On Inclusion Properties of Two Versions of Orlicz-Morrey Spaces

TL;DR

This paper investigates the inclusion relations between two versions of Orlicz-Morrey spaces on , providing comparisons and extending results to their weak variants, with key insights from characteristic function norms.

Contribution

It offers a detailed comparison of the inclusion properties of two definitions of Orlicz-Morrey spaces and extends the analysis to weak spaces, clarifying their relationships.

Findings

Inclusion criteria for the two versions are established.

Norms of characteristic functions of balls are crucial for the analysis.

Results are extended to weak Orlicz-Morrey spaces.

Abstract

There are two versions of Orlicz-Morrey spaces (on $\mathbb{R}^n$), defined by Nakai in 2004 and by Sawano, Sugano, and Tanaka in 2012. In this paper we discuss the inclusion properties of these two spaces and compare the results. Computing the norms of the characteristic functions of balls in $\mathbb{R}^n$ is one of the keys to our results. Similar results for weak Orlicz-Morrey spaces of both versions are also obtained.

Equations

$$\|f\|_{(\phi,\Phi,B(a,r))}:=\inf\Bigl{\{}b>0:\frac{\phi(|B(a,r)|)}{|B(a,r)|}\int_{B(a,r)}\Phi\Bigl{(}\frac{|f(x)|}{b}\Bigr{)}dx\leq 1\Bigr{\}}$$

$$\Vert f{\Vert }_{{\mathcal{M}}_{\psi ,\Psi }({\mathbb{R}}^n)}:=\underset{a\in {\mathbb{R}}^n,\text{leavevmode}r>0}{\sup }\psi (|B(a,r)|)\Vert f{\Vert }_{(\Psi ,B(a,r))}<\infty ,$$

$$\Vert f{\Vert }_{L_{{\varphi }_1,{\Phi }_1}({\mathbb{R}}^n)}\le C\Vert f{\Vert }_{L_{{\varphi }_2,{\Phi }_2}({\mathbb{R}}^n)},$$

$$\Vert f{\Vert }_{{\mathcal{M}}_{{\psi }_1,{\Psi }_1}({\mathbb{R}}^n)}\le C\Vert f{\Vert }_{{\mathcal{M}}_{{\psi }_2,{\Psi }_2}({\mathbb{R}}^n)}$$

$$\frac{1}{|B(a,r)|}{\int }_{B(a,r)}{\Psi }_1\left(\frac{|f(x)|}{C_{1}b}\right)dx$$

$$=\frac{1}{|B(a,r)|}{\int }_{B(a,r)}{\Psi }_2\left(\frac{|f(x)|}{b}\right)dx\le 1.$$

$${\Vert \frac{f}{C_1}\Vert }_{({\Psi }_1,B(a,r))}=\inf A_{({\Psi }_1,B(a,r))}\le \inf A_{({\Psi }_2,B(a,r))}=\Vert f{\Vert }_{({\Psi }_2,B(a,r))},$$

$$\Vert f{\Vert }_{{\mathcal{M}}_{{\psi }_1,{\Psi }_1}({\mathbb{R}}^n)}$$

$$\le \underset{a\in {\mathbb{R}}^n,\text{leavevmode}r>0}{\sup }{C}_1{C}_2{\psi }_2(|B(a,r)|)\Vert f{\Vert }_{({\Psi }_2,B(a,r))}$$

$$=C\Vert f{\Vert }_{{\mathcal{M}}_{{\psi }_2,{\Psi }_2}({\mathbb{R}}^n)}.$$

$$\frac{{\psi }_1(|B(a,r)|)}{{\Psi }_1^{-1}(1)}=\Vert {\chi }_{B(a,r)}{\Vert }_{{\mathcal{M}}_{{\psi }_1,{\Psi }_1}({\mathbb{R}}^n)}\le C\Vert {\chi }_{B(a,r)}{\Vert }_{{\mathcal{M}}_{{\psi }_2,{\Psi }_2}({\mathbb{R}}^n)}=\frac{C{\psi }_2(|B(a,r)|)}{{\Psi }_2^{-1}(1)},$$

$$\Vert f{\Vert }_{{\mathcal{M}}_{{\psi }_1,\Psi }({\mathbb{R}}^n)}\le C\Vert f{\Vert }_{{\mathcal{M}}_{{\psi }_2,\Psi }({\mathbb{R}}^n)}$$

Full text

**Keywords: Inclusion property, Orlicz-Morrey spaces, weak Orlicz-Morrey spaces.

MSC 2010: Primary 46E30; Secondary 46B25, 42B35.**

1 Introduction

Orlicz-Morrey spaces are generalizations of Orlicz spaces and Morrey spaces (on $\mathbb{R}^{n}$). There are two versions of Orlicz-Morrey spaces: one is defined by Nakai [Gala, Nakai1] and another by Sawano, Sugano, and Tanaka [Gala, Sawano]. We shall discuss both of them here. In particular, we are interested in the inclusion properties of these spaces.

