Fourier Integrals in Classical Analysis [2ed.] 1107120071, 978-1-107-12007-5, 9781108234733, 1108234739

Table of contents :
Content: Cover
Series page
Title page
Copyright information
Dedication
Table of contents
Preface to the Second Edition
Preface to the First Edition
0 Background
0.1 Fourier Transform
0.2 Basic Real Variable Theory
0.3 Fractional Integration and Sobolev Embedding Theorems
0.4 Wave Front Sets and the Cotangent Bundle
0.5 Oscillatory Integrals
Notes
1 Stationary Phase
1.1 Stationary Phase Estimates
The One-Dimensional Case
Stationary Phase in Higher Dimensions
1.2 Fourier Transform of Surface-carried Measures
Notes
2 Non-homogeneous Oscillatory Integral Operators 2.1 Non-degenerate Oscillatory Integral Operators2.2 Oscillatory Integral Operators Related to the Restriction Theorem
2.3 Riesz Means in Rn
2.4 Nikodym Maximal Functions and Maximal Riesz Means in R2
Notes
3 Pseudo-differential Operators
3.1 Some Basics
3.2 Equivalence of Phase Functions
3.3 Self-adjoint Elliptic Pseudo-differential Operators on Compact Manifolds
Notes
4 The Half-wave Operator and Functions of Pseudo-differential Operators
4.1 The Half-wave Operator
4.2 The Sharp Weyl Formula
4.3 Smooth Functions of Pseudo-differential Operators
Notes 5 Lp Estimates of Eigenfunctions5.1 The Discrete L2 Restriction Theorem
Application: Unique Continuation for the Laplacian
5.2 Estimates for Riesz Means
5.3 More General Multiplier Theorems
Notes
6 Fourier Integral Operators
6.1 Lagrangian Distributions
6.2 Regularity Properties
Sharpness of Results
6.3 Spherical Maximal Theorems: Take 1
Notes
7 Propagation of Singularities and Refined Estimates
7.1 Wave Front Sets Redux
7.2 Propagation of Singularities
7.3 Improved Sup-norm Estimates of Eigenfunctions
7.4 Improved Spectral Asymptotics
Notes 8 Local Smoothing of Fourier Integral Operators8.1 Local Smoothing in Two Dimensions and Variable Coefficient Nikodym Maximal Theorems
Orthogonality Arguments in Two Dimensions
Variable Coefficient Nikodym Maximal Functions
8.2 Local Smoothing in Higher Dimensions
Orthogonality Arguments in Higher Dimensions
8.3 Spherical Maximal Theorems Revisited
Notes
9 Kakeya-type Maximal Operators
9.1 The Kakeya Maximal Operator and the Kakeya Problem
9.2 Universal Bounds for Kakeya-type Maximal Operators
9.3 Negative Results in Curved Spaces Negative Results for Oscillatory Integrals Arising in Curved Spaces9.4 Wolff's Bounds for Kakeya-type Maximal Operators
Auxiliary Maximal Function Bounds
Wolff's Bounds for Nikodym Maximal Functions in Rn
9.5 The Fourier Restriction Problem and the Kakeya Problem
Notes
Appendix Lagrangian Subspaces of T∗Rn
References
Index of Notation
Index

Citation preview

C A M B R I D G E T R AC T S I N M AT H E M AT I C S General Editors ´ S , W. F U LTO N , F. K I RWA N , B. BOLLOBA P. S A R NA K , B . S I M O N , B . TOTA RO 210 Fourier Integrals in Classical Analysis, Second Edition

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C A M B R I D G E T R AC T S I N M AT H E M AT I C S G E N E R A L E D I TO R S ´ S , W. F U LTO N , F. K I RWA N , B. BOLLOBA P. S A R NA K , B . S I M O N , B . TOTA RO A complete list of books in the series can be found at www.cambridge.org/mathematics. Recent titles include the following: 176. The Monster Group and Majorana Involutions. By A. A. IVANOV 177. A Higher-Dimensional Sieve Method. By H. G. DIAMOND, H. HALBERSTAM, and W. F. GALWAY 178. Analysis in Positive Characteristic. By A. N. KOCHUBEI ´ MATHERON 179. Dynamics of Linear Operators. By F. BAYART and E. 180. Synthetic Geometry of Manifolds. By A. KOCK 181. Totally Positive Matrices. By A. PINKUS 182. Nonlinear Markov Processes and Kinetic Equations. By V. N. KOLOKOLTSOV 183. Period Domains over Finite and p-adic Fields. By J.-F. DAT, S. ORLIK, and M. RAPOPORT ´ 184. Algebraic Theories. By J. AD AMEK , J. ROSICK Y´ , and E. M. VITALE 185. Rigidity in Higher Rank Abelian Group Actions I: Introduction and Cocycle Problem. By A. KATOK and V. NIT¸ IC A˘ 186. Dimensions, Embeddings, and Attractors. By J. C. ROBINSON 187. Convexity: An Analytic Viewpoint. By B. SIMON 188. Modern Approaches to the Invariant Subspace Problem. By I. CHALENDAR and J. R. PARTINGTON 189. Nonlinear Perron–Frobenius Theory. By B. LEMMENS and R. NUSSBAUM 190. Jordan Structures in Geometry and Analysis. By C.-H. CHU 191. Malliavin Calculus for L´evy Processes and Infinite-Dimensional Brownian Motion. By H. OSSWALD 192. Normal Approximations with Malliavin Calculus. By I. NOURDIN and G. PECCATI 193. Distribution Modulo One and Diophantine Approximation. By Y. BUGEAUD 194. Mathematics of Two-Dimensional Turbulence. By S. KUKSIN and A. SHIRIKYAN 195. A Universal Construction for Groups Acting Freely on Real Trees. By I. CHISWELL and ¨ T. MULLER 196. The Theory of Hardy’s Z-Function. By A. IVI C´ 197. Induced Representations of Locally Compact Groups. By E. KANIUTH and K. F. TAYLOR 198. Topics in Critical Point Theory. By K. PERERA and M. SCHECHTER 199. Combinatorics of Minuscule Representations. By R. M. GREEN ´ 200. Singularities of the Minimal Model Program. By J. KOLL AR 201. Coherence in Three-Dimensional Category Theory. By N. GURSKI 202. Canonical Ramsey Theory on Polish Spaces. By V. KANOVEI, M. SABOK, and J. ZAPLETAL 203. A Primer on the Dirichlet Space. By O. EL - FALLAH, K. KELLAY, J. MASHREGHI, and T. RANSFORD 204. Group Cohomology and Algebraic Cycles. By B. TOTARO 205. Ridge Functions. By A. PINKUS 206. Probability on Real Lie Algebras. By U. FRANZ and N. PRIVAULT 207. Auxiliary Polynomials in Number Theory. By D. MASSER 208. Representations of Elementary Abelian p-Groups and Vector Bundles. By D. J. BENSON 209. Non-homogeneous Random Walks. By M. MENSHIKOV, S. POPOV, and A. WADE 210. Fourier Integrals in Classical Analysis (Second Edition). By CHRISTOPHER D. SOGGE

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Fourier Integrals in Classical Analysis Second Edition CHRISTOPHER D. SOGGE The Johns Hopkins University

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University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 4843/24, 2nd Floor, Ansari Road, Daryaganj, Delhi – 110002, India 79 Anson Road, #06-04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107120075 DOI: 10.1017/9781316341186 c Cambridge University Press 1993 First edition  c Christopher D. Sogge 2017 Second edition  This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1993 Second edition 2017 A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Sogge, Christopher D. (Christopher Donald), 1960– Title: Fourier integrals in classical analysis / Christopher D. Sogge, The Johns Hopkins University. Description: Second edition. | Cambridge : Cambridge University Press, 2017. | Series: Cambridge tracts in mathematics ; 210 | Includes bibliographical references and index. Identifiers: LCCN 2017004380 | ISBN 9781107120075 (hardback : alk. paper) Subjects: LCSH: Fourier series. | Fourier integral operators. | Fourier analysis. Classification: LCC QA404.S64 2017 | DDC 515/.723–dc23 LC record available at https://lccn.loc.gov/2017004380 ISBN 978-1-107-12007-5 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

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To my family

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Contents

Preface to the Second Edition Preface to the First Edition

page xi xiii

0

Background 0.1 Fourier Transform 0.2 Basic Real Variable Theory 0.3 Fractional Integration and Sobolev Embedding Theorems 0.4 Wave Front Sets and the Cotangent Bundle 0.5 Oscillatory Integrals Notes

1 1 9 22 30 37 41

1

Stationary Phase 1.1 Stationary Phase Estimates 1.2 Fourier Transform of Surface-carried Measures Notes

42 42 51 56

2

Non-homogeneous Oscillatory Integral Operators 2.1 Non-degenerate Oscillatory Integral Operators 2.2 Oscillatory Integral Operators Related to the Restriction Theorem 2.3 Riesz Means in Rn 2.4 Nikodym Maximal Functions and Maximal Riesz Means in R2 Notes

57 58

Pseudo-differential Operators 3.1 Some Basics 3.2 Equivalence of Phase Functions 3.3 Self-adjoint Elliptic Pseudo-differential Operators on Compact Manifolds Notes

97 97 104

3

60 70 76 96

110 115

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viii 4

Contents The Half-wave Operator and Functions of Pseudo-differential Operators 4.1 The Half-wave Operator 4.2 The Sharp Weyl Formula 4.3 Smooth Functions of Pseudo-differential Operators Notes

116 117 126 133 135

5

Lp Estimates of Eigenfunctions 5.1 The Discrete L2 Restriction Theorem 5.2 Estimates for Riesz Means 5.3 More General Multiplier Theorems Notes

137 138 151 155 160

6

Fourier Integral Operators 6.1 Lagrangian Distributions 6.2 Regularity Properties 6.3 Spherical Maximal Theorems: Take 1 Notes

162 162 170 187 194

7

Propagation of Singularities and Refined Estimates 7.1 Wave Front Sets Redux 7.2 Propagation of Singularities 7.3 Improved Sup-norm Estimates of Eigenfunctions 7.4 Improved Spectral Asymptotics Notes

195 195 199 204 216 232

8

Local Smoothing of Fourier Integral Operators 8.1 Local Smoothing in Two Dimensions and Variable Coefficient Nikodym Maximal Theorems 8.2 Local Smoothing in Higher Dimensions 8.3 Spherical Maximal Theorems Revisited Notes

233

Kakeya-type Maximal Operators 9.1 The Kakeya Maximal Operator and the Kakeya Problem 9.2 Universal Bounds for Kakeya-type Maximal Operators 9.3 Negative Results in Curved Spaces 9.4 Wolff’s Bounds for Kakeya-type Maximal Operators 9.5 The Fourier Restriction Problem and the Kakeya Problem Notes

268 268 278 285 294 310 314

9

234 253 263 266

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Contents Appendix: Lagrangian Subspaces of T ∗ Rn References Index of Notation Index

ix 317 319 331 333

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Preface to the Second Edition

In the twenty plus years since I wrote the first edition of this book there have been many important developments in harmonic and microlocal analysis. For instance, very recently there has been seminal work on multilinear methods through the breakthrough of Bennett, Carbery and Tao [1], the related oscillatory integral estimates of Bourgain and Guth [1] and the decoupling estimates of Bourgain and Demeter [1]. These works and others have led to many applications, including the proof of sharp local smoothing estimates. I have chosen not to include these, partly because they represent a fast moving target, but moreover because they would have added considerably to the book’s length. My goal for the first as well as the current edition was to provide a self-contained but by no means exhaustive introduction to harmonic and microlocal analysis. To this end, I added two new chapters. One, which ties in with the earlier material on Fourier integral operators, presents H¨ormander’s theory of propagation of singularities and uses this to prove the Duistermaat–Guillemin theorem, which is a triumph of microlocal methods, saying that for generic manifolds one can obtain an improvement in the error term for the Weyl law. This chapter includes material that perhaps should have been included in the first edition and augments the microlocal analysis portion of the book. I also added a new chapter on what could now be considered classical harmonic analysis. It concerns results related to the Kakeya conjecture, including the “bush” method of Bourgain and the “hairbrush” method of Wolff for obtaining Kakeya and Nikodym maximal estimates, as well as the relationship between the Fourier restriction problem and the Kakeya conjecture. Most of the results presented in this chapter were being developed as the first edition of my book was being completed. I also added a bit of new material to the chapters in the first edition, including Bourgain’s counterexample to certain oscillatory integral estimates, which

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xii

Preface to the Second Edition

showed that Stein’s oscillatory integral theorem was sharp, and I also made a few corrections to the earlier text. This turned out to be a more difficult undertaking than I anticipated due to the fact that the first edition of the book was written in the age of floppy disks and although I had saved the output files on several of them, I did not do the same for the source files for the first edition. To my delight, Cambridge University Press came to the rescue by re-typesetting all of the old material. They did a wonderful job. I am very grateful to Sam Harrison and Clare Dennison and the typesetters at Cambridge University Press for this. I am also very grateful to Sam Harrison for suggesting that a new edition of my book be written and for his kind encouragement. As was the case for the first edition, I am also very grateful for the assistance I have received from many of my colleagues. In particular, I would like to thank Changxing Miao and his group for going through the new material and for their suggestions, and I am especially grateful to Yakun Xi for his thorough proofreading of the material prepared by Cambridge University Press as well as the two new chapters. The role that Sam Harrison and Yakun Xi played in this project has been invaluable to me. Finally, I am also grateful to the students at Johns Hopkins University for their feedback and suggestions when the new material was presented in courses over the last couple of years. Lutherville

C. D. Sogge

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Preface to the First Edition

Except for minor modifications, this monograph represents the lecture notes of a course I gave at UCLA during the winter and spring quarters of 1991. My purpose in the course was to present the necessary background material and to show how ideas from the theory of Fourier integral operators can be useful for studying basic topics in classical analysis, such as oscillatory integrals and maximal functions. The link between the theory of Fourier integral operators and classical analysis is of course not new, since one of the early goals of microlocal analysis was to provide variable coefficient versions of the Fourier transform. However, the primary goal of this subject was to develop tools for the study of partial differential equations and, to some extent, only recently have many classical analysts realized its utility in their subject. In these notes I attempted to stress the unity between these two subjects and only presented the material from microlocal analysis that would be needed for the later applications in Fourier analysis. I did not intend for this course to serve as an introduction to microlocal analysis. For this the reader should be referred to the excellent treatises of H¨ormander [5], [7] and Treves [1]. In addition to these sources, I also borrowed heavily from Stein [4]. His work represents lecture notes based on a course that he gave at Princeton while I was his graduate student. As the reader can certainly tell, this course influenced me quite a bit and I am happy to acknowledge my indebtedness. My presentation of the overlapping material is very similar to his, except that I chose to present the material in the chapter on oscillatory integrals more geometrically, using the cotangent bundle. This turns out to be useful in dealing with Fourier analysis on manifolds and it also helps to motivate some results concerning Fourier integral operators, in particular the local smoothing estimates at the end of the monograph. Roughly speaking, the material is organized as follows. The first two chapters present background material on Fourier analysis and stationary

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xiv

Preface to the First Edition

phase that will be used throughout. The next chapter deals with nonhomogeneous oscillatory integrals. It contains the L2 restriction theorem for the Fourier transform, estimates for Riesz means in Rn , and Bourgain’s circular maximal theorem. The goal of the rest of the monograph is mainly to develop generalizations of these results. The first step in this direction is to present some basic background material from the theory of pseudo-differential operators, emphasizing the role of stationary phase. After the chapter on pseudo-differential operators comes one dealing with the sharp Weyl formula of H¨ormander [4], Avakumoviˇc [1], and Levitan [1]. I followed the exposition in H¨ormander’s paper, except that the Tauberian condition in the proof of the Weyl formula is stated in terms of L∞ estimates for eigenfunctions. In the next chapter, this slightly different point of view is used in generalizing some of the earlier results from Fourier analysis in Rn to the setting of compact manifolds. Finally, the last two chapters are concerned with Fourier integral operators. First, some background material is presented and then the mapping properties of Fourier integral operators are investigated. This is all used to prove some recent local smoothing estimates for Fourier integral operators, which in turn imply variable coefficient versions of Stein’s spherical maximal theorem and Bourgain’s circular maximal theorem. It is a pleasure to express my gratitude to the many people who helped me in preparing this monograph. First, I would like to thank everyone who attended the course for their helpful comments and suggestions. I am especially indebted to D. Grieser, A. Iosevich, J. Johnsen, and H. Smith who helped me both mathematically and in proofreading. I am also grateful to M. Cassorla and R. Strichartz for their thorough critical reading of earlier versions of the manuscript. Lastly, I would like to thank all of my collaborators for the important role they have played in the development of many of the central ideas in this course. In this regard, I am particularly indebted to A. Seeger and E. M. Stein. This monograph was prepared using AMS-TEX. The work was supported in part by the NSF and the Sloan foundation. Sherman Oaks

C. D. Sogge

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0 Background

The purpose of this chapter and the next is to present the background material that will be needed. The topics are standard and a more thorough treatment can be found in many excellent sources, such as Stein [2] and Stein and Weiss [1] for the first half and H¨ormander [7, Vol. 1] for the second. We start out by rapidly going over basic results from real analysis, including standard theorems concerning the Fourier transform in Rn and Calder´on–Zygmund theory. We then apply this to prove the Hardy–Littlewood– Sobolev inequality. This theorem on fractional integration will be used throughout and we shall also present a simple argument showing how the n-dimensional theorem follows from the original one-dimensional inequality of Hardy and Littlewood. This type of argument will be used again and again. Finally, in the last two sections we give the definition of the wave front set of a distribution and compute the wave front sets of distributions which are given by oscillatory integrals. This will be our first encounter with the cotangent bundle and, as the monograph progresses, this will play an increasingly important role.

0.1 Fourier Transform Given f ∈ L1 (Rn ), we define its Fourier transform by setting  fˆ (ξ ) =

Rn

e−ix,ξ  f (x) dx.

(0.1.1)

Given h ∈ Rn , let (τh f )(x) = f (x + h). Notice that τ−h e−i·,ξ  = eih,ξ  e−i·,ξ  and so (τh f )∧ (ξ ) = eih,ξ  fˆ (ξ ).

(0.1.2)

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2

Background

In a moment, we shall see that we can invert (0.1.1) (for appropriate f ) and that we have the formula  eix,ξ  fˆ (ξ ) dξ . (0.1.3) f (x) = (2π )−n Rn

Thus, the Fourier transform decomposes a function into a continuous sum of characters (eigenfunctions for translations). Before turning to Fourier’s inversion formula (0.1.3), let us record some elementary facts concerning the Fourier transform of L1 functions. Theorem 0.1.1 (1) || fˆ ||∞ ≤ || f ||1 . (2) If f ∈ L1 , then fˆ is uniformly continuous. Theorem 0.1.2 (Riemann–Lebesgue) If f ∈ L1 (Rn ), then fˆ (ξ ) → 0 as ξ → ∞, and hence, fˆ ∈ C0 (Rn ). Theorem 0.1.1 follows directly from the definition (0.1.1). To prove Theorem 0.1.2, one first notices from an explicit calculation that the result holds when f is the characteristic function of a cube. From this one derives Theorem 0.1.2 via a limiting argument. Even though fˆ is in C0 , the integral (0.1.3) will not converge for general f ∈ L1 . However, for a dense subspace we shall see that the integral converges absolutely and that (0.1.3) holds. Definition 0.1.3 The set of Schwartz-class functions, S(Rn ), consists of all φ ∈ C∞ (Rn ) satisfying sup |xγ ∂ α φ(x)| < ∞, (0.1.4) x

for all multi-indices

α, γ .1

We give S the topology arising from the semi-norms (0.1.4). This makes S a Fr´echet space. Notice that the set of all compactly supported C∞ functions, C0∞ (Rn ), is contained in S. Let Dj = 1i ∂x∂ j . Then we have: ˆ ). Also, Theorem 0.1.4 If φ ∈ S, then the Fourier transform of Dj φ is ξj φ(ξ ˆ the Fourier transform of xj φ is −Dj φ. 1 Here α = (α , . . . , α ), γ = (γ , . . . , γ ) while xγ = xγ1 · · · xγn and ∂ α = (∂/∂x )α1 · · · n n n 1 1 1 1 (∂/∂xn )αn .

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0.1 Fourier Transform

3

Proof To prove the second assertion we differentiate (0.1.1) to obtain  ˆ Dj φ(ξ ) = e−ix,ξ  (−xj )φ(x) dx, since the integral converges uniformly. If we integrate by parts, we see that   ˆ ) = −Dj e−ix,ξ  · φ(x) dx = e−ix,ξ  Dj φ(x) dx, ξj φ(ξ which is the first assertion. Notice that Theorem 0.1.4 implies the formula  ˆ ) = e−ix,ξ  Dα ((−x)γ φ(x)) dx. ξ α Dγ φ(ξ

(0.1.5)

 If we set C = (1 + |x|)−n−1 dx, then this leads to the estimate ˆ )| ≤ C sup(1 + |x|)n+1 |Dγ (xα φ(x))|. sup |ξ γ Dα φ(ξ ξ

(0.1.6)

x

Inequality (0.1.6) of course implies that the Fourier transform maps S into itself. However, much more is true: Theorem 0.1.5 The Fourier transform φ → φˆ is an isomorphism of S into S whose inverse is given by Fourier’s inversion formula (0.1.3). The proof is based on a couple of lemmas. The first is the multiplication formula for the Fourier transform: Lemma 0.1.6 If f , g ∈ L1 then  Rn

 fˆ g dx =

Rn

f gˆ dx.

The next is a formula for the Fourier transform of Gaussians: Lemma 0.1.7



−ix,ξ  e−ε|x|2 /2 dx = (2π/ε)n/2 e−|ξ |2 /2ε . Rn e

The first lemma is easy to prove. If we apply (0.1.1) and Fubini’s theorem, we see that the left side equals       f (y)e−ix,y dy g(x) dx = e−ix,y g(x) dx f (y) dy  = gˆ f dy.

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4

Background

It is also clear that Lemma 0.1.7 must follow from the special case where n = 1. But  ∞  ∞ 1 2 2 2 e−t /2 e−itτ dt = e−τ /2 e− 2 (t+iτ ) dt −∞ −∞  ∞ 2 −τ 2 /2 =e e−t /2 dt −∞

=

√ 2 2π e−τ /2 .

In the second step we have used Cauchy’s theorem. If we make the change of variables ε1/2 s = t in the last integral, we get the desired result. Proof of Theorem 0.1.5 We must prove that when φ ∈ S,  ˆ ) dξ . φ(x) = (2π )−n eix,ξ  φ(ξ By the dominated convergence theorem, the right side equals  ˆ )e−ε|ξ |2 /2 dξ . lim (2π )−n eix,ξ  φ(ξ ε→0

If we recall (0.1.2), then we see that this equals  2 −n/2 lim (2π ε) φ(x + y)e−|y| /2ε dy. ε→0

 2 Finally, since (2π )−n/2 e−|y| /2 dy = 1, it is easy to check that the last limit is φ(x). If for f , g ∈ L1 we define convolution by  (f ∗ g)(x) = f (x − y)g(y) dy, then another fundamental result is: Theorem 0.1.8 If φ, ψ ∈ S then   (2π )n φψ dx = φˆ ψˆ dξ , ˆ )ψ(ξ ˆ ), (φ ∗ ψ)∧ (ξ ) = ψ(ξ ∧ −n ˆ ˆ ). (φψ) (ξ ) = (2π ) (φ ∗ ψ)(ξ

(0.1.7) (0.1.8) (0.1.9)

ˆ Then the Fourier inversion formula To prove (0.1.7), set χ = (2π )−n ψ. implies that χˆ = ψ. Consequently, (0.1.7) follows from Lemma 0.1.6. We leave the other two formulas as exercises.

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0.1 Fourier Transform

5

We shall now discuss the Fourier transform of more general functions. First, we make a definition. Definition 0.1.9 The dual space of S is S . We call S the space of tempered distributions. Definition 0.1.10 If u ∈ S , we define its Fourier transform uˆ ∈ S by setting, for all φ ∈ S, ˆ uˆ (φ) = u(φ). (0.1.10) Notice how Lemma 0.1.6 says that when u ∈ L1 , Definition 0.1.10 agrees with our previous definition of uˆ . Using Fourier’s inversion formula for S, one can check that u → uˆ is an isomorphism of S . If u ∈ L1 and uˆ ∈ L1 , we conclude that the inversion formula (0.1.3) must hold for almost all x. Theorem 0.1.11 If u ∈ L2 then uˆ ∈ L2 and

ˆu 22 = (2π )n u 22

(Plancherel’s theorem).

Furthermore, Parseval’s formula holds whenever φ, ψ ∈ L2 :   φψ dx = (2π )−n φˆ ψˆ dξ .

(0.1.11)

(0.1.12)

Proof Choose uj ∈ S satisfying uj → u in L2 . Then, by (0.1.7),

ˆuj − uˆ k 22 = (2π )n uj − uk 22 → 0. Thus, uˆ j converges to a function v in L2 . But the continuity of the Fourier transform in S forces v = uˆ . This gives (0.1.11), since (0.1.11) is valid for each uj . Since we have just shown that the Fourier transform is continuous on L2 , (0.1.12) follows from the fact that we have already seen that it holds when φ and ψ belong to the dense subspace S. Since, for 1 ≤ p ≤ 2, f ∈ Lp can be written as f = f1 + f2 with f1 ∈ L1 , f2 ∈ L2 , 2 . A much better it follows from Theorem 0.1.1 and Theorem 0.1.11 that fˆ ∈ Lloc result is: Theorem 0.1.12 (Hausdorff–Young) Let 1 ≤ p ≤ 2 and define p by 1/p + 1/p = 1. Then, if f ∈ Lp it follows that fˆ ∈ Lp and

fˆ p ≤ (2π )n/p f p . Since we have already seen that this result holds for p = 1 and p = 2, this follows from:

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6

Background

Theorem 0.1.13 (M. Riesz interpolation theorem) Let T be a linear map from Lp0 ∩ Lp1 to Lq0 ∩ Lq1 satisfying

Tf qj ≤ Mj f pj ,

j = 0, 1,

(0.1.13)

with 1 ≤ pj , qj ≤ ∞. Then, if for 0 < t < 1, 1/pt = (1 − t)/p0 + t/p1 , 1/qt = (1 − t)/q0 + t/q1 ,

Tf qt ≤ (M0 )1−t (M1 )t f pt ,

f ∈ L p0 ∩ L p1 .

(0.1.14)

Proof If pt = ∞ the result follows from H¨older’s inequality since then p0 = p1 = ∞. So we shall assume that pt < ∞. By polarization it then suffices to show that      Tfgdx ≤ M 1−t M t f p g q , (0.1.15) t 1 0   t when f and g vanish outside of a set of finite measure and take on a finite  N number of values, that is, f = m j=1 aj χEj , g = k=1 bk χFk , with Ej ∩ Ej = ∅ and Fk ∩ Fk = ∅ if j  = j and k  = k and |Ej |, | Fk | < ∞ for all j and k. We may also assume f pt , g q t  = 0 and so, if we divide both sides by the norms, it suffices to prove (0.1.15) when f pt = g q t = 1. Next, if aj = eiθj |aj | and bk = eiψk |bk |, then, assuming qt > 1, we set fz =

m 

|aj |α(z)/α(t) eiθj χEj ,

j=1

gz =

N 

|bk |(1−β(z))/(1−β(t)) eiψk χFk ,

k=1

where α(z) = (1 − z)/p0 + z/p1 and β(z) = (1 − z)/q0 + z/q1 . If qt =  1 then we modify the definition by taking gz ≡ g. It then follows that F(z) = Tfz gz dx is entire and bounded in the strip 0 ≤ Re(z) ≤ 1. Also, F(t) equals the left side of (0.1.15). Consequently, by the three-lines lemma,2 we would be done if we could prove | F(z)| ≤ M0 ,

Re(z) = 0,

| F(z)| ≤ M1 ,

Re(z) = 1.

To prove the first inequality, notice that for y ∈ R, α(iy) = 1/p0 + iy(1/p1 − 1/p0 ). Consequently, | fiy |p0 = |eiargf · | f |iy(1/p1 −1/p0 ) · | f |pt /p0 |p0 = | f |pt . 2 See, for example, Stein and Weiss [1, p. 180].

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0.1 Fourier Transform

7



Similar considerations show that |giy |q0 = |g|qt . Applying H¨older’s inequality and (0.1.13) gives | F(iy)| ≤ Tfiy q0 giy q

0

q /q 0

≤ M0 fiy p0 giy q = M0 f pptt /p0 g qt 0

t

= M0 .

Since a similar argument gives the other inequality, we are done. Later on it will be important to know how the Fourier transform behaves under linear changes of variables. If T : Rn → Rn is a linear bijection and u ∈ S (Rn ) ∩ C(Rn ), we can define its pullback under T by T ∗ u = u ◦ T. Note that a change of variables gives  ∗ (T u)(φ) = u(Tx)φ(x) dx  = u(y) |det T −1 | φ(T −1 y) dy = u(|det T −1 | φ(T −1 · )), so, for general u ∈ S (Rn ), we define the pullback using the left and right sides of this equality. Theorem 0.1.14 With the above notation (T ∗ u)∧ = |det T|−1 (t T −1 )∗ uˆ .

(0.1.16)

We leave the proof as an exercise. As a consequence we have: Corollary 0.1.15 If u ∈ S (Rn ) is homogeneous of degree σ , then uˆ is homogeneous of degree −n − σ . Proof u being homogeneous of degree σ means that if Mt x = tx, then Mt∗ u = ∗ u ˆ . If we replace t by 1/t tσ u. So, by Theorem 0.1.14, tσ uˆ = (Mt∗ u)∧ = t−n M1/t ∗ −n−σ uˆ . this means that Mt uˆ = t Remark Notice that if Re σ < −n, then uˆ is continuous. Using this and Theorem 0.1.4, the reader can check that if u is homogeneous and in C∞ (Rn \0), then so is uˆ . Let us conclude this section by presenting the Poisson summation formula. If g is a function on Rn , then we shall say that g is periodic (with period 2π) if g(x + 2πm) = g(x) for all m ∈ Zn . Given, say, φ ∈ S(Rn ), there are two ways that one can construct a periodic function out of φ. First, one could set

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8

Background

  ix,m . ˆ g = m∈Zn φ(x + 2π m); or one could take g = (2π )−n m∈Zn φ(m)e ∞ Notice that both series converge uniformly to a periodic C function in view ˆ The Poisson summation formula says that the of the rapid decrease of φ and φ. two periodic extensions are the same. Theorem 0.1.16 If φ ∈ S(Rn ) then   ix,m ˆ φ(m)e φ(x + 2πm) = (2π )−n . m∈Zn

m∈Zn

In particular, we have the Poisson summation formula:   ˆ φ(2πm) = (2π )−n φ(m). m∈Zn

(0.1.17)

m∈Zn

To prove this result, let Tn = 2π(Rn /Zn ). Then, if we set Q = [−π , π ]n , it is  clear that the series m∈Zn φ(x + 2π m) = g converges uniformly in the L1 (Q) norm. Thus, for k ∈ Zn , its Fourier coefficients are given by    −ix,k g(x) dx = e−ix,k φ(x + 2π m) dx gk = e Q

=

 m∈Zn

 =

Rn

Q −ix,k

e

m∈Zn

φ(x + 2πm) dx =

Q



m∈Zn

e−ix,k φ(x) dx Q+2πm

ˆ e−ix,k φ(x) dx = φ(k).

 ix,m = g ˆ On the other hand, if we set (2π )−n m∈Zn φ(m)e ˜ , then the series also 1 converges uniformly in L (Q). Its Fourier coefficients are   ix,m ˆ dx g˜ k = e−ix,k (2π )−n φ(m)e Q −n

= (2π )

 m∈Rn

= (2π )−n



m∈Zn



ˆ φ(m)

eix,m−k dx Q

ˆ ˆ φ(m) · (2π )n δk,m = φ(k).

m∈Zn

Thus, since g and g˜ have the same Fourier coefficients, we would be done if we could prove:  Lemma 0.1.17 If μ is a Borel measure on Tn satisfying Tn e−ix,k dμ(x) = 0 for all k ∈ Zn , then μ = 0. To prove this, we first notice that, by the Stone–Weierstrass theorem, trigonometric polynomials are dense in C(Tn ), since they form an algebra

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0.2 Basic Real Variable Theory

9

that separates  points and is closed under complex conjugation. Our hypothesis implies that Tn P(x) dμ(x) = 0 whenever P is a trigonometric polynomial. The approximation property then implies that Tn f (x) dμ(x) = 0 for any f ∈ C(Tn ). By the Riesz representation theorem μ = 0. Using the Poisson summation theorem one can recover basic facts about Fourier series. For instance:  Theorem 0.1.18 If g ∈ L2 (Tn ) then (2π )−n k∈Zn gk eix,k converges to g in the L2 norm and we have Parseval’s formula   |g|2 dx = (2π )−n |gk |2 . Tn

k∈Zn

  Conversely, if |gk |2 < ∞, then (2π )−n k∈Zn gk eix,k converges to an L2 function with Fourier coefficients gk . Proof If g ∈ C∞ (Tn ) then the Poisson summation formula implies that, if we identify Tn and Q as above, then for x ∈ Q,   (2π )−n gk eix,k = g(x + 2π k). k∈Zn

k∈Zn

Hence, if g ∈ C∞ (Tn ), its Fourier series converges to g uniformly, since one can check by integration by parts that gk = O(|k|−N ) for any N. Consequently,     gk gk eix,k−k  dx = (2π )−n |gk |2 . |g|2 dx = (2π )−2n Tn

Tn

Thus, the map sending g ∈ L2 (Tn ) to its Fourier coefficients gk ∈ 2 (Zn ) is an isometry. It is also unitary since the range contains the dense subspace 1 (Zn ).

0.2 Basic Real Variable Theory In this section we shall study two basic topics in real variable theory: the boundedness of the Hardy–Littlewood maximal function and the boundedness of certain Fourier-multiplier operators. Since the Hardy–Littlewood maximal theorem is simpler and since a step in its proof will be used in the proof of the multiplier theorem, we shall start with it. 1 , we If ωn denotes the volume of the unit ball B in Rn , then, given f ∈ Lloc define the Hardy–Littlewood maximal function associated to f by  dy Mf (x) = sup | f (x − ty)| . (0.2.1) ω n t>0 B

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10

Background

If B(x, t) denotes the ball of radius t centered at x then of course   dy 1 = f (y) dy, Mt f (x) = f (x − ty) ωn |B(x, t)| B(x,t) B

(0.2.2)

so in (0.2.1) we are taking the supremum of the mean values of | f | over all balls centered at x. We have used the notation in (0.2.1) to be consistent with some generalizations to follow. Theorem 0.2.1 (Hardy–Littlewood maximal theorem) If 1 < p ≤ ∞ then

Mf Lp (Rn ) ≤ Cp f Lp (Rn ) .

(0.2.3)

Furthermore, M is not bounded on L1 ; however, |{x : Mf (x) > α}| ≤ Cα −1 f L1 (Rn ) .

(0.2.4)

As a consequence, we obtain Lebesgue’s differentiation theorem: 1 , then for almost every x Corollary 0.2.2 If f ∈ Lloc  dy lim f (x − ty) = f (x). t→0 B ωn

(0.2.5)

Before proving the Hardy–Littlewood maximal theorem, let us give the simple argument showing how it implies the corollary. First, it is clear that, in order to prove (0.2.5), it suffices to consider only compactly supported f . Hence, we may assume f ∈ L1 (Rn ) and that f is real valued. Next, let us set f ∗ (x) = | lim sup Mt f (x) − lim inf Mt f (x)|. t→0

t→0

For g ∈ L1 , g∗ (x) ≤ 2Mg(x). Consequently, (0.2.4) gives |{x : g∗ (x) > α}| ≤ 2Cα −1 g L1 . To finish matters, we use the fact that, given ε > 0, any f ∈ L1 can be written as f = g + h with h ∈ C(Rn ) and g L1 < ε. Clearly h∗ ≡ 0, and so |{x : f ∗ (x) > α}| = |{x : g∗ (x) > α}| ≤ 2Cα −1 ε. Since ε is arbitrary, we conclude that f ∗ = 0 almost everywhere, which of course gives (0.2.5). Turning to the theorem, we leave it as an exercise for the reader that if f = χB then Mf L1 = +∞. On the other hand, to prove the substitute, (0.2.4), we shall require:

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0.2 Basic Real Variable Theory

11

Lemma 0.2.3 (Wiener covering lemma) Let E ⊂ Rn be measurable and suppose that E ⊂ Bj , where {Bj } are balls satisfying supj diam Bj = C0 < +∞. Then there is a disjoint subcollection Bjk such that  |Bjk |. (0.2.6) |E| ≤ 5n k

Proof First choose Bj1 so that diam Bj1 ≥ 12 C0 . We now proceed inductively. If disjoint balls Bj1 , . . . , Bjk have been selected we choose, if possible, a ball Bjk+1

satisfying diam Bjk+1 ≥ 12 supj {diam Bj : Bj Bjl = ∅, l = 1, . . . , k}.  In this way we get a collection of disjoint balls {Bjk }. If k |Bjk | = +∞ then  (0.2.6) clearly holds, so we shall assume that |Bjk | is finite. If B∗jk denotes the ball with the same center as Bjk but five times the radius we claim that  E⊂ B∗jk . (0.2.7) k

  This of course gives (0.2.6) since |E| ≤ |B∗jk | ≤ 5n |Bjk |. To prove the claim it suffices to show that Bj ⊂ B∗jk if Bj is one of the balls in the covering. This is trivial if Bj is one of the Bjk , so we shall assume  that this is not the case. To proceed, notice that |Bjk | → 0 since |Bjk | < ∞. With this in mind, let k be the first integer for which diam Bjk+1 < 12 diam Bj . It then follows from the construction that Bj must intersect one of the balls Bj1 , . . . , Bjk . For, if not, it should have been picked instead of Bjk+1 since its

diameter is twice as large. Finally, if Bj Bjl  = ∅ then since diam Bjl ≥ 12 diam Bj , it follows that Bj ⊂ B∗jl . Proof of (0.2.4) For a given a α > 0, let Eα = {x : Mf (x) > α}. It then follows that given x ∈ Eα there is a ball Bx centered at x such that  | f (y)| dy > α|Bx |. Bx

Applying the covering lemma, we can choose points xk ∈ Eα such that the Bxk  are disjoint and |Bxk | ≥ 5−n |Eα |. Thus,   n n −1 |Bxk | ≤ 5 α | f (y)| dy. |Eα | ≤ 5 k

Bxk

Finally, since the balls Bxk are disjoint, we get (0.2.4) with C = 5n . To prove the remaining inequality (0.2.3) we shall need an interpolation theorem. To state it we need to make a few definitions.

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12

Background

Definition 0.2.4 Let T be a mapping from Lp (Rn ), 1 ≤ p ≤ ∞, to measurable functions in Rn . Then, if 1 ≤ q < ∞, we say that T is weak-type (p, q) if |{x : |Tf (x)| > α}| ≤ C(α −1 f Lp )q . For q = ∞, we say that T is weak-type (p, ∞) if

Tf L∞ ≤ C f Lp . Note that, by Chebyshev’s inequality, if T : Lp → Lq is bounded then it is weak-type (p, q). We also define Lp1 (Rn ) + Lp2 (Rn ) as all f which can be written as f = f1 + f2 with f1 ∈ Lp1 and f2 ∈ Lp2 . As an exercise notice that for p1 < p < p2 , Lp (Rn ) ⊂ Lp1 (Rn ) + Lp2 (Rn ). Theorem 0.2.5 (Marcinkiewicz) Suppose that 1 < r ≤ ∞. Let T be a mapping from L1 (Rn ) + Lr (Rn ) to the space of measurable functions that satisfies |T(f + g)| ≤ |Tf | + |Tg|. Then, if T is both weak-type (1, 1) and weak-type (r, r) it follows that whenever 1 < p < r

Tf Lp ≤ Cp f Lp ,

f ∈ Lp (Rn ).

Clearly this implies (0.2.3) since T = M is sub-additive and both weak-type (1, 1) and weak-type (∞, ∞). Proof of Theorem 0.2.5 Given a measurable function g, let m(α) be the distribution function associated to g, that is, m(α) = |{x : |g(x)| > α}|. Then, if p < ∞ and g ∈ Lp (Rn ), it follows that   ∞  p p |g(y)| dy = − α dm(α) = p Rn

0

∞

α p−1 m(α) dα,

(0.2.8)

(0.2.9)

0

while for g ∈ L∞

g L∞ = inf{α : m(α) = 0}.

(0.2.10)

To prove the theorem, let us first suppose that r = ∞. Dividing T by a constant if necessary, we may assume that Tf L∞ ≤ f L∞ . Then, we define f1 (x) by setting f1 (x) = f (x) if | f (x)| ≥ α/2 and zero otherwise. It then follows that |Tf (x)| ≤ |Tf1 (x)| + α/2. Thus, {x : |Tf (x)| > α} ⊂ {x : |Tf1 (x)| > α/2},

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0.2 Basic Real Variable Theory

13

which, by our assumption that T is weak-type (1, 1), means that  |{x : |Tf1 (x)| > α/2}| ≤ C(α/2)−1 | f1 | dx  | f | dx, = 2Cα −1 | f |>α/2

if C is the weak (1, 1) operator norm. Taking g = |Tf (x)| and applying (0.2.9) gives   ∞ |Tf |p dx = p α p−1 |{x : |Tf (x)| > α}| dα

 0 ∞ p−1 −1 ≤p α | f | dx dα. 2Cα | f |>α/2

0

 2| f (x)|

Since α p−2 dα = (p − 1)−1 |2f (x)|p−1 , the last quantity is equal to 0 p p p Cp | f | dx with Cp = 2p p/(p − 1). This gives the conclusion when r = ∞. If r < ∞ and we set f2 = f − f1 then we need to use the fact that {x : |Tf (x)| > α} ⊂ {x : |Tf1 (x)| > α/2} ∪ {x : |Tf2 (x)| > α/2}. Thus, m(α) = |{x : |Tf (x)| > α}| ≤ |{x : |Tf1 (x)| > α/2}| + |{x : Tf2 (x)| > α/2}|. By assumption, there are constants C1 and Cr such that   (0.2.11) m(α) ≤ C1 (α/2)−1 | f1 | dx + Crr (α/2)−r | f2 |r dx   | f | dx + (2Cr )r α −r | f |r dx. = 2C1 α −1 | f |>α/2

| f |≤α/2

To use this, first notice that we have already argued that

   ∞ p−1 −1 α α | f | dx dα = 2p−1 (p − 1)−1 | f |p dx. 0

On the other hand,   ∞ p−1 −r α α 0

| f |>α/2

| f |≤α/2





| f | dx dα = 2 r

p−r

|f |

r

Rn

= 2p−r (r − p)−1

∞

|f | Rn

α

p−1−r

dα dx

| f |p dx.

Putting these two observations together and applying (0.2.11) gives   p p |Tf | dx ≤ Cp | f |p dx, p

with Cp = 2p p[C1 /(p − 1) + Crr /(r − p)].

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14

Background

We now turn to the study of multiplier operators. Given a function m ∈ S we define the Fourier-multiplier operator:  eix,ξ  m(ξ )fˆ (ξ ) dξ , f ∈ S. (0.2.12) Tm f (x) = (2π )−n Rn

Using Plancherel’s theorem, one sees that

Tm f 22 = (2π )−n mfˆ 22 , and, hence, Tm is bounded on L2 if and only if m ∈ L∞ . As an exercise, the reader should verify that any linear operator that is bounded on L2 (Rn ) and commutes with translations must be of the form (0.2.12) with m ∈ L∞ . For p  = 2, the problem of characterizing the multipliers m for which Tm : Lp → Lp is much more subtle. For instance, if n = 1 and m(ξ ) = −π isgn ξ , then the inverse Fourier transform of m is 1/x. Hence, in this case, Tm is not / L1 (R), if f = χ[0,1] . This operator is called the Hilbert bounded on L1 , as Tm f ∈ transform, and we shall see that the next theorem will imply that it is bounded on Lp for all 1 < p < ∞. To state the hypotheses we need to introduce some notation. If − is minus the Laplacian, − = −(∂/∂x1 )2 − · · · − (∂/∂xn )2 , then Theorem 0.1.4 implies  eix,ξ  |ξ |2 fˆ (ξ ) dξ . − f (x) = (2π )−n Rn

With this in mind, for s ∈ C, we define operators (I − )s/2 : S → S by  s/2 −n eix,ξ  (1 + |ξ |2 )s/2 fˆ (ξ ) dξ . (0.2.13) (I − ) f (x) = (2π ) Rn

p

Finally, for 1 ≤ p ≤ ∞ and s ∈ R, we define the Sobolev spaces Ls (Rn ) as all u ∈ S for which (I − )s/2 u is a function and

u Lsp (Rn ) = (I − )s/2 u Lp (Rn ) < ∞. A useful observation is that

u 2L2 (Rn ) s

−n

= (2π )

(0.2.14)

 |ˆu(ξ )|2 (1 + |ξ |)s dξ .

(0.2.15)

We can now state the multiplier theorem: Theorem 0.2.6 Let m ∈ L∞ (Rn ). Assume further that, for some integer s > n/2,  sup λ−n λ|α| Dα β( · /λ)m( · ) 2L2 (Rn ) < ∞ (0.2.16) 0≤|α|≤s λ>0

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0.2 Basic Real Variable Theory

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whenever β ∈ C0∞ (Rn \0). It then follows that for 1 < p < ∞

Tm f Lp (Rn ) ≤ Cp f Lp (Rn ) .

(0.2.17)

|{x : |Tm f (x)| > α}| ≤ Cα −1 f L1 (Rn ) .

(0.2.18)

Furthermore, for α > 0

When we say that β ∈ C0∞ (Rn \0) we are abusing the notation a bit. We mean that β is a C0∞ (Rn ) function that is supported in Rn \0. Similar notation will be used throughout. Remarks Notice that (0.2.16) holds if m ∈ C∞ (Rn \0) ∩ L∞ (Rn ) and |Dγ m(ξ )| ≤ Cγ |ξ |−|γ |

∀γ .

In particular, if m is homogeneous of degree zero and in C∞ (Rn \0), then by the remark after Corollary 0.1.15, Tm must be bounded on Lp (Rn ) for all 1 < p < ∞. Thus, if K ∈ C∞ (Rn \0) ∩ S is homogeneous of degree −n and we define the principal-value convolution  f (x − y)K(y) dy, Tf (x) = P.V.(f ∗ K)(x) = lim ε→0 |y|>ε

then T is bounded on

Lp

for 1 < p < ∞.

Returning to the theorem, notice that Tm : L2 (Rn ) → L2 (Rn ). Furthermore,   Tm f g dx = f Tm g dx. Since m satisfies the same hypotheses, by duality, it suffices to prove (0.2.17) for 1 < p < 2. Moreover, by the Marcinkiewicz interpolation theorem, we see that we need only prove the weak-type (1, 1) estimate (0.2.18). The key tool in the proof of the weak-type estimate is the Calder´on–Zygmund decomposition of L1 functions: Lemma 0.2.7 (Calder´on–Zygmund lemma) Let f ∈ L1 (Rn ) and α > 0. Then we can decompose f : ∞  bk , (0.2.19) f = g+ 1

where

g 1 +

∞ 

bk 1 ≤ 3 f 1 ,

(0.2.20)

almost everywhere,

(0.2.21)

1

|g(x)| < 2n α

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16

Background

and for certain non-overlapping cubes Qk bk (x) = 0 if x ∈ / Qk ∞ 

 bk dx = 0,

and

|Qk | ≤ α −1 f 1 .

(0.2.22) (0.2.23)

1

Proof of Lemma 0.2.7 We start out by dividing Rn into a lattice of cubes of volume > α −1 f 1 . Thus, if Q is one of the cubes in the lattice  (0.2.24) |Q|−1 | f | dx < α. Q

Divide each cube into equal non-overlapping cubes and let Q11 , Q12 , . . . be the resulting cubes for which (0.2.24) no longer holds, that is,  | f | dx ≥ α. (0.2.25) |Q1k |−1 2n

Q1k



Notice that α|Q1k | ≤

| f | dx < 2n α|Q1k |

(0.2.26)

Q1k

by (0.2.24) and the fact that, if Q1k was obtained by dividing Q, then 2n |Q1k | = |Q|. We set  f dx, x ∈ Q1k , g(x) = |Q1k |−1 Q1k

b1k (x) = f (x) − g(x),

x ∈ Q1k ,

and

b1k (x) = 0,

x∈ / Q1k .

(0.2.27)

Next, we consider all the cubes that are not among the {Q1k }. By construction, each one satisfies (0.2.24). We divide each one as before into 2n subcubes  and let Q21 , Q22 , . . . be the resulting ones for which |Q2k |−1 Q2k | f |dx ≤ α. We extend the definition (0.2.27) for these cubes. Continuing this procedure we get non-overlapping cubes Qjk and functions bjk which we rearrange in a sequence. If the definition of g is extended by setting g(x) = f (x) for x ∈ /  = Qk then  (0.2.19) holds. Furthermore, since Qk |g| dx ≤ Qk | f | dx it follows from the triangle inequality that   (|g| + |bk |) dx ≤ 3 | f | dx, Qk

Qk

which leads to (0.2.20) since the cubes Qk are non-overlapping and g = f on /  there are arbitrarily small c . Also, (0.2.21) holds when x ∈ ; while if x ∈ cubes containing x over which the mean value of | f | is < α. Thus |g| < α almost everywhere in c and so (0.2.21) holds. The cancellation property (0.2.22)

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0.2 Basic Real Variable Theory

17

follows from the construction, and, finally, (0.2.23) follows from the fact that the cubes Qk satisfy the analog of (0.2.26). Proof of (0.2.18) Choose ψ(ξ ) ∈ C0∞ (B(0, 2)) which equals 1 when |ξ | ≤ 1. Then, if we let β(ξ ) = ψ(ξ ) − ψ(2ξ ), it follows that β ∈ C0∞ (Rn \0) and 1=

∞ 

β(2−j ξ ),

ξ  = 0.

(0.2.28)

−∞

Set mλ (ξ ) = β(ξ )m(λξ ); then scaling (0.2.16) gives  Rn

|(I − )s/2 mλ (ξ )|2 dξ ≤ C.

(0.2.29)

Or, by (0.2.15), if Kˆ λ = mλ then  |Kλ (x)|2 (1 + |x|2 )s dx ≤ C, Rn

which implies the important estimates  |Kλ (x)| dx ≤ C(1 + R)n/2−s .

(0.2.30)

{x:max |xj |>R}

Since ξj mλ (ξ ) satisfies the same type of estimates as mλ (ξ ) it also follows that  |∇Kλ (x)| dx ≤ C, which leads to

 |Kλ (x + y) − Kλ (x)| dx ≤ C|y|.

The main step in the proof of (0.2.18) is to show that if f = g + Calder´on–Zygmund decomposition of f then, if Kˆ = m,    |Tm bk | dx = |K ∗ bk | dx ≤ C |bk | dx. x∈Q / ∗k

x∈Q / ∗k

(0.2.31) 

bk is the

(0.2.32)

Here Q∗k is the cube with the same center as Qk but twice the side-length. After possibly making a translation, we may assume that Qk = {x : max |xj | ≤ R}.

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18

Background

Notice that, by (0.1.16), the Fourier transform of λn Kλ (λx) is mλ (ξ/λ). By (0.2.28), this means that K(x) =

∞ 

2nj K2 j (2 j x),

−∞

with convergence in S . To use this first notice that, since bk vanishes outside Qk , (0.2.30) gives    n |λ Kλ (λ · ) ∗ bk | dx ≤ |λn Kλ (λ(x − y))||bk (y)| dxdy x∈Q / ∗k

y∈Qk x∈Q / ∗k



≤ bk 1

{x:max |xj |>λR}

|Kλ (x)| dx

≤ C(Rλ)n/2−s bk 1 . Since s > n/2, this provides favorable estimates when the scale λ−1 is smaller than R.  To handle the other case, we need to use (0.2.31). First we notice that since bk = 0 it follows that  Kλ (λ · ) ∗ bk = {Kλ (λ(x − y)) − Kλ (λx)} bk (y) dy, and so (0.2.31) leads to  |λn Kλ (λ · ) ∗ bk | dx x∈Q / ∗k



≤ y∈Qk

 x∈Q / ∗k

λn |Kλ (λ(x − y)) − Kλ (λx)| |bk (y)| dxdy

≤ C(Rλ) bk 1 . Putting these two estimates together and applying the triangle inequality gives ⎛ ⎞    |K ∗ bk | dx ≤ C bk 1 · ⎝ (2 j R)n/2−s + 2 j R⎠ x∈Q / ∗k

2 j R≥1

2 j R α/2}| ≤ Cα −2 g 22 ≤ C α −1 f 1 .

(0.2.33)

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0.2 Basic Real Variable Theory

19

If ∗ = ∪Q∗k then

|∗ | ≤ 2n α −1 f 1 ,  while (0.2.32) implies that if b = bk

(0.2.34)

|{x ∈ / ∗ : |Tm b(x)| > α/2}|  |K ∗ bk | dx ≤ C α −1 f 1 . ≤ 2α −1 k

x∈Q / ∗k

(0.2.35)

Finally, since {x : |Tm f (x)| > α} ⊂ {x : |Tm g(x)| > α/2} ∪ {x : |Tm b(x)| > α/2}, (0.2.33)–(0.2.35) give (0.2.18). It is easier to prove the H¨older continuity of the operators in Theorem 0.2.6. Recall that f is said to be H¨older continuous of order 0 < γ < 1 if | f (x) − f (y)| < ∞. |x − y|γ x=y

| f |γ = sup

This norm is related to another that involves the dyadic decomposition of the Fourier transform that was used in the proof of Theorem 0.2.6. More precisely, if β is the function occurring there, let us define fj by fˆj (ξ ) = β(2−j ξ )fˆ (ξ ) and set

f Lip(γ ) = sup 2γ j fj L∞ . j

Then we have: Lemma 0.2.8 If 0 < γ < 1 and f is H¨older continuous of order γ then there is a constant 0 < Cγ < ∞ such that Cγ−1 | f |γ ≤ f Lip(γ ) ≤ Cγ | f |γ .

(0.2.36)

Proof Let fj be as above. Clearly, | fj | ≤ 2−γ j f Lip(γ ) . Furthermore, since fˆj (ξ ) vanishes when |ξ | > const.2 j , it follows that we have the bound | fj | ≤ C2(1−γ )j

f Lip(γ ) . Therefore the mean value theorem gives  | fj (x) − fj (y)| | f (x) − f (y)| ≤ j

⎛

≤ C f Lip(γ ) ⎝2



2−γ j + |x − y|

2−j 1 it follows from Parseval’s formula that    f g dx = Sj f Sl g dx.

(2−j+l ξ ) = 0

{j,l:|j−l|≤1}

Applying the Schwarz inequality and then H¨older’s inequality therefore gives      f g dx ≤ 3 Sf p Sg p ,   if, as usual, 1/p + 1/p = 1. Since we are assuming that Sg p ≤ Cp g p , if we take the supremum over all g with g p = 1, the left side becomes f p , which together with the previous remark implies f p ≤ C Sf p . To finish the proof and show

Sf p ≤ Cp f p ,

(0.2.37 )

1 < p < ∞,

it is convenient to use Rademacher functions. These functions, r0 (t), r1 (t), . . ., are defined on the unit interval [0, 1] in the following manner. First we set r0 (t) = 1 for 0 ≤ t ≤ 12 and r0 (t) = −1 for 12 < t < 1. We then extend r0 periodically, that is, r0 (t) = r0 (1 + t), and set rj (t) = r0 (2 j t). One sees immediately that the rj are orthonormal on [0, 1]. The important fact is that  if F(t) = aj rj (t) ∈ L2 ([0, 1]) then F ∈ Lp ([0, 1]) for all 0 < p < ∞ and we have Khintchine’s inequality 

1/2 −1 2 |aj | ≤ Ap F Lp ([0,1]) (0.2.38) Ap F Lp ([0,1]) ≤ F L2 ([0,1]) = for some Ap < ∞. For a proof of this see Stein [2, Appendix D]. Using (0.2.38) and the multiplier theorem, it is not hard to prove (0.2.37 ). We first notice that if we define multiplier operators Tt by (Tt f )∧ (ξ ) =

∞ 

rj (t)β(2−j ξ )fˆ (ξ ) = mt (ξ )fˆ (ξ ),

j=0

1 γ then ( 0 |Sj f (x)|2 )1/2 ≤ Ap ( 0 |Tt f (x)|p dt)1/p . But |Dξ mt (ξ )| ≤ Cγ |ξ |−|γ | with constants Cγ independent of t. Consequently, Theorem 0.2.6 implies ∞ 

∞  Rn

0

|Sj f (x)|

 2 p/2

dx ≤ Ap

 1 0

Rn

|Tt f (x)|p dxdt ≤ Cp f pp . 

