Pseudodifferential and Singular Integral Operators: An Introduction with Applications

De Gruyter Graduate Lectures Abels • Pseudodifferential and Singular Integral Operators

Helmut Abels

Pseudodifferential and Singular Integral Operators An Introduction with Applications

De Gruyter

Mathematics Subject Classification 2010: Primary: 35J48, 35K41, 35S05, 47G30, 47F05, 46E35, 42B20, 42B37, 42B15, 42B10.

ISBN: 978-3-11-025030-5 e-ISBN: 978-3-11-025031-2 Library of Congress Cataloging-in-Publication Data Abels, H. (Helmut) Pseudodifferential and singular integral operators : an introduction with applications / by Helmut Abels. p. cm. – (De Gruyter textbook) Includes bibliographical references and index. ISBN 978-3-11-025030-5 (hardcover : alk. paper) – ISBN 978-3-11-025031-2 (e-book) 1. Pseudodifferential operators. 2. Integral operators. I. Title. QA329.7.A24 2012 515′.94–dc23 2011041884

Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the internet at http://dnb.d-nb.de.

© 2012 Walter de Gruyter GmbH & Co. KG, 10785 Berlin/Boston Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen °° Printed on acid-free paper Printed in Germany www.degruyter.com

Preface

The purpose of this book is to give a self-contained introduction to the theory of pseudodifferential operators and singular integral operators. The exposition is written in such a way that it should be accessible for third year Bachelor students and Master students with a basic education in analysis of functions in several variables and integration theory. A basic knowledge on functional analysis is recommended. In particular, bounded linear operators between Banach and Fréchet spaces will be used in the whole book. The most important facts for the purposes of this book are summarized in Appendix A.3. The book is based on several lecture series held by the author at Darmstadt University of Technology, the International Max Planck Research School Mathematics in the Sciences Leipzig, the University of Leipzig, and the University of Regensburg between the years 2003 and 2011. It is divided into three parts, each consisting of two chapters, and the appendix and an introduction. Part I is devoted to the Fourier transformation, tempered distributions and pseudodifferential operators. First in Chapter 2 all basic facts on Fourier transformation, tempered distributions and Sobolev spaces that are needed throughout the book are discussed. The chapter is the basis for all other chapters in the book. Then in Chapter 3 the basic calculus of pseudodifferential operators on Rn is presented. In particular, compositions, adjoints, and mapping properties of pseudodifferential operators are discussed. Moreover, applications to elliptic pseudodifferential equations are studied as well. In Part II, namely Chapter 4 and Chapter 5, an introduction to the theory of singular integral operators is given. First the theory for translation invariant operators is presented in Chapter 4. Moreover, several examples are discussed and the important Mikhlin multiplier theorem is proved. Then, in Chapter 5, an extension of the theory to a class of not necessarily translation invariant operators is studied. Furthermore, the results are extended to Banach-space-valued functions and a variant of the Mikhlin multiplier theorem for Hilbert-space-valued functions is proved. Afterwards applications to pseudodifferential operators are discussed. To this end, the essential step is to prove suitable estimates for the kernel of the pseudodifferential operators, which has several other consequences too. The major parts of Chapter 4 and Chap-

vi

Preface

ter 5 are independent of the basic theory of pseudodifferential operators presented in Chapter 3. Only for the application to pseudodifferential operators in the Sections 5.4 and 5.5 some of the results in Chapter 3 are needed. Finally, Part III consists of applications of the results of the first two parts. First in Chapter 6 an introduction to the theory of Bessel potential and Besov spaces is given, which can partly be seen as an application of the (Hilbert-space-valued) Mikhlin multiplier theorem. Then in Chapter 7 several applications of the Mikhlin multiplier theorem and the results on pseudodifferential operators are discussed. In particular, solvability of resolvent equations for differential operators and their application to abstract parabolic evolution equations are presented. Since the results are based on different parts of the books, the sections can be used independently. The book is written in such a way that the chapters and results are partly independent. Only the content of Chapter 2 is needed for all chapters. The interested reader and lecturer can make different choices depending in the interest. The Chapters 3 and 4 are completely independent. Each can be used for an introductory course in pseudodifferential operators, singular integral operators, respectively. In Chapter 5 the Sections 5.1–5.3 are only based on Chapter 4. Only the Sections 5.4 and 5.5 need results from the theory of pseudodifferential operators. Chapter 6 is independent of the Chapter 3–5 if one takes the Hilbert-space-valued Mikhlin multiplier theorem for granted. Finally, Chapter 7 is based on different parts of the book, which is indicated in each section. We note that Chapter 2 (without Section 2.8), Chapter 3, Sections 6.2 and 6.6 (only for the Hölder–Zygmund spaces, i.e., s > 0; p D q D 1), and Section 7.3 were essentially the content of a two-hours lecture series held at the University of Regensburg and Darmstadt University of Technology. Moreover, a lecture series on singular integral operators at the International Max Planck Research School Mathematics in the Sciences Leipzig was mostly based on the content of Chapters 4–6 and Sections 7.1 and 7.2. Finally, I like to thank everybody who helped me to write and improve this book. In particular, I am grateful to Alexander Huber, Dominik Köppl, Lars Müller, Christine Pfeuffer, and Alexander Voitovich, for their careful proofreading of the manuscript and previous versions of the lecture notes. Finally, I am indebted to my wife and my kids for all the patience and support during the time the book was written. Regensburg, October 2011

Helmut Abels

Contents

Preface

v

1

Introduction

1

I

Fourier Transformation and Pseudodifferential Operators

2

Fourier Transformation and Tempered Distributions

9

2.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2 Rapidly Decreasing Functions – S.Rn / . . . . . . . . . . . . . . . . . . . . . . .

13

2.3 Inverse Fourier Transformation and Plancherel’s Theorem . . . . . . . . .

15

2.4 Tempered Distributions and Fourier Transformation . . . . . . . . . . . . .

20

2.5 Fourier Transformation and Convolution of Tempered Distributions .

23

2.6 Convolution on S 0 .Rn / and Fundamental Solutions . . . . . . . . . . . . . .

25

2.7 Sobolev and Bessel Potential Spaces . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.8 Vector-Valued Fourier-Transformation . . . . . . . . . . . . . . . . . . . . . . . .

30

2.9 Final Remarks and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

3

2.9.1

Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

2.9.2

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

Basic Calculus of Pseudodifferential Operators on Rn

40

3.1 Symbol Classes and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . .

40

3.2 Composition of Pseudodifferential Operators: Motivation . . . . . . . . .

45

3.3 Oscillatory Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

3.4 Double Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

viii

Contents

3.5 Composition of Pseudodifferential Operators . . . . . . . . . . . . . . . . . . .

54

3.6 Application: Elliptic Pseudodifferential Operators and Parametrices .

57

3.7 Boundedness on Cb1 .Rn / and Uniqueness of the Symbol . . . . . . . . .

63

3.8 Adjoints of Pseudodifferential Operators and Operators in .x; y/-Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

3.9 Boundedness on L2 .Rn / and L2 -Bessel Potential Spaces . . . . . . . . .

68

3.10 Outlook: Coordinate Transformations and PsDOs on Manifolds . . . .

74

3.11 Final Remarks and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

3.11.1 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

3.11.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

II

Singular Integral Operators

4

Translation Invariant Singular Integral Operators

85

4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

4.2 Main Result in the Translation Invariant Case . . . . . . . . . . . . . . . . . .

87

4.3 Calderón–Zygmund Decomposition and the Maximal Operator . . . .

91

4.4 Proof of the Main Result in the Translation Invariant Case . . . . . . . . .

95

4.5 Examples of Singular Integral Operators . . . . . . . . . . . . . . . . . . . . . . 100 4.6 Mikhlin Multiplier Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.7 Outlook: Hardy spaces and BMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.8 Final Remarks and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5

4.8.1

Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.8.2

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Non-Translation Invariant Singular Integral Operators

122

5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.2 Extension to Non-Translation Invariant and Vector-Valued Singular Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.3 Hilbert-Space-Valued Mikhlin Multiplier Theorem . . . . . . . . . . . . . . 129

Contents

ix

5.4 Kernel Representation of a Pseudodifferential Operator . . . . . . . . . . . 133 5.5 Consequences of the Kernel Representation . . . . . . . . . . . . . . . . . . . . 140 5.6 Final Remarks and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.6.1 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.6.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

III 6

Applications to Function Space and Differential Equations Introduction to Besov and Bessel Potential Spaces

149

6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.2 A Fourier-Analytic Characterization of Hölder Continuity . . . . . . . . . 150 6.3 Bessel Potential and Besov Spaces – Definitions and Basic Properties 153 6.4 Sobolev Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 6.5 Equivalent Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.6 Pseudodifferential Operators on Besov Spaces . . . . . . . . . . . . . . . . . . 164 6.7 Final Remarks and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 6.7.1 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 6.7.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 7

Applications to Elliptic and Parabolic Equations

171

7.1 Applications of the Mikhlin Multiplier Theorem . . . . . . . . . . . . . . . . 7.1.1 Resolvent of the Laplace Operator . . . . . . . . . . . . . . . . . . . . . 7.1.2 Spectrum of Multiplier Operators with Homogeneous Symbols 7.1.3 Spectrum of a Constant Coefficient Differential Operator . . .

171 171 174 177

7.2 Applications of the Hilbert-Space-Valued Mikhlin Multiplier Theorem 180 7.2.1 Maximal Regularity of Abstract ODEs in Hilbert Spaces . . . . 180 7.2.2 Hilbert-Space Valued Bessel Potential and Sobolev Spaces . . 185 7.3 Applications of Pseudodifferential Operators . . . . . . . . . . . . . . . . . . . 7.3.1 Elliptic Regularity for Elliptic Pseudodifferential Operators . 7.3.2 Resolvents of Parameter-Elliptic Differential Operators . . . . . 7.3.3 Application of Resolvent Estimates to Parabolic Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

186 186 188 193

x

Contents

7.4 Final Remarks and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

IV A

7.4.1

Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

7.4.2

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Appendix Basic Results from Analysis

199

A.1 Notation and Functions on Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 A.2 Lebesgue Integral and Lp -Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 A.3 Linear Operators and Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 A.4 Bochner Integral and Vector-Valued Lp -Spaces . . . . . . . . . . . . . . . . . 209 A.5 Fréchet Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 A.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Bibliography

217

Index

221

Chapter 1

Introduction One of the leading ideas in the theory of pseudodifferential operators is to reduce the study of properties of a linear differential operator X c˛ .x/@˛x ; P D j˛jm

which is a polynomial in the derivatives @x D .@x1 ; : : : ; @xn / with constants c˛ depending on x, to its symbol X c˛ .x/.i /˛ ; p.x; / D j˛jm

which is a polynomial in the phase variable  2 Rn with constants depending on the space variable x 2 Rn . Then the question arises how properties of the differential operator p.x; Dx / – as for example invertibility or the spectrum of p.x; Dx / – are related to properties of the symbol p.x; /. The main tool in the theory is the Fourier transformation Z e ix f .x/dx;  2 Rn ; (1.1) F Œf ./ WD fO./ WD Rn

which is defined for suitable functions f W Rn ! C. The considerations in the following will be formal in order to sketch some main ideas. But they can be made precise under suitable assumptions. The identity (1.1) can be interpreted as inner product of the function f and the function x 7! e ix : Z F Œf ./ D f .x/eix dx D .f; e i   /L2 .Rn / (1.2) Rn

for fixed  2 Rn , where .  ;  / denotes the inner product on L2 .Rn /. Hence fO./ can be interpreted as the contribution of the complex (multidimensional) oscillation x 7! e ix to the function, where the phase variable  D .1 ; : : : ; n / 2 Rn describes the frequency of the oscillation. But the last identity in (1.2) is formal since x 7! e ix … L2 .Rn /. Knowing the Fourier transform g./ D fO./, the function f can be reconstructed with the aid of the inverse Fourier transformation Z 1 1 eix g./d  for all x 2 Rn F Œg.x/ WD .2/n Rn

2

Chapter 1 Introduction

(under certain assumptions on f ). Indeed it holds that F 1 ŒF Œf   F 1 ŒfO D f

(Inversion formula)

for suitable f . Here F 1 Œg can be interpreted as infinitesimal linear combination of x 7! e ix with coefficients g./ (and a correction factor 1=.2/n ). Using the inversion formula, Z X 1 ˛ c˛ .x/@x eix u./d O  P u.x/ D .2/n Rn j˛jm Z X 1 D c˛ .x/ .i /˛ eix u./d O  .2/n Rn j˛jm Z 1 e ix p.x; /u./d O  for all x 2 Rn ; D .2/n Rn where we have used @xj eix D i j eix and therefore @˛x eix D .i /˛ eix . This motivates the definition of the symbol p.x; / of P . In the following let p.x; / D p./ be independent of x 2 Rn . Then O D F 1 Œp./F Œu./: P u D F 1 Œp./u./ Hence application of P to u acts as multiplication of u./ O with the symbol p./. This suggests that inversion of P corresponds to multiplication with 1=p./ on the side of the Fourier transformed functions. Therefore we define Z 1 O 1 1 1 O f ./d  Qf .x/ WD F Œp./ f ./.x/ D eix n .2/ Rn p./ assuming that p./ ¤ 0 for all  2 Rn . Then F ŒQf  D p./1 fO./ since F F 1 D I (inversion formula) and therefore PQf D F 1 Œp./F ŒQf  D F 1 Œp./p./1 fO D F 1 ŒfO./ D f and similarly QPf D f , i.e., Q is indeed the inverse of P . Of course Q is no longer a differential operator; it belongs to the class of pseudodifferential operators, which are defined as Z 1 eix q.x; /fO./d  for all x 2 Rn ; q.x; Dx /f .x/ WD .2/n Rn where q.x; / is a suitable function, which is not necessarily a polynomial in . In the case that the coefficients of P depend on x the inverse of P is not determined that easy. If we define analogously to the constant coefficient case Z 1 1 fO./d ; e ix Qf .x/ WD n .2/ Rn p.x; /

3

Chapter 1 Introduction

then PQf .x/ D

1 .2/n

Z Rn

P

 e ix

 1 fO./d  p.x; /

for all x 2 Rn :

Because of the product rule,   1 1 ix D .P eix / C r.x; / P e p.x; / p.x; / 1 D p.x; / C r.x; / D 1 C r.x; / p.x; / where the remainder term r.x; / consists of terms where respect to x at least once. Hence

1 p.x;/

is differentiated with

PQf D I C r.x; Dx /; where r.x; Dx / ¤ 0 if p.x; / is not independent of x. But in some sense r.x; Dx / is of lower order (order less than 0) and is of minor role for many purposes. Here Q is called parametrix of P . In order to make the formal considerations above rigorous, one needs to study mapping properties of differential and pseudodifferential operators between suitable function spaces. In the simplest case one studies mapping properties between so-called (Lq -)Sobolev spaces Wqm .Rn / D ¹f 2 Lq .Rn / W @˛x f 2 Lq .Rn / for all j˛j  mº ; where the derivative @˛x f has to be understood in a generalized sense (i.e., in the sense of distributions). Moreover, Lq .Rn / is the space of all measurable f W Rn ! C such that Z 1 q q jf .x/j dx < 1: kf kLq .Rn / D Rn

The natural question arises whether a solution u 2 Lq .Rn / of a differential (or pseudodifferential) equation of order m p.x; Dx /u.x/ D f .x/

for all x 2 Rn

(1.3)

belongs to Wqm .Rn /. This is expected (if p.x; Dx / is elliptic) since p.x; Dx /W Wqm .Rn / ! Lq .Rn / if p.x; Dx / is a differential operator of order m with bounded and measurable coefficients. In the case that q D 2 and p.x; / D p./ is independent of x (and elliptic) this is a consequence of Plancherel’s theorem, which states that F W L2 .Rn / ! L2 .Rn / is a linear isomorphism and Z Z 1 2 O jf ./j d  D jf .x/j2 dx: .2/n Rn Rn

4

Chapter 1 Introduction

This is directly related to the fact that a so-called Fourier multiplication operator f 7! m.Dx /f D F 1 Œm./fO./ is bounded on L2 .Rn / if and only if mW Rn ! C is essentially bounded. If 1 < q < 1, q ¤ 2, and p.x; / D p./, the result is still true; but its proof is much more involved. It follows from the theory of (translation invariant) singular integral operators, which give sufficient conditions for a Fourier multiplication operator m.Dx / as above to be bounded on Lq .Rn / for every 1 < q < 1. In particular, the result is a consequence of the so-called Mikhlin multiplier theorem. Under its assumptions one has a kernel representation Z m.Dx /f D k.x  y/f .y/ dy for all x … supp f Rn

for a kernel kW Rn n ¹0º ! C that satisfies jk.z/j  C jzjn ;

jrk.z/j  C jzjn1

for all z ¤ 0:

Here the term singular integral operator comes from the fact that k is not necessarily in L1 .Rn /. (Note that e.g. jzjn … L1 .Rn /.) This makes the analysis of these kind of operators delicate and fascinating. This theory is treated in Chapter 4. In the case that p.x; / depends on x 2 Rn one can prove the same kind of “regularity result” for a solution u 2 Lq .Rn / of (1.3) (under suitable assumptions on p). But in order to prove it one needs to show that the parametrix Q above maps Lq .Rn / to Wqm .Rn /. Already in the case q D 2 the result is no longer an easy consequence of the Plancherel theorem. In Chapter 3 we will prove this fact using that the class of pseudodifferential operators is closed under compositions and (formal) adjoints. The proof in the case 1 < q < 1, q ¤ 2, is even more involved. To this end one needs an extension of the results on translation invariant singular integral operators in Chapter 4 to a class of singular integral operators, which are not necessarily translations invariant. This extension is done in Chapter 5. To apply this result it is essential that also pseudodifferential operators q.x; Dx / (of the class treated in this book) admit a kernel representation of the form Z k.x; x  y/f .y/ dy for all x … supp f; q.x; Dx /f .x/ D Rn

where the kernel satisfies sup jk.x; z/j  C jzjn ;

x2Rn

which is also shown in Chapter 5.

sup jrk.x; z/j  C jzjn1

x2Rn

for all z ¤ 0;

5

Chapter 1 Introduction

One advantage of studying differential equations with the aid of pseudodifferential operators is that the equations can be treated on a large class of function spaces simultaneously. More precisely, if one knows that a pseudodifferential operators maps between certain function spaces one can directly apply the result to the parametrix Q above and obtain a corresponding regularity result for the solution u of the differential equation above. In Chapter 6 we will give a brief introduction to the scales of so-called Bessel potential and Besov spaces. They include the Lp -Sobolev spaces above (if 1 < p < 1) as well as the classical Hölder spaces ´ μ jf .x/  f .y/j ˛ n 0 n <1 ; C .R / D u 2 Cb .R / W sup jx  yj˛ x¤y ® ¯ C m;˛ .Rn / D u 2 Cbm .Rn / W @˛x u 2 C ˛ .Rn / for all j˛j  m ; where 0 < ˛ < 1, m 2 N0 . The class of pseudodifferential operators treated in this book will have natural mapping properties on the full scales of Besov and Bessel potential spaces. Therefore one obtains results on elliptic regularity on all of these spaces. In particular this implies classical regularity results on Hölder spaces. This and further applications are treated in Chapter 7. In this chapter we will also study resolvent equations u.x/ C p.x; Dx /u.x/ D f .x/

for all x 2 Rn ;

where  2 C is called spectral parameter. We will obtain results on unique solvability of this equation for  in a suitable sector of the complex plain, provided a certain condition on parameter ellipticity is satisfied. Moreover, the solution u will be estimated in terms of the right-hand side f and the spectral parameter . Such kind of results are import in order to study the associated (parabolic) evolution equation @ t u.x; t / C p.x; Dx /u.x; t / D f .x; t / for all t > 0; x 2 Rn ; u.x; 0/ D u0 .x/

for all x 2 Rn :

This will also be briefly discussed in Chapter 7. Finally, we note that good solvability and regularity results of linear differential equations in Lq -Sobolev spaces for a general 1 < q < 1 (and not only for q D 2) are important for applications to non-linear differential equations. Having results for a general 1 < q < 1 one can get sharper results for the non-linear equations since one can apply Sobolev embedding results more flexibly. But applications to non-linear differential equations are beyond the scope of this book. Moreover, pseudodifferential operator methods can also be used to study differential equations in sufficiently smooth domains   Rn together with suitable boundary conditions. But also these applications are beyond the scope of this introduction. We refer to Grubb [11] for an introduction to this topic.

Part I

Fourier Transformation and Pseudodifferential Operators

Chapter 2

Fourier Transformation and Tempered Distributions Summary In this chapter the Fourier transformation is introduced and their most important properties for the study of pseudodifferential operators are proved. This includes Plancherel’s Theorem, the Fourier Inversion Theorem and Fourier Transformation of tempered distributions. Moreover, the relation between smoothness properties of a function and decay properties of its Fourier transform are discussed. This will be essential for the study of regularity questions for partial differential equations and the later study of function spaces.

Learning targets  Introduction to the Fourier transformation and its most important properties.  The Plancherel Theorem and the Fourier inverse.  Learn about the relation between smoothness of functions and decay properties of its Fourier transform.  Get to know the concept of (tempered) distributions and distributional derivatives.  Get the necessary knowledge for the study of pseudodifferential operators and their applications to partial differential equations.

2.1 Definition and Basic Properties The Fourier transformation is a powerful tool for analyzing functions on Rn and solving linear partial differential equations on Rn . Given f 2 L1 .Rn / we define the Fourier transform of f by Z eix f .x/dx (2.1) fO./ WD F Œf ./ WD Rn

for  2 Rn . Since jeix f .x/j D jf .x/j for all  2 Rn , we have eix f .x/ 2 L1 .Rn / with respect to x and (2.1) is well-defined. The operator F is called Fourier transformation. Sometimes we will write e.g. Fx7! in order to denote that the Fourier

10

Chapter 2 Fourier Transformation and Tempered Distributions

transformation is taken with respect to the variable x and the Fourier transform depends on . The following theorem summarizes several important properties of the Fourier transformation, which can be proved using only its definition: Theorem 2.1.

1. F W L1 .Rn / ! Cb0 .Rn / is a linear mapping such that kF Œf kC 0 .Rn /  kf kL1 .Rn / : b

2. If f W Rn ! C is a continuously differentiable function such that f 2 L1 .Rn / and @j f 2 L1 .Rn /, then F Œ@xj f  D i j F Œf  D i j fO:

(2.2)

3. If f 2 L1 .Rn / such that xj f 2 L1 .Rn /, then fO./ is continuously partial differentiable with respect to j and @j fO./ D F Œixj f .x/:

(2.3)

4. Let f 2 L1 .Rn / and .y f /.x/ WD f .x C y/, y 2 Rn , denote the translation of f by y. Then F Œy f ./ D e iy fO./ for all  2 Rn . 5. Let f 2 L1 .Rn / and let ." f /.x/ WD f ."x/, " > 0, denote the dilation of f by ". Then F Œ" f ./ D "n fO.="/ D "n ."1 fO/./: Proof. 1. First of all kfOkL1 .Rn / D sup jfO./j  sup 2Rn

2Rn

Z Rn

jeix f .x/jdx D kf kL1 .Rn / :

Hence fO is bounded. Let g.x; / D e ix f .x/. Then  7! g.x; / is continuous for every fixed x 2 Rn , x 7! g.x; / is integrable for every  2 Rn , and jg.x; /j  jf .x/j 2 L1 .Rn /. Therefore we can apply Theorem A.3 which implies the continuity of f . 2. Use integration by parts, cf. Exercise 2.50. 3. Use Theorem A.4, cf. Exercise 2.50. 4. Use the change-of-variables theorem, cf. Exercise 2.50. 5. Use the change-of-variables theorem, cf. Exercise 2.50.

11

Section 2.1 Definition and Basic Properties

!

The relation (2.2) is the fundamental property of the Fourier transformation from the point of view of differential equations. Because of the factor i in (2.2), we define 1 Dxj D @xj ; i

Dx D .Dx1 ; : : : ; Dxn /:

Then (2.2) is equivalent to F ŒDxj f  D j F Œf  D j fO:

Moreover, if we express a linear differential operator P with constant coefficients as Pf D

X

c˛ Dx f

j˛jm

with some constants c˛ 2 C, then F ŒPf  D

X

c˛  ˛ fO:

j˛jm

Hence application of a linear differential operator P to f corresponds to multiplication of fO with the polynomial p./ WD

X

c˛  ˛ :

j˛jm

The function p./ is called the symbol of P . Moreover, we write P D p.Dx /. Remark 2.2. If f 2 L1 .Rn / is continuously differentiable and @xj f 2 L1 .Rn / for all j D 1; : : : ; n, then fO./ and j fO./ are bounded functions. Hence .1 C jj/jfO./j  C

,

jfO./j 

C ; 1 C jj

which means that fO./ decays as jj1 as jj ! 1. More generally, if f 2 C k .Rn / such that @˛x f 2 L1 .Rn / for all j˛j  k, then one can show in the same way that jfO./j 

C : .1 C jj/k

12

Chapter 2 Fourier Transformation and Tempered Distributions

For short: Differentiability of f implies a polynomial decay of fO as jj ! 1. On the other hand, if .1 C jxj/k f .x/ 2 L1 .Rn / for some k 2 N, we can apply Theorem 2.1.3 successively to conclude that fO 2 C k .Rn /. Hence faster decay of f .x/ as jxj ! 1 yields higher differentiability of fO. Finally, the convolution of two functions f 2 L1 .Rn /, g 2 Lp .Rn /, where 1  p  1, is defined by Z .f  g/.x/ WD f .x  y/g.y/dy for almost all x 2 Rn : (2.4) Rn

Using Fubini’s theorem one can show that .f  g/.x/ exists for almost all x 2 Rn and kf  gkLp .Rn /  kf kL1 .Rn / kgkLp .Rn /

for all f 2 L1 .Rn /; g 2 Lp .Rn /; (2.5)

cf. [1, Section 2.12] or [29, Theorem 8.14 and Excercise 4, Chapter 8]. In the following we will often use the abbreviation kf kp WD kf kLp .Rn /

for all f 2 Lp .Rn /; 1  p  1:

For convolutions and Fourier transformation, we have the following simple rule: Lemma 2.3. Let f; g 2 L1 .Rn /, then fO./g./ O D F Œf  g ./ for all  2 Rn : Proof. First of all, f .x  y/g.y/ 2 L1 .Rn  Rn / with respect to .x; y/ since Z Z Z jf .x  y/g.y/jdydx D jf .z/g.y/jd.y; z/ kf  gk1  Rn

Rn

Rn Rn

D kf k1 kgk1 : by Fubini’s theorem and the change-of-variable theorem. Hence Z Z e ix f .x  y/g.y/dydx F Œf  g./ D Rn Rn  Z Z ix e .y f /.x/dx g.y/dy D n Rn ZR D O eiy fO./g.y/dy D fO./g./; Rn

where we have used Fubini’s theorem and the fourth statement of Theorem 2.1.

Section 2.2 Rapidly Decreasing Functions – S.Rn /

13

2.2 Rapidly Decreasing Functions – S.Rn / The space of rapidly decreasing, smooth functions will play a central role since Fourier transformation is an isomorphism on it. Its definition can be motivated as follows: Let f 2 C01 .Rn /. Then @˛x f 2 L1 .Rn / for all ˛ 2 N0 . As seen in Remark 2.2, fO./ decays faster than .1 C jxj/k for any k 2 N. Moreover, .1 C jxj/k f 2 L1 .Rn / for all k 2 N. Hence fO 2 Cbk .Rn / for all k 2 N, i.e., fO 2 Cb1 .Rn / D T1 k n 1 n O kD1 Cb .R /. But f ./ does not have compact support if f 2 C0 .R / unless f  0, cf. Exercise 2.55. If f 2 C01 .Rn /, then fO belongs to the following function space. Definition 2.4. The space S.Rn / of all rapidly decreasing smooth functions is the set of all smooth f W Rn ! C such that for all ˛ 2 N0n and N 2 N0 there is a constant C˛;N such that j@˛x f .x/j  C˛;N .1 C jxj/N

(2.6)

uniformly in x 2 Rn . If f 2 S.Rn / and m 2 N, we define the semi-norm: jf jm;S WD

sup

sup jx ˛ @ˇx f .x/j:

j˛jCjˇ jm x2Rn

We note that f 2 S.Rn / are also called Schwartz functions. 2 Obviously, C01 .Rn / S.Rn /. The inclusion is strict since f .x/ D ejxj 2 S.Rn / n C01 .Rn /. Moreover, S.Rn / Cb1 .Rn /. If f; g 2 S.Rn /, then fg 2 S.Rn /. Moreover, x ˛ f 2 S.Rn / for all ˛ 2 N0n and f 2 S.Rn /. This follows from the following more general statement: 1 .Rn / be the set of all smooth polynomially bounded functions, Lemma 2.5. Let Cpoly i.e., the set of all smooth f W Rn ! C such that for all ˛ 2 N0n there exist m˛ 2 N0 and C˛ > 0 with

j@˛x f .x/j  C˛ .1 C jxj/m˛

for all x 2 Rn :

1 .Rn / and g 2 S.Rn / we have fg 2 S.Rn /. Then for every f 2 Cpoly

Proof. This easily follows from the product rule, the Leibniz formula, respectively. See Exercise 2.58. We note that S.Rn / are Fréchet spaces with respect to the semi-norms j  jm;S , m 2 N. This can be easily proved using the completeness of Cbk .Rn / for any k 2 N0 .

14

Chapter 2 Fourier Transformation and Tempered Distributions

Remark 2.6. At first sight, it may be more natural to use jf j0m;S WD

sup

sup .1 C jxj/k j@˛ f .x/j

kCj˛jm x2Rn

as semi-norms on S.Rn / since they are more closely related to the inequality (2.6). But the definition of j  jm;S is more convenient when dealing with Fourier transformation, which will be demonstrated in the proof of the next lemma. Moreover, it does not matter if we use the semi-norms j  jm;S or j  j0m;S – the semi-norms are equivalent in the following sense: For every m 2 N0 there is an k.m/ 2 N0 such that jf j0m;S  Cm jf jk.m/;S

0 and jf jm;S  Cm jf j0k.m/;S

for all f 2 S.Rn /. (Actually, in this special case we can simply choose k.m/ D m.) 1 Finally, we note that we can replace .1 C jxj/ in (2.6) by hxi WD .1 C jxj2 / 2 since p hxi  .1 C jxj/  2hxi: For our purposes, a fundamental property of S.Rn / is the following: Lemma 2.7. F W S.Rn / ! S.Rn / is a linear mapping. Moreover, for all m 2 N0 there is some Cm > 0 such that jfOjm;S  Cm jf jmCnC1;S

for all f 2 S.Rn /:

I.e., F W S.Rn / ! S.Rn / is bounded. Proof. First of all, if f 2 S.Rn /, Z   kf k1 D .1 C jxj/n1 .1 C jxj/nC1 jf .x/j dx RZn .1 C jxj/n1 dxjf jnC1;S D C jf jnC1;S ; C Rn

where C depends only on the dimension n. Thus jfOj0;S D kfOk1  kf k1  C jf jnC1;S by Theorem 2.1.1 and the previous estimate. Because of Theorem 2.1.2/3,  ˛ Dˇ fO./ D F ŒDx˛ .x ˇ f .x//: Hence ˇ k ˛ D fOk1  C jDx˛ .x ˇ f .x//jnC1;S

(2.7)

15

Section 2.3 Inverse Fourier Transformation and Plancherel’s Theorem

by (2.7). Using the Leibniz formula (A.1), Dx˛ .x ˇ f .x//

! X ˛ .Dx x ˇ /.Dx˛ f .x//: D ˛



Since Dx x ˇ is polynomial of degree less than jˇj, jDx˛ .x ˇ f .x//jnC1;S  C˛;ˇ jf jj˛jCjˇ jCnC1: Collecting all estimates and taking the supremum over ˛; ˇ 2 N0n with j˛j C jˇj  m, we finally conclude jfOjm;S  Cm jf jmCnC1;S with arbitrary m 2 N0 where Cm depends only on n and m. Hence F Œf  2 S.Rn / for all f 2 S.Rn /. 2 =2

Example 2.8. Let f .x/ D e jxj 2

, x 2 Rn . Then fO./ D .2/n=2 ejj

2 =2

.

2

Proof. Since f .x/ D e x1 =2    exn =2 and eix D e ix1 1    eixn n , Z Z 2 ix1 1 x12 =2 O e e dx1    eix1 1 e xn =2 dxn D g. O 1 /    g. O n/ f ./ D R

R

2

where g.x/ D e x =2 , x 2 R. Hence it is sufficient to consider the case n D 1. Because of Theorem 2.1.3, g./ O is continuously differentiable and gO 0 ./ D d x 2 =2 F Œixg.x/. Moreover, xg.x/ D dx e D g 0 .x/. Therefore, using Theorem 2.1.2, O  2 R; gO 0 ./ D i F Œg 0 .x/ D  g./; p R x 2 =2 and g.0/ O D Re dx D 2. Hence gO is uniquely determined by the latter p 2 initial value problem, which has the unique solution g./ O D 2e  =2 . An alternative proof using the Cauchy integral formula can be found in [27, Example 1.7].

2.3 Inverse Fourier Transformation and Plancherel’s Theorem In the following we will show that the Fourier transformation F W S.Rn / ! S.Rn / is invertible and that its inverse is given by Z 1 1 eix g./d ; g.x/ L WD F Œg.x/ WD .2/n Rn

16

Chapter 2 Fourier Transformation and Tempered Distributions

which is well-defined for all g 2 L1 .Rn /. F 1 is called inverse Fourier transformation. We note that F 1 Œg.x/ D .2/n F Œg.x/:

(2.8)

Sometimes we will write e.g. F71 Œg in order to denote that the inverse Fourier !x transformation is taken with respect to the variable  and the inverse Fourier transform depends on x. 2 If f .x/ D e jxj =2 is the function discussed in Example 2.8, then 2 2 F 1 ŒfO./ D .2/n=2 F Œejj =2 .x/ D e jxj =2 D f .x/:

Hence F 1 ŒfO D f in this special f . In general we have: Lemma 2.9 (Inversion Formula). Let f 2 S.Rn /. Then f .x/ D F 1 ŒfO.x/ for all x 2 Rn . In particular, F W S.Rn / ! S.Rn / is a linear isomorphism. Proof. First of all, F

1

ŒfO.x/ D

1 .2/n

Z Rn



Z Rn

e

i.xy/

f .y/dy d :

Since e i.xy/ f .y/ … L1 .R2n / as function in .y; /, we cannot apply Fubini’s 2 2 theorem directly. Therefore we introduce a factor " ./ WD e " jj =2 , " > 0, which assures absolute integrability. Since lim"!0 " ./ D 1 for all  2 Rn and fO./ 2 L1 .Rn /, we have by Lebesgue’s and Fubini’s theorem  Z Z 1 1 O i.xy/ F Œf .x/ D lim e " ./f .y/dy d  "!0 .2/n Rn Rn Z 1 2 2 e i.xy/ e" jj f .y/d.y; /: D lim "!0 .2/n R2n Using the substitution  D =" and y D x C "z, Z 1 2 2 ei.xy/ e" jj =2 f .y/d.y; / n .2/ R2n  Z Z 1 iz jj2 =2 e e d f .x C "z/dz D .2/n Rn Rn Z 1 2 D ejzj =2 f .x C "z/dz n=2 .2/ Rn because of Example 2.8. Since f is continuous, we get Z Z 1 1 2 jzj2 =2 e f .x C "z/dz D f .x/ ejzj =2 dz D f .x/ lim n=2 n=2 "!0 .2/ .2/ Rn Rn by Lebesgue’s theorem on dominated convergence. Hence F 1 ŒfO.x/ D f .x/, which proves the lemma.

17

Section 2.3 Inverse Fourier Transformation and Plancherel’s Theorem 2

2

Remark 2.10. The technique of inserting a rapidly decreasing factor like e" jj =2 , " > 0, and passing to the limit afterwards will be fundamental in the following. It is the basis of the definition of the oscillatory integrals in Section 3.3 below. Theorem 2.11 (Plancherel’s Theorem). For every f; g 2 S.Rn / Z Z 1 fO./g./d f .x/g.x/dx D O : .2/n Rn Rn

(2.9)

In particular, 2 kf kL 2 .Rn / D

1 2 kfOkL 2 .Rn / : .2/n

and F extends to a linear isomorphism F W L2 .Rn / ! L2 .Rn /. Proof. Using Fubini’s theorem, it is easy to check that Z Z 1 F Œf ./g./d O D f .x/F 1 Œg.x/dx O .2/n Rn Rn O which implies (2.9). In for all f; g 2 S.Rn /. Because of Lemma 2.9, g D F 1 Œg, particular, if f D g, 1 kF Œf k22 D kf k22 : .2/n Since C01 .Rn / is dense in L2 .Rn / and obviously C01 .Rn / S.Rn / L2 .Rn /, F can be extended by continuity to a bounded linear operator F W L2 .Rn / ! L2 .Rn /. By (2.8) the same is true for F 1 . Moreover, F 1 ŒF Œf  D f for all f 2 L2 .Rn / since this is true for all f 2 S.Rn /. Remark 2.12. Because of the factor .2/n in the definition of the inverse Fourier transformation and in (2.9), we introduce the common notation

d ¯ WD

!

d .2/n

Then F 1 Œg.x/ D

Z Rn

e ix g./d ¯

(2.10)

18

Chapter 2 Fourier Transformation and Tempered Distributions

and

Z

Z

Rn

f .x/g.x/dx D

Rn

fO./g./ O d: ¯

In the following all integrals with respect to a phase variable ; ; : : : will be taken with respect to the scaled Lebesgue measure d; ¯ d ; ¯ : : : . All integrals with respect to a space variable x; y; z; : : : will be integrals using the usual Lebesgue measure dx; dy; dz; : : : . Then all constants are part of the measure and we do not have to worry about them. Lemma 2.13 (Fourier multipliers on L2 .Rn /). Let mW Rn ! C be a measurable functions. Then m.Dx /f WD F 1 Œm./fO./ for all f 2 L2 .Rn / is a well-defined bounded operator m.Dx /W L2 .Rn / ! L2 .Rn / if and only if m 2 L1 .Rn /. Moreover, km.Dx /kL.L2 .Rn // D kmkL1 .Rn / . Proof. If m 2 L1 .Rn /, obviously m./fO./ 2 L2 .Rn / for every f 2 L2 .Rn / because of Theorem 2.11. Moreover, km.Dx /f kL2 .Rn / D

1 .2/

n 2

km./fOk2 

1 n

.2/ 2

kmk1 kfOk2 D kmk1 kf k2 :

Hence m.Dx /W L2 .Rn / ! L2 .Rn / is a bounded linear operator and km.Dx /kL.L2 .Rn //  kmk1 : For the converse implication it is sufficient to prove that a multiplication operator M W L2 .Rn / ! L2 .Rn /, .M fO/./ D m./fO./ is bounded if and only if m 2 L1 .Rn /. Hence, if m.Dx /W L2 .Rn / ! L2 .Rn / is a bounded linear operator, then F m.Dx /F 1 W L2 .Rn / ! L2 .Rn / is also a bounded linear operator, where .F m.Dx /F 1 g/./ D m./g./ for all g 2 L2 .Rn /. Moreover, ²Z ³ 1 n kf kL1 .Rn / D sup f ./h./d  W h 2 L .R /; khkL1 .Rn / D 1 n ³ ²ZR 1 n jf ./jjh./jd  W h 2 L .R /; khkL1 .Rn / D 1 D sup Rn

19

Section 2.3 Inverse Fourier Transformation and Plancherel’s Theorem

for every f 2 L1 .Rn /, cf. (A.4), and Z 2 2 km./g./kL2 .Rn / D jm./j2 jg./j2 d   kM k2L.L2 .Rn // kgkL 2 .Rn / : Rn

Hence 2 kmkL 1 .Rn /

Z 2

D kjmj kL1 .Rn / D

sup h2L1 .Rn /WkhkL1 .Rn / 1

Z

D

sup g2L2 .Rn /WkgkL2 .Rn / 1

Rn

Rn

jm./j2 jh./j d 

jm./j2 jg./j2 d   kM k2L.L2 .Rn // ; 1

where we have used that h 2 L1 .Rn / if and only if g./ WD jh./j 2 2 L2 .Rn /. Therefore m 2 L1 .Rn / and km.Dx /kL.L2 .Rn // kmkL1 .Rn / . Example 2.14. 1. If m./ D F Œk, where k 2 L1 .Rn /, then m 2 Cb0 .Rn / L1 .Rn / due to Lemma 2.3. Moreover, m.Dx /f D k  f and km.Dx /f k2  kmk1 kf k2  kkk1 kf k2 : (Note that this inequality also follows from (2.5).) i

2. Let mj ./ D jjj , j D 1; : : : ; n. Then kmj k1 D 1 and Rj WD mj .Dx /W L2 .Rn / ! L2 .Rn / is a bounded operator. Rj is called Riesz operator. It will be shown that Z xj  yj f .y/dy D “kj  f ”; Rj f D lim cn "!0 n jx  yjnC1 R nC1

where kj .z/ D zj =jzjnC1 , cn D   2 . nC1 2 /, cf. Example 4.17 below. Note that the kernel kj … L1 .Rn /. These operators are typical examples of so-called singular integral operators. If n D 1, m1 ./ D i sign  and R1 is also called Hilbert transformation. 3. The symbol of  is jj2 , i.e.,  f D F 1 Œjj2fO./. Hence formally the inverse of  is given by the operator . /1 f D F 1 ŒfO./=jj2  and therefore   1 1 i j i k O @xj @xk . / f D F f ./ D Rj Rk f; jj jj where Rj ; Rk are the Riesz operator defined above. Hence, if f 2 L2 .Rn /, then @j @k . /1 f 2 L2 .Rn / in some sense; but the derivatives do not have exist in the classical sense. (They have to be understood as distributional derivatives, which will be defined in the next section.) More details can be found in Section 4.1.

20

Chapter 2 Fourier Transformation and Tempered Distributions

Corollary 2.15. Let f; g 2 S.Rn /. Then f  g 2 S.Rn /, where the convolution f  g is defined as in (2.4). Moreover, f  g D g  f and for every ˛ 2 N0n @˛x .f  g/.x/ D .@˛x f /  g.x/ D f  .@˛x g/.x/

for all x 2 Rn :

Proof. Because of Lemma 2.3, F Œf  g./ D fO./g./ O for all  2 Rn ; where fO./g./ O 2 S.Rn / due to Lemma 2.5. Moreover, due to (2.5) f g 2 L2 .Rn /. Hence Theorem 2.11 yields f  g D F 1 ŒfOg O

in L2 .Rn /;

O 2 S.Rn / due to Lemma 2.9. Furthermore, the last identity implies where F 1 ŒfOg 1 f  g D F ŒgO fO D g  f since f; g 2 S.Rn / are arbitrary. Similarly, @˛x .f  g/.x/ D F 1 Œ.i /˛ fOg.x/ O D F 1 Œ.i /˛ fO  g.x/ D .@˛x f /  g.x/; for all x 2 Rn , where we have used (2.2). Interchanging f and g finally gives @˛x .f  g/.x/ D f  .@˛x g/.x/ for all x 2 Rn .

2.4 Tempered Distributions and Fourier Transformation For many applications the space S.Rn / is to small. Therefore we introduce: Definition 2.16. The space of tempered distributions is S 0 .Rn / WD .S.Rn //0 – the space of linear and bounded functionals f W S.Rn / ! C. A sequence fk 2 S 0 .Rn / converges to f 2 S 0 .Rn / if and only if lim hfk ; 'i D hf; 'i

k!1

for all ' 2 S.Rn /:

Here hf; 'i WD f .'/, ' 2 S.Rn / denotes the duality product. Remark 2.17. 1. If f W S.Rn / ! C is a linear mapping, then by definition f is bounded if and only if there is some m 2 N0 and a constant C > 0 such that jhf; 'ij  C j'jm;S

for all ' 2 S.Rn /:

21

Section 2.4 Tempered Distributions and Fourier Transformation

2. If f W Rn ! C is a measurable function such that hxiN f .x/ 2 L1 .Rn / for some N 2 N0 , then f is naturally identified with a tempered distribution Ff 2 S.Rn /0 defined by Z f .x/'.x/dx for all ' 2 S.Rn /: (2.11) hFf ; 'i WD Rn

Moreover, jhFf ; 'ij  khxiN f kL1 .Rn / j'j0N;S  C khxiN f k1 j'jN;S ;

' 2 S.Rn /;

where j  j0N;S is as in Remark 2.6. We note that the mapping f 7! Ff is one-to-one because of the fundamental lemma of the calculus of variation, cf. Lemma A.7. Definition 2.18. Let F 2 S 0 .Rn /. Then F is called a regular tempered distribution if there is some f 2 L1loc .Rn / such that hxiN f .x/ 2 L1 .Rn / for some N 2 N0 and F D Ff , where Ff is as in (2.11). In the following we will identify a regular (tempered) distribution F D Ff with the measurable function f – identifying functions that coincide almost everywhere as usual. Not every distribution is a regular distribution as the following example shows, cf. Exercise 2.66. Example 2.19. The famous delta distribution ı0 2 S 0 .Rn / is defined by hı0 ; 'i WD '.0/

for all ' 2 S.Rn /:

hıx ; 'i WD '.x/

for all ' 2 S.Rn /;

More generally,

defines a tempered distribution for every x 2 Rn . One of the most important properties of distributions is that they can be differentiated infinitely many times (in a generalized sense). Definition 2.20. Let f 2 S 0 .Rn /. Then the distributional derivative @˛x f 2 S 0 .Rn / of f is the distribution defined by h@˛x f; 'i WD .1/j˛j hf; @˛x 'i

for ' 2 S.Rn /:

1 .Rn /, then the product fg 2 S 0 .Rn / is defined by Moreover, if g 2 Cpoly

hfg; 'i D hf; g'i

for all ' 2 S.Rn /:

22

Chapter 2 Fourier Transformation and Tempered Distributions

Remark 2.21. 1. The adjoint T 0 W Y 0 ! X 0 of a linear operator T W X ! Y is defined by hT 0 y 0 ; xi WD hy 0 ; T xi for y 0 2 Y 0 and x 2 X. Hence the derivative @˛x W S 0 .Rn / ! S 0 .Rn / is the adjoint of .1/j˛j @˛x W S.Rn / ! S.Rn /. 2. If f W Rn ! C is a k-times continuously differentiable function with hxiN @˛x f 2 L1 .Rn / for all j˛j  k, then f can be naturally considered as a tempered distribution, cf. Remark 2.17.1, Moreover, the distributional derivatives up to order k coincides with the usual derivative, i.e. Z Z f .x/@˛x '.x/dx D @˛x f .x/'.x/dx h@˛x f; '.x/i D .1/j˛j Rn

Rn

for all j˛j  k and ' 2 S.Rn / due to integration by parts. 3. Since every measurable function f W Rn ! C such that hxiN f .x/ 2 L1 .Rn / for some N 2 N0 can be considered as a distribution, it can be differentiated infinitely many times in the sense of distributions although it may even not be continuous. Example 2.22. 1. Let f be the Heaviside function, i.e. f .x/ D 1 for x 0 and f .x/ D 0 for x < 0. Then f 2 S 0 .R/ and the distributional derivative f 0 is Z Z 1 0 0 hf ; 'i D  f .x/' .x/dx D  ' 0 .x/dx D '.0/ D hı0 ; 'i R

0

for all ' 2 S.R/. Hence f 0 D ı0 is the delta distribution. 2. Let f 2 L1 .R/ be such that there are x1 < x2 <    < xn with f j.xj ;xj C1 / D fj j.xj ;xj C1 / for some fj 2 Cb1 .Œxj ; xj C1 / for all j D 1; : : : ; n  1. Then @x f D

n1 X

fj0  .xj ;xj C1 / C

j D1

n1 X

Œf .xj /ıxj

j D1

in the sense of S 0 .Rn /, where Œf .xj / WD fj .xj /  fj 1 .xj /, cf. Exercise 2.61. In particular, if additionally f 2 C 0 .R/, then @x f is a regular tempered distribution and @x f D

n1 X

fj0  .xj ;xj C1 / :

j D1

3. By definition the distributional derivative of the delta distribution is h@˛x ı0 ; 'i D .1/j˛j hı0 ; @˛x 'i D .1/j˛j .@˛x '/.0/ for all ' 2 S.Rn /.

Section 2.5 Fourier Transformation and Convolution of Tempered Distributions

23

The support of a continuous function f W Rn ! C is defined as the set supp f D ¹x 2 Rn W f .x/ ¤ 0º;

(2.12)

which is a closed set be definition. This definition does make much sense for measurable functions if functions coinciding on zero sets are identified. Moreover, the definition does not make sense for a general distribution since a distribution is not defined in a single point. The proper substitute for the definition is: Definition 2.23. Let F 2 S 0 .Rn / and let U  Rn be open. Then F vanishes on U , for short: F jU  0 if hF; 'i D 0 for all ' 2 C01 .U /: The support of F is defined as ¯ ® supp F D x 2 Rn W there is no " > 0 such that F jB" .x/  0 If f 2 L1loc.Rn / (with hxiN f .x/ 2 L1 .Rn / for some N 2 N), then f vanish on U if Ff vanishes on U . Moreover, supp f WD supp Ff . Here ' 2 C01 .U / is extended by zero to a function in C01 .Rn /. Consequently, supp F is the smallest closed set such that F vanishes on Rn n supp F . We note that, if f 2 L1 .Rn / is continuous (with hxiN f .x/ 2 L1 .Rn / for some N 2 N), then the definition of supp f in (2.12) coincides with the definition of supp f due to Definition 2.23, cf. Exercise 2.67. Example 2.24. Let ıx 2 S 0 .Rn /, x 2 Rn , be as in Example 2.19. Then supp ıx D ¹xº since hıx ; 'i D 0 for any ' 2 C01 .Rn / with x … supp '.

2.5 Fourier Transformation and Convolution of Tempered Distributions Most operations on S.Rn / directly carry over to S 0 .Rn / by duality. In particular, this holds true for the Fourier transformation.

24

Chapter 2 Fourier Transformation and Tempered Distributions

Definition 2.25. Let f 2 S 0 .Rn /. The Fourier transform F Œf  and its Fourier inverse F 1 Œf  of f are defined as the distributions hF Œf ; 'i WD hf; F Œ'i

for all ' 2 S.Rn /;

hF 1 Œf ; 'i WD hf; F 1 Œ'i for all ' 2 S.Rn /: Remark 2.26. We note that the definition of F on S 0 .Rn / is consistent with the definition of F on L1 .Rn / in the following sense: If f 2 L1 .Rn / and Ff 2 S 0 .Rn / is the associated tempered distribution due to (2.11), then Z f ./'./ O d hF ŒFf ; 'i D hFf ; F Œ'i D n R Z Z Z fO.x/'.x/ dx D eix f ./ d '.x/ dx D Rn

Rn

Rn

D hFfO ; 'i for all ' 2 S.Rn /, i.e., F ŒFf  D FfO . – Indeed this calculation is a motivation for the definition of F on S 0 .Rn /. The analogous statement holds for F 1 . Proposition 2.27. The Fourier transformation F W S 0 .Rn / ! S 0 .Rn / is a continuous linear mapping. Moreover, F W S 0 .Rn / ! S 0 .Rn / is a linear isomorphism with inverse F 1 . Proof. Since F W S.Rn / ! S.Rn / is a continuous linear operator and hF Œf ; 'i D hf; F Œ'i D .f ı F /.'/; F Œf  D f ı F W S.Rn / ! C is a continuous linear operator. Moreover, if fk ! f in S 0 .Rn / as k ! 1, then hF Œfk ; 'i D hfk ; F Œ'i ! hf; F Œ'i D hF Œf ; 'i k!1

! S 0 .Rn / is continuous. for all ' 2 S.R Hence F 1 Finally, F has the same properties as F and obviously F 1 ŒF Œf  D f for all 0 n f 2 S .R /. n /.

W S 0 .Rn /

Remark 2.28. As differentiation, the Fourier transformation is defined by duality, i.e., F W S 0 .Rn / ! S 0 .Rn / is defined as the adjoint of F W S.Rn / ! S.Rn /. Example 2.29. The Fourier transform of the delta distribution ı0 can be simply calculated: Z hF Œı0 ; 'i D hı0 ; F Œ'i D F Œ'.0/ D '.x/dx D h1; 'i Rn

for all ' 2 S.R /. Hence F Œı D 1. n

Section 2.6 Convolution on S 0 .Rn / and Fundamental Solutions

25

We conclude with a useful technical lemma: Lemma 2.30. For every f 2 S 0 .Rn / there is some N 2 N0 such that hxi2N hDx i2N f 2 L2 .Rn /; where hDx i2N f WD F 1 Œhi2N fO for all f 2 S 0 .Rn /. We note that the products hxi2N f and hi2N fO are well-defined in S 0 .Rn / for 1 .Rn /. More precisely, for any any f 2 S 0 .Rn / since hxi2N 2 Cb1 .Rn / Cpoly s s 2 R the function hi is a smooth function satisfying j@˛ his j  Cs;˛ .1 C jj/sj˛j

for all  2 Rn

(2.13)

for all ˛ 2 N0n and some Cs;˛ > 0. This estimate can be proved by considering the homogeneous function f W RnC1 n ¹0º ! R defined by fs .; t / D j.t; /js , cf. Exercise 2.51. Proof of Lemma 2.30. See Exercise 2.65. Corollary 2.31. For every f 2 S 0 .Rn / there is a sequence .fk /k2N0 C01 .Rn / such that fk !k!1 f in S 0 .Rn /. Proof. See Exercise 2.68.

2.6 Convolution on S 0 .Rn / and Fundamental Solutions Next we define the convolution of f 2 S 0 .Rn / and g 2 S.Rn /. To this end we note that Z Z .f  g/.x/'.x/ dx D .g  f /.x/'.x/ dx n Rn ZR Z f .y/g.x  y/'.x/ dy dx D n n ZR R Z D f .y/ g.x  y/'.x/ dx dy n Rn ZR D f .y/.gQ  '/.y/ dy Rn

Q D g.x/ for all for all f; g; ' 2 S.Rn / because of Fubini’s Theorem, where g.x/ x 2 Rn . The same conclusion holds if f 2 L1loc .Rn / with hxiN f .x/ 2 L1 .Rn / for some N 2 N.

