# Vector-valued multilinear singular integrals with nonsmooth kernels and commutators on generalized weighted Morrey space

## Abstract

In this paper, we prove weighted norm inequalities for vector-valued multilinear singular integrals with nonsmooth kernels and commutators on generalized weighted Morrey space.

## Introduction

Let T be a multilinear operator defined on the m-fold product of Schwarz spaces and taking values in the space of tempered distributions,

$$T:\mathcal{S}\bigl(R^{n}\bigr)\times\cdots\times\mathcal{S} \bigl(R^{n}\bigr)\rightarrow \mathcal{S'} \bigl(R^{n}\bigr).$$

In  the multilinear operator T satisfying the following conditions was studied:

1. (1)

There exists a function K defined off the diagonal $$x=y_{1}=\cdots=y_{m}$$ in $$(R^{n})^{m+1}$$ such that

$$T(f_{1},\ldots,f_{m}) (x)= \int_{(R^{n})^{m}}K(x,y_{1},\ldots ,y_{m})f_{1}(y_{1}) \cdots f_{m}(y_{m})\,dy_{1}\cdots dy_{m}$$
(1.1)

for all $$x\notin\bigcap_{j=1}^{m}\operatorname{supp}f_{j}$$.

2. (2)

There exists $$C>0$$ such that

$$\bigl\vert K(y_{0},y_{1},\ldots,y_{m}) \bigr\vert \leqslant\frac{C}{(\sum_{k,l=0}^{m} \vert y_{k}-y_{l} \vert )^{mn}}.$$
(1.2)
3. (3)

For some $$\epsilon>0$$, there exists $$C>0$$ such that

$$\bigl\vert K(y_{0},y_{1},\ldots,y_{j}, \ldots,y_{m})-K\bigl(y_{0},y_{1},\ldots ,y'_{j},\ldots,y_{m}\bigr) \bigr\vert \leqslant\frac{C \vert y_{j}-y'_{j} \vert ^{\epsilon}}{(\sum_{k,l=0}^{m} \vert y_{k}-y_{l} \vert )^{mn+\epsilon}},$$
(1.3)

provided that $$0\leqslant j\leqslant m$$ and $$|y_{j}-y'_{j}|\leqslant \frac{1}{2}\mathrm{max}_{0\leqslant k\leqslant m}|y_{j}-y_{k}|$$.

4. (4)

There exist $$1\leqslant q_{1},\ldots,q_{m}<\infty$$ such that

$$T:L^{q_{1}}\times\cdots\times L^{q_{m}}\rightarrow L^{q}$$

is bounded, where $$\frac{1}{q}=\frac{1}{q_{1}}+\cdots+\frac{1}{q_{m}}$$.

In  it is proved that

$$T:L^{q_{1}}\times\cdots\times L^{q_{m}}\rightarrow L^{q},$$

where $$\frac{1}{q}=\frac{1}{q_{1}}+\cdots+\frac{1}{q_{m}}$$ and $$1< q_{j}<\infty$$ for all $$j=1,\ldots,m$$, and

$$T:L^{q_{1}}\times\cdots\times L^{q_{m}}\rightarrow L^{q,\infty},$$

where $$1\leqslant q_{1},\ldots,q_{m}<\infty$$ and $$\frac{1}{q}=\frac {1}{q_{1}}+\cdots+\frac{1}{q_{m}}$$. In particular,

$$T: L^{1}\times\cdots\times L^{1}\rightarrow L^{\frac{1}{m},\infty}.$$

Let $$\overrightarrow{b}=(b_{1},\ldots,b_{m})\in(\mathrm{BMO})^{m}$$ be a locally integrable vector function. The commutator of $$\overrightarrow {b}$$ and m-linear Calderón–Zygmund operator T, denoted $$T_{\sum \overrightarrow{b}}$$, was introduced by Pérez and Torres  and defined as

$$T_{\sum\overrightarrow{b}}(f_{1},\ldots,f_{m})=\sum _{j=1}^{m}T_{b_{j}}^{j}(f_{1}, \ldots,f_{m}),$$

where

$$T_{b_{j}}^{j}(\overrightarrow{f})=b_{j}T(f_{1}, \ldots,f_{j},\ldots ,f_{m})-T(f_{1}, \ldots,b_{j}f_{j},\ldots,f_{m}).$$

The iterated commutator $$T_{\prod\overrightarrow{b}}$$ is defined by

$$T_{\prod\overrightarrow{b}}(f_{1},\ldots,f_{m})= \bigl[b_{1},\ldots ,\bigl[b_{m-1},[b_{m},T]_{m} \bigr]_{m-1}\cdots\bigr]_{1}.$$

To clarify the notation, if T is associated in the usual way with a kernel K satisfying (1.1)–(1.3), then, formally,

$$T_{\sum\overrightarrow{b}}(\overrightarrow{f}) (x)= \int_{(R^{n})^{m}}\sum_{j=1}^{m} \bigl(b_{j}(x)-b_{j}(y_{j}) \bigr)K(x,y_{1},\ldots ,y_{m})f_{1}(y_{1}) \cdots f_{m}(y_{m})\,dy_{1}\cdots dy_{m}$$

and

$$T_{\prod\overrightarrow{b}}(\overrightarrow{f}) (x)= \int _{(R^{n})^{m}}\prod_{j=1}^{m} \bigl(b_{j}-b_{j}(y_{j}) \bigr)K(x,y_{1},\ldots ,y_{m})f_{1}(y_{1}) \cdots f_{m}(y_{m})\,dy_{1}\cdots dy_{m}.$$

The theory of multiple weight associated with m-linear Calderón–Zygmund operators was developed by Lerner et al. . Let $$1< p_{j}<\infty$$ for $$j=1,\ldots,m$$, $$\frac{1}{p}=\frac {1}{p_{1}}+\cdots+\frac{1}{p_{m}}$$, and $$\overrightarrow {p}=(p_{1},\ldots,p_{m})$$. They showed that if $$\overrightarrow{w}\in A_{\overrightarrow{p}}$$ (see the definition in the next section), then

$$\bigl\Vert T(\overrightarrow{f}) \bigr\Vert _{L^{p}(v_{\overrightarrow{w}})}\leqslant C \prod_{j=1}^{m} \Vert f_{j} \Vert _{L^{p_{j}}(w_{j})}.$$

If $$1\leqslant p_{j}<\infty$$ for $$j=1,\ldots,m$$ and at least one $$p_{j}=1$$, then they also proved that

$$\bigl\Vert T(\overrightarrow{f}) \bigr\Vert _{L^{p,\infty}(v_{\overrightarrow {w}})}\leqslant C \prod_{j=1}^{m} \Vert f_{j} \Vert _{L^{p_{j}}(w_{j})}.$$

Let $$1< p_{j}<\infty$$, $$j=1,\ldots,m$$, and $$1< p<\infty$$ with $$\frac {1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}$$, Pérez and Torres proved that if $$\overrightarrow{b}\in(\mathrm{BMO})^{m}$$, then

$$\bigl\Vert T_{\sum\overrightarrow{b}}(\overrightarrow{f}) \bigr\Vert _{L^{p}}\leqslant C\sum_{j=1}^{m} \Vert b_{j} \Vert _{\mathrm{BMO}}\prod _{j=1}^{m} \Vert f_{j} \Vert _{L^{p_{j}}}.$$

In  the weighted $$L^{p}$$-version of bounds is also obtained: for all $$\overrightarrow{w}\in A_{\overrightarrow{p}}$$,

$$\bigl\Vert T_{\sum\overrightarrow{b}}(\overrightarrow{f}) \bigr\Vert _{L^{p}(v_{\overrightarrow{w}})}\leqslant C\sum_{j=1}^{m} \Vert b_{j} \Vert _{\mathrm{BMO}}\prod _{j=1}^{m} \Vert f_{j} \Vert _{L^{p_{j}}(w_{j})}.$$

As for $$T_{\sum\overrightarrow{b}}$$, a strong-type bound for $$T_{\prod \overrightarrow{b}}$$ was also established by Pérez et al. .

The vector-valued multilinear operator $$T_{\gamma}$$ associated with the operator T was first studied by Grafakos and Martell . For $$\gamma>0$$, the vector-valued multilinear operator $$T_{\gamma}$$ is defined by

\begin{aligned} T_{\gamma}(\overrightarrow{f}) (x) =& \bigl\vert T(f_{1}, \ldots,f_{m}) (x) \bigr\vert _{\gamma } \\ =& \Biggl(\sum_{k=1}^{\infty} \bigl\vert T(f_{1k},\ldots,f_{mk}) (x) \bigr\vert ^{\gamma} \Biggr)^{1/\gamma}, \end{aligned}

where $$f_{i}=\{f_{ik}\}_{k=1}^{\infty}$$ for $$i=1,\ldots,m$$. Let $$\frac {1}{m}< p<\infty$$, $$\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{i}}$$ with $$1< p_{1},\ldots,p_{m}<\infty$$, $$\frac{1}{m}<\gamma<\infty$$, and $$\frac{1}{\gamma}=\frac{1}{\gamma_{1}}+\cdots+\frac{1}{\gamma_{i}}$$ with $$1<\gamma_{1},\ldots,\gamma_{m}<\infty$$. Grafakos and Martell proved that

$$\bigl\Vert T_{\gamma}(\overrightarrow{f}) \bigr\Vert _{L^{p}(R^{n})}\leqslant C\prod_{j=1}^{m} \bigl\Vert |f_{j}|_{\gamma_{j}} \bigr\Vert _{L^{p_{j}}(R^{n})}.$$
(1.4)

Later, Cruz-Uribe et al.  proved that if $$\frac {1}{m}\leqslant p<\infty$$, $$\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac {1}{p_{i}}$$ with $$1< p_{1},\ldots,p_{m}<\infty$$, $$\frac{1}{m}<\gamma<\infty$$, and $$\frac{1}{\gamma}=\frac{1}{\gamma_{1}}+\cdots+\frac{1}{\gamma _{i}}$$ with $$1<\gamma_{1},\ldots,\gamma_{m}<\infty$$, then

$$\bigl\Vert T_{\gamma}(\overrightarrow{f}) \bigr\Vert _{L^{p,\infty}(R^{n})}\leqslant C\prod_{j=1}^{m} \bigl\Vert |f_{j}|_{\gamma_{j}} \bigr\Vert _{L^{p_{j}}(R^{n})}.$$
(1.5)

They also obtained the weighted $$L^{p}$$-versions of (1.4) and (1.5), but their results are not the multiple weighted estimates obtained by Lerner et al. .

