]) .

(10.54)

and

526

H¨ ormander operators

Proof. Let a1 =

s X

wi ,

a2 =

i=1

s X

wj0 .

j=1

Then  Dλ (c1 a1 + c2 a2 ) = Dλ c1

s X

wi + c2

i=1

=

s X

s X

 wj0  = Dλ

j=1

s X

!
0

c1 wi + c2 wj

i=1

s s X X
λi c1 wi + cwj0 = c1 λi wi + c2 λi wj0 = c1 Dλ (a1 ) + c2 Dλ (a2 ) .

i=1

i=1

i=1

Also   s s n X X X [Dλ (a1 ) , Dλ (a2 )] =  λi wi , λj wj0  = λi+j [wi , wj ] , i=1

j=1

i,j=1

and since [wi , wj ] ∈ Wi+j if i + j 6 s and [wi , wj ] = 0 if i + j > s we have X X [Dλ (a1 ) , Dλ (a2 )] = λi+j [wi , wj ] = Dλ ([wi , wj ]) = Dλ ([a1 , a2 ]) . i+j6s

i+j6s

We stress that our construction of the free nilpotent Lie algebra gq,s depends on the choice of the weights pi assigned to each symbol σi . We will specialize this choice later in order to obtain the free stratified Lie algebra and the free stratified Lie algebra of type II. From now on, with a small abuse of notation, we will denote the elements of gq,s , that is cosets of the form g + hs , simply by g. Remark 10.49 For fixed q, s and weights pi , 1 6 i 6 q let us consider the free nilpotent Lie algebra gq,s . We know that gq,s is spanned by the elements σ[I] with |I| 6 s. Among such multiindices we can choose a set B = {I1 , . . . , IM } that includes the multiindices (i) with 1 6 i 6 q, so that |I1 | 6 |I2 | 6 · · · 6 |IM | and  simple σ[I] I∈B is a base of gq,s as a vector space. Since every a ∈ gq,s can be written uniquely as M X a= ui σ[Ii ] i=1

we can use the map (u1 , . . . , uM ) 7→

M X

ui σ[Ii ]

i=1

to identify gq,s with RM and transfer the structure of homogeneous Lie algebra from PM PM gq,s to RM . Namely, for u, v ∈ RM , consider a = i=1 ui σ[Ii ] , b = i=1 vi σ[Ii ] , and write M X [a, b] = ci σ[Ii ] . i=1

Lifting and approximation

527

so that we can set [u, v] = (c1 , c2 , . . . , cM ) . PM Similarly, for a given u ∈ RM , let a = i=1 ui σ[Ii ] , since σ[Ii ] ∈ W|Ii | we have Dλ (a) =

M X

λ|Ii | ui σ[Ii ]

i=1

so that we can set   Dλ (u) = λ|I1 | u1 , λ|I2 | u2 , . . . , λ|IM | uM . From now on we will denote the coordinates in RM either as ui ,with 1 6 i 6 M or as uI with I ∈ B. Similarly the canonical base of RM will be denoted either as ei with 1 6 i 6 M or as eI with I ∈ B. Observe that if I ∈ B the elements σ[I] of gq,s have been identified with eI . Example 10.50 Let Σ = {σ1 , σ2 } with p1 = p2 = 1. We want to construct g2,2 : the free nilpotent Lie algebra of step 2 with 2 generators. There are only three linearly independent elements of weight 6 2: σ1 , σ2 , σ(1,2) . It follows that g2,2 can be identified with R3 . Let u, v ∈ R3 . Since [σ1 , σ1 ] = 0; [σ2 , σ2 ] = 0     σ1 , σ(1,2) = 0; σ2 , σ(1,2) = 0;   σ(1,2) , σ(1,2) = 0 we have   u1 σ1 + u2 σ2 + u3 σ(1,2) , v1 σ1 + v2 σ2 + v3 σ(1,2) = [u1 σ1 , v2 σ2 ] + [u2 σ2 , v1 σ1 ] = (u1 v2 − u2 v1 ) σ(1,2) . Hence [(u1 , u2 , u3 ) , (v1 , v2 , v3 )] = (0, 0, u1 v2 − u2 v1 ) . Moreover
Dλ (u1 , u2 , u3 ) = λu1 , λu2 , λ2 u3 . It is immediate to check that this Lie algebra is isomorphic to the Lie algebra of left invariant vector fields on the Heisenberg group H1 . This algebra (see Example 3.25) is spanned by the vector fields X1 =

∂ ∂ ∂ ∂ ∂ + 2x2 ; X2 = − 2x1 ; X3 = ∂x1 ∂x3 ∂x2 ∂x3 ∂x3

defined in R3 . Indeed, since [X1 , X2 ] = −4X3 , if is enough consider the linear map
Ψ : g2,2 → Lie H1 defined by
Ψ (σ1 ) = X1 ; Ψ (σ2 ) = X2 ; Ψ σ(1,2) = −4X3 .

528

10.5.2

H¨ ormander operators

Construction of the underlying homogeneous group of gq,s

In the next step we would like to see gq,s as the Lie algebra of a homogeneous Lie group. In order to understand the problem and identify the candidate algebraic structure of the Lie group we are looking for, let us reverse our reasoning for a moment. Assume we do have a homogeneous group G with group operation , let g be its Lie algebra and consider the exponential map Exp : g → G Exp : X 7→ Exp (X) = exp (X) (0) Let X, Y ∈ g. In Chapter 3 (see (3.25)) we proved that exp (Y ) (x) = x  Exp (Y ). Hence for x = Exp (X) = exp (X) (0) we have exp (Y ) exp (X) (0) = Exp (X)  Exp (Y ) . Under suitable conditions, by the Baker-Campbell-Hausdorff formula (see Theorem 9.24, there exists one (and only one) vector field S (X, Y ), computable in some way starting with X, Y , such that exp (Y ) exp (X) (0) = exp (S (X, Y )) (0) = Exp (S (X, Y )) .

(10.55)

If we define X ◦ Y = S (X, Y ), by Proposition 9.72 this ◦ is a group operation. Moreover, by (10.55) we have Exp (X ◦ Y ) = Exp (X)  Exp (Y ) , so that the mapping Exp : (g, ◦) → (G, ) becomes a Lie group isomorphism. We can now come back to our real situation, where (G, ) does not exist yet, and the identity (10.55) cannot be used, because it has been established, in the nilpotent case (see Theorem 9.24), for vector fields X, Y and not for the elements of an abstract Lie algebra. Nevertheless, the group (gq,s , ◦) actually exists and by the previous reasoning should be isomorphic to the desired group (G, ). Hence, the path is now drawn: starting with the group structure (gq,s , ◦) in RM , we have to check that this is actually a homogeneous group, we have to construct its Lie algebra and to check that it is isomorphic to gq,s . So, let us start with the free nilpotent Lie algebra gq,s , which by Remark 10.49 can be identified with RM . Using the notation introduced in Chapter 9 let S (u, v) = u + v +

+∞ X

Ck (u, v)

k=2

be the formal series involved in BCH formula. Recall that Ck is a homogeneous Lie polynomial of degree k. By Proposition 10.44, when k > s and u, v ∈ RM we have Ck (u, v) = 0 so that we can define u ◦ v = S (u, v) = u + v +

s X k=2

Ck (u, v) .

Lifting and approximation

529

Remark 10.51 Let S (u, v) = (S1 (u, v) , S2 (u, v) , . . . , SM (u, v)). It is a simple computation to check that every Si (u, v) is actually a polynomial of degree at most s in the variables ui , vi . Ps Proposition 10.52 The binary operation u ◦ v = S (u, v) = u + v + k=2 Ck (u, v) and the dilations Dλ induce in RM a structure of homogeneous group G. The inverse of a element u ∈ G is the opposite −u. Proof. In Proposition 9.72 we proved that S (u, v) as a formal series satisfies the properties of a group operation with inverse −u. Since in the current setting the Lie algebra is nilpotent, the aforementioned formal series reduces to a polynomial and the identities between formal series to identities between polynomials in the variables u and v. Let us check that Dλ is a group automorphism, that is Dλ (u) ◦ Dλ (v) = Dλ (u ◦ v) . Using the definition of ◦, this is equivalent to Dλ (u) + Dλ (v) +

+∞ X

Ck (Dλ (u) , Dλ (v)) = Dλ

u+v+

k=2

s X

! Ck (u, v) ,

k=2

and since Dλ is linear it is clearly enough to check that for every k Ck (Dλ u, Dλ v) = Dλ Ck (u, v) . Moreover, since Ck is a Lie polynomial it is enough to check this for repeated commutators. This follows iterating (10.54). Example 10.53 Let us construct, using Proposition 10.52, the homogeneous group associated to the Lie algebra g2,2 of Example 10.50. Since C2 (u, v) = 12 [u, v] in this case the group operation is given by 1 [(u1 , u2 , u3 ) , (v1 , v2 , v3 )] 2 1 = (u1 + v1 , u2 + v2 , u3 + v3 ) + (0, 0, u1 v2 − u2 v1 ) 2   1 = u1 + v1 , u2 + v2 , u3 + v3 + (u1 v2 − u2 v1 ) . 2

(u1 , u2 , u3 ) ◦ (v1 , v2 , v3 ) = (u1 , u2 , u3 ) + (v1 , v2 , v3 ) +

Observe that in this case up to a rescaling the group is the Heisenberg group H1 . Now we will show that Lie (G), that is the Lie algebra of left invariant vector fields over G, is isomorphic to gq,s so that gq,s can be interpreted as the Lie algebra of a homogeneous group. To do this let us consider the map Ξ : gq,s → Lie (G) PM

that associates to σ = j=1 vj σ[Ij ] ∈ gq,s the unique left invariant vector field Xv = Ξ (v) that agrees with v = (v1 , . . . , vM ) at the origin.

530

H¨ ormander operators

Proposition 10.54 We have the following: (i) the map Ξ is an isomorphism of Lie algebras. (ii) the vector fields Yi = Ξ (σi ), 1 6 i 6 q are homogeneous of degree pi .
(iii) for every multiindex I we have Y[I] = Ξ σ[I] so that Lie (G) is spanned by the vector fields Y i and their commutators up to step s. (iv) the vector fields Y[I] |I|6s are free up to step s. Proof. We already proved in Proposition 3.26 that Ξ is linear and bijective. To show point i) we have to check that Ξ is compatible with the Lie brackets of the two algebras: Y[v,w] = [Yv , Yw ] .

(10.56)

Recall that [v, w] is computed in gq,s while [Yv , Yw ] = Yv Yw − Yw Yv . Clearly it is enough to check that the two vector fields in (10.56) agree at the origin. Let f be a smooth function on RM , then by (3.9) d Yv f (x) = . [f (x ◦ tv)] dt t=0 that is " d Yv f (x) = f dt

s

X 1 x + tv + [x, tv] + Ck (x, tv) 2 k=3   1 = ∇f (x) · v + [x, v] + D (x, v) 2

!#

t=0

where s X d . Ck (x, tv) dt t=0 k=3 d Ck (x, tv) t=0 that do not vanish are the itWe remark that the only terms in dt erated commutators that contain tv exactly once. In turn this implies that these commutators contain x more than once, so that D (x, v) =

∂D (0, v) = 0, ∂xi

(10.57)

while we also have D (0, v) = 0. Now we can compute [Yv , Yw ] f (0) = Yv Yw f (0) − Yw Yv f (0) . By the above computations we have Yv Yw f (0) = v · ∇ (Yw f ) (0). Moreover,    1 ∇Yw f (x) = ∇ ∇f (x) · w + [x, w] + D (x, w) 2     1 1 = Hf (x) w + [x, w] + D (x, w) + ∇f (x) · ∇ w + [x, w] + D (x, w) 2 2

Lifting and approximation

531

where Hf denotes the Hessian of f . Using (10.57) and the fact that [x, w] is linear in the variable x we easily obtain 1 Yv Yw f (0) = v · Hf (0) w + ∇f (0) · [v, w] . 2 Similarly we obtain 1 Yw Yv f (0) = w · ∇Yv f (0) = w · Hf (0) v + ∇f (0) · [w, v] 2 and therefore 1 1 Yv Yw f (0) − Yw Yv f (0) = ∇f (0) · [v, w] − ∇f (0) · [w, v] 2 2 = ∇f (0) · [v, w] = Y[v,w] f (0) . Since f is generic this shows that the vector fields Yv Yw − Yw Yv and Y[v,w] agree at the origin. Point ii) follows from Theorem 3.29 since Dλ (σi ) = λpi σi . Point iii) is a consequence of (10.56) and the fact that, since gq,s is spanned by the elements σ[I] with |I| 6 s, the same is true for Lie (G). Point iv) easily follows from the isomorphism of the Lie algebras proved in point (i).  The next proposition contains the key property of the Y[I] which makes the approximation procedure applicable. Proposition 10.55 For every u ∈ RM we have the following X X ∂ uI Y[I] = uI . ∂uI I∈B

I∈B

Proof. First of all observe that in the identification of gq,s with RM for every I ∈ B we have identified σ[I] with the coordinate uI . It follows that Y[I] is the ∂ at the origin. Let f be a smooth left invariant vector fields that agrees with ∂u I M function on R , by (3.9) we have " !# s X d d = [f (u ◦ teI )] f u + teI + Ck (u, teI ) Y[I] f (u) = dt dt t=0 k=2 t=0 " s # X ∂f d = (u) + ∇f (u) Ck (u, teI ) . ∂uI dt k=2

t=0

Hence s X d X ∂f + ∇f (u) uI Ck (u, teI ) . uI Y[I] f (u) = uI ∂uI dt I∈B I∈B I∈B k=2 t=0 d Observe now that the commutators appearing in dt Ck (u, teI ) t=0 that contain teI more than once vanish. Suppose that a given commutator contains teI exactly once. Then X uI [u, [u, . . . , [u, teI ]]] = t [u, [u, . . . , [u, u]]] = 0 X

X

I∈B

It follows that

P

I∈B

uI Y[I] f (u) =

P

I∈B

∂f uI ∂u (u). I

532

H¨ ormander operators

The structure of homogeneous group associated to the Lie algebra gq+1,s and the properties of the vector fields Yi allows us to prove, at last, Theorem 10.32. Proof of Theorem 10.32. We start constructing the free nilpotent Lie algebra gq+1,s with q + 1 generators, that we label σ0 , σ1 , . . . , σq , with the choice pi = 1 for 1 6 j 6 q and p0 = 2. We have already remarked that the dimension of a free Lie algebra is uniquely determined by the step, the number of generators and the weights (see Remark 10.15), it follows that the dimension of gq+1,s is exactly N2 (q, s). By Proposition 10.52
and Proposition 10.54 there is a structure of homogeneous group G = RN , ◦, Dλ so that Lie (G) is isomorphic to gq+1,s . Let Yi , i = 1, . . . , q+1 as in Theorem 10.54 and relabel Yq+1 with Y0 . These vector fields are free up to the step s and span RN . This shows (i). Moreover Yi is homogeneous of degree 1 when 1 6 i 6 q and is homogeneous of degree 2 when i = 0. It follows that for every multiindex I, Y[I] is homogeneous of degree |I|. This gives (ii). To see point (iii) observe that
the vector fields Y[I] agree with σ[I] at the origin. If I ∈ B this ∂ means that Y[I] 0 = ∂u . Point (iv) has been proved in Proposition 10.55. I In the next proposition we address the case of vector fields of weight 1, that is no drift term of weight 2. The proof of Theorem 10.31 is then straightforward. Proposition 10.56 Let pi = 1 for 1 6 j 6 q then RM , ◦, Dλ Proposition 10.52 is a Carnot group. Proof. First of all we show that the decomposition gq,s = stratified Lie algebra of step s, that is Wk = [Wk−1 , W1 ]

Ls

k=1




as defined in

Wk gives rise to a

for k = 2, . . . , s

and [Wk−1 , W1 ] = {0} for k > s. By the definition of Wk it is enough to show that Vk = [Vk−1 , V1 ]. Since every g ∈ Vk is a linear combination of iterated commutator e[I] of weight k it is enough to show that e[I] ∈ [Vk−1 , V1 ] with |I| = k. To see this, let I = (i1 , i2 , . . . , ik ) = (i1 , I 0 ). Since |I 0 | = k − 1 and (i1 ) = 1 we have     e[I] = ei1 , e[I 0 ] = − e[I 0 ] , ei1 ∈ [Vk−1 , V1 ] . The isomorphism of Lie algebras described in Proposition 10.52 shows that also the Lie algebra of left invariant vector fields on RM is stratified and that its first layer is spanned by the vector fields Y1 , Y2 , . . . , Yq .

Lifting and approximation

10.6

533

Notes

The main results in sections 2 and 3, about lifting and approximation, are originally contained in the paper by Rothschild-Stein [142, Part II]. The idea of attacking the study of general H¨ ormander vector fields making use of the theory already developed in the case of homogeneous groups had been explicitly declared by Folland in [85, p. 162], as a motivation for the development of that theory, and its roots date back to the paper [91] by Folland-Stein, where this idea had been already developed in a particular case. Several authors have given alternative proofs of Rothschild-Stein’s lifting and approximation theorem. In 1978 H¨ ormander-Melin [108], and Goodman [101], independently, presented alternative proofs of the lifting theorem and a pointwise version of the approximation theorem. Folland [86], 1977, aiming to present a more transparent proof of the lifting theorem, actually proved a different result, of independent interest. He considered the special case of H¨ormander vector fields defined in the whole Rn , whose Lie algebra is nilpotent and homogeneous, assumptions which are fulfilled for instance by X1 = ∂x1 ; X2 = xk1 ∂x2 in R2 . In this situation he proved that the vector fields can be lifted to (not necessarily free), left invariant and homogeneous vector fields on a Carnot group, without the necessity of introducing a remainder. In other words, in this special case the relation (10.1) simplifies to ei (f (Θη (·))) = (Yi f ) (Θη (·)) . X Moreover, differently from what happens in the case of general H¨ormander vector fields, Folland’s lifting is global. Some new and interesting applications of Folland’s lifting theorem have been given by Biagi-Bonfiglioli [6], 2017, and [7], in the explicit construction of a global homogeneous fundamental solution for a class of homogeneous (but not translation invariant) H¨ormander operators on Rn , therefore generalizing Folland’s result in [85]. In 1999 Christ-Nagel-Stein-Wainger in [64, §22] proved a more general version of the lifting theorem, because they also consider “weighted” vector fields. Assume that X1 , . . . , Xq is a system of vector fields in an open set Ω ⊆ Rn , and each Xi has an assigned “weight” expressed by an integer pi > 1. We say that a commutator [Xi , Xj ] has weight pi + pj , and so on. Assume that X1 , . . . , Xq satisfy H¨ormander’s condition at weighted step s, that is the commutators of weighted length 6 s are enough to span Rn at every point of Ω. Then there exists an integer N = n + m e1 , . . . , X eq in RN , which are free up to weighted and, locally, lifted vector fields X step s and still satisfy H¨ ormander’s condition at weighted step s. The authors also prove an approximation result. This more general theorem in particular covers the case of a H¨ ormander operator with drift, where X1 , . . . , X1 have weight 1 while X0 has weight 2. (This case is not explicitly carried out in Rothschild-Stein’s paper).

534

H¨ ormander operators

In 2005 Bonfiglioli-Uguzzoni in [18] proved the following lifting theorem for Carnot groups. Let X1 , . . . , Xq be the generators of a Carnot group G in Rn , which is not free. (The Heisenberg groups Hn for n > 2 are an example of nonfree Carnot groups). Then there exists a higher dimensional free Carnot group G0 e1 , . . . , X eq of G0 are a lifting in RN for some N > n such that the generators X of X1 , . . . , Xq . Like for Folland’s 1977 result in [86], here vector fields are lifted directly to the generators of a Carnot group, without the necessity of a remainder. Hence, this result is not covered by the original one by Rothschild-Stein. Differently from Folland, here the lifted vector fields are free, and the starting vector fields are already left invariant. So, this result and Folland’s one are not comparable. In this chapter we have basically followed the approach of H¨ormander-Melin: section 10.2 and the first two subsections of section 10.3 can be seen as a detailed exposition of the paper [108], as already presented in the paper [32], where this approach is used in the context of nonsmooth H¨ormander vector fields, that is vector fields with coefficients of class C s−1 (Ω), satisfying H¨ormander’s condition at step s in Ω. However, we have also proved in the following parts of the chapter some further results related to the map Θ and the quasidistance induced by the lifted vector fields which are necessary to get the full statement of the results proved by RothschildStein in [142, Part II], formulated in their language. We note that the version of the approximation theorem that we have presented here (Theorem 10.30, taken from [108]) is somewhat more general than the one originally proved in [142, Part II], since the vector fields Y[I] need not be left invariant on a homogeneous group; they only need to satisfy (10.41). The comparison between volumes of balls in the original and lifted space (Theorem 10.40) is a result originally due to Nagel-Stein-Wainger [131] and, independently and with a different technique, to S´ anchez-Calle [144]. More precisely, the result e ((x0 , 0) , δ), while the result proved in [131] contains only the lower bound on B proved in [144] contains both the inequalities but does not cover the case of a drift term X0 in the definition of control distance. A proof of the full result that we have covered can be found in the paper by Jerison [110]. In the special case of a family of H¨ ormander vector fields defined in the whole Rn which are 1-homogeneous with respect to a family of nonisotropic dilations (but not necessarily left invariant with respect to any group structure), this result has been proved to hold globally in Rn , in [10].

Chapter 11

Sobolev and H¨ older estimates for general H¨ ormander operators

11.1

Introduction and general overview

The goal of this chapter is the proof of existence, local a priori estimates and regularity results, in Sobolev or H¨ older spaces modeled on the vector fields Xi and their distance, for H¨ ormander operators. Let us first recall and fix some notation. Let X0 , X1 , . . . , Xq be a system of H¨ ormander vector fields in some domain Ω ⊂ Rn , where the drift X0 (if present) has weight 2 while the other vector fields Xi have weight 1. If k·k is a function space norm, we will set X

j

D u = kXI uk for j = 1, 2, 3, . . . |I|=j

0

D u = kuk . For instance, q X

2

D u = kX0 uk + kXi Xj uk . i,j=1 1

We recall that the Sobolev norms 1, 2, 3, . . . , by

k,p WX

kukW k,p (Ω) X

(Ω) are defined, for p ∈ (1, ∞) and k =

k X

j

D u p = L (Ω) j=0

where the derivatives Xi u, XI u must be intended in weak sense. α (Ω) are defined, for α ∈ (0, 1] and k = The H¨ older seminorms and norms CX 1, 2, . . . , letting   |u (x) − u (y)| |u|C α (Ω) = sup : x, y ∈ Ω, x 6= y α X d (x, y) kukC α (Ω) = |u|C α (Ω) + kukL∞ (Ω) X

kukC k,α (Ω) = X

k X

j

D u α . C (Ω) j=0

X

1 The definitions and basic properties of Sobolev and H¨ older spaces induced by a system of vector fields have been given in Chapter 2, sections 2.1 and 2.2.

535

536

H¨ ormander operators

Here d is the (weighted) control distance induced by X0 , X1 , . . . , Xq , and the derivatives Xj u, XI u must be intended as intrinsic derivatives: Xj u (x) =

d u (γ (t))/t=0 dt

where γ (t) is an integral line of the vector field Xj with γ (0) = x. k,p k,α To compare the derivatives Xi u appearing in the definitions of WX or CX spaces, recall that continuous intrinsic derivatives are also derivatives in weak sense (see Proposition 2.22). α For α = 1 the space CX (Ω) is called Lip (Ω). More properties of these function spaces will be discussed in sections 11.5.1 and 11.5.3, respectively. The main results of this chapter are those obtained when the drift is lacking, that is for operators L=

q X

Xi2

i=1

where every vector field Xi is weighted 1. In this context, we will prove the following: Pq Theorem 11.1 (Interior regularity for sum of squares) Let L = i=1 Xi2 be a H¨ ormander operator without drift in Ω. Then: (a) for any domains Ω0 b Ω00 b Ω, nonnegative integer k and p ∈ (1, ∞) there k,p exists a constant c such that for any u ∈ D0 (Ω), if Lu ∈ WX (Ω) then k+2,p u ∈ WX,loc (Ω) and the following holds: o n kukW k+2,p (Ω0 ) 6 c kLukW k,p (Ω00 ) + kukLp (Ω00 ) ; X

X

0

00

(b) for any domains Ω b Ω b Ω, nonnegative integer k and α ∈ (0, 1) there k,α exists a constant c such that for any u ∈ D0 (Ω), if Lu ∈ CX (Ω) then k+2,α 0 u ∈ CX,loc (Ω ) and the following holds: n o kukC k+2,α (Ω0 ) 6 c kLukC k,α (Ω00 ) + kukL∞ (Ω00 ) . X

X

This result can be seen as a deep extension of the subelliptic estimates proved in Chapter 5: any distributional solution of Lu = f can be locally regularized, in the scale of Sobolev and H¨ older spaces induced by the vector fields, with an exact gain of two derivatives. This fact is perfectly analogous to what we have proved in Chapter 8 on homogeneous groups (Theorems 8.5 and 8.7). We will prove Theorem 11.1 throughout Theorems 11.43, 11.45, 11.58, 11.59, 11.62. If the operator L contains also a drift term X0 , that is L=

q X

Xi2 + X0

i=1

with X0 , X1 , . . . , Xq satisfying H¨ ormander’s condition in Ω, then we just prove the basic interior estimate (k = 0), that is:

Sobolev and H¨ older estimates for general H¨ ormander operators

537

Theorem 11.2 (Interior estimates for H¨ ormander operators) Let L = Pq 2 X + X be a H¨ o rmander operator in Ω. Then: 0 i=1 i (a) for any domains Ω0 b Ω00 b Ω and p ∈ (1, ∞) there exists a constant c such 2,p that for any u ∈ D0 (Ω), if Lu ∈ Lp (Ω) then u ∈ WX,loc (Ω) and n o kukW 2,p (Ω0 ) 6 c kLukLp (Ω00 ) + kukLp (Ω00 ) ; X

(b) for any domains Ω0 b Ω00 b Ω and α ∈ (0, 1) there exists a constant c such 2,α α that for any u ∈ D0 (Ω), if Lu ∈ CX (Ω) then u ∈ CX,loc (Ω) and n o kukC 2,α (Ω0 ) 6 c kLukC α (Ω00 ) + kukL∞ (Ω00 ) . X

X

This result will be proved in Theorems 11.43, 11.58, 11.62. The reason why we do not prove higher order estimates also for operators with drift is related to the highly technical difficulties hidden in the details of some of the results proved in sections 11.2 and 11.3. If the drift is present, the proof of a useful representation formula for higher order derivatives, like the one contained in Theorem 11.28, seems to be a challenging open problem. In turn, this fact depends on a technical result (see Theorem 11.19) that allows to exchange fractional integral operators and differentiation with respect to vector fields. If the drift is lacking (see Remark 11.20) this result simplifies to a form that enables us to prove suitable representation formulas for higher order derivatives. Remark 11.3 (Operators with lower order terms) In both the above theorems, we could also add to the operator L lower order terms, that is consider L1 u = Lu +

q X

ci (x) Xi u + c0 (x) u.

i=1

Then the same results hold, with natural assumptions on the coefficients ci : k,∞ k,α • for Theorem 11.1: (a) ci ∈ WX (Ω) ; (b) ci ∈ CX (Ω). ∞ α • for Theorem 11.2: (a) ci ∈ L (Ω); (b) ci ∈ CX (Ω). The proofs of these more general results could be achieved by a routine modification of the proofs that we will give in this chapter, and we leave the details to the interested reader. Finally, we will also prove (section 11.6) a global solvability result (on bounded domains), under the mild extra assumption of not total degeneration of the operator: Theorem 11.4 (Solvability in Sobolev and H¨ older spaces) Let L be a H¨ ormander operator which in a bounded domain Ω ⊂ Rn is uniformly not toPq 2 tally degenerate (see (6.1)), that is i=1 bij (x) > c0 > 0 in Ω, where Xi = Pn j=1 bij (x) ∂xj . Then:

538

H¨ ormander operators 2,p (a) for any f ∈ Lp (Ω) (for some p ∈ (1, ∞)) there exists u ∈ WX,loc (Ω)∩L1 (Ω) satisfying (in the distributional and then in the strong sense) the equation Lu = f and such that

kukL1 (Ω) 6 C kf kL1 (Ω)

(11.1)

with C = CΩ,L as in Proposition 6.1. The function u also satisfies the a priori estimate n o kukW 2,p (Ω0 ) 6 c kf kLp (Ω) + kukL∞ (Ω00 ) X

for any domains Ω0 b Ω00 b Ω; 2,α α (b) for any f ∈ CX (Ω) (for some α ∈ (0, 1)) there exists u ∈ CX,loc (Ω)∩L1 (Ω) satisfying (in the distributional and then in the classical sense) the equation Lu = f . The function u also satisfies (11.1) and the a priori estimate n o kukC 2,α (Ω0 ) 6 c kf kC α (Ω) + kukL∞ (Ω00 ) X

X

for any domains Ω0 b Ω00 b Ω. The general strategy of the proof of the a priori estimates appearing in Theorems 11.1-11.2 has been already sketched at the beginning of Chapter 10, where an important part of the necessary machinery, the lifting and approximation, has been developed for this scope. See Theorem 10.6 for a summary of the relevant properties. Actually, for the proof of Theorems 11.1-11.2 we will combine this technique with the existence and properties of a global homogeneous fundamental solution for homogeneous left invariant H¨ ormander operators on groups, presented in Chapter 6. For the convenience of the reader, in the next theorem we summarize some of its properties (see Theorems 6.18, 6.20, 6.32, and Corollary 6.31). Theorem 11.5 (Homogeneous fundamental solution on groups) Let G and Yi (i = 0, 1, . . . , q) be as in Theorem 10.6 and assume that the homogeneous dimenPq sion of G is Q > 3. Then the operator L = i=1 Yi2 + Y0 has a unique fundamental solution Γ such that:
(a) Γ ∈ C ∞ RN \ {0} ; (b) Γ, Yi Γ, Yi Yj Γ (for i, j = 1, 2 . . . , q) and Y0 Γ are homogeneous of degree (2 − Q), (1 − Q), −Q, −Q respectively, hence satisfy bounds: |Γ (u)| 6 |Yi Γ (u)| 6 |Yi Yj Γ (u)| + |Y0 Γ (u)| 6

c Q−2

;

Q−1

;

kuk c kuk c

Q

kuk

for any u ∈ G, i, j = 1, . . . , q, for some constant c = c (G);

Sobolev and H¨ older estimates for general H¨ ormander operators

(c) for every test function f and every v ∈ RN , Z
f (v) = (Lf ∗ Γ) (v) = Γ u−1 ◦ v Lf (u) du;

539

(11.2)

RN

moreover, for every i, j = 1, . . . , q, there exist constants αij such that Z
Yi Yj f (v) = PV Yi Yj Γ u−1 ◦ v Lf (u)du + αij Lf (v); (11.3) RN

(d) For any differential operator Z 2 homogeneous of degree 2 on G (not necessarily translation invariant) we have Z Z Z 2 Γ ( u) du = Z 2 Γ ( u) dσ(u) = 0 for every R > r > 0 r 0 is still an operator of type λ; ei T and T X ei are operators of type (3) if T is an operator of type λ > 1, then X λ − 1. These requirements lead to the following Definition 11.7 (Kernels of type λ) We say that k(ξ, η) is a kernel of type λ
e ξ, R ), for some integer λ (typically we will use λ = 0, 1, 2), if for (over the ball B
e ξ, R , every positive integer m we can write, for ξ, η ∈ B k(ξ, η) ≡ k 0 (ξ, η) + k 00 (ξ, η) (H ) m X ≡ ai (ξ)bi (η)Di Γ(·) + a0 (ξ)b0 (η)D0 Γ(·) (Θ(η, ξ)) i=1

+

(H m X

) a0i (ξ)b0i (η)Di0 Γ∗ (·)

+

a00 (ξ)b00 (η)D00 Γ∗ (·)

(Θ(η, ξ))

i=1

where: 
 e ξ, R (i = 0, 1, . . . Hm ); • ai , bi , a0i , b0i ∈ C0∞ B • for i = 1, . . . , Hm , Di and Di0 are homogeneous differential operators of degree 6 2 − λ (so that Di Γ and Di0 Γ∗ are homogeneous functions of degree > λ − Q); • D0 and D00 are differential operators (generally nonhomogeneous but) such that D0 Γ and D00 Γ∗ have m (weighted) derivatives with respect to the vector fields Yi (i = 0, 1, . . . , q). Moreover, the coefficients of the differential operators Di , Di0 for i = 0, 1, . . . , Hm possibly depend also on the variables ξ, η, in such a way that the joint dependence on (ξ, η, u) is smooth. In order to simplify notation, we will not always express explicitly this dependence of the coefficients of Di on ξ, η. Only when it is necessary we will write, for instance, ai (ξ)bi (η)Diξ,η Γ(Θ(η, ξ)) to recall this dependence. Remark 11.8 Let us clarify the motivation of the various ingredients appearing in the previous definition, which is rather involved. (i) The necessity of considering kernels modeled both on Γ and Γ∗ clearly dee0 : for an operator sum of squares the pends on the presence of the drift X 00 term k is not necessary. (ii) Actually, the previous definition could be simplified replacing the functions Di Γ, Di Γ∗ with unspecified homogeneous functions of degree > λ − Q. However, we are interested in keeping track of the dependence of our kernels on the fundamental solutions Γ, Γ∗ , in view of some subsequent generalizations of this theory, which will be described in Chapter 12. (iii) The presence of the cutoff functions is necessary in view of the local nature of the map Θ(η, ξ).

544

H¨ ormander operators

(iv) The requirement on the differential operators D0 and D00 amounts to asking that their coefficients vanish at the origin of an order large enough to make the functions D0 Γ, D00 Γ∗ smooth enough at the origin (whereas Γ and Γ∗ are singular). (v) The dependence of the coefficients of Di , Di0 on the variables ξ, η comes from the fact that, starting with the prototype kernel a (ξ) Γ∗ (Θ (η, ξ)) b (η) ei -derivatives with respect to ξ, which introduces a dependence we will take X on η in view of the approximation formula (11.5); then we will transpose the kernel, transforming this dependence on η in a dependence on ξ, then ei derivatives with respect to ξ, introducing a new we will again compute X dependence on η. Remark 11.9 Observe that if a smooth function c (ξ, η, u) is D (λ)-homogeneous with respect to u, then also its partial derivatives with respect to ξ or η have the same homogeneity. In particular,     ∂ ξ,η ∂ ξ,η Di Di Γ(·), Γ(·) ∂ξi ∂ηi have the same homogeneity as Diξ,η Γ(·). Proposition 11.10 (Growth of kernels of type > 1) If k (ξ, η) is a kernel of type λ then for every ξ, η it satisfies a bound c . (11.11) |k (ξ, η)| 6 Q−λ kΘ(η, ξ)k In particular, if λ > 1 the kernel is integrable and satisfies: Z Z sup |k (ξ, η)| dη + sup |k (ξ, η)| dξ 6 c. η

ξ

Proof. The bound |Di Γ (u)| 6 kukcQ−λ for Di homogeneous of degree 2−λ, immediately implies the bound on k. Let now λ > 1. Since k is compactly supported both eρ (ξ, r) have measure comparable in ξ and η and, for ρ (ξ, η) = kΘ(η, ξ)k, the balls B Q to r , by Lemma 7.5, for some large R and any ξ, Z Z c |k (ξ, η)| dη 6 dη 6 cRλ . Q−λ eρ (ξ,R) kΘ(η, ξ)k B

Definition 11.11 (Operators of type λ) We say that T is an operator of type
e λ > 1 (over the ball B ξ, R ) if k(ξ, η) is a kernel of type λ and Z T f (ξ) = k(ξ, η) f (η) dη e B

Sobolev and H¨ older estimates for general H¨ ormander operators

545


e ξ, R ). By the previous remark the integral converges and T f is a for f ∈ C0∞ (B bounded function. We say that T is an operator of type 0 if k(ξ, η) is a kernel of type 0 and Z T f (ξ) = PV k(ξ, η) f (η) dη + α (ξ) f (ξ) , e B

where α, the multiplicative part of T , is a smooth function of the form Z α (ξ) = a (ξ) Dξ Γ (u) ω (u) dσ (u) kuk=1

e ξ, R ), ω is a smooth function depending on the coefficients of the where a ∈ C0∞ (B homogeneous vector fields Yi , Dξ is a homogeneous differential operator of degree 6 1 smoothly depending on ξ, and the principal value integral exists. Explicitly, this principal value is defined by: Z Z k(ξ, η) f (η) dη = lim k(ξ, η) f (η) dη PV


ε→0

e B

kΘ(η,ξ)k>ε

for some homogeneous norm k·k on the group. With reference to Definition 11.7, we will call the k 0 , k 00 “kernels of type λ modeled on Γ, Γ∗ ”, respectively. Analogously, we will sometimes speak of operators of type λ modeled on Γ or Γ∗ , to denote that the kernel has this special form. It is useful to note that the above definition immediately implies the: Proposition 11.12 (i) The sum of an operator of type λ with an operator of type λ0 > λ is an operator of type λ. (ii) A kernel k of type λ > 0 can always be written as the sum of a kernel of type λ + 1 and a finite sum of kernels ai (ξ)bi (η)Di Γ(Θ (η, ξ)) or ai (ξ)bi (η)Di Γ∗ (Θ (η, ξ)) with Di homogeneous of degree 2 − λ. The finite sum of kernels which appear in point (ii) of the above proposition can be called “the principal part of k”. A common operation on operators is transposition:
e ξ, R , we will denote Definition 11.13 If T is an operator of type λ > 0 over B by T ∗ the transpose operator, defined by Z Z f (ξ) T ∗ g (ξ) dξ = g (ξ) T f (ξ) dξ e B



e ξ, R for any f, g ∈ C0∞ B

e B




.

