From Harnack inequality to heat kernel estimates on metric measure spaces and applications

Abstract

Aim of this short note is to show that a dimension-free Harnack inequality on an infinitesimally Hilbertian metric measure space where the heat semigroup admits an integral representation in terms of a kernel is suffcient to deduce a sharp upper Gaussian estimate for such kernel. As intermediate step, we prove the local logarithmic Sobolev inequality (known to be equivalent to a lower bound on the Ricci curvature tensor in smooth Riemannian manifolds). Both results are new also in the more regular framework of $RCD(K,\infty )$ spaces.

References in corpus (4)

• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces
• Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure
• Li-Yau and Harnack type inequalities in $RC{D}^{\ast }(K,N)$ metric measure spaces

Cited by in corpus (2)

• The equality case in Cheeger's and Buser's inequalities on $\mathsf{RCD}$ spaces
• Functional inequalities for the heat flow on time-dependent metric measure spaces