Spaces of Pointwise Multipliers on Morrey Spaces and Weak Morrey Spaces

Abstract

The spaces of pointwise multipliers on Morrey spaces are described in terms of Morrey spaces, their preduals, and vector-valued Morrey spaces introduced by Ho. This paper covers weak Morrey spaces as well. The result in the present paper completes the characterization of the earlier works of the first author’s papers written in 1997 and 2000, as well as Lemarié-Rieusset’s 2013 paper. As a corollary, the main result in the present paper shows that different quasi-Banach lattices can create the same vector-valued Morrey spaces. The goal of the present paper is to provide a complete picture of the pointwise multiplier spaces.

1. Introduction

The aim of this note is to consider spaces of pointwise multipliers on Morrey spaces and weak Morrey spaces. Our results supplement the ones in [1,2,3,4]. We state our main results in Section 2. Section 1 is devoted to the formulation of the results.

We denote by $L^0\left({\mathbb{R}}^n\right)$ the space of all measurable functions from ${\mathbb{R}}^n$ to $\mathbb{R}$ or $\mathbb{C}$. Let $E_1,E_2\subset L^0\left({\mathbb{R}}^n\right)$ be linear subspaces. We say that a function $g\in L^0\left({\mathbb{R}}^n\right)$ is a pointwise multiplier from $E_1$ to $E_2$, if the pointwise multiplication $f·g$ is in $E_2$ for any $f\in E_1$. We denote by $\mathrm{PWM}(E_1,E_2)$ the set of all pointwise multipliers from $E_1$ to $E_2$. We abbreviate this as $\mathrm{PWM}(E,E)$ to $\mathrm{PWM}\left(E\right)$.

For $p\in (0,\infty ]$, $L^p\left({\mathbb{R}}^n\right)$ denotes the usual Lebesgue space equipped with the norm ${\parallel ·\parallel }_{L^p}$. It is well known by Hölder’s inequality that:

$${\parallel f·g\parallel }_{L^{p_2}}\le {\parallel f\parallel }_{L^{p_1}}{\parallel g\parallel }_{L^{p_3}}(f\in L^{p_1}\left({\mathbb{R}}^n\right),g\in L^{p_3}\left({\mathbb{R}}^n\right))$$

for $1/p_2=1/p_1+1/p_3$ with $p_j\in (0,\infty ]$, $j=1,2,3$, so that $p_1\ge p_2$. This shows that:

$$\mathrm{PWM}(L^{p_1}\left({\mathbb{R}}^n\right),L^{p_2}\left({\mathbb{R}}^n\right))\supset L^{p_3}\left({\mathbb{R}}^n\right).$$

Conversely, we can show the reverse inclusion by using the uniform boundedness theorem or the closed graph theorem, that is,

$$\mathrm{PWM}(L^{p_1}\left({\mathbb{R}}^n\right),L^{p_2}\left({\mathbb{R}}^n\right))=L^{p_3}\left({\mathbb{R}}^n\right).$$

(1)

In particular, if $p_1=p_2=p$, then:

$$\mathrm{PWM}\left(L^p\left({\mathbb{R}}^n\right)\right)=L^{\infty }\left({\mathbb{R}}^n\right).$$

(2)

Meanwhile, if $p_1<p_2$, then:

$$\mathrm{PWM}(L^{p_1}\left({\mathbb{R}}^n\right),L^{p_2}\left({\mathbb{R}}^n\right))=\left\{0\right\}$$

(3)

since $L_{\mathrm{loc}}^{p_1}\left({\mathbb{R}}^n\right)$ is not included in $L_{\mathrm{loc}}^{p_2}\left({\mathbb{R}}^n\right)$. Proofs of (1) and (2) can be found in the work of Maligranda and Persson [5], Proposition 3 and Theorem 1. See also [4]. We do not prove (3) directly in this paper, but we mention that (3) is a direct consequence in Section 2. The goal of this note is to generalize this observation to Morrey spaces motivated by the works [2,3,4,6]. For $p\in (0,\infty )$ and $\lambda \in [0,n]$, the (classical/strong) Morrey space $L^{p,\lambda }\left({\mathbb{R}}^n\right)$ is defined as the space of $f\in L^0\left({\mathbb{R}}^n\right)$ such that:

$${\parallel f\parallel }_{L^{p,\lambda }}=\underset{Q\in \mathcal{Q}}{sup}{\left(\frac{1}{{\left|Q\right|}^{\frac{\lambda }{n}}}{\int }_Q{\left|f\left(y\right)\right|}^{p}dy\right)}^{1/p}<\infty ,$$

(4)

where $\mathcal{Q}$ stands for the set of all cubes in ${\mathbb{R}}^n$ whose edges are parallel to the coordinate axes. The parameter p serves to describe the local integrability of functions, while $\lambda$ describes the growth of $\underset{Q}{\int }{\left|f\left(y\right)\right|}^{p}dy$ in comparison with $\left|Q\right|$. It is easy to see that $L^{p,\lambda }\left({\mathbb{R}}^n\right)$ is a quasi-Banach space, which is subject to the scaling law ${\parallel f(t·)\parallel }_{L^{p,\lambda }}=t^{-\frac{n-\lambda }{p}}{\parallel f\parallel }_{L^{p,\lambda }}$ for all $f\in L^{p,\lambda }\left({\mathbb{R}}^n\right)$ and $t>0$. The notation $L^{p,\lambda }\left({\mathbb{R}}^n\right)$ was used, for instance, by Peetre [7]. The weak Morrey space $\mathrm{w}{L}^{p,\lambda }\left({\mathbb{R}}^n\right)$ is defined by a routine procedure: The weak Morrey space $\mathrm{w}{L}^{p,\lambda }\left({\mathbb{R}}^n\right)$ is the set of all measurable functions $f\in L^0\left({\mathbb{R}}^n\right)$ for which ${\parallel f\parallel }_{\mathrm{w}{L}^{p,\lambda }}=\underset{\lambda >0}{sup}\lambda \parallel {\chi }_{(\lambda ,\infty ]}{\left(\right|f\left|\right)\parallel }_{L^{p,\lambda }}$ is finite, where ${\chi }_A$ stands for the characteristic function of the set A.

To describe various properties of functions in $L^{p,\lambda }\left({\mathbb{R}}^n\right)$, it is sometimes convenient to use the notation ${\mathcal{M}}_q^p\left({\mathbb{R}}^n\right)$. Let $0<q\le p\le \infty$. Recall that for an $L_{\mathrm{loc}}^q\left({\mathbb{R}}^n\right)$-function f, its Morrey norm ${\parallel f\parallel }_{{\mathcal{M}}_q^p}$ is defined by:

$${\parallel f\parallel }_{{\mathcal{M}}_q^p}\equiv \underset{Q\in \mathcal{Q}}{sup}{\left|Q\right|}^{\frac{1}{p}-\frac{1}{q}}{\left({\int }_Q{\left|f\left(y\right)\right|}^q\mathrm{d}y\right)}^{\frac{1}{q}}.$$

(5)

The Morrey space ${\mathcal{M}}_q^p\left({\mathbb{R}}^n\right)$ is the set of all $L^q\left({\mathbb{R}}^n\right)$-locally integrable functions f for which the norm ${\parallel f\parallel }_{{\mathcal{M}}_q^p}$ is finite. Once again, by the routine procedure, we define the weak Morrey space $\mathrm{w}{\mathcal{M}}_q^p\left({\mathbb{R}}^n\right)$ as the set of all measurable functions $f\in L^0\left({\mathbb{R}}^n\right)$ for which ${\parallel f\parallel }_{\mathrm{w}{\mathcal{M}}_q^p}=\underset{\lambda >0}{sup}\lambda \parallel {\chi }_{(\lambda ,\infty ]}{\left(\right|f\left|\right)\parallel }_{{\mathcal{M}}_q^p}$ is finite. The parameter q describes the local integrability of functions. As is seen from the scaling law ${\parallel f(t·)\parallel }_{{\mathcal{M}}_q^p}=t^{-\frac{n}{p}}{\parallel f\parallel }_{{\mathcal{M}}_q^p}$ for all $f\in {\mathcal{M}}_q^p\left({\mathbb{R}}^n\right)$ and $t>0$, the parameter p in the Morrey space ${\mathcal{M}}_q^p\left({\mathbb{R}}^n\right)$ describes the global integrability. We remark that some authors swap the role of p and q; see [6] for example.

By (4) and (5), we have:

$$L^{q,\lambda }\left({\mathbb{R}}^n\right)={\mathcal{M}}_q^p\left({\mathbb{R}}^n\right),\mathrm{if}\lambda =n(1-q/p)\mathrm{or}\mathrm{equivalently}p=\frac{qn}{n-\lambda }.$$

Let $0<p<\infty$. It is noteworthy that $L^{p,0}\left({\mathbb{R}}^n\right)={\mathcal{M}}_p^p\left({\mathbb{R}}^n\right)=L^p\left({\mathbb{R}}^n\right)$ and that $L^{p,n}\left({\mathbb{R}}^n\right)={\mathcal{M}}_p^{\infty }\left({\mathbb{R}}^n\right)=L^{\infty }\left({\mathbb{R}}^n\right)$, so that Morrey spaces generalize Lebesgue spaces.