Clearly, if k (ξ, η) is the kernel of T , then k (η, ξ) is the kernel of T ∗ . It is useful to note that:

546

H¨ ormander operators

Proposition 11.14 (Transposition of operators of type λ) If T is an operator
e ξ, R , modeled on Γ or Γ∗ , then T ∗ is an operator of type λ, of type λ > 0 over B modeled on Γ∗ , Γ, respectively. In particular, the transpose of an operator of type λ is still an operator of type λ. b b Proof. Let D be any differential operator on the group G. For any f ∈ C0∞ (G) , let f (u) = f (−u) and let D be the differential \ operator defined by the identity b b Df = D f .

(11.12)

b Clearly, if D is homogeneous of some degree β, the same is true for D; if DΓ or DΓ∗ b haveb m (weighted) derivatives with respect to Yi (i = 0, 1, . . . , q), the same is true ∗ d b for DΓ or DΓ . Also, recalling that Γ∗ (u) = Γ(−u), applying (11.12) to f = Γ∗ we have DΓ∗ = DΓ. Then, if (H ) m X k 0 (ξ, η) = ai (ξ)bi (η)Di Γ(·) + a0 (ξ)b0 (η)D0 Γ(·) (Θ(η, ξ)) i=1

is a kernel of type λ modeled on Γ, since Θ (ξ, η) = −Θ (η, ξ), (see (10.39)), ) (H b b m X k 0 (η, ξ) = ai (η)bi (ξ)Di Γ∗ (·) + a0 (ξ)b0 (η)D0 Γ∗ (·) (Θ(η, ξ)) i=1

is a kernel of type λ modeled on Γ∗ . Analogously one can prove the converse. We have now to deal with the relations between operators of type λ and the ei . The first result is the differential operators represented by the vector fields X following: Theorem 11.15 (Composition of derivatives and operators of type λ) ek T and T X ek (k = 1, . . . , q) are Suppose T is an operator of type λ > 1. Then X e e operators of type λ − 1. If λ > 2, then X0 T and T X0 are operators of type λ − 2. In ek T is an operator of type zero with particular if T is an operator of type 1, then X the multiplicative part given by Z αk (ξ) = a (ξ) D1ξ Γ (u) nk dσ (u) (11.13) kuk=1

PN j=1 bkj (u) νj , with Yk = j=1 bkj (u) ∂uj ξ and D1 is a homogeneous differential operator

for a suitable cutoff function a, nk =

PN

and ν is the outer normal on kuk = 1, of degree 1, smoothly depending on the variable ξ. Analogously, if T is an operator e0 T is an operator of type zero with the multiplicative part given by of type 2, then X Z α0 (ξ) = a (ξ) D0ξ Γ (u) n0 dσ (u) kuk=1

with the analogous meaning of a, n0 , and D0ξ homogeneous differential operator of degree 1, smoothly depending on the variable ξ.

Sobolev and H¨ older estimates for general H¨ ormander operators

547

To prove this, we start with the following two lemmas:
e ξ, R , then X ej k(·, η) (ξ) Lemma 11.16 If k(ξ, η) is a kernel of type λ over B e0 k(·, η) (ξ) is a kernel of type λ − 2. (j = 1, . . . , q) is a kernel of type λ − 1 and X Proof. This basically follows by the definition of kernel of type λ and Theorem 10.6. It is enough to consider the part k 0 of the kernel of T , the proof for k 00 being completely analogous (see Definition 11.7). Also, it is enough to show the assertion ej k(·, η) (ξ) (j = 1, . . . , q), since the case j = 0 is analogous. We have to for X compute: ) (H m X ξ ξ e Xj ai (·)bi (η)D Γ (Θ(η, ·)) + a0 (·)b0 (η)D Γ (Θ(η, ·)) (ξ) 0

i

i=1

(the ξ-dependence of the coefficients of the differential operators Diξ will be relevant in the present computation). Now,   ej ai (·)bi (η)Dξ Γ (Θ(η, ·)) (ξ) X i     = ai (ξ)bi (η) Yj Diξ Γ (Θ(η, ξ)) + ai (ξ)bi (η) Rη,j Diξ Γ (Θ(η, ξ)) (11.14)   ξ ξ e e + ai (ξ)bi (η) Xj Di Γ (Θ(η, ξ)) + Xj ai (ξ)bi (η)Di Γ (Θ(η, ξ)) ≡ A + B + C + D. For i = 1, . . . , Hm , the kernels Diξ Γ are homogeneous functions of degree > λ−Q on ξ G,  the term Yj Di Γ appearing in A is homogeneous of degree > λ−Q−1 while  hence ej Dξ Γ appearing in C (see Remark 11.9) and Dξ Γ in D are homogeneous of X i i   ξ degree > λ−Q. On the other hand, the kernel Rη,j Di Γ in B is not homogeneous, but Rη,j has weight > 0 (see Definition 10.24). Let us write X Rη,j = cη,J (u) ∂uJ , J∈B

where u = (uJ )J∈B denotes the canonical coordinates on G (see Chapter 10, section 10.3). For any fixed integer K > 0, we can write the Taylor expansion of cη,J (u) up to order K, and then rearrange the monomials with respect to their (weighted) homogeneities, getting a sum of terms of the kind cJk (η) pk (u) where pk (u) is a k-homogeneous pure monomial and cJk (η) 6= 0 only if pk (u) ∂uJ is homogeneous of degree > 0 (by definition of operator of weight > 0), plus a remainder cJ (η, u) (where we collect the remainder of Taylor formula and the monomials which have Euclidean degree 6 K but weighted homogeneity > K) that satisfies   K cJ (η, u) = o kuk for u → 0   uniformly in η. Hence in the term B, the function Rη,j Diξ Γ (u) is a finite sum of terms cJk (η) pk (u)∂uJ Diξ Γ (u)

548

H¨ ormander operators

  plus a remainder, where pk ∂uJ Diξ Γ = Di0ξ Γ is still homogeneous of degree > λ−Q while, choosing K large enough, the remainder X cJ (η, u) ∂uJ Diξ Γ(u) J

is as smooth as we want. This shows that the sum in (11.14) obeys the definition of kernel of type λ − 1. Also, splitting   ej a0 (·)b0 (η)Dξ Γ (Θ(η, ·)) (ξ) k0 (ξ, η) ≡ X 0 as in (11.14) one can check that if D0ξ Γ (u) has m (weighted) derivatives with respect to the vector fields Yi (i = 0, 1, . . . , q), then an analogous property holds for k0 , with m replaced by m − 1. Remark 11.17 From the above proof we read, in particular, that if ai (ξ)bi (η)Di Γ(Θ (η, ξ)) is a term in the sum defining a kernel of type λ > 1, then also ai (ξ)bi (η) (Rη,j Di Γ) (Θ (η, ξ)) (where Rη,j is a remainder of weight > 0) is a kernel of type λ.
e ξ, R , then X ei T (i = Lemma 11.18 If T is an operator of type λ > 1 over B e 1, . . . , q) is an operator of type λ − 1. If λ > 2, then X0 T is an operator of type λ − 2. Proof. With reference to Definition 11.7, it is enough to consider the part k 0 of the kernel of T , the proof for k 00 being completely analogous. So, let us consider the ei T (i = 1, . . . , q), where T has kernel k 0 . If λ > 2, the result immediately operator X 
 e ξ, R , follows by the previous lemma. If λ = 1, then for any f ∈ C ∞ B 0

Z T f (ξ) = e (ξ,R) B

a(ξ)b(η)D1ξ Γ (Θ(η, ξ)) f (η) dη + T 0 f (ξ)

where T 0 is an operator of type 2, and D1ξ is a 1-homogeneous differential operator (actually, we could have a finite sum of terms of this kind, see Proposition 11.12). ei T 0 is an operator of type 1, so we are left to show that the We already know that X first term is an operator of type 0. To do this, we use a distributional
argument, e=B e ξ, R and for any which will be repeated several times in the following. Let B   e , let us compute ω ∈ C∞ B 0

Z

Z e ∗ ω (ξ) X a(ξ)b(η)D1ξ Γ(Θ(η, ξ))f (η) dηdξ i e e B B Z Z ∗ e = lim Xi ω (ξ) a(ξ)b(η)ϕε (Θ(η, ξ)) D1ξ Γ(Θ(η, ξ))f (η) dηdξ ε→0

e B

e B

Sobolev and H¨ older estimates for general H¨ ormander operators

549


where ϕε (u) = ϕ (Dε−1 (u)) and ϕ ∈ C0∞ RN , ϕ (u) = 0 for kuk < 1/2, ϕ (u) = 1 for kuk > 1. Applying Theorem 10.6 we have Z Z ∗ e Xi ω (ξ) a(ξ)b(η)ϕε (Θ(η, ξ)) D1ξ Γ(Θ(η, ξ))f (η) dηdξ e e ZB ZB e ∗ ω (ξ) a(ξ)ϕε (Θ(η, ξ)) Dξ Γ(Θ(η, ξ))dξdη = b(η)f (η) X i 1 e e B B Z Z ei a(ξ)ϕε (Θ(η, ξ)) Dξ Γ(Θ(η, ξ))dξdη = b(η)f (η) ω (ξ) X 1 e e (11.15) B B Z Z   ξ ei D Γ(Θ(η, ξ))dξdη + b(η)f (η) ω (ξ) a(ξ)ϕε (Θ(η, ξ)) X 1 e e B B Z Z i h  + b(η)f (η) ω (ξ) a(ξ) (Yi + Rη,i ) ϕε D1ξ Γ (Θ(η, ξ)) dξdη e B

e B

≡ Aε + Bε + Cε . Now, for ε → 0, Z Aε →

Z ei a(ξ)D1 Γ(Θ(η, ξ))dξdη b(η)f (η) ω (ξ) X e e B B Z Z = f (η) F1 ω (η) dη = ω (η) F1∗ f (η) dη e B

(11.16)

e B

where F1 is an operator of type 1, and F1∗ , its transpose, is still an operator of type 1 (see Proposition 11.14). As for the second term, Z Z   ei Dξ Γ(Θ(η, ξ))dξdη Bε → b(η)f (η) ω (ξ) a(ξ) X 1 e e B B Z Z (11.17) = f (η) F2 ω (η) dη = ω (η) F2∗ f (η) dη e B

e B

where, by Remark 11.9, F2 is an operator of type 1, and the same is true for F2∗ by Proposition 11.14. For the last term we have Z Z Cε = b(η)f (η) ω (ξ) a(ξ) (ϕε Yi D1 Γ) (Θ(η, ξ)) dξdη e e B B Z Z + b(η)f (η) ω (ξ) a(ξ) (ϕε Rη,i D1 Γ) (Θ(η, ξ)) dξdη e e (11.18) ZB ZB + b(η)f (η) ω (ξ) a(ξ) [(D1 Γ) (Yi + Rη,i ) ϕε ] (Θ(η, ξ)) dξdη e B

≡

Cε1

+

e B

Cε2

+

Cε3 .

Now, Cε1



Z →

Z



ω (ξ) PV a(ξ)Yi D1 Γ(Θ(η, ξ))b(η)f (η) dη dξ e B Z = ω (ξ) Sf (ξ) dξ e B

(11.19)

e B

with S operator of type 0. To check that this principal value actually exists we have to exploit the vanishing property of the function Yi D1 Γ (u) (see Theorem 11.5,

550

H¨ ormander operators

(d)). With the change of variables Θ(η, ξ) = u, by Theorem 10.6, (c), we have dξ = c (η) (1 + O (kuk)) du and letting ω e (u) = (ωa) (Θ(η, ·)−1 (u)) we get Z Z (ωa) (ξ) (ϕε Yi D1 Γ) (Θ(η, ξ)) dξ = c (η) (e ω ϕε Yi D1 Γ) (u) (1 + O (kuk)) du e B Z Z = c (η) (ϕε Yi D1 Γ) (u) [e ω (u) − ω e (0)] du + c (η) ω e (0) (ϕε Yi D1 Γ) (u) du Z + c (η) (e ω ϕε Yi D1 Γ) (u) O (kuk) du = αε + βε + γε with Z αε → c (η)

(Yi D1 Γ) (u) [e ω (u) − ω e (0)] du

which is absolutely converging since [e ω (u) − ω e (0)] = O (|u|) = O (kuk); βε ≡ 0 by Corollary 6.31; Z γε → c (η) (e ω Yi D1 Γ) (u) O (kuk) du which is still absolutely converging. Next, Z  Z Cε2 → ω (ξ) a(ξ)Rη,i D1 Γ(Θ(η, ξ))b(η)f (η) dη dξ e e B B Z = ω (ξ) F3 f (ξ) dξ

(11.20)

e B

with F3 operator of type 1 (see Remark 11.17). To handle Cε3 , let us perform again the change of variables u = Θ(η, ξ) which, by Theorem 10.6 (c) gives Z Z 3 Cε = (bf ) (η) ω e (u) (D1 Γ) (u) (Yi + Rη,i ) ϕε (u)c (η) (1 + O (kuk)) dudη kuk 0, hence recalling that ϕε (u) vanishes for kuk < ε/2 we can write X n η
−αj |Rη,i ϕε (u)| = cj (u) ε ∂uj ϕ (Dε−1 u) j=1 6

n n X X η

c (u) kuk−αj ∂uj ϕ (Dε−1 u) 6 c ∂uj ϕ (Dε−1 u) j j=1

j=1


for kuk < ε, while for kuk > ε we have ϕε (u) = 1 hence ∂uj ϕ (Dε−1 u) = 0.

Sobolev and H¨ older estimates for general H¨ ormander operators

551


Since D1 Γ is (1 − Q)-homogeneous, the change of variables D 1ε u = v, gives   Z Z 1 3 ω e (D (ε) v) Yi ϕ (v) + O (1) × Cε = (bf ) (η) ε e B kvk< R ε × c (η) ε1−Q D1η Γ (v) (1 + O (ε kvk)) εQ dvdη Z

Z

ω e (0) Yi ϕ (v) D1η Γ (v) dvdη e ZB Z Z = (ωabcf ) (η) Yi ϕ (v) D1η Γ (v) dvdη ≡ (ωabcf ) (η) αi (η) dη,

→

(11.21)

(bcf ) (η)

e B

e B

which is the integral of ω times the multiplicative part of an operator of type 0. It is worthwhile (although not logically necessary to prove the theorem) to realize that the quantity αi (η) appearing in (11.21) actually does not depend on the function ϕ. Namely, recalling that Yi ϕ (v) is supported in the spherical shell 1/2 6 kvk 6 1, with ϕ (u) = 1 for kuk = 1 and ϕ (u) = 0 for kuk = 1/2, an integration by parts gives Z Yi ϕ (v) D1η Γ (v) dv 1/26kvk61 Z Z =− ϕ (v) Yi D1η Γ (v) dv + D1η Γ (v) ni dσ (v) kvk=1

1/26kvk61

with ni =

PN

j=1 bij

(u) νj , where Yi =

PN

j=1 bij

(u) ∂uj and ν is the outer normal on

kvk = 1. The vanishing property of the kernel Yi D1ξ Γ (·) implies that if ϕ is a radial function the first integral vanishes. Therefore Z αi (η) = D1η Γ (v) ni dσ (v) kvk=1

which also shows that αi (η) smoothly depends on η. By (11.15), (11.16), (11.17), (11.19), (11.20), (11.21) we have therefore proved that ei T f (ξ) = F1∗ f (ξ) + F2∗ f (ξ) + F3 f (ξ) + Sf (ξ) + α (ξ) (abcf ) (ξ) X which is an operator of type 0. This completes the proof of the first statement of e0 T is an operator of type the Lemma. The proof of the fact that if λ > 2 then X λ − 2 is completely analogous. Conclusion of the proof of Theorem 11.15. We have already proved the ek T and X e0 T in Lemma 11.18. The assertions about T X ek , T X e0 just assertion on X follow coupling together the results about the derivative and the transpose of  ∗ an ek = operator of type λ. Namely, let T be an operator of type λ > 1, then T X e ∗ T ∗ . By Proposition 11.14, T ∗ is an operator of type λ, while recalling that X k  ∗ e ∗ = −X ek + c we have T X e k = −X ek T ∗ + cT ∗ . Now −X ek T ∗ is an operator of X k ek T ∗ + cT ∗ by Proposition type λ − 1 by Lemma 11.18, hence the same is true for −X e 11.12 and the same is true for T Xk by Proposition 11.14.

552

H¨ ormander operators

Let us come to the second important result of this section, which gives a way ei and an to somewhat exchange the order of composition between a derivative X operator of type λ, as anticipated in the introduction of this chapter: Theorem 11.19 (i) If F is an operator of type λ > 1, then for i = 1, 2 . . . , q there exist operators Fik (k = 0, 1, . . . , q) of type λ, and operators Pi of type λ + 1 such that ei F = X

q X

ek + Fi0 + Pi X e0 . Fik X

(11.22)

k=1

With possibly different operators Fik we can also write q X

ei F = X

ek + Fi0 + Pi L. e Fik X

(11.23)

k=1

(ii) If P is an operator of type λ > 2, there exist operators Fk (k = 0, 1, . . . , q) of type λ − 1 and an operator P0 of type λ such that e0 P = X

q X

ek + F0 + P0 X e0 . Fk X

(11.24)

k=1

With possibly different operators Fk we also have e0 P = X

q X

ek + F0 + P0 L. e Fk X

(11.25)

k=1

Remark 11.20 If our system of H¨ ormander vector fields does not contain a drift e0 , the previous commutation formulas simplify to term X ei F = X

q X

ek + Fi0 (i = 1, . . . , q) Fik X

k=1

with Fik (k = 1, . . . , q) of type λ, if F in any operator of type λ > 1. To establish the previous theorem we will need a way to express ξ-derivatives of the integral kernel in terms of η-derivatives of the kernel, in order to integrate by parts. This will involve the use of right invariant vector fields on the group G. R Throughout the following, we will denote by Y[I] the right invariant vector field on G which agrees with Y[I] at the origin. We have the following: b Lemma 11.21 For any function f defined on G, let f (u) = f (−u) (recall that −u = u−1 ); then for every multiindexbI [ R Y[I] f = −Y[I] f.

\ Proof. Let us define the vector fields Y [I] by b Y [I] f = −Y[I] f ,

(11.26)

(11.27)

Sobolev and H¨ older estimates for general H¨ ormander operators

553

then, for any a ∈ G, denoting by La , Ra the corresponding operators of left and right translation, respectively (acting on functions), and noting that d b   b −1 Ra f (u) = Ra f (−u) = f (−u ◦ a) = f (−a ◦ u) = f (−a ◦ u) = L−a f (u) we have

\

b b  d L−a f =\ −L−a Y[I] f Y [I] Ra f = −Y[I] Ra f = −Y[I] [ b  = L−a −Y[I] f = L−a Y [I] f = Ra Y [I] f ,

hence Y [I] are right invariant vector fields. Also, note that for any vector field b P Y = aj (u) ∂uj we have Y f (0) = − (Y f ) (0) because b X X
aj (u) ∂uj [f (−u)] = − aj (u) ∂uj f (−u) implies Y f (u) = b X
aj (0) ∂uj f (0) = − (Y f ) (0) Y f (0) = − hence by (11.27) we know that Y k f (0) = Yk f (0). Therefore Y k is the right invariant vector field which coincides with Yk at the origin, that is Y k = YkR , and the Lemma is proved. Next, we can prove the following: Lemma 11.22 For any f ∈ C0∞ (G) and η, ξ in a neighborhood of ξ0 , we can write, for any i = 1, . . . , s, k = 1, . . . , ki (recall s is the step of the Lie algebra)   
 e[I] [f (Θ (·, ξ))] (η) = − Y R f (Θ (η, ξ)) + R[I],ξ 0 f (Θ (η, ξ)) , (11.28) X [I]
0 where R[I],ξ is a vector field of weight > − |I| + 1 smoothly depending on ξ. Proof. By (11.5) and (11.26), b b i  hb ξ e e X[I]\ [f (Θ (·, ξ))] (η) = X[I] f (Θ (ξ , ·)) (η) = Y[I] f + R[I] f (Θ (ξ , η)) b   ξ R (11.29) f (Θ (ξ , η)) = − Y[I] f (Θ (ξ , η)) + R[I]     ξ 0 R = − Y[I] f (Θ (η, ξ)) + (R[I] ) f (Θ (η, ξ)) , b     ξ 0 ξ where (R[I] ) f (u) = R[I] f (−u) is obviously a differential operator of weight > − |I| + 1. This proves (11.28). Let now

n o e[I] X

I∈B

a basis of RN at every point of a neighborhood of ξ, made

e0 , X e1 , . . . , X eq , with |I| 6 s. Let us consider also the correby N commutators of X  sponding basis Y[I] I∈B consisting in left invariant vector fields on G. By Theorem N

10.31 (iii) or 10.32 (iii), this basis coincides with the canonical basis {Yi }i=1 of the Lie algebra of G and therefore has the structure specified in Theorem 3.29. We have the following:

554

H¨ ormander operators

e[I] , |I| 6 s, there exist smooth functions Lemma 11.23 For any vector field X {aηIJ (u)}J∈B having weight > max {|J| − |I| , 0} on G and smoothly depending on η, such that for any f ∈ C0∞ (G), one can write e[I] [f (Θ (η, ·))] (ξ) X X η
(11.30) e[J] [f (Θ (·, ξ))] (η) + Rξ,η,[I] f (Θ (η, ξ)) = aIJ (Θ (η, ξ)) X J∈B

where Rξ,η,[I] are vector fields of weight > − |I| + 1, smoothly depending on ξ, η. Proof. By (11.5) we know that e[I] [f (Θ (η, ·))] (ξ) X

= Y[I] f + Rη,[I] f (Θ (η, ξ)) ≡ Zη,[I] f (Θ (η, ξ)) ,

(11.31)

where Z
η,[I] is a vector field of weight > − |I|, smoothly depending on η. To rewrite Zη,[I] f in the suitable form, we start from the identity X ∂ Zη,[I] = cηIJ (u) , ∂uJ J∈B,|J|>|I|

where

cηIJ

has weight > |J| − |I| and smoothly depends on η and the identity X ∂ ∂ R + dIJ (u) . (11.32) Y[I] = ∂uI ∂uJ J∈B,|J|>|I|

where dIJ are homogeneous of degree |J| − |I| (see Theorem 3.29). Inverting the triangular system (11.32), we obtain X ∂ R = fIJ (u) Y[J] , ∂uI J∈B,|J|>|I|

where each fIJ (u) is homogeneous of degree |J| − |I|. Then   X X η R Zη,[I] = cIJ (u)  fJK (u) Y[K]  J∈B,|J|>|I|

X

≡

K∈B,|K|>|J|

bηIL

(11.33)

R (u) Y[L]

L∈B,|L|>|I|

where bηIL has weight > |L| − |I|

(11.34)

and smoothly depends on η. By Lemma 11.22, then X
R Zη,[I] f (Θ (η, ξ)) = bηIL (Θ (η, ξ)) Y[L] f (Θ (η, ξ)) L∈B,|L|>|I|

X

=

e[L] [f (Θ (·, ξ))] (η) −bηIL (Θ (η, ξ)) X

L∈B,|L|>|I|

+

X L∈B,|L|>|I|

  0 bηIL R[L],ξ f (Θ (η, ξ)) ,

(11.35)

Sobolev and H¨ older estimates for general H¨ ormander operators

555

0 where R[L],ξ is a differential operator of weight > − |L| + 1, hence by (11.34) the differential operator on G X 0 has weight > − |I| + 1 (11.36) Rξ,η,[I] ≡ bηIL R[L],ξ L∈B,|L|>|I|

and depends smoothly on ξ, η. Collecting (11.31), (11.34), (11.35), (11.36), the Lemma is proved, with aηIJ (u) = −bηIJ (u). With this lemma in hand, we can now prove the following intermediate result towards Theorem 11.19: Theorem 11.24 (i) Let T be an operator of type λ > 1. For i = 1, . . . , q, there exist operators T i of type λ and (for J ∈ B) operators TJi of type λ + |J| − 1, such that X ei T = e[J] + T i . X TJi X (11.37) J∈B

(ii) Let T be an operator of type λ > 2. There exists an operator T 0 of type λ − 1, and (for J ∈ B, |J| > 2) an operator TJ0 of type λ + max {|J| − 2, 0}, such that X e0 T = e[J] + T 0 . X TJ0 X (11.38) J∈B

Proof. First of all, it is enough to consider the part k 0 of the kernel of T , the proof for k 00 being completely analogous (see Definition 11.7). (i) If T is an operator of type λ > 1 with kernel k 0 , by Proposition 11.12 we can write it as a finite sum of operators of the kind Z T f (ξ) = a (ξ) DΓ (Θ (η, ξ)) b (η) f (η) dη + T 0 f (ξ) where DΓ is homogeneous of degree λ − Q and T 0 is an operator of degree λ + 1. ei T 0 is an operator of type λ, it has already the form T i required by the Since X theorem, hence it is enough to prove that Z e Xi a (ξ) Dξ Γ (Θ (η, ξ)) b (η) f (η) dη can be rewritten in the form X

e[J] f (ξ) + T i f (ξ) TJi X

J∈B

with TJi , T i operators of type λ + |J| − 1 and λ, respectively. Next, we have to distinguish two cases.

556

H¨ ormander operators

Case 1: λ > 2. In this case we can move the derivative inside the integral: Z ei X

Z ei a (ξ) Dξ Γ (Θ (η, ξ)) b (η) f (η) dη a (ξ) Dξ Γ (Θ (η, ξ)) b (η) f (η) dη = X Z ei Dξ )Γ (Θ (η, ξ)) b (η) f (η) dη + a (ξ) (X Z ei [DΓ (Θ (η, ·))] (ξ) b (η) f (η) dη + a (ξ) X ≡ A1 (ξ) + A2 (ξ) + B (ξ) .

The terms A1 (ξ) and A2 (ξ) are operators of type λ (for A2 (ξ), see Remark 11.9), while applying Lemma 11.23 with |I| = 1 we get Z B (ξ) =

a (ξ)

X

e[J] [DΓ (Θ (·, ξ))] (η) b (η) f (η) dη aηiJ (Θ (η, ξ)) X

J∈B

Z +

a (ξ) (Rξ,η,i DΓ) (Θ (η, ξ)) b (η) f (η) dη

≡ C (ξ) + D (ξ) where Rξ,η,i are vector fields of weight > 0, smoothly depending on ξ, η, and the aηiJ have weight > |J| − 1. By Remark 11.17, D (ξ) is an operator of type λ acting e ∗ = −X e[J] + c[J] with c[J] smooth functions, on f . As to C (ξ), since X [J]

C (ξ) = − a (ξ)

XZ

e[J] [aη (Θ (·, ξ)) b (·)] (η) DΓ (Θ (η, ξ)) f (η) dη X iJ

J∈B

+ a (ξ)

XZ

(aηiJ DΓ) (Θ (η, ξ)) c[J] (η) b (η) f (η) dη

J∈B

− a (ξ)

XZ

e[J] f (η) dη. (aηiJ DΓ) (Θ (η, ξ)) b (η) X

J∈B

The first two terms in the last expression are still operators of type λ (or better) applied to f (for the first term, recall that aηiJ are smooth functions, hence the e[J] [aη (Θ (·, ξ)) b (·)] (η) is locally bounded for every |J|), while the third function X iJ term is a sum of operators of type λ + |J| − 1 (due to the weight of the function e[J] f, as required by the theorem. aηiJ ) applied to X Case 2: λ = 1. In this case we have to compute the derivative of the integral in distributional sense, as already done in the proof of Lemma 11.18 (although no singular integral will appear in the final result, due to the shift of the derivative on the function f ): with the same meaning of ϕε , let us compute

Sobolev and H¨ older estimates for general H¨ ormander operators

557

Z ei X Z =

a (ξ) ϕε (Θ (η, ξ)) DΓ (Θ (η, ξ)) b (η) f (η) dη


ei a (ξ) ϕε Dξ Γ (Θ (η, ξ)) b (η) f (η) dη X Z   ei Dξ Γ (Θ (η, ξ)) b (η) f (η) dη + a (ξ) ϕε (Θ (η, ξ)) X Z X η e[J] [(ϕε DΓ) (Θ (·, ξ))] (η) b (η) f (η) dη + a (ξ) aiJ (Θ (η, ξ)) X J∈B

Z +

a (ξ) (Rξ,η,i (ϕε DΓ)) (Θ (η, ξ)) b (η) f (η) dη

≡ Aε (ξ) + Bε (ξ) + Cε (ξ) + Dε (ξ) . Now, Z Aε (ξ) + Bε (ξ) →

ei a (ξ) DΓ (Θ (η, ξ)) b (η) f (η) dη X Z   ei Dξ Γ (Θ (η, ξ)) b (η) f (η) dη, + a (ξ) X

which is an operator of type λ applied to f . As to Cε (ξ), integrating by parts we have Z X η e[J] f (η) dη Cε (ξ) = − a (ξ) aiJ (Θ (η, ξ)) b (η) (ϕε DΓ) (Θ (η, ξ)) X J∈B

Z −

a (ξ)

X

e[J] [aη (Θ (·, ξ)) b] (η) [(ϕε DΓ) (Θ (η, ξ))] f (η) dη X iJ

J∈B

Z +

a (ξ)

X

c[J] (η) aηiJ (Θ (η, ξ)) (bf ) (η) [(ϕε DΓ) (Θ (η, ξ))] dη

J∈B

so that Z Cε (ξ) → −

a (ξ)

X

e[J] f (η) dη aηiJ (Θ (η, ξ)) b (η) DΓ (Θ (η, ξ)) X

J∈B

Z −

a (ξ)

X

e[J] [aη (Θ (·, ξ)) b] (η) DΓ (Θ (η, ξ)) f (η) dη X iJ

J∈B

Z +

a (ξ)

X

c[J] (η) aηiJ (Θ (η, ξ)) (bf ) (η) DΓ (Θ (η, ξ)) dη,

J∈B

where the first term is a sum of operators of type λ + |J| − 1 (due to the weight of e[J] f, while the other two terms are sums of operators the function aηiJ ) applied to X of type λ (or better) applied to f . Finally, Z Dε (ξ) = a (ξ) [(Rξ,η,i ϕε ) DΓ] (Θ (η, ξ)) b (η) f (η) dη Z + a (ξ) [ϕε (Rξ,η,i DΓ)] (Θ (η, ξ)) b (η) f (η) dη ≡ Dε1 (ξ) + Dε2 (ξ)

558

H¨ ormander operators

where Dε2 (ξ) →

Z a (ξ) [Rξ,η,i DΓ] (Θ (η, ξ)) b (η) f (η) dη,

which is an operator of type λ applied to f, while Dε1 (ξ) → 0, since Rξ,η,i ϕε is bounded uniformly in ε (reasoning like in estimate of the term Cε3 in the proof of Lemma 11.18) and supported in {kΘ (η, ξ)k < ε}. (ii) Let now T be an operator of type λ > 2 with kernel k 0 . As in the case (i), it is enough to prove that Z e0 a (ξ) DΓ (Θ (η, ξ)) b (η) f (η) dη, X where DΓ is homogeneous of degree λ − Q, can be rewritten in the form X e[J] f (ξ) + T 0 f (ξ) TJ0 X J∈B

TJ0 , T 0

operators of type λ + max(|J| − 2, 0) and λ − 1, respectively. Let us with consider only the case λ > 3, the case λ = 2 being handled with the modification seen in (i), Case 2. Z e0 a (ξ) Dξ Γ (Θ (η, ξ)) b (η) f (η) dη X Z e0 a (ξ) Dξ Γ (Θ (η, ξ)) b (η) f (η) dη + = X Z   e0 Dξ Γ (Θ (η, ξ)) b (η) f (η) dη + a (ξ) X Z X e[J] [DΓ (Θ (·, ξ))] (η) b (η) f (η) dη + a (ξ) a0J (Θ (η, ξ)) X J∈B

Z +

a (ξ) (Rξ,0 DΓ) (Θ (η, ξ)) b (η) f (η) dη

≡ A (ξ) + B (ξ) + C (ξ) + D (ξ) . where Rξ,0 is now a differential operator of weight > −1, and the a0J have weight > max {|J| − 2, 0} . Then A (ξ) , B (ξ) are still operators of type λ, applied to f ; D (ξ) is an operator of type λ − 1, applied to f . Moreover, XZ   e[J] a0 (Θ (·, ξ)) b (·) (η) DΓ (Θ (η, ξ)) f (η) dη C (ξ) = − a (ξ) X J J∈B

+ a (ξ)

XZ

a0J (Θ (η, ξ)) DΓ (Θ (η, ξ)) c[J] (η) b (η) f (η) dη

J∈B

− a (ξ)

XZ

e[J] f (η) dη a0J (Θ (η, ξ)) DΓ (Θ (η, ξ)) b (η) X

J∈B

where the first two terms are still operators of type λ (or better) applied to f , while e|J| f. So the third is the sum of operators of type λ + max {|J| − 2, 0} applied to X we are done.

Sobolev and H¨ older estimates for general H¨ ormander operators

559

We can now proceed to the: Proof of Theorem 11.19. Let us start proving (11.22). First, given an operator F of type λ > 1, we can apply Theorem 11.24 writing X ei F = e[J] + F i X TJi X J∈B

with F i operator of type λ, and TJi , operators of type λ + |J| − 1. Now, let us e[J] appearing in this identity. If |J| = 1, the term is consider one of the terms TJi X e[J] can be written as already in the form required by the Theorem. If |J| = 2, then X e1 , X e2 , . . . , X eq , and (possibly) X e0 . Then T i X e a combination of commutators of X J [J] i e e i e i contains terms TJ Xh Xj and (possibly) the term TJ X0 , where TJ is an operator of type λ + 1. Then:   i e eh X ej = T i X e e TJi X J h Xj = TJ 0 Xj , where by Theorem 11.15, TJi 0 is an operator of type λ; hence in this case all the terms are in the form allowed by the Theorem. e[J] as a sum of differential monomials Finally, if |J| > 2, it is enough to write X e eJ 0 : it is always XJ 0 and then look at the last term of the differential operator X 00 eJ 0 either as X eJ 00 X ej with |J | = |J| − 1 and j = 1, . . . , q, or as possible to rewrite X eJ 00 X e0 with |J 00 | = |J| − 2. In the first case, we have X   i eJ 0 = T i X e e e TJi X J J 00 Xj = TJ 000 Xj , with TJi 000 operator of type λ (recall that TJi is an operator of type λ + |J| − 1), which is already in the proper form; in the second case, we have   i e eJ 0 = T i X e e TJi X J J 00 X0 = PJ X0 with PJi operator of type λ + 1 (since TJi is of type λ + |J| − 1 and |J 00 | = |J| − 2). Hence also in this case every term is in the proper form. This proves (11.22). This also implies (11.23) since we can rewrite ! q q q X X X ei F = ek + Fi0 + Pi X e0 = ek + Fi0 + Pi L e− ek2 X Fik X Fik X X =

k=1 q h X

k=1



ek Fik − Pi X

i

k=1

ek + Fi0 + Pi L e X

k=1

and since , Fi0 are of type λ while Pi is of type λ + 1, by Lemma 11.18 h  Fiki e Fik − Pi Xk is another operator of type λ, and (11.23) is proved. Analogously one can prove (11.24) and (11.25), exploiting point (ii) of Theorem 11.24, so we are done.

