Title: Sharp gradient estimate for heat kernels on $RCD^*(K,N)$ metric measure spaces,
Abstract: In this paper, we will establish an elliptic local Li-Yau gradient estimate
for weak solutions of the heat equation on metric measure spaces with
generalized Ricci curvature bounded from below. One of its main applications is
a sharp gradient estimate for the logarithm of heat kernels. These results seem
new even for smooth Riemannian manifolds.