Advances in Analysis: The Legacy of Elias M. Stein (PMS-50) [Course Book ed.] 9781400848935

Table of contents :
Contents
Preface
Chapter One. Selected Theorems by Eli Stein
Chapter Two. Eli’s Impact: A Case Study
Chapter Three On Oscillatory Integral Operators in Higher Dimensions
Chapter Four. Hölder Regularity for Generalized Master Equations with Rough Kernels
Chapter Five. Extremizers of a Radon Transform Inequality
Chapter Six. Should We Solve Plateau’s Problem Again?
Chapter Seven. Averages along Polynomial Sequences in Discrete Nilpotent Lie Groups: Singular Radon Transforms
Chapter Eight. Internal DLA for Cylinders
Chapter Nine. The Energy Critical Wave Equation in 3D
Chapter Ten. On the Bounded L2 Curvature Conjecture
Chapter Eleven. On Div-Curl for Higher Order
Chapter Twelve. Square Functions and Maximal Operators Associated with Radial Fourier Multipliers
Chapter Thirteen. Problems of Harmonic Analysis Related to Finite-Type Hypersurfaces in R3, and Newton Polyhedra
Chapter Fourteen. Multi-Linear Multipliers Associated to Simplexes of Arbitrary Length
Chapter Fifteen. Diagonal Estimates for Bergman Kernels in Monomial-Type Domains
Chapter Sixteen. On the Singularities of the Pluricomplex Green’s Function
Chapter Seventeen. Smoothness of Spectral Multipliers and Convolution Kernels in Nilpotent Gelfand Pairs
Chapter Eighteen. On Eigenfunction Restriction Estimates and L4-Bounds for Compact Surfaces with Nonpositive Curvature
List of Contributors
Index

Citation preview

Advances in Analysis

Princeton Mathematical Series Editors: Phillip A. Griffiths and John N. Mather 1. The Classical Groups by Hermann Weyl 8. Theory of Lie Groups: I by Claude Chevalley 9. Mathematical Methods of Statistics by Harald Cramér 14. The Topology of Fibre Bundles by Norman Steenrod 17. Introduction to Mathematical Logic, Vol. I by Alonzo Church 19. Homological Algebra by Henri Cartan and Samuel Eilenberg 28. Convex Analysis by R. Tyrrell Rockafellar 30. Singular Integrals and Differentiability Properties of Functions by Elias M. Stein 32. Introduction to Fourier Analysis on Euclidean Spaces by Elias M. Stein and Guido Weiss 33. Étale Cohomology by James S. Milne 35. Three-Dimensional Geometry and Topology, Volume 1 by William P. Thurston. Edited by Silvio Levy 36. Representation Theory of Semisimple Groups: An Overview Based on Examples by Anthony W. Knapp 38. Spin Geometry by H. Blaine Lawson, Jr., and Marie-Louise Michelsohn 43. Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals by Elias M. Stein 44. Topics in Ergodic Theory by Yakov G. Sinai 45. Cohomological Induction and Unitary Representations by Anthony W. Knapp and David A. Vogan, Jr. 46. Abelian Varieties with Complex Multiplication and Modular Functions by Goro Shimura 47. Real Submanifolds in Complex Space and Their Mappings by M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild 48. Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane by Kari Astala, Tadeusz Iwaniec, and Gaven Martin 49. A Primer on Mapping Class Groups by Benson Farb and Dan Margalit 50. Advances in Analysis: The Legacy of Elias M. Stein edited by Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger

Advances in Analysis: The Legacy of Elias M. Stein

Edited by Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

c 2014 by Princeton University Press Copyright
Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved Library of Congress Cataloging-in-Publication Data Advances in analysis : the legacy of Elias M. Stein / edited by Charles Fefferman, Alexandru D. Ionescu, D.H. Phong, and Stephen Wainger. pages cm. — (Princeton mathematical series ; 50) Summary: "Princeton University’s Elias Stein was the first mathematician to see the profound interconnections that tie classical Fourier analysis to several complex variables and representation theory. His fundamental contributions include the Kunze-Stein phenomenon, the construction of new representations, the Stein interpolation theorem, the idea of a restriction theorem for the Fourier transform, and the theory of Hp Spaces in several variables. Through his great discoveries, through books that have set the highest standard for mathematical exposition, and through his influence on his many collaborators and students, Stein has changed mathematics. Drawing inspiration from Stein’s contributions to harmonic analysis and related topics, this volume gathers papers from internationally renowned mathematicians, many of whom have been Stein’s students. The book also includes expository papers on Stein’s work and its influence.The contributors are Jean Bourgain, Luis Caffarelli, Michael Christ, Guy David, Charles Fefferman, Alexandru Ionescu, David Jerison, Carlos Kenig, Sergiu Klainerman, Loredana Lanzani, Sanghyuk Lee, Lionel Levine, Akos Magyar, Detlef Müller, Camil Muscalu, Alexander Nagel, D. H. Phong, Malabika Pramanik, Andrew Raich, Fulvio Ricci, Keith Rogers, Andreas Seeger, Scott Sheffield, Luis Silvestre, Christopher Sogge, Jacob Sturm, Terence Tao, Christoph Thiele, Stephen Wainger, and Steven Zelditch” — Provided by publisher. Includes bibliographical references and index. ISBN 978-0-691-15941-6 (hardback) 1. Mathematical analysis—Congresses. I. Stein, Elias M., 1931– honouree. II. Fefferman, Charles, 1949– editor of compilation. III. Ionescu, Alexandru Dan, 1973– editor of compilation. IV. Phong, Duong H., 1953– editor of compilation. V. Wainger, Stephen, 1936– editor of compilation. QA299.6.A38 2014 515—dc23 2013020363 British Library Cataloging-in-Publication Data is available This book has been composed in Times Printed on acid-free paper. ∞ Typeset by S R Nova Pvt Ltd, Bangalore, India Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

To Eli

Contents

Preface

ix

Chapter 1 Selected Theorems by Eli Stein

1

Charles Fefferman Chapter 2 Eli’s Impact: A Case Study

35

Charles Fefferman Chapter 3 On Oscillatory Integral Operators in Higher Dimensions

47

Jean Bourgain Chapter 4 Hölder Regularity for Generalized Master Equations with Rough Kernels

63

Luis Caffarelli and Luis Silvestre Chapter 5 Extremizers of a Radon Transform Inequality

84

Michael Christ Chapter 6 Should We Solve Plateau’s Problem Again?

108

Guy David Chapter 7 Averages along Polynomial Sequences in Discrete Nilpotent Lie Groups: Singular Radon Transforms

146

Alexandru D. Ionescu, Akos Magyar, and Stephen Wainger Chapter 8 Internal DLA for Cylinders

189

David Jerison, Lionel Levine, and Scott Sheffield Chapter 9 The Energy Critical Wave Equation in 3D

215

Carlos Kenig Chapter 10 On the Bounded L2 Curvature Conjecture

224

Sergiu Klainerman Chapter 11 On Div-Curl for Higher Order

Loredana Lanzani and Andrew S. Raich

245

viii

CONTENTS

Chapter 12 Square Functions and Maximal Operators Associated with Radial Fourier Multipliers

273

Sanghyuk Lee, Keith M. Rogers, and Andreas Seeger Chapter 13 Problems of Harmonic Analysis Related to Finite-Type Hypersurfaces in R3 , and Newton Polyhedra

303

Detlef Müller Chapter 14 Multi-Linear Multipliers Associated to Simplexes of Arbitrary Length

346

Camil Muscalu, Terence Tao, and Christoph Thiele Chapter 15 Diagonal Estimates for Bergman Kernels in Monomial-Type Domains

402

Alexander Nagel and Malabika Pramanik Chapter 16 On the Singularities of the Pluricomplex Green’s Function

419

D. H. Phong and Jacob Sturm Chapter 17 Smoothness of Spectral Multipliers and Convolution Kernels in Nilpotent Gelfand Pairs

436

Fulvio Ricci Chapter 18 On Eigenfunction Restriction Estimates and L4 -Bounds for Compact Surfaces with Nonpositive Curvature

447

Christopher D. Sogge and Steve Zelditch List of Contributors

463

Index

465

Preface This is the proceedings of the conference “Analysis and Applications,” held in May 2011 in honor of the 80th birthday of Elias M. Stein. The reader will find here an extraordinary range of deep mathematics, covering most of the main trends in harmonic analysis, partial differential equations, and allied fields. All bear the stamp of Stein’s broad and original view of the subject. It is a pleasure to dedicate this book to him. We are grateful to the American Institute of Mathematics, the Clay Mathematics Institute, the National Science Foundation, and the Princeton University Mathematics Department for the financial support that made the conference possible. Many people worked hard to make the conference a success. We are especially grateful to Scott Kenney, Gale Sandor, and Carol DiSanto for solving many practical problems. Our thanks go also to Vickie Kearn, Quinn Fusting, and Nathan Carr at Princeton University Press for invaluable help in producing this book. We mourn the passing of Frances Wroblewski of the Princeton Mathematics Department; she also contributed invaluable help. We are deeply grateful to the speakers and participants who made the conference “Analysis and Applications” a major success. Most of all, we are grateful to Eli Stein. C. Fefferman, A. D. Ionescu, D. H. Phong, S. Wainger Fall 2012

Advances in Analysis

Chapter One Selected Theorems by Eli Stein Charles Fefferman INTRODUCTION The purpose of this chapter is to give the general reader some idea of the scope and originality of Eli Stein’s contributions to analysis∗ . His work deals with representation theory, classical Fourier analysis, and partial differential equations. He was the first to appreciate the interplay among these subjects, and to preceive the fundamental insights in each field arising from that interplay. No one else really understands all three fields; therefore, no one else could have done the work I am about to describe. However, deep understanding of three fields of mathematics is by no means sufficient to lead to Stein’s main ideas. Rather, at crucial points, Stein has shown extraordinary originality, without which no amount of work or knowledge could have succeeded. Also, large parts of Stein’s work (e.g., the fundamental papers [26, 38, 41, 59] on complex analysis in tube domains) don’t fit any simple one-paragraph description such as the one above. It follows that no single mathematician is competent to present an adequate survey of Stein’s work. As I attempt the task, I am keenly aware that many of Stein’s papers are incomprehensible to me, while others were of critical importance to my own work. Inevitably, therefore, my survey is biased, as any reader will see. Fortunately, S. Gindikin provided me with a layman’s explanation of Stein’s contributions to representation theory, thus keeping the bias (I hope) within reason. I am grateful to Gindikin for his help, and also to Y. Sagher for a valuable suggestion. For purposes of this chapter, representation theory deals with the construction and classification of the irreducible unitary representations of a semisimple Lie group. Classical Fourier analysis starts with the Lp -boundedness of two fundamental operators, the maximal function.
x+h 1 |f (y)|dy, f ∗ (x) = sup h>0 2h x−h and the Hilbert transform 1
→0+ π

Hf (x) = lim


|x−y|>


f (x)dy . x−y

Finally, we shall be concerned with those problems in partial differential equations that come from several complex variables. ∗ Reprinted from Essays on Fourier Analysis in Honor of Elias M. Stein, Princeton University Press, 1995.

2

CHARLES FEFFERMAN

COMPLEX INTERPOLATION Let us begin with Stein’s work on interpolation of operators. As background, we state and prove a clasical result, namely the M. Riesz Convexity Theorem. Suppose X, Y are measure spaces, and suppose T is an operator that carries functions on X to functions on Y. Assume T is bounded from Lp0 (X) to Lr0 (Y ), and from Lp1 (X) to Lr1 (Y ). (Here, p0 , p1 , r0 , r1 ∈ [1, ∞].) , 1r = rt1 + (1−t) , Then T is bounded from Lp (X) to Lr (Y ) for p1 = pt1 + (1−t) p0 r0 0 ≤ t ≤ 1. The Riesz Convexity Theorem says that the points ( p1 , 1r ) for which T is bounded from Lp to Lr from a convex region in the plane. A standard application is the Hausdorff-Young inequality: We take T to be the Fourier transform on Rn , and note that T is obviously bounded from L1 to L∞ , and from L2 to L2 . Therefore, T  is bounded from Lp to the dual class Lp for 1 ≤ p ≤ 2.  The idea of the proof of the Riesz Convexity Theorem is to estimate Y (Tf )·g for  f ∈ Lp and g ∈ Lr . Say f = F eiφ and g = Geiψ with F, G ≥ 0 and φ, ψ real. Then we can define analytic families of functions fz , gz by setting fz = F az+b eiφ , gz = Gcz+d eiψ , for real a, b, c, d to be picked in a moment. Define
(z) = (Tfz )gz . (1) Y

Evidently,  is an analytic function of z. For the correct choice of a, b, c, d we have 







|fz |p0 = |f |p

and

|gz |r0 = |g|r

|fz |p1 = |f |p

and

|gz |r1 = |g|r

fz = f

and

gz = g

when

Re z = 0;

(2)

when

Re z = 1;

(3)

when

z = t.

(4)

From (2) we see that fz Lp0 , gz Lr  ≤ C for Re z = 0. So the definition (1) and 0 the assumption T : Lp0 → Lr0 show that |(z)| ≤ C 

for Re z = 0.

(5)

Similarly, (3) and the assumption T : Lp1 → Lr1 imply |(z)| ≤ C 

for Re z = 1.

(6)

Since  is analytic, (5) and (6) imply |(z)| ≤ C  for 0 ≤ Re z ≤ 1 by the maximum principle for a strip. In particular, |(t)| ≤ C  . In view of (4), this  means that | Y (Tf )g| ≤ C  , with C  determined by f Lp and gLr  . Thus, T is bounded from Lp to Lr , and the proof of the Riesz Convexity Theorem is complete. This proof had been well-known for over a decade, when Stein discovered an amazingly simple way to extend its usefulness by an order of magnitude. He realized that an ingenious argument by Hirschman [H] on certain multiplier operators

3

SELECTED THEOREMS BY ELI STEIN

on Lp (Rn ) could be viewed as a Riesz Convexity Theorem for analytic families of operators. Here is the result. Stein Interpolation Theorem. Assume Tz is an operator depending analytically on z in the strip 0 ≤ Re z ≤ 1. Suppose Tz is bounded from Lp0 to Lr0 when Re z = 0, and from Lp1 toLr1 when Re z = 1. Then Tt is bounded from Lp to Lr , , 1r = rt1 + (1−t) and 0 ≤ t ≤ 1. where p1 = pt1 + (1−t) po r0 Remarkably, the proof of the theorem comes from that of the Riesz Convexity  of the alphabet. Instead of taking (z) =  Theorem by adding a single letter (Tf )g as in (1), we set (z) = z z y Y (Tz fz )gz . The proof of the Riesz Convexity Theorem then applies with no further changes. Stein’s Interpolation Theorem is an essential tool that permeates modern Fourier analysis. Let me just give a single application here, to illustrate what it can do. The example concerns Cesaro summability of multiple Fourier integrals. We define an operator TαR on functions on Rn by setting  α |ξ |2 fˆ(ξ ). T
αR f (ξ ) = 1 − R2 + Then TαR f 

Lp (Rn )

≤ Cαp f 

Lp (Rn )

,

if

  1   − 1 < α . p 2 n − 1

(7)

This follows immediately from the Stein Interpolation Theorem. We let α play the role of the complex parameter z, and we interpolate between the elementary cases p = 1 and p = 2. Inequality (7), due to Stein, was the first non-trivial progress on spherical summation of multiple Fourier series.

REPRESENTATION THEORY 1 Our next topic is the Kunze-Stein phenomenon, which links the Stein Interpolation Theorem to representations of Lie groups. For simplicity we restrict attention to G = SL(2, R), and begin by reviewing elementary Fourier analysis on G. The irreducible unitary representations of G are as follows: The principal series, parametrized by a sign σ = ±1 and a real parameter t; The discrete series, parametrized by a sign σ = ±1 and an integer k ≥ 0; and The complementary series, parametrized by a real number t ∈ (0, 1). We don’t need the full description of these representations here. The irreducible representations of G give rise to a Fourier transform. If f is a function on G, and U is an irreducible unitary representation of G, then we define
f (g)Ug dg, fˆ(U ) = G

4

CHARLES FEFFERMAN

where dg denotes Haar measure on the group. Thus, fˆ is an operator-valued function defined on the set of irreducible unitary representations of G. As in the Euclidean case, we can analyze convolutions in terms of the Fourier transform. In fact, f
∗ g = fˆ · gˆ

(8)

as operators. Moreover, there is a Plancherel formula for G, which asserts that
f 2L2 (G) = fˆ(U )2Hilbert-Schmidt dµ(U ) for a measure µ (the Plancherel measure). The Plancherel measure for G is known, but we don’t need it here. However, we note that the complementary series has measure zero for the Plancherel measure. These are, of course, the analogues of familiar results in the elementary Fourier analysis of Rn . Kunze and Stein discovered a fundamental new phenomenon in Fourier analysis on G that has no analogue on Rn . Their result is as follows. Theorem (Kunze-Stein Phenomenon). There exists a uniformly bounded representation Uσ,τ of G, parametrized by a sign σ = ±1 and a complex number τ in a strip , with the following properties. (A) The Uσ,τ all act on the same Hilbert space H. (B) For fixed σ = ±1, g ∈ G, and ξ , η ∈ H , the matrix element (Uσ,τ )g ξ, η is an analytic function of τ ∈ . (C) The Uσ,τ for Re τ = 12 are equivalent to the representations of the principal series. (D) The U+1,τ for suitable τ are equivalent to the representations of the complementary series. (See [14] for the precise statement and proof, as well as Ehrenpreis-Mautner [EM] for related results.) The Kunze-Stein Theorem suggests that analysis on G resembles a fictional version of classical Fourier analysis in which the basic exponential ξ  −→ exp(iξ · x) is a bounded analytic function on strip |Im ξ | ≤ C, uniformly for all x. As an immediate consequence of the Kunze-Stein Theorem, we can give an analytic continuation of the Fourier transform for G. In fact, we set fˆ(σ, τ ) =  f (g)(U σ,τ )g dg for σ = ±1, τ ∈ . G Thus, f ∈ L1 (G) implies fˆ(σ, ·) analytic and bounded on . So we have continued analytically the restriction of fˆ to the principal series. It is as if the Fourier transform of an L1 function on (−∞, ∞) were automatically analytic in a strip. If f ∈ L2 (G), then fˆ(σ, τ ) is still defined on the line {Re τ = 12 }, by virtue of the Plancherel formula and part (C) of the Kunze-Stein Theorem. Interpolating between L1 (G) and L2 (G) using the Stein Interpolation Theorem, we see that f ∈  Lp (G)(1 ≤ p < 2) implies fˆ(σ, ·) analytic and satisfying an Lp -inequality on a strip p . As p increases from 1 to 2, the strip p shrinks from to the line {Re τ = 12 }. Thus we obtain the following results. Corollary 1. If f ∈ Lp (G)(1 ≤ p < 2), then fˆ is bounded almost everywhere with respect to the Plancherel measure.

SELECTED THEOREMS BY ELI STEIN

5

Corollary 2. For 1 ≤ p < 2 we have the convolution inequality f ∗ gL2 (G) ≤ Cp f Lp (G) gL2 (G) . To check Corollary 1, we look separately at the principal series, the discrete  series, and the complementary series. For the principal series, we use the Lp inequality established above for the analytic function τ  −→ fˆ(σ, τ ) on the strip 

p . Since an Lp -function analytic on a strip p is clearly bounded on an interior line {Re τ = 12 }, it follows at once that fˆ is bounded on the principal series. Regarding the discrete series Uσ,k we note that  1/p   p µσ,k fˆ(Uσ,k)  ≤ f Lp (G) (9) σ,k

for suitable weights µσ,k and for 1 ≤ p ≤ 2. The weights µσ,k amount to the Plancherel measure on the discrete series, and (9) is proved by a trivial interpolation, just like the standard Hausdorff-Young inequality. The boundedness of the fˆ(Uσ,k ) is immediate from (9). Thus the Fourier transform fˆ is bounded on both the principal series and the discrete series, for f ∈ Lp (G) (1 ≤ p < 2). The complementary series has measure zero with respect to the Plancherel measure, so the proof of Corollary 1 is complete. Corollary 2 follows trivially from Corollary 1, the Plancherel formula, and the elementary formula (8). This proof of Corollary 2 poses a significant challenge. Presumably, the corollary holds because the geometry of G at infinity is so different from that of Euclidean space. For example, the volume of the ball of radius R in G grows exponentially as R → ∞. This must have a profound impact on the way mass piles up when we take convolutions on G. On the other hand, the statement of Corollary 2 clearly has nothing to do with cancellation; proving the corollary for two arbitrary functions f , g is the same as proving it for |f | and |g|. When we go back over the proof of Corollary 2, we see cancellation used crucially, e.g., in the Plancherel formula for G; but there is no explicit mention of the geometry of G at infinity. Clearly there is still much that we do not understand regarding convolutions on G. The Kunze-Stein phenomenon carries over to other semisimple groups, with profound consequences for representation theory. We will continue this discussion later in the chapter. Now, however, we turn our attention to classical Fourier analysis.

CURVATURE AND THE FOURIER TRANSFORM One of the most fascinating themes in Fourier analysis in the last two decades has been the connection between the Fourier transform and curvature. Stein has been the most important contributor to this set of ideas. To illustrate, I will pick out two of his results. The first is a “restriction theorem,” i.e., a result on the restriction fˆ| of the Fourier transform of a function f ∈ Lp (Rn ) to a set of measure zero. If  p > 1, then the standard inequality fˆ ∈ Lp (Rn ) suggests that fˆ should not even be

6

CHARLES FEFFERMAN

well-defined on , since has measure zero. Indeed, if is (say) the x-axis in the plane R2 , then we can easily find functions f (x1 , x2 ) = ϕ(x1 )ψ(x2 ) ∈ Lp (R2 ) for which fˆ| is infinite everywhere. Fourier transforms of f ∈ Lp (R2 ) clearly cannot be restricted to straight lines. Stein proved that the situation changes drastically when is curved. His result is as follows. Stein’s Restriction Theorem. Suppose is the unit circle, 1 ≤ p < 87 , and f ∈ C0∞ (R2 ). Then we have the a priori inequality fˆ| L2 ≤ Cp f Lp (R2 ) , with Cp depending only on p. Using this a priori inequality, we can trivially pass from the dense subspace C0∞ to define the operator f  −→ fˆ| for all f ∈ Lp (R2 ). Thus, the Fourier transform of f ∈ Lp (p < 87 ) may be restricted to the unit circle. Improvements and generalizations were soon proven by other analysts, but it was Stein who first demonstrated the phenomenon of restriction of Fourier transforms. Stein’s proof of his restriction theorem is amazingly simple. If µ denotes uniform measure on the circle ⊂ R2 , then for f ∈ C0∞ (R2 ) we have

|fˆ|2 = (fˆµ)(fˆ) = f ∗ µ, ˆ f ≤ f Lp f ∗ µ ˆ Lp  . (10)

R2

The Fourier transform µ(ξ ˆ ) is a Bessel function. It decays like |ξ |−1/2 at infinity, a fact intimately connected with the curvature of the circle. In particular, µˆ ∈ Lq for 4 < q ≤ ∞, and therefore f ∗ µ ˆ Lp ≤ Cp f Lp for 1 ≤ p < 87 , by the usual elementary estimates for convolutions. Putting this estimate back into (10), we see  that |fˆ|2 ≤ Cp f 2Lp , which proves Stein’s Restriction Theorem. The Stein Restriction Theorem means a lot to me personally, and has strongly influenced my own work in Fourier analysis. The second result of Stein’s relating the Fourier transform to curvature concerns the differentiation of integrals on Rn . n . For x ∈ Rn and Theorem. Suppose f ∈ Lp (Rn ) with n ≥ 3 and p > n−1 r > 0, let F(x, r) denote the average of f on the sphere of radius r centered at x. Then limr→0 F (x, r) = f (x) almost everywhere.

The point is that unlike the standard Lebesgue Theorem, we are averaging f over a small sphere instead of a small ball. As in the restriction theorem, we are seemingly in trouble because the sphere has measure zero in Rn , but the curvature of the sphere saves the day. This theorem is obviously closely connected to the smoothness of solutions of the wave equation. The proof of the above differentiation theorem relies on an R Elementary Tauberian Theorem. Suppose that limR→0 R1 0 F (r)dr exists and  ∞ dF 2  1 R 0 r| dr | dr < ∞. Then limR→0 F (R) exists, and equals limR→0 R 0 F (r)dr. This result had long been used, e.g., to pass from Cesaro averages of Fourier series to partial sums. (See Zygmund [Z].) On more than one occasion, Stein has shown the surprising power hidden in the elementary Tauberian Theorem. Here we

7

SELECTED THEOREMS BY ELI STEIN



apply it to F (x, r) for a fixed x. In fact, we have F (x, r) = f (x + ry)dµ(y), with µ equal to normalized surface measure on the unit sphere, so that the Fourier transforms of F and f are related by Fˆ (ξ, r) = fˆ(ξ )µ(rξ ˆ ) for each fixed r. Therefore, assuming f ∈ L2 for simplicity, we obtain 2  2



∞ 
∞
 ∂   ∂  F (x, r) dx dr r  F (x, r) dr dx = r   ∂r n n ∂r R R 0 0
=




∞

r R

0





=

Rn

2


 ∂  Fˆ (ξ, r) dξ dr =   n ∂r

∞ 0

Rn


0

∞

 2 ∂   r  µ(rξ ˆ ) |fˆ(ξ )|2 drdξ ∂r

 2
∂  r  µ(rξ ˆ ) dr |fˆ(ξ )|2 dξ = (const.) |fˆ(ξ )|2 dξ < ∞. n ∂r R

(Here we make crucial use of curvature, which causes µˆ to decay at infinity, so that ∞ the integral in curly brackets converges.) It follows that 0 r| ∂r∂ F (x, r)|2 dr < ∞  R for almost every x ∈ Rn . On the other hand, R1 0 F (x, r)dr is easily seen to be the convolution of f with a standard approximate identity. Hence the usual R Lebesgue differentiation theorem shows that limR→0 R1 0 F (x, r)dr = f (x) for almost every x. So for almost all x ∈ Rn , the function F (x, r) satisfies the hypotheses of the elementary Tauberian theorem. Consequently,
1 R F (x, r)dr = f (x) lim F (x, r) = lim r→0 R→0 R 0 almost everywhere, proving Stein’s differentiation theorem for f ∈ L2 (Rn ). n , we repeat the above arguTo prove the full result for f ∈ Lp (Rn ), p > n−1 ment with surface measure µ replaced by an even more singular distribution on Rn . Thus we obtain a stronger conclusion than asserted, when f ∈ L2 . On the other hand, for f ∈ L1+ε we have a weaker result than that of Stein, namely Lebesgue’s differentiation theorem. Interpolating between L2 and L1+ε , one obtains the Stein differentiation theorem. The two results we picked out here are only a sample of the work of Stein and others on curvature and the Fourier transform. For instance, J. Bourgain has dramatic results on both the restriction problem and spherical averages. We refer the reader to Stein’s address at the Berkeley congress [128] for a survey of the field. H P -SPACES Another essential part of Fourier analysis is the theory of H p -spaces. Stein transformed the subject twice, once in a joint paper with Guido Weiss, and again in a joint paper with me. Let us start by recalling how the subject looked before Stein’s work. The classical theory deals with analytic functions F (z) on the unit

8

CHARLES FEFFERMAN

disc. Recall that F belongs to H p (0 < p < ∞) if the norm F H p ≡ limr→1−  2π ( 0 |F (reiθ )|p dθ )1/p is finite. The classical H p -spaces serve two main purposes. First, they provide growth conditions under which an analytic function tends to boundary values on the unit circle. Secondly, H p serves as a substitute for Lp to allow basic theorems on Fourier series to extend from 1 < p < ∞ to all p > 0. To prove theorems about F ∈ H p , the main tool is the Blaschke product B(z) = v eiθv

zv − z , 1 − z¯ v z

(11)

where {z v } are the zeroes of the analytic function F in the disc, and θv are suitable phases. The point is that B(z) has the same zeroes as F , yet it has absolute value 1 on the unit circle. We illustrate the role of the Blaschke product by sketching the proof of the Hardy-Littlewood maximal theorem for H p . The maximal theorem says that F ∗ Lp ≤ Cp F H p for 0 < p < ∞, where F ∗ (θ ) = supz∈ (θ) |F (z)|, and (θ ) is the convex hull of eiθ and the circle of radius 12 about the origin. This basic result is closely connected to the pointwise convergence of F (z) as z ∈ (θ ) tends to eiθ . To prove the maximal theorem, we argue as follows. First suppose p > 1. Then we don’t need analyticity of F . We can merely assume that F is harmonic, and deduce the maximal theorem from real variables. In fact, it is easy to show that F arises as the Poisson integral of an Lp function f on the unit circle. The maximal theorem for f , a standard theorem of real variables,  θ+h 1 says that Mf Lp ≤ Cp f Lp , where Mf (θ ) = suph>0 ( 2h θ−h |f (t)|dt). It is quite simple to show that F ∗ (θ ) < CMf (θ ). Therefore F ∗ Lp ≤ CMf Lp < C  F H p , and the maximal theorem is proven for H p (p > 1). If p ≤ 1, then the problem is more subtle, and we need to use analyticity of F (z). Assume for a moment that F has no zeroes in the unit disc. Then for 0 < q < p, we can define a single-valued branch of (F (z))q , which will belong to H p/q since F ∈ H p . Since p˜ ≡ pq > 1, the maximal theorem for H p˜ is already known. Hence, maxz∈ (θ) |(F (z))q | ∈ Lp/q , with norm p/q
π p/q p q dθ ≤ Cp,q F q H p/q = Cp,q F H p . max |(F (z)) | −π

z∈ (θ)

∗

That is, F Lp ≤ Cp,q F H p , proving the maximal theorem for functions without zeroes. To finish the proof, we must deal with the zeroes of an F ∈ H p (p ≤ 1). We bring in the Blaschke product B(z), as in (11). Since B(z) and F (z) have the same zeroes and since |B(z)| = 1 on the unit circle, we can write F (z) = G(z)B(z) with G analytic, and |G(z)| = |F (z)| on the unit circle. Thus, GH p = F H p . Inside the circle, G has no zeroes and |B(z)| ≤ 1. Hence |F | ≤ |G|, so  max |F (z)|Lp ≤  max |G(z)|Lp ≤ Cp GH p = Cp F H p , z∈ (θ)

z∈ (θ)

by the maximal theorem for functions withour zeroes. The proof of the maximal theorem is complete. (We have glossed over difficulties that should not enter an expository paper.)

SELECTED THEOREMS BY ELI STEIN

9

Classically, H p theory works only in one complex variable, so it is useful only for Fourier analysis in one real variable. Attempts to generalize H p to several complex variables ran into a lot of trouble, because the zeroes of an analytic function F (z 1 , . . . z n ) ∈ H p form a variety V with growth conditions. Certainly V is much more complicated than the discrete set of zeroes {z v } in the disc. There is no satisfactory substitute for the Blaschke product. For a long time, this blocked all attempts to extend the deeper properties of H p to several variables. Stein and Weiss [13] realized that several complex variables was the wrong generalization of H p for purposes of Fourier analysis. They kept clearly in mind what H p -spaces are supposed to do, and they kept an unprejudiced view of how to achieve it. They found a version of H p theory that works in several variables. The idea of Stein and Weiss was very simple. They viewed the real and imaginary parts of an analytic function on the disc as the gradient of a harmonic function. In several variables, the gradient of a harmonic function is a system u = (u1 , u2 , . . . , un ) of functions on Rn that satisfies the Stein-Weiss Cauchy-Riemann equations  ∂uk ∂uj ∂uk = , = 0. (12) ∂xk ∂xj ∂xk k In place of the Blaschke product, Stein and Weiss used the following simple obseru|p = (u21 + u22 + · · · + u2n )p/2 is vation. If u = (u1 , . . . , un ) satisfies (12), then | . We sketch the simple proof of this fact, then explain subharmonic for p > n−2 n−1 how an H p theory can be founded on it. u|  = 0 and calculate (| u|p ) To see that | u|p is subharmonic, we first suppose | ∂uj in coordinates that diagonalize the symmetric matrix ( ∂xk ) at a given point. The result is (| u|p ) = p| u|p−2 {|w|  2 | u|2 − (2 − p)| v |2 },

(13)

k with wk = ∂u and vk = uk wk . ∂xk n Since k=1 wk = 0 by the Cauchy-Riemann equations, we have  2       2  wj  ≤ (n − 1) |wj |2 = (n − 1)|w|  2 − (n − 1)|wk |2 , |wk | =   j =k  j =k

i.e., |wk |2 ≤ n−1 |w|  2 . Hence | v |2 ≤ (maxk |wk |2 )| u|2 ≤ ( n−1 )|w|  2 | u|2 , so the n n n−2 expression in curly brackets in (13) is non-negative for p ≥ n−1 , and | u|p is subharmonic. So far, we know that | u|p is subharmonic where it isn’t equal to zero. Hence for 0 < r < r(x) we have | u(x)|p ≤ Aυ|y−x|=r | u(y)|p ,

(14)

provided | u(x)|  = 0. However, (14) is obvious when | u(x)| = 0, so it holds for any x. That is, | u|p is a subharmonic function for p ≥ n−2 , as asserted. n−1 Now let us see how to build an H p theory for Cauchy-Riemann systems, based on subharmonicity of | u|p . To study functions on Rn−1 (n ≥ 2), we regard Rn−1 as

10

CHARLES FEFFERMAN

the boundary of Rn+ = {(x1 , . . . , xn )|xn > 0}, and we define H p (Rn+ ) as the space of all Cauchy-Riemann systems (u1 , u2 , . . . , un ) for which the norm
p  uH p = sup | u(x1 , . . . , xn−1 , t)|p dx1 · · · dxn−1 Rn−1

t>0

is finite. For n = 2 this definition agrees with the usual H p -spaces for the upper half-plane. Next we show how the Hardy-Littlewood maximal theorem extends from the disc to Rn+ . u(y, t)| for x ∈ Rn−1 . Define the maximal function M( u)(x) = sup|y−x| 1). For small h > 0, the function Fh (x, t) = n−2 | u(x, t + h)| n−1 (x ∈ Rn−1 , t ≥ 0) is subharmonic on Rn+ and continuous up to the boundary. Therefore, Fh (x, t) ≤ P.I.(fh ),

(15) n−2 n−1

where P.I. is the Poisson integral and fh (x) = Fh (x, 0) = | u(x, h)| . By definition of the H p -norm, we have  
n−1 p |fh (x)|p˜ dx ≤  uH p , with p˜ = p > 1. (16) n−2 Rn−1 On the other hand, since the Poisson integral arises by convoling with an approximate identity, one shows easily that sup P.I.(fh )(y, t) ≤ Cfh∗ (x)

(17)

|y−x|0


|x−y| 1. Hence (15), (16), and (17) show that p˜ 
sup Fh (y, t)

x∈Rn−1

|y−x| 0; (c) Rotations (z, w)  −→ (U z, w) for unitary (n − 1) × (n − 1) matrices U . The multiplication law in (a) makes H into a nilpotent Lie group, the Heisenberg group. Translation-invariance of the Seigel domain allows us to pick the basic complex vector-fields L1 · · · Ln−1 to be translation-invariant on H . After we make a suitable choice of metric, the operators L and
b become translation- and rotation-invariant, and homogeneous with respect to the dilations δt . Therefore, the solution1 of
b w = α should have the form of a convolution w = K ∗ α on the Heisenberg group. The convolution kernel K is homogeneous with respect to the dilations δt and invariant under rotations. Also, since K is a fundamental solution, it satisfies
b K = 0 away from the origin. This reduces to an elementary ODE after we take the dilation- and rotation-invariance into account. Hence one can easily find K explicitly and thus solve the
b -equation for the Siegel domain. To derive sharp regularity theorems for
b , we combine the explicit fundamental solution with the Knapp-Stein theorem on singular integrals on the Heisenberg group. For instance, if
b w ∈ L2 , then Lj Lk w, L¯ j Lk w, Lj L¯ k w, and L¯ j L¯ k w all belong to L2 . To see this, we write
b w = α,

w = K ∗ α,

Lj Lk w = (Lj Lk K) ∗ α,

and note that Lj Lk K has the critical homogeneity and integral 0. Thus Lj Lk K is a singular integral kernel in the sense of Knapp and Stein, and it follows that Lj Lk w ≤ Cα. For the first time, nilpotent Lie groups have entered into the ¯ study of ∂-problems. Folland and Stein viewed their results on the Heisenberg group not as ends in themselves, but rather as a tool to understand general strongly pseudoconvex CR-manifolds. A CR-manifold M is a generalization of the boundary of a smooth domain D ⊂ Cn . For simplicity we will take M = ∂D here. The key idea is that near any point w in a strongly pseudoconvex M, the CR-structure for M is very nearly equivalent to that of the Heisenberg group H via a change of coordinates w : M → H . More precisely, w carries w to the origin, and it carries the CRstructure on M to a CR-structure on H that agrees with the usual one at the origin. Kohn’s work showed that
b w = α has a solution if we are in complex dimension > 2. In two complex dimensions,
b w = α has no solution for most α. We assume dimension > 2 here.

1

18

CHARLES FEFFERMAN

Therefore, if w = K ∗ α is our known solution of
b w = α on the Heisenberg group, then it is natural to try
K(w (z))α(w)dw (27) w(z) = M

as an approximate solution of
b w = α on M. (Since w and α are sections of bundles, one has to explain carefully what (27) really means.) If we apply
b to the w defined by (27), then we find that
b w = α − Eα,

(28)

−1/2

where E is a sort of Heisenberg version of (−) . In particular, E gains smoothness, so that (I − E)−1 can be constructed modulo infinitely smoothing operators by means of a Neumann series. Therefore (27) and (28) show that the full solution of
b w = α is given (modulo infinitely smoothing errors) by 
w(z) = K(w (z))(E k α)(w)dw, (29) k

M

from which one can deduce sharp estimates to understand completely
−1 b on M. The process is analogous to the standard method of “freezing coefficients” to solve variable-coefficient elliptic differential equations. Let us see how the sharp results are stated. As on the Heisenberg group, there are smooth, complex vector fields Lk that span the tangent vectors of type (0, 1) locally. Let Xj be the real and imaginary parts of the Lk . In terms of the Xj we define “non-Euclidean” versions of standard geometric and analytic concepts. Thus, the non-Euclidean ball B(z, ρ) may be defined as an ellipsoid with principal axes of length ∼ ρ in the codimension 1 hyperplane spanned by the Xj , and length ∼ ρ 2 perpendicular to that hyperplane. In terms of B(z, ρ), the non-Euclidean Lipschitz spaces α (M) are defined as the set of functions u for which |u(z) − u(w)| < Cρ α for w ∈ B(z, ρ). (Here, 0 < α < 1. There is a natural extension to all α > 0.) The non-Euclidean Sobolev spaces Sm,p (M) consist of all distributions u for which all Xj1 Xj2 · · ·Xjs u ∈ Lp (M) for 0 ≤ s ≤ m. Then the sharp results on
b are as follows. If
b w = α and α ∈ Sm,p (M), then w ∈ Sm+2,p (M) for m ≥ 0, 1 < p < ∞. If
b w = α and α ∈ α (M), then Xj Xk w ∈ α (M) for 0 < α < 1 (say). For additional sharp estimates, and for comparisons between the non-Euclidean and standard function spaces, we refer the reader to [67]. To prove their sharp results, Folland and Stein developed the theory of singular integral operators in a non-Euclidean context. The Cotlar-Stein lemma proves the crucial results on L2 -boundedness of singular integrals. Additional difficulties arise from the non-commutativity of the Heisenberg group. In particular, standard singular integrals or pseudodifferential operators commute modulo lower-order errors, but non-Euclidean operators are far from commuting. This makes more difficult the passage from Lp estimates to the Sobolev spaces Sm,p (M). ¯ let me explain the remarkable paper Before we continue with Stein’s work on ∂, of Rothschild-Stein [72]. It extends the Folland-Stein results and viewpoint to general Hörmander operators L = jN=1 Xj2 + X0 . Actually, [72] deals with systems

SELECTED THEOREMS BY ELI STEIN

19

whose second-order part is j Xj2 , but for simplicity we restrict attention here to L. In explaining the proofs, we simplify even further by supposing X0 = 0. The goal of the Rothschild-Stein paper is to use nilpotent groups to write down an explicit parametrix for L and prove sharp estimates for solutions of Lu = f . This ambitious hope is seemingly dashed at once by elementary examples. For instance, take L = X12 + X22 with ∂ ∂ (30) , X2 = x on R2 . ∂x ∂y Then X1 and [X1 , X2 ] span the tangent space, yet L clearly cannot be approximated by translation-invariant operators on a nilpotent Lie group in the sense of Folland-Stein. The trouble is that L changes character completely from one point to another. Away from the y-axis {x = 0}, L is elliptic, so the only natural nilpotent group we can reasonably use is R2 . On the y-axis, L degenerates, and evidently cannot be approximated by a translation-invariant operator on R2 . The problem is so obviously fatal, and its solution by Rothschild and Stein so simple and natural, that [72] must be regarded as a gem. Here is the idea: Suppose we add an extra variable t and “lift” X1 and X2 in (30) to vector fields ∂ ∂ ∂ X˜ 1 = (31) , X˜ 2 = x + on R3 . ∂x ∂y ∂t X1 =

Then the Hörmander operator L˜ = X˜ 12 +X˜ 22 looks the same at every point of R3 , and may be readily understood in terms of nilpotent groups as in Folland-Stein [67]. In particular, one can essentially write down a fundamental solution and prove sharp estimates for L˜ −1 . On the other hand, L˜ reduces to L when acting on functions ˜ = f imply sharp u(x, y, t) that do not depend on t. Hence, sharp results on Lu results on Lu = f . Thus we have the Rothschild-Stein program: First, add new variables and lift the given vector fields X1 , · · · , XN to new vector fields X˜ 1 · · · X˜ N whose underlying structure does not vary from point to point. Next, approximate L˜ = 1N X˜ 2 by a translation-invariant operator Lˆ = 1N Yj2 on a nilpotent Lie group N . Then ˆ and use it to write down an approximate analyze the fundamental solution of L, ˜ From the approximate solution, derive sharp estimates fundamental solution for L. ˜ = f . Finally, descend to the original equation Lu = f by for solutions of Lu restricting attention to functions u, f that do not depend on the extra variables. To carry out the first part of their program, Rothschild and Stein prove the following result. Theorem A. Let X1 · · · XN be smooth vector fields on a neighborhood of the origin in Rn . Assume that the Xj and their commutators [[[Xj1 , Xj2 ], Xj3 ] · · · Xjs ] of order up to r span the tangent space at the origin. Then we can find smooth vector fields X˜ 1 · · · X˜ N on a neighborhood U˜ of the origin in Rn+m with the following properties. (a) The X˜ j and their commutators up to oder r are linearly independent at each point of U˜ , except for the linear relations that follow formally from the antisymmetry of the bracket and the Jacobi identity.

20

CHARLES FEFFERMAN

(b) The X˜ j and their commutators up to order r span the tangent space of U˜ . (c) Acting on functions on Rn+m that do not depend on the last m coordinates, the X˜ j reduce to the given Xj . Next we need a nilpotent Lie group N appropriate to the vector fields X˜ 1 · · · X˜ N . The natural one is the free nilpotent group NNr of step r on N generators. Its Lie algebra is generated by Y1 · · · YN whose Lie brackets of order higher than r vanish, but whose brackets of order ≤ r are linearly independent, except for relations forced by antisymmetry of brackets and the Jacobi identity. We regard the Yj as translation-invariant vector fields on NNr . It is convenient to pick a basis {yα }α∈A for the Lie algebra of NNr , consisting of Y1 · · · YN and some of their commutators. On NNr we form the Hörmander operator Lˆ = 1N Yj2 . Then Lˆ is translationinvariant and homogeneous under the natural dilations on NNr . Hence Lˆ −1 is given by convolution on NNr with a homogeneous kernel K(·) having a weak singularity at the origin. Hypoellipticity of Lˆ shows that K is smooth away from the origin. ˆ = f very well. Thus we understand the equation Lu We want to use Lˆ to approximate L˜ at each point y ∈ U˜ . To do so, we have to identify a neighborhood of y in U˜ with a neighborhood of the origin in NNr . This ˜ The idea is to use exhas to be done just right, or else Lˆ will fail to approximate L. ponential coordinates on both U˜ and NNr . Thus, if x = exp(α∈A tα Yα )(identity) ∈ NNr , then we use (tα )α∈A as coordinates for x. Similarly, let (X˜ α )α∈A be the commutators of X˜ 1 · · · X˜ N analogous to the Yα , and let y ∈ U˜ be given. Then given a nearby point x = exp(α∈A tα X˜ α )y ∈ U˜ , we use (tα )α∈A as coordinates for x. Now we can identify U˜ with a neighborhood of the identity in NNr , simply by identifying points with the same coordinates. Denote the identification by y : U˜ → NNr , and note that y (y) = identity. In view of the identification y , the operators Lˆ and L˜ live on the same space. The next step is to see that they are approximately equal. To formulate this, we need some bookkeeping on the nilpotent group NNr . Let {δt }t>0 be the natural dilations on NNr . If ϕ ∈ C0∞ (NNr ), then write ϕt for the function x  −→ ϕ(δt x). When ϕ is fixed and t is large, then ϕt is supported in a tiny neighborhood of the identity. Let D be a differential operator acting on functions on NNr . We say that D has “degree” at most k if for each ϕ ∈ C0∞ (NNr ) we have |D(ϕt )| = O(t k ) for large, positive t. According to this definitions, Y1 , . . . YN have degree 1 while [Yj , Yk ] has degree 2, and the degree of a(x)[Yj , Yk ] depends on the behavior of a(x) near the identify. Now we can say in what sense L˜ and Lˆ are approximately equal. The crucial result is as follows. ˜ Theorem B. Under the map −1 y , the vector field Xj pulls back to Yj + Zy,j , where Zy,j is a vector field on NNr of “degree” ≤ 0. Using Theorem B and the map y , we can produce a parametrix for L˜ and prove that it works. In fact, we take ˜ K(x, y) = K(y x),

(32)

21

SELECTED THEOREMS BY ELI STEIN

ˆ For fixed y, we want to know that where K is the fundamental solution of L. ˜ L˜ K(x, y) = δy (x) + E(x, y),

(33)

where δy (·) is the Dirac delta-function and E(x, y) has only a weak singularity at x = y. To prove this, we use y to pull back to NNr . Recall that L˜ = 1N X˜ j2 while Lˆ = 1N Yj2 . Hence by Theorem B, L˜ pulls back to an operator of the form Lˆ + Dy , with Dy having “degree” at most 1. Therefore (33) reduces to proving that ˆ (Lˆ + Dy )K(x) = δid. (x) + E(x),

(34)

ˆ where Eˆ has only a weak singularity at the identity. Since LK(x) = δid. (x), (34) means simply that Dy K(x) has only a weak singularity at the identity. However, this is obvious from the smoothness and homogeneity if K(x), and from the fact ˜ y) is an approximate fundamental solution that Dy has degree ≤ 1. Thus, K(x, ˜ for L. ˜ one can From the explicit fundamental solution for the lifted operator L, “descend” to deal with the original Hörmander operator L in two different ways. a. Prove sharp estimates for the lifted problems, then specialize to the case of functions that don’t depend on the extra variables. ˜ to obb. Integrate out the extra variables from the fundamental solution for L, tain a fundamental solution for L. Rothschild and Stein used the first approach. They succeeded in proving the estimate

 



N 2

  Xj + X0 u X0 uLp (U ) + Xj Xk uLp (U ) ≤ Cp

p

j =1 L (V )

+Cp uLp (V )

for 1 < p < ∞ and U ⊂⊂ V .

(35)

This is the most natural and the sharpest estimate for Hörmander operators. It was new even for p = 2. Rothschild and Stein also proved sharp estimates in spaces analogous to the α and Sm,p of Folland-Stein [67], as well as in standard Lipschitz and Sobolev spaces. We omit the details, but we point out that commuting derivatives past a general Hörmander operator here requires additional ideas. Later, Nagel, Stein, and Wainger [119] returned to the second approach (“b” above) and were able to estimate the fundamental solution of a general Hörmander operator. This work overcomes substantial problems. In fact, once we descend from the lifted problem to the original equation, we again face the difficulty that Hörmander operators cannot be modelled directly on nilpotent Lie groups. So it isn’t even clear how to state a theorem on the fundamental solution of a Hörmander operator. Nagel, Stein, and Wainger [119], realized that a family of non-Euclidean “balls” BL (x, ρ) associated to the Hörmander operator L plays the basic role. They defined the BL (x, ρ) and proved their essential properties. In particular, they saw that the family of balls survives the projection from the lifted problem back to the original equation, even though the nilpotent Lie

22

CHARLES FEFFERMAN

group structure is destroyed. Non-Euclidean balls had already played an important part in Folland-Stein [67]. However, it was simple in [67] to guess the correct family of balls. For general Hörmander operators L the problem of defining and controlling non-Euclidean balls is much more subtle. Closely related results appear also in [FKP], [FS]. Let us look first at a nilpotent group such as NNr , with its family of dilations {δt }t>0 . Then the correct family of non-Euclidean balls BNN r (x, ρ) is essentially dictated by translation and dilation-invariance, starting with a more or less arbitrary harmless “unit ball” BNN r (identity, 1). Recall that the fundamental solution for  2 Lˆ = N 1 Yj on NNr is given by a kernel K(x) homogeneous with respect to the δt . Estimates that capture the size and smoothness of K(x) may be phrased entirely in terms of the non-Euclidean balls BNN r (x, ρ). In fact, the basic estimate is as follows. Cm ρ 2−m vol BNN r (0, ρ)  ρ for x ∈ BNN r (0, ρ)\BNN r 0, and m ≥ 0. (36) 2 Next we associate non-Euclidean balls to a general Hörmander operator. For  2 X as in our discussion of Rothschild-Stein [72]. One simplicity, take L = N j 1 definition of the balls BL (x, ρ) involves a moving particle that starts at x and travels along the integral curve of Xj1 for time t1 . From its new position x  the particle then travels along the integral curve of Xj2 for time t2 . Repeating the process finitely many times, we can move the particle from its initial position x to a final position y in a total time t = t1 + · · · + tm . The ball BL (x, ρ) consists of all y that can be reached in this way in time t < ρ. For instance, if L is elliptic, then BL (x, ρ) is2 essentially the ordinary (Euclidean) ball about x of radius ρ. If we take Lˆ = N 1 Yj on NN,r , then the balls BLˆ (x, ρ) behave naturally under translations and dilations; hence they are essentially the same as the BNN,r (x, ρ) appearing in (36). NagelWainger-Stein analyzed the relations between BLˆ (x, ρ), BL˜ (x, ρ) and BL (x, ρ) for an arbitrary Hörmander operator L. (Here L˜ and Lˆ are as in our previous discussion of Rothschild-Stein.) This allowed them to integrate out the extra variables in the ˜ to derive the following sharp estimates from (36). fundamental solution of L, |Yj1 Yj2 . . . Yjm K(x)| ≤

Theorem. Suppose X1 · · · XN and their repeated commutators span the tangent space. Also, suppose we are in dimension greater than 2. Then the solution of   2 X )u = f is given by u(x) = K(x, y)f (y)dy with ( N 1 j |Xj1 · · · Xjm K(x, y)| ≤

Cm ρ 2−m (volBL (y, ρ))

f or

 ρ x ∈ BL (y, ρ)\BL y, 2

and m ≥ 0. Here the Xji act either in the x- or the y- variable. ¯ Let us return from Hörmander operators to the ∂-problems on strongly pseudon . Greiner and Stein derived sharp estimates for the Neuconvex domains D ⊂ C mann Laplacian
w = α in their book [78]. This problem is hard, because two

23

SELECTED THEOREMS BY ELI STEIN

different families of balls play an important role. On the one hand, the standard (Euclidean) balls arise here, because
is simply the Laplacian in the interior of D. On the other hand, non-Euclidean balls (as in Folland-Stein [67]) arise on ∂D, because they are adapted to the non-elliptic boundary Conditions for
. Thus, any understanding of
requires notions that are natural with respect to either family of ¯ We say that a smooth balls. A key notion is that of an allowable vector field on D. vector field X is allowable if its restriction to the boundry ∂D lies in the span of the complex vector fields L1 , . . . , Ln−1 , L¯ 1 . . . L¯ n−1 . Here we have retained the ¯ notation of our earlier discussion of ∂-problems. At an interior point, an allowable vector field may point in any direction, but at a boundary point it must be in the natural codimension-one subspace of the tangent space of ∂D. Allowable vector fields are well-suited both to the Euclidean and the Heisenberg balls that control
. The sharp estimates of Greiner-Stein are as follows. Theorem. Suppose
w = α on a strictly pseudoconvex domain D ⊂ Cn . If α p p belongs to the Sobolev space Lk , then w belongs to Lk−1 (1 < p < ∞). Moreover, p ¯ belongs if X and Y are allowable vector fields, then XY w belongs to Lk . Also, Lw p ¯ to Lk+1 if L is a smooth complex vector field of type (0, 1). Similarly, if α belongs to the Lipschitz space Lip(β) (0 < β < 1), then the gradient of w belongs to Lip(β) ¯ belongs to Lip(β) if L is a smooth complex vector as well. Also the gradient of Lw field of type (0, 1); and XY w belongs to Lip(β) for X and Y allowable vector fields. These results for allowable vector fields were new even for L2 . We sketch the proof. Suppose
w = α. Ignoring the boundary conditions for a moment, we have w = α in D, so w = Gα + P.I.(w) ˜

(37)

where w˜ is defined on ∂D, and G, P.I. denote the standard Green’s operator and Poisson integral, respectively. The trouble with (37) is that we know nothing about w˜ so far. The next step is to bring in the boundary condition for
w = α. According to Calderón’s work on general boundary-value problems, (37) satisfies the ¯ ∂-Neumann boundary conditions if and only if Aw˜ = {B(Gα)}|∂D

(38)

for a certain differential operater B on D, and a certain pseudodifferential operator A on ∂D. Both A and B can be determined explicitly from routine computation. Greiner and Stein [78] derive sharp regularity theorems for the pseudodifferential equation Aw˜ = g, and then apply those results to (38) in order to understand w˜ in terms of α. Once they know sharp regularity theorems for w, ˜ formula (37) gives the behavior of w. Let us sketch how Greiner-Stein analyzed Aw˜ = g. This is really a system of n pseudodifferential equations for n unknown functions (n = dim Cn ). In a suitable frame, one component of the system decouples from the rest of the problem (modulo negligible errors) and leads to a trivial (elliptic) pseudodifferential equation.

24

CHARLES FEFFERMAN

The non-trivial part of the problem is a first-order system of (n − 1) pseudodifferential operators for (n − 1) unknowns, which we write as
+ w # = α # .

(39)

Here α # consists of the non-trivial components of {B(Gα)}|∂D , w # is the unknown, and
+ may be computed explicitly. Greiner and Stein reduce (39) to the study of the Kohn-Laplacian
b . In fact, they produce a matrix
− of first-order pseudodifferential operators similar to
+ , and then show that
−
+ =
b modulo negligible errors.2 Applying
− to (39) yields
b w # =
− α # + negligible.

(40)

From Folland-Stein [67] one knows an explicit integral operator K that inverts
b modulo negligible errors. Therefore, w # = K
− α # + negligible.

(41)

Equations (37) and (41) express w in terms of α as a composition of various explicit operators, including: the Poisson integral; restriction to the boundary;
− ; K; G. Because the basic notion of allowable vector fields is well-behaved with respect to both the natural families of balls for
w = α, one can follow the effect of each of these very different operators on the relevant function spaces without losing information. To carry this out is a big job. We refer the reader to [78] for the rest of the story. There have been important recent developments in the Stein program for several complex variables. In particular, we refer the reader to Phong’s paper in Essays in Fourier Analysis in Honor of Elias M. Stein (Princeton University Press, 1995) for a discussion of singular Radon transforms; and to Nagel-Rosay-Stein-Wainger [131], D.-C. Chang-Nagel-Stein [132], and [McN], [Chr], [FK] for the solution of ¯ the ∂-problems on weakly pseudoconvex domains of finite type in C2 . Particularly in several complex variables we are able to see in retrospect the fundamental interconnections among classical analysis, representation theory, and partial differential equations, which Stein was the first to perceive. I hope this chapter has conveyed to the reader the order of magnitude of Stein’s work. However, let me stress that it is only a selection, picking out results which I could understand and easily explain. Stein has made deep contributions to many other topics, e.g., Limits of sequences of operators Extension of Littlewood-Paley Theory from the disc to Rn Differentiability of functions on sets of positive measure Fourier analysis on RN when N → ∞ Function theory on tube domains Analysis of diffusion semigroups Pseudodifferential calculus for subelliptic problems. The list continues to grow. 2

This procedure requires significant changes in two complex variables, since then
b isn’t invertible.

SELECTED THEOREMS BY ELI STEIN

25

Bibliography of E.M. Stein [1] “Interpolation of linear operators.” Trans. Amer. Math. Soc. 83 (1956), 482–492. [2] “Functions of exponential type.” Ann. of Math. 65 (1957), 582–592. [3] “Interpolation in polynomial classes and Markoff’s inequality.” Duke Math. J. 24 (1957), 467–476. [4] “Note on singular integrals.” Proc. Amer. Math. Soc. 8 (1957), 250–254. [5] (with G. Weiss) “On the interpolation of analytic families of operators acting on H p spaces.” Tohoku Math. J. 9 (1957), 318–339. [6] (with E. H. Ortrow) “A generalization of lemmas of Marcinkiewicz and Fine with applications to singular integrals.” Annali Scuola Normale Superiore Pisa 11 (1957), 117–135. [7] “A maximal function with applications to Fourier series.” Ann. of Math. 68 (1958), 584–603. [8] (with G. Weiss) “Fractional integrals on n-dimensional Euclidean space.” J. Math. Mech. 77 (1958), 503–514. [9] (with G. Weiss) “Interpolation of operators with change of measures.” Trans. Amer. Math. Soc. 87 (1958), 159–172. [10] “Localization and summability of multiple Fourier series.” Acta Math. 100 (1958), 93–147. [11] “On the functions of Littlewood-Paley, Lusin, Marcinkiewicz.” Trans. Amer. Math. Soc. 88 (1958), 430–466. [12] (with G. Weiss) “An extension of a theorem of Marcinkiewicz and some of its applications.” J. Math. Mech. 8 (1959), 263–284. [13] (with G. Weiss) “On the theory of harmonic functions of several variables I, The theory of H p spaces.” Acta Math. 103 (1960), 25–62. [14] (with R. A. Kunze) “Uniformly bounded representations and harmonic analysis of the 2 × 2 real unimodular group.” Amer. J. Math. 82 (1960), 1–62. [15] “The characterization of functions arising as potentials.” Bull. Amer. Math. Soc. 67 (1961), 102–104; II, 68 (1962), 577–582. [16] “On some functions of Littlewood-Paley and Zygmund.” Bull. Amer. Math. Soc. 67 (1961), 99–101. [17] “On limits of sequences of operators.” Ann. of Math. 74 (1961), 140–170.

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[18] “On the theory of harmonic functions of several variables II. Behavior near the boundary.” Acta Math. 106 (1961), 137–174. [19] “On certain exponential sums arising in multiple Fourier series.” Ann. of Math. 73 (1961), 87–109. [20] (with R. A. Kunze) “Analytic continuation of the principal series.” Bull. Amer. Math. Soc. 67 (1961), 543–546. [21] “On the maximal ergodic theorem.” Proc. Nat. Acad. Sci. 47 (1961), 1894– 1897. [22] (with R. A. Kunze) “Uniformly bounded representations II. Analytic continuation of the principal series of representations of the n × n complex unimodular groups.” Amer J. Math. 83 (1961), 723–786. [23] (with A. Zygmund) “Smoothness and differentiability of functions.” Ann. Sci. Univ. Budapest, Sectio Math., III-IV (1960–61), 295–307. [24] “Conjugate harmonic functions in several variables.” Proceedings of the International Congress of Mathematicians, Djursholm-Linden, Institut MittagLeffler (1963), 414–420. [25] (with A. Zygmund) “On the differentiability of functions.” Studia Math. 23 (1964), 248–283. [26] (with G. and M. Weiss) “H p -classes of holomorphic functions in tube domains.” Proc. Nat. Acad. Sci. 52 (1964), 1035–1039. [27] (with B. Muckenhoupt) “Classical expansions and their relations to conjugate functions.” Trans. Amer. Math. Soc. 118 (1965), 17–92. [28] “Note on the boundary values of holomorphic functions.” Ann. of Math. 82 (1965), 351–353. [29] (with S. Wainger) “Analytic properties of expansions and some variants of Parseval-Plancherel formulas.” Arkiv. Math., Band 5 37 (1965), 553–567. [30] (with A. Zygmund) “On the fractional differentiability of functions.” London Math. Soc. Proc. 34A (1965), 249–264. [31] “Classes H 2 , multiplicateurs, et fonctions de Littlewood-Paley.” Comptes Rendues Acad. Sci. Paris 263 (1966), 716–719; 780–781; also 264 (1967), 107–108. [32] (with R. Kunze) “Uniformly bounded representations III. Intertwining operators.” Amer. J. Math. 89 (1967), 385–442. [33] “Singular integrals, harmonic functions and differentiability properties of functions of several variables.” Proc. Symp. Pure Math. 10 (1967), 316–335.

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[34] “Analysis in matrix spaces and some new representations of SL(N, C).” Ann. of Math. 86 (1967), 461–490. [35] (with A. Zygmund) “Boundedness of translation invariant operators in Hölder spaces and Lp spaces.” Ann. of Math. 85 (1967), 337–349. [36] “Harmonic functions and Fatou’s theorem.” In Proceeding of the C.I.M.E. Summer Course on Homogeneous Bounded Domains, Cremonese, 1968. [37] (with A. Koranyi) “Fatou’s theorem for generalized halfplanes.” Annali di Pisa 22 (1968), 107–112. [38] (with G. Weiss) “Generalizations of the Cauchy-Riemann equations and representations of the rotation group.” Amer. J. Math. 90 (1968), 163–196. [39] (with A. Grossman and G. Loupias) “An algebra of pseudodifferential operators and quantum mechanics in phase space.” Ann. Inst. Fourier, Grenoble 18 (1968), 343–368. [40] (with N. J. Weiss) “Convergence of Poisson integrals for bounded symmetric domains.” Proc. Nat. Acad. Sci. 60 (1968), 1160–1162. [41] “Note on the class L log L.” Studia Math. 32 (1969), 305–310. [42] (with A. W. Knapp) “Singular integrals and the principal series.” Proc. Nat. Acad. Sci. 63 (1969), 281–284. [43] (with N. J. Weiss) “On the convergence of Poisson integrals.” Trans. Amer. Math. Soc. 140 (1969), 35–54. [44] Singular integrals and differentiability properties of functions. Princeton Mathematical Series, 30. Princeton University Press, 1970. [45] (with A. W. Knapp) “The existence of complementary series.” In Problems in Analysis. Princeton University Press, 1970. [46] Topics in harmonic analysis related to the Littlewood-Paley theory. Annals of Mathematics Studies, 103. Princeton University Press, 1970. [47] “Analytic continuation of group representations.” Adv. Math. 4 (1970), 172–207. [48] “Boundary values of holomorphic functions.” Bull. Amer. Math. Soc. 76 (1970), 1292–1296. [49] (with A. W. Knapp) “Singular integrals and the principal series II.” Proc. Nat. Acad. Sci. 66 (1970), 13–17. [50] (with S. Wainger) “The estimating of an integral arising in multipier transformations.” Studia Math. 35 (1970), 101–104.

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[51] (with G. Weiss) Introduction to Fourier analysis on Euclidean spaces. Princeton University press, 1971. [52] (with A. Knapp) “Intertwining operators for semi-simple groups.” Ann. of Math. 93 (1971), 489–578. [53] (with C. Fefferman) “Some maximal inequalities.” Amer. J. Math. 93 (1971), 107–115. [54] “Lp boundedness of certain convolution operators.” Bull. Amer. Math. Soc. 77 (1971), 404–405. [55] “Some problems in harmonic analysis suggested by symmetric spaces and semi-simple groups.” Proceedings of the International Congress of Mathematicians, Paris: Gauthier-Villers 1 (1971), 173–189. [56] “Boundary behavior of holomorphic functions of several complex variables.” Princeton Mathematical Notes. Princeton University Press, 1972. [57] (with A. Knapp) Irreducibility theorems for the principal series. (Conference on Harmonic Analysis, Maryland) Lecture Notes in Mathematics, No. 266. Springer Verlag, 1972. [58] (with A. Koranyi) “H 2 spaces of generalized half-planes.” Studia Math. XLIV (1972), 379–388. [59] (with C. Fefferman) “H p spaces of several variables.” Acta Math. 129 (1972), 137–193. [60] “Singular integrals and estimates for the Cauchy-Riemann equations.” Bull. Amer. Math. Soc. 79 (1973), 440–445. ¯ [61] “Singular integrals related to nilpotent groups and ∂-estimates.” Proc. Symp. Pure Math. 26 (1973), 363–367. [62] (with R. Kunze) “Uniformly bounded representations IV. Analytic continuation of the principal series for complex classical groups of types Bn , Cn , Dn .” Adv. Math. 11 (1973), 1–71. [63] (with G. B. Folland) “Parametrices and estimates for the ∂¯b complex on strongly pseudoconvex boundaries.” Bull. Amer. Math. Soc. 80 (1974), 253–258. [64] (with J. L. Clerc) “Lp multipliers for non-compact symmetric spaces.” Proc. Nat. Acad. Sci. 71 (1974), 3911–3912. [65] (with A. Knapp) “Singular integrals and the principal series III.” Proc. Nat. Acad. Sci. 71 (1974), 4622–4624. [66] (with G. B. Folland) “Estimates for the ∂¯b complex and analysis on the Heisenberg group.” Comm. Pure and Appl. Math. 27 (1974), 429–522.

