Choquet integrals, Hausdorff content and fractional operators

TL;DR

This paper extends the boundedness results of fractional integral and maximal operators to weak Choquet and Choquet-Morrey spaces with respect to Hausdorff content, providing new insights into fractional operators in these function spaces.

Contribution

It introduces new boundedness results for fractional integral operators on weak Choquet and Choquet-Morrey spaces, extending prior work and offering novel findings for $I_{\alpha}$.

Findings

Fractional integral operator $I_{\alpha}$ is bounded on weak Choquet spaces.

Fractional maximal operator $M_{\alpha}$ is bounded on these spaces.

New results for $I_{\alpha}$ in these function spaces.

Abstract

It is shown 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. These results are extensions of the previous works due to Adams, Orobitg and Verdera, and Tang. The results for the fractional integral operator $I_{\alpha}$ are essentially new.