# Sharp gradient estimate for heat kernels on $RCD^*(K,N)$ metric measure spaces

## TL;DR

This paper establishes a sharp gradient estimate for heat kernels on $RCD^*(K,N)$ spaces, extending classical results to a broader class of metric measure spaces with lower Ricci curvature bounds.

## Contribution

It introduces a novel elliptic Li-Yau gradient estimate for weak solutions of the heat equation on $RCD^*(K,N)$ spaces, applicable even in smooth Riemannian manifolds.

## Key findings

- Sharp gradient estimate for heat kernels derived
- Results applicable to non-smooth metric measure spaces
- Extends classical heat kernel estimates to $RCD^*(K,N)$ 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.