Minimal weak upper gradients in Newtonian spaces based on quasi-Banach function lattices (Q2860894)

This paper studies Newtonian spaces, i.e., generalizations of first-order Sobolev spaces on metric measure spaces, based on quasi-Banach function lattices. The only assumption on the underlying metric measure space \((\mathcal P,d,\mu)\) is that the measure is a \(\sigma\)-finite Borel measure and that every ball has positive and finite measure.NEWLINENEWLINEA linear space \(X=X(\mathcal P,\mu)\) is said to be a quasi-Banach function lattice over \((\mathcal P,\mu)\) if there exists a quasi-norm \(\|\cdot\|_{X}\) that determines the set \(X\), \(\|\cdot\|_{X}\) satisfies the lattice property (\(|u|\leq |v|\) a.e.\ implies \(\|u\|_{X}\leq \|v\|_{X}\)) and the Riesz-Fischer property (if \(u_{n}\geq 0\) a.e., then \(\|\sum_{n=1}^{\infty}u_{n}\|_{X}\leq \sum_{n=1}^{\infty}c^{n}\|u_{n}\|_{X} \)). In this paper, the quasi-norm is also assumed to be continuous (if \(\|u_{n}-u\|_{X}\rightarrow 0\), then \(\|u_{n}\|_{X}\rightarrow \|u\|_{X}\)).NEWLINENEWLINEThe Newtonian spaces are defined by using an approach based on (weak) upper gradients. The main theorem of the paper states that minimal weak upper gradients exist in the very general setting of quasi-Banach function lattices. Then, under the additional assumption that Lebesgue's differentiation theorem holds, several representation formulae for minimal weak upper gradients are provided.