Aspects of Sobolev-Type InequalitiesThis 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