Kwapień's property (K) in Lipschitz-free spaces
Abstract
We prove that a Lipschitz-free space F(M) fails Kwapień's property (K) whenever its underlying metric space M contains a bi-Lipschitz copy of C× K, where C⊂R is closed with positive measure and K contains a Cantor metric space. Consequently, this holds for every subset of Rn, ngeqslant2, whose closure has positive measure, and for a compact planar set of Assouad dimension one. These Lipschitz-free spaces embed into no strongly weakly compactly generated space and no L1-space; in the separable case their duals are not isomorphic to ℓ∞. Combining our result with a decomposition of Aliaga and Medina, we show that F(Rn×Zm) is strongly weakly compactly generated if and only if it has (K), and this occurs precisely when nleqslant1. In particular, F(R2) does not embed isomorphically into F(R×Z), answering a question of Aliaga. The planar examples nevertheless have the Dunford--Pettis property and Pełczyński's property (V*) and are weakly sequentially complete.
Related
- The Ball-Covering Property in Lipschitz-Free Spaces
- The Dunford-Pettis property for Lipschitz-free spaces over ℝn
- Structural consequences of the Schur p-property for Lipschitz-free p-spaces
- Isometric copies of ℓ∞n in Lipschitz-free spaces over finite metric spaces
- Lipschitz-free spaces over separable Banach spaces and their complementability in the bidual
BibTeX
@misc{kania2026kwapieńs,
title = {{Kwapień's property (K) in Lipschitz-free spaces}},
author = {Tomasz Kania and Jerzy Kąkol},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36194},
eprint = {2610.36194},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36194},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Kwapień's property (K) in Lipschitz-free spaces
AU - Tomasz Kania
AU - Jerzy Kąkol
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.36194
UR - https://arxiv.org/abs/2610.36194
AB - We prove that a Lipschitz-free space F(M) fails Kwapień's property (K) whenever its underlying metric space M contains a bi-Lipschitz copy of C× K, where C⊂R is closed with positive measure and K contains a Cantor metric space. Consequently, this holds for every subset of R^n, ngeqslant2, whose closure has positive measure, and for a compact planar set of Assouad dimension one. These Lipschitz-free spaces embed into no strongly weakly compactly generated space and no L_1-space; in the separable case their duals are not isomorphic to ℓ_∞. Combining our result with a decomposition of Aliaga and Medina, we show that F(R^n×Z^m) is strongly weakly compactly generated if and only if it has (K), and this occurs precisely when nleqslant1. In particular, F(R^2) does not embed isomorphically into F(R×Z), answering a question of Aliaga. The planar examples nevertheless have the Dunford--Pettis property and Pełczyński's property (V^*) and are weakly sequentially complete.
ER -