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

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.

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 |