Inclusion Properties of Henstock-Orlicz Spaces

ABSTRACT

Moreover, research on the Orlicz space and inclusion properties that applicable in several spaces has been carried out, such as the generalization of Orlicz spaces (Ebadian & Jabbari, 2021), some characterization of generalized Orlicz spaces (Ferreira et al., 2020), some properties of Discrete Orlicz space (Prayoga et al., 2020), some necessary and sufficient conditions for inclusion properties on Orlicz spaces and another type of this spaces (Masta et al., 2016), the inclusion properties of two version of Orlicz-Morrey spaces (Masta et al., 2017b), the inclusion property on Orlicz-Morrey Spaces (Masta et al., 2017a), and the inclusion properties of discrete Morrey spaces (Prayoga et al., 2020).
Furthermore, research on Henstock-Orlicz spaces was carried out by several mathematicians such as about the definition of ℋ-Orlicz and its dense space , the theory of ℋ-Orlicz with respect to vector measure (Kalita et al., 2020), and the relatively weakly compactness of some ℋ -Orlicz spaces of Henstock-Gel'fand integrable function .
Based on the aforementioned studies, the inclusion properties of several functional spaces such as Orlicz space, weak-Orlicz space, Morrey space, Orlicz-Morrey space, and Discrete Morrey space, and others, have been proven. Moreover, several mathematicians have made observations in the ℋ-Orlicz spaces. Therefore, here we are interested to study about inclusions relation in the ℋ-Orlicz spaces. The aims of this research is to obtain some sufficient and necessary conditions for the inclusion properties on ℋ-Orlicz spaces.

B. METHODS
The research method in this article begins with a literature study on inclusion properties from books and scientific journals, especially those related to the ℋ-Orlicz Spaces and inclusion properties on some functional spaces. The reference also have been published at least the last 15 years. Reference sources were obtained from digital search engines such as Google Scholar, and from journals subscribed to by the institution, as well as scientific books from the institutional library.
Next, in particular, it is preceded by studying the Henstock-Kurzweil spaces, the Orlicz spaces, and the ℋ-Orlicz spaces and then examining the properties of inclusions. Moreover, several definitions and lemmas are needed in the discussion related to the results of this article. Definition 1.1. (Royden & Fitzpatrick, 2010) Let be a set, then a collection of subsets of is is called a σ-algebra if it satisfies the following properties: Furthermore, elements of the σ-algebra are called measurable sets. In this case, is a measurable set. Then an ordered pair ( , ), where X is a set and is a σ-algebra over X, is called a measurable space. Next, let be a measure on X, then ( , , ) is called a measure space. In some literature, notation of a measure space notation is simplified to ( , ). Moreover, a function between two measurable spaces is called a measurable function if the pre-image of every measurable set is measurable. Lemma 1.1. (Masta et al., 2016) Suppose that Φ is a Young function and Lemma 1.2. (Masta et al., 2016) Let Φ be a Young function, then Φ(Ct) ≤ C Φ(t) for t > 0 and 0 ≤ C ≤ 1.
From the definitions and lemmas above, first, we studied the similarities between the definitions of the Orlicz spaces and the ℋ-Orlicz spaces which both utilize the Young's function. Then we studied the definition and properties of Young's function in each spaces. The second, we studied the inclusion properties in various functional spaces. For the last step, based on the definition of ℋ-Orlicz spaces and its norm (given in section C), we formulated a hypothesis regarding the inlcusion properties. By deductive proof, we proven the hypothesis and stated it as theorem. Finally, sufficient and necessary conditions will be found for inclusion in the ℋ-Orlicz spaces.
Therefore, ‖ ‖ ℋ . ∎ From Theorem 1.1, we obtain that if ≤ then ℋ (ℝ ) ⊆ ℋ (ℝ ) with ‖ ‖ ℋ ≤ ‖ ‖ ℋ for every ∈ ℋ (ℝ ). For this result, the converse also holds. Moreover, to prove this converse, we need to use the characteristic function of the ball in ℝ . In addition, the following lemma's are needed. Lemma 1.3. (Masta et al., 2016) Let Φ be a Young function, ∈ ℝ , > 0 and ( , ) is a ball in ℝ . Then where ( ( , )) is the measure of the ball ( , ) with centered at a and radius r.
Next, we come to the main result of this article. The sufficient and necessary conditions for inclusion properties of ℋ-Orlicz spaces are proved in the following theorem. Theorem 1.3. Let Φ and be Young Functions and > 0. Then the following statements are equivalent.
Proof. First, we will show that (1) implies (2), then (2) implies (3). The proof is similar as the proof of Theorem 1.1. In breafly, suppose that ∈ ℋ (ℝ ), we have ( ) ∫ ψ ( ‖ ‖ ℋ ) ℝ ≤ 1. By (1), we obtain So, ∈ ℋ Φ (ℝ ) . As consequently, (2) also holds. Furthermore, because We also obtain (2) implies (3). Next, we have to show that (3) indicates (1) to complete the proof. Now, assume that (3) holds, by Lemma 1.3 and (3), we have 1 and we obtain Φ −1 ( 1 ( ( , )) ) ≥ ψ −1 ( 1 ( ( , )) ) for arbitrary ∈ ℝ and > 0. By Lemma 1, we have Φ ( 1 ( ( , )) ) ≤ ψ ( 1 ( ( , )) ), since ∈ ℝ and > 0 are arbitrary, then for every > 0 we obtained Φ( ) ≤ ( ). ∎ Theorem 1.3 explains that inclusion property can be satisfied in the ℋ-Orlicz spaces. This is showed by being satisfied of sufficient and necessary conditions for inclusion properties in the ℋ-Orlicz space which includes the relationship between the inclusion of the two Young functions and the inclusion of the ℋ-Orlicz space with respect to each Young function, also about the inclusion of the norm of a function in the ℋ-Orlicz space respect to each Young function. With the proof of the theorem, it can enrich the development of theory about ℋ-Orlicz spaces, especially regarding the properties that satisfied in these spaces, and develop a theory that inclusion properties can also be applied to other functional spaces such as ℋ-Orlicz spaces.
For further research, it can be study about the conditions that can cause the inclusion relation to be equal realtion and also inclusion relation between ℋ-Orlicz spaces with three or more Young functions. In addition, it is possible to study inclusion properties in other types of ℋ-Orlicz spaces.