560

11.3

H¨ ormander operators

Parametrix and representation formulas

Throughout this section we will make extensive use of computations on operators of type λ. To make more readable our formulas, we will keep using the symbols S, F, P, which recall us the words “Singular integral”, “Fractional integral”, and “Parametrix” (with possibly some indices) to denote operators of type 0, 1, 2, respectively. In order to prove representation formulas for second order derivatives, we start with the following parametrix identities.
e Given a, b ∈ C ∞ (B e ξ, R ) such that ab = a, Theorem 11.25 (Parametrix of L) 0
e ξ, R , there exist F1 , F2 , operators of type 1 and P1 , P2 , operators of type 2 in B given by Z b(η) f (η) dη P1 f (ξ) = a(ξ) Γ∗ (Θ(η, ξ)) c (η) e ZB b (ξ) P2 f (ξ) = a (η) Γ(Θ(η, ξ)) f (η) dη c(ξ) Be (where c is the function appearing in Theorem 10.6 (c)) such that: e ∗ P1 + F1 and aI = P2 L e + F2 aI = L

(11.39)

where I is the identity. Moreover, F1 is modeled on Γ∗ and F2 is modeled on Γ.
e ∗ P1 f, for f ∈ C ∞ (B e ξ, R ). As in the proof of Lemma Proof. Let us compute L 0
e ξ, R ) let us evaluate 11.18 we use a distributional argument: for ω ∈ C0∞ (B Z

Z e (ξ) P1 f (ξ) dξ = lim Lω

ε→0

e B

e (ξ) P ε f (ξ) dξ Lω 1

e B

where P1ε f (ξ) = a(ξ)

Z

ϕε (Θ(η, ξ)) Γ∗ (Θ(η, ξ))

e B

b(η) f (η) dη c (η)

with ϕε as in the proof of Lemma 11.18. In the next integration by parts, we use the following identity: e ∗ (f g) = (L e ∗ f )g + f (L e ∗ g) L +2

q X h=1

eh f X eh g + df g + g X

q X h=1

eh f + f eh X

q X h=1

eh g lh X

(11.40)

Sobolev and H¨ older estimates for general H¨ ormander operators

561

for suitable smooth functions d, eh , lh (which follows by direct computation from e ∗ = −X ei + ci ). Then X i Z  Z Z bf (η) ∗ ε e e a(ξ)ϕε (Θ(η, ξ)) Γ (Θ(η, ξ))Lω (ξ) dξ dη Lω (ξ) P1 f (ξ) dξ = e e e c (η) B B B Z  Z bf (η) e ∗ ((ϕε Γ∗ ) (Θ(η, ·))a) (ξ) ω (ξ) dξ dη L = e e c (η) B ZB  Z bf (η) e ∗ [(ϕε Γ∗ ) (Θ(η, ·))] (ξ) ω (ξ) dξ dη = a(ξ)L e c (η) e B B ! Z X Z q   bf (η) ∗ eh a(ξ) + a(ξ)eh (ξ) X eh [(ϕε Γ ) (Θ(η, ·))] (ξ) ω (ξ) dξ dη 2X + e e c (η) B B h=1 ! ! Z Z q X bf (η) ∗ ∗ e e L a(ξ) + d (ξ) a (ξ) + Xh a(ξ)lh (ξ) (ϕε Γ ) (Θ(η, ξ))ω (ξ) dξ dη + e e c (η) B B h=1 ≡ Aε + Bε + Cε .

Now, as ε → 0, Z

Z

Cε →

e ∗ a(ξ) + d (ξ) a (ξ) + L

f (η) e B

e B

h=1

Z

Z

=

q X

∗

f (η)P ω (η) dη = e B

! ! b (η) ∗ e ω (ξ) dξ dη Xh a(ξ)lh (ξ) Γ (Θ(η, ξ)) c (η)

ω (ξ) P f (ξ)dξ e B

where e ∗ a(ξ) + d (ξ) a (ξ) + L

P ω (η) = e B

P ∗ f (ξ) =

q X

! b (η) e ω (ξ) dξ Xh a(ξ)lh (ξ) Γ∗ (Θ(η, ξ)) c (η) h=1 ! q X ∗ e a(ξ) + d (ξ) a (ξ) + eh a(ξ)lh (ξ) Γ∗ (Θ(η, ξ)) b (η) f (η)dη L X c (η)

Z

Z e B

h=1

∗

are operators of type 2, with P modeled on Γ∗ . As to Aε we have Z Aε = e B

bf (η) c (η)

Z

Z +

a(ξ) e B

e B q X

a(ξ) [L∗ (ϕε Γ∗ (·))] (Θ(η, ξ)) ω (ξ) dξ !

[Yi Rη,i + Rη,i Yi + Rη,i Rη,i ] + Rη,0

!) ∗

(ϕε Γ (·)) (Θ(η, ξ)) ω (ξ) dξ

i=1

with the change of variables u = Θ(η, ξ) in the ξ-integral ! Z Z
∗ ∗ −1 = bf (η) L (ϕε Γ ) (u) (aω) Θ(η, ·) (u) (1 + O (kuk)) du dη kuk 2d(ξ e 0 , ξ), ξ0 , ξ, η with d(ξ |K(ξ0 , η) − K(ξ, η)| + |K(η, ξ0 ) − K(η, ξ)| 6 C

e 0 , ξ) d(ξ e 0 , η)Q+1−` d(ξ

(11.49)

where the constant C has the form n o ∇u W ξ,η (u) + ∇ξ W ξ,η (u) + ∇η W ξ,η (u) c· sup e (ξ,R) kuk=1,ξ,η∈B with c independent of K. The same conclusions holds for the kernel K 0 (ξ, η) = W ξ,η (Θ (η, ξ)) φ (kΘ (η, ξ)k) if φ is as in (11.47). In particular, the functions K, K 0 are fractional integral kernels of exponents β = 1, ν = ` in the sense of Definition 7.10. Proof. Point (i) is easy, by the homogeneity of W and the equivalence between de and ρ (see Proposition 10.38), since K(ξ, η) = kΘ (η, ξ)k

`−Q

W ξ,η (u0 )

for some u0 with ku0 k = 1, hence 0 0 `−Q sup sup W ξ ,η (u) · kΘ (η, ξ)k e (ξ,R) kuk=1 ξ 0 ,η 0 ∈B 0 0 e η)`−Q . 6c· sup sup W ξ ,η (u) · d(ξ, 0 0 e (ξ,R) kuk=1 ξ ,η ∈B

|K(ξ, η)| 6

e ξ0 ). The condition d(η, e ξ0 ) > 2d(ξ, e ξ0 ) To prove (ii), fix ξ0 , η, and let r = 21 d(η, er (ξ0 ). Applying Theorem 1.56 (“Lagrange means that ξ is a point ranging in B theorem”) to the function f (ξ) = K (ξ, η) we can write ! q X e e e e |f (ξ) − f (ξ0 )| 6 cd (ξ, ξ0 ) sup Xi f (ζ) + d (ξ, ξ0 ) sup X0 f (ζ) . e 0 ,r) i=1 ζ∈B(ξ

e 0 ,r) ζ∈B(ξ

Noting that e e ·,η η (Θ (η, ζ))) (ζ) Xi K (·, η) (ζ) 6 |(Yi W + Ri W ) (Θ (η, ζ))| + (X iW and recalling that, by Remark 11.9, ∇ζ W ζ,η (u) has the same homogeneity in u as e (ξ0 , r) , W ζ,η (u), we get, for ζ ∈ B 0 0 c e + sup Xi K (·, η) (ζ) 6 ∇u W ξ ,η (u) Q−`+1 0 0 e ρ (ζ, η) kuk=1,ζ ,η ∈B (ξ,R) 0 0 c + sup ∇ζ W ζ ,η (u) Q−` e (ξ,R) ρ (ζ, η) kuk=1,ζ 0 ,η 0 ∈B o n 0 0 0 0 c 6 sup . ∇u W ζ ,η (u) + ∇ζ 0 W ζ ,η (u) Q−`+1 e 0 0 e (ξ,R) d (ξ0 , η) kuk=1,ζ ,η ∈B

Sobolev and H¨ older estimates for general H¨ ormander operators

571

Similarly e X0 K (·, η) (ζ) 6

sup e kuk=1,ζ 0 ,η 0 ∈B

(ξ,R)

o n 0 0 0 0 ∇u W ζ ,η (u) + ∇ζ 0 W ζ ,η (u)

c , Q−`+2 e d (ξ0 , η)

hence |K(ξ0 , η) − K(ξ, η)| 6 C de(ξ, ξ0 )

1 de(ξ0 , η)

Q−`+1

+

de(ξ, ξ0 ) de(ξ0 , η)

Q−`+2

! 6C

de(ξ, ξ0 ) Q−`+1 de(ξ0 , η)

with C=c

sup e (ξ,R) kuk=1,ζ 0 ,η 0 ∈B

o n 0 0 0 0 ∇u W ζ ,η (u) + ∇ζ 0 W ζ ,η (u) .

To get the analogous bound for |K (η, ξ) − K (η, ξ0 )| , it is enough to apply the pree η) = W f ξ,η (Θ (η, ξ)) with W f ξ,η (u) = W η,ξ (u−1 ). vious estimate to the function K(ξ, 0 This completes the proof of (ii) for K. As to the kernel K , multiplication for the cutoff function φ just adds in the above computations an innocuous term ei (φ (kΘ (η, ·)k)) (ξ) = (Yi Φ + Rη,i Φ) (Θ (η, ξ)) X with Φ (u) = φ (kuk). kΘ (η, ξ)k < R0 .

This function is bounded and nonzero only if

R0 2


2 appearing in the expansion (11.46). By (11.48), we can write ai (ξ)bi (η)Di Γ (Θ (η, ξ)) = ai (ξ)bi (η)K (η, ξ) + ai (ξ)bi (η)Kreg (η, ξ) where:

• the kernel ai (ξ)bi (η)Kreg (η, ξ) is Lip0 B ξ, R , hence the correspond
e ξ, R ) into itself continuously and ing integral operator maps Lp (B
e ξ, R ) into itself continuously; C α (B • the kernel K (η, ξ), by Proposition 11.32 and the initial remarks in this section, satisfies the assumptions of Theorem 7.28 with β = 1, ν = `, α0 = Q, hence by Proposition 7.11 the same is true for the kernel ai (ξ)bi (η)K (η, ξ), and by Theorem 7.28 the operator I` with this kernel satisfies kI` f kLq (Be(ξ,δ0 R)) 6 c kf kLp (Be(ξ,δ1 R))  
e ξ, δ0 R , the last in. Then, for ai , bi supported in B with q = p `+Q Q equality can be rewritten as kI` f kLq (Be(ξ,R)) 6 c kf kLp (Be(ξ,R)) .

572

H¨ ormander operators

Moreover, since p < q we can write kI` f kLp (Be(ξ,R)) 6 cRQ( p − q ) kI` f kLq (Be(ξ,R)) 1

1

= cR`/Q kI` f kLq (Be(ξ,R)) 6 cR kf kLp (Be(ξ,R))

(11.50)

Also, by Theorem 7.14 and Remark 7.16 we have kI` f kC δ (Be(ξ,R)) 6 c kf kC α (Be(ξ,R))

(11.51)

for any α, δ ∈ (0, 1) , in particular for δ = α ∈ (0, 1). We have so far proved Theorem 11.29 for λ > 1, that is: the operator with kernel kF (ξ, η) in (11.46) satisfies the assertions of our theorem. We are left to prove the continuity assertions of the theorem for the singular integral with kernel kS (ξ, η) = a1 (ξ)b1 (η)D1 Γ(Θ(η, ξ)) appearing in (11.46). Recall that D1 Γ (u) is a −Q homogeneous function; it is also a 2-homogeneous derivative of a 2 − Q function, which is locally integrable and therefore defines a distribution; hence by Corollary 6.31 the function D1 Γ (u) has vanishing mean, that is Z D1 Γ (u) du = 0 for every 0 < r1 < r2 , (11.52) r1 2

ei [1 − ϕε ] · X ei k (ξ, ·) (η) u(η)dη, X

e ∗ [1 − ϕε ] (η) u(η)dη, k (ξ, ·) L

Z d (η) k(ξ, η) [1 − ϕε (η)] u(η)dη,

A4 (ξ) = A5 (ξ) = A6 (ξ) =

ε e B(ξ,r 1 ),ρ(ξ,η)> 2 Z q X ε e h=1 B(ξ,r1 ),ρ(ξ,η)> 2 q Z X

h=1

ε e B(ξ,r 1 ),ρ(ξ,η)> 2

eh (k(ξ, ·)) (η) u(η)dη, [1 − ϕε (η)]eh (η) X eh [1 − ϕε (·)] u(η)dη. k(ξ, η)lh (η) X

Recall that, for i = 1, . . . , q, e c |k(ξ, η)| 6 ρ(ξ,η)c Q−1 ; Xi k (ξ, ·) (η) 6 ρ(ξ,η) Q; e∗ c L k(ξ, ·) (η) 6 ρ(ξ,η) Q+1 ; e e∗ Xi (1 − ϕε ) (η) 6 εc ; L (1 − ϕε ) (η) 6 εc2 ; and the derivatives of (1 − ϕε ) are supported in the annulus 2ε 6 ρ(ξ, η) 6 ε. These bounds follow by (11.11) and Proposition 11.16. Let C > 0 be such that 1 ρ(ξ, η) 6 de(ξ, η) 6 Cρ(ξ, η) (11.66) C (see Remark 11.35 for this equivalence), and let k0 be the nonnegative integer such that 2k0 −1 ε < 2Cr 1 6 2k0 ε e (this is possible for ε < 4Cr1 ). Then, since B (ξ, r) is comparable to rQ , Z |u (η)| |A1 (ξ)| 6 c dη Q+1 ε k0 ε ρ(ξ, η) 2 0 such that in V = B (x0 , r0 ) we can lift the vector fields e0 , X e1 , . . . , X eq , defined in a neighborhood U of (x0 , 0) in RN for X0 , X1 , . . . , Xq to X some N = n + m, and free up to step s in U and, for every r ∈ (0, r0 ), for every y∈Ω e n o B ((x , 0) , r) 0 m e , s ∈ R : (y, s) ∈ B ((x0 , 0) , r) 6 c1 |B (x0 , r)| and for every y ∈ B (x0 , δ0 r), e n o B ((x , 0) , r) 0 m e . s ∈ R : (y, s) ∈ B ((x0 , 0) , r) > c2 |B (x0 , r)|

(11.68)

Here |A| denotes the Lebesgue measure of the set A in Rk for the suitable dimension k (which is m, N, n for the three sets involved in the inequalities). We then have the following: Theorem 11.41 (Relation between lifted and unlifted Sobolev spaces) With the same notation of the previous theorem, for any function u (x) of n variables defined in B (x0 , r0 ), let us define the corresponding function u e of N variables u e (ξ) = u e (x, h) = u (x) . e (ξ0 , r0 ) be the lifted ball of B (x0 , r0 ) , with ξ0 = (x0 , 0) ∈ RN . Then for every Let B integer k = 1, 2, . . . and every r 6 r0 there exist constants c1 , c2 (independent of u) such that c2 kukLp (B(x0 ,δ0 r)) 6 ke ukLp (B(ξ e 0 ,r)) 6 c1 kukLp (B(x0 ,r)) c2 kukW k,p (B(x0 ,δ0 r)) 6 ke ukW k,p (B(ξ e 0 ,r)) 6 c1 kukW k,p (B(x0 ,r)) X

f X

where δ0 < 1 is the number appearing in Theorem 11.40.

X

Sobolev and H¨ older estimates for general H¨ ormander operators

585

Proof. The inequalities in the first line follow by Theorem 11.40 and recalling that e (ξ0 , r) on Rn (see Proposition 10.39). Actually, B (x0 , r) is the projection of B Z p p = |u (x)| dxdh ke ukLp B(ξ ( e 0 ,r)) e B(ξ0 ,r) ! Z Z p = |u (x)| dh dx e B(x0 ,r) {s∈Rm :(x,s)∈B((x 0 ,0),r)} Z n o e ((x0 , 0) , r) |u (x)|p dx = s ∈ Rm : (x, s) ∈ B B(x0 ,r) e B ((x0 , 0) , r) Z p 6 c1 |u (x)| dx. |B (x0 , r)| B(x0 ,r) Also, Z

p

ke ukLp

e 0 ,r)) (B(ξ

n o e ((x0 , 0) , r) |u (x)|p dx s ∈ Rm : (x, s) ∈ B B(x0 ,δ0 r) e B ((x0 , 0) , r) Z p |u (x)| dx. > c2 |B (x0 , r)| B(x0 ,δ0 r)

>

The second statement in the theorem then follows by the first one since, recalling that ei = Xi + X

N −n X

cij (x, h) ∂hj

j=1

ei u ei u (x) = (Xi u) (x) = X g we have X e (ξ) = X i u (ξ) and



g

e 6 c1 kXi ukLp (B(x0 ,r)) e p = X

Xi u i u e 0 ,r)) e 0 ,r)) Lp (B(ξ L (B(ξ

e e p > c2 kXi ukLp (B(x0 ,δ0 r))

Xi u e 0 ,r)) L (B(ξ where these inequalities can be obviously extended to higher order derivatives. 11.5.2

A priori estimates in Sobolev spaces

We can now come to the
2,p e Theorem 11.42 (Local WX e estimates for the lifted operator) Let B ξ, R be as before. There exists
r0 < R and for every r 6 r0 there exists c > 0 such that eρ ξ, r ) we have for any u ∈ W 2,p (B e X



e kukW 2,p (Beρ (ξ,r/2)) 6 c Lu

eρ (ξ,r)) Lp (B

f X

The constants c, r0 depend on

n oq ei X

i=0

 + kukLp (Beρ (ξ,r)) .

, p, R; c also depends on r.

586

H¨ ormander operators

Proof.
The proof is
similar to that of Theorem 11.39. Let r0 be
such that eρ ξ, r0 ⊂ B eρ ξ, r ). Let ϕ e ξ, R/2 and, for some r 6 r0 , let u ∈ W 2,p (B B e X be a cutoff function as in Lemma 11.36,

eρ ξ, σ 0 r . eρ ξ, σr ≺ ϕ ≺ B B
2,p e Then, by Corollary 2.10, uϕ ∈ WX,0 e (Bρ ξ, r ) and using a density argument we can apply Theorem 11.34 to ϕu, writing  

e + kϕukW 1,p (Beρ (ξ,r)) . kϕukW 2,p (Beρ (ξ,r)) 6 c L (ϕu) f X

eρ (ξ,r)) Lp (B

f X

For 1 6 i, j 6 q, from the above inequality we get

e e

Xi Xj u p e L (Bρ ) 

1 1

e

e 6 c Lu kukLp (Be0 )

p e0 +

Du p e0 + ρ (1 − σ)r (1 − σ)2 r2 L (Bρ ) L (Bρ ) 

1

e kukLp (Be0 ) + kukLp (Be0 ) + + Du e0 ) ρ ρ (1 − σ)r Lp (B ρ  

1 1

e

e 6 c Lu kukLp (Be0 )

p e0 +

Du p e0 + ρ (1 − σ)r (1 − σ)2 r2 L (Bρ ) L (Bρ )
eρ = B eρ ξ, σr and where we assumed r < 1 and for ease of notation we set B
eρ ξ, σ 0 r . Multiplying both sides for (1 − σ)2 r2 we get eρ0 = B B

e e X u (1 − σ)2 r2 X i j eρ ) Lp (B   (11.69)





e

e 6 c (1 − σ)2 r2 Lu

p e0 + (1 − σ)r Du

p e0 + kukLp (Beρ0 ) L (Bρ ) L (Bρ ) Next, we compute

q



X 2 2 e 2 2 2 e e i u = (1 − σ) r Lu − X (1 − σ) r X0 u p

eρ )

p L (B eρ ) i=1 L (B (11.70)   q

X



e ei X ej u 6 (1 − σ)2 r2 Lu + .

X eρ ) eρ ) Lp (B Lp (B i,j=1 Combining (11.69) and (11.70), we have

e2 u p (11.71) (1 − σ)2 r2 D eρ ) L (B  





e

e 6 c (1 − σ)2 r2 Lu

p e0 + (1 − σ)r Du

p e0 + kukLp (Beρ0 ) . L (Bρ ) L (Bρ )



e Adding (1 − σ)r Du to both sides of (11.71) gives eρ ) Lp (B



e2

e (1 − σ)2 r2 D u + (1 − σ)r Du

p e eρ ) Lp (B L (Bρ )   (11.72)





e

e 0 0 6 c (1 − σ 0 )2 r2 Lu + (1 − σ )r Du + kuk

p e0

p e0 e ) Lp (B ρ L (Bρ ) L (Bρ )

Sobolev and H¨ older estimates for general H¨ ormander operators

587

and by Theorem 11.39, 



e 6 c (1 − σ 0 )2 r2 Lu

 c 0 + δΦ + Φ + kuk 2 0 e ) . Lp (B e0 ) ρ δ Lp (B ρ
1 Taking at the left hand side the supremum for σ ∈ 2 , 1 and choosing δ small enough, we obtain   2 e Φ2 + Φ1 6 c r Lu + kukLp (Beρ (ξ,r)) , eρ (ξ,r )) Lp (B then



e

e2 + r Du r2 D u p

p e e L (Bρ (ξ,r/2)) L (Bρ (ξ,r/2))   2 e 6 c r Lu p + kukLp (Beρ (ξ,r)) , eρ (ξ,r )) L (B and therefore  

e + kuk kukW 2,p (Beρ (ξ,r/2)) 6 c (r) Lu

p e eρ (ξ,r )) , Lp (B f L (Bρ (ξ,r )) X which is the desired result. We are now left to the last step, namely coming back to the original (unlifted) domain. The idea is simply: apply the estimates in the lifted space to a function u (ξ) which actually only depends on the original variables x: 2,p Theorem 11.43 (Interior WX -estimates for H¨ ormander operators) For 0 any domain Ω b Ω and p ∈ (1, ∞) there exists a constant c such that for any 2,p (Ω), u ∈ WX o n kukW 2,p (Ω0 ) 6 c kLukLp (Ω) + kukLp (Ω) . X

Proof. Fix x ∈ Ω0 b Ω and let R be such that B (x, R) ⊂ Ω and all the previous construction can be performed. Let r0 < R be as in Theorem 11.42, and let u ∈ 2,p WX (B (x, r0 )).

For r < r0 small enough by Theorem 11.41 the function u e belongs 2,p to WX B ξ, r , so we can apply Theorem 11.42 to u e , writing ρ e  

e u p ke ukW 2,p (Beρ (ξ,r/2)) 6 c Le + ke ukLp (Beρ (ξ,r)) . eρ (ξ,r )) f L (B X e and since Le e u = Lu, we Then, by Theorem 11.41, the equivalence between ρ and d, get, for suitable numbers δ1 < δ2 < 1 < k, kukW 2,p (B(x,δ1 r)) 6 c ke ukW 2,p (Beρ (ξ0 ,r/2)) ukW 2,p (B(ξ e 0 ,δ2 r)) 6 c ke X f f X X   o n

e 6 c Le u p + ke ukLp (Be(ξ,kr)) 6 c kLukLp (B(x,kr)) + kukLp (B(x,kr)) . e (ξ,kr )) L (B k

Let us now cover Ω0 with a finite number of balls {BX (xi , δ1 ri )}i=1 ; then collecting the above inequalities we get n o kukW 2,p (Ω0 ) 6 c kLukLp (Ω) + kukLp (Ω) . X

588

H¨ ormander operators

Remark 11.44 (Lifting is a local procedure) We want to stress the following point, related to the strategy of the proof of a priori estimates which we have just completed: lifting is a local procedure. This means that if we have a system of smooth vector fields satisfying H¨ ormander’s condition in some domain Ω ⊂ Rn , then every point of Ω has a neighborhood where we can perform the lifting and approximation procedure. The dimension of the lifted space RN , and therefore the structure of the homogeneous group G and its homogeneous dimension Q, only depends on the number q and the step s that we have fixed once and for all (see Remark 1.26). This does not mean that we assume that at any point a fixed number of commutators is needed to fulfill H¨ ormander’s condition, but only that a finite upper bound for the step exists in the whole Ω.The local nature of the procedure also means that, once we have proved some kind of estimate in the space of lifted variables, we have to come back to the system of vector fields projecting these estimates on a small neighborhood of some point of Ω and then glue together our local estimates in Ω. This gluing, instead, cannot be done at the level of lifted variables. We now turn to the particular case of H¨ ormander operators without drift. For these operators, we are going to extend the estimate of Theorem 11.43 to higher order derivatives. We will prove the following: Theorem 11.45 (Interior Sobolev regularity for sum of squares) Let L=

q X

Xi2

i=1

be a H¨ ormander operator without drift in Ω. For any domain Ω0 b Ω, nonnegative 2,p (Ω), integer k and p ∈ (1, ∞) there exists a constant c such that for any u ∈ WX k,p k+2,p if Lu ∈ WX (Ω) then u ∈ WX (Ω0 ) and the following holds: n o kukW k+2,p (Ω0 ) 6 c kLukW k,p (Ω) + kukLp (Ω) . (11.73) X

X

As a first step we consider the lifted operator.
e ξ, R be as in the previous section, p ∈ (1, ∞) and k a Proposition 11.46 Let B n oq ei positive integer. There exists a constant c depending on X , p, k, R such that i=0
e ξ, R/2 ), for every u ∈ C0∞ (B  

e kukW k+2,p (Be(ξ,R/2)) 6 c Lu p + kukW k+1,p (Be(ξ,R/2)) . e (ξ,R/2)) f f L (B X X

e ξ, R/2 . Then by Theorem e ξ, R ) such that a = 1 in B Proof. Fix a ∈ C0∞ (B
e ξ, R/2 ) and any multiindex I with |I| = k + 2 we can 11.28, for any u ∈ C0∞ (B
e ξ, R , write, in B X X 0 eI u = X eI (au) = eJ Lu e + SI Lu e + eK u + S 0 u X SIJ X SIK X I |J|6k

|K|6k+1

Sobolev and H¨ older estimates for general H¨ ormander operators

589


0 e ξ, R . Now, let us where all the S... , S... are operators of type 0 over the ball B
e ξ, R ) norms of both sides of the previous identity. Applying Theorem take Lp (B 11.29 and keeping in mind the support of u we get ) (



e

e 6 c Lu k,p + kukW k+1,p (Be(ξ,R/2))

XI u p e e (ξ,R/2)) f L (B (ξ,R/2)) W f (B X X which implies the assertion. We can now prove the following:
k,p e e Proposition 11.47 (WX e estimates for L) Let B ξ, R be as before, p ∈ (1, ∞), and k be a positive integer. For any r small enough, depending on R and k, there
k+2,p e k+1 exists c > 0 such that for any u ∈ WX ( B ξ, 2 r ) we have ρ e ( )



e + kukLp (Beρ (ξ,2k+1 r)) . kukW k+2,p (Beρ (ξ,r/2)) 6 c Lu k,p eρ (ξ,2k+1 r )) f W f (B X X

e ξ, R ⊂ B eρ ξ, R0 . For r 6 2−k−2 R0 , let u ∈ Proof. Let R0 be such that B
k+2,p e WX (Bρ ξ, 2k+1 r ) and let ϕ be a cutoff function as in Corollary 11.37, satisfying e

eρ ξ, r/2 ≺ ϕ ≺ B eρ ξ, r . B
k+2,p e Then by Corollary 2.10 uϕ ∈ WX,0 (Bρ ξ, r ), hence using a density argument we e can apply Proposition 11.46 to ϕu, writing ( )

e

kϕukW k+2,p (Beρ (ξ,r)) 6 c L (ϕu) k,p + kϕukW k+1,p (Beρ (ξ,r)) . eρ (ξ,r )) f f W f (B X X X From the above inequality and Corollary 11.37 iii) we get kukW k+2,p (Beρ (ξ,r/2)) f ( X q

X

e

e e 6 c ϕLu +

k,p e

Xi ϕXi u k,p e W f (Bρ (ξ,r )) W f (Bρ (ξ,r )) X X i=1 )



e + uLϕ

k,p e Wf (Bρ (ξ,r))

+ kϕukW k+1,p (Beρ (ξ,r)) f X

X

c c c

e + k+1 kukW k+1,p (Beρ (ξ,r)) + k+2 kukW k,p (Beρ (ξ,r)) 6 k Lu

k,p e f f r r r W f (Bρ (ξ,r )) X X X ) (

1 1

e Lu + k+2 kukW k+1,p (Beρ (ξ,r)) . 6c

k,p e k f r r W f (Bρ (ξ,r )) X X By iteration these inequalities give (



e kukW k+2,p (Beρ (ξ,r/2)) 6 c (r, k) Lu

f X

)

k,p e Wf (Bρ (ξ,2k r)) X

+ kukW 2,p (Beρ (ξ,2k r)) f X

590

H¨ ormander operators

which combined with the basic estimate of Theorem 11.42 give ) (



e kukW k+2,p (Beρ (ξ,r/2)) 6 c (r, k) Lu k,p + kukLp (Beρ (ξ,2k+1 r)) . f W X (Beρ (ξ,2k+1 r)) f X

We can now come to the Proof of Theorem 11.45. By Proposition 11.47 and Theorem 11.41, reasoning exactly like in the proof of Theorem 11.43 we can prove the estimate (11.73) under k+2,p the stronger assumption that u ∈ WX (Ω). It remains to prove the regularization 2,p k,p result, that is the fact that if u ∈ WX (Ω) and Lu ∈ WX (Ω), then actually k+2,p u ∈ WX,loc (Ω) and (11.73) holds. We will prove the assertion by induction on k > 0. For k = 0 there is nothing to prove, so let us assume the assertion up to k,p 2,p (Ω). Then, (Ω) and Lu ∈ WX k − 1, and let us prove it for k. So assume u ∈ WX k−1,p k+1,p in particular, Lu ∈ WX (Ω) and by the inductive assumption u ∈ WX (Ω). k+2,p We have to prove that u ∈ WX,loc (Ω). Fixing a point x inside Ω, we can perform the lifting procedure in a small neighborhood of ξ = (x, 0). In view of Theorem
k+2,p e 11.41, if we prove that u e ∈ WX ξ, r ) for r small enough, this will imply that ( B e k+2,p u ∈ WX (BX (x, r0 )) for some possibly smaller r0 , and by the genericity of x this will imply the assertion. So the situation is the following: we know that

k+1,p e e u ∈ W k,p (B e ξ, r ) u e ∈ WX (B ξ, r ) and Le e e X

for some small r, and we want to prove that
k+2,p e u e ∈ WX (B ξ, r0 ) e

e ξ, r ), φ = 1 in B e ξ, r/2 , so for some possibly smaller r0 . Fix a function φ ∈ C0∞ (B
k+1,p e that φe u ∈ WX,0 (B ξ, r ) by Corollary 2.10. For any multiindex I with |I| = k +1 e let us write, for the function φe u, the representation formula of Theorem 11.28 
 e ξ, r (this holds for functions in C0∞ B and then, by density, for functions in

e ξ, r )): given a ∈ C ∞ (B e ξ, r ) (we can take a = φ) for any multiindex W k+1,p (B 0

e X,0

I with |I| = k + 1 we have X X
0 e I φ2 u eJ L e (φe e (φe eK (φe X e = SIJ X u) + SI L u) + SIK X u) + SI0 (φe u) ≡ GI |J|6k−1

|K|6k

(11.74) 0 where all the S... and S... are operators of type 0 and each of them has the ej F for some j = 1, . . . , q and some operator F of type 1. Since form S = X

k+1,p e e u ∈ W k,p (B e ξ, r ), we have u e ∈ WXe (B ξ, r ) and Le e X e (φe eu + u e +2 L u) = φLe eLφ

q X i=1


ei u ei φ ∈ W k,p (B e ξ, r ). X eX e X

Sobolev and H¨ older estimates for general H¨ ormander operators

591

ej F... for Let us rewrite (11.74) expressing every operator S... of type 0 in the form X some j and some operator F... of type 1 (the possibility of doing so is contained in the statement of Theorem 11.28): X ej FIJ X eJ L e (φe e j FI L e (φe GI = X u) + X u) IJ I |J|6k−1

+

X

0 ej 0 FIK eK (φe ej 0 FI0 (φe X X u) + X u) . IK I

|K|6k

Next, we apply Theorem 11.19 which, for operators without drift (see Remark 11.20), assures that for any operator F of type 1 and any i = 1, 2 . . . q there exist operators Fik (k = 1, . . . , q) of type 1 such that q X ei F = ek + Fi0 . X Fik X k=1

Therefore we can write (omitting and simplifying some indices) X eJ L e (φe e (φe e (φe e (φe GI = FX u) + F L u) + F Xj L u) + F L u) |J|6k

X

+

eK (φe ej (φe FX u) + F (φe u) + F X u)

|K|6k+1

ei derivative of GI and, recalling that X ei F becomes an We can then take one X operator S of type 0, write X ei GI = eJ L e (φe e (φe e (φe e (φe X SX u) + S L u) + SXj L u) + S L u) |J|6k

X

+

eK (φe ej (φe SX u) + S (φe u) + S X u) .

|K|6k+1 k+1,p e WX (B e



k,p e e (φe Since u e ∈ ξ, r ), L u) ∈ WX e (B ξ, r ) and S is continuous on

ei GI ∈ Lp (B e ξ, r ). By (11.74), this implies that e ξ, r ), we read that X Lp (B
1,p e eI u for any multiindex I with |I| = k + 1, X e ∈ WX (B ξ, r/2 ) that is u e ∈ e 
 k+2,p e W B ξ, r/2 . e X

11.5.3

Some properties of H¨ older spaces

To prove a priori estimates and regularization results in the scale of H¨older spaces, as we have done for Sobolev spaces, we first need to establish some properties of these function spaces. The definition and basic properties of H¨older spaces induced by a system of H¨ ormander vector fields have been introduced in Chapter 2, section 2.2. As already seen for Sobolev spaces, here we need some more properties of these spaces, related to interpolation inequalities, cutoff functions, and the relation between H¨ older spaces in the space of lifted variables and in the original one. We start recalling the statement of Proposition 2.18, which contains a number of useful inequalities which will be used over and over in the following.

592

H¨ ormander operators

Proposition 11.48 (Basic inequalities for H¨ older norms) (i) For any f ∈ C01 (B (x, R))  X q 1−α |f |C α (B(x,R)) 6 cR · sup |Xi f | + R sup |X0 f | .

