# New characterizations of Ricci curvature on RCD metric measure spaces

## TL;DR

This paper establishes new local characterizations of Ricci curvature in RCD metric measure spaces using gradient estimates and a novel iteration technique rooted in non-smooth Bakry-Émery theory.

## Contribution

It extends existing gradient estimate results to the L^1 case and introduces a new local analysis object for Ricci curvature in RCD spaces.

## Key findings

- L^p-gradient estimate implies L^1-gradient estimate for p>2
- Provides a new proof of von Renesse-Sturm theorem
- Introduces a local Ricci curvature characterization method

## 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 Savar\'e'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 ${\rm RCD}$ space in a local way. The argument is a new iteration technique based on non-smooth Bakry-\'Emery theory, which is a new method to study the curvature dimension condition of metric measure spaces.