Equivalence of two BV classes of functions in metric spaces, and existence of a Semmes family of curves under a $1$-Poincar\'e inequality

TL;DR

This paper proves the equivalence of two BV function classes in metric spaces under certain conditions and establishes the existence of a Semmes family of curves when a space supports a 1-Poincaré inequality.

Contribution

It demonstrates the equivalence of Martio's and Miranda Jr.'s BV classes in doubling metric measure spaces supporting a 1-Poincaré inequality and proves the existence of a Semmes family of curves.

Findings

Two BV classes coincide under specified conditions.

Existence of Semmes family of curves in such metric spaces.

Supports analysis of function spaces in metric measure spaces.

Abstract

We consider two notions of functions of bounded variation in complete metric measure spaces, one due to Martio and the other due to Miranda~Jr. We show that these two notions coincide, if the measure is doubling and supports a $1$-Poincar\'e inequality. In doing so, we also prove that if the measure is doubling and supports a $1$-Poincar\'e inequality, then the metric space supports a \emph{Semmes family of curves} structure.