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

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 $1$-Poincaré 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 $\mathrm{BV}$ functions in terms of a near-diagonal energy in this general setting.

To appear in Potential Analysis

References in corpus (1)

• Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,\infty) metric measure spaces

Cited by in corpus (4)

• The Total Variation Flow in Metric Random Walk Spaces
• Heat content asymptotics for sub-Riemannian manifolds
• Heat content and horizontal mean curvature on the Heisenberg group
• Short-time heat content asymptotics via the wave and eikonal equations