Proper inclusions of Morrey spaces

TL;DR

This paper investigates the precise nature of inclusions among Morrey and weak Morrey spaces, establishing their properness and providing necessary conditions, thereby refining existing mathematical understanding of these function spaces.

Contribution

It proves that all inclusions between Morrey and weak Morrey spaces are proper and offers necessary conditions, refining previous inclusion results.

Findings

Inclusions between Morrey and weak Morrey spaces are proper.

Proper inclusion between a Morrey space and a weak Morrey space is demonstrated.

Necessary conditions for each inclusion are established.

Abstract

In this paper, we prove that the inclusions between Morrey spaces, between weak Morrey spaces, and between a Morrey space and a weak Morrey space are all proper. The proper inclusion between a Morrey space and a weak Morrey space is established via the unboundedness of the Hardy-Littlewood maximal operator on Morrey spaces of exponent 1. In addition, we also give a necessary condition for each inclusion. Our results refine previous inclusion properties studied in [Gunawan et al, \emph{Math. Nachr.} {\bf 290} (2017), 332--340].

Equations

$$\Vert f{\Vert }_{{\mathcal{M}}_q^p}:=\underset{a\in {\mathbb{R}}^d,r>0}{\sup }|B(a,r){|}^{\frac{1}{q}}{\left(\frac{1}{|B(a,r)|}{\int }_{B(a,r)}|f(y){|}^{p}dy\right)}^{\frac{1}{p}}<\infty .$$

$$L^q={\mathcal{M}}_q^q\subseteq {\mathcal{M}}_q^{p_2}\subseteq {\mathcal{M}}_q^{p_1}\subseteq {\mathcal{M}}_q^1$$

$$\Vert f{\Vert }_{w{\mathcal{M}}_q^p}:=\underset{\gamma >0}{\sup }\Vert \gamma {\chi }_{\{|f|>\gamma \}}{\Vert }_{{\mathcal{M}}_q^p}$$

$$w{\mathcal{M}}_q^{p_2}\subseteq w{\mathcal{M}}_q^{p_1}.$$

$$w{\mathcal{M}}_q^{p_2}\subseteq {\mathcal{M}}_q^{p_1}.$$

$$\frac{d{p}_1}{q}<\beta <\frac{d{p}_2}{q}$$

$$d-\beta =\frac{d(q-p_1)+d(q-p_2)}{2q}>0.$$

$${\int }_{B(0,k+1)\setminus B(0,k)}g(x)dx=|B(0,r_k)\setminus B(0,k)|.$$

$$f(x):={\chi }_{B(0,1)}(x)+\sum _{k=1}^{\infty }{\chi }_{B(0,r_k)\setminus B(0,k)}(x).$$

$${\int }_{B(a,r)}|f(x){|}^{p}dx\le {\int }_{B(0,r)}|f(x){|}^{p}dx$$

$${\int }_{B(0,r)}|f(x){|}^{p}dx={\int }_{B(0,r)}|f(x)|dx\le {\int }_{B(0,2r)}g(x)dx,$$

$${\int }_{B(0,r)}|f(x){|}^{p}dx\le {\int }_{B(0,2r)}|x{|}^{-\beta }dx=C{r}^{d-\beta }$$

$${\int }_{B(0,r)}|f(x){|}^{p}dx\ge {\int }_{B(0,r)\setminus B(0,1)}|x{|}^{-\beta }dx=C(r^{d-\beta }-1)\ge C\left(1-\frac{1}{2^{d-\beta }}\right)r^{d-\beta }.$$

$$|B(0,r){|}^{\frac{1}{q}-\frac{1}{p_1}}{\left({\int }_{B(0,r)}|f(x){|}^{p_1}dx\right)}^{\frac{1}{p_1}}\le C{r}^{\frac{d}{q}-\frac{d}{p_1}}{r}^{\frac{d}{p_1}-\frac{\beta }{p_1}}=C{r}^{\frac{d}{q}-\frac{\beta }{p_1}}\le C.$$

