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.

1 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 [1] the multilinear operator T satisfying the following conditions was studied:

• (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) 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) 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) 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 [1] 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 [2] 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. [3]. 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 [3] 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. [4].

The vector-valued multilinear operator $T_{\gamma}$ associated with the operator T was first studied by Grafakos and Martell [5]. 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. [6] 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. [3].

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 [7] 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. [3].

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. [9] 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. [10] 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 [11] 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 [12] 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. [13] 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 [14] 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 [15]. 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:

• (i) when all$p_{j}>1$, there exists a constantCsuch that

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

• (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

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

2 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 [3])

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

([3])

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

• (i) $\overrightarrow{w}\in A_{\overrightarrow{p}}$;

• (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

([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

([15] (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 [16]. 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 [17].

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 [15].

The following results were obtained by Chen et al.

Theorem A

([13])

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:

• (i) If$1< p_{j}<\infty$, $j=1,\ldots,m$, then

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

• (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

([13])

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

([13])

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

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

3 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.