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 $γ\colon [a,b] \to \mathcal{X}$ is a map of bounded variation if and only if $f \circ γ$ 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 $γ$. 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.

Figures (5)

• Figure 1: Dyadic tree for $N=5$.
• Figure 2: A Lipschitz function $f$ and functions $M_1$ (orange), $M_2$ (blue), and $M_3$ (green) constructed from it via \ref{['MartingaleExampleFormula']}.
• Figure 3: The graphs $G_1$, $G_2$, and $G_3$.
• Figure 4: Two vertices and corresponding atoms for $N=2$.
• Figure 5: Illustration to the proof of the analog of \ref{['BoundindMartingaleDifference']}

Theorems & Definitions (23)

• Proposition 1.1: Simplification of Theorem $1.2$ in OT2025
• Definition 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Proposition 2.1: Proposition $15.7$ in DavidSemmes1997
• Example 2.2
• Lemma 2.3