Local Li-Yau's estimates on $RC{D}^{\ast }(K,N)$ metric measure spaces

Abstract

In this paper, we will study the (linear) geometric analysis on metric measure spaces. We will establish a local Li-Yau's estimate for weak solutions of the heat equation and prove a sharp Yau's gradient gradient for harmonic functions on metric measure spaces, under the Riemannian curvature-dimension condition $RC{D}^{\ast }(K,N).$

The last version. We add the RCD in the title to make the paper's content more specific, and add some 3 references and correct some typoes. To appear in Calc. Var. PDE

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• Weyl's law on $RC{D}^{\ast }(K,N)$ metric measure spaces