Boundedness and Compactness for the Iterated Commutators of Bilinear Fractional Maximal Operators on Weighted Morrey Spaces

Abstract

Let ${\mathcal{M}}_{\alpha }$ be the bilinear fractional maximal operator. In this paper, we prove that the commutators ${\mathcal{M}}_{\alpha ,b}^i$ in the i-th entry $(i=1,2)$ and the bilinear iterated commutators ${\mathcal{M}}_{\alpha ,\overrightarrow{b}}$ of ${\mathcal{M}}_{\alpha }$ are bounded operators from product weighted Morrey spaces $L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)\times L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)$ to weighted Morrey spaces $L^{q,\kappa }\left(v_{\overrightarrow{w}}^q\right)$, provided that $b\in \mathrm{BMO}\left({\mathbb{R}}^{\mathrm{n}}\right)$ and $\overrightarrow{b}=(b_1,b_2)\in \mathrm{BMO}\left({\mathbb{R}}^{\mathrm{n}}\right)\times \mathrm{BMO}\left({\mathbb{R}}^{\mathrm{n}}\right)$. Furthermore, by using the techniques of function decompositions and the Fréchet–Kolmogorov theorem on weighted Morrey spaces, the compactness of ${\mathcal{M}}_{\alpha ,b}^i(i=1,2)$ and ${\mathcal{M}}_{\alpha ,\overrightarrow{b}}$ are also established whenever $b\in \mathrm{CMO}\left({\mathbb{R}}^{\mathrm{n}}\right)$ and $\overrightarrow{b}=(b_1,b_2)\in \mathrm{CMO}\left({\mathbb{R}}^{\mathrm{n}}\right)\times \mathrm{CMO}\left({\mathbb{R}}^{\mathrm{n}}\right)$, where $\mathrm{CMO}\left({\mathbb{R}}^{\mathrm{n}}\right)$ denotes the closure of $C_c^{\infty }\left({\mathbb{R}}^n\right)$ in the $\mathrm{BMO}\left({\mathbb{R}}^{\mathrm{n}}\right)$ topology.

1. Introduction

For $0<\alpha <2n$, $f, g\in L_{\mathrm{loc}}^p\left({\mathbb{R}}^n\right)$, the bilinear fractional integral ${\mathcal{I}}_{\alpha }$ and the bilinear fractional maximal operator ${\mathcal{M}}_{\alpha }$ are defined, respectively, by

$${\mathcal{I}}_{\alpha }(f,g)\left(x\right)={\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(y\right)g\left(z\right)}{{\left(\right|x-y|+|x-z\left|\right)}^{2n-\alpha }}}dydz$$

and

$${\mathcal{M}}_{\alpha }(f,g)\left(x\right)=\underset{B\ni x}{sup}{\displaystyle \frac{1}{{\left|B\right|}^{2-\frac{\alpha }{n}}}}{\int }_B{\int }_B\left|f\left(y\right)\right|\left|g\left(z\right)\right|dydz,$$

where the supremum is taken over all balls B containing x in ${\mathbb{R}}^n$. It is easy to see that

$${\mathcal{M}}_{\alpha }(f,g)\left(x\right)\lesssim I_{\alpha }\left(\right|f|,|g\left|\right)\left(x\right),$$

(1)

where the implicit constant depends only on n and $\alpha$. Kenig and Stein in [1], Grafakos and Kalton in [2] considered the $L^p$-boundedness of ${\mathcal{I}}_{\alpha }$ and ${\mathcal{M}}_{\alpha }$, respectively. For $b\in L_{\mathrm{Loc}}^1\left({\mathbb{R}}^n\right)$, we define the linear commutators of the bilinear fractional integral ${\mathcal{I}}_{\alpha }$ and the bilinear fractional maximal operator ${\mathcal{M}}_{\alpha }$ as follows

$$\begin{array}{cc}\hfill {\mathcal{I}}_{\alpha ,b}^1(f,g)\left(x\right)& ={\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{b\left(y\right)-b\left(x\right)}{{\left(\right|x-y|+|x-z\left|\right)}^{2n-\alpha }}}f\left(y\right)g\left(z\right)dydz,\hfill \\ \hfill {\mathcal{I}}_{\alpha ,b}^2(f,g)\left(x\right)& ={\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{b\left(z\right)-b\left(x\right)}{{\left(\right|x-y|+|x-z\left|\right)}^{2n-\alpha }}}f\left(y\right)g\left(z\right)dydz,\hfill \end{array}$$

and

$$\begin{array}{c}\hfill {\mathcal{M}}_{\alpha ,b}^1(f,g)\left(x\right)=\underset{B\ni x}{sup}{\displaystyle \frac{1}{{\left|B\right|}^{2-\frac{\alpha }{n}}}}{\int }_B{\int }_B|b\left(x\right)-b\left(y\right)|\left|f\left(y\right)g\left(z\right)\right|dydz,\\ \hfill {\mathcal{M}}_{\alpha ,b}^2(f,g)\left(x\right)=\underset{B\ni x}{sup}{\displaystyle \frac{1}{{\left|B\right|}^{2-\frac{\alpha }{n}}}}{\int }_B{\int }_B|b\left(x\right)-b\left(z\right)|\left|f\left(y\right)g\left(z\right)\right|dydz.\end{array}$$

For $\overrightarrow{b}=\left(b_1,b_2\right)\in \mathrm{BMO}\times \mathrm{BMO}:={\mathrm{BMO}}^2$, we define the iterated commutator of the bilinear fractional integral ${\mathcal{I}}_{\alpha }$ and the maximal iterated commutator of ${\mathcal{M}}_{\alpha }$ as

$${\mathcal{I}}_{\alpha ,\overrightarrow{b}}(f,g)\left(x\right)={\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{\left(b_1\left(x\right)-b_1\left(y\right)\right)\left(b_2\left(x\right)-b_2\left(z\right)\right)f\left(y\right)g\left(z\right)}{{\left(\right|x-y|+|x-z\left|\right)}^{2n-\alpha }}}dydz$$

and

$${\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)\left(x\right)=\underset{B\ni x}{sup}{\displaystyle \frac{1}{{\left|B\right|}^{2-\frac{\alpha }{n}}}}{\int }_B{\int }_B\left|b_1\left(x\right)-b_1\left(y\right)\right|\left|b_2\left(x\right)-b_2\left(z\right)\right|\left|f\left(y\right)g\left(z\right)\right|dydz.$$

The weighted boundedness of the bilinear fractional integral ${\mathcal{I}}_{\alpha }$ and the bilinear fractional maximal operator ${\mathcal{M}}_{\alpha }$ on Lebesgue Spaces are obtained by Moen [3]. He also introduced a multilinear fractional type of weight $A_{(\overrightarrow{p},q)}$ (see Definition 6) and gave a characterization of that class. The multiple fractional type weights $A_{(\overrightarrow{p},q)}$ are somewhat different from the weights in [4]. But it is a natural generalization of the classical $A_{p,q}$ weights in [5].

In 1978, Uchiyama [6] improved the boundedness result by establishing compactness for Calderón-Zygmund operators with symbols where $b\in \mathrm{CMO}$, where CMO denotes the closure of $C_c^{\infty }$ in the topology of BMO. Hereafter, we write BMO and CMO instead of $\mathrm{BMO}\left({\mathbb{R}}^n\right)$ and $\mathrm{CMO}\left({\mathbb{R}}^n\right)$, respectively. The boundedness and compactness of commutators of classical operators have attracted the attention of many scholars (see [7,8,9,10,11], for example). In [12], Chen and Xue established strongly weighted bounded results for the linear commutators ${\mathcal{I}}_{\alpha ,b}^i$ on Lebesgue spaces. Subsequently, Xue [13] also proved the following weighted strong type estimates for the iterated commutators ${\mathcal{I}}_{\alpha ,\overrightarrow{b}}$ and ${\mathcal{M}}_{\alpha ,\overrightarrow{b}}$ with $A_{(\overrightarrow{p},q)}$ weights on Lebesgue spaces. Furthermore, Chen and Wu [14] established an important improvement over the original results of Chen and Xue concerning strong-type weighted norm inequalities for multilinear commutators as follows.

Theorem 1

([14]). Let $0<\alpha <2n$, $b\in \mathrm{BMO}$ and $\overrightarrow{b}=\left(b_1,b_2\right)\in {\mathrm{BMO}}^2$. For $1<p_1,p_2<\infty$, ${\displaystyle \frac{1}{p}}={\displaystyle \frac{1}{p_1}}+{\displaystyle \frac{1}{p_2}}$ and ${\displaystyle \frac{1}{q}}={\displaystyle \frac{1}{p}}-{\displaystyle \frac{\alpha }{n}}$, if  $\overrightarrow{w}=\left(w_1,w_2\right)\in A_{(\overrightarrow{p},q)}$, then there exists a constant $C>0$ such that

$$\begin{array}{cc}\hfill {∥{\mathcal{I}}_{\alpha ,b}^i(f,g)∥}_{L^q\left(v_{\overrightarrow{w}}^q\right)}\le & C{∥b∥}_{\mathrm{BMO}}{∥f∥}_{L^{p_1}\left(w_1^{p_1}\right)}{∥g∥}_{L^{p_2}\left(w_2^{p_2}\right)},\hfill \\ \hfill {∥{\mathcal{I}}_{\alpha ,\overrightarrow{b}}(f,g)∥}_{L^q\left(v_{\overrightarrow{w}}^q\right)}\le & C{\parallel \overrightarrow{b}\parallel }_{{\mathrm{BMO}}^2}{\parallel f\parallel }_{L^{p_1}\left(w_1^{p_1}\right)}{\parallel g\parallel }_{L^{p_2}\left(w_2^{p_2}\right)},\hfill \end{array}$$

where $v_{\overrightarrow{w}}=w_1{w}_2$, $i=1,2$.

Remark 1.

From Theorem 1, we can obtain the following conclusions:

$$\begin{array}{cc}\hfill {∥{\mathcal{M}}_{\alpha ,b}^i(f,g)∥}_{L^q\left(v_{\overrightarrow{w}}^q\right)}\le & C{∥b∥}_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1}\left(w_1^{p_1}\right)}{\parallel g\parallel }_{L^{p_2}\left(w_2^{p_2}\right)},\hfill \\ \hfill {∥{\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)∥}_{L^q\left(v_{\overrightarrow{w}}^q\right)}\le & C{\parallel \overrightarrow{b}\parallel }_{{\mathrm{BMO}}^2}{\parallel f\parallel }_{L^{p_1}\left(w_1^{p_1}\right)}{\parallel g\parallel }_{L^{p_2}\left(w_2^{p_2}\right)},\hfill \end{array}$$

with the same conditions as in Theorem 1 and $i=1,2$.

In [15], the compactness on weighted Lebesgue spaces of the bilinear fractional maximal linear commutators ${\mathcal{M}}_{\alpha ,b}^i$ and the bilinear fractional maximal iterated commutator ${\mathcal{M}}_{\alpha ,\overrightarrow{b}}$ was established, where $b\in \mathrm{CMO}$ and $\overrightarrow{b}\in \mathrm{CMO}\times \mathrm{CMO}$. According to ([14], Lemma 3.1), this result can be further improved as follows.

Theorem 2.

Let $0<\alpha <2n$, $1<p_1,p_2<\infty$, $1<q<\infty$, ${\displaystyle \frac{1}{p}}={\displaystyle \frac{1}{p_1}}+{\displaystyle \frac{1}{p_2}}$and ${\displaystyle \frac{1}{q}}={\displaystyle \frac{1}{p}}-{\displaystyle \frac{\alpha }{n}}$. If $\overrightarrow{w}\in A_{(\overrightarrow{p},q)}$, $b\in CMO$and $\overrightarrow{b}=\left(b_1,b_2\right)\in {\mathrm{CMO}}^2$, where $v_{\overrightarrow{w}}=w_1{w}_2$, then ${\mathcal{M}}_{\alpha ,b}^i$and ${\mathcal{M}}_{\alpha ,\overrightarrow{b}}$are compact from $L^{p_1}\left(w_1^{p_1}\right)\times L^{p_2}\left(w_2^{p_2}\right)$to $L^q\left(v_{\overrightarrow{w}}^q\right)$.

As we all known, it was Morrey [16] who first introduced the classical Morrey space $L^{p,\lambda }$ in 1938 and applied it to the study of elliptic partial differential equations. Subsequently, Morrey spaces have found numerous applications in the field of partial differential equations, such as applications in the Navier-Stokes equations [17], Schrödinger equations [18], etc. In 2009, Komori and Shirai [19] considered a weighted generalization and introduced the following weighted Morrey spaces $L^{p,\kappa }\left(w\right)$.

Definition 1

(Weighted Morrey space [19]). For $1\le p<\infty$, $0<\kappa <1$ and a weight $w$, the weighted Morrey space is defined as follows:

$$L^{p,\kappa }\left(w\right):=\left\{f\in L_{\mathrm{loc}}^p\left(w\right):{\parallel f\parallel }_{L^{p,\kappa }\left(w\right)}<\infty \right\},$$

where

$${\parallel f\parallel }_{L^{p,\kappa }\left(w\right)}=\underset{B}{sup}{\left({\displaystyle \frac{1}{w{\left(B\right)}^{\kappa }}}{\int }_B{\left|f\left(t\right)\right|}^{p}w\left(t\right)dt\right)}^{\frac{1}{p}}$$

and the supremum is taken over all balls B in ${\mathbb{R}}^n$.

Remark 2.

We give some explanation about the weighted Morrey spaces.

(i) 

If $w\equiv 1$and $\kappa =\lambda /n$with $0<\lambda <n$, then $L^{p,\kappa }\left(w\right)=L^{p,\lambda }\left({\mathbb{R}}^n\right)$, which is the classical Morrey space.

(ii) 

Let w satisfies the doubling condition (6) in Section 3. If $\kappa =0$, $L^{p,0}\left(w\right)=L^p\left(w\right)$. If  $\kappa =1$, $L^{p,1}\left(w\right)=L^{\infty }\left(w\right)$ by the Lebesgue differentiation theorem with respect to w; see [20] for more details.

In order to handle the fractional case, Komori and Shirai [19] also introduced weighted Morrey spaces with two weights.

Definition 2

(Weighted Morrey space with two weights [19]). Assume that $1\le p<\infty$ and $0<\kappa <1$. For a pair of weights $(\mu ,\lambda ),$ the two-weighted Morrey space is defined as

$$L^{p,\kappa }(\mu ,\lambda )=\left\{f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^p\left(\mu \right):{\parallel f\parallel }_{L^{p,\kappa }(\mu ,\lambda )}<\infty \right\},$$

where

$${\parallel f\parallel }_{L^{p,\kappa }(\mu ,\lambda )}=\underset{B}{sup}{\left({\displaystyle \frac{1}{\lambda {\left(B\right)}^{\kappa }}}{\int }_B{\left|f\left(t\right)\right|}^p\mu \left(t\right)dt\right)}^{{\displaystyle \frac{1}{p}}}$$

and the supremum is taken over all balls B in ${\mathbb{R}}^n$. When $\mu =\lambda$, we use the simplified notation $L^{p,\kappa }\left(\mu \right)$.

Moreover, the Morrey space boundedness of the bilinear fractional integral ${\mathcal{I}}_{\alpha }$ and the bilinear fractional maximal operator ${\mathcal{M}}_{\alpha }$ has been studied by several authors; see [21,22,23,24], for example. By using a refined function decomposition technique coupled with sharp $A_{(\overrightarrow{p},q)}$ weight inequalities, Tao and Gao [25] established the boundedness of the bilinear fractional integral ${\mathcal{I}}_{\alpha }$ and the bilinear fractional maximal operator ${\mathcal{M}}_{\alpha }$ on weighted Morrey spaces.

