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

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 $\mathbb{R}^n$.


Lemma 1.2 Let Y be a Young function and
for every + ∈ ℳ #,Z (ℝ ).

Many authors have been culminating important observations about inclusion
properties of function spaces, see (Jiménez-Vargas et al., 2018;Maligranda and Matsuoka, 2015;Masta et al., 2018;Masta, 2018;Taqiyuddin and Masta, 2018;Diening, and Růžička, 2007), etc. Reports in literature  obtained sufficient and necessary conditions for inclusion of strong Orlicz-Morrey spaces of all versions. In the same paper ,  also proved the sufficient and necessary conditions for inclusion properties of weak Orlicz -Morrey spaces of Nakai 's and Sawano -Sugano -Tanaka's versions.
In this paper, we would like to obtain the inclusion properties of weak Orlicz-Morrey space Iℳ #,* (ℝ ) of Deringoz-Guliyev-Samko's version, and compare it with the result for Nakai's and Sawano-Sugano-Tanaka's versions.

METHODS
To obtain the sufficient and necessary conditions for inclusion properties of ℳ #,* (ℝ ), we used the similar methods in (Gunawan et al., , 2018Masta et al., 2018 ;Masta , 2018 ;Osançlıol , 2014 ), which pay attention to the characteristic functions of open balls in ℝ , in the following lemma. .
For weak Orlicz-Morrey spaces, we have the following lemma. Lemma 1.4 Let Y be a Young function, $ ∈ " # , and > 0, then there exists a constant > 0 such that .
Proof. Since Θ is a Young function and $ ∈ " # , by Lemmas 1.2 and 1.3, we have .
In this paper, the letter was used for constants that may change from line to line, while constants with subscripts, such as , , do not change in different lines.

RESULTS AND DISCUSSION
First, we reproved sufficient and necessary conditions for inclusion properties of Orlicz-Morrey space ℳ #,* (ℝ ) in the following theorem.
Teorema 2.1 Then the following statements are equivalent:
We have shown the sufficient and necessary conditions for the inclusion relation between weak Orlicz-Morrey space Iℳ #,* (ℝ ). In the proof of the inclusion property, we used the norm of characteristic function on ℝ . The inclusion properties of weak Orlicz-Morrey space Iℳ #,* (ℝ ) (Theorem 2.2) and weak Orlicz-Morrey space Iℳ f,g (ℝ ) of Sawano-Sugano-Tanaka 's version (Theorem 3.9 in previous report ) generalized the inclusion properties of weak Morrey spaces and resulted weak Morrey spaces in literature 2017 b ). Meanwhile , the inclusion properties of weak Orlicz -Morrey space IW h,i (ℝ ) of Nakai's version (Theorem 3.4 in literature  generalized the unique inclusion properties of weak Orlicz spaces in other report (Masta et al., 2016).
Comparing Theorem 2.2 and Theorem 3.9 in Masta et al. (2018 ), we say that the condition on the Young function for the inclusion of the weak Orlicz-Morrey space I ℳ f,g (ℝ ) is simpler than that for the weak Orlicz-Morrey space Iℳ #,* (ℝ ).

CONCLUSION
This article has discussed the third version of weak Orlicz-Morrey space, which is an enlargement of third version of (strong) Orlicz-Morrey space. This study also investigated 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 ℝ .

AUTHORS' NOTE
The author(s) declare(s) that there is no conflict of interest regarding the publication of this article. Authors confirmed that the data and the paper are free of plagiarism.