Multilinear fractional integral operators on generalized weighted Morrey spaces

Abstract

Let $I_{\alpha ,m}$ be multilinear fractional integral operator and let $(b_1,\dots ,b_m)\in {(BMO)}^m$. In this paper, the estimates of $I_{\alpha ,m}$, the m-linear commutators $I_{\alpha ,m}^{\mathrm{\Sigma }b}$ and the iterated commutators $I_{\alpha ,m}^{\mathrm{\Pi }b}$ on the generalized weighted Morrey spaces are established.

MSC:42B35, 42B20.

1 Introduction and results

The classical Morrey spaces were introduced by Morrey [1] in 1938, have been studied intensively by various authors, and together with weighted Lebesgue spaces play an important role in the theory of partial differential equations; they appeared to be quite useful in the study of local behavior of the solutions of elliptic differential equations and describe local regularity more precisely than Lebesgue spaces. See [2–4] for details. Moreover, various Morrey spaces have been defined in the process of this study. Mizuhara [5] introduced the generalized Morrey space $M_{\phi }^p$; Komori and Shirai [6] defined the weighted Morrey spaces $L^{p,\kappa }(\omega )$; Guliyev [7] gave the concept of generalized weighted Morrey space $M_{\phi }^p(\omega )$, which could be viewed as an extension of both $M_{\phi }^p$ and $L^{p,\kappa }(\omega )$. The boundedness of some operators on these Morrey spaces can be seen in [5–9].

Let ${\mathbb{R}}^n$ be the n-dimensional Euclidean space, ${({\mathbb{R}}^n)}^m={\mathbb{R}}^n\times \cdots \times {\mathbb{R}}^n$ be the m-fold product space ($m\in \mathbb{N}$), and let $\overrightarrow{f}=(f_1,\dots ,f_m)$ be a collection of m functions on ${\mathbb{R}}^n$. Given $\alpha \in (0,mn)$ and $(b_1,\dots ,b_m)\in {(BMO)}^m$. We consider the multilinear fractional integral operators $I_{\alpha ,m}$ defined by

$$I_{\alpha ,m}(\overrightarrow{f})(x)={\int }_{{({\mathbb{R}}^n)}^m}\frac{f_1(y_1)\cdots f_m(y_m)}{{(|x-y_1|+\cdots +|x-y_m|)}^{mn-\alpha }}d{y}_1\cdots d{y}_m.$$

(1.1)

The corresponding m-linear commutators $I_{\alpha ,m}^{\mathrm{\Sigma }b}$ and the iterated commutators $I_{\alpha ,m}^{\mathrm{\Pi }b}$ defined by, respectively,

$$I_{\alpha ,m}^{\mathrm{\Sigma }b}(\overrightarrow{f})(x)=\sum _{i=1}^m{\int }_{{({\mathbb{R}}^n)}^m}\frac{(b_i(x)-b_i(y_i)){\prod }_{j=1}^m{f}_j(y_j)}{{(|x-y_1|+\cdots +|x-y_m|)}^{mn-\alpha }}d{y}_1\cdots d{y}_m$$

(1.2)

and

$$I_{\alpha ,m}^{\mathrm{\Pi }b}(\overrightarrow{f})(x)={\int }_{{({\mathbb{R}}^n)}^m}\frac{{\prod }_{i=1}^m(b_i(x)-b_i(y_i))f_i(y_i)}{{(|x-y_1|+\cdots +|x-y_m|)}^{mn-\alpha }}d{y}_1\cdots d{y}_m.$$

(1.3)

As is well known, multilinear fractional integral operator was first studied by Grafakos [10], subsequently, by Kenig and Stein [11], Grafakos and Kalton [12]. In 2009, Moen [13] introduced weight function $A_{\overrightarrow{P},q}$ and gave weighted inequalities for multilinear fractional integral operators; In 2013, Chen and Wu [14] obtained the weighted norm inequalities for the multilinear commutators $I_{\alpha ,m}^{\mathrm{\Sigma }b}$ and $I_{\alpha ,m}^{\mathrm{\Pi }b}$. More results of the weighted inequalities for multilinear fractional integral and its commutators can be found in [15–17].

The aim of the present paper is to investigate the boundedness of multilinear fractional integral operator and its commutator on the generalized weighted Morrey spaces. Our results can be formulated as follows.

Theorem 1.1 Let $m\ge 2$ and let $0<\alpha <mn$. Suppose $1/p={\sum }_{i=1}^{m}1/p_i$, $1/q_i=1/p_i-\alpha /mn$, and $1/q={\sum }_{i=1}^{m}1/q_i=1/p-\alpha /n$, $\overrightarrow{\omega }=({\omega }_1,\dots ,{\omega }_m)$ satisfy the $A_{\overrightarrow{p},q}$ condition with ${\omega }_1^{q_1},\dots ,{\omega }_m^{q_m}\in A_{\mathrm{\infty }}$, and $\overrightarrow{{\phi }_k}=({\phi }_{k1},\dots ,{\phi }_{km})$, $k=1,2$, satisfy the condition

$${\int }_s^{\mathrm{\infty }}\frac{{ess\hspace{0.17em}inf}_{r<t<\mathrm{\infty }}{\prod }_{i=1}^m{\phi }_{1i}(x,t){({\omega }_i^{p_i}(B(x,t)))}^{\frac{1}{p_i}}}{{\prod }_{i=1}^m{({\omega }_i^{p_i}(B(x,r)))}^{\frac{1}{p_i}}}\frac{dr}{r^{1-\alpha }}\le C{\phi }_2(x,s),$$

(1.4)

where ${\phi }_2={\prod }_{i=1}^m{\phi }_{2i}$, ${\nu }_{\overrightarrow{\omega }}={\prod }_{i=1}^m{\omega }_i$. If $p_1,\dots ,p_m\in (1,\mathrm{\infty })$, then there exists a constant C independent of $\overrightarrow{f}$ such that

$${\parallel I_{\alpha ,m}\overrightarrow{f}\parallel }_{M_{{\phi }_2}^q({\nu }_{\overrightarrow{\omega }}^q)}\le C\prod _{i=1}^m{\parallel f_i\parallel }_{M_{{\phi }_{1i}}^{p_i}({\omega }_i^{p_i})};$$

(1.5)

If $p_1,\dots ,p_m\in [1,\mathrm{\infty })$, and $min\{p_1,\dots ,p_m\}=1$, then there exists a constant C independent of $\overrightarrow{f}$ such that

$${\parallel I_{\alpha ,m}\overrightarrow{f}\parallel }_{W{M}_{{\phi }_2}^q({\nu }_{\overrightarrow{\omega }}^q)}\le C\prod _{i=1}^m{\parallel f_i\parallel }_{M_{{\phi }_{1i}}^{p_i}({\omega }_i^{p_i})}.$$

(1.6)

Theorem 1.2 Let $m\ge 2$ and let $0<\alpha <mn$. Suppose $p_1,\dots ,p_m\in (1,\mathrm{\infty })$ with $1/p={\sum }_{i=1}^{m}1/p_i$, $1/q_i=1/p_i-\alpha /mn$ and $1/q={\sum }_{i=1}^{m}1/q_i=1/p-\alpha /n$, $\overrightarrow{\omega }=({\omega }_1,\dots ,{\omega }_m)$ satisfy the $A_{\overrightarrow{p,}q}$ condition with ${\omega }_1^{p_1},\dots ,{\omega }_m^{p_m}\in A_{\mathrm{\infty }}$, ${\nu }_{\overrightarrow{\omega }}={\prod }_{i=1}^m{\omega }_i$, and $\overrightarrow{{\phi }_k}=({\phi }_{k1},\dots ,{\phi }_{km})$, $k=1,2$, satisfy the condition

$${\int }_s^{\mathrm{\infty }}{(1+ln\frac{r}{s})}^m\frac{{ess\hspace{0.17em}inf}_{r<t<\mathrm{\infty }}{\prod }_{i=1}^m{\phi }_{1i}(x,t){({\omega }_i^{p_i}(B(x,t)))}^{\frac{1}{p_i}}}{{\prod }_{i=1}^m{({\omega }_i^{p_i}(B(x,r)))}^{\frac{1}{p_i}}}\frac{dr}{r^{1-\alpha }}\le C{\phi }_2(x,s),$$

(1.7)

where ${\phi }_2={\prod }_{i=1}^m{\phi }_{2i}$, ${\nu }_{\overrightarrow{\omega }}={\prod }_{i=1}^m{\omega }_i$. If $(b_1,\dots ,b_m)\in {(BMO)}^m$, then there exists a constant $C>0$ independent of $\overrightarrow{f}$ such that

$${\parallel I_{\alpha ,m}^{\mathrm{\Sigma }b}(\overrightarrow{f})\parallel }_{M_{{\phi }_2}^q({\nu }_{\overrightarrow{\omega }}^q)}\le C\prod _{i=1}^m{\parallel b_i\parallel }_{\ast }{\parallel f_i\parallel }_{M_{{\phi }_{1i}}^{p_i}({\omega }_i^{p_i})};$$

(1.8)

and

$${\parallel I_{\alpha ,m}^{\mathrm{\Pi }b}(\overrightarrow{f})\parallel }_{M_{{\phi }_2}^q({\nu }_{\overrightarrow{\omega }}^q)}\le C\prod _{i=1}^m{\parallel b_i\parallel }_{\ast }{\parallel f_i\parallel }_{M_{{\phi }_{1i}}^{p_i}({\omega }_i^{p_i})}.$$

(1.9)

2 Definitions and preliminaries

A weight ω is a nonnegative, locally integrable function on ${\mathbb{R}}^n$. Let $B=B(x_0,r_B)$ denote the ball with the center $x_0$ and radius $r_B$. For any ball B and $\lambda >0$, λB denotes the ball concentric with B whose radius is λ times as long. For a given weight function ω and a measurable set E, we also denote the Lebesgue measure of E by $|E|$ and set weighted measure $\omega (E)={\int }_E\omega (x)dx$.