And since the same argument can be used to estimate ( done.

2 1/2 , j 1 and

1 1 1 = 1− − r p q for some 1 < p < q < ∞, then

Ir f Lq (Rn ) ≤ Cp,q f Lp (Rn ) .

(0.3.2)

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0.3 Fractional Integration and Sobolev Embedding Theorems

23

Notice how this result is related to Young’s inequality since the kernel |y|−n/r just misses belonging to Lr (Rn ). The proof will be based on a sequence of lemmas. The first is: Lemma 0.3.3 If 1 ≤ p < r then

1−p/r

Ir f ∞ ≤ Cp,r f p/r p f ∞

,

Proof We first notice that for any R > 0   −n/r |Ir f (x)| ≤ |y| | f (x − y)| dy + |y|R

(0.3.3)

|y|−n/r | f (x − y)| dy

  ≤ C Rn−n/r f ∞ + (Rn−np /r )1/p f p   = C Rn/r f ∞ + R−n/q f p . If we choose R so that the last two terms agree, that is,

Rn/p = Rn/r Rn/q = f p / f ∞ , p/r

1−p/r

then the terms inside the parentheses both equal f p f ∞

.

To use the Calder´on–Zygmund decomposition we need:  Lemma 0.3.4 Let b ∈ L1 be supported in a cube Q and satisfy b dx = 0. Then 

1/r x∈Q / ∗

|Ir b|r dx

≤ Cr b 1 .

(0.3.4)

Proof We may assume that Q is the cube of side-length R centered at the origin. Then for x ∈ / Q∗ , the double of Q, we have    |Ir b(x)| ≤  |x − y|n/r − |x|−n/r  |b(y)| dy ≤ CR|x|−1−n/r b 1 , by the mean value theorem. Integrating this inequality gives us (0.3.4) since

1/r  −r−n |x| dx = C/R. x∈Q / ∗

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24

Background The last lemma is:

Lemma 0.3.5 Ir is weak-type (1, r): |{x : |Ir f (x)| > γ }| ≤ Cr (γ −1 f 1 )r .

(0.3.5)

 Proof Write f = g + bk as in Lemma 0.2.7. To simplify the calculations we may assume that f 1 = 1. Then by (0.3.3) with p = 1, 1/r

1−1/r

|Ir g| ≤ g 1 g ∞ Define α by Cα 1/r = γ /2. Then

|{x : |Ir f (x)| > γ }| ≤ |{x : If ∗ = ∪Q∗k , then



≤ Cα 1−1/r = Cα 1/r . 

|Ir bk (x)| > γ /2}|.

|∗ | ≤ 2n α −1 = C γ −r ,

while, by (0.3.4), |{x ∈ / ∗ :



|Ir bk (x)| > γ /2}|1/r

1/r   ≤ (γ /2)−1 |Ir bk (x)|r dx ≤ C γ −1 . x∈ / ∗

If we combine the last three inequalities, we get (0.3.5). Proof of Theorem 0.3.2 We use the argument that was used to prove the Marcinkiewicz interpolation theorem. We may assume that f p = 1 and shall use  ∞

Ir f qq = q γ q−1 m(γ ) dγ , 0

where m(γ ) = |{x : |Ir f (x)| > γ }|. To estimate m(γ ) we, as before, set f = f0 +f1 , where f0 = f when | f | > α and 0 otherwise. Then, by (0.3.3) and our assumptions regarding the exponents, 1−p/r



Ir f1 ∞ ≤ Cp,r f1 p/r p f1 ∞



≤ Cp,r α 1−p/r = Cp,r α p/q .

We now choose α so that γ /2 = Cp,r α p/q . Then m(γ ) ≤ |{x : |Ir f0 (x)| > γ /2}| ≤ Cr (γ −1 f0 1 )r by (0.3.5). This implies that

Ir f qq



∞

≤C

γ q−1 (γ −1 f0 1 )r dγ .

0

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0.3 Fractional Integration and Sobolev Embedding Theorems

25

−1 γ /2)q/p , recall the definition But, if we make the change of variables α = (Cp,r of f0 , and apply Minkowski’s integral inequality, we find that this integral is ⎫r ⎧  1/r ⎬ ⎨   | f (x)| ≤ C α −1+p−rp/q dα | f (x)| dx . ⎭ ⎩ 0

But q > r by assumption, and hence  1/r | f (x)|

α −1+p−rp/q dα

= | f (x)|p(1/r−1/q) = | f (x)|p−1 ,

0 q

which leads to the desired inequality Ir f q ≤ C since we are assuming that

f p = 1. Remark Historically, Sobolev proved the n-dimensional version of the Hardy–Littlewood–Sobolev inequality using the one-dimensional version which is due to Hardy and Littlewood. Let us give a simple argument showing how the n-dimensional version follows from Young’s inequality and the one-dimensional version. Variations on this argument will be used later. First, we write x = (x , xn ), where x = (x1 , . . . , xn−1 ). Then, if xn is fixed, we can use Minkowski’s integral inequality to get

Ir f ( · , xn ) Lq (Rn−1 )   ∞    ≤  Rn−1

−∞

But





Rn−1

−n/r

Rn−1

|(x , xn − yn )|

f (y) |x − y|

− nr ·r

dx



1/r

q 1/q  dy  dx dyn . 

= (Cn |xn − yn |−1 )1/r .

So Young’s inequality and the above yield  ∞ |xn − yn |−1/r f ( · , xn ) Lp (Rn−1 ) dyn .

Ir f ( · , xn ) Lq (Rn−1 ) ≤ C −∞

Raising this to the qth power, integrating, and applying the one-dimensional fractional integration theorem gives

Ir f Lq (Rn )  ∞    ≤C 

q

1/q  |xn − yn |

f ( · , xn ) Lp (Rn−1 ) dyn  dxn −∞ −∞  ∞

1/p p ≤C

f ( · , xn ) Lp (Rn−1 ) dxn = C f Lp (Rn ) . ∞

−1/r

−∞

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26

Background

Let us now see that this argument also yields a couple of results that will be useful later on. Both involve mixed-norm spaces Lr Lp (R1+d ), where, if r is finite, we set  ∞ 1/r

f (t, · ) rLp (Rd ) dt

f Lr Lp (R1+d ) = −∞

and extend the definition to the case where r = ∞ in the obvious way. Our first result, as the reader may verify, follows immediately from the argument that we just gave: Theorem 0.3.6 Let W be an integral operator with kernel K(t, x; t , y), where x, y ∈ Rd and t, t ∈ R, i.e.,  WF(t, x) = K(t, x; t , y) F(t , y) dt dy. R1+d

Suppose that for every fixed t, t the associated integral operator  Wt,t f (x) = K(t, x; t , y) f (y) dy Rd

maps C0∞ (Rd ) into C(Rd ) and satisfies

Wt,t f Lq (Rd ) ≤ C0 |t − t |−σ f Lp (Rd ) ,

f ∈ C0∞ (Rd ),

for some 1 ≤ p ≤ q ≤ ∞, with 0 < σ < 1 and C0 < ∞ fixed. Then if 1 < r < s < ∞ and σ = 1 − (1/r − 1/s), it follows that

WF Ls Lq (R1+d ) ≤ C F Lr Lp (R1+d ) ,

with C = C0 Cr,s

if Cr,s is the constant in the one-dimensional Hardy–Littlewood inequality (0.3.2). If we use a “TT ∗ argument” we can deduce the following: Corollary 0.3.7 Suppose U(t) : C0∞ (Rd ) → C(Rd ), is a family of linear operators depending continuously on the parameter t ∈ R. Suppose further that (0.3.6)

U(t) L2 (Rd )→L2 (Rd ) ≤ 1, and that

U(t)U ∗ (t ) L1 (Rd )→L∞ (Rd ) ≤ M|t − t |−σ ,

(0.3.7)

where U ∗ (t ) denotes the adjoint of U(t ). Then if 2 < q, s < ∞ and σ=

2σ 2 + , q s

(0.3.8)

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0.3 Fractional Integration and Sobolev Embedding Theorems

27

we have for F ∈ C0∞ (R1+d )

WF Ls Lq (R1+d ) ≤ Cs ,s M if

 WF =

∞

−∞

∗



1− 2q

F Ls Lq (R1+d ) ,





U(t)U (t )F(t , · ) dt , or

t

−∞

(0.3.9)

U(t)U ∗ (t )F(t , · ) dt .

Furthermore, in this case we also have  1 1 −

U( · )f Ls Lq (R1+d ) ≤ Cs ,s M 2 q f L2 (Rd ) , as well as

  

∞ −∞

(0.3.10)

  U(t)U ∗ (t )F(t , · ) dt 

Ls2 Lq2 (R1+d )

≤

Cs ,s1 Cs ,s2 M 1

1− q1 − q1 1 2

2

F

s

q

L 1 L 1 (R1+d )

,

(0.3.11)

if, as above, Cr,s is the constant in the one-dimensional Hardy–Littlewood inequality (0.3.2) and if 2 < qj , sj < ∞ and

σ=

2σ 2 + , qj sj

j = 1, 2.

To prove the corollary we first note that (0.3.6) implies that

U(t)U ∗ (t ) L2 (Rd )→L2 (Rd ) ≤ 1. By Theorem 0.1.13, if we interpolate between this estimate and (0.3.7) we deduce that

U(t)U ∗ (t ) Lq (Rd )→Lq (Rd ) ≤ M

1− 2q

|t − t |

−σ (1− 2q )

=M

1− 2q

|t − t |−1+( s − s ) ,

1

1

assuming for the equality that (0.3.8) is valid. Using this, we see that (0.3.9) follows immediately from Theorem 0.3.6. To prove (0.3.10) we note that, by H¨older’s inequality and (0.3.9), we have  ∞ 2   U ∗ (t )F(t , · ) dt  2 d  L (R ) −∞  ∞   ∞  = U(t)U ∗ (t )F(t , x) dt F(t, x) dxdt −∞ Rd −∞  ∞    ≤ U(t)U ∗ (t )F(t , · ) dt  s q 1+d F Ls Lq (R1+d ) L L (R

−∞

≤ Cs ,s M

1− 2q

)

F 2Ls Lq (R1+d ) .

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28

Background

This implies (0.3.10) by duality. Since (0.3.10) and this dual version imply (0.3.11), the proof of the corollary is complete. Let us conclude this section by using the Hardy–Littlewood–Sobolev inequality to also deduce the following: Theorem 0.3.8 (Sobolev embedding theorem) p

(1) If 1 < p ≤ q < ∞ and 1/p − 1/q = s/n then Ls (Rn ) ⊂ Lq (Rn ) and the inclusion is continuous. p p (2) If s > n/p and 1 ≤ p < ∞ then Ls (Rn ) ⊂ L∞ (Rn ) and any u ∈ Ls (Rn ) can be modified on a set of measure zero such that it is continuous. For a given s let Ks (x) = (2π )−n

 Rn

eix,ξ  (1 + |ξ |2 )−s/2 dξ .

To prove the theorem we shall need the following result. Lemma 0.3.9 If 0 < s then Ks is a function. Moreover, if 0 < s < n then given any N there is a constant CN such that |Ks (x)| ≤ CN |x|−n+s (1 + |x|)−N ,

(0.3.12)

and Ks = O((1 + |x|)−N ) for all N if s > n. Proof We first prove (0.3.12) for |x| ≥ 1. If we integrate by parts we obtain3  −n −2m Ks (x) = (2π ) |x| eix,ξ  (−)m (1 + |ξ |2 )−s/2 dξ . Rn

Since |(−)m (1 + |ξ |2 )−s/2 | ≤ Cm (1 + |ξ |)−s−2m is integrable when m > n/2, we conclude that (0.3.12) holds for |x| ≥ 1. The real issue, though, is to see that the inequality holds for x near the origin. To handle this case, let ρ ∈ C0∞ (Rn ) equal one near the origin. For a given R we have       (2π )−n eix,ξ  ρ(ξ/R)(1 + |ξ |2 )−s/2 dξ     (1 + |ξ |)−s dξ ≤ C Rn−s . ≤C |ξ | n/2       (2π )−n eix,ξ  (1 − ρ(ξ/R)) (1 + |ξ |2 )−s/2 dξ     |x|−2m |ξ |−s−2m dξ ≤C |ξ |>C0 R

= C |x|−2m Rn−s−2m . If we choose R = |x|−1 the two terms balance and both are O(|x|−n+s ), which gives us (0.3.12). The last part of the lemma follows from a similar integration by parts argument which is left to the reader. Proof of Theorem 0.3.8 We start out with (1). We must show that

u Lq (Rn ) ≤ C (I − )s/2 u Lp (Rn ) .

(0.3.13)

If s = 0 then p = q and the result is trivial, so we assume that s > 0. If we set v = (I − )s/2 u, then proving (0.3.13) amounts to showing that

(I − )−s/2 v Lq (Rn ) ≤ C v Lp (Rn ) .

(0.3.7 )

But our assumption that 1 < p < q < ∞ implies that 0 < s < n. Thus we can apply Lemma 0.3.9 to see that

1 1 1 −n/r |Ks (x)| ≤ C|x| , = 1− − . r p q Consequently, since (I − )−s/2 v = Ks ∗ v, (0.3.7 ) follows from the Hardy– Littlewood–Sobolev inequality. To prove (2), we notice that the lemma implies that for s > n/p, Ks ∈ Lp (Rn ). Thus convolution with Ks is a bounded operator from Lp to L∞ . This implies that  

u( · + y) − u( · ) ∞ = (I − )−s/2 [v( · + y) − v( · )]∞ ≤ C v( · + y) − v( · ) p .

By assumption, v = (I − )s/2 u ∈ Lp , and since v( · + y) − v( · ) p → 0 as y → 0, we conclude that u can be modified on a set of measure zero so that the resulting function is continuous.

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30

Background

0.4 Wave Front Sets and the Cotangent Bundle In this section we shall go over the basics from the spectral analysis of singularities. It follows from results of the first section that if u is a compactly supported distribution (written as u ∈ E ), then u is smooth if and only if uˆ is rapidly decreasing. However, if u is not smooth, it is possible that uˆ is rapidly decreasing in some directions. Thus only certain high-frequency components of the Fourier transform may contribute to the singularities of u. The notion of wave front set will make all of this precise and will be important for later results. If u ∈ E (Rn ) then sing supp u, the singular support of u, is the set of all x ∈ Rn such that x has no open neighborhood on which the restriction of u is C∞ . The wave front set of u will consist of certain (x, ξ ) ∈ Rn × Rn \0 with x ∈ sing supp u. To describe the frequency component we first define (u) ⊂ Rn \0 as the closed cone consisting of all η ∈ Rn \0 such that η has no conic neighborhood in which |ˆu(ξ )| ≤ CN (1 + |ξ |)−N ,

N = 1, 2, . . . .

By a conic neighborhood of a subset of Rn \0 we mean an open set N which contains the subset and has the property that if ξ ∈ N then so is λξ for all λ > 0. Returning to the definition of (u), it follows that u ∈ E is in C0∞ if and only if (u) = ∅. One should think of sing supp u as measuring the location of the singularities of u and (u) as measuring their direction. This is consistent with the following: Lemma 0.4.1 If φ ∈ C0∞ (Rn ) and u ∈ E (Rn ), then (φu) ⊂ (u). Proof If v = φu then −n

vˆ (ξ ) = (2π )

(0.4.1)

 ˆ − η)ˆu(η) dη. φ(ξ

Since u ∈ E , uˆ is a smooth function satisfying |ˆu(η)| ≤ C(1 + |η|)m , for some m. To use this, we first notice that if ξ is outside of a fixed conic neighborhood of (u) and η is inside a slightly smaller conic neighborhood

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then |ξ − η| ≥ c(|ξ | + |η|) for some c > 0. And so in this case ˆ − η)ˆu(η)| ≤ CN (1 + |ξ | + |η|)−N (1 + |η|)m |φ(ξ ≤ CN (1 + |ξ | + |η|)−N+m . On the other hand, if η is outside of a fixed small conic neighborhood of (u) then for any ξ ˆ − η)ˆu(η)| ≤ CN (1 + |ξ − η|)−N (1 + |η|)−N . |φ(ξ Combining these two observations leads to (0.4.1) since   −N+m dη + (1 + |ξ − η|)−N (1 + |η|)−N dη (1 + |ξ | + |η|) is O(|ξ |−N+m+n + |ξ |−N+n ) if N is large. Next, if X ⊂ Rn is open, we define for u ∈ D (X) (the dual of C0∞ (X)) and x ∈ X, ! (φu), φ ∈ C0∞ (X), φ(x)  = 0. x (u) = φ

As an exercise, one should verify that Lemma 0.4.1 implies that (φj u) → x (u) if φj (x)  = 0 and supp φj → {x}. Definition 0.4.2 If u ∈ D (X), then the wave front set of u is given by WF(u) = {(x, ξ ) ∈ X × Rn \0 : ξ ∈ x (u)}. It is clear that the projection of WF(u) onto X is sing supp u. This implies that WF(u) is closed. Furthermore, it is not hard to use Lemma 0.4.1 to prove the following: Proposition 0.4.3 If u ∈ E (Rn ) then the projection of WF(u) onto the frequency component is (u). This result and the remark preceding it show that WF(u) contains all the information in sing supp u and (u). The advantage of using WF(u) is that, unlike (u), WF(u) is invariant under changes of variables. Before proving this and other properties of the wave front set, let us give some simple examples. First of all, if u ∈ S (Rn ) is homogeneous and C∞ away from the origin, it follows that WF(u) = {(0, ξ ) : ξ ∈ supp uˆ \{0}}. In particular, in R if u = δ0 (x), the Dirac delta distribution, or if u = P.V. 1/x then WF(u) = {(0, ξ ) : ξ ∈ R\0}. On the other hand, 1 u = (x + i0)−1 = lim (x + iε)−1 = P.V. − π iδ0 (x), ε→0+ x

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32

Background

has Fourier transform equal to −2πiχ[0,∞) and so in this case the wave front set is {(0, ξ ) : ξ > 0}. Another example is given in the following result. Theorem 0.4.4 Let V be a subspace of Rn and u = u0 dS, where u0 ∈ C0∞ (V) and dS is Lebesgue measure on V. Then WF(u) = supp u × (V ⊥ \0). Proof Applying Theorem 0.1.14, we see that we may assume that V = {x : x1 = · · · = xj = 0}. Then if φ ∈ C0∞ and if we set x = (x , x ), x = (x1 , . . . , xj ), it follows that  (φu)∧ (ξ ) = e−ix ,ξ  φ(x , 0)u0 (x ) dx . This formula implies the result. For the right side is a rapidly decreasing / WF(u) for any ξ  = 0, that is, WF(u) ⊂ function of ξ , and hence (x, ξ , ξ ) ∈ supp u × V ⊥ \0. Also, if φ(x , 0)u0 (x )  = 0 then its Fourier transform must be nonzero at some ξ0 which implies that (φu)∧ (ξ0 , ξ ) is a nonzero constant for all ξ and hence no ξ = (0, ξ ) can have a conic neighborhood on which (φu)∧ is rapidly decreasing. Since this gives the other inclusion, supp u × V ⊥ \0 ⊂ WF(u), we are done. As our first application of wave front analysis we present the following theorem of H¨ormander on the multiplication of distributions. Theorem 0.4.5 Let u, v ∈ D (X) and suppose that (x, ξ ) ∈ WF(u) ⇒ (x, −ξ ) ∈ / WF(v). Then, the product uv is well defined and WF(uv) ⊂ {(x, ξ + η) : (x, ξ ) ∈ WF(u), (x, ξ ) ∈ WF(v)} ∪ WF(u) ∪ WF(v).

(0.4.2)

Proof By using a C∞ partition of unity consisting of functions with small support, we may assume that u, v ∈ E (Rn ) and that moreover ξ ∈ (u) ⇒ −ξ ∈ / (v). It then follows that

 (ˆu ∗ vˆ )(ξ ) =

uˆ (ξ − η)ˆv(η) dη

(0.4.3)

is an absolutely convergent integral. For if η belongs to a small conic neighborhood of (v) then for fixed ξ , uˆ (ξ − η) = O(|η|−N ), while if η

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is outside of this small conic neighborhood of (v), vˆ (η) = 0(|η|−N ) by definition. These observations imply that (0.4.3) converges absolutely and is tempered since |ˆu(ξ )|, |ˆv(ξ )| ≤ C(1+|ξ |)m for some m. Thus, uv is well defined and in E by Fourier’s inversion formula. To prove (0.4.2) we write uˆ = uˆ 0 + uˆ 1 and vˆ = vˆ 0 + vˆ 1 , where uˆ 0 , vˆ 0 ∈ S and uˆ 1 and vˆ 1 vanish outside of small conic neighborhoods of (u) and (v), respectively. It then follows that the Fourier transform of uv is   −n uˆ j (ξ − η)ˆvk (η) dη. (2π ) j,k=0,1

If both j = k = 0 then this convolution is rapidly decreasing in all directions. If j = 0, k = 1 or j = 1, k = 0 it follows from the proof of Lemma 0.4.1 that it is rapidly decreasing outside of a small conic neighborhood of (v) or (u), respectively. Furthermore, by construction, the remaining term where j = k = 1 vanishes for ξ outside of a small conic neighborhood of (u) + (v). Together, these observations lead to (uv) ⊂ (u) ∪ (v) ∪ ((u) + (v)), which in turn implies (0.4.2). We now turn to the very important change of variables formula for wave front sets. We assume that κ :X→Y is a diffeomorphism between open subsets of Rn . If  is a subset of Y × Rn \0 we define the pullback of  by κ ∗  = {(x, ξ ) : (κ(x), (t κ )−1 ξ ) ∈ }.

(0.4.4)

As a good exercise the reader should verify that the linear change of variables formula for the Fourier transform, (0.1.16), implies that if κ : Rn → Rn is linear then (0.4.5) WF(κ ∗ u) = κ ∗ WF(u), u ∈ D (Y). The main theorem of this section is that the same is true for any diffeomorphism if we extend the definition of pullbacks in the natural way by setting     dκ −1   ∗ −1 (κ u)(φ) = u(), (y) = φ(κ (y)) det (y). dy Theorem 0.4.6 As above let κ : X → Y be a diffeomorphism between open sets in Rn . Then formula (0.4.5) holds.

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34

Background

Proof Since the problem is local there is no loss of generality in assuming that u ∈ E . Since (κ −1 )∗ κ ∗ = Identity, it suffices to show that WF(κ ∗ u) ⊂ κ ∗ WF(u).

(0.4.5 )

We clearly may assume that 0 ∈ X, κ(0) = 0, and that κ (0) = Identity. Then if 1 is a small conic neighborhood of (u) we shall prove that (κ ∗ (φu)) ⊂ 1

(0.4.5 )

if φ ∈ C0∞ (Rn ) is supported in a small neighborhood of the origin. This would imply (0.4.5 ).  To prove this, we fix β ∈ C0∞ ({ξ : 12 < |ξ | < 2}) satisfying β(2−j ξ ) = 1,  ξ  = 0, and set βj (ξ ) = β(2−j ξ ) for j = 1, 2, . . . and β0 (ξ ) = 1− ∞ 1 βj (ξ ). Then if ρ = κ ∗ φ  ∞  ρ(x) eiκ(x),η uˆ (η)βj (η) dη. (0.4.6) κ ∗ (φu)(x) = (2π )−n j=0

As before we can write uˆ = uˆ 0 + uˆ 1 , where uˆ 0 ∈ S and supp uˆ 1 ⊂ 0 , with 0 being a closed subset of 1 satisfying (u)  0  1 . Since it is not hard to check that the Fourier transform of the analog of (0.4.6) where uˆ is replaced by uˆ 0 is rapidly decreasing in all directions, it suffices to show that     −ix,ξ  iκ(x),η  e ρ(x)ˆu1 (η)βj (η) dηdx e  ≤ CN (2 j + |ξ |)−N ,

ξ∈ / 1 , j = 1, 2, . . . .

(0.4.7)

To prove this we set R = 2 j and write each term as  n ei[Rκ(x),η−x,ξ ] ρ(x)ˆu1 (Rη)β1 (η) dηdx R  ei(R+|ξ |)(x,ξ ,η) ρ(x)ˆu1 (Rη)β1 (η) dηdx, = Rn where  = [Rκ(x), η − x, ξ ]/(R + |ξ |). Notice that the integrand has fixed compact support. Furthermore, our assumption that κ (0) = Identity implies that κ(x) = x + O(|x|2 ). Hence, if ρ is supported in a small neighborhood of 0, then the same considerations that were used in the proof of Lemma 0.4.1 show that, if ξ ∈ / 1 , then |Rη − ξ + O(|x|Rη)| R + |ξ | |Rη| + |ξ | − O(|x|) ≥ c, ≥ c0 R + |ξ |

|∇x | =

some c > 0,

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on the support of ρ(x)ˆu1 (Rη)β1 (η). Since |ˆu1 (Rη)| ≤ CRm , for some m, our result is a consequence of the following: Lemma 0.4.7 Let a ∈ C0∞ (Rn ) and assume that  ∈ C∞ satisfies |∇x | ≥ c > 0 on supp a. Then for all λ > 1,      eiλ(x) a(x) dx  ≤ CN λ−N , N = 1, 2, . . . , (0.4.8)   where CN depends only on c if  and a belong to a bounded subset of C∞ and a is supported in a fixed compact set. Proof of Lemma 0.4.7 Given x0 ∈ supp a there is a direction ν ∈ Sn−1 such that |ν, ∇| ≥ c/2 on some ball centered at x0 . Thus, by compactness, we can choose a partition of unity αj ∈ C0∞ consisting of a finite number of terms and  corresponding unit vectors νj such that αj (x) = 1 on supp a and |νj , ∇| ≥ c/2 on supp αj . If we set aj (x) = αj (x)a(x), it suffices to prove that for each j      eiλ(x) aj (x) dx  ≤ CN λ−N .   After possibly changing coordinates we may assume that νj = (1, 0, . . . , 0) which means that |∂/∂x1 | ≥ c/2 on supp aj . If we let L(x, D) =

∂ 1 , iλ∂/∂x1 ∂x1

then L(x, D)eiλ(x) = eiλ(x) . Consequently, if  ∂  1 L∗ = L∗ (x, D) = ∂x1 iλ∂/∂x1 is the adjoint, then 

 iλ(x)

e

aj (x) dx =

eiλ(x) (L∗ )N aj (x) dx.

Since our assumptions imply that (L∗ )N aj = O(λ−N ), the result follows.

Let us finish this section by seeing that Theorem 0.4.6 implies that WF(u) is a well-defined subset of the cotangent bundle of X and that, moreover, we can define wave front sets of distributions on a smooth manifold X. We shall say that X is a C∞ manifold if it is a Hausdorff space for which there is a countable collection of open sets ν ⊂ X together with ˜ ν ⊂ Rn satisfying homeomorphisms κν : ν → 

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36 (i)

Background ν

ν = X;

(ii) κν ◦ κν−1 : κν (ν ∩ ν ) → κν (ν ∩ ν ) is C∞ . Note that the mapping in (ii) is between open subsets of Rn . The structure is ˜ ν ⊂ Rn are called local coordinates of course not unique. Also, y = κν (x) ∈  of x in the local coordinate patch ν . This C∞ structure allows one to define C∞ functions on X in the natural way. We say that u is C∞ , or u ∈ C∞ (X), if for every ν, the function on ˜ ν , u(κν−1 (y)) is C∞ .  Next, t is said to be a tangent vector at x if t is a continuous linear operator on C∞ , sending real functions to R, and having the property that if x ∈ ν then there is vector tν ∈ Rn such that n   ∂  ˜ ν ). t(u ◦ κν ) = tjν u(y) whenever u ∈ C0∞ ( y=κν (x) ∂yj j=1

Thus, t annihilates constant functions, and the vector space of all tangent vectors at x, Tx M, has dimension n. Notice that if x ∈ ν ∩ ν and if we set κ = κν ◦ κν−1 : κν (ν ∩ ν ) → κν (ν ∩ ν ), then, by the chain rule, we must have t(u ◦ κν ) =

n  j=1

where



tjν



 ∂  u(Y) Y=κν (x) ∂Yj

tν = κ (y)tν ,

when u ∈ C0∞ (κν (ν ∩ ν )),

y = κν (x) ∈ κν (ν ∩ ν ).

Thus, if we let TX =

(0.4.9)



Tx X x∈X and use the coordinates (x, t) → (κν (x), tν ), x ∈ ν , the tangent bundle becomes a C∞ manifold of dimension 2n. In view of Theorem 0.4.6, it is also natural to consider the cotangent bundle which we now define. For each x ∈ X, the dual of Tx X, Tx∗ X, is a vector space of the same dimension. The cotangent bundle  T ∗X = Tx∗ X x∈X

C∞

manifold of dimension 2n having the structure induced by the local is the coordinates (x, ξ ) → (κν (x), ξ ν ),

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if x ∈ ν and ξ ν is the unique vector in Rn such that t, ξ  = tν , ξ ν , whenever t is a tangent vector at x and (κν (x), tν ) are the corresponding local coordinates in TX. Notice that this implies the following transformation law for cotangent vectors:

ξ ν = (t κ (y))−1 ξ ν , since we require



y = κν (x),

(0.4.9 )



tν , ξ ν  = tν , ξ ν  = t, ξ .

Thus, if (y, η) ∈ κν (ν ∩ν )×Rn are the local coordinates of (x, ξ ) ∈ T ∗ (ν ∩ ν ), it follows that (Y, ζ ) = (κ(y), (t κ (y))−1 η) are its local coordinates in κν (ν ∩ ν ) × Rn . This is of course consistent with the change of variables formula (0.4.5) for wave front sets. As a consequence, if X ⊂ Rn is open and u ∈ D (Rn ) then WF(u) is a well-defined subset of T ∗ X\0 = {(x, ξ ) ∈ T ∗ X : ξ  = 0}. Moreover, if X is a C∞ manifold and u ∈ D (X), then Theorem 0.4.6 implies that κν∗ WF(u ◦ κν−1 ) ⊂ T ∗ X\0 is independent of ν.4

0.5 Oscillatory Integrals In this section we shall study oscillatory integrals of the form   iφ(x,θ) e a(x, θ ) dθ ≡ lim eiφ(x,θ) ρ(εθ )a(x, θ ) dθ , Iφ (x) = RN

ε→0 RN

(0.5.1)

where, in the definition, ρ ∈ C0∞ (RN ) equals one near the origin. If φ and a satisfy certain conditions we shall see that the definition is independent of ρ. Here we assume that x ∈ X where X is an open subset of Rn with n possibly different from N. φ is called the phase function and we always assume that there is an open cone  ⊂ RN \0 that contains suppθ a so that, for (x, θ ) ∈ X ×, φ(x, λθ ) = λφ(x, θ ) if dφ  = 0.

λ > 0,

(0.5.2)

Here dφ denotes the differential of φ with respect to all of the variables. In addition to this we shall assume that the amplitude a(x, θ ) is a symbol of order m. That is, we assume that for all multi-indices α, γ , |Dγx Dαθ a(x, θ )| ≤ Cα,γ (1 + |θ |)m−|α|

(0.5.3)

4 If κ : X → Y is a smooth map then we define the pullback, κ ∗ η, of η ∈ T ∗ Y to be ξ = κ ∗ η ∈ κ(x) Tx∗ X if t, ξ  = κ∗ t, η whenever t ∈ Tx X and κ∗ t ∈ Tκ(x) Y is its push-forward.

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38

Background

whenever x belongs to a fixed compact subset of X and θ ∈ RN . Theorem 0.5.1 If φ and a are as above then Iφ ∈ D (X) and WF(Iφ ) ⊂ {(x, φx (x, θ )) : (x, θ ) ∈ X × 

and

φθ (x, θ ) = 0}.

(0.5.4)

Remark The reader should verify that this result contains the observation about wave front sets of homogeneous distributions as well as the theorem about the wave front sets of C∞ densities on subspaces of Rn . Proof We first show that Iφ ∈ D (X). Let βj be the dyadic bump functions occurring in the proof of Theorem 0.4.6. Let  j Iφ (u) = eiφ(x,θ) βj (θ )a(x, θ )u(x)dxdθ . Then if K is a fixed relatively compact subset of X it suffices to show that for any M there is a k(M) such that, for j = 1, 2, . . .,  j sup|Dα u|, u ∈ C0∞ (K). (0.5.5) |Iφ (u)| ≤ CM 2−Mj |α|≤k(M)

But for R = 2 j j Iφ (u) = RN

 eiRφ(x,θ) β1 (θ )a(x, Rθ )u(x)dxdθ.

Since (0.5.3) implies that |Dγx Dαθ [β1 (θ )a(x, Rθ )]| ≤ Cα,γ Rm ,

x ∈ K,

(0.5.5) follows from Lemma 0.4.7 and our assumption that dφ  = 0. Notice that (0.5.5) also shows that the definition of Iφ , (0.5.1), is independent of ρ since we are assuming that ρ equals one near the origin and hence the difference of any two such functions would be in C0∞ (RN \0). We now turn to the proof of the assertion about the wave front set. If u is as above we must show that  I(ξ ) = ei[φ(x,θ)−x,ξ ] u(x)a(x, θ ) dθ dx is rapidly decreasing when ξ is outside of an open cone 1 which contains {φx (x, θ ) : (x, θ ) ∈ supp u × , φθ (x, θ ) = 0}. As in the proof of Theorem 0.4.6, this amounts to showing that for such ξ      i[λφ(x,θ)−x,ξ ]  u(x)β1 (θ )a(x, λθ ) dxdθ  ≤ CM (λ + |ξ |)−M (0.5.6) e 

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0.5 Oscillatory Integrals

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for any M. But if we set (x, θ ) = then |∇x,θ | ≈

λφ(x, θ ) − x, ξ  , λ + |ξ |

|λφx (x, θ ) − ξ | + λ|φθ (x, θ )| ≥ c > 0, λ + |ξ |

(0.5.7)

in the support of u(x)β1 (θ )a(x, λθ ). To prove this we notice that by homogeneity it suffices to see that (0.5.7) holds when λ|θ | + |ξ | = λ. If θ = 0, |λφx (x, θ ) − ξ | = |ξ |, while, if θ  = 0 and φθ = 0 then |λφx (x, θ ) − ξ | ≥ / 1 . Since we assume that dφ  = 0 these two c (|λφx (x, θ )| + |ξ |) since ξ ∈ observations lead to (0.5.7). Finally, since (0.5.7) holds, (0.5.6) follows from Lemma 0.4.7. Most of the time we shall deal with a more narrow class of phase functions: Definition 0.5.2 φ is called a non-degenerate (homogeneous) phase function if (0.5.2) is satisfied and if the differentials d(∂φ/∂θj ), j = 1, . . . , N, are linearly independent. Notice that the extra hypothesis implies that φ = {(x, θ ) ∈ X ×  : φθ (x, θ ) = 0},

(0.5.8)

is an n-dimensional C∞ submanifold. Moreover, the map φ  (x, θ ) → (x, φx (x, θ )) ∈ T ∗ X\0

(0.5.9)

is a diffeomorphism. Thus, φ = {(x, φx (x, θ )) : (x, θ ) ∈ X × 

and

φθ (x, θ ) = 0}

(0.5.10)

is a smooth n-dimensional (immersed) submanifold of T ∗ X\0. It is actually a special type of submanifold. To describe this let σ = dξ ∧dx = n ∗ j=1 dξj ∧ dxj be the standard symplectic form on T X\0. Thus if (t, τ ) and ∗ (t , τ ) are two tangent vectors to T X\0 at a point (x, ξ ) then the value of the symplectic form σ here is t , τ  − t, τ . So the matrix corresponding to the form is

0 −I . I 0 If W is a subspace of this 2n-dimensional tangent space then we set W ⊥ = {v : σ (w, v) = 0, ∀w ∈ W}. Notice that W = W ⊥ implies that dim W = n since σ is a non-degenerate two form.

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40

Background

Definition 0.5.3 A smooth (immersed) submanifold  of T ∗ X\0 is called a Lagrangian submanifold, or Lagrangian, if given (x, ξ ) ∈  we have W = W ⊥ , where W is the tangent space to  at (x, ξ ). Thus  must be n-dimensional. Furthermore, the reader can check that an n-dimensional submanifold is Lagrangian if and only if σ ((t, τ ), (t , τ )) = 0 for any two tangent vectors. Proposition 0.5.4 If φ is a non-degenerate phase function it follows that φ is a Lagrangian submanifold of T ∗ X\0.  Proof Since ω = ξ · dx = nj=1 ξj dxj satisfies dω = σ it suffices to show that the restriction of the canonical one form, ω, to φ vanishes identically. But the pullback of ω to φ under the map κ in (0.5.9) is φx · dx = dφ − φθ · dθ. This vanishes identically on φ , since, by Euler’s homogeneity relations, φ = φθ , θ ), φ must vanish identically on φ . Since κ is a diffeomorphism, it follows that ω vanishes identically on φ . We shall call Iφ (x) a Lagrangian distribution if φ is Lagrangian. We remark in passing that it is not necessary for φ to be non-degenerate in order for Iφ (x) to be Lagrangian. For instance, if we set Rn+1  θ = (θ , θn+1 ) and define φ(x, θ ) =  x, θ  in some cone avoiding the θn+1 axis then φ is Lagrangian, while φ of course cannot be a non-degenerate phase function since it is independent of θn+1 . Remark Notice that φ is homogeneous. We shall also encounter Lagrangian submanifolds that are C∞ sections of T ∗ X. A C∞ section is a submanifold of the form {(x, ψ(x))} with ψ ∈ C∞ . Since the pullback of ω under the diffeomorphism x → (x, ψ(x)) is ψ · dx it follows from Poincar´e’s lemma that the section is Lagrangian if and only if locally ψ(x) = ∇ϕ(x) for some C∞ function ϕ. For ψ · dx is closed if and only if ψ has this form. When we study non-homogeneous oscillatory integrals of the form  eiλϕ(x,y) a(x, y)u(y) dy, Tλ u(x) = Rn

the (non-homogeneous) Lagrangian associated to ϕ will play a role that is similar to those associated to homogeneous phase functions.

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Notes

41

Notes As we pointed out before, most of the material in this chapter is standard so we refer the reader to Stein [2] , Stein and Weiss [1], and H¨ormander [7] for historical comments. On the other hand, the argument showing how the n-dimensional Hardy–Littlewood–Sobolev inequality follows from the one-dimensional version is taken from Strichartz [1], where it was used to prove a sharp embedding theorem for the wave operator in Rn . Corollary 0.3.7 was stated in this manner in Keel and Tao [1]; however, similar formulations had occurred earlier, for instance in Ginibre and Velo [1] and Lindblad and Sogge [1]. Similarly, the argument behind its proof appeared earlier in many places such as these two works, the first edition of this book, and before that in many articles on Fourier restriction phenomena, such as Strichartz [3] and Tomas [1]. On the other hand, Keel and Tao were able to obtain the important and much harder endpoint versions of the inequalities in Corollary 0.3.7 where the exponents s is allowed to equal 2 provided that q = ∞.

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1 Stationary Phase

In this chapter we continue to present background material that will be needed later. We start out in Section 1 by going over basic results from the theory of stationary phase, including the stationary phase formula. This will play an important role in everything to follow, especially in the composition theorems for pseudo-differential and Fourier integral operators and in the proof of estimates for oscillatory integral operators. In the next section we use the stationary phase formula to compute the Fourier transform of measures carried on smooth hypersurfaces with non-vanishing Gaussian curvature. The first application of this result is a classical theorem of Hardy and Littlewood, and Hlawka on the distribution of lattice points in Rn .

1.1 Stationary Phase Estimates Stationary phase is of central importance in classical analysis since integrals of the form  I(λ) = eiλ(y) a(y) dy, λ > 0, (1.1.1) Rn

arise naturally in many problems. The purpose of this section is to study such integrals when  ∈ C∞ is real and a ∈ C0∞ . If  is not constant, the factor eiλ(y) becomes highly oscillatory as λ → ∞, and, on account of this, one expects I(λ) to decay as λ becomes large. Stationary phase is a tool that allows one to make rather precise estimates for these integrals when the phase function, , satisfies natural conditions involving its derivatives. The simplest result is Lemma 0.4.7, which says that I(λ) = O(λ−N )

∀N

if

∇  = 0.

(1.1.2)

On the other hand, if ∇ = 0 somewhere but the determinant of the n × n matrix (∂ 2 /∂yj ∂yk ) never vanishes, then I(λ) = O(λ−n/2 ).

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1.1 Stationary Phase Estimates

43

The One-Dimensional Case For simplicity, let us first see that this is the case when n = 1. Specifically, we shall consider one-dimensional oscillatory integrals involving phase functions with non-degenerate critical points. For later use, it will be convenient to work with amplitudes that involve the parameter λ. The natural condition to place on such functions a(λ, y) is that they always vanish when y does not belong to a fixed compact set, and that  γ α   ∂  ∂  a(λ, y) ≤ Cαγ (1 + λ)−α , (1.1.3)  ∂y ∂y for all α, γ . Under these conditions we have following. Theorem 1.1.1 Suppose that (0),  (0) = 0, and  (y)  = 0 on supp a(λ, · )\{0}. Set  ∞ I(λ) = eiλ(y) a(λ, y) dy. 0

Then if

 (0)  = 0

and α = 0, 1, 2, . . . ,  α   ∂  −1/2−α   .  ∂λ I(λ) ≤ Cα (1 + λ)

Example The Bessel functions  −1 Jk (λ) = (2π )

2π

(1.1.4)

e iλ sin θ eikθ dθ , k = 0, 1, . . . ,

0

are a model case. Notice that Theorem 1.1.1 implies that |Jk (λ)| ≤ Ck λ−1/2 . To prove (1.1.4) notice that, by the product rule, α k  α!  ∞  j ∂ ∂ I(λ) = eiλ(y) i(y) . a(λ, y) dy. ∂λ j! k! 0 ∂λ j+k=α

But the hypotheses on  and Taylor’s theorem imply that there is a nonzero C∞ function η such that (y) = y2 η(y) on the support of a, and, hence, α = y2α ηα . Thus, by (1.1.3), one sees that (1.1.4) is implied by the following:

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Stationary Phase

Lemma 1.1.2 (Van der Corput) Let  be as in Theorem 1.1.1. Then for k = 0, 1, 2, . . .  ∞    iλ(y) k  e y a(λ, y) dy ≤ Ck λ−1/2−k/2 . (1.1.5)  0

Proof Let Ik denote the integral in (1.1.5). To estimate it we fix a C∞ function ρ(y) that equals 1 when y < 1 but equals 0 for y > 2. Then for δ > 0,  ∞  ∞   Ik = eiλ yk a(λ, y)ρ(y/δ) dy + eiλ yk a(λ, y) 1 − ρ(y/δ) dy 0

0

= I + II. The first integral is easy to handle. Just by taking absolute values we get  2δ |I| ≤ C yk dy = C δ 1+k . 0

To estimate the second one, we shall need to integrate by parts. As in the proof of Lemma 0.4.7, let ∂ 1 L∗ (y, D) = . ∂y iλ Then for N = 0, 1, 2, . . .     N  k    |II| =  eiλ · L∗ (y, D) y a(λ, y) 1 − ρ(y/δ) dy   N  k    ∗ ≤ (1.1.6) y a(λ, y) 1 − ρ(y/δ)  dy.  L (y, D) y>δ

   1  1     (y)  ≤ C y ,

However,

and so the product rule for differentiation implies that the last integrand in (1.1.6) is majorized by   λ−N max yk−2N , yk−N δ −N for any N. However, if the support of a is contained in [−c, c], where c < ∞, this means that, for N ≥ k + 2, we can dominate |II| by  c  λ−N yk−2N + yk−N δ −N dy ≤ Cλ−N δ 1+k−2N . δ

Putting together our estimates, we conclude that   |Ik | ≤ C δ 1+k + λ−N δ 1+k−2N .

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1.1 Stationary Phase Estimates

45

However, the right side is smallest when the two summands agree, that is, δ = λ−1/2 , which gives |Ik | ≤ Cλ−1/2−k/2 as desired. For many problems it is useful to have a variable coefficient version of Theorem 1.1.1. Suppose that (x, y) is a C∞ real phase function such that  y (0, 0) = 0, but

 yy (0, 0)  = 0.

Then, by the implicit function theorem, there must be a smooth solution y(x) to the equation (1.1.7)  y (x, y(x)) = 0, with y(0) = 0, when x is small enough. For such phase functions, we shall study oscillatory integrals  ∞ eiλ(x,y) a(λ, x, y) dy, (1.1.8) I(x, λ) = −∞

C∞

function having small enough y-support so that y(x) is where a now is a the only solution to (1.1.7). Then if, in addition, a satisfies     ∂ α ∂ β1 ∂ β2   a (λ, x, y) ≤ Cαβ (1 + λ)−α , (1.1.9)    ∂λ ∂x ∂y we have the following result. Corollary 1.1.3 Let a and  be as above. Then for α, β = 0, 1, 2, . . .    ∂ α ∂ β    −iλ(x,y(x)) I(x, λ)  ≤ Cαβ (1 + λ)−1/2−α , (1.1.10) e    ∂λ ∂x when x is small. Proof First, set

"(x, y) = (x, y) − (x, y(x)), 

and note that this phase function equals zero when y = y(x). Then, if β = 0, the quantity we need to estimate is  ∂ k  α!  ∂ j  ˜ a(λ, x, y) dy. eiλ(x, y) · j! k! ∂λ ∂λ j+k=α

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46

Stationary Phase

However, by Theorem 1.1.1 and (1.1.9), λk times such a summand is O(λ−1/2−j ), which implies (1.1.10) for this special case. " has a zero of order two To handle the cases involving nonzero β, note that  in the y variable when y = y(x). On account of this, one sees that the above arguments show that the inequalites for arbitrary β follow from (1.1.5). To finish the section we remark that the proof of the Van der Corput lemma can be adapted to show the following variant of (1.1.5): If (j) (0) = 0 for 0 ≤ j ≤ m − 1 but (m) (0)  = 0 then  ∞    iλ(y)  e a(y) dy ≤ Cλ−1/m (1.1.11)  0

provided that a has small enough support. We leave the proof as an exercise for the interested reader. Stationary Phase in Higher Dimensions We shall now consider n-dimensional oscillatory integrals  eiλ(y) a(λ, y) dy, λ > 0, I(λ) = Rn

(1.1.12)

which involve C∞ functions  and a, with  being real-valued and a having compact support. Our main task will be to extend Theorem 1.1.1. We will be working with phase functions  having non-degenerate critical points. Recall that y0 is said to be a non-degenerate critical point if ∇(y0 ) = 0, but

  det ∂ 2 /∂yj ∂yk  = 0

when y = y0 .

(1.1.13)

Notice that non-degenerate critical points must be isolated, since, by Taylor’s theorem, if we let H be the Hessian matrix in (1.1.13), then near a non-degenerate critical point y0 , # $ (y) = 12 H(y − y0 ), (y − y0 ) + O(|y − y0 |3 ), and hence ∇(y) = H(y − y0 ) + O(|y − y0 |2 ). Finally, we shall say that  is a non-degenerate phase function if all of its critical points are non-degenerate.

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1.1 Stationary Phase Estimates

47

As before, we shall work with amplitudes a(λ, y) whose y-support is contained in a fixed compact set, and now we shall also require that       ∂ α ∂ γ    ≤ Cαγ (1 + λ)−α , a(λ, y)  ∂λ  ∂y for all α and γ . Our main result then is the following. Theorem 1.1.4 Suppose that a is as above, (0) = 0, and 0 is a non-degenerate critical point of . Set  eiλ(y) a(λ, y) dy. I(λ) = Rn

Then, if ∇(y)  = 0 on supp a(λ, · )\{0},   α   ∂ −n/2−α   .  ∂λ I(λ) ≤ Cα (1 + λ)

(1.1.14)

We shall prove (1.1.14) by first establishing the result in the model case where the phase function equals a non-degenerate quadratic form   Q(y) = 12 y21 + · · · + y2j − y2j+1 − · · · − y2n . (1.1.15) Q is obviously non-degenerate since the Hessian of Q is a diagonal matrix where the diagonal entries equal 1 or −1. Proposition 1.1.5 If Q and a are as above α  ∂ eiλQ(y) a(λ, y) dy = O(λ−n/2−|α| ). ∂λ

(1.1.16)

By repeating the arguments in the previous section, however, it is easy to see that (1.1.16) would be a consequence of the following result. Lemma 1.1.6 If, for a given multi-index α, we set yα = yα1 1 · · · yαn n , then      eiλQ(y) a(λ, y)yα dy ≤ Cα (1 + λ)−(n+|α|)/2 . (1.1.17)   Furthermore, if |α| is odd, then the integral is O((1 + λ)−(n+|α|+1)/2 ). Proof We shall use an induction argument. In Lemma 1.1.2 we saw that (1.1.17) holds when n = 1. Therefore, let us now assume that the result is true when the dimension equals n − 1.

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48

Stationary Phase For the induction, we first write the integral in (1.1.17) as  ∞   iλQ(y ) iλy21 /2 α1 e e a(λ, y1 , y )y1 dy1 (y )α dy , Rn−1

−∞

where we have set Q(y ) = Q(y)−y21 /2 (which of course is also a non-degenerate function). By Lemma 1.1.2, however, we can control the inner integral. In fact, (1.1.5) implies that the inner integral equals a function a˜ (λ, y ) that has compact support in the y variable and satisfies     ∂ j ∂ γ   a ˜ (λ, y )  ≤ Cjγ (1 + λ)−(1+α1 )/2−j .    ∂λ ∂y But, by the induction hypothesis, this means that     (1+α )/2  iλQ(y ) α  −[(n−1)+|α |]/2 1 λ e a ˜ (λ, y )(y ) dy ,   ≤ C(1 + λ) Rn−1

which implies (1.1.17). To prove the assertion when |α| is odd one argues as above but uses the estimate that if α1 is odd,  ∞ 2 eiλy1 /2 a(λ, y1 , y )yα1 1 dy1 = O(λ−1/2−(α1 +1)/2 ). −∞

We leave the proof of this as an exercise for the reader. The next thing we shall do is to show that Theorem 1.1.4 can be deduced from Proposition 1.1.5. To do so we will make use of the following result, which is the classical Morse lemma. Lemma 1.1.7 Suppose that  has a non-degenerate critical point at the origin and that (0) = 0. Then near y = 0 there is a smooth change of variables, y →" y, such that, in the new coordinates,  2  (" y) = 12 " y1 + · · · +" y2j −" y2j+1 − · · · −" y2n . Proof After making an initial (linear) change of variables, we can always assume that ⎛ ⎞ 1 0 ⎜ ⎟ .. ⎜ ⎟ . ⎜ ⎟ ⎟   ⎜ 1 ⎜ ⎟ 2 ∂ /∂yj ∂yk = ⎜ ⎟ ⎜ ⎟ −1 ⎜ ⎟ ⎜ .. ⎟ . ⎝ ⎠ 0 −1

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1.1 Stationary Phase Estimates

49

when y = 0. Then, since ∂/∂y1 = 0 and ∂ 2 /∂y21  = 0 at the origin, it follows from the implicit function theorem that there must be a smooth function y1 = y1 (y ) solving the equation ∂ (y1 , y ) = 0. ∂y1 However, if we make the change of variables y → (y1 − y1 , y ), we can always assume that y1 = 0, that is, ∂ (0, y ) = 0. ∂y1 But, by Taylor’s theorem, this means that we are now in the situation where (y) = (0, y ) + c(y)y21 /2, with c being a smooth function that is positive near the origin. Finally, if we then let  " y1 (y) = c(y) y1 , it follows that "(y ), y21 +  (y) = 12 " which by induction establishes the lemma. The Morse lemma and (1.1.16) imply Theorem 1.1.4, since we can rewrite the oscillatory integral there,  I(λ) = eiλ(y) a(λ, y) dy, as

 eiλQ("y) " a(λ," y) d" y,

where " a is a function having the same properties as a and Q is a quadratic form as in (1.1.15). (Actually, we have cheated since this change of variables only works in a neighborhood of 0; however, since ∇  = 0 when y  = 0, (1.1.2) implies that we can always assume that the support of a is sufficiently small.) Next we will want to state a variable coefficient version of Theorem 1.1.4. Suppose that (x, y) is a real C∞ function satisfying ∇y (0, 0) = 0,   2 det ∂ /∂yj ∂yk  = 0

when x, y = 0.

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50

Stationary Phase

Then, as above, the implicit function theorem implies that there is a smooth solution to the equation ∇y (x, y(x)) = 0 (1.1.18) when x is small. Since (1.1.18) holds, y(x) is called a stationary point of . For such functions we will consider oscillatory integrals  eiλ(x,y) a(λ, x, y) dy, I(x, λ) = Rn

where we assume that a has small enough y-support so that y(x) is the only solution to (1.1.18) and, in addition, satisfies  α γ1 γ2   ∂  ∂ ∂   ≤ Cαγ (1 + λ)−α a x, y) (λ,  ∂λ  ∂x ∂y for all α, γj . Corollary 1.1.8 Suppose that a and  are as above. Then for every α = 0, 1, 2, . . . and multi-index γ  α γ    ∂ ∂ −iλ(x,y(x))  I(x, λ)  ≤ Cαγ (1 + λ)−n/2−α . (1.1.19) e  ∂λ ∂x The proof of this result is the same as the one-dimensional version, Corollary 1.1.3. Let us conclude this section by showing that the estimates we have obtained for oscillatory integrals are sharp. To do so we notice that Lemma 0.1.7 implies the formula  ∞

−∞

eiλx

Suppose that Q(x) =

1 2

2 /2

dx = (λ/2πi)−1/2 .



 2 x12 + · · · + xj2 − xj+1 − · · · − xn2 ,

then eiλQ is even. Taking this into account, one can see that the above formula together with (1.1.2) and Proposition 1.1.5 implies that, for λ > 0,  πi eiλQ(x) η(x) dx = (λ/2π )−n/2 η(0)e 4 sgnQ + O(λ−n/2−1 ), Rn

where Q denotes the Hessian of Q. But, by using the Morse lemma, we find this gives that   −1/2 πi sgn eiλ(x) η(x) dx = (λ/2π )−n/2 eiλ(0) η(0)det  (0) e4 Rn

+ O(λ−n/2−1 ),

(1.1.20)

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1.2 Fourier Transform of Surface-carried Measures

51

if  has a non-degenerate critical point at 0 and η has small support. Notice that, in this case, sgn  is constant on supp η, so the first term on the right side is well defined. Similar considerations show that the estimates on the derivatives of oscillatory integrals are also sharp.

1.2 Fourier Transform of Surface-carried Measures Suppose that S is a smooth hypersurface in Rn and let dσ be the induced Lebesgue measure on S. Then if β ∈ C0∞ (Rn ) we can form the compactly supported measures dμ(x) = β(x) dσ (x). The main goal of this section is to study the Fourier transform of this measure, that is,  ' )= dμ(ξ

e−ix,ξ  dμ(x),

(1.2.1)

S

when the Gaussian curvature of S never vanishes. The chief result will be that, under this assumption, the decay of the Fourier transform in all directions is O(|ξ |−(n−1)/2 ). The key role that curvature plays is evident from the ' would have no decay in the observation that if S were a hyperplane then dμ normal direction (cf. Theorem 0.4.4). Locally, we can always choose coordinates so that S is the graph of a smooth function h(x ). That is, S = {(h(x ), x ) : x = (x2 , · · · , xn ) ∈ Rn−1 }. If S is parameterized in this way dσ (x) =

1 + |∇h|2 dx

and the curvature is   K = (1 + |∇h|2 )−(n+1)/2 det ∂ 2 h/∂xj ∂xk . Also, ' )= dμ(ξ

 Rn−1

(1.2.2)

  e−i(h(x ),x ), ξ  β (h(x ), x ) 1 + |∇h(x )|2 dx .

