A Note on Inclusion Properties of Weighted Orlicz Spaces

TL;DR

This paper establishes precise conditions for when one weighted Orlicz space is included in another, improving upon previous results by utilizing characteristic function norms of Euclidean balls.

Contribution

It provides necessary and sufficient conditions for inclusion relations between weighted Orlicz spaces, extending Osan extcirlioul's 2014 findings.

Findings

Derived new inclusion criteria for weighted Orlicz spaces.

Utilized characteristic function norms of Euclidean balls in proofs.

Extended previous inclusion results with complete conditions.

Abstract

In this paper we present sufficient and necessary conditions for inclusion relation between two weighted Orlicz spaces which complete the Osan\c{c}liol result in 2014. One of the keys to prove our results is to use the norm of the characteristic functions of the balls in $\mathbb{R}^n$.

Equations

$$\Vert f{\Vert }_{L_{\Phi }({\mathbb{R}}^n)}:=\inf \left\{b>0:{\int }_{{\mathbb{R}}^n}\Phi \left(\frac{|f(x)|}{b}\right)dx\le 1\right\}.$$

$$\|f\|_{wL_{\Phi}(\mathbb{R}^{n})}:=\inf\left\{{b>0:\mathop{\sup}\limits_{t>0}\Phi(t)\mu\left(\Bigl{\{}x\in\mathbb{R}^{n}:\frac{|f(x)|}{b}>t\Bigr{\}}\right)\leq 1}\right\}<\infty.$$

$$\Vert f{\Vert }_{L_{\Phi }^u({\mathbb{R}}^n)}:=\Vert uf{\Vert }_{L_{\Phi }({\mathbb{R}}^n)}=\inf \left\{b>0:{\int }_{{\mathbb{R}}^n}\Phi \left(\frac{|u(x)f(x)|}{b}\right)dx\le 1\right\}.$$

$${\int }_{{\mathbb{R}}^n}{\Phi }_1\left(\frac{|u(x)f(x)|}{C\Vert f{\Vert }_{L_{{\Phi }_2}^u({\mathbb{R}}^n)}}\right)dx\le {\int }_{{\mathbb{R}}^n}{\Phi }_2\left(\frac{C|u(x)f(x)|}{C\Vert f{\Vert }_{L_{{\Phi }_2}^u({\mathbb{R}}^n)}}\right)dx={\int }_{{\mathbb{R}}^n}{\Phi }_2\left(\frac{|u(x)f(x)|}{\Vert f{\Vert }_{L_{{\Phi }_2}^u({\mathbb{R}}^n)}}\right)dx\le 1.$$

$$\frac{1}{\Phi_{1}^{-1}\Bigl{(}\frac{1}{|B(a,r)|}\Bigr{)}}=\left\|\frac{\chi_{B(a,r)}}{u}\right\|_{L_{\Phi_{1}}^{u}(\mathbb{R}^{n})}\leq C\left\|\frac{\chi_{B(a,r)}}{u}\right\|_{L_{\Phi_{2}}^{u}(\mathbb{R}^{n})}=\frac{C}{\Phi_{2}^{-1}\Bigl{(}\frac{1}{|B(a,r)|}\Bigr{)}}.$$

$$\Phi_{1}\Bigl{(}\frac{t_{0}}{C}\Bigr{)}\leq\Phi_{2}(t_{0}),$$

$$\frac{u(x)}{C}\le \Vert T_{x}f{\Vert }_{L_{\Phi }^u({\mathbb{R}}^n)}\le Cu(x).$$

$$\int_{{\mathbb{R}}^{n}}\Phi\Bigl{(}\frac{|u(v)f(v)|}{\|f\|_{L_{\Phi}^{u}(\mathbb{R}^{n})}}\Bigr{)}dv\leq 1.$$

$$\displaystyle\int_{{\mathbb{R}}^{n}}\Phi\Bigl{(}\frac{|u(y)T_{x}f(y)|}{u(x)\|f\|_{L_{\Phi}^{u}(\mathbb{R}^{n})}}\Bigr{)}dy$$

$$\displaystyle=\int_{{\mathbb{R}}^{n}}\Phi\Bigl{(}\frac{|u(v+x)f(v)|}{u(x)\|f\|_{L_{\Phi}^{u}(\mathbb{R}^{n})}}\Bigr{)}dv$$

$$\displaystyle\leq\int_{{\mathbb{R}}^{n}}\Phi\Bigl{(}\frac{|u(v)u(x)f(v)|}{u(x)\|f\|_{L_{\Phi}^{u}(\mathbb{R}^{n})}}\Bigr{)}dv$$

