Derivations and Sobolev functions on extended metric-measure spaces

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 \((X,d)\) 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 \(\operatorname{Lip}_b(X,\tau,{\sf d})\) of bounded \(\tau\)-continuous \(d\)-Lipschitz functions on \(X\).
• Several notions of Lipschitz derivations on \(X\), defined in duality with \(\operatorname{Lip}_b(X,\tau,d)\).
• 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 Savaré. En route, we obtain some results of independent interest, among which are:

• A Lipschitz-constant-preserving extension result for \(\tau\)-continuous \(d\)-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 \(W^{1,p}(\mathbb X)\).