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

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