发表机构
Public University of Navarre; Universidad de La Rioja; University of Warwick; University of Jyväskylä（纳瓦拉公立大学; 拉里奥哈大学; 华威大学; 于韦斯屈莱大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文刻画了拟Banach空间中每条Lipschitz曲线可微的充要条件为同构于具有Radon–Nikodým性质的Banach空间，并研究$\ell_p$（$0<p<1$）中解耦坐标构造下的几乎处处可微性。
AI 中文摘要
Banach 空间中 Lipschitz 曲线的微分问题可追溯至 20 世纪 30 年代初的 Tamarkin，并且是经典 Banach 空间理论中 Radon–Nikodým 性质的起源之一。本文刻画了那些使得每条 Lipschitz 曲线 $F\to\mathbb{R}\to X$ 都存在可微点的拟 Banach 空间 $X$，并证明该性质成立当且仅当 $X$ 同构于一个具有 Radon–Nikodým 性质的 Banach 空间，从而证实了 Kalton 于 2008 年私下传达给第一作者的一个猜想。等价地，每个非局部凸的拟 Banach 空间都允许一条无处可微的 Lipschitz 曲线。我们的结果是通过局部凸性关于 Lipschitz 曲线无穷小振荡的定量刻画获得的。受此障碍的启发，我们进而研究 $\ell_p$（$0<p<1$）中的正可微性现象，并识别出自然的解耦坐标构造，在这些构造下 Lipschitz 正则性仍然蕴含几乎处处可微性。我们还将此行为与 $\ell_p$ 上自然度量和拟度量的度量可微性进行对比。
英文摘要
The problem of differentiating Lipschitz curves in Banach spaces goes back to Tamarkin in the early 1930s and is one of the origins of the Radon--Nikodým property in classical Banach space theory. In this article we characterize those quasi-Banach spaces $X$ for which every Lipschitz curve $F\to\mathbb{R}\to X$ admits a point of differentiability and prove that this property holds if and only if $X$ is isomorphic to a Banach space with the Radon--Nikodým property, thus substantiating a conjecture of Kalton, communicated personally to the first-named author in 2008. Equivalently, every nonlocally convex quasi-Banach space admits a nowhere differentiable Lipschitz curve. Our result is obtained from a quantitative characterization of local convexity in terms of the infinitesimal oscillation of Lipschitz curves. Motivated by this obstruction, we then investigate positive differentiability phenomena in $\ell_p$, $0<p<1$, and identify natural decoupled-coordinate constructions for which Lipschitz regularity nevertheless implies almost-everywhere differentiability. We also contrast this behavior with metric differentiability for the natural metric and the quasi-metric on $\ell_p$.
