A Note on Inclusions of Discrete Morrey Spaces

TL;DR

This paper investigates inclusion relations among discrete Morrey spaces, providing necessary conditions, generalizations, and properness of these relations, along with connections to classical Morrey spaces and their weak type variants.

Contribution

It introduces new necessary conditions for inclusion, generalizes known properties, and explores the properness and connections of these relations in discrete Morrey spaces.

Findings

Established necessary inclusion conditions.

Proved a generalization of p-summable sequence spaces.

Showed all inclusion relations are proper.

Abstract

We give a necessary condition for inclusion relations between discrete Morrey spaces which can be seen as a complement of the results in \cite{GKS,HS2}. We also prove another inclusion property of discrete Morrey spaces which can be viewed as a generalization of the inclusion property of the spaces of $p$-summable sequences. Analogous results for weak type discrete Morrey spaces is also presented. In addition, we show that each of these inclusion relations is proper. Some connections between inclusion properties of discrete Morrey spaces and those of Morrey spaces are also discussed.

Equations

$$\underset{m\in \mathbb{Z},N\in {\mathbb{N}}_0}{\sup }|S_{m,N}{|}^{\frac{p}{q}-1}\sum _{j\in S_{m,N}}|x_j{|}^p<\infty ,$$

$$\Vert x{\Vert }_{{\ell }_q^p}:=\underset{m\in \mathbb{Z},N\in {\mathbb{N}}_0}{\sup }|S_{m,N}{|}^{\frac{1}{q}-\frac{1}{p}}{\left(\sum _{j\in S_{m,N}}|x_j{|}^p\right)}^{\frac{1}{p}}.$$

$${\ell }_q^{p_2}\subseteq {\ell }_q^{p_1}.$$

$${\ell }_{q_2}^{p_2}\subseteq {\ell }_{q_1}^{p_1}⟹q_2\le q_1$$

$${\ell }^{p_1}\subseteq {\ell }^{p_2}$$

$$\Vert x{\Vert }_{{\ell }_{\infty }^p}:=\underset{m\in \mathbb{Z},N\in {\mathbb{N}}_0}{\sup }|S_{m,N}{|}^{-\frac{1}{p}}{\left(\sum _{m\in \mathbb{Z},N\in {\mathbb{N}}_0}|x_j{|}^p\right)}^{\frac{1}{p}}<\infty .$$

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

$$\left(\frac{q}{p_1}-1\right)w+\frac{2q}{p_1}<v<\left(\frac{q}{p_2}-1\right)w+2.$$

$$x_j:=\{\begin{array}{ll}1,& |j|=0,1,2,\dots ,2^{v+w}\\ 1,& |j|=2^{k(v+w)},2^{k(v+w)}-2^{kw},2^{k(v+w)}-2(2^{kw}),2^{k(v+w)}-3(2^{kw}),\\ & \dots ,2^{k(v+w)}-(2^{k(v-2)})(2^{kw}),\mathrm{where}k\in \mathbb{N}\cap [k_0,\infty )\\ 0,& \mathrm{otherwise}.\end{array}$$

$$\sum _{|i|\le 2^{n(v+w)}}|x_i{|}^p=\left(1+2^{v+w+1}+2\sum _{k=k_0}^n(1+2^{k(v-2)})\right)\sim 2^{n(v-2)}$$

$$|S_{0,2^{n(v+w)}}{|}^{\frac{1}{q}-\frac{1}{p_1}}{\left(\sum _{j\in S_{0,2^{n(v+w)}}}|x_j{|}^{p_1}\right)}^{\frac{1}{p_1}}$$

$$\ge (3\cdot 2^{n(v+w)}{)}^{\frac{1}{q}-\frac{1}{p_1}}{2}^{\frac{n(v-2)}{p_1}}$$

$$=3^{\frac{1}{q}-\frac{1}{p_1}}{2}^{n\left(\frac{v+w}{q}-\frac{w}{p_1}-\frac{2}{p_1}\right)}.$$

$$\Vert x{\Vert }_{{\ell }_q^{p_1}}\gtrsim \underset{n\in \mathbb{N}}{\sup }{2}^{n\left(\frac{(v+w)}{q}-\frac{w}{p_1}-\frac{2}{p_1}\right)}=\infty ,$$

$${\ell }_n:=\max \{\ell \in \{0,1,2,\dots ,2^{n(v-2)-1}\}:2^{nw}\ell \le N_n\}.$$

$$|S_{m_n,N_n}{|}^{\frac{1}{q}-\frac{1}{p_2}}{\left(\sum _{j\in S_{m_n,N_n}}|x_j{|}^{p_2}\right)}^{\frac{1}{p_2}}$$

$$\lesssim (2^{nw}{\ell }_n{)}^{\frac{1}{q}-\frac{1}{p_2}}(4{\ell }_n{)}^{\frac{1}{p_2}}$$

$$=4^{\frac{1}{p_2}}{2}^{\frac{nw}{q}-\frac{nw}{p_2}}({\ell }_n{)}^{\frac{1}{q}}\lesssim 2^{n\left(\frac{v+w-2}{q}-\frac{w}{p_2}\right)}.$$

