This book focuses on Poincar Nash and other Sobolevtype inequalities andtheir applications to the Laplace and heat diffusion equations on Riemannianmanifolds. A... » 

This book focuses on Poincar Nash and other Sobolevtype inequalities andtheir applications to the Laplace and heat diffusion equations on Riemannianmanifolds. Applications covered include the ultracontractivity of the heatdiffusion semigroup Gaussian heat kernel bounds the RozenblumLiebCwikelinequality and elliptic and parabolic Harnack inequalities. Emphasis is placedon the role of families of local Poincar and Sobolev inequalities. The textprovides the first self contained account of the equivalence between theuniform parabolic Harnack inequality on the one hand and the conjunction ofthe doubling volume property and Poincar s inequality on the other. «