An Introduction to Pseudo-Differential Operators

M W Wong

• Publisher: WSPC
• Year: 2014
• eBook ISBN: 9789814583107
• Edition: 3
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

The aim of this third edition is to give an accessible and essentially self-contained account of pseudo-differential operators based on the previous edition. New chapters notwithstanding, the elementary and detailed style of earlier editions is maintained in order to appeal to the largest possible group of readers. The focus of this book is on the global theory of elliptic pseudo-differential operators on L p ( R n ).

The main prerequisite for a complete understanding of the book is a basic course in functional analysis up to the level of compact operators. It is an ideal introduction for graduate students in mathematics and mathematicians who aspire to do research in pseudo-differential operators and related topics.

Contents:

• Introduction, Notation and Preliminaries
• Differentiation of Integrals Depending on Parameters
• The Convolution
• The Fourier Transform
• Tempered Distributions
• Symbols, Pseudo-Differential Operators and Asymptotic Expansions
• A Partition of Unity and Taylor's Formula
• The Product of Two Pseudo-Differential Operators
• The Formal Adjoint of a Pseudo-Differential Operator
• The Parametrix of an Elliptic Pseudo-Differential Operator
• L p -Boundedness of Pseudo-Differential Operators
• The Sobolev Spaces H s,p, -∞<s<∞, 1≤ p <∞
• Closed Linear Operators
• Minimal and Maximal Pseudo-Differential Operators
• Global Regularity of Elliptic Partial Differential Equations
• Weak Solutions of Pseudo-Differential Equations
• Gårding's Inequality
• Strong Solutions of Pseudo-Differential Equations
• One-Parameter Semigroups Generated by Pseudo-Differential Operators
• Fredholm Operators
• Fredholm Pseudo-Differential Operators
• Symmetrically Global Pseudo-Differential Operators
• Spectral Invariance of Symmetrically Global Pseudo-Differential Operators

Readership: Graduate students and researchers in the fields of analysis and differential equations.
Key Features:

• The existing books on pseudo-differential operators serve well as references, but are not suitable as textbooks. They are too difficult for students to learn the basics in order to do research in the field expeditiously
• The third edition of this book contains new topics that give a much better perspective of the global theory of elliptic pseudo-differential operators than the earlier editions
• While the topics in the competing book by M A Shubin are mainly given in the context of L 2 ( R n ), a unique feature of this book is that the emphasis has been put on the theory of pseudo-differential operators on L p ( R n ), where p is between 1 and infinity/li>

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Preface
• 1. Introduction, Notation and Preliminaries
• 2. Differentiation of Integrals Depending on Parameters
• 3. The Convolution
• 4. The Fourier Transform
• 5. Tempered Distributions
• 6. Symbols, Pseudo-Differential Operators and Asymptotic Expansions
• 7. A Partition of Unity and Taylor’s Formula
• 8. The Product of Two Pseudo-Differential Operators
• 9. The Formal Adjoint of a Pseudo-Differential Operator
• 10. The Parametrix of an Elliptic Pseudo-Differential Operator
• 11. Lp-Boundedness of Pseudo-Differential Operators
• 12. The Sobolev Spaces Hs,p, −∞ < s < ∞, 1 ≤ p < ∞
• 13. Closed Linear Operators
• 14. Minimal and Maximal Pseudo-Differential Operators
• 15. Global Regularity of Elliptic Partial Differential Equations
• 16. Weak Solutions of Pseudo-Differential Equations
• 17. Gåxding’s Inequality
• 18. Strong Solutions of Pseudo-Differential Equations
• 19. One-Parameter Semigroups Generated by Pseudo-Differential Operators
• 20. Fredholm Operators
• 21. Fredholm Pseudo-Differential Operators
• 22. Symmetrically Global Pseudo-Differential Operators
• 23. Spectral Invariance of Symmetrically Global Pseudo-Differential Operators
• Bibliography
• Index