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

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.