ESTIMATES OF LOGARITHMIC SOBOLEV CONSTANT - AN IMPROVEMENT OF BAKRY-EMERY CRITERION

This paper is mainly devoted to estimate the logarithmic Sobolev (abbr ev. L.S.) constant for diffusion operators on manifold or in R(d). In most cases, we study the lower bounds but a generalization to [A. Korz eniowski, J. Funct. Anal. 71 (1987), 363-370, Theorem 1] for the upper bound is also presented (Theorem 1.5). Based on a simple observation (due to [J.-D. Deuschel and D. W. Stroock, J. Funct. Anal. 92 (1990), 30-48]) of the comparison between the L.S. constants for different pot entials, the powerful Bakry-Emery criterion for the L.S. inequality is improved considerably in the paper, especially for the manifolds with non-positive sectional curvatures (Theorem 1.3(1)). In terms of our n otation: beta(r)=inf(p(x,p)greater than or equal to r) inf(X epsilon T x(M),parallel to X parallel to=1)(Ricc - Hess(V))(X, X), where rho(x, p) is the distance between x and an arbitrary fixed point p epsilon M, the improvement can be roughly stated as follows. The condition ''inf (r greater than or equal to 0) beta(r) > 0'' for which the criterion i s available is now replaced by ''sup(r greater than or equal to 0) bet a(r)>0.'' (C) 1997 Academic Press.