An Isometric Representation for the Lipschitz-Free Space of Length Spaces Embedded in Finite-Dimensional Spaces
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912640659357696 |
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| author | Flores, Gonzalo |
| author_facet | Flores, Gonzalo |
| contents | For a domain $Ω$ in a finite-dimensional space $E$, we consider the space $M=(Ω,d)$ where $d$ is the intrinsic distance in $Ω$. We obtain an isometric representation of the space $\mathrm{Lip}_{0}(M)$ as a subspace of $L^{\infty}(Ω;E^{*})$ and we use this representation in order to obtain the corresponding isometric representation for the Lipschitz-free space $\mathcal{F}(M)$ as a quotient of the space $L^{1}(Ω;E)$. We compare our result with those existent in the literature for bounded domains with Lipschitz boundary, and for convex domains, which can be then deduced as a corollaries of our result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13538 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Isometric Representation for the Lipschitz-Free Space of Length Spaces Embedded in Finite-Dimensional Spaces Flores, Gonzalo Functional Analysis Primary 46B04, 46B10, Secondary 46F10 For a domain $Ω$ in a finite-dimensional space $E$, we consider the space $M=(Ω,d)$ where $d$ is the intrinsic distance in $Ω$. We obtain an isometric representation of the space $\mathrm{Lip}_{0}(M)$ as a subspace of $L^{\infty}(Ω;E^{*})$ and we use this representation in order to obtain the corresponding isometric representation for the Lipschitz-free space $\mathcal{F}(M)$ as a quotient of the space $L^{1}(Ω;E)$. We compare our result with those existent in the literature for bounded domains with Lipschitz boundary, and for convex domains, which can be then deduced as a corollaries of our result. |
| title | An Isometric Representation for the Lipschitz-Free Space of Length Spaces Embedded in Finite-Dimensional Spaces |
| topic | Functional Analysis Primary 46B04, 46B10, Secondary 46F10 |
| url | https://arxiv.org/abs/2411.13538 |