Aspects of Sobolev-Type Inequalities (London Mathematical Society Lecture Note Series, Series Number 289)

This book, first published in 2001, 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. It is suitable to be used as an advanced graduate textbook and will also be a useful source of information for graduate students and researchers in analysis on manifolds, geometric differential equations, Brownian motion and diffusion on manifolds, as well as other related areas. Focusing On Poincaré, Nash And Other Sobolev-type Inequalities And Their Applications To The Laplace And Heat Diffusion Equations On Riemannian Manifolds, This Text Is An Advanced Graduate Book That Will Also Suit Researchers. 1 Sobolev Inequalities In R[superscript N] 7 -- 1.1 Sobolev Inequalities 7 -- 1.1.2 The Proof Due To Gagliardo And To Nirenberg 9 -- 1.1.3 P = 1 Implies P [greater Than Or Equal] 1 10 -- 1.2 Riesz Potentials 11 -- 1.2.1 Another Approach To Sobolev Inequalities 11 -- 1.2.2 Marcinkiewicz Interpolation Theorem 13 -- 1.2.3 Proof Of Sobolev Theorem 1.2.1 16 -- 1.3 Best Constants 16 -- 1.3.1 The Case P = 1: Isoperimetry 16 -- 1.3.2 A Complete Proof With Best Constant For P = 1 18 -- 1.3.3 The Case P> 1 20 -- 1.4 Some Other Sobolev Inequalities 21 -- 1.4.1 The Case P> N 21 -- 1.4.2 The Case P = N 24 -- 1.4.3 Higher Derivatives 26 -- 1.5 Sobolev -- Poincare Inequalities On Balls 29 -- 1.5.1 The Neumann And Dirichlet Eigenvalues 29 -- 1.5.2 Poincare Inequalities On Euclidean Balls 30 -- 1.5.3 Sobolev -- Poincare Inequalities 31 -- 2 Moser's Elliptic Harnack Inequality 33 -- 2.1 Elliptic Operators In Divergence Form 33 -- 2.1.1 Divergence Form 33 -- 2.1.2 Uniform Ellipticity 34 -- 2.1.3 A Sobolev-type Inequality For Moser's Iteration 37 -- 2.2 Subsolutions And Supersolutions 38 -- 2.2.1 Subsolutions 38 -- 2.2.2 Supersolutions 43 -- 2.2.3 An Abstract Lemma 47 -- 2.3 Harnack Inequalities And Continuity 49 -- 2.3.1 Harnack Inequalities 49 -- 2.3.2 Holder Continuity 50 -- 3 Sobolev Inequalities On Manifolds 53 -- 3.1.1 Notation Concerning Riemannian Manifolds 53 -- 3.1.2 Isoperimetry 55 -- 3.1.3 Sobolev Inequalities And Volume Growth 57 -- 3.2 Weak And Strong Sobolev Inequalities 60 -- 3.2.1 Examples Of Weak Sobolev Inequalities 60 -- 3.2.2 (s[superscript [theta] Subscript R, S])-inequalities: The Parameters Q And V 61 -- 3.2.3 The Case 0