Dimension-Free Harnack Inequalities on $$\hbox {RCD}(K, \infty )$$ Spaces

Abstract

The dimension-free Harnack inequality for the heat semigroup is established on the \(\mathrm{{RCD}}(K,\infty )\) space, which is a non-smooth metric measure space having the Ricci curvature bounded from below in the sense of Lott–Sturm–Villani plus the Cheeger energy being quadratic. As its applications, the heat semigroup entropy-cost inequality and contractivity properties of the semigroup are studied, and a strong-enough Gaussian concentration implying the log-Sobolev inequality is also shown as a generalization of the one on the smooth Riemannian manifold.