Lipschitz Functions on Quasiconformal Trees

TL;DR

This paper characterizes Lipschitz free spaces of quasiconformal trees, showing they have Lipschitz dimension 1, and introduces a geometric tree-like decomposition applicable to broader metric spaces.

Contribution

It identifies Lipschitz free spaces of quasiconformal trees and generalizes decomposition techniques to metric spaces with a tree-like structure.

Findings

Lipschitz free spaces of quasiarcs are identified up to linear isomorphism.

Quasiconformal trees have Lipschitz dimension 1.

Any weak quasiarc can be decomposed into rectifiable and purely unrectifiable parts.

Abstract

We first identify (up to linear isomorphism) the Lipschitz free spaces of quasiarcs. By decomposing quasiconformal trees into quasiarcs as done in an article of David, Eriksson-Bique, and Vellis, we then identify the Lipschitz free spaces of quasiconformal trees and prove that quasiconformal trees have Lipschitz dimension 1. Generalizing the aforementioned decomposition, we define a geometric tree-like decomposition of a metric space. Our results pertaining to quasiconformal trees are in fact special cases of results about metric spaces admitting a geometric tree-like decomposition. Furthermore, the methods employed in our study of Lipschitz free spaces yield a decomposition of any (weak) quasiarc into rectifiable and purely unrectifiable subsets, which may be of independent interest.