Fractional integrals on variable Hardy-Morrey spaces (Q519952)

In this paper the authors generalize a result of \textit{M. H. Taibleson} and \textit{G. Weiss} [Astérisque 77, 67--151 (1980; Zbl 0472.46041)]. Using atomic decomposition they characterize boundedness of the fractional integral operator \[ I_\alpha f(x)=\frac1{\gamma_\alpha}\int_{\mathbb R^n}\frac{f(y)}{| x-y| ^{n-\alpha}}\,dy \] (\(0<\alpha<n\), \(\gamma_\alpha=\pi^{n/2}2^\alpha\Gamma(\alpha/2)/\Gamma((n-\alpha)/2)\)) on variable Hardy-Morrey spaces. To obtain these results they first derive vector-valued fractional maximal inequalities on variable Morrey spaces. For related results on Lebesgue function spaces with variable exponents see a book of \textit{D. V. Cruz-Uribe} and \textit{A. Fiorenza} [Variable Lebesgue spaces. Foundations and harmonic analysis, Applied and Numerical Harmonic Analysis. New York, NY: Birkhäuser/Springer ix, 312 p. (2013; Zbl 1268.46002)].