The classical $A_p$ weight theory was first introduced by Muckenhoupt in the study of weighted $L^p$ boundedness of Hardy-Littlewood maximal functions in [18]. A weight ω is said to belong to $A_p$ for $1<p<\mathrm{\infty }$, if there exists a constant C such that for every ball $B\subset {\mathbb{R}}^n$,

$$(\frac{1}{|B|}{\int }_B\omega (x)dx){\left(\frac{1}{|B|}{\int }_B\omega {(x)}^{1-p^{\prime }}dx\right)}^{p-1}\le C,$$

(2.1)

where $p^{\prime }$ is the dual of p such that $1/p+1/p^{\prime }=1$. The class $A_1$ is defined by replacing the above inequality with

$$\frac{1}{|B|}{\int }_{B}w(y)dy\le C\cdot \underset{x\in B}{ess\hspace{0.17em}inf}w(x)\text{for every ball }B\subset {\mathbb{R}}^n.$$

(2.2)

A weight ω is said to belong to $A_{\mathrm{\infty }}$ if there are positive numbers C and δ so that

$$\frac{\omega (E)}{\omega (B)}\le C{\left(\frac{|E|}{|B|}\right)}^{\delta }$$

(2.3)

for all balls B and all measurable $E\subset B$. It is well known that

$$A_{\mathrm{\infty }}=\bigcup _{1\le p<\mathrm{\infty }}{A}_p.$$

(2.4)

We need another weight class $A_{p,q}$ introduced by Muckenhoupt and Wheeden in [19]. A weight function ω belongs to $A_{p,q}$ for $1<p<q<\mathrm{\infty }$ if there is a constant $C>0$ such that, for every ball $B\subset {\mathbb{R}}^n$,

$${\left(\frac{1}{|B|}{\int }_B\omega {(x)}^{q}dx\right)}^{1/q}{\left(\frac{1}{|B|}{\int }_B\omega {(x)}^{-p^{\prime }}dx\right)}^{p^{\prime }}\le C.$$

(2.5)

When $p=1$, ω is in the class $A_{1,q}$ with $1<q<\mathrm{\infty }$ if there is a constant $C>0$ such that, for every ball $B\subset {\mathbb{R}}^n$,

$${\left(\frac{1}{|B|}{\int }_B\omega {(x)}^{q}dx\right)}^{1/q}\left(\underset{x\in B}{ess\hspace{0.17em}sup}\frac{1}{\omega (x)}\right)\le C.$$

(2.6)

Let us recall the definition of multiple weights. For m exponents $p_1,\dots ,p_m$, we write $\overrightarrow{p}=(p_1,\dots ,p_m)$. Let $p_1,\dots ,p_m\in [1,\mathrm{\infty })$, $1/p={\sum }_{i=1}^{m}1/p_i$, and let $q>0$. Given $\overrightarrow{\omega }=({\omega }_1,\dots ,{\omega }_m)$, set ${\nu }_{\overrightarrow{\omega }}={\prod }_{i=1}^m{\omega }_i$. We say that $\overrightarrow{\omega }$ satisfies the $A_{\overrightarrow{p},q}$ condition if it satisfies

$$\underset{B}{sup}{\left(\frac{1}{|B|}{\int }_B{\nu }_{\overrightarrow{\omega }}{(x)}^{q}dx\right)}^{1/q}\prod _{i=1}^m{\left(\frac{1}{|B|}{\int }_B{\omega }_i{(x)}^{-p_i^{\prime }}dx\right)}^{1/p_i^{\prime }}\le C.$$

(2.7)

When $p_i=1$, ${(\frac{1}{|B|}{\int }_B{\omega }_i{(x)}^{-p_i^{\prime }}(x)dx)}^{1/p_i^{\prime }}$ is understood as ${({inf}_{x\in B}{\omega }_i(x))}^{-1}$.

Lemma 2.1 [13, 14]

Let $0<\alpha <mn$, and $p_1,\dots ,p_m\in [1,\mathrm{\infty })$, let $1/p={\sum }_{k=1}^{m}1/p_k$, and let $1/q=1/p-\alpha /n$. If $\overrightarrow{\omega }\in A_{\overrightarrow{p},q}$, then

$${\nu }_{\overrightarrow{\omega }}^q\in A_{mq}\mathit{\text{and}}{\omega }_i^{-p_i^{\prime }}\in A_{m{p}_i^{\prime }}\mathit{\text{for }}i=1,\dots ,m,$$

(2.8)

where ${\nu }_{\overrightarrow{\omega }}={\prod }_{i=1}^m{\omega }_i$.

Lemma 2.2 [20]

Let $m\ge 2$, $q_1,\dots ,q_m\in [1,\mathrm{\infty })$ and $q\in (0,\mathrm{\infty })$ with $1/q={\sum }_{i=1}^{m}1/q_i$. Assume that ${\omega }_1^{q_1},\dots ,{\omega }_m^{q_m}\in A_{\mathrm{\infty }}$ and ${\nu }_{\overrightarrow{\omega }}={\prod }_{i=1}^m{\omega }_i$. Then for any ball B, there exists a constant $C>0$ such that

$$\prod _{i=1}^m{\left({\int }_B{\omega }_i{(x)}^{q_i}dx\right)}^{q/q_i}\le C{\int }_B{\nu }_{\overrightarrow{\omega }}{(x)}^{q}dx.$$

(2.9)

Let $1\le p<\mathrm{\infty }$, let φ be a positive measurable function on ${\mathbb{R}}^n\times (0,\mathrm{\infty })$, and let ω be a nonnegative measurable function on ${\mathbb{R}}^n$. Following [7], we denote by $M_{\phi }^p(\omega )$ the generalized weighted Morrey space and the space of all functions $f\in L_{\mathrm{loc}}^p(\omega )$ with finite norm

$${\parallel f\parallel }_{M_{\phi }^p(w)}=\underset{x\in {\mathbb{R}}^n,r>0}{sup}\frac{1}{\phi (x,r)}{\left(\frac{1}{w(B(x,r))}{\parallel f\parallel }_{L^p(\omega ,B(x,r))}^p\right)}^{1/p},$$

(2.10)

where

$${\parallel f\parallel }_{L^p(\omega ,B(x,r))}={\int }_{B(x,r)}{|f(y)|}^{p}w(y)dy.$$

Furthermore, by $W{M}_{\phi }^p(\omega )$ we denote the weak generalized weighted Morrey space of all function $f\in W{M}_{\phi }^p(\omega )$ for which

$${\parallel f\parallel }_{W{M}_{\phi }^p(w)}=\underset{x\in {\mathbb{R}}^n,r>0}{sup}\frac{1}{\phi (x,r)}{\left(\frac{1}{w(B(x,r))}{\parallel f\parallel }_{W{L}^p(\omega ,B(x,r))}^p\right)}^{1/p},$$

(2.11)

where

$${\parallel f\parallel }_{W{L}^p(\omega ,B(x,r))}=\underset{t>0}{sup}t{\left(\omega \left(\{y\in B(x,r):|f(y)|>t\}\right)\right)}^{\frac{1}{p}}.$$

• (1) If $\omega =1$ and $\phi (x,r)=r^{\frac{\lambda -n}{p}}$ with $0<\lambda <n$, then $M_{\phi }^p(\omega )=L^{p,\lambda }$ is the classical Morrey space.

• (2) If $\phi (x,r)=\omega {(B(x,r))}^{\frac{\kappa -1}{p}}$, then $M_{\phi }^p(\omega )=L^{p,\kappa }(\omega )$ is the weighted Morrey space.

• (3) If $\phi (x,r)=\nu {(B(x,r))}^{\frac{\kappa }{p}}\omega {(B(x,r))}^{-\frac{1}{p}}$, then $M_{\phi }^p(\omega )=L^{p,\kappa }(\nu ,\omega )$ is the two weighted Morrey space.

• (4) If $\omega =1$, then $M_{\phi }^p(\omega )=M_{\phi }^p$ is the generalized Morrey space.

• (5) If $\phi (x,r)=\omega {(B(x,r))}^{-\frac{1}{p}}$, then $M_{\phi }^p(\omega )=L^p(\omega )$.

