Heat Kernel Bounds for Second Order Elliptic Operators on Riemannian Manifolds

Abstract

1. Introduction. We investigate the relationship between ultracontractive bounds on heat kernels, weighted Sobolev inequalities, and logarithmic Sobolev inequalities, in a more general context than previously. We find conditions which imply these bounds for a wide class of second order elliptic operators on manifolds. We apply our result to LaplaceBeltrami operators on manifolds with cusps, and to operators on a manifold whose coefficients become degenerate or infinite on the boundary of the manifold. In an earlier paper with B. Simon [11] we introduced the notion of intrinsic ultracontractivity for Schrodinger operators on RN and for the Dirichlet Laplacian on certain regions Q in RN. Kusuoka and Stroock [13] independently discovered the same phenomena for a somewhat different class of operators using a probabilistic intuition. At the end of their paper they remarked that their ideas could be extended to manifolds with