By using the maximum principle and analysis of heat semigroups, Harnack inequalities are studied for log-Sobolev functions. From this, some lower bound estimates of the log-Sobolev constant are presented by using the spectral gap inequality and the coupling method. The resulting inequalities either recover or improve the corresponding ones proved by Chung and Yau. Especially, Harnack inequalities and estimates of log-Sobolev constants can be dimension-free.
"Harnack Inequalities for Log-Sobolev Functions and Estimates of Log-Sobolev Constants." Ann. Probab. 27 (2) 653 - 663, April 1999. https://doi.org/10.1214/aop/1022677381