Let us recall the definition and some properties of $BMO$. A locally integrable function b is said to be in $BMO$ if

$$\underset{B\subset {\mathbb{R}}^n}{sup}\frac{1}{|B|}{\int }_B|b(x)-b_B|dx={\parallel b\parallel }_{\ast }<\mathrm{\infty },$$

where $b_B={|B|}^{-1}{\int }_{B}b(y)dy$.

Lemma 2.3 (John-Nirenberg inequality; see [21])

Let $b\in BMO$. Then for any ball $B\subset {\mathbb{R}}^n$, there exist positive constants $C_1$ and $C_2$ such that for all $\lambda >0$,

$$\left|\{x\in B:|b(x)-b_B|>\lambda \}\right|\le C_1|B|exp(-C_2\lambda /{\parallel b\parallel }_{\ast }).$$

(2.12)

By Lemma 2.3, it is easy to get the following.

Lemma 2.4 Suppose $\omega \in A_{\mathrm{\infty }}$ and $b\in BMO$. Then for any $p\ge 1$ we have

$${(\frac{1}{\omega (B)}{\int }_B{|b(x)-b_B|}^p\omega (x)dx)}^{1/p}\le C{\parallel b\parallel }_{\ast }.$$

(2.13)

Lemma 2.5 [22]

Let $b\in BMO$, $1\le p<\mathrm{\infty }$, and $r_1,r_2>0$. Then

$${\left(\frac{1}{|B(x_0,r_1)|}{\int }_{B(x_0,r_1)}{|b(y)-b_{B(x_0,r_2)}|}^{p}dy\right)}^{\frac{1}{p}}\le C{\parallel b\parallel }_{\ast }(1+|ln\frac{r_1}{r_2}\left|\right),$$

(2.14)

where $C>0$ is independent of f, $x_0$, $r_1$, and $r_2$.

By Lemma 2.4 and Lemma 2.5, it is easily to prove the following results.

Lemma 2.6 Suppose $\omega \in A_{\mathrm{\infty }}$ and $b\in BMO$. Then for any $1\le p<\mathrm{\infty }$ and $r_1,r_2>0$, we have

$${(\frac{1}{\omega (B(x_0,r_1))}{\int }_{B(x_0,r_1)}{|b(x)-b_{B(x_0,r_2)}|}^p\omega (x)dx)}^{1/p}\le C{\parallel b\parallel }_{\ast }(1+|ln\frac{r_1}{r_2}\left|\right).$$

(2.15)

We also need the following result.

Lemma 2.7 [23]

Let f be a real-valued nonnegative function and measurable on E. Then

$${(\underset{x\in E}{ess\hspace{0.17em}inf}f(x))}^{-1}=\underset{x\in E}{ess\hspace{0.17em}sup}\frac{1}{f(x)}.$$

(2.16)

At the end of this section, we list some known results about weighted norm inequalities for the multilinear fractional integrals and their commutators.

Lemma 2.8 [13]

Let $m\ge 2$ and let $0<\alpha <mn$. Suppose $1/p=1/p_1+\cdots +1/p_m$, $1/q=1/p-\alpha /n$, $\overrightarrow{\omega }=({\omega }_1,\dots ,{\omega }_m)$ satisfies the $A_{\overrightarrow{p},q}$ condition. If $p_1,\dots ,p_m\in (1,\mathrm{\infty })$, then there exists a constant C independent of $\overrightarrow{f}=(f_1,\dots ,f_m)$ such that

$${\parallel I_{\alpha ,m}\overrightarrow{f}\parallel }_{L^q({\nu }_{\overrightarrow{\omega }}^q)}\le C\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i})}.$$

(2.17)

If $p_1,\dots ,p_m\in [1,\mathrm{\infty })$, and $min\{p_1,\dots ,p_m\}=1$, then there exists a constant C independent of $\overrightarrow{f}$ such that

$${\parallel I_{\alpha ,m}\overrightarrow{f}\parallel }_{W{L}^q({\nu }_{\overrightarrow{\omega }}^q)}\le C\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i})},$$

(2.18)

where ${\nu }_{\overrightarrow{\omega }}={\prod }_{i=1}^m{\omega }_i$.

Lemma 2.9 [14]

Let $m\ge 2$, let $0<\alpha <mn$ and let $(b_1,\dots ,b_m)\in {(BMO)}^m$. For $1<p_1,\dots ,p_m<\mathrm{\infty }$, $1/p=1/p_1+\cdots +1/p_m$, and $1/q=1/p-\alpha /n$, if $\overrightarrow{\omega }\in A_{\overrightarrow{p},q}$, then there exists a constant $C>0$ such that

$${\parallel I_{\alpha ,m}^{\mathrm{\Sigma }b}(\overrightarrow{f})\parallel }_{L^q({\nu }_{\overrightarrow{\omega }}^q)}\le C\prod _{i=1}^m{\parallel b_i\parallel }_{\ast }{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i})};$$

(2.19)

and

$${\parallel I_{\alpha ,m}^{\mathrm{\Pi }b}(\overrightarrow{f})\parallel }_{L^q({\nu }_{\overrightarrow{\omega }}^q)}\le C\prod _{i=1}^m{\parallel b_i\parallel }_{\ast }{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i})},$$

(2.20)

where ${\nu }_{\overrightarrow{\omega }}={\prod }_{i=1}^m{\omega }_i$.

3 Proof of Theorem 1.1

We first prove the following conclusions.

Theorem 3.1 Let $m\ge 2$ and let $0<\alpha <mn$. Suppose $1/p={\sum }_{i=1}^{m}1/p_i$, $1/q_i=1/p_i-\alpha /mn$, and $1/q={\sum }_{i=1}^{m}1/q_i=1/p-\alpha /n$, $\overrightarrow{\omega }=({\omega }_1,\dots ,{\omega }_m)$ satisfy the $A_{\overrightarrow{P},q}$ condition with ${\omega }_1^{q_1},\dots ,{\omega }_m^{q_m}\in A_{\mathrm{\infty }}$. If $p_1,\dots ,p_m\in (1,\mathrm{\infty })$, then there exists a constant C independent of $\overrightarrow{f}$ such that

$$\begin{array}{rl}{\parallel I_{\alpha ,m}\overrightarrow{f}\parallel }_{L^q({\nu }_{\overrightarrow{\omega }}^q,B(x_0,s))}\le & C\prod _{i=1}^m{\left({\omega }_i^{q_i}(B(x_0,s))\right)}^{\frac{1}{q_i}}\\ \times {\int }_{2s}^{\mathrm{\infty }}\left(\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}.\end{array}$$

(3.1)

If $p_1,\dots ,p_m\in [1,\mathrm{\infty })$, and $min\{p_1,\dots ,p_m\}=1$, then there exists a constant C independent of $\overrightarrow{f}$ such that

$$\begin{array}{rl}{\parallel I_{\alpha ,m}\overrightarrow{f}\parallel }_{W{L}^q({\nu }_{\overrightarrow{\omega }}^q,B(x_0,s))}\le & C\prod _{i=1}^m{\left({\omega }_i^{q_i}(B(x_0,s))\right)}^{\frac{1}{q_i}}\\ \times {\int }_{2s}^{\mathrm{\infty }}\left(\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }},\end{array}$$

(3.2)

where ${\nu }_{\overrightarrow{\omega }}={\prod }_{i=1}^m{\omega }_i$.

Proof We represent $f_i$ as $f_i=f_i^0+f_i^{\mathrm{\infty }}$, where $f_i^0=f_i{\chi }_{B(x_0,2s)}$, $i=1,\dots ,m$, and ${\chi }_{B(x_0,2s)}$ denotes the characteristic function of $B(x_0,2s)$. Then

$$\begin{array}{rl}\prod _{i=1}^m{f}_i(y_i)& =\prod _{i=1}^m(f_i^0(y_i)+f_i^{\mathrm{\infty }}(y_i))\\ =\sum _{{\alpha }_1,\dots ,{\alpha }_m\in \{0,\mathrm{\infty }\}}{f}_1^{{\alpha }_1}(y_1)\cdots f_m^{{\alpha }_m}(y_m)\\ =\prod _{i=1}^m{f}_i^0(y_i)+{\mathrm{\Sigma }}^{\prime }{f}_1^{{\alpha }_1}(y_1)\cdots f_m^{{\alpha }_m}(y_m),\end{array}$$

where each term of ${\mathrm{\Sigma }}^{\prime }$ contains at least one ${\alpha }_i\ne 0$. Since $I_{\alpha ,m}$ is an m-linear operator,

$$\begin{array}{rl}{\parallel I_{\alpha ,m}\overrightarrow{f}\parallel }_{L^q({\nu }_{\overrightarrow{\omega }}^q,B(x_0,s))}\le & C{\parallel I_{\alpha ,m}(f_1^0,\dots ,f_m^0)\parallel }_{L^q({\nu }_{\overrightarrow{\omega }}^q,B(x_0,s))}\\ +C{\mathrm{\Sigma }}^{\prime }{\parallel I_{\alpha ,m}(f_1^{{\alpha }_1},\dots ,f_m^{{\alpha }_m})\parallel }_{L^q({\nu }_{\overrightarrow{\omega }}^q,B(x_0,s))}\\ =& J^{0,\dots ,0}+{\mathrm{\Sigma }}^{\prime }{J}^{{\alpha }_1,\dots ,{\alpha }_m}\end{array}$$

