Weighted estimates for fractional bilinear Hardy operators on variable exponent Morrey–Herz space

Abstract

In this article, we analyze the boundedness for the fractional bilinear Hardy operators on variable exponent weighted Morrey–Herz spaces ${M\dot{K}^{\alpha (\cdot ),\lambda}_{q,p(\cdot)}(w)}$. Similar estimates are obtained for their commutators when the symbol functions belong to BMO space with variable exponents.

1 Introduction

Back in 1920, Hardy [1] defined an operator for a locally integrable $g\in \mathbb{R}^{n}$ known as the Hardy operator,

$$ \mathscr{H} g(y)=y^{-1} \int _{0}^{y}g(z)\,dz ,\quad y>0,$$

(1)

and established the following inequality:

$$ \Vert \mathscr{H} g \Vert _{L^{p}(\mathbb{R}^{+})}\leq p^{\prime} \Vert g \Vert _{L^{p}( \mathbb{R}^{+})},\quad \infty >p>1,$$

(2)

where $p^{\prime}=p/(p-1)$ is shown to be the best possible constant. At a later stage, Faris [2] gave an n-dimensional extension of (1) of which the equivalent form is given by

$$ Hg(y)= \bigl\vert B \bigl(0, \vert y \vert \bigr) \bigr\vert ^{-1} \int _{B(0, \vert y \vert )}g(z)\,dz ,$$

(3)

where $|B(0,|y|)|$ is the Lebesgue measure of the ball $B(0,|y|)$ in n-dimensional Euclidean space $\mathbb{R}^{n}$. Recently [3], it was shown that H satisfies

$$ \Vert Hg \Vert _{L^{p}(\mathbb{R}^{n})}\leq p^{\prime} \Vert g \Vert _{L^{p}(\mathbb{R}^{n})}, \quad 1< p\leq \infty ,$$

(4)

where $p^{\prime }$ is declared a sharp constant. Inequalities (2) and (4) were recently extended to power weighted Lebesgue spaces in [4] and [5] where sharp constants depend upon the weight indices. Inequalities (2) and (4) are known as strong-type $(p,p)$ Hardy inequalities because in these inequalities the Hardy operator maps $L^{p}$ to $L^{p}$. The authors in [6] have established the weak-type $(p,p)$ Hardy inequalities in which the Hardy operator maps $L^{p}$ to $L^{p,\infty}$. However, it was shown that the optimal constant for weak-type Hardy inequalities is 1, which is less than $p/(p-1)$. Subsequently, the sharp constants for weak-type Hardy inequalities on Morrey-type spaces were obtained in [7] and [8]. Likewise the sharp constant for the high-dimensional fractional Hardy operator [9]

$$ H_{\beta}g(y)= \bigl\vert B \bigl(0, \vert y \vert \bigr) \bigr\vert ^{\frac{\beta}{n}-1} \int _{B(0, \vert y \vert )}g(z)\,dz , \quad 0\leq \beta < n, $$

(5)

on Lebesgue spaces was not fixed until 2015. Zhao and Lu [10] solved this problem by extending the Bliss results for the one-dimensional fractional Hardy operator. The boundedness of the Hardy operator $H_{\beta}$ has been shown in [10], and the following inequality has been established:

$$ \Vert H_{\beta }g \Vert _{L^{q}(\mathbb{R}^{n})}\leq A \Vert g \Vert _{L^{p}(\mathbb{R}^{n})},$$

(6)

where

$$ A= \biggl(\frac{p'}{q} \biggr)^{1/q} \biggl( \frac{n}{q\beta} \cdot B \biggl( \frac{n}{q\beta},\frac{n}{q'\beta} \biggr) \biggr)^{-\beta /n}. $$

For $g_{1},g_{2},\ldots,g_{m}\in L_{\mathrm{loc}}^{1}(\mathbb{R}^{n})$ and $m\in \mathbb{N}$, the m-linear Hardy operator was introduced by Z. Fu and L. Grafakos in [4], written as follows:

$$ H(g_{1},\ldots,g_{m})=\frac{1}{ \vert x \vert ^{nm}} \int _{ \vert (y_{1},\ldots,y_{m}) \vert < \vert x \vert } \prod_{i=1}^{m}g_{i}(y_{i})\,dy _{1}, \ldots,dy_{m}. $$

They also developed the sharp bounds for the m-linear Hardy operator. The 2-linear operator is known as the bilinear operator. The authors of [11] used the commutator of the bilinear Hardy operator

$$ \begin{aligned}[b] & \bigl[b_{i},H^{i} \bigr](g_{1},\ldots,g_{m}) (z)=b_{i}(z)H(g_{1}, \ldots,g_{m}) (z)-H(g_{1},\ldots,g_{i-1},g_{i}b_{i},g_{i+1}, \ldots,g_{m}) (z)\end{aligned} $$

and obtained the boundedness of bilinear commutators generated by the bilinear Hardy operator. Later on, in [12] A. Hussain defined the fractional m-linear p-adic Hardy operator. Now, in the present paper, we introduce the definition of the fractional m-linear Hardy operators as

$$\begin{aligned}& H_{\beta}(g_{1},\ldots,g_{m})= \frac{1}{ \vert x \vert ^{nm-\beta}} \int _{ \vert (y_{1},\ldots,y_{m}) \vert < \vert x \vert } \prod_{i=1}^{m}g_{i}(y_{i})\,dy _{1}, \ldots,dy_{m},\\& H^{*}_{\beta}(g_{1},\ldots,g_{m})= \int _{ \vert (y_{1},\ldots,y_{m}) \vert > \vert x \vert } \frac{1}{{ \vert y \vert ^{nm-\beta}}}\prod _{i=1}^{m}{g_{i}(y_{i})}\,dy _{1}, \ldots,dy_{m}, \end{aligned}$$

where $y=(y_{1},y_{2},\ldots,y_{m})$. We also introduce a definition for the commutator of fractional m-linear Hardy operators

$$\begin{aligned}& [b,H_{\beta}](g_{1},\ldots,g_{m}) (z)=\sum _{i=1}^{m} \bigl[b_{i},H^{i}_{\beta} \bigr](g_{1},\ldots,g_{m}) (z),\\& \bigl[b,H^{*}_{\beta} \bigr](g_{1}, \ldots,g_{m}) (z)=\sum_{i=1}^{m} \bigl[b_{i},H^{*i}_{ \beta} \bigr](g_{1},\ldots,g_{m}) (z),\\& \begin{aligned}[b] & \bigl[b_{i},H^{i}_{\beta} \bigr](g_{1},\ldots,g_{m}) (z)=b_{i}(z)H_{\beta}(g_{1}, \ldots,g_{m}) (z)-H_{ \beta}(g_{1}, \ldots,g_{i-1},g_{i}b_{i},g_{i+1}, \ldots,g_{m}) (z),\end{aligned} \\& \begin{aligned}[b] & \bigl[b_{i},H^{*i}_{\beta} \bigr](g_{1},\ldots,g_{m}) (z)=b_{i}(z)H^{*}_{\beta}(g_{1}, \ldots,g_{m}) (z)-H^{*}_{ \beta}(g_{1}, \ldots,g_{i-1},g_{i}b_{i},g_{i+1}, \ldots,g_{m}) (z).\end{aligned} \end{aligned}$$

Hardy inequalities have been the main focus of interest in various monographs [13, 14]. The optimal bounds for Hardy-type inequalities are established only in a few cases and the research in this area is an active part of modern analysis. Some recent publications in this area include [15, 16]. Besides this, sharp constants for Hardy-type inequalities on the product of some function spaces have also been exposed in [17]. Essential reviews of Hardy operators on various function spaces include [4, 5, 18–20].

The work in [21] sparked the idea of generalizing function spaces. The variable Lebesgue space $L^{p(\cdot )}$ was initially presented by Rákosník and Kováčik in [22]. Following that, the creation of variable exponent Lebesgue spaces began, as did the exploration of the boundedness of numerous operators, notably the maximum operator on the variable exponent Lebesgue space $L^{p(\cdot )}$ [23, 24]. In recent times, the theory of generalized function spaces has piqued the interest of researchers working in several domains of mathematical analysis, including image processing [25], electrorheological fluid modeling [26], and partial differential equations [27].

Furthermore, Izuki proposed variable exponent Herz spaces $\dot{K}_{q}^{\alpha ,p(\cdot )}$ in [28]. After this, Drihemn and Almeida [29] proposed a revised definition of Herz spaces that included α as a variable exponent. However, in [30], the Herz space with all exponents as variables was developed and investigated. Variable exponent Morrey–Herz spaces $M\dot{K}^{\alpha ,\lambda}_{q,p(\cdot )}$ appeared for the first time in [31]. The generalized concept of Morrey–Herz spaces was provided in [31] by substituting the exponent α with $\alpha (\cdot )$. A few significant thoughts in this regard were given in [32, 33]. Its weighted theory formulation based upon the Muckenhoupt weights [34] is a recent accomplishment in the field of variable exponent function spaces. Cruz-Uribe in [35] introduced the boundedness for the Hardy–Littlewood maximal operator M,

$$ Mg(t)=\sup_{E:\mathrm{ball}, t\in E}\frac{1}{ \vert E \vert } \int _{E} \bigl\vert g(z) \bigr\vert \,dz , $$

on the variable exponent weighted Lebesgue space $L^{p(\cdot )}(w)$. Hästö and Diening demonstrated in [36] the equivalence between the continuity criteria of M on $L^{p(\cdot )}(w)$ and the Muckenhoupt condition. However, weighted variable exponent Morrey–Herz spaces are set out and investigated in [11, 37].

In the present paper, we will study the boundedness of the m-linear fractional Hardy operator on variable exponent weighted Herz–Morrey space. Furthermore, we also discuss the boundedness of commutators generated by the m-linear fractional Hardy operator on variable exponent weighted Herz–Morrey space. It is worth mentioning that we study Banach spaces and Muckenhoupt weights. We consequently expand a few results introduced in [18]. To control the continuity criteria of the m-linear fractional Hardy operator, we will utilize the boundedness of the fractional integral defined as

$$ I_{\beta}(g) (t)= \int _{\mathbb{R}^{n}}\frac{g(z)}{ \vert t-z \vert ^{n-\beta}}\,dz . $$

The boundedness of the Riesz potential on variable exponent Lebesgue spaces is reported in [38]. The boundedness of the fractional integral operator on weighted Herz spaces is introduced by Noi and Izuki [39].

This article contains four sections. The next section includes some lemmas and definitions. In the third section, we provide some important lemmas which are used in Sect. 4 to get our main results.

