Tilings of the Hyperbolic Space and Lipschitz Functions
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908761856147456 |
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| author | Bargetz, Christian Luggin, Franz Russo, Tommaso |
| author_facet | Bargetz, Christian Luggin, Franz Russo, Tommaso |
| contents | We use a special tiling for the hyperbolic $d$-space $\mathbb{H}^d$ for $d=2,3,4$ to construct an (almost) explicit isomorphism between the Lipschitz-free space $\mathcal{F}(\mathbb{H}^d)$ and $\mathcal{F}(P)\oplus\mathcal{F}(\mathcal{N})$ where $P$ is a polytope in $\mathbb{R}^d$ and $\mathcal{N}$ a net in $\mathbb{H}^d$ coming from the tiling. This implies that the spaces $\mathcal{F}(\mathbb{H}^d)$ and $\mathcal{F}(\mathbb{R}^{d})\oplus \mathcal{F}(\mathcal{M})$ are isomorphic for every net $\mathcal{M}$ in $\mathbb{H}^d$. In particular, we obtain that, for $d=2,3,4$, $\mathcal{F}(\mathbb{H}^d)$ has a Schauder basis. Moreover, using a similar method, we also give an explicit isomorphism between $\mathrm{Lip}(\mathbb{H}^{d})$ and $\mathrm{Lip}(\mathbb{R}^d)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_04201 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tilings of the Hyperbolic Space and Lipschitz Functions Bargetz, Christian Luggin, Franz Russo, Tommaso Functional Analysis 46B03, 51M10 (Primary), and 46B20, 46E15, 26B35, 20F55 (Secondary) We use a special tiling for the hyperbolic $d$-space $\mathbb{H}^d$ for $d=2,3,4$ to construct an (almost) explicit isomorphism between the Lipschitz-free space $\mathcal{F}(\mathbb{H}^d)$ and $\mathcal{F}(P)\oplus\mathcal{F}(\mathcal{N})$ where $P$ is a polytope in $\mathbb{R}^d$ and $\mathcal{N}$ a net in $\mathbb{H}^d$ coming from the tiling. This implies that the spaces $\mathcal{F}(\mathbb{H}^d)$ and $\mathcal{F}(\mathbb{R}^{d})\oplus \mathcal{F}(\mathcal{M})$ are isomorphic for every net $\mathcal{M}$ in $\mathbb{H}^d$. In particular, we obtain that, for $d=2,3,4$, $\mathcal{F}(\mathbb{H}^d)$ has a Schauder basis. Moreover, using a similar method, we also give an explicit isomorphism between $\mathrm{Lip}(\mathbb{H}^{d})$ and $\mathrm{Lip}(\mathbb{R}^d)$. |
| title | Tilings of the Hyperbolic Space and Lipschitz Functions |
| topic | Functional Analysis 46B03, 51M10 (Primary), and 46B20, 46E15, 26B35, 20F55 (Secondary) |
| url | https://arxiv.org/abs/2402.04201 |