This publication, first released in 2001, specializes in Poincaré, Nash and different Sobolev-type inequalities and their purposes to the Laplace and warmth diffusion equations on Riemannian manifolds. functions lined comprise the ultracontractivity of the warmth diffusion semigroup, Gaussian warmth kernel bounds, the Rozenblum-Lieb-Cwikel inequality and elliptic and parabolic Harnack inequalities. Emphasis is put on the function of households of neighborhood Poincaré and Sobolev inequalities. The textual content offers the 1st self contained account of the equivalence among the uniform parabolic Harnack inequality, at the one hand, and the conjunction of the doubling quantity estate and Poincaré's inequality at the different. it truly is compatible for use as a complicated graduate textbook and also will be an invaluable resource of data for graduate scholars and researchers in research on manifolds, geometric differential equations, Brownian movement and diffusion on manifolds, in addition to different similar components.
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