Theorem 3

([25]). Let $0<\alpha <2n$, $0<\kappa <1$, $1<p_1,p_2<\infty$, ${\displaystyle \frac{1}{p}}={\displaystyle \frac{1}{p_1}}+{\displaystyle \frac{1}{p_2}}$, ${\displaystyle \frac{1}{q_1}}={\displaystyle \frac{1}{p_1}}-{\displaystyle \frac{\alpha }{2n}}$, ${\displaystyle \frac{1}{q_2}}={\displaystyle \frac{1}{p_2}}-{\displaystyle \frac{\alpha }{2n}}$ and ${\displaystyle \frac{1}{q}}={\displaystyle \frac{1}{q_1}}+{\displaystyle \frac{1}{q_2}}={\displaystyle \frac{1}{p}}-{\displaystyle \frac{\alpha }{n}}$, $\overrightarrow{w}=\left(w_1,w_2\right)\in A_{(\overrightarrow{p},q)}$, $w_1^{q_1},w_2^{q_2}\in A_{\infty }$ and $v_{\overrightarrow{w}}=w_1{w}_2$. Then

$${∥{\mathcal{I}}_{\alpha }(f,g)∥}_{L^{q,\kappa }\left(v_{\overrightarrow{w}}^q\right)}\lesssim {\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)},$$

(2)

$${∥{\mathcal{M}}_{\alpha }(f,g)∥}_{L^{q,\kappa }\left(v_{\overrightarrow{w}}^q\right)}\lesssim {\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}.$$

(3)

Here and in what follows, $A_{\infty }$ denotes the class of Muckenhoupt weights (see Definition 5 in Section 2 for the precise definition)

The boundedness and compactness of many classical operators on Morrey spaces in harmonic analysis have also been extensively studied; see, e.g., [26,27,28,29,30,31,32,33,34]. Recently, Ding and Mei [35] gave the boundedness and compactness of the commutators ${\mathcal{I}}_{\alpha ,b}^i$ and ${\mathcal{I}}_{\alpha ,\overrightarrow{b}}$ on product Morrey spaces. In [36], the boundedness and compactness of commutators of the bilinear Hardy-Littlewood maximal operator on Morrey spaces were studied by Wang, Zhou, and Teng.

Therefore, one may ask whether the conclusions of Theorem 2 still hold if the weighted Lebesgue spaces extend to the weighted Morrey spaces. To this question, we provide a positive answer. In addition, we also get that the commutators are all bounded on weighted Morrey spaces.

The purpose of this paper is to obtain the boundedness and compactness of the bilinear fractional maximal linear commutators ${\mathcal{M}}_{\alpha ,b}^i$ and the bilinear fractional maximal iterated commutator ${\mathcal{M}}_{\alpha ,\overrightarrow{b}}$ on weighted Morrey spaces. This paper establishes the following main results.

Theorem 4

(Boundedness of ${\mathcal{M}}_{\alpha ,b}^i$ and ${\mathcal{M}}_{\alpha ,\overrightarrow{b}}$). Let $0<\alpha <2n$, $0<\kappa <1$, $1<p_1,p_2<\infty$, $1<q<\infty$, ${\displaystyle \frac{1}{p}}={\displaystyle \frac{1}{p_1}}+{\displaystyle \frac{1}{p_2}}$, ${\displaystyle \frac{1}{q_1}}={\displaystyle \frac{1}{p_1}}-{\displaystyle \frac{\alpha }{2n}}$, ${\displaystyle \frac{1}{q_2}}={\displaystyle \frac{1}{p_2}}-{\displaystyle \frac{\alpha }{2n}}$ and ${\displaystyle \frac{1}{q}}={\displaystyle \frac{1}{q_1}}+{\displaystyle \frac{1}{q_2}}={\displaystyle \frac{1}{p}}-{\displaystyle \frac{\alpha }{n}}$. If $\overrightarrow{w}=(w_1,w_2)\in A_{(\overrightarrow{p},q)}$, $w_1^{q_1},w_2^{q_2}\in A_{\infty }$, $b\in \mathrm{BMO}$ and $\overrightarrow{b}=\left(b_1,b_2\right)\in \mathrm{BMO}\times \mathrm{BMO}=:{\mathrm{BMO}}^2$, where $v_{\overrightarrow{w}}=w_1{w}_2$, $i=1,2$, then

$$\begin{array}{cc}\hfill {∥{\mathcal{M}}_{\alpha ,b}^i\left(f,g\right)∥}_{L^{q,\kappa }\left(v_{\overrightarrow{w}}^q\right)}\lesssim & {∥b∥}_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)},\hfill \\ \hfill {∥{\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)∥}_{L^{q,\kappa }\left(v_{\overrightarrow{w}}^q\right)}\lesssim & {\parallel \overrightarrow{b}\parallel }_{{\mathrm{BMO}}^2}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}.\hfill \end{array}$$

For clarity, we explain the notation for the weighted Morrey spaces used in Theorems 4 and 5. The symbol $L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)$ denotes a two-weighted Morrey space as defined in Definition 2, where the parameter ${\displaystyle \frac{\kappa p_1}{q_1}}$ is the Morrey-type smoothness index for this space.

Theorem 5

(Compactness of ${\mathcal{M}}_{\alpha ,b}^i$ and ${\mathcal{M}}_{\alpha ,\overrightarrow{b}}$). Let $0<\alpha <2n$, $0<\kappa <1$, $1<p_1,p_2<\infty$, $1<q<\infty$, ${\displaystyle \frac{1}{p}}={\displaystyle \frac{1}{p_1}}+{\displaystyle \frac{1}{p_2}}$, ${\displaystyle \frac{1}{q_1}}={\displaystyle \frac{1}{p_1}}-{\displaystyle \frac{\alpha }{2n}}$, ${\displaystyle \frac{1}{q_2}}={\displaystyle \frac{1}{p_2}}-{\displaystyle \frac{\alpha }{2n}}$ and ${\displaystyle \frac{1}{q}}={\displaystyle \frac{1}{q_1}}+{\displaystyle \frac{1}{q_2}}={\displaystyle \frac{1}{p}}-{\displaystyle \frac{\alpha }{n}}$. If $\overrightarrow{w}\in A_{(\overrightarrow{p},q)}$, $w_1^{q_1},w_2^{q_2}\in A_{\infty }$, $b\in CMO$ and $\overrightarrow{b}=\left(b_1,b_2\right)\in {\mathrm{CMO}}^2$, where $v_{\overrightarrow{w}}=w_1{w}_2$ and $\mathrm{CMO}\left({\mathbb{R}}^{\mathrm{n}}\right)$ denotes the closure of $C_c^{\infty }\left({\mathbb{R}}^n\right)$ in the $\mathrm{BMO}\left({\mathbb{R}}^{\mathrm{n}}\right)$ topology, then ${\mathcal{M}}_{\alpha ,b}^i$ and ${\mathcal{M}}_{\alpha ,\overrightarrow{b}}$ are compact from $L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)\times L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)$ to $L^{q,\kappa }\left(v_{\overrightarrow{w}}^q\right)$.

Remark 3.

By Remark 2, if $\kappa =0$, $L^{p_1,0}\left(w_1^{p_1},1\right)=L^{p_1}\left(w_1^{p_1}\right)$, $L^{p_2,0}\left(w_2^{p_2},1\right)=L^{p_2}\left(w_2^{p_2}\right)$and $L^{q,0}\left(v_{\overrightarrow{w}}^q\right)=L^q\left(v_{\overrightarrow{w}}^q\right)$. If  $\overrightarrow{w}\in A_{(\overrightarrow{p},q)}$, $b\in CMO$ and $\overrightarrow{b}\in {\mathrm{CMO}}^2$, where $v_{\overrightarrow{w}}=w_1{w}_2$, ${\mathcal{M}}_{\alpha ,b}^i$ and ${\mathcal{M}}_{\alpha ,\overrightarrow{b}}$ are compact operators from $L^{p_1}\left(w_1^{p_1}\right)\times L^{p_2}\left(w_2^{p_2}\right)$ to $L^q\left(v_{\overrightarrow{w}}^q\right)$, and also bounded provided that $b\in \mathrm{BMO}$ and $\overrightarrow{b}\in {\mathrm{BMO}}^2$, with the same indexes as Theorem 2.

For the unweighted setting, if $w_1=w_2=1$, $L^{p_1,\frac{\kappa p_1}{q_1}}\left(1,1\right)=L^{p_1,\frac{\kappa p_1}{q_1}}\left({\mathbb{R}}^n\right)$, $L^{p_2,\frac{\kappa p_2}{q_2}}\left(1,1\right)=L^{p_2,\frac{\kappa p_2}{q_2}}\left({\mathbb{R}}^n\right)$, ${\mathcal{M}}_{\alpha ,b}^i$and ${\mathcal{M}}_{\alpha ,\overrightarrow{b}}$are also bounded operators from $L^{p_1,\frac{\kappa p_1}{q_1}}\left({\mathbb{R}}^n\right)\times L^{p_2,\frac{\kappa p_2}{q_2}}\left({\mathbb{R}}^n\right)$to $L^{q,\kappa }$provided that $b\in \mathrm{BMO}$and $\overrightarrow{b}\in {\mathrm{BMO}}^2$. If $b\in \mathrm{CMO}$and $\overrightarrow{b}\in {\mathrm{CMO}}^2$, then ${\mathcal{M}}_{\alpha ,b}^i$and ${\mathcal{M}}_{\alpha ,\overrightarrow{b}}$are also compact from $L^{p_1,\frac{\kappa p_1}{q_1}}\left({\mathbb{R}}^n\right)\times L^{p_2,\frac{\kappa p_2}{q_2}}\left({\mathbb{R}}^n\right)$to $L^{q,\kappa }\left({\mathbb{R}}^n\right)$.

While our proof strategy follows a known framework, extending boundedness and compactness results from weighted Lebesgue spaces to the product weighted Morrey setting introduces several nontrivial obstacles that require new technical ideas. First, the product weighted Morrey space $L^{p_1,\frac{\kappa p_1}{q_1}}(w_1^{p_1},w_1^{q_1})\times L^{p_2,\frac{\kappa p_2}{q_2}}(w_2^{p_2},w_2^{q_2})$ has a two-layered structure: each weight $w_i$ appears with distinct exponents $p_i$ and $q_i$, governing local integrability and global decay, respectively. Tracking their interplay and how they combine via the operator to yield the target weight $v_{\overrightarrow{w}}^q$ in $L^{q,\kappa }\left(v_{\overrightarrow{w}}^q\right)$ complicates every estimate. Second, applying the Fréchet–Kolmogorov theorem in this setting is not straightforward. Verifying the decay and equicontinuity conditions demands a delicate regional decomposition of the commutator, together with careful applications of Bernoulli’s inequality, the doubling property of weights, and Hölder’s inequality, not merely pointwise or in $L^q$. Third, decomposing the symbol $b\in \mathrm{CMO}$ as $b=(b-b_{2B}){\chi }_{2B}+(b-b_{2B}){\chi }_{{\mathbb{R}}^n\backslash 2B}+b_{2B}$ forces estimates that respect the local–global balance of the Morrey norm. Handling integrals over nested balls whose radii are linked to the Morrey parameter $\kappa$ leads to inequalities more subtle than those in the Lebesgue case [25,35].

This paper is organized as follows. In Section 3, we first characterize of the class $A_{(\overrightarrow{p},q)}$ and some facts about weights $A_p$. Then we prove the boundedness boundedness of the associated operators. In Section 4, we establish the compactness results.

Throughout this paper, we adopt the following notations. For $p\in [1,\infty )$, The dual exponent of p is denoted by $p^{\prime }$, which satisfies $p^{\prime }={\displaystyle \frac{p}{p-1}}$. Given a Lebesgue measurable set $E\subset {\mathbb{R}}^n$, $\left|E\right|$ will denote the Lebesgue measure of E, and ${\chi }_E$ denotes its characteristic function. Let $B=B(x,r)$ be a ball in ${\mathbb{R}}^n$ centered at x with radius r, $\lambda B$ denotes the ball with the sanme center as B and side length $\lambda$ times the side length of B. C always denotes a positive constant independent of the main parameters involved, but whose value may differ from line to line. We use the symbol $A\lesssim B$ to indicate that there exists a positive constant C such that $A\le CB$. More notations appearing in this paper; see

Table 1. Summary of exponents and their relationships.

Symbol	Description	Definition/Relationship
n	Dimension	$n\in \mathbb{N}$
$\alpha$	Fractional order	$0<\alpha <2n$
$\kappa$	Morrey space parameter	$0<\kappa <1$
$p_1,p_2$	Input exponents (Lebesgue)	$1<p_1,p_2<\infty$
p	Harmonic mean of $p_1,p_2$	$\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$
$q_1$	Intermediate exponent for $w_1$	$\frac{1}{q_1}=\frac{1}{p_1}-\frac{\alpha }{2n}$
$q_2$	Intermediate exponent for $w_2$	$\frac{1}{q_2}=\frac{1}{p_2}-\frac{\alpha }{2n}$
q	Output exponent (Morrey)	$\frac{1}{q}=\frac{1}{q_1}+\frac{1}{q_2}=\frac{1}{p}-\frac{\alpha }{n}$
$\overrightarrow{w}$	Weight pair	$\overrightarrow{w}=(w_1,w_2)$
$v_{\overrightarrow{w}}$	Product weight	$v_{\overrightarrow{w}}=w_1{w}_2$

for details.

2. Preliminaries

In this section, we collect the definitions, notation that will be used throughout the paper.

2.1. Function Spaces

In this paper, our work will be built on the weighted Morrey space with two weights.

Definition 3

(Weighted Morrey space with two weights [19]). Assume that $1\le p<\infty$ and $0<\kappa <1$. For a pair of weights $(\mu ,\lambda )$, the two-weighted Morrey space is defined as

$$L^{p,\kappa }(\mu ,\lambda )=\left\{f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^p\left(\mu \right):{\parallel f\parallel }_{L^{p,\kappa }(\mu ,\lambda )}<\infty \right\},$$

where

$${\parallel f\parallel }_{L^{p,\kappa }(\mu ,\lambda )}=\underset{B}{sup}{\left({\displaystyle \frac{1}{\lambda {\left(B\right)}^{\kappa }}}{\int }_B{\left|f\left(t\right)\right|}^p\mu \left(t\right)dt\right)}^{{\displaystyle \frac{1}{p}}}$$

and the supremum is taken over all balls B in ${\mathbb{R}}^n$. When $\mu =\lambda$, we use the simplified notation $L^{p,\kappa }\left(\mu \right).$

Next, let us recall the definition of bounded mean oscillation space $\mathrm{BMO}\left({\mathbb{R}}^{\mathrm{n}}\right)$, as well as its subspace $\mathrm{CMO}\left({\mathbb{R}}^{\mathrm{n}}\right)$.

Definition 4

(BMO and CMO). A locally integrable function b belongs to the space $\mathrm{BMO}\left({\mathbb{R}}^n\right)$ if

$${\parallel b\parallel }_{\mathrm{BMO}}:=\underset{B}{sup}{\displaystyle \frac{1}{\left|B\right|}}{\int }_B|b\left(x\right)-b_B|dx<\infty ,$$

where $b_B:={\displaystyle \frac{1}{\left|B\right|}}{\int }_{B}b\left(y\right)dy$. The space $\mathrm{CMO}\left({\mathbb{R}}^n\right)$ is defined as the closure of $C_c^{\infty }\left({\mathbb{R}}^n\right)$ in the $\mathrm{BMO}\left({\mathbb{R}}^n\right)$ topology.

The following property about $\mathrm{BMO}$ functions is classical, it will also play an impotant role in our proof of main theorems.

Lemma 1

([37]). Let $1<q<\infty$ and $w\in A_{\infty }$, then the following inequality holds

$${\left({\displaystyle \frac{1}{w\left(B\right)}}{\int }_B{\left|b\left(x\right)-b_B\right|}^{q}w\left(x\right)dx\right)}^{1/q}\le C{\parallel b\parallel }_{\mathrm{BMO}}.$$

(4)

2.2. Weights

Definition 5

(Muckenhoupt $A_p$ class). For $1<p<\infty$, a weight w belongs to $A_p$ if

$${\left[w\right]}_{A_p}:=\underset{B}{sup}\left({\displaystyle \frac{1}{\left|B\right|}}{\int }_{B}w\right){\left({\displaystyle \frac{1}{\left|B\right|}}{\int }_B{w}^{1-p^{\prime }}\right)}^{p/p^{\prime }}<\infty ,$$

where $p^{\prime }=p/(p-1)$. For $p=1$, $w\in A_1$ if there exists $C>0$ such that $Mw\left(x\right)\le Cw\left(x\right)$ for almost every $x\in {\mathbb{R}}^n$. We also denote $A_{\infty }:={\bigcup }_{p\ge 1}{A}_p$.

In order to prove our main Theorem, we need some facts about the weight $A_{(\overrightarrow{p},q)}$ and $A_p$. The weight class $A_{(\overrightarrow{p},q)}$, mentioned earlier, was introduced by Moen [3] independently.