$$\displaystyle=\int_{{\mathbb{R}}^{n}}\Phi\Bigl{(}\frac{|u(v)f(v)|}{\|f\|_{L_{\Phi}^{u}(\mathbb{R}^{n})}}\Bigr{)}dv$$

$$\le 1.$$

$$\Vert T_{x}f{\Vert }_{L_{\varphi ,\Phi }^u({\mathbb{R}}^n)}\le Cu(x),$$

$$\displaystyle\int_{{\mathbb{R}}^{n}}\Phi\Bigl{(}\frac{|u(x)f(v)|}{\mathop{\sup}\limits_{v\in{\mathbb{R}}^{n}}u(-v)\|T_{x}f\|_{L_{\Phi}^{u}(\mathbb{R}^{n})}}\Bigr{)}dv$$

$$\displaystyle\leq\int_{{\mathbb{R}}^{n}}\Phi\Bigl{(}\frac{|u(v+x)f(v)|}{\|T_{x}f\|_{L_{\Phi}^{u}(\mathbb{R}^{n})}}\Bigr{)}dv$$

$$\displaystyle\leq\int_{{\mathbb{R}}^{n}}\Phi\Bigl{(}\frac{|u(y)f(y-x)|}{\|T_{x}f\|_{L_{\Phi}^{u}(\mathbb{R}^{n})}}\Bigr{)}dy$$

$$\displaystyle=\int_{{\mathbb{R}}^{n}}\Phi\Bigl{(}\frac{|u(y)L_{x}f(y)|}{\|T_{x}f\|_{L_{\Phi}^{u}(\mathbb{R}^{n})}}\Bigr{)}dy$$

$$\le 1.$$

$$\Vert f{\Vert }_{L_{{\Phi }_1}^{u_1}({\mathbb{R}}^n)}\le C_1\Vert f{\Vert }_{L_{{\Phi }_2}^{u_1}({\mathbb{R}}^n)}\le C_1{C}_2\Vert f{\Vert }_{L_{{\Phi }_2}^{u_2}({\mathbb{R}}^n)}.$$

$$\frac{u_1(x)}{C}\le \Vert T_{x}f{\Vert }_{L_{{\Phi }_1}^{u_1}({\mathbb{R}}^n)}\le C\Vert T_{x}f{\Vert }_{L_{{\Phi }_2}^{u_2}({\mathbb{R}}^n)}\le C{u}_2(x),$$

Full text

**Keywords: Inclusion property, Weighted Lebesgue spaces, Weighted Orlicz spaces.

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

1 Introduction

Orlicz spaces are generalization of Lebesgue spaces which were firstly introduced by Z. W. Birnbaum and W. Orlicz in 1931 (see [Luxemburg, Rao]). Let us first recall the definition of Orlicz spaces. Let $\Phi:[0,\infty)\rightarrow[0,\infty)$ be a Young function [that is, $\Phi$ is convex, $\lim\limits_{t\to 0}\Phi(t)=0=\Phi(0)$, left-continuous and $\lim\limits_{t\to\infty}\Phi(t)=\infty$], the Orlicz space $L_{\Phi}(\mathbb{R}^{n})$ is the set of measurable functions $f:{\mathbb{R}}^{n}\rightarrow\mathbb{R}$ such that $\int_{{\mathbb{R}}^{n}}\Phi(a|f(x)|)dx<\infty$ for some $a>0$. The space $L_{\Phi}({\mathbb{R}}^{n})$ is a Banach space equipped with the norm

[TABLE]

Meanwhile, for $\Phi$ is a Young function, the weak Orlicz space $wL_{\Phi}(\mathbb{R}^{n})$ is the set of all measurable functions $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ such that

[TABLE]

Now, we move to the weighted Orlicz spaces and weighted weak Orlicz spaces. Let $\Phi$ be a Young function and $u$ is a weight on ${\mathbb{R}}^{n}$ (i.e $u:\mathbb{R}^{n}\rightarrow(0,\infty)$ is a measurable function), the weighted Orlicz space $L_{\Phi}^{u}(\mathbb{R}^{n})$ is the set of all measurable functions $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ such that $uf\in L_{\Phi}(\mathbb{R}^{n})$. Note that, the space $L_{\Phi}^{u}(\mathbb{R}^{n})$ is a Banach space equipped with the norm

[TABLE]

Similar with weighted Orlicz spaces, for a Young function $\Phi$ and a weight $u$ on ${\mathbb{R}}^{n}$, the weighted weak Orlicz space $wL_{\Phi}^{u}(\mathbb{R}^{n})$ is the set of all measurable functions $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ such that $\|f\|_{wL_{\Phi}^{u}(\mathbb{R}^{n})}:=\|uf\|_{wL_{\Phi}(\mathbb{R}^{n})}<\infty$.

