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 mani fold 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 diame ter of the manifold. If the boundary of the manifold is either empty o r convex, then the logarithmic Sobolev constant for Brownian motion is not less than GRAPHICS Next, the gradient estimates of heat semigroup s (including the Neumann heat semigroup and the Dirichlet one) are stu died by using coupling method together with a derivative formula modif ied from [11]. The resulting estimates recover or improve those given in [7,21] for harmonic functions.