Local Li-Yau's estimates on $RCD^*(K,N)$ metric measure spaces

TL;DR

This paper extends Li-Yau's estimates and Yau's gradient bounds to metric measure spaces satisfying the RCD^*(K,N) condition, advancing geometric analysis in non-smooth settings.

Contribution

It establishes local Li-Yau's estimates and sharp Yau's gradient bounds for harmonic functions on RCD^*(K,N) spaces, a significant generalization of classical results.

Findings

Proved local Li-Yau's estimate for heat equation solutions.

Established sharp Yau's gradient bounds for harmonic functions.

Extended geometric analysis tools to RCD^*(K,N) metric measure spaces.

Abstract

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 $RCD^*(K,N).$