(11.75)

B(x,R)

i=1 B(x,R)

α (ii) For any couple of functions f, g ∈ CX (B (x, R)), one has

|f g|C α (B(x,R)) X

6 |f |C α (B(x,R)) kgkL∞ (B(x,R)) + |g|C α (B(x,R)) kf kL∞ (B(x,R)) X

(11.76)

X

and kf gkC α (B(x,R)) 6 kf kC α (B(x,R)) kgkC α (B(x,R)) . X

X

X

(11.77)

(iii) Moreover, if both f and g vanish at least at a point of B (x, R), then |f g|C α (B(x,R)) 6 2Rα |f |C α (B(x,R)) |g|C α (B(x,R)) . X

X

X

(11.78)

The following relations hold between H¨ older seminorms on different domains: |f |C α (Ω1 ) 6 |f |C α (Ω2 ) if Ω1 ⊂ Ω2 ; X

X

α |f |C α (B(x,r)) = |f |C α (Ω2 ) if B (x, r) ⊂ Ω2 and f ∈ CX,0 (B (x, r)) . X

X

Moreover, if B (xi , r) (i = 1, 2, · · · , k) is a finite family of balls of the same radius r, such that ∪ki=1 B (xi , r) is connected and ∪ki=1 B (xi , 2r) ⊂ Ω, then α for any f ∈ CX (Ω), kf kC α (∪k X

i=1 B(xi ,r))

6c

k X

kf kC α (B(xi ,2r)) X

(11.79)

i=1

with c depending on the family of balls, but not on f . 2,α (B (x, R)) and 0 < r 6 r0 , (iv) There exists r0 > 0 such that for any f ∈ CX,0 we have the following interpolation inequality: kX0 f kL∞ (B(x,R)) 6 rα/2 |X0 f |C α (B(x,R)) + X

2 kf kL∞ (B(x,R)) . r

(11.80)

As we have seen in Chapter 2, section 2.2, in H¨older spaces we do not have density results of smooth functions as in Sobolev spaces. This forces to modify something in the general strategy of the proof of a priori estimates, interpolation inequalities, and so on. First of all, we need the following Proposition 11.49 The representation formula for second derivatives contained
in 2,α e Theorem 11.26 and the a priori estimate (11.61) still hold for u ∈ C0, e (B ξ, R ). X e Under the assumption that our H¨ ormander operator L does not contain the drift e0 , the representation formula for higher order derivatives contained in Theterm X
e ξ, R ). orem 11.28 still holds for u ∈ C k,α (B e 0,X

Sobolev and H¨ older estimates for general H¨ ormander operators

593

Proof. By Proposition 2.22, intrinsic derivatives (as in the definition of the spaces k,α k,p CX e ) are also weak derivative (as in the definition of the spaces WX e ), therefore

k,α e k,p e C0,Xe (B ξ, R ) ⊂ WXe (B ξ, R ) for p ∈ [1, ∞] . As already noted, the representation formulas in Theorem

11.26 and e ξ, R ) to W k,p (B e ξ, R ), and then Theorem 11.28 extend, by density, from C0∞ (B e 0,X hold as almost everywhere equalities. This fact applies in particular to functions
k,α e in C0, ( B ξ, R ) for which, however, an equality almost everywhere implies an e X equality everywhere. This
shows that the aforementioned representation formulas
k,α e 2,α e ξ, R ) or u ∈ C ( B ξ, R ), respectively. From these reprehold for u ∈ C0, ( B e e 0,X X sentation formulas the a priori estimates (11.61) follow as in the proof of Theorem 11.34. We now need some results about cutoff functions and interpolation inequalities for H¨ older norms in the space of lifted variables, analogous to those we have proved for Sobolev spaces in section 11.5.1. However, these will be somewhat harder to establish. Note that, differently from what we have done for Sobolev spaces, here 2,α we need to prove these inequalities for functions CX , and not just for smooth functions. We start with the following: Proposition 11.50 (Interpolation inequality for test functions) Fornevery o ei , α ∈ (0, 1), there exist constants γ > 1 and c > 0, depending on α, R and X
2,α e such that for every ε ∈ (0, 1) and every f ∈ C0, e (B ξ, R/2 ), X

c

e

e (11.81) 6 ε Lf + γ kf kL∞ (Be(ξ,R/2))

Xl f α e

∞ e ε C f(B (ξ,R/2)) L (B (ξ,R/2)) X for l = 1, . . . , q. To prove this proposition, we need the following
e ξ, R and α ∈ (0, 1). Lemma 11.51 Let F be an operator of type λ > 1 over B n o ei , such Then there exist positive constants γ > 1 and c, depending on α and X
2,α e that for every f ∈ CX,0 (B ξ, R ) and ε ∈ (0, 1)

c

e

e 6 ε Lf + γ kf kL∞ (Be(ξ,R)) . (11.82)

∞ e

F Lf α e ε C f(B (ξ,R)) L (B (ξ,R)) X e is replaced by any differential operator of weight Moreover, (11.82) still holds if L e e e two, like Xi Xj or X0 . Proof. It is enough to prove the lemma for F of type 1, the other cases are easier. Let Z e = e (η) dη, F Lf k (ξ, η) Lf e (ξ,R) B

594

H¨ ormander operators

e (ξ, ε/2) ≺ ζε ≺ with k kernel of type 1, and let ζε be a cutoff function such that B e (ξ ε). We split F L e as follows B Z e (ξ) = e (η) dη F Lf k (ξ, η) Lf e (ξ,R) B Z Z e e (η) dη = k (ξ, η) [1 − ζε (η)] Lf (η) dη + k (ξ, η) ζε (η) Lf e e d(η,ξ)>ε/2 d(η,ξ)6ε Z Z ∗ e e (η) dη = L [k (ξ, ·) (1 − ζε (·))] (η) f (η) dη + k (ξ, η) ζε (η) Lf e d(η,ξ)>ε/2

e d(η,ξ)6ε

≡ I (ξ) + II (ξ) . e ∗ [k (ξ, ·) (1 − ζε (·))] (η) and observe that for a suitable γ > 1 Let hε (ξ, η) = L e ε Xi h (·, η) (ξ) 6 cε−γ for i = 0, 1, . . . , q. By Theorem 1.56 (Lagrange Theorem), it follows that |hε (ξ1 , η) − hε (ξ2 , η)| 6 cR ε−γ de(ξ1 , ξ2 ) and therefore Z

|hε (ξ1 , η) − hε (ξ2 , η)| |f (η)| dη e e 6 cε−γ B R kf kL∞ (B eR ) d (ξ1 , ξ2 ) .

|I (ξ1 ) − I (ξ2 )| 6

Also, since Z |I (ξ)| 6 e d(η,ξ)>ε/2

e cε−γ |f (η)| dη 6 cε−γ B R kf kL∞ (B e (ξ,R))

we obtain kIkC α (Be(ξ,R)) 6 cR ε−γ kf kL∞ (Be(ξ,R)) for any α ∈ (0, 1) . f X

Let us consider II (ξ), and let kε (ξ, η) = k (ξ, η) ζε (η). By definition of kernel of type 1 and keeping into account the support of kε , for any fixed δ ∈ (0, 1), the kernel satisfies: 1−Q 1−δ−Q |kε (ξ, η)| 6 cde(ξ, η) 6 cεδ de(ξ, η) ;

de(ξ, ξ1 ) |kε (ξ, η) − kε (ξ1 , η)| 6 c Q de(ξ1 , η)

de(ξ, ξ1 ) de(ξ1 , η)

! 1−δ−Q

6 cεδ de(ξ, η)

de(ξ, ξ1 ) de(ξ1 , η)

!

for de(ξ1 , η) > 2de(ξ, ξ1 ) . According to Definition 7.10, kε is a fractional integral kernel of exponents β = 1, ν = 1 − δ and by Theorem 7.14 and Remark 7.16, we have



e kIIkC α (Be(ξ,R)) 6 cεδ Lf

∞ e f L (B (ξ,R)) X

Sobolev and H¨ older estimates for general H¨ ormander operators

595

for every α < 1 − δ. Therefore, for every α ∈ (0, 1) there exist δ, γ > 0 such that



e

e 6 cεδ Lf + cε−γ kf kL∞ (Be(ξ,R)) ,

F Lf α e

∞ e C f(B (ξ,R)) L (B (ξ,R)) X which implies the Lemma.

Proof of Proposition 11.50. By Theorem 11.25 and (11.42), we can write e + F f, af = P Lf and, for l = 1, . . . , q, el (af ) = Fl Lf e + Sl f X
e ξ, R . If we assume where P, F, S are operators of type 2, 1, 0, respectively, over B

e ξ, R/2 , then, for f ∈ C 2,α (B e ξ, R/2 ) (recalling Proposition 11.49) a = 1 on B X,0 we obtain e + Ff f = P Lf el f = Fl Lf e + Sl f X and inserting the first identity in the second one el f = Fl Lf e + Sl P Lf e + Sl F f. X Therefore, by Theorem 11.29 and Lemma 11.51, we get (with all the norms taken
e ξ, R/2 ) over B  



e

e

e

Xl f α 6 Fl Lf

α + c P Lf

α + kF f kCX α Cf CX CX X (11.83) o n

e −γ 6 c ε Lf ∞ + ε kf kL∞ + kF f kC α . L

X

Next, we note that an operator F of type 1, with the language of Chapter 7, is a fractional integral operator with exponents β = ν = 1, hence by Theorem 7.14 and Remark 7.16 we have kF f kα 6 c kf kL∞ , which inserted in (11.83) completes the proof of our assertion. Starting with Proposition 11.50 we can now prove a version of interpolation inequality for functions which do not vanish at the boundary of the domain.
eρ ξ, R there exists Lemma 11.52 (Cutoff functions) For any 0 < s < r, ξ ∈ B
ϕ ∈ C0∞ RN with the following properties: eρ (ξ, s) ≺ ϕ ≺ B eρ (ξ, r); (i) B (ii) for every k = 0, 1, . . . there exists a constant ck (independent of r, s, ϕ) such that

k ck

D ϕ ∞ e ; (11.84) 6 k L (Bρ (ξ,R)) (r − s)

k ck

D ϕ α e 6 (11.85) k+1 C f(Bρ (ξ,R)) X (r − s)

596

H¨ ormander operators


 α e (iii) For any f ∈ CX e Bρ ξ, R

e

f Xi ϕ

Cα f X

(Beρ (ξ,R))

6

c (r − s)

2

kf kC α (Beρ (ξ,R)) f X

(11.86)

e

f X0 ϕ

e ξ,R)) Cα B f( ρ (



e e , f X X ϕ i j

e ξ,R)) Cα B f( ρ (

6

X

X

c (r − s)

3

kf kC α (Beρ (ξ,R)) . f X

eρ (ξ, σr) , B eρ (ξ, σ 0 r) replaced by Proof. Let us apply Lemma 11.36 with the balls B e e the balls Bρ (ξ, s) , Bρ (ξ, r) respectively. Then the corresponding cutoff function ϕ satisfies (i) and c c e e e e , ; X Xi ϕ 6 0 ϕ + Xi Xj ϕ 6 (r − s) (r − s)2 hence an obvious iteration gives (11.84). Let us show that this implies (11.85). Recalling the equivalence between de and ρ, applying Lagrange theorem to XI ϕ for
eρ ξ, R , |I| = k we can write, for every ξ1 , ξ2 ∈ B e eI ϕ (ξ2 ) XI ϕ (ξ1 ) − X ! q

X

e e

e e + ρ (ξ1 , ξ2 ) X0 XI ϕ ∞ 6 cρ (ξ1 , ξ2 ) ·

Xi XI ϕ ∞ e eρ (ξ,R)) L (B L (Bρ (ξ,R)) i=1   ρ (ξ1 , ξ2 ) ρ (ξ1 , ξ2 ) 6c . · 1+ k+1 (r − s) (r − s) by (11.84). Next, let us distinguish two cases. If ρ (ξ1 , ξ2 ) 6 r − s we have e eI ϕ (ξ2 ) 6 c ρ (ξ1 , ξ2 ) . XI ϕ (ξ1 ) − X k+1 (r − s) If ρ (ξ1 , ξ2 ) > r − s then

e eI ϕ (ξ2 ) 6 X eI ϕ (ξ1 ) + X eI ϕ (ξ2 ) 6 2 eI ϕ XI ϕ (ξ1 ) − X

X

∞ e L (Bρ (ξ,R)) c ρ (ξ1 , ξ2 ) ρ (ξ1 , ξ2 ) · =c . 62 k+1 (r − s) (r − s)k (r − s) In any case, e XI ϕ

6 cR1−α ·

1

, k+1 (r − s) which gives (11.85). Finally, (iii) follows from (ii) for k = 1, 2 by (11.77). Cα f X

(Beρ (ξ,R))

We can now prove Theorem 11.53 (Interpolation inequality) There exist positive constants c, R,
eρ ξ, R ), 0 < r < R, 0 < δ < 1/3, γ such that for any f ∈ C 2,α (B

c

e kDf kC α (Beρ (ξ,r)) 6 δ Lf + γ

∞ e eρ (ξ,R)). γ kf kL∞ (B δ (R − r) L (Bρ (ξ,R)) The constants c, R, γ depend on α, {Xi } ; γ is as in Proposition 11.50.

Sobolev and H¨ older estimates for general H¨ ormander operators

597


eρ ξ, R ), 0 < t < s 6 R and ζ is the cutoff function with Proof. If f ∈ C 2,α (B

eρ ξ, t ≺ ζ ≺ B eρ ξ, s , constructed in Lemma 11.52, applying Proposition 11.50 B to f ζ we get:

c

e (ζf ) ∞ kDf kC α (Beρ (ξ,t)) 6 kD (ζf )kC α (Beρ (ξ,s)) 6 ε L + γ kf kL∞ (Beρ (ξ,s)) eρ (ξ,s)) ε L (B (11.87) where



e

L (ζf ) ∞ e L (Bρ (ξ,s)) q



X

e

e e

e 6 ζ Lf +2 + f Lζ

∞ e

Xj ζ Xj f ∞ e

∞ e L (Bρ (ξ,s)) L (Bρ (ξ,s)) L (Bρ (ξ,s)) j=1

c c

e + 6 Lf kDf kL∞ (Beρ (ξ,s)) +

∞ e eρ (ξ,s)) . 2 kf kL∞ (B s−t L (Bρ (ξ,s)) (s − t) Next, we insert the last inequality in (11.87), choosing ε = δ (s − t) /c and get:



e kDf kC α (Beρ (ξ,t)) 6 δ kDf kL∞ (Beρ (ξ,s)) + δ Lf

∞ e f L (Bρ (ξ,s)) X   δ 1 +c kf kL∞ (Beρ (ξ,s)) + (s − t) δ γ (s − t)γ Let ψ (t) = kDf kC α (Beρ (ξ,t)) . Then, for R fixed once and for all (small enough), f X ϑ < 1/3 fixed, and any δ < ϑ we get, since γ > 1,

c

e ψ (t) 6 ϑψ (s) + γ kf k + δ

Lf ∞ e ∞ eρ (ξ,R)) γ L (B δ (s − t) L (Bρ (ξ,R)) for any 0 < t < s < R, and by Lemma 8.55 we get ψ (r) 6

c

e kf k + δ

Lf ∞ e ∞ e γ L (Bρ (ξ,R)) δ γ (R − r) L (Bρ (ξ,R))

for any 0 < r < R. α (B (x, R)) and Next, we
are going to study the relation between the spaces CX ξ, R ). The result we will prove is the following:

α e CX e (B

Theorem older spaces) Let
11.54 (Relation between lifted and unlifted H¨ e ξ, R be a lifted ball, with ξ = (x, 0). If f is a function defined in B (x, R) and B
e ξ, R , then the following fe(x, h) = f (x) is regarded as a function defined on B inequalities hold (whenever the right-hand side is finite):



e 6 kf kC α (B(x,R)) ,

f α e X C f(B (ξ,R)) X



kf kC α (B(x,δ0 t)) 6 c fe α for 0 < t < s < R (11.88) e (ξ,s)) X C f (B X

598

H¨ ormander operators

where c also depends on R, s, t and δ0 < 1 is the number appearing in Theorem 11.40. Moreover,

e e ei fe (11.89) 6 kXi1 Xi2 · · · Xik f kC α (B(x,R)) ,

Xi1 Xi2 · · · X

α e k X C f(B (ξ,R)) X

e e e e (11.90) kXi1 Xi2 · · · Xik f kC α (B(x,δ0 t)) 6 c X i1 Xi2 · · · Xik f α e (ξ,s)) X C f(B X for 0 < t < s < R and ij = 0, 1, . . . , q. The above result relies on the integral characterization of H¨older spaces introduced in section 7.8, in the context of locally doubling spaces, and the relation between the volumes of lifted and unlifted balls (see Theorem 11.40). Let us first recall the aforementioned integral characterization of H¨older functions. In the next definition and proposition we come back to the domain Ω of the original variables. Definition 11.55 For B (x, 2R) ⊂ Ω, f ∈ L1 (B (x, R)), α ∈ (0, 1), 0 < t < s 6 R, let Z 1 |f (y) − c| dy. Mα f ≡ Mα,B(x,t),B(x,s) f = sup inf α x∈B(x,t),r6s−t c∈R r |B (x, r)| B(x,r) α (B (x, R)) then Mα f 6 |f |C α (B(x,R)) . Observe that if f ∈ CX

With the notation of Theorem 7.38, let Ω0 = B (x, t), Ω1 = B (x, s), 3κ = s − t. Then we have the following: Proposition 11.56 For B (x, 2R0 ) ⊂ Ω, R < R0 , α ∈ (0, 1) , 0 < t < s 6 R, if f ∈ L1 (B (x, s)) is a function such that Mα,Bt ,Bs (f ) < ∞, then there exists a α function f ∗ , a.e. equal to f , such that f ∗ ∈ CX (B (x, t)) and for x, y ∈ B (x, s) α

|f ∗ (x) − f ∗ (y)| 6 cMα f · d (x, y) if 2d (x, y) 6 s − t.

α (B(x,R)) Proof of Proposition 11.54. The first inequality |fe|C α (Be(ξ,R)) 6 |f |CX f X

immediately follows by the inequality d (x, y) 6 de((x, h) , (y, k)) for every (x, h) , (y, k) (see Proposition 10.39), since e f (x, h) − fe(y, k) = |f (x) − f (y)| α 6 |f |C α (B(x,R)) d (x, y) 6 |f |C α (B(x,R)) de((x, h) , (y, k)) . X

X

To prove the second one, let 0 < t < s < R, x ∈ B (x, δ0 t) , where δ0 is the number
e ξ, δ0 t on in Theorem 11.40, r 6 s − t and ξ = (x, 0). Since the projection of B
e ξ, δ0 t . By Theorem 11.40 we have, for Rn is B (x, δ0 t), there exists ξ = (x, h) ∈ B

Sobolev and H¨ older estimates for general H¨ ormander operators

599

every k ∈ R, 1 e B (ξ, r)

Z

1

e B(ξ,r)

e f (η) − k dη

Z

!

Z |f (y) − k|

= e (ξ, r) B

B(x,r)

1 > e B (ξ, r)

Z

c > |B (x, r)|

Z

B(x,δ0 r)

dh dy e {h:(y,h)∈B(ξ,r) } n o e (ξ, r) dy |f (y) − k| h : (y, h) ∈ B

c |f (y) − k| dy > |B (x, δ0 r)| B(x,δ0 r)

Z |f (y) − k| dy B(x,δ0 r)

where in the last inequality we used the doubling condition (see Theorem 9.1). Choosing k = f (x) = fe(ξ) we have Z 1 |f (y) − k| dy inf α k∈R r |B (x, δ0 r)| B(x,δ r) 0 Z (11.91) 1 e e (ξ) dη 6 c fe 6 c f (η) − f α e (ξ,s)) C f(B e e (ξ, r) B(ξ,r) r α B X
where the last inequality holds because r 6 s − t, de ξ, ξ < δ0 t gives the inclusion

e ξ, s . e (ξ, r) ⊂ B e ξ, δ0 t + s − t ⊂ B B Hence (11.91) gives 1 α x∈B(x,t),r6δ0 (s−t) k∈R r |B (x, r)| sup

Z

inf

B(x,r)

|f (y) − k| dy 6 c fe

e ξ,s)) Cα B f( (

.

X

By Proposition 11.56 there exists a function f ∗ , a.e. equal to f , such that f ∗ ∈ α (B (x, t)) and for every x, y ∈ B (x, δ0 s) CX α |f ∗ (x) − f ∗ (y)| 6 cd (x, y) fe α e (ξ,s)) C f(B X if 2d (x, y) 6 δ0 (s − t) . On the other hand, if 2d (x, y) > δ0 (s − t) ,



|f ∗ (x) − f ∗ (y)| 6 |f ∗ (x)| + |f ∗ (y)| 6 2 kf kL∞ B(x,δ0 s) 6 2 fe

e (ξ,s)) L∞ B (B

and we conclude



kf ∗ kC α (B(x,δ0 t)) 6 c fe X

e ξ,s)) Cα B f( (

.

X

Finally, note that f ∗ = f everywhere because we already know that f is continuous, since the control distance induces the Euclidean topology, f (x) = fe(x, h) and fe is e d-continuous, hence Euclidean continuous. g ei fe = X Now, inequalities (11.89) and (11.90) also follow, because X i f , hence the same reasoning can be iterated to higher order derivatives.

600

H¨ ormander operators

11.5.4

A priori estimates in H¨ older spaces

We start with the following H¨ older estimates in the space of lifted variables. With respect to what already proved in Theorem 11.34, here the function u is not necessarily compactly supported; moreover, in the right hand side of the estimate we do not have first order derivatives, and u appears only through its L∞ norm, instead α of CX e norm. 2,α e Theorem 11.57 (Local CX e estimates for L) There exist r0 , c, β > 0 such that, 2,α e for every u ∈ CXe (B(ξ, r0 )), 0 < t < s < r0 , ( )

c

e kukC 2,α (Beρ (ξ,t)) 6 + kukL∞ (Beρ (ξ,s)) ,

Lu α e f (s − t)β C f(Bρ (ξ,s)) X X n oq n oq ei ei where r0 , c depend on R, X , α; β depends on X and α. i=1

Proof. If u ∈

2,α e CX e (B

i=0




ξ, R ) with R small enough, t < R, s = (t + R) /2 and ζ is

eρ ξ, s , we can apply Theorem 11.34 eρ ξ, t ≺ ζ ≺ B a cutoff function such that B to uζ (thanks to Proposition 11.49), getting, by Lemma 11.52, kukC 2,α (Beρ (ξ,t)) f (X



e (uζ) 6 c L

e ξ,s)) Cα B f( ρ (

q

X

e +

Xl (uζ)

e ξ,s)) Cα B f( ρ (

l=1

X

)


eρ ξ, s to simplify the notation) (omitting the ball B (





e

e 6 c kζkC α Lu

α + kDζkC α kDukC α + Lζ Cf

f X

f X

X

+ kζDukL∞ + kuDζkL∞

L∞

f X

q X e + Xl (uζ) l=1

( 6c

+ kuζkC α (Beρ (ξ,s)) f X

X

e kukL∞ + uLζ

Cα f X

) Cα f

+ kuζkL∞ + |uζ|C α

f X

X

1 1 1

e +

Lu α + 2 kDukC α 2 kukL∞ f s−t X Cf (s − t) (s − t) X ) q X e e + uLζ α + Xl (uζ) α + |uζ|C α . Cf X

Cf

l=1

f X

X

e C α , |X el (uζ) |C α , |uζ|C α . Applying (11.75), Lemma Next, we handle the terms |uLζ| f X

11.52 and (11.80) we can write, for some small η to be chosen later:

      e

e e e uLζ α 6 R1−α D uLζ

∞ + R X 0 uLζ ∞ Cf

L

X

( 6 R1−α

c

L

c

)

2 kDukL∞ + 3 kukL∞ (s − t) (s − t) ) (

  2

2−α α/2 e e e + uLζ +R η X0 uLζ η L∞ Cα f X

Sobolev and H¨ older estimates for general H¨ ormander operators

601

Now:     e e e0 Lζ e e + uX e X u Lζ 6 X0 uLζ 0 Cα Cα Cα f f f



X    X  X c

e

e 1−α e e e e 6 X u X Lζ u X Lζ + R + R X u

∞

D

0 0 0 0 3 ∞ L L Cα (s − t) f X ( )

c c c

e 1−α 6 3 X0 u α + R 4 kDukL∞ + 5 kukL∞ Cf (s − t) (s − t) (s − t) X ( )

c c

e + R2−α 4 X0 u ∞ + 6 kukL∞ L (s − t) (s − t) e The terms |uζ|C α , X l (uζ) α can be bounded analogously, getting a similar exf X

Cf X

k

pression with smaller exponents in the terms c/ (s − t) . Collecting the above estimates and allowing
the constant c to depend on R we get, for small (s − t), (omitting e the ball Bρ ξ, s to simplify notation) ) (

1 1 1

e + kukC 2,α (Beρ (ξ,t)) 6 c

Lu α + 2 kDukC α 3 kukL∞ f f s−t X Cf X (s − t) (s − t) X c + 2 kukL∞ η (s − t) ) (

1 1 1

e α/2 + cη 4 X0 u α + 4 kDukL∞ + 6 kukL∞ Cf (s − t) (s − t) (s − t) X 4

For a small ε to be chosen later, we now pick η such that cη α/2 / (s − t) = ε, and write: kukC 2,α (Beρ (ξ,t)) f ( X )

1 1 1

e 6c + kukL∞

Lu α + 2 kDukC α 2+8/α f s−t X Cf (s − t) ε2/α (s − t) X

(11.92)

+ ε kukC 2,α . f X

Next, we apply Theorem 11.53 to write kDukC α (Beρ (ξ,s)) 6 δ kukC 2,α (Beρ (ξ,R)) + f X

f X

c eρ (ξ,R)) γ kukL∞ (B δ γ (R − s)

(11.93)

However, by our choice of t, s, R, we have s − t = R − s = R−t 2 . We insert (11.93) in (11.92) with δ such that cδ/(s − t)2 = ε, and get:

c

e kukC 2,α (Beρ (ξ,t)) 6

Lu α e f R−t C f(Bρ (ξ,R)) X X c + 2ε kukC 2,α (Beρ (ξ,R)) + 0 kukL∞ (Beρ (ξ,R)) β0 f X εα (R − t)

602

H¨ ormander operators

Letting ψ (t) = kukC 2,α (Beρ (ξ,t)) and choosing ε such that 2ε = ϑ < 1/3, we can f X rewrite the last inequality as !

c

e ψ (t) 6 ϑψ (R) + + kukL∞ (Beρ (ξ,R))

Lu α e β0 C f(Bρ (ξ,R)) (R − t) X and by Lemma 8.55 we get kukC 2,α (Beρ (ξ,t)) 6 f X

c (R − t)

β0

!



e

Lu

e ξ,R)) Cα B f( ρ (

+ kukL∞ (Beρ (ξ,R))

X

for R small enough, with c depending also on an upper bound for R. 2,α Theorem 11.58 (Interior CX -estimates for L) For any domain Ω0 b Ω and 2,α (Ω), α ∈ (0, 1) there exists a constant c such that for any u ∈ CX o n kukC 2,α (Ω0 ) 6 c kLukC α (Ω) + kukL∞ (Ω) . X

X

Proof. Fix x ∈ Ω
0 and R such that in B (x, R) ⊂ Ω all the previous construction e ξ, R and so on) can be performed. Combining Theorems 11.57 and (lifting to B ] e u = (Lu), 11.54 we can write, for 0 < t < s < s1 < R, recalling that Le kukC 2,α (B(x,δ0 t)) X

) (



e + ke ukL∞ (Be(ξ,s1 )) u α 6 c ke ukC 2,α (Be(ξ,s)) 6 c Le e (ξ,s1 )) f C f (B X X o n ukL∞ (B(x,s1 )) 6 c kLukC α (B(x,s1 )) + ke f X

with c also depending on the numbers t, s1 , which now are fixed so that δ0 t = s1 /2 ≡ k S r0 . Next, let us choose a family of balls B (xi , r0 ) in Ω such that B (xi , r0 /2) i=1

is connected and Ω0 ⊂

k S i=1

B (xi , r0 /2) ⊂

k S

B (xi , 2r0 ) ⊂ Ω. Then, by Proposition

i=1

11.48 (iii), we have kukC 2,α (Ω0 ) 6 kukC 2,α (∪B(xi ,r0 /2)) 6 c X

k X

X

kukC 2,α (B(xi ,r0 )) X

i=1

6c

k n o n o X kLukC α (B(xi ,2r0 )) + kukL∞ (B(xi ,2r0 )) 6 c kLukC α (Ω) + kukL∞ (Ω) . X

X

i=1

So we are done. Also for H¨ older estimates, we now come to the particular case of H¨ormander operators without drift, and for these operators we want to extend the result of Theorem 11.58 to higher order derivatives. We will prove the following:

Sobolev and H¨ older estimates for general H¨ ormander operators

603

Theorem 11.59 (Interior H¨ older regularity for sum of squares) Let L=

q X

Xi2

i=1

be a H¨ ormander operator without drift in Ω. For any domain Ω0 b Ω, nonnegative 2,α integer k and α ∈ (0, 1) there exists a constant c such that for any u ∈ CX (Ω), if k,α k+2,α 0 Lu ∈ CX (Ω) then u ∈ CX (Ω ) and the following holds: n o kukC k+2,α (Ω0 ) 6 c kLukC k,α (Ω) + kukL∞ (Ω) . X

X

Toward this aim, we start proving the local estimate on higher order derivative in the space of lifted variables:
k,α e ξ, R be as Theorem 11.60 (CX estimates for the lifted operator) Let B e before, let α ∈ (0, 1) and let k be a positive integer. There exists r0 > 0 such
that k+2,α e for every r 6 r0 there exists c > 0 such that for any u ∈ CX (Bρ ξ, 2k+1 r ) we e have  

e + kukL∞ (Beρ (ξ,2k+1 r)) . kukC k+2,α (Beρ (ξ,r/2)) 6 c Lu k,α eρ (ξ,2k+1 r )) X CX (B Note that, once Theorem 11.60 is established, Theorem 11.59 follows by the same argument used to prove Theorem 11.58 by Theorem 11.57.
e ξ, R ), Proof. By Theorem 11.28 (with Proposition 11.49), for a fixed a ∈ C0∞ (B
k,α e any multiindex I with |I| > 3 and u ∈ CX,0 e (B ξ, R ) we have eI (au) = X

X

eJ Lu e + SI Lu e + SIJ X

|J|6|I|−2 0 SIK

X

0 eK u + SI0 u SIK X

|K|6|I|−1

SI0

are operators of type 0 and each of them has the form and where SIJ , SI , ej F for some j = 1, . . . , q and some operator F of type 1. If we choose a = 1 X
e ξ, R/2 , taking C α norms of both sides, by Theorem 11.29 we get, for every on B e X
k,α e u ∈ C (B ξ, R/2 ) e X,0



e

XI u

e (ξ,R/2)) Cα (B f

6c

X

+

 

X



|J|6|I|−2

X



e e

XJ Lu

e (ξ,R/2)) Cα (B f

e (ξ,R/2)) Cα (B f

X



e

XK u

e (ξ,R/2)) Cα (B f

|K|6|I|−1



e + Lu

X

X

+ kukC α (Be(ξ,R/2)) f X

  

that is, letting |I| = k + 2 for some k = 1, 2, . . . , ( )



e kukC k+2,α (Be(ξ,R/2)) 6 c Lu k,α + kukC k+1,α (Be(ξ,R/2)) . f f C X X (Be(ξ,R/2)) f X

(11.94)

604

H¨ ormander operators

By iteration, this yields (



e kukC k+2,α (Be(ξ,R/2)) 6 c Lu + kukC 2,α (Be(ξ,R/2))

k,α e f f C X X (B (ξ,R/2))

) (11.95)

f X

which, by Theorem 11.57, gives ( )



e + kukL∞ (Be(ξ,R/2)) . kukC k+2,α (Be(ξ,R/2)) 6 c Lu k,α f C X (Be(ξ,R/2))

(11.96)

f X



eρ ξ, R0 ⊂ B e ξ, R and, for r 6 2−k−2 R0 ≡ r0 and u ∈ Let R0 be such that B 


2,α e eρ ξ, r/2 ≺ ζ ≺ B eρ ξ, r , we CX Bρ ξ, 2k+1 r , pick a cutoff function ζ with B e apply (11.95) to uζ, getting, by Lemma 11.52, kukC k+2,α (Beρ (ξ,r/2)) 6 kuζkC k+2,α (Beρ (ξ,r)) f f X ) ( X

e

6 c L (uζ) k,α + kuζkL∞ (Beρ (ξ,r)) eρ (ξ,r )) C f (B X ( q

X

e

e e e 6 c ζ Lu +2

k,α e

Xi uXi Lζ k,α e C f (Bρ (ξ,r )) C f (Bρ (ξ,r )) X X i=1 )

e + kuζkL∞ (Beρ (ξ,r)) + uLζ

k,α e C f (Bρ (ξ,r )) X ) (

1 1

e 6c + k+4 kukC k+1,α (Beρ (ξ,r)) .

Lu k,α e f rk+1 r C f (Bρ (ξ,r )) X X By iteration these estimates give ( )



e kukC k+2,α (Beρ (ξ,r/2)) 6 c (r, k) Lu k,α + kukC 2,α (Beρ (ξ,2k r)) f f C X X (Beρ (ξ,2k r)) f X

which combined with the basic estimate in Theorem 11.57 yields ( )



e + kukL∞ (Beρ (ξ,2k+1 r)) . kukC k+2,α (Beρ (ξ,r/2)) 6 c (r, k) Lu k,α f C X (Beρ (ξ,2k+1 r)) f X

Proof of Theorem 11.59. Exploiting Theorem 11.60 and following the same argument used to prove Theorem 11.58 by Theorem 11.57 one can prove the a k+2,α priori estimates, assuming first that u ∈ CX (Ω). Once the a priori estimates are established, the proof of the regularization result can be achieved repeating k,p 2,α verbatim the proof of Theorem 11.45, replacing WX (Ω) with CX (Ω) and the p α continuity of operators of type 0 on L with the continuity on CX .

Sobolev and H¨ older estimates for general H¨ ormander operators

11.6

605

Smoothing of distributional solutions and solvability in H¨ older or Sobolev spaces

Let us briefly summarize the results that we have achieved so far in this chapter. We have proved that for any H¨ ormander operator L in a bounded domain Ω of Rn , 0 any Ω b Ω: 2,p • if u ∈ WX (Ω) for some p ∈ (1, ∞), then u satisfies the local a priori estimate n o kukW 2,p (Ω0 ) 6 c kLukLp (Ω) + kukLp (Ω) ; X

2,α • if u ∈ CX (Ω) for some α ∈ (0, 1), then u satisfies the local a priori estimate n o kukC 2,α (Ω0 ) 6 c kLukC α (Ω) + kukL∞ (Ω) , X

X

Moreover, when L is an operator without drift, the previous results can be extended to higher order regularity results. However, so far, two very natural questions have not been answered yet: (1) If u is just a distributional solution to Lu = f in Ω and f is a function in 2,p α (Ω), can we say that that u is actually a function, in WX,loc Lp (Ω) or CX (Ω) 2,α or CX,loc (Ω) , respectively? α (Ω), can we say (2) For an assigned function f belonging to Lp (Ω) or CX 2,p 2,α that a solution u to Lu = f actually exists, in WX,loc (Ω) or CX,loc (Ω) , respectively? In this section we will give positive answer to both the questions and this, combined with the a priori estimates proved in the previous section, will complete the proof of the results announced in the introduction of this chapter. We start with the following local result in the space of lifted variables. Proposition 11.61 (Local solvability in the lifted variables) For R small
eR = B e ξ, R , the following hold: enough and a fixed ball B eR ), p ∈ (1, ∞), there exists u ∈ W 2,p (B eR ) such that (a) for every g ∈ Lp (B e X e = g a.e. in B eR Lu eR ), α ∈ (0, 1), there exists u ∈ C 2,α (B eR ) such that (b) for every g ∈ C α (B e X,0

e X

e = g in B eR . Lu eR ) such that fn → f in Lp ; extend eR ) and let fn ∈ C ∞ (B Proof. (a) Let f ∈ Lp (B 0 e e and L e∗ f, fn to zero outside BR . Applying Theorem 11.25 (with the roles of L e e exchanged), we can write the parametrix formula in B4R : aI = LP + F with P e4R . Choosing a ∈ C ∞ (B e4R ) with operator of type 2 and F of operator type 1 over B 0 e2R we can write a = 1 in B e fn + F fn in B e2R . fn = LP (11.97)   2,p e Let un = P fn , by Proposition 11.30, P f, P fn ∈ WX B4R and e   2,p e un → P f ≡ u in WX B4R . e

606

H¨ ormander operators

Also e n = fn − F fn → f − F f in Lp (B e4R ). Lu e + F f in Lp (B e2R ). We have Hence passing to the limit in (11.97) we get f = Lu 2,p e e eR ). therefore solved in WXe (BR ) the equation Lu = f − F f with assigned f ∈ Lp (B To conclude the proof of the solvability result, it is enough to show that the integral equation f − Ff = g p

eR ) has always a solution f ∈ Lp (B eR ). Actually, we with an assigned g ∈ L (B eR ), let T : will prove that this is true for R small enough. For a fixed g ∈ Lp (B p e p e L (BR ) → L (BR ) be the operator such that T f = g + F f and let us show that, for R small enough, T is a contraction. Actually, as we have seen in section 11.4, the kernel of F can be written as the sum of fractional integral kernels, getting fractional integral operators I` satisfying (see (11.50)) the estimates 1 1 kI` f kLp (Be(ξ,R)) 6 cRQ( p − q ) kf kLp (Be(ξ,R)) (11.98)     with ` = 1, 2, . . . , and q = p `+Q > p 1+Q plus a regular (Lipschitz continuous, Q Q

compactly supported) kernel, getting an integral operator Ireg satisfying kIreg f kL∞ (Be(ξ,R)) 6 c kf kLp (Be(ξ,R)) .

