NEW CHARACTERIZATIONS OF RICCI CURVATURE ON RCD METRIC MEASURE SPACES

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 Savare'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 RCD space in a local way. In the proof we adopt an iteration technique based on non-smooth Bakry-Emery theory, which is a new method to study the curvature dimension condition of metric measure spaces.