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[67] “Singular integrals, old and new.” In Colloquium Lectures of the 79th Summer Meeting of the American Mathematical Society, August 18–22, 1975. American Mathematical Society, 1975. [68] “Necessary and sufficient conditions for the solvability of the Lewy equation.” Proc. Nat. Acad. Sci. 72 (1975), 3287–3289. [69] “Singular integrals and the principal series IV.” Proc. Nat. Acad. Sci. 72 (1975), 2459–2461. [70] “Singular integral operators and nilpotent groups.” In Proceedings of the C.I.M.E., Differential Operators on Manifolds. Edizioni, Cremonese, 1975: 148–206. [71] (with L. P. Rothschild) “Hypoelliptic differential operators and nilpotent groups.” Acta Math. 137 (1976), 247–320. [72] (with A. W. Knapp) “Intertwining operators for SL(n, r).” Studies in Math. Physics. E. Lieb, B. Simon and A. Wightman, eds. Princeton University Press, 1976: 239–267. [73] (with S. Wainger) “Maximal functions associated to smooth curves.” Proc. Nat. Acad. Sci. 73 (1976), 4295–4296. [74] “Maximal functions: Homogeneous curves.” Proc. Nat. Acad. Sci. 73 (1976), 2176–2177. [75] “Maximal functions: Poisson integrals on symmetric spaces.” Proc. Nat. Acad. Sci. 73 (1976), 2547–2549. [76] “Maximal functions: Spherical means.” Proc. Nat. Acad. Sci. 73 (1976), 2174–2175. ¯ [77] (with P. Greiner) “Estimates for the ∂-Neumann problem.” Mathematical Notes 19. Princeton University Press, 1977. [78] (with D. H. Phong) “Estimates for the Bergman and Szegö projections.” Duke Math. J. 44 (1977), 695–704. [79] (with N. Kerzman) “The Szegö kernels in terms of Cauchy-Fantappie kernels.” Duke Math. J. 45 (1978), 197–224. [80] (with N. Kerzman) “The Cauchy kernels, the Szegö kernel and the Riemann mapping function.” Math. Ann. 236 (1978), 85–93. [81] (with A. Nagel and S. Wainger) “Differentiation in lacunary directions.” Proc. Nat. Acad. Sci. 73 (1978), 1060–1062. [82] (with A. Nagel) “A new class of pseudo-differential operators.” Proc. Nat. Acad. Sci. 73 (1978), 582–585.

30

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[83] (with N. Kerzman) “The Szegö kernel in terms of the Cauchy-Fantappie kernels.” In Proceeding of the Conference on Several Complex Variables, Cortona, 1977. 1978. [84] (with P. Greiner) “On the solvability of some differential operators of the type
b .” In Proceedings of the Conference on Several Complex Variables, Cortona, 1978. [85] (with S. Wainger) “Problems in harmonic analysis related to curvature.” Bull. Amer. Math. Soc. 84 (1978), 1239–1295. [86] (with R. Gundy) “H p theory for the poly-disc.” Proc. Nat. Acad. Sci. 76 (1979), 1026–1029. [87] “Some problems in harmonic analysis.” Proc. Symp. Pure and Appl. Math. 35 (1979), Part I, 3–20. [88] (with A. Nagel and S. Wainger) “Hilbert transforms and maximal functions related to variable curves.” Proc. Symp. Pure and Appl. Math. 35 (1979), Part I, 95–98. [89] (with A. Nagel) “Some new classes of pseudo-differential operators.” Proc. Symp. Pure and Appl. Math. 35 (1979), Part II, 159–170. [90] “A variant of the area integral.” Bull. Sci. Math. 103 (1979), 446–461. [91] (with A. Nagel) “Lectures on pseudo-differential operators: Regularity theorems and applications to non-elliptic problems.” Mathematical Notes 24. Princeton University Press, 1979. [92] (with A. Knapp) “Intertwining operators for semi-simple groups II.” Invent. Math. 60 (1980), 9–84. [93] “The differentiability of functions in Rn .” Ann. of Math. 113 (1981), 383– 385. [94] “Compositions of pseudo-differential operators.” In Proceedings of Journées Equations aux derivées partielles, Saint-Jean de Monts, Juin 1981, Sociéte Math. de France, Conférence #5, 1–6. [95] (with A. Nagel and S. Wainger) “Boundary behavior of functions holomorphic in domains of finite type.” Proc. Nat. Acad. Sci. 78 (1981), 6596–6599. [96] (with A. Knapp) “Some new intertwining operators for semi-simple groups.” In Non-commutative harmonic analysis on Lie groups, Colloq. MarseilleLuminy, 1981. Lecture Notes in Mathematics, no. 880. Springer Verlag, 1981. [97] (with M. H. Taibleson and G. Weiss) “Weak type estimates for maximal operators on certain H p classes.” Rendiconti Circ. mat. Pelermo, Suppl. n. 1 (1981), 81–97.

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[98] (with D. H. Phong) “Some further classes of pseudo-differential and singular integral operators arising in boundary value problems, I, Composition of operators.” Amer. J. Math. 104 (1982), 141–172. [99] (with D. Geller) “Singular convolution operators on the Heisenberg group.” Bull. Amer. Math. Soc. 6 (1982), 99–103. [100] (with R. Fefferman) “Singular integrals in product spaces.” Adv. Math. 45 (1982), 117–143. [101] (with G. B. Folland) “Hardy spaces on homogeneous groups.” Mathematical Notes 28. Princeton University Press, 1982. [102] “The development of square functions in the work of A. Zygmund.” Bull. Amer. Math. Soc. 7 (1982). [103] (with D. M. Oberlin) “Mapping properties of the Radon transform.” Indiana Univ. Math. J. 31 (1982), 641–650. [104] “An example on the Heisenberg group related to the Lewy operator.” Invent. Math. 69 (1982), 209–216. [105] (with R. Fefferman, R. Gundy, and M. Silverstein) “Inequalities for ratios of functionals of harmonic functions.” Proc. Nat. Acad. Sci. 79 (1982), 7958– 7960. [106] (with D. H. Phong) “Singular integrals with kernels of mixed homogeneities.” (Conference in Harmonic Analysis in honor of Antoni Zygmund, Chicago, 1981), W. Beckner, A. Calderón, R. Fefferman, P. Jones, eds. Wadsworth, 1983. [107] “Some results in harmonic analysis in Rn , for n → ∞.” Bull. Amer. Math. Soc. 9 (1983), 71–73. [108] “An H 1 function with non-summable Fourier expansion.” In Proceedings of the Conference in Harmonic Analysis, Cortona, Italy, 1982. Lecture Notes in Mathematics, no. 992. Springer Verlag, 1983. [109] (with R. R. Coifman and Y. Meyer) “Un nouvel espace fonctionel adapté a l’étude des opérateurs définis pour des intégrales singulières.” In Proceedings of the Conference in Harmonic Analysis, Cortona, Italy, 1982. Lecture Notes in Mathematics, no. 992. Springer Verlag, 1983. [110] “Boundary behaviour of harmonic functions on symmetric spaces: Maximal estimates for Poisson integrals,” Invent. Math. 74 (1983), 63–83. [111] (with J. O. Stromberg) “Behavior of maximal functions in Rn for large n.” Arkiv f. math. 21 (1983), 259–269. [112] (with D. H. Phong) “Singular integrals related to the Radon transform and boundary value problems.” Proc. Nat. Acad. Sci. 80 (1983), 7697–7701.

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[113] (with D. Geller) “Estimates for singular convolution operators on the Heisenberg group.” Math. Ann. 267 (1984), 1–15. [114] (with A. Nagel) “On certain maximal functions and approach regions.” Adv. Math. 54 (1984), 83–106. [115] (with R. R. Coifman and Y. Meyer) “Some new function spaces and their applications to harmonic analysis.” J. Funct. Anal. 62 (1985), 304–335. [116] “Three variations on the theme of maximal functions.” (Proceedings of the Seminar on Fourier Analysis, El Escorial, 1983) Recent Progress in Fourier Analysis. I. Peral and J. L. Rubio de Francia, eds. [117] Appendix to the paper “Unique continuation. . . .” Ann. of Math. 121 (1985), 489–494. [118] (with A. Nagel and S. Wainger) “Balls and metrics defined by vector fields I: Basic properties.” Acta Math. 155 (1985), 103–147. [119] (with C. Sogge) “Averages of functions over hypersurfaces.” Invent. Math. 82 (1985), 543–556. [120] “Oscillatory integrals in Fourier analysis.” In Beijing lectures on Harmonic analysis. Annals of Mathematics Studies, 112. Princeton University Press, 1986. [121] (with D. H. Phong) “Hilbert integrals, singular integrals and Radon transforms II.” Invent. Math. 86 (1986), 75–113. [122] (with F. Ricci) “Oscillatory singular integrals and harmonic analysis on nilpotent groups.” Proc. Nat. Acad. Sci. 83 (1986), 1–3. [123] (with F. Ricci) “Homogeneous distributions on spaces of Hermitian matricies.” Jour. Reine Angw. Math. 368 (1986), 142–164. [124] (with D. H. Phong) “Hilbert integrals, singular integrals and Radon transforms I.” Acta Math. 157 (1986), 99–157. [125] (with C. D. Sogge) “Averages over hypersurfaces: II.” Invent. math. 86 (1986), 233–242. [126] (with M. Christ) “A remark on singular Calderón-Zygmund theory.” Proc. Amer. Math. Soc. 99, 1 (1987), 71–75. [127] “Problems in harmonic analysis related to curvature and oscillatory integrals.” Proc. Int. Congress of Math., Berkeley 1 (1987), 196–221. [128] (with F. Ricci) “Harmonic analysis on nilpotent groups and singular integrals I.” J. Funct. Anal. 73 (1987), 179–194. [129] (with F. Ricci) “Harmonic analysis on nilpotent groups and singular integrals II.” J. Funct. Anal. 78 (1988), 56–84.

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[130] (with A. Nagel, J. P. Rosay, and S. Wainger) “Estimates for the Bergman and Szegö kernels in certain weakly pseudo-convex domains.” Bull. Amer. Math. Soc. 18 (1988), 55–59. ¯ [131] (with A. Nagel and D. C. Chang) “Estimates for the ∂-Neumann problem for pseudoconvex domains in C2 of finite type.” Proc. Nat. Acad. Sci. 85 (1988), 8771–8774. [132] (with A. Nagel, J. P. Rosay, and S. Wainger) “Estimates for the Bergman and Szegö kernels in C2 .” Ann. of Math. 128 (1989), 113–149. [133] (with D. H. Phong) “Singular Radon transforms and oscillatory integrals.” Duke Math. J. 58 (1989), 347–369. [134] (with F. Ricci) “Harmonic analysis on nilpotent groups and singular integrals III.” J. Funct. Anal. 86 (1989), 360–389. [135] (with A. Nagel and F. Ricci) “Fundamental solutions and harmonic analysis on nilpotent groups.” Bull. Amer. Math. Soc. 23 (1990), 139–143. [136] (with A. Nagel and F. Ricci) “Harmonic analysis and fundamental solutions on nilpotent Lie groups in Analysis and P.D.E.” A collection of papers dedicated to Mischa Cotlar. Marcel Decker, 1990. [137] (with C. D. Sogge) “Averages over hypersurfaces, smoothness of generalized Radon transforms.” J. d’ Anal. Math. 54 (1990), 165–188. [138] (with S. Sahi) “Analysis in matrix space and Speh’s representations.” Invent. Math. 101 (1990), 373–393. [139] (with S. Wainger) “Discrete analogues of singular Radon transforms.” Bull. Amer. Math. Soc. 23 (1990), 537–544. [140] (with D. H. Phong) “Radon transforms and torsion.” Duke Math. J. (Int. Math. Res. Notices) #4 (1991), 44–60. [141] (with A. Seeger and C. Sogge) “Regularity properties of Fourier integral operators.” Ann. of Math. 134 (1991), 231–251. [142] (with J. Stein) “Stock price distributions with stochastic volatility: an analytic approach.” Rev. Fin. Stud. 4 (1991), 727–752. [143] (with D. C. Chang and S. Krantz) “Hardy spaces and elliptic boundary value problems.” In the Madison Symposium on Complex Analysis, Contemp. Math. 137 (1992), 119–131.

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Other References [BGS] D. Burkholder, R. Gundy, and M. Silverstein. “A maximal function characterization of the class H p .” Trans. Amer. Math. Soc. 157 (1971): 137–153. [Ca] A. P. Calderón. “Commutators of singular integral operators.” Proc. Nat. Acad. Sci. 53 (1965): 1092–1099. [Chr] M. Christ. “On the ∂¯b -equation and Szegö projection on a CR manifold.” In Proceedings, El Escorial Conference on Harmonic Analysis 1987. Lecture Notes in Mathematics, no. 1384. Springer Verlag, 1987. [Co] M. Cotlar. “A unified theory of Hilbert transforms and ergodic theory.” Rev. Mat. Cuyana I (1955): 105–167. [CZ] A. P. Calderón and A. Zygmund. “On higher gradients of harmonic functions.” Studia Math. 26 (1964): 211–226. [EM] L. Ehrenpreis and F. Mautner. “Uniformly bounded representations of groups.” Proc. Nat. Acad. Sci. 41 (1955): 231–233. [FK] C. Fefferman and J. J. Kohn. “Estimates of kernels on three-dimensional CR manifolds.” Rev. Mat. Iber. 4, no. 3 (1988): 355–405. [FKP] C. Fefferman, J. J. Kohn, and D. Phong. “Subelliptic eigenvalue problems.” In Proceedings, Conference in Honor of Antoni Zygmund. Wadsworth, 1981. [FS] C. Fefferman and A. Sanchez-Calle. “Fundamental solutions for second order subelliptic operators.” Ann. of Math. 124 (1986): 247–272. [GN] I. M. Gelfand and M. A. Neumark. Unitäre Darstellungen der Klassischen Gruppen. Akademie Verlag, 1957. [H] I. I. Hirschman, Jr. “Multiplier transformations I.” Duke Math. J. 26 (1956): 222–242; “Multiplier transformations II.” Duke Math. J. 28 (1961): 45–56. [McN] J. McNeal, “Boundary behaviour of the Bergman kernel function in C2 .” Duke Math. J. 58 (1989): 499–512. [Z] A. Zygmund. Trigonometric Series. Cambridge University Press, 1959.

Chapter Two Eli’s Impact: A Case Study Charles Fefferman In [7], we discussed some of Eli Stein’s contributions to analysis. Here, by picking out a single striking example, we illustrate the continuing powerful influence of Eli’s ideas. We start by recalling Eli’s ideas on Littlewood-Paley theory, as well as several major developments in pure and applied mathematics, to which those ideas gave rise. We then discuss the remarkable recent work of Gressman and Strain [9–11] on the Boltzmann equation, and explain in particular its connection to Eli’s work. Before Eli, Littlewood-Paley theory was one of the deepest parts of the classical study of Fourier series in one variable. Eli found the right viewpoint to develop Littlewood-Paley theory on Rn . He went on to develop Littlewood-Paley theory on any compact Lie group, and then in any setting in which there is a reasonable heat kernel. Eli realized that there is a deep connection between ideas in LittlewoodPaley theory and the ∂-problems in several complex variables. Together with several co-authors (Folland, Greiener, Nagel, Ricci, Rothschild, . . . ) he carried out analysis on nilpotent groups and applied that analysis to partial differential equations and several complex variables. By his writing, his teaching and his collaborations (see, e.g., [15–23] and [28]), Eli has disseminated these ideas, to the extent that they are now part of the viewpoint of every well-informed analyst.∗ As we shall see, those ideas have had striking impact in unexpected places. Let us begin by sketching how Littlewood-Paley theory looked before Eli transformed it. Let f (x) be a real-valued function defined on Rn (before Eli, n = 1). Let f
(ξ ) be the Fourier transform of f . We introduce a partition of unity 1=

∞ 

χk (ξ ) on Rn \{0}, where

k=−∞

• each χk is supported on {2k−1 ≤ |ξ | ≤ 2k+1 } and • |∂ α χk (ξ )| ≤ Cα 2−k|α| for any k, α, ξ.

Charles Fefferman was partially supported by NSF DMS-0901040. ∗ It is impossible to mention Eli’s role in disseminating ideas without taking note of the remarkable series of analysis textbooks by Stein and Shakarchi [24–27]; see the book review [8].

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We then define a function fk (x) on Rn by setting f
k (ξ ) = χk (ξ )f
(ξ ). The Littlewood-Paley function G(f ) on Rn is defined by the formula  12  ∞  2 |fk (x)| for x ∈ Rn . G(f )(x) = k=−∞

One of the main results of Littlewood-Paley theory asserts that, for 1 < p < ∞, we have the Littlewood-Paley Theorem. f ∈ Lp (Rn ) if and only if G(f ) ∈ Lp (Rn ). Moreover, cf Lp (Rn ) ≤ G(f )Lp (Rn ) ≤ Cf Lp (Rn ) , where c and C depend only on p and n. The classical version of the above theorem (in one dimension) as well as several related results is due to Littlewood, Paley, Lusin, Marcinkiewicz, and Zygmund; see [31]. Its proof was based on complex variables. An essential tool was the Blaschke product   z − zν iθν e · B(z) = 1 − zν z ν and the Blaschke factorization (z) · B(z) F (z) = F  has no zeros. where B(z) is a Blaschke product and F Here, the z ν are the zeros of a given analytic function F on the unit disc. Given a function f on R, we pass to the Poisson integral U (x+iy), defined on the upper half-plane R2+ ; and then pass to the conjugate harmonic function V (x + iy). Thus, F = U + iV is analytic in the upper half-plane. In terms of the analytic function F , we define the Littlewood-Paley functions  12  ∞  2 y|F (x + iy)| dy (x ∈ R), g(f )(x) = 

0 

S(f )(x0 ) =

|F (z)| dxdy 2

 12

(x0 ∈ R),

z=x+iy∈Γ (x0 )

where Γ (x0 ) denotes the sector in Figure 2.1. In addition, one introduces another Littlewood-Paley function gλ∗ (f ) on R, depending on a parameter λ. The functions g(f ), S(f ), gλ∗ (f ) are strongly tied to complex variables and therefore they can be controlled using the Blaschke factorization. On the other hand, g(f ), S(f ), gλ∗ (f ) are close enough to G(f ) that they can be used to prove the Littlewood-Paley Theorem. The above analysis was a tour-de-force, but it was restricted to the onedimensional case because there is no higher-dimensional analogue of the Blaschke product.

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ELI’S IMPACT

Γ (x0) R2+ x0 Figure 2.1

Enter Eli! Eli viewed Littlewood-Paley theory as an application of the theory of singular integrals. From the theory of singular integrals, we recall the Calderón-Zygmund decomposition: Given a function f ∈ L1 (Rn ) and a number λ > 0, we can decompose f into a “good” function g and a “bad” function b : f = g + b; where

• g ∈ L2 (Rn ) with an estimate Rn |g(x)|2 dx ≤ Cλf L1 (Rn ) • b is supported in pairwise disjoint cubes Qν (ν ≥ 1), and has integral zero on each Qν . Moreover,

• Q |b(x)|dx ≤ Cλ|Qν | ν

for each ν

and •

 ν

|Qν | ≤

C f L1 (Rn ) . λ

Here, |Q| denotes the volume of a given cube Q, and C denotes a constant depending only on the dimension n. The Calderón-Zygmund decomposition was used to analyze singular integral operators, such as the Riesz transforms ( ∂x∂ j )(−x )−1/2 on Lp (Rn ). Eli saw that the Calderón-Zygmund decomposition can be used to understand the ∗ ∗ Littlewood-Paley functions g(f ), S(f  ), g∗ λ (f ), G(f ), because g(b), S(b), gλ (b), G(b) are easily estimated outside ν Qν , where b is the “bad function” in the Calderón-Zygmund decomposition, and Q∗ν is related to Qν as in Figure 2.2. Eli’s work gave the first real understanding (pun intended) of Littlewood-Paley theory. See [15, 16]. In the late 1960s, Eli showed that Littlewood-Paley theory could be generalized further; it extends to compact Lie groups, and indeed to any setting in which there is a reasonable heat kernel. See [18]. Eli then turned his attention to developing the Littlewood-Paley theory relevant to complex analysis on the unit ball in Cn . He saw the right point of view,

38

CHARLES FEFFERMAN

Qν∗

Qν

Figure 2.2

from which complex analysis on strictly pseudoconvex domains is closely analogous to basic potential theory on Rn . To introduce Eli’s analogy, we prepare the way with a few remarks: After a linear fractional transformation, the unit sphere in Cn+1 can be viewed as a nilpotent Lie group Hn , the Heisenberg group. A point of Hn has the form (z, t), with z ∈ Cn , t ∈ R. The group law is given by (z, t) · (z  , t  ) = (z + z  , t + t  + I m(z · z  )). The natural dilations on Hn are given by Sλ : (z, t) → (λz, λ2 t)

for λ > 0.

Therefore, if (z, t)−1 · (z  , t  ) = (z  , t  ) in Hn , then the natural “distance” between (z, t) and (z  , t  ) is d((z, t), (z  , t  )) ≈ |z  | + |t  | /2 . 1

(0.1)

Now we can discuss Eli’s analogy between basic potential theory on Rn and complex analysis on strictly pseudoconvex domains. The Group is Rn for basic potential theory, and Hn for complex analysis. The Basic PDE in Rn is the Laplace equation u = f . For complex analysis, the basic equations are the ∂-problems ∂u = α, ∂ b u = α, the ∂-Neumann problem, and the Kohn-Laplacian equation
b u = α. The Fundamental Solutions are as follows. For basic potential theory, u = f is solved by the familiar formula f (y)dy (n > 2). u(x) = cn n |x − y|n−2 R For complex analysis, the equation
b u = α is solved by u(x) = K(x, y)α(y)dy, Hn

where K(x, y) behaves like a negative power of the non-Euclidean distance d(x, y) on Hn ; see (0.1).

39

ELI’S IMPACT

Sharp Estimates for Solutions. In basic potential theory, sharp estimates arise from Calderón-Zygmund singular integral operators. For complex analysis, one needs analogues of singular integral operators on the Heisenberg group Hn . To control such operators on L2 , one cannot simply invoke the Fourier transform as in the Euclidean case. Rather, Eli used his non-commutative version of Cotlar’s lemma on sums of almost orthogonal operators. Fortunately, Knapp and Stein had demonstrated the usefulness of this approach in their study of principal series representations of semisimple Lie groups. It would take us too far afield to discuss these matters further in this paper. That’s only the beginning of the story. Eli’s analogy extends to many classes of domains in Cn , and to many other related PDEs. See [19–22] for these ideas. Our purpose here is to show that Eli’s ideas continue to exert a profound influence. To illustrate, it would be natural to discuss, e.g., wavelets; or Coifman’s ideas on imbedding large data sets into a low-dimensional Euclidean space (as well as the profound extension of those ideas due to Amit Singer); or the remarkable work of Klainerman-Rodnianski-Szeftel on general relativity. In this chapter, we will instead concentrate on another fundamental problem on which Eli’s ideas have recently exerted a major impact. We now discuss The Boltzmann Equation The Boltzmann equation describes statistically a gas of particles confined in, say, a 3-dimensional torus T3 = R3 /Z3 . Let x ∈ T3 denote the position of a particle; let v ∈ R3 denote its velocity; and let t ∈ [0, ∞) denote the time. We write F (v, x, t) to denote the density of particles per unit volume in (v, x)space R3 × T3 at time t. Next, we describe what happens to the particles. There are two relevant phenomena: • Transport: A particle at position x and velocity v at time t will be at position x + v · t and velocity v at time t + t. • Elastic Binary Collisions: A particle at position x and having velocity v may collide at time t with another particle with velocity v ∗ at position x. After the collision, the two particles at x have velocities v  and v∗ , respectively. By conservation of energy and momentum, we have v =

v + v∗ 1 + |v − v∗ |σ 2 2

and v + v∗ 1 − |v − v∗ |σ 2 2 for a vector σ in the unit sphere S 2 . v∗ =

40

CHARLES FEFFERMAN

Let θ be the angle between the vectors v  − v∗ and v − v∗ (or, equivalently, between σ and v − v∗ ). The Boltzmann equation asserts that ∂t F + v · x F = Q(F, F ), where, for each fixed (x, t), Q(F, G)(v) = dv∗ dσ B(v − v∗ , σ ) · [F∗ G − F∗ G], R3

S2

and we have set G = G(v), G = G(v  ), F∗ = F (v∗ ), F∗ = F (v∗ ). Maxwell computed the kernel B(v − v∗ , σ ), assuming that particles interact by a potential given by a negative power of the distance. He found that B(v − v∗ , σ ) ≈ |v − v∗ |γ |θ |−2−2s , with γ > −3, 0 < s < 1. Note that the singularity in σ ∈ S 2 is not locally integrable, and the factor |v − v∗ |γ is not integrable at infinity. The vast majority of work on the Boltzmann equation before about the year 2000 assumed that B(v − v∗ , σ ) is (at least) integrable with respect to σ ∈ S 2 . We now know that the physically interesting case has fundamentally different behavior. The Boltzmann equation has a 5-parameter family of equilibrium solutions   −|v − v0 |2 3 . F (v, x, t) = ρ(2π T )− 2 exp 2T Here, ρ = particle density and T = temperature are positive real numbers, while v0 = bulk velocity is a vector in R3 . In general, we start with a given initial particle density F0 (v, x), and solve the Boltzmann equation with initial condition F (v, x, 0) = F0 (v, x). This gives rise to the following Great Unsolved Problem Prove (or disprove) that any physically reasonable initial F0 (v, x) gives rise to a Boltzmann solution F (v, x, t) that converges to one of the above equilibrium solutions as t −→ ∞. Decide how rapidly the convergence takes place. This problem has been the subject of much work over many years. We mention here the contributions of Alexandre, Arkeryd, Carleman, Desvillettes, diPerna, Guo, Hilbert, Levermore, Lions, Liu, Mouhot, Ukai, Villani, Wennberg, Yang and Yu; see [1, 3–6, 12–14, 30] and the references therein. This is of course an incomplete list; see however [30]. Our purpose here is to discuss dramatic recent progress due to P. Gressman and R. Strain [9]. We take B(v − v∗ , σ ) ≈ |v − v∗ |γ |θ |−2−2s as before. We restrict attention here to the parameter range γ + 2s ≥ 0. For each such γ , s, we have the following result.

41

ELI’S IMPACT

Theorem 1 (Gressman-Strain). Let F 0 (v, x) be a positive initial particle density, 2 3 close enough to g = (2π )− 2 exp −|v| in a suitable norm. 2 Suppose that F0 (v, x)dvdx = 1, T3 ×R3 vF0 (v, x)dvdx = 0, and T3 ×R3 |v|2 F0 (v, x)dvdx = 1. T3 ×R3

Then there exists a positive solution F (v, x, t) of the Boltzmann equation with initial condition F (v, x, 0) = F0 (v, x), such that F (·, ·, t) −→ g exponentially fast as t −→ ∞. Thus, initial data close to equilibrium lead to a Boltzmann solution that tends exponentially fast to equilibrium as time tends to infinity. This represents dramatic progress, because it is the first result of its type in which one makes assumptions only on the initial data, without assuming existence or favorable properties of solutions at nonzero times. See also Alexandre, Morimoto, Ukai, Xu and Yang [2]. We assumed above that γ + 2s ≥ 0. For physically relevant γ , s with γ + 2s < 0, there are analogous results, but they are more complicated to state, and the convergence to equilibrium is subexponential. See [9]. A fundamental idea in the proof of Gressman and Strain is to carry out analysis and define a Littlewood-Paley function in a non-Euclidean setting, adapted to the Boltzmann equation and to the particular equilibrium solution g. To see the main ideas, we write √ for small f. F = g + gf The Boltzmann equation becomes ∂t f + v · x f + Lf = Γ (f, f ), 1 √ √ Γ (f, h) = g − 2 Q( gf, gh) √ √ Lf = −Γ (f, g) − Γ ( g, f ).

where and (0.2)

We now give a highly oversimplified discussion of the Gressman-Strain analysis of the above equation. We want to use energy estimates. The plan is to multiply the Boltzmann equation (0.2) by f and integrate. We hope that does some good. We find that 1 d 2 f (·, ·, t)L2 + f (v, x, t)v · x f (v, x, t)dvdx + f Lf dvdx 2 dt = f Γ (f, f )dvdx.

42

CHARLES FEFFERMAN

Now

f (v, x, t)v · x f (v, x, t)dvdx   1 v · x |f (v, x, t)|2 dvdx = 0, = 2 R3 ×T3

as we see from integration by parts. So 1 d f 2L2 + f Lf dvdx = f Γ (f, f )dvdx. 2 dt

(0.3)

Suppose we could find a norm f X such that f Lf dvdx ≥ cf 2X and

f Γ (f, f )dvdx ≤ Cf L2 f 2X .

Then our energy identity (0.3) would tell us that

If Cf L2
0. Proof. We prove the lemma in two steps. Step 1. We show first that Bp0 ≤ C(δ0 , A).

(6.6)

Assume p ≥ p0 is a dyadic integer and fix i1 , . . . , iK ∈ I such that the supremum in (6.2) is attained. Then, using self-adjointness and (6.1), we write p

p

p

Bp2 = (S1,i1 + S2,i2 + . . . + SK,iK )2  2p

2p

≤ S1,i1 + . . . + SK,iK  + 2

K−1


p

p

p

Sm,im (Sm+1,im+1 + . . . + SK,iK )

m=1

≤ B2p + 2

K−1


γm,p .

(6.7)

m=1

We estimate also γm,2p . For any jm , . . . , jK ∈ I 2p

p

2p

p

Sm,jm (Sm+1,jm+1 + . . . + SK,jK )  ≤ Sm,jm (Sm+1,jm+1 + . . . + SK,jK )2  +2

K−1
m =m+1

p

p

p

Sm ,jm (Sm +1,jm +1 + . . . + SK,jK )

≤ Bp γm,p + 2

K−1
m =m+1

γm ,p ,

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IONESCU, MAGYAR, AND WAINGER

using (6.1) and the identity 2p

p

2p

p

Sm+1,jm+1 + . . . + SK,jK = (Sm+1,jm+1 + . . . + SK,jK )2 −

K−1


p

p

p

Sm ,jm (Sm +1,jm +1 + . . . + SK,jK )

m =m+1 p −(Sm +1,jm +1

p

p

+ . . . + SK,jK )Sm ,jm .

Thus, for any m = 1, . . . , K and any dyadic integer p ≥ p0 K−1


γm,2p ≤ Bp γm,p + 2

γm ,p .

(6.8)

m =m+1

We use now inequalities (6.4), (6.7), and (6.8) to prove (6.6). Let L = L(δ0 ) =

∞


2−δ0 m .

m=0

Let p1 ≥ p0 denote the smallest dyadic integer for which Bp1 ≤ (100LA)p1 . Such p1 exists because Bp ≤ K, using (6.1). The bound (6.6) follows if p1 = p0 . Otherwise we have, for any dyadic integer p ∈ [p0 , p1 ) and any m = 1, . . . , K, Bp > (100LA)p ; Bp2 ≤ B2p + 2

K−1


γm,p ;

(6.9)

m=1

γm,2p ≤ Bp γm,p + 2

K−1


γm ,p .

m =m

It follows from the second equation of (6.9) and (6.4) that Bp20 ≤ B2p0 + 4ALBp0 . Using the first equation of (6.9) it follows that Bp20 ≤ 2B2p0 . Using the third equation of (6.9) and (6.4) it follows that γm,2p0 ≤ Bp0 2A2−δ0 m Bp0 + 4ALBp0 2−δ0 m ≤ 2−δ0 m B2p0 (8A), using Bp0 ≥ 2L and Bp20 ≤ 2B2p0 . More generally, we prove by induction that for any dyadic integer p ∈ [p0 , p1 ) and any m = 1, . . . , K Bp2 ≤ 2B2p

and

γm,2p ≤ 2−δ0 m B2p (4A)2p .

(6.10)

This was already proved above for p = p0 . Assume p ∈ [2p0 , p1 ) is a dyadic integer. It follows from the second inequality in (6.9) and the induction hypothesis that Bp2 ≤ B2p + 2L(4A)p Bp .

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NILPOTENT LIE GROUPS

Since Bp > (100LA)p , this gives the first inequality in (6.10). Using the third inequality in (6.9) and the induction hypothesis, γm,2p ≤ Bp 2−δ0 m Bp (4A)p + 2 · 2−δ0 m LBp (4A)p ≤ 2−δ0 m B2p (4A)2p , using Bp2 ≤ 2B2p and Bp ≥ 2L. By induction, this completes the proof of (6.10). Recall now that Bp1 ≤ (100LA)p1 . Thus, using only the first inequality in (6.10), Bp1 /2 ≤ 21/2 (100LA)p1 /2 Bp1 /4 ≤ 21/2 21/4 (100LA)p1 /4 ... l

l

Bp1 /2l ≤ 21/2 21/4 · . . . · 21/2 (100LA)p1 /2 . The bound (6.6) follows by letting 2l = p1 /p0 . Step 2. We prove now the bound (6.5). It follows from (6.4) and (6.6) that Bp0 ≤ A

γm,p0 ≤ A 2−δ0 m

and



for m = 1, . . . , K,

(6.11)



for some constant A = A (δ0 , A). We would like to prove that, for some constant A = A (A , δ0 ), Bp0 /2 ≤ A

and

γm,p0 /2 ≤ A 2−δ0 m/4

for m = 1, . . . , K.

(6.12)

We would then be able to prove (6.5) by repeating this step finitely many times. We may assume p0 ≥ 2 and look at Bp0 /2 . Fix i1 , . . . , iK ∈ I which attain the supremum in the definition of Bp0 /2 and write p /2

p /2

p

p

0 0 )2  ≤ S1,i0 1 + . . . + SK,i  Bp20 /2 = (S1,i0 1 + . . . + SK,i K K

+2

K−1


p /2

p /2

p /2

0 0 0 Sm,i (Sm+1,i + . . . + SK,i ) m K m+1

(6.13)

m=1

≤ A + 2

K−1


p /2

p /2

0 0 Sm,im (Sm+1,i + . . . + SK,i ), K m+1

m=1

using (6.1). Let Q=

sup

sup

p /2

m=1,...,K−1 jm ,...,jK ∈I

p /2

0 0 2δ0 m/4 Sm,jm (Sm+1,j + . . . + SK,j ). K m+1

(6.14)

Fix m, jm , . . . , jK such that the supremum in (6.14) is attained. Then we have p /2

p /2

0 0 + . . . + SK,j ) Q = 2δ0 m/4 Sm,jm (Sm+1,j K m+1

≤ 2δ0 m/4

8m
m =m+1

p /2

p /2

p /2

0 0 Sm,jm Sm0 ,jm  + 2δ0 m/4 Sm,jm (S8m+1,j + . . . + SK,j ). K 8m+1

(6.15) Now, using the second inequality in (6.11) and the definition of Q in (6.14), (6.1), selfadjointness, and the hypothesis Sm,0 = 0, Sm,jm Sm0 ,jm 2 ≤ Sm,jm Sm0 ,jm Sm,jm  ≤ A 2−δ0 m , p /2

p

184

IONESCU, MAGYAR, AND WAINGER

and p /2

p /2

p

p

0 0 Sm,jm (S8m+1,j + . . . + SK,j )2 K 8m+1 0 0 ≤ Sm,jm (S8m+1,j + . . . + SK,j )+2 K 8m+1




≤ A 2−δ0 m + 2




m ≥8m

p /2

p /2

p /2

0 Sm0 ,jm (Sm0 +1,jm +1 + . . . + SK,j ) K



Q2−δ0 m ≤ 2−δ0 m (A + 2LQ).

m ≥8m

Therefore, it follows from (6.15) and the last two inequalities that $ $ √ Q ≤ 2δ0 m/4 2−δ0 m/2 A (7m) + 2δ0 m/4 2−δ0 m/2 A + 2LQ ≤ Cδ0 A + 2LQ. It follows that Q ≤ C(δ0 , A ). In view of the definition (6.14), this proves the second inequality in (6.12). The first inequality in (6.12) follows from (6.13). This completes the proof of the lemma. We will need a version of this lemma for non-selfadjoint operators. Lemma 6.2. Assume that H is a Hilbert space, Sm ∈ L(H ), m = 1, . . . , K, and Sm  ≤ 1,

m = 1, . . . , K.

(6.16)

Let I = {0, 1},

Sm,0 = Sm ,

Sm,1 = 0.

For any dyadic integer p we define Dp = p = D

sup

∗ ∗ (S1,i1 S1,i )p + . . . + (SK,iK SK,i )p , 1 K

sup

∗ ∗ (S1,i S )p + . . . + (SK,i S )p . 1 1,i1 K K,iK

i1 ,...,iK ∈I i1 ,...,iK ∈I

(6.17)

For any m = 1, . . . , K − 1 and dyadic integer p we define µm,p =  µm,p =

sup

∗ ∗ ∗ (Sm,im Sm,i )[(Sm+1,im+1 Sm+1,i )p + . . . + (SK,iK SK,i )p ], m m+1 K

sup

∗ ∗ ∗ (Sm,i S )[(Sm+1,i S )p + . . . + (SK,i S )p ]. m m,im m+1 m+1,im+1 K K,iK

im ,...,iK ∈I im ,...,iK ∈I

(6.18) Assume that p0 + 1), µm,p0 ≤ A2−δ0 m (Dp0 + 1) and  µm,p0 ≤ A2−δ0 m (D

m = 1, . . . , K − 1, (6.19) for some dyadic integer p0 and some numbers A ≥ 1 and δ0 > 0. Then S1 + . . . + SK  ≤ C(δ0 , A, p0 ).

(6.20)

Remark 6.1. A simplified version of the lemma, which is used in this chapter, is the following: assume that H is a Hilbert space, Sm ∈ L(H ), m = 1, . . . , K, and

185

NILPOTENT LIE GROUPS

let Sm,0 = Sm , Sm,1 = 0. Assume that, for all m = 1, . . . , K, Sm  ≤ 1,

sup m∈{1,...,K}

sup

∗ ∗ ∗ Sm,i [(Sm+1,im+1 Sm+1,i )p0 + . . . + (SK,iK SK,i )p0 ] ≤ A2−δ0 m , m m+1 K

sup

∗ ∗ Sm,im [(Sm+1,i S )p0 + . . . + (SK,i S )p0 ] ≤ A2−δ0 m . m+1 m+1,im+1 K K,iK

im ,...,iK ∈I im ,...,iK ∈I

(6.21) Then S1 + . . . + SK  ≤ C(δ0 , A, p0 ). Proof of Lemma 6.2. We apply Lemma 6.1 to the operators Sm Sm∗ and Sm∗ Sm . It follows that there are constants A ≥ 1 and δ > 0 depending only on δ0 , A, P0 such that 1 ≤ A, D1 + D

µm,1 +  µm,1 ≤ A2−δm ,

m = 1, . . . , K.

(6.22)

For any m = 1, . . . , K − 1 let νm =  νm =

sup

∗ ∗ ∗ Sm,i [(Sm+1,im+1 Sm+1,i ) + . . . + (SK,iK SK,i )], m m+1 K

sup

∗ ∗ Sm,im [(Sm+1,i S ) + . . . + (SK,i S )]. m+1 m+1,im+1 K K,iK

im ,...,iK ∈I im ,...,iK ∈I

Clearly, for any m = 1, . . . , K − 1, νm2 ≤ D1 µm,1 ,

1   νm2 ≤ D µm,1 .

Therefore, using (6.22), νm ≤ 2A2−δm/2 , νm + 

m = 1, . . . , K.

(6.23)

Clearly, S1 + . . . + SK 2 ≤ S1 S1∗ + . . . + SK Sk∗  + 2

K−1


∗ Sm (Sm+1 + . . . + SK∗ ).

m=1

Since D1 ≤ A, for (6.20) it suffices to prove that ∗ + . . . + SK∗ ) ≤ A 2−δm/8 , Sm (Sm+1

m = 1, . . . , K − 1.

(6.24)

Let Q= = Q

sup

sup

∗ ∗ 2δm/8 Sm,im (Sm+1,i + . . . + SK,i ), m+1 K

sup

sup

∗ 2δm/8 Sm,i (Sm+1,im+1 + . . . + SK,iK ). m

m=1,...,K−1 im ,...,iK ∈I m=1,...,K−1 im ,...,iK ∈I

Fix m, im , . . . , iK such that the supremum in the definition of Q is attained. Then Q ≤ 2δm/8

8m
m =m+1

∗ ∗ Sm,im Sm∗  ,im  + 2δm/8 Sm,im (S8m+1,i + . . . + SK,i ). 8m+1 K

(6.25)

186

IONESCU, MAGYAR, AND WAINGER

For any m ∈ [m + 1, 8m] ∩ Z we have, using (6.23), Sm,im Sm∗  ,im  ≤ Sm,im Sm∗  ,im Sm ,im 1/2 ≤  νm1/2 ≤ 2A2−δm/4 . Using Sm  ≤ 1 and the definitions, it follows that ∗ ∗ + . . . + SK,i )2 Sm,im (S8m+1,i 8m+1 K ∗ ∗ ≤ Sm,im (S8m+1,i + . . . + SK,i )(S8m+1,i8m+1 + . . . + SK,iK ) 8m+1 K

≤ νm + 2

K
m =8m+1

≤ νm + 2

K


Sm∗  ,im (Sm +1,im +1 + . . . + SK,iK )  −δm /8 . Q2

m =8m+1

Therefore, using (6.23) and (6.25), 1/2 ). Q ≤ C(δ, A)(1 + Q A similar argument shows that  ≤ C(δ, A)(1 + Q1/2 ), Q and the desired bound (6.24) follows.

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[8] M. Christ, Hilbert transforms along curves I: nilpotent groups, Ann. Math. 122/3 (1985), 575–596. [9] M. Christ, A. Nagel, E. M. Stein, and S. Wainger, Singular and maximal Radon transforms: analysis and geometry, Ann. Math. 150 (1999), 489–577. [10] H. Davenport, Cubic forms in thirty-two variables, Phil. Trans. R. Soc. Lond. A 251 (1959), 193–232. [11] J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies 116, North-Holland Publishing Co., Amsterdam (1985). [12] A. D. Ionescu, A. Magyar, E. M. Stein, and S. Wainger, Discrete Radon transforms and applications to ergodic theory, Acta Math. 198 (2007), 231–298. [13] A. D. Ionescu and S. Wainger, Lp boundedness of discrete singular Radon transforms, J. Amer. Math. Soc. 19 (2006), 357–383. [14] A. Leibman, Convergence of multiple ergodic averages along polynomials of several variables, Israel J. Math. 146 (2005), 303–315. [15] A. I. Malcev, On a class of homogeneous spaces, Izvestia Acad. Nauk SSSR Ser. Math. 13 (1949), 9–32. [16] A. Magyar, E. M. Stein, and S. Wainger, Maximal operators associated to discrete subgroups of nilpotent Lie groups, J. Anal. Math. 101 (2007), 257–312. [17] D. Oberlin, Two discrete fractional integrals, Math. Res. Lett. 8 (2001), 1–6. [18] L. Pierce, A note on twisted discrete singular Radon transforms, Math. Res. Lett. 17 (2010), 701–720. [19] L. Pierce, Discrete fractional Radon transforms and quadratic forms, Duke Math. J. (to appear). [20] F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals I. Oscillatory integrals, J. Funct. Anal. 73 (1987), 179–194. [21] F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals II. Singular kernels supported on manifolds, J. Funct. Anal. 78 (1988), 56–84. [22] J. L. Rubio de Francia, A Littlewood–Paley inequality for arbitrary intervals, Revista Matematica Iberoamericana 1 (1985), 1–14. [23] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, Princeton (1993). [24] E. M. Stein and S. Wainger, Discrete analogues of singular Radon transforms, Bull. Amer. Math. Soc. 23 (1990), 537–544.

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[25] E. M. Stein and S. Wainger, Discrete analogues in harmonic analysis I: 2 estimates for singular Radon transforms, Amer. J. Math. 121 (1999), 1291–1336. [26] E. M. Stein and S. Wainger, Two discrete fractional integral operators revisited, J. Analyse Math. 87 (2002), 451–479.

Chapter Eight Internal DLA for Cylinders David Jerison, Lionel Levine, and Scott Sheffield

8.1 INTRODUCTION Internal Diffusion-Limited Aggregation (internal DLA) is a random lattice growth model. Consider the two-dimensional lattice, Z × Z. In the case of a single source at the origin, the random occupied set A(T ) of T lattice sites is defined inductively as follows. Let A(1) be the singleton set containing the origin. Given A(T − 1), start a random walk in Z × Z at the origin. Then A(T ) := {n} ∪ A(T − 1) where n ∈ Z×Z is the first site reached by the random walk that is not in A(T −1). In this chapter, we will discuss the continuum limit of internal DLA, which is governed by a deterministic fluid flow equation known as Hele-Shaw flow. Our main focus will be on fluctuations. In [JLS11] we characterized the average fluctuations of the model just described in terms of a close relative of the Gaussian Free Field, defined below. In this chapter we will prove the analogous results for the lattice cylinder. In the case of the cylinder, the fluctuations are described in terms of the Gaussian Free Field exactly. We will also state without proof an almost sure bound on the maximum fluctuation in the case of the cylinder analogous to the case of the planar lattice proved in [JLS12a]. The main tools used in the proofs are martingales. As we shall see, the martingale property in this context is the counterpart in probability theory of well-known conservation laws for Hele-Shaw flow. The internal DLA model was introduced in 1986 by Meakin and Deutch [MD86] to describe chemical processes such as electropolishing, etching, and corrosion. Think of the occupied region as a blob of fluid. Figure 8.1 depicts a simulation of a cluster (blob) of size one million in dimension 2. At each step a corrosive molecule is introduced at a source, which in this simulation is a single point at the origin. The corrosive particle wanders at random through the fluid until it reaches the fluid-metal boundary, where it eats away a tiny portion of metal and enlarges slightly the fluid region. The question that concerned Meakin and Deutch was the smoothness of the surface that is being polished, that is, how irregular the boundary is. Figure 8.2 is a close-up picture of the boundary fluctuations. David Jerison was partially supported by NSF grant DMS-1069225. Lionel Levine was supported by NSF grant DMS-1105960. Scott Sheffield was partially supported by NSF grant DMS-0645585.

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Figure 8.1 Internal DLA cluster A(T ) with T = 106 sites in Z2 .

Figure 8.2 Detail of boundary of the 1 million particle cluster or blob.

Figure 8.1 suggests that the limit shape from a point source is a disk. Indeed, in 1992, Lawler, Bramson, and Griffeath [LBG92] proved that the rescaled limit shape of internal DLA from a point source is a ball in any dimension. In 1995, Lawler [Law95] proved almost sure bounds on the cluster of the form B(r − Cr 1/3 ) ∩ Zd ⊂ A(T ) ⊂ B(r + Cr 1/3 ), where T is the volume of the ball of radius r and C is a dimensional constant. On the other hand, the numerical√simulations of Meakin and Deutch predicted fluctuations, on average, of size O( log r) in dimension 2 and O(1) in dimension 3. They made their predictions based on small values of T , but much larger simulations are now possible and give the same results. The theorems we will describe are consistent with the size of fluctuations predicted by Meakin and Deutch and reveal deeper structure, namely, that the fluctuations obey a central limit theorem. The Fourier coefficients of the fluctuations tend to independent Gaussians, whose variance we can compute. This gives a heuristic explanation of numerical results on average fluctuations and many other predictions, such as, what should be the best possible bound on maximum fluctuations. In 2010, Asselah and Gaudillière [AG10] improved the power in Lawler’s bound in dimensions greater than 2. Later in 2010, Asselah and Gaudillière [AG10a, AG10b] and the present authors [JLS12a, JLS12b] independently proved logarithmic bounds on the maximum fluctuation.

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Theorem 1. (Maximum Fluctuations) There is a dimensional constant Cd , such that almost surely for sufficiently large r, B(r − C2 log r) ∩ Z2 ⊂ A(T ) ⊂ B(r + C2 log r) with T = π r 2 . Moreover, for d ≥ 3,

B(r − Cd log r) ∩ Zd ⊂ A(T ) ⊂ B(r + Cd log r) where T is the volume of the ball of radius r. The maximum fluctuations represent the worst case along the entire circumference as opposed to the average fluctuations observed by Meakin and Deutch. Whether one considers the average or the worst case, the model produces remarkably smooth surfaces — even more smooth in dimension 3 than in dimension 2. Before going any further, we should add a disclaimer. Despite their superficial similarity, the internal DLA model and the Diffusion-Limited Aggregation (DLA) model introduced by Witten and Sander [WS81] are very different. DLA is a model of particle deposition, in which a seed particle is placed at the origin in a lattice. Particles follow a random walk starting at infinity and attach to the existing cluster the first time they are adjacent to it. The particles form a cluster of fractal character and the continuum limit is very far from deterministic. In their 1986 article, Meakin and Deutch refer to the work of Witten and Sander and explain that the internal DLA model is better behaved than DLA and intended to describe quite different physical phenomena, ones that do not exhibit chaos. The Hele-Shaw model is also highly relevant to DLA, but it is the complement of the cluster that is interpreted as the fluid region. Thus the fluid region shrinks. When fluid is sucked away, the HeleShaw equation is ill-posed, and the methods of partial differential equations no longer apply except at very short time scales. Instead, algebraic methods are used. The subject is of great interest in statistical physics and has a direct connection with random matrices, but it is not the subject of this chapter. This chapter discusses various aspects of several works of the authors [JLS12a, JLS12b, JLS11]. Rather than prove any of the theorems in those papers, which concern Zd , we prove two central limit theorems (Theorems 3 and 4) in which the set Z2 of [JLS11] is replaced by the lattice cylinder (Z/N Z)×Z. In the next section, we state our theorems in this new geometric setting. In the third section we explain the relationship between internal DLA and Hele-Shaw flow. Sections 8.4 and 8.5 give complete proofs of two central limit theorems for fluctuations of internal DLA on cylinders. We discuss the work of Levine and Peres concerning the relationship of internal DLA with the obstacle problem in Section 8.6. In the last section we make a few further remarks about the theorems of [JLS12a, JLS12b, JLS11], the effects of geometry on the problem, higher-dimensional questions, and questions related to more general random walks.

8.2 MAIN RESULTS FOR THE CYLINDER In this section we state our main results in the case of the two-dimensional cylinder rather than the single source model in the plane which is carried out in [JLS12a]. We will make a comparison at the end of the chapter.

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Figure 8.3 The symmetric difference of AN (T ) and {y ≤ T /N 2 } is the thin, ragged band at the top with early points in black above the line y = T /N 2 and late points in gray below. The bar at the bottom is the region y ≤ 0.

Consider the cyclic group ZN = Z/NZ, whose elements will typically be denoted n1 = 1, 2, . . . , N. In the lattice cylinder ZN × Z, define the set A(0) = {n = (n1 , n2 ) ∈ ZN × Z : n2 ≤ 0}. For integers T > 0, the set A(T ) of lattice points is defined inductively, with source at n2 = −∞. Equivalently, given the set A(T − 1), start a random walk in ZN × Z at one of the sites (n1 , 0), n1 = 1, . . . , N, with equal probability. A(T )\A(T − 1) consists of the site at which the random walk exits A(T − 1) for the first time. Denote A+ (T ) = A(T )\A(0). A theorem analogous to Theorem 1, stated in a slightly more precise form, is Theorem 2. Given 0 < y1 and a < ∞, there is a constant C depending only on y1 , and a such that with probability 1 − N −a , for all y, 0 ≤ y ≤ y1 , {n : n2 ≤ yN − C log N } ∩ (ZN × Z) ⊂ A(T ) ⊂ {n : n2 ≤ yN + C log N } with T = yN 2 . Next, we scale A(T ) by the factor 1/N to obtain a subset AN (T ) of T × R with T = R/Z. For n = (n1 , n2 ), n1 = 1, . . . , N, n2 ∈ Z, and 0 < x ≤ 1 representing x ∈ T, let QN (n) = {(x, y) ∈ T × R : n1 − 1 < Nx ≤ n1 , n2 − 1 < Ny ≤ n2 }.

(1)

QN (n) is the square of sidelength 1/N with n/N at its upper right corner. Define   AN (T ) = QN (n); A+ QN (n). (2) N (T ) = n∈A(T )

n∈A+ (T )

2 Thus A+ N (T ) is the occupied subset of T×R+ consisting of T squares of area 1/N . We define a discrepancy function DN,T by

DN,T (x, y) = N (1AN (T ) − 1{y≤T /N 2 } ).

(3)

Figure 8.3 gives a closer look at the discrepancy between A(T ) and the expected strip by distinguishing early and late sites relative to the time T = yN 2 . DN,T takes

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on the values ±N and 0. DN,T > 0 means that Q is early relative to the time T . DN,T < 0 means that Q is late. The factor N in the definition of DN,T is the appropriate normalization so that the limit exists in the sense of distributions as N → ∞. Informally, our next theorem says that DN,T (x, y) → D(x)δ(y − y0 )

(4)

in the sense of distributions with ∞  ak bk D(x) ∼ √ cos(2π kx) + √ sin(2π kx) k k k=1 and ak and bk independent, normally distributed random variables with mean zero and variance 1. The random variable D(x) is not defined for individual values of x. For each x, the variance, ∞  1 (cos2 (2π x) + sin2 (2π x)) = ∞. k k=1 The precise statement of the√theorem uses duality and involves weight factors that are merely asymptotic to c/ k. Let H0 be the Sobolev space of functions η on T × R+ satisfying η(x, 0) = 0 and square norm equal to the Dirichlet integral,  ∞ 1 |∇η(x, y)|2 dxdy.

η 2H0 = 0

0

The restriction of H0 to the circle y = y0 is the Sobolev space H 1/2 . Its dual is the space of H −1/2 distributions on T with dual norm given by  1  f (x)η(x, y0 ) dx : η H0 ≤ 1 .

f {y0 } = sup 0

Fix an integer K, and consider test functions ϕ ∈ C ∞ (T × R) of the form  ϕ(x, y) = αk (y)e2πikx . |k|≤K

Assume that for each k, αk is supported in the annulus 0 < c1 ≤ |y| ≤ c2 , and the ϕ is real-valued, i.e., α−k = αk . Theorem 3. Let T = y0 N 2 . Then as N → ∞,  DN,T (ϕ) := DN,T (x, y)ϕ(x, y) dxdy T×R

tends in law to a normally distributed random variable with mean zero and variance  mk |αk (y0 )|2 Sy20 (ϕ) := ϕ(·, y0 ) 2{y0 } = 0 0) and early points (LN < 0) are indicated in gray and black, respectively. (b) The shade of gray indicates the size of LN (x, y).

8.3 IDLA AND HELE-SHAW FLOW In this section, we give a heuristic description of the relationship between internal DLA and the Hele-Shaw model. This section contains no proofs, only formal derivations. The proof that the deterministic limit of internal DLA is Hele-Shaw flow, given in 2009 by Levine and Peres [LP10], proceeds via a discrete version of a classical obstacle problem. We will discuss their work in slightly more detail in Section 8.6. Recall that in internal DLA from a single source in Z2 a particle takes a random walk from the origin in A(T ) until the first time it exits. Then it stops and augments the cluster to form A(T + 1). It stops at sites y at unit distance from A(T ) with probability pT (y), and this first exit probability is the discrete harmonic measure. In other words, it satisfies  v(y)pT (y) (8) v(0) = y∈Z2

for every function v : Z2 → R satisfying Lv(x) = 0,

for all x ∈ A(T ),

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where L is the discrete Laplacian defined by 1 Lf (x) = [f (x + e1 ) + f (x − e1 ) + f (x + e2 ) + f (x − e2 )] − f (x). 4 This suggests that the deterministic continuum limit of the growth process is governed by harmonic measure. Indeed, the continuum limit of the random walk is Brownian motion, and, according to Kakutani’s theorem, the hitting probability of Brownian motion starting from a point of a domain is the harmonic measure relative to that point. The continuum process in which a region grows proportionally to its harmonic measure is known as Hele-Shaw flow. Hele-Shaw flow describes the flow of fluid between two nearby parallel plates. The occupied region is essentially two-dimensional, so it is modeled by an open set t ⊂ R2 at time t. Given a domain 0 at time t = 0, fluid is pumped in at the origin so that the area grows at a uniform speed, |t | = t + |0 |. The pressure p(x, t) satisfies p(x, t) = −δ in t and p(x, t) = 0 on ∂t . The Hele-Shaw equation governing the growth says that the normal velocity of the boundary of t is |∇p|. Since p is Green’s function for t with pole at the origin, Hele-Shaw’s equation can also be expressed as saying that the growth of the domain is proportional to its harmonic measure. The correspondence with the discrete case is 0 ↔ A(T0 ), t ↔ A(T1 ), |0 | = T0 /N 2 , |t | = T1 /N 2 , and t = (T1 − T0 )/N 2 . One way to solve the Hele-Shaw equation is to solve instead for  t u(x, t) = p(x, s)ds. (9) 0

It’s well-known (cf. [GV06]) that for each fixed t, u solves an obstacle problem as follows. Choose γ (x, t) to be a function on R2 solving γ = tδ + 10 − 1. Let w solve the obstacle problem w(x, t) = inf{f : f ≤ 0, f ≥ γ }. Although w depends on the choice of γ , the set t = {x ∈ R2 : w(x, t) > γ (x, t)} and the function u(x, t) = w(x, t) − γ (x, t) ≥ 0 are independent of the choice of γ . On t , u = −γ = 1 − tδ − 10 , and on ct , u = 0. In fact, u = 1t − 10 − tδ 2

(10)

in all of R . Conversely, starting from u, differentiate (10) with respect to t, to obtain ∂ u(x, t) = V σt − δ ∂t where V is the normal velocity of the boundary of t and σt is the arc length measure of ∂t . Define ∂ p(x, t) = u(x, t). (11) ∂t

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Then ∂ u(x, t) = p(x, t) = −δ + |∇p|σt , ∂t and hence p = (∂/∂t)u is the pressure for a Hele-Shaw fluid cell with normal velocity V = |∇p|. The formulas above yield conservation laws,  ∂ v(x) dx (12) v(0) = ∂t t for every harmonic function v. We derive (12) in integrated form by multiplying (10) by v and integrating to obtain     (v)u dx = v(u) dx = v dx − v dx − tv(0). 0= R2

R2

t

0

(One sees formally that the integration by parts has no boundary terms because u vanishes to second order on ∂t and is identically zero outside.) These formulas are also known as quadrature formulas [GV06]. We have now come nearly full circle. Let ωt be the harmonic measure of t with respect to the origin, defined by the property  v(x)ωt (dx) v(0) = ∂t

for every harmonic function in t with, say, continuous boundary values. Then ωt = |∇p|σt = V σt , where V is the normal velocity of ∂t , and    ∂ v(0) = v(x) dx = v(x)V σt (dx) = v(x)ωt (dx), ∂t t ∂t ∂t The discrete analogue is the equation we started with, (8). For any fixed discrete harmonic function v, define  v(n). M(T ) = n∈A(T )

If v(0) = 0, then (8) implies that the conditional expectation of M(T + 1) given A(T ) is  v(y)pT (y) = v(0) = 0. (13) E(M(T + 1)|A(T )) = y∈Z2

In other words, M is a martingale. Martingales of this type for various choices of v are the main tools in the proofs of theorems about fluctuations. The martingale property is an immediate consequence of the discrete version of Kakutani’s theorem. The continuum theorems won’t be necessary to us; they just help us to gain intuition. Finally, we carry out a heuristic derivation that suggests the form of the central limit theorems concerning fluctuations. Suppose that the boundary is given by a perturbation of the disk, in polar coordinates,  αk eikθ r < R + f (θ ), f (θ ) = k∈Z

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with α−k = αk . We calculate the linearization of Hele-Shaw flow for perturbations of the disk. The Hadamard variational formula says that the (first order in ) change in the gradient of Green’s function is minus the radial derivative of the harmonic extension of f ,    kαk ∂  k ikθ  αk (r/R) e  =− eikθ . −  ∂r R k∈Z

r=R

k∈Z

The minus sign is very important. When f (θ ) > 0, the perturbation is farther from the origin than the location Reiθ on the circle and the harmonic measure is smaller than average, and fewer particles than average accumulate near Reiθ . This deterministic aspect of the process keeps the shape close to circular. Next, we guess as to the stochastic ingredients of the evolution. We propose that the modes vary independently. We expect that for some constant c > 0, k > 0, t = π r 2, dr (14) + c dBk (ρ) = −kαk dρ + c dBk , (ρ = log r) dαk = −kαk r with independent white noise (derivative dBk of a Brownian motion Bk ) of equal amplitude in each mode. The term −kαk dρ represents the deterministic drift back towards the disk coming from the calculation above. With c = 1, this is the stochastic differential equation that yields the Gaussian Free Field. In [JLS12a] we find instead that the stochastic differential equation turns out to be dαk = −(k + 1)αk dρ + dBk ,

ρ = log r.

The fact that k is replaced by k + 1 is related to the curvature of the boundary. The circumference of the circle increases with r, so there is room for more particles at the larger radius, and the modes decrease slightly more than given in the rough calculation above. On the other hand, in the case of the cylinder, the circumference of the boundary circle of reference remains constant, and we show in this chapter that we get exactly the Gaussian Free Field. 8.4 PROOF OF THEOREM 3 Note first that if g(x) = e2πikx , k > 0 and  g(x) sinh(2π ky)/ sinh(2π ky0 ), u(x, y) = g(x)e−2πk(y−y0 ) ,

0 ≤ y ≤ y0 , y0 ≤ y < ∞,

then the restriction norm

g H 1/2 = inf{ v H0 : v(x, y0 ) = g(x)} is achieved by the harmonic extension u. This is proved by computing  ∞ 1 |∇u|2 dxdy = (4π k)/(1 − e−4πky0 ) = 1/mk 0

0

so that formula (5) holds.

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Divide the outcomes of the cluster growth A(T ) into the three events. Event 1, with probability at least 1 − N −100 , is the event that the conclusion of Theorem 2 holds, or, put another way, DN,T is supported in the set F = {(x, y) : |y − T /N 2 | ≤ C(log N )/N } for all T ≤ C1 N 2 . Event 2 is the event that DN,T is supported in y ≤ C2 for all T ≤ C1 N 2 , but Event 1 does not hold. Thus Event 2 has probability at most N −100 . Event 3 is the complement of Events 1 and 2. To estimate the probability of Event 3, we recall from [JLS12a] that thin tentacles are rare events. Lemma A of [JLS12a] can be stated in a nearly equivalent form as follows. Denote B(n) = {m ∈ ZN × Z : n2 − N/2 ≤ m2 < n2 + N/2}. This is a cylinder with about N 2 lattice sites. Lemma 5. (Thin tentacles) There are positive absolute constants C0 , b > 0, and c > 0 such that for all n ∈ ZN × Z with n2 ≥ N , P{n ∈ A(T ) and #(A(T ) ∩ B(n)) ≤ bN 2 } ≤ C0 e−cN

2

/ log N

.

(15)

This lemma implies that for C2 sufficiently large relative to C1 , Event 3 has 2 probability at most O(e−cN / log N ). Indeed, suppose there is n ∈ A(T ) such that n2 > C2 N for some T ≤ C1 N 2 . Then since #A(T ) = T , and A(T ) is connected, for at least one n ∈ A(T ) with n2 ≥ N , #B(n ) ∩ A(T ) ≤ (2/C2 )C1 N 2 . Thus if C1 /C2 < b, Lemma 5 applies to B(n ) and Event 3 has probability at most 2 C0 C2 e−cN / log N . On Event 1 we will replace ϕ by a harmonic function. For |k| ≤ K  N , define q(k, N ) ≥ 0 by 1 − cos(2π k/N ) = cosh(q/N) − 1.

(16)

q(k, N ) = 2π |k| + O(1/N 2 ).

(17)

It follows that

Define for n ∈ Z/NZ × Z ψ0 (n, T , N) =



αk (T /N 2 )e2πin1 /N e(q/N)(n2 −T /N) .

(18)

0 1, after a single step in the process, a toppling of the site x, the new heights are 1 at x and (µ(x) − 1)/4 extra at each of the four adjacent sites. This process continues until every site has at most

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Figure 8.6 Divisible sandpile sum of two overlapping disks 1A + 1B .

height 1. One can show that the process stops in a finite number of steps for finitely supported µ. Moreover, the final height function, denoted ν(x), is independent of the order in which the toppling occurs. The divisible sandpile reflects the average behavior of the random walk. Figure 8.6 depicts starting from the same µN as in the preceding picture, Figure 8.5. The resemblance is already striking for N ≈ 200. Levine and Peres show that the random process is well-approximated by the deterministic sandpile process by means of an auxiliary function they call the odometer function u(x). The function u(x) records the (fractional) number of particles donated by the site x in the course of the deterministic process. The word odometer reminds us that u(x) does not represent a net loss of particles, but rather the total quantity donated without subtracting the number received. It is not hard to check that u solves the discrete Laplace equation Lu(x) = ν(x) − µ(x). Moreover, u can be obtained from the solution to the discrete obstacle problem as follows. Lemma 9. (Levine-Peres) Fix any γ (x) satisfying Lγ = µ − 1. Let w solve the obstacle problem w(x) = inf{f (x) : Lf ≤ 0,

f ≥ γ }.