(11.99)

From (11.98) we deduce, for any ` = 1, 2, . . . , Q

Q

kI` f kLp (Be(ξ,R)) 6 cRQ( p − p Q+1 ) kf kLp (Be(ξ,R)) = cR p ( Q+1 ) kf kLp (Be(ξ,R)) 1

1

1

while (11.99) gives Q

Q

kIreg f kLp (Be(ξ,R)) 6 cR p kIreg f kL∞ (Be(ξ,R)) 6 cR p kf kLp (Be(ξ,R)) Collecting these inequalities we have o n 1 Q Q kF f kLp (Be(ξ,R)) 6 cR p ( Q+1 ) + cR p kf kLp (Be(ξ,R)) and choosing R small enough we conclude 1 kf1 − f2 kLp (Be(ξ,R)) . 2 Then, by the Banach-Caccioppoli fixed point theorem, T has a unique fixed point eR ), that is there exists f ∈ Lp (B eR ) such that in Lp (B kT f1 − T f2 kLp (BeR ) = kF (f1 − f2 )kLp (BeR ) 6

f = g + F f. p

eR ) we can first get an f ∈ Lp (B eR ) such that f − F f = g, Hence, for a fixed g ∈ L (B 2,p e e = f − F f = g. and then get u ∈ WXe (BR ) such that Lu e3R ) and let fn (b) Let us now prove the result in the H¨ older case. Let f ∈ C α (B e X,0

e3R ), so be the mollified of f as in Theorem 2.20 (f, fn extended to zero outside B 0 e that fn → f in CXe . We can still write (11.97) in B2R and, letting un = P fn , by

Sobolev and H¨ older estimates for general H¨ ormander operators

607

2,α e 2,α e Proposition 11.30 we have P f, P fn ∈ CX e (B4R ) with un → P f ≡ u in CX e (B4R ). 2,α e Note that actually u ∈ C (B4R ), where the compact support follows by definition e X,0

of operator of type 2. Also, by Theorem 2.20, e n = fn − F fn → f − F f in C 0 (B e2R ). Lu e X Hence taking uniform limits in (11.97) we have found that u satisfies e2R ). e + F f in C 0 (B f = Lu e X 2,α e e We have therefore solved in CX e (B2R ) the equation Lu = f − F f with assigned α e f ∈ CX,0 e (B3R ). We have now to solve the integral equation f − F f = g. Here α a minor problem arises since we cannot apply the fixed point theorem to CX,0 e , α which is not a Banach space as CXe is. Let us first show that, for R small enough e4R ). For a fixed g ∈ C α (B e4R ), let T : the integral equation is solvable in C α (B α e CX e (B4R )

α e CX e (B4R )

e X

e X

→ be such that T f = g + F f and let us show that, for R small enough, T is a contraction. Reasoning as above, we exploit the fact that F is the sum of fractional integral operators I` satisfying now (see (11.51)) an estimate kI` f kC δ (Be4R ) 6 c kf kC α (Be4R )

(11.100)

for any α, δ ∈ (0, 1), plus a regular (Lipschitz continuous, compactly supported) kernel, getting an integral operator Ireg satisfying the same estimate. From (11.100) we deduce, for δ = (α + 1) /2 and any ` = 1, 2, . . . , |I` f |C α (Be4R ) 6 cRδ−α |I` f |C δ (Be4R ) 6 cRδ−α kf kC α (Be4R ) . Recall that the integral operator I` maps any function to a compactly supported function, and for a compactly supported function I` f we have kI` f kL∞ (Be4R ) 6 Rα |I` f |C α (Be4R ) 6 cRδ kf kC α (Be4R ) so that  1−α kI` f kC α (Be4R ) 6 c Rδ−α + Rδ kf kC α (Be4R ) 6 cR 2 kf kC α (Be4R ) . Therefore we have kF f kC α (Be4R ) 6 cR

1−α 2

kf kC α (Be4R )

e4R ), and choosing R small enough we conclude that T is a contraction on C α (B α e α e so that for every g ∈ CX ( B ) there exists f ∈ C ( B ) such that f = g+ 4R 4R e e X F f . In order to use this f to find u by the previous procedure, we would need e3R , which is not generally true. However, we can f compactly supported in B assume that the cutoff functions which enter the definition of F are supported e3R (this is consistent with the requirement a = 1 in B e2R that we have made in B α e at the beginning of the proof). This implies that for every f ∈ CX e (B4R ) we have e3R ). Also, solving by Neumann series the equation f = g + F f we see F f ∈ C α (B e X,0

608

H¨ ormander operators

α α α e e e that if g ∈ CX,0 e (B2R ) e (B3R ). In particular, for a fixed g ∈ CX,0 e (B3R ) then f ∈ CX,0 α e e we can first get an f ∈ CX,0 e (B3R ) such that f − F f = g in B2R , and then get 2,α e e = f − F f = g. u ∈ u ∈ C (B2R ) such that Lu e X

Thanks to the previous result, we can now prove the following Theorem 11.62 (Smoothing of distributional solutions) Let u ∈ D0 (Ω) be a distributional solution to Lu = f in Ω. Then: 2,p (a) if f ∈ Lp (Ω) for some p ∈ (1, ∞) , then u ∈ WX,loc (Ω) and it is a strong solution to Lu = f in Ω. 2,α α (b) if f ∈ CX (Ω) for some α ∈ (0, 1) , then u ∈ CX,loc (Ω) and it is a classical solution to Lu = f in Ω. Proof. We will prove that each point of Ω has a neighborhood where u belongs 2,p 2,α α to WX or CX , respectively, according to the belonging of f to Lp (Ω) or CX (Ω). This will imply the assertion. We want to link the statement of this theorem to that of Proposition 11.61, but to do this we need a way to transfer distributions from the original space to that of lifted variables,
and viceversa. Let B (x, R) ⊂ Ω ⊂ Rn e ξ, R ⊂ RN and let T ∈ D0 (B (x, R)) . We be control ball which can be lifted to B
e ξ, R . Let us take R small enough so that we can apply Proposition 11.61 in B

e ξ, R ), we have e ξ, R ) requiring that, for any φ ∈ D(B define Te ∈ D0 (B  D E  Z Te, φ = T, φ (·, t) dt . (11.101)
R e ξ, R ), and Note that ϕ (x) = RN −n φ (x, t) dt is in D (B (x, R)) whenever φ ∈ D(B

e ξ, R ). e ξ, R ) implies ϕk → ϕ in D (B (x, R)). Hence, Te ∈ D0 (B φk → φ in D(B So, let u ∈ D0 (Ω) be a distributional solution to the equation Lu = f in Ω with f ∈ α Lp (Ω) for some p ∈ (1, ∞) or f ∈ CX (Ω) for some α ∈ (0, 1). Then T = u/B(x,R) is a distributional solution to LT = f in B (x, R). We claim that Te, defined as above, e Te = fe where we let fe(x, t) = f (x) (recall that f is is a distributional solution to L
e ξ, R ), a function). Namely, for any φ ∈ D(B  D E  Z ∗ ∗ e e e T , L φ = T, L φ (·, t) dt . On the other hand, recalling that ek = Xk + X e∗ = X∗ − X k k

m X j=1 m X j=1

ukj (x, t1 , t2 , . . . , tj−1 ) ∂tj , ukj (x, t1 , t2 , . . . , tj−1 ) ∂tj

Sobolev and H¨ older estimates for general H¨ ormander operators

609

(see Theorem 10.6) we can compute e∗ = L

q  X

ek∗ X

2

e0∗ = L∗ − +X

k=1

 q X m X k=1



(Xk∗ ukj ) (x, t1 , t2 , . . . , tj−1 ) ∂tj

j=1

 2   m  X −2 ukj (x, t1 , t2 , . . . , tj−1 ) Xk∗ ∂tj +  ukj (x, t1 , t2 , . . . , tj−1 ) ∂tj    j=1 j=1 m X

−

m X

u0j (x, t1 , t2 , . . . , tj−1 ) ∂tj

j=1

≡ L∗ +

m X

ψj (x, t1 , . . . , tj−1 ) ∂tj − 2

j=1

q m X X

ukj (x, t1 , t2 , . . . , tj−1 ) Xk∗ ∂tj

j=1 k=1

hence, since each derivative ∂tj is multiplied by a coefficient which does not depend on tj , Z Z m Z X e ∗ φ (x, t) dt = L∗ φ (x, t) dt + L ∂tj (ψj (x, t) φ (x, t)) dt j=1

−2

q Z m X X

∂tj (ukj (x, t) Xk∗ φ (x, t)) dt

j=1 k=1 ∗

=L

Z φ (x, h) dh

because if (ψφ)R (x, t) is a smooth function, compactly supported with respect to t for R every x, then ∂tj (ψφ) (x, t) dt = 0. We conclude, since φ (·, t) dt ∈ D (B (x, R))   Z  Z D E  e ∗ φ = T, L∗ φ (·, t) dt = f, φ (·, t) dt Te, L Z Z Z   = f (x) φ (x, t) dtdx = feφ (x, t) dtdx
e Te = fe in B e ξ, R , in the distributional sense, so the claim is which means that L proved. Next, we note that:
e ξ, R ) (Theorem 11.41); • if f ∈ Lp (Ω) then fe ∈ Lp (B
α α e • if f ∈ CX (Ω) then fe ∈ CX (B ξ, R ) (Theorem 11.54). We can then apply Proposition 11.61, getting: 2,p e e u = fe a.e. in B eR ; • if f ∈ Lp (Ω), a function u e ∈ WX e (BR ) such that Le 2,α e α e u = fe in B eR . • if f ∈ CX (Ω), a function u e ∈ CX e (BR ) such that Le

e = Te − u eU e = 0 in B eR and, since Therefore, the distribution U e solves the equation L ∞ e e e e +u L is hypoelliptic, actually U ∈ C (BR ). But then the distribution Te = U e 2,p e 2,α e is the sum of a smooth function and a function in WXe (BR ) or CXe (BR ), and is eR/2 ) or C 2,α (B eR/2 ), respectively. We have therefore therefore a function in W 2,p (B e X

e X

610

H¨ ormander operators

regularized the lifted distribution Te to a function belonging to the proper function space. Next, we claim that the function Te is actually independent of t. To
see this, e ξ, R/4 ), recalling the definition (11.101) of Te, let us compute, for φ ∈ D(B   Z D E D E ∂tj Te, φ = − Te, ∂tj φ = − T, ∂tj φ (·, t) dt = − hT, 0i = 0. Finally, we want to recognize that the original distribution T is also a function, with T (x) = Te (x, t) = Te (x) a ball smaller than B (x, R/4). To see this,  at least in
 e recall that for every φ ∈ D B ξ, R/4 Z

 D E  Z e e T (x) φ (x, t) dxdt = T , φ = T, φ (·, t) dt .

Apply this identity to φ (x, t) = φ1 (x) φ2 (t) where φ2 is fixed once and for all, 0 6 φ2 6 1, φ2 (t) = 1 for |t| < δ and some δ > 0, φ1 is any function
in D (B (x, R/8)) e ξ, R/4 . Then we get and δ is small enough so that B (x, R/8) × {|t| < δ} ⊂ B  Z  Z Z e T (x) φ1 (x) dx φ2 (t) dt = T, φ1 (·) φ2 (t) dt = c hT, φ1 i with c =

R

φ2 (t) dt > 0, and finally Z hT, φ1 i = Te (x) φ1 (x) dx for every φ1 ∈ D (B (x, R/8))

that is T (x) = Te (x) for a.e. x ∈ B (x, R/8) . But then: 2,p e • if f ∈ Lp (Ω) and therefore Te ∈ WX e (BR/8 ), then by Theorem 11.41 T ∈ 2,p WX (B (x, δ0 R/8));

2,α e • if f ∈ C α (Ω) and therefore Te ∈ CX e (BR/8 ), then by Theorem 11.54 T ∈ 2,α CX (B (x, δ0 R/16)),

so we are done. Finally, exploiting the smoothing result we can prove the global solvability result announced in the introduction: α Proof of Theorem 11.4. If f belongs to Lp (Ω) or to CX (Ω), in particular 1 f ∈ L (Ω). By Proposition 6.1 and our assumption (6.1) we can apply Proposition 6.2 to the measure dν = ψ (x) dx and find a Radon measure µ satisfying in the distributional sense Lµ = f and

|µ| (Ω) 6 C kf kL1 (Ω) . In particular, µ ∈ D0 (Ω) and applying Theorem 11.62 we get that µ is actually 2,p 2,α a function u ∈ WX,loc (Ω) or u ∈ CX,loc (Ω), respectively, with the corresponding local a priori estimate proved in Theorems 11.43 and 11.58. In particular, |µ| (Ω) = kukL1 (Ω) and (11.1) follows.

Sobolev and H¨ older estimates for general H¨ ormander operators

11.7

611

Notes

The core of this chapter, namely a priori Lp -estimates for solutions to equations Lu = f for operators L of H¨ ormander’s type, consists in results originally proved by Rothschild-Stein [142], and most of this chapter, in particular sections 11.2 and 11.3, can be seen as a detailed exposition of material taken from that paper. The present exposition however takes advantage of other sources, which we would like to acknowledge here. (1) We have systematically adopted the language (which is standard in the context of elliptic PDEs) of a priori estimates, that is estimates of the size of the solution in some Banach space. This requires the use of density results, interpolation inequalities, cutoff functions and some techniques of Sobolev spaces which were not present in [142]. Interpolation inequalities for Sobolev norms have been firstly proved, in the context of H¨ormander vector fields, by Bramanti-Brandolini in [26] (for homogeneous groups) and [25] (for the general case), in the framework of more general nonvariational operators structured on H¨ ormander vector fields. (2) In [142] all the proofs are written for the operator without drift. Here we have carried out in detail also the drift case, as far as possible. The necessary adaptations of results in sections 11.2-11.3 were firstly worked out by Bramanti-Zhu in [41]. (3) Since at the time of the paper [142] the distance induced by a system of vector fields had not been studied yet in detail, a priori estimates in H¨older spaces defined with respect to the control distance were not covered in [142]. These kinds of estimates, in the more general context of nonvariational operators structured on H¨ ormander vector fields, were firstly studied by Bramanti-Brandolini [28], where some of the properties of these spaces that we have collected in section 11.5.3 were firstly proved. Proposition 2.22 is taken from [19, Lemma 2.1]. (4) We have taken advantage of the language and results about the geometry of vector fields developed in Chapter 10, section 10.4, and originally contained in the papers by Nagel-Stein-Wainger [131] and Sanchez-Calle [144], which came almost ten years later [142]. On the one hand, the equivalence of Rothschild-Stein quasidistance ρ with the control distance de of the lifted vector fields turns out to be useful; on the other hand, Theorem 10.40 (comparison of volumes of lifted and unlifted metric balls) is a key tool in the proof of Theorem 11.54, a crucial result in the proof of a priori estimates in H¨ older spaces, originally contained in [28].

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Chapter 12

Nonvariational operators constructed with H¨ ormander vector fields

12.1

Introduction

This last chapter is somewhat different from the rest of the book, both in content and in style. As for the content, it is less “classical” than the rest of the book, since here we will not study H¨ ormander operators, but some special classes of degenerate elliptic operators with nonsmooth coefficients which, however, are constructed with H¨ ormander vector fields. The study of these classes of operators started more than 30 years later than H¨ ormander’s founding paper [107]. So here we are going to present some more recent results, which also represent a significant application of all the ideas and techniques previously developed. In particular, the special care that we have often taken, in the previous chapters, in keeping under control the dependence of the constants or proving uniform estimates (with respect to some parameters) will now show its importance. As to the style, this chapter will be a bit less self-contained than the previous parts of the book, in two senses. First, this chapter will go through some general lines quite similar to those of Chapter 11, so the reader is required to have a fresh knowledge of facts, procedures and notation used in Chapter 11, which will not be systematically recalled; also, some portions of proofs that here apply and are identical (o strictly similar) to proofs in Chapter 11 will not be repeated here. We can say that in this chapter the proofs are written in order to point out the necessary changes with respect to the previous chapter. Second, a few abstract results from real analysis in locally doubling spaces, in the same spirit of those developed in Chapter 7 and which have not been used in the previous parts of the book, will be stated without proof and used in this chapter.

12.1.1

Nonvariational operators structured on H¨ ormander vector fields

In Chapter 11 we have shown that precise regularity estimates can be proved for operators of the kind “sum of squares of H¨ ormander vector fields”, both in Sobolev and H¨ older spaces structured on the vector fields and the metric they induce. Let 613

614

H¨ ormander operators

us write once again these estimates, for an operator L=

q X

Xi2

i=1

where X1 , . . . , Xq is a system of H¨ ormander vector fields in a bounded domain Ω ⊂ Rn . Then (see Theorem 11.1): (a) For any domain Ω0 b Ω, nonnegative integer k and p ∈ (1, ∞) there exists 2,p k,p a constant c such that for any u ∈ WX (Ω), if Lu ∈ WX (Ω) then u ∈ k+2,p 0 WX (Ω ) and the following holds: n o kukW k+2,p (Ω0 ) 6 c kLukW k,p (Ω) + kukLp (Ω) . (12.1) X

X

0

(b) For any domain Ω b Ω, nonnegative integer k and α ∈ (0, 1) there exists 2,α k,α a constant c such that for any u ∈ CX (Ω), if Lu ∈ CX (Ω) then u ∈ k+2,α 0 CX (Ω ) and the following holds: n o kukC k+2,α (Ω0 ) 6 c kLukC k,α (Ω) + kukL∞ (Ω) . (12.2) X

X

Now, these results have an amazing similarity with those which are known to hold for uniformly elliptic operators in nondivergence form, Eu =

n X

aij (x) uxi xj

i,j=1 n

where the matrix of coefficients {aij }i,j=1 is symmetric, uniformly positive definite in Ω and bounded: n X ν|ξ|2 6 aij (x)ξi ξj 6 ν −1 |ξ|2 (12.3) i,j=1 n

for a.e. x ∈ Ω, every ξ ∈ R , some constant ν > 0. More precisely, confining for the moment to the basic a priori estimate for k = 0, if the coefficients aij are H¨older continuous in Ω then a priori estimates of type n o kukC 2,α (Ω0 ) 6 c kEukC α (Ω) + kukL∞ (Ω) are the content of the classical Schauder theory (see [100, Chapter 6]) while if the coefficients aij are uniformly continuous in Ω then a priori estimates n o kukW 2,p (Ω0 ) 6 c kEukLp (Ω) + kukLp (Ω) are contained in the classical theory by Agmon-Douglis-Nirenberg [2] (see also [100, Chapter 8]). The same result has been proved later under the relaxed assumption of aij belonging to Sarason’s class of functions with vanishing mean oscillation (V M O), by Chiarenza-Frasca-Longo [59], [60]. This requirement amounts to asking a sort of uniform continuity in integral sense, expressed by the condition η (r) ≡ max ηaij (r) → 0 as r → 0, i,j

Nonvariational operators constructed with H¨ ormander vector fields

615

where, for any locally integrable function f , we set ! Z 1 f (y) − fB dy . ηf (r) = sup sup ρ |Bρ (x)| Bρ (x) x ρ 0 such that {x ∈ Ω : d (x, y) < 2ε for some y ∈ Ω1 } ⊂ Ω2 . 1

(12.6)

1

For a function u ∈ L (Ω2 ), and 0 < r < ε, set Z 1 |u(x) − uB | dx, ηu,Ω1 (r) = sup sup t6r x0 ∈Ω1 |B (x0 , t)| B(x0 ,t) 1 Note that the domain of the function η u,Ω1 depends on ε (and therefore on Ω2 ), while its value ηu,Ω1 (r) at a particular r where it is defined only depends on Ω1 and u.

616

H¨ ormander operators −1

where uB = |B (x0 , t)|

R B(x0 ,t)

u. We say that u ∈ BM O (Ω1 , Ω2 ) if

kukBM O(Ω1 ,Ω2 ) = sup ηu,Ω1 (r) < ∞. r6ε

We say that u ∈ V M O (Ω1 , Ω2 ) if u ∈ BM O (Ω1 , Ω2 ) and ηu,Ω1 (r) → 0 as r → 0. The function ηu,Ω1 will be called the V M O local modulus of u over Ω1 . Note that its behavior for r → 0 is independent of Ω2 . We say that a ∈ V M Oloc (Ω) if for every choice of Ω1 , Ω2 and ε > 0 as in (12.6), we have that a ∈ V M O (Ω1 , Ω2 ). More explicitly, this means that for any fixed Ω0 b Ω, the function Z 1 ηu,Ω0 (r) = sup sup |u(x) − uB | dx, t6r x0 ∈Ω0 |B (x0 , t)| B(x0 ,t) is finite for r small enough and vanishes for r → 0. Let us now state precisely the results that we will prove in this chapter. As usual, let X0 , X1 , . . . , Xq be a family of H¨ ormander vector fields in a bounded domain n Ω ⊂ R . The drift X0 , if present, will have weight 2 with respect to the other vector fields Xi , of weight one. We consider a symmetric matrix of real functions q {aij }i,j=1 defined in Ω. The main results of this chapter are those obtained when the drift is lacking: Theorem 12.2 (Interior regularity for operators without drift) Let X1 , . . . , Xq be a system of H¨ ormander vector fields in a bounded domain Ω ⊂ Rn and q let {aij }i,j=1 be a symmetric matrix of real functions defined in Ω, satisfying the boundedness and uniform positivity condition in Ω, expressed by q X ν|ξ|2 6 aij (x)ξi ξj 6 ν −1 |ξ|2 (12.7) i,j=1

for some constant ν > 0, every ξ ∈ Rq and a.e. x ∈ Ω. Let q X L= aij (x) Xi Xj .

(12.8)

i,j=1

Then (a) For every domain Ω0 b Ω, nonnegative integer k, p ∈ (1, ∞) , if aij ∈ k,∞ WX (Ω) or, for k = 0, aij ∈ V M Oloc (Ω) , there exists a constant c such 2,p k,p k+2,p that for any u ∈ WX (Ω), if Lu ∈ WX (Ω) then u ∈ WX (Ω0 ) and the following holds: n o kukW k+2,p (Ω0 ) 6 c kLukW k,p (Ω) + kukLp (Ω) . (12.9) X

X

0

(b) For every domain Ω b Ω, nonnegative integer k, α ∈ (0, 1), if aij ∈ k,α 2,α CX (Ω), there exists a constant c such that for any u ∈ CX (Ω), if Lu ∈ k,α k+2,α CX (Ω) , then u ∈ CX (Ω0 ) and the following holds: n o kukC k+2,α (Ω0 ) 6 c kLukC k,α (Ω) + kukL∞ (Ω) . X

X

(12.10)

Nonvariational operators constructed with H¨ ormander vector fields

617

q

The constant c in the a priori estimates depends on Ω, Ω0 , {Xi }i=1 , ν, k; morek,∞ over, in (a) it depends on p and the WX (Ω) norms of the coefficients aij (or, if α k = 0, the V M O moduli ηaij ,Ω0 ); in (b) it depends on α and the CX norms of the coefficients aij . The a priori estimates (12.9) and (12.10) will be proved respectively in section 12.4 and section 12.3. The regularization of the solution will be addressed in section 12.6. The assumptions on the coefficients aij made in point (a) for k > 1 are consistent 1,∞ with the V M O assumption made for k = 0 in view of the fact that WX (Ω) ⊂ V M Oloc (Ω), as we will prove in Proposition 12.6. If the operator L contains also a drift term X0 , then we will content of proving the basic interior estimate (k = 0), that is: Theorem 12.3 (Interior estimates for operators with drift) Let X0 , X1 , . . . , Xq be a system of H¨ ormander vector fields in a bounded domain Ω ⊂ Rn , let q {aij }i,j=1 be as in the previous theorem, and consider L=

q X

aij (x) Xi Xj + X0 .

(12.11)

i,j=1

(a) For every domain Ω0 b Ω and p ∈ (1, ∞), if aij ∈ V M Oloc (Ω) there exists 2,p a constant c such that for any u ∈ WX (Ω) o n kukW 2,p (Ω0 ) 6 c kLukLp (Ω) + kukLp (Ω) . X

α (Ω) there exists a (b) For every domain Ω0 b Ω and α ∈ (0, 1) , if aij ∈ CX 2,α constant c such that for any u ∈ CX (Ω) o n kukC 2,α (Ω0 ) 6 c kLukC α (Ω) + kukL∞ (Ω) . X

X

The constant c in the a priori estimates has the same dependence described in the previous theorem. Point (a) will be proved in section 12.4, point (b) section 12.3. Remark 12.4 (Operators with lower order terms) In both the above theorems, we could also add to the operator L lower order terms, that is consider q X L1 u = Lu + ci (x) Xi u + c0 (x) u. i=1

Then the same results hold, under the natural assumptions on the coefficients ci . For Theorem 12.2: k,∞ (a) ci ∈ WX (Ω) ;

k,α (b) ci ∈ CX (Ω) .

For Theorem 12.3: (a) ci ∈ L∞ (Ω) ;

α (b) ci ∈ CX (Ω) .

As in the previous chapter, the proofs of these more general results could be achieved by a routine modification of the proofs, and we omit the details.

618

H¨ ormander operators

Remark 12.5 (Operators with a0 (x) X0 ) In the drift case, we could also consider the family of operators written as q X L= aij (x) Xi Xj + a0 (x) X0 (12.12) i,j=1 α where the function a0 has the same regularity of the aij (V M Oloc (Ω) or CX (Ω), respectively), and satisfies the nondegeneracy and boundedness condition

ν 6 |a0 (x)| 6 ν −1 . The same results hold true in this case. In the H¨ older case, we can simply rewrite the equation q X aij (x) Xi Xj + a0 (x) X0 = f (x) i,j=1

as q X aij (x) f (x) Xi Xj + X0 = a (x) a 0 (x) i,j=1 0 a (x)

and note that the new coefficients aij0 (x) are still H¨ older continuous. So, in this case, actually the family of operators (12.12) coincides with (12.11). In the V M O case this procedure is no longer possible, but we could modify our proof to take into account this variable coefficients. This would generate longer and more involved formulas, but no real new problem. For the sake of simplicity, we will not give the details of this extension. 1,∞ We end this section proving the embedding WX (Ω) ⊂ V M Oloc (Ω) announced after the statement of Theorem 12.2.

Proposition 12.6 Let X1 , . . . , Xq be a system of H¨ ormander vector fields in a bounded domain Ω ⊂ Rn (without drift X0 ). For every Ω0 b Ω there exists a constant c > 0 depending on the vector fields and on the domains Ω and Ω0 , such that for every ball BX (x0 , 3r) ⊂ Ω0 we have Z 1 |f (x) − fBr | dx 6 cr kf kW 1,∞ (Ω) . X |Br | Br 1,∞ In particular WX (Ω) ⊂ V M Oloc (Ω). 1,∞ Proof. Let f ∈ WX (Ω). For a fixed Ω0 b Ω and ε > 0 small enough, let fε be the mollified of f , defined as in Lemma 2.8. Since fε is smooth, by Theorem 1.54 we can write, for any ball BX (x0 , 3r) ⊂ Ω0 and x ∈ Br ≡ BX (x0 , r), fε (x) − (fε ) = |fε (x) − fε (x)| 6 √q2r · sup |Xfε | Br

BX (x0 ,3r)

(for some point x ∈ Br ). Averaging over Br we also have Z 1 fε (x) − (fε ) dx 6 √q2r · sup |Xfε | Br |Br | Br BX (x0 ,3r)

Nonvariational operators constructed with H¨ ormander vector fields

619

Recall that, by Proposition 2.12, kfε kW 1,∞ (Ω0 ) 6 c kf kW 1,∞ (Ω) , hence X X Z 1 fε (x) − (fε ) dx 6 cr kf k 1,∞ . Br WX (Ω) |Br | Br However, the mollified fε converges to f in L1 (Ω) as ε → 0, hence we conclude Z 1 |f (x) − fBr | dx 6 cr kf kW 1,∞ (Ω) . X |Br | Br In particular this implies that for Ω0 b Ω and r small enough ηf,Ω0 (r) 6 cr kf kW 1,∞ (Ω) . X

Remark 12.7 The embedding result proved in the previous proposition, which will be enough for our purposes, is far from being sharp. Indeed the Poincar´e inequality for vector fields proved by Jerison (see [110]), gives 1,Q WX (Ω) ⊂ V M Oloc (Ω) ,

if |Br | > crQ for r > 0 small enough. This is the case if we are considering the space of lifted variables and Q is the corresponding homogeneous dimension. Actually, by Poincar´e’s inequality, for every Ω0 b Ω there exist c > 0 and λ > 1 such that for every x ∈ Ω0 , Br = Br (x) with r small enough, Z Z cr 1 |f (x) − fBr | dx 6 |Xf (x)| dx 6 c kXf kLQ (Bλr ) |Br | Br |Br | Bλr and 1 ηa (r) ≡ sup 0 |B r (x)| x∈Ω

Z Br (x)

|f (x) − fBr | dx 6 c sup kXf kLQ (Bλr (x)) ≡ ε (r) x∈Ω0

where, by the absolute continuity of Lebesgue integral, ε (r) → 0 for r → 0+ , under 1,Q the assumption a ∈ WX (Ω). 12.2

Operators of type λ and representation formulas

We now define various differential operators that we will handle in the following. Our main interest is to study the operator L=

q X

aij (x)Xi Xj + X0 ,

i,j=1

under the assumptions of Theorem 12.3. For any x ∈ Ω we can apply Theorem 10.6 ei which are (lifting theorem) to the vector fields Xi , obtaining new vector fields X free up
to weight s and satisfy H¨ ormander’s condition
of step s in a neighborhood e ξ, R of ξ = (x, 0) ∈ RN . For ξ = (x, t) ∈ B e ξ, R , set B e aij (x, t) = aij (x),

620

H¨ ormander operators

and let q X

e= L

ei X ej + X e0 e aij (ξ)X

(12.13)

i,j=1


e ξ, R . Next, we “freeze” the coefficients e be the lifted operator , defined in B aij of
e e e L at some point ξ0 ∈ B ξ, R (while we do not freeze the vector fields Xi ), and consider the frozen lifted operator : e0 = L

q X

ei X ej + X e0 . e aij (ξ0 )X

(12.14)

i,j=1

e 0 we will consider the approximating operator, defined on the homogeTo study L neous group G: L0 =

q X

e aij (ξ0 )Yi Yj + Y0

i,j=1

and its transpose: L∗0 =

q X

e aij (ξ0 )Yi Yj − Y0

i,j=1

where i } are the left invariant vector fields on the group G associated to {Xi } n {Yo e and Xi . Since the coefficients e aij (ξ0 ) are now constant, the operators L0 , L∗0 are left invariant and 2-homogeneous on the homogeneous group G. We can apply to these operators the theory developed in Chapter 6: by Proposition 6.7, they are homogeneous left invariant H¨ ormander operators on G, so that, by Theorem 6.18, they possess a global homogeneous fundamental solution. Moreover, L0 , L∗0 belong to the class Lν (see Definition 6.8), where ν is the constant appearing in (12.7), therefore the fundamental solution satisfies suitable uniform estimates, which depend on the coefficients e aij only through the number ν, but do not depend on
e ξ, R . The following theorem collects precisely the particular frozen point ξ0 ∈ B the properties of this fundamental solution that we will need. Theorem 12.8 (Homogeneous fundamental solution
on groups) With the e ξ, R the operator L0 has a above notation and assumptions, for every ξ0 ∈ B unique fundamental solution Γ (ξ
0 ; ·) such that: (a) Γ (ξ0 ; ·) ∈ C ∞ RN \ {0} ; (b) letting, from now on, Γi (ξ0 ; u) = Yi [Γ(ξ0 ; ·)] (u) and Γij (ξ0 ; u) = Yi Yj [Γ(ξ0 ; ·)] (u) we have that Γ (ξ0 ; ·) , Γi (ξ0 ; ·) , Γij (ξ0 ; ·) (for i, j = 1, . . . , q) and Γ0 (ξ0 ; ·) are homogeneous of degree (2 − Q), (1 − Q) , −Q, −Q respectively, and for

Nonvariational operators constructed with H¨ ormander vector fields

621

every u ∈ G satisfy, with some c = c(G, ν), |Γ (ξ0 ; u)| 6 |Γi (ξ0 ; u)| 6 |Γij (ξ0 ; u)| + |Γ0 (ξ0 ; u)| 6

c Q−2

;

Q−1

;

kuk c kuk c

Q

kuk

;

(c) for every test function f and every v ∈ RN , Z
f (v) = (L0 f ∗ Γ) (v) = Γ ξ0 ; u−1 ◦ v L0 f (u) du;

(12.15)

RN

moreover, for every i, j = 1, . . . , q, there exist constants αij (ξ0 ) such that Z
Γij ξ0 ; u−1 ◦ v L0 f (u)du + αij (ξ0 ) L0 f (v); Yi Yj f (v) = PV RN

(12.16)

actually, Z Yj Γ (ξ0 ; ·)

αij (ξ0 ) = kwk=1

N X

! bik νk

(w) dσ (w) ,

(12.17)

k=1

so that |αij (ξ0 )| 6 c (G, ν) . (d) for any differential operator D2 homogeneous of degree 2 on G (not necessarily translation invariant) and for every R > r > 0, we have Z Z 2 D Γ (ξ0 ; u) du = D2 Γ ( ξ0 ; u) dσ(u) = 0; r λ − Q); D0 and D00 are differential operators such that D0 Γ(ξ0 ; ·) and D00 Γ∗ (ξ0 ; ·) have m (weighted) derivatives with respect to the vector fields Yi (i = 0, 1, . . . , q). Moreover, the coefficients of the differential operators Di , Di0 for i = 0, 1, . . . , Hm possibly depend also on the variables ξ, η, in such a way that the joint dependence on (ξ, η, u) is smooth. Definition 12.10 (Frozen operators of type λ) We say that T (ξ0 ) is a frozen
e ξ, R if k(ξ0 ; ξ, η) is a frozen kernel of type λ and operator of type λ > 1 over B Z T (ξ0 ) f (ξ) = k(ξ0 ; ξ, η) f (η) dη e B

e ξ, R ). We say that T (ξ0 ) is a frozen operator of type 0 if k(ξ0 ; ξ, η) for f ∈ C0∞ (B is a frozen kernel of type 0 and Z T (ξ0 ) f (ξ) = PV k(ξ0 ; ξ, η) f (η) dη + α (ξ0 , ξ) f (ξ) ,


e B

Nonvariational operators constructed with H¨ ormander vector fields

where the principal value is defined by: Z Z PV k(ξ0 ; ξ, η) f (η) dη = lim ε→0

e B

623

k(ξ0 ; ξ, η) f (η) dη

kΘ(η,ξ)k>ε

for some homogeneous norm k·k on the group, and the function α has the form: Z α (ξ0 , ξ) = a (ξ) Dξ Γ (ξ0 ; u) ω (u) dσ (u) , kuk=1

e ξ, R ), ω is a smooth function depending on the coefficients of the where a ∈ C0∞ (B homogeneous vector fields Yi , Dξ is a homogeneous differential operator of degree 6 1 smoothly depending on ξ.


