Aspects of Sobolev-Type Inequalities

Product Description


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.


Review


'The book is very well written and organized. it contains so many comments and explanations that both experts and non-experts on the subject may enjoy reading it.' Zentralblatt für Mathematik

'... a well-written and self-contained account of the topic.' EMS Newsletter

'[This book] constitues a valuable addition to the modern theory of inequalities.' Bulletin of the Belgian Mathematical Society


Book Description


This book, first published in 2001, is a concise introduction to analysis on manifolds focusing on functional inequalities and their applications to the solution of the heat diffusion equation. It gives a detailed treatment of recent advances and would be suitable for use as an advanced graduate textbook, as well as a reference for graduates and researchers.