The unit normals to S at (h(x ), x ) are the two antipodal points ν ∈ Sn−1 satisfying # $ (1.2.3) ∇x ν, (h(x ), x ) = 0.

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Stationary Phase

Let  be the function inside the gradient, that is, # $ (ν, x ) = ν, (h(x ), x ) .

(1.2.4)

Then, if ∇h = 0, ν = (1, 0, . . . , 0) would be one of the normals, and, furthermore, in this case we would have the identity     ∂ 2 /∂xj ∂xk = ∂ 2 h/∂xj ∂xk . By a rotation argument and (1.2.2) we can conclude that if S has non-vanishing curvature   det ∂ 2 (ν, x )/∂xj ∂xk  = 0 (1.2.5) when ν is normal to S at (h(x ), x ). From this and the implicit function theorem we see that the Gauss map which sends a point x ∈ S to its normal ν(x) ∈ Sn−1 is a local diffeomorphism when K  = 0 on S. Having made these remarks, we can now see that the following result follows from Corollary 1.1.8 and Theorem 1.1.4. Theorem 1.2.1 As above, let S be a smooth hypersurface in Rn with non-vanishing Gaussian curvature and dμ a C0∞ measure on S. Then ' )| ≤ C(1 + |ξ |)−(n−1)/2 . |dμ(ξ

(1.2.6)

Moreover, suppose that  ⊂ Rn \0 is the cone consisting of all ξ that are normal to some point x ∈ S belonging to a fixed relatively compact neighborhood N of supp dμ. Then, α   ∂ ' ) = O (1 + |ξ |)−N ∀N, if ξ ∈ dμ(ξ / , ∂ξ  ' )= dμ(ξ e−ixj ,ξ  aj (ξ ), if ξ ∈ , (1.2.7) where the (finite) sum is taken over all points xj ∈ N having ξ as a normal and  α   ∂    ≤ Cα (1 + |ξ |)−(n−1)/2−|α| . a (ξ ) (1.2.8) j  ∂ξ  Remark Note that (1.1.20) implies that the estimate (1.2.6) is sharp. That is, if dμ  = 0, there must be a nonempty open cone  ⊂ Rn such that, for large ξ , one has the bounds from below   ' dμ(ξ ) ≥ c(1 + |ξ |)−(n−1)/2 , ξ ∈ , for some c > 0, provided that dμ has sufficiently small support so that the sum in (1.2.7) involves only one term.

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Corollary 1.2.2 Let χ (x) denote the characteristic function of the unit ball in Rn . Then |χ(ξ ˆ )| ≤ C(1 + |ξ |)−(n+1)/2 . (1.2.9) Proof Let β(r) ∈ C∞ (R) equal one when r > 12 and zero when r < 14 . Then, if we let χ1 (x) = β(|x|)χ (x) and χ0 (x) = χ (x) − χ1 (x), it follows that

χ0 ∈ C0∞ (Rn ).

But, then (1.1.2) implies that

|χˆ 0 (ξ )| ≤ CN (1 + |ξ |)−N for any N. Consequently, we are only left with showing that |χˆ 1 (ξ )| ≤ C(1 + |ξ |)−(n+1)/2 . If we use polar coordinates, x = rω, r > 0, ω ∈ Sn−1 , we get that  −ix,ξ  χˆ 1 (ξ ) = dx ( ) β(|x|)e |x|∈

 =

1

1 4 ,1

β(r)

1/4



−irω,ξ 

e Sn−1

dσ (ω) rn−1 dr.

However, (1.2.7) implies that the inner integral equals e−ir|ξ | a1 (rξ ) + eir|ξ | a2 (rξ ), where



∂ ∂ξ

α

  aj (ξ ) = O (1 + |ξ |)−(n−1)/2−|α| .

But this, along with a simple integration by parts argument, gives that     1  −ir|ξ | n−1  β(r)e a1 (rξ ) r dr ≤ C(1 + |ξ |)−(n−1)/2 · (1 + |ξ |)−1    1/4 = C(1 + |ξ |)−(n+1)/2 , and since the other term satisfies the same estimate, we are done. We are now ready to give the first application of the stationary phase method, which will be a classical result of Hardy and Littlewood, and Hlawka concerning the distribution of lattice points in Rn .

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54

Stationary Phase

Theorem 1.2.3 Let N(λ) denote the number of lattice points j ∈ Zn satisfying |j| ≤ λ. Then, if B is the unit ball in Rn ,   N(λ) = Vol (B) · λn + O λn−2+2/(n+1) . (1.2.10) Proof The strategy will be to estimate N(λ) indirectly by comparing it to a slightly smoother function of λ. With this in mind, let us fix a non-negative C∞ bump function β that is supported in B( 12 ) and satisfies  β(y) dy = 1. (1.2.11) Here B(λ) denotes the ball {x : |x| ≤ λ} in Rn . Next, if χλ denotes the characteristic function of B(λ), then for a number ε to be specified later, we let    χ "λ (ε, x) = ε−n β( · /ε) ∗ χλ (x) = ε−n β((x − y)/ε) χλ (y) dy and

" N (ε, λ) =



χ "λ (ε, j).

(1.2.12)

j∈Zn

To compare N˜ and N note that the support properties of β imply that χλ (x) = χ "λ (ε, x) when |x| ∈ / [λ − ε, λ + ε],  and, since N(λ) = j∈Zn χλ (j), this implies " N (ε, λ − ε) ≤ N (λ) ≤ " N (ε, λ + ε).

(1.2.13)

To estimate N˜ we need to use the following form of the Poisson summation formula for Rn , which follows from (0.1.17) via a change of scale:   f (j) = fˆ (2πj). j∈Zn

j∈Zn

ˆ ) the formula and Since the Fourier transform of χ "λ (ε, x) equals χˆ λ (ξ ) · β(εξ (1.2.12) give that  ˆ " N (ε, λ) = χˆ λ (2πj)β(2π εj) j∈Zn

= λn



ˆ χ(2π ˆ λj)β(2π εj),

(1.2.14)

j∈Zn

where χ is the characteristic function of the unit ball. But, if we recall that  fˆ (0) = f dx,

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1.2 Fourier Transform of Surface-carried Measures we see that (1.2.14) implies that " N (ε, λ) = Vol (B) · λn + λn



55

ˆ χ(2π ˆ λj)β(2π εj).

(1.2.15)

j ∈ Zn j = 0

However, by Corollary 1.2.2, we have the estimate |χ(ξ ˆ )| ≤ C(1 + |ξ |)−(n+1)/2 , and since β ∈ C0∞ it follows that ˆ )| ≤ C(1 + |ξ |)−N |β(ξ

for any

N.

This means that we can majorize the second summand in (1.2.15) by λn times  (1 + |λj|)−(n+1)/2 (1 + |εj|)−N j ∈ Zn j = 0

 ≈

|ξ |≥1

(

≤C λ

(1 + |λξ |)−(n+1)/2 (1 + |εξ |)−N dξ =

−(n+1)/2 −(n−1)/2

ε

−(n+1)/2 −n

+ (λ/ε)

ε

 )

1≤|ξ |≤ε −1

 +

|ξ |≥ε−1

= 2Cλ−(n+1)/2 ε−(n−1)/2 . Putting everything together, we get " N (ε, λ) = Vol (B) · λn + O (λ(n−1)/2 ε−(n−1)/2 ). But, if we consider this result with λ replaced by λ ± ε and note that (λ ± ε)n = λn + O(ελn−1 ), then we see that (1.2.13) implies the estimate   N(λ) = Vol (B) · λn + O ελn−1 + λ(n−1)/2 ε−(n−1)/2 . The remainder term is minimized when ελn−1 = λ(n−1)/2 ε−(n−1)/2 , that is, n−1 ε = λ− n+1 . And for this choice of ε one has (1.2.10). Remark The reader may check that one can never have N(λ) = Vol (B)λn + O(λn−2 ). Also, notice that, if one considers the standard Laplacian on the n-torus n (∂/∂θ )2 = , then the eigenvalues of Tn = S1 × · · · × S1 , that is, j=1 j − are all integers that are the sum of the squares of n integers. Thus, (1.2.10) implies that the number of eigenvalues of – that are ≤ λ2 equals Vol (B)λn + O(λn−2+2/(n+1) ). Later, we shall see that a weaker result holds for all Riemannian manifolds, and the proof of this result will be similar in spirit to the proof of Theorem 1.2.3.

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56

Stationary Phase

Notes The material presented here is standard and we have followed the expositions of Beals, Fefferman, and Grossman [1], Stein [4], and H¨ormander [8]. Most of the material from Section 1.2 was first proved by Hlawka [1], including the theorem on the distribution of lattice points that extended earlier results of Hardy and Littlewood.

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2 Non-homogeneous Oscillatory Integral Operators

Recall that in the last chapter we studied non-homogeneous oscillatory integrals of the form  eiλφ(y) a(y) dy. I(λ) = Rn

If φ has a non-degenerate critical point at 0 and a is a C∞ cutoff function having small support, we saw that |I(λ)| ≈ λ−n/2

as

λ → +∞,

whenever a(0)  = 0. In this chapter we shall study some natural generalizations of these results, where the integrals now will take their values in Lp spaces. Specifically, we shall consider operators of the form  eiλφ(x,y) a(x, y)f (y) dy, λ > 0, Tλ f (x) = Rn

where now a is a smooth cutoff function and φ ∈ C∞ (Rm × Rn ) is real. The most basic result occurs when m = n: If φ is non-degenerate in the sense that the mixed Hessian satisfies the non-degeneracy condition   ∂ 2φ det  = 0, ∂xj ∂yk then we shall find that

Tλ f L2 (Rn ) ≤ Cλ−n/2 f L2 (Rn ) . This result obviously has the same flavor as the estimates for I(λ), and, in fact, one can see that, for every λ, there are functions for which Tλ f 2 / f 2 ≈ (1 + λ)−n/2 if a is non-trivial. There are many natural situations where the non-degeneracy condition is not met. For instance, if φ(x, y) = |x−y|, then one can check that the mixed Hessian

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58

Non-homogeneous Oscillatory Integral Operators

only has rank (n − 1). Oscillatory integral operators with phase functions of this type will also be studied. Later on we shall see that the estimates obtained can be used to prove sharp bounds for the Lp norm of eigenfunctions on a Riemannian manifold as well as some related results.

2.1 Non-degenerate Oscillatory Integral Operators The main result of this section is the following. Theorem 2.1.1 Suppose that φ is a real C∞ phase function satisfying the non-degeneracy condition   ∂ 2φ det = 0 (2.1.1) ∂xj ∂yk on the support of a(x, y) ∈ C0∞ (Rn × Rn ). Then for λ > 0,     iλφ(x,y)  e a(x, y)f (y) dy  ≤ Cλ−n/2 f L2 (Rn ) .   Rn

(2.1.2)

L2 (Rn )

If we let Tλ be the operator in (2.1.2), then

Tλ f L∞ ≤ C f L1 . Therefore, by applying the M. Riesz interpolation theorem, we get the following result. Corollary 2.1.2 If 1 ≤ p ≤ 2 then

Tλ f Lp (Rn ) ≤ Cλ−n/p f Lp (Rn ) ,

(2.1.3)

if p denotes the conjugate exponent. Remark Notice that the phase function φ = x, y satisfies the hypotheses of Theorem 2.1.1. Furthermore, since (2.1.3) implies that   √ √    eix,y a(x/ λ, y/ λ)f (y) dy  ≤ C f p ,   p

we see that (2.1.3) implies the Hausdorff–Young inequality:

fˆ p ≤ C f p .

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2.1 Non-degenerate Oscillatory Integral Operators

59

Before proving the theorem it is illustrative to restate the non-degeneracy condition in an equivalent form. Let Cφ = {(x, φx (x, y), y, −φy (x, y))} ⊂ T ∗ Rn × T ∗ Rn be the canonical relation associated to the non-homogeneous phase function φ. Then by the remark at the end of Section 0.5, Cφ is Lagrangian with respect to the symplectic form dξ ∧ dx − dη ∧ dy. Moreover, the condition (2.1.1) is equivalent to the condition that the two projections  (x, φx (x, y)) ∈ T ∗ Rn

Cφ

 (y, −φy (x, y)) ∈ T ∗ Rn

be local diffeomorphisms (i.e., have surjective differentials). In results to follow we shall encounter variations on this geometric condition. It of course means that   ∇x [φ(x, y) − φ(x, z)] ≈ |y − z|, |y − z| small. (2.1.1 ) This is what we shall use in the proof of Theorem 2.1.1 and it of course just follows from the fact that   * + ∂ 2 φ(x, y) ∇x φ(x, y) − φ(x, z) = (y − z) + O(|y − z|2 ). ∂xj ∂yk Proof of Theorem 2.1.1 By using a smooth partition of unity we can decompose a(x, y) into a finite number of pieces each of which has the property that (2.1.1 ) holds on its support. So there is no loss of generality in assuming that |∇x [φ(x, y) − φ(x, z)]| ≥ c|y − z| for some c > 0. To use this we notice that

(2.1.4)



Tλ f 22 where

on supp a,

=

Kλ (y, z)f (y)f (z) dydz,

(2.1.5)

 Kλ (y, z) =

Rn

eiλ[φ(x,y)−φ(x,z)] a(x, y)a(x, z) dx.

However, (2.1.4) and Lemma 0.4.7 imply that |Kλ (y, z)| ≤ CN (1 + λ|y − z|)−N ∀N.

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Non-homogeneous Oscillatory Integral Operators

Consequently, by Young’s inequality, the operator with kernel Kλ sends L2 into itself with norm O(λ−n ). This along with (2.1.5) yields

Tλ f 22 ≤ Cλ−n f 22 , as desired.

2.2 Oscillatory Integral Operators Related to the Restriction Theorem As we pointed out in the introduction, there are very important situations where the non-degeneracy hypothesis of Theorem 2.1.1 is not fulfilled. For instance, if φ(x, y) = |x − y| then the mixed Hessian of φ cannot have full rank n since for fixed x the image of y → φx (x, y) is Sn−1 (and hence this map can not be a submersion). In fact, one can check that for this phase function the differentials of the projections from Cφ to T ∗ Rn have corank 1 everywhere. Nonetheless, since the image of y → φx (x, y) has non-vanishing Gaussian curvature, we shall see that the oscillatory integrals Tλ associated to this phase function map Lp (Rn ) → Lq (Rn ) with norm O(λ−n/q ) for certain p and q. Moreover, we shall actually prove a stronger result that says that certain oscillatory integral operators sending functions of n−1 variables to functions of n variables satisfy the same type of estimates. At the end of the section we shall show that,in higher odd dimensions, the results that we obtain that are due to Stein are optimal in the sense that one can write down phase functions that saturate the bounds. This simple but striking counterexample is due to Bourgain. There are also negative results for even dimensions n ≥ 4, but they do not quite match up with the positive results in Stein’s oscillatory integral theorem. As before, these oscillatory integrals will be of the form  (2.2.1) Tλ f (z) = eiλφ(z,y) a(z, y)f (y) dy, except, now, a ∈ C0∞ (Rn × Rn−1 ) and φ is real and C∞ in a neighborhood of supp a. Thus, in what follows, z shall always denote a vector in Rn and y one in Rn−1 . The canonical relation associated to φ is now a subset of T ∗ Rn × T ∗ Rn−1 . The hypotheses in the oscillatory theorem will be based on the properties of the projections of Cφ into T ∗ Rn−1 and the fibers of T ∗ Rn . First, the non-degeneracy hypothesis is that rank dT ∗ Rn−1 ≡ 2(n − 1),

(2.2.2)

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2.2 Oscillatory Related to the Restriction Theorem

61

if T ∗ Rn−1 : Cφ → T ∗ Rn−1 is the natural projection. Thus, (2.2.2) says that the differentials of this projection must have full rank everywhere, or, to put it another way, the mixed Hessian always has maximal rank; that is,   ∂ 2φ rank ≡ n − 1. (2.2.2 ) ∂yj ∂zk Assumption (2.2.2) of course is an analog of the hypothesis in Theorem 2.1.1 and it is enough to guarantee that Tλ : Lp (Rn−1 ) → Lq (Rn ) with norm O(λ−(n−1)/q ) if q ≥ 2 and p ≥ q . To get better results where the norm is O(λ−n/q ) a curvature hypothesis is needed. To state it, we first notice that, since Cφ = {(z, φz (z, y), y, −φy (z, y))}, (2.2.2) and the constant rank theorem imply that, for every z0 ∈ suppz a, the image of y → φz (z0 , y), Sz0 = Tz0∗ Rn (Cφ ) = {φz (z0 , y) : (z0 , φz (z0 , y), y, −φy (z0 , y)) ∈ Cφ } is a C∞

(immersed) hypersurface in Tz∗0 Rn . We can identify Tz∗0 Rn

(2.2.3)

with Rn and

the curvature hypothesis is just that Sz0 ⊂ Tz∗0 Rn

has everywhere non-vanishing Gaussian curvature.

(2.2.4)

Since (0.4.9 ) says that changes of coordinates induce changes of coordinates in the cotangent bundle that are linear in the fibers, one concludes that, like (2.2.2), (2.2.4) is an invariant condition. We just pointed out that (2.2.2) is a condition involving second derivatives of the phase function; momentarily we shall see that (2.2.4) is equivalent to a condition involving third derivatives of the phase function. If the two conditions (2.2.2) and (2.2.4) are met, then we say that the phase function satisfies the Carleson–Sj¨olin condition. The main result of this section concerns oscillatory integral operators with such phase functions. Theorem 2.2.1 Let Tλ be as in (2.2.1) and suppose that the non-degeneracy hypothesis (2.2.2) and the curvature hypothesis (2.2.4) both hold. Then

Tλ f Lq (Rn ) ≤ Cp λ−n/q f Lp (Rn−1 ) if q =

n+1 n−1 p

(2.2.5)

and

(1) 1 ≤ p ≤ 2 for n ≥ 3; (2) 1 ≤ p < 4 for n = 2. Before turning to the proof, let us go over a couple of important consequences that will help illustrate the hypotheses. The first one is the restriction theorem for the Fourier transform.

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62

Non-homogeneous Oscillatory Integral Operators

Corollary 2.2.2 Suppose that S ⊂ Rn , n ≥ 2, is a C∞ hypersur-face with everywhere non-vanishing Gaussian curvature. Then if dσ is Lebesgue measure on S and if dμ = βdσ with β ∈ C0∞ , it follows that 1/r     fˆ (ξ )r dμ(ξ ) ≤ Cs f Ls (Rn ) , f ∈ S(Rn ), (2.2.6) S

provided that r = (1) 1 ≤ s ≤ (2) 1 ≤ s
0, provided again that (x , t ) is close to (x, t). So in this case, Lemma 0.4.7 implies that a stronger estimate holds, where in the right side of (2.2.13) we may replace (n − 1)/2 by any power. Proof of Theorem 2.2.1, Part (2) Here n = 2 and we must show that for q = 3p and 1 ≤ p < 4   ∞   eiλφ(z,y) a(z, y)f (y) dy q 2 ≤ Cp λ−2/q f Lp (R) . (2.2.14)  −∞

L (R )

To take advantage of the fact that q > 4, we write  2 eiλ[φ(z,y)+φ(z,y )] a(z, y)a(z, y )f (y)f (y ) dydy . (Tλ f (z)) =

(2.2.15)

We would like to use the non-degenerate oscillatory integral theorem to estimate the Lq/2 (R2 ) norm of this quantity. However, the mixed Hessian of the phase function (z; y, y ) = φ(z, y) + φ(z, y ) has determinant   φ  z1 y φz1 y   φz2 y φz 2 y

    = φz 1 y (z, y)φz 2 y (z, y ) − φz 1 y (z, y )φz 2 y (z, y), 

(2.2.16)

and so the assumptions are not verified as the determinant vanishes on the diagonal y = y . On the other hand, the Carleson–Sj¨olin assumption (2.2.8)

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Non-homogeneous Oscillatory Integral Operators

implies that

     ∂ 2   det  ≥ c|y − y |  ∂z∂(y, y ) 

(2.2.17)

for some c > 0 if |y − y | is small. In fact, at the diagonal, the y derivative of (2.2.16) equals the determinant in (2.2.8). There is no loss of generality in assuming that (2.2.17) holds whenever the above integrand is nonzero. To exploit (2.2.17) as well as the fact that the integrand in (2.2.15) is symmetric in (y, y ), it is convenient to make the change of variables u = (y − y , y + y ). Then since |du/dy| = 2, it follows that     ∂ 2     ≥ c|u1 |. det  ∂z∂u 

(2.2.17 )

Also, since (z; u) is an even function of the diagonal variable u1 , it must be a C∞ function of u21 . So we make the change of variables   v = 12 u21 , u2 . Then since |dv/du| = |u1 |, it follows that     ∂ 2    det  ≥ c.  ∂z∂v 

(2.2.17 )

We can now apply Corollary 2.1.2. If r = q/2 and if r is conjugate to r , it follows that

Tλ f 2Lq (R2 ) = (Tλ f )2 Lr (R2 )      i(z,v) −1  = e f (y)f (y ) |dv/d(y, y )| dv   ≤ Cλ−2/r ≤ Cλ



−4/q



r

1/r    f (y)f (y ) |dv/d(y, y )|−1 r dv



r

−1+α

| f (y)| | f (y )| |y − y | r



dydy

1/r ,

with α = 2 − r. Since +−1 * +−1 * − (p/r) , α = p/r (2.2.14) follows from applying the Hardy–Littlewood–Sobolev inequality.

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2.2 Oscillatory Related to the Restriction Theorem

67

Let us conclude this section by showing that the bounds in Theorem 2.2.1 cannot be improved in odd dimensions n ≥ 3 in the sense that there are phase functions for which the associated oscillatory integrals saturate the bounds in (2.2.5). Let us start out with the construction for n = 3 since that is the simplest case. In this case and for dimensions higher than n = 3, we shall use the symmetric matrix

1 s A(s) = , (2.2.18) s s2 which depends on the real parameter s. Observe that the matrix consisting of the derivatives of each component satisfies det A (s) ≡ −1,

(2.2.19)

Rank A(s) ≡ 1.

(2.2.20)

while, on the other hand, Using this matrix we define the phase function on z = (z1 , z2 , z3 ) ∈ R3 and y = (y1 , y2 ) ∈ R2 as follows: # $ φ(z, y) = z · y + 12 A(z3 )y, y , z = (z1 , z2 ). (2.2.21) the non-degeneracy hypothesis (2.2.2 ) is satisfied because Clearly 2 ∂ φ/∂zj ∂yk is the identity matrix. The curvature hypothesis (2.2.4) also holds due to (2.2.19). Indeed, for every fixed z = z0 the hypersurfaces in (2.2.3) are the graphs of the non-degenerate symmetric quadratic form Q(z3 ) = A (z3 ), i.e., (2.2.22) Sz = (y1 , y2 , y1 y2 + z3 y22 ), which has non-vanishing principal curvatures of opposite signs. Next, choose now 0 ≤ a ∈ C0∞ (R) satisfying a(s) = 1 for |s| ≤ 1 and define the oscillatory integral operator  Tλ f (z) = eiλφ(z,y) a(z, y) f (y) dy, with a(z, y) = a(|z|) a(|y|). (2.2.23) By what we just observed, this operator satisfies the hypotheses of Theorem 2.2.1 when n = 3. On the other hand, if we just pick any f ∈ C∞ (R2 ) satisfying f ≥ 0, and f (y) = 1, if |y| ≤ 1/2, (2.2.24) we claim that for small |z3 | we have 1

|Tλ f (z)| ≈ λ− 2 , if

z ∈ {z ∈ R2 : dist(z , Vz3 )
0 and zero otherwise. Here t+ If f ∈ S (and thus fˆ ∈ S) it follows from Fourier’s inversion theorem that δ Sλ f (x) → f (x) for every x as λ → +∞. In this section we shall consider the convergence of the Riesz means of Lp functions. To apply the oscillatory integral theorems of the previous section, we shall assume that the “cospheres” associated to q,

 = {ξ : q(ξ ) = 1},

(2.3.2)

have non-vanishing Gaussian curvature. Note that, since q is smooth and ∇q  = 0,  is a C∞ hypersurface. The assumption regarding the curvature of  is equivalent to   ∂ 2q ≡ n − 1. (2.3.3) rank ∂ξj ∂ξk The most important case is when q(ξ ) = |ξ |. For a given 1 ≤ p ≤ ∞ we define the critical index for Lp (Rn ):  ,    δ(p) = max n  12 − 1p  − 12 , 0 .

(2.3.4)

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2.3 Riesz Means in Rn

71

Note that δ(p) > 0 ⇔ p ∈ / [2n/(n + 1), 2n/(n − 1)]. It is known that a necessary condition for Sλδ f → f

in Lp

when

f ∈ Lp , p  = 2,

is that δ > δ(p). When δ(p) = 0 this is a theorem of C. Fefferman. The other cases follow from the fact that the kernel of Sλδ is in Lp only when δ > δ(p) for 1 ≤ p ≤ 2n/(n + 1). (This can be seen from Lemma 2.3.3 below.) Theorem 2.3.1 Let Sδ = S1δ . Then if the cospheres  associated to q(ξ ) have non-vanishing Gaussian curvature, and + * 2(n+1) + * ∪ n−1 , ∞ , or (1) n ≥ 3 and p ∈ 1, 2(n+1) n+3 (2) n = 2 and 1 ≤ p ≤ ∞, it follows that

Sδ f Lp (Rn ) ≤ Cp,δ f Lp (Rn )

when

δ > δ(p).

(2.3.5)

Corollary 2.3.2 If p is as in Theorem 2.3.1 then Sλδ f → f

in Lp (Rn ),

when f ∈ Lp (Rn ) and δ > δ(p). The proof of the corollary is easy. First (2.3.5) implies that the means Sλδ are uniformly bounded on Lp when δ > δ(p). Next, given f ∈ Lp and ε > 0 there is a g ∈ S such that f − g p < ε, and hence Sλδ f − Sλδ f p = O(ε). Since both g and gˆ are in S, Fourier’s inversion theorem and the uniform boundedness of Sλδ : Lp → Lp imply that Sλδ g → g in Lp . This implies the assertion, since, by Minkowski’s inequality, one sees that Sλδ f − f p = O(3ε) when λ is large. The proof of Theorem 2.3.1 requires knowledge of the convolution kernel of the operator Sδ :  K δ (x) = (2π )−n eix,ξ  (1 − q(ξ ))δ+ dξ . Rn

This kernel will be a sum of two terms, each involving an oscillatory factor and an amplitude. To describe the phases, note that our assumptions imply that given x ∈ Rn \0, there are exactly two points ξ1 (x), ξ2 (x) ∈  such that the differential of the map   ξ → x, ξ  vanishes when ξ = ξj (x). In fact, ξj (x) are just the two points in  with normal x. We now define (2.3.6) ψj (x) = x, ξj (x).

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Non-homogeneous Oscillatory Integral Operators

Since ξj (x) is homogeneous of degree zero and smooth in Rn \0, it follows that ψj is also smooth and it is of course homogeneous of degree one. Note that in the model case where q = |ξ |, we would have ψj = ±|x|. Lemma 2.3.3 The kernel of Sδ can be written as K δ (x) =

a1 (x)eiψ1 (x) (1 + |x|)

n+1 2 +δ

+

a2 (x)eiψ2 (x) (1 + |x|)

n+1 2 +δ

  + O (1 + |x|)−n−1 ,

where the aj are bounded from below near infinity and satisfy      ∂ α  ∂x aj (x) ≤ Cα |x|−|α| ∀α.

(2.3.7)

(2.3.8)

The proof of the lemma is a straightforward application of the stationary phase method. First, let η ∈ C∞ (R) equal zero near the origin but equal one on [1/2, ∞). We then claim that the difference of K δ and  " K δ (x) = (2π )−n eix,ξ  η(q(ξ )) (1 − q(ξ ))δ+ dξ is O((1 + |x|)−n−1 ). Clearly the Fourier transform of χ (ξ ) = (1 − η(q(ξ ))) is O((1 + |x|)−N ) for all N, since it is compactly supported and is also smooth due to the fact that it equals one near ξ = 0. Therefore, to verify the claim we just need to show that  + * eix,ξ  χ (ξ ) 1 − (1 − q(ξ ))δ+ dξ = O((1 + |x|)−n−1 ). If |x| ≤ 1 the bound is obvious; otherwise split the integral into two pieces:  + * eix,ξ  χ (|x|ξ ) χ (ξ ) 1 − (1 − q(ξ ))δ+ dξ    * + + eix,ξ  1 − χ (|x|ξ ) χ (ξ ) 1 − (1 − q(ξ ))δ+ dξ = I + II. Since the integrand is O(|ξ |) clearly I = O(|x|−n−1 ). One obtains the same bounds for II using a simple integration by parts argument and the fact that  ∂ α  * +  1 − χ (|x|ξ ) 1 − (1 − q(ξ ))δ+  = O(|ξ |1−|α| ) ∀α,  ∂ξ if |x| > 1. To finish the proof, we shall show that " K δ can be written as the sum of the first two terms in the right side of (2.3.7). If we recall that Theorem 1.2.1 says that the inverse Fourier transform of surface measure on the cosphere satisfies  b1 (x)eiψ1 (x) b2 (x)eiψ2 (x) −n eix,ω dσ (ω) = + , (2π ) (1 + |x|)(n−1)/2 (1 + |x|)(n−1)/2 

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73

for functions bj satisfying (2.3.8), then it follows that " K δ is the sum of two terms. The first is  b1 (ρx)eiρψ1 (x) " η(ρ)(1 − ρ)δ+ dρ (1 + |ρx|)(n−1)/2  (1 + |x|)(n−1)/2 b1 ((1 − τ )x) eiψ1 (x) = " η(1 − τ )τ+δ eiτ ψ1 (x) dτ , (1 + |x|)(n−1)/2 (1 + (1 − τ )|x|)(n−1)/2 where " η is η times a smooth function coming from the polar coordinates. δ Since |ψ1 (x)| ≥ c|x| for some c > 0, and since the Fourier transform of ρ+ is homogeneous of degree −δ − 1 and smooth away from the origin, it follows that the last integral is of the form a˜ 1 (x)/(1 + |x|)1+δ with a˜ 1 satisfying (2.3.8). Since the other term has the same form, we are done. To apply Lemma 2.3.3 we shall use a scaling argument and the following. , ( )Lemma 2.3.4 If a(x, y) is in C0∞ and supp a(x, y) ⊂ (x, y) : |x − y| ∈ 12 , 2 then     iλψj (x−y) −n/q  a(x, y)f (y) dy 

f Lp (Rn ) , (2.3.9)  q n ≤ Cq λ  ne R

if q =

n+1 n−1 p

L (R )

and

(1) 1 ≤ p ≤ 2 for n ≥ 3; (2) 1 ≤ p < 4 for n = 2. Furthermore, for a given ψj the constants depend only on the size of finitely many derivatives of a. To verify this result we need to check that ψj (x, y) = ψj (x − y) satisfies the n × n Carleson–Sj¨olin condition in Corollary 2.2.3. This is easy since, by construction, ∇x ψj (x, y) = ξj (x − y). This implies that for every x0 , Sx0 = {∇x ψj (x0 , y)} = , and, hence, the curvature condition holds. Since the Gauss map Sn−1 →  is a local diffeomorphism, it follows that, for fixed y, the differential of x → ξj (x−y) ∈  has constant rank n − 1. So the non-degeneracy condition is also satisfied. In view of these facts, Lemma 2.3.4 follows from Corollary 2.2.3. Notice that p < q in (2.3.9). So if we use H¨older’s inequality and the fact that a vanishes for y outside of a compact set, we get     iλψj (x−y)   e a(x, y)f (y) dy ≤ Cq λ−n/q f Lp (Rn ) . (2.3.9 )   Rn

Lp (Rn )

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74

Non-homogeneous Oscillatory Integral Operators , ( )To apply this, we fix β ∈ C0∞ (R) supported in s : s ∈ 12 , 2 and satisfying ∞ −l −∞ β(2 s) = 1, s > 0. Since, by Young’s inequality convolution with a kernel that is O((1+|x|)−n−1 ) is bounded on all Lp spaces, let us abuse notation a bit and let here K δ denote here the sum of the first two terms in (2.3.7). We then define Klδ (x) = β(2−l |x|)K δ (x), l = 1, 2, 3, . . . ,  δ and K0δ (x) = K δ (x) − ∞ 1 Kl (x). We claim that if q is as in (2.3.9 ), then we have the following estimate for convolution with these dyadic kernels:

Klδ ∗ f Lq (Rn ) ≤ C2[δ(q)−δ]l f Lq (Rn ) .

(2.3.10)

The estimate for l = 0 is easy since K0δ is bounded and supported in the ball of radius 2. Since this implies that K0δ is in L1 , Young’s inequality implies that convolution with K0δ is bounded. To prove the estimate for l > 0 we note that the Lq → Lq operator norm of convolution with Klδ equals the norm of convolution with the dilated kernels " Klδ (x) = λn Klδ (λx),

λ = 2l .

But this function equals λ

n−1 2 −δ

β(|x|)

n−1 a1 (λx)eiλψ1 (x) = λ 2 −δ a1 (λ, x)eiλψ1 (x) (λ−1 + |x|)(n+1)/2+δ

∂ α plus a similar term involving ψ2 . Since (2.3.8) implies that |( ∂x ) a1 (λ, x)| ≤ Cα for every α, (2.3.9 ) implies that

" Klδ ∗ f q ≤ Cq λ

n−1 n 2 − q −δ

f q .

n But n−1 2 − q = δ(q) and so (2.3.10) follows. Finally, by summing a geometric series we conclude that (2.3.10) implies that for q as above

Sδ f q ≤ Cq,δ f q ,

δ > δ(q),

since, as we pointed out before, convolution with the error term in (2.3.7) is bounded on all Lp spaces. This proves (2.3.5) for q ≥ 2(n + 1)/(n − 1), n ≥ 3, and q > 4 for n = 2. The remaining cases follow from duality and interpolating with the trivial inequality Sδ f 2 ≤ f 2 . If we drop the assumption that  has non-vanishing Gaussian curvature, it becomes much harder to analyze the Lp mapping properties of Sδ . This is because the stationary phase methods used before break down, and, except in special cases, it is impossible to compute K δ . Nonetheless, it is not hard to prove the following.

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Theorem 2.3.5 Assume only that q(ξ ) is homogeneous of degree one and both C∞ and nonnegative in Rn \0. Then, if Sδ is the Riesz mean associated to q, it follows that

Sδ f L1 (Rn ) ≤ C f L1 (Rn ) ,

δ > δ(1) =

n−1 . 2

(2.3.11)

Proof The operator we are trying to estimate is  eix,ξ  (1 − q(ξ ))δ+ fˆ (ξ ) dξ . Sδ f (x) = (2π )−n Rn

If β is as above and we now let  δ −n eix,ξ  β(2l (1 − q(ξ )))(1 − q(ξ ))δ fˆ (ξ ) dξ , l = 1, 2, . . . , Sl f (x) = (2π ) S0δ f (x) = Sδ f (x) −

∞ 

Slδ f (x),

1

then S0δ : L1 → L1 since, by the proof of Lemma 2.3.3, its kernel is O((1 + |x|)−n−1 ). Hence we would be done if we could show that, for any ε > 0,

Slδ f 1 ≤ Cε 2−[δ−

n−1 2 −ε]l

f 1 .

(2.3.11 )

To prove this we now let Klδ be the kernel of this operator. Then, for every N,  , Klδ (x) = (2π )−n |x|−2N eix,ξ  (−)N β(2l (1 − q(ξ )))(1 − q(ξ ))δ dξ . . /   Since (−)N β(2l (1 − q(ξ )))(1 − q(ξ ))δ = O 22Nl 2−δl , it follows that  −2N  δ  K (x) ≤ CN 2−δl x/2l  , l

and so, for large N and fixed ε > 0,   δ  K (x) dx ≤ CN,ε 2−Nl . l |x|>2(1+ε)l

The last inequality says that Klδ (x) is essentially supported in the ball of Klδ is Klδ times the characteristic radius 2(1+ε)l centered at the origin. In fact, if " function of this ball, then convolution with the difference has norm O(2−lN ). Klδ ∗ f , then it suffices to show that this operator satisfies the So if " Slδ f = "  bounds in (2.3.11 ). But if we write f = fj , where the fj are supported in Slδ fj is supported non-overlapping cubes Qj of side-length 2(1+ε)l it follows that " ∗ in Qj , the cube with the same center but five times the side-length. The cubes

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Non-homogeneous Oscillatory Integral Operators

Q∗j have finite overlap and so

" Slδ f 1 ≤ C



" Slδ fj L1 (Q∗ ) . j

j

Since the difference between Slδ and " Slδ has norm O(2−lN ), this implies that the desired result must follow from

Slδ f L1 (B(2(1+ε)l )) ≤ C2−[δ−

n−1 2 −ε]l

f L1 (Rn ) .

(2.3.11 )

But the Schwarz inequality implies that n

Slδ f L1 (B(2(1+ε)l )) ≤ 2 2 (1+ε)l Slδ f L2 (Rn ) . We can estimate the right side since    2   

Slδ f 22 = (2π )−n β 2l (1 − q(ξ )) (1 − q(ξ ))δ  | fˆ (ξ )|2 dξ  −2lδ ˆ 2

f ∞ · dξ ≤ C2 {ξ : 1−q(ξ )∈[2−l−1 ,2−l+1 ]} ≤ C2−2lδ 2−l f 21 .

Combining the last two inequalities gives (2.3.11 ).

2.4 Nikodym Maximal Functions and Maximal Riesz Means in R2 We shall show here that, in two dimensions, the maximal Riesz means operators   S∗δ f (x) = sup Sλδ f (x) λ>0

enjoy the same mapping properties as Sδ when p > 2. Theorem 2.4.1 If p > 2, δ > δ(p), and  = {ξ : q(ξ ) = 1} has non-vanishing Gaussian curvature

S∗δ f Lp (R2 ) ≤ Cp,δ f Lp (R2 ) . Remark If one repeats the arguments used in the proof of Corollary 2.3.2, one sees that Sλδ f (x) → f (x) almost everywhere as λ → ∞, if f ∈ Lp (R2 ), p ≥ 2, and δ > δ(p).

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The proof of the theorem will be based on interpolating with estimates that arise in the special case where p = 4:

S∗δ f L4 (R2 ) ≤ Cδ f L4 (R2 ) ,

δ > 0.

(2.4.1)

The proof of this inequality is based on first dominating S∗δ f by the Hardy–Littlewood maximal function and certain square functions involving the half-wave operators associated to q(ξ ):    itQ −2 eix,ξ  eitq(ξ ) fˆ (ξ ) dξ . e f (x) = (2π ) (2.4.2) R2

Estimating the square functions is the most difficult part of the proof and it is done in two steps. First, there is an orthogonality argument that uses elementary wave front analysis and involves breaking the operators eitQ into simpler ones whose mapping properties are much easier to analyze. After this step, we dominate the pieces that arise from this decomposition by means of a “Nikodym maximal operator.”1 This auxiliary maximal operator involves averages over thin tubular neighborhoods of rays on light cones associated to Q. At the end of this section we shall see that our arguments can be modified to show that the following related maximal operator      sup  f (x − ty) dσ (y) t>0



is bounded on Lp (R2 ) for all p > 2. In subsequent chapters we shall return to variations on this result and we shall make use of techniques used in its proof. Turning to the proof, let us first see how, using the Hardy–Littlewood maximal function and the Littlewood–Paley estimate, we can deduce (2.4.1) from the seemingly weaker inequality where the supremum is only taken over λ ∈ [1, 2]. To do this we let η ∈ C∞ (R) be as in the proof of Lemma 2.3.3. Then if we set    Tλδ f (x) = (2π )−2 eix,ξ  η q(ξ/λ) (1 − q(ξ/λ))δ+ fˆ (ξ ) dξ , it follows that the absolute value of Rδλ f (x) = Sλδ f (x) − Tλδ f (x) is point-wise dominated by the Hardy–Littlewood maximal function independently of λ since, as we showed in the proof of Lemma 2.3.3, the Fourier transform of (1 − η(q(ξ )))(1 − q(ξ ))δ+ is O((1 + |x|)−n−1 ). Therefore, if we use the 1 Following the terminology in C´ordoba [1], in the first edition of this book, we called these

“Kakeya maximal operators.” Following Bourgain’s seminal work, Bourgain [2], it has been customary to call them “Nikodym maximal operators” as we are now doing. We shall encounter their cousins, the “Kakeya maximal operators,” in Chapter 8 and explain the terminology.

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Non-homogeneous Oscillatory Integral Operators

Hardy–Littlewood maximal theorem, we conclude that (2.4.1) would be a consequence of (2.4.1 )

T∗δ f 4 ≤ Cδ f 4 , δ > 0. Notice that, since η vanishes near the origin, there must be an integer k0 such that (2.4.3) supp η(q(ξ )) (1 − q(ξ ))δ+ ⊂ {ξ : 2−k0 < |ξ | < 2k0 }. To exploit this, we use a Littlewood–Paley decomposition of f : f=

∞ 

fj ,

fˆj (ξ ) = β(2−j |ξ |)fˆ (ξ ),

where

−∞

with β

∈ C0∞ (R)

satisfying

supp β ⊂

∞ 

*1 + 2,2 ,

It then must follow that Tλδ f = Tλδ



β(2−j s) = 1,

s > 0.

−∞





( ) if λ ∈ 2k , 2k+1 .

fj

|j−k|≤k0 +2

Using this we get ∞   δ  T f (x)4 ≤ ∗

=

  δ T f (x)4

sup

k k+1 k=−∞ λ∈[2 , 2 ] ∞  

λ

 δ Tλ

sup

k k+1 k=−∞ λ∈[2 , 2 ]



 4  fj (x) .



|j−k|≤k0 +2

Based on this, we claim that (2.4.1 ) follows from the localized estimate      (2.4.4)  sup Tλδ f (x) ≤ Cδ f 4 , k ∈ Z. 4

λ∈[2k , 2k+1 ]

This claim is not difficult to verify. In fact, since, by Theorem 0.2.10, 

( | fk |2 )1/2 4 ≤ C f 4 , we see that (2.4.4) yields  ∞      4   δ 4     sup Tλ f (x) dx ≤ sup Tλδ fj (x) dx λ>0 λ∈[2k ,2k+1 ] k=−∞ |j−k|≤k0 +2  ∞  | fk |4 dx ≤C k=−∞

≤C

   ∞

| fk |

2

 1 ·4 2

dx ≤ C



 | f |4 dx.

k=−∞

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79

By dilation invariance, (2.4.4) must follow from the special case where k = 0. Furthermore, if we again use the fact that the difference between Tλδ f (x) and Sλδ f (x) is pointwise dominated by the Hardy–Littlewood maximal function and recall (2.4.3), we deduce that this in turn would be a consequence of  , ( )     sup Sλδ f (x) ≤ Cδ f 4 , if supp fˆ ⊂ ξ : |ξ | ∈ 2−k0 −2 , 2k0 +2 . 4

λ∈[1,2]

(2.4.1 ) This completes the first reduction. We now turn to the main step in the proof which involves the use of the half-wave operators eitQ defined in (2.4.2). To use them we need to know about the Fourier transform of the distribution τ−δ = |τ |δ χ(−∞,0] ∈ S (R). By results in Section 0.1, the Fourier transform must be homogeneous of degree −δ − 1 and C∞ away from the origin. This is all that will be used in the proof; however, we record that the Fourier transform is actually the distribution cδ (t + i0)−δ−1 , where

cδ = ieiδπ/2 (δ + 1),

# $ $ (t + i0)−δ−1 , ψ = lim (t + iε)−δ−1 , ψ ,

#

ε→0+

Using this we first notice that Sλδ f

−2 −δ

= (2π )

λ

 R2

= (2π )−1 cδ λ−δ



ψ ∈ S(R).

eix,ξ  (λ − q(ξ ))δ+ fˆ (ξ ) dξ ∞ −∞

e−iλt (t + i0)−δ−1 eitQ f dt.

Based on this we make the decomposition δ Sλδ f = Sλ,0 f+

∞ 

δ Sλ,k f,

k=1

where, if β is as above, then for k = 1, 2, . . .  ∞ δ f = (2π )−1 cδ λ−δ e−iλt β(2−k |t|)(t + i0)−δ−1 eitQ f dt. Sλ,k −∞

(2.4.5)

These are Fourier multiplier operators, of course, and the multiplier,  k  −δk behaves like 2 ρ 2 (λ − q(ξ )) , with ρ a fixed C0∞ (R) function. In particular, mδλ,k becomes an increasingly singular function around λ · , but, on the other hand, its L∞ norm is decreasing like 2−kδ as k → +∞. Taking this into account, we naturally expect that for any ε > 0     δ f (x)| ≤ Cε 2−(δ−ε)k f 4 , for f as in (2.4.1 ). (2.4.6)  sup |Sλ,k mδλ,k ,

λ∈[1,2]

4

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Non-homogeneous Oscillatory Integral Operators

By summing a geometric series this of course implies (2.4.1 ). The inequality is trivial when k = 0. In fact, the Fourier transform of the  −k −δ−1 must be smooth since the latter is distribution [1 − ∞ k=1 β(2 t)](t − i0) δ in E . Consequently, mλ,0 must be a multiplier which is smooth away from ξ = 0. Therefore, since we are assuming that fˆ has fixed compact support as in (2.4.1 ), the special case where k = 0 in (2.4.6) follows from the Hardy–Littlewood maximal theorem. To estimate the terms involving k > 0 we shall use the following elementary result. Lemma 2.4.2 Suppose that F is C1 (R). Then, if p > 1 and 1/p + 1/p = 1, 

1/p 

1/p | F(λ)|p dλ · | F (λ)|p dλ . sup | F(λ)|p ≤ | F(0)|p + p λ

To prove the lemma one just writes  λ  λ d | F(λ)|p = | F(0)|p + | F|p−1 F ds | F(s)|p ds = | F(0)|p + p dλ 0 0 and then uses H¨older’s inequality to estimate the last term. ∞ To apply the lemma ( in)the special case where p = 2, we now fix ρ ∈ C0 (R) satisfying supp ρ ⊂ 12 , 4 and ρ = 1 on [1, 2]. Then, using (2.4.5) and H¨older’s inequality, we conclude that the left side of (2.4.6) is majorized by 2 1/2         ρ(λ)e−iλt β(2−k |t|)(t + i0)−δ−1 eitQ f dt dλ   4   d 2 1/2      × (ρ(λ)e−iλt )β(2−k |t|)(t + i0)−δ−1 eitQ f dt dλ   . 4 dλ

Plancherel’s theorem implies that this expression is controlled by  dt 1/2    |eitQ f |2 2+2δ   4 |t| |t|∈[2k−1 , 2k+1 ]   1/2  dt   × |eitQ f |2 2δ  . 4 |t| |t|∈[2k−1 , 2k+1 ] Thus (2.4.6) would be a consequence of the estimate  

1/2    dt   |eitQ f |2 1+ε     |t|∈[2k , 2k+1 ] |t| 4

≤ Cε f 4 ,

supp fˆ ⊂ {ξ : |ξ | ∈ [1, 2]},

ε > 0.

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By taking complex conjugates one sees that we need only estimate the expression involving t integration over [2k , 2k+1 ]. Moreover, by a change of scale argument, one sees that this in turn is equivalent to proving the following result. Proposition 2.4.3 For ε > 0 and τ > 1 there is a constant Cε for which  1/2    2    itQ 2   |e f | dt    1  4 2 L (R )

ε

≤ Cε τ f L4 (R2 ) ,

supp fˆ ⊂ {ξ : |ξ | ∈ [τ , 2τ ]}.

(2.4.7)

Remark In Chapter 6 we shall see that the operators  eix,ξ  eitq(ξ ) (1 + |ξ |)−σ fˆ (ξ ) dξ (2π )−2 R2

are bounded on L4 (R2 ) if and only σ ≥ 14 . On the other hand, (2.4.7) implies that for any σ > 0 this expression is in L4 (L2 ([1, 2])). Let us introduce some notation that will be used in the proof of the proposition. First, if ρ and β are as above and if we set  eix,ξ  eitq(ξ ) β(|ξ |/τ )fˆ (ξ ) dξ , Fτ f (x, t) = ρ(t) R2

then it is clear that (2.4.7) would be a consequence of  

1/2      2 ≤ Cε τ ε f L4 (R2 ) . |Fτ f (x, t)| dt     4 2

(2.4.7 )

L (R )

The first step in the proof of this inequality is to decompose the square function inside the L4 norm for fixed x. To do this we write  Fτj f , Fτ f = j

where

 Fτj f (x, t) = ρ(t)

R2

eix,ξ  eitq(ξ ) α(τ −1/2 q(ξ ) − j)β(|ξ |/τ )fˆ (ξ ) dξ

for some function α ∈ C0∞ (R) satisfying supp α ⊂ [−1, 1]

and

∞ 

α(s − j) ≡ 1.

j=−∞

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Non-homogeneous Oscillatory Integral Operators j

Note that there are O(τ 1/2 ) nonzero terms Fτ f . Moreover, if “"” denotes the partial Fourier transform with respect to the t variable and if we set  j −1 " Fτ f (x, t) = (2π ) eitη (Fτj f )"(x, η)dη, j η∈Iτ ( ) Iτj = τ 1/2 j − 2τ 1/2 , τ 1/2 j + 2τ 1/2 , (2.4.8) then we have the following. Lemma 2.4.4 Given any N, there is a uniform constant CN such that if we set j j "τj f (x, t), then Rτ f (x, t) = Fτ f (x, t) − F  

1/2      j 2 ≤ CN τ −N f L4 (R2 ) . |Rτ f (x, t)| dt    4 2  L (R )

j

Proof The Fourier transform of Fτ f (x, t) is (2π )2 fˆ (ξ ) times  mτj (ξ , η) = β(|ξ |/τ ) eit(q(ξ )−η) ρ(t)α(τ −1/2 q(ξ ) − j) dt. R

+ * / τ 1/2 j − τ 1/2 , τ 1/2 j + τ 1/2 . So, for Note that α(τ −1/2 q(ξ ) − j) = 0 if q(ξ ) ∈ j j η outside of Iτ , the double of this interval, mτ and all of its derivatives are j −N . But this implies that the kernel of Rτ , O (τ + |(ξ , η)|)  −1 ei[x−y,ξ +tη] mτj (ξ , η) dξ dη, (2π ) j (ξ ,η)∈R2 ×(Iτ )c

is O(τ −N (1 + |t|)−1 (1 + |x − y|)−N ). Using this one deduces the lemma. By applying the lemma twice and using Plancherel’s theorem one gets  1/2    |Fτ f (x, t)|2 dt   4   1/2    "τj f (x, t)|2 dt | F ≤  + CN τ −N f 4 4

j

  1/2    "τj f (x, t)|2 dt ≤ 4 |F  + CN τ −N f 4 4

j

  1/2    ≤ 4 |Fτj f (x, t)|2 dt  + CN τ −N f 4 . 4

j j

On account of this and the fact that Fτ f (x, t) vanishes for t ∈ / apply H¨older’s inequality to conclude that

(2.4.7 )

(

)

1 2,4

, we can

would follow from the

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2.4 Nikodym Maximal Functions and Maximal Riesz Means in R2 purely L4 estimate ⎛ ⎞1/2        ⎝ j 2⎠ |Fτ f |       j

≤ Cε τ ε f L4 (R2 ) .

83

(2.4.7 )

L4 (R2 ×R)

So far we have used a decomposition of the Fourier integral operator F with respect to the radial frequency variables. Now we must make a decomposition involving the angular variables. To make this decomposition, we shall need to define homogeneous partitions of unity of R2 \0 that depend on the scale τ . Specifically, we choose C∞ functions χτν , ν = 1, . . . , N(τ ) ≈ τ 1/2 , satisfying  ν ν ν χτ = 1 and having the following properties. First, the χτ are to be homogeneous of degree zero and satisfy the uniform estimates |Dγ χτν (ξ )| ≤ Cγ τ |γ |/2 for all γ when |ξ | = 1. Furthermore, if unit vectors ξτν are chosen so that χτν (ξτν )  = 0, the χτν are to have the natural support properties associated to these estimates, that is, χτν (ξ ) = 0 if |ξ | = 1 and

|ξ − ξτν | ≥ Cτ −1/2 .

Using these functions we put   ατν,j (t, ξ ) = ρ(t) χτν (ξ ) α τ −1/2 q(ξ ) − j β(|ξ |/τ ),

(2.4.9)

and define the decomposed operators  eix,ξ  eitq(ξ ) ατν,j (t, ξ )fˆ (ξ ) dξ . Fτν,j f (x, t) = It then follows that If we set



ν,j ν Fτ f

R2

j

= Fτ f .

Qν,j = {ξ : ατν,j (t, ξ )  = 0,

some t},

we record, for later use, that Qν,j is comparable to a square of size τ 1/2 × τ 1/2 and that the sets {Qν,j }j,ν have finite overlap. Moreover, a key observation is ν,j that the Fourier transform of (x, t) → Fτ f (x, t) is concentrated on ν,j = {(ξ , q(ξ )) ∈ R3 \0 : ξ ∈ Qν,j }. This is consistent with the results of Section 0.5 since the wave front set of (x, t) → eitQ f must be contained in {(x, t, ξ , η) : η = q(ξ )}. To make our statement about the Fourier transform more precise, let ∈ ν,j C0∞ (R) equal one near the origin. Then the difference, Rτ f (x, t), between

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Non-homogeneous Oscillatory Integral Operators

ν,j

Fτ f (x, t) and "τν,j f (x, t) = (2π )−3 F

j (ξ ,η)∈R2 ×Iτ

ei[x,ξ +tη]

 η − q(ξ )  

× satisfies



|η|ε

Fτν,j f

∧

(ξ , η) dξ dη

−N

f 4 .

Rν,j τ f L4 (R3 ) ≤ CN τ

(2.4.10)

One proves this using the argument which was used to estimate the other ν,j error term. First of all, we notice that the Fourier transform of Fτ f (x, t) is 2 (2π ) fˆ (ξ ) times  mν,j (ξ , η) = eit(q(ξ )−η) ατν,j (t, ξ )dt, τ R

ν,j

and hence the kernel of Rτ must be   η − q(ξ )   −1 ei[x−y,ξ +tη] 1 − χI j (η) (2π ) mν,j τ (ξ , η) dξ dη. τ |η|ε But if the integrand is nonzero, it follows that |q(ξ )−η| ≥ c(τ +|η|)ε , for some ν,j c> implies that for such (ξ , η), mτ and all of its derivatives are   0. Since this−N , the estimate (2.4.10) follows as before. O (τ + |(ξ , η)|) In view of (2.4.10), the desired inequality would follow from showing that ⎛ ⎞1/2        2  ⎝  F˜ τν,j f  ⎠  ≤ Cε τ ε f L4 (R2 ) . (2.4.11)    ν  4 2  j L (R ×R)

The key observation behind this inequality is that, if has small enough support and if we let   ∗ν,j = {(ξ , η) : dist (ξ , η), ν,j ≤ τ ε }, ν,j then the Fourier transform of (x, t) → F˜ τ f (x, t) vanishes outside of ∗ν,j . To exploit this we shall use the following geometric lemma.

Lemma 2.4.5 If ε > 0 is as in the definition of ∗ν,j , there is a uniform constant C so that, if j and j are fixed,  χ∗ν,j +∗ (ξ , η) ≤ Cτ ε .

{(ν,ν ): arg ξτν , arg ξτν ∈[0,π/2]}

ν ,j

Here ∗ν,j + ∗ν ,j denotes the algebraic sum of the two sets, and if we identify R2 and C in the usual way, arg ξτν is the argument of the unit vector ξτν .