Remark 12.11 (Multiplicative part of operators of type 0) Note that, by (12.18), the multiplicative part α (ξ0 , ξ) of frozen operators of type 0 satisfies |α (ξ0 , ξ)| 6 c (G, ν, Xi ) . Proceeding exactly like in section 11.3 (and so relying on the results of section 11.2) we have the following:
e 0 ) Given a, b ∈ C ∞ (B e ξ, R ) such that Theorem 12.12 (Parametrix of L 0 ab = a, there exist Fij (ξ0 ), F0 (ξ0 ), Fij∗ (ξ0 ), F0∗ (ξ0 ) , frozen operators of type 1
e ξ, R ) such that: and P (ξ0 ), P ∗ (ξ0 ), frozen operators of type 2 (over B e ∗0 P ∗ (ξ0 ) + aI = L e0 + aI = P (ξ0 )L

q X

e aij (ξ0 ) Fij∗ (ξ0 ) + F0∗ (ξ0 ) ;

i,j=1 q X

e aij (ξ0 ) Fij (ξ0 ) + F0 (ξ0 )

(12.19)

(12.20)

i,j=1

where I denotes the identity. More precisely, Z b(η) f (η) dη P ∗ (ξ0 ) f (ξ) = a(ξ) Γ∗ (ξ0 ; Θ(η, ξ)) c (η) e B Z b (ξ) P (ξ0 )f (ξ) = a (η) Γ (ξ0 ; Θ(η, ξ)) f (η) dη c(ξ) Be where c is the function appearing in Theorem 10.6 point 3. em X el and use Theorem Starting with (12.20) we can now take second derivatives X 11.19 in order to exchange derivatives and integral operators. The presence of both the drift and the coefficients e ahk (ξ0 ) introduces a complication in the resulting identities, since, by Theorem 11.19, if F (ξ0 ) is a frozen operator of type λ > 1, then for i = 1, 2 . . . q there exist frozen operators Fik (ξ0 ) (k = 0, 1, . . . , q) of type λ, and frozen operators Pi (ξ0 ) of type λ + 1 such that ei F (ξ0 ) = X

q X k=1

ek + Fi0 (ξ0 ) + Pi (ξ0 )X e0 . Fik (ξ0 )X

624

H¨ ormander operators

However, we now have to rewrite   q X e e e e Pi (ξ0 )X0 = Pi (ξ0 ) L0 − e ahk (ξ0 ) Xh Xk h,k=1 q X

e0 − = Pi (ξ0 )L

ek e ahk (ξ0 ) Fih (ξ0 )X

h,k=1

with Fih (ξ0 ) frozen operators of type λ. A careful computation then gives: em X el u in terms of L e 0 u) Given a ∈ Theorem 12.13 (Representation of X

∞ e e ξ, R , for any m, l = 1, 2, . . . , q, there exist frozen operaC0 (B ξ, R ), ξ0 ∈ B

e ξ, R ) e tors over the ball B ξ, R , such that for any u ∈ C0∞ (B em X el (au) = Slm (ξ0 ) L e0 u + X

q X

0 ek u + Slm Slm,k (ξ0 ) X (ξ0 ) u

k=1

+

q X

( e aij (ξ0 )

i,j=1

q X

q X

ij ek u + Slm,k (ξ0 ) X

k=1

0ij ek u e ahk (ξ0 ) Slm,h (ξ0 ) X

(12.21)

h,k=1

) +

ij Flm

e 0 u + S ij (ξ0 ) u (ξ0 ) L lm

(as usual, operators denoted by S are of type 0 and operators denoted by F are of type em F (ξ0 ) 1). Moreover, each of the frozen operators of type 0 has the form S (ξ0 ) = X for some frozen operator F (ξ0 ) of type 1 and some m = 1, . . . , q. If the drift is lacking the previous formula simplifies as follows: em X el (au) = Slm (ξ0 ) L e0 u + X +

q X

q X

0 ek u + Slm Slm,k (ξ0 ) X (ξ0 ) u

k=1 q X

( e aij (ξ0 )

i,j=1

) ij Slm,k

ek u + (ξ0 ) X

ij Slm

(ξ0 ) u .

k=1

em X el u in terms of Lu e we need repreIn order to prove a priori estimates on X e e e e sentation formulas of Xm Xl u in terms of Lu and not L0 u. Writing: h i e 0 u (ξ) = Lu e (ξ) + L e 0 u (ξ) − Lu e (ξ) L e (ξ) + = Lu

q X

ei X ej u (ξ) [e aij (ξ0 ) − e aij (ξ)] X

i,j=1

and inserting this identity in (12.21) we get:

Nonvariational operators constructed with H¨ ormander vector fields

625

em X el u in terms of Lu) e Theorem 12.14 (Representation of X Under the same assumptions and with same notation of the previous theorem, ! q X em X el (au) = Slm (ξ0 ) Lu e + Slm (ξ0 ) ei X ej u X [e aij (ξ0 ) − e aij (·)] X i,j=1

+

+

q X

0 ek u + Slm Slm,k (ξ0 ) X (ξ0 ) u

k=1 q X

( e aij (ξ0 )

i,j=1

+

ij Flm

q X

ij Slm,k

(12.22)

ek u + (ξ0 ) X

k=1

e + (ξ0 ) Lu

q X

0ij ek u e ahk (ξ0 ) Slm,h (ξ0 ) X

h,k=1

q X

ij Flm

)  ij eh X ek u + S (ξ0 ) u . (ξ0 ) [e ahk (ξ0 ) − e ahk (·)] X lm 

h,k=1

If the drift is lacking the previous formula simplifies as follows: q X

em X el (au) = Slm (ξ0 ) Lu e + X

  ei X ej u Slm (ξ0 ) [e aij (ξ0 ) − e aij (·)] X

i,j=1

+

+

q X

0 ek u + Slm Slm,k (ξ0 ) X (ξ0 ) u

k=1 q X

( e aij (ξ0 )

i,j=1

q X

) ij e (ξ0 ) Xk u + Slm (ξ0 ) u .

ij Slm,k

k=1

Remark 12.15 The representation formulas of the above theorem have a cumbersome aspect, due to the presence of the coefficients e aij (ξ0 ) which appear several times as multiplicative factors. Anyway, if we agree to leave implicitly understood in the symbol of frozen operators the possible multiplication by the coefficients e aij , our formulas (12.21) and (12.22) assume the following more compact form: em X el (au) = Slm (ξ0 ) L e0 u + X

q X

ek u + S 0 (ξ0 ) u Sklm (ξ0 ) X lm

(12.23)

k=1

and, respectively, em X el (au) = Slm (ξ0 ) Lu e + Slm (ξ0 ) X

q X

! ei X ej u [e aij (ξ0 ) − e aij (·)] X

i,j=1

+

q X

Sklm

ek u + (ξ0 ) X

0 Slm

(12.24)

(ξ0 ) u.

k=1 α p In the proof of a priori estimates, when we will take CX e or L norms of both sides of these identities, the multiplicative factors e ahj will be simply bounded by α ∞ taking, respectively, the CX or the L norms of the e a hj ; hence leaving these factors e implicitly understood is harmless.

626

H¨ ormander operators

α e e The above theorem is well suited to the proof of CX e estimates for Xi Xj u, since in this case the coefficients e aij are assumed H¨older continuous, hence the quantity [e aij (ξ0 ) − e aij (·)] is uniformly small on small balls. In order to prove Lp estimate e e for Xi Xj u under the assumptions of V M O coefficients e aij we need to unfreeze the point ξ0 in the previous identities. This, however, requires a new notion of integral operator:

Definition 12.16 (Variable kernels and operators of type λ) We say that
e k(ξ, η) is a variable kernel of type λ over B ξ, R , for some nonnegative integer λ (we will use λ = 0, 1, 2), if k(ξ, η) is obtained by a frozen kernel k(ξ0 ; ξ, η) of type λ letting ξ0 = ξ. Explicitly, k(ξ, η) = k 0 (ξ, η) + k 00 (ξ, η) (H ) m X = ai (ξ)bi (η)Di Γ(ξ; ·) + a0 (ξ)b0 (η)D0 Γ(ξ; ·) (Θ(η, ξ)) i=1

+

(H m X

) a0i (ξ)b0i (η)Di0 Γ∗ (ξ; ·)

+

a00 (ξ)b00 (η)D00 Γ∗ (ξ; ·)

(Θ(η, ξ))

i=1

where the meaning of the symbols is the same as in
Definition 12.9. We say that T e ξ, R if k(ξ, η) is a variable kernel of is a variable operator of type λ > 1 over B type λ and Z T f (ξ) = k(ξ, η) f (η) dη e B

e ξ, R ). We say that T is a variable operator of type 0 if k(ξ, η) is for f ∈ C0∞ (B a variable kernel of type 0 and Z T f (ξ) = PV k(ξ, η) f (η) dη + α (ξ, ξ) f (ξ) ,


e B

where α (ξ0 , ξ) is the multiplicative part of a frozen operator of type 0. We will also need the following: Definition 12.17 (Commutator of multiplication and integral operator)

e e ξ, R ), If T is a variable operator of type λ over a ball B ξ, R , and a ∈ L∞ (B then we define the commutator operator [a, T ] as: [a, T ] f = aT f − T (af )
e ξ, R ). for any f ∈ C0∞ (B With this language and notation, let us start again with (12.22). If we rewrite it as a pointwise identity in ξ, and then let ξ0 = ξ we get the following:

Nonvariational operators constructed with H¨ ormander vector fields

627

em X el u by variable operators) Given Theorem 12.18
(Representation of X e ξ, R ), for any m, l = 1, 2, . . . , q, there exist variable operators over a ∈ C0∞ (B

e ξ, R , such that for any u ∈ C ∞ (B e ξ, R ) B 0 em X el (au) = Slm Lu e + X

q X

ei X ej u + [e aij , Slm ] X

i,j=1

+

q X

( e aij

i,j=1

q X

0 ek u + Slm Slm,k X u

k=1

ij ek u + Slm,k X

k=1

q X

q X

0ij ek u + F ij Lu e e ahk Slm,h X lm

(12.25)

h,k=1

) q h i X ij ij e e e ahk , Flm Xh Xk u + Slm u . + h,k=1

Where the operators denoted by S are of type 0, the operators denoted by F are of e (which are no longer frozen at type 1 and e aij are the coefficients of the operator L ξ0 ). Remark 12.19 Leaving understood in the symbol of variable operator the possible multiplication by the coefficients e aij we can write the above representation formula (see Remark 12.15) as em X el (au) = Slm Lu e + X

q X i,j=1

12.3

ei X ej u + [e aij , Slm ] X

q X

0 ek u + Slm Slm,k X u.

k=1

H¨ older estimates

Starting with the representation formula contained in Theorem 12.14 we can now substantially repeat the proof seen in section 11.5.4 (based on the results of sec2,α tion 11.5.3) and prove CX estimates for nonvariational operators structured on H¨ ormander vector fields. The first step in the proof of H¨ older estimates is contained in the following:
e ξ, R be as before. There exist r < R and c > 0 such that Theorem 12.20 Let B
2,α e for every u ∈ CX,0 e (B ξ, r ), ) ( q



X

e e

e

e 6 c Lf α + + kf kC α (Be(ξ,r))

Xk Xh u α e

Xl f α e e (ξ,r )) f C f(B (ξ,r )) C f(B C f(B (ξ,r )) X X X X l=1 n o ei , α, ν, and ke where c and r depend on R, X aij kC α (Be(ξ,R)) .

e ξ, R ) such that a = 1 in B e ξ, R/2 and write Proof. We fix a ∈ C0∞ (B the representation formula (12.22) (or its simplified counterpart (12.24)) for u ∈
2,α e CX,0 ( B ξ, R ) and ξ0 = ξ. Note that, by the arguments seen in Chapter 11 (see the e
e ξ, R/2 ) proof of Proposition 11.49) this identity holds not only for every u ∈ C ∞ (B 0

628

H¨ ormander operators


2,α e α but for any u ∈ CX,0 e norms of both sides, exploiting e (B ξ, R/2 ). Then, taking CX α the continuity of frozen operators of type 0 (or 1) on CX e , the compact support, and using (11.77) we get: ( q



X 

e



e e ei X e j u 6 c Lu + aij (ξ) − e aij (·) X

e

Xm Xl u

Cα f

Cα f

X

X

+

q X

X

)

e

Xk u

k=1

Cα f

i,j=1

Cα f

(12.26)

+ kukC α

f X

X


2,α e This holds for any u ∈ CX,0 e (B ξ, R/2 ), with a constant possibly depending on
2,α e R. We will apply it to functions u ∈ CX,0 e (B ξ, r ) for r < R/2 small enough. To ei X ej u in the right-hand side of the last inequality, we handle the term involving X
  2,α e e e now exploit the fact that, for u ∈ CX,0 aij (ξ) − e aij (·) e (B ξ, r ), both Xi Xj u and e
e ξ, r ; then (11.78) implies vanish at a point of B   e e α ei X ej u aij (·) X 6 2r |e a | , X X u aij (ξ) − e e α e α ij i j e C f(B (ξ,r )) e ξ,r )) Cα B C f(B (ξ,r )) X f( ( X X while obviously





ei X ej f aij (ξ) − e aij (·) X

e

e (ξ,r )) L∞ (B



e e 6 rα |e aij |C α (Be(ξ,r)) X i Xj f

e (ξ,r )) L∞ (B

.

This allows to get q

X 

 ei X ej u aij (ξ) − e aij (·) X

e

e ξ,R/2)) Cα B f( (

i,j=1

X

6 2rα

q X i,j=1

|e aij |C α (Be(ξ,r)) f X



e e

Xi Xj f

e ξ,r )) Cα B f( ( X

which inserted in (12.26) gives, for r small enough,

e e

Xk Xh u α e C f(B (ξ,r )) X ) ( q



X

e

e 6 c Lf α + + kf kC α (Be(ξ,r))

Xl f α e e (ξ,r )) f C f(B C f(B (ξ,r )) X X X l=1 (this is the classical “Korn’s trick” used to prove Schauder estimates for uniformly elliptic operators with H¨ older continuous coefficients). This theorem is the analog of Theorem 11.34, which in Chapter 11 has been the 2,α starting point to prove CX -estimates. The subsequent steps are exactly the same that we have carried out in section 11.5.4:

Nonvariational operators constructed with H¨ ormander vector fields

629

(1) Using cutoff functions and interpolation inequalities we prove (see Theorem 11.57) that there exist r0 < r and c, β > 0 such that, for every 2,α e u ∈ CX e (B(ξ, r0 )), 0 < t < s < r0 , ) (

c

e + kukL∞ (Beρ (ξ,s)) , kukC 2,α (Beρ (ξ,t)) 6

Lu α e f (s − t)β C f(Bρ (ξ,s)) X X ei }q , α; β depends on {X ei }q and α. where r0 , c depend on R, {X i=1 i=0 (2) Applying the relation between H¨ older spaces in the lifted and original space (Proposition 11.54) and a covering argument, we prove (see Theorem 11.58) that for any domain Ω0 b Ω and α ∈ (0, 1) there exists a constant c such 2,α that for any u ∈ CX (Ω), n o kukC 2,α (Ω0 ) 6 c kLukC α (Ω) + kukL∞ (Ω) . X

X

α This completes the proof of the CX -part of Theorem 12.3. We are now interested in establishing higher order estimates for operators without drift.
We start recalling an identity proved in Theorem 11.28 for u ∈ e ξ, R ) that can be easily to u ∈ C 2,α (B(ξ, e R)). C0∞ (B e X,0

e Theorem 12.21 (Higher order representation formulas) Assume that L
∞ e e does not contain the drift term X0 . Given a ∈ C0 (B ξ, R ), for any multiindex I
e ξ, R ) we have with |I| > 2 and u ∈ C0∞ (B X X 0 eI (au) = eJ L e 0 u + SI L e0 u + eK u + SI0 u X SIJ X SIK X (12.27) |J|6|I|−2

|K|6|I|−1

ej F for some j = 1, . . . , q and where S denotes operators of type 0 of the form S = X some operator F of type 1. In the above identity the first sum is
empty if |I| = 2. 2,α e The same representation formula still holds for u ∈ CX,0 e (B ξ, R ). We can now come to the
e ξ, R ) such that a ≡ 1 in Proof of Theorem 12.2 (b). We fix a ∈ C0∞ (B

e ξ, R/2 and write (12.27) for u ∈ C k+2,α (B e ξ, R/2 ) and ξ0 = ξ. Inserting in B e X,0 this identity the relation q X   e e ei X ej u, L0 u = Lu + e aij (ξ) − e aij (·) X i,j=1


e ξ, R/2 : we get, in B eI u = X

X

eJ Lu e + SIJ X

X

eJ SIJ X

   ei X ej u e aij (ξ) − e aij (·) X

i,j=1 |J|6|I|−2

|J|6|I|−2

e + + SI Lu

q X

q X i,j=1

   ei X ej u + SI e aij (ξ) − e aij (·) X

X |K|6|I|−1

0 eK u + S 0 u. SIK X I

630

H¨ ormander operators

α Next, we take CX e norms of both sides of the identity and exploit the continuity of α operators of type 0 on CX e , getting, for |I| = k, 

 X

e

e e

XI u C α (Be(ξ,R)) 6 c

XJ Lu α e f  C (B (ξ,R)) X |J|6k−2

+

q X

f X

  X 

e  ei X ej u aij (ξ) − e aij (·) X

XJ e

e ξ,R)) Cα B f( (

i,j=1 |J|6k−2

X



e + Lu

e ξ,R)) B Cα f( (

+





ei X ej u aij (ξ) − e aij (·) X

e

e ξ,R)) B Cα f( (

i,j=1

X

+

q X

X



e

XK u

X

e ξ,R)) Cα B f( (

|K|6k−1

+ kukC α (Be(ξ,R)) f X

X

Observe that,

 

ei X ej u aij (·) X aij (ξ) − e

e

e ξ,R)) Cα B f( (

 

.

(12.28)





e e X u 6 2 ke aij kC α (Be(ξ,R)) X i j

e ξ,R)) Cα B f( (

f X

X

X

6 c kukC k−1,α (Be(ξ,R)) , f X

since k > 2. Also,   X 

e  ei X ej u aij (ξ) − e aij (·) X

XJ e

e ξ,R)) B Cα f( ( X

  X   eJ X ei X ej u aij (·) X 6 aij (ξ) − e

α e

e C f(B (ξ,R)) X |J|6k−2

    X

e

eJ X ei X ej u . + aij X

XJ1 e 2 e ξ,R)) Cα B f( ( X |J |+|J |6k−2

|J|6k−2

1

(12.29)

2

|J1 |>1

Reasoning as in the proof of Theorem 12.20,   X 

 eJ X ei X ej u aij (ξ) − e aij (·) X

e

e ξ,R)) Cα B f( (

|J|6k−2

X

α

6 2R |e aij |C α (Be(ξ,R)) f X

e ξ,R)) Cα B f( (

|J|6k−2

6 2Rα |e aij |C α (Be(ξ,R)) f X

  X

e ei X ej u

XJ X

(12.30)

X

k X

h

D u α e . C (B (ξ,R)) f X

h=2

On the other hand,

   

e eJ X ei X ej u aij X

XJ1 e

2

X

e ξ,R)) Cα B f( (

|J1 |+|J2 |6k−2 |J1 |>1

6 c ke aij kC k−2,α (Be(ξ,R)) f X

X

k−1 X h=2

h

D u α e . C (B (ξ,R)) f X

(12.31)

Nonvariational operators constructed with H¨ ormander vector fields

631

Inserting (12.30) and (12.31) in (12.29) and then in (12.28) we get, summing for |I| = k, (



k

e

D u α e 6 c + kukC k−1,α (Be(ξ,R))

Lu k−2,α e C f(B (ξ,R)) f Cf X (B (ξ,R)) X X ) q k X X

h α

D u α e + 2R |e aij |C α (Be(ξ,R)) C f(B (ξ,R)) f X X i,j=1 h=2
2,α e and then for u ∈ CX,0 e (B ξ, r ) with r < R/2 small enough, ) (



e + kukC k−1,α (Be(ξ,r)) , kukC k,α (Be(ξ,r)) 6 c Lu k−2,α f f Cf X X (Be(ξ,r)) X which is (11.94). Starting from this we can proceed by the same reasoning used in Chapter 11, proof of Theorem 11.60 and get a priori estimates (12.10), with the constants also depending on ke aij kC k−2,α (Be(ξ,R)) , that is on kaij kC k−2,α (Ω) . Hence X

f X

k+2,α Theorem 12.2 (b) is proved, assuming u ∈ CX,loc (Ω). The regularization result contained in Theorem 12.2 (b) will be proved in section 12.6.

12.4

Lp estimates

Our aim is now to prove Theorem 12.2 (a) and Theorem 12.3 (a), that is a priori Lp estimates. As in the previous chapter, we first prove local estimates for smooth functions with small support, in the space of lifted variables:
e ξ, R be as before, and p ∈ (1, ∞). There exist positive Theorem 12.22 Let B ei }q , p, R such that for every u ∈ constants r0
< R and c, depending on {X i=0 ∞ e C0 (B ξ, r0 ),  

2

e

D u p e 6 c Lu p + kDukLp (Be(ξ,r0 )) + kukLp (Be(ξ,r0 )) . L (B (ξ,r0 )) e (ξ,r0 )) L (B q

The constants depend on {Xi }i=1 , ν, p and the V M O moduli of aij , ηaij ,Ω0 . To present the
general strategy of the proof recall that by Theorem
12.18 if e ξ, R ) e ξ, R ) is a fixed cutoff function, then for any u ∈ C ∞ (B a ∈ C0∞ (B 0 q q X X 0 em X el (au) = Slm Lu e + ei X ej u + ek u + Slm X [e aij , Slm ] X Slm,k X u i,j=1

+

q X

( e aij

i,j=1

q X k=1

ij ek u + Slm,k X

k=1 q X

0ij ek u + F ij Lu e e ahk Slm,h X lm

(12.32)

h,k=1

) q h i X ij ij eh X ek u + S u + e ahk , Flm X lm h,k=1

where S... , F... are variable operators of type 0 and 1, respectively. Assume the following result, that we will prove in the following sections:

632

H¨ ormander operators

Theorem 12.23 (Lp continuity of variable operators) Let T be a variable
e ξ, R , and let p ∈ (1, ∞). Then: operator of type 0 or 1 over the ball B  q ei (i) there exists c > 0, depending on p, R, X , and ν, such that i=0 kT ukLp (Be(ξ,r)) 6 c kukLp (Be(ξ,r))
e ξ, r ) and r 6 R; for every u ∈ Lp (B (ii) for every a ∈ V M O
X,loc (Ω) , any ε > 0, there exists r 6 R such that for e ξ, r ), every u ∈ Lp (B k[e a, T ] ukLp (Be(ξ,r)) 6 ε kukLp (Be(ξ,r))  q ei where e a (x, h) = a (x) . The number r depends on p, R, X , ν, ηa,Ω0 , i=0 and ε (see Definition 12.1 for the definition of V M OX,loc (Ω) and ηa,Ω0 ). With this in hand, we can give the
e ξ, r ) norms of both sides of (12.32) Proof of Theorem 12.22. Taking Lp (B (with r 6 R/2
to be chosen later) and
applying the previous theorem, (i), for a = 1 e ξ, R/2 and u ∈ C ∞ (B e ξ, r ) we get, for some constant also depending on in B 0 ke ahk kL∞ ,:  q 

X

e

e e ei X ej u Lu 6 c + aij , Slm ] X X X u

p e

[e

p e

m l p e  L (B (ξ,r )) L (B (ξ,r )) L (B (ξ,r )) i,j=1 q

X

e +

Xk u k=1

e (ξ,r )) Lp (B

+ kukLp (Be(ξ,r)) +

q q

h

i X X

ij eh X ek u + ahk , Flm X

e

i,j=1 h,k=1

 

e (ξ,r ))  Lp (B

.

Applying also point (ii) of the previous theorem we
get that for any ε > 0 there e ξ, r0 ) then exists r0 small enough such that if u ∈ C0∞ (B 

2

e

D u p e 6 c + kDukLp (Be(ξ,r0 ))

Lu p e L (B (ξ,r0 )) L (B (ξ,r0 )) o

+ kukLp (Be(ξ,r)) + ε D2 u Lp (Be(ξ,r0 )) which for ε and r0 small enough gives  

2

e

D u p e 6 c Lu + kDuk + kuk

e (ξ,r0 )) e (ξ,r0 )) , Lp (B Lp (B L (B (ξ,r0 )) e (ξ,r0 )) Lp (B which is the assertion. Analogously to what we have seen in the previous section for H¨older estimates, the subsequent steps are exactly the same that we have carried out in Chapter 11, section 11.5.2, relying on the results about Sobolev spaces proved in section 11.5.1:

Nonvariational operators constructed with H¨ ormander vector fields

633

(1) Using cutoff functions and interpolation inequalities we prove that (see Theorem 11.42) there exists r1 < r0
and for any r 6 r1 there exists c > 0 2,p e such that for any u ∈ WX e (Bρ ξ, r ) we have  

e + kukLp (Beρ (ξ,r)) . kukW 2,p (Beρ (ξ,r/2)) 6 c Lu eρ (ξ,r )) f Lp (B X ei }q , p, r0 , ν and ηa ,Ω0 . The constants c, r1 depend on {X ij i=0 2,p (2) Applying the relation between Lp and WX -spaces in the lifted and original space (Theorem 11.41) and a covering argument, we prove that (see Theorem 11.43) for any domain Ω0 b Ω and p ∈ (1, ∞) there exists a constant c 2,p such that for any u ∈ WX (Ω), n o kukW 2,p (Ω0 ) 6 c kLukLp (Ω) + kukLp (Ω) . X

This completes the proof of the Lp -part of Theorem 12.3. Next, we come to the Proof Theorem 12.2 (a). The argument is similar to the one followed for the proof of higher order estimates in H¨ older spaces (Theorem 12.2 (b)) but with the
e ξ, R ) unfreezing procedure used in the proof of Theorem 12.18. We fix a ∈ C0∞ (B
e ξ, R/2 , for any multiindex I with |I| > 2 and u ∈ such that a = 1 in B
∞ e C0 (B ξ, r ), with r 6 R/2 to be chosen later, we write formula (12.27) X X 0 eI u (ξ) = eJ L e 0 u (ξ) + SI L e 0 u (ξ) + eK u (ξ) + S 0 u (ξ) . X SIJ X SIK X I |J|6|I|−2

|K|6|I|−1

Inserting in this identity the relation e 0 u = Lu e + L

q X

ei X ej u, [e aij (ξ0 ) − e aij (·)] X

i,j=1




e ξ, r : we get, in B eI u = X

X

eJ Lu e + SIJ X

X

  eJ [e ei X ej u SIJ X aij (ξ0 ) − e aij (·)] X

i,j=1 |J|6|I|−2

|J|6|I|−2 q X

e + + SI Lu

q X

  ei X ej u + SI [e aij (ξ0 ) − e aij (·)] X

i,j=1

=

X

0 eK u + SI0 u SIK X

|K|6|I|−1

eJ Lu e + SIJ X

q X

X

i,j=1 |J|6|I|−2

|J|6|I|−2

X

 SIJ



X

 eJ e eJ X ei X ej u −X a X 1 ij 2

|J1 |+|J2 |=|J| |J1 |>1

 e e e + [e aij (ξ0 ) − e aij (·)] XJ Xi Xj u e + + SI Lu

q X i,j=1

  ei X ej u + SI [e aij (ξ0 ) − e aij (·)] X

X |K|6|I|−1

0 eK u + S 0 u. SIK X I

634

H¨ ormander operators

Letting finally ξ0 = ξ we get q X X eI u = eJ Lu e − X SIJ X q X

X

SIJ



  eJ e eJ X ei X ej u X a X ij 1 2

i,j=1 |J|6|I|−2 |J1 |+|J2 |=|J| |J1 |>1

|J|6|I|−2

+

X

X

eJ X ei X ej u [e aij , SIJ ] X

i,j=1 |J|6|I|−2

e + + SI Lu

q X

X

ei X ej u + [e aij , SI ] X

i,j=1

0 eK u + SI0 u SIK X

|K|6|I|−1

where all the S... are variable operators of type 0. Next, we take Lp -norms of both sides of the identity and exploit Theorem 12.23 (i) getting, for |I| = k, 



e

e 6 c Lu

XI u p e

k−2,p e  L (B (ξ,R)) Wf (B (ξ,R)) X +

q X



e ke aij kW k−2,∞ (Be(ξ,R)) kukW k−1,p (Be(ξ,R)) + Lu

f X

i,j=1

+ kukW k−1,p (Be(ξ,R)) +

e (ξ,R)) Lp (B

f X

q X



eJ X ei X ej u aij , SIJ ] X

[e

X

X

e (ξ,R)) Lp (B

i,j=1 |J|6|I|−2 q

X

ei X ej u + aij , SI ] X

[ e

i,j=1

 

e (ξ,R))  Lp (B

.

Then, for any ε > 0 fixed, by Theorem 12.23 (ii) there exists r 6 R/2 depending
e ξ, r ), on the V M O moduli of e aij such that, for u ∈ C0∞ (B (



k

e

D u p e + kukW k−1,p (Be(ξ,r)) 6 c Lu

k−2,p e L (B (ξ,r )) f Wf X (B (ξ,r)) X )

2

+ ε kukW k,p (Be(ξ,r)) + ε D u Lp (Be(ξ,r)) f X with c also depending on ke aij kW k−2,∞ (Be(ξ,R)) . Choosing ε and r small enough we X conclude ) (



e kukW k,p (Be(ξ,r)) 6 c Lu k−2,p + kukW k−1,p (Be(ξ,r)) . X f Wf X (Be(ξ,r)) X This is the analog of Proposition 11.46 proved in Chapter 11 for operators sum of squares of H¨ ormander vector fields. Starting with this, one can now repeat exactly the subsequent steps of the proof of Theorem 11.45 and, with the use of cutoff k,p functions (Corollary 11.37), approximation of WX e -functions by smooth functions (Corollary 2.10), the relation between lifted and unlifted Sobolev spaces (Theorem 11.41) and a covering argument, prove a priori estimates (12.9), assuming u ∈ k+2,p WX,loc (Ω). The regularity result stated in Theorem 12.2 (a) will be proved in section 12.6.

Nonvariational operators constructed with H¨ ormander vector fields

12.5

635

Lp continuity of variable operators of type 0 and their commutators

To complete the proof of Theorem 12.2 (a) we are left to prove Theorem 12.23, that is the Lp continuity of variable operators of type 0, and the fact that the commutator of such an operator with the multiplication for a V M O ∩ L∞ function has small operator norm if we restrict to small balls. Here two main issues are involved. The first is that of reducing the study of variable operators of type 0 to that of frozen operators of type 0, which are completely analogous to the operators of type 0 that we have studied in Chapter 11, section 11.4. This involves a classical procedure originally designed to expand singular integral kernels of the kind k (x, x − y) (which are not of convolution type) in series of spherical harmonics, so reducing to the study of a series of singular kernels of convolution type. Although the technique in itself is completely classical, for the sake of completeness we will present this procedure in detail. However, to make more readable the exposition, in section 12.5.1 we will just state the required results, postponing their proofs to the last section in this chapter. We stress that the possibility of implementing this technique it in our context depends on the existence of uniform estimates (with respect to ξ0 ) on the derivatives of the fundamental solution Γ (ξ0 ; ·) , which we have proved in Chapter 6 and recalled in Theorem 12.8 (f). The second issue is that of proving the required commutator estimate for a variable operator of type 0. After reducing this problem to the analogous one for frozen operators of type 0 (by means of the aforementioned expansion in spherical harmonics), the result is proved by an abstract theorem from real analysis, which will be discussed (without proof) in section 12.5.2. In section 12.5.3 both these tools will be applied to the proof of Theorem 12.23. 12.5.1

Spherical harmonics expansions

We are now going to introduce a few basic facts of the classical theory of spherical harmonics, that is the restriction to the unit sphere of homogeneous harmonic polynomials. Let us fix some notation. We will denote with SN the unit sphere  x ∈ R
N : |x| = 1 and with dS the surface measure on SN . We will use the space L2 SN of square integrable (complex valued) functions equipped with the scalar product Z hf, giL2 = f (θ) g (θ)dS (θ) . SN

Let Pk be the space of homogeneous polynomials of degree k in RN and let Hk = {P ∈ Pk : ∆P = 0} be the subspace of harmonic homogeneous polynomials. The space of spherical harmonics of degree k is the restriction Hk to the unit sphere  Hk = P|SN : P ∈ Hk .

636

H¨ ormander operators

Note that Hk is
finite dimensional and set dk = dim Hk . If we regard Hk as a subspace L2 SN , we can define an orthonormal basis of Hk , n o (k) Yj . j=1,...,dk
(k) Even though the functions in L2 SN are complex valued, the functions Yj can be chosen real valued. (k) All the functions Yj (for j = 1, . . . , dk and k = 0, 1, 2, 3 . . .) are spherical harmonics. The next theorem collects for future reference all the results that will be useful in the following: Theorem 12.24 (Properties of spherical harmonics) The spherical harmonics n o (k) Yj j=1,...,dk k=0,1...,+∞


form a complete orthonormal system in L2 SN . Moreover, there exists a constant cN , only depending on the dimension N , such that (i) the dimension dk satisfies the bound: dk 6 cN k N −2 for every k = 1, 2, 3 . . . (ii) the spherical harmonics and their derivatives satisfy the following bounds: for k = 1, 2, 3 . . . , N

kYk kL∞ (SN ) 6 cN k 2 −1 N

k∇Yk kL∞ (SN ) 6 cN k 2 . (iii) expanding in series of spherical harmonics a smooth function f , the Fourier coefficients Z (k) b fj,k = f (θ) Yj (θ) dS (θ) SN

(recall that

(k) Yj

are real valued, so the complex conjugate is not necessary in the definition of fbj,k ) satisfy the following bounds: for every m = 1, 2, 3 . . . there exists cN,m > 0 such that for k > 1 and j = 1, . . . , dk

b cN,m fj,k 6 2m D2m f L2 (SN ) . k
(iv) If f ∈ C ∞ SN then f (θ) =

dk +∞ X X

(k) fbj,k Yj (θ) for every θ ∈ SN

k=0 j=1

and the series converges uniformly in the sense that

m dk

X X

bj,k Y (k) − f lim f = 0. j

m→+∞

k=0 j=1

∞ N L

(S )

The above theorem will be proved in section 12.7. The completeness of the system of spherical harmonic will be proved in Theorem 12.40; point (i) will be

Nonvariational operators constructed with H¨ ormander vector fields

637

proved in (12.65); (ii) is contained in Lemma 12.43 (5) and Proposition 12.45; (iii) is Proposition 12.41 and (iv) is Proposition 12.44. 12.5.2

The commutator theorem

Although we will not prove the main result that we will present here, we want to give to the reader at least a general picture of the situation. Let us consider a singular integral operator on RN Z T f (x) = PV k (x, y) f (y) dy. RN

Since we are only giving a first overview of the subject, for the moment we do not need to be more precise than this. Consider the commutator of T with the multiplication operator by a function a ∈ L∞ : [a, T ] f (x) = a (x) T f (x) − T (af ) (x) Z = PV k (x, y) [a (x) − a (y)] f (y) dy.

(12.33)

If we already know that T is bounded on Lp with norm kT kp−p , then we can write k[a, T ] f kp 6 kaT f kp + kT (af )kp 6 2 kT kp−p kak∞ kf kp .

(12.34)

In other words, the commutator [a, T ] is obviously a bounded operator on Lp . However, this estimate is very rough, as one can see letting a (x) ≡ 1. In this case, the estimate (12.34) gives k[a, T ] f kp 6 2 kT kp−p kf kp while from (12.33) we read that the operator [a, T ] is identically zero! More generally, the Lp − Lp norm of the commutator [a, T ] should be small whenever the function a has small oscillations, in some sense (but not necessarily small absolute size) since this means that the kernel of the commutator operator, k (x, y) [a (x) − a (y)] , is small, in some sense. We are interested in the case of a
nonsmooth function a having, however, bounded mean oscillation: a ∈ BM O RN , that is Z 1 f (y) − fB(x,r) dy < ∞. kf k∗ ≡ sup sup x∈RN r>0 |B (x, r)| B(x,r) A famous result due to Coifman-Rochberg-Weiss [69] states that if T is a classical Calder´ on-Zygmund singular integral in RN , that is its kernel is k (x, y) = K (x − y) with K homogeneous of degree −N, smooth outside the
origin, with vanishing integral on the surface of the unit ball, and a ∈ BM O RN , then k[a, T ] f kLp (RN ) 6 cp,T kak∗ kf kLp (RN )

(12.35)

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H¨ ormander operators

for every p ∈ (1, ∞). So, the operator norm of the commutator is controlled by the BM O seminorm of a, which is small as soon as a has small oscillations. Note that for a general a ∈ BM O (in particular, possibly unbounded), the mere convergence of the integral defining [a, T ] f is not trivial. However,
it can be proved. For our purposes it is not restrictive to assume a ∈ L∞ RN (which makes trivial the existence of the integral); the point is that we want to bound the operator norm of the commutator with kak∗ and not with kakL∞ (RN ) . The first application of this theorem to a priori estimates for PDEs was given by Chiarenza-Frasca-Longo [59], [60], dealing with nonvariational uniformly elliptic operators with V M O coefficients. This application relies on the following consequence of (12.35):
if a ∈ V M O RN , for every ε > 0 there exists r > 0 such that for f ∈ C0∞ (B (x, r)) we have k[a, T ] f kLp (B(x,r)) 6 ε kf kLp (B(x,r)) .

