CHOQUET INTEGRALS, HAUSDORFF CONTENT AND FRACTIONAL OPERATORS

Abstract

We show that the fractional integral operator $I_{\alpha }$ , $0<\alpha <n$ , and the fractional maximal operator $M_{\alpha }$ , $0\le \alpha <n$ , are bounded on weak Choquet spaces with respect to Hausdorff content. We also investigate these operators on Choquet–Morrey spaces. The results for the fractional maximal operator $M_\alpha $ are extensions of the work of Tang [‘Choquet integrals, weighted Hausdorff content and maximal operators’, Georgian Math. J.18(3) (2011), 587–596] and earlier work of Adams and Orobitg and Verdera. The results for the fractional integral operator $I_{\alpha }$ are essentially new.