Let u be the odometer function starting from µ. Then u = w − γ. The fact that the discrete function u tends to its continuum counterpart depends ultimately on estimating the difference between the fundamental solution of L and .

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8.7 FURTHER REMARKS AND QUESTIONS Theorem 1 is proved in [JLS12a] in dimension 2 and in [JLS12b] in dimensions d ≥ 3. The proof uses martingales associated to the discrete analogue of Green’s function with a pole at a point near the putative boundary, either inside or outside. These martingales take values larger than the expected value if there are extra points in the cluster near the pole, and the martingale is smaller than its expected value if there are fewer then the typical number of occupied sites in the cluster near the pole. The central limit theorem is inadequate to the task of estimating large deviations of the martingale. Instead one uses the parametrization of the martingale by Brownian motion. The lemma concerning thin tentacles, Lemma 5, is crucial as well. The proof also involves an iteration of successively better estimates on the inner and outer deviations of the shape. The proof of Theorem 2 is analogous to that of Theorem 1, just as Theorems 3 and 4 are similar to the corresponding theorems for Z2 . In this chapter, we chose to treat the case of the cylinder so as to identify a case in which the fluctuations are described exactly by the Gaussian Free Field. In [JLS11] we carried out the case of the disk. One difference with the case of the cylinder is that it’s somewhat harder to construct suitable discrete harmonic functions approximating z k . Furthermore, the estimates require variants for averages with respect to discrete harmonic functions of van der Corput’s theorem counting lattice points in disks. We have not yet carried out the case d = 3, although we believe it follows from very similar methods. The technical difficulty is that it requires variants of theorems, stronger theorems than van der Corput’s, concerning the number of lattice points in a ball in 3-space, along the lines of improvements due to Vinogradov (see [IKKN04]). Whereas the square Dirichlet norm is  2π  ∞  dr 2 |∇f (x, y)| dxdy = (|r∂r f |2 + |∂θ f |2 ) dθ 2 r R 0 0  ∞ dr = 2π (|r∂r fk |2 + |kfk |2 ) r k∈Z 0 in which f (x, y) =



fk (r)eikθ ,

k∈Z

the square of the norm of the Gaussian random field representing fluctuations from a source at the origin in Z2 is  ∞ dr 2π (|r∂r fk |2 + |(|k| + 1)fk |2 ) . r 0 k∈Z

In general, we expect that the random field will reflect the curvature of the deterministic region. But even in this simple case, the norm is expressed in terms of non-local (pseudo-differential) operators. The expression for the norm in Theorem 3 at distance y0 , starting from the (exactly straight) boundary of y ≤ 0, involves the factor (1 − e−4π |k|y0 ). Thus in some

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average sense, the influence of deterministic behavior at y = 0 attenuates at an exponential rate at y = y0 . It would be nice to understand this better. At the same time one can ask about the mixing time, that is, given a known boundary at one time, how long do we need to wait before that configuration is mostly forgotten? One can also ask questions about random walks other than the standard one. The first author1 supervised work on this subject in the summer of 2008 by a high school student, Max Rabinovich [R08]. He adapted the methods of Levine and Peres to the hexagonal lattice. His key observation is that the same methods work, provided one can approximate the discrete fundamental solution by the analogue of the Newtonian potential. At first it appears that the estimates need to be good to second order at infinity which they are not for the hexagonal lattice. But on closer inspection, what is required are error estimates for the difference of fundamental solutions, as compared to the gradient of the Newtonian potential. The gradient is of order 1/r d−1 as r → ∞ and the error term is one order better, 1/r d , which is two orders better than 1/r d−2 as required. Rabinovich’s theorem applies to all random walks on Zd given by a finitely supported probability measure p on Zd such that the random walk moves from x to x + α with probability p(α) and  p(α)α = 0. α∈Zd

This condition means that the random walk has no drift. It remained to consider walks with drift. James Propp proposed the specific example of a walk that moves East or North, each with probability 1/2. If the source is the origin, this fills a cluster in the first quadrant. If there are T particles, it is natural to rescale by parabolic scaling: u = x + y, and v = x − y are replaced by U = (x + y)/T 2/3 , V = (x − y)/T 1/3 . Then in parallel with the work of Levine and Peres, one expects the cluster to be associated with an obstacle problem based on parabolic operators in the (U, V ) variables as treated by Caffarelli, Petrosyan, and Shahgholian [CPS04]. After this lecture, Cyrille Lucas [Lu12] carried out this program and in the process established the existence of a so-called heat ball. Take the limiting (and indeed simplest case) of the Hele-Shaw flow in which 0 shrinks to a point. Then t is the Euclidean ball of volume t. The conservation law (12) in integrated form can be written  1 v(x) dx v(0) = vol B B for any harmonic function v. Of course, this is just the well-known mean value property for harmonic functions. The domain analogous to the ball for the heat operator is a set H ⊂ {(x, t) : t ≥ 0} of area 1 such that  v(x, t) dxdt v(0, 0) = H

where v satisfies the adjoint or backwards heat equation [(∂/∂x)2 + (∂/∂t)]v(x, t) = 0. 1

The author thanks Pavel Etingof for suggesting this problem.

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Evidently, any parabolic dilation HR = {(Rx, R 2 t) : (x, t) ∈ H } of H satisfies  1 v(x, t) dxdt. v(0, 0) = 3 R HR Many weighted averages of v produce v(0, 0). This one is interesting because the weight is constant, proportional to Lebesgue measure. As mentioned in the lecture, the question that remains open is the regularity of the boundary of H . The discrete construction of the divisible sandpile gives an approximation to the continuum set H . The theorems of [CPS04] give a criterion involving approximations to H . Their criterion would imply that the boundary of H is smooth if there were a practical bound on the constants involved. It would also be interesting to show that H is convex, which looks rather obvious from the sandpile approximation. A typical approach would be to realize H as the level set of a log concave function. However, the odometer function u associated with H is not log concave. On the other hand, we have numerical evidence that u/ h is log 2 concave, in which h is the standard fundamental solution, t −1/2 e−x /4t . This would imply that H = {u/ h > 0} is convex. It is not hard to show in the Propp example that the discrete analogue of u/ h is log concave in the x direction. Unfortunately, one does find numerically a very few sites near the boundary at which log concavity fails slightly in the t direction. So at least in the Propp example, it’s hard to see how a combinatorial proof could succeed.

REFERENCES [AG10] A. Asselah and A. Gaudillière, A note on the fluctuations for internal diffusion limited aggregation. arXiv:1004.4665. [AG10a] A. Asselah and A. Gaudillière, From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models. arXiv:1009.2838. [AG10b] A. Asselah and A. Gaudillière, Sub-logarithmic fluctuations for internal DLA. arXiv:1011.4592. [AG11] A. Asselah and A. Gaudillière, Lower bounds on fluctuations for internal DLA. arXiv:1111.4233. [Bro71] B. M. Brown, Martingale central limit theorems, Ann. Math. Statist. 42 (1971), 59–66. [CPS04] L. Caffarelli, A. Petrosyan, and H. Shahgholian, Regularity of a free boundary in parabolic potential theory, J. Amer. Math. Soc. 17, no. 4 (2004), 827–869. [DF91] P. Diaconis and W. Fulton, A growth model, a game, an algebra, Lagrange inversion, and characteristic classes, Rend. Sem. Mat. Univ. Pol. Torino 49, no. 1 (1991), 95–119.

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[FU96] Y. Fukai and K. Uchiyama, Potential kernel for two-dimensional random walk, Ann. Probab. 24, no. 4 (1996), 1979–1992. [GQ00] J. Gravner and J. Quastel, Internal DLA and the Stefan problem, Ann. Probab. 28, no. 4 (2000), 1528–1562. [GV06] B. Gustafsson and A. Vasiliev, Conformal and potential analysis in HeleShaw cells, Birkhäuser Verlag, 2006. [H88] E. Haeusler, On the rate of convergence in the central limit theorem for martingales with discrete and continuous time, Ann. Probab. 16, no. 1 (1988), 275–299. [HB70] C. C. Heyde and B. M. Brown, On the departure from normality of a certain class of martingales, Ann. Math. Statist. 41 (1970), 2161–2165. [HH80] P. Hall and C. C. Heyde, Martingale limit theory and its application, Academic Press, 1980. [IKKN04] A. Ivi´c, E. Krätzel, M. Kühleitner, and W. G. Nowak, Lattice points in large regions and related arithmetic functions: recent developments in a very classic topic, Elementare und analytische Zahlentheorie, Schr. Wiss. Ges. Johann Wolfgang Goethe Univ. Frankfurt am Main, 20, 89–128, 2006. [JLS09] D. Jerison, L. Levine, and S. Sheffield, Internal DLA: slides and audio, Midrasha on probability and geometry: the mathematics of Oded Schramm. http://iasmac31.as.huji.ac.il:8080/groups/midrasha_14/weblog/855d7/images/ bfd65.mov, 2009. [JLS11] D. Jerison, L. Levine, and S. Sheffield, Internal DLA and the Gaussian free field. arXiv:1101.0596. [JLS12a] D. Jerison, L. Levine, and S. Sheffield, Logarithmic fluctuations for internal DLA. J. Amer. Math. Soc. 25 (2012), 271–301. arXiv:1010.2483. [JLS12b] D. Jerison, L. Levine, and S. Sheffield, Internal DLA in higher dimensions. arXiv:1012.3453. [KS04] G. Kozma and E. Schreiber, An asymptotic expansion for the discrete harmonic potential, Electron. J. Probab. 9, no. 1 (2004), 1–17. arXiv:math/0212156. [Law95] G. F. Lawler, Subdiffusive fluctuations for internal diffusion limited aggregation, Ann. Probab. 23, no. 1 (1995), 71–86. [LBG92] G. F. Lawler, M. Bramson and D. Griffeath, Internal diffusion limited aggregation, Ann. Probab. 20, no. 4 (1992), 2117–2140. [LP10] L. Levine and Y. Peres, Scaling limits for internal aggregation models with multiple sources, J. d’Analyse Math. 111 (2010), 151–219. arXiv:0712.3378.

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[Lu12] C. Lucas, The limiting shape for drifted internal diffusion limited aggregation is a true heat ball. arXiv:1202.2032. [MD86] P. Meakin and J. M. Deutch, The formation of surfaces by diffusionlimited annihilation, J. Chem. Phys. 85 (1986), 2320. [R08] Maxim Rabinovich, On the scaling limit of a divisible sandpile model. http://math.mit.edu/news/summer/rsiabstracts.html. [She07] S. Sheffield, Gaussian free fields for mathematicians, Probab. Theory Related Fields, 139, no. 3–4 (2007), 521–541. arXiv:math/0312099. [WS81] T. A. Witten and L. M. Sander, Diffusion-limited aggregation, a kinetic critical phenomenon, Phys. Rev. Lett. 47, no. 19 (1981), 1400–1403.

Chapter Nine The Energy Critical Wave Equation in 3D Carlos Kenig In this chapter we will discuss the energy critical nonlinear wave equation in 3 space dimensions. We start by a review of the linear wave equation  2  ∂t w − w = h (LW ) w|t=0 = w0   ∂t w|t=0 = w1 .

(1)

We write the solution: w(t) = S(t)(w0 , w1 ) + D(t)(h), where S(t) denotes the solution of the homogeneous problem (h = 0) and D(t) the solution of the inhomogeneous one ((w0 , w1 ) = (0, 0)). One of the main properties of the linear wave equation is the finite speed propagation:   If supp (w0 , w1 ) ∩ B(x0 , a) = ∅, supp h ∩ ∪0≤t≤a B(x0 , a − t) × {t} = ∅, then w ≡ 0 on ∪0≤t≤a B(x0 , a − t) × {t}. An important estimate (Strichartz estimate) is (see [16]):   1 2 8 4 wLx,t ≤ C (w0 , w1 )H˙ 1 ×L2 + D h 3 . Lx,t

The energy critical nonlinear wave equation in the focusing case is:  2 5 3  ∂t u − u = u , x ∈ R , t ∈ R (N LW ) u|t=0 = u0 ∈ H˙ 1 (R3 )   ∂t u|t=0 = u1 ∈ L2 (R3 ).

(2)

The defocusing case has −u5 in the right-hand side.   (NLW) is called energy critical because 11 u xλ , λt is also a solution and this λ2 leaves unchanged the H˙ 1 × L2 norm.

Supported in part by NSF grant DMS-0968472.

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9.1 SMALL DATA THEORY FOR (NLW) If (u0 , u1 )H˙ 1 ×L2 is small, ∃! solution u, defined for all time, such that u ∈ C((−∞, +∞); H˙ 1 × L2 ) ∩ L8xt , which scatters, i.e., t→±∞

± (u(t), ∂t u(t)) − S(t)(u± 0 , u1 )H˙ 1 ×L2 −→ 0, ± 2 ˙1 for (u± 0 , u1 ) ∈ H × L . Moreover, for any data (u0 , u1 ) ∈ H˙ 1 × L2 , we have short time existence and hence there exists a maximal I = (−T − (u), T+ (u)) . interval of existence The energy E(u) = 12 |∇u(t)|2 + 12 |∂t u(t)|2 − 16 |u(t)|6 is constant for t ∈ I . In the defocusing case, − 16 becomes 16 . (For these facts see [23], [34], etc.) In the defocusing case work of Struwe, Grillakis, Shatah-Struwe, Kapitanski, Bahouri-Shatah (80s-90s) proves that for any (u0 , u1 ) ∈ H˙ 1 × L2 , the solution exists globally and scatters ([37], [17], [34], [18], [3]). In the focusing case this fails. Levine (1974) showed that if E(u0 , u1 ) ≤ 0, then T− , T+ < ∞ [22]. (This is done by obstruction.) Recently, Krieger-Schlag-Tataru ([21]) constructed solutions for which T+ < ∞. Also, in the focusing case, the elliptic equation admits a non-negative solution W (ground-state), which solves W + W 5 = 0. This elliptic equation has been much studied in connection with the Yamabe problem in differential geometry (see [1], [40], [39], [31], etc.). W has the explicit form 1 W (x) =

 12 . 2 1 + |x|3

W is the unique non-negative solution of the elliptic equation (Gidas-Ni-Nirenberg 79 [15]) and the only radial H 1 solution (Pohozaev 65 [32]). W is a global in time solution of (NLW), which we call a soliton. It does not scatter to a linear solution. W is a “non-dispersive” solution. We now recall some results for (NLW) in the last few years. Theorem 1 ([19]). If E(u) < E(W ) then: i) If ∇u0  < ∇W , we have global existence, scattering. ii) If ∇u0  > ∇W , we have T+ , T− < ∞. The case ∇u0  = ∇W  is impossible. A strengthening of this result is: Theorem 2 ([10]). If sup0 5/2 in Theorem 1.1 to s > 2. It turns out, see [31], that these results are sharp in the general class of quasilinear wave equations of type (1.4). To do better one needs to take into account the special structure of the Einstein equations and rely on a class of estimates which go beyond Strichartz, namely the so-called bilinear estimates.10 In the case of semilinear wave equations, such as wave maps, Maxwell-KleinGordon, and Yang-Mills, the first results which make use of bilinear estimates go back to [17–19]. In the particular case of the Maxwell-Klein-Gordon and YangMills equation the main observation was that, after the choice of a special gauge (Coulomb gauge), the most dangerous nonlinear terms exhibit a special, null structure for which one can apply the bilinear estimates derived in [17]. With the help of 10 Note that no such result, i.e., well-posedness for s = 2, is presently known for either scalar equations of the form (1.6) or systems of the form (1.4).

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these estimates one was able to derive a well-posedness result, in the flat Minkowski space R1+3 , for the exponent s = sc + 1/2 = 1, where sc = 1/2 is the critical Sobolev exponent in that case.11 To carry out a similar program in the case of the Einstein equations one would need, at the very least, the following crucial ingredients: A. Provide a coordinate condition, relative to which the Einstein vacuum equations verify an appropriate version of the null condition. B. Provide an appropriate geometric framework for deriving bilinear estimates for the null quadratic terms appearing in the previous step. C. Construct an effective progressive wave representation F (parametrix) for solutions to the scalar linear wave equation
g φ = F , derive appropriate bounds for both the parametrix and the corresponding error term E = F −
g F , and use them to derive the desired bilinear estimates. Note that the last two steps are to be implemented using only hypothetical L2 bounds for the space-time curvature tensor, consistent with the conjectured result. To start with, it is not at all clear what should be the correct coordinate condition, or even if there is one for that matter. Remark 1.3. As mentioned above, the only known structural condition related to the classical null condition, called the weak null condition, tied to wave coordinates, fails the test. Indeed, the following simple system in Minkowski space verifies the weak null condition and yet, according to [31], it is ill posed for s = 2.
φ = 0,


ψ = φ · φ.

Coordinate conditions, such as spatial harmonic,12 also do not seem to work. We rely instead on a Coulomb type condition, for orthonormal frames, adapted to a maximal foliation. Such a gauge condition appears naturally if we adopt a Yang-Mills description of the Einstein field equations using Cartan’s formalism of moving frames,13 see [7]. It is important to note nevertheless that it is not all a priori clear that such a choice would do the job. Indeed, the null form nature of the Yang-Mills equations in the Coulomb gauge is only revealed once we commute the resulting equations with the projection operator P on the divergence free vectorfields. Such an operation is natural in that case, since P commutes with the flat d’Alembertian. In the case of the Einstein equations, however, the corresponding commutator term [
g , P] generates14 a whole host of new terms and it is quite a miracle that they can all be treated by an extended version of bilinear estimates. This corresponds precisely to the s = 2 exponent in the case of the Einstein vacuum equations. Maximal foliation together with spatial harmonic coordinates on the leaves of the foliation would be the coordinate condition closest in spirit to the Coulomb gauge. 13 We would like to thank L. Anderson for pointing out to us the possibility of using such a formalism as a potential bridge to [18]. 14 Note also that additional error terms are generated by projecting the equations on the components of the frame. 11 12

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At an even more fundamental level, the flat Yang-Mills equations possess natural energy estimates based on the time symmetry of the Minkowski space. There are no such timelike Killing vectorfields in curved space. We have to rely instead on the future unit normal to the maximal foliation t whose deformation tensor is non-trivial. This leads to another class of nonlinear terms which have to be treated by a novel trilinear estimate. Remark 1.4. In addition to the ingredients mentioned above, we also need a mechanism of reducing the proof of the conjecture to small data, in an appropriate sense. Indeed, even in the flat case, the Coulomb gauge condition cannot be globally imposed for large data. In fact [19] relied on a cumbersome technical device based on local Coulomb gauges, defined on domain of dependence of small balls. We rely instead on a variant of the gluing construction of [11], [12].

10.2 STATEMENT OF THE MAIN RESULTS 10.2.1 Maximal Foliations In this section, we recall some well-known facts about maximal foliations (see, for example, the introduction in [10]). We assume the space-time (M, g) to be foliated by the level surfaces t of a time function t. Let T denote the unit normal to t , and let k denote the second fundamental form of t , i.e., kab = −g(Da T , eb ), where ea , a = 1, 2, 3 denotes an arbitrary frame on t and Da T = Dea T . We assume that the t foliation is maximal, i.e., we have: trg k = 0

(2.1)

where g is the induced metric on t . The constraint equations on t for a maximal foliation are given by: ∇ a kab = 0,

(2.2)

where ∇ denotes the induced covariant derivative on t , and Rscal = |k|2 .

(2.3) −2

Also, we denote by n the lapse of the t-foliation, i.e., n the following elliptic equation on t :

= −g(Dt, Dt). n satisfies

n = n|k|2 .

(2.4)

Finally, we recall the structure equations of the maximal foliation: ∇0 kab = Ra 0 b 0 − n−1 ∇a ∇b n − kac kb c ,

(2.5)

∇a kbc − ∇b kac = Rc0ab ,

(2.6)

Rab − kac k c b = Ra0b0 .

(2.7)

and

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10.2.2 Main Theorem We recall below the definition of the volume radius on a general Riemannian manifold M. Definition 2.1. Let Br (p) denote the geodesic ball of center p and radius r. The volume radius rvol (p, r) at a point p ∈ M and scales ≤ r is defined by rvol (p, r) = inf 

r ≤r

|Br  (p)| , r 3

with |Br | the volume of Br relative to the metric on M. The volume radius rvol (M, r) of M on scales ≤ r is the infimum of rvol (p, r) over all points p ∈ M. Our main result is the following: Theorem 2.2 (Main theorem). Let (M, g) be an asymptotically flat solution to the Einstein vacuum equations (1.2) together with a maximal foliation by space-like hypersurfaces t defined as level hypersurfaces of a time function t. Assume that the initial slice (0 , g, k) is such that the Ricci curvature Ric ∈ L2 (0 ), ∇k ∈ L2 (0 ), and 0 has a strictly positive volume radius on scales ≤ 1, i.e., rvol (0 , 1) > 0. Then, there exists a time T = T (RicL2 (0 ) , ∇kL2 (0 ) , rvol (0 , 1)) > 0 and a constant C = C(RicL2 (0 ) , ∇kL2 (0 ) , rvol (0 , 1)) > 0 such that the following control holds on 0 ≤ t ≤ T : 2 2 ≤ C, ∇kL∞ ≤ C and inf rvol (t , 1) ≥ RL∞ [0,T ] L (t ) [0,T ] L (t )

0≤t≤T

1 . C

Remark 2.3. Note that the dependence on RicL2 (0 ) , ∇kL2 (0 ) in the main theorem can be replaced by dependence on RL2 (0 ) where R denotes the space-time curvature tensor.15 Indeed this follows from the following well-known L2 estimate (see section 8 in [22]): 

1 |∇k|2 + |k|4 ≤ 4 0

 |R|2

(2.8)

0

and the Gauss equation relating Ric to R. 10.2.3 Reduction to Small Initial Data We first need an appropriate covering of 0 by harmonic coordinates. This is obtained using the following general result based on Cheeger-Gromov convergence of Riemannian manifolds. 15

Here and in what follows the notations R, R will stand for the Riemann curvature tensors of t and M, while Ric, Ric and Rscal , Rscal will denote the corresponding Ricci and scalar curvatures.

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Theorem 2.4 ([1] or Theorem 5.4 in [34]). Given c1 > 0, c2 > 0, c3 > 0, there exists r0 > 0 such that any 3-dimensional, complete, Riemannian manifold (M, g) with RicL2 (M) ≤ c1 and volume radius at scales ≤ 1 bounded from below by c2 , i.e., rvol (M, 1) ≥ c2 , verifies the following property: Every geodesic ball Br (p) with p ∈ M and r ≤ r0 admits a system of harmonic coordinates x = (x1 , x2 , x3 ) relative to which we have (1 + c3 )−1 δij ≤ gij ≤ (1 + c3 )δij , and



(2.9)

 |∂ 2 gij |2 |g|dx ≤ c3 .

r

(2.10)

Br (p)

We consider  > 0 which will be chosen as a small universal constant. We apply theorem 2.4 to the Riemannian manifold 0 . Then, there exists a constant: r0 = r0 (RicL2 (0 ) , ∇kL2 (0 ) , rvol (0 , 1), ) > 0 such that every geodesic ball Br (p) with p ∈ 0 and r ≤ r0 admits a system of harmonic coordinates x = (x1 , x2 , x3 ) relative to which we have: (1 + )−1 δij ≤ gij ≤ (1 + )δij , and



 |∂ 2 gij |2 |g|dx ≤ .

r Br (p)

Now, by the asymptotic flatness of 0 , the complement of its end can be covered by the union of a finite number of geodesic balls of radius r0 , where the number N0 of geodesic balls required only depends on r0 . In particular, it is therefore enough to obtain the control of R, k and rvol (t , 1) of Theorem 2.2 when one restricts to the domain of dependence of one such ball. Let us denote this ball by Br0 . Next, we rescale the metric of this geodesic ball by:

2 2 , , r0  > 0. gλ (t, x) = g(λt, λx), λ = min R2L2 (Br ) ∇k2L2 (Br ) 0

16

0

Brλ0

be the rescaled versions of R, k and Br0 . Then, in view of our Let Rλ , kλ and choice for λ, we have: √ Rλ L2 (Brλ ) = λRL2 (Br0 ) ≤ , 0

∇kλ L2 (Brλ ) = 0

√ λ∇kL2 (Br0 ) ≤ ,

and ∂ gλ L2 (Brλ ) 2

0

16

√ = λ∂ 2 gL2 (Br0 ) ≤

 λ ≤ . r0

Since in what follows there is no danger to confuse the Ricci curvature Ric with the scalar curvature R, we use the shorthand R to denote the full curvature tensor Ric.

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Note that Brλ0 is the rescaled version of Br0 . Thus, it is a geodesic ball for gλ of radius rλ0 ≥ 1 ≥ 1. Now, considering gλ on 0 ≤ t ≤ 1 is equivalent to considering g on 0 ≤ t ≤ λ. Thus, since r0 , N0 and λ depend only on RL2 (0 ) , ∇kL2 (0 ) , rvol (0 , 1) and , Theorem 2.2 is equivalent to the following theorem: Theorem 2.5 (Main theorem, version 2). Let (M, g) be an asymptotically flat solution to the Einstein vacuum equations (1.2) together with a maximal foliation by space-like hypersurfaces t defined as level hypersurfaces of a time function t. Let B be a geodesic ball of radius one in 0 , and let D its domain of dependence. Assume that the initial slice (0 , g, k) is such that: 1 . 2 Let Bt = D ∩ t the slice of D at time t. Then, there exists a small universal constant 0 > 0 such that if 0 <  < 0 , then the following control holds on 0 ≤ t ≤ 1: 1 2 2  , ∇kL∞   and inf rvol (Bt , 1) ≥ . RL∞ [0,1] L (Bt ) [0,1] L (Bt ) 0≤t≤1 4 RL2 (B) ≤ , ∇kL2 (B) ≤  and rvol (B, 1) ≥

Notation. In the statement of Theorem 2.5, and in the rest of the chapter, the notation f1  f2 for two real positive scalars f1 , f2 means that there exists a universal constant C > 0 such that: f1 ≤ Cf2 . Theorem 2.5 is not yet in suitable form for our proof since some of our constructions will be global in space and may not be carried out on a subregion B of 0 . Thus, we glue a smooth asymptotically flat solution of the constraint equations (1.3) outside of B, where the gluing takes place in an annulus just outside B. This can be achieved using the construction in [11], [12]. We finally get an asymptotically flat solution to the constraint equations, defined everywhere on 0 , which agrees with our original data set (0 , g, k) inside B. We still denote this data set by (0 , g, k). It satisfies the bounds: 1 . 4 Remark 2.6. Notice that the gluing process in [11] and [12] requires the kernel of a certain linearized operator to be trivial. This is achieved by conveniently choosing the asymptotically flat solution to (1.3) that is glued outside of B to our original data set. This choice is always possible since the metrics for which the kernel is nontrivial are nongeneric (see [5]). RL2 (0 ) ≤ 2, ∇kL2 (0 ) ≤ 2 and rvol (0 , 1) ≥

Remark 2.7. Assuming only L2 -bounds on R and ∇k is not enough to carry out the construction in [11] and [12]. However, the authors do not investigate issues of minimal regularity assumptions, and a closer look at the construction should allow to decrease the needed regularity. Remark 2.8. Since k2L4 (0 ) ≤ RicL2 we deduce that kL2 (B)   1/2 on the geodesic ball B of radius one. Furthermore, asymptotic flatness is compatible with

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a decay of |x|−2 at infinity, and in particular with k in L2 (0 ). So we may assume that the gluing process is such that the resulting k satisfies: kL2 (0 )  . Finally, we have reduced Theorem 2.2 to the case of a small initial data set: Theorem 2.9 (Main theorem, version 3). Let (M, g) be an asymptotically flat solution to the Einstein vacuum equations (1.2) together with a maximal foliation by space-like hypersurfaces t defined as level hypersurfaces of a time function t. Assume that the initial slice (0 , g, k) is such that: 1 . 2 Then, there exists a small universal constant 0 > 0 such that if 0 <  < 0 , then the following control holds on 0 ≤ t ≤ 1: RL2 (0 ) ≤ , kL2 (0 ) + ∇kL2 (0 ) ≤  and rvol (0 , 1) ≥

2 2 2  , kL∞ + ∇kL∞   and inf rvol (t , 1) ≥ RL∞ [0,1] L (t ) [0,1] L (t ) [0,1] L (t )

0≤t≤1

1 . 4

10.2.4 Strategy of the Proof The proof of Theorem 2.9 consists in four steps. Step A (Yang-Mills formalism) We first cast the Einstein vacuum equations within a Yang-Mills formalism. This relies on the Cartan formalism of moving frames. The idea is to give up on a choice of coordinates and express instead the Einstein vacuum equations in terms of the connection 1-forms associated to moving orthonormal frames, i.e., vectorfields eα , which verify, g(eα , eβ ) = mαβ = diag(−1, 1, 1, 1). The connection 1-forms (they are to be interpreted as 1-forms with respect to the external index µ with values in the Lie algebra of so(3, 1)), defined by the formulas (Aµ )αβ = g(Dµ eβ , eα ),

(2.11)

Dµ Fµν + [Aµ , Fµν ] = 0,

(2.12)

verify the equations where, denoting (Fµν )αβ := Rαβµν ,   (Fµν )αβ = Dµ Aν − Dν Aµ − [Aµ , Aν ] αβ .

(2.13)

In other words we can interpret the curvature tensor as the curvature of the so(3, 1)valued connection 1-form A. Note also that the covariant derivatives are taken only with respect to the external indices µ, ν and do not affect the internal indices α, β. We can rewrite (2.12) in the form,
g Aν − Dν (Dµ Aµ ) = Jν (A, DA) where, Jν = Dµ ([Aµ , Aν ]) − [Aµ , Fµν ].

(2.14)

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Observe that the equations (2.12) and (2.13) look just like the Yang-Mills equations on a fixed Lorentzian manifold (M, g) except, of course, that in our case A and g are not independent but connected rather by (2.11), reflecting the quasilinear structure of the Einstein equations. Just as in the case of [17], which establishes the well-posedness of the Yang-Mills equation in Minkowski space in the energy norm (i.e., s = 1), we rely in an essential manner on a Coulomb type gauge condition. More precisely, we take e0 to be the future unit normal to the t foliation and choose e1 , e2 , e3 an orthonormal basis to t , in such a way that we have, essentially, div A = ∇ i Ai = 0, where A is the spatial component of A. It turns out that A0 satisfies an elliptic equation while each component Ai = g(A, ei ), i = 1, 2, 3 verifies an equation of the form,
g Ai = −∂i (∂0 A0 ) + Aj ∂j Ai + Aj ∂i Aj + l.o.t.

(2.15)

with l.o.t. denoting nonlinear terms which can be treated by more elementary techniques (including nonsharp Strichartz estimates).

Step B (Bilinear and trilinear estimates). To eliminate ∂i (∂0 A0 ) in (2.15), we need to project (2.15) onto divergence free vectorfields with the help of a non-local operator which we denote by P. In the case of the flat Yang-Mills equations, treated in [17], this leads to an equation of the form,
Ai = P(Aj ∂j Ai ) + P(Aj ∂i Aj ) + l.o.t. where both terms on the right can be handled by bilinear estimates. In our case we encounter, however, three fundamental differences with the flat situation of [17]. • To start with the operator P does not commute with
g . It turns out, fortunately, that the terms generated by commutation can still be estimated by a larger class of bilinear estimates which includes contractions with the curvature tensor. • All energy estimates used in [17] are based on the standard timelike Killing vectorfield ∂t . In our case the corresponding vectorfield e0 = T (the future unit normal to t ) is not Killing. This leads to another class of trilinear error terms. • The main difference with [17] is that we now need bilinear and trilinear estimates for solutions of wave equations on background metrics which possess only limited regularity. This last item is a major problem, both conceptually and technically. On the conceptual side we need to rely on a more geometric proof of bilinear estimates based on a plane wave representation formula17 for solutions of scalar wave equations,
g φ = 0. 17

We follow the proof of the bilinear estimates outlined in [23] which differs substantially from that of [17] and is reminiscent of the null frame space strategy used by Tataru in his fundamental paper [51].

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The proof of the bilinear estimates rests on the representation formula18   ∞ ω φf (t, x) = eiλ u(t,x) f (λω)λ2 dλdω S2

(2.16)

0

where f represents schematically the initial data,19 and where the eikonal equation,20 gαβ ∂α ω u ∂β ω u = 0,

ω

u is a solution of (2.17)

with appropriate initial conditions on 0 and dω the area element of the standard sphere in R3 . Remark 2.10. Note that (2.16) is a parametrix for a scalar wave equation. The lack of a good parametrix for a covariant wave equation forces us to develop a strategy based on writing the main equation in components relative to a frame, i.e., instead of dealing with the tensorial wave equation (2.14) directly, we consider the system of scalar wave equations (2.15). Unlike in the flat case, this scalarization procedure produces several terms which are potentially dangerous, and it is fortunate that they can still be controlled by the use of an extended21 class of bilinear estimates. Step C (Control of the parametrix). To prove the bilinear and trilinear estimates of Step B, we need in particular to control the parametrix at initial time (i.e., restricted to the initial slice 0 )   ∞ ω eiλ u(0,x) f (λω)λ2 dλdω (2.18) φf (0, x) = S2

0

and the error term corresponding to (2.16)   ∞ ω eiλ u(t,x) (
g ω u)f (λω)λ3 dλdω (2.19) Ef (t, x) =
g φf (t, x) = i S2

0

i.e., φf is an exact solution of
g only in flat space in which case
g ω u = 0. This requires the following four substeps: C1 Make an appropriate choice for the equation satisfied by ω u(0, x) on 0 , and control the geometry of the foliation of 0 by the level surfaces of ω u(0, x). C2 Prove that the parametrix at t = 0 given by (2.18) is bounded in L(L2 (R3 ), L2 (0 )) using the estimates for ω u(0, x) obtained in C1. C3 Control the geometry of the foliation of M given by the level hypersurfaces of ω u. 18

(2.16) actually corresponds to the representation formula for a half-wave. The full representation formula corresponds to the sum of two half-waves. 19 Here f is in fact at the level of the Fourier transform of the initial data and the norm λf  L2 (R3 ) corresponds, roughly, to the H 1 norm of the data. 20 In the flat Minkowski space ω u(t, x) = t ± x · ω. 21 Such as contractions between the Riemann curvature tensor and derivatives of solutions of scalar wave equations.

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C4 Prove that the error term (2.19) satisfies the estimate Ef L2 (M) ≤ C λf L2 (R3 ) using the estimates for ω u and
g ω u proved in C3. To achieve Step C3 and Step C4, we need, at the very least, to control
g ω u in L∞ . This issue was first addressed in the sequence of papers [24]–[26] where an L∞ bound for
g ω u was established, depending only on the L2 -norm of the curvature flux along null hypersurfaces. The proof required an interplay between both geometric and analytic techniques and had all the appearances of being sharp, i.e., we don’t expect an L∞ bound for
g ω u which requires bounds on less than two derivatives in L2 for the metric.22 To obtain the L2 -bound for the Fourier integral operator E defined in (2.19), we need, of course, to go beyond uniform estimates for
g ω u. The classical L2 bounds for Fourier integral operators of the form (2.19) are not at all economical in terms of the number of integration by parts which are needed. In our case the total number of such integration by parts is limited by the regularity properties of the function
g ω u. To get an L2 -bound for the parametrix at initial time (2.18) and the error term (2.19) within such restrictive regularity properties we need, in particular: • In Step C1 and Step C3, a precise control of derivatives of ω u and
g ω u with respect to both ω as well as with respect to various directional derivatives.23 To get optimal control we need, in particular, a very careful construction of the initial condition for ω u on 0 and then sharp space-time estimates of Ricci coefficients, and their derivatives, associated to the foliation induced by ω u. • In Step C2 and Step C4, a careful decomposition of the Fourier integral operators (2.18) and (2.19) in both λ and ω, similar to the first and second dyadic decomposition in harmonic analysis, see [40], as well as a third decomposition, which in the case of (2.19) is done with respect to the space-time variables relying on the geometric Littlewood-Paley theory developed in [26]. Below, we make further comments on Steps C1–C4: (1) The choice of u(0, x, ω) on 0 in Step C1. Let us note that the typical choice u(0, x, ω) = x · ω in a given coordinate system would not work for us, since we don’t have enough control on the regularity of a given coordinate system within our framework. Instead, we need to find a geometric definition of u(0, x, ω). A natural choice would be
g u = 0 on 0 which by a simple computation turns out to be the following simple variant of the minimal surface equation24     ∇u ∇u ∇u div =k , on 0 . |∇u| |∇u| |∇u| Classically, this requires, at the very least, the control of R in L∞ . Taking into account the different behavior in tangential and transversal directions with respect to the level surfaces of ω u. 24 In the time symmetric case k = 0, this is exactly the minimal surface equation. 22 23

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239

Unfortunately, this choice does not allow us to have enough control of the derivatives of u in the normal direction to the level surfaces of u. This forces us to look for an alternate equation for u:     1 ∇u ∇u ∇u =1− +k , on 0 . div |∇u| |∇u| |∇u| |∇u| This equation turns out to be parabolic in the normal direction to the level surfaces of u, and allows us to obtain the desired regularity in Step C1. On closer inspection it is related with the well-known mean curvature flow on 0 . (2) How to achieve Step C3. The regularity obtained in Step C1, together with null transport equations tied to the eikonal equation, elliptic systems of Hodge type, the geometric Littlewood-Paley theory of [26], sharp trace theorems, and an extensive use of the structure of the Einstein equations, allows us to propagate the regularity on 0 to the space-time, thus achieving Step C3. (3) The regularity with respect to ω in Steps C1 and C3. The regularity with respect to x for u is clearly limited as a consequence of the fact that we only assume L2 -bounds on R. On the other hand, R is independent of the parameter ω, and one might infer that u is smooth with respect to ω. Surprisingly, this is not at all the case. Indeed, the regularity in x obtained for u in Steps C1 and C3 is better in directions tangent to the level hypersurfaces of u. Now, the ω derivatives of the tangential directions have nonzero normal components. Thus, when differentiating the structure equations with respect to ω, tangential derivatives to the level surfaces of u are transformed in nontangential derivatives which in turn severely limits the regularity in ω obtained in Steps C1 and C3. (4) How to achieve Steps C2 and C4. Let us note that the classical arguments for proving L2 -bounds for Fourier operators are based either on a T T ∗ argument, or a T ∗ T argument, which requires several integration by parts either with respect to x for T ∗ T , or with respect to (λ, ω) for T T ∗ . Both methods would fail by far within the regularity for u obtained in Step C1 and Step C3. This forces us to design a method which allows us to take advantage both of the regularity in x and ω. This is achieved using in particular the following ingredients: • geometric integrations by parts taking full advantage of the better regularity properties in directions tangent to the level hypersurfaces of u, and • the standard first and second dyadic decomposition in frequency space, with respect to both size and angle (see [40]), an additional decomposition in physical space relying on the geometric Littlewood-Paley projections of [26] for Step C4, as well as another decomposition involving frequency and angle for Step C2. Even with these precautions, at several places in the proof, one encounters log-divergences which have to be tackled by ad hoc techniques taking full advantage of the structure of the Einstein equations.

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Step D (Sharp L4 (M) Strichartz estimates). We note that the parametrix constructed in Step C needs to also be used to prove a sharp L4 (M) Strichartz estimate. Indeed the proof of several bilinear estimates of Step B reduces to the proof of a sharp L4 (M) Strichartz estimate for the parametrix (2.16) with λ localized in a dyadic shell. More precisely, let j ≥ 0, and let ψ a smooth function on R3 supported in

Let φf,j λ ∼ 2j

1 ≤ |ξ | ≤ 2. 2 denote the parametrix (2.16) with an additional frequency localization   φf,j (t, x) =

S2

∞

eiλ

ω

u(t,x)

ψ(2−j λ)f (λω)λ2 dλdω.

(2.20)

0

We will need the sharp25 L4 (M) Strichartz estimate j

φf,j L4 (M)  2 2 ψ(2−j λ)f L2 (R3 ) .

(2.21)

The standard procedure for proving26 (2.21) is based on a T T ∗ argument which reduces it to an L∞ estimate for an oscillatory integral with a phase involving ω u. This is then achieved by the method of stationary phase which requires quite a few integrations by parts. In fact, the standard argument would require, at the least,27 that the phase function u = ω u verifies ∂t,x u ∈ L∞ , ∂t,x ∂ω2 u ∈ L∞ .

(2.22)

This level of regularity is, unfortunately, incompatible with the regularity properties of solutions to our eikonal equation (2.17). In fact, based on the estimates for ω u derived in step C3, we are only allowed to assume ∂t,x u ∈ L∞ , ∂t,x ∂ω u ∈ L∞ .

(2.23)

We are thus forced, see [47], to follow an alternative approach28 to the stationary phase method inspired by [37] and [38], which minimizes the number of integration by parts which are needed. Remark 2.11. To reemphasize that the special structure of the Einstein equations is of fundamental importance in deriving our result we would like to stress that the bilinear estimates are needed not only to treat the terms of the form Aj ∂j Ai and Aj ∂j Ai mentioned above (which are also present in flat space) but also to derive energy estimates for solutions to
g φ = F . Moreover, a new type of trilinear estimate is required to get L2 -bounds for the curvature tensor R. All these depend in an essential way on the Einstein equations. 25

Note in particular that the corresponding estimate in the flat case is sharp. Note that the procedure we describe would prove not only (2.21) but the full range of mixed Strichartz estimates. 27 The regularity (2.22) is necessary to make sense of the change of variables involved in the stationary phase method. 28 We refer to the approach based on the overlap estimates for wave packets derived in [37] and [38] in the context of Strichartz estimates, respectively, for C 1,1 and H 2+ metrics. Note, however, that our approach does not require a wave packet decomposition. 26

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241

REFERENCES [1] M. T. Anderson, Cheeger-Gromov theory and applications to general relativity. In The Einstein equations and the large scale behavior of gravitational fields, 347–377. Birkhäuser, Basel, 2004. [2] H. Bahouri, J.-Y. Chemin, Équations d’ondes quasilinéaires et estimation de Strichartz. Amer. J. Math. 121 (1999), 1337–1777. [3] H. Bahouri, J.-Y. Chemin, Équations d’ondes quasilinéaires et effet dispersif . IMRN 21 (1999), 1141–1178. [4] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations Part II: The KDVequation, GAFA 3, no. 3 (1993), 209-262. [5] R. Beig, P. T. Chru´sciel, and R. Schoen, KIDs are non-generic. Ann. Henri Poincaré, 6(1):155–194, 2005. [6] Y. C. Bruhat, Theoreme d’Existence pour certains systemes d’equations aux derivees partielles nonlineaires., Acta Math. 88 (1952), 141-225. [7] E. Cartan, Sur une généralisation de la notion de courbure de Riemann et les espaces torsion. C. R. Acad. Sci. (Paris) 174, 593-595, 1922. [8] D. Christodoulou, Bounded variation solutions of the spherically symmetric einstein-scalar field equations, Comm. Pure and Appl. Math 46, 1131-1220, 1993. [9] D. Christodoulou, The instability of naked singularities in the gravitational collapse of a scalar field, Ann. of Math. 149, 183-217, 1999. [10] D. Christodoulou and S. Klainerman, The global nonlinear stability of the Minkowski space, volume 41 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1993. [11] J. Corvino, Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Commun. Math. Phys. 214, 137-189, 2000. [12] J. Corvino, R. Schoen, On the asymptotics for the vacuum Einstein constraint equations, Jour. Diff. Geom. 73, 185-217, 2006. [13] A. Fischer, J. Marsden, The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system, I. Comm. Math. Phys. 28 (1972), 138. [14] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure and Appl. Math. 18, 697-715, 1965.

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[15] T. J. R. Hughes, T. Kato and J. E. Marsden, Well-posed quasi-linear secondorder hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal. 63, 1977, 273-394 [16] S. Klainerman, PDE as a unified subject, Proceeding of Visions in Mathematics, GAFA 2000 (Tel Aviv 1999). Geom Funct. Anal. 2000, Special Volume, Part 1, 279-315. [17] S. Klainerman, M. Machedon, Space-Time Estimates for Null Forms and the Local Existence Theorem, Communications on Pure and Applied Mathematics 46, 1221-1268 (1993). [18] S. Klainerman, M. Machedon, Finite energy solutions of the Maxwell-KleinGordon equations, Duke Math. Journal 74, 19-44 (1994). [19] S. Klainerman, M. Machedon, Finite energy solutions for the Yang-Mills equations in R1+3 , Annals of Math. 142 (1995), 39-119. [20] S. Klainerman, I. Rodnianski, Improved local well-posedness for quasi-linear wave equations in dimension three, Duke Math. Journal 117 (2003), no. 1, 1-124. [21] S. Klainerman, I. Rodnianski, Rough solutions to the Einstein vacuum equations, Annals of Mathematics 161 (2005), 1143–1193. [22] S. Klainerman, I. Rodnianski, On a break-down criterion in general relativity, J. Amer. Math. Soc. 23 (2010), 345-382. [23] S. Klainerman, I. Rodnianski, Bilinear estimates on curved space-times, J. Hyperbolic Differ. Equ. 2 (2) (2005), 279-291. [24] S. Klainerman, I. Rodnianski, Casual geometry of Einstein vacuum spacetimes with finite curvature flux, Inventiones 159 (2005), 437-529. [25] S. Klainerman, I. Rodnianski, Sharp trace theorems on null hypersurfaces, GAFA 16, no. 1 (2006), 164-229. [26] S. Klainerman, I. Rodnianski, A geometric version of Littlewood-Paley, GAFA 16, no. 1 (2006), 126-163. [27] S. Klainerman, I. Rodnianski, On the radius of injectivity of null hypersurfaces, J. Amer. Math. Soc. 21 (2008), 775-795. [28] S. Klainerman, I. Rodnianski, J. Szeftel, Overview of the proof of the bounded L2 curvature conjecture, arXiv:1204.1772, 127 p., 2012. [47] S. Klainerman, I. Rodnianski, J. Szeftel, An L4t L4x Strichartz estimate for the wave equation on a rough background. Work in progress. [30] J. Krieger, W. Schlag, Concentration compactness for critical wave maps, Monographs of the European Mathematical Society, 2012.

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[48] T. Tao, Global regularity of wave maps I-V−−, preprints. [49] D. Tataru, Strichartz estimates for second order hyperbolic operators with non smooth coefficients, J.A.M.S. 15 (2002), no. 2, 419-442. [50] D. Tataru, Strichartz estimates for operators with non smooth coefficients and the nonlinear wave equation, Amer. J. Math. 122 (2000), 349–376. [51] D. Tataru, Local and global results for wave maps I, Comm. PDE 23 (1998), no. 9-10, 1781–1793. [52] K. Uhlenbeck, Connections with Lp bounds on curvature, Commun. Math. Phys. 83, 31-42, 1982. [53] Q. Wang, Improved breakdown criterion for Einstein vacuum equation in CMC gauge, To appear in Comm. Pure Appl. Math., arXiv:1004.2938.

Chapter Eleven On Div-Curl for Higher Order Loredana Lanzani and Andrew S. Raich 11.1 INTRODUCTION In 2004 Stein and the first named author discovered a connection [LS] between the celebrated Gagliardo-Nirenberg inequality [G]-[N] for functions
f
Lr (Rn ) ≤ C
∇f
L1 (Rn ) ,

r = n/(n − 1)

(1)

and a recent estimate of Bourgain and Brezis [BB2] for divergence-free vector fields as proved by Van Schaftingen [VS1]
Z
Lr (Rn ) ≤ C
Curl Z
L1 (Rn ) ,

r = n/(n − 1),

div Z = 0.

(2)

Such connection is provided by the exterior derivative operator acting on differential forms on Rn with (say) smooth and compactly supported coefficients d :
q (Rn ) →
q+1 (Rn ),

0 ≤ q ≤ n.

(3)

It was proved in [LS] that the inequality
u
Lr (Rn ) ≤ C(
du
L1 (Rn ) +
d ∗ u
L1 (Rn ) ),

r = n/(n − 1)

(4)

holds for any form u of degree q other than q = 1 (unless d ∗ u = 0) and q = n − 1 (unless du = 0). Note that (1) is the case q = 0, whereas (2) is the case q = 1 specialized to d ∗ u = 0. Since those earlier results, div/curl-type phenomena have been studied both in the Euclidean and non-Euclidean settings [Am], [BV], [HP-1], [HP-2], [M], [MM], [Mi], [VS4], [CV], [Y]. In [VS2] and the recent works [BB3], [VS3], [VS5], differential conditions of higher order have been considered for the first time in such context. (By contrast, the exterior derivative in (3) is defined in terms of differential conditions of order 1.) The goal of the present chapter is to produce a new class of differential operators of order k (where k is any given positive integer) that satisfy an appropriate analogue of (4) and contain the operators introduced in [BB3], [VS2] and [VS3]; since the conditions d ◦ d = 0;

d∗ ◦ d∗ = 0

(5)

Loredana Lanzani was supported by a National Science Foundation IRD plan, and in part by awards DMS-0700815 and DMS-1001304 Andrew S. Raich was supported in part by the National Science Foundation, award DMS-0855822

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play an important role in the proof of (4), the new operators should satisfy (5) as well. We achieve this goal in a number of ways, beginning with: Theorem 1.1. If u ∈ Cq∞ (Rn ) has compact support, then
u
W k−1,r ≤ C(
T u
L1 +
T ∗ u
L1 ),

r = n/(n − 1)

whenever q is neither 1 (unless T ∗ u = 0) nor n − 1 (unless T u = 0), where  

j I ∂ k uI  dx L . L T u :=  k  ∂x j |I |=q |L|=q+1

(6)

(7)

j =1,...,n

Here and in the sequel, W a,p (Rn ) denotes the Sobolev space consisting of a,p a-times differentiable functions in the Lebesgue class Lp (Rn ) (and Wq (Rn ) will denote the space of q-forms with coefficients in W a,p (Rn )), while CAB ∈ {−1, 0,+1} is the sign of the permutation that carries the ordered set AB = {a1 , . . . , a , b1 , . . . , bq } to the label C = (c1 , . . . , c+q ), if these have identical content, and is otherwise zero. Note that when k = 1 then T = d and inequality (6) is indeed (4). Another such complex, again involving a differential condition of order k ≥ 1, is obtained by embedding Rn isometrically in a larger space RN . (The choice of “inflated” dimension N will be discussed later.) The resulting operators act on “hybrid Rn -to-RN ” spaces of forms whose coefficients are trivial extensions to RN of functions defined on Rn ; to distinguish such spaces from the classical Sobolev a,p spaces Wq (RN ) (to which they are by necessity transversal) we will use the notation qa,p (RN ), W

0 ≤ q ≤ N,

q∞,c (RN ), C

0≤q≤N

and we will write

to indicate a dense subspace of smooth “compactly supported” forms. These operators, which we denote T1,ℵ , map ∞,c q∞,c (RN ) → C q+1 (RN ), T1,ℵ : C

0 ≤ q ≤ N.

The label ℵ refers to a choice of an ordering for the set of all k-th order derivatives in Rn , and so in practice we define a finite family {T1,ℵ }ℵ of such complexes. (We use the subscript “1” in T1,ℵ to specify that T1,ℵ maps q-forms to (q + 1)-forms, a distinction that will be needed later on.) The explicit definition of T1,ℵ will be given in the next section; what matters here is that these operators satisfy a more general version of (6) in the sense that the following inequality implies (6) but the converse is not true: q∞ (RN ) has compact support, then Theorem 1.2. If U ∈ C ∗ U
L1 ),
U
W k−1,r ≤ C(
T1,ℵ U
L1 +
T1,ℵ

r = n/(n − 1)

∗ whenever q is neither 1 (unless T1,ℵ U = 0) nor N − 1 (unless T1,ℵ U = 0).

(8)

247

DIV-CURL FOR HIGHER ORDER

Theorem 1.1 recaptures an L1 -duality estimate of Bourgain and Brezis: Theorem 1.3 ([BB3]). Let k ≥ 1. For every vector field u ∈ L1 (Rn ; Rn ) if n
∂ k uj j =1

∂xjk

=0

in the sense of distributions, then

uj · hj ≤ C
u
L1
∇h
Ln ,

(9)

j = 1, . . . , n

(10)

Rn

for any h ∈ (W 1,n ∩ L∞ )(Rn ; Rn ), where the constant C only depends on the dimension of the space n and on the order k. On the other hand, Theorem 1.2 was motivated by a recent result of van Schaftingen: Theorem 1.4 ([VS3]). Given k ≥ 1 and n ≥ 2, let  n−1+k . m := k

(11)

For any vector field g = (gα )α∈S(n,k) ∈ L1 (Rn ; Rm ) if


∂ k gα =0 ∂x α α∈S(n,k) in the sense of distributions, then

gα · hα ≤ C
g
L1
∇h
Ln ,

(12)

α ∈ S(n, k)

(13)

Rn

for any h ∈ (W 1,n ∩ L∞ )(Rn ; Rm ), where the constant C only depends on the dimension of the space n and on the order k. Here S(n, k) denotes the set of k-multi-indices in Rn : 

n
αt = k . S(n, k) = α = (α1 , . . . , αn ) αt ∈ {0, 1, . . . , k},

(14)

t=1

A key ingredient in the proof of Theorems 1.1 and 1.2 is the fact that the Hodge Laplacians for these operators, namely
T := T T ∗ + T ∗ T : Cq∞,c (Rn ) → Cq∞,c (Rn ), 0 ≤ q ≤ n and ∗ ∗  1,ℵ := T1,ℵ T1,ℵ q∞,c (RN ) → C q∞,c (RN ), 0 ≤ q ≤ N
+ T1,ℵ T1,ℵ : C

satisfy a uniform Legendre-Hadamard condition which in turn yields elliptic estimates.

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LANZANI AND RAICH

Rather surprisingly, it turns out that in fact there is a larger class of such operators, mapping ∞,c q∞,c (RN ) → C q+ (RN ), T,ℵ : C

0≤q≤N

where the label  now runs over all the elements of what we call the set of admissible degree increments, which is a subset of {1, . . . , k} determined by n (the dimension of the source space) and k (the order of differentiation): for any n ≥ 2 and k ≥ 2, the set of admissible degree increments contains at least two distinct elements:  = 1 (discussed earlier) and also  = k. Each admissible degree increment in turn determines an “inflated dimension” N (in particular N will change with ). However the situation for  = 1 differs from the case  = 1 in two important respects: the crucial condition (5) will hold only for odd , and if  = 1 the Hodge Laplacian for T,ℵ will fail to be uniformly elliptic (even for  odd): as a result there is no analog of (6). Instead, we show that for any admissible degree increment (thus also for  = 1), the operators T,ℵ satisfy L1 -duality estimates that are similar in spirit, and indeed are equivalent to (13); see Theorem 2.3 for the precise statement. A further class of operators which contains our very first example T , see (7), can be defined in terms of T,ℵ and of the aforementioned embedding: Rn → RN . Such operators map ∞,c (Rn ) T,ℵ : Cq∞,c (Rn ) → Cq+

and satisfy div-curl and/or L1 -duality estimates that are stated solely in terms of the pq (RN ) and W qa,p (RN ), source space Rn rather than the “hybrid Rn -to-RN ” spaces L see Theorem 2.4 and (77). (Of course, if  = 1 such operators are non-trivial only for n ≥ .) We need to explain the reason for our choice to keep track, through the label ℵ, of the orderings of S(n, k): this has to do with the notion of invariance. One would like to know whether the identity (15) T,ℵ  ∗ F =  ∗ T,ℵ F q∞ (RN ) and for some non-trivial class of diffeomorphisms holds for any F ∈ C  : RN → RN of class C k+1 : it is in this context that the choice of ℵ may be relevant. In the case k = 1 our construction gives N = n with ℵ spanning the set of all permutations of {1, . . . , n}, and since k is 1 there is only the admissible degree increment  = 1. As a result, for k = 1 we have  

ℵ(j )I ∂UI  L L T1,ℵ U =   dx , ℵ ∈ (1, . . . , n). ∂x j |I |=q |L|=q+1 j =1,...,n

In particular one has T1,ℵ0 = T = d

249

DIV-CURL FOR HIGHER ORDER

for exactly one permutation ℵ0 (the identity) which therefore determines an invariant operator. On the other hand it is easy to check that for any ℵ = ℵ0 the operators T1,ℵ fail to be invariant. No such phenomenon exists for k ≥ 2: there is no choice of ℵ (nor ) that makes T,ℵ invariant and (15) fails even in the case when  originates from a rotation of Rn . It can be verified that T,ℵ , too, is not invariant because if k ≥ 2 the identity T,ℵ ψ ∗ = ψ ∗ T,ℵ

(16)

fails for any  and for any ℵ, already for ψ a rotation of R . Finally, we point out that our results can be rephrased in terms of the canceling and cocanceling conditions of [VS4]: within that framework, our results provide new classes of differential operators of arbitrary order that are canceling and/or cocanceling, with the size of the admissible degree increments acting as an indicator of the canceling property. See the remarks in Section 11.4. This chapter is organized as follows: in Section 11.2 we introduce the notion of admissible degree increment, we describe the “hybrid Rn -to-RN ” Sobolev spaces qa,p (RN ) in term, of the embedding, and we define the operators T,ℵ and T,ℵ W and discuss their basic properties (adjoints; uniform ellipticity). The L1 -duality estimates for T,ℵ and for T,ℵ are stated in Theorems 2.3 and 2.4, and the precise statements of (8) and of (6) are given in Theorem 2.8 and in (77). All the proofs are deferred to Section 11.3. Section 11.4 contains some remarks and a few questions. n

11.1.1 Notation As customary, we let
q (Rn ) denote the space of q-forms:


n I n
q (R ) = f = fI dx fI : R → R ,

0≤q≤n

(17)

I ∈I(n,q)

where I(n, q) denotes the set of q-labels for Rn :   I(n, q) = I = (i1 , . . . , iq ) | it ∈ {1, . . . , n}, it < it+1

(18)

and dx I = dxi1 ∧ · · · ∧ dxiq . When q = n the expression above is the volume form and we use the notation dV . We will regard the label set I(n, q) as canonically ordered (alphabetical ordering). Letting i : Rn → RN denote the isometric embedding mentioned above and defined in (26), the “hybrid Rn -to-RN ” subspace of
q (RN ) (consisting of those q-forms whose coefficients are trivial extensions to RN of functions defined on Rn ) is more precisely described as follows


N I  FI dz FI ◦ i ◦ π = FI , 0 ≤ q ≤ N (19)
q (R ) := F = I ∈I(N,q)

where π : RN → Rn

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LANZANI AND RAICH

is chosen so that (π ◦ i)(x) = x,

for all

x ∈ Rn .

(20)

As a result the reverse composition i ◦ π : RN → RN

(21)

is a projection. We will denote the Hodge-star operators for each of
q (Rn ) and
q (RN ), respectively, by ∗n and ∗N ; note that we have q (RN ) →
N−q (RN ), ∗N :


0 ≤ q ≤ N.

(22)

11.2 STATEMENTS 11.2.1 Admissible degree increments Given three integers: i. n ≥ 2 (the dimension of the source space), ii. k ≥ 1 (the order of the differential condition), and iii. 1 ≤  ≤ k, we say that  is an admissible degree increment for the pair (n, k) if and only if the polynomial equation   N n−1+k = (23)  k has a solution N that satisfies the following two conditions: N ∈ Z+ , N ≥ n − 1 +  .

(24)

Note that the pair (n, 1) (that is, k = 1) has exactly one admissible degree increment, namely  = 1, and in this case equation (23) has the unique solution: N = n. On the other hand, for k ≥ 2 any pair (n, k) will have at least two admissible degree increments ( = 1, k) and possibly more, for instance: the pair (n, k) = (2, 9) has (exactly) four admissible degree increments, namely  = 1, 2, 3, 9; similarly, the pair (n, k) := (2, 29) has (at least)  = 1, 2, 29. For any admissible degree increment, we consider the embedding i : Rn → RN

(25)

i(x1 , . . . , xn ) = (z 1 , . . . , z N ) := (x1 , . . . , xn , 0, . . . , 0)

(26)

defined as follows where N = N (n, k, ) is as in (23) and (24). We let i also denote the embedding of k-multi-indices i : S(n, k) → S(N, k) that is canonically induced by (26), namely i(α1 , . . . , αn ) := (α1 , . . . , αn , 0, . . . , 0) ∈ S(N, k)

(27)

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DIV-CURL FOR HIGHER ORDER

and adopt the notation iS(n, k) := {iα | α ∈ S(n, k)}
S(N, k).

(28)

|iS(n, k)| = m

(29)

We have with m = m(n, k) as in (11), and so there are m!-many distinct orderings of iS(n, k). By the definition of N the set of labels I(N, ) also has cardinality m and we will think of each ordering of iS(n, k) as a one-to-one correspondence ℵ : iS(n, k) → I(N, );

ℵ−1 : I(N, ) → iS(n, k).

(30)

11.2.2 Hybrid function spaces Given an integer a ≥ 0 and given p, p ≥ 1 such that 1/p + 1/p = 1, we first set (q = 0) p (RN ) := {F : RN → R | F ◦ i ◦ π = F, F ◦ i ∈ Lp (Rn , dV )} (31) L where i is as in (26) and π(z 1 , . . . , z n , . . . , z N ) = (x, . . . , xn ) := (z 1 , . . . , z n )

(32)

satisfies (20), and

  p (RN ), λ ∈ S(N, s), 0 ≤ s ≤ a .  a,p (RN ) := F : RN → R | ∂ λ F ∈ L W

 a,p (RN ) then it follows from Note that if F ∈ W F ◦i◦π =F that ∂F = 0, ∂z t

for any t = n + 1, . . . , N,

(33)

which in turn grants ∂sF =0 (34) ∂z λ for any 1 ≤ s ≤ a and for any λ ∈ S(N, s) \ iS(n, s), so that these spaces are more precisely described as follows:   p (RN ), β ∈ S(n, s), 0 ≤ s ≤ a .  a,p (RN ) = F : RN → R | ∂ iβ F ∈ L W pq (RN ) of any degree As customary, these definitions are extended to forms F ∈ L a,p N  0 ≤ q ≤ N (resp. F ∈ Wq (R ), 0 ≤ q ≤ N ) by requiring that
p (RN ) (resp. FI ∈ W  a,p (RN )) FI dz I has FI ∈ L F = I ∈I(N,q)

for any I ∈ I(N, q). We observe for future reference that identity (20) grants  s ∂ F ∂ s (F ◦ i) (35) ◦ i = ∂z iβ ∂x β  a,p (RN ). for any β ∈ S(n, s) and for any F ∈ W

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Lemma 2.1. For any 0 ≤ q ≤ N ; for any p ≥ 1 and for any integer a ≥ 1 the following properties hold: 2q (RN ) is a Hilbert space with inner product i. L

F, GL := ∗n i ∗ ∗N (F ∧ ∗N G).

(36)

Rn

pq (RN ) is a Banach space with norm ii. L  1/p

  p/2
F
Lpq :=  ∗n i ∗ ∗N (F ∧ ∗N F )  .

(37)

Rn

qa,2 (RN ) is a Hilbert space with inner product iii. W
(F, G)W := D iβ F, D iβ GL

(38)

0≤s≤a β∈S(n,s)

where we have set




D iβ F :=

(∂ iβ FI )dz I .

(39)

I ∈I(N,q)

qa,p (RN ) is a Banach space with norm W 

iv.

1/p


iβ p 
∂ HI
Lp 
H
Wqa,p :=   

.

(40)

I ∈I(N,q) β∈S(n,s) 0≤s≤a

v.

The set q∞,c (RN ) := C

is dense in (40)).

