The book is based on an advanced university lecture course providing an extensive study of basic properties of pseudo-differential operators and some of their principal applications, with particular emphasis on the Nash-Moser theorem. The exposition is written in a very attractive way, which should appeal to experts in the field as well as to PhD students (or even gifted undergraduates) with some basic knowledge of elementary functional analysis, Fourier analysis and, perhaps, the theory of distributions. The book is self-contained, and the authors have taken a lot of trouble to make their exposition as reader-friendly as possible.

The book is divided into three chapters (plus an introductory Chapter 0). Chapter I is an exposition of the theory of pseudo-differential operators (the authors call this part a ‘minimal theory’, but it is in fact quite comprehensive). The material includes the concept of a symbol, its use in operator calculus, the action of operators on Sobolev spaces and the invariance under change of variables. Chapter II is divided into three themes, covering (among other topics) the Littlewood-Paley theory of dyadic decomposition of distributions (with such interesting facts as a characterisation of the Hölder and Sobolev spaces), ‘micro-local analysis’ and some energy estimates. The last chapter, ‘The implicit function theorems’ treats the role of implicit functions in elliptic problems, examples of applications of fixed-point theorems to semilinear hyperbolic systems, and a thorough and comprehensive exposition of the Nash-Moser theorem on the existence and properties of a solution to the equation Φ(u) = Φ(u0) + f.

Reviewer:

lp