PSEUDODIFFERENTIAL OPERATORS

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