2 Notations and definitions

The letter C is used throughout this paper to represent a constant, and the value of the constant may differ from one line to the next. We consider a set S which is nonempty and measurable in $\mathbb{R}^{n}$, and $\chi _{S}$ stands for the characteristic function of S, where $|S|$ denotes the Lebesgue measure. Let us begin by defining variable exponent Lebesgue spaces using the basic articles and books [22, 24, 40, 41].

Definition 2.1

Suppose we have a measurable function $q(\cdot ):\mathbb{R}^{n}\rightarrow [1,\infty ]$. Moreover, $L^{q{(\cdot )}}(\mathbb{R}^{n})$ represents a Lebesgue space where the variable exponent is a set of every measurable function g, i.e.,

$$ F_{q}(g)= \int _{\mathbb{R}^{n}} \bigl( \bigl\vert g(x) \bigr\vert \bigr)^{q(x)}\,dx < \infty . $$

The space $L^{q{(\cdot )}}(\mathbb{R}^{n})$ becomes a Banach space with the following norm:

$$ \Vert g \Vert _{L^{q(\cdot )}}=\inf \biggl\{ \sigma >0:F_{q} \biggl(\frac{g}{\sigma} \biggr)= \int _{\mathbb{R}^{n}} \biggl(\frac{ \vert g(x) \vert }{\sigma} \biggr)^{q(x)}\,dx \leq 1 \biggr\} . $$

Definition 2.2

We denote by $P (\mathbb{R}^{n})$ the set of all measurable functions $q(\cdot ):\mathbb{R}^{n}\rightarrow (1,\infty )$ such that

$$ 1< q_{-}\leq q(x)\leq q_{+}< \infty , $$

where

$$ q_{-}:=\operatorname*{essinf}_{x\in \mathbb{R}^{n}}q(x), \qquad q_{+}:= \operatorname*{esssup}_{x\in \mathbb{R}^{n}}q(x). $$

Definition 2.3

Let $q(\cdot )$ be a real-valued function on $\mathbb{R}^{n}$. We state that:

• (i) $\mathscr{C}^{\log}_{\mathrm{{loc}}}(\mathbb{R}^{n})$ is a set consisting of all local log Hölder continuous functions $q(\cdot )$ fulfilling

  $$ \bigl\vert q(x)-q(y) \bigr\vert \lesssim \frac{-C}{\log ( \vert x-y \vert )},\qquad \vert y-x \vert < \frac{1}{2},\quad x,y \in \mathbb{R}^{n}; $$

• (ii) if $q(\cdot )\in \mathscr{C}^{\log}_{0}(\mathbb{R}^{n})$, then it satisfies the following condition at the origin:

  $$ \bigl\vert q(x)-q(0) \bigr\vert \lesssim \frac{C}{\log ( \vert \frac{1}{ \vert x \vert }+e \vert )},\quad x\in \mathbb{R}^{n}; $$

• (iii) if $q(\cdot )\in \mathscr{C}^{\log}_{\infty}(\mathbb{R}^{n})$, then it fulfills the following inequality at infinity:

  $$ \bigl\vert q(x)-q_{\infty} \bigr\vert \leq \frac{C_{\infty}}{\log ( \vert x \vert +e)}, \quad x\in \mathbb{R}^{n}; $$

• (iv) $\mathscr{C}^{\log}=\mathscr{C}^{\log}_{\mathrm{{loc}}}\cap \mathscr{C}^{\log}_{ \infty}$ represents the set of every global log Hölder continuous function $q(\cdot )$.

It was shown in [42] that if $q(\cdot )\in P (\mathbb{R}^{n})\cap \mathscr{C}^{\log}( \mathbb{R}^{n})$, then M is bounded on $L^{q(\cdot )}(\mathbb{R}^{n})$.

Assume that $w({x})$ is a weight function on $\mathbb{R}^{n}$ that is both nonnegative and locally integrable. Let $L^{q(\cdot )}(w)$ be the space of all complex-valued functions f on $\mathbb{R}^{n}$ such that $fw^{\frac{1}{q(\cdot )}}\in L^{q(\cdot )}(\mathbb{R}^{n})$. The space $L^{q(\cdot )}(w)$ is a Banach function space with respect to the norm

$$ \Vert f \Vert _{L^{q(\cdot )}(w)}= \bigl\Vert fw^{\frac{1}{q(\cdot )}} \bigr\Vert _{L^{q(\cdot )}}. $$

In [34], Benjamin Muckenhoupt proposed $A_{p}$ weights theory with $1< p<\infty$ on $\mathbb{R}^{n}$. Noi and Izuki recently expanded the Muckenhoupt $A_{p}$ class by using p as a variable in [39, 43].

Definition 2.4

Assume $q(\cdot )\in P (\mathbb{R}^{n})$. A weight w is said to be an $A_{q(\cdot )}$ weight if

$$ \sup_{B}{ \vert B \vert ^{-1}} \bigl\Vert w^{-1/q(\cdot )}\chi _{B} \bigr\Vert _{L^{q^{\prime}( \cdot )}} \bigl\Vert w^{1/q(\cdot )}\chi _{B} \bigr\Vert _{L^{q(\cdot )}}< \infty . $$

It was proved in [44] that $w\in A_{q(\cdot )}$ if and only if M is bounded on $L^{q(\cdot )}$ space.

Remark 2.5

[39] If $q(\cdot ),p(\cdot )\in P (\mathbb{R}^{n})\cap \mathscr{C}^{\log}( \mathbb{R}^{n})$ and $q(\cdot )\le p(\cdot )$, then we have

$$ A_{1}\subset A_{q(\cdot )}\subset A_{p(\cdot )}. $$

Definition 2.6

Suppose $\beta \in (0,n)$ and $p_{1}(\cdot ),p_{2}(\cdot )\in P (\mathbb{R}^{n})$ in such a way that $\frac{1}{p_{1}(x)}=\frac{1}{p_{2}(x)}+\frac{\beta}{n}$. A weight w is called $A(p_{1}(\cdot ),p_{2}(\cdot ))$ weight if

$$ \vert B \vert ^{\frac{\beta}{n}-1} \Vert \chi _{B} \Vert _{(L^{p_{1}(\cdot )}(w^{p_{1}( \cdot )}))^{\prime }} \Vert \chi _{B} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \le C. $$

Definition 2.7

[39] If $\beta \in (0,n)$, $p_{1}(\cdot ),p_{2}(\cdot )\in P (\mathbb{R}^{n})$, and $\frac{1}{p_{1}(x)}=\frac{1}{p_{2}(x)}+\frac{\beta}{n}$, then $w\in A_{(p_{1}(\cdot ),p_{2}(\cdot ))}$ if and only if $w^{p_{2}(\cdot )}\in A_{1+p_{2}(\cdot )/p_{1}^{\prime }(\cdot )}$.

The variable exponent weighted Morrey–Herz space $M\dot{K}^{\alpha (\cdot ),\lambda}_{q,p(\cdot )}(w)$ is defined now. Let $\chi _{k}=\chi _{A_{k}}$, $B_{k}=\{x\in \mathbb{R}^{n}:|x|\leq 2^{k}\}$, and $A_{k}=B_{k}\setminus B_{k-1}$ for $k\in \mathbb{Z}$.

Definition 2.8

[19] Let $0< q<\infty$, $0\leq \lambda <\infty$, $\alpha (\cdot ):\mathbb{R}^{n}\rightarrow \mathbb{R}$ with $\alpha (\cdot )\in L^{\infty}(\mathbb{R}^{n})$, $p(\cdot )\in P (\mathbb{R}^{n})$, and let w be a weight on $\mathbb{R}^{n}$. Then $M\dot{K}^{\alpha (\cdot ),\lambda}_{q,p(\cdot )}(w)$ is defined by

$$ M\dot{K}^{\alpha (\cdot ),\lambda}_{q,p(\cdot )}({w})= \bigl\{ f\in L^{p( \cdot )}_{\mathrm{loc}} \bigl(\mathbb{R}^{n}\setminus \{0\},w \bigr): \Vert f \Vert _{M\dot{K}^{ \alpha (\cdot ),\lambda}_{q,p(\cdot )}(w)}< \infty \bigr\} , $$

where

$$ \Vert f \Vert _{M\dot{K}^{\alpha (\cdot ),\lambda}_{q,p(\cdot )}({w})}=\sup_{k_{0} \in{Z}}2^{-k_{0}\lambda } \Biggl(\sum_{k=-\infty}^{k_{0}}2^{k\alpha ( \cdot )q } \Vert f\chi _{k} \Vert ^{q }_{L^{p (\cdot )}(w)} \Biggr)^{1/q}. $$

When $\lambda =0$, we have a weighted Herz space with variable exponent $\dot{K}^{\alpha (\cdot )}_{q,p(\cdot )}(w)$.

3 Main lemmas

This section presents several relevant lemmas that will aid in the proof of our main boundedness result.

Lemma 3.1

[45] If Y is a Banach function space, then:

• (1) The associated space $Y^{\prime }$ should also be a Banach function space;

• (2) $\|\cdot \|_{Y}$ and $\|\cdot \|_{(Y^{\prime })^{\prime }}$ are parallel;

• (3) (generalized Hölder inequality) if $f\in Y^{\prime }$ and $g\in Y$, then

  $$ \int _{\mathbb{R}^{n}} \bigl\vert g(x)f(x) \bigr\vert \leq \Vert f \Vert _{Y^{\prime }} \Vert g \Vert _{Y}. $$

Lemma 3.2

[46] Consider a Banach function space Y. M is weakly bounded on Y, such that

$$ \Vert \chi _{\{Mf>\rho \}} \Vert _{Y} \lesssim \rho ^{-1} \Vert f \Vert _{Y} $$

is true for $\rho >0$ and every $f\in Y$. Then we have

$$ \sup_{B:\mathrm{ball}}\frac{1}{ \vert B \vert } \Vert \chi _{B} \Vert _{Y} \Vert \chi _{B} \Vert _{Y^{\prime }}< \infty . $$

Lemma 3.3

[39] For all balls B and for a Banach function space Y, we have

$$ 1\leq \frac{1}{ \vert B \vert } \Vert \chi _{B} \Vert _{Y} \Vert \chi _{B} \Vert _{Y^{\prime }}. $$

Lemma 3.4

[39] If Y is a Banach function space and M is bounded on $Y^{\prime }$, then for any $E\subset \mathbb{R}^{n}$ and $S\subset E$, there exists a constant $\delta \in (0,1)$ such that

$$ \frac{ \Vert \chi _{S} \Vert _{Y}}{ \Vert \chi _{E} \Vert _{Y}}\lesssim \biggl( \frac{ \vert S \vert }{ \vert E \vert } \biggr)^{\delta}. $$

Lemma 3.5

[47] We have:

• (1) $Y(\mathbb{R}^{n},W)$ is a Banach function space with respect to the norm

  $$ \Vert f \Vert _{Y(\mathbb{R}^{n},W)}= \Vert fw \Vert _{Y}, $$

  where

  $$ Y \bigl(\mathbb{R}^{n},W \bigr)=\{f\in \mathbb{M}:fW\in Y\}; $$

• (2) $Y^{\prime }(\mathbb{R}^{n},W^{-1})$ is also a Banach function space.

