Pseudodifferential and Singular Integral Operators: An Introduction with Applications

by Helmut Abels

Overview

This textbook provides a self-contained and elementary introduction to the modern theory of pseudodifferential operators and their applications to partial differential equations.

In the first chapters, the necessary material on Fourier transformation and distribution theory is presented. Subsequently the basic calculus of pseudodifferential operators on the n-dimensional Euclidean space is developed. In order to present the deep results on regularity questions for partial differential equations, an introduction to the theory of singular integral operators is given - which is of interest for its own. Moreover, to get a wide range of applications, one chapter is devoted to the modern theory of Besov and Bessel potential spaces. In order to demonstrate some fundamental approaches and the power of the theory, several applications to wellposedness and regularity question for elliptic and parabolic equations are presented throughout the book. The basic notation of functional analysis needed in the book is introduced and summarized in the appendix.

The text is comprehensible for students of mathematics and physics with a basic education in analysis.

Product Details

ISBN-13:	9783110250305
Publisher:	De Gruyter
Publication date:	12/22/2011
Series:	De Gruyter Textbook
Pages:	232
Product dimensions:	6.70(w) x 9.30(h) x 0.50(d)
Age Range:	18 Years

About the Author

Helmut Abels, University of Regensburg, Germany.

Table of 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'(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

2.9.1 Further Reading 33

2.9.2 Exercises 34

3 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

3.5 Composition of Pseudodifferential Operators 54

3.6 Application: Elliptic Pseudodifferential Operators and Parametrices 57

3.7 Boundedness on C_b^∞ (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 Calderon-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

4.8.1 Further Reading 118

4.8.2 Exercises 118

5 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

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 Applications to Function Space and Differential Equations

6 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 171

7.1.1 Resolvent of the Laplace Operator 171

7.1.2 Spectrum of Multiplier Operators with Homogeneous Symbols 174

7.1.3 Spectrum of a Constant Coefficient Differential Operator 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 186

7.3.1 Elliptic Regularity for Elliptic Pseudodifferential Operators 186

7.3.2 Resolvents of Parameter-Elliptic Differential Operators 188

7.3.3 Application of Resolvent Estimates to Parabolic Initial Value Problems 193

7.4 Final Remarks and Exercises 194

7.4.1 Further Reading 194

7.4.2 Exercises 195

IV Appendix

A Basic Results from Analysis 199

A.1 Notation and Functions on Rⁿ 199

A.2 Lebesgue Integral and L^p-Spaces 201

A.3 Linear Operators and Dual Spaces 206

A.4 Bochner Integral and Vector-Valued L^p-Spaces 209

A.5 Fréchet Spaces 212

A.6 Exercises 216

Bibliography 217

Index 221