For the sequence $$\{\overrightarrow{f_{k}}\}_{k=1}^{\infty }=(f_{1k},\ldots,f_{mk})_{k=1}^{\infty}$$, the vector-valued versions of the commutators $$T_{\sum\overrightarrow{b},\gamma}$$ and $$T_{\prod \overrightarrow{b},\gamma}$$ are defined by

$$T_{\sum\overrightarrow{b},\gamma}(\overrightarrow{f}) (x)= \bigl\vert T_{\sum \overrightarrow{b},\gamma}( \overrightarrow{f}) \bigr\vert _{\gamma}(x)= \Biggl(\sum _{k=1}^{\infty} \bigl\vert T_{{\sum}\overrightarrow{b}}(f_{1k}, \ldots ,f_{mk}) (x) \bigr\vert ^{\gamma} \Biggr)^{1/\gamma}$$

and

$$T_{\prod\overrightarrow{b},\gamma}(\overrightarrow{f}) (x)= \bigl\vert T_{\prod \overrightarrow{b},\gamma}( \overrightarrow{f}) \bigr\vert _{\gamma}(x)= \Biggl(\sum _{k=1}^{\infty} \bigl\vert T_{{\prod}\overrightarrow{b}}(f_{1k}, \ldots ,f_{mk}) (x) \bigr\vert ^{\gamma} \Biggr)^{1/\gamma}.$$

In 2008, Tang  established weighted norm inequalities for the commutators of a vector-valued multilinear operator, but his results are not the multiple weighted estimates obtained by Lerner et al. .

In this paper, we consider T associated with the kernel satisfying a weaker regularity conditions introduced in [8, 9]. Let $$\{A_{t}\}_{t>0}$$ be a class of integral operators that play the role of an approximation of the identity. We always assume that the operators $$A_{t}$$ are associated with kernels $$a_{t}(x,y)$$ in the sense that, for all $$f\in\bigcup_{p\in[1,\infty]}L^{p}$$ and $$x\in R^{n}$$,

$$A_{t}f(x)= \int_{R^{n}}a_{t}(x,y)f(y)\,dy$$

and that the kernels $$a_{t}(x,y)$$ satisfy the condition

$$\bigl\vert a_{t}(x,y) \bigr\vert \leqslant h_{t}(x,y):=t^{-n/s}h \biggl(\frac { \vert x-y \vert }{t^{1/s}} \biggr),$$
(1.6)

where s is a positive fixed constant, and h is a positive bounded decreasing function such that for some $$\eta>0$$,

$$\lim_{r\rightarrow\infty}r^{n+\eta}h\bigl(r^{s} \bigr)=0.$$
(1.7)

Recall that the jth transpose $$T^{*,j}$$ of the m-linear operator T is defined as

$$\bigl\langle T^{*,j}(f_{1},\ldots,f_{m}),g \bigr\rangle =\bigl\langle T(f_{1},\ldots ,f_{j-1},g,f_{j+1}, \ldots,f_{m}),f_{j}\bigr\rangle$$

for all $$f_{1},\ldots,f_{m},g$$ in $$\mathcal{S}(R^{n})$$. Note that the kernel $$K^{*,j}$$ of $$T^{*,j}$$ is related to the kernel K of T via the identity

$$K^{*,j}(x,y_{1},\ldots,y_{j-1},y_{j},y_{j+1}, \ldots ,y_{m})=K(y_{j},y_{1}, \ldots,y_{j-1},x,y_{j+1},\ldots,y_{m}).$$

If an m-linear operator T maps a product of Banach spaces $$X_{1}\times\cdots\times X_{m}$$ to another Banach space X, then the transpose $$T^{*,j}$$ maps $$X_{1}\times\cdots\times X_{j-1}\times X\times X_{j+1}\times\cdots\times X_{m}$$ to $$X_{j}$$. Moreover, the norms of T and $$T^{*,j}$$ is equal. To maintain uniform notation, we occasionally denote T by $$T^{*,0}$$ and K by $$K^{*,0}$$.

### Assumption 1

For each $$i\in\{1,\ldots,m\}$$, there exists an operator $$\{A_{t}^{(i)}\}_{t>0}$$ with kernels $$a_{t}^{(i)}(x,y)$$ that satisfy conditions (1.6)–(1.7) with constants s, η, and that for every $$j\in\{0,1,\ldots,m\}$$, there exist kernels $$K_{t}^{*,j,(i)}(x,y_{1},\ldots,y_{m})$$ such that

\begin{aligned}[b] \bigl\langle T^{*,j}\bigl(f_{1},\ldots,A_{t}^{(i)}f_{i}, \ldots,f_{m}\bigr),f\bigr\rangle & = \int_{R^{n}} \int_{(R^{n})^{m}}K^{*,j,(i)}_{t}(x,y_{1}, \ldots ,y_{m})\\&\quad\times f_{1}(y_{1})\cdots f_{m}(y_{m})g(x)\,dy_{1}\cdots dy_{m}\,dx\end{aligned}
(1.8)

for all $$f_{1},\ldots,f_{m}$$ in $$\mathcal{S}(R^{n})$$ with $$\bigcap_{k=1}^{m}\operatorname{supp} (f_{k})\cap\operatorname{supp} (g)=\emptyset$$. Moreover, there exist a function $$\phi\in C(R)$$ with $$\operatorname{supp} \phi\subset[-1,1]$$ and constants $$\epsilon>0$$ and A such that for all $$j\in\{0,1,\ldots,m\}$$ and $$i\in\{1,\ldots,m\}$$, we have

\begin{aligned}[b] \bigl\vert K^{*,j}(x,y_{1},\ldots,y_{m})-K^{*,j,(i)}(x,y_{1}, \ldots,y_{m}) \bigr\vert &\leqslant\frac{A}{(\sum_{k=1}^{m} \vert x-y_{k} \vert )^{mn}}\sum _{k=1,k\neq i}^{m}\phi \biggl(\frac{ \vert y_{i}-y_{k} \vert }{t^{1/s}} \biggr)\\&\quad+\frac{At^{\epsilon /s}}{(\sum_{k=1}^{m} \vert x-y_{k} \vert )^{mn+\epsilon}}\end{aligned}
(1.9)

whenever $$t^{1/s}\leqslant|x-y_{i}|/2$$.

If T satisfies Assumption 1, then we will say that T is an m-linear operator with generalized Calderón–Zygmund kernel K. We denote the set of functions K satisfying (1.8) and (1.9) with parameters m, A, s, η, and ϵ by m-$$\operatorname{GCZK}(A,s,\eta,\epsilon )$$. We say that T is of class m-$$\operatorname{GCZO}(A,s,\eta,\epsilon)$$ if T has an associated kernel K in m-$$\operatorname{GCZK}(A,s,\eta,\epsilon)$$.

### Assumption 2

There exist operators $$\{B_{t}\}_{t>0}$$ with kernels $$b_{t}(x,y)$$ that satisfy conditions (1.6) and (1.7) with constants s and η. Let

$$K_{t}^{(0)}(x,y_{1},\ldots,y_{m})= \int_{R^{n}}K(z,y_{1},\ldots ,y_{m})b_{t}(x,z)\,dz.$$

We assume that the kernels $$K_{t}^{(0)}(x,y_{1},\ldots,y_{m})$$ satisfy the following estimates: there exist a function $$\phi\in C(R)$$ with $$\operatorname{supp}\phi\subset[-1,1]$$ and constants $$\epsilon>0$$ and A such that

$$\bigl\vert K_{t}^{(0)}(x,y_{1}, \ldots,y_{m}) \bigr\vert \leqslant\frac{A}{(\sum_{k=1}^{m} \vert x-y_{k} \vert )^{mn}}$$

whenever $$2t^{1/s}\leqslant\min_{1\leqslant j\leqslant m}|x-y_{j}|$$, and

\begin{aligned} \bigl\vert K(x,y_{1},\ldots,y_{m})-K^{(0)}_{t} \bigl(x',y_{1},\ldots,y_{m}\bigr) \bigr\vert & \leqslant\frac{A}{(\sum_{k=1}^{m} \vert x-y_{k} \vert )^{mn}}\sum_{k=1,k\neq i}^{m} \phi \biggl(\frac{ \vert y_{i}-y_{k} \vert }{t^{1/s}} \biggr)\\&\quad+\frac{At^{\epsilon /s}}{(\sum_{k=1}^{m} \vert x-y_{k} \vert )^{mn+\epsilon}}\end{aligned}

whenever $$2|x-x'|\leqslant t^{1/s}$$ and $$2t^{1/s}\leqslant\max_{1\leqslant j\leqslant m}|x-y_{i}|$$.

Throughout this paper, we always assume that the m-linear operator T satisfies the following assumption.

### Assumption 3

There exist $$p_{1},\ldots,p_{m}\in[1,\infty)$$ and $$p\in(0,\infty)$$ with $$1/p=\sum_{j=1}^{m}1/p_{j}$$ such that T maps $$L^{p_{1}}(R^{n})\times\cdots\times L^{p_{m}}(R^{n})$$ to $$L^{p}(R^{n})$$.

