Atomic Decomposition for Mixed Morrey Spaces

Abstract

In this paper, we consider some norm estimates for mixed Morrey spaces considered by the first author. Mixed Lebesgue spaces are realized as a special case of mixed Morrey spaces. What is new in this paper is a new norm estimate for mixed Morrey spaces that is applicable to mixed Lebesgue spaces as well. An example shows that the condition on parameters is optimal. As an application, the Olsen inequality adapted to mixed Morrey spaces can be obtained.

1 Introduction

We obtain some decomposition results for mixed Lebesgue spaces and mixed Morrey spaces. Let us first recall the definition of mixed Lebesgue spaces. Let $0<q_1,q_2,\ldots ,q_n \le \infty$ be constants. Write $\mathbf {q}=(q_1,q_2,\ldots ,q_n)$. Then define the mixed Lebesgue norm $\Vert \cdot \Vert _{L^{\mathbf {q}}}$ by

$$\begin{aligned} \Vert f\Vert _{L^{\mathbf {q}}} \equiv \left( \int _{{{\mathbb {R}}}} \cdots \left( \int _{{{\mathbb {R}}}} \left( \int _{{\mathbb R}}|f(x_1,x_2,\ldots ,x_n)|^{q_1}\mathrm{d}x_1 \right) ^{\frac{q_2}{q_1}}\mathrm{d}x_2 \right) ^{\frac{q_3}{q_2}} \cdots \mathrm{d}x_n \right) ^{\frac{1}{q_n}}. \end{aligned}$$

A natural modification for $x_i$ is made when $q_i=\infty$. We define the mixed Lebesgue space $L^{\mathbf {q}}({{\mathbb {R}}}^n)$ to be the set of all measurable function f on ${\mathbb {R}}^n$ with $\Vert f\Vert _{L^{\mathbf {q}}}<\infty$. Here and below we use the notation

$$\begin{aligned} \mathbf {q}=(q_1,q_2,\ldots ,q_n), \quad \mathbf {q}^{*}=(q_1^*,q_2^*,\ldots ,q_n^*), \quad \mathbf {t}=(t_1,t_2,\ldots ,t_n) \end{aligned}$$

to denote the vectors in ${{\mathbb {R}}}^n$. The aim of this paper is to develop a theory of decompositions based on the following boundedness of the maximal operator.

Here and below, for $0 \le a \le b \le \infty , a \le \mathbf {q} \le b$ means that $a \le qi \le b$ for all $i = 1, 2,\ldots ,n.$

Theorem 1

Assume that

$$\begin{aligned} 1\le t_k<\min \{q_1, \ldots , q_k\}\le \infty \quad (k=1, \ldots , n). \end{aligned}$$

Define

$$\begin{aligned} M^{(\mathbf {t})}f(x)=\sup _{Q \in {{\mathcal {Q}}}} \frac{\chi _Q(x)}{\Vert \chi _Q\Vert _{L^{\mathbf {t}}}}\Vert f\chi _Q\Vert _{L^{\mathbf {t}}} \end{aligned}$$

for a measurable function f. Then for all measurable functions f

$$\begin{aligned} \Vert M^{(\mathbf {t})}f\Vert _{L^{\mathbf {q}}} \lesssim \Vert f\Vert _{L^{\mathbf {q}}}. \end{aligned}$$

For $0<p<\infty$, and $0<\mathbf {q}<\infty$ satisfying

$$\begin{aligned} \frac{n}{p} \le \sum \limits _{j=1}^n \frac{1}{q_j} \end{aligned}$$

recall that mixed Morrey spaces are defined by the norm given by

$$\begin{aligned} \Vert f\Vert _{{{\mathcal {M}}}^p_{\mathbf {q}}} \equiv \sup _{Q \in {\mathcal D}({{\mathbb {R}}}^n)} |Q|^{\frac{1}{p}-\frac{1}{n}\sum \nolimits _{j=1}^n \frac{1}{q_j}}\Vert f\chi _Q\Vert _{L^{\mathbf {q}}} \end{aligned}$$

for measurable functions $f:{{\mathbb {R}}}^n \rightarrow {{\mathbb {C}}}$, where ${{\mathcal {D}}}({{\mathbb {R}}}^n)$ denotes the set of all dyadic cubes. We denote by ${{\mathcal {Q}}}({{\mathbb {R}}}^n)$ the set of all cubes whose edges are parallel to the coordinate axes. If there is no confusion, we substitute ${{\mathcal {D}}}$ and ${{\mathcal {Q}}}$ for ${{\mathcal {D}}}({{\mathbb {R}}}^{n})$ and ${{\mathcal {Q}}}({{\mathbb {R}}}^{n})$, respectively.

Using Theorem 1, we seek to prove the following decomposition result about the functions in mixed Morrey spaces.

This result extends [26, Chapter 8, Lemma 5]

Theorem 2

Suppose that the parameters $p,\mathbf {q},s,\mathbf {t}$ satisfy

$$\begin{aligned}&1<p<s<\infty , \quad 1<\max \{q_1, \ldots , q_k\}<t_k<\infty \quad (k=1, \ldots , n),\\&\frac{n}{p} \le \sum \limits _{j=1}^n \frac{1}{q_j}, \quad \frac{n}{s} \le \sum \limits _{j=1}^n \frac{1}{t_j}. \end{aligned}$$

Assume that $\{a_j\}_{j=1}^\infty \subset {\mathcal M}^s_{\mathbf {t}}({{\mathbb {R}}}^n),$ $\{\lambda _j\}_{j=1}^\infty \subset [0,\infty ),$ and $\{Q_j\}_{j=1}^\infty \subset {\mathcal Q}({{\mathbb {R}}}^n)$ fulfill

$$\begin{aligned} \Vert a_j\Vert _{{{\mathcal {M}}}^s_{\mathbf {t}}} \le |Q_j|^{\frac{1}{s}}, \quad \mathrm{supp}(a_j) \subset Q_j, \quad \left\| \sum \limits _{j=1}^\infty \lambda _j \chi _{Q_j}\right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}} <\infty . \end{aligned}$$

(1)

Then $\displaystyle f=\sum \nolimits _{j=1}^\infty \lambda _j a_j$ converges in ${{\mathcal {S}}}'({{\mathbb {R}}}^n) \cap L^{\mathbf {q}}_\mathrm{loc}({{\mathbb {R}}}^n)$ and satisfies

$$\begin{aligned} \Vert f\Vert _{{{\mathcal {M}}}^p_{\mathbf {q}}} \le C_{p,\mathbf {q},s,\mathbf {t}} \left\| \sum \limits _{j=1}^\infty \lambda _j\chi _{Q_j}\right\| _{{\mathcal M}^p_{\mathbf {q}}}. \end{aligned}$$

(2)

The next assertion concerns the decomposition of functions in ${{\mathcal {M}}}^p_{\mathbf {q}}({{\mathbb {R}}}^n)$. Hereafter, we write ${{\mathbb {N}}}_0={{\mathbb {N}}} \cup \{0\}$. For $d \in {\mathbb {N}}_0$, denote by ${\mathcal P}_d({{\mathbb {R}}}^n)$ the set of all polynomial functions with degree less than or equal to d, so that ${{\mathcal {P}}}({{\mathbb {R}}}^n) \equiv \bigcup \nolimits _{d=0}^\infty {{\mathcal {P}}}_d({{\mathbb {R}}}^n)$. It is clear that ${{\mathcal {P}}}_{-1}({{\mathbb {R}}}^n)=\{0\}$. Let $K \in {\mathbb {N}}_0$. The set ${{\mathcal {P}}}_K({{\mathbb {R}}}^n)^\perp$ denotes the set of measurable function f for which $\langle \cdot \rangle ^K f \in L^1({\mathbb R}^n)$ and $\displaystyle \int _{{{\mathbb {R}}}^n}x^\alpha f(x)\mathrm{d}x=0$ for any $\alpha \in {{\mathbb {N}}}_0^n$ with $|\alpha | \le K$, where $\langle \cdot \rangle =(1+|\cdot |^2)^{\frac{1}{2}}$. Such a function f is said to satisfy the moment condition of order K. In this case, one also writes $f \perp {{\mathcal {P}}}_K({\mathbb R}^n)$.

One writes $\mathbf {q}<\mathbf {t}$ if $q_j<t_j$ for each $j=1,2,\ldots ,n$.

The following theorem is a consequence of the paper [11].

Theorem 3

Suppose that the real parameters $p,\mathbf {q},K$ satisfy

$$\begin{aligned} 1<p<\infty , \quad 1<\mathbf {q}<\infty , \quad \frac{n}{p} \le \sum \limits _{j=1}^n \frac{1}{q_j}, \quad K \in {{\mathbb {N}}}_0 \cap \left( \frac{n}{q_0}-n-1, \infty \right) , \end{aligned}$$

where $q_0=\min (q_1, \ldots , q_n)$. Let $f \in {\mathcal M}^p_{\mathbf {q}}({{\mathbb {R}}}^n)$. Then there exists a triplet $\{a_j\}_{j=1}^\infty \subset L^\infty ({{\mathbb {R}}}^n) \cap {\mathcal P}_K^\perp ({{\mathbb {R}}}^n)$, $\{\lambda _j\}_{j=1}^\infty \subset [0,\infty )$, and $\{Q_j\}_{j=1}^\infty \subset {{\mathcal {Q}}}({\mathbb R}^n)$ such that $f=\sum \nolimits _{j=1}^\infty \lambda _j a_j$ in ${{\mathcal {S}}}'({{\mathbb {R}}}^n)$ and that, for any $v>0$

$$\begin{aligned} |a_j| \le \chi _{Q_j}, \quad \left\| \left( \sum \limits _{j=1}^\infty (\lambda _j\chi _{Q_j})^{v} \right) ^{\frac{1}{v}}\right\| _{{\mathcal M}^p_{\mathbf {q}}} \le C_v\Vert f\Vert _{{{\mathcal {M}}}^p_{\mathbf {q}}}. \end{aligned}$$

(3)

Here the constant $C_v>0$ is independent of f.

We rephrase Theorems 2 and 3 in the case of mixed Lebesgue spaces.

Corollary 1

Suppose that the parameters $\mathbf {q},\mathbf {t}$ satisfy

$$\begin{aligned} 1<\max \{q_1, \ldots , q_k\}<t_k<\infty \quad (k=1, \ldots , n). \end{aligned}$$

Assume that $\{a_j\}_{j=1}^\infty \subset L^{\mathbf {t}}({\mathbb R}^n)$, $\{\lambda _j\}_{j=1}^\infty \subset [0,\infty )$, and $\{Q_j\}_{j=1}^\infty \subset {{\mathcal {Q}}}({{\mathbb {R}}}^n)$ fulfill

$$\begin{aligned} \Vert a_j\Vert _{L^{\mathbf {t}}} \le |Q_j|^{\frac{1}{n}\sum \nolimits _{k=1}^n \frac{1}{t_k}}, \quad \mathrm{supp}(a_j) \subset Q_j, \quad \left\| \sum \limits _{j=1}^\infty \lambda _j \chi _{Q_j}\right\| _{L^{\mathbf {q}}} <\infty . \end{aligned}$$

Then $\displaystyle f=\sum \nolimits _{j=1}^\infty \lambda _j a_j$ converges in $L^{\mathbf {q}}({{\mathbb {R}}}^n)$ and satisfies

$$\begin{aligned} \Vert f\Vert _{L^{\mathbf {q}}} \le C_{p,q,s,t} \left\| \sum \limits _{j=1}^\infty \lambda _j\chi _{Q_j}\right\| _{L^{\mathbf {q}}}. \end{aligned}$$

Corollary 2

Let $1<\mathbf {q}<\infty$ and $K \in {{\mathbb {N}}}_0 \cap \left( \frac{n}{q}-n-1,\infty \right)$. Let $f \in L^{\mathbf {q}}({{\mathbb {R}}}^n)$. Then there exists a triplet $\{a_j\}_{j=1}^\infty \subset L^\infty ({{\mathbb {R}}}^n) \cap {\mathcal P}_K^\perp ({{\mathbb {R}}}^n),$ $\{\lambda _j\}_{j=1}^\infty \subset [0,\infty ),$ and $\{Q_j\}_{j=1}^\infty \subset {{\mathcal {Q}}}({\mathbb R}^n)$ such that $f=\sum \nolimits _{j=1}^\infty \lambda _j a_j$ in $L^{\mathbf {q}}({{\mathbb {R}}}^n)$ and that, for any $v>0$

$$\begin{aligned} |a_j| \le \chi _{Q_j}, \quad \left\| \left( \sum \limits _{j=1}^\infty (\lambda _j\chi _{Q_j})^{v} \right) ^{\frac{1}{v}}\right\| _{L^{\mathbf {q}}} \le C_v\Vert f\Vert _{L^{\mathbf {q}}}. \end{aligned}$$

Here the constant $C_v>0$ is independent of f.

