A Bakry-\'Emery criterion for weighted contractivity and $L^2$-Hardy inequalities

TL;DR

This paper establishes a link between weighted contractivity of symmetric Markov diffusion semigroups and $L^2$-Hardy inequalities, providing a Bakry-Émery criterion and applications.

Contribution

It introduces a Bakry-Émery type criterion characterizing weighted contractivity via $L^2$-Hardy inequalities for Markov semigroups.

Findings

Weighted contractivity is equivalent to $L^2$-Hardy inequalities.

A Bakry-Émery criterion for weighted contractivity is provided.

Applications of the criterion are discussed.

Abstract

We show a symmetric Markov diffusion semigroup satisfies a weighted contractivity condition if and only if a $L^2$-Hardy inequality holds, and we give a Bakry-\'Emery type criterion for the former. We then give some applications.