26

Chapter 2 Fourier Transformation and Tempered Distributions

Therefore we define: Definition 2.32. Let f 2 S 0 .Rn /, g 2 S.Rn /. Then the convolution f  g 2 S 0 .Rn / of f and g is defined by hf  g; 'i D hf; gQ  'i

for all ' 2 S.Rn /

where g.x/ Q D g.x/ for all x 2 Rn . Remark 2.33. 1. If f is a regular tempered distribution, then the calculation above shows that the definition of f  g in the sense of Definition 2.32 coincides with Z f .x  y/g.y/ dy for all x 2 Rn : .f  g/.x/ D Rn

2. One can show that f  g is regular tempered distribution with f  g 2 C 1 .Rn / given by .f  g/.x/ D hf; Qx gi for all x 2 Rn ; where Qx g.y/ D g.y  x/ for all y 2 Rn , cf. Exercise 2.69. Example 2.34. Let ı0 be as in Example 2.19. Then Z g.y  0/'.y/ dy hı0  g; 'i D hı0 ; gQ  'i D Rn

for all g; ' 2 S.Rn /. Hence ı0  g is a regular tempered distribution and ı0  g D g

for all g 2 S.Rn /:

Finally, the calculus for derivatives and the Fourier transformation of convolutions on S.Rn / easily extends to convolutions with elements in S 0 .Rn /: Lemma 2.35. Let f 2 S 0 .Rn /; g 2 S.Rn /. Then @˛x .f  g/ D .@˛x f /  g D f  .@˛x g/;

F Œf  g D fOgO

in S 0 .Rn / for all ˛ 2 N0n . Proof. See Exercise 2.71. Example 2.36. The so-called fundamental solution of the Laplace equation N is defined by ´ c2 log jxj if n D 2; N.x/ D (2.14) nC2 cn jxj if n 3;

27

Section 2.7 Sobolev and Bessel Potential Spaces

1 1 where c2 D  4 , cn D  .n2/! , where !n D 2 n=2 =. n2 / is the surface area n of the unit sphere. Then N 2 L1loc .Rn / and .1 C jxj/M N.x/ 2 L1 .Rn / for some M 2 N0 . Hence N defines a regular tempered distribution. Moreover, one can show that

N D

N X j D1

@2xj N D ı0

in S 0 .Rn /;

cf. e.g. [28, Example 5.58]. Now, if we define u D N  f for some f 2 S.Rn /, then u D . N /  f D ı0  f D f

in S 0 .Rn /

because of Lemma 2.35 and Example 2.34. Hence we can solve the Laplace equation n /. in S 0 .Rn / for any right-hand Pside f 2 S.R ˛ More generally, if P D j˛jm c˛ @x is a differential operator with constant coefficients c˛ 2 C, m 2 N0 , then E 2 S 0 .Rn / is called a fundamental solution for P if PE D ı0 : Again, if u D E  f for some f 2 S.Rn /, then P u D f in S 0 .Rn / by the same calculation as before.

2.7 Sobolev and Bessel Potential Spaces Definition 2.37. Let 1  p  1, m 2 N0 . Then the (Lp -)Sobolev space of order m (on Rn ) is defined by Wpm .Rn / D ¹f 2 Lp .Rn / W @˛x f 2 Lp .Rn / for all j˛j  mº: Moreover, we define kf kWpm .Rn / D

´P 1 p . j˛jm k@˛x f kLp .Rn / / p maxj˛jm k@˛x f kL1 .Rn /

if 1  p < 1 if p D 1:

Here @˛x f has to be understood in the sense of S 0 .Rn / and @˛x f 2 Lp .Rn / if @˛x f is a regular distribution in Lp .Rn /, i.e., there is some g˛ 2 Lp .Rn / such that Z h@˛x f; 'i D .1/j˛j g˛ .x/@˛x '.x/ dx Rn

for all ' 2 S.Rn /. As usual we identify g˛ and @˛x f .

28

Chapter 2 Fourier Transformation and Tempered Distributions

The fundamental relation (2.2) generalizes to functions in W2m .Rn / as follows: Theorem 2.38. Let f 2 W2m .Rn /, m 2 N0 . Then for any j˛j  m we have F Œ@˛x f ./ D .i /˛ fO./ for almost all  2 Rn : Proof. Let f 2 W2m .Rn /. Then g˛ WD @˛x f 2 L2 .Rn / for any j˛j  m and therefore gO ˛ 2 L2 .Rn / by Plancherel’s Theorem. Moreover, Z Z Z 1 ˛ j˛j gO ˛ ./'./ d  D @x f .x/'.x/ L dx D .1/ f .x/@˛x '.x/ L dx .2/n Rn Rn Rn Z 1 fO./.i /˛ './ d  D .1/j˛j .2/n Rn Z 1 .i /˛ fO./'./ d  D .2/n Rn for any ' 2 S.Rn / by Plancherel’s Theorem, the definition of distributional derivatives in Lp .Rn / and Theorem 2.1. Since ' 2 S.Rn / is arbitrary, this implies F Œ@˛x f ./ D .i /˛ fO./ for almost all  2 Rn . As a consequence we obtain that for any f 2 W2m .Rn / Z X 1 2 j ˛ j2 jfO./j2 d ; kf kW m .Rn / D n 2 .2/ Rn j˛jm

where cm him 

X

j ˛ j2  Cm him

for all  2 Rn

j˛jm

for some constants cm ; Cm > 0. This gives a motivation for the following definition: Definition 2.39. Let s 2 R. Then the (L2 -)Bessel potential space H2s .Rn / is defined as H2s .Rn / WD ¹u 2 S 0 .Rn /W hDx is u 2 L2 .Rn /º; kukH2s WD kuks;2 WD khDx is uk2 ; 1

where hi WD .1 C jj2 / 2 and hDx is f D F 1 Œhis fO

for all f 2 S 0 .Rn /:

Often we will write H s .Rn / instead of H2s .Rn /. 1 Because of (2.13), his 2 Cpoly .Rn /. Therefore his fO./ 2 S.Rn / for all f 2 S.Rn / because of Lemma 2.5. By duality his fO 2 S 0 .Rn / for all f 2 S 0 .Rn /. Therefore hDx is W S 0 .Rn / ! S 0 .Rn / is well-defined above.

29

Section 2.7 Sobolev and Bessel Potential Spaces

Remark 2.40. By definition hDx is is an isomorphism from H2s .Rn / to L2 .Rn /. Hence H2s .Rn / normed by k  k2;s is Banach space. Moreover, .u; v/H2s WD .hDx is u; hDx is v/L2 ;

u; v 2 H2s .Rn /

is an inner product on H2s .Rn / and kuk22;s D .u; u/H2s . Thus H2s .Rn / is a Hilbert space. Moreover, W2m .Rn / is a Hilbert space with inner product X Z m @˛x f .x/@˛x g.x/ dx .f; g/W2 .Rn / D j˛jm

Rn

In Corollary 3.42 below we will show that S.Rn / is dense in H2s .Rn / for all s 2 R. Finally, H2s1 .Rn /  H2s2 .Rn / for s1 s2 since kuks2 ;2 D khDx is2 uk2 D khDx is2 s1 hDx is1 uk2  C khDx is1 uk2 D C kuks1 ;2 ; where we have used that his2 s1 2 L1 .Rn / and therefore hDx is2 s1 W L2 .Rn / ! L2 .Rn / is a bounded operator. The parameter s 2 R of H2s .Rn / determines the regularity of the functions u 2 H2s .Rn /. (“How many derivatives of u are in L2 .Rn /?”) More precisely, we have for s 2 N0 : Lemma 2.41. Let m 2 N0 . Then W2m .Rn / D H2m .Rn / with equivalent norms. Proof. If f 2 H2m .Rn /, then Dx˛ f D q˛ .Dx /hDx im f; where q˛ ./ WD  ˛ him 2 L1 .Rn / if j˛j  m. Hence X

kDx˛ f k22

12

 Cm khDx im f k2 D Cm kf km;2

j˛jm

due to Lemma 2.13. Conversely, if f 2 W2m .Rn /, @˛x f 2 L2 .Rn / for all j˛j  m. Moreover, X j 1 hi2 D C : j hi hi hi n

hi D

j D1

Thus him D

N X kD1

pk ./qk ./;

30

Chapter 2 Fourier Transformation and Tempered Distributions

where qk ./ is a polynomial of order m and pk 2 L1 .Rn /, N D N.m/ 2 N. Hence N

X



kf km;2 D khDx im f k2 D

pk .Dx /qk .Dx /f

kD1

 Cm

N X

2

0 kqk .Dx /f k2  Cm kf kW2m ;

kD1

where we have used that pk .Dx /W L2 .Rn / ! L2 .Rn / are bounded linear operators and that kqk .Dx /f k2  C kf kW2m since qk .Dx / is a differential operator of order m. More generally on defines: Definition 2.42. Let s 2 R and let 1 < p < 1. Then the (Lp -)Bessel potential space Hps .Rn / of order s is defined by ® ¯ Hps .Rn / WD f 2 S 0 .Rn / W hDx is f 2 Lp .Rn / kf kHps .Rn / WD khDx is f kLp .Rn / : In Section 6.3, it will be shown that Wpm .Rn / D Hpm .Rn / for any m 2 N0 , 1 < p < 1. The proof is similar to the proof of Lemma 2.41. But one has to use the Mikhlin Multiplier Theorem, which will be proved in Section 4.6 below, instead of Plancherel’s Theorem/Lemma 2.13.

2.8 Vector-Valued Fourier-Transformation

i

This section will only be needed in Section 4.6 for the formulation and the proof of the Hilbert-space valued Mikhlin Multiplier Theorem and later applications based on that. Therefore it can be skipped for the first reading.

For the following the reader should be familiar with the basic properties of the Bochner integral, cf. Appendix A.4. For many applications it is important to extend the Fourier transformation to vector-valued functions f W Rn ! X, where X is a Banach space. As in the scalar case, i.e., X D C, we define Z e ix f .x/dx 2 X (2.15) fO./ WD F Œf ./ WD Rn Z 1 fL.x/ WD F 1 Œf .x/ WD eix f ./d  2 X .2/n Rn

Section 2.8 Vector-Valued Fourier-Transformation

31

for  2 Rn and f 2 L1 .Rn I X/, where the integral above is defined as a Bochner integral. A lot of identities for the usual (scalar-valued) Fourier transformation directly carry over to the vector-valued case by using: Z 0 O hf ./; x iX;X 0 D eix hf .x/; x 0 iX;X 0 dx D F Œfx 0 ./ (2.16) Rn

for all  2 Rn ; x 0 2 X 0 , where fx0 .x/ WD hf .x/; x 0 iX;X 0 for all x 2 Rn due to (A.8). In particular we have: Theorem 2.43. Let X be a Banach space. 1. F W L1 .Rn I X/ ! Cb0 .Rn I X/ is a linear mapping such that kF Œf kC 0 .Rn IX/  kf kL1 .Rn IX/ : b

(2.17)

2. If f W Rn ! X is a continuously differentiable function such that f 2 L1 .Rn I X/ and @xj f 2 L1 .Rn I X/ for some j 2 ¹1; : : : ; nº, then F Œ@xj f  D i j F Œf  D i j fO: 3. If f 2 L1 .Rn I X/ such that xj f 2 L1 .Rn I X/, then fO./ is continuously partially differentiable with respect to j and @j fO./ D F Œixj f .x/: 4. Let f 2 L1 .Rn I X/ and let ." f /.x/ WD f ."x/, " > 0, x 2 Rn , denote the dilation of f by ". Then F Œ" f ./ D "n fO.="/ D "n .1=" fO/./: 5. Let f 2 L1 .Rn I X/ and .y f /.x/ WD f .x C y/, y 2 Rn , denote the translation of f by y. Then F Œy f ./ D e iy fO./ for all  2 Rn . 6. If f 2 L1 .Rn I X/, g 2 L1 .Rn /, then fO./g./ O D F Œf  g ; where

Z .f  g/.x/ D

Rn

f .x  y/g.y/dy

denotes the convolution of f and g, cf. Lemma A.16.

32

Chapter 2 Fourier Transformation and Tempered Distributions

Proof. 1. The estimate (2.17) follows directly from the definition of F Œf  and (A.7). The continuity of  7! fO./ follows from Z 0 kfO. 0 /  fO./kX  je ix  eix jkf .x/kX d.x/ Rn

and Theorem A.3 as in the scalar case. 2. The statement can be reduced to the statement for scalar functions as follows: For every fixed x 0 2 X 0 let g.x/ WD hf .x/; x 0 iX;X 0 for all x 2 Rn . Then g 2 L1 .Rn / is continuously differentiable with respect to xj and @xj g D h@xj f .x/; x 0 iX;X 0 for all x 2 Rn . Hence hF Œ@xj f ./; x 0 iX;X 0 D F Œ@xj g./ D i j F Œg./ D hi j fO./; x 0 iX;X 0 for all  2 Rn , where we have used (2.16). Since x 0 2 X 0 was arbitrary, we conclude F Œ@xj f ./ D i j fO./ for all  2 Rn : 3. The proof is done in the same way as in the scalar case using an appropriate version of Theorem A.4, cf. Exercise A.25. 4./5./6. The identities can be reduced to the corresponding identities in the scalar case using (2.16) as before. As in the scalar case we define: Definition 2.44. The space S.Rn I X/ is the set of all smooth f W Rn ! X such that for all ˛ 2 N0n and N 2 N0 there is a constant C˛;N such that k@˛x f .x/kX  C˛;N .1 C jxj/N

(2.18)

uniformly in x 2 Rn . If f 2 S.Rn I X/ and m 2 N, we define the semi-norm: jf jm;S WD

sup

sup kx ˛ @ˇx f .x/kX

j˛jCjˇ jm x2Rn

As before we have: Lemma 2.45. F W S.Rn I X/ ! S.Rn I X/ is a bounded linear mapping. Proof. The proof is the same as for Lemma 2.7. Lemma 2.46 (Inversion Formula). Let f 2 S.Rn I X/. Then f .x/ D F 1 ŒfO.x/ for all x 2 Rn . In particular, F W S.Rn I X/ ! S.Rn I X/ is a linear isomorphism.

33

Section 2.9 Final Remarks and Exercises

Proof. The proof of Theorem 2.9 directly carries over to the vector-valued situation. Alternatively, one can reduce the identity f .x/ D F 1 ŒfO.x/ for all x 2 Rn to the scalar situation by multiplying with x 0 2 X 0 and using (2.16). Plancherel’s theorem does not carry over to X-valued functions for a general Banach space X. To this end one has to make sense of the pointwise products in (2.9) O respectively. This can be done if X is a between f .x/ and g.x/, fO./ and g./, Hilbert space. Theorem 2.47 (Plancherel’s Theorem). Let H be a complex Hilbert space. Then for every f; g 2 S.Rn I H / Z Z 1 .f .x/; g.x//H dx D .fO./; g.// O (2.19) H d : .2/n Rn Rn In particular, 2 kf kL 2 .Rn IH / D

1 2 kfOkL 2 .Rn IH / : .2/n

and F extends to a linear isomorphism F W L2 .Rn I H / ! L2 .Rn I H /. Proof. Using Fubini’s theorem, it is easy to check that Z Z 1 .F Œf ./; g.// O d  D .f .x/; F 1 Œg.x// O H H dx n .2/n Rn R for all f; g 2 S.Rn I H /. Because of Lemma 2.46, g D F 1 Œg, O which implies (2.19). In particular, if f D g, 1 2 2 kF Œf kL 2 .Rn IH / D kf kL2 .Rn IH / : .2/n Since C01 .Rn I H / is dense in L2 .Rn I H / and obviously C01 .Rn I H / S.Rn I H /, F can be extended by continuity to a bounded operator F W L2 .Rn I H / !L2 .Rn I H /. By (2.8) the same is true for F 1 . Moreover, F 1 ŒF Œf  D f for all f 2 L2 .Rn I H / since this is true for all f 2 S.Rn I H /.

2.9 Final Remarks and Exercises 2.9.1 Further Reading In this chapter we gave a brief introduction to the most important facts on Fourier transformation and distribution theory that are needed in this book. There are many good books on harmonic analysis, where the Fourier transformation is treated in

34

Chapter 2 Fourier Transformation and Tempered Distributions

greater detail, cf. e.g. the monographs by Katznelson [19] and Stein and Weiss [33]. Among the many monographs on distribution theory we recommend the classical book by Hörmander [14] and the introduction by Walter [41]. Both books treat distributions from the viewpoint of applications. Mathematically, the theory of distributions is closely connected to the theory of topological vector space. A classical introduction to that topic is given by Treves [36]. A good introduction to Fourier transformation, distribution theory, and Sobolev spaces in relation to partial differential equations is also contained in introductions to partial differential equations by Renardy and Rogers [28] and Jacob [18].

2.9.2 Exercises Exercise 2.48. A function f W Rn n ¹0º ! C is called homogeneous of degree d 2 R if f .rx/ D r d f .x/ for all r > 0; x ¤ 0. 1. Let f W Rn n ¹0º ! C be continuous and homogeneous of degree d 2 R. Prove that there is a constant C > 0 (depending on f ) such that jf .x/j  C jxjd ;

for all x 2 Rn n ¹0º:

Determine the smallest possible C . 2. Let f W Rn n ¹0º ! C be k-times continuously differentiable and homogeneous of degree d 2 R. Prove that @˛x f is homogeneous of degree d  j˛j for all j˛j  k. Moreover, conclude that j@˛x f .x/j  C˛ jxjd j˛j ;

for all x 2 Rn n ¹0º;

for all j˛j  k, where C˛ depends on ˛ and f . Exercise 2.49. Prove the statements of Theorem 2.1 in detail. Exercise 2.50. Calculate the Fourier transform of the following functions: ´ eax if x 0; f1 .x/ D 0 else; f2 .x/ D e ajxj ; f3 .x/ D Œa;a .x/; where a > 0. Compare the properties of the functions fj (continuity, differentiability, analyticity and decay as jxj ! 1) with corresponding properties of fOj .

35

Section 2.9 Final Remarks and Exercises

Exercise 2.51. Let hi D some Cs;˛ > 0 such that

p

1 C jj2 . Prove that for any s 2 R, ˛ 2 N0n there is

j@˛ his j  Cs;˛ .1 C jj/sj˛j

for all  2 Rn : m

Hint. Consider the function f .a; x/ WD .a2 C jxj2 / 2 , where a 2 R, x 2 Rn . Use that f is homogeneous of degree m, i.e., f .ra; rx/ D r m f .a; x/ for all r > 0, a 2 R, x 2 Rn . Exercise 2.52. Let A 2 Rnn be an invertible matrix and f 2 L1 .Rn /. Prove that F Œf ı A./ D

1 fO.AT /; jdet Aj

where A is identified with the mapping x 7! Ax and AT WD .A1 /T . Exercise 2.53. We consider the partial differential equation .1  /u D f

in Rn

(2.20)

for u; f 2 S.Rn /. 1. Determine the symbol of 1  , i.e., find p./ such that .1  /u D O F 1 Œp./u./. 2. Prove that for every f 2 S.Rn / there is a unique solution u D .1  /1 f 2 S.Rn / of (2.20). Moreover, show that .1  /1 W S.Rn / ! S.Rn / is bounded. 3. Let H2s .Rn / D S.Rn / where

k  ks;2

be the Bessel potential space of order s 0,

kf ks;2 WD khis fO./k2 ; 1

with hi WD .1 C jj2 / 2 . Prove that .1  /1 extends to a bounded linear operator .1  /1 W H2s .Rn / ! H2sC2 .Rn /. Exercise 2.54.

1. Prove that S.Rn / Lp .Rn / for all 1  p  1.

2. Show that for every m 2 N there are constants Cm ; cm such that cm jf jm;S  jf j0m;S  Cm jf jm;S :

36

Chapter 2 Fourier Transformation and Tempered Distributions

Exercise 2.55. Let f 2 C01 .R/ such that supp f  BR .0/. Prove that fO./ D

Z

R R

e ix f .x/dx;

 2 C;

is a holomorphic function in C. Moreover, jfO./j  C eRj Im j . In particular, fO./,  2 R, is real analytic and supp fO is not compact unless fO./  0. Exercise 2.56. Let u 2 L2 .Rn /. Prove that u possesses a weak derivative @xj u 2 L2 .Rn / if and only if i j uO 2 L2 .Rn /. In this case F Œ@xj u./ D i j u./ O

almost everywhere:

(2.21)

Hint. Multiply (2.21) mit F Œ', ' 2 C01 .Rn /. Exercise 2.57. Prove that for every ˛ 2 N0n and m 2 N there are constants 0 > 0 such that Cm;˛ ; Cm;˛ jx ˛ f jm;S  Cm;˛ jf jmCj˛j;S

0 j@˛x f jm;S  Cm;˛ jf jmCj˛j;S

uniformly in f 2 S.Rn /. 1 .Rn / be as in Lemma 2.5. Exercise 2.58. Let Cpoly 1 .Rn / and define .Ma f /.x/ WD a.x/f .x/ for all f 2 S.Rn /. 1. Let a 2 Cpoly Prove that Ma W S.Rn / ! S.Rn / is a bounded linear operator.

2. Prove that every differential operator X Lu.x/ D c˛ .x/Dx˛ u;

u 2 S.Rn /;

j˛jm 1 .Rn / defines a continuous linear operator LW S.Rn / ! with c˛ 2 Cpoly S.Rn /.

Exercise 2.59 (A Simple Functional Calculus). Let pW Rn ! C be measurable and set † WD p.Rn /. For all F 2 L1 .†/ let F .p.Dx //f WD F 1 ŒF .p.//fO./ for all f 2 L2 .Rn /: 1. Prove that for all F 2 L1 .†/ we have F .p.Dx // 2 L.L2 .Rn // and the mapping ˆW L1 .†/ 3 F 7! F .p.Dx // 2 L.L2 .Rn //

37

Section 2.9 Final Remarks and Exercises

is linear. Moreover, show that for every Fj 2 L1 .†/, j D 1; 2, it holds that F1 .p.Dx // ı F2 .p.Dx // D .F1  F2 /.p.Dx //. 2. Prove that for all  2 C n † we have .  p.Dx //1 2 L.L2 .Rn // and .  p.Dx //.  p.Dx //1 f D .  p.Dx //1 .  p.Dx //f D f for all f 2 S.Rn /, where p.Dx /f D F 1 Œp./fO./ for all f 2 S.Rn /. 3. For which  2 C there exists .  /1 in the sense above? Exercise 2.60. Let .Mf /.x/ WD F 1 Œm./fO, where m 2 L1 .Rn /. Prove that M W L2 .Rn / ! L2 .Rn / is invertible if and only if m1 ./ 2 L1 .Rn /. Remark. X D L1 .Rn / and X D L.L2 .Rn // are Banach algebras with unit, i.e., X is Banach space and a ring with unit such that .xy/ D .x/y D x.y/,  2 C, and kxykX  kxkX kykX for all x; y 2 X: Exercise 2.61 (Distributional Derivative). Let f 2 L1 .R/ be such that there are x1 < x2 < ::: < xn with f j.xj ;xj C1 / D fj j.xj ;xj C1 / for some fj 2 Cb1 .Œxj ; xj C1 / for all j D 1; :::n  1. Prove that the distributional derivative @x f of f in S 0 .R/ is given by @x f D

n1 X

fj0

 .xj ;xj C1 / C

j D1

n1 X

Œf .xj /ıxj

j D1

where Œf .xj / WD fj .xj /  fj 1 .xj / and for every x 2 R hıx ; i D .x/

for all  2 S.R/:

Exercise 2.62. Let p be a polynomial on Rn such that p.x/ D c˛ 2 C. Calculate F Œp in the sense of S 0 .Rn /.

P

j˛jm c˛ x

˛,

Exercise 2.63. Let s > n2 . Prove that H2s .Rn / ,! Cb0 .Rn /. Exercise 2.64. Let .Tu /.x 0 / WD u.x 0 ; 0/, x 0 D .x1 ; : : : ; xn1 / 2 Rn , be the restriction of u 2 S.Rn / to the hyper-plane ¹.x 0 ; 0/W x 0 2 Rn1 º. Prove that for s > 12 kT uk

s 1 2

H2

.Rn1 /

 Cs kukH2s .Rn / ;

for u 2 S.Rn /:

38

Chapter 2 Fourier Transformation and Tempered Distributions s 12

Hence the T extends to a bounded operator T W H2s .Rn / ! H2 operator) if s > 12 .

.Rn1 / (trace

Hint. Use ju. Q 0 ; 0/j2 D

Z

0 1

.@n u. Q 0 ; xn /u. Q 0 ; xn / C u. Q 0 ; xn /@n u. Q 0 ; xn //dxn ;

where u. Q 0 ; xn / D Fx 0 7! 0 Œu.  ; xn / is the partial Fourier transform of u. Exercise 2.65.

1. Prove that for any k 2 N0 there is a Ck > 0 such that jf j00k;S WD

sup j˛jCjˇ jk

kx ˛ Dxˇ f k2  Ck khxik hDx ik f k2

for all f 2 S.Rn /. Hint. First prove for fixed N 2 N0 by mathematical induction that sup j˛jk;jˇ jN

kx ˛ Dxˇ f k2  Ck;N sup kx ˛ hDx iN f k2 : j˛jk

2. Conclude that for any f 2 S 0 .Rn / there are N 2 N and C; C 0 > 0 such that jhf; 'ij  C khxi2N hDx i2N 'k2 for all ' 2 S.Rn /. Hint. Use the Riesz representation theorem for L2 .Rn /. 3. Prove that hxi2N hDx i2N f 2 L2 .Rn / for some N 2 N0 . Remark. More precisely, one can prove that \ H2s1 ;s2 .Rn /; S 0 .Rn / D S.Rn / D s1 ;s2 2R

[

H2s1 ;s2 .Rn /;

s1 ;s2 2R

where H2s1 ;s2 .Rn / D ¹f 2 S 0 .Rn / W hxis2 hDx is1 f 2 L2 .Rn /º are weighted Bessel potential spaces and hxim W H2s1 ;s2 Cm .Rn / ! H2s1 ;s2 .Rn / hDx im W H2s1 Cm;s2 .Rn / ! H2s1 ;s2 .Rn / are isomorphisms.

39

Section 2.9 Final Remarks and Exercises

Exercise 2.66. Prove that the delta distribution is not a regular distribution. Hint. Consider a suitable sequence of test functions .'j /j 2N with 'j .0/ D 1 for all j 2 N. Exercise 2.67. Let f 2 L1 .Rn / be continuous with hxiN f .x/ 2 L1 .Rn / for some N 2 N. Prove that supp f defined by (2.12) coincides with the definition of supp f due to Definition 2.23. Exercise 2.68. Prove that for every f 2 S 0 .Rn / there is a sequence .fk /k2N C01 .Rn / such that fk !k!1 f in S 0 .Rn /. Hint. Use Lemma 2.30 and the density of C01.Rn / in L2 .Rn /. Exercise 2.69. Let f 2 S 0 .Rn /, g 2 S.Rn /. Prove that f  g is a regular 1 tempered distribution satisfying f  g 2 Cpoly .Rn / and f  g.x/ D hf; Qx gi for all x 2 Rn ; where Qx g.y/ D g.x  y/ for all x; y 2 Rn . Hint. Use that C01 .Rn / is dense in L2 .Rn / and Lemma 2.30. Exercise 2.70. Let ıx , x 2 Rn , be as in Example 2.66. Show that ıx  g D x g

for all g 2 S.Rn /;

where .x g/.y/ D g.x C y/ for all x; y 2 Rn . Exercise 2.71. Let f 2 S 0 .Rn /; g 2 S.Rn /. Prove that @˛x .f  g/ D .@˛x f /  g D f  .@˛x g/;

F Œf  g D fOgO

in S 0 .Rn / for all ˛ 2 N0n . Exercise 2.72. Prove Lemmas 2.45 and 2.46 in detail.

Chapter 3

Basic Calculus of Pseudodifferential Operators on Rn Summary In this chapter we present a first introduction to pseudodifferential operators with m symbols in the class S1;0 , which is an important special case of the more general Hörm mander class S;ı studied in more advanced text books. The main goals are to prove the most important mapping properties and to show that the class of pseudodifferential operators are closed under compositions and has formal adjoints. Moreover, asymptotic formulae for the symbol of the composition and the formal adjoints are proven, which provide important information for applications, e.g. to elliptic operators. To this end oscillatory integrals are introduced and used.

Learning targets m and the definition of a pseudodifferential  Understand the basic symbol class S1;0 operator.

 Learn the fundamental calculus of pseudodifferential operators, in particular compositions and adjoints.  Learn the basic mapping properties of pseudodifferential operators on S.Rn /, S 0 .Rn /, Cb1 .Rn /, and between L2 -Bessel potential spaces.  Get to know first applications to elliptic pseudodifferential operators, their parametrices and elliptic regularity.

3.1 Symbol Classes and Basic Properties m . There are many We now introduce the basic pseudodifferential symbol class S1;0 other more general or modified symbol classes, which are used in the literature and research for different purposes. But the following symbol class is the most simple and most common one. It is a natural symbol class that contains the symbols of differential operators with smooth coefficients and inverses of elliptic symbols. m .RN  Rn / is the vector-space of Definition 3.1. Let m 2 R, n; N 2 N. Then S1;0 N n all smooth functions pW R  R ! C such that

j@˛ @ˇx p.x; /j  C˛;ˇ .1 C jj/mj˛j

(3.1)

41

Section 3.1 Symbol Classes and Basic Properties

holds for all ˛ 2 N0n , ˇ 2 N0N , where C˛;ˇ is independent of x 2 RN ;  2 Rn . The function p is called pseudodifferential symbol and m is called the order of p. Moreover, [ 1 m .RN  Rn / WD S1;0 .RN  Rn / and S1;0 m2R 1 S1;0 .RN

n

 R / WD

\

m S1;0 .RN  Rn /:

m2R m instead of S m .RN  Rn /. For short we also write S1;0 1;0

Remark 3.2. In the following we usually deal with the case N D n. But sometimes it is useful to have defined the symbol classes for general N; n 2 N. m .Rn  Rn / is a symbol, then If p 2 S1;0

Z p.x; Dx /f .x/ WD OP.p/f .x/ WD

Rn

eix p.x; /fO./d¯ 

for all x 2 Rn (3.2)

defines the associated pseudodifferential operator, where f W Rn ! C is a suitable function. If f 2 S.Rn /, then fO 2 S.Rn / and therefore p.x; /fO./ 2 S.Rn / with respect to  for every fixed x 2 Rn due to Lemma 2.5. Therefore the integral in (3.2) exists and p.x; Dx /f is well-defined. In the following we will prove that p.x; Dx /W S.Rn / ! S.Rn / is a continuous mapping. But before we prove this fact, we discuss some examples and make some simple observations. P Example 3.3. 1. Let p.x; / D j˛jm c˛ .x/ ˛ , x;  2 Rn , be a polynomial in  of order m 2 N0 with smooth coefficients c˛ 2 Cb1 .Rn / for all j˛j  m. Then m p 2 S1;0 .Rn  Rn / and p.x; Dx /f WD

X

c˛ .x/Dx˛ f

j˛jm

for every f 2 S.Rn /. Hence every linear differential operator with smooth and bounded coefficients is a pseudodifferential operator. In particular the Laplacian D @2x1 C    C @2xn is a pseudodifferential operator with symbol jj2 . p 2. The function hi WD 1 C jj2 is a pseudodifferential symbol of order 1, see Exercise 2.51. Since 1 C jj2 is the symbol of 1  , the associated pseudodifferential operator Z q eix 1 C jj2 fO./d hDx if D ¯ Rn

Chapter 3 Basic Calculus of Pseudodifferential Operators on Rn

42

can be considered as the square root of 1  . For short: hDx i D

p 1  . m

m for every m 2 R and hD im D .1  / 2 , cf. More generally, him 2 S1;0 x Exercise 2.51 m .RN  Rn /, which is related to the We can define a sequence of semi-norms on S1;0 family of inequalities (3.1) in a natural way. Let .m/

jpjk

WD max

sup

j˛j;jˇ jk x2RN ;2Rn

jD˛ Dxˇ p.x; /j.1 C jj/mCj˛j

(3.3)

for k 2 N. Here sup x2RN ;2Rn

jD˛ Dxˇ p.x; /j.1 C jj/mCj˛j

is the smallest constant C˛;ˇ such that (3.1) holds for all x 2 RN ;  2 Rn and fixed ˛; ˇ 2 N0 . m It is not difficult to check that S1;0 with these semi-norms is a Fréchet space. We refer to Appendix A.5 for some basic facts on Fréchet spaces. Remark 3.4. In the p literature the function .1 C jj/ in the estimates (3.1) is often replaced by hi D 1 C jj2 . This can be done without changing the symbol classes since q p q 1 C jj2  .1 C jj/  2 1 C jj2 ; cf. proof of Lemma 3.7 below. Using hi instead of .1 C jj/, the notation becomes a bit shorter. m

Proposition 3.5. Let pj 2 S1;0j .Rn  Rn /, mj 2 R, j D 1; 2, and let p.x; / WD m1 Cm2 p1 .x; /p2 .x; / for all x;  2 Rn . Then p 2 S1;0 .Rn  Rn /. Moreover, for any k 2 N0 there is some Ck depending only on k and n such that .m1 Cm2 /

jpjk

.m1 /

 Ck jp1 jk

.m2 /

jp2 jk

:

The proposition is a simple consequence of the Leibniz formula, cf. Exercise 3.53. The main result of this section is: m Theorem 3.6. Let p 2 S1;0 .Rn  Rn /, m 2 R, be a pseudodifferential symbol. Then

p.x; Dx /W S.Rn / ! S.Rn / is a bounded mapping. More precisely, for every k 2 N there is some Ck > 0 such that .m/

jp.x; Dx /f jk;S  Ck jpjk jf jmC2.nC1/Ck;S

for all f 2 S.Rn /:

43

Section 3.1 Symbol Classes and Basic Properties

Proof. Since f 2 S.Rn /, fO 2 S.Rn / due to Lemma 2.7. Then Z 1 sup jp.x; Dx /f .x/j  sup hin1 jhimp.x; /jjhinCmC1 fO./j d  n Rn x2Rn x2Rn .2/ Z 1 nCmC1 O f k1  hin1 d jpj.m/ 0 khi .2/n Rn  Cm jpj.m/ jfOjmCnC1;S  C jpj.m/ jf jmC2nC2;S (3.4) 0

0

by Lemma 2.7 and Lemma A.9 in the appendix, where Cm depends only on ma and the dimension. In order to estimate the derivatives, we calculate Z eix p.x; /fO./d @xj .p.x; Dx /f .x// D @xj ¯ n R Z Z D e ix p.x; /i j fO./d e ix @xj p.x; /fO./d ¯ ¯ C Rn

Rn

D p.x; Dx /.@xj f /.x/ C .@xj p/.x; Dx /f .x/; where we have applied Theorem A.4 to interchange integration and differentiation. Hence using (3.4) first with f replaced by @xj f and then p replaced by @xj p, we obtain .m/ .m/ sup j@xj .p.x; Dx /f .x/j  C.jpj0 j@xj f jmC2nC2;S C j@xj pj0 jf jmC2nC2;S /

x2Rn

.m/

 C jpj1 jf jmC2nC3;S :

(3.5)

Similarly, Z ixj p.x; Dx /f .x/ D D

Z

Rn Rn

.@j e ix /p.x; /fO./d ¯ eix p.x; /@j fO./d ¯ C

Z Rn

eix .@j p/.x; /fO./d ¯

D p.x; Dx /.ixj f .x// C .@j p/.x; Dx /f and therefore .m/

.m1/

sup jxj p.x; Dx /f .x/j  C.jpj0 jxj f jmC2nC2;S C j@j pj0

x2Rn

jf jmC2nC2;S /

 C jpj.m/ 1 jf jmC2nC3;S

(3.6) .m1/

by (3.4), where we note that @j p.x; / is of order m  1 and j@j pj0 by the definition of the semi-norms.

.m/

 jpj1

Chapter 3 Basic Calculus of Pseudodifferential Operators on Rn

44

Using (3.5) and (3.6), one can easily prove by mathematical induction that sup jx ˛ @ˇx p.x; Dx /f .x/j  C˛;ˇ jpj.m/ jf jmC2.nC1/Cj˛jCjˇ j;S j˛jCjˇ j

x2Rn

uniformly in x 2 Rn for all ˛; ˇ 2 N0n . Hence .m/ jp.x; Dx /f jk;S  Ck jpjk jf jmC2.nC1/Ck;S ;

which proves the theorem. Finally, we prove the following simple but important inequality: 1

Lemma 3.7 (Peetre’s inequality). Let hi WD .1 C jj2 / 2 ,  2 Rn . Then for all s 2 R his  2jsj h  ijsj h is ;

; 2 Rn :

Proof. First of all we have hi2 D .1 C jj2 /  .1 C jj/2  .1 C jj/2 C .1  jj/2 D 2.1 C jj2 /: Hence hi  .1 C jj/ 

p

2hi for all  2 Rn :

(3.7)

In the case s 0 the triangle inequality implies 1 C jj  1 C j  j C j j  .1 C j  j/.1 C j j/ and therefore his  .1 C j  j/s .1 C j j/s  2s h  is h is by (3.7) for all ; 2 Rn . If s < 0, we can use the previous inequality with interchanged role of  and and s replaced by s to conclude h is  2s h  is his ; which is equivalent to his  2jsj h is h  ijsj :

Section 3.2 Composition of Pseudodifferential Operators: Motivation

45

3.2 Composition of Pseudodifferential Operators: Motivation Because of Theorem 3.6, the composition of two pseudodifferential operators p1 .x; Dx / and p2 .x; Dx / is a well-defined bounded operator p1 .x; Dx /p2 .x; Dx /W S.Rn / ! S.Rn /: The natural question arises if this operator is again a pseudodifferential operator, i.e., 1 .Rn  Rn / such that if there is a symbol p 2 S1;0 p.x; Dx / D p1 .x; Dx /p2 .x; Dx /: If this is the case, it is of interest how the symbol p.x; / is related to the symbols p1 .x; / and p2 .x; /. The behavior of pseudodifferential operators under composition is of particular interest for calculation inverses or at least to approximate inverses of pseudodifferential operators, which are also called parametrices. In order to motivate the following sections, we calculate the composition of p1 .x; Dx / and p2 .x; Dx / formally, ignoring all technical difficulties. First of all, “ p1 .x; Dx /g.x/ D

ei.xy/ p1 .x; /g.y/dy d¯

and “ p2 .x; Dx /f .x/jxDy D

e i.yz/ p2 .y; /f .z/dz d¯ :

Hence we get for g.y/ D p2 .x; Dx /f jxDy p1 .x; Dx /p2 .x; Dx /f .x/ “  “ i.xy/ i.yz/ p1 .x; / p2 .y; /f .z/dz d¯  dy d

e D e ¯ ZZZZ D e i.xz/ ei.xy/./ p1 .x; /p2 .y; /f .z/dy d¯ dz d: ¯ Using the substitution x 0 D y  x and  0 D  , we obtain p1 .x; Dx /p2 .x; Dx /f “  “ i.xz/ ix 0  0 0 0 0 0 p1 .x;  C  /p2 .x C x ; /dx d e f .z/dz d¯  D e ¯

Chapter 3 Basic Calculus of Pseudodifferential Operators on Rn

46

Hence formally the symbol of p1 .x; Dx /p2 .x; Dx / is “ 0 0 .p1 #p2 /.x; / WD eix  p1 .x;  C  0 /p2 .x C x 0 ; /dx 0 d ¯ 0:

(3.8)

But the main problem is that the latter integral in general does not exists in the classical sense. We will define it as so called oscillatory integral: “ 0 0 Os- eix  p1 .x;  C  0 /p2 .x C x 0 ; /dx 0 d ¯ 0 “ 0 0

."x 0 ; " 0 /e ix  p1 .x;  C  0 /p2 .x C x 0 ; /dx 0 d WD lim ¯ 0; "!0

where 2 S.Rn  Rn / with .0; 0/ D 1. In the following section we prove that the oscillatory integral are well-defined for suitable integrands. Moreover, we will show several results, which will justify our formal calculations above.

3.3 Oscillatory Integrals n n Definition 3.8. The space of amplitudes Am  .R R /, m;  2 R, is the set of smooth n n functions aW R  R ! C such that

j@˛ @yˇ a.y; /j  C˛;ˇ .1 C j j/m .1 C jyj/ uniformly in y; 2 Rn for all ˛; ˇ 2 N0n . Moreover, let WD jajAm  ;k

max

sup .1 C j j/m .1 C jyj/ j@˛ @yˇ a.y; /j;

j˛jCjˇ jk y;2Rn

k 2 N;

be the associated sequence of monotone increasing semi-norms. n n It is not difficult to check that Am  .R  R / is a Fréchet space. Moreover, we have m .Rn  Rn / Am .Rn  Rn / with continuous embedding. S1;0 0 n n n n Theorem 3.9. Let a 2 Am  .R  R /, m;  2 R, and let 2 S.R  R / with

.0; 0/ D 1. Then “ “ Os- e iy a.y; /dy d¯ WD lim

."y; " /eiy a.y; /dy d¯

"!0

exists and “ Os- e iy a.y; /dy d¯

“ 0 0 D e iy hyi2l hD i2l Œh i2l hDy i2l a.y; /dy d¯ ; (3.9)

47

Section 3.3 Oscillatory Integrals

Aussage ueber Majorante vom Integrand? where l; l 0 2 N0 are chosen such that 2l > n C m and 2l 0 > n C  and the integrand is in L1 .Rn  Rn /. In particular, the definition does not depend on the choice of and ˇ ˇ “ ˇ ˇ ˇOs- eiy a.y; /dy d¯ ˇ  Cm; jajAm ;2.lCl 0 / ; (3.10) ˇ  ˇ where Cm; > 0 is independent of a. ˇ

Proof. Using Dy˛ e iy D . /˛ eiy and D eiy D .y/ˇ eiy for ˛; ˇ 2 N0n , we have h i2l hDy i2l eiy D e iy

0

0

and hyi2l hD i2l e iy D e iy :

(3.11)

Since ."y; " / 2 S.Rn  Rn / for fixed " > 0, we can integrate by parts and obtain “ I" WD

."y; " /eiy a.y; /dy d¯

“ D e iy h i2l hDy i2l . ."y; " /a.y; // dy d

¯ “ 0 0 D eiy hyi2l hD i2l Œh i2l hDy i2l . ."y; " /a.y; //dy d

¯ On the other hand, ¹ ."y; " /º0<"<1  ¹ " .y; /º0<"<1 is bounded in A00 .Rn  Rn / D Cb1 .R2n /, lim"!0 ."y; " / D 1 uniformly on compact sets, and

lim"!0 @y˛ @ˇ " .y; / D 0 uniformly in .y; / 2 Rn  Rn if .˛; ˇ/ ¤ 0. Hence there are constants C˛;ˇ independent of both 0 < " < 1 and a 2 Am  such that j@y˛ @ˇ . " .y; /a.y; // j  C˛;ˇ jajAm h im hyi :  ;j˛jCjˇ j

(3.12)

s .Rn  Rn /, Moreover, since his 2 S1;0

j@˛ h is j  Cs;˛ h isj˛j :

(3.13)

Combining (3.12) and (3.13), there are constants Cl;˛ independent of 0 < " < 1 such that h im2l hyi : j@˛ Œh i2l hDy i2l . " .y; /a.y; //j  Cl;˛ jajAm  ;2lCj˛j Consequently there are constants Cl;l 0 independent of 0 < " < 1 and a such that 0

0

jhyi2l hD i2l Œh i2l hDy i2l . " .y; /a.y; //j m2l 0 h i hyi2l  Cl;l 0 jajAm  ;2.lCl /

0

(3.14)

Chapter 3 Basic Calculus of Pseudodifferential Operators on Rn

48

0

If we now choose 2l > nCm and 2l 0 > nC , then h im2l hyi2l 2 L1 .Rn Rn /, cf. Lemma A.9. Hence Lebesgue theorem on dominated convergence, ."y;" / !"!0 1, and @y˛ @ˇ ."y; " / !"!0 0 for .˛; ˇ/ ¤ 0 imply “ lim I" D

"!0

0

0

eiy hyi2l hD i2l Œh i2l hDy i2l a.y; /dy d¯ :

Thus the limit in the definition of the oscillatory integral exists and (3.9) holds. Moreover, the representation (3.9) shows that the definition does not depend on the choice of . Finally, passing to the limit " ! 0 in (3.14), (3.10) follows from (3.9) and Lemma A.9 in the appendix. n n Corollary 3.10. Let aj 2 Am  .R  R / be a bounded sequence such that

lim @y˛ @ˇ aj .y; / D @y˛ @ˇ a.y; /

j !1

for every y; 2 Rn

n n for all ˛; ˇ 2 N0n and some a 2 Am  .R  R /. Then

“ “ lim Os- e iy aj .y; /dy d¯ D Os- eiy a.y; /dy d¯ :

j !1

Proof. The assumptions imply that 0

0

lim hyi2l hD i2l Œh i2l hDy i2l aj .y; /

j !1

0

0

D hyi2l hD i2l Œh i2l hDy i2l a.y; / for every y; 2 Rn . Moreover, (3.14) implies   0 0 jhyi2l hD i2l Œh i2l hDy i2l ."y; " /aj .y; "/ j 0

m2l 0 h i hyi2l :  Cl;l 0 jaj jAm  ;2.lCl /

Since the sequence aj is bounded in Am 0  C uniformly in j 2 N.  , jaj jAm  ;2.lCl / Hence the representation (3.9) and Lebesgue’s theorem on dominated convergence imply the statement of the corollary. Example 3.11. Let u 2 Cb1 .Rn /. Then a.y; / D eix u.y/ 2 A00 .Rn  Rn / and “ Os- e i.xy/ u.y/dy d

¯

49

Section 3.3 Oscillatory Integrals

is well-defined. We can calculate the oscillatory integral explicitly: If we choose

.y; / D .y/ . /, where 2 S.Rn / with .0/ D 1, then  Z “ Z i.xy/ i.xy/ u.y/dy d¯ D lim ." /d

Os- e ."y/u.y/ e ¯ dy "!0 Z x  y dy ."y/u.y/"n F 1 Œ  D lim "!0 " Z .".x  "y 0 //u.x  "y 0 /F 1 Œ .y 0 /dy 0 D lim "!0 Z D u.x/F 1 Œ .y 0 /dy 0 D u.x/F ŒF 1 Œ .0/ D u.x/ .0/ D u.x/ due to Theorem 2.1.5 and since lim"!0 .".x"y 0 //u.x"y 0 / D Thus formally we have F 1 ŒF Œu D u for all u 2 Cb1 .Rn /.

.0/u.x/ D u.x/.

n n n Lemma 3.12. Let a 2 Am  .R  R /, m;  2 R, and let ˛ 2 N0 . Then

“ “ iy ˛ Os- e y a.y; /dy d¯ D Os- e iy D˛ a.y; /dy d¯ ; “ “ Os- e iy ˛ a.y; /dy d¯ D Os- e iy Dy˛ a.y; /dy d¯ : n n ˛ Proof. First of all we note that D˛ a.y; /; Dy˛ a.y; / 2 Am  .R  R /, y a.y; / 2

.Rn  Rn /, and ˛ a.y; / 2 AmCj˛j .Rn  Rn /. Therefore the oscillatory Am  Cj˛j integrals are well-defined. We only prove the first identity since the proof of the second is done in the same way. Moreover, it is sufficient to consider the case j˛j D 1. (Then the general case follows by mathematical induction.) If j˛j D 1, then y ˛ D yj for 1  j  n. Moreover, we choose in the definition 2 of the oscillatory integral as .y; / D ej.y;/j =2 . Then “

."y; " /e

iy

“

yj a.y; /dy d¯ D 

."y; " /.Dj eiy /a.y; /dy d¯

“ D eiy Dj . ."y; " /a.y; // dy d : ¯

Using Dj ."y; " / D i "2 j ."y; " /, we obtain Dj . ."y; " /a.y; // D ."y; " /Dj a.y; / C i "2 ."y; " / j a.y; /:

Chapter 3 Basic Calculus of Pseudodifferential Operators on Rn

50

Therefore “ “ iy

."y; " /e yj a.y; /dy d¯ D

."y; " /e iy Dj a.y; /dy d¯

“ 2 C i"

."y; " /eiy j a.y; /dy d¯ : Passing to the limit " ! 0 yields the first equality. nCk Theorem 3.13 (Fubini’s theorem for oscillatory integrals). Let a 2 Am   .R nCk /, m;  2 R, n; k 2 N. Then R “ 0 0 n n b.y; / WD Os- e iy  a.y; y 0 ; ; 0 /dy 0 d

¯ 0 2 Am  .R  R /;

where integration is with respect to Rk  Rk , and “ 0 0 @y˛ @ˇ b.y; / D Os- eiy  @y˛ @ˇ a.y; y 0 ; ; 0 /dy 0 d

¯ 0: Moreover,

(3.15)

ZZZZ 0 0 Ose iyiy  a.y; y 0 ; ; 0 /dydy 0 d

¯ d

¯ 0  “  “ 0 0 D Os- eiy Os- e iy  a.y; y 0 ; ; 0 /dy 0 d

¯ 0 dy d¯ :

Proof. Because of Peetre’s inequality and . ; 0 / D . ; 0/C.0; 0 /, .y; y 0 / D .y; 0/C .0; y 0 /, h. ; 0 /im h.y; y 0 /i  2jmjCjj h im hyi h 0 ijmj hy 0 ijj : jmj

ˇ

Hence @y˛ @ a.y;  ; ;  / 2 Ajj .R2k / with respect to .y 0 ; 0 / and m  j@y˛ @ˇ a.y;  ; ;  /jAjmj .R2k /;j  Cj;;m jajAm 2.kCn/ /;j Cj˛jCjˇ j h i hyi  .R j j

for arbitrary j 2 N. Therefore (3.10) implies ˇ “ ˇ ˇ ˇ 0ˇ ˇ Os- eiy 0  0 @˛ @ˇ a.y; y 0 ; ; 0 /dy 0 d

¯ ˇ y  ˇ m   Cm; jajAm 2.kCn/ /;2.lCl 0 /Cj˛jCjˇ j h i hyi ;  .R

where 2l > jmj C k and 2l 0 > j j C k. Moreover, we choose l; l 0 so large that 2l > m C n and 2l 0 > n C  . Because of the representation (3.9) and Theorem A.4, ˇ we can take @y˛ @ out of the oscillatory integral, which proves (3.15). Thus m  j@y˛ @ˇ b.y; /j  Cm jajAm 2.nCk/ /;2.lCl 0 /Cj˛jCjˇ j h i hyi :  .R

51

Section 3.4 Double Symbols 0

0

n n 2l hD i2l Œh i2l hD i2l b.y; / 2 Hence b.y; / 2 Am  y  .R  R /. Thus hyi L1 .Rn  Rn / because of Theorem 3.9. Moreover, “ 0 0 0 0 b.y; / D e iy  hy 0 i2l hD 0 i2l Œh 0 i2l hDy 0 i2l a.y; y 0 ; ; 0 /dy 0 d

¯ 0

and therefore 0

0

hyi2l hD i2l Œh i2l hDy i2l b.y; / “ 0 0 0 0 D e iy  .hyihy 0 i/2l .hD ihD 0 i/2l Œ.h ih 0 i/2l .hDy ihDy 0 i/2l ady 0 d

¯ 0; where 0

0

.hyihy 0 i/2l .hD ihD 0 i/2l Œ.h ih 0 i/2l .hDy ihDy 0 i/2l a.y; y 0 ; ; 0 / 2 L1 .R2.nCk/ /: Hence we can apply Fubini’s theorem and get  “  “ iy iy 0  0 0 0 0 0 a.y; y ; ; /dy d

Os- e dy d

Os- e ¯ ¯ Z 0 0 0 0 D e iyiy  .hyihy 0 i/2l .hD ihD 0 i/2l d.y 0 y; ; 0 /  Œ.h ih 0 i/2l .hDy ihDy 0 i/2l a .2/nCk ZZZZ 0 0 D Oseiyiy  .h ih 0 i/2l .hDy ihDy 0 i/2l a.y; y 0 ; ; 0 /dy 0 d

¯ ¯ 0 dy d

ZZZZ 0 0 D Ose iyiy  a.y; y 0 ; ; 0 /dy 0 d

¯ ¯ 0 dy d ; where we have used Lemma 3.12.