Theorem 3 is a special case of Theorem 4 to follow, which concerns the decomposition of Hardy-mixed Morrey spaces. Based on [21], we define Hardy-mixed Morrey spaces. For $0<\mathbf {q}, p<\infty$ satisfying $\displaystyle \frac{n}{p} \le \sum \nolimits _{j=1}^n \frac{1}{q_j}$, the Hardy-mixed Morrey space $H{{\mathcal {M}}}^p_{\mathbf {q}}({{\mathbb {R}}}^n)$ is defined as the set of any $f \in {{\mathcal {S}}}'({{\mathbb {R}}}^n)$ for which the quasi-norm $\Vert f\Vert _{H{{\mathcal {M}}}^p_{\mathbf {q}}} =\left\| \sup _{t>0}|e^{t \Delta }f|\right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}}$ is finite, where $e^{t\Delta }f$ stands for the heat extension of f;

$$\begin{aligned} e^{t\Delta }f(x)= \left\langle \frac{1}{\sqrt{(4\pi t)^n}} \exp \left( -\frac{|x-\cdot |^2}{4t}\right) ,f \right\rangle \quad (t>0,\,x \in {{\mathbb {R}}}^n). \end{aligned}$$

See [28] for the equivalent norms of the Hardy–Morrey spaces. We rephrase Theorems 2 and 3 in full generality in terms of Hardy-mixed Morrey spaces. The following result is again a consequence of the paper [11].

Theorem 4

Suppose that the real parameters $p,\mathbf {q},K$ satisfy

$$\begin{aligned} 0<p<\infty , \quad 0<\mathbf {q}<\infty , \quad \frac{n}{p} \le \sum \limits _{j=1}^n \frac{1}{q_j}, \quad K \in {{\mathbb {N}}}_0 \cap \left( \frac{n}{q_0}-n-1, \infty \right) , \end{aligned}$$

where $q_0=\min (q_1, \ldots , q_n)$. Let $f \in H{\mathcal M}^p_{\mathbf {q}}({{\mathbb {R}}}^n)$. Then there exists a triplet $\{a_j\}_{j=1}^\infty \subset L^\infty ({{\mathbb {R}}}^n) \cap {\mathcal P}_K^\perp ({{\mathbb {R}}}^n),$ $\{\lambda _j\}_{j=1}^\infty \subset [0,\infty ),$ and $\{Q_j\}_{j=1}^\infty \subset {{\mathcal {Q}}}({\mathbb R}^n)$ such that $f=\sum \nolimits _{j=1}^\infty \lambda _j a_j$ in ${{\mathcal {S}}}'({{\mathbb {R}}}^n)$ and that, for any $v>0,$

$$\begin{aligned} |a_j| \le \chi _{Q_j}, \quad \left\| \left( \sum \limits _{j=1}^\infty (\lambda _j\chi _{Q_j})^{v} \right) ^{\frac{1}{v}}\right\| _{{\mathcal M}^p_{\mathbf {q}}} \le C_{v}\Vert f\Vert _{H{{\mathcal {M}}}^p_{\mathbf {q}}}. \end{aligned}$$

(4)

Here the constant $C_v$ is a constant that is independent on v but not on f.

We remark that Theorems 2 and 4 are the special cases of the results in [11].

Theorem 2 has the following counterpart.

Theorem 5

Suppose that the parameters $p,\mathbf {q},s,\mathbf {t}$ satisfy

$$\begin{aligned}&1<p<s<\infty , \quad 0<\max \{1,q_1, \ldots , q_k\}<t_k<\infty \quad (k=1, \ldots , n),\\&\frac{n}{p} \le \sum \limits _{j=1}^n \frac{1}{q_j}, \quad \frac{n}{s} \le \sum \limits _{j=1}^n \frac{1}{t_j}. \end{aligned}$$

Write $v({\mathbf {q}}) \equiv \min \{1,q_1,\ldots ,q_n\}$ and $\displaystyle d_q=\left[ n\left( \frac{1}{v(\mathbf {q})}-1\right) \right]$. Assume that a triple

$$\begin{aligned} (\{a_j\}_{j=1}^\infty ,\{\lambda _j\}_{j=1}^\infty ,\{Q_j\}_{j=1}^\infty ) \in ({{\mathcal {M}}}^s_{\mathbf {t}}({{\mathbb {R}}}^n) \cap {\mathcal P}_{d_q}^\perp ({{\mathbb {R}}}^n)) \times [0,\infty ) \times {\mathcal Q}({{\mathbb {R}}}^n) \end{aligned}$$

fulfills

$$\begin{aligned} \Vert a_j\Vert _{{{\mathcal {M}}}^s_{\mathbf {t}}} \le |Q_j|^{\frac{1}{s}}, \quad \mathrm{supp}(a_j) \subset Q_j, \quad \left\| \left( \sum \limits _{j=1}^\infty ( \lambda _j \chi _{Q_j})^{v({\mathbf {q}})} \right) ^{\frac{1}{v({\mathbf {q}})}}\right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}} <\infty . \end{aligned}$$

Then $\displaystyle f=\sum \nolimits _{j=1}^\infty \lambda _j a_j$ converges in ${{\mathcal {S}}}'({{\mathbb {R}}}^n)$ and satisfies

$$\begin{aligned} \Vert f\Vert _{H{{\mathcal {M}}}^p_{\mathbf {q}}} \le C_{p,\mathbf {q},s,\mathbf {t}} \left\| \left( \sum \limits _{j=1}^\infty (\lambda _j\chi _{Q_j})^{v({\mathbf {q}})} \right) ^{\frac{1}{v({\mathbf {q}})}}\right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}}. \end{aligned}$$

Remark that in [14] Jia and Wang considered the case of $q_i=q\le 1$ for $i= 1,2,\ldots ,n$. We also remark that Theorems 4 and 5 with $q_i=q=p\le 1$ for $i=1,2,\ldots ,n$ are included in [10, Theorems 2.1 and 2.2]. Theorem 2 is new and even in Theorem 3–5 we do not have to postulate $\mathbf {q} \le 1$. Concerning ${{\mathcal {M}}}^p_{\mathbf {q}}({{\mathbb {R}}}^n)$ and $H{\mathcal M}^p_{\mathbf {q}}({{\mathbb {R}}}^n)$ when $\mathbf {q}>1$, we have the following assertion:

Proposition 1

Let $1<p<\infty$ and $1<\mathbf {q}<\infty$ satisfy

$$\begin{aligned} \frac{n}{p} \le \sum \limits _{j=1}^n \frac{1}{q_j}. \end{aligned}$$

• (1) If $f \in {{\mathcal {M}}}^p_{\mathbf {q}}({{\mathbb {R}}}^n),$ then $f \in H{{\mathcal {M}}}^p_{\mathbf {q}}({{\mathbb {R}}}^n)$.

• (2) If $f \in H{{\mathcal {M}}}^p_{\mathbf {q}}({{\mathbb {R}}}^n),$ then f can be represented by a locally integrable function and the representative belongs to ${{\mathcal {M}}}^p_{\mathbf {q}}({{\mathbb {R}}}^n)$.

We elaborate a detailed proof of Proposition 1 in Sect. 3.

As an application of Theorem 2, we can reprove the following Olsen inequality about the fractional integral operator $I_\alpha$, where $I_\alpha \,(0<\alpha <n)$ is defined by

$$\begin{aligned} I_\alpha f(x) = \int _{{\mathbb R}^n}\frac{f(y)}{|x-y|^{n-\alpha }}\,\mathrm{d}y. \end{aligned}$$

The following result is known:

Proposition 2

[18, Theorem 1.11] Suppose that the parameters $\alpha ,p,\mathbf {q},s,\mathbf {t}$ satisfy

$$\begin{aligned} 1<p<s<\infty , \quad 1<\mathbf {q}<\mathbf {t}<\infty , \quad \frac{n}{p} \le \sum \limits _{j=1}^n \frac{1}{q_j}, \quad \frac{n}{s} \le \sum \limits _{j=1}^n \frac{1}{t_j} \end{aligned}$$

and

$$\begin{aligned} \frac{1}{p}-\frac{\alpha }{n}=\frac{1}{s}, \quad \frac{q_j}{p}=\frac{t_j}{s} \quad (j=1,2,\ldots ,n). \end{aligned}$$

Then $I_\alpha$ is bounded from ${{\mathcal {M}}}^p_{\mathbf {q}}({\mathbb R}^n)$ to ${{\mathcal {M}}}^s_{\mathbf {t}}({{\mathbb {R}}}^n)$.

Based upon Proposition 2, we can prove the following result.

Theorem 6

Suppose that the parameters $\alpha ,p,\mathbf {q},p^*,\mathbf {q}^{\,*},s,\mathbf {t}$ satisfy

$$\begin{aligned}&1<p,p^*,s<\infty , \quad 1<\mathbf {q},\mathbf {q}^{\,*},\mathbf {t}<\infty ,\nonumber \\&\quad \frac{n}{p} \le \sum \limits _{j=1}^n \frac{1}{q_j}, \quad \frac{n}{p^*} \le \sum \limits _{j=1}^n \frac{1}{q_j^*}, \quad \frac{n}{s} \le \sum \limits _{j=1}^n \frac{1}{t_j},\nonumber \\&\quad \max \{t_1, \ldots , t_j\}<q_j^*, \quad \frac{1}{p}>\frac{\alpha }{n}, \quad \frac{1}{p^*} \le \frac{\alpha }{n}, \end{aligned}$$

(5)

for each $j=1,2,\ldots ,n,$ and that

$$\begin{aligned} \frac{1}{s}=\frac{1}{p^*}+\frac{1}{p}-\frac{\alpha }{n}, \quad \dfrac{t_j}{s}=\dfrac{q_j}{p} \quad (j=1,2,\ldots ,n). \end{aligned}$$

(6)

Then for all $f \in {{\mathcal {M}}}^p_{\mathbf {q}}({{\mathbb {R}}}^n)$ and $g \in {{\mathcal {M}}}^{p^*}_{\mathbf {q}^{\,*}}({{\mathbb {R}}}^n)$

$$\begin{aligned} \Vert g \cdot I_{\alpha }f\Vert _{{{\mathcal {M}}}_{\mathbf {t}}^{s}} \le C \Vert g \Vert _{{{\mathcal {M}}}^{p^*}_{\mathbf {q}^{\,*}}} \cdot \Vert f \Vert _{{\mathcal M}^{p}_{\mathbf {q}}}, \end{aligned}$$

where the constant C is independent of f and g.

This result recaptures [23, Proposition 1.8] as the special case of $q_i = q$ and $t_i = t$ for all $i = 1,2,\ldots ,m$. Note that a detailed calculation in [22, p. 6] shows that Theorem 6 is not just a combination of Proposition 2 and Lemma 1.

Lemma 1

Suppose that the parameters $p,\mathbf {q},p^*,\mathbf {q}^{\,*},s,\mathbf {t}$ satisfy

$$\begin{aligned}&1<p,p^*,s<\infty , \quad 1<\mathbf {q},\mathbf {q}^{\,*},\mathbf {t}<\infty ,\\&\frac{n}{p} \le \sum \limits _{j=1}^n \frac{1}{q_j}, \quad \frac{n}{p^*} \le \sum \limits _{j=1}^n \frac{1}{q_j^*}, \quad \frac{n}{s} \le \sum \limits _{j=1}^n \frac{1}{t_j}. \end{aligned}$$

Assume

$$\begin{aligned} \frac{1}{s}=\frac{1}{p^*}+\frac{1}{p}, \quad \frac{1}{t_j}=\frac{1}{q_j^*}+\frac{1}{q_j}. \end{aligned}$$

Then

$$\begin{aligned} \Vert f \cdot g\Vert _{{{\mathcal {M}}}^s_{\mathbf {t}}} \le \Vert f\Vert _{{\mathcal M}^{p}_{\mathbf {q}}} \Vert g\Vert _{{{\mathcal {M}}}^{p^*}_{\mathbf {q}^{\,*}}} \quad (f \in {{\mathcal {M}}}^p_{\mathbf {q}}({{\mathbb {R}}}^n), g \in {\mathcal M}^{p^*}_{\mathbf {q}^{\,*}}({{\mathbb {R}}}^n)). \end{aligned}$$

We can prove this lemma easily by using Hölder’s inequality. So we omit the proof. We write $\infty '=1$ and $s'=\frac{s}{s-1}$ for $1<s< \infty$. We have the following proposition:

Proposition 3

In addition to the assumption in Theorem 6, suppose that $u \in (1,\infty ]$ satisfies $u'<\min \{q_1,q_2,\ldots ,q_n, p\}$. Let $\varOmega \in L^{u}({\mathbb {S}}^{n-1})$ be homogeneous of degree zero, that is, $\varOmega$ satisfies, for any $\lambda >0,$ $\varOmega (\lambda x)=\varOmega (x)$. Then,

$$\begin{aligned} \left\| g \cdot I_{\varOmega ,\alpha } (f) \right\| _{{\mathcal M}^s_{\mathbf {t}}}\le C\left\| g\right\| _{{\mathcal M}^{p^*}_{\mathbf {q}^{\,*}}} \left\| \varOmega \right\| _{L^{u}\left( {\mathbb {S}}^{n-1} \right) } \left\| f \right\| _{{\mathcal M}^p_{\mathbf {q}}}, \end{aligned}$$

where

$$\begin{aligned} I_{\varOmega ,\alpha }f(x)\equiv \int _{{\mathbb {R}}^{n}} \dfrac{\varOmega (x-y)}{|x-y|^{n-\alpha }} f(y)\mathrm{d}y. \end{aligned}$$

Proposition 3 is a direct consequence of Theorem 6, the next lemma and the boundedness of the Hardy–Littlewood maximal operator M.