$$|S_{m_n,N_n}{|}^{\frac{1}{q}-\frac{1}{p_2}}{\left(\sum _{j\in S_{m_n,N_n}}|x_j{|}^{p_2}\right)}^{\frac{1}{p_2}}\lesssim 1.$$

$$|S_{0,N}{|}^{\frac{1}{q}-\frac{1}{p_2}}{\left(\sum _{j\in S_{0,N}}|x_j{|}^{p_2}\right)}^{\frac{1}{p_2}}\le {\left(\sum _{j\in S_{0,2^{k_0(v+w)}}}|x_j{|}^{p_2}\right)}^{\frac{1}{p_2}}=C{2}^{\frac{k_0(v-2)}{p_2}}.$$

$$2^{n(v+w)}<N<2^{(n+1)(v+w)}.$$

$$|S_{0,N}{|}^{\frac{1}{q}-\frac{1}{p_2}}{\left(\sum _{j\in S_{0,N}}|x_j{|}^{p_2}\right)}^{\frac{1}{p_2}}$$

$$\le C{\left(2^{n(v+w)}+1\right)}^{\frac{1}{q}-\frac{1}{p_2}}{2}^{\frac{(n+1)(v-2)}{p_2}}$$

$$\le C{2}^{n\left(\frac{w}{q}-\frac{w}{p_2}+\frac{v}{q}-\frac{2}{p_2}\right)}.$$

$$|S_{0,N}{|}^{\frac{1}{q}-\frac{1}{p_2}}{\left(\sum _{j\in S_{0,N}}|x_j{|}^{p_2}\right)}^{\frac{1}{p_2}}\lesssim 1.$$

$$\underset{m\in \mathbb{Z},N\in {\mathbb{N}}_0}{\sup }|S_{m,N}{|}^{\frac{1}{q}-\frac{1}{p_2}}{\left(\sum _{j\in S_{m,N}}|x_j{|}^{p_2}\right)}^{\frac{1}{p_2}}\lesssim 1,$$

$$\Vert x{\Vert }_{{\ell }^{\infty }}:=\underset{j\in \mathbb{Z}}{\sup }|x_j|<\infty .$$

$$\Vert x{\Vert }_{{\ell }_q^p}\ge |S_{m,0}{|}^{\frac{1}{q}-\frac{1}{p}}|x_m|.$$

$$\Vert y{\Vert }_{{\ell }^{\infty }}=\underset{j\in \mathbb{Z}}{\sup }|y_j|=1<\infty .$$

$$\Vert y{\Vert }_{{\ell }_q^p}=\underset{m\in \mathbb{Z},N\in \mathbb{N}}{\sup }|S_{m,N}{|}^{\frac{1}{q}-\frac{1}{p}}{\left(\sum _{j\in S_{m,N}}|y_j{|}^p\right)}^{\frac{1}{p}}=\underset{N\in {\mathbb{N}}_0}{\sup }(2N+1{)}^{\frac{1}{q}-\frac{1}{p}}(2N+1{)}^{\frac{1}{p}}=\underset{N\in {\mathbb{N}}_0}{\sup }(2N+1{)}^{\frac{1}{q}}=\infty .$$

$$\Vert x{\Vert }_{{\ell }^{\infty }}=\underset{j\in \mathbb{Z}}{\sup }|x_j|\le \Vert x{\Vert }_{{\ell }_{\infty }^p}.$$

$$\Vert x{\Vert }_{{\ell }_{\infty }^p}$$

$$\le \underset{m\in \mathbb{Z},N\in {\mathbb{N}}_0}{\sup }|S_{m,N}{|}^{-\frac{1}{p}}|S_{m,N}{|}^{\frac{1}{p}}\Vert x{\Vert }_{{\ell }^{\infty }}$$

$$=\Vert x{\Vert }_{{\ell }^{\infty }}.$$

Full text

**MSC (2010): 42B35, 46A45, 46B45.

Key words: Discrete Morrey spaces, inclusion properties, Morrey spaces.**

1 Introduction

Let $1\leq p\leq q<\infty$. The discrete Morrey space $\ell^{p}_{q}=\ell^{p}_{q}(\mathbb{Z})$, introduced in [3], is defined to be the set of all sequences $x=(x_{j})_{j\in\mathbb{Z}}$ such that

[TABLE]

where $\mathbb{N}_{0}:=\mathbb{N}\cup\{0\}$, $S_{m,N}:=\{m-N,\ldots,m,\ldots,m+N\}$, and $|S_{m,N}|=2N+1$. This space is a Banach space with the norm

[TABLE]

We remark that, for $p=q$, we have $\ell^{p}_{p}=\ell^{p}$. Moreover, it is shown in [3] that $\ell^{p}$ is a proper subset of $\ell^{p}_{q}$ whenever $1\leq p<q<\infty$. In the same paper, the authors also prove that if $1\leq p_{1}\leq p_{2}\leq q$, then

[TABLE]

In addition, it is shown that

[TABLE]

whenever $1\leq p_{1}\leq q_{1}<\infty$, $1\leq p_{2}\leq q_{2}<\infty$, and $p_{1}\leq p_{2}$. Recently, these inclusion results are extended to discrete Morrey spaces on $\mathbb{Z}^{d}$ in [7]. Moreover, the authors in [7] give a necessary condition for inclusion relations of discrete Morrey spaces on $\mathbb{Z}^{d}$. They also prove that the embedding (1.2) is never compact.