When T is of class m-$$\operatorname{GCZO}(A,s,\eta,\epsilon)$$ and satisfies Assumption 3, Duong et al.  proved that the multilinear singular integral operator T is bounded from $$L^{p_{1}}(w)\times\cdots \times L^{p_{m}}(w)$$ to $$L^{p}(w)$$, where $$w\in A_{p_{0}}$$ with $$p_{0}=\min(p_{1},\ldots,p_{m})>1$$. Grafakos et al.  obtained that T maps $$L^{p_{1}}(w_{1})\times\cdots\times L^{p_{m}}(w_{m})$$ to $$L^{p}(v_{\overrightarrow{w}})$$ ($$L^{p,\infty }(v_{\overrightarrow{w}})$$) for $$\overrightarrow{w}\in A_{\overrightarrow{p}}$$. For the boundedness of commutator generated by a BMO function, Anh and Duong  established that $$T_{\sum\overrightarrow{b}}$$ is bounded from $$L^{p_{1}}(w_{1})\times \cdots\times L^{p_{m}}(w_{m})$$ to $$L^{p}(v_{\overrightarrow{w}})$$ for $$\overrightarrow{w}\in\prod_{j=1}^{m}A_{p_{j}}$$ with $$p_{j}>1$$, $$j=1,\ldots ,m$$. Chen and Wu  proved that $$T_{\sum\overrightarrow {b}}$$ is bounded from $$L^{p_{1}}(w_{1})\times\cdots\times L^{p_{m}}(w_{m})$$ to $$L^{p}(v_{\overrightarrow{w}})$$ for $$w\in A_{\overrightarrow{p}}$$, $$\overrightarrow{b}\in\mathrm{BMO}^{m}$$.

On the other hand, for the vector-valued Calderón–Zygmund operator $$T_{\gamma}$$ in m-$$\operatorname{GCZO}(A,s,\eta,\epsilon)$$ satisfying Assumption 3. Chen et al.  proved that $$T_{\gamma}$$ is bounded from $$L^{p_{1}}(w_{1})\times\cdots\times L^{p_{m}}(w_{m})$$ to $$L^{p}(v_{\overrightarrow{w}})$$ ($$L^{p,\infty}(v_{\overrightarrow{w}})$$) for $$w\in A_{\overrightarrow{p}}$$. They also obtained the boundedness of commutators $$T_{\sum\overrightarrow{b},\gamma}$$ and $$T_{\prod \overrightarrow{b},\gamma}$$ from $$L^{p_{1}}(w_{1})\times\cdots\times L^{p_{m}}(w_{m})$$ to $$L^{p}(v_{\overrightarrow{w}})$$ for $$w\in A_{\overrightarrow{p}}$$, $$\overrightarrow{b}\in\mathrm{BMO}^{m}$$. He and Zhou  extended the results of Chen et al. to weighted Morrey spaces. They proved that $$T_{\gamma}$$ is bounded from $$L^{p_{1},\theta}(w_{1})\times\cdots\times L^{p_{m},\theta}(w_{m})$$ to $$L^{p,\theta}(v_{\overrightarrow{w}})$$ ($$WL^{p,\theta}(v_{\overrightarrow {w}})$$) for $$\overrightarrow{w}\in\prod_{j=1}^{m}A_{p_{j}}$$ with $$p_{j}>1$$, $$j=1,\ldots,m$$, where $$0<\theta<1$$. They also obtained the boundedness of the lth-order iterated BMO commutator $$T_{\prod \overrightarrow{b}_{\sigma},\gamma}$$ in weighted Morrey spaces.

The generalized weighted Morrey space $$(L^{p}(w),L^{q})^{\alpha}$$ was introduced by Feuto . Moreover, he showed that the Calderón–Zygmund operators, Marcinkiewicz operators, the maximal operators associated with Bochner–Riesz operators, and their commutators are bounded on $$(L^{p}(w),L^{q})^{\alpha}$$.

Inspired by the works mentioned, in this paper, we prove weighted norm inequalities for vector-valued multilinear singular integrals with nonsmooth kernels and commutators on generalized weighted Morrey spaces. We state our main results as follows.

### Theorem 1.1

LetTbe a multilinear operator inm-$$\operatorname{GCZO}(A,s,\eta,\epsilon)$$with kernelKsatisfying Assumption 2. Let$$p\leqslant\alpha< q\leqslant \infty$$, $$p_{1},\ldots,p_{m}\in[1,\infty)$$with$$1/p=\sum_{j=1}^{m}1/p_{j}$$, and$$\gamma_{1},\ldots,\gamma_{m}\in(1,\infty)$$with$$1/\gamma=\sum_{j=1}^{m}1/\gamma_{j}$$. Then for$$\overrightarrow {w}=(w_{1},\ldots,w_{m})\in A_{\overrightarrow{p}}$$, we have:

1. (i)

when all$$p_{j}>1$$, there exists a constantCsuch that

$$\bigl\Vert T_{\gamma}(\overrightarrow{f}) \bigr\Vert _{(L^{p}(v_{\overrightarrow {w}}),L^{q})^{\alpha}}\leqslant C\prod_{j=1}^{m} \bigl\Vert |f_{j}|_{\gamma _{j}} \bigr\Vert _{(L^{p_{j}}(w_{j}),L^{qp_{j}/p})^{\alpha p_{j}/p}};$$
2. (ii)

when some$$p_{j}=1$$, there exists a constantCsuch that

$$\bigl\Vert T_{\gamma}(\overrightarrow{f}) \bigr\Vert _{(L^{p,\infty}(v_{\overrightarrow {w}}),L^{q})^{\alpha}}\leqslant C\prod_{j=1}^{m} \bigl\Vert |f_{j}|_{\gamma _{j}} \bigr\Vert _{(L^{p_{j}}(w_{j}),L^{qp_{j}/p})^{\alpha p_{j}/p}}.$$

### Theorem 1.2

LetTbe a multilinear operator inm-$$\operatorname{GCZO}(A,s,\eta,\epsilon)$$with kernelKsatisfying Assumption 2. Let$$p\leqslant\alpha< q\leqslant \infty$$, $$p_{1},\ldots,p_{m}\in(1,\infty)$$with$$1/p=\sum_{j=1}^{m}1/p_{j}$$, and$$\gamma_{1},\ldots,\gamma_{m}\in(1,\infty)$$with$$1/\gamma=\sum_{j=1}^{m}1/\gamma_{j}$$. If$$\overrightarrow {w}=(w_{1},\ldots,w_{m})\in A_{\overrightarrow{p}}$$and$$\overrightarrow {b}\in\mathrm{BMO}^{m}$$, then there exists a constantCsuch that

$$\bigl\Vert T_{\sum\overrightarrow{b},\gamma}(\overrightarrow{f}) \bigr\Vert _{(L^{p}(v_{\overrightarrow{w}}),L^{q})^{\alpha}}\leqslant C\sum_{j=1}^{m} \Vert b_{j} \Vert _{\mathrm{BMO}}\prod _{i=1}^{m} \bigl\Vert |f_{i}|_{\gamma _{i}} \bigr\Vert _{(L^{p_{i}}(w_{i}),L^{qp_{i}/p})^{\alpha p_{i}/p}}.$$

### Theorem 1.3

LetTbe a multilinear operator inm-$$\operatorname{GCZO}(A,s,\eta,\epsilon)$$with kernelKsatisfying Assumption 2. Let$$p\leqslant\alpha< q\leqslant \infty$$, $$p_{1},\ldots,p_{m}\in(1,\infty)$$with$$1/p=\sum_{j=1}^{m}1/p_{j}$$, and$$\gamma_{1},\ldots,\gamma_{m}\in(1,\infty)$$with$$1/\gamma=\sum_{j=1}^{m}1/\gamma_{j}$$. If$$\overrightarrow {w}=(w_{1},\ldots,w_{m})\in A_{\overrightarrow{p}}$$and$$\overrightarrow {b}\in\mathrm{BMO}^{m}$$, then there exists a constantCsuch that

$$\bigl\Vert T_{\prod\overrightarrow{b},\gamma}(\overrightarrow{f}) \bigr\Vert _{(L^{p}(v_{\overrightarrow{w}}),L^{q})^{\alpha}}\leqslant C\prod_{j=1}^{m} \Vert b_{j} \Vert _{\mathrm{BMO}}\prod _{i=1}^{m} \bigl\| |f_{i}|_{\gamma _{i}} \bigr\| _{(L^{p_{i}}(w_{i}),L^{qp_{i}/p})^{\alpha p_{i}/p}}.$$

## Some preliminaries and notations

For a measurable set E, we define $$|E|$$ as the Lebesgue measure of E, and $$\chi_{E}$$ as the characteristic function of E; $$Q(x,r)$$ denotes the cube centered at x with the sidelength r, $$aQ(x,r)=Q(x,ar)$$, and $$\overrightarrow{p}=(p_{1},\ldots,p_{m})$$. For any number $$r>0$$, $$r\overrightarrow{p}=(rp_{1},\ldots,rp_{m})$$. For a locally integrable function f, $$f_{Q}$$ denotes the average $$f_{Q}=\frac{1}{|Q|}\int_{Q}f(x)\,dx$$. The letter C will denote a constant not necessarily the same at each occurrence.

