Structural consequences of the Schur p-property for Lipschitz-free p-spaces

@misc{albiac2026structural,
  title = {{Structural consequences of the Schur $p$-property for Lipschitz-free $p$-spaces}},
  author = {Fernando Albiac and José L. Ansorena and Marek Cúth},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10747},
  eprint = {2609.10747},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10747},
}

TY  - UNPB
TI  - Structural consequences of the Schur p-property for Lipschitz-free p-spaces
AU  - Fernando Albiac
AU  - José L. Ansorena
AU  - Marek Cúth
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.10747
UR  - https://arxiv.org/abs/2609.10747
AB  - Let 0<p<q≤ 1. We show that the Schur p-property provides a powerful structural principle for Lipschitz-free p-spaces. Our main result asserts that ℱ_p(M) has the Schur p-property for every q-metric space M; when M is compact, it has the strong Schur p-property, with a constant depending only on p and q. As consequences, ℱ_p(M) is ℓ_p-saturated, contains no isomorphic copy of an infinite-dimensional r-Banach space for p<r≤ 1, and every bounded operator from a p-Banach space into ℱ_p(M) is either compact or fixes a copy of ℓ_p. These results settle Questions 6.1, 6.2, 6.5, and 6.6 from our recent work [F. Albiac, J. L. Ansorena, J. B\'ıma and M. Cúth, Lipschitz free p-spaces for 0<p<1 in the light of the Schur p-property and the compact reduction, J. Geom. Anal. 36 (2026), Paper No. 54] on the Schur p-property, together with several related problems. They also confirm a prediction of Kalton and the first-named author from 2009: no infinite-dimensional q-Banach space has the p-Lipschitz lifting property. Further applications reveal a sharp contrast between the linear and Lipschitz structures of nonlocally convex spaces.
ER  -