Lemma 2

[12] If $1<u\le \infty ,$ then we have

$$\begin{aligned} |I_{\varOmega , \alpha } f(x)|\le C\left\| \varOmega \right\| _{L^{u}({\mathbb {S}}^{n-1})} |I_{\alpha }F(x)|, \end{aligned}$$

where $F(x)\equiv M\left( |f|^{u'} \right) (x)^{\frac{1}{u'}}$.

Hardy-mixed Morrey spaces admit a characterization by using the grand maximal operator. To formulate the result, we recall the following two fundamental notions [25].

• (1) Topologize ${{\mathcal {S}}}({{\mathbb {R}}}^n)$ by norms $\{p_N\}_{N \in {{\mathbb {N}}}}$ given by

  $$\begin{aligned} p_N(\varphi ) \equiv \sum \limits _{|\alpha |\le N}\sup _{x\in {\mathbb R}^n}(1+|x|)^N |\partial ^{\alpha }\varphi (x)| \end{aligned}$$

  for each $N \in {{\mathbb {N}}}$. Define ${{\mathcal {F}}}_N\equiv \{\varphi \in {{\mathcal {S}}}({{\mathbb {R}}}^n):p_N(\varphi )\le 1\}$.

• (2) Let $f \in {{\mathcal {S}}}'({{\mathbb {R}}}^n)$. The grand maximal operator ${{\mathcal {M}}}f$ is given by

  $$\begin{aligned} {{\mathcal {M}}}f(x)\equiv \sup \{|t^{-n}\psi (t^{-1}\cdot )*f(x)| \,:\,t>0, \, \psi \in {{\mathcal {F}}}_N\}\quad (x \in {{\mathbb {R}}}^n), \end{aligned}$$

  (7)

  where we choose and fix a large integer N.

The following proposition can be proved.

Proposition 4

Let $0<\mathbf {q}<\infty , 0<p<\infty ,$ and $\frac{n}{p}\le \sum \nolimits _{j=1}^n\frac{1}{q_j}$. Then

$$\begin{aligned} \Vert {{\mathcal {M}}}f\Vert _{{{\mathcal {M}}}^p_{\mathbf {q}}} \sim \Vert f\Vert _{H{\mathcal M}^p_{\mathbf {q}}} \end{aligned}$$

for all $f \in {{\mathcal {S}}}'({{\mathbb {R}}}^n)$.

When $p \le 1$ and $q_1=q_2=\cdots =q_n$, this proposition is contained in [14]. Here for the sake of convenience, we give the proof of Proposition 4 in Sect. 3.

We plan to prove our results in the following manner. First of all, we elaborate the proof of Theorem 1 in Sect. 2. Next, we concentrate on Theorem 2 in Sect. 4.1. Subsequently, based on the argument of the proof of Theorem 2, we prove Theorem 5 in Sect. 4.2. Necessary lemmas for the proofs are stated in each subsection. Finally, Sect. 5 is devoted to the proof of Theorem 6.

2 Proof of Theorem 1

We invoke a result due to Bagby [2].

Lemma 3

Let $1<q_1,\ldots ,q_m<\infty$ and $1<p<\infty$. For $i=1,2\ldots ,m,$ let $(\varOmega _i, \mu _i)$ be $\sigma$-finite measure spaces, and $\varOmega =\varOmega _1 \times \cdots \times \varOmega _m$. For $f \in L^0({\mathbb {R}}^n\times \varOmega ),$

$$\begin{aligned} \int _{{\mathbb {R}}^n}\left\| Mf(x,\cdot )\right\| _{L^{(q_1,\ldots ,q_m)}}^p\mathrm{d}x \lesssim \int _{{\mathbb {R}}^n}\left\| f(x,\cdot )\right\| _{L^{(q_1,\ldots ,q_m)}}^p\mathrm{d}x. \end{aligned}$$

The following lemma is used in the induction step (see [18, (12)]).

Here and below, for $t > 0$ and $j = 1,2,\ldots ,n$, we denote by $M_j$ the 1-dimensional maximal operator which acts on the j-th variable and write $M_j^{(t)}f=(M_{j}[|f|^{t}])^\frac{1}{t}$

Lemma 4

Let $\mathbf {q}=(q_1,q_2,\ldots ,q_n) \in (1, \infty )^n$ and let

$$\begin{aligned} t_n \in [1,\min \{q_1,q_2,\ldots ,q_n\}). \end{aligned}$$

Then

$$\begin{aligned} \left\| M^{(t_n)}_{n} f\right\| _{L^{\mathbf {q}}} \lesssim \Vert f\Vert _{L^{\mathbf {q}}} \end{aligned}$$

for all $f \in L^{\mathbf {q}}({{\mathbb {R}}}^n)$.

For the proof we use the following notation for $h \in L^0({\mathbb R}^n)$:

$$\begin{aligned} \left\| h\right\| _{L^{(q_1,\ldots ,q_m)}}(x_{m+1}, \ldots , x_n) \equiv \left\| \left[ \Vert h\Vert _{L^{(q_1,\ldots ,q_{m-1})}} \right] \right\| _{L^{(q_{m})}} (x_{m+1}, \ldots , x_n) \end{aligned}$$

and when $m=1$, we define

$$\begin{aligned} \Vert h\Vert _{L^{(q_1)}}(x_2, \ldots ,x_n) \equiv \left( \int _{{\mathbb {R}}}|h(x_1, \ldots , x_n)|^{q_1} \mathrm{d}x_1 \right) ^{\frac{1}{q_1}}. \end{aligned}$$

Proof

Thanks to Lemma 3, we obtain

$$\begin{aligned} \left\| M^{(t_n)}_{n} f\right\| _{L^{\mathbf {q}}}^{q_n}&=\int _{{\mathbb {R}}} \left\| M^{(t_n)}_{n}f(\cdot ,x_n)\right\| _{L^{(q_1, \ldots , q_{n-1})}}^{q_n} \mathrm{d}x_n\\&=\int _{{\mathbb {R}}} \left\| M_{n} [|f|^{t_n}](\cdot ,x_n)\right\| _{L^{\left( \frac{q_1}{t_n}, \ldots , \frac{q_{n-1}}{t_n}\right) }}^{\frac{q_n}{t_n}} \mathrm{d}x_n\\&\lesssim \int _{{\mathbb {R}}} \left\| [|f(\cdot ,x_n)|^{t_n}]\right\| _{L^{\left( \frac{q_1}{t_n}, \ldots , \frac{q_{n-1}}{t_n}\right) }}^{\frac{q_n}{t_n}} \mathrm{d}x_n =\Vert f\Vert _{L^{\mathbf {q}}}^{q_n}. \end{aligned}$$

Thus, we obtain the desired result. $\square$

Proof of of Theorem 1

We start with a preliminary observation for maximal operators. Let $x \in {{\mathbb {R}}}^n$. Let $Q=I_1 \times \cdots \times I_n$ where each $I_j$ is an interval in ${\mathbb {R}}$ with same length. Then,

$$\begin{aligned}&\frac{\chi _Q(x)}{\Vert \chi _Q\Vert _{L^{\mathbf {t}}}}\Vert f\chi _Q\Vert _{L^{\mathbf {t}}} =\frac{\bigotimes _{j=1}^n\chi _{I_j}(x)}{\prod \nolimits _{j=1}^n|I_j|^{\frac{1}{t_j}}} \left\| f\chi _{\prod \limits _{j=1}^nI_j}\right\| _{L^{\mathbf {t}}}\\&\quad =\frac{\bigotimes _{j=2}^n\chi _{I_j}(x_2, \ldots , x_n)}{\prod \nolimits _{j=2}^n|I_j|^{\frac{1}{t_j}}}\\&\qquad \times \left\| \left[ \left( \frac{\chi _{I_1}(x_1)}{|I_1|}\int |f(y_{1}, \cdot )|^{t_1}\chi _{I_1}(y_1)\mathrm{d}y_1 \right) ^{\frac{1}{t_1}} \right] \chi _{\prod \limits _{j=2}^nI_j}\right\| _{L^{(t_2, \ldots , t_n)}}\\&\quad \le \frac{\bigotimes _{j=2}^n\chi _{I_j}(x_2, \ldots , x_n)}{\prod \nolimits _{j=2}^n|I_j|^{\frac{1}{t_j}}} \left\| \left[ M^{(t_1)}_{1}f \right] \chi _{\prod \limits _{j=2}^nI_j}\right\| _{L^{(t_2, \ldots , t_n)}}.\\ \end{aligned}$$

Continuing this procedure, we have

$$\begin{aligned} \frac{\chi _Q(x)}{\Vert \chi _Q\Vert _{L^{\mathbf {t}}}}\Vert f\chi _Q\Vert _{L^{\mathbf {t}}} \le M_n^{(t_n)} \cdots M_1^{(t_1)} (f) (x). \end{aligned}$$

Thus, it follows that

$$\begin{aligned} M^{(\mathbf {t})}f(x)\le M_n^{(t_n)} \cdots M_1^{(t_1)} (f) (x). \end{aligned}$$

Therefore, it suffices to show that

$$\begin{aligned} \left\| M_n^{(t_n)} \cdots M_1^{(t_1)} (f)\right\| _{L^{\mathbf {q}}} \lesssim \Vert f\Vert _{L^{\mathbf {q}}}. \end{aligned}$$

(8)

We proceed by induction on n. For $n=1$, the result follows by the classical case of the boundedness of the Hardy–Littlewood maximal operator.

Suppose that the result holds for $n=m-1$ with $m>1$ in ${\mathbb {N}}$: assume that

$$\begin{aligned} \Vert M_{m-1}^{(t_{m-1})} \cdots M_{1}^{(t_1)} h\Vert _{L^{(q_1, \ldots , q_{m-1})}} \lesssim \Vert h\Vert _{L^{(q_1, \ldots , q_{m-1})}} \end{aligned}$$

for $1<t_k<\min \{q_1, \ldots , q_k\}<\infty$ for each $k=1, \ldots , m-1$, and for $h \in L^0({{\mathbb {R}}}^{m-1})$. Since $t_m<\min \{q_1, \ldots , q_m\}$, for $g \in L^0({\mathbb {R}}^m)$, thanks to Lemma 4 we have

$$\begin{aligned} \left\| M_m^{(t_m)} g\right\| _{L^{(q_1, \ldots , q_m)}}&= \left\| \left[ \left\| M_m^{(t_m)}g\right\| _{L^{(q_1,\ldots ,q_{m-1})}}\right] \right\| _{L^{(q_m)}}\\&=\left\| \left[ \left\| M_m[|g|^{t_m}]\right\| _{L^{\left( \frac{q_1}{t_m},\ldots ,\frac{q_{m-1}}{t_m}\right) }}\right] \right\| _{L^{\left( \frac{q_m}{t_m}\right) }}^{\frac{1}{t_m}} \\&\lesssim \left\| \left[ \left\| g\right\| _{L^{(q_1,\ldots ,q_{m-1})}}\right] \right\| _{L^{(q_m)}} = \Vert g\Vert _{L^{(q_1, \ldots , q_m)}}. \end{aligned}$$

Thus, by the induction assumption, letting $g=M_{m-1}^{(t_{m-1})} \cdots M_1^{(t_1)} (f)$ in the above, we obtain

$$\begin{aligned} \left\| M_m^{(t_m)} \cdots M_1^{(t_1)} (f)\right\| _{L^{(q_1, \ldots , q_m)}}&= \left\| M_m^{(t_m)}\left[ M_{m-1}^{(t_{m-1})} \cdots M_1^{(t_1)} (f)\right] \right\| _{L^{(q_1, \ldots , q_m)}}\\&\lesssim \left\| M_{m-1}^{(t_{m-1})} \cdots M_1^{(t_1)} (f)\right\| _{L^{(q_1, \ldots , q_m)}}\\&= \left\| \left\| M_{m-1}^{(t_{m-1})} \cdots M_1^{(t_1)} (f)\right\| _{L^{(q_1,\ldots ,q_{m-1})}}\right\| _{L^{(q_m)}}\\&\lesssim \left\| \left\| f\right\| _{L^{(q_1,\ldots ,q_{m-1})}}\right\| _{L^{(q_m)}} \lesssim \Vert f\Vert _{L^{(q_1, \ldots , q_m)}}. \end{aligned}$$

Hence, inequality (8) holds for any dimension n. We obtain the desired result. $\square$

One can show that the condition

$$\begin{aligned} t_k<\min \{q_1,q_2,\ldots ,q_k\} \end{aligned}$$

is sharp.