Remark 3.6

Consider $p(\cdot )\in P (\mathbb{R}^{n})$. When we compare Lebesgue space $L^{p(\cdot )}(w^{p(\cdot )})$ and $L^{p^{\prime }(\cdot )}(w^{-p^{\prime }(\cdot )})$ with $Y(\mathbb{R}^{n},W)$, we get:

• 1: if $w=W$ and $Y=L^{p(\cdot )}(\mathbb{R}^{n})$, then $L^{p(\cdot )}(w^{p(\cdot )})=L^{p(\cdot )}(\mathbb{R}^{n},w)$;

• 2: if $w^{-1}=W$ and $Y=L^{p'(\cdot )}(\mathbb{R}^{n})$, then $L^{p'(\cdot )}(\mathbb{R}^{n},w^{-1})=L^{p'(\cdot )}(w^{-p^{\prime }( \cdot )})$.

By the result of Lemma 3.5, we obtain

$$ L^{p'(\cdot )}\bigl(\mathbb{R}^{n},w^{-1}\bigr)= \bigl(L^{p(\cdot )}\bigl(\mathbb{R}^{n},w\bigr) \bigr)^{ \prime }=L^{p'(\cdot )}\bigl(w^{-p^{\prime }(\cdot )}\bigr)= \bigl(L^{p(\cdot )}\bigl(w^{p( \cdot )}\bigr)\bigr)^{\prime}. $$

Lemma 3.7

[48] Let $p(\cdot )\in P (\mathbb{R}^{n})\cap \mathscr{C}^{\log}( \mathbb{R}^{n})$ and $w^{p_{2}(\cdot )}\in A_{p_{2}(\cdot )}$ admit $w^{-p^{\prime }_{2}(\cdot )}\in A_{p^{\prime }_{2}(\cdot )}$. Then there exist $\delta _{11},\delta _{22}\in (0,1)$ such that

$$ \frac{ \Vert \chi _{S} \Vert _{(L^{P_{2}(\cdot )}w^{p_{2}(\cdot )})^{\prime}} }{ \Vert \chi _{E} \Vert _{(L^{P_{2}(\cdot )}w^{p_{2}(\cdot )})^{\prime}} } \lesssim \biggl(\frac{ \vert S \vert }{ \vert E \vert } \biggr)^{\delta _{22}},\qquad \frac{ \Vert \chi _{S} \Vert _{(L^{P_{1}(\cdot )}w^{p_{1}(\cdot )})^{\prime}} }{ \Vert \chi _{E} \Vert _{(L^{P_{1}(\cdot )}w^{p_{1}(\cdot )})^{\prime}} } \lesssim \biggl( \frac{ \vert S \vert }{ \vert E \vert } \biggr)^{\delta _{11}}$$

(7)

for every ball E and for each measurable set $S\subset E$.

Lemma 3.8

[39] Let $p(\cdot )\in P (\mathbb{R}^{n})\cap \mathscr{C}^{\log}( \mathbb{R}^{n})$, $0<\beta <\frac{n}{p_{+}}$, and $\frac{1}{q(\cdot )}=\frac{1}{p(\cdot )}-\frac{\beta}{n}$. Then $I^{\beta}$ is bounded from $L^{p(\cdot )}(w^{P(\cdot )})$ to $L^{q(\cdot )}(w^{q(\cdot )})$ for $w\in A(p(\cdot ),q(\cdot ))$.

Lemma 3.9

[49] Assume that ${q}(\cdot )\in P (\mathbb{R}^{n})$. Then for all $b\in \mathit{BMO}$ and all $j,i\in{Z}$ with $j>i$ we have

$$\begin{aligned}& C^{-1} \Vert b \Vert _{\mathit{BMO}}\leq \sup _{\mathtt{B}:\mathrm{Ball}} \frac{1}{ \Vert \chi _{\mathtt{B}} \Vert _{L^{\mathrm{q}(\cdot )}}} \bigl\Vert (b-b_{ \mathtt{B}})\chi _{\mathtt{B}} \bigr\Vert _{L^{\mathrm{q}(\cdot )}} \leq C \Vert b \Vert _{\mathit{BMO}}, \end{aligned}$$

(8)

$$\begin{aligned}& \bigl\Vert (b-b_{{B}_{i}})\chi _{{B}_{j}} \bigr\Vert _{L^{{q}(\cdot )}}\leq C(j-i) \Vert b \Vert _{\mathit{BMO}} \Vert \chi _{\mathtt{B}_{j}} \Vert _{L^{{q}(\cdot )}}. \end{aligned}$$

(9)

4 Main results

Lemma 4.1

If $w^{p_{2}(\cdot )},w^{p_{1}(\cdot )}\in A_{1}$, $q(\cdot )\in P (\mathbb{R}^{n})\cap \mathscr{C}^{\log}( \mathbb{R}^{n})$, and $p(\cdot )$ is such that $\frac{1}{q(\cdot )}=\frac{1}{p(\cdot )}-\frac{\beta}{n}$ with $\frac{1}{p(\cdot )}=\frac{1}{p_{1}(\cdot )}+\frac{1}{p_{2}(\cdot )}$, then

$$ \begin{aligned}[b] \Vert \chi _{B_{k}} \Vert _{L^{q(\cdot )}(w^{q(\cdot )})}\leq C2^{k(2n-\beta )} \Vert \chi _{B_{k}} \Vert _{(L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{\prime }}^{-1} \Vert \chi _{B_{k}} \Vert _{(L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{\prime }}^{-1}.\end{aligned} $$

Proof

We assume that $f=\chi _{B_{k}}$ and use the definition of $I_{\beta}$

$$\begin{aligned}& I_{\beta}(\chi _{B_{k}}) (x) \geq C 2^{k\beta}\chi _{B_{k}}(x),\\& \chi _{B_{k}}(x)\leq C 2^{-k\beta}I_{\beta}(\chi _{B_{k}}) (x). \end{aligned}$$

Applying the norm on both sides and using the results of Lemmas 3.2 and 3.8, we obtain

$$ \begin{aligned}[b] \Vert \chi _{B_{k}} \Vert _{L^{q(\cdot )}(w^{q(\cdot )})}&\leq C 2^{-k\beta} \Vert I_{ \beta} \chi _{B_{k}} \Vert _{L^{q(\cdot )}(w^{q(\cdot )})} \\ & \leq C 2^{-k\beta} \Vert \chi _{B_{k}} \Vert _{L^{p(\cdot )}(w^{p(\cdot )})} \\ & \leq C 2^{-k\beta} \Vert \chi _{B_{k}} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}( \cdot )})} \Vert \chi _{B_{k}} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \\ & \leq C2^{k(2n-\beta )} \Vert \chi _{B_{k}} \Vert _{(L^{p_{1}(\cdot )}(w^{p_{1}( \cdot )}))^{\prime }}^{-1} \Vert \chi _{B_{k}} \Vert _{(L^{p_{2}(\cdot )}(w^{p_{2}( \cdot )}))^{\prime }}^{-1}.\end{aligned} $$

(10)

 □

Proposition 4.2

[11] Let $p(\cdot )\in P (\mathbb{R}^{n})$, $0< q<\infty$, and $0\leq \lambda <\infty$. If $\alpha (\cdot )\in L^{\infty}(\mathbb{R}^{n})\cap \mathscr{C}^{\log}( \mathbb{R}^{n})$, then

$$ \begin{aligned}& \bigl\Vert H_{\beta}(f_{1},f_{2}) \cdot \chi _{k} \bigr\Vert ^{q }_{M \dot{K}^{\alpha (\cdot ),\lambda}_{q ,p (\cdot )}(w^{p (\cdot )})} \\ &\quad =\sup_{k_{0}\in{Z}}2^{-k_{0}\lambda q }\sum _{k=-\infty}^{k_{0}}2^{k \alpha (\cdot )q } \bigl\Vert H_{\beta}(f_{1},f_{2})\cdot \chi _{k} \bigr\Vert ^{q }_{L^{p (\cdot )}(w^{p (\cdot )})} \\ &\quad \thickapprox \max \Biggl\{ \underset{k_{0}< 0}{\sup _{k_{0}\in{Z}}}\, 2^{-k_{0} \lambda q }\sum_{k=-\infty}^{k_{0}}2^{k\alpha (0)q } \bigl\Vert H_{\beta}(f_{1},f_{2}) \cdot \chi _{k} \bigr\Vert ^{q }_{L^{p (\cdot )}(w^{p (\cdot )})}, \\ &\quad \quad \underset{k_{0}\geq 0}{\sup_{k_{0}\in{Z}}}\, 2^{-k_{0}\lambda q } \Biggl(\sum_{k=-\infty}^{-1}2^{k\alpha (0)q } \bigl\Vert H_{\beta}(f_{1},f_{2}) \cdot \chi _{k} \bigr\Vert ^{q }_{L^{p (\cdot )}(w^{p (\cdot )})} \\ &\quad \quad{}+\sum_{k=0}^{k_{0}}2^{k\alpha (\infty )q } \bigl\Vert H_{\beta}(f_{1},f_{2}) \cdot \chi _{k} \bigr\Vert ^{q }_{L^{p (\cdot )}(w^{p (\cdot )})} \Biggr) \Biggr\} . \end{aligned} $$