By a weight we always mean a positive locally integrable function. We say that a weight w belongs to the class $$A_{p}$$ for $$1< p<\infty$$ if there is a constant C such that for all cubes Q,

$$\biggl(\frac{1}{ \vert Q \vert } \int_{Q}w(y)\,dy \biggr) \biggl(\frac{1}{ \vert Q \vert } \int _{Q}w(y)^{\frac{1}{p-1}}\,dy \biggr)^{p-1} \leqslant C.$$

In particular case, when $$p=1$$, it is understood as

$$\biggl(\frac{1}{ \vert Q \vert } \int_{Q}w(y)\,dy \biggr)\leqslant C\inf_{x\in Q}w(x).$$

If $$w\in A_{p}$$, then there exist positive constants δ and C such that

$$\frac{w(E)}{w(Q)}\leqslant C \biggl(\frac{ \vert E \vert }{ \vert Q \vert } \biggr)^{\delta}$$
(2.1)

for any measurable subset E of a ball Q. Since the classes $$A_{p}$$ increase with respect to p, we write $$A_{\infty}=\bigcup_{p\geqslant 1}A_{p}$$.

### Definition 2.1

(Multiple weights )

Let $$\overrightarrow{p}=(p_{1},\ldots,p_{m})$$ and $$1/p=1/p_{1}+\cdots +1/p_{m}$$ with $$1\leqslant p_{1},\ldots,p_{m}<\infty$$. Given$$\overrightarrow{w}=(w_{1},\ldots,w_{m})$$ with each $$w_{j}$$ being nonnegative measurable, set

$$v_{\overrightarrow{w}}=\prod_{j=1}^{m}w_{j}^{p/p_{j}}.$$

We say that $$\overrightarrow{w}$$ satisfies the $$A_{\overrightarrow{p}}$$ condition and write $$\overrightarrow{w}\in A_{\overrightarrow{p}}$$ if

$$\sup_{Q} \biggl(\frac{1}{ \vert Q \vert } \int_{Q}v_{\overrightarrow{w}}(x)\,dx \biggr)^{1/p} \prod_{j=1}^{m} \biggl( \frac{1}{ \vert Q \vert } \int _{Q}w_{j}(x)^{1-p'_{j}}\,dx \biggr)^{1/p'_{j}}< \infty,$$

where the supremum is taken over all cubes $$Q\subset R^{n}$$, and the term $$(\frac{1}{|Q|}\int_{Q}w_{j}(x)^{1-p'_{j}}\,dx )^{1/p'_{j}}$$ is understood as $$(\inf_{Q}w_{j})^{-1}$$ when $$p_{j}=1$$.

### Lemma 2.2

()

Let$$1\leqslant p_{1},\ldots ,p_{m}<\infty$$and$$\overrightarrow{w}=(w_{1},\ldots,w_{m})$$. Then the following statements are equivalent:

1. (i)

$$\overrightarrow{w}\in A_{\overrightarrow{p}}$$;

2. (ii)

$$w_{j}^{1-p'_{j}}\in A_{mp'_{j}}$$, $$j=1,\ldots,m$$, and$$v_{\overrightarrow{w}}\in A_{mp}$$,

where$$w_{j}^{1-p'_{j}}$$is understood as$$w_{j}^{1/m}\in A_{1}$$in the case$$p_{j}=1$$.

### Lemma 2.3

()

Let$$1\leqslant p_{1},\ldots ,p_{m}<\infty$$and$$\overrightarrow{w}=(w_{1},\ldots,w_{m})\in A_{\overrightarrow{p}}$$. Then there exists$$r>1$$such that$$\overrightarrow{w}\in A_{\overrightarrow{p}/r}$$.

To prove the results for commutators, we recall the definition and some basic properties of BMO function spaces. We say a locally integrable function b is in BMO if

$$\Vert b \Vert _{\mathrm{BMO}}:=\sup_{Q} \frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert b(y)-b_{Q} \bigr\vert \,dy< \infty.$$

For $$b\in\mathrm{BMO}$$, $$1< p<\infty$$, we have $$\|b\|_{\mathrm{BMO}}=\|b\| _{\mathrm{BMO}^{p}}$$, where

$$\Vert b \Vert _{\mathrm{BMO}^{p}}:=\sup_{Q} \biggl( \frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert b(x)-b_{Q} \bigr\vert ^{p}\,dx \biggr)^{1/p},$$

and for all cubes Q, if $$w\in A_{\infty}$$, then by (2.1) and the John–Nirenberg inequality we have

$$\biggl(\frac{1}{w(Q)} \int_{Q} \bigl\vert b(x)-b_{Q} \bigr\vert ^{p}w(x)\,dx \biggr)^{1/p}\leqslant C \Vert b \Vert _{\mathrm{BMO}}.$$
(2.2)

For all nonnegative integers k, by simple calculation we get

$$\vert b_{2^{k+1}Q}-b_{Q} \vert \leqslant C(k+1) \Vert b \Vert _{\mathrm{BMO}}.$$
(2.3)

### Definition 2.4

( (Generalized weighted Morrey space))

Let $$1\leqslant p\leqslant\alpha\leqslant q\leqslant\infty$$, and let w be a weight. The space $$(L^{p}(w),L^{q})^{\alpha }:=(L^{p}(w),L^{q})^{\alpha}(R^{n})$$ is defined as the set of all measurable functions f satisfying $$\|f\|_{(L^{p}(w),L^{q})^{\alpha }}<\infty$$, where

$$\Vert f \Vert _{(L^{p}(w),L^{q})^{\alpha}}:=\sup_{r>0} \Vert f \Vert _{(L^{p}(w),L^{q})^{\alpha},r}< \infty$$

with

$$\Vert f \Vert _{(L^{p}(w),L^{q})^{\alpha},r}:= \biggl[ \int_{R^{n}} \bigl( \bigl(w\bigl(B(y,r)\bigr) \bigr)^{1/\alpha-1/p-1/q} \Vert f\chi_{B(y,r)} \Vert _{L^{p}(w)} \bigr)^{q}\,dy \biggr]^{1/q}$$

for $$r>0$$, with the usual modification when $$q=\infty$$. When $$w=1$$, the space $$(L^{p},L^{q})^{\alpha}$$ was introduced in . For $$p<\alpha$$ and $$q=\infty$$, the space $$(L^{p}(w),L^{q})^{\alpha}$$ is the weighted Morrey space $$L^{q,\theta}(w)$$ with $$\theta=1/p-1/\alpha$$ defined by Komori and Shirai .

The weak space $$(L^{p,\infty}(w),L^{q})^{\alpha}$$ is defined with

$$\Vert f \Vert _{(L^{p,\infty}(w),L^{q})^{\alpha},r}:= \biggl[ \int_{R^{n}} \bigl( \bigl(w\bigl(B(y,r)\bigr) \bigr)^{1/\alpha-1/p-1/q} \Vert f\chi_{B(y,r)} \Vert _{L^{p,\infty }(w)} \bigr)^{q}\,dy \biggr]^{1/q}.$$

When $$p=1$$, the space $$(L^{1,\infty}(w),L^{q})^{\alpha}$$ was introduced in .

The following results were obtained by Chen et al.

### Theorem A

()

LetTbe a multilinear operator inm-$$\operatorname{GCZO}(A,s,\eta,\epsilon)$$with kernelKsatisfying Assumption 2. Let$$1\leqslant p_{1},\ldots,p_{m}<\infty$$with$$1/p=\sum_{j=1}^{m}1/p_{j}$$. Then for$$\overrightarrow{w}=(w_{1},\ldots ,w_{m})\in A_{\overrightarrow{p}}$$, we have:

1. (i)

If$$1< p_{j}<\infty$$, $$j=1,\ldots,m$$, then

$$\bigl\Vert T_{\gamma}(\overrightarrow{f}) \bigr\Vert _{L^{p}(v_{\overrightarrow {w}})}\leqslant C\prod_{j=1}^{m} \bigl\Vert |f_{j}|_{\gamma_{j}} \bigr\Vert _{L^{p_{j}}(w_{j})}.$$
2. (ii)

If$$1\leqslant p_{j}<\infty$$, $$j=1,\ldots,m$$, and at least one$$p_{j}=1$$, then

$$\bigl\Vert T_{\gamma}(\overrightarrow{f}) \bigr\Vert _{L^{p,\infty}(v_{\overrightarrow {w}})}\leqslant C\prod_{j=1}^{m} \bigl\Vert |f_{j}|_{\gamma_{j}} \bigr\Vert _{L^{p_{j}}(w_{j})}.$$

### Theorem B

()

LetTbe a multilinear operator inm-$$\operatorname{GCZO}(A,s,\eta,\epsilon)$$with kernelKsatisfying Assumption 2. Let$$1< p_{1},\ldots,p_{m}<\infty$$with$$1/p=\sum_{j=1}^{m}1/p_{j}$$. If$$\overrightarrow{w}=(w_{1},\ldots,w_{m})\in A_{\overrightarrow{p}}$$and$$\overrightarrow{b}\in\mathrm{BMO}^{m}$$, then there exists a constantCsuch that

$$\bigl\Vert T_{\sum\overrightarrow{b},\gamma}(\overrightarrow{f}) \bigr\Vert _{L^{p}(v_{\overrightarrow{w}})}\leqslant C\sum_{j=1}^{m} \Vert b_{j} \Vert _{\mathrm{BMO}}\prod _{i=1}^{m} \bigl\Vert |f_{i}|_{\gamma_{i}} \bigr\Vert _{L^{p_{i}}(w_{i})}.$$

### Theorem C

()

LetTbe a multilinear operator inm-$$\operatorname{GCZO}(A,s,\eta,\epsilon)$$with kernelKsatisfying Assumption 2. Let$$1< p_{1},\ldots,p_{m}<\infty$$with$$1/p=\sum_{j=1}^{m}1/p_{j}$$. If$$\overrightarrow{w}=(w_{1},\ldots,w_{m})\in A_{\overrightarrow{p}}$$and$$\overrightarrow{b}\in\mathrm{BMO}^{m}$$, then there exists a constantCsuch that

$$\bigl\Vert T_{\prod\overrightarrow{b},\gamma}(\overrightarrow{f}) \bigr\Vert _{L^{p}(v_{\overrightarrow{w}})}\leqslant C\prod_{j=1}^{m} \Vert b_{j} \Vert _{\mathrm{BMO}}\prod _{i=1}^{m} \bigl\Vert |f_{i}|_{\gamma_{i}} \bigr\Vert _{L^{p_{i}}(w_{i})}.$$

