Functions of bounded variation on complete and connected one-dimensional metric spaces

TL;DR

This paper extends the concept of functions of bounded variation (BV) to complete, connected one-dimensional metric spaces with finite Hausdorff measure, establishing equivalence with Miranda's definition and analyzing conditions for finite perimeter.

Contribution

It generalizes BV functions to a broader metric space setting and clarifies the necessity of doubling and Poincaré conditions in perimeter characterization.

Findings

BV functions are equivalent to Miranda's definition in this setting.

Conditions like doubling and Poincaré are essential for finite perimeter results.

Examples demonstrate the importance of these conditions.

Abstract

In this paper, we study functions of bounded variation on a complete and connected metric space with finite one-dimensional Hausdorff measure. The definition of BV functions on a compact interval based on pointwise variation is extended to this general setting. We show this definition of BV functions is equivalent to the BV functions introduced by Miranda. Furthermore, we study the necessity of conditions on the underlying space in Federer's characterization of sets of finite perimeter on metric measure spaces. In particular, our examples show that the doubling and Poincar\'e inequality conditions are essential in showing that a set has finite perimeter if the codimension one Hausdorff measure of the measure-theoretic boundary is finite.