Calabuig, J. M.; Rodríguez, J.; Sánchez-Pérez, E. A. On the structure of \(L^{1}\) of a vector measure via its integration operator. (English) Zbl 1183.46029 Integral Equations Oper. Theory 64, No. 1, 21-33 (2009). Let \(m\) be a countably additive \(X\)-valued vector measure, where \(X\) is a Banach space. We have that \(L^r(m)\) embeds continuously into \(L^1(m)\) for \(r\geq 1\); let this be denoted by \(L^r(m)\hookrightarrow L^1(m)\). The integration operator \(I_m^1\) is the mapping sending \(f\in L_1(m)\) to \(\int f\,dm\in X\). \(I_m^r\) stands for the integration operator restricted to \(L^r(m)\). The main issue considered and solved is the following: When is there a control measure \(\lambda\) for \(m\) such that \(L^r(m)\hookrightarrow L^r(\lambda)\hookrightarrow L^1(m)\)? It turns out that this happens exactly when some Hölder-like inequalities are fulfilled, which again happens exactly when \(I_m^r\) is \(r\)-concave.When \(E\) is an order continuous Banach function space with weak order unit, there is a (non-unique) representing vector measure \(m\) such that \(E\) is order isomorphic to \(L^1(m)\). It is proved that \(E\) is order isomorphic to the \(L^1\) space of a non-negative scalar measure if and only if for every representing vector measure \(m\) and every \(1\leq p<\infty\), \(I_m^1\) is positive \(p\)-summing. Reviewer: Olav Nygaard (Kristiansand) Cited in 3 Documents MSC: 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46G10 Vector-valued measures and integration Keywords:Banach function space; integration operator; \(p\)-concave operator; positive \(p\)-summing operator; vector measure PDFBibTeX XMLCite \textit{J. M. Calabuig} et al., Integral Equations Oper. Theory 64, No. 1, 21--33 (2009; Zbl 1183.46029) Full Text: DOI