Skip to main content

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 [1] 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 [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:

  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 [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:

  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

([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:

  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

([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

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

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.

References

  1. 1.

    Grafakos, L., Torres, R.H.: Multilinear Calderón–Zygmund theory. Adv. Math. 165(1), 124–164 (2002)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Pérez, C., Torres, R.: Sharp maximal function estimates for multilinear singular integrals. Contemp. Math. 320, 323–331 (2003)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Lerner, A.K., Ombrosi, S., Pérez, C., Torres, R.H., Trujillo-González, R.: New maximal functions and multiple weights for the multilinear Calderón–Zygmund theory. Adv. Math. 220(4), 1222–1264 (2009)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Pérez, C., Pradolini, G., Torres, R., Trujillo-González, R.: End-point estimates for iterated commutators of multilinear singular integrals. Bull. Lond. Math. Soc. 46, 26–42 (2014)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Grafakos, L., Martell, J.M.: Extrapolation of weighted norm inequalities for multivariable operators and applications. J. Geom. Anal. 14, 19–46 (2004)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Cruz-Uribe, D., Martell, J.M., Pérez, C.: Extrapolation from \(A_{\infty}\) weights and applications. J. Funct. Anal. 213(2), 412–439 (2004)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Tang, L.: Weighted estimates for vector-valued commutators of multilinear operators. Proc. R. Soc. Edinb., Sect. A 138, 897–922 (2008)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Duong, X.T., Grafakos, L., Yan, L.X.: Multilinear operators with non-smooth kernels and commutators of singular integrals. Trans. Am. Math. Soc. 362, 2089–2113 (2010)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Duong, X.T., Gong, G., Grafakos, L., Li, J., Yan, L.X.: Maximal operator for multilinear singular integrals with non-smooth kernels. Indiana Univ. Math. J. 58, 2517–2541 (2009)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Grafakos, L., Liu, L.G., Yang, D.C.: Multiple weighted norm inequalities for maximal multilinear singular integrals with non-smooth kernels. Proc. R. Soc. Edinb., Sect. A 141, 755–775 (2011)

    Article  Google Scholar 

  11. 11.

    Anh, B.T., Duong, X.T.: On commutators of vector BMO functions and multilinear singular integrals with non-smooth kernels. J. Math. Anal. Appl. 371, 80–94 (2010)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Chen, S.Q., Wu, H.X.: Multiple weighted estimates for commutators of multilinear singular integrals with non-smooth kernels. J. Math. Anal. Appl. 396, 888–903 (2012)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Chen, D.X., Zou, D., Mao, S.Z.: Multiple weighted estimates for vector-valued multilinear singular integrals with non-smooth kernels and its commutators. J. Funct. Spaces Appl. (2013). https://doi.org/10.1155/2013/363916

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    He, S., Zhou, J.: Vector-valued maximal multilinear Calderón–Zygmund operator with nonsmooth kernel on weighted Morrey space. J. Pseudo-Differ. Oper. Appl. 8, 213–239 (2017)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Feuto, J.: Norm inequalities in generalized Morrey spaces. J. Fourier Anal. Appl. 4, 896–909 (2014)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Feut, J., Fofana, I., Koua, K.: Integral fractional mean functions on spaces of homogeneous type. Afr. Diaspora J. Math. 9, 8–30 (2010)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Komori, Y., Shirai, S.: Weighted Morrey spaces and a singular integral operator. Math. Nachr. 282, 219–231 (2009)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

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

Availability of data and materials

This paper is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this paper are included in the papers under a Creative Commons licence, unless indicated otherwise in a credit line to the material. If the material is not included in the papers under a Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Funding

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

Author information

Affiliations

Authors

Contributions

All authors contributed equality and significantly in writing this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Jiang Zhou.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhao, N., Zhou, J. Vector-valued multilinear singular integrals with nonsmooth kernels and commutators on generalized weighted Morrey space. J Inequal Appl 2020, 180 (2020). https://doi.org/10.1186/s13660-020-02447-0

Download citation

MSC

  • 42B20
  • 42B35

Keywords

  • Vector-valued multilinear singular integrals
  • Nonsmooth kernels
  • Generalized weighted Morrey space
  • Commutators