Li-Yau and Harnack type inequalities in $RC{D}^{\ast }(K,N)$ metric measure spaces

Abstract

Metric measure spaces satisfying the reduced curvature-dimension condition $C{D}^{\ast }(K,N)$ and where the heat flow is linear are called $RC{D}^{\ast }(K,N)$-spaces. This class of non smooth spaces contains Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature bounded below by $K$ and dimension bounded above by $N$. We prove that in $RC{D}^{\ast }(K,N)$-spaces the following properties of the heat flow hold true: a Li-Yau type inequality, a Bakry-Qian inequality, the Harnack inequality.

21 pages

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• Hamilton's Gradient Estimates and A Monotonicity Formula for Heat Flows on Metric Measure Spaces
• Second order differentiation formula on compact $RC{D}^{\ast }(K,N)$ spaces
• Singular Weyl's law with Ricci curvature bounded below
• Sharp Heat Kernel Bounds and Entropy in Metric Measure Spaces
• From Harnack inequality to heat kernel estimates on metric measure spaces and applications
• $(K,N)$-convexity and the curvature-dimension condition for negative $N$
• Sharp Spectral Gaps on Metric Measure Spaces
• Hölder continuity of tangent cones in RCD(K,N) spaces and applications to non-branching
• Dimension free Harnack inequalities on $\RCD(K, \infty)$ spaces
• Noncompact RCD(0,N) spaces with linear volume growth
• The Measure Preserving Isometry Groups of Metric Measure Spaces
• Sharp gradient estimate for heat kernels on $RC{D}^{\ast }(K,N)$ metric measure spaces