标题： 有界变分函数和度量测度空间中的Lipschitz代数 显示英文标题

标题： Functions of bounded variation and Lipschitz algebras in metric measure spaces

摘要： 给定一个在度量测度空间$({\mathrm X},{\mathsf d},\mathfrak m)$上定义的局部Lipschitz函数的单位代数$\mathscr A$，我们研究两个相关的有界变分函数概念及其关系：通过用$\mathscr A$中的元素在能量上逼近得到的空间${\mathrm BV}_{\mathrm H}({\mathrm X};\mathscr A)$，以及通过涉及与$\mathscr A$对偶作用的导出算子的分部积分公式定义的空间${\mathrm BV}_{\mathrm W}({\mathrm X};\mathscr A)$。 我们的主要结果提供了代数$\mathscr A$的一个充分条件，使得${\mathrm BV}_{\mathrm H}({\mathrm X};\mathscr A)$与标准的度量 BV 空间${\mathrm BV}_{\mathrm H}({\mathrm X})$相同，这对应于将$\mathscr A$取为所有局部 Lipschitz 函数的集合。 我们的结果适用于几个感兴趣的案例，例如配备光滑函数代数的欧几里得空间和黎曼流形，或者配备柱面函数代数的巴拿赫空间和 Wasserstein 空间。 对于指数为$p\in(1,\infty)$的度量 Sobolev 空间${\mathrm H}^{1,p}$，类似的结果之前已被多位不同的作者获得。 显示英文摘要

摘要： Given a unital algebra $\mathscr A$ of locally Lipschitz functions defined over a metric measure space $({\mathrm X},{\mathsf d},\mathfrak m)$, we study two associated notions of function of bounded variation and their relations: the space ${\mathrm BV}_{\mathrm H}({\mathrm X};\mathscr A)$, obtained by approximating in energy with elements of $\mathscr A$, and the space ${\mathrm BV}_{\mathrm W}({\mathrm X};\mathscr A)$, defined through an integration-by-parts formula that involves derivations acting in duality with $\mathscr A$. Our main result provides a sufficient condition on the algebra $\mathscr A$ under which ${\mathrm BV}_{\mathrm H}({\mathrm X};\mathscr A)$ coincides with the standard metric BV space ${\mathrm BV}_{\mathrm H}({\mathrm X})$, which corresponds to taking as $\mathscr A$ the collection of all locally Lipschitz functions. Our result applies to several cases of interest, for example to Euclidean spaces and Riemannian manifolds equipped with the algebra of smooth functions, or to Banach and Wasserstein spaces equipped with the algebra of cylinder functions. Analogous results for metric Sobolev spaces ${\mathrm H}^{1,p}$ of exponent $p\in(1,\infty)$ were previously obtained by several different authors.