(12.36)

Let us discuss how the last statement can be derived from (12.35). This is not a completely trivial fact, since the estimate (12.35) is global in nature, while (12.36) is local. However, one can exploit the fact that, differently from what happens for any BM O function, a V M O function can be approximated by smooth functions in BM O seminorm. Namely, it can be proved (see [145, Theorem 1, Lemma 1]) that for any ε > 0, there exists a uniformly continuous function aε such that kaε − ak∗ < ε and ηaε (r) 6 ηa (r) , for r small enough. Also, we note that for x ∈ B (x, r) and f ∈ C0∞ (B (x, r)) the value of Z [a, T ] f (x) = PV k (x, y) [a (x) − a (y)] f (y) dy only depends on the values of a (y) for y ∈ B (x, r). This implies that if a is another function which coincides with a inside B (x, r), then [a, T ] f (x) = [a, T ] f (x) for x ∈ B (x, r) . Then, for a fixed ε > 0 and aε as above, let us modify the function aε outside the ball B (x, r) just putting it constant along every ray exiting from x, that is, we let   ( y−x r for y ∈ / B (x, r) aε x + |y−x| aε (y) = aε (y) for y ∈ B (x, r) . We can write: k[a, T ] f kLp (B(x,r)) 6 k[a − aε , T ] f kLp (B(x,r)) + k[aε , T ] f kLp (B(x,r)) where by (12.35) k[a − aε , T ] f kLp (B(x,r)) 6 c kaε − ak∗ kf kLp (B(x,r)) 6 cε kf kLp (B(x,r))

Nonvariational operators constructed with H¨ ormander vector fields

639

while k[aε , T ] f kLp (B(x,r)) = k[aε , T ] f kLp (B(x,r)) 6 c kaε k∗ kf kLp (B(x,r)) By definition of aε , it easy to see that the seminorm kaε k∗ is controlled by ηa (r), the V M O modulus of the function a at r, so that for r small enough we can also write k[aε , T ] f kLp (B(x,r)) 6 ε kf kLp (B(x,r)) and k[a, T ] f kLp (B(x,r)) 6 cε kf kLp (B(x,r)) . Note that this technique requires that the function a (which in the application to PDEs represents the coefficients aij of the equation) is originally defined in the whole space; then we approximate globally a with a continuous function that we will later modify outside the ball B (x, r). So, a drawback of this technique is that one needs either to assume that the V M O coefficients of the equation are defined on the whole space (which is a bit unnatural if one is studying the equation on a bounded domain), or one has to prove an extension theorem for V M O functions, from a domain to the whole space. A generalization of Coifman-Rochberg-Weiss theorem to spaces of homogeneous type (quasimetric doubling spaces) has been proved by Bramanti-Cerutti [34] and this extension was firstly applied to nonvariational operators structured on H¨ ormander vector fields by Bramanti-Brandolini [26], following the path of Chiarenza-Frasca-Longo. Another version of the commutator theorem, in locally homogeneous spaces (in particular, locally doubling metric spaces) has been proved by Bramanti-Zhu [40], relying on the result on homogeneous spaces proved in [34]. Namely, the following holds (see [40, Thm. 7.1]): Theorem 12.25 (Commutators of local singular integrals) Let Ω1 b Ω2 b Ω and, for some x ∈ Ω1 and B (x, R) ⊂ Ω2 , let K (x, y) and K ∗ (x, y) = K (y, x) satisfy the assumptions of Theorem 7.35 in the ball B (x, R), let e (x, y) = a0 (x) K (x, y) b0 (y) K where a0 , b0 are cutoff functions as in Theorem 7.35, and let Z e T f (x) = PV K(x, y)f (y) dµ(y), B(x,R)

so that, by the previous theory, for every p ∈ (1, ∞) kT f kLp (B(x,R)) 6 cT,p kf kLp (B(x,R)) . For a ∈ V M O (Ω1 , Ω2 ) ∩ L∞ (Ω2 ) (see Definition 12.1), let Ca f (x) = T (af ) (x) − a (x) T f (x) , then for any p ∈ (1, ∞) , any ε > 0 there exists r > 0, such that for any f ∈ Lp (B (x, r)) we have kCa f kLp (B(x,r)) 6 εcT kf kLp (B(x,r)) .

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H¨ ormander operators

The constant r depends on p, Ω1 , Ω2 and the V M O (Ω1 , Ω2 ) modulus of a, but not on T . The above result allows to apply the technique of Chiarenza-Frasca-Longo without having to assume the function a globally defined: it is a result directly established in a local form. 12.5.3

Proof of Lp continuity of variable operators and their commutators

In this section we will prove the two assertions contained in Theorem 12.23, regarding a variable operator T of type 0 and its commutator [a, T ] with a function a ∈ V M O ∩ L∞ . To begin with, let us study the relation between the V M O modulus of the original function a (x) defined in Ω and the V M O modulus of its lifted version e a (ξ). We have the following Proposition 12.26 Let a ∈ V M
Oloc (Ω) and let Ω0 b Ω. For a fixed x ∈ Ω0 , let e ξ, R (with ξ = (x, 0)) and the function us consider the lifted ball B e a (ξ) = e a(x, t) = a(x). e and for a suitable δ < 1, letting B e1 = Then e a
belongs to the class V M Oloc (B), e ξ, δR there exist constants c, C > 1 such that B ηea,Be1 (r) 6 cηa,Ω0 (Cr). To prove this result we need the following John-Nirenberg-type theorem in locally doubling spaces: Theorem 12.27 For every p ∈ (1, +∞) there exist constants c, C > 1 such that, letting !1/p Z 1 (p) p |f (y) − fB(x,t) | dy ηa,Ω1 (r) ≡ sup µ(B(x, t)) B(x,t) x∈Ω1 ,t6r (p)

we have, for r small enough, ηa,Ω1 (r) 6 cηa,Ω1 (Cr) . The above theorem is contained in [39, Thm. 4.5]. Proof of Proposition 12.26. We will use the comparison between the volume of lifted and unlifted ball (see Theorem 11.40). We will also exploit the fact that 2 Z Z Z 1 1 2 f (ξ) − 1 f (η) dη dξ 6 |f (ξ) − c| dξ |E| E |E| E |E| E for every c ∈ R. Fix x0 ∈ Ω0 ∩ B (x, R) with B (x, R) b Ω. For ξ = (x, h),

Nonvariational operators constructed with H¨ ormander vector fields

641

ξ0 = (x0 , h0 ) let us write, for a constant c to be chosen later 2 Z Z 1 1 e a (ξ) − e a (η) dη dξ e B(ξ e 0 ,r) e (ξ0 , r) B(ξ B (ξ0 , r) e 0 ,r) B Z 1 2 6 |e a (ξ) − c| dξ B(ξ e e 0 ,r) B (ξ0 , r) Z n o 1 2 e (ξ0 , r) dx k : (x, k) ∈ B |a (x) − c| = e (ξ0 , r) B(x0 ,r) B Z c1 2 6 |a (x) − aB | dx |B (x0 , r)| B(x0 ,r) (2)

(2)

where we have chosen c = aB . This shows that ηea,Be (r) 6 c1 ηa,Ω0 (r). On the other 1

(2)

hand, by the previous theorem, ηa,Ω0 (r) 6 cηa,Ω0 (Cr), while by H¨older inequality (2)

ηea,Be1 (r) 6 ηea,Be (r). Hence the assertion follows. 1

Let us now proceed with
the proof of Theorem 12.23. Recall that a variable e ξ, r is written as operator of type 0 on B Z T f (ξ) = PV k(ξ, η) f (η) dη + α (ξ, ξ) f (ξ) , e B

where α (ξ, ξ) is a bounded measurable function, which by Remark 12.11 satisfies the estimate |α (ξ, ξ)| 6 c for some constant c depending on the vector fields, the group G and the ellipticity constant ν. The multiplicative part f (ξ) 7−→ α (ξ, ξ) f (ξ)

e ξ, r ) into Lp (B e ξ, r ), with operator norm bounded by this clearly maps Lp (B constant. Moreover, this part of the operator T does not affect the commutator [a, T ] (since it cancels). As to the integral part of T , let us split the variable kernel as k(ξ, η) = k 0 (ξ, η) + k 00 (ξ, η). It is enough to prove our result for k 0 . Let us expand it as k 0 (ξ, η) =

H X

ai (ξ)bi (η)Diξ,η Γ (ξ; Θ (η, ξ)) + a0 (ξ)b0 (η)D0ξ,η Γ(ξ; Θ (η, ξ))

i=1

≡ kU (ξ, η) + kB (ξ, η)

(12.37)

where the kernels Diξ,η Γ(ξ; u) (for i = 1, . . . , H) are homogeneous of some degree > −Q and satisfy the cancellation property, and D0ξ,η Γ(ξ; u) is bounded in u and smooth in ξ, η. The kernels kU , kB are “unbounded” and “bounded”, respectively. The operator with kernel kB is obviously Lp continuous. Let us show that it also satisfies the desired commutator estimate. This is a consequence of the following general fact which is an easy consequence of H¨older inequality.

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H¨ ormander operators

Lemma 12.28 Let T be an integral operator defined on a ball Br by Z T f (x) = k (x, y) f (y) dy Br

where k (x, y) is a bounded measurable kernel. Then, for 1 < p < ∞, kT f kLp (Br ) 6 Ck,p |Br | kf kLp (Br ) and for any a ∈ L∞ (Br ) , k[a, T ] f kLp (Br ) 6 Ck,p |Br | kakL∞ (Br ) kf kLp (Br ) . For the integral operator TkB with kernel kB (ξ, η) we have
e k[a, TkB ] f kLp (Be(ξ,r)) 6 c B ξ, r kak∞ kf kp
e and the quantity c B ξ, r kak∞ can be made smaller than any ε > 0 for r small enough. Hence the operator TkB satisfies the assertion of Theorem 12.23 (ii). To handle the kernel kU (ξ, η) in (12.37) we write kU (ξ, η) =

H X

ai (ξ)bi (η)Diξ,η Γ (ξ; Θ (η, ξ))

i=1

=

H X

ai (ξ)bi (η)Diξ Γ (ξ; Θ (η, ξ)) +

H X

i=1

e ξ,η Γ (ξ; Θ (η, ξ)) ai (ξ)bi (η)D i

i=1

(12.38) where for every u ∈ G we set Diξ Γ (ξ; u) ≡ Diξ,ξ Γ (ξ; u) , e ξ,η Γ (ξ; u) ≡ Dξ,η Γ (ξ; u) − Dξ,ξ Γ (ξ; u) . D i i i Clearly we can consider separately each of the above terms. We start with the terms that include Diξ Γ. Since u 7→ Diξ Γ (ξ; u) is homogeneous of some degree −αi > −Q on the group G, if we introduce in G the homogeneous norm kuk = λ ⇐⇒ D1/λ (u) = 1 for u 6= 0 k0k = 0, (see Proposition 3.10 (c)) we can write Diξ Γ (ξ; u) =

Diξ Γ (ξ; u0 ) α kuk i

with u0 = D1/kuk (u) ∈ SN (where RN = G). Then, expanding the function u0 7→ Diξ Γ (ξ; u0 ) in series of spherical harmonics (see §12.5.1) we obtain Diξ Γ (ξ; u) =

dk ∞ X X k=0 j=1

(k)

cξi,kj (ξ)

Yj (u0 ) . α kuk i

Nonvariational operators constructed with H¨ ormander vector fields

643

The kernels (k)

Yj (u0 ) Ki,kj (u) = α kuk i

(12.39)

are −αi -homogeneous kernels which on the sphere SN coincide with the spherical harmonics. The functions cξi,kj (·) are the Fourier coefficients of this expansion. By Theorem 12.24 we know that: dk 6 cN k N −2 for every k = 1, 2, . . .

(12.40)

and for any positive integer m there exists cN,m such that

c

ξ N,m ξ ci,kj (ξ) 6 2m D2m Di Γ (ξ; ·) 2 N . k L (S ) In turn, we can bound uniformly the derivatives D2m Diξ Γ (ξ; ·) by (12.18), getting c (N, m, ν, X) ξ . (12.41) ci,kj (ξ) 6 k 2m Here the (smooth) dependence of the differential operators Diξ on the variable ξ is harmless since these operators are finitely many (i = 1, . . . , H in (12.37)). Then, let Z dk ∞ X X cξi,kj (ξ) ai (ξ)bi (η)Ki,kj (Θ (η, ξ)) f (η) dη Si f (ξ) = PV e B (ξ,r ) k=0 j=1 where Ki,kj is as in (12.39) and Z Si,kj f (ξ) = ai (ξ)bi (η)Ki,kj (Θ (η, ξ)) f (η) dη. e (ξ,r ) B Recall that every kernel Ki,kj (u) is homogeneous of some degree > −Q, so some of these Si,kj will be fractional integral operators, others will be singular integral operators. In this case we could write, for every p ∈ (1, ∞), kSi f kLp (Be(ξ,r)) 6

dk ∞ X X

·

ci,kj (·) ∞ L k=0 j=1

kSi,kj f kLp (Be(ξ,r))

(12.42)

kSi,kj (e af ) − e a · Si,kj f kLp (Be(ξ,r)) .

(12.43)

(Be(ξ,r))

and kSi (e af ) − e a · Si f kLp (Be(ξ,r)) 6 6

dk ∞ X X

·

ci,kj (·) ∞ L k=0 j=1

(Be(ξ,r))

The problem is then reduced to establishing Lp continuity of the operators Si,kj (which are similar to frozen operators of type λ) and the corresponding commutator estimate, with some control on the operator norms, in terms of j, k. After that, using the bounds (12.40), (12.41) we will prove the convergence of the series. Let us concentrate on the study of the operators Si,kj . Let us distinguish the fractional

644

H¨ ormander operators

and singular integral case, corresponding to the case αi < Q or αi = Q in (12.39), respectively. In the fractional integral case, Si,km has the same properties of a frozen operator of type λ > 1, with the kernel Di Γ(ξ0 ; ·) replaced by (k)

Ki,kj (u) =

Yj (u0 ) . α kuk i

By Proposition 11.32 we can say that the kernel of Si,km satisfies the growth condition and the mean value inequality of a fractional integral kernel, with constants bounded by c · sup |∇u Ki,kj (u)| , kuk=1

(with c depending on the vector fields); in turn, by Theorem 12.24 (ii) we know that sup |∇u Ki,kj (u)| 6 cN k N/2 . kuk=1

Hence we can conclude, as in Chapter 11, section 11.4 kSi,kj f kLp (Be(ξ,r)) 6 c · k N/2 kf kLp (Be(ξ,r)) for i = 1, . . . , H,

(12.44)

where we have also taken into account Remark 7.36. However, we can also apply directly to Si,kj the general result about fractional integrals proved in Chapter 7 (see Theorem 7.28) and improve the previous estimate to kSi,kj f kLq (Be(ξ,r)) 6 c · k N/2 kf kLp (Be(ξ,r)) for i = 1, . . . , H, for a certain q > p. This is useful in order to get the desired commutator estimate for Si,kj . Namely, the following general fact is helpful: Lemma 12.29 Let T be an integral operator defined on a ball Br by Z T f (x) = k (x, y) f (y) dy Br

and assume that, for some q > p, kT f kLq (Br ) 6 CT kf kLp (Br ) . Then for any a ∈ L∞ (Br ) , 1

k[a, T ] f kLp (Br ) 6 2CT |Br | p

− q1

kakL∞ (Br ) kf kLp (Br ) .

Proof. By H¨ older inequality, 1

k[a, T ] f kLp (Br ) 6 k[a, T ] f kLq (Br ) |Br | p

− q1 1

6 2 kakL∞ (Br ) kT f kLq (Br ) |Br | p

− q1 1

6 2CT kakL∞ (Br ) kf kLp (Br ) |Br | p

− q1

.

Nonvariational operators constructed with H¨ ormander vector fields

645

By the previous lemma we can write 1

k[a, Si,kj ] f kLp (Be(ξ,r)) 6 ck N/2 |Br | p

− q1

kakL∞ (Be(ξ,r)) kf kLp (Be(ξ,r)) .

This means that for any ε > 0 and r > 0 small enough (dependent on ε, a but independent from i, k, j) we have k[e a, Si,kj ] f kLp (Be(ξ,r)) 6 ε · k N/2 kf kLp (Be(ξ,r)) .

(12.45)

Finally, let us consider the singular integral case, when (k)

Ki,kj (u) =

Yj

(u0 )

kuk

Q

.

Again by Proposition 11.32 and Theorem 12.24 (ii) we can say that the kernel of Si,kj satisfies the growth condition and the mean value inequality of a singular integral kernel, with constants bounded by sup |∇u Ki,kj (u)| 6 cN k N/2 . kuk=1

In this case we have also to check the vanishing property of the kernel. Note that Z Z (k) Ki,kj (u) dS (u) = Yj (θ) dS (θ) . Sn

Sn

(k)

Since Yj for k > 0 is orthogonal to spherical harmonics of order 0 (constant functions) we have Z Ki,kj (u) dS (u) = 0 Sn

for k = 1, 2, . . . Observe that the Fourier coefficient corresponding to the zero order term vanishes since Diξ,η Γ (ξ; ·) satisfies the vanishing property. We can then conclude that (12.44) holds also in this case. As to the commutator estimate, we now apply Theorem 12.25 and conclude that, for a ∈ V M Oloc and every ε > 0, there exists r (also depending on the V M O modulus of a) such that (12.45) still holds. We can now use the results about the operators Si,kj and the bounds on the coefficients cξi,kj (ξ) in the spherical harmonic expansion, to conclude our computation. By (12.42), (12.40), (12.41) we can write, for any positive integer m and a constant c = c (N, m, ν, X) kSi f kLp (Be(ξ,r)) 6

dk ∞ X X

·

ci,kj (·) ∞ L k=0 j=1 ∞ X N −2−N/2−2m

6c

k

k=0

(Be(ξ,r))

kSi,kj f kLp (Be(ξ,r))

kf kLp (Be(ξ,r)) .

646

H¨ ormander operators

The series converges for m large enough, only depending on the dimension N . Hence kSi f kLp (Be(ξ,r)) 6 c kf kLp (Be(ξ,r)) . An analogous computation starting from (12.45) shows that for every a ∈ V M Oloc any ε > 0, there exists r small enough such that k[e a, Si ] f kLp (Be(ξ,r)) 6 ε kf kLp (Be(ξ,r)) . e ξ,η Γ. We will show that It remains to consider the terms in (12.38) that contain D i these terms give rise to fractional integrals and can be handled, as we already have done for the other terms, using Theorem 7.28 and Lemma 12.29. For u ∈ G let us write e ξ,η ξ,η ξ,η 0 ξ,ξ Di Γ (ξ; u) = Di Γ (ξ; u) − Di Γ (ξ; u) 6 |η − ξ| · ∇η Di Γ (ξ; u) for some point η 0 between η and ξ. Since u 7→ Diξ,η Γ (ξ; u) is homogeneous of some 0 degree −αi > −Q also u 7→ ∇η Diξ,η Γ (ξ; u) has the same homogeneity so that, by (12.18) c ξ,η 0 ∇η Di Γ (ξ; u) 6 α kuk i for some constant c independent of ξ and η 0 . Moreover, |η − ξ| 6 cde(ξ, η) 6 cρ (ξ, η) so that e ξ,η D Γ (ξ; Θ (η, ξ)) 6 i

c

αi −1 .

ρ (ξ, η)

This means that our kernel satisfies the growth condition of a fractional integrals. Applying Theorem 7.28 to Z e ξ,η Γ (ξ; Θ (η, ξ)) f (η) dη Sei f (ξ) = ai (ξ)bi (η)D i e B (ξ,r ) we obtain

e

Si f

e (ξ,r )) Lp (B



6 c Sei f

e (ξ,r )) Lq (B

6 c kf kLp (Be(ξ,r))

for a some q > p. Lemma 12.29 provides the required commutator estimate and the proof of Theorem 12.23 is completed. 12.6

Regularization of solutions

It remains to prove, for operators without drift, the regularization results contained in Theorem 12.2. More precisely, having already proved the a priori estimates k+2,p contained in Theorem 12.2 under the assumptions u is either in WX,loc (Ω) or in k+2,α CX,loc (Ω), we want to prove that:

Nonvariational operators constructed with H¨ ormander vector fields

647

k,∞ (a) for every domain Ω0 b Ω, p ∈ (1, ∞), positive integer k, if aij ∈ WX (Ω), 2,p k,p k+2,p 0 u ∈ WX (Ω) and Lu ∈ WX (Ω) then u ∈ WX (Ω ), k,α (b) for every domain Ω0 b Ω, α ∈ (0, 1), positive integer k, if aij ∈ CX (Ω), 2,α k,α k+2,α u ∈ CX (Ω) and Lu ∈ CX (Ω) then u ∈ CX (Ω0 ). It is clearly enough to prove the corresponding regularization results, in local form, in the space of lifted variables, and then apply the results already used in Chapter 11 about the relation between function spaces in the original variables and in the lifted variables. eR ≡ B e (ξ0 , R) Let us start with the H¨ older case (b), writing on a lifted ball B the representation formula (12.23)

em X el (au) = Slm (ξ0 ) L e0 u + X

q X

0 ek u + Slm Sklm (ξ0 ) X (ξ0 ) u

(12.46)

k=1 2,α e which holds for any u ∈ CX,0 e (Br ), r < R. We already know that the operators

0 Slm (ξ0 ) , Sklm (ξ0 ) , Slm (ξ0 ) satisfy the continuity estimate

kS (ξ0 ) f kC α (Ber ) 6 CR,ν kf kC α (Ber )

(12.47)

α er ), where the constant CR,ν also depends on the ellipticity for any f ∈ CX,0 (B constant ν in (12.7). We are going to see (12.46) as an identity involving a suitable integral operator, to which apply the Banach-Caccioppoli fixed point theorem. To this aim, for a fixed 2,α e u ∈ CX,0 (BR ) let

e + Gl,m = Slm (ξ0 ) Lu

q X

0 ek u + Slm Sklm (ξ0 ) X (ξ0 ) u

(12.48)

k=1 α e hence, by (12.47), Gl,m ∈ CX (BR ) and

 em X el (au) = Gl,m + Slm (ξ0 )  X

q X

 ei X ej u . [e aij (ξ0 ) − e aij (·)] X

i,j=1

er ) such that Next, for a number r < R to be fixed later, pick another β ∈ C0∞ (B e β = 1 in Br/2 , and write   q X em X el (au) = βGl,m + βSlm (ξ0 )  ei X ej u . βX [e aij (ξ0 ) − e aij (·)] X (12.49) i,j=1 α α e e Let CX,∗ e (Br ) be the Banach space of CX e (Br ) functions vanishing on the boundary er , and let us define the operator of B  q×q  q×q α α e e T : CX,∗ → CX,∗ e (Br ) e (Br )

648

H¨ ormander operators

 q×q q α er ) ( B which to any F = (Fij )i,j=1 ∈ CX,∗ associates e   q X (T (F ))lm = βGl,m + βSlm (ξ0 )  [e aij (ξ0 ) − e aij (·)] Fij  . i,j=1

Theorem 12.30 For r > 0 small enough, the operator T is a contraction of q×q  er ) in itself. C α (B e X,∗

α e e Proof. Since Fij ∈ CX,∗ e (Br ), Fij can be extended to zero in BR , hence q X

α eR ), [e aij (ξ0 ) − e aij (·)] Fij ∈ CX,0 (B

i,j=1

and by (12.47),  Slm (ξ0 ) 

q X

 α e [e aij (ξ0 ) − e aij (·)] Fij  ∈ CX (BR )

i,j=1

and  βSlm (ξ0 ) 



q X

α e [e aij (ξ0 ) − e aij (·)] Fij  ∈ CX,∗ e (Br ).

i,j=1 α e α e Since also Gl,m ∈ CX (BR ) and βGl,m ∈ CX,∗ e (Br ), we conclude that T maps q×q  α e CX,∗ in itself. In order to show that T is a contraction, let e (B (ξ0 , r)) q×q  e (ξ0 , r)) . We have, by (12.47). F (1) , F (2) ∈ C α (B e X,∗



T F (1) − T F (2) 

e 0 ,r)) Cα (B(ξ f

q×q

X



q q   X X

(1) (2)

βSlm (ξ0 ) 6 [e aij (ξ0 ) − e aij (·)] Fij − Fij

i,j=1 l,m=1

e 0 ,r)) Cα (B(ξ f X

6

q X

q X

 

(1) (2) c [e aij (ξ0 ) − e aij (·)] Fij − Fij

e 0 ,r)) Cα (B(ξ f

l,m=1 i,j=1

X



6 c ω (r) F (1) − F (2) 

e 0 ,r)) Cα (B(ξ f

q×q

X

with ω (r) =

sup i,j=1,2,...,q

α |e aij |C α (B(ξ e 0 ,r)) r . f X

 q×q α e Hence for r small enough T is a contraction of CX,∗ in itself. e (Br )

Nonvariational operators constructed with H¨ ormander vector fields

649

1,α e 1,α e Next, denoting by CX,∗ e (Br ) the Banach space of CX e (Br ) functions f which ei f , on ∂ B er , we can prove the following vanish, together with their derivatives X similar result: 2,α e e ∈ C 1,α (B eR ) and r is small enough, the Theorem 12.31 If u ∈ CX,0 aij , Lu e (BR ), e e X q×q  1,α operator T is also a contraction of CX,∗ in itself. e (B (ξ0 , r))

Proof. We already know that βGl,m

e + = βSlm (ξ0 ) Lv

q X

ek v + βS 0 (ξ0 ) v ∈ C α (B er (ξ0 )). βSklm (ξ0 ) X lm X,0

k=1

eh (βGl,m ) ∈ C α (B er ), let us compute To show that also X e X,∗ (   eh Slm (ξ0 ) Lu e + e e Xh (βGl,m ) = Xh β Gl,m + β X +

q X

) eh Sklm X

ek u + (ξ0 ) X

0 eh Slm X

(ξ0 ) u

k=1

exploiting Theorem 11.19 (and the fact that every operator of type 0 in our representation formula is actually the derivative of an operator of type 1) ( q !   X s 0 e + eh β Gl,m + β es + Slm Slm (ξ0 ) X (ξ0 ) Lu = X s=1

+

+

q X

q X

k=1

s=1

q X

s Slm,k

! 0 ek u e (ξ0 ) Xs + Slm,k (ξ0 ) X ! )

s Slm

es + (ξ0 ) X

0 Slm

(ξ0 ) u

s=1 s (note that the operators Slm actually depend also on h). Recalling that u ∈ 2,α f 1,α e e CX,0 e ((B R ) and Lu ∈ CX e ((BR ), by (12.47) we get that the quantity in the curly α e brackets belongs to C (BR (ξ0 )), hence by our choice of β, e X

eh (βGl,m ) ∈ C α (B er ) ⊂ C α (B er ). X e e X,0 X,∗ As to the other term of T (F ), βTlm (ξ0 )

X q i,j=1

 [e aij (ξ0 ) − e aij (·)] Fij ,

650

H¨ ormander operators

1,α e 1,α e for e aij ∈ CX e (BR ), Fij ∈ CX,∗ e (Br ) we can compute, again by Theorem 11.19:

 X  q e Xk βSlm (ξ0 ) [e aij (ξ0 ) − e aij (·)] Fij i,j=1

=β

 X q

s es + S 0 (ξ0 ) Slm (ξ0 ) X lm

 X q

s=1

 [e aij (ξ0 ) − e aij (·)] Fij

i,j=1

ek β)Slm (ξ0 ) + (X

X q

 [e aij (ξ0 ) − e aij (·)] Fij

i,j=1

=β

X q

s Slm (ξ0 )

X q

s=1

 es Fij [e aij (ξ0 ) − e aij (·)] X

−

0 + Slm (ξ0 )

s Slm (ξ0 )

s=1

i,j=1

X q

q X

 X q   es e X aij Fij i,j=1

 [e aij (ξ0 ) − e aij (·)] Fij

i,j=1

X  q   ek β Slm (ξ0 ) [e aij (ξ0 ) − e aij (·)] Fij . + X i,j=1

Since, under our assumptions, the functions: q X

es Fij , [e aij (ξ0 ) − e aij (·)] X

i,j=1

q X

es e (X aij )Fij and

i,j=1

q X

[e aij (ξ0 ) − e aij (·)] Fij

i,j=1

α α e e belong to CX,∗ e (B(ξ0 , R)), by (12.47) and our choice of β we e (B(ξ0 , r)) ⊂ CX,0 conclude

 X  q α e e [e aij (ξ0 ) − e aij (·)] Fij ∈ CX,∗ Xk βTlm (ξ0 ) e (B(ξ0 , r)), i,j=1

1,α e hence T maps CX,∗ e (Br ) in itself. Let us show that T is also a contraction in 1,α e CX,∗ e (Br ). We already know that



T F (1) − T F (2)

α (B(ξ e 0 ,r))q×q CX



6 cω (r) F (1) − F (2) α e 0 ,r)))q×q (CX (B(ξ

(12.50)

Nonvariational operators constructed with H¨ ormander vector fields

651

so let us compute ek T F (1) − X ek T F (2) X X X q q   s es F (1) − X es F (2) =β Slm (ξ0 ) [e aij (ξ0 ) − e aij (·)] X ij ij −

s=1 q X

i,j=1

s Slm

(ξ0 )

s=1

+

0 Slm

X q

  (1) (2) e Xs e aij Fij − Fij

i,j=1

(ξ0 )

X q

[e aij (ξ0 ) − e aij (·)]



(1) Fij

−

(2) Fij

 

i,j=1

ek β)Slm (ξ0 ) + (X

X q

  (1) (2) [e aij (ξ0 ) − e aij (·)] Fij − Fij

i,j=1

≡ A + B + C + D. Applying again (12.47),

e (1) e (2) kAkC α (Ber (ξ0 )) 6 cω (r) XF − XF

α e ij ij f C f(B(ξ0 ,r)) X X

(1) (2) 6 cω (r) Fij − Fij 1,α , e

(12.51)

C f (B(ξ0 ,r)) X

kCkC α (B(ξ e 0 ,r)) + kDkC α (B er (ξ0 )) f X

f X

kBkC α (B(ξ e 0 ,r)) 6 c f X



(1) (2) 6 cω (r) Fij − Fij

e 0 ,r)) Cα (B(ξ f

,

(12.52)

X

q

X

e aij

Xe

e 0 ,r)) Cα B(ξ f(

s,i,j=1

X



(1) (2)

Fij − Fij

e 0 ,r)) Cα B(ξ f(

.

X

1,α e To complete the bound on B, let us note that if g ∈ CX,∗ e (Br ) we have

kgk∞ 6

sup e 0 ,r) ξ,η∈B(ξ

|g (ξ) − g (η)| 6 |g|C α (B(ξ e 0 ,r)) (2r)

α

1,α e and applying (2.19) (g is in CX,0 e (BR )),

|g|C α (B(ξ e 0 ,r)) =

|g (ξ) − g (η)| e 1−α , 6 sup Xg (2r) α e (ξ, η) e 0 ,r) dX e 0 ,r) ξ,η∈B(ξ B(ξ sup

hence   α 1−α kgkC α (B(ξ e 0 ,r)) 6 kgkC 1,α (B(ξ e 0 ,r)) (2r) + (2r) f X

f X

and kBkC α (B(ξ e 0 ,r))

(12.53)

f X

6c

q

X

e

aij

Xs e s,i,j=1



e 0 ,r)) Cα (B(ξ f X

α

(2r) + (2r)

1−α



(1) (2)

Fij − Fij

e 0 ,r)) C 1,α (B(ξ f X

.

652

H¨ ormander operators

From (12.50), (12.51), (12.52), (12.53) we deduce

that for r small enough



(1)

(1) (2)  q×q  − F (2)  1,α

T F − T F 1,α q×q 6 δ F e 0 ,r) C f (B(ξ

e 0 ,r) C f (B(ξ

X

X

with δ < 1, and we are done. By the Banach-Caccioppoli fixed point theorem, the previous theorems imply α e that the operator T has a unique fixed point in CX,∗ e (B(ξ0 , r) and a unique fixed e 0 , r); by (12.49) this implies that u ∈ C 3,α (B er ), that is the regupoint in C 1,α (B(ξ e X,∗

e X

larization result in the space of lifted variables for k = 1. Iteration gives the general case. The Lp case (a) can be dealt along similar lines, exploiting operators with variables kernels and commutators of such operators with V M O functions. We now start with representation formula (12.25), rewritten according to Remark 12.19 in the shorter form q q X X em X el (au) = Slm Lu e + ei X ej u + ek u + S 0 u X [e aij , Slm ] X Slm,k X lm i,j=1

k=1

0 where the variable operators Slm , Slm,k , Slm satisfy the continuity estimate kSf kLp (Ber ) 6 CR,ν kf kLp (Ber ) p e er ) such that β = 1 in B er/2 , and write for any f ∈ L (Br ), r < R. Let β ∈ C ∞ (B 0

em X el (au) = βSlm Lu e +β βX

q X

ei X ej u + β [e aij , Slm ] X

i,j=1 2,p e For a fixed u ∈ WX e (BR ) let

e + Gl,m = Slm Lu

q X

0 ek u + βSlm Slm,k X u

k=1

q X

0 ek u + Slm Sklm X u

(12.54)

k=1

eR ) and hence Gl,m ∈ Lp (B em X el (au) = βGl,m + β βX

q X

ei X ej u. [e aij , Slm ] X

i,j=1 q×q

 er ) Now, let us define the operator T : Lp (B  q×q q er ) F = (Fij )i,j=1 ∈ Lp (B associates T (F ) = βGl,m + β

q X

→



(12.55)

q×q er ) Lp (B which to any

[e aij , Slm ] Fij .

i,j=1

Exploiting the commutator theorem (Theorem 12.23 (ii)), it is now straightforward  q×q er ) to check that for r > 0 small enough, the operator T is a contraction of Lp (B in itself. The smallness of r depends on the V M O moduli of the coefficients e aij , 1,∞ which we are assuming in WX , hence a fortiori in V M O (see Proposition 12.6). e The next step is to establish that, under suitable assumptions, T is also a contraction  q×q eR ) of W 1,p (B in itself. This requires the following Sobolev embedding: e X

Nonvariational operators constructed with H¨ ormander vector fields

653

eR and every p ∈ Proposition 12.32 (Sobolev embedding) For a fixed ball B 1,p e [1, +∞) there exists a constant c (R, p) such that for every f ∈ WX e (BR ) compactly eR , one has supported in B kf kLq (BeR ) 6 c kf kW 1,p (BeR ) f X

with q = p



1+Q Q



and Q homogeneous dimension in the lifted space. (Recall that e0 is lacking). we are assuming that the drift term X e (ξ0 , 2R), for a fixed cutoff a = 1 in B eR and any Proof. Let us write, on the ball B ∞ e f ∈ C0 (BR ), the parametrix formula e + Ff = af = P Lf

q  X

q  X ei X ei f + F f = ei f + F f PX Fi X i=1

i=1

with P, F, Fi operators of type 2, 1, 1, respectively. Then, recall that, by Theorem 7.28, the operators of type 1, Fi , F are fractional integral operators, mapping f (ξ0 , 2R)) into Lq ((B eR ) with q = p ((1 + Q)/Q), hence continuously Lp (B kf kLq (BeR ) 6 c kf kW 1,p (BeR ) f X

(where we exploited the compact support of f ). By density (Theorem 2.9), the 1,p e e estimate holds for every f ∈ WX e (BR ) compactly supported in BR . 2,p e 1,∞ e 1,p e e Theorem 12.33 If u ∈ WX aij ∈ WX e (BR ), e e (BR ), Lu ∈ WX e (BR ) and r is  q×q 1,p e small enough, the operator T is also a contraction of WX in itself. e (BR ) 2,p e 1,p e e Proof. For a fixed u ∈ WX e (BR ) such that Lu ∈ WX e (BR ), exploiting Theorem eh (βGl,m ) ∈ Lp (B eR ). Also, 11.19 one can check that X  X  q eh T (F ) = X eh (βGl,m ) + X eh β X [e aij , Slm ] Fij



eh β eh (βGl,m ) + X =X

i,j=1 q X

[e aij , Slm ] Fij

i,j=1

+β

q  X

q h  i X eh e eh Slm Fij X aij Slm Fij + β e aij , X

i,j=1

i,j=1



eh (βGl,m ) + X eh β =X

q X

[e aij , Slm ] Fij + β

i,j=1

q  X

 eh e X aij Slm Fij

i,j=1

q q X X     h h el Fij + β +β e aij , Slm,l X e aij , Slm Fij i,j,l=1

i,j,l=1

654

H¨ ormander operators

where we have exploited again Theorem 11.19. The commutator theorem shows that



q q q X X X

    h h e eh β)

(X e aij , Slm Fij e aij , Slm,l Xl Fij + β [e aij , Slm ] Fij + β

p e

i,j=1 i,j,l=1 i,j,l=1

L (Br )

is bounded by ε kFij kW 1,p (Ber ) for r small enough. Also, f X



q  

X eh e

β X a S F ij lm ij

i,j=1

6c

q

X

e

aij

Xh e

er ) L∞ ( B

i,j=1

er ) Lp (B

kβSlm Fij kLp (Ber ) .

We apply H¨ older inequality and Proposition 12.32 to every r < R, getting, for some constant c depending on R but not on r, p1 − q1 e kβSlm Fij kLp (Ber ) 6 c B kβSlm Fij kLq (Ber ) r 6 cr1/q kβSlm Fij kW 1,p (Ber ) 6 cr1/q k Fij kW 1,p (Ber ) f X

f X

where in the last inequality we applied Proposition 11.31, so that, for r small enough, kβSlm Fij kLp (Ber ) 6 ε k Fij kW 1,p (Ber ) . f X

q×q 1,p e This shows that T is a contraction of WX ( B ) in itself, for r small enough. R e 

This completes the proof of the regularization result in the scale of Sobolev spaces, for k = 1. Iteration gives the general case. 12.7

Proof of the estimates on spherical harmonics

In this section we will prove Theorem 12.24. For convenience of the reader, we recall here some notation. We denote by Pk the space of homogeneous polynomials of degree k in RN and by Hk the subspace of Pk of harmonic homogeneous polynomials. The space of spherical harmonics of degree k is given by  Hk = P|SN : P ∈ Hk n o (k) be an orthonormal basis of Hk , as a and dk = dim Hk . We let Yj j=1,...,dk o n
(k) subspace L2 SN and we consider the set of spherical harmonics Yj j=1,...,dk , k=0,1...,+∞

which is not restrictive to assume real valued.

