Pseudo-differential operators and the Nash-Moser theorem. (Opérateurs pseudo-différentiels et théorème de Nash-Moser.) (French) Zbl 0791.47044

Alinhac, Serge; Gérard, Patrick

Savoirs Actuels. Paris: InterEditions. Paris: Editions du CNRS. 188 p. (1991).

As the authors state in the introduction, this is an elementary presentation of the theorem of Nash-Moser, and related topics. The organization of this book is as follows. Chapter 0 gathers the basics on the theory of distributions. On Chapter 1 the authors present a theory of classical pseudo-differential operators, with emphasis on symbolic calculus. Chapter 2 is dedicated to Littlewood-Paley theory, microlocal analysis, and energy estimates. Finally, Chapter 3 is divided in three sections, covering implicit function theorems, the use of fixed point theorems, and Nash-Moser theorem. Every chapter closes with a commentary and a list of problems.
Although the book is aimed at students, or newcomers to the subject, the careful and knowledgeable presentation makes its reading a treat for anybody interested in the interplay of ideas coming from the theory of PDE, Harmonic analysis and Operator theory.
The problems presented at the end of each chapter are a special and welcome feature. They are for the most part nontrivial and in occasions down right difficult. The latter are marked \(^*\) and altogether, they make for a great and unusual addition to a book of this nature. The authors should be congratulated for putting the time and effort required to compile such a good problem selection.
The book closes with a list of notations and a short bibliography classified as prerequisites, readings at a comparable level, and advanced reading.

Reviewer: J.Alvarez (Las Cruces)

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

Keywords:

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