 

F =



pq (RN ) L




  q (RN ) FI ◦ i ∈ Cc∞ (Rn ) FI dz I ∈
(41) 

I ∈I(N,q)

qa,p (RN )) with respect to the norm (37) (resp. (resp. W

q∞,c (RN ) is said to converge in the sense of the space A sequence { j }j ⊂ C N ∞,c N   Dq (R ) to ∈ Cq (R ), see [A], provided the following conditions are satisfied: i. There is a set K  Rn such that Supp(( j − ) ◦ i) ⊂ K for each j. ii. For any 0 ≤ s < ∞ and for each β ∈ S(n, s) we have ∂ β (( j )I ◦ i) ∂ β ( I ◦ i) = j →∞ ∂x β ∂x β lim

for any I ∈ I(N, q).

uniformly in K

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DIV-CURL FOR HIGHER ORDER

q∞,c (RN ) with respect There exists a locally convex topology on the vector space C to which a linear functional L is continuous if, and only if, L( j ) → L( ) whenq (RN ). ever j → in the sense of the space D  q−a,p (RN ) of For any 1 ≤ p, p  < ∞ with 1/p + 1/p  = 1, the dual space W a,p q (RN ) is identified (in the usual fashion, see e.g., [A, III.3.8 – III.3.12]) with a W closed subspace of the Cartesian product a 
N n−1+j p N ( q )M  where M = M(n, a) := (L (R )) j j =0 

q−a,p (RN ) we have qa,p (RN ) and G ∈ W and from this it follows that for any F ∈ W |F, GL| ≤
F
Wqa,p
G
W −a,p , q

(42)

see again [A]. Note that since q∞,c (RN ) ∩ Cq∞,c (RN ) = {0} C rq (RN ) and Lrq (RN ) are transversal; the same is true for W qa,p (RN ) and the spaces L a,p Wq (RN ) and for the respective dual spaces. 11.2.3 Operators and their adjoints For ℵ as in (30) and for any admissible degree increment , we define a kth-order   differential condition via the action

ℵ(iα)I ∂ k FI 
FI dz I → L (43) dz L  iα  ∂z I ∈I(N,q) I ∈I(N,q) L∈I(N, q+) α∈S(n,k)

where N is as in (23) and (24) and q ∈ {0, 1, . . . , N }. Here iα is as in (28) and ℵ is the correspondence (30). This action produces a differential operator T,ℵ that maps ∞,c (RN ), 0 ≤ q ≤ N. (44) T, ℵ : Cq∞,c (RN ) → Cq+ It follows from (35) that the action (43) also determines an operator ∞,c q∞,c (RN ) → C q+ (RN ), T, ℵ : C

0 ≤ q ≤ N.

(45)

Now observe that (20) grants that the pullback by π maps q∞,c (RN ), π ∗ : Cq∞,c (Rn ) → C see (32). On the other hand, it is immediate to check that q∞,c (RN ) → Cq∞,c (Rn ). i∗ : C On account of these observations we see that the action (43) produces a third operator T,ℵ that maps ∞,c T,ℵ : Cq∞,c (Rn ) → Cq+ (Rn ),

0≤q≤n

(46)

and is defined as follows: T,ℵ := i ∗ T,ℵ π ∗ .

(47)

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LANZANI AND RAICH

Note that T,ℵ acts non-trivially only for n ≥ .

(48)

Condition (48) may be viewed in two different ways: as a constraint on the size of the degree increment  relative to the pair (n, k) (however, note that (48) is satisfied by  = 1 for any pair (n, k)) or as a constraint on the size of n relative to k (and in this case, imposing the constraint n ≥ k ensures that (48) holds for all admissible degree increments). p,2 p,2 In the following, ·, · denotes the duality in Wq (RN ) (resp. Wq (Rn )). Proposition 2.2. Let  be an admissible degree increment for (n, k). The formal adjoint of T,ℵ on Wqa,2 (RN ) is ∗ := (−1)k+q(N−−q) ∗N T,ℵ ∗N , T,ℵ

0 ≤ q ≤ N.

(49)

∞,c That is, for any F ∈ Cq∞,c (RN ) and for any G ∈ Cq+ (RN ) we have ∗  T,ℵ F, G  =  F, T,ℵ G .

(50)

qa,2 (RN ) is The formal adjoint of T,ℵ on W ∗ := (−1)k+q(N−−q) ∗N T,ℵ ∗N , T,ℵ

0 ≤ q ≤ N.

(51)

∞,c q+ q∞,c (RN ) and for any G ∈ C (RN ) we have That is, for any F ∈ C ∗  T,ℵ F, G L =  F, T,ℵ G L .

(52)

Suppose further that n ≥ . Then, the formal adjoint of T,ℵ on Wqa,2 (Rn ) is ∗ := (−1)k+q(n−−q) ∗n T,ℵ ∗n T,ℵ

0 ≤ q ≤ n.

(53)

∞,c that is, for any f ∈ Cq∞,c (Rn ) and for any g ∈ Cq+ (Rn ) we have ∗  T,ℵ f, g  =  f, T,ℵ g.

(54)

11.2.4 Estimates. Theorem 2.3. Let n ≥ 2 and k ≥ 1 be given. Let  ∈ {1, . . . , k} be any admissible degree increment for the pair (n, k), and let N be a solution of (23) that satisfies (24).

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DIV-CURL FOR HIGHER ORDER

1q (RN ), if For any 0 ≤ q ≤ N −  and for any F ∈ L T, ℵ F = 0 in the sense of distributions, then F, H L ≤ C
F 
L1
∇H
Ln q q

(55)

(56)

N  1,n ∞ for any H ∈ (L q ∩ Wq )(R ).

1p (RN ), if For any  ≤ p ≤ N and for any G ∈ L T,∗ ℵ G = 0 in the sense of distributions, then G, KL ≤ C
G
 1p
∇K
L np L

(57)

(58)

N  1,n ∞ for any K ∈ (L p ∩ Wp )(R ).

 depends only on n and k. The constant C Theorem 2.4. Let n ≥ 2 and k ≥ 1 be given. Let  ∈ {1, . . . , k} be an admissible degree increment for the pair (n, k) such that n ≥ . For any 0 ≤ q ≤ n −  and for any f ∈ L1q (Rn ), if T,ℵ f = 0

(59)

in the sense of distributions, then |f, h| ≤ C
f
L1q
∇h
Lnq

(60)

1,n n for any h ∈ (L∞ q ∩ Wq )(R ).

For any  ≤ p ≤ n and for any g ∈ L1p (Rn ), if ∗ T,ℵ g=0

(61)

|g, h| ≤ C
g
L1p
∇h
Lnp

(62)

in the sense of distributions, then

1,n n for any h ∈ (L∞ p ∩ Wp )(R ).

The constant C depends only on n and k. We have Theorem 1.4 ⇐⇒ Theorem 2.3 ⇒ Theorem 2.4 ⇒ Theorem 1.3.

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11.2.5 Hodge systems Concerning the compatibility conditions for the Hodge system for each of T,ℵ and T,ℵ , we have 

 2k
2  ℵ(iβ)ℵ(iα)I ∂ FI  M (T,ℵ ◦ T,ℵ ) F = (1 + (−1) ) M  dz ∂z iα ∂z iβ M∈I(N,q+2) I ∈I(N,q)


α,β∈S(n,k)

so in particular T,ℵ ◦ T,ℵ = 0 ⇐⇒  is odd.

(63)

A similar computation shows that the same is true for T,ℵ , so in the sequel we will often pay special attention to the admissible degree increment  = 1. ∗ be given by (51) and set Lemma 2.5. Let T,ℵ ∗ ∗  ,ℵ := T,ℵ T,ℵ
+ T,ℵ T,ℵ .

If




H =

(64)

q∞,c (RN ), HI dz I ∈ C

I ∈I(N,q)

then


∂ 2k HI MI ℵ(iα) dz M C ℵ(iβ) iα ∂z iβ ∂z M,I ∈I(N,q)

,ℵ H = (−1)k+N


(65)

α,β∈S(n,k)

where




MI ℵ(iα) C ℵ(iβ) =

L L ℵ(iα)I · ℵ(iβ)M +

L∈I(N,q+)




M I ℵ(iα)K · ℵ(iβ)K .

(66)

K∈I(N,q−)

In particular, for  = 1 we have 1,ℵ H =






∂ 2k HI dz I . iα ∂z iα ∂z I ∈I(N,q)

1,ℵ HI ) dz I = (−1)k+N (


I ∈I(N,q)

(67)

α∈S(n,k)

Let T,ℵ be given by (53) (assume n ≥ ) and set ∗ ∗
,ℵ := T,ℵ T,ℵ + T,ℵ T,ℵ .

If h=




(68)

hI dz I ∈ Cq∞,c (Rn ),

I ∈I(n,q)

then
∂ 2k hI MI Cℵ(iα) dx M ℵ(iβ) α ∂x β ∂x M,I ∈I(n,q)


,ℵ h = (−1)k+n

α,β ∈ (π◦ℵ−1 )(I(n,))

(69)

257

DIV-CURL FOR HIGHER ORDER

where




MI Cℵ(iα) ℵ(iβ) =

L L ℵ(iα)I · ℵ(iβ)M +

L∈I(n,q+)




M I ℵ(iα)K · ℵ(iβ)K .

(70)

K∈I(n,q−)

In particular, for  = 1 we have
∂ 2k hI

1,ℵ h = (
1,ℵ hI ) dx I = (−1)k+n dx I . α ∂x α ∂x I ∈I(n,q) I ∈I(n,q)

(71)

α∈ (π◦ℵ−1 )({1,...,n})

Corollary 2.6. For any 0 ≤ q ≤ N and for any choice of the correpondence 1,ℵ satisfies the Legendre-Hadamard condition in the following ℵ, the operator
sense:   
MI  Re  (72) Cℵ(iα)ℵ(iβ) ξ iα ξ iβ ζI ζ M  ≥ C |ξ |2k |ζ |2 I,M∈I(N,q) α,β∈S(n,k) N

for any ξ ∈ i(Rn ) and for any ζ ∈ C( q ) . See [DKPV]. Indeed, by (67) we have that the coordinate-based representation 1,ℵ is independent of the choice of ℵ and furthermore of
   

  MI ℵ(iα)ℵ(iβ) Re  ξ iα ξ iβ ζI ζ M  =  ξ12α1 · · · ξn2αn  |ζ |2 C α∈S(n,k)

I,M∈I(N,q) α,β∈S(n,k)

and if ξ ∈ i(Rn ) then




ξ12α1 · · · ξn2αn ≥ C|ξ |2k

α∈S(n,k)

where C = C(n, k). On the other hand, the coordinate-based representation of
1,ℵ does depend on the choice of ℵ, see (67), and so does the uniform ellipticity of
1,ℵ ; for instance, if ℵ is chosen so that (π ◦ ℵ−1 )({1, 2, . . . , n}) = = {(k, 0, . . . , 0), (1, k − 1, 0, . . . , 0), . . . , (1, 0, . . . , 0, k − 1)} ⊂ S(n, k), then
1,ℵ has symbol  

n


 ξ12 ξj2(k−1)  |ζ |2 ,

n

ξ ∈ Rn , η ∈ C(q )

j =1

which fails to be uniformly elliptic (take, e.g., ξ := (0, 1, . . . , 1)). Choosing instead (π ◦ ℵ−1 )({1, 2, . . . , n}) = {(k, 0, . . . , 0), (0, k, 0, . . . , 0), . . . , (0, . . . , 0, k)}

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LANZANI AND RAICH

(corresponding to the example T discussed in the Introduction) leads to an operator
1,ℵ which is easily verified to be uniformly elliptic, as we have n


ξj2k ≥ n1−k |ξ |2k .

j =1

Lemma 2.7. We have that 1,ℵ: C q∞,c (RN ) → C q∞,c (RN )
q (RN ) we have is invertible for any 1 < p < ∞. For any ϕ ∈ L p



Wq2k,p 
ϕ
Lpq

(73)

−1 ϕ. where :=
1,ℵ 1q+1 (RN ) and G ∈ L 1q−1 (RN ) satisfy the hyTheorem 2.8. Suppose that F ∈ L potheses of Theorem 2.3. Let ∗ −1 (T1,ℵ q (RN ), 0 ≤ q ≤ N Z=
F + T1,ℵ G) ∈
1,ℵ

be the solution of the Hodge system for T1,ℵ with data (F, G), that is:  T1,ℵ Z = F ∗ Z = T1,ℵ

Then

(74)

(75)

G.

 
Z
Wqk−1,r ≤ C
F
L1q+1 +
G
L1q−1 , for r = n/(n − 1)

(76)

whenever q is neither 1 (unless G = 0) nor N − 1 (unless F = 0). We have: Theorem 2.3 ( = 1;

1 ≤ p, q ≤ N − 1) ⇐⇒ Theorem 2.8

For those choices of ℵ that give rise to a uniformly elliptic
1,ℵ , an analogous result holds for  T1,ℵ h = f, T1,ℵ f = 0 (77) ∗ ∗ h = g, T1,ℵ g=0 T1,ℵ which turns out to be equivalent to Theorem 2.4 ( = 1). We omit the statement. We remark in closing that for  ≥ 2 there is no analog of (67). Indeed, setting {λ0 } := {ℵ(iα)} ∩ {ℵ(iβ)} and  := {ℵ(iα)} \ {λ0 } ; {ℵ(iα)}

 := {ℵ(iβ)} \ {λ0 } {ℵ(iβ)}

(where the brackets { } indicate that the (ordered) label J is being regarded as an (unordered) set {J }), it can be proved that 2
ℵ(iβ) MI ℵ(iα)ℵ(iβ) C = (1 + (−1)(−|{λ0 }|) )  ℵ(iα) · 
·  ℵ(iα)I .



λ0 ℵ(iα)

λ0 ℵ(iβ)

ℵ(iβ)M

(78)

259

DIV-CURL FOR HIGHER ORDER

,ℵ does depend on the choice In particular, the coordinate-based representation of
MI  = 0 whenever α = β, of the representation ℵ, and it is no longer true that Cℵ(iα)ℵ(iβ) even for odd .

11.3 PROOFS Proof of Lemma 2.1. Conclusions i. and ii. are an immediate consequence of the (classical) theory for Rn combined with the readily verified identities: r/2 



r  |FI ◦ i|2 (x) dV (x) (79)
F
Lr = q

I ∈I(N,q) Rn

and






F, GL =

I ∈I(N,q)

(FI ◦ i)(x) · (GI ◦ i)(x) dV (x) .

(80)

I ∈I(N,q) Rn

rq (RN ), let q∞,c (RN ) in L To prove the density of C
rq (RN ) FI dz I ∈ L F = I ∈I(N,q)

rq (RN ), for any I ∈ I(N, q) we have that FI ◦ i ∈ be given. By the definition of L r n L (R ), and so there is {fj,I }j ∈N ⊂ Cc∞ (Rn ) such that
fj,I − FI ◦ i
Lr (Rn ) → 0 as j → ∞. Define Fj =




Fj,I dz I ,

(81)

Fj,I := fj,I ◦ π,

I ∈I(N,q)

Then, using (20), we see that Fj,I ◦ i ◦ π = Fj,I

and

Fj,I ◦ i = fj,I ∈ Cc∞ (Rn )

hold for any I ∈ I(N, q), and from these it follows that q∞,c (RN ). {Fj }j ∈N ⊂ C Moreover, on account of (79) and (81), there is C = C(r, N) such that

Fj − F
rLr ≤ C
fj,I − FI ◦ i
rLr (Rn ) → 0 as j → ∞, q

I ∈I(N,q)

as desired. The conclusions concerning the Sobolev spaces follow from the theory a,p for Wq (Rn ) via (34). ∞,c Proof of Proposition 2.2. Let F ∈ Cq∞,c (RN ) and G ∈ Cq+ be given. One has   k

ℵ(iα)I ∂ FI  L GL  dV . T,ℵ F ∧ ∗N G =  iα ∂z I ∈I(N,q) L∈I(N,q+) α∈S(n,k)

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Integrating both sides of this identity and then further integrating the right-hand side by parts k-many times we find  



ℵ(iα)I ∂ k GL  dV . (82) L FI T,ℵ F, G = (−1)k  iα  ∂z I ∈I(N,q) L∈I(N,q+) RN

α∈S(n,k)

On the other hand, a computation that requires manipulating the coefficients Jℵ(iα)K gives   k

ℵ(iα)I ∂ GL  L FI F ∧ ∗N (∗N T,ℵ ∗N G) = (−1)q(N−q)+q   dV . ∂z iα I ∈I(N,q) L∈I(N,q+) α∈S(n,k)

Identity (50) is now obtained by integrating the two sides of the identity above and comparing with (82) after having adjusted the multiplicative constants as in (49). Note that since D λ T,ℵ F = T,ℵ D λ F

for any multi-index λ

where D λ F ∈ Cq∞,c (RN ) is defined as in (39), the same argument also shows that ∗ G. D λ T,ℵ F, D λ G = D λ F, D λ T,ℵ

The proofs of (52) and of (54) follow in a similar fashion. Theorem 1.4 ⇒ Theorem 2.3. Let  be an admissible degree increment and let 0 ≤ q ≤ N − . Suppose that
q (RN ) FI dz I ∈
F = I ∈I(N,q)

and




H =

q (RN ) HI dz I ∈


I ∈I(N,q)

satisfy the hypotheses of Theorem 2.3. Fix an arbitrary I0 ∈ I(N, q), and choose (any) L0 ∈ I(N, q + ) so that I0 ⊆ L0 . (The hypothesis: q≤N− grants q +≤N and so at least one such L0 must exist.) With I0 and L0 fixed as above, define hL0 =(hLα 0 )α∈S(n,k) and g L0 = (gαL0 )α∈S(n,k) via
ℵ(iα)I hLα 0 = L 0 HI ◦ i, α ∈ S(n, k); I ∈I(N,q)

gαL0 =


I ∈I(N,q)

Lℵ(iα)I FI ◦ i, 0

α ∈ S(n, k).

261

DIV-CURL FOR HIGHER ORDER

We claim that g L0 satisfies condition (12) in Theorem 1.4: to this end, note that by (35) we have   k
∂ k g L0
∂ FI   α = Lℵ(iα)I  ◦ i = [T, ℵ F ]L0 ◦ i = 0 0 α ∂x ∂z iα I ∈I(N,q) α∈S(n,k) α∈S(n,k)

where the last identity is due to the hypothesis (55). It thus follows from Theorem 1.4 that

g L0 · hL0 ≤ C
g L0
L1 (Rn )
∇hL0
Ln (Rn ) α0 α0 Rn

where α0 ∈ S(n, k) is uniquely determined by I0 and L0 via iα0 := ℵ−1 (L0 \ I0 ) (note that L0 \ I0 ∈ I(N, ).) But for α0 as above we have 0 )I Lℵ(iα = 0 ⇐⇒ I = I0 0

and so 0 )I0 gαL00 = Lℵ(iα FI0 ◦ i, 0

and

0 )I0 hLα00 = Lℵ(iα HI0 ◦ i. 0

On the other hand, it is immediate to verify that
g L0
L1 (Rn ) 
F
L1q (RN ) , and



gαL00

Rn


∇hL0
Ln (Rn ) 
∇H
Lnq (RN ) ,

·

hLα00

=

(FI0 ◦ i) · (HI0 ◦ i)(x)dV (x). Rn

Since I0 ∈ I(N, q) had been fixed arbitrarily, we have proved that   ≤ C
F
L1 (RN )
∇H
Ln (RN ) ◦ i) · (H ◦ i) (x)dV (x) (F I I q q n

(83)

R

holds for any I ∈ I(N, q), for any 0 ≤ q ≤ N − , and for any admissible degree increment . Inequality (56) follows from (83) and the coordinate-based representation for ·, ·L, see (80). (We remark that in the special case q = 0, the proof follows along these very same lines by defining gα := F ◦ i for each α ∈ S(n, k).) In order to prove (58), it suffices to apply (56) to: F := ∗N G and H := ∗N K (with q := N − p). Theorem 2.3 ⇒ Theorem 1.4. Let  be any admissible degree increment for (n, k) and let ℵ be any one-to-one correspondence: iS(n, k) → I(N, ). Suppose that g and h satisfy the hypotheses of Theorem 1.4; without loss of generality we may assume that gα , hα ∈ C0∞ (Rn ), α ∈ S(n, k). Choose q := N −  and define F and

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LANZANI AND RAICH

H in
N− (RN ) via 

I ∈ I(N, N − )

(84)



I ∈ I(N, N − )

(85)

II FI := (1,...,N) gα ◦ π, II HI := (1,...,N) hα ◦ π,

where I  := {1, . . . , N} \ I ∈ I(N, ), and α ∈ S(n, k) is uniquely determined by I and by ℵ via iα = ℵ−1 (I  ). Since π ◦ i is the identity on Rn , we have 

II FI ◦ i = (1,...,N) gα ∈ C0∞ (Rn )

FI ◦ i ◦ π = FI , ∞,c N− so F ∈ C (RN ) and

N∞,c (RN ). T,ℵ F = [T,ℵ F ]N dz 1 ∧ · · · ∧ dz N ∈ C Using (35) and (84) we find



[T,ℵ F ]N ◦ i = 




ℵ(iα) ℵ(iα) 1,...,N

α∈S(n,k)

=

 ∂k Fℵ(iα)  ◦ i ∂z iα




∂ k gα =0 ∂x α α∈S(n,k)

where the last identity is due to the hypothesis (12). Now observe that if G ∈ q (RN ), then
G = 0 ⇐⇒ GI ◦ i = 0 for each I ∈ I(N, q). Combining all of the above we obtain T,ℵ F = 0 so that Theorem 2.3 grants 
L1 (RN )
∇H
Ln (RN ) . |F, H L| ≤ C
F q q But since (π ◦ i)(x) = x for all x ∈ Rn it follows from (80), (84), and (85) that


gα · hα ;
F
L1q (RN ) =
g
L1 ;
∇H
Lnq (RN ) =
∇h
n F, H L = α∈S(n,k)

and so




gα · h α α∈S(n,k)

≤ C
g
 L1
∇h
Ln

(86)

is true for any h ∈ (L∞ ∩ W 1,n )(Rn , Rm ). Now fix α0 ∈ S(n, k) arbitrarily and define hˆ α := δα0 α hα ,

α ∈ S(n, k)

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DIV-CURL FOR HIGHER ORDER

where δα0 α denotes the Kroenecker symbol. Then hˆ ∈ (L∞ ∩ W 1,n )(Rn , Rm ) and so by applying (86) to hˆ we obtain


ˆ ˆ  g · h α α ≤ C
g
L1
∇ h
Ln . α∈S(n,k) However,


ˆ gα · hα = gα0 · hα0 α∈S(n,k)

and

ˆ Ln ≤
∇h
Ln ,
∇ h


 so (13) is true for any choice of α0 ∈ S(n, k), with C := C. Theorem 2.3 ⇒ Theorem 2.4. Let  be an admissible degree increment such that n ≥ , let ℵ be any one-to-one correspondence: iS(n, k) → I(N, ) and let 0 ≤ q ≤ n− be given. Suppose that
fI dx I ∈ L1q (Rn ) f = I ∈I(n,q)

satisfies the hypotheses of Theorem 2.4; without loss of generality we may assume that f ∈ Cq∞,c (Rn ). By the definition of T,ℵ , see (47), we have  

ℵ(iα)I  ∂ k (fI ◦ π )  T,ℵ f = L ◦ i dx L  iα ∂z I ∈I(n,q) L∈I(n,q+) ℵ(iα)∈I(n,)

and applying (35) we obtain



 k
∂ fI  L  Lℵ(iα)I T,ℵ f =  dx = 0 ∂x α I ∈I(n,q) L∈I(n,q+)


ℵ(iα)∈I(n,)

where the last identity is due to the hypothesis (59). Fix I0 ∈ I(n, q) and choose (any) L0 ∈ I(n, q + ) so that I0 ⊆ L0 . (The hypothesis q ≤ n −  grants q +  ≤ n, so at least one such L0 must exist.) Note that since  ≤ n ≤ N we have I(n, ) ⊆ I(N, ), so with I0 and L0 fixed as above, we may define
L ∞,c (RN ) FJ 0 dz J ∈ C F L0 = J ∈I(N,)

via

  FJL0

:=

LJ 0I fI ◦ π

I ∈I(n,q)

0

for J ∈ I(n, ) for J ∈ I(N, ) \ I(n, ).

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LANZANI AND RAICH

Applying (51) with q :=  we obtain (ignore the factor (−1)k+q(N−−q) ) L0
∂ k Fℵ(iα)

∗ F L0 = T,ℵ

α∈S(n,k)

∂z iα

0∞,c (RN ) ∈C

and by the definition of F L0 this is further simplified to


∗ F L0 = T,ℵ

Lℵ(iα)I 0

I ∈I(n,q) ℵ(iα)∈I(n,)

∂ k (fI ◦ π ) . ∂z iα

Note that on account of (20) and of (35) we have


∗ (T,ℵ F L0 ) ◦ i =

Lℵ(iα)I 0

I ∈I(n,q) ℵ(iα)∈I(n,)

∂ k fI = [T,ℵ f ]L0 = 0. ∂x α

∗ 0 (RN ) and so But T,ℵ F L0 ∈
∗ ∗ F L0 ) ◦ i = 0 ⇐⇒ T,ℵ F L0 = 0, (T,ℵ

see (19). Thus ∗ F L0 = 0 T,ℵ

and by Theorem 2.3 we conclude that  L0
L1
∇H
Ln |F L0 , H L| ≤ C
F  

(87)

∞,c (RN ). Now set is true for any H ∈ C J0 := L0 \ I0 ∈ I(n, ) and let




h=

1,n n hI dx I ∈ (L∞ q ∩ Wq )(R )

I ∈I(n,q)

be given (without loss of generality we may assume that h ∈ Cq∞,c (Rn )). Define
Hˆ J dz J ∈
 (RN ) Hˆ = J ∈I(N,)

with Hˆ J = δJ0 J




LJ 0I hI ◦ π,

J ∈ I(N, ),

I ∈I(n,q)

where δJ0 J is the Kroenecker symbol. Then Hˆ J ◦ i ◦ π := Hˆ J

and Hˆ J ◦ i ∈ Cc∞ (Rn ),

so that ∞,c (RN ). Hˆ ∈ C

J ∈ I(N, )

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DIV-CURL FOR HIGHER ORDER

Note that F L0 , Hˆ L =

fI0 · hI0 ,

and


∇ Hˆ
Ln (RN ) 
∇h
Ln (Rn ) .

Rn

Moreover, by the definition of F L0 we have
F L0
L1 (RN ) 
f
L1 (Rn ) . Thus, applying (87) to Hˆ we conclude that

fI · hI ≤ C
f 
L1 (Rn )
∇h
Ln (Rn ) 0 0 n

(88)

R

is true for any I0 ∈ I(n, q), for any 0 ≤ q ≤ n −  and for any h ∈ Cq∞,c (Rn ), and this in turn implies (60). In order to prove (62), it suffices to apply (60) to: f := ∗n g (with q := n − p). Theorem 2.4 ⇒ Theorem 1.3. We claim that, in fact, Theorem 1.3 is equivalent to ∗ in Theorem 2.4 in the special case:  = 1; q = 1 and for the statement for T,ℵ specific choices of the ordering ℵ. Indeed it is easy to see that, for  = 1 and q = 1, (87) and (53) give ∗ f = (−1)k+n T1,ℵ

n
j =1

∂ k fj ∂x

f =

, π ◦ℵ−1 (j )

n


fj dxj ∈
1 (Rn ).

j =1

Choosing now any ordering ℵ : iS(n, k) ↔ I(N, 1) such that π ◦ ℵ−1 (j ) = (0, . . . , 0, k, 0, . . . , 0),

j = 1, . . . , n

(where, in the expression above, it is understood that k occupies the j -th position) we obtain ∗ f = T1,ℵ

n
∂ k uj j =1

∂xjk

,

uj := (−1)k+n fj , j = 1, . . . , n.

The equivalence of the two statements is now apparent. Proof of Lemma 2.5. The proof of (65) and (66) is a computation that uses (43) ∗ , which is obalong with the following coordinate-based representation for T,ℵ tained from (51): ∗ H = (−1)k+N T,ℵ




ℵ(iβ)V

I

V ∈I(N,q−) I ∈I(N,q) β∈S(n,k)

∂ k HI V dz . ∂z iβ

To prove (67) we examine (66) in the case  = 1: MI ℵ(iα) MI MI C ℵ(iβ) = Aℵ(iα) ℵ(iβ) + Bℵ(iα) ℵ(iβ)

(89)

266

LANZANI AND RAICH

where




MI A ℵ(iα) ℵ(iβ) :=

L L ℵ(iα)I · ℵ(iβ)M ,

(90)

L∈I(N,q+1) MI ℵ(iα) B ℵ(iβ) :=




M I ℵ(iα)K · ℵ(iβ)K

(91)

K∈I(N,q−1)

and distinguish two cases: α = β; and α = β. MI Suppose first that α = β. In this case we claim that C ℵ(iα) ℵ(iβ) = 0. The proof of this claim rests on the following: Remark 3.1. The truth value of the following three (combined) conditions on ℵ, α, β, I , and M: ℵ(iα) ∈ / {I }; ℵ(iβ) ∈ / {M}; {ℵ(iα)} ∪ {I } = {ℵ(iβ)} ∪ {M}

(92)

is equivalent1 to the truth value of ℵ(iα) ∈ {M}; ℵ(iβ) ∈ {I }; {M} \ {ℵ(iα)} = {I } \ {ℵ(iβ)}.

(93)

We postpone the proof of Remark 3.1 and continue with the proof of Lemma 2.5; to this end we claim that if α = β then (92) holds

⇐⇒

MI A ℵ(iα)ℵ(iβ) = 0.

MI Indeed, since α, β, I , and M are fixed, the summation that defines A ℵ(iα)ℵ(iβ) , see (90), consists of at most one term, that is, L0 L0 MI A ℵ(iα)ℵ(iβ) = ℵ(iβ)M · ℵ(iα)I

for at most one choice of L0 ∈ I(N, q+1), and it’s easy to see that (92) holds if, and L0 L0 ·ℵ(iα)I = 0 only if, there is exactly one choice of L0 ∈ I(N, q +1) such that ℵ(iβ)M and in such case we have ℵ(iβ)M MI A ℵ(iα)ℵ(iβ) = ℵ(iα)I .

(94)

Similar considerations grant, again for α = β, that we also have (93) holds

⇐⇒

MI ℵ(iα)ℵ(iβ) = 0 B

MI and if B ℵ(iα)ℵ(iβ) = 0 there is a unique choice of K0 ∈ I(N, q − 1) such that ℵ(iβ)M MI M I ℵ(iα) B ℵ(iβ) = ℵ(iα)K0 ℵ(iβ)K0 = − ℵ(iα)I .

Combining all of the above, we conclude that if α = β then either MI MI A ℵ(iα)ℵ(iβ) = Bℵ(iα)ℵ(iβ) = 0 or MI MI A ℵ(iα)ℵ(iβ) = − Bℵ(iα)ℵ(iβ) . 1

If q = 0 or q = N − 1, then (92) is equivalent to (93) in the sense that each is false.

(95)

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DIV-CURL FOR HIGHER ORDER

In either case it follows that MI ℵ(iα)ℵ(iβ) = 0 C

whenever α = β .

(96)

Suppose next that α = β; in this case (90) and (91) become


MI A ℵ(iα) ℵ(iα) =

L L ℵ(iα)I · ℵ(iα)M ,

(97)

L∈I(N,q+1) MI ℵ(iα) B ℵ(iα) =




M I ℵ(iα)K · ℵ(iα)K

(98)

K∈I(N,q−1)

and since α, I , and M are fixed, each of these summations consists of at most one term, that is, L0 L0 MI A ℵ(iα) ℵ(iα) = ℵ(iα)I · ℵ(iα)M ;

MI M I ℵ(iα) B ℵ(iα) = ℵ(iα)K0 · ℵ(iα)K0

for at most one choice for each of L0 ∈ I(N, q + 1) and K0 ∈ I(N, q − 1). In particular we see that I = M

⇒

MI MI A ℵ(iα) ℵ(iα) = Bℵ(iα) ℵ(iα) = 0 .

(99)

On the other hand, for I = M we have MM M 2 ℵ(iα) B ℵ(iα) = (ℵ(iα)K0 )

L0 2 MM A ℵ(iα) ℵ(iα) = (ℵ(iα)M ) ;

for at most one choice of L0 and of K0 . We now further distinguish between MM ℵ(iα) ∈ {I } and ℵ(iα) ∈ / {I }. If ℵ(iα) ∈ {I } then we have A ℵ(iα) ℵ(iα) = 0 (because MM  / {I } the L’s do not have repeated terms) and Bℵ(iα) ℵ(iα) = 1. If, instead, ℵ(iα) ∈ MM MM = 1 and B = 0. All then we find by the same token that A ℵ(iα) ℵ(iα) ℵ(iα) ℵ(iα) together this gives

0 for M = I MI ℵ(iα) = (100) C ℵ(iα) 1 for M = I . Combining (96) and (100) we obtain (67). The proofs of (69) – (71) are obtained in a similar fashion; in this case (49) grants ∗ h = (−1)k+n T,ℵ




ℵ(iβ)V

I

V ∈I(n,q−) I ∈I(n,q) β∈(π◦ℵ−1 )(I(n,))

∂ k hI dx V . ∂x β

Proof of Remark 3.1. If α = β and the three conditions in (92) hold, then it follows at once that the first two conditions in (93) are true; by the first condition in (92) we have {I } = {I } ∩ {ℵ(iα)}c ; combining this identity with the third condition in (92) we obtain {I } = ({ℵ(iβ)} ∪ {M}) ∩ {ℵ(iα)}c , and since α = β then {ℵ(iβ)} ∩ {ℵ(iα)}c = {ℵ(iβ)}, and it follows that {I } = {ℵ(iβ)} ∪ {Q0 },

{Q0 } := {M} ∩ {ℵ(iα)}c .

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By the second condition in (92) we have {ℵ(iβ)} ∩ {Q0 } = ∅ and so {I } \ {ℵ(iβ)} = {Q0 }. On the other hand, since we have proved that ℵ(iα) ∈ {M} is true, then we have {M} = ({M} ∩ {ℵ(iα)}) ∪ ({M} ∩ {ℵ(iα)}c ) = {ℵ(iα)} ∪ {Q0 } and obviously {ℵ(iα)} ∩ {Q0 } = ∅, so {M} \ {ℵ(iα)} = {Q0 } which shows that the third condition in (93) is true, as well. Suppose, conversely, that α = β and that the three conditions in (93) hold. Then ˙ denotes disjoint ˙ 0 } (where ∪ the first condition in (93) grants {M} = {ℵ(iα)}∪{P ˙ 0 }, and union); similarly, the second condition in (93) grants {I } = {ℵ(iβ)}∪{S it follows from the third condition in (93) that S0 = P0 . Note that in particular / {P0 }; since α = β, it follows that the first two conditions ℵ(iα) ∈ / {P0 } and ℵ(iβ) ∈ in (92) hold. But these (and the above) considerations in turn imply {ℵ(iα)} ∪ {I } = {ℵ(iα)} ∪ {ℵ(iβ)} ∪ P0 = {ℵ(iβ)} ∪ {M} which shows that the third condition in (92) is true, as well. Proof of Lemma 2.7. The proof is easily reduced to the classical theory via Corollary 2.6 along with (35) and the coordinate-based representations for
·
Ln , see (79). See [CZ], [SR, pg. 62], [S, VI.5], and [T, 13.6]. Theorem 2.3 ( = 1) ⇒ Theorem 2.8. Without loss of generality we may assume: ∞,c ∞,c q−1 q∞,c (RN ). Write q+1 (RN ); G ∈ C (RN ), so that Z ∈ C F ∈C Z =X+Y where

and



T1,ℵ X

=

F

∗ X T1,ℵ

=

0

T1,ℵ Y

=

0

∗ Y T1,ℵ

=

G.



(101)

(102)

We claim that
X
Wqk−1,r ≤ C
F
L1q+1 ,

r := n/(n − 1)

(103)


Y
Wqk−1,r ≤ C
G
L1q−1 ,

r := n/(n − 1).

(104)

and Note that if Y solves (102) then X := ∗N Y solves (101) with F := ∗N G, and so it suffices to prove (103) for F and X as in (101). By duality, this is equivalent to proving iβ D X, ϕL ≤ C
F
L1
ϕ
Ln (105) q+1

q

269

DIV-CURL FOR HIGHER ORDER

q∞,c (RN ). Let for any β ∈ S(n, s) with 0 ≤ s ≤ k − 1, and for any ϕ ∈ C q∞,c (RN ) be as in Lemma 2.7. Note that ∈C ∗ ∗ T1,ℵ D iβ X = D iβ T1,ℵ X = 0.

T1,ℵ D iβ X = D iβ T1,ℵ X = D iβ F ;

By (52) and the above considerations it follows that D iβ X, ϕL = D iβ F, T1,ℵ L = F, D iβ T1,ℵ L. ∞,c q+1 By Theorem 2.3 ( = 1; H := D iβ T1,ℵ ∈ C (RN )) we have iβ D X, ϕL ≤ C
F
L1
∇D iβ T1,ℵ
Ln ≤ C
F
L1

 2k,n , Wq q+1 q+1 q+1

and it follows from Lemma 2.7 (with p := n) that iβ D X, ϕL ≤ C
F
L1
ϕ
Ln q q+1 as desired. Theorem 2.8 ⇒ Theorem 2.3 ( = 1; 1 ≤ q, p ≤ N − 1). We show that (56) 1q (RN ) holds for any q in the range 1 ≤ q ≤ N − 1. Suppose that T1,ℵ F = 0, F ∈ L N  1,n ∞ and let H ∈ (L q ∩ Wq )(R ). Without loss of generality we may assume: ∞,c ∞,c N q−1 q (R ). Let X ∈ C (RN ) be the solution of (101) with data F . H, F ∈ C Then, by Hölder inequality (42) we have F, H L = X, T∗ H L 
X
 k−1,r
T∗ H
 −(k−1),n 1,ℵ 1,ℵ Wq Wq and it follows from the latter and Theorem 2.8 that F, H L 
F
L1
T∗ H
 −(k−1),n . 1,ℵ W q q−1

Now observe that if we integrate the expression ∗ T1,ℵ H, ζ L

by parts (k − 1)-times and then apply Hölder inequality, we obtain ∗ H, ζ L| ≤
∇H
Lnq
ζ
W k−1,r |T1,ℵ q−1

and this leads to the conclusion of the proof of (56) as ∗ ∗ H
W −(k−1),n = sup T1,ℵ H, ζ L .
T1,ℵ q−1


ζ
W k−1,r ≤1 q−1

11.4 CONCLUDING REMARKS 1. The proof of Theorems 2.3 and 2.8 rely on the specific choice of the embedding i : Rn → RN only to the extent that (20), in fact (21), and (34) hold. This suggests that Theorems 2.3, 2.4, and 2.8 should hold in the more general context of an isometrically embedded manifold M(n) → RN .

270

LANZANI AND RAICH

2. If q ≥ N −  + 1 or p ≤  − 1 then one of the two compatibility conditions (55) and (57) holds trivially and in this case the conclusion of Theorems 2.3 and 2.8 are easily seen to be false: if k = 1 and T1,ℵ0 = d (exterior derivative) substitute inequalities hold provided the “defective” datum belongs to a suitable (proper) subspace of L1 , namely the real Hardy space H 1 (Rn ), see [LS]. We do not know whether substitute inequalities hold when k ≥ 2. 3. In the context of [VS4] our results say the following: ∞,c q∞,c (RN ) to E := C q+1 • T1,ℵ is canceling from V := C (RN ) for any 0 ≤ q ≤ N − 2, see [VS4, Theorem 1.3]. ∞,c ∗ q∞,c (RN ) to E := C q−1 • T1,ℵ is canceling from V := C (RN ) for any 2 ≤ q ≤ N , see [VS4, Theorem 1.3]. • For any admissible degree increment  and for any 0 ≤ q ≤ N − , ∞,c q∞,c (RN ) to E := C q+ (RN ), see [VS4, T,ℵ is cocanceling from V := C Propositions 2.1 and 2.2]. ∗ • For any admissible degree increment  and for any  ≤ q ≤ N , T,ℵ ∞,c ∞,c N N   is cocanceling from V := Cq (R ) to E := Cq− (R ), see [VS4, Propositions 2.1 and 2.2]. • The class T,ℵ has similar properties with V = Cq∞ (Rn ) and E = ∞ (Rn ). Cq±

In particular, T1,ℵ and T1,ℵ , as well as their adjoints, are new examples of canceling operators of arbitrary order k.

REFERENCES [A] Adams, R. A., Sobolev spaces, Academic Press, New York (1975). [Am] Amrouche, C. and Nguyen, H.-H., New estimates for the div-curl operators and elliptic problems with L1 -data in the whole space and in the half space, J. Diff. Eq. 250, 3150–3195 (2010). [BB1] Bourgain, J. and Brezis H., On the equation divY = f and application to control of phases, J. Amer. Math. Soc. 16, 393–426 (2003). [BB2] Bourgain, J. and Brezis H., New estimates for the Laplacian, the div-curl and related Hodge systems, C. R. Math. Acad. Sci. Paris 338, 539–543 (2004). [BB3] Bourgain, J. and Brezis H., New estimates for elliptic equations and Hodge type systems, J. Eur. Math. Soc. 9, 277–315 (2007). [BV] Brezis, H. and Van Schaftingen, J., Boundary estimates for elliptic systems with L1 -data, Cal. Var. PDE 30, 369–388 (2007). [CV] Chanillo, S. and van Shaftingen, J., Subelliptic Bourgain-Brezis estimates on groups, Math. Res. Lett. 16, 235–263 (2009).

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[CZ] Calderón, A. and Zygmund, A., On the existence of certain singular integrals, Acta Math. 88, 85–139 (1952). [DKPV] Dahlberg, B., Kenig, C., Pipher, J. and Verchota, G., Area integral estimates for higher order equations and systems, Ann. Inst. Fourier (Grenoble) 47 (1997) no. 5, 1425–1461. [G] Gagliardo, E., Ulteriori propriet’ di alcune classi di funzioni di pi’ variabili, Ricerche Mat. 8, 24–51 (1959). [HP-1] Hounie, J. and Picon, T., Local Gagliardo-Nirenberg estimates for elliptic systems of vector fields, Math. Res. Lett. 18, no. 4, 791–804 (2011). [HP-2] Hounie, J. and Picon, T., Local L1 estimates for elliptic systems of complex vector fields, to appear in Proc. AMS. [LS] Lanzani, L. and Stein, E. M., A note on div-curl inequalities, Math. Res. Lett. 12, 57–61 (2005). [M] Mazya, V., Estimates for differential operators of vector analysis involving L1 -norm, J. Math. Soc. 12, 221–240 (2010). [Mi] Mironescu, P., On some inequalities of Bourgain, Brezis, Maz’ya, and Shaposhnikova related to L1 vector fields, C. R. Math. Acad. Sci. Paris 348, 513–515 (2010). [MM] Mitrea, M. and Mitrea, I., A remark on the regularity of the div-curl system, Proc. Am. Math. Soc. 137, 1729–1733 (2009). [Mu] Munkres, J. R., Analysis on manifolds, Westview Press (1990). [N] Nirenberg, L., On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa 13, 115–162 (1959). [S] Stein, E. M., Harmonic Analysis, Princeton University Press (1993). [SR] Saint Raymond, X., Elementary introduction to the theory of pseudodifferential operators, CRC Press, Boca Raton (1991). [T] Taylor, M. E., Partial differential equations III, Springer Verlag (1996). [VS1] Van Schaftingen, J., Estimates for L1 -vector fields, C. R. Math. Acad. Sci. Paris 338, 23–26 (2004). [VS2] Van Schaftingen, J., Estimates for L1 vector fields with a second order condition, C. R. Math. Acad. Sci. Paris 339, 181–186 (2004). [VS3] Van Schaftingen, J., Estimates for L1 -vector fields under higher order differential conditions, J. Eur. Math. Soc. 10, 867–882 (2008).

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[VS4] Van Schaftingen, J., Limiting Sobolev inequalities for vector fields and canceling linear differential operators, to appear in J. Eur. Math. Soc. 15, 877–921 (2013). [VS5] Van Schaftingen, J., Limiting fractional and Lorentz space estimates of differential forms, Proc. Am. Math. Soc. 138, 235–240 (2010). [Y] Yung, P.-L., Sobolev inequalities for (0, q)-forms on CR manifolds of finite type, Math. Res. Lett. 17, 177–196 (2010).

Chapter Twelve Square Functions and Maximal Operators Associated with Radial Fourier Multipliers Sanghyuk Lee, Keith M. Rogers, and Andreas Seeger We begin with an overview on square functions for spherical and Bochner–Riesz means which were introduced by Eli Stein, and discuss their implications for radial multipliers and associated maximal functions. We then prove new endpoint estimates for these square functions, for the maximal Bochner–Riesz operator, and for more general classes of radial Fourier multipliers.

OVERVIEW Square functions. The classical Littlewood–Paley functions on Rd are defined by 2 1/2
 ∞ ∂   g[f ] =  Pt f  t dt ∂t 0 where (Pt )t>0 is an approximation of the identity defined by the dilates of a “nice” kernel (for example, (Pt ) may be the Poisson or the heat semigroup). Their significance in harmonic analysis and many important variants and generalizations have been discussed in Stein’s monographs [38], [39], [44], in the survey [45] by Stein and Wainger, and in the historical article [43]. Here we focus on Lp -bounds for two square functions introduced by Stein, for which (Pt ) is replaced by a family of operators with rougher kernels or multipliers. The first is generated by the generalized spherical means β

At f (x) =

1 (β)

 |y|≤1

(1 − |y|2 )β−1 f (x − ty) dy

defined a priori for Reβ > 0. The definition can be extended to Reβ ≤ 0 by analytic continuation; for β = 0 we recover the standard spherical means. In [41] Stein used (a variant of) the square function 2
 ∞  ∂ 1/2  β  Gβ f =  At f  t dt ∂t 0 Supported in part by NRF grant 2012008373, ERC grant 277778, MINECO grants MTM2010-16518, SEV-2011-0087, and NSF grant 1200261.

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to prove Lp -estimates for the maximal function supt>0 |At f |, in particular he d established pointwise convergence for the standard spherical means when p > d−1 and d ≥ 3; see also [45]. The second square function 2
 ∞ ∂ 1/2   , Gα f =  Rtαf  t dt ∂t 0 generated by the Bochner–Riesz means 
1 |ξ |2 α  Rtα f (x) = 1− 2 f (ξ ) eix,ξ  dξ , d (2π ) |ξ |≤t t was introduced in Stein’s 1958 paper [37] and used to control the maximal funcα−1/2+ε f | for f ∈ L2 in order to prove almost everywhere contion supt>0 |Rt vergence for Bochner–Riesz means of both Fourier integrals and series (see also Chapter VII in [46]). Later, starting with the work of Carbery [3], it was recognized that sharp Lp bounds for Gα with p > 2 imply sharp Lp -bounds for maximal functions associated with Bochner–Riesz means and then also maximal functions associated with more general classes of radial Fourier multipliers ([4], [13]). In [45], Stein and Wainger posed the problem of investigating the relationships between various square functions. Addressing this problem, Sunouchi [48] (in one dimension) and Kaneko and Sunouchi [23] (in higher dimensions) used Plancherel’s theorem to establish among other things the uniform pointwise equivalence Gα f (x) ≈ Gβ f (x),

β =α−

d−2 . 2

(1)

In view of this remarkable result we shall consider Gα only. Implications for radial multipliers. We recall Stein’s point of view for proving results for Fourier multipliers from Littlewood–Paley theory. Suppose the convolution operator T is given by T f = hf where h satisfies the assumptions of the Hörmander multiplier theorem. That is, if ϕ is a radial nontrivial C ∞ function with compact support away from the origin, and L2α (Rd ) is the usual Sobolev space, it is assumed that supt>0 ϕh(t · )L2α is finite for some α > d/2. Under this assumption T is bounded on Lp for 1 < p < ∞ ([21], [39], [55]). In Chapter IV of the monograph [39], Stein approached this result by establishing the pointwise inequality ∗ [f ], g[Tf ](x) ≤ C sup ϕh(t · )L2α gλ(α)

(2)

t>0

where g is a standard Littlewood–Paley function and gλ∗ is a tangential variant of ∗ [f ]p
f p for g which does not depend on the specific multiplier. As gλ(α) 2 ≤ p < ∞ and α > d/2, this proves the theorem since (under certain nondegeneracy assumptions on the generating kernel) one also has g[f ]p ≈ f p for 1 < p < ∞. A similar point of view was later used for radial Fourier multipliers. Let m be a bounded function on R+ , let ϕ◦ ∈ C0∞ (1, 2), and let Tm be defined by  (3) T m f (ξ ) = m(|ξ |)f (ξ ) .

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The work of Carbery, Gasper, and Trebels [5] yields an analogue of (2) for radial multipliers in which the gλ∗ -function is replaced with a robust version of Gα which has the same Lp -boundedness properties as Gα . A variant of their argument, given by Carbery in [4], shows that one can work with Gα itself and so there is a pointwise estimate g[Tm f ](x) ≤ C sup ϕ◦ m(t·)L2α (R) Gα f (x)

(4)

t>0

where again g is a suitable standard Littlewood–Paley function. Lp mapping properties of Gα together with (4) have been used to prove essentially sharp estimates for radial convolution operators, with multipliers in localized Sobolev spaces. However it was not evident whether (4) could also be used to capture endpoint results, for radial multipliers in the same family of spaces. We shall address this point in §12.1 below. Carbery [4] also obtained a related pointwise inequality for maximal functions, sup |Tm(t·) f (x)| ≤ Cm ◦ exp L2α (R) Gα f (x) ,

(5)

t>0

which for p ≥ 2 yields effective Lp -bounds for maximal operators generated by radial Fourier multipliers from such bounds for Gα ; see also Dappa and Trebels [13] for similar results. Only little is currently known about maximal operators for radial Fourier multipliers in the range p < 2; cf. Tao’s work [50], [51] for examples and for partial results in two dimensions. Lp -bounds for Gα . We now discuss necessary conditions and sufficient conditions on p ∈ (1, ∞) for the validity of the inequality Gα f p
f p ;

(6)

here the notation A
B is used for A ≤ CB with an unspecified constant. By (4) it is necessary for (6) that α > 1/2 since for L2α (R) to be imbedded in L∞ we need α > 1/2. For 1 < p < 2 the inequality can only hold if α > α(p) ˜ = d( p1 − 12 ) + 12 . This is seen by writing
 ∞ 2
2 α−1 dt 1/2 tα (ξ ) = α |ξ | 1 − |ξ | |Ktα ∗ f |2 where K . (7) Gα f = t t2 t2 + 0 Then, for a suitable Schwartz function η, with  η vanishing near 0 and compactly supported in a narrow cone, and for t ∼ 1 and large x in an open cone, we have Ktα ∗ η(x) = cα t d eit|x| |tx|−

d−1 2 −α

+ Et (x)

(8)

where Et are lower order error terms. This leads to 1/2
 2 |Ktα ∗ η|2 dt ∈ Lp (Rd ) =⇒ α > α(p) ˜ . 1

Note that the oscillation for large x in (8) plays no role here. Concerning positive results for p ≤ 2, the L2 -bound for α > 1/2 follows from Plancherel and was already observed in [37]. The case 1 < p ≤ 2, α > d+1 2 is covered by the Calderón–Zygmund theory for vector-valued singular integrals,

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and analytic interpolation yields Lp -boundedness for 1 < p < 2, α > α(p), ˜ see [47], [22]. There is also an endpoint result for α = α(p), ˜ indeed one can use the arguments by Fefferman [14] for the weak type endpoint inequalities for Stein’s gλ∗ function to prove that Gα(p) is of weak type (p, p) for 1 < p < 2 (Henry Dappa, ˜ personal communication, see also Sunouchi [48] for the case d = 1). The range 2 < p < ∞ is more interesting, since now the oscillation of the kernel Ktα plays a significant role, and, in dimensions d ≥ 2, the problem is closely related to the Fourier restriction and Bochner–Riesz problems. A necessary condition for p > 2 can be obtained by duality. Inequality (6) for p > 2 implies that for all b ∈ L2 ([1, 2]) and η as above  2 
 1/2   b(t)Ktα ∗ η dt 
|b(t)|2 dt . (9)  1

p

[1,2]

Ktα

If we again split as in (8), and prove suitable upper bounds for the expression involving the error terms, then we see that, for R  1,  p      b(|x|)   d−1 +α  dx < ∞,  |x|≥R  |x| 2 which leads to the necessary condition α > d( p1 − 12 ) = d( 12 − p1 ). It is conjectured that (6) holds for 2 < p < ∞ if and only if α > α(p) = max{d( 12 − p1 ), 12 }. For d = 1 this can be shown in several ways, and the estimate follows from Calderón–Zygmund theory (one such proof is in [48]). The full conjecture for d = 2 was proved by Carbery [3], and a variable coefficient generalization of his result was later obtained in [27]. The partial result for p > 2d+2 d−1 which relies on the Stein–Tomas restriction theorem is in Christ [9] and in [30]. A better range (unifying the cases d = 2 and d ≥ 3) was recently obtained by the authors [25]; that is, inequality (6) holds for α > d(1/2 − 1/p) and d ≥ 2 in the range 2 + 4/d < p < ∞. This extends previous results on Bochner–Riesz means by the first author [24] and relies on Tao’s bilinear adjoint restriction theorem [52]. Motivated by a still open problem of Stein [42], the authors also proved a related , weighted inequality in [25], namely for d ≥ 2, 1 ≤ q < d+2 2   d [Gα f (x)]2 w(x) dx
|f (x)|2 Wq w(x) dx, α > , 2q where Wq is an explicitly defined operator which is of weak type (q, q) and bounded on Lr with q < r ≤ ∞. This is an analogue of a result by Carbery and the third author in two dimensions [6] and extends a weighted inequality by Christ [9] in higher dimensions. One might expect that recent progress by Bourgain and Guth [2] on the Bochner–Riesz problem will lead to further improvements in the ranges of these results but this is currently open. By the equivalence (1) one can interpret the boundedness of Gα as a regularity result for spherical means and then for solutions of the wave equation. By a somewhat finer analysis in conjunction with the use of the Fefferman–Stein #-function [17] the authors obtained an Lp (L2 ) endpoint result, local in time, in fact not just for the wave equation, but also for other dispersive equations.

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RADIAL FOURIER MULTIPLIERS

Namely, if γ > 0, d ≥ 2, 2 + 4/d < p < ∞, then 
  

1

−1

 it (− )γ /2 2 1/2   e p , f  dt 
f Bs,p p


1 s 1 1 =d − − . γ 2 p 2

p

(10) p

Here Bs,p is the Besov space which strictly contains the Sobolev space Ls for p > 2. Concerning endpoint estimates, many such results for Bochner–Riesz multipliers and variants had been previously known (cf. [10], [11], [12], [33], [34], [49]). For the Bochner–Riesz means Rtλ with the critical exponent λ(p) = d(1/2−1/p)−1/2, Tao [50] showed that if for some p1 > 2d/(d − 1) the Lp1 -boundedness holds for all λ > λ(p1 ), then one also has a bound in the limiting case, for p1 < p < ∞,

λ(p) namely Rt maps Lp,1 to Lp , and Lp → Lp ,∞ . In contrast no positive result for Gd/2−d/p seems to have been known, even for the version with dilations restricted to (1/2, 2). It should be emphasized that, despite the pointwise equivalence of the two square functions in (1), the sharp regularity result (10) does not imply a corresponding endpoint bound for Gd/2−d/p (in fact the latter is not bounded on Lp ). In this chapter we will prove a sharp result for Gd/2−d/p in the restricted open range of the Stein–Tomas adjoint restriction theorem, and obtain related results for maximal operators and Fourier multipliers.

12.1 ENDPOINT RESULTS Theorem 1.1. Let d ≥ 2,

2(d+1) d−1

< p < ∞ and α = d( 12 − p1 ). Then

Gα f p ≤ Cf Lp,2 .

(11)

Here Lp,q denotes the Lorentz space. We note that the Lp → Lp boundedness fails; moreover Lp,2 cannot be replaced by a larger space Lp,ν for ν > 2. This can be shown by the argument in (9), namely, if b ∈ L2 ([1, 2]), then the function

− d +1  b(| · |)(1 + | · |) p 2 belongs to Lp ,2 but not necessarily to Lp ,r for r < 2. The p,2 space L has occured earlier in endpoint results related to other square functions, see [31], [35], [53]. The pointwise bound (5) and Theorem 1.1 yield a new bound for maximal functions, in particular for multipliers in the Sobolev space L2d/2−d/p which are compactly supported away from the origin. This Sobolev condition is too restrictive to give any endpoint bound for the maximal Bochner–Riesz operator. However such a result can be deduced from a related result on maximal functions Mm f (x) = sup |F −1 [m(t| · |)f](x)| t>0

with m compactly supported away from the origin. Our assumptions involve the 2 (which is L2α when q = 2) and thus the following result seems to Besov space Bα,q be beyond the scope of a square function estimate when q  = 2.

278

LEE, ROGERS, AND SEEGER

Theorem 1.2. Let d ≥ 2, 2(d+1) < p < ∞, α = d( 12 − p1 ) and p ≤ q ≤ ∞. d−1 2 . Assume that m is supported in (1/2, 2) and that m belongs to the Besov space Bα,q Then 2 f  p,q . Mm f Lp ≤ CmBα,q L

We apply this to the Bochner–Riesz maximal operator R∗λ defined by R∗λ f (x) = sup |Rtλ f (x)|. t>0

= uλ (t)+mλ (t) where mλ is supported in (1/2, 2) and uλ ∈ C0∞ (R). Then the maximal function Muλ f is pointwise dominated by the Hardy–Littlewood maximal function and thus bounded on Lp for all p > 1. The function mλ belongs 2 and Theorem 1.2 with q = ∞ yields a maximal to the Besov space Bλ+1/2,∞ version of (the dual of) Christ’s endpoint estimate in [11]. Split (1−t 2 )λ+

Corollary 1.3. Let d ≥ 2,

2(d+1) d−1

< p < ∞, and λ = d( 12 − p1 ) − 12 . Then

R∗λ f p ≤ Cf Lp,1 . We now consider operators Tm with radial Fourier multipliers, as defined in (3), which do not necessarily decay at ∞. The pointwise bounds (4), Theorem 1.1 , and duality yield optimal Lp → Lp,2 estimates in the range 1 < p < 2(d+1) d+3 for Hörmander type multipliers with localized L2α conditions in the critical case α = d( p1 − 12 ). This demonstrates the effectiveness of Stein’s point of view in (2) and (4). The following more general theorem is again beyond the scope of a square function estimate. We use dilation invariant assumptions involving localizations of 2 . We note that in [33] it had been left open whether one could Besov spaces Bα,q use endpoint Sobolev space or Besov spaces with q > 1 in (12) below. , α = d( p1 − 12 ) and p ≤ q ≤ ∞. Let Theorem 1.4. Let d ≥ 2, 1 < p < 2(d+1) d+3 ∞ ϕ◦ be a nontrivial C0 function supported in (1, 2). Assume 2 < ∞. sup ϕ◦ m(t · )Bα,q

(12)

t>0





Then Tm maps Lp to Lp,q and Lp ,q to Lp . It is not hard to see that the assumption (12) is independent of the choice of the particular cutoff ϕ◦ . The result is sharp as Tm does not map Lp to Lp,r for r < q. This can be seen by considering some test multipliers of Bochner–Riesz type. Indeed, let 1 be a radial C ∞ function, with 1 (x) = 1 for 2−1/2 ≤ |x| ≤ 21/2 and supported in {1/2 < |x| < 2} and similarly let χ be a radial C ∞ function compactly supported away from the origin and so that χ (ξ ) = 1 in a neighborhood of the unit sphere. Set (now with p < 2)  ∞ d( 1 − 1 )− 1 1 (2j η) dη . cj (1 − |ξ − η|2 )+ p 2 2 2j d  m(ξ ) = χ (ξ ) j =1

279

RADIAL FOURIER MULTIPLIERS

2 We first remark that if we write m(ξ ) = m◦ (|ξ |), then m◦ ∈ Bα,q (R) if and only 2 d if m ∈ Bα,q (R ) (here we use that m◦ is compactly supported away from the origin). Now considering the explicit formula for the kernel of Bochner–Riesz 2 (Rd ) if and only if means (cf. (41) below) it is easy to see that m ∈ Bd/p−d/2,q q −1 p,q {cj }∞ is satisj =1 belongs to  ; moreover the necessary condition F [m] ∈ L q fied if and only if {cj } belongs to  . These considerations show the sharpness of Theorem 1.4 and also the sharpness of Theorem 1.2. p For the operator Tm acting on the subspace Lrad , consisting of radial Lp functions, the estimate corresponding to Theorem 1.4 has been known to be true in 2d . In fact Garrigós and the third author [18] obthe optimal range 1 < p < d+1 tained an actual characterization of classes of Hankel multipliers which yields, for p ≤ q ≤ ∞, 

 2d . Tm Lprad →Lp,q ≈ sup F −1 φ(| · |)m(t| · |) Lp,q (Rd ) if 1 < p < d +1 t>0 p

This easily implies the Lrad → Lp,q boundedness under assumption (12), see ) with the smaller [18]. Similarly, if in Theorem 1.4 we replace the range (1, 2d+2 d+3 p-range (1, 2d−2 ) (applicable only in dimension d ≥ 4) the result follows from the d+1 characterization of radial Lp Fourier multipliers acting on general Lp functions in a recent article by Heo, Nazarov, and the third author [19]. There it is proved that for p ≤ q ≤ ∞, 

 2d − 2 . (13) Tm Lp →Lp,q ≈ sup F −1 φ(| · |)m(t| · |) Lp,q (Rd ) if 1 < p < d +1 t>0 The remainder of this chapter is devoted to the proofs of the above theorems. They are mostly based on ideas in [19]. It remains an interesting open problem to extend the range of (13), in particular to prove such a result for some p > 1 in dimensions two and three. Moreover it would be interesting to prove the above theorems beyond the Stein–Tomas range.

12.2 CONVOLUTION WITH SPHERICAL MEASURES In this section we prove an inequality for convolutions with spherical measures acting on functions with a large amount of cancellation. It can be used to obtain results such as Theorem 1.4 for radial multipliers which are compactly supported away from the origin. To formulate this inequality let η be a Schwartz function on Rd and let ψ be a radial C ∞ function with compact support in {x : |x| ≤ 1} and such that (ξ ) = u(|ξ |) ψ vanishes of order 10d at the origin. For j ≥ 1 let Ij = [2j , 2j +1 ] and denote by σr the surface measure on the sphere of radius r which is centered at the origin. Thus the norm of σr as a measure is O(r d−1 ). We recall the Bessel function formula  σr (ξ ) = r d−1 J (r|ξ |) with J (s) = c(d)s −

d−2 2

J d−2 (s) , 2

(14)

280

LEE, ROGERS, AND SEEGER

which implies | σr (ξ )|
r d−1 (1 + r|ξ |) of ψ, we have

− d−1 2

. In view of the assumed cancellation


ψ ∗ σr ∞ = O(r (d−1)/2 ).

(15)

In what follows let ν be a probability measure on [1, 2]. We will need to work with functions with values in the Hilbert space H = L2 (R+ , drr ) and write  2
 ∞  dr 1/2   |Ft (r, ·)|2 dν(t) . F Lp (L1 (H)) =  p r 1 0 . Then Proposition 2.1. Let 1 ≤ p < 2(d+1) d+3     2
  1/p p   ψ ∗ σrt ∗ η ∗ Ft,j (r, ·) dr dν(t)
2j d Fj Lp (L1 (H)) .  p

Ij

1

j ≥1

j ≥1

The measure ν is used here to unify the proofs of Theorems 1.2 and 1.4. For our applications we are only interested in two such measures. For Theorems 1.1 and 1.4 we take for ν the Dirac measure at t = 1 (and consequently in this case we can set σrt = σr and eliminate all t-integrals in the proofs below). For the application to Theorem 1.2 we take for ν the Lebesgue measure on [1, 2]. We first give a proof for the Lp -bound of each term in the j -sum, which uses standard arguments ([14], [15]). . Then Lemma 2.2. Let 1 ≤ p ≤ 2(d+1) d+3     2    ψ ∗ σrt ∗ Ft (r, ·) dr dν(t)
2j d/2 F Lp (L1 (H)) .  2

Ij

1

Proof. We use Plancherel’s theorem and then the Stein–Tomas restriction theorem (ξ ) = u(|ξ |), we get [54]. With J as in (14) so that  σr (ξ ) = r d−1 J (r|ξ |) and ψ from the restriction theorem 2  2   ψ ∗ σrt ∗ Ft (r, ·) dr dν(t)  2

Ij

1





=c 


|u(ρ)|2

S d−1

  

2

2




1

2 t (r, ρξ ) dr dν(t) dσ (ξ ) ρ d−1 dρ (rt)d−1 J (rtρ)F

Ij

1

 2d  |u(ρ)|2 ρ p −d−1 

 




2



2  (rt)d−1 J (rtρ)Ft (r, ·) dr dν(t) dρ p

Ij

 2 1/2 2 2d    |u(ρ)|2 ρ p −d−1  r d−1 J (rtρ)Ft (r, ·) dr  dρ dν(t) . p

Ij

1

In the last step we have used Minkowski’s integral inequality. We claim that, for fixed x ∈ Rd and t ∈ [1, 2],    2    −d−1  2 2d d−1 Ft (r, x)2 r d−1 dr , p |u(ρ)| ρ r J (rtρ)Ft (r, x) dr  dρ
 Ij

Ij

(16)

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RADIAL FOURIER MULTIPLIERS

with the implicit constant uniform in x, t, and the lemma follows by substituting this in the previous display. To see (16) we first notice that for a radial H (w) = H◦ (|w|) we have  (ξ ). H◦ (r)r d−1 J (r|ξ |) dr = cd H Thus, if we take H x,t (w) = χIj (|w|)Ft (|w|, x), the left-hand side of (16) is a constant multiple of  2d x,t (tξ )|2 dξ  (ξ )|2 |ξ | p −2d |H |ψ    x,t (ξ )|2 dξ = c  |H x,t (w)|2 dw = c |Ft (r, x)|2 r d−1 dr ,
|H Ij

 vanishes of high order at the and we are done. In the inequality we used that ψ origin. If we fix j and assume that FQ,t (r, ·) is supported for all r in a cube Q of side 2 length 2j then the expression 1 Ij ψ ∗ σrt ∗ FQ,t (r, ·)dr dν(t) is supported in a similar slightly larger cube. From this it quickly follows that   2    ψ ∗ σrt ∗ FQ,t (r, ·) dr dν(t)  p

Ij

1

 j d/p 
2 

2




∞

|FQ,t (r, ·)|2

dr 1/2

  dν(t) .

p r This estimate is however insufficient to prove Proposition 2.1 for p > 1. We shall also need the following orthogonality lemma. 1

0

Lemma 2.3. Let J1 , J2 ⊂ (0, ∞) be intervals and let E1 , E2 be compact sets in Rd with dist(E1 , E2 ) ≥ M ≥ 1. Suppose that for every r ∈ Ji , the function x  → fi (r, x) is supported in Ei . Then, for t1 , t2 ∈ [1, 2],       ψ ∗ σr1 t1 ∗ f1 (r1 , ·), ψ ∗ σr2 t2 ∗ f2 (r2 , ·) dr1 dr2   J1

J2


M−

d−1 2

2   i=1

2 1




|fi (r, y)|2 r d−1 dr

1/2

 dy .

