Third Version of Weak Orlicz--Morrey Spaces and Its Inclusion Properties

TL;DR

This paper introduces and analyzes the third version of weak Orlicz--Morrey spaces, focusing on their inclusion properties and generalizations of earlier space versions, with key estimates on quasi-norms.

Contribution

It provides the first detailed study of the third version of weak Orlicz--Morrey spaces, including necessary and sufficient conditions for their inclusion relations.

Findings

Established inclusion relations between different weak Orlicz--Morrey spaces.

Derived conditions based on quasi-norm estimates of characteristic functions.

Extended the understanding of space generalizations and their properties.

Abstract

Orlicz--Morrey spaces are generalizations of Orlicz spaces and Morrey spaces which were first introduced by Nakai. There are three versions of Orlicz--Morrey spaces, i.e: Nakai's (2004), Sawano--Sugano--Tanaka's (2012), and Deringoz--Guliyev--Samko's (2014) versions. On this article we will discuss the third version of weak Orlicz--Morrey space which is seen as an enlargement of third version of (strong) Orlicz--Morrey space. Similar to its first version and second version, the third version of weak Orlicz-Morrey space is considered as a generalization of weak Orlicz spaces, weak Morrey spaces, and generalized weak Morrey spaces. In this study, we will investigate some properties of the third version of weak Orlicz--Morrey spaces, especially the sufficient and necessary conditions for inclusion relations between two these spaces. One of the keys to get our result is to estimate the quasi-norm of characteristics function of open balls in $\mathbb{R}^n$.