A note on boundedness of operators in Grand Grand Morrey spaces

TL;DR

This paper introduces grand grand Morrey spaces, extending the framework of Morrey spaces, and proves boundedness of key operators like Hardy-Littlewood maximal and Calderón-Zygmund operators within these spaces.

Contribution

It presents the concept of grand grand Morrey spaces and a reduction lemma that links their operator boundedness to that in classical Morrey spaces.

Findings

Boundedness of Hardy-Littlewood maximal operator in grand grand Morrey spaces

Boundedness of Calderón-Zygmund operators in grand grand Morrey spaces

Reduction lemma connecting boundedness in grand grand Morrey and Morrey spaces

Abstract

In this note we introduce grand grand Morrey spaces, in the spirit of the grand Lebesgue spaces. We prove a kind of \textit{reduction lemma} which is applicable to a variety of operators to reduce their boundedness in grand grand Morrey spaces to the corresponding boundedness in Morrey spaces. As a result of this application, we obtain the boundedness of the Hardy-Littlewood maximal operator and Calder\'on-Zygmund operators in the framework of grand grand Morrey spaces.