Ji

Proof. We follow [19] and apply Parseval’s identity and polar coordinates in ξ . Then, 

ψ ∗ σr1 t1 ∗ f1 (r1 , ·), ψ ∗ σr2 t2 ∗ f2 (r2 , ·)   2  = c |ψ (ξ )|  σr1 t1 (ξ ) σr2 t2 (ξ ) f1 (r1 , y1 )f2 (r2 , y2 )eiξ,y2 −y1  dy1 dy2 dξ 

|u(ρ)|2 (r1 t1 )d−1 J (r1 t1 ρ)(r2 t2 )d−1 J (r2 t2 ρ) =c  × f1 (r1 , y1 )f2 (r2 , y2 )J (ρ|y1 − y2 |) dy1 dy2 ρ d−1 dρ,

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LEE, ROGERS, AND SEEGER

so that the left-hand side of the desired inequality is equal to a constant multiple of    2 |u(ρ)| (r1 t1 )d−1 J (r1 t1 ρ)f1 (r1 , y1 ) dr1 J1

 ×

(r2 t2 )d−1 J (r2 t2 ρ)f2 (r2 , y2 ) dr2 J (ρ|y1 − y2 |) dy1 dy2 ρ d−1 dρ . (17) J2 y

Now define two radial kernels by Hi i (w) = fi (|w|, yi )χJi (|w|) so that the expression (17) can be written as a constant times   1y1 (t1 ξ )H 2y2 (t2 ξ ) J (|ξ ||y1 − y2 |) dy1 dy2 dξ. (ξ )|2 H (18) |ψ Then, using the decay for Bessel functions and the M-separation assumption, |J (|ξ ||y1 − y2 |)|
(1 + ρM)−

d−1 2

,

yi ∈ Ei , i = 1, 2.

By the Cauchy–Schwartz inequality, the left-hand side of the desired inequality is thus bounded by 1/2 
   (ξ )|2 |ψ yi 2  | H (t ξ )| dξ dy i d−1 i yi ∈Rd (1 + |ξ |M) 2 i=1,2      d−1 H yi  dy ,
M− 2 i 2 i=1,2

y∈Rd

and by Plancherel’s theorem this is  
 d−1
M− 2 i=1,2


M

− d−1 2

wi ∈Rd

 
 i=1,2

|fi (|w|, y)|2 χJi (|w|)dw

|fi (r, y)|2 r d−1 dr

1/2

1/2

 dy

 dy ,

Ji

and so we are done. Proof of Proposition 2.1. The case p = 1 is trivial and we assume p > 1 in what follows. For z ∈ Zd consider the cube qz of all x with z i ≤ xi < z i + 1 for i = 1, . . . , d. Let  2
 ∞  2 dr 1/2   dν(t), γj,z (f ) = sup  η(x − y)Fj,t (r, y) dy  r x∈qz 1 0 and since η is a Schwartz function it is straightforward to verify that, for every j ,  1/p   2
 ∞
dr 1/2   p |γj,z (f )|
 |Fj,t (r, ·)|2 dν(t) , (19) p r 1 0 d z∈Z

with the implicit constant independent of j . If γj,z (f )  = 0 we set  −1 bj,z,t (r, x) = [γj,z (f )] χqz (x) η(x − y)Fj,t (r, y) dy

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RADIAL FOURIER MULTIPLIERS

and if γj,z (f ) = 0 we set bj,z,t = 0. Then  2
 ∞ dr 1/2 sup |bj,z,t (r, x)|2 dν(t) ≤ 1. r 1 0 d x∈qz

(20)

z∈Z

Let



2

Vj,z (x) = 1

 ψ ∗ σrt ∗ bj,z,t (r, x) dr dν(t). Ij

In view of (19) it suffices to show that for arbitrary functions z  → γj,z on Zd we , have, for 1 < p < 2(d+1) d+3  
1/p   γj,z Vj,z 
|γj,z |p 2j d (21)  p

j ≥1 z∈Zd

j ≥1 z∈Zd

where the implicit constant is independent of the specific choices of the bj,z,t (satisfying (20) with bj,z,t supported in qz ). Let µd denote the measure on N × Zd given by µd (E) = 2j d #{z ∈ Zd : (j, z) ∈ E} . j ≥1

Then (21) expresses the L (Z ×N, µd ) → Lp (Rd ) boundedness of an operator T . In the open p-range it suffices by real interpolation to show that T maps Lp,1 (Zd × N, µd ) to Lp,∞ (Rd ). This amounts to checking the restricted weak-type inequality     meas x :  Vj,z  > λ
λ−p 2j d #(Ej ) (22) p

d

j ≥1 z∈Ej

j ≥1

where Ej are finite subsets of Zd . Now for each (j, z) the term Vj,z is supported on a ball C2j +1 and therefore the entire sum is supported on a set of measure  of radius jd 2 #(E
j ). Thus the estimate (22) holds for λ ≤ 10. Assume now that j ≥1 λ > 10. We decompose Rd into dyadic “half open” cubes of sidelength 2j and let Qj be the collection of these 2j -cubes. For each Q ∈ Qj let Q∗ be the cube with same center as Q but sidelength 2j +5 . Note that for z ∈ Q the term Vj,z is supported in Q∗ . Letting

Qj (λ) := {Q ∈ Qj : #(Ej ∩ Q) > λp } and

=

 

Q∗ ,

j Q∈Qj (λ)

we have the favorable estimate #(Ej ∩ Q) |Q| ≤ 25 2j d meas() ≤ 25 λp j ≥1 j ≥1 Q∈Qj (λ)


λ−p

j ≥1

2j d #(Ej ) .

Q∈Qj (λ)

284

LEE, ROGERS, AND SEEGER

Thus the remaining estimates need only involve the “good” part of Ej ;  Ejλ = Q ∩ Ej . Q∈Qj \Qj (λ)

Note that every subset of diameter C2j , with C > 1, contains
C d λp points in Ejλ . Letting Vj,z , Vj = z∈Ejλ

it remains to show that     Vj (x) > λ
λ−p 2j d #(Ej ) . meas x :  j ≥1

This will follow from

j ≥1

 2 2p   Vj  ≤ Cλ d+1 log λ 2j d #(Ej )  2

j ≥1

2(d+1) d+3

and Tshebyshev’s inequality since, for p < λ

(23)

j ≥1

2p d+1 −2

and λ > 1,

log λ ≤ Cp λ−p .

Proof of (23). Setting N (λ) = 10 log2 λ, we treat the sums over j ≤ N (λ) and j > N(λ) separately. Using the Cauchy–Schwartz inequality for the first sum,  2   Vj 
log(λ) Vj 22 + Vj 22  2

j ≥1

j ≤N(λ)

+ Since the expression

j >N(λ)





   Vj , Vk .

(24)

j >N(λ) N(λ) λ ≤ µd (j, x) : 2− p sup |fk (x)|B > λ k   p 2j d dx ≤ λ−p sup |fk (x)|B dx, = k

j : 2j d < p supk |fk (x)|B λ−p

which yields (31) for q = ∞. By complex interpolation (with fixed p) we obtain (31) for p ≤ q ≤ ∞. As an immediate consequence of Lemma 2.4 we obtain a Lorentz space version of Proposition 2.1 which is the main ingredient in the proof of Theorem 1.2. and p ≤ q ≤ ∞. Then Corollary 2.5. Let 1 < p < 2(d+1) d+3  jd  2     2− p ψ ∗ η ∗ σrt ∗ Fj,t (r, ·) dr dν(t)  1

j ≥1

 


2

Ij




1

j ≥1

q

|Fj,t |H

1/q

Lp,q

  dν(t) . p

A preparatory result. For the proof of Theorems 1.1 and 1.4 we shall need a more technical variant of the corollary which is compatible with atomic decompositions. In what follows we let ν be Dirac measure at t = 1 so that the integrals in t disappear. Let  ≥ 1 and for z ∈ Zd let Rz = {x : 2 zi ≤ xi < 2 (zi + 1), i = 1, . . . , d} ; these sets form a grid of disjoint cubes with sidelength 2 covering Rd . In the following proposition we use the conclusion of Proposition 2.1 as our hypothesis. Proposition 2.6. Suppose that, for some p1 ∈ (1, 2),   
1/p1   p ψ ∗ σr ∗ η ∗ Fj (r, ·) dr 
2j d Fj L1p1 (H) .  p1

j ≥+2 Ij

j ≥1

Let bj,z ∈ L2 (H) with bj,z L2 (H) ≤ 1, let βj (z) ∈ C, and define 
   βj (z) ψ ∗ η ∗ σr ∗ χRz bj,z (r, ·) dr . Sj βj (x) = Ij

z

Then, for 1 < p < p1 and p ≤ q ≤ ∞,  

p/q 1/p   2−j d/p Sj βj  p,q ≤ Cp 2(d(1/p−1/2)−ε(p)) |βj (z)|q ,  L

j ≥+2

where ε(p) =

(d−1)p1 2

( p1 −

z∈Zd 1 ). p1

j ≥1

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LEE, ROGERS, AND SEEGER

Proof. We argue as in [20], Prop. 3.1. First note that  
1/p1   Sj βj 
2d(1/p1 −1/2) 2j d |βj (z)|p1 .  p1

j ≥+2

(32)

z

j ≥1

Indeed, by hypothesis the left-hand side is dominated by a constant times p1 1/p1
   2j d  βj (z)χRz bj,z  p j ≥1




≤

L 1 (H)

z

2j d

z

j ≥1

1/p1  p |βj (z)|p1 χRz bj,z L1p1 (H)

and after using Hölder’s inequality on each Rz and the L2 normalization of bj,z we obtain (32). There is a better L1 bound. Note that for r ≈ 2j the term ψ ∗ σr ∗ bj,z (r, ·) is supported on an annulus with radius ≈ 2j and width 2 . We use the Cauchy– Schwartz inequality on this annulus and then (15) and estimate      ψ ∗η∗ βj (z) σr ∗ (bj,z (r, ·)χRz ) dr   z

j ≥+2






j ≥+2




|βj (z)|

Ij

z




2

|βj (z)|

z

Ij



j (d−1)

2

ψ ∗ σr ∗ (bj,z (r, ·)χRz )1 dr (2 2j (d−1) )1/2 ψ ∗ σr ∗ (bj,z (r, ·)χRz )2 dr |βj (z)|

 bj,z (r, ·)2 dr , Ij

z

j ≥1

1

Ij





j ≥+2 /2



and by Cauchy–Schwartz on Ij and the normalization assumption on bj,z we get     Sj βj 
2/2 2j d |βj (z)| . (33)  j ≥+2

1

j ≥1

z

Now Lemma 2.4 is used to interpolate (32) and (33) and the assertion follows.

12.3 PROOF OF THEOREM 1.2 We start with a simple fact on Besov spaces, namely if ζ is a C ∞ function supported on a compact subinterval of (0, ∞) then 2 (Rd )
gB 2 (R) , ζ (| · |)g(| · |)Bα,q α,q

α > 0.

(34)

To see this, note that the corresponding inequality with Sobolev spaces L2α , α = 0, 1, 2, . . . is true by direct computation, and then (34) follows by real interpolation.

289

RADIAL FOURIER MULTIPLIERS

Next if F −1 [m(| · |)](x) = κ(|x|) we can use polar coordinates to see that ∞ 
q/2 1/q  2 2α+d−1 m(| · |)B 2 (Rd ) ≈ |κ(r)| r dr ; (35) α,q j =0

Ij

here, as in §12.2, Ij = [2j , 2j +1 ] for j ≥ 1, and I0 = (0, 2]. We shall first prove a dual version of a bound for a maximal operator where the dilations are restricted to [1, 2]. , α = d( p1 − 12 ), p ≤ q ≤ ∞. Then, Proposition 3.1. Let d ≥ 2, 1 < p < 2(d+1) d+3 2 with support in (1/2, 2), for m ∈ Bα,q  2    2     2  Tm(t·) ft dt  p,q
mBα,q |ft | dt  . (36)  L

1

1

p

 is supported in {1/8 ≤ |ξ | ≤ 8} and Proof. Let φ be a radial C -function so that φ equal to one in {1/4 ≤ |ξ | ≤ 4}. Then Tm(t·) ft = Tm(t·) (φ ∗ ft ) for 1 ≤ t ≤ 2. Also  ∞ Tm(t·) f = κ(r)t 1−d σrt ∗ φ ∗ f dr, 1 ≤ t ≤ 2, ∞

0

where κ is bounded and smooth, and the right-hand side of (35) is finite with α =  vanishing of high d/p − d/2. We may split φ = ψ ∗ η where ψ ∈ Cc∞ with ψ order at the origin. It then suffices to show that   2  ∞   κ(r)t 1−d ψ ∗ σrt ∗ ft dr dt  p,q  1




L

2

∞

Ij

j =1

 q/2 1/q   2   2 2d/p dr |κ(r)| r |ft | dt  .  p r 1

(37)

This estimate follows by applying Corollary 2.5. Take ν to be Lebesgue measure on [1, 2], use the tensor product Fj,t (r, x) = 2j d/p χIj (r)κ(r) t 1−d ft (x), and observe that Fj Lp (L1 (H)) can be estimated by the right-hand side of (37). We also need a standard “orthogonality” estimate, in Lorentz spaces. Lemma 3.2. Let {βk }k∈Z a family of L1 -functions, satisfying (i) supk βk L1 (Rd ) < ∞,  k (ξ )| < ∞. (ii) supξ k∈Z |β Then  
1/p   p βk ∗ fk  p,q
fk Lp,q , 1 < p < 2, p ≤ q ≤ ∞,  L

k

and




(38)

k

p

βk ∗ f Lp,q

1/p


f Lp,q ,

2 < p < ∞, 1 ≤ q ≤ p.

(39)

k

Here the functions {fk } are allowed to have values in a Hilbert space H (and f

may have values in H ).

290

LEE, ROGERS, AND SEEGER

Proof. By duality (38) and (39) are equivalent. To see (38) we define md to be the product measure on Rd × Z of Lebesgue measure on Rd and counting measure on Z. Define an  operator P acting on functions (x, k)  → fk (x), letting F = {fk }, by P F = k βk ∗ fk . By assumption (i) P maps the space L1 (Rd × Z, md ) to L1 (Rd ) and by the almost orthogonality assumption (ii) it maps L2 (Rd × Z, md ) to L2 (Rd ). Hence by real interpolation P maps Lp,q (Rd × Z, md ) to Lp,q (Rd ) for all 1 < p < 2 and q > 0. Let Ek,m (F ) = {x : |fk (x)|H > 2m }. If p ≤ q we have, by the triangle inequality in q/p ,  q 1
 p q F Lp,q (m ;H) ≈ 2mq  meas(Ek,m (F )) d m

≤



k

k

 q p  1
1/p  p q p p 2mq meas(Ek,m (F )) ≈ fk Lp,q (H) ,

m

k

where for q = ∞ we make the usual modification. This proves (38). Proof of Theorem 1.2, conclusion. Now let 2(d+1) < p < ∞ and p ≤ q ≤ ∞. Let d−1 −k   φ be as above and define Lk by Lk f (ξ ) = φ (2 ξ )f(ξ ). We may then estimate
  1/p  sup |Tm(2k t·) Lk f |p . Mm f p ≤ k∈Z

1≤t≤2

p

For every k ∈ Z,    sup |Tm(2k t·) Lk f | ≤ CmB 2 Lk f Lp,q ; p d/2−d/p,q 1≤t≤2

this follows for k = 0 by duality from Proposition 3.1, and then for general k by scaling. By Lemma 3.2 1/p
  Lk f p p,q
f Lp,q L k∈Z

and combining the estimates we are done.

12.4 PROOFS OF THEOREMS 1.1 AND 1.4 Many endpoint bounds for convolution operators on Lebesgue spaces can be obtained by interpolation involving a Hardy space estimate and an L2 estimate; this idea goes back to [40], [17]. In some instances it has been advantageous to use Hardy space or BMO methods such as atomic decompositions or the Fefferman– Stein #-maximal function directly on Lp to prove theorems which cannot immediately be obtained by interpolation (see for example endpoint questions treated in [32], [46], [25], [19], [29]). We formulate such a result suitable for application in the proofs of Theorems 1.1 and 1.4. In order to give a unified treatment we need to consider vector-valued operators.

291

RADIAL FOURIER MULTIPLIERS

Let H1 , H2 be Hilbert spaces. We consider translation invariant operators mapping L2 (H1 ) to L2 (H2 ), with convolution kernels having values in the space L(H1 , H2 ) of bounded operators from H1 to H2 . On the Fourier transform side,  (ξ ) ∈ H2 ,  (ξ ) = M(ξ )f(ξ ) where f(ξ ) ∈ H1 , Tf the operators are given by Tf 2 d with supξ |M(ξ )|L(H1 ,H2 ) < ∞. If S is an L (R ) convolution operator with scalar kernel (and multiplier) and H is a Hilbert space, then S extends to a bounded operator on L2 (Rd , H), denoted temporarily by S ⊗ I dH . If T is as before with L(H1 , H2 )-valued kernel, then (S ⊗ I dH2 )T = T (S ⊗ I dH1 ). With a slight abuse of notation we shall continue to write S for either S ⊗ I dH2 or S ⊗ I dH1 . We need to formulate a hypothesis which will be used for convolution operators with multipliers compactly supported away from the origin. Hypothesis 4.1. Let 1 < p < 2, p ≤ q ≤ ∞, ε > 0, and A > 0. We say that the kernel K satisfies Hyp(p, q, ε, A) if for every  ≥ 0 one can split the kernel into a short and long range contribution lg K = Ksh  + K

so that the following properties hold: (i) K is supported in {x : |x| ≤ 2+10 }.  sh (ii) supξ ∈Rd F[K ](ξ )|L(H1 ,H2 ) ≤ A. (iii) For every family of L2 functions {az }z∈Zd , with supp (az ) ∈ Rz and supz az L2 (H1 ) ≤ 1, and for γ ∈ p (Zd ) the inequality  
1/p 1 1  Klg ∗ (γ (z)az ) p,q ≤ A2(d( p − 2 )−ε) |γ (z)|p  sh

L

z

z

holds. Theorem 4.2. Given p ∈ (1, 2), p ≤ q ≤ ∞, ε > 0, and A > 0 suppose that

Kk , k ∈ Z are L(H1 , H2 )-valued kernels satisfying hypothesis Hyp(p, q, ε, A). Define the convolution operator Tk by  k Tk f (x) = 2kd K (2k (x − y))f (y)dy.

Let η be a scalar Schwartz function with  η supported in {ξ : 1/4 ≤ |ξ | ≤ 4} and  let ηk = 2kd η(2k ·). Then the operator f  → k∈Z ηk ∗ Tk f , initially defined on H1 valued Schwartz functions with compact Fourier support away from the origin, extends to an operator acting on all f ∈ Lp (H1 ) so that the inequality     ηk ∗ Tk f  p,q ≤ Cp Af Lp (H1 )  k

L

(H2 )

holds. The proof of Theorem 4.2 is by now quite standard, but for completeness we include it in Appendix A below. Given Theorem 4.2 we now show how it can be used to deduce Theorems 1.1 and 1.4 from the results in §12.2.

292

LEE, ROGERS, AND SEEGER

Remark 4.3. We actually prove a slightly more general result: Assuming that 2d ) then the the estimate of Proposition 2.1 holds for some exponent p1 ∈ (1, d+1 conclusion of Theorem 1.4 holds for 1 < p < p1 and the conclusion of Theorem 1.1 holds for p1 < p < ∞. A similar remark also applies to Theorem 1.2. Proof of Theorem 1.1. With p1 as in Remark 4.3, by duality and changes of variables t = 2k s, it is enough to show that, for 1 < p < p1 and α = d( p1 − 12 ),   2 
 2  |ξ |2
ds 1/2  |ξ |2 α−1  ds      F −1 2k 2 1 − 2k 2 |fs |2 fs  . (40)   p,2
 + L p 2 s 2 s s s 1 k∈Z 1  is supported in {1/4 ≤ |ξ | ≤ 4} with φ (ξ ) = 1 in {1/3 ≤ Let φ be such that φ |ξ | ≤ 3}. Let Jα (ρ) = ρ −

d−2 2 −α

J d−2 +α (ρ) 2

(41)

α−1 so that F[Jα (t| · |)](ξ ) = cα t −d (1 − |ξ |2 /t 2 )+ (see Chapter VII of [46]). In particular J0 = J as in (14). Let φk = 2kd φ(2k ·). Then (40) follows from  2 
 2  ds 1/2  ds      Jα (s|y|)fs (· − y)dy  p,2
 φk ∗ |fs |2  . (42)  L p s s 1 1 k∈Z

The reduction of (40) to (42) involves incorporating irrelevant powers of s in the definition of fs and an application of standard estimates for vector-valued singular α−1 away from the unit sphere. integrals ([39]) to handle the contribution of (1−|ξ |2 )+ We omit the details.  and ψ is a radial We now split φ = η ∗ ψ ∗ ψ where  η has the same support as φ ∞  C0 function supported in {x : |x| ≤ 1/10}, furthermore ψ vanishes to order 10d at the origin. If H1 = L2 ([1, 2], dr) then we wish to apply Theorem 4.2 with the H 1 valued kernel Kk ≡ K (independent of k) defined by  2  ∞ K(x), v = v(s) Jα (sr)ψ ∗ σr (x)dr ds . (43) 1

0

We define the corresponding short range kernel K by letting the r-integral in (43) lg extend over [0, 2+2 ] and the long range kernel K by letting the r-integral extend over (2+2 , ∞). Clearly the support condition (i) in Hypothesis 4.1 holds. Note that d/p −d/2 > 1/2 for p < 2d/(d + 1). Thus to check condition (ii) of Hypothesis 4.1 it suffices to verify that 2 1/2
 2   2+2   (ξ )2 Jα (rs)ψ σr (ξ )dr  ds ≤ Aα , α > 1/2 . sup  sh

ξ ∈Rd

1

0

(ξ ) = u(|ξ |), this reduces to Writing ψ 2 1/2
 2   2+2   sup |u(ρ)|2 Jα (rs)J (rρ)r d−1 dr  ds
Aα .  ρ>0

1

0

(44)

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RADIAL FOURIER MULTIPLIERS

We may take the r-integral over [1, 2+2 ] since the estimate for the contribution for r ∈ [0, 1] is immediate. We use the standard asymptotic expansions for the modified Bessel-function Jα , Jα (u) = u−

d−1 2 −α

1 

  + iu  − −iu + O(|u|−2 ) , u−n cn,α e + cn,α e

u ≥ 1,

(45)

n=0

and also the analogous expansion for J = J0 . If we consider only the leading terms in both asymptotic expansions we are led to bound   +2 2 1/2 |u(ρ)|2
2  2  eir(±s±ρ) r −α dr  ds
Aα , α > 1/2 , sup  d−1 2 ρ>0 ρ 1 1 which follows from Plancherel’s theorem on R. The other terms with lower order or nonoscillatory error terms are similar or more straightforward. Note that we also use |u(ρ)| ≤ ρ 10d for ρ ∈ (0, 1). This establishes condition (ii) in Hypothesis 4.1. Finally we verify condition (iii). Let {az }z∈Zd be L2 (H1 ) functions with supz az L2 (H1 ) ≤ 1, supported on 2 -cubes with disjoint interiors. We then need to show that  2      ψ ∗ ψ ∗ σr ∗ γ (z) Jα (sr)az (s, ·)ds  p,2  j ≥+2 Ij


A2(d( p − 2 )−ε) 1

1




L

1

z

|γ (z)|p

1/p .

(46)

z

Setting cj,z =


  Ij

Rz

  

2 1

2 dr 1/2  Jα (sr)az (s, x)ds  dx r

we may apply Proposition 2.6 for q = 2 with βj (z) = 2j d/p γ (z)cj,z ,

and

−1 bj,z (r, x) = χIj (r)cj,z



2

Jα (sr)az (s, x)ds

1

if cj,z  = 0, and bj,z = 0 if cj,z = 0. We can then dominate the left-hand side of (46) by a constant times

p/2 1/p d d 2( p − 2 −ε(p)) |βj (z)|2 z

j

with ε(p) > 0 for p < p1 . We are only left to show that for fixed z
1/2 |βj (z)|2
|γ (z)| j

where the implicit constant is uniform in z. This estimate follows from   2  2 2 dr   22j d/p Jα (sr)az (s, x)ds  |az (s, x)|2 ds
 r 1 1 I j j ≥+2

(47)

294

LEE, ROGERS, AND SEEGER

and integration over x ∈ Rz . To see (47) we use again the asymptotics (45). The estimate for the oscillatory terms (with n = 0, 1) becomes

 2j d/p

r

2

 

Ij

j ≥+2

2

−2α−2n−d 

e

±isr − d−1 2 −α−n

s

 2  az (s, x)ds  dr


1

2

|az (s, x)|2 ds,

1

and since α = d/p − d/2 it suffices to show    

2

 2  e±isr v(s)az (s, x)dt  dr


1

2

|az (s, x)|2 ds

1

with sups |v(s)| ≤ C. But this is an immediate consequence of Plancherel’s d−1 theorem. Lastly, if in (47) we put the error term O((sr)−α− 2 −2 ) for Jα (sr), the resulting expression can be easily estimated by 

∞

r 2

−3



2

dr

|az (s, x)|ds

2




1

2

|az (s, x)|2 ds .

1

This concludes the proof of (47), and thus the proof of Theorem 1.1. Proof of Theorem 1.4. We apply Theorem 4.2 with H1 = H2 = C. It is easy to see that it suffices to show that, for α = d(1/p − 1/2),  

  2 f p , F −1 mk (2−k | · |) η(2−k ·)f  p,q
sup mk Bα,q  L

k∈Z

k

2 (R) supported in (1/2, 2) and η is a radial where mk are functions in Bα,q Schwartz function with  η supported in the annulus {1/4 < |ξ | < 4}. Now write F −1 [mk (| · |)](x) = κk (|x|). Using polar coordinates and (34) we see that
 dr q/2 1/q 2 , |κk (r)|2 r 2d/p
mk Bα,q α = d(1/p − 1/2) , (48) r Ij j ≥1

and of course sup0 0 for p < p1 . Finally, by (48) p/q 1/p
1/p

2 |βk,j (z)|q
|γ (z)|p mk Bd(1/p−1/2),q , z

z

j

which completes the proof.

APPENDIX A. PROOF OF THEOREM 4.2 By normalization we may assume that Hypothesis Hyp(p, q, ε, A) holds with A = 1. We use atomic decompositions in Lp which are constructed from square functions, based on the ideas by Chang and Fefferman [8]. A convenient and useful form is given by an 2 -valued version of Peetre’s maximal square function (cf. [28], [55]),
1/2 Sf (x) = sup |Lk f (x + y)|2H1 , k

|y|≤100d·2−k

where Lk f = φk ∗ f , with φk = 2kd φ(2k ·), and φ is a radial Schwartz function  supported in {ξ : 1/5 < |ξ | < 5}. Then with φ Sf p ≤ Cp f Lp (H1 ) ,

1 < p < ∞.

We closely follow the argument in [20]. Choose φ by splitting the function η in the statement of Theorem 4.2 as η =ψ ∗φ

296

LEE, ROGERS, AND SEEGER

where ψ is a radial C0∞ -function with support in {x : |x| < 1/4} whose Fourier transform vanishes to order 10d at the origin. We set ψk = 2kd ψ(2k ·), then ηk = ψk ∗ φk and we have η k Tk f = ψk Tk Lk f . k

k

For k ∈ Z, we tile R by the dyadic cubes of sidelength 2−k and write L(Q) = −k if the sidelength of a dyadic cube Q is 2−k . For each n ∈ Z, let d

n = {x : Sf (x) > 2n }. Let Qn−k be the set of all dyadic cubes of sidelength 2−k which have the property that |Q ∩ n | ≥ |Q|/2 but |Q ∩ n+1 | < |Q|/2. Let ∗n = {x : Mχn (x) > 100−d } with M the Hardy–Littlewood maximal operator. The set ∗n is open, contains n , and satisfies |∗n |
|n |. Let Wn be the set of all dyadic cubes W for which the 50-fold dilate of W is contained in ∗n and W is maximal with respect to this property. The collection {W } forms a Whitney-type decomposition of ∗n . The interiors of the Whitney cubes are disjoint. For each W ∈ Wn we denote by W ∗ the tenfold dilate of W ; the dilates {W ∗ : W ∈ Wn } have still bounded overlap. Note that each Q ∈ Qn−k is contained in a unique W ∈ Wn . For W ∈ Wn , set (Lk f )χQ , ak,W,n = Q∈Qn−k Q⊂W

and for any dyadic cube W define ak,W =



ak,W,n .

n:W ∈Wn

The functions ak,W,n can be considered as “atoms,” but without the usual normalization. For fixed n one has ak,W,n 2L2 (H1 )
22n meas(n ). (49) W ∈Wn

k

Indeed (arguing as in [8]) the left-hand side is equal to  |Lk f (x)|2H1 dx Q∈Wn

≤2

k

Q∈Qn−k



Q∈Wn



≤2

Q

k

n \n+1

Q∈Qn−k

sup√ |Lk f (x + y)|2H1 dx

Q∩(n \n+1 ) |y|≤2−k d

Sf (x)2 ≤ 2 meas(n )22(n+1) .

sh , be the convolution operator with kernels 2kd K (2k ·) and Let Tk, , Tk, k,sh 2kd K (2k ·), respectively. The desired estimate will follow once we establish lg

k,lg

297

RADIAL FOURIER MULTIPLIERS

the short range inequality    k

≥0

  sh ψk ∗ Tk, ak,W 



Lp (H2 )

W ∈∪n Wn L(W )=−k+


Sf p

(50)

and for fixed  ≥ 0 the long range inequality    k

  lg ψk ∗ Tk, ak,W 



Lp,q (H2 )

W ∈∪n Wn L(W )=−k+


2−ε Sf p .

(51)

Proof of (50). We prove that for 1 < r < 2 and for fixed n ∈ Z    k

≥0

W ∈Wn L(W )=−k+

r  sh ψk ∗ Tk, ak,W,n  r

L (H2 )

≤ Cr 2nr meas(n ).

(52)

By “real interpolation” (cf. Lemma 2.2 in [19]) it follows that the stronger estimate    n

k

≥0

W ∈Wn L(W )=−k+

p  sh ψk ∗ Tk, ak,W,n  p

L (H2 )






2np meas(n )

n

 p holds and this implies (50) since n 2np meas(n )
Sf p . Since the expression inside the norm in (52) is supported in ∗n we see that the left-hand side of (52) is dominated by   meas(∗n )1−r/2  k

≥0

W ∈Wn L(W )=−k+

r  sh ψk ∗ Tk, ak,W,n  2

L (H2 )

.

(53)

The convolution operators with kernel ψk are almost orthogonal and thus we can dominate the left-hand side of (53) by a constant times meas(∗n )1−r/2


   k

≥0

W ∈Wn L(W )=−k+

2  sh Tk, ak,W,n  2

r/2

L (H2 )

.

(54)

sh Now, for each W with L(W ) = −k + , the function Tk, ak,W,n is supported in ∗ ∗ the expanded cube W . The cubes W with W ∈ j have bounded overlap, and therefore the expression (54) is


meas(∗n )1−r/2


k

≥0

W ∈Wn L(W )=−k+

 2  sh  Tk, ak,W,n  2

L (H2 )

r/2 .

(55)

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LEE, ROGERS, AND SEEGER

Now we have for fixed W   sh T ak,W,n  k,

By (49) we have k

≥0

L2 (H2 )


ak,W,n L2 (H1 ) .

  2 ak,W,n 2
ak,W,n 2
22n meas(n ). 2

W ∈Wn L(W )=−k+

W ∈Wn

k

Since meas(∗n )
meas(n ) it follows that the right-hand side of (55) is dominated by a constant times meas(n )2nr which then yields (52) and finishes the proof of the short range estimate. Proof of (51). We use  the first estimate in Lemma 3.2, with βk = ψk , and the H2 lg valued functions Fk = Tk, ak,W . We then see that (51) follows from W: L(W )=−k+

   k

W: L(W )=−k+

p  lg Tk, ak,W  p,q L

(H2 )


2−εp



meas(n )2np .

(56)

n

By rescaling and assumption (iii) in Definition 4.1 we have for every k 
1/p  1 1   lg p ak,W  p,q
2−ε 2(−k)d( p − 2 ) ak,W L2 (H1 ) . Tk, L

W: L(W )=−k+

(H2 )

W: L(W )=−k+

Thus in order to finish the proof we need the inequality p p 2(−k)d(1− 2 ) ak,W L2 (H1 )
2np meas(n ) . k

W: L(W )=−k+

(57)

n

For fixed k and fixed W , the functions ak,W,n , n ∈ Z live on disjoint sets (since the dyadic cubes of sidelength 2−k are disjoint and each such cube is in exactly one family Qn−k ). Therefore
1/2 ak,W,n 2L2 (H1 ) ak,W L2 (H1 )
n

and thus we can bound the left-hand side of (57) by p p 2(−k)d(1− 2 ) ak,W,n L2 (H1 ) n

k∈Z

≤

W ∈Wn : L(W )=−k+


n

≤

n

k∈Z



meas(W )

1−p/2


W ∈Wn : L(W )=−k+

meas(∗n )1−p/2

k


k

W ∈Wn



ak,W,n 2L2 (H1 )

W ∈Wn : L(W )=−k+

ak,W,n 2L2 (H1 )

p/2

;

p/2

299

RADIAL FOURIER MULTIPLIERS

here we used the disjointness of Whitney cubes in Wn . By (49) the last displayed expression is bounded by meas(∗n )1−p/2 (22n meas(n ))p/2
2np meas(n ) C n

n

which gives (57).

REFERENCES [1] J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), no. 2, 147–187. [2] J. Bourgain, L. Guth, Bounds on oscillatory integral operators based on multilinear estimates, Geom. Funct. Anal. 21 (2011), no. 6, 1239–1295. [3] A. Carbery, The boundedness of the maximal Bochner–Riesz operator on L4 (R2 ), Duke Math. J. 50 (1983), 409–416. [4] ———, Radial Fourier multipliers and associated maximal functions, Recent progress in Fourier analysis (El Escorial, 1983), 49–56, North-Holland Math. Stud., 111, North-Holland, Amsterdam, 1985. [5] A. Carbery, G. Gasper, W. Trebels, Radial Fourier multipliers of Lp (R2 ), Proc. Nat. Acad. Sci. U.S.A. 81 (1984), no. 10, Phys. Sci., 3254–3255. [6] A. Carbery, A. Seeger, Weighted inequalities for Bochner–Riesz means in the plane, Q. J. Math. 51 (2000), no. 2, 155–167. [7] L. Carleson, P. Sjölin, Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44 (1972), 287–299. [8] S.Y.A. Chang, R. Fefferman, A continuous version of duality of H 1 and BMO on the bidisc, Annals of Math. 112 (1980), 179–201. [9] M. Christ, On almost everywhere convergence of Bochner–Riesz means in higher dimensions, Proc. Amer. Math. Soc. 95 (1985), 16–20. [10] ———, Weak type (1,1) bounds for rough operators, Ann. of Math. (2) 128 (1988), no. 1, 19–42. [11] ———, Weak type endpoint bounds for Bochner–Riesz multipliers, Rev. Mat. Iberoamericana 3 (1987), no. 1, 25–31. [12] M. Christ, C. D. Sogge, The weak type L1 convergence of eigenfunction expansions for pseudodifferential operators, Invent. Math. 94 (1988), no. 2, 421–453. [13] H. Dappa, W. Trebels, On maximal functions generated by Fourier multipliers, Ark. Mat. 23 (1985), no. 2, 241–259.

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[14] C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36. [15] ———, A note on spherical summation multipliers. Israel J. Math. 15 (1973), 44–52. [16] C. Fefferman, E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115. [17] ———, H p spaces of several variables, Acta Math. 129 (1972), no. 3–4, 137–193. [18] G. Garrigós, A. Seeger, Characterizations of Hankel multipliers, Math. Ann. 342 (2008), no.1, 31–68. [19] Y. Heo, F. Nazarov, A. Seeger, Radial Fourier multipliers in high dimensions, Acta Math. 206 (2011), no. 1, 55–92. [20] ———, On radial and conical Fourier multipliers, J. Geom. Anal. 21 (2011), no. 1, 96–117. [21] L. Hörmander, Estimates for translation invariant operators in Lp spaces, Acta Math. 104 (1960), 93–140. [22] S. Igari, S. Kuratsubo, A sufficient condition for Lp -multipliers, Pacific J. Math. 38 (1971), 85–88. [23] M. Kaneko, G. Sunouchi, On the Littlewood–Paley and Marcinkiewicz functions in higher dimensions, Tôhoku Math. J. (2) 37 (1985), no. 3, 343–365. [24] S. Lee, Improved bounds for Bochner–Riesz and maximal Bochner–Riesz operators, Duke Math. J. 122 (2004), 205–232. [25] S. Lee, K. M. Rogers, A. Seeger, Improved bounds for Stein’s square functions, Proc. London Math. Soc. 104 (2012), no. 6, 1198–1234. [26] S. Lee, A. Seeger, Lebesgue space estimates for a class of Fourier integral operators associated with wave propagation, Math. Nach., to appear. [27] G. Mockenhaupt, A. Seeger, C. D. Sogge, Local smoothing of Fourier integral operators and Carleson-Sjölin estimates, J. Amer. Math. Soc. 6 (1993), no. 1, 65–130. [28] J. Peetre, On spaces of Triebel-Lizorkin type, Ark. Mat. 13 (1975), 123–130. [29] M. Pramanik, K. M. Rogers, A. Seeger, A Calderón–Zygmund estimate with applications to generalized Radon transforms and Fourier integral operators, Studia Math. 202 (2011), no. 1, 1–15. [30] A. Seeger, On quasiradial Fourier multipliers and their maximal functions, J. Reine Angew. Math. 370 (1986), 61–73.

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[31] ———, Estimates near L1 for Fourier multipliers and maximal functions, Arch. Math. (Basel) 53 (1989), no. 2, 188–193. [32] ———, Remarks on singular convolution operators, Studia Math. 97 (1990), 91–114. [33] ———, Endpoint estimates for multiplier transformations on compact manifolds, Indiana Univ. Math. J. 40 (1991), 471–533. [34] ———, Endpoint inequalities for Bochner–Riesz multipliers in the plane, Pacific J. Math. 174 (1996), 543–553. [35] A. Seeger, T. Tao, Sharp Lorentz space estimates for rough operators. Math. Ann. 320 (2001), no. 2, 381–415. [36] A. Seeger, T. Tao, J. Wright, Endpoint mapping properties of spherical maximal operators, J. Inst. Math. Jussieu 2 (2003), no. 1, 109–144. [37] E. M. Stein, Localization and summability of multiple Fourier series, Acta Math. 100 (1958), 93–147. [38] ———, Topics in harmonic analysis related to the Littlewood–Paley theory. Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. [39] ———, Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. [40] ———, Lp boundedness of certain convolution operators, Bull. Amer. Math. Soc. 77 (1971), 404–405. [41] ———, Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 7, 2174–2175. [42] ———, Some problems in harmonic analysis, in Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 1, 3–20, Amer. Math. Soc., Providence, R.I. [43] ———, The development of square functions in the work of A. Zygmund, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 359–376. [44] ———, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S. Murphy. Princeton Mathematical Series, No. 43. Princeton University Press, Princeton, N.J., 1993. [45] E. M. Stein, S. Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1239–1295.

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[46] E. M. Stein, G. Weiss, Introduction to Fourier analysis in Euclidean spaces, Princeton University Press, 1971. [47] G. Sunouchi, On the Littlewood–Paley function g ∗ of multiple Fourier integrals and Hankel multiplier transformations, Tôhoku Math. J. (2) 19 (1967), 496–511. [48] ———, On the functions of Littlewood–Paley and Marcinkiewicz, Tôhoku Math. J. (2) 36 (1984), no. 4, 505–519. [49] T. Tao, Weak-type endpoint bounds for Riesz means, Proc. Amer. Math. Soc. 124 (1996), no. 9, 2797–2805. [50] ———, The weak-type endpoint Bochner–Riesz conjecture and related topics, Indiana Univ. Math. J. 47 (1998), 1097–1124. [51] ———, On the maximal Bochner-Riesz conjecture in the plane for p < 2, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1947–1959. [52] ———, A sharp bilinear restriction estimate for paraboloids, Geom. Funct. Anal. 13 (2003), 1359–1384. [53] T. Tao, J. Wright, Endpoint multiplier theorems of Marcinkiewicz type, Rev. Mat. Iberoamericana 17 (2001), no. 3, 521–558. [54] P. A. Tomas, Restriction theorems for the Fourier transform. Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 1, pp. 111–114, Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979. [55] H. Triebel, Theory of function spaces. Monographs in Mathematics, 78. Birkhäuser Verlag, Basel, 1983.

Chapter Thirteen Problems of Harmonic Analysis Related to Finite-Type Hypersurfaces in R3 , and Newton Polyhedra Detlef Müller 13.1 INTRODUCTION Let S be a smooth, finite type hypersurface in R3 with Riemannian surface measure dσ, and consider the compactly supported measure dµ := ρdσ on S, where 0 ≤ ρ ∈ C0∞ (S). The problems on which I shall essentially focus are the following ones:. A. Find, if possible, optimal uniform decay estimates for the Fourier transform of the surface carried measure dµ.
B. If we denote by At the averaging operator At f (x) := S f (x − ty)dµ(y), determine for which exponents p the associated maximal operator Mf (x) := sup |At f (x)| t>0 p

3

is bounded on L (R ). For instance, if S is a Euclidean sphere centered at the origin, then M is the spherical maximal operator studied first by Stein [61]. C. Determine the range of exponents p for which a Fourier restriction estimate 1/2  2 ˆ |f (x)| dµ(x) ≤ Cf Lp (R3 ) S

holds true. I shall explain how these questions can be answered in an (almost) complete way in terms of Newton polyhedra associated to the given surface S. All these results are based on joint work with Ikromov, and in parts also with Kempe [37–42]. Problem A is a classical question about estimates for oscillatory integrals, and there exists a huge body of results on it, in particular for convex hypersurfaces (some references, also to higher dimensional results, will follow). The other two problems had first been formulated by Stein: the study of maximal averages along hypersurfaces has been initiated in Stein’s work on the spherical maximal function [61], and also the idea of Fourier restriction goes back to him (cf. §13.5). Their great importance for the study of partial differential equations became clear through Strichartz’s article, [63], and in the PDE-literature their dual versions are often called Strichartz estimates. Of course, these and related problems, such as the question of optimal sub-level estimates and integrability indices, make perfect sense also in higher dimensions

304

DETLEF MÜLLER

(and possibly higher co-dimension), but only partial answers are known to most of these problems in this case. The reason for this is that strong information on the resolution of singularities is available for analytic functions of two variables, for instance, by means of Puiseux series expansions of roots, whereas the situation for multivariate functions of more than two variables is substantially more complex. Nevertheless, there has been a lot of progress on the question as to how to construct more elementary and “concrete” resolutions of singularities for real analytic multivariate functions, giving more detailed information than what Hironaka’s celebrated theorem [35] on the resolution of singularities would yield, for instance, in work by Bierstone and P. D. Milman [7], [8], Sussmann [64], Parusi´nski [53], [54], Greenblatt [28], [30], Collins, Greenleaf, and Pramanik [15], among others. These techniques have already led to a very good understanding of, for instance, the sublevel estimation problem for slices of the surfaces in direction to the Gaussian normal, and the related determination of critical integrability indices, in independent work by Greenblatt [28], [30], and also Collins, Greenleaf, and Pramanik [15], by rather different methods. Yet another approach has been developed by Denef, Nicaise, and Sargos [21] in order to estimate oscillatory integrals in higher dimensions. These encouraging results give rise to the hope that, eventually, substantial progress might also be possible in higher dimensions for the deeper problems A–C, but at present, only partial answers are available, for restricted classes of surfaces (see, e.g., [31], [33]). What makes these problems indeed a lot harder than the sub-level set problem is that it will surely require estimates of oscillatory integrals with phase functions depending also on small parameters, not just a fixed phase function. This leads to the fundamental problem of stability of the estimates in Problem A under small perturbation; I shall briefly come back to this later. Before returning to the two-dimensional case, let me address some results on the three problems A–C which have been obtained for particular classes of hypersurfaces also in higher dimensions. The case of convex hypersurface of finite line type has been studied quite intensively, for instance in work by Randol [57], Svensson [65], Schulz [59], Bruna, Nagel, and Wainger [12], and Cowling, Disney, Mauceri, and myself [17] concerning problem A, and for instance by Nagel, Seeger, and Wainger [52] and Iosevich, Sawyer, and Seeger [46] on problem B. Various particular classes of nonconvex hypersurfaces have been examined too, for instance in work by Iosevich and Sawyer [44], Cowling and Mauceri [18], [19], and by Sogge and Stein [60], where the method of damping (with powers of the Gaussian curvature) had been introduced in order to derive partial answers to problem B for very general hypersurfaces in Rn . However, from now on I shall concentrate on the two-dimensional case, and give only occasionally a few references to work in higher dimensions. Since all three problems can be seen as “classical” by now, there is an abundance of literature associated to them, so that it would seem impossible to give due credit to everyone who has made contributions, and I apologize in advance to everyone whose work is not mentioned, due to lack of space or my personal ignorance. Many further references can be found in the cited articles, and for a more detailed account of the

305

FINITE-TYPE HYPERSURFACES

state of the theory and its historical development until roughly a decade ago, I refer, for instance, to Stein’s monograph [62]. It is obvious that all three problems A–C can be localized to sufficiently small neighborhoods of given points x 0 on S. Observe also that the problems A and B are invariant under translations and rotations of the ambient space, so that we may replace the surface S by any suitable image under a Euclidean motion of R3 . We may thus assume that x 0 = (0, 0, 0), and that S is the graph S = {(x1 , x2 , φ(x1 , x2 )) : (x1 , x2 ) ∈ },

(1.1)

of a smooth function φ defined on a sufficiently small neighborhood  of the origin, such that φ(0, 0) = 0,

∇φ(0, 0) = 0.

(1.2)

The situation is different for problem B, since dilations do not commute with translations, so that we are only allowed to work with linear transformations of the ambient space. We shall therefore study the maximal operator M under the following transversality assumption on S. Assumption 1.1. The affine tangent plane x + Tx S to S through x does not pass / Tx S for every x ∈ S, through the origin in R3 for every x ∈ S. Equivalently, x ∈ so that 0 ∈ / S, and x is transversal to S for every point x ∈ S. Notice that this assumption allows us to find a linear change of coordinates in R3 so that in the new coordinates S can locally be represented as the shifted graph of a function φ as before, more precisely, S = {(x1 , x2 , 1 + φ(x1 , x2 )) : (x1 , x2 ) ∈ },

(1.3)

where φ satisfies again (1.2). Our transversality assumption is natural in this context. Indeed, various examples show that if it is not satisfied, then the behavior of the corresponding maximal function may change drastically. Observe also that if φ is flat, i.e., if all derivatives of φ vanish at the origin, and if ρ(x 0 ) > 0, then it is well-known and easy to see that the maximal operator M is Lp -bounded if and only if p = ∞, so that this case is of no interest. In a similar way, also the problems A and C will become of quite a different nature. We shall therefore assume that φ is non-flat, i.e., of finite type. Correspondingly, we shall always assume that the hypersurface S is of finite type, in the sense that every tangent plane has finite order of contact. Recall that in the study of a convex hypersurface, a standard assumption used by many authors is that the surface is of finite line type, which means that every tangent line has finite order of contact, which is stronger than our assumption. The chapter will be organized as follows: In the next paragraph, I shall briefly review some basic concepts concerning Newton polyhedra, adaptedness of coordinates and the notion of height, which have been introduced by Arnol’d (cf. [2], [3]) and his school, most notably Varchenko [66]. These concepts had originally been studied for real analytic functions φ, but

306

DETLEF MÜLLER

as shown in [39], can be extended to smooth, finite type functions φ. Problems A to C will then be discussed subsequently in §13.3−§13.5. Since many of our proofs are quite elaborate, my main goal will be to give a kind of guided tour through the basic structure of the proofs, hoping that this might also be helpful for everyone interested in studying some of the proofs in more detail.

13.2 NEWTON POLYHEDRA AND ADAPTED COORDINATES Let me first recall some basic notions from [39], [66]. If φ is given as before, consider the associated Taylor series φ(x1 , x2 ) ∼

∞ 

cα1 ,α2 x1α1 x2α2

α1 ,α2 =0

of φ centered at the origin. The set  T (φ) := (α1 , α2 ) ∈ N2 : cα1 ,α2 =



1 ∂ α1 ∂ α2 φ(0, 0)  = 0 α1 ! α2 ! 1 2

will be called the Taylor support of φ at (0, 0). We shall always assume that T (φ)  = ∅, i.e., that the function φ is of finite type at the origin. The Newton polyhedron N (φ) of φ at the origin is defined to be the convex hull of the union of all the quadrants (α1 , α2 ) + R2+ in R2 , with (α1 , α2 ) ∈ T (φ). The associated Newton diagram Nd (φ) in the sense of Varchenko [66] is the union of all compact faces of the Newton polyhedron; here, by a face, we shall mean an edge or a vertex. We shall use coordinates (t1 , t2 ) for points in the plane containing the Newton polyhedron, in order to distinguish this plane from the (x1 , x2 )-plane. The Newton distance, or shorter distance d = d(φ) between the Newton polyhedron and the origin in the sense of Varchenko, is given by the coordinate d of the point (d, d) at which the bi-sectrix t1 = t2 intersects the boundary of the Newton polyhedron. The principal face π(φ) of the Newton polyhedron of φ is the face of minimal dimension containing the point (d, d). Deviating from the notation in [66], we shall call the series  cα1 ,α2 x1α1 x2α2 φpr (x1 , x2 ) := (α1 ,α2 )∈π(φ)

the principal part of φ. In case that π(φ) is compact, φpr is a mixed homogeneous polynomial; otherwise, we shall consider φpr as a formal power series. Note that the distance between the Newton polyhedron and the origin depends on the chosen local coordinate system in which φ is expressed. By a local coordinate system at the origin I shall mean a smooth coordinate system defined near the origin which preserves 0. The height of the smooth function φ is defined by h(φ) := sup{dy },

307

FINITE-TYPE HYPERSURFACES

N (φ) 1/κ2 Nd (φ) π(φ) d(φ)

1/κ1

d(φ)

Figure 13.1 Newton polyhedron.

where the supremum is taken over all local coordinate systems y = (y1 , y2 ) at the origin, and where dy is the distance between the Newton polyhedron and the origin in the coordinates y. A given coordinate system x is said to be adapted to φ if h(φ) = dx . In [39] we proved that one can always find an adapted local coordinate system in two dimensions, thus generalizing the fundamental work by Varchenko [66] who worked in the setting of real-analytic functions φ (see also [56]). Recall also that if the principal face of the Newton polyhedron N (φ) is a compact edge, then it lies on a unique “principal line” L := {(t1 , t2 ) ∈ R2 : κ1 t1 + κ2 t2 = 1}, with κ1 , κ2 > 0. By permuting the coordinates x1 and x2 , if necessary, we shall always assume that κ1 ≤ κ2 . The weight κ = (κ1 , κ2 ) will be called the principal weight associated to φ. It induces dilations δr (x1 , x2 ) := (r κ1 x1 , r κ2 x2 ), r > 0, on R2 , so that the principal part φpr of φ is κ-homogeneous of degree one with respect to these dilations, i.e., φpr (δr (x1 , x2 )) = rφpr (x1 , x2 ) for every r > 0, and d=

1 1 = . κ 1 + κ2 |κ|

(2.1)

It can then easily be shown (cf. Proposition 2.2 in [39]) that φpr can be factorized as φpr (x1 , x2 ) = cx1ν1 x2ν2

M 

q

p

(x2 − λl x1 )nl ,

(2.2)

l=1

with M ≥ 1, distinct non-trivial “roots” λl ∈ C \ {0} of multiplicities nl ∈ N \ {0}, and trivial roots of multiplicities ν1 , ν2 ∈ N at the coordinate axes. Here, p and q have no common divisor, and κ2 /κ1 = p/q.

308

DETLEF MÜLLER

More generally, if κ = (κ1 , κ2 ) is any weight with 0 < κ1 ≤ κ2 such that the line Lκ := {(t1 , t2 ) ∈ R2 : κ1 t1 + κ2 t2 = 1} is a supporting line to the Newton polyhedron N (φ) of φ, then the κ-principal part of φ  cα1 ,α2 x1α1 x2α2 φκ (x1 , x2 ) := (α1 ,α2 )∈Lκ

is a non-trivial polynomial which is κ-homogeneous of degree 1 with respect to the dilations associated to this weight as before, and which can be factorized in a similar way as in (2.2). By definition, we then have φ(x1 , x2 ) = φκ (x1 , x2 ) + terms of higher κ-degree.

(2.3)

Adaptedness of a given coordinate system can be verified by means of the following criterion (see [39]): Denote by m(φpr ) := ord S 1 φpr the maximal order of vanishing of φpr along the unit circle S 1 centered at the origin. The homogeneous distance of a κ-homogeneous polynomial P (such as P = φpr ) is given by dh (P ) := 1/(κ1 + κ2 ) = 1/|κ|. Notice that (dh (P ), dh (P )) is just the point of intersection of the line given by κ1 t1 + κ2 t2 = 1 with the bi-sectrix t1 = t2 . The height of P can then be computed by means of the formula h(P ) = max{m(P ), dh (P )}.

(2.4)

In [39] (Corollary 4.3 and Corollary 2.3), we proved the following characterization of adaptedness of a given coordinate system: Proposition 2.1. The coordinates x are adapted to φ if and only if one of the following conditions is satisfied: (a) The principal face π(φ) of the Newton polyhedron is a compact edge, and m(φpr ) ≤ d(φ). (b) π(φ) is a vertex. (c) π(φ) is an unbounded edge. These conditions had already been introduced by Varchenko, who has shown that they are sufficient for adaptedness when φ is analytic. We also note that in case (a) we have h(φ) = h(φpr ) = dh (φpr ). Moreover, it / N; in can be shown that (a) applies whenever π(φ) is a compact edge and κ2 /κ1 ∈ this case we even have m(φpr ) < d(φ) (cf. [39], Corollary 2.3). 13.2.1 Construction of adapted coordinates In the case where the coordinates (x1 , x2 ) are not adapted to φ, the previous results show that the principal face π(φ) must be a compact edge, that m := κ2 /κ1 ∈ N, and that m(φpr ) > d(φ). One easily verifies that this implies that p = m, q = 1 in (2.2), and that there is at least one, non-trivial real root x2 = λl x1 of φpr of multiplicity nl = m(φpr ) > d(φ). Indeed, one can show that this root is unique.

309

FINITE-TYPE HYPERSURFACES

Putting b1 := λl , we shall denote the corresponding root x2 = b1 x1 of φpr as its principal root. Changing coordinates y1 := x1 ,

y2 := x2 − b1 x1m ,

we arrive at a “better” coordinate system y = (y1 , y2 ). Indeed, this change of coordinates will transform φpr into a function φ pr , where the principal face of φ pr will be a horizontal half-line at level t2 = m(φpr ), so that d(φpr ) > d(φ), and ˜ > d(φ), if φ˜ expresses φ is the coordinates y correspondingly one finds that d(φ) (cf. [39]). Somewhat oversimplifying, by iterating this procedure, we essentially arrive at Varchenko’s algorithm for the construction of an adapted coordinate system (cf. [39] for details). In conclusion, one can show that there exists a smooth real-valued function ψ (which we may choose as the so-called principal root jet of φ) of the form ψ(x1 ) = cx1m + O(x1m+1 )

(2.5)

with c  = 0, defined on a neighborhood of the origin such that an adapted coordinate system (y1 , y2 ) for φ is given locally near the origin by means of the (in general non-linear) shear y1 := x1 ,

y2 := x2 − ψ(x1 ).

(2.6)

In these adapted coordinates, φ is given by φ a (y) := φ(y1 , y2 + ψ(y1 )).

(2.7)

Example 2.2. φ(x1 , x2 ) := (x2 − x1m )n + x1 . Assume that > mn. Then the coordinates are not adapted. Indeed, φpr (x1 , x2 ) = (x2 − x1m )n , d(φ) = 1/(1/n + 1/(mn)) = mn/(m + 1) and m(φpr ) = n > d(φ). Adapted coordinates are given by y1 := x1 , y2 := x2 − x1m , in which φ is expressed by φ a (y) = y2n + y1 . Remark 2.3. An alternative proof of Varchenko’s theorem on the existence of adapted coordinates for analytic functions φ of two variables has been given by Phong, J. Sturm, and Stein in [56], by means of Pusieux series expansions of the roots of φ. Let us finally observe that when m = κ2 /κ1 = 1 in the first step of Varchenko’s algorithm, then a linear change of coordinates of the form y1 = x1 , y2 = x2 − b1 x1 ˜ Since all of our problems A–C are invariant will transform φ into a function φ. under such linear changes of coordinates, by replacing our original coordinates ˜ we may in the sequel always assume without loss (x1 , x2 ) by (y1 , y2 ) and φ by φ, of generality that either our coordinates (x1 , x2 ) are adapted, or they are not adapted and m = κ2 /κ1

is an integer

≥ 2.

(2.8)

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DETLEF MÜLLER

m

N (φa )

h(φ)

d(φ)

N (φ)

mn

Figure 13.2 φ(x1 , x2 ) := (x2 − x1m )n + x1

l

( > mn).

A linear, non-adapted coordinate system for which (2.8) holds true will be called linearly adapted to φ.

13.3 PROBLEM A: DECAY OF THE FOURIER TRANSFORM OF THE SURFACE CARRIED MEASURE dµ Observe first that in view of (1.1), we may write

µ(ξ ) as an oscillatory integral  e−i(ξ3 φ(x1 ,x2 )+ξ1 x1 +ξ2 x2 ) η(x) dx, ξ ∈ R3 ,

µ(ξ ) =: J (ξ ) = 

where η ∈ C0∞ (). Since ∇φ(0, 0) = 0, the complete phase in this oscillatory integral will have no critical point on the support of η unless |ξ1 | + |ξ2 | |ξ3 |, provided  is chosen sufficiently small. Integrations by parts then show that

µ(ξ ) = O(|ξ |−N ) as |ξ | → ∞, for every N ∈ N, unless |ξ1 | + |ξ2 | |ξ3 |. We may thus focus on the latter case. In this case, by writing λ = −ξ3 and ξj = sj λ, j = 1, 2, we are reduced to estimating two-dimensional oscillatory integrals of the form  I (λ; s) := eiλ(φ(x1 ,x2 )+s1 x1 +s2 x2 ) η(x1 , x2 ) dx1 dx2 , where we may assume without loss of generality that λ 1, and that s = (s1 , s2 ) ∈ R2 is sufficiently small, provided that η is supported in a sufficiently small neighborhood of the origin. The complete phase function is thus a small, linear perturbation of the function φ. If s
= 0, then the function I (λ; 0) is given by an oscillatory integral of the form eiλφ(x) η(x) dx, and it is well-known ([4], [6]) that for any analytic phase function φ defined on a neighborhood of the origin in Rn such that φ(0) = 0, such

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FINITE-TYPE HYPERSURFACES

an integral admits an asymptotic expansion as λ → ∞ of the form ∞  n−1 

aj,k (φ)λ−rk log(λ)j ,

(3.1)

k=0 j =0

provided the support of η is sufficiently small. Here, the rk form an increasing sequence of rational numbers consisting of a finite number of arithmetic progressions, which depends only on the zero set of φ, and the aj,k are distributions with respect to the cut-off function η. The proof is based on Hironaka’s theorem. Let us come back to the case n = 2. Following [66] (with a slight modification), we next define what we like to call Varchenko’s exponent ν(φ) ∈ {0, 1} : If there exists an adapted local coordinate system y near the origin such that the principal face π(φ a ) of φ, when expressed by the function φ a in the new coordinates, is a vertex, and if h(φ) ≥ 2, then we put ν(φ) := 1; otherwise, we put ν(φ) := 0. We remark [40] that the first condition is equivalent to the following one: If y is any adapted local coordinate system at the origin, then either π(φ a ) is a ) = d(φ a ). a vertex, or a compact edge and m(φpr Varchenko [66] has shown that the leading exponent in (3.1) is given by r0 = 1/ h(φ), and ν(φ) is the maximal j for which aj,k (φ)  = 0. This implies in particular that |I (λ; 0)| ≤ Cλ− h(φ) log(λ)ν(φ) , 1

λ 1,

(3.2)

and this estimate is sharp in the exponents. Subsequently, Karpushkin [47] proved that this estimate is stable under sufficiently small analytic perturbations of φ (analogous results are known to be wrong in higher dimensions [66]). In particular, we find that J (λ; s) satisfies the same estimate (3.2) for |s| sufficiently small, so that we obtain the following uniform estimate for µ, ˆ |

µ(ξ )| ≤ C(1 + |ξ |)− h(φ) log(2 + |ξ |)ν(φ) , 1

ξ ∈ R3 ,

(3.3)

provided the support of ρ is sufficiently small. In [40], we proved, by a quite different method, that Karpushkin’s result remains valid for smooth, finite type functions φ, at least for linear perturbations, which led to the following: Theorem 3.1. Let S = graph(φ) be as before, and assume that φ is smooth and of finite type. Then estimate (3.3) holds true provided the support of ρ is sufficiently small. The special case where ξ = (0, 0, ξ3 ) is normal to S at the origin is due to Greenblatt [29]. One can also show that this estimate is sharp in the exponents even when φ is not analytic, except for the case where the principal face π(φ a ) is an unbounded edge. Indeed, if π(φ a ) is compact, then [40] I (λ; 0) Cλ− h(φ) log(λ)ν(φ) 1

as λ → +∞, where C is a non-zero constant. However, when π(φ a ) is unbounded then the following examples, due to Iosevich and Sawyer [45], show that there may

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DETLEF MÜLLER

be a different behavior in general: Let φ(x1 , x2 ) := x22 + e−1/|x1 | , with α > 0; then α

|I (λ; 0)|

1 as λ → +∞, λ1/2 log λ1/α

whereas ν(φ) = 0. It appears likely that even the full analog of Karpushkin’s theorem, i.e., stability of estimate (3.2) under arbitrary small, smooth perturbations, holds true in two dimensions. 13.3.1 Outline of some main ideas of the proof Our proof of Theorem 3.1 is based on a certain decomposition scheme for the given surface S, related to the Newton polyhedra of φ, respectively φ a , which has been inspired by the work of Phong and Stein, in combination with re-scaling arguments. Since the same type of decompositions plays an important role also in the more involved proofs of our results on the other problems B and C (which require also further, more subtle refinements of them), I shall outline some of the major ideas used in the proof subsequently. We first notice that the case where h(φ) < 2 is covered by Duistermaat’s work [24] (notice that Duistermaat proves estimates of the form (3.3) without the presence of a logarithmic factor log(2 + |ξ |), even for a wider class of phase functions). Note that the required estimates can also be derived easily from the normal forms in Theorem 5.8. I shall therefore subsequently assume that h := h(φ) ≥ 2. In many situations, one can reduce the problem to a one-dimensional one by means of van der Corput’s lemma, respectively the following (not quite straightforward) consequence of it, whose formulation goes back to J. E. Björk (see [22]) and G. I. Arhipov [1]. Lemma 3.2. Assume that f is a smooth real valued function defined on an interval I ⊂ R which is of polynomial type n ≥ 2 (n ∈ N), i.e., there are positive constants c1 , c2 > 0 such that c1 ≤

n 

|f (j ) (s)| ≤ c2

for every s ∈ I.

j =2

Then for λ ∈ R,

     eiλf (s) g(s) ds  ≤ CgC 1 (I ) (1 + |λ|)−1/n ,   I

where the constant C depends only on the constants c1 and c2 . 13.3.1.1 The case where the coordinates are adapted to φ Let us write d = d(φ), and recall that here h = d, since the coordinates are adapted. Assume for instance that the principal face π(φ) is a compact edge.

