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.
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