Catching jumps of metric-valued mappings with Lipschitz functions

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 $\gamma :[a,b]\to \mathcal{X}$ is a map of bounded variation if and only if $f\circ \gamma$ is a function of bounded variation for every Lipschitz function $f:\mathcal{X}\to \mathbb{R}$. 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.

arXiv comment17 pages, 5 figures