Theorem 4.3

Let $0< q,q_{1},q_{2}<\infty$, $q(\cdot )\in P (\mathbb{R}^{n})\cap \mathscr{C}^{\log}( \mathbb{R}^{n})$, and $p(\cdot )$ be such that $\frac{1}{q(\cdot )}=\frac{1}{p(\cdot )}-\frac{\beta}{n}$ with $\frac{1}{p(\cdot )}=\frac{1}{p_{1}(\cdot )}+\frac{1}{p_{2}(\cdot )}$. Also, let $w^{p_{2}(\cdot )},w^{p_{1}(\cdot )}\in A_{1}$, let $\lambda =\lambda _{1}+\lambda _{2}$, and let $\alpha (\cdot )\in \mathscr{C}^{\log}(\mathbb{R}^{n})\cap L^{\infty}( \mathbb{R}^{n})$ be a log Hölder continuous function at the origin satisfying $\alpha (0)=\alpha _{1}(0)+\alpha _{2}(0)$, $\alpha (\infty )=\alpha _{1}(\infty )+\alpha _{2}(\infty )$ with $\alpha (0)\leq \alpha (\infty )< n\delta _{11}+\lambda +n\delta _{22}$, where $\delta _{22},\delta _{11}\in (0,1)$ are the constants arising in (7). Then

$$ \bigl\Vert H_{\beta}(f_{1},f_{2}) \bigr\Vert _{M\dot{K}^{\alpha (\cdot ),\lambda}_{q,q(\cdot )}(w^{q( \cdot )})}\leq C \Vert f_{1} \Vert _{M\dot{K}^{\alpha _{1}(\cdot ),\lambda _{1}}_{q_{1},p_{1}( \cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2} \Vert _{M\dot{K}^{\alpha _{2}(\cdot ), \lambda _{2}}_{q_{2},p_{2}(\cdot )}(w^{p_{2}(\cdot )})}. $$

Proof

For each $f_{1},f_{2}\in {M\dot{K}^{\alpha (\cdot ),\lambda}_{q_{1},p_{1}( \cdot )}(w^{p_{1}(\cdot )})}$, if we express $f_{1j}=f_{1}\cdot \chi _{j}=f_{1}\cdot \chi _{A_{j}}$ and $f_{2j}=f_{2}\cdot \chi _{j}=f_{2}\cdot \chi _{A_{j}}$ for any $j\in \mathbb{Z}$, then we have

$$\begin{aligned}& f_{1}(x)=\sum_{j=-\infty}^{\infty}f_{1}(x) \cdot \chi _{j}(x)=\sum_{j=- \infty}^{\infty}f_{1j}(x),\\& f_{2}(x)=\sum_{j=-\infty}^{\infty}f_{2}(x) \cdot \chi _{j}(x)=\sum_{j=- \infty}^{\infty}f_{2j}(x). \end{aligned}$$

The generalized Hölder inequality gives

$$ \begin{aligned}[b] \bigl\vert H_{\beta}(f_{1},f_{2}) (x)\cdot \chi _{k}(x) \bigr\vert &\leq \frac{1}{ \vert x \vert ^{2n-\beta}} \int _{B_{k}} \int _{B_{k}} \bigl\vert f_{1}(t_{1}) \bigr\vert \bigl\vert f_{2}(t_{2}) \bigr\vert \,dt _{1}\,dt _{2} \cdot \chi _{k}(x) \\ &=\frac{1}{ \vert x \vert ^{2n-\beta}} \int _{B_{k}} \bigl\vert f_{1}(t_{1}) \bigr\vert |\,dt _{1} \int _{B_{k}}f_{2}(t_{2})\,dt _{2} \cdot \chi _{k}(x) \\ &\leq C2^{-2kn}\sum_{j=-\infty}^{k} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert \chi _{j} \Vert _{({L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{\prime }}} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \\ &\quad {} \times \Vert \chi _{j} \Vert _{({L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{\prime }}}2^{k \beta} \chi _{k}(x).\end{aligned} $$

(11)

Utilizing Lemmas 3.7 and 3.2, we acquire

$$ \begin{aligned}[b] & \bigl\Vert H_{\beta}(f_{1},f_{2})\cdot \chi _{k} \bigr\Vert _{L^{q(\cdot )}{(w^{q(\cdot )})}} \\ &\quad \leq C2^{k\beta}\sum_{j=-\infty}^{k} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert \chi _{j} \Vert _{({L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{\prime }}} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \\ &\quad\quad {} \times \Vert \chi _{j} \Vert _{({L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{ \prime }}}2^{-2kn} \Vert \chi _{k} \Vert _{L^{q(\cdot )}{(w^{q(\cdot )})}} \\ &\quad \leq C\sum_{j=-\infty}^{k} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \\ &\quad\quad {} \times \Vert \chi _{j} \Vert _{({L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{ \prime }}} \Vert \chi _{j} \Vert _{({L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{\prime }}}2^{-k(2n- \beta )} \Vert \chi _{k} \Vert _{L^{q(\cdot )}{(w^{q(\cdot )})} }.\end{aligned} $$

(12)

To move forward, we use Lemma 4.1 and (12), and we obtain

$$ \begin{aligned}[b] &\bigl\Vert H_{\beta}(f_{1},f_{2}) \cdot \chi _{k} \bigr\Vert _{L^{q(\cdot )}{(w^{q(\cdot )})}} \\ &\quad \leq C\sum _{j=-\infty}^{k} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \Vert \chi _{j} \Vert _{({L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{ \prime }}} \Vert \chi _{j} \Vert _{({L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{\prime }}} \\ &\quad \quad {} \times \Vert \chi _{k} \Vert ^{-1}_{({L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{\prime }}} \Vert \chi _{k} \Vert ^{-1}_{({L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{\prime }}} \\ &\quad \leq C\sum_{j=-\infty}^{k} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \frac{ \Vert \chi _{j} \Vert _{(L^{p_{2}(\cdot )}{(w^{p_{2}(\cdot )})})^{\prime }}}{ \Vert \chi _{k} \Vert _{(L^{p_{2}(\cdot )}{(w^{p_{2}(\cdot )})})^{\prime }}} \frac{ \Vert \chi _{j} \Vert _{(L^{p_{1}(\cdot )}{(w^{p_{1}(\cdot )})})^{\prime }}}{ \Vert \chi _{k} \Vert _{(L^{p_{1}(\cdot )}{(w^{p_{1}(\cdot )})})^{\prime }}} \\ &\quad \leq C\sum_{j=-\infty}^{k}2^{n\delta _{11}(j-k)}2^{n\delta _{22}(j-k)} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \\ &\quad \leq C\sum_{j=-\infty}^{k}2^{(n\delta _{11}+n\delta _{22})(j-k)} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} .\end{aligned} $$

(13)

In the remainder of the proof, in order to calculate $\|f_{1j}\|_{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})}$ and $\|f_{2j}\|_{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})}$, we consider the following two cases.

Case 1: For $j<0$,

$$ \begin{aligned}[b] \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})}&=2^{-j\alpha _{1}(0)} \bigl(2^{j\alpha _{1}(0)q_{1}} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})}^{q_{1}} \bigr)^{\frac{1}{q_{1}}} \\ & \leq 2^{-j\alpha _{1}(0)} \Biggl(\sum_{i=-\infty}^{j}2^{i\alpha _{1}(0)q_{1}} \Vert f_{1i} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})}^{q_{1}} \Biggr)^{ \frac{1}{q_{1}}} \\ & \leq 2^{j(\lambda _{1}-\alpha _{1}(0))}2^{-j\lambda _{1}} \Biggl( \sum _{i=-\infty}^{j}2^{i\alpha _{1}(\cdot )q_{1}} \Vert f_{1i} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})}^{q_{1}} \Biggr)^{\frac{1}{q_{1}}} \\ & \leq C2^{j(\lambda _{1}-\alpha _{1}(0))} \Vert f_{1} \Vert _{M\dot{K}^{\alpha _{1}(\cdot ), \lambda _{1}}_{q_{1},p_{1}(\cdot )}(w^{p_{1}(\cdot )})}.\end{aligned} $$

Similarly

$$ \begin{aligned}[b] \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})}&\leq C2^{j(\lambda _{2}- \alpha _{2}(0))} \Vert f_{2} \Vert _{M\dot{K}^{\alpha _{2}(\cdot ),\lambda _{2}}_{q_{2},p_{2}(\cdot )}(w^{p_{2}(\cdot )})}.\end{aligned} $$

Here we are using $\alpha (0)=\alpha _{1}(0)+\alpha _{2}(0)$ and $\lambda =\lambda _{1}+\lambda _{2}$:

$$ \begin{aligned}[b] &\Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \\ &\quad \leq C2^{j(\lambda _{1}-\alpha _{1}(0))}2^{j(\lambda _{2}-\alpha _{2}(0))} \Vert f_{1} \Vert _{M\dot{K}^{\alpha _{1}(\cdot ),\lambda _{1}}_{q_{1},p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2} \Vert _{M\dot{K}^{\alpha _{2}(\cdot ),\lambda _{2}}_{q_{2},p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \\ &\quad = C2^{j(\lambda -\alpha (0))} \Vert f_{1} \Vert _{M\dot{K}^{\alpha _{1}(\cdot ), \lambda _{1}}_{q_{1},p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2} \Vert _{M\dot{K}^{ \alpha _{2}(\cdot ),\lambda _{2}}_{q_{2},p_{2}(\cdot )}(w^{p_{2}(\cdot )})}.\end{aligned} $$

Case 2: For $j\geq 0$,

$$ \begin{aligned}[b] \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})}&=2^{-j\alpha _{1}( \infty )} \bigl(2^{j\alpha _{1}(\infty )q_{1}} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})}^{q_{1}} \bigr)^{\frac{1}{q_{1}}} \\ & \leq 2^{-j\alpha _{1}(\infty )} \Biggl(\sum_{i=0}^{j}2^{i\alpha _{1}( \infty )q_{1}} \Vert f_{1i} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})}^{q_{1}} \Biggr)^{\frac{1}{q_{1}}} \\ & \leq 2^{j(\lambda _{1}-\alpha _{1}(\infty ))}2^{-j\lambda _{1}} \Biggl(\sum _{i=-\infty}^{j}2^{i\alpha _{1}(\cdot )q_{1}} \Vert f_{1i} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})}^{q_{1}} \Biggr)^{\frac{1}{q_{1}}} \\ & \leq C2^{j(\lambda _{1}-\alpha _{1}(\infty ))} \Vert f_{1} \Vert _{M\dot{K}^{ \alpha _{1}(\cdot ),\lambda _{1}}_{q_{1},p_{1}(\cdot )}(w^{p_{1}(\cdot )})}.\end{aligned} $$

(14)

Similarly

$$ \begin{aligned}[b] \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})}&\leq C2^{j(\lambda _{2}- \alpha _{2}(\infty ))} \Vert f_{2} \Vert _{M\dot{K}^{\alpha _{2}(\cdot ),\lambda _{2}}_{q_{2},p_{2}(\cdot )}(w^{p_{2}(\cdot )})}.\end{aligned} $$

