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

created by durandcar on 11 Sep 2018

[BibTeX]

Preprint

Inserted: 11 sep 2018
Last Updated: 11 sep 2018

Year: 2018

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 $1$-Poincaré inequality. In doing so, we also prove that if the measure is doubling and supports a $1$-Poincaré inequality, then the metric space supports a Semmes family of curves structure.