An inclusion property of Orlicz-Morrey spaces

The Orlicz-Morrey spaces Lϕ,Φ (where Φ is a Young function and ϕ is a parameter for the Morrey spaces) are generalizations of Orlicz spaces and Morrey spaces. Inclusion properties between Orlicz spaces LΦ and between Morrey spaces ℳ ψ p are well known. In this study we will investigate the inclusion relation between Orlicz-Morrey spaces Lϕ1, Φ1 and Lϕ2, Φ2 with respect to Young functions Φ1, Φ2 and parameters ϕ1, ϕ2. Also, we give sufficient and necessary conditions for the inclusion property of these spaces, which are obtained through norm estimates for the characteristic functions of balls in ℝn. In addition, we shall give a sufficient and necessary condition for generalized Hölder’s inequality.


Introduction
Orlicz-Morrey spaces are generalizations of Orlicz spaces and Morrey spaces. There are two versions of Orlicz-Morrey spaces. One is defined by Nakai [2,9] and another by Sawano, Sugano, and Tanaka [2,12]. Here we are interested in studying the inclusion property of Orlicz-Morrey spaces which were introduced by Nakai. First, we recall the definiton of Young functions (see [11,9]). A function Φ : [0, ∞) → [0, ∞) is called a Young function if Φ is convex, left-continuous, lim t→0 = Φ(t) = Φ(0) = 0, and lim t→∞ Φ(t) = ∞. Let G 1 be the set of all functions φ : (0, ∞) → (0, ∞) such that φ(r) is nondecreasing but φ(r) r is nonincreasing. For φ 1 , φ 2 ∈ G 1 , we write φ 1 ∼ φ 2 if there exists a constant C > 1 such that for all t > 0. Let Φ be a Young function and φ ∈ G 1 . The Orlicz-Morrey space L φ,Φ (R n ) is the set of measurable functions f ∈ L 1 loc (R n ) such that for every open ball B in R n , the following is finite. We use the notation B to denote the family of all open balls B in R n , and |B| for its Lebesgue measure. L φ,Φ (R n ) is a Banach space with respect to the norm The is almost decreasing and t n p ψ(t) is almost increasing) and Φ(t) = t p we have L φ,Φ (R n ) = M p ψ (R n ), the generalized Morrey space introduced by Nakai in 1994. Gunawan et al. [3] have proved an inclusion property of generalized Morrey spaces: for all t > 0 and some C > 0. On the other hand, inclusion properties between Orlicz spaces L Φ (R n ) and between weak Orlicz spaces wL Φ (R n ) are well known (see [5,6]).
Motivated by these results, the purpose of this study is to get the inclusion property of Orlicz-Morrey spaces L φ,Φ (R n ).
The main results are presented in Section 2. In particular, Theorem 2.2 contains a necessary and sufficient condition for the inclusion relation between Orlicz-Morrey spaces. In Section 3, we have also given sufficent and necessary conditions for generalized Hölder's inequality.
To prove the results, we will use the same method as in [3] and [6] which pay attention to the characteristic functions of balls in R n and use the inverse function of Φ, namely Φ −1 (s) := inf{r ≥ 0 : Φ(r) > s}.
In the following, we recall several lemmas which will be used in the next section.
where |B(a, r)| denotes the volume of B(a, r). Proof.
Let f be an element of L φ,Φ (R n ) and take an arbitrary > 0, then there exists b > 0 such for any open ball B ∈ B. 3

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The
We can prove the Lemma 2.1 by using similar arguments in the proof of [9, Proposition 3.2].

Generalized Hölder Inequality
In the following, we will give sufficient and necessary conditions for generalized Hölder inequality on Orlicz-Morrey spaces. To get the result we will give attention to estimate the norm of the characteristic function of ball in R n . Proof.
On the other hand, we have This shows that for every open ball B ∈ B. Hence, we conclude that   ).
Since B(a, r) is arbitrary, we get for s, t > 0.