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

Abstract

The dimension-free Harnack inequality for the heat semigroup is established on the RCD(K , ∞) 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 strongenough Gaussian concentration implying the log-Sobolev inequality is also shown as a generalization of the one on the smooth Riemannian manifold.

Heat Kernel Bounds on Metric Measure Spaces and Some Applications

“…In a forthcoming paper, the second named author will use a dimensional free Harnack inequality (cf. [38,23]) to investigate heat kernel bounds on RCD(K, ∞) spaces (cf. [1,3,5]).…”

The Li–Yau inequality and heat kernels on metric measure spaces

“…On RCD * (K, ∞) spaces, a dimension free Harnack inequality for the heat semigroup was obtained by Li [31]. On a RCD * (K, N) space (X, d, µ) with N < ∞ and µ being a probability measure, Garofalo and Mondino [20] established the Li-Yau inequality for K = 0, and Harnack inequalities for general K ∈ R. Although it was required µ(X) = 1, it is easy to see that their results work for general cases as soon as µ(X) < ∞.…”

Functional inequalities for the heat flow on time‐dependent metric measure spaces

“…Similarly one deduces Bochner from the reverse local logarithmic Sobolev bound. Indeed by (15), it holds by the same argument as above…”

“…In particular, they proved that both the Bochner inequality (without dimensional term) and the L 2 -gradient estimate are equivalent to the synthetic Ricci bound CD(K, ∞); and they deduced the local Poincaré inequality and the logarithmic Harnack inequality. Savaré [18] extended the powerful self-improvement property of Bochner's inequality to mm-spaces and utilized it to deduce the L 1 -gradient estimate; based on the latter, H. Li [15] proved the dimension-independent Harnack inequality which, in turn, implies the logarithmic Harnack inequality.…”

“…Theorem 3.2. Assume that one of the local logarithmic Sobolev inequalities,(14) or(15), holds. Then the pointwise dynamic Bochner holds for t, that is, for all…”