Proposition 5

In Theorem 1, for each $k=1,2,\ldots ,n,$ the condition $t_k<\min \{q_1,q_2,\ldots ,q_k\}$ can not be removed.

Proof

We induct on n. The base case $n=1$ is clear since the Hardy–Littlewood maximal operator is bounded on $L^p({{\mathbb {R}}})$ if and only if $p>1$. Assume that the conclusion of Proposition 5 is true for $n=m-1$ and that $M^{(t_1,t_2,\ldots ,t_m)}$ is bounded on $L^{(q_1,q_2,\ldots ,q_m)}({{\mathbb {R}}}^m)$. Let $h \in L^{(t_1,t_2,\ldots ,t_{m-1})}({{\mathbb {R}}}^{m-1})$ and $N \in {\mathbb N}$. Then

$$\begin{aligned} \chi _{[-N,N]^m}&\left( M^{(t_1,t_2,\ldots ,t_{m-1})}\left[ \chi _{[-N,N]^{m-1}}h\right] \otimes \chi _{[-N,N]}\right) \\&\quad \le M^{(t_1,t_2,\ldots ,t_{m})}\left[ (\chi _{[-N,N]^{m-1}}h) \otimes \chi _{[-N,N]}\right] . \end{aligned}$$

Consequently,

$$\begin{aligned}&(2N)^{\frac{1}{q_m}} \left\| \chi _{[-N,N]^{m-1}} M^{(t_1,t_2,\ldots ,t_{m-1})}\left[ \chi _{[-N,N]^{m-1}}h\right] \right\| _{L^{(q_1,q_2,\ldots ,q_{m-1})}} \\&\quad = \left\| \chi _{[-N,N]^m} M^{(t_1,t_2,\ldots ,t_{m-1})}\left[ \chi _{[-N,N]^{m-1}}h\right] \otimes \chi _{[-N,N]} \right\| _{L^{(q_1,q_2,\ldots ,q_m)}}\\&\quad \le \left\| M^{(t_1,t_2,\ldots ,t_{m-1})}\left[ \chi _{[-N,N]^{m-1}}h\right] \otimes \chi _{[-N,N]} \right\| _{L^{(q_1,q_2,\ldots ,q_m)}}\\&\quad \le C \left\| \left( \chi _{[-N,N]^{m-1}}h\right) \otimes \chi _{[-N,N]} \right\| _{L^{(q_1,q_2,\ldots ,q_m)}}\\&\quad \le C (2N)^{\frac{1}{q_m}} \Vert h\Vert _{L^{(q_1,q_2,\ldots ,q_{m-1})}}. \end{aligned}$$

So, we are led to

$$\begin{aligned}&\left\| \chi _{[-N,N]^{m-1}} M^{(t_1,t_2,\ldots ,t_{m-1})}[\chi _{[-N,N]^{m-1}}h] \right\| _{L^{(q_1,q_2,\ldots ,q_{m-1})}}\\&\quad \le C \Vert h\Vert _{L^{(q_1,q_2,\ldots ,q_{m-1})}}. \end{aligned}$$

Letting $N \rightarrow \infty$, we obtain

$$\begin{aligned} \left\| M^{(t_1,t_2,\ldots ,t_{m-1})}h \right\| _{L^{(q_1,q_2,\ldots ,q_{m-1})}} \le C \Vert h\Vert _{L^{(q_1,q_2,\ldots ,q_{m-1})}}. \end{aligned}$$

By the induction assumption, we have $t_k<\min \{q_1,q_2,\ldots ,q_k\}$ for all $k=1,2,\ldots ,m-1$. If we start from the inequality

$$\begin{aligned}&{ \chi _{[-N,N]^m} \left( \chi _{[-N,N]} \otimes M^{(t_1,t_2,\ldots ,t_{m-1})}\left[ \chi _{[-N,N]^{m-1}}h\right] \right) }\\&\quad \le M^{(t_1,t_2,\ldots ,t_{m})}\left[ \chi _{[-N,N]} \otimes \left( \chi _{[-N,N]^{m-1}}h\right) \right] , \end{aligned}$$

and argue similarly, we obtain

$$\begin{aligned} \left\| M^{(t_2,t_3,\ldots ,t_{m})}h \right\| _{L^{(q_2,q_3,\ldots ,q_{m})}} \le C \Vert h\Vert _{L^{(q_2,q_3,\ldots ,q_{m})}}. \end{aligned}$$

Thus $t_m<\min (q_2,q_3,\ldots ,q_m)$ by the induction assumption. It remains to show that $t_m<q_1$. To this end, we consider the function of the form:

$$\begin{aligned} f(x_1,x_2,\ldots ,x_m)=\sum \limits _{j=-\infty }^\infty \chi _{\left( [jN,(j+1)N] \times [-N,N]^{m-1}\right) }(x_1,x_2,\ldots ,x_m) h_{j}(x_m), \end{aligned}$$

where $h_j \in L^{q_m}({\mathbb {R}})$. Then for all $(x_1,x_2,\ldots ,x_m)$

$$\begin{aligned}&{ \chi _{\left( {{\mathbb {R}}} \times [-N,N]^{m-1}\right) }(x_1,x_2,\ldots ,x_m)M^{(\mathbf {t})}f(x_1,x_2,\ldots ,x_m) }\\&\quad \ge \sum \limits _{j=-\infty }^\infty \chi _{\left( [jN,(j+1)N] \times [-N,N]^{m-1}\right) }(x_1,x_2,\ldots ,x_m) M^{(t_m)}\left[ \chi _{[-N,N]}h_{j}\right] (x_m). \end{aligned}$$

We abbreviate

$$\begin{aligned} H_m(x)\equiv M^{(t_m)}\left[ \chi _{[-N,N]}h_{j}\right] (x_m). \end{aligned}$$

Hence, we obtain

$$\begin{aligned}&\left\| \sum \limits _{j=-\infty }^\infty \chi _{\left( [jN,(j+1)N] \times [-N,N]^{m-1}\right) }M^{(t_m)}\left[ \chi _{[-N,N]}h_{j}\right] \right\| _{L^{\mathbf {q}}}\\&\quad = \left\| \left( \sum \limits _{j=-\infty }^\infty \int _{{\mathbb {R}}} \chi _{[jN,(j+1)N]} \left( H_m(\cdot _m) \right) ^{q_1} \mathrm{d}x_1 \right) ^\frac{1}{q_1} \chi _{[-N,N]^{m-1}} \right\| _{L^{(q_2, \ldots , q_m)}}\\&\quad \sim (2N)^{\frac{1}{q_1}+\cdots +\frac{1}{q_{m-1}}} \left\| \left( \sum \limits _{j=-\infty }^\infty \left( H_m(\cdot _m) \right) ^{q_1} \right) ^\frac{1}{q_1} \right\| _{L^{q_m}}. \end{aligned}$$

In the same way, we deduce

$$\begin{aligned}&{ \left\| \chi _{\left( {{\mathbb {R}}} \times [-N,N]^{m-1}\right) }M^{(\mathbf {t})}f \right\| _{L^{\mathbf {q}}} }\\&\quad \lesssim (2N)^{\frac{1}{q_1}+\cdots +\frac{1}{q_{m-1}}} \left\| \left( \sum \limits _{j=-\infty }^\infty \left( |\chi _{[-N,N]}h_{j}(\cdot _m)| \right) ^{q_1} \right) ^\frac{1}{q_1} \right\| _{L^{q_m}}, \end{aligned}$$

since $M^{(\mathbf {t})}$ is bounded. Thus, letting $N\rightarrow \infty$, we obtain

$$\begin{aligned} \left\| \{M^{(t_m)}h_{j}\}_{j=-\infty }^\infty \right\| _{L^{q_m}(\ell ^{q_1})} \le \left\| \{h_{j}\}_{j=-\infty }^\infty \right\| _{L^{q_m}(\ell ^{q_1})}. \end{aligned}$$

This forces $q_1>t_m$ (see [25, p. 75, § 5.1]). $\square$

3 Proof of Propositions 1 and 4

3.1 Proof of Proposition 1

To prove Proposition 1, we need the description of the (pre) dual spaces of mixed Morrey spaces [19]. Recall that when $1<p<\infty$ and $1<\mathbf {q}<\infty$ satisfy

$$\begin{aligned} \frac{n}{p} \le \sum \limits _{j=1}^n \frac{1}{q_j}, \end{aligned}$$

then the predual space ${{\mathcal {H}}}^{p'}_{\mathbf {q}{\,'}}({\mathbb R}^n)$ of the mixed Morrey space ${{\mathcal {M}}}^p_{\mathbf {q}}({\mathbb R}^n)$ is given by

$$\begin{aligned} {{\mathcal {H}}}^{p'}_{\mathbf {q}{\,'}}({{\mathbb {R}}}^n) = \left\{ g=\sum \limits _{j=1}^\infty \mu _j b_j\,:\, \{\mu _j\}_{j=1}^\infty \in \ell ^1({{\mathbb {N}}}), \, \text{ each } b_j \text{ is } \text{ a } (p',\mathbf {q}{\,'})\text{-block } \right\} . \end{aligned}$$

Here by “a $(p',\mathbf {q}{\,'})$-block” we mean an $L^{\mathbf {q}{\,'}}({{\mathbb {R}}}^n)$-function supported on a cube Q with $L^{\mathbf {q}{\,'}}({{\mathbb {R}}}^n)$-norm lesser or equal to $|Q|^{\frac{1}{n}\sum \nolimits _{j=1}^n \frac{1}{q_j'}-\frac{1}{p'}}$. The norm of ${{\mathcal {H}}}^{p'}_{\mathbf {q}{\,'}}({{\mathbb {R}}}^n)$ is defined by

$$\begin{aligned} \Vert g\Vert _{{{\mathcal {H}}}^{p'}_{\mathbf {q}{\,'}}} = \inf \sum \limits _{j=1}^\infty |\mu _j|, \end{aligned}$$

where $\inf$ is over all admissible expressions above. A fundamental fact about this space is that ${\mathcal H}^{p'}_{\mathbf {q}{\,'}}({{\mathbb {R}}}^n)$ is separable, that the dual of ${{\mathcal {H}}}^{p'}_{\mathbf {q}{\,'}}({{\mathbb {R}}}^n)$ is cannonically identified with ${{\mathcal {M}}}^{p}_{\mathbf {q}}({{\mathbb {R}}}^n)$ and that

$$\begin{aligned} \Vert f\Vert _{{{\mathcal {M}}}^p_{\mathbf {q}}} = \sup \left\{ \Vert f \cdot g\Vert _{L^1} \,:\,\Vert g\Vert _{{{\mathcal {H}}}^{p'}_{\mathbf {q}{\,'}}}=1\right\} . \end{aligned}$$

Proposition 1 was investigated by Long [16] and Zorko [29] when $q_j=q$ for all $j=1,\ldots ,n$; see [15] as well. We refer to [1, 9], and [19] for more recent characterizations of the predual spaces.

Example 1

Suppose that $1\le t_k'<\min (q_1',q_2',\ldots ,q_k')<\infty$. If we let $\kappa$ be the operator norm of the maximal operator $M^{(\mathbf {t}{\,'})}$ on $L^{\mathbf {q}{\,'}}({{\mathbb {R}}}^n)$, whose finiteness is guaranteed by Theorem 1, then we obtain $\kappa ^{-1}\chi _{Q}M^{(\mathbf {t}{\,'})}g$ is a $(p',\mathbf {q}{\,'})$-block modulo a multiplicative constant for any $(p',{\mathbf {q}{\,'}})$-block g. Indeed, it is supported on a cube Q and it satisfies

$$\begin{aligned} \left\| \kappa ^{-1}\chi _{Q}M^{(\mathbf {t}{\,'})}g \right\| _{L^{\mathbf {q}{\,'}}}&\le \Vert \chi _{Q}g\Vert _{L^{\mathbf {q}{\,'}}}= \Vert g\Vert _{L^{\mathbf {q}{\,'}}}\le |Q|^{\frac{1}{n}\sum \nolimits _{j=1}^n \frac{1}{q_j'}-\frac{1}{p'}}. \end{aligned}$$

Proof of Proposition 1

• (1) Denote by $B(R)=\{x \in {{\mathbb {R}}}^n\,:\,|x|<R\}$ for $R>0$. Since

  $$\begin{aligned} \Vert f\Vert _{L^1(B(R))} \le CR^{-\frac{n}{p}+n}\Vert f\Vert _{{{\mathcal {M}}}^p_{\mathbf {q}}}, \end{aligned}$$

  we have $f \in {{\mathcal {S}}}'({{\mathbb {R}}}^n)$. As is described in [6], we have a pointwise estimate $|e^{t\Delta }f| \le Mf$, where M denotes the Hardy–Littlewood maximal operator. Since M is shown to be bounded in [4], we have $f \in H{\mathcal M}^p_{\mathbf {q}}({{\mathbb {R}}}^n)$.

