Pseudo-Differential Operators and Markov Processes Volume 1. Fourier Analysis and Semigroups

✍ Scribed by Niels Jacob, N. Jacob

• Publisher: ICP
• Year: 2001
• Tongue: English
• Leaves: 517
• Category: Library

✦ Synopsis

After recalling essentials of analysis — including functional analysis, convexity, distribution theory and interpolation theory — this book handles two topics in detail: Fourier analysis, with emphasis on positivity and also on some function spaces and multiplier theorems; and one-parameter operator semigroups with emphasis on Feller semigroups and Lp-sub-Markovian semigroups. In addition, Dirichlet forms are treated. The book is self-contained and offers new material originated by the author and his students.

✦ Table of Contents

Contents
Preface
Notation
General Notation
Functions and Distributions
Measures and Integrals
Spaces of Functions, Measures and Distributions
Some Families of Functions
Norms, Scalar Products and Seminorms
Notation from Functional Analysis, Operators
Introduction: Pseudo Differential Operators and Markov Processes
Part I. Fourier Analysis and Semigroups
Chapter 1. Introduction
Chapter 2. Essentials from Analysis
2.1 Calculus Results
2.2 Some Topology
2.3 Measure Theory and Integration
2.4 Convexity
2.5 Analytic Functions
2.6 Functions and Distributions
2.7 Some Functional Analysis
2.8 Some Interpolation Theory
Chapter 3. Fourier Analysis and Convolution Semigroups
3.1 The Fourier Transform in S(R^n)
3.2 The Fourier Transform in L^p(R^n), 1 <= p <= 2
3.3 The Fourier Transform in S'(R^n)
3.4 The Paley-Wiener-Schwartz Theorem
3.5 Bounded Borel Measures and Positive Definite Functions
3.6 Convolution Semigroups and Negative Definite Functions
3.7 The Lévy-Khinchin Formula for Continuous Negative Definite Functions
3.8 Laplace and Stieltjes Transform, and Completely Monotone Functions
3.9 Bernstein Functions and Subordination of Convolution Semigroups
3.10 Some Function Spaces related to Continuous Negative Definite Functions
3.11 Besov Spaces and Triebel-Lizorkin Spaces
3.12 Fourier Multiplier Theorems
3.13 Notes to Chapter 3
Chapter 4. One Parameter Semigroups
4.1 Strongly Continuous Operator Semigroups
4.2 Analytic Semigroups
4.3 Subordination in the Sense of Bochner for Operator Semigroups
4.4 Perturbations and Approximations
4.5 Generators of Feller Semigroups
4.6 Sub-Markovian Semigroups and their Generators
4.7 Dirichlet Forms and Generators of Sub-Markovian Semigroups
4.8 Extending Feller Semigroups, Resolvents and their Generators
4.9 Notes to Chapter 4
Bibliography
1-7
8-19
20-31
32-42
43-54
55-65
66-75
76-87
88-99
100-111
112-123
124-135
136-147
148-159
160-170
171-182
183-194
195-206
207-219
220-232
233-245
246-258
259-270
271-282
283-295
296-308
309-317
Author Index
Subject Index