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 are not concerned 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 center this book around Lie groups, but could easily have pushed it in several other directions as it interacts with the theory of second order partial differential operators, and probability theory, as well as with group theory.
Author(s): Nicholas T. Varopoulos, L. Saloff-Coste, T. Coulhon
Series: Cambridge Tracts in Mathematics
Publisher: CUP
Year: 1993
Language: English
Pages: 168
CONTENTS......Page 5
Preface......Page 7
Foreword......Page 9
1 Sobolev inequalities in R^n......Page 13
2 Sobolev inequalities and the heat equation on Lie groups......Page 15
3 Harnack's principle......Page 17
4 A guide to this book......Page 18
1 Introduction, notation......Page 20
2 Hardy-Littlewood-Sobolev theory......Page 21
3 Converses to the Hardy-Littlewood-Sobolev theory......Page 24
4 Localizations......Page 29
5 Symmetric submarkovian semigroups......Page 32
References and Comments......Page 37
1 Hormander's condition and hypoellipticity......Page 38
Uniformity matters......Page 39
2 Harnack inequalities......Page 40
Uniformity matters......Page 44
3 The exponential map......Page 46
4 Carnot-Caratheodory distances......Page 51
References and Comments......Page 53
1 Some remarkable properties of nilpotent Lie groups......Page 54
2 Examples......Page 55
3 Harnack inequalities for nilpotent Lie groups......Page 57
4 Estimates of the heat kernel......Page 60
5 Estimates of the volume......Page 62
6 Sobolev's theorem......Page 67
7 Sobolev inequalities......Page 68
References and Comments......Page 73
1 Estimates of the volume......Page 75
2 Proof of the Key Lemma......Page 77
3 Local scaling of the Harnack inequality......Page 79
4 The case of unimodular Lie groups......Page 81
5 The general case......Page 82
References and Comments......Page 84
1 Introduction......Page 86
2 Distance and volume growth function on a group......Page 87
3 The main results for superpolynomial groups......Page 89
4 Comparison of Dirichlet forms and finite variance......Page 93
5 Nilpotent finitely generated groups......Page 96
6 Kesten's conjecture......Page 97
References and Comments......Page 98
1 Main results......Page 100
2 Dimension theory for symmetric submarkovian operators......Page 101
3 Comparison of Dirichlet forms......Page 106
4 Volume growth and polynomial decay of convolution powers......Page 111
5 The case of superpolynomial growth......Page 115
References and Comments......Page 117
1 Preliminaries......Page 118
2 Polynomial growth Lie groups......Page 119
3 Harnack inequalities for polynomial growth groups......Page 126
4 Exponential growth Lie groups......Page 131
References and Comments......Page 134
1 Local theory......Page 135
2 An inequality of Hardy and some consequences......Page 140
3 A Sobolev inequality again......Page 144
References and Comments......Page 149
1 Geometry of Lie groups and quasiregular maps......Page 150
2 Picard theorems on Lie groups......Page 151
3 Brownian motion on covering manifolds and random walks on groups......Page 153
4 Dimension at infinity of a covering manifold......Page 155
5 Quasiregular maps and compact manifolds......Page 158
References and Comments......Page 159
Bibliography......Page 160
Index......Page 168
