2 General introduction
be effectively solved by real students, as experience of the teaching at ENS
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
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
), 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