A function $\Phi:[0,\infty)\to[0,\infty)$ is called a Young function if $\Phi$ is convex, left-continuous, $\Phi(0)=0$, and $\lim\limits_{t\to\infty}\Phi(t)=\infty$. Given two Young functions $\Phi,\Psi$, we write $\Phi\prec\Psi$ if there exists a constant $C>0$ such that $\Phi(t)\leq\Psi(Ct)$ for all $t>0$.

Let $G_{1}$ be the set of all functions $\phi:(0,\infty)\rightarrow(0,\infty)$ such that $\phi(r)$ is nondecreasing but $\frac{\phi(r)}{r}$ is nonincreasing. For a Young function $\Psi$, we also define $G_{2}$ to be the set of all functions $\psi:(0,\infty)\rightarrow(0,\infty)$ such that $\psi(r)$ is nondecreasing but for any $s>0$, $\frac{\psi((r+s)^{n})}{\Psi^{-1}((\frac{r+s}{s})^{n})}$ is nonincreasing.

For $\phi_{1},\leavevmode\nobreak\ \phi_{2}:(0,\infty)\rightarrow(0,\infty)$, we write $\phi_{1}\preceq\phi_{2}$ if there exists a constant $C>0$ such that $\phi_{1}(t)\leq C\phi_{2}(t)$ for all $t>0$. If $\phi_{1}\preceq\phi_{2}$ and $\phi_{2}\preceq\phi_{1}$, then we write $\phi_{1}\approx\phi_{2}$.

Let $\Phi$ be a Young function and $\phi\in G_{1}$. The Orlicz-Morrey spaces $L_{\phi,\Phi}(\mathbb{R}^{n})$ (of Nakai’s version) is the set of measurable functions $f\in L^{1}_{\rm loc}(\mathbb{R}^{n})$ such that for every $a\in\mathbb{R}^{n}$ and $r>0$, the following quantity

[TABLE]

is finite. We use the notation $B(a,r)$ to denote the open ball in $\mathbb{R}^{n}$ centered at $a$ with radius $r$, and $|B(a,r)|$ for its Lebesgue measure. The Orlicz-Morrey spaces $L_{\phi,\Phi}(\mathbb{R}^{n})$ is a Banach space with respect to the norm $\|f\|_{L_{\phi,\Phi}(\mathbb{R}^{n})}:=\sup\limits_{a\in\mathbb{R}^{n},\leavevmode\nobreak\ r>0}\|f\|_{(\phi,\Phi,B(a,r))}.$

For $\phi(r)=r$, the space $L_{\phi,\Phi}(\mathbb{R}^{n})$ is the Orlicz space $L_{\Phi}(\mathbb{R}^{n})$. Meanwhile, for $\Phi(r)=r^{p}$ and $\phi(r)=r^{1-\frac{\lambda}{n}}$ where $0\leq\lambda\leq n$, the space $L_{\phi,\Phi}(\mathbb{R}^{n})$ reduces to the Morrey space $L_{p,\lambda}(\mathbb{R}^{n})$.

Now, let $\Psi$ be a Young function and $\psi\in G_{2}$. Sawano, Sugano, and Tanaka defined the Orlicz-Morrey space ${\mathcal{M}}_{\psi,\Psi}(\mathbb{R}^{n})$ to be the set of measurable functions $f\in L^{1}_{loc}(\mathbb{R}^{n})$ such that

[TABLE]

where \|f\|_{(\Psi,B(a,r))}:=\inf\bigl{\{}{b>0:\frac{1}{|B(a,r)|}\int_{B(a,r)}\Psi\bigl{(}\frac{|f(x)|}{b}\bigr{)}dx\leq 1}\bigr{\}}. Notice here that $\psi(|B(a,r)|)$ dominates of the growth $\|f\|_{(\Psi,B(a,r))}$.

For the Young function $\Psi(t)=t^{p}$ ($1\leq p<\infty$), the spaces ${\mathcal{M}}_{\psi,\Psi}(\mathbb{R}^{n})$ are recognized as the generalized Morrey spaces ${\mathcal{M}}^{p}_{\psi}(\mathbb{R}^{n})$.

Recently, Gunawan et al. [Gunawan] presented a sufficient and necessary condition for the inclusion relation between generalized Morrey spaces, as in the following theorem.

