2 General introduction

be effectively solved by real students, as experience of the teaching at ENS

has shown.

It was our wish that this text should also be useful to researchers as

a simple and self-contained introduction to subjects with which they are

unfamiliar.

The dual purpose of these notes led us to keep them short, sometimes

at the expense of a certain denseness of the text (which we believe is essen-

tially accessible to motivated students). In particular, we had in mind our

many colleagues in 'applied mathematics' who wish to use the Nash-Moser

theorem in their research or to keep themselves up to date on microlocal

analysis, without delving into the arcana of the specialist literature: they

will be able to read the desired chapters independently of each other.

The choice of the material presented is a matter of personal taste and

of the fields of research of the authors who, incidentally, believe that certain

difficult (nonlinear) problems cannot be solved without a sufficient knowl-

edge of pseudo-differential operators.

The authors are indebted to numerous mathematicians (cited in the

commentaries) who have inspired them to present the subjects dealt with,

and, in particular, to L. Hormander, to whom the mathematical contents of

Chapter I and Section III.C are largely due. The Bibliography at the end

of the book indicates the sources used.

While presenting important concepts which are the true starting points

for numerous recent developments, we have sought to end up with real the-

orems: microlocal elliptic regularity; propagation of singularities; existence

of solutions of quasilinear hyperbolic systems; existence of isometric embed-

dings; the Nash-Moser theorem. The plan of the book is as follows.

In Chapter I we present the 'minimal' theory of pseudo-differential op-

erators, in a global context (on E

n

), which turns out to be very nice in

practice. The main points here are the notion of the symbol, the symbolic

calculus for operators, the action in Sobolev spaces and the invariance under

change of coordinates. The text presents only a few concrete applications

and the most technical proofs are brought together in the appendix, in order

to enable the reader to obtain an overall view of the subject. The exercises

in Chapter I, which are particularly numerous, provide an introduction to a

number of variants of the theory proposed and present several applications

notably to the analysis on compact manifolds.

Chapter II brings together three themes. Section A presents the Little-

wood-Paley theory of 'dyadic decomposition' of distributions: this system-

atizes the natural division of the space of frequencies £ according to their

size |£|, associated with the classical symbolic calculus of Chapter I. This