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2.4 Nikodym Maximal Functions and Maximal Riesz Means in R2

85

We postpone the proof of this result. It is based on the fact that our assumption that the cospheres associated to q(ξ ) have non-vanishing Gaussian curvature implies that the cones {(ξ , q(ξ ))} ⊂ R3 \0 have one non-vanishing principal curvature. This allows the overlap to be finite rather than just O(τ 1/2 ). To apply this result, by symmetry, we may clearly restrict the ν summation in (2.4.11) to indices as in Lemma 2.4.5. It then follows that the fourth power of the resulting expression can be estimated in the following way using Plancherel’s theorem and the overlap lemma: ⎛ ⎞1/2  4      ⎝ "τν,j f |2 ⎠  F |     ν   j 4 2         ν,j ν ,j  "τ f F "τ f  dxdt F =    j,j  ν,ν  2       ν,j ∧ ν ,j ∧   " " ∗ ∗ ≤  (χν,j +ν ,j ) (ξ , η)(Fτ f ) ∗ (Fτ f )  dξ dη  j,j  ν,ν     "ν,j "ν ,j 2 ≤ Cτ ε Fτ f Fτ f  dxdt j,j ν,ν

⎛ ⎞1/2   4       F "τν,j f 2 ⎠  = Cτ ε ⎝  .    j ν  4

Consequently, if we use (2.4.10) again, we conclude that we would be done if we could show that ⎛ ⎞1/2       ⎝  ν,j 2 ⎠  Fτ f ≤ C log τ f L4 (R2 ) . (2.4.12)      4 3  ν,j L (R )

This completes the orthogonality arguments. The proof of (2.4.12) will be based on estimating the L2 (R3 ) → L2 (R2 ) norm of a “Nikodym maximal function” which involves averages over rays on light cones associated to q(ξ ). To use the Nikodym maximal estimates to follow, we shall need to make use of a variant of Littlewood–Paley theory which involves a decomposition of the Fourier transform into functions supported in a lattice of cubes. To describe ν,j what is needed, we first recall that the ξ -support of the symbols ατ of the ν,j operators Fτ are sets Qν,j all of which are comparable to cubes of side-length

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86

Non-homogeneous Oscillatory Integral Operators

τ 1/2 . In particular, each intersects at most a fixed number of cubes in a τ 1/2 lattice of R2 . With this in mind, we let α be as above and set fˆm (ξ ) = α(τ −1/2 ξ1 − m1 )α(τ −1/2 ξ2 − m2 )fˆ (ξ ),

m = (m1 , m2 ) ∈ Z2 .

 Thus m∈Z2 fm = f . In addition, if for a given (ν, j), we let Iν,j ⊂ Z2 be ν,j those m for which Fτ fm is not identically zero, it follows from the above discussion that Card (Iν,j ) ≤ C,

(2.4.13)

with C being a uniform constant. Also since the sets Qν,j have finite overlap there is an absolute constant C such that Card {(ν, j) : m ∈ Iν,j } ≤ C

∀ m ∈ Z2 .

(2.4.14)

We now turn to (2.4.12). Let Kτν,j (x, t; y) =

 R2

eix−y,ξ  eitq(ξ ) ατν,j (t, ξ ) dξ

ν,j

be the kernel of Fτ . In a moment we shall see that  |Kτν,j (x, t; y)| dy ≤ C R2

(2.4.15)

uniformly. Thus, the Schwarz inequality and (2.4.13) give 2        |K ν,j (x, t; y)| dy  f (y) m τ    m∈Iν,j   | fm (y)|2 |Kτν,j (x, t; y)| dy. ≤C

 ν,j  F f (x, t)2 ≤ C τ

m∈Iν,j

If we now use (2.4.14) we see that, for a given g(x, t),        ν,j 2  |Fτ f (x, t)| g(x, t) dxdt   ν,j      2 ν,j |Kτ (x, t; y)| |g(x, t)| dxdt dy. | fm (y)| sup ≤C m

ν,j

Since the square of the left side of (2.4.12) is dominated by the supremum over all g 2 = 1 of the last quantity, by applying the Schwarz inequality, we see

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2.4 Nikodym Maximal Functions and Maximal Riesz Means in R2

87

that we would be done if we could prove  2 1/2    |Kτν,j (x, t; y)| g(x, t) dxdt dy sup R2 ν,j

≤ C| log τ |3/2 g L2 (R3 ) ,

(2.4.16)

as well as the following. Lemma 2.4.6 For 2 ≤ p ≤ ∞, 

1/2       2 | fm |    m 

≤ C f Lp (R2 ) .

Lp (R2 )

Proof of Lemma 2.4.6 When p = 2 the inequality holds because of Plancherel’s theorem. If we apply a vector-valued version of the M. Riesz interpolation theorem (which follows from the same proof) we conclude that it suffices to prove the inequality for p = ∞. By dilation invariance we may take τ = 1. Then, if Q = [−π, π]2 and if αˇ is the inverse Fourier transform of α,  1/2 | fm (0)|2 m∈Z2

    =  m∈Z2

R2

2 1/2 −im,y  f (y)α(−y ˇ ˇ dy 1 )α(−y 2 )e

2 1/2      ≤ ˇ n1 − y1 )α(2π ˇ n2 − y2 )e−im,y dy .  f (y − 2πn)α(2π n∈Z2 m∈Z2

Q

But the terms in the absolute values of the last expression are the Fourier coefficients of the function there. So Parseval’s formula says that the last expression is

1/2    2   f (y − 2πn)α(2π ˇ n1 − y1 )α(2π ˇ n2 − y2 ) dy 2π n∈Z2

Q

1/2    2   α(2πn ˇ ≤ C f ∞ · ˇ n2 − y2 ) dy . 1 − y1 )α(2π n∈Z2

Q

Since αˇ ∈ S the last sum converges and this finishes the proof. To prove (2.4.15) and (2.4.16) let us introduce some notation. Given a direction (cos θ , sin θ ) ∈ S1 we let γθ ⊂ R3 be the ray defined by γθ = {(x, t) : x + tq ξ (cos θ , sin θ ) = 0}.

(2.4.17)

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88

Non-homogeneous Oscillatory Integral Operators

Clearlythese light rays depend smoothly on θ and our assumption that rank  ∂2q ∂ξi ∂ξk = 1 implies that γθ  = γθ if θ  = θ . We also remark that, by the Euler homogeneity relations, the union of all the rays is the dual cone to {(ξ , q(ξ ))} ⊂ R3 \0. The estimate about the kernels that we require is the following lemma. ν,j

Lemma 2.4.7 If ξτν are the unit vectors occurring in the definition of Fτ and if (cos θν , sin θν ) = ξτν , then given N there is an absolute constant CN for which  −N  |Kτν,j (x, t; y)| ≤ CN τ 1 + τ 1/2 dist (x − y, t), γθν . (2.4.18) Note that (2.4.18) implies that, for fixed y, the kernels are essentially supported in a tubular neighborhood of width τ 1/2 around γθν + (y, 0). With this in mind one gets (2.4.15). ν,j

Proof of Lemma 2.4.7 To prove (2.4.18), we recall that ατ has ξ -support contained in a sector ν of R2 \0 of angle O(τ −1/2 ). Thus, since q ξ is homogeneous of degree zero, |q ξ (ξ ) − q ξ (ξτν )| ≤ Cτ −1/2 ,

ξ ∈ suppξ ατν,j .

This implies that for some c > 0 |∇ξ [x − y, ξ  + tq(ξ )]| ≥ c dist((x − y, t), γθν ), provided that dist ((x − y, t), γθν ) is larger than a fixed constant times τ −1/2 . ∂ α ν,j ) ατ | ≤ Cα τ −|α|/2 and By applying Lemma 0.4.7 we get (2.4.18), since |( ∂ξ ν,j

|{suppξ ατ }| ≤ Cτ . Using (2.4.18), one sees that (2.4.16) follows from the following Nikodym maximal theorem. Proposition 2.4.8 Let γθ be defined by (2.4.17). If, for a given 0 < δ < 12 , we let Rθ = {(x, t) ∈ R2 × [0, 1] : dist((x, t), γθ ) < δ}, it follows that  1  2 1/2    sup g(y − x, t) dxdt dy ≤ C| log δ|3/2 g L2 (R3 ) . R2 θ |Rθ | Rθ Proof of Proposition 2.4.8 First we fix a ∈ C0∞ (R2 ) satisfying 0 ≤ aˆ and also ρ ∈ C0∞ (R) vanishing outside a small neighborhood of θ = 0. We set aδ (θ; t, ξ ) = ρ(θ ) χ[0,1] (t) a(δξ ),

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2.4 Nikodym Maximal Functions and Maximal Riesz Means in R2

89

where χ[0,1] denotes the characteristic function of [0, 1]. Then it suffices to prove that Aθ g(y) −2

= (2π )

 R3

= (2π )−2 satisfies



  R



R2

ei[y−x,ξ +tqξ (cos θ , sin θ), ξ ] aδ (θ; t, ξ ) g(x, t) dξ dxdt

R2

ei[y,ξ +tqξ (cos θ , sin θ), ξ ] aδ (θ; t, ξ )" g(ξ , t) dξ dt

    sup |Aθ g(y)| θ

L2 (R2 )

≤ C| log δ|3/2 g L2 (R3 ) .

(2.4.19)

Here “"” denotes the partial Fourier transform with respect to x. Just as before, to apply Lemma 2.4.2, we need to make a couple of reductions. First, if we define dyadic operators  ei[y,ξ +tqξ (cos θ,sin θ),ξ ] Aτθ g(y) = (2π )−2 ×β(|ξ |/τ ) aδ (θ; t, ξ )" g(ξ , t) dξ dt, where β is as above, then it suffices to prove that     sup |Aτθ g(y)| 2 2 ≤ C log τ g L2 (R3 ) , L (R )

θ

This is because Aθ =



τ > 2.

(2.4.19 )

k

1 0.

(3.2.2)

Clearly ϕ = x − y, ξ  satisfies (3.2.2), and we have seen that in this case Pϕ is a pseudo-differential operator of order m. Our main result is that this is true for other phase functions as well. Theorem 3.2.1 Suppose that ϕ is as above and that ϕ(x, y, ξ ) = x − y, ξ  + O(|x − y|2 |ξ |).

(3.2.3)

Then, if P(x, y, ξ ) ∈ Sm , Pϕ is a pseudo-differential operator of order m, and, moreover, if we set P(x, ξ ) = P(x, x, ξ ), it follows that Pϕ − P(x, D) is a pseudo-differential operator of order (m − 1). Conversely, given a pseudo-differential operator P there is an operator Pϕ such that P − Pϕ is a smoothing operator. By (3.2.3) we of course mean that  ∂ α * +  ϕ(x, y, ξ ) − x − y, ξ   ≤ Cα |x − y|2 |ξ |1−|α| ,  ∂ξ for every α. Before turning to the proof, let us first see that this result implies that compactly supported pseudo-differential operators are invariant under changes of coordinates. Specifically, suppose that  and κ are open subsets of Rn and that κ :  → κ is a diffeomorphism; then one can define a map sending functions in  to functions in κ by setting uκ (x) = (u ◦ κ −1 )(x) = u(κ −1 (x)),

x ∈ .

Our next result says that there is also a push-forward map for pseudo-differential operators. Corollary 3.2.2 Let κ, , and κ be as above. Then if P(x, ξ ) ∈ Sm vanishes for x not belonging to a compact subset of , there is a pseudo-differential operator Pκ (x, D) that is compactly supported in κ such that, modulo smoothing operators,   y = κ(x). (3.2.4) Pκ (x, D)uκ (y) = P(x, D)u(x),

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106

Pseudo-differential Operators

Furthermore, if κ (x) = (∂κ/∂x) is the Jacobian matrix, Pκ (κ(x), (t κ (x))−1 ξ ) − P(x, ξ ) ∈ Sm−1 ,

(3.2.5)

and, thus Pκ (κ(x), ξ ) − P(x, t κ (x)ξ ) is of order m − 1. To prove this we first put P(x, y, ξ ) = ρ(x − y)P(x, ξ ), where ρ ∈ C0∞ equals one near the origin. Then, modulo a smoothing error, P(x, D)u equals  −n eix−y,ξ  P(x, y, ξ )u(y) dξ dy (2π )  eix−y,ξ  P(x, y, ξ )uκ (κ(y)) dξ dy. = (2π )−n However, if we let x˜ = κ(x), then, by changing variables in the integrations, we see that this integral equals (2π )−n times  −1 −1 t eiκ (˜x)−κ (z), κ (x)η ,  −1   × P(κ −1 (˜x), κ −1 (z), t κ (x)η)  ∂κ∂z  ∂κ ∂x uκ (z) dηdz. When z = x˜ the expression inside the braces equals P(x, t κ (x)η). Therefore, since  ∂κ(x)   κ −1 (˜x) − κ −1 (z) = x˜ − z + O(|˜x − z|2 ), ∂x Theorem 3.2.1 implies the result. Proof of Theorem 3.2.1 Let ϕ0 (x, y, ξ ) = ϕ(x, y, ξ ) and ϕ1 (x, y, ξ ) = x − y, ξ . Then for 0 ≤ t ≤ 1 set ϕt (x, y, ξ ) = (1 − t)ϕ0 + tϕ1 and  eiϕt (x,y,ξ ) P(x, y, ξ )u(y) dξ dy, (Pt u)(x) = (2π )−n where P(x, y, ξ ) ∈ Sm is as in (3.2.1). Since ϕ0 = ϕ satisfies (3.2.2) and (3.2.3), we can assume that |∇ξ ϕt | ≥ c|x − y|

on supp P(x, y, ξ ).

(3.2.6)

We may have to decrease the support of P near the diagonal; however, since (3.2.6) holds for t = 0, P0 − Pϕ would then be smoothing. Next, notice that   d j −n [i(ϕ1 − ϕ0 )] j eiϕt P(x, y, ξ )u(y) dξ dy. Pt u = (2π ) dt The symbol here, [i(ϕ1 − ϕ0 )] j P(x, y, ξ ),

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3.2 Equivalence of Phase Functions

107

is in Sm+j , but recall that we are assuming that ϕ1 − ϕ0 = O(|x − y|2 |ξ |). Thus, by the arguments of the previous section, when t = 1, the operator with this compound symbol is actually a pseudo-differential operator of order m − j, j and for all t the kernel of dtd j Pt becomes arbitrarily smooth, as j → ∞, since (3.2.6) holds. To use this, set  (−1) j  d  j  Pt  . Qj = j! dt t=1 Then, by Taylor’s formula, P0 =

k−1 



1

Qj + (−1) /k! k

tk−1

0

0

 d k dt

Pt dt.

Thus, if we let P(x, D) be a pseudo-differential operator defined by the formal  series ∞ 0 Qj , it follows that P − P0 has a smoothing kernel. Furthermore, if we let Q(x, D) have symbol P(x, x, ξ ), then P0 −Q(x, D) is a pseudo-differential operator of order m − 1. This proves the first part of the theorem. To prove the converse assertion one simply repeats the argument reversing the roles of ϕ0 and ϕ1 . Then, since it is clear that (3.2.2) implies that Lemma 3.1.2 can be extended to include operators of the form (3.2.1), one sees that any pseudo-differential operator can be written as an operator Pϕ , modulo a smoothing error. Since the composition of pseudo-differential operators is a pseudo-differential operator, it follows that, if Pϕ is as above and if Q(x, D) is a pseudo-differential operator of order μ, then Q(x, D) ◦ Pϕ is an operator of the form (3.2.1), which, modulo an operator of order m + μ − 1, equals  eiϕ(x,y,ξ ) Q(x, ξ )P(x, y, ξ )u(y) dξ dy. (2π )−n In the next chapter we shall need to know more about the lower order terms. One could evidently find this by a closer examination of the proof of Theorem 3.1.1; however, it is easier to do things directly. We shall assume that the symbol Q(x, ξ ) vanishes when x ∈ / K, K compact, since this is all we shall need. The first step in computing a composition formula for Q ◦ Pϕ is to notice that when φ is real and C∞ and ∇φ(x)  = 0

on K,

then Van der Corput’s lemma can be used to compute * + e−iλφ Q(x, D) eiλφ f , λ large,

(3.2.7)

(3.2.8)

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Pseudo-differential Operators

when f is smooth. To see this we first introduce the notation φ(x) − φ(y) = ∇φ(x), x − y − φ2 (x, y).

(3.2.9)

Clearly then φ2 (x, y) = O(|x − y|2 ). To use this, notice that we can write the quantity in (3.2.8) as (2π )−n



ei[x−y,ξ −λ(φ(x)−φ(y))] Q(x, ξ )f (y) dξ dy.

(3.2.10)

Since we are assuming (3.2.7), the only significant contributions in this integral occur when |ξ | ≈ λ. Therefore, we may assume that Q(x, ξ ) vanishes when |ξ | ∈ / [c1 λ, c2 λ] for certain cj > 0, because if we multiplied Q by the appropriate cutoff function, the difference between the analog of (3.2.10) and the above integral would be O(λ−N ) for all N. Next observe that we can make a change of variables to rewrite (3.2.10) as −n

(2π )



ei[x−y,ξ −λφ2 (x,y)] Q(x, ξ + λ∇φ(x))f (y) dξ dy.

We use Taylor’s formula to write Q(x, ξ + λ∇φ(x)) =

 1  ∂ α Q(x, λ∇φ(x))ξ α + RN,λ (x, ξ ), α! ∂ξ

|α| 0 is small and λ = ν(ν + n − 1) is large, N(λ + ε) − N(λ − ε) = dν ≈ ν n−1 ≈ λ(n−1)/2 . Since this holds for all small ε, it is clear that in this case the estimates for the remainder term in the Weyl formula cannot be improved. We now turn to the proof of Theorem 4.2.1. Recall that N(λ) is the trace of  the partial summation operator Sλ f = λj ≤λ Ej f , that is,  N(λ) = M

Sλ (x, x) dx.

Thus, the formula would follow if we could show that, if dx agrees with Lebesgue measure in a local coordinate system around x ∈ M, then the kernels satisfy  dξ + O(λn−1 ) (4.2.2) Sλ (x, x) = (2π )−n {ξ : p(x,ξ )≤λ} n−1

= c(x)λ + O(λ n

with c(x) = (2π )−n

),

 {ξ : p(x,ξ )≤1}

dξ .

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128

The Half-wave Operator

To try to prove this, we recall that if χ(−∞,λ] denotes the characteristic function of the interval (−∞, λ], then  ∞ −1 (χ(−∞,λ] )∧ (t)eitP dt. Sλ = (2π ) −∞

The Fourier transform of the characteristic function of (−∞, 0] is the distribution 1 1 = π δ0 (t) + i , (4.2.3) i(t + i0)−1 = i lim t ε→0+ t + iε and so Sλ = (2π )−1



∞ −∞

i(t + i0)−1 e−iλt eitP dt.

(4.2.4)

Note that (t + i0)−1 is only singular at t = 0; therefore, since Theorem 4.1.2 allows us to compute the kernel of eitP very precisely when |t| < ε, it might be reasonable to compare Sλ with  ∞ −1 " Sλ = (2π ) ρ(t)i(t ˆ + i0)−1 e−iλt eitP dt, (4.2.5) −∞

if ρˆ is the Fourier transform of some ρ ∈ S and satisfies ρ(t) ˆ = 1,

|t| < ε/2,

and

ρ(t) ˆ = 0,

|t| > ε.

If we let χ "(−∞,λ] = χ(−∞,λ] ∗ ρ, note that the “approximate summation operators” are given by " Sλ f =

∞ 

χ "(−∞,λ] (λj ) Ej f .

(4.2.5 )

1

Let us now use Theorem 4.1.2 to compute " Sλ (x, x). We assume that coordinates are chosen so that, around x ∈ M, dx agrees with Lebesgue measure. If this is the case then (4.2.5) implies that  " ρ(t) ˆ i(t + i0)−1 q(t, x, x, ξ ) eit(p(x,ξ )−λ) dξ dt Sλ (x, x) = (2π )−n−1  ∞ −1 +(2π ) ρ(t) ˆ i(t + i0)−1 R(t, x, x) e−iλt dt, (4.2.6) −∞

where q and R are as in Theorem 4.1.2. However, since R is C∞ , Corollary 0.1.15 implies that the last term in (4.2.6) is O(1). To prove that the first term on the right side of (4.2.6) is equal to c(x)λn + O(λn−1 ), note that Taylor’s formula implies that q(t, x, x, ξ ) = q(0, x, x, ξ ) + tr(t, x, x, ξ )

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4.2 The Sharp Weyl Formula

129

where r ∈ S0 . However, it only contributes O(λn−1 ) to the error since (4.2.3) gives  ρ(t) ˆ (t + i0)−1 tr(t, x, ξ ) eit(p(x,ξ )−λ) dtdξ  = ρ(t) ˆ r(t, x, ξ ) eit(p(x,ξ )−λ) dtdξ  = rˆ (λ − p(x, ξ ), x, ξ ) dξ , where, with an abuse of notation, rˆ ( · , x, ξ ) denotes the Fourier transform of ρ( ˆ · )r( · , x, ξ ). But the last integral is clearly dominated by  (4.2.7) (1 + |λ − p(x, ξ )|)−N dξ = O(λn−1 ). (The last estimate is easy to prove after recalling that ξ → p(x, ξ ) is positive and homogeneous of degree one in Rn \0.) By putting together these estimates, we see that we have shown that  " ρ(t) ˆ i(t + i0)−1 Sλ (x, x) = (2π )−n−1 × q(0, x, x, ξ ) eit(p(x,ξ )−λ) dξ dt + O(λn−1 ).

(4.2.8)

However, (4.1.9) implies that the main term here is  −n−1 ρ(t) ˆ i(t + i0)−1 eit(p(x,ξ )−λ) dξ dt (2π )    χ(−∞,0] ∗ ρ (p(x, ξ ) − λ) dξ = (2π )−n    n −n χ(−∞,0] ∗ (ρˆ − 1)∨ (p(x, ξ ) − λ) dξ . (4.2.9) = c(x)λ + (2π ) d ˆ To estimate the last term, note that (ρˆ − 1)∨ (s) = ds ψ(s), where ψ(t) = (1 − ρ(t))/it. ˆ Since ψ is a bounded function which is rapidly decreasing at infinity, and since, by (4.2.3),   χ(−∞,0] ∗ (ρˆ − 1)∨ (s) = −ψ(s),

one can use (4.2.7) to see that the last term in (4.2.9) is O(λn−1 ). Finally, since, by (4.1.9), q(0, x, x, ξ ) − 1 = O(|ξ |−1 ), it is clear that this argument also gives  ρ(t) ˆ (t + i0)−1 (q − 1) eit(p−λ) dξ dt = O(λn−1 ), and this means that we have proved the desired estimate " S(x, x) = c(x)λn + O(λn−1 ).

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130

The Half-wave Operator

Therefore, we would be done if we could show that |Sλ (x, x) − " Sλ (x, x)| ≤ Cλn−1 ,

x ∈ M.

(4.2.10)

To prove this, we note that (4.2.4) and (4.2.5) imply that the Fourier transform of the function λ → Sλ (x, x) − " Sλ (x, x) vanishes when |t| < ε/2. To exploit this we shall use the following Tauberian lemma. Lemma 4.2.3 Let g(λ) be a piecewise continuous tempered function of R. Assume that for λ > 0 |g(λ + s) − g(λ)| ≤ C(1 + λ)a ,

0 < s ≤ 1.

(4.2.11)

Then, if gˆ (t) = 0, when |t| ≤ 1, one must have |g(λ)| ≤ C(1 + λ)a .

(4.2.12)

Remark Clearly, the assumption that gˆ (t) vanish for small t is needed. For, if g(λ) = am λm + · · · + a0 is a polynomial of degree m, then (4.2.11) holds for a = m − 1, while clearly (4.2.12) cannot hold if am  = 0. On the other hand, in this case gˆ is, in some sense, concentrated at the origin, since sing supp gˆ = {0}. Proof of Lemma 4.2.3 Let  G(λ) =

λ

λ+1

g(s) ds.

Then G is absolutely continuous and, except on possibly a set of measure zero, G exists and satisfies |G (λ)| = |g(λ + 1) − g(λ)| ≤ C(1 + λ)a . It is also clear that the Fourier transform of G vanishes in [−1, 1]. Next since (4.2.11) and the triangle inequality imply that |g(λ)| ≤ |G(λ)| + C(1 + λ)a , it follows that (4.2.12) would follow from the estimate |G(λ)| ≤ C (1 + λ)a . To prove this last inequality let η ∈ C∞ satisfy η(t) = 0, when |t| < 12 , and ˆ η(t) = 1, for |t| > 1, and let ψ be defined by ψ(t) = (it)−1 η(t). Then ψ is bounded and rapidly decreasing at infinity. Consequently, like G, G ∗ ψ is in

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4.2 The Sharp Weyl Formula

131

S (R) and we have G ∗ ψ = G since their Fourier transforms agree. Therefore, the estimate for G gives  a |ψ(s)| (1 + |s|)a ds |G(λ)| = |(G ∗ ψ)(λ)| ≤ C(1 + λ) ≤ C(1 + λ)a , in view of the rapid decrease of ψ. If we apply the lemma to g(λ) = Sλ (x, x) − " Sλ (x, x), then, since we have already observed that gˆ vanishes near 0, we need only prove the following two estimates: |Sλ+s (x, x) − Sλ (x, x)| ≤ C(1 + λ)n−1 , Sλ (x, x)| ≤ C(1 + λ)n−1 , |" Sλ+s (x, x) − "

0 < s ≤ 1,

(4.2.13)

0 0, χ ≥ 0, and χ(t) ˆ = 0 unless |t| ≤ ε, we define  χ (λj − λ) Ej f . (4.2.15 ) χ "λ f = j

(Such a function always exists, since if the function ρ in (4.2.5) is real, then χ (λ) = (ρ(λ/4))2 has the right properties.) It is useful to have the L2 norm on the left side, since orthogonality and the fact that χ (0)  = 0 imply that (4.2.18) would be a consequence of the analogous estimates for the approximate operators:

" χλ f 2 ≤ C(1 + λ)(n−1)/2 f 1 .

(4.2.18 )

Moreover, the above arguments also apply here since χ(t) ˆ = 0 when |t| ≥ ε and  ∞ χ(t) ˆ e−itλ eitP dt. (4.2.19) χ "λ = (2π )−1 −∞

Next, notice that if χ "λ (x, y) is the kernel for χ "λ , then  χ (λj − λ)ej (x)ej (y). χ "λ (x, y) = j

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4.3 Smooth Functions of Pseudo-differential Operators

133

Hence, it is easy to see from the Schwarz inequality that the L1 (M) → L2 (M) operator norm of χ "λ satisfies  2 χλ (x, y)|2 dx

" χλ L1 (M)→L2 (M) ≤ sup |" y∈M

= sup



2 χ (λj − λ) |ej (y)|2

y∈M j

≤ χ L∞ (R) · sup χ "λ (y, y).

(4.2.20)

y∈M

In the last inequality we have used the fact that χ ≥ 0. Finally, if we let c(t, x, ξ ) = χˆ (t)q(t, x, x, ξ ), where q ∈ S0 is the symbol in the parametrix for eitP , then (4.2.19) and Theorem 4.1.2 give    ∞      −itλ    cˆ (λ − p(x, ξ ), x, ξ ) dξ  +  χˆ (t)e R(t, x, x) dt |" χλ (x, x)| ≤  n R −∞  −N ≤ CN (1 + |λ − p(x, ξ )|) dξ + O(1) ≤ C(1 + λ)n−1

(by (4.2.7)).

Combining this with (4.2.20) completes the proof. Remark Lemma 4.2.4 is a generalization of the (L1 , L2 ) restriction theorem for the Fourier transform in Rn . In fact, duality and a scaling argument show that the latter is equivalent to the uniform estimates      −n eix,ξ  fˆ (ξ ) dξ  ∞ n ≤ C(1 + λ)(n−1)/2 f L2 (Rn ) . (2π ) |ξ |∈[λ,λ+1]

L (R )

The next chapter will be devoted to proving a generalization of the full restriction theorem under the assumption that the principal symbol p(x, ξ ) satisfies certain natural curvature conditions. These estimates will allow us to extend the Tauberian argument used above to handle other situations, such as proving estimates for Riesz means on M.

4.3 Smooth Functions of Pseudo-differential Operators Let m ∈ C∞ (R) belong to the symbol class Sμ , that is, assume that   d α   m(λ) ≤ Cα (1 + |λ|)μ−α  dλ

∀ α.

(4.3.1)

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134

The Half-wave Operator

Then, using the spectral decomposition of P, we can define an operator m(P) that sends C∞ (M) to C∞ (M) by m(P)f =

∞ 

m(λj ) Ej f ,

(4.3.2)

j=1

if P is as above. Using the ideas in the proof of the sharp Weyl formula, we shall see that m(P) is actually a pseudo-differential operator. Theorem 4.3.1 Let P ∈ cl1 (M) be elliptic and self-adjoint with respect to a positive C∞ density dx. Then, if m ∈ Sμ , m(P) is a pseudo-differential operator of order μ, and the principal symbol of m(P) is m(p(x, ξ )). As before, one can see that the same result holds for pseudo-differential operators of arbitrary positive order as well. Also, since Theorem 3.1.6 says that zero order pseudo-differential operators are bounded on Lp , we also have the following. Corollary 4.3.2 Let P be as above. Then if m ∈ S0 ,

m(P)f Lp (M) ≤ Cp f Lp (M) ,

1 < p < ∞.

(4.3.3)

Proof of Theorem 4.3.1 We assume that local coordinates are chosen as in Theorem 4.1.2, and, as in the proof of Theorem 4.2.1, we fix ρ ∈ S(R) satisfying ρ(t) ˆ = 1, |t| ≤ ε/2, and ρ(t) ˆ = 0, |t| > ε. Then, using the half-wave operator, we make the decomposition  ∞  ∞   itP itP 1 − ρ(t) ˆ m(t)e ˆ ρ(t) ˆ m(t)e ˆ dt + (2π )−1 dt m(P) = (2π )−1 −∞

−∞

=" m(P) + r(P), where, if ψ is defined by ψˆ = 1 − ρ, ˆ   " m(λ) = m ∗ ρ (λ) and

  r(λ) = m ∗ ψ (λ).

Since m satisfies (4.3.1), m(t) ˆ is C∞ away from t = 0, and rapidly decreasing at ∞; thus r(λ) ∈ S. Since the kernel of r(P) equals r(P)(x, y) =

∞ 

r(λj ) ej (x)ej (y),

1

one can, therefore, use the crude estimates from Section 3.3 for the size of the derivatives of the eigenfunctions ej (x) together with the Weyl formula to see that r(P)(x, y) is C∞ .

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Notes

135

On the other hand, Theorem 4.1.2 implies that, in our local coordinate system, the kernel of " m(P) equals  −n−1 ρ(t) ˆ m(t) ˆ q(t, x, y, ξ ) eitp(y,ξ ) eiϕ(x,y,ξ ) dξ dt (2π )  ˆ m(t) ˆ R(t, x, y) dt. + (2π )−1 ρ(t) Since R is C∞ and since ρ(t) ˆ m(t) ˆ ∈ E (R), the second term must be in C∞ (M × M). Thus, we would be done if we could show that  ρ(t) ˆ m(t) ˆ q(t, x, y, ξ ) eitp(y,ξ ) eiϕ(x,y,ξ ) dξ dt (2π )−n−1 (4.3.4) is the kernel of a pseudo-differential operator with principal symbol m(p(x, ξ )). To see this we note that, as before, we can write q(t, x, y, ξ ) = q(0, x, y, ξ ) + tr(t, x, y, ξ ), ˆ is singular at the origin, we would, therefore, expect that where r ∈ S0 . Since m the main term in (4.3.4) would be  ρ(t) ˆ m(t) ˆ q(0, x, y, ξ ) eitp(y,ξ ) eiϕ(x,y,ξ ) dξ dt (2π )−n−1  −n " m(p(y, ξ )) q(0, x, y, ξ ) eiϕ(x,y,ξ ) dξ . = (2π ) Since, q(0, x, x, ξ ) − 1 ∈ S−1 , Theorem 3.2.1 implies that this is the kernel of a pseudo-differential operator having principal symbol " m(p(x, ξ )), and since m−" m ∈ S, we conclude that this kernel has the desired form. Thus, we would be done if we could show that  ρ(t) ˆ m(t) ˆ tr(t, x, y, ξ ) eitp eiϕ dξ dt (2π )−n−1 is the kernel of a pseudo-differential operator of order ≤ μ − 1. But this too follows from Theorem 3.2.1 after checking that  ∞ (2π )−1 ρ(t) ˆ m(t) ˆ tr(t, x, y, ξ ) eitp(y,ξ ) dt ∈ Sμ−1 . −∞

Notes The parametrix construction for the half-wave operator is taken from H¨ormander [4]. This was a modification of the related construction of Lax [1] which used generating functions. We have used Lax’s construction at the end of Section 4.1 to construct parametrices for strictly hyperbolic differential operators. As the reader can tell, Lax’s

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136

The Half-wave Operator

approach is somewhat more elementary, but, since the phase functions in the Lax construction need not be linear in t, it is harder to use in the study of eigenvalues and eigenfunctions. The proof of the sharp Weyl formula is from H¨ormander [4], except that here the Tauberian argument uses the bounds for the spectral projection operators in Lemma 4.2.4 that are due to Sogge [2] and Christ and Sogge [1]. The Tauberian arguments that were used in the proof go back to Avakumoviˇc [1] and Levitan [1], where they were used to prove the sharp Weyl formula for second order elliptic self-adjoint differential operators. The material from Section 4.3 is due to Strichartz [2] and Taylor [1].

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5 Lp Estimates of Eigenfunctions

In Chapter 2 we saw that if  ⊂ Rn is a compact C∞ hypersurface with non-vanishing curvature then 1/2  | fˆ (ξ )|2 dσ (ξ ) ≤ C f Lp (Rn ) , 1 ≤ p ≤ 2(n+1) n+3 . 

If, in addition, say,  = {ξ : q(ξ ) = 1}, where q is homogeneous of degree one, C∞ , and positive in Rn \ 0, this inequality is equivalent to 1/2  | fˆ (ξ )|2 dξ ≤ Cε 1/2 f Lp (Rn ) , 0 < ε < 12 . {ξ : q(ξ )∈[1,1+ε]}

If we define projection operators −n

χλ f (x) = (2π )

 {ξ : q(ξ )∈[λ,λ+1]}

eix,ξ  fˆ (ξ ) dξ ,

then taking ε = 1/λ and applying a scaling argument shows that the last inequality is equivalent to the uniform estimate

χλ f L2 (Rn ) ≤ C(1 + λ)δ(p) f Lp (Rn ) , with

1≤p≤

2(n+1) n+3 ,

  δ(p) = n 1p − 12  − 12 .

Notice that, for this range of exponents, δ(p) agrees with the critical index for Riesz summation (see Section 2.3). The operators χλ of course are the translation-invariant analogs of the spectral projection operators that were introduced in the proof of the sharp Weyl formula. The main goal of this chapter is to show that these operators satisfy the same bounds as their Euclidean versions under the assumption that the cospheres associated to the principal symbols x = {ξ : p(x, ξ ) = 1} ⊂ Tx∗ M \ 0 have everywhere non-vanishing curvature. Since δ(1) = (n − 1)/2 this is the natural extension of the estimate (4.2.18).

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138

Lp Estimates of Eigenfunctions

After establishing this “discrete L2 restriction theorem” we shall give a few applications. First, we shall show that the special case having to do with spherical harmonics can be used to give a simple proof of a sharp unique continuation theorem for the Laplacian in Rn . Then we shall see how the Tauberian arguments that were used in the proof of the sharp Weyl formula can be adapted to help prove sharp multiplier theorems for functions of pseudo-differential operators. Specifically, we shall see that estimates for Riesz means and the H¨ormander multiplier theorem carry over to this setting.

5.1 The Discrete L2 Restriction Theorem Let M be a C∞ compact manifold of dimension n ≥ 2. We assume that P = P(x, D) ∈ cl1 (M) is self-adjoint, with principal symbol p(x, ξ ) positive on T ∗ M \ 0. Then if χλ are the spectral projection operators defined in (4.2.15), we have the following result. Theorem 5.1.1 Assume that, for each x ∈ M, the cospheres {ξ : p(x, ξ ) = 1} ⊂ Tx∗ M \ 0 have everywhere non-vanishing curvature. Then if δ(p) = n 1p − 12  − 12 and λ > 0

χλ f L2 (M) ≤ C(1 + λ)δ(p) f Lp (M) ,

1≤p≤

χλ f L2 (M) ≤ C(1 + λ)(n−1)(2−p)/4p f Lp (M) ,

2(n+1) n+3 ,

2(n+1) n+3

(5.1.1)

≤ p ≤ 2.

(5.1.2)

Furthermore, these estimates are sharp. If we use Theorem 3.3.1, then we see that the dual versions of these inequalities yield the following estimates for the “size” of eigenfunctions on compact Riemannian manifolds. Corollary 5.1.2 Let g be the Laplace–Beltrami operator on a compact C∞ Riemannian manifold (M, g). Then, if {λj } are the eigenvalues of −g , and if  one defines projection operators Rλ f = √λj ∈[λ,λ+1] Ej f , one has the sharp estimates

Rλ f Lq (M) ≤ C(1 + λ)δ(q) f L2 (M) ,

Rλ f Lq (M) ≤ C(1 + λ)

(n−1)(2−q )/4q

2(n+1) n−1

f L2 (M) ,

≤ q ≤ ∞,

2≤q≤

2(n+1) n−1 .

In particular, if {ej } is an orthonormal basis corresponding to the spectral decomposition of −g , then

ej Lq (M) ≤ C(1 + λj )δ(q)/2 ,

ej Lq (M) ≤ C(1 + λj )

(n−1)(2−q )/8q

2(n+1) n−1

,

≤ q ≤ ∞,

2≤q≤

2(n+1) n−1 .

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5.1 The Discrete L2 Restriction Theorem

139

Proof of Theorem 5.1.1 We shall first prove the positive results and then show that the estimates cannot be improved. Since we have already seen that (5.1.1) holds when p = 1 and since χλ : L2 → L2 with norm one, the M. Riesz interpolation theorem implies that we need only prove the special case where p = 2(n + 1)/(n + 3); that is, it suffices to show that

χλ f L2 (M) ≤ C(1 + λ)δ(p) f Lp (M) ,

when p =

2(n+1) n+3 .

(5.1.3)

To prove this we shall use the idea from the proof of Lemma 4.2.4 of proving an equivalent version of this inequality that involves operators whose kernels can be computed very explicitly. The operators χ "λ used in the proof of Lemma 4.2.4 have kernels that are badly behaved in the diagonal {x = y}. So, in the present context, it is convenient to modify their definitions slightly. To do this we let ε > 0 be as in Theorem 4.1.2. Thus, in suitable local coordinate systems, eitP has a parametrix of the form (4.1.16) as long as |t| < ε. Then for 0 < ε0 < ε to be specified later we fix χ ∈ S(R) satisfying χ (0) = 1

and

χˆ (t) = 0 if t ∈ / (ε0 /2, ε0 ),

and define approximate projection operators  χ "λ f = χ (P − λ)f = χ (λj − λ) Ej f . j

Then, as before, orthogonality arguments show that the uniform bounds (5.1.3) hold if and only if

" χλ f L2 (M) ≤ C(1 + λ)δ(p) f Lp (M) ,

p=

2(n+1) n+3 ,

λ > 0.

(5.1.3 )

Let Q(t) and R(t) be as in Theorem 4.1.2. Then   1 1 χ "λ f = ˆ dt. Q(t)f e−itλ χˆ (t) dt + R(t)f e−itλ χ(t) 2π 2π Since R(t) has a C∞ kernel, the last term has a kernel that is O(λ−N ) for any N and hence it satisfies much better bounds than those in (5.1.3 ). Therefore, if we work in local coordinates so that dx agrees with Lebesgue measure, it suffices to show that  ei[ϕ(x,y,ξ )+tp(y,ξ )] e−itλ Tλ f (x) = (2π )−n−1 × χˆ (t) q(t, x, y, ξ ) f (y) dξ dtdy satisfies the bounds in (5.1.3 ) if ϕ and q are as in (4.1.16).

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140

Lp Estimates of Eigenfunctions

Notice that since C−1 |x − y| ≤ |∇ξ ϕ(x, y, ξ )| ≤ C|x − y| for some constant C, it follows that on the support of the integrand  * + ∇ξ ϕ(x, y, ξ ) + tp(y, ξ )   = 0 if |x − y| ∈ / [C0−1 ε0 , C0 ε0 ] for some constant C0 . Therefore, by Theorem 0.5.1, if we let a(t, x, y, ξ ) = η(x, y)χˆ (t)q(t, x, y, ξ ), where η is a smooth function which equals 1 when / [(2C0 )−1 ε0 , 2C0 ε0 ], it follows |x − y| ∈ [C0−1 ε0 , C0 ε0 ] and 0 when |x − y| ∈ that the difference between Tλ f and  −n−1 " ei[ϕ(x,y,ξ )+tp(y,ξ )] e−itλ a(t, x, y, ξ ) f (y) dξ dtdy Tλ f (x) = (2π ) has a kernel with norm O(λ−N ) for any N. On account of these reductions, we would be done if we could show that

" Tλ f L2 ≤ C(1 + λ)δ(p) f Lp ,

p=

(5.1.3 )

2(n+1) n+3 .

The proof of this follows immediately from the estimates for non-homogeneous oscillatory integrals and the following result. Lemma 5.1.3 Let a(t, x, y, ξ ) ∈ S0 be as above and set  Kλ (x, y) = ei[ϕ(x,y,ξ )+tp(y,ξ )] e−iλt a(t, x, y, ξ ) dξ dt,

λ > 1.

Then, if ε0 is chosen small enough we can write Kλ (x, y) = λ

n−1 2

aλ (x, y)eiλψ(x,y) ,

(5.1.4)

where ψ and aλ have the following properties. First, the phase function ψ is real and C∞ and satisfies the n × n Carleson–Sj¨olin condition on supp Kλ . In addition, aλ ∈ C∞ with uniform bounds     α ∂x,y aλ (x, y) ≤ Cα . Remark Note that since we are assuming that χ (0) = 1 it follows that χ "λ eλ = eλ , if eλ is an eigenfunction with eigenvalue λ. Therefore, by (5.1.4),  n−1 2 aλ (x, y) eiλψ(x,y) eλ (y) dy + Rλ eλ (x), eλ (x) = λ where ψ and aλ are as above and the trivial error satisfies Rλ L1 →L∞ = O(λ−N ) for any N. If we apply Corollary 2.2.3 we see from the lemma that the oscillatory integral operator with kernel Kλ is bounded from L2 → L  n−1  n−1  n−1  O λ 2 λ−n 2(n+1) = O λ 2(n+1) .

2(n+1) n−1

with norm

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5.1 The Discrete L2 Restriction Theorem

141

Since the n × n Carleson–Sj¨olin condition is symmetric, the adjoint of " Tλ must also satisfy this condition. So, by duality, we see that the operators " Tλ are bounded from L

2(n+1) n+3

n−1

to L2 with norm O(λ 2(n+1) ). But δ(p) =

n−1 2(n+1)

when

2(n+1) n+3

p= and hence the proof of Theorem 5.1.1 would be complete once Lemma 5.1.3 has been established. Proof of Lemma 5.1.3 The proof will have two steps. First we shall show that Kλ is of the form (5.1.4), and then we shall show that the phase function satisfies the n × n Carleson–Sj¨olin condition. The first thing we notice is that, if p(y, ξ ) ∈ / [λ/2, 2λ],    eit[p(y,ξ )−λ] a(t, x, y, ξ ) dt = O (|p(y, ξ )| + λ)−N for any N. But p(y, ξ ) ≈ |ξ |. So if β ∈ C0∞ (Rn \ 0) equals one when |ξ | ∈ [C−1 , C], with C sufficiently large, the difference between Kλ and  ei[ϕ(x,y,ξ )+tp(y,ξ )] e−iλt β(ξ/λ) a(t, x, y, ξ ) dξ dt (5.1.5) is C∞ , with all derivatives being O(λ−N ) for any N. On account of this, it suffices to show that we can write (5.1.5) as in (5.1.4). But if we set (x, y; t, ξ ) = ϕ(x, y, ξ ) + t(p(y, ξ ) − 1) and aλ (x, y; t, ξ ) = β(ξ )a(t, x, y, λξ ) then, after making a change of variables, we can rewrite (5.1.5) as  n eiλ (x,y;t,ξ ) aλ (x, y; t, ξ ) dξ dt. λ

(5.1.5 )

Notice that aλ is compactly supported and that all of its derivatives are bounded independently of λ. So we can use stationary phase to evaluate this integral if the Hessian with respect to the n + 1 variables (ξ , t) is non-degenerate on supp aλ . By symmetry it suffices to show that this is the case when ξ lies on the ξn axis. If we write ξ = (ξ1 , . . . , ξn−1 ) then ⎞ ⎛ ∂2p ∂2ϕ O(ε0 ) O(ε0 ) t ∂ξ 2 + ∂ξ 2 ⎟ ⎜ ⎟ ⎜ ∂2 ⎟ ⎜ = ⎟ , on supp aλ . ⎜ O(ε ) · · · O(ε ) 0 p 0 0 2 ξ n ⎟ ⎜ ∂(ξ , t) ⎠ ⎝ O(ε0 ) · · · O(ε0 )

p ξn

0

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The O(ε0 ) terms are elements of the matrix ∂2 ∂ξ ∂(ξn , t) or its transpose, and each of these is O(ε0 ) since t ≈ ε0 and, by (4.1.3), ∂ 2 ϕ/∂ξj ∂ξk = O(|x − y|2 ) = O(ε02 ) on supp aλ . Because of this, we claim that   ∂2   (5.1.6)  ≈ ε0n−1 , on supp aλ , det ∂(ξ , t)2 if ε0 > 0 is sufficiently small. To see this we first notice that, by homogeneity and our assumption that ξ lies on the ξn axis, we have p ξn (y, ξ ) = p(y, ξ )/ξn ≈ 1. Additionally, our curvature hypothesis implies that ∂ 2 p/∂ξ 2 is non-singular. By using these two facts and (4.1.3) again, one sees that (5.1.6) must be valid if ε0 > 0 is small enough. Furthermore, since ϕ satisfies (4.1.3) one can modify the argument before the proof of Lemma 2.3.3 to see that, if ε0 is sufficiently small, then, on supp aλ , (ξ , t) → has a unique stationary point for every fixed (x, y). Hence Corollary 1.1.8 implies that (5.1.5 ) can be written as in (5.1.4). What remains to be checked is that the phase function satisfies the n × n Carleson–Sj¨olin condition. That is, we must show that, for every fixed x, Sx = {∇x ψ(x, y) : aλ (x, y)  = 0} ⊂ Tx∗ X

(5.1.7)

is a C∞ hypersurface with everywhere non-vanishing Gaussian curvature, and also that ∂ 2ψ (5.1.8) ≡ n − 1 on supp aλ . rank ∂x∂y Since the adjoint of " Tλ has a similar form, the proof of (5.1.7) also gives that, for each y, . / Sy = ∇y ψ(x, y) : aλ (x, y)  = 0 ⊂ Ty∗ X has everywhere non-vanishing Gaussian curvature. This, together with (5.1.7) and (5.1.8), implies that the n × n Carleson–Sj¨olin condition is satisfied. To prove (5.1.7) and (5.1.8), let us first compute ψ. We note that ∇ξ ,t

= (ϕξ (x, y, ξ ) + tp ξ (y, ξ ), p(y, ξ ) − 1).

Thus, if (t(x, y), ξ(x, y)) is the solution to the equations 0 ϕξ (x, y, ξ ) + tp ξ (y, ξ ) = 0, p(y, ξ ) = 1,

(5.1.9)

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it follows from Corollary 1.1.8 that our phase function must be given by ψ(x, y) =

(x, y; t(x, y), ξ(x, y)) = ϕ(x, y, ξ(x, y)).

(5.1.10)

To check that the curvature condition is satisfied, notice that ∇x ψ(x, y) = ϕx (x, y, ξ(x, y)) +

∂ξ(x, y) ϕξ (x, y, ξ(x, y)). ∂x

However, the first half of (5.1.9) implies that ∂ξ(x, y) ∂ξ(x, y) ϕξ (x, y, ξ(x, y)) = −t(x, y) pξ (y, ξ(x, y)) ∂x ∂x ∂ = −t(x, y) {p(y, ξ(x, y))}. ∂x Since the second half of (5.1.9) implies that the last expression vanishes identically we conclude that ∇x ψ(x, y) = ϕx (x, y, ξ(x, y)). But this along with the eikonal equation (4.1.6) implies that p(x, ∇x ψ(x, y)) = p(y, ξ(x, y)). So if we use the second half of (5.1.9) again, we conclude that the hyper-surfaces in (5.1.7) are just the cospheres x = {ξ : p(x, ξ ) = 1}, which have non-vanishing curvature by assumption. "(x − y) is To verify (5.1.8) we fix y and let p(ξ ) = p(y, ξ ). Then if ψ determined by the analog of (5.1.9) and (5.1.10) where ϕ is replaced by the Euclidean phase function x − y, ξ  and p(y, ξ ) by p(ξ ) we have already "(x − y)/∂x∂y ≡ n − 1. But seen (in the proof of Lemma 2.3.4) that rank ∂ 2 ψ "(x − y) + O(|x − y|2 ) as x → y, which implies that ∂ 2 ψ/∂x∂y = ψ(x, y) = ψ "(x − y)/∂x∂y + O(1). From this and the fact that ψ " is homogeneous of ∂ 2ψ 2 degree one, we conclude that rank ∂ ψ/∂x∂y ≥ n − 1 on supp aλ if ε0 is small enough. However, since we have just seen that y → ∇x ψ(x, y) is contained in the hypersurface x , the rank can be at most n − 1 and so we get (5.1.8). Remark The proof of Lemma 5.1.3 shows that the operators χ "λ inherit their structure from the wave group eitP . In fact, if C ⊂ T ∗ (M × R) \ 0 × T ∗ M \ 0 is the canonical relation associated to the wave group, which is given in (4.1.21), it follows that the canonical relation associated to ψ satisfies Cψ ⊂ {(x, ξ , y, η) : (x, t, ξ , p(x, ξ ), y, η) ∈ C, for some 0 < t < ε and (x, ξ ) such that p(x, ξ ) = 1}.

(5.1.11)

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To see this, notice that, for x close to y, there is a unique positive small time t such that (x, t, y) is in the singular support of the kernel of eitP . But, if we call this time t(x, y), it follows from Euler’s homogeneity relations, ϕ = ϕξ , ξ , p = p ξ , ξ , along with (5.1.9) and (5.1.10) that ψ(x, y) = −t(x, y). We should  also point out that this implies that, when P = −g , ψ is just minus the Riemannian distance between x and y. Proof of sharpness We first prove that for 1 ≤ p ≤ 2 lim sup sup λ

−n( p1 − 12 )+ 21

λ→+∞ f ∈Lp (M)

χλ f 2 ≥ c,

f p

(5.1.12)

for some c > 0. This of course implies that the estimate (5.1.1) must be sharp. ˆ ) = 0 if To prove (5.1.12) we fix β ∈ S(R) satisfying β(1)  = 0 and β(τ τ∈ / [−ε, ε], where ε is as in Theorem 4.1.2. We then fix x0 ∈ M and define  β(λj /λ) ej (x0 )ej (x). fλ (x) = j

Notice that this is the kernel of β(P/λ) evaluated at (x0 , x). So, if we argue as in Section 4.3, we conclude that, given N, there must be an absolute constant CN such that | fλ (x)| ≤ CN λn (1 + λ dist(x, x0 ))−N . Consequently,

fλ p ≤ C0 λn−n/p .

On the other hand, since β(1)  = 0, we conclude that there must be a c0 > 0 such that, if λ is large enough,  |β(λj /λ)|2 |ej (x0 )|2 ej 22

χλ fλ 22 = λj ∈[λ,λ+1]



≥ c0

|ej (x0 )|2 .

λj ∈[λ,λ+1]

However,



|M| · sup

x0 ∈M λ ∈[λ,λ+1]

|ej (x0 )|2

j





≥

|ej (x)|2 dx ≥ N(λ + 1) − N(λ).

(5.1.12 )

M λ ∈[λ,λ+1] j

Combining this with the upper bounds for the norm of fλ yields

χλ f 2 ≥ cλ−n+n/p {N(λ + 1) − N(λ)}1/2 f ∈Lp f p sup

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for some c > 0. However, since N(λ) ≈ λn , lim sup λ−(n−1) {N(λ + 1) − N(λ)} > 0, λ→+∞

and hence (5.1.12) follows from (5.1.12 ). The proof that the other estimate in Theorem 5.1.1 is sharp is more difficult and requires a special choice of local coordinates in M. Before specifying this, let us first notice that it suffices to show that, for large enough λ,

" χλ f 2 (n−1) ≥ cλ p

f

p f ∈L sup



1 1 2p − 4



, 1 ≤ p ≤ 2,

(5.1.13)

if χ "λ are the approximate spectral projection operators occurring in the proof of Theorem 5.1.1. To prove this lower bound we shall use the last remark. That is, we shall use the fact that −ψ(x, y) = t(x, y), where, if x is close but not equal to y, t(x, y) is the unique small positive number such that M t(x,y) (x, ξ ) = y for some ξ ∈ Tx∗ M \ 0. Here t : T ∗ M \ 0 → T ∗ M \ 0 is the flow out for time t along the Hamilton vector field associated to p(x, ξ ), and M : T ∗ M \ 0 → M is the natural projection operator. Keeping this in mind, there is a natural local coordinate system vanishing at given x0 ∈ M which is adapted to ψ(x, y). This is just given by κ(z) = y if M |y| (x0 , ξ ) = z with ξ = y/|y|.

(5.1.14)

Here ξ denotes the coordinates in Tx∗0 M \ 0 which are given by an initial choice  of local coordinates around x0 (see Section 0.4). Note that, if P = −g , the local coordinates would just be geodesic normal coordinates around x0 and so one should think of (5.1.14) as the natural “polar coordinates” associated to p(x, ξ ). They are of course well defined for z near x0 and C∞ (away from possibly z = x0 ) since, for a given initial choice of local coordinates vanishing at x0 , M t (x0 , ξ ) = tp ξ (x0 , ξ ) + O(t2 ), and since, by assumption, p ξ ξ has maximal rank n − 1. The reason that these coordinates are useful for proving (5.1.13) is that, in these coordinates, ψ(0, y) = −|y|. Moreover, using the group property of t , one sees that ψ(x, y) = −(x1 − y1 ) and − ∇x ψ(x, y) = ∇y ψ(x, y) = (1, 0), if x = (x1 , 0), y = (y1 , 0),

and

0 < y1 < x1 .

(5.1.15)

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From this and Taylor’s theorem we deduce that if x = (x2 , . . . , xn ) and 0 < y1 < x1 then ψ(x, y) = −(x1 − y1 ) + O(|x − y |2 ).

(5.1.15 )

Since ψ is smooth away from the diagonal, the constants in the error term remain bounded if x1 − y1 is larger than a fixed constant. We can now deduce (5.1.13). We first fix a nontrivial α ∈ C0∞ (R+ ) and set αλ (y) = α(y1 /ε1 ) α(λ1/2 |y |/ε1 ) where 0 < ε1 & ε0 , with ε0 being the number occurring in the definition of χ "λ . If we then define fλ (y) = e−iλy1 αλ (y), we shall show that, if ε1 is small enough, we can obtain favorable lower bounds for " χλ fλ 2 . Notice that, for large λ, fλ is supported in a small neighborhood of 1 size O(λ− 2 ) about the positive x1 axis, which, by construction, is the projection of a bicharacteristic for p(x, ξ ). To establish the lower bound, let C0 be the constant occurring in the proof of (5.1.3) and set λ = {x : ε0 /2C0 ≤ x1 ≤ 2C0 ε0 , |x | ≤ ε1 λ−1/2 }. Then, if ε1 is sufficiently small, given any N there is a constant CN such that, if aλ (x, y) is as in Lemma 5.1.3, then 2      λ n−1 2 eiλψ(x,y) aλ (x, y)fλ (y)dy dx ≤ " χλ fλ 2L2 (M) + CN λ−N .   λ

But the left side can be rewritten as   eiλ[(ψ(x,y)−y1 )−(ψ(x,"y)−"y1 )] × αλ (y) αλ (" y) aλ (x, y) aλ (x," y) dyd" ydx. λn−1 λ

Notice that, by (5.1.15 ), the term in the exponential is O(ε12 ) on the support of the integrand. Using this and the fact that the aλ belong to a bounded subset of C∞ shows that, for small enough ε1 (compared to ε0 ), the last expression is bounded from below by a fixed positive constant times  |aλ (x, 0)|2 dx − Cε12n+1 |λ |, ε12n λn−1 (λ−(n−1)/2 )2 λ

where C is a fixed constant. However, since q(0, x, x, ξ ) = 1 + O(|ξ |−1 ), it follows from the proof of Lemma 5.1.3 that, if ε0 is small and λ large, then

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there must be positive constants such that  |aλ (x, 0)|2 dx ≥ c|λ | ≥ c λ−(n−1)/2 . λ

Thus, if λ is large and if ε1 is small, we reach the desired conclusion that

" χλ fλ 2 ≥ c0 λ−(n−1)/4 ,

some

c0 > 0.

