Lipschitz functions on unions and quotients of metric spaces

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.