(3.3)

and

$$\begin{array}{rl}{\parallel I_{\alpha ,m}\overrightarrow{f}\parallel }_{W{L}^q({\nu }_{\overrightarrow{\omega }}^q,B(x_0,s))}\le & C{\parallel I_{\alpha ,m}(f_1^0,\dots ,f_m^0)\parallel }_{W{L}^q({\nu }_{\overrightarrow{\omega }}^q,B(x_0,s))}\\ +C{\mathrm{\Sigma }}^{\prime }{\parallel I_{\alpha ,m}(f_1^{{\alpha }_1},\dots ,f_m^{{\alpha }_m})\parallel }_{W{L}^q({\nu }_{\overrightarrow{\omega }}^q,B(x_0,s))}\\ =& K^{0,\dots ,0}+{\mathrm{\Sigma }}^{\prime }{K}^{{\alpha }_1,\dots ,{\alpha }_m}.\end{array}$$

(3.4)

Then by (2.17), if $1<p_i<\mathrm{\infty }$, $i=1,\dots ,m$, we get

$$J^{0,\dots ,0}\le C\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,2s))}.$$

(3.5)

By (2.18), if $min\{p_1,\dots ,p_m\}=1$, then

$$K^{0,\dots ,0}\le C\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,2s))}.$$

(3.6)

Applying Hölder’s inequality, for $1\le p_i\le q_i<\mathrm{\infty }$, $i=1,\dots ,m$, we have

$$\begin{array}{rl}1& \le {\left(\frac{1}{|B|}{\int }_B{\omega }_i{(y_i)}^{p_i}d{y}_i\right)}^{\frac{1}{p_i}}{\left(\frac{1}{|B|}{\int }_B{\omega }_i{(y_i)}^{-p_i^{\prime }}d{y}_i\right)}^{\frac{1}{p_i^{\prime }}}\\ \le {\left(\frac{1}{|B|}{\int }_B{\omega }_i{(y_i)}^{q_i}d{y}_i\right)}^{\frac{1}{q_i}}{\left(\frac{1}{|B|}{\int }_B{\omega }_i{(y_i)}^{-p_i^{\prime }}d{y}_i\right)}^{\frac{1}{p_i^{\prime }}}\end{array}$$

for any ball $B\subset {\mathbb{R}}^n$. Then

$${|B(x_0,2s)|}^{m-\frac{\alpha }{n}}\le \prod _{i=1}^m{\left({\omega }_i^{q_i}(B(x_0,2s))\right)}^{\frac{1}{q_i}}{\left({\omega }_i^{-p_i^{\prime }}(B(x_0,2s))\right)}^{\frac{1}{p_i^{\prime }}}.$$

Thus, for $1\le p_i<\mathrm{\infty }$,

$$\begin{array}{r}\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,2s))}\\ \le C\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,2s))}\cdot {|B(x_0,2s)|}^{m-\frac{\alpha }{n}}{\int }_{2s}^{\mathrm{\infty }}\frac{dr}{r^{mn-\alpha +1}}\\ \le C\prod _{i=1}^m{\left({\omega }_i^{q_i}(B(x_0,2s))\right)}^{\frac{1}{q_i}}{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,2s))}{\left({\omega }_i^{-p_i^{\prime }}(B(x_0,2s))\right)}^{\frac{1}{p_i^{\prime }}}{\int }_{2s}^{\mathrm{\infty }}\frac{dr}{r^{mn-\alpha +1}}\\ \le C\prod _{i=1}^m{\left({\omega }_i^{q_i}(B(x_0,2s))\right)}^{\frac{1}{q_i}}\\ \cdot {\int }_{2s}^{\mathrm{\infty }}\left(\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{-p_i^{\prime }}(B(x_0,r))\right)}^{\frac{1}{p_i^{\prime }}}\right)\frac{dr}{r^{mn-\alpha +1}}.\end{array}$$

From (2.7) and Lemma 2.2 we get

$$\begin{array}{rl}\prod _{i=1}^m{\left({\omega }_i^{-p_i^{\prime }}(B(x_0,r))\right)}^{\frac{1}{p_i^{\prime }}}& \le C{|B(x_0,r)|}^{\frac{1}{q}+{\sum }_{i=1}^m\frac{1}{p_i^{\prime }}}{\left({\int }_{B(x_0,r)}{\nu }_{\overrightarrow{\omega }}{(x)}^{q}dx\right)}^{-\frac{1}{q}}\\ \le C{|B(x_0,r)|}^{m-\frac{\alpha }{n}}\prod _{i=1}^m{\left({\omega }_i^{q_i}(B(x_0,r))\right)}^{-\frac{1}{q_i}}.\end{array}$$

(3.7)

Using Hölder’s inequality,

$${\left(\frac{1}{|B|}{\int }_B{\omega }_i{(y)}^{p_i}dy\right)}^{\frac{1}{p_i}}\le {\left(\frac{1}{|B|}{\int }_B{\omega }_i{(y)}^{q_i}dy\right)}^{\frac{1}{q_i}}.$$

Note that $1/q_i=1/p_i-\alpha /mn$, then

$${\left({\omega }_i^{q_i}(B(x_0,r))\right)}^{-\frac{1}{q_i}}\le C{r}^{\alpha /m}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}.$$

(3.8)

Then for $1\le p_i<\mathrm{\infty }$, $i=1,\dots ,m$,

$$\begin{array}{rl}\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,2s))}\le & \prod _{i=1}^m{\left({\omega }_i^{q_i}(B(x_0,2s))\right)}^{\frac{1}{q_i}}\\ \times {\int }_{2s}^{\mathrm{\infty }}\left(\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}.\end{array}$$

(3.9)

This gives $J^{0,\dots ,0}$ and $K^{0,\dots ,0}$ are majored by

$$\prod _{i=1}^m{\left({\omega }_i^{q_i}(B(x_0,2s))\right)}^{\frac{1}{q_i}}\cdot {\int }_{2s}^{\mathrm{\infty }}\left(\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}.$$

(3.10)

For the other term, let us first consider the case when ${\alpha }_1=\cdots ={\alpha }_m=\mathrm{\infty }$. For any $x\in B(x_0,s)$, $y\in B(x_0,2^{j+1}s)\setminus B(x_0,2^{j}s)$, we have $|x-y_i|\approx |x-y_j|$ for $i\ne j$. Then

$$\begin{array}{r}|I_{\alpha ,m}(f_1^{\mathrm{\infty }},\dots ,f_m^{\mathrm{\infty }})(x)|\\ \le C{\int }_{{({\mathbb{R}}^n\setminus B(x_0,2s))}^m}\frac{|f_1(y_1)\cdots f_m(y_m)|}{{(|x-y_1|+\cdots +|x-y_m|)}^{mn-\alpha }}d{y}_1\cdots d{y}_m\\ \le C\sum _{j=1}^{\mathrm{\infty }}{\int }_{{(B(x_0,2^{j+1}s)\setminus B(x_0,2^{j}s))}^m}\frac{|f_1(y_1)\cdots f_m(y_m)|}{{(|x-y_1|+\cdots +|x-y_m|)}^{mn-\alpha }}d{y}_1\cdots d{y}_m\\ \le C\sum _{j=1}^{\mathrm{\infty }}\prod _{i=1}^m{\int }_{B(x_0,2^{j+1}s)\setminus B(x_0,2^{j}s)}\frac{|f_i(y_i)|}{{|x-y_i|}^{n-\frac{\alpha }{m}}}d{y}_i\\ \le C\sum _{j=1}^{\mathrm{\infty }}\prod _{i=1}^m\left({\left(2^{j+1}s\right)}^{-n+\frac{\alpha }{m}}{\int }_{B(x_0,2^{j+1}s)}|f_i(y_i)|d{y}_i\right).\end{array}$$

Applying Hölder’s inequality, it can be found that ${sup}_{x\in B(x_0,s)}|I_{\alpha ,m}(f_1^{\mathrm{\infty }},\dots ,f_m^{\mathrm{\infty }})(x)|$ is less than

$$C\sum _{j=1}^{\mathrm{\infty }}\prod _{i=1}^m\left({\left(2^{j+1}s\right)}^{-n+\frac{\alpha }{m}}{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,2^{j+1}s))}{\left({\omega }_i^{-p_i}\left(B(x_0,2^{j+1}s)\right)\right)}^{\frac{1}{p_i}}\right).$$

Hence,

$$\begin{array}{r}\underset{x\in B(x_0,s)}{sup}|I_{\alpha ,m}(f_1^{\mathrm{\infty }},\dots ,f_m^{\mathrm{\infty }})(x)|\\ \le C\sum _{j=1}^{\mathrm{\infty }}{\int }_{2^{j+1}s}^{2^{j+2}s}{\left(2^{j+2}s\right)}^{-nm+\alpha -1}\left(\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,2^{j+1}s))}{\left({\omega }_i^{-p_i}\left(B(x_0,2^{j+1}s)\right)\right)}^{\frac{1}{p_i}}\right)dr\\ \le C\sum _{j=1}^{\mathrm{\infty }}{\int }_{2^{j+1}s}^{2^{j+2}s}\left(\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,2^{j+1}s))}{\left({\omega }_i^{-p_i}\left(B(x_0,2^{j+1}s)\right)\right)}^{\frac{1}{p_i}}\right)\frac{dr}{r^{mn-\alpha +1}}\\ \le C{\int }_{2s}^{\mathrm{\infty }}\left(\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{-p_i}(B(x_0,r))\right)}^{\frac{1}{p_i}}\right)\frac{dr}{r^{mn-\alpha +1}}.\end{array}$$

