Schur $p$-性质对Lipschitz-free $p$-空间的结构性后果

中文摘要 AI 辅助

设 $0<p<q\leq 1$。我们证明Schur $p$-性质为Lipschitz-free $p$-空间提供了一个强有力的结构原理。我们的主要结果断言：对每个 $q$-度量空间 $M$，$\mathcal{F}_p(M)$ 具有Schur $p$-性质；当 $M$ 紧致时，它具有强Schur $p$-性质，且常数仅依赖于 $p$ 和 $q$。作为推论，$\mathcal{F}_p(M)$ 是 $\ell_p$-饱和的，不包含 $p<r\leq 1$ 时无限维 $r$-Banach 空间的同构拷贝，并且从 $p$-Banach 空间到 $\mathcal{F}_p(M)$ 的每个有界算子要么是紧算子，要么固定一个 $\ell_p$ 的拷贝。这些结果解决了我们近期工作 [F. Albiac, J. L. Ansorena, J. B\\'ima 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] 中关于Schur $p$-性质的问题6.1、6.2、6.5和6.6，以及若干相关问题。它们还证实了Kalton和第一作者在2009年的预测：没有无限维 $q$-Banach 空间具有 $p$-Lipschitz 提升性质。进一步的应用揭示了非局部凸空间的线性结构与Lipschitz结构之间的鲜明对比。

英文摘要

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\'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.

发表机构

• Public University of Navarre（纳瓦拉公立大学）
• Universidad de La Rioja（拉里奥哈大学）
• Charles University（查理大学）

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