Let $0<q_i\le p_i<\infty$, $i=1,2$. We consider the space of pointwise multipliers from ${\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right)$ to ${\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right)$. A direct consequence of the closed graph theorem is that there exists a constant $M>0$ such that, for $f\in {\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right)$ and $g\in \mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$,

$${\parallel f·g\parallel }_{{\mathcal{M}}_{q_2}^{p_2}}\le M{\parallel f\parallel }_{{\mathcal{M}}_{q_1}^{p_1}}.$$

(6)

One naturally defines a norm on $\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$ by:

$${\parallel g\parallel }_{\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1},{\mathcal{M}}_{q_2}^{p_2})}\equiv inf\{M>0:\left(6\right)\mathrm{holds}\mathrm{for}\mathrm{all}f\in {\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right)\}$$

for $g\in \mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$. In the following, unless otherwise stated, the equality:

$$\mathrm{PWM}(E_1,E_2)=E_3$$

tacitly means the norm equivalence, that is a function $g\in L^0\left({\mathbb{R}}^n\right)$ belongs to $E_3$ if and only if $g\in \mathrm{PWM}(E_1,E_2)$, and in this case:

$${\parallel g\parallel }_{\mathrm{PWM}(E_1,E_2)}\sim {\parallel g\parallel }_{E_3},$$

where the implicit constants in ∼ do not depend on g. It follows from the scaling law of Morrey spaces that:

$${\parallel g(t·)\parallel }_{\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1},{\mathcal{M}}_{q_2}^{p_2})}=t^{-\frac{n}{p_2}+\frac{n}{p_1}}{\parallel g\parallel }_{\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1},{\mathcal{M}}_{q_2}^{p_2})}$$

for all $g\in \mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$.

An easy consequence of Hölder’s inequality is that:

$${\parallel f·g\parallel }_{L^{p_2,{\lambda }_2}}\le {\parallel f\parallel }_{L^{p_1,{\lambda }_1}}{\parallel g\parallel }_{L^{p_3,{\lambda }_3}},$$

if $p_j\in (0,\infty )$ and ${\lambda }_j\in [0,n]$, $j=1,2,3$ satisfy $1/p_2=1/p_1+1/p_3$ and ${\lambda }_2/p_2={\lambda }_1/p_1+{\lambda }_3/p_3$. This shows that:

$$\mathrm{PWM}(L^{p_1,{\lambda }_1}\left({\mathbb{R}}^n\right),L^{p_2,{\lambda }_2}\left({\mathbb{R}}^n\right))\supset L^{p_3,{\lambda }_3}\left({\mathbb{R}}^n\right).$$

(7)

Therefore, the aim of this note is to investigate the difference between the two spaces above. It is important to note that the scaling laws considered above force the parameters $p_1,p_2,p_3$ to satisfy ${\lambda }_2/p_2={\lambda }_1/p_1+{\lambda }_3/p_3$.

In this paper, we describe $\mathrm{PWM}(L^{p_1,{\lambda }_1}\left({\mathbb{R}}^n\right),L^{p_2,{\lambda }_2}\left({\mathbb{R}}^n\right))$ for all parameters $p_j\in (0,\infty )$ and ${\lambda }_j\in [0,n)$, $j=1,2$. Of interest is the case where ${\lambda }_2<{\lambda }_1$, since we already specified $\mathrm{PWM}(L^{p_1,{\lambda }_1}\left({\mathbb{R}}^n\right),L^{p_2,{\lambda }_2}\left({\mathbb{R}}^n\right))$ in the case ${\lambda }_1\le {\lambda }_2$ in our earlier paper [3].

Theorem 1

([3], Corollary 2.4). Let $p_i\in (0,\infty )$ and ${\lambda }_i\in [0,n)$, $i=1,2$. Then:

$$\begin{array}{c}\mathrm{PWM}(L^{p_1,{\lambda }_1}\left({\mathbb{R}}^n\right),L^{p_2,{\lambda }_2}\left({\mathbb{R}}^n\right))\hfill \\ \hfill \left\{\begin{array}{cc}=\left\{0\right\},\hfill & p_1<p_{2}orn+({\lambda }_1-n)\frac{p_2}{p_1}<{\lambda }_2,\hfill \\ =L^{\infty }\left({\mathbb{R}}^n\right),\hfill & p_2\le p_{1}and{\lambda }_2=n+({\lambda }_1-n)\frac{p_2}{p_1},\hfill \\ =L^{p_3,{\lambda }_3}\left({\mathbb{R}}^n\right),\hfill & p_2<p_{1}and{\lambda }_1\le {\lambda }_2<n+({\lambda }_1-n)\frac{p_2}{p_1},\hfill \\ ⫌L^{p_3,{\lambda }_3}\left({\mathbb{R}}^n\right),\hfill & p_2<p_{1}and0<{\lambda }_1\frac{p_2}{p_1}\le {\lambda }_2<{\lambda }_1,\hfill \\ ⫌\left\{0\right\},\hfill & p_2\le p_{1}and0\le {\lambda }_2<{\lambda }_1\frac{p_2}{p_1},\hfill \end{array}\right.\end{array}$$

where $p_3=p_1{p}_2/(p_1-p_2)$ and ${\lambda }_3=(p_1{\lambda }_2-p_2{\lambda }_1)/(p_1-p_2)$.

Let $p_i\in (0,\infty )$, $i=1,2$. As the endpoint cases of ${\lambda }_1=n$ or/and ${\lambda }_2=n$, we have:

$$\mathrm{PWM}(L^{p_1,{\lambda }_1}\left({\mathbb{R}}^n\right),L^{p_2,{\lambda }_2}\left({\mathbb{R}}^n\right))\left\{\begin{array}{cc}=\left\{0\right\},\hfill & 0\le {\lambda }_1<{\lambda }_2=n,\hfill \\ =L^{p_2,{\lambda }_2}\left({\mathbb{R}}^n\right),\hfill & 0\le {\lambda }_2\le {\lambda }_1=n.\hfill \end{array}\right.$$

We rephrase Theorem 1 as follows:

Theorem 2.

Let $0<q_i\le p_i<\infty$, $i=1,2$. Then:

$$\begin{array}{c}\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))\hfill \\ \hfill \left\{\begin{array}{cc}=\left\{0\right\},\hfill & q_1<q_2\mathrm{or}{p}_1<p_2,\hfill \\ =L^{\infty }\left({\mathbb{R}}^n\right),\hfill & q_2\le q_{1}and{p}_1=p_2,\hfill \\ ={\mathcal{M}}_{q_3}^{p_3}\left({\mathbb{R}}^n\right),\hfill & q_2<q_{1}and{p}_1{q}_2/q_1\le p_2<p_1,\hfill \\ ⫌{\mathcal{M}}_{q_3}^{p_3}\left({\mathbb{R}}^n\right),\hfill & q_2<q_1<p_{1}and1/(1/p_1+1/q_2-1/q_1)\le p_2<p_1{q}_2/q_1,\hfill \\ ⫌\left\{0\right\},\hfill & q_2\le q_1<p_{1}and{p}_2<1/(1/p_1+1/q_2-1/q_1),\hfill \end{array}\right.\end{array}$$

where $q_3=q_1{q}_2/(q_1-q_2)$ and $p_3=p_1{p}_2/(p_1-p_2)$.

We have notation for the scale $L^{p,\lambda }\left({\mathbb{R}}^n\right)$ analogous to the scale ${\mathcal{M}}_q^p\left({\mathbb{R}}^n\right)$. We may also replace ${\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right)$ and/or ${\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right)$ by $\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right)$ and/or $\mathrm{w}{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right)$ to define the corresponding multiplier spaces. According to [1], we have a counterpart of Theorem 1 to weak Morrey spaces: we can replace $L^{p_i,{\lambda }_i}\left({\mathbb{R}}^n\right)$ by $\mathrm{w}{L}^{p_i,{\lambda }_i}\left({\mathbb{R}}^n\right)$ in Theorem 1 and ${\mathcal{M}}_{q_i}^{p_i}\left({\mathbb{R}}^n\right)$ by $\mathrm{w}{\mathcal{M}}_{q_i}^{p_i}\left({\mathbb{R}}^n\right)$ in Theorem 2. As for weak Morrey spaces, the following results were obtained in [1].

Theorem 3

([8], Corollary 3). The same conclusion as Theorem 1 remains valid if we replace $L^{p_k,{\lambda }_k}\left({\mathbb{R}}^n\right)$ by $\mathrm{w}{L}^{p_k,{\lambda }_k}\left({\mathbb{R}}^n\right)$ for $k=1,2,3$. As a result, the same conclusion as Theorem 2 remains valid if we replace ${\mathcal{M}}_{q_k}^{p_k}\left({\mathbb{R}}^n\right)$ by $\mathrm{w}{\mathcal{M}}_{q_k}^{p_k}\left({\mathbb{R}}^n\right)$ for $k=1,2,3$.

It is interesting to compare these results with the following endpoint cases:

$$\begin{array}{c}\mathrm{PWM}(L^{\infty }\left({\mathbb{R}}^n\right),{\mathcal{M}}_q^p\left({\mathbb{R}}^n\right))={\mathcal{M}}_q^p\left({\mathbb{R}}^n\right),\\ \mathrm{PWM}(L^{\infty }\left({\mathbb{R}}^n\right),\mathrm{w}{\mathcal{M}}_q^p\left({\mathbb{R}}^n\right))=\mathrm{w}{\mathcal{M}}_q^p\left({\mathbb{R}}^n\right),\\ \mathrm{PWM}({\mathcal{M}}_q^p\left({\mathbb{R}}^n\right),L^{\infty }\left({\mathbb{R}}^n\right))=\mathrm{PWM}(\mathrm{w}{\mathcal{M}}_q^p\left({\mathbb{R}}^n\right),L^{\infty }\left({\mathbb{R}}^n\right))=\left\{0\right\}\end{array}$$

for all $0<q\le p<\infty$.

The goal of this note is to give complete characterizations of:

$$\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$$

including

$$\mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right)),$$

$$\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),\mathrm{w}{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$$

and:

$$\mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),\mathrm{w}{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right)).$$

Here are tables of the characterization of these spaces. For example, in

Table 1. $\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$.

	${\mathit{p}}_1<{\mathit{p}}_2$	${\mathit{p}}_1={\mathit{p}}_2$	${\mathit{p}}_1>{\mathit{p}}_2$
$q_1<q_2$	Theorem 2	Theorem 2	Theorem 2
$q_1=q_2$	Theorem 2	Theorem 2	Theorem 4
$q_1>q_2$	Theorem 2	Theorem 2	Theorem 4

, we deal with the case of $p_1>p_2$ and $q_1>q_2$ in Theorem 4 to follow.

The remaining part of this paper is organized as follows: In Section 2, we present our main results summarized as Table 1,

Table 2. $\mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$.