Let $u_{1},u_{2}:\mathbb{R}^{n}\rightarrow(0,\infty)$, we denote $u_{1}\preceq u_{2}$ if there exists a constant $C>0$ such that $u_{1}(x)\leq Cu_{2}(x)$ for all $x\in\mathbb{R}^{n}$. Note that, if $u_{1}\preceq u_{2}$ then $\|f\|_{L_{\Phi}^{u_{1}}(\mathbb{R}^{n})}\leq C\|f\|_{L_{\Phi}^{u_{2}}(\mathbb{R}^{n})}$ and $\|f\|_{wL_{\Phi}^{u_{1}}(\mathbb{R}^{n})}\leq C\|f\|_{wL_{\Phi}^{u_{2}}(\mathbb{R}^{n})}.$

The study of Lebesgue spaces and Orlicz spaces has been studied by many researchers in the last few decades (see [Kufner, Luxemburg, Lech, Masta1, Orlicz, Taqi], etc.). In 1989, Maligranda [Lech] discussed inclusion properties of Orlicz spaces. Later in 2016, Masta* et al.* [Masta1] obtained sufficient and necessary conditions for inclusion relation between two Orlicz spaces and between two weak Orlicz spaces by using different technique from Maligranda. Moreover, they have found that two Orlicz spaces and two weak Orlicz spaces can be compared with respect to Young functions for any measurable set, although the Lebesgue space $L_{p}$ are not comparable with respect to the number $p$.

On the other hand, Osançliol [Alen] have proved sufficient and necessary conditions for inclusion relation between two weighted Orlicz spaces, as in the following theorem.

Theorem 1.1**.**

[Alen]* Let $\Phi$ be a continuous Young function satisfying the $\bigtriangleup_{2}$ condition [that is, there exists $K>0$ such that $\Phi(2t)\leq K\Phi(t)$ for all $t\geq 0$], and $u_{1},u_{2}$ are measurable functions such that $u_{i}(x+y)\leq u_{i}(x)\cdot u_{i}(y)$ for every $x,y\in{\mathbb{R}}^{n}$, where $i=1,2$. Then the following statements are equivalent:*

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

(2)* $L_{\Phi}^{u_{2}}(\mathbb{R}^{n})\subseteq L_{\Phi}^{u_{1}}(\mathbb{R}^{n})$.*

(3)* There exists a constant $C>0$ such that $\|f\|_{L_{\Phi}^{u_{1}}(\mathbb{R}^{n})}\leq C\|f\|_{L_{\Phi}^{u_{2}}(\mathbb{R}^{n})}$, for every $f\in L_{\Phi}^{u_{2}}(\mathbb{R}^{n})$.*

Related result for weak type of Orlicz spaces can be found in [Masta4].

In this paper, we are interested in studying the inclusion properties of weighted Orlicz spaces. In connection with Theorem 1.1, we shall prove inclusion relation between weighted Orlicz spaces with respect to Young functions $\Phi_{1},\Phi_{2}$ and weights $u_{1},u_{2}$.

To achieve our purpose, we will use the similar methods in [Gunawan, Masta1, Masta2, Masta3, Alen] which pay attention to the characteristic functions of open balls in ${\mathbb{R}}^{n}$. Next, we recall some lemmas which will be used later in next section.

Lemma 1.2**.**

[Nakai1]* Suppose that $\Phi$ is a Young function and $\Phi^{-1}(s):=\inf\{r\geq 0:\Phi(r)>s\}$. We have*

(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 $0\leq s<\infty$.*

Lemma 1.3**.**

[Masta3]* Let $\Phi_{1},\Phi_{2}$ be Young functions. For any $s>0$, if there exists $C_{1},C_{2}>0$ such that $\Phi^{-1}_{2}(s)\leq C_{1}\Phi^{-1}_{1}(C_{2}s)$, then we have $\Phi_{1}(\frac{t}{C_{1}})\leq C_{2}\Phi_{2}(t)$ for $t=\Phi^{-1}_{2}(s).$*

In this paper, the letter $C$ will be used for constants that may change from line to line, while constants with subscripts, such as $C_{1},C_{2}$, do not change in different lines.

2 Results

First,we will investigate the inclusion properties of weighted Orlicz spaces with respect to distinct Young functions $\Phi_{1}$ and $\Phi_{2}$. For getting the result, we give attention to estimate the norm of the characteristic function of open ball in ${\mathbb{R}}^{n}$ as in the following lemma.

