This book corresponds to a graduate course which has been taught many times at the universities Paris-Sud, Paris-Nord and Paris 7, and other places. The aim of this text is to give the foundation of what is nowadays called microlocal analysis in the C∞ framework, as it was created in the sixties and seventies by Kohn-Nirenberg, Maslov and Hörmander. Our presentation follows essentially the one given by Hörmander [Hö2]; as for the symplectic geometry, we have been inspired by the lecture notes of Duistermaat [D].

This subject is of growing importance, with a range of applications going beyond the original problems of linear partial differential equations. In particular the link with quantum mechanics is now firmly established, and there is a growing number of books covering more or less specialized parts of the theory. We believe that a short monograph concentrating on the basic principles could be of value, not only for the graduate student, but also for the mathematician who wants to get quickly into the subject and to understand its basic mechanisms. For this, the classical PDE framework seemed to us the most suitable one. The basic principles of microlocal analysis are essentially only two: integration by parts and the method of stationary phase. Compared with the article [HÖ2] and many of the other presentations, we have insisted even more on the stationary phase method, which appears already in the development of the theory of pseudodifferential operators and also (as in Melin-Sjöstrand [MS]) in the proof of the equivalence of phase functions in the global theory of Fourier integral operators.