Substituting (3.7) and (3.8) into the above, we obtain

$$\begin{array}{r}\underset{x\in B(x_0,s)}{sup}|T(f_1^{\mathrm{\infty }},\dots ,f_m^{\mathrm{\infty }})(x)|\\ \le C{\int }_{2s}^{\mathrm{\infty }}\left(\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}.\end{array}$$

(3.11)

Using Hölder’s inequality,

$${\left({\int }_{B(x_0,2s)}{\nu }_{\overrightarrow{\omega }}{(x)}^{q}dx\right)}^{\frac{1}{q}}\le C\prod _{i=1}^m{\left({\omega }_i^{q_i}(B(x_0,2s))\right)}^{\frac{1}{q_i}}.$$

(3.12)

From (3.11) and (3.12) we know $J^{\mathrm{\infty },\dots ,\mathrm{\infty }}$ and $K^{\mathrm{\infty },\dots ,\mathrm{\infty }}$ are not greater than (3.10) for $1\le p_i<\mathrm{\infty }$, $i=1,\dots ,m$.

Now we consider the case where exactly τ of the ${\alpha }_i$ are ∞ for some $1\le \tau <m$. We only give the arguments for one of the cases. The rest is similar and can easily be obtained from the arguments below by permuting the indices. Then for any $x\in B(x_0,s)$,

$$\begin{array}{r}|I_{\alpha ,m}(f_1^{\mathrm{\infty }},\dots ,f_{\tau }^{\mathrm{\infty }},f_{\tau +1}^0,\dots ,f_m^0)(x)|\\ \le C{\int }_{{({\mathbb{R}}^n\setminus B(x_0,2s))}^{\tau }}{\int }_{{(B(x_0,2s))}^{m-\tau }}\frac{|f_1(y_1)\cdots f_m(y_m)|}{{(|x-y_1|+\cdots +|x-y_m|)}^{mn-\alpha }}d{y}_1\cdots d{y}_m\\ \le C\prod _{i=\tau +1}^m{\int }_{B(x_0,2s)}|f_i(y_i)|d{y}_i\\ \times \sum _{j=1}^{\mathrm{\infty }}\frac{1}{{|B(x_0,2^{j+1}s)|}^{m-\alpha /n}}{\int }_{{(B(x_0,2^{j+1}s)\setminus B(x_0,2^{j}s))}^{\tau }}|f_1(y_1)\cdots f_{\tau }(y_{\tau })|d{y}_1\cdots d{y}_{\tau }\\ \le C\prod _{i=\tau +1}^m{\int }_{B(x_0,2s)}|f_i(y_i)|d{y}_i\cdot \sum _{j=1}^{\mathrm{\infty }}\frac{1}{{|B(x_0,2^{j+1}s)|}^{m-\alpha /n}}\prod _{i=1}^{\tau }{\int }_{B(x_0,2^{j+1}s)\setminus B(x_0,2^{j}s)}|f_i(y_i)|d{y}_i\\ \le C\sum _{j=1}^{\mathrm{\infty }}\prod _{i=1}^m{\left(2^{j+1}s\right)}^{-n+\alpha /m}{\int }_{B(x_0,2^{j+1}s)}|f_i(y_i)|d{y}_i.\end{array}$$

Similar to the estimates for $J^{\mathrm{\infty },\dots ,\mathrm{\infty }}$, we get

$$\begin{array}{r}\underset{x\in B(x_0,s)}{sup}|I_{\alpha ,m}(f_1^{\mathrm{\infty }},\dots ,f_{\tau }^{\mathrm{\infty }},f_{\tau +1}^0,\dots ,f_m^0)(x)|\\ \le C{\int }_{2s}^{\mathrm{\infty }}\left(\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}.\end{array}$$

(3.13)

Then $J^{\mathrm{\infty },\dots ,\mathrm{\infty },0,\dots ,0}$ and $K^{\mathrm{\infty },\dots ,\mathrm{\infty },0,\dots ,0}$ are all less than

$$\prod _{i=1}^m{\left({\omega }_i^{q_i}(B(x_0,2s))\right)}^{\frac{1}{q_i}}\cdot {\int }_{2s}^{\mathrm{\infty }}\left(\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}.$$

(3.14)

Combining the above estimates, the proof of Theorem 3.1 is completed. □

Now, we can give the proof of Theorem 1.1. From the definition of generalized weighted Morrey space, the norm of $I_{\alpha ,m}(\overrightarrow{f})$ on $M_{{\phi }_2}^q({\nu }_{\overrightarrow{\omega }}^q)$ equals

$$\underset{x\in R^n,r>0}{sup}{\phi }_2{(x,s)}^{-1}{(\frac{1}{{\nu }_{\overrightarrow{\omega }}^q(B(x,s))}{\int }_{B(x,s)}{|I_{\alpha ,m}(\overrightarrow{f})(y)|}^q{\nu }_{\overrightarrow{\omega }}^q(y)dy)}^{1/q}.$$

(3.15)

By Lemma 2.2 we have

$${({\int }_{B(x,s)}{\nu }_{\overrightarrow{\omega }}^q(x)dx)}^{-\frac{1}{q}}\le C\prod _{i=1}^m{({\int }_{B(x,s)}{\omega }_i^{q_i}(x)dx)}^{-\frac{1}{q_i}}.$$

(3.16)

Combining (3.1) and (3.16),

$$\begin{array}{r}{(\frac{1}{{\nu }_{\overrightarrow{\omega }}^q(B(x,s))}{\int }_{B(x,s)}{|I_{\alpha ,m}(\overrightarrow{f})(y)|}^q{\nu }_{\overrightarrow{\omega }}^q(y)dy)}^{1/q}\\ \le {\int }_{2s}^{\mathrm{\infty }}\left(\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}.\end{array}$$

(3.17)

Since $f_i\in M_{{\phi }_{1i}}^{p_i}({\omega }_i^{p_i})$, from Lemma 2.7 and the fact ${\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x,r))}$ are all non-decreasing functions of r, we get

$$\begin{array}{rl}\frac{{\prod }_{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x,r))}}{{ess\hspace{0.17em}inf}_{r<t<\mathrm{\infty }}{\prod }_{i=1}^m{\phi }_{1i}(x,t){({\omega }_i^{p_i}(B(x,t)))}^{\frac{1}{p_i}}}\le & \underset{0<r<t<\mathrm{\infty }}{ess\hspace{0.17em}sup}\frac{{\prod }_{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x,r))}}{{\prod }_{i=1}^m{\phi }_{1i}(x,t){({\omega }_i^{p_i}(B(x,t)))}^{\frac{1}{p_i}}}\\ \le & \underset{t>0,x\in {\mathbb{R}}^n}{ess\hspace{0.17em}sup}\frac{{\prod }_{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x,t))}}{{\prod }_{i=1}^m{\phi }_{1i}(x,t){({\omega }_i^{p_i}(B(x,t)))}^{\frac{1}{p_i}}}\\ \le & C\prod _{i=1}^m{\parallel f_i\parallel }_{M_{{\phi }_{1i}}^{p_i}({\omega }_i^{p_i})}.\end{array}$$

(3.18)

Then

$$\begin{array}{r}{\int }_s^{\mathrm{\infty }}\left(\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }^{p_i},B(x,r))}{\left({\omega }_i^{p_i}(B(x,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}\\ ={\int }_s^{\mathrm{\infty }}\frac{{\prod }_{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x,r))}}{{ess\hspace{0.17em}inf}_{r<t<\mathrm{\infty }}{\prod }_{i=1}^m{\phi }_{1i}(x,t){({\omega }_i^{p_i}(B(x,t)))}^{\frac{1}{p_i}}}\\ \times \frac{{ess\hspace{0.17em}inf}_{r<t<\mathrm{\infty }}{\prod }_{i=1}^m{\phi }_{1i}(x,t){({\omega }_i^{p_i}(B(x,t)))}^{\frac{1}{p_i}}}{{\prod }_{i=1}^n{({\omega }_i^{p_i}(B(x,r)))}^{\frac{1}{p_i}}}\frac{dr}{r^{1-\alpha }}\\ \le C\prod _{i=1}^m{\parallel f_i\parallel }_{M_{{\phi }_{1i}}^{p_i}({\omega }_i^{p_i})}{\int }_s^{\mathrm{\infty }}\frac{{ess\hspace{0.17em}inf}_{r<t<\mathrm{\infty }}{\prod }_{i=1}^m{\phi }_{1i}(x,t){\omega }_i{(B(x,t))}^{\frac{1}{p_i}}}{{\prod }_{i=1}^n{\omega }_i{(B(x,r))}^{\frac{1}{p_i}}}\frac{dr}{r^{1-\alpha }}.\end{array}$$

(3.19)

By (1.4) we get

$${\int }_s^{\mathrm{\infty }}\left(\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x,r))}{\left({\omega }_i^{p_i}(B(x,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}\le C{\phi }_2(x,s)\prod _{i=1}^m{\parallel f_i\parallel }_{M_{{\phi }_{1i}}^{p_i}({\omega }_i^{p_i})}.$$

(3.20)

Combining (3.15), (3.17), and (3.20), then

$${\parallel I_{\alpha ,m}\overrightarrow{f}\parallel }_{M_{{\phi }_2}^q({\nu }_{\overrightarrow{\omega }}^q)}\le C\prod _{i=1}^m{\parallel f_i\parallel }_{M_{{\phi }_{1i}}^{p_i}({\omega }_i^{p_i})}.$$

This completes the proof of first part of Theorem 1.1.

