Catching jumps of metric-valued mappings with Lipschitz functions

TL;DR

This paper investigates when metric-valued mappings can be characterized by their composition with Lipschitz functions, highlighting the importance of continuity and providing examples where the characterization holds or fails.

Contribution

It demonstrates that the continuity assumption is crucial for the characterization of bounded variation in metric spaces, with specific results for ultrametric spaces and counterexamples.

Findings

Characterization holds for ultrametric spaces without continuity.

Counterexamples in $oldsymbol{ ext{l}_2}$, infinite trees, and Laakso spaces.

Continuity is essential for the characterization in general metric spaces.

Abstract

It follows from recent results of V. Bakhtin, R. Oleinik, and the second named author that, given a metric space $\mathcal{X}$, a continuous map $\gamma\colon [a,b] \to \mathcal{X}$ is a map of bounded variation if and only if $f \circ \gamma$ is a function of bounded variation for every Lipschitz function $f\colon\mathcal{X} \to \mathbb{R}$. In this note, we show that the continuity assumption is of crucial importance: for many interesting examples of metric spaces there are no analogs of that characterization without the continuity assumption on $\gamma$. The interesting examples are: $\ell_2$, infinite metric trees, and Laakso-type spaces. However, for ultrametric spaces the said characterization holds without any continuity assumptions.