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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.08762 |
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| _version_ | 1866911992765218816 |
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| author | Oleinik, Roman D. Tyulenev, Alexander I. |
| author_facet | Oleinik, Roman D. Tyulenev, Alexander I. |
| contents | Let $\operatorname{X}:=(\operatorname{X},\operatorname{d})$ be an arbitrary metric space. For each $p \in [1,\infty]$, we prove that a map $γ:[a,b] \to \operatorname{X}$ is $p$-absolutely continuous if and only if, for every Lipschitz function $h:\operatorname{X} \to \mathbb{R}$, the post-composition $h \circ γ$ is a $p$-absolutely continuous function. Furthermore, if $\operatorname{X}$ is complete and separable, then, for each $p \in (1,\infty)$, we show that the equivalence class (up to $\mathcal{L}^{1}$-a.e. equality) of a Borel map $γ:[a,b] \to \operatorname{X}$ belongs to the Sobolev $W_{p}^{1}([a,b],\operatorname{X})$-space if and only if, for every Lipschitz function $h:\operatorname{X} \to \mathbb{R}$, the equivalence class (up to $\mathcal{L}^{1}$-a.e. equality) of the post-composition $h \circ γ$ belongs to the Sobolev $W_{p}^{1}([a,b],\mathbb{R})$-space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08762 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterization of AC and Sobolev curves via Lipschitz post-compositions Oleinik, Roman D. Tyulenev, Alexander I. Functional Analysis Let $\operatorname{X}:=(\operatorname{X},\operatorname{d})$ be an arbitrary metric space. For each $p \in [1,\infty]$, we prove that a map $γ:[a,b] \to \operatorname{X}$ is $p$-absolutely continuous if and only if, for every Lipschitz function $h:\operatorname{X} \to \mathbb{R}$, the post-composition $h \circ γ$ is a $p$-absolutely continuous function. Furthermore, if $\operatorname{X}$ is complete and separable, then, for each $p \in (1,\infty)$, we show that the equivalence class (up to $\mathcal{L}^{1}$-a.e. equality) of a Borel map $γ:[a,b] \to \operatorname{X}$ belongs to the Sobolev $W_{p}^{1}([a,b],\operatorname{X})$-space if and only if, for every Lipschitz function $h:\operatorname{X} \to \mathbb{R}$, the equivalence class (up to $\mathcal{L}^{1}$-a.e. equality) of the post-composition $h \circ γ$ belongs to the Sobolev $W_{p}^{1}([a,b],\mathbb{R})$-space. |
| title | Characterization of AC and Sobolev curves via Lipschitz post-compositions |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2408.08762 |