Catching jumps of metric-valued mappings with Lipschitz functions

Fuente: arXiv
Saved in:
Bibliographic Details
Main Authors: Stolyarov, Dmitriy, Tyulenev, Alexander
Format: Preprint
Published: 2026
Subjects:
Online Access:
Tags: Add Tag
No Tags, Be the first to tag this record!
_version_ 1866908865536196608
author Stolyarov, Dmitriy
Tyulenev, Alexander
author_facet Stolyarov, Dmitriy
Tyulenev, Alexander
contents 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.
format Preprint
id arxiv_https___arxiv_org_abs_2603_03869
institution arXiv
publishDate 2026
record_format arxiv
spellingShingle Catching jumps of metric-valued mappings with Lipschitz functions
Stolyarov, Dmitriy
Tyulenev, Alexander
Classical Analysis and ODEs
Metric Geometry
Probability
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.
title Catching jumps of metric-valued mappings with Lipschitz functions
topic Classical Analysis and ODEs
Metric Geometry
Probability
url https://arxiv.org/abs/2603.03869