3.4 Double Symbols The composition p1 .x; Dx /p2 .x; Dx / calculated in Section 3.2 is an example of a pseudodifferential operator in more general form – a pseudodifferential operator with a double symbol: ZZZZ 0 0 00 0 e i.xx /Ci.x x / p.x; ; x 0 ;  0 /u.x 00 /dx 00 d p.x; Dx ; x; Dx /u D ¯ 0 dx 0 d ¯ for u 2 S.Rn /, where the integrals have to be understood as iterated integrals. Here m1 ;m2 p.x; ; x 0 ;  0 / D p1 .x; /p2 .x 0 ;  0 / 2 S1;0 .Rn  Rn  Rn  Rn / is defined as follows:

Chapter 3 Basic Calculus of Pseudodifferential Operators on Rn

52

Definition 3.14. Let m; m0 2 R. Then the space of double pseudodifferential symbols m;m0 .Rn  Rn  Rn  Rn / is the space of all smooth functions pW Rn  Rn  Rn  S1;0 n R ! C such that 0

0

0

0

jD˛ Dxˇ D˛0 Dxˇ0 p.x; ; x 0 ;  0 /j  C˛;ˇ ;˛ 0 ;ˇ 0 .1 C jj/mj˛j .1 C j 0 j/m j˛ j uniformly in x; ; x 0 ;  0 2 Rn for arbitrary ˛; ˇ; ˛ 0 ; ˇ 0 2 N0n. max.m;m0 ;mCm0 /

0

m;m Note that S1;0 .Rn  Rn  Rn  Rn / A0 .R2n  R2n /. The statements on composition of two pseudodifferential operators will be a corollary to the following more general theorem: 0

m;m .Rn  Rn  Rn  Rn / be a double Theorem 3.15. Let m; m0 2 R and let p 2 S1;0 symbol. Then “ mCm0 .Rn  Rn /: pL .x; / WD Os- eiy p.x;  C ; x C y; /dy d¯ 2 S1;0

Moreover, pL .x; / 

X 1 @˛ Dx˛0 p.x; ; x 0 ;  0 /jx0 Dx; 0 D ˛Š n

˛2N0

in the sense that pL .x; / 

X 1 mCm0 N 1 @˛ D ˛0 p.x; ; x 0 ;  0 /jx0 Dx; 0 D 2 S1;0 .Rn  Rn / ˛Š  x

j˛jN

for all N 2 N0 . Proof. First of all, let ax; .y; / WD p.x;  C ; x C y; / for all x; y; ; 2 Rn . Using Peetre’s inequality, j@˛ @yˇ ax; .y; /j D j@˛ @yˇ p.x;  C ; x C y; /j 0

0

 C˛;ˇ h C imj˛j him  C˛;ˇ h C im him 0

 C˛;ˇ 2jmj h ijmj himCm : mCm . Therefore Hence ax; .y; / 2 Ajmj 0 with jax; jAjmj ;jmjC2nC2  C.1 C jj/ 0

0

ˇ “ ˇ ˇ ˇ 0 jpL .x; /j D ˇˇ Os- eiy p.x;  C ; x C y; /dy d¯ ˇˇ  C.1 C jj/mCm (3.16) because (3.10).

53

Section 3.4 Double Symbols

Q 2n 2n Since p.x; ; x 0 ; / 2 Am Q D max.m1 ; m2 ; m1 Cm2 /, with respect 0 .R R /, m 0 2n Q 2n  R2n / with to .x; x /; .; / 2 R , we also have p.x;  C ; x C y; / 2 Am 0 .R 2n respect to .x; y/; .; / 2 R . Hence we can apply (3.15) to conclude that “ (3.17) @˛ @ˇx pL .x; / D Os- eiy @˛ @ˇx Œp.x;  C ; x C y; / dy d : ¯

Combining (3.16) and (3.17) yields j@˛ @ˇx pL .x; /j ˇ ˇ “ ˇ ˇ iy ˛ ˇ ˇ @ @x Œp.x;  C ; x C y; / dy d

D ˇ Os- e ¯ ˇˇ 0

 C.1 C jj/mCm j˛j :

(3.18) (3.19)

In order to prove the asymptotic expansion, we use the Taylor series expansion: X ˛ p˛ .x; ; x C y; / ˛Š j˛jN X ˛ Z 1 .1   /N p˛ .x;  C  ; x C y; /d; C .N C 1/ ˛Š 0

p.x;  C ; x C y; / D

j˛jDN C1

where p˛ .x; ; y; / D pL .x; / D

@˛ p.x; ; y; /.

Hence

“ X 1 Os- eiy ˛ p˛ .x; ; x C y; /dy d¯

˛Š j˛jN “ X 1 C .N C 1/ Os- eiy ˛ r˛ .x; ; y; /dy d¯ ; ˛Š j˛jDN C1

where

Z r˛ .x; ; y; / D

0

1

.@˛ p/.x;  C  ; x C y; /.1   /N d:

Because of Lemma 3.12 and Example 3.11, “ “ iy ˛

p˛ .x; ; x C y; /dy d¯ D Os- eiy Dy˛ p˛ .x; ; x C y; /dy d¯

Os- e D @˛ Dy˛ p.x; ; y; /jyDx;D : Therefore it remains to estimate r˛ .x; ; y; /. As in the beginning of the proof,  j@ˇ @y .@˛ Dy˛ p/.x;  C  ; x C y; / j 0

 C˛;ˇ ; 2jmj .1 C j j/jmj .1 C jj/mCm j˛j 0

 C˛;ˇ ; 2jmj .1 C j j/jmj .1 C jj/mCm j˛j ;

54

Chapter 3 Basic Calculus of Pseudodifferential Operators on Rn

where C˛;ˇ; does not depend on  2 Œ0; 1. Hence ¹p.x;  C   ; x C  ; /º0 1 is n n uniformly bounded in Ajmj 0 .R  R / as amplitudes in .y; /. Therefore we conclude jmj r˛ .x; ;  ;  / 2 A0 .Rn  Rn / and   0 j@ˇ @y .Dy˛ r˛ /.x; ; ; y/ j  C˛;ˇ ; 2jmj .1 C j j/jmj .1 C jj/mCm j˛j : This implies ˇ “ ˇ ˇ ˇ iy ˛ ˇ Os- e

r˛ .x; ; ; y/dy d¯ ˇˇ ˇ ˇ “ ˇ ˇ ˇ 0 D ˇˇ Os- e iy Dy˛ r˛ .x; ; ; y/dy d¯ ˇˇ  C˛ .1 C jj/mCm j˛j

(3.20)

ˇ 

because of (3.10). Finally, the derivatives @ @x r˛ .x; / are estimated in the same way as before.

3.5 Composition of Pseudodifferential Operators With the aid of the oscillatory integrals, we can make the formal calculations in Section 3.2 rigorous. m .Rn  Rn /, then First of all, if p 2 S1;0  Z Z i.xy/ e p.x; /u.y/dy d p.x; Dx /u D ¯ “ 0 D Os- e ix  p.x; /u.x C x 0 /dx 0 d (3.21) ¯ for all u 2 S.Rn /. The proof is left to the reader as an exercise, cf. Exercise 3.58. Using this representation and Theorem 3.13, we easily get p1 .x; Dx /p2 .x; Dx /u  “  “ ix 0  ix 00  0 0 0 0 00 00 0 p1 .x; / Os- e p2 .x C x ;  /u.x C x C x /dx d D Os- e ¯ dx 0 d ¯ ZZZZ 0 00 0 D Ose ix ix  p1 .x; /p2 .x C x 0 ;  0 /u.x C x 0 C x 00 /dx 00 d ¯ 0 dx 0 d ¯ ZZZZ 0 0 D Ose ix iy p1 .x;  0 C /p2 .x C x 0 ;  0 /u.x C y/dx 0 d ¯ 0 dy d

¯  “  “ 0 0 D Os- eiy Os- e ix  p1 .x;  0 C /p2 .x C x 0 ;  0 /dx 0 d

¯ u.x C y/dy d¯  0 “ 0 D Os- e iy p1 #p2 .x;  0 /u.x C y/dy d¯ ;

55

Section 3.5 Composition of Pseudodifferential Operators

where we have used D    0 and y D x 0 C x 00 , Theorem 3.13, and p1 #p2 is defined as in (3.8). m

Theorem 3.16. Let pj 2 S1;0j .Rn  Rn /, j D 1; 2, be two pseudodifferential symm1 Cm2 .Rn  Rn / such that bols. Then there is some p1 #p2 2 S1;0 p1 .x; Dx /p2 .x; Dx / D .p1 #p2 /.x; Dx /: Moreover, p1 #p2 has the following asymptotic expansion: p1 #p2 .x; / 

X 1 @˛ p1 .x; /Dx˛ p2 .x; / ˛Š  n

(3.22)

˛2N0

in the sense that p1 #p2 .x; / 

X 1 m1 Cm2 N @˛ p1 .x; /Dx˛ p2 .x; / 2 S1;0 .Rn  Rn / ˛Š 

j˛j
for all N 2 N. m1 ;m2 .Rn  Rn  Rn  Proof. Let p.x; ; x 0 ;  0 / D p1 .x; /p2 .x 0 ;  0 /. Then p 2 S1;0 n R / and p1 #p2 .x; / D pL .x; /. Therefore the theorem is a consequence of Theorem 3.15.

Hence the composition of two pseudodifferential operators is again a pseudodifferential operator. In particular, the asymptotic expansion implies: p1 #p2 .x; / D p1 .x; /p2 .x; / C r.x; /;

(3.23)

m1 Cm2 1 .Rn  Rn /. Hence the symbol of the composition cowhere r.x; / 2 S1;0 incides with the product of the symbols modulo a term of lower order. This is the essential fact needed for the construction of a parametrix to an elliptic pseudodifferential operator in Section 3.6. Moreover, we note that, if p2 .x; / D p2 ./ is independent of x, we simply have p1 .x; Dx /p2 .Dx / D OP.p1 .x; /p2 .// since F Œp2 .Dx /u./ D p2 ./u./. O Furthermore, if p1 .x; Dx / is a differential operator of order m 2 N0 with coefficients in Cb1 .Rn /, then

p1 .x; Dx /p2 .x; Dx / D .p1 #p2 /.x; Dx /; where .p1 #p2 /.x; / WD

X 1 @˛ p1 .x; /Dx˛ p2 .x; /: ˛Š 

j˛jm

Chapter 3 Basic Calculus of Pseudodifferential Operators on Rn

56

Hence the asymptotic expansion of p1 #p2 .x; / consists only of finitely many terms. This identity is an easy consequence of the Leibniz formula, cf. Exercise 3.56. Alternatively, it can be observed by modifying the proof of Theorem 3.15. (In this case there is no remainder term r˛; if N > m.) Because of (3.23), we also obtain: m

Corollary 3.17. Let pj 2 S1;0j .Rn  Rn /, j D 1; 2, be two pseudodifferential symm1 Cm2 1 .Rn  Rn / such that bols. Then there is some r 2 S1;0 Œp1 .x; Dx /; p2 .x; Dx / D r.x; Dx /; where ŒA; B D AB  BA denotes the commutator of two operators. Proof. See Exercise 3.57. Another application of the asymptotic expansion (3.22) is: Theorem 3.18. Let '; 2 Cb1 .Rn / such that dist.supp '; supp / > 0 and let m .Rn  Rn /. Then there is some q 2 S 1 .Rn  Rn / such that p 2 S1;0 1;0 q.x; Dx /f .x/ D '.x/p.x; Dx /. f /.x/

for all f 2 S.Rn /; x 2 Rn :

Proof. First of all, we have that '.x/p.x; Dx /. f /.x/ D p1 .x; Dx /p2 .x; Dx /f .x/ where p1 .x; / D '.x/p.x; /, p2 .x; / D m some q 2 S1;0 .Rn  Rn / such that

for all f 2 S.Rn /;

.x/. Hence by Theorem 3.16 there is

q.x; Dx /f .x/ D '.x/p.x; Dx /. f /.x/

for all f 2 S.Rn /:

Moreover, by (3.22) q.x; / 

X 1 @˛ p1 .x; /Dx˛ p2 .x; /; ˛Š  n

˛2N0

D ;. Hence q 2 where @˛ p1 .x; /Dx˛ p2 .x; /  0 since supp ' \ supp mN n n 1 n S1;0 .R  R / for any N 2 N0 . This implies q 2 S1;0 .R  Rn /. Remark 3.19. The statement of the last theorem is important in order to define pseudodifferential operators on manifolds, cf. Section 3.10 below. It is also used to prove that pseudodifferential operators are pseudo-local, cf. Theorem 5.25 below.

Section 3.6 Application: Elliptic Pseudodifferential Operators and Parametrices

57

3.6 Application: Elliptic Pseudodifferential Operators and Parametrices m .Rn  Rn /, m 2 R, is called elliptic if there are Definition 3.20. A symbol p 2 S1;0 C; R > 0 such that

jp.x; /j C jjm

for all jj R; x 2 Rn

(3.24)

Example 3.21. 1. Let p./ D jj2 be the symbol of  . Then p is an elliptic symbol of order 2. Moreover, q./ D him , m 2 R, is an elliptic symbol of order m. 2. Let A.x/ 2 Cb1 .Rn /nn be a matrix that is uniformly positive definite, i.e., there is some c > 0 such that  T A.x/ cjj2 ;

for all x;  2 Rn :

Then p.x; / D  T A.x/ is an elliptic symbol of order 2. m .Rn  Rn /, m 2 R, be an elliptic symbol and R > 0 as Lemma 3.22. Let p 2 S1;0 in (3.24). Then

q.x; / WD where

2 Cb1 .Rn / such that

m ./p.x; /1 2 S1;0 .Rn  Rn /;

./ D 1 for jj R C 1 and

./ D 0 for jj  R.

Proof. Since q.x; / D ./ D 0 for jj  R, q.x; / is obviously smooth in .x; / and it suffices to consider jj R. Because of the chain rule, @j p.x; /1 D p.x; /2 @j p.x; / and the same identity holds with @j replaced by @xj . Using (3.24), j@j p.x; /1 j  C jj2m him1  C him1 and j@xj p.x; /1 j  C jj2m him  C him for all jj R. In the same way one can easily prove by mathematical induction that j@˛ @ˇx p.x; /1 j  C˛;ˇ himj˛j

Chapter 3 Basic Calculus of Pseudodifferential Operators on Rn

58

for all jj R. On the other hand qW Rn  Rn ! C is smooth and therefore 0 j@˛ @ˇx q.x; /j  C˛;ˇ  C˛;ˇ himj˛j

for all jj  R C 1. Since q.x; / D p.x; /1 for jj R C 1, we conclude that for every ˛; ˇ 2 N0n there is some C˛;ˇ > 0 such that j@˛ @ˇx q.x; /j  C˛;ˇ himj˛j for all  2 Rn . m Corollary 3.23. Let p 2 S1;0 .Rn  Rn / be an elliptic symbol. Then there is some m .Rn  Rn / such that q 2 S1;0

p.x; Dx /q.x; Dx / D I C r.x; Dx /;

q.x; Dx /p.x; Dx / D I C r 0 .x; Dx /

1 .Rn  Rn /. with r; r 0 2 S1;0

Proof. Let q be defined as in Lemma 3.22. Then, because of Theorem 3.16, p.x; Dx /q.x; Dx / D .pq/.x; Dx / C r.x; Q Dx /; 1 where rQ 2 S1;0 .Rn  Rn /. Moreover, p.x; /q.x; / D 1 for all jj R C 1. Hence 1 .Rn  Rn / and p.x; /q.x; /  1 2 S1;0

p.x; Dx /q.x; Dx / D I C r.x; Dx / 1 .Rn  Rn /. The identity with r.x; / D p.x; /q.x; /  1 C r.x; Q / 2 S1;0

q.x; Dx /p.x; Dx / D I C r 0 .x; Dx / 1 .Rn  Rn / is proved the same way. with r 0 2 S1;0

One can even strengthen the last corollary to: m .Rn  Rn /, m 2 R. Then the following conditions are Theorem 3.24. Let p 2 S1;0 equivalent:

1. p is elliptic. m .Rn Rn / such that p.x; D /q.x; D / D I Cr.x; D /, 2. There is some q 2 S1;0 x x x 1 n where r 2 S1;0 .R  Rn /. 0 m 2 S1;0 .Rn  Rn / such that 3. For every N 2 N there are qN ; qN

p.x; Dx /qN .x; Dx / D I C rN .x; Dx /; 0 0 .x; Dx /p.x; Dx / D I C rN .x; Dx /; qN 0 N 2 S1;0 .Rn  Rn /. where rN ; rN

Section 3.6 Application: Elliptic Pseudodifferential Operators and Parametrices

59

For the proof of the theorem we will need: 1 .Rn  Rn /. Then Theorem 3.25 (Uniqueness of the Symbol). Let p; q 2 S1;0 n p.x; Dx /u D q.x; Dx /u for all u 2 S.R / implies p.x; / D q.x; /.

The theorem will be proved in the next section. Proof of Theorem 3.24. 1. implies 2.: This is a consequence of Corollary 3.23. 2. implies 1.: Since p.x; Dx /q.x; Dx / D OP .p.x; /q.x; // C r.x; Q Dx / and p.x; Dx /q.x; Dx / D I C r.x; Dx / 1 .Rn  Rn /, we obtain with r; rQ 2 S1;0 1 .Rn  Rn /; p.x; /q.x; /  1 D r.x; /  r.x; Q / 2 S1;0

where we have used Theorem 3.25. In particular jp.x; /q.x; /  1j  C hi1 for all x;  2 Rn . Hence there is some R > 0 such that jp.x; /q.x; /  1j  12 for all jj R and x 2 Rn . Thus j.p.x; /q.x; //1 j  2 for all jj R and x 2 Rn . Therefore we finally conclude that jp.x; /1 j D j.p.x; /q.x; //1 jjq.x; /j  2C him m .Rn  Rn /. This implies (3.24). for all x 2 Rn ; jj R since q 2 S1;0 m .Rn  Rn /, 1. implies 3.: Because of Corollary 3.23, there are some q 2 S1;0 0 2 S 1 .Rn  Rn / such that r; r 1;0

p.x; Dx /q.x; Dx / D I  r.x; Dx /;

q.x; Dx /p.x; Dx / D I  r 0 .x; Dx /:

0 m 2 S1;0 .Rn  Rn / be such that Now let qN ; qN N 1 X

qN .x; Dx / WD q.x; Dx /

k

r.x; Dx / ;

0 qN .x; Dx /

kD0

WD

N 1 X

r 0 .x; Dx /k q.x; Dx /:

kD0

N .Rn  Rn / by r .x; D / WD Because of Theorem 3.16, we can define rN 2 S1;0 x N N r.x; Dx / and get

p.x; Dx /qN .x; Dx / D .I  r.x; Dx //

N 1 X kD0

r.x; Dx /k D I  r.x; Dx /N ;

Chapter 3 Basic Calculus of Pseudodifferential Operators on Rn

60 Similarly,

0 qN .x; Dx /p.x; Dx / D

N 1 X

r 0 .x; Dx /k .I  r 0 .x; Dx // D I  r 0 .x; Dx /N ;

kD0 0 2 S N .Rn  Rn / is defined by r 0 .x; D / WD r 0 .x; D /N . where rN x x 1;0 N 3. implies 2.: This is obvious.

One can improve the statement of Theorem 3.24 in the following way:1 m Proposition 3.26. Let p 2 S1;0 .Rn  Rn /, m 2 R. Then p is elliptic if and only if m n there is some q1 2 S1;0 .R  Rn / such that

p.x; Dx /q1 .x; Dx / D I C r1 .x; Dx /

(3.25)

1 .Rn  Rn /. with r 2 S1;0

In order to prove this statement, the following lemma is needed: m

Lemma 3.27. Let pj 2 S1;0j .Rn  Rn / with m1    mj ! 1 as j ! 1. m1 Then there is some p 2 S1;0 .Rn  Rn / such that p.x; / 

1 X

pj .x; / W, p.x; / 

j D1

N 1 X

mN pj .x; / 2 S1;0 .Rn  Rn /

(3.26)

j D1

for all N 2 N. Proof of Proposition 3.26. The “if”-part follows directly from Theorem 3.24 since 1 S N for any N 2 N. S1;0 1;0 Conversely, if p is elliptic and q.x; Dx / is as in Corollary 3.23, we can define m .Rn  Rn / by q1 2 S1;0 q1 .x; Dx / WD q.x; Dx /q 0 .x; Dx /; where q 0 .x; / 

1 X

r #k .x; /;

kD0

r is the same as in the proof of Theorem 3.24, and k r #k .x; / WD r#    #r.x; / 2 S1;0 .Rn  Rn /: „ ƒ‚ … k-times 1

This improvement is optional and can be skipped on first reading.

Section 3.6 Application: Elliptic Pseudodifferential Operators and Parametrices

61

The existence of q 0 follows from Lemma 3.27. Then for any N 2 N 1 NX p.x; Dx /q1 .x; Dx / D p.x; Dx /q.x; Dx / r #k .x; Dx / j D0

C p.x; Dx /q.x; Dx /rN .x; Dx / 0 .x; Dx /; D p.x; Dx /qN .x; Dx / C rN 0 N where qN .x; Dx / is as in the proof of Theorem 3.24 and rN 2 S1;0 .Rn Rn /. Hence

p.x; Dx /q1 .x; Dx / D I C r 00 .x; Dx /; N .Rn  Rn /. Because of the uniqueness of the symbol of a pseudowith r 00 2 S1;0 differential operator, cf. Theorem 3.25, r 00 is independent of N . Since N 2 N was 1 arbitrary, we conclude that r 00 2 S1;0 .Rn  Rn /.

Proof of Lemma 3.27. Let

2 C 1 .Rn / such that 0  ´ 0 if jj  1; ./ D 1 if jj 2:

./  1 and

We will define p by p.x; / WD

1 X

."j /pj .x; /

(3.27)

j D1

for a suitable sequence ."j /j 2N such that 1 "j "j C1 > 0 for all j 2 N and limj !1 "j D 0. To this end we observe that for any " > 0 ´ 0 if jj  "1 ; (3.28) ."/ D 1 if jj 2"1 : Moreover, it is easy to prove that for any ˛ 2 N0n n ¹0º j@˛ ."/j  C˛ "j˛j ; @˛ ."/ D 0 if jj  "1 or jj 2"1 uniformly in " > 0 and  2 Rn for some C˛ > 0. For any ˛ ¤ 0 we have that @˛ ."/ ¤ 0 only if "1 < jj < 2"1 . Furthermore, if 0 < "  1, then "  2jj1  4.1 C jj/1

for all "1 < jj < 2"1 :

Chapter 3 Basic Calculus of Pseudodifferential Operators on Rn

62

Hence for any ˛ 2 N0n there is some C˛ > 0 such that j@˛ ."/j  C˛ .1 C jj/j˛j

for all  2 Rn :

0 .Rn  Rn / is bounded. Therefore This means that ¹ ."  / W 0 < "  1º S1;0 Proposition 3.5 implies that for any j 2 N

p";j .x; / WD

."/pj .x; /

for all x;  2 Rn ; 0 < "  1 m

defines a set of symbols, which are bounded in S1;0j .Rn  Rn /. Hence for any j 2 N .m /

there is some constant Cj independent of 0 < "  1 such that jp";j jj j  Cj . Therefore   j@˛ @ˇx . ."/pj .x; //j  Cj .1 C jj/mj j˛j  Cj .1 C jj/1 .1 C jj/mj C1j˛j for all x;  2 Rn , j˛j; jˇj  j , and 0 < "  1. Now we choose "j 2 .0; 1 such that limj !1 "j D 0 and Cj "j  2j Since

for all j 2 N:

."j / D 0 if .1 C jj/1 "j , we have j@˛ @ˇx . ."/pj .x; //j  2j .1 C jj/mj C1j˛j

(3.29)

for all x;  2 Rn , j˛j; jˇj  j . Moreover, let p be defined by (3.27). Then p is well-defined and p 2 C 1.Rn  Rn / since for every  2 Rn the sum in (3.27) is finite m and because of (3.28) and "j !j !1 0. In order to prove that p 2 S1;0 .Rn  Rn / n be arbitrary and choose j 2 N such that j max.j˛j; jˇj/ and let ˛; ˇ 2 N0 0 0 mj0 C 1  m1 . Then p.x; / D q.x; / C r.x; /; q.x; / WD

jX 0 1

where

."j /pj .x; /;

r.x; / D

j D1

1 X

."j /pj .x; /:

j Dj0 m

m1 Since the sum in the definition of q is finite and p"j ;j 2 S1;0j , we have q 2 S1;0 . In particular we have

j@˛ @ˇx q.x; /j  C˛;ˇ .1 C jj/m1 j˛j

for all x;  2 Rn

with arbitrary ˛; ˇ 2 N0n. Moreover, for any ˛; ˇ 2 N0n with j˛j; jˇj  j0 we have j@˛ @ˇx r.x; /j  

1 X j Dj0 1 X j Dj0

j@˛ @ˇx pj;"j .x; /j 2j .1 C jj/mj C1j˛j  .1 C jj/m1 j˛j

Section 3.7 Boundedness on Cb1 .Rn/ and Uniqueness of the Symbol

63

uniformly in x;  2 Rn due to (3.29). Hence we conclude that j@˛ @ˇx p.x; /j  C˛;ˇ .1 C jj/m1 j˛j

for all x;  2 Rn

m1 for any j˛j; jˇj  j0 . Since j0 2 N was arbitrary, p 2 S1;0 .Rn  Rn /. In order to prove (3.26), we use that

p.x; / 

N 1 X

pj .x; / D

j D1

N 1 X

1 X

j D0

j DN

. ."j /  1/pj .x; / C

."j /pj .x; /:

Here 0

p .x; / WD

1 X

."j /pj .x; /

j DN

is defined in the same way as p in (3.27) just with the series starting with pN instead mN of p1 . Hence the first part implies that p0 2 S1;0 .Rn  Rn /. On the other hand N 1 X

1 . ."j /  1/pj 2 S1;0 .Rn  Rn /

j D0

since

."j /  1 2 C01 .Rn /. Consequently, p

N 1 X

mN pj 2 S1;0 .Rn  Rn /;

j D1

which finishes the proof.

3.7 Boundedness on Cb1 .Rn / and Uniqueness of the Symbol As seen above,

“ 0 p.x; Dx /u.x/ D Os- e ix  p.x; /u.x C x 0 /dx 0 d ¯

(3.30)

for all u 2 S.Rn /. Here the oscillatory integral is well defined for all u 2 Cb1 .Rn /. Therefore we can extend the definition of p.x; Dx / to Cb1 .Rn /. m .Rn  Rn /. Then p.x; D / defined by (3.30) for u 2 Theorem 3.28. Let p 2 S1;0 x 1 n Cb .R / is a bounded linear operator

p.x; Dx /W Cb1 .Rn / ! Cb1 .Rn /:

Chapter 3 Basic Calculus of Pseudodifferential Operators on Rn

64

Proof. Consider ax .x 0 ; / D p.x; /u.x C x 0 / for all x; x 0 ;  2 Rn . Since for every ˛; ˇ 2 N0n j@˛x0 @ˇ p.x; /u.x C x 0 /j  C˛;ˇ jpj.m/ kukC j˛j .Rn / him ; jˇ j b

.m/ n m we have ax .x 0 ; / 2 Am 0 with jax jA0 ;k  C jpjk kukCbk uniformly in x 2 R . Thus 0 sup jp.x; Dx /u.x/j  C sup jax jAm 0 ;2.lCl /

x2Rn



x2Rn .m/ C 0 jpj2.lCl 0 / kukC 2.lCl 0 / .Rn / b

(3.31)

by Theorem 3.9, where 2l > m C n, 2l 0 > n. In order to estimate @˛x p.x; Dx /u we 2n  R2n / and therefore observe that a.x; x 0 ; ;  0 / WD p.x; /u.x C x 0 / 2 Am 0 .R “ 0 @xj p.x; Dx /u D Os- eix  .@xj p/.x; /u.x C x 0 /dx 0 d ¯ “ 0 C Os- e ix  p.x; /.@xj u/.x C x 0 /dx 0 d ¯ D .@xj p/.x; Dx /u C p.x; Dx /@xj u by (3.15) in Theorem 3.13. Applying this formula successively and using (3.31), it .m/ follows immediately that kp.x; Dx /ukC k .Rn /  C jpj2.lCl 0/Ck kukC 2.lCl 0 /Ck .Rn / , b

which proves the theorem.

b

Remark 3.29. Using this extension, one can show that eix .p.x; Dx /e i: /.x/ D p.x; /

(3.32)

for all x;  2 Rn , cf. Exercise 3.62. With this identity it is now easy to prove the uniqueness of the symbol of pseudodifferential operators:

Proof of Theorem 3.25. Because of (3.32), we only need to prove p.x; Dx /u D q.x; Dx /u for all u 2 Cb1 .Rn /:

(3.33)

To this end let u 2 Cb1 .Rn / and u" .x/ WD ."x/u.x/ for all x 2 Rn , " > 0 and some 2 S.Rn / with .0/ D 1. Then u" 2 S.Rn / and @˛x u" .x/ !"!0 @˛x u.x/

for every x 2 Rn ; ˛ 2 N0n :

Section 3.8 Adjoints of Pseudodifferential Operators and Operators in .x; y/-Form

65

Therefore Corollary 3.10 implies “ 0 0 p.x; Dx /u" .x/ D Os- eix  p.x; /u" .x C x 0 / dx 0 d ¯ 0 !"!0 p.x; Dx /u.x/ „ ƒ‚ … DWa";x; .x 0 ; 0 /

for every x 2 Rn since ˇ ˇ @˛x @ 0 a";x; .x 0 ;  0 / !"!0 @˛x @ 0 ax; .x 0 ;  0 / for all x 0 ;  2 Rn ; ˛; ˇ 2 N0n n with ax; .x 0 ;  0 / D p.x; /u.x Cx 0 / and since .a";x; /0<"1 is bounded in Am 0 .R  n n R / for every x;  2 R , which follows from the boundedness of .u" /0<"1 in Cb1 .Rn /. In the same way, we obtain the corresponding statement for q.x; Dx /. Therefore

p.x; Dx /u.x/ D lim p.x; Dx /u" .x/ D lim q.x; Dx /u" .x/ D q.x; Dx /u.x/ "!0

"!0

for every x 2 Rn since p.x; Dx /u" D q.x; Dx /u" due to u" 2 S.Rn /. Since u 2 Cb1 .Rn / was arbitrary, (3.33) follows and the theorem is proved. More generally, if P W Cb1 .Rn / ! Cb1 .Rn / is an arbitrary bounded linear map, then we can define a symbol of P by p.x; / WD eix .P e i: /.x/ We note that one can use the relation (3.32) to determine the symbol of the composition p1 .x; Dx /p2 .x; Dx / without using Theorem 3.16, cf. Exercise 3.67

3.8 Adjoints of Pseudodifferential Operators and Operators in .x; y/-Form Definition 3.30. Let A; A W S.Rn / ! S.Rn /. Then A is called formal adjoint of A if .Au; v/L2 .Rn / D .u; A v/L2 .Rn /

for all u; v 2 S.Rn /:

The adjoint of an operator plays an important role in many purposes. (For example: solvability of equations). In the following, we will prove that every pseudodifferential operator p.x; Dx / possesses a formal adjoint p  .x; Dx /, which is again a pseudodifferential operator (of the same order). This will allow us to extend p.x; Dx /, up to now only defined on S.Rn /, to a linear operator defined on S 0 .Rn /.

Chapter 3 Basic Calculus of Pseudodifferential Operators on Rn

66

Now we calculate the formal adjoint of p.x; Dx /: “ eix p.x; /u./ O d ¯ v.x/dx

.p.x; Dx /u; v/L2 .Rn / D “

e ix p.x; /v.x/dx u./ O d ¯

D Z D

Z eix p.x; /v.x/dx d; u./ O ¯

where we have used Fubini’s theorem. – Note that v; uO 2 S.Rn /, which implies that O 2 L1 .Rn  Rn / with respect to .x; /. e ix p.x; /u./v.x/ m .Rn  Rn /, m 2 R, and v 2 S.Rn /. Then Lemma 3.31. Let p 2 S1;0

Z w./ WD

Rn

eix p.x; /v.x/dx 2 S.Rn /:

Proof. See Exercise 3.64. – It is also a consequence of Lemma 3.33 below. Hence we can use .F Œu; v/L2 D .2/n .u; F 1 Œv/L2 and get “



.p.x; Dx / v/.x/ D

e i.xy/ p.y; /v.y/dy d¯ :

(3.34)

This operator is a pseudodifferential operator in the so called y-form (also called Rform), which is a special case of operators of the form “ p.x; Dx ; x/u WD

e i.xy/ p.x; y; /u.y/dy d¯ 

(3.35)

m .R2n  Rn /, i.e., for all u 2 S.Rn /, where p 2 S1;0

j@˛ @ˇx @y p.x; y; /j  C˛;ˇ; .1 C jj/mj˛j for all ˛; ˇ; 2 N0n. These operators are called pseudodifferential operators in .x; y/form and p.x; Dx / is called pseudodifferential operator in x-form. As before we obtain “ 0 p.x; Dx ; x/u D Os- e ix  p.x; x C x 0 ; /u.x C x 0 /dx 0 d ¯ for all u 2 S.Rn /.

Section 3.8 Adjoints of Pseudodifferential Operators and Operators in .x; y/-Form

67

m .R2n Rn /, m 2 R. Then p.x; D ; x/u D p .x; D /u Theorem 3.32. Let p 2 S1;0 x x L for all u 2 Cb1 .Rn /, where

“ m pL .x; / D Os- e iy p.x; x C y;  C /dy d

.Rn  Rn /: ¯ 2 S1;0 Moreover, pL .x; / 

X 1 @˛ Dy˛ p.x; y; /jyDx ˛Š n

˛2N0

in the sense that for all N 2 N0 pL .x; / 

X 1 mN 1 @˛ D ˛ p.x; y; /jyDx 2 S1;0 .Rn  Rn /: ˛Š  y

j˛jN

Proof. Using “ 0 u.x C x / D Os- ei.xCx y/ u.y/dy d¯

0

due to Example 3.11, we get ZZZZ 0 0 e ix  ei.xCx y/ p.x; x C x 0 ; /u.y/dy d¯ dx 0 d p.x; Dx ; x/u D Os¯  “  “ 0 D Os- e i.xy/ Os- eix ./ p.x; x C x 0 ; /dx 0 d ¯ u.y/dy d

¯ “ D Os- e i.xy/ pL .x; /u.y/dy d¯ : Hence application of Theorem 3.15 finishes the proof. As a direct consequence we obtain: m .R2n Rn /, m 2 R. Then p.x; D ; x/W S.Rn / ! S.Rn / Lemma 3.33. Let p 2 S1;0 x is a bounded linear operator. Moreover, if the definition (3.35) is replaced by “ 0 p.x; Dx ; x/u WD Os- eix  p.x; x C x 0 ; /u.x C x 0 /dx 0 d ¯

for u 2 Cb1 .Rn /, then p.x; Dx ; x/W Cb1 .Rn / ! Cb1 .Rn / is a bounded operator. Proof. Since p.x; Dx ; x/u D pL .x; Dx /u for all u 2 Cb1 .Rn /, the lemma is a consequence of the corresponding statements for pL .x; Dx /.

Chapter 3 Basic Calculus of Pseudodifferential Operators on Rn

68

m .Rn  Rn /, then the formal adjoint of p.x; D / is Corollary 3.34. If p 2 S1;0 x p  .x; Dx / where “  m .Rn  Rn /: p .x; / D Os- e iy p.x C y;  C /dy d

¯ 2 S1;0

Moreover, p  .x; / 

X 1 @˛ Dx˛ p.x; / ˛Š n

˛2N0

in the sense that for every N 2 N0 X 1 mN 1 @˛ D ˛ p.x; / 2 S1;0 p  .x; /  .Rn  Rn /: ˛Š  x j˛jN

Proof. The corollary is a direct consequence of (3.34) and Theorem 3.32. m Definition 3.35. Let p 2 S1;0 .Rn  Rn /. Then we define p.x; Dx /W S 0 .Rn / ! 0 .Rn / by S

hp.x; Dx /u; vi WD hu; p  .x; Dx /vi u 2 S 0 .Rn /; v 2 S.Rn /: Since S.Rn /  S 0 .Rn / (identifying functions with functional in the way described in Section 2.4), it is important to notice that, if u 2 S.Rn /, hp.x; Dx /u; vi D hu; p .x; Dx /vi 

D .u; p .x; Dx /v/L2 .Rn / D

Z Rn

p.x; Dx /u.x/v.x/ dx

for all v 2 S.Rn /. I.e., the definition of p.x; Dx /u in sense of S 0 .Rn / coincides with the first definition of p.x; Dx /u for u 2 S.Rn /.

3.9 Boundedness on L2 .Rn/ and L2-Bessel Potential Spaces The main result of this section is: 0 Theorem 3.36. Let p 2 S1;0 .Rn  Rn /. Then p.x; Dx / (first defined on S.Rn /) extends to a bounded operator p.x; Dx /W L2 .Rn / ! L2 .Rn /.

The theorem will be important for later applications to partial differential and pseudodifferential operator equations. In Section 5.5 we will prove that zero order pseudodifferential operators are also bounded on Lp .Rn / for every 1 < p < 1. But for this purpose the theory of singular integral operators is needed, which is not the case if p D 2. The proof of Theorem 3.36 is divided into several parts.

Section 3.9 Boundedness on L2 .Rn / and L2 -Bessel Potential Spaces

69

n1 .Rn  Rn /. Lemma 3.37. Theorem 3.36 holds if p 2 S1;0 n1 .Rn  Rn /, then p.x; / 2 L1 .Rn / with respect to  and thereProof. If p 2 S1;0 fore Z Z Z p.x; Dx /f .x/ D eix p.x; /fO./d D e i.xy/ p.x; /d¯ f .y/dy ¯ n n n R R ZR D k.x; x  y/f .y/dy Rn

Œp.x; /.z/. This for all f 2 S.Rn / by Fubini’s theorem, where k.x; z/ WD F71 !z kernel satisfies ˇZ ˇ ˇ ˇ .n1/ ˛ iz ˛ ˇ e @ p.x; /d¯  ˇˇ  C˛ jpjj˛j jz k.x; z/j D ˇ Rn

.n1/ since j@˛ p.x; /j  jpjj˛j hin1j˛j 2 L1 .Rn / with respect to . Hence .n1/

j.1 C jzj2 /n k.x; z/j  C jpj2n and therefore

g.z/ WD sup jk.x; z/j  C.1 C jzj2 /n 2 L1 .Rn /: x2Rn

Thus kp.x; Dx /f kL2 .Rn /

Z

C





g.x  y/jf .y/jdy

n

R

L2 .Rn /

0

 C kgkL1 .Rn / kf kL2 .Rn / because of (2.5). Now the statement is a consequence of Lemma A.10 since S.Rn / is dense in L2 .Rn /. m .Rn  Rn / with m > 0. Lemma 3.38. Theorem 3.36 holds for p 2 S1;0

Proof. In order to prove kp.x; Dx /f k2  C kf k2 for f 2 S.Rn /, it is sufficient to show kp  .x; Dx /p.x; Dx /f k2  C kf k2 since kp.x; Dx /f k22 D .p  .x; Dx /p.x; Dx /f; f /  kp .x; Dx /p.x; Dx /f k2 kf k2 : m .Rn  Rn /, then But, if p 2 S1;0

p  .x; Dx /p.x; Dx / D p 0 .x; Dx /

Chapter 3 Basic Calculus of Pseudodifferential Operators on Rn

70

2m .Rn  Rn /. Hence, using the previous lemma, we obtain by mathewith p 0 2 S1;0 matical induction that

kp.x; Dx /f kL2 .Rn /  C kf kL2 .Rn / mk for all p 2 S1;0 with mk D .n C 1/=2k , k 2 N. This proves the lemma since for every m > 0 there is some k 2 N such that mk > m. Again the statement of the lemma follows now from the density of S.Rn / in L2 .Rn /, cf. Lemma A.10.

In order to finish the proof of Theorem 3.36, we need 0 .Rn  Rn / with p.x; / 2 R for all x;  2 Rn and F 2 Lemma 3.39. If p 2 S1;0 1 0 .Rn  Rn /. C .R/, then F .p.x; // 2 S1;0

Proof. First of all jp.x; /j  R for some R > 0 and F is bounded on compact sets. Hence jF .p.x; //j  supjzjR jF .z/j. Moreover, j@j F .p.x; //j  sup jF 0 .z/jj@j p.x; /j  C hi1 ; jzjR

j@xj F .p.x; //j  sup jF 0 .z/jj@xj p.x; /j  C: jzjR

Finally, the estimate of j@˛ @ˇx F .p.x; //j for arbitrary ˛; ˇ 2 N0n can be proved using a mathematical induction and the chain rule as before. 0 .Rn  Rn /, then jp.x; /j  M for all x;  2 Rn , Proof of Theorem 3.36. If p 2 S1;0 .0/

where M WD jpj0 0. Therefore 0 .Rn  Rn / p0 .x; / WD M 2  p.x; /p.x; / 2 S1;0 1

and p 0 .x; / 0. Now let F 2 C 1 .R/ be defined by F .t / D .1 C t / 2 for t 0. 0 .Rn  Rn / and Then q.x; / WD F .p 0 .x; // 2 S1;0 q  .x; Dx /q.x; Dx /f D OP.F .p0 .x; //2 /f C r.x; Dx /f D .1 C M 2 /f  OP.p.x; /p.x; //f C r.x; Dx /f D .1 C M 2 /f  p  .x; Dx /p.x; Dx /f C r 0 .x; Dx /f 1 .Rn  Rn / because of Theorem 3.16 and Corolfor f 2 S.Rn /, where r; r 0 2 S1;0 lary 3.34. Hence 2 kp.x; Dx /f kL 2 .Rn /

 .p .x; Dx /p.x; Dx /f; f /L2 .Rn / C .q  .x; Dx /q.x; Dx /f; f /L2 .Rn / 2 0  .1 C M 2 /kf kL 2 .Rn / C .r .x; Dx /f; f /:

Section 3.9 Boundedness on L2 .Rn / and L2 -Bessel Potential Spaces

71

1 .Rn  Rn /, kr 0 .x; D /f k  C kf k due to Lemma 3.38. Hence Since r 0 2 S1;0 x 2 2 kp.x; Dx /f k22  .1 C M 2 C C /kf k22 for all f 2 S.Rn /, which implies the result since S.Rn / is dense in L2 .Rn /.

Remark 3.40. Checking the previous proofs, it can be observed that the estimates of .0/ kp.x; Dx /kL.L2 .Rn // depend only on jpjk for some suitably large k 2 N (and not on p directly). Since the mapping p 7! p.x; Dx / is linear, we have .0/

kp.x; Dx /kL.L2 .Rn //  C jpjk :

(3.36)

.0/

.0/

This can be seen as follows: Let q.x; / WD p.x; /=jpjk . Then jqjk D 1 and kq.x; Dx /kL.L2 .Rn //  C .0/

0 .Rn  Rn /. Hence multiplication by jpj yields (3.36). independently of p 2 S1;0 k

Recall that the L2 -Bessel potential spaces and their norms are defined by H2s .Rn / D ¹u 2 S 0 .Rn /W hDx is u 2 L2 .Rn /º; kukH2s .Rn /  kuks;2 WD khDx is uk2 ; where s 2 Rn . With the aid of the L2 -continuity of zero order pseudodifferential operators it is now easy to prove the following theorem on continuity of pseudodifferential operators between L2 -Bessel potential spaces: m .Rn  Rn /. Then p.x; D /W H sCm .Rn / ! H s .Rn /. Theorem 3.41. Let p 2 S1;0 x 2 2 Moreover, there is some k 2 N0 such that

kp.x; Dx /kL.H sCm .Rn /;H s .Rn //  Cs;m jpj.m/ k 2

2

m for all p 2 S1;0 .Rn  Rn /:

Proof. First of all, we note that, if u 2 L2 .Rn / S 0 .Rn /, p.x; Dx /u is defined in the sense of S 0 .Rn /, cf. Definition 3.35, and m D 0, then jhp.x; Dx /u; vij D j.u; p  .x; Dx /v/L2 .Rn / j  kukL2 .Rn / kp  .x; Dx /kL2 .Rn / kvkL2 .Rn / for all v 2 S.Rn / due to Theorem 3.36 and Corollary 3.34. Therefore p.x; Dx /u 2 L2 .Rn / Š L2 .Rn /0 by the Riesz representation theorem. Now we consider general s; m 2 R. Since hDx isCm W H2sCm .Rn / ! L2 .Rn / and hDx is W L2 .Rn / ! H2s .Rn / are linear isomorphisms, p.x; Dx /W H2sCm .Rn / ! H2s .Rn / is a linear bounded operator if and only if q.x; Dx / WD hDx is p.x; Dx /hDx ism W L2 .Rn / ! L2 .Rn /

Chapter 3 Basic Calculus of Pseudodifferential Operators on Rn

72

0 .Rn  Rn /. Hence the first is a bounded operator. Because of Theorem 3.16, q 2 S1;0 statement is a consequence of Theorem 3.36. The second statement is a consequence of Remark 3.40 and the fact that the mapping m1 m2 m1 Cm2  S1;0 3 .p1 ; p2 / 7! p1 #p2 2 S1;0 S1;0

is bounded applied to p1 .x; / D his , p2 .x; / D p.x; /hism . More precisely, this implies kp.x; Dx /kL.H sCm .Rn /;H s .Rn // 2

2

0  Cs;m kq.x; Dx /kL.L2 .Rn //  Cs;m jqjk

.0/

.sm/ 0  Cs;m jh  is j.s/ jpj.m/ jh  ism jn.k/  Cs;m jpj.m/ n.k/ n.k/ n.k/

for some n.k/ 2 N, where k is as in (3.36). Corollary 3.42. Let s 2 Rn . Then S.Rn / is dense in H2s .Rn /. Proof. Because of the definition of H2s .Rn /, hDx is W H2s .Rn / ! L2 .Rn / is an isomorphism. Moreover, because of Theorem 3.6, hDx is W S.Rn / ! S.Rn / for any s 2 R. Now, since C01 .Rn / S.Rn / are dense in L2 .Rn /, cf. Theorem A.6, S.Rn / D hDx is S.Rn / is dense in H s .Rn /. m .Rn  Rn / be an elliptic symbol Corollary 3.43. (Elliptic regularity) Let p 2 S1;0 S and f 2 H2s .Rn /, s; m 2 R. Then, if u 2 H21 .Rn / WD s2R H2s .Rn / is a solution of the pseudodifferential equation

p.x; Dx /u D f; then u 2 H2sCm .Rn /. Proof. Let u 2 H21 .Rn /. Then u 2 H2sCmN .Rn / for some N 2 N. By the same m .Rn  Rn / such that construction as in Theorem 3.24, there is some qN .x; / 2 S1;0 qN .x; Dx /p.x; Dx / D I C rN .x; Dx /; N where rN 2 S1;0 .Rn  Rn /. Therefore

qN .x; Dx /f D qN .x; Dx /p.x; Dx /u D u C rN .x; Dx /u: Since qN .x; Dx /f 2 H2sCm .Rn / and rN .x; Dx /u 2 H2sCm .Rn / by Theorem 3.41, we have u 2 H2sCm .Rn /. Remark 3.44. The latter corollary says that the solution u is “as smooth as the righthand side allows”. This property is also called elliptic regularity.

Section 3.9 Boundedness on L2 .Rn / and L2 -Bessel Potential Spaces

73

Lemma 3.45. Let s > n2 . Then there is some Cs > 0 such that kukC 0 .Rn /  Cs khDx is ukL2 .Rn / ;

kukL2 .Rn /  Cs sup jhxis u.x/j

b

x2Rn

for all u 2 S.Rn /. In particular, H2s .Rn / ,! Cb0 .Rn /. Proof. First of all, by Theorem 2.1 and the Cauchy–Schwarz inequality, O L1 .Rn /  kh  is kL2 .Rn / kh  is uk O L2 .Rn / D Cs khDx is ukL2 .Rn / kukC 0 .Rn /  kuk b

for all u 2 S.Rn /, where we have used that khis kL2 .Rn / < 1 because of s > n2 and Lemma A.9. This implies H2s .Rn / ,! Cb0 .Rn / since S.Rn / is dense in H2s .Rn /. Similarly as before kukL2 .Rn /  kh  is ukL1 .Rn / kh  is kL2 .Rn / D Cs khis ukL1 .Rn / for any u 2 S.Rn /. As a consequence we obtain: Theorem 3.46. Let k 2 N0 and s > k C n2 . Then H2s .Rn / ,! Cbk .Rn /. Proof. First of all max sup j@˛x f .x/j  C

j˛jk x2Rn

X

k@˛x f kH sk .Rn /  C 0 kf kH s .Rn /

j˛jk

for all f 2 S.R /. Since S.R / is dense in H2s .Rn /, the statement follows. n

n

We conclude the section with a useful technical corollary. Corollary 3.47. Let juj00k;S WD

sup j˛jCjˇ jk

kx ˛ Dxˇ ukL2 .Rn /

j  j00k;S ,

S.Rn /,

for u 2 k 2 N0 . Then k 2 N, is a decreasing sequence of semi-norms on S.Rn / which is equivalent to the semi-norms j  jk;S defined above. More precisely, juj00k;S  Ck jujkC2n;S

and

jujk;S  Ck juj00kC2n;S

for all u 2 S.Rn / and k 2 N0 . Proof. Since hDx i2n D .1  /n is a differential operator of order 2n, jujk;S D

sup j˛jCjˇ jk

kx ˛ Dxˇ uk1  C

by Lemma 3.45. Similarly, since juj00k;S D

sup j˛jCjˇ jk

sup j˛jCjˇ jk

hxi2n

kx ˛ Dxˇ uk2  C

khDx i2n x ˛ Dxˇ uk2  Ck juj00kC2n;S

D .1 C jxj2 /n is a polynomial of order 2n, sup

j˛jCjˇ jk

khxi2n x ˛ Dxˇ uk1  Ck jujkC2n;S :

Chapter 3 Basic Calculus of Pseudodifferential Operators on Rn

74

3.10 Outlook: Coordinate Transformations and Pseudodifferential Operators on Manifolds In this section we will show that for every suitable smooth diffeomorphism W Rn ! Rn the operator QW S.Rn / ! S.Rn / defined by Qu.x/ WD .p.x; Dx /w/..x//;

w.x/ WD u. 1 .x//; u 2 S.Rn /

(3.37)

is again a pseudodifferential operator. This is important to obtain a definition of pseudodifferential operators on a smooth compact manifold in a way that is essentially independent of the choice of local charts. For the following we denote Qu D   p.x; Dx / ;1 u; where .  v/.x/ WD v..x// and . ;1 u/.x/ WD u. 1 .x//. More precisely, we assume that W Rn ! Rn is a smooth function such that @xj  2 Cb1 .Rn /n for all j D 1; : : : ; n and 0 < c  jdet D.x/j  C < 1

(3.38)

for all x 2 Rn and some constants c; C > 0. In particular, this implies that  1 W Rn ! Rn is a again smooth and @xj  1 2 Cb1 .Rn / for all j D 1; : : : ; n. In the following rx  denotes the total derivative of W Rn ! Rn . We assume for simplicity that 1 sup jr.x/  I j  : n 2 x2R

(3.39)

For the discussion of the general case we refer to [21, Theorem 6.3]. Here j  j is any matrix norm, which is induced by some vector norm. The main result of this section is: m .Rn  Rn /, m 2 R, let W Rn ! Rn be as above and Theorem 3.48. Let p 2 S1;0 n / ! S.Rn / be defined by (3.37). Then there is some q 2 S m .Rn Rn / such QW S.R 1;0 that Qu D q.x; Dx /u for all u 2 S.Rn /. Moreover, q.x; Dx / has the asymptotic expansion

q.x; / 

X 1 @˛ Dy˛ q.x; Q y; /jyDx ; ˛Š n

where

˛2N0

q.x; Q y; / D p..x/; A.x; y/T /jdet A.x; y/j1 jdet ry .y/j and Z 1 ry .x C t .y  x// dt for all x; y close enough A.x; y/ D 0

Section 3.10 Outlook: Coordinate Transformations and PsDOs on Manifolds

75

in the sense that for any N 2 N0 X 1 mN 1 q.x; /  Q y; /jyDx 2 S1;0 .Rn  Rn /: @˛ D ˛ q.x; ˛Š  y j˛jN

In particular, we have q.x; / D p..x/; .ry .x//T / C r.x; /

m1 with r 2 S1;0 :

Proof. First of all, (3.39) ensures that jA.x; y/  I j 

1 2

for all x; y 2 Rn :

Therefore A.x; y/T D .A.x; y/1 /T exists for all x; y 2 Rn . Moreover, let 2 S.Rn / with .0/ D 1 and " ./ D ."/ for all " > 0,  2 Rn . Under these assumptions we have Z d.y; / Qu D lim e i. .x/y/ " ./p..x/; /u. 1 .y// "!0 Rn Rn .2/n Z d.x 0 ; / 0 D lim e i. .x/ .x // " ./p..x/; /u.x 0 /jdet rx 0 .x 0 /j "!0 Rn Rn .2/n for all u 2 S.Rn /. On the other hand, since Z 1  0 0 0 rx 0 .x C t .x  x // dt .x  x 0 / D A.x 0 ; x/.x  x 0 /; .x/  .x / D 0

we obtain by the change of variable  D A.x 0 ; x/T

Z 1 0 lim ei.xx / " .A.x 0 ; x/T /q.x; Q x 0 ; /u.x 0 / d.x 0 ; / Qu D .2/n "!0 Rn Rn “ D lim Os- e iy " .A.x C y; x/T /q.x; Q x C y; /u.x C y/ dx 0 d

¯ "!0

due to ..x/  .x 0 //   D .x  x 0 /  A.x 0 ; x/T , where q.x; Q x 0 ; / D p..x/; A.x 0 ; x/T /jdet A.x; x 0 /j1 jdet rx0 .x 0 /j: Since ¹ " .A.x; x 0 /T / W " 2 .0; 1/º is bounded in A00 .Rn  Rn / with respect to .x 0 ; / and

" .A.x 0 ; x/T / !"!0 1 for all x; x 0 ; 2 Rn ; @˛x0 @ˇ " .A.x 0 ; x/T / !"!0 0 for all x; x 0 ; 2 Rn ; j˛j C jˇj > 0; we can apply Corollary 3.10 to conclude that “ Q x C y; /u.x C y/ d.x 0 ; /: Qu.x/ D Os- e iy q.x; Finally, an application of Theorem 3.32 finishes the proof.