• (2) Let $f \in H{{\mathcal {M}}}^p_{\mathbf {q}}({{\mathbb {R}}}^n)$. Then $\{e^{t\Delta }f\}_{t>0}$ is a bounded set of ${\mathcal M}^p_{\mathbf {q}}({{\mathbb {R}}}^n)$, which admits a separable predual as we have seen. Therefore, there exists a sequence $\{t_j\}_{j=1}^\infty$ decreasing to 0 such that $\{e^{t_j\Delta }f\}_{j=1}^\infty$ converges to a function g in the weak-* topology of ${{\mathcal {M}}}^p_{\mathbf {q}}({{\mathbb {R}}}^n)$. Meanwhile, it can be shown that $\lim _{t \downarrow 0}e^{t\Delta }f=f$ in the topology of ${{\mathcal {S}}}'({{\mathbb {R}}}^n)$ [21]. Since the weak-* topology of ${\mathcal M}^p_{\mathbf {q}}({{\mathbb {R}}}^n)$ is stronger than the topology of ${{\mathcal {S}}}'({{\mathbb {R}}}^n)$, it follows that $f=g \in {\mathcal M}^p_{\mathbf {q}}({{\mathbb {R}}}^n)$.

3.2 Proof of Proposition 4

The proof is similar to Hardy spaces with variable exponents [5, 17]. We content ourselves with stating two fundamental estimates (13) and (14).

We define the (discrete) maximal function with respect to $e^{t\Delta }$ by

$$\begin{aligned} M_{\mathrm{heat}}f(x) \equiv \sup _{j\in {{\mathbb {Z}}}} |e^{2^j\Delta }f(x)| \quad (x \in {{\mathbb {R}}}^n). \end{aligned}$$

(9)

Recall that, for $f\in {{\mathcal {S}}}'({{\mathbb {R}}}^n)$, the grand maximal function is defined by

$$\begin{aligned} {{\mathcal {M}}} f(x)\equiv \sup \{|t^{-n}\psi (t^{-1}\cdot )*f(x)| \,:\,t>0, \, \psi \in {\mathcal F}_N\} \quad (x \in {{\mathbb {R}}}^n), \end{aligned}$$

(10)

where ${{\mathcal {F}}}_N$ is given by

$$\begin{aligned} {{\mathcal {F}}}_N\equiv \{\varphi \in {{\mathcal {S}}}({\mathbb R}^n):p_N(\varphi )\le 1\}. \end{aligned}$$

(11)

Suppose that we are given an integer $K \gg 1$. We write

$$\begin{aligned} M^*_{\mathrm{heat}} f(x) \equiv \sup _{j \in {{\mathbb {Z}}}} \left( \sup _{y \in {{\mathbb {R}}}^n} \frac{|e^{2^j\Delta }f(y)|}{(1+4^j|x-y|^2)^K}\right) \quad (x \in {{\mathbb {R}}}^n). \end{aligned}$$

(12)

The next lemma connects $M^*_{\mathrm{heat}}$ with $M_{\mathrm{heat}}$ in terms of the usual Hardy–Littlewood maximal function M.

Lemma 5

( [17, Lemma 3.2], [20, §4]) For $0<\theta <1,$ there exists $K_\theta$ so that for all $K \ge K_\theta ,$ we have

$$\begin{aligned} M^*_{\mathrm{heat}} f(x) \le C M^{(\theta )}[M_{\mathrm{heat}} f](x) =C M\left[ \sup _{k \in {{\mathbb {Z}}}}|e^{2^k\Delta }f|^\theta \right] (x)^{\frac{1}{\theta }} \quad (x \in {{\mathbb {R}}}^n) \end{aligned}$$

(13)

for any $f\in {{\mathcal {S}}}'({{\mathbb {R}}}^n),$ where $M^{(\theta )}$ is the powered maximal operator given by

$$\begin{aligned} M^{(\theta )}g(x)\equiv M[|g|^\theta ](x)^{\frac{1}{\theta }} \quad (x \in {{\mathbb {R}}}^n) \end{aligned}$$

for measurable functions g.

In the course of the proof of [17, Theorem 3.3], it can be shown that

$$\begin{aligned} {{\mathcal {M}}}f(x) \sim \sup _{\tau \in {{\mathcal {F}}}_N, \, j \in {{\mathbb {Z}}}}|\tau ^j*f(x)| \lesssim M^*_{\mathrm{heat}} f(x) \end{aligned}$$

(14)

once we fix an integer $K \gg 1$ and $N \gg 1$.

With the fundamental pointwise estimates (13) and (14), Proposition 4 can be proved with ease. We omit the details.

4 Proofs of Theorems 2–5

4.1 Proof of Theorem 2

By decomposing $Q_j$ suitably, we may suppose each $Q_j$ is dyadic.

To prove this, we resort to the duality. For the time being, we assume that there exists $N \in {{\mathbb {N}}}$ such that $\lambda _j=0$ whenever $j \ge N$. Let us assume in addition that $a_j$ are non-negative. Fix a non-negative $(p',\mathbf {q}{\,'})$-block $g \in {{\mathcal {H}}}^{p'}_{\mathbf {q}{\,'}}({{\mathbb {R}}}^n)$ with the associated cube Q.

Assume first that each $Q_j$ contains Q as a proper subset. If we group j’s such that $Q_j$ are identical, we can assume that $Q_j$ is the jth dyadic parent of Q for each $j \in {{\mathbb {N}}}$. Then by the Hölder inequality [3]

$$\begin{aligned} \int _{{{\mathbb {R}}}^n}f(x)g(x)\,\mathrm{d}x = \sum \limits _{j=1}^\infty \lambda _j \int _{Q}a_j(x)g(x)\,\mathrm{d}x \le \sum \limits _{j=1}^\infty \lambda _j \Vert a_j\Vert _{L^{\mathbf {q}}(Q)}\Vert g\Vert _{L^{\mathbf {q}{\,'}}(Q)} \end{aligned}$$

from $f=\sum \nolimits _{j=1}^\infty \lambda _j a_j$. Due to the size condition of $a_j$ and g, we obtain

$$\begin{aligned} \int _{{{\mathbb {R}}}^n}f(x)g(x)\,\mathrm{d}x&\le \sum \limits _{j=1}^\infty \lambda _j |Q|^{\frac{1}{n}\sum \nolimits _{j=1}^n \frac{1}{q_j}-\frac{1}{s}}|Q_j|^{\frac{1}{s}}|Q|^{\frac{1}{n}\sum \nolimits _{j=1}^n \frac{1}{q_j'}-\frac{1}{p'}}\\&\le \sum \limits _{j=1}^\infty \lambda _j |Q|^{\frac{1}{p}-\frac{1}{s}}|Q_j|^{\frac{1}{s}}. \end{aligned}$$

Note that

$$\begin{aligned} \left\| \sum \limits _{j=1}^\infty \lambda _j \chi _{Q_j}\right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}} \ge \left\| \lambda _{j_0} \chi _{Q_{j_0}}\right\| _{{\mathcal M}^p_{\mathbf {q}}} = |Q_{j_0}|^{\frac{1}{p}}\lambda _{j_0} \end{aligned}$$

for each $j_0$. Consequently, it follows from the condition $p<s$ that

$$\begin{aligned} \int _{{{\mathbb {R}}}^n}f(x)g(x)\,\mathrm{d}x&\le \sum \limits _{j=1}^\infty |Q|^{\frac{1}{p}-\frac{1}{s}}|Q_j|^{\frac{1}{s}-\frac{1}{p}} \left\| \sum \limits _{j=1}^\infty \lambda _j \chi _{Q_j}\right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}} \le C \left\| \sum \limits _{j=1}^\infty \lambda _j \chi _{Q_j}\right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}}. \end{aligned}$$

Conversely assume that Q contains each $Q_j$. Then by the Hölder inequality

$$\begin{aligned} \int _{{{\mathbb {R}}}^n}f(x)g(x)\,\mathrm{d}x&= \sum \limits _{j=1}^\infty \lambda _j \int _{Q_j}a_j(x)g(x)\,\mathrm{d}x\le \sum \limits _{j=1}^\infty \lambda _j \Vert a_j\Vert _{L^{\mathbf {t}}(Q_j)}\Vert g\Vert _{L^{\mathbf {t}{\,'}}(Q_j)}. \end{aligned}$$

Thanks to the condition of $a_j$, we obtain

$$\begin{aligned} \int _{{{\mathbb {R}}}^n}f(x)g(x)\,\mathrm{d}x&\le \sum \limits _{j=1}^\infty \lambda _j |Q_j|^{\frac{1}{n}\sum \nolimits _{j=1}^n \frac{1}{t_j}-\frac{1}{s}}|Q_j|^{\frac{1}{s}}\Vert g\Vert _{L^{\mathbf {t}{\,'}}(Q_j)}. \end{aligned}$$

Thus, in terms of the maximal operator $M^{(\mathbf {t'})}$ defined in Theorem 1, we obtain

$$\begin{aligned} \int _{{{\mathbb {R}}}^n}f(x)g(x)\,\mathrm{d}x&\le \sum \limits _{j=1}^\infty \lambda _j |Q_j|\times \inf _{y \in Q_j}M^{(\mathbf {t}{\,'})}g(y)\\&\lesssim \int _{{{\mathbb {R}}}^n} \left( \sum \limits _{j=1}^\infty \lambda _j \chi _{Q_j}(y)\right) M^{(\mathbf {t}{\,'})}g(y)\,\mathrm{d}y\\&\le \int _{{{\mathbb {R}}}^n} \left( \sum \limits _{j=1}^\infty \lambda _j \chi _{Q_j}(y)\right) \chi _{Q}(y)M^{(\mathbf {t}{\,'})}g(y)\,\mathrm{d}y. \end{aligned}$$

Hence, by Example 1, we obtain

$$\begin{aligned} \displaystyle \int _{{{\mathbb {R}}}^n}f(x)g(x)\,\mathrm{d}x \le \kappa \left\| \sum \limits _{j=1}^\infty \lambda _j \chi _{Q_j} \right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}}. \end{aligned}$$

This is the desired result. Finally, we can remove the assumption that $\lambda _j=0$ for large j by the monotone convergence theorem. Thus, the proof is complete.

4.2 Proof of Theorem 5

Recall again that the grand maximal operator ${{\mathcal {M}}}$ was given by

$$\begin{aligned} {{\mathcal {M}}}f(x)= \sup \{ |\varphi _t*f(x)| \,:\,\varphi \in {\mathcal F}_N, \, t>0\} \quad (x \in {{\mathbb {R}}}^n). \end{aligned}$$

Then we know that

$$\begin{aligned} {{\mathcal {M}}}a_j(x) \le C \left( \chi _{3Q_j}(x)Ma_j(x)+(M\chi _{Q_j}(x))^{\frac{n+d_q+1}{n}} \right) , \end{aligned}$$

(15)

where $\displaystyle d_q=\left[ n\left( \frac{1}{v(\mathbf {q})}-1\right) \right]$ and $v(\mathbf {q})=\min (1, q_1, \ldots , q_n)$. See [17, (5.2)] for more details. The first term can be controlled by an argument similar to Theorem 2. The second term can be handled by using the Fefferman–Stein maximal inequality for mixed Morrey spaces [18].

Proposition 6

Let $1<\mathbf {q}, p<\infty ,$ $\frac{n}{p} \le \sum \nolimits _{j=1}^n \frac{1}{q_j},$ and $1<r \le \infty$. Then

$$\begin{aligned} \left\| \left( \sum \limits _{j=1}^\infty (Mf_j)^r \right) ^{1/r}\right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}} \le C \left\| \left( \sum \limits _{j=1}^\infty |f_j|^r \right) ^{1/r}\right\| _{{\mathcal M}^p_{\mathbf {q}}} \end{aligned}$$

for all sequences of measurable functions $\{f_j\}_{j=1}^\infty$.

See [24, Theorem 2.2], [27, Lemma 2.5] for the case of classical Morrey spaces.