## Proof of the main results

### Proof of Theorem 1.1

(i) Let $$\{f_{1k},\ldots,f_{mk}\} _{k=1}^{\infty}$$ be any smooth vector-valued functions. For any $$Q=Q(y,r)\in R^{n}$$, We split each $$\overrightarrow {f_{k}}=\overrightarrow{f_{k}}^{0}+\overrightarrow{f_{k}}^{\infty}$$, where $$\{\overrightarrow{f_{k}}^{0}\}_{k=1}^{\infty}=\{\overrightarrow {f_{k}}\chi_{Q^{*}}\}_{k=1}^{\infty}=\{f_{1k}\chi_{Q^{*}},\ldots ,f_{mk}\chi_{Q^{*}}\}_{k=1}^{\infty}$$ and $$Q^{*}=8Q$$. Then

\begin{aligned} \prod_{j=1}^{m}f_{jk}(y_{j}) =& \prod_{j=1}^{m}\bigl(f_{jk}^{0}(y_{j})+f_{jk}^{\infty}(y_{j}) \bigr) =\sum_{\alpha_{1},\ldots,\alpha_{m}}f_{1k}^{\alpha_{1}}(y_{1}) \cdots f_{mk}^{\alpha_{m}}(y_{m}) \\ =&\prod_{j=1}^{m}f_{jk}^{0}(y_{j})+ \sum_{\alpha_{1},\ldots,\alpha _{m}\in\{0,\infty\}}f_{1k}^{\alpha_{1}}(y_{1}) \cdots f_{mk}^{\alpha _{m}}(y_{m})+\prod _{j=1}^{m}f_{jk}^{\infty}(y_{j}), \end{aligned}

where $$\alpha_{1},\ldots,\alpha_{m}$$ are not all equal to 0 or ∞ at the same time. Hence, for $$x\in Q(y,r)$$, we have

\begin{aligned} \bigl\vert T_{\gamma}(f_{1},\ldots,f_{m}) (x) \bigr\vert =& \biggl\vert T_{\gamma}\bigl(f^{0}_{1}, \ldots ,f^{0}_{m}\bigr) (x)\\ &{}+\sum _{\alpha_{1},\ldots,\alpha_{m}\in\{0,\infty\} }T_{\gamma}\bigl(f^{\alpha_{1}}_{1}, \ldots,f^{\alpha_{m}}_{m}\bigr) (x)+T_{\gamma } \bigl(f^{\infty}_{1},\ldots,f^{\infty}_{m} \bigr) \biggr\vert \\ \leqslant& \bigl\vert T_{\gamma}\bigl(f^{0}_{1}, \ldots,f^{0}_{m}\bigr) (x) \bigr\vert \\ &{}+ \biggl\vert \sum_{\alpha _{1},\ldots,\alpha_{m}\in\{0,\infty\}}T_{\gamma}\bigl(f^{\alpha _{1}}_{1}, \ldots,f^{\alpha_{m}}_{m}\bigr) (x) \biggr\vert + \bigl\vert T_{\gamma}\bigl(f^{\infty }_{1},\ldots,f^{\infty}_{m} \bigr) \bigr\vert \\ :=&I+\mathit{II}+\mathit{III}. \end{aligned}

We first estimate III. Taking $$t=(2r)^{s}$$, since $$x,z\in Q$$ and $$y_{j}\in R^{n}\setminus Q^{*}$$, for all $$j=1,\ldots,m$$, we have

$$\vert y_{j}-z \vert \geqslant \vert y_{j}-r \vert - \vert z-r \vert >7r>2t^{1/s}.$$

Hence $$\phi(|y_{j}-z|/t^{1/s})=0$$. By Assumption 2 we have

$$\bigl\vert K(x,y_{1},\ldots,y_{m})-K_{t}^{(0)}(x,y_{1}, \ldots,y_{m}) \bigr\vert \leqslant \frac{At^{\epsilon/s}}{(\sum_{k=1}^{m} \vert x-y_{k} \vert )^{mn+\epsilon}}.$$

Then for any $$x\in Q$$, by Assumption 2, we have

\begin{aligned} \mathit{III} \leqslant& \Biggl(\sum_{k=1}^{\infty} \Biggl( \int_{(R^{n}\setminus Q^{*})^{m}} \bigl\vert K(x,\overrightarrow{y})-K_{t}^{(0)}(x, \overrightarrow {y}) \bigr\vert \prod_{j=1}^{m} \bigl\vert f_{jk}(y_{j}) \bigr\vert \, d \overrightarrow{y} \Biggr)^{\gamma } \Biggr)^{1/\gamma} \\ &{}+ \Biggl(\sum_{k=1}^{\infty} \Biggl( \int_{(R^{n}\setminus Q^{*})^{m}} \bigl\vert K_{t}^{(0)}(x, \overrightarrow{y}) \bigr\vert \prod_{j=1}^{m} \bigl\vert f_{jk}(y_{j}) \bigr\vert \,d \overrightarrow{y} \Biggr)^{\gamma} \Biggr)^{1/\gamma} \\ \leqslant&C \Biggl(\sum_{k=1}^{\infty} \Biggl(\sum_{v=1}^{\infty} \int _{(8^{v+1}Q\setminus8^{v}Q)^{m}}\frac{At^{\epsilon/s}}{(\sum_{k=1}^{m} \vert x-y_{k} \vert )^{mn+\epsilon}}\prod _{j=1}^{m} \bigl\vert f_{jk}(y_{j}) \bigr\vert \,d\overrightarrow{y} \Biggr)^{\gamma} \Biggr)^{1/\gamma} \\ &{}+ \Biggl(\sum_{k=1}^{\infty} \Biggl(\sum _{v=1}^{\infty} \int _{(8^{v+1}Q\setminus8^{v}Q)^{m}}\frac{A}{(\sum_{k=1}^{m} \vert x-y_{k} \vert )^{mn}}\prod _{j=1}^{m} \bigl\vert f_{jk}(y_{j}) \bigr\vert \,d\overrightarrow {y} \Biggr)^{\gamma} \Biggr)^{1/\gamma} \\ \leqslant&C \Biggl(\sum_{k=1}^{\infty} \Biggl(\sum_{v=1}^{\infty} \biggl( \frac{ \vert Q^{*} \vert ^{\epsilon/s}}{(8^{v+1} \vert Q \vert ^{1/n})^{mn+\epsilon}}+\frac {1}{(8^{v+1} \vert Q \vert ^{1/n})^{mn}} \biggr) \int_{(8^{v+1}Q)^{m}}\prod_{j=1}^{m} \bigl\vert f_{jk}(y_{j}) \bigr\vert \, d \overrightarrow{y} \Biggr)^{\gamma} \Biggr)^{1/\gamma} \\ \leqslant&C\sum_{v=1}^{\infty} \frac{1}{ \vert 8^{v+1}Q \vert ^{m}}\prod_{j=1}^{m} \Biggl(\sum_{k=1}^{\infty} \biggl( \int _{8^{v+1}Q} \bigl\vert f_{jk}(y_{j}) \bigr\vert \,dy_{j} \biggr)^{\gamma_{j}} \Biggr)^{1/\gamma _{j}} \\ \leqslant&C\sum_{v=1}^{\infty}\prod _{j=1}^{m}\frac{1}{ \vert 8^{v+1}Q \vert } \int _{8^{v+1}Q} \vert f_{j} \vert _{\gamma_{j}}(y_{j})\,dy_{j}, \end{aligned}

and the Hölder inequality gives

$$\int_{8^{v+1}Q} \vert f_{j} \vert _{s_{j}}(y_{j})\,dy_{j}\leqslant C \biggl( \int _{8^{v+1}Q}\bigl( \vert f_{j} \vert _{s_{j}}(y_{j})\bigr)^{p_{j}}w_{j}(y_{j})\,dy_{j} \biggr)^{1/p_{j}} \biggl( \int_{8^{v+1}Q}w_{j}(y_{j})^{1-p'_{j}}\,dy_{j} \biggr)^{1/p'_{j}}.$$

By the definition of $$A_{\overrightarrow{p}}$$ we obtain

$$\mathit{III}\leqslant C\sum_{v=1}^{\infty} \frac{1}{(\int _{8^{v+1}Q}v_{\overrightarrow{w}})^{1/p}}\prod_{j=1}^{m} \bigl\Vert \vert f_{j} \vert _{\gamma_{j}} \chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{j}}(w_{j})}.$$