76

Chapter 3 Basic Calculus of Pseudodifferential Operators on Rn

Next we define pseudodifferential operators on smooth compact manifolds. To this end we recall: Definition 3.49. M is a smooth compact manifold if M is a topological compact space and the following conditions hold: S 1. There are finitely many open sets 1 ; : : : ; N M such that M D jND1 j . 2. For every j 2 ¹1; : : : ; N º there is an open set Uj Rn and a continuous bijective function j W j ! Uj with continuous inverse, called chart. 3. For every j; k 2 ¹1; : : : ; N º such that j;k WD j \ k ¤ ; the mapping j;k W k .j;k / ! j .j;k / can be extended to some C 1 -diffeomorphism Qj;k W Vx ! Vy for some open Vx k .j;k /, Vy j .j;k /. Let M be a smooth compact manifold. Then a function f W M ! C is called smooth if for every j D 1; : : : ; N as above fj WD f ıj1 2 C 1 .j .j //. Moreover, if  M is open C01 ./ D ¹f 2 C 1 .M / W supp f º: Before we define pseudodifferential operators on a compact manifold we need the definition of certain remainder operators: Definition 3.50. Let M be a smooth compact manifold as above and let P W C 1 .M / ! C 1 .M / be a linear operator. Then P has a C 1 -kernel representation if for every j; k 2 ¹1; : : : ; N º there is some Kj;k 2 C 1 .Uk  Uj / such that for every u 2 C01 .j / Z .P u/.k1 .x// D Kj;k .x; x 0 /uj .x 0 / dx 0 for all x 2 Uk ; Uj

where uj .x/ D u.j1 .x// in Uj . For the following let ˆj , j D 1; : : : ; N be a smooth partition of unity subordinate to j , j D 1; : : : ; N , i.e., 0  ˆj 2 C01 .M / with supp ˆj  j ;

N X

ˆj .x/ D 1 for all x 2 M (3.40)

j D1

Moreover, let ‰j 2 C01 .M / with supp ‰j  j , j D 1; : : : ; N , be such that ‰j .x/ D 1 for all x 2 supp ˆj ; j D 1; : : : ; N:

(3.41)

77

Section 3.11 Final Remarks and Exercises

Finally, we define 'j ;

j

2 C01 .Uj / by

'j WD ˆj ı j1 ;

j

D ‰j ı j1 :

(3.42)

Definition 3.51. Let M be a smooth compact manifold as above and let P W C 1 .M / ! C 1 .M / be a linear operator. Then P is a pseudodifferential operator m .M / with symbols .p /N of class S1;0 j j D1 with respect to the charts j , j D 1; : : : ; N , if the following holds true: Let ˆj ; ‰j 2 C 1 .M /, j D 1; : : : ; N satisfy (3.40) and (3.41) and let .P u/.x/ D

N X

.ˆj P ‰j u/.x/ C

j D1

N X

.ˆj P .1  ‰j /u/.x/

for all x 2 M: (3.43)

j D1

Then .ˆj P .1  ‰j /u/ has a C 1 -kernel representation and, using 'j ; (3.42), ˆj P ‰j u can be written in the form .ˆj P ‰j u/.j1 .x// D 'j pj .x; Dx /.

j uj /.x/

j

defined in

for all x 2 Uj ;

where uj .x/ D u.j1 .x//, x 2 Uj , for all u 2 C 1 .M /, j D 1; : : : ; N . Remark 3.52. If P is a differential operator, supp P .1  ‰j /u  supp.1  ‰j /. Therefore .ˆj P .1  ‰j /u/.x/ D 0 for all x 2 M . Hence .P u/.x/ D

N X

.ˆj P ‰j u/.x/

for all x 2 M

j D1

in this case. With the aid of Theorem 3.48 and Theorem 3.18 one prove that the definition of a pseudodifferential operator is independent of the choice of charts. We refer to [21] for details.

3.11 Final Remarks and Exercises 3.11.1 Further Reading The main part of this chapter is based on parts of the first two chapters of the monograph by Kumano-Go [21], where a more general class of pseudodifferential operators m , 0  ı    1 is studied. Here p 2 S m if pW Rn  Rn ! C with symbols in S;ı ;ı is smooth and satisfies for every ˛; ˇ 2 N0n the estimate j@˛ @ˇx p.x; /j  C˛;ˇ .1 C jj/mj˛jCıjˇ j

for all x;  2 Rn

Chapter 3 Basic Calculus of Pseudodifferential Operators on Rn

78

and for some C˛;ˇ > 0. Alternative elementary introductions to the theory of pseudodifferential operators were given by Raymond [27] and Wong [44]. More details of pseudodifferential operators and further applications can be found e.g. in the monographs by Grubb [11] and Kumano-Go [21].

3.11.2 Exercises m

Exercise 3.53. Let pj 2 S1;0j .Rn  Rn /, mj 2 R, j D 1 and let p.x; / WD m1 Cm2 .Rn  Rn / and that p1 .x; /p2 .x; / for all x;  2 Rn . Prove that p 2 S1;0 .m1 Cm2 /

for every k 2 N one has jpjk only on k and n.

.m1 /

 Ck jp1 jk

.m2 /

jp2 jk

, where Ck depends

Exercise 3.54. Let pW Rn Rn ! C be a smooth function which is homogeneous of degree m 2 R in  for jj 1, i.e. p.x; r/ D r m p.x; / for all jj 1 and ˇ r 1. Moreover, let @˛ @x p 2 Cb1 .Rn / with respect to x for all ˛; ˇ 2 N0n . m .Rn  Rn /. Prove that p 2 S1;0 Exercise 3.55. Let 2 S.Rn / with .0/ D 1 and let " .x/ WD ."x/, " > 0. Prove that ¹ " W 0 < " < 1º is bounded in Cb1 .Rn /. P Exercise 3.56. Let p1 .x; / WD j˛jm1 c˛ .x/ ˛ be the symbol of a differenm2 .Rn  Rn /. In this special case it is easy to prove tial operator and let p2 2 S1;0 that the composition of the associated operators is a pseudodifferential operator. Moreover, it is an elementary calculation to determine the symbol of the composition. More precisely, prove that p1 .x; Dx /p2 .x; Dx / D .p1 #p2 /.x; Dx /; where X 1 ˇ @ p.x; /Dxˇ q.x; /: p1 #p2 .x; / D ˇŠ  jˇ jm1

In order to prove the statement one may use the identity ! 1 ˛ ˛ˇ D @ˇ  ˛ :  ˇŠ ˇ Exercise 3.57 (Commutator of Pseudodifferential Operators). m m1 Cm2 1 Let pj 2 S1;0j .Rn  Rn /, j D 1; 2. Show that there is some r 2 S1;0 n n .R  R / such that p1 .x; Dx /p2 .x; Dx /  p2 .x; Dx /p1 .x; Dx / D r.x; Dx /:

79

Section 3.11 Final Remarks and Exercises

Exercise 3.58. Prove (3.21). m .Rn  Rn /, m 2 R, be an elliptic symbol and let Exercise 3.59. Let p 2 S1;0 m" .Rn  Rn / for some " > 0. Prove that p.x; / C q.x; / is an elliptic q 2 S1;0 symbol of order m. m .Rn Rn /, m 2 R, be a symbol that is homogeneous Exercise 3.60. Let p 2 S1;0 of degree m for jj 1, i.e.,

p.x; r/ D r m p.x; /

for all r; jj 1:

Show that p is elliptic if and only if p.x; / ¤ 0 for all jj D 1, x 2 Rn and inf

x2Rn ;jjD1

jp.x; /j > 0:

m .Rn  Rn / is called polyhomogeneous if p Exercise 3.61. A symbol p 2 S1;0 admits a symbolic expansion

p.x; / 

1 X

pk .x; /

kD0

W, p.x; / 

N X

mN 1 pk .x; / 2 S1;0 .Rn  Rn /; 8N 2 N;

kD0 mk .Rn  Rn / are homogeneous of degree m  k for jj 1. where pk 2 S1;0 Moreover, p0 .x; / is called principal symbol of p.x; /.

1. Show that p.x; /q.x; / is a polyhomogeneous symbol of order m1 C m2 if p and q are polyhomogeneous symbols of order m1 , m2 , respectively and that p.x; /q.x; / 

1 X X

pj .x; /ql .x; /;

kD0 j ClDk

P1

P where p.x; /  j D0 pj .x; / and q.x; /  1 lD0 ql .x; / and pj and ql are homogeneous of degree m1  j , m2  l, respectively, in jj 1. 2. Show that p#q.x; / is a polyhomogeneous symbol of order m1 C m2 if p and q are polyhomogeneous symbols of order m1 , m2 , respectively, and that .p#q/.x; / 

1 X

X

kD0 j˛jCj ClDk

1 ˛ @ pj .x; /Dx˛ ql .x; /: ˛Š 

Chapter 3 Basic Calculus of Pseudodifferential Operators on Rn

80

Exercise 3.62 (Symbol of a Pseudodifferential Operator). m .Rn  Rn /, m 2 R. Show that for all x;  2 Rn Let p 2 S1;0 e ix .p.x; Dx /e i: /.x/ D p.x; /: Exercise 3.63. We consider P WD y W Cb1 .Rn / ! Cb1 .Rn /, y 2 Rn , where .y u/.x/ D u.x C y/. Determine the symbol of y defined by p.x; / WD e ix .P e i: /.x/: m .Rn  Rn / hold for some m 2 R? Does p 2 S1;0 m .Rn  Rn /, m 2 R, and v 2 S.Rn /. Prove that Exercise 3.64. Let p 2 S1;0

Z w./ WD

Rn

eix p.x; /v.x/dx 2 S.Rn /:

Exercise 3.65 (Adjoints and Operators in .x; y/-Form). P ˛ 1. Let p.x; Dx / WD j˛jm c˛ .x/Dx be a differential operator with c˛ 2 1 n Cb .R /. Prove by an elementary calculation that .p.x; Dx /u; v/L2 .Rn / D .u; q.Dx ; x/v/L2 .Rn /

for all u; v 2 S.Rn /;

where q.Dx ; x/v D

X

Dx˛ .c˛ .x/v.x//:

j˛jm m .R2n  Rn / be a symbol of an operator in .x; y/-form such 2. Let p 2 S1;0 that p.x; x; / D 0 for all x;  2 Rn . Prove that p.x; Dx ; x/ D pL .x; Dx /; m1 .Rn  Rn /. where pL 2 S1;0

Exercise 3.66. Let ' 2 S.Rn / with '.0/ D 1. Prove that for every f 2 S.Rn / lim '."x/f D lim '."Dx /f D lim '."Dx /'."x/f D f

"!0

"!0

"!0

1 Hint. Show that '."x/ !"!0 1 in S1;0 .Rn  Rn /.

Exercise 3.67. Calculate eix p1 .x; Dx /p2 .x; Dx /e i: without using Theorem 3.16.

in S.Rn /:

81

Section 3.11 Final Remarks and Exercises

Exercise 3.68. Let X s , s 2 R, be a scale of Banach spaces such that (i) S.Rn /  X s  S 0 .Rn / with continuous embeddings, where functions and regular distributions are identified in the standard way. (ii) hDx im W X sCm ! X s is a bounded linear operator for all s; m 2 R, 0 .Rn  Rn / and s 2 R. (iii) p.x; Dx /W X s ! X s for all p 2 S1;0

Prove that (a) X s D ¹u 2 S 0 .Rn / W hDx is u 2 X 0 º and X s  X t if t  s with continuous embedding. m .Rn  Rn /, s; m 2 Rn . (b) p.x; Dx /W X sCm ! X s for all p 2 S1;0 m .Rn  Rn / is an elliptic symbol and (c) If p 2 S1;0

p.x; Dx /u D f with f 2 X s and u 2 X 1 WD

S

s2R X

s

, then u 2 X sCm .

Remark. A scale of Banach spaces X s , s 2 R, satisfying the conditions (i)– (iii) above is also called microlocalizable. As seen above H2s .Rn /, s 2 R, is a microlocalizable scale of Banach space. The same is true for Hps .Rn / and the s .Rn / as will be shown in Section 5.5 and Section 6.6. so-called Besov spaces Bp;q 1 .Rn  Rn /. Prove that for any u 2 C 1 .Rn / Exercise 3.69. Let p 2 S1;0 b

Z

Z Rn

p.x; Dx /u.x/'.x/ dx D

Rn

u.x/p  .x; Dx /'.x/ dx

for all ' 2 S.Rn /;

where p.x; Dx /u.x/ is defined as in (3.30). Hence p.x; Dx /u defined as in (3.30) coincides with the definition in the sense of S 0 .Rn /. Hint. Use the approximation in the proof of Theorem 3.25.

Part II

Singular Integral Operators

Chapter 4

Translation Invariant Singular Integral Operators Summary In this chapter we give a first introduction to the theory of singular integral operators on Rn . These operators arise naturally in the regularity study of elliptic and parabolic equations. We will first study the case of translation invariant singular integral operators, which can be described by a multiplication operator on the Fourier transform side. While the continuity on L2 .Rn / easily follows from Plancherel’s theorem, the proof of continuity on Lp .Rn / for 1 < p < 1, p ¤ 2, is more involved. To this end we use the so-called Calderón–Zygmund decomposition to show that the operators are bounded from L1 .Rn / to the weak Lebesgue-space L1weak .Rn / and apply the Marcinkiewicz interpolation theorem. Based on this result, we will prove the Mikhlin multiplier theorem, which is a convenient tool for proving Lp -boundedness of Fourier multiplication operators.

Learning targets  Get to know basic examples of singular integral operators and the nature of their singular kernels.  Learn the Marcinkiewicz interpolation theorem, the maximal operators and their applications to singular integral operators.  Understand relations between the kernel of a singular integral operator and its Fourier multiplier.  Learn the Mikhlin multiplier theorem.

4.1 Motivation Let u 2 H 2 .Rn / be a solution of the elliptic partial differential equation  u D f

in Rn

for some f 2 L .R / \ L .R /, 1  p  1, where u D 2

n

p

n

Pn

(4.1)

2 j D1 @xj u.

Question. Does @xj @xk u 2 Lp .Rn / hold for every j; k D 1; : : : ; n? If 1 < p < 1, the answer is positive. – This will follow from the continuity of the so-called Riesz operators or alternatively from the Mikhlin multiplier theorem below.1 The same question arises if in (4.1) the Laplace operator is replaced by a 1

In the case p D 1 and p D 1 the answer is in general negative, cf. Section 4.7.

86

Chapter 4 Translation Invariant Singular Integral Operators

general elliptic differential (or pseudodifferential) operator with (sufficiently) smooth coefficients. But for the following we will only consider the Laplace equation for simplicity. In order to get a representation of @xj @xk u, we apply the Fourier transformation to (4.1) and obtain: O D jj2 u./

n X j D1

F Œ@2xj u./ D F Œ u./ D fO./ for almost all  2 Rn

because of Theorem 2.38. Hence F Œ@xj @xk u./ D j k u./ O D

i j i j O f ./ for almost all  2 Rn n ¹0º; jj2

which yields @xj @xk u D F

1



 i j i j O f ./ : jj jj

Therefore @xj @xk u D mj;k .Dx /f , where mj;k .Dx /f D F 1 Œmj;k ./fO./

for all f 2 L2 .Rn /

i i

and mj;k ./ D jjj jjj for all  ¤ 0 and j; k D 1; : : : ; n. Operators of this form are called Fourier multiplication operators. Since mj;k 2 L1 .Rn / for all j; k D 1; : : : ; n, Plancherel’s theorem (Theorem 2.11), Lemma 2.13, respectively, implies kmj;k .Dx /f kL2 .Rn /  kmj;k kL1 .Rn / kf kL2 .Rn /

for all f 2 L2 .Rn /;

i.e., mj;k .Dx / 2 L.L2 .Rn // and kmj;k .Dx /kL.L2 .Rn //  kmj;k kL1 .Rn / . Such kind of Fourier multiplication operators are important examples of (translation invariant) singular integral operators. In order to understand the nature of singular integral operators, a description with a so-called kernel kW Rn n ¹0º ! C such that Z mj;k .Dx /f .x/ D k.x  y/f .y/ dy for all x … supp f Rn

is important. The existence of such a kernel for mj;k .Dx / will be shown in Section 4.6. But to see the nature of such kernels, we calculate the second derivatives of a solution of (4.1), which is defined with the aid of the fundamental solution of the Laplace equation N defined by ´ if n D 2; c2 log jxj N.x/ D nC2 cn jxj if n 3; 1 1 , cn D  .n2/! , and !n D 2 n=2 =. n2 / is the surface area of the where c2 D 2 n unit sphere, cf. (2.14). To this end, we assume that f 2 Lp .Rn /, 1 < p < 1, with

87

Section 4.2 Main Result in the Translation Invariant Case

supp f compact. Then

Z

v.x/ D N  f .x/ D

Rn

N.x  y/f .y/ dy

is well-defined for all x 2 L1 .Rn / and we have for every x … supp f Z @xj @xk v.x/ D .@xj @xk N /.x  y/f .y/ dy; Rn

where

 @xj @xk N.x/ D cn ˛n

ıj k xj xk  n nC2 jxjn jxj

 for all x ¤ 0

and all j; k D 1; : : : ; n, where ˛2 D 1 and ˛n D 2  n if n 3, cf. Exercise 4.39. Here @xj @xk N is the so-called kernel of the operator f 7! @xj @xk N  f . We note that @xj @xk N is homogeneous of degree n, i.e., .@xj @xk N /.rx/ D r n @xj @xk N.x/ In particular, this implies that Z Z j.@xj @xk N /.x/j dx D B1 .0/

for all x ¤ 0; r > 0:

ˇ  ˇ ˇ x ˇˇ dx jxj ˇ.@xj @xk N / jxj ˇ B1 .0/ ˇ  ˇ Z Z 1 ˇ x ˇˇ 1 ˇ r dr D ˇ.@xj @xk N / jxj ˇ d.x/ D C1; @B1 .0/ 0 n ˇ

where we have used Theorem A.8. We note that j@xj @xk N.x/j behaves like jxjn … L1 .B1 .0//. Here x 7! jxjn is just “at the borderline” of being integrable since jxjs 2 L1 .B1 .0// if and only if s > n. Hence we cannot apply kf  gkLp .Rn /  kf kL1 .Rn / kgkLp .Rn /

for allf 2 L1 .Rn /; g 2 Lp .Rn /

to conclude that @xj @xk v 2 Lp .Rn /. Nevertheless we will be able to show that Tf WD @xj @xk N  f can be extended to a bounded linear operator on Lp .Rn / for any 1 < p < 1. A suitable condition on the kernel of the operator, which will be the Hörmander condition below, will be essential.

4.2 Main Result in the Translation Invariant Case We consider operators of the form Tf .x/ D F 1 ŒKO fO.x/ that satisfy the following conditions:

for all f 2 S.Rn /

(4.2)

88

Chapter 4 Translation Invariant Singular Integral Operators

Assumption 4.1.

1. KO 2 L1 .Rn /.

2. There is some k 2 L1loc .Rn n ¹0º/ such that for every f 2 C01 .Rn / Z Tf .x/ D k.x  y/f .y/ dy for almost all x … supp f Rn

and k satisfies the Hörmander condition Z jk.x  y/  k.x/jdx  BK jxj>2jyj

for all y 2 Rn

(4.3)

(4.4)

for some BK 2 .0; 1/. Remark 4.2. 1. First of all, by Plancherel’s theorem, Lemma 2.13, respectively, and the assumption KO 2 L1 .Rn / the operator T extends to a bounded linear operator T 2 L.L2 .Rn //. For the extension, (4.3) holds for any f 2 L2 .Rn / with compact support, which can be proved by approximation with C01 .Rn /functions. 2. T is translation invariant in the sense that T commutes with translations, i.e, T h D h T for all h 2 Rn , where .h f /.x/ D f .x C h/ for all x 2 Rn , since b fO./ D h Tf T .h f / D F 1 Œeih K./

for all f 2 S.Rn /:

Hörmander’s condition is some kind of weak integrability and smoothness condition. It is satisfied e.g. if jrk.z/j  C jzjn1

for all z ¤ 0

because of the next lemma. An example of a kernel, which satisfies the latter zj , j D 1; : : : ; n, which is (up to multiplication with condition, is k.z/ D jzjnC1 a constant) the kernel of the so-called Riesz operators if n 2 and the Hilbert transform if n D 1. We will discuss examples and sufficient conditions for the conditions above in the next sections. Lemma 4.3. Let kW Rn n ¹0º ! C be a continuously differentiable function that satisfies jrz k.z/j  C jzjn1

for all z ¤ 0:

Then k satisfies (4.4). Proof. First of all, Z k.x  y/  k.x/ D 

0

1

y  rk.x  ty/ dt:

(4.5)

89

Section 4.2 Main Result in the Translation Invariant Case

Therefore, if jxj > 2jyj, then jk.x  y/  k.x/j  sup jrz k.x  ty/jjyj t2Œ0;1

 C jxjn1 jyj since jx  tyj 12 jxj for all t 2 Œ0; 1. Hence Z Z jk.x  y/  k.x/j dx  C jxj>2jyj

jxj>2jyj

(4.6)

jxjn1 dxjyj  C 0

uniformly in y ¤ 0. Our main result for this class of operators is: Theorem 4.4. Let T be as in (4.2) satisfying the assumptions above. Then for every t >0 j¹x W jTf .x/j > t ºj 

C1 kf kL1 .Rn / t

for all f 2 L1 .Rn / \ L2 .Rn /:

(4.7)

Moreover, T extends to a bounded linear operator T W Lp .Rn / ! Lp .Rn / for all 1 < p < 1. Remark 4.5. As can be seen in Section 4.7, T will in general not be continuous on L1 .Rn / under the conditions above. The so-called weak type-.1; 1/ estimate (4.7) is a weaker substitute. It implies that T extends to a bounded linear operator T W L1 .Rn / ! L1weak.Rn /, where L1weak .Rn / D ¹f W Rn ! C measurable W kf kL1

weak

kf kL1

weak

< 1º;

D sup t j¹x W jf .x/j > t ºj ; t>0

cf. Exercise 4.41 for details.2 Because of Z j¹x W jf .x/j > t ºj 

¹xWjf .x/j>tº

p

kf kLp .Rn / jf .x/jp dx  p t t

(4.8)

for all t > 0 and any 1  p < 1, we have L1 .Rn /  L1weak .Rn /. The inclusion is strict since e.g. f .x/ D jxjn … L1 .Rn / but f 2 L1weak .Rn /, which can be easily checked. The usefulness of the weak L1 -space comes from the Marcinkiewicz interpolation theorem, presented below. 2

Here k  kL1 is only a quasi-norm, i.e, it satisfies all conditions of a norm accept for the triangle weak  C.kf kL1 C kgkL1 / for some C 1, cf. inequality, which is replaced by kf C gkL1 weak weak weak Exercise 4.40. But boundedness of operators with respect to quasi-norms is defined in the same way as with respect to norms.

90

Chapter 4 Translation Invariant Singular Integral Operators

The structure of the proof is as follows: 1. First of all, because of the Marcinkiewicz interpolation theorem, cf. Theorem 4.12 below, the Lp -continuity of T for 1 < p < 2 follows from (4.7) and T 2 L.L2 .Rn //. Moreover, the Lp -continuity for 2 < p < 1 will be proved with a duality argument. 2. The proof of (4.7) is based on the so-called Calderón–Zygmund decomposition of a function f 2 L1 .Rn /, namely f D g C b, where the “good part” g is bounded by t and the “bad part” b has zero mean value on a family of nonoverlapping cubes .Qj /j 2N , N  N. Finally, let us note the following simply estimate following from the Hörmander condition (4.4), which will be used to estimate the “bad part” b of the Calderón–Zygmund decomposition on suitable cubes Qj : Lemma 4.6. Let Rk 2 L1loc .Rn n ¹0º/ satisfy (4.4). Then for every a 2 L1 .Rn / with supp a  Q and Q a.x/ dx D 0 Z Rn ne Q

jk  a.x/j dx  BK kakL1 .Rn / ;

p p e D Q 2 n denotes the cube with same center as Q and 2 n times the sidewhere Q length of Q, cf. Figure 4.1.

Proof. Since k  a commutes with translations, wepcan always reduce to the case that e D Q 2 n and y 2 Q imply jxj > 2jyj. the center of Q is the origin. Then x … Q

p

Q2 n p

n`

Q `

p

e D Q2 Figure 4.1. Choice of Q

n

(for n D 2).

Section 4.3 Calderón–Zygmund Decomposition and the Maximal Operator

Therefore Z Rn ne Q

Z jk  a.x/j dx  

ˇ ˇZ ˇ ˇ ˇ .k.x  y/  k.x//a.y/ dy ˇ dx ˇ ˇ e

Rn nQ

Z Z Q

 BK

91

Q

jxj>2jyj

jk.x  y/  k.x/jja.y/j dx dy

Z

Q

ja.y/j dy;

which proves the statement. Remark 4.7 (For readers interested in distribution theory). We note that an equivalent description of the assumptions above is that Tf .x/ D K  f .x/ D hK; f .x  /iS 0 .Rn /;S.Rn / ;

f 2 S.Rn /;

O 2 S 0 .Rn / and KjRn n¹0º D k, where k 2 L1 .Rn n¹0º/ satisfies where K D F 1 ŒK loc (4.4). Here KjRnn¹0º D k means that Z hK; 'i D k.x/'.x/ dx for all ' 2 C01.Rn n ¹0º/: Rn

4.3 Calderón–Zygmund Decomposition and the Hardy–Littlewood Maximal Operator Let us start with some notation: In the following let ƒk D 2k Zk , k 2 N0 , and let Dk , k 2 Z, denote the set of all “dyadic cubes” with side length 2k meaning the collection of all (closed) cubes S Q with corners on neighboring points of the lattice ƒk . Moreover, let D D k2Z Dk . Finally, if Q is an arbitrary cube, then Q˛ , ˛ > 0, denotes the cube with same center as Q and ˛ times the side-length of Q. Moreover, we say that two cubes Q; Q 0 are non-overlapping if jQ \ Q0 j D 0. The main result of this section and a key step in the proof of (4.7) is: Theorem 4.8 (Calderón–Zygmund Decomposition). Let f 2 L1 .Rn / and let t > 0. Then there are disjoint measurable sets F;  such that Rn D F [  and 1. jf .x/j  t for almost every x 2 F , S 2.  D j 2N , where Qj , j 2 N  N, are non-overlapping dyadic cubes and Z 1 jf .y/j dy  2n t: (4.9) t< jQj j Qj

92

Chapter 4 Translation Invariant Singular Integral Operators

Moreover, if f D g C b, where ´ g.x/ D

f .x/ R 1 jQj j

Qj

f .y/ dy

if x 2 F; if x 2 Qj ;

then 1. jg.x/j  2n t almost every in Rn , R 2. b.x/ D 0 for every x 2 F and Qj b.x/ dx D 0 for each j 2 N . Proof of Theorem 4.8 (Part 1). Let C t0 for given t > 0 be the set of all Q 2 D satisfying the condition Z 1 t< jf .x/j dx jQj Q and let C t be the subset of all Q 2 C t0 that are maximal with respect to inclusion in C t0 . Every Q 2 C t0 is contained in some Q0 2 C t since 1 jQj  kf kL1 .Rn / for all Q 2 C t0 : t Next, if Q 2 C t \ Dk and Q Q0 2 Dk1 , then by the maximality of Q we have Q0 … C t0 , i.e., Z 1 jf .x/j dx  t: jQ0 j Q0 Moreover, since jQ0 j D 2n jQj, we get Z Z 2n 1 jf .x/j dx  jf .x/j dx  2n t t< jQj Q jQ0 j Q0

for all Q 2 C t :

Hence C t D ¹Qj W j 2 N º, where Qj ; j 2 N  N0 ; are non-overlapping and (4.9) is satisfied for all j 2 N .S R 1 Now let F WD Rn n j 2N Qj . If x 2 F , then jQj Q f .y/ dy  t for every Q 2 D such that x 2 Q. Hence it remains to prove that jf .x/j  t for almost every x 2 F . To this end, we need Lebesgue’s differentiation theorem, which is a corollary to the weak type .1; 1/ estimate of the so-called Hardy–Littlewood maximal operator, presented next. Before continuing the proof of Theorem 4.8, we define a dyadic version of the Hardy–Littlewood maximal operator, namely for f 2 L1loc .Rn / Z 1 jf .y/j dy for all x 2 Rn : .Md f /.x/ D sup 3j 3 jQ Q x2Q2D Then Md is a sub-linear mapping from L1 .Rn / to the space of measurable functions. It even holds the following weak type .1; 1/-estimate.

Section 4.3 Calderón–Zygmund Decomposition and the Maximal Operator

93

Lemma 4.9. There is a constant C depending only on the dimension such that j¹x W Md f .x/ > t ºj 

C kf kL1 .Rn / t

for every t > 0 and f 2 L1 .Rn /. Proof. Let t > 0 and f 2 L1 .Rn / be given. Moreover, let x 2 E t WD ¹y W Md f .y/ > t º: Then there is some cube Q 2 D such that x 2 Q and Z 1 jf .y/j dy > t: jQ3 j Q3 Let k 2 N0 be such that Q 2 Dk . Since Q3 consists of exactly 3n dyadic cubes with the same side-length as Q, there is at least one cube Q0 2 Dk with Q0 \ Q ¤ ; such that Z 1 t jf .y/j dy > n DW t 0 : jQ0 j Q0 3 Now we use the family of cubes C t 0 D ¹Qj W j 2 N º, N  N, constructed in the 0  Q 2 C 0 for first part of the proof of Theorem 4.8 with t replaced by t 0 . Then QS j t 3 some j 2 N and x 2 Q  Qj . Since x 2 E t was arbitrary, E t  j 2N Qj3 . Thus jE t j 

X j 2N

jQj3 j

D

X j 2N

Z 3n X kf k1 3 jQj j  0 jf .x/j dx  Cn t t Qj n

j 2N

since .Qj /j 2N are non overlapping. Remark 4.10. A usual version of the Hardy–Littlewood maximal operator is defined as Z 1 .Mf /.x/ WD sup jf .y/j dy for all x 2 Rn ; f 2 L1loc .Rn /; x2Q jQj Q where the supremum is taking over all cubes in Rn containing x. But this variant and the dyadic variant Md are comparable in the sense that there are constants c; C > 0 such that c.Md f /.x/  .Mf /.x/  C.Md f /.x/

for all x 2 Rn ; f 2 L1loc .Rn /: (4.10)

Actually, the first inequality with c D 1 is obvious. In order to verify the second inequality, let Q 0 be a cube containing x. Then there is a dyadic cube Q 2 Dk

94

Chapter 4 Translation Invariant Singular Integral Operators

containing x with 2.k1/n  jQ0 j  2k n . Hence Q 0 Q3 and jQ3 j D 3n 2k n  6n jQ0 j. Therefore Z Z 1 6n jf .y/j dy  jf .y/j dy; jQ0 j Q0 jQ3 j Q3 which shows the second inequality since Q0 with x 2 Q0 was arbitrary. Because of (4.10) and Lemma 4.9, we obtain C kf k1 j¹x W Mf .x/ > t ºj  t 1 .Rn /, where C depends only on the dimension. for every t > 0 and f 2 L

(4.11)

Corollary 4.11 (Lebesgue’s Differentiation Theorem). Let f 2 L1loc .Rn /. Then Z 1 f .y/ dy for almost all x 2 Rn : f .x/ D lim jQj!0;x2Q jQj Q Proof. First of all, we can assume without loss of generality that f 2 L1 .Rn /. Otherwise, we replace f by fR WD f jBR .0/ for an arbitrary R > 0 and prove the result for fR , which implies the statement for f . First we show that the limit on the right-hand side exists almost everywhere. To this end let ˇ ˇ Z Z ˇ ˇ 1 1 ˇ f .y/ dy  lim inf f .y/ dy ˇˇ : Rf .x/ D ˇ lim sup jQj jQj!0;x2Q jQj jQj!0;x2Q

Q

Q

Now let f 2 L1 .Rn / Obviously, Rf  0 for every continuous f 2 and t > 0. Then for every " > 0 there is some continuous f 0 2 L1 .Rn / with kf  f 0 k1  ". Then L1 .Rn /.

Rf .x/ D R.f  f 0 /.x/  2M.f  f 0 /.x/ and therefore " kf  f 0 k1 C : t t Since " > 0 was arbitrary, j¹x W Rf .x/ > t ºj D 0 and, since t > 0 was arbitrary too, Rf .x/ D 0 almost everywhere. Hence Z 1 lim f .y/ dy jQj!0;x2Q jQj Q j¹x W Rf .x/ > t ºj  j¹x W 2M.f  f 0 /.x/ > t ºj  C

exists for almost every x 2 Rn . In order to prove the statement of the corollary, we define ˇ ˇ Z ˇ ˇ 1 f .y/ dy  f .x/ˇˇ: R0 f .x/ D ˇˇ lim jQj!0;x2Q jQj Q By similar arguments as before, we conclude that R0 f .x/ D 0 almost everywhere.

Section 4.4 Proof of the Main Result in the Translation Invariant Case

95

Proof of Theorem 4.8 (Part S 2). It only remains to prove that jf .x/j  t for almost n every x 2 F WD R n j 2N Qj . First of all, if x 2 F , then for every Q 2 D with x2Q Z 1 jf .x/j dx  t: jQj Q Hence, choosing a sequence Qk 2 D with x 2 Qk and jQk j !k!1 0, we obtain jf .x/j  t for every x 2 F by Corollary 4.11. The statements for g; b follow immediately.

4.4 Proof of the Main Result in the Translation Invariant Case As noted above, we will first prove (4.7). Then we prove the Marcinkiewicz interpolation, i.e., Theorem 4.12 below, which implies the statement of Theorem 4.4 for 1 < p  2. The case p > 2 is then proved by duality. Proof of (4.7). In order to prove the weak type .1; 1/ estimate (4.7), let f 2 L1 .Rn /\ L2 .Rn / and let f .x/ D g.x/ C b.x/ be the Calderón–Zygmund decomposition of f due to Theorem 4.8 for given t > 0. Then ¹x W jTf .x/j > t º  ¹x W jT g.x/j > t =2º [ ¹x W jT b.x/j > t =2º and therefore j¹x W jTf .x/j > t ºj  j¹x W jT g.x/j > t =2ºj C j¹x W jT b.x/j > t =2ºj : It is sufficient to estimate each term on the right-hand side separately. Moreover, we will use that Z XZ X jQj j  jf .x/j dx D jf .x/j dx  kf kL1 .Rn / : (4.12) t jj D t j 2N

j 2N

Qj



In order to estimate T g, we use that jg.x/j  2n t for almost every x 2 Rn , f .x/ D g.x/ for x 2 F , (4.12), (4.8), and that T 2 L.L2 .Rn //, which yields Z Z 4 O 2 4 2 j¹x W jT g.x/j > t =2ºj  2 jT g.x/j dx  2 kKk1 jg.x/j2 dx t t  Z 2 2 2 O  C kKk1 t t jf .x/j dx C t jj F

 C t 1 kf kL1 .Rn / :

96

Chapter 4 Translation Invariant Singular Integral Operators

In order to estimate T b, we apply Lemma 4.6 to bj .x/ WD b.x/ Qj .x/, where Z T bj .x/ D k.x  y/bj .y/ dy for all x … Qj ; Qj

p

ej D Q2 n , by the assumption on the kernel k. Thus, if Q j Z Z jT bj .x/j dx  BK kbj kL1 .Rn /  2BK jf .x/j dx: Rn ne Qj Qj P P On the other hand, since b 2 L2 .Rn /, j 2N bj and therefore j 2N T bj converge in L2 .Rn / to b and T b, respectively, (if N is infinite). The same holds true for almost everywhere convergence for a suitable subsequence. Hence X jT b.x/j  jT bj .x/j almost everywhere j 2N

and

Z Rn ne 

eD where 

S

j 2N

jT b.x/j dx  2BK

XZ j 2N

ej . Finally, Q

e C2 j¹x W jT b.x/j > t =2ºj  jj t

Qj

jf .x/j dx  2Bk kf k1 ;

Z Rn ne 

jT b.x/j dx 

C kf k1 ; t

where we have used that Z X X 3n X 3n ej j  3n Q  jQ jQj j  jf .x/j dx  jj kf k1 t t Qj j 2N

j 2N

j 2N

This finishes the proof of (4.7). In order to finish the proof of Theorem 4.7, we need: Theorem 4.12 (Marcinkiewicz Interpolation Theorem). Suppose that 1 < r  1. Let T be a sub-additive mapping from L1 .Rn / C Lr .Rn / to the vector space of measurable functions on Rn , which is of weak type .1; 1/ and .r; r/, i.e, q

.t I Tf / WD j¹x W jTf .x/j > t ºj  Cq

kf kLq .Rn / tq

(4.13)

for q D 1 and q D r if r < 1 and kTf kL1 .Rn /  C1 kf kL1 .Rn / if r D 1. Then kTf kLp .Rn /  Cp kf kLp .Rn /

for all f 2 Lp .Rn /

and all 1 < p < r, where Cp depends only on C1 ; Cr , p, and r.

97

Section 4.4 Proof of the Main Result in the Translation Invariant Case

Remark 4.13. Note that, because of (4.8) .t I g/ D j¹x W jg.x/j > t ºj 

q kgkL q .Rn /

tq

:

(4.14)

Hence, if T 2 L.Lq .Rn //, the (4.13) holds. Hence (4.13) is a weaker condition than T 2 L.Lq .Rn //. For the following proof we will use that for any measurable gW Rn ! K and 1  p<1 Z Z 1 p jg.x/j dx D p t p1 .t I g/ dt: (4.15) Rn

0

A proof can be found in [29, Theorem 8.16]. We note that this identity can be easily verified if g is a simple function. The statement for a general g can be reduced to the latter case by approximating jgj monotonically by simple functions from below. Proof of Theorem 4.12. First we consider the case r < 1. Let f 2 Lp .Rn / and consider the distribution function .t I Tf /, t > 0, defined as above. For given t > 0 we define f D f1 C f2 by ´ f .x/ if jf .x/j > t; f1 .x/ D 0 else: Then f1 2 L1 .Rn / and f2 2 Lr .Rn / since Z Z p jf1 .x/j dx D jf1 .x/jp jf1 .x/j1p dx  t 1p kf kLp .Rn / Rn

Rn

and similarly Z Z r jf2 .x/j dx D Rn

jf2 .x/jr p jf2 .x/jp dx  t r p kf kLp .Rn / : p

Rn

Now, since jTf .x/j  jTf1 .x/j C jTf2 .x/j, we have ¹x W jTf .x/j > t º  ¹x W jTf1 .x/j > t =2º [ ¹x W jTf2 .x/j > t =2º : Therefore .t I Tf /  .t =2I Tf1 / C .t =2I Tf2 / Z Z C1 Cr  jf1 .x/j dx C jf2 .x/jr dx t =2 Rn .t =2/r Rn Z Z 2r Cr 2C1 jf .x/j dx C r jf .x/jr dx; D t t ¹jf .x/j>tº ¹jf .x/jtº

98

Chapter 4 Translation Invariant Singular Integral Operators

where we have used the weak type .1; 1/ and .r; r/ estimate and (4.13). Now we combine this estimate with (4.15). To this end we calculate Z

1

t

p1 1

Z

t

 ¹jf j>tº

0

jf .x/j dx

Z

Z jf .x/j jf .x/j t p2 dt dx Rn 0 Z 1 jf .x/jjf .x/jp1 dx D p  1 Rn

dt D

since p > 1 and similarly Z

1

t p1 t r



Z ¹jf jtº

0

jf .x/jr dx

Z

Z 1 jf .x/jr t p1r dt dx n R jf .x/j Z 1 D jf .x/jr jf .x/jpr dx r  p Rn

dt D

since p < r. Altogether kTf kLp .Rn /  Cp kf kLp .Rn /

for all f 2 Lp .Rn /:

Finally, if r D 1, we assume for simplicity that C1 D 1. – Otherwise replace T by 1 T . – Then we use the same splitting of f as before, but cut at height t =2 instead C1 of t . Hence jTf2 .x/j  2t since kT kL.L1 .Rn //  1. Therefore ¹x W jTf .x/j > t º  ¹x W jTf1 .x/j > t =2º and .t; Tf /  .t =2; Tf1 /. The rest of the proof is done as before having only the first term, cf. Exercise 4.45. Remark 4.14. Let 1 < p < r be the constants in the proof above. Note that Cp ! 1 if p ! 1 or p ! r < 1, which is natural since T is not necessarily bounded on L1 .Rn / and Lr .Rn / if r < 1. Proof of Theorem 4.4. Because of (4.7), T 2 L.L2 .Rn //, and (4.14), the Marcinkiewicz interpolation theorem for r D 2 implies that T 2 L.Lp .Rn // for every 1 < p < 2. Finally, for the case p > 2 we use that Z Rn

Z Tf .x/g.x/ dx D D

Z

Rn Rn

O fO./g./ K./ O d ¯ D e g.x/ dx f .x/T

Z Rn

O g./ fO./K./ O d ¯

Section 4.4 Proof of the Main Result in the Translation Invariant Case

99

O fO./ for all f 2 S.Rn /. Moreover, ef D F 1 ŒK./ for all f; g 2 S.Rn /, where T 1 .Rn / with supp f \ supp g D ; we have for every f; g 2 C0 Z Z Z Tf .x/g.x/ dx D k.x  y/f .y/ dy g.x/ dx Rn Rn Rn Z Z D f .y/ k.x  y/g.x/ dx dy n Rn ZR eg.x/ dx: f .x/T D Rn

Hence for every g 2 C01 .Rn / Z eg.x/ D T k.y  x/f .y/ dy

for almost all x … supp g;

Rn

Q where the kernel k.z/ D k.z/, z ¤ 0, satisfies the Hörmander condition again. e 2 L.Lq .Rn // for all 1 < q  2. This implies T 2 L.Lq .Rn // for all Therefore T 2  q < 1 because of ˇZ ˇ ˇ ˇ ˇ kTf kLq .Rn / D sup Tf .x/g.x/ dx ˇˇ ˇ 0

g2Lq .Rn /;kgkq0 D1

D

0

sup

g2Lq .Rn /;kgkq0 D1

ˇZ ˇ ˇ ˇ

Rn

Rn

ˇ ˇ e g.x/ dx ˇ  C kf kLq .Rn / f .x/T ˇ

for all f 2 Lp .Rn / due to (A.4). Finally, since kMf k1  kf k1 , (4.11) and Theorem 4.12 yield as a by-product the following important theorem on Lp -continuity of the maximal operator: Theorem 4.15. Let 1 < p  1. Then there is some constant Cp > 0 such that kMf kLp .Rn /  Cp kf kLp .Rn /

for all f 2 Lp .Rn /:

The same holds true for Md instead of M . Remark 4.16. Let T be as in Assumption 4.1. Going through the proof of Theorem 4.4 one can easily verify that for every 1 < p < 1 and every R > 0 there is some constant Cp .R/ such that kT kL.Lp .Rn //  Cp .R/ O L1 .Rn /  R. for every T satisfying (4.4) with some BK  R and satisfying kKk

100

Chapter 4 Translation Invariant Singular Integral Operators

4.5 Examples of Singular Integral Operators An important class of examples of (translation invariant) singular integral operators arises by convolution with principle value distributions, which are defined as follows: Let k 2 L1loc.Rn n ¹0º/ be such that Z jk.x/j dx  C1 for all r > 0 (4.16) r <jxj2r

and ´ R j r <jxj
for all 0 < r < R; exists;

(4.17)

where C1 ; C2 > 0 are some fixed constants. Then we define K D p:v: k 2 S 0 .Rn / as Z Z k.x/'.x/ dx WD lim k.x/'.x/ dx (4.18) hK; 'iS 0 .Rn /;S.Rn / WD p:v: "!0 jxj>"

Rn

for all ' 2 S.Rn /. Example 4.17. Let j D 1; : : : ; n. Then the Riesz operators Rj are defined by Z yj Rj f .x/ WD p:v: cn f .x  y/ dy for all f 2 S.Rn /; nC1 n jyj R where cn D  

nC1 2

. nC1 2 /. Here k.x/ D cn

satisfies (4.17) because of: Z Z k.x/ dx D  r <jxj
r <jxj
xj jxjnC1 Z

k.x/ dx D 

r <jxj
k.x/ dx D 0

since k.x/ D k.x/. Moreover, k satisfies (4.16) because of Z

Z r <jxj2r

jk.x/j dx 

r <jxj<2r

jxjn dx D !n

Z

2r

s 1 ds

r

D !n .ln 2r  ln r/ D !n ln 2 for any r > 0 due to Theorem A.8, where !n denotes the surface measure of @B1 .0/ in Rn .

101

Section 4.5 Examples of Singular Integral Operators

Lemma 4.18. Let k 2 L1loc .Rn n ¹0º/ and K be as above. Then the limit in (4.18) exists and K 2 S 0 .Rn /. Proof. Let

Z l WD p:v:

Z

jxj<1

k.x/ dx WD lim

r !0C r <jxj<1

k.x/ dx;

which exists due to (4.17). Then Z Z Z k.x/'.x/ dx D '.0/ k.x/ dx C k.x/.'.x/  '.0// dx jxj>" "<jxj<1 "<jxj<1 Z C k.x/'.x/ dx; jxj>1

where the last integrals exist because of Z Z   jk.x/'.x/j dx  sup jxjj'.x/j jxj>1

x2Rn

   sup jxjjj'.x/j x2Rn

jxj>1 1 X

jk.x/j dx jxj Z

2k

kD0

   2C1 sup jxjjj'.x/j :

2k jxj<2kC1

jk.x/j dx

x2Rn

Moreover, Z 1 Z X jk.x/.'.x/  '.0//j dx  jxj<1

kD0

2k1 jxj<2k

 kr'k1

jk.x/jj'.x/  '.0/j dx

1 Z X 2k1 jxj<2k

kD0 1 X

 C1 kr'k1

jk.x/jjxj dx

2k D 2C1 kr'k1

kD0

since j'.x/  '.0/j  kr'k1 jxj. Hence we can use Lebesgue’s theorem on dominated convergence to conclude Z Z k.x/'.x/ dx D l'.0/ C k.x/.'.x/  '.0// dx p:v: jxj>" jxj<1 Z C k.x/'.x/ dx; jxj>1

All these principal value distributions give rise to singular integral operators provided that Hörmander’s condition is satisfied:

102

Chapter 4 Translation Invariant Singular Integral Operators

Theorem 4.19. Assume that k 2 L1loc .Rn n ¹0º/ satisfies (4.16), (4.17), and (4.4). Then K defined by (4.18) satisfies Assumption 4.1. Hence Z Tf .x/ WD p:v: k.y/f .x  y/ dy for all f 2 S.Rn / (4.19) Rn

extends to a bounded linear operator T W Lp .Rn / ! Lp .Rn / for every 1 < p < 1. Proof. It only remains to show that KO 2 L1 .Rn /. To this end we consider the truncated kernels ´ k.x/ if " < jxj < R; R k" .x/ D (4.20) 0 else: Then R O Tf D lim lim k"R  f D lim lim F 1 Œkc " f "!0 R!1

"!0 R!1

R 1 n for all f 2 S.Rn /. Hence, if we show that .kc " /0<"1;R1 is bounded in L .R /, then

b fO for all f 2 S.Rn / Tf D F 1 ŒK

(4.21)

b 2 L1 .Rn / because of the following argument: for some K By the weak--compactness of bounded closed sets in L1 .Rn /, for every " > 0 there is a sequence Rk;" !k!1 1 such that R b b" k " k;" *k!1 K

in L1 .Rn /;

b " /0<"1 is again bounded in L1 .Rn /. Hence there is a sequence "j !j !1 where .K 0 such that b b "j * K j !1 K

in L1 .Rn /:

But this implies Z Z Rk;" b k "j j ./fO./g./ Tf .x/g.x/ dx D lim lim O d ¯ j !1 k!1 Rn Rn Z Z b "j ./fO./g./ b fO./g./ K K./ O d O d D lim ¯ D ¯ j !1 Rn

Rn

for all f; g 2 S.Rn / since fO./g./ O 2 L1 .Rn /. This implies (4.21).

103

Section 4.5 Examples of Singular Integral Operators

R 1 n Hence it remains to prove that .kc " /0<"1;R1 is bounded in L .R /: First of all, r because of the Hörmander’s condition, we have for every 0 < jyj  2 , 0 < r < R Z Z Z R R jkr .x  y/  kr .x/j dx  2 jk.x/j dx C 2 jk.x/j dx Rn r <jxj
 4C1 C BK : First let  2 Rn be such that " 

2 jj

(4.22)

 R. Then we decompose

c R R n br kc " ./ D k" ./ C kr ./ for all  2 R ; where r D 1. Then

2 jj .

 Moreover, we set y D  jj 2 , which implies jyj D

r 2

and e iy D

1 c 1 R R iy jkc /j D jF ŒkrR  krR .  y/./j r ./j D jkr ./.1  e 2Z 2 1 1 jkrR .x/  krR .x  y/j dx  2C1 C BK  2 Rn 2

(4.23)

due to (4.22) since 0 < jyj  2r . On the other hand (4.16) implies Z jxj
jk.x/jjxj dx D 

1 Z X kD0 1 X kD0

2k1 r <jxj2k r

2k r

jk.x/jjxj dx

Z 2k1 r <jxj2k r

jk.x/j dx  2C1 r

for all r > 0. Thus

ˇZ ˇ ˇZ ˇ ˇ ˇ ˇ ˇ r ix e ˇ ˇ ˇ jk" ./j  ˇ k.x/.e  1/ dx ˇ C ˇ k.x/ dx ˇˇ "<jxj
(4.24)

since rjj  2. Hence 1 R jkc " ./j  .2 C 4/C1 C C2 C BK 2

for all " 

2  R: jj

R If  2 Rn such that R < 2 , then one can estimate kc " ./ directly using (4.24) with jj R < ", then kc R instead of r. Finally, if  2 Rn such that 0 < 2 " ./ can be estimated jj by using (4.23) with r replaced by ".

104

Chapter 4 Translation Invariant Singular Integral Operators

An important special case for singular integral operators of the form above is when k 2 L1loc .Rn n ¹0º/ is homogeneous of degree n. Then   .x/ x jxjn for all x ¤ 0; k.x/ D D n jxj jxj where .x/ is homogeneous of degree 0, satisfies .x/ D k.x/ for all jxj D 1 and is integrable on @B1 .0/. Moreover, k satisfies (4.16)–(4.17) if and only if Z k.x/ d.x/ D 0; @B1 .0/

where  denotes the surface measure on @B1 .0/, and the corresponding convolution operator T as in (4.19) is invariant with respect to dilations, i.e., T .ır f / D ır .Tf / for all f 2 S.Rn /; r > 0; where .ır f /.x/ D f .rx/ for all x 2 Rn , r > 0, cf. Exercise 4.42. In order to formulate a sufficient condition that k satisfies the Hörmander condition, we introduce the L1 -modulus of continuity for : Z !1 .; t / D sup j.x C h/  .x/j d.x/ for all 0 < t < 1: h2Rn ;jhjt

@B1 .0/

Theorem 4.20. Let W Rn n ¹0º ! C be homogeneous of degree 0 such that  2 R L1 .@B1 .0//, @B1 .0/ .x/ d.x/ D 0, and Z

1

!1 .; t / 0

Then

Z Tf .x/ D p:v:

Rn

dt < 1: t

.y/ f .x  y/ dy jyjn

(4.25)

for all f 2 S.Rn /

satisfies the Assumption 4.1 and extends to a bounded linear operator T W Lp .Rn / ! Lp .Rn / for all 1 < p < 1. Moreover, if .x/ D .x/ for all x ¤ 0, then c ./ D m./fO./ for all f 2 S.Rn /; Tf where i m./ D  2

Z @B1 .0/

.x/ sign.x  / d.x/:

If n D 1, (4.26) has to be understood as m./ D  2i

P

˙ .˙1/ sign.˙1 

(4.26) /.

105

Section 4.5 Examples of Singular Integral Operators

Remark 4.21. We note that m in (4.26) is homogeneous of degree 0, cf. Exercise 4.43. Moreover, if  2 C ˛ .@B1 .0// for some 0 < ˛ < 1, i.e., j.x/  .y/j  C jx  yj˛ for all x; y 2 @B1 .0/ and some C > 0, then !1 .; t /  C area.@B1 .0//t ˛ and therefore

Z

1 0

In particular, if  2

dt  C0 !1 .; t / t

C 1 .@B1 .0//,

Z

1 0

for all t > 0

t 1C˛ dt D

C0 < 1: ˛

then (4.25) is satisfied.

Proof of Theorem 4.20. For the first part it is sufficient to show that k.x/ D .x/jxjn satisfies the Hörmander condition (4.4). To this end we use (4.25) and (4.6) for jxjn instead of k.x/ to conclude that for every y ¤ 0: Z jk.x  y/  k.x/j dx jxj>2jyj Z Z j.x  y/  .x/j dx C j.x/.jx  yjn  jxjn /j dx  n jx  yj jxj>2jyj jxj>2jyj Z 1Z Z 0 0 0 dt j.tx  y/  .tx /j d.x / C C jyj j.x/jjxjn1 dx  t 2jyj jx 0 jD1 jxj>2jyj   Z 1 Z 1 jyj dt  C C kkL1 .@B1 .0// jyj !1 r 2 dr t t 2jyj 2jyj Z 1 ds C C kkL1 .@B1 .0// : !1 .s/ D s 0 Finally, we have to prove that m./ D F ŒK in S 0 .Rn /, where K is the principal value distribution with kernel k.x/ D .x/jxjn and m is as in (4.26). To this end let k"R .x/ be the truncated kernel as in (4.20). Since k.x/ D .x/jxjn is odd, Z 1 R ./ D k R .x/.e ix  e ix / dx kc " 2 Rn " Z .x/ sin.ix  / dx D i n "<jxj
106

Chapter 4 Translation Invariant Singular Integral Operators

R1 RR Here 0 sin.at / dtt D limR!1 0 sin.at / dtt is understood as an improper Riemann integral and Z 1 Z 1 dt dt D sign.a/ for all a ¤ 0: sin.at / sin t t t 0 0 Moreover, we have used that Z

1

sin t 0

 dt D : t 2

cf. e.g. [22, Chapter 13,§3, Exercise 2]. Example 4.22. 1. If n D 1, there is only one singular integral operator, which satisfies the assumptions of Theorem 4.20 including .x/ D .x/, up to a constant multiple. This corresponds to .t / D 1 sign t , k.t / D 1t , respectively, namely Hf .x/ D p:v:

1 

Z R

f .x  y/

dy y

for all f 2 S.Rn /:

This operator is called Hilbert transform. Because of (4.26),

b

Hf ./ D i sign./fO./ for all f 2 S.Rn /: 2. If n 2, one obtains a family of operators similar to the Hilbert transform by choosing h .x/ D cn x  h, where jhj D 1 and cn is chosen such that Z Z 2 jy  hj d.y/ D jx1 j d.x/ D : cn @B1 .0/ @B1 .0/ nC1

This gives cn D   2 . nC1 2 /, cf. e.g. [31, Chapter 3, §3, Theorem 5]. Choosing h D ej , j D 1; : : : ; n gives the Riesz operators Z yj Rj f .x/ D p:v: cn f .x  y/ dy for all f 2 S.Rn /: nC1 Rn jyj They can be expressed via Fourier transformation as

b

Rj f ./ D i

j O f ./ for all f 2 S.Rn / jj

because of Theorem 4.20 and the identity Z  h mh ./ WD cn x  h sign.x  / d.x/ D 2 jj @B1 .0/

(4.27)

107

Section 4.6 Mikhlin Multiplier Theorem

for all ; h 2 Rn ;  ¤ 0. One can prove the last identity as follows: Since  7! mh ./ is homogeneous of degree 0, it is sufficient to consider the case jj D 1. For any jj D 1 fixed, let l.h/ WD mh ./ for all h 2 C n . Then lW C n ! C is linear. Therefore there is some z 2 C n such that l.h/ D z  h for all h 2 C n . Moreover, jl.h/j  1 and Z  l./ D cn jx  j d.x/ D 1 2 @B1 .0/ by the choice of cn . Hence z D , which implies (4.27).