Finally, since

fλ p ≤ Cλ−(n−1)/2p , we conclude that (5.1.13) must hold. Application: Unique Continuation for the Laplacian A special case of Corollary 5.1.2 concerns spherical harmonics. If M = Sn−1 , n ≥ 3, and g = S is the usual Laplacian on Sn−1 the eigenvalues of the conformal Laplacian, −S +[(n−2)/2]2 , are {(k+(n−2)/2)2 }, k = 0, 1, 2, . . . . The eigenspace corresponding to the kth eigenvalue is called the space of spherical harmonics of degree k and it has dimension ≈ kn−2 for large k. If we let Hk be the projection onto this eigenspace, then Corollary 5.1.2, duality, and the fact that Hk = Hk ◦ Hk give the bounds

Hk f Lp (Sn−1 ) ≤ C(1 + k)1−2/n f Lp (Sn−1 ) ,

p=

2n n+2 .

(5.1.16)

Notice that for this value of p we have 1/p − 1/p = 2/n, so the exponents in (5.1.16) correspond to the dual exponents in the classical Sobolev inequality for the Laplacian in Rn :

u Lp (Rn ) ≤ C u Lp (Rn ) ,

p=

2n n+2 ,

u ∈ C0∞ (Rn ).

We claim that (5.1.16) along with the Hardy–Littlewood–Sobolev inequality yield a weighted version of this. Namely, if p and p are as above there is a uniform constant C for which

|x|−τ u Lp (Rn ) ≤ C |x|−τ u Lp (Rn ) , if

1 u ∈ C0∞ (Rn \ 0) and dist (τ , Z + n−2 2 ) = 2.

(5.1.17)

The condition on τ is of course related to the spectrum of the conformal Laplacian on Sn−1 . Before proving (5.1.17) let us see how it yields the following unique continuation theorem.

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Theorem 5.1.4 Let n ≥ 3 and let X be a connected open subset of Rn p containing the origin. Suppose that Dα u ∈ Lloc (X) for |α| ≤ 2 where p = 2n/(n + 2). Assume further that |u| ≤ |Vu|

(5.1.18)

n/2

for some potential V ∈ Lloc (X). Then if u vanishes of infinite order at the origin in the Lp mean, that is, if for all N  2n |u|p dx = O(RN ), p = n+2 , (5.1.19) |x| 0 since the argument for τ < 0 is similar. Following the proof that the one-dimensional Hardy–Littlewood–Sobolev inequality implies the higher-dimensional versions, we first fix t and estimate the Lp norm over Sn−1 :  ∞ ∞

Tτ f (t, · ) Lp (dω) ≤

mτ (t, s; k)Hk f (s, · ) Lp (dω) ds, −∞ k=0

where mτ (t, s; k)  ∞ -−1 , 1 n−2 = ei(t−s)η (η − i(k + n−2 − τ ))(η + i(k + + τ )) dη. 2 2 2π −∞ 1 Since we are assuming that |k + n−2 2 − τ | ≥ 2 , it follows that, for any N,  −N . |mτ (t, s; k)| ≤ CN (1 + k)−1 1 + |k + n−2 2 − τ | · |t − s|

Hence, if we use the estimates for spherical harmonics (5.1.16), we get

Tτ f (t, · ) Lp (dω)  ∞ ∞ −N ≤C (1 + k)−2/n (1 + |k + n−2 2 − τ | · |t − s|) −∞ k=0

× f (s, · ) Lp (dω) ds. But our assumption on τ implies that if N > 2 ∞ 

−N (1 + k)−2/n (1 + |k + n−2 = O(|t − s|−1+2/n ). 2 − τ | · |t − s|)

k=0

Hence we get

Tτ f (t, · ) Lp (dω) ≤ C



∞ −∞

|t − s|−1+2/n f (s, · ) Lp (dω) ds,

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which leads to (5.1.21) after an application of the Hardy–Littlewood–Sobolev inequality (i.e., Theorem 0.3.6) as 1/p − 1/p = 2/n.

5.2 Estimates for Riesz Means In this section we shall study the Riesz means of index δ ≥ 0 associated to P(x, D):  Sλδ f (x) = (1 − λj /λ)δ Ej f . (5.2.1) λj ≤λ

As in Rn , these operators can never be uniformly bounded in Lp (M), p  = 2, when δ ≤ δ(p) if δ(p) is the critical index . / δ(p) = max n| 1p − 12 | − 12 , 0 . (5.2.2) On the other hand, we shall prove the following positive result which extends Euclidean estimates in Section 2.3. Theorem 5.2.1 Let P(x, D) ∈ cl1 (M) be positive and self-adjoint. Then there is a uniform constant Cδ such that for all λ > 0

Sλδ f L1 (M) ≤ Cδ f L1 (M) ,

δ>

n−1 2 .

(5.2.3)

If we also assume that the cospheres associated to P, x = {ξ : p(x, ξ ) = 1} ⊂ Tx∗ M \ 0 have non-vanishing Gaussian curvature, then for p ∈ [1, 2(n + 1)/(n + 3)] ∪ [2(n + 1)/(n − 1), ∞] and λ > 0

Sλδ f Lp (M) ≤ Cp,δ f Lp (M) ,

δ > δ(p).

(5.2.4)

As a consequence of this result, we get that, for p < ∞ as in the theorem, → f in the Lp topology if δ > δ(p). Notice that, for the exponents in both (5.2.3) and (5.2.4), δ(p) = n| 1p − 12 |− 12 . Hence, by Lemma 4.2.4, Theorem 5.1.1, and duality, both inequalities are a consequence of the following. Sλδ f

Proposition 5.2.2 Suppose that P(x, D) ∈ cl1 is positive and self-adjoint. Suppose also that for a given 1 ≤ p < 2 there is a uniform constant C such that

χλ f L2 (M) ≤ C(1 + λ)n(1/p−1/2)−1/2 f Lp (M) ,

λ ≥ 0.

(5.2.5)

Then (5.2.4) holds.

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Proof of Proposition 5.2.2 Since the Fourier transform of τ−δ is cδ (t + i0)−δ−1 , we can write  Sλδ f = (2π )−1 cδ λ−δ eitP f e−itλ (t + i0)−δ−1 dt. To use the parametrix for eitP we let ε be as in Theorem 4.1.2 and fix ρ ∈ C0∞ (R) which equals 1 for |t| < ε/2 and 0 for |t| > ε. We then set Sλδ f = " Sλδ f + Rδλ f , where " Sλδ f = (2π )−1 cδ λ−δ



eitP f e−itλ ρ(t)(t + i0)−δ−1 dt.

We would be done if we could show that, for δ > δ(p),

" Sλδ f p ≤ Cδ f p ,

(5.2.6)

Rδλ f p

(5.2.7)

≤ Cδ f p .

We first estimate the remainder term and in fact show that it satisfies a much stronger estimate:

Rδλ f 2 ≤ C(1 + λ)[n(1/p−1/2)−1/2]−δ f p .

(5.2.7 )

This is not difficult. We note that, for δ ≥ 0, the Fourier transform of (1−ρ(t))(t +i0)−δ−1 is bounded and rapidly decreasing at infinity. This means that  rδ (λ − λj ) Ej f Rδλ f = λ−δ j

for some function rδ satisfying |rδ (λ)| ≤ CN (1 + |λ|)−N for all N. Hence, for large N,

Rδλ f 22 ≤ CN λ−2δ

∞  (1 + |λ − k|)−N χk f 22 k=1

∞  ≤ C λ−2δ (1 + |λ − k|)−N (1 + k)2[n(1/p−1/2)−1/2] f 2p

k=1 −2[δ−n(1/p−1/2)+1/2]

≤ C (1 + λ)

f 2p ,

as desired. To estimate the main term we decompose " Sλδ as in the proof of Theo ∞ k rem 2.4.1. To this end, we fix β ∈ C0 (R \ 0) satisfying ∞ k=−∞ β(2 s) = 1, s  = 0. We then write  δ δ " " Sλ,0 f+ f, Sλδ f = " Sλ,k k≥1

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5.2 Estimates for Riesz Means where for k = 1, 2, . . . δ " Sλ,k f = (2π )−1 cδ λ−δ



153

eitP f e−itλ β(λ2−k t) ρ(t) (t + i0)−δ−1 dt.

δ f ≡ 0 if 2k is larger than a fixed multiple of λ. Note that " Sλ,k We claim that (5.2.5) and the finite propagation speed of the singularities of (eitP )(x, y) can be used to prove that, for any ε > 0, δ

" Sλ,k f p ≤ Cε 2−k[δ−n(1/p−1/2)+1/2−ε] f p .

(5.2.8)

By summing a geometric series this leads to (5.2.6). Actually the estimate for k = 0 just follows from Theorem 4.3.1. This is because δ " Sλ,0 f = mδλ,0 (P)f ,

where mδλ,0 (τ ) is the convolution of (1 − τ/λ)δ+ with λ−1 η(τ/λ) if η is the ∞  inverse Fourier transform of (1 − β(2−k τ )) ∈ C0∞ (R). Hence, for any N, k=1

 ∂ α    mδλ,0 (τ ) ≤ Cα,N λ−|α| (1 + τ/λ)−N .  ∂τ Thus, if p > 1 the estimate is a special case of (4.3.3). Since pseudo-differential operators of order −1 are bounded on L1 , the case of p = 1 follows via Theorem 4.3.1 and the easy fact that the pseudo-differential operators with symbol mδλ,0 (p(x, ξ )) are uniformly bounded on L1 . To handle the terms with k ≥ 1 in (5.2.8) we write  δ −1 −δ " Q(t)f e−itλ β(λ2−k t) ρ(t) (t + i0)−δ−1 dt Sλ,k f = (2π ) cδ λ  + (2π )−1 cδ λ−δ R(t)f e−itλ β(λ2−k t) ρ(t) (t + i0)−δ−1 dt, where Q(t) is the parametrix for eitP . Since the kernel of R(t) is C∞ one sees that the kernel of the second operator is O(2−kN ) for any N. Hence it suffices to show that the integral operator with kernel  δ ei[ϕ(x,y,ξ )+t(p(y,ξ )−λ)] (x, y) = (2π )−n−1 cδ λ−δ Hλ,k × q(t, x, y, ξ ) β(λ2−k t)ρ(t) (t + i0)−δ−1 dξ dt satisfies the bounds in (5.2.8).

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154

A main step in the proof of this is to show that, if ε > 0 is fixed, then for every N  δ |Hλ,k (x, y)| dy, |x−y|>λ−1 2k(1+ε)  δ |Hλ,k (x, y)| dx ≤ Cε,N 2−kN . (5.2.9) |x−y|>λ−1 2k(1+ε)

If we could prove this inequality then, by repeating the arguments in the proof of Theorem 2.3.5, we would find that (5.2.8) would follow from showing that one always has δ

" Sλ,k f Lp (B(x0 ,λ−1 2k )) ≤ C2−k[δ−n(1/p−1/2)+1/2] f Lp (M) ,

(5.2.8 )

if B(x0 , r) denotes the ball of radius τ around x0 with respect to a fixed smooth metric. But, if we use H¨older’s inequality, we can dominate the left side by δ (2k /λ)n(1/p−1/2) " Sλ,k f L2 (M) . δ f = mδ (λ − P)f where |mδ (τ )| ≤ Next, we observe that " Sλ,k λ,k λ,k CN 2−kδ (1 + |2k τ/λ|)−N for any N. Hence (5.2.5) gives δ

" Sλ,k f 2L2 (M)

≤ C2−2kδ

∞  (1 + 2k λ−1 |λ − j|)−N (1 + j)2[n(1/p−1/2)−1/2] f 2p j=0

−2kδ 2[n(1/p−1/2)−1/2]

≤C 2

λ

(λ/2k ) f 2p .

So, if we combine this with the last estimate we conclude that the left side of (5.2.8 ) is always majorized by 2−kδ (2k /λ)n(1/p−1/2) λn(1/p−1/2)−1/2 (λ/2k )1/2 f p = 2−k[δ−n(1/p−1/2)+1/2] f p , as desired. To finish matters and prove (5.2.9) we let aδλ,k ( · , x, y, ξ ) denote the inverse Fourier transform of t → cδ λ−δ q(t, x, y, ξ )β(λ2−k t)ρ(t)(t + i0)−δ−1 . Then one sees that γ

|Dατ Dξ aδλ,k (τ , x, y, ξ )| ≤ Cαγ N 2−kδ (2k /λ)α (1 + |2k τ/λ|)−N (1 + |ξ |)−|γ | . (5.2.10) But  δ (x, y) = (2π )−n Hλ,k

eiϕ(x,y,ξ ) aδλ,k (p(y, ξ ) − λ, x, y, ξ ) dξ ,

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5.3 More General Multiplier Theorems

155

and so (5.2.10) (and the fact that |∇ξ ϕ| ≥ c|x − y| on the support of the symbol) gives the bounds δ |Hλ,k (x, y)| ≤ CN 2−k λn |λ2−k (x − y)|−N ,

which of course yield (5.2.9). Remark One could also prove (5.2.4) in a more constructive way by combining arguments from the proofs of Theorems 5.1.1 and 2.3.1. In fact, the stationary phase arguments that were used in the proof of Lemma 5.1.3 show that the kernel of " Sλδ is O(λ−N ) outside of a fixed neighborhood of the diagonal, while near the diagonal it is of the form   " Sλδ (x, y) = λ(n−1)/2−δ eiλψ1 (x,y) aδλ,1 (x, y) + e−iλψ2 (y,x) aδλ,2 (x, y) , where ψj is as in Lemma 5.1.3 and |Dαx,y aδλ,j (x, y)| ≤ Cα (dist (x, y))−(n+1)/2−δ−|α| . Since the phase functions satisfy the n × n Carleson–Sj¨olin condition one can therefore break up the kernel dyadically near the diagonal and argue as in the proof of Theorem 2.3.1 to obtain (5.2.4).

5.3 More General Multiplier Theorems Suppose that m ∈ L∞ (R). Fix a function β ∈ C0∞ ((1/2, 2)) satisfying ∞ j −∞ β(2 τ ) = 1, τ > 0, and suppose also that  ∞ sup λ−1 |λα Dατ (β(τ/λ)m(τ ))|2 dτ < ∞, 0 ≤ α ≤ s, (5.3.1) λ>0

−∞

where s is an integer > n/2. Then if p(ξ ) is homogeneous of degree one, positive, and C∞ in Rn \ 0 it follows that m(p(ξ )) satisfies the hypotheses in the H¨ormander multiplier theorem, Theorem 0.2.6. Therefore  (m(p(D))f )(x) = (2π )−n eix,ξ  m(p(ξ ))fˆ (ξ ) dξ Rn

Lp (Rn )

for all 1 < p < ∞. is a bounded operator on The purpose of this section is to extend this result to the setting of compact manifolds of dimension n ≥ 2. Theorem 5.3.1 Let m ∈ L∞ (R) satisfy (5.3.1). Then if P(x, D) ∈ positive and self-adjoint it follows that

m(P)f Lp (M) ≤ Cp f Lp (M) ,

1 < p < ∞.

1 cl (M)

is

(5.3.2)

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Proof Since the complex conjugate of m satisfies the same hypotheses we need only prove (5.3.2) for exponents 1 < p ≤ 2. This will allow us to exploit orthogonality and also reduce (5.3.2) to showing that m(P) is weak-type (1,1): μ{x : |m(P)f (x)| > α} ≤ Cα −1 f 1 .

(5.3.2 )

Here μ(E) denotes the dx measure of E ⊂ M. Since m(P) is bounded on L2 , (5.3.2 ) implies (5.3.2) by the Marcinkiewicz interpolation theorem. The proof of the weak-type estimate will involve a splitting of m(P) into two pieces: a main piece to which the Euclidean arguments apply, plus a remainder which can be shown to satisfy much better bounds than are needed using the estimates for the spectral projection operators. Specifically, if ρ ∈ C0∞ (R) is as in the proof of Theorem 5.2.1, we write m(P) = " m(P) + r(P), where " m(P) = (m ∗ ρ)(P) ˇ =

1 2π

 ˆ dt. eitP ρ(t) m(t)

To estimate the remainder we define for λ = 2 j , j = 1, 2, . . ., mλ (τ ) = β(τ/λ)m(τ ). Then we put  1 eitP (1 − ρ(t)) m rλ (P) = ˆ λ (t) dt, 2π  r2k (P) is a bounded and rapidly decreasing and notice that r0 (P) = r(P) − k≥1

function of P. Hence r0 (P) is bounded from L1 to any Lp space. Therefore, we would have

r(P)f 2 ≤ C f 1 ,

(5.3.3)

which is much stronger than the analog of (5.3.2 ), if we could show that

rλ (P)f 2 ≤ Cλn/2−s f 1 ,

λ = 2 j,

j = 1, 2 . . . .

(5.3.3 )

To prove this we use the L1 → L2 bounds for the spectral projection operators to get

rλ (P)f 22 ≤

∞ 

rλ (P)χk f 22

k=0 ∞ 

≤C

sup

k=0 τ ∈[k,k+1]

|rλ (τ )|2 (1 + k)n−1 f 21 .

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5.3 More General Multiplier Theorems

157

Hence (5.3.3 ) would be a consequence of ∞ 

sup

k=0 τ ∈[k,k+1]

|rλ (τ )|2 (1 + k)n−1 ≤ Cλn−2s .

(5.3.4)

We claim that this follows from our assumption (5.3.1). The first thing to / [λ/2, 2λ] both " mλ (τ ) and rλ (τ ) notice, though, is that since mλ (τ ) = 0 for τ ∈ / [λ/4, 4λ]. Hence (5.3.4) would follow are O((1 + |τ | + |λ|)−N ) for any N if τ ∈ from  sup |rλ (τ )|2 ≤ Cλ1−2s . (5.3.4 ) k∈[λ/4,4λ] τ ∈[k,k+1]

But if we use the fundamental theorem of calculus and the Cauchy–Schwarz inequality, we find that we can dominate this by    1 2 2 |m ˆ λ (t)(1 − ρ(t))|2 dt |rλ (τ )| dτ + |rλ (τ )| dτ = 2π  1 |tm ˆ λ (t)|2 |(1 − ρ(t))|2 dt. + 2π Recall that ρ = 1 for |t| < ε/2, so a change of variables shows that this is majorized by  1 −1−2s |ts m ˆ λ (t/λ)|2 dt λ 2π  = λ−1−2s |Dsτ (λmλ (λτ ))|2 dτ  , = λ1−2s · λ−1 |λs Dsτ (β(τ/λ)m(τ ))|2 dτ . By (5.3.1) the expression inside the braces is bounded independently of λ, giving us (5.3.4 ). Since we have established (5.3.3), it suffices to show that " m(P) = m(P) − r(P) is weak-type (1,1): μ{x : |" m(P)f (x)| > α} ≤ Cα −1 f 1 .

(5.3.5)

But if we argue as before, this would follow from showing that the integral operator with kernel  ei[ϕ(x,y,ξ )+tp(y,ξ )] ρ(t)m(t)q(t, ˆ x, y, ξ ) dξ dt K(x, y) = (2π )−n−1  m(p(y, ξ ), x, y, ξ ) dξ (5.3.6) = (2π )−n eiϕ(x,y,ξ ) "

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158

is weak-type (1,1). Here we have abused notation a bit by letting " m(τ , x, y, ξ ) denote the inverse Fourier transform of t → ρ(t)m(τ ˆ ) × q(t, x, y, ξ ), that is,   " m(τ , x, y, ξ ) = [ρ( · )q( · , x, y, ξ )]∨ ∗ m (τ ). (5.3.7) Notice that, if we define " mλ (τ , x, y, ξ ) in a similar manner, then this function is C∞ and, moreover, (5.3.1) and the fact that q ∈ S0 imply   sup |Dγx Dαξ (" mλ (p(y, λξ ), x, y, λξ ))|2 dξ < ∞. (5.3.8) λ 0≤|α|≤s Rn

If we let



Kλ (x, y) =

eiϕ(x/λ,y/λ,λξ ) " mλ (p(y/λ, λξ ), x/λ, y/λ, λξ ) dξ ,

we claim that this gives the bounds  |Kλ (x, y)| dx ≤ C(1 + R)n/2−s , {x : |x−y|>R}  |Kλ (x, y0 ) − Kλ (x, y1 )| dx ≤ C|y0 − y1 |.

(5.3.9)

Before proving these, though, let us see why they imply that  Tf (x) = K(x, y)f (y) dy  is weak-type (1,1). We let f = g + ∞ on–Zygmund k=1 bk be the Calder´ decomposition of f at level α given in Lemma 0.2.7. Let Qk ⊃ supp bk be the cube associated to bk . Then, since T is bounded on L2 , if one repeats the arguments at the end of the proof of Theorem 0.2.6, one concludes that the weak-type boundedness of T would be a consequence of the estimates  |Tbk (x)| dx ≤ C bk 1 , (5.3.5 ) x∈Q / ∗k

if Q∗k denotes the double of Qk . To prove this, we may assume that Qk is centered at the origin and we let R be its side-length. Then using the first estimate in (5.3.9) we get       λn Kλ (λx, λy)bk (y) dy dx   x∈Q / ∗k



≤ bk 1 sup y

{x:|x−y|>R}

|λn Kλ (λx, λy)| dx ≤ C(λR)n/2−s bk 1 .

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5.3 More General Multiplier Theorems  Using the cancellation property bk dy = 0, we have 

159

 λ Kλ (λx, λy) bk (y) dy = n

λn [Kλ (λx, λy) − Kλ (λx, 0)] bk (y) dy.

This and the second inequality in (5.3.9) lead to       λn Kλ (λx, λy)bk (y) dy dx   x∈Q / ∗k



≤



y∈Qk x∈Q / ∗k

λn |Kλ (λx, λy) − Kλ (λx, 0)| |bk (y)| dxdy

≤ C(λR) bk 1 .  nj j j But K(x, y) = ∞ j=1 2 K2 j (2 x, 2 y) + K0 (x, y), where K0 is bounded and vanishes when |x − y| is larger than a fixed constant. Therefore, if we combine the last two estimates we conclude that     |Tbk (x)| dx ≤ C bk 1 · (2 j R)n/2−s + 2 jR x∈Q / ∗k

2 j R≥1

2 j R 1  3 |ξ β (−Dj − hj (ξ ))αj uˆ (ξ )|2 dξ ≤ Cα R2(m+n/4) , R/2≤|ξ |≤2R

|α| = |β|,

or, equivalently,  R/2≤|ξ |≤2R

|ξ |2|α| |Dα v(ξ )|2 dξ ≤ Cα R2(m+n/4) .

By rescaling, we see that vR (ξ ) = v(Rξ )/Rm−n/4 satisfy the uniform estimates  1/2≤|ξ |≤2

|Dα vR (ξ )|2 dξ ≤ Cα

∀α.

By the Sobolev embedding theorem, this implies that |Dα vR (ξ )| ≤ Cα when |ξ | = 1, or, equivalently, |Dα v| ≤ Cα (1 + |ξ |)−m+n/4−|α| .

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We need one other result for the proof of the equivalence of phase function theorem. Recall that in Section 0.5 we saw that every Lagrangian section of T ∗ Rn is locally the graph of the gradient of a C∞ function. A similar result holds for homogeneous Lagrangian submanifolds. Proposition 6.1.2 Let γ0 = (x0 , ξ0 ) ∈  ⊂ T ∗ X\0, with  being a C∞ conic Lagrangian manifold. Then local coordinates vanishing at x0 can be chosen such that (1)   (x, ξ ) → ξ is a local diffeomorphism, (2) and there is a unique real homogeneous H ∈ C∞ such that, near (x0 , ξ0 ),  = {(H (ξ ), ξ )}. Let us first assume that (1) holds and then see that this implies (2). This is easy, for, near γ0 ,  = {(φ(ξ ), ξ )} for some φ which is homogeneous of degree zero and C∞ near ξ0 . We saw in the proof of Proposition 0.5.4 that the  canonical one form ω = ξj dxj must vanish identically on . This means that if φj denotes the jth coordinate,  ξj dφj (ξ ) = 0.  Or, if we set H(ξ ) = ξj φj (ξ ), then  dH(ξ ) = φj (ξ )dξj , that is, φj (ξ ) = ∂H(ξ )/∂ξj , giving us (2). To prove that local coordinates can be chosen so that (1) holds we need an elementary result from the theory of symplectic vector spaces whose proof will be given in the appendix. Lemma 6.1.3 If V0 and V1 are two Lagrangian subspaces of T ∗ Rn one can always find a third Lagrangian subspace V that is transverse to both V0 and V1 . Recall that two C∞ submanifolds Y, Z of a C∞ manifold X are said to intersect transversally at x0 ∈ Y ∩ Z if Tx0 X = Tx0 Y + Tx0 Z. Proof of (1) We first choose local coordinates y so that γ0 = (0, ε1 ) with ε1 = (1, 0, . . . , 0). The tangent plane to  at γ0 , V0 , must be a Lagrangian plane. If it is transverse to the plane W = {(y, ξ1 )} = {(y, dy1 )}, we can take y as our coordinates since the transversality of V0 and W is equivalent to (1). If not, we use Lemma 6.1.3 to pick a Lagrangian plane V that is transverse to both V0 and V1 = {(0, ξ )}. The transversality of V and V1 means that V is a section passing through γ0 and hence V = {(y, d(y1 + Q(y)))} for some real quadratic form Q.

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Fourier Integral Operators

If we now take x1 = y1 + Q(y), xj = yj , j = 2, . . . , n, as our new coordinates, it follows that, in these coordinates, the tangent plane at γ0 , V0 , and V = {(x, dx1 )} are transverse, giving us (1). We now come to the main result. Recall that the homogeneous phase function φ(x, θ ) is said to be non-degenerate if dφ  = 0 and when φθ = 0 the N differentials d(∂φ/∂θj ) are linearly independent. We saw before that this implies that φ = {(x, θ ) : φθ (x, θ ) = 0} is a C∞ submanifold of X × (RN \0) and that  = {(x, φx (x, θ )) : (x, θ ) ∈ φ } is Lagrangian. Theorem 6.1.4 Let φ be a non-degenerate phase function in an open conic neighborhood of (x0 , θ0 ) ∈ Rn × (RN \0). Then, if a ∈ Sμ (Rn × RN ), μ = m + n/4−N/2, is supported in a sufficiently small conic neighborhood  of (x0 , θ0 ) it follows that  −(n+2N)/4 eiφ(x,θ) a(x, θ ) dθ (6.1.3) u(x) = (2π ) RN

is in I m (Rn , ) with  as above. If we also assume that coordinates are chosen so that  = {(H (ξ ), ξ )}, then πi



eiH(ξ ) uˆ (ξ ) − (2π )n/4 a(x, θ )| det φ |−1/2 e 4 sgn φ ∈ Sm−n/4−1

(6.1.4)

for |ξ | > 1 near ξ0 = φx (x0 , θ0 ), where (x, θ ) is the solution of φθ (x, θ ) = 0, φx (x, θ ) = ξ , and

φxx φxθ φ = . φθx φθθ Conversely, every u ∈ I m (Rn , ) with WF(u) contained in a small neighborhood of (x0 , ξ0 ) can be written as (6.1.3) modulo C∞ . One should notice that this result contains the equivalence of phase function theorem for pseudo-differential operators, Theorem 3.2.1, since the phase function ϕ(x, y, ξ ) there and the Euclidean phase function x − y, ξ  both m (Rn × parameterize the trivial Lagrangian {(x, x, ξ , −ξ )}. Similarly, if u ∈ Icomp n R , ), with  the trivial Lagrangian, it follows that u(x, y) is the kernel of a pseudo-differential operator of order m. So this clarifies the remark made earlier that the order of the distribution kernel of a pseudo-differential operator agrees with the order of the pseudo-differential operator. Using (6.1.4) we can define ellipticity. We say that u ∈ I m (X, ) is elliptic if, when coordinates are chosen so that  = {(H (ξ ), ξ )}, the absolute value of either of the terms on the left side of (6.1.4) is bounded from below by |ξ |m−n/4 for large ξ . Since det φ is homogeneous of degree −(N − n) this just

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means that a(x, θ ) is bounded from below by |θ |m+n/4−N/2 when |θ | is large and (x, θ ) ∈ φ . Proof of Theorem 6.1.4 We may assume that a(x, θ ) vanishes when x is outside of a compact set and that coordinates have been chosen so that  = {(H (ξ ), ξ )}. We shall then use stationary phase to evaluate  ei[φ(x,θ)−x,ξ +H(ξ )] a(x, θ ) dθdx. (6.1.5) eiH(ξ ) uˆ (ξ ) = (2π )−(n+2N)/4 The first thing we must check is that the Hessian (with respect to the variables of integration) of the phase function is non-degenerate, that is, det φ  = 0. But this follows from the fact that the maps φ  (x, θ ) → (x, φx (x, θ )) ∈ 

and

  (x, ξ ) → ξ

are both diffeomorphisms. Since φθ = 0 on φ , this means that the map   (x, θ ) → (φx , φθ ) has surjective differential on φ and hence in , if this set is small enough. But this is just the statement that det φ  = 0. Since the stationary points, depending on the parameter ξ , are non-degenerate, we can assume, after possibly contracting , that, for every ξ near ξ0 , there is a unique stationary point. Since we may also assume that φx  = 0 in  it follows that the difference between (6.1.5) and  ei[φ(x,θ)−x,ξ +H(ξ )] β(θ/|ξ |)a(x, θ ) dθ dx (6.1.5 ) (2π )−(n+2N)/4 is rapidly decreasing if β ∈ C0∞ (Rn \0) equals one for |θ | ∈ [C−1 , C] with C sufficiently large—in particular, large enough so that β(θ/|ξ |) = 1 at a stationary point. Next, if we set λ = |ξ | and ω = ξ/|ξ |, we can rewrite (6.1.5 ) as  −(n+2N)/4 N eiλ[φ(x,θ)−x,ω+H(ω)] β(θ )a(x, λθ ) dθdx. (2π ) λ Notice that the integration is over a fixed compact set and that, at a stationary point, we have φθ = 0, x = H (ω), and hence, by Euler’s homogeneity relations, φ(x, θ ) = φθ (x, θ ), θ  = 0

and

x, ω = H (ω), ω = H(ω).

Thus one reaches the conclusion that the phase function always vanishes at the stationary point. Therefore, the stationary phase formula (1.1.20) tells us that the difference between (6.1.5 ) and πi

(2π )n/4 λ(N−n)/2 a(x, λθ )| det φ (x, θ )|−1/2 e 4 sgn φ



(6.1.6)

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is in Sm−n/4−1 , since (m + n/4 − N/2) + (N − n)/2 = m − n/4. But det φ is homogeneous of degree −(N − n) and so λ(N−n)/2 | det φ (x, θ )|−1/2 = | det φ (x, λθ )|−1/2 . Hence, (6.1.6) is the second term in (6.1.4), which gives us the first half of the theorem. To prove the converse, we use Propositions 6.1.1 and 6.1.2 to see that it suffices to consider u ∈ I m (Rn , ) having the property that v = uˆ eiH ∈ Sm−n/4 is supported in a small conic neighborhood of ξ0 . We then let (x, θ ) = ∂φ/∂x and put πi



a0 (x, θ ) = (2π )−n/4 v ◦ (x, θ )| det φ |1/2 e− 4 sgn φ ∈ Sm+(n−2N)/4 . If we then define u0 by the analog of (6.1.3) where a(x, θ ) is replaced by a0 (x, θ ), it follows that u − u0 ∈ I m−1 . Continuing, we construct uj from aj (x, θ ) ∈ Sm+(n−2N)/4−j so that u − (u0 + · · · + uj ) ∈ I m−j−1 (Rn , ). If we then  pick a ∼ aj , it then follows that (6.1.4) holds mod C∞ . Let us end this discussion by making a few miscellaneous remarks concerning the many ways that one can write Lagrangian distributions. First, if u is given by an oscillatory integral as in (6.1.3), one can add as many θ variables as one wishes and get a similar expression involving the new variables. In fact, if Q(θN+1 , . . . , θN ) is a non-degenerate quadratic form "(" θ ) = φ(θ ) + in N − N variables, one checks that the phase function φ Q(θN+1 , . . . , θN )/|θ|, defined in the region |(θN+1 , . . . , θN )| < |θ |, is also non-degenerate and parameterizes . So by Theorem 6.1.4 we can express " and a symbol a(x, " u by an oscillatory integral involving φ θ ) ∈ Sm+n/4−N /2 . A more interesting construction involves the reduction of theta variables. As a preliminary, we need an observation about the rank of the Hessian of the phase function with respect to the theta variables. Specifically, let  :   (x, ξ ) → x and φ : φ  (x, θ ) → x denote projection onto the base variable, and let κ : φ  (x, θ ) → (x, φx (x, θ )) ∈ . Then, if κ(x0 , θ0 ) = (x0 , ξ0 ), we can (x , θ ) with the rank of d (x , ξ ). In fact, since κ compare the rank of φθθ 0 0  0 0 is locally a diffeomorphism and φ is n-dimensional—both because we are assuming that φ is non-degenerate—we have the formula dim Ker dφ = n − rank d .

(6.1.7)

But if a tangent vector is in Ker dφ it must necessarily be of the form v =   2 j vj ∂/∂θj . And it must also satisfy j vj ∂ φ/∂θj ∂θk = 0, k = 1, . . . , N, since this is a necessary and sufficient condition for v to be a tangent vector to φ .

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(x , θ ), Since the dimension of such tangent vectors at (x0 , θ0 ) is N − rank φθθ 0 0 (6.1.7) implies the following:

Proposition 6.1.5 If φ is non-degenerate and (x0 , φx (x0 , θ0 )) = (x0 , ξ0 ) ∈  we have (x0 , θ0 ) = n − rank d (x0 , ξ0 ). (6.1.8) N − rank φθθ Let us now see that this result implies that we can reduce the number of theta (x , θ ). We may assume that variables if rank d is large. Let r = rank φθθ 0 0 φθ θ (x0 , θ0 ) is invertible where θ = (θ1 , . . . , θN−r ) and θ = (θN−r+1 , . . . , θN ). By the implicit function theorem, near (x0 , ξ0 ), there is a unique solution θ = g(x, θ ) to the equation φθ (x, θ , θ ) = 0. Clearly g must be smooth and θ = (θ , θ − g(x, θ )) and define homogeneous of degree one in θ . We then let " " by φ "(x, " " = 0 if and only if " θ = 0 we conclude that φ θ ) = φ(x, θ ). Since φ " θ " = 0 and φ " = 0 when " φ θ = 0. Therefore, if we set " x," θ θ θ ," "(x, " ψ(x, θ ) = φ(x, θ , g(x, θ )) = φ θ , 0), we get

" |" ψxθ =φ xθ˜ θ =0

and

ψθ θ = " θ" |θ =0 . θ θ˜ "

These conditions imply that ψ is non-degenerate. Furthermore, the first " = condition, along with Euler’s homogeneity relations and the fact that φ x" θ 0 when " θ = 0, gives that ψx = φx when θ" = 0. Consequently, ψ also parameterizes the Lagrangian  near (x0 , ξ0 ). Therefore, if u(x) is given by (6.1.3), with a(x, θ ) having small enough support, there must be a symbol b ∈ Sm+n/4−(N−r)/2 such that, modulo C∞ ,  u(x) = (2π )−[n+2(N−r)]/4 eiψ(x,θ ) b(x, θ ) dθ . (6.1.9) RN−r

Notice that the order of the symbol here has increased by r/2 from the order of the symbol a(x, θ ) in (6.1.3) to compensate for the fact that, in (6.1.9), the integration involves r fewer variables. m (X, ) then we can From this we see that if rank d ≡ r and if u ∈ Icomp write u as a finite sum of oscillatory integrals of the form (6.1.9) modulo C∞ . If d has constant rank then we say that u ∈ I m (X, ) is conormal since  is the conormal bundle of Y =  (), which must be an r-dimensional smooth submanifold of X by the constant rank theorem. A concrete example that illustrates the remarks about the reduction of theta variables concerns the Fourier integral operators eitP when the cospheres associated to the principal symbol of P, {ξ : p(x, ξ ) = 1} ⊂ Tx∗ X, have non-vanishing Gaussian curvature. Under this assumption, we saw in the proof

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of Lemma 5.1.3 that the phase functions t (x, y, ξ ) = ϕ(x, y, ξ )+tp(y, ξ ) satisfy rank ∂ 2 t /∂ξj ∂ξk ≡ n − 1 for small nonzero times t. This means that, modulo a smoothing error, for such times one can write   (eitP f )(x) =

∞

M −∞

eiψt (x,y,θ) at (x, y, θ )f (y) dθ dy

for some at (x, y, θ ) ∈ S(n−1)/2 . The symbol and phase function depend smoothly on the time parameter when t ranges over compact subintervals of [−ε, ε]\0; however, not as t → 0 since eitP |t=0 is the identity operator and hence cannot be expressed by oscillatory integrals involving one theta  variable. In the special case where P = −g one can use (4.1.21) to see that, for t > 0, ψt (x, y, θ ) must be a non-vanishing function of (x, t, y) times θ (|t|− dist (x, y)), where dist (x, y) denotes the distance with respect to the Riemannian metric g. The solution to the Cauchy problem for ∂ 2 /∂t2 − g of course can be written in a similar form, involving two oscillatory integrals both having this same phase function.

6.2 Regularity Properties Here we shall study the mapping properties of a special class of Fourier integral operators. If X and Y are C∞ manifolds, then we shall say that an integral operator F with kernel F(x, y) ∈ I m (X × Y, ) is a Fourier integral operator of order m if  ⊂ {(x, y, ξ , η) ∈ T ∗ (X × Y)\0 : ξ  = 0, η  = 0}. We shall usually write things, though, in terms of the associated canonical relation, C = {(x, ξ , y, η) : (x, y, ξ , −η) ∈ } ⊂ (T ∗ X\0) × (T ∗ Y\0),

(6.2.1)

and from now on use the notation F ∈ I m (X, Y; C). The reason that it is more natural to express things in terms of the canonical relation, rather than the Lagrangian associated to the distribution kernel, will become more apparent in the composition formula below and the formulation of various hypotheses concerning the operators. Notice that, since  is Lagrangian with respect to the symplectic from σX + σY , the minus sign in (6.2.1) implies that C must  be Lagrangian with respect to the symplectic form σX − σY = dξj ∧ dxj −  dηk ∧ dyk in (T ∗ X\0) × (T ∗ Y\0). In this section we shall study the mapping properties of Fourier integral operators whose canonical relation is locally the graph of a canonical transformation—or locally a canonical graph for short. By this we mean that if γ0 = (x0 , ξ0 , y0 , η0 ) ∈ C then there must by a symplectomorphism χ defined

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near (y0 , η0 ) so that, near γ0 , C is of the form {(x, ξ , y, η) : (x, ξ ) = χ (y, η)}.

(6.2.2)

Notice that this forces dim X = dim Y. In addition, this condition is equivalent to the condition that either of the natural projections C → T ∗ X\0 or C → T ∗ Y\0 (and hence both) are local diffeomorphisms. Clearly, if C is locally a canonical graph then the projections are local diffeomorphisms. To see the converse we notice that if, say, C → T ∗ Y\0 is a local diffeomorphism, then, near γ0 , we can use (y, η) as coordinates for C and hence the canonical relation must locally be of the form (6.2.2). The fact that χ must then be canonical is a consequence of the fact that σX − σY vanishes identically on C which forces σY = χ ∗ (σX ). Let us also, for future use, express this condition in terms of phase functions that locally parameterize C. If φ(x, y, θ ) is a non-degenerate phase function parameterizing , then, by (6.2.1), C = {(x, φx (x, y, θ ), y, −φy (x, y, θ )) : (x, y, θ ) ∈ φ }. We claim that C being locally a canonical graph is equivalent to the condition that   φxθ φxy (6.2.3) = 0 on φ . det φθy φθθ To see this, we notice that this is the Jacobian of (y, θ ) → (φx , φθ ). So if (6.2.3) holds, then we can solve the equations φθ (x, y, θ ) = 0,

ξ = φx (x, y, θ )

with respect to y, θ and hence use (x, ξ ) as local coordinates on φ . Thus, (6.2.3) is equivalent to the condition that the projection C → T ∗ X\0 is a local diffeomorphism, which establishes the claim. Having characterized the hypothesis, let us turn to the main result. Theorem 6.2.1 Let X and Y be n-dimensional C∞ manifolds and let F ∈ I m (X, Y; C), with C being locally the graph of a canonical transformation. Then 2 (X) if m ≤ 0, 2 (1) F : Lcomp (Y) → Lloc p p (2) F : Lcomp (Y) → Lloc (X) if 1 < p < ∞ and m ≤ −(n − 1)| 1p − 12 |, (3) F : Lipcomp (Y, α) → Liploc (X, α) if m ≤ −(n − 1)/2.

Furthermore, none of these results can be improved if F is elliptic and if corank dX×Y = 1 somewhere, where X×Y : C → X × Y is the natural projection operator.

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Notice that the non-degeneracy hypothesis that C be locally a canonical graph is the homogeneous version of the non-degeneracy hypothesis in the non-degenerate oscillatory integral theorem, Theorem 2.1.1. In both cases, the non-degeneracy hypothesis is equivalent to the condition that the projection from the canonical relation to T ∗ X be non-singular. Also, by the discussion at the end of the previous section, the last condition in the theorem means that sing supp F(x, y) contains a C∞ hypersurface. We shall first prove the most important part, the L2 estimate of Eskin and H¨ormander. To do this we shall need the following composition theorem. μ

m (X, Y; C ) and G ∈ I Theorem 6.2.2 Let F ∈ Icomp 1 comp (Y, Z; C2 ). Then, if C1 is locally the graph of a canonical transformation, F ◦ G ∈ I m+μ (X, Z; C) where

C = C1 ◦ C2 = {(x, ξ , z, ζ ) : (x, ξ , y, η) ∈ C1 and (y, η, z, ζ ) ∈ C2

for some (y, η) ∈ T ∗ Y\0}.

Also, if both F and G are elliptic, then so is F ◦ G. By taking adjoints (see below), one sees of course that the same result holds if we instead assume that C2 is locally a canonical graph. Remark In Chapter 4 we saw that, for small t, the canonical relation of the operator e−itP is C = {(x, t, ξ , τ , y, η) : (x, ξ ) = t (y, η), τ = −p(x, ξ )}, where t is the canonical transformation defined by flowing along the Hamilton vector field associated to p(x, ξ ) for time t. However, Theorem 6.2.2 shows that if this holds for small time then it must hold for all times since e−i(t+s)P = e−itP ◦ e−isP has canonical relation C ◦ Cs = {(x, t, ξ , τ , y, η) : (x, ξ ) = t (y, η), τ = −p(x, ξ )} ◦{(y, η, z, ζ ) : (y, η) = s (z, ζ )} = {(x, t, ξ , τ , y, η) : (x, ξ ) = t+s (y, η), τ = −p(x, ξ )}. Hence if C has the stated form for t in [−ε, ε] then it must also have this form for t in [−ε, ε] + [−ε, ε] = [−2ε, 2ε], and by iterating this one concludes that C is as above for all time. If we assume the composition theorem for the moment, it is easy to prove part (1) of Theorem 6.2.1. After perhaps breaking up the operator we 0 (X, Y; C) with C as in (6.2.2). But then F ∗ ∈ may assume that F ∈ Icomp 0 ∗ Icomp (Y, X; C ) where C ∗ = {(y, η, x, ξ ) : (x, ξ , y, η) ∈ C} = {(y, η, x, ξ ) : (x, ξ ) = χ (y, η)}.

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Hence, C ◦ C ∗ must be the trivial relation {(y, η, y, η)} and therefore Theorems 6.2.2 and 6.1.4 imply that F ∗ F must be a pseudo-differential operator of order 0. So the L2 boundedness of pseudo-differential operators of order 0 gives   |F u|2 dx =

F ∗ Fu u dy ≤ F ∗ Fu 2 u 2 ≤ C u 22 ,

proving (1). Notice that Theorem 6.2.2 implies that PF ∈ I m+μ (X, Y; C) if P is a pseudo-differential operator of order μ on X. Using this and Theorem 6.2.1 one obtains the following regularity theorem. Corollary 6.2.3 Let F ∈ I m (X, Y; C) be as in Theorem 6.2.1. Then p

p

(1) F : Lcomp (Y) → Lloc,m−αp (X), if 1 < p < ∞ and αp = (n − 1)|1/p − 1/2|, (2) F : Lipcomp (Y, α) → Liploc (X, α − α∞ ), with α∞ = (n − 1)/2. As before, all of these results are sharp if F is elliptic and corank dX×Y = 1 somewhere. Notice that the Corollary says that, compared to the L2 estimates, one in general loses (n − 1)|1/p − 1/2| derivatives in Lp and (n − 1)/2 derivatives in the pointwise sense. Proof of Theorem 6.2.2 We may assume that C1 is parameterized by a non-degenerate phase function φ(x, y, θ ) that is defined in a conic region of (X × Y) × (RN1 \0) and that C2 is parameterized by a non-degenerate phase function ϕ(y, z, σ ) defined in a conic region of (Y × Z) × (RN2 \0). It then follows that C1 ◦ C2 = {(x, φx , z, −ϕz ) : φθ (x, y, θ ) = 0, ϕσ (y, z, σ ) = 0, φy (x, y, θ ) = ϕy (y, z, σ )}.

Equivalently, if we set " = ((|θ |2 + |σ |2 )1/2 y, θ , σ ), and define the homogeneous phase function (x, z, ") = φ(x, y, θ ) + ϕ(y, z, σ ), we have

C1 ◦ C2 = {(x,  x , z, − x ) : (x, z, ") ∈  }.

Notice that  is defined in a conic region of (X × Z) × (RN \0), where N = N1 + N2 + n, with n = dim Y = dim X. We claim that  is non-degenerate. This

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will show that C1 ◦ C2 is a smooth canonical relation that is Lagrangian with respect to σX − σZ . To prove the claim we must show that the differentials d(∂/∂"j ), j = 1, . . . , N, are linearly independent on  . One can rephrase this as the requirement that the N × (N + n + dim Z) matrix ∂ 2 /∂"∂(", x, z) have full rank N here. This in turn just means that ⎞ ⎛ + ϕ φyy ϕyσ φyx ϕyz yy φyθ ⎟ ⎜ φθy φθθ 0 φθx 0 ⎠ (6.2.4) ⎝ 0 ϕσ σ 0 ϕσ z ϕσ y has full rank when φθ = 0, ϕσ = 0, and φy + ϕy = 0. But we have already seen that C1 being locally a canonical graph is equivalent to the condition that the (N1 + n) × (N1 + n) submatrix   φyx φyθ φθθ φθx be non-singular when φθ = 0. In addition, the fact that ϕ is non-degenerate forces (ϕσ y ϕσ σ ϕσ z ) to have full rank N2 . By combining these two facts, and noting the form of the matrix, one sees that (6.2.4) must have rank N1 + n + N2 = N, giving us the claim. It is now a simple matter to finish the proof. Using a partition of unity, we may assume that a1 (x, y, θ ) ∈ Sm−N1 /2+n/2 and a2 (y, z, σ ) ∈ Sμ−N2 /2+(n+nZ )/4 are supported in small compactly based cones in (Rn × Rn ) × (RN1 \0) and (Rn × RnZ ) × (RN2 \0), respectively. Here nZ = dim Z. We must show that if  F(x, y) = eiφ(x,y,θ) a1 (x, y, θ ) dθ, RN1  G(y, z) = eiϕ(y,z,σ ) a2 (y, z, σ ) dσ , RN2

then

 (F ◦ G)(x, z) =

F(x, y)G(y, z) dy   = ei[φ(x,y,θ)+ϕ(y,z,σ )] a1 (x, y, θ )a2 (y, z, σ ) dθ dσ dy

defines an element of I m (X, Z; C). If the wave front sets of y → F(x, y) and y → G(y, z) are always disjoint, the inner product defines a C0∞ kernel and the result trivially holds. If not—that is, if C  = ∅—and if we assume in addition,

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as we may, that the aj have small conic supports it is clear from Theorem 0.4.5 that the inner product is well defined. To show that (F ◦ G)(x, z) has the desired form we first notice that, if β ∈ C0∞ (R\0) equals one for s ∈ [C−1 , C] with C sufficiently large, and if we set a(x, z; y, θ, σ ) = β(|θ |/|σ |)a1 (x, y, θ )a2 (y, z, σ ), then the difference between (F ◦ G)(x, z) and   ei[φ(x,y,θ)+ϕ(y,z,σ )] a(x, z; y, θ , σ ) dydθ dσ

(6.2.5)

is C∞ . To prove this one just notices that, on the support of the integrand, if φy + ϕy = 0 then |θ | and |σ | must be comparable. Hence, if C above is large enough |∇y (φ + ϕ)| must be bounded below by a multiple of |θ | + |σ | on the support of a − a1 a2 which leads to the claim. To finish, we change variables and write (6.2.5) as  ei(x,z;") a(x, z; ")(|θ |2 + |σ |2 )−n/2 d". (6.2.5 ) RN

Note that since a1 and a2 are symbols of order m − N1 /2 + n/2 and μ − N2 /2 + (n + nZ )/4, respectively, and since their product is therefore a symbol of order m + μ − (N1 + N2 )/2 + 3n/4 + nZ /4 in a conic region where |θ | and |σ | are comparable, we conclude that the symbol in (6.2.5 ) is of order m + μ − N/2 + (n + nZ )/4. Hence (6.2.5 ) defines an element of I m+μ (X, Z; C), by Theorem 6.1.4. We now turn to the proof of parts (2) and (3) of Theorem 6.2.1. It will be convenient to represent the Fourier integrals there in a manner which is close to that of pseudo-differential operators. To do this we shall use the following result. m (Rn , Rn ; C) where C is a canonical graph. Proposition 6.2.4 Let F ∈ Icomp Then, modulo C∞ , one can write the kernel of F as the sum of finitely many terms, each of which in appropriate local coordinate systems is of the form  −n ei[ϕ(x,η)−y,η] b(x, η) dη. (6.2.6) (2π ) Rn

Each term is of course in I m (Rn , Rn ; C), and the symbols are compactly supported in x and are in Sm . Moreover, on supp b, det(∂ 2 ϕ/∂x∂η)  = 0.

(6.2.7)

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Proof The first step is to see that local coordinates can be chosen so that C is given by a generating function. Using Lemma 6.1.3 and the fact that x0 = {(y, η) : (x0 , ξ , y, η) ∈ C for some ξ } ⊂ T ∗ Y\0 must be Lagrangian, we can adapt the proof of Proposition 6.1.2 to see that local y coordinates can be chosen so that C  (x, ξ , y, η) → (x, η) is a diffeomorphism, if, as we may, we assume that C is small enough. This means that ξ and y are functions of (x, η) on C. To use this, we notice that since the canonical one form vanishes on C we have 0 = ξ , dx − η, dy = ξ , dx − d(y, η) + y, dη. Therefore, if we set S(x, η) = y, η, we must have ∂S/∂x = ξ , ∂S/∂η = y, i.e., C = {(x, ∂S/∂x, ∂S/∂η, η)}. By Theorem 6.1.4, this means that, if we set ϕ(x, η) = S(x, −η), and, if WF(F(x, y)) is sufficiently small, we can write, modulo C∞ ,  ei[ϕ(x,η)−y,η] a(x, y, η) dη (6.2.8) F(x, y) = (2π )−n Rn

for some symbol a ∈ Sm . But Theorem 6.1.4 also implies that, if we set a0 (x, η) = a(x, ϕη (x, η), η) ∈ Sm , then  (2π )−n ei[ϕ(x,η)−y,η] (a(x, y, η) − a0 (x, η)) dη ∈ I m−1 . Continuing, we can choose aj ∈ Sm−j so that   aj (x, η)) dη ∈ I m−N . (2π )−n ei[ϕ(x,η)−y,η] (a(x, y, η) − j 2, when proving estimates involving Lp for p < 2, it turns out to be more convenient to use the adjoint of the microlocal representation (6.2.6) of the kernels. However, by the above discussion, if we

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apply Proposition 6.2.4 to F ∗ and then take adjoints, we see that it suffices to show that if b(y, ξ ) ∈ Sm vanishes for y outside of a compact set of Rn and if ϕ(y, ξ ) satisfies (6.2.7), then     i[x,ξ −ϕ(y,ξ )]   e b(y, ξ )f (y) dξ dy   p n L (R )

≤ C f Lp (Rn ) ,

m ≤ −(n − 1)|1/p − 1/2|.

(6.2.9)

We may assume that b vanishes for |ξ | < 10. Then if β ∈ C0∞ ((1/2, 2))  −k satisfies ∞ −∞ β(2 s) = 1, s > 0, we set  Fλ f (x) = ei[x,ξ −ϕ(y,ξ )] bλ (y, ξ )f (y) dξ dy, where bλ (y, ξ ) = β(|ξ |/λ)b(y, ξ ). If we call F the operator in (6.2.9) then we of course have  F= F2 k .

(6.2.10)

k≥1

To prove the Lipschitz estimates (3), we shall use the following. Proposition 6.2.5 Let b ∈ Sm be as above. Then for λ > 1 we have the uniform bounds (6.2.11)

Fλ f L1 (Rn ) ≤ Cλm+(n−1)/2 f L1 (Rn ) . Furthermore, the constant C remains bounded if b has fixed support and belongs to a bounded subset of Sm . Before proving (6.2.11), let us see how it implies the Lipschitz estimates. In fact, if we define the Littlewood–Paley operators Pλ by (Pλ f )∧ (ξ ) = β(|ξ |/λ)fˆ (ξ ), and recall the definition of Lip (α) from Section 0.2, we conclude that part (3) follows by duality from the uniform estimates

Pλ1 FPλ2 f L1 (Rn ) ≤ Cλα1 λ−α 2 f L1 (Rn ) ,

if m = −(n − 1)/2.

(6.2.12)

But using the fact that, on supp b, c|ξ | ≤ |ϕy (y, ξ )| ≤ c−1 |ξ |, one can argue as in the proof of Lemma 2.4.4 to see that there is a uniform constant C0 such −N −1 that Pλ1 FPλ2 (L1 ,L1 ) ≤ CN λ−N 1 λ2 for any N unless C0 ≤ λ1 /λ2 ≤ C0 . The

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operators Pλ are uniformly bounded on L1 and so we now see that (6.2.12) must be a consequence of

Pλ Ff 1 ≤ C f 1 ,

m = −(n − 1)/2.

But if we recall the form of F, we see that  Pλ F = Pλ

(6.2.12 )

F2 k ,

1/4≤λ/2k ≤4

and, therefore, (6.2.11) implies (6.2.12 ) and hence part (3) of the theorem. Before moving on, let us notice that (6.2.11) can be used to prove everything else except for the endpoint estimate in (6.2.9). In fact, the L2 estimate for Fourier integral operators shows that the dyadic operators Fλ are bounded on L2 with norm O(λm ). So, by applying the M. Riesz interpolation theorem, we conclude that (6.2.11) yields the dyadic estimates

Fλ f Lp (Rn ) ≤ Cλm+(n−1)|1/p−1/2| f Lp (Rn ) . Consequently, for m strictly smaller than −(n − 1)|1/p − 1/2|, we get (6.2.9) by summing a geometric series. To prove the endpoint estimate, we shall need to know more precise information about the kernels of the operators Fλ than is given in Proposition 6.2.5. In addition, we shall need to use some basic facts from Hardy space theory. Using the Hardy space H 1 , we shall be able to prove the sharp estimate (6.2.9) by interpolating between the L2 estimate and an estimate involving L1 . This of course will be reminiscent of the proof of the multiplier theorem in Section 0.2. First, we define the Hardy space H 1 (Rn ). It consists of all L1 functions which have the additional property that all of their Riesz transforms, Rj f , also belong to L1 . Here Rj is defined by (Rj f )∧ (ξ ) = fˆ (ξ )ξj /|ξ |,

j = 1, . . . , n.