For II, without loss of generality, we assume that $$\alpha_{i}=\infty$$ for $$i=1,\ldots,l$$ and $$\alpha_{j}=0$$ for $$j=l+1,\ldots,m$$. For $$x\in Q(y,r)$$, by Assumption 2 we get

\begin{aligned} & \bigl\vert T_{\gamma}\bigl(f_{1k}^{\infty}, \ldots,f_{lk}^{\infty },f_{(l+1)k}^{0}, \ldots,f_{mk}^{0}\bigr) (x) \bigr\vert \\ &\quad\leqslant \Biggl(\sum_{k=1}^{\infty} \Biggl( \int_{(R^{n}\setminus Q^{*})^{m}} \bigl\vert K(x,\overrightarrow{y})-K_{t}^{(0)}(x, \overrightarrow {y}) \bigr\vert \prod_{j=1}^{m} \bigl\vert f_{jk}(y_{j}) \bigr\vert \,d \overrightarrow{y} \Biggr)^{\gamma } \Biggr)^{1/\gamma} \\ &\qquad{}+ \Biggl(\sum_{k=1}^{\infty} \Biggl( \int_{(R^{n}\setminus Q^{*})^{m}} \bigl\vert K_{t}^{(0)}(x, \overrightarrow{y}) \bigr\vert \prod_{j=1}^{m} \bigl\vert f_{jk}(y_{j}) \bigr\vert \,d \overrightarrow{y} \Biggr)^{\gamma} \Biggr)^{1/\gamma} \\ &\quad\leqslant C \Biggl(\sum_{k=1}^{\infty} \Biggl(\prod_{j=l+1}^{m} \int _{Q^{*}} \bigl\vert f_{jk}(y_{j}) \bigr\vert \,dy_{j} \biggl( \int_{(R^{n}\setminus Q^{*})^{l}} \biggl(\frac{At^{\epsilon/s}}{(\sum_{j=1}^{l} \vert x-y_{j} \vert )^{mn+\epsilon}}+\frac{A}{(\sum_{j=1}^{l} \vert x-y_{j} \vert )^{mn}} \biggr) \biggr) \\ &\qquad{}\times\prod_{j=1}^{l} \bigl\vert f_{jk}(y_{j}) \bigr\vert \,dy_{j} \Biggr)^{\gamma} \Biggr)^{1/\gamma} \\ &\quad\leqslant C\sum_{v=1}^{\infty} \biggl( \frac{ \vert Q^{*} \vert ^{\epsilon /s}}{(8^{v+1} \vert Q \vert ^{1/n})^{mn+\epsilon}}+\frac {1}{(8^{v+1} \vert Q \vert ^{1/n})^{mn}} \biggr) \\ &\qquad{}\times\prod_{j=l+1}^{m} \Biggl(\sum _{k=1}^{\infty} \biggl( \int _{Q^{*}} \bigl\vert f_{jk}(y_{j}) \bigr\vert \,dy_{j} \biggr)^{\gamma_{j}} \Biggr)^{1/\gamma _{j}} \prod_{j=1}^{l} \Biggl(\sum _{k=1}^{\infty} \biggl( \int _{8^{v+1}Q} \bigl\vert f_{jk}(y_{j}) \bigr\vert \,dy_{j} \biggr)^{\gamma_{j}} \Biggr)^{1/\gamma _{j}} \\ &\quad\leqslant C\sum_{v=1}^{\infty} \frac{1}{ \vert 8^{v+1}Q \vert ^{m}}\prod_{j=l+1}^{m} \int_{Q^{*}} \vert f_{j} \vert _{\gamma_{j}}(y_{j})\,dy_{j}\prod _{j=1}^{l} \int_{8^{v+1}Q} \vert f_{j} \vert _{\gamma_{j}}(y_{j})\,dy_{j} \\ &\quad\leqslant C\sum_{v=1}^{\infty}\prod _{j=1}^{m}\frac{1}{ \vert 8^{v+1}Q \vert } \int _{8^{v+1}Q} \vert f_{j} \vert _{\gamma_{j}}(y_{j})\,dy_{j}, \end{aligned}

and by the Hölder inequality and the definition of $$A_{\overrightarrow {p}}$$ we have

$$\mathit{II}\leqslant C\sum_{v=1}^{\infty} \frac{1}{(\int _{8^{v+1}Q}v_{\overrightarrow{w}})^{1/p}}\prod_{j=1}^{m} \bigl\Vert \vert f_{j} \vert _{\gamma_{j}} \chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{j}}(w_{j})}.$$

Combining the estimates of II and III, we obtain

\begin{aligned}[b] \bigl\vert T_{\gamma}(\overrightarrow{f}) (x) \bigr\vert &\leqslant C \bigl\vert T_{\gamma }\bigl(\overrightarrow{f}^{0}\bigr) \bigr\vert \\&\quad+C\sum_{v=1}^{\infty} \frac{1}{(\int _{8^{v+1}Q}v_{\overrightarrow{w}})^{1/p}}\prod_{j=1}^{m} \bigl\Vert \vert f_{j} \vert _{\gamma_{j}} \chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{j}}(w_{j})}.\end{aligned}
(3.1)

Taking the $$L^{p}(v_{\overrightarrow{w}})$$ norms on the cube $$Q(y,r)$$ of both sides of (3.1), by Theorem A(i) we get

\begin{aligned}[b] \bigl\Vert T_{\gamma}(\overrightarrow{f})\chi_{Q(x,r)} \bigr\Vert _{L^{p}(v_{\overrightarrow{w}})}&\leqslant C\prod_{j=1}^{m} \bigl\Vert \vert f_{j} \vert _{\gamma_{j}} \chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{j}}(w_{j})}\\&\quad+C\sum _{v=1}^{\infty}\frac{(\int_{Q}v_{\overrightarrow{w}})^{1/p}}{(\int _{8^{v+1}Q}v_{\overrightarrow{w}})^{1/p}}\prod _{j=1}^{m} \bigl\Vert \vert f_{j} \vert _{\gamma_{j}}\chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{j}}(w_{j})}.\end{aligned}
(3.2)

Multiplying both sides of (3.2) by $$v_{\overrightarrow{w}}(Q)^{1/\alpha -1/q-1/p}$$, by Lemmas 2.2 and 2.3 we get

\begin{aligned} &v_{\overrightarrow{w}}(Q)^{1/\alpha-1/q-1/p} \bigl\Vert T_{\gamma }( \overrightarrow{f})\chi_{Q(x,r)} \bigr\Vert _{L^{p}(v_{\overrightarrow{w}})} \\ &\quad\leqslant C\sum_{v=0}^{\infty} \frac{1}{8^{nk\delta(1/\alpha-1/q)}} v_{\overrightarrow{w}}\bigl(8^{v+1}Q \bigr)^{1/\alpha-1/q-1/p}\prod_{j=1}^{m} \bigl\Vert \vert f_{j} \vert _{\gamma_{j}} \chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{j}}(w_{j})} \end{aligned}

For $$\sum_{j=1}^{m}p/p_{j}=1$$, by the Hölder inequality

\begin{aligned} & \bigl\Vert v_{\overrightarrow{w}}(Q)^{1/\alpha-1/q-1/p} \bigr\Vert T_{\gamma }(\overrightarrow{f})\chi_{Q(x,r)}\|_{L^{p}(v_{\overrightarrow{w}})} \|_{L^{q}(R^{n})} \\ &\quad\leqslant C\sum_{v=0}^{\infty} \frac{1}{8^{nk\delta(1/\alpha -1/q)}}\prod_{j=1}^{m} \bigl\Vert w_{j}\bigl(8^{v+1}Q\bigr)^{p/\alpha p_{j}-1/p_{i}-p/qp_{j}} \bigl\Vert \vert f_{j} \vert _{\gamma_{j}} \chi_{8^{v+1}Q}\bigr\| _{L^{p_{j}}(w_{j})}\bigr\| _{L^{qp_{j}/p}(R^{n})}. \end{aligned}

Note that $$\sum_{v=0}^{\infty}\frac{1}{8^{nk\delta(1/\alpha-1/q)}}$$ converges. Hence

$$\bigl\Vert T_{\gamma}(\overrightarrow{f}) \bigr\Vert _{(L^{p}(v_{\overrightarrow {w}}),L^{q})^{\alpha}}\leqslant C\prod_{j=1}^{m} \bigl\Vert |f_{j}|_{\gamma _{j}} \bigr\Vert _{(L^{p_{j}}(w_{j}),L^{qp_{j}/p})^{\alpha p_{j}/p}}.$$

(ii) For any $$\lambda>0$$, by (3.1) and Theorem A(ii) we have

\begin{aligned} &\lambda v_{\overrightarrow{w}}\bigl(x\in Q(y,r): \bigl\vert T_{\gamma}( \overrightarrow {f}) (x) \bigr\vert >\lambda\bigr)^{1/p} \\ &\quad\leqslant C\prod_{j=1}^{m} \bigl\Vert \vert f_{j} \vert _{\gamma_{j}}\chi _{8^{v+1}Q} \bigr\Vert _{L^{p_{j}}(w_{j})}+C\sum _{v=1}^{\infty}\frac{(\int _{Q}v_{\overrightarrow{w}})^{1/p}}{(\int_{8^{v+1}Q}v_{\overrightarrow {w}})^{1/p}}\prod _{j=1}^{m} \bigl\Vert \vert f_{j} \vert _{\gamma_{j}}\chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{j}}(w_{j})}. \end{aligned}

Hence

\begin{aligned} & \bigl\Vert T_{\gamma}(\overrightarrow{f})\chi_{Q(y,r)} \bigr\Vert _{L^{p,\infty }(v_{\overrightarrow{w}})} \\ &\quad\leqslant C\prod_{j=1}^{m} \bigl\Vert \vert f_{j} \vert _{\gamma_{j}}\chi _{8^{v+1}Q} \bigr\Vert _{L^{p_{j}}(w_{j})}+C\sum _{v=1}^{\infty}\frac{(\int _{Q}v_{\overrightarrow{w}})^{1/p}}{(\int_{8^{v+1}Q}v_{\overrightarrow {w}})^{1/p}}\prod _{j=1}^{m} \bigl\Vert \vert f_{j} \vert _{\gamma_{j}}\chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{j}}(w_{j})}. \end{aligned}

Multiplying both sides of this inequality by $$v_{\overrightarrow {w}}(Q)^{1/\alpha-1/q-1/p}$$ and applying a similar method to (i), we complete the proof of Theorem 1.1. □