	${\mathit{p}}_1<{\mathit{p}}_2$	${\mathit{p}}_1={\mathit{p}}_2$	${\mathit{p}}_1>{\mathit{p}}_2$
$q_1<q_2$	Proposition 1, 1 and 2.	Proposition 1, 2.	Proposition 1, 2.
$q_1=q_2$	Proposition 1, 1.	Propositions 1, 3.	Theorem 6
$q_1>q_2$	Proposition 1, 1.	Proposition 2	Theorem 6

,

Table 3. $\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),\mathrm{w}{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$.

	${\mathit{p}}_1<{\mathit{p}}_2$	${\mathit{p}}_1={\mathit{p}}_2$	${\mathit{p}}_1>{\mathit{p}}_2$
$q_1<q_2$	Proposition 3, 1.	Proposition 3, 1.	Proposition 3, 1.
$q_1=q_2$	Proposition 3, 1.	Proposition 3, 2.	Theorem 7
$q_1>q_2$	Proposition 3, 1.	Proposition 3, 2.	Theorem 7

and

Table 4. $\mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),\mathrm{w}{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$.

	${\mathit{p}}_1<{\mathit{p}}_2$	${\mathit{p}}_1={\mathit{p}}_2$	${\mathit{p}}_1>{\mathit{p}}_2$
$q_1<q_2$	Theorem 3	Theorem 3	Theorem 3
$q_1=q_2$	Theorem 3	Theorem 3	Theorem 8
$q_1>q_2$	Theorem 3	Theorem 3	Theorem 8

. Section 3 deals with preliminary and general facts of the multiplier spaces. Section 4 is devoted to the proof of the results summarized in the tables above.

2. Main Results

2.1. Characterization of $\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$

To characterize the pointwise multiplier space $\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$, we recall a couple of notions in [9,10].

A quasi-Banach (resp. Banach) lattice on ${\mathbb{R}}^n$ is a nonzero quasi-Banach (resp. Banach) space $(E,\parallel ·\parallel )$ contained in $L^0\left({\mathbb{R}}^n\right)$ such that ${\parallel f\parallel }_E\le {\parallel g\parallel }_E$ holds for all $f,g\in E$ such that $\left|f\right|\le \left|g\right|$. Let $u\in (0,\infty )$. For a quasi-Banach lattice $E\subset L^0\left({\mathbb{R}}^n\right)$, we define its u-convexification $E^u$ by:

$$E^u\equiv {\{f:|f|}^u\in {E\},\parallel f\parallel }_{E^u}\equiv {\left({\parallel \left|f\right|}^u{\parallel }_E\right)}^{1/u}.$$

For example, ${\left(L^1\left({\mathbb{R}}^n\right)\right)}^p=L^p\left({\mathbb{R}}^n\right)$.

We next recall the notion of block spaces introduced by Long [10].

Definition 1.

Let $1\le q\le p<\infty$. A function $A\in L^0\left({\mathbb{R}}^n\right)$ is a $(p,q)$-block if there exists a cube Q that supports A and:

$${\parallel A\parallel }_{L^{q^{\prime }}}\le {\left|Q\right|}^{\frac{1}{q^{\prime }}-\frac{1}{p^{\prime }}},$$

(8)

where $p^{\prime }$ and $q^{\prime }$ stand for the conjugate exponent of p and q, respectively. If we need to specify Q, then we say that b is a $(p,q)$-block supported on Q. Let $1\le q\le p<\infty$, and define the block space ${\mathcal{H}}_{q^{\prime }}^{p^{\prime }}\left({\mathbb{R}}^n\right)$ as the set of all $f\in L^{p^{\prime }}\left({\mathbb{R}}^n\right)$ for which f is realized as the sum ${\displaystyle f=\sum _{j=1}^{\infty }{\tau }_j{A}_j}$ with some ${\left\{{\tau }_j\right\}}_{j=1}^{\infty }\in {\ell }^1\left(\mathbb{N}\right)$ and some sequence ${\left\{A_j\right\}}_{j=1}^{\infty }$ of $(p,q)$-blocks. Define the norm ${\parallel f\parallel }_{{\mathcal{H}}_{q^{\prime }}^{p^{\prime }}}$ for $f\in {\mathcal{H}}_{q^{\prime }}^{p^{\prime }}\left({\mathbb{R}}^n\right)$ as:

$${\parallel f\parallel }_{{\mathcal{H}}_{q^{\prime }}^{p^{\prime }}}\equiv \underset{\tau }{inf}{\parallel \tau \parallel }_{{\ell }^1},$$

(9)

where $\tau ={\left\{{\tau }_j\right\}}_{j=1}^{\infty }$ runs over all admissible expressions as above.

Finally, to state our result, we recall the definition of vector-valued Morrey spaces proposed by Ho [9]. Let $E\left({\mathbb{R}}^n\right)\subset L^0\left({\mathbb{R}}^n\right)$ be a quasi-Banach lattice, and let $p>0$. Then, the E-based vector-valued Morrey space ${\mathcal{M}}_E^p\left({\mathbb{R}}^n\right)$ is the set of all $f\in L^0\left({\mathbb{R}}^n\right)$ for which:

$${\parallel f\parallel }_{{\mathcal{M}}_E^p}\equiv \underset{Q\in \mathcal{Q}}{sup}{\left|Q\right|}^{\frac{1}{p}}\frac{\parallel {\chi }_Q{f\parallel }_E}{\parallel {\chi }_Q{\parallel }_E}$$

is finite.

Recall that a quasi-Banach lattice E enjoys the Fatou property if $\underset{j\in \mathbb{N}}{sup}{f}_j\in E$ and $\underset{j\to \infty }{lim}{\parallel f_j\parallel }_E=∥\underset{j\in \mathbb{N}}{lim}{f}_j∥$ for any sequence ${\left\{f_j\right\}}_{j=1}^{\infty }$ in E satisfying $0\le f_1\le f_2\le \cdots$. We make a brief remark on the relation among these notions introduced above.

Remark 1.

If $\parallel {\chi }_Q{\parallel }_E={\left|Q\right|}^{\frac{1}{p}}$ for all cubes Q and if E has the Fatou property, then a simple observation shows ${\mathcal{M}}_E^p\left({\mathbb{R}}^n\right)=E\left({\mathbb{R}}^n\right)$ with the equivalence of norms. In particular, If $E\left({\mathbb{R}}^n\right)={\mathcal{H}}_{q^{\prime }}^{p^{\prime }}\left({\mathbb{R}}^n\right)$, then ${\mathcal{M}}_E^{p^{\prime }}\left({\mathbb{R}}^n\right)=E\left({\mathbb{R}}^n\right)$.

We provide a complete picture of the description of $\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$.

Theorem 4.

Let $0<q_i\le p_i<\infty$, $i=1,2$:

• If $q_1<q_2$ or $p_1<p_2$, then $\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))=\left\{0\right\}$;
• If $q_1\ge q_2$ and $p_1=p_2$, then $\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))=L^{\infty }\left({\mathbb{R}}^n\right)$;
• If $q_1\ge q_2$ and $p_1>p_2$, then $\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))={\mathcal{M}}_{X^{q_2}}^{p_3}\left({\mathbb{R}}^n\right),$ where $p_3$ and X are given by:

  $$p_3=\frac{p_1{p}_2}{p_1-p_2},X={\mathcal{H}}_{{\left(\frac{q_1}{q_2}\right)}^{\prime }}^{{\left(\frac{p_1}{q_2}\right)}^{\prime }}\left({\mathbb{R}}^n\right).$$
  In particular,

  $$\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_1^{p_2}\left({\mathbb{R}}^n\right))={\mathcal{M}}_{{\mathcal{H}}_{q_1^{\prime }}^{p_1^{\prime }}}^{p_3}\left({\mathbb{R}}^n\right),$$

  where $p_3$ is defined by $p_3=\frac{p_1{p}_2}{p_1-p_2}$.

It is significant that Theorem 4 does not require $\frac{q_1}{p_1}\ge \frac{q_2}{p_2}$, unlike Theorem 2. We give an equivalent form using the scale $L^{p,\lambda }\left({\mathbb{R}}^n\right)$.

Theorem 5.

Let $p_i\in (0,\infty )$ and ${\lambda }_i\in [0,n)$, $i=1,2$:

• If $p_1<p_2$ or $\frac{p_1}{n-{\lambda }_1}<\frac{p_2}{n-{\lambda }_2}$, then $\mathrm{PWM}(L^{p_1,{\lambda }_1}\left({\mathbb{R}}^n\right),L^{p_2,{\lambda }_2}\left({\mathbb{R}}^n\right))=\left\{0\right\};$
• If $p_1\ge p_2$ and $\frac{p_1}{n-{\lambda }_1}=\frac{p_2}{n-{\lambda }_2}$, then $\mathrm{PWM}(L^{p_1,{\lambda }_1}\left({\mathbb{R}}^n\right),L^{p_2,{\lambda }_2}\left({\mathbb{R}}^n\right))=L^{\infty }\left({\mathbb{R}}^n\right)$;
• If $p_1\ge p_2$ and $\frac{p_1}{n-{\lambda }_1}>\frac{p_2}{n-{\lambda }_2}$, then $\mathrm{PWM}(L^{p_1,{\lambda }_1}\left({\mathbb{R}}^n\right),L^{p_2,{\lambda }_2}\left({\mathbb{R}}^n\right))={\mathcal{M}}_{X^{p_2}}^{v_3}\left({\mathbb{R}}^n\right),$ where $v_3$ and X are given by:

  $$\frac{n-{\lambda }_1}{p_{1}n}+\frac{1}{v_3}=\frac{n-{\lambda }_2}{p_{2}n},X\left({\mathbb{R}}^n\right)={\mathcal{H}}_{{\left(\frac{p_1}{p_2}\right)}^{\prime }}^{{\left(\frac{p_{1}n}{p_2(n-{\lambda }_1)}\right)}^{\prime }}\left({\mathbb{R}}^n\right).$$

We prove Theorem 4 in Section 4.1.