Similarly, the norm of $I_{\alpha ,m}(\overrightarrow{f})$ on $W{M}_{{\phi }_2}^p({\nu }_{\overrightarrow{\omega }}^q)$ equals

$$\underset{x\in R^n,r>0}{sup}{\phi }_2{(x,s)}^{-1}{\left(\frac{1}{{\nu }_{\overrightarrow{\omega }}^q(B(x,s))}{\parallel I_{\alpha ,m}(\overrightarrow{f})\parallel }_{W{L}^q({\nu }_{\overrightarrow{\omega }}^q,B(x,s))}^q\right)}^{1/q}.$$

(3.21)

Combining (3.2) and (3.16),

$$\begin{array}{r}{\left(\frac{1}{{\nu }_{\overrightarrow{\omega }}^q(B(x,s))}{\parallel I_{\alpha ,m}(\overrightarrow{f})\parallel }_{W{L}^q({\nu }_{\overrightarrow{\omega }}^q,B(x,s))}^q\right)}^{1/q}\\ \le C{\int }_s^{\mathrm{\infty }}\left(\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x,r))}{\left({\omega }_i^{p_i}(B(x,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}.\end{array}$$

(3.22)

Substituting (3.20) into (3.22),

$${\left(\frac{1}{{\nu }_{\overrightarrow{\omega }}^q(B(x,s))}{\parallel I_{\alpha ,m}(\overrightarrow{f})\parallel }_{W{L}^q({\nu }_{\overrightarrow{\omega }}^q,B(x,s))}^q\right)}^{1/q}\le C{\phi }_2(x,s)\prod _{i=1}^m{\parallel f_i\parallel }_{M_{{\phi }_{1i}}^{p_i}({\omega }_i^{p_i})}.$$

(3.23)

Then

$${\parallel I_{\alpha ,m}\overrightarrow{f}\parallel }_{W{M}_{{\phi }_2}^q({\nu }_{\overrightarrow{\omega }}^q)}\le C\prod _{i=1}^m{\parallel f_i\parallel }_{M_{{\phi }_{1i}}^{p_i}({\omega }_i^{p_i})}.$$

This completes the proof of second part of Theorem 1.1.

4 Proof of Theorem 1.2

Theorem 4.1 Let $m\ge 2$ and let $0<\alpha <mn$. Suppose $1/p={\sum }_{i=1}^{m}1/p_i$, $1/q_i=1/p_i-\alpha /mn$, and $1/q={\sum }_{i=1}^{m}1/q_i=1/p-\alpha /n$, $\overrightarrow{\omega }=({\omega }_1,\dots ,{\omega }_m)$ satisfy the $A_{\overrightarrow{p},q}$ condition with ${\omega }_1^{q_1},\dots ,{\omega }_m^{q_m}\in A_{\mathrm{\infty }}$, ${\nu }_{\overrightarrow{\omega }}={\prod }_{i=1}^m{\omega }_i$. If $p_1,\dots ,p_m\in (1,\mathrm{\infty })$, $(b_1,\dots ,b_m)\in {(BMO)}^m$, then there exists a constant C independent of $\overrightarrow{f}$ such that

$$\begin{array}{r}{\parallel I_{\alpha ,m}^{\mathrm{\Sigma }b}\overrightarrow{f}\parallel }_{L^q({\nu }_{\overrightarrow{\omega }}^q,B(x_0,s))}\\ \le C\prod _{i=1}^m{\parallel b_i\parallel }_{\ast }{\left({\omega }_i^{q_i}(B(x_0,s))\right)}^{\frac{1}{q_i}}\\ \times {\int }_{2s}^{\mathrm{\infty }}{(1+ln\frac{r}{s})}^m\left(\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}\end{array}$$

(4.1)

and

$$\begin{array}{r}{\parallel I_{\alpha ,m}^{\mathrm{\Pi }b}\overrightarrow{f}\parallel }_{L^q({\nu }_{\overrightarrow{\omega }}^q,B(x_0,s))}\\ \le C\prod _{i=1}^m{\parallel b_i\parallel }_{\ast }{\left({\omega }_i^{q_i}(B(x_0,s))\right)}^{\frac{1}{q_i}}\\ \times {\int }_{2s}^{\mathrm{\infty }}{(1+ln\frac{r}{s})}^m\left(\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }},\end{array}$$

(4.2)

where ${\nu }_{\overrightarrow{\omega }}={\prod }_{i=1}^m{\omega }_i$.

Proof We will give the proof for $I_{\alpha ,m}^{\mathrm{\Pi }b}$ because the proof for $I_{\alpha ,m}^{\mathrm{\Sigma }b}$ is very similar but easier. Moreover, for simplicity of the expansion, we only present the case $m=2$.

We represent $f_i$ as $f_i=f_i^0+f_i^{\mathrm{\infty }}$, where $f_i^0=f_i{\chi }_{B(x_0,2s)}$, $i=1,2$, and ${\chi }_{B(x_0,2s)}$ denotes the characteristic function of $B(x_0,2s)$. Then

$$\begin{array}{rl}{\parallel I_{\alpha ,2}^{\mathrm{\Pi }b}(\overrightarrow{f})\parallel }_{L^q({\nu }_{\overrightarrow{\omega }}^q,B(x_0,s))}\le & C{\left({\int }_{B(x_0,s)}\right|I_{\alpha ,2}^{\mathrm{\Pi }b}(f_1^0,f_2^0)(x){|}^q{\nu }_{\overrightarrow{\omega }}^q(x)dx)}^{\frac{1}{q}}\\ +C{\left({\int }_{B(x_0,s)}\right|I_{\alpha ,2}^{\mathrm{\Pi }b}(f_1^0,f_2^{\mathrm{\infty }})(x){|}^q{\nu }_{\overrightarrow{\omega }}^q(x)dx)}^{\frac{1}{q}}\\ +C{\left({\int }_{B(x_0,s)}\right|I_{\alpha ,2}^{\mathrm{\Pi }b}(f_1^{\mathrm{\infty }},f_2^0)(x){|}^q{\nu }_{\overrightarrow{\omega }}^q(x)dx)}^{\frac{1}{q}}\\ +C{\left({\int }_{B(x_0,s)}\right|I_{\alpha ,2}^{\mathrm{\Pi }b}(f_1^{\mathrm{\infty }},f_2^{\mathrm{\infty }})(x){|}^q{\nu }_{\overrightarrow{\omega }}^q(x)dx)}^{\frac{1}{q}}\\ =& I+II+III+IV.\end{array}$$

(4.3)

Since $I_{\alpha ,2}^{\mathrm{\Pi }b}$ bounded from $L^{p_1}({\omega }_1^{p_1})\times L^{p_2}({\omega }_2^{p_2})$ to $L^q({\nu }_{\overrightarrow{\omega }}^q)$, we get

$${\left({\int }_{B(x_0,s)}\right|I_{\alpha ,2}^{\mathrm{\Pi }b}(f_1^0,f_2^0)(x){|}^q{\nu }_{\overrightarrow{\omega }}^q(x)dx)}^{\frac{1}{q}}\le C\prod _{i=1}^2{\parallel b_i\parallel }_{\ast }{\parallel f_i\parallel }_{L^{p_i}({\omega }^{p_i},B(x_0,2s))}.$$

Then by (3.9) we get

$$\begin{array}{rl}I\le & C\prod _{i=1}^2{\parallel b_i\parallel }_{\ast }{\left({\omega }_i^{q_i}(B(x_0,s))\right)}^{\frac{1}{q_i}}\\ \cdot {\int }_{2s}^{\mathrm{\infty }}\left(\prod _{i=1}^2{\parallel f_i\parallel }_{L^{p_i}({\omega }^{p_i},B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}.\end{array}$$

(4.4)

Owing to the symmetry of II and $III$, we only estimate II. Taking ${\lambda }_i={(b_i)}_{B(x_0,s)}$, then

$$\begin{array}{rl}{I}_{\alpha ,2}^{\mathrm{\Pi }b}(f_1^0,f_2^{\mathrm{\infty }})(x)=& (b_1(x)-{\lambda }_1)(b_2(x)-{\lambda }_2)I_{\alpha ,2}(f_1^0,f_2^{\mathrm{\infty }})(x)\\ -(b_1(x)-{\lambda }_1)I_{\alpha ,2}(f_1^0,(b_2-{\lambda }_2)f_2^{\mathrm{\infty }})(x)\\ -(b_2(x)-{\lambda }_2)I_{\alpha ,2}((b_1-{\lambda }_1)f_1^0,f_2^{\mathrm{\infty }})(x)\\ +I_{\alpha ,2}((b_1-{\lambda }_1)f_1^0,(b_2-{\lambda }_2)f_2^{\mathrm{\infty }})(x)\\ =& I{I}_1+I{I}_2+I{I}_3+I{I}_4.\end{array}$$

