Global Pseudo-differential Calculus on Euclidean Spaces (Pseudo-Differential Operators)

✍ Scribed by Fabio Nicola, Luigi Rodino

• Publisher: Birkhäuser Basel
• Year: 2010
• Tongue: English
• Leaves: 317
• Series: Pseudo-Differential Operators volume 4
• Edition: 1st Edition.
• Category: Library

✦ Synopsis

This book presents a global pseudo-differential calculus in Euclidean spaces, which includes SG as well as Shubin classes and their natural generalizations containing Schroedinger operators with non-polynomial potentials. This calculus is applied to study global hypoellipticity for several pseudo-differential operators. The book includes classic calculus as a special case. It will be accessible to graduate students and of benefit to researchers in PDEs and mathematical physics.

✦ Table of Contents

Cover......Page 1
Pseudo-Differential Operators, 4......Page 3
Global Pseudo-differential Calculus on Euclidean Spaces......Page 4
ISBN 9783764385118......Page 5
Table of Contents
......Page 6
Preface......Page 10
Introduction......Page 13
0.1 Basic Facts and Notation......Page 21
0.2 Function Spaces and Fourier Transform......Page 23
0.3 Identities and Inequalities for Factorials and Binomial Coefficients
......Page 25
Summary......Page 27
1.1 Symbol Classes......Page 31
1.2.1 Action on S......Page 35
1.2.2 Adjoint and Transposed Operator. Action on S......Page 41
1.2.3 Composition of Operators......Page 43
1.3 Global Regularity......Page 46
1.3.1 Hypoellipticity and Construction of the Parametrix......Page 47
1.4 Boundedness on L2......Page 52
1.5 Sobolev Spaces......Page 53
1.6 Fredholm Properties......Page 57
1.6.1 Abstract Theory......Page 58
1.6.2 Pseudo-Differential Operators......Page 62
1.7 Anti-Wick Quantization......Page 63
1.7.1 Short-Time Fourier Transform and Anti-Wick Operators......Page 64
1.7.2 Relationship with the Weyl Quantization......Page 66
1.7.3 Applications to Boundedness on L2 and Almost Positivity of Pseudo-Differential Operators......Page 70
1.7.4 Sobolev Spaces Revisited......Page 72
1.8 Quantizations of Polynomial Symbols......Page 74
Notes......Page 76
Summary......Page 79
2.1 Γ-Pseudo-Differential Operators......Page 82
2.2 Γ-Elliptic Differential Operators; the Harmonic Oscillator......Page 89
2.3 Asymptotic Integration and Solutions of Exponential Type......Page 94
2.4 H-Polynomials......Page 100
2.5 Quasi-Elliptic Polynomials......Page 106
2.6 Multi-Quasi-Elliptic Polynomials......Page 109
2.7 ΓP -Pseudo-Differential Operators......Page 118
2.8 Lp-Estimates......Page 128
Summary......Page 141
3.1 G-Pseudo-Differential Calculus......Page 144
3.2 Polyhomogeneous G-Operators......Page 149
3.3 G-Elliptic Ordinary Differential Operators......Page 157
3.4 Other Classes of Globally Regular Operators......Page 160
Notes......Page 162
Summary......Page 165
4.1 Unbounded Operators in Hilbert spaces......Page 167
4.2 Pseudo-Differential Operators in L2: Realization and Spectrum......Page 170
4.3 Complex Powers......Page 176
4.3.1 The Resolvent Operator......Page 178
4.3.2 Proof of Theorem 4.3.6......Page 185
4.4 Hilbert-Schmidt and Trace-Class Operators......Page 189
4.5 Heat Kernel......Page 205
4.6 Weyl Asymptotics......Page 207
Notes......Page 211
Summary......Page 215
5.1 Non-Commutative Residue for Γ-Operators......Page 218
5.2 Trace Functionals for G-Operators......Page 225
5.3 Dixmier Traceability for General Pseudo-Differential Operators......Page 233
Notes......Page 236
Summary......Page 239
6.1 The Function Spaces Sμν (Rd)......Page 241
6.2 Γ-Operators and Semilinear Harmonic Oscillators......Page 253
6.3 G-Pseudo-Differential Operators on Sμν (Rd)......Page 268
6.4 A Short Survey on Travelling Waves......Page 283
6.5 Semilinear G-Equations......Page 287
Notes......Page 297
Bibliography......Page 299
Index......Page 313