One of the aims of this present paper is to give an alternative proof of a necessary condition of inclusion of discrete Morrey spaces. In particular, we show that $p_{1}\leq p_{2}$ is necessary for the inclusion relation (1.2). Besides (1.2), we also discuss another inclusion property of discrete Morrey spaces which generalizes

[TABLE]

for every $p_{1}\leq p_{2}$, namely $\ell^{p_{1}}_{q_{1}}\subseteq\ell^{p_{2}}_{q_{2}}$ whenever $p_{1}\leq p_{2}$ and $\frac{q_{1}}{q_{2}}\leq\frac{p_{1}}{p_{2}}$. We prove this result by using the fact that $\ell^{\infty}$ contains $\ell^{p}_{q}$ for all $1\leq p\leq q<\infty$.

We also see later that $\ell^{\infty}$ coincides with the space $\ell^{p}_{\infty}$ where $p<\infty$. Here, $\ell^{p}_{\infty}$ is defined to be the set of all sequences $x=(x_{j})_{j\in\mathbb{Z}}$ for which

[TABLE]

In addition to the results for discrete Morrey spaces, we give the corresponding results for weak type discrete Morrey spaces in Section 2.2. The proper inclusion relation between a discrete Morrey space and its weak type is given in Section 2.3.

In Section 3, we discuss the relation between discrete Morrey spaces and ‘continuous’ Morrey spaces on $\mathbb{R}$. Recall that, for $1\leq p\leq q<\infty$, the Morrey space $\mathcal{M}^{p}_{q}(\mathbb{R}^{n})$ is defined to be the set of all functions $f\in L^{p}_{\rm loc}(\mathbb{R}^{n})$ such that the norm

[TABLE]

is finite. We apply the result in [8] to recover the inclusion properties of discrete Morrey spaces from those of Morrey spaces. We also reprove some necessary condition for the inclusion relations between Morrey spaces in [2] by combining the results in Section 2 and [8]. Our main result in this section is a necessary condition for the proper inclusion relation between weak type discrete Morrey spaces and discrete Morrey spaces.

Let us mention some previous works related to this paper. The inclusion properties of discrete Morrey spaces, their weak type spaces and their generalization are initially studied in [3]. The analogous results on Morrey spaces ${\mathcal{M}}^{p}_{q}(\mathbb{R}^{n})$ can be found in [1, 2, 6, 9, 10, 11]. The boundedness of the Hardy-Littlewood maximal operator on discrete Morrey spaces is investigated in [4].

Throughout this paper, we denote by $C$ a positive constant which is independent of the sequence $x=(x_{j})_{j\in\mathbb{Z}}$ and its value may be different from line to line. We write $A\lesssim B$ if there exists a positive constant $C$ such that $A\leq CB$. Meanwhile, $A\gtrsim B$ means $B\lesssim A$. In addition, we denote by $A\sim B$ if $A\lesssim B$ and $B\lesssim A$.

2 Main results

In this section, we consider three types of inclusion properties of discrete Morrey spaces.

2.1 Inclusion property of discrete Morrey spaces

Our first result is a necessary condition for the inclusion property of the first kind.

Theorem 2.1**.**

Let $1\leq p_{1}\leq q<\infty$ and $1\leq p_{2}\leq q<\infty$. If $\ell^{p_{2}}_{q}\subseteq\ell^{p_{1}}_{q}$, then $p_{1}\leq p_{2}$.

Remark 2.2*.*

As mentioned before, a similar result for discrete Morrey spaces on $\mathbb{Z}^{d}$ can be found in [7]. Here, we give a different proof of the necessary condition for this inclusion property.

Proof.

Assume to the contrary that $1\leq p_{2}<p_{1}<q$. Choose $v,w\in\mathbb{N}$ such that

[TABLE]

Let $k_{0}$ be the smallest positive integer such that $\displaystyle 1-\frac{1}{2^{2k_{0}}}>\frac{1}{2^{v+w-1}}$. Define $x=(x_{j})_{j\in\mathbb{Z}}$ by

[TABLE]

Let $n\in\mathbb{N}\cap[k_{0},\infty)$. Observe that, for every $1\leq p<\infty$ we have

[TABLE]

Therefore, for $p=p_{1}$, we have

[TABLE]

According to (2.1), we see that $\displaystyle\frac{(v+w)}{q}-\frac{w}{p_{1}}-\frac{2}{p_{1}}>0$, so that $2^{n\left(\frac{(v+w)}{q}-\frac{w}{p_{1}}-\frac{2}{p_{1}}\right)}\to\infty$ as $n\rightarrow\infty$. Combining this with (2.1), we have

[TABLE]

which tells us that $x\notin\ell^{p_{1}}_{q}$.

