Newtonian spaces based on quasi-Banach function lattices (Q2818770)

Given a metric measure space \(\mathcal{P}\) \(=\) \((\mathcal{P},d,\mu)\) (equipped with a metric \(d\) and a \(\sigma\)-finite regular Borel measure \(\mu\)), one denotes by \(\mathcal{M}( \mathcal{P},\mu) \) the set of all extended real-valued \(\mu\)-measurable functions on \(\mathcal{P}\) and by \(X=X( \mathcal{P},\mu) \) a quasi-Banach function lattice (that is, a linear space of equivalence classes of functions from \(\mathcal{M} ( \mathcal{P},\mu) \) endowed with a quasi-norm \(\| \cdot\| _{X}\). In this context, one can introduce first-order Sobolev-type spaces using the concept of \(X\)-weak upper gradient à la \textit{J. Heinonen} and \textit{P. Koskela} [Proc. Natl. Acad. Sci. USA 93, No. 2, 554--556 (1996; Zbl 0842.30016); Acta Math. 181, No. 1, 1--61 (1998; Zbl 0915.30018)]. Precisely, for \(u\in\mathcal{M}( \mathcal{P},\mu) \) one puts \(\| u\| _{N^{1} X}=\| u\| _{X}+\inf_{g}\| g\| _{X}\), where the infimum is taken over all upper gradients \(g\) of \(u\). By definition, the Newtonian space based on \(X\) is the space \(N^{1}X=\{ u\in\mathcal{M} ( \mathcal{P},\mu) :\| u\| _{N^{1}X}<\infty\} \). The paper under review develops the basic theory of Newtonian spaces using tools such as Sobolev capacity and modulus of a curve family. Among the results obtained are the absolute continuity of Newtonian functions along curves and the completeness of Newtonian spaces.