(4.5)

Similar to the estimate of (3.13), for any $x\in B(x_0,s)$ we can deduce

$$\begin{array}{r}\underset{x\in B(x_0,s)}{sup}|I_{\alpha ,2}(f_1^0,f_2^{\mathrm{\infty }})(x)|\\ \le C{\int }_{2s}^{\mathrm{\infty }}\left(\prod _{i=1}^2{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}.\end{array}$$

(4.6)

By Lemma 2.1 we know ${\nu }_{\overrightarrow{\omega }}^q\in A_{\mathrm{\infty }}$. Applying Hölder’s inequality and (2.13), we have

$$\begin{array}{r}{({\int }_{B(x_0,s)}{\left|(b_1(x)-{\lambda }_1)(b_2(x)-{\lambda }_2)\right|}^q{\nu }_{\overrightarrow{\omega }}^q(x)dx)}^{\frac{1}{q}}\\ \le C\prod _{i=1}^2{({\int }_{B(x_0,s)}{|b_i(x)-{\lambda }_i|}^{2q}{\nu }_{\overrightarrow{\omega }}^q(x)dx)}^{\frac{1}{2q}}\le C\prod _{i=1}^2{\parallel b_i\parallel }_{\ast }\cdot {\left({\nu }_{\overrightarrow{\omega }}^q(B(x_0,s))\right)}^{\frac{1}{q}}.\end{array}$$

(4.7)

Then by (4.6), (4.7), and (3.12), we have

$$\begin{array}{r}{({\int }_{B(x_0,s)}{|I{I}_1|}^q{\nu }_{\overrightarrow{\omega }}^q(x)dx)}^{\frac{1}{q}}\\ \le {({\int }_{B(x_0,s)}{\left|(b_1(x)-{\lambda }_1)(b_2(x)-{\lambda }_2)\right|}^q{\nu }_{\overrightarrow{\omega }}^q(x)dx)}^{\frac{1}{q}}\underset{x\in B(x_0,s)}{sup}|I_{\alpha ,2}(f_1^0,f_2^{\mathrm{\infty }})(x)|\\ \le C\prod _{i=1}^2{\parallel b_i\parallel }_{\ast }{\left({\omega }_i^{q_i}(B(x_0,s))\right)}^{\frac{1}{q_i}}\\ \cdot {\int }_{2s}^{\mathrm{\infty }}\left(\prod _{i=1}^2{\parallel f_i\parallel }_{L^{p_i}({\omega }_i,B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}.\end{array}$$

(4.8)

For any $x\in B(x_0,s)$, we have

$$\begin{array}{r}|I_{\alpha ,2}(f_1^0,(b_2-{\lambda }_2)f_2^{\mathrm{\infty }})(x)|\\ \le C{\int }_{B(x_0,2s)}{\int }_{{\mathbb{R}}^n\setminus B(x_0,2s)}\frac{|f_1(y_1)(b_2(y_2)-{\lambda }_2)f_2(y_2)|}{{(|x-y_1|+|x-y_2|)}^{2n-\alpha }}d{y}_{1}d{y}_2\\ \le C\sum _{j=1}^{\mathrm{\infty }}{\left(2^{j+1}s\right)}^{-2n+\alpha }{\int }_{B(x_0,2s)}|f_1(y_1)|d{y}_1{\int }_{B(x_0,2^{j+1}s)}|(b_2(y_2)-{\lambda }_2)f_2(y_2)|d{y}_2.\end{array}$$

(4.9)

Note that

$${\int }_{B(x_0,2s)}|f_1(y_1)|d{y}_1\le C{\parallel f_1\parallel }_{L^{p_1}({\omega }_1^{p_1},B(x_0,2s))}{\left({\omega }_1^{-p_1^{\prime }}(B(x_0,2s))\right)}^{\frac{1}{p_1^{\prime }}}$$

(4.10)

and

$$\begin{array}{r}{\int }_{B(x_0,2^{j+1}s)}|(b_2(y_2)-{\lambda }_2)f_2(y_2)|d{y}_2\\ \le C{\parallel f_2\parallel }_{L^{p_2}({\omega }_2^{p_2},B(x_0,2^{j+1}s))}{\parallel b_2(\cdot )-{\lambda }_2\parallel }_{L^{p_2^{\prime }}({\omega }_2^{-p_2^{\prime }},B(x_0,2^{j+1}s))}.\end{array}$$

(4.11)

Then

$$\begin{array}{r}\underset{x\in B(x_0,s)}{sup}|I_{\alpha ,2}(f_1^0,(b_2-{\lambda }_2)f_2^{\mathrm{\infty }})(x)|\\ \le C\sum _{j=1}^{\mathrm{\infty }}{\left(2^{j+1}s\right)}^{-2n+\alpha }\prod _{i=1}^2{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,2^{j+1}s))}\\ \times {\left({\omega }_1^{-p_1}\left(B(x_0,2^{j+1}s)\right)\right)}^{\frac{1}{p_1}}{\parallel b_2(\cdot )-{\lambda }_2\parallel }_{L^{p_2^{\prime }}({\omega }_2^{-p_2^{\prime }},B(x_0,2^{j+1}s))}\\ \le C{\int }_{2s}^{\mathrm{\infty }}\prod _{i=1}^2{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_1^{-p_1^{\prime }}(B(x_0,r))\right)}^{\frac{1}{p_1^{\prime }}}\\ \times {\parallel b_2(\cdot )-{\lambda }_2\parallel }_{L^{p_2^{\prime }}({\omega }_2^{-p_2^{\prime }},B(x_0,r))}\frac{dr}{r^{2n+1-\alpha }}.\end{array}$$

(4.12)

From Lemma 2.1 we know ${\omega }_2^{-p_2^{\prime }}\in A_{2p_2^{\prime }}$, then by Lemma 2.4 we get

$$\begin{array}{r}{\parallel b_2(\cdot )-{\lambda }_2\parallel }_{L^{p_2^{\prime }}({\omega }_2^{-p_2^{\prime }},B(x_0,r))}\\ \le C{({\int }_{B(x_0,r)}{|b_2(z)-{\lambda }_2|}^{p_2^{\prime }}{\omega }_2^{-p_2^{\prime }}(z)dz)}^{\frac{1}{p_2^{\prime }}}\\ \le C(1+|ln\frac{r}{s}\left|\right){\parallel b_2\parallel }_{\ast }({\omega }_2^{-p_2^{\prime }}{(B(x_0,r))}^{\frac{1}{p_2^{\prime }}}.\end{array}$$

(4.13)

By (3.7) and (3.8) we have

$$\prod _{i=1}^2{\left({\omega }_i^{-p_i^{\prime }}(B(x_0,r))\right)}^{\frac{1}{p_i}}\le C{|B(x_0,r)|}^2\prod _{i=1}^2{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}.$$

(4.14)

From (4.12), (4.13), and (4.14) we can deduce

$$\begin{array}{r}\underset{x\in B(x_0,s)}{sup}|I_{\alpha ,2}(f_1^0,(b_2-{\lambda }_2)f_2^{\mathrm{\infty }})(x)|\\ \le C{\parallel b_2\parallel }_{\ast }{\int }_{2s}^{\mathrm{\infty }}(1+ln\frac{r}{s})\left(\prod _{i=1}^2{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}.\end{array}$$

(4.15)

Applying (2.13) and (3.12) we have

$$\begin{array}{rl}{({\int }_{B(x_0,s)}{|b_1(x)-{\lambda }_1|}^q{\nu }_{\overrightarrow{\omega }}^q(x)dx)}^{\frac{1}{q}}\le & C{\parallel b_1\parallel }_{\ast }{\left({\nu }_{\overrightarrow{\omega }}^q(B(x_0,s))\right)}^{\frac{1}{q}}\\ \le & C{\parallel b_1\parallel }_{\ast }\prod _{i=1}^2{\left({\omega }_i^{q_i}(B(x_0,r))\right)}^{\frac{1}{q_i}}.\end{array}$$

(4.16)

Then by (4.15) and (4.16),

$$\begin{array}{r}{({\int }_{B(x_0,s)}{|I{I}_2|}^q{\nu }_{\overrightarrow{\omega }}^q(x)dx)}^{\frac{1}{q}}\\ \le {({\int }_{B(x_0,s)}{|b_1(x)-{\lambda }_1|}^q{\nu }_{\overrightarrow{\omega }}^q(x)dx)}^{\frac{1}{q}}\underset{x\in B(x_0,s)}{sup}|I_{\alpha ,2}(f_1^0,(b_2-{\lambda }_2)f_2^{\mathrm{\infty }})(x)|\\ \le C\prod _{i=1}^2{\parallel b_i\parallel }_{\ast }{\omega }_i{(B(x_0,s))}^{\frac{1}{p_i}}\\ \times {\int }_{2s}^{\mathrm{\infty }}(1+ln\frac{r}{s})\left(\prod _{i=1}^2{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}.\end{array}$$