$$|B(0,r){|}^{\frac{1}{q}-\frac{1}{p_1}}{\left({\int }_{B(0,r)}|f(x){|}^{p_1}dx\right)}^{\frac{1}{p_1}}\le C{r}^{\frac{d}{q}-\frac{d}{p_1}}{\left({\int }_{B(0,r)}|f(x){|}^{p_1}dx\right)}^{\frac{1}{p_1}}\le C{r}^{\frac{d}{q}}\le C.$$

$$|B(0,r){|}^{\frac{1}{q}-\frac{1}{p_2}}{\left({\int }_{B(0,r)}|f(x){|}^{p_2}dx\right)}^{\frac{1}{p_2}}\ge C{r}^{\frac{d}{q}-\frac{d}{p_2}}{r}^{\frac{d-\beta }{p_2}}=C{r}^{\frac{d}{q}-\frac{\beta }{p_2}}.$$

$$\underset{a\in {\mathbb{R}}^d,r>0}{\sup }|B(a,r){|}^{\frac{1}{q}-\frac{1}{p_2}}{\left({\int }_{B(a,r)}|f(x){|}^{p_2}dx\right)}^{\frac{1}{p_2}}$$

$$\ge C\underset{r>2}{\sup }{r}^{\frac{d}{q}-\frac{\beta }{p_2}}=\infty .$$

$${\chi }_{\{|f|>\gamma \}}=\{\begin{array}{ll}0,& \gamma \ge 1,\\ f,& \gamma \in (0,1).\end{array}$$

$$\Vert f{\Vert }_{w{\mathcal{M}}_q^{p_2}}=\underset{\gamma \in (0,1)}{\sup }\gamma \Vert {\chi }_{\{|f|>\gamma \}}{\Vert }_{{\mathcal{M}}_q^{p_2}}=\underset{\gamma \in (0,1)}{\sup }\gamma \Vert f{\Vert }_{{\mathcal{M}}_q^{p_2}}=\Vert f{\Vert }_{{\mathcal{M}}_q^{p_2}}=\infty ,$$

$$\Vert f{\Vert }_{{\mathcal{M}}_q^p}=\Vert |f{|}^p{\Vert }_{{\mathcal{M}}_{\frac{q}{p}}^1}^{\frac{1}{p}}$$

$$\Vert f{\Vert }_{w{\mathcal{M}}_q^p}=\Vert |f{|}^p{\Vert }_{w{\mathcal{M}}_{\frac{q}{p}}^1}^{\frac{1}{p}}$$

$$\Vert f{\Vert }_{{\mathcal{M}}_q^p}=\underset{B}{\sup }{\left(|B{|}^{\frac{p}{q}-1}{\int }_B|f(x){|}^{p}dx\right)}^{\frac{1}{p}}=\Vert |f{|}^p{\Vert }_{{\mathcal{M}}_{\frac{q}{p}}^1}^{\frac{1}{p}}.$$

$$\Vert |f{|}^p{\Vert }_{w{\mathcal{M}}_{\frac{q}{p}}^1}^{\frac{1}{p}}=\underset{\gamma >0}{\sup }{\gamma }^{\frac{1}{p}}\Vert {\chi }_{\{|f{|}^p>\gamma \}}{\Vert }_{{\mathcal{M}}_{\frac{q}{p}}^1}^{\frac{1}{p}}=\underset{\gamma >0}{\sup }{\gamma }^{\frac{1}{p}}\Vert {\chi }_{\{|f|>{\gamma }^{\frac{1}{p}}\}}{\Vert }_{{\mathcal{M}}_q^p}=\Vert f{\Vert }_{w{\mathcal{M}}_q^p},$$

$$Mf(x):=\underset{r>0}{\sup }\frac{1}{|B(x,r)|}{\int }_{B(x,r)}|f(y)|dy,x\in {\mathbb{R}}^d.$$

$$\Vert Mg{\Vert }_{w{\mathcal{M}}_r^1}\le C\Vert g{\Vert }_{{\mathcal{M}}_r^1},$$

