Lipschitz-free spaces and purely 1-unrectifiable metric spaces
Abstract
The Lipschitz-free space ℱ(M) is a canonical linearization of a complete metric space M whose topological dual is the space of Lipschitz functions on M. We review the properties of ℱ(M) when the underlying space M is purely 1-unrectifiable, that is, it contains no bi-Lipschitz copy of a subset of ℝ with positive measure. For compact M, this is equivalent to several Banach space properties of ℱ(M), including the Radon-Nikodým and Schur properties or admitting a predual. We shall see how the study of locally flat Lipschitz functions on M reveals these equivalences, and describe a technique that allows most of them to be transferred to the non-compact setting. This manuscript is an expository text based on results by the author in collaboration with C. Gartland, C. Petitjean and A. Procházka, originally published in a Trans. Amer. Math. Soc. paper, and corresponds to a lecture delivered at the Second Winter School in Geometric Measure Theory at Westlake University, Hangzhou, on February 2026.
Related
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BibTeX
@misc{aliaga2026lipschitzfree,
title = {{Lipschitz-free spaces and purely 1-unrectifiable metric spaces}},
author = {Ramón J. Aliaga},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.02918},
eprint = {2606.02918},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.02918},
}
RIS
TY - UNPB
TI - Lipschitz-free spaces and purely 1-unrectifiable metric spaces
AU - Ramón J. Aliaga
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2606.02918
UR - https://arxiv.org/abs/2606.02918
AB - The Lipschitz-free space ℱ(M) is a canonical linearization of a complete metric space M whose topological dual is the space of Lipschitz functions on M. We review the properties of ℱ(M) when the underlying space M is purely 1-unrectifiable, that is, it contains no bi-Lipschitz copy of a subset of ℝ with positive measure. For compact M, this is equivalent to several Banach space properties of ℱ(M), including the Radon-Nikodým and Schur properties or admitting a predual. We shall see how the study of locally flat Lipschitz functions on M reveals these equivalences, and describe a technique that allows most of them to be transferred to the non-compact setting. This manuscript is an expository text based on results by the author in collaboration with C. Gartland, C. Petitjean and A. Procházka, originally published in a Trans. Amer. Math. Soc. paper, and corresponds to a lecture delivered at the Second Winter School in Geometric Measure Theory at Westlake University, Hangzhou, on February 2026.
ER -