Pseudodifferential and Singular Integral Operators

This book provides a self-contained and elementary introduction to the modern theory of pseudodifferential operators and their application to partial differential equations. It presents the necessary material on Fourier transformation and distribution theory, the basic calculus of pseudodifferential operators on the n-dimensional Euclidean space, an introduction to the theory of singular integral operators, the modern theory of Besov and Bessel potential spaces, and several applications to wellposedness and regularity question for elliptic and parabolic equations. The basic notation of functional analysis needed in the book is introduced and summarized in the appendix.

Autorentext
Helmut Abels, University of Regensburg, Germany.

Klappentext

The purpose of this textbook is to give a self-contained introduction to the theory of pseudodifferential operators and singular integral operators. Moreover, several applications to elliptic and parabolic equations are given and connections to the theory of function spaces are discussed. The book is divided into three parts, each consisting of two chapters, and the appendix and an introduction. Part I is devoted to the Fourier transformation, tempered distributions and pseudodifferential operators. In Part II an introduction to the theory of singular integral operators is given. Finally, Part III consists of applications of the results of the first two parts.

The book is based on several lecture series held by the author at Darmstadt University of Technology, the International Max Planck Research School Mathematics in the Sciences Leipzig, the University of Leipzig, and the University of Regensburg. The exposition is written in such a way that it should be accessible for third year Bachelor students and Master students with a basic education in analysis of functions in several variables, integration theory, and a basic knowledge in functional analysis.