Definition 6

(Class of $A_{(\overrightarrow{p},q)}$ [3]). Let $1\le p_1,p_2\le \infty$, ${\displaystyle \frac{1}{p}}={\displaystyle \frac{1}{p_1}}+{\displaystyle \frac{1}{p_2}}$, $\overrightarrow{p}=\left(p_1,p_2\right)$ and $0<q\le \infty$. One says that a vector of weights $\overrightarrow{w}=\left(w_1,w_2\right)$ is in the multiple weights class $\overrightarrow{w}\in A_{(\overrightarrow{p},q)}$ if it satisfies

$${\left[\overrightarrow{w}\right]}_{A_{(\overrightarrow{p},q)}}:=\underset{B}{sup}{\left({\displaystyle \frac{1}{\left|B\right|}}{\int }_B{v}_{\overrightarrow{w}}^q\right)}^{{\displaystyle \frac{1}{q}}}{\left({\displaystyle \frac{1}{\left|B\right|}}{\int }_B{w}_1^{-p_1^{\prime }}\right)}^{{\displaystyle \frac{1}{p_1^{\prime }}}}{\left({\displaystyle \frac{1}{\left|B\right|}}{\int }_B{w}_2^{-p_2^{\prime }}\right)}^{{\displaystyle \frac{1}{p_2^{\prime }}}}<\infty ,$$

where $v_{\overrightarrow{w}}=w_1{w}_2$ and $w_i(i=1,2)$ are nonnegative and locally integrable functions on ${\mathbb{R}}^n$. When $q=\infty$, ${\left({\displaystyle \frac{1}{\left|B\right|}}{\int }_B{v}_{\overrightarrow{w}}^q\right)}^{{\displaystyle \frac{1}{q}}}$ is understood as $\underset{x\in B}{esssup}{v}_{\overrightarrow{w}}$. Moreover, when $p_i=1$, ${\left({\displaystyle \frac{1}{\left|B\right|}}{\int }_B{w}_i^{-p_i^{\prime }}\right)}^{{\displaystyle \frac{1}{p_i^{\prime }}}}$ is understood as ${\left(\mathrm{ess}{inf}_B{w}_i\right)}^{-1}$.

We will need the following characterization of multiple weights, which was established by Iida [38].

Lemma 2

([38]). Let $1\le p_1,p_2\le \infty$, ${\displaystyle \frac{1}{p}}={\displaystyle \frac{1}{p_1}}+{\displaystyle \frac{1}{p_2}}$ and $0<q\le \infty$. A multiple weights $\overrightarrow{w}\in A_{(\overrightarrow{p},q)}$ if and only if

(i) $v_{\overrightarrow{w}}^q\in A_{1+q\left(2-\frac{1}{p}\right)}$, where $v_{\overrightarrow{w}}=w_1{w}_2$;

(ii) $w_i^{-p_i^{\prime }}\in A_{1+p_i^{\prime }{s}_i}(i=1,2)$, where $s_i={\displaystyle \frac{1}{q}}+2-{\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{p_i^{\prime }}}(i=1,2)$.

An analogous statement holds for $q=\infty$, if we regard the condition $v_{\overrightarrow{w}}\in A_{1+q\left(2-\frac{1}{p}\right)}$as the condition $v_{\overrightarrow{w}}^{-\frac{1}{2-\frac{1}{p}}}\in A_1$.

To handle the product weight $v_{\overrightarrow{w}}$, we need the following estimate which controls its $L^q$ norm by the $L^{q_1}$ and $L^{q_2}$ norms of $w_1$ and $w_2$, respectively.

Lemma 3

([39]). Let $1\le q_1,q_2<\infty$, $0<q<\infty$, ${\displaystyle \frac{1}{q}}={\displaystyle \frac{1}{q_1}}+{\displaystyle \frac{1}{q_2}}$ and $w_1^{q_1},w_2^{q_2}\in A_{\infty }$, $v_{\overrightarrow{w}}=w_1{w}_2$, then for any ball B in ${\mathbb{R}}^n$, there exists an absolute constant $C>0$ such that

$${\left({\int }_B{w}_1^{q_1}\right)}^{{\displaystyle \frac{q}{q_1}}}{\left({\int }_B{w}_2^{q_2}\right)}^{{\displaystyle \frac{q}{q_2}}}\le C{\int }_B{v}_{\overrightarrow{w}}^q.$$

(5)

The classical Muckenhoupt weights enjoy the following doubling condition and reverse Hölder type inequality; see [40], for example.

Lemma 4

([40]). If $w\in A_p$ and $1\le p<\infty$, then for any fixed ball B in ${\mathbb{R}}^n$, we have

$$w\left(\lambda B\right)\le {\lambda }^{np}{\left[w\right]}_{A_p}w\left(B\right).$$

(6)

Lemma 5

(Reverse Hölder property of $A_{\infty }$ weights [40]). If $w\in A_{\infty }=\bigcup _{1\le p<\infty }{A}_p$, then for any B in ${\mathbb{R}}^n$ and any set $E\subset B$, there exists a constant $\theta >0$ such that

$${\displaystyle \frac{w\left(E\right)}{w\left(B\right)}}\le C{\left({\displaystyle \frac{\left|E\right|}{\left|B\right|}}\right)}^{\theta }.$$

(7)

3. Proof of Boundedness

In order to prove Theorem 4, we will use the techniques of function decomposition and the properties of weights.

Proof of Theorem 4. 

Let $B=B(x_0,r)\subset {\mathbb{R}}^n$ be a fixed ball. We perform a standard truncation:

$$f=f{\chi }_{2B}+f{\chi }_{{\left(2B\right)}^c}=:f^0+f^{\infty },g=g{\chi }_{2B}+g{\chi }_{{\left(2B\right)}^c}=:g^0+g^{\infty }.$$

(8)

Since ${\mathcal{M}}_{\alpha ,b}^1$ is a sub-linear operator, we can split ${\mathcal{M}}_{\alpha ,b}^1\left(f,g\right)$ as the following four part to estimate by Minkowski’s inequality,

$$\begin{array}{cc}\hfill & {\left({\displaystyle \frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\kappa }}}{\int }_B{\left|{\mathcal{M}}_{\alpha ,b}^1\left(f,g\right)\left(x\right)\right|}^q{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{1/q}\hfill \\ \hfill \lesssim & {\left({\displaystyle \frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\kappa }}}{\int }_B{\left|{\mathcal{M}}_{\alpha ,b}^1\left(f^0,g^0\right)\left(x\right)\right|}^q{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{1/q}\hfill \\ \hfill & +{\left({\displaystyle \frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\kappa }}}{\int }_B{\left|{\mathcal{M}}_{\alpha ,b}^1\left(f^0,g^{\infty }\right)\left(x\right)\right|}^q{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{1/q}\hfill \\ \hfill & +{\left({\displaystyle \frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\kappa }}}{\int }_B{\left|{\mathcal{M}}_{\alpha ,b}^1\left(f^{\infty },g^0\right)\left(x\right)\right|}^q{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{1/q}\hfill \\ \hfill & +{\left({\displaystyle \frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\kappa }}}{\int }_B{\left|{\mathcal{M}}_{\alpha ,b}^1\left(f^{\infty },g^{\infty }\right)\left(x\right)\right|}^q{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{1/q}\hfill \\ \hfill =:& A_1+A_2+A_3+A_4.\hfill \end{array}$$

Regarding $A_1$, the condition $\overrightarrow{w}\in A_{(\overrightarrow{p},q)}$ yields $v_{\overrightarrow{w}}^q\in A_{1+q(2-\frac{1}{p})}$. Hence, it follows from the weighted $L^p$ boundedness for ${\mathcal{M}}_{\alpha ,b}^i$ in Remark 1, (5) and (6) that

$$\begin{array}{cc}\hfill A_1& \lesssim \frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\frac{\kappa }{q}}}{\parallel {\mathcal{M}}_{\alpha ,b}^1\left(f^0,g^0\right)\parallel }_{L^q\left(v_{\overrightarrow{w}}^q\right)}\hfill \\ \hfill & \lesssim \frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\frac{\kappa }{q}}}{\parallel b\parallel }_{\mathrm{BMO}}\parallel f^0{\parallel }_{L^{p_1}\left(w_1^{p_1}\right)}{\parallel g^0\parallel }_{L^{p_2}\left(w_2^{p_2}\right)}\hfill \\ \hfill & =\frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\frac{\kappa }{q}}}{\parallel b\parallel }_{\mathrm{BMO}}{\left({\int }_{2B}{\left|f\left(x\right)\right|}^{p_1}{w}_1^{p_1}\left(x\right)dx\right)}^{\frac{1}{p_1}}{\left({\int }_{2B}{\left|g\left(x\right)\right|}^{p_2}{w}_2^{p_2}\left(x\right)dx\right)}^{\frac{1}{p_2}}\hfill \\ \hfill & \lesssim \frac{w_1^{q_1}{\left(2B\right)}^{\frac{\kappa }{q_1}}{w}_2^{q_2}{\left(2B\right)}^{\frac{\kappa }{q_2}}}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\frac{\kappa }{q}}}{\parallel b\parallel }_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\hfill \\ \hfill & \lesssim {\left(\frac{v_{\overrightarrow{w}}^q\left(2B\right)}{v_{\overrightarrow{w}}^q\left(B\right)}\right)}^{\frac{\kappa }{q}}{\parallel b\parallel }_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\hfill \\ \hfill & \lesssim {\parallel b\parallel }_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}.\hfill \end{array}$$

(9)

Consider now $A_2$. To estimate $A_2$, we need to use the properties of ${\mathcal{M}}_{\alpha }$, so it’s natural to write

$$\begin{array}{cc}\hfill |{\mathcal{M}}_{\alpha ,b}^1\left(f^0,g^{\infty }\right)\left(x\right)|& \le \left|b\left(x\right)-b_B\right||{\mathcal{M}}_{\alpha }\left(f^0,g^{\infty }\right)\left(x\right)|+|{\mathcal{M}}_{\alpha }\left(\left(b_B-b\right)f^0,g^{\infty }\right)\left(x\right)|\hfill \\ & =:A_{21}+A_{22}.\hfill \end{array}$$

For any $x\in B$, using (1), we have

$$\begin{array}{cc}\hfill |{\mathcal{M}}_{\alpha }(f^0,g^{\infty })\left(x\right)|\lesssim & |{\mathcal{I}}_{\alpha }\left(\right|f^0|,|g^{\infty }\left|\right)\left(x\right)|\hfill \\ \hfill \lesssim & {\int }_{{\left(2B\right)}^c}{\int }_{2B}{\displaystyle \frac{|f^0\left(y\right)\left|\right|g^{\infty }\left(z\right)|}{{\left(\right|x-y|+|x-z\left|\right)}^{2n-\alpha }}}dydz\hfill \\ \hfill \lesssim & {\int }_{2B}\left|f\left(y\right)\right|dy\sum _{j=1}^{\infty }{\displaystyle \frac{1}{|2^{j+1}{B|}^{2-\frac{\alpha }{n}}}}{\int }_{2^{j+1}B\backslash 2^{j}B}\left|g\left(z\right)\right|dz\hfill \\ \hfill \lesssim & \sum _{j=1}^{\infty }{\displaystyle \frac{1}{|2^{j+1}{B|}^{2-\frac{\alpha }{n}}}}{\int }_{2^{j+1}B}\left|f\left(y\right)\right|dy{\int }_{2^{j+1}B}\left|g\left(z\right)\right|dz.\hfill \end{array}$$

Applying Hölder’s inequality and the fact that $\overrightarrow{w}\in A_{(\overrightarrow{p},q)}$ and (5), we obtain

$$\begin{array}{cc}\hfill & |{\mathcal{M}}_{\alpha }\left(f^0,g^{\infty }\right)\left(x\right)|\hfill \\ \hfill \lesssim & \sum _{j=1}^{\infty }{\displaystyle \frac{w_1^{q_1}{\left(2^{j+1}B\right)}^{\frac{\kappa }{q_1}}{w}_1^{q_2}{\left(2^{j+1}B\right)}^{\frac{\kappa }{q_2}}}{|2^{j+1}{B|}^{2-\frac{\alpha }{n}}}}{\left({\int }_{2^{j+1}B}{w}_1^{-p_1^{\prime }}\left(y\right)dy\right)}^{\frac{1}{p_1^{\prime }}}\hfill \\ \hfill & \times {\left({\int }_{2^{j+1}B}{w}_2^{-p_2^{\prime }}\left(y\right)dy\right)}^{\frac{1}{p_2^{\prime }}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\hfill \\ \hfill \lesssim & \sum _{j=1}^{\infty }{\displaystyle \frac{w_1^{q_1}{\left(2^{j+1}B\right)}^{\frac{\kappa }{q_1}}{w}_1^{q_2}{\left(2^{j+1}B\right)}^{\frac{\kappa }{q_2}}}{|2^{j+1}{B|}^{2-\frac{\alpha }{n}}}}\hfill \\ \hfill & \times {\displaystyle \frac{|2^{j+1}{B|}^{2+\frac{1}{q}-\frac{1}{p}}}{v_{\overrightarrow{w}}^q{\left(2^{j+1}B\right)}^{\frac{1}{q}}}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\hfill \\ \hfill \lesssim & {\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\sum _{j=1}^{\infty }{\displaystyle \frac{{\left(w_1^{q_1}{\left(2^{j+1}B\right)}^{\frac{q}{q_1}}{w}_1^{q_2}{\left(2^{j+1}B\right)}^{\frac{q}{q_2}}\right)}^{\frac{\kappa }{q}}}{v_{\overrightarrow{w}}^q{\left(2^{j+1}B\right)}^{\frac{1}{q}}}}\hfill \\ \hfill \lesssim & {\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\sum _{j=1}^{\infty }{v}_{\overrightarrow{w}}^q{\left(2^{j+1}B\right)}^{\frac{\kappa -1}{q}}.\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill & {\left({\displaystyle \frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\kappa }}}{\int }_B{A}_{21}^q{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{1/q}\hfill \\ \hfill \lesssim & \sum _{j=1}^{\infty }{\left({\displaystyle \frac{v_{\overrightarrow{w}}^q\left(B\right)}{v_{\overrightarrow{w}}^q\left(2^{j+1}B\right)}}\right)}^{\frac{1-\kappa }{q}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\hfill \\ \hfill & \times {\left({\displaystyle \frac{1}{v_{\overrightarrow{w}}^q\left(B\right)}}{\int }_B{|b\left(x\right)-b_B|}^q{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{\frac{1}{q}}.\hfill \end{array}$$

(10)

Note that $v_{\overrightarrow{w}}^q\in A_{1+q(2-\frac{1}{p})}\subset A_{\infty }$. Thus, by (7), there exsits $\theta >0$ such that

$${\displaystyle \frac{v_{\overrightarrow{w}}^q\left(B\right)}{v_{\overrightarrow{w}}^q\left(2^{j+1}B\right)}}\le C{\left({\displaystyle \frac{\left|B\right|}{|2^{j+1}B|}}\right)}^{\theta }.$$

(11)

By Lemma 1, we get

$$\begin{array}{cc}\hfill & {\left(\frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\kappa }}{\int }_B{A}_{21}^q{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{1/q}\hfill \\ \hfill \lesssim & {\parallel b\parallel }_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\sum _{j=1}^{\infty }{\left(\frac{1}{2^{jn}}\right)}^{\frac{(1-\kappa )\theta }{q}}\hfill \\ \hfill \lesssim & {\parallel b\parallel }_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)},\hfill \end{array}$$

(12)

where the last series converges due to $\theta >0$ and $0<\kappa <1$.