Let us show Theorem 5. Using Proposition 4 and (15), we have

$$\begin{aligned} \Vert f\Vert _{H{{\mathcal {M}}}^p_{\mathbf {q}}}&\sim \Vert {{\mathcal {M}}}f\Vert _{{\mathcal M}^p_{\mathbf {q}}} \le \left\| \sum \limits _{j=1}^\infty \lambda _j {{\mathcal {M}}}a_j \right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}}\\&\lesssim \left\| \sum \limits _{j=1}^\infty \lambda _j \left( \chi _{3Q_j}Ma_j+(M\chi _{Q_j})^{\frac{n+d_q+1}{n}} \right) \right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}}\\&\lesssim \left\| \sum \limits _{j=1}^\infty \lambda _j \chi _{3Q_j}Ma_j \right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}} + \left\| \sum \limits _{j=1}^\infty \lambda _j(M\chi _{Q_j})^{\frac{n+d_q+1}{n}} \right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}} \equiv I_1+I_2. \end{aligned}$$

First, we consider $I_1$. The proof is similar to Theorem 2. For the sake of completeness, we supply the proof. Thanks to decomposing $Q_j$ suitably, we may suppose each $Q_j$ is dyadic. We will use duality again. We assume that there exists $N \in {{\mathbb {N}}}$ such that $\lambda _j=0$ whenever $j \ge N$. Let $\displaystyle r=\frac{p}{v(\mathbf {q})}$ and $\displaystyle \mathbf {w}=\frac{\mathbf {q}}{v(\mathbf {q})}$, so that $r, \mathbf {w}>1$. Then,

$$\begin{aligned} \left\| \sum \limits _{j=1}^\infty \lambda _j \chi _{3Q_j}Ma_j \right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}}&\le \left\| \left( \sum \limits _{j=1}^\infty \left[ \lambda _j \chi _{3Q_j}Ma_j \right] ^{v(\mathbf {q})} \right) ^{\frac{1}{v(\mathbf {q})}} \right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}}\\&= \left\| \sum \limits _{j=1}^\infty \left[ \lambda _j \chi _{3Q_j}Ma_j \right] ^{v(\mathbf {q})} \right\| _{{\mathcal M}^r_{\mathbf {w}}}^{\frac{1}{v(\mathbf {q})}}. \end{aligned}$$

Fix a non-negative $(r',\mathbf {w}{\,'})$-block $g \in {\mathcal H}^{r'}_{\mathbf {w}{\,'}}({{\mathbb {R}}}^n)$ with the associated cube Q. Assume first that each $Q_j$ contains Q as a proper subset. If we group j’s such that $Q_j$ are identical, we can assume that $Q_j$ is the jth dyadic parent of Q for each $j \in {{\mathbb {N}}}$. Then,

$$\begin{aligned}&\int _{{\mathbb {R}}^n} \sum \limits _{j=1}^\infty \left[ \lambda _j \chi _{3Q_j}(x)Ma_j(x) \right] ^{v(\mathbf {q})} g(x) \mathrm{d}x = \sum \limits _{j=1}^\infty \lambda _j^{v(\mathbf {q})} \int _{Q} \left[ Ma_j(x) \right] ^{v(\mathbf {q})} g(x) \mathrm{d}x\\&\quad \le \sum \limits _{j=1}^\infty \lambda _j^{v(\mathbf {q})} \left\| \left[ Ma_j \right] ^{v(\mathbf {q})} \right\| _{L^{\mathbf {w}}(Q)} \Vert g\Vert _{L^{\mathbf {w}{\,'}}(Q)}\\&\quad \le \sum \limits _{j=1}^\infty \lambda _j^{v(\mathbf {q})} \left\| Ma_j \right\| _{L^{^{\mathbf {q}}}(Q)}^{v(\mathbf {q})} \Vert g\Vert _{L^{\mathbf {w}{\,'}}(Q)}\\&\quad \le \sum \limits _{j=1}^\infty \lambda _j^{v(\mathbf {q})} \left[ \left\| Ma_j \right\| _{L^{\mathbf {t}}(Q)} |Q|^{\frac{1}{n}\sum \nolimits _{j=1}^n\left( \frac{1}{q_j}-\frac{1}{t_j}\right) } \right] ^{v(\mathbf {q})} \Vert g\Vert _{L^{\mathbf {w}{\,'}}(Q)}. \end{aligned}$$

Using the boundedness of the Hardy–Littlewood maximal operator on ${{\mathcal {M}}}^{s}_{\mathbf {t}}({{\mathbb {R}}}^n)$, we have

$$\begin{aligned}&\int _{{\mathbb {R}}^n} \sum \limits _{j=1}^\infty \left[ \lambda _j \chi _{3Q_j}(x)Ma_j(x) \right] ^{v(\mathbf {q})} g(x) \mathrm{d}x\\&\quad \le \sum \limits _{j=1}^\infty \lambda _j^{v(\mathbf {q})} \left[ \left\| Ma_j \right\| _{L^{\mathbf {t}}(Q)} |Q|^{\frac{1}{n}\sum \nolimits _{j=1}^n\left( \frac{1}{q_j}-\frac{1}{t_j}\right) } \right] ^{v(\mathbf {q})} \Vert g\Vert _{L^{\mathbf {w}{\,'}}(Q)}\\&\quad \le \sum \limits _{j=1}^\infty \lambda _j^{v(\mathbf {q})} |Q|^{\frac{1}{n}\sum \nolimits _{j=1}^n\frac{1}{w_j}} \left[ |Q|^{-\frac{1}{s}} \left\| Ma_j \right\| _{{\mathcal M}^{s}_{\mathbf {t}}} \right] ^{v(\mathbf {q})} \Vert g\Vert _{L^{\mathbf {w}{\,'}}(Q)} \\&\quad \lesssim \sum \limits _{j=1}^\infty \lambda _j^{v(\mathbf {q})} |Q|^{\frac{1}{n}\sum \nolimits _{j=1}^n\frac{1}{w_j}} \left[ |Q|^{-\frac{1}{s}} \left\| a_j \right\| _{{{\mathcal {M}}}^{s}_{\mathbf {t}}} \right] ^{v(\mathbf {q})} \Vert g\Vert _{L^{\mathbf {w}{\,'}}(Q)}. \end{aligned}$$

Thus, using the size condition of $a_j$ and g, we obtain

$$\begin{aligned}&\left( \int _{{\mathbb {R}}^n} \sum \limits _{j=1}^\infty \left[ \lambda _j \chi _{3Q_j}(x)Ma_j(x) \right] ^{v(\mathbf {q})} g(x) \mathrm{d}x \right) ^{\frac{1}{v(\mathbf {q})}}\\&\quad \lesssim \left( \sum \limits _{j=1}^\infty \lambda _j^{v(\mathbf {q})} |Q|^{\frac{1}{n}\sum \nolimits _{j=1}^n\frac{1}{w_j}} \left[ |Q|^{-\frac{1}{s}} |Q_j|^{\frac{1}{s}} \right] ^{v(\mathbf {q})} |Q|^{\frac{1}{n}\sum \nolimits _{j=1}^n\frac{1}{w_j'}-\frac{1}{r'}} \right) ^{\frac{1}{v(\mathbf {q})}}\\&\quad = |Q|^{\frac{1}{p}-\frac{1}{s}} \left( \sum \limits _{j=1}^\infty \left[ \lambda _j |Q_j|^{\frac{1}{s}} \right] ^{v(\mathbf {q})} \right) ^{\frac{1}{v(\mathbf {q})}}. \end{aligned}$$

Note that

$$\begin{aligned} \left\| \left( \sum \limits _{j=1}^\infty ( \lambda _j \chi _{Q_j})^{v({\mathbf {q}})} \right) ^{\frac{1}{v({\mathbf {q}})}} \right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}} \ge \left\| \lambda _{j_0} \chi _{Q_{j_0}} \right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}} = \lambda _{j_0}|Q_{j_0}|^{\frac{1}{p}} \end{aligned}$$

for each $j_0 \in {\mathbb {N}}$. Thus,

$$\begin{aligned}&\left( \int _{{\mathbb {R}}^n} \sum \limits _{j=1}^\infty \left[ \lambda _j \chi _{3Q_j}(x)Ma_j(x) \right] ^{v(\mathbf {q})} g(x) \mathrm{d}x \right) ^{\frac{1}{v(\mathbf {q})}}\\&\quad \lesssim \sum \limits _{k=1}^\infty |Q|^{\frac{1}{p}-\frac{1}{s}} |Q_k|^{\frac{1}{s}-\frac{1}{p}} \left\| \left( \sum \limits _{j=1}^\infty ( \lambda _j \chi _{Q_j})^{v({\mathbf {q}})} \right) ^{\frac{1}{v({\mathbf {q}})}} \right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}}\\&\quad \sim \left\| \left( \sum \limits _{j=1}^\infty ( \lambda _j \chi _{Q_j})^{v({\mathbf {q}})} \right) ^{\frac{1}{v({\mathbf {q}})}} \right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}}. \end{aligned}$$

Conversely assume that Q contains each $Q_j$. Then by the Hölder inequality and the boundedness of the Hardy–Littlewood maximal operator on $L^{\mathbf {t}}({{\mathbb {R}}}^n)$,

$$\begin{aligned}&\int _{{\mathbb {R}}^n} \sum \limits _{j=1}^\infty \left[ \lambda _j \chi _{3Q_j}(x)Ma_j(x) \right] ^{v(\mathbf {q})} g(x) \mathrm{d}x\\&\quad = \sum \limits _{j=1}^\infty \lambda _j^{v(\mathbf {q})} \int _{3Q_j} \left[ Ma_j(x) \right] ^{v(\mathbf {q})} g(x) \mathrm{d}x\\&\quad \le \sum \limits _{j=1}^\infty \lambda _j^{v(\mathbf {q})} \left\| \left[ Ma_j \right] ^{v(\mathbf {q})} \right\| _{L^{\mathbf {\tau }}(3Q_j)} \Vert g\Vert _{L^{\mathbf {\tau }{\,'}}(3Q_j)} \quad \left( \mathbf {\tau }=\frac{\mathbf {t}}{v(\mathbf {q})}\right) \\&\quad \le \sum \limits _{j=1}^\infty \lambda _j^{v(\mathbf {q})} \left\| Ma_j \right\| _{L^{\mathbf {t}}(3Q_j)}^{v(\mathbf {q})} \Vert g\Vert _{L^{\mathbf {\tau }{\,'}}(3Q_j)}\\&\quad \le \sum \limits _{j=1}^\infty \lambda _j^{v(\mathbf {q})} \left\| a_j \right\| _{L^{\mathbf {t}}}^{v(\mathbf {q})} \Vert g\Vert _{L^{\mathbf {\tau }{\,'}}(3Q_j)}. \end{aligned}$$

Considering the condition of $a_j$, we obtain

$$\begin{aligned}&\int _{{\mathbb {R}}^n} \sum \limits _{j=1}^\infty \left[ \lambda _j \chi _{3Q_j}(x)Ma_j(x) \right] ^{v(\mathbf {q})} g(x) \mathrm{d}x\\&\quad \lesssim \sum \limits _{j=1}^\infty \lambda _j^{v(\mathbf {q})} \left[ |Q_j|^{\frac{1}{n}\sum \nolimits _{j=1}^n \frac{1}{t_j}-\frac{1}{s}} \left\| a_j \right\| _{{{\mathcal {M}}}^s_{\mathbf {t}}} \right] ^{v(\mathbf {q})} \Vert g\Vert _{L^{\mathbf {\tau }{\,'}}(3Q_j)}\\&\quad \le \sum \limits _{j=1}^\infty \lambda _j^{v(\mathbf {q})} \left[ |Q_j|^{\frac{1}{n}\sum \nolimits _{j=1}^n \frac{1}{t_j}} \right] ^{v(\mathbf {q})} \Vert g\Vert _{L^{\mathbf {\tau }{\,'}}(3Q_j)}. \end{aligned}$$

Thus, in terms of the maximal operator $M^{(\mathbf {t'})}$ defined in Theorem 1, we obtain

$$\begin{aligned}&\left( \int _{{\mathbb {R}}^n} \sum \limits _{j=1}^\infty \left[ \lambda _j \chi _{3Q_j}(x)Ma_j(x) \right] ^{v(\mathbf {q})} g(x) \mathrm{d}x \right) ^{\frac{1}{v(\mathbf {q})}}\\&\quad \le \left( \sum \limits _{j=1}^\infty \lambda _j^{v(\mathbf {q})} |Q_j|\times \inf _{y \in Q_j}M^{(\mathbf {\tau }{\,'})}g(y) \right) ^{\frac{1}{v(\mathbf {q})}}\\&\quad \le \left( \int _{{{\mathbb {R}}}^n} \left( \sum \limits _{j=1}^\infty \left( \lambda _j \chi _{Q_j}(y)\right) ^{v(\mathbf {q})} \right) M^{(\mathbf {\tau }{\,'})}g(y)\,\mathrm{d}y \right) ^{\frac{1}{v(\mathbf {q})}}\\&\quad \le \left( \int _{{{\mathbb {R}}}^n} \left( \sum \limits _{j=1}^\infty \left( \lambda _j \chi _{Q_j}(y)\right) ^{v(\mathbf {q})} \right) \chi _{Q}(y)M^{(\mathbf {\tau }{\,'})}g(y)\,\mathrm{d}y \right) ^{\frac{1}{v(\mathbf {q})}}. \end{aligned}$$

