Isomorphisms between spaces of Lipschitz functions

TL;DR

This paper establishes isomorphisms between Lipschitz function spaces over various metric and Banach spaces, revealing structural similarities across different mathematical contexts.

Contribution

It introduces new tools for proving isomorphisms of Lipschitz function spaces and demonstrates their applicability to diverse spaces like Euclidean, nilpotent groups, and classical Banach spaces.

Findings

Lip_0(\u211a^d) f f Lip_0(\u211a^d) for all d

fLip_0(\u0393) f f Lip_0(G) for certain nilpotent groups

fLip_0(\u2113_p) f f Lip_0(L_p) for 1 f p < f f infinity

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.