This book focuses on Poincaré, Nash and other Sobolev-type inequalities and their applications to the Laplace and heat diffusion equations on Riemannian manifolds. Applications covered include the ultracontractivity of the heat diffusion semigroup, Gaussian heat kernel bounds, the Rozenblum-Lieb-Cwikel inequality and elliptic and parabolic Harnack inequalities. Emphasis is placed on the role of families of local Poincaré and Sobolev inequalities. The text provides the first self contained account of the equivalence between the uniform parabolic Harnack inequality, on the one hand, and the conjunction of the doubling volume property and Poincaré's inequality on the other.
Author(s): Laurent Saloff-Coste
Series: London Mathematical Society Lecture Note Series
Edition: 1
Publisher: Cambridge University Press
Year: 2001
Language: English
Pages: 202
Front Cover......Page 1
Series Titles......Page 2
Title......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 10
Introduction ......Page 12
1.1.1 Introduction ......Page 18
1.1.2 The proof due to Gagliardo and to Nirenberg ......Page 20
1.1.3 p = 1 implies p > 1 . ......Page 21
1.2.1 Another approach to Sobolev inequalities . ......Page 22
1.2.2 Marcinkiewicz interpolation theorem . ......Page 24
1.3.1 The case p = 1: isoperimetry ......Page 27
1.3.2 A complete proof with best constant for p = 1 ......Page 29
1.3.3 The case p > 1 . . . ......Page 31
1.4.1 The case p > n .. . ......Page 32
1.4.2 The case p = n . ......Page 35
1.4.3 Higher derivatives ......Page 37
1.5.1 The Neumann and Dirichlet eigenvalues . ......Page 40
1.5.2 Poincare inequalities on Euclidean balls . . ......Page 41
1.5.3 Sobolev-Poincare inequalities . ......Page 42
2.1.1 Divergence form . ......Page 44
2.1.2 Uniform ellipticity . ......Page 45
2.1.3 A Sobolev-type inequality for Moser's iteration ......Page 48
2.2.1 Subsolutions . ......Page 49
2.2.2 Supersolutions . . ......Page 54
2.2.3 An abstract lemma . ......Page 58
2.3.1 Harnack inequalities ......Page 60
2.3.2 Holder continuity . . ......Page 61
3.1.1 Notation concerning Riemannian manifolds ......Page 64
3.1.2 Isoperimetry . ......Page 66
3.1.3 Sobolev inequalities and volume growth ......Page 68
3.2.1 Examples of weak Sobolev inequalities ......Page 71
3.2.2 (Se,)-inequalities: the parameters q and v ......Page 72
3.2.3 The case 0 < q < oo ......Page 74
3.2.4 The case q = oo . ......Page 77
3.2.5 The case -oo < q < 0 . ......Page 79
3.2.6 Increasing p . ......Page 81
3.2.7 Local versions ......Page 83
3.3.1 Pseudo-Poincare inequalities ......Page 84
3.3.2 Pseudo-Poincare technique: local version . ......Page 86
3.3.3 Lie groups ......Page 88
3.3.4 Pseudo-Poincare inequalities on Lie groups ......Page 90
3.3.5 Ricci > 0 and maximal volume growth ......Page 93
3.3.6 Sobolev inequality in precompact regions . ......Page 96
4.1.1 Nash inequality implies ultracontractivity ......Page 98
4.1.2 The converse ......Page 102
4.2.1 The Gaffney-Davies L2 estimate . : ......Page 104
4.2.2 Complex interpolation . ......Page 106
4.2.3 Pointwise Gaussian upper bounds . ......Page 109
4.2.4 On-diagonal lower bounds . ......Page 110
4.3.1 The Schrodinger operator A - V . ......Page 114
4.3.2 The operator TV = 0-1V . .. ......Page 116
4.3.3 The Birman-Schwinger principle . ......Page 120
5.1 Scale-invariant Harnack principle . ......Page 122
5.2.1 Local Sobolev inequalities and volume growth ......Page 124
5.2.2 Mean value inequalities for subsolutions . ......Page 130
5.2.3 Localized heat kernel upper bounds ......Page 133
5.2.4 Time-derivative upper bounds . ......Page 138
5.2.5 Mean value inequalities for supersolutions ......Page 139
5.3 Poincare inequalities ......Page 141
5.3.1 Poincare inequality and Sobolev inequality ......Page 142
5.3.2 Some weighted Poincare inequalities ......Page 144
5.3.3 Whitney-type coverings . ......Page 146
5.3.4 A maximal inequality and an application . ......Page 150
5.3.5 End of the proof of Theorem 5.3.4 . ......Page 152
5.4.1 An inequality for log u . ......Page 154
5.4.2 Harnack inequality for positive supersolutions ......Page 156
5.4.3 Harnack inequalities for positive solutions ......Page 157
5.4.4 Holder continuity ......Page 160
5.4.5 Liouville theorems . ......Page 162
5.4.6 Heat kernel lower bounds ......Page 163
5.4.7 Two-sided heat kernel bounds . ......Page 165
5.5 The parabolic Harnack principle ......Page 166
5.5.1 Poincare, doubling, and Harnack . ......Page 168
5.5.2 Stochastic completeness ......Page 172
5.5.3 Local Sobolev inequalities and the heat equation . ......Page 175
5.5.4 Selected applications of Theorem 5.5.1 . ......Page 179
5.6.1 Unirnodular Lie groups . ......Page 183
5.6.2 Homogeneous spaces ......Page 186
5.6.3 Manifolds with Ricci curvature bounded below ......Page 187
5.7 Concluding remarks . ......Page 191
Bibliography ......Page 194
Index ......Page 200
Back Cover......Page 202