So the norm is given by

f H 1 = f L1 (Rn ) +

n 

Rj f L1 (Rn ) .

j=1

Notice that, since (Rj f )∧ (ξ ) must be continuous if it belongs to L1 , it follows that fˆ (0) must vanish—in other words, if f ∈ H 1 then f must have mean value zero. There is a decomposition f ∈ H 1 which is similar to the Calder´on–Zygmund decomposition of f ∈ L1 . However, in the present setting, the decomposition

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involves elements which possess the salient features of the good and bad functions—namely, they are bounded, localized, and have mean value zero. Specifically, given f ∈ H 1 , one can write  αQ aQ , (6.2.13) f= Q

where the sum ranges over all the dyadic cubes coming from a fixed lattice in Rn ,  |αQ | ≈ f H 1 , (6.2.14) Q

and the individual atoms satisfy  supp aQ ⊂ Q,

aQ dx = 0,

aQ L∞ ≤ |Q|−1 .

(6.2.15) (6.2.16)

This is called the atomic decomposition of H 1 and a proof can be found, for instance, in Coifman and Weiss [1]. Using (6.2.13)–(6.2.16) and more detailed information about the kernels of the dyadic operators Fλ , we shall prove the following substitute for an L1 → L1 estimate. Proposition 6.2.6 If m = −(n − 1)/2 and if F is as above

Ff L1 (Rn ) ≤ C f H 1 (Rn ) .

(6.2.17)

Furthermore, the constants remain bounded if a has fixed support and belongs to a bounded subset S−(n−1)/2 . Momentarily, we shall see that this implies (6.2.9) if we apply the following interpolation lemma which, for later use, we state in more generality than is needed now. Lemma 6.2.7 Suppose that when 0 ≤ Re(z) ≤ 1, Tz is a family of linear operators on Rn that has the property that, whenever f and g are fixed simple functions that vanish outside of a set of finite measure, the map  Tz f g dx z→ Rn

is bounded and analytic in the strip 0 < Re(z) < 1 and continuous in the closure. Then:

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(1) If, for Re(z) = j = 0, 1, Tz f qj ≤ Cj f pj , with 1 ≤ pj , qj ≤ ∞, and if, for 0 < t < 1 we define pt and qt by 1/pt = (1 − t)/p0 + t/p1 and 1/qt = (1 − t)/q0 + t/q1 , it follows that

Tt f qt ≤ C01−t C1t f pt .

(6.2.18)

(2) If, instead, we have Tz f L1 ≤ C0 f H 1 when Re(z) = 0, then (6.2.18) holds with pt and qt satisfying 1/qt = (1 − t) + t/q1 and 1/pt = (1 − t) + t/p1 , if Tz f q1 ≤ C1 f p1 for Re(z) = 1. Part (1) of the interpolation theorem is due to Stein and follows from modifying the argument that was used to prove the M. Riesz interpolation theorem, Theorem 0.1.13.  In fact, if one changes the definition of F(z) in the proof to be F(z) = Tz fz gz dx, and then repeats the arguments, one gets (6.2.18). The complex interpolation theorem for H 1 is due to Fefferman and Stein [1]. The proof uses an argument that reduces (2) to (1) via the duality of H 1 and BMO and the characterization of Lp in terms of the sharp function of Fefferman and Stein. To see how the interpolation lemma implies (6.2.9), we set αp = (n − 1)|1/p − 1/2| and then put  2 ei[x,ξ −ϕ(y,ξ )] b(y, ξ )|ξ |αp −(n−1)(1−z)/2 f (y) dξ dy. F z f (x) = e(z−2+2/p) Notice that when Re(z) = t the symbols involved belong to a bounded subset of S−(n−1)(1−t)/2 . So we have

F z f L1 ≤ C f H 1 ,

Re(z) = 0,

F f L2 ≤ C f L2 ,

Re(z) = 1,

z

which gives the sharp estimate (6.2.9) as F t = F, t = 2(1 − 1/p). We now turn to the proof of Proposition 6.2.6. In the course of the proof we shall prove an estimate that implies Proposition 6.2.5. To prove the H 1 estimate (6.2.17), we see by (6.2.13) and (6.2.14) that it suffices to check that for an individual atom

FaQ L1 ≤ C.

(6.2.17 )

It is easy to see that this inequality holds when Q is a cube of side-length ≥ 1. First, since the symbol b(y, ξ ) vanishes for y outside of a fixed compact set, one checks that the kernel of F satisfies |F (x, y)| ≤ CN |x|−N for any N when |x| is larger than some fixed constant depending on b and ϕ. Hence, to prove (6.2.17 ), it suffices to check that the inequality holds when the L1 norm

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is taken over a fixed ball B. But if we use the L2 boundedness of F along with (6.2.16) we see that

FaQ L1 (B) ≤ |B|n/2 FaQ L2 (Rn ) ≤ C aQ L2 (Q) ≤ C|Q|−1/2 . Since the right side is bounded when Q is large, we may assume from now on that Q has side-length R = 2−j ≤ 1. We shall now construct a small exceptional set NQ on which we can obtain favorable estimates for the L1 norm of FaQ using the Schwarz inequality and the L2 boundedness of Fourier integral operators. To do this, we first choose for λ = 2k unit vectors ξλν , ν = 1, . . . , N(λ), such that |ξλν − ξλν | ≥ c0 λ−1/2 , ν  = ν , for some positive constant c0 and such that balls of radius λ−1/2 centered at ξλν cover Sn−1 . We observe that N(λ) ≈ λ(n−1)/2 .

(6.2.19)

For a fixed y ∈ Q, we let Iλ,ν = {x : |(x − ϕξ (y, ξλν )), ξλν | ≤ λ−1 y

and

ν −1/2 |⊥ }, λ,ν (x − ϕξ (y, ξλ ))| ≤ λ

(6.2.20)

ν where ⊥ λ,ν denotes the projection onto the subspace orthogonal to ξλ . Thus y Iλ,ν is a rectangle centered at ϕξ (y, ξλν ) with n − 1 sides of length λ−1/2 and one of length λ−1 . The thinnest side points in the direction ξλν . We now define y

Nλ = and, if the side-length of Q is R, NQ =

N(λ)  ν=1



y

Iλ,ν ,

y

NR−1 .

(6.2.21)

y∈Q

Since |Iλ,ν | = λ−(n+1)/2 it follows immediately that y

|NQ | ≤ CR(n+1)/2 R−(n−1)/2 = CR.

(6.2.22)

Note that, if ξ ∈ Sn−1 and |ξ − ξλν | ≤ λ−1/2 , then ϕξ (y, ξ ) belongs to a fixed y dilate of Iλ,ν (depending on ϕ). Hence the singular support of x → F(x, y) is basically contained in NQ , since the latter is a subset of y = {x : x = ϕξ (y, ξ )

for some (y, ξ ) ∈ supp b}.

In fact, if the rank of the differential of the projection C → Rn ×Rn is maximal, NQ is basically a tubular neighborhood of width R around y . If the rank is

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not maximal, the exceptional set may be larger than a tubular neighborhood of this size. Let us first estimate FaQ on the exceptional set. Since F is of order −(n − 1)/2 it follows that F(I − )(n−1)/4 is bounded on L2 and hence

FaQ L1 (NQ ) ≤ CR1/2 FaQ 2 ≤ R1/2 (I − )−(n−1)/4 aQ 2 . But, if we let pn = 2n/(2n − 1) so that n(1/pn − 1/2) − (n − 1)/2, the Hardy–Littlewood–Sobolev inequality gives

(I − )−(n−1)/4 aQ 2 ≤ C aQ pn ≤ R−n+n/pn = CR−1/2 . Combining these two inequalities leads to the favorable estimate

FaQ L1 (NQ ) ≤ C. Now we shall prove the inequality off of the exceptional set:

FaQ L1 (Rn \NQ ) ≤ C.

(6.2.23)

If Fλ are the kernels of the dyadic pieces of F we claim that this would be a consequence of the following estimates valid for y, y ∈ Q:  |Fλ (x, y)| dx ≤ C(Rλ)−1 if λ > R−1 , n R \NQ (6.2.24) λ ≤ R−1 . Rn |Fλ (x, y) − Fλ (x, y )| dx ≤ CRλ if Clearly these two estimates yield (6.2.23). In fact, the first one gives

F2k aQ L1 (Rn \NQ ) ≤ C(R2k )−1

if

R2k > 1.

While, if we use the mean value zero property (6.2.15), we get that if y ∈ Q  F2k aQ (x) = (F2k (x, y) − F2k (x, y ))aQ (y) dy. and hence the second estimate implies

F2k aQ L1 (Rn \NQ ) ≤ CR2k

if

R2k < 1.

Recalling (6.2.10), one sees that (6.2.23) follows by combining these two estimates and summing a geometric series. To prove (6.2.24) it is necessary to introduce homogeneous partitions of unity of Rn \0 that depend on the scale λ. Specifically, we choose C∞ functions  χλν , ν = 1, . . . , N(λ), satisfying ν χλν = 1 and having the following additional properties. First, the χλν are to be homogeneous of degree zero and satisfy the uniform estimates |Dγ χλν (ξ )| ≤ Cγ λ|γ |/2

∀λ

when

|ξ | = 1.

(6.2.25)

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Second, χλν (ξλν )  = 0 and the χλν are to have the natural support properties associated to (6.2.25), that is, χλν (ξ ) = 0

if |ξ | = 1 and

|ξ − ξλν | ≥ Cλ−1/2 .

These are just the n-dimensional versions of the partitions of unity used to prove the maximal theorems in Section 2.4. We now define the new kernels Fλν (x, y), 1 ≤ ν ≤ N(λ), by Fλν (x, y) =



ei[x,ξ −ϕ(y,ξ )] bνλ (y, ξ ) dξ ,

where

bνλ (y, ξ ) = χλν (ξ )bλ (y, ξ ).

(6.2.26) The idea behind this decomposition is that the χλν have the largest possible angular support so that ξ → ϕ behaves like a linear function in the appropriate scale on supp bνλ . By Euler’s homogeneity relations, the closest linear approximation of supp χλν  ξ → ϕ(y, ξ ) should be ϕξ (y, ξλν ), ξ , and, in fact, if we let rλν (y, ξ ) = ϕ(y, ξ ) − ϕξ (y, ξλν ), ξ  denote the difference, we claim that the following estimates are valid in supp bνλ if N ≥ 1: |(∇ξ , ξλν )N rλν (y, ξ )| ≤ CN λ−1 |ξ |1−N ,

|Dαξ rλν (y, ξ )| ≤ CN min{λ−1/2 , |ξ |1−N },

(6.2.27) |α| = N.

(6.2.28)

Thus, by considering the case N = 1, one sees that this remainder term is much better behaved in the “radial direction” ξλν , as opposed to the “angular n directions” ⊥ λ,ν (R ). To prove (6.2.27), it suffices, by homogeneity, to show that the same estimate holds when ξ ∈ Sn−1 ∩ supp χλν . But when ξλν = ξ , Euler’s formula gives rλν (y, ξλν ) = 0, or, equivalently, ∇ξ rλν (y, ξλν ) = 0. But ∇ξ rλν is homogeneous of degree zero and hence (∇ξ , ξλν )N ∇ξ rλν (y, ξλν ) = 0, which leads to the desired estimate (∇ξ , ξλν )N rλν (y, ξ ) = O(λ−1 ) since the first two terms in the Taylor expansion around ξλν must vanish and dist (ξ , ξλν ) ≤ Cλ−1/2 for ξ ∈ Sn−1 ∩ supp χλν . To prove (6.2.28), we notice that the estimate trivially holds for N > 1 as |ξ |1−N ≤ Cλ−1/2 for ξ ∈ supp bνλ . So (6.2.28) would follow from the estimate ∇ξ rλν (y, ξ ) = ϕξ (y, ξ )−ϕξ (y, ξλν ) = O(λ−1/2 ). Since this term is homogeneous of degree zero, we need only show that the estimate holds for ξ ∈ Sn−1 ∩ supp χλν , which is apparent.

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We shall use (6.2.27)–(6.2.28) and an integration by parts argument to estimate the kernels Fλν (x, y). After performing a rotation, we may assume that ξλν = (1, 0, . . . , 0),

⊥ λ,ν (ξ ) = (0, ξ2 , . . . , ξn ) = (0, ξ ).

From the integration by parts, we split ξ = (ξ1 , ξ ) and define the self-adjoint operator Lλν = (I − λ2 ∂ 2 /∂ξ12 )(I − λ∇ξ , ∇ξ ). Our assumptions on the symbol together with (6.2.25) and (6.2.26) imply that (6.2.29) |(Lλν )N bνλ (y, ξ )| ≤ CN λ−(n−1)/2 . Furthermore, (6.2.27) and (6.2.28) imply that the same estimate holds if we replace bνλ by ν " bνλ (y, ξ ) = eirλ (y,ξ ) bνλ (y, ξ ). So we integrate by parts and obtain  ν ν ν bνλ (y, ξ ) dξ , Fλ (x, y) = HN,λ (x, y) ei(x−ϕξ (y,ξλ )),ξ  (Lλν )N " where the weight factor is ν (x, y) HN,λ −N  −N  1 + |λ1/2 (x − ϕξ (y, ξλν )) |2 . = 1 + |λ(x − ϕξ (y, ξλν ))1 |2

Since the symbol of the last integrand is supported inside a region of volume O(λ(n+1)/2 ) and since it satisfies the estimates in (6.2.29), we conclude that ν |Fλν (x, y)| ≤ CN λHN,λ (x, y).

Consequently, we get, for fixed y,  |Fλν (x, y)| dx ≤ Cλ−(n−1)/2 Rn

(6.2.30)

(6.2.31)

uniformly in ν, which implies Proposition 6.2.5 in view of (6.2.19). To prove the first estimate in (6.2.24), we notice that, given N and y ∈ Q,  ν |λHN,λ (x, y)| dx ≤ CN λ−(n−1)/2 (Rλ)−N , λ > R−1 . (6.2.32) Rn \NQ

This is because if λ is large compared to R−1 and if N is larger than N , then ν (x, y) is O((Rλ)−N ) times H ν HN,λ N−N ,λ (x, y) in the complement of NQ . Clearly (6.2.32) leads to the first estimate by summing over ν.

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The second estimate in (6.2.24) follows immediately from  |∇y Fλν (x, y)| dx ≤ Cλ · λ−(n−1)/2 . Rn

But this follows from the proof of (6.2.31) as Dαy Fλν (x, y) behaves like λFλν (x, y) when |α| = 1. This finishes the proof of the positive results in Theorem 6.2.1. Sharpness of Results Let us now see that the estimates in Theorem 6.2.1 cannot be improved for elliptic F ∈ I m (X, Y; C) if dX×Y has full rank somewhere, where X×Y : C → X × Y is the natural projection operator. Note that if it does have full rank somewhere then the singular support of the kernel is as large as it can be in the sense that it must contain a piece of a C∞ hypersurface. One sees from the bounds for |NQ | in the proof how this enters into the Lp estimates. Returning to the issue at hand, we may assume that C is the graph of a canonical transformation and that x0 belongs to the projection onto X. It then follows that x0 = {(y, η) : (x0 , ξ , y, η) ∈ C

some ξ } ⊂ T ∗ Y\0

must be Lagrangian. Moreover, if corank dX×Y ≡ 1, then x0 is the conormal bundle of a smooth hypersurface Sx0 ⊂ Y. By choosing appropriate local coordinates we may always assume that X = Y = Rn , x0 = 0, Sx0 ⊂ Sn−1 , and x0 = {(η/|η|, η)}. If we then let  ei[y,ξ +|ξ |] (1 + |ξ |2 )−μ/2 dξ , fμ (y) = (2π )−n Rn

it follows from the composition theorem for Fourier integrals—taking Z to be the empty set—that Ffμ (x) is an elliptic Lagrangian distribution of order m − μ + n/4 whose wave front set is contained in the conormal bundle of the origin—that is, {(0, ξ )}. In other words, in some nonempty open cone based at the origin, there must be a c > 0 such that as x → 0, |F fμ (x)| > c|x|μ−m−n . Thus, we have p / Lloc if μ − m − n ≤ −n/p. (6.2.33) Ffμ ∈ On the other hand, using stationary phase (cf. Lemma 2.3.3) one sees that for y near Sn−1 | fμ (y)| ≈ (dist(y, Sn−1 ))μ−(n+1)/2 ,

μ < (n + 1)/2,

meaning that fμ ∈ Lp ⇐⇒ μ > (n + 1)/2 − 1/p.

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Substituting this into (6.2.33) shows that F cannot be bounded on Lp for p ≥ 2 if (m + n + 1/p − (n + 1)/2) > n/p, that is, −(n − 1)(1/2 − 1/p) < m. This shows that the estimates in Theorem 6.2.1 are sharp for p ≥ 2. By duality, the same holds for p < 2 and one can adapt the above arguments to see that the Lipschitz estimates are also sharp. Remark If X = Y is a compact C∞ manifold of dimension n and if P ∈ cl1 is elliptic then, for a given t, e−itP ∈ I 0 (X, X; Ct ) where Ct is the canonical relation described in the remark after Theorem 6.2.2. Since Ct is a canonical graph we conclude that

e−itP f Lp (X) ≤ C f Lαp (X), p

αp = (n − 1)|1/p − 1/2|,

e−itP f Lip(α−(n−1)/2) ≤ C f Lip(α) ,

(6.2.34)

where C remains bounded for t in any compact time interval. We claim that, for all but a discrete set of times t, these estimates cannot be improved. In view of the above discussion, this amounts to showing that the differential of Ct → X × Y must have full rank somewhere unless t belongs to a discrete exceptional set. To see that this is the case, we note that the condition τ = −p(x, ξ ) in the full canonical relation of e−itP means that if x, ξ  − φ(y, t, ξ ) is a (local) phase function for the operator, then φt = p(φξ (y, t, ξ ), ξ ) = p(x, ξ ). Let us suppose that dX×Y does not have full rank for a given time t0 . We shall then see how these facts imply that for all t near t0 the rank of Ct → X × Y has to be maximal somewhere. To show this, we first note that, given x0 , we can always choose ξ0 ∈ x0 = {ξ ∈ Tx∗0 X\0 : p(x0 , ξ ) = 1} so that this cosphere has all n − 1 principal curvatures positive at ξ0 . If coordinates are chosen so that ξ0 = (0, . . . , 0, 1), this is equivalent to the statement that (∂ 2 p/∂ξj ∂ξk )1≤j,k≤n−1 is positive definite at (x0 , ξ0 ). If y0 is then fixed so that (x0 , ξ0 , y0 , φy (y0 , t0 , ξ0 )) ∈ Ct0 , let n + r denote the rank of the differential of the projection at this point. In view of Proposition 6.1.5, after perhaps performing a rotation in the first n − 1 variables, we may assume that φξ ξ (y0 , t0 , ξ0 ) is invertible if ξ = (ξ , ξ ), ξ = (ξ1 , . . . , ξr ), and φξ j ξk (y0 , t0 , ξ0 ) = 0 if either j or k is > r. (In the case where r = 0 one would modify this in the obvious way taking ξ = ξ .) To make use of this, we notice that at (y0 , t, ξ0 ) we must have   φξ ξ + O(t − t0 ) O(t − t0 ) . (6.2.35) φξ ξ = O(t − t0 ) (t − t0 )p ξ ξ + O((t − t0 )2 )

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But this matrix must have the largest possible rank, n − 1, for t close to t0 , because the (n − r) × (n − r) Hessian of the homogeneous function ξ → p(x0 , 0, ξ ) must have largest rank n − r − 1, due to the fact that {ξ : (0, ξ ) ∈ x0 } has non-vanishing Gaussian curvature. One reaches this conclusion about the rank of φξ ξ after noting that

At Bt det ≈ tm−r , t small, Ct Dt if At = A + O(t), with A being a non-singular r × r matrix, Bt , Ct = O(t) and if Dt = tD + O(t2 ) with D being an (m − r) × (m − r) non-singular matrix. Since Proposition 6.1.5 says that the rank of the differential of the projection of Ct at γ0 = (φξ (y0 , t, ξ0 ), ξ0 , y0 , φy (y0 , t, ξ0 )) ∈ Ct is n plus the rank of φξ ξ (y0 , t, ξ0 ), we conclude that this differential must have maximal rank 2n − 1 at γ0 , which finishes the proof. Similar remarks apply to the regularity properties of solutions to strictly  hyperbolic differential equations. Specifically, let L(x, t, Dx,t ) = Dtm + m j=1 Pj m−j

(x, t, Dx )Dt be a strictly hyperbolic differential operator of order m. Then if {f }m−1 j=0 are the data for the Cauchy problem 0 Lu = 0, j ∂t u|t=0 = fj , we have, for instance, that the solution satisfies

u

Lp (X)

≤C

m−1  j=0

fj Lp

αp −j

(X).

4 Moreover, if m j=1 (r − λj (x, t, ξ )) is a factorization of the principal symbol of L and if, for every t, at least one of the roots λj is a nonzero function of T ∗ X\0—that is, elliptic—this result cannot be improved.

6.3 Spherical Maximal Theorems: Take 1 In this section we shall present some maximal theorems that are related to Stein’s spherical maximal theorem. The latter is the higher-dimensional version of the circular maximal theorem proved at the end of Section 2.4. It says that, if n ≥ 3 and p > n/(n − 1), p 1/p     sup  f (x − ty) dσ (y) dx ≤ Cp f Lp (Rn ) , f ∈ S. (6.3.1) Rn t>0

Sn−1

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Notice that the spherical means operators that are involved are Fourier integral operators of order −(n − 1)/2. This is because their distribution kernels are t−n δ0 (1 − |x − y|/t), where δ0 is the Dirac delta distribution. Consequently, we can write the spherical mean operator corresponding to the dilation t as   ∞ −1 t−n eiθ[1−|x−y|/t] f (y) dθ dy. (2π ) Rn −∞

Since there is just one theta variable, the order must be 1/2 − 2n/4 = −(n − 1)/2 as claimed. Notice also that the phase function satisfies (6.2.3) and hence the spherical means operators are a smooth family of Fourier integral operators of order −(n − 1)/2, each of which is locally a canonical graph. In this section we shall always deal with Fourier integral operators Ft ∈ m Icomp (X, Y; Ct ), t ∈ I, where X and Y are C∞ manifolds of a common dimension n and I ⊂ R is an interval. We say that {Ft } is a smooth family of operators in m if there are finitely many non-degenerate phase functions φt,j (x, y, θ ) and Icomp symbols at,j (x, y, θ ) so that, for t ∈ I, we can write   eiφt,j (x,y,θ) at,j (x, y, θ )f (y) dθ dy. Ft f (x) = j

Y R

Nj

We require that both the φt,j and the at,j be smooth functions of t with values in S1 (j ) and Sm−Nj /2+n/2 (j ), respectively, where j is an open conic subset of (X × Y) × (RNj \0). In addition, we require that the symbols be supported in j and vanish for (x, y) not belonging to a fixed compact subset of X × Y. m , by which we Usually, we shall deal with smooth bounded families in Icomp mean that, in addition to the above, we have the following uniform bounds for (t, x, y, θ ) ∈ I × j : γ

|Dx,y,t Dαθ φt,j (x, y, θ )| ≤ Cαγ (1 + |θ|)−1−|α| , γ |Dx,y,t Dαθ at,j (x, y, θ )| ≤ Cαγ (1 + |θ|)m−(Nj −n)/2−|α| .

(6.3.2)

We can now state the main result of this section. Later on we shall see that it can be used to recover Stein’s theorem (6.3.1). m (X, Y; C ), t ∈ [1, 2], be a smooth family of Theorem 6.3.1 Let Ft ∈ Icomp t m . Assume Fourier integral operators that belongs to a bounded subset of Icomp also that, for every t ∈ [1, 2], Ct is locally a canonical graph. Then if αp = (n − 1)|1/p − 1/2|, we have for p > 1 1/p 

sup |Ft f (x)|p dx

≤ C f Lp ,

m < −αp − 1/p.

(6.3.3)

t∈[1,2]

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6.3 Spherical Maximal Theorems: Take 1

189

In particular, if n ≥ 3 and p > n/(n − 1), the maximal inequality holds if m = −(n − 1)/2. The proof involves a straightforward application of Lemma 2.4.2. We fix  β ∈ C0∞ ((1/2, 2)) satisfying β(2−k s) = 1, s > 0, and define for k = 1, 2, . . .   eiφt,j (x,y,θ) at,j (x, y, θ )β(|θ |/2k )f (y) dθdy. Fk,t f (x) = j

 Then, if we set F0,t = Ft − ∞ k=1 Fk,t , it follows that F0,t has a bounded compactly supported kernel and hence the maximal operator associated to it is trivially bounded on all Lp spaces. On account of this, (6.3.3) would follow from showing that for k = 1, 2, . . .  1/p sup |Fk,t f (x)|p dx

≤ C2k(m+αp +1/p) f Lp .

(6.3.3 )

t∈[1,2]

But if we apply Lemma 2.4.2, we can dominate the pth power of the left side by  |Fk,1 f (x)|p dx X

+

  2 1

1/p    1/p p 2  d  Fk,t f (x) dxdt |Fk,t f (x)| dxdt .   X X dt 1 p

Theorem 6.2.1 implies that the first term satisfies the desired bounds and also that

1/p   p/p p |Fk,t f (x)| dx ≤ C 2k(m+αp ) f p X

 p−1 = C 2k(m+αp ) f p .

m , we can also Since (6.3.2) implies that 2−k dtd Fk,t is a bounded family in Icomp estimate the last factor: p 1/p    d  Fk,t f (x) dx ≤ C2k(m+αp +1) f p .   dt X

If we integrate in t and combine these two estimates, we get (6.3.3 ). The last part of the theorem just follows from the fact that, for n ≥ 3, αp + 1/p is smaller than (n − 1)/2 precisely when p > n/(n − 1). Notice that when n = 2 there are no exponents for which this is true. In fact, for p < 2, αp + 1/p is larger than 1/2, while for p ≥ 2, αp + 1/p ≡ 1/2. This explains why maximal theorems like the circular maximal theorem are harder to prove than

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their higher-dimensional counterpart. Momentarily, we shall see that the last statement in Theorem 6.3.1 cannot hold in this level of generality when n = 2. An additional condition is needed that takes into account the t dependence of the canonical relations Ct . To make this more precise, let us state a corollary of Theorem 6.3.1. This deals with averaging over hypersurfaces given by Sx,t = {y ∈ Rn : t (x, y) = 0}. Here, the defining function is assumed to be a smooth function of (t, x, y) ∈ [1, 2] × Rn × Rn . We also assume that both ∇x t and ∇y t never vanish, and, moreover, that the Monge–Ampere matrix associated to t is non-singular:   0 ∂t /∂y det ∂t  = 0. (6.3.4) ∂ 2 t ∂x

∂x∂y

If this is the case, we say that the family of surfaces satisfies the rotational curvature condition of Phong and Stein, and we have the following: Corollary 6.3.2 Let Sx,t be as above and let dσx,t denote Lebesgue measure on the hypersurface. Fix η(x, y) ∈ C0∞ (Rn × Rn ) and (for f ∈ S) set  At f (x) = η(x, y)f (y) dσx,t (y). Sx,t

Then, if n ≥ 3, it follows that

sup |At f (x)| Lp (Rn ) ≤ Cp f Lp (Rn ) ,

if p > n/(n − 1).

(6.3.5)

t∈[1,2]

To see that this follows from the last part of Theorem 6.3.1, we write   ∞ −1 eiθt (x,y) η0 (x, y)f (y) dθ dy, At f (x) = (2π ) −∞

−(n−1)/2

where η0 is a C∞ function times η. This is a bounded family in Icomp . By (6.2.3), each operator is locally a canonical graph since the Monge–Ampere condition is satisfied, and hence (6.3.5) is just a special case of Theorem 6.3.1. There are two extreme cases that should be pointed out. One is when Sx,t = {x} + t · S, where S is a fixed hypersurface. In this case the operators are (essentially) translation-invariant and (6.3.4) is satisfied if and only if the hypersurface S has non-vanishing Gaussian curvature. The other extreme case is when the rotational curvature hypothesis is fulfilled because of the way the surfaces change with x and y and not because of the presence of Gaussian

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6.3 Spherical Maximal Theorems: Take 1

191

curvature. For instance, if t (x, y) = x, y − t,

(6.3.6)

the operator At involves averaging over the hyperplane Sx,t = {y : x, y = t}. If η(x, y) vanishes near the origin, this family still satisfies the curvature hypothesis (6.3.4) despite the fact that the Gaussian curvature and all the principal curvatures vanish identically on Sx,t . Remark The last example shows how the above results cannot extend in this level of generality to two dimensions. For if t is given by (6.3.6) and 0 ≤ η ∈ C0∞ is nontrivial then (6.3.5) can never hold for a finite p if n = 2. This is because, given ε > 0, there is a measurable subset ε ⊂ [−1, 1] × [−1, 1] satisfying |ε | < ε but having the property that a unit line segment can be continously moved in ε until its orientation is reversed. Thus, ε contains a unit line segment in every direction. If one takes f to be the characteristic function of an appropriate translate and dilate of this Kakeya set ε , the left side is ≈ 1 while the right side is < ε1/p , providing a counterexample to the possibility of the two-dimensional theorem. Let us conclude this section by showing how Theorem 6.3.1 can be used to prove maximal theorems involving smooth hypersurfaces that shrink to a point as t → 0. Specifically, we assume that " Sx,y ⊂ Rn is a family of C∞ hypersurfaces that depend smoothly on the parameters (x, t) ∈ Rn × [0, 1]. We then put Sx,t = x + t" Sx,y . Our assumptions are that: (i) Given any subinterval [a, 1] with a > 0 the surfaces Sx,t satisfy the rotational curvature hypothesis in Corollary 6.3.2, and (ii) the “initial hypersurface” " Sx,0 always has everywhere nonvanishing Gaussian curvature. Corollary 6.3.3 Let η ∈ C0∞ (Rn × Rn ) and suppose that Sx,t satisfies the two conditions described above. Then if n ≥ 3 and if dσx,t now denotes Lebesgue measure on " Sx,t we have        f (x − ty)η(x, y) dσx,t (y) p n  sup 0 0 smaller than the injectivity radius T, we let At f (x) denote the average of f over the geodesic sphere of radius t around x. It then follows that sup0 T,

if ξ ∈ U0c .

By the lower semicontinuity of L : S∗ M → S∗ M, for every ξ ∈ U0c there must be a δ(ξ ) > 0 such that L(y, η) > T

if

|y − x0 | < δ(ξ ) and

|η − ξ | < δ(ξ ), η ∈ Sn−1 . (7.3.16)

For a given ξ ∈ Sn−1 and δ > 0 define the spherical cap V(ξ , δ) = {η ∈ Sn−1 : |η − ξ | < δ}.

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7.3 Improved Sup-norm Estimates of Eigenfunctions We then have U0c ⊂



209

V(ξ , δ(ξ )).

ξ ∈U0c

Since U0c is compact, by the Heine–Borel theorem, we can find a finite collection of points ξj , j = 1, . . . , N, in U0c such that U0c ⊂ V(ξ1 , δ(ξ1 )) ∪ · · · ∪ V(ξN , δ(ξN )). If δ = min δ(ξj ), 1≤j≤N

then, by (7.3.16), we have L(y, η) > T

if

|x0 − y| < δ

and

η ∈ U0c .

(7.3.17)

By the first part of (7.3.15) and the Urysohn lemma we can find a function B ∈ C∞ (Sn−1 ) satisfying 0 ≤ B ≤ 1 and B(ξ ) = 1 if

ξ ∈ U1c

and

B(ξ ) = 0 for

ξ ∈ U0.

Choose a function ψ ∈ C0∞ (Rn ) satisfying 0 ≤ ψ ≤ 1 and ψ(y) = 1

|x0 − y| ≤ δ/2

if

and

supp ψ ⊂ {y : |x0 − y| < δ}.

By (7.3.17) if we set B(y, η) = ψ(y)B(η/|η|), η ∈ Rn \0, then we have L(y, η) > T If we let

if (y, η) ∈ supp B ∩ S∗ M.

(7.3.18)

  b(y, η) = ψ(y) 1 − B(η/|η|) ,

then we have B(x, D) + b(x, D) = ψ(x).

(7.3.19)

Since 1 − B(η) = 0 in U1c , by the second part of (7.3.15) we also have  |b(x, ξ )|2 dξ ≤ Cn T −1 , (7.3.20) p(x,ξ )≤1

where Cn depends only on the dimension. Having set things up, we are now in a position to prove Lemma 7.3.3. We shall take Nx0 ,T to be the neighborhood on M of x0 , which is defined in our

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coordinate system by the open ball of radius δ/2 about x0 . Let us fix a bump function β ∈ C0∞ (R) satisfying β(t) = 1,

if |t| ≤ 1/2,

and

β(t) = 0

if |t| ≥ 1.

By Fourier’s inversion theorem, we then can write the left side of (7.3.13) as ∞ 

χ (T(λ − λj ))|ej (x)|2

j=1

=

1 2π T +

 itλ β(t)χ(t/T)e ˆ

1 2πT



∞ 

e−iλj t |ej (x)|2 dt

(7.3.21)

j=1



 1 − β(t) χˆ (t/T)eitλ

∞ 

e−iλj t |ej (x)|2 dt

j=1

= I + II. We can estimate the first term, I, on all of M if we use (4.2.13) which is equivalent to the uniform bounds  |ej (x)|2 ≤ C(1 + λ)n−1 , λ ≥ 0. (7.3.22) λj ∈[λ,λ+1]

To use this, we note that the inverse Fourier transform,

T,

of

t → β(t)χˆ (t/T) satisfies |

T (τ )| ≤ CN (1 + |τ |)

−N

,

for each N = 1, 2, . . . , with CN independent of T ≥ 1. Consequently, |I| = T

∞  

−1 

T (λ − λj )|ej (x)|

 

2

j=1 ∞ 

≤ T −1 CN

(1 + |λ − λj |)−N |ej (x)|2

j=1

≤ CT

−1 n−1

λ

,

if we assume N > n and use (7.3.22) in the last step. As a result, we would be done if we could show that the other term, II, in the right hand side enjoys these same bounds for large enough λ when |x − x0 | ≤ δ/2. Since the bump function defining the pseudo-differential cutoffs, b(x, D) and B(x, D) equals one when |x − x0 | ≤ δ/2, it suffices to see that (ψ(x))2 times

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7.3 Improved Sup-norm Estimates of Eigenfunctions

211

II is dominated by the right side of (7.3.13) when λ is sufficiently large. Since the kernel of the half-wave operator, eitP , restricted to the diagonal is given by ∞ 

eitλj |ej (x)|2 ,

j=1

if we use (7.3.8), (7.3.19) and (7.3.21), we find that II multiplied by ψ 2 can be decomposed as 1 2π T + +



T



−T

1 2π T 1 2π T

1 + 2π T

   1 − β(t) χˆ (t/T)eitλ B ◦ e−itP ◦ B∗ (x, x) dt 

T



   itλ 1 − β(t) χ(t/T)e ˆ B ◦ e−itP ◦ b∗ (x, x) dt

−T T



   itλ 1 − β(t) χ(t/T)e ˆ b ◦ e−itP ◦ B∗ (x, x) dt

(7.3.23)

−T T



−T

   itλ 1 − β(t) χ(t/T)e ˆ b ◦ e−itP ◦ b∗ (x, x) dt,

with B∗ and b∗ denoting the adjoints of B(x, D) and b(x, D), respectively. By (7.3.17) and Corollary 7.2.3 and the fact that β equals one near the origin, we know that the integrands of each of the first three terms in (7.3.23) are smooth functions of (t, x). Therefore, if we integrate by parts in t we conclude that each is majorized by CT,N,B λ−N , for any N = 1, 2, . . . . Therefore, we would have (7.3.13) if we could show that the last term is dominated by the right side of this inequality when λ is sufficiently large. To verify this, we shall use the fact that the inverse Fourier transform, ρT , of t → T −1 (1 − β(t)) χˆ (t/T) satisfies |ρT (τ )| ≤ CN (1 + |τ |)−N ,

(7.3.24)

for every N = 1, 2, 3, . . . with constants independent of T ≥ 1. To use this, we observe that since the kernel of e−itP is ∞ 

e−itλj ej (x)ej (y),

j=1

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we have the identity 

∞   2  b ◦ e−itP ◦ b∗ (x, x) = e−itλj b(x, D)ej (x) . j=1

Consequently the last term in (7.3.23) equals ∞ 

 2 ρT (λ − λj )b(x, D)ej (x) .

j=1

As a result, if we recall (7.3.20), use (7.3.24) and repeat the argument that was used to bound I in (7.3.21), we conclude that we would have the remaining estimate that the last term in (7.3.23) is dominated by the right side of (7.3.13) if we could establish the following generalization of (7.3.22). Proposition 7.3.4 Let (M, g) be given and fix a coordinate patch  and a relatively compact open subset ω of . Assume that local coordinates are chosen so that dV equals Lebesgue measure in local coordinates on . Suppose A ∈ cl0 (M) has principal symbol a0 (x, ξ ) and that (x, ξ ) ∈ ess supp A implies that x ∈ ω. Then for λ ≥ 1 we have  |A(x, D)ej (x)|2 λj ∈[λ,λ+1]

≤ Cλn−1

 p(x,ξ )≤1

|a0 (x, ξ )|2 dξ + CA λn−2 ,

(7.3.25)

where C depends on  and ω but not on A. Proof of Proposition 7.3.4 The proof is a variation of the arguments in §4.2 that established (7.3.22). Similar to what we did there, let us choose a bump function ρ ∈ S(R) satisfying ρ ≥ 0, ρ(0) = 1,

and

supp ρˆ ⊂ [−#, #]

where ε > 0 is chosen so that we can use Theorem 4.1.2 to obtain a parametrix for eitP (x, y) when y ∈ ω and |t| ≤ ε. We first claim that we would have (7.3.25) if we could prove the “smoothed-out version” ∞ 

ρ(λ − λj )|A(x, D)ej (x)|2

j=1

≤ Cλn−1

 p(x,ξ )≤1

|a0 (x, ξ )|2 dξ + CA λn−2 .

(7.3.25 )

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7.3 Improved Sup-norm Estimates of Eigenfunctions

213

Indeed since ρ is nonnegative and ρ(0) = 1, it follows that we can find a δ > 0 so that  |A(x, D)ej (x)|2 λj ∈[λ,λ+δ]

is bounded by twice the left side of (7.3.25 ). Therefore, if this inequality were valid, it would imply the analog of (7.3.25) where instead of summing over λj ∈ [λ, λ + 1] we sum over λj ∈ [λ, λ + δ]. However, if we add up O(δ −1 ) such estimates it is clear that we would obtain (7.3.25), which means that we have reduced matters to proving (7.3.25 ). If we repeat arguments that were just given, we can use Fourier’s inversion formula to see that ∞ 

ρ(λ − λj )|A(x, D)ej (x)|2

j=1

=

1 2π



ε

−ε

  itλ A ◦ e−itP ◦ A∗ (x, x) dt. ρ(t)e ˆ

(7.3.26)

If we recall Theorem 4.1.2, we can write e−itP f for |t| ≤ ε as a Fourier integral  (7.3.27) Q(t)f (x) = (2π )−n eiϕ(x,y,ξ )−itp(y,ξ ) q(t, x, y, ξ ) f (y) dξ dy plus a smoothing error if f ∈ C0∞ (ω), where ϕ satisfies (4.1.3 ) and q is a classical zero-order symbol satisfying (4.1.9), i.e., q(0, x, x, ξ ) − 1 ∈ S−1 .

(7.3.28)

Since A(x, D) is a pseudo-differential operator and hence has the trivial canonical relation, it follows from the composition theorem, Theorem 6.2.2, that (A ◦ e−itP ◦ A∗ )f takes a similar form for such f . Indeed there must be a zero-order symbol qA so that, modulo a smoothing error,  (A ◦ e−itP ◦ A∗ )f = (2π )−n eiϕ(x,y,ξ )−itp(y,ξ ) qA (t, x, y, ξ ) f (y) dξ dy. Since A does not depend on t, if we set t = 0 here, it follows from Theorem 3.2.1 and (7.3.28) that we must have qA (0, x, x, ξ ) − |a0 (x, ξ )|2 ∈ S−1 , due to the fact that the principal symbol of A ◦ A∗ is |a0 (x, ξ )|2 . Therefore, by (7.3.26) and our assumptions regarding ess supp A, modulo a term that is rapidly decreasing in λ (with constants depending on A), the left

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side of (7.3.25 ) equals   2 it(λ−p(x,ξ )) ρ(t)|a ˆ dtdξ (2π )−n−1 0 (x, ξ )| e n R R   ρ(t) ˆ r−1 (t, x, ξ )eit(λ−p(x,ξ )) dtdξ + (2π )−n−1 Rn R   −n−1 + (2π ) ρ(t) ˆ tr0 (t, x, ξ )eit(λ−p(x,ξ )) dtdξ

(7.3.29)

Rn R

= I + II + III, where rj ∈ S j for j = −1 and j = 0 in II and III, respectively. In proving this we are using the fact that qA (t, x, x, ξ ) = qA (0, x, x, ξ ) + tr0 (t, x, ξ ) where r0 is as above and the fact that qA (0, x, x, ξ ) agrees with |a0 (x, ξ )|2 modulo a symbol of order −1. For the first term, we have the formula  −n ρ(λ − p(x, ξ )) |a0 (x, ξ )|2 dξ , I = (2π ) and, since ρ ∈ S(R), it is clear that I is majorized by the first term in the right side of (7.3.25 ). Consequently, we would complete the proof of Proposition 7.3.4 if we could show that the other two terms here, II and III, are dominated by the second term in the right side of (7.3.25 ). We could do this by more or less repeating arguments from Chapter 4. However, to motivate the more delicate arguments that will arise in the next section, let us see that we can use polar coordinates to reduce matters to the following simple result, which will also be of use there. Lemma 7.3.5 Assume that β ∈ C0∞ (R) and α(t, r) ∈ C∞ (R × R) satisfies |∂t ∂rk α(t, r)| ≤ Cjk (1 + |r|)μ−k , j

Then if μ ≥ 0  ∞    

∞

−∞ −∞

β(t)α(t, r)e

it(λ−r)

j, k = 1, 2, 3, . . . .

  dtdr  ≤ C|λ|μ , |λ| ≥ 1.

(7.3.30)

(7.3.31)

Furthermore, if μ ≥ 1 and α(0, r) ≡ 0, we have    

∞



∞

−∞ −∞

β(t)α(t, r)e

it(λ−r)

  dtdr  ≤ C|λ|μ−1 , |λ| ≥ 1.

(7.3.32)

The constants C depend only on β, μ and finitely many of the constants in (7.3.30).

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7.3 Improved Sup-norm Estimates of Eigenfunctions

215

Before giving the simple proof, let us complete the proof of Proposition 7.3.4 by giving the simple argument showing that the lemma implies that the terms II and III in (7.3.29) are each O(λn−2 ). If we let a = r−1 or a = tr0 we can rewrite the dξ integrals there using polar coordinates, ξ = rω, ω ∈ Sn−1 , r ≥ 0. Indeed since p(x, ξ ) = rp(x, ω), with p(x, ω) > 0, we have   ∞ −itp(x,ξ ) e a(t, x, ξ ) dξ = e−itp(x,ω)r a(t, x, rω)dωrn−1 dr Rn Sn−1 0

  ∞ dω e−itr rn−1 a(t, x, rω/p(x, ω)) dr. (7.3.33) = (p(x, ω))n Sn−1 0 In showing that II and III are OA (λn−2 ), we may assume that the symbols r0 and r−1 vanish near ξ = 0, since the contributions coming from small frequencies are O(1). If we then let α(t, r) denote the term inside the parenthesis in the right side of (7.3.33) when a = r−1 it is clear that (7.3.30) is valid with μ = n − 2, leading to the aforementioned bound for II. If we let a = tr0 , then we have α(0, r) ≡ 0 and (7.3.30) holding with μ = n − 1. So here we get the desired bound for III from the lemma, if we now use (7.3.32). Thus, to finish the proof of Proposition 7.3.4, we just need to prove the lemma. Proof of Lemma 7.3.5 The first inequality is straightforward. We note that, by a simple integration by parts argument, the inner integral in (7.3.31) satisfies    ∞  β(t)α(t, r)eit(λ−r) dt  ≤ CN (1 + |r|)μ (1 + |λ − r|)−N , −∞

for each N = 1, 2, . . . . This leads to (7.3.31), for if μ ≥ 0 and N > μ + 1  ∞ (1 + |r|)μ (1 + |λ − r|)−N dr ≤ C|λ|μ , |λ| ≥ 1. −∞

To prove (7.3.32) we note that because of the assumption that α(0, r) ≡ 0 we can write β(t)α(t, r) = tβ(t)" α (t, r) where " α satisfies (7.3.30). Since d −itr e , dr if we integrate by parts in r, we find that the left side of (7.3.32) is dominated by the analog of (7.3.31) with α replaced by ∂r α(t, r). The latter satisfies the analog of (7.3.30) with μ replaced by μ − 1. Therefore, (7.3.32) follows from what we have just done. te−itr = it

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7.4 Improved Spectral Asymptotics In this section we shall prove the main result of this chapter, the Duistermaat– Guillemin theorem. It says that if the set of periodic geodesics has measure zero then one can improve the bounds for the error term in the Weyl formula, i.e., Theorem 4.2.1. The hypothesis means that |C| = 0 if, as in (7.3.5), C ⊂ S∗ M denotes the points belonging to periodic geodesics for the flow on S∗ M. Theorem 7.4.1 (Duistermaat–Guillemin theorem) Let N(λ) be the number of  eigenvalues of P = −g , counted with respect to multiplicity, which are ≤ λ. Suppose that |C| = 0. (7.4.1)  n Then if ωn denotes the volume of the unit ball in R and Vol M = M dV, where c = (2π )−n ωn Vol M.

N(λ) = cλn + o(λn−1 ),

(7.4.2)

1

As we mentioned before, in local coordinates dV = |g| 2 dx, where |g| = det (gjk (x)). Since  1 dξ = |g| 2 ωn , p(x,ξ )≤1

if

5 6 6 n p(x, ξ ) = 7 g jk (x)ξj ξk 

j,k=1

denotes the principal symbol of −g , it is clear that the constant c in (7.4.2) agrees with the earlier one, (4.2.1). In the proof of Theorem 4.2.1, (4.2.13) played a critical role. As we mentioned before, this inequality is equivalent to (7.3.22), which, after integrating against dV, yields N(λ + 1) − N(λ) = O(λn−1 ).

(7.4.3)

This is a bound for the number of eigenvalues lying in unit bands. Clearly, this estimate is necessary in order for the earlier “sharp Weyl formula” N(λ) = cλn + O(λn−1 )

(7.4.4)

to be valid. Also, as we noted before, for the standard sphere Sn we cannot improve upon (7.4.4) since if we replace N(λ + 1) in (7.4.3) by N(λ + ε) with √ ε we still get a difference that is ≈ λn−1 if λ = k(k + n − 1), k ∈ N, is one of the distinct eigenvalues for the sphere. Of course, for the sphere, C is the full unit cotangent bundle and so (7.4.1) fails dramatically.

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217

A key step in the proof of the improvement (7.4.2) over (7.4.4) is that if we assume (7.4.1) we can obtain o(λn−1 ) bounds for the number of eigenvalues in intervals about λ that shrink as λ → ∞. Specifically, we have the following natural result. Proposition 7.4.2 Let (M, g) be given and assume that (7.4.1) is valid. Then there is a uniform constant C = C(M, g) such that for every ε > 0 there is a constant (ε) < ∞ such that N(λ + ε) − N(λ) ≤ Cελn−1 ,

λ ≥ (ε).

if

(7.4.5)

Remark We note that (7.4.5) is a very natural estimate since it says that the number of eigenvalues satisfying λ < λj ≤ λ + ε is bounded by the volume of the corresponding annulus {ξ ∈ Rn : λ < |ξ | ≤ λ + ε} in Euclidean space if λ is sufficiently large. As we just noted, for the sphere, this small scale spectral synthesis is impossible, since, in this case, we can only obtain unit scale bounds. If E(λ,λ+ε] denotes projection onto the (λ, λ + ε] band of the spectrum of P, i.e.,  E(λ,λ+ε] f = Ej f , λj ∈(λ,λ+ε]

then its kernel is E(λ,λ+ε] (x, y) =



ej (x)ej (y).

λj ∈(λ,λ+ε]

Consequently, we can rewrite the left side of (7.4.5) as N(λ + ε) − N(λ) = Trace E(λ,λ+ε]   = E(λ,λ+ε] (x, x) dV = M



|ej (x)|2 dV.

(7.4.6)

M λ ∈(λ,λ+ε] j

The proof of Proposition 7.3.2 shows that, if we take ε = T −1 there, then we actually can obtain analogous pointwise estimates for the kernel of E(λ,λ+ε] on the diagonal if we assume a stronger condition than (7.4.1). Specifically, for every ε > 0, we have that E(λ,λ+ε] (x, x) ≤ Cελn−1 for large λ if we assume that |Lx | = 0 for every x ∈ M. We will not be able to prove such a strong estimate under the assumption (7.4.1); however, we will be able to obtain the trace estimate (7.4.5) if we use the following variant of the propagation of singularities results from §7.2 for the trace of the half-wave kernel.

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Lemma 7.4.3 Let A, B ∈

0 cl (M).





Then

A ◦ eit

t→

√

−g

 ◦ B∗ (x, x) dV

M

is smooth at t = t0 if there is no T ∗ M\0  (x, ξ ) ∈ ess supp A ∩ ess supp B so that t0 (x, ξ ) = (x, ξ ). Proof Using finite partitions of unity of T ∗ M\0 involving symbols of elements of cl0 (M), we can write   Aν and B = Bμ , A= ν

μ

with each Aν and Bμ having essential supports contained in small conic neighborhoods of some (xν , ξν ) ∈ T ∗ M\0 and (yμ , ημ ) ∈ T ∗ M\0, respectively. So to prove the lemma, it suffices to prove the assertion when A has essential support in a small conic neighborhood N1 ⊂ T ∗ M\0 of a point (x, ξ ) and B has essential support in a small conic neighborhood N2 ⊂ T ∗ M\0 of a point (y, η). If (x, ξ/p(x, ξ ))  = (y, η/p(y, η)) (the projection of the two points onto S∗ M), then after perhaps shrinking the two neighborhoods we can assume that N1 ∩ N2 = ∅. But then    ∞ √   A ◦ eit −g ◦ B∗ (x, x) dV = eitλj Aej (x) Bej (x) dV M

M j=1

=

  ∞

eitλj B∗ Aej (x) ej (x) dV

M j=1

 =



(B∗ A) ◦ eit

√

−g 

(x, x) dV

M

is smoothing for all times as B∗ A is smoothing. For the remaining case where (x, ξ/p(x, ξ )) = (y, η/p(y, η)) we may assume that N1 and N2 are the same conic neighborhood, N , of (x, ξ ). If t0 (x, ξ )  = (x, ξ ), then if, as we may assume, N is small enough, by Corollary 7.2.3, √   (t, x) → A ◦ eit −g ◦ B∗ (x, x)

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7.4 Improved Spectral Asymptotics

219

is smooth at t = t0 for all x ∈ M. Thus, in this remaining case we also have that √   Trace A ◦ eit −g ◦ B∗ is smooth at t = t0 . We can now use Lemma 7.4.3 and arguments from the proof of Proposition 7.3.2 to prove Proposition 7.4.2. Proof of Proposition 7.4.2 If 0 < ε & 1 is as in (7.4.5), we shall let T = ε−1 . We then consider CT = {(x, ξ ) ∈ S∗ M : t (x, ξ )  = (x, ξ ), ∀0 < |t| ≤ T}c . By our earlier discussion of the lower semicontinuity of the length functional, L(x, ξ ) for the geodesic flow, CT must be a closed subset of S∗ M. It also must be of measure zero by our assumption (7.4.1) since it is a subset of C. By our earlier arguments, we can find pseudo-differential operators B, b ∈ 0 (M) such that Id = B(x, D) + b(x, D), ess supp b contains a small neighborcl hood of CT and, if b0 (x, ξ ) is the principal symbol of b,  |b0 (x, ξ )|2 dξ dx ≤ ε, (7.4.7) p(x,ξ )≤1

while if (x, ξ ) ∈ ess supp B, t (x, ξ )  = (x, ξ ), ∀ 0 < |t| ≤ T.

(7.4.8)

To use this, fix a nonnegative ρ ∈ S(R) satisfying ρ(t) ≥ 1, |t| ≤ 1 and supp ρˆ ⊂ [−1, 1]. By (7.4.6) we then have   N(λ + ε) − N(λ) = |ej (x)|2 dV M λ ∈(λ,λ+ε] j

≤

  ∞

ρ(ε−1 (λ − λj )) |ej (x)|2 dV

M j=1

=

ε 2π

 

T

M −T

 √  itλ −it −g e (x, x) dtdV, ρ(εt)e ˆ

using, in the last step, Fourier’s inversion formula, our support assumptions on ρˆ and the fact that T = ε −1 .

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Fix β ∈ C0∞ (R) satisfying β(t) = 1, |t| ≤ 1/2, but β(t) = 0, for |t| ≥ 1. We can then use this cutoff function to split the last expression into two pieces to deduce that   T  √  ε itλ −it −g e (x, x) dtdV β(t)ρ(εt)e ˆ N(λ + ε) − N(λ) ≤ 2π M −T   T  √  ε itλ −it −g e (x, x) dtdV (1 − β(t)) ρ(εt)e ˆ + 2π M −T = I + II. Let ε ∈ S(R) be defined by ˆ ε (t) = ρ(εt)β(t). ˆ Then with bounds independent of 0 < ε < 1, we have |

ε (τ )| ≤ CN (1 + |τ |)

−N

,

N = 1, 2, 3, . . . .

Note that I is the integral over M of ε

∞ 

ε (λ − λj )|ej (x)|

2

,

j=1

which, by the preceding inequality is majorized by each N by ε

∞ 

(1 + |λ − λj |)−N |ej (x)|2 .

j=1

Consequently, if we take N > n and use (7.3.22), we find that |I| ≤ Cελn−1 . As a result of this, the proof of Proposition 7.4.2 would be complete if we could show that the other term above, II, also enjoys these bounds when λ is large enough. Similar to what was done before, we use our pseudo-differential cutoffs to decompose it as follows:  √   1 ε(1 − β(t)) ρ(εt) ˆ eiλt B ◦ e−it −g (x, x) dVdt 2π  √   1 ε(1 − β(t)) ρ(εt) ˆ eiλt b ◦ e−it −g ◦ B∗ (x, x) dVdt + 2π  √   1 ε(1 − β(t)) ρ(εt) ˆ eiλt b ◦ e−it −g ◦ b∗ (x, x) dVdt + 2π = I + II + III. By (7.4.8) and Lemma 7.4.3, |I| + |II| ≤ CN,ε (1 + |λ|)−N , for any N.

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221

To handle the remaining term, III, we note that if hε denotes the inverse Fourier transform of t → ε(1 − β(t)) ρ(εt), ˆ then we have that |hε (τ )| ≤ CN (1 + |τ |)−N , N = 1, 2, . . . , where the CN are independent of 0 < ε < 1. Since III =

  ∞ M j=1

hε (λ − λj ) |b(x, D)ej (x)|2 dV,

if we integrate (7.3.25) over M and use (7.4.7), we conclude that there must be a constant C, which is also independent of 0 < ε < 1, and a constant Cb,ε , depending on b and ε, so that |III| ≤ Cελn−1 + Cb,ε λn−2 . Thus, III also satisfies the bounds in (7.4.5), which completes the proof. We now are in a position to set up the proof of Theorem 7.4.1. Note that the Fourier transform of the characteristic function, χ[−λ,λ] , of [−λ, λ] equals  λ  λ sin λt itτ e dτ = cos tτ dτ = 2 . t −λ −λ  Consequently, if λ is not an eigenvalue of −g , we can rewrite the Weyl counting function as follows:  N(λ) = |ej (x)|2 dV λj ≤λ M

=

  ∞ M j=1

χ[−λ,λ] (λj )|ej (x)|2 dV

 ∞  1 eitλj |ej (x)|2 dVdt χˆ [−λ,λ] (t) 2π j=1  1 sin λt  it√−g  = (x, x) dVdt. e π t =

In view of (7.4.5), in proving (7.4.2), as we shall assume in what follows, we may make this assumption about λ. As we noted before, there must be a δ > 0 so that t (x, ξ )  = (x, ξ ), if (x, ξ ) ∈ S∗ M,

and

0 < |t| ≤ 2δ.

