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

Niko Marola, Michele Miranda Jr., Nageswari Shanmugalingam

Published 2014-05-01, updated 2015-07-18Version 2

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.