Here we are using $\alpha (\infty )=\alpha _{1}(\infty )+\alpha _{2}(\infty )$ and $\lambda =\lambda _{1}+\lambda _{2}$:

$$ \begin{aligned}[b] &\Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \\ &\quad \leq C2^{j(\lambda _{1}-\alpha _{1}(\infty ))}2^{j(\lambda _{2}- \alpha _{2}(\infty ))} \Vert f_{1} \Vert _{M\dot{K}^{\alpha _{1}(\cdot ),\lambda _{1}}_{q_{1},p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2} \Vert _{M\dot{K}^{\alpha _{2}(\cdot ),\lambda _{2}}_{q_{2},p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \\ &\quad = C2^{j(\lambda -\alpha (\infty ))} \Vert f_{1} \Vert _{M\dot{K}^{\alpha _{1}(\cdot ), \lambda _{1}}_{q_{1},p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2} \Vert _{M\dot{K}^{ \alpha _{2}(\cdot ),\lambda _{2}}_{q_{2},p_{2}(\cdot )}(w^{p_{2}(\cdot )})}.\end{aligned} $$

With the use of the definition of variable exponent Herz–Morrey space and Proposition 4.2, we can get the following inequality:

$$\begin{aligned} &\bigl\Vert H_{\beta}(f_{1},f_{2}) \bigr\Vert _{M\dot{K}^{\alpha (\cdot ),\lambda}_{q,q(\cdot )}(w^{q( \cdot )})} \\ &\quad =\sup_{k_{0}\in{Z}}2^{-k_{0}\lambda } \Biggl(\sum_{k=- \infty}^{k_{0}}2^{k\alpha (\cdot )q} \bigl\Vert H_{\beta}(f_{1},f_{2})\cdot \chi _{k} \bigr\Vert ^{q}_{L^{q(\cdot )}(w^{q(\cdot )})} \Biggr)^{\frac{1}{q}} \\ &\quad \leq \max \Biggl\{ \underset{k_{0}< 0}{\sup_{k_{0}\in{Z}}}\, 2^{-k_{0} \lambda } \Biggl(\sum_{k=-\infty}^{k_{0}}2^{k\alpha (0)q} \bigl\Vert H_{\beta}(f_{1},f_{2}) \cdot \chi _{k} \bigr\Vert ^{q}_{L^{q(\cdot )}(w^{q(\cdot )})} \Biggr)^{\frac{1}{q}}, \\ &\quad \quad \underset{k_{0}\geq 0}{\sup_{k_{0}\in{Z}}} \, 2^{-k_{0}\lambda } \Biggl( \Biggl(\sum_{k=-\infty}^{-1}2^{k\alpha (0)q} \bigl\Vert H_{\beta}(f_{1},f_{2}) \cdot \chi _{k} \bigr\Vert ^{q}_{L^{q(\cdot )}(w^{q(\cdot )})} \Biggr)^{\frac{1}{q}} \\ &\quad \quad {} + \Biggl(\sum_{k=0}^{k_{0}}2^{k\alpha (\infty )q} \bigl\Vert H_{\beta}(f_{1},f_{2}) \cdot \chi _{k} \bigr\Vert ^{q}_{L^{q(\cdot )}(w^{q(\cdot )})} \Biggr)^{\frac{1}{q}} \Biggr) \Biggr\} \\ &\quad =\max \{Y_{1},Y_{2}+Y_{3} \}, \end{aligned}$$

(15)

where

$$\begin{aligned}& Y_{1}=\underset{k_{0}< 0}{\sup_{k_{0}\in{Z}}}\, 2^{-k_{0}\lambda } \Biggl( \sum_{k=-\infty}^{k_{0}}2^{k\alpha (0)q} \bigl\Vert H_{\beta}(f_{1},f_{2}) \cdot \chi _{k} \bigr\Vert ^{q}_{L^{q(\cdot )}(w^{q(\cdot )})} \Biggr)^{\frac{1}{q}},\\& Y_{2}=\underset{k_{0}\geq 0}{\sup _{k_{0}\in{Z}}}\, 2^{-k_{0}\lambda } \Biggl(\sum _{k=-\infty}^{-1}2^{k\alpha (0)q} \bigl\Vert H_{\beta}(f_{1},f_{2}) \cdot \chi _{k} \bigr\Vert ^{q}_{L^{q(\cdot )}(w^{q(\cdot )})} \Biggr)^{\frac{1}{q}},\\& Y_{3}=\underset{k_{0}\geq 0}{\sup _{k_{0}\in{Z}}}\, 2^{-k_{0}\lambda } \Biggl(\sum _{k=0}^{k_{0}}2^{k\alpha (\infty )q} \bigl\Vert H_{\beta}(f_{1},f_{2}) \cdot \chi _{k} \bigr\Vert ^{q}_{L^{q(\cdot )}(w^{q(\cdot )})} \Biggr)^{\frac{1}{q}}. \end{aligned}$$

First, we will find the estimate of $Y_{1}$. Since $\alpha (0)\leq \alpha ({\infty})< n\delta _{11}+n\delta _{22}+ \lambda$,

$$ \begin{aligned}[b] Y_{1}&\leq C\underset{k_{0}< 0}{\sup_{k_{0}\in{Z}}}\, 2^{-k_{0}\lambda } \Biggl(\sum _{k=-\infty}^{k_{0}}2^{k\alpha (0)q} \Biggl(\sum _{j=- \infty}^{k} 2^{(-n\delta _{22}-n\delta _{11})(k-j)} \Vert f_{1} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \Biggr)^{q} \Biggr)^{ \frac{1}{q}} \\ & \leq C\underset{k_{0}< 0}{\sup_{k_{0}\in{Z}}}\, 2^{-k_{0}\lambda } \Biggl(\sum_{k=-\infty}^{k_{0}}2^{k\alpha (0)q} \Biggl(\sum_{j=- \infty}^{k} 2^{(-n\delta _{11}-n\delta _{22})(k-j)}2^{j(\lambda - \alpha (0))} \Vert f_{1} \Vert _{M\dot{k}^{\alpha _{1}(\cdot ),\lambda _{1}}_{q_{1},p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \\ &\quad {}\times \Vert f_{2} \Vert _{M\dot{k}^{\alpha _{2}(\cdot ),\lambda _{2}}_{q_{2},p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \Biggr)^{q} \Biggr)^{\frac{1}{q}} \\ & \leq C \Vert f_{1} \Vert _{M\dot{k}^{\alpha _{1}(\cdot ),\lambda _{1}}_{q_{1},p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2} \Vert _{M\dot{k}^{\alpha _{2}(\cdot ),\lambda _{2}}_{q_{2},p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \\ &\quad {}\times\underset{k_{0}< 0}{ \sup_{k_{0}\in{Z}}}\, 2^{-k_{0}\lambda } \Biggl(\sum _{k=- \infty}^{k_{0}}2^{k\lambda q} \Biggl(\sum _{j=-\infty}^{k} 2^{(n\delta _{11}+n \delta _{22}+\lambda -\alpha (0))(j-k)} \Biggr)^{q} \Biggr)^{ \frac{1}{q}} \\ &\leq C \Vert f_{1} \Vert _{M\dot{k}^{\alpha _{1}(\cdot ),\lambda _{1}}_{q_{1},p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2} \Vert _{M\dot{k}^{\alpha _{2}(\cdot ),\lambda _{2}}_{q_{2},p_{2}(\cdot )}(w^{p_{2}(\cdot )})} .\end{aligned} $$

The estimate of $Y_{2}$ is the same as that of $Y_{1}$. Finally, we approximate $Y_{3}$:

$$\begin{aligned} Y_{3}&\leq C\underset{k_{0} \geq 0}{\sup_{k_{0}\in{Z}}}\, 2^{-k_{0} \lambda } \Biggl(\sum _{k=0}^{k_{0}}2^{k\alpha (\infty )q} \Biggl(\sum _{j=- \infty}^{k} 2^{(-n\delta _{22}-n\delta _{11})(k-j)} \Vert f_{1} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \Biggr)^{q} \Biggr)^{ \frac{1}{q}} \\ & \leq C\underset{k_{0}\geq 0}{\sup_{k_{0}\in{Z}}}\, 2^{-k_{0}\lambda } \Biggl(\sum_{k=0}^{k_{0}}2^{k\alpha (\infty )q} \Biggl(\sum_{j=- \infty}^{k} 2^{(-n\delta _{11}-n\delta _{22})(k-j)}2^{j(\lambda - \alpha (\infty ))} \Vert f_{1} \Vert _{M\dot{k}^{\alpha _{1}(\cdot ),\lambda _{1}}_{q_{1},p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \\ &\quad {}\times \Vert f_{2} \Vert _{M\dot{k}^{\alpha _{2}(\cdot ),\lambda _{2}}_{q_{2},p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \Biggr)^{q} \Biggr)^{\frac{1}{q}} \\ & \leq C \Vert f_{1} \Vert _{M\dot{k}^{\alpha _{1}(\cdot ),\lambda _{1}}_{q_{1},p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2} \Vert _{M\dot{k}^{\alpha _{2}(\cdot ),\lambda _{2}}_{q_{2},p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \\ &\quad {}\times \underset{k_{0}\geq 0}{\sup_{k_{0}\in{Z}}}\, 2^{-k_{0}\lambda } \Biggl( \sum _{k=0}^{k_{0}}2^{k\lambda q} \Biggl(\sum _{j=-\infty}^{k} 2^{(n \delta _{11}+n\delta _{22}+\lambda -\alpha (\infty ))(j-k)} \Biggr)^{q} \Biggr)^{\frac{1}{q}} \\ &\leq C \Vert f_{1} \Vert _{M\dot{k}^{\alpha _{1}(\cdot ),\lambda _{1}}_{q_{1},p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2} \Vert _{M\dot{k}^{\alpha _{2}(\cdot ),\lambda _{2}}_{q_{2},p_{2}(\cdot )}(w^{p_{2}(\cdot )})} . \end{aligned}$$

Putting the approximations of $Y_{1}$, $Y_{2}$, and $Y_{3}$ into (15) yields the required outcome. □

Theorem 4.4

Let $q_{1}$, $q_{2}$, q, $p_{1}(\cdot )$, $p_{2}(\cdot )$, $p(\cdot )$, w, $\alpha (\cdot )$, and β be as in Theorem 4.3. Additionally, if $\alpha (\infty )\geq \alpha (0)>\lambda -n\delta$, where $\delta \in (0,1)$ is a constant arising in (3.4), then

$$ \bigl\Vert H^{*}_{\beta}(f_{1},f_{2}) \bigr\Vert _{M\dot{K}^{\alpha (\cdot ),\lambda}_{q,q(\cdot )}(w^{q( \cdot )})}\leq C \Vert f_{1} \Vert _{M\dot{K}^{\alpha _{1}(\cdot ),\lambda _{1}}_{q_{1},p_{1}( \cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2} \Vert _{M\dot{K}^{\alpha _{2}(\cdot ), \lambda _{2}}_{q_{2},p_{2}(\cdot )}(w^{p_{2}(\cdot )})}. $$