(7.4.9)

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To exploit this, let us now choose a bump function β satisfying β ∈ C0∞ (R),

β(t) = 1 if |t| ≤ δ/2, but β(t) = 0 if |t| ≥ δ.

(7.4.10)

Then for a fixed ε & δ we can decompose the Weyl counting function in the following manner:  sin λt  it√−g  1 (x, x) dVdt β(t) e N(λ) = π t    1 sin λt  it√−g  + (x, x) dVdt 1 − β(t) β(εt) e π t (7.4.11)    sin λt  it√−g  1 + (x, x) dVdt 1 − β(εt) e π t = I + II + III, assuming, as we may, say, that ε < δ/10 so that (1 − β(εt))(1 − β(t)) = (1 − β(εt)). Recall that we are trying to prove that for every ε > 0 |N(λ) − (2π )−n ωn Vol M λn | ≤ ελn−1 ,

if λ ≥ (ε),

(7.4.12)

with (ε) depending on 0 < ε & 1. We first claim that the last two terms, II and III, are both O(ελn−1 ) “errors” if λ is large enough. To handle the last piece, which involves very large times, we shall use Proposition 7.4.2, which we noted before is a special case of the result we are trying to prove. To handle the second term, II, which involves medium and large times, we shall use a simple modification of the proof of this proposition and invoke the propagation of singularities results for traces. So for both II and III we shall need to make use of our hypothesis that |C| = 0. The final step in the proof of (7.4.12) will be to show that if I is as in (7.4.11) we have I = (2π )−n ωn λn + O(λn−2 ).

(7.4.13)

The proof of this will be postponed until the end of the section. Based on a careful study of the nature of the “big” singularity of the half-wave kernel at t = 0, we shall actually see that (7.4.13) holds for any (M, g). So, not unexpectedly, unlike the bounds for II and III, the local estimate (7.4.13) does not require the assumption that |C| = 0. As we just mentioned, to handle the third term in the right side of (7.4.11), we shall make use of Proposition 7.4.3. We first note that the function

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7.4 Improved Spectral Asymptotics

223

t → t−1 (1 − β(t)) has a Fourier transform  ∞ h(τ ) = e−itτ t−1 (1 − β(t)) dt, −∞

that satisfies

|h(τ )| ≤ CN (1 + |τ |)−N , N = 1, 2, 3, . . . .

(7.4.14)

Therefore, if we use Euler’s formula to rewrite the sin λt factor using complex exponentials, we see that  ∞  ε −1 iλt (εt) (1 − β(εt))e eiλj t |ej (x)|2 dVdt III = 2π i − =i

ε 2π i



∞  

j=1

(εt)−1 (1 − β(εt))e−iλt

∞ 

eiλj t |ej (x)|2 dVdt

j=1 ∞  

h((λj − λ)/ε) |ej (x)|2 dV − i

j=1 M

h((λj + λ)/ε) |ej (x)|2 dV.

j=1 M

Thus, by (7.4.14), we have ∞    −N 1 + ε−1 |λ − λj | |ej (x)|2 dV |III| ≤ CN j=1 M



= CN



−N

1 + ε−1 |λ − λj |

+ CN

|ej (x)|2 dV M

λj ∈[λ−1,λ+1]







1+ε

−1

−N

(7.4.15)



|λ − λj |

|ej (x)|2 dV, M

λj ∈[λ−1,λ+1] /

for every N ∈ N, with constants independent of our ε. Using (7.3.22) again we see that by taking N to be large enough, we can dominate the last term by O(εm λn−1 ) for any m ∈ N. To handle the other term in the right side of (7.4.15) we note that we can write [λ − 1, λ + 1] as the disjoint union over k ∈ Z of the intervals Ik = (λ + εk, λ + ε(k + 1)] ∩ [λ − 1, λ + 1]. Consequently this term is majorized by      (1 + |k|)−N |ej (x)|2 dV = (1 + |k|)−N · #{j : λj ∈ Ik }. k

λj ∈Ik M

k

By (7.4.5), there is a uniform constant C so that if Ik is nonempty #{j : λj ∈ Ik } ≤ Cελn−1 ,

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provided that λ is large enough (depending on ε). Since the sum over k ∈ Z of (1 + |k|)−N is finite if N > 1, we conclude that there is a uniform constant C such that for every 0 < ε & 1 we can find a number (ε) < ∞ so that if III is as in (7.4.11) |III| ≤ Cελn−1 ,

if λ ≥ (ε).

(7.4.16)

It is a bit less technical to show that the other “error term,” II, in our decomposition (7.4.11) enjoys similar bounds. As we mentioned before, we shall do this by adapting the proof of Proposition 7.4.2. If T = ε −1 , let us use the exact same decomposition Id = B(x, D) + b(x, D) involving zero-order classical pseudo-differential operators satisfying (7.4.7) and (7.4.8). We then can decompose II in (7.4.11) as the sum  √  sin λt  1 (1 − β(t)) β(εt) B ◦ eit −g (x, x) dVdt π t  √  sin λt  1 (1 − β(t)) β(εt) + b ◦ eit −g ◦ B∗ (x, x) dVdt π t  √  sin λt  1 (1 − β(t)) β(εt) + b ◦ eit −g ◦ b∗ (x, x) dVdt π t = I + II + III. Because of (7.4.10), the integrands vanish if |t| ≥ δ/ε. So, if as we may, we assume that 0 < δ < 1, we can repeat the arguments used to bound the corresponding terms in the proof of Proposition 7.4.2 and use (7.4.8) and Lemma 7.4.3 to see that both I and II are Ob,T,N (λ−N ) for any N. To handle the remaining piece, III, we note that if ε denotes the inverse Fourier transform of t → t−1 (1 − β(t)) β(εt), then we have |

ε (τ )| ≤ CN (1 + |τ |)

−N

for any N ∈ N with constants independent of 0 < ε < 1. Repeating the first part of the argument used to control III above shows that we here have III = i

∞ 

 ε (λ − λj )

j=1 ∞ 

−i

j=1

|b(x, D)ej (x)|2 dV M

 ε (λ + λj )

|b(x, D)ej (x)|2 dV, M

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7.4 Improved Spectral Asymptotics

225

and so, for each N ∈ N we have ∞   |III| ≤ CN (1 + |λ − λj |)−N |b(x, D)ej (x)|2 dV. j=1 M

If we use Proposition 7.3.4 and our assumption (7.4.7), we conclude that there must be a uniform constant C, independent of 0 < ε < 1 and another one Cb,T , which depends on b and ε = T −1 , so that we have |III| ≤ Cελn−1 + Cb,ε λn−2 . If we combine this with the more favorable bounds for the other terms, I and II in the decomposition of II, we conclude that, as for III in (7.4.11), we have that there must be a uniform constant C such that for each 0 < ε < 1 we can find a finite (ε) so that |II| ≤ Cελn−1 ,

if

λ ≥ (ε).

(7.4.17)

In view of these two bounds, (7.4.16) and (7.4.17) for the two “error terms” in the decomposition (7.4.11) of N(λ), after replacing ε by a fixed multiple of it, we would have (7.4.12), and hence Theorem 7.4.1, if we could establish the bound (7.4.13) for the “local term” in the decomposition. We shall actually prove something a bit stronger, which is a pointwise variant that yields (7.4.13) after integrating over M. Proposition 7.4.4 Let (M, g) be given. Assume that 0 < δ < 1 is chosen so that (7.4.9) is valid and let β ∈ C0∞ (R) satisfy (7.4.10). Fix a coordinate patch  in M and local coordinates in which dV becomes dx. Then if 0 is a fixed relatively compact subset of  we have   1 sin λt  it√−g   (x, x) dt − (2π )−n ωn λn  ≤ Cλn−2 , x ∈ 0 . β(t) e π t (7.4.18) To prove (7.4.18), it will be convenient to use a different parametrix for the half-wave operator than the one in (7.3.27). This is because, although the phase function of the Fourier integral operator in (7.3.27) is linear in t, which has been very useful in Chapter 4 and earlier sections here, in order to prove (7.4.18) we require very precise information about the symbols and their t-derivatives at t = 0. Obtaining this information using the symbol in the Fourier integral representation in (7.3.27) is quite technical, but becomes much easier if we write the Fourier integral in an equivalent form using results from Chapter 6. The small price to pay, though, is that the resulting phase functions will be a bit harder to work with.

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If we use Proposition 6.2.4 we see that we can rewrite the Fourier integral Q(−t) given by (7.3.27) as one with a phase of the form φ(t, x, ξ ) − y, ξ . More specifically, if we assume that 0 is as in the Proposition 7.4.4, for f ∈  C0∞ (0 ) we can see that we can write eitP f , with P = −g , as  −n eiφ(t,x,ξ )−iy,ξ  a(t, x, ξ ) f (y) dξ dy, (7.4.19) (2π ) modulo a smoothing error if |t| is small, where a is a zero order symbol and for each fixed such t, φ(t, x, ξ ) is a generating function for the canonical relation Ct of the half-wave operator. Since eitP is the identity operator when t = 0 and since (∂t − iP)eitP = 0, we must have that the phase function of the Fourier integral satisfies φ(0, x, ξ ) = x, ξ  ∂φ = p(x, ∇x φ), ∂t

(7.4.20)

with p(x, ξ ) being the principal symbol of P. Since  itP  e (x, y)|t=0 = δ(x − y), we conclude that the symbol a of the Fourier integral operator in (7.4.19) must satisfy a(0, x, ξ ) ≡ 1.

(7.4.21) ∞

This symbol a(t, x, ξ ) is a classical symbol of order zero, a ∼ j=0 a−j where the a−j are homogeneous in ξ of order −j and solve successive transport equations, just as in §4.1. Repeating arguments from this section shows that the main transport equation forces us to have 1 ∂t a0 (0, x, ξ ) = p0 (x, ξ ), i

(7.4.22)

 if P(x, ξ ) = ∞ j=0 p1−j (x, ξ ) is the corresponding asymptotic expansion for P involving symbols that are homogeneous of 1−j. Thus, p1 (x, ξ ) is the principal symbol p(x, ξ ), and p0 is the next term in the expansion. Unlike the principal part, the second term in the expansion of the symbol of a classical pseudo-differential operator does not have good invariance properties. It has a cousin, though, the subprincipal symbol, which does. Specifically if  Q ∈ clm (M) has an expansion in local coordinates Q(x, ξ ) ∼ ∞ j=0 qm−j (x, ξ )

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7.4 Improved Spectral Asymptotics

227

involving terms that are homogeneous of degree m − j, then its subprincipal symbol is defined as 1  ∂ 2 qm . 2i ∂xj ∂ξj n

sub Q = qm−1 −

(7.4.23)

j=1

Clearly sub Q is homogeneous of degree m − 1, and one can also show that is invariantly defined, although we shall not need to use the latter. A simple calculation using the Kohn–Nirenberg formula, (3.1.3), shows that if A and B are classical pseudo-differential operators with principal symbols aprin and bprin , respectively, then sub (A ◦ B) = (sub A) · bprin + aprin · (sub B) +

1 {aprin , bprin }, 2i

where the last term denotes the Poisson bracket that we encountered in §7.1. In particular, we have that  sub (−g ) = 2p(x, ξ ) sub −g .  But sub g = 0 and therefore sub −g vanishes as well. It is simple to check the former in our local coordinate system where |g| ≡ 1, since we have   g = g jk (x)∂j ∂k + (∂j g jk )∂k . We have gone through this minor digression to see that, as sub P = 0, we can use (7.4.23) to rewrite (7.4.22) 1  ∂ 2 p(x, ξ ) mod S−1 , 2 ∂xj ∂ξj n

∂t a(0, x, ξ ) =

(7.4.24)

j=1

where a denotes the full symbol of the Fourier integral in (7.4.19). This will turn out to be useful since the right side of this formula will naturally arise in a polar coordinates argument that is akin to the simple one at the end of the last section. By (7.4.21) and (7.4.24), we know the first order Taylor expansion about t = 0 of the symbol of the Fourier integral representation (7.4.19). By (7.4.20) we know the same for the phase. However, to carry out calculations precise enough to prove Proposition 7.4.4, it turns out that we need to know the second order Taylor coefficient for the phase as well. To obtain this, we note that by (7.4.20) we have that ∂φ (0, x, ξ ) = p(x, ξ ). ∂t

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Therefore, by differentiating the second part of (7.4.20) we find that  ∂p ∂p ∂ 2 φ  ∂p ∂ 2 φ = , = ∂ξj ∂xj ∂t ∂ξj ∂xj ∂t2 n

n

j=1

j=1

if t = 0,

using the second part of (7.4.20) again in the last equality. We can write this more succinctly as ∂p(x, ξ ) ∂p(x, ξ ) ∂ 2φ (0, x, ξ ) = · . 2 ∂ξ ∂x ∂t From this we infer that near t = 0 we can write φ(t, x, ξ ) − x, ξ  = tψ(t, x, ξ ) 1 ∂p(x, ξ ) ∂p(x, ξ ) ψ(0, x, ξ ) = p(x, ξ ), ∂t ψ(0, x, ξ ) = · . 2 ∂ξ ∂x This will be useful since when x ∈ 0 and |t| ≤ δ, we can write   it√−g  e (x, x) = (2π )−n eitψ(t,x,ξ ) a(t, x, ξ ) dξ mod C∞ .

(7.4.25)

(7.4.26)

The t-dependence of ψ makes analyzing, with the precision required, the singularity at t = 0 of this expression a bit complicated. We shall get around this difficulty by using, for each t, a “polar coordinate” system adapted to this phase in order that we can ultimately use Lemma 7.3.5 from the end of the last section. Let us start by recalling that if  is the defining function for a surface , i.e.,  = {ξ : (ξ ) = 0} and ∇  = 0 if  = 0, then δ() is the Leray measure on , which is defined by the formula δ((ξ )) = dS/|∇(ξ )|, where dS is the induced Lebesgue measure. Taking  = τ − ψ(t, x, ξ ), since ψ(t, x, rω) = rψ(t, x, ω), we can use polar coordinates to write   rn−1 δ r − τ/ψ(t, x, ω) drdω ψ(t, x, ω)   τ n−1 δ r − τ/ψ(t, x, ω) drdω. = n (ψ(t, x, ω))

δ(τ − ψ)rn−1 drdω =

Therefore, if, say h ∈ C∞ (Rn ), |t| is small and τ > 0, we have    dω n−1 h τ ω/ψ(t, x, ω) . h, δ(τ − ψ) = τ n−1 (ψ(t, x, ω))n S

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229

Consequently, for small |t|, we obtain the useful formula  ∞ −n (2π ) eitτ a, δ(τ − ψ) dτ 0 −n



= (2π )

−n

∞

0 

= (2π )

Sn−1

eitτ a(t, x, τ ω/ψ(t, x.ω))

itτ ψ(t,x,ω)

e

τ n−1 dωdτ (ψ(t, x, ω))n

a(t, x, τ ω)τ n−1 dτ dω

 = (2π )−n eitψ(t,x,ξ ) a(t, x, ξ ) dξ   √ = eit −g (x, x) mod C∞ . In other words, if

α(t, x, τ ) =  a, δ(τ − ψ) , then for small |t| we have, modulo smooth errors  ∞  it√−g  e (x, x) = eitτ α(t, x, τ ) dτ .

(7.4.27)

(7.4.28)

0

Note that since ∂τ χ[0,∞) = δ, we can also rewrite α here as  ∂ α(t, x, τ ) = a(t, x, ξ ) dξ . ∂τ ψ 0.

(7.4.30)

Also, for t = 0, we have α(0, x, τ ) =  1, δ(τ − p(x, ξ )) ,

(7.4.31)

and ∂t α(0, x, τ ) = 0 modulo terms of order ≤ n − 2.

(7.4.32)

Proof of Lemma 7.4.5 Since a is a symbol of order 0, (7.4.30) follows immediately from (7.4.27). So does (7.4.31), after recalling that a(0, x, ξ ) ≡ 1 and ψ(0, x, ξ ) = p(x, ξ ). To prove (7.4.32), we use (7.4.29) to get    ∂ ∂ χ[0,∞) τ − ψ a dξ ∂t α(t, x, τ ) = ∂τ ∂t  )   ∂ ( χ[0,∞) τ − ψ ∂t a dξ − a ∂ψ = , δ(τ − ψ) . (7.4.33) ∂t ∂τ

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If we use the last part of (7.4.25), we find that when t = 0 9 8  1 dS ∂p ∂p ∂ψ , −a ∂t , δ(τ − ψ) = − 2 p(x,ξ )=τ ∂x ∂ξ |∇ξ p(x, ξ )|  n  1 ∂ 2p =− dξ , 2 p(x,ξ ) 2t. Because of this, and the fact that here we also have

u( · , t) L2 (Rn ) ≡ f L2 (Rn ) , one concludes that u(x, t) can, in general, only

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2 (Rn × R) if f is in L2 (Rn ). It is not also hard to see that if α = be in Lloc p (n − 1)|1/p − 1/2|, then, for 1 < p < 2 one can in general only say that p u(x, t) ∈ Lαp ,loc (Rn × R) if f ∈ Lp (Rn ), which is just a trivial consequence of Theorem 6.2.1. For 2 < p < ∞, the situation is much different. For every p there is an p ε = ε(p, n) such that u(x, t) ∈ Lε−αp ,loc (Rn × R) if f ∈ Lp (Rn ). Since we saw p

at the end of Section 6.2 that, in general, u(·, t) only belongs to L−αp ,loc (Rn ) √

if f ∈ Lp (Rn ), we conclude that, when measured in Lp , p > 2, eit − f has much better properties as a distribution in (x, t), rather than in x alone. In two dimensions this local smoothing estimate just follows from inequality (2.4.27); however, we shall see that this estimate holds for variable coefficient Laplacians in all dimensions. The class of Fourier integral operators we shall deal with satisfy the so-called cinematic curvature condition. This is just the natural homogeneous version of the Carleson–Sj¨olin condition for non-homogeneous oscillatory integral operators that was stated in Section 2.2.

8.1 Local Smoothing in Two Dimensions and Variable Coefficient Nikodym Maximal Theorems Before we can state the local smoothing estimates we need to go over the hypotheses. From now on Y and Z are to be C∞ paracompact manifolds of dimension n and n + 1, respectively. As usual, we assume that n ≥ 2. We shall consider a class of Fourier integral operators1 I μ−1/4 (Z, Y; C), which is determined by the properties of the canonical relation C. Notice that our assumptions imply that C ⊂ T ∗ Z\0 × T ∗ Y\0 is a conic submanifold of dimension 2n + 1. To guarantee nontrivial local regularity properties of operators F ∈ μ−1/4 (Z, Y; C), we shall impose conditions on C that are based on the I properties of the following three projections:  T ∗ Y\0

C ↓ Z



Tz∗0 Z\0.

(8.1.1)

1 To be consistent with the convention regarding the orders of Fourier integral operators given

in Chapter 6, we prefer to state things in terms of order μ − 1/4 so that when the operators are written in terms of oscillatory integrals with n theta variables, such as when one uses a generating function, the symbols will have order μ. The possible confusion arises from the fact that Z and Y have different dimensions.

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8.1 Local Smoothing in Two Dimensions

235

The condition has two parts: first, a natural non-degeneracy condition that involves the first two projections, and, second, a condition involving the principal curvatures of the images of the projection of C onto the fibers of T ∗ Z\0. To describe the first condition, let T ∗ Y and Z denote the first two projections in (8.1.1). We require that they both be submersions, that is, rank dT ∗ Y ≡ 2n,

(8.1.2)

rank dZ ≡ n + 1.

(8.1.3)

Together, these make up the non-degeneracy requirement. As a side remark, let us point out that if Y and Z were of the same dimension, then, as we saw in the Chapter 6, (8.1.2) would imply that C is locally a canonical graph. Also, in this case the differential in (8.1.3) would automatically be surjective; however, since we are assuming that the dimensions are different, (8.1.2) does not imply that Z is a submersion. In order to describe the curvature condition, let z0 ∈ Z C and let Tz∗ Z be 0 the projection of C onto the fiber Tz∗0 Z\0. Then, clearly, z0 = Tz∗ Z (C) 0

(8.1.4)

is always a conic subset of Tz∗0 Z\0. In fact, z0 is a smooth immersed hypersurface in Tz∗0 Z\0. In order to see this, note that the first assumption, (8.1.2), implies that the differential of the projection of C onto the whole space T ∗ Z\0 must have constant rank 2n + 1. (See formulas (8.1.7) and (8.1.2 ) below.) Furthermore, since the differential of C → T ∗ Z\0 splits into the differential in the Z direction and in the fiber direction, we see that in view of (8.1.3), rank dTz∗ Z ≡ n 0

(8.1.5)

and therefore, by the constant rank theorem, z0 is a smooth conic n-dimensional hypersurface. Now in addition to the non-degeneracy assumption we shall impose the following. Cone condition : For every ζ ∈ z0 , n − 1 principal curvatures

(8.1.6)

do not vanish. Since z0 is conic, this is the maximum number of curvatures that can be nonzero. Clearly (8.1.6) does not depend on the choice of local coordinates in Z since changes of variables in Z induce changes of variables in the cotangent bundle that are linear in the fibers.

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If (8.1.2), (8.1.3), and (8.1.6) are all met then we say that C satisfies the cinematic curvature condition. This condition is of course related to the Carleson–Sj¨olin condition since the non-degeneracy condition is just the homogeneous analog of (2.2.2) and the cone condition is just the homogeneous replacement for the curvature condition (2.2.4). Let us now see how our assumptions can be reformulated in two useful ways, if we use local coordinates. First, we note that we can use the proof of Proposition 6.2.4 to see that (8.1.2) and (8.1.3) imply that, near a given (z0 , ζ0 , y0 , η0 ) ∈ C, local coordinates can be chosen so that C is given by a generating function. Specifically, there is a phase function ϕ(z, η) such that C is parameterized by ϕ(z, η) − y, η, that is, C can be written (locally) as {(z, ϕz (z, η), ϕη (z, η), η) : η ∈ Rn \0 in a conic neighborhood of η0 }. (8.1.7) In this case, condition (8.1.2) becomes ≡ n, rank ϕzη

(8.1.2 )

which of course means that if we fix z0 , then, as before, z0 = {ϕz (z0 , η) : η ∈ Rn \0 in a conic neighborhood of η0 } ⊂ Tz∗0 Z\0 must be a smooth conic submanifold of dimension n. So if z0  ζ = ϕz (z0 , η) and θ ∈ Sn is normal to z0 at ζ , it follows that ±θ are the unique directions for which ∇η ϕz (z0 , η), θ  = 0. (8.1.8) The condition that n − 1 principal curvatures be nonzero at ζ then is just that  rank

∂2  ϕz (z0 , η), θ  = n − 1, ∂ηj ∂ηk

if

η, θ are as in (8.1.8).

(8.1.6 )

It is also convenient to give a formulation that is in the spirit of the wave equation. This involves a splitting of the z variables into “timelike” and “spacelike” directions. If, as above, we work locally, then (8.1.2 ) guarantees that we can choose coordinates z = (x, t) ∈ Rn × R vanishing at a given point z0 such that, first of all, rank ϕxη ≡ n, (8.1.9) and second ϕt  = 0,

if

η  = 0.

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8.1 Local Smoothing in Two Dimensions

237

In other words, 0 must be of the form 0 = {(ϕx (0, η), q(ϕx (0, η))}, for some q satisfying q(ξ )  = 0 if ξ  = 0. This is because, if (ξ , τ ) are the variables dual to (x, t), then 0 does not intersect the τ axis. The cone condition, (8.1.6), just translates here to the condition that rank q ξ ξ ≡ n − 1. Since x,t must have the same form for small (x, t), we see that local coordinates can always be chosen so that C is of the form C = {(x, t, ξ , τ , y, η) : (x, ξ ) = χt (y, η), τ = q(x, t, ξ )  = 0},

(8.1.10)

where, by (8.1.9), χt and

is a canonical transformation rank q ξ ξ ≡ n − 1.

(8.1.2 ) (8.1.6 )

We can now state the main result of this chapter. Theorem 8.1.1 Suppose that F ∈ I μ−1/4 (Z, Y; C) where, as before, C satisfies the non-degeneracy condition (8.1.2), (8.1.3), and the cone condition (8.1.6). p p Then F : Lcomp (Y) → Lloc (Z) if μ ≤ −(n − 1)(1/2 − 1/p) + ε and ε < ε(p), with ⎧ 1 ⎪ 4 ≤ p < ∞, ⎪ ⎨ 2p , ε(p) = (8.1.11)   ⎪ ⎪ ⎩ 1 1 − 1 , 2 < p ≤ 4. 2

2

p

Remark This result is not sharp at least for the model case where F = √ n it − R e , t  = 0, where  = Rn denotes the Euclidean Laplacian with n = 2, due to results of Wolff [5], Bourgain and Demeter [1] and others. A natural conjecture would be that for p ≥ 2n/(n − 1) and F is as above one should be able to take ε(p) = 1/p in the theorem. If this result were true, one could use it to prove sharp estimates for Riesz means in Rn by estimating the operators in (2.4.5) using Minkowski’s integral inequality and a scaling argument. Also, one can use the counterexample that was used to prove the sharpness of Theorem 6.2.1 to see that for 2 < p < 2n/(n − 1) there cannot be local smoothing of all orders < 1/p, and in fact the best possible result would just follow from interpolating with the trivial L2 estimate if the conjecture for 2n/(n − 1) were true. By counterexamples in §9.3, √ one cannot have local it −g , t  = 0, for certain smoothing of all orders < 1/p if n ≥ 3 and F = e Laplace–Beltrami operators g . Different arguments are needed to handle two dimensions versus higher dimensions. So we shall prove the two-dimensional case in this section and

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then turn to the case of higher dimensions in Section 2. Before turning to the proofs, though, let us state a corollary. If M is either a C∞ compact manifold of dimension n or Rn we consider the Cauchy problem ⎧ ⎪ ⎪((∂/∂t)2 − g )u(x, t) = 0, ⎨ (8.1.12) ⎪ ⎪ ⎩u = f , (∂/∂t)u = g. t=0

t=0

Here g is a Laplace–Beltrami operator that is assumed to be the usual Laplacian in the Euclidean case. By the results at the end of Section 6.2, p p p it follows that if f ∈ Lα (M) and g ∈ Lα−1 (M) then u(·, t) ∈ Lα−αp (M) if αp = (n − 1)|1/p − 1/2|. Furthermore, this result is sharp for all nonzero times in the Euclidean and all but a discrete set of times in the compact case. Since the canonical relation for the solution to the wave equation has the form  jk g (x)ξj ξk , the solution operator satisfies the cinematic (8.1.10) with q = curvature condition. Therefore, Theorem 8.1.1 gives local smoothing for the wave equation. Corollary 8.1.2 Let u be the solution to the Cauchy problem (8.1.12). Then, if I ⊂ R is a compact interval and if ε < ε(p),   p p

u Lp (M×I) ≤ C f Lα (M) + g L (M) , 2 < p < ∞. α−αp +ε

α−1

p

p

Similarly, eitP belongs to Lα−αp +ε,loc (M × R) if f ∈ Lα (M), provided that P ∈ 1 cl (M) is self-adjoint and elliptic and the cospheres x = {ξ ∗ Tx M\0 all have non-vanishing Gaussian curvature.

: p(x, ξ ) = ±1} ⊂

Orthogonality Arguments in Two Dimensions We now turn to the first step in the proof of the local smoothing estimates for n = 2. Since local coordinates can be chosen so that C is of the form (8.1.7), we can use the proof of Proposition 6.1.4 to conclude that we may assume that F is of the form  eiϕ(x,t,η) a(x, t, η)fˆ (η) dη, Ff (x, t) = R2

where the phase function ϕ satisfies (8.1.2 ) and (8.1.6 ). Since we are now dealing with the two-dimensional case, a is assumed to be a symbol of order zero that vanishes for x ∈ R2 and t ∈ R outside fixed compact sets and η outside a fixed neighborhood of the η2 axis. With this normalization, we shall show

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8.1 Local Smoothing in Two Dimensions

239

that, for ε > 0,

Ff L4 (R3 ) ≤ Cε f L4

1/8+ε (R

2)

.

(8.1.13)

In proving this, we may assume that f is supported in a fixed compact set since the kernel K(x, t; y) of F is O(|y|−N ) for sufficiently large y. Since α4 = 1/4, (8.1.13) implies the two-dimensional local smoothing for 4 L . The other estimates follow by analytic interpolation using

Ff L2 (R3 ) ≤ C f L2 (R2 )

Ff Lp (R3 ) ≤ C f Lαp

p (R

2)

.

Both of these estimates follow from Theorem 6.2.1 since, for fixed times t, f → Ff ( · , t) is a Fourier integral operator of order zero that is locally a canonical graph. Following the proof of (2.4.27), we fix β ∈ C0∞ ((1/2, 2)) and set aλ (x, t, η) = β(|η|/λ)a(x, t, η). If we define  Fλ f (x, t) = eiϕ(x,t,η) aλ (x, t, η)fˆ (η) dη, then, by summing a geometric series, we see that (8.1.13) must follow from the estimates

Fλ f L4 (R3 ) ≤ Cε λ1/8+ε f L4 (R2 ) ,

λ > 1, ε > 0.

(8.1.14)

To prove (8.1.14) we need to use the angular decomposition that was used in the proof of Theorem 6.2.1. So, if we identify R2 and C in the obvious way, 1/2 then for ν = 0, 1 . . . , [log λ1/2 ] we let ηλν = e2π iν/λ . Then, we choose functions  χλν ∈ C∞ (R2 \0) which are homogeneous of degree zero, satisfy ν χλν (η) = 1, and have the property that χλν (ηλν )  = 0 but χλν (η) = 0 if |η| = 1 and |η − ηλν | ≥ Cλ−1/2 , and, lastly, |∂ α χλν (η)| ≤ Cα λ|α|/2 , for every α if |η| = 1. As before, using the χλν we make an angular decomposition of the operators by setting  F ν f (x, t) = eiϕ(x,t,η) χ ν (η)aλ (x, t, η)fˆ (η) dη. λ

λ

The square of the left side of (8.1.14) is dominated by      Fλν1 f Fλν2 f  2 3 .  k

|ν1 −ν2 |≈2k

L (R )

Since we are assuming that a has small conic support the summation only involves indices k < log λ1/2 . We shall need to make a further decomposition based on k. To this end, we  fix ρ ∈ C0∞ ((−1, 1)) satisfying ρ(u) = 1 for |u| < 1/4 and j∈Z ρ(u − j) ≡ 1.

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We then set



ν,j

ν,j

Fλ,k f (x, t) =

eiϕ(x,t,η) aλ,k (x, t, η)fˆ (η) dη,

with   ν,j aλ,k (x, t, η) = χλν (η) ρ λ−1 2k ϕt (x, t, η) − j aλ (x, t, η).

(8.1.15)

Note that this decomposition involves localizing τ = ϕt to intervals of height ≈ λ2−k . These are larger than the intervals of height λ1/2 that were used in Section 2.4 to prove the constant coefficient local smoothing estimates. Since ϕt is not assumed to be constant in x and t, this different localization is needed for the integration by parts arguments that are to follow. j Let “∼” denote the partial Fourier transform in the t variable. Then if Iλ,k is the interval of length 2−k λ1+ε and center j2−k λ, the proof of Lemma 2.4.4 shows that  ∼  1 ν,j ν,j ν,j iτ t Fλ,k (f )(x, t) = e (f ) (x, τ ) dτ + Rλ,k (f )(x, t), F λ,k j 2π τ ∈Iλ,k ν,j

ν f = where Rλ,k has an L4 → L4 norm which is O(λ−N ). Thus, since Fλ,k  ν,j j≈2k Fλ,k (f ), one can use Plancherel’s theorem in the t variable, as in (2.4.28), to reach the desired conclusion

Fλ f 2L4 (R3 )

⎛ 2 ⎞1/2            ⎟   ν ,j ν ,j ε k/2 ⎜ 1 1 2 2  ≤ Cλ 2 ⎝ Fλ,k (f )Fλ,k (f ) ⎠      k  j1 ,j2 |ν −ν |≈2k  1

2

L2 (R3 )

+CN λ−N f 2L4 (R2 ) . On account of this, the desired inequality, (8.1.14), would follow if we could show that there is a fixed constant C, independent of λ and k, for which ⎛ 2 ⎞1/2           ⎟  ⎜  ν ,j ν ,j 1 1 2 2  Fλ,k (f )Fλ,k (f ) ⎠  ⎝      j1 ,j2 |ν1 −ν2 |≈2k 

L2 (R3 )

ε

−k 1/2 1/2

≤ Cλ (2 λ

)

f 2L4 (R2 ) .

(8.1.16)

This is the main inequality. Notice that, in going from (8.1.14) to (8.1.16), we have lost a factor of 2k/4 in the L4 bounds.

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241

To prove (8.1.16), we shall need to exploit the cone condition. Recall that this condition says that, if O is a small conic neighborhood of suppη a, then x,t = {(ϕx (x, t, η), ϕt (x, t, η)) : η ∈ O} is a C∞ conic hypersurface in R3 \0 which has the property that at every point (ξ , τ ) ∈ x,t one principal curvature is nonzero. To use this, let us define subsets ν,j

ν,j

x,t = {(ϕx (x, t, η), ϕt (x, t, η)) : η ∈ supp aλ,k },

(8.1.17)

which depend on the scale λ and also on k. By the non-degeneracy hypothesis, ν,j (8.1.9), x,t is contained in a sector around ϕx (x, t, ηλν ), ϕt (x, t, ηλν ) ) which has angle ≈ λ−1/2 . Here ηλν are the unit vectors occurring in the definition of χλν . ν,j ν,j The sets x,t have height ≈ λ2−k . So x,t is basically a λ1/2 × λ2−k rectangle ν,j in the tangent plane to x,t at (ϕx (x, t, ηλν ), ϕt (x, t, ηλν )). On account of this, it is a geometrical fact that, if we consider algebraic sums of the sets in (8.1.17), then if |ν − ν | = dist ((ν1 , ν2 ), (ν1 , ν2 )) is larger than a fixed constant,  ν ,j1 ν ,j2 ν ν  ν ,j ν ,j dist (x,t1 1 + x,t2 2 , x,t1 + x,t2 ) ≥ cλ(ηλν1 + ηλν2 ) − (ηλ1 + ηλ2 ), (8.1.18) where if, as we are assuming, a has small conic support, the constant c > 0 depends only on.ϕ. To see this, let us fix / “heights” τ1 , τ2 ≈ λ so that the cross ν ,j ν ,j sections " x,tl l = (ξ , τ ) ∈ x,tl l : τ = τl are nonempty. Here we are assuming, ν,j as we may, that 0 < ϕt . It is not hard to check that the sets x,t have been . / ν ,j ν ,j chosen to have the largest height so that (ξ , τ ) ∈ x,t1 1 + x,t2 2 : τ = τ1 + τ2 ν ,j ν ,j is contained in a tubular neighborhood of width O(2k ) around " x,t1 1 + " x,t2 2 . Using the curvature of x,t one can prove that if |ν − ν | is larger than a fixed constant    ν ,j1 ν ,j2 ν ν  ν ,j ν ,j dist " x,t1 1 + " x,t2 2 , " x,t1 + " x,t2 ≈ λ(ηλν1 + ηλν2 ) − (ηλ1 + ηλ2 ), which, by the previous observations, leads to (8.1.18), since λ|(ηλν1 + ηλν2 ) − ν

ν

(ηλ1 + ηλ2 )| ≥ 2k |ν − ν |. Next, we claim that (8.1.18) leads to the bounds

  ν ,j1 ν ,j2   ν ,j ν ,j  Fλ,k1 1 (f )Fλ,k2 2 (f ) Fλ,k1 (f )Fλ,k2 (f ) dxdt ≤ CN λ−N f 44 ,

if |ν − ν | ≥ λε . (8.1.19)

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One sees this by first noticing that the left side equals the absolute value of 

ν ,j

ν ,j

1 1 2 2 ei aλ,k (x, t, η)aλ,k (x, t, ξ )



ν1 ,j1 ν2 ,j2 × aλ,k (x, t, η )aλ,k (x, t, ξ ) fˆ (η)fˆ (ξ ) fˆ (η )fˆ (ξ ) dηdξ dη dξ ,

with  = ϕ(x, t, η) + ϕ(x, t, ξ ) − ϕ(x, t, η ) − ϕ(x, t, ξ ). To estimate this term we need the following lower bounds for the gradient of the phase function on the support of the integrand:     ν ν   |∇x,t | ≥ cλ ηλν1 + ηλν2 − ηλ1 + ηλ2   ν  ν ≈ λ λ−1/2 |ν − ν | · max |ηλj − ηλk | , |ν − ν | ≥ c0 (8.1.20)    ν 2 ν |∇x | ≥ c1 (η + ξ ) − (η + ξ ) − c2 λ max |ηλj − ηλk | . (8.1.21) Here cj are fixed positive constants. The first inequality in the first lower bound is equivalent to (8.1.18), while the second inequality there follows from the fact that the unit vectors {ηλν } are separated by angle ≈ λ−1/2 . To prove (8.1.21) one notices that, by (x, t, η) · η. Hence we can write homogeneity, ϕx (x, t, η) = ϕxη   (x, t, ηλν1 ) (η + ξ ) − (η + ξ ) ∇x  = ϕxη   (x, t, η) − ϕxη (x, t, ηλν1 ) η + ϕxη   (x, t, ξ ) − ϕxη (x, t, ηλν1 ) ξ . + · · · − ϕxη  = The first term has absolute value ≥ c1 |(η + ξ ) − (η + ξ )| because det ϕxη ν1 0. Also, both η → (ϕxη (x, t, η) − ϕxη (x, t, ηλ ))η and its first order derivatives vanish when η = ηλν1 . Thus, this term is O(λ · (dist(ηλν1 , η/|η|))2 ) = ν

ν

O(λ · (max|ηλj − ηλk |)2 ). Since the other error terms satisfy the same estimates, (8.1.21) follows. In addition to the lower bounds, we also need the following upper bounds for the derivatives of  on the support of the integrand in (8.1.19):     ν 2 ν α | ≤ Cα (η + ξ ) − (η + ξ ) + λ max |ηλj − ηλk | . |∂x,t

(8.1.22)

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243

Let us first see that this holds when α = 0. To do so, we use Euler’s identity to write # $  = ϕη (x, t, ηλν1 ), ((η + ξ ) − (η + ξ )) $ # + (ϕη (x, t, η) − ϕη (x, t, ηλν1 )), η $ # + · · · − (ϕη (x, t, ξ ) − ϕη (x, t, ηλν1 )), ξ . The first term can be estimated by |(η + ξ ) − (η + ξ )|, while the proof of ν

ν

(8.1.21) shows that the other terms are O(λ · (max |ηλj − ηλk |)2 ). Since we have only used homogeneity, the same argument shows that derivatives in (x, t) of  must satisfy the same type of estimates. The last ingredient is the fact that (8.1.15) implies that ν,j

α |∂x,t aλ,k (x, t, η)| ≤ Cα 2k|α| .

(8.1.23)

By putting (8.1.20)–(8.1.23) together, one gets the claim (8.1.19) via a partial integration. In fact, we can dominate the left side of (8.1.19) by CN λ−N f 42 , which leads to (8.1.19) since we are assuming that f has fixed compact support. Returning to (8.1.16), one sees after an application of (8.1.19) and the Schwarz inequality that the left side is majorized by ⎛ ⎞1/2   2      ν,j λε ⎝ |Fλ,k (f )|2 ⎠  + f 2L4 (R2 ) .    ν,j  4 3 L (R )

This means that we would be done if we could show that ⎛ ⎞1/2       ⎝  ν,j 2⎠ |Fλ,k (f )| ≤ Cλε (2−k λ1/2 )1/4 f L4 (R2 ) .      ν,j  4 3 L (R )

This is the main reduction. To be able to apply the variable coefficient Nikodym maximal estimates that are to follow, we need to make one more reduction, which this time is much easier since it relies only on the fact that ϕt  = 0. The point of this step will be to reduce matters to a square function involving operators whose kernels, for fixed y and ν, are concentrated in small tubular neighborhoods of curves in R2 × R. To this end, if ρ is as in (8.1.15), we define Fourier multiplier operators, Pm , acting on functions of three variables by setting  m ∧ P g (ξ , τ ) = ρ(λ−1/2 τ − m)ˆg(ξ , τ ).

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Using the definition (8.1.15) and the proof of Lemma 2.4.4 one sees that, if j ν,j is fixed, there must be an integer m(j) such that Pm Fλ,k has Lp (R2 ) → Lp (R3 ) norm O(λ−N ) if |m − m(j)| ≥ λ1/2 2−k λε . To prove this claim, one merely uses the fact that the symbol in (8.1.15) vanishes if ϕt does not belong to an interval of length 2−k λ which depends only on j. Using this claim, one repeats the arguments that lead to (8.1.16) to verify that ⎛ ⎞1/2        ⎝ ν,j |Fλ,k (f )|2 ⎠      4 3  ν,j L (R ) ⎛ ⎞1/2         ν,j ≤ Cλε (2−k λ1/2 )1/4 ⎝ |Pm Fλ,k (f )|2 ⎠  + C f L4 (R2 ) .    ν,j,m  4 3 L (R )

Combining this with the last inequality means that we would be done if we could prove that ⎛ ⎞1/2        ⎝ ν,j |Pm Fλ,k (f )|2 ⎠  ≤ Cλε f L4 (R2 ) . (8.1.24)     4 3  ν,j,m L (R )

To prove this, we recall that suppp ρ ⊂ (−1, 1) and ρ(u) = 1 for |u| < 14 . So, ν,j,m ν,j if we define aλ,k = aλ,k (x, t, η)ρ((λ−1/2 ϕt (x, t, η) − m)/8) and then set  ν,j,m ν,j,m Fλ,k (f )(x, t) = eiϕ(x,t,η) aλ,k (x, t, η)fˆ (η) dη, then

 m ν,j  P F − Pm F ν,j,m  λ,k

λ,k

Lp →Lp

= O(λ−N ).

Consequently, (8.1.24) must follow from ⎛ ⎞1/2        ⎝ m ν,j,m 2⎠ |P Fλ,k (f )| ≤ Cλε f L4 (R2 ) .      4 3  ν,j,m

(8.1.24 )

L (R )

To prove this, a couple of observations are in order. We first notice that if m ν,j,m is fixed then there are O(1) indices j for which Fλ,k  = 0. Also, the symbols, ν,j,m

aλ,k (x, t, η), of these operators vanish if ϕt ∈ / [λ1/2 m − 8λ1/2 , λ1/2 m + 8λ1/2 ]. In addition, if we let . / Q(x, t, ν, m) = η : χλν (η)ρ((λ−1/2 ϕt (x, t, η) − m)/8)  = 0 ,

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245

then this set is comparable to a cube of side-length λ1/2 . In particular, each one intersects at most C0 cubes in a given λ1/2 lattice of cubes in R2 . Since ϕt is smooth, C0 can be chosen to be independent of the relatively compact set K = {(x, t) : a(x, t, η)  = 0}. With this in mind, we set fˆμ (η) = ρ(λ−1/2 η1 − μ1 )ρ(λ−1/2 η2 − μ2 )fˆ (η),

μ = (μ1 , μ2 ) ∈ Z2 .

 Thus f = fμ . In addition, if, for a given (x, t, ν, m), we let I(x, t, ν, m) ⊂ Z2 ν,j,m denote those μ for which Fλ,k fμ (x, t)  = 0, it follows from the properties of the Q(x, t, ν, m) that   Card I(x, t, ν, m) ≤ C1 ,

(8.1.25)

where C1 is an absolute constant. Similarly, for fixed ν and (x, t) ∈ K there must be a uniform constant C2 such that . / Card m : μ ∈ I(x, t, ν, m) ≤ C2

∀ μ ∈ Z2 .

(8.1.26)

Finally, we let J (ν) denote those μ for which β(|η|/λ)χλν (η)× ρ(λ−1/2 η1 − μ1 )ρ(λ−1/2 η2 −η2 ) does not vanish identically and note that μ belongs to only finitely many of the sets J (ν) since supp β(|η|/λ)χλν (η) is contained in the set {η : |η| ≈ λ, and |ηλν − η/|η| | ≤ Cλ−1/2 }. ν,j,m ν,j,m We now turn to (8.1.24 ). Let Kλ,k (x, t; y) be the kernel of Pm Fλ,k . We shall see that we have the uniform bounds 

ν,j,m

|Kλ,k (x, t; y)| dy ≤ C.

(8.1.27)

ν,j,m

Thus, since, for fixed m, Fλ,k vanishes for all but finitely many j, the Schwarz inequality, (8.1.25), and (8.1.26) yield  m,j

2          ν,j,m ν,j,m |Pm Fλ,k f (x, t)|2 ≤ C fμ (y) |Kλ,k (x, t; y)| dy    μ∈I(x,t,ν,m) m,j   μ∈J (ν)    ν,j,m ≤ C | fμ (y)|2 |Kλ,k (x, t; y)| dy m,j

≤ C



μ∈I(x,t,ν,m) μ∈J (ν)



μ∈J (ν)

ν,j,m

| fμ (y)|2 sup |Kλ,k (x, t; y)| dy. m,j

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From this we see that for a given g(x, t)     2     m ν,j,m P F    λ,k f (x, t) g(x, t) dxdt    ν,j,m 1 0   ν,j,m 2 | fμ (y)| sup sup |Kλ,k (x, t; y)| |g(x, t)| dxdt dy. ≤C μ

ν

R3 j,m

(8.1.28)  We have already seen in Lemma 2.4.6 that ( μ | fμ |2 )1/2 4 ≤ C f 4 . Therefore, since the left side of (8.1.24 ) is dominated by the supremum over all g 2 = 1 of the left side of (8.1.28), we would be done if we could prove the maximal inequality ⎛ 2 ⎞1/2       ν,j,m ⎝ sup  sup |Kλ,k (x, t; y)| g(x, t) dxdt dy⎠   2 3 R ν R j,m ≤ C| log λ|3/2 g L2 (R3 ) .

(8.1.29)

To prove the missing inequalities (8.1.27) and (8.1.29) we need to introduce some notation. If N ⊂ R3 × R2 \0 is a small conic neighborhood of supp aλ , we define the smooth curves γy,n = {(x, t) : ϕη (x, t, η) = y, (x, t, η) ∈ N }.

(8.1.30)

Then one estimate we need is the following. Lemma 8.1.3 If ηλν are the unit vectors occurring in the definition of Fλν , then, given any N,  −N ν,j,m . (8.1.31) |Kλ,k (z, y)| ≤ CN λ 1 + λ1/2 dist (z, γy,ηλν ) −1/2

ˆ 1/2 (s − t))eimλ (t−s) . it Proof Since the kernel of Pm is δ0 (x, y)λ1/2 ρ(λ ν,j,m ν,j,m suffices to show that the kernel " Kλ,k (x, t; y) of Fλ,k satisfies the estimates in (8.1.31). But  ν,j,m ν,j,m " Kλ,k (x, t; y) = ei[ϕ(x,t,η)−y,η] aλ,k (x, t, η) dη. R2

After using a finite partition of unity, we may assume that the (x, t)-support of  = 0, the symbol is small. Since we are assuming that ϕt  = 0 and hence ϕtη it follows from the implicit function theorem that γy,ηλν is of the form (x(t), t)

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247

near K = supp x,t a if this set is small enough. Also, we are assuming that det  = 0, so if K is small enough there must be a c > 0 such that ϕxη  * + ∇η ϕ(x, t, η) − y, η  ≥ c|x − x(t)|, (x, t) ∈ K, η = ην . λ But ∇η ϕ is homogeneous of degree zero and therefore |∇η ϕ(x, t, η) − ∇η ϕ(x, t, ηλν )| ≤ Cλ−1/2 ,

η ∈ supp χλν .

Combining these two bounds shows that there is a c > 0 such that |∇η [ϕ(x, t, η) − y, η]| ≥ c dist ((x, t), γy,ηλν ), provided that dist((x, t), γy,ηλν ) is larger than a fixed multiple of λ−1/2 . Since ν,j,m

∂ηα aλ,k (x, t, η) = O(λ−|α|/2 ) and since, for fixed (x, t), this symbol vanishes ν,j,m for η outside a set of measure O(λ), the desired estimate for " Kλ,k follows from integration by parts. It is clear that (8.1.31) implies the uniform L1 estimates (8.1.27) for the kernels. Using (8.1.31) again, we shall see that the maximal estimates (8.1.29) follow from the Nikodym maximal estimates in the next subsection. Variable Coefficient Nikodym Maximal Functions For later use in proving the higher-dimensional local smoothing estimates, and since the result is of independent interest, we shall state a maximal theorem that is more general than the one needed to prove (8.1.29). We now assume that Z and Y are as in Theorem 8.1.1, with the dimension of Y being n ≥ 2 and dim Z = n + 1. To state the hypotheses in an invariant way, let C satisfy the non-degeneracy conditions (8.1.2) and (8.1.3). It then follows that γy,η = {z ∈ Z : (z, ζ , y, η) ∈ C, some ζ }, is a C∞ immersed curve in Z that depends smoothly on the parameters (y, η) ∈ T ∗ Y (C). Let us fix a smooth metric on Z and define the “δ-tube” Rδy,η = {z : dist (z, γy,η ) < δ}. The variable coefficient maximal theorem then is the following. Theorem 8.1.4 Let C satisfy the non-degeneracy conditions (8.1.2) and (8.1.3) as well as the cone condition (8.1.6). Then, if 0 < δ < 12 and

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α ∈ C0∞ (Y × Z) 2 ⎞1/2     1   ⎝ sup  α(y, z)g(z) dz  dy⎠ δ   δ Y η∈ ∗ (C ) Vol(Ry,η ) Ry,η ⎛



Ty Y

≤ Cδ −

n−2 2

3

| log δ| 2 g L2 (Z) .

(8.1.32)

Since the canonical relation associated to the operator in the proof of the two-dimensional version of Theorem 8.1.1 is given by (8.1.7) (with z = (x, t)), it is clear from (8.1.31) that (8.1.32) implies (8.1.29) by taking δ = λ−1/2 , 2λ−1/2 , . . . . Before turning to the proof, let us state one more consequence. If (Y, g) is a compact Riemannian manifold, then for a given y ∈ Y and θ ∈ Ty Y, let γy,θ (t) be the geodesic starting at y in the direction θ which is parameterized by arclength. Then, if we fix 0 < T < ∞ and let Rδy,η = {(x, t) : dist (x, γy,θ (t)) < δ, 0 ≤ t ≤ T}, one can use the Legendre transform, mapping TY → T ∗ Y, (see, e.g., Sternberg [1]) to see that a special case of Theorem 8.1.4 is    n−2 3 1   |g(z)| dxdt  2 ≤ Cδ − 2 | log δ| 2 g L2 (Y×R) . sup δ L (Y) θ Vol(Ry,θ ) Rδy,θ Proof of Theorem 8.1.4 For the sake of clarity, let us first present the arguments in the case where n = 2 and then explain the modifications that are needed to handle the higher-dimensional case. We write z = (x, t) in what follows. We may work locally and assume that γy,η is of the form (8.1.30) where of course we are now assuming that N ⊂ R3 × R2 \0. We may assume that (0, 0, 0; 1, 0) ∈ N . Also, by (8.1.2 ) and (8.1.6 ), we may assume for the sake of convenience that coordinates have been chosen so that ⎛ ⎞ 1 0 ∂2 = ⎝ 0 1 ⎠ and ϕz  = 0 at z = 0, η = (1, 0). (8.1.33) ϕzη ∂η22 3 0 0 To proceed, we fix a ∈ C0∞ (R2 ) satisfying aˆ ≥ 0. Then for α(z, θ ) ∈ C0∞ supported in a small neighborhood of (0, 0) we put αδ (z, θ ; η) = α(z, θ )a(δη)

(8.1.34)

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8.1 Local Smoothing in Two Dimensions and define



Aδ g(y, θ ) =





R3 R2

ei[ϕη (z,cos θ,sin θ),η−y,η] αδ (z, θ ; η) g(z) dηdz.

249

(8.1.35)

This operator dominates the averaging operator in (8.1.32) if the bump function there has small support and if δ is replaced by a multiple of δ (depending on a). So it suffices to show that 1/2  sup |Aδ g(y, θ )|2 dy ≤ C| log δ|3/2 g L2 (R3 ) . (8.1.36) R2 θ

To prove this we shall want to use the following special case of Lemma 2.4.4: 

1/2 

1/2 | F|2 ds | F |2 ds sup | F(θ )|2 ≤ 2 θ∈R

if F ∈ C1 (R) and

F(0) = 0.

(8.1.37)

Before applying this, though, we need to break up the operators Aδ . To do this,  as before, we let β ∈ C0∞ (R\0) satisfy k β(2−k s) = 1, s  = 0. We then define dyadic operators  Aλδ g(y, θ ) = ei[ϕη (z,cos θ,sin θ),η−y,η] β(|η|/λ)αδ (z, θ ; η) g(z) dηdz. Then, using Plancherel’s theorem one sees that (8.1.36) would follow from Schwarz’s inequality and   sup |Aλ g(y, θ )| 2 2 ≤ C log λ g 2 3 , λ > 2. (8.1.36 ) L (R ) δ L (R ) θ

We need to make one more reduction. If we set (z; η, θ ) = ϕη (z, cos θ , sin θ ), η,

(8.1.38)

it is based on the observation that (8.1.8), (8.1.33), and homogeneity imply that, at z = 0,  z3 θ = 0  z3 θθ

when θ = 0 ⇐⇒ (1, 0) = ±η/|η|,

 = 0 when θ = 0 and (1, 0) = ±η/|η|.

Hence, (8.1.33) also implies that, for ±η/|η| = (1, 0) and θ ∈ suppθ αδ ,      ∂ 2    ≈ |η| |θ |. det  ∂z∂(η, θ )  More generally, if we change variables (η, θ ) → (η, ") = (η, θ − arcsin(η2 /|η|)),

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where arcsin is the inverse with values in [−π/2, π/2], we get      ∂ 2    ≈ |η| |"|, det  ∂z∂(η, ") 

(8.1.39)

provided that η is outside a conic neighborhood of the η2 axis such that  is C∞ in these coordinates. On the other hand, if (θ, η) ∈ suppθ,η αδ is in a sufficiently small neighborhood of the η2 axis, | det ∂ 2 /∂z∂(η, θ )| ≈ |η|. Another important fact, which follows from the homogeneity of ϕ, is that ∂  = 0 when " = 0. ∂" Based on (8.1.39) and (8.1.40) we set for k = 1, 2, . . .  λ,k ei[ϕη (z,cos θ,sin θ),η−y,η] Aδ g(y, θ ) =

(8.1.40)

×β(2−k λ1/2 ")β(|η|/λ)αδ (z, θ ; η) g(z) dηdz, (8.1.41) 

λ,k λ and Aλ,0 k≥1 Aδ . Since " is bounded, there are O(log λ) terms and δ = Aδ − hence (8.1.36 ) would follow from the uniform estimates   sup |Aλ,k g(y, θ )| 2 2 ≤ C g 2 3 , k = 0, 1, 2, . . . . (8.1.36 ) L (R ) δ L (R ) θ

We can now invoke (8.1.37). By applying it and Schwarz’s inequality we see that (8.1.36 ) would be a consequence of the estimates ⎞1/2 ⎛ 2   j    ∂ ⎠ ⎝ Aλ,k  δ g(y, θ ) dθ dy  ∂θ ≤ C(λ−1/4 2−k/2 )1−2j g L2 (R3 ) ,

j = 0, 1.

