Harnack inequalities for log-Sobolev functions and estimates of log-Sobolev constants

By using the maximum principle and analysis of heat semigroups, Harnack ine qualities are studied for log-Sobolev functions. From this, some lower boun d 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. Especial ly, Harnack inequalities and estimates of log-Sobolev constants can be dime nsion-free.