We combine Theorems 2 and 4 to have a nontrivial coincidence of function spaces.

Corollary 1.

Let $q_i\in (0,\infty )$ and $p_i\in [q_i,\infty )$, $i=1,2$. Assume that $q_1>q_2$ and $p_1>p_2.$ Write $q_3=\frac{q_1{q}_2}{q_1-q_2}$, $p_3=\frac{p_1{p}_2}{p_1-p_2}$ and $X\left({\mathbb{R}}^n\right)={\mathcal{H}}_{{\left(\frac{q_1}{q_2}\right)}^{\prime }}^{{\left(\frac{p_1}{q_2}\right)}^{\prime }}\left({\mathbb{R}}^n\right)$. If $\frac{q_1}{p_1}\ge \frac{q_2}{p_2}$, then ${\mathcal{M}}_{q_3}^{p_3}\left({\mathbb{R}}^n\right)={\mathcal{M}}_{X^{q_2}}^{p_3}\left({\mathbb{R}}^n\right).$

A remark about Corollary 1 may be in order.

Remark 2.

Let $X\left({\mathbb{R}}^n\right)$ be as in Corollary 1, and let $Y\left({\mathbb{R}}^n\right)=L^{q_3}\left({\mathbb{R}}^n\right)$. Corollary 1 reveals that ${\mathcal{M}}_Y^{p_3}\left({\mathbb{R}}^n\right)={\mathcal{M}}_{X^{q_2}}^{p_3}\left({\mathbb{R}}^n\right),$ although $X{\left({\mathbb{R}}^n\right)}^{q_2}\ne Y\left({\mathbb{R}}^n\right)$.

2.2. Characterization of $\mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$

Once we prove Theorem 4, we can pass the results above from ${\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right)$ to $\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right)$ with ease if $0<q_2<q_1<\infty$. To describe the multiplier space $\mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$, we will recall the definition given in [11,12]:

Definition 2.

• ([11], Definition 1.4.1) Let $f:{\mathbb{R}}^n\to \mathbb{C}$ be a measurable function. Then, define its decreasing rearrangement $f^{\ast }$ by:

  $$f^{\ast }\left(t\right)=inf\{\lambda >0:|\left\{\right|f|>\lambda \}|\le t\};$$
• ([11], Definition 1.4.6) Let $1<p,q<\infty$. The Lorentz space $L^{p,q}$ is the set of all measurable functions $f:{\mathbb{R}}^n\to \mathbb{C}$ for which:

  $${\parallel f\parallel }_{L^{p,q}}={\left({\int }_0^{\infty }{\left(t^{\frac{1}{p}}{f}^{\ast }\left(t\right)\right)}^q\frac{dt}{t}\right)}^{\frac{1}{q}}$$

  is finite;
• ([12], Definition 2.3) Let $1<q\le p<\infty$. A measurable function b is said to be a $(p^{\prime };q^{\prime },1)$-block if there exists a cube Q such that:

  $$\mathrm{supp}\left(b\right)\subset {Q,\parallel b\parallel }_{L^{q^{\prime },1}}\le {\left|Q\right|}^{\frac{1}{p}-\frac{1}{q}};$$

  (10)
• ([12], Definition 2.3) Let $1<q\le p<\infty$. The space ${\mathcal{H}}_{q^{\prime },1}^{p^{\prime }}\left({\mathbb{R}}^n\right)$ is the set of all $L^p\left({\mathbb{R}}^n\right)$-functions f for which there exist a sequence ${\left\{{\lambda }_j\right\}}_{j=1}^{\infty }\in {\ell }^1\left(\mathbb{N}\right)$ and a sequence ${\left\{b_j\right\}}_{j=1}^{\infty }$ of $(p^{\prime };q^{\prime },1)$-blocks for which:

  $$f=\sum _{j=1}^{\infty }{\lambda }_j{b}_j$$

  (11)

  in $L^p\left({\mathbb{R}}^n\right)$. For $f\in {\mathcal{H}}_{q^{\prime },1}^{p^{\prime }}\left({\mathbb{R}}^n\right)$, one defines:

  $${\parallel f\parallel }_{{\mathcal{H}}_{q^{\prime },1}^{p^{\prime }}}=inf\sum _{j=1}^{\infty }\left|{\lambda }_j\right|,$$

  where inf is over all possible decompositions in $\left(11\right)$.

Concerning Lorentz spaces, a couple of remarks may be in order:

Remark 3.

Let $0<p,p_1,p_2,q,q_1,q_2\le \infty$:

• Let G be a measurable set in ${\mathbb{R}}^n$. Then:

  $$\parallel {\chi }_G{\parallel }_{L^{p,q}}={\left(\frac{p}{q}\right)}^{\frac{1}{q}}{\left|G\right|}^{\frac{1}{p}},$$

  where we understand ${(p/q)}^{1/q}=1$ for $q=\infty$. See [11], Example 1.4.8;
• Assume that:

  $$\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2},\frac{1}{q}=\frac{1}{q_1}+\frac{1}{q_2}.$$
  Then:

  $${\parallel f·g\parallel }_{L^{p,q}}\le e^{\frac{1}{p}}{\parallel f\parallel }_{L^{p_1,q_1}}{\parallel g\parallel }_{L^{p_2,q_2}}$$

  for all $f\in L^{p_1,q_1}\left({\mathbb{R}}^n\right)$ and $g\in L^{p_2,q_2}\left({\mathbb{R}}^n\right)$, or equivalently:

  $${\parallel g\parallel }_{\mathrm{PWM}(L^{p_1,q_1},L^{p,q})}\le e^{\frac{1}{p}}{\parallel g\parallel }_{L^{p_2,q_2}}$$

  for all $g\in L^{p_2,q_2}\left({\mathbb{R}}^n\right)$. See [8], p. 6, Corollary 3, for the precise constant;
• We have an equivalent expression if $p>1$: For all $f\in L^0\left({\mathbb{R}}^n\right)$,

  $$\begin{array}{ccc}\hfill & & {\parallel f\parallel }_{\mathrm{w}{L}^p}={\parallel f\parallel }_{L^{p,\infty }}\hfill \\ \hfill & & \sim sup\left\{{\left|E\right|}^{\frac{1}{p}-1}{\parallel f\parallel }_{L^1\left(E\right)}:E is a measurable set with \left|E\right|\in (0,\infty )\right\}.\hfill \end{array}$$

  (12)
  See [11], Exercise 1.1.12.

Theorem 6.

Let $0<q_i\le p_i<\infty$, $i=1,2$. If $p_1>p_2$ and $q_1\ge q_2$, then:

$$\mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))={\mathcal{M}}_{X^{q_2}}^{p_3}\left({\mathbb{R}}^n\right),$$

where $p_3$ and X are given by:

$$p_3=\frac{p_1{p}_2}{p_1-p_2},X={\mathcal{H}}_{{\left(\frac{q_1}{q_2}\right)}^{\prime },1}^{{\left(\frac{p_1}{q_2}\right)}^{\prime }}\left({\mathbb{R}}^n\right).$$

We prove Theorem 6 in Section 4.2.

The special case of $p_1=q_1>p_2=q_2$ deserves attention.

Corollary 2.

In addition to the assumption in Theorem 6, we let $p_1=q_1>p_2=q_2$. Then:

$$\mathrm{PWM}(\mathrm{w}{L}^{p_1}\left({\mathbb{R}}^n\right),L^{p_2}\left({\mathbb{R}}^n\right))={\mathcal{M}}_{X^{p_2}}^{p_3}\left({\mathbb{R}}^n\right),$$

where $p_3$ and X are given by:

$$p_3=\frac{p_1{p}_2}{p_1-p_2},X={\mathcal{H}}_{{\left(\frac{p_1}{p_2}\right)}^{\prime },1}^{{\left(\frac{p_1}{p_2}\right)}^{\prime }}\left({\mathbb{R}}^n\right).$$

We complement Corollary 2.

Proposition 1.

Let $0<q_i\le p_i<\infty$, $i=1,2$. If either one of the following conditions holds, then:

$$\mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))=\left\{0\right\}:$$

• $p_1<p_2$;
• $q_1<q_2$;
• $p_1=p_2$ and $q_1=q_2$.

We prove Proposition 1 in Section 4.3.

If $p_1=p_2$ and $q_1>q_2$, then we have something similar to the case of classical Morrey spaces.

Proposition 2.

Let $0<q_i\le p_i<\infty$, $i=1,2$. Assume $p_1=p_2$ and $q_1>q_2$. Then:

$$\mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))=L^{\infty }\left({\mathbb{R}}^n\right).$$

We prove Proposition 2 in Section 4.4.

2.3. Characterization of $\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),\mathrm{w}{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$

Next, we pass from ${\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right)$ to $\mathrm{w}{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right)$.

Theorem 4 allows us to characterize $\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),\mathrm{w}{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$.

Theorem 7.

Let $0<q_i\le p_i<\infty$, $i=1,2$, satisfy $p_1>p_2$ and $q_1\ge q_2$. Define $p_3$ by:

$$\frac{1}{p_3}=\frac{1}{p_2}+\frac{1}{q_2^{\prime }}-\frac{1}{p_1}.$$

Then. a function $h\in L^0\left({\mathbb{R}}^n\right)$ belongs to $\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),\mathrm{w}{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$ if and only if ${\chi }_{E}h\in {\mathcal{M}}_{{\mathcal{H}}_{q_1^{\prime }}^{p_1^{\prime }}}^{p_3}\left({\mathbb{R}}^n\right)$ for all measurable sets E with $\left|E\right|\in (0,\infty )$ and:

$$sup\{|E{|}^{\frac{1}{q_2}-1}\parallel {\chi }_{E}h{\parallel }_{{\mathcal{M}}_{{\mathcal{H}}_{q_1^{\prime }}^{p_1^{\prime }}}^{p_3}}:E is a measurable set with |E|\in (0,\infty )\}<\infty .$$

In this case,

$${\parallel h\parallel }_{\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1},\mathrm{w}{\mathcal{M}}_{q_2}^{p_2})}\sim {sup\left\{\right|E|}^{\frac{1}{q_2}-1}\parallel {\chi }_E{h\parallel }_{{\mathcal{M}}_{{\mathcal{H}}_{q_1^{\prime }}^{p_1^{\prime }}}^{p_3}}:E is a measurable set with \left|E\right|\in (0,\infty )\}.$$

We prove Theorem 7 in Section 4.5.

