Complete Manifolds and Log-Sobolev Inequalities for Subelliptic Operators with Functional Inequality

Abstract

We give the best constant with weights being powers of distance from the origin in weighted Sobolev inequality. Complete manifolds with non-negative curvature and Sobolev inequalities, convergence to equilbrium with logarithmic Sobolev constant on manifolds with Ricci curvature bounded below are considered .We also show logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Riemannian geometry on the path space ,with Hardy type inequalities on complete Riemannian manifolds are studied .We establish the functional inequalities for empty essential spectrum and the Poincare ́ inequality on loop spaces.The complete manifolds supporting a weighted Sobolev type inequality , functional inequality and a class of processes on the path space over a compact and non compact Riemannian manifold with unbounded diffusion are established . Nash and log- Sobolev inequalities for hypoelliptic and subelliptic operators satisfying a generalized curvature dimension inequalities are characterized.