Proof

By utilizing Hölder’s inequality, we obtain

$$\begin{aligned}& \begin{aligned}[b] \bigl\vert H^{*}_{\beta}(f_{1},f_{2}) (x)\cdot \chi _{k}(x) \bigr\vert &\leq \int _{R^{n} \setminus B_{k}} \int _{R^{n}\setminus B_{k}}\frac{1}{ \vert t \vert ^{2n-\beta}} \bigl\vert f_{1}(t_{1}) \bigr\vert \bigl\vert f_{2}(t_{2}) \bigr\vert \,dt _{1}\,dt _{2} \cdot \chi _{k}(x) \\ &\leq C\sum_{j=k+1}^{\infty}2^{-j(2n-\beta )} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \\ & \quad {}\times \Vert \chi _{j} \Vert _{({L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{\prime }}} \Vert \chi _{j} \Vert _{({L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{\prime }}}\chi _{k}(x),\end{aligned} \\& \begin{aligned}[b] \bigl\Vert H^{*}_{\beta}(f_{1},f_{2}) (x)\cdot \chi _{k} \bigr\Vert _{L^{q(\cdot )}{(w^{q(\cdot )})}}& \leq C\sum _{j=k+1}^{\infty}2^{-j(2n-\beta )} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \\ &\quad {}\times \Vert \chi _{j} \Vert _{({L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{\prime }}} \Vert \chi _{j} \Vert _{({L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{\prime }}} \Vert \chi _{k} \Vert _{L^{q(\cdot )}(w^{q}(\cdot ))}.\end{aligned} \end{aligned}$$

(16)

In the light of inequality (10), we acquire

$$ \begin{aligned}[b] \bigl\Vert H^{*}_{\beta}(f_{1},f_{2}) (x)\cdot \chi _{k} \bigr\Vert _{L^{q(\cdot )}{(w^{q(\cdot )})}}& \leq C\sum _{j=k+1}^{\infty} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \\ &\quad {}\times \Vert \chi _{j} \Vert ^{-1}_{L^{q(\cdot )}(w^{q}(\cdot ))} \Vert \chi _{k} \Vert _{L^{q( \cdot )}(w^{q}(\cdot ))} \\ &\leq C\sum_{j=k+1}^{\infty}2^{n\delta (k-j)} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})},\end{aligned} $$

(17)

where we used the result of Lemma 3.7.

We will use the same procedure as in Theorem 4.3 to obtain

$$ \bigl\Vert H^{*}_{\beta}(f_{1},f_{2}) \bigr\Vert ^{q}_{M\dot{K}^{\alpha (\cdot ),\lambda}_{q,q(\cdot )}(w^{q( \cdot )})}=\max \{Z_{1},{Z_{2}+Z_{3}} \}, $$

(18)

where

$$\begin{aligned}& Z_{1}=\underset{k_{0}< 0}{\sup_{k_{0}\in{Z}}}\, 2^{-k_{0}\lambda q} \sum_{k=- \infty}^{k_{0}}2^{k\alpha (0)q} \bigl\Vert H^{*}_{\beta}(f_{1},f_{2}) (x)\cdot \chi _{k} \bigr\Vert ^{q}_{L^{q(\cdot )}(w^{q(\cdot )})}, \\& Z_{2}=\underset{k_{0}\geq 0}{\sup _{k_{0}\in{Z}}}\, 2^{-k_{0}\lambda q} \sum_{k=-\infty}^{-1}2^{k\alpha (0)q} \bigl\Vert H^{*}_{\beta}(f_{1},f_{2}) (x) \cdot \chi _{k} \bigr\Vert ^{q}_{L^{q(\cdot )}(w^{q(\cdot )})},\\& Z_{3}=\underset{k_{0}\geq 0}{\sup _{k_{0}\in{Z}}}\, 2^{-k_{0}\lambda q} \sum_{k=0}^{k_{0}}2^{k\alpha (\infty )q} \bigl\Vert H^{*}_{\beta}(f_{1},f_{2}) (x) \cdot \chi _{k} \bigr\Vert ^{q}_{L^{q(\cdot )}(w^{q(\cdot )})}. \end{aligned}$$

The boundedness of $Z_{l} \ (l=1,2,3)$ is alike to that of $Y_{l} \ (l=1,2,3)$ of Theorem 4.3. Here we are close to our result. □

Theorem 4.5

Let $0< q,q_{1},q_{2}<\infty$, $q(\cdot )\in P (\mathbb{R}^{n})\cap \mathscr{C}^{\log}( \mathbb{R}^{n})$, and $p(\cdot )$ be such that $\frac{1}{q(\cdot )}=\frac{1}{p(\cdot )}-\frac{\beta}{n}$ with $\frac{1}{p(\cdot )}=\frac{1}{p_{1}(\cdot )}+\frac{1}{p_{2}(\cdot )}$. Also, let $w^{p_{2}(\cdot )},w^{p_{1}(\cdot )}\in A_{1}$, let $\lambda =\lambda _{1}+\lambda _{2}$, and let $\alpha (\cdot )\in \mathscr{C}^{\log}(\mathbb{R}^{n})\cap L^{\infty}( \mathbb{R}^{n})$ be a log Hölder continuous function at the origin satisfying $\alpha (0)=\alpha _{1}(0)+\alpha _{2}(0)$, $\alpha (\infty )=\alpha _{1}(\infty )+\alpha _{2}(\infty )$ with $\alpha (0)\leq \alpha (\infty )< n\delta _{11}+\lambda +n\delta _{22}$, where $\delta _{22},\delta _{11}\in (0,1)$ are the constants arising in (7). Then $[b,H_{\beta}]$ is bounded from ${M\dot{K}^{\alpha _{1}(\cdot ),\lambda _{1}}_{q_{1},p_{1}(\cdot )}(w^{p_{1}( \cdot )})}\times {M\dot{K}^{\alpha _{2}(\cdot ),\lambda _{2}}_{q_{2},p_{2}( \cdot )}(w^{p_{2}(\cdot )})}$ to ${M\dot{K}^{\alpha (\cdot ),\lambda}_{q,q(\cdot )}(w^{q(\cdot )})}$, where ${b}=(b_{1},b_{2})$ and $b_{1},b_{2}\in \mathit{BMO}$.

Proof

We have

$$\begin{aligned}& \begin{aligned}[b] & \bigl\vert [b_{1},H_{\beta}](f_{1},f_{2}) (z)\cdot \chi _{k}(z) \bigr\vert \\ &\quad \leq \frac{1}{ \vert z \vert ^{2n-\beta}} \int _{B_{k}} \int _{B_{k}} \bigl\vert f_{1}(t_{1})f_{1}(t_{2}) \bigl(b_{1}(z)-b(t_{1}) \bigr) \bigr\vert \,dt _{1}\,dt _{2} \cdot \chi _{k}(z) \\ &\quad \leq C 2^{-k(2n-\beta )}\sum_{j=-\infty}^{k} \int _{B_{k}} \int _{B_{k}} \bigl\vert f_{1}(t)f_{2}(t) \bigl(b_{1}(z)-(b_{1})_{B_{j}}+(b_{1})_{B_{j}}-b_{1}(t_{1}) \bigr) \bigr\vert \,dt _{1}\,dt _{2} \cdot \chi _{k}(z) \\ &\quad \leq C 2^{-k(2n-\beta )}\sum_{j=-\infty}^{k} \int _{B_{k}} \int _{B_{k}} \bigl\vert f_{2}(t_{2})f_{1}(t_{1}) \bigl(b_{1}(z)-(b_{1})_{B_{j}} \bigr) \bigr\vert \,dt _{1}\,dt _{2} \cdot \chi _{k}(z) \\ &\quad \quad {}+C 2^{-k(2n-\beta )}\sum_{j=-\infty}^{k} \int _{B_{k}} \int _{B_{k}} \bigl\vert f_{2}(t_{2})f_{1}(t_{1}) \bigl(b_{1}(t_{1})-(b_{1})_{B_{j}} \bigr) \bigr\vert \,dt _{1}\,dt _{2} \cdot \chi _{k}(z) \\ &\quad =I+II,\end{aligned} \\& \begin{aligned}[b] I= C 2^{-k(2n-\beta )}\sum _{j=-\infty}^{k} \int _{B_{k}} \int _{B_{k}} \bigl\vert f_{2}(t_{2})f_{1}(t_{1}) \bigl(b_{1}(z)-(b_{1})_{B_{j}} \bigr) \bigr\vert \,dt _{1}\,dt _{2} \cdot \chi _{k}(z). \end{aligned} \end{aligned}$$

By utilizing the Hölder inequality, we obtain

$$ \begin{aligned}[b] I&\leq C2^{-k(2n-\beta )}\sum _{j=-\infty}^{k} \bigl\vert \bigl(b_{1}(z)-(b_{1})_{B_{j}} \bigr) \cdot \chi _{k}(z) \bigr\vert \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert \chi _{j} \Vert _{({L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{\prime }}} \\ &\quad {}\times \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \Vert \chi _{j} \Vert _{({L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{\prime }}}.\end{aligned} $$

Utilizing Lemmas 3.7, 3.2, and 3.9, we acquire

$$\begin{aligned}& \begin{aligned}[b] & \Vert I \Vert _{L^{q(\cdot )}{(w^{q(\cdot )})}} \\ &\quad \leq C2^{k\beta}\sum_{j=-\infty}^{k} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})}\Vert \chi _{j} \Vert _{({L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{\prime }}} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \\ &\quad \quad {}\times \Vert \chi _{j} \Vert _{({L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{ \prime }}}2^{-2kn} \bigl\Vert \bigl(b_{1}(z)-(b_{1})_{B_{j}} \bigr)\chi _{k} \bigr\Vert _{L^{q(\cdot )}{(w^{q(\cdot )})}} \\ &\quad \leq C2^{-k(2n-\beta )}\sum_{j=-\infty}^{k} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \Vert \chi _{j} \Vert _{({L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{ \prime }}} \\ &\quad \quad {}\times \Vert \chi _{j} \Vert _{({L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{\prime }}}(k-j) \Vert b_{1} \Vert _{\mathit{BMO}} \Vert \chi _{k} \Vert _{L^{q(\cdot )}{(w^{q(\cdot )})} },\end{aligned} \end{aligned}$$