We shall show that $x\in\ell^{p_{2}}_{q}$. For every $n\in\mathbb{N}\cap[k_{0},\infty)$, define $m_{n}:=2^{n(v+w)}-2^{n(v+w-2)-1}$ and let $N_{n}\in\{0,1,2,\ldots,2^{n(v+w-2)-1}\}$. We also define

[TABLE]

Observe that

[TABLE]

Since $\frac{(v+w-2)}{q}-\frac{w}{p_{2}}<0$, we have

[TABLE]

If $N\leq 2^{k_{0}(v+w)}$, then

[TABLE]

On the other hand, if $N>2^{k_{0}(v+w)}$, then there exists $n\in\mathbb{N}$ such that

[TABLE]

Hence we obtain

[TABLE]

Note that (2.1) implies $\displaystyle\frac{w}{q}-\frac{w}{p_{2}}+\frac{v}{q}-\frac{2}{p_{2}}<0$. As a consequence of this inequality and (2.1), we have

[TABLE]

Combining (2.6) and (2.9), we get

[TABLE]

which means that $\|x\|_{\ell^{p_{2}}_{q}}<\infty$. Hence, $x\in\ell^{p_{2}}_{q}$ but $x\notin\ell^{p_{1}}_{q}$. This contradicts $\ell^{p_{2}}_{q}\subseteq\ell^{p_{1}}_{q}$. ∎

Remark 2.3*.*

Note that the condition $p_{1}\leq p_{2}$ and $q_{2}\leq q_{1}$ is not a necessary condition for the inclusion $\ell^{p_{2}}_{q_{2}}\subseteq\ell^{p_{1}}_{q_{1}}$ because $\ell^{p_{2}}_{p_{2}}\subseteq\ell^{p_{1}}_{p_{1}}$ when $p_{2}<p_{1}$.

We now move on to the inclusion property of the second kind. As a preparation, we show that $\ell^{\infty}$ contains $\ell^{p}_{q}$ where $1\leq p\leq q<\infty.$ Recall that $\ell^{\infty}=\ell^{\infty}(\mathbb{Z})$ is the set of all sequences $x=(x_{j})_{j\in\mathbb{Z}}$ such that

[TABLE]

Theorem 2.4**.**

If $1\leq p\leq q<\infty$, then (i) $\ell^{p}_{q}\subset\ell^{\infty}$ and (ii) $\ell^{p}_{\infty}=\ell^{\infty}.$ Moreover, the inclusion $\ell^{p}_{q}\subset\ell^{\infty}$ is proper.

Remark 2.5*.*

The inclusion $\ell^{p}_{q}\subset\ell^{\infty}$ is obtained in [8]. For the reader’s convenience, we provide our proof and also show that the inclusion is proper.

Proof.

(i) Suppose that $1\leq p\leq q<\infty$. Take an arbitrary $x\in\ell^{p}_{q}$. Note that for every $m\in\mathbb{Z}$, we have

[TABLE]

Because $|S_{m,0}|=1$, we obtain $\|x\|_{\ell^{\infty}}=\sup_{j\in\mathbb{Z}}|x_{j}|\leq\|x\|_{\ell^{p}_{q}}$.

Next we shall show that $\ell^{\infty}\setminus\ell^{p}_{q}\neq\emptyset$. Define $(y_{j})_{j\in\mathbb{Z}}$ by $y_{j}:=1$ for every $j\in\mathbb{Z}$. We obtain

[TABLE]

On the other hand,

[TABLE]

Thus, $y\in\ell^{\infty}\setminus\ell^{p}_{q}$.

(ii) Now, take an arbitrary $x\in\ell^{p}_{\infty}$. For every $m\in\mathbb{Z}$, $\|x\|_{\ell^{p}_{\infty}}\geq|S_{m,0}|^{-\frac{1}{p}}|x_{m}|$ holds. Since $|S_{m,0}|=1$, we have

[TABLE]

Meanwhile, take an arbitrary $x\in\ell^{\infty}$. Observe that

[TABLE]

Therefore, $x\in\ell^{p}_{\infty}$. Hence, $\ell^{p}_{\infty}=\ell^{\infty}.$ ∎

Using the relation between $\ell^{p}_{q}$ and $\ell^{\infty}$ in Theorem 2.4, we shall prove the inclusion property of discrete Morrey spaces of the second type.

Theorem 2.6**.**

Let $1\leq p_{1}\leq q_{1}<\infty$ and $1\leq p_{2}\leq q_{2}<\infty$. If $p_{1}\leq p_{2}$ and $\frac{q_{1}}{q_{2}}\leq\frac{p_{1}}{p_{2}}$, then $\ \ell^{p_{1}}_{q_{1}}\subseteq\ell^{p_{2}}_{q_{2}}$ with

[TABLE]

Moreover, $\ell^{p_{2}}_{q_{2}}\setminus\ell^{p_{1}}_{q_{1}}\neq\emptyset$ whenever ($p_{1}<p_{2}$ and $\frac{q_{1}}{q_{2}}\leq\frac{p_{1}}{p_{2}}$) or ($p_{1}\leq p_{2}$ and $\frac{q_{1}}{q_{2}}<\frac{p_{1}}{p_{2}}$).

