Vector calculus on weighted reflexive Banach spaces (Q6623333)

The setting of the paper is Sobolev spaces on metric measure spaces \((X, d, \mu)\) for the particular case that \(X=\mathbb B\) is a separable Banach space, \(d\) is the metric derived from the norm of \(\mathbb B\) and \(\mu\) is a finite measure on the Borel sets of \(\mathbb B\); such a pair \((\mathbb B, \mu)\) is referred to as a weighted Banach space. The idea of the paper is to compare the Sobolev space \(W^{1,p} (\mathbb B, \mu)\) for the metric space \(\mathbb B\), defined in terms of test plans and upper gradients, cf. [\textit{L.~Ambrosio} et al., Invent. Math. 195, No.~2, 289--391 (2014; Zbl 1312.53056)], with more ``linear'' variants that refer to the linear Banach space structure of \(\mathbb B\) rather than its metric structure. Thus, the paper also contributes to the recent topic in functional analysis to characterise the linear structure of a Banach space by its purely metric structure.\N\NThe main theorem is Theorem~3.1 characterising functions \(f\in W^{1,p} (\mathbb B, \mu)\) by means of the distributional divergence if \(\mathbb B\) has the Radon-Nikodym property. If \(\mathbb B\) is even reflexive, there is another characterisation by means of weak differentials. The accompanying Theorem~3.2 presents a description of the minimal weak upper gradient under the same assumptions on \(\mathbb B\). Finally, Theorem~3.3 identifies the tangent bundle \(T_\mu \mathbb B\).\N\NTo formulate these results precisely, a whole catalogue of symbols and notation is needed, and the first half of the paper is devoted to introducing these preliminaries, while the second half displays the diligently crafted detailed proofs and further comments.