Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations (Operator Theory: Advances and Applications)

✍ Scribed by Bert-Wolfgang Schulze, Man Wah Wong

• Publisher: Birkhäuser Basel
• Year: 2010
• Tongue: English
• Leaves: 294
• Series: Operator Theory: Advances and Applications
• Edition: 1st Edition.
• Category: Library

✦ Synopsis

This volume is an outgrowth of the international workshop entitled "Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations" held at York University on August 4–8, 2008. It consists of the expository paper based on the 6-hour minicourse given by Professor Bert-Wolfgang Schulze, and sixteen papers based on lectures given at the workshop and on invitations. While the focus is on the current developments of pseudo-differential operators in the context of complex analysis and partial differential equations, other topics related to the analysis, applications and computations of pseudo-differential operators are featured.

✦ Table of Contents

Title page......Page 4
Copyright Page......Page 5
Table of Contents
......Page 6
Preface......Page 8
1. Introduction......Page 9
2. Interior and Boundary Symbols for Differential Operators......Page 10
3. Inverses of Boundary Symbols......Page 17
4. Pseudo-Differential Boundary Value Problems......Page 33
5. Ellipticity of Boundary Value Problems......Page 44
6. The Anti-Transmission Property......Page 53
References......Page 56
1. SG Pseudo-Differential Operators......Page 59
2. L2 Spectral Invariance......Page 62
3. Lp Spectral Invariance......Page 63
References......Page 64
1. Introduction......Page 66
2.1. Edge-Degenerate Families on a Smooth Compact Manifold......Page 67
2.2. Continuity in Schwartz Spaces......Page 71
2.3. Leibniz Products and Remainder Estimates......Page 73
3.1. Weighted Cylindrical Spaces......Page 84
3.2. Elements of the Calculus......Page 85
References......Page 86
1. Introduction......Page 88
2. FIO Reduction of Wt......Page 90
3. Perturbations of Tensor Products of Harmonic Oscillators......Page 93
References......Page 95
1. Introduction......Page 98
2. Weyl’s Lemma for the Dunkl Laplacian......Page 102
3. Application: Converse Mean Value......Page 104
References......Page 107
1. Introduction......Page 108
2. Decompositions of Functions......Page 110
3. Higher-Order Pompeiu Operators......Page 113
4. Higher-Order Poisson Kernels and Homogeneous Equations......Page 114
5. Inhomogeneous Equations......Page 125
References......Page 134
1. Introduction......Page 136
2. Preliminaries......Page 137
3. Review of Polyharmonic Green Functions and Dirichlet Problems for the n-Poisson Equation......Page 138
4. A Class of Integral Operators Related to Dirichlet Problems......Page 141
4.1. Properties of the Operators Gk,lm,nf......Page 142
5. Dirichlet Problem for the Generalized Higher-Order Poisson Equation......Page 145
References......Page 147
Schwarz, Riemann, Riemann–Hilbert Problems and Their Connections in Polydomains......Page 149
1. Introduction......Page 150
2.1. The Formulation of the Problem for C2......Page 153
2.2. The Problem Formulations for Cn......Page 155
2.3. The Schwarz Problem for Polydomains......Page 159
2.4.2. Higher-Dimensional Space......Page 160
2.5. The Schwarz Problem Without Solvability Conditions......Page 161
2.6. A Necessary and Sufficient Condition for the Boundary Values of Holomorphic Functions on the Torus Domains......Page 162
3.1. The Plemelj–Sokhotkzi Formula......Page 163
3.2. The Formulation of the Riemann Problem......Page 164
3.3. The Homogeneous Riemann Problem......Page 165
4. The Riemann–Hilbert Problem......Page 166
4.1. The Well-Posed Formulation of the Riemann–Hilbert Problem for Polydomains......Page 167
4.2. Solution of the Problem......Page 168
5. The Connection......Page 170
References......Page 171
1. Introduction......Page 173
2. Multilinear Rihaczek Transforms and Multilinear Pseudo-Differential Operators......Page 176
3. The Moyal Identity and L2-Boundedness......Page 177
4. Lp-Boundedness, 1 ≤ p ≤ 2......Page 178
5. Multilinear Wigner Transforms and Multilinear Weyl Transforms......Page 179
6. A Basic Connection......Page 182
7. Lp-Boundedness, 1 ≤ p < ∞......Page 183
References......Page 185
1. Introduction......Page 187
2. A Characterization of Nuclear Operators on Lp(μ), 1 ≤ p < ∞......Page 188
3. Calculus of the Trace on Lp(μ)......Page 192
References......Page 199
1. Introduction......Page 200
2. The Main Results......Page 202
3. Trace Class Norm Inequalities for Two-Wavelet Multipliers......Page 207
4. The Generalized Landau–Pollak–Slepian Operator......Page 212
References......Page 215
1. Introduction......Page 217
2. Hilbert–Schmidt Operators......Page 218
3. Lp-Boundedness and Lp-Compactness, 1 ≤ p < ∞......Page 219
4. Almost Diagonalization......Page 223
References......Page 225
1. Introduction......Page 226
2. Preliminaries......Page 228
3. Pseudo-Differential Operators with Symbols in Modulation Spaces......Page 232
References......Page 235
1. Introduction......Page 238
1.1. What is Phase Space?......Page 239
1.3. Wave Equations, Modes, Group Velocity, the Stationary Phase Approximation and the Phase-Space Approximation
......Page 240
2. Transforming Differential Equations into Phase Space......Page 241
2.1.1. Schr¨odinger Free Particle Equation
......Page 245
3. Phase-Space Approximation......Page 246
3.2. Differential Equations for the Approximations......Page 248
3.3.2. Diffusion Equation with Drift......Page 249
3.3.4. Linearized KdV Equation
......Page 250
4. Exact Differential Equation for a Mode......Page 251
References......Page 252
1. Introduction......Page 254
2. Two-Window Spectrogram and Cohen Kernel......Page 257
3. Integrated Spectrogram: Motivations......Page 262
4. Boundedness of the Integrated Spectrogram......Page 264
5. Basic Properties of the Integrated Spectrogram......Page 266
References......Page 270
1. Introduction......Page 272
2. Time-Time Distribution for the Discrete Wavelet Transform......Page 274
3. Conclusions......Page 277
References......Page 279
1. Introduction......Page 280
2. The Stockwell Transform and the Morlet Wavelet Transform......Page 282
3.1. Power Spectral Density......Page 283
3.3. Phase-Locking Value......Page 284
3.4. Remarks......Page 285
4.1. Simulations......Page 286
4.2. An Application in Magnetoencephalography
......Page 289
References......Page 292