Proof.

Take an arbitrary $x\in\ell^{p_{1}}_{q_{1}}$. Then for every $m\in\mathbb{Z}$ and $N\in\mathbb{N}_{0}$, we have

[TABLE]

where we use Theorem 2.4 and the definition of $\ell^{p_{1}}_{q_{1}}$. Since $\frac{1}{q_{2}}-\frac{p_{1}}{q_{1}p_{2}}\leq 0$, we get

[TABLE]

and hence $x\in\ell^{p_{2}}_{q_{2}}$ with

[TABLE]

We now prove the second part of the theorem. Note that our assumption implies $q_{1}<q_{2}$. Define $(y_{j})_{j\in\mathbb{Z}}$ where $y_{0}:=1$ dan $y_{j}:=|j|^{-\frac{1}{q_{2}}}$ for every $j\in\mathbb{Z}-\{0\}$. Then, for every $N\in\mathbb{N}_{0}$ and for $k=1,2$, we have

[TABLE]

Therefore

[TABLE]

Meanwhile, we obtain

[TABLE]

Thus, $y\in\ell^{p_{2}}_{q_{2}}\setminus\ell^{p_{1}}_{q_{1}}$. This completes the proof. ∎

As a corollary of Theorem 2.6, we obtain the following result.

Theorem 2.7**.**

If $1\leq p\leq q_{1}<\infty$ and $1\leq p\leq q_{2}<\infty$ then $q_{1}\leq q_{2}$ if and only if $\ell^{p}_{q_{1}}\subseteq\ell^{p}_{q_{2}}$ with $\|\cdot\|_{\ell^{p}_{q_{2}}}\leq\|\cdot\|_{\ell^{p}_{q_{1}}}$. Furthermore, $\ell^{p}_{q_{1}}$ is a proper subset of $\ell^{p}_{q_{2}}$ whenever $1\leq p\leq q_{1}<q_{2}<\infty$.

Proof.

In view of Theorem 2.6, we only need to prove that the inclusion $\ell^{p}_{q_{1}}\subseteq\ell^{p}_{q_{2}}$ with $\|\cdot\|_{\ell^{p}_{q_{2}}}\leq\|\cdot\|_{\ell^{p}_{q_{1}}}$ imply $q_{1}\leq q_{2}$. Let $K\in\mathbb{N}$ and take an example $x=(x_{j})_{j\in\mathbb{Z}}$ where $x_{j}:=1$ for $|j|\leq K$ and $x_{j}:=0$ for $|j|>K$. Observe that

[TABLE]

Since $2K+1\geq 1$, we conclude that $q_{1}\leq q_{2}$. ∎

Remark 2.8*.*

The argument in the proof of Theorem 2.7 cannot be applied to ‘continuous’ Morrey spaces $\mathcal{M}^{p}_{q_{k}}(\mathbb{R}^{n})$, $k=1,2$, because the case $|B(a,r)|<1$ occurs in the formula of the $\mathcal{M}^{p}_{q_{k}}$-norm.

2.2 Incusion property of weak type discrete Morrey spaces

Let us recall the definition of weak type discrete Morrey spaces.

Definition 2.9**.**

Let $1\leq p\leq q<\infty$. The weak type discrete Morrey spaces $w\ell^{p}_{q}$ is defined to be the set of all sequences $x=(x_{j})_{j\in\mathbb{Z}}$ for which the quasi-norm

[TABLE]

is finite.

The quasi-norm in Definition 2.9 can be rewritten as

[TABLE]

where $1_{\{j\in\mathbb{Z}:|x_{j}|>\gamma\}}:=(1_{\{j\in\mathbb{Z}:|x_{j}|>\gamma\}}(j))_{j\in\mathbb{Z}}$ and

[TABLE]

Note that, for $p=q$, the space $w\ell^{p}_{p}$ is the weak type $\ell^{p}$ space. It is shown in [3, Example 3.1] that the space $\ell^{p}$ is a proper subset of $w\ell^{p}_{p}$. More general, the discrete Morrey space $\ell^{p}_{q}$ is a subset of $w\ell^{p}_{q}$ (see [3, Theorem 3.2]).

For the inclusion between weak type discrete Morrey spaces, it is proven in [3] that $w\ell^{p_{2}}_{q}\subseteq w\ell^{p_{1}}_{q}$ whenever $1\leq p_{1}\leq p_{2}\leq q<\infty$. Now we show that $p_{1}\leq p_{2}$ is a necessary condition for these inclusions.

Theorem 2.10**.**

Let $1\leq p_{1}\leq q<\infty$ and $1\leq p_{2}\leq q<\infty$. If $w\ell^{p_{2}}_{q}\subseteq w\ell^{p_{1}}_{q}$, then $p_{1}\leq p_{2}$.

Proof.

