Characterizations of Sets of Finite Perimeter Using Heat Kernels in Metric Spaces

The overarching goal of this paper is to link the notion of sets of finite perimeter (a concept associated with N1,1-spaces) and the theory of heat semigroups (a concept related to N1,2-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 BV functions in terms of a near-diagonal energy in this general setting.