Global Pseudo-differential Calculus on Euclidean Spaces (eBook)

Fabio Nicola, Luigi Rodino (Autoren)

eBook Download: PDF

2011 | 2010
X, 306 Seiten
Springer Basel (Verlag)
978-3-7643-8512-5 (ISBN)

Infos

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.

Inhalt

Table of Contents 6
Preface 9
Introduction 11
Background Material 18
0.1 Basic Facts and Notation 18
0.2 Function Spaces and Fourier Transform 20
0.3 Identities and Inequalities for Factorials and Binomial Coefficients 22
Chapter 1 Global Pseudo-Differential Calculus 24
Summary 24
1.1 Symbol Classes 28
1.2 Basic Calculus 32
1.2.1 Action on S 32
1.2.2 Adjoint and Transposed Operator. Action on S 38
1.2.3 Composition of Operators 40
1.3 Global Regularity 43
1.3.1 Hypoellipticity and Construction of the Parametrix 44
1.4 Boundedness on L2 49
1.5 Sobolev Spaces 50
1.6 Fredholm Properties 54
1.6.1 Abstract Theory 55
1.6.2 Pseudo-Differential Operators 59
1.7 Anti-Wick Quantization 60
1.7.1 Short-Time Fourier Transform and Anti-Wick Operators 61
1.7.2 Relationship with the Weyl Quantization 63
1.7.3 Applications to Boundedness on L2 and Almost Positivity of Pseudo-Differential Operators 67
1.7.4 Sobolev Spaces Revisited 69
1.8 Quantizations of Polynomial Symbols 71
Notes 73
Chapter 2 G-Pseudo-Differential Operators and H-Polynomials 76
Summary 76
2.1 G-Pseudo-Differential Operators 79
2.2 G-Elliptic Differential Operators the Harmonic Oscillator86
2.3 Asymptotic Integration and Solutions of Exponential Type 91
2.4 H-Polynomials 97
2.5 Quasi-Elliptic Polynomials 103
2.6 Multi-Quasi-Elliptic Polynomials 106
2.7 GP -Pseudo-Differential Operators 115
2.8 Lp-Estimates 125
Chapter 3 G-Pseudo-Differential Operators 137
Summary 137
3.1 G-Pseudo-Differential Calculus 140
3.2 Polyhomogeneous G-Operators 145
3.3 G-Elliptic Ordinary Differential Operators 153
3.4 Other Classes of Globally Regular Operators 156
Notes 158
Chapter 4 Spectral Theory 160
Summary 160
4.1 Unbounded Operators in Hilbert spaces 162
4.2 Pseudo-Differential Operators in L2: Realization and Spectrum 165
4.3 Complex Powers 171
4.3.1 The Resolvent Operator 173
4.3.2 Proof of Theorem 4.3.6 180
4.4 Hilbert-Schmidt and Trace-Class Operators 184
4.5 Heat Kernel 200
4.6 Weyl Asymptotics 202
Notes 206
Chapter 5 Non-Commutative Residue and Dixmier Trace 209
Summary 209
5.1 Non-Commutative Residue for G-Operators 212
5.2 Trace Functionals for G-Operators 219
5.3 Dixmier Traceability for General Pseudo-Differential Operators 227
Notes 230
Chapter 6 Exponential Decay and Holomorphic Extension of Solutions 232
Summary 232
6.1 The Function Spaces Sµ. (Rd) 234
6.2 G-Operators and Semilinear Harmonic Oscillators 246
6.3 G-Pseudo-Differential Operators on Sµ. (Rd) 261
6.4 A Short Survey on Travelling Waves 276
6.5 Semilinear G-Equations 280
Notes 290
Bibliography 292
Index 305

Produkt-Details

• Erscheint lt. Verlag: 30.1.2011
• Reihe/Serie: Pseudo-Differential Operators
• Reihe/Serie: Pseudo-Differential Operators
• Zusatzinfo: X, 306 p.
• Verlagsort: Basel
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Technik
• Schlagworte: Calculus • differential equation • hypoelliptic operator • Partial differential equations • pseudo-differential calculus • Schrödinger operator
• ISBN-10: 3-7643-8512-X / 376438512X
• ISBN-13: 978-3-7643-8512-5 / 9783764385125