4.6 Mikhlin Multiplier Theorem Theorem 4.23 (Mikhlin Multiplier Theorem). Let mW Rn n ¹0º ! C be an .n C 2/-times continuously differentiable function such that j@˛ m./j  Ajjj˛j

(4.28)

for all  ¤ 0 and j˛j  n C 2. Then for every 1 < p < 1 the operator m.Dx / defined by m.Dx /f D F 1 Œm./fO./

for all f 2 S.Rn /

extends to a bounded linear operator m.Dx /W Lp .Rn / ! Lp .Rn /:

(4.29)

Remark 4.24. 1. We note that the conclusion (4.29) holds even under the weaker assumption that (4.28) holds for all j˛j  Œ n2  C 1, cf. e.g. [3, Theorem 6.1.6]. 2. We note that every .n C 2/-times continuously differentiable function mW Rn n ¹0º ! C that is homogeneous of degree 0, i.e., m./ D m./ for all  > 0;  ¤ 0, automatically satisfies (4.28), cf. Exercise 4.46. In order to prove Theorem 4.23, we will use the so-called dyadic partition of unity, which is nowadays a standard tool in the theory of function spaces and is frequently used to analyze mapping properties of certain operators. Usually a dyadic partition of unity on Rn n ¹0º is a partition of unity 'j ./, j 2 Z on Rn n ¹0º such that supp 'j ./  ¹ 2 Rn W c2j  jj  C 2j º

for all j 2 Z:

Here c; C > 0 are some suitable fixed numbers often chosen to be c D 12 ; C D 2, which we will do in the following. Such a partition can be easily constructed by choosing some non-negative 2 C01 .Rn / such that ./ > 0 if and only if 12 < jj < 2. Then defining j ./ WD .2j /, j 2 Z, we have X ˆ./ WD j ./ > 0 for all  ¤ 0; j 2Z

108

Chapter 4 Translation Invariant Singular Integral Operators

where we note that for each  ¤ 0 the sum above contains at most two non-vanishing terms. Hence 'j ./ WD ˆ./1

j ./

defines a partition of unity with the desired properties. Moreover, in this case 'j ./ D '0 .2j / since ˆ.2j / D ˆ./, which implies that j@˛ 'j ./j  C k@˛ '0 kL1 .Rn / 2j˛jj

(4.30)

for all ˛ 2 N0n and j 2 Z. Now the idea of the proof of Theorem 4.23 is to decompose X m./ D mj ./ for all  ¤ 0; j 2Z

where mj ./ D 'j ./m./. Then j@˛ mj ./j



X 0ˇ ˛

! ˛ ˛ˇ ˇ j@ 'j ./jj@ m./j  C 2j j˛j ˇ

(4.31)

because of (4.28), (4.30), and since 2j 1  jj  2j C1 on supp 'j . For each part mj ./, we have Z mj .Dx /f .x/ D F 1 Œmj ./fO./.x/ D kj .x  y/f .y/ dy; Rn

where 1 n kj .x/ WD F71 !x Œmj  2 Cb .R /

since mj ./ has compact support, cf. Remark 2.2. Hence formally XZ kj .x  y/f .y/ dy; m.Dx /f .x/ D n j 2Z R

where it remains to show that the sum on the right-hand side converges for x … supp f , that X k.z/ WD kj .z/ for z ¤ 0 j 2Z

converges to a function satisfying j@˛z k.z/j  C jzjnj˛j

for all z ¤ 0; j˛j  1;

109

Section 4.6 Mikhlin Multiplier Theorem

and that for every f 2 S.Rn / Z k.x  y/f .y/ dy m.Dx /f .x/ D Rn

for almost all x … supp f:

To this end, we need some suitable uniform estimates of kj .z/. These are a consequence of the following lemma: Lemma 4.25. Let N 2 N0 and let gW Rn ! C be an N -times continuously differentiable function with compact support. Then ˇ jF 1 Œg.x/j  CN jsupp gjjxjN sup k@ gkL1 .Rn /

(4.32)

jˇ jDN

for all x ¤ 0, where CN is independent of g. Proof. Let ˇ 2 N0n with jˇj D N . Then .ix/ˇ F 1 Œg.x/ D F 1 Œ@ g.x/ for all x 2 Rn ˇ

and therefore jx ˇ jjF 1 Œg.x/j  k@ˇ gkL1 .Rn /  jsupp gj sup k@ˇ gkL1 .Rn / : jˇ jDN

Since ˇ 2 N0n with jˇj D N was arbitrary, jxjN jF 1 Œg.x/j  jsupp gj sup k@ gkL1 .Rn / ˇ

jˇ jDN

for all x 2 Rn , which completes the proof. Corollary 4.26. Let m be as in the assumptions of Theorem 4.23 and let mj ./ D m./'j ./, j 2 Z, where 'j ; j 2 Z, is the dyadic partition of unity of Rn n ¹0º as Œm  satisfies above. Then kj .x/ WD F71 !x j j@˛z kj .z/j  C˛ 2j.nCj˛jM / jzjM

for all z ¤ 0; j 2 Z

and all M D 0; : : : ; n C 2, ˛ 2 N0n , where C˛ is independent of j; z. Proof. First of all,  @˛z kj .z/ D F 1 .i /˛ mj ./ .z/

for all z 2 Rn ; ˛ 2 N0n ; j 2 Z;

(4.33)

110

Chapter 4 Translation Invariant Singular Integral Operators

where j@ˇ



! ˇ ˇ   j.@ .i /˛ /@ mj ./j 0ˇ X  C˛;ˇ jjj˛jjˇ jCjj 2j jj ¹W2j 1 jj2j C1 º ./

X  .i /˛ mj ./ j 

0ˇ



0 2j.j˛jjˇ j/ C˛;ˇ

for all jˇj  n C 2. Hence, applying Lemma 4.25 to .i /˛ mj with N D M D 0; : : : ; n C 2, we obtain j@˛z kj .z/j  Cn;˛ jsupp 'j j2j.j˛jM / jzjM  Cn;˛ 2j.nCj˛jM / jzjM for all z ¤ 0, j 2 Z, which finishes the proof. Proposition 4.27. Let m be as in the assumptions of Theorem 4.23. Then there is some k 2 C 1 .Rn n ¹0º/ such that j@˛z k.z/j  C jzjnj˛j and

for all z ¤ 0; j˛j  1

(4.34)

for all x … supp f; f 2 S.Rn /:

(4.35)

Z

m.Dx /f .x/ D

Rn

k.x  y/f .y/ dy

P Proof. We will show that j 2Z @˛z kj .z/ converges absolutely and uniformly on every compact subset of Rn n P ¹0º to a function k.z/ satisfying (4.34). The main idea of the proof is to split the sum j 2Z kj .z/ for given z ¤ 0 into the two parts X

@˛z kj .z/

and

2j jzj1

X

@˛z kj .z/

2j >jzj1

and to show convergence and the estimate (4.34) separately. For the first sum we use (4.33) with j˛j  1 and M D 0. Then X X j@˛z kj .z/j  C 2j.nCj˛j/  C 0 jzjnj˛j ; j ld jzj1

j ld jzj1

where ld denotes the logarithm with respect to basis 2. For the second sum we apply (4.33) with j˛j  1 and M D n C j˛j C 1 and obtain X X j@˛z kj .z/j  C 2j jzjnj˛j1  C 0 jzjnj˛j : ld jzj1 <j

ld jzj1 <j

111

Section 4.6 Mikhlin Multiplier Theorem

P Hence j 2Z @˛z kj .z/ converges absolutely and uniformly on every closed subset of Rn n ¹0º to a function k.z/ that satisfies (4.34) for all j˛j  1. Finally, it remains to show that k.z/ satisfies (4.35). First of all, X mj ./fO./ for all  2 Rn ; f 2 S.Rn / m./fO./ D P

j 2Z

P

j 2Z jmj ./j is uniformly bound1 n O ed with respect to  and f ./ 2 L .R /, the sum on the right-hand side converges in L1 .Rn / to the left-hand side by Lebesgue’s theorem on dominated convergence. Hence X XZ m.Dx /f .x/ D mj .Dx /f .x/ D kj .x  y/f .y/ dy;

since

j 2Z 'j ./ is locally finite.

Moreover, since

n j 2Z R

j 2Z

for every f 2 S.Rn / since Fourier transformation is a bounded linear operator from L1 .Rn / to Cb0 .Rn /. Therefore it only remains to interchange the summation and integration in last term above provided that x … supp f . But this can be done since P k .z/ converges absolutely and uniformly on every closed subset of Rn n ¹0º j 2Z j as shown above. Hence (4.35) follows. Proof of Theorem 4.23. We can apply Theorem 4.4 since (4.34) and Lemma 4.3 imply (4.4). Therefore m.Dx / extends to a bounded linear operator from Lp .Rn / to Lp .Rn / for all 1 < p < 1 due to Theorem 4.4. Analyzing the proof of the Mikhlin multiplier theorem, one easily shows the following refinement: Proposition 4.28. Let m be as in Theorem 4.23 and let A > 0 be as in (4.28). Then for every 1 < p < 1 there is some constant Cp > 0, independent of A, such that km.Dx /kL.Lp .Rn //  Cp A:

(4.36)

Proof. First of all, one easily observes that the constant C in (4.34) depends only on A and not on m directly. Hence from the proof of Lemma 4.3 it easily follows that T D m.Dx / satisfies (4.4) with some BK depending only on A. Due to Remark 4.16 km.Dx /kL.Lp .Rn //  Cp0 .A/ for some Cp .A/ depending only in A (and p) and not directly on m, where 1 < p < 1 is arbitrary. Now one can observe that Cp0 .A/ can be chosen as Cp0 .A/ D Cp A, where Cp D Cp0 .1/ as follows: If m satisfies (4.28) for some A > 0, then m0 ./ WD m./=A satisfies (4.28) with A replaces by 1. Hence km0 .Dx /kL.Lp .Rn //  Cp0 .1/

,

km.Dx /kL.Lp .Rn //  Cp0 .1/A:

112

Chapter 4 Translation Invariant Singular Integral Operators

4.7 Outlook: Hardy spaces and BMO In this section we discuss briefly how to prove continuity of translation invariant operators on the so-called Hardy space H 1 .Rn / and the space of functions of bounded mean oscillations BMO.Rn /, which are the proper substitutes of L1 .Rn / and L1 .Rn / in the theory of singular integral operators and regularity theory of elliptic equations. This section is optional and can be skipped.

In this section we address the question whether singular integral operators like the Riesz and Hilbert transformations are continuous on Lp .Rn / in the limit case p D 1 or p D 1. The answer is negative in general as the following examples show: 1. Let 1 Hf .x/ D p:v: 

Z

1 1

f .x  t /

dt t

be the Hilbert transformation. Then for f D Œa;b 2 L1 .R/ \ L1 .R/, a < b, Hf .x/ D

1 p:v: 

Z

xb xa

f .x  t /

1 xa dt D ln t  xb

for all x > b, where Hf … L1 .R/ since     ba xa xa 1 1 D ln D 1CO CO 2 xb xb x xb x2

as x ! 1:

Moreover, Hf … L1 .R/. i 2. Let f 2 L1 .Rn / and let Rj f .x/ D F 1 Œ jjj fO./.x/, j D 1; : : : ; n, be the Riesz transformation. Then Z 1 n Rj f 2 L .R / ) f .x/ dx D 0: (4.37) Rn

i In order to prove (4.37), one observes Rj f 2 L1 .Rn / implies that jjj fO./ 2 i C 0 .Rn /. But, since fO./ 2 C 0 .Rn / too and jjj is discontinuous at 0, R O Rn f .x/ dx D f .0/ D 0 necessarily.

Hence one sees on one hand that, if f 2 L1 .R/, then Hf can have some logarithmic 1 .R/. On the other hand, if R f 2 L1 .Rn /, then singularities even if f 2 L1 .R/\L j R f has the cancellation property Rn f .x/ dx D 0. As we will see, the Hardy space H 1 .Rn /, which is a subset of L1 .Rn / and the space of functions with bounded mean

i

113

Section 4.7 Outlook: Hardy spaces and BMO

oscillations BMO.Rn /, which contains L1 .Rn /, are natural substitutes of L1 .Rn / and L1 .Rn / in the context of singular integral operators. Before we come to the definition of the spaces, let us point out that a central point of the definition and analysis of the spaces H 1 .Rn / and BMO.Rn / are the following variants of the classical maximal operator Z 1 jf .x/j dx for f 2 L1loc .Rn /: Mf .x/ D sup jQj Q x2Q 1 n 1. RFor the analysis of H 1 .Rn / we define for  a given ˆ 2 C0 .R / with x n ˆ t , t > 0, the associated maximal Rn ˆ.x/ dx ¤ 0 and ˆ t .x/ D t operator Mˆ as

.Mˆ f /.x/ D sup jˆ t  f .x/j t>0

for all x 2 Rn ; f 2 L1loc .Rn /:

Note that the absolute value in the definition of Mˆ is outside the convolution integral. Hence certain cancellation properties of f can be taken into account, which is not the case for Mf , where the absolute value is inside the integral. 2. For the definition and analysis of BMO.Rn / one replaces Mf by the sharp function Z 1 f # .x/ D sup jf .x/  fQ j dx; f 2 L1loc .Rn /; jQj Q x2Q R 1 where fQ D jQj Q f .y/ dy denotes the mean value of f on Q and the supremum is taken over all cubes containing x. – Note that f #  0 if and only if f is a constant. With these variants of the maximal operator we come to: R Definition 4.29. Let ˆ 2 C01 .Rn / with Rn ˆ.x/ dx ¤ 0 be given and let 1  p  1. Then ® ¯ H p .Rn / D f 2 L1loc .Rn / W Mˆ f 2 Lp .Rn / is the Hardy space with integral exponent p. Moreover, ® ¯ BMO.Rn / D f 2 L1loc.Rn / W f # 2 L1 .Rn / ; where two functions differing by a constant are identified. The norms of H p .Rn / and BMO.Rn / are defined as kf kH p .Rn / WD kMˆ f kLp .Rn / ;

kf kBMO.Rn / WD kf ] kL1 .Rn / :

We will see later that the definition of H p .Rn / does not depend on the choice of ˆ.

114

Chapter 4 Translation Invariant Singular Integral Operators

First of all, note that jf .x/j  jMˆ f .x/j since Z lim ˆ t  f .x/ D f .x/ ˆ.y/ dy t!0

almost everywhere

Rn

by Corollary 4.11. Hence H p .Rn /  Lp .Rn / for all 1  p  1: R Moreover, since for every ˆ 2 C01 .Rn / with Rn ˆ.x/ dx ¤ 0 there is a constant C > 0 such that Mˆ f .x/  CMf .x/;

(4.38)

cf. [32, Chapter II, §2.1, Proposition], we obtain: Corollary 4.30. Let 1 < p  1. Then H p .Rn / D Lp .Rn /. Proof. If f 2 Lp .Rn /, then Mf 2 Lp .Rn / by the boundedness of the maximal operator on Lp .Rn / with p > 1, cf. Theorem 4.15. Hence (4.38) shows that f 2 H p .Rn / and therefore Lp .Rn /  H p .Rn /. Since we have seen above that H p .Rn /  Lp .Rn /, the corollary follows. But for the limit case p D 1, H 1 .Rn / ¨ L1 .Rn / as the following characterization shows: Theorem 4.31 (Atomic decomposition of H 1 .Rn /). Let f 2 L1loc .Rn /, then f 2 H 1 .Rn / if and only if f D

1 X

k ak

kD0

with

1 X

jk j < 1

kD0

where k 2 C and ak , k 2 N0 , are H 1 -atoms. Here a function aW Rn ! K is called H 1 -atom if there is some ball B such that 1. supp a  B, 1 , 2. kakL1 .Rn /  jBj R 3. Rn a.x/ dx D 0.

Moreover, kf kH 1 .Rn / is equivalent to inf

1 °X kD0

jk j W f D

1 X kD0

± k ak ; where ak are H 1 -atoms; k 2 C :

115

Section 4.7 Outlook: Hardy spaces and BMO

For a proof we refer to [32, Chapter III, §2, Theorem 2]. Corollary 4.32. Let f 2 H 1 .Rn /. Then Proof. Since f D

P1

kD0 k ak

Rn

f .x/ dx D 0.

converges in L1 .Rn /, we have

Z Rn

R

f .x/ dx D

1 X kD0

Z k

Rn

ak .x/ dx D 0

for any f 2 H 1 .Rn /. The following lemma shows the special role of H 1 -atoms: Lemma 4.33. Let a be an H 1 -atom as above and let T be a linear operator satisfying Assumption 4.1. Then there is some constant Cn depending only on the dimension such that kT akL1 .Rn /  Cn kT kL.L2 .Rn // C BK where BK is the constant in (4.4). p Proof. Let Q be a cube such that B  Q and jQj  . n/n jBj, where B is the ball for which the conditions of an H 1 -atom are satisfied. Then by Lemma 4.6 Z jT a.x/j dx  BK kakL1 .Rn /  BK kakL1 .Rn / jBj  BK ; Rn ne Q p

e D Q 2 n . On the other hand, where Q Z 1 1 e 2 kT akL2 .Rn /  Cn0 jBj 2 kT kL.L2 .Rn // kakL2 .Rn / jT a.x/j dx  jQj e Q 1

1

 Cn jBj 2 kT kL.L2 .Rn // jBj 2 kakL1 .Rn /  Cn kT kL.L2 .Rn // ; which proves the lemma. Corollary 4.34. Let T be a linear operator satisfying Assumption 4.1 and denote its eW L1 .Rn / ! L1 .Rn /, cf. Exercise 4.41, extension to a bounded linear operator T weak 1 n 1 n again by T . Then T W H .R / ! L .R / is a bounded linear operator. P1  a , where ak are H 1 -atoms and Proof. Let f 2 H 1 .Rn / and f D kD0 PN k k P1 1 n 1 n kD0 jk j  C kf kH 1 .Rn / . Then fN WD kD0 k ak 2 H .R / \ L .R / con1 n 1 n 1 verges to f in H .R / ,! L .R /. In particular, TfN !N !1 Tf in Lweak .Rn /

116

Chapter 4 Translation Invariant Singular Integral Operators

and there is a subsequence .fNj /j 2N such that limj !1 TfNj .x/ D Tf .x/ almost everywhere. Hence Z kTf kL1 .Rn /  lim inf jTfNj .x/j dx D lim inf kTfNj kL1 .Rn / j !1

j !1

Rn

by the Lemma of Fatou and lim sup kTfN kL1 .Rn /  N !1

1 X

jk jkT ak kL1 .Rn /

kD0

   C Cn kT kL.L2 .Rn // C BK kf kH 1 .Rn /

because of Lemma 4.33. This implies the statement. A famous result by C. Fefferman is the following duality of H1.Rn / and BMO.Rn /: Theorem 4.35. H 1 .Rn /0 Š BMO.Rn /. More precisely, J W BMO.Rn / ! H 1 .Rn /0 defined by Z f .x/g.x/ dx for all f 2 BMO.Rn /; g 2 H 1 .Rn / hJf; giH 1 .Rn /0 ;H 1 .Rn / D Rn

is a linear isomorphism. We refer to [32, Chapter IV, §1.2] for a proof. But we note that, using the atomic decomposition of H 1 .Rn /-functions, one easily obtains the following inclusion: Let f 2 BMO.Rn /. PThen for every g 2 H 1 .Rn / there is a decomposition f D P 1 1 kD0 k ak with kD0 jk j  C kf kH 1 .Rn / . Therefore ˇZ ˇ X ˇZ ˇ 1 ˇ ˇ ˇ ˇ ˇ ˇ ˇ f .x/g.x/ dx ˇ  jk j ˇ ak .x/g.x/ dx ˇˇ ˇ Rn

kD0

Bk

ˇZ ˇ  C kf kH 1 .Rn / ˇˇ





ˇ ˇ dx ˇˇ

ak .x/ g.x/  gBk Z ˇ ˇ 1 ˇg.x/  gB ˇ dx  C kf kH 1 .Rn / k jBk j Bk Bk

 C kf kH 1 .Rn / kg # kL1 .Rn / D C kf kH 1 .Rn / kgkBMO.Rn / ; where Bk denotes the ball for which the conditions of an H 1 -atom are satisfied for ak and gBk the mean-value of R g on Bk . Hence g induces naturally an element Jg in H 1 .Rn /0 . – Note that, since Rn f .x/ dx D 0, the functional defined above does not depend on the choice of the constant for g. Corollary 4.36. Let T be a linear operator satisfying Assumption 4.1. Then T (first defined e.g. on L1 .Rn / \ L2 .Rn /) can be extended to a bounded linear operator T 2 L.L1 .Rn /; BMO.Rn //.

117

Section 4.7 Outlook: Hardy spaces and BMO

Proof. First let f 2 L1 .Rn / \ L2 .Rn /. Then ˇ ˇZ ˇ ˇ ˇ sup Tf .x/g.x/ dx ˇˇ D sup ˇ n 1 2 1 2 g2H \L WkgkH 1 D1

R

g2H \L WkgkH 1 D1

ˇZ ˇ ˇ ˇ

Rn

ˇ ˇ ˇ

f .x/T  g.x/ dx ˇ

where T  is dual operator of T , which satisfies the Assumptions 4.1 again, cf. proof of Theorem 4.4. Hence ˇ ˇZ ˇ ˇ ˇ Tf .x/g.x/ dx ˇˇ sup ˇ g2H 1 \L2 WkgkH 1 D1



Rn

sup g2H 1 \L2 WkgkH 1 D1

kf kL1 .Rn / kT  gkL1 .Rn /  C kf kL1 .Rn /

because of Corollary 4.34 applied to T  . Therefore Tf 2 H 1 .Rn /0 Š BMO.Rn / and kTf kBMO.Rn /  C kf kL1 .Rn /

for every f 2 L1 .Rn / \ L2 .Rn /

due to (4.35). Now we can extend T to a bounded linear operator T W L1 .Rn / ! BMO.Rn / as follows: Given f 2 L1 .Rn / we have fk WD f Bk *k!1 f

in L1 .Rn /;

where f Bk 2 L2 .Rn /. Hence Tf 2 BMO.Rn / can be (uniquely) extended from L1 .Rn / \ L2 .Rn / to L1 .Rn / by weak--continuity: Z Z Tfk .x/g.x/ dx D lim fk .x/T  g.x/ dx hTf; giBMO;H 1 WD lim k!1 Rn

k!1 Rn

for all g 2 H 1 .Rn /, where fk *k!1 f in L1 .Rn / and T g 2 L1 .Rn / by Corollary 4.34. It is not difficult to show that the definition of Tf for f 2 L1 .Rn / is independent of the choice of the sequence .fk /k2N  L1 .Rn / \ L2 .Rn / with fk *k!1 f in L1 .Rn /. We note that it is not difficult to estimate kTf kBMO.Rn /  C kf kL1 .Rn / for f 2 L .Rn / with compact support directly, cf. e.g. [32, Page 156]. Finally, we note the following nice characterization: 1

Theorem 4.37. Let Rj f D F 1 Πjjj fO./, j D 1; : : : ; n, be the Riesz operators. Then i

1. f 2 H 1 .Rn / if and only if f 2 L1 .Rn / and Rj f 2 L1 .Rn / for all j D 1; : : : ; n. 2. f 2 BMO.Rn /P if and only if there are some gj 2 L1 .Rn /, j D 0; : : : ; n such that f D g0 C jnD1 Rj gj .

118

Chapter 4 Translation Invariant Singular Integral Operators

For a proof we refer to the references given in [32, Chapter III, §4.3] and [32, Chapter IV, §6.4]. Corollary 4.38. Let m.Dx /f D F 1 Œm./fO./ be a multiplier operator satisfying the conditions of Theorem 4.23. Then m.Dx / 2 L.H 1 .Rn // \ L.BMO.Rn //. Proof. First let f 2 H 1 .Rn /. Then m.Dx /f 2 L1 .Rn / because of Corollary 4.34 and Theorem 4.23. Hence it remains to prove that Rj m.Dx /f 2 L1 .Rn /. Since Rj m.Dx /f D F

1



 i j O m./f ./ D F 1 Œmj ./fO./ jj

i

where jjj and m satisfy (4.28), mj satisfies (4.28) and therefore Rj m.Dx / satisfies the Assumption 4.1. Thus Rj m.Dx /f 2 L1 .Rn / if f 2 H 1 .Rn / by Corollary 4.34. Finally, the boundedness on BMO.Rn / is proved in a similar manner.

4.8 Final Remarks and Exercises 4.8.1 Further Reading The results of this chapter are based on the monographs by Garcia-Cuerva and Rubio de Francia [7] and Stein [32], where the interested reader will find many further results. Moreover, we also recommend the books by Duoandikoetxea [5] and Grafakos [8, 9] for further studies. A more detailed discussion and characterizations of translation invariant operators can be found in [33, Chapter I, Section 3]. For a more detailed study of Hardy space and BMO.Rn / we refer to [32].

4.8.2 Exercises Exercise 4.39. Let f 2 Lp .Rn / with supp f compact and Z v.x/ D N  f .x/ D N.x  y/f .y/ dy for all x 2 Rn ; Rn

where N is the fundamental solution of the Laplace equation as in (2.14). 1. Show that v.x/ is well defined for every x 2 Rn . 2. Show that v.x/ is twice differentiable in every x … supp f and Z @xj @xk v.x/ D .@xj @xk N /.x  y/f .y/ dy Rn

119

Section 4.8 Final Remarks and Exercises

for every j; k D 1; : : : ; n where   ıj k xj xk  n @xj @xk N.x/ D cn ˛n jxjn jxjnC2

for all x ¤ 0

and all j; k D 1; : : : ; n, where ˛2 D 1 and ˛n D 2  n if n 3. Exercise 4.40. Let L1weak .Rn / and k  kL1

weak

be defined as in Remark 4.5.

1. Prove that there is a constant C > 1 such that kf C gkL1

weak

 C.kf kL1

weak

C kgkL1

weak

/ for all f; g 2 L1weak .Rn /:

Moreover, show that kf kL1

weak

D jkf kL1

weak

for all f 2 L1weak .Rn /;  2 C;

0, and kf kL1 D 0 if and only if f .x/ D 0 almost everykf kL1 weak weak where. Remark. This means that k  kL1 is a so-called quasi-norm and L1weak .Rn / weak becomes a quasi-normed linear space.. 2. Prove that L1weak .Rn / is complete in the following sense: If .fk /k2N L1weak .Rn / is a Cauchy sequence, i.e., kfk  fj kL1

n weak .R /

!k;j !1 0;

then there is some f 2 L1weak .Rn / such that fk !k1 f in L1weak .Rn /, i.e., kfk  f kL1

n weak .R /

!k;j !1 0:

Moreover, there is a subsequence .fkj /j 2N , kj !j !1 1 such that fkj .x/ !j !1 f .x/ for almost every x 2 Rn . Furthermore, the limit f is unique in the sense that, if additionally fk !k!1 f 0 in L1weak .Rn /, then f .x/ D f 0 .x/ for almost every x 2 Rn . Exercise 4.41. Let T be as in Theorem 4.4. Prove that there is a unique bounded ejL1 .Rn /\L2 .Rn / D T . Here eW L1 .Rn / ! L1 .Rn / such that T linear operator T weak 1 n 1 n eW L .R / ! L T weak .R / is bounded if there is some constant M > 0 such that e  M kf kL1 for all f 2 L1 .Rn /. kT f kL1 weak

Hint. Use Exercise 4.40.

120

Chapter 4 Translation Invariant Singular Integral Operators

Exercise 4.42. Let k 2 L1loc.Rn n ¹0º/ be homogeneous of degree n, i.e., k.x/ D n k.x/ for all x ¤ 0,  > 0. Prove that k satisfies (4.16) and (4.17) if and only if Z k.x/ d .x/ D 0; @B1 .0/

where  denotes the surface measure on @B1 .0/. Moreover, prove that the corresponding convolution operator T as in (4.19) is invariant with respect to dilations, i.e., T .ır f / D ır .Tf / for all f 2 S.Rn /; r > 0; where .ır f /.x/ D f .rx/ for all x 2 Rn , r > 0. Exercise 4.43. Prove that m./ defined by (4.26) is homogeneous of degree 0. Exercise 4.44 (Properties of the Dyadic Partition of Unity). Let 'j , j 2 Z, be a dyadic partition of unity on Rn n ¹0º as in Section 4.6. Moreover, let f W Rn ! C, m 2 R. Prove that there is some constant C > 0 such that jf ./j  C jjm

for all  2 Rn n ¹0º

if and only if there is some C 0 > 0 such that sup j'j ./f ./j  C 0 2j m

2Rn

for all j 2 Z:

Exercise 4.45. Proof Theorem 4.12 in the case r D 1 in detail. Exercise 4.46. Let mW Rn n ¹0º ! C be .n C 2/-times continuously differentiable and homogeneous of degree 0, i.e., m./ D m./ for all  > 0,  ¤ 0. Prove that m satisfies the assumptions of the Mikhlin multiplier theorem, i.e., Theorem 4.23. Hint. Use that @˛ m./ is homogeneous of degree j˛j. Moreover, show that the constant in (4.28) can be chosen as C D max sup jjj˛jj@˛ m./j: j˛jnC2 ¤0

Exercise 4.47. Let mj 2 C N .Rn n ¹0º/, j D 1; 2, such that j@˛ mj ./j  Cj jjsj j˛j

for all  2 Rn n ¹0º; j˛j  N

121

Section 4.8 Final Remarks and Exercises

for some Cj > 0 and sj 2 R. Prove that there is some constant C0 > 0 such that m./ D m1 ./m2 ./ satisfies j@˛ m./j  C0 jjs1 Cs2 j˛j

for all  2 Rn n ¹0º; j˛j  N:

Conclude that the product of two function mj satisfying the condition of the Mikhlin multiplier theorem satisfies these conditions again.

Chapter 5

Non-Translation Invariant Singular Integral Operators Summary In this chapter we first generalize the main result on translation invariant singular integral operators of the last chapter to the non-translation invariant case and to Banachspace valued functions. The result is conditional in the sense that one needs continuity of the operator on some Lp0 -space with p0 > 1. We extend the Mikhlin multiplier theorem to the case of Hilbert-space-valued functions. Moreover, we prove continuity of zero-order pseudodifferential operators on Lp .Rn /, 1 < p < 1. To this end it is essential to get a representation of the Schwartz kernel of the pseudodifferential operator by a smooth function away from the diagonal, which implies the Hörmander condition. Finally, further applications of the kernel representation are discussed including pseudo-locality.

Learning targets  Learn how the results in the scalar translation invariant case of the previous chapter can be generalized.  See how the Mikhlin multiplier theorem generalizes to Hilbert space valued functions.  Get a deeper understanding of the pseudo-local nature of pseudodifferential operators, their kernel representation and their mapping properties.

i

For this chapter the reader should be familiar with the result of Appendix A.4 on Bochner integrals and Bochner spaces. Moreover, the results of the Sections 5.4 and 5.5 are based on the results of Chapter 3.

5.1 Motivation The result due to Theorem 4.4 can be used to get a priori estimates of solutions to elliptic partial differential equations in Lp -Sobolev spaces: E.g., if X a˛ Dx˛ u D f a.Dx /u WD j˛jD2m

123

Section 5.1 Motivation

P for some u; f 2 S.Rn /, where m 2 N0 , a./ D j˛jD2m a˛  ˛ , a˛ 2 C, satisfies ja./j c0 jj2m for all  2 Rn and some c0 > 0, then for every 1 < p < 1 there is some Cp > 0 such that X kDx˛ ukp  Cp kf kp j˛jD2m

independently of u; f 2 S.Rn /. This is a consequence of the Mikhlin multiplier theorem since   ˛  ˛ 1 O Dx u D F f ./ ; a./ where a./1  ˛ is a bounded and homogeneous functions of degree 0, cf. Section 7.1 for a more detailed discussion. But the singular integral operators studied in the previous sections are all translation invariant, i.e., the operators commute with the translation operators y defined by .y f /.x/ D f .x C y/ for all x; y 2 Rn . Therefore these results cannot be used directly to study e.g. variable coefficient elliptic partial differential equations like, X a.x; Dx /u D a˛ .x/Dx˛ u D f j˛jD2m

for suitable u; f , where the coefficients a˛ .x/ are functions depending on x. Since a.x; Dx / does not commute with translations in general, its inverse a.x; Dx /1 , if it exists, does not commute with translations in general too. Therefore we study in the following operators T , which admit a representation Z k.x; x  y/f .y/ dy for all x … supp f Tf .x/ D Rn

for a suitable function k 2 L1loc.Rn  .Rn n ¹0º// that satisfies a variant of the Hörmander condition (4.4) to be specified later. We will even generalize this further by considering vector-valued functions f W Rn ! X0 and operator-valued kernels kW Rn  .Rn n ¹0º/ ! L.X0 ; X1 /, where X0 ; X1 are arbitrary Banach spaces. In particular the result applies to pseudodifferential operators Z p.x; Dx /f D eix p.x; /fO./d; f 2 S.Rn /; ¯ Rn

0 .Rn  Rn /. To this end it is essential that p.x; Dx / has a with symbol p 2 S1;0 representation by a kernel in the form Z k.x; x  y/f .y/ dy for all x … supp f; (5.1) p.x; Dx /f .x/ D Rn

124

Chapter 5 Non-Translation Invariant Singular Integral Operators

where f 2 S.Rn / for a suitable locally integrable function kW Rn  .Rn n ¹0º/ ! C, cf. Theorem 5.12 below. The generalization to vector-valued functions is e.g. motivated by the study of abstract ordinary differential equations d u.t / C Au.t / D f .t / in X for all t > 0; dt u.0/ D 0; where u; f W Œ0; 1/ ! X and AW D.A/  X ! X is an unbounded operator on X. Formally, u can be calculated with the aid of Fourier transformation: Let e0 u and e0 f denote the extensions of u; f , resp., by 0 for t < 0. Then d .e0 u/.t / C A.e0 u/.t / D .e0 f /.t / dt and therefore

b

b

for all t 2 R

b

i  e0 u. / C Ae0 u. / D e0 f . / for all  2 R; where b g . / D F t7! Œg denotes the Fourier transformed in time of g.t /. Hence

b

1 u.t / D F71 !t Œ.i  C A/ e0 f . / for all t > 0

and

b

d 1 u.t / D F71 !t Œi  .i  C A/ e0 f . / for all t > 0 dt provided that i  .i  C A/1 exits for all  ¤ 0. Hence estimating the Lp .RC I X/d norm of dt u in terms of the Lp .RC I X/-norm of f reduces to the study of certain operator-valued Fourier multiplier operators/singular integral operators.

5.2 Extension to Non-Translation Invariant and Vector-Valued Singular Integral Operators In the following let X0 ; X1 be Banach spaces and let T be a linear operator satisfying the following assumptions: Assumption 5.1. Let T W Lp0 .Rn I X0 / ! Lp0 .Rn I X1 / be a bounded linear operator for some 1 < p0  1, where X0 ; X1 are Banach spaces. Moreover, we assume that there is a locally integrable kernel kW Rn  .Rn n ¹0º/ ! L.X0 ; X1 / such that for every f 2 Lp0 .Rn I X0 / with compact support Z Tf .x/ D k.x; x  y/f .y/ dy for almost every x … supp f Rn

Section 5.2 Extension to Non-Translation Invariant Singular Integral Operators

and that k satisfies the Hörmander condition Z kk.x; x  y/  k.x; x/kL.X0 ;X1 / dx  BK jxj>2jyj

for all y 2 Rn :

125

(5.2)

Similar to Section 4.2 the condition (5.2) is a consequence of the following stronger condition: Lemma 5.2. Let kW Rn  .Rn n ¹0º/ ! L.X0 ; X1 / be a locally integrable function that is continuously differentiable in the second variable and satisfies krz k.x; z/kL.X0 ;X1 /  C jzjn1

for almost every x 2 Rn ; z ¤ 0:

(5.3)

Then k satisfies (5.2). Proof. The proof is almost the same as the proof of Lemma 4.3: We use that Z k.x; x  y/  k.x; x/ D 

0

1

y  rk.x; x  ty/ dt:

Therefore, if jxj > 2jyj, then kk.x; x  y/  k.x; x/kL.X0 ;X1 /  sup krz k.x; x  ty/kL.X0 ;X1 / jyj t2Œ0;1

 C jxjn1 jyj since jx  tyj 12 jxj for all t 2 Œ0; 1. Hence Z

Z jxj>2jyj

kk.x; x  y/  k.x; x/kL.X0 ;X1 / dx  C

jxj>2jyj

jxjn1 dxjyj  C 0

uniformly in y ¤ 0. As before we have as a simply consequence of the latter condition the following L1 -estimate: 1 n Lemma R 5.3. Let T be as above. Then for every a 2 L .R I X0 / with supp a  Q and Q a.x/ dx D 0

Z Rn ne Q

kT a.x/kX1 dx  BK kakL1 .Rn IX0 / ;

p p e D Q2 n denotes the cube with same center as Q and 2 n times the sidewhere Q length of Q, cf. Figure 4.1.

126

Chapter 5 Non-Translation Invariant Singular Integral Operators

The proof is identical with the proof of Lemma 4.6, just replacing j  j by the corresponding norms k  kZ , Z D X0 ; X1 ; L.X0 ; X1 /. Analogously to scalar functions one defines for f 2 L1 .Rn I X/ the maximal operator Z 1 kf .y/kX dy; .Mf /.x/ D sup x2Q jQj Q where the supremum is taken over all cubes Q Rn containing x. Moreover, the construction of the Calderón–Zygmund can be done in the same way since the construction is only based on the size of the mean-value of jf .x/j, kf .x/kX , respectively. Hence again one only replaces the absolute value jj by the corresponding norms kkX . Therefore all the results of Section 4.3 directly carry over to vector-valued functions f 2 L1 .Rn I X/. In particular, one has the weak-type .1; 1/-estimate of the maximal operator j¹x 2 Rn W j.Mf /.x/j > t ºj 

C kf kL1 .Rn IX/ t

for all t > 0 and Lebesgue’s differentiation theorem Z 1 f .x/ D lim f .y/ dy in X for almost every x 2 Rn ; x2Q;jQj!0 jQj Q where one uses (as in the scalar case) that continuous, integrable functions f W Rn ! X are dense in L1 .Rn I X/. Using this, an analogous version of the Calderón–Zygmund decomposition stated in Theorem 4.8 holds for f 2 L1 .Rn I X/ again with the obvious replacements of j  j by k  kX . More precisely, we have: Theorem 5.4. Let f 2 L1 .Rn I X0 /, where X0 is a Banach space, and let t > 0. Then there are disjoint measurable sets F;  such that Rn D F [  and 1. kf .x/kX0  t for almost every x 2 F , S 2.  D j 2N , where Qj , j 2 N  N, are non-overlapping dyadic cubes and 1 t< jQj j

Z Qj

kf .y/kX0 dy  2n t:

Moreover, if f D g C b, where ´ g.x/ D

f .x/ R 1 jQj j

Qj

f .y/ dy

if x 2 F; if x 2 Qj ;

Section 5.2 Extension to Non-Translation Invariant Singular Integral Operators

127

then 1. kg.x/kX0  2n t almost every in Rn , R 2. b.x/ D 0 for every x 2 F and Qj b.x/ dx D 0 for each j 2 N . Based on this, we obtain our second main results: Theorem 5.5. Let T be as in Assumption 5.1. Then j¹x 2 Rn W kTf .x/kX1 > t ºj 

C1 kf kL1 .Rn IX0 / t

for all t > 0

(5.4)

and for all f 2 L1 .Rn I X0 / \ Lp0 .Rn I X0 /. Moreover, T extends to a bounded linear operator T W Lp .Rn I X0 / ! Lp .Rn I X1 / for all 1 < p  p0 . Proof. Again the main step consists in proving (5.4). The proof is similar to the corresponding part of the proof of Theorem 4.8. For the convenience of the reader we repeat parts of the proof with the necessary modifications: Let f 2 L1 .Rn I X/ \ Lp0 .Rn I X/ and let f .x/ D g.x/ C b.x/ be the Calderón– Zygmund decomposition of f due to Theorem 5.4 for given t > 0. As in the scalar case j¹x W kTf .x/kX1 > t ºj  j¹x W kT g.x/kX1 > t =2ºj C j¹x W kT b.x/kX1 > t =2ºj and it is sufficient to estimate each term separately. In order to estimate T g, we use that kg.x/kX1  2n t for almost every x 2 Rn , f .x/ D g.x/ for x 2 F , t jj  kf kL1 .Rn IX0 / , and that T 2 L.Lp0 .Rn //: Z Z 2p0 2p0 p0 p0 kT g.x/kX dx  C kg.x/kX dx j¹x W kT g.x/kX1 > t =2ºj  p 1 0 t 0 Rn t p0 Rn  Z t p0 1 kf .x/kX0 dx C t p0 jj  Cp0 t p0 F

 Cp0 t 1 kf kL1 .Rn IX0 / ; where we have used (4.14) for x 7! kg.x/kX0 . In order to estimate T b, we apply Lemma 5.3 to bj .x/ WD b.x/ Qj .x/, j 2 N , where Z k.x  y/bj .y/ dy for almost all x … Qj T bj .x/ D Qj

p 2 n

ej D Q by the assumption on the kernel k. Thus, if Q , j Z Z kT bj .x/kX1 dx  BK kbj kL1 .Rn IX0 /  2BK kf .x/kX0 dx: Rn ne Qj Qj

128

Chapter 5 Non-Translation Invariant Singular Integral Operators

P P On the other hand, since b 2 Lp0 .Rn I X0 /, j 2N bj and therefore j 2N T bj converge in Lp0 .Rn I X0 // to b and T b, respectively (if N is infinite). Hence kT b.x/kX1 

X

kT bj .x/kX1

almost everywhere

j 2N

as before and therefore Z XZ kT b.x/kX1 dx  2BK kf .x/kX0 dx  2Bk kf kL1 .Rn IX0 / ; Rn ne  Q j j 2N eD where 

S

j 2N

ej . Finally, Q

Z ˇ® ¯ˇ C 2 ˇ x W kT b.x/kX > t =2 ˇ  jj e kT b.x/kX1 dx  kf kL1 .Rn IX0 / C 1 t Rn ne t  e  C kf kL1 .Rn IX / , which finishes the proof of (5.4). Finally, we apply the since t jj 0 vector-valued variant of the Marcinkiewicz interpolation theorem, cf. Theorem 5.6 below, to finish the proof. Theorem 5.6 (Marcinkiewicz Interpolation Theorem). Suppose that 1 < r  1 and that X0 ; X1 are Banach spaces. Let T be a sub-additive mapping from L1 .Rn I X0 / C Lr .Rn I X0 / to the vector space of strongly measurable functions on Rn with values in X1 , which is of weak type .1; 1/ and .r; r/, i.e, q

kf kLq .Rn IX0 / ˇ® ¯ˇ .t I Tf / WD ˇ x W kTf .x/kX1 > t ˇ  Cq tq

(5.5)

for all f 2 Lq .Rn I X0 /, for q D 1 and q D r if r < 1 and kTf kL1 .Rn IX1 /  C1 kf kL1 .Rn IX0 / if r D 1 for some Cq > 0. Then kTf kLp .Rn IX1 /  Cp kf kLp .Rn IX0 /

for all f 2 Lp .Rn I X0 /

for all 1 < p < r, where Cp depends only on C1 ; Cr , p, and r. Proof. The proof in the scalar case carries almost literally over to the vector-valued case since the estimates are only based on the size of the functions Tf and f . One only has to replace j  j by k  kXj for j D 0; 1. Alternatively, one can apply the following argument: Let x0 2 X0 with kx0 kX0 D 1 be arbitrary and consider the mapping Mx0 from L1 .Rn / C Lr .Rn / to the space of measurable (scalar) functions defined by Mx0 g.x/ D kT .gx0 /.x/kX1

for all g 2 L1 .Rn / C Lr .Rn /:

Section 5.3 Hilbert-Space-Valued Mikhlin Multiplier Theorem

129

Then Mx0 satisfies the conditions of the scalar Marcinkiewicz interpolation theorem, i.e., Theorem 4.12, with constants independent of x0 . Hence for every 1 < p < r there is some Cp (independent of x0 ) such that kT .gx0 /kLp .Rn IX1 / D kMx0 gkLp .Rn /  Cp kgkLp .Rn / D Cp kgx0 kLp .Rn IX0 / for all g 2 Lp .Rn /. This implies kTf kLp .Rn IX1 /  Cp kf kLp .Rn IX0 / for all simple functions f W Rn ! X0 . Since simple functions are dense in Lp .Rn I X0 /, the statement of the theorem follows. Remark 5.7. In contrast to the result in the translation-invariant and scalar case, Theorem 5.5 is a conditional result based on the fact that it is already known that T is continuous between the Lp0 -spaces. Even for p0 D 2 this is for a given nontranslation invariant operator T in general a non-trivial fact to prove. Even in the case X0 D X1 D C there are some operators T , which satisfy all the remaining conditions of Assumption 5.1, but fail to be continuous on Lp .Rn / for any 1 < p < 1, cf. e.g. [32, Chapter VII, Section 1.2]. We note that a characterization of operators T satisfying T 2 L.Lp0 .Rn // is given by the famous T 1-Theorem, cf. e.g. [5, Chapter 9] and [32, Chapter VII,Section 3] Finally, we note that, even if T commutes with translations and can therefore be written as b fO./; Tf D F 1 ŒK./ b 2 L1 .Rn I L.X0 ; X1 // does not imply T 2 L.L2 .Rn I X0 /; L2 .Rn I X1 // in genK eral unless X0 and X1 are Hilbert spaces, cf. e.g. introduction of [42].

5.3 Hilbert-Space-Valued Mikhlin Multiplier Theorem The goal of this section is to generalize the Mikhlin multiplier theorem to the case of Hilbert space-valued functions. This generalization will have applications to abstract parabolic evolution equations and to the theory of Bessel potential and Besov spaces in Chapter 6. More precisely, the main result of this section is: Theorem 5.8. Let mW Rn n ¹0º ! L.X0 ; X1 / be an .n C 2/-times continuously differentiable function such that k@˛ m./kL.X0 ;X1 /  C jjj˛j

(5.6)

130

Chapter 5 Non-Translation Invariant Singular Integral Operators

for all  ¤ 0 and j˛j  n C 2, where X0 ; X1 are complex Banach spaces. Then there is a continuously differentiable kernel kW Rn n ¹0º ! L.X0 ; X1 / such that Z m.Dx /f D k.x  y/f .y/ dy for x … supp f; (5.7) Rn

for every f 2 S.Rn I X0 / and k satisfies k@˛z k.z/kL.X0 ;X1 /  C jzjnj˛j

for all z ¤ 0; j˛j  1:

(5.8)

In particular, if X0 D H0 ; X1 D H1 are Hilbert spaces, then m.Dx / extends to a bounded linear operator m.Dx /W Lp .Rn I H0 / ! Lp .Rn I H1 / for all 1 < p < 1:

(5.9)

Remark 5.9. 1. As in the scalar case the conclusion (5.9) holds under the weaker assumption that (5.6) holds for all j˛j  Œ n2  C 1, cf. e.g. [3, Theorem 6.1.6]. To this end one verifies (5.2) directly instead of proving (5.8) first and applying Lemma 5.2. 2. As in the scalar case every function mW Rn n ¹0º ! L.X0 ; X1 / that is .n C 2/times continuously differentiable and homogeneous of degree 0 automatically satisfies (5.6). In the following let .'j /j 2Z be the same dyadic partition of unity on Rn n ¹0º as in Section 4.6 and let mj ./ WD m./'j ./ for all j 2 Z,  2 Rn as in the scalar case. Then 1 n kj .x/ WD F71 !x Œmj .x/ 2 Cb .R I L.X0 ; X1 //

since mj ./ 2 L1 .Rn I L.X0 ; X1 // has compact support and because of Theorem 2.43. Hence formally XZ kj .x  y/f .y/ dy; m.Dx /f D j 2Z

Rn

where it remains to show that the sum on the right-hand side converges for x … supp f and that X kj .z/ k.z/ D j 2Z

converges for all z ¤ 0 to a function satisfying (5.8). To this end, we prove the same kind of estimates for kj .z/ as in the scalar case:

Section 5.3 Hilbert-Space-Valued Mikhlin Multiplier Theorem

131

Lemma 5.10. Let Z be a complex Banach space, N 2 N0 , and let gW Rn ! Z be an N -times continuously differentiable function with compact support. Then kF 1 Œg.x/kZ  CN j supp gjjxjN sup k@ˇ gkL1 .Rn IZ/ jˇ jDN

uniformly in x ¤ 0 and g. Proof. The proof is literally the same as the proof of Lemma 4.25. Corollary 5.11. Let m be as in the assumptions of Theorem 4.23 and let mj ./ D m./'j ./, j 2 Z, where 'j ; j 2 Z, is the dyadic decomposition of unity of Rn n ¹0º as in Section 4.6. Then kj .x/ D F71 Œm .x/ satisfies !x j k@˛z kj .z/kL.X0 ;X1 /  C˛ 2j.nCj˛jM / jzjM

for all z ¤ 0; j 2 Z

(5.10)

and all M D 0; : : : ; n C 2, ˛ 2 N0n , where C˛ is independent of j; z. Proof. The proof is literally the same is the proof of Corollary 4.26. P Proof of Theorem 4.23. Firstly, we will show that j 2Z @˛z kj .z/ converges absolutely and uniformly on every compact subsetPof Rn n ¹0º to a function k.z/ satisfying (5.8). As in the scalar case we split the sum j 2Z kj .z/ into the two parts X

@˛z kj .z/

X

and

2j jzj1

@˛z kj .z/

2j >jzj1

and to show convergence and the estimate (5.8) separately. For the first sum we use (5.10) with j˛j  1 and M D 0. Then X X k@˛z kj .z/kZ  C 2j.nCj˛j/  C 0 jzjnj˛j ; j ld jzj1

j ld jzj1

where Z D L.X0 ; X1 / and ld denotes the logarithm with respect to basis 2. For the second sum we apply (5.10) with j˛j  1 and M D n C j˛j C 1 and obtain X X k@˛z kj .z/kZ  2 2j jzjnj˛j1  C 0 jzjnj˛j : ld jzj1 <j

ld jzj1 <j

P Hence j 2Z @˛z kj .z/ converges absolutely and uniformly on every closed subset of Rn n ¹0º to a function k.z/ that satisfies (5.8) for all j˛j  1. Finally, it remains to show that k.z/ satisfies (5.7). First of all, X mj ./fO./ for all f 2 S.Rn I X0 / m./fO./ D j 2Z

132 since

Chapter 5 Non-Translation Invariant Singular Integral Operators

P

j 2Z 'j ./

is locally finite for every  2 Rn . Moreover, since X

kmj ./kL.X0 ;X1 /

j 2N

is uniformly bounded with respect to  and fO 2 L1 .Rn I X0 /, the sum on the righthand side converges in L1 .Rn I X1 / to the left-hand side by Lebesgue’s theorem on dominated convergence. Hence X mj .Dx /f .x/ m.Dx /f .x/ D j 2Z

D

XZ

n j 2Z R

kj .x  y/f .y/ dy

for all f 2 S.Rn I X0 /

since Fourier transformation is bounded operator from L1 .Rn I X1 / to Cb0 .Rn I X1 /, cf. Theorem 2.43. Therefore it only remains to interchange the summation and integration in last term above provided that x … supp f . But this can be done since P n j 2Z kj .z/ converges absolutely and uniformly on every closed subset of R n ¹0º as shown above. Hence (5.7) follows. If X0 D H0 ; X1 D H1 are Hilbert spaces, then the Fourier transformation is an isomorphism on L2 .Rn I Hj /, j D 1; 2, due to Theorem 2.47, and m.Dx / extends to a bounded linear operator from L2 .Rn I H0 / to L2 .Rn I H1 / because of: Z 2 2 km.Dx /f kL d D km./fO./kH ¯ 2 .Rn IH / 1 1 n RZ 2 2 kfO./kH C d ¯ D C kf kL 2 .Rn IH / ; 0 0 Rn

where we have used (5.6). Hence we are in the position to apply Theorem 5.5 with p0 D 2 since (5.8) and Lemma 5.2 imply (5.2). Therefore m.Dx / extends to a bounded linear operator from Lp .Rn I H0 / to Lp .Rn I H1 / for all 1 < p  2. In order to get the statement for 2 < p < 1 we argue again by duality. Because of Z Z .m.Dx /f .x/; g.x//H1 dx D .m./fO./; g.// O ¯ H1 d Rn Rn Z .fO./; m./ g.// O D ¯ H0 d n R Z .f .x/; m .Dx /g.x//H0 dx D Rn

where m ./ D m./ 2 L.H1 ; H0 / denotes the adjoint of m./ for all  2 Rn . Here we have used (2.19). Now m ./ satisfies again (5.6) with H0 ; H1 interchanged.

