Li-Yau and Harnack type inequalities in $RCD^*(K,N)$ metric measure spaces

TL;DR

This paper establishes Li-Yau, Bakry-Qian, and Harnack inequalities for heat flow in $RCD^*(K,N)$ spaces, extending classical geometric analysis results to a broad class of non-smooth metric measure spaces.

Contribution

It proves fundamental heat flow inequalities in $RCD^*(K,N)$-spaces, generalizing known results from smooth Riemannian manifolds to non-smooth settings.

Findings

Li-Yau inequality established in $RCD^*(K,N)$-spaces

Bakry-Qian inequality proven for these spaces

Harnack inequality demonstrated in the non-smooth setting

Abstract

Metric measure spaces satisfying the reduced curvature-dimension condition $CD^*(K,N)$ and where the heat flow is linear are called $RCD^*(K,N)$-spaces. This class of non smooth spaces contains Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature bounded below by $K$ and dimension bounded above by $N$. We prove that in $RCD^*(K,N)$-spaces the following properties of the heat flow hold true: a Li-Yau type inequality, a Bakry-Qian inequality, the Harnack inequality.