Lipschitz Functions on Unions and Quotients of Metric Spaces

TL;DR

This paper proves that unions and quotients of certain metric spaces with finite Nagata dimension and specific Lipschitz free space properties also have Lipschitz free spaces isomorphic to L^1, using geometric and decomposition methods.

Contribution

It establishes that unions and quotients of metric spaces with finite Nagata dimension and L^1 Lipschitz free spaces also have L^1 Lipschitz free spaces, providing new geometric insights.

Findings

Union of such metric spaces has Lipschitz free space isomorphic to L^1

Union of quasiconformal trees has Lipschitz free space isomorphic to L^1

Lipschitz dimension of unions and quotients is 1

Abstract

Given a finite collection $\{X_i\}_{i\in I}$ of metric spaces, each of which has finite Nagata dimension and Lipschitz free space isomorphic to $L^1$, we prove that their union has Lipschitz free space isomorphic to $L^1$. The short proof we provide is based on the Pelczy\'nski decomposition method. A corollary is a solution to a question of Kaufmann about the union of two planar curves with tangential intersection. A second focus of the paper is on a special case of this result that can be studied using geometric methods. That is, we prove that the Lipschitz free space of a union of finitely many quasiconformal trees is isomorphic to $L^1$. These geometric methods also reveal that any metric quotient of a quasiconformal tree has Lipschitz free space isomorphic to $L^1$. Finally, we analyze Lipschitz light maps on unions and metric quotients of quasiconformal trees in order to prove that the Lipschitz dimension of any such union or quotient is equal to 1.