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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2408.08762 |
| Etiquetas: |
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Tabla de Contenidos:
- 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.