We supplement Theorem 7 by considering the case of $p_1\le p_2$.

Proposition 3.

Let $0<q_i\le p_i<\infty$, $i=1,2$:

• Assume $p_1<p_2$ or $q_1<q_2$. Then:

  $$\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),\mathrm{w}{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))=\left\{0\right\};$$
• Assume $p_1=p_2$ and $q_1\ge q_2$. Then:

  $$\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),\mathrm{w}{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))=L^{\infty }\left({\mathbb{R}}^n\right).$$

We prove Proposition 3 in Section 4.6.

2.4. Characterization of $\mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),\mathrm{w}{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$

Finally, we pass both ${\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right)$ and ${\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right)$ to $\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right)$ and $\mathrm{w}{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right)$, respectively. The proof is a mere combination of Theorems 6 and 7. Therefore, we omit the detail again.

Theorem 8.

Let $0<q_i\le p_i<\infty$, $i=1,2$, satisfy $p_1>p_2$ and $q_1\ge q_2$. Define $p_3$ by:

$$\frac{1}{p_3}=\frac{1}{p_2}+\frac{1}{q_2^{\prime }}-\frac{1}{p_1}.$$

Then $h\in L^0\left({\mathbb{R}}^n\right)$ belongs to $\mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),\mathrm{w}{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$ if and only if ${\chi }_{E}h\in {\mathcal{M}}_{{\mathcal{H}}_{q_1^{\prime },1}^{p_1^{\prime }}}^{p_3}\left({\mathbb{R}}^n\right)$ for all measurable sets E with $\left|E\right|\in (0,\infty )$ and:

$$sup\{{|E|}^{\frac{1}{q_2}-1}\parallel {\chi }_{E}h{\parallel }_{{\mathcal{M}}_{{\mathcal{H}}_{q_1^{\prime },1}^{p_1^{\prime }}}^{p_3}}:E is a measurable set with |E|\in (0,\infty )\}<\infty$$

and in this case:

$${\parallel h\parallel }_{\mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1},\mathrm{w}{\mathcal{M}}_{q_2}^{p_2})}\sim {sup\left\{\right|E|}^{\frac{1}{q_2}-1}\parallel {\chi }_E{h\parallel }_{{\mathcal{M}}_{{\mathcal{H}}_{q_1^{\prime },1}^{p_1^{\prime }}}^{p_3}}:E is a measurable set with |E|\in (0,\infty )\}.$$

In particular, $h\in L^0\left({\mathbb{R}}^n\right)$ belongs to $\mathrm{PWM}(\mathrm{w}{L}^{p_1}\left({\mathbb{R}}^n\right),\mathrm{w}{L}^{p_2}\left({\mathbb{R}}^n\right))$ if and only if, for all measurable sets E with $\left|E\right|\in (0,\infty )$, ${\chi }_{E}h\in {\mathcal{M}}_{{\mathcal{H}}_{p_1^{\prime },1}^{p_1^{\prime }}}^{p_1^{\prime }}\left({\mathbb{R}}^n\right)$ and

$$sup{\left\{\right|E|}^{\frac{1}{p_2}-1}\parallel {\chi }_{E}h{\parallel }_{{\mathcal{M}}_{{\mathcal{H}}_{p_1^{\prime },1}^{p_1^{\prime }}}^{p_1^{\prime }}}:E is a measurable set with |E|\in (0,\infty )\}<\infty$$

and in this case:

$${\parallel h\parallel }_{\mathrm{PWM}(\mathrm{w}{L}^{p_1},\mathrm{w}{L}^{p_2})}\sim sup\{{\left|E\right|}^{\frac{1}{p_2}-1}\parallel {\chi }_E{h\parallel }_{{\mathcal{M}}_{{\mathcal{H}}_{p_1^{\prime },1}^{p_1^{\prime }}}^{p_1^{\prime }}}:E is a measurable set with \left|E\right|\in (0,\infty )\}.$$

In the above, the implicit constants do not depend on h.

In Theorem 8, the case of $p_1\le p_2$ is covered in Theorem 3.

It seems to make sense to compare Theorems 7 and 8 with an existing result. Let $p_1=q_1$ and $p_2=q_2$ in Theorems 7 and 8.

Corollary 3.

Let $0<p_2<p_1<\infty$. Then:

$$\mathrm{PWM}(L^{p_1}\left({\mathbb{R}}^n\right),\mathrm{w}{L}^{p_2}\left({\mathbb{R}}^n\right))=\mathrm{PWM}(\mathrm{w}{L}^{p_1}\left({\mathbb{R}}^n\right),\mathrm{w}{L}^{p_2}\left({\mathbb{R}}^n\right))=\mathrm{w}{L}^{\frac{p_1{p}_2}{p_1-p_2}}\left({\mathbb{R}}^n\right).$$

(13)

In [8], Corollary 3, the first author showed the second equality in (13). We reprove Corollary 3 by the use of Theorems 7 and 8 in Section 4.7.

3. Preliminaries

For the proof of the theorems in the present paper, we use a scaling property. Arithmetic shows that the following scaling property holds:

Lemma 1.

([5], (g) p. 326) Let $E_1$ and $E_2$ be quasi-Banach lattices, and let $u>0$. Then:

$$\mathrm{PWM}(E_1^u,E_2^u)=\mathrm{PWM}{(E_1,E_2)}^u.$$

We move on to the convexification of E-based Morrey spaces. Actually, as the next lemma shows, E-based Morrey spaces are closed under the convexification of quasi-Banach lattices.

Lemma 2.

Let $E\subset L^0\left({\mathbb{R}}^n\right)$ be a quasi-Banach lattice and $p,u>0$. Then: ${\left({\mathcal{M}}_E^{\frac{p}{u}}\left({\mathbb{R}}^n\right)\right)}^u={\mathcal{M}}_{E^u}^p\left({\mathbb{R}}^n\right)$.

Proof. 

For $f\in L^0\left({\mathbb{R}}^n\right)$, a direct computation shows:

$$\begin{array}{c}\hfill {\parallel f\parallel }_{{\mathcal{M}}_{E^u}^p}=\underset{Q\in \mathcal{Q}}{sup}{\left|Q\right|}^{\frac{1}{p}}\frac{\parallel {\chi }_Q{f\parallel }_{E^u}}{\parallel {\chi }_Q{\parallel }_{E^u}}=\underset{Q\in \mathcal{Q}}{sup}{\left({\left|Q\right|}^{\frac{u}{p}}\frac{\parallel {\chi }_Q{\left|f\right|}^u{\parallel }_E}{\parallel {\chi }_Q{\parallel }_E}\right)}^{\frac{1}{u}}={\left({\parallel \left|f\right|}^u{\parallel }_{{\mathcal{M}}_E^{\frac{p}{u}}}\right)}^{\frac{1}{u}}={\parallel f\parallel }_{{\left({\mathcal{M}}_E^{\frac{p}{u}}\right)}^u}.\end{array}$$

□

We also investigate how ${\mathcal{M}}_E^p\left({\mathbb{R}}^n\right)$ inherits the dilation property from E.

Lemma 3.

We have ${\parallel f(t·)\parallel }_{{\mathcal{M}}_E^p}=t^{-\frac{n}{p}}{\parallel f\parallel }_{{\mathcal{M}}_E^p}$ for all $f\in {\mathcal{M}}_E^p\left({\mathbb{R}}^n\right)$ and $t>0$ as long as E is subject to the scaling law ${\parallel g(t·)\parallel }_E=t^{-\frac{n}{u}}{\parallel g\parallel }_E$ for some $u>0$ and for all $g\in E$ and $t>0$.

Proof. 

The proof is straightforward, and we omit the detail. □

Remark that Lemma 3 is not used for the proof of the main results in the present paper. However, Lemma 3 allows us to compare the scaling laws in the function spaces in question.

In Section 2, we introduced block spaces together with some of their variants. We recall that these spaces can be identified with the Köthe dual of Morrey spaces.

If E is a Banach lattice, then recall that its “Köthe dual” $E^{\prime }$ is defined in $L^0\left({\mathbb{R}}^n\right)$ by the set of all $g\in L^0\left({\mathbb{R}}^n\right)$ such that:

$${\parallel g\parallel }_{E^{\prime }}\equiv sup\left\{{\parallel f·g\parallel }_{L^1}:f\in L^0\left({\mathbb{R}}^n\right),{\parallel f\parallel }_E\le 1\right\}<\infty .$$

(14)

We can specify the Köthe dual of Morrey spaces as follows:

Lemma 4.

• Let $1\le q\le p<\infty$. Then, the Köthe dual of ${\mathcal{M}}_q^p\left({\mathbb{R}}^n\right)$ is ${\mathcal{H}}_{q^{\prime }}^{p^{\prime }}\left({\mathbb{R}}^n\right)$ with the coincidence of norms;
• Let $1<q\le p<\infty$. Then, the Köthe dual of $\mathrm{w}{\mathcal{M}}_q^p\left({\mathbb{R}}^n\right)$ is isomorphic to ${\mathcal{H}}_{q^{\prime },1}^{p^{\prime }}\left({\mathbb{R}}^n\right)$ with the equivalence of norms.

Lemma 4 is a culmination of what we proved in various papers. See [13], Theorem 3.1, for 1. with $q=1$, and see [14], Theorem 4.1, for example, for 1. with $1<q<\infty$, while 2. was proven in [12], Theorem 2.7.

A direct consequence of Lemma 4 is that we have:

$$\parallel {\chi }_Q{\parallel }_{{\mathcal{H}}_{q^{\prime }}^{p^{\prime }}}={\left|Q\right|}^{\frac{1}{p^{\prime }}}$$

(15)

for all cubes Q.