Nonvariational operators constructed with H¨ ormander vector fields

12.7.1

655

The Laplace-Beltrami operator on the unit sphere

As we shall see, spherical harmonics turn out to be eigenfunctions of a symmetric differential operator defined on the sphere: the Laplace-Beltrami operator. Since eigenfunctions associated to different eigenvalues are orthogonal we will show that it is possible to expand, in a very natural way, functions in L2 of the unit sphere in terms of eigenfunctions of the Laplace-Beltrami operator. A function f defined on the unit sphere SN can be extended by homogeneity to a function defined on RN \ {0}. Unless noted otherwise we adopt the convention of extending f to be homogeneous of degree 0, that is   x f (x) = f . (12.56) |x| We start with the following Definition 12.34 (Laplace-Beltrami operator on Sn ) For a function f ∈
2 N C S we define ∆SN f = ∆f |SN N

where f , originally defined on S , has been extended to RN \ {0} by (12.56). In other words ∆SN f is the restriction to the unit sphere of the ordinary Laplacian, evaluated on the function f extended by homogeneity.
In the next lemma we show that ∆SN is “symmetric” in L2 SN .
Lemma 12.35 Let f, g ∈ C 2 SN . Then Z Z ∆SN f (θ) g (θ) dS (θ) = f (θ) ∆SN g (θ) dS (θ) . SN

SN

Proof. First of all we observe that since f is homogeneous of degree 0, then ∆f (x)  is homogeneous of degree −2. Let now D = x ∈ RN : 12 6 |x| 6 2 , by Green’s identity  Z Z  ∂f ∂g [∆f (x) g (x) − f (x) ∆g (x)] dx = g (x) (x) − f (x) (x) dS (x) ∂n ∂n D ∂D where dS is the surface measure and n the outer normal to ∂D. Since f and g are ∂g ∂f and ∂n vanish, so that homogeneous of degree 0, both ∂n Z Z ∆f (x) g (x) dx = f (x) ∆g (x) dx. D

D

Let us compute, exploiting the homogeneities of ∆f and g, Z Z 2Z ∆f (x) g (x) dx = ∆f (rθ) g (rθ) dS (θ) rN −1 dr 1 2

D

Z

SN

2

Z

= 1 2

r−2 ∆f (θ) g (θ) dS (θ) rN −1 dr

SN

Z = cN SN

∆SN f (θ) g (θ) dS (θ) .

656

H¨ ormander operators

A similar computation gives Z Z f (x) ∆g (x) dx = cN SN

D

f (θ) ∆SN g (θ) dS (θ)

and since cN 6= 0 we are done. 12.7.2

Homogeneous polynomials and spherical harmonics

A remarkable property of homogeneous polynomials is contained in the following Proposition 12.36 For every P ∈ Pk there exist homogeneous harmonic polynomials Pj ∈ Hk−2j , for j = 0, . . . , m, such that 2

2m

P (x) = P0 (x) + |x| P1 (x) + · · · + |x|

Pm (x)

(12.57)

(where m is the greatest integer such that 2m 6 k). Remark 12.37 Note that (12.57) says in particular that on the surface |x| = 1 any homogeneous polynomial can be written as a sum of homogeneous harmonic polynomials. Let P ∈ Pk , then P (x) =

X

cα xα

|α|=k

where the summation is over all multiindices α = (α1 , . . . , αN ) of length |α| = αN 1 α1 + · · · + αN = k and as usual xα = xα 1 · · · xN . We introduce the differential operator X ∂ |α| P (∂) = cα α1 . N ∂x1 · · · ∂xα N |α|=k

To prove the above proposition we need the following: Lemma 12.38 For P, Q ∈ Pk let: hP, Qi = P (∂) Q.

(12.58)

Then equation (12.58) defines a scalar product in Pk . Proof. The fact hP, Qi is linear in the first argument and conjugate linear in the second one is immediate from the definition. Also observe that for |α| = |β| = k,    0 α 6= β

α β ∂ |α| β1 βN x · · · x = x ,x = αN 1 N 1 α! α = β ∂xα 1 · · · ∂xN P α where α! = α1 ! . . . αN !. If follows that for P (x) = and Q (x) = |α|=k cα x P P α |α|=k dα x we have hP, Qi = |α|=k α!cα dα so that hP, Qi = hQ, P i. Also, since X 2 hP, P i = α! |cα | |α|=k

we see that hP, P i = 0 if and only if P ≡ 0.

Nonvariational operators constructed with H¨ ormander vector fields

657

Proof of Proposition 12.36. We start proving the following claim. If we set n o 2 2 |x| Pk−2 = Q ∈ Pk : Q (x) = |x| P (x) for some P ∈ Pk−2 , then, for every k > 2, 2

Pk = Hk ⊕ |x| Pk−2

(12.59)

where the direct sum ⊕ is intended with respect to the scalar product defined in the previous Lemma. It is clearly enough to show that the orthogonal complement 2 (in PD ∈ Pk and assume that for every P ∈ Pk−2 we k ) of |x| P E k−2 is Hk . Let Q  2

2

have |x| P, Q = 0. Since |x| P (∂) = P (∂) ∆ and ∆Q ∈ Pk−2 , we have D E 2 0 = |x| P, Q = P (∂) ∆Q = hP, ∆Qi

which, choosing P = ∆Q, implies ∆Q = 0. Conversely, let Q ∈ Hk and P ∈ Pk−2 . Then, reading the above computation in the reverse order we see that Q is in the 2 orthogonal complement of |x| Pk−2 in Pk , so (12.59) is proved. Let now P ∈ Pk . By (12.59) there exist P0 ∈ Hk and Q0 ∈ Pk−2 such that 2

P (x) = P0 (x) + |x| Q0 (x) . Applying again (12.59) to Q0 (x) and iterating we obtain 2

2m−2

P (x) = P0 (x) + |x| P1 (x) + · · · + |x|

Pm−1 (x) + |x|

2m

Qm (x) .

with Pj ∈ Hk−2j , and Qm ∈ Pk−2m . Since every polynomial of degree < 2 is harmonic choosing m so that k − 2m 6 1 gives (12.57). We can now prove that spherical harmonics are eigenfunctions of the LaplaceBeltrami operator. This in turn implies that the spaces Hk are mutually orthogonal. Proposition 12.39 Let Y ∈ Hk , then for every θ ∈ SN we have ∆SN Y (θ) = −k (N + k − 2) Y (θ) . Moreover if Y1 ∈ Hk and Y2 ∈ Hm with k 6= m then Z Y1 (θ) Y2 (θ) dS (θ) = 0.

(12.60)

(12.61)

In the proof of the above result we
will use Euler identity for homogeneous functions: namely, if f ∈ C 1 RN \ {0} is homogeneous of degree α ∈ R then, ∇f (x) · x = αf (x) for every x ∈ RN \ {0} .

(12.62)

This follows differentiating f (λx) = λα f (x) with respect to λ and letting λ = 1. Proof. Since Y is the restriction to SN of a polynomial P ∈ Hk we have   x k . P (x) = |x| Y |x|

658

H¨ ormander operators

Hence,    x k 0 = ∆P = ∆ |x| Y |x|           x x x k k k + 2∇ |x| · ∇ Y + |x| ∆ Y . (12.63) = ∆ |x| Y |x| |x| |x| Recalling that the Laplacian of a radial function f (x) = g (|x|) is   N −1 0 00 ∆f (x) = g (ρ) + g (ρ) , ρ ρ=|x| a direct computation shows that   k k−2 ∆ |x| = k (k + N − 2) |x|   k k−2 ∇ |x| = k |x| x.   x is homogeneous of degree 0, by Euler identity (12.62) we Since the function Y |x| have         X x ∂ x k k−2 ∇ |x| · ∇ Y = k |x| xi = 0. Y |x| ∂xi |x| Recalling that      x x = ∆SN Y , ∆ Y |x| |x| by (12.63) we obtain     x x k k−2 0 = k (k + N − 2) |x| Y + |x| ∆SN Y |x| |x| which is (12.60) with θ = x/ |x| and |x| = 1. Finally, (12.60) and Lemma 12.35 easily give (12.61). We now compute the dimension dk of the eigenspaces Hk . Clearly, dk = dim Hk . In view of (12.59) we also have dim Hk = dim Pk − dim Pk−2 . Since a base for Pk is given by the monomials xα with |α| = k, the dimension of Pk is given by the number of different ways  we can write  k as a sum of N positive k+N −1 numbers, which is known to be equal to . Hence k (k + N − 1)! (k + N − 3)! (k + N − 3)! − = (2k + N − 2) . k! (N − 1)! (k − 2)! (N − 1)! (N − 2)!k! Thus, when k → +∞ and N is fixed, 2 dk ∼ k N −2 . (12.64) (N − 2)! In particular, for every N there exists cN such that dk =

dk 6 cN k N −2 for every k = 1, 2, 3 . . .

(12.65)

Nonvariational operators constructed with H¨ ormander vector fields

(k)

659

(k)

Theorem 12.40 For every fixed k, let Y1 , . . . Ydk be an orthonormal base of Hk . Then the spherical harmonics n o (k) Yj j=1,...,dk k=0,1,...,+∞


form a complete orthonormal system in L2 SN . n o (k) Proof. The fact that Yj j=1,...,dk is an orthonormal system comes from the k=0,1,...+∞

choice of

(k) (k) Y1 , . . . Ydk

and Proposition 12.39. It remains to prove that this system n o
(k) 2 is complete, that is that the linear span of Yj SN . j=1,...,dk is dense in L k=0,1,...+∞
2 N Since by Weierstrass theorem polynomials are dense n inoL S , it is enough to show (k) that every polynomial is in the linear span of Yj j=1,...,dk . This is a simple k=0,1,...+∞

consequence of (12.57) that restricted on the sphere says that every homogeneous polynomial is a sum of spherical harmonics (see Remark 12.37). Proposition 12.41 Let

n o (k) Yj


as above, let f ∈ C ∞ SN and define

j=1,...,dk k=0,1,...+∞

Z fbj,k = SN

(k)

f (θ) Yj

(θ) dS (θ) .

For every positive integers m, k and j = 1, . . . , dk 1 b m fj,k 6 2m k(∆SN ) f kL2 (SN ) . k In particular, for every m there exists cN,m > 0 such that for k > 1 and j = 1, . . . , dk

c b fj,k 6 2m D2m f L2 (SN ) , k where here D2m denote Euclidean derivatives of order 2m. Proof. Using Proposition 12.39 and Lemma 12.35 we can write Z Z −1 (k) (k) f (θ) ∆SN Yj (θ) dS (θ) fbj,k = f (θ) Yj (θ) dS (θ) = k (N + k − 2) SN SN Z −1 (k) = ∆ N f (θ) Yj (θ) dS (θ) k (N + k − 2) SN S  m Z −1 (k) m = (∆SN ) f (θ) Yj (θ) dS (θ) k (N + k − 2) SN
(k) so that, since N > 2 and Yj are normalized in L2 SN

1 1 b

(k) m m fj,k 6 m k(∆SN ) f kL2 (SN ) Yj 2 N 6 2m k(∆SN ) f kL2 (SN ) . k [k (N + k − 2)] L (S )

660

H¨ ormander operators

12.7.3

Zonal spherical harmonics and bounds on the spherical harmonics

Let us fix θ0 ∈ SN and consider the linear functional defined on Hk given by Y 7→ Y (θ0 ) .
Since Hk is a finite dimensional subspace of L2 SN , we can use its self-duality to (k) ensure that there exists a spherical harmonic Zθ0 ∈ Hk such that Z (k) Y (θ0 ) = Y (θ) Zθ0 (θ)dS (θ) . SN

(k)

Definition 12.42 Zθ0 is called the zonal spherical harmonic of degree k and center θ0 . (k)

In the next lemma we collect a few remarkable properties of Zθ0 . Lemma 12.43 (k) (k) (k) (1) Let Y1 , . . . , Ydk be an orthonormal base for Hk , and assume Yj real valued. Then dk X (k) (k) (k) Zθ0 (θ) = Yi (θ) Yi (θ0 ) . (12.66) i=1 (k) Zθ0

(k) Zθ0

(k)

(2) is real and satisfies (θ) = Zθ (θ0 ). (3) If R is a rotation in RN then (k) (k) ZRθ0 (Rθ) = Zθ0 (θ) . (4) For every θ, θ0 ∈ SN we have dk (k) Zθ0 (θ) 6 ωN where ωN is the measure of the surface of the unit sphere. (5) For every θ ∈ SN rd N −2 (k) k 6 cN k 2 Yi (θ) 6 ωN (k)

Proof. 1. Since Zθ0 ∈ Hk , there exist coefficients ci (θ0 ) such that (k)

Zθ0 (θ) =

dk X

(k)

ci (θ0 ) Yi

(θ) .

i=1 (k)

Moreover, by the definition of Zθ0 (θ) , Z (k) (k) (k) Yj (θ0 ) = Yj (θ) Zθ0 (θ)dS (θ) =

SN d k X i=1

=

dk X i=1

Z ci (θ0 ) SN

(k)

Yj

(k)

(θ) Yi

(θ) dS (θ)

ci (θ0 )δij = cj (θ0 ) = cj (θ0 )

(12.67)

(12.68)

Nonvariational operators constructed with H¨ ormander vector fields

(k)

(since Yj

661

are real), which gives (12.66) (k)

2. Since Yi (12.66) satisfies

(k)

are real functions, clearly also Zθ0 (θ) is real and therefore by (k)

(k)

Zθ0 (θ) = Zθ (θ0 ) .
3. Let Y ∈ Hk , since also Y R−1 θ ∈ Hk (because the Laplacian is rotation invariant), we have Z Z
(k) (k) Y (θ) ZRθ0 (Rθ) dS (θ) = Y R−1 θ ZRθ0 (θ) dS (θ) SN SN
= Y R−1 Rθ0 = Y (θ0 ) . (k)

(k)

Thus, ZRθ0 (Rθ) represents the same functional on Hk as Zθ0 (θ) and therefore (k)

(k)

ZRθ0 (Rθ) = Zθ0 (θ). (k)

4. First of all observe that by point 3, Zθ0 (θ0 ) is independent of θ0 . Hence (k) Zθ0

1 (θ0 ) = ωN

Z SN

(k) Zθ

1 (θ) dS (θ) = ωN

Z SN

dk X dk (k) 2 . (12.69) Yi (θ) dS (θ) = ωN i=1

(k)

Since Zθ0 ∈ Hk , by the definition of zonal spherical harmonic we have Z (k) (k) (k) Zθ0 (θ) = Zθ0 (σ) Zθ (σ)dS (σ) . SN

Hence,

(k) (k) Zθ0 (θ) 6 Zθ0

L2 (SN )



(k) It remains to compute Zθ0



(k)

Zθ

L2 (SN )

. (k)

L2 (SN )

. Using point 1, orthonormality of Yi

and

(12.69) we have dk

2

X dk (k)

(k) 2 (k) . = Yi (θ0 ) = Zθ0 (θ0 ) =

Zθ0 2 ω L N i=1

(k) Hence Zθ0 (θ) 6

dk ωN

.

(k)

∈ Hk , then r Z

(k) dk (k)

(k) (k) (k) . Yi (σ) Zθ0 (σ) dS (σ) 6 Yi 2 N Zθ0 2 N 6 Yi (θ0 ) = ωN L (S ) L (S ) N 5. Let Yi

S

Finally, by (12.65) we have r

dk 6 ωN

s

N −2 cN k N −2 = c0N k 2 . ωN

662

H¨ ormander operators

n o (k) By Theorem 12.40 Yj is a complete orthonormal system in j=1,...,dk k=0,1,...+∞

L2 SN so that for f ∈ L2 SN the series dk +∞ X X

(k) fbj,k Yj (θ)

k=0 i=1

converges to f in L2 sense. We are going to show that for a smooth function, this expansion is uniformly convergent. In particular, it will converge in any norm
p N L S .
Proposition 12.44 Let f ∈ C ∞ SN and let Z (k) b fj,k = f (θ) Yj (θ) dS (θ) . SN

Then f (θ) =

dk +∞ X X

(k) fbj,k Yj (θ)

(12.70)

k=0 j=1

and the series converges uniformly.
Proof. Since the above series converges to f in L2 SN it is enough to show that it converges uniformly. By Proposition 12.41, (12.68) and (12.65) we have for every k>1 dk X cN,m

N −2 (k) b fj,k Yj (θ) 6 2m D2m f L2 (SN ) cN k 2 c0N k N −2 k j=1 3

6 CN,m,f · k 2 N −3−2m which gives a convergent series for m > N − 1. This means that the series (12.70) converges totally and therefore uniformly. We end this section with an estimate on the derivatives of spherical harmonics: Proposition 12.45 There exists cN > 0 such that for k = 1, 2, 3 . . . N

k∇Yk kL∞ (SN ) 6 cN k 2 . Proof. Let Yk ∈ Hk for some k > 1, then for some Pk ∈ Hk we can write, for every x ∈ RN   x −k = |x| Pk (x) . (12.71) Yk (x0 ) ≡ Yk |x|



∂

∂

In order to bound ∂x Y we will start proving a bound on P .

k k ∂x i i L∞ (SN )

By the divergence theorem and since ∆Pk = 0 we can write Z Z ∂Pk 2 (x)dS (x) = |∇Pk (x)| dx. Pk (x) ∂n N |x| 0 and k = 1, 2, 3, . . . there exists φc,k ∈ D (Ω) such that |T (φc,k )| > c kφc,k kC k (Ω0 ) . Letting ψc,k = φc,k / |T (φc,k )| , then 1 with |T (ψc,k )| = 1. c ∞ Then the sequence {ψk,k }k=1 converges to zero in D (Ω) (note that supp φj ⊂ Ω0 for every j) while T (ψk,k ) does not converge to zero, which is a contradiction since T ∈ D0 (Ω) . kψc,k kC k (Ω0 )
0 and a nonnegative integer N such that |T (f )| 6 cpN (f ) ∀f ∈ E (Ω) .

(A.6)

Proof. Assume first that (A.6) holds for some integer N0 and let fj → 0 in E (Ω). Then pN (fj ) → 0 as j → +∞, and by (A.6) T (fj ) → 0. Hence (since we already know that T is linear), T ∈ E 0 (Ω). Now, let T ∈ E 0 (Ω) and, by contradiction, assume (A.6) does not hold. Then for every c > 0 and N = 1, 2, 3, . . . there exists ψc,N ∈ E (Ω) such that (reasoning like in the proof of Proposition A.15) pN (ψc,N )
0 such that for every f ∈ E (Ω), Dα f (a) = 0 ∀α with |α| 6 N =⇒ hT, f i = 0. Proof of the Lemma. We can assume a = 0. Since T ∈ E 0 (Ω), by Proposition A.20 there exist c, N > 0 such that |T (f )| 6 cpN (f ) ∀f ∈ E (Ω) . For this integer N , let f ∈ E (Ω) such that Dα f (0) = 0 for every α with |α| 6 N . Let U1 , U2 be open neighborhoods of the origin with U1 b U2 b Ω. Just to simplify notation, assume we can take U1 = {|x| < 1} , U2 = {|x| < 2} . Let g ∈ C ∞ (Rn ) be such that g = 0 in U1 and g = 1 outside U2 . Define gk (x) = g (kx), so that gk (x) = 0 for |x| < 1/k and gk (x) = 1 for |x| > 2/k. Then supp (f gk ) ∩ supp T = ∅, so (by Exercise A.25) hT, f gk i = 0 and |hT, f i| = |hT, f − f gk i| 6 cpN (f − f gk ) . We are left to prove that pN (f − f gk ) → 0 as k → +∞. Now, (f − f gk ) (x) = 0 for |x| > 2/k hence   2 α pN (f − f gk ) = max |D (f (1 − gk )) (x)| : |x| 6 , |α| = 0, 1, . . . , N . k

674

H¨ ormander operators

Also, X

Dα (f (1 − gk )) =

cα1 α2 Dα1 f · Dα2 (1 − gk )

|α1 |+|α2 |=|α|

kD

α2

(1 − gk )k∞ 6 k |α2 | kDα2 (1 − g)k∞ .

Moreover, since Dα f (0) = 0 when |α| 6 N , if |x| 6 2/k we can write |Dα f (x)| = |Dα f (x) − Dα f (0)| 6 |x| sup |∇Dα f (y)| 2 |y|6 k

so that sup sup |Dα f (x)| 6

2 |x|6 k

|α|=j

2 sup k |x|6 2

k

sup |Dα f (x)| |α|=j+1

and iteratively  N −j+1 2 sup sup sup |D f (x)| 6 k |x|6 2 |x|6 2 |α|=j α

k

k

6

c k N −j+1

sup

|Dα f (x)|

sup |α|=N +1

|Dα f (x)| =

sup

|x|61 |α|=N +1

c (f ) . k N −j+1

Therefore pN (f − f gk ) 6 c

max

k |α2 |

|α1 |+|α2 |6N k N −|α1 |+1

6

c → 0, k

so we are done. Proof of Theorem A.27. We keep assuming a = 0. Since supp T is compact, by Proposition A.26 actually T ∈ E 0 (Ω). Let N be the integer given by the previous lemma. For f ∈ E (Ω) let us write its Taylor expansion of order N at 0 in the form f (x) = PN (x) + h (x) α

with h ∈ E (Ω) satisfying D h (0) = 0 for every α with |α| 6 N . Then hT, hi = 0 by the above lemma, hence * N + N X X X Dα f (0) X Dα f (0) hT, f i = hT, PN i = T, xα = hT, xα i . α! α! k=0 |α|=k

k=0 |α|=k

Note that |α|

hDα δ0 , f i = (−1)

hδ0 , Dα f i = (−1)

|α|

Dα f (0)

hence hT, f i =

N X |α| X (−1) hT, xα i hDα δ0 , f i α!

k=0 |α|=k

so that T =

N X X k=0 |α|=k

|α|

cα Dα δ0 with cα =

(−1) α!

where the cα are well defined constants since T ∈ E 0 (Ω).

hT, xα i

Short summary of distribution theory

675

Theorem A.29 (Local structure of distributions) Let T ∈ D0 (Ω) and let Ω0 be an open set compactly contained in Ω. There exist a compactly supported continuous function g ∈ C0 (Ω) and a multiindex α such that for every φ ∈ D (Ω) Z |α| α hT, φi = hD g, φi = (−1) g (x) Dα φ (x) dx. Ω0

In other words: every distribution is, locally, a derivative of a compactly supported continuous function. Proof. Let k0 be an integer such that Ωk0 k Ω0 (where {Ωk } is the sequence of domains which appears in Definition A.19). By Proposition A.15, there exist c > 0 and an integer N (which is not restrictive to assume > k0 , so that ΩN k Ω0 ) such that |hT, φi| 6 c kφkC N (Ω0 ) 6 cpN (φ)

∀φ ∈ D (Ω0 ) .

(A.7)

Fix φ ∈ D (Ω0 ) . For every multiindex α with 0 6 |α| 6 N take x0 ∈ Ω0 such that |Dα φ (x0 )| = kDα φk∞ . For a fixed a ∈ ∂Ω0 (so that Dα φ (a) = 0) we can apply Lagrange theorem on the segment [a, x0 ] (since φ can be thought as defined and smooth in the whole space, it is not important that this segment be totally contained in Ω0 ) and find y0 such that kDα φk∞ = |Dα φ (x0 ) − Dα φ (a)| 6 |x0 − a| |∇Dα φ (y0 )| so that max kDα φk∞ 6 diam (Ω0 ) max kDα φk∞

|α|=j

|α|=j+1

and iteratively pN (φ) = max kDα φk∞ 6 c (diam (Ω0 )) |α|6N

N

max kDα φk∞

|α|=N

(A.8)

Next, note that for any φ ∈ D (Ω0 ) , if Q is a cube {maxi=1,...,n |xi | < R} containing Ω0 we can represent Z x1 Z x2 Z xn ∂n φ (x1 , x2 , . . . , xn ) = ... φ (t1 , t2 , . . . , tn ) dt1 . . . dtn . −R −R −R ∂x1 ∂x2 . . . ∂xn This also holds for Dα φ, so we can bound Z x1 Z x2 Z xn ∂n α kDα φk∞ 6 sup ... D φ (t , t , . . . , t ) 1 2 n dt1 . . . dtn x −R −R −R ∂x1 ∂x2 . . . ∂xn



∂n α

6 D φ ∂x1 ∂x2 . . . ∂xn L1 (Ω0 ) so that max kDα φk∞ 6

|α|=N

max kDα φkL1

|α|=N +n

676

H¨ ormander operators

which together with (A.8) and (A.7) gives, letting M = N + n, |hT, φi| 6 c max kDα φkL1 for every φ ∈ D (Ω0 ) .

(A.9)

|α|=M

Let n o DM (D (Ω0 ) ) = {Dα φ}|α|=M : φ ∈ D (Ω0 ) (an element of DM (D (Ω0 ) ) is a vector of functions). The linear functional ψ : DM (D (Ω0 ) ) → R ψ : {Dα φ}|α|=M 7→ hT, φi is well defined because, for φ1 , φ2 ∈ D (Ω0 ) , {Dα φ1 }|α|=M = {Dα φ2 }|α|=M =⇒ φ1 = φ2 , and is continuous in L1 norm by (A.9). By the Hahn-Banach theorem, it can be extended to a linear continuous functional, which we still call ψ, ψ : L1 (Ω0 )

m

→R

where m = card (α : |α| = M ) . By Riesz’ representation theorem, there exist
m m functions {hα }|α|=M , hα ∈ L∞ (Ω0 ) , which represent this functional on L1 (Ω0 ) , and in particular on DM (D (Ω0 ) ): for every φ ∈ D (Ω0 ) we have * +   X Z X M α α α hα D φ = (−1) D hα , φ hT, φi = ψ {D φ}|α|=M = |α|=M

Ω0

|α|=M

which shows that on Ω0 we can write X M T = (−1) Dα hα for suitable hα ∈ L∞ (Ω0 ) . |α|=M

Next, we can rewrite this representation in terms of derivatives of continuous functions. To do this, let us first note that, since we will always evaluate hDα hα , φi for φ supported inside Ω0 , we can also extend the functions hα to zero outside Ω0 without changing hDα hα , φi. Then we can define the functions Z x1 Z x2 Z xn gα (x1 , . . . xn ) = ... hα (t1 , t2 , . . . , tn ) dt1 . . . dtn −R

−R

−R

with R as above. We obviously have that gα is continuous (in the whole space) and ∂ n gα = hα ∂x1 ∂x2 . . . ∂xn in weak (and then in distributional) sense. Therefore in Ω0   X ∂n M T = Dα (−1) gα ∂x1 ∂x2 . . . ∂xn

(A.10)

|α|=M

which is the sum of m derivatives of the continuous functions (−1)

M

gα .

Short summary of distribution theory

677

Finally, we can show that the expression (A.10) can be rewritten as a derivative of a single continuous function. It is enough to prove the assertion for the case of two continuous functions (and then iterate the reasoning): we will show that if T coincides in Ω0 with Dα1 g1 + Dα2 g2 with g1 , g2 functions in C0 (Ω) then it can be rewritten as Dα1 +α2 g for another function g ∈ C0 (Ω) . First of all, given a function f ∈ C0 (Ω) a multiindex α = (a1 , . . . , an ), we want to construct a function I α f ∈ C (Ω) such that Dα I α f = f. It is enough to integrate ai times with respect to the i-th variable. More precisely, let us define the operators Z xi f (x1 , . . . , xi−1 , t, . . . , xn ) dt Ii f (x) = −R

Iik f (x) = Ii (Ii (. . . (Ii f (x)))) if k is a positive integer, | {z } k times

Ii0 f

(x) = f (x)

and, for a multiindex α = (a1 , . . . , an ) , I α f (x) = I1a1 (I2a2 (. . . (Inan f (x)))) . Then I α f is (more than) continuous, and Dα I α f = f. Therefore we can write, in the distributional sense, T = Dα1 g1 + Dα2 g2 = Dα1 +α2 (I α2 g1 + I α1 g2 ) in Ω0 where g = I α2 g1 + I α1 g2 is continuous in Ω. Actually, for any φ ∈ D (Ω0 )

α1 +α2 



 D g, φ = Dα1 +α2 (I α2 g1 ) , φ + Dα1 +α2 (I α1 g2 ) , φ = (−1)

|α1 |

hDα2 (I α2 g1 ) , Dα1 φi + (−1)

= (−1)

|α1 |

hg1 , Dα1 φi + (−1)

|α2 |

|α2 |

hDα1 (I α1 g2 ) , Dα2 φi

hg2 , Dα2 φi

= hDα1 g1 + Dα2 g2 , φi = hT, φi . Finally, since Ω0 b Ω we can multiply g for a cutoff function χ ∈ D (Ω) such that χ = 1 in a neighborhood of Ω0 without changing the representation of T in Ω0 . The new function gχ belongs to C0 (Ω) , so we are done. We end this section with the following result, which has been used in the proof of Theorem 6.33. It strengthens in a nontrivial way the easy results about converging sequences of distributions which we have previously stated. The standard proof of this result makes use of Banach-Steinhaus’ theorem in Fr´echet spaces. The proof that we will give (taken from [78]) is more self-contained and, as a consequence, a bit longer than the standard one.

678

H¨ ormander operators ∞

Theorem A.30 Let {Tj }j=1 ⊂ D0 (Ω) such that the (numerical) sequence hTj , φi is convergent for every φ ∈ D (Ω) . Then the limit T (φ) = lim hTj , φi j→∞

actually defines a distribution T . Proof. The linearity of the limit T is obvious; the problem is to prove (A.1). By ∞ contradiction, assume T ∈ / D0 (Ω). Then there exists {φj }j=1 ⊂ D (Ω) such that φj → 0 in D (Ω) but T (φj ) does not converge to zero. Passing to a subsequence, if necessary, we can assume that |T (φj )| > c > 0 for every j. Also, since φj → 0 in D (Ω) there exists Ω0 b Ω such that supp φj ⊂ Ω0 and, possibly passing to another subsequence, kφj kC j (Ω0 )
2 (this is possible because T φej → +∞) D E D E   and Tn1 such that Tn1 , φei1 > 2 (this is possible because Tj , φei1 → T φei1 ). We now proceed inductively. to have already chosen φeik and Tnk for 1 6 n Assume o k < j. We choose φeij from φej such that:



e

φij

C j (Ω0 )


j, and φei

C j (Ω0 )

D E Tnk , φeij
T, φeik + j + 1

6

1 2j );

(A.12)

(A.13)

16k

679

(A.14)

16k Tnj , φeij − Tnj , φeik Tnj , φeik − k=1 ∞ X

k=j+1

1 = j. 2k−j k=j+1

 We have found a ψ ∈ D (Ω) such that Tnj , ψ → +∞, which is a contradiction. >j+1−

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Index

A priori estimates, 394, 577, 585, 600 Adjoint map, 11 Approximating operator, 620 Approximation, 479, 483, 506, 510, 511 Approximation in H¨ older spaces, 86

on homogeneous groups, 141 Doubling property, 104, 400 Exponential map, 116 Exponential of a vector field, 2 Exterior normal, 42

Ball-box theorem, 404 BCH formula, 408, 409, 414–416, 466, 469 BMO space, 616

Finite difference, 157 Formal Taylor expansion, 410 Fourier transform, 193 Fractional derivative, 199 integral, 301, 316, 345 integral kernel, 299 Free nilpotent Lie algebra, 523 Free vector fields, 482, 487 Frobenius theorem, 50 Frozen kernel, 622 operator, 622 Fundamental solution, 251, 264, 343

Calder´ on-Zygmund decomposition, 319 Canonical homogeneous H¨ ormander operator with drift, 135 sublaplacian, 125, 254 Canonical basis (of a homogeneous Lie algebra), 110 Canonical coordinates, 499 Carnot group, 125 Characterization of sublaplacians, 184 Commutator, 4, 202, 626, 637, 639 Connectivity, 19, 24, 128 Control ball, 56, 400 Control distance, 17, 125, 402 Convolution, 118, 142 Cutoff functions, 384, 578, 595

Gauge, 98 balls, 104 quasidistance, 103 Geometric control theory, xxv Global H¨ older estimates, 380 homogeneous fundamental solution, 264 invertibility, 449 Sobolev estimates, 360 Sobolev regularity, 339 Graded Lie algebra, 136 Groups of Heisenberg type, 144, 286

Derivative of a convolution, 120 of a distribution, 668 Dirichlet problem, 132 Distance d, 402, 458 d∗ , 402, 458 Distribution, 668 691

692

H¨ ormander operators

H¨ older continuity, 305, 306 estimates, 378, 627 space, 80, 298 H¨ ormander operator, 9, 14 vector fields, 8 H¨ ormander’s condition, 8, 12 condition of step s, 13 theorem, xvi, 187 Heat type operators, 149, 288 Heisenberg group, 143, 285 Homogeneous differential operator, 106 function, 95 group, 94 H¨ ormander operator, 255 Lie algebra, 115 norm, 98 stratified Lie algebra, 124 stratified Lie algebra of type II, 135 Homotopy, 439 lifting, 440 Hypoellipticity, xiv, 153, 174, 187, 232, 234 Inequalities for H¨ older norms, 84 Interior estimates, 537, 587, 602, 617 regularity, 536, 588, 603, 616 Interpolation inequality, 371, 382, 385, 394, 580, 583, 593, 596 Intrinsic derivative, 536 Kernel of type 0, 354 of type α > 0, 346 of type λ, 543 Kohn Laplacian, xxiii Kolmogorov-Fokker-Planck operator, xxii, 137, 147, 286 Korn’s trick, 628 Lagrange Theorem, 36, 37 Laplace-Beltrami operator, 655 Lebesgue differentiation theorem, 314 Left invariant differential operator, 106 vector field, 108

Lie algebra, 8 Lie algebra of left invariant vector fields, 109 Lie group, 94 Lie polynomial, 469 Lifted operator, 620 Lifting, 439, 479, 483, 493 Lipschitz space, 298 Local Lp estimates, 373 Campanato spaces, 327 diffeomorphism, 438 fundamental solution, 263 H¨ older estimates, 386, 392 H¨ older regularity, 341 homogeneous dimension, 296 maximal function, 313 Sobolev estimates, 370 Sobolev regularity, 340 Localized kernel, 299 Localized subelliptic estimate, 229 Locally doubling metric spaces, 295 Map Θ, 509 Marchaud inequality, 169 Marcinkievicz interpolation theorem, 333 Maximal function, 345 Maximum principles, 37, 39, 41 Mean value inequality, 272 Mollifier, 72, 121, 197 Nilpotent Lie algebra, 115 Nonvariational operator, 615 Normal family, 244 Operator of order m, 200, 232 Operators of type λ, 544, 566, 619 Parametrix, 174, 560, 623 Path, 438 Poincar´e inequality, 129, 140, 188 Principal value distribution, 276 Propagation of maxima, 41, 48 Quasidistance, 103 Quasiexponential maps, 26, 417 Rank of a Lie algebra, 8 Regularity estimates, 154, 187

Index

693

Regularity of control balls, 464 Regularization, 646 Representation formula, 281, 283, 379, 564, 565, 624, 625, 627, 629 Right invariant differential operator, 106 vector field, 108

Transpose, 213, 546 of a vector field, 111 operator, 107 Two-sided fundamental solution, 269

Segment property, 462 Simple operator, 212 Singular integral, 301, 309, 325 integral kernel, 299 Sobolev embedding, 351, 353, 653 Sobolev regularity, 362, 366 Sobolev space, 68, 195 Solvability, 253, 340, 341, 373, 386, 537, 605 Space of homogeneous type, 104 Spherical harmonics, 635 Standard commutator, 10 Stratified Lie algebra, 123 Strong segment property, 463 Structure of balls, 404, 438 Subelliptic estimates, 191, 217, 222, 238 Sublaplacian, 143, 157, 184 Suboptimal bases, 420 System of H¨ ormander vector fields, 13

Vanishing mean property, 280 Variable kernel, 626 Vitali covering Lemma, 312 VMO space, 615 Volume of d∗ -ball, 405 Volume of control balls, 56, 400

Uniform subelliptic estimates, 257

Weak (1, 1) estimate, 318 compactness, 70 derivative, 67 segment property, 463 solution, 132 Weight of a vector field, 503 Weighted control distance, 21 vector fields, 10 Weights, 501 Weyl’s Lemma, xiii Young’s inequality, 119

Tempered distributions, 194 Test functions, 667

Zonal spherical harmonics, 660