As in Example 1, ${\kappa }^{-1}\chi _QM^{(\mathbf {\tau }{\,'})}g$ is a $(r', \mathbf {w}{\,'})$-block as long as $\kappa$ is the operator norm of $M^{(\mathbf {\tau }{\,'})}$ on $L^{\mathbf {q}{\,'}}({{\mathbb {R}}}^n)$. Hence, we obtain

$$\begin{aligned}&\left( \int _{{\mathbb {R}}^n} \sum \limits _{j=1}^\infty \left[ \lambda _j \chi _{3Q_j}(x)Ma_j(x) \right] ^{v(\mathbf {q})} g(x) \mathrm{d}x \right) ^{\frac{1}{v(\mathbf {q})}}\\&\quad \lesssim {\kappa }^{\frac{1}{v({\mathbf {q}})}} \left\| \left( \sum \limits _{j=1}^\infty ( \lambda _j \chi _{Q_j})^{v({\mathbf {q}})} \right) \right\| _{{\mathcal M}^r_{\mathbf {w}}}^{\frac{1}{v({\mathbf {q}})}} = {\kappa }^{{\frac{1}{v({\mathbf {q}})}}} \left\| \left( \sum \limits _{j=1}^\infty ( \lambda _j \chi _{Q_j})^{v({\mathbf {q}})} \right) ^{\frac{1}{v({\mathbf {q}})}} \right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}}. \end{aligned}$$

Next, we consider $I_2$. Put

$$\begin{aligned} u=\frac{n+d_q+1}{n}p, \quad \mathbf {v}=\frac{n+d_q+1}{n}\mathbf {q}. \end{aligned}$$

Then, by Proposition 6 and the embedding $\ell {}^{v(\mathbf {q})} \hookrightarrow \ell ^1$, we have

$$\begin{aligned} I_2 = \left\| \left[ \sum \limits _{j=1}^\infty \lambda _j(M\chi _{Q_j})^{\frac{n+d_q+1}{n}} \right] ^{\frac{n}{n+d_q+1}} \right\| _{{\mathcal M}^u_{\mathbf {v}}}^{\frac{n+d_q+1}{n}}&\lesssim \left\| \sum \limits _{j=1}^\infty \lambda _j\chi _{Q_j} \right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}}\\&\le \left\| \left( \sum \limits _{j=1}^\infty ( \lambda _j \chi _{Q_j})^{v({\mathbf {q}})} \right) ^{\frac{1}{v({\mathbf {q}})}} \right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}}. \end{aligned}$$

Thus, we obtain the desired result.

4.3 Proof of Theorem 4

We outline the proof of Theorem 4 since this is similar to [13]. As in [21, Exercise 3.34], if $0<r<1$ and $f \in {{\mathcal {S}}}'({{\mathbb {R}}}^n) \cap L^1_{\mathrm{loc}}({{\mathbb {R}}}^n)$ satisfies $f \in {{\mathcal {M}}}^p_{\mathbf {q}}({{\mathbb {R}}}^n)$, then we can find $\{a_j\}_{j=1}^\infty \subset L^\infty ({{\mathbb {R}}}^n) \cap {{\mathcal {P}}}_L^\perp ({{\mathbb {R}}}^n)$ and a sequence $\{Q_j\}_{j=1}^\infty$ of cubes:

• (1) $\mathrm{supp}(a_j) \subset Q_j,$

• (2) $f=\sum \nolimits \nolimits _{j=1}^\infty a_j$ in ${{\mathcal {S}}}'({\mathbb R}^n)$,

• (3) $\displaystyle \left\{ \sum \nolimits _{j=1}^\infty (\Vert a_j\Vert _{L^\infty } \chi _{Q_j})^r \right\} ^{\frac{1}{r}} \lesssim {{\mathcal {M}}}f.$

Using this inequality, we can prove Theorem 4.

Proof of Theorem 4

Let $f \in H{{\mathcal {M}}}^p_{\mathbf {q}}({{\mathbb {R}}}^n)$. Then we consider the decomposition:

$$\begin{aligned} e^{t\Delta }f=\sum \limits _{Q \in {{\mathcal {D}}}}\lambda ^t_Q a^t_Q \end{aligned}$$

in the topology of ${\mathcal {S}}'({{\mathbb {R}}}^n)$, where $a^t_Q \in {{\mathcal {P}}}_K^\perp ({{\mathbb {R}}}^n)$, $\lambda ^t_Q \ge 0$ and

$$\begin{aligned} |a^t_Q| \le \chi _{3Q}, \quad \left\| \sum \limits _{Q \in {\mathcal D}} \lambda ^t_Q\chi _{3Q} \right\| _{{{\mathcal {M}}}^p_{\mathbf {q}}} \lesssim \Vert {{\mathcal {M}}}[e^{t\Delta }f] \Vert _{{{\mathcal {M}}}^p_{\mathbf {q}}} \lesssim \Vert {{\mathcal {M}}}f \Vert _{{{\mathcal {M}}}^p_{\mathbf {q}}}. \end{aligned}$$

Due to the weak-* compactness of the unit ball of $L^\infty ({\mathbb R}^n)$, there exists a sequence $\{t_l\}_{l=1}^\infty$ that converges to 0 such that

$$\begin{aligned} \lambda _Q=\lim _{l \rightarrow \infty }\lambda ^{t_l}_Q, \quad a_Q=\lim _{l \rightarrow \infty }a^{t_l}_Q \end{aligned}$$

exist for all $Q \in {{\mathcal {D}}}$ in the sense that

$$\begin{aligned} \lim _{l \rightarrow \infty } \int _{{{\mathbb {R}}}^n}a^{t_l}_Q(x)\varphi (x)\mathrm{d}x = \int _{{{\mathbb {R}}}^n}a_Q(x)\varphi (x)\mathrm{d}x \end{aligned}$$

for all $\varphi \in L^1({{\mathbb {R}}}^n)$. We claim

$$\begin{aligned} f=\sum \limits _{Q \in {{\mathcal {D}}}}\lambda _Q a_Q \end{aligned}$$

in the topology of ${\mathcal {S}}'({{\mathbb {R}}}^n)$. Let $\varphi \in {{\mathcal {S}}}({\mathbb {R}}^n)$ be a test function. Then we have

$$\begin{aligned} \langle f,\varphi \rangle = \lim _{l \rightarrow \infty } \langle e^{t_l\Delta }f,\varphi \rangle&= \lim _{l \rightarrow \infty } \sum \limits _{Q \in {{\mathcal {D}}}}\lambda ^{t_l}_Q \int _{{\mathbb R}^n}a^{t_l}_Q(x)\varphi (x)\mathrm{d}x \end{aligned}$$

from the definition of the convergence in ${{\mathcal {S}}}'({\mathbb R}^n)$. Once we fix m, we have

$$\begin{aligned} |\lambda ^{t_l}_Q| \lesssim \frac{\Vert {{\mathcal {M}}}f\Vert _{{\mathcal M}^p_{\mathbf {q}}}}{\Vert \chi _{[0,2^{-m})^n}\Vert _{{{\mathcal {M}}}^p_{\mathbf {q}}}} \end{aligned}$$

(16)

and

$$\begin{aligned} \left| \int _{{{\mathbb {R}}}^n}a^{t_l}_Q(x)\varphi (x)\mathrm{d}x\right| \le \int _{3Q} |\varphi (x)|\mathrm{d}x. \end{aligned}$$

Since

$$\begin{aligned} \sum \limits _{Q \in {{\mathcal {D}}}_m} \frac{\Vert {\mathcal M}f\Vert _{{{\mathcal {M}}}^p_{\mathbf {q}}}}{\Vert \chi _{[0,2^{-m})^n}\Vert _{{\mathcal M}^p_{\mathbf {q}}}} \int _{3Q} |\varphi (x)|\mathrm{d}x =3^n \frac{\Vert {{\mathcal {M}}}f\Vert _{{{\mathcal {M}}}^p_{\mathbf {q}}}}{\Vert \chi _{[0,2^{-m})^n}\Vert _{{{\mathcal {M}}}^p_{\mathbf {q}}}} \Vert \varphi \Vert _{L^1} <\infty , \end{aligned}$$

we are in the position of using the Fubini theorem to have

$$\begin{aligned} \sum \limits _{m\in {\mathbb {Z}}} \int _{{{\mathbb {R}}}^n} \left( \sum \limits _{Q \in {{\mathcal {D}}}_m} \lambda ^{t_l}_Q a^{t_l}_Q(x)\right) \varphi (x)\mathrm{d}x = \sum \limits _{m\in {\mathbb {Z}}} \sum \limits _{Q \in {{\mathcal {D}}}_m} \lambda ^{t_l}_Q \int _{{{\mathbb {R}}}^n} a^{t_l}_Q(x) \varphi (x)\mathrm{d}x. \end{aligned}$$

With this in mind, let us set

$$\begin{aligned} a_{m,l}\equiv \sum \limits _{Q \in {{\mathcal {D}}}_m}\lambda ^{t_l}_Q \int _{{{\mathbb {R}}}^n}a^{t_l}_Q(x)\varphi (x)\mathrm{d}x \end{aligned}$$

for each $m \in {{\mathbb {Z}}}$ and $l \in {{\mathbb {N}}}$. Then we have

$$\begin{aligned} |a_{m,l}| \le C2^{\frac{n m}{p}}\Vert {{\mathcal {M}}}f\Vert _{{\mathcal M}^p_{\mathbf {q}}} \Vert \varphi \Vert _{L^{1}} \quad (m \in {{\mathbb {Z}}}) \end{aligned}$$

(17)

thanks to (16).

Let $m \in {{\mathbb {Z}}}$. Then we have

$$\begin{aligned} a_{m,l}&= \sum \limits _{Q \in {{\mathcal {D}}}_m} \lambda _Q^{t_l} \int _{3Q}a_Q^{t_l}(x)\varphi (x)\mathrm{d}x\\&= \sum \limits _{Q \in {{\mathcal {D}}}_m} \lambda _Q^{t_l} \int _{3Q}a_Q^{t_l}(x) \left( \varphi (x)-\sum \limits _{|\beta | \le K} \frac{1}{\beta !}\partial ^{\beta }\varphi (c(Q))(x-c(Q))^{\beta } \right) \mathrm{d}x \end{aligned}$$

since $a_Q^{t_l} \in {{\mathcal {P}}}_K^\perp ({{\mathbb {R}}}^n)$. Thus, by the mean-value theorem, we have

$$\begin{aligned} |a_{m,l}| \le C(\varphi ) \sum \limits _{Q \in {{\mathcal {D}}}_m} |\lambda _Q^{t_l}|\ell (Q)^{n+K+1} \sup _{y \in 3Q} \frac{1}{1+|y|^{n+1}}. \end{aligned}$$

(18)

Here $C(\varphi )$ is a constant depending on $\varphi$.

Meanwhile, for each ${\tilde{m}} \in {{\mathbb {Z}}}^n$, we have

$$\begin{aligned} \left\| \sum \limits _{Q \in {{\mathcal {D}}}_m, |c(Q)-{\tilde{m}}| \le n} \lambda _Q^{t_l}\chi _Q\right\| _{{{\mathcal {M}}}^p_{q_0}} \lesssim \Vert {{\mathcal {M}}}f\Vert _{{{\mathcal {M}}}^p_{\mathbf {q}}}, \end{aligned}$$

which implies

$$\begin{aligned} \left\| \sum \limits _{Q \in {{\mathcal {D}}}_m, |c(Q)-{\tilde{m}}| \le n} \lambda _Q^{t_l}\chi _Q\right\| _{L^{q_0}} \lesssim \Vert {\mathcal M}f\Vert _{{{\mathcal {M}}}^p_{\mathbf {q}}} \end{aligned}$$

or equivalently

$$\begin{aligned} \left( \sum \limits _{Q \in {{\mathcal {D}}}_m, |c(Q)-{\tilde{m}}| \le n} 2^{-m n}|\lambda _Q^{t_l}|^{q_0} \right) ^{\frac{1}{q_0}} \lesssim \Vert {{\mathcal {M}}}f\Vert _{{{\mathcal {M}}}^p_{\mathbf {q}}}. \end{aligned}$$

Since $\ell ^{q_0}({{\mathbb {Z}}}^n) \hookrightarrow \ell ^1({\mathbb Z}^n)$,

$$\begin{aligned} \sum \limits _{Q \in {{\mathcal {D}}}_m, |c(Q)-{\tilde{m}}| \le n}|\lambda _Q^{t_l}| \lesssim 2^{\frac{m n}{q_0}} \Vert {\mathcal M}f\Vert _{{{\mathcal {M}}}^p_{\mathbf {q}}}. \end{aligned}$$

Combining this estimate with (18), we obtain

$$\begin{aligned} |a_{m,l}|&\lesssim \sum \limits _{{\tilde{m}} \in {\mathbb Z}^n} \sum \limits _{Q \in {{\mathcal {D}}}_m, |c(Q)-{\tilde{m}}| \le n} |\lambda _Q^{t_l}|\ell (Q)^{n+K+1} \sup _{y \in 3Q} \frac{1}{1+|y|^{n+1}} \nonumber \\&\sim \sum \limits _{{\tilde{m}} \in {{\mathbb {Z}}}^n} \sum \limits _{Q \in {{\mathcal {D}}}_m, |c(Q)-{\tilde{m}}| \le n} \frac{|\lambda _Q^{t_l}|\ell (Q)^{n+K+1}}{1+|{\tilde{m}}|^{n+1}} \nonumber \\&\lesssim 2^{\frac{m n}{q_0}-(n+K+1)m} \Vert {{\mathcal {M}}}f\Vert _{{\mathcal M}^p_{\mathbf {q}}}. \end{aligned}$$

