Title: Derivations and Sobolev functions on extended metric-measure spaces Show Chinese title

Title: 导数和扩展度量测度空间上的Sobolev函数

Abstract: We investigate the first-order differential calculus over extended metric-topological measure spaces. The latter are quartets $\mathbb X=(X,\tau,{\sf d},\mathfrak m)$, given by an extended metric space $(X,{\sf d})$ together with a weaker topology $\tau$ (satisfying suitable compatibility conditions) and a finite Radon measure $\mathfrak m$ on $(X,\tau)$. The class of extended metric-topological measure spaces encompasses all metric measure spaces and many infinite-dimensional metric-measure structures, such as abstract Wiener spaces. In this framework, we study the following classes of objects: - The Banach algebra ${\rm Lip}_b(X,\tau,{\sf d})$ of bounded $\tau$-continuous ${\sf d}$-Lipschitz functions on $X$. - Several notions of Lipschitz derivations on $X$, defined in duality with ${\rm Lip}_b(X,\tau,{\sf d})$. - The metric Sobolev space $W^{1,p}(\mathbb X)$, defined in duality with Lipschitz derivations on $X$. Inter alia, we generalise both Weaver's and Di Marino's theories of Lipschitz derivations to the extended setting, and we discuss their connections. We also introduce a Sobolev space $W^{1,p}(\mathbb X)$ via an integration-by-parts formula, along the lines of Di Marino's notion of Sobolev space, and we prove its equivalence with other approaches, studied in the extended setting by Ambrosio, Erbar and Savar\'{e}. En route, we obtain some results of independent interest, among which are: - A Lipschitz-constant-preserving extension result for $\tau$-continuous ${\sf d}$-Lipschitz functions. - A novel and rather robust strategy for proving the equivalence of Sobolev-type spaces defined via an integration-by-parts formula and those obtained with a relaxation procedure. - A new description of an isometric predual of the metric Sobolev space $W^{1,p}(\mathbb X)$. Show Chinese abstract

Abstract: 我们研究扩展度量-拓扑测度空间上的的一阶微分演算。 后者是由一个扩展度量空间$(X,{\sf d})$以及一种较弱的拓扑$\tau$（满足适当的兼容条件）和一个有限的Radon测度$\mathfrak m$在$(X,\tau)$上构成的四元组$\mathbb X=(X,\tau,{\sf d},\mathfrak m)$。 扩展度量-拓扑测度空间类包括所有度量测度空间和许多无限维的度量测度结构，例如抽象Wiener空间。 在这个框架中，我们研究以下对象类： - 有界$\tau$连续${\sf d}$-Lipschitz 函数在$X$上的巴拿赫代数${\rm Lip}_b(X,\tau,{\sf d})$。 - 几种关于${\rm Lip}_b(X,\tau,{\sf d})$对偶定义的$X$上的 Lipschitz 导出。 - 度量 Sobolev 空间$W^{1,p}(\mathbb X)$，在与$X$上的 Lipschitz 导数对偶中定义。 此外，我们还将 Weaver 和 Di Marino 的 Lipschitz 导数理论推广到扩展设置，并讨论它们之间的联系。 我们还通过一个分部积分公式引入了一个 Sobolev 空间$W^{1,p}(\mathbb X)$，沿着 Di Marino 的 Sobolev 空间的概念，并证明了它与其他方法的等价性，在扩展设置中由 Ambrosio、Erbar 和 Savaré 进行了研究。 在此过程中，我们获得了一些具有独立兴趣的结果，其中包括： - 一个保持 Lipschitz 常数的扩展结果，适用于$\tau$连续的${\sf d}$Lipschitz 函数。 - 一种新颖且相当稳健的策略，用于证明通过分部积分公式定义的 Sobolev 类空间与通过松弛过程得到的空间之间的等价性。 - 对度量 Sobolev 空间$W^{1,p}(\mathbb X)$的等距预共轭的一种新描述。