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

Abstract

Let $0<p<q\leq 1$. We show that the Schur $p$-property provides a powerful structural principle for Lipschitz-free $p$-spaces. Our main result asserts that $\mathcal{F}_p(M)$ 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, $\mathcal{F}_p(M)$ is $\ell_p$-saturated, contains no isomorphic copy of an infinite-dimensional $r$-Banach space for $p<r\leq 1$, and every bounded operator from a $p$-Banach space into $\mathcal{F}_p(M)$ is either compact or fixes a copy of $\ell_p$. These results settle Questions 6.1, 6.2, 6.5, and 6.6 from our recent work [F. Albiac, J. L. Ansorena, J. B\'{\i}ma and M. C\'uth, 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.