Characterizations of sets of finite perimeter using heat kernels in metric spaces

TL;DR

This paper establishes a novel connection between sets of finite perimeter and heat semigroup behavior in metric measure spaces, providing new characterizations of BV functions under broad conditions.

Contribution

It introduces a new characterization of finite perimeter sets and BV functions using heat kernels in metric spaces with doubling measures and Poincaré inequalities.

Findings

Characterization of finite perimeter sets via short-time heat semigroup behavior

New BV function characterization through near-diagonal energy

Applicable to metric spaces with doubling measure and Poincaré inequality

Abstract

The overarching goal of this paper is to link the notion of sets of finite perimeter (a concept associated with $N^{1,1}$-spaces) and the theory of heat semigroups (a concept related to $N^{1,2}$-spaces) in the setting of metric measure spaces whose measure is doubling and supports a $1$-Poincar\'e inequality. We prove a characterization of sets of finite perimeter in terms of a short time behavior of the heat semigroup in such metric spaces. We also give a new characterization of ${\rm BV}$ functions in terms of a near-diagonal energy in this general setting.