On the other hand, when $x\in B(x_0,r)$, $y\in 2B(x_0,r)$ and $z\in {\left(2B(x_0,r)\right)}^c$, we observe that $|x-y|+|x-z|\sim |x_0-y| + |x_0-z|$. In fact, since $|x_0-x|\le r\le |x_0-y|$, so

$$|x-y|+|x-z| \le |x_0-y| + |x_0-z|+2|x-x_0| \le 3(|x_0-y|+|x_0-z\left|\right).$$

Similarly, using $|x_0-x|\le r\le |x-z|$, it yields that

$$|x_0-y|+|x_0-z|\le |x-y|+|x-z|+2|x-x_0|\le 3\left(\right|x-y|+|x-z\left|\right).$$

Hence, we get

$$\begin{array}{cc}\hfill A_{22}& =|{\mathcal{M}}_{\alpha }\left(\left(b_B-b\right)f^0,g^{\infty }\right)\left(x\right)|\hfill \\ \hfill & \lesssim {\int }_{{\left(2B\right)}^c}{\int }_{2B}{\displaystyle \frac{|b\left(y\right)-b_B\left|\right|f\left(y\right)\left|\right|g\left(z\right)|}{{\left(\right|x-y|+|x-z\left|\right)}^{2n-\alpha }}}dydz\hfill \\ \hfill & \lesssim {\int }_{2B}|b\left(y\right)-b_B\left|\right|f\left(y\right)|dy\sum _{j=1}^{\infty }{\displaystyle \frac{1}{|2^{j+1}{B|}^{2-\frac{\alpha }{n}}}}{\int }_{2^{j+1}B\setminus 2^{j}B}\left|g\left(z\right)\right|dz.\hfill \end{array}$$

(13)

Note that $w_i^{-p_i^{\prime }}\in A_{1+p_i^{\prime }{s}_i}\subset A_{\infty }$. By Hölder’s inequality and Lemma 1, we can get

$$\begin{array}{cc}\hfill & {\int }_{2B}|b\left(y\right)-b_B\left|\right|f\left(y\right)|dy\hfill \\ \hfill \lesssim & {\int }_{2B}|b\left(y\right)-b_{2B}\left|\right|f\left(y\right)|dy+{\int }_{2B}|b_{2B}-b_B\left|\right|f\left(y\right)|dy\hfill \\ \hfill \lesssim & {\left({\int }_{2B}{|b\left(y\right)-b_B|}^{p_1^{\prime }}{w}_1^{-p_1^{\prime }}\left(y\right)dy\right)}^{\frac{1}{p_1^{\prime }}}{\left({\int }_{2B}{\left|f\left(y\right)\right|}^{p_1}{w}_1^{p_1}\left(y\right)dy\right)}^{\frac{1}{p_1}}\hfill \\ \hfill & +{\parallel b\parallel }_{\mathrm{BMO}}{\int }_{2B}\left|f\left(y\right)\right|dy\hfill \\ \hfill \lesssim & w_1^{-p_1^{\prime }}{\left(2B\right)}^{\frac{1}{p_1^{\prime }}}{\parallel b\parallel }_{\mathrm{BMO}}{\left({\int }_{2B}{\left|f\left(y\right)\right|}^{p_1}{w}_1^{p_1}\left(y\right)dy\right)}^{\frac{1}{p_1}}.\hfill \end{array}$$

For $\overrightarrow{w}=(w_1,w_2)\in A_{(\overrightarrow{p},q)}$, applying Hölder’s inequality and Lemma 2, we have

$$\begin{array}{cc}\hfill A_{22}\lesssim & {\parallel b\parallel }_{\mathrm{BMO}}{w}_1^{-p_1^{\prime }}{\left(2B\right)}^{\frac{1}{p_1^{\prime }}}{\left({\int }_{2B}{\left|f\left(y\right)\right|}^{p_1}{w}_1^{p_1}\left(y\right)dy\right)}^{\frac{1}{p_1}}\hfill \\ \hfill & \times \sum _{j=1}^{\infty }\frac{1}{|2^{j+1}{B|}^{2-\frac{\alpha }{n}}}{\int }_{2^{j+1}B\backslash 2^{j}B}\left|g\left(z\right)\right|dz\hfill \\ \hfill \le & {\parallel b\parallel }_{\mathrm{BMO}}\sum _{j=1}^{\infty }\frac{w_1^{-p_1^{\prime }}{\left(2B\right)}^{\frac{1}{p_1^{\prime }}}}{|2^{j+1}{B|}^{2-\frac{\alpha }{n}}}{\left({\int }_{2B}{\left|f\left(y\right)\right|}^{p_1}{w}_1^{p_1}\left(y\right)dy\right)}^{\frac{1}{p_1}}\hfill \\ \hfill & \times {\int }_{2^{j+1}B\backslash 2^{j}B}\left|g\left(z\right)\right|w\left(z\right)w^{-1}\left(z\right)dz\hfill \\ \hfill \lesssim & {\parallel b\parallel }_{\mathrm{BMO}}\sum _{j=1}^{\infty }\frac{w_1^{-p_1^{\prime }}{\left(2B\right)}^{\frac{1}{p_1^{\prime }}}}{|2^{j+1}{B|}^{2-\frac{\alpha }{n}}}{\left({\int }_{2^{j+1}B}{\left|f\left(y\right)\right|}^{p_1}{w}_1^{p_1}\left(y\right)dy\right)}^{\frac{1}{p_1}}\hfill \\ \hfill & \times {\left({\int }_{2^{j+1}B}{\left|g\left(z\right)\right|}^{p_2}{w}_2^{p_2}\left(y\right)dz\right)}^{\frac{1}{p_2}}{\left({\int }_{2^{j+1}B}{w}_2^{-p_2^{\prime }}\left(y\right)dy\right)}^{\frac{1}{p_2^{\prime }}}\hfill \\ \hfill \lesssim & {\parallel b\parallel }_{\mathrm{BMO}}\sum _{j=1}^{\infty }\frac{w_1^{-p_1^{\prime }}{\left(2^{j+1}B\right)}^{\frac{1}{p_1^{\prime }}}{w}_2^{-p_2^{\prime }}{\left(2^{j+1}B\right)}^{\frac{1}{p_2^{\prime }}}}{|2^{j+1}{B|}^{2-\frac{\alpha }{n}}}{\left(\frac{\left|2B\right|}{|2^{j+1}B|}\right)}^{\frac{\theta }{p_1^{\prime }}}\hfill \\ \hfill & \times w_1^{q_1}{\left(2^{j+1}B\right)}^{\frac{\kappa }{q_1}}{w}_2^{q_2}{\left(2^{j+1}B\right)}^{\frac{\kappa }{q_2}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\hfill \\ \hfill \lesssim & {\parallel b\parallel }_{\mathrm{BMO}}\sum _{j=1}^{\infty }\frac{w_1^{q_1}{\left(2^{j+1}B\right)}^{\frac{\kappa }{q_1}}{w}_2^{q_2}{\left(2^{j+1}B\right)}^{\frac{\kappa }{q_2}}}{|2^{j+1}{B|}^{2-\frac{\alpha }{n}}}{\left(\frac{\left|2B\right|}{|2^{j+1}B|}\right)}^{\frac{\theta }{p_1^{\prime }}}\hfill \\ \hfill & \times \frac{|2^{j+1}{B|}^{2+\frac{1}{q}-\frac{1}{p}}}{v_{\overrightarrow{w}}^q{\left(2^{j+1}B\right)}^{\frac{1}{q}}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\hfill \\ \hfill \lesssim & {\parallel b\parallel }_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\hfill \\ \hfill & \times \sum _{j=1}^{\infty }{\left(\frac{\left|2B\right|}{|2^{j+1}B|}\right)}^{\frac{\theta }{p_1^{\prime }}}\frac{{\left(w_1^{q_1}{\left(2^{j+1}B\right)}^{\frac{q}{q_1}}{w}_2^{q_2}{\left(2^{j+1}B\right)}^{\frac{q}{q_2}}\right)}^{\frac{\kappa }{q}}}{v_{\overrightarrow{w}}^q{\left(2^{j+1}B\right)}^{\frac{1}{q}}}\hfill \\ \hfill \lesssim & {\parallel b\parallel }_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\hfill \\ \hfill & \times \sum _{j=1}^{\infty }{\left(\frac{\left|2B\right|}{|2^{j+1}B|}\right)}^{\frac{\theta }{p_1^{\prime }}}{v}_{\overrightarrow{w}}^q{\left(2^{j+1}B\right)}^{\frac{\kappa -1}{q}}.\hfill \end{array}$$

Thus, for $0<\kappa <1$, $\theta >0$ and $1<p_1<\infty$, the Minkowski inequality gives us that

$$\begin{array}{cc}\hfill & {\left(\frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\kappa }}{\int }_B{A}_{22}^q{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{1/q}\hfill \\ \hfill \lesssim & {\parallel b\parallel }_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\hfill \\ \hfill & \times {\left(\frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\kappa }}{\int }_B{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{\frac{1}{q}}\sum _{j=1}^{\infty }{\left(\frac{\left|2B\right|}{|2^{j+1}B|}\right)}^{\frac{\theta }{p_1^{\prime }}}{v}_{\overrightarrow{w}}^q{\left(2^{j+1}B\right)}^{\frac{\kappa -1}{q}}\hfill \\ \hfill \lesssim & {\parallel b\parallel }_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\hfill \\ \hfill & \times \sum _{j=1}^{\infty }{\left(\frac{\left|2B\right|}{|2^{j+1}B|}\right)}^{\frac{\theta }{p_1^{\prime }}}{\left(\frac{v_{\overrightarrow{w}}^q\left(B\right)}{v_{\overrightarrow{w}}^q\left(2^{j+1}B\right)}\right)}^{\frac{1-\kappa }{q}}\hfill \\ \hfill \lesssim & {\parallel b\parallel }_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\hfill \\ \hfill & \times \sum _{j=1}^{\infty }{\left(\frac{\left|2B\right|}{|2^{j+1}B|}\right)}^{\frac{\theta }{p_1^{\prime }}}{\left(\frac{\left|B\right|}{|2^{j+1}B|}\right)}^{\frac{(1-\kappa )\theta }{q}}\hfill \\ \hfill \lesssim & {\parallel b\parallel }_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\sum _{j=1}^{\infty }{\left(\frac{1}{2^{jn}}\right)}^{\theta (1-\frac{1}{p_1}+\frac{1-\kappa }{q})}\hfill \\ \hfill \lesssim & {\parallel b\parallel }_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}.\hfill \end{array}$$

(14)

Hence,

$$A_2\lesssim {\parallel b\parallel }_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}.$$

Similar to $A_2$, we divide ${\mathcal{M}}_{\alpha ,b}^1\left(f^{\infty },g^0\right)$ into $A_{31}$ and $A_{32}$ to estimate $A_3$ as follows,

$$\begin{array}{cc}\hfill |{\mathcal{M}}_{\alpha ,b}^1\left(f^{\infty },g^0\right)\left(x\right)| \le & \left|b\left(x\right)-b_B\right||{\mathcal{M}}_{\alpha }\left(f^{\infty },g^0\right)\left(x\right)|+|{\mathcal{M}}_{\alpha }\left(\left(b_B-b\right)f^{\infty },g^0\right)\left(x\right)|\hfill \\ \hfill =:& A_{31}+A_{32}.\hfill \end{array}$$

Similar to the approach outlined in (10) and (12) for handling $A_{21}$, we get

$$A_{31}\lesssim \left|b\left(x\right)-b_B\right|{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\sum _{j=1}^{\infty }{v}_{\overrightarrow{w}}^q{\left(2^{j+1}B\right)}^{\frac{\kappa -1}{q}}.$$

Since $b\in \mathrm{BMO}$, a straightforward calculation gives

$$|b_{2^{j+1}B}-b_B|\le C(j+1){\parallel b\parallel }_{\mathrm{BMO}}.$$

(15)

Thus, by the estimate (15), similar to estimate (13) and (14) of $A_{22}$, we get

$$\begin{array}{cc}\hfill A_{32}\lesssim & {\int }_{2B}{\int }_{{\left(2B\right)}^c}\frac{|b\left(y\right)-b_B\left|\right|f\left(y\right)\left|\right|g\left(z\right)|}{{\left(\right|x-y|+|x-z\left|\right)}^{2n-\alpha }}dydz\hfill \\ \hfill \lesssim & \sum _{j=1}^{\infty }{\int }_{2B}{\int }_{2^{j+1}B\backslash 2^{j}B}\frac{|b\left(y\right)-b_B\left|\right|f\left(y\right)\left|\right|g\left(z\right)|}{{\left(\right|x-y|+|x-z\left|\right)}^{2n-\alpha }}dydz\hfill \\ \hfill \lesssim & \sum _{j=1}^{\infty }\frac{1}{|2^{j+1}{B|}^{2-\frac{\alpha }{n}}}{\int }_{2B}\left|g\left(z\right)\right|dz{\int }_{2^{j+1}B}\left(|b\left(y\right)-b_{2^{j+1}B}|+|b_{2^{j+1}B}-b_B|\right)\left|f\left(y\right)\right|dy\hfill \\ \hfill \lesssim & {\parallel b\parallel }_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\sum _{j=1}^{\infty }(j+1)v_{\overrightarrow{w}}^q{\left(2^{j+1}B\right)}^{\frac{\kappa -1}{q}},\hfill \end{array}$$

(16)

which shows that

$$\begin{array}{cc}\hfill & {\left(\frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\kappa }}{\int }_B{A}_{32}^q{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{1/q}\hfill \\ \hfill \lesssim & {\parallel b\parallel }_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\hfill \\ \hfill & \times {\left(\frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\kappa }}{\int }_B{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{\frac{1}{q}}\sum _{j=1}^{\infty }(j+1)v_{\overrightarrow{w}}^q{\left(2^{j+1}B\right)}^{\frac{\kappa -1}{q}}\hfill \\ \hfill \lesssim & {\parallel b\parallel }_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\sum _{j=1}^{\infty }(j+1){\left(\frac{v_{\overrightarrow{w}}^q\left(B\right)}{v_{\overrightarrow{w}}^q\left(2^{j+1}B\right)}\right)}^{\frac{1-\kappa }{q}}\hfill \\ \hfill \lesssim & {\parallel b\parallel }_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\sum _{j=1}^{\infty }(j+1){\left(\frac{\left|B\right|}{|2^{j+1}B|}\right)}^{\frac{(1-\kappa )\theta }{q}}\hfill \\ \hfill \lesssim & {\parallel b\parallel }_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\sum _{j=1}^{\infty }(j+1){\left(\frac{1}{2^{jn}}\right)}^{\theta \left(\frac{1-\kappa }{q}\right)}\hfill \\ \hfill \lesssim & {\parallel b\parallel }_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}.\hfill \end{array}$$

(17)

Hence,

$$A_3\lesssim {\parallel b\parallel }_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}.$$

Finally, we turn to the term $A_4$. The estimate for $|{\mathcal{M}}_{\alpha }(f^{\infty },g^{\infty })\left(x\right)|$ is similar to the method used for $|{\mathcal{M}}_{\alpha }(f^0,g^{\infty })\left(x\right)|$. We outline the argument as follows.

$$\begin{array}{cc}\hfill & |{\mathcal{M}}_{\alpha }\left(f^{\infty },g^{\infty }\right)\left(x\right)|\hfill \\ \hfill & \lesssim |I_{\alpha }\left(\right|f^{\infty }|,|g^{\infty }\left|\right)\left(x\right)|\hfill \\ \hfill & \lesssim {\int }_{{\left(2B\right)}^c\times {\left(2B\right)}^c}\frac{\left|f\right(z\left)\right|\left|g\right(z\left)\right|}{{\left(\right|x-y|+|x-z\left|\right)}^{2n-\alpha }}dydz\hfill \\ \hfill & \lesssim {\int }_{{(2B\times 2B)}^c}\frac{\left|f\right(z\left)\right|\left|g\right(z\left)\right|}{{\left(\right|x-y|+|x-z\left|\right)}^{2n-\alpha }}dydz\hfill \\ \hfill & \lesssim \sum _{j=1}^{\infty }{\int }_{2^j(2B\times 2B)\backslash 2^{j-1}(2B\times 2B)}\frac{\left|f\right(z\left)\right|\left|g\right(z\left)\right|}{{\left(\right|x-y|+|x-z\left|\right)}^{2n-\alpha }}dydz\hfill \\ \hfill & \lesssim \sum _{j=1}^{\infty }\frac{1}{|2^{j+1}{B|}^{2-\frac{\alpha }{n}}}{\int }_{2^{j+1}B}\left|f\left(y\right)\right|dy{\int }_{2^{j+1}B}\left|g\left(z\right)\right|dz\hfill \\ \hfill & \lesssim {\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\sum _{j=1}^{\infty }{v}_{\overrightarrow{w}}^q{\left(2^{j+1}B\right)}^{\frac{\kappa -1}{q}}.\hfill \end{array}$$

(18)

Similarly, the following estimate for $|{\mathcal{M}}_{\alpha }\left(\left(b_{2B}-b\right)f^{\infty },g^{\infty }\right)\left(x\right)|$ can also be obtained.

$$\begin{array}{cc}\hfill & |{\mathcal{M}}_{\alpha }\left(\left(b_{2B}-b\right)f^{\infty },g^{\infty }\right)\left(x\right)|\hfill \\ \hfill \lesssim & {\parallel b\parallel }_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\sum _{j=1}^{\infty }(j+1)v_{\overrightarrow{w}}^q{\left(2^{j+1}B\right)}^{\frac{\kappa -1}{q}}.\hfill \end{array}$$

(19)