$$\Vert Mg{\Vert }_{{\mathcal{M}}_r^1}\le C^{\prime }\Vert Mg{\Vert }_{w{\mathcal{M}}_r^1}$$

$$\Vert Mg{\Vert }_{{\mathcal{M}}_r^1}\le C\Vert g{\Vert }_{{\mathcal{M}}_r^1}$$

$$\Vert f{\Vert }_{w{\mathcal{M}}_q^p}\le \Vert f{\Vert }_{{\mathcal{M}}_q^p}\le {\left(\frac{q{\omega }_{d-1}}{d(q-p)|B(0,1)|}\right)}^{\frac{1}{p}}\Vert f{\Vert }_{w{\mathcal{M}}_q^p},$$

$$f(x)$$

$$\le \frac{|B(0,|x|){|}^{\frac{1}{p}-\frac{1}{q}}\Vert f{\Vert }_{w{\mathcal{M}}_q^p}}{|B(0,|x|){|}^{\frac{1}{p}}}$$

Full text

**MSC (2010): 42B35, 46E30

Keywords: Morrey spaces, weak Morrey spaces, inclusion properties.**

1 Introduction

Morrey spaces were first introduced by C.B. Morrey in [7] in relation to the study of the solution of certain elliptic partial differential equations. For $1\leq p\leq q<\infty$, the Morrey space ${\mathcal{M}}^{p}_{q}={\mathcal{M}}^{p}_{q}({\mathbb{R}}^{d})$ is defined to be the set of all $f\in L^{p}_{\rm loc}({\mathbb{R}}^{d})$ such that

[TABLE]

Here, $B(a,r)$ is an open ball centered at $a$ with radius $r$, and $|B(a,r)|$ denotes its Lebesgue measure. Notice that, when $p=q$, one can recover the Lebesgue space $L^{p}=L^{p}({\mathbb{R}}^{d})$ as the special case of ${\mathcal{M}}^{p}_{q}$. See [9] for various spaces related to Morrey spaces. Many researchers have proved the boundedness of classical integral operators on Morrey spaces and their generalizations. See, for instance, [1, 2] and the references therein.

Concerning the Hardy-Littlewood maximal operator (defined in Section 3), one may prove its boundedness on Morrey spaces using the inclusion ${\mathcal{M}}_{q}^{p}\subseteq{\mathcal{M}}^{1}_{q}$. In general, we have the following inclusions of Morrey spaces

[TABLE]

provided that $1\leq p_{1}\leq p_{2}\leq q<\infty$. These inclusions may be obtained by applying Hölder’s inequality. Note that, for $1\leq p_{2}<q<\infty$, we have $f(x):=|x|^{-\frac{d}{q}}\in{\mathcal{M}}^{p_{2}}_{q}\setminus{\mathcal{M}}^{q}_{q}$. This tells us that the inclusion ${\mathcal{M}}^{q}_{q}\subseteq{\mathcal{M}}^{p_{2}}_{q}$ is proper for $1\leq p_{2}<q<\infty$.

Besides the ‘strong’ Morrey spaces, we also have weak Morrey spaces whose definitions are given as follows:

Definition 1.1**.**

Let $1\leq p\leq q<\infty$. A measurable functions $f$ on ${\mathbb{R}}^{d}$ is said to belong to the weak Morrey space $w{\mathcal{M}}^{p}_{q}=w{\mathcal{M}}^{p}_{q}({\mathbb{R}}^{d})$ if the quasi-norm

[TABLE]

is finite.

Note that, by using the inequality $\gamma\chi_{\{|f|>\gamma\}}\leq|f|$ for every $\gamma>0$, we have ${\mathcal{M}}^{p}_{q}\subseteq w{\mathcal{M}}^{p}_{q}$. The inclusion properties of weak Morrey spaces, generalized Morrey spaces, generalized weak Morrey spaces, and their necessary conditions were discussed in [4]. In particular, for the case of Morrey spaces and weak Morrey spaces, the results can be stated as follows:

Theorem 1.2**.**

[4]* For $1\leq p_{1}\leq p_{2}\leq q<\infty$, the following inclusion holds:*

[TABLE]

Further, if $p_{1}<p_{2}$, then

[TABLE]

In addition to the above inclusion relations of Morrey spaces, we have the following theorems.

