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.
