Pseudo-differential operators and the Nash–Moser theorem. Transl. from the French by Stephen S. Wilson. (English) Zbl 1121.47033

Alinhac, Serge; Gérard, Patrick

Graduate Studies in Mathematics 82. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-3454-1/hbk). vii, 168 p. (2007).

This book is the English translation of the French edition [“Opérateurs pseudo-différentiels et théorème de Nash–Moser” (Savoirs Actuels, InterEditions/Editions du CNRS, Paris) (1991; Zbl 0791.47044)]. As stated by the authors, it is an elementary presentation of pseudodifferential operators and the Nash–Moser theorem, as well as related topics. After reviewing in Chapter 0 some fundamental results needed in the text, in Chapter 1 the authors introduce the reader to the classical theory of pseudodifferential operators and their symbolic calculus. Littlewood–Paley theory, wave-front sets and energy estimates are presented in Chapter 2, and in the final Chapter 3, implicit function theorems, fixed point methods and the Nash–Moser theorem are covered.
At the end of each chapter, one finds commentaries and a most useful list of exercises, divided into two classes: those elementary and intended to help the reader in assimilating the material, and those that are more advanced and difficult to go deeper into the subject. The book ends with a carefully prepared bibliography, divided into three parts: books which should be read beforehand, papers which are accessible at the level of this book, and, finally, complementary papers that are at the research level. The authors made a great accomplishment in writing this beautiful and elegant text, of great value to both students and researchers.

Reviewer: Alberto Parmeggiani (Bologna)

MSC:

47G30	Pseudodifferential operators
35S05	Pseudodifferential operators as generalizations of partial differential operators
42B25	Maximal functions, Littlewood-Paley theory
46F10	Operations with distributions and generalized functions
46F05	Topological linear spaces of test functions, distributions and ultradistributions
46-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
35-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
58J40	Pseudodifferential and Fourier integral operators on manifolds
58-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis

Keywords:

Nash-Moser theorem; pseudodifferential operators; symbolic calculus; Littlewood-Paley theory; microlocal analysis; energy estimates; fixed-point theorems

Citations:

Zbl 0791.47044