Theorem 1.3**.**

Let $1\leq p_{1}<p_{2}<q<\infty$. Then each of the following inclusions is proper:

1. (i)

${\mathcal{M}}^{p_{2}}_{q}\subseteq{\mathcal{M}}^{p_{1}}_{q}$; 2. (ii)

$w{\mathcal{M}}^{p_{2}}_{q}\subseteq{\mathcal{M}}^{p_{1}}_{q}$; 3. (iii)

$w{\mathcal{M}}^{p_{2}}_{q}\subseteq w{\mathcal{M}}^{p_{1}}_{q}$.

Theorem 1.4**.**

Let $1\leq p\leq q<\infty$. Then the inclusion ${\mathcal{M}}^{p}_{q}\subseteq w{\mathcal{M}}^{p}_{q}$ is proper.

Remark 1.5*.*

The claim about the proper inclusion ${\mathcal{M}}^{p_{2}}_{q}\subseteq{\mathcal{M}}^{p_{1}}_{q}$ is stated in [4, p. 2] without proof. We shall see the detailed explanation of this claim in the proof of Theorem 1.3(i). In [4, Remark 2.4], the authors refer to [3] for the proper inclusion between the generalized Morrey space $L^{1,\phi}$ and the corresponding weak type space $wL^{1.\phi}$, where $\phi(t)=t^{-1}\log(3+t)$. Since $L^{1,\phi}\neq\mathcal{M}^{1}_{q}$ for this choice of $\phi$, Theorem 1.4 can be seen as a complement of the result in [3].

We also obtain the following necessary conditions for inclusion of Morrey spaces and weak Morrey spaces which can be seen as a refinement of some necessary conditions given in [4].

Theorem 1.6**.**

Let $1\leq p_{i}\leq q_{i}<\infty$ for $i=1,2$. Then the following implications hold:

1. (i)

${\mathcal{M}}^{p_{2}}_{q_{2}}\subseteq{\mathcal{M}}^{p_{1}}_{q_{1}}$* implies $q_{1}=q_{2}$ and $p_{1}\leq p_{2}$;* 2. (ii)

$w{\mathcal{M}}^{p_{2}}_{q_{2}}\subseteq w{\mathcal{M}}^{p_{1}}_{q_{1}}$* implies $q_{1}=q_{2}$ and $p_{1}\leq p_{2}$;* 3. (iii)

$w{\mathcal{M}}^{p_{2}}_{q_{2}}\subseteq{\mathcal{M}}^{p_{1}}_{q_{1}}$* implies $q_{1}=q_{2}$ and $p_{1}<p_{2}$.*

Remark 1.7*.*

A necessary and sufficient condition for inclusion of Morrey spaces on a bounded domain can be found in [10, Theorem 2.1] and [11]. The case of Morrey spaces on ${\mathbb{R}}^{d}$ is mentioned in [6, Eq. (3.9)] and the authors refer to [12, Satz 1.6]. However, we do not have the access to the paper, so that we do not know how the proof goes. See also [6, Corollary 3.14] for weighted version of Theorem 1.6. Here we present a proof of the necessary and sufficient condition for the inclusion property, which is different from and simpler than that in [10].

The organization of this paper is as follows. In the next section, we prove that for $1\leq p_{1}<p_{2}<q<\infty$ the set ${\mathcal{M}}^{p_{1}}_{q}\setminus{\mathcal{M}}^{p_{2}}_{q}$ is not empty. By the same example, we also show that for $1\leq p_{1}<p_{2}<q$ the inclusion $w{\mathcal{M}}^{p_{2}}_{q}\subseteq w{\mathcal{M}}^{p_{1}}_{q}$ is proper. In Section 3, we give the proof of Theorem 1.4 using the unboundedness of the Hardy-Littlewood maximal operator on Morrey spaces of exponent 1. The proof of Theorem 1.6 is given in the last section. Throughout this paper, we denote by $C$ a positive constant which is independent of the function $f$ and its value may be different from line to line.

