Estimates of Logarithmic Sobolev Constant: An Improvement of Bakry–Emery Criterion

This paper is mainly devoted to estimate the logarithmic Sobolev (abbrev. L.S.) constant for diffusion operators on manifold or in Rd. In most cases, we study the lower bounds but a generalization to [A. Korzeniowski,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 potentials, 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 notation:β(r)=infρ(x, p)⩾r infX∈Tx(M), ‖X‖=1(Ricc−HessV)(X, X), whereρ(x, p) is the distance betweenxand an arbitrary fixed pointp∈M, the improvement can be roughly stated as follows. The condition “infr⩾0 β(r)>0” for which the criterion is available is now replaced by “supr⩾0 β(r)>0.”