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Article Dans Une Revue Transactions of the American Mathematical Society Année : 2022

Purely 1-Unrectifiable Metric Spaces and Locally Flat Lipschitz Functions

Résumé

For any compact metric space M , we prove that the locally flat Lipschitz functions separate points (of M) uniformly if and only if M is purely 1-unrectifiable, resolving a problem posed by Weaver in 1999. We subsequently use this geometric characterization to answer several questions in Lipschitz analysis. Notably, it follows that the Lipschitz-free space F (M) over a compact metric space M is a dual space if and only if M is purely 1unrectifiable. Furthermore, for any complete metric space M , we deduce that pure 1-unrectifiability actually characterizes some well-known Banach space properties of F (M) such as the Radon-Nikodým property and the Schur property. A direct consequence is that any complete, purely 1-unrectifiable metric space isometrically embeds into a Banach space with the Radon-Nikodým property. Finally, we provide a solution to a problem of Whitney from 1935 by finding a rectifiability-based characterization of 1-critical compact metric spaces, and we use this characterization to prove the following: a bounded turning tree fails to be 1-critical if and only each of its subarcs has σ-finite Hausdorff 1-measure.
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Dates et versions

hal-03242635 , version 1 (31-05-2021)

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Ramón J Aliaga, Chris Gartland, Colin Petitjean, Antonín Procházka. Purely 1-Unrectifiable Metric Spaces and Locally Flat Lipschitz Functions. Transactions of the American Mathematical Society, 2022, 375 (5), pp.3529-3567. ⟨10.1090/tran/8591⟩. ⟨hal-03242635⟩
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