2 The proof of Theorem 1.3

We shall first prove Theorem 1.3 (i) by constructing a function which belongs to ${\mathcal{M}}^{p_{1}}_{q}$ but not to ${\mathcal{M}}^{p_{2}}_{q}$, for $1\leq p_{1}<p_{2}<q<\infty$.

Proof of Theorem 1.3 (i).

Let $1\leq p_{1}<p_{2}<q<\infty$ and $\beta:=\frac{d(p_{1}+p_{2})}{2q}$. Then we have

[TABLE]

and

[TABLE]

Define $g(x):=\chi_{B(0,1)}(x)+\chi_{\mathbb{R}^{n}\setminus B(0,1)}(x)|x|^{-\beta}$. Then, for each $k\in\mathbb{N}$, we choose $r_{k}\in(k,k+1)$ such that

[TABLE]

Next define

[TABLE]

We shall show that $f\in{\mathcal{M}}^{p_{1}}_{q}\setminus{\mathcal{M}}^{p_{2}}_{q}$. First observe that

[TABLE]

for every $1\leq p<\infty,\ a\in\mathbb{R}^{d},\ r>0$. Now, for $1\leq p<\infty$ and $r>2$, we have

[TABLE]

so

[TABLE]

and

[TABLE]

Therefore, by substituting $p=p_{1}$ into (2.3) and recalling (2.1), we have

[TABLE]

On the other hand, for each $r\leq 2$, we have

[TABLE]

By combining (2.5) and (2.6) we conclude that $f\in{\mathcal{M}}^{p_{1}}_{q}$.

Meanwhile, by substituting $p=p_{2}$ into (2.4), we have

[TABLE]

Since $\frac{d}{q}-\frac{\beta}{p_{2}}>0$, we have

[TABLE]

Thus $f\notin{\mathcal{M}}^{p_{2}}_{q}$, and we are done. ∎

Theorem 1.3 (ii) and (iii) are proved by using the function $f$ from the proof of Theorem 1.3 (i) and its relation with the characteristic function of its level set. The detailed proof goes as follows:

Proof of Theorem 1.3 (ii)-(iii).

For $1\leq p_{1}<p_{2}<q<\infty$, let $f$ be defined by (2.2). Observe that

[TABLE]

This together with the fact that $f\notin{\mathcal{M}}^{p_{2}}_{q}$ gives

[TABLE]

and hence $f\in{\mathcal{M}}^{p_{1}}_{q}\setminus w{\mathcal{M}}^{p_{2}}_{q}$. Thus we have shown that $w{\mathcal{M}}^{p_{2}}_{q}\subseteq{\mathcal{M}}^{p_{1}}_{q}$ is a proper inclusion. Since ${\mathcal{M}}^{p_{1}}_{q}\subseteq w{\mathcal{M}}^{p_{1}}_{q}$, we also have $f\in w{\mathcal{M}}^{p_{1}}_{q}\setminus w{\mathcal{M}}^{p_{2}}_{q}$, so the inclusion (iii) is proper. ∎

3 The proof of Theorem 1.4

In order to prove Theorem 1.4, we need the following lemma.

Lemma 3.1**.**

Let $1\leq p\leq q<\infty$. Then

[TABLE]

for every $f\in{\mathcal{M}}^{p}_{q}$ and

[TABLE]

for every $f\in w{\mathcal{M}}^{p}_{q}$.

Proof.

We calculate

[TABLE]

By applying the first identity for $\chi_{\{|f|>\gamma^{\frac{1}{p}}\}}$, we have

[TABLE]

as desired. ∎

We also use the following fact about the unboundedness of the Hardy-Littlewood maximal operator $M$ on Morrey spaces of exponent 1. The operator $M$ maps a locally integrable function $f$ to $Mf$ which is given by

[TABLE]

Lemma 3.2**.**

The Hardy-Littlewood maximal operator $M$ is not bounded on the Morrey space ${\mathcal{M}}^{1}_{q}$ for $1<q<\infty$.

