Dimension free Harnack inequalities on $\RCD(K, \infty)$ spaces

TL;DR

This paper proves a dimension-free Harnack inequality for heat semigroups on $ cd(K, fty)$ spaces, extending key functional inequalities and concentration results to non-smooth metric measure spaces with Ricci curvature bounds.

Contribution

It establishes the first dimension-free Harnack inequality on $ cd(K, abla)$ spaces, generalizing classical results from smooth Riemannian manifolds to non-smooth settings.

Findings

Proved dimension-free Harnack inequality for heat semigroup on $ cd(K, fty)$ spaces.

Derived entropy-cost inequalities and contractivity properties of the heat semigroup.

Showed Gaussian concentration implies the log-Sobolev inequality in this setting.

Abstract

The dimension free Harnack inequality for the heat semigroup is established on the $\RCD(K,\infty)$ 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.