# Introduction to Pseudodifferential and Fourier Integral Operators

## Pseudodifferential Operators

• François Treves
Book

Part of the The University Series in Mathematics book series (USMA)

1. Front Matter
Pages i-xxvii
2. François Treves
Pages 1-81
3. François Treves
Pages 83-128
4. François Treves
Pages 129-216
5. François Treves
Pages 217-237
6. François Treves
Pages 239-299
7. Back Matter
Pages xxix-xxxix

### Introduction

I have tried in this book to describe those aspects of pseudodifferential and Fourier integral operator theory whose usefulness seems proven and which, from the viewpoint of organization and "presentability," appear to have stabilized. Since, in my opinion, the main justification for studying these operators is pragmatic, much attention has been paid to explaining their handling and to giving examples of their use. Thus the theoretical chapters usually begin with a section in which the construction of special solutions of linear partial differential equations is carried out, constructions from which the subsequent theory has emerged and which continue to motivate it: parametrices of elliptic equations in Chapter I (introducing pseudodifferen­ tial operators of type 1, 0, which here are called standard), of hypoelliptic equations in Chapter IV (devoted to pseudodifferential operators of type p, 8), fundamental solutions of strongly hyperbolic Cauchy problems in Chap­ ter VI (which introduces, from a "naive" standpoint, Fourier integral operators), and of certain nonhyperbolic forward Cauchy problems in Chapter X (Fourier integral operators with complex phase). Several chapters-II, III, IX, XI, and XII-are devoted entirely to applications. Chapter II provides all the facts about pseudodifferential operators needed in the proof of the Atiyah-Singer index theorem, then goes on to present part of the results of A. Calderon on uniqueness in the Cauchy problem, and ends with a new proof (due to J. J. Kohn) of the celebrated sum-of-squares theorem of L. Hormander, a proof that beautifully demon­ strates the advantages of using pseudodifferential operators.

### Keywords

Cauchy problem differential equation differential operator equation partial differential equation proof pseudodifferential operators theorem

#### Authors and affiliations

• François Treves
• 1
1. 1.Rutgers UniversityNew BrunswickUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4684-8780-0
• Copyright Information Springer-Verlag US 1980
• Publisher Name Springer, Boston, MA
• eBook Packages
• Print ISBN 978-1-4684-8782-4
• Online ISBN 978-1-4684-8780-0