313

FINITE-TYPE HYPERSURFACES

By decomposing R2 into the half-spaces R2± := R × R± , we may also assume that the integration in J (ξ ) takes place over one of these half-spaces only, say, R2+ . Let κ be the principal weight, with associated dilations δr (x1 , x2 ) = (r κ1 x1 , r κ2 x2 ). Recall also that then φpr = φκ is δr -homogeneous of degree 1. We fix a suitable smooth cut-off function χ on R2 supported in an annulus A on which |x| ∼ 1, such that the functions χk := χ ◦ δ2k form a partition of unity, and then decompose J (ξ ) =

∞ 

Jk (ξ ),

k=k0

where

 Jk (ξ ) :=

R2+

e−i(ξ3 φ(x)+ξ1 x1 +ξ2 x2 ) η(x)χk (x) dx.

Scaling by δ2−k , we see that  −k k −kκ1 −kκ2 −k|κ| Jk (ξ ) = 2 e−i(2 ξ3 φ (x)+2 ξ1 x1 +2 ξ2 x2 ) η(δ2−k (x))χ (x) dx,

(3.4)

R2+

with φ k (x) := 2k φ(δ2−k x). Notice that in view of (2.3), φ k (x) = φκ (x) + error term. We claim that given any point x 0 ∈ A, we can find a unit vector e ∈ R2 and j some j ∈ N with 2 ≤ j ≤ h such that ∂e φκ (x 0 )  = 0, where ∂e denotes the partial derivative in direction of e. Indeed, if ∇φκ (x 0 )  = 0, then the homogeneity of φκ and Euler’s homogeneity relation imply that rank (D 2 φκ (x 0 )) ≥ 1. Therefore, we can find a unit vector e ∈ R2 such that ∂e2 φκ (x 0 )  = 0. And, if ∇φκ (x 0 ) = 0, then by Euler’s homogeneity relation we have φκ (x 0 ) = 0 as well. Thus the function φκ vanishes in x 0 of order j ≥ 2. But, in view of Proposition 2.1, then j ≤ m(φpr ) ≤ d = h, which verifies the claim. For k ≥ k0 sufficiently large we can thus apply van der Corput’s lemma to the integration along lines parallel to the direction e in the integral defining Jk (ξ ) near the point x 0 . Applying Fubini’s theorem and a partition of unity argument, we thus obtain |Jk (ξ )| ≤ CηC 3 (R2 ) 2−k|κ| (1 + 2−k |ξ3 |)−1/j ≤ CηC 3 (R2 ) 2−k|κ| (1 + 2−k |ξ |)−1/M , where M denotes the maximal j that arises in this context. Summation in k then yields the following estimates:  if M|κ| > 1 , (1 + |ξ |)−1/M ,  |J (ξ )| ≤ CηC 3 (R2 ) (1 + |ξ |)−1/M log(2 + |ξ |), if M|κ| = 1 ,  if M|κ| < 1 . (1 + |ξ |)−|κ| ,

(3.5)

(3.6)

However, since we are assuming that π(φ) is a compact edge, we have that 1/|κ| = d(φ) = h, and moreover M ≤ h. This implies |κ|M ≤ 1. Since we

314

DETLEF MÜLLER

have seen that here ν(φ) = 1 if and only if M = m(φpr ) = h, i.e., if and only if M|κ| = 1, we obtain estimate (3.3). 13.3.1.2 The case where the coordinates are not adapted to φ Here, in a first step, we may reduce to a narrow neighborhood of the principal root. Indeed, away from the principal root of φpr , we can argue in the same way as before, since the multiplicity of any real root of φpr different from the principal root is bounded by d ≤ h. I.e., we can reduce to a narrow κ-homogeneous neighborhood of the curve x2 = b1 x1m , of the form |x2 − b1 x1m | ≤ εx1m ,

(3.7)

say by means of a function ρ1 (x) := χ0 ((x2 −b1 x1m )/(εx1m )), where χ0 is a suitable smooth bump function supported in the interval [−1, 1] and ε > 0 is sufficiently small. I.e., in place of J (ξ ), it suffices to estimate J ρ1 (ξ ), where we write  e−i(ξ3 φ(x1 ,x2 )+ξ1 x1 +ξ2 x2 ) η(x) χ (x) dx J χ (ξ ) := R2+

if χ is any integrable function. Second Step: Domain decomposition into “homogeneous” domains Dl and transition domains El . In order to study the contribution J ρ1 (ξ ) by the domain (3.7), we change to the adapted coordinates y and essentially re-write   y  a 2 ρ1 e−i(ξ3 φ (y1 ,y2 )+ξ1 y1 +ξ2 ψ(y1 )+ξ2 y2 ) η(y) ˜ χ˜ 0 (3.8) J (ξ ) = m dy, 2 εy R+ 1 where η˜ and χ˜ 0 have properties similar to η, respectively χ0 . Let us denote the vertices of the Newton polyhedron N (φ a ) by (Al , Bl ), l = 0, . . . , n, where we assume that they are ordered so that Al−1 < Al , l = 1, . . . , n, with associated compact edges given by the intervals γl := [(Al−1 , Bl−1 ), (Al , Bl )], l = 1, . . . , n. The unbounded horizontal edge with left endpoint (An , Bn ) will be denoted by γn+1 . To each of these edges γl , we associate the weight κ l = (κ1l , κ2l ), so that γl is contained in the line Ll := {(t1 , t2 ) ∈ R2 : κ1l t1 + κ2l t2 = 1}. For l = n + 1, we have κ1n+1 := 0, κ2n+1 = 1/Bn . We denote by al :=

κ2l , κ1l

l = 1, . . . , n

the reciprocal of the slope of the line Ll . For l = n+1, we formally set an+1 := ∞. If l ≤ n, the κ l -principal part φκal of φ a corresponding to the supporting line Ll is of the form  Nα  A φκal (y) = cl y1 l−1 y2Bl (3.9) y2 − clα y1al α

(cf. [38]).

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FINITE-TYPE HYPERSURFACES

1/κ22

γ1

N (φa )

(A0 , B0 ) (A1 , B1 ) γ2

(A2 , B2 )

γn

(An , Bn ) γn+1

1/κ21

Figure 13.3 Edges and weights.

Remark 3.3. When φ is analytic, then this expression is linked to the Puiseux series expansion of roots of φ as follows [55] (compare also [39]): We may then factorize  (y2 − r(y1 )), φ a (y1 , y2 ) = U (y1 , y2 )y1ν1 y2ν2 r

where the product is indexed by all non-trivial roots r = r(y1 ) of φ a (which may also be empty) and where U (0, 0)  = 0. Moreover, these roots can be expressed in a small neighborhood of 0 as Puiseux series α

al 1l

al

α ···α

α1 ···αp−1 ···lp

al

r(y1 ) = clα11 y1 1 + clα11l2α2 y1 1 2 + · · · + cl11···lp p y1 1

+ ··· ,

where α ···α

β

α ···α

cl11···lp p−1  = cl11···lp p−1 α ···α

γ

for

β = γ ,

α ···α

p−2 , al11···lp p−1 > al11···lp−1

α ···α

with strictly positive exponents al11···lp p−1 > 0 and non-zero complex coefficients α ···α cl11···lp p  = 0, and where we have kept enough terms to distinguish between all the non-identical roots of φ a . The leading exponents in these series are the numbers a1 < a2 < · · · < an . One can therefore group the roots into the clusters of roots [l], l = 1, . . . , n, where the l’th cluster [l] consists of all roots with leading exponent al . Correspondingly, we can decompose φ a (y1 , y2 ) = U (y1 , y2 )y1ν1 y2ν2

n  l=1

[l] (y1 , y2 ),

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DETLEF MÜLLER

where [l] (y1 , y2 ) :=



(y2 − r(y1 )).

r∈[l] l

l

Observe the following: If δsl (x1 , x2 ) = (s κ1 x1 , s κ2 x1 ), s > 0, denote the dilations associated to the weight κ l , and if r ∈ [l1 ] is a root in the cluster [l1 ], then one l easily checks that for y = (y1 , y2 ) in a bounded set we have δsl y2 = s κ2 y2 and l α a l r(δsl y1 ) = s al1 κ1 cl11 y1 1 (1 + O(s ε )) as s → 0, for some ε > 0. Consequently,  a κ l α al  if l1 < l ,  −s l1 1 cl11 y1 1 , l l l ε αl al κ 2 δs y2 − r(δs y1 ) = (1 + O(s )) s (y2 − cl y1 ), if l1 = l ,  l  s κ2 y2 , if l1 > l. This shows that the κ l -principal part of φ a is given by   ν1 + l l |[l1 ]|  1 1 φκal = Cl y1 y2 (y2 − clα1 y1al )Nl,α1 ,

(3.10)

α1

where Nl,α1 denotes the number of roots in the cluster [l] with leading term clα1 y1al . A look at the Newton polyhedron reveals that the exponents of y1 and y2 in (3.10) can be expressed in terms of the vertices (Aj , Bj ) of the Newton polyhedron:   ν1 + |[l1 ]|al1 = Al−1 , ν2 + |[l1 ]| = Bl . l1 l

Notice also that n   (y2 − clα1 y1al )Nl,α1 = ([l] )κ l . α1

l=1

Comparing this with (3.9), the close relation between the Newton polyhedron of φ a and the Puiseux series expansion of roots becomes evident, and accordingly we say that the edge γl := [(Al−1 , Bl−1 ), (Al , Bl )] is associated to the cluster of roots [l]. Next we choose the integer l0 ≥ 1 such that a1 < · · · < al0 −1 ≤ m < al0 < · · · < al < al+1 < · · · < an . Since the original coordinates x were assumed to be non-adapted, the vertex (Al0 −1 , Bl0 −1 ) will lie strictly above the bisectrix, i.e., Al0 −1 < Bl0 −1 . Let us consider the case where the principal face of the Newton polyhedron of φ a is a compact edge (the other cases require modified arguments). We choose λ > l0 so that the edge γλ = [(Aλ−1 , Bλ−1 ), (Aλ , Bλ )] is the principal face π(φ a ) of the Newton polyhedron of φ a (cf. Figure 3, where λ = 3). In the case where a ) = d(φ a ), it can easily be seen that by running Varchenko’s algorithm one m(φpr step further, we can pass to new adapted coordinates in which the principal face is a vertex. a ) < d(φ a ), so that ν(φ) = 0. We therefore may assume that m(φpr

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FINITE-TYPE HYPERSURFACES

We shall narrow down the domain (3.7) to a neighborhood Dλ of the principal root jet of the form |x2 − ψ(x1 )| ≤ Nλ x1aλ ,

(3.11) λ

where Nλ is a constant to be chosen later. This domain is κ -homogeneous in the adapted coordinates y. To this end, following an idea from [55], we decompose the difference set of the domains (3.7) and (3.11) into the domains Dl := {(x1 , x2 ) : εl x1al < |x2 − ψ(x1 )| ≤ Nl x1al },

l = l0 , . . . , λ − 1,

and the intermediate domains a

El := {(x1 , x2 ) : Nl+1 x1 l+1 < |x2 − ψ(x1 )| ≤ εl x1al },

l = l0 , . . . , λ − 1,

al Nl0 x1 0

as well as El0 −1 := {(x1 , x2 ) : < |x2 − ψ(x1 )| ≤ ε1 x1m }. Here, the εl > 0 are small and the Nl > 0 are large parameters to be chosen suitably. Observe that the domain Dla := {(y1 , y2 ) : εl y1al < |y2 | ≤ Nl y1al } corresponding to Dl is κ l -homogeneous in the adapted coordinates y given by (2.2), and contains the cluster of roots [l], if εl and Nl are chosen sufficiently small, respectively large, while the domain Ela corresponding to El can be viewed as a domain of transition between two different homogeneities. The contributions by the domain Dl to J ρ1 (ξ ) can again be estimated in a similar way as we did in the adapted case, by using dyadic decompositions and subsequent re-scalings by means of the dilations δrl associated to the weight κ l . More precisely, the corresponding k’th term will be given by the following analogue of (3.4)  −kκ l −kκ l −kκ l −k k −k|κ l | e−i(2 ξ3 φ (y)+ξ2 ψ(2 1 y1 )+2 1 ξ1 y1 +2 2 ξ2 y2 ) η(δ2l −k y)χ (y) dy, Jk (ξ ) = 2 R2+

where φ k (y) = φκal (y) + error term. Notice, however, that since 1 − mκ1l > κ2l − mκ1l > 0, the contribution of the non-linearity ψ to the complete phase of the corresponding oscillatory integrals may be large, compared to the term containing φ k , so that we are only allowed to apply van der Corput’s estimate to the integration with respect to the variable y2 if we want to reduce to one-dimensional oscillatory integrals! We therefore need a control on the multiplicities of roots of ∂22 φκal at points y 0 in the corresponding annulus A not lying on the y1 axis (note that the latter corresponds to the principal root jet in the coordinates y). Indeed, it can be shown (cf. Proposition 2.3 (b) in [38]) that these multiplicities are bounded by dh (φκal )−2, where dh (φκal ) denotes the homogeneous distance of φκal , and it is evident from the geometry of the Newton polyhedron of φ a that dh (φκal ) < d(φ a ) = h, so that for every point y 0 in A ∩ Dl there is some j ∈ {2, . . . , h − 1} such that j

∂2 φκal (y 0 )  = 0. As for the contributions by the domains El , here we perform a separate dyadic decomposition in both variables y1 and y2 , so that we geometrically decompose El

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into dyadic rectangles of size 2−j × 2−k , and then re-scale in both variables so that these rectangles become the standard cube, say, [1, 2] × [1, 2]. a that Now, the estimates of Section 11 in [38] show that the phase functions φj,k one obtains after these re-scalings satisfy the estimate a ∂22 φj,k (y 0 )  = 0

for every y 0 ∈ [1, 2] × [1, 2].

Since h ≥ 2, this clearly suffices to obtain the necessary order of decay of the Fourier transform of these dyadic pieces. Moreover, scaling back to the original dyadic rectangles, a careful analysis of the dependency of the corresponding estimates on the parameters j, k shows that it is indeed possible to sum theses estimates and obtain the same type of estimate for the contributions by the domains El as for the domains Dl , even without logarithmic factor. Note that, so far, we have always been able to reduce our estimations to van der Corput’s lemma, respectively, Lemma 3.2, i.e., to one-dimensional oscillatory integrals. 13.3.1.3 Third Step: Study of the contribution by the homogenous domain Dλ containing the principal root jet What remains to be estimated is the contribution of the domain (3.11) to J (ξ ). As we shall see, the study of this domain will in certain cases require the estimation of genuinely 2-dimensional oscillatory integrals, and a reduction to the onedimensional case is no longer possible. We recall that according to our convention a ) < d(φ a ) = h, m(φpr

(3.12)

so that ν(φ) = 0. In the adapted coordinates y, the domain (3.11) is given by |y2 | ≤ Nλ y1aλ , and we can cover it by a finite number of κ λ -homogeneous subdomains of the form |y2 − cy1aλ | ≤ ε0 y1aλ , where c ∈ [−Nλ , Nλ ], and where, for a given c, we may choose ε0 > 0 suitably small. Writing ψ(x1 ) = x1m ω(x1 ), with a smooth function ω satisfying ω(0)  = 0, we can thus reduce to estimating oscillatory integrals of the form   y − cy aλ  2 1 eiF (y,ξ ) ρ η(y) dy, (3.13) J c (ξ ) = ε0 y1aλ R2+ with a phase function F (y, ξ ) := ξ3 φ a (y) + ξ1 y1 + ξ2 y1m ω(y1 ) + ξ2 y2 depending on ξ ∈ R3 . Arguing in a similar way as in the case of adapted coordinates, and recalling that a = φκaλ , we may again perform a dyadic decomposition and re-scale by means φpr of the dilations δrλ , in order to write J c (ξ ) =

∞  k=k0

Jk (ξ ),

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FINITE-TYPE HYPERSURFACES

where −|κ λ |k



−k

Jk (ξ ) = 2

ei2

ξ3 Fk (y,s)

ρ

 y − cy aλ  2 1 η(δ2λ−k y) χ (y) dy, ε0 y1aλ

(3.14)

where a Fk (y, s) := φpr (y1 , y2 ) + s1 y1 + S2 y1m ω(2−κ1 k y1 ) + s2 y2 + error, λ

where s := (s1 , s2 , S2 ) is given by λ

s1 := 2(1−κ1 )k

ξ1 , ξ3

λ

s2 := 2(1−κ2 )k

ξ2 , ξ3

S2 := 2(κ2 −mκ1 )k s2 . λ

λ

Note that 2 ≤ m < aλ = κ2λ /κ1λ and k 1, so that |S2 | |s2 |, and that y1 ∼ 1 and

|y2 − cy1aλ |
ε0

for y in the support of the integrand of Jk (ξ ). Recall also that we are assuming that |ξ | ∼ |ξ3 |. One is thus led to the estimation of oscillatory integrals depending on certain parameters (here s1 , s2 , S2 ) which may have various relative sizes. The case where |S2 | ≥ M for some sufficiently large constant M 1 can easily be treated by means of van der Corput’s lemma applied to the y1 - integration, so let us assume that |S2 | < M. Then |s2 | 1, provided we have chosen k0 sufficiently large. We may also easily reduce to the case where j

j

a (1, c) = ∂2 φκaλ (1, c) = 0 for 1 ≤ j < h, ∂2 φpr

(3.15)

for otherwise an integration by parts in y2 (if j = 1) or a simple application of Lemma 3.2 yields a suitable estimate as before. The case where c > 0 can easily be reduced to the case c = 0 by performing another change of variables y2  → y2 + cy1aλ in the integral defining Jk (ξ ). Indeed, one can show that our assumption (3.15) implies that necessarily aλ = κ2λ /κ1λ ∈ N (cf. Corollary 3.2 (iii) in [38]), and one checks that the new coordinates are again adapted to φ. a a (1, 0)  = 0, for otherwise φpr So, let us assume that c = 0. Then necessarily φpr would have a root of multiplicity at least h at (1, 0), which would contradict (3.12). a (1, 0) = 1, we can then write Assuming without loss of generality that φpr (compare [38], Subsection 9.1) a (y1 , y2 ) = y2B Q(y1 , y2 ) + y1n , φpr

where Q is a κ λ -homogeneous polynomial such that Q(1, 0)  = 0, and where B ≥ h > 2. Recall that |S2 | < M, so that |s2 | 1. Moreover, the case where |s1 | ≥ N for some sufficiently large constant N 1 can easily be dealt with by means of an integration by parts in y1 , so let us assume that |s1 | < N. It will then suffice to show that, given any point (s10 , S20 ) ∈ [−M, M] × [−N, N ] and any point y10 ∼ 1, there exist a neighborhood U of (s10 , S20 ), a neighborhood V

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of y10 , and some σ > 1/ h, such that we have an estimate of the form 2−k|κ| (3.16) (1 + 2−k |ξ |)σ for every (s1 , S2 ) ∈ U, provided the function χ in the definition of Jk (ξ ) is supported in V and ε0 and k are chosen sufficiently small, respectively large. Summing over all k, this will clearly imply an estimate as in (3.3), even without logarithmic factor. To this end, first notice that for (s1 , S2 ) ∈ U and k sufficiently large, the function Fk (y, s) can be viewed as a small C ∞ -perturbation of the function |Jk (ξ )|


Fpr (y) := y2B Q(y1 , y2 ) + s10 y1 + S20 ω(0)y1m + y1n . Thus, if ∇Fpr (y10 , 0)  = 0, then we obtain (3.16), with σ = 1, simply by integration by parts. Assume therefore that (y10 , 0) is a critical point of Fpr . Then y10 is a critical point of the polynomial function g(y1 ) := s10 y1 + S20 ω(0)y1m + y1n , which comprises all terms of Fpr depending on the variable y1 only. Note that the exponents satisfy 1 < m < n, since n = 1/κ1λ > κ2λ /κ1λ > m. It is then easy to see that g  and g  cannot vanish simultaneously at the given point y10 , so that there are positive constants c1 , c2 > 0 and a compact neighborhood V of (y10 , 0) such that c1 ≤

3 

|g (j ) (y1 )| ≤ c2

for every y1 ∈ V .

j =2 j

This implies an analogous estimate for the partial derivatives ∂1 Fk (y1 , y2 , s) of Fk , uniformly for (s1 , S2 ) ∈ U and the range of y2 that we consider, provided we choose U and ε0 sufficiently small. Applying the van der Corput type estimate in Lemma 3.2, we thus obtain estimate (3.16), with σ = 1/3, so that we are done provided h > 3. Notice also that if g  (y10 )  = 0, then by the same type of argument we see that (3.16) holds true with σ = 1/2 > 1/ h. We may thus finally assume that 2 < h ≤ 3, and that g  (y10 ) = g  (y10 ) = 0. In this case we have κ2λ 1 = h ≤ 3 and > m ≥ 2, λ λ κ1 + κ2 κ1λ so that 1/κ2λ < 9/2. Note that B ≤ 1/κ2λ is a positive integer, and h ≤ B < 9/2, so that either B = 4 or B = 3. We translate the critical point (y10 , 0) of Fpr to the origin by considering the function 1 Fpr (z) := Fpr (y10 + z 1 , z 2 ) − g(y10 ) = z 2B Q(y10 + z 1 , z 2 ) + g (3) (y10 ) z 13 + . . . . 6 

1 < 2. It is easy to see that this function has height h := h(Fpr ) given by h= 1/3+1/B We can therefore again apply Duistermaat’s results in [24] to the corresponding part of the oscillatory integral Jk (ξ ) and obtain estimate (3.16), with σ = 1/ h > 1/ h. Note here that the estimates in [24] are stable under small perturbations.

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FINITE-TYPE HYPERSURFACES

13.4 PROBLEM B: LP - BOUNDEDNESS OF THE MAXIMAL OPERATOR M ASSOCIATED TO THE HYPERSURFACE S. We next turn to Problem B. As has already been mentioned in the introduction, the first, fundamental result on this problem is due to Stein [61], who proved that for n ≥ 3, the spherical maximal function is bounded on Lp (Rn ) for every p > n/(n − 1). The analogous estimate in dimension n = 2 turned to require even deeper methods and was later established by Bourgain [9]. These results became the starting point for intensive studies of various classes of maximal operators associated to subvarieties. Stein’s monograph [62] is an excellent reference to many of these developments. From these early works, the influence of geometric properties of S on the validity of Lp -estimates for the maximal operator M became evident. For instance, Greenleaf [34] proved that M is bounded on Lp (Rn ) if n ≥ 3 and p > n/(n − 1), provided S has everywhere non-vanishing Gaussian curvature. In contrast, the case where the Gaussian curvature vanishes at some point is still wide open, and complete answers are available at present only for finite type curves in the plane [43], and finite type hypersurfaces in R3 and p > 2 (cf. Theorem 4.1). Let me come back to our hypersurface S in R3 . Recall from the introduction that we assume that S satisfies the transversality condition of §13.1, so that, by localizing to a small neighborhood of a given point x 0 ∈ S and applying a suitable linear change of coordinates, we may assume that x 0 = 0 and that S is given as the graph S = graph(1 + φ), where φ is a smooth, finite type function defined on a neighborhood of the origin and satisfying φ(0, 0) = 0, ∇φ(0, 0) = 0. We then define the height of S at x 0 by h(x 0 , S) := h(φ). It is easily seen that this notion is invariant under affine linear changes of coordinates in the ambient space R3 . Recall also that Mf (x) := supt>0 |At f (x)|, where At denotes the averaging operator over the t-dilate of S given by  At f (x) := f (x − ty)ρ(y) dσ (y), t > 0. S

We can then state our main result from [38] (Theorems 1.2, 1.3), which gives an almost complete answer to the question of Lp -boundedness of the maximal operator M when p > 2 : Theorem 4.1. Assume the hypersurface S satisfies the transversality Assumption 1.1. (i) If the measure ρdσ is supported in a sufficiently small neighborhood of x 0 , then M is bounded on Lp (R3 ) whenever p > max{h(x 0 , S), 2}. (ii) If M is bounded on Lp (R3 ) for some p > 1, and if ρ(x 0 ) > 0, then p ≥ h(x 0 , S). Moreover, if S is analytic at x 0 , then p > h(x 0 , S).

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13.4.1 Related quantities: contact index and sublevel growth In [44], Iosevich and Sawyer had discovered a very interesting connection between the behavior of maximal functions M associated to hypersurfaces S (in Rn ) and an integrability index associated to the hypersurface. In order to describe this, assume that dH (x) := dist (H, x) denotes the distance from a point x on S to a given hyperplane H. In particular, if x 0 ∈ S, then dT ,x 0 (x) := dist (x 0 + Tx 0 S, x) will denote the distance from x ∈ S to the affine tangent plane to S at the point x 0 . The following result has been proved in [44] in arbitrary dimensions n ≥ 2 and without requiring Assumption 1.1. Theorem 4.2 (Iosevich-Sawyer). If the maximal operator M is bounded on Lp (Rn ), where p > 1, then  dH (x)−1/p ρ(x) dσ (x) < ∞ (4.1) S

for every affine hyperplane H in Rn which does not pass through the origin. Moreover, it was conjectured in [44] that for p > 2 the condition (4.1) is indeed necessary and sufficient for the boundedness of the maximal operator M on Lp , at least if for instance S is compact and ρ > 0. Remark 4.3. Notice that condition (4.1) is easily seen to be true for every affine hyperplane H which is nowhere tangential to S, so that it is in fact a condition on affine tangent hyperplanes to S only. Moreover, if Assumption 1.1 is satisfied, then there are no affine tangent hyperplanes which pass through the origin, so that in this case it is a condition on all affine tangent hyperplanes. Moreover, it is not very hard to prove (cf. [38]) that if S is a smooth hypersurface of finite type in R3 , then, for every p < h(x 0 , S),  dT ,x 0 (x)−1/p dσ (x) = ∞ for every p < h(x 0 , S) (4.2) S∩U

and every neighborhood U of x 0 . And, if S is analytic near x 0 , then (4.2) holds true also for p = h(x 0 , S). Notice that this result does not require Assumption 1.1. As an immediate consequence of Theorem 4.1, Theorem 4.2, and (4.2) we obtain Corollary 4.4. Assume that S ⊂ R3 is of finite type and satisfies Assumption 1.1, and let x 0 ∈ S be a fixed point. Moreover, let p > 2. Then, if S is analytic near x 0 , there exists a neighborhood U ⊂ S of the point 0 x such that for any ρ ∈ C0∞ (U ) with ρ(x 0 ) > 0 the associated maximal operator M is bounded on Lp (R3 ) if and only if condition (4.1) holds for every affine hyperplane H in R3 which does not pass through the origin. If S is only assumed to be smooth near x 0 , then the same conclusion holds true, with the possible exception of the exponent p = h(x 0 , S). This confirms the conjecture by Iosevich and Sawyer in our setting for analytic S, and for smooth, finite-type S with the possible exception of the exponent

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FINITE-TYPE HYPERSURFACES

p = h(x 0 , S). For the critical exponent p = h(x 0 , S), if S is not analytic near x 0 , examples show that unlike in the analytic case it may happen that M is bounded 0 on Lh(x ,S) (see, e.g., [45]), and the conjecture remains open for this value of p. In view of these results, it is natural to define the uniform contact index γu (x 0 , S) of the hypersurface S at the point x 0 ∈ S as the supremum over the set of all γ for which there exists an open neighborhood U of x 0 in S such that the estimate  dH (x)−γ dσ (x) < ∞ U ∩S

holds true for every affine hyperplane H in Rn . If we restrict ourselves in this definition to the affine tangent hyperplane H = x 0 + Tx 0 S at the point x 0 , we shall call the corresponding index the contact index γ (x 0 , S) of the hypersurface S at the point x 0 ∈ S. Here, we shall always assume that ρ(x 0 )  = 0. Note that if we change coordinates so that x 0 = 0 and S is the graph of φ near the origin, where φ satisfies (1.2), then the contact index is just the supremum over all γ such that there is some neighborhood U of the origin so that |φ|−γ ∈ L1 (U ), so that the contact index agrees with the critical integrability index of φ as defined for instance in [15]. A closely related quantity is the sublevel growth rate σ (φ), defined as the supremum over all σ > 0 such that there is some constant Cσ > 0 so that |{x ∈ U : |φ(x)| < ε}| ≤ Cσ εσ

for every ε > 0.

Indeed, it can easily be shown by means of Chebyshev’s inequality (cf. [15]) that σ (φ) = γ (x 0 , S),

if x 0 = 0 and S = graph(φ).

In analogy with Arnol’d’s notion of “singularity index” [2], let us finally introduce the uniform oscillation index βu (x 0 , S) of the hypersurface S at the point x 0 ∈ S as the supremum over the set of all β such that  (ξ )| ≤ Cβ (1 + |ξ |)−β , |ρdσ for all ρ supported in a sufficiently small neighborhood of x 0 . If we restrict directions ξ to the normal to S at x 0 , then the corresponding decay rate will be called the oscillation index β(x 0 , S) at x 0 . Combining our results with results from [56] (compare also [27]), we easily obtain the following result for smooth, finite type hypersurfaces S in R3 : Corollary 4.5. Let x 0 ∈ S ⊂ R3 be a given point so that ρ(x 0 ) > 0. Then βu (x 0 , S) = β(x 0 , S) = γu (x 0 , S) = γ (x 0 , S) = 1/ h(x 0 , S). Remarks 4.6. (a) In dimension n ≥ 3, the corresponding identities may fail to be true. Indeed, Varchenko had already observed in [66] (Example 3) that for graphs S of functions φ of three variables the oscillation index β(0, S) may differ from 1/ h(0, S). Moreover, it is easy to give examples of functions φ of three variables where the contact index γ = γ (0, S) is strictly smaller than the oscillation index β = β(0, S). For instance, this applies to φ(x1 , x2 , x3 ) := x32 − (x12 + x22 ), where the method of stationary phase shows that β = 3/2, whereas the factorization

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φ(x1 , x2 , x3 ) = (x3 − x12 + x22 )(x3 + x12 + x22 ) shows that γ = 1 (compare counterexample 8.1 in [15]). However, the proof of Theorem 1.6 in [31] shows that the oscillation index β can only be different from the contact index γ when 1/γ is an odd integer. To the best of my knowledge, it is not known whether the identities βu (x 0 , S) = β(x 0 , S) and γu (x 0 , S) = γ (x 0 , S) may persist in higher dimensions. Indeed, this question represents a special case of a question posed by Arnol’d, namely whether the oscillation index of a given function φ is semicontinuos in the following sense: Suppose φ(x, s) is a phase function which depends on x in a small neighborhood of the origin in Rn and some small parameters s ∈ Rk . Is there a small neighborhood of (0, 0) ∈ Rn × Rk so that the oscillation index of φ(x, s) at x 0 is greater or equal to that of φ(x, 0) at the origin for every (x 0 , s 0 ) in this neighborhood? That question had been answered to the negative in dimensions n ≥ 3 by Varchenko [66], and to the positive for analytic functions depending on two variables by Karpushkin [47]. For linear perturbations of a given function φ, our Theorem 3.1 shows that Arnol’d’s conjecture holds true even for smooth, finite type functions φ, and it seems open whether this kind of stability may still hold true even in higher dimensions, since all counterexamples to Arnol’d’s conjecture which have been found hitherto have non-linear perturbation terms. Further important papers dealing with this stability question for general perturbations are, e.g., [13], [56]. (b) Greenblatt [28], [30], and independently Collins, Greenleaf, and Paramanik [15], have devised (quite different) algorithms of resolution of singularities which in principle allow us to compute the contact index also in higher dimensions. (c) If p ≤ 2, then examples (see, e.g., [46]) show that neither the notion of height nor that of contact index will determine the range of exponents p for which the maximal operator M is Lp -bounded. Our work (in progress) on this case seems to indicate a certain conjecture how to express this range in terms of Newton polyhedra. Moreover, for certain surfaces this conjecture relates to fundamental open problems in Fourier analysis, such as the conjectured reverse square function estimate for the cone multiplier (see, e.g., [10], [49]).

13.4.2 A few hints on the proof of Theorem 4.1 Since the proof in [38] is quite involved, I shall here just try to indicate some of the main steps, grossly oversimplifying some of the arguments. In the first part of the proof, we basically follow the first two steps of the scheme that has been outlined in the preceding paragraph. Recall that in these steps, it had effectively been possible to reduce the problem to a one-dimensional one. The same applies basically to the estimation of the maximal operator M, because in the

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FINITE-TYPE HYPERSURFACES

first two steps, we have had a sufficiently good control on the multiplicity of roots by the height h. In these arguments, the following result plays an analogous role for problem B as the van der Corput type Lemma 3.2 played for problem A. Let U be an open neighborhood of the point x 0 ∈ R2 , and let φpr ∈ C ∞ (U, R) such that ∂2m φpr (x10 , x20 )  = 0,

(4.3)

where m ≥ 2. Let φ = φpr + φr , ∞

where φr ∈ C (U, R) is a sufficiently small perturbation. Denote by Sε the surface in R3 given by Sε := {(x1 , x2 , 1 + εφ(x1 , x2 )) : (x1 , x2 ) ∈ U }, with ε > 0, and consider the averaging operators  ε f (x − ty)ψ(y) dσ (y), At f (x) := Sε

where dσ denotes the surface measure and ψ ∈ C0∞ (Sε ) is a non-negative cut-off function. Define the associated maximal operator by Mε f (x) := sup |Aεt f (x)|. t>0

Proposition 4.7. Assume that φpr satisfies (4.3) and that the neighborhood U of the point x 0 is sufficiently small. Then there exist numbers M ∈ N, δ > 0, such that for every φr ∈ C ∞ (U, R) with φr C M < δ and every p > m there exists a positive constant Cp such that for ε > 0 sufficiently small the maximal operator Mε satisfies the following a priori estimate: Mε f p ≤ Cp ε−1/p f p ,

f ∈ S(R3 ) .

(4.4)

This result can be reduced to the one-dimensional case, i.e., the study of maximal functions associated to curves in the plane. Indeed, just consider the “fan” of all hyperplanes passing through the x1 -axis. This fan will fibre the given surface into a family of curves, and our dilations leave each of these planes invariant. The proof for the analogous result for plane curves then basically follows Iosevich’s approach in [43]. The case where m = 2 is indeed the most difficult one; the cases where m ≥ 3 can then easily be reduced to this case by means of dyadic decompositions and re-scalings. Note that if m = 2, then the corresponding maximal operator in the plane behaves in a similar way as the “circular” maximal function studied by Bourgain [9]. An alternative approach to Bourgain’s one had later been given by Mockenhaupt, Seeger, and Sogge [50], [51] based on suitable local smoothing estimates for classes of Fourier integral operators. This approach is stable under small perturbations of the given curve, which is exactly what is needed for our purposes. Indeed, in our applications of Proposition 4.7, we even need to replace φ(x1 , x2 ) by φ(x1 , x2 − ψε (x1 )), where the function ψε may blow up like O(ε−δ ) for some δ ∈ [0, 1], and it turns out that this is still admissible, i.e., the estimate (4.4) remains valid also in this case.

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Now, if the coordinates are adapted to φ (a similar argument will apply to the first step when thecoordinates are not adapted), then, in analogy with the decomposition J (ξ ) = ∞ k=k0 Jk (ξ ), we can dyadically decompose the surface S as in Subsection 13.3.1.1, and accordingly decompose At f (y, y3 ) =

∞ 

Akt f (y, y3 ),

k=k0

where one finds that Akt f (y, y3 ) = 2−k|κ|

 R2

f (y − tδ2−k (x), y3 − t (1 + 2−k φ k (x))) η(δ2−k x)χ (x) dx,

with φ k is as before. We denote by Mk the maximal operator associated to the averaging operators Akt . Observe next that the scaling operators T k , defined by T k f (y, y3 ) := 2 p

k|κ| p

f (δ2k (y), y3 ),

3

act isometrically on L (R ), and  −k k k −k|κ| (T At T )f (y, y3 ) = 2 f (y −tx, y3 −t (1+2−k φ k (x))) η(δ2−k (x))χ (x) dx, R2

so that we are reduced to estimating the maximal operator associated to this family of averaging operators. This, in turn, can be accomplished by means of Proposition 4.7, where we choose ε := 2−k . The resulting estimates can then be summed over k, and we arrive at the desired estimation for M in this case. Assume next that the coordinates x are not adapted to φ. Eventually, we then again arrive at the situation studied in the third step (compare Subsection 13.3.1.3), i.e., we need to estimate the contribution of the domain Dλ to the maximal operator M. Again, one can indeed reduce to a small subdomain of the form |x2 − ψ(x1 ) − cx1aλ | ≤ ε0 x1aλ ,

with x1 > 0,

and also assume that (3.15) holds true at the point (1, c) = (1, 0). Recall that we may then write a φpr (y1 , y2 ) = y2B Q(y1 , y2 ) + y1n ,

where Q is a κ λ -homogeneous polynomial such that Q(1, 0)  = 0, and B ≥ h > 2. Again, in this situation, a reduction to the one-dimensional case is no longer possible. It turns out that this is the most difficult case, which requires various further ideas. The heart of the matter are in fact rather precise estimations of certain classes of two-dimensional oscillatory integrals depending on small parameters (cf. §5 in [38]). Indeed, in order to understand the behavior of φ a as a function of y2 , for y1 fixed, we decompose φ a (y1 , y2 ) = φ a (y1 , 0) + θ (y1 , y2 ), and write the complete phase  for J (ξ ) in adapted coordinates (y1 , y2 ) in the form F (y, ξ ) = (ξ3 φ a (y1 , 0) + ξ1 y1 + ξ2 ψ(y1 )) + (ξ3 θ (y1 , y2 ) + ξ2 y2 ).

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Notice that φ a (y1 , 0) = y1n ρ(y1 ),

ψ(y1 ) = y1m1 ω(y1 ),

θκ λ (y1 , y2 ) = y2B Q(y1 , y2 ),

where θκ λ denotes the κ λ -principal part of θ, and where ρ and ω do not vanish at y1 = 0. Now, by means of a dyadic decomposition and re-scaling using the κ λ -dilations λ {δr }r>0 we would like to reduce our considerations as before to the domain where y1 ∼ 1. In this domain, |y2 | 1, so that θκ λ (y) ∼ y2B Q(y1 , 0). What leads to problems is that the “error term” θr := θ − θκ λ , which consists of terms of higher κ λ -degree than θκ λ , may nevertheless contain terms of lower y2 -degree lj < B of l n the form cj y2j y1 j , provided nj is sufficiently large (this corresponds to a “fine splitting of roots” of θ when φ is analytic). After scaling the k-th dyadic piece in our decomposition by δ2λ−k in order to achieve that y1 ∼ 1 and |y2 |
ε0 , such terms will have small coefficients compared to the one of y2B Q(y1 , y2 ), but for |y2 | very small they may nevertheless become dominant and have to be taken into account. In order to resolve this problem, we apply a further domain decomposition by means of a suitable stopping time argument, into homogeneous domains D  and transition domains E  , oriented, in some sense, at the level sets of ∂2 φ a , which in turn again are chopped up into dyadic, respectively bi-dyadic, pieces. After rescaling, the contributions of these pieces to the maximal operators can eventually be estimated by means of oscillatory integral techniques in two variables. More precisely, it turns out that what is needed are uniform estimates for various classes of oscillatory integrals of the form  eiλF (x,σ,δ) a(x, δ) dx, (λ > 0), J (λ, σ, δ) := R2

with a phase function F of the form F (x1 , x2 , σ, δ) := f1 (x1 , δ) + σf2 (x1 , x2 , δ), and an amplitude a defined for x in some open neighborhood of the origin in R2 with compact support in x. The functions f1 , f2 are assumed to be real-valued and depend, like the function a, smoothly on x and on small real parameters δ1 , . . . , δν , which form the vector δ := (δ1 , . . . , δν ) ∈ Rν . σ denotes another small real parameter. With a slight abuse of language we shall say that ψ is compactly supported in some open set U ⊂ R2 if there is a compact subset K ⊂ U such that supp ψ(·, δ) ⊂ K for every δ. To give an idea as to which type of oscillatory integrals we need to estimate, let us remark that the most difficult instance are “oscillatory integrals of degenerate Airy type,” for which we have the following result: Theorem 4.8. Assume that |∂1 f1 (0, 0)| + |∂12 f1 (0, 0)| + |∂13 f1 (0, 0)|  = 0

and

∂1 ∂2 f2 (0, 0, 0)  = 0,

and that there is some m ≥ 2 such that ∂2l f2 (0, 0, 0) = 0 for l = 1, . . . , m − 1

and

∂2m f2 (0, 0, 0)  = 0.

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Then there exist a neighborhood U ⊂ R2 of the origin and constants ε, ε  > 0 such that for any amplitude a which is compactly supported in U the following estimate C (4.5) |J (λ, σ, δ)| ≤ 1 +ε λ 2 |σ |(lm +cm ε) holds true uniformly for |σ | + |δ| < ε , where lm := m−3 and cm := 2 for m ≥ 6. lm := 2(2m−3)

1 6

and cm := 1 for m < 6, and

Observe that the order of decay O(λ−1/2−ε ) in (4.5) is just what we need in order to apply the usual method to control the maximal operator on L2 by means of Sobolev’s embedding theorem applied to the scaling variable t > 0. Remark 4.9. If the Fourier transform of the surface carried measure µ decays more slowly than O(|ξ |−1/2−ε ), then one cannot directly apply Sobolev’s embedding theorem in order to control the maximal function on L2 . This happens in many situations when the Gaussian curvature of the hypersurface S vanishes. One method to overcome this problem is to use suitable damping factors in the amplitude of the corresponding oscillatory integrals, for instance powers of the Gaussian curvature, and combine this with a suitable complex interpolation argument. This technique had been introduced in [60], and also been used in our first approach to our problems in [37]. However, the choice of a suitable damping factor can become quite a tricky task, and we believe that the techniques in [38], which avoid damping factors and rather rely on suitable decompositions of the given surface in combination with re-scaling arguments, are simpler and more straightforward. For an approach based on damping methods, we refer to [32], where most of those cases are treated by damping techniques which can be reduced to a onedimensional problem.

13.5 PROBLEM C: FOURIER RESTRICTION TO THE HYPERSURFACE S. We finally turn to the last problem, namely the determination of the range of exponents p for which an Lp -L2 Fourier restriction estimate 1/2  2 ˆ |f (x)| dµ(x) ≤ Cf Lp (R3 ) , f ∈ S(R3 ), (5.1) S

holds true. The idea of Fourier restriction goes back to Stein, and a first instance of this concept is the sharp Lp -Lq Fourier restriction estimate for the circle in the plane by Zygmund [67], who extended earlier work by Fefferman and Stein [25] (see also Hörmander [36], [14] for estimates on more general oscillatory integral operators). For subvarieties of higher dimension, the first fundamental result was obtained (in various steps) for Euclidean spheres S n−1 by Stein and Tomas, who eventually

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FINITE-TYPE HYPERSURFACES

proved that an Lp -L2 Fourier restriction estimate holds true for S n−1 , n ≥ 3, if and only if p ≥ 2(2/(n − 1) + 1) (cf. [62] for the history of this result). Even though I shall not pursue that problem here, let me briefly remind that a more general and even substantially deeper problem is to determine the exact range of exponents p and q for which an Lp -Lq restriction estimate  1/q |fˆ(x)|q dσ (x) ≤ Cf Lp (Rn ) S n−1

holds true for spheres. It is conjectured that this is the case if and only if p  > 2n/(n − 1) and p  ≥ q(2/(n − 1) + 1), and there has been a lot of very deep work on this problem by many mathematicians, including Bourgain, Wolff, Vargas, Vega, Katz, Tao, Keel, Lee, and most recently Bourgain and Guth [11], which has led to important progress, but the problem is still open in dimensions n ≥ 3 and represents one of the major challenges in Euclidean harmonic analysis, bearing various deep connections with other important open problems, such as the BochnerRiesz conjecture, the Kakeya conjecture, and Sogge’s local smoothing conjecture for solutions to the wave equation. I refer to Stein’s book [62] for more information on these topics and their history until 1993, and various related essays by T. Tao which can be found on his Web page. Coming back to the restriction estimate (5.1) for our hypersurface S in R3 , we begin with the case where there exists a linear coordinate system which is adapted to the function φ. For this case, a complete answer had been given in [40] (for analytic hypersurfaces, partial results had been obtained before by Magyar [48]) : Theorem 5.1. Assume that, after applying a suitable linear change of coordinates, the coordinates (x1 , x2 ) are adapted to φ. We then define the critical exponent pc by pc := 2h(φ) + 2, 

(5.2) 

where p denotes the exponent conjugate to p, i.e., 1/p + 1/p = 1. Then there exists a neighborhood U ⊂ S of the point x 0 such that for every nonnegative density ρ ∈ C0∞ (U ) the Fourier restriction estimate (5.1) holds true for every p such that 1 ≤ p ≤ pc .

(5.3)

Moreover, if ρ(x 0 )  = 0, then the condition (5.3) on p is also necessary for the validity of (5.1). In many cases, this result is an immediate consequence of Theorem 3.1 and Greenleaf’s classical restriction estimate in [34], which I shall state here for the special case of hypersurfaces only.  )|
|ξ |−1/ h . Then the restriction Theorem 5.2 (Greenleaf). Assume that |dµ(ξ estimate (5.1) holds true for every p ≥ 1 such that p ≥ 2(h + 1). Indeed, observe that a problem arises only when Varchenko’s exponent ν(φ) equals one in (3.3). In this case, a direct application of Greenleaf’s result yields only the range 1 ≤ p < pc .

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To capture also the endpoint p = pc , recall from Subsection 13.3.1.1 that we ∞ had effectively decomposed the measure µ into a dyadic sum µ = k=k0 µk , where µk = (χk ⊗ 1)µ, and where µˆ k (ξ ) = Jk (ξ ) is given by (3.4). Moreover, we had estimated Jk (ξ ) in (3.5), which in particular yields the estimate | µk (ξ )| ≤ C2−k|κ| (1 + 2−k |ξ3 |)−1/ h . The measures µk are supported in dyadic annuli of “radius” 2−k , which are images of an annulus of radius of size one under the dilations δ2−k associated to the principal weight κ, so that we cannot directly apply Greenleaf’s restriction estimate. However, it is important to notice that our estimates for µ k (ξ ) do not carry a logarithmic factor yet (that only arose for µ(ξ ˆ ) in certain cases through summation over the k), and a simple re-scaling argument can then be applied to derive the following uniform restriction estimate for the family of measures µk :  |fˆ(x)|2 dµk (x) ≤ C 2 f 2pc . (5.4) Fix a cut-off function χ˜ ∈ C0∞ (R2 ) supported in an annulus centered at the origin such that χ˜ = 1 on the support of χ , and define dyadic frequency decomposition operators k by    ˜ 2k x  ) fˆ(x  , x3 ), k f (x) := χ(δ

 f (x)|2 dµk (x), where we have written (x1 , x2 ) = x  . Then |fˆ(x)|2 dµk (x) = | k and setting p := pc , we see that (5.4) yields in fact  2   |fˆ(x)|2 dµk (x) ≤ C 2  k f p ,

for any k ≥ k0 . In combination with Minkowski’s inequality, this implies  1/2  1/2  1/2   |fˆ(x)|2 dµ(x) |fˆ(x)|2 dµk (x) ≤  = k f 2p  k≥k0



 = C 

k≥k0

 

|k f (x)|p dx

2/p

p/2 1/p   

k≥k0

 1/2         ≤C |k f (x)|2      k≥k0 

,

Lp (R3 )

since p < 2. Estimate (5.1) for p = pc follows thus by means of Littlewood-Paley theory. From now on, we shall therefore always make the following Assumption 5.3. There is no linear coordinate system which is adapted to φ. According to our discussion in Subsection 13.2.1, we may then also assume that the coordinates x are linearly adapted to φ, and that there are adapted coordinates

331

FINITE-TYPE HYPERSURFACES 1/κ2

∆(m)

N (φa ) hr (φ) + 1

π(φ)

d+1

m+1 L 1/κ1

Figure 13.4 r-height.

y of the form y1 = x1 , y2 = x2 − ψ(x1 ), where ψ(x1 ) = x1m ω(x1 ),

with ω(0)  = 0 and m ≥ 2.

(5.5)

a

φ (y) will again denote φ when expressed in these adapted coordinates, and we shall use the notions introduced for the study of the Newton polyhedron of φ a in Subsection 13.3.1.2. Consider the line parallel to the bi-sectrix (m) := {(t, t + m + 1) : t ∈ R}. For any edge γl ⊂ Ll := {(t1 , t2 ) ∈ R2 : κ1l t1 + κ2l t2 = 1} define hl by (m) ∩ Ll = {(hl − m, hl + 1)}, i.e., hl =

1 + mκ1l − κ2l , κ1l + κ2l

(5.6)

and define the restriction height, or short, r-height, of φ by   r max hl . h (φ) := max d, {l=1,...,n+1:al >m}

Remarks 5.4. (a) For L in place of Ll and κ in place of κ l , one has m = κ2 /κ1 and d = 1/(κ1 + κ2 ), so that one gets d in place of hl in (5.6). (b) Since m < al , we have hl < 1/(κ1l + κ2l ), hence hr (φ) < h(φ). It is easy to see by Remark 5.4 (a) that the r-height admits the following geometric interpretation: By following Varchenko’s algorithm (cf. Subsection 8.2 of [38]), one realizes that the principal line L of N (φ) is a supporting line also for the Newton

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DETLEF MÜLLER

polyhedron of φ a , which intersects N (φ a ) in a compact face, either in a single vertex, or a compact edge. I.e., the intersection contains at least one and at most two vertices of N (φ a ), and we choose (Al0 −1 , Bl0 −1 ) as the one with smallest second coordinate. Then l0 is the smallest index l such that γl has a slope smaller than the slope of L, i.e., al0 −1 ≤ m < al0 . We may thus consider the augmented Newton polyhedron N r (φ a ) of φ a , which is the convex hull of the union of N (φ a ) with the half-line L+ ⊂ L with right endpoint (Al0 −1 , Bl0 −1 ). Then hr (φ) + 1 is the second coordinate of the point at which the line (m) intersects the boundary of N r (φ a ). The main result from [41], [42] then reads as follows. Theorem 5.5. Let φ  = 0 be real analytic, and assume that there is no linear coordinate system adapted to φ. Then there exists a neighborhood U ⊂ S of x 0 = 0 such that for every non-negative density ρ ∈ C0∞ (U ), the Fourier restriction estimate (5.1) holds true for every p ≥ 1 such that p ≥ pc := 2hr (φ) + 2. Remarks 5.6. (a) An application of Greenleaf’s result would imply, at best, that the condition p  ≥ 2h(φ) + 2 is sufficient for (5.1) to hold, which is a strictly stronger condition than p  ≥ pc . (b) A. Seeger recently informed me that in a preprint, which regrettably had remained unpublished, Schulz [58] had already observed this kind of phenomenon for particular examples of surfaces of revolution. (c) It can be shown that the number m is well-defined, i.e., it does not depend on the chosen linearly adapted coordinate system x. Example 5.7. φ(x1 , x2 ) := (x2 − x1m )n ,

n, m ≥ 2.

The coordinates (x1 , x2 ) are not adapted. Adapted coordinates are y1 := x1 , y2 := x2 − x1m , in which φ is given by φ a (y1 , y2 ) = y2n . Here κ1 =

1 , mn

κ2 =

d := d(φ) = and

nm 1 = < n, κ 1 + κ2 m+1

 pc =

1 , n

2d + 2,

if n ≤ m + 1,

2n,

if n > m + 1 .

On the other hand, h := h(φ) = n, so that 2h + 2 = 2n + 2 > pc . An analogous theorem holds true even for smooth, finite type functions φ, under an additional Condition (R) which, roughly speaking, requires that whenever the Newton diagram suggests that a root with leading term given by the principal root jet ψ(x1 ) should have multiplicity B, then indeed such a root of multiplicity B does

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FINITE-TYPE HYPERSURFACES

exist (this is a condition on the behavior of flat terms). Condition (R) is always satisfied when φ is real-analytic. For example, Condition (R) would hold true for φg (x1 , x2 ) = (x2 − x12 − f (x1 ))2 , for every flat smooth function f (x1 ) (i.e., f (j ) (0) = 0 for every j ∈ N). On the other hand, (R) is not satisfied for φb (x1 , x2 ) := (x2 − x12 )2 + f (x1 ), unless f vanishes identically. I also like to mention that there is a more invariant description of the notion of r-height (cf. Proposition 1.9 in [41]), somewhat in the spirit of Varchenko’s definition of height, but I refrain from stating it here since this would require the introduction of further, somewhat technical notions. 13.5.1 Necessity of the condition p ≥ 2hr (φ) + 2 In order to better understand the meaning of the notion of r-height, let me present the proof of the necessity of the condition p  ≥ 2hr (φ) + 2 for the validity of the Fourier restriction estimate (5.1) when ρ(x 0 )  = 0. The proof will be based on a modified Knapp-type argument. Let γl be any edge of N (φ a ) with al > m, and choose the weight κ l such that γl lies on the line Ll given by κ1l t1 + κ2l t2 = 1. Consider the region l

l

Dεa := {y ∈ R2 : |y1 | ≤ εκ1 , |y2 | ≤ εκ2 },

ε > 0,

in adapted coordinates y. In the original coordinates x, it corresponds to l

l

Dε := {x ∈ R2 : |x1 | ≤ εκ1 , |x2 − ψ(x1 )| ≤ εκ2 }. Assume that ε is sufficiently small. Since   l l φ a (εκ1 y1 , εκ2 y2 ) = ε φκal (y1 , y2 ) + O(εδ ) for some δ > 0, we have that |φ a (y)| ≤ Cε for every y ∈ Dεa , i.e., |φ(x)| ≤ Cε

for every x ∈ Dε .

(5.7)

Moreover, for x ∈ Dε , l

l

l

|x2 | ≤ εκ2 + |ψ(x1 )|
εκ2 + εmκ1 . Since m ≤ al = κ2l /κ1l , we find that l

|x2 |
εmκ1 , l

so that we may assume that Dε is contained in the box where |x1 | ≤ εκ1 , |x2 | ≤ l εmκ1 . Choose fε such that       x3 x1 x2 . χ χ0 f ε (x1 , x2 , x3 ) = χ0 0 l l ε εκ1 εmκ1

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DETLEF MÜLLER

Then by (5.7) we see that f ε (x1 , x2 , φ(x1 , x2 )) ≥ 1 on Dε , hence, if ρ(0)  = 0, then 1/2  l l |f ε |2 ρdσ ≥ |Dε |1/2 = ε(κ1 +κ2 )/2 . S 

Since fε p  ε((1+m)κ1 +1)/p , we find that the restriction estimate can hold true only if l

p ≥ 2

(1 + m)κ1l + 1 = 2hl + 2, κ1l + κ2l

where we recall that hl = (1 + mκ1l − κ2l )/(κ1l + κ2l ). Notice that the argument still works if we replace the previous line Ll by the line L associated to the weight κ, and φκal by φκa . Since here mκ1 = κ2 , this leads to the condition p  ≥ 2d + 2, so that altogether necessarily   p  ≥ 2 max d, max hl + 2 = 2hr (φ) + 2. l:a l >m

13.5.2 Sufficiency of the condition p ≥ 2hr (φ) + 2: I. Some key steps in the proof when h lin (φ) ≥ 2 Let us assume that we are working in linearly adapted coordinates x, so that d := d(φ) = h lin := h lin (φ). In the preceding discussions of problems A and B, it had been natural to distinguish between the cases where h := h(φ) < 2 and where h ≥ 2, since in the latter case, in many situations a reduction to a one-dimensional situation had been possible by means of the van der Corput type Lemma 3.2. For similar reasons, in the discussion of Problem C it appears natural to distinguish between the cases where d < 2 and where d ≥ 2. In addition, when d ≥ 2, it turns out that the case where d ≥ 5 can be handled in a somewhat simplified way compared to the case where 2 ≤ d < 5. We shall therefore assume in this section that d ≥ 5. In a first step, we can again localize to the narrow κ-homogeneous subdomain (3.7) of the curve x2 = b1 x1m given by |x2 − cx1m | ≤ εx1m , by means of a cut-off function ρ1 . Indeed, the technique of proof that we used in the case of adapted coordinates can essentially be carried over to the domain complementary to (3.7) without major new ideas, since one can show that the Fourier transforms of the corresponding dyadic pieces µk of the measure µ satisfy estimates of the form | µk (ξ )| ≤ C2−k|κ| (1 + 2−k |ξ3 |)−1/d . Notice also that hr := hr (φ) ≥ d. Let us again assume for instance that the principal face of the Newton polyhedron of φ a is a compact edge. Using the same notation as in Section 13.3, we choose again λ > l0 so that the edge γλ = [(Aλ−1 , Bλ−1 ), (Aλ , Bλ )] is the principal face π(φ a ) of the Newton polyhedron of φ a .

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FINITE-TYPE HYPERSURFACES

In a second step, we again narrow down the domain (3.7) to the neighborhood Dpr := Dλ of the principal root jet given by (3.11), where |x2 − ψ(x1 )| ≤ Nλ x1aλ , again by decomposing the difference set of the domains (3.7) and (3.11) into the domains Dl := {(x1 , x2 ) : εl x1al < |x2 − ψ(x1 )| ≤ Nl x1al },

l = l0 , . . . , λ − 1,

and the intermediate domains a

El := {(x1 , x2 ) : Nl+1 x1 l+1 < |x2 − ψ(x1 )| ≤ εl x1al }, as well as El0 −1 := {(x1 , x2 ) :

al Nl0 x1 0

l = l0 , . . . , λ − 1,

< |x2 − ψ(x1 )| ≤ ε1 x1m }.

Contribution by the domains El . Let us denote by µEl the contribution of the transition domains El to the measure µ. We then decompose µEl bi-dyadically w.r. to the adapted coordinates y as µ El =



µj,k ,

j,k

so that µj,k is supported where y1 = x1 ∼ 2−j and y2 = x2 − ψ(x1 ) ∼ 2−k . Observe that this a “curved rectangle” in our original coordinates x. In a similar way as in the case of adapted coordinates, we would like to localize to these curved rectangles by means of Littlewood-Paley theory in order to reduce to uniform restriction estimates for the family of measure µj,k , i.e.,  |f |2 dµj,k ≤ Cf 2Lp (R3 ) , (5.8) S

for p ≤ pc . Clearly, because of the non-linearity ψ(x1 ), this is not possible by means of Littlewood-Paley techniques in the variables x1 and x2 , but it turns out that we can use the variables x1 and x3 to accomplish this. Indeed, one can show that φ a (y) = cl y1Al y2Bl (1 + small error)

on

Ela ,

which in turn implies that on the domains El , respectively Ela (recall that Ela represents El in the adapted coordinates y), the conditions y1 ∼ 2−j , y2 ∼ 2−k are equivalent to the conditions x1 ∼ 2−j

and

φ(x) ∼ 2−(Al j +Bl k)

(cf. Lemma 6.1 in [41]). Working in the coordinates y, after re-scaling of the measures µj,k to get normalized measures νj,k supported on a surface Sj,k where y1 ∼ 1 ∼ y2 , by means of the formula above one eventually finds that Sj,k is a small perturbation of the limiting surface S∞ := {(y1 , y1m ω(0), cy1Al y2Bl ) : y1 ∼ 1 ∼ y2 }.

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DETLEF MÜLLER

But |∂(cy1Al y2Bl )/∂y2 | ∼ 1, since Bl ≥ 1, which shows that S∞ , and hence also Sj,k , is a smooth hypersurface with one non-vanishing principal curvature (with respect to y1 ) of size ∼ 1. This implies that | νj,k (ξ )| ≤ C(1 + |ξ |)−1/2 , uniformly in j and k. Applying Greenleaf’s restriction theorem to these measures, and scaling these estimates back, we eventually arrive (in a not completely trivial way) at the estimates (5.8). It is important to observe here that Greenleaf’s results imply restriction theorems for p ≥ 2(1 + 2) = 6, which is sufficient for our purposes, since pc ≥ 2d + 2, where d ≥ 2. Contribution by the domains Dl . Let us next turn to the domains Dl . After dyadic decomposition of the domain Dl in the adapted coordinates y by means of the κ l -dilations and suitable re-scaling, the re-scaled measure νk corresponding to the measures µk turns out to be of the form  l l l νk , f  := f (y1 , 2(mκ1 −κ2 )k y2 + y1m ω(2−κ1 k y1 ), φ k (y)) η(y) ˜ dy, and by means of a finite partition of unity, we may assume that the amplitude η˜ is supported in a sufficiently thin set U (c0 ), on which y1 ∼ 1

and

|y2 − c0 y1al | ≤ εy1al .

This measure νk is supported in a variety Sk which in the limit as k → ∞ tends to the variety S∞ := {g∞ (y1 , y2 ) := (y1 , ω(0)y1m , φκal (y)) : (y1 , y2 ) ∈ U (c0 )}, since mκ1l − κ2l < al κ1l − κ2l = 0 and since φ k tends to φκal . Here, c0 is fixed with |c0 | ≤ Nl . Again, we have to prove uniform restriction estimates for the family of measures νk . Depending on c0 , different cases may arise. Case 1. ∂2 φκal (1, c0 )  = 0. Then we may use z 2 := φκal (y1 , y2 ) in place of y2 as a new coordinate for S∞ (which thus is a hypersurface), and since y1 ∼ 1 on U (c0 ), we find that S∞ , hence also Sk , is a hypersurface with one non-vanishing principal curvature. Then we may essentially argue as for the domains El . Case 2. ∂2 φκal (1, c0 ) = 0, but ∂1 φκal (1, c0 )  = 0. In this case, since φκal is a κ l -homogenous polynomial, by Euler’s homogeneity relation we have also φκal (1, c0 )  = 0. One can then show that one can fibre the variety S∞ into the family of curves γc (y1 ) := g∞ (y1 , cy1al ) = (y1 , ω(0)y1m , φκal (y1 , cy1al )), for c sufficiently close to c0 , and one finds that the curve γc0 (y1 ) = 1/κ l

(y1 , ω(0)y1m , b0 y1 1 ) has non-vanishing torsion, since b0  = 0. The same applies then to the curves γc , and for k sufficiently large, we obtain the analogous results for the varieties Sk .

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FINITE-TYPE HYPERSURFACES

This allows us to decompose the measure dνk as a direct integral of measures dc supported on curves γcl with non-vanishing torsion. We may thus apply Drury’s Fourier restriction theorem for curves with non-vanishing torsion (cf. Theorem 2 in [23] and [5], [20]) to the measures dc and obtain uniform estimates   12 ≤ Cp f Lp (R3 ) , |fˆ|2 dc when p  > 7 and 2 ≤ p  /6. Since we assume here that pc ≥ 2(d + 1) > 2(5+1) = 12, these estimates, after re-scaling to the measures µk , yield the desired restriction estimates for the contributions by the domains Dl . Notice that it is here that we need the condition d = h lin > 5. Case 3. ∂2 φκal (1, c0 ) = 0 and ∂1 φκal (1, c0 ) = 0. Then Euler’s homogeneity relation implies that also φκal (1, c0 ) = 0, so that φκal has a real root of multiplicity B ≥ 2 at (1, c0 ), and one finds that φκal (y1 , y2 ) = y2Bl (y2 − c0 y1al )B Q(y1 , y2 ),

(5.9)

where Q is a κ -homogenous smooth function such that Q(1, c0 )  = 0 and Q(1, 0)  = 0. One can also prove that B < d/2. l

We can then essentially follow the Stein-Tomas method for proving Lp -L2 restriction estimates. We localize to frequencies of size  > 1 by putting ξ   ν ν k (ξ ), k (ξ ) := χ1  where χ1 is a smooth bump function supported where |ξ | ∼ 1. We claim that the measures νk satisfy the following estimates, uniformly in k ≥ k0 , provided k0 is sufficiently large and ε sufficiently small:

Indeed,

−1/B  ; ν k ∞ ≤ C

(5.10)

νk ∞

(5.11)

≤ C

2−1/B

.