### Proof of Theorem 1.2

It suffices to prove that $$T^{j}_{b_{j},\gamma},b_{j}\in\mathrm{BMO}$$. For $$Q=Q(y,r)$$, $$x\in Q$$, we can write

\begin{aligned} T^{j}_{b_{j},\gamma}(\overrightarrow{f}) (x) =&T^{j}_{b_{j},\gamma }( \overrightarrow{f}\chi_{Q^{*}}) (x)+\sum_{\alpha_{1},\ldots,\alpha _{m}\in\{0,\infty\}} \bigl(b_{j}(x)T_{\gamma}\bigl(f_{1k}^{\alpha_{1}}, \ldots ,f_{jk}^{\alpha_{j}},\ldots,f_{mk}^{\alpha_{m}} \bigr) \\ &{}-T_{\gamma}\bigl(f_{1k}^{\alpha_{1}}, \ldots,b_{j}f_{jk}^{\alpha _{j}}, \ldots,f_{mk}^{\alpha_{m}}\bigr) (x) \bigr) \\ &{}+b_{j}(x)T_{\gamma}\bigl(f_{1k}^{\infty}, \ldots,f_{jk}^{\infty},\ldots ,f_{mk}^{\infty} \bigr)-T_{\gamma}\bigl(f_{1k}^{\infty}, \ldots,b_{j}f_{jk}^{\infty }, \ldots,f_{mk}^{\infty}\bigr) (x) \\ =&I'+\mathit{II}'+\mathit{III}', \end{aligned}

where $$\alpha_{1},\ldots,\alpha_{m}$$ are not all equal to 0 or ∞ at the same time. For III′, we have

\begin{aligned} \bigl\vert \mathit{III}' \bigr\vert \leqslant& \bigl\vert \bigl(b_{j}(x)-b_{Q}\bigr)|T_{\gamma} \bigl(f_{1k}^{\infty},\ldots ,f_{jk}^{\infty}, \ldots,f_{mk}^{\infty}\bigr) \bigr\vert + \bigl\vert T_{\gamma}\bigl(f_{1k}^{\infty }, \ldots,(b_{j}-b_{Q})f_{jk}^{\infty}, \ldots,f_{mk}^{\infty}\bigr) (x) \bigr\vert \\ \leqslant& \bigl\vert b_{j}(x)-b_{Q} \bigr\vert \sum_{v=1}^{\infty}\frac{1}{(\int _{8^{v+1}Q}v_{\overrightarrow{w}})^{1/p}} \prod_{i=1}^{m} \bigl\Vert \vert f_{i} \vert _{\gamma_{i}}\chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{i}}(w_{i})} \\ &{}+ \bigl\vert T_{\gamma}\bigl(f_{1k}^{\infty}, \ldots ,(b_{j}-b_{Q}+b_{8^{v+1}Q}-b_{8^{v+1}Q})f_{jk}^{\infty}, \ldots ,f_{mk}^{\infty}\bigr) (x) \bigr\vert . \end{aligned}

Similarly to III in (i), we have

\begin{aligned} & \bigl\vert T_{\gamma}\bigl(f_{1k}^{\infty}, \ldots ,(b_{j}-b_{Q}+b_{8^{v+1}Q}-b_{8^{v+1}Q})f_{jk}^{\infty}, \ldots ,f_{mk}^{\infty}\bigr) (x) \bigr\vert \\ &\quad\leqslant \bigl\vert T_{\gamma}\bigl(f_{1k}^{\infty}, \ldots ,(b_{j}-b_{8^{v+1}Q})f_{jk}^{\infty}, \ldots,f_{mk}^{\infty}\bigr) (x) \bigr\vert \\ &\qquad{}+ \bigl\vert T_{\gamma}\bigl(f_{1k}^{\infty}, \ldots ,(b_{8^{v+1}Q}-b_{Q})f_{jk}^{\infty}, \ldots,f_{mk}^{\infty}\bigr) (x) \bigr\vert \\ &\quad\leqslant\sum_{v=1}^{\infty} \frac{ \vert b_{8^{v+1}Q}-b_{Q} \vert }{(\int _{8^{v+1}Q}v_{\overrightarrow{w}})^{1/p}}\prod_{i=1}^{m} \bigl\Vert \vert f_{i} \vert _{\gamma_{i}} \chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{i}}(w_{i})} \\ &\qquad{}+\sum_{v=1}^{\infty} \frac{1}{ \vert 8^{v+1}Q \vert ^{m}} \int _{(8^{v+1}Q)^{m}}\prod_{i=1,i\neq j}^{m} \vert f_{i} \vert _{\gamma _{i}}(y_{i}) \vert f_{j} \vert _{\gamma _{j}}(y_{j}) \bigl\vert b_{j}(y_{j})-b_{8^{v+1}Q} \bigr\vert \,d \overrightarrow{y}. \end{aligned}

Since $$\overrightarrow{w}\in A_{\overrightarrow{p}}$$, we can select suitable $$r>1$$ such that $$\overrightarrow{w}\in A_{\overrightarrow {p}/r}$$ by Lemma 2.3, and by the Hölder inequality and Lemma 2.2 we have

\begin{aligned} &\sum_{v=1}^{\infty}\frac{1}{ \vert 8^{v+1}Q \vert ^{m}} \int_{(8^{v+1}Q)^{m}}\prod_{i=1,i\neq j}^{m} \vert f_{i} \vert _{\gamma_{i}}(y_{i}) \vert f_{j} \vert _{\gamma _{j}}(y_{j}) \bigl\vert b_{j}(y_{j})-b_{8^{v+1}Q} \bigr\vert \,d \overrightarrow{y} \\ &\quad\leqslant\sum_{v=1}^{\infty} \frac{1}{ \vert 8^{v+1}Q \vert ^{m/r}} \Biggl(\prod_{i=1,i\neq j}^{m} \int_{8^{v+1}Q} \vert f_{i} \vert _{\gamma_{i}}^{r}\,dy_{i} \Biggr)^{1/r} \biggl( \int_{8^{v+1}Q} \bigl( \vert f_{j} \vert _{\gamma _{j}}(y_{j}) \bigl\vert b_{j}(y_{j})-b_{8^{v+1}Q} \bigr\vert \bigr)^{r} \biggr)^{1/r} \\ &\quad\leqslant\sum_{v=1}^{\infty} \frac{1}{ \vert 8^{v+1}Q \vert ^{m/r}}\prod_{i=1,i\neq j}^{m} \biggl( \int_{8^{v+1}Q} \vert f_{i} \vert _{\gamma _{i}}^{p_{i}}w_{i}(y_{i})\,dy_{i} \biggr)^{1/p_{i}} \biggl( \int _{8^{v+1}Q}w_{i}(y_{i})^{-r/(p_{i}-r)} \biggr)^{(p_{i}-r)/p_{i}r} \\ &\qquad{}\times \biggl( \int _{8^{v+1}Q} \bigl\vert b_{j}(y_{j})-b_{8^{v+1}Q} \bigr\vert ^{p_{j}r/(p_{j}-r)}w_{j}(y_{j})^{-r/(p_{j}-r)}\,dy_{j} \biggr)^{(p_{i}-r)/p_{i}r}\\ &\qquad{}\times \biggl( \int_{8^{v+1}Q} \vert f_{j} \vert _{\gamma _{j}}^{p_{j}}w_{j}(y_{j})\,dy_{j} \biggr)^{1/p_{j}} \\ &\quad\leqslant C \Vert b_{j} \Vert _{\mathrm{BMO}}\sum _{v=1}^{\infty}\frac{1}{(\int _{8^{v+1}Q}v_{\overrightarrow{w}})^{1/p}}\prod _{i=1}^{m} \bigl\Vert \vert f_{i} \vert _{\gamma_{i}}\chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{i}}(w_{i})}. \end{aligned}

Hence we have

\begin{aligned} \bigl\vert \mathit{III}' \bigr\vert \leqslant& \bigl\vert b_{j}(x)-b_{Q} \bigr\vert \sum _{v=1}^{\infty}\frac{1}{(\int _{8^{v+1}Q}v_{\overrightarrow{w}})^{1/p}}\prod _{i=1}^{m} \bigl\Vert \vert f_{i} \vert _{\gamma_{i}}\chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{i}}(w_{i})} \\ &{}+\sum_{v=1}^{\infty} \frac{ \vert b_{8^{v+1}Q}-b_{Q} \vert }{(\int _{8^{v+1}Q}v_{\overrightarrow{w}})^{1/p}}\prod_{i=1}^{m} \bigl\Vert \vert f_{i} \vert _{\gamma_{i}} \chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{i}}(w_{i})} \\ &{}+ \Vert b_{j} \Vert _{\mathrm{BMO}}\sum _{v=1}^{\infty}\frac{1}{(\int _{8^{v+1}Q}v_{\overrightarrow{w}})^{1/p}}\prod _{i=1}^{m} \bigl\Vert \vert f_{i} \vert _{\gamma_{i}}\chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{i}}(w_{i})}. \end{aligned}

For II′, we now consider $$\alpha_{i}=\infty$$ for $$i=1,\ldots,l$$ and $$\alpha_{j}=0$$ for $$j=l+1,\ldots,m$$. There are two cases:

\begin{aligned} &\bigl\vert b_{j}(x)T_{\gamma}\bigl(f_{1k}^{\infty}, \ldots,f_{jk}^{\infty},\ldots ,f_{lk}^{\infty},f_{(l+1)k}^{0}, \ldots,f_{mk}^{0}\bigr)\\&\quad-T_{\gamma } \bigl(f_{1k}^{\infty},\ldots,b_{j}f_{jk}^{\infty}, \ldots,f_{lk}^{\infty },f_{(l+1)k}^{0}, \ldots,f_{mk}^{0}\bigr) (x) \bigr\vert \end{aligned}

or

\begin{aligned} &\bigl\vert b_{j}(x)T_{\gamma}\bigl(f_{1k}^{\infty}, \ldots,f_{lk}^{\infty },f_{(l+1)k}^{0}, \ldots,f_{jk}^{0},\ldots,f_{mk}^{0} \bigr)\\&\quad-T_{\gamma }\bigl(f_{1k}^{\infty}, \ldots,f_{lk}^{\infty},f_{(l+1)k}^{0}, \ldots ,b_{j}f_{jk}^{0}, \ldots,f_{mk}^{0}\bigr) (x) \bigr\vert .\end{aligned}