Combining the above estimates, we get

$${∥{\mathcal{M}}_{\alpha ,b}^1\left(f,g\right)∥}_{L^{q,\kappa }\left(v_{\overrightarrow{w}}^q\right)}\lesssim {∥b∥}_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}.$$

What’s more, it immediately follows that

$${∥{\mathcal{M}}_{\alpha ,b}^2\left(f,g\right)∥}_{L^{q,\kappa }\left(v_{\overrightarrow{w}}^q\right)}\lesssim {∥b∥}_{\mathrm{BMO}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}.$$

Next, we prove the boundedness of the bilinear fractional maximal iterated commutator ${\mathcal{M}}_{\alpha ,\overrightarrow{b}}$ on weighted Morrey spaces. Under the same assumptions, it suffices to verify the following inequalities produced by the standard function decomposition (8):

$$\begin{array}{c}\hfill D_1:={\left(\frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\kappa }}{\int }_B{\left|{\mathcal{M}}_{\alpha ,\overrightarrow{b}}\left(f^0,g^0\right)\left(x\right)\right|}^q{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{1/q}\\ \hfill \lesssim {\parallel \overrightarrow{b}\parallel }_{{\mathrm{BMO}}^2}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)},\end{array}$$

(20)

$$\begin{array}{c}\hfill D_2:={\left(\frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\kappa }}{\int }_B{\left|{\mathcal{M}}_{\alpha ,\overrightarrow{b}}\left(f^0,g^{\infty }\right)\left(x\right)\right|}^q{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{1/q}\\ \hfill {\parallel \overrightarrow{b}\parallel }_{{\mathrm{BMO}}^2}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)},\end{array}$$

(21)

$$\begin{array}{c}\hfill D_3:={\left(\frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\kappa }}{\int }_B{\left|{\mathcal{M}}_{\alpha ,\overrightarrow{b}}\left(f^{\infty },g^0\right)\left(x\right)\right|}^q{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{1/q}\\ \hfill \lesssim {\parallel \overrightarrow{b}\parallel }_{{\mathrm{BMO}}^2}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)},\end{array}$$

(22)

$$\begin{array}{c}\hfill D_4:={\left(\frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\kappa }}{\int }_B{\left|{\mathcal{M}}_{\alpha ,\overrightarrow{b}}\left(f^{\infty },g^{\infty }\right)\left(x\right)\right|}^q{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{1/q}\\ \hfill \lesssim {\parallel \overrightarrow{b}\parallel }_{{\mathrm{BMO}}^2}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}.\end{array}$$

(23)

In the same way as estimate $A_1$ in (9) and using the weighted boundedness for ${\mathcal{M}}_{\alpha ,\overrightarrow{b}}$ in Remark 1, we can get the proof of (20), namely

$$D_1\lesssim {\parallel \overrightarrow{b}\parallel }_{{\mathrm{BMO}}^2}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}.$$

To estimate $D_2$ in (21), we still need to switch the commutator ${\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f^0,g^{\infty })$ into fractional maximal operators, so we divided it into the following parts:

$$\begin{array}{cc}\hfill |{\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f^0,g^{\infty })|\le & |b_1\left(x\right)-b_{1,B}\left|\right|b_2\left(x\right)-b_{2,B}\left|\right|{\mathcal{M}}_{\alpha }(f^0,g^{\infty })\left(x\right)|\hfill \\ \hfill & +|b_2\left(x\right)-b_{2,B}\left|\right|{\mathcal{M}}_{\alpha }((b_{1,B}-b_1)f^0,g^{\infty })\left(x\right)|\hfill \\ \hfill & +|b_1\left(x\right)-b_{1,B}\left|\right|{\mathcal{M}}_{\alpha }(f^0,(b_{2,B}-b_2)g^{\infty })\left(x\right)|\hfill \\ \hfill & +|{\mathcal{M}}_{\alpha }((b_{1,B}-b_1)f^0,(b_{2,B}-b_2)g^{\infty })\left(x\right)|\hfill \\ \hfill =:& D_{21}+D_{22}+D_{23}+D_{24}.\hfill \end{array}$$

First, similar to the estimates of $A_{21}$ in (12), we have

$$\begin{array}{cc}\hfill & {\left(\frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\kappa }}{\int }_B{D}_{21}^q{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{1/q}\hfill \\ \hfill \lesssim & \sum _{j=1}^{\infty }{\left(\frac{v_{\overrightarrow{w}}^q\left(B\right)}{v_{\overrightarrow{w}}^q\left(2^{j+1}B\right)}\right)}^{\frac{1-\kappa }{q}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\hfill \\ \hfill & \times {\left(\frac{1}{v_{\overrightarrow{w}}^q\left(B\right)}{\int }_B|b_1\left(x\right)-b_{1,B}{|}^q{|b_2\left(x\right)-b_{2,B}|}^q{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{\frac{1}{q}}\hfill \\ \hfill \lesssim & {\parallel \overrightarrow{b}\parallel }_{{\mathrm{BMO}}^2}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}.\hfill \end{array}$$

Then, similar to (14) for $A_{22}$ and (16) and (17) for $A_{32}$, we obtain, respectively,

$$\begin{array}{cc}\hfill {\left(\frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\kappa }}{\int }_B{D}_{22}^q{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{1/q}\lesssim & {\parallel \overrightarrow{b}\parallel }_{{\mathrm{BMO}}^2}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\hfill \\ \hfill & \times \sum _{j=1}^{\infty }{\left(\frac{\left|2B\right|}{|2^{j+1}B|}\right)}^{\frac{\theta }{p_1^{\prime }}}{\left(\frac{v_{\overrightarrow{w}}^q\left(B\right)}{v_{\overrightarrow{w}}^q\left(2^{j+1}B\right)}\right)}^{\frac{1-\kappa }{q}}\hfill \\ \hfill \lesssim & {\parallel \overrightarrow{b}\parallel }_{{\mathrm{BMO}}^2}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\left(\frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\kappa }}{\int }_B{D}_{23}^q{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{1/q}\lesssim & {\parallel \overrightarrow{b}\parallel }_{{\mathrm{BMO}}^2}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\hfill \\ \hfill & \times \sum _{j=1}^{\infty }(j+1){\left(\frac{v_{\overrightarrow{w}}^q\left(B\right)}{v_{\overrightarrow{w}}^q\left(2^{j+1}B\right)}\right)}^{\frac{1-\kappa }{q}}\hfill \\ \hfill \lesssim & {\parallel \overrightarrow{b}\parallel }_{{\mathrm{BMO}}^2}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}.\hfill \end{array}$$

Note that

$$\begin{array}{cc}\hfill & |{\mathcal{M}}_{\alpha }((b_{1,B}\left(x\right)-b_1)f^0,(b_{2,B}\left(x\right)-b_2)g^{\infty })\left(x\right)|\hfill \\ \hfill \lesssim & {\int }_{{\left(2B\right)}^c}{\int }_{2B}\frac{|b_{1,B}\left(x\right)-b_1\left(y\right)\left|\right|b_{2,B}\left(x\right)-b_2\left(z\right)\left|\right|f\left(y\right)\left|\right|g\left(z\right)|}{{\left(\right|x-y|+|x-z\left|\right)}^{2n-\alpha }}dydz\hfill \\ \hfill \lesssim & {\int }_{2B}|b_1\left(y\right)-b_{1,2B}\left|\right|f\left(y\right)|dy\sum _{j=1}^{\infty }\frac{1}{|2^{j+1}{B|}^{2-\frac{\alpha }{n}}}{\int }_{2^{j+1}B\backslash 2^{j}B}|b_2\left(z\right)-b_{2,2B}\left(x\right)\left|\right|g\left(z\right)|dz\hfill \\ \hfill \lesssim & {\parallel \overrightarrow{b}\parallel }_{{\mathrm{BMO}}^2}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\hfill \\ \hfill & \times \sum _{j=1}^{\infty }(j+1){\left(\frac{\left|2B\right|}{|2^{j+1}B|}\right)}^{\frac{\theta }{p_1^{\prime }}}{\left(\frac{v_{\overrightarrow{w}}^q\left(B\right)}{v_{\overrightarrow{w}}^q\left(2^{j+1}B\right)}\right)}^{\frac{1-\kappa }{q}}.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill {\left(\frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\kappa }}{\int }_B{D}_{24}^q{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{1/q}\lesssim & {\parallel \overrightarrow{b}\parallel }_{{\mathrm{BMO}}^2}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\hfill \\ \hfill & \times \sum _{j=1}^{\infty }(j+1){\left(\frac{1}{2^{jn}}\right)}^{\theta (1-\frac{1}{p_1}+\frac{1-\kappa }{q})}\hfill \\ \hfill \lesssim & {\parallel \overrightarrow{b}\parallel }_{{\mathrm{BMO}}^2}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}.\hfill \end{array}$$

Then, we get

$$D_2\lesssim {\parallel \overrightarrow{b}\parallel }_{{\mathrm{BMO}}^2}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}.$$

Similar results hold for $D_3$ and $D_4$, the details of which are omitted here. Combining the estimates above, we obtain

$${∥{\mathcal{M}}_{\alpha ,\overrightarrow{b}}\left(f,g\right)∥}_{L^{q,\kappa }\left(v_{\overrightarrow{w}}^q\right)}\lesssim {\parallel \overrightarrow{b}\parallel }_{{\mathrm{BMO}}^2}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}.$$

We complete the proof of Theorem 4. □

4. Proof of Compactness

This section is devoted to establishing the compactness of commutators of bilinear fractional maximal operators on weighted Morrey spaces. The proof requires the following auxiliary lemmas.

First, we list some facts of the commutators of ${\mathcal{M}}_{\alpha }$, which are obvious, we omit their proofs here.

Lemma 6.

If $0<\alpha <2n$, $0<\kappa <1$, $1<p<\infty$and $b,\nu \in \mathrm{BMO}$. Then

(i) 

$|{\mathcal{M}}_{\alpha ,b}^i(f,g)\left(x\right)-{\mathcal{M}}_{\alpha ,\nu }^i(f,g)\left(x\right)|\le {\mathcal{M}}_{\alpha ,b-\nu }^i(f,g)\left(x\right)$;

(ii) 

$\parallel {\mathcal{M}}_{\alpha ,b}^i(f,g){\chi }_{E_A}{\parallel }_{L^{p,\kappa }\left(w\right)}\le \parallel {\mathcal{M}}_{\alpha ,\nu }^i(f,g){\chi }_{E_A}{\parallel }_{L^{p,\kappa }\left(w\right)}+{\parallel {\mathcal{M}}_{\alpha ,b-\nu }^i(f,g){\chi }_{E_A}\parallel }_{L^{p,\kappa }\left(w\right)}$, where $E_A:=\{x\in {\mathbb{R}}^n:\left|x\right|>A\}$, for every $A>0$;

(iii) 

For each $h\in {\mathbb{R}}^n$,

$$\begin{array}{cc}\hfill & \parallel {\mathcal{M}}_{\alpha ,b}^i(f,g)(·+h)-{\mathcal{M}}_{\alpha ,b}^i{(f,g)(·)\parallel }_{L^{p,\kappa }\left(w\right)}\hfill \\ \hfill \le & \parallel {\mathcal{M}}_{\alpha ,\nu }^i(f,g)(·+h)-{\mathcal{M}}_{\alpha ,\nu }^i{(f,g)(·)\parallel }_{L^{p,\kappa }\left(w\right)}\hfill \\ \hfill & +\parallel {\mathcal{M}}_{\alpha ,b-\nu }^i{(f,g)\parallel }_{L^{p,\kappa }\left(w\right)}+{\parallel {\mathcal{M}}_{\alpha ,b-\nu }^i(f,g)(·+h)\parallel }_{L^{p,\kappa }\left(w\right)}.\hfill \end{array}$$

For the sake of obtaining our compact results, we also need the following estimates for maximal operators.

Lemma 7.

Let $0\le \alpha <2n$, $b\in {\mathcal{C}}_c^{\infty }\left({\mathbb{R}}^n\right)$and $x,h\in {\mathbb{R}}^n$. Consider concentric balls $B_1:=B\left(x_0,r\right)$and $B_2:=B\left(x_0,r+\left|h\right|\right)$such that $x\in B_1$. Then

$$\begin{array}{cc}\hfill & {\int }_{B_2}{\int }_{B_2}\left|\frac{{\chi }_{B_1}\left(y\right){\chi }_{B_1}\left(z\right)}{{\left|B_1\right|}^{2-\frac{\alpha }{n}}}-\frac{{\chi }_{B_2}\left(y\right){\chi }_{B_2}\left(z\right)}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}}\right||b(x+h)-b\left(y\right)\parallel f\left(y\right)\parallel g\left(z\right)|dydz\hfill \\ \hfill \lesssim & \left|h\right|{\mathcal{M}}_{\alpha }(f,g)\left(x\right)+{\left|h\right|}^{\frac{1}{s^{\prime }}}{\mathcal{M}}_{\alpha ,s}(f,g)\left(x\right),\hfill \end{array}$$

where $s>1$, $0<\alpha s<2n$(equivalently, $1<s<2n/\alpha$), $\frac{1}{s}+\frac{1}{s^{\prime }}=1$and ${\mathcal{M}}_{\alpha ,s}(f,g)\left(x\right)={\left({\mathcal{M}}_{\alpha s}{\left(\right|f|}^s{,\left|g\right|}^s)\left(x\right)\right)}^{\frac{1}{s}}$.

Proof. 

Let’s consider the following two cases: $r\le \left|h\right|$ and $r>\left|h\right|$.

Case I: $r\le \left|h\right|$. For any $y\in B_2$, a simple calculation gives us

$$\begin{array}{cc}\hfill \left|b\right(x+h)-b(y\left)\right|\lesssim & {|y-x-h|\parallel \nabla b\parallel }_{L^{\infty }}\hfill \\ \hfill \lesssim & \left(\right|y-x_0|+|x-x_0{|+|h\left|\right)\parallel \nabla b\parallel }_{L^{\infty }}\hfill \\ \hfill \lesssim & {(r+|h|+r+|h\left|\right)\parallel \nabla b\parallel }_{L^{\infty }}\hfill \\ \hfill \lesssim & {\left|h\right|\parallel \nabla b\parallel }_{L^{\infty }}.\hfill \end{array}$$

Then

$$\begin{array}{cc}\hfill & {\int }_{B_2}{\int }_{B_2}\left|\frac{{\chi }_{B_1}\left(y\right){\chi }_{B_1}\left(z\right)}{{\left|B_1\right|}^{2-\frac{\alpha }{n}}}-\frac{{\chi }_{B_2}\left(y\right){\chi }_{B_2}\left(z\right)}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}}\right||b(x+h)-b\left(y\right)|\left|f\left(y\right)\right|\left|g\left(z\right)\right|dydz\hfill \\ \hfill \lesssim & {\left|h\right|\parallel \nabla b\parallel }_{L^{\infty }}\frac{1}{|B_1{|}^{2-\frac{\alpha }{n}}}{\int }_{B_1}{\int }_{B_1}\left|f\left(y\right)\right|\left|g\left(z\right)\right|dydz\hfill \\ \hfill & +{\left|h\right|\parallel \nabla b\parallel }_{L^{\infty }}\frac{1}{|B_2{|}^{2-\frac{\alpha }{n}}}{\int }_{B_2}{\int }_{B_2}\left|f\left(y\right)\right|\left|g\left(z\right)\right|dydz\hfill \\ \hfill \lesssim & {\left|h\right|\parallel \nabla b\parallel }_{L^{\infty }}{\mathcal{M}}_{\alpha }(f,g)\left(x\right).\hfill \end{array}$$

