Two characterization of BV functions on Carnot groups via the heat semigroup

TL;DR

This paper introduces two new ways to characterize BV functions and sets of finite perimeter in Carnot groups using the heat semigroup, extending Euclidean results to a non-commutative setting.

Contribution

It provides the first two characterizations of BV functions in Carnot groups via heat semigroup behavior, including a rectifiability assumption for the second.

Findings

Characterizations hold in Carnot groups similar to Euclidean spaces.

The second characterization requires rectifiability of the reduced boundary.

Results are established for Step 2 Carnot groups.

Abstract

In this paper we provide two different characterizations of sets with finite perimeter and functions of bounded variation in Carnot groups, analogous to those which hold in Euclidean spaces, in terms of the short-time behaviour of the heat semigroup. The second one holds under the hypothesis that the reduced boundary of a set of finite perimeter is rectifiable, a result that presently is known in Step 2 Carnot groups.