General introduction

3

very simple theory allows one to rapidly obtain interesting properties of

composite functions in Sobolev and Holder spaces. Section B presents the

concept of the wave front set and its links with pseudo-differential operators:

this time it is a matter of the conical division of the space of frequencies £

according to their directions £ G

Sn~1,

associated with the classical symbol

homogeneities. Finally, Section C deals with hyperbolic energy inequali-

ties for which pseudo-differential operators turn out to be an effective tool.

Thus, Chapter II serves to present very useful applications of the 'dry the-

ory' of Chapter I, while preparing the material and the concepts which will

be needed in Chapter III.

The final chapter discusses certain problems of a nonlinear nature which

arise in geometry or in analysis and which may be reduced to perturbation

problems. The plan of this chapter reflects the various situations which

one may encounter: 'elliptic' situations in which the usual Banach implicit

function theorem suffices; 'fixed-point' situations, such as one often finds in

nonlinear hyperbolic problems or again in the isometric embedding problem;

and, finally, situations where the 'loss of derivatives' is too great and a

Nash-Moser technique has to be used. The Nash-Moser theorem relies

completely on the acquisition of a priori 'tame' inequalities; the reader who

is already familiar with a priori inequalities (presented in Chapter I and

Section III.C) will grasp the concept of 'tame' estimates through its clear

link with Littlewood-Paley theory and the paradifferential calculus of J.-

M. Bony (Section II.A).

This establishes the underlying cohesion of this book, which can be

schematized in the accompanying diagram.