Pseudodifferential and Singular Integral Operators (eBook)

lt;!doctype html public "-//w3c//dtd html 4.0 transitional//en"> Helmut Abels, University of Regensburg, Germany.

Contents 8
Preface 6
1 Introduction 12
I Fourier Transformation and Pseudodifferential Operators 18
2 Fourier Transformation and Tempered Distributions 20
2.1 Definition and Basic Properties 20
2.2 Rapidly Decreasing Functions – P (Rn) 
24 
2.3 Inverse Fourier Transformation and Plancherel’s Theorem 26
2.4 Tempered Distributions and Fourier Transformation 31
2.5 Fourier Transformation and Convolution of Tempered Distributions 34
2.6 Convolution on on P'(Rn) and Fundamental Solutions 36
2.7 Sobolev and Bessel Potential Spaces 38
2.8 Vector-Valued Fourier-Transformation 41
2.9 Final Remarks and Exercises 44
2.9.1 Further Reading 44
2.9.2 Exercises 45
3 Basic Calculus of Pseudodifferential Operators on Rn 
51 
3.1 Symbol Classes and Basic Properties 51
3.2 Composition of Pseudodifferential Operators: Motivation 56
3.3 Oscillatory Integrals 57
3.4 Double Symbols 62
3.5 Composition of Pseudodifferential Operators 65
3.6 Application: Elliptic Pseudodifferential Operators and Parametrices 68
3.7 Boundedness on Cb8 (Rn) and Uniqueness of the Symbol 
74 
3.8 Adjoints of Pseudodifferential Operators and Operators in (x, y )-Form 76
3.9 Boundedness on L2(Rn) and L2-Bessel Potential Spaces 
79 
3.10 Outlook: Coordinate Transformations and PsDOs on Manifolds 85
3.11 Final Remarks and Exercises 88
3.11.1 Further Reading 88
3.11.2 Exercises 89
II Singular Integral Operators 94
4 Translation Invariant Singular Integral Operators 96
4.1 Motivation 96
4.2 Main Result in the Translation Invariant Case 98
4.3 Calderón-Zygmund Decomposition and the Maximal Operator 102
4.4 Proof of the Main Result in the Translation Invariant Case 106
4.5 Examples of Singular Integral Operators 111
4.6 Mikhlin Multiplier Theorem 118
4.7 Outlook: Hardy spaces and BMO 123
4.8 Final Remarks and Exercises 129
4.8.1 Further Reading 129
4.8.2 Exercises 129
5 Non-Translation Invariant Singular Integral Operators 133
5.1 Motivation 133
5.2 Extension to Non-Translation Invariant and Vector-Valued Singular Integral Operators 135
5.3 Hilbert-Space-Valued Mikhlin Multiplier Theorem 140
5.4 Kernel Representation of a Pseudodifferential Operator 144
5.5 Consequences of the Kernel Representation 151
5.6 Final Remarks and Exercises 154
5.6.1 Further Reading 154
5.6.2 Exercises 155
III Applications to Function Space and Differential Equations 158
6 Introduction to Besov and Bessel Potential Spaces 160
6.1 Motivation 160
6.2 A Fourier-Analytic Characterization of Holder Continuity 161
6.3 Bessel Potential and Besov Spaces – Definitions and Basic Properties 164
6.4 Sobolev Embeddings 171
6.5 Equivalent Norms 173
6.6 Pseudodifferential Operators on Besov Spaces 175
6.7 Final Remarks and Exercises 179
6.7.1 Further Reading 179
6.7.2 Exercises 179
7 Applications to Elliptic and Parabolic Equations 182
7.1 Applications of the Mikhlin Multiplier Theorem 182
7.1.1 Resolvent of the Laplace Operator 182
7.1.2 Spectrum of Multiplier Operators with Homogeneous Symbols 185
7.1.3 Spectrum of a Constant Coefficient Differential Operator 188
7.2 Applications of the Hilbert-Space-Valued Mikhlin Multiplier Theorem 191
7.2.1 Maximal Regularity of Abstract ODEs in Hilbert Spaces 191
7.2.2 Hilbert-Space Valued Bessel Potential and Sobolev Spaces 196
7.3 Applications of Pseudodifferential Operators 197
7.3.1 Elliptic Regularity for Elliptic Pseudodifferential Operators 197
7.3.2 Resolvents of Parameter-Elliptic Differential Operators 199
7.3.3 Application of Resolvent Estimates to Parabolic Initial Value Problems 204
7.4 Final Remarks and Exercises 205
7.4.1 Further Reading 205
7.4.2 Exercises 206
IV Appendix 208
A Basic Results from Analysis 210
A.1 Notation and Functions on Rn 
210 
A.2 Lebesgue Integral and Lp-Spaces 212
A.3 Linear Operators and Dual Spaces 217
A.4 Bochner Integral and Vector-Valued Lp-Spaces 220
A.5 Fréchet Spaces 223
A.6 Exercises 227
Bibliography 228
Index 232