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arXiv · 1809.09957

Isomorphisms between spaces of Lipschitz functions

Leandro Candido·Marek Cúth·Michal Doucha
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Abstract

We develop tools for proving isomorphisms of normed spaces of Lipschitz functions over various doubling metric spaces and Banach spaces. In particular, we show that $\operatorname{Lip}_0(\mathbb{Z}^d)\simeq\operatorname{Lip}_0(\mathbb{R}^d)$, for all $d\in\mathbb{N}$. More generally, we e.g. show that $\operatorname{Lip}_0(\Gamma)\simeq \operatorname{Lip}_0(G)$, where $\Gamma$ is from a large class of finitely generated nilpotent groups and $G$ is its Mal'cev closure; or that $\operatorname{Lip}_0(\ell_p)\simeq\operatorname{Lip}_0(L_p)$, for all $1\leq p<\infty$. We leave a large area for further possible research.

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Leandro Candido, Marek Cúth, Michal Doucha. 2018-09-26. Isomorphisms between spaces of Lipschitz functions. https://doi.org/10.1016/j.jfa.2019.02.003

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