(4.17)

Similarly, we also have

$$\begin{array}{r}{({\int }_{B(x_0,s)}{|I{I}_3|}^p{\nu }_{\overrightarrow{\omega }}(x)dx)}^{\frac{1}{p}}\\ \le C\prod _{i=1}^2{\parallel b_i\parallel }_{\ast }{\omega }_i{(B(x_0,s))}^{\frac{1}{p_i}}\\ \times {\int }_{2s}^{\mathrm{\infty }}(1+ln\frac{r}{s})\left(\prod _{i=1}^2{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}.\end{array}$$

(4.18)

For any $x\in B(x_0,s)$, with the same method of estimate for (4.15) we have

$$\begin{array}{r}|I_{\alpha ,2}((b_1-{\lambda }_1)f_1^0,(b_2-{\lambda }_2)f_2^{\mathrm{\infty }})(x)|\\ \le C\sum _{j=1}^{\mathrm{\infty }}{\left(2^{j+1}s\right)}^{-2n+\alpha }\prod _{i=1}^2{\int }_{B(x_0,2^{j+1}s)}|(b_i(y_i)-{\lambda }_i)f_i(y_i)|d{y}_i\\ \le C{\int }_{2s}^{\mathrm{\infty }}\prod _{i=1}^2{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r)}\cdot {\parallel b_i(\cdot )-{\lambda }_i\parallel }_{L^{p_i^{\prime }}({\omega }_i^{-p_i^{\prime }},B(x_0,r))}\frac{dr}{r^{2n-\alpha +1}}\\ \le C\prod _{i=1}^2{\parallel b_i\parallel }_{\ast }{\int }_{2s}^{\mathrm{\infty }}{(1+ln\frac{r}{s})}^2\\ \times \left(\prod _{i=1}^2{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}.\end{array}$$

(4.19)

Then

$$\begin{array}{r}{({\int }_{B(x_0,s)}{|I{I}_4|}^q{\nu }_{\overrightarrow{\omega }}^q(x)dx)}^{\frac{1}{q}}\\ \le C{\left({\nu }_{\overrightarrow{\omega }}^q(B(x_0,s))\right)}^{\frac{1}{q}}\underset{x\in B(x_0.s)}{sup}|I_{\alpha ,2}((b_1-{\lambda }_1)f_1^0,(b_2-{\lambda }_2)f_2^{\mathrm{\infty }})(x)|\\ \le C\prod _{i=1}^2{\parallel b_i\parallel }_{\ast }{\omega }_i{(B(x_0,s))}^{\frac{1}{p_i}}\\ \times {\int }_{2s}^{\mathrm{\infty }}{(1+ln\frac{r}{s})}^2\left(\prod _{i=1}^2{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}.\end{array}$$

(4.20)

Then combining (4.8), (4.17), (4.18), and (4.20) we get

$$\begin{array}{r}{({\int }_{B(x_0,s)}{|II|}^q{\nu }_{\overrightarrow{\omega }}^q(x)dx)}^{\frac{1}{q}}\\ \le C\prod _{i=1}^2{\parallel b_i\parallel }_{\ast }{\omega }_i{(B(x_0,s))}^{\frac{1}{p_i}}\\ \times {\int }_{2s}^{\mathrm{\infty }}{(1+ln\frac{r}{s})}^2\left(\prod _{i=1}^2{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}.\end{array}$$

(4.21)

Finally, we still decompose $I_{\alpha ,2}^{\mathrm{\Pi }b}(f_1^{\mathrm{\infty }},f_2^{\mathrm{\infty }})(x)$ as follows:

$$\begin{array}{rl}{I}_{\alpha ,2}^{\mathrm{\Pi }b}(f_1^{\mathrm{\infty }},f_2^{\mathrm{\infty }})(x)=& (b_1(x)-{\lambda }_1)(b_2(x)-{\lambda }_2)I_{\alpha ,2}(f_1^{\mathrm{\infty }},f_2^{\mathrm{\infty }})(x)\\ -(b_1(x)-{\lambda }_1)I_{\alpha ,2}(f_1^{\mathrm{\infty }},(b_2-{\lambda }_2)f_2^{\mathrm{\infty }})(x)\\ -(b_2(x)-{\lambda }_2)I_{\alpha ,2}((b_1-{\lambda }_1)f_1^{\mathrm{\infty }},f_2^{\mathrm{\infty }})(x)\\ +I_{\alpha ,2}((b_1-{\lambda }_1)f_1^{\mathrm{\infty }},(b_2-{\lambda }_2)f_2^{\mathrm{\infty }})(x)\\ =& I{V}_1+I{V}_2+I{V}_3+I{V}_4.\end{array}$$

(4.22)

Because each term $I{V}_j$ is completely analogous to $I{I}_j$, $j=1,2,3,4$, being slightly different, we get the following estimate without details:

$$\begin{array}{r}{({\int }_{B(x_0,s)}{|IV|}^q{\nu }_{\overrightarrow{\omega }}^q(x)dx)}^{\frac{1}{q}}\\ \le C\prod _{i=1}^2{\parallel b_i\parallel }_{\ast }{\omega }_i{(B(x_0,s))}^{\frac{1}{p_i}}\\ \times {\int }_{2s}^{\mathrm{\infty }}{(1+ln\frac{r}{s})}^2\left(\prod _{i=1}^2{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}.\end{array}$$

(4.23)

Summing up the above estimates, (4.2) is proved for $m=2$. □

In the following we give the proof of Theorem 1.2. From (3.16) and (4.2),

$$\begin{array}{r}{(\frac{1}{{\nu }_{\overrightarrow{\omega }}^q(B(x,s))}{\int }_{B(x,s)}{|I_{\alpha ,m}^{\mathrm{\Pi }b}(\overrightarrow{f})(y)|}^q{\nu }_{\overrightarrow{\omega }}^q(y)dy)}^{1/q}\\ \le C\prod _{i=1}^m{\parallel b_i\parallel }_{\ast }{\int }_{2s}^{\mathrm{\infty }}{(1+ln\frac{r}{s})}^m\\ \times \left(\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}.\end{array}$$

(4.24)

Since $\overrightarrow{{\phi }_k}$, $k=1,2$, satisfy the condition (1.7), and $f_i\in M_{{\phi }_{1i}}^{p_i}({\omega }_i^{p_i})$, by (3.18) we get

$$\begin{array}{r}{\int }_{2s}^{\mathrm{\infty }}{(1+ln\frac{r}{s})}^m\left(\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}{\left({\omega }_i^{p_i}(B(x_0,r))\right)}^{-\frac{1}{p_i}}\right)\frac{dr}{r^{1-\alpha }}\\ ={\int }_s^{\mathrm{\infty }}\frac{{\prod }_{i=1}^m{\parallel f_i\parallel }_{L^{p_i}({\omega }_i^{p_i},B(x_0,r))}}{{ess\hspace{0.17em}inf}_{r<t<\mathrm{\infty }}{\prod }_{i=1}^m{\phi }_{1i}(x,t){({\omega }_i^{p_i}(B(x,t)))}^{\frac{1}{p_i}}}\\ \times {(1+ln\frac{r}{s})}^m\frac{{ess\hspace{0.17em}inf}_{r<t<\mathrm{\infty }}{\prod }_{i=1}^m{\phi }_{1i}(x,t){({\omega }_i^{p_i}(B(x,t)))}^{\frac{1}{p_i}}}{{\prod }_{i=1}^m{({\omega }_i^{p_i}(B(x,t)))}^{\frac{1}{p_i}}}\frac{dr}{r^{1-\alpha }}\\ \le C\prod _{i=1}^m{\parallel f_i\parallel }_{M_{{\phi }_{1i}}^{p_i}({\omega }_i^{p_i})}{\int }_s^{\mathrm{\infty }}{(1+ln\frac{r}{s})}^m\frac{{ess\hspace{0.17em}inf}_{r<t<\mathrm{\infty }}{\prod }_{i=1}^m{\phi }_{1i}(x,t){({\omega }_i^{p_i}(B(x,t)))}^{\frac{1}{p_i}}}{{\prod }_{i=1}^m{({\omega }_i^{p_i}(B(x,t)))}^{\frac{1}{p_i}}}\frac{dr}{r^{1-\alpha }}\\ \le C{\phi }_2(x,s)\prod _{i=1}^m{\parallel f_i\parallel }_{M_{{\phi }_{1i}}^{p_i}({\omega }_i^{p_i})}.\end{array}$$

(4.25)

Combining (4.24) and (4.25), we have

$${\parallel I_{\alpha ,m}^{\mathrm{\Pi }b}\overrightarrow{(f)}\parallel }_{M_{{\phi }_2}^q({\nu }_{\overrightarrow{\omega }}^q)}\le C\prod _{i=1}^m{\parallel b_i\parallel }_{\ast }{\parallel f_i\parallel }_{M_{{\phi }_{1i}}^{p_i}({\omega }_i^{p_i})}.$$