Intrinsic perimeter, compactness and Poincar\'e inequality for SBV functions in Carnot-Carath\'eodory spaces

TL;DR

This paper develops an intrinsic perimeter measure for rectifiable sets in Carnot-Carathéodory spaces, establishing compactness and Poincaré inequalities for SBV functions under certain geometric conditions.

Contribution

It introduces a new intrinsic perimeter measure and proves fundamental inequalities for SBV functions in equiregular Carnot-Carathéodory spaces with property R.

Findings

Established an intrinsic perimeter measure for rectifiable sets

Proved a compactness result for SBV functions

Derived a Poincaré inequality under property R

Abstract

By introducing an intrinsic perimeter measure for intrinsic countably rectifiable sets, we prove a compactness result and a Poincar\'e inequality for special functions with bounded variation in equiregular Carnot-Carath\'eodory spaces which satisfy an additional natural assumption, called property $\mathcal R$.