Mixed Morrey spaces

TL;DR

This paper introduces mixed Morrey spaces, explores their fundamental properties, and studies the boundedness of key operators, extending classical results and establishing new inequalities in these generalized function spaces.

Contribution

It defines mixed Morrey spaces, extends classical properties to these spaces, and proves boundedness of important operators, including maximal, fractional, and singular integrals.

Findings

Boundedness of the iterated maximal operator in mixed Morrey spaces.

Extension of classical properties to mixed Morrey spaces.

Establishment of weighted inequalities and vector-valued maximal inequalities.

Abstract

We introduce mixed Morrey spaces and show some basic properties. These properties extend the classical ones. We investigate the boundedness in these spaces of the iterated maximal operator, the fractional integtral operator and singular integral operator. Furthermore, as a corollary, we obtain the boundedness of the iterated maximal operator in classical Morrey spaces. We also establish a version of the Fefferman--Stein vector-valued maximal inequality and some weighted inequalities for the iterated maximal operator in mixed Lebesgue spaces. We point out some errors in the proof of the existing literature.