New characterizations of Ricci curvature on RCD metric measure spaces

Abstract

We prove that on a large family of metric measure spaces, if the $L^p$-gradient estimate for heat flows holds for some $p>2$, then the $L^1$-gradient estimate also holds. This result extends Savaré's result on metric measure spaces, and provides a new proof to von Renesse-Sturm theorem on smooth metric measure spaces. As a consequence, we propose a new analysis object based on Gigli's measure-valued Ricci tensor, to characterize the Ricci curvature of $\mathrm{RCD}$ space in a local way. The argument is a new iteration technique based on non-smooth Bakry-Émery theory, which is a new method to study the curvature dimension condition of metric measure spaces.

arXiv commentThe first title of this paper was `Self-improvement of gradient estimate of heat flows on metric measure spaces'

Cited by in corpus (1)

• Optimal transport, gradient estimates, and pathwise Brownian coupling on spaces with variable Ricci bounds