ξ   (mκ l −κ l )k −κ l k m  e−i[ξ1 y1 +ξ2 (2 1 2 y2 +y1 ω(2 1 y1 ))+ξ3 φk (y)] η(y) (ξ ) = χ ˜ dy, ν 1 k  which, in the limit as k → ∞, simplifies as ξ   m a  (ξ ) = χ e−i[ξ1 y1 +ξ2 ω(0)y1 +ξ3 φκ l (y)] η(y) ν ˜ dy. 1 ∞  Now, if |ξ3 | ≥ c|(ξ1 , ξ2 )|, then an application of van der Corput’s lemma to the −1/B  (cf. (5.9)), and if |ξ3 | |(ξ1 , ξ2 )|, integration in y2 yields |ν ∞ (ξ )|
|ξ3 |  we may apply van der Corput’s lemma to the y1 -integration and obtain |ν ∞ (ξ )|
−1/2 . Since B ≥ 2, and because van der Corput’s estimates are stable under |(ξ1 , ξ2 )| small perturbations, we thus obtain (5.10).  (x1 , x2 , x3 ) is given by In order to verify (5.11), observe that ν∞  3 (F −1 χ1 )((x1 − y1 ), (x2 − ω(0)y1m ), (x3 − φκal (y1 , y2 ))η(y) ˜ dy1 dy2 ,

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hence, by a change of coordinates,       z1 z1   (x1 , x2 , x3 )| ≤ |2 ρ(z 1 )ρ  x3 −φκal x1 − , y2 η1 x1 − , y2 dz 1 dy2 |, |ν∞   where ρ and η1 are suitable, non-negative Schwartz functions, and η1 localizes again to U (c0 ). However, since |∂2B φκal (y1 , y2 ))|  1 on the domain of integration, classical sublevel estimates, originating in work by van der Corput [16] (see also [1], and [13],[26]), essentially imply that the integral with respect to y2 can be estimated by O(−1/B ), uniformly in y1 and . Interpolating the estimates (5.10) and (5.11), and applying the standard Stein-Tomas argument (see, for instance, [34]), it is easily seen that we can sum the corresponding estimates over all dyadic  1, and we obtain the Lp -L2 restriction estimate 1/2  2

≤ Cp f Lp |f | dνk whenever p  > 4B, uniformly in k, for k sufficiently large. Since B < d/2, we have pc ≥ 2d + 2 > 4B, so that the range p > 4B does include the critical value p = pc . Scaling back to the measures µk , we find that also the original measures µk satisfy a uniform restriction estimate 1/2  |f |2 dµk ≤ Cp f Lp , where Cp does not depend on k, provided p  ≥ 2hl + 2. However, this applies to pc , since hr (φ) ≥ hl . Finally, observe that we can achieve our dyadic decomposition into the measures µk by means of a dyadic decomposition in the variable x1 , so that these uniform estimates allow to sum over all k by means of Littlewood-Paley theory applied to variable x1 ! What remains to be understood is the contribution by the domain Dpr = Dλ given by |x2 − ψ(x1 )| ≤ Nλ x1aλ . In this domain, the upper bound B < d/2 for the multiplicity B of real roots will in general no longer be valid, as examples show, not even the weaker condition B < hr (φ)/2, which would still suffice for the previous argument. In order to resolve this problem, we apply again a further domain decomposition by means of a stopping time argument. Notice first that if we are to proceed as for the case l < λ, a major problem arises a (1, c0 ) = 0; in all other cases we can essentially argue only in Case 3 where ∇φpr as before. We therefore devise the stopping time argument essentially as follows: We put φ (1) := φ a . If Case 3 does not appear for any choice of c0 , then we stop our algorithm with φ (1) , and are done. Otherwise, if Case 3 applies to c0 , so that c0 y1aλ is a root of φκaλ , then we define new coordinates z in place of y by putting z 1 := x1

and

z 2 := x2 − ψ(x1 ) − c0 x1aλ ,

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FINITE-TYPE HYPERSURFACES

and express φ by φ (2) in the coordinates z. Again, if Case 3 does not appear (for φ (2) in place of φ (1) ) in the corresponding z-domain, we stop our algorithm. We also stop the algorithm when no further fine splitting of roots does occur, i.e., when there is a root with leading term ψ(x1 )+c0 x1aλ of φ which has the same multiplicity as the trivial root z 2 = 0 of the principal part of φ (2) . Otherwise, we continue in an analogous way. This algorithm will stop after a finite number of steps, and eventually leads to a further domain decomposition of Dpr into “homogeneous” domains D(l) and transition domains E(l) , which can eventually be treated by methods similar to those applied for the domains El and Dl . 13.5.3 Sufficiency of the condition p  ≥ 2hr (φ) + 2: II. Brief sketch of some ideas of the proof when h lin (φ) < 2 This case turns out to be by far more difficult than the case where h lin (φ) ≥ 5, and its discussion requires numerous further methods and ideas which I can sketch only very briefly. A first observation is that hr (φ) = d when d = h lin (φ) < 2, so that pc = 2d + 2. The starting point of our analysis is the following local normal form of our function φ. It is closely related to the classification of singularities in [2] and [24]. Theorem 5.8. If h lin (φ) < 2, then locally φ is of the form φ(x1 , x2 ) = b(x1 , x2 )(x2 − ψ(x1 ))2 + b0 (x1 ).

(5.12)

Here b, b0 , and ψ are smooth, and ψ is again the principal root jet, and either (a) b(0, 0)  = 0, and either b0 is flat (singularity of type A∞ ), or of finite type n, i.e., b0 (x1 ) = x1n β(x1 ), where β(0)  = 0 (singularity of type An−1 ); or (b) b(0, 0) = 0 and b(x1 , x2 ) = x1 b1 (x1 , x2 ) + x22 b2 (x2 ), with b1 (0, 0)  = 0 (singularity of type D). Let us just consider the case where φ is of finite type An−1 (type D can be treated in a rather similar way). In a first step, by making use of these normal forms in order to estimate certain two-dimensional oscillatory integrals that arise in estimating the Fourier transforms of surface carried measures, we can again reduce to the domain (3.7), where |x2 − cx1m | ≤ εx1m . In a second step, if κ denotes again the principal weight associated to the principal edge of N (φ), we again perform a dyadic decomposition of the measure µ over the domain (3.7), and re-scale in a suitable way. This leads to a phase function  2 φ(x, δ) := b(δ1 x1 , δ2 x2 ) x2 − x1m ω(δ1 x1 ) + δ0 x1n β(δ1 x1 ), where δ = (δ0 , δ1 , δ2 ) = (2−(nκ1 −1)k , 2−κ1 k , 2−κ2 k ) are small parameters which tend to 0 as k tends to infinity, and where b(δ1 x1 , δ2 x2 ) ∼ b(0, 0)  = 0 and β(0)  = 0.

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What we then need to prove is the following: Proposition 5.9. Given any point v = (v1 , v2 ) such that v1 ∼ 1 and v2 = v1m ω(0), there exists a neighborhood V of v in (R+ )2 such that for every cut-off function η ∈ D(V ), the measure νδ given by  νδ , f  := f (x, φ(x, δ)) η(x1 , x2 ) dx satisfies a restriction estimate 1/2  2

|f | dνδ ≤ Cp,η f Lp (R3 ) , whenever p  ≥ 2d + 2, provided δ is sufficiently small. In order to prove this proposition, we again perform a dyadic decomposition, this time with respect to the x3 -variable. By means of Littlewood-Paley theory, it then turns out that it is sufficient to prove uniform restriction estimates for the following family of measures  νδ,j , f  := f (x, φ(x, δ)) χ (22j φ(x, δ))η(x1 , x2 ) dx, of the form



|f |2 dνδ,j

1/2 ≤ Cp,η f Lp (R3 ) .

Here, χ (x3 ) is supported on a set where |x3 | ∼ 1. If 22j δ0 1, then it turns out that this localization means in fact again a localization to a curved rectangle where |x1 − v1 | < ε

and

|x2 − x1m ω(δ1 x1 )| ∼ 2−j ,

but in other cases, it has another meaning. It turns out that these estimates require a refined spectral decomposition of the measures νδ,j , namely a dyadic decomposition in every dual coordinate ξ1 , ξ2 , ξ3 . Slightly cheating, for every triple  = (λ1 , λ2 , λ3 ) of dyadic numbers λi = 2−ki , we therefore define the functions νj by ξ  ξ  ξ  1 2 3  χ1 χ1 ν ν δ,j (ξ ), j (ξ ) = χ1 λ1 λ2 λ3   and choose χ1 (s) supported where |s| ∼ 1 so that νδ,j =  νj , where summation is essentially over all these dyadic triples . We have here suppressed the dependency of νj on the small parameters δ. Note that |ξi | ∼ λi on the support of ν λ . j

For every fixed , we then essentially follow again the Stein-Tomas approach,   by estimating ν j ∞ and νj ∞ . This requires the distinction of various cases, depending on the relative sizes, of λ1 , λ2 , and λ3 . In the end, it turns out that the most difficult case is where λ1 ∼ λ2 ∼ λ3 and 22j δ0 ∼ 1, and the main result to be proven in this case is the following.

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Proposition 5.10. Let φ be of type An−1 , with m = 2 and finite n ≥ 5. Then   1 |f |2 dνj ≤ C 2 7 j f 2L14/11 (R3 ) , (5.13) 2≤λ1 ∼λ2 ∼λ3 ≤26j

S

for all j ∈ N sufficiently big, say j ≥ j0 , where the constant C does neither depend on δ, nor on j. The proof of this result requires yet further refinements. Indeed, the Fourier transform of νj is an oscillatory integral with complete phase of the form (y; δ, j, ξ ) = ξ1 y1 + ξ2 y12 ω(δ1 y1 ) + ξ3 σy1n β(δ1 y1 ) +2−j ξ2 y2 + ξ3 b (y, δ, j ) y22 , where σ := 22j δ0 ∼ 1 and |b (y, δ, j )| ∼ 1. Notice that if |ξ1 | ∼ |ξ2 | ∼ |ξ2 |, then φ may have degenerate critical points, with non-vanishing third derivatives, with respect to the variable y1 , so that we encounter oscillatory integrals of “Airy type.” This case requires a further dyadic frequency decomposition with respect to the distance to certain “Airy cones.” For the quite comprehensive details, I refer to [41]. The case where 2 ≤ d < 5 turns out to be the most delicate one, since on the one hand, we can no longer apply Drury’s restriction theorem, which necessitates an even more refined domain decomposition, and furthermore, somewhat subtle interpolation arguments are needed in many situations in order to handle the endpoint p = pc (cf. [42]).

REFERENCES ˇ [1] Arhipov, G. I., Karacuba, A. A., and Cubarikov, V. N., Trigonometric integrals. Izv. Akad. Nauk SSSR Ser. Mat., 43 (1979), 971–1003, 1197 (Russian); English translation in Math. USSR-Izv., 15 (1980), 211–239. [2] Arnol’d, V. I., Remarks on the method of stationary phase and on the Coxeter numbers. Uspekhi Mat. Nauk, 28 (1973),17–44 (Russian); English translation in Russian Math. Surveys, 28 (1973), 19–48. [3] Arnol’d, V. I., Gusein-Zade, S. M., and Varchenko, A. N., Singularities of differentiable maps. Vol. II, Monodromy and asymptotics of integrals, Monographs in Mathematics, 83. Birkhäuser, Boston Inc., Boston, MA, 1988. [4] Atiyah, M., Resolution of singularities and division of distributions. Comm. Pure Appl. Math., 23 (1970), 145–150. [5] Bak, J. G., Oberlin, D. M., and Seeger, A., Restriction of Fourier transforms to curves and related oscillatory integrals. Amer. J. Math., 131 (2) (2009), 277–311. [6] Bernstein, I. N., and Gelfand, S. I., Meromorphy of the function P λ . Funktsional. Anal. i Priložen., 3 (1) (1969), 8485.

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[7] Bierstone, E., and Milman, P. D., Semianalytic and subanalytic sets. Inst. Hautes Études Sci. Publ. Math., 67 (1988), 5–42. [8] Bierstone, E., and Milman, P. D., Arc-analytic functions. Invent. Math., 101 (1990), 411–424. [9] Bourgain, J., Averages in the plane over convex curves and maximal operators. J. Analyse Math., 47 (1986), 69–85. [10] Bourgain, J., Estimates for cone multipliers. Geometric aspects of functional analysis (Israel, 1992–1994), 4160, Oper. Theory Adv. Appl. 77 (1995), 4160. [11] Bourgain, J., and Guth, L., Bounds on oscillatory integral operators. C. R. Acad. Sci. Paris, Ser. I, 349 (2011), 137–141. [12] Bruna, J., Nagel, A., and Wainger, S., Convex hypersurfaces and Fourier transforms. Ann. of Math. (2), 127 (1988), no. 2, 333–365. [13] Carbery, C., Christ, M., and Wright, J., Multidimensional van der Corput and sublevel set estimates. J. Amer. Math. Soc., 12 (1999), no. 4, 981–1015. [14] Carleson, L., and Sjölin, P., Oscillatory integrals and a multiplier problem for the disc. Studia Math., 44 (1972), 287–299. [15] Collins, T., Greenleaf, A., and Pramanik, M., A multi-dimensional resolution of singularities with applications to analysis. Preprint, arXiv:1007.0519. [16] van der Corput, J. G., Zahlentheoretische Abschätzungen. Math. Ann., 84 (1921), 53–79. [17] Cowling, M., Disney, S., Mauceri, G., and Müller, D., Damping oscillatory integrals. Invent. Math., 101 (1990), 237–260. [18] Cowling, M., and Mauceri, G., Inequalities for some maximal functions. II. Trans. Amer. Math. Soc., 296 (1986), no. 1, 341–365. [19] Cowling, M., and Mauceri, G., Oscillatory integrals and Fourier transforms of surface carried measures. Trans. Amer. Math. Soc., 304 (1987), no.1, 53–68. [20] Dendrinos, S., and Müller, D., Uniform estimates for the local restriction of the Fourier transform to curve. Trans. Amer. Math. Soc., to appear. [21] Denef, J., Nicaise, J., and Sargos, P., Oscillatory integrals and Newton polyhedra. J. Anal. Math., 95 (2005), 147–172. [22] Domar, Y., On the Banach algebra A(G) for smooth sets  ⊂ Rn . Comment. Math. Helv., 52 (1997), no. 3, 357–371. [23] Drury, S. W., Restrictions of Fourier transforms to curves. Ann. Inst. Fourier., 35 (1985), 117–123.

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[24] Duistermaat, J. J., Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math., 27 (1974), 207–281. [25] Fefferman, C., Inequalities for strongly singular convolution operators. Acta Math. (1970), 9–36. [26] Grafakos, L., Modern Fourier analysis. Graduate Texts in Mathematics 250. Springer, New York, 2009. [27] Greenblatt, M., Newton polygons and local integrability of negative powers of smooth functions in the plane. Trans. Amer. Math. Soc., 358 (2006), no. 2, 657–670. [28] Greenblatt, M., An elementary coordinate-dependent local resolution of singularities and applications. J. Funct. Anal., 255 (2008), no. 8, 1957–1994. [29] Greenblatt, M., The asymptotic behavior of degenerate oscillatory integrals in two dimensions. J. Funct. Anal., 257 (2009), no. 6, 1759–1798. [30] Greenblatt, M., Resolution of singularities, asymptotic expansions of integrals and related phenomena. J. Anal. Math., 111 (2010), 221–245. [31] Greenblatt, M., Oscillatory integral decay, sublevel set growth, and the Newton polyhedron. Math. Ann., 346 (2010), 857–895. [32] Greenblatt, M., Lp boundedness of maximal averages over hypersurfaces in R3 . Preprint. [33] Greenblatt, M., Maximal averages over hypersurfaces and the Newton polyhedron. Preprint, arXiv:1002.0109. [34] Greenleaf, A., Principal curvature and harmonic analysis. Indiana Univ. Math. J., 30 (4) (1981), 519–537. [35] Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero I, II. Ann. Math. (2), 79 (1964), 109–326. [36] Hörmander, L., Oscillatory integrals and multipliers on FLp. Ark. Mat., 11 (1973), 1–11. [37] Ikromov, I. A., Kempe, M., and Müller, D., Damped oscillatory integrals and boundedness of maximal operators associated to mixed homogeneous hypersurfaces. Duke Math. J., 126 (2005), no. 3, 471–490. [38] Ikromov, I. A., Kempe, M., and Müller, D., Estimates for maximal functions associated to hypersurfaces in R3 and related problems of harmonic analysis. Acta Math., 204 (2010), 151–271. [39] Ikromov, I. A., and Müller, D., On adapted coordinate systems. Trans. Amer. Math. Soc., 363 (2011), no. 6, 2821–2848.

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[40] Ikromov, I. A., and Müller, D., Uniform estimates for the Fourier transform of surface carried measures in R3 and an application to Fourier restriction. J. Fourier Anal. Appl., 17 (2011), no. 6, 1292–1332. [41] Ikromov, I. A., and Müller, D., Lp -L2 Fourier restriction for hypersurfaces in R3 : Part I. Preprint; ArXiv: http://arxiv.org/abs/1208.6090. [42] Ikromov, I. A., and Müller, D., Lp -L2 Fourier restriction for hypersurfaces in R3 : Part II. In preparation. [43] Iosevich, A., Maximal operators associated to families of flat curves in the plane. Duke Math. J., 76 (1994), no. 2, 633–644. [44] Iosevich, A., and Sawyer, E., Oscillatory integrals and maximal averages over homogeneous surfaces. Duke Math. J., 82 (1996), no. 1, 103–141. [45] Iosevich, A., and Sawyer, E., Maximal averages over surfaces. Adv. Math., 132 (1997), no. 1, 46–119. [46] Iosevich, A., Sawyer, E., and Seeger, A., On averaging operators associated with convex hypersurfaces of finite type. J. Anal. Math., 79 (1999), 159–187. [47] Karpushkin, V. N., A theorem on uniform estimates for oscillatory integrals with a phase depending on two variables. Trudy Sem. Petrovsk., 10 (1984), 150–169, 238 (Russian); English translation in J. Soviet Math., 35 (1986), 2809–2826. [48] Magyar, A., On Fourier restriction and the Newton polygon. Proceedings Amer. Math. Soc., 137 (2009), 615–625. [49] Mockenhaupt, G., A note on the cone multiplier. Proc. Amer. Math. Soc., 117 (1993), no. 1, 145–152. [50] Mockenhaupt, G., Seeger, A., and Sogge, C. D., Wave front sets, local smoothing and Bourgain’s circular maximal theorem. Ann. of Math. (2), 136 (1992), no. 1, 207–218. [51] Mockenhaupt, G., Seeger, A., and Sogge, C. D., Local smoothing of Fourier integral operators and Carleson-Sjölin estimates. J. Amer. Math. Soc., 6 (1993), no. 1, 65–130. [52] Nagel, A., Seeger, A., and Wainger, S., Averages over convex hypersurfaces. Amer. J. Math., 115 (1993), no. 4, 903–927. [53] Parusi´nski, A., Subanalytic functions. Trans. Amer. Math. Soc., 344 (1994), 583–595. [54] Parusi´nski, A., On the preparation theorem for subanalytic functions. New developments in singularity theory (Cambridge 2000), Nato Sci. Ser. II Math. Phys. Chem., 21, Kluwer Academic Publisher, Dordrecht (2001), 193–215.

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[55] Phong, D. H., and Stein, E. M., The Newton polyhedron and oscillatory integral operators. Acta Math., 179 (1997), no. 1, 105–152. [56] Phong, D. H., Stein, E. M., and Sturm, J. A., On the growth and stability of real-analytic functions. Amer. J. Math., 121 (1999), no. 3, 519–554. [57] Randol, B., On the asymptotic behavior of the Fourier transform of the indicator function of a convex set. Trans. Amer. Math. Soc., 139 (1969), 279–285. [58] Schulz, H., On the decay of the Fourier transform of measures on hypersurfaces, generated by radial functions, and related restriction theorems. Unpublished preprint, 1990. [59] Schulz, H., Convex hypersurfaces of finite type and the asymptotics of their Fourier transforms. Indiana Univ. Math. J., 40 (1991), 1267–1275. [60] Sogge, C. D., and Stein, E. M., Averages of functions over hypersurfaces in Rn . Invent. Math., 82 (1985), no. 3, 543–556. [61] Stein, E. M., Maximal functions. I. Spherical means. Proc. Nat. Acad. Sci. U.S.A., 73 (1976), no. 7, 2174–2175. [62] Stein, E.M., Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series 43. Princeton University Press, Princeton, NJ, 1993. [63] Strichartz, R. S., Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J., 44 (1977), 705–714. [64] Sussmann, H. J.,. Real-analytic desingularization and subanalytic sets: an elementary approach. Trans. Amer. Math. Soc., 317 (1990), 417–461. [65] Svensson, I., Estimates for the Fourier transform of the characteristic function of a convex set. Ark. Mat., 9 (1971), 11–22. [66] Varchenko, A. N., Newton polyhedra and estimates of oscillating integrals. Funkcional. Anal. i Priložen, 10 (1976), 13–38 (Russian); English translation in Funktional Anal. Appl., 18 (1976), 175–196. [67] Zygmund, A., On Fourier coefficients and transforms of functions of two variables. Studia Math., 9 (1974), 189–201.

Chapter Fourteen Multi-Linear Multipliers Associated to Simplexes of Arbitrary Length Camil Muscalu, Terence Tao, and Christoph Thiele 14.1 INTRODUCTION The present chapter is a natural continuation of our previous work in [12] and [13]. In those articles we studied the Lp boundedness properties of a tri-linear operator T3 defined by the formula
T3 (f1 , f2 , f3 )(x) = (1) f1 (ξ1 )f2 (ξ2 )f3 (ξ3 )e2πix(ξ1 +ξ2 +ξ3 ) dξ1 dξ2 dξ3 ξ1 0 so that for all constants 0 < εm < ε0 , 1 ≤ m ≤ N , there exists a unique function G(z; p1 , · · · , pN ) ∈ P SH (M, ω) ∩ C α (M¯ \ {p1 , · · · , pN }), which satisfies the equation (1.1), the boundary condition (1.2), and the following asymptotics near each pole pm ,   n  G(z; p1 , · · · , pN ) = εm log  |fj m (z)|2  + O(1). (2.1) j =1

Here α is any constant satisfying 0 < α < 1. In general, it is not possible to choose ε0 to be arbitrary. In fact, if δ > 0 is given, then it is easy to construct (M, ω) and p ∈ M, and local holomorphic functions f1 , ..., fn , for which the maximal ε0 is less than δ. For example, let M = X × D where (X, ω) is a compact Kähler manifold with unit volume, and D ⊆ C is the unit disk. Let p = (x, 0) ∈ X × {0} and choose local coordinates z 1 , ..., z n on X centered at x. Suppose G is a Green’s function on M with singularity ε log (|w|2 + |z 1 |2k + · · · + |z n |2k ), where w is a coordinate on C, centered at 0 ∈ C. Then G(0, z 1 , ..., z n ) ∈ P SH (X, ω) has Lelong number εk so, by a result of Demailly [D], its Monge-Ampère mass is at least εk. Thus ε ≤ k1 . However, the restriction on ε0 can be removed for strongly pseudoconvex manifolds, i.e., manifolds M admitting a C 2 function ρ with ∂M = {ρ = 0}, ¯ > 0. In this case, we have the following solution of the Dirichlet and i∂ ∂ρ problem: ¯ Let fj m , 1 ≤ Theorem 2. Let ω be a non-negative smooth (1, 1)-form on M. j ≤ n, 1 ≤ m ≤ N be as in the previous theorem. Assume that M is strongly pseudoconvex. Then for any function ϕb ∈ C 2 (∂M), and any constant εm > 0, 1 ≤ m ≤ N , there exists a unique function G(z; p1 , · · · , pN ) ∈ P SH (M, ω) ∩ C α (M¯ \ {p1 , · · · , pN }), which satisfies the equation (1.1), the asymptotics (2.1) near each pole pm , and the Dirichlet boundary condition limz→∂M G(z; p1 , · · · , pN ) = ϕb .

(2.2)

Again α is any constant satisfying 0 < α < 1. Theorem 2 is closest to earlier results of B. Guan [Gb], Blocki [B1], and Lempert [L]. The regularity properties of the pluricomplex Green’s function are much more precise in these earlier works [Gb], [B1], and [L], but we gain here in generality, including the property that local holomorphic singularities can be assigned arbitrarily. We shall make use of this latter fact in the construction of new geodesic rays in the space of Kähler potentials (see section §16.5).

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In general, the case of Kähler manifolds with boundary is quite different from the case of Kähler manifolds without boundary. Nevertheless, the idea used in the proof of Theorem 1 may also be used to yield results for singular Monge-Ampère equations on compact manifolds without boundary. The simplest example is the following: Theorem 3. Let (X, ω) be a compact Kähler manifold with unit volume. Let p ∈ X, let f ∈ C ∞ (X) be positive, and assume X f ωn = 1. Let δp be the Dirac measure concentrated at p. Then for ε > 0 sufficiently small, there exists a unique ϕ ∈ P SH (X, ω) ∩ C 1,α (X\{p}) satisfying ϕ = ε log |z|2 + C 1,α (X) near z = p and


i ¯ ω + ∂ ∂ϕ 2

n = (1 − ε)f ωn + εδp .

(2.3)

As Coman-Guedj [CG] have shown, there are examples of Kähler manifolds for which ε must be strictly smaller than one. ¯ n = µ on a compact Kähler manifold We observe that the equation (ω + 2i ∂ ∂ϕ) without boundary has been solved by Berman, Boucksom, Guedj, and Zeriahi [BBGZ], when µ is a measure which does not charge pluripolar sets. The important case of measures µ which charge pluripolar sets remains open. A general result is that of Ahag et al. [ACCH], who show that, on a hyperconvex domain in Cn , a non-negative measure is a complex Monge-Ampère measure if it is dominated by a Monge-Ampère measure. The regularity and precise singularities of the solutions are however still obscure at this moment. Theorem 3 provides another example of the solvability of a Monge-Ampère equation with measures charging pluripolar sets. In fact, it can be seen from its proof in section §16.4 that more general formulations are possible, with the singularity p replaced by the complex variety Z = {s1 (z) = · · · = sk (z) = 0}, 1 ≤ k < n, where the sα (z) are sections of a holomorphic vector bundles satisfying some non-degeneracy conditions. A related and important problem is to provide flexible characterizations of the unbounded functions for which the Monge-Ampère measure is well-defined (see, e.g., Cegrell [C] and Blocki [B2]).

16.3 PROOF OF THEOREMS 1 AND 2 In this section we give the proof of Theorems 1 and 2. It will be seen that the argument does not depend essentially on N , so we set N = 1 and drop the index m to lighten the notation. The bulk of the work is the proof of Theorem 1, and we can just indicate at the end the easy modifications for Theorem 2. The key lemmas are the following: Lemma 1. Denote by (M, p, f ) the data consisting of the Kähler manifold (M, ω), the point p, and the given local holomorphic functions fj (z), 1 ≤ α ≤ n with p as their only common zero. Then there exist a complex manifold X  = X (M, p, f )

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PLURICOMPLEX GREEN’S FUNCTION

and a holomorphic map π  : X → M sending ∂X to ∂M, with the following properties: (i) There is a closed, non-negative (1, 1)-form  on X , an effective divisor E  , and an ε > 0 with i  − ε ∂ ∂¯ log hE  > 0 2

(3.1)

for some smooth metric hE  on O(−E  ). (ii) The restriction π = π  |X¯  \E  is a biholomorphism π : X¯  \ E  → M¯ \ p with   n  i (3.2) π∗  = ω + ε ∂ ∂¯ ψ(z) log |fj (z)|2 + 1 − ψ(z) 2 j =1 where ψ(z) is a function which is 1 in a neighborhood of p, and which is compactly supported in another such neighborhood. Lemma 2. Let X be a complex manifold with smooth boundary of dimension n, equipped with a non-negative closed form  , with  − ε 2i ∂ ∂¯ log hE  > 0 for some effective divisor E  supported away from ∂M, some smooth metric hE  on O(−E  ), and some ε > 0. Then there exists a unique function  ∈ P SH (X ,  )∩L∞ (X )∩ C α (X¯  \E  ) which solves the Dirichlet problem n
i ¯  ∂X = 0. (3.3)  + ∂ ∂ = 0 on X , 2 Here α is any constant satisfying 0 < α < 1. The theorem follows readily from the two lemmas. Let  be the function given by Lemma 2 applied to the complex manifold X  = X (M, p, f ) and the non¯ n = 0 on X . Set ϕ =  ◦ π . Since π negative form  of (3.2). Then ( + 2i ∂ ∂)   ¯ n=0 is a biholomorphism between X \E and M \p, this implies (π∗  + 2i ∂ ∂φ) on M \p, i.e.,  n    n  i |fj (z)|2 + (1 − ψ(z)) + ϕ(z) . (3.4) 0 = ω + ∂ ∂¯ ε ψ(z) log 2 j =1 We can now set



G(z; p) = ε ψ(z) log

n 

 |fj (z)|2 + (1 − ψ(z)) + ϕ(z) − ε.

(3.5)

j =1

¯ n = 0 on M \ p, vanishes on ∂M, Clearly it satisfies the equation (ω + 2i ∂ ∂G) and has the desired asymptotics near p. It is ω-plurisubharmonic on M \ p, and bounded from above. Thus it extends to an ω-plurisubharmonic function on M. This shows that G(z; p) satisfies all the desired properties, and the existence part of the theorem is proved.

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˜ ∈ P SH (M, ω) satisfy (1.1), (1.2), and (2.1). Let To prove the uniqueness, let G   n  ˜ − ε ψ(z) log ˜ =G (3.6) |fj (z)|2 + (1 − ψ(z)) + ε.  j =1

˜ ∈ P SH ( , X \E  )∩C α (X \E  ). Since it is bounded, it extends to a funcThen  ˜ ∈ P SH ( , X )∩C α (X \E  )∩ tion, which by abuse of notation, will be denoted  i ¯  n ∞  L (X). Moreover, ( + 2 ∂ ∂ ) = 0. This is certainly true on X \E  and, since ˜ =  and, hence,  is bounded, is true on all of X . By Lemma 2, we have  ˜ = G. G Thus it suffices to establish Lemma 1 and Lemma 2. We begin with the proof of Lemma 1, which we break into several steps. Some of these are well-known, but we did not find the version that we needed in the literature, so we have provided a complete derivation. 16.3.1 Blow-ups of complex manifolds We start by recalling the construction of the blow-up of a complex manifold along a smooth submanifold and some of its basic properties. Let W be a complex manifold of dimension n, and let Z ⊂ W be a submanifold of dimension d < n. Let N = T W|Z /T Z be the normal bundle of Z and let E = P(N ). Let W  = (W \Z) ∪ E

(3.7)

and define π : W  → W by extending the map π : E → Z to be the identity map on W \Z. The set W  has a natural complex structure for which π : W  → W is a holomorphic map. The complex structure on W  is defined as follows: First, we require that W \Z ⊆ W  is an open set and the inclusion W \Z → W  is holomorphic. Next, we let Uα be a collection of coordinate balls in W such that Z ⊆ ∪α Uα and such that Uα ∩ Z = {(x α , y α ) ∈ Uα : y α = 0}. On Uα ∩ Uβ we have z β = φαβ (z α ) function between open subsets of Cn . The with φαβ : Uαβ → U  βα a biholomorphic 

11 D12 derivative Dφαβ = D D21 D22 is an invertible n×n matrix, where D11 has size d ×d, and D22 has size (n − d) × (n − d). Note that D22 (p) is invertible if p ∈ Uα ∩ Z = {z α ∈ Uα : y α = 0}. In fact, D22 is the isomorphism of the normal bundle of {y α = 0} ⊆ Uαβ to the normal bundle of {y β = 0} ⊆ Uβα induced by the biholomorphic map φαβ . Let us spell out this point in more detail. We can write

β

j

yi = Ai yjα , 1 ≤ i, j ≤ n − d,

(3.8)

for certain (non-unique) holomorphic functions Alk on Uαβ . This follows from the fact that φαβ takes the x α -axis (i.e., the set {y α = 0}) to the x β -axis (i.e., the set j {y β = 0}). Then D22 (p) = (Ai (p)) is an invertible matrix. Define a complex manifold Uα as follows: Uα = {(z, t) : z ∈ Uα , t ∈ Pn−d−1 : yi tj = ti yj , 1 ≤ i, j ≤ n − d}.

(3.9)

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Let fα : Uα → π −1 (Uα ) ⊆ W  be the bijective map  z ify  = 0, (z, t)  → t1 ∂ + · · · + tn−d ∂ ify = 0. ∂y1 yn−d

(3.10)

We wish to use the maps fα to give W  a complex structure. To do this, we must   → Uβα are holoshow that the change of coordinate maps fαβ = fβ−1 ◦ fα : Uαβ morphic. We proceed as follows: fαβ (z α , t α ) = (z β , t β ) where z β = φαβ (z α ) and β

tj

β

ti

β

=

yj

β

yi

=

Akj ykα Aki ykα

=

Akj tkα Aki tkα

β

.

(3.11) β

If yi  = 0 for some i, then the first equality implies that ti  = 0 so we can take β

β

ti = 1 and tj = If

β yi

β yj β yi

, which is holomorphic.

= 0 for all i, then we make use of the fact that D22 is invertible on Z β

β

Ak y α

so there exists i such that Aki tkα  = 0. We take ti = 1 and tj = Ajk ykα which is i k holomorphic. Thus we see that W  is a complex manifold. Now let π : W  → W be as above. Let p ∈ E and q = π(p). Then the discussion above shows that there exist a coordinate neighborhood  of p ∈ W  and coordinates (ζ0 , ...., ζd , θ1 , ..., θn−d−1 ) centered at p with the following properties: 1) 2) 3) 4)

ζi = z i ◦ π for some coordinate functions z j on W . E ∩  = {ζ0 = 0}. (θ1 |E , ..., θn−d−1 |E ) is a set of local coordinates of π −1 (p) centered at p ∈ E. If p ∈ Uα then ζ0 |yjα for all j and yjα0 /ζ0 is nowhere vanishing for some j0 . The last condition says that E ∩  = {yjα0 = 0}.

This blow-up process can be iterated: if Z  ⊆ W  is a smooth subvariety then we can construct W  = BL(Z  , W  ) and we have maps W  → W  → W . We say that W  is an iterated blow-up. If π : W  → W is an iterated blow-up, the exceptional divisor is by definition the smallest effective divisor E ⊆ W  such that W  \E → W \π(E) is an isomorphism. 16.3.2 Analytic spaces Let X be a set. We say that X is an analytic space if there is a complex manifold W (called an ambient manifold) with X ⊂ W and satisfying the following property: for every p ∈ X, there are an open set p ∈ U ⊂ W and functions f1 , · · · , fr : U → C such that U ∩ X = {w ∈ W ; f1 (w) = · · · = fr (w) = 0}.

(3.12)

We denote by Xreg the subset of X where X is locally a complex manifold, and by Xsing its complement in X. A Kähler metric on X is by definition a Kähler metric on Xreg which is the restriction of a Kähler metric on an ambient manifold W .

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Lemma 3. Let (M, p, f ) be data consisting of a complex manifold M, a point p ∈ M, and local holomorphic functions f1 (z), · · · , fn (z) defined in a neighborhood of p, and with p as their only common zero. Then we can associate to these data an analytic space X = X(M, p, f ) with the following properties: (i) There is a biholomorphism between X\X0 and M\p, for some subset X0 of X which is biholomorphically equivalent to CPn−1 . (ii) Let ψ(z) be a cut-off function which is 1 in a coordinate chart around p in M, and 0 outside another such chart. Then for all δ small enough, the pull-back to X\X0 of the form ωδ defined as   n  i ωδ = ω + δ ∂ ∂¯ ψ(z) log |fj (z)|2 + 1 − ψ(z) (3.13) 2 j =1 defined on M \p extends to a Kähler form on X. Proof of Lemma 3. Given the data (M, p, f ), let U be a coordinate neighborhood centered at p in M, define a space V by V = {(z 1 , ..., z n ), (y1 , ..., yn ) ∈ U × CPn−1 : yi fj (z) = yj fi (z)},

(3.14)

and let π : V → U be the projection. We then define X = X(M, p, f ) by X = (M\{p} ∪ V )/ ∼

(3.15)

where, for m ∈ M and v ∈ V , we say m ∼ v if π(v) = m. Note that the fiber X0 of V above the point p is the entire projective space CPn−1 . We claim that X is an analytic space. To show that X is an analytic space, we must find a complex manifold W such that X ⊆ W and such that X is locally defined by the simultaneous vanishing of a finite collection of holomorphic functions. Let B ⊆ Cn be a small open ball centered at the origin and let Z ⊆ U × Cn be the smooth manifold Z = {(z, ξ ) : z ∈ U, ξ ∈ B : f1 (z) − ξ1 = · · · = fn (z) − ξn = 0}.

(3.16)

If B is sufficiently small, then the image of the map Z → U is compactly supported in U . Thus Z ⊆ M × B is a smooth submanifold whose image, when projected to M, lies in a relatively compact subset of U . Finally, let W = BL(Z, M × B). Thus W is locally defined by W = {(z, ξ, y) ∈ U × B × CPn−1 : yi (fj (z) − ξj ) = yj (fi (z) − ξi )}.

(3.17)

Then W is a smooth manifold and X ⊆ W is defined by ξ1 = · · · = ξn = 0. This shows X is an analytic space. We can define a Kähler metric on X as follows. Let ω be a Kähler metric on M. Extend ω to a Kähler metric on M × B. Choose ψ ∈ C ∞ (U ) with the property that ψ equals one in a neighborhood p and support(ψ) ⊆ U is compact. Then the composition of the map U × B → U with ψ is a smooth function on U × B which, by abuse of notation, is again denoted ψ. Taking supp(ψ) ⊆ U to be a sufficiently

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large compact set, we may assume ψ = 1 on Z. Let   δ  ωδ = ω + i ∂ ∂¯ (1 − ψ) + ψ log |fj (z) − ξj |2 2 on W \E = (M × B)\Z and let
 i ¯ 2 ∂ ∂ log |y| = ω + δωFS ωδ = ω + δ 2

(3.18)

(3.19)

on the open neighborhood {ψ = 1}o of E (i.e., the interior of the closed set {ψ = 1}). Here ωFS is the Fubini-Study metric on PCn−1 . The two definitions are consistent, and define ωδ , a smooth (1,1) form on W . Since ω + δωFS > 0 in a fixed (independent of δ) neighborhood of E, we find that ωδ > 0 on all of W for δ sufficiently small. Thus ωδ is a Kähler metric on W and its restriction to X is a Kähler metric on X. The proof of Lemma 3 is complete. For general functions f1 (z), · · · , fn (z), the space X(M, p, f ) is not smooth. In order to obtain a smooth manifold, we use Hironaka’s theorem on resolution of singularities. One version of this theorem is the following: Theorem 4. Let W be a complex manifold and let X ⊂ W be a complex analytic space. Then there exist an iterated blow-up W  of W , with corresponding holomorphic map π : W  → W with the following property: let E be the exceptional divisor, and set X = π −1 (X)\E.

(3.20)

Then X  is a smooth manifold and the map π : X  → X is surjective. Moreover, E  = E ∩ X = π −1 (Xsing )

(3.21)

is a divisor with normal crossings, and π : X  \E  → Xreg is an isomorphism. 16.3.3 Metrics on blow-ups The following is the key property of blow-ups that we need: Lemma 4. Let W be a smooth complex manifold and Z ⊂ W a smooth submanifold. Let π : W  → W be the blow-up of W with center Z, and let E ⊂ W be the exceptional divisor. If ω is any Kähler metric on W , then there exists a Hermitian metric hE on O(−E) so that, for any compact subset K of W  , there exists εK > 0 with i (3.22) π ∗ ω − ε ∂ ∂¯ log hE > 0 2 for all 0 < ε < εK . Proof of Lemma 4. Let {Uα } ⊆ W  be a locally finite collection of coordinate neighborhoods which cover Z, and let z α = (x α , y α ) be coordinates on Uα with the property: Uα ∩ Z = {y α = 0}. Choose ψα ∈ Cc∞ (Uα ) such that 0 ≤ ψα ≤ 1 and ψ =

r α=1

ψα = 1 on Z.

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Let f be a section of O(−E) over an open set . This means that f is a holomorphic function on  which vanishes on E ∩ . Let |f |2hE =

(1 − ψ) +



|f |2 . α 2 α 2 α ψα (|y1 | + · · · + |yn−d | )

(3.23)

Observe that this makes sense: we may assume that E ∩   = ∅ and that  is a small open set such that ψ ◦ π = 1 on . Choose coordinates (ζ, θ ) as in section 16.3.1. Thus ζ = (ζ0 , ..., ζd ) and θ = (θ1 , ..., θn−d−1 ). Since f vanishes on E we have f = gζ0 for some holomorphic function g and |f |2hE = 

|f |2 |g|2  = . α 2 α α α 2 2 2 α ψα (|y1 | + · · · + |yn−d | ) α ψα (|y1 /ζ0 | + · · · + |yn−d /ζ0 | ) (3.24)

This shows that hE is a well-defined smooth metric on O(−E). Next we claim that (π ∗ ω − 2i ε∂ ∂¯ log hE )(p) > 0 for all p ∈ E and sufficiently small ε > 0. To see this, fix p ∈ E. If F (ζ, θ ) is smooth in a neighborhood of {p}, then 
A B (3.25) ∂i ∂j¯ F = t ¯ B D where A has size (d + 1) × (d + 1), B has size (d + 1) × (n − d − 1), and D has size (n − d − 1) × (n − d − 1). We see that in these coordinates
 A 0 ω(p) = (3.26) 0 0 with A > 0. Now we write −i∂ ∂¯ log hE (p) =




X t ¯ Y

 Y . D

(3.27)

Since ψ is independent of θj we have  ∂2 α log ψα (p)(|y1α /ζ0 |2 + · · · + |yn−d /ζ0 |2 ). (3.28) ∂θj ∂ θ¯k α n−d−1 Finally we observe that i,j =1 Di j¯ dθi ∧ dθj¯ is the pullback of a Fubini-Study metric with respect to a holomorphic map whose deriviative has maximal rank. Thus D > 0. Di j¯ =

The claim now follows from the following linear algebra fact: Lemma 5. Let A, X be (d + 1) × (d + 1) Hermitian matrices and D an (n − d − 1) × (n − d − 1) Hermitian matrix. Let Y be a (d + 1) × (n − d − 1) matrix. Assume A > 0 and D > 0. Then for λ > 0 sufficiently large we have 

X Y A 0 > 0. (3.29) M(λ) = λ + t¯ 0 0 Y D

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PLURICOMPLEX GREEN’S FUNCTION

Proof. We may assume A to be diagonal with positive diagonal entries a0 , ..., ad . Then det(M(λ)) is a polynomial of degree d, with real coefficients in λ whose leading coefficient is a0 · · · ad > 0. Thus det M(λ) > 0 for λ > 0 sufficiently large. The same argument shows that the determinants of all the square submatrices of M(λ), which are situated in the lower right corner of M(λ), also have positive determinants for λ sufficiently large. This proves Lemma 5. Since (π ∗ ω − 2i ε∂ ∂¯ log hE ) > 0 on Z, we see that there is an open neighborhood U of Z ∩ K on which (π ∗ ω − 2i ε∂ ∂¯ log hE ) > 0 for all sufficiently small ε > 0. Choosing ε > 0 sufficiently small, we have (π ∗ ω − 2i ε∂ ∂¯ log hE ) > 0 on all of K. This proves Lemma 4. The resolution of singularities in Hironaka’s theorem will usually require an iteration of blow-ups. Thus we need the following generalization of Lemma 4: Lemma 6. Let W be a compact complex manifold and let π : W  → W be an iterated blow-up of W . Let E ⊂ W  be the exceptional divisor. If ω is any Kähler metric on W , then there exist an effective divisor E  on W  supported on E, and a Hermitian metric hE  on O(−E  ) so that there exists ε > 0 with i π ∗ ω − ε ∂ ∂¯ log hE  > 0. 2

(3.30)

Proof of Lemma 6. Let π : W  → W be the composition of two blow-ups, π = π2 ◦ π1 , with π2 : W  = W2 → W1 , π1 = W1 → W . Apply Lemma 4 to π1 and the Kähler metric ω on W . If E1 is the exceptional divisor of π1 , we obtain a metric hE1 on the line bundle O(−E1 ) on W1 with (π1 )∗ ω − ε1 2i ∂ ∂¯ log hE1 > 0 on W1 . Apply next Lemma 4 to π2 and the Kähler metric (π1 )∗ ω − ε1 2i ∂ ∂¯ log hE1 on W1 . We obtain then a metric hE2 on O(−E2 ), where E2 is the exceptional divisor of π2 , with π2∗ (π1∗ ω−ε1 2i ∂ ∂¯ log hE1 )−ε2 2i ∂ ∂¯ log hE2 > 0 on W2 . We can take ε1 = n11 and ε2 = n11n2 for n1 and n2 large enough integers. We can then write
 i ¯ i 1 i ¯ ∗ ∗ π2 π1 ω − ε1 ∂ ∂ log hE1 − ε2 ∂ ∂¯ log hE2 = π ∗ ω − ∂ ∂ log hn2 (π2 )∗ E1 hE2 2 2 n1 n2 2 (3.31) so the lemma holds in this case with the line bundle given by O(−E2 )⊗ π2∗ O(−n2 E1 ). Clearly the argument extends to any finite number of blow-ups, and the lemma is proved. The preceding lemma can be extended to the case of complex analytic sets: Lemma 7. Let X be a complex analytic set, and ω a Kähler metric on X. Let π : X → X be a resolution of singularities and E ⊂ X the exceptional divisor. Then there is a divisor E  on X , whose support is contained in E, a Hermitian metric hE  on the line bundle O(−E  ), and an ε > 0 such that π ∗ ω − ε 2i ∂ ∂¯ log hE  > 0. Proof of Lemma 7. By definition of a Kähler metric on X, the metric ω extends to a Kähler metric on an ambient space W of X. By definition of resolution of singularities, the map π extends to a map π : W  → W , with X ⊂ W  , and

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W  an iterated blow-up. The metric on the line bundle O(−E  ) on W  obtained from Lemma 6 applied to W and the Kähler form ω restricts to a metric on the line bundle O(−E  ) on X  with the desired property. We can now give the proof of Lemma 1: We apply Lemma 3, to obtain the analytic space X = X(M, p, f ) and the Kähler form ωδ with the properties stated there. The space X is only an analytic set, but we can apply Theorem 4 to obtain a resolution of singularities π : X → X. We can then apply Lemma 7 to obtain a line bundle O(−E  ) on X  with π ∗ (ωδ ) − ε 2i ∂ ∂¯ log hE  > 0 on X . Set  = π ∗ ωδ .

(3.32)

Since the resolution of singularities is a biholomorphism of X \E  to Xreg , and Xreg contains (a biholomorphic image of) M \p, the Kähler form  = π ∗ (ωδ ) retains the same expression (3.13) on M \ p, and is hence given by the expression (3.2). The proof of Lemma 1 is complete. It remains only to establish Lemma 2. In the special case when the background form  is actually strictly positive, this has been proved by X.X. Chen [Ch] and Blocki [B4]. But for our purposes, it is essential to allow degeneracies in  , as such degeneracies arise due to blow-ups. The full Lemma 2, allowing for degeneracies, is actually the main result of [PS4], stated there as Theorem 2. The desired solution of the homogeneous complex Monge-Ampère equation (3.3) is obtained as a C α limit on compact subsets of X\E of solutions of elliptic equations where the righthand side tends to 0. The key estimate is the following pointwise C 1 estimate ([PS4], Theorem 1) |∇ϕ(z)| ≤ C1 exp(C2 ϕ(z)) for the solutions of the Dirichlet problem for the equation 
i ¯ n ω + ∂ ∂ϕ = F (z, ϕ)ωn 2

(3.33)

(3.34)

where ω is a Kähler form on X . Here C1 and C2 are strictly positive constants which depend only on upper bounds for infX ϕ, supX ×[infX ϕ,∞) F , 1 supX ×[infX ϕ,∞) (|∇F n | + |∂ϕ F n1 |), ϕC 1 (∂X ) and a lower bound for the holomorphic bisectional curvature of ω. The point of the estimate (3.33) is that it does not require an upper bound for supX ϕ. It is used to obtain the existence of convergent subsequences in C α (X \E  ) for any 0 < α < 1 of solutions ϕ of the equation (3.34) which may tend to +∞ along a divisor E  . The proof of (3.33) makes essential use of a differential inequality for solutions of complex Monge-Ampère equations due to Blocki [B4]. We refer to [PS4] for the complete details. The proof of the main theorem is complete. 16.3.4 The case of strongly pseudoconvex manifolds We give now the proof of Theorem 2. First we observe that the theorem can be reduced to the case of boundary value 0 and ω strictly positive. Indeed, for any boundary value ϕ, we can pick an extension ϕˆ of ϕ to M¯ with the property that i∂ ∂¯ ϕˆ > 0. This can be done by choosing any

PLURICOMPLEX GREEN’S FUNCTION

431

¯ Next, set ωˆ = ω + i ∂ ∂¯ ϕ, extension, and adding a large positive multiple of i∂ ∂ρ. ˆ 2 ˆ and G = G − ϕ. ˆ The equation (1.1) can be rewritten as n
n
i i ¯ ˆ = ωˆ + ∂ ∂¯ G = 0 on M \ {p1 , · · · , pN }. (3.35) ω + ∂ ∂G 2 2 ˆ ∈ P SH (M, ω) with boundary value 0, then So if we can solve this equation for G ˆ the function G = G + ϕˆ ∈ P SH (M, ω) ˆ is a solution of the original problem. Next, assume that the boundary value is 0 and the form ω is strictly positive. We note that the restriction to εj small in the proof of Theorem 1 is just due to the requirement that the form ωδ of (3.13) be strictly positive. But in the present case, ¯ with A large for any δ, it suffices to replace the form ωδ by the form ωδ + A(δ)i∂ ∂ρ enough, in order to obtain a form which is strictly positive. The rest of the proof applies verbatim. The proof of Theorem 2 is complete.

16.4 PROOF OF THEOREM 3 Let X  = BL(X, p), the blow-up of the point p, and let π : X → X be the projection map. Choose, as before, a cut-off function ψ which is supported in a neighborhood of the point p. Then for ε sufficiently small, i ¯ (4.1) log |z|2 + (1 − ψ)) ωε = ω + ε∂ ∂(ψ 2 extends to a smooth Kähler metric ω on the smooth manifold X . Consider the equation on X
 i ¯  n  = cf (π ∗ ω)n (4.2) ω + ∂ ∂ϕ 2 for a function ϕ  ∈ P SH (X , ω ) with c a normalization constant so that both sides have the same total volume. This equation can be rewritten as 
i ¯  n = cF (ω )n (4.3) ω + ∂ ∂ϕ 2 ∗

n

is a smooth non-negative function. A careful examination of where F ≡ f (π(ωω) )n Yau’s treatment [Y1] of equations of the form (4.3) shows that a priori upper bounds for ϕ  C 0 (X ) and for ω ϕ  C 0 (X ) do not require a strictly positive lower bound for F . Thus the equation (4.3) admits a generalized solution ϕ ∈ P SH (X  , ω ) ∩ C 1,α (X ) for any 0 < α < 1. Restricting to X \E we get n
i ¯ 2  (4.4) ω + ∂ ∂(ε(ψ log |z| + 1 − ψ) + ϕ ) = cf ωn on X\{p}. 2 Thus, if we let ϕ = εψ log |z|2 + (1 − ψ) + ϕ  , we get 
i ¯ n ω + ∂ ∂ϕ = εδp + cf ωn 2 which implies c = 1 − ε. The proof of Theorem 3 is complete.

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Clearly the proof extends to the cases of local singularities fj (z), when the analytic set X = X(M, p, f ) is a smooth manifold. It is easy to formulate conditions on the fj (z) which would insure this property, but we leave this to the interested reader.

16.5 GEODESICS IN THE SPACE OF KÄHLER POTENTIALS Let (X, ω0 ) be a compact Kähler manifold without boundary. A well-known conjecture of Yau [Y2] is that the existence of a Kähler form in the class [ω0 ] with constant scalar curvature should be equivalent to the stability of (X, [ω0 ]) in the sense of geometric invariant theory. Suitable notions of stability have been proposed by Tian [T] and Donaldson [D98, D02] (see also [PS2], [PSSW] for some other notions of stability, and [PS5] for a survey). In particular, in [D98], Donaldson introduces a notion of stability based on the behavior of the K-energy functional of Mabuchi near infinity along geodesic rays in the space of Kähler potentials. Such rays have been constructed from test configurations (see [PS2], [PS3], [CT], [SZ], [RWN], and also [AT] in the analytic category, using the Cauchy-Kowalevski theorem). Here we illustrate Theorem 1 by constructing certain new rays, exploiting the fact that local singularities can be prescribed near infinity. More precisely, the space K of Kähler potentials is defined by   i ¯ >0 . (5.1) K = ϕ ∈ C ∞ (X); ωϕ ≡ ω0 + ∂ ∂ϕ 2 It carries a natural Riemannian structure defined by the L2 norm on Tϕ (K) with respect to the volume form ωϕn . A path (−T , 0]  t → ϕ(·, t) is a geodesic if and only if it satisfies the equation ¯

˙ k¯ ϕ˙ = 0 ϕ¨ − gϕj k ∂j ϕ∂

(5.2)

where gϕj k¯ is the metric corresponding to the Kähler form ωϕ . A key observation due to Donaldson [D98] and Semmes [Se] is that this equation is equivalent to the homogeneous complex Monge-Ampère equation


i ¯ π ∗ ω0 + ∂ ∂ 2

n+1 =0

(5.3)

on the manifold M = X × {e−T < |w| < 1}, for the function  defined by (z, w) = ϕ(z, log |w|),

(5.4)

and where ∂ is now with respect to both z and w. The end points of the geodesic paths in K correspond to Dirichlet boundary conditions for the equation (5.3) on M. Generalized geodesics will correspond to generalized solutions of the

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equation (5.3) in the sense of pluripotential theory, which are invariant under the rotation w → eiθ w. For the purpose of stability, we are particularly interested in geodesic rays, which correspond to T = ∞, and the manifold M is given by M = X × D × , with D × = {0 < |w| < 1} being the pointed disk. We compactify M into Mˆ = X × D, ¯ 2 is a Kähler by adjoining the central fiber X0 = X × {0}. Then ω + 2i ∂ ∂|w| metric on M. Let p1 , · · · , pN ∈ M be any N distinct points in the central fiber, i.e., π(pα ) = 0 ∈ C, where π : M → D is the projection on the second factor. For each α, let Uα be a neighborhood of pα in X, and let f1α (z, w), · · · , fn+1,α (z, w) be any n + 1 holomorphic functions on Uα × D, with the property that their only common zero is at (pα , 0) and with nj=1 |fj α (z, w)|2 invariant under the rotation w → eiθ w. Theorem 1 gives then a function G(z, w; p1 , · · · , pN ) with the prescribed 2 ¯ + G))n+1 = 0 on singularities (2.1) and satisfying the equation (ω + 2i ∂ ∂(|w| × X × D . We may choose the blow-ups in the proof of Theorem 1 to be equivariant under the above rotation. The function G(z, w; p1 , · · · , pN ) must also be invariant under rotation, so |w|2 +G defines a generalized geodesic ray in the space of Kähler metrics. Because of the singularities at (pα , 0), the rays are not trivial (i.e., ϕ is not constant along the ray). They are different from those previously obtained in the literature.

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[CG] Coman, D. and V. Guedj, “Quasi-plurisubharmonic Green’s functions,” J. Math. Pures Appl. 92 5 (2009) 456–475. [Ch] Chen, X. X., “The space of Kähler metrics,” J. Differential Geom. 56 (2000) 189–234. [CKNS] Caffarelli, L., J. J. Kohn, L. Nirenberg, and J. Spruck, “The Dirichlet problem for non-linear second order elliptic equations II. Complex MongeAmpère equations and uniformly elliptic equations,” Commun. Pure Appl. Math. 38 (1985) 209–252. [CT] Chen, X. X. and Y. Tang, “Test configurations and geodesic rays,” arXiv:0707.4149. [D] Demailly, J. P., “Complex analytic and differential geometry,” book available online at the author’s website. [D02] Donaldson, S., “Scalar curvature and stability of toric varieties,” J. Differential Geom. 59 (2002) no. 2, 289–349. [D98] Donaldson, S., “Symmetric spaces, Kähler geometry, and Hamiltonian dynamics,” Amer. Math. Soc. Transl. 196 (1999) 13–33. [Gb] Guan, B., “The Dirichlet problem for the complex Monge-Ampère equation and regularity of the pluricomplex Green’s function,” Commun. Anal. Geom. 6 (1998) no. 4, 687–703. [L] Lempert, L., “Solving the degenerate complex Monge-Ampère equation with one concentrated singularity,” Math. Ann. 263 (1983) 515–532. [PS2] Phong, D. H. and J. Sturm, “The Monge-Ampère operator and geodesics in the space of Kähler potentials,” Invent. Math. 166 (2006) 125–149, arXiv: math/0504157. [PS2] Phong, D. H. and J. Sturm, “On stability and the convergence of the KählerRicci flow,” J. Differential Geom. 72 (2006) no. 1, 149–168. [PS3] Phong, D. H. and J. Sturm, “Test configurations and geodesics in the space of Kähler potentials,” J. Symplectic Geom. 5 (2007) no. 2, 221–247, arXiv: math/0606423. [PS4] Phong, D. H. and J. Sturm, “The Dirichlet problem for degenerate complex Monge-Ampère equations,” Comm. Anal. Geom. 18 (2010) no. 1, 145–170, arXiv: 0904.1898. [PS5] Phong, D. H. and J. Sturm, “Lectures on stability and constant scalar curvature,” Current Developments in Mathematics Volume 2007 (2009) 101–176, arXiv: 0801.4179.

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[PSSW] Phong, D. H., J. Song, J. Sturm, and B. Weinkove, “The Kähler-Ricci flow and the ∂¯ operator on vector fields,” J. Differential Geom. 81 (2009) no. 3, 631–647. [RWN] Ross, J. and D. Witt Nystrom, “Analytic test configurations and geodesic rays,” arXiv:1101.1612. [Se] Semmes, S., “Complex Monge-Ampère equations and symplectic manifolds,” Amer. J. Math. 114 (1992) 495–550. [Si] Sibony, N., “Quelques problemes de prolongements de courants an analyse complexe,” Duke Math. J. 52 (1985) no. 1, 157–197. [SZ] Song, J. and S. Zelditch, “Test configurations, large deviations and geodesic rays on toric varieties,” arXiv:0712.3599. [T] Tian, G., “Kähler-Einstein metrics with positive scalar curvature,” Invent. Math. 130 (1997) 1–37. [Y1] Yau, S. T., “On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I,” Comm. Pure Appl. Math. 31 (1978) 339–411. [Y2] Yau, S. T., “Open problems in geometry,” Proc. Symposia Pure Math. 54 (1993) 1–28.

Chapter Seventeen Smoothness of Spectral Multipliers and Convolution Kernels in Nilpotent Gelfand Pairs Fulvio Ricci

17.1 INTRODUCTION Most of the problems of harmonic analysis on the Heisenberg group Hn have to do, in one way or another, with the left-invariant sublaplacian L, introduced by G. Folland and E. M. Stein in 1974 [11]. Often L appears in combination with the central derivative T = ∂t , like in the Folland-Stein operators L + iαT , which include the Kohn Laplacian on the boundary on the Siegel domain, or in the Laplace-Beltrami operators L + aT 2 , a > 0, of the left-invariant Riemannian metrics on Hn which are also invariant under the action of the unitary group Un . A widely studied subject is the analysis of operators m(L) defined by multipliers m(ξ ) on the positive half-line [12, 14, 18], or of the operators m(L, i −1 T ), involving the joint spectral analysis of (the self-adjoint extensions of) L and i −1 T , and with m(ξ, λ) defined on their joint L2 -spectrum (the Heisenberg fan) [16, 17, 22, 23]. Obviously, the operators m(L, i −1 T ) inherit Un -invariance from their generators L and T , i.e., their convolution kernels k are radial in the Cn -variable (as customary, we identify Hn with Cn × R). It is a basic fact that, conversely, every L2 bounded convolution operator on Hn by a radial distribution is a multiplier operator m(L, i −1 T ) for some bounded Borel function m on the Heisenberg fan [1, 22]. This correspondence between a radial distribution on Hn and the corresponding multiplier on the Heisenberg fan is completely analogous to the correspondence between a radial distribution k on Rn and the multiplier m of the Laplacian on ˆ ) = m(|ξ |2 )), but also between a general the positive half-line (i.e., such that k(ξ n distribution k on R and its Fourier transform kˆ = m. Various results in the literature show that the correspondence between a radial distribution on Hn and its multiplier m has many properties in common with the Fourier transform in Rn . It follows from a result of A. Hulanicki [14] that if a multiplier m on the positive half-line is the restriction of a Schwartz function, then the operator m(L) on Hn has a Schwartz convolution kernel. This result was later extended in [23] to operators of the form m(L, i −1 T ), where m is the restriction to the Heisenberg fan of a Schwartz function on R2 . Using dyadic decompositions, similar statements can be derived for certain classes of multipliers. For instance, if m is the restriction to the fan of a smooth

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Mihlin-Hörmander multiplier, i.e., satisfying the condition |∂ξ ∂λl m(ξ, λ)| ≤ Cj l (|ξ | + |λ|)−j −l , j

∀ j, l ,

∀ (ξ, λ)  = 0 ,

(1.1)

then k is a radial Calderón-Zygmund kernel, smooth away from the origin and satisfying the differential inequalities |∂zα ∂z¯ ∂tl k(z, t)| ≤ Cαβl (|z| + |t| 2 )−n−1−|α|−|β|−2l , β

1

∀ α, β, l ,

∀ (z, t)  = 0 , (1.2) together with the cancellation conditions adapted to nonisotropic dilations. Another result of the same type was proved in [16], but this time also containing the opposite implication: a radial distribution k on Hn is a smooth flag kernel1 with singularities on the t-axis if and only if the corresponding multiplier m is the restriction of a smooth flag multiplier on R2 with singularities on the ξ -axis. It is natural to ask if opposite implications also hold in the other situations considered above. Precisely, given the identity m(L, i −1 T )f = f ∗ k , consider the following questions: (i) assuming that k is a radial smooth Calderón-Zygmund kernel satisfying (1.2) and the appropriate cancellation conditions, is m the restriction to the fan of a smooth function on R2 satisfying (1.1)? (ii) assuming that k is a radial Schwartz function, is m the restriction of a Schwartz function on R2 ? A positive answer to question (ii) has been given in [1]. Using dyadic decompositions, it also implies a positive answer to (i). Due to the topology of the Heisenberg fan (which will be presented in the next section), (ii) is a harder question than (i), as well as (i) is harder than the analogous question for flag kernels answered in [16]. The same questions can be posed, in principle, whenever we have a finite family of homogeneous, self-adjoint, commuting, left-invariant differential operators on a homogenous nilpotent Lie group N . A natural situation to consider is the one where the given operators are characterized by the property of being invariant under the action of a given compact group K of automorphisms of N . More precisely, we assume that they generate the whole algebra D(N )K of left-invariant and K-invariant differential operators on N . This is the context of a special class of Gelfand pairs, which we call nilpotent Gelfand pairs for simplicity and denote2 by (N, K). The automorphism group K is assumed to be compact and large enough to verify either of the following two equivalent conditions: 1. the algebra D(N )K is commutative; 2. the convolution algebra L1 (N )K of K-invariant integrable functions on N is commutative. 1 2

We adopt here the terminology of flag kernels and multipliers, later introduced in [19, 20]. The correct notation for the Gelfand pair should be (K
N, K).

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Once we assign a generating set D = {D1 , . . . , Dd } of self-adjoint elements of D(N )K , the operators m(D1 , . . . , Dd ), with m bounded on their joint spectrum D ⊂ Rd of the Dj , coincide with the L2 -bounded convolution operators on N with a K-invariant kernel. It is true on every nilpotent Gelfand pair that, if m is the restriction to D of a Schwartz function on Rd , then the convolution kernel k of m(D1 , . . . , Dd ) is a K-invariant Schwartz function [2, 7, 14, 23]. An equivalent formulation of the same statement can be given in terms of the spherical transform on (N, K), i.e., the Gelfand transform for the commutative Banach algebra L1 (N )K . In fact, the set D is a natural model of the Gelfand spectrum of this algebra, and, if m and k are as above, m coincides with the Gelfand transform Gk of k. So the above statement can be rephrased as follows: if Gk extends to a Schwartz function on Rd , then k ∈ S(N )K . We are then interested in opposite implication: k ∈ S(N )K =⇒ Gk extends to a Schwartz function on Rd .

(S)

Property (S) has been proved in [2] to be true for all nilpotent Gelfand pairs in which N = Hn . More instances in which (S) holds have been considered in [1, 7]. In [8] we posed the question of the validity of (S) for general nilpotent Gelfand pairs, conjecturing that it should hold in all cases. The present state of the art is that (S) is known to be true for all nilpotent Gelfand pairs in which3 n/[n, n] is irreducible under the action of K (Vinberg’s condition) [8–10]. The general strategy of proof is based on a bootstrapping argument, whose steps are determined by the level of complexity of the pairs involved, where the “complexity” depends on the structure of N and the way K acts on the two layers [n, n] and n/[n, n] of the Lie algebra. The starting level is that of pairs in which N is abelian, and the next is that of pairs in which N is a Heisenberg group. The subsequent levels of complexity are determined by the rank of the action of K on [n, n] and Vinberg’s condition is assumed. In the following sections we sketch the main ideas of this program. After a brief discussion of the early steps, we will focus on a sample family of pairs with increasing rank.

17.2 THE EARLY STEPS The simplest class of nilpotent Gelfand pairs consists of those in which N = Rn and K is any closed subgroup of On [1, 2]. For such pairs, spherical analysis reduces to a modified version of Fourier analysis for K-invariant functions. Let P(Rn )K be the algebra of K-invariant polynomials on Rn , with ρ1 , . . . , ρd a finite set of real generators. We denote by ρ the map ρ = (ρ1 , . . . , ρd ) : Rn −→ Rd . 3

It must be remarked here that, in any nilpotent Gelfand pair, n is at most step-two [3].

