Nagata Dimension and Lipschitz Extensions Into Quasi-Banach Spaces
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| Format: | Preprint |
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2024
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| _version_ | 1866917582576025600 |
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| author | Bíma, Jan |
| author_facet | Bíma, Jan |
| contents | Given two metric spaces $\mathcal N \subseteq \mathcal M$ in inclusion and $0<p\leq 1$, we wish to determine the smallest constant $\mathfrak{t}_p (\mathcal N, \mathcal M)$ such that any Lipschitz map $f: \mathcal N \to Z$ into any $p$-Banach space $Z$ can be extended to a Lipschitz map $f' : \mathcal M \to Z$ satisfying $\operatorname{Lip} f' \leq \mathfrak{t}_p (\mathcal N, \mathcal M)\cdot \operatorname{Lip} f$. In this article, we prove that if $\mathcal N$ has finite Nagata dimension at most $d$ with constant $γ$, then $\mathfrak{t}_p (\mathcal N, \mathcal M) \lesssim_p γ\cdot (d+1)^{1/p -1} \cdot \log (d+2)$ for all $0<p\leq 1$. We show that examples of spaces with finite Nagata dimension include doubling spaces, as well as minor-excluded metric graphs. We also establish that the constant $\mathfrak{t}_p (\mathcal N, \mathcal M)$ generally increases as $p$ approaches zero. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_03189 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nagata Dimension and Lipschitz Extensions Into Quasi-Banach Spaces Bíma, Jan Functional Analysis 54C20, 46A16 (Primary) 54F45, 05C63, 05C83 (Secondary) Given two metric spaces $\mathcal N \subseteq \mathcal M$ in inclusion and $0<p\leq 1$, we wish to determine the smallest constant $\mathfrak{t}_p (\mathcal N, \mathcal M)$ such that any Lipschitz map $f: \mathcal N \to Z$ into any $p$-Banach space $Z$ can be extended to a Lipschitz map $f' : \mathcal M \to Z$ satisfying $\operatorname{Lip} f' \leq \mathfrak{t}_p (\mathcal N, \mathcal M)\cdot \operatorname{Lip} f$. In this article, we prove that if $\mathcal N$ has finite Nagata dimension at most $d$ with constant $γ$, then $\mathfrak{t}_p (\mathcal N, \mathcal M) \lesssim_p γ\cdot (d+1)^{1/p -1} \cdot \log (d+2)$ for all $0<p\leq 1$. We show that examples of spaces with finite Nagata dimension include doubling spaces, as well as minor-excluded metric graphs. We also establish that the constant $\mathfrak{t}_p (\mathcal N, \mathcal M)$ generally increases as $p$ approaches zero. |
| title | Nagata Dimension and Lipschitz Extensions Into Quasi-Banach Spaces |
| topic | Functional Analysis 54C20, 46A16 (Primary) 54F45, 05C63, 05C83 (Secondary) |
| url | https://arxiv.org/abs/2402.03189 |