When $E_1$ and $E_2$ are both homogeneous in the sense that the translation operator induces isomorphism, we can mollify $\mathrm{PWM}(E_1,E_2)$. Furthermore, in this case, by the next lemma, we see that the functions in $\mathrm{PWM}(E_1,E_2)$ do not increase the local integrability of the functions.

Lemma 5.

Let $E_1,E_2$ be Banach lattices, which are translation invariant in the sense that ${\parallel h(·-y)\parallel }_{E_j}={\parallel h\parallel }_{E_j}$ for all $h\in E_j$, $j=1,2$. Assume that $E_1$ and $E_2$ enjoy the Fatou property and that $E_2\subset L_{\mathrm{loc}}^u\left({\mathbb{R}}^n\right)$ for some $u\in (0,\infty )$:

• ${\chi }_{{[0,1]}^n}\in E_1\cap E_2$.
• The space $\mathrm{PWM}(E_1,E_2)$ is a translation-invariant Banach lattice, and any element in $\mathrm{PWM}(E_1,E_2)$ is almost everywhere finite;
• If $f\in L^1\left({\mathbb{R}}^n\right)$ and $g\in \mathrm{PWM}(E_1,E_2)$, then $f\ast g\in \mathrm{PWM}(E_1,E_2)$ and:

  $${\parallel f\ast g\parallel }_{\mathrm{PWM}(E_1,E_2)}\le {\parallel f\parallel }_{L^1}{\parallel g\parallel }_{\mathrm{PWM}(E_1,E_2)}.$$

  (16)
  In particular, for almost all $x\in {\mathbb{R}}^n$,

  $${\int }_{{\mathbb{R}}^n}\left|g(x-y)f\left(y\right)\right|dy<\infty .$$

  (17)
• If $\mathrm{PWM}(E_1,E_2)\ne \left\{0\right\}$, then ${\chi }_{{[-1,1]}^n}\in \mathrm{PWM}(E_1,E_2)$.
• $\mathrm{PWM}(E_1,E_2)\subset L_{\mathrm{loc}}^u\left({\mathbb{R}}^n\right)$.
• If there exists a function $f\in E_1\setminus L_{\mathrm{loc}}^u\left({\mathbb{R}}^n\right)$, then $\mathrm{PWM}(E_1,E_2)=\left\{0\right\}$.

Proof. 

• We concentrate on $E_1$; $E_2$ can be dealt with similarly. Let $f\in E_1$ be a nonzero function. By truncation, the linearity of $E_1$, and the lattice property of $E_1$, we may assume that $f={\chi }_F$ for some bounded measurable set F. Notice that:

  $$g_N:=\frac{1}{N^n}\sum _{k_1=1}^N\sum _{k_2=1}^N\cdots \sum _{k_n=1}^{N}f\left(·-\frac{(k_1,k_2,\dots ,k_n)}{N}\right)\in E_1$$

  satisfies $\parallel g_N{\parallel }_{E_1}\le {\parallel f\parallel }_{E_1}$ due to the translation invariance and the triangle inequality. Since $g_N\to {\chi }_{{[0,1]}^n}\ast f$ in the topology of $L^1\left({\mathbb{R}}^n\right)$ as $N\to \infty$, by the Fatou property of $E_1$, ${\chi }_{{[0,1]}^n}\ast f\in E_1$. Since $\parallel {\chi }_{{[0,1]}^n}{\ast f\parallel }_{L^1}=\left|F\right|>0$, it follows that ${\chi }_{{[0,1]}^n}\ast f$ is a nonzero continuous function. By the translation invariance and the lattice property of $E_1$, it follows that ${\chi }_{{[0,1]}^n}\in E_1$;
• Let $g\in \mathrm{PWM}(E_1,E_2)$ and $y\in {\mathbb{R}}^n$. Then:

  $${\parallel g(·-y)f\parallel }_{E_2}={\parallel f(·+y)g\parallel }_{E_2}\le {\parallel g\parallel }_{\mathrm{PWM}(E_1,E_2)}{\parallel f(·+y)\parallel }_{E_1}={\parallel g\parallel }_{\mathrm{PWM}(E_1,E_2)}{\parallel f\parallel }_{E_1}$$

  for all $f\in E_1$. Thus, we see that $g(·-y)\in \mathrm{PWM}(E_1,E_2)$ and that:

  $${\parallel g(·-y)\parallel }_{\mathrm{PWM}(E_1,E_2)}\le {\parallel g\parallel }_{\mathrm{PWM}(E_1,E_2)}.$$
  Likewise, if we swap the role of g and $g(·-y)$, then we have:

  $${\parallel g\parallel }_{\mathrm{PWM}(E_1,E_2)}\le {\parallel g(·-y)\parallel }_{\mathrm{PWM}(E_1,E_2)}.$$
  Thus, $\mathrm{PWM}(E_1,E_2)$ is translation invariant. Since $E_2$ is a Banach lattice, we see that $\mathrm{PWM}(E_1,E_2)$ is a Banach lattice. To check that any element $g\in \mathrm{PWM}(E_1,E_2)$ is finite almost everywhere, we only need to show that $g{\chi }_{{[-1,1]}^n}$ is finite almost everywhere. Assume otherwise; $F:=\{x\in {[-1,1]}^n:|g\left(x\right)|=\infty \}$ has a positive measure. Then, ${\chi }_F\in \mathrm{PWM}(E_1,E_2)$ since ${\chi }_F\le \left|g\right|\in \mathrm{PWM}(E_1,E_2)$. Thus, ${\chi }_F={\chi }_F·{\chi }_{{[-1,1]}^n}\in E_2$. This implies that $\parallel {\chi }_F{\parallel }_{E_2}\in (0,\infty )$. However, this is a contradiction since $\infty >\parallel g{\chi }_{{[-1,1]}^n}{\parallel }_{E_2}\ge {\parallel \infty {\chi }_F\parallel }_{E_2}=\infty$;
• We prove:

  $$\left|f\right|\ast \left|g\right|\in \mathrm{PWM}(E_1,E_2){\mathrm{and}\parallel \left|f\right|\ast \left|g\right|\parallel }_{\mathrm{PWM}(E_1,E_2)}\le {\parallel f\parallel }_{L^1}{\parallel g\parallel }_{\mathrm{PWM}(E_1,E_2)},$$

  which is slightly stronger than (16). For $h\in E_1$, we have:

  $$\begin{array}{cc}\hfill {∥h·\left|f\right|\ast \left|g\right|∥}_{E_2}& \le {\int }_{{\mathbb{R}}^n}{∥h(·)g(·-y)f\left(y\right)∥}_{E_2}dy\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^n}{∥h(·+y)g(·)f\left(y\right)∥}_{E_2}dy\hfill \\ \hfill & \le {\parallel g\parallel }_{\mathrm{PWM}(E_1,E_2)}{\int }_{{\mathbb{R}}^n}{∥h(·+y)f\left(y\right)∥}_{E_1}dy\hfill \\ \hfill & ={\parallel g\parallel }_{\mathrm{PWM}(E_1,E_2)}{\int }_{{\mathbb{R}}^n}{∥h(·+y)∥}_{E_1}\left|f\left(y\right)\right|dy\hfill \\ \hfill & ={\parallel g\parallel }_{\mathrm{PWM}(E_1,E_2)}{\int }_{{\mathbb{R}}^n}{∥h∥}_{E_1}\left|f\left(y\right)\right|dy\hfill \\ \hfill & ={\parallel g\parallel }_{\mathrm{PWM}(E_1,E_2)}{\parallel f\parallel }_{L^1}{\parallel h\parallel }_{E_1}.\hfill \end{array}$$
  Finally, (17) is a consequence of 2. and the fact that:

  $${\int }_{{\mathbb{R}}^n}\left|f(·-y)g\left(y\right)\right|dy=\left|f\right|\ast \left|g\right|\in \mathrm{PWM}(E_1,E_2);$$
• If $\mathrm{PWM}(E_1,E_2)\ne \left\{0\right\}$, then by the lattice property of $\mathrm{PWM}(E_1,E_2)$, there exists a nonzero and non-negative function $g\in \mathrm{PWM}(E_1,E_2)$. By 1., ${\chi }_{{[-R,R]}^n}\ast g\in$$\mathrm{PWM}(E_1,E_2)\setminus \left\{0\right\}$. If we choose $R\ge 1$ large enough, then ${\chi }_{{[-R,R]}^n}\ast g\ge \kappa {\chi }_{{[-1,1]}^n}$ for some $\kappa >0$. Due to the lattice property of $\mathrm{PWM}(E_1,E_2)$, we obtain ${\chi }_{{[-1,1]}^n}\in \mathrm{PWM}(E_1,E_2)$;
• By 2., the lattice property, and the translation invariance of $E_1$, ${\chi }_K\in E_1$ for all compact sets K. Thus, if $f\in \mathrm{PWM}(E_1,E_2)$, then ${\chi }_{K}f\in E_2\subset L_{\mathrm{loc}}^u\left({\mathbb{R}}^n\right)$;
• Assume $\mathrm{PWM}(E_1,E_2)\ne \left\{0\right\}$. By translation, we may assume $f{\chi }_{{[-1,1]}^n}\notin L_{\mathrm{loc}}^u\left({\mathbb{R}}^n\right)$. Meanwhile, by 3., $|f{\chi }_{{[-1,1]}^n}|\in E_2\subset L_{\mathrm{loc}}^u\left({\mathbb{R}}^n\right)$. This is a contradiction.

□

4. Proof of the Main Results

4.1. Proof of Theorem 4

The proof of Theorem 4 is not so long. Furthermore, the statements in Theorem 4, 1. and 2. are already included in Theorem 2. Therefore, we consider 3. solely. First, assume that $p_2=q_2=1$. In this case, we need to find a description of $\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),L^1\left({\mathbb{R}}^n\right))$. According to [5], this is nothing but the Köthe dual of ${\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right)$. In this case, it remains to note that ${\mathcal{H}}_{q_1^{\prime }}^{p_1^{\prime }}\left({\mathbb{R}}^n\right)={\mathcal{M}}_{{\mathcal{H}}_{q_1^{\prime }}^{p_1^{\prime }}}^{p_1^{\prime }}\left({\mathbb{R}}^n\right)$ thanks to Remark 1 and that $p_1^{\prime }=\frac{p_3}{q_2}=p_3$.