Case II: $r>\left|h\right|$. By the triangle inequality tells us that

$$\begin{array}{cc}\hfill & {\int }_{B_2}{\int }_{B_2}\left|\frac{{\chi }_{B_1}\left(y\right){\chi }_{B_1}\left(z\right)}{{\left|B_1\right|}^{2-\frac{\alpha }{n}}}-\frac{{\chi }_{B_2}\left(y\right){\chi }_{B_2}\left(z\right)}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}}\right||b(x+h)-b\left(y\right)|\left|f\left(y\right)\right|\left|g\left(z\right)\right|dydz\hfill \\ \hfill \le & {\int }_{B_2}{\int }_{B_2}\left|\frac{{\chi }_{B_1}\left(y\right){\chi }_{B_1}\left(z\right)}{{\left|B_1\right|}^{2-\frac{\alpha }{n}}}-\frac{{\chi }_{B_1}\left(y\right){\chi }_{B_1}\left(z\right)}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}}\right||b(x+h)-b\left(y\right)|f\left(y\right)\left|\right|g\left(z\right)|dydz\hfill \\ \hfill & +{\int }_{B_2}{\int }_{B_2}\left|\frac{{\chi }_{B_1}\left(y\right){\chi }_{B_1}\left(z\right)}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}}-\frac{{\chi }_{B_1}\left(y\right){\chi }_{B_2}\left(z\right)}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}}\right||b(x+h)-b\left(y\right)|\left|f\left(y\right)\right|\left|g\left(z\right)\right|dydz\hfill \\ \hfill & +{\int }_{B_2}{\int }_{B_2}\left|\frac{{\chi }_{B_1}\left(y\right){\chi }_{B_2}\left(z\right)}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}}-\frac{{\chi }_{B_2}\left(y\right){\chi }_{B_2}\left(z\right)}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}}\right||b(x+h)-b\left(y\right)|\left|f\left(y\right)\right|\left|g\left(z\right)\right|dydz\hfill \\ \hfill =:& E_1+E_2+E_3.\hfill \end{array}$$

For any $y\in B_1$, we simply have

$$\begin{array}{cc}\hfill \left|b\right(x+h)-b(y\left)\right|\lesssim & {|y-x-h|\parallel \nabla b\parallel }_{L^{\infty }}\hfill \\ \hfill \lesssim & \left(\right|y-x_0|+|x-x_0{|+|h\left|\right)\parallel \nabla b\parallel }_{L^{\infty }}\hfill \\ \hfill \lesssim & {(r+r+|h\left|\right)\parallel \nabla b\parallel }_{L^{\infty }}\hfill \\ \hfill \lesssim & {r\parallel \nabla b\parallel }_{L^{\infty }}.\hfill \end{array}$$

For $E_1$, since the above estimate, we obtain

$$\begin{array}{cc}\hfill E_1& =\left|\frac{{\left|B_2\right|}^{2-\frac{\alpha }{n}}-{\left|B_1\right|}^{2-\frac{\alpha }{n}}}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}{\left|B_1\right|}^{2-\frac{\alpha }{n}}}\right|{\int }_{B_1}{\int }_{B_1}|b(x+h)-b\left(y\right)|\left|f\left(y\right)\right|\left|g\left(z\right)\right|dydz\hfill \\ \hfill & \lesssim {r\parallel \nabla b\parallel }_{L^{\infty }\left({\mathbb{R}}^n\right)}\left|\frac{{\left|B_2\right|}^{2-\frac{\alpha }{n}}-{\left|B_1\right|}^{2-\frac{\alpha }{n}}}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}}\right|\frac{1}{{\left|B_1\right|}^{2-\frac{\alpha }{n}}}{\int }_{B_1}{\int }_{B_1}\left|f\left(y\right)\right|\left|g\left(z\right)\right|dydz.\hfill \end{array}$$

We now claim the following inequality holds

$$\left|\frac{{\left|B_2\right|}^{2-\frac{\alpha }{n}}-{\left|B_1\right|}^{2-\frac{\alpha }{n}}}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}}\right|\lesssim \frac{\left|h\right|}{r}.$$

(24)

In fact, we have

$$\begin{array}{cc}\hfill \left|\frac{{\left|B_2\right|}^{2-\frac{\alpha }{n}}-{\left|B_1\right|}^{2-\frac{\alpha }{n}}}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}}\right|=& \left|1-{\left(\frac{|B_1|}{|B_2|}\right)}^{2-\frac{\alpha }{n}}\right|=\left|{\left[{\left(\frac{r}{r+\left|h\right|}\right)}^{2-\frac{\alpha }{n}}\right]}^n-1\right|\hfill \\ \hfill \le & \left|{\left[{\left(\frac{r+\left|h\right|}{r}\right)}^{2-\frac{\alpha }{n}}\right]}^n-1\right|\le \left|{\left[{\left(1+\frac{\left|h\right|}{r}\right)}^{2-\frac{\alpha }{n}}\right]}^n-1\right|.\hfill \end{array}$$

When $0<2-\frac{\alpha }{n}\le 1$, by the Bernoulli inequality, we have

$$\begin{array}{cc}\hfill \left|{\left[{\left(1+\frac{\left|h\right|}{r}\right)}^{2-\frac{\alpha }{n}}\right]}^n-1\right|\le & \left|{\left(1+\left(2-\frac{\alpha }{n}\right)\frac{\left|h\right|}{r}\right)}^n-1\right|\hfill \\ \hfill \lesssim & \left|1+\left(2n-\alpha \right)\frac{\left|h\right|}{r}-1\right|\lesssim \frac{\left|h\right|}{r}.\hfill \end{array}$$

On the other hand, for $1<2-\frac{\alpha }{n}\le 2$, a similar calculation yields

$$\begin{array}{cc}\hfill \left|{\left[{\left(1+\frac{\left|h\right|}{r}\right)}^{2-\frac{\alpha }{n}}\right]}^n-1\right|=& \left|{\left[\left(1+\frac{\left|h\right|}{r}\right){\left(1+\frac{\left|h\right|}{r}\right)}^{1-\frac{\alpha }{n}}\right]}^n-1\right|\hfill \\ \hfill \le & \left|{\left(1+\frac{\left|h\right|}{r}\right)}^n{\left(1+\left(1-\frac{\alpha }{n}\right)\frac{\left|h\right|}{r}\right)}^n-1\right|\hfill \\ \hfill \lesssim & \left|{\left(1+\left(1-\frac{\alpha }{n}\right)\frac{\left|h\right|}{r}\right)}^n-1\right|\hfill \\ \hfill \lesssim & \left|1+\left(n-\alpha \right)\frac{\left|h\right|}{r}-1\right|\lesssim \frac{\left|h\right|}{r}.\hfill \end{array}$$

Therefore, our claim (24) holds. It implies that

$$E_1\lesssim {\left|h\right|\parallel \nabla b\parallel }_{L^{\infty }}{\mathcal{M}}_{\alpha }(f,g)\left(x\right).$$

Applying Hölder’s inequality with $s\in (1,\infty )$ and $\frac{1}{s}+\frac{1}{s^{\prime }}=1$ yields

$$\begin{array}{cc}\hfill \left|b\right(x+h)-b(y\left)\right|=& {|b(x+h)-b\left(y\right)|}^{\frac{1}{s^{\prime }}}{|b(x+h)-b\left(y\right)|}^{\frac{1}{s}}\hfill \\ \hfill \lesssim & {|y-x-h|}^{\frac{1}{s^{\prime }}}{\parallel b\parallel }_{L^{\infty }}^{\frac{1}{s}}{\parallel \nabla b\parallel }_{L^{\infty }}^{\frac{1}{s^{\prime }}}\hfill \\ \hfill \lesssim & r^{\frac{1}{s^{\prime }}}{\parallel b\parallel }_{L^{\infty }}^{\frac{1}{s}}{\parallel \nabla b\parallel }_{L^{\infty }}^{\frac{1}{s^{\prime }}}.\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill E_2\lesssim & \frac{1}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}}{\int }_{B_2\backslash B_1}{\int }_{B_1}|b(x+h)-b\left(y\right)|\left|f\left(y\right)\right|\left|g\left(z\right)\right|dydz\hfill \\ \hfill \lesssim & \frac{r^{\frac{1}{s^{\prime }}}}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}}{\int }_{B_2\backslash B_1}\left|g\left(z\right)\right|dz{\int }_{B_2}\left|f\left(y\right)\right|dy\hfill \\ \hfill \lesssim & r^{\frac{1}{s^{\prime }}}{\left(\frac{|B_2|-|B_1|}{|B_2|}\right)}^{\frac{1}{s^{\prime }}}{\left(\frac{1}{|B_2{|}^{2-\frac{\alpha s}{n}}}{\int }_{B_2}{\int }_{B_2}{\left|f\left(y\right)\right|}^s{\left|g\left(z\right)\right|}^{s}dydz\right)}^{\frac{1}{s}}\hfill \\ \hfill \lesssim & r^{\frac{1}{s^{\prime }}}{\left(\frac{{(r+|h\left|\right)}^n-r^n}{{(r+|h\left|\right)}^n}\right)}^{\frac{1}{s^{\prime }}}{\left({\mathcal{M}}_{\alpha s}{\left(\right|f|}^s{,\left|g\right|}^s)\left(x\right)\right)}^{\frac{1}{s}}\hfill \\ \hfill \lesssim & r^{\frac{1}{s^{\prime }}}{\left(\frac{\left|h\right|}{r}\right)}^{\frac{1}{s^{\prime }}}{\mathcal{M}}_{\alpha ,s}(f,g)\left(x\right)\hfill \\ \hfill =& {\left|h\right|}^{\frac{1}{s^{\prime }}}{\mathcal{M}}_{\alpha ,s}(f,g)\left(x\right).\hfill \end{array}$$

By the same argument as for $E_2$, we obtain

$$E_3\lesssim {\left|h\right|}^{\frac{1}{s^{\prime }}}{\mathcal{M}}_{\alpha ,s}(f,g)\left(x\right).$$

Thus, we get the desired result. □

The proof of Theorem 5 relies on the following characterization of strongly precompact subsets in $L^{q,\kappa }\left(w\right)$, which is a variant of Fréchet–Kolmogorov theorem on weighted Morrey spaces.

Lemma 8

([41]). Let $0<\kappa <1$, $1<p<\infty$, $w\in A_p$, $\mathcal{F}$ be a subset in $L^{p,\kappa }\left(w\right)$. Then $\mathcal{F}$ is strongly pre-compact in $L^{p,\kappa }\left(w\right)$ if and only if the following three conditions are satisfied:

(1) 

$\underset{f\in \mathcal{F}}{sup}{\parallel f\parallel }_{L^{p,\kappa }\left(w\right)}<\infty$;

(2) 

$\underset{A\to \infty }{lim}{∥f{\chi }_{E_A}∥}_{L^{p,\kappa }\left(w\right)}=0$, uniformly in $f\in \mathcal{F}$, where $E_A=\left\{x\in {\mathbb{R}}^n:\left|x\right|>A\right\}$;

(3) 

$\underset{\left|h\right|\to 0}{lim}{\parallel f(·+h)-f(·)\parallel }_{L^{p,\kappa }\left(w\right)}=0$, uniformly in $f\in \mathcal{F}$.

In order to check Fréchet–Kolmogorov theorem above for bilinear fractional maximal operators, the following results are necessary.

Lemma 9

([9]). Let $1<p<\infty$, if $w\in A_p$, we have

$$\underset{A\to \infty }{lim}{\int }_{\left|x\right|>A}\frac{w\left(x\right)}{{\left|x\right|}^{np}}dx=0\mathrm{and}\underset{A\to \infty }{lim}{\int }_{\left|x\right|>A}\frac{w^{1-p^{\prime }}\left(x\right)}{{\left|x\right|}^{n{p}^{\prime }}}dx=0.$$

Now, we start the proof of compactness.

Proof of Theorem 5. 

We consider only the compactness of ${\mathcal{M}}_{\alpha ,b}^1$ and ${\mathcal{M}}_{\alpha ,\overrightarrow{b}}$, as the proof for ${\mathcal{M}}_{\alpha ,b}^2$ is similar. Without loss of generality, it may be assumed that F and G are sets on $L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)$ and $L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)$, respectively. We need to show the set $\mathcal{F}=\{{\mathcal{M}}_{\alpha ,b}^1(f,g):f\in F,g\in G\}$ is strongly pre-compact on $L^{q,\kappa }\left(v_{\overrightarrow{w}}^q\right)$ when $b\in \mathrm{CMO}$. According to a density argument, if $b\in \mathrm{CMO}$, $ϵ>0$, then there exists a sequence of functions $b_{ϵ}\in {\mathcal{C}}_c^{\infty }$ such that

$${∥b-b_{ϵ}∥}_{\mathrm{BMO}}<ϵ.$$

(25)

Thus, by Lemma 6 and (25), to prove Theorem 5, it suffices to establish the strong precompactness of $\mathcal{F}$ in $L^{q,\kappa }\left(v_{\overrightarrow{w}}^q\right)$ for $b\in {\mathcal{C}}_c^{\infty }$. By Lemma 8, we only need to verify that conditions (1)–(3) hold uniformly in $\mathcal{F}$ for $b\in {\mathcal{C}}_c^{\infty }$. The proof is carried out in two steps, note that condition (1) is an immediate consequence of Theorem 4.

Step I. $\mathcal{F}$ satisfies condition (2) in Lemma 8.

As is well known, the uncentered and centered maximal operators control each other. Therefore, it suffices to consider the centered fractional maximal operator. Without loss of generality, we assume that ${\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}={\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}=1$. For any $x\in \{x:|x|>A\}$, $suppb\subset B(0,R)$, we have

$$\begin{array}{cc}\hfill \left|{\mathcal{M}}_{\alpha ,b}^1(f,g)\left(x\right)\right|& \lesssim \underset{r>0}{sup}\frac{1}{{\left|B(x,r)\right|}^{2-\frac{\alpha }{n}}}{\int }_{B(x,r)}{\int }_{B(x,r)}\left|b\left(y\right)\right|\left|f\left(y\right)\right|\left|g\left(z\right)\right|dydz\hfill \\ \hfill & \lesssim \underset{r>0}{sup}\frac{{\parallel b\parallel }_{L^{\infty }\left({\mathbb{R}}^n\right)}}{{\left|B(x,r)\right|}^{2-\frac{\alpha }{n}}}{\int }_{B(0,R)}\left|f\left(y\right)\right|dy{\int }_{B(x,r)}\left|g\left(z\right)\right|dz.\hfill \end{array}$$

Write

$$V_1^R\left(x\right):=\frac{1}{{\left|B(x,r)\right|}^{2-\frac{\alpha }{n}}}{\int }_{B(0,R)}\left|f\left(y\right)\right|dy{\int }_{\left|z\right|\le \left|x\right|}\left|g\left(z\right)\right|dz$$

and

$$V_2^R\left(x\right):=\frac{1}{{\left|B(x,r)\right|}^{2-\frac{\alpha }{n}}}{\int }_{B(0,R)}\left|f\left(y\right)\right|dy{\int }_{\left|z\right|>\left|x\right|}\left|g\left(z\right)\right|dz.$$

Since $\left|x\right|>A\ge max\{2R,1\}$ and $B(x,r)\cap suppb\ne \varnothing$, we have $r>\left|x\right|-R>\left|x\right|/2$. Applying Hölder’s inequality, it yields that

$$\begin{array}{cc}\hfill V_1^R\left(x\right)\lesssim & \frac{1}{{\left|x\right|}^{2n-\alpha }}{\int }_{B(0,R)}\left|f\left(y\right)\right|dy{\int }_{B(0,|x\left|\right)}\left|g\left(z\right)\right|dz\hfill \\ \hfill \lesssim & \frac{1}{{\left|x\right|}^{2n-\alpha }}{w}_1^{q_1}{\left(B(0,R)\right)}^{\frac{\kappa }{q_1}}{w}_2^{q_2}{\left(B\right(0,\left|x\right|\left)\right)}^{\frac{\kappa }{q_2}}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}\hfill \\ \hfill & \times {\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}{\left({\int }_{B(0,R)}{w}_1^{-p_1^{\prime }}\left(y\right)dy\right)}^{\frac{1}{p_1^{\prime }}}{\left({\int }_{B(0,|x\left|\right)}{w}_2^{-p_2^{\prime }}\left(z\right)dz\right)}^{\frac{1}{p_2^{\prime }}}\hfill \\ \hfill \lesssim & \frac{1}{{\left|x\right|}^{2n-\alpha }}{w}_1^{q_1}{\left(B(0,R)\right)}^{\frac{\kappa }{q_1}}{w}_2^{q_2}{\left(B\right(0,\left|x\right|\left)\right)}^{\frac{\kappa }{q_2}}\hfill \\ \hfill & \times {\left({\int }_{B(0,R)}{w}_1^{-p_1^{\prime }}\left(y\right)dy\right)}^{\frac{1}{p_1^{\prime }}}{\left({\int }_{B(0,|x\left|\right)}{w}_2^{-p_2^{\prime }}\left(z\right)dz\right)}^{\frac{1}{p_2^{\prime }}}.\hfill \end{array}$$