(19)

Since $K+1>n\left( \frac{1}{q_0}-1\right)$, we obtain

$$\begin{aligned} n+K+1>\frac{n}{q_0}. \end{aligned}$$

Thus by (17) and (19), we obtain

$$\begin{aligned} |a_{m,l}| \lesssim \min \left( 2^{\frac{m n}{q_0}-(n+K+1)m},2^{\frac{m n}{p}}\right) . \end{aligned}$$

Since

$$\begin{aligned} \sum \limits _{m=-\infty }^\infty \min \left( 2^{\frac{m n}{q_0}-(n+K+1)m},2^{\frac{m n}{p}}\right) \lesssim 1, \end{aligned}$$

we are in the position of using the Lebesgue convergence theorem to have

$$\begin{aligned} \lim _{l \rightarrow \infty }\sum \limits _{m=-\infty }^\infty a_{m,l} = \sum \limits _{m=-\infty }^\infty \left( \lim _{l \rightarrow \infty }a_{m,l}\right) . \end{aligned}$$

That is,

$$\begin{aligned} \langle f,\varphi \rangle = \lim _{l \rightarrow \infty } \langle e^{t_l\Delta }f,\varphi \rangle = \sum \limits _{m=-\infty }^\infty \left( \lim _{l \rightarrow \infty } \sum \limits _{Q \in {\mathcal D}_m}\lambda ^{t_l}_Q \int _{{{\mathbb {R}}}^n}a^{t_l}_Q(x)\varphi (x)\mathrm{d}x \right) . \end{aligned}$$

Hence, using Fubini’s theorem again, we obtain

$$\begin{aligned} \langle f,\varphi \rangle&= \sum \limits _{m=-\infty }^\infty \left( \lim _{l \rightarrow \infty } \int _{{{\mathbb {R}}}^n} \left( \sum \limits _{Q \in {{\mathcal {D}}}_m}\lambda ^{t_l}_Q a^{t_l}_Q(x)\right) \varphi (x)\mathrm{d}x \right) \\&= \sum \limits _{m=-\infty }^\infty \sum \limits _{Q \in {{\mathcal {D}}}_m} \lim _{l \rightarrow \infty } \left( \int _{{{\mathbb {R}}}^n} \lambda ^{t_l}_Q a^{t_l}_Q(x)\varphi (x)\mathrm{d}x \right) \\&= \sum \limits _{m=-\infty }^\infty \sum \limits _{Q \in {{\mathcal {D}}}_m} \int _{{{\mathbb {R}}}^n} \lambda _Q a_Q(x)\varphi (x)\mathrm{d}x = \left< \sum \limits _{Q \in {{\mathcal {D}}}}\lambda _Q a_Q,\varphi \right>. \end{aligned}$$

Consequently, we obtain the desired result. $\square$

5 Proof of Theorem 6

First, we prove two lemmas. We invoke an estimate from [7, Lemma 2.2] and [8, Lemma 2.1].

Lemma 6

There exists a constant depending only on n and $\alpha$ such that, for every cube Q,  we have $I_\alpha \chi _Q(x) \ge C\ell (Q)^\alpha \chi _Q(x)$ for all $x \in Q$.

To prove the next estimate, we use Proposition 2. We invoke another estimate from [13, Lemma 4.2].

Lemma 7

Let $K=0,1,2,\ldots$ Suppose that A is an $L^\infty ({\mathbb R}^n) \cap {{\mathcal {P}}}_{K}^{\bot }({{\mathbb {R}}}^n)$-function supported on a cube Q. Then,

$$\begin{aligned} |I_\alpha A(x)| \le C_{\alpha ,K}\Vert A\Vert _{L^\infty } \ell (Q)^\alpha \sum \limits _{k=1}^\infty \frac{1}{2^{k(n+K+1-\alpha )}}\chi _{2^kQ}(x) \quad (x \in {{\mathbb {R}}}^n). \end{aligned}$$

(20)

Now we prove Theorem 6. We may assume that $f \in L^\infty _{\mathrm{c}}({{\mathbb {R}}}^n)$ is a positive measurable function in view of the positivity of the integral kernel. We decompose f according to Theorem 3 with $K>\alpha -\frac{n}{p^*}-1$; $f=\sum \nolimits _{j=1}^\infty \lambda _j a_j$, where $\{Q_j\}_{j=1}^\infty \subset {{\mathcal {D}}}({{\mathbb {R}}}^n)$, $\{a_j\}_{j=1}^\infty \subset L^\infty ({{\mathbb {R}}}^n) \cap {\mathcal P}_{K}^{\bot }({{\mathbb {R}}}^n)$ and $\{\lambda _j\}_{j=1}^\infty \subset [0,\infty )$ fulfill (3). Then by Lemma 7, we obtain

$$\begin{aligned} |g(x)I_\alpha f(x)| \le C \sum \limits _{j=1}^\infty \sum \limits _{k=1}^\infty \frac{\lambda _j}{2^{k(n+K+1-\alpha )}} \left( \ell (Q_j)^\alpha |g(x)|\chi _{2^kQ_j}(x)\right) . \end{aligned}$$

Therefore, we conclude

$$\begin{aligned} \Vert g \cdot I_\alpha f\Vert _{{{\mathcal {M}}}^{s}_{\mathbf {t}}} \le C \Vert g\Vert _{{{\mathcal {M}}}^{p^*}_{\mathbf {q}^{\,*}}} \left\| \sum \limits _{j=1}^\infty \sum \limits _{k=1}^\infty \frac{\lambda _j\ell (2^kQ_j)^{\alpha -\frac{n}{p^*}}}{2^{k(n+K+1)}} \cdot \frac{\ell (2^kQ_j)^{\frac{n}{p^*}}}{\Vert g\Vert _{{\mathcal M}^{p^*}_{\mathbf {q}^{\,*}}}} |g|\chi _{2^kQ_j} \right\| _{{\mathcal M}^{s}_{\mathbf {t}}}. \end{aligned}$$

For each $(j,k) \in {{\mathbb {N}}} \times {{\mathbb {N}}}$, write

$$\begin{aligned} \kappa _{jk} \equiv \frac{\lambda _j\ell (2^kQ_j)^{\alpha -\frac{n}{p^*}}}{2^{k(n+K+1)}}, \quad b_{jk} \equiv \frac{\ell (2^kQ_j)^{\frac{n}{p^*}}}{\Vert g\Vert _{{\mathcal M}^{p^*}_{\mathbf {q}^{\,*}}}} |g|\chi _{2^kQ_j}. \end{aligned}$$

Then,

$$\begin{aligned} \sum \limits _{j=1}^\infty \sum \limits _{k=1}^\infty \frac{\lambda _j\ell (2^kQ_j)^{\alpha -\frac{n}{p^*}}}{2^{k(n+K+1)}} \cdot \frac{\ell (2^kQ_j)^{\frac{n}{p^*}}}{\Vert g\Vert _{{\mathcal M}^{p^*}_{\mathbf {q}^{\,*}}}} |g|\chi _{2^kQ_j} = \sum \limits _{j,k=1}^\infty \kappa _{jk}b_{jk}, \end{aligned}$$

each $b_{jk}$ is supported on a cube $2^kQ_j$ and

$$\begin{aligned} \Vert b_{jk}\Vert _{{{\mathcal {M}}}^{p^*}_{\mathbf {q}^{\,*}}} \le \ell (2^kQ_j)^{\frac{n}{p^*}}. \end{aligned}$$

Observe that $\chi _{2^kQ_j} \le 2^{kn}M\chi _{Q_j}$. Hence, if we choose $1<\theta$ so that

$$\begin{aligned} K>\alpha -\frac{n}{p^*}-1+\theta n-n, \end{aligned}$$

then we have

$$\begin{aligned}&{ \left\| \sum \limits _{j=1}^\infty \sum \limits _{k=1}^\infty \kappa _{jk}\chi _{2^kQ_j} \right\| _{{{\mathcal {M}}}^{s}_{\mathbf {t}}} }\\&\quad = \left\| \sum \limits _{j=1}^\infty \sum \limits _{k=1}^\infty \frac{\lambda _j\ell (2^kQ_j)^{\alpha -\frac{n}{p^*}}}{2^{k(n+K+1)}} \chi _{2^kQ_j} \right\| _{{{\mathcal {M}}}^{s}_{\mathbf {t}}} \\&\quad = \left\| \sum \limits _{j=1}^\infty \lambda _j \ell (Q_j)^{\alpha -\frac{n}{p^*}}(M\chi _{Q_j})^\theta \right\| _{{{\mathcal {M}}}^{s}_{\mathbf {t}}}\\&\quad \le C \left\| \sum \limits _{j=1}^\infty \left( M\left[ \lambda _j{}^{\frac{1}{\theta }} \ell (Q_j)^{\frac{1}{\theta }\left( \alpha -\frac{n}{p^*}\right) } \chi _{Q_j}\right] \right) ^\theta \right\| _{{{\mathcal {M}}}^{s}_{\mathbf {t}}}\\&\quad \le C \left( \left\| \left\{ \sum \limits _{j=1}^\infty \left( M\left[ \lambda _j{}^{\frac{1}{\theta }} \ell (Q_j)^{\frac{1}{\theta }\left( \alpha -\frac{n}{p^*}\right) } \chi _{Q_j}\right] \right) ^\theta \right\} ^{\frac{1}{\theta }} \right\| _{{{\mathcal {M}}}^{\theta s}_{\theta \mathbf {q}}} \right) ^{\theta }. \end{aligned}$$

By virtue of Proposition 6, the Fefferman–Stein inequality for mixed Morrey spaces, with $f_j=\lambda _j{}^{\frac{1}{\theta }} \ell (Q_j)^{\frac{1}{\theta }\left( \alpha -\frac{n}{p^*}\right) }\chi _{Q_j}$, we can remove the maximal operator and we obtain

$$\begin{aligned} \Vert g \cdot I_\alpha f\Vert _{{{\mathcal {M}}}^{s}_{\mathbf {t}}} \le C\Vert g\Vert _{{{\mathcal {M}}}^{p^*}_{\mathbf {q}^{\,*}}} \left\| \sum \limits _{j=1}^\infty \lambda _j \ell (Q_j)^{\alpha -\frac{n}{p^*}}\chi _{Q_j} \right\| _{{\mathcal M}^{s}_{\mathbf {t}}}. \end{aligned}$$

We distinguish two cases here.

• (1) If $\alpha =\frac{n}{p^*}$, then $p=s$ and $\mathbf {q}=\mathbf {t}$. Thus, we can use (3).

• (2) If $\alpha >\frac{n}{p^*}$, then, by Proposition 2 and Lemma 6, we obtain

  $$\begin{aligned} \left\| \sum \limits _{j=1}^\infty \lambda _j \ell (Q_j)^{\alpha -\frac{n}{p^*}}\chi _{Q_j} \right\| _{{\mathcal M}^{s}_{\mathbf {t}}}&\le C \left\| I_{\alpha -\frac{n}{p^*}} \left[ \sum \limits _{j=1}^\infty \lambda _j \chi _{Q_j}\right] \right\| _{{{\mathcal {M}}}^{s}_{\mathbf {t}}}\\&\le C \left\| \sum \limits _{j=1}^\infty \lambda _j \chi _{Q_j} \right\| _{{{\mathcal {M}}}^{p}_{\mathbf {q}}}. \end{aligned}$$

  Thus, we are still in the position of using (3).

Consequently, we obtain

$$\begin{aligned} \left\| \sum \limits _{j=1}^\infty \sum \limits _{k=1}^\infty \kappa _{jk}\chi _{2^kQ_j} \right\| _{{{\mathcal {M}}}^{s}_{\mathbf {t}}} \lesssim \left\| f \right\| _{{{\mathcal {M}}}^{p}_{\mathbf {q}}}<\infty . \end{aligned}$$

(21)

Observe also that $p^*>s$ and that $\mathbf {q}^{\,*}>\mathbf {t}$. Thus, by Theorem 2 and (21), it follows that

$$\begin{aligned} \Vert g \cdot I_\alpha f\Vert _{{{\mathcal {M}}}^{s}_{\mathbf {t}}}&\le C\Vert g\Vert _{{{\mathcal {M}}}^{p^*}_{\mathbf {q}^{\,*}}} \left\| \sum \limits _{j=1}^\infty \sum \limits _{k=1}^\infty \kappa _{jk}\chi _{2^kQ_j} \right\| _{{{\mathcal {M}}}^{s}_{\mathbf {t}}} \le C\Vert g\Vert _{{{\mathcal {M}}}^{p^*}_{\mathbf {q}^{\,*}}} \left\| f \right\| _{{{\mathcal {M}}}^{p}_{\mathbf {q}}}. \end{aligned}$$