We just consider the following case, the other case being completely analogous:

\begin{aligned} & \bigl\vert b_{j}(x)T_{\gamma}\bigl(f_{1k}^{\infty}, \ldots,f_{jk}^{\infty},\ldots ,f_{lk}^{\infty},f_{(l+1)k}^{0}, \ldots,f_{mk}^{0}\bigr)\\ &\qquad{}-T_{\gamma } \bigl(f_{1k}^{\infty},\ldots,b_{j}f_{jk}^{\infty}, \ldots,f_{lk}^{\infty },f_{(l+1)k}^{0}, \ldots,f_{mk}^{0}\bigr) (x) \bigr\vert \\ &\quad\leqslant \bigl\vert \bigl(b_{j}(x)-b_{Q} \bigr)T_{\gamma}\bigl(f_{1k}^{\infty},\ldots ,f_{jk}^{\infty},\ldots,f_{lk}^{\infty},f_{(l+1)k}^{0}, \ldots ,f_{mk}^{0}\bigr) \bigr\vert \\ &\qquad{}+ \bigl\vert T_{\gamma}\bigl(f_{1k}^{\infty}, \ldots,(b_{j}-b_{Q})f_{jk}^{\infty }, \ldots,f_{lk}^{\infty},f_{(l+1)k}^{0}, \ldots,f_{mk}^{0}\bigr) (x) \bigr\vert \\ &\quad\leqslant \bigl\vert b_{j}(x)-b_{Q} \bigr\vert \sum_{v=1}^{\infty}\frac{1}{(\int _{8^{v+1}Q}v_{\overrightarrow{w}})^{1/p}} \prod_{i=1}^{m} \bigl\Vert \vert f_{i} \vert _{\gamma_{i}}\chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{i}}(w_{i})} \\ &\qquad{}+\sum_{v=1}^{\infty} \frac{ \vert b_{8^{v+1}Q}-b_{Q} \vert }{(\int _{8^{v+1}Q}v_{\overrightarrow{w}})^{1/p}}\prod_{i=1}^{m} \bigl\Vert \vert f_{i} \vert _{\gamma_{i}} \chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{i}}(w_{i})} \\ &\qquad{}+\sum_{v=1}^{\infty} \frac{1}{ \vert 8^{v+1}Q \vert ^{m}}\prod_{i=l+1}^{m} \int_{Q^{*}} \vert f_{i} \vert _{\gamma_{i}}(y_{i})\,dy_{i}\\ &\qquad{}\times \int _{(8^{v+1}Q)^{m}}\prod_{i=1,i\neq j}^{l} \vert f_{i} \vert _{\gamma _{i}}(y_{i}) \vert f_{j} \vert _{\gamma _{j}}(y_{j}) \bigl\vert b_{j}(y_{j})-b_{8^{v+1}Q} \bigr\vert \, d \overrightarrow{y}. \end{aligned}

Since $$\overrightarrow{w}\in A_{\overrightarrow{p}}$$, we can select suitable $$r>1$$ such that $$\overrightarrow{w}\in A_{\overrightarrow {p}/r}$$ by Lemma 2.3, and by the Hölder inequality and Lemma 2.2 we have

\begin{aligned} &\sum_{v=1}^{\infty}\frac{1}{ \vert 8^{v+1}\sqrt{n}Q \vert ^{m}} \prod_{i=l+1}^{m} \int_{Q^{*}} \vert f_{i} \vert _{\gamma_{i}}(y_{i})\,dy_{i}\\ &\qquad{}\times \int _{(8^{v+1}\sqrt{n}Q)^{m}}\prod_{i=1,i\neq j}^{l} \vert f_{i} \vert _{\gamma _{i}}(y_{i}) \vert f_{j} \vert _{\gamma_{j}}(y_{j}) \bigl\vert b_{j}(y_{j})-b_{8^{v+1}\sqrt {n}Q} \bigr\vert \, d \overrightarrow{y} \\ &\quad\leqslant\sum_{v=1}^{\infty} \frac{1}{ \vert 8^{v+1}\sqrt{n}Q \vert ^{m}}\prod_{i=1,i\neq j}^{m} \int_{8^{v+1}\sqrt{n}Q} \vert f_{i} \vert _{\gamma _{i}}(y_{i})\,dy_{i}\\ &\qquad{}\times \int_{8^{v+1}\sqrt{n}Q} \vert f_{j} \vert _{\gamma _{j}}(y_{j}) \bigl\vert b_{j}(y_{j})-b_{8^{v+1}\sqrt{n}Q} \bigr\vert \,dy_{j} \\ &\quad\leqslant C \Vert b_{j} \Vert _{\mathrm{BMO}}\sum _{v=1}^{\infty}\frac{1}{(\int _{8^{v+1}\sqrt{n}Q}v_{\overrightarrow{w}})^{1/p}}\prod _{i=1}^{m} \bigl\Vert \vert f_{i} \vert _{\gamma_{i}}\chi_{8^{v+1}\sqrt{n}Q} \bigr\Vert _{L^{p_{i}}(w_{i})}. \end{aligned}

Hence we have

\begin{aligned} \bigl\vert T^{j}_{b_{j},\gamma}(\overrightarrow{f}) (x) \bigr\vert \leqslant & \bigl\vert T^{j}_{b_{j},\gamma}( \overrightarrow{f}\chi _{Q^{*}}) (x) \bigr\vert \\ &{} + \bigl\vert b_{j}(x)-b_{Q} \bigr\vert \sum _{v=1}^{\infty}\frac{1}{(\int _{8^{v+1}Q}v_{\overrightarrow{w}})^{1/p}}\prod _{i=1}^{m} \bigl\Vert \vert f_{i} \vert _{\gamma_{i}}\chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{i}}(w_{i})} \\ &{}+\sum_{v=1}^{\infty} \frac{ \vert b_{8^{v+1}Q}-b_{Q} \vert }{(\int _{8^{v+1}Q}v_{\overrightarrow{w}})^{1/p}}\prod_{i=1}^{m} \bigl\Vert \vert f_{i} \vert _{\gamma_{i}} \chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{i}}(w_{i})} \\ &{}+ \Vert b_{j} \Vert _{\mathrm{BMO}}\sum _{v=1}^{\infty}\frac{1}{(\int _{8^{v+1}Q}v_{\overrightarrow{w}})^{1/p}}\prod _{i=1}^{m} \bigl\Vert \vert f_{i} \vert _{\gamma_{i}}\chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{i}}(w_{i})}. \end{aligned}

Taking the $$L^{p}(v_{\overrightarrow{w}})$$ norms on the cube $$Q(y,r)$$ of both sides of this inequality, by Theorem B and Lemmas 2.2 and 2.3 we have

\begin{aligned} & \bigl\Vert T^{j}_{b_{j},\gamma}(\overrightarrow{f}) \chi_{Q(y,r)} \bigr\Vert _{L^{p}(v_{\overrightarrow{w}})} \\ &\quad\leqslant C \Vert b_{j} \Vert _{\mathrm{BMO}}\prod _{i=1}^{m} \bigl\Vert \vert f_{i} \vert _{\gamma_{i}}\chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{i}}(w_{i})}\\ &\qquad{}+ \Vert b_{j} \Vert _{\mathrm{BMO}}\sum _{v=1}^{\infty}\frac{k(\int_{Q}v_{\overrightarrow {w}})^{1/p}}{(\int_{8^{v+1}Q}v_{\overrightarrow{w}})^{1/p}}\prod _{i=1}^{m} \bigl\Vert \vert f_{i} \vert _{\gamma_{i}}\chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{i}}(w_{i})}. \end{aligned}

Multiplying both sides of this inequality by $$v_{\overrightarrow {w}}(Q)^{1/\alpha-1/q-1/p}$$, by (2.1) and Lemma 2.2 we get

\begin{aligned} &v_{\overrightarrow{w}}(Q)^{1/\alpha-1/q-1/p} \bigl\Vert T^{j}_{b_{j},\gamma }( \overrightarrow{f})\chi_{Q(y,r)} \bigr\Vert _{L^{p}(v_{\overrightarrow {w}})} \\ &\quad\leqslant C\sum_{v=0}^{\infty} \frac{(k+1) \Vert b_{j} \Vert _{\mathrm {BMO}}}{8^{nk\delta(1/\alpha-1/q)}} v_{\overrightarrow{w}}\bigl(8^{v+1}Q \bigr)^{1/\alpha-1/q-1/p}\prod_{i=1}^{m} \bigl\Vert \vert f_{i} \vert _{\gamma_{i}} \chi_{8^{v+1}Q} \bigr\Vert _{L^{p_{i}}(w_{i})}. \end{aligned}

By a proof similar to that of Theorem 1.1(i) we have

$$\bigl\Vert T_{\sum\overrightarrow{b},\gamma}(\overrightarrow{f}) \bigr\Vert _{(L^{p}(v_{\overrightarrow{w}}),L^{q})^{\alpha}}\leqslant C\sum_{j=1}^{m} \Vert b_{j} \Vert _{\mathrm{BMO}}\prod _{i=1}^{m} \bigl\Vert |f_{i}|_{\gamma _{i}} \bigr\Vert _{(L^{p_{i}}(w_{i}),L^{qp_{i}/p})^{\alpha p_{i}/p}}.$$

Thus we complete the proof of Theorem 1.2. □

The proof of Theorem 1.3 also uses very similar arguments, and hence we omit the details.

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### Acknowledgements

The authors would like to express their thanks to the referees for valuable advices regarding the previous version of this paper.

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The research was supported by National Natural Science Foundation of China (Grant Nos. 11661075 and 11826202).

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