Theorem 1.1**.**

Let $1\leq p_{1}\leq p_{2}<\infty$ and $\psi_{1},\psi_{2}\in G_{2}$. Then the following statements are equivalent:

(a)* $\psi_{1}\preceq\psi_{2}$.*

(b)* ${\mathcal{M}}^{p_{2}}_{\psi_{2}}(\mathbb{R}^{n})\subseteq{\mathcal{M}}^{p_{1}}_{\psi_{1}}(\mathbb{R}^{n})$.*

(c)* There exists a constant $C>0$ such that $\|f\|_{{\mathcal{M}}^{p_{1}}_{\psi_{1}}(\mathbb{R}^{n})}\leq C\|f\|_{{\mathcal{M}}^{p_{2}}_{\psi_{2}}(\mathbb{R}^{n})}$ for every $f\in{\mathcal{M}}^{p_{2}}_{\psi_{2}}(\mathbb{R}^{n})$.*

In the same paper, Gunawan et al. also gave a necessary and sufficient condition for the inclusion relation between generalized weak Morrey spaces.

Meanwhile, the inclusion relation between Orlicz spaces $L_{\Phi}({\mathbb{R}}^{n})$ and between weak Orlicz spaces $wL_{\Phi}({\mathbb{R}}^{n})$ are known (see [Lech, Masta1]). In 2016, Masta et al. [Masta2] also obtained the inclusion properties of Orlicz-Morrey space $L_{\phi,\Phi}(\mathbb{R}^{n})$ of Nakai’s version, as in the following theorem.

Theorem 1.2**.**

Let $\Phi_{1},\Phi_{2}$ be Young functions and $\phi_{1},\phi_{2}\in G_{1}$ such that $\phi_{1}\approx\phi_{2}$. Then the following statements are equivalent:

(1)* $\Phi_{1}\prec\Phi_{2}$.*

(2)* $L_{\phi_{2},\Phi_{2}}(\mathbb{R}^{n})\subseteq L_{\phi_{1},\Phi_{1}}(\mathbb{R}^{n})$.*

(3)* There exists a constant $C>0$ such that*

[TABLE]

for every $f\in L_{\phi_{2},\Phi_{2}}(\mathbb{R}^{n})$.

Remark 1.3**.**

Note that the relation $\Phi_{1}\prec\Phi_{2}$ is a necessary and sufficient condition for the inclusion relation between Orlicz-Morrey spaces of Nakai’s version. For $\phi_{1}(t)=\phi_{2}(t)=t$, Theorem 1.2 reduces to Theorem 3.4(a) in [Lech]. Furthermore, for $\phi_{1}(t)=\phi_{2}(t)=t$ and $w_{1}(x)=w_{2}(x)=1$, Theorem 1.2 complements Corollary 2.11 in [Alen], which states that $\Phi_{1}\prec\Phi_{2}$ is a sufficient condition for inclusion relation between Orlicz spaces. Related results about inclusion properties of Orlicz-Morrey spaces can be found in [Kita].

In this paper, we would like to obtain the inclusion properties of Orlicz-Morrey spaces ${\mathcal{M}}_{\psi,\Psi}(\mathbb{R}^{n})$ of Sawano-Sugano-Tanaka’s version, and compare it with the result for Nakai’s version. In addition, we will also prove similar results for weak Orlicz-Morrey spaces of both versions. With our results, we can see what parameters are significance in the inclusion properties for both versions.

To prove the results, we will use the same method as in [Gunawan, Masta1, Masta2], that is by computing the norms of the characteristic functions of balls in ${\mathbb{R}}^{n}$. We also employ the properties of the inverse function of $\Phi$, which are presented in the following lemma.

Lemma 1.4**.**

[Masta2, Nakai1, Rao]* Suppose that $\Phi$ is a Young function and $\Phi^{-1}$ denotes its inverse, which is given by $\Phi^{-1}(s):=\inf\{r\geq 0:\Phi(r)>s\}$ for every $s\geq 0$. Then the followings hold:*

(1)* $\Phi^{-1}(0)=0$.*

(2)* $\Phi^{-1}(s_{1})\leq\Phi^{-1}(s_{2})$ for $s_{1}\leq s_{2}$.*

(3)* $\Phi(\Phi^{-1}(s))\leq s\leq\Phi^{-1}(\Phi(s))$ for $0\leq s<\infty$.*

(4)* If, for some constants $C_{1},C_{2}>0$, we have $\Phi_{2}^{-1}(s)\leq C_{1}\Phi_{1}^{-1}(C_{2}s)$, then $\Phi_{1}(\frac{t}{C_{1}})\leq C_{2}\Phi_{2}(t)$ for $t=\Phi_{2}^{-1}(s)$.*

Throughout this paper, the letter $C$ denotes a constant that may vary in values from line to line. To keep track of some constants, we use subscripts, such as $C_{1}$ and $C_{2}$.

