Aspects of Sobolev-Type Inequalities

by Laurent Saloff-Coste

• ISBN13: 9780521006071
• ISBN10: 0521006074
• Format: Paperback
• Copyright: 2001-12-10
• Publisher: Cambridge University Press

Summary

This book focuses on Poincar_, 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 Poincar_ 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 Poincar_'s inequality on the other.

Table of Contents

Preface	ix
Introduction	1	(6)
Sobolev inequalities in Rn	7	(26)
Sobolev inequalities	7	(4)
Introduction	7	(2)
The proof due to Gagliardo and to Nirenberg	9	(1)
p = 1 implies p ≥ 1	10	(1)
Riesz potentials	11	(5)
Another approach to Sobolev inequalities	11	(2)
Marcinkiewicz interpolation theorem	13	(3)
Proof of Sobolev theorem 1.2.1	16	(1)
Best constants	16	(5)
The case p = 1: isoperimetry	16	(2)
A complete proof with best constant for p = 1	18	(2)
The case p > 1	20	(1)
Some other Sobolev inequalities	21	(8)
The case p > n	21	(3)
The case p = n	24	(2)
Higher derivatives	26	(3)
Sobolev-Poincare inequalities on balls	29	(4)
The Neumann and Dirichlet eigenvalues	29	(1)
Poincare inequalities on Euclidean balls	30	(1)
Sobolev-Poincare inequalities	31	(2)
Moser's elliptic Harnack inequality	33	(20)
Elliptic operators in divergence form	33	(5)
Divergence form	33	(1)
Uniform ellipticity	34	(3)
A Sobolev-type inequality for Moser's iteration	37	(1)
Subsolutions and supersolutions	38	(11)
Subsolutions	38	(5)
Supersolutions	43	(4)
An abstract lemma	47	(2)
Harnack inequalities and continuity	49	(4)
Harnack inequalities	49	(1)
Holder continuity	50	(3)
Sobolev inequalities on manifolds	53	(34)
Introduction	53	(7)
Notation concerning Riemannian manifolds	53	(2)
Isoperimetry	55	(2)
Sobolev inequalities and volume growth	57	(3)
Weak and strong Sobolev inequalities	60	(13)
Examples of weak Sobolev inequalities	60	(1)
(S&thetas;r,s)-inequalities: The parameters q and v	61	(2)
The case 0 < q < ∞	63	(3)
The case q = ∞	66	(2)
The case -∞ < q < 0	68	(2)
Increasing p	70	(2)
Local versions	72	(1)
Examples	73	(14)
Pseudo-Poincare inequalities	73	(2)
Pseudo-Poincare technique: local version	75	(2)
Lie groups	77	(2)
Pseudo-Poincare inequalities on Lie groups	79	(3)
Ricci ≥ 0 and maximal volume growth	82	(3)
Sobolev inequality in precompact regions	85	(2)
Two applications	87	(24)
Ultracontractivity	87	(6)
Nash inequality implies ultracontractivity	87	(4)
The converse	91	(2)
Gaussian heat kernel estimates	93	(10)
The Gaffney-Davies L2 estimate	93	(2)
Complex interpolation	95	(3)
Pointwise Gaussian upper bounds	98	(1)
On-diagonal lower bounds	99	(4)
The Rozenblum-Lieb-Cwikel inequality	103	(8)
The Schrodinger operator Δ - V	103	(2)
The operator Tv = Δ -1V	105	(4)
The Birman-Schwinger principle	109	(2)
Parabolic Harnack inequalities	111	(71)
Scale-invariant Harnack principle	111	(2)
Local Sobolev inequalities	113	(17)
Local Sobolev inequalities and volume growth	113	(6)
Mean value inequalities for subsolutions	119	(3)
Localized heat kernel upper bounds	122	(5)
Time-derivative upper bounds	127	(1)
Mean value inequalities for supersolutions	128	(2)
Poincare inequalities	130	(13)
Poincare inequality and Sobolev inequality	131	(2)
Some weighted Poincare inequalities	133	(2)
Whitney-type coverings	135	(4)
A maximal inequality and an application	139	(2)
End of the proof of Theorem 5.3.4	141	(2)
Harnack inequalities and applications	143	(12)
An inequality for log u	143	(2)
Harnack inequality for positive supersolutions	145	(1)
Harnack inequalities for positive solutions	146	(3)
Holder continuity	149	(2)
Liouville theorems	151	(1)
Heat kernel lower bounds	152	(2)
Two-sided heat kernel bounds	154	(1)
The parabolic Harnack principle	155	(17)
Poincare, doubling, and Harnack	157	(4)
Stochastic completeness	161	(3)
Local Sobolev inequalities and the heat equation	164	(4)
Selected applications of Theorem 5.5.1	168	(4)
Examples	172	(8)
Unimodular Lie groups	172	(3)
Homogeneous spaces	175	(1)
Manifolds with Ricci curvature bounded below	176	(4)
Concluding remarks	180	(2)
Bibliography	182	(6)
Index	188