(19)

$$\begin{aligned}& \begin{aligned} II&= 2^{-k(2n-\beta )}\sum _{j=-\infty}^{k} \int _{B_{k}} \int _{B_{k}} \bigl\vert f_{2}(t_{2})f_{1}(t_{1}) \bigl(b_{1}(t_{1})-(b_{1})_{B_{j}} \bigr) \bigr\vert \,dt _{1}\,dt _{2} \cdot \chi _{k}(z) \\ &=2^{-k(2n-\beta )}\sum_{j=-\infty}^{k} \int _{B_{k}} \biggl\vert f_{2}(t_{2})\,dt _{2} \int _{B_{k}}f_{1}(t_{1}) \bigl(b_{1}(t_{1})-(b_{1})_{B_{j}} \bigr) \biggr\vert \,dt _{1}\cdot \chi _{k}(z) \\ &\leq C2^{-k(2n-\beta )}\sum_{j=-\infty}^{k} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \bigl\Vert \bigl(b_{1}(t_{1})-(b_{1})_{B_{j}} \bigr)\chi _{j} \bigr\Vert _{({L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{ \prime }}} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \\ &\quad {}\times \Vert \chi _{j} \Vert _{({L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{\prime }}}\cdot \chi _{k}(z),\end{aligned} \\& \begin{aligned}[b] \Vert II \Vert _{L^{q(\cdot )}{(w^{q(\cdot )})}}&\leq C2^{-k(2n-\beta )}\sum_{j=- \infty}^{k} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \bigl\Vert \bigl(b_{1}(t_{1})-(b_{1})_{B_{j}} \bigr) \chi _{j} \bigr\Vert _{({L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{\prime }}} \\ &\quad {}\times \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \Vert \chi _{j} \Vert _{({L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{\prime }}} \bigl\Vert \chi _{k}(x) \bigr\Vert _{L^{q(\cdot )}{(w^{q(\cdot )})}} \\ & \leq C2^{-k(2n-\beta )}\sum_{j=-\infty}^{k} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert b_{1} \Vert _{\mathit{BMO}} \Vert \chi _{j} \Vert _{({L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{ \prime }}} \\ &\quad {}\times \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})}\Vert \chi _{j} \Vert _{({L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{\prime }}} \bigl\Vert \chi _{k}(x) \bigr\Vert _{L^{q(\cdot )}{(w^{q(\cdot )})}}. \end{aligned} \end{aligned}$$

(20)

From inequalities (19) and (20), we have

$$ \begin{aligned}[b] & \bigl\Vert [b_{1},H_{\beta}](f_{1},f_{2}) (z)\cdot \chi _{k} \bigr\Vert _{L^{q(\cdot )}{(w^{q(\cdot )})} } \\ &\quad \leq C2^{-k(2n-\beta )}\sum_{j=-\infty}^{k}(k-j) \Vert b_{1} \Vert _{\mathit{BMO}} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \Vert \chi _{j} \Vert _{({L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{\prime }}} \\ &\quad \quad {}\times \Vert \chi _{j} \Vert _{({L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{\prime }}} \Vert \chi _{k} \Vert _{L^{q(\cdot )}{(w^{q(\cdot )})} }. \end{aligned} $$

To move forward, we use Theorem 4.3:

$$ \begin{aligned}[b] & \bigl\Vert [b_{1},H_{\beta}](f_{1},f_{2}) (z)\cdot \chi _{k} \bigr\Vert _{L^{q(\cdot )}{(w^{q(\cdot )})} } \\ &\quad \leq C\sum_{j=-\infty}^{k}2^{(n\delta _{11}+n\delta _{22})(j-k)}(k-j) \Vert b_{1} \Vert _{\mathit{BMO}} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} .\end{aligned} $$

(21)

With the use of the definition of variable exponent Herz–Morrey space and Proposition 4.2 we get the following inequality:

$$ \begin{aligned}[b] &\bigl\Vert [b_{1},H_{\beta}](f_{1},f_{2}) (z) \bigr\Vert _{M\dot{K}^{\alpha (\cdot ), \lambda}_{q,q(\cdot )}(w^{q(\cdot )})} \\ &\quad =\sup_{k_{0}\in{Z}}2^{-k_{0} \lambda } \Biggl(\sum_{k=-\infty}^{k_{0}}2^{k\alpha (\cdot )q} \bigl\Vert [b_{1},H_{ \beta}](f_{1},f_{2}) (z)\cdot \chi _{k} \bigr\Vert ^{q}_{L^{q(\cdot )}(w^{q(\cdot )})} \Biggr)^{\frac{1}{q}} \\ &\quad \leq \max \Biggl\{ \underset{k_{0}< 0}{\sup_{k_{0}\in{Z}}}\, 2^{-k_{0} \lambda } \Biggl(\sum_{k=-\infty}^{k_{0}}2^{k\alpha (0)q} \bigl\Vert [b_{1},H_{ \beta}](f_{1},f_{2}) (z)\cdot \chi _{k} \bigr\Vert ^{q}_{L^{q(\cdot )}(w^{q(\cdot )})} \Biggr)^{\frac{1}{q}}, \\ &\quad \quad \underset{k_{0}\geq 0}{\sup_{k_{0}\in{Z}}}\, 2^{-k_{0}\lambda } \Biggl( \Biggl(\sum_{k=-\infty}^{-1}2^{k\alpha (0)q} \bigl\Vert [b_{1},H_{\beta}](f_{1},f_{2}) (z) \cdot \chi _{k} \bigr\Vert ^{q}_{L^{q(\cdot )}(w^{q(\cdot )})} \Biggr)^{\frac{1}{q}} \\ &\quad\quad {} + \Biggl(\sum_{k=0}^{k_{0}}2^{k\alpha (\infty )q} \bigl\Vert [b_{1},H_{\beta}](f_{1},f_{2}) (z) \cdot \chi _{k} \bigr\Vert ^{q}_{L^{q(\cdot )}(w^{q(\cdot )})} \Biggr)^{\frac{1}{q}} \Biggr) \Biggr\} \\ &\quad =\max \{A_{1},A_{2}+A_{3} \},\end{aligned} $$

(22)

where

$$\begin{aligned}& A_{1}=\underset{k_{0}< 0}{\sup_{k_{0}\in{Z}}}\, 2^{-k_{0}\lambda } \Biggl( \sum_{k=-\infty}^{k_{0}}2^{k\alpha (0)q} \bigl\Vert [b_{1},H_{\beta}](f_{1},f_{2}) (z) \cdot \chi _{k} \bigr\Vert ^{q}_{L^{q(\cdot )}(w^{q(\cdot )})} \Biggr)^{\frac{1}{q}},\\& A_{2}=\underset{k_{0}\geq 0}{\sup _{k_{0}\in{Z}}}\, 2^{-k_{0}\lambda } \Biggl(\sum _{k=-\infty}^{-1}2^{k\alpha (0)q} \bigl\Vert [b_{1},H_{\beta}](f_{1},f_{2}) (z) \cdot \chi _{k} \bigr\Vert ^{q}_{L^{q(\cdot )}(w^{q(\cdot )})} \Biggr)^{\frac{1}{q}},\\& A_{3}=\underset{k_{0}\geq 0}{\sup _{k_{0}\in{Z}}}\, 2^{-k_{0}\lambda } \Biggl(\sum _{k=0}^{k_{0}}2^{k\alpha (\infty )q} \bigl\Vert [b_{1},H_{\beta}](f_{1},f_{2}) (z) \cdot \chi _{k} \bigr\Vert ^{q}_{L^{q(\cdot )}(w^{q(\cdot )})} \Biggr)^{\frac{1}{q}}. \end{aligned}$$

In the remainder of the proof, we use the same calculation as in Theorem 4.3. We obtain the following result:

$$ \begin{aligned}[b] & \bigl\Vert [b_{1},H_{\beta}](f_{1},f_{2}) \bigr\Vert _{M\dot{K}^{\alpha (\cdot ),\lambda}_{q,q(\cdot )}(w^{q( \cdot )})}\leq C \Vert b_{1} \Vert _{\mathit{BMO}} \Vert f_{1} \Vert _{M\dot{K}^{\alpha _{1}(\cdot ), \lambda _{1}}_{q_{1},p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2} \Vert _{M \dot{K}^{\alpha _{2}(\cdot ),\lambda _{2}}_{q_{2},p_{2}(\cdot )}(w^{p_{2}( \cdot )})}.\end{aligned} $$

Similarly, we can easily estimate the following result:

$$ \begin{aligned}[b] & \bigl\Vert [b_{2},H_{\beta}](f_{1},f_{2}) \bigr\Vert _{M\dot{K}^{\alpha (\cdot ),\lambda}_{q,q(\cdot )}(w^{q( \cdot )})}\leq C \Vert b_{2} \Vert _{\mathit{BMO}} \Vert f_{1} \Vert _{M\dot{K}^{\alpha _{1}(\cdot ), \lambda _{1}}_{q_{1},p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2} \Vert _{M \dot{K}^{\alpha _{2}(\cdot ),\lambda _{2}}_{q_{2},p_{2}(\cdot )}(w^{p_{2}( \cdot )})}. \end{aligned} $$

 □

Theorem 4.6

Let $q_{1}$, $q_{2}$, q, $p_{1}(\cdot )$, $p_{2}(\cdot )$, $p(\cdot )$, w, $\alpha (\cdot )$, and β be as in Theorem 4.3. Additionally, if $\alpha (\infty )\geq \alpha (0)>\lambda -n\delta$, where $\delta \in (0,1)$ is a constant arising in (3.4), then $[b,H^{*}_{\beta}]$ is bounded from ${M\dot{K}^{\alpha _{1}(\cdot ),\lambda _{1}}_{q_{1},p_{1}(\cdot )}(w^{p_{1}( \cdot )})}\times {M\dot{K}^{\alpha _{2}(\cdot ),\lambda _{2}}_{q_{2},p_{2}( \cdot )}(w^{p_{2}(\cdot )})}$ to ${M\dot{K}^{\alpha (\cdot ),\lambda}_{q,q(\cdot )}(w^{q(\cdot )})}$, where ${b}=(b_{1},b_{2})$ and $b_{1},b_{2}\in \mathit{BMO}$.

