Nakai, Eiichi Generalized fractional integrals on Orlicz-Morrey spaces. (English) Zbl 1118.42005 Kato, Mikio (ed.) et al., Proceedings of the international symposium on Banach and function spaces, Kitakyushu, Japan, October 2–4, 2003. Yokohama: Yokohama Publishers (ISBN 4-946552-14-6/pbk). 323-333 (2004). Given a function \(f\) on \(\mathbb R^n\), let \[ I_\rho f (x)=\int_{R^n} f (y) \frac{\rho (| x-y| )}{| x-y| ^n}\, dy. \] If \(\rho (| x-y| )=| x-y| ^\alpha\), this operator coincides with the Riesz potential, which acts on \(L^p\)-functions according to the Hardy-Littlewood-Sobolev theorem. The paper contains a generalization of this theorem for operators \(I_\rho\) acting on Orlicz-Morrey spaces provided that \(\rho\) obeys some specific conditions.For the entire collection see [Zbl 1063.46002]. Reviewer: Boris Rubin (Baton Rouge) Cited in 2 ReviewsCited in 81 Documents MSC: 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 26A33 Fractional derivatives and integrals 42B35 Function spaces arising in harmonic analysis Keywords:Orlicz-Morrey spaces; Fractional integrals PDFBibTeX XMLCite \textit{E. Nakai}, in: Proceedings of the international symposium on Banach and function spaces, Kitakyushu, Japan, October 2--4, 2003. Yokohama: Yokohama Publishers. 323--333 (2004; Zbl 1118.42005)