# Proper inclusions of Morrey spaces

## TL;DR

This paper investigates the precise nature of inclusions among Morrey and weak Morrey spaces, establishing their properness and providing necessary conditions, thereby refining existing mathematical understanding of these function spaces.

## Contribution

It proves that all inclusions between Morrey and weak Morrey spaces are proper and offers necessary conditions, refining previous inclusion results.

## Key findings

- Inclusions between Morrey and weak Morrey spaces are proper.
- Proper inclusion between a Morrey space and a weak Morrey space is demonstrated.
- Necessary conditions for each inclusion are established.

## Abstract

In this paper, we prove that the inclusions between Morrey spaces, between weak Morrey spaces, and between a Morrey space and a weak Morrey space are all proper. The proper inclusion between a Morrey space and a weak Morrey space is established via the unboundedness of the Hardy-Littlewood maximal operator on Morrey spaces of exponent 1. In addition, we also give a necessary condition for each inclusion. Our results refine previous inclusion properties studied in [Gunawan et al, \emph{Math. Nachr.} {\bf 290} (2017), 332--340].