Fundamentals of algebraic microlocal analysis. (English) Zbl 0924.35001

Kato, Goro; Struppa, Daniele C.

Pure and Applied Mathematics, Marcel Dekker. 217. New York, NY: Marcel Dekker, Inc. x, 296 p. (1999).

This book is an excellent and concise introduction to the algebraic theory of systems of differential equations with analytic coefficients, as developed by the Japanese school of M. Sato. The authors start out by giving a self-contained description of hyperfunctions, and proceed to further study the sheaf of singularities of hyperfunctions, known as the sheaf of microfunctions. Here is the point in which the word “microlocal” from the title enters. All this is preliminary to the algebraic treatment of systems of differential equations, namely the study of the so-called \({\mathcal D}\)-modules, i.e. sheaf of modules over the sheaf \({\mathcal D}\) of variable coefficients differential operators. In the last chapter, they go back to an analytic viewpoint, where the fundamental Sato’s structure theorem for systems of differential equations is established.
The treatment of these matters requires knowledge in the Theory of Pseudodifferential Operators, Homological Algebra and Algebraic Geometry, which is provided by the authors, in a very concise and efficient way, any time it has to be used. The authors provide, throughout the book, a large number of examples and, at the end of each chapter, historical notes on the subject, that give the reader a more solid comprehension of the motivations behind scenes and a more solid grasp of the techinques. An extensive and updated list of references is given, which makes the book also very useful as source of references.
The book is a valuable contribution to make this important area of mathematical research more accessible to graduate students, researchers working in the areas of Partial Differential Equations and Algebraic Geometry, and to physicists.

Reviewer: A.Parmeggiani (Bologna)

MSC:

35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations
35A27	Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
58J15	Relations of PDEs on manifolds with hyperfunctions
32C38	Sheaves of differential operators and their modules, \(D\)-modules
32A45	Hyperfunctions

Keywords:

systems of differential equations with analytic coefficients; sheaf of microfunctions; \({\mathcal D}\)-modules