Next, we assume that $p_2>q_2=1$. Then by the definition of ${\mathcal{M}}_1^{p_2}\left({\mathbb{R}}^n\right)$, $g\in L^0\left({\mathbb{R}}^n\right)$ belongs to $\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))=\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_1^{p_2}\left({\mathbb{R}}^n\right))$ if and only if ${\left|Q\right|}^{\frac{1}{p_2}-1}{\chi }_{Q}g\in \mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),L^1\left({\mathbb{R}}^n\right))$ for each $Q\in \mathcal{Q}$ and fulfills:

$$\underset{Q\in \mathcal{Q}}{sup}{\left|Q\right|}^{\frac{1}{p_2}-1}{\parallel {\chi }_{Q}g\parallel }_{\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1},L^1)}<\infty .$$

According to the previous paragraph, this is equivalent to ${\left|Q\right|}^{\frac{1}{p_2}-1}{\chi }_{Q}g\in {\mathcal{H}}_{q_1^{\prime }}^{p_1^{\prime }}\left({\mathbb{R}}^n\right)$ for each $Q\in \mathcal{Q}$ and $\underset{Q\in \mathcal{Q}}{sup}{\left|Q\right|}^{\frac{1}{p_2}-1}\parallel {\chi }_Q{g\parallel }_{{\mathcal{H}}_{q_1^{\prime }}^{p_1^{\prime }}}=\underset{Q\in \mathcal{Q}}{sup}{\left|Q\right|}^{\frac{1}{p_3}}\frac{\parallel {\chi }_Q{g\parallel }_{{\mathcal{H}}_{q_1^{\prime }}^{p_1^{\prime }}}}{\parallel {\chi }_Q{\parallel }_{{\mathcal{H}}_{q_1^{\prime }}^{p_1^{\prime }}}}<\infty$, i.e., $g\in {\mathcal{M}}_{{\mathcal{H}}_{q_1^{\prime }}^{p_1^{\prime }}}^{p_3}\left({\mathbb{R}}^n\right)$.

We handle the general case. Let $L>0$. According to Lemma 1,

$$g\in \mathrm{PWM}({\mathcal{M}}_{L{q}_1}^{L{p}_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{L{q}_2}^{L{p}_2}\left({\mathbb{R}}^n\right))$$

if and only if ${\left|g\right|}^L\in \mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$. Therefore, from Lemma 2 and what we proved in the previous paragraph, we deduce:

$$\begin{array}{cc}\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))& ={\left(\mathrm{PWM}({\mathcal{M}}_{\frac{q_1}{q_2}}^{\frac{p_1}{q_2}}\left({\mathbb{R}}^n\right),{\mathcal{M}}_1^{\frac{p_2}{q_2}}\left({\mathbb{R}}^n\right))\right)}^{q_2}\hfill \\ & ={\left({\mathcal{M}}_X^{\frac{p_3}{q_2}}\left({\mathbb{R}}^n\right)\right)}^{q_2}\hfill \\ & ={\mathcal{M}}_{X^{q_2}}^{p_3}\left({\mathbb{R}}^n\right).\hfill \end{array}$$

The proof is therefore complete.

4.2. Proof of Theorem 6

In the proof of Theorem 4, we may replace ${\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right)$ by $\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right)$. Then, accordingly, we have to replace ${\mathcal{H}}_{{\left(\frac{q_1}{q_2}\right)}^{\prime }}^{{\left(\frac{p_1}{q_2}\right)}^{\prime }}\left({\mathbb{R}}^n\right)$ by ${\mathcal{H}}_{{\left(\frac{q_1}{q_2}\right)}^{\prime },1}^{{\left(\frac{p_1}{q_2}\right)}^{\prime }}\left({\mathbb{R}}^n\right)$. Thus, the proof is similar to Theorem 4.

4.3. Proof of Proposition 1

We may assume $q_1,q_2>1$ by the scaling argument by Lemma 1:

• Since $q_1,q_2>1$, we may regard $\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right)$ and ${\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right)$ as Banach spaces as in (12). Assume:

  $$\mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))\ne \left\{0\right\}.$$
  Then, ${\chi }_{{[-1,1]}^n}\in \mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$ by virtue of Lemma 5, 4.. This implies $\parallel f·{\chi }_{{[-1,1]}^n}{\parallel }_{{\mathcal{M}}_{q_2}^{p_2}}\le {\parallel g\parallel }_{\mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1},{\mathcal{M}}_{q_2}^{p_2})}{\parallel f\parallel }_{\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}}$ for all $f\in \mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right)$. If we substitute $f(t·)$ instead of f into this condition, we obtain $\parallel f·{\chi }_{{[-t,t]}^n}{\parallel }_{{\mathcal{M}}_{q_2}^{p_2}}\le t^{\frac{n}{p_2}-\frac{n}{p_1}}{\parallel g\parallel }_{\mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1},{\mathcal{M}}_{q_2}^{p_2})}{\parallel f\parallel }_{\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}}$. Since $p_2>p_1$, if we let $t\to \infty$, then we have ${\parallel f\parallel }_{{\mathcal{M}}_{q_2}^{p_2}}=0$ for all $f\in \mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right)$. This is a contradiction;
• Let $r\in (q_1,q_2)$. According to [15], p. 67 (see also [3], Theorem 2.2 and Remark 2.3, and [16], Theorem 4.9), there exists $f\in {\mathcal{M}}_r^{p_1}\left({\mathbb{R}}^n\right)\setminus L^{q_2}\left({\mathbb{R}}^n\right)$ such that $\mathrm{supp}\left(f\right)\subset {[0,1]}^n$. Thus, we are in the position of using Lemma 5, 6. to have the conclusion;
• By virtue of Lemma 5, 4., if:

  $$\mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))=\mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right))\ne \left\{0\right\},$$

  then ${\chi }_{{[-1,1]}^n}\in \mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right))$. Then, for $f\in \mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right)$ and $r>0$,

  $$\begin{array}{cc}\parallel {\chi }_{{[-r,r]}^n}{f\parallel }_{{\mathcal{M}}_{q_1}^{p_1}}& =r^{\frac{n}{p_1}}{\parallel {\chi }_{{[-1,1]}^n}f(r·)\parallel }_{{\mathcal{M}}_{q_1}^{p_1}}\hfill \\ & \le \parallel {\chi }_{{[-1,1]}^n}{\parallel }_{\mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1},{\mathcal{M}}_{q_1}^{p_1})}{r}^{\frac{n}{p_1}}{\parallel f(r·)\parallel }_{\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}}\hfill \\ & =\parallel {\chi }_{{[-1,1]}^n}{\parallel }_{\mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1},{\mathcal{M}}_{q_1}^{p_1})}{\parallel f\parallel }_{\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}}.\hfill \end{array}$$
  Letting $r\to \infty$, we obtain:

  $${\parallel f\parallel }_{{\mathcal{M}}_{q_1}^{p_1}}\le \parallel {\chi }_{{[-1,1]}^n}{\parallel }_{\mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1},{\mathcal{M}}_{q_1}^{p_1})}{\parallel f\parallel }_{\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}}.$$
  This implies $\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right)\subset {\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right)$. This is impossible; see [16,17] as well as [18], Section 4.

4.4. Proof of Proposition 2

Thanks to Theorem 2 and the embedding:

$${\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right)\subset \mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right)\subset {\mathcal{M}}_{q_2}^{p_1}\left({\mathbb{R}}^n\right),$$

we have:

$$\begin{array}{cc}{L}^{\infty }\left({\mathbb{R}}^n\right)& =\mathrm{PWM}({\mathcal{M}}_{q_2}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))\hfill \\ & \subset \mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))\hfill \\ & \subset \mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))\hfill \\ & =L^{\infty }\left({\mathbb{R}}^n\right).\hfill \end{array}$$

4.5. Proof of Theorem 7

We may assume $q_1,q_2>1$ by Lemma 1. The proof of Theorem 7 is a direct combination of Theorem 4 and Lemma 6 below.

Lemma 6.

Let $1<q_i\le p_i<\infty$, $i=1,2$. Assume $p_1>p_2$ and $q_1\ge q_2$. Define $r>0$ by:

$$\frac{1}{r}=\frac{1}{p_2}-\frac{1}{q_2}+1.$$

Then, $f\in L^0\left({\mathbb{R}}^n\right)$ belongs to $\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),\mathrm{w}{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$ if and only if, for all measurable sets E with $\left|E\right|\in (0,\infty )$, ${\chi }_{E}f\in \mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_1^r\left({\mathbb{R}}^n\right))$ and:

$$sup\{{|E|}^{\frac{1}{q_2}-1}\parallel {\chi }_{E}f{\parallel }_{\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1},{\mathcal{M}}_1^r)}:E is a measurable set with |E|\in (0,\infty )\}<\infty .$$

In this case,

$$\begin{array}{c}{\displaystyle {\parallel f\parallel }_{\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1},\mathrm{w}{\mathcal{M}}_{q_2}^{p_2})}}\hfill \\ \sim sup\{{|E|}^{\frac{1}{q_2}-1}\parallel {\chi }_{E}f{\parallel }_{\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1},{\mathcal{M}}_1^r)}:E is a measurable set with |E|\in (0,\infty )\}.\hfill \end{array}$$

Once Lemma 6 is established, Theorem 4 immediately gives the proof of Theorem 7. Therefore, we concentrate on Lemma 6.

Proof of Lemma 6.

