On estimation of the logarithmic Sobolev constant and gradient estimates of heat semigroups

This paper presents some explicit lower bound estimates of logarithmic Sobolev constant for diffusion processes on a compact Riemannian manifold with negative Ricci curvature. Let Ric greater than or equal to - K for some K > 0 and d, D be respectively the dimension and the diameter of the manifold. If the boundary of the manifold is either empty or convex, then the logarithmic Sobolev constant for Brownian motion is not less than GRAPHICS Next, the gradient estimates of heat semigroups (including the Neumann heat semigroup and the Dirichlet one) are studied by using coupling method together with a derivative formula modified from [11]. The resulting estimates recover or improve those given in [7,21] for harmonic functions.