一种检测跳跃的随机利普希茨函数
A random Lipschitz function detecting jumps
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中文总结 AI 辅助
该研究围绕可检测跳跃的随机利普希茨函数展开，探究度量空间具备利普希茨函数捕捉映射总变差性质的充要条件，完善了相关度量空间的刻画结论。
中文摘要 AI 辅助
近期，A.秋连涅夫与笔者研究了满足如下条件的度量空间类$\mathcal{X}$：对每个映射$γ\colon [0,1]\to\mathcal{X}$，都存在一个$1$-利普希茨函数$F\colon \mathcal{X}\to \mathbb{R}$可捕捉$γ$的总变差，即满足$\operatorname{V}_{F\circγ} \gtrsim \operatorname{V}_γ$。在这篇短注中，我们证明：度量空间$\mathcal{X}$具有该性质，当且仅当在$\mathcal{X}$上存在一个随机$1$-利普希茨函数$f$，使得对$\mathcal{X}$中的任意$x$与$y$，都有$\mathbb{E} |f(x) - f(y)| \gtrsim ρ(x,y)$成立。
英文摘要
Recently, A. Tyulenev and the author studied the class of metric spaces $\mathcal{X}$ such that for every mapping $γ\colon [0,1]\to\mathcal{X}$, there exists a $1$-Lipschitz function $F\colon \mathcal{X}\to \mathbb{R}$ that catches the total variation of $γ$, i.e. such that $\operatorname{V}_{F\circγ} \gtrsim \operatorname{V}_γ$. In this short note, we show that a metric space $\mathcal{X}$ enjoys this property if and only if there exists a random $1$-Lipschitz function $f$ on $\mathcal{X}$ such that $\mathbb{E} |f(x) - f(y)| \gtrsim ρ(x,y)$ for every $x$ and $y$ in $\mathcal{X}$.
