Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations (eBook)

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