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Book review

Author(s) Varopoulos, N.Th.
Saloff-Coste, L.
Coulhon, T.
Title Analysis and Geometry on Groups
Publisher Cambridge University Press
Year of publication 1992
Reviewed by Daniel Vacaretu

The book "Analysis and Geometry on Groups" is primarily an advanced research monograph. One of the aims of this book is to study Sobolev inequalities on Lie groups.

After the first chapter, an introductory one, the authors describe some basic properties of the sums of squares of vector fields (Hormander's theorem and a local Harnack inequality), study the sublaplacian associated with a Hormander system of left invariant vector fields on a nilpotent Lie group, and then consider unimodular connected Lie groups yielding in the case of polynomyal volume growth, two-side Gaussian estimates for the heat kernel, optimal Sobolev inequalities, and Harnack's principle. Chapter IX gives up the study of the heat kernel and concentrates on Sobolev inequalities for non-unimodular Lie groups. Finally, Chapter X contains various geometric application of the above theory.

Each chapter has a References and comments section and at its end, the book contains a good list of references which is a proof of the very important author's contribution in the development of these subjects.

The book is designed for specialists in functional analysis, Lie groups, Markov chains or potential theory so that I suggest its acquisition.