This is an announcement for the paper “Isomorphisms between spaces of Lipschitz functions” by Leandro Candidohttps://arxiv.org/search/math?searchtype=author&query=Candido%2C+L, Marek Cúthhttps://arxiv.org/search/math?searchtype=author&query=C%C3%BAth%2C+M, Michal Douchahttps://arxiv.org/search/math?searchtype=author&query=Doucha%2C+M.
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.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1809.09957