Lipschitz extension and Lipschitz-free spaces over nets in normed spaces

TL;DR

This paper investigates the structure of Lipschitz and Lipschitz-free spaces over nets in normed spaces, establishing isomorphisms and extension properties that advance understanding of their geometric and functional properties.

Contribution

It proves that Lipschitz-free spaces over nets in Banach spaces are isomorphic to their countable $ ext{l}_1$-sums, extending previous results and answering open questions.

Findings

Lipschitz-free space over a net in a Banach space is isomorphic to its countable $ ext{l}_1$-sum.

The Lipschitz space over $ ext{Z}_{ ext{l}_1}$ is isomorphic to that over $ ext{l}_1$.

The space $ ext{Lip}_0(N_X)$ contains a complemented copy of $ ext{Lip}_0(X)$.

Abstract

We consider subsets $S$ of a metric space $M$ such that Lipschitz mappings defined on $S$ can be extended to Lipschitz mappings on $M$, and we show that the union of such subsets has the same property under appropriate geometric conditions. We then derive several consequences to the isomorphic structure and classification of Lipschitz and Lipschitz-free spaces. Our main result is that the Lipschitz-free space $\mathcal{F}(M)$ is isomorphic to its countable $\ell_1$-sum when $M$ is either a net $N_X$ in any Banach space $X$ or the integer grid $\mathbb{Z}_{\ell_1}$ in $\ell_1$. We also prove that the Lipschitz space $\mathrm{Lip}_0(\mathbb{Z}_{\ell_1})$ is isomorphic to $\mathrm{Lip}_0(\ell_1)$ and that $\mathrm{Lip}_0(N_X)$ contains a complemented copy of $\mathrm{Lip}_0(X)$, among other results. This answers questions raised by Albiac, Ansorena, C\'uth and Doucha and Candido, C\'uth and Doucha, respectively, and extends previous results by the same authors as well as H\'ajek and Novotn\'y.