The Li-Yau inequality and heat kernels on metric measure spaces

Let (X, d, mu) be an RCD* (K, N) space with K is an element of R and N is an element of [1, infinity). Suppose that (X, d) is connected, complete and separable, and supp mu = X. We prove that the Li-Yau inequality for the heat flow holds true on (X, d, mu) when K >= 0. A Baudoin-Garofalo inequality and Harnack inequalities for the heat flow are established on (X, d, mu) for general K is an element of H. Large time behaviors of heat kernels are also studied. (C) 2014 Elsevier Masson SAS. All rights reserved.