zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Elementary introduction to the theory of pseudodifferential operators. (English) Zbl 0847.47035
Studies in Advanced Mathematics. Boca Raton, FL: CRC Press. viii, 108 p. $ 39.95 (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.


MSC:
47G30Pseudodifferential operators
35S05General theory of pseudodifferential operators