Observe that for $z\in B(x,r)$, we get $\left|z\right|\le \left|x\right|+r\le 3r$. Similarly, we have

$$\begin{array}{cc}\hfill V_2^R\left(x\right)\lesssim & {\int }_{B(0,R)}\left|f\left(y\right)\right|dy{\int }_{\left|z\right|>\left|x\right|}\frac{\left|g\right(z\left)\right|}{{\left|z\right|}^{2n-\alpha }}dz\hfill \\ \hfill \lesssim & {\int }_{B(0,R)}\left|f\left(y\right)\right|dy\sum _{l=1}^{\infty }\frac{1}{(2^{l-1}{\left|x\right|)}^{2n-\alpha }}{\int }_{\left|z\right|\le 2^l\left|x\right|}\left|g\left(z\right)\right|dz\hfill \\ \hfill \lesssim & \sum _{l=1}^{\infty }\frac{w_1^{q_1}{\left(B(0,R)\right)}^{\frac{\kappa }{q_1}}{w}_2^{q_2}\left(B\right(0,2^l{\left|x\right|\left)\right)}^{\frac{\kappa }{q_2}}}{(2^{l-1}{\left|x\right|)}^{2n-\alpha }}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}\hfill \\ \hfill & \times {\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}{\left({\int }_{B(0,R)}{w}_1^{-p_1^{\prime }}\left(y\right)dy\right)}^{\frac{1}{p_1^{\prime }}}{\left({\int }_{B(0,2^l|x\left|\right)}{w}_2^{-p_2^{\prime }}\left(z\right)dz\right)}^{\frac{1}{p_2^{\prime }}}\hfill \\ \hfill \lesssim & \sum _{l=1}^{\infty }\frac{w_1^{q_1}{\left(B(0,R)\right)}^{\frac{\kappa }{q_1}}{w}_2^{q_2}\left(B\right(0,2^l{\left|x\right|\left)\right)}^{\frac{\kappa }{q_2}}}{(2^{l-1}{\left|x\right|)}^{2n-\alpha }}\hfill \\ \hfill & \times {\left({\int }_{B(0,R)}{w}_1^{-p_1^{\prime }}\left(y\right)dy\right)}^{\frac{1}{p_1^{\prime }}}{\left({\int }_{B(0,2^l|x\left|\right)}{w}_2^{-p_2^{\prime }}\left(z\right)dz\right)}^{\frac{1}{p_2^{\prime }}}.\hfill \end{array}$$

Therefore, we get

$$\begin{array}{cc}\hfill \left|{\mathcal{M}}_{\alpha ,b}^1(f,g)\left(x\right)\right|& \lesssim \sum _{l=0}^{\infty }\frac{w_1^{q_1}{\left(B(0,R)\right)}^{\frac{\kappa }{q_1}}{w}_2^{q_2}\left(B\right(0,2^l{\left|x\right|\left)\right)}^{\frac{\kappa }{q_2}}}{(2^l{\left|x\right|)}^{2n-\alpha }}\hfill \\ \hfill & \times {\left({\int }_{B(0,R)}{w}_1^{-p_1^{\prime }}\left(y\right)dy\right)}^{\frac{1}{p_1^{\prime }}}{\left({\int }_{B(0,2^l|x\left|\right)}{w}_2^{-p_2^{\prime }}\left(z\right)dz\right)}^{\frac{1}{p_2^{\prime }}}.\hfill \end{array}$$

Thus, for any fixed ball $\tilde{B}\subset {\mathbb{R}}^n$, by the Minkowski inequality, we obtain

$$\begin{array}{cc}\hfill & {\left(\frac{1}{v_{\overrightarrow{w}}^q{\left(\tilde{B}\right)}^{\kappa }}{\int }_{\tilde{B}}{\left|{\mathcal{M}}_{\alpha ,b}^1(f,g)\left(x\right)\right|}^q{\chi }_{\left\{\right|x|>A\}}\left(x\right)v_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{\frac{1}{q}}\hfill \\ \hfill \lesssim & \sum _{j=1}^{\infty }{\left(\frac{1}{v_{\overrightarrow{w}}^q{\left(\tilde{B}\right)}^{\kappa }}{\int }_{\tilde{B}\bigcap \{2^{j-1}A<\left|x\right|\le 2^{j}A\}}{\left|{\mathcal{M}}_{\alpha ,b}^1(f,g)\left(x\right)\right|}^q{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{\frac{1}{q}}\hfill \\ \hfill \lesssim & {\left(\frac{1}{v_{\overrightarrow{w}}^q{\left(\tilde{B}\right)}^{\kappa }}{\int }_{\tilde{B}\bigcap \{2^{j-1}A<\left|x\right|\le 2^{j}A\}}{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{\frac{1}{q}}\sum _{j=1}^{\infty }\sum _{l=0}^{\infty }\frac{w_1^{q_1}{\left(B(0,R)\right)}^{\frac{\kappa }{q_1}}{w}_2^{q_2}{\left(B(0,2^{j+l}A)\right)}^{\frac{\kappa }{q_2}}}{{\left(2^{j+l-1}A\right)}^{2n-\alpha }}\hfill \\ \hfill & \times {\left({\int }_{B(0,R)}{w}_1^{-p_1^{\prime }}\left(y\right)dy\right)}^{\frac{1}{p_1^{\prime }}}{\left({\int }_{B\left(0,2^{j+l}A\right)}{w}_2^{-p_2^{\prime }}\left(z\right)dz\right)}^{\frac{1}{p_2^{\prime }}}\hfill \\ \hfill \lesssim & v_{\overrightarrow{w}}^q{\left(\tilde{B}\cap \{2^{j-1}A<\left|x\right|\le 2^{j}A\}\right)}^{\frac{1-\kappa }{q}}\sum _{j=1}^{\infty }\sum _{l=0}^{\infty }\frac{w_1^{q_1}{\left(B(0,2^{j+l}A)\right)}^{\frac{\kappa }{q_1}}{w}_2^{q_2}{\left(B(0,2^{j+l}A)\right)}^{\frac{\kappa }{q_2}}}{{\left(2^{j+l-1}A\right)}^{2n-\alpha }}\hfill \\ \hfill & \times {\left(\frac{R}{2^{j+l}A}\right)}^{n\theta (\frac{\kappa }{q_1}+\frac{1}{p_1^{\prime }})}{\left({\int }_{B\left(0,2^{j+l}A\right)}{w}_1^{-p_1^{\prime }}\left(y\right)dy\right)}^{\frac{1}{p_1^{\prime }}}{\left({\int }_{B\left(0,2^{j+l}A\right)}{w}_2^{-p_2^{\prime }}\left(z\right)dz\right)}^{\frac{1}{p_2^{\prime }}}\hfill \\ \hfill \lesssim & \sum _{j=1}^{\infty }\sum _{l=0}^{\infty }{\left(\frac{v_{\overrightarrow{w}}^q\left(B(0,2^{j}A)\right)}{v_{\overrightarrow{w}}^q\left(B(0,2^{j+l}A)\right)}\right)}^{\frac{1-\kappa }{q}}\frac{1}{{\left(2^{j+l-1}A\right)}^{2n-\alpha }}{\left(\frac{R}{2^{j+l}A}\right)}^{n\theta (\frac{\kappa }{q_1}+\frac{1}{p_1^{\prime }})}\hfill \\ \hfill & \times {\left({\int }_{B\left(0,2^{j+l}A\right)}{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{\frac{1}{q}}{\left({\int }_{B\left(0,2^{j+l}A\right)}{w}_1^{-p_1^{\prime }}\left(y\right)dy\right)}^{\frac{1}{p_1^{\prime }}}{\left({\int }_{B\left(0,2^{j+l}A\right)}{w}_2^{-p_2^{\prime }}\left(z\right)dz\right)}^{\frac{1}{p_2^{\prime }}}\hfill \\ \hfill \lesssim & \sum _{j=1}^{\infty }\sum _{l=0}^{\infty }{\left(\frac{v_{\overrightarrow{w}}^q\left(B(0,2^{j}A\right)}{v_{\overrightarrow{w}}^q\left(B(0,2^{j+l}A)\right)}\right)}^{\frac{1-\kappa }{q}}\frac{{\left(2^{j+l}A\right)}^{n(2+\frac{1}{q}-\frac{1}{p})}}{{\left(2^{j+l-1}A\right)}^{2n-\alpha }}{\left(\frac{R}{2^{j+l}A}\right)}^{n\theta (\frac{\kappa }{q_1}+\frac{1}{p_1^{\prime }})}\hfill \\ \hfill \lesssim & \sum _{j=1}^{\infty }\sum _{l=0}^{\infty }{\left(\frac{v_{\overrightarrow{w}}^q\left(B(0,2^{j}A\right)}{v_{\overrightarrow{w}}^q\left(B(0,2^{j+l}A)\right)}\right)}^{\frac{1-\kappa }{q}}\frac{1}{2^{ln\theta (\frac{1}{p_1^{\prime }}+\frac{\kappa }{q_1})}}\frac{1}{2^{jn\theta (\frac{1}{p_1^{\prime }}+\frac{\kappa }{q_1})}}{\left(\frac{R}{A}\right)}^{n\theta (\frac{\kappa }{q_1}+\frac{1}{p_1^{\prime }})}\hfill \\ \hfill \lesssim & {\left(\frac{R}{A}\right)}^{n\theta (\frac{\kappa }{q_1}+\frac{1}{p_1^{\prime }})},\hfill \end{array}$$

where $\theta >0$, $0<\kappa <1$, $q_1,p_1^{\prime }>0$ and the implict constant in the last inequality is independent of the ball $\tilde{B}$. Since $R>0$ is a fixed constant reltated to the compact support of b and the exponent $n\theta (\frac{\kappa }{q_1}+\frac{1}{p_1^{\prime }})$ is strictly positive, thus, as $A\to \infty$, the term ${\left(\frac{R}{A}\right)}^{n\theta (\frac{\kappa }{q_1}+\frac{1}{p_1^{\prime }})}$ tends to zero. Then, for $1<q<\infty$, we have

$$\underset{A\to \infty }{lim}{\left(\frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\kappa }}{\int }_B{\left|{\mathcal{M}}_{\alpha ,b}^1(f,g)\left(x\right)\right|}^q{\chi }_{\left\{\right|x|>A\}}\left(x\right)v_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{\frac{1}{q}}=0$$

holds, where $f\in F$ and $g\in G$.

Step II. $\mathcal{F}$ satisfies condition (3) in Lemma 8.

Our goal is to prove the following estimate

$$\underset{\left|h\right|\to 0}{lim}{∥{\mathcal{M}}_{\alpha ,b}^1(f,g)(·+h)-{\mathcal{M}}_{\alpha ,b}^1(f,g)(·)∥}_{L^{q,\kappa }\left(v_{\overrightarrow{w}}^q\right)}=0.$$

Without loss of generality, consider fixed points $x,h\in {\mathbb{R}}^n$ satisfying $\left|h\right|<1$, we assume that

$${\mathcal{M}}_{\alpha ,b}^1(f,g)(x+h)\le {\mathcal{M}}_{\alpha ,b}^1(f,g)\left(x\right)\mathrm{and}{\mathcal{M}}_{\alpha ,b}^1(f,g)\left(x\right)<\infty .$$

Therefore, for any $ϵ\in (0,1)$, there is a ball $B_1=B\left(x_0,r\right)\ni x$ such that

$$(1-ϵ){\mathcal{M}}_{\alpha ,b}^1(f,g)\left(x\right)\lesssim \frac{1}{{\left|B_1\right|}^{2-\frac{\alpha }{n}}}{\int }_{B_1}{\int }_{B_1}|b\left(x\right)-b\left(y\right)\parallel f\left(y\right)\parallel g\left(z\right)|dydz.$$

(26)

Since $x+h\in B\left(x_0,r+\left|h\right|\right):=B_2$, we get

$$\begin{array}{c}\hfill {\mathcal{M}}_{\alpha ,b}^1(f,g)(x+h)\ge \frac{1}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}}{\int }_{B_2}{\int }_{B_2}|b(x+h)-b\left(y\right)|\left|f\left(y\right)\right|\left|\mathrm{g}\left(z\right)\right|dydz.\end{array}$$

(27)

We also have

$$\begin{array}{cc}\hfill & |{\mathcal{M}}_{\alpha ,b}^1(f,g)(x+h)-{\mathcal{M}}_{\alpha ,b}^1(f,g)\left(x\right)|\hfill \\ \hfill =& (1-ϵ){\mathcal{M}}_{\alpha ,b}^1(f,g)\left(x\right)-{\mathcal{M}}_{\alpha ,b}^1(f,g)(x+h)+ϵ{\mathcal{M}}_{\alpha ,b}^1(f,g)\left(x\right).\hfill \end{array}$$

Applying the inequalities (26) and (27), we can get

$$\begin{array}{cc}\hfill & (1-ϵ){\mathcal{M}}_{\alpha ,b}^1(f,g)\left(x\right)-{\mathcal{M}}_{\alpha ,b}^1(f,g)(x+h)\hfill \\ \hfill \lesssim & \frac{1}{{\left|B_1\right|}^{2-\frac{\alpha }{n}}}{\int }_{B_1}{\int }_{B_1}|b\left(x\right)-b\left(y\right)|\left|f\left(y\right)\right|\left|g\left(z\right)\right|dydz\hfill \\ \hfill & -\frac{1}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}}{\int }_{B_2}{\int }_{B_2}|b(x+h)-b\left(y\right)|\left|f\left(y\right)\right|\left|g\left(z\right)\right|dydz\hfill \\ \hfill \lesssim & \frac{1}{{\left|B_1\right|}^{2-\frac{\alpha }{n}}}{\int }_{B_1}{\int }_{B_1}|b(x+h)-b\left(x\right)|\left|f\left(y\right)\right|\left|g\left(z\right)\right|dydz\hfill \\ \hfill & +\frac{1}{{\left|B_1\right|}^{2-\frac{\alpha }{n}}}{\int }_{B_1}{\int }_{B_1}|b(x+h)-b\left(y\right)|\left|f\left(y\right)\right|\left|g\left(z\right)\right|dydz\hfill \\ \hfill & -\frac{1}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}}{\int }_{B_2}{\int }_{B_2}|b(x+h)-b\left(y\right)|\left|f\left(y\right)\right|\left|g\left(z\right)\right|dydz\hfill \\ \hfill \lesssim & {\left|h\right|\parallel \nabla b\parallel }_{L^{\infty }\left({\mathbb{R}}^n\right)}{\mathcal{M}}_{\alpha }(f,g)\left(x\right)\hfill \\ \hfill & +{\int }_{B_2}{\int }_{B_2}\left|\frac{{\chi }_{B_1}\left(y\right){\chi }_{B_1}\left(z\right)}{{\left|B_1\right|}^{2-\frac{\alpha }{n}}}-\frac{{\chi }_{B_2}\left(y\right){\chi }_{B_2}\left(z\right)}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}}\right||b(x+h)-b\left(y\right)|\left|f\left(y\right)\right|\left|\mathrm{g}\left(z\right)\right|dydz.\hfill \end{array}$$

To optimize the bound, we let $ϵ=\left|h\right|$ and apply the Lemma 7,

$$\begin{array}{cc}\hfill & \left|{\mathcal{M}}_{\alpha ,b}^1(f,g)(x+h)-{\mathcal{M}}_{\alpha ,b}^1(f,g)\left(x\right)\right|\hfill \\ \hfill \lesssim & \left|h\right|{\mathcal{M}}_{\alpha }(f,g)\left(x\right)+{\left|h\right|}^{\frac{1}{s^{\prime }}}{\mathcal{M}}_{\alpha ,s}(f,g)\left(x\right)+\left|h\right|{\mathcal{M}}_{\alpha ,b}^1(f,g)\left(x\right).\hfill \end{array}$$

Note that for $0<\alpha <2n$, $s>1$,

$$\begin{array}{cc}\hfill & {∥{\mathcal{M}}_{\alpha ,s}(f,g)∥}_{L^{q,\kappa }\left(v_{\overrightarrow{w}}^q\right)}={∥{\mathcal{M}}_{\alpha s}{\left(\right|f|}^s{,\left|g\right|}^s)∥}_{L^{\frac{q}{s},\kappa }\left(v_{\overrightarrow{w}}^q\right)}^{\frac{1}{s}}\hfill \\ \hfill \lesssim & {\parallel \left|f\right|}^s{\parallel }_{L^{\frac{p_1}{s},\frac{\kappa p_1}{q_1}}\left(w_1^{\frac{p_1}{s}},w_1^{q_1}\right)}^{\frac{1}{s}}{\parallel \left|g\right|}^s{\parallel }_{L^{\frac{p_2}{s},\frac{\kappa p_2}{q_2}}\left(w_2^{\frac{p_2}{s}},w_2^{q_2}\right)}^{\frac{1}{s}}\hfill \\ \hfill =& {\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}.\hfill \end{array}$$

