Local Li-Yaus 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-Yaus estimate for weak solutions of the heat equation and prove a sharp Yaus gradient gradient for harmonic functions on metric measure spaces, under the Riemannian curvature-dimension condition $RCD^*(K,N).$