Catching jumps of metric-valued mappings with Lipschitz functions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908865536196608 |
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| 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 |
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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 |