Proof

We have

$$\begin{aligned}& \begin{aligned}[b] & \bigl\vert \bigl[b_{1},H^{*}_{\beta} \bigr](f_{1},f_{2}) (z)\cdot \chi _{k}(z) \bigr\vert \\ &\quad \leq \int _{ \mathbb{R}^{n}\setminus B_{k}} \int _{\mathbb{R}^{n}\setminus B_{k}} \frac{1}{ \vert t \vert ^{2n-\beta}} \bigl\vert f_{1}(t_{1})f_{1}(t_{2}) \bigl(b_{1}(z)-b(t_{1}) \bigr) \bigr\vert \,dt _{1}\,dt _{2} \cdot \chi _{k}(z) \\ &\quad \leq C \sum_{j=k+1}^{\infty} 2^{-j(2n-\beta )} \int _{\mathbb{R}^{n} \setminus B_{k}} \int _{\mathbb{R}^{n}\setminus B_{k}} \bigl\vert f_{1}(t_{1})f_{2}(t_{2}) \bigl(b_{1}(z)-(b_{1})_{B_{j}}+(b_{1})_{B_{j}}-b_{1}(t_{1}) \bigr) \bigr\vert \,dt _{1}\,dt _{2} \\ &\quad \quad {} \cdot \chi _{k}(z) \\ &\quad \leq C \sum_{j=k+1}^{\infty} 2^{-j(2n-\beta )} \int _{\mathbb{R}^{n} \setminus B_{k}} \int _{\mathbb{R}^{n}\setminus B_{k}} \bigl\vert f_{2}(t_{2})f_{1}(t_{1}) \bigl(b_{1}(z)-(b_{1})_{B_{j}} \bigr) \bigr\vert \,dt _{1}\,dt _{2} \cdot \chi _{k}(z) \\ &\quad\quad {} +C \sum_{j=k+1}^{\infty} 2^{-j(2n-\beta )} \int _{\mathbb{R}^{n} \setminus B_{k}} \int _{\mathbb{R}^{n}\setminus B_{k}} \bigl\vert f_{2}(t_{2})f_{1}(t_{1}) \bigl(b_{1}(t_{1})-(b_{1})_{B_{j}} \bigr) \bigr\vert \,dt _{1}\,dt _{2} \cdot \chi _{k}(z) \\ &\quad =III+IV,\end{aligned} \\& \begin{aligned}[b] III= C \sum_{j=k+1}^{\infty} 2^{-j(2n-\beta )} \int _{\mathbb{R}^{n} \setminus B_{k}} \int _{\mathbb{R}^{n}\setminus B_{k}} \bigl\vert f_{2}(t_{2})f_{1}(t_{1}) \bigl(b_{1}(z)-(b_{1})_{B_{j}} \bigr) \bigr\vert \,dt _{1}\,dt _{2} \cdot \chi _{k}(z). \end{aligned} \end{aligned}$$

By utilizing the Hölder inequality, we obtain

$$ \begin{aligned}[b] III&\leq C\sum_{j=k+1}^{\infty} 2^{-j(2n-\beta )} \int _{\mathbb{R}^{n} \setminus B_{k}} \int _{\mathbb{R}^{n}\setminus B_{k}} \bigl\vert \bigl(b_{1}(z)-(b_{1})_{B_{j}} \bigr) \cdot \chi _{k}(z) \bigr\vert \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \\ & \quad{} \times \Vert \chi _{j} \Vert _{({L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{\prime }}} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})}\Vert \chi _{j} \Vert _{({L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{\prime }}}.\end{aligned} $$

Utilizing Lemmas 3.7, 3.2, and 3.9, we acquire

$$\begin{aligned}& \begin{aligned}[b] & \Vert III \Vert _{L^{q(\cdot )}{(w^{q(\cdot )})}} \\ &\quad \leq C \sum_{j=k+1}^{\infty} 2^{-j(2n-\beta )} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert \chi _{j} \Vert _{({L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{\prime }}} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \\ &\quad \quad {}\times \Vert \chi _{j} \Vert _{({L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{ \prime }}} \bigl\Vert \bigl(b_{1}(z)-(b_{1})_{B_{j}} \bigr)\chi _{k} \bigr\Vert _{L^{q(\cdot )}{(w^{q(\cdot )})}} \\ &\quad \leq C \sum_{j=k+1}^{\infty} 2^{-j(2n-\beta )} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \Vert \chi _{j} \Vert _{({L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{ \prime }}} \\ &\quad \quad {}\times \Vert \chi _{j} \Vert _{({L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{\prime }}}(k-j) \Vert b_{1} \Vert _{\mathit{BMO}} \Vert \chi _{k} \Vert _{L^{q(\cdot )}{(w^{q(\cdot )})} },\end{aligned} \end{aligned}$$

(23)

$$\begin{aligned}& \begin{aligned} IV&= \sum_{j=k+1}^{\infty} 2^{-j(2n-\beta )} \int _{\mathbb{R}^{n} \setminus B_{k}} \int _{\mathbb{R}^{n}\setminus B_{k}} \bigl\vert f_{2}(t_{2})f_{1}(t_{1}) \bigl(b_{1}(t_{1})-(b_{1})_{B_{j}} \bigr) \bigr\vert \,dt _{1}\,dt _{2} \cdot \chi _{k}(z) \\ &=\sum_{j=k+1}^{\infty} 2^{-j(2n-\beta )} \int _{\mathbb{R}^{n} \setminus B_{k}} \int _{\mathbb{R}^{n}\setminus B_{k}} \bigl\vert f_{2}(t_{2})f_{1}(t_{1}) \bigl(b_{1}(t_{1})-(b_{1})_{B_{j}} \bigr) \bigr\vert \,dt _{2}\,dt _{1} \cdot \chi _{k}(z) \\ &\leq C\sum_{j=k+1}^{\infty} 2^{-j(2n-\beta )} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \bigl\Vert \bigl(b_{1}(t_{1})-(b_{1})_{B_{j}} \bigr)\chi _{j} \bigr\Vert _{({L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{ \prime }}} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \\ & \quad {}\times \Vert \chi _{j} \Vert _{({L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{\prime }}}\cdot \chi _{k}(x),\end{aligned} \\& \begin{aligned}[b] \Vert IV \Vert _{L^{q(\cdot )}{(w^{q(\cdot )})}}&\leq C\sum_{j=k+1}^{\infty} 2^{-j(2n- \beta )} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \bigl\Vert \bigl(b_{1}(t_{1})-(b_{1})_{B_{j}} \bigr) \chi _{j} \bigr\Vert _{({L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{\prime }}} \\ &\quad {}\times \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \Vert \chi _{j} \Vert _{({L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{\prime }}} \bigl\Vert \chi _{k}(x) \bigr\Vert _{L^{q(\cdot )}{(w^{q(\cdot )})}} \\ & \leq C\sum_{j=k+1}^{\infty} 2^{-j(2n-\beta )} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert b_{1} \Vert _{\mathit{BMO}} \Vert \chi _{j} \Vert _{({L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{ \prime }}} \\ &\quad {}\times \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})}\Vert \chi _{j} \Vert _{({L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{\prime }}} \bigl\Vert \chi _{k}(x) \bigr\Vert _{L^{q(\cdot )}{(w^{q(\cdot )})}}. \end{aligned} \end{aligned}$$

(24)

From inequalities (23) and (24), we have

$$ \begin{aligned}[b] & \bigl\Vert \bigl[b_{1},H^{*}_{\beta} \bigr](f_{1},f_{2}) (z)\cdot \chi _{k} \bigr\Vert _{L^{q(\cdot )}{(w^{q(\cdot )})} } \\ &\quad \leq C\sum_{j=k+1}^{\infty} 2^{-j(2n-\beta )}(k-j) \Vert b_{1} \Vert _{\mathit{BMO}} \Vert f_{1j} \Vert _{L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )})} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )})} \Vert \chi _{j} \Vert _{({L^{p_{1}(\cdot )}(w^{p_{1}(\cdot )}))^{\prime }}} \\ &\quad \quad {}\times \Vert \chi _{j} \Vert _{({L^{p_{2}(\cdot )}(w^{p_{2}(\cdot )}))^{\prime }}} \Vert \chi _{k} \Vert _{L^{q(\cdot )}{(w^{q(\cdot )})} } .\end{aligned} $$

With the use of Proposition 4.2 we get the following inequality:

$$ \begin{aligned}[b] &\bigl\Vert \bigl[b_{1},H^{*}_{\beta} \bigr](f_{1},f_{2}) \bigr\Vert _{M\dot{K}^{\alpha (\cdot ), \lambda}_{q,q(\cdot )}(w^{q(\cdot )})} \\ &\quad =\sup_{k_{0}\in{Z}}2^{-k_{0} \lambda } \Biggl(\sum_{k=-\infty}^{k_{0}}2^{k\alpha (\cdot )q} \bigl\Vert \bigl[b_{1},H^{*}_{ \beta} \bigr](f_{1},f_{2}) (z)\cdot \chi _{k} \bigr\Vert ^{q}_{L^{q(\cdot )}(w^{q(\cdot )})} \Biggr)^{\frac{1}{q}} \\ &\quad \leq \max \Biggl\{ \underset{k_{0}< 0}{\sup_{k_{0}\in{Z}}}\, 2^{-k_{0} \lambda } \Biggl(\sum_{k=-\infty}^{k_{0}}2^{k\alpha (0)q} \bigl\Vert \bigl[b_{1},H^{*}_{ \beta} \bigr](f_{1},f_{2}) (z)\cdot \chi _{k} \bigr\Vert ^{q}_{L^{q(\cdot )}(w^{q(\cdot )})} \Biggr)^{\frac{1}{q}}, \\ &\quad \quad \underset{k_{0}\geq 0}{\sup_{k_{0}\in{Z}}}\, 2^{-k_{0}\lambda } \Biggl( \Biggl(\sum_{k=-\infty}^{-1}2^{k\alpha (0)q} \bigl\Vert \bigl[b_{1},H^{*}_{ \beta} \bigr](f_{1},f_{2}) (z)\cdot \chi _{k} \bigr\Vert ^{q}_{L^{q(\cdot )}(w^{q(\cdot )})} \Biggr)^{\frac{1}{q}} \\ &\quad \quad {} + \Biggl(\sum_{k=0}^{k_{0}}2^{k\alpha (\infty )q} \bigl\Vert \bigl[b_{1},H^{*}_{ \beta} \bigr](f_{1},f_{2}) (z)\cdot \chi _{k} \bigr\Vert ^{q}_{L^{q(\cdot )}(w^{q(\cdot )})} \Biggr)^{\frac{1}{q}} \Biggr) \Biggr\} \\ &\quad =\max \{B_{1},B_{2}+B_{3} \}.\end{aligned} $$

(25)

We can easily find the estimates of $B_{i}\ (i = 1, 2, 3)$ by similar methods to the ones in Theorem 4.5. □