Assume to the contrary that $p_{1}>p_{2}$. Let $x=(x_{j})_{j\in\mathbb{Z}}$ be defined by (2.2). Since $x\in\ell^{p_{2}}_{q}$ and $\ell^{p_{2}}_{q}\subseteq w\ell^{p_{2}}_{q}$, we have $x\in w\ell^{p_{2}}_{q}$. If we can prove that $x\notin w\ell^{p_{1}}_{q}$, then we obtain a contradiction, so we must have $p_{1}\leq p_{2}$. Now we show that $x\notin w\ell^{p_{1}}_{q}$. Observe that

[TABLE]

Since either $x_{j}=0$ or $x_{j}=1$, we have

[TABLE]

Consequently,

[TABLE]

as desired. ∎

Similar with the (strong type) discrete Morrey spaces, we also show the inclusion property of weak type discrete Morrey spaces of the second kind.

Theorem 2.11**.**

Let $1\leq p_{1}\leq q_{1}<\infty$ and $1\leq p_{2}\leq q_{2}<\infty$. If $p_{1}\leq p_{2}$ and $\frac{q_{1}}{q_{2}}\leq\frac{p_{1}}{p_{2}}$, then $\ w\ell^{p_{1}}_{q_{1}}\subseteq w\ell^{p_{2}}_{q_{2}}$ with

[TABLE]

Moreover, $w\ell^{p_{2}}_{q_{2}}\setminus w\ell^{p_{1}}_{q_{1}}\neq\emptyset$ whenever ($p_{1}<p_{2}$ and $\frac{q_{1}}{q_{2}}\leq\frac{p_{1}}{p_{2}}$) or ($p_{1}\leq p_{2}$ and $\frac{q_{1}}{q_{2}}<\frac{p_{1}}{p_{2}}$).

Proof.

Let $x\in w\ell^{p_{1}}_{q_{1}}$. According to Theorem 2.6, we have

[TABLE]

which tells us that $x\in w\ell^{p_{2}}_{q_{2}}$. Hence, $w\ell^{p_{1}}_{q_{1}}\subseteq w\ell^{p_{2}}_{q_{2}}$ with $\|\cdot\|_{w\ell^{p_{2}}_{q_{2}}}\leq\|\cdot\|_{w\ell^{p_{1}}_{q_{1}}}$.

Now we prove the second part of this theorem. By our assumption, we have $q_{1}<q_{2}$. Let $y=(y_{j})_{j\in\mathbb{Z}}$ be defined as in the proof of Theorem 2.6. Note that $y\in\ell^{p_{2}}_{q_{2}}$. Therefore,

[TABLE]

Hence, $y\in w\ell^{p_{2}}_{q_{2}}$. On the other hand, for every $N\in\mathbb{N}$, we have

[TABLE]

Consequently,

[TABLE]

Hence, $y\notin w\ell^{p_{1}}_{q_{1}}$. Thus, $w\ell^{p_{2}}_{q_{2}}\setminus w\ell^{p_{1}}_{q_{1}}\neq\emptyset$. ∎

Necessary and sufficient conditions for the inclusion property of weak type discrete Morrey spaces of the second kind are presented in the following theorem.

Theorem 2.12**.**

Let $1\leq p\leq q_{1}<\infty$ and $1\leq p\leq q_{2}<\infty$. Then $q_{1}\leq q_{2}$ if and only if $w\ell^{p}_{q_{1}}\subseteq w\ell^{p}_{q_{2}}$ with $\|\cdot\|_{w\ell^{p}_{q_{2}}}\leq\|\cdot\|_{w\ell^{p}_{q_{1}}}$. Furthermore, $w\ell^{p}_{q_{1}}$ is a proper subset of $w\ell^{p}_{q_{2}}$ whenever $q_{1}<q_{2}$.

Proof.

If $q_{1}\leq q_{2}$, then by taking $p_{1}=p_{2}=p$ in Theorem 2.11, we get $w\ell^{p}_{q_{1}}\subseteq w\ell^{p}_{q_{2}}$ with $\|\cdot\|_{w\ell^{p}_{q_{2}}}\leq\|\cdot\|_{w\ell^{p}_{q_{1}}}$. Conversely, assume that $w\ell^{p}_{q_{1}}\subseteq w\ell^{p}_{q_{2}}$ with $\|\cdot\|_{w\ell^{p}_{q_{2}}}\leq\|\cdot\|_{w\ell^{p}_{q_{1}}}$. Let $x=(x_{j})_{j\in\mathbb{Z}}$ be defined by

[TABLE]

Then

[TABLE]

Therefore, $(2K+1)^{\frac{1}{q_{2}}-\frac{1}{q_{1}}}\leq 1$. Since $2K+1\geq 1$, we have $\displaystyle\frac{1}{q_{2}}-\frac{1}{q_{1}}\leq 0$, so that $q_{1}\leq q_{2}$. The second part of this theorem follows from Theorem 2.11 by taking $p_{1}=p_{2}=p$. ∎

2.3 Inclusion relation between discrete Morrey spaces and weak type discrete Morrey spaces

Theorem 2.13**.**

Let $1\leq p_{1}<p_{2}\leq q<\infty$. Then the inclusion $w\ell^{p_{2}}_{q}\subseteq\ell^{p_{1}}_{q}$ is proper.

Proof.