Remark 3.3*.*

Lemma 3.2 is a consequence of a necessary condition of the boundedness of $M$ on generalized Orlicz-Morrey spaces given in [8, Corollary 5.3]. The Morrey space ${\mathcal{M}}^{p}_{q}$ in this paper is recognized as the Orlicz-Morrey space $L^{(\Phi,\phi)}$ with $\Phi(t)=t^{p}$ and $\phi(t)=t^{-\frac{1}{q}}$. Based on [8, Corollary 5.3], the maximal operator $M$ is bounded on $L^{(\Phi,\phi)}$ if and only if $\Phi\in\nabla_{2}$ (that is, $\Phi(r)\leq\frac{1}{2k}\Phi(kr)$ for some $k\geq 1$). Clearly $\Phi(t)=t\notin\nabla_{2}$.

Now, we are ready to prove Theorem 1.4.

Proof of Theorem 1.4.

Let $1\leq p\leq q$. If $p=q$, then $f(x):=|x|^{-\frac{d}{q}}\in w{\mathcal{M}}^{p}_{q}\setminus{\mathcal{M}}^{p}_{q}$. So assume that $p<q$ and write $r=\frac{q}{p}$. In view of Lemma 3.1, it suffices for us to prove that ${\mathcal{M}}^{1}_{r}\subset w{\mathcal{M}}^{1}_{r}$ properly. Suppose to the contrary that ${\mathcal{M}}^{1}_{r}=w{\mathcal{M}}^{1}_{r}$. Since the Hardy-Littlewood maximal operator $M$ is bounded from $\mathcal{M}^{1}_{r}$ to $w\mathcal{M}^{1}_{r}$, we obtain

[TABLE]

for every $g\in{\mathcal{M}}^{1}_{r}$. Meanwhile, by the Closed Graph Theorem, there must exist a constant $C^{\prime}>0$ such that

[TABLE]

for every $g\in{\mathcal{M}}^{1}_{r}$. Combining the two inequalities, we obtain

[TABLE]

for every $g\in{\mathcal{M}}^{1}_{r}$. This tells us that $M$ is bounded on ${\mathcal{M}}^{1}_{r}$, which contradicts Lemma 3.2. Therefore, $w{\mathcal{M}}^{1}_{r}\setminus{\mathcal{M}}^{1}_{r}\neq\emptyset$, as desired. ∎

To conclude this section, we write a proposition which gives us a subset of weak Morrey spaces with norm equivalence between the Morrey norm $\|\cdot\|_{{\mathcal{M}}^{p}_{q}}$ and the weak Morrey quasi-norm $\|\cdot\|_{w{\mathcal{M}}^{p}_{q}}$.

Proposition 3.4**.**

Let $1\leq p<q<\infty$. Suppose that $f$ is a positive radial decreasing function in $w\mathcal{M}^{p}_{q}(\mathbb{R}^{d})$. Then $f\in\mathcal{M}^{p}_{q}(\mathbb{R}^{d})$ with

[TABLE]

that is, $\|f\|_{w{\mathcal{M}}^{p}_{q}}\sim\|f\|_{{\mathcal{M}}^{p}_{q}}$.

Proof.

Recall that, since $\gamma\chi_{\{|f|>\gamma\}}\leq|f|$ for every $\gamma>0$, we have $\|f\|_{w{\mathcal{M}}^{p}_{q}}\leq\|f\|_{{\mathcal{M}}^{p}_{q}}$. Next, let $x\in\mathbb{R}^{d}$. Since $\{y\in B(0,|x|):f(y)>f(x)\}=B(0,|x|)$, we have

[TABLE]

By combining the last estimate and

[TABLE]

where $\omega_{d-1}$ is the surface area of the unit sphere $\mathbb{S}^{d-1}$, we get

[TABLE]

Hence $\|f\|_{w{\mathcal{M}}^{p}_{q}}\sim\|f\|_{{\mathcal{M}}^{p}_{q}}$. ∎

4 The proof of Theorem 1.6

Proof of Theorem 1.6 (i).

