Fine properties of metric space-valued mappings of bounded variation in metric measure spaces

TL;DR

This paper investigates the properties of bounded variation (BV) mappings from metric measure spaces into metric spaces, comparing two notions of BV, and analyzing the fine structure of jump sets in proper metric target spaces.

Contribution

It establishes the equivalence of two BV notions for Banach space targets and explores the detailed structure of jump sets for BV mappings into proper metric spaces.

Findings

Two BV notions coincide for Banach space targets with comparable energies.

Jump points in proper metric space targets have at least two and at most k_0 jump values.

Preimages of balls around jump values have lower density at least gamma.

Abstract

Here we consider two notions of mappings of bounded variation (BV) from the metric measure space into the metric space; one based on relaxations of Newton-Sobolev functions, and the other based on a notion of AM-upper gradients. We show that when the target metric space is a Banach space, these two notions coincide with comparable energies, but for more general target metric spaces, the two notions can give different function-classes. We then consider the fine properties of BV mappings (based on the AM-upper gradient property), and show that when the target space is a proper metric space, then for a BV mapping into the target space, co-dimension $1$-almost every point in the jump set of a BV mapping into the proper space has at least two, and at most $k_0$, number of jump values associated with it, and that the preimage of balls around these jump values have lower density at least $\gamma$ at that point. Here $k_0$ and $\gamma$ depend solely on the structural constants associated with the metric measure space, and jump points are points at which the map is not approximately continuous.