Lemma 2.1**.**

[Ning, Masta1]* Let $\Phi$ be a Young function, $a\in{\mathbb{R}}^{n}$, and $r>0$ be arbitrary. Then we have \left\|\frac{\chi_{B(a,r)}}{u}\right\|_{L_{\Phi}^{u}(\mathbb{R}^{n})}=\|\chi_{B(a,r)}\|_{\L_{\Phi}(\mathbb{R}^{n})}=\frac{1}{\Phi^{-1}\bigl{(}\frac{1}{|B(a,r)|}\bigr{)}} where $|B(a,r)|$ denotes the volume of open ball $B(a,r)$ centered at $a\in{\mathbb{R}}^{n}$ with radius $r>0$.*

Now we come to the inclusion relation between $L_{\Phi_{1}}^{u}(\mathbb{R}^{n})$ and $L_{\Phi_{2}}^{u}(\mathbb{R}^{n})$ with respect to Young functions $\Phi_{1},\Phi_{2}$. Given two Young functions $\Phi_{1},\Phi_{2}$, we write $\Phi_{1}\prec\Phi_{2}$ if there exists a constant $C>0$ such that $\Phi_{1}(t)\leq\Phi_{2}(Ct)$ for all $t>0$.

Theorem 2.2**.**

Let $\Phi_{1},\Phi_{2}$ be Young functions and $u:\mathbb{R}^{n}\rightarrow(0,\infty)$ be a measurable function. Then the following statements are equivalent:

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

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

(3)* There exists a constant $C>0$ such that $\|f\|_{L_{\Phi_{1}}^{u}(\mathbb{R}^{n})}\leq C\|f\|_{L_{\Phi_{2}}^{u}(\mathbb{R}^{n})}$, for every $f\in L_{\Phi_{2}}^{u}(\mathbb{R}^{n})$.*

Proof. Assume that (1) holds. Suppose that $f\in L_{\Phi_{2}}^{u}(\mathbb{R}^{n})$. Observe that

[TABLE]

By definition of $\|\cdot\|_{L_{\Phi}^{u}(\mathbb{R}^{n})}$, we have $\|f\|_{L_{\Phi_{1}}^{u}(\mathbb{R}^{n})}\leq C\|f\|_{L_{\Phi_{2}}^{u}(\mathbb{R}^{n})}.$ This proves that $L_{\Phi_{2}}^{u}(\mathbb{R}^{n})\subseteq L_{\Phi_{1}}^{u}(\mathbb{R}^{n})$.

Next, since $(L_{\Phi_{1}}^{u}(\mathbb{R}^{n}),L_{\Phi_{2}}^{u}(\mathbb{R}^{n}))$ 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 now that (3) holds. By Lemma 2.1, we have

[TABLE]

Since \frac{1}{\Phi_{1}^{-1}\Bigl{(}\frac{1}{|B(a,r)|}\Bigr{)}}\leq\frac{C}{\Phi_{2}^{-1}\Bigl{(}\frac{1}{|B(a,r)|}\Bigr{)}} is equivalent to $C\Phi_{1}^{-1}(\frac{1}{|B(a,r)|})\geq\Phi_{2}^{-1}(\frac{1}{|B(a,r)|})$ for arbitrary $a\in\mathbb{R}^{n}$ and $r>0$, by Lemma 1.3, we have

[TABLE]

for $t_{0}=\Phi_{2}^{-1}(\frac{1}{|B(a,r)|})$. Since $a\in\mathbb{R}^{n}$ and $r>0$ are arbitrary, we conclude that $\Phi_{1}(t)\leq\Phi_{2}(Ct)$ for every $t>0$.∎

Remark 2.3**.**

For $u(x)=1$, Theorem 2.2 reduces to Theorem 2.5 in [Masta1].

Next, we also give the sufficient and necessary conditions for inclusion relation between weighted Orlicz spaces $L_{\Phi_{1}}^{u_{1}}(\mathbb{R}^{n})$ and $L_{\Phi_{2}}^{u_{2}}(\mathbb{R}^{n})$ with respect to Young functions $\Phi_{1},\Phi_{2}$ and weights $u_{1},u_{2}$. To get the result, we need the following lemma.

