Analysis and Geometry on Groups

Nicholas T. Varopoulos; L. Saloff-Coste; T. Coulhon

doi:10.1017/CBO9780511662485

The geometry and analysis that is discussed in this book extends to classical results for general discrete or Lie groups, and the methods used are analytical but have little to do with what is described these days as real analysis. Most of the results described in this book have a dual formulation; they have a 'discrete version' related to a finitely generated discrete group, and a continuous version related to a Lie group. The authors chose to centre this book around Lie groups but could quite easily have pushed it in several other directions as it interacts with opetators, and probability theory, as well as with group theory. This book will serve as an excellent basis for graduate courses in Lie groups, Markov chains or potential theory.

"The book is very concise and contains a great wealth of ideas and results...Each chapter contains a small section, 'References and comments', in which the authors, in their own way, introduce the reader to the brief history of the subject and its bibliography." A. Hulanicki, Mathematical Reviews

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Analysis and Geometry on Groups

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

Preface
pp vii-viii

Foreword
pp ix-xii

I - Introduction
pp 1-7

II - Dimensional inequalities for semigroups of operators on the Lp spaces
pp 8-25

III - Systems of vector fields satisfying Hörmander's condition
pp 26-41

IV - The heat kernel on nilpotent Lie groups
pp 42-62

V - Local theory for sums of squares of vector fields
pp 63-73

VI - Convolution powers on finitely generated groups
pp 74-87

VII - Convolution powers on unimodular compactly generated groups
pp 88-105

VIII - The heat kernel on unimodular Lie groups
pp 106-122

IX - Sobolev inequalities on non-unimodular Lie groups
pp 123-137

X - Geometric applications
pp 138-147

Bibliography
pp 148-155

Index
pp 156-156

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Analysis and Geometry on Groups

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