(28)

Hence, by (3) and Theorem 4, for any $s>1$, we have

$$\begin{array}{cc}\hfill & {∥{\mathcal{M}}_{\alpha ,b}^1(f,g)(·+h)-{\mathcal{M}}_{\alpha ,b}^1(f,g)(·)∥}_{L^{q,\kappa }\left(v_{\overrightarrow{w}}^q\right)}\hfill \\ \hfill \lesssim & \left(\left|h\right|+{\left|h\right|}^{\frac{1}{s^{\prime }}}\right){\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\lesssim \left|h\right|+{\left|h\right|}^{\frac{1}{s^{\prime }}}.\hfill \end{array}$$

Hence, we obtain the following desired estimate

$$\underset{\left|h\right|\to 0}{lim}{∥{\mathcal{M}}_{\alpha ,b}^1(f,g)(·+h)-{\mathcal{M}}_{\alpha ,b}^1(f,g)(·)∥}_{L^{q,\kappa }\left(v_w^q\right)}=0.$$

Next, we will prove the compactness of ${\mathcal{M}}_{\alpha ,\overrightarrow{b}}$. Suppose that $\overrightarrow{b}\in \mathrm{CMO}\times \mathrm{CMO}$. For any $ϵ>0$, there exists ${\overrightarrow{b}}^{ϵ}=\left(b_1^{ϵ},b_2^{ϵ}\right)\in {\mathcal{C}}_c^{\infty }\times {\mathcal{C}}_c^{\infty }$ such that ${∥b_i-b_i^{ϵ}∥}_{\mathrm{BMO}}<ϵ,i=1,2$, then by Theorem 4, we obtain

$$\begin{array}{cc}\hfill & {∥{\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)-{\mathcal{M}}_{\alpha ,{\overrightarrow{b}}^{ϵ}}(f,g)∥}_{L^{q,\kappa }\left(v_{\overrightarrow{w}}^q\right)}\hfill \\ \hfill \le & {∥{\mathcal{M}}_{\alpha ,\left(b_1-b_1^{ϵ},b_2\right)}(f,g)∥}_{L^{q,\kappa }\left(v_w^q\right)}+{∥{\mathcal{M}}_{\alpha ,\left(b_1,b_2-b_2^{ϵ}\right)}(f,g)∥}_{L^{q,\kappa }\left(v_w^q\right)}\le Cϵ.\hfill \end{array}$$

Thus, to prove the compactness of ${\mathcal{M}}_{\alpha ,\overrightarrow{b}}$ on $L^{q,\kappa }\left(v_{\overrightarrow{w}}^q\right)$ for any $\overrightarrow{b}\in \mathrm{CMO}\times \mathrm{CMO}$, it suffices to show that ${\mathcal{M}}_{\alpha ,\overrightarrow{b}}$ is compact for any $\overrightarrow{b}\in {\mathcal{C}}_c^{\infty }\times {\mathcal{C}}_c^{\infty }$. For arbitrary subsets $F\subset L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)$ and $G\subset L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right),$ let

$$\mathcal{G}=\left\{{\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g):f\in F,g\in G\right\}.$$

Then, we shall prove that for any $\overrightarrow{b}\in {\mathcal{C}}_c^{\infty }\times {\mathcal{C}}_c^{\infty }$, $\mathcal{G}$ satisfies the conditions (1)–(3) of Lemma 8. By Theorem 4, condition (1) holds immediately.

Next we will prove that $\mathcal{G}$ satisfies condition (2) in Lemma 8. Assume that $b_i\in {\mathcal{C}}_c^{\infty }$ and $\mathrm{supp}{b}_i\subset B(0,R)$, $i=1,2$. For any $\left|x\right|>A\ge max\{2R,1\}$, using Hölder’s inequality and Lemma 5 yields

$$\begin{array}{cc}\hfill & \left|{\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)\left(x\right)\right|\hfill \\ \hfill \lesssim & \underset{r>0}{sup}\frac{1}{{\left|B(x,r)\right|}^{2-\frac{\alpha }{n}}}{\int }_{B(0,R)}{\int }_{B(0,R)}\left|b_1\left(y\right)∥b_2\left(z\right)∥f\left(y\right)\parallel g\left(z\right)\right|dydz\hfill \\ \hfill \lesssim & \underset{r>0}{sup}\frac{{∥b_1∥}_{L^{\infty }}{∥b_2∥}_{L^{\infty }}}{{\left|B(x,r)\right|}^{2-\frac{\alpha }{n}}}{\int }_{B(0,R)}{\int }_{B(0,R)}|f\left(y\right)\parallel g\left(z\right)|dydz\hfill \\ \hfill \lesssim & \frac{w_1^{q_1}{\left(B(0,R)\right)}^{\frac{\kappa }{q_1}}{w}_2^{q_2}{\left(B(0,R)\right)}^{\frac{\kappa }{q_2}}}{{\left|x\right|}^{2n-\alpha }}{\parallel f\parallel }_{L^{p_1,\frac{\kappa p_1}{q_1}}\left(w_1^{p_1},w_1^{q_1}\right)}{\parallel g\parallel }_{L^{p_2,\frac{\kappa p_2}{q_2}}\left(w_2^{p_2},w_2^{q_2}\right)}\hfill \\ \hfill & \times {\left({\int }_{B(0,R)}{w}_1^{-p_1^{\prime }}\left(y\right)dy\right)}^{\frac{1}{p_1^{\prime }}}{\left({\int }_{B(0,R)}{w}_2^{-p_2^{\prime }}\left(z\right)dz\right)}^{\frac{1}{p_2^{\prime }}}\hfill \\ \hfill \lesssim & {\left(\frac{R}{\left|x\right|}\right)}^{2n-\alpha }{v}_{\overrightarrow{w}}^q{\left(B(0,R)\right)}^{\frac{\kappa -1}{q}},\hfill \end{array}$$

where the second from last step is due to $r>\left|x\right|-R>\left|x\right|/2$ and we used the fact that $\overrightarrow{w}\in A_{(\overrightarrow{p},q)}$ and Lemma 3 in the last inequality. Using Lemma 2, $1\le q<\infty$ and $1<1+q\left(2-\frac{1}{p}\right)=q\left(2-\frac{\alpha }{n}\right)<\infty ,$ we can get $v_{\overrightarrow{w}}^q\in A_{q\left(2-\frac{\alpha }{n}\right)}$. Hence, we have for any fixed ball $\tilde{B}$,

$$\begin{array}{cc}\hfill & {\left(\frac{1}{v_{\overrightarrow{w}}^q{\left(\tilde{B}\right)}^{\kappa }}{\int }_{\tilde{B}}{\left|{\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)\left(x\right)\right|}^q{\chi }_{\left\{\right|x|>A\}}\left(x\right)v_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{\frac{1}{q}}\hfill \\ \hfill \lesssim & \sum _{j=0}^{\infty }{\left(\frac{1}{v_{\overrightarrow{w}}^q{\left(\tilde{B}\right)}^{\kappa }}{\int }_{\tilde{B}\cap \{2^{j-1}A<|x|\le 2^{j}A\}}{\left(\frac{R}{\left|x\right|}\right)}^{(2n-\alpha )q}{v}_{\overrightarrow{w}}^q{\left(B(0,R)\right)}^{\kappa -1}{v}_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{\frac{1}{q}}\hfill \\ \hfill \lesssim & \sum _{j=0}^{\infty }{v}_{\overrightarrow{w}}^q{\left(B(0,R)\right)}^{\frac{\kappa -1}{q}}{\left(\frac{R}{2^{j}A}\right)}^{2n-\alpha }{\left(\frac{v_{\overrightarrow{w}}^q(\tilde{B}\cap \left\{\right|x|\le 2^{j}A\})}{v_{\overrightarrow{w}}^q{\left(\tilde{B}\right)}^{\kappa }}\right)}^{\frac{1}{q}}\hfill \\ \hfill \lesssim & \sum _{j=0}^{\infty }{\left(\frac{R}{2^{j}A}\right)}^{2n-\alpha }{\left(\frac{v_{\overrightarrow{w}}^q(\tilde{B}\cap \left\{\right|x|\le 2^{j}A\})}{v_{\overrightarrow{w}}^q\left(B(0,R)\right)}\right)}^{\frac{1-\kappa }{q}}\hfill \\ \hfill \lesssim & \sum _{j=0}^{\infty }{\left(\frac{R}{2^{j}A}\right)}^{2n-\alpha }{\left(\frac{2^{j}A}{R}\right)}^{\frac{1-\kappa }{q}q(2n-\alpha )}\hfill \\ \hfill \lesssim & \sum _{j=0}^{\infty }{\left(\frac{R}{2^{j}A}\right)}^{(2n-\alpha )\kappa },\hfill \end{array}$$

where we used the doubling condition in Lemma 4 in the fourth inequality. Therefore, together with Lemma 9 yields that

$$\underset{A\to \infty }{lim}{\left(\frac{1}{v_{\overrightarrow{w}}^q{\left(B\right)}^{\kappa }}{\int }_B{\left|{\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)\left(x\right)\right|}^q{\chi }_{\left\{\right|x|>A\}}\left(x\right)v_{\overrightarrow{w}}^q\left(x\right)dx\right)}^{\frac{1}{q}}=0.$$

Thus, condition (2) of Lemma 8 holds for all $f\in F$ and $g\in G$.

Finally, we will prove that the set $\mathcal{G}$ satisfies condition (3) of Lemma 8, that is, to show that G is uniformly equicontinuous. It suffices to verify that for any $ϵ\in (0,1)$, $f\in F$ and $g\in G,$ the following estimate holds

$${∥{\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)(·+h)-{\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)(·)∥}_{L^q\left(v_w^{q,\kappa }\right)}\lesssim ϵ,$$

(29)

where $\left|h\right|$ is sufficiently small and depends only on $ϵ$. Consider two fixed points $x,h\in {\mathbb{R}}^n$ satisfying $\left|h\right|<1$, without loss of generality, we assume

$${\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)(x+h)\le {\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)\left(x\right)\mathrm{and}{\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)\left(x\right)<\infty .$$

Therefore, let $ϵ\in (0,1)$, there exists a ball $B_1=B\left(x_0,r\right)\ni x$ such that

$$\begin{array}{cc}\hfill & (1-ϵ){\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)\left(x\right)\hfill \\ \hfill \lesssim & \frac{1}{{\left|B_1\right|}^{2-\frac{\alpha }{n}}}{\int }_{B_1}{\int }_{B_1}\left|b_1\left(x\right)-b_1\left(y\right)∥b_2\left(x\right)-b_2\left(z\right)∥f\left(y\right)\parallel \mathrm{g}\left(z\right)\right|dydz.\hfill \end{array}$$

(30)

Since $x+h\in B\left(x_0,r+\left|h\right|\right)=:B_2$, we infer that

$$\begin{array}{cc}\hfill & {\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)(x+h)\hfill \\ \hfill \ge & \frac{1}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}}{\int }_{B_2}{\int }_{B_2}\left|b_1(x+h)-b_1\left(y\right)∥b_2(x+h)-b_2\left(z\right)∥f\left(y\right)\parallel g\left(z\right)\right|dydz.\hfill \end{array}$$

(31)

Applying the above inequalities (30) and (31), we have

$$\begin{array}{cc}\hfill & (1-ϵ){\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)\left(x\right)-{\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)(x+h)\hfill \\ \hfill \lesssim & \frac{1}{{\left|B_1\right|}^{2-\frac{\alpha }{n}}}{\int }_{B_1}{\int }_{B_1}\left|b_1\left(x\right)-b_1\left(y\right)∥b_2\left(x\right)-b_2\left(z\right)∥f\left(y\right)\parallel \right|g\left(z\right)\mid dydz\hfill \\ \hfill & -\frac{1}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}}{\int }_{B_2}{\int }_{B_2}\left|b_1(x+h)-b_1\left(y\right)\right|\left|b_2(x+h)-b_2\left(z\right)\parallel f\left(y\right)\parallel g\left(z\right)\right|dydz\hfill \\ \hfill \lesssim & \frac{1}{{\left|B_1\right|}^{2-\frac{\alpha }{n}}}{\int }_{B_1}{\int }_{B_1}\left|b_1(x+h)-b_1\left(x\right)\right|\left|b_2(x+h)-b_2\left(x\right)\right||f\left(y\right)\parallel g\left(z\right)|dydz\hfill \\ \hfill & +\frac{1}{{\left|B_1\right|}^{2-\frac{\alpha }{n}}}{\int }_{B_1}{\int }_{B_1}\left|b_1(x+h)-b_1\left(x\right)\right|\left|b_2(x+h)-b_2\left(z\right)\parallel f\left(y\right)\parallel g\left(z\right)\right|dydz\hfill \\ \hfill & +\frac{1}{{\left|B_1\right|}^{2-\frac{\alpha }{n}}}{\int }_{B_1}{\int }_{B_1}\left|b_2(x+h)-b_2\left(x\right)\right|\left|b_1(x+h)-b_1\left(y\right)\parallel f\left(y\right)\parallel g\left(z\right)\right|dydz\hfill \\ \hfill & +\frac{1}{{\left|B_1\right|}^{2-\frac{\alpha }{n}}}{\int }_{B_1}{\int }_{B_2}\left|b_1(x+h)-b_1\left(y\right)∥b_2(x+h)-b_2\left(z\right)∥f\left(y\right)\parallel g\left(z\right)\right|dydz\hfill \\ \hfill & -\frac{1}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}}{\int }_{B_2}{\int }_{B_2}\left|b_1(x+h)-b_1\left(y\right)\right|\left|b_2(x+h)-b_2\left(z\right)\parallel f\left(y\right)\parallel g\left(z\right)\right|dydz.\hfill \end{array}$$

Consider the fact $\left|b_i(x+h)-b_i\left(x\right)\right|\lesssim \left|h\right|\parallel \nabla b_i{\parallel }_{L^{\infty }}$ for $i=1,2$, we have

$$\begin{array}{cc}\hfill & (1-ϵ){\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)\left(x\right)-{\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)(x+h)\hfill \\ \hfill \lesssim & {\left|h\right|}^2{∥\nabla b_1∥}_{L^{\infty }}{∥\nabla b_2∥}_{L^{\infty }}{\mathcal{M}}_{\alpha }(f,g)\left(x\right)\hfill \\ \hfill & +\left|h\right|{∥\nabla b_1∥}_{L^{\infty }}{∥b_2∥}_{L^{\infty }}{\mathcal{M}}_{\alpha }(f,g)\left(x\right)+\left|h\right|{∥b_1∥}_{L^{\infty }}{∥\nabla b_2∥}_{L^{\infty }}{\mathcal{M}}_{\alpha }(f,g)\left(x\right)\hfill \\ \hfill & +{\int }_{B_2}{\int }_{B_2}\left|\frac{{\chi }_{B_1}\left(y\right){\chi }_{B_1}\left(z\right)}{{\left|B_1\right|}^{2-\frac{\alpha }{n}}}-\frac{{\chi }_{B_2}\left(y\right){\chi }_{B_2}\left(z\right)}{{\left|B_2\right|}^{2-\frac{\alpha }{n}}}\right|\left|b_1(x+h)-b_2\left(y\right)\right|\hfill \\ \hfill & \times \left|b_2(x+h)-b_2\left(z\right)\right||f\left(y\right)\parallel g\left(z\right)|dydz.\hfill \end{array}$$

Similar to the proof of Lemma 7. Let $ϵ=\left|h\right|$, then for any $s>1$, we obtain

$$\begin{array}{cc}\hfill & \left|{\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)(x+h)-{\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)\left(x\right)\right|\hfill \\ \hfill \lesssim & \left|h\right|{\mathcal{M}}_{\alpha }(f,g)\left(x\right)+{\left|h\right|}^{\frac{1}{s^{\prime }}}{\mathcal{M}}_{\alpha ,s}(f,g)\left(x\right)+\left|h\right|{\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)\left(x\right).\hfill \end{array}$$

Thus, applying Lemma 2, Theorem 4 and (28), for any $s>1$, we get

$${∥{\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)(·+h)-{\mathcal{M}}_{\alpha ,\overrightarrow{b}}(f,g)(·)∥}_{L^{q,\kappa }\left(v_w^q\right)}\lesssim \left|h\right|+{\left|h\right|}^{\frac{1}{s^{\prime }}}.$$

Thus, we conclude that (29) holds for whenever $f\in F$ and $g\in G$. This completes the proof of Theorem 5. □