2 Inclusion Properties of Orlicz-Morrey Spaces

As mentioned earlier, the key to our results is knowing the norms of the characteristic balls in $\mathbb{R}^{n}$. Here is the first one on ${\mathcal{M}}_{\psi,\Psi}(\mathbb{R}^{n})$:

Lemma 2.1**.**

[Gala]* For every $a\in\mathbb{R}^{n}$ and $r>0$, we have $\|\chi_{B(a,r)}\|_{{\mathcal{M}}_{\psi,\Psi}(\mathbb{R}^{n})}=\frac{\psi(|B(a,r)|)}{\Psi^{-1}(1)}$.*

Our first theorem gives equivalent statements for the inclusion relation between Orlicz-Morrey spaces of Sawano-Sugano-Tanaka’s version.

Theorem 2.2**.**

Let $\Psi_{1},\Psi_{2}$ be Young functions such that $\Psi_{1}\prec\Psi_{2}$ and $\psi_{1},\psi_{2}\in G_{2}$. Then the following statements are equivalent:

(1)* $\psi_{1}\preceq\psi_{2}$.*

(2)* ${\mathcal{M}}_{\psi_{2},\Psi_{2}}(\mathbb{R}^{n})\subseteq{\mathcal{M}}_{\psi_{1},\Psi_{1}}(\mathbb{R}^{n})$.*

(3)* There exists a constant $C>0$ such that*

[TABLE]

for every $f\in{\mathcal{M}}_{\psi_{2},\Psi_{2}}(\mathbb{R}^{n})$.

Proof. Let us first prove that (1) implies (2). Let $f\in{\mathcal{M}}_{\psi_{2},\Psi_{2}}(\mathbb{R}^{n})$. Recall that $\Psi_{1}\prec\Psi_{2}$ means that there exists a constant $C_{1}>0$ such that $\Psi_{1}(t)\leq\Psi_{2}(C_{1}t)$ for every $t>0$. For every $a\in\mathbb{R}^{n}$ and $r>0$, let A_{(\Psi_{1},B(a,r))}=\bigl{\{}{b>0:\frac{1}{|B(a,r)|}\int_{B(a,r)}\Psi_{1}\bigl{(}\frac{|f(x)|}{C_{1}b}\bigr{)}dx\leq 1}\bigr{\}} and A_{(\Psi_{2},B(a,r))}=\bigl{\{}{b>0:\frac{1}{|B(a,r)|}\int_{B(a,r)}\Psi_{2}\bigl{(}\frac{|f(x)|}{b}\bigr{)}dx\leq 1}\bigr{\}}. Thus, for any $b\in A_{(\Psi_{2},B(a,r))}$, we have

[TABLE]

Hence it follows that $b\in A_{(\Psi_{1},B(a,r))}$, and so we conclude that $A_{(\Psi_{2},B(a,r))}\subseteq A_{(\Psi_{1},B(a,r))}$. Accordingly, we have

[TABLE]

and this holds for every $a\in\mathbb{R}^{n}$ and $r>0$.

Now there exists $C_{2}>0$ such that $\psi_{1}(s)\leq C_{2}\psi_{2}(s)$ for every $s>0$. Combining this with the previous estimate, we obtain

[TABLE]

This proves that ${\mathcal{M}}_{\psi_{2},\Psi_{2}}(\mathbb{R}^{n})\subseteq{\mathcal{M}}_{\psi_{1},\Psi_{1}}(\mathbb{R}^{n})$.

Next, since $({\mathcal{M}}_{\psi_{2},\Psi_{2}}(\mathbb{R}^{n}),{\mathcal{M}}_{\psi_{1},\Psi_{1}}(\mathbb{R}^{n}))$ is a Banach pair, it follows from [Krein, Lemma 3.3] that (2) and (3) are equivalent. It thus remains to show that (3) implies (1).

Assume that (3) holds. Let $a\in\mathbb{R}^{n}$ and $r>0$. By Lemma 2.1, we have

[TABLE]

whence $\psi_{1}(|B(a,r)|)\leq\frac{C\Psi_{1}^{-1}(1)}{\Psi_{2}^{-1}(1)}\psi_{2}(|B(a,r)|)$. Since $a\in\mathbb{R}^{n}$ and $r>0$ are arbitrary, we get $\psi_{1}(t)\leq C_{1}\psi_{2}(t)$ for every $t>0$, where $C_{1}=\frac{C\Psi_{1}^{-1}(1)}{\Psi_{2}^{-1}(1)}$. ∎

Corollary 2.3**.**

Let $\Psi$ be a Young function and $\psi_{1},\psi_{2}\in G_{2}$. Then the following statements are equivalent:

(1)* $\psi_{1}\preceq\psi_{2}$.*

(2)* ${\mathcal{M}}_{\psi_{2},\Psi}(\mathbb{R}^{n})\subseteq{\mathcal{M}}_{\psi_{1},\Psi}(\mathbb{R}^{n})$.*

(3)* There exists a constant $C>0$ such that*

[TABLE]

for every $f\in{\mathcal{M}}_{\psi_{2},\Psi}(\mathbb{R}^{n})$.