Traces of vanishing Hölder spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915654642171904 |
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| author | Mohanta, Kaushik Mudarra, Carlos Oikari, Tuomas |
| author_facet | Mohanta, Kaushik Mudarra, Carlos Oikari, Tuomas |
| contents | For an arbitrary subset $E\subset\mathbb{R}^n,$ we introduce and study the three vanishing subspaces of the Hölder space $\dot{C}^{0,ω}(E)$ consisting of those functions for which the ratio $|f(x)-f(y)|/ω(|x-y|)$ vanishes, when $(1)$ $|x-y|\to 0$ , $(2)$ $|x-y|\to\infty$ or $(3)$ $\min(|x|,|y|)\to\infty.$ We prove that the Whitney extension operator maps each of these vanishing subspaces from $E$ to the corresponding vanishing spaces defined on the whole ambient space $\mathbb{R}^n.$ In fact, this follows as the zeroth order special case of a more general problem involving higher order derivatives. As a consequence, we obtain complete characterizations of approximability of Hölder functions $\dot{C}^{0,ω}(E)$ by Lipschitz and boundedly supported functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_14156 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Traces of vanishing Hölder spaces Mohanta, Kaushik Mudarra, Carlos Oikari, Tuomas Classical Analysis and ODEs Functional Analysis 26B05, 26B35, 42B35, 46E35, 46T20 For an arbitrary subset $E\subset\mathbb{R}^n,$ we introduce and study the three vanishing subspaces of the Hölder space $\dot{C}^{0,ω}(E)$ consisting of those functions for which the ratio $|f(x)-f(y)|/ω(|x-y|)$ vanishes, when $(1)$ $|x-y|\to 0$ , $(2)$ $|x-y|\to\infty$ or $(3)$ $\min(|x|,|y|)\to\infty.$ We prove that the Whitney extension operator maps each of these vanishing subspaces from $E$ to the corresponding vanishing spaces defined on the whole ambient space $\mathbb{R}^n.$ In fact, this follows as the zeroth order special case of a more general problem involving higher order derivatives. As a consequence, we obtain complete characterizations of approximability of Hölder functions $\dot{C}^{0,ω}(E)$ by Lipschitz and boundedly supported functions. |
| title | Traces of vanishing Hölder spaces |
| topic | Classical Analysis and ODEs Functional Analysis 26B05, 26B35, 42B35, 46E35, 46T20 |
| url | https://arxiv.org/abs/2401.14156 |