Section 5.4 Kernel Representation of a Pseudodifferential Operator

133

Thus we can apply the first part to conclude that m .Dx / is a bounded linear operator 0 0 from Lp .Rn I H1 / to Lp .Rn I H0 / for all 2 < p < 1 and p1 C p10 D 1. Now we use that for every f 2 Lp .Rn I H /, 1  p < 1, and any Hilbert space H ˇZ ˇ ˇ ˇ ˇ sup .f .x/; g.x//H dx ˇˇ ; kf kLp .Rn IH / D ˇ 0

Rn

g2Lp .Rn IH /;kgkp0 D1

where 1 < p 0  1 such that p1 C p10 D 1. Here “ ” follows from the Hölder inequality and the Cauchy–Schwarz inequality for .  ;  /H . Moreover, “” can be shown by choosing ´ p2 if f .x/ ¤ 0; f .x/kf .x/kH g.x/ D 0 else: Therefore we can conclude as in the scalar case that ˇZ ˇ ˇ sup km.Dx /f kLp .Rn IH1 / D ˇ p0 n g2L



sup

g2Lp0 .Rn IH1 /WkgkLp0 D1

.R IH1 /WkgkLp0 D1

Rn

.m.Dx /f .x/; g.x//H1

ˇ ˇ dx ˇˇ

kf kLp .Rn IH0 / km .Dx /gkLp0 .Rn IH0 /

 km .Dx /kL.Lp0 .Rn IH1 /;Lp0 .Rn IH0 // kf kLp .Rn IH0 / for all f 2 S.Rn I H0 /. Since S.Rn I H0 / is dense in Lp .Rn I H0 /, (5.9) follows in the case 2 < p < 1.

5.4 Kernel Representation of a Pseudodifferential Operator A classical theorem due to Schwartz states that for every continuous linear operator T W S.Rn / ! S 0 .Rn / there is a unique K 2 S 0 .Rn  Rn / such that hT u; viS.Rn / D hK; u ˝ viS.Rn Rn /

for all u; v 2 S.Rn /;

(5.11)

where .u ˝ v/.x; y/ D u.x/v.y/, cf. e.g. [36, Corollary toRTheorem 51.6]. Here K 2 S 0 .Rn  Rn / is called kernel of T . Formally “T u.x/ D Rn K.x; y/u.y/dy”. Obviously, this result applies to pseudodifferential operators since p.x; Dx /W m .Rn  Rn / and m 2 R arbitrary. S.Rn / ! S.Rn / S 0 .Rn / for p 2 S1;0 m n n For the case that p 2 S1;0 .R  R / with m  n  1, we have seen in the proof of Lemma 3.37 that Z k.x; x  y/f .y/dy for all f 2 S.Rn /; p.x; Dx /f D Rn

134

Chapter 5 Non-Translation Invariant Singular Integral Operators

where k.x; z/ D F71 Œp.x; /.z/ is a continuous function in z 2 Rn . This shows !z that in this case the kernel of T D p.x; Dx / due to (5.11) is the continuous function K.x; y/ D k.x; x  y/ for all x; y 2 Rn : In general the kernel of a pseudodifferential operator does not have to be continuous. For example if p.x; / D 1 and therefore p.x; Dx / D I , the kernel is the distribution K 2 S 0 .Rn  Rn / given by Z u.x/v.x/dx for all u; v 2 S.Rn /: hK; u ˝ viS.Rn Rn / D Rn

Formally “K.x; y/ D ı0 .x  y/” and p.x; Dx /u.x/ D ı0  u.x/ WD hı0 ; u.x  /iS.Rn / D u.x/

for all u 2 S.Rn /:

The main result of this section gives a precise statement of the form of the Schwartz kernel of a pseudodifferential operator away from the diagonal ¹.x; x/ W x 2 Rn º. m .Rn  Rn /, m 2 R. Then there is a smooth function Theorem 5.12. Let p 2 S1;0 n n kW R  .R n ¹0º/ ! C such that Z k.x; x  y/u.y/ dy for all x … supp u (5.12) p.x; Dx /u.x/ D Rn

for all u 2 S.Rn /. Moreover, for every ˛; ˇ 2 N0n ; N 2 N0 , k satisfies 8 nmj˛j hziN ˆ
if n C m C j˛j > 0; if n C m C j˛j D 0; if n C m C j˛j < 0

(5.13)

uniformly in x; z 2 Rn ; z ¤ 0. In particular, we have that Z n k.x; x  y/u.y/v.x/d.x; y/; hp.x; Dx /u; viS.R / D Rn Rn

for all u; v 2 S.Rn / with supp u\supp v D ; and the Schwartz kernel K 2 S 0 .Rn  Rn / of p.x; Dx / is a smooth function on ¹.x; y/ 2 Rn  Rn W x ¤ yº. For the following we need a dyadic partition of unity on Rn , which can be constructed as follows: Let '0 2 C01 .Rn / be such that '0 ./ D 1 for jj  1 and '0 ./ D 0 for jj 2. Moreover, let 'j ./ D '0 .2j /  '0 .2j C1 / for j 2 N. Then supp 'j  ¹ 2 Rn W 2j 1  jj  2j C1 º

for all j 2 N;

supp '0  B2 .0/;

135

Section 5.4 Kernel Representation of a Pseudodifferential Operator

and

1 X

'j ./ D

j D0

k X

'j ./ D '0 .2k / D 1

j D0

for  2 Rn and k 2 N such that jj  2k . Hence .'j /j 2N0 , is a partition of unity on Rn subordinated to the dyadic rings ¹ 2 Rn W 2j 1  jj  2j C1 º, j 2 N, and B2 .0/. Remark 5.13. The asymptotic behavior of a function f ./ as jj ! 1 can be described with the aid of this partition of unity an alternative way as follows: jf ./j  C him

,

sup j'j ./f ./j  C 0 2j m

2Rn

for all j 2 N0

where m 2 R and C 0 > 0 does not depend on j , cf. Exercise 5.28. 1 .Rn  Rn / since ' 2 C 1 .Rn /. Moreover, Obviously, 'j 2 S1;0 j 0 N X

'j ./ D '0 .2N / !N !1 1 pointwise for all  2 Rn and

j D0 N X

@˛ 'j ./ !N !1 0

uniformly for all  2 Rn if ˛ ¤ 0

j D0

since for every  2 Rn there are at most two non-zero terms in the sums and @˛ 'j ./ D 2j˛j.j 1/ @˛ '1 .2j C1 / for all ˛ 2 N0n ; j 2 N:

(5.14)

Hence 1 X

'j ./fO D fO

j D0

and

1 X

'j .Dx /f D f

in S.Rn /

j D0

P since j jND0 'j ./fO  fOj00k;S !N !1 0 for all k 2 N by Lebesgue’s theorem, where j  j00k;S , k 2 N, is the equivalent sequence of semi-norms on S.Rn / which was defined in Corollary 3.47 by replacing k  k1 with k  k2 . With the aid of the dyadic decomposition we decompose a pseudodifferential operator as p.x; Dx /f D

1 X j D0

p.x; Dx /'j .Dx /f D

1 X j D0

pj .x; Dx /f

(5.15)

136

Chapter 5 Non-Translation Invariant Singular Integral Operators

1 .Rn  Rn / and the series for all f 2 S.Rn /, where pj .x; / D p.x; /'j ./ 2 S1;0 converges in S.Rn / since p.x; Dx /W S.Rn / ! S.Rn / is continuous. Moreover, since pj .x; / is compactly supported in , Z kj .x; x  y/f .y/dy; (5.16) pj .x; Dx /f D Rn

where kj .x; z/ D F71 Œp .x; /.z/, similarly as in the proof of Lemma 3.37. !z j m .Rn  Rn /, m 2 R, and let k .x; z/ be defined as above. Lemma 5.14. Let p 2 S1;0 j Then

j@ˇx @˛z kj .x; z/j  C˛;ˇ ;M jzjM 2j.nCmM Cj˛j/

for all z ¤ 0; j 2 N0

(5.17)

and all ˛; ˇ 2 N0n, M 2 N0 , where C˛;ˇ ;M does not depend on j 2 N0 and z ¤ 0. Proof. First of all, Z z  @ˇx Dz˛ kj .x; z/

D

Rn

eix D Œ ˛ @ˇx pj .x; /d¯ 

for all ˛; ˇ; 2 N0n. We now make direct estimates on the above integral. Firstly, the integrand is supported in the ball ¹jj  2j C1 º, which has volume bounded by a multiple of 2nj . Secondly, since the support is even contained in the set ¹2j 1  jj  2j C1 º (when j ¤ 0) and c2j  hi  C 2j if 2j 1  jj  2j C1 , ! X 0 0  ˛ ˇ jD Œ @x pj .x; /j  jD . ˛ @ˇx p.x; //jjD 'j ./j 0 0 0  X 0 0  C˛;ˇ ; himCj˛jj j ¹2j 1 jj2j C1 º ./  2j.jjj j/ 0 0 



0 j.mCj˛jjj/ C˛;ˇ ; 2 ˇ

mCj˛j

due to the symbol estimates of  ˛ @x p.x; / 2 S1;0

jz  Dxˇ Dz˛ kj .x; z/j  C˛;ˇ ; 2j.nCmCj˛jM /

.Rn  Rn /. Hence whenever j j D M:

Taking the maximum over all with j j D M , gives (5.17), and proves the lemma. Proof of Theorem 5.12. First of all, because of (5.15) and (5.16) p.x; Dx /u.x/ D

1 Z X n j D0 R

kj .x; x  y/u.y/ dy

(5.18)

Section 5.4 Kernel Representation of a Pseudodifferential Operator

137

for all x 2 Rn ; u 2 S.Rn /. We will prove (5.12) and (5.13) by showing that 1 X

@˛z @ˇx kj .x; z/

j D0

converges absolutely and uniformly with respect to .x; / 2 Rn  .Rn n B" .0// for every " > 0 to a function k.x; z/ satisfying (5.13). First let 0 < jzj  1. Then we split the sum into X X @˛z @ˇx kj .x; z/ and @˛z @ˇx kj .x; z/: 2j jzj1

2j >jzj1

In order to estimate the first sum, we use Lemma 5.14 with M D 0 and obtain X

j@˛z @ˇx kj .x; z/j

 C˛;ˇ

1 bld.jzj X /c

2j.nCmCj˛j/

j D0

2j jzj1

8 .mCnCj˛j/ if m C n C j˛j > 0; ˆ
where ld D log 2 . For the second term we use Lemma 5.14 with M > n C m C j˛j and estimate X

1 X

j@˛z @ˇx kj .x; z/j  C˛;ˇ jzjM

2j.nCmCj˛jM /

j Dbld.jzj1 /cC1

2j >jzj1

 C˛;ˇ jzj.nCmCj˛j/ : Finally, if jzj 1, we choose M > max.n C m C j˛j; N / in Lemma 5.14 to conclude 1 X

j@˛z @ˇx kj .x; z/j

 C˛;ˇ ;M jzj

M

 C˛;ˇ ;M jzj

M

j D0

1 X

2j.nCmCj˛jM /

j D0 0  C˛;ˇ;M jzjmnj˛jN :

P Hence j1D0 kj .x; z/ converges absolutely and uniformly with respect to .x; / 2 Rn  .Rn n B" .0// for every " > 0 to a function k.x; z/ satisfying (5.13). Using the uniform convergence and (5.18), we conclude that (5.12) holds for all x 2 Rn with dist.x; supp u/ " for arbitrary " > 0. Hence (5.12) follows for all x … supp u. In the case m < 0, we can get a kernel representation for p.x; Dx / without the restriction “x … supp f ”. (I.e., including the diagonal ¹.x; x/ W x 2 Rn º.)

138

Chapter 5 Non-Translation Invariant Singular Integral Operators

m .Rn  Rn / with m < 0. Then there is a smooth function Theorem 5.15. Let p 2 S1;0 kW Rn  .Rn n ¹0º/ ! C such that k.x;  / 2 L1 .Rn / for all x 2 Rn and Z p.x; Dx /u.x/ D k.x; x  y/u.y/ dy for all x 2 Rn ; u 2 S.Rn /: (5.19) Rn

Moreover, for every ˛; ˇ 2 N0n ; N 2 N0 , k satisfies (5.13). Proof. See Exercise 5.27. m Corollary 5.16. Let p 2 S1;0 .Rn  Rn / with m < 0. Then there is some C > 0 such that

kp.x; Dx /f kLq .Rn /  C kf kLq .Rn /

for all f 2 S.Rn /; 1  q  1:

Proof. Let k be as in Theorem 5.15 and let g.z/ WD supx2Rn jk.x; z/j for all z ¤ 0. Because of (5.13), we have that Z Z kgkL1 .Rn / D sup jk.x; z/j dz  C jzjnm .1 C jzj/N dz < 1 Rn x2Rn

Rn

since n  m > n, where N is chosen such that m  N < 0. Hence Z g.x  y/ju.y/j dy for all x 2 Rn ; u 2 S.Rn /: jp.x; Dx /u.x/j  Rn

Hence kp.x; Dx /ukLp .Rn /  kg  jujkLp .Rn /  kgkL1 .Rn / kukLp .Rn / for all u 2 S.Rn / due to (2.5). Note that in the statement of the corollary the limit cases q D 1; 1 are included. On the other hand, it is assumed that p is of negative order. In the case of a zero order pseudodifferential operator, we will show boundedness on Lq .Rn / for all 1 < q < 1 in the next section. In the case m D 1, the kernel is even smooth: 1 Theorem 5.17. Let p 2 S1;0 .Rn  Rn / and let k be as in Theorem 5.15. Then k can be extended to a smooth function KW Rn  Rn ! C such that for any ˛; ˇ 2 N0n , N 2 N0 there is some C˛;ˇ ;N > 0 such that

j@˛x @ˇz K.x; z/j  C˛;ˇ ;N .1 C jzj/N

for all x; z 2 Rn :

(5.20)

Here K.x; z/ D F71 !z Œp.x; /.z/

for all x; z 2 Rn :

Moreover, p.x; Dx / is a smoothing operator in the sense that p.x; Dx /W S 0 .Rn / ! C 1 .Rn /:

(5.21)

Section 5.4 Kernel Representation of a Pseudodifferential Operator

139

1 .Rn  Proof. First of all, p.x;  / 2 S.Rn / for every fixed x 2 Rn since p 2 S1;0 Rn /. Hence

K.x; z/ WD F 1 Œp.x;  /.z/

for all x; z 2 Rn

is well-defined and K.x;  / 2 S.Rn / for all x 2 Rn . Now, if kj .x; z/ D F71 !z Œp.x; /'j ./;

j 2 N0 ;

are as in the proof of Theorem 5.12, then X K.x;  / D kj .x;  / for all x 2 Rn ; j 2N0

where the series converges in S.Rn / for every fixed x 2 Rn since p.x;  / 2 S.Rn / for every fixed x 2 Rn . Hence p.x; Dx /u.x/ D

1 Z X n j D0 R

Z kj .x; x  y/u.y/ dy D

Rn

K.x; x  y/u.y/ dy

for all x 2 Rn and u 2 S.Rn / by Lebesgue’s theorem on dominated convergence, where kj are as above. Since also Z p.x; Dx /u.x/ D k.x; x  y/u.y/ dy for all x … supp u; u 2 S.Rn /; Rn

k.x; z/ D K.x; z/ for all x 2 Rn ; z ¤ 0. Hence (5.20) follows from Theorem 5.12 since (5.13) holds for any m 2 R. (See also Exercise 5.29.) In order to prove (5.21) let f 2 S 0 .Rn /. Then there is some N 2 N such that 0 f WD hxi2N hDx i2N f 2 L2 .Rn /, cf. Exercise 2.65. Moreover, let .f"0 /0<"1  C01 .Rn / be such that f"0 !"!0 f 0 in L2 .Rn /. Hence f" WD hDx i2N .hi2N f"0 / !"!0 f

in S 0 .Rn /;

where f" 2 C01 .Rn / S.Rn /. Therefore Z K.x; x  y/hDy i2N .hyi2N f"0 .y//dy p.x; Dx /f" .x/ D n R Z D .hDz i2N K/.x; x  y/hyi2N f"0 .y/dy; Rn

where the right-hand side converges uniformly on all compact subsets of Rn with respect to x as " ! 0, since jhDz i2N k.x; x  y/j  CN hx  yin  CN0 hxijnj hyin 2 L2 .Rn /;

140

Chapter 5 Non-Translation Invariant Singular Integral Operators

where hyin 2 L2 .Rn /, and f"0 !"!0 f in L2 .Rn /. This implies that Z Z p.x; Dx /f .x/ D .hDz i2N k/.x; x  y/hyi2N f 0 .y/dy; Rn

Rn

since lim hp.x; Dx /f" ; 'i D hp.x; Dx /f; 'i

"!0

for all ' 2 C01 .Rn /:

From this one can conclude p.x; Dx /f 2 C 1 .Rn / in a straight forward manner using (5.20). Remark 5.18. One can even characterize pseudodifferential operators of order 1 by the kernel representation with some K satisfying (5.20). More precisely, if Z K.x; x  y/ dy for all u 2 S.Rn /; x 2 Rn ; P u.x/ D Rn

1 .Rn  Rn / is given by where K satisfies (5.20), then P D p.x; Dx / where p 2 S1;0

p.x; / D Fz7! Œk.x; z/./

for all x;  2 Rn ;

cf. Exercise 5.30.

5.5 Consequences of the Kernel Representation 0 .Rn  Rn / and 1 < q < 1. Then p.x; D / extends to Theorem 5.19. Let p 2 S1;0 x a bounded linear operator p.x; Dx /W Lq .Rn / ! Lq .Rn /.

Proof. Because of Theorem 3.36, we know that p.x; Dx / 2 L.L2 .Rn //. Moreover, because of Theorem 5.12, there is a kernel k such that (5.12) holds and k satisfies j@˛x k.x; z/j  C jzjnj˛j

for all x 2 Rn ; z ¤ 0; j˛j D 1;

i.e., k satisfies (5.3). Hence k satisfies the Hörmander condition (5.2) due to Lemma 5.2. Therefore p.x; Dx / satisfies all the assumptions of Theorem 5.5. Hence p.x; Dx / extends to a bounded linear operator p.x; Dx /W Lq .Rn / ! Lq .Rn /

for all 1 < q  2:

The statement for 2 < q < 1 is proved by duality similarly as in the proof of Theorem 4.4 and Theorem 5.8. We use that Z Z p.x; Dx /f .x/g.x/ dx D f .x/p .x; Dx /g.x/ dx for all f; g 2 S.Rn / Rn

Rn

Section 5.5 Consequences of the Kernel Representation

141

0 .Rn  Rn / is as in Corollary 3.34. Now p  .x; D / extends to a where p  2 S1;0 x 0 bounded linear operator on Lq .Rn / by the first part, where q1 C q10 D 1. Hence ˇ ˇZ ˇ ˇ ˇ sup p.x; Dx /f .x/g.x/ dx ˇˇ kp.x; Dx /f kLq .Rn / D ˇ 0

Rn

g2Lq .Rn /WkgkLq0 D1



sup

g2Lq0 .Rn /WkgkLq0 D1

kf kLq .Rn / kp  .x; Dx /gkLq0 .Rn /

 kp  .x; Dx /kL.Lq0 .Rn // kf kLq .Rn / for all f 2 S.Rn / due to (A.4). Thus p.x; Dx / extends to a bounded operator on Lq .Rn /. As a direct consequence we obtain: m .Rn  Rn / and let 1 < q < 1, m 2 R. Then p.x; D / Theorem 5.20. Let p 2 S1;0 x extends to a bounded linear operator

p.x; Dx /W HqsCm .Rn / ! Hqs .Rn /: Proof. The theorem is proved in the same way as Theorem 3.41. It also follows from Exercise 3.68. Remark 5.21. As in the case q D 2, cf. Theorem 3.41, one can observe from the proof of Theorem 5.20 and its preceding results that there is a constant Cs;m;q > 0 .m/ that depends only on jpjk for some sufficiently large k 2 N such that kp.x; Dx /kL.H sCm .Rn /;H s .Rn //  Cs;m;q : q

q

0 >0 By the same argument as in Remark 3.40 one observes that there are some Cs;m;q and k 2 N independent of p such that 0 jpj.m/ kp.x; Dx /kL.H sCm .Rn /;H s .Rn //  Cs;m;q k q

q

since the mapping p 7! p.x; Dx / is linear. With the aid of the kernel representation (5.12) one can prove many other refined mapping properties of pseudodifferential operators. First of all we have: Theorem 5.22. Let '; 2 Cb1 .Rn / such that dist.supp '; supp / > 0 and let m .Rn  Rn /. Then there is some k 2 C 1 .Rn  Rn / such that p 2 S1;0 Z '.x/.p.x; Dx /. f //.x/ D k.x; x  y/f .y/ dy for all x 2 Rn ; f 2 S.Rn /; Rn

where k satisfies (5.20) for any N 2 N, ˛; ˇ 2 N0n. In particular '.x/p.x; Dx / .x/W S 0 .Rn / ! C 1 .Rn /:

142

Chapter 5 Non-Translation Invariant Singular Integral Operators

Proof. By Theorem 3.18 '.x/.p.x; Dx /. f //.x/ D q.x; Dx /f .x/

for all f 2 S.Rn /;

1 where q 2 S1;0 .Rn  Rn /. Moreover, due to Theorem 5.17, there is some smooth n  Rn ! C such that (5.20) holds, kW R Z q.x; Dx / D k.x; x  y/f .y/ dy for all x 2 Rn ; f 2 S.Rn /; Rn

and q.x; Dx

/W S 0 .Rn /

! C 1 .Rn /.

As a direct consequence we obtain that pseudodifferential operators are pseudolocal and elliptic pseudodifferential operators are hypo-elliptic. To this end we need the notion of the singular support of distribution: Definition 5.23. Let F 2 S 0 .Rn / and let U  Rn be open. Then F is smooth on U if there is some f 2 C 1 .U / such that Z hF; 'i D f .x/'.x/ dx for all ' 2 C01 .U /: Rn

Moreover, the singular support of F is defined by sing supp F D ¹x 2 Rn W there is no " > 0 such that F is smooth on B" .x/º : Consequently, sing supp F is the smallest set such that F is smooth on Rn n sing supp F . Definition 5.24. Let P W S 0 .Rn / ! S 0 .Rn / be a linear operator. Then P is called pseudo-local if sing supp P u  sing supp u for all u 2 S 0 .Rn /: Moreover, P is called hypo-elliptic if sing supp P u D sing supp u for all u 2 S 0 .Rn /: m .Rn  Rn / for some m 2 R. Then p.x; D / is pseudoTheorem 5.25. Let p 2 S1;0 x local. If p is additionally elliptic, then p.x; Dx / is hypo-elliptic.

Proof. Let u 2 S 0 .Rn / and let x … sing supp u. Then by the definition of sing supp u there is some " > 0 such that u is smooth on B" .x/, i.e., there is some v 2 C 1 .B" .x// such that Z hu; 'i D v.y/'.y/ dy for all ' 2 C01 .B" .x//: Rn

143

Section 5.6 Final Remarks and Exercises

Now let ; 2 C01 .B" .x// with

u D v 2 C01 .Rn / and

 1 on B"=2 .x/ and  1 in supp . Then

.x/p.x; Dx /u.x/ D

.x/.p.x; Dx /. u//.x/ C

.x/.p.x; Dx /..1  /u//.x/

D

.x/.p.x; Dx /. v//.x/ C

.x/.p.x; Dx /..1  /u//.x/

for all x 2 Rn , where p.x; Dx /. v/ 2 S.Rn / since v 2 C01 .Rn /. Moreover, because of Theorem 5.22, p.x; Dx /..1  /u/ 2 C 1 .Rn /; Therefore p.x; Dx /u is smooth on B"=2 .x/ since

 1 on B"=2 .x/. Hence

x … sing supp p.x; Dx /u; which shows that sing supp p.x; Dx /u  sing supp u since x … sing supp u was arbitrary. m .Rn  Rn / Now, if p is additionally elliptic, then there is a parametrix q1 2 S1;0 such that q1 .x; Dx /p.x; Dx / D I C r1 .x; Dx / 1 .Rn  Rn / because of Proposition 3.26. This implies where r1 2 S1;0

u D q1 .x; Dx /p.x; Dx /u  r1 .x; Dx /u; where sing supp q1 .x; Dx /p.x; Dx /u  sing supp p.x; Dx /u by the first part and r1 .x; Dx /u 2 C 1 .Rn / because of Theorem 5.17.

5.6 Final Remarks and Exercises 5.6.1 Further Reading Section 5.2 is based on the results of [32, Chapter I], where an even more general result is presented. Moreover, many further results on singular integral operators, including the T 1-theorem and mapping properties of pseudodifferential operators can be found in [32]. The T 1-theorem and singular integral operators in general are also discussed in Duoandikoetxea [5]. For further results on the Mikhlin multiplier theorem for Banach-space-valued functions, we refer to Weis [42] and Denk et al. [4] and the references given there. A further discussion of hypo-ellipticity can be found in the books by Kumano-Go [21], Jacob [18], and Hörmander [12, 13].

144

Chapter 5 Non-Translation Invariant Singular Integral Operators

5.6.2 Exercises Exercise 5.26. Let T 2 L.Lp0 .Rn // be as in Assumption 5.1. Show with the aid of the results in Section 4.7 that T 2 L.H 1 .Rn /; L1 .Rn //. Exercise 5.27 (Kernel of a PsDOs of Negative Order). m .Rn  Rn / with m < 0. Prove that Let p 2 S1;0 Z p.x; Dx /f .x/ D

Rn

k.x; x  y/f .y/dy

for all x 2 Rn ; f 2 S.Rn /;

where k satisfies (5.13). Hint. Prove that

1 X

sup jkj .x; z/j n

j D0 x2R

converges in L1 .Rn / with respect to z, where kj are as in Section 5.4. Exercise 5.28 (Properties of the Dyadic Partition of Unity). Let .'j /j 2N0 be the dyadic partition of unity on Rn from Section 5.4. 1. Make a sketch of 'j , j 2 N0 for n D 1. 2. Let f W Rn ! C, m 2 R. Prove that there is some constant C > 0 such that jf ./j  C him ;

for all  2 Rn

if and only if there is some C 0 > 0 such that sup j'j ./f ./j  C 0 2j m

2Rn

for all j 2 N0 :

3. Prove that for all ˛ 2 N0n there is some constant C˛ > 0 such that j@˛ 'j ./j  C˛ min.2j j˛j ; hij˛j / for all j 2 N0 and  2 Rn . m 4. Prove that for all f 2 S 0 .Rn / and p 2 S1;0 .Rn  Rn /, m 2 R,

p.x; Dx /f D

1 X

pj .x; Dx /f

in S 0 .Rn /

j D0

where pj .x; / D p.x; /'j ./ for all x;  2 Rn , j 2 N0 .

145

Section 5.6 Final Remarks and Exercises

m .Rn / with m < n  N for some N 2 N . Exercise 5.29. Let p 2 S1;0 0 Prove that the kernel k in Theorem 5.12 can be extended (uniquely) to some K 2 C N .Rn  Rn /

Exercise 5.30. Let K satisfy (5.20) for any N 2 N, ˛; ˇ 2 N0n , and let Z K.x; x  y/ dy for all u 2 S.Rn /; x 2 Rn : P u.x/ D Rn

1 .Rn  Rn / is given by Prove that P D p.x; Dx / where p 2 S1;0

p.x; / D Fz7! Œk.x; z/./

for all x;  2 Rn :

Part III

Applications to Function Space and Differential Equations

Chapter 6

Introduction to Besov and Bessel Potential Spaces Summary The purpose of this chapter is to give a brief introduction to the modern theory of function spaces. More precisely, we study the scales of Besov and Bessel potential spaces, which generalize both the classical Hölder spaces and the Sobolev spaces. To this end a suitable Fourier analytic characterization using a dyadic partition unity in Fourier space is used. We study the relations between Besov and Bessel potential spaces and their relation to Sobolev and Hölder spaces. Finally, boundedness of pseudodifferential operators on these spaces is shown.

Learning targets  Understand the relations between Besov and Bessel potential space.  Learn about Fourier analytic characterizations of weak derivatives and Hölder continuity.  Get some basic ideas from the modern theory of function spaces. In this chapter the Hilbert-space-valued Mikhlin multiplier theorem (Theorem 5.8) will be needed. Moreover, for Section 6.6 the results of Chapter 3 and Section 5.4 are needed.

6.1 Motivation In this section we will address the following question: How to measure regularity of functions? There are two classical approaches: 1. Hölder Spaces C s .Rn /: In this case regularity is measured by differences, resp. the modulus of continuity of a function f W Rn ! C defined by !.t I f / WD

sup

x2Rn ;jhjt

jf .x C h/  f .x/j:

Then f 2 C s .Rn /, 0 < s < 1, if and only if f W Rn ! C is bounded and there is a constant C > 0 such that !.t I f /  C t s

for all t > 0:

i

150

Chapter 6 Introduction to Besov and Bessel Potential Spaces

2. Sobolev Spaces Wpm .Rn /: In this case regularity is measured in terms of (generalized) derivatives in Lp -spaces: f 2 Wpm .Rn /, m 2 N0 , 1  p  1 if and only if @˛x f 2 Lp .Rn / for all j˛j  m. Moreover, we have introduced: 3. Bessel Potential Spaces Hps .Rn /: In this case regularity is measured through a certain decay on the Fourier transform side: Recall f 2 Hps .Rn /, s 2 R; 1 < p < 1, if and only if s hDx is f D F 1 Œ.1 C jj2 / 2 fO./ 2 Lp .Rn /:

We have already seen in Lemma 2.41 that Hpm .Rn / D Wpm .Rn / for m 2 N0 and p D 2. In Theorem 6.8 below it will be proven that this is also the case if 1 < p < 1. Hence the Bessel potential spaces are a generalization of the Sobolev spaces for 1 < p < 1. But the following questions arise:

How are Hölder spaces and Bessel potential spaces related?

How is Hölder continuity related to decay properties on the Fourier transform side?

6.2 A Fourier-Analytic Characterization of Hölder Continuity First of all, we recall from Theorem 2.1: 1. If f W Rn ! C is a continuously differentiable function such that f 2 L1 .Rn / and @xj f 2 L1 .Rn /, then F Œ@xj f  D i j F Œf  D i j fO./ for all  2 Rn :

(6.1)

2. If f 2 L1 .Rn / such that xj f 2 L1 .Rn /, then fO./ is continuously partially differentiable with respect to j and @j fO./ D F Œixj f .x/./:

(6.2)

These identities show the duality between differentiability of f and decay of fO./ as jj ! 1 as well as decay of f for jxj ! 1 and differentiability of fO. In the following we will use a dyadic partition of unity 'j ./, j 2 N0 (on Rn ). This is a partition of unity 'j ./, j 2 N0 , on Rn with 'j 2 C01 .Rn / such that supp 'j  ¹ 2 Rn W 2j 1  jj  2j C1 º

for all j 1:

(6.3)

The partition of unity can be constructed such that supp '0 B2 .0/, 'j ./ D '1 .2j C1 / for all j 1 and (6.3) holds, cf. Section 5.4 for details. Then we have j@˛ 'j ./j  C k@˛ '1 kL1 .Rn / 2j˛jj

for all ˛ 2 N0n ; j 1:

(6.4)

Section 6.2 A Fourier-Analytic Characterization of Hölder Continuity

151

Moreover, we note that f .x/ D

1 X

'j .Dx /f .x/

for all x 2 Rn ; f 2 S.Rn /

j D0

and 'j .Dx /f D F 1 Œ'j ./fO./ D 'Lj  f for all f 2 S.Rn /, j 2 N0 , where 'Lj D F 1 Œ'j  and 'Lj .x/ D 2.j 1/n 'L1 .2j 1 x/

for all j 2 N and x 2 Rn :

(6.5)

Furthermore 'j .Dx /f D F 1 Œ'j ./fO 2 S 0 .Rn / is well-defined for all f 2 S 0 .Rn /, j 2 N0 , and one has 'j .Dx /f 2 C 1 .Rn / due to Remark 2.33. It is easy to check that for f 2 Lp .Rn /, 1  p  1, we have 'j .Dx /f .x/ D 'Lj  f .x/

for all x 2 Rn :

Finally, we note that   'j .Dx /f D 'j 1 .Dx / C 'j .Dx / C 'j C1 .Dx / 'j .Dx /f

(6.6)

for all f 2 S.Rn /; j 2 N0 , since 'j 1 C 'j C 'j C1  1 on supp 'j (where '1  0). Using this decomposition, we obtain the following characterization of Hölder continuous functions. Theorem 6.1. Let 0 < s < 1. Then f 2 C s .Rn / if and only if f 2 L1 .Rn / and kf kCs WD sup 2js k'Lj  f kL1 .Rn / < 1: j 2N0

Moreover, k  kCs is an equivalent norm on C s .Rn /. Proof. First let f 2 C s .Rn /. Then sup jf .x  y/  f .x/j  kf kC s jyjs

x2Rn

for all y 2 Rn . Because of (6.5), we have k'Lj kL1 .Rn /  C;

kr 'Lj kL1 .Rn /  C 2j

for all j 2 N0 :

(6.7)

152

Chapter 6 Introduction to Besov and Bessel Potential Spaces

Moreover,

Z Rn

Z 'Lj .y/dy D

Rn

'L1 .y/dy D F Œ'L 1 .0/ D '1 .0/ D 0

for all j 1. For the following we denote D F 1 Œ'1 .2/ and j n .2j x/ for all x 2 Rn , j 2 N. Hence 2 Z 'j .Dx /f .x/ D f .x  y/ 2j .y/dy n ZR .f .x  y/  f .x// 2j .y/dy D since

2j .x/

Rn

R Rn

D

(6.8)

D 0. Therefore Z k'j .Dx /f k1  kf kC s jyjs 2j .y/dy Rn Z D 2js kf kC s jzjs .z/dz D C 2js kf kC s 2j .y/ dy

Rn

for all j 2 N and f 2 S.Rn /. The latter inequality implies kf kCs  C kf kC s since also k'0 .Dx /f k1  C kf k1 . Conversely, let f 2 L1 .Rn / be such that kf kCs < 1. Now, if jyj  1, X X .fj .x  y/  fj .x// C .fj .x  y/  fj .x//; f .x  y/  f .x/ D 2j jyj1

2j >jyj1

where fj D 'j .Dx /f . In order to estimate the first sum, we use the mean value theorem to conclude that jfj .x  y/  fj .x/j  jyjkrfj k1

for all x; y 2 Rn :

(6.9)

Moreover, since @xk fj D @xk 'j 1 .Dx /fj C @xk 'j .Dx /fj C @xk 'j C1 .Dx /fj ; due to (6.6) and k@xk 'l .Dx /gkL1 .Rn /  k@xk 'Ll kL1 .Rn / kgkL1 .Rn /  C 2l kgkL1 .Rn / for general l 2 N0 , g 2 L1 .Rn /, we obtain X X sup jfj .x  y/  fj .y/j  C x2Rn

2j jyj1

jyjkrfj k1

2j jyj1

 C jyj

X

2j jyj1

2j.1s/ kf kCs  C jyjs kf kCs :

153

Section 6.3 Bessel Potential and Besov Spaces – Definitions and Basic Properties

The second sum can be simply estimated by X X ˇ ˇ ˇfj .x  y/  fj .y/ˇ  2 sup x2Rn

2j >jyj1

kfj k1

2j >jyj1

 2kf kCs

X

2js D C jyjs kf kCs

2j >jyj1

Altogether kf kC s  C kf kCs . Remark 6.2. Because of (6.7) and (2.5), we get k'j .Dx /f kLp .Rn /  k'Lj kL1 .Rn / kf kLp .Rn /  C kf kLp .Rn / ; j

kr'j .Dx /f kLp .Rn /  kr 'Lj kL1 .Rn / kf kLp .Rn /  C 2 kf kLp .Rn /

(6.10) (6.11)

for any f 2 Lp .Rn /, 1  p  1, j 2 N0 .

6.3 Bessel Potential and Besov Spaces – Definitions and Basic Properties Theorem 6.1 gives a motivation for the following definition of the Besov space s .Rn /. Bpq s .Rn / is defined Definition 6.3. Let s 2 R; 1  p; q  1. Then the Besov space Bpq by s s Bpq .Rn / WD ¹f 2 S 0 .Rn / W kf kBpq .Rn / < 1º;

where s kf kBpq .Rn / WD

1 X

q

2jsq k'j .Dx /f kLp .Rn /

q1

if q < 1;

j D0 js s kf kBpq .Rn / WD sup 2 k'j .Dx /f kLp .Rn / j 2N0

if q D 1:

s Here the exponent s is the order of Bpq .Rn /, p is called integration exponent, and q is called summation exponent.

Remark 6.4. 1. The exponents s and p play the same role as for the Bessel potential space Hps .Rn / and the Sobolev space Wps .Rn / (if s 2 N0 ). The third s .Rn / on a finer scale exponent q measures the regularity of a function f 2 Bpq than s as will be seen below.

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Chapter 6 Introduction to Besov and Bessel Potential Spaces

s 2. Because of Theorem 6.1, C s .Rn / D B11 .Rn / for 0 < s < 1. More generally, s Cs .Rn / WD B11 .Rn /, s > 0, are called Hölder–Zygmund spaces. s .Rn / if and only if 3. Note that f 2 Bpq

.'j .Dx /f /j 2N0 2 `sq .N0 I Lp .Rn //: Here `sq .MI X/, M  Z, is the space of all X-valued sequences x D .xj /j 2M such that ´P 1 . j 2M .2js kxj kX /q / q if q < 1; kxk`sq .MIX/ WD if q D 1: supj 2M 2js kxj kX Moreover, we set `q .MI X/ WD `0q .MI X/. Of course .xj /j 2M 2 `sq .MI X/ if and only if .2js xj /j 2M 2 `q .MI X/. 4. Using Plancherel’s theorem, it is not difficult to show that s B22 .Rn / D H2s .Rn /:

The proof is left to the reader as an exercise, cf. Exercise 6.24. – But the statement will also follow from Corollary 6.13 below. Some simple relations between the Besov spaces are summarized in the following: Lemma 6.5. Let s 2 R, 1  p; q1 ; q2  1, and let " > 0. Then s s Bpq .Rn / ,! Bpq .Rn / if q1  q2 ; 1 2

sC" s Bp1 .Rn / ,! Bp1 .Rn /:

Proof. First of all, we have 1 X

q kaj kX2



 sup

j 2N0

j D0

for every a 2

`q1 .N

0 I X/,

q q  kaj kX2 1

1 X j D0

q

q

kaj kX1  kak`q21 .N0 IX/ < 1

if 1  q1  q2 < 1 which implies

`q1 .N0 I X/ ,! `q2 .N0 I X/; `sq1 .N0 I X/ ,! `sq2 .N0 I X/

(6.12)

if 1  q1  q2  1. This yields the first embedding in the statement of the lemma. s The second embedding follows from `sC" 1 .N0 I X/ ,! `1 .N0 I X/ because of k.aj /j 2N0 k`s1 .N0 IX/ D 

1 X

2sj kaj kX

j D0 1 X j D0

for all .aj /j 2N0 2 `sC" 1 .N0 /.

2"j



sup 2.sC"/j kaj kX D C" k.aj /j 2N0 k`sC" .N

j 2N0

1

0 IX/

155

Section 6.3 Bessel Potential and Besov Spaces – Definitions and Basic Properties

Remark 6.6. The lemma shows that q measures regularity of f on a finer scale s .Rn / ,! B s 0 .Rn / with arbitrary 1  than s, meaning that, if s > s 0 , then Bpq pq2 1 q1 ; q2  1. Here we note that sequences in `q1 .N0 / have to “decay faster to zero” than sequences in `q2 .N0 / in some sense if q1 < q2 . Recall that the Bessel potential spaces Hps .Rn /, s 2 R, 1 < p < 1, are defined by ® ¯ Hps .Rn / WD f 2 S 0 .Rn / W hDx is f 2 Lp .Rn / kf kHps .Rn / WD khDx is f kLp .Rn / ; 1

where hi WD .1 C jj2 / 2 and hDx is f D F 1 Œhis fO

for all f 2 S 0 .Rn /:

Moreover, we will often use j@˛ his j  Cs;˛ .1 C jj/sj˛j

(6.13)

for all ˛ 2 N0n and some Cs;˛ > 0, cf. (2.13). Remark 6.7. By definition hDx is W Hps .Rn / ! Lp .Rn / is a linear isomorphism with inverse hDx is . Since S.Rn / is dense in Lp .Rn / and hDx is W S.Rn / ! S.Rn /, S.Rn / is dense in Hps .Rn / for any s 2 R, 1 < p < 1. As a consequence of the Mikhlin multiplier theorem we obtain: Theorem 6.8. Let m 2 N0 and let 1 < p < 1. Then Hpm .Rn / D Wpm .Rn / with equivalent norms. Proof. We first prove the inclusion Hpm .Rn /  Wpm .Rn /. Let f 2 S.Rn /. Then   ˇ ˇ 1 ˇ O 1 .i / m O hi f ./ : @x f D F Œ.i / f ./ D F him Hence, in order to obtain k@ˇx f kLp .Rn /  Cp khDx im f kLp .Rn / D Cp kf kHpm .Rn /

(6.14)

ˇ

for all  2 Rn . for ˇ 2 N0n with jˇj  m, we apply Theorem 4.23 to mˇ ./ D .i/ him Therefore we have to verify (4.28) for mˇ . To this end, we use (6.13) and j@˛ .i /ˇ j  C˛;ˇ jjjˇ jj˛j

for all  ¤ 0; ˛ 2 N0n :

(6.15)

Moreover, .1 C jj/mj˛j  jjjˇ jj˛j if jˇj  m. Therefore j@˛ mˇ ./j  C˛;ˇ jjj˛j

for all  ¤ 0; ˛ 2 N0n

follows from (6.15), (6.13), and the following claim:

(6.16)

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Chapter 6 Introduction to Besov and Bessel Potential Spaces

Claim. Let s1 ; s2 2 R, N 2 N and let m1 ; m2 W Rn n ¹0º ! C be N -times continuously differentiable satisfying j@˛ mj ./j  C jjsj j˛j

for all  ¤ 0; j˛j  N; j D 1; 2 0

for some C > 0. Then there is some C > 0 such that j@˛ .m1 ./m2 .//j  C 0 jjs1 Cs2 j˛j

(6.17)

for all j˛j  N and  ¤ 0. Proof of Claim. The claim follows directly from the Leibniz formula. Because of (6.16), the conditions of Theorem 4.23 are satisfied and (6.14) follows for all jˇj  m, which proves Hpm .Rn / ,! Wpm .Rn / since S.Rn / is dense in Hpm .Rn /. Hence it remains to prove Wpm .Rn /  Hpm .Rn / with continuous embedding. If m D 2k, k 2 N0 , is even, then him D .1 C jj2 /k is a polynomial of degree m. Therefore hDx im is a differential operator of order m and X khDx im f kLp .Rn /  C k@˛x f kLp .Rn / for all f 2 Wpm .Rn /; j˛jm

which proves the embedding in this case. If m D 2k C 1, k 2 N0 , is odd, then   n n X X j2 j 1 1 m m 2k hi D hi hi C hi2k j ; C D hi2 hi2 hi hi j D1

j D1

where hi2k and hi2k j are polynomials of degree at most 2k C 1. Hence khDx im f kLp .Rn /  C

n X X

kmj .Dx /@˛x f kLp .Rn /

for all f 2 Wpm .Rn /;

j˛jm j D0

where m0 ./ D hi1 and mj ./ D j hi1 , j D 1; : : : ; n. Therefore it remains to verify the Mikhlin condition (4.28) for mj ./. If j D 0, then (4.28) for m./ D m0 ./ follows from (6.13) with s D 1 because of hi1j˛j  jjj˛j. If j D 1; : : : ; n, then (4.28) follows for m./ D mj ./ from (6.13) with s D 1, (6.15) with ˇ D ej , and (6.17). In order to get a sharp comparison of Besov and Bessel potential spaces we prove: Theorem 6.9. Let s 2 R, 1 < p < 1. Then there are constants c; C > 0 such that 1

X 12



22js j'j .Dx /f .x/j2

ckf kHps .Rn / 

j D0

for all f 2 Hps .Rn /.

Lp .Rn /

 C kf kHps .Rn /

Section 6.3 Bessel Potential and Besov Spaces – Definitions and Basic Properties

157

Remark 6.10. Because of the latter equivalent norm on Hps .Rn /, one defines more s .Rn /, s 2 R; 1 < p; q < 1, as generally the Triebel–Lizorkin space Fpq s s .Rn / WD ¹f 2 S 0 .Rn / W kf kFpq Fpq .Rn / < 1º; 1

 X q1



s .Rn / WD

kf kFpq 2qjs j'j .Dx /f .x/jq p n : L .R /

j D0

s .Rn /. Finally, we note that Hence Theorem 6.9 shows that Hps .Rn / D Fp2





s kf kFpq .Rn / D .'j .Dx /f /j 2N Lp .Rn I`s .N0 // : q

We refer to the monographs [37, 39, 38, 40] for results on these spaces. s Proof of Theorem 6.9. First we will show that kf kFp2 .Rn /  C kf kHps .Rn / for all s n f 2 Hp .R /. To this end we define a mapping

QW S.Rn / Lp .Rn / ! Lp .Rn I `2 .N0 // by .Qg/.x/ WD .2js 'j .Dx /hDx is g.x//j 2N0 2 `2 .N0 / for all x 2 Rn : Then O .x/ .Qg/.x/ D F71 !x Œm./g./

for all x 2 Rn ;

where m./ 2 L.C; `2 .N0 // is defined by m./a WD .2js 'j ./his /j 2N0 a

for all a 2 C;  2 Rn :

In order to show that Q extends to a bounded linear operator QW Lp .Rn / ! Lp .Rn I `2 .N0 // for all 1 < p < 1;

(6.18)

we verify the condition of Theorem 4.23: k@˛ m./k2L.C;`2 .N

0 //

D

1 X

22js j@˛ .'j ./his /j2

j D0

 C˛;s 22js hi2s2j˛j supp 'j ./  C˛;s jj2j˛j for all  ¤ 0, ˛ 2 N0n , where we have used that 2j 1  jj  2j C1 on supp 'j if j 1 and j@˛ .'j ./his /j  C˛;s hisj˛j

158

Chapter 6 Introduction to Besov and Bessel Potential Spaces

uniformly in j 2 N0 , which follows from (6.13), (6.4), and the product rule. Hence (6.18) follows and therefore s s s kf kFp2 .Rn / D kQhDx i f kLp .Rn I`q .N0 //  C khDx i f kLp .Rn / D C kf kHps .Rn /

for all f 2 Hps .Rn /. Note that we have shown that e Hps .Rn / ! Lp .Rn I `sq .N0 // QW is bounded, where e D .2js QhDx is f /j 2N0 D .'j .Dx /f /j 2N0 : Qf

(6.19)

Conversely, we define a mapping RW S.Rn I `2 .N0 // Lp .Rn I `2 .N0 // ! Lp .Rn / by .Ra/.x/ D

1 X

2js e 'j .Dx /hDx is aj .x/

for all x 2 Rn ; a 2 S.Rn I `2 .N0 //:

j D0

Here e 'j ./ D 'j 1 ./ C 'j ./ C 'j C1./ for all  2 Rn , j 2 N0 , where '1  0. 'j ./ D 1 on supp 'j . Then – Note that 'Qj ./'j ./ D 'j ./ for all  2 Rn since e .Ra/.x/ D F71 Oj ./ !x Œm./a where m./ 2 L.`2 .N0 /; C/ is defined by m./a D

1 X

2js e 'j ./his aj

for all .aj /j 2N0 2 `2 .N0 /:

j D0

Similarly, as before k@˛ m./k2L.`2 .N0 /;C/ 

1 X

22js j@˛ .'Qj ./his /j2

j D0

 C˛;s

1 X

22js hi2s2j˛j ¹2j 2jj2j C2 º C C ¹jj8º

j D0



0 C˛;s jj2j˛j ;

where we have used that for each  2 Rn at most four terms in the sum above are 'j  ¹2j 2  jj  2j C2 º. Hence, non-zero and that 22js  C his on supp e applying Theorem 4.23 once more, we obtain that R extends to a bounded operator RW Lp .Rn I `2 .N0 // ! Lp .Rn / for all 1 < p < 1:

159

Section 6.3 Bessel Potential and Besov Spaces – Definitions and Basic Properties

Now we apply R to aj D 2js 'j .Dx /f , j 2 N0 . Then Ra D

1 X

2js e 'j .Dx /hDx is 2js 'j .Dx /f D hDx is f

j D0

since

P1

'j .Dx /'j .Dx /g j D0 e

D

P1

j D0 'j .Dx /g

D g for all g 2 S.Rn /. Thus

kf kHps .Rn / D khDx is f kLp .Rn / D kRakLp .Rn /  C kakLp .Rn I`2 .N0 // s D C k.2js 'j .Dx /f /j 2N0 kLp .Rn I`2 .N0 // D C kf kFp2 .Rn /

for all f 2 Hps .Rn /, which proves the lemma. – Finally, we note that the previous estimates imply that e Lp .Rn I `s2 .N0 // ! Hps .Rn / RW is bounded, where e j /j 2N WD hDx is R.2js aj /j 2N D R.a 0 0

1 X

e 'j .Dx /aj

(6.20)

j D0

and therefore RQ QQ D I on Hps .Rn /. Remark 6.11. We note that the last proof shows that Hps .Rn / is a retract of the space Lp .Rn I `s2 .N0 // due to the following definition: Definition 6.12. A Banach space X is called a retract of a Banach space Y if there are bounded linear operators RW Y ! X and QW X ! Y such that RQ D idX . e Q e are defined by (6.19) and (6.20), then R eQ e D I on Hps .Rn / as seen above. If R, – We note that the mappings are independent of p and s. – Moreover, using the same s .Rn / is a retract of `q .N I Lp .Rn //. e and Q e it is easy to show that Bpq mappings R 0 This observation is important for characterizing the so-called real and complex inters polation spaces of Hps .Rn / and Bpq .Rn /, cf. e.g. [3] or [37]. Corollary 6.13. Let 1 < p < 1, s 2 R. Then s s .Rn / ,!Hps .Rn / ,! Bp2 .Rn / Bpp

if 1 < p  2;

(6.21)

s .Rn / Bp2

if 2  p < 1:

(6.22)

,!Hps .Rn /

,!

s Bpp .Rn /

s .Rn / for all s 2 R. In particular, H2s .Rn / D B22

160

Chapter 6 Introduction to Besov and Bessel Potential Spaces

Proof. The statements follows from Theorem 6.9 and the embeddings `q .N0 I Lp .Rn // ,! Lp .Rn I `q .N0 // if 1  q  p  1 p

q

n

q

n

(6.23)

p

L .N0 I ` .R // ,! ` .R I L .N0 // if 1  p  q  1;

(6.24)

as well as from (6.12). Here (6.23) follows from Z k.fj /j 2N0 kLp .Rn I`q .N0 // D 

1 X

jfj .x/j

q

pq

Rn j D0 1 X q1



jfj jq p L q .Rn / j D0

dx

 p1

1

X

1

q D

jfj jq p j D0

D

1 X

q

kfj kLp .Rn /

L q .Rn /

q1

;

j D0

where we have used the triangle inequality for series. (Alternatively one uses the more general integral Minkowski’s inequality). The inequality (6.24) is proved analogously. Combining Corollary 6.13 and Lemma 6.5, one easily derives: Corollary 6.14. For every s 2 R, " > 0, 1 < p < 1, and 1  q  1 we have s .Rn / ,! Hps" .Rn /: HpsC" .Rn / ,! Bpq

6.4 Sobolev Embeddings We start with a Sobolev-type embedding theorem for Besov spaces. Theorem 6.15. Let s; s1 2 R with s  s1 , 1  q1  q  1, and 1  p1  p  1 such that s

n n  s1  : p p1

Then s .Rn / Bps11 q1 .Rn / ,! Bpq

(6.25)

Proof. As first step we prove that n

n

k'k .Dx /f kLq .Rn /  C 2k. p  q / kf kLp .Rn /

(6.26)

for all 1  p  q  1, k 2 N0 , f 2 Lp .Rn /. First of all, we note that k'k .Dx /f kLp .Rn / D k'Lk  f kLp .Rn /  k'Lk kL1 .Rn / kf kLp .Rn /  C kf kLp .Rn /

161

Section 6.4 Sobolev Embeddings

for all f 2 Lp .Rn /, 1  p  1, k 2 N0 due to (6.7), which implies (6.26) in the case p D q. Moreover, Z k'k .Dx /f kL1 .Rn /  sup j'L k .x  y/jjf .y/j dy x2Rn

Rn

n

 k'Lk kLp0 .Rn / kf kLp .Rn /  C 2k p kf kLp .Rn / because of n

k'Lk kLp0 .Rn / D 2.k1/ p k'L 1 kLp0 .Rn /

for all k 2 N:

This implies (6.26) in the case q D 1. Now (6.26) in the general case follows from 1 p

p

kf kLq .Rn /  kf kL1q.Rn / kf kLq p .Rn / for 1  p  q  1 and (6.26) in the cases q D p; 1. Next we use that 'k .Dx /f D 'Qk .Dx /'k .Dx /f with 'Qk .Dx / D 'k1 .Dx / C 'k .Dx / C 'kC1 .Dx /, '1 .Dx / D 0. Here 'Qk .Dx /f satisfies the same estimates above as 'k .Dx /. Moreover, let f 2 Bps11 q1 .Rn / with p; p1 ; s; s1 ; q; q1 as in the theorem. Then 2sk k'k .Dx /f kLp .Rn / D 2sk k'Qk .Dx /'k .Dx /f kLp .Rn / n k. pn  p /

 C 2sk 2

1

k'k .Dx /f kLp1 .Rn /  C 2s1 k k'k .Dx /f kLp1 .Rn /

for all k 2 N0 . Taking the `q .N0 / norm with respect to k 2 N0 and using `q1 .N0 / ,! `q .N0 / finishes the proof. Corollary 6.16. Let s; s1 2 R with s  s1 and 1  p1  p  1 such that s

n n : < s1  p p1

Then Hps11 .Rn / ,! Hps .Rn /:

(6.27)

Proof. See Exercise 6.27. We note that (6.27) also holds if s  pn D s1  pn1 , cf. [3, Theorem 6.51]. The proof in this case is more involved.