Let $x\in w\ell^{p_{2}}_{q}$. We shall show that there exists $C>0$ such that

[TABLE]

for every $m\in\mathbb{Z}$ and $N\in\mathbb{N}_{0}$. Observe that

[TABLE]

where

[TABLE]

and $R>0$ is chosen later. The estimate for $I_{1}(R)$ is

[TABLE]

Meanwhile, by using Definition 2.9 and $p_{1}<p_{2}$, we have

[TABLE]

Combining (2.13) and (2.3) and taking $\displaystyle R:=\left(\frac{p_{1}}{p_{2}-p_{1}}\right)^{\frac{1}{p_{2}}}\frac{\|x\|_{w\ell^{p_{2}}_{q}}}{(2N+1)^{\frac{1}{q}}}$, we get

[TABLE]

Hence, (2.12) follows by taking $p_{1}$-th root of (2.15) and multiplying by $|S_{m,N}|^{\frac{1}{q}-\frac{1}{p_{1}}}$. Since (2.12) holds for arbitrary $m\in\mathbb{Z}$ and $N\in\mathbb{N}_{0}$, we have $x\in\ell^{p_{1}}_{q}$ with

[TABLE]

as desired.

We shall now prove that the inclusion is proper. Choose $v,w\in\mathbb{N}$ such that

[TABLE]

Let $k_{0}$ be the smallest positive integer such that $\displaystyle 1-\frac{1}{2^{2k_{0}}}>\frac{1}{2^{v+w-1}}$. Define $x=(x_{j})_{j\in\mathbb{Z}}$ by the formula

[TABLE]

By the same calculation as in the proof of Theorem 2.1, we obtain $x\in\ell^{p_{1}}_{q}$ but $x\notin\ell^{p_{2}}_{q}$. Moreover, repeating the calculation in the proof of Theorem 2.10, we obtain $\|x\|_{w\ell^{p_{2}}_{q}}\geq\frac{1}{2}\|x\|_{\ell^{p_{2}}_{q}}$, so that $x\notin w\ell^{p_{2}}_{q}$. This shows that $x\in\ell^{p_{1}}_{q}\setminus w\ell^{p_{2}}_{q}$. Thus, we conclude that the inclusion $w\ell^{p_{2}}_{q}\subseteq\ell^{p_{1}}_{q}$ is proper. ∎

3 Relation between discrete Morrey spaces and their continuous counterpart

In this section, we discuss the relation between the inclusion property of discrete Morrey spaces and that of Morrey spaces. In particular, we reprove the inclusion property of discrete Morrey spaces by using the inclusion property of Morrey spaces, and we recover some necessary conditions for the inclusion property of Morrey spaces. We also give a necessary condition for the proper inclusion relation between discrete Morrey spaces and weak type discrete Morrey spaces. Our results are based on some recent results in [8].

We now recall some definitions and notation. Let $1\leq p\leq q<\infty$. The Morrey space $\mathcal{M}^{p}_{q}(\mathbb{R})$ is defined to be the set of all functions $f\in L^{p}_{\rm loc}(\mathbb{R})$ for which

[TABLE]

is finite. The weak Morrey space $w\mathcal{M}^{p}_{q}(\mathbb{R})$ is defined to be the set of all measurable functions $f$ on $\mathbb{R}$ such that the quasi-norm

[TABLE]

is finite. From these definitions, it is clear that $\mathcal{M}^{p}_{q}(\mathbb{R})\subseteq w\mathcal{M}^{p}_{q}(\mathbb{R})$. The relation between $\ell^{p}_{q}$ and $\mathcal{M}^{p}_{q}(\mathbb{R})$ is given as follows. For every sequence $x=(x_{j})_{j\in\mathbb{Z}}$, define a function $\overline{x}:\mathbb{R}\to[0,\infty)$ by

[TABLE]

Then, the discrete Morrey space $\ell^{p}_{q}$ can be realized as a closed subspace of $\mathcal{M}^{p}_{q}(\mathbb{R})$ in the following sense.

Theorem 3.1**.**

[8, Theorem 2.1]* Let $1\leq p\leq q<\infty$. Then, there exist positive constants $C_{1}$ and $C_{2}$ such that*

[TABLE]

for every $x\in\ell^{p}_{q}$ and

[TABLE]

for every sequence $y=(y_{j})_{j\in\mathbb{Z}}$.

The analogous result for weak type discrete Morrey spaces is presented in the following.

Theorem 3.2**.**

[8, Theorem 2.3]* Let $1\leq p\leq q<\infty$. Then, there exist positive constants $C_{1}$ and $C_{2}$ such that for every $x\in w\ell^{p}_{q}$ and for every sequence $y=(y_{j})_{j\in\mathbb{Z}}$ the inequalities*

[TABLE]

and

[TABLE]

hold.

We apply Theorems 3.1 and 3.2 with inclusion of Morrey spaces to recover the inclusion of discrete Morrey spaces obtained in [3].

Corollary 3.3**.**

[3]* Let $1\leq p_{1}\leq p_{2}\leq q<\infty$. Then $\ell^{p_{2}}_{q}\subseteq\ell^{p_{1}}_{q}$ and $w\ell^{p_{2}}_{q}\subseteq w\ell^{p_{1}}_{q}$.*

Proof.