(8.1.42)

However, (8.1.40) implies that, on the supports of the symbols (∂/∂θ)ϕη (z, cos θ , sin θ ), η = O(λ1/2 2k ) and hence (∂/∂θ)Aλ,k δ behaves like

λ1/2 2k Aλ,k δ , so we shall only prove the estimate for j = 0. It turns out to be easier to prove the estimate for the adjoint operator because this allows us to use the Fourier transform. Specifically, if “ " ” denotes the partial Fourier transform with respect to y, then the desired estimate is equivalent to    eiϕη (z,cos θ,sin θ),η β(2−k λ1/2 ")    f (η, θ ) dηdθ  ≤ Cλ−1/4 2−k/2 f L2 . ×β(|η|/λ)αδ (z, θ ; η)"  L2 (dz)

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8.1 Local Smoothing in Two Dimensions

251

However, if we make the change of variables η → λη, θ → ", this in turn would be a consequence of the following. Lemma 8.1.5 Let bk (z; η, ") satisfy |∂zα bk | ≤ Cα ∀ α, as well as bk = 0 if either |z| > 1, |η| ∈ / [1, 2], or " ∈ / [2k−1 λ−1/2 , 2k λ−1/2 ] when k = 1, 2, . . . , [2π log λ1/2 ] −1/2 ] when k = 0. Then, if we also assume that bk vanishes in a and " ∈ / [0, λ neighborhood of the η2 axis (so that (z; η, ") is C∞ on supp bk ), it follows that  eiλ(z;η,") bk (z; η, ")f (η, ") dηd" T λ,k f (z) = R3

satisfies

T λ,k f L2 (R3 ) ≤ Cλ−1 (λ−1/2 2−k )1/2 f L2 (R3 ) .

(8.1.43)

On the other hand, if r(z; η, θ ) = 0 for |η| ∈ / [1, 2] and η outside a small neighborhood of the η2 axis, or for θ outside a small neighborhood of the origin, then  Rλ f (z) =

R3

eiλ(z;η,θ) r(z; η, θ )f (η, θ ) dηdθ ,

satisfies Rλ f L2 (R3 ) ≤ Cλ−3/2 f L2 (R3 ) . Proof Let us first prove the estimates for the operators T λ,k . We use a modification of the argument used in the proof of the Carleson–Sj¨olin theorem (Theorem 2.2.1, (2)). We first note that, after perhaps using a smooth partition of unity, we may assume that bk has small support. Then the square of the left side of (8.1.43) equals   H λ,k (η, "; η , " )f (η, ")f (η , " ) dηd"dη d" , (8.1.44) R3 R3

with H λ,k =

 R3





eiλ[(z;η,")−(z;η ," )] bk (z; η, ")bk (z; η , " ) dz.

To estimate this kernel, we note that (8.1.40) implies that, for " ≥ 0,  is a C1 function of "2 . But (8.1.39) implies that, in the coordinates where " ≥ 0 is replaced by "2 , the Hessian of  is non-singular. This and the fact that  is C1 in (η, "2 ) implies that there must be a c > 0 such that  * + ∇z (z; η, ") − (z; η , " )  ≥ c|(η − η , "2 − (" )2 )| on the support of the symbol if bk has small enough support. Consequently, since both  and the bk are uniformly smooth in the z variables, a partial integration gives the bounds |H λ,k | ≤ CN (1 + λ|η − η |)−N (1 + λ|"2 − (" )2 |)−N ,

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252

Local Smoothing of Fourier Integral Operators

for any N. But if k  = 0, |"2 − (" )2 | ≈ λ−1/2 2k |" − " | on the support of the kernel and hence the L1 norm with respect to either of the pairs of variables is O(λ−2 2−k λ−1/2 ). One reaches the same conclusion for k = 0 if one recalls that in this case the kernel vanishes for |"| > λ−1/2 . Using these estimates, one concludes that (8.1.44) must be ≤ Cλ−2 2−k λ−1/2 f 22 , which finishes the proof of (8.1.43). If we recall that our assumptions imply that det ∂ 2 (z; η, θ )/∂z∂(η, θ )  = 0 on supp r, modifications of this argument give the estimate for Rλ . The proof of the higher-dimensional maximal estimates follows the same lines. First, if ω(θ) are local coordinates near ω(0) = (1, 0, . . . , 0) ∈ Sn−1 , then one sets   ei[ϕη (z,ω(θ)),η−y,η] αδ (z, θ ; η) g(z) dηdz, Aδ g(y, θ ) = Rn+1 Rn

where now αδ (z, θ ; η) = α(z, θ )a(δη) with α ∈ C0∞ supported near the origin and a ∈ C0∞ (Rn ) satisfying aˆ ≥ 0. The maximal inequality would then follow from     sup |Aδ g(y, θ )| 2 n ≤ Cδ −(n−2)/2 | log δ|3/2 g L2 (Rn+1 ) . L (R )

θ

This time we shall want to use the following higher-dimensional version of (8.1.37): sup | F(θ )|2 ≤ Cn−1

θ∈Rn−1

 |α|+|β|≤n−1



|∂θα F|2 dθ

1/2 

β

1/2

|∂θ F|2 dθ

,

(8.1.37 )

if F(θ ) = 0 when θj = 0 for some 1 ≤ j ≤ n − 1. To apply it, as before, we must break up the operators. First, if Aλδ are the dyadic operators, it suffices to show that the associated maximal operators send L2 (Rn+1 ) → L2 (Rn ) with norm O(λ(n−2)/2 log λ). To see this, we note as before that (8.1.2 ) and (8.1.6 ) imply that (z; η, θ ) = ϕη (z, ω(θ)), η satisfies rank ∂ 2 /∂z∂(η, θ ) = n + 1 unless ω(θ) and η are colinear. So if  is the complement of a conic neighborhood of (1, 0, . . . , 0) = ω(0) that contains {ω(θ) : θ ∈ suppθ α}, and if we set  ei(z;η,θ) β(|η|/λ)χ (η)αδ (z, θ ; η)g(z) dηdz, Rλδ g(y, θ ) =

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8.2 Local Smoothing in Higher Dimensions then



253

1/2 |∂θα Rλδ g(y, θ )|2 dθ dy ≤ Cλ|α|−1/2 g 2 .

This follows from the argument used to estimate the remainder terms in Lemma 8.1.5. By (8.1.37 ) the maximal operator associated to Rλδ has L2 bound Aλδ be the difference between Aλδ and Rλδ , it suffices O(λ(n−2)/2 ). So, if we let " to show that   sup |" Aλδ g(y, θ )|2 ≤ Cλ(n−2)/2 log λ g 2 . (8.1.45) θ

If we let (θ , η) = min± dist (ω(θ ), ±η/|η|), then, by the above discussion, we may assume that (θ , η) is small on the support of the symbol, " αδ , of " Aλδ , and, hence, |∂θα | ≤ Cα ||1−|α| there. With this in mind, we then set for k = 1, 2, . . . " Aλ,k δ g(y, θ )  αδ (z, θ ; η) g(z) dηdz, = ei(z;η,θ) β(2−k λ1/2 (θ , η))β(|η|/λ)"  "λ "λ,k and " Aλ,0 k≥1 Aδ . Then arguing as before shows that (8.1.45) follows δ = Aδ − from the estimates 

1/2 α"λ,k 2 |∂θ Aδ g(y, θ )| dθ dy ≤ C(λ−1/4 2−k/2 )1−2|α| g 2 . 1/2 2k )|α|" Aλ,k Aλ,k Since ∂θα" δ ≈ (λ δ , the arguments for α = 0 give the bounds for general α. To prove this, just like before, one estimates the adjoint operator in order to use the Fourier transform. Since we are assuming that ϕ satisfies (8.1.2 ) and (8.1.6 ), the arguments that were used to handle the two-dimensional case can easily be modified to show that this operator satisfies the desired estimates. 

8.2 Local Smoothing in Higher Dimensions In this section we shall prove the local smoothing estimates in Theorem 8.1.1 corresponding to n ≥ 3. The orthogonality arguments are much simpler here because we can use a variable coefficient version of Strichartz’s L2 restriction theorem for the light cone in Rn+1 . Applying this L2 → Lq local smoothing theorem, we use a variation of the arguments in Section 5.2 that showed how, in the favorable range of exponents for the Lp → L2 spectral projection theorem, one could deduce sharp Lp → Lp estimates for Riesz means from Lp → L2 estimates for the spectral projection operators.

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254

Local Smoothing of Fourier Integral Operators

To prove the higher-dimensional Lp → Lp local smoothing theorem, in addition to the Nikodym maximal estimates just proved, we shall need the following sharp L2 → Lq local smoothing estimates whose straightforward proof will be given at the end of this section. Theorem 8.2.1 Suppose that F ∈ I μ−1/4 (Z, Y; C) and that C satisfies the non-degeneracy assumptions (8.1.2) – (8.1.3) and the cone condition (8.1.6). q 2 (Y) → Lloc (Z) if 2(n + 1)/(n − 1) ≤ q < ∞ and μ ≤ −n(1/2 − Then F : Lcomp 1/q) + 1/q. Remark Using the Sobolev embedding theorem and the L2 bound-edness of Fourier integral operators, it is not hard to see that if one just assumes (8.1.2) q 2 (Y) → Lloc (Z) if μ ≤ −n(1/2 − 1/q) and 2 ≤ q < and (8.1.3) then F : Lcomp ∞. Thus, Theorem 8.2.1 says that, under cinematic curvature, there is local smoothing of order 1/q for q as in the theorem. Using the counterexample that was used to prove the sharpness of Theorem 6.2.1, one can see that the above L2 → Lq local smoothing theorem is sharp. In the model case, where C is the canonical relation for the solution to the wave equation in Rn , it is equivalent to Strichartz’s restriction theorem

1/2  2 dξ |ˆg(ξ , |ξ |)| ≤ C g Lp (Rn+1 ) , p = 2(n+1) n+3 . n |ξ | R The reason for this is that, by duality and Plancherel’s theorem, the last inequality is equivalent to    dξ  i[x,ξ +t|ξ |] ˆ  f (ξ ) 1/2   ne |ξ | Lq (Rn+1 ) R ≤ C f L2 (Rn ) ,

q = p =

2(n+1) (n−1) .

However, since −n(1/2 − 1/q) − 1/q = −1/2 for this value of q, one can use the model case of Theorem 8.2.1, together with Littlewood–Paley theory (cf. the proof of Theorem 8.2.1 below) and scaling arguments, to deduce the last inequality. Orthogonality Arguments in Higher Dimensions ∈ C0∞ (( 12 , 2))

be as before. Then, if a(z, η) is a symbol of order zero We let β that vanishes for z outside of a compact set in Rn+1 , we define dyadic operators  eiϕ(z,η) aλ (z, η)fˆ (η) dη, aλ (z, η) = β(|η|/λ)a(z, η). Fλ f (z) = Rn

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8.2 Local Smoothing in Higher Dimensions

255

If we assume that ϕ satisfies (8.1.2 ) and (8.1.6 ), then, by summing a geometric series, it suffices to show that for ε > 0

Fλ f L4 (Rn+1 ) ≤ Cε λ(n−2)/4 λ1/8+ε f L4 (Rn ) .

(8.2.1)

To prove this we need to make an angular decomposition of these operators. So we let {χλν }, ν = 1, . . . , N(λ) ≈ λ(n−1)/2 , be the homogeneous partition of unity that was used in the proof of Theorem 6.2.1. We then put  ν Fλ f (z) = eiϕ(z,η) χλν (η)aλ (z, η)fˆ (η) dη. We now make one further decomposition so that the resulting operators will have symbols that have η-supports that are comparable to λ1/2 cubes. To this end, if ρ ∈ C0∞ ((−1, 1)) are the functions that were used in the two-dimensional orthogonality arguments, we set  ν,j ν,j Fλ f (z) = eiϕ(z,η) aλ (z, η)fˆ (η) dη, with ν,j

aλ (z, η) = χλν (η)ρ(λ−1/2 |η| − j)aλ (z, η). A couple of important observations are in order. First, if ν,j

ν,j

Qλ = suppη aλ (z, η),

(8.2.2)

ν,j

then the Qλ are all comparable to cubes of side-length λ1/2 which are contained in the annulus {η : |η| ≈ λ}. This, along with the fact that the symbols satisfy the natural estimates associated to this support property, ν,j

|∂zγ ∂ηα aλ (z, η)| ≤ Cα,γ (1 + |η|)−|α|/2 ≈ λ−|α|/2 ,

(8.2.3)

makes the decomposed operators much easier to handle. The square function that will be used in the proof of (8.2.1) will involve ν,j operators that are related to the Fλ operators. In fact, the main step in the proof of the orthogonality argument is to establish the following result which, for reasons of exposition, is stated in more generality than what is needed here. Proposition 8.2.2 Fix ε > 0. Then, given any N, there are finite M(N) and CN such that whenever Q ⊂ Rn+1 is a cube of side-length λ−1/2−ε and

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256

Local Smoothing of Fourier Integral Operators

2(n + 1)/(n − 1) ≤ q < ∞

Fλ f Lp (Q)

⎛ ⎞1/2         ν,j  − q1 − n4 +n( 21 − 1q )− 1q 2 ≤ CN |Q| λ |Fλ, f | ⎠  ⎝   ||≤M(N)  ν,j 

Lq (Q)

+CN λ−N f Lq (Rn ) . Here ν,j Fλ, f (z) =

(8.2.4) 

ν,j eiϕ(z,η) aλ, (z, η)fˆ (η) dη,

with symbols satisfying ν,j

aλ, (z, η) = 0

ν,j

if η ∈ / Qλ ,

ν,j

|∂ηα aλ, (z, η)| ≤ Cα λ−|α|/2

∀ α.

(8.2.5)

If the phase function ϕ is fixed, the constants in (8.2.4) and (8.2.5) depend on only finitely many of those in (8.2.3). ν,j

ν,j

The first operator Fλ,0 will just be an oscillatory factor times Fλ , while, for  ≥ 1, the operators in (8.2.4) will involve derivatives of the symbol and the phase function. Turning to the proof, it is clear that (8.2.4) would be a consequence of the following uniform upper bounds valid for z ∈ Q:

Fλ f Lq (Q) ⎛ ⎜ ≤ CN ⎝λ

− 4n

λ

n( 21 − 1q )− 1q

⎛





⎝

||≤M(N)

ν,j

⎞

⎞1/2 ν,j

|Fλ, f (z)|2 ⎠

⎟ + λ−N f Lq (Rn ) ⎠ . (8.2.4 )

We may assume that 0 ∈ Q and we shall prove the estimate for z = 0. ν,j ν,j ν,j To proceed, given ν, j for which Qλ is nonempty, we choose ηλ ∈ Qλ and set  ν,j ν,j ν,j c (z) = ei[ϕ(z,η)−ϕ(z,ηλ )] a (z, η)fˆ (η) dη. λ

λ

Note that

ν,j

ν,j

|∂zα {ei[ϕ(z,η)−ϕ(z,ηλ )] aλ (z, η)}| = O(λ|α|/2 ). So, if we let

ν,j

ν,j

cλ, (z) = ∂z cλ (z),

(8.2.6)

 = (1 , . . . , n ),

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8.2 Local Smoothing in Higher Dimensions it follows that



ν,j

cλ (z) =

||≤M(N)

257

ν,j

z cλ, (0) + RM (z),

where, if M is large enough and if z ∈ Q (and hence |z| = O(λ−1/2−ε )), |RM (z)| ≤ CN λ−N f q . Notice that Fλ f (z) =



ν,j iϕ(z,ηλ ) cν,j (z). ν,j e λ

ν,j

Fλ, f (z) = λ−||/2 ∂z



Therefore, if we set ν,j

ν,j ei[ϕ(z,η)−ϕ(z,ηλ )] aλ (z, η)fˆ (η) dη,

then since |z |λ||/2 ≤ 1, the left side of (8.2.4 ) is dominated by Cλ−N f q plus        iϕ(z,ην,j ) ν,j   λ F e f (0) . λ,    q ||≤M  ν,j L (Q)

ν,j |η − ηλ |

ν,j suppη aλ

ν,j

Since ≤ for η ∈ it is clear that the symbol of Fλ, satisfies (8.2.5). Therefore, by the last majorization, we would be done if we had the following discrete version of the L2 → Lq local smoothing theorem. Cλ1/2

ν,j

Lemma 8.2.3 Let Q and ηλ be as above. Then if Q is contained in a small relatively compact neighborhood of suppη aλ (so that ϕ is well defined)   ⎛ ⎞1/2      ν,j n n( 1 − 1 )− 1  eiϕ(z,ηλ ) cν,j  ≤ Cλ− 4 λ 2 q q ⎝ |cν,j |2 ⎠ .    p  ν,j ν,j L (Q)

Proof As before we assume 0 ∈ Q. If we then normalize the phase function by setting φ(z, η) = ϕ(z, η) − ϕ(0, η), then the desired estimate is equivalent to the following:   ⎛ ⎞1/2     iφ(z,ην,j ) ν,j  n n( 1 − 1 )− 1  λ c  e ≤ Cλ− 4 λ 2 q q ⎝ |cν,j |2 ⎠ .    ν,j  p ν,j

(8.2.7)

L (Q)

To see that this is a corollary of the L2 → Lq local smoothing theorem, we ν,j ν,j let Qλ be the cube of side-length λ1/2 in Rn that is centered at ηλ . If λ is sufficiently large and if we let 0 1−1 ν,j

ν,j

bλ (z, η) =

ν,j Qλ

ei[φ(z,η)−φ(z,ηλ

)]

dη

· χQν,j (η), λ

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258

Local Smoothing of Fourier Integral Operators

it follows that, for z ∈ Q, ν,j

ν,j

|bλ (z, η)| ≤ C|Qλ |−1 = Cλ−n/2 . ν,j

ν,j

To see this, note that φ(z, η) − φ(z, ηλ ) = 0 when η = ηλ and ∇η φ(z, η) = ν,j O(λ−1/2−ε) for z in Q. Hence, |φ(z, η) − φ(z, ηλ )| < 12 for large λ, giving the estimate. Similar considerations show that ν,j

|∂zα bλ (z, η)| ≤ Cα λ−n/2 λ|α|/2 ,

z ∈ Q.

(8.2.8)

These estimates are relevant since the quantity inside the Lq norm in (8.2.7) equals   ν,j eiφ(z,η) bλ (z, η)cν,j dη. ν,j

So, if we write 

∂ ν,j bλ (u1 , 0, . . . , 0, η) du1 ∂u  z01 1 zn+1 ∂ ∂ ν,j +··· + ··· ··· bλ (u, η) du, ∂u ∂u 1 n+1 0 0

ν,j ν,j bλ (z, η) = bλ (0, η) +

z1

then it follows that the left side of (8.2.7) is dominated by      ν,j   ν,j   eiφ(z,η) bλ (0, η)c dη + ···   q  ν,j L (Q)        ∂   ∂ ν,j ν,j   eiφ(z,η) + ··· · · · b (u, η)c dη λ   ∂u1 ∂un+1   ν,j

− 12 −ε

|uk |≤λ

du.

Lq (Q)

ν,j

Since bλ vanishes unless |η| ≈ λ, Theorem 8.2.1 and Plancherel’s theorem imply that this is majorized by      1 1 1  ν,j n( 2 − q )− q  ν,j  λ bλ (0, η)c  + ···   ν,j  2 L (dη)         1 1 1 ∂ ∂ ν,j n( − )− ν,j   ··· +λ 2 q q · · · b (u, η)c du. λ   ∂u1 ∂un+1   − 1 −ε 2

|uk |≤λ

ν,j

L2 (dη)

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8.2 Local Smoothing in Higher Dimensions

259

ν,j

However, using (8.2.8) and the fact that the sets Qλ have finite overlap shows that this in turn is ⎞1/2 ⎛  ν,j n( 21 − 1q )− 1q − n2 ⎝ λ |cν,j |2 |Qλ |⎠ ≤ Cλ ν,j

⎛ = C λ

− n4

λ

n( 21 − 1q )− 1q



⎝

⎞1/2 |cν,j |2 ⎠

,

ν,j

as desired. End of proof of Theorem 8.1.1 Returning to (8.2.1), let us fix a lattice of cubes in Rn+1 of side-length λ−1/2−ε . We then consider the special case of (8.2.4) corresponding to q = 4. If we raise this inequality to the fourth power and then sum over the cubes that intersect suppz aλ , we conclude that, for large enough M,

Fλ f L4 (Rn+1 )

⎛ ⎞1/2        n+1 ε n n−1   ν,j + − 2 ≤ Cλ 8 4 λ 4 λ 4 |Fλ, f | ⎠  + C f L4 (Rn ) ⎝   ||≤M  ν,j  4 n+1 L (R ) ⎛ ⎞1/2       ⎝ ν,j 2 ⎠  = Cλ(n−1)/8+ε/4 |Fλ, f | + C f L4 (Rn ) .     ||≤M  ν,j  4 n+1 L (R

)

Next, if ρ is as above, then we define fˆμ (η) = ρ(λ−1/2 η1 − μ1 ) · · · ρ(λ−1/2 ηn − μn )fˆ (η). As before,



which supp

μ∈Zn fμ = f , and, if for a given ν, j we ν,j ρ(λ−1/2 η − μ) ∩ Qλ  = ∅, we have

ν,j

let Iλ ⊂ Zn be those μ for

ν,j

Card Iλ ≤ C. Conversely, we also have, for a given fixed μ, ν,j

Card {(ν, j) : μ ∈ Iλ } ≤ C. The final ingredient in the proof is that if  ν,j ν,j Kλ, (z, y) = ei[ϕ(z,η)−y,η] aλ, (z, η) dη

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260

Local Smoothing of Fourier Integral Operators ν,j

denotes the kernel of Fλ, , then, since the symbols satisfy (8.2.5), the proof of Lemma 8.1.3 gives −N  ν,j , (8.2.9) |Kλ, (z, y)| ≤ CN λn/2 1 + λ1/2 dist (z, γy,ην,j ) 

λ

ν,j |Kλ, (z, y)| dy ≤ C

where γy,η is given by (8.1.30). In particular, uniformly. If we put these observations together, we see that, if g(z) ≥ 0,   ν,j |Fλ, f (z)|2 g(z) dz Rn+1 ν,j

≤C



 Rn μ

 | fμ (y)|2 sup ν,j

Rn+1

 ν,j |Kλ, (z, y)|g(z)dz dy.

 But ( | fμ |2 )1/2 4 ≤ C f 4 . So, if we take the supremum over all g as above satisfying g 2 = 1, we conclude that (8.2.1) would now follow from  2 1/2    ν,j sup  |Kλ, (z, y)| g(z) dz dy Rn ν,j

Rn+1

≤ Cλ(n−2)/4 | log λ|3/2 g L2 (Rn+1 ) . Since this follows from Theorem 8.1.4, by taking δ = λ−1/2 , 2λ−1/2 , . . . , we are done. Proof of Theorem 8.2.1 Let a(x, t, η) be a symbol of order μq = −n(1/2 − 1/q) + 1/q which vanishes for (x, t) outside of a compact set K ⊂ Rn+1 or |η| ≤ 10. Then we must show that      eiϕ(x,t,η) a(x, t, η)fˆ (η) dη   Lq (Rn+1 )

2(n+1) n−1

≤ C f L2 (Rn ) ,

≤ q < ∞,

(8.2.10)

provided that the canonical relation C associated to this Fourier integral operator satisfies (8.1.10), (8.1.2 ), and (8.1.6 ). The first step is to notice that it suffices to make dyadic estimates. Specifically, if β ∈ C0∞ ((1/2, 2)) is the Littlewood–Paley bump function used before, and if we set aλ (x, t, η) = β(|η|/λ)a(x, t, η), then we claim that (8.2.10) follows from the uniform estimates     2(n+1)  eiϕ(x,t,η) aλ (x, t, η)fˆ (η) dη   q n+1 ≤ C f L2 (Rn ) , n−1 ≤ q ≤ ∞. L (R

)

(8.2.10 )

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8.2 Local Smoothing in Higher Dimensions

261

To see this, we let Lj be the Littlewood–Paley operators in Rn+1 defined by (Lj g)∧ (ζ ) = β(|ζ |/2 j )ˆg(ζ ). Then, since ϕt  = 0 for η  = 0, one sees that there must be an absolute constant C0 such that, if Fλ is the dyadic operator in (8.2.10 ), then

Lj Fλ f Lq (Rn+1 ) ≤ CN 2−Nj λ−N f L2 (Rn )

∀ N,

/ [C0−1 , C0 ]. if 2 j /λ ∈

(8.2.11)

To apply this, we notice that, if F is the operator in (8.2.10), then, by Littlewood–Paley theory and the fact that q > 2, we have ⎛ ⎞1/2  ⎛ ⎞1/2  ∞  ∞       |Lj Ff |2 ⎠  ≤ Cq ⎝

Lj Ff 2q ⎠ .

Ff q ≤ Cq ⎝   j=0  j=0  q



Note that F = F2k . So, if fj is defined by fˆj (η) = β(|η|/2 j )fˆ (η), we can use (8.2.11) to see that, if (8.2.10 ) were true, then the last term would be majorized by ⎛ ⎞1/2 ∞   2⎠ ⎝

fj 2 + 2−Nj f 2 ≤ C f 2 . j

j=0

Since we have verified the claim we are left with proving (8.2.10 ). We shall actually prove the dual version

Fλ∗ g L2 (Rn ) ≤ C g Lp (Rn+1 ) , However, since 

|Fλ∗ g(y)|2 dy =



1≤p≤

2(n+1) n+3 .

(8.2.10 )

Fλ Fλ∗ g(x, t) g(x, t) dxdt

≤ Fλ Fλ∗ g Lp (Rn+1 ) g Lp (Rn+1 ) , this in turn would follow from

Fλ Fλ∗ g Lp (Rn+1 ) ≤ C g Lp (Rn+1 ) ,

1≤p≤

2(n+1) n+3 .

(8.2.12)

Recall that Fλ is of order μq − 1/4 with μq = −n|1/q − 1/2| + 1/q. Thus, since the canonical relation of F is given by (8.1.10), the composition theorem implies that Fλ Fλ∗ is a Fourier integral operator of order 2μq − 1/2 with canonical relation C ◦ C ∗ = {(x, t, ξ , τ , y, s, η, σ ) : (x, ξ ) = χt ◦ χs−1 (y, η), τ = q(x, t, ξ ), σ = q(y, s, η)}.

(8.2.13)

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So, if we assume, as we may, that a(x, t, η) is supported in a small conic set, it follows that the kernel Fλ Fλ∗ is of the form  ei[φ(x,t,s,η)−y,η] bλ (x, t, s, η) dη, Rn

modulo C∞ , where bλ ∈ S2μq vanishes unless |η| ≈ λ and t − s is small. Consequently, if we let αλ (x, t, s, η) = λ−2μq bλ (x, t, s, η) ∈ S0 , then we see that (8.2.12) would be a consequence of the following estimates. Lemma 8.2.4 Let φ be as above and suppose that α ∈ S0 vanishes unless |t − s| is small. Then, if we define the dyadic operators  ei[φ(x,t,s,η)−y,η] β(|η|/λ)α(x, t, s, η)g(y, s) dηdyds, Gλ g(x, t) = it follows that

Gλ g Lp (Rn+1 ) ≤ Cλ

2n( 21 − p1 )− p2

g Lp (Rn+1 ) ,

1≤p≤

2(n+1) n+3 .

(8.2.14)

Furthermore, the constants remain bounded if α as above belongs to a bounded subset of S0 . Proof Since the symbol of Gλ vanishes unless |η| ≈ λ, the kernel must be O(λn ). Hence

Gλ g L∞ (Rn+1 ) ≤ Cλn g L1 (Rn+1 ) . So, if we apply the M. Riesz interpolation theorem, we find that (8.2.14) would follow from the other endpoint estimate:

Gλ g Lp (Rn+1 ) ≤ Cλ g Lp (Rn+1 ) ,

p=

2(n+1) n+3 .

(8.2.14 )

To prove this we need to use (8.2.13) to read off the properties of the phase function φ. First, since χt ◦χs−1 = Identity when t = s, it follows that (x, y, η) → φ(x, t, t, η) − y, η must parameterize the trivial Lagrangian. In other words, φ(x, t, s, η) = x, η

if t = s.

Also, since τ = q(x, t, ξ ) in (8.2.13), it follows that, when t = s, φt = q(x, t, η). So we conclude further that φ(x, t, s, η) = x, η + (t − s)q(x, s, η) + (t − s)2 r(x, t, y, s, η).

(8.2.15)

Using this we can estimate Gλ . More precisely, we claim that if we set  ei[φ(x,t,s,η)−y,η] β(|η|/λ)α(x, t, s, η)f (y) dηdy, Tt,s f (x) =

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8.3 Spherical Maximal Theorems Revisited

263

and if α is as above, then

Tt,s f L∞ (Rn ) ≤ Cλ

n+1 2

|t − s|−

n−1 2

f L1 (Rn ) .

(8.2.16)

This estimate is relevant because we also have

Tt,s f L2 (Rn ) ≤ C f L2 (Rn ) , since the zero order Fourier integral operators Tt,s have canonical relations that are canonical graphs. By interpolating between these two estimates we get n−1

Tt,s f Lp (Rn ) ≤ Cλ|t − s|− n+1 f Lp (Rn ) ,

p=

2(n+1) n+3 .

And since 1−(1/p−1/p ) = (n−1)/(n+1) for this value of p, we get (8.2.14 ) by applying the Hardy–Littlewood–Sobolev inequality—or, more precisely, Theorem 0.3.6. The proof of (8.2.16) just uses stationary phase. Assuming that t − s > 0, we can make a change of variables and write the kernel of Tt,s as  * + x−y (t − s)−n ei  t−s ,η+q(x,s,η)+(t−s)r(x,t,y,s,η)     ×β |η|/λ(t − s) α x, t, s, η/(t − s) dη. (8.2.17) To estimate the integral we first recall that, by assumption, rank q ηη ≡ n − 1. So, if N is a small conic neighborhood of suppη α and if t is close to s, the sets Sx,t,y,s = {η ∈ N : q(x, s, η) + (t − s)r(x, t, y, s, η) = ±1} have non-vanishing Gaussian curvature. In addition, since q  = 0 for η  = 0, it follows that q η  = 0 there as well. Consequently, for t close to s ) ( x−y ∇η  , η + q(x, s, η) + (t − s)r(x, t, y, s, η)  = 0, t−s unless |x − y| ≈ |t − s|. (8.2.18) If we put these two facts together and use the polar coordinates associated to Sx,t,y,s , we conclude that Theorem 1.2.1 implies that (8.2.17) must be    n+1  n+1 n−1 O (t − s)−n |λ(t − s)| 2 = O λ 2 |t − s|− 2 . Since this of course implies (8.2.16), we are done.

8.3 Spherical Maximal Theorems Revisited Using the local smoothing estimates in Theorem 8.1.1 we can improve many of the maximal theorems in Section 6.3. We shall deal here with smooth families

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Local Smoothing of Fourier Integral Operators

of Fourier integrals Ft ∈ I m (X, X; Ct ), t ∈ I, having the property that if we set Ff (x, t) = Ft f (x), then F ∈ I m−1/4 (X × I, X; C), where the full canonical relation C satisfies the non-degeneracy conditions and the cone condition described in Section 8.1. Our main result then is the following. Theorem 8.3.1 Let X be a smooth n-dimensional manifold and suppose that Ft ∈ I m (X, X; Ct ), t ∈ [1, 2], is a smooth family of Fourier integral operators that m . Suppose further that the full canonical belongs to a bounded subset of Icomp relation C ⊂ T ∗ (X × [1, 2])\0 × T ∗ X\0 associated to this family satisfies the non-degeneracy conditions (8.1.2) – (8.1.3) and the cone condition (8.1.6). Then, if ε(p) is as in Theorem 8.1.1 and 2 < p < ∞,    sup |Ft f (x)| p ≤ Cm,p f Lp (X) , L (X) t∈[1,2]

if m < −(n − 1)( 12 − 1p ) − 1p + ε(p).

(8.3.1)

If we define Fk,t as in the proof of Theorem 6.3.1, then Theorem 8.1.1 yields p  2   d  j 1/p   Fk,t f (x) dxdt  X dt 1 ≤ Cε 2kj 2k[m+(n−1)(1/2−1/p)−ε] f p ,

ε < ε(p).

By substituting this into the proof of Theorem 6.3.1, we get (8.3.1). Remark A reasonable conjecture would be that for p ≥ 2n/(n − 1) the mapping properties of the maximal operators associated to the family of operators should be essentially the same as the mapping properties of the individual operators, at least when n = 2. By this we mean that for p ≥ 2n/(n − 1) (8.3.1) should actually hold for all m < −(n − 1)(1/2 − 1/p). This of course would follow from showing that there is local smoothing of all orders < 1/p for this range of exponents. Using Theorem 8.3.1 we can give an important extension of Corollary 6.3.2 that allows us to handle the case of n = 2 under the assumption of cinematic curvature. Specifically, if we consider averaging operators  At f (x) = f (y) η(x, y) dσx,t (y), η ∈ C0∞ , Sx,t

associated to C∞ curves Sx,t in the plane that vary smoothly with the parameters, then we have the following result.

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8.3 Spherical Maximal Theorems Revisited

265

Corollary 8.3.2 Let C ⊂ T ∗ (R2 × [1, 2])\0 × T ∗ R2 \0 be the conormal bundle of the C∞ hypersurface S = {(x, t, y) : y ∈ Sx,t } ⊂ (R2 × [1, 2]) × R2 . Then if C satisfies the non-degeneracy condition (8.1.2) and the cone condition (8.1.6)    sup |At f (x)| p 2 ≤ Cp f p 2 , p > 2. (8.3.2) t∈[1,2]

L (R )

L (R )

Note that, as we saw in Section 6.3, At is a Fourier integral operator of order Also, when n = 2, −(n − 1)(1/2 − 1/p) − 1/p = −1/2. So (8.3.2) follows from Theorem 8.3.1 since, if we set Ff (x, t) = At f (x), then F is a conormal operator whose canonical relation is the conormal bundle of S. It is not hard to adapt the counterexample that was used to show that the circular maximal operator can never be bounded on Lp (R2 ) when p ≤ 2 and see that the same applies to (8.3.2). Also, as we pointed out in Section 6.3, the Kakeya set precludes the possibility of Corollary 8.3.2 holding without the assumption of cinematic curvature. More precisely, we saw that (8.3.2) cannot hold for any finite exponents if Sx,t = {y : x, y = t}. But S = {(x, t, y) : x, y = t} is a subspace of R5 and hence the cone condition cannot hold for the rotating lines operators, since, if C is the conormal bundle of S, then the images of the projection of C onto the fibers of T ∗ (R2 × R) are just subspaces, meaning that the cones in (8.1.6) are just linear subspaces that of course cannot have any non-vanishing principal curvatures. We can also estimate maximal theorems corresponding to curves in R2 which shrink to a point. Specifically, we now let − 12 .

Sx,t , Sx,t = x + t" where " Sx,t are C∞ curves depending smoothly on (x, t) ∈ R2 × [0, 1]. If we define new averaging operators by setting  f (x − ty) dσx,t (y), At f (x) = " Sx,t

where dσx,t denotes Lebesgue measure on " Sx,t then we have the following result. Corollary 8.3.3 Let C ⊂ T ∗ (R2 × (0, 1])\0 × T ∗ R2 \0 be the conormal bundle of S = {(x, t, y) : y ∈ Sx,t , t > 0} ⊂ (R2 × (0, 1]) × R2 .

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Local Smoothing of Fourier Integral Operators

Assume that C satisfies the non-degeneracy condition (8.1.2) and the cone condition (8.1.6). Then, if the initial curves " Sx,0 have non-vanishing curvature,    sup |At f (x)|  p 2 ≤ Cp f p 2 , p > 2. L (R ) L (R ) 0 0,

f Lp (Rn ) ,

if 1 ≤ p ≤ n

and q = (n − 1)p ,

(9.5.2)

and consequently Kakeya sets must have full Hausdorff dimension. Before turning to the details of the proof, let us reframe the hypothesis and conclusion of the theorem in an equivalent way. First, we recall that, by duality arguments that were used in §2.2, (9.5.1) holds if and only if the Fourier extension operator enjoys the bounds     2n eix·ω g(ω) dω q n ≤ Cq g Lq (Sn−1 ) , q > n−1 , (9.5.1 )  Sn−1

L (R )

with dω being surface measure on Sn−1 . Second, we note that (9.5.2) holds if and only if for 0 < δ < 1/2 we have the uniform bounds

fδ∗ Ln (Sn−1 ) ≤ Cε δ −ε f Ln (Rn ) ,

∀ ε > 0.

(9.5.2 )

Indeed, by interpolation with the trivial estimate

fδ∗ L∞ (Sn−1 ) ≤ Cδ −(n−1) f L1 (Rn ) ,

(9.5.3)

we see that (9.5.2 ) yields (9.5.2), and (9.5.2 ) is a special case of (9.5.2). Also, by Proposition 9.1.5, the last statement in Theorem 9.5.1 is also a consequence of (9.5.2 ).

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9.5 The Fourier Restriction Problem and the Kakeya Problem

311

Thus, in order to prove the theorem it suffices to show that the Fourier extension bounds in (9.5.1 ) yield (9.5.2 ). We really only need to do this for n ≥ 3 since we saw in Chapter 2 that when n = 2 both are valid; however, the argument works equally well there as well. The two-dimensional version of (9.5.2 ) is due to C´ordoba and we shall now present a simple duality result, also due to C´ordoba, that will be used in the proof. n Proposition 9.5.2 Let p ≥ n−1 . Suppose that there is a constant C so that for every 0 < δ < 1/2 and every δ-separated set {ωj }N j=1 and aj ≥ 0, j = 1, . . . , N, we have   n−1   −σ   1 ≤ Cδ p

a p , (9.5.4) a j Tδ (ωj )   p n L (R )

whenever Tδ (ωj ) as in (9.1.7) is a δ-tube about a unit line segment pointing in the direction ωj for each j = 1, 2, . . . , N. Then if p is conjugate to the Lebesgue exponent in (9.5.4)

fδ∗ Lp (Sn−1 ) ≤ C δ −σ f Lp (Rn ) .

(9.5.5)

Remark One usually will take the ω to be a maximal δ-separated set in which case N ≈ δ −(n−1) . Also, the special case of (9.5.4) corresponding to aj ≡ 1, n and σ = ε, then is equivalent to the statement that p = n−1      1Tδ (ωj )   

n

L n−1 (Rn )

≤ Cε δ −ε .

(9.5.4 )

If the tubes happened to not overlap this estimate would be trivial since the number of summands is O(δ −(n−1) ) and each tube has volume comparable to δ n−1 . The point of the conjectured bounds (9.5.4 ) is that arbitrary tubes corresponding to δ-separated directions can only have minor overlapping as measured in the inequality. One can check that a modification of the proof of Proposition 9.1.5 shows that (9.5.4 ) would imply that Kakeya subsets of Rn must always have Hausdorff dimension equal to n, thus solving this Kakeya problem. Proof of Proposition 9.5.2 Fix a maximal δ-separated set {ωj } of Sn−1 . Note that there must be a constant C so that if dist(ω, ωj ) ≤ δ then we have fδ∗ (ω) ≤ Cfδ∗ (ωj ). This is because any δ-tube about a unit line segment γω in the direction ω can be covered by O(1) tubes about line segments in the direction ωj if dist(ωj , ω) ≤ δ.

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Kakeya-type Maximal Operators

From this we deduce that  

fδ∗ Lp (Sn−1 ) ≤

{ω∈Sn−1 : dist(ω,ωj )≤δ}

j

≤C



p 1/p fδ∗ (ω) dω



δ n−1 (fδ∗ (ωj ))p

1/p = Cδ

n−1 p

j



(fδ∗ (ωj ))p

1/p .

j

Since fδ∗ (ωj ) > 0, by the duality of p and p , we can find aj ≥ 0 with a p = 1 so that

1/p   (fδ∗ (ωj ))p = aj fδ∗ (ωj ). j

If we combine this with the preceding inequality, we conclude that we can choose δ-tubes Tδ (ωj ) in the direction ωj so that  n−1  1 aj | f | dx

fδ∗ Lp (Sn−1 ) ≤ Cδ p |Tδ (ωj )| Tδ (ωj ) j

  − n−1 aj 1Tδ (ωj ) | f | dx, ≤ C δ p using the fact that |Tδ (ωj )| ≈ δ n−1 and that 1 − 1/p = 1/p in the last step. Since, by H¨older’s inequality, the right side is dominated by    − n−1    δ p  a 1

f Lp (Rn ) , j Tδ (ωj )   j



Lp (Rn )

it is clear that (9.5.4) implies (9.5.5) as a p = 1. The reader can check that a similar simple duality argument shows that (9.5.5) implies (9.5.4), meaning that the two estimates are equivalent; however, this will not be needed for the proof of Theorem 9.5.1. Proof of Theorem 9.5.1 Recall that our task is equivalent to showing that (9.5.1 ) yields (9.5.2 ). To use the proposition it is convenient to note that, by a simple change of scale argument, the former is equivalent to

Tλ g Lq (Rn ) ≤ Cq δ 2n/q g Lq (Sn−1 ) , 2n n−1

if



 Tλ g (y) =

(9.5.1 )

< q < ∞, δ = λ−1/2 ∈ (0, 1/2),

 eiλy·ω g(ω) dω, Sn−1

δ = λ−1/2 .

(9.5.6)

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9.5 The Fourier Restriction Problem and the Kakeya Problem

313

We shall use this numerology that δ = λ−1/2 throughout the proof. It has occurred several times in our exposition, starting in §2.4. If we repeat the arguments from the end of §9.3, we see that this implies the vector-valued version  

1/2 

1/2        2 2n/q  2    |T g | ≤ C δ |g | , λ  q      Lq (Rn )



Lq (Sn−1 )



2n n−1

< q < ∞,

δ = λ−1/2 ∈ (0, 1/2).

(9.5.7)

The constant in (9.5.7) can be taken to be a multiple of the one in (9.5.1 ) for each q. To proceed, chose a δ-separated subset {ωj } of Sn−1 and let {Tδ (ωj )} be δ-tubes about unit line segments pointing in these directions, i.e., .   / Tδ (ωj ) = y ∈ Rn : dist y, {tωj + xj : |t| ≤ 1/2} < δ for certain xj ∈ Rn . Choose 0 ≤ ρ ∈ C0∞ (R) satisfying ρ(0) = 1. Then if C is chosen large enough we have      eiλy·ω e−iλxj ·ω ρ Cλ1/2 dist(ω, ωj ) dω   Sn−1

≥ cλ−

n−1 2

if y ∈ Tδ (ωj )

,

and δ = λ−1/2 ,

for some uniform c > 0. One deduces this from the fact that, if C is large enough we have, say,   λ|(y − xj ) · (ω − ωj )| < 1/2, if y ∈ Tδ (ωj ) and ρ Cλ1/2 dist(ω, ωj )  = 0. Because of this lower bound, if for a given bj ≥ 0 we take   fj (ω) = bj e−iλxj ·ω ρ Cλ1/2 dist(ω, ωj ) , then we have |Tλ fj (y)| ≥ cδ n−1 bj 1Tδ (ωj ) (y), and, therefore,  j

1/2 |Tλ fj (y)|

2

≥ cδ

n−1



if δ = λ−1/2 ,

1/2

b2j 1Tδ (ωj ) (y)

,

(9.5.8)

j

assuming that C is as above. In addition to the above, since the ωj are δ-separated with δ = λ−1/2 , we can choose C large enough so that the subsets of Sn−1 , {supp ρ(Cλ1/2 dist( · , ωj ))},

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Kakeya-type Maximal Operators

are disjoint. In this case, due to the fact each has measure ≈ λ− deduce that  



1/q   2 1/2  q − n−1   2 λ | f | ≈ b j j   Lq (Sn−1 )

j

n−1 q

=δ



q

n−1 2

= δ n−1 , we

(9.5.9)

1/q .

bj

If we use (9.5.7) along with (9.5.8) and (9.5.9), we see that   2  3n−1   q 1/q δ n−1  ( bj 1Tδ (ωj ) )1/2 Lq (Rn ) ≤ Cq δ q bj , 2n n−1

< q < ∞.

If we relabel aj = b2j and p = q/2, this can be rewritten as   aj 1T

δ (ωj )

 



Lp (Rn )

≤ Cp δ

−2(n−1)+ 3n−1 p

a p , n n−1

< p < ∞. (9.5.10)

Since the aj ≥ 0 here are arbitrary, by Proposition 9.5.2, (9.5.10) implies that we have the bounds,

fδ∗ Lp (Sn−1 ) ≤ Cp δ = Cp δ

−2(n−1)+ 3n−1 − n−1 p p −2(n−1)+ 2n p

f Lp (Rn )

f Lp (Rn ) ,

(9.5.11)

1 < p < n,

for the Kakeya maximal function. Observe that as p increases to n, 2n/p increases to 2(n − 1), due to the fact that n/(n−1) is the conjugate exponent for n. Thus, (9.5.11) implies that, given ε > 0, we can find an exponent p(ε) ∈ (1, n) so that

fδ∗ Lp (Sn−1 ) ≤ Cp δ −ε f Lp (Rn ) ,

if p(ε) ≤ p < n.

(9.5.12)

If we interpolate with the trivial bounds

fδ∗ L∞ (Sn−1 ) ≤ f L∞ (Rn ) , we deduce that (9.5.12) yields (9.5.2 ), which completes the proof.

Notes The connection between the Hausdorff dimensions of Kakeya sets, Kakeya maximal function bounds and restriction estimates is due to Bourgain [2]. In this paper

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Notes

315

he also introduced the bush argument and proved new Kakeya and Nikodym maximal function bounds, including the fact that the bounds in Theorems 9.2.1 and 9.2.2 are valid for p ≤ 7/3 when n = 3, which was a significant improvement over what the bush argument gives, i.e., p ≤ 2 in this case. The Euclidean estimates (9.2.7) for Nikodym maximal functions are due to Christ, Duoandikoetxea, and Rubio de Francia [1], and they also are a consequence of earlier estimates of Christ [2] and Drury [1]. The generalization to the Riemannian case is due to Minicozzi and Sogge [1]; however, as we have noted, this result is a straightforward consequence of the earlier bush argument from Bourgain [2]. The geometric counterexamples in §9.3 are also due to Minicozzi and Sogge [1], but they were done independently later in a slightly different context by Wisewell [1] and Bourgain and Guth [1]. Moreover, in the latter paper positive results for oscillatory integral operators involving the convexity hypothesis described in §9.3 were obtained for p = 10/3 that are optimal in view of the counterexamples, and they also similarly proved optimal variable Nikodym maximal function estimates in four dimensions. Earlier related work was done by Lee [1] and others. The improved Kakeya maximal estimates for p ≤ n+2 2 in Theorem 9.4.1 is due to Wolff [3]. We followed his proof for the most part, except that we replaced his “induction on scales” argument with the one involving the auxiliary maximal functions that were introduced in Sogge [7]. The latter paper also showed that when n = 3 Wolff’s results for Nikodym maximal functions extend to hyperbolic space in the three-dimensional case, and results were obtained also for certain types of three-dimensional manifolds of variable curvature that are stronger than the ones given in §9.2. The use of the auxiliary maximal functions was natural for these problems since one could not use scaling arguments. Miao, Yang, and Zheng [1] showed that one could also use this method to give a proof of Wolff’s Kakeya maximal function bounds for R3 using the auxiliary maximal function method. Xi [1] extended these results for the constant curvature case to higher dimensions by formulating the auxiliary maximal function as in (9.4.16). This turns out to be an essentially equivalent version of the maximal function used in three dimensions by Sogge [7], but the formulation in the latter obscured what was needed to make the argument work in higher dimensions. Wolff’s bounds for Kakeya and Nikodym maximal functions and for the size of the Hausdorff dimension of Kakeya and Nikodym sets remain the best known in three and four dimensions, but in higher dimensions there have been improvements, such as those in Bourgain [4] which connected Kakeya problems to arithmetic combinatorics. Further improvements on Kakeya problems for dimensions n ≥ 5 are in Katz and Tao [2]. Also, Katz, Laba, and Tao [1] showed that in the important three-dimensional case, Kakeya subsets of R3 have upper Minkowski dimension d for a certain d > 5/2, which improved the bounds of Wolff for the upper Minkowski dimension but not for the lower Minkowski dimension or the Hausdorff dimension.

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Appendix Lagrangian Subspaces of T ∗ Rn

Here we shall prove Lemma 6.1.3. That is, we shall show that, if V0 and V1 are two Lagrangian subspaces of T ∗ Rn , then there must be a third Lagrangian subspace V that is transverse to both. To do this we need to introduce some terminology. We shall call 2n linearly independent unit vectors {ej }nj=1 , {ej }nj=1 a symplectic basis if, for all j, k = 1, . . . n, σ (ej , ek ) = σ (εj , εk ) = 0 and σ (εj , ek ) = −σ (ek , εj ) − δj,k .

(A.0.1)

Here σ is the symplectic form on T ∗ Rn . The standard symplectic basis of course is when ej and εj are the unit vectors along the xj and ξj axes, respectively. The main step in the proof of Lemma 6.1.3 is to show that a partial symplectic basis can always be extended to a full symplectic basis. 1 2 and {εj }nj=1 are linearly independent Proposition A.0.1 Suppose that {ej }nj=1 unit vectors satisfying (A.0.1). Then one can choose additional unit vectors so that e1 , . . . , en , ε1 , . . . , εn becomes a full symplectic basis.

Proof Let us first assume that n1 < n2 . We then claim that we can choose a unit vector en1 +1 so that {ej }n11 +1 , {εj }n12 satisfy (A.0.1) and are linearly independent. To do this we notice that, since σ is non-degenerate, we can always choose a unit vector e so that σ (e, ej ) = 0, 1 ≤ j ≤ n1 Furthermore, if xe +

and

 1≤j≤n1

σ (εk , e) = δk,n1 +1 , 1 ≤ k ≤ n2 .

xj ej +



ξj εj = 0

1≤j≤n2

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318

Appendix

then taking the σ product with εn1 +1 shows that x = 0. Hence, setting en1 +1 = e gives us the claim. Since the same argument applies to the case where n2 < n1 we may assume that n1 = n2 . If then n1 = n we are done, otherwise we can argue as above to see that we can choose a unit vector e so that σ (e, ej ) = σ (e, εk ) = 0, 1 ≤ j, k ≤ n1 . From this, we conclude as before that {ej }n11 +1 , {εj }n11 must be a partial symplectic basis if we now set en1 +1 = e. Since we can next add a unit vector εn1 +1 to our partial symplectic basis, we complete the proof by induction. Proof of Lemma 6.1.3 It suffices to show that we can choose a symplectic basis e1 , . . . , en , ε1 , . . . , εn of T ∗ Rn so that V0 is spanned by ε1 , . . . , εn and V1 is spanned by ek+1 , . . . , en , ε1 , . . . , εk . For we can then take V to be the subspace ,  V= xj ej + ξj εj : x = ξ , ξ = 0, / with x = (x1 , . . . , xk ), x = (xk+1 , . . . , xn ) . To verify this assertion we first choose a basis ε1 , . . . , εk for the subspace V0 ∩ V1 and extend it to a full basis ε1 , . . . , εn of V0 . If we restrict σ to V0 × V1 , we see that it gives a duality between V0 /(V0 ∩ V1 ) and V1 /(V0 ∩ V1 ). This is because both V0 and V1 are Lagrangian and hence Vj⊥ = Vj , j = 0, 1. On account of this we can choose ek+1 , . . . , en ∈ V1 so that σ (εi , ej ) = δi,j if k + 1 ≤ i, j ≤ n. If we then extend the partial symplectic basis {εj }nj=1 , {ej }nj=k+1 to a full symplectic basis, using Proposition A.0.1, we are done.

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Wiener, N. [1] The ergodic theorem, Duke Math. J. 5 (1939), 1–18. Wisewell, S. [1] Kakeya sets of curves, Geom. Funct. Anal. 15 (2005), 1319–62. Wolff, T. [1] Unique continuation for |u| ≤ V|∇u| and related problems, Revista Math. Iber. 6 (1990), 155–200. [2] A property of measures in Rn and an application to unique continuation, Geom. Funct. Anal. 2 (1992), 225–84. [3] An improved bound for Kakeya type maximal functions, Revista Math. 11 (1993) 651–674. [4] Recent work connected with the Kakeya problem, Prospects in Mathematics, Princeton, NJ, 1996 AMS, 129–62. [5] Local smoothing type estimates on Lp for large p, Geom. Funct. Anal. 10 (2000), 1237–88. [6] A sharp bilinear cone restriction estimate Ann. Math. 153 (2001), 661–98. [7] Lectures on harmonic analysis. With a foreword by Charles Fefferman and preface by Izabella Laba. Edited by Laba and Carol Shubin University Lecture Series 29, American Mathematical Society, Providence, RI, 2003. Xi, Y. [1] On Kakeya-Nikodym type maximal inequalities, Trans. Amer. Math. Soc., to appear. Zygmund, A. [1] On a theorem of Marcinkiewicz concerning interpolation of operators, J. Math. Pure Appl. 35 (1956), 223–48. [2] On Fourier coefficients and transforms of two variables, Studia Math. 50 (1974), 189–201. [3] Trigonometric series, Cambridge University Press, Cambridge, 1979.

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Index of Notation

S 2 Dα 3 B(x, r) 10  14 p Ls 14 Lip (γ ) 19 E 30 (u) 30 sing supp 30 D 31 WF(u) 31 (t + i0)δ 32, 79 ˜ ν 35 ν ,  T ∗ X 36 φ 39 W ⊥ 39 φ 40 Cφ 59 δ(p) 70 Sλδ 70, 151 S∗δ 76 P(x, D) 98

Sm 98  P ∼ Pj 100 m cl (M) 110 Ej , εj , ej (x) 112 N(λ) 112 t 123 g , 125 χλ f 131 m(P) 134 I m (X, ) 163 C 170 I m (X × Y; C) 170 R(u) 198 Char Q 197 ess supp A 203 S∗ M 205 γω 269 fδ∗ 270 Tδ (γω,x ) 270 1E 271 dimH E 274 fδ∗∗ 277 dg (x, y) 283

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Index

amplitude, 37 Besicovitch set, 268 bicharacteristics, 123 Calder´on–Zygmund decomposition, 15 canonical one form, 40 canonical relation, 59, 170 canonical transformation, 123 Carleson–Sj¨olin condition, 61 n × n condition, 62 characteristic set, 197 cinematic curvature, 236 circular maximal theorem, 93 cone condition, 235, 236 conormal distribution, 169 cospheres, 137 cotangent bundle, 36 unit, 205 critical index, 70 δ-separated, 270 Duistermaat–Guillemin theorem, 216 Egorov’s theorem, 201 eikonal equation, 119 equivalence of phase function theorem, 104, 166 essential support, 203 Euler’s homogeneity relations, 40 formal series, 100 Fourier integral operators, 169 Fourier transform, 1 fractional integral, 22 H¨older continuity, 19 half-wave operator, 79, 117 Hamilton vector field, 123 flow, 123

Hardy space, 178 Hardy–Littlewood maximal theorem, 10 Hardy–Littlewood–Sobolev inequality, 22 Hausdorff dimension, 274 Hausdorff measures, 273 Hausdorff–Young inequality, 5 Hessian matrix, 46, 57 Kakeya maximal operator, 270 Kakeya set, 268 Khintchine’s inequality, 21 Kohn–Nirenberg theorem, 99 Lagrangian distribution, 40, 162, 163 Lagrangian submanifolds, 40 Littlewood–Paley operators, 20 local smoothing, 233, 253, 254 Marcinkiewicz interpolation theorem, 12, 272 maximal δ-separated subset, 270 Minkowski dimension, 270 lower, 269 upper, 269 Morse lemma, 48 Nikodym maximal function, 277, 283 Nikodym maximal theorem, 88, 247, 277 Nikodym set, 277, 288 oscillatory integral operators, 57 non-degenerate, 57 oscillatory integrals, 37, 42 parametrix, 102, 117 phase function, 37 non-degenerate, 39, 46, 57 Plancherel’s theorem, 5 Poisson summation formula, 8 principal symbol, 110 Propagation of singularities theorem, 201

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334 pseudo-differential operator, 97 classical, 110 elliptic, 110 properly supported, 163 pseudo-local, 98 pullback, 7, 33 restricted weak-type estimate, 272 restriction theorem, 62, 254 discrete version, 138 Riesz interpolation theorem, 6 Riesz means, 70, 151 maximal, 76 rotational curvature, 190 sharp Weyl formula, 116 singular support, 30 smoothing operator, 98 Sobolev spaces, 14 spectral function, 113

Index spectral projection operator, 131 spherical maximal theorems, 187 stationary point, 50 strictly hyperbolic, 125 subprincipal symbol, 227 symbol, 98 symplectic form, 39 TT ∗ argument, 26 tangent space, 36 transport equation, 120 Unique Continuation for Laplacian, 147 Van der Corput lemma, 44 wave front set, 31 weak-type estimate, 12 restricted, 272 Young’s inequality, 22

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