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New characterizations of Ricci curvature on RCD metric measure spaces (1702.00740v2)

Published 2 Feb 2017 in math.FA, math.AP, and math.MG

Abstract: We prove that on a large family of metric measure spaces, if the $Lp$-gradient estimate for heat flows holds for some $p>2$, then the $L1$-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.

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