Riesz type potential operators in generalized grand Morrey spaces

TL;DR

This paper introduces generalized grand Morrey spaces on quasimetric measure spaces and proves boundedness results for various operators, including Hardy-Littlewood maximal, Calderón-Zygmund, and Riesz potential operators, extending classical results.

Contribution

It develops a reduction lemma for operator boundedness in generalized grand Morrey spaces, enabling new boundedness results for classical operators.

Findings

Boundedness of Hardy-Littlewood maximal operator established.

Boundedness of Calderón-Zygmund potential operators demonstrated.

Riesz potential operators are bounded in both homogeneous and nonhomogeneous generalized grand Morrey spaces.

Abstract

In this paper we introduce generalized grand Morrey spaces in the framework of quasimetric measure spaces, in the spirit of the so-called grand Lebesgue spaces. We prove a kind of reduction lemma which is applicable to a variety of operators to reduce their boundedness in generalized 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 as well as the boundedness of Calder\'on-Zygmund potential type operators. Boundedness of Riesz type potential operators are also obtained in the framework of homogeneous and also in the nonhomogeneous case in generalized grand Morrey spaces.