It follows from the inclusion ${\mathcal{M}}^{p_{2}}_{q_{2}}\subseteq{\mathcal{M}}^{p_{1}}_{q_{1}}$ that

[TABLE]

for every $r>0$. Therefore

[TABLE]

for every $r>0$, which implies that $q_{1}=q_{2}$. Now choose \epsilon\in\bigl{(}0,\min\{\frac{dp_{1}}{q_{1}},\frac{dp_{2}}{q_{2}}\}\bigr{)}. For $j\in{\mathbb{N}}$, define $h_{j}(x):=\chi_{\{j\leq|x|\leq j+j^{-\epsilon}\}}(x)$, and for $K\in{\mathbb{N}}$ write $f(x):=\chi_{\{0\leq|x|<1\}}(x)+\sum_{j=1}^{K}h_{j}(x)$. Then

[TABLE]

Meanwhile, for each $L\in\mathbb{N},\ L\leq K$, we observe that

[TABLE]

Hence,

[TABLE]

By combining (4), (4.2), $q_{1}=q_{2}$, and $\|f\|_{{\mathcal{M}}^{p_{1}}_{q_{1}}}\leq C\|f\|_{{\mathcal{M}}^{p_{2}}_{q_{2}}}$, we get

[TABLE]

As this holds for every $K\in\mathbb{N}$, we conclude that $p_{1}\leq p_{2}$. ∎

Remark 4.1*.*

Note that the difference between the proof of Theorem 1.6 (i) and [4, Remark 3.4] is that we do not assume $p_{1}\leq p_{2}$.

Proof of Theorem 1.6 (ii).

By arguing as in the proof of Theorem 1.6 (i) and using the identities $\|\chi_{B(0,r)}\|_{w{\mathcal{M}}^{p_{1}}_{q_{1}}}=|B(0,r)|^{\frac{1}{q_{1}}}$ and $\|\chi_{B(0,r)}\|_{w{\mathcal{M}}^{p_{2}}_{q_{2}}}=|B(0,r)|^{\frac{1}{q_{2}}}$, we have $q_{1}=q_{2}$. Assume to the contrary that $p_{1}>p_{2}$. Define $f$ by (2.2). By a similar argument as in the proof of Theorem 1.3 (ii)-(iii), we have $f\in w{\mathcal{M}}^{p_{2}}_{q_{2}}$ but $f\notin w{\mathcal{M}}^{p_{1}}_{q_{1}}$, which contradicts $w{\mathcal{M}}^{p_{2}}_{q_{2}}\subseteq w{\mathcal{M}}^{p_{1}}_{q_{1}}$. Hence $p_{1}\leq p_{2}$. ∎

Remark 4.2*.*

Observe that unlike [4, Theorem 4.4 and Remark 4.5], the condition $p_{1}\leq p_{2}$ is not assumed in Theorem 1.6 (ii).

Proof of Theorem 1.6 (iii).

Since ${\mathcal{M}}^{p_{2}}_{q_{2}}\subseteq w{\mathcal{M}}^{p_{2}}_{q_{2}}$, we have ${\mathcal{M}}^{p_{2}}_{q_{2}}\subseteq{\mathcal{M}}^{p_{1}}_{q_{1}}$. Therefore, by virtue of Theorem 1.6 (ii), we have $q_{1}=q_{2}$ and $p_{1}\leq p_{2}$. Now, assume to the contrary that $p_{1}=p_{2}$. According to Theorem 1.4, there exists $f_{0}\in w{\mathcal{M}}^{p_{2}}_{q_{2}}$ such that $f_{0}\notin{\mathcal{M}}^{p_{1}}_{q_{1}}$. This contradicts $w{\mathcal{M}}^{p_{2}}_{q_{2}}\subseteq{\mathcal{M}}^{p_{1}}_{q_{1}}$. Thus $p_{1}<p_{2}$, as desired. ∎

Acknowledgement. The first and second authors are supported by P3MI – ITB Program 2017. We would like to thank the referees for their useful comments on the earlier version of this paper.