162

Chapter 6 Introduction to Besov and Bessel Potential Spaces

6.5 Equivalent Norms The following theorem is the direct generalization of Theorem 6.1 for general Besov s .Rn / with 0 < s < 1. spaces Bpq Theorem 6.17. Let 0 < s < 1 and let 1  p; q  1. Then there are constants c; C > 0 (depending only on s; p; q; n) such that Z s ckf kBpq .Rn /  kf kLp .Rn / C

1

0

!p .t I f /q dt t sq t

 q1

s  C kf kBpq .Rn /

(6.28)

if q < 1 and s ckf kBpq .Rn /  kf kLp .Rn / C sup

t>0

!p .t I f / s  C kf kBpq .Rn / ts

(6.29)

if q D 1, where !p .t I f / WD sup kf .  C h/  f kLp .Rn / jhjt

is the Lp -modulus of continuity of f . Remark 6.18. We refer to [3, Theorem 6.2.5] for a more general statement in the case s > 0. See also Exercise 6.29. Proof of Theorem 6.17. We will only prove the case q < 1 since the proof in the case q D 1 is a simple variant of the proof of Theorem 6.1. First of all, since t 7! !p .t I f / is a monotone increasing function and t is proportional to 2j on Œ2j C1 ; 2j , Z

1 0

1 X !p .t I f /q dt  C 2sj q !p .2j I f /q t sq t j D0

and 1 X

2sj q !p .2j I f /q  C

j D0

Moreover, Z 1 1

!p .t I f /q dt q  2q kf kLp .Rn / t sq t

Z

Z

2 0

1 1

!p .t I f /q dt : t sq t

t sq1 dt D Cq kf kLp .Rn / q

163

Section 6.5 Equivalent Norms

since !p .t I f /  2kf kLp .Rn / . Hence we can replace the middle term in (6.28) by kf kLp .Rn / C

1 X

2sj q !p .2j I f /q

q1

:

j D0 s .Rn / we denote f D First we prove the second inequality in (6.28). For f 2 Bpq k 'k .Dx /f . Then

kfk .: C h/  fk kLp .Rn /  jhjkrfk kLp .Rn / due to (6.9) and therefore !p .t I fk /  t krfk kLp .Rn / D t kr'k .Dx /e ' k .Dx /fk kLp .Rn /  C t 2k ke ' k .Dx /fk kLp .Rn / because of (6.11), where e ' k .Dx / D 'k1 .Dx / C 'k .Dx / C 'kC1 .Dx /, k P 2 N0 , and '1 .Dx / D 0. On the other hand, !p .t; fk /  2kfk kLp .Rn / and f D 1 kD0 fk . Therefore 2sj !p .2j I f /  C

1 X

2sj min.1; 2j Ck /ke ' k .Dx /fk kLp .Rn /



j D0

 C0

1 X

2s.j k/ min.1; 2j Ck /2sk k'k .Dx /fk kLp .Rn / :

j D0

Now, defining aj D C 0 2sj min.1; 2j / for j 2 Z and bj D 2sj k'k .Dx /fj kLp .Rn / if j 0 and bj D 0 else, we see that 2sj !p .2j I f /  .a  b/j , where X

.a  b/j D

aj k bk

for all j 2 Z

k2Z

is the convolution of two sequences. Hence 1 X

2sj q !p .2j I f /q

q1

s  ka  bk`q .Z/  kak`1 .Z/ kbk`q .Z/  C kf kBpq .Rn / ;

j D0

where a 2 `1 .Z/ since s 2 .0; 1/. Here we have used the discrete convolution inequality ka  bk`r .Z/  kak`1 .Z/ kbk`r .Z/ , cf. (A.5). In order to prove the first inequality in (6.28), we use that Z .f .x  2j z/  f .x// .z/ dz; 'j .Dx /f D Rn

164

Chapter 6 Introduction to Besov and Bessel Potential Spaces

cf. (6.8). Therefore Z k'j .Dx /f kLp .Rn /  

n ZR

Rn

kf .   2j z/  f kLp .Rn / j .z/j dz !p .2j jzjI f /j .z/j dz:

and 1 X

q 2sj q k'j .Dx /f kL p .Rn /

j D1

q1

Z 

1 X Rn j D1

2sj q !p .2j jzjI f /q

Z

q1

j .z/j dz

1 !p .t jzjI f /q dt q s jzj j .z/j dz C .jzjt /sq t 0 Rn Z 1 1 Z !p .t I f /q dt q jzjs j .z/j dz DC t sq t 0 Rn 1 Z 1 !p .t I f /q dt q D Cs ; t sq t 0 Z

1

P where we can estimate j1D0 2sj q !p .2j jzjI f /q by the corresponding integral by the same arguments as in the beginning of the proof. Moreover, we have used that Z 1 Z 1 dt dt f .at / f .t / D for all a > 0: t t 0 0 Since also k'0 .Dx /f kLp .Rn /  C kf kLp .Rn / , the proof is finished.

6.6 Pseudodifferential Operators on Besov Spaces Finally, we show that pseudodifferential operators have natural mapping properties between Besov spaces. m .Rn  Rn /, m 2 R, and let s 2 R, 1  q; r  1. Then Theorem 6.19. Let p 2 S1;0 sCm n s .Rn / is a bounded linear operator. p.x; Dx /W Bqr .R / ! Bqr

The proof is based on: m .Rn Rn /, m 2 R, and let p .x; D / D p.x; D /' .D /, Lemma 6.20. Let p 2 S1;0 j x x j x where .'j /j 2N0 is defined as above. Then

kpj .x; Dx /kL.Lq .Rn //  C 2j m ;

j 2 N0 ;

for every 1  q  1, where C does not dependent on j .

(6.30)

165

Section 6.6 Pseudodifferential Operators on Besov Spaces

Proof. As seen in Section 5.4, pj .x; Dx /f .x/ D

Z Rn

for allx 2 Rn ;

kj .x; x  y/f .y/dy

where the kj satisfy j@ˇx @˛z kj .x; z/j  C˛;ˇ ;M jzjM 2j.nCmM Cj˛j/

for all z ¤ 0; j 2 N0

(6.31)

and all ˛; ˇ 2 N0n, M 2 N0 , where C˛;ˇ ;M does not depend on j 2 N0 and z ¤ 0, cf. (5.17). According to these estimates Z  Z Z j.nCm/ n1 j.m1/ jkj .x; z/jdz  C 2 dz C jzj 2 dz : Rn

jzj2j

jzj>2j

The first part of the estimate comes from (6.31) for M D 0, and the second part comes from (6.31) for M D n C 1. Hence we get by a simple calculation, using e.g. Theorem A.8, Z jkj .x; z/jdz  C 2j m ; Rn

which proves (6.30) since

Z



kj .  ;   y/f .y/dy



Rn

Lq .Rn /

Z  sup

x2Rn

Rn

jkj .x; z/jdzkf kLq .Rn /

due to (2.5). Remark 6.21. If pj .Dx ; x/ D 'j .Dx /p.Dx ; x/ is the corresponding operator in y-form, then Z Z Z i.xy/ e pj .y; /f .y/d¯ dy D kj .y; x  y/f .y/dy; pj .Dx ; x/f D Rn

Rn

Rn

where kj .y; z/ D F71 Œp .y; /.z/ is the same function as for the operator in x!z j form. Moreover, kpj .Dx ; x/kL.Lq .Rn //  C 2j m ;

j 2 N0

(6.32)

holds for 1  q  1. This inequality is proved in precisely the same way as (6.30). m .Rn  Rn /, m 2 R, and let p .x; D / be defined as Lemma 6.22. Let p 2 S1;0 x k above. Then for every l 2 N there is some Ck > 0 such that

k'j .Dx /pk .x; Dx /kL.Lq .Rn //  Cl 2min.j;k/m 2jj kjl and all 1  q  1.

for all j; k 2 N0 ; (6.33)

166

Chapter 6 Introduction to Besov and Bessel Potential Spaces

Proof. Similarly as in the proof of Lemma 2.41,  l n X X 1 j 1D C  D p .˛/ ./ ˛ ; j 1 C jj2 1 C jj2 j D1

l .Rn  Rn /. Hence where p .˛/ 2 S1;0

'j .Dx /pk .x; Dx / D 'j .Dx / D

X

(6.34)

j˛jl

X

p.˛/ .Dx /Dx˛ pk .x; Dx /

j˛jl

p

.˛/

.Dx /'j .Dx /Dx˛ pk .x; Dx /

j˛jl l .Rn  Rn / and since 'j .Dx / and p .˛/ .Dx / commute. Since p .˛/ ./ 2 S1;0

Dx˛ pk .x; Dx / D q˛ .x; Dx /'k .Dx / DW q˛;k .x; Dx / mCj˛j

for some q˛ 2 S1;0

.Rn  Rn /, we can now use (6.30) for pj.˛/ .Dx / D p .˛/ .Dx /'j .Dx /

and q˛;k .x; Dx / to get k'j .Dx /pk .x; Dx /kL.Lq .Rn // X  kp .˛/ .Dx /'j .Dx /kL.Lq .Rn // kq˛;k .x; Dx /kL.Lq .Rn //  C 2j l  2k.lCm/ ; j˛jl

which proves (6.33) for the case k  j . In order to prove the case k > j , we use some kind of symmetry in k and j . First of all, we have p.x; Dx / D .p.x; Dx / / D .p  .x; Dx // D q.Dx ; x/ for q.y; / D p .y; /. Hence 'j .Dx /pk .x; Dx / D 'j .Dx /q.Dx ; x/'k .Dx / X qj .Dx ; x/Dx˛ p.˛/ .Dx /'k .Dx /: D j˛jl

Now using (6.30), (6.32) for qj .Dx ; x/Dx˛ , we conclude in the same way as before that k'j .Dx /pk .x; Dx /kL.Lq .Rn // X  kqj .Dx ; x/Dx˛ kL.Lq .Rn // kp .˛/ .Dx /'k .Dx /kL.Lq .Rn // j˛jl

 C 2kl  2j.lCm/ ; which proves (6.33) for the case k > j .

167

Section 6.6 Pseudodifferential Operators on Besov Spaces

Proof of Theorem 6.19. First of all, f D

1 X

in S 0 .Rn /;

fj

j D0

where fj D 'j .Dx /f , f 2 S 0 .Rn /. Since supp 'j \ supp 'k D ; for jj  kj > 1, p.x; Dx /f D D

1 X kD0 1 X

pk .x; Dx /f D

1 X

pk .x; Dx /.fk1 C fk C fkC1 /

kD0

pk .x; Dx /fQk ;

kD0 sCm .Rn /, then where fQk D fk1 C fk C fkC1 and we have set f1  0. If f 2 Bqr

k.2.sCm/k fQk /k2N0 k`r .N0 ;Lq .Rn // D k.2.sCm/k .fk1 C fk C fkC1 //k2N0 k`r .N0 ;Lq .Rn //  C k.2.sCm/k 'k .Dx /f /k2N0 k`r .N0 ;Lq .Rn // D C kf kB sCm : qr

Now using (6.33), 2sj k'j .Dx /pk .x; Dx /fQk kLq  C 2sj Cmin.j;k/mjj kjlk.sCm/ 2k.sCm/ kfQk kLq ; where 2sj Cmin.j;k/mjj kjlk.sCm/ ´ 2j.sl/Ck.ls/ D 2jj kj.ls/ D k.mCsCl/Cj.mCsCl/ D 2jj kj.mCsCl/ 2

if j k; if j < k:

Choosing l 2 N so large that l  s 1 and m C s C l 1, 2sj k'j .Dx /p.x; Dx /f kLq  C

1 X

2jj kj 2k.sCm/ kfQk kLq :

kD0

Now let aj D 2jj j , bj D 2.sCm/j kfQj kLq for j 2 N0 and aj D bj D 0 for j 2 Z n N0 . Then s  C k.aj /j 2Z  .bj /j 2Z k`r .Z/ kp.x; Dx /f kBqr

 C k.aj /j 2Z k`1 .Z/ k.bj /j 2Z k`r .Z/  C k.2.sCm/j fQj /j 2N0 k`r .N0 ;Lq .Rn //  C kf kB sCm ; qr

168

Chapter 6 Introduction to Besov and Bessel Potential Spaces

where .cj /j 2Z D .aj /j 2Z  .bj /j 2Z is the convolution of .aj /j 2Z and .bj /j 2Z defined by cj D

1 X

aj k bk

for all j 2 Z

kD1

and we have used (A.5). As an application we obtain a result on elliptic regularity for the Besov spaces s .Rn /: Bpq m .Rn  Rn / be an elliptic symbol and f 2 B s .Rn /, Corollary 6.23. Let p 2 S1;0 pq S 1 r s; m 2 R, 1  p; q  1. Then, if u 2 Bp1 .Rn / WD r 2R Bp1 .Rn / is a solution of the pseudodifferential equation

p.x; Dx /u D f; sCm then u 2 Bpq .Rn /.

Proof. The proof is the similar to the proof of Corollary 6.23 and left to the reader, cf. Exercise 6.30.

6.7 Final Remarks and Exercises 6.7.1 Further Reading The systematic study of Bessel potential and Besov spaces is strongly connected to the interpolation theory of Banach spaces. For a first introduction to interpolation theory we recommend the book by Lunardi [25]. A classical introduction to interpolation theory was written by Bergh and Löfström [3], where also Bessel potential and Besov spaces are studied in more detail. The results of this chapter are partly based in [3, Chapter 6]. Much more comprehensive results on these and further function space can be found in the monographs by Triebel, e.g. [37], which is a classical reference, and [39, 38, 40].

6.7.2 Exercises Exercise 6.24. Prove that for any s 2 R s B22 .Rn / D H2s .Rn /:

Hint. Use Plancherel’s theorem.

169

Section 6.7 Final Remarks and Exercises

Exercise 6.25. Show that for any 1  p  1 n

p .Rn / ,! C 0 .Rn / Bp1

with continuous embedding. n

j Hint. Use that 'j .Dx /f D 'Lj  f , where k'Lj kLq .Rn /  C 2 q0 uniformly in j 2 Z.

Exercise 6.26. Let 1 .Rn / D Bpq

\

s 1 Bpq .Rn / and Bpq .Rn / D

s2R

[

s Bpq .Rn /;

s2R

where 1  p; q  1. Prove that 1 1 Bpq .Rn / D Bpq .Rn /; 1 2

1 1 Bpq .Rn / D Bpq .Rn / 1 2

for any 1  p; q1 ; q2  1. Exercise 6.27. Let s; s1 2 R with s  s1 and 1  p1  p  1 such that s

n n < s1  : p p1

Prove that Hps11 .Rn / ,! Hps .Rn /: Hint. Use the relations between Besov and Bessel potential spaces.

Exercise 6.28. Let s 2 R, 1 < p < 1, 1  q  1 and let k 2 N0 . Prove that s .Rn / f 2 Bpq

,

sk @˛x f 2 Bpq .Rn / for all j˛j  k;

f 2 Hps .Rn /

,

@˛x f 2 Hpsk .Rn / for all j˛j  k:

s .Rn / and H s .Rn / are equivalent to Moreover, prove that the norms on Bpq p

X j˛jk

k@˛x f kBpq sk .Rn / ;

X j˛jk

k@˛x f kHpsk .Rn / ; resp:

170

Chapter 6 Introduction to Besov and Bessel Potential Spaces

Exercise 6.29. Let 0 < s < 1, k 2 N0 , and let 1 < p < 1; 1  q  1. Prove that there are constants c; C (depending only on s; p; q; n; k) such that ckf kB sCk .Rn /  pq

X

k@˛x f kLp .Rn /

C

j˛jk

X Z j˛jDk

1

0

!p .t I @˛x f /q dt q1 t sq t

 C kf kB sCk .Rn / pq

if q < 1 and ckf kB sCk .Rn /  pq

X

k@˛x f kLp .Rn / C

j˛jk

sup t>0;j˛jDk

!p .t I @˛x f / ts

 C kf kB sCk .Rn / pq

if q D 1, where !p .t I f / D supjhjt kf .  C h/  f kLp .Rn / : Exercise 6.30. Prove Corollary 6.23.

Chapter 7

Applications to Elliptic and Parabolic Equations Summary In this chapter we present several applications of the results of the previous chapter. We discuss applications of the Mikhlin multiplier theorem in the scalar and Hilbertspace-valued version and applications of the theory of pseudodifferential operators. On one hand we study unique solvability of resolvent equations for parameter-elliptic differential operators in different situations and obtain suitable estimates of the resolvents. These estimates are important for applications to the corresponding parabolic equation, which are also briefly discussed. On the other hand we discuss results on elliptic regularity for elliptic pseudodifferential equations. Moreover, an application to Hilbert-space-valued Bessel potential spaces is also presented.

Learning targets  Learn about typical applications of the Mikhlin multiplier theorem and results on mapping properties of pseudodifferential operators to resolvent equations.  Understand some relations between resolvent estimates, the vector-valued Mikhlin multiplier theorem and results on maximal regularity of abstract parabolic equations.  See how parametrices and results on boundedness of pseudodifferential operator between function spaces can be used to obtain results on elliptic regularity.

7.1 Applications of the Mikhlin Multiplier Theorem The applications of this section are based on the Theorem 4.23. Further results from the Chapters 4 and 5 are not needed.

7.1.1 Resolvent of the Laplace Operator In the section we study the partial differential equation .  /u D f where D

Pn

2 j D1 @xj

in Rn ;

(7.1)

is the Laplace operator and  2 C. For the following we

i

172

Chapter 7 Applications to Elliptic and Parabolic Equations

†

ı

Figure 7.1. Sector †ı with opening angle ı 2 .0; /.

define open sectors †ı , 0 < ı < , by †ı D ¹z 2 C n ¹0º W jarg zj < ıº ; cf. Figure 7.1. For applications to the corresponding parabolic equation @ t u.x; t /  u.x; t / D f .x; t / u.x; 0/ D u0 .x/

for all x 2 Rn ; t > 0; for all x 2 Rn

unique solvability of (7.1) for  2 C n .1; 0 together with the estimates shown below are important, cf. Sections 7.2.1 and 7.3.3. The main result of this section is: Lemma 7.1. Let 0 < ı <  and 1 < p < 1. Then .  /W Wp2 .Rn / ! Lp .Rn / is invertible and .  /1 2 L.Lp .Rn // for every  2 †ı . Moreover, there is some constant Cı;p > 0 such that Cı;p ; jj  Cı;p

k.  /1 kL.Lp .Rn //  kr 2 .  /1 kL.Lp .Rn //

(7.2) (7.3)

uniformly in  2 †ı . Proof. If f 2 S.Rn / and  2 †ı , then a simple calculation shows  O  f ./ 1 1 D m .Dx /f for all f 2 S.Rn /; .  / f D F  C jj2 where m ./ D . C jj2 /1 . In order to estimate the Lp .Rn /-norm of .  /1 f we estimate @˛ m ./. To this end we define m .; r/ D .e i r 2 C jj2 /1

for all  2 Œı; ı; r 2 R;  2 Rn :

Section 7.1 Applications of the Mikhlin Multiplier Theorem

173

Then m .; r/ is a smooth and homogeneous function of degree 2 with respect to .; r/ 2 RnC1 n ¹0º. Therefore for every ˛ 2 N0n the function @˛ m .; r/ is homogeneous of degree 2  j˛j and j@˛ m .; r/j  Cı;˛ j.; r/j2j˛j

for all .; r/ ¤ 0;  2 Œı; ı;

where Cı;˛ D

sup j.;r /jD1; 2Œı;ı

j@˛ m .; r/j < 1:

Since m ./ D m .; r/ if  D e i r 2 2 †ı , we conclude that for every ˛ 2 N0n there is some Cı;˛ > 0 such that 1

j@˛ m ./j  Cı;˛ .jj 2 C jj/2j˛j  Cı;˛ jj1 jjj˛j uniformly in  2 †ı ,  2 Rn . Hence m ./ satisfies the condition of the Mikhlin multiplier theorem (4.28) with A D maxj˛jN C2 Cı;˛ jj1 . Therefore km .Dx /kL.Lp .Rn // 

Cı;p jj

uniformly in  2 †ı

because of Proposition 4.28, where 1 < p < 1 is arbitrary. This proves (7.2). In order to show (7.3) for m .Dx / instead of .  /1 one uses that   j k O 1 f ./ D m ;j k .Dx /f; @xj @xk m .Dx /f D F  C jj2  

j k where m ;j k ./ D  Cjj 2 . Then

m ;j k .; r/ D 

j k i e r 2 C jj2

is a homogeneous function of degree 0, which is smooth in .; r/ ¤ 0. Therefore by the same arguments as before 1

j@˛ m ;j k ./j  Cı;˛ .jj 2 C jj/j˛j  Cı;˛ jjj˛j uniformly in 2†ı ;  2Rn. Hence m ;j k satisfies (4.28) with ADmaxj˛jN C2 Cı;˛ . Thus m ;j k .Dx / D @xk @xj m .Dx / extends to a bounded linear operator on Lp .Rn /. In the same way one shows that also @xj m .Dx / extends to a bounded linear operator on Lp .Rn / for every j D 1; : : : ; n. This implies that m .Dx /W Lp .Rn / ! Wp2 .Rn / is bounded and kr 2 m .Dx /kLp .Rn /  Cp;ı

for all  2 †ı :

174

Chapter 7 Applications to Elliptic and Parabolic Equations

Moreover, for every f 2 S.Rn / m .Dx /.  /f D .  /m .Dx /f D f and therefore .  /m .Dx /u D u for all u 2 Wp2 .Rn / since S.Rn / is dense in Wp2 .Rn / D Hp2 .Rn /. Hence . /W Wp2 .Rn / ! Lp .Rn / is onto. Furthermore, if . /u D 0 for some u 2 Wp2 .Rn /, then for every g 2 S.Rn / Z Z Z 0D ..  /u/m .Dx /g dx D u.  /.m .Dx /g/ dx D ug dx: Rn

Rn

Rn

Thus u  0, which shows that .  / is injective, and .  /1 D m .Dx /.

7.1.2 Spectrum of Multiplier Operators with Homogeneous Symbols Let aW Rn n ¹0º ! C N N , N 2 N, be an .n C 2/-times continuously differentiable function that is homogeneous of degree ˛ > 0 such that a./ is invertible for all  ¤ 0. Moreover, set [ .a.// [ ¹0º; K WD 2Rn n¹0º

where .a.// denotes the spectrum of a./ 2 C N N Š L.C N /. Then K is a closed cone (i.e., rK D K for all r > 0) since a./ is homogeneous of degree ˛ > 0 and therefore [ [ [ r ˛ .a.// D r .a.//: KD jjD1;r 0

r 0

jjD1

Furthermore, let a.Dx /u D F 1 Œa./u./ O

for all u 2 S.Rn /:

Examples of operators satisfying the assumptions above are principal parts of parameter-elliptic differential operators with constant coefficients, which will be used in Section 7.1.3 below. Moreover, a./ D jj˛ for ˛ > 0 satisfies the assumptions above. Since the symbol of  is jj2 , a.Dx / can be considered as . /˛=2 . Theorem 7.2. Let a.Dx /, K be as above and let 1 < p < 1. Then a.Dx / extends to a bounded linear operator a.Dx /W Hp˛ .Rn /N ! Lp .Rn /N

175

Section 7.1 Applications of the Mikhlin Multiplier Theorem

for every 1 < p < 1. Moreover, the spectrum of a.Dx / as an unbounded operator on Lp .Rn /N is contained in K. Furthermore, for every closed cone K 0 with K \ K 0 D ¹0º and every 1 < p < 1 there is a constant CK 0 ;p such that k.  a.Dx //1 kL.Lp .Rn /N / 

CK 0 ;p jj

(7.4)

for all  2 K 0 n ¹0º. Finally, for every s 2 Œ0; m there is a constant CK 0 ;p;s such that k.  a.Dx //1 kL.Lp .Rn /N ;Hps .Rn /N /  CK 0 ;p;s .1 C jj/

ms m

(7.5)

for all  2 K 0 with jj 1. Remark 7.3. The existence of the resolvent for all  in a sufficiently large cone K 0 and a resolvent estimate of the form (7.4) are important to solve the associated d . We refer to Section 7.2.1 and Secparabolic equation, where  is replaced by dt tion 7.3.3. Proof. First we show that a.Dx /W Hp˛ .Rn /N ! Lp .Rn /N is a bounded operator. To this end, we use that   1 a./ ˛ hi u./ O D m.Dx /hDx i˛ f; a.Dx /u D F hi˛ where m./ D a./hi˛ . Hence it is sufficient to show that m./ satisfies (4.28). Since a./ is homogeneous of degree ˛, there is some constant C > 0 such that j@ a./j  C jj˛jˇ j ˇ

for all jˇj  n C 2;  ¤ 0:

Hence the latter estimate and ˇ j@ hi˛ j  C˛;ˇ hi˛jˇ j  C˛;ˇ jj˛jˇ j

for all  ¤ 0

cf. (2.13) with s D ˛, imply (4.28) if one applies the follow statement: Claim 1. Let s1 ; s2 2 R, N 2 N and let m1 ; m2 W Rn n ¹0º ! C be N -times continuously differentiable functions satisfying j@ mj ./j  C jjsj jˇ j ˇ

for all jˇj  N

for some C > 0. Then there is some C 0 > 0 such that j@ .m1 ./m2 .//j  C 0 jjs1 Cs2 jˇ j ˇ

for all jˇj  N . Proof of Claim 1. The claim follows directly from the Leibniz formula (A.1).

(7.6)

176

Chapter 7 Applications to Elliptic and Parabolic Equations

Thus a.Dx /W Hp˛ .Rn /N ! Lp .Rn /N is a bounded linear operator. In order to prove existence of .  a.Dx //1 and the corresponding estimates, we define m ./ D .  a.//1 for all  2 Rn ,  2 K n ¹0º. Claim 2. There is some constant CK 0 > 0 such that for all ˇ 2 N0n with jˇj  n C 2 j@ˇ .  a.//1 j  CK 0

jjjˇ j jj C jj˛

for all  2 K 0 n ¹0º;  ¤ 0:

Proof of Claim 2. The proof is done by mathematical induction. First let jˇj D 0. We define m .; r/ D .e i jrjs C a.//1 for .; r/ 2 RnC1 n ¹0º and  2 Œ;  with e i 2 K 0 . Then m .; r/ is a continuous function in .; r/ 2 RnC1 n ¹0º that is homogeneous of degree ˛. – Note that a./ has a continuous extension to  D 0 with a.0/ D 0 since a is homogeneous of degree ˛ > 0. – Hence C C  ˛ ; j.r; /j˛ jrj C jj˛

jm .; r/j 

where C D supj.;r /jD1;e i 2K 0 jm .; r/j. Putting  D ei r 2 K 0 , the claim follows in the case jˇj D 0. Next we assume that the claim is proved for jˇj  k and some k 2 N0 . Let jˇj D k C 1 and ˇ D ˇ 0 C ej for some j 2 ¹1; : : : ; nº. Then ˇ0

@ .  a.//1 D @ ˇ



.  a.//1 @j a./.  a.//1



since d .A.t //1 D A.t /1 A0 .t /A.t /1 dt

(7.7)

for every continuously differentiable mapping AW R ! C N N , which follows from differentiating I D A.t /A.t /1 . Hence X

j@ .  a.//1 j  C ˇ

j@ .  a.//1 jj@ı @j a./jj@ .  a.//1 j 



CıCDˇ 0

 C0

X CıCDˇ

jjjˇ j jjjjjıjjj1C˛  C .jj C jj˛ /2 jj C jj˛

for all  ¤ 0;  2 K 0 n ¹0º, where ; ı; 2 N0n above and we have used the statement of the claim for jˇj  k. This proves the claim.

Section 7.1 Applications of the Mikhlin Multiplier Theorem

177

Now the claim implies j@ .  a.//1 j  ˇ

CK 0 jˇ j jj jj

for all  2 K 0 n ¹0º;  ¤ 0

and all jˇj  n C 2. Hence Proposition 4.28 yields that m .Dx / extends to a bounded operator m .Dx /W Lp .Rn /N ! Lp .Rn /N for every 1 < p < 1 with operator norm estimated as in (7.4). Finally, it remains to show that m .Dx /W Lp .Rn /N ! H ˛ .Dx /N and that m .Dx / D .  a.Dx //1 . To this end we use that for every s 2 Œ0; ˛, " > 0 j@ˇ .his .  a.//1 /j X C j@ his jj@ˇ  .  a.//1 j 0ˇ

 Cs0

X 0ˇ

hisjj

jjjˇ jCjj s˛ 00  Cs;";ˇ jj ˛ jjjˇ j jj C jj˛

(7.8)

uniformly in  2 K 0 with jj " due to (2.13). Hence Theorem 4.23 implies that hDx i˛ m .Dx /W Lp .Rn /N ! Lp .Rn /N is bounded and therefore m .Dx /W Lp .Rn /N ! Hp˛ .Rn /N is bounded for all 1 < p < 1. Moreover, (7.8) and Proposition 4.28 implies (7.4) and (7.5). Finally, since .  a.Dx //m .Dx /f D F 1 Œ.  a.//.  a.//1 fO./ D f and m .Dx /.  a.Dx //f D F 1 Œ.  a.//1 .  a.//fO./ D f for every f 2 S.Rn /N and S.Rn / is dense in Hp˛ .Rn /, we obtain m .Dx / D .  a.Dx //1

for every  2 K 0 n ¹0º:

7.1.3 Spectrum of a Constant Coefficient Differential Operator In the following let a.Dx / WD

X j˛jm

a˛ Dx˛

178

Chapter 7 Applications to Elliptic and Parabolic Equations

be a (matrix-valued) differential operator with constant coefficients a˛ 2 C N N , where m; N 2 N. Moreover, let X a0 .Dx / WD a˛ Dx˛ j˛jDm

be the principal part of a.Dx / and a0 ./ D Moreover, for every 0 < ı <  let

P

j˛jDm a˛ 

˛

be the principal symbol.

†ı D ¹z 2 C n ¹0º W jarg zj < ıº as before. We assume that a.Dx / is parameter-elliptic, i.e., [

.a0 .// †ı n ¹0º

(7.9)

jjD1

for some 0 < ı < . With the aid of Theorem 7.2 we are able to prove: Theorem 7.4. Let a.Dx / be as above, 0 < ı <  be as in (7.9), 1 < p < 1, and let AW Hpm .Rn /N ! Lp .Rn /N be defined by Au D a.Dx /u

for all u 2 Hpm .Rn /N :

Then there is some R > 0 such that .A/ †ı [ BR .0/: Moreover, for every  2 .0;   ı/ there is some constant C > 0 such that k. C A/1 kL.Lp .Rn /N / 

C 1 C jj

for every  2 † with jj R. The proof is based on the following lemma: Lemma 7.5. Let AW D.A/  X ! X be an unbounded linear operator such that . C A/1 exists for all  2 † with jj R and k. C A/1 kL.X/ 

C 1 C jj

for all  2 † ; jj R;

holds for some R > 0,  2 .0; /. Moreover, let BW D.B/ ! X be an unbounded linear operator such that D.B/ D.A/ and kB. C A/1 kL.X/ ! 0 for jj ! 1 in † :

(7.10)

Section 7.1 Applications of the Mikhlin Multiplier Theorem

179

†ı n BR .0/

R  .A/

Figure 7.2. Sketch of possible spectrum of A.

Then there is some R0 > 0 such that  C A C B is invertible for all  2 † with jj R 0 . Moreover, the estimate k. C A C B/1 kL.X/ 

C 1 C jj

for all  … †ı ; jj R0

holds. Proof. The proof is based on the identity . C A C B/u D . C A C B/u D .I C B. C A/1 /. C A/u for all u 2 D.A/ and a Neumann series argument, cf. Exercise 7.24. Proof of Theorem 7.4. For the following let A0 W D.A0 / Lp .Rn /N ! Lp .Rn /N and BW D.B/ Lp .Rn /N ! Lp .Rn /N be defined by X A0 u D a0 .Dx /u; Bu D a˛ Dx˛ u j˛j<m

for all u 2 D.A0 / WD D.B/ WD Hpm .Rn /N . Now, if we choose “A D A0 ”, K D †ı and K 0 D † in Theorem 7.2, then . C A0 /1 exists for all  2 K 0 n ¹0º and k. C A0 /1 kL.Lp .Rn /N / 

C jj

for all  2 † :

In order to apply Lemma 7.5, we use that kB. C A0 /1 ukLp .Rn /  C k. C A0 /1 ukHpm1 .Rn /N 

C0

1

.1 C jj/ m

kukLp .Rn /N

180

Chapter 7 Applications to Elliptic and Parabolic Equations

for all  2 † with jj 1 due to (7.5). Hence kB. C A0 /1 kL.Lp .Rn /N / ! 0 as jj ! 1 where  2 † . Therefore Lemma 7.5 implies the statement of the theorem. Remark 7.6. The result of Theorem 7.4 holds also if Lp .Rn / and Hp˛ .Rn / are replaced by Hps .Rn / and HpsC˛ .Rn / since hDx is and a.Dx / commute, cf. Exercise 7.25.

7.2 Applications of the Hilbert-Space-Valued Mikhlin Multiplier Theorem

i

The applications of this section are based on Theorem 5.8. Further results from Chapter 5 and no results on pseudodifferential operators are not needed.

7.2.1 Maximal Regularity of Abstract ODEs in Hilbert Spaces We consider the following abstract ordinary differential equation: d u.t / C Au.t / D f .t / in H for all t > 0; dt u.0/ D 0 in H;

(7.11) (7.12)

where u; f W Œ0; 1/ ! H are suitable functions, AW D.A/  H ! H is closed and densely defined operator with dense range, and H is a Hilbert space. Theorem 7.7. Assume that .CA/1 exists for all  2 C with Re  > 0 and satisfies the estimates k. C A/1 kL.H / 

C jj

for all Re > 0

(7.13)

for some constant C > 0. Then for every 1 < p < 1 and every f 2 Lp .0; 1I H / d p there is a unique u 2 Lp loc .Œ0; 1/I D.A// with dt u; Au 2 L .0; 1I H / solving (7.11) and (7.12). Moreover, there is some constant Cp > 0 independent of f and u such that



d

u

C kAukLp .0;1IH /  Cp kf kLp .0;1IH / : (7.14)

dt p L .0;1IH /

Section 7.2 Applications of the Hilbert-Space-Valued Mikhlin Multiplier Theorem

181

Proof. First we prove existence. To this end we first assume that f 2 C01 ..0; 1/I H / and define vW R ! H by 1 O 0 . /.t / for all t 2 R; v.t / D e 0 t F71 !t Œ.i  C 0 C A/ g

where g 0 .t / D e  0 t f .t / for all t 2 R and some 0 > 0 and f is extended by zero for negative t . Then v.t / 2 C 1 .RI H / since .i  C 0 C A/1 gO 0 . /; i  .i  C 0 C A/1 gO 0 . / 2 L1 .RI H / with respect to  due to (7.13) and gO 0 2 S.RI H /. Moreover, d v.t / D F 1 Œ.i  C 0 /.i  C A/1 gO 0 . /.t / dt D e 0 t F 1 Œ.I  A.i  C 0 C A/1 /gO 0 . /.t / D f .t /  Av.t / for all t 2 R. In particular u WD vjŒ0;1/ solves (7.11). In order to show v.0/ D u.0/ D 0 we use that Z  t 0 t 1 e v.t / D e e itt .i  C 0 C A/1 gO 0 . / d  2 R Z 0 0 t 1 De eit .i  0 C  C 0 C A/1 gO 0 . 0  i / d  0 2 RCi Z 0 0 t 1 De e it .i  0 C  C 0 C A/1 gO 0 . 0  i / d  0 2 R for all  0, t 2 R, where gO 0 .  i / D F t7! Œe t e  0 t f .t / D gO C 0 . /: Here we have used that the integrand above is an analytic function in  with Im  0. Therefore 1 v.t / D e . C 0 /t F71 O 0 C . /.t / !t Œ.i  C  C 0 C A/ g

for all  > 0, which shows that v is independent of the choice of 0 > 0. Hence (7.13) implies sup ke . C 0 /t v.t /kH  t2R

Z R

k.i  C  C 0 C A/1 k2L.H / d 

 C.0 /

Z

1

e

2. 0 C /t

0

 C.0 /kf kL2 .0;1IH /

2 kf .t /kH

1

 12 dt

2

kgO 0 kL2 .RIH /

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Chapter 7 Applications to Elliptic and Parabolic Equations

for all  0. Passing to the limit  ! 1 we conclude that v.t / D 0 for all t < 0. Hence v.0/ D u.0/ D 0 and u solves (7.11)-(7.12). Now the main step is to show the a priori estimate (7.14) for u. To this end we use that  1 O 0 . / ; Au.t / D e 0 t F71 !t A.i  C 0 C A/ g where it remains to estimate m 0 . / D A.i  C 0 C A/1 . Claim. For every k 2 N0 there is a constant Ck > 0 independent of 0 > 0 such that k@k m 0 . /kL.H /  Ck j jk

for all  ¤ 0:

(7.15)

Proof of Claim. We prove the claim by mathematical induction. If k D 0, we use that A.i  C 0 C A/1 D I  .i  C 0 /.i  C 0 C A/1 :

(7.16)

Hence (7.15) for k D 0 follows from (7.13). Next assume that (7.15) holds for all k  k 0 for some k 0 2 N0 . Then 0

0

@k C1 A.i  C 0 C A/1 D @k .A.i  C 0 C A/1 i.i  C 0 C A/1 /  1 0 D @k A.i  C 0 C A/1 .I  A.i  C 0 C A/1 / ;  where we have used (7.7) with C N N replaced by L.H /, which is proved the same way. Hence using the product rule and (7.15) for k  k 0 one derives (7.15) for k 0 C 1 in a straight-forward manner. Using (7.15) and Theorem 4.28, we conclude that for every 1 < p < 1 there is a constant Cp > 0 such that ke  0  AvkLp .RIH /  Cp ke  0  f kLp .RIH / : Since (7.15) is independent of 0 , Cp is independent of 0 > 0 too. Hence, passing to the limit 0 ! 0 and using that v is independent of 0 > 0, we conclude that kAukLp .0;1IH /  Cp kf kLp .0;1IH / : d u instead of Au. Therefore we Because of (7.11), the same estimate holds also for dt have proven the existence of a solution u as in the theorem if f 2 C01 .0; 1I H /. Now the existence of a solution for a general f 2 Lp .0; 1I H / can be shown by approximating f by a sequence fk 2 C01 .0; 1I H / and considering the limit k ! 1 for the corresponding solutions uk using the uniform bound (7.14). Finally, it remains to prove uniqueness. Therefore we assume that u is a solution as in the theorem with f  0. Then



Z t





1 d d 0



p u. / d   t u

ku.t /kH D

dt Lp .0;1IH / 0 d H

Section 7.2 Applications of the Hilbert-Space-Valued Mikhlin Multiplier Theorem

183

for all t 0 by the Hölder inequality. Hence e  0 t u.t / 2 L1 .0; 1I H / for every 0 > 0. Therefore we define ´ e 0 t u.t / if t > 0; v.t / D 0 else d v; Av 2 L1 .RI H / and for some 0 > 0. Then v; dt

d v D Av  0 v dt

in L1 .RI H /:

Hence taking the Fourier transformation of this equation gives .i  C 0 C A/v. O / D 0 for all  2 R: O / D 0 for all  2 R, Since i  C0 CA is invertible for all  2 R, we conclude that v. which yields v  0 and therefore u  0. Remark 7.8 (Maximal Regularity in Banach Spaces). In Theorem 7.7 it is essential that H is a Hilbert space and not a general Banach space. Nevertheless the result can be generalized to the case that H is a so-called UMDspace, which is a certain subclass of Banach spaces. But to this end the resolvent estimate (7.13) has to be replaced by the assumption that the operator family ® ¯ . C A/1 W Re  > 0  L.H / is R-bounded, which is a stronger assumption than the uniform boundedness of the operator norms. (If H is a Hilbert space, uniform boundedness is equivalent to Rboundedness.) We refer to Denk et al. [4] and Weis [42] for details and further references. Example 7.9. Theorem 7.7 applies for example to A D  with domain D.A/ D H22 .Rn / and H D L2 .Rn /, where (7.13) follows from Lemma 7.1. Here the restriction to p D 2 is needed to obtain Hilbert spaces. Therefore Theorem 7.7 implies that for every f 2 Lp .0; 1I L2 .Rn //, 1 < p < 1, there is a unique solution u 2 L2loc .Œ0; 1/I H 2 .Rn // with @ t u; u 2 Lp .0; 1I L2 .Rn // solving the heat equation @ t u  u D f uj tD0 D 0

in Lp .0; 1I L2 .Rn //; in L2 .Rn /:

A similar results holds for A D a.Dx /, where a.Dx / is as in Section 7.1.2.

184

Chapter 7 Applications to Elliptic and Parabolic Equations

P Now, if A D a.Dx / D j˛jm c˛ Dx˛ is a constant coefficient differential operator on Rn as in Section 7.1.3, then Theorem 7.7 does not apply directly since (7.13) only holds for jj R for some R > 0. Nevertheless one can obtain an analogous result on any finite time interval: P Theorem 7.10. Let a.Dx / D j˛jm c˛ Dx˛ be as in Section 7.1.3. Then for every 0 < T < 1 and every f 2 Lp .0; T I L2 .Rn /N / with 1 < p < 1 there is a solution u 2 Wp1 .0; T I L2 .Rn /N / \ Lp .0; T I H2m .Rn /N / such that @ t u C a.Dx /u D f u.0/ D 0

in Lp .0; T I L2 .Rn /N /; 2

n N

in L .R / :

(7.17) (7.18)

Proof. Let R > 0, A be as in Theorem 7.4 and let ! > 0 we such that ! C †  † n BR .0/. Moreover, we define A! W D.A! /  L2 .Rn /N ! L2 .Rn /N by A! u D .! C a.Dx //u

for all u 2 D.A! / WD H2m .Rn /N :

Then . C A! /1 D . C ! C a.Dx //1 exists for all  2 † and (7.13) holds for A! . Therefore Theorem 7.7 implies that for every g 2 Lp .0; 1I L2 .Rn /N / there is a unique solution v 2 L2loc.Œ0; 1/I H2m .Rn /N / with @ t v; A! v 2 Lp .0; 1I L2 .Rn /N / solving @ t v C .! C a.Dx //v D g v.0/ D 0

in Lp .0; 1I L2 .Rn /N /; in L2 .Rn /N :

Now let ´ g.t / D

e!t f .t / if 0 < t < T; 0 else

for some f 2 Lp .0; T I L2 .Rn /N / with 0 < T < 1. Then g 2 Lp .0; 1I L2 .Rn /N / and the corresponding solution v above exists. Hence, if we set u.t / WD e !t v.t / for all t 2 .0; T /, then u 2 Wp1 .0; T I L2 .Rn /N / \ Lp .0; T I H2m .Rn /N / and u solves (7.17) and (7.18) because of @ t u.t / D e !t @ t v.t / C !e!t v.t / D e !t .g.t /  a.Dx /v.t // D f .t /  a.Dx /u.t / for almost all 0 < t < T . Remark 7.11. Since a.Dx / and . C a.Dx //1 commute with hDx is for any s 2 R, one can replace L2 .Rn /N and H m .Rn /N by H s .Rn /N and H sCm .Rn /N in Theorem 7.10, cf. Exercise 7.25.

185

Section 7.2 Applications of the Hilbert-Space-Valued Mikhlin Multiplier Theorem

7.2.2 Hilbert-Space Valued Bessel Potential and Sobolev Spaces Let X be a Banach space. Similarly to the scalar case, we define the Sobolev space Wpm .Rn I X/ WD ¹f 2 Lp .Rn I X/ W @˛x f 2 Lp .Rn I X/ for all j˛j  mº X kf kWpm .Rn IX/ WD k@˛x f kLp .Rn IX/ ; j˛jm

where m 2 N0 , 1  p  1. Here the (vector-valued) distributional derivative @˛x f 2 S 0 .Rn I X/ D L.S.Rn /I X/ is defined by Z ˛ ˛ j˛j h@x f; 'i WD .@x f /.'/ WD .1/ f .x/@˛x '.x/ dx 2 X Rn

for all ' 2 S.Rn / and we say @˛x f 2 Lp .Rn I X/ if there is some g˛ 2 Lp .Rn I X/, which is identified with @˛x f , such that Z g˛ .x/'.x/ dx in X; for all ' 2 S.Rn /: h@˛x f; 'i D Rn

Moreover, we define the so-called Bessel potential space of order s 0 and integral exponent p 2 .1; 1/ as Hps .Rn I X/ WD S.Rn I X/

k  kH s .Rn IX / p

kf kHps .Rn IX/ WD khDx is f kLp .Rn IX/ D kF 1 Œhis fO./kLp .Rn IX/ 1

As before hi D .1 C jj2 / 2 . If p D 2 and X D H is a Hilbert space, then it follows easily from Plancherel’s theorem that W2m .Rn I H / D H2m .Rn I H / as in the proof of Lemma 2.41. This even holds true for 1 < p < 1: Lemma 7.12. Let H be a Hilbert space, 1 < p < 1, and let m 2 N0 . Then Wpm .Rn I H / D Hpm .Rn I H / with equivalent norms. Proof. We first prove the embedding Hpm .Rn I H / ,! Wpm .Rn I H /. Let f 2 S.Rn I H /. Then @ˇx f

DF

1

  ˇ 1 .i / m O O Œ.i / f ./ D F hi f ./ him ˇ

Hence in order to obtain k@ˇx f kLp .Rn IH /  Cp khDx im f kLp .Rn IH / D Cp kf kHpm .Rn IH /

(7.19)

186

Chapter 7 Applications to Elliptic and Parabolic Equations ˇ

for ˇ 2 N0n with jˇj  m we apply Theorem 4.23 to mˇ ./ D .i/ . Therefore one him has to verify (4.28) for m D mˇ , which is done in the proof of Theorem 6.8. Hence Theorem 4.23 implies that Hpm .Rn I H / ,! Wpm .Rn I H / since S.Rn I H / is dense in Hpm .Rn I H / by definition. Hence it remains to prove Wpm .Rn I H / ,! Hpm .Rn I H /. If m D 2k, k 2 N0 , is even, then him D .1 C jj2 /k is a polynomial of degree m. Therefore hDx im is a differential operator of order m and X k@˛x f kLp .Rn IH / ; khDx im f kLp .Rn IH /  C j˛jm

which proves the embedding in this case. If m D 2k C 1, k 2 N0 , is odd, then   n n X X j2 j 1 1 m m 2k hi D hi C C D hi hi2k j ; 2 2 hi hi hi hi j D1

j D1

where hi2k and hi2k j are polynomials of degree at most 2k C 1. Hence khDx im f kLp .Rn IH /  C

n X X

kmj .Dx /@˛x f kLp .Rn IH / ;

j˛jm j D0

where m0 ./ D hi1 and mj ./ D j hi1 , j D 1; : : : ; n. Hence it remains to verify the Mikhlin condition (4.28) for mj ./, which is also done in the proof of Theorem 6.8.

7.3 Applications of Pseudodifferential Operators

i

The applications of this section are based on the results for pseudodifferential operators in Chapter 3 and Sections 5.4 and 5.5. But no results on Bochner spaces, vector-valued Fourier transformation and the Mikhlin multiplier theorem are needed.

7.3.1 Elliptic Regularity for Elliptic Pseudodifferential Operators In the following we consider regularity properties of solutions u of the equation p.x; Dx /u D f

in S 0 .Rn /;

(7.20)

m .Rn  Rn / is elliptic. We have the following result on elliptic regularwhere p 2 S1;0 ity:

187

Section 7.3 Applications of Pseudodifferential Operators

m .Rn  Rn /, m 2 R, be elliptic and let 1 < q < 1. Theorem 7.13. Let p 2 S1;0 Moreover, let u 2 Hqr .Rn / be a solution of (7.20) for some f 2 Hqs .Rn /, where r; s 2 R. Then u 2 HqsCm .Rn /. Moreover, there is some constant Cr;s;q > 0 independent of u and f such that

kukH sCm .Rn /  Cr;s;q .kf kHqs .Rn / C kukHqr .Rn / /

(7.21)

q

Proof. The proof is similar to the proof of Corollary 3.43. To this end we choose 0 m N 2 N such that s C m  r C N . Moreover, let qN 2 S1;0 .Rn  Rn / be the (left) parametrix from Theorem 3.24. Then (7.20) implies 0 0 .x; Dx /f  rN .x; Dx /u; u D qN 0 N where rN 2 S1;0 .Rn  Rn /. Hence Theorem 5.20 implies that 0 .x; Dx /u 2 Hqr CN .Rn / ,! HqsCm .Rn /; rN

0 qN .x; Dx /f 2 HqsCm .Rn /;

which implies u 2 Hqs .Rn /. The norm estimate (7.21) follows from 0 0 kukH sCm .Rn /  Cs;q .kqN .x; Dx /f kH sCm .Rn / C krN .x; Dx /ukH rCN .Rn / / q

q

q

0 .kf kHqs .Rn / C kukHqr .Rn / /;  Cr;s;q

which finishes the proof. With the aid of the Sobolev-type embedding u 2 Hqr .Rn / ,! HqQrQ " .Rn /

(7.22)

for any " > 0, where rQ D r  nq C nqQ and 1 < q  qQ < 1, cf. Corollary 6.14, one can even prove: Lemma 7.14. Let u; f be as in the assumptions of Theorem 7.13. If additionally f 2 HqQsQ .Rn / for some qQ 2 .q; 1/ and sQ 2 R, then u 2 HqQsQCm .Rn /. Proof. Let f 2 HqQsQ .Rn / additionally. Then by (7.22) we have u 2 HqQrQ " .Rn /. Therefore we can apply Theorem 7.13 with r; s; q replaced by r; Q sQ ; qQ to conclude that u 2 HqQsQCm .Rn /. Remark 7.15. In the statement of Theorem 7.13 one can replace Bessel potential s .Rn /, which were introduced in Chapter 6, cf. spaces Hqs .Rn / by Besov spaces Bqt Exercise 7.26. In particular, if m 2 N0 and u 2 Cb0 .Rn / is a solution of (7.20) with f 2 C ˛ .Rn / for some 0 < ˛ < 1, then u 2 C m;˛ .Rn / because of C m;˛ .Rn / D mC˛ ˛ .Rn /, C ˛ .Rn / D B11 .Rn / due to Theorem 6.1 and Exercise 6.29. B11

188

Chapter 7 Applications to Elliptic and Parabolic Equations

The statement of Theorem 7.13 on elliptic regularity can also be localized. To this end we say that u; v 2 S 0 .Rn / coincide on an open set U Rn if hu; 'i D hv; 'i

for all ' 2 C01 .U /:

m .Rn  Rn /, m 2 R, be elliptic, s 2 R, and let 1 < q < Theorem 7.16. Let p 2 S1;0 S 1 1. Moreover, let u 2 Hq .Rn / D r 2R Hqr .Rn / be a solution of (7.20) for some f 2 Hq1 .Rn /. Moreover, assume that there is some g 2 Hqs .Rn /, s 2 R, such that f and g coincides on some open set U  Rn . Then for every open bounded set V with V U there is some v 2 HqsCm .Rn / such that u and v coincide on V .

Proof. Let V be an open bounded set such that V U and let r 2 R be such that u 2 Hqr Cm .Rn /. The statement of the theorem follows from: Claim. For every k 2 N0 and every bounded open set V with V U there is some min.r Ck;s/ v 2 Hq .Rn / such that u and v coincide. Proof of Claim. If k D 0, the statement is trivial. Now assume that the claim holds for some k 2 N0 and let V be a bounded open set with V U . Moreover, let 2 C01 .Rn / such that  1 on V and supp U . Then the assumption applied to some open and bounded W supp with W U implies that v WD u 2 Hqmin.r Ck;sCm/ .Rn /. If r C k s C m, then v 2 Hqmin.r CkC1;sCm/ .Rn / and the statement follows. Hence it only remains to consider the case r C k < s C m. Now we use that (7.20) implies p.x; Dx /v D p.x; Dx /u C Œp.x; Dx /; u D f C Œp.x; Dx /; u D

g C Œp.x; Dx /; u

in S 0 .Rn / since f and g coincide on U , where Œp.x; Dx /;  is a pseudodifferential operator of order m  1 because of Corollary 3.17. Hence g C Œp.x; Dx /; u 2 Hqs .Rn / C Hqr CkmC1 .Rn / D Hqmin.r CkmC1;s/ .Rn / min.r CkC1;sCm/

and Theorem 7.13 applied to v implies v 2 Hq follows.

.Rn /. Hence the claim

7.3.2 Resolvents of Parameter-Elliptic Differential Operators In the following we study the resolvent equation . C a.x; Dx //u D f in S 0 .Rn /; P where  2 C and a.x; / D j˛jm c˛ .x/ ˛ , c˛ 2 Cb1 .Rn /, m 2 N. As seen in Section 7.2.1 such a resolvent equation is import to study the corresponding parabolic equation. A further application will be presented in Section 7.3.3 below.

189

Section 7.3 Applications of Pseudodifferential Operators

We denote by a0 .x; / D

P

j˛jDm c˛ .x/

˛

the principal symbol of a.x; Dx /.

