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Lipschitz functions on quasiconformal trees

Volume 262 / 2023

David Freeman, Chris Gartland Fundamenta Mathematicae 262 (2023), 153-203 MSC: Primary 51F30; Secondary 30L05, 28A15, 28A78, 46B20. DOI: 10.4064/fm273-3-2023 Published online: 25 April 2023


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.


  • David FreemanUniversity of Cincinnati Blue Ash College
    Blue Ash, OH 45236, USA
  • Chris GartlandTexas A&M University
    College Station, TX 77843, USA

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