Weighted boundedness of the Hardy-Littlewood maximal and Calder\'on-Zygmund operators on Orlicz-Morrey and weak Orlicz-Morrey spaces

TL;DR

This paper extends the weighted boundedness results of Hardy-Littlewood and Calderón-Zygmund operators from Lebesgue spaces to Orlicz-Morrey and weak Orlicz-Morrey spaces, including proofs of weak-weak modular inequalities.

Contribution

It introduces the boundedness of these operators on Orlicz-Morrey and weak Orlicz-Morrey spaces, generalizing known results and encompassing various classical spaces as special cases.

Findings

Weighted boundedness on Orlicz-Morrey spaces established

Weak-weak modular inequality proved for these operators

Results include classical spaces as special cases

Abstract

For the Hardy-Littlewood maximal and Calder\'on-Zygmund operators, the weighted boundedness on the Lebesgue spaces are well known. We extend these to the Orlicz-Morrey spaces. Moreover, we prove the weighted boundedness on the weak Orlicz-Morrey spaces. To do this we show the weak-weak modular inequality. The Orlicz-Morrey space and its weak version contain weighted Orlicz, Morrey and Lebesgue spaces and their weak versions as special cases. Then we also get the boundedness for these function spaces as corollaries.