Lemma 2.4**.**

Let $u:\mathbb{R}^{n}\rightarrow(0,\infty)$ be a measurable function such that $u(x+y)\leq u(x)\cdot u(y)$ for every $x,y\in{\mathbb{R}}^{n}$. If $\Phi$ is a Young function, then:

(1)* For all $f\in L_{\Phi}^{u}(\mathbb{R}^{n})$ and for all $x\in\mathbb{R}^{n}$, we have $\|T_{x}f\|_{L_{\Phi}^{u}(\mathbb{R}^{n})}\leq u(x)\|f\|_{L_{\Phi}^{u}(\mathbb{R}^{n})},$ where $T_{x}f(y)=f(y-x)$.*

(2)* If $f\in L_{\Phi}^{u}(\mathbb{R}^{n})$ and $f\neq 0$, then there exists a constant $C>0$ (depends on $f$) such that*

[TABLE]

Proof.

(1) Let $f\in L_{\Phi}^{u}(\mathbb{R}^{n})$ and $T_{x}f(y)=f(y-x)$, then

[TABLE]

Observe that (by setting $v:=y-x$), we have

[TABLE]

This shows that $\|T_{x}f\|_{L_{\Phi}^{u}(\mathbb{R}^{n})}\leq u(x)\|f\|_{L_{\Phi}^{u}(\mathbb{R}^{n})}$.

(2) Let $f\in L_{\Phi}^{u}(\mathbb{R}^{n})$ and $f\neq 0$, then there exist a constant $C>0$ (depends on $f$) such that $\|f\|_{L_{\Phi}^{u}(\mathbb{R}^{n})}\leq C.$ By Lemma 2.4 (1), then we have

[TABLE]

for every $x\in{\mathbb{R}}^{n}$.

Since $f(x)\neq 0$ for every $x\in{\mathbb{R}}^{n}$,we have $\|T_{x}f\|_{L_{\Phi}^{u}(\mathbb{R}^{n})}>0$. Observe that

[TABLE]

This shows that $\frac{u(x)\|f\|_{{}_{L_{\Phi}(\mathbb{R}^{n})}}}{\mathop{\sup}\limits_{v\in R^{n}}u(-v)}\leq\|T_{x}f\|_{L_{\Phi}^{u}(\mathbb{R}^{n})}$.

Choose, C:=\mathop{\max}\Bigl{\{}C_{1},\frac{\mathop{\sup}\limits_{v\in{\mathbb{R}}^{n}}u(-v)}{\|f\|_{L_{\Phi}(\mathbb{R}^{n})}}\Bigr{\}}. Hence, we conclude that $\frac{u(x)}{C}\leq\|T_{x}f\|_{L_{\Phi}^{u}(\mathbb{R}^{n})}\leq Cu(x)$, as desired. ∎

Now, we present the sufficient and necessary conditions for the inclusion properties of weighted Orlicz spaces $L_{\Phi_{1}}^{u_{1}}(\mathbb{R}^{n})$ and $L_{\Phi_{2}}^{u_{2}}(\mathbb{R}^{n})$ with respect to Young functions $\Phi_{1},\Phi_{2}$ and weights $u_{1},u_{2}$.

Theorem 2.5**.**

Let $\Phi_{1},\Phi_{2}$ be Young functions such that $\Phi_{1}\prec\Phi_{2}$ and $u_{1},u_{2}$ are measurable functions such that $u_{i}(x+y)\leq u_{i}(x)\cdot u_{i}(y)$ for every $x,y\in{\mathbb{R}}^{n}$, where $i=1,2$. Then the following statements are equivalent:

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

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

(3)* There exists a constant $C>0$ such that $\|f\|_{L_{\Phi_{1}}^{u_{1}}(\mathbb{R}^{n})}\leq C\|f\|_{L_{\Phi_{2}}^{u_{2}}(\mathbb{R}^{n})}$, for every $f\in L_{\Phi_{2}}^{u_{2}}(\mathbb{R}^{n})$.*

Proof.

Assume that (1) holds. Let $f$ be an element of $L_{\Phi_{2}}^{u_{2}}(\mathbb{R}^{n})$. Since $\Phi_{1}\prec\Phi_{2}$ and $u_{1}\preceq u_{2}$, there exists constants $C_{1},C_{2}>0$ such that $\Phi_{1}(t)\leq\Phi_{2}(C_{1}t)$ for all $t>0$ and $u_{1}(x)\leq C_{2}u_{2}(x)$ for every $x\in{\mathbb{R}}^{n}$. Using a similar argument in the proof of Theorem 2.2 we have

[TABLE]

As before, we have that (2) and (3) are equivalent. It thus remains to show that (3) implies (1). Assume that (3) holds. By Lemma 2.4, we have

[TABLE]

for every $x\in\mathbb{R}^{n}$. So, we obtain $u_{1}\preceq u_{2}$.∎

Note that, for $\Phi_{1}(x)=\Phi_{2}(x)$ for every $x\in{\mathbb{R}}^{n}$, Theorem 2.5 reduces to Theorem 1.1.