Elementary introduction to the theory of pseudodifferential operators.

*(English)*Zbl 0847.47035
Studies in Advanced Mathematics. Boca Raton, FL: CRC Press. viii, 108 p. (1991).

This very nice book contains a truly elementary introduction to some classical facts on pseudodifferential operators. A graduate student with a standard background in analysis would certainly be able to read it without any difficulty.

Let’s go through the chapters of this book. The first one, “Fourier transformation and Sobolev spaces”, sets forth notations and basic material. Chapter Two is devoted to “pseudodifferential symbols”, and introduces the reader to the notion of oscillatory integral with a quadratic phase function. The proofs are carefully written, and the exercises, which provide some supplementary information, are solvable by a student. The third chapter deals with pseudodifferential operators, and establishes some of the main properties of continuity. The notes after each chapter introduce the reader to the bibliography; they enter into the most advanced literature on the topic, providing the reader with an overview of applications of this theory. Some theorems on local solvability are presented in the last chapter, using pseudodifferential operators as a tool.

The only criticism one could make is that the scope of applications described here is very limited and will not necessarily convince the reader that this theory is truly important for the understanding of partial differential equations. On the other hand, the proofs are so carefully written, the notes and exercises so rich in information, that this book should be definitely recommended to anybody interested in a first attempt to understand the theory of pseudodifferential operators.

Let’s go through the chapters of this book. The first one, “Fourier transformation and Sobolev spaces”, sets forth notations and basic material. Chapter Two is devoted to “pseudodifferential symbols”, and introduces the reader to the notion of oscillatory integral with a quadratic phase function. The proofs are carefully written, and the exercises, which provide some supplementary information, are solvable by a student. The third chapter deals with pseudodifferential operators, and establishes some of the main properties of continuity. The notes after each chapter introduce the reader to the bibliography; they enter into the most advanced literature on the topic, providing the reader with an overview of applications of this theory. Some theorems on local solvability are presented in the last chapter, using pseudodifferential operators as a tool.

The only criticism one could make is that the scope of applications described here is very limited and will not necessarily convince the reader that this theory is truly important for the understanding of partial differential equations. On the other hand, the proofs are so carefully written, the notes and exercises so rich in information, that this book should be definitely recommended to anybody interested in a first attempt to understand the theory of pseudodifferential operators.

Reviewer: N.Lerner (Rennes) (MR 94b:47066)

##### MSC:

47G30 | Pseudodifferential operators |

35S05 | Pseudodifferential operators as generalizations of partial differential operators |