Our proof is an alternative to that in [3]. Let $x\in\ell^{p_{2}}_{q}$. Then, it follows from (3.1) that $\overline{x}\in\mathcal{M}^{p_{2}}_{q}$. Since $p_{1}\leq p_{2}$, we have $\mathcal{M}^{p_{2}}_{q}\subseteq\mathcal{M}^{p_{1}}_{q}$, so that $\overline{x}\in\mathcal{M}^{p_{1}}_{q}$. According to (3.1), we have $x\in\ell^{p_{1}}_{q}$. Thus, $\ell^{p_{2}}_{q}\subseteq\ell^{p_{1}}_{q}$. To prove that $w\ell^{p_{2}}_{q}\subseteq w\ell^{p_{1}}_{q}$, let $y\in w\ell^{p_{2}}_{q}$. Then, as a consequence of (3.3), we have $\overline{y}\in w\mathcal{M}^{p_{2}}_{q}$. The inclusion $w\mathcal{M}^{p_{2}}_{q}\subseteq w\mathcal{M}^{p_{1}}_{q}$ implies $\overline{y}\in w\mathcal{M}^{p_{1}}_{q}$. By virtue of (3.4), we have $y\in w\ell^{p_{1}}_{q}$. Thus, $w\ell^{p_{2}}_{q}\subseteq w\ell^{p_{1}}_{q}$. ∎

As a consequence of Theorems 2.1, 2.10, 3.1, and 3.2, we may recover a necessary condition for the inclusion property of Morrey spaces and that of weak Morrey spaces obtained in [2].

Corollary 3.4**.**

[2, Theorem 1.6]* Let $1\leq p_{1}\leq q<\infty$ and $1\leq p_{2}\leq q<\infty$.*

If $\mathcal{M}^{p_{2}}_{q}\subseteq\mathcal{M}^{p_{1}}_{q}$, then $p_{1}\leq p_{2}$. 2. 2.

If $w\mathcal{M}^{p_{2}}_{q}\subseteq w\mathcal{M}^{p_{1}}_{q}$, then $p_{1}\leq p_{2}$.

Proof.

Let $x\in\ell^{p_{2}}_{q}$. Then, according to Theorem 3.1, we have $\overline{x}\in\mathcal{M}^{p_{2}}_{q}$. Since $\mathcal{M}^{p_{2}}_{q}\subseteq\mathcal{M}^{p_{1}}_{q}$, we obtain $\overline{x}\in\mathcal{M}^{p_{1}}_{q}$. This fact and (3.2) imply $x\in\ell^{p_{1}}_{q}$. Therefore, $\ell^{p_{2}}_{q}\subseteq\ell^{p_{1}}_{q}$. Consequently, by virtue of Theorem 2.1, we conclude that $p_{1}\leq p_{2}$. 2. 2.

Let $x\in w\ell^{p_{2}}_{q}$. By virtue of Theorem 3.2, we have $\overline{x}\in w\mathcal{M}^{p_{2}}_{q}$, so that $\overline{x}\in w\mathcal{M}^{p_{1}}_{q}$. Combining this with (3.4), we obtain $x\in w\ell^{p_{1}}_{q}$. Consequently, $\ell^{p_{2}}_{q}\subseteq\ell^{p_{1}}_{q}$. Thus, the conclusion follows from Theorem 2.10.

∎

Finally, we present our main result in this section, namely a necessary condition for the proper inclusion relation between weak type discrete Morrey spaces and discrete Morrey spaces.

Theorem 3.5**.**

Let $1\leq p_{1}\leq q<\infty$ and $1\leq p_{2}\leq q<\infty$. If the inclusion $w\ell^{p_{2}}_{q}\subseteq\ell^{p_{1}}_{q}$ is proper, then $p_{1}<p_{2}$.

Remark 3.6*.*

Note that Theorem 3.5 can be viewed as a complement to Theorem 2.13. The analogous result in the case of Morrey spaces can be found in [2, 5].

Proof of Theorem 3.5.

Assume to the contrary that $p_{1}\geq p_{2}$. Let $x\in\ell^{p_{1}}_{q}\setminus w\ell^{p_{2}}_{q}$. It follows from (3.1) that $\overline{x}\in\mathcal{M}^{p_{1}}_{q}$. Since $p_{1}\geq p_{2}$, we have $\mathcal{M}^{p_{1}}_{q}\subseteq\mathcal{M}^{p_{2}}_{q}$, so that $\overline{x}\in\mathcal{M}^{p_{2}}_{q}$. Combining this with (3.2), we get $x\in\ell^{p_{2}}_{q}$. Consequently, $x\in w\ell^{p_{2}}_{q}$. This contradicts the fact that $x\in\ell^{p_{1}}_{q}\setminus w\ell^{p_{2}}_{q}$. Thus, we must have $p_{1}<p_{2}$. ∎

Acknowledgement. The first and second authors are supported by P3MI–ITB Research and Innovation Program 2018.