Let $h\in L^0\left({\mathbb{R}}^n\right)$. Thanks to (12), $h\in \mathrm{w}{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right)$ if and only if ${\chi }_{E}h\in {\mathcal{M}}_1^{p_2}\left({\mathbb{R}}^n\right)$ for all measurable sets E with $\left|E\right|\in (0,\infty )$ and:

$$\underset{Q}{sup}\underset{E}{sup}{\left|Q\right|}^{\frac{1}{p_2}-\frac{1}{q_2}}{\left|E\right|}^{\frac{1}{q_2}-1}\parallel {\chi }_{E\cap Q}{h\parallel }_{L^1}=\underset{E}{sup}{\left|E\right|}^{\frac{1}{q_2}-1}{\parallel {\chi }_{E}h\parallel }_{{\mathcal{M}}_1^r}<\infty ,$$

where E moves over all measurable sets with $0<\left|E\right|<\infty$. Therefore, supposing that E moves over all measurable sets with $\left|E\right|\in (0,\infty )$, we obtain:

$$\begin{array}{cc}{\parallel f\parallel }_{\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1},\mathrm{w}{\mathcal{M}}_{q_2}^{p_2})}& =sup\{\parallel f·g{\parallel }_{\mathrm{w}{\mathcal{M}}_{q_2}^{p_2}}:g\in {\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),\parallel g{\parallel }_{{\mathcal{M}}_{q_1}^{p_1}}=1\}\hfill \\ & \sim \underset{E}{sup}{sup\left\{\right|E|}^{\frac{1}{q_2}-1}\parallel f·g·{\chi }_E{\parallel }_{{\mathcal{M}}_1^r}:g\in {\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right){,\parallel g\parallel }_{{\mathcal{M}}_{q_1}^{p_1}}=1\}\hfill \\ & =\underset{E}{sup}{\left|E\right|}^{\frac{1}{q_2}-1}{\parallel f{\chi }_E\parallel }_{\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1},{\mathcal{M}}_1^r)},\hfill \end{array}$$

as required. □

4.6. Proof of Proposition 3

• Suppose $p_1<p_2$. We can go through the same argument as Proposition 1, 2. to conclude that by using the function:

  $${\chi }_{B(x_0,r)}{|·-x_0|}^{-\frac{n(p_1+p_2)}{2p_1{p}_2}}\in {\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right)\setminus \mathrm{w}{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right)$$

  for some $x_0\in {\mathbb{R}}^n$ and $r>0$. If $q_1<q_2$, then we take $r_1,r_2$ so that $q_1<r_1<r_2<q_2$. Then, we have:

  $$\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),\mathrm{w}{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))\subset \mathrm{PWM}(\mathrm{w}{\mathcal{M}}_{r_1}^{p_1}\left({\mathbb{R}}^n\right),{\mathcal{M}}_{r_2}^{p_2}\left({\mathbb{R}}^n\right))=\left\{0\right\}$$

  thanks to Proposition 1, 2.;
• It is clear that:

  $$L^{\infty }\left({\mathbb{R}}^n\right)\subset \mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),\mathrm{w}{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right)).$$
  Thus, it suffices to show that:

  $$\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),\mathrm{w}{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))\subset L^{\infty }\left({\mathbb{R}}^n\right).$$
  To this end, let $g\in \mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1}\left({\mathbb{R}}^n\right),\mathrm{w}{\mathcal{M}}_{q_2}^{p_2}\left({\mathbb{R}}^n\right))$. Then:

  $${\left|Q\right|}^{\frac{1}{p_2}-\frac{1}{q_2}}\parallel g{\chi }_Q{\parallel }_{\mathrm{w}{L}^{q_2}}\le {\parallel g\parallel }_{\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1},\mathrm{w}{\mathcal{M}}_{q_2}^{p_2})}{\left|Q\right|}^{\frac{1}{p_1}}={\parallel g\parallel }_{\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1},\mathrm{w}{\mathcal{M}}_{q_2}^{p_2})}{\left|Q\right|}^{\frac{1}{p_2}}.$$
  Thus, by the Lebesgue differentiation theorem, we obtain:

  $${\parallel g\parallel }_{L^{\infty }}\le {\parallel g\parallel }_{\mathrm{PWM}({\mathcal{M}}_{q_1}^{p_1},\mathrm{w}{\mathcal{M}}_{q_2}^{p_2})},$$

  as required.

4.7. Proof of Corollary 3

Theorems 7 and 8 can be shown to recover this result as follows:

• Thanks to the fact that $\parallel {\chi }_Q{\parallel }_{{\mathcal{H}}_{p_1^{\prime }}^{p_1^{\prime }}}={\parallel {\chi }_Q\parallel }_{L^{p_1^{\prime }}}$ and the Fatou property of ${\mathcal{H}}_{p_1^{\prime }}^{p_1^{\prime }}\left({\mathbb{R}}^n\right)$ established in [14], ${\mathcal{M}}_{{\mathcal{H}}_{p_1^{\prime }}^{p_1^{\prime }}}^{p_1^{\prime }}\left({\mathbb{R}}^n\right)$ coincides with $L^{p_1^{\prime }}\left({\mathbb{R}}^n\right)$. Thus, according to [11], Exercise 1.4.14, we see that $\mathrm{PWM}(L^{p_1}\left({\mathbb{R}}^n\right),\mathrm{w}{L}^{p_2}\left({\mathbb{R}}^n\right))=\mathrm{w}{L}^{\frac{p_1{p}_2}{p_1-p_2}}\left({\mathbb{R}}^n\right)$;
• Using Lemma 4, we deduce:

  $$\begin{array}{cc}& sup\{|E{|}^{\frac{1}{p_2}-1}\parallel {\chi }_{E}h{\parallel }_{{\mathcal{H}}_{p_1^{\prime },1}^{p_1^{\prime }}}:E is a measurable set with |E|\in (0,\infty )\}\\ & \gtrsim sup\{{|E|}^{\frac{1}{p_2}-1}\parallel {\chi }_{E}h{\parallel }_{L^{p_1^{\prime }}}:E is a measurable set with |E|\in (0,\infty )\}\\ \hfill & \sim {\parallel h\parallel }_{\mathrm{w}{L}^{\frac{p_1{p}_2}{p_1-p_2}}}.\hfill \end{array}$$
  Let $r_1$ be a number slightly less than $p_1$, so that $r_1^{\prime }$ is slightly larger than $p_1^{\prime }$. Define $v_1$ by:

  $$\frac{1}{p_1^{\prime }}=\frac{1}{v_1}+\frac{1}{r_1^{\prime }}.$$
  Thanks to Remark 3,

  $$\parallel {\chi }_E{h\parallel }_{L^{p_1^{\prime },1}}\lesssim \parallel {\chi }_E{\parallel }_{L^{v_1,1}}{\parallel h\parallel }_{\mathrm{w}{L}^{r_1^{\prime }}}\sim {\left|E\right|}^{\frac{1}{v_1}}{\parallel h\parallel }_{\mathrm{w}{L}^{r_1^{\prime }}}$$

  for all $h\in \mathrm{w}{L}^{p_1}\left({\mathbb{R}}^n\right)$. Using this estimate and Remark 1, we have:

  $$\begin{array}{cc}\parallel {\chi }_E{h\parallel }_{{\mathcal{M}}_{{\mathcal{H}}_{p_1^{\prime },1}^{p_1^{\prime }}}^{p_1^{\prime }}}& \sim \parallel {\chi }_E{h\parallel }_{{\mathcal{H}}_{p_1^{\prime },1}^{p_1^{\prime }}}\hfill \\ & \sim sup\{\parallel {\chi }_{E}gh{\parallel }_{L^1}:g\in \mathrm{w}{\mathcal{M}}_{p_1}^{p_1}\left({\mathbb{R}}^n\right),\parallel g{\parallel }_{\mathrm{w}{\mathcal{M}}_{p_1}^{p_1}}\le 1\}\hfill \\ & =sup\{\parallel {\chi }_{E}gh{\parallel }_{L^1}:g\in \mathrm{w}{L}^{p_1}\left({\mathbb{R}}^n\right),\parallel g{\parallel }_{\mathrm{w}{L}^{p_1}}\le 1\}\hfill \\ & \lesssim \parallel {\chi }_E{h\parallel }_{\mathrm{w}{L}^{p_1^{\prime }}}\hfill \\ & \lesssim {\left|E\right|}^{\frac{1}{v_1}}{\parallel {\chi }_{E}h\parallel }_{\mathrm{w}{L}^{r_1^{\prime }}}.\hfill \end{array}$$
  Thus, it follows from the embedding $L^{r_1^{\prime }}\left({\mathbb{R}}^n\right)\hookrightarrow \mathrm{w}{L}^{r_1^{\prime }}\left({\mathbb{R}}^n\right)$ that:

  $$\begin{array}{cc}\hfill & sup\{|E{|}^{\frac{1}{p_2}-1}\parallel {\chi }_{E}h{\parallel }_{{\mathcal{H}}_{p_1^{\prime },1}^{p_1^{\prime }}}:E is a measurable set with |E|\in (0,\infty )\}\hfill \\ \hfill & \lesssim sup\{|E{|}^{\frac{1}{p_2}+\frac{1}{v_1}-1}\parallel {\chi }_{E}h{\parallel }_{L^{r_1^{\prime }}}:E is a measurable set with |E|\in (0,\infty )\}\hfill \\ \hfill & =sup\{|E{|}^{\frac{p_1-p_2}{p_1{p}_2}-\frac{1}{r_1^{\prime }}}\parallel {\chi }_{E}h{\parallel }_{L^{r_1^{\prime }}}:E is a measurable set with |E|\in (0,\infty )\}.\hfill \end{array}$$
  Invoking [11], Exercise 1.4.14, once again, one obtains:

  $${sup\left\{\right|E|}^{\frac{1}{p_2}-1}\parallel {\chi }_E{h\parallel }_{{\mathcal{H}}_{p_1^{\prime },1}^{p_1^{\prime }}}:E is a measurable set with \left|E\right|\in (0,\infty )\}\lesssim {\parallel h\parallel }_{\mathrm{w}{L}^{\frac{p_1{p}_2}{p_1-p_2}}}.$$
  Thus, Theorems 7 and 8 can recover the result in [8].