Definition 7.17. Let ı 2 .0; / and let a.x; / be a differential symbol of order m as above. Then a is said to be (uniformly) parameter-elliptic on †ı if ¹a0 .x; / W x 2 Rn ; jj D 1º †ı We note that, since 0 … †ı , a uniformly elliptic differential symbol satisfies ja0 .x; /j c > 0 for all x 2 Rn ; jj D 1 for some c > 0. Example 7.18. Let a.x; / D jj2 . Then m D 2 and a0 .x; / D 1 for all x 2 Rn and jj D 1. Hence a is parameter-elliptic on †ı for every ı 2 .0; /. Moreover, if A.x/ 2 Cb1 .Rn /nn is a real symmetric matrix, which is uniformly positive definite, i.e.,  T A.x/ c > 0, then a.x; / D  T A.x/ is parameterelliptic on †ı for every ı 2 .0; /. Finally, if n D 1, a.x; / D i  is uniformly elliptic on †ı only if ı 2 .0; =2/. Proposition 7.19. Let a be parameter-elliptic on †ı , ı 2 .0; /, and let a0 .x; / be its principal symbol. Then for every ˛; ˇ 2 N0n there is some C˛;ˇ > 0 such that 1

j@˛ @ˇx . C a0 .x; //1 j  C˛;ˇ .1 C jj m C jj/mj˛j

(7.23)

for all  2 †ı , x;  2 Rn , such that jj C jjm 1. Proof. The proof relies essentially on the following statement: Claim. For all ˛; ˇ 2 N0n there is some C˛;ˇ > 0 such that j@˛ @ˇx . C a0 .x; //1 j  C˛;ˇ

(7.24)

for all  2 †ı and  2 Rn with max.jjm ; jj/ D 1. Proof of Claim. First let max.jjm ; jj/ D jjm D 1. Since c˛ 2 Cb1 .Rn /, the set A WD ¹a0 .x; / W x 2 Rn ; jj D 1º is bounded. Moreover, A †ı D C n †ı . Hence dist.A; @†ı / > 0 since A is compact, @†ı is closed, and A \ @†ı D ;. Thus j. C a0 .x; //1 j 

1 dist.A; @†ı /

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Chapter 7 Applications to Elliptic and Parabolic Equations

for all jj D 1, x 2 Rn , and  2 †ı . Thus j@z . C a0 .x; //1 j D j. C a0 .x; //2 @z a0 .x; /j  C˛;ˇ for all jj D 1; x 2 Rn ;  2 †ı , where z D xj or z D j , j D 1; : : : ; n. Similarly it is easy to prove by mathematical induction that j@˛ @ˇx . C a0 .x; //1 j  C˛;ˇ

for all jj D 1; x 2 Rn ;  2 †ı ;

for arbitrary ˛; ˇ 2 N0n since all derivatives of a0 .x; / are uniformly bounded on jj D 1, x 2 Rn . Now we consider the case max.jjm ; jj/ D jj D 1. Because of the homogeneity of the principal symbol, i.e., a0 .x; r/ D r m a0 .x; / for r > 0, ¹a0 .x; / W x 2 Rn ; jj  1º [ r m  ¹a0 .x; / W x 2 Rn ; jj D 1º †ı [ ¹0º; D 0r 1

S where 0r1 r m  ¹a0 .x; / W x 2 Rn ; jj D 1º is a compact set since it is the image of Œ0; 1  ¹a0 .x; / W x 2 Rn ; jj D 1º under the multiplication .r; z/ 7! r m  z. Now let A WD ¹a0 .x; / W x 2 Rn ; jj  1º: Then A C n †ı and dist.A; @†ı \ ¹jj D 1º/ > 0. Thus j. C a0 .x; //1 j 

1 dist.A; @†ı \ ¹jj D 1º/

and therefore one can show as before j@˛ @ˇx . C a0 .x; //1 j  C˛;ˇ for all jj  1, x 2 Rn , and  2 †ı with jj D 1. This completes the proof of the claim. Finally, let a .x; / WD . C a0 .x; //1 . We note that a .x; / is homogeneous 1 of degree m with respect to . m ; / in the sense that ar m .x; r/ D r m a .x; /

for all r > 0; x;  2 Rn ;  2 †ı :

(7.25)

ˇ

This implies that @˛ @x a .x; / is homogeneous of degree m  j˛j with respect to 1

. m ; /, i.e., .@˛ @ˇx ar m /.x; r/ D r mj˛j @˛ @ˇx a .x; /

for all r > 0; x;  2 Rn ;  2 †ı ;

191

Section 7.3 Applications of Pseudodifferential Operators

by differentiating (7.25). Replacing .; / by .r m ; r 1 /, we have    @˛ @ˇx a .x; / D r mj˛j @˛ @ˇx ar m x; : r 1

Now choosing r D max.jj; jj m /, (7.24) implies 1

1

j@˛ @ˇx . C a0 .x; //1 j  C max.jj; jj m /mj˛j  C 0 .jj m C jj/mj˛j for all x;  2 Rn and  2 †ı , which proves the proposition. Lemma 7.20. Let a.x; / be a parameter-elliptic symbol on †ı of order m, ı 2 .0; /, as defined above and let 1 < p < 1, s 2 R. Then A D a.x; Dx /W D.A/ ! Hps .Rn / with D.A/ D HpsCm .Rn / Hps .Rn / is a closed and densely defined linear operator. Moreover, there are some C; R > 0 such that  C A is invertible for all  2 †ı with jj R > 0 and k. C A/1 kL.Hps .Rn //  C jj1 : Proof. Let  2 †ı with jj 1. Moreover, let a .x; / D . C a0 .x; //1 and let m0 2 Œ0; m. Then for all ˛; ˇ 2 N0n mj˛j  1 mm0 0 j@˛ @ˇx a .x; /j  C˛;ˇ 1 C jj m C jj  C˛;ˇ jj m him j˛j (7.26) 0

m .Rn  Rn / with for all  2 †ı with jj 1 and x;  2 Rn . Hence a 2 S1;0 .m0 /

ja jk

 Ck jj

mm0 m

uniformly in  2 †ı with jj 1

for every k 2 N. Therefore ka .x; Dx /kL.H s .Rn /;H sCm0 .Rn //  C jj p

mm0 m

p

(7.27)

uniformly in  2 †ı with jj 1 because of Theorem 5.20. Moreover, because of the asymptotic expansion stated in Theorem 3.16 and the fact that all terms depend in a bounded manner on p1 and p2 . C a0 .x; Dx //a .x; Dx / D a .x; Dx / C a0 .x; Dx /a .x; Dx / D I C r .x; Dx /; where for every k 2 N0 there are n.k/ 2 N0 and Ck > 0 such that .0/

.m/

.mC1/

jr jk  Ck ja0 jn.k/ jja jn.k/

1

 Ck jj m :

(7.28)

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Chapter 7 Applications to Elliptic and Parabolic Equations

– Alternatively one can use the following argument: Since a0 .x; Dx / is a differential operator, X 1 ˛ r .x; / D @ a0 .x; /Dx˛ a .x; /: ˛Š  j˛jm;˛¤0

We note that one can prove (7.28) directly using this identity. – Thus 1

kr .x; Dx /kL.Hps .Rn //  C jr j.0/  C 0 jj m k for all jj 1, where k is as in Remark 5.21 for m D 0. Furthermore, . C a.x; Dx //a .x; Dx / D I C r .x; Dx / C R 0 ; P where R 0 WD j˛j<m c˛ .x/Dx˛ a .x; Dx / satisfies 1

kR 0 kL.Hps .Rn //  C ka .x; Dx /kL.H s .Rn /;H sCm1 .Rn //  C jj m p

p

m0

due to (7.27) for D m  1. Hence there is some R > 0 such that kr .x; Dx / C R 0 kL.Hps .Rn //  jj R. Therefore I C r .x; Dx / C R 0 is invertible and

1 2

for all

k.I C r .x; Dx / C R 0 /1 kL.Hps .Rn //  2 for all jj R, cf. Theorem A.11. Thus . C a.x; Dx //a .x; Dx /.I C r .x; Dx / C R 0 /1 D I for all  2 †ı ; jj R. Therefore  C a.x; Dx / is surjective and has a continuous right inverse. By the same arguments one shows that also a .x; Dx /. C a.x; Dx // D I C R 00 ; 1

where kR 00 kL.H sCm .Rn //  C jj m for all  2 †ı , jj 1. This shows that  C p a.x; Dx / is injective if jj R,  2 †ı and R 1 is sufficiently large. Therefore  C a.x; Dx / is invertible for all jj R,  2 †ı and k. C a.x; Dx //1 kL.Hps .Rn //  ka .x; Dx /kL.Hps .Rn // k.I C r .x; Dx / C R 0 /1 kL.Hps .Rn //  C jj1 : This proves the theorem. Remark 7.21. The last theorem implies that the spectrum of a.x; Dx / is contained in a key-hole region .C n †ı / [ BR .0/ for some R > 0, cf. Figure 7.2, and that the resolvent . C a.x; Dx //1 decays as jj1 as jj ! 1 in †ı . The same result holds if Hps .Rn / and HpsCm .Rn / are replaced by Besov spaces s sCm .Rn /, where s 2 R and 1  p; q  1 are arbitrary. Bpq .Rn / and Bpq

193

Section 7.3 Applications of Pseudodifferential Operators

7.3.3 Application of Resolvent Estimates to Parabolic Initial Value Problems The study of abstract parabolic initial value problems d u.t / C Au.t / D 0 dt u.0/ D u0

in X for t > 0

(7.29)

in X;

(7.30)

where AW D.A/  X ! X is a suitable unbounded operator (e.g.  ) can be reduced to the resolvent equation . C A/u D f

in X:

This is the content of the theory of analytic semi-groups, cf. e.g. [26],[28]. We only state one of the main results Theorem 7.22. Let X be a Banach space and let AW D.A/  X ! X be a linear and closed operator such that D.A/ is dense in X. If  C A is invertible for every  2 0 C †ı , †ı D ¹z 2 C n ¹0º W jarg zj < ıº; where ı 2 .=2; / and 0 2 R, and satisfies k. C A/1 kL.X/ 

C j  0 j

for all  2 0 C †ı ;

(7.31)

then there is a family of bounded linear operators .T .t // t0  L.X/ such that 1. T .t /T .s/ D T .t C s/ for all t; s 0, T .0/ D I . 2. .0; 1/ 3 t 7! T .t / 2 L.X/ is differentiable and t > 0, where T .t / 2 L.X; D.A// for all t > 0.

d T .t / dt

D AT .t / for every

3. lim t!0 T .t /u0 D u0 in X for all u0 2 X. 4. kT .t /kL.X/ Cjt jkAT .t /kL.X/  C e 0 t for all t 0 and some constant C 1. Here .T .t // t0 is a so-called (strongly continuous) analytic semi-group. This theorem is a consequence of the characterization of analytic semi-groups, cf. Pazy [26, Theorem 3.1, Chapter 1] or Renargy and Rogers [28, Theorem 11.17]. With the aid of the semi-group one can easily solve (7.29)-(7.30) as follows: For u0 2 X let u.t / D T .t /u0 for all t 0. Then uW .0; 1/ ! X is differentiable by the last theorem and d u.t / D AT .t /u0 D Au.t / for all t > 0: dt

194

Chapter 7 Applications to Elliptic and Parabolic Equations

Moreover, u.0/ D lim t!0C u.t / D lim t!0C T .t /u0 D u0 . In particular we obtain solvability of the parabolic initial value problem @ t u.x; t / C a.x; Dx /u.x; t / D 0 u.x; 0/ D u0 .x/

for all t > 0; x 2 Rn ; for all x 2 Rn ;

where a.x; Dx / is a parameter-elliptic differential operator as in Lemma 7.20 because of: Corollary 7.23. Let A D a.x; Dx / and ı 2 .=2; / be as in Lemma 7.20 and let s 2 R, 1 < p < 1. Then there is some 0 > 0 such that A satisfies the conditions of Theorem 7.22 with X D Hps .Rn /, D.A/ D HpsCm .Rn /. In particular, A generates a strongly continuous, analytic semi-group T .t /, t 0 on Hps .Rn /, which satisfies kT .t /kL.Hps .Rn // C t kAT .t /kL.Hps .Rn //  Cs e 0 t

for all t 0:

Proof. Just choose some 0 > 0 such that 0 C †ı  †ı n BR .0/. Then apply Theorem 7.22. s .Rn / for any s 2 Rn , The analogous result holds in the case of Besov spaces Bpq 1  p; q  1.

7.4 Final Remarks and Exercises 7.4.1 Further Reading An introduction to semi-group theory can be found in the books by Pazy [26] and Renardy and Rogers [28, Chapter 11]. Further results on abstract evolution equations and applications of this theory are treated in the monographs by Amann [2] and Lunardi [24]. We refer to Denk et al. [4] and Weis [42] for further applications of Banach-spacevalued versions of the Mikhlin multiplier theorem and to maximal regularity questions for abstract evolution equations. For applications to (pseudo-)differential equations on manifolds and domains with boundary we refer to Grubb [11]. Even more advanced applications to parameterelliptic pseudodifferential boundary value problems can be found in Grubb [10]. A lot of further applications in various directions are treated in [12, 13, 18, 21]. Non-smooth pseudodifferential operators and their applications can be found in Taylor [34, 35]. Applications of pseudodifferential operators to Markov processes are presented in the monographs by Jacob [15, 16, 17].

195

Section 7.4 Final Remarks and Exercises

7.4.2 Exercises Exercise 7.24 (Resolvent Estimates and Perturbations). Let AW D.A/  X ! X be an unbounded linear operator such that . C A/1 exists for all  2 † with jj R and k. C A/1 kL.X/ 

C jj

for all  2 † ; jj R;

holds for some R > 0,  2 .0; /. Moreover, let BW D.B/ ! X be an unbounded linear operator such that D.B/  D.A/ and kB. C A/1 kL.X/ ! 0 for jj ! 1 in † :

(7.32)

1. Prove that there is some R 0 > 0 such that  C A C B is invertible for all  2 † with jj R 0 . Moreover, the estimate k. C A C B/1 kL.X/ 

C jj

for all  2 † ; jj R0

holds. Hint. Use . C A C B/u D . C A C B/u D .I C B. C A/1 /. C A/u for all u 2 D.A/ and a Neumann series argument. 2. Now let X D L2 .Rn / and D.A/ D H2m .Rn /, m > 0. Moreover, let 0 BW H2m .Rn / ! L2 .Rn / be a bounded linear operator for some 0 < m0 < m. Prove that (7.32) holds. Hint. Show and use the interpolation inequality mm0

m0

kukH m0  kukL2m kukHmm 2

2

for all u 2 H2m .Rn /:

Exercise 7.25. Let a.Dx / be as in Section 7.1.3, 0 < ı <  be as in (7.9), 1 < p < 1, s 2 R, and let AW HpsCm .Rn /N ! Hps .Rn /N be defined by Au D a.Dx /u

for all u 2 HpsCm .Rn /N :

1. Prove that there is some R > 0 such that .A/ †ı [ BR .0/:

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Chapter 7 Applications to Elliptic and Parabolic Equations

Moreover, prove that for every  2 .ı; / there is a constant C > 0 such that k. C A/1 kL.Hps .Rn /N / 

C 1 C jj

for every  2 † with jj R. 2. Prove that for every f 2 Lp .0; T I H2s .Rn /N / with 1 < p < 1 there is a solution u 2 Wp1 .0; T I H2s .Rn /N / \ Lp .0; T I H2sCm .Rn /N / such that @ t u C a.Dx /u D f u.0/ D 0

in Lp .0; T I H2s .Rn /N /; in H2s .Rn /N :

m Exercise 7.26. Let p 2 S1;0 .Rn  Rn /, m 2 R, be elliptic and let 1  r .Rn / be a solution of (7.20) for some q; q1 ; q2  1. Moreover, let u 2 Bqq 1 s n sCm f 2 Bqq .R / and r; s 2 R. Then u 2 Bqq .Rn /. Moreover, there is a con2 2 stant C D Cr;s;q;q1 ;q2 > 0 independent of u and f such that s kukB sCm .Rn /  C.kf kBqq qq2

2

.Rn /

r C kukBqq

1

.Rn / /:

Part IV

Appendix

Appendix A

Basic Results from Analysis Summary In this appendix we introduce some basic notation and summarize some basic results on calculus and integration on Rn and linear functional analysis. Moreover, we briefly discuss the Bochner integral and its most important properties for the purpose of this book. Finally, we give a short introduction to Fréchet spaces. Although we will not need any deep result from the theory of Fréchet spaces/topological vector spaces, it will be useful for the study of pseudodifferential operators to know some of the basic facts.

A.1 Notation and Functions on Rn In this section we briefly summarize some basic notation and results from calculus in Rn . Throughout the book N will denote the set of all natural numbers zero not included and N0 WD N [ ¹0º. Moreover, R and C denote the set of real and complex numbers, respectively, and K stands for R or C. Constants appearing in inequalities will usually be denoted by C sometimes marked with an index as e.g. C˛ to denote that C depends on ˛. In sequences of inequalities all constants will simply be denoted by C although they may change from line to line. The characteristic function of a set A is denoted by A , i.e., ´ 1 if x 2 A;

A .x/ D 0 else: Moreover, Br .x0 / D ¹x 2 Rn W jx  x0 j < rº for any r > 0, x0 2 Rn . We will make use of multi-indices, which will keep the notation (relatively) short. A multi-index is a vector ˛ D .˛1 ; : : : ; ˛n / 2 N0n . Given ˛ 2 N0n we define the length of ˛ as j˛j D ˛1 C    C ˛n and its factorial as ˛Š D ˛1 Š    ˛n Š. For x 2 Rn and ˛ 2 N0n we define x ˛ D x1˛1    xn˛n . Then x ˛ is a polynomial of degree j˛j and an arbitrary polynomial pW Rn ! C of order m 2 N0 can be written as X c˛ x ˛ p.x/ D j˛jm

P

with coefficients c˛ 2 C. Here j˛jm denotes summation with respect to all multiindices ˛ 2 N0n with length j˛j  m. Moreover, if ˛; ˇ 2 N0n , we write ˛  ˇ if and only if ˛j  ˇj for all j D 1; : : : ; n.

200

Appendix A Basic Results from Analysis

The so-called binomial coefficients are defined by ! ˛ ˛Š D ˇ ˇŠ.˛  ˇ/Š   if ˇ  ˛ and ˇ˛ D 0 otherwise. Then it is easy to check that ! ! ! ˛n ˛ ˛1  : D ˇ ˇ1 ˇn Using this notation and the fundamental relation ! ! ! n n nC1 C D k kC1 kC1 for n; k 2 N0 , one can prove by mathematical induction that ! X ˛ ˛ x ˇ y ˛ˇ .x C y/ D ˇ ˇ ˛

for all x; y 2 Rn , which generalizes the well-known binomial formula in one dimension. In the following let U  Rn be an open set. Recall that the space C k .U / consists of all k-times differentiable f W U ! K with continuous derivatives up to T functions 1 .U / D k n j˛j .U /, then we define order k and C k2N C .U /. If ˛ 2 N0 and u 2 C @˛x u.x/ D

@j˛j u.x/ @˛x11 : : : @˛xnn

for all x 2 U:

If u depends on several variables, say x and y, @˛x u and @y˛ u denote the derivative defined above with respect to x and y, respectively. Using the product rule of differentiation, one can prove the Leibniz formula: ! X ˛ v/.x/ (A.1) .@ˇx u/.x/.@˛ˇ @˛x .uv/.x/ D x ˇ ˇ ˛

for all ˛ 2 N0n and u; v 2 C j˛j .Rn /. Moreover, we recall Taylor’s formula: Theorem A.1. Let u 2 C k .Rn /, k 2 N. Then for any x and y 2 Rn one has X y˛ X y˛ Z 1 u.x C y/ D @˛x u.x/ C k .1  t /k1 @˛x u.x C ty/dt: ˛Š ˛Š 0 j˛j
j˛jDk

Proof. See e.g. [27, Chapter 1, Theorem 1.1].

Section A.2 Lebesgue Integral and Lp -Spaces

201

Finally, if   Rn is a domain, we denote by Cbk ./, k 2 N0 , the space of all ˛ f 2 C k ./ with bounded derivatives up to order k such that T1@ f khas a continuous 1 extension on  for all j˛j  k. Moreover, Cb ./ WD kD0 Cb ./. It is wellknown from basic courses in functional analysis that Cbk ./ equipped with the norm kf kC k ./ D sup sup j@˛x f .x/j b

j˛jk x2

is a Banach space, i.e., a complete normed vector space. Moreover, if U  Rn is open, then C01 .U / denotes the set of all f 2 C 1 .U / such that supp f D ¹x 2 U W f .x/ ¤ 0º is a compact subset of U . We note that for any compact K Rn and any open U  Rn with K  U , there is some ' 2 C01 .U / such that '  1 on K; cf. e.g. [11, Theorem 2.13] or [23, Lemma 2.19]. If ' 2 C01 .U /, we will identified it with its extension by zero to a function in 1 C0 .Rn /.

A.2 Lebesgue Integral and Lp -Spaces Recall that a function f W Rn ! R is Lebesgue-integrable or just integrable if f is measurable and Z jf .x/jdx < 1: (A.2) Rn

Here and in the following all integrals are taken with respect to the Lebesgue measure on Rn . More generally, a function f W Rn ! C is Lebesgue-integrable if and only if Re f and Im f are Lebesgue-integrable. Then Z Z Z f .x/dx WD Re f .x/dx C i Im f .x/dx: Rn

Rn

Rn

The space of all Lebesgue-integrable functions f W Rn ! K (with K D R or K D C) is denoted by L1 .Rn /. It is a linear vector space, which can be normed by Z jf .x/jdx kf kL1 .Rn / WD kf k1 WD Rn

if functions which differ on a set of measure zero are identified. L1 .Rn / is complete with respect to the norm k  k1 .

202

Appendix A Basic Results from Analysis

Throughout the book many functions will be at least continuous, in most cases even infinitely differentiable. If f and g are continuous and f D g almost everywhere, i.e., they differ on a set of measure zero, then f .x/ D g.x/ for every x 2 Rn . Therefore we do not have to pay attention on the possible null-sets on which the functions possibly differ if we only work with continuous functions. We need the following theorems from integration theory: Theorem A.2 (Lebesgue’s Theorem on Dominated Convergence). Let fk 2 L1 .Rn /, k 2 N, be sequence of functions such that lim fk .x/ D f .x/ almost everywhere;

k!1

jfk .x/j  g.x/

almost everywhere;

where g 2 L1 .Rn /. Then f 2 L1 .Rn / and Z Z fk .x/dx D lim k!1 Rn

Rn

f .x/dx:

As an application of Lebesgue’s theorem on dominated convergence one can prove the following two theorems on parameter-dependent integrals, which will often be used. Theorem A.3. Let U Rm be an open set and a 2 U . Moreover, let f W Rn  U ! C be a function such that: 1. For all x 2 Rn the function t 7! f .x; t / is continuous in a. 2. For all t 2 U the function x 7! f .x; t / is integrable on Rn . 3. There is some F 2 L1 .Rn / such that jf .x; t /j  F .x/ for almost all x 2 Rn and all t 2 U . Then the function

Z U 3 t 7! g.t / WD

Rn

f .x; t /dx 2 C

is continuous in a. Theorem A.4. Let I R be an open interval and f W Rn  I ! C be such that 1. For every x 2 Rn the function t 7! f .x; t / is differentiable on I . 2. For every t 2 I the function x 7! f .x; t / is integrable on Rn . 3. There is some F 2 L1 .Rn / such that j@ t f .x; t /j  F .x/ for almost all x 2 Rn .

Section A.2 Lebesgue Integral and Lp -Spaces

203

Z

Then I 3 t 7! g.t / WD

Rn

f .x; t /dx 2 C

is a differentiable function on I and for all t 2 I Z 0 @ t f .x; t /dx: g .t / D Rn

The proofs of these theorems are left to the reader as an exercise. Alternatively, see [6, §11, Satz 1/Satz 2]. Theorem A.5 (Fubini’s Theorem). Let f 2 L1 .Rk  Rl /, k; l 2 N. Then f .x;  / 2 L1 .Rl / for almost all x 2 Rk , Z x 7! f .x; y/dy 2 L1 .Rk /; Rk

and

Z

Z

Z Rk Rl

f .x; y/d.x; y/ D

Rk

Rl

 f .x; y/dy dx:

Conversely, if f .x;  / 2 L1 .Rl / for almost all x 2 Rk and x 7! L1 .Rk /, then f 2 L1 .Rk  Rl / and (A.3) holds.

R Rk

(A.3) f .x; y/dy 2

We will also use the change-of-variable theorem in the following case: Z Z f .ˆ.x//jdet Dˆ.x/j dx D f .y/ dy Rn

Rn

for all f 2 L1 .Rn / and C 1 -diffeomorphisms ˆW Rn ! Rn . In particular, we have Z Z f .Ax C b/jdet Ajdx D f .y/dy Rn

Rn

for all f 2 L1 .Rn /, b 2 Rn , and A 2 Rnn with det A ¤ 0. We will also use the spaces Lp .Rn /, 1  p < 1, which consists of all measurable functions f W Rn ! K such that Z kf kLp .Rn / WD kf kp WD

p

Rn

jf .x/j dx

 p1

< 1:

Moreover, ® ¯ L1 .Rn / WD f W Rn ! K measurable W kf kL1 .Rn / < 1 kf kL1 .Rn / WD kf k1 WD ess sup jf .x/j WD x2Rn

inf

sup jf .x/j:

E Rn ;jE jD0 x2Rn nE

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Appendix A Basic Results from Analysis

Here jEj denotes the Lebesgue measure of E and the functions in L1 .Rn / are called essentially bounded. All these spaces are Banach spaces (again identifying functions which coincide almost everywhere). Moreover, we have the Hölder inequality kfgkL1 .Rn /  kf kLp .Rn / kgkLp0 .Rn /

0

for all f 2 Lp .Rn /; g 2 Lp .Rn /;

where 1  p  1 and 1  p 0  1 is determined by 1 D p1 C p10 with the convention 1 D 0”, i.e., p 0 D 1 if p D 1 and p 0 D 1 if p D 1. Throughout the book p0 will “1 always be defined by the latter equation. Furthermore, we note that L2 .Rn / is also a Hilbert space. Its norm and inner product are given by: Z kf kL2 .Rn / WD kf k2 WD .f; g/L2 .Rn / WD

1 2

Z

Rn

Rn

jf .x/j dx

2

;

f .x/g.x/dx

for all f; g 2 L2 .Rn /. The inner product of L2 .Rn / will play a fundamental role. If U Rn is open, we define the vector space p Lloc .U / D ¹f W U ! K measurable W f jK 2 Lp .K/ for all K U compactº

of all functions f W U ! K that are locally in Lp . Moreover, we say that f W U ! K is locally integrable if f 2 L1loc.U /. We will frequently use the following results: Lemma A.6. Let 1  p < 1. Then C01 .Rn / is dense in Lp .Rn /, i.e., for every f 2 Lp .Rn / there is a sequence fk 2 C01 .Rn /, k 2 N, such that limk!1 kf  fk kp D 0. Proof. See e.g. [1, Satz 2.10] or [6, §10 Satz 3]. Theorem A.7 (Fundamental Lemma of the Calculus of Variations). Let U  Rn be an open set and let f 2 L1loc.U / such that Z U

f .x/'.x/ dx D 0 for all ' 2 C01 .U /:

Then f .x/ D 0 for almost all x 2 U . Proof. See e.g. [11, Lemma 3.2] or [1, Lemma 2.21], where f 2 L1 ./ is assumed.

Section A.2 Lebesgue Integral and Lp -Spaces

205

Moreover, we note that for any f 2 Lp .Rn /, 1  p  1 ˇ ˇZ ˇ ˇ ˇ; ˇ kf kLp .Rn / D sup f .x/g.x/ dx ˇ ˇ g2Lp0 .Rn /;kgkp0 D1

Rn

(A.4)

where 1  p 0  1 such that p1 C p10 D 1. Here “ ” in (A.4) follows from the Hölder inequality. Moreover, if p < 1, “” can be shown by choosing ´ f .x/jf .x/jp2 if f .x/ ¤ 0; g.x/ D 0 else: The case p D 1 follows from choosing ´ 1

f .x/ g.x/ D jAj A jf .x/j 0

if f .x/ ¤ 0; else;

where A Rn is a measurable set of positive measure such that jf .x/j kf k1  " for almost all x 2 A and " > 0 is arbitrary. Theorem A.8. Let K.r; R/ D ¹x 2 Rn W r < jxj < Rº for some 0  r < R  1. Then f 2 L1 .K.r; R// if and only if g.s; '/ D f .r'/

for all s 2 .r; R/; ' 2 @B1 .0/

is measurable and satisfies Z

R r

Z @B1 .0/

jg.s; '/j d' s n1 ds < 1:

If this is the case Z

Z K.r;R/

f .x/ dx D

R r

Z f .r'/ d' s n1 ds: @B1 .0/

Here “ds” denotes integration with respect to the surface measure on @B1 .0/. A proof can be found e.g. in [20, Section 9.3 and “Aufgabe 11.8”]. The following simple lemma will often be used: Lemma A.9. Let s > n. Then hxis 2 L1 .Rn / and .1 C jxj/s 2 L1 .Rn /, where 1 hxi D .1 C jxj2 / 2 for all x 2 Rn . Proof. The lemma can easily be proved by Theorem A.8. An alternative approach can be found in [27, Lemma 1.3].

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Appendix A Basic Results from Analysis

Finally, we will need a discrete version of the Lebesgue spaces Lp once in a while: ® ¯ `p .Z/ D a D .aj /j 2Z W aj 2 K for all j 2 Z; kak`p .Z/ < 1 ´P 1 . j 2Z jaj jp / p if 1  p < 1; kak`p .Z/ D supj 2Z jaj j if p D 1; where 1  p  1. Moreover, the convolution a  b of a D .ak /k2Z 2 `1 .Z/ and b D .bk /k2Z 2 `r .Z/, 1  r  1 is defined by X akj bj for all k 2 Z: .a  b/k D j 2Z

Similarly to the convolution inequality for functions on Rn , cf. (2.5), we have ka  bk`r .Z/  kak`1 .Z/ kbk`r .Z/ :

(A.5)

The inequality can be proved in the same way as for the usual convolution, i.e., (2.5), using Hölder’s inequality.

A.3 Linear Operators and Dual Spaces In this section we summarize some basic facts from linear functional analysis, which will be needed in this book. We refer to [1, 30, 43, 45] for a systematic introduction. In the following let X; Y be Banach spaces. Then a linear mapping AW X ! Y is called bounded if there is some C > 0 such that kAxkY  C kxkX

for all x 2 X:

The set of bounded linear operators AW X ! Y is denoted by L.X; Y /. Moreover, we set L.X/ WD L.X; X/. We note that a linear mapping AW X ! Y is bounded if and only if the operator norm kT xkY kAkL.X;Y / D sup kT xkY D sup kxkX D1 x¤0 kxkX is finite. With this norm L.X; Y / becomes a Banach spaces. Moreover, we note that A is bounded if and only if A is continuous. (This also follows from Theorem A.23 below.) We will frequently use the following extension result: Lemma A.10. Let X; Y be Banach spaces, D  X be a dense linear subspace, and let T W D ! Y be a linear mapping that is bounded, i.e., there is some C > 0 such that kT xkY  C kxkX

for all x 2 D:

e 2 L.X; Y / such that T x D e Then there is a unique T T x for all x 2 D.

207

Section A.3 Linear Operators and Dual Spaces

Often the extension e T is again denoted by T . We refer to [1, Ü3.3] for a proof. More generally, an unbounded (linear) operator AW D.A/  X ! Y is a linear mapping AW D.A/ ! Y , where D.A/ is a linear subspace of X. – Note that every bounded linear operator is also an unbounded linear operator with D.A/ D X. – Furthermore, AW D.A/  X ! Y is called closed if for every sequence .xn /n2N D.A/ the following implication holds: xn !n!1 x

in X; Axn !n!1 y

in Y

)

x 2 D.A/; Ax D y:

We note that, if D.A/ D X, then AW X ! Y is closed if and only if A is bounded because of the closed graph theorem and since X; Y are Banach spaces. Furthermore, we note that, if AW D.A/  X ! Y is closed, then D.A/ equipped with the norm kxkD.A/ D kxkX C kAxkX

for all x 2 D.A/

is a Banach space. Then AW D.A/ ! Y is even a bounded operator.1 We call a linear operator AW D.A/  X ! Y invertible if it is bijective and A1 W Y ! X is bounded. We note that, since X; Y are Banach spaces, AW D.A/  X ! Y is invertible if and only if AW D.A/  X ! Y is closed and bijective. If X is a complex Banach space, then the spectrum of an unbounded linear operator AW D.A/  X ! X is defined as .A/ D ¹ 2 C W .I  A/W D.A/  X ! X is not invertibleº and .A/ WD C n .A/ is called the resolvent set of A. R .A/ WD .  A/1

for all  2 .A/

is called resolvent of A. A very useful and often used criterion for invertibility is: Theorem A.11 (Neumann Series). Let X be a Banach space and A 2 L.X/ such that kAkL.X/ < 1. Then I AW X ! X is invertible and .I  A/1 D

1 X

Ak ;

kD0

where the series converges in L.X/. A proof can be found in [1, Section 3.7] or [46, Section 1.23]. As a consequence one obtains that the set of invertible bounded linear operators in L.X; Y / is open. More precisely, 1

It is essential that here the norm of D.A/ is used and not the norm of X !

208

Appendix A Basic Results from Analysis

Lemma A.12. Let A 2 L.X; Y /, where X; Y are Banach spaces and let B 2 L.X; Y / be such that 1 : kA  BkL.X;Y / < 1 kA kL.Y;X/ Then B is invertible and B 1 D A1 .I  .A  B/A1 /1 ; where .I  .A  B/A1 /1 D

1 X

..A  B/A1 /k :

kD0

The proof is left to the reader as an exercise. Furthermore, we recall that the dual space X 0 of a Banach space X is the set all bounded linear operators x 0 W X ! K, i.e., X 0 D L.X; K/. The elements of X 0 are called (linear) functionals. Moreover, hx 0 ; xi  hx 0 ; xiX 0 ;X WD x 0 .x/ denotes the duality product of x 2 X, x 0 2 X 0 . Because of the Hahn–Banach theorem, X 0 separates points of X, i.e., for every x 2 X n ¹0º there is some x 0 2 X 0 such that hx 0 ; xi ¤ 0. Therefore one has the equivalence xD0

,

hx 0 ; xi D 0 for all x 0 2 X 0 :

If H is a Hilbert space, then J W H ! H 0 is defined by hJ x; yiH 0 ;H WD .y; x/H

for all x; y 2 H

is a (conjugate) linear isomorphism due to the Riesz representation theorem. In particular, this applies to L2 .Rn /: For every F 2 L2 .Rn /0 there is a unique f 2 L2 .Rn / such that Z hF; giL2 .Rn /0 ;L2 .Rn / D g.x/f .x/ dx for all g 2 L2 .Rn /: Rn

More generally, if 1  p < 1, then for every F 2 Lp .Rn /0 there is a unique 0 f 2 Lp .Rn / (where p1 C p10 D 1 as usual) such that Z g.x/f .x/ dx for all g 2 Lp .Rn /: hF; giLp .Rn /0 ;Lp .Rn / D Rn

A proof can be found in [1, Satz 4.12] or [23, Section 2.14]. Since L1 .Rn / is separable, it follows that bounded sets in L1 .Rn / Š .L1 .Rn /0 are weak- sequentially compact, i.e., for every bounded sequence .fk /k2N L1 .Rn / there is a subsequence .fkj /j 2N and some f 2 L1 .Rn / such that Z Z fkj .x/g.x/ dx !j !1 f .x/g.x/ dx for all g 2 L1 .Rn /: Rn

Rn

The theorem follows e.g. from [1, Satz 6.5]. This property will be used once in a while throughout the book. But no further results on weak(-) convergence will be needed.

Section A.4 Bochner Integral and Vector-Valued Lp -Spaces

209

A.4 Bochner Integral and Vector-Valued Lp -Spaces For some extensions and applications of the theory of singular integral operators the Bochner integral and Lp -spaces for Banach-space-valued functions will be needed. This concerns the Chapters 5 and 6 and the applications in Section 7.2.

A more detailed introduction to Bochner integrals can be found in [45]. Definition A.13. Let X be a Banach space and let / be a measure space. Then P.M; N a mapping f W M ! X is called simple if f D kD1 ˛k Ak for some measurable Ak  M and ˛k 2 X. Moreover, f W M ! X is called strongly measurable if there is a sequence fk W M ! X of simple functions fk , k 2 N, such that fk .x/ !k!1 f .x/ in X for almost every x 2 M . Moreover, f is called Bochner integrable if there is a sequence fk W M ! X, k 2 N, of simple functions such that Z lim kfk .x/  f .x/kX d.x/ D 0: (A.6) k!1 M

If f D

PN

kD1 ˛k Ak

is a simple function, then the X-valued integral is defined by

Z M

f .x/ d.x/ WD

N X

˛k .Ak / 2 X:

kD1

If f is  and fk , k 2 N, are simple functions such that (A.6) holds, RBochner integrable then M fk .x/ d.x/ k2N is a Cauchy sequence in X and we define the Bochner integral as Z Z f .x/ d.x/ WD lim fk .x/ d.x/: k!1 M

M

As in the scalar case, i.e., X D K, the definition is independent of choice of the sequence .fk /k2N . A useful criterion is the following: Theorem A.14 (Bochner). Let f W M ! X. Then f is Bochner integrable if and only if f is strongly measurable and x 7! kf .x/kX is integrable. Moreover,

Z Z





f .x/ d.x/  kf .x/kX d.x/: (A.7)

M

X

M

We refer to [45, Theorem 1, Chapter V, Section 5] for a proof.

i

210

Appendix A Basic Results from Analysis

From the definitions it easily follows that, if f W M ! X is Bochner integrable and A 2 L.X; Y /, then Af W M ! Y is Bochner integrable and Z Z A f .x/ d.x/ D Af .x/ d.x/: M

w0

X0

In particular, if 2 is integrable and

Z

M

and f W M ! X is Bochner integrable, then x 7! hf .x/; w0 i

f .x/ d.x/; w

0



M

Z D

M

hf .x/; w 0 i d.x/:

(A.8)

Here hw; w 0 i D w 0 .w/ denotes the duality product of w 2 X and w 2 X 0 . Using (A.7), one can generalize Lebesgue’s theorem on dominated convergence to Bochner integrals: Theorem A.15. Let fk ; f 2 L1 .M I X/, k 2 N0 , be such that 1. limk!1 fk .x/ D f .x/ in X for -almost every x 2 M , 2. kfk .x/kX  g.x/ for -almost every x 2 M for some g 2 L1 .M /. Then Z lim

k!1 M

Z fk .x/ d.x/ D

f .x/ d.x/: M

Proof. By the assumptions we have kfk .x/  f .x/kX !k!1 0 for almost every x 2 M and kfk .x/  f .x/kX  2g.x/

for almost every x 2 M:

Hence by Lebesgue’s theorem on dominated convergence (for scalar integrals) and (A.7)

Z Z Z





fk .x/ d.x/  f .x/ d.x/  kfk .x/  f .x/kX d.x/ !k!1 0;

M

M

X

M

which proves the theorem. Finally, if f 2 L1 .Rn I X/; g 2 L1 .Rn /, we define the convolution f  g 2 by Z f .x  y/g.y/ dy for almost every x 2 Rn : .f  g/.x/ D

L1 .Rn I X/

Rn

Section A.4 Bochner Integral and Vector-Valued Lp -Spaces

211

Lemma A.16. For every f 2 L1 .Rn I X/; g 2 L1 .Rn / the convolution f  g is well-defined and f  g 2 L1 .Rn I X/. Moreover, kf  gkL1 .Rn IX/  kf kL1 .Rn IX/ kgkL1 .Rn / :

(A.9)

Proof. First of all, if f is a simple function, then f  g is of the form .f  g/.x/ D

N X

gk .x/˛k

for almost all x 2 Rn

kD1

for some gk 2 L1 .Rn / and ˛k 2 X for all k D 1; : : : ; N . Hence f  g is a strongly measurable in this case. Moreover, Z Z Z kf  g.x/kX dx  kf .x  y/kX jg.y/j dy dx Rn

Rn

Rn

D kf kL1 .Rn IX/ kgkL1 .Rn / by Fubini’s theorem and the transformation formula. In particular this implies (A.9). Moreover, f  g is Bochner integrable by Theorem A.14. Now let f be Bochner integrable and let fk be as in (A.6). Then Z kfk .x  y/  f .x  y/kX jg.y/j dy kfk  g.x/  f  g.x/kX  M

Rn ,

for almost every x 2 which implies Z Z kfk  g.x/  f  g.x/kX dx  kfk .x/  f .x/kX dxkgkL1 .Rn / !k!1 0: Rn

Rn

Since fk  g is Bochner integrable by the first part, there are simple functions fk0 such that Z 1 kfk  g.x/  fk0 kX dx  : k Rn Altogether Z Z 1 0 kf  g.x/  fk kX dx  C kfk .x/  f .x/kX dxkgkL1 .Rn / !k!1 0: k Rn Rn Hence f  g is Bochner integrable. Finally (A.9) is proved in the same way as in the first part. Similarly to the scalar case we define for 1  p  1 the vector-valued Lebesgue spaces or Bochner spaces as Lp .M I X/ WD ¹f W M ! X strongly measurable W kf kX 2 Lp .M /º



kf kLp .M IX/ WD kf kX Lp .M / for all f 2 Lp .M I X/:

212

Appendix A Basic Results from Analysis

We note that simple functions are dense in Lp .M I X/ if 1  p < 1. Moreover, C01 .Rn I X/ is dense in Lp .Rn I X/ for any 1  p < 1 as in the scalar case. This can be proved easily by approximating a given f 2 Lp .Rn I X/ first by simple functions and approximating the simple functions by functions in C01 .Rn I X/. The last approximation can be done using that C01 .Rn / is dense in Lp .Rn / if p < 1. In the following we restrict ourselves for simplicity to subsets of Rn together with the Lebesgue measure. Furthermore, if U  Rn is open, we define as in the scalar case Lp loc .U I X/ D ¹f W U ! X strongly measurable W

f jK 2 Lp .KI X/ for all K U compactº:

Definition A.17. Let f 2 L1 .a; bI X/ WD L1 ..a; b/I X/, where 1  a < b  1. Then g 2 L1 .a; bI X/ is called weak derivative of f if Z a

Z

b

g.t /'.t / dt D 

b

f .t /' 0 .t / dt

a

for all ' 2 C01 ..a; b//:

Note that, if g 2 L1 .a; bI X/ is a weak derivative of f 2 L1 .a; bI X/, then x 7! gw .x/ WD hg.x/; wi is a weak derivative of the scalar function x 7! fw .x/ D hf .x/; wi for every w 2 X 0 . In particular, the weak derivative of f is unique if it exists. Therefore it will for simplicity be denoted by f 0 . Finally, we define the vector-valued Sobolev space of first order as: ® ¯ Wp1 .a; bI X/ WD f 2 Lp .a; bI X/ W f has a weak derivative f 0 2 Lp .a; bI X/ ; kf kWp1 .a;bIX/ WD kf kLp .a;bIX/ C kf 0 kLp .a;bIX/ : Moreover, we denote by C k .Œa; bI X/, k 2 N0 , the Banach space of all k-times differentiable functions f W Œa; b ! X equipped with a standard norm, e.g. kf kC k .Œa;bIX/ D and C 1 .Œa; bI X/ D

T k2N0

sup x2Œa;b;j D0;:::;k

kf .j / .x/kX

C k .Œa; bI X/.

A.5 Fréchet Spaces In the study of pseudodifferential operators one frequently works with different spaces of smooth functions with certain properties. They all have in common that they cannot be normed to become a Banach space (with their natural topology); but they are socalled Fréchet spaces.

213

Section A.5 Fréchet Spaces

As an example we consider the space Cb1 .Rn / of all smooth and bounded functions f W Rn ! C with bounded derivatives. Then Cb1 .Rn / D

1 \

Cbk .Rn /:

kD0

In contrast to Cb1 .Rn / the spaces Cbk .Rn /, k 2 N0 , are Banach spaces equipped with the norm kf kC k D sup sup j@˛x f .x/j: b

j˛jk x2Rn

Roughly speaking: Since in spaces of smooth functions all (infinitely many) derivatives have to be controlled for completeness, they cannot be normed by a single norm such that the space is complete usually. But these spaces can be “normed” by an infinite sequence of (semi-)norms, e.g. f 2 Cb1 .Rn / if and only if kf kC k is finite for b all k 2 N. Definition A.18. Let V be a (complex or real) linear space over K. Then a mapping W V ! Œ0; 1/ is a semi-norm if 1. .rf / D jrj.f / for all f 2 V and r 2 K. 2. .f C g/  .f / C .g/ for all f; g 2 V . If m , m 2 N, is a sequence of semi-norms on a linear space V and fj 2 V , j 2 N, we say that .fj /j 2N is a Cauchy sequence if limi;j !1 m .fi  fj / D 0 for all m 2 N. Moreover, we say that .fj /j 2N converges to f 2 V if and only if limj !1 m .fj  f / D 0 for all m 2 N. Definition A.19. A linear space V is called a Fréchet space if there is a sequence of semi-norms .m /m2N satisfying the following conditions: 1. .m /m2N is an increasing sequence: m .f /  mC1 .f / for all m 2 N and f 2V. 2. .m /m2N separates points: for any f ¤ 0 there is some m 2 N such that m .f / ¤ 0. 3. V is complete: any Cauchy sequence .fj /j 2N converges to some f 2 V . Obviously, every Banach space X with norm k  kX is a Fréchet space with (semi-) norms m D k  kX for all m 2 N.

214

Appendix A Basic Results from Analysis

We note that Cb1 .Rn / is a Fréchet space, cf. Exercise A.27. Moreover, we have: Lemma A.20. The space S.Rn /, cf. Section 2.2, together with the j  jm;S , m 2 N, is a Fréchet space. The proof is left as exercise for the interested reader. We just note that it mainly remains to check the completeness of S.Rn /, which can be done by using the completeness of Cb1 .Rn /. Remark A.21 (For readers interested in topology). Every Fréchet space .V; .m /m2N / is a complete metric space .V; d / where d.f; g/ WD

1 X mD1

2m

m .f  g/ 1 C m .f  g/

for all f; g 2 V:

(A.10)

Then convergence with respect to the semi-norms coincides with convergence in the metric sense. Since every metric space carries a natural topology which is defined by the neighborhood basis ³ ² 1 ; l 2 N; f0 2 V; Ul .f0 / D f 2 V W d.f  f0 / < l every Fréchet space is a topological space. Equipped with this topology, the operations of multiplication by scalars and addition of vectors are continuous, i.e., V is a linear topological space. Moreover, every Fréchet space is locally convex, i.e., there is a neighborhood basis at the origin that consists of convex sets. Finally, the Fréchet spaces can be characterized as the locally convex linear topological space that can be equipped with a metric and are complete. See e.g. [36]. Definition A.22. Let V and W be Fréchet spaces with semi-norms .m /m2N and .m /m2N . Then a linear mapping T W V ! W is bounded if for every m 2 N there is some k D k.m/ 2 N and some Cm such that m .Tf /  Cm k.m/ .f / for all f 2 V: The set of all bounded linear operators T W V ! W is denoted by L.V; W / and L.V / WD L.V; V /. Theorem A.23. Let V and W be Fréchet spaces and T W V ! W be a linear mapping. Then T is bounded if and only if T is continuous in the sense that limj !1 fj D f implies limj !1 Tfj D Tf . In particular, if W is a Banach space, T W V ! W is continuous if and only if there is some m 2 N and C > 0 such that kTf kW  Cm .f /

for all f 2 V:

215

Section A.5 Fréchet Spaces

Proof. Let m and m , m 2 N, denote the semi-norms of V and W , respectively. First, let T be bounded and .fj /j 2N V be a sequence such that limj !1 fj D f in V , i.e., limj !1 m .fj  f / D 0 for all m 2 N. Since T is bounded for every m 2 N there exists some k.m/ 2 N such that m .T g/  Cm k.m/ .g/ for all g 2 V . Hence m .Tfj  Tf / D m .T .fj  f //  Cm k.m/ .fj  f / ! 0 as j ! 1; where m 2 N is arbitrary. This means limj !1 Tfj D Tf in W . Thus T is continuous. Conversely, let T be continuous. Assume that T is not bounded. Then there is some m0 2 N and some sequence .fj /j 2N V such that for every C > 0 and k 2 N there is some j0 2 N0 with m0 .Tfj0 / > Ck .fj0 /: In particular, we can choose a subsequence .fjk /k2N , such that m0 .Tfjk / > kk .fjk /: Then fQk WD fjk =m0 .Tfjk /, k 2 N, is a sequence such that m .fQk /  k .fQk / <

1 k

for all m  k 2 N:

Thus limk!1 m .fQk / D 0 for all m 2 N, i.e., limk!1 fQk D 0 in V . Hence limk!1 T fQk D 0 because of the continuity of T . On the other hand m0 .T fQk / D

m0 .T fQk / D 1 for all k 2 N; m .T fQk / 0

which contradicts limk!1 fQk D 0. Finally, we note that a bilinear mapping BW U  V ! W is bounded with respect to some Frechét spaces .U; .k /k2N /; .V; .k /k2N /; .W; .k /k2N / if for every k 2 N there are some Ck > 0 and some n.k/ 2 N such that k .B.u; v//  Ck n.k/ .u/n.k/ .v/

for all u 2 U; v 2 V:

As for linear mappings B is bounded if and only if B is continuous.

216

Appendix A Basic Results from Analysis

A.6 Exercises Exercise A.24. Proof Lemma A.12. Exercise A.25. Let I R be an open interval, X be a Banach space, and let f W Rn  I ! X be such that: 1. For every x 2 Rn the function I 3 t 7! f .x; t / 2 X is differentiable on I . 2. For every t 2 I the function Rn 3 x 7! f .x; t / 2 X is integrable on Rn . 3. There is some F 2 L1 .Rn / such that k@ t f .x; t /kX  F .x/ for almost all x 2 Rn . Prove that

Z I 3 t 7! g.t / WD

Rn

f .x; t /dx 2 X

is a differentiable function on I and for all t 2 I Z 0 @ t f .x; t /dx: g .t / D Rn

Exercise A.26. Let .V; .m /m2N / be a Fréchet space. 1. Prove that d defined in (A.10) is a metric. 2. Prove that limj !1 d.fj ; f / if and only if m .fj  f / D 0 for all m 2 N. Exercise A.27. Prove that Cb1 .Rn / equipped with the norms m D k  kCbm , m 2 N0 , is a Fréchet space. – It can be used that Cbk .Rn / is a Banach space for every k 2 N0 . Exercise A.28. A subset A of a Fréchet space V is called bounded if supf 2A m .f / < 1 for every m 2 N. – Prove that T W V ! W is bounded if and only if T .A/ is bounded for every bounded A V . Exercise A.29. Let Vk , k 2 N, be a countable family of Fréchet spaces such that T VkC1  Vk for all k 2 N with continuous embeddings. Prove that V WD k2N Vk is a Fréchet space. V is called projective limes of Vk , k 2 N.

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Index

BMO.Rn / 113 s .Rn / 153 Bpq ˛ C .Rn / 5 Cs .Rn / 154 C m;˛ .Rn / 5 1 .Rn / 13 Cpoly H s .Rn / 28 Hps .Rn / 30 L1 -modulus of continuity 104 L1weak .Rn / 89, 119 p Lloc 204 m S1;0 .RN  Rn / 40 Wp1 .a; bI X / 212 Wpm .Rn / 27 n n Am  .R  R / 46 0 n S .R / 20 S.Rn / 13 †ı 178 `p .Z/ 206 `sq .MI X / 154 H 1 -atoms 114 H p .Rn / 113 amplitudes 46 Besov space 153 Bessel potential space 28, 30 Bochner integrable 209 Bochner integral 209 Bochner spaces 211 Calderón–Zygmund decomposition 91, 126 convolution 20

convolution of functions 12, 211 convolution of sequences 206 convolution with tempered distribution 26 delta distribution 21 derivative of a distribution 21 dilation of a function 10 distribution, regular tempered 21 distributional derivative 21 distributions, tempered 20 dual space of a Banach space 208 duality product 208 dyadic cubes 91 dyadic partition of unity 107, 134, 150 elliptic pseudodifferential symbol 57 Fourier multiplication operators 86 Fourier transform of a tempered distribution 24 Fourier transformation 9 Fourier transformation on S 0 .Rn/ 24 Fourier transformation, vector-valued 30 functions of bounded mean oscillations 113 functions, smooth and polynomially bounded 13 fundamental solution 26 Hölder inequality 204 Hölder–Zygmund spaces 154 Hörmander condition 88 Hardy space 113 Heaviside function 22 Hilbert transform 19, 106

222 homogeneous function 34 hypo-elliptic 142 inverse Fourier of a tempered distribution 24 inverse Fourier transformation 16 Inversion Formula 16, 32 Lebesgue spaces, vector-valued 211 Lebesgue’s differentiation theorem 94 Lebesgue’s theorem on dominated convergence 210 linear operator, bounded 206 linear operator, closed 207 linear operator, unbounded 207 locally integrable 204 manifold, smooth and compact 76 Marcinkiewicz interpolation theorem 96 Marcinkiewicz interpolation theorem, vector-valued 128 maximal operator 92, 93 Mikhlin multiplier theorem 107 Mikhlin multiplier theorem, Hilbertspace-valued 129 parameter-elliptic 178, 189 Peetre’s inequality 44 Plancherel’s Theorem 17, 33 principle value distributions 100 product with tempered distribution 21 pseudo-local 142 pseudodifferential operator 41

Index pseudodifferential operator, .x; y/form 66 pseudodifferential operator, x-form 66 pseudodifferential operator, y-form 66 pseudodifferential symbol 40 quasi-norm 119 quasi-normed linear space 119 rapidly decreasing functions 13 resolvent of a linear operator 207 resolvent set of a linear operator 207 retract 159 Riesz operator 19, 100, 106 Schwartz functions 13 semi-group, analytic 193 sharp function 113 singular support 142 Sobolev space 27 Sobolev space, vector-valued 212 spectrum of a linear operator 207 strongly measurable 209 support of a continuous function 23 support of a distribution 23 translation invariant operator 88 translation of a function 10 Triebel–Lizorkin space 157 weak derivative 212 weak type-.1; 1/ estimate 89 weak- compactness in L1 .Rn / 208