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For j ≤ d, we set Dj = ρj (i −1 ∂x ) . Then D = {D1 , . . . , Dd } generates D(Rn )K . The following facts can be easily proved by Fourier analysis. Proposition 2.1. (i) The joint spectrum of the operators Dj is the set D = ρ(Rn ). (ii) If m is a bounded Borel function on D , then m(D1 , . . . , Dd )f = f ∗ k , where kˆ = m ◦ ρ. Since k is Schwartz and K-invariant if and only if the same holds for m ◦ ρ, then Property (S) is an immediate consequence of the following variant of G. Schwarz’s theorem [21]. Lemma 2.2 ([2]). Given any f ∈ S(Rn )K , there exists g ∈ S(Rd ) such that f = g ◦ ρ. Passing now to the next level of complexity, we take N = Hn . These pairs have been studied in [2]. We summarize the argument in the special case of radial functions, i.e., with K = Un . Following the notation of the Introduction, we take D = {L, i −1 T }, where L=−

n
(Xj2 + Yj2 ) j =1

is the sublaplacian and T the central derivative. Then D (the Heisenberg fan) is the union of the infinitely many closed half-lines in the (ξ, λ)-plane, exiting from the origin into the right half-plane ξ > 0 with slopes 0 and ±(n + 2l)−1 , l ∈ N. Consider first the problem of local smooth extendability of the spherical transform Gf of a radial Schwartz function f on Hn to a neighborhood in R2 of a given point (ξ, λ) of D . Due to the fact that the two operators in D are homogeneous of the same degree under the natural automorphic dilations of Hn , and that this homogeneity is reflected by the conic form of D , it is very easy to see that Gf depends smoothly on the scaling parameter along each half-line in the fan. Hence local smooth extendability is possible near each point of D which does not belong to the “singular” half-line λ = 0. From this one can conclude that if, for every N , Gf (ξ, λ) = o(λN )

(λ → 0)

(2.1)

uniformly on compact sets in ξ , then Gf admits a Schwartz extension to R2 . In order to remove assumption (2.1), we invoke the following lemma from [13].

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Lemma 2.3. Let f ∈ S(Hn )Un . There exists a sequence of functions gk ∈ S(Hn )Un such that Ggk only depends on the ξ variable and, for every m, m
1 k f = ∂t gk + ∂tm+1 rm , k! k=0

(2.2)

with rm ∈ S(Hn )Un . Reading (2.2) on the spherical transform side as an identity on D , it hints at the existence of a Taylor development of Gf on the singular half-line. Precisely, adapting the Whitney extension theorem to Schwartz jets, we can say that there exists a Schwartz function m on R2 with Taylor expansion in λ at λ = 0 dictated by (2.2) for ξ ≥ 0. Now we can say that, if k ∈ S(Hn )Un is the convolution kernel of m(L, i −1 T ) (so that Gk = m), then G(f − m) satisfies (2.1), and hence admits a Schwartz extension. This gives the conclusion. We mention that there are many proper subgroups K of Un that make (Hn , K) a nilpotent Gelfand pair. They are all classified in [4]. Examples of this kind are • K = Tn (“polyradial” functions, depending on |z 1 |2 , . . . , |z n |2 , t). A minimal D consists of the operators Dj = −(Xj2 + Yj2 ) , 1 ≤ j ≤ n ,

Dn+1 = i −1 T .

• K = SOn × T (functions depending on |z|2 , |z 2 |2 , t). Here we have  2  2

D1 = L , D2 =  (Xj2 − Yj2 ) +  (Xj Yj + Yj Xj ) , D3 = i −1 T . j

j

In the general setting of pairs (Hn , K), the application of the Whitney extension theorem requires Lemma 2.2 concerning pairs with N abelian. This is a first instance of the use of a bootstrapping procedure.

17.3 SPHERICAL ANALYSIS IN A SPECIAL CASE: HERMITIAN-FREE STEP-TWO GROUPS Denote by hfn = v ⊕ z the step-two, stratified, real Lie algebra with first layer v = Cn and second layer z = [v, v] consisting of skew-Hermitian n × n matrices. The Lie bracket of v, w ∈ Cn (regarded as column vectors) is given by 1 (vw∗ − wv ∗ ) . 2 More abstractly, hfn is the step-two Lie algebra freely generated by Cn over R, modulo the relations [iv, iw] = [v, w] for v, w ∈ Cn . Given k ∈ Un and (v, z) ∈ Cn ⊕ un , we set [v, w] =

k(v, z) = (kv, kzk ∗ ) .

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This gives an action of Un by automorhisms on hfn , and hence on the connected and simply connected Lie group H Fn with Lie algebra hfn . It is then convenient to recognize z as the vector space underlying the Lie algebra un of Un , and the action of K on z as the adjoint action. It is also natural to decompose

z = sun ⊕ iRI = z0 ⊕ z . It is well-known that (H Fn , Un ) is a nilpotent Gelfand pair [5, 15, 24]. For n = 1, H Fn is the three-dimensional Heisenberg group H1 . In the same way as the Heisenberg group Hn is identified with the Shilov boundary of the Siegel domain in Cn+1 , H Fn is identified with the Shilov boundary of the symmetric domain of elements (v, A) ∈ Cn × Cn×n satisfying the inequality 1 (A − A∗ ) > |v|2 I . 2i We describe the main aspects of spherical analysis on (N, K) = (H Fn , Un ). The Un -invariant functions on H Fn are those which can be expressed as functions of the 2n fundamental invariants ρj = v ∗ z j −1 v ,

(1 ≤ j ≤ n) ,

ρn+j = tr (iz)j ,

(1 ≤ j ≤ n) ,

(3.1)

cf. [8]. By symmetrization, each ρj produces a self-adjoint generator Dj of D(N )K . In particular, D1 is the sublaplacian, Dn+1 is the derivative in the direction of z , and Dn+2 is the Laplacian on z. We set D = (D1 , . . . , D2n ). The spherical transform of a function f ∈ L1 (N )K is given by integration against the bounded spherical functions, i.e., the bounded K-invariant eigenfunctions ϕ of the Dj ’s with ϕ(0, 0) = 1. Each bounded spherical function is identified by the 2n-tuple of its eigenvalues ξj , and the set D of these 2n-tuples is, at the same time, the Gelfand spectrum of L1 (N )K and the joint L2 -spectrum of the Dj ’s [2, 6]. By Fourier analysis on the center of N (or by representation theory of K
N ), each bounded spherical functions has the form   ∗ ∗ itr (τ ∗ kzk ∗ ) ψ(kv)e dk = ψ(kv)eitr ((k τ k) z) dk , (3.2) ϕ(v, z) = K

K

for some τ ∈ z. So we can naturally associate (in a non-injective way) to each ϕ the K-orbit Kτ in z. We may also assume that τ is in the Cartan subalgebra h of un consisting of diagonal matrices τ = diag(iτ1 , . . . , iτn ). Then τ is uniquely determined if we further impose that it belongs to the positive Weyl chamber h+ , defined by τ1 ≥ τ2 ≥ · · · ≥ τn .

(3.3)

We distinguish the elements τ of h+ according to the set of equalities in (3.3) or, equivalently, to the set of positive roots of (un , h) which vanish at τ , and regard this as a notion of singularity of the given element τ . The regular elements are those with all strict inequalities, and the “most singular” elements are those in z .

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17.4 REDUCTION TO QUOTIENT PAIRS The type of singularity of a point in h+ is important to establish local homeomorphisms between D and spectra of pairs of lower rank. To explain this, suppose that τ ∈ h+ has τ1 = · · · = τp1 > τp1 +1 = · · · = τp1 +p2 > · · · · · · > τp1 +···+pk−1 +1 = · · · = τn . Then the centralizer of τ in K is the subgroup K τ = Up1 × Up2 × · · · × Upk , and the normal space in z to the orbit Kτ at τ is ⊥  ad(un )τ = up1 ⊕ up2 ⊕ · · · ⊕ upk = zτ . The quotient Lie algebra nτ = hfn /ad(un )τ is then isomorphic to

v ⊕ zτ = hfp1 ⊕ hfp2 ⊕ · · · ⊕ hfpk , where p1 + p2 + · · · pk = n. We obtain in this way a quotient pair (N τ , K τ ) which is the direct product of the pairs (H Fpl , Upl ) for l = 1, . . . , k. For each l we denote by Dl ⊂ R2pl the joint spectrum of the operators in Dl constructed from the invariants (3.1) in dimension τ pl . Then the Gelfand spectrum of L1 (N τ )K is τ =

k

Dl ⊂ R2n .

l=1

We describe the relations between spherical functions for (N, K) and (N τ , K τ ), and the implications concerning (= D ) and  τ . Recall that points in  represent bounded spherical function for (N, K), and each spherical function is associated to a K-orbit in z, hence to a point in h+ . Similarly on  τ . This defines two continuous maps u :  −→ h+ ,

uτ :  τ  −→ h+ .

The notion of singularity can then be transferred, via the map u, from elements of h+ to elements of . Lemma 4.1 ([10]). Given τ ∈ h+ , let p1 , . . . , pk , N τ , K τ ,  τ be as above. (i) The maps u, uτ are continuous. (ii) Let ψ be a spherical function for (N τ , K τ ), and denote by π τ the canonical projection from N to N τ . Then  τ ϕ = ψ = (ψ ◦ π τ )(kv, kzk ∗ ) dk (4.1) K

is a spherical function for (N, K). (iii) The map τ , regarded as a function from  τ to , is continuous and surjective.

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(iv) Given a neighborhood V of τ in h+ , let (V ) = u−1 (V ) ⊂  and  τ (V ) = (uτ )−1 (V ) ⊂  τ . Then (V ) and  τ (V ) are open in  and  τ , respectively, and, if V is small enough, τ is a homeomorphism of  τ (V ) onto (V ). (v) There exist smooth functions ,  from R2n to itself such that |τ (V ) = τ and |(V ) = ( τ )−1 . Of course, when τ has top singularity, (N τ , K τ ) = (N, K) and the above statement becomes a triviality. The interest in what we are doing is then restricted to the set of points which are regular or have intermediate singularity. Assume therefore that this is the case. Formula (4.1) induces a correspondence between the two spherical transforms, G on (N, K) and G τ on (N τ , K τ ). In fact, let Rτ be the operator that projects functions f ∈ L1 (N )K into K τ -invariant functions on N τ by integrating over cosets of ad(un )τ ,  Rτ f (v, ζ ) = f (v, ζ + s) ds . (4.2) ad(un )τ

Then R satisfies the identity τ

G τ (Rτ f ) = (Gf ) ◦ τ . So, if we know that Property (S) is satisfied by the pair (N τ , K τ ), we can conclude that, for all f ∈ S(N )K , Gf ◦ τ extends to a function g ∈ S(R2n ), and by (v) of Lemma 4.1, g ◦  is a smooth extension of Gf restricted to (V ). On the basis of the following statement, the hypothesis that Property (S) is verified by the pair (N τ , K τ ) becomes an inductive hypothesis on the rank n of the pair (N, K). Lemma 4.2 ([10]). Suppose that two nilpotent Gelfand pairs (N1 , K1 ) and (N2 , K2 ) satisfy Property (S). Then also the product pair (N1 × N2 , K1 × K2 ) satisfies Property (S). In order to formulate the conclusion of this argument, we must first present some properties of the set  of most singular points in . Notice that a spherical function ϕ as in (3.2) belongs to  if and only if ϕ(v, z) = ϕ0 (v)eiτ tr z with ϕ0 radial and τ ∈ R. If this is the case, ϕ(v, ˜ t) = ϕ0 (v)eiτ t is a spherical function for the quotient pair (N , K) ∼ = (Hn , Un ), where N is the quotient of N with Lie algebra n = hfn /z0 ∼ =v⊕z. An equivalent condition for ϕ to be in  is that Dj ϕ = 0 for all the differential operators Dj derived from (3.1), except for D1 (the sublaplacian) and Dn+1 (the derivative in the direction of z ). This implies that, when viewed as a subset of ,  is the intersection of  itself with the coordinate subspace (ξ1 , ξn+1 ), and that this set is the Heisenberg fan. In analogy with (2.1), we then say that the spherical transform Gf of a function f vanishes of infinite order on  if, for every N ,   Gf (ξ ) = O |ξ˜ |N as ξ˜ = (ξ2 , . . . , ξn , ξn+2 , . . . , ξ2n ) → 0 , (4.3)

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uniformly on compact sets with respect to (ξ1 , ξn+1 ). We also denote by S0 (N )K the space of K-invariant Schwartz functions for which this holds. Proposition 4.3 ([10]). Assume that Property (S) is satisfied by all proper quotient pairs (Nt , Kt ) of (N, K). Then, for every f ∈ S0 (N )K , Gf extends to a Schwartz function on R2n . 17.5 JETS ON

The final step consists in removing the restriction (4.3), and we do this by means of a Hadamard-type formula, (5.1) below, which plays the same role of (2.2) in the Heisenberg case. The proof of (5.1) is in large part algebraic, relying on decomposition formulas of finite dimensional representations of K, and is altogether the most technically involved part of the whole subject. In order to formulate the statement, we must introduce some additional nota = {D2 , . . . , Dn , Dn+2 , . . . , D2n } from tion. We separate the set of operators D

If dj is the order

α for a monomial in the elements of D. {D1 , Dn+1 }. We write D αn+2 α2n

in the variables of z0 and D

α = D2α2 · · · Dnαn Dn+2 · · · D2n , we set of Dj ∈ D
|α|∼ = dj αj . j

Proposition 5.1 ([9, 10]). Given f ∈ S(N )K , there is a family of functions gα ∈ S(N )K , with α ∈ N2n−2 such that Ggα only depends on (ξ1 , ξn+1 ) and, for every m ∈ N,

1

α gα + ∂zγ rγ , (5.1) D f = α! |α| ≤m |γ |=m+1 ∼

where the derivatives are taken in directions of z0 and the functions rγ in the remainder term are Schwartz. γ ∂z

Once this is established, we can apply Property (S) for the pair (N , K) to state that there exist Schwartz extensions ψα of the spherical transforms Ggα to the (ξ1 , ξn+1 )-plane. Then (5.1) says that, on D ,
1  m/dmax +1  Gf (ξ ) = ξ α + O |

ξ| (

ξ → 0) , Ggα (ξ1 , ξn+1 )

α! |α| ≤m ∼

where

ξ = (ξ2 , . . . , ξn , ξn+2 , . . . , ξ2n ) and dmax = maxj dj . Since |α| ≤ m/dmax implies that |α|∼ ≤ m, we have that, for every m ,
1  m +1  Gf (ξ ) = (

ξ → 0) . ξ α + O |

ξ| Ggα (ξ1 , ξn+1 )

α! |α|≤m Applying Whitney’s extension theorem again, we produce a Schwartz function u with Taylor expansion
1 u∼ ξα . Ggα (ξ1 , ξn+1 )

α! 2n−2 α∈N

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445

We can repeat the argument used in Section 17.2 for the pair (Hn , Un ) to obtain the inductive statement that provided that Property (S) holds for all pairs (H Fk , Uk ) with k < n, then it holds for (H Fn , Un ). Since we know that (S) holds for n = 1, we have shown that Property (S) holds in fact for all pairs (H Fn , Un ).

REFERENCES [1] F. Astengo, B. Di Blasio, F. Ricci, Gelfand transforms of polyradial Schwartz functions on the Heisenberg group, J. Funct. Anal., 251 (2007), 772–791. [2] F. Astengo, B. Di Blasio, F. Ricci, Gelfand pairs on the Heisenberg group and Schwartz functions, J. Funct. Anal., 256 (2009), 1565–1587. [3] C. Benson, J. Jenkins, G. Ratcliff, On Gelfand pairs associated with solvable Lie groups, Trans. Amer. Math. Soc., 321 (1990), 85–116. [4] C. Benson, G. Ratcliff, A classification of multiplicity free actions, J. Algebra, 181 (1996), 152–186. [5] G. Carcano, A commutativity condition for algebras of invariant functions, Boll. Un. Mat. Ital. 7 (1987), 1091–1105. [6] F. Ferrari Ruffino, The topology of the spectrum for Gelfand pairs on Lie groups, Boll. Un. Mat. Ital., 10 (2007), 569–579. [7] V. Fischer, F. Ricci, Gelfand transforms of SO(3)-invariant Schwartz functions on the free nilpotent group N3,2 , Ann. Inst. Fourier Gren., 59 (2009), no. 6, 2143–2168. [8] V. Fischer, F. Ricci, O. Yakimova, Nilpotent Gelfand pairs and spherical transforms of Schwartz functions I. Rank-one actions on the centre, Math. Zeitschr. DOI 10.1007/s00209-011-0861-3. [9] V. Fischer, F. Ricci, O. Yakimova, Nilpotent Gelfand pairs and spherical transforms of Schwartz functions II. Taylor expansions on singular sets, to appear in Lie Groups: Structure, Actions and Representations, in honor of J. A. Wolf on the occasion of his 75th birthday. [10] V. Fischer, F. Ricci, O. Yakimova, Nilpotent Gelfand pairs and spherical transforms of Schwartz functions III. Isomorphisms between Schwartz spaces under Vinberg’s condition, in preparation. [11] G. Folland, E. M. Stein, Estimates for the ∂¯b -complex and analysis on the Heisenberg group, Comm. Pure Appl. Math., 27 (1974), 429–522. [12] G. Folland, E. M. Stein, Hardy Spaces on Homogeneous Groups, Princeton University Press, Princeton, N.J., 1982.

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[13] D. Geller, Fourier analysis on the Heisenberg group. I. Schwartz space, J. Funct. Anal., 36 (1980), 205–254. [14] A. Hulanicki, A functional calculus for Rockland operators on nilpotent Lie groups, Studia Math., 78 (1984), 253–266. [15] J. Lauret, Gelfand pairs attached to representations of compact Lie groups, Transf. Groups, 5 (2000), 307–324. [16] D. Müller, F. Ricci, E. M. Stein, Marcinkiewicz multipliers and multiparameter structure on Heisenberg(-type) groups, I, Inv. Math., 119 (1995), 199–233. [17] D. Müller, F. Ricci, E. M. Stein, Marcinkiewicz multipliers and multiparameter structure on Heisenberg(-type) groups, II, Math. Z., 221 (1996), 267–291. [18] D. Müller, E. M. Stein, On spectral multipliers for Heisenberg and related groups, J. Math. Pures Appl., 73 (1994), 413–440. [19] A. Nagel, F. Ricci, E. M. Stein, Singular integrals with flag kernels and analysis on quadratic CR manifolds, J. Funct. Anal., 181 (2001), 29–118. [20] A. Nagel, F. Ricci, E. M. Stein, S. Wainger, Singular integrals with flag kernels on homogeneous groups: I, to appear in Revista Mat. Iberoam., arXiv 1108.0177. [21] G.W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology, 14 (1975), 63–68. [22] R. Strichartz, Lp -harmonic analysis and Radon transforms on the Heisenberg group, J. Funct. Anal., 96 (19915), 350–406. [23] A. Veneruso, Schwartz kernels on the Heisenberg group, Boll. Unione Mat. Ital., 6 (2003), 657–666. [24] E. B. Vinberg, Commutative homogeneous spaces and co-isotropic symplectic actions, Russian Math. Surveys, 56 (2001), 1–60.

Chapter Eighteen On Eigenfunction Restriction Estimates and L4 -Bounds for Compact Surfaces with Nonpositive Curvature Christopher D. Sogge and Steve Zelditch 18.1 INTRODUCTION Let (M, g) be a compact two-dimensional Riemannian manifold without boundary. We shall assume throughout that the curvature of (M, g) is everywhere nonpositive. If
g is the Laplace-Beltrami operator associated with the metric g, then we are concerned with certain size estimates for the eigenfunctions −
g eλ (x) = λ2 eλ (x),

x ∈ M.

Thus
we are normalizing things so that eλ is an eigenfunction of the first order operator −
g with eigenvalue λ. If eλ is also normalized to have L2 -norm one, we are interested in various size estimates for the eλ which are related to how concentrated they may be along geodesics. If  denotes the space of all unit-length geodesics in M then our main result is the following “restriction theorem” for this problem. Theorem 1.1. Assume that (M, g) is as above. Then given ε > 0 there is a λ(ε) < ∞ so that 1/2  1 |eλ |2 ds ≤ ελ 4 eλ L2 (M) , λ > λ(ε), (1.1) sup γ ∈

γ

with ds denoting arc-length measure on γ , and L2 (M) being the Lebesgue space with respect to the volume element dVg for (M, g). Earlier, Burq, Gérard, and Tzvetkov [3] showed that for any 2-dimensional compact boundaryless Riemannian manifold one has  1/2 1 2 |eλ | ds ≤ Cλ 4 eλ L2 (M) , (1.2) γ

with C independent of γ ∈ . The first such estimates were somewhat weaker ones of Reznikov [13] for hyperbolic surfaces, which inspired this current line of research. The estimate (1.2) is sharp for the round sphere S 2 because of the highest The authors were supported in part by the NSF. Some of this research was carried out while the first author was visiting Zhejiang University in Hangzhou, China, and he wishes to thank his colleagues there for their kindness. We also wish to thank W. Minicozzi III for many very helpful discussions.

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weight spherical harmonics (see [3], [19]). Burq, Gérard, and Tzvetkov [3] also showed that 1/4  1 |eλ |4 ds ≤ Cλ 4 eλ L2 (M) , γ ∈ , γ

and so by interpolating with this result and (1.1) one concludes that when M has 1 nonpositive curvature supγ ∈ eλ Lp (γ ) /eλ L2 (M) = o(λ 4 ) for 2 ≤ p < 4. An interesting but potentially difficult problem would be to show that this remains true under this hypothesis for the endpoint p = 4. Theorem 1.1 is related to certain Lp -estimates for eigenfunctions. In [17] the first author proved that for any compact Riemannian manifold of dimension 2 one has for λ ≥ 1, eλ Lp (M) ≤ Cλ 2 ( 2 − p ) eλ L2 (M) , 1

1

1

2 ≤ p ≤ 6,

(1.3)

6 ≤ p ≤ ∞.

(1.4)

and eλ Lp (M) ≤ Cλ2( 2 − p )− 2 eλ L2 (M) , 1

1

1

These estimates are also sharp for the round sphere S 2 (see [16]). The first estimate, (1.3), is sharp because of the highest weight spherical harmonics, and thus, like (1.1) or (1.2), it measures concentration of eigenfunction mass along geodesics. The second estimate, (1.4), is sharp due to the zonal functions on S 2 , which concentrate at points. The sharp variants of (1.3) and (1.4) (with different exponents) for manifolds with boundary were obtained by H. Smith and the first author in [15], and it would be interesting to obtain analogues of the results in the present chapter for this setting, but this appears to be difficult. In the last decade there have been several results showing that, for typical (M, g), (1.4) can be improved for p > 6 (see [21], [22]) to bounds of the form 1 1 1 eλ Lp (M) /eλ L2 (M) = o(λ 2 ( 2 − p ) ) for fixed p > 6. Recently, Hassell and Tacey [9], following Bérard’s [1] earlier estimate for p = ∞, showed that for 1 1 1 √ fixed p > 6 this ratio is O(λ2( 2 − p )− 2 / log λ), which influenced the present work. Also, in [23] the authors showed that if the geodesic flow is ergodic, which is automatically the case if the curvature of M is negative, then (1.1) holds for a density one sequence of eigenfunctions. Except for some special cases of an arithmetic nature (e.g., Zygmund [27] or Spinu [24]) there have been few cases showing that (1.3) can be improved for Lebesgue exponents with 2 < p < 6. In [19], using in part results from Bourgain [2], it was shown that eλ Lp (M) /eλ L2 (M) = o(λ 2 ( 2 − p ) ) 1

1

1

for some 2 < p < 6 if and only if 1

sup eλ L2 (γ ) /eλ L2 (M) = o(λ 4 ).

γ ∈

Thus, we have the following corollary to Theorem 1.1.

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NONPOSITIVE CURVATURE

Corollary 1.2. As above, let (M, g) be a compact 2-dimensional manifold with nonpositive curvature. Then, if ε > 0 and 2 < p < 6 are fixed there is a λ(ε, p) < ∞ so that eλ Lp (M) ≤ ελ 2 ( 2 − p ) eλ L2 (M) , 1

1

1

λ > λ(ε, p).

We remark that an interesting open problem would be to obtain this type of result for the case of p = 6. It is valid for the standard torus T2 = R2 /Z2 since Zygmund [27] showed that there one has eλ L4 (T2 ) /eλ L2 (T2 ) = O(1) and the classical theorem of Gauss about lattice points in the plane yields eλ L∞ (T2 ) / 1 eλ L2 (T2 ) = O(λ 4 ). Since p = 6 is the exponent for which concentration at points and concentration along geodesics are both relevant, proving a general result along the lines of Corollary 1.2 would presumably have to take into account both of these phenomena. One expects, though, such a result for p = 6 should be valid when M has negative curvature. This result seems to be intimately related to the problem of trying to determine when one has the endpoint improvement for the 1 restriction problem, i.e., supγ ∈ eλ L4 (γ ) /eλ L2 (M) = o(λ 4 ). In [19] the first author showed that if γ0 ∈  is not part of a periodic geodesic then 1

eλ L2 (γ0 ) /eλ L2 (M) = o(λ 4 ). The proof involved an estimate involving the wave equation associated with
g and a bit of microlocal (wavefront) analysis. The main step in proving Theorem 1.1 is to see that this remains valid as well if γ0 is part of a periodic orbit under the above curvature assumptions. We shall be able to do this by lifting the wave equation for (M, g) up to the corresponding one for its universal cover, which by a classi˜ with the metric g˜ cal theorem of Hadamard [7] and von Mangolt [26] is (R2 , g), being the pullback of g via a covering map, which can be taken to be expx0 for any x0 ∈ M. By identifying solutions of wave equations for (M, g) with “peri˜ we are able to obtain the necessary bounds using a bit of odic” ones for (R2 , g) ˜ Fortunately for us, wavefront analysis and the Hadamard parametrix for (R2 , g). by a classical volume comparison theorem of Günther [6], the leading coefficient of the Hadamard parametrix has favorable size estimates under our curvature assumptions. (It is easy to see that the contribution of the lower order terms in the Hadamard parametrix to (1.1) are straightforward to handle.)

18.2 PROOF OF GEODESIC RESTRICTION BOUNDS Since the space of all unit-length geodesics is compact, in order to prove (1.1), it suffices to show that, given γ0 ∈  and ε > 0, one can find a neighborhood N (γ0 , ε) of γ0 in  and a number λ(γ0 , ε) so that  1 |eλ |2 ds ≤ ελ 2 eλ 2L2 (M) , γ ∈ N (γ0 , ε), λ > λ(γ0 , ε). (2.1) γ

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SOGGE AND ZELDITCH

In proving this we may assume that the injectivity radius of (M, g) is ten or more. We recall also that, given x0 ∈ M, the exponential map at x0 , expx0 : Tx0 M  R2 → M is a universal covering map. We shall take x0 to be the midpoint of our unit-length geodesic γ0 . We also shall work in geodesic polar coordinates about x0 . ˜ If g˜ is the pullback to R2 of the metric g via the covering map then (R2 , g) is a Riemannian universal cover of (M, g). Like (M, g) it also has nonpositive curvature. Additionally, rays t → t (cos θ, sin θ ), t ≥ 0, through the origin are geodesics for g. ˜ Such a ray is the lift of the unit speed geodesic starting at x0 , which in our local coordinate system has the initial tangent vector (cos θ, sin θ ). Note that in these coordinates vanishing at x0 , t → t (cos θ, sin θ ), |t| ≤ 10 are also geodesics for g. We may assume further that we have 1 1 (2.2) γ0 = {(t, 0) : − ≤ t ≤ }. 2 2 To prove (2.1) it will be convenient to fix a real-valued even function χ ∈ S(R) having the property that χ (0) = 1 and χ(t) ˆ = 0, |t| ≥ 14 , where χˆ denotes the Fourier transform of χ . We then have that for T > 0
χ (T ( −
g − λ))eλ = eλ , and, therefore, to prove (2.1), it suffices to show that if T is large and fixed then there is a neighborhood1 N = N (γ0 , T ) of γ0 so that   
2  χ (T ( −
g − λ))f 2 ds ≤ CT −1 λ 12 f 2 2 γ ∈ N, L (M) + CT ,N f L2 (M) , γ

(2.3) where C (but not CT ,N ) is a uniform constant depending on (M, g) but independent of T and N . To prove (2.3), we shall be able to use the wave equation as  √
1 χˆ (t/T )e−itλ eit −
g f dt χ (T ( −
g − λ))f = 2π T R  T /4
1 = χ(t/T ˆ )e−itλ cos t −
g f dt π T −T /4
(2.4) +χ (T ( −
g + λ))f, using the fact that χˆ (t) is even and supported in |t| ≤ 14 . Since the kernel of the last term satisfies
α χ (T ( −
g + λ))(x, y)| ≤ CT ,N λ−N (2.5) |∂x,y for any N in compact subsets of any local coordinate system, to prove (2.3) it suffices to show that 2  T /4     1
  1 −itλ  χˆ (t/T )e cos t −
g f dt  ds ≤ CT −1 λ 2 + CT ,N πT γ

−T /4

×f L2 (M) ,

γ ∈ N (γ0 , T ).

(2.6)

We can use the topology of S ∗ M to define these neighborhoods, since every γ ∈  can be uniquely identified with an element (y, ξ ) ∈ S ∗ M, with y being the midpoint of γ and ξ being the direction of γ at y. 1

451

NONPOSITIVE CURVATURE

If γ0 is not part of a periodic geodesic of period ≤ T , then we can easily prove (2.6) just by using wavefront analysis and arguments that are similar to the proof of the Duistermaat-Guillemin theorem [5]. This was done in [19], but we shall repeat the argument here for the sake of completeness and since it motivates what is needed to handle the argument when γ0 is a portion of a periodic geodesic of period ≤ T . To handle the latter case we shall exploit the relationship between solutions of the wave equation on (M, g) of the form  (∂t2 −
g )u(t, x) = 0, (t, x) ∈ R+ × M (2.7) u(0, · ) = f, ∂t u(0, · ) = 0, and certain ones on (R, g) ˜  ˜ x), ˜ (t, x) ˜ ∈ R + × R2 (∂t2 −
g˜ )u(t, (2.8) u(0, ˜ · ) = f˜, ∂t u(0, ˜ · ) = 0.  
Note that u(t, x) = cos(t −
g )f (x) is the solution of (2.7). To describe the relationship between the two equations we shall use the deck transformations associated with our universal covering map p = expx0 : R2 → M.

(2.9)

˜ α : R2 → R2 , is a deck transformation if Recall that an automorphism for (R2 , g), p ◦ α = p. In this case we shall write α ∈ Aut(p). In the case where T2 is the standard two-torus, each α would just be translation in R2 with respect to some j ∈ Z2 . ˜ the translate of x˜ by Motivated by this if x˜ ∈ R2 and α ∈ Aut(p), let us call α(x) α. then we recall a set D ⊂ R2 is called a fundamental domain of our universal covering p if every point in R2 is the translate of exactly one point in D. Of course there are infinitely many fundamental domains, but we may assume that ours is relatively compact, connected, and contains the ball of radius 2 centered at the origin in view of our assumption about the injectivity radius of (M, g). We can then think of our unit geodesic γ0 = {(t, 0) : |t| ≤ 12 } (written in geodesic polar coordinates as above) both as one in (M, g) and one in the fundamental domain which is of the same form. Likewise, a function f (x) on M is uniquely identified by ˜ on D if we set fD (x) ˜ = f (x), where x˜ is the unique point in D∩p −1 (x). one fD (x) ˜ to Using fD we can define a “periodic extension,” f˜, of f to R2 by defining f˜(y) ˜ if x˜ = y˜ modulo Aut(p), i.e., if (x, ˜ α) ∈ D × Aut(p) are the be equal to fD (x) unique pair so that y˜ = α(x). ˜ Note then that f˜ is periodic with respect to Aut(p) since we necessarily have that f˜(x) ˜ = f˜(α(x)) ˜ for every α ∈ Aut(p). We can now describe the relationship between the wave equations (2.7) and (2.8). First, if (f (x), 0) is the Cauchy data in (2.7) and (f˜(x), ˜ 0) is the periodic extension ˜ then the solution u(t, ˜ x) ˜ to (2.8) must also be a periodic function of x˜ to (R2 , g), since g˜ is the pullback of g via p and p = p ◦ α. As a result, we have that the solution to (2.7) must satisfy u(t, x) = u(t, ˜ x) ˜ if x˜ ∈ D and p(x) ˜ = x. Another way of saying this is that if f˜ is the pullback of f via p and t is fixed then u(t, ˜ ·)

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SOGGE AND ZELDITCH

solving (2.8) must be the pullback of u(t, · ) in (2.7). Thus, periodic solutions to (2.8) correspond uniquely to solutions of (2.7). In other words, we have the important formula for the wave kernels 

  cos(t −
g˜ (x, ˜ α(y)), ˜ (2.10) cos(t −
g )(x, y) = α∈Aut(p)

if x˜ and y˜ are the unique points in D for which p(x) ˜ = x and p(y) ˜ = y. Note that the sum in (2.10) only has finitely many nonzero terms for a given (x, y, t) since, by the finite propagation speed for
g˜ = ∂t2 −
g˜ , the summands in ˜ α(y)) ˜ > t. For instance, if x = y = x0 the number the right all vanish when dg˜ (x, ˜ ≤ t} of nontrivial terms would equal the cardinality of p−1 (x0 ) ∩ {x˜ ∈ R2 : |x| ˜ = |x|. ˜ Despite where |x| ˜ denotes the Euclidean length, due to the fact that dg˜ (0, x) this, the number of nontrivial terms will grow exponentially in t if the curvature is bounded from above by a fixed negative constant. To see this, let us review one last thing before focusing more closely on the proof of our restriction-estimate. As we shall see, even though there can be an exponentially growing number of nontrivial terms in the right-hand side of (2.10), which could create havoc for our proofs if we are not careful, this turns out to be related to something that will actually be beneficial for our calculations. These facts are related to the fact that in the geodesic polar coordinates we are ˜ the metric g˜ takes the form using, (t cos θ, t sin θ ), t > 0, θ ∈ (−π, π], for (R2 , g), ds 2 = dt 2 + A2 (t, ξ ) dθ 2 ,

(2.11)

where we may assume that A(t, θ ) > 0 for t > 0. Consequently, the volume element in these coordinates is given by dVg (t, θ ) = A(t, θ ) dtdθ,

(2.12)

and by Günther’s [6] comparison theorem if the curvature of (M, g) and hence that ˜ is nonpositive, we have of (R2 , g) A(t, θ ) ≥ t.

(2.13)

Furthermore, if one assumes that the curvature is ≤ −κ 2 , with κ > 0 then one has 1 sinh(κt). (2.14) κ Since the volume element for two-dimensional Euclidean space in polar coordinates is t dtdθ and that of the hyperbolic plane with constant curvature −κ 2 is 1 sinh(κt) dtdθ, Günther’s volume comparison theorem says that in geodesic poκ lar coordinates the volume element for spaces of nonpositive curvature is at least that of R2 with the flat metric, while if the curvature is bounded above by −κ 2 the volume element is at least that of the hyperbolic plane of constant curvature −κ 2 . In the latter case, as we warned, the number of nontrivial terms in the sum in the right side of (2.10) will be at least bounded below by a multiple of eκt as t → +∞. Let us now turn to the proof of (2.6) and hence Theorem 1.1. Given γ ∈  we let T ∗ γ ⊂ T ∗ M and S ∗ γ ⊂ S ∗ M be the cotangent and unit cotangent bundles over γ , respectively. Thus, if (x, ξ ) ∈ T ∗ γ then ξ is a tangent vector to γ at x A(t, θ ) ≥

453

NONPOSITIVE CURVATURE

if T ∗ M  ξ → ξ ∈ T M is the standard musical isomorphism, which, in local

j coordinates, sends ξ = (ξ1 , ξ2 ) ∈ Tx∗ M to ξ = (ξ 1 , ξ 2 ) with ξ = k g j k (x)ξk . ∗ ∗ Then if t : S M → S M denotes geodesic flow in the unit cotangent bundle over M, and (x, ξ ) ∈ S ∗ γ we let L(x, ξ ) be the minimal t > 0 so that t (x, ξ ) = (x, ξ ) and define it to be +∞ if no such time t exists. Then if γ is not part of a periodic geodesic this quantity is +∞ on S ∗ γ , and if it is then it is constant on S ∗ γ and equal to the minimal period of the geodesic, (γ ) (which must be larger than ten because of our assumptions). Note also that L(x, ξ ) can also be thought of as a function on S ∗ M, and that, in this case, it is lower semicontinuous. Recall that we are working in geodesic polar coordinates vanishing at x0 , the midpoint of γ0 , and that γ0 is of the form (2.2) in these coordinates. Let us choose β ∈ C0∞ (R) equal to one on [− 34 , 34 ] but 0 outside [−1, 1]. We then let bε (x, D) and Bε (x, D) be zero-order pseudodifferential operators which in the above local coordinates have symbols bε (x, ξ ) = β(|x|)β(ξ2 /ε|ξ |),

and Bε (x, ξ ) = β(|x|)(1 − β(ξ2 /ε|ξ |)),

respectively. Our first claim is that if ε > 0 and γ ∈  are fixed, then we can find a neighborhood N (γ0 , ε) of γ0 so that  T /4   2
   Bε ◦ cos(t −
g )f  dsdt ≤ CT ,ε f 2L2 (M) , γ ∈ N (γ0 , ε), −T /4

γ

(2.15) which, by an application of the Schwartz inequality, would yield part of (2.6), namely,  T /4   2
  1 χ(t/T ˆ )e−iλt Bε ◦ cos(t −
g )f dt  ds ≤ CT ,ε f 2L2 (M) ,  γ π T −T /4 γ ∈ N (γ0 , ε).

(2.16)

If Rγ denotes the restriction to γ ∈ , then (2.15) follows from the fact that the operator
f → Rγ (A ◦ cos(t −
g )f ), regarded as an operator from C ∞ (M) → C ∞ (γ × [−T /4, T /4]), is a Fourier integral operator of order zero which is locally a canonical graph (i.e., nondegenerate) if supp A(x, ξ ) ∩ S ∗ γ = ∅, and hence a bounded operator from L2 (M) to L2 (γ × [−T /4, T /4]). since Bε (x, ξ ) vanishes on a neighborhood of S ∗ γ0 , we conclude that this is the case A = Bε for γ ∈  close to γ0 , which gives us (2.15). is a theorem of HörmanThe L2 -boundedness of nondegenerate Fourier integrals
der [10], while the observation about Rγ (A ◦ cos(t −
g )) is one of Tataru [25]. It √ is also easy to check the latter, because, for fixed t, eit −
g : C ∞ (M) → C ∞ (M) is a nondegenerate Fourier integral operator, and, therefore, one needs only to verify the assertion when t = 0, in which case it is an easy calculation using any parametrix for the half-wave operator. The estimate (2.16) holds for any γ0 ∈ . Let us now argue that if (γ0 ), the period of γ0 , is larger than T or if γ0 is not part of a periodic geodesic, then we also

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SOGGE AND ZELDITCH

have favorable bounds if Bε is replaced by bε , with ε > 0 sufficiently
small. To do this, we recall that the wave front set of the kernel of bε ◦ cos(t −
g ) ◦ bε∗ is contained in  (x, t, ξ, τ ; y, −η) : ±t (x, ξ ) = (y, η), (2.17) g j k (x)ξj ξk , (x, ξ ), (y, η) ∈ supp bε . τ2 = To exploit this, let Wγ be the operator 

1  T /4
χ(t/T ˆ )e−iλt bε ◦ cos(t −
g )f dt . Wγ f = Rγ π T −T /4

(2.18)

Our goal then is to show that, under the present assumption that (γ0 ) > T , Wγ L2 (M)→L2 (γ ) ≤ CT − 2 λ 4 + CT ,bε 1

1

for γ ∈  belonging to some neighborhood N (γ0 , T , ε) of γ0 . This is equivalent to showing that the dual operator Wγ∗ : L2 (γ ) → L2 (M) with the same norm, and since  Wγ∗ g2L2 (M) = Wγ Wγ∗ g g ds ≤ Wγ Wγ∗ gL2 (γ ) gL2 (γ ) , γ

we would be done if we could show that

 1 Wγ Wγ∗ gL2 (γ ) ≤ CT −1 λ 2 + CT ,bε gL2 (γ ) .

(2.19)

But, by Euler’s formula, the kernel of 4Wγ Wγ∗ is K|γ ×γ , where K(x, y), x, y ∈ M is the kernel of the operator

bε ◦ ρ(T ( −
g − λ)) ◦ bε∗ + bε ◦ ρ(T ( −
g + λ)) ◦ bε∗

+2bε ◦ χ (T ( −
g − λ))χ (T ( −
g + λ)) ◦ bε∗ , if ρ(τ ) = (χ (τ ))2 . The last two terms satisfy bounds like those in (2.5) (with constant depending on T and bε ), and the first term is  T /2
  1 ρ(t/T ˆ )e−iλt bε ◦ cos(t −
g ) ◦ bε∗ (x, y) dt. (2.20) π T −T /2 We are using the fact that ρˆ = χˆ ∗ χˆ is supported in [− 12 , 12 ]. In view of (2.17), if ε > 0 is sufficiently small, since we are assuming that (γ0 )√> T , it follows that we can find a neighborhood N of γ0 in M so that (bε ◦ cos(t −
g ) ◦ bε∗ )(x, y) is smooth on N × N when t ≥ 2. Thus, on N × N the difference between (2.19) and  T /2


1 β(t/5) ρ(t/T )e−itλ bε ◦ cos(t −
g ) ◦ bε∗ (x, y) dt K(x, y) = π T −T /2 is OT ,bε (1). But, by using the Hadamard parametrix (see below), one finds that |K(x, y)| ≤ CT −1 λ 2 (dg (x, y))− 2 + Cbε ,T (1 + λ (1 + λdg (x, y))− 2 ), 1

1

3

x, y ∈ N , (2.21) for some uniform constant C, which is independent of ε, T and λ. Since, by Young’s inequality, the integral operator with kernel K|γ ×γ is bounded from

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NONPOSITIVE CURVATURE

L2 (γ ) → L2 (γ ) with norm bounded by CT −1 λ + Cbε ,T if γ ⊂ N , we get (2.19), which finishes the proof that (2.3) holds provided that (γ0 ) > T . The above argument used the fact that if (γ0 ) > T , with T fixed, then if ε > 0 / supp bε for is small enough and (x, ξ ) ∈ supp bε with x ∈ γ0 , then t (x, ξ ) ∈ 2 < |t| ≤ T /2. In effect, this allowed us to cut the effect of loops though γ0 of its extension of length T from our main calculation, since they were all transverse. If γ0 ∈  is part of a periodic geodesic of period ≤ T , i.e., (γ0 ) ≤ T , then this need not be true. On the other hand, if T is fixed and (x, ξ ) is as above, then for sufficiently small ε we will have   j (γ0 ) − 2, j (γ0 ) + 2 . (2.22) / supp bε , if x ∈ γ0 , and t ∈ / ±t (x, ξ ) ∈ 1 2

j ∈Z

Note that our assumption that the injectivity radius of (M, g) is 10 or more implies that (γ0 ) ≥ 10. To exploit this, we shall use (2.10) which relates the wave kernel for (M, g) with the one for its universal cover using the covering map given by p = expx0 with x0 being the midpoint of γ0 . Note that the points α(0), α ∈ Aut(p) exactly correspond to geodesic loops through x0 , with looping time being equal to the distance from α(0) to the origin in R2 . Just a few of these correspond to smooth loops through x0 along the periodic geodesic containing γ0 . Since we are assuming that we are ˜ working with local coordinates on (M, g) and global geodesic polar ones on (R2 , g) so that γ0 is of the form (2.2), the automorphisms with this property are exactly the αj ∈ Aut(p), j ∈ Z for which   (2.23) αj (0) = j (γ0 ), 0 . Note that Gγ0 = {αj }j ∈Z is a cyclic subgroup of Aut(p) with generator α1 , which is the stabilizer group for the lift of periodic geodesic containing γ0 . Consequently, we can choose ε > 0 small enough and a neighborhood N of γ0 in M so that2
     bε ◦ cos(t −
g˜ ) ◦ bε∗ (x, ˜ α(y)) ˜ ∈ C ∞ N × N × j (γ0 ) − 2, j (γ0 ) + 2 , if Aut(p)  α ∈ / Gγ0 .

(2.24)

Therefore, by (2.22)–(2.24), if we repeat the arguments that were used to prove (2.19), we conclude that we would have    ∞ 2
 1  χ(t/T ˆ )e−iλt bε ◦ cos(t −
g )f dt  ds  π T −∞ γ 2  1 1 (2.25) ≤ CT − 4 λ 4 + CT ,bε f 2L2 (M) , γ ∈ N (γ0 , T ), 2

We should point out that we are abusing notation a bit in
(2.24). The last factor denotes the kernel ˜ α(y)) ˜ composed on the left and of the integral operator on M with kernel H (x, y) = (cos t −
g˜ )(x, right by bε and bε∗ , respectively, and as before we are identifying points x in M with their cousins x˜ in the fundamental domain. In the coordinate systems we are using, though, both are the same when we are close to γ0 .

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for some neighborhood N (γ0 , T ) in , if we could show that if the αj are as in (2.23) and  ∞   1 β (s − j (γ0 ))/5 ρ(s/T ˆ )e−isλ K(x, y) = πT −∞ {j ∈Z+ : j t (γ0 )≤T /2}
  ˜ αj (y)) × bε ◦ cos s −
g˜ ◦ bε∗ (x, ˜ ds, (2.26) then

 1 1 1 1 3  |K(x, y)| ≤ CT −1 λ 2 (dg (x, y))− 2 + T − 2 λ 2 + CT ,bε (1 + λ (1+ λdg (x, y))− 2 ) , x, y ∈ N ,

(2.27)

with N being some neighborhood in M of γ0 (depending on T ). The second term in the right side of this inequality did not occur in the previous steps. It comes from the terms in (2.26) with j  = 0. Also, the fact that (2.27) yields (2.25) just follows from an application of Young’s inequality. To prove (2.27), it suffices to see that we can find N as above so that   
     ˜ αj (y)) ˆ )e−isλ bε ◦ cos s −
g˜ ◦ bε∗ (x, ˜ ds   β (s − j (γ0 ))/5 ρ(s/T  − 12 1  ˜ j (y)) ˜ ˜ αj (y)), ˜ eκdg (x,α + CT ,bε , ≤ Cλ 2 max dg (x, x, y ∈ N , 0  = |j |(γ0 ) ≤ T ,

(2.28)

assuming that the curvature of (M, g) is everywhere ≤ −κ , κ ≥ 0, while for j = 0, we have 
    β(s/5)ρ(s/T ˆ )e−isλ bε ◦ cos s −
g ◦ bε∗ (x, y) ds  2

 1 1 3  ≤ Cλ 2 (dg (x, y))− 2 + CT ,bε 1 + λ (1 + λdg (x, y))− 2 ,

x, y ∈ N .

(2.29)

Note that dg˜ (x, ˜ αj (y)) ˜ ∈ [j (γ0 ) − 1, j (γ0 ) + 1] when x, y ∈ γ0 and hence ˜ αj (y)) ˜ ≥ |j | when x, y ∈ N with N being a small neighborhood of γ0 in dg˜ (x, M. We shall assume that this is the case in what follows. We then get (2.27) by summing over j . (Observe that if the curvature is assumed to be bounded below by a negative constant, we get something a bit stronger than (2.27) where in the 1 second term we may replace T − 2 by T −1 .) Both (2.28) and (2.29) are routine consequences of stationary phase and the Hadamard parametrix for the wave equation. To prove (2.29) let φ(x, y) denote geodesic normal coordinates of y about x. Then if |t| ≤ 5, by the Hadamard parametrix (see [11] or [20]) and the composition calculus for Fourier integral operators (see Chapter 6 in [18]) 
  eiφ(x,y)·ξ ±it|ξ | aε (x, y, ξ ) dξ + Oε (1), bε ◦ cos(t −
g ) ◦ bε∗ (x, y) = ±

R2

(2.30) 0 depends on −
g and bε but satisfies where aε ∈ S1,0

|aε | ≤ C,

α and |∂x,y ∂ξσ aε | ≤ Cεασ (1 + |ξ |)−|σ | .

(2.31)

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The first constant is independent of C and only depends on the size of the symbol of bε , which is ≤ β4L∞ (R) . Recall (see [18]) the following fact about the Fourier transform of a density times Lebesgue measure on the circle S 1 = { = (cos θ, sin θ )},  2π 1 3 eiw· aε (x, y, ) dθ = |2π w|− 2 e±i|w| aε (x, y, ±w) + Oε (|w|− 2 ), 0

±

|w| ≥ 1,

(2.32)

where the constants for the last term depend on the size of finitely many constants in (2.32). Since |φ(x, y)| = dg (x, y), if we combine (2.30) and (2.31), we find ˆ ), then when dg (x, y) ≥ λ−1 , that, modulo a Oε (1) term, if ψ(s) = β(s/5)ρ(s/T the quantity in (2.29) is the sum over ± of a fixed multiple of  ∞   − 12 ˆ − r) + ψ(λ ˆ + r) e±irdg (x,y) aε (x, y, ±rφ(x, y))r 12 dr ψ(λ (dg (x, y)) 0  ∞   − 32 ˆ − r)| + |ψ(λ ˆ + r)|) (1 + r)− 12 dr . +Oε (dg (x, y)) (|ψ(λ 0 1 ˆ )| ≤ CN (1 + |τ |)−N By (2.31), the first term is O(aε ∞ (λdg (x, y))− 2 ), since |ψ(τ − 32 −1/2 for any N. Since the last term is Oε (λ (dg (x, y)) ), we have established (2.29) when dg (x, y) ≥ λ−1 . The fact that it is also O(λ)+Oε (1) is a simple consequence of (2.30) and (2.31) which gives the bounds for dg (x, y) ≤ λ−1 and concludes the proof of (2.29). To prove
(2.29) we can exploit the fact that, unlike the case of t = 0, if t  = 0 then cos t −
g : C ∞ (M) → C ∞ (M) is a conormal Fourier integral operator with singular support of codimension one. Based on this and (2.17) we deduce that if (x, t, ξ, τ ; y, η) is in the wave front set of
˜ αj (y)), ˜ j  = 0, (cos(t −
g˜ ))(x,

and both x and y are on γ0 , then both ξ and η must be on the first coordinate axis. Therefore, since the symbol, bε (x, ξ ), of bε equals one when x ∈ γ0 and ξ is in a conic neighborhood of this axis (depending on ε), we conclude that there must be a neighborhood N of γ0 in M so that

    bε ◦ cos(t −
g˜ ) ◦ bε∗ (x, ˜ − cos t −
g˜ (x, ˜ ∈ C ∞ (N × N ), ˜ αj (y)) ˜ αj (y)) 0  = |j |(γ0 ) ≤ T . Because of this, we would have the remaining inequality, (2.28), if we could show that  
    ˜ αj (y)) ˆ )e−isλ cos s −
g˜ (x, ˜ ds   β((s − j (γ0 ))/5)ρ(s/T  − 12 1  ˜ j (y)) ˜ ˜ αj (y)), ˜ eκdg (x,α + CT ≤ Cλ 2 max dg (x, x, y ∈ N , 0  = |j |(γ0 ) ≤ T .

(2.33)

To prove this, we shall use the fact that on (R2 , g) ˜ we can use the Hadamard parametrix even for large times. Recall that the Hadamard parametrix says that if

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we set E0 (t, x) = (2π )−2

 R2

eix·ξ cos(t|ξ |) dξ,

t and define Eν , ν = 1, 2, 3, . . . recursively by 2Eν (t, x) = t 0 Eν−1 (s, x)ds, ν = 1, 2, 3, . . . , then there are functions wν ∈ C ∞ (R2 × R2 ) so that we have N
  wν (x, y) Eν (t, dg˜ (x, y)) + RN (t, x, y), cos(t −
g˜ ) (x, y) = ν=0 2 L∞ loc (R × R

where for n = 2, RN ∈ × R2 ) if N ≥ 10. We are abusing the notation a bit by putting Eν (t, r) equal to the radial function Eν (t, x) for some |x| = r. The Eν , ν = 1, 2, 3, . . . , are Fourier integrals of order −ν; for instance,  t sin t|ξ | −2 eix·ξ dξ. E1 (t, x) = (2π ) 2|ξ | R2 As a result of this, we would have (2.33) if we could show that      ˜ j (y))·ξ ˜ ˜ αj (y)) ˜ β((s − j (γ0 ))/5)ρ(s/T ˆ ) e−iλs ei(x−α cos(s|ξ |) dξ ds  w0 (x,  − 12 1  ˜ j (y)) ˜ ˜ αj (y)), ˜ eκdg (x,α , ≤ Cλ 2 max dg (x,

j = 1, 2, . . . ,

(2.34)

as well as     ˆ )e−isλ Eν (s, dg (x, ˜ αj (y)) ˜ ds  ≤ Cν ,  β((s − j (γ0 ))/5)ρ(s/T 0  = j (γ0 ) ≤ T , ν = 1, 2, 3, . . . .

(2.35)

Here we are using the fact that |wν (x, y)| ≤ CT for |x|, |y| ≤ T . If we repeat the stationary phase argument that was used to prove (2.29), we see that the left side of (2.34) is dominated by a fixed constant times ˜ αj (y)) ˜ (dg˜ (x, ˜ αj (y))) ˜ −2 , λ 2 w0 (x, 1

1

and, consequently, we would have (2.29) if − 12   1 ˜ j (y)) ˜ w0 (x, ˜ αj (y))(d ˜ ˜ αj (y))) ˜ − 2 ≤ C max dg (x, ˜ αj (y)), ˜ eκdg (x,α (2.36) g˜ (x, assuming, as above, that the curvature of M is ≤ −κ 2 , κ ≥ 0. The last inequality comes from the fact that in geodesic normal coordinates about x, we have  − 1 w0 (x, y) = det gij (y) 4 , (see [1], [8] or §2.4 in [20]). If y has geodesic polar coordinates (t, θ ) about x, then √ t = dg˜ (x, y), and if A(t, θ ) is as in (2.12), we conclude that w0 (x, y) = t/A(t, θ ), and therefore (2.36) follows from Günther’s comparison estimate (2.14) if −κ 2 < 0 and (2.13) if κ = 0. The second estimate (2.35) is elementary and left for the reader, who can check 1 that the terms are actually O(λ 2 −ν ). (This is also just a special case of Lemma 3.5.3 in [20].) This completes the proof of (2.33), and, hence, that of Theorem 1.1.

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18.3 CONCLUDING REMARKS It is straightforward to see that the proof of Theorem 1.1 shows that one can strengthen our main estimate (1.1) in a natural way. Specifically, if γ0 is a periodic geodesic of length (γ0 ) and if we define the δ-tube about γ to be Tδ (γ0 ) = {y ∈ M : distg (y, γ0 ) < δ}, with δ > 0 fixed, then there is a uniform constant Cδ so that whenever ε > 0 we have for large λ  1 1 |eλ |2 ds ≤ ελ 2 eλ 2L2 (Tδ (γ0 )) + Cγ0 ,δ,ε eλ 2L2 (M) . (3.1) (γ0 ) γ0 Thus, (1.1) essentially lifts to the cylinder R2 /Gγ0 , with, as above, Gγ0 , being the ˜ stabilizer group for the lift of γ0 to the universal cover (R2 , g). To prove this, we as before write I = Bε + bε , with bε (x, ξ ) equal to one near T ∗ γ0 but supported in a small conic neighborhood of this set. Since the analog of (2.16) is valid, i.e.,  T /4   2
 1  χˆ (t/T )e−iλt Bε ◦ cos(t −
g )f dt  ds ≤ CT ,ε,γ0 f 2L2 (M) , (3.2)  γ0 π T −T /4 it suffices to show that 2    T /4   1  
1 −iλt  χ(t/T ˆ )e bε ◦ cos t −
g eλ dt  ds  (γ0 ) γ0 T −T /4 is dominated by the right side of (3.1). If Kε (x, s), x ∈ M, s ∈ γ0 denotes the kernel of this operator then, if δ > 0 and T are fixed, it follows that |Kε (x, s)| ≤ Cγ0 ,T ,δ ,

x∈ / Tδ (γ0 ),

(3.3)

provided that bε is supported in a sufficiently small conic neighborhood of T ∗ γ0 . This is a simple consequence of the fact that when bε is as above, by (2.17), bε ◦ 
/ Tδ (γ0 ), s ∈ γ0 and |t| ≤ T . Since (2.21) is cos t −
g (x, s) is smooth when x ∈ valid, we conclude that there is a uniform constant C so that for large λ we have 2    T /4   1  
1 −iλt  χˆ (t/T )e bε ◦ cos t −
g eλ dt  ds (γ0 ) γ0  T −T /4 ≤ CT −1 λ 2 eλ 2L2 (Tδ (γ0 )) + CT ,δ,γ0 eλ 2L2 (M) , 1

(3.4)

which along with (3.2) gives us (3.1). This is because we can dominate the quantity in (3.4) by the sum of the corresponding expression where eλ is replaced by 1Tδ (γ0 ) eλ and 1Tδc (γ0 ) eλ and use (2.21) and our earlier arguments to show that the first of these terms is dominated by the first term in the right side of (3.4) if λ is large, while the second such term is dominated by the last term in the right side of (3.4) on account of (3.3). We would also like to point out that it seems likely that one should be able to take the parameter T in the proof of either (1.1) or (3.1) to be a function of λ.

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This would also require that the parameter ε also be a function of λ, and thus the argument would be more involved. It would not be surprising if, as in Bérard [1] or Hassell and Tacey [9], one could take T to be ≈ log λ, in which case the L2 restriction bounds in Theorem 1.1 and the L4 -estimates in Corollary 1.2 could also 1 1 be improved to be O(λ 4 (log λ)−δ1 ) and O(λ 8 (log λ)−δ2 ), respectively, for some δj > 0. It is doubtful that these bounds would be optimal, though–indeed if a difficult conjecture of Rudnick and Sarnak [14] were valid, both would be O(λε ) for any ε > 0. One of the main technical issues in carrying out the analysis when T depends on λ would be to determine the analog of (2.15) in this case. One would also have to take into account more carefully size estimates for the coefficients wν , ν > 0, in the Hadamard parametrix, but Bérard [1] carried out an analysis of these that would seem to be sufficient if T ≈ log λ. On the other hand, we have argued here that the w0 coefficient is very well behaved, and so perhaps there could be further grounds for improvement.

REFERENCES [1] P. H. Bérard, On the wave equation on a compact manifold without conjugate points, Math. Z. 155 (1977), 249–276. [2] J. Bourgain, Geodesic restrictions and Lp -estimates for eigenfunctions of Riemannian surfaces, Linear and complex analysis, 27–35, Amer. Math. Soc. Tranl. Ser. 2, 226, Amer. Math. Soc., Providence, RI, 2009. [3] N. Burq, P. Gérard, and N. Tzvetkov, Restriction of the Laplace-Beltrami eigenfunctions to submanifolds, Duke Math. J. 138 (2007), 445–486. [4] E. Cartan, Leçons sur la Geométrie de Espaces de Riemann (2nd ed.) Paris, Gauthier-Villars, 1946. [5] J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), 39–79. [6] P. Günther, Einige Sätze über das Volumenelement eines Riemannschen Raumes, Publ. Math. Debrecen 7 (1960), 78–93. [7] J. Hadamard, Les surfaces à courbures opposeés et leurs géodésiques, J. Math. Pures Appl. 4 (1898), 27–73. [8] J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover Publications, New York, 1953. [9] A. Hassell and M. Tacey, personal communication. [10] L. Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), 79–183. [11] L. Hörmander, The analysis of linear partial differential operators. III. Pseudodifferential operators, Springer-Verlag, Berlin, 1985.

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[12] P. Kröger, An extension of Günther’s volume comparison theorem, Math. Ann. 329 (2004), 593–596. [13] A. Reznikov, Norms of geodesic restrictions for eigenfunctions on hyperbolic surfaces and representation theory, arXiv:math.AP/0403437. [14] Z. Rudnick and P. Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys 161 (1994), 195–213. [15] H. Smith and C. D. Sogge, On the Lp norm of spectral clusters for compact manifolds with boundary, Acta Math. 198 (2007), 107–153. [16] C. D. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. J. 53 (1986), 43–65. [17] C. D. Sogge, Concerning the Lp norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988), 123–138. [18] C. D. Sogge, Fourier integrals in classical analysis, Cambridge Tracts in Math., Cambridge University. Press, Cambridge, 1993. [19] C. D. Sogge, Kakeya-Nikodym averages and Lp -norms of eigenfunctions, arXiv:0907.4827 to appear, Tohoku Math. J (centennial edition). [20] C. D. Sogge, Hangzhou lectures on eigenfunctions of the Laplacian, in preparation, www.mathematics.jhu.edu/sogge/zju. [21] C. D. Sogge, J. Toth, and S. Zelditch, About the blowup of quasimodes, on Riemannian manifolds, arXiv:0908.0688 to appear, J. Geom. Anal. [22] C. D. Sogge and S. Zelditch, Riemannian manifolds with maximal eigenfunction growth, Duke Math. J. 114 (2002), 387–437. [23] C. D. Sogge and S. Zelditch, Concerning the L4 norms of typical eigenfunctions on compact surfaces, arXiv:1011.0215 to appear. [24] F. Spinu, The L4 norm of the Eisenstein series, Thesis, Princeton University 2003, www.math.jhu.edu/~fspinu. [25] D. Tataru, On the regularity of boundary traces for the wave equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 26 (1998), 185–206. [26] H. von Mangolt, Über diejenigen Punkte auf positiv gekrummten Flächen, welche die Eigenschaft haben, dass die von ihnen ausgehended geodätischen Linien nie aufhörn, kürzeste linien zu sein, J. Reine Angew. Math. 91 (1881), 23–52. [27] A. Zygmund, On Fourier coefficients and transforms of two variables, Studia Math. 50 (1974), 189–201.

List of Contributors Jean Bourgain: School of Mathematics, Institute for Advanced Study Luis Caffarelli: Department of Mathematics, University of Texas at Austin Michael Christ: Department of Mathematics, University of California at Berkeley Guy David: Département de Mathématiques, Université de Paris-Sud (Orsay) Charles Fefferman: Department of Mathematics, Princeton University Alexandru D. Ionescu: Department of Mathematics, Princeton University David Jerison: Department of Mathematics, Massachusetts Institute of Technology Carlos Kenig: Department of Mathematics, University of Chicago Sergiu Klainerman: Department of Mathematics, Princeton University Loredana Lanzani: Department of Mathematics, University of Arkansas Sanghyuk Lee: School of Mathematical Sciences, Seoul National University Lionel Levine: Department of Mathematics, Cornell University Akos Magyar: Department of Mathematics, University of British Columbia Detlef Müller: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel Camil Muscalu: Department of Mathematics, Cornell University Alexander Nagel: Department of Mathematics, University of Wisconsin-Madison Duong H. Phong: Department of Mathematics, Columbia University Malabika Pramanik: Department of Mathematics, University of British Columbia Andrew R. Raich: Department of Mathematical Sciences, University of Arkansas Fulvio Ricci: Scuola Normale Superiore, Pisa Keith M. Rogers: Instituto de Ciencias Matemáticas, Madrid Andreas Seeger: Department of Mathematics, University of Wisconsin-Madison Scott Sheffield: Department of Mathematics, Massachusetts Institute of Technology Luis Silvestre: Department of Mathematics, University of Chicago Christopher D. Sogge: Department of Mathematics, Johns Hopkins University Jacob Sturm: Department of Mathematics, Rutgers University Terence Tao: Department of Mathematics, University of California at Los Angeles Christoph Thiele: Hausdorff Center for Mathematics, University of Bonn Stephen Wainger: Department of Mathematics, University of Wisconsin-Madison (recently retired) Steven Zelditch: Department of Mathematics, Northwestern University

Index adapted coordinates, 307 amnesic solutions, 135 atoms, 295 balls, non-Euclidean, 22 Bergman kernel, 402 Blowup: of complex manifolds, 424; of non-linear waves, 216 Bochner-Riesz multipliers, 48, 274 Boltzmann equation, 40 Cauchy-Riemann equations, 15; of Stein-Weiss, 9 currents, 108; integral, 112 diffusion-limited aggregation, 189 domains, 15: decoupled, 406; of model monomial type, 404; pseudoconvex, 17; Reinhart, 413; strongly pseudoconvex, 16 Einstein equations, 226 extremizers, 85; radial, 94 fluctuations, 191 forms, hybrid, 249 function spaces, hybrid, 251 HP spaces, 7 Hadamard parametrix, 456 Heisenberg fan, 436 Heisenberg group, 17 homology, in Reifenberg’s problem, 110 Hörmander operators, 18 interpolation, complex 2; Stein’s theorem on, 3 integrals, oscillatory, 49, 171, 319 Kakeya sets, 51 Kumze-Stein phenomenon, 3

Littlewood-Paley functions, 35, 273 mass, 113 master equations, 63 maximal function, 1, 303 multipliers: Bochner-Riesz (see Bochner-Riesz multipliers); multilinear, 346; radial, 273 Newton polyhedra, 303 nilpotent Gelfand pairs, 437 nilpotent Lie groups, 13, 15, 146 orthogonality, lemmas on, 180; of Cotlar-Stein, 14 Plateau’s problem, 108 Radon transform, 84 regularity, Hölder, 80 representation theory, 3, 12 resolution of singularities, 427 restriction theorems, 5, 47, 303, 328 sandpile, 208 scattering of nonlinear waves, 216 sets, Almgren minimal, 108 size of currents, 114 smash sum, 208 solitons, 216 solutions, viscosity, 68 spaces, analytic, 425 symmetrization, 95; inverse, 97 symmetry, ellipsoidal, 102 tiles, 372 trees, rooted, 354 varifolds, 111