Book review

The main purpose of this book is to present some recent results on Fuchsian partial differential equations, including essential original contributions of the authors. The basic tool used in the treatment of these equations is the Mellin transformation, which is presented in details in the second chapter of the book.

In order to make the book almost self-contained, a preliminary chapter, Chapter I, Introduction, is included, containing the basic definitions and results from the theory of distributions, needed in the rest of the book, some of them with direct proofs or with proofs given as exercises with hints and other without proofs, but with pertinent references.

The presentation of the Mellin transformation is given in Chapter II, Mellin distributions and the Mellin transformation. In contrast to the classical presentations of Laplace and Mellin transformations, the attention is focussed on the local properties of Mellin transformations, i.e. those properties which are preserved under multiplication by cut-off functions of various types. The Mellin distributions are defined as the elements of the dual of the space <formula>, where <formula> and o denotes substitution in distributions, with the natural topology induced by S(R^n) (the space of rapidly decreasing functions). The Mellin transformation is a linear continuous isomorphism between <formula> and S'. Similar to the Fourier transformation <formula>, for <formula> and <formula>. This chapter contains a detailed study of Mellin distributions and transformation, including the Paley-Wiener theorem and examples of Mellin transformations of some functions. The chapter ends with the presentation of the modified Cauchy and Hilbert transformations in dimension 1, modified in the sense that the Cauchy transformation is not considered to the real axis R but relative to a fixed pure imaginary line <formula>, fixed.

The theory of Fuchsian differential (both ordinary and partial) Fuchsian equations is presented in Chapter III. Fuchsian type differential operators. It should be mentioned that the Fuchsian singular operators considered in this book are not of Baouendi-Goulaouic type, but they arise naturally rather that Laplace-Beltrami operators on manifolds with corners and fall within the global theories developed by R. Melrose, Analysis on manifolds with corners, Lectures Notes M.I.T. Press, Preprint 1988.

The book ends with an Appendix. Generalized smooth functions and theory of resurgent functions of Jean Ecalle, establishing connections between the theories of Mellin and Borel transformations and of resurgent functions of J. Ecalle, with applications to nonlinear Euler equation and to elliptic Fuchsian operators.

The reading of the book requieres only knowledges up to the level of the first 3 years of university studies in mathematics, with very few exceptions (the division theorem and the Fourier transform of convolution).

The result is a fine book, containing many original results of the authors, scattered before through journal papers, which will be of great interest to all working in integral transformations, distribution theory, ordinary and partial differential equations, as well as in their applications.