Local Li-Yau's estimates on $RCD^*(K,N)$ metric measure spaces
  In this paper, we will study the (linear) geometric analysis on metric measure spaces. We will establish a local Li-Yau's estimate for weak solutions of the heat equation and prove a sharp Yau's gradient gradient for harmonic functions on metric measure spaces, under the Riemannian curvature-dimension condition $RCD^*(K,N).$ 
Hui-Chun Zhang, Xi-Ping Zhu
math.DG math.MG