Banach--Kadec--Paley定理的球面版本
Sphere version of Banach--Kadec--Paley theorem
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中文总结 AI 辅助
本文引入几乎Lipschitz嵌入概念，证明单位球几乎Lipschitz嵌入等价于线性嵌入，从而球的几何完全决定L_p空间的线性结构，并给出Hölder几何的锐利结果。
中文摘要 AI 辅助
我们引入了度量空间之间几乎Lipschitz嵌入的概念，并将其应用于经典\\(L_p\\)-空间单位球的定量几何。我们的主要结果是经典Banach--Kadec--Paley定理的球面版本：对于\\(1\le p,q<\infty\\)，单位球\\(S_{L_q}\\)几乎Lipschitz嵌入到\\(L_p\\)中当且仅当\\(L_q\\)线性嵌入到\\(L_p\\)中。因此，\\(L_p\\)-空间球的几乎Lipschitz几何完全决定了\\(L_p\\)-空间的底层线性结构。证明依赖于球面映射连续模的定量估计。这些估计源自Kalton--Randrianarivony的集中不等式、Mendel--Naor的度量余类型不等式以及Naor的锐度量\\(X_p\\)不等式。它们还产生了关于\\(L_p\\)-空间球Hölder几何的若干锐利结果。
英文摘要
We study almost Lipschitz embeddings in the quantitative geometry of unit spheres of classical \(L_p\)-spaces. Our main result is a sphere version of the classical Banach--Kadec--Paley theorem: for \(1\le p,q<\infty\), the unit sphere \(S_{L_q}\) admits an almost Lipschitz embedding into \(L_p\) if and only if \(L_q\) admits a linear embedding into \(L_p\). Consequently, the almost Lipschitz geometry of \(L_p\)-spheres completely determines the underlying linear structure of \(L_p\)-spaces. The proof relies on quantitative estimates for moduli of continuity of sphere mappings. These estimates are derived from Kalton--Randrianarivony's concentration inequalities, Mendel--Naor's metric cotype inequality, and Naor's sharp metric \(X_p\) inequality. They also yield several sharp results on the Hölder geometry of \(L_p\)-spheres.
发表机构
- School of Mathematical Sciences, Xiamen University（厦门大学数学科学学院）
- School of Mathematics and Statistics, Chongqing University of Posts and Telecommunications（重庆邮电大学数学与统计学院）
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