**Graduate Studies in Mathematics**

Volume: 82;
2007;
168 pp;
Hardcover

MSC: Primary 35;
Secondary 47

**Print ISBN: 978-0-8218-3454-1
Product Code: GSM/82**

List Price: $45.00

AMS Member Price: $36.00

MAA Member Price: $40.50

**Electronic ISBN: 978-1-4704-2112-0
Product Code: GSM/82.E**

List Price: $42.00

AMS Member Price: $33.60

MAA Member Price: $37.80

#### Supplemental Materials

# Pseudo-differential Operators and the Nash–Moser Theorem

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*Serge Alinhac; Patrick Gérard*

Translated by Stephen S. Wilson

This book presents two essential and apparently unrelated subjects. The
first, microlocal analysis and the theory of pseudo-differential
operators, is a basic tool in the study of partial differential equations
and in analysis on manifolds. The second, the Nash–Moser theorem,
continues to be fundamentally important in geometry, dynamical systems, and
nonlinear PDE.

Each of the subjects, which are of interest in their own right as well as
for applications, can be learned separately. But the book shows the deep
connections between the two themes, particularly in the middle part, which
is devoted to Littlewood–Paley theory, dyadic analysis, and the
paradifferential calculus and its application to interpolation
inequalities.

An important feature is the elementary and self-contained character
of the text, to which many exercises and an introductory Chapter
\(0\) with basic material have been added. This makes the book
readable by graduate students or researchers from one subject who are
interested in becoming familiar with the other. It can also be used
as a textbook for a graduate course on nonlinear PDE or geometry.

#### Readership

Graduate students and research mathematicians interested in microlocal analysis, PDE, and geometry.

#### Reviews & Endorsements

The authors made a great accomplishment in writing this beautiful and elegant text, of great value to both students and researchers.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Pseudo-differential Operators and the Nash-Moser Theorem

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface to the English edition vii8 free
- General introduction 110 free
- Chapter 0. Notation and review of distribution theory 514
- Chapter I. Pseudo- differential operators 1524
- § 1. Introduction 1524
- §2. Symbols 2029
- §3. Pseudo- differential operators in S and S' 2433
- §4. Composition of operators 2837
- §5. Action of pseudo- differential operators and Sobolev spaces 2938
- §6. Operators in an open subset of R[sup(n)] 3443
- §7. Operators on a manifold 3746
- §8. Appendix 4150
- Commentary on Chapter I 4958
- Exercises for Chapter I 5059

- Chapter II. Nonlinear dyadic analysis, microlocal analysis, energy estimates 7786
- Chapter III. Implicit function theorems 121130
- Bibliography 161170
- Main notation introduced 165174
- Index 167176
- Back Cover Back Cover1178