Weighted estimates for vector-valued multilinear square function

Abstract

Let T be the multilinear square operator, respectively with certain smooth kernels and non-smooth kernels defined in (Xue and Yan in J. Math. Anal. Appl. 422:1342-1362, 2015) and (Hormozi et al. in arXiv preprint, 2015), and let $T^{*}$ be its corresponding maximal operator. In this paper, we prove the vector-valued weighted norm boundedness for T and $T^{*}$ and also establish multiple weighted inequalities for their corresponding iterated commutator generated by the vector-valued multilinear operator and BMO function.

1 Introduction

The importance of the multilinear Littlewood-Paley g-function and related multilinear Littlewood-Paley type estimates was shown in PDE and other fields, one can see the works by Coifman et al. [3, 4], David and Journe [5], and also by Fabes et al. [6–8]. Moreover, a class of multilinear square functions was considered in [8], which was used for Kato’s problem.

Recently, Xue et al. [9] introduced the multilinear-Paley g-function with a convolution-type kernel in the following way:

$$ g({\vec{f}}) (x)= \Biggl( \int_{0}^{\infty}\Biggl\vert \frac{1}{t^{mn}} \int _{(\mathbb{R}^{n})^{m}}\psi\biggl(\frac{y_{1}}{t},\ldots,\frac{y_{m}}{t} \biggr) \prod_{j=1}^{m}f_{j}(x-y_{j}) \, d\vec{y}\Biggr\vert ^{2}\frac{dt}{t} \Biggr)^{1/2}, $$

and obtained the strong $L^{p_{1}}(\omega_{1})\times\cdots\times L^{p_{m}}(\omega _{m})$ to $L^{p}(v_{\vec{\omega}})$ boundedness and the weak type results. Later, Xue and Yan [1] studied a class of multilinear square functions associated with the following more general non-convolution-type kernels.

Definition 1

(Integral smooth condition of C-Z type I) (see [1])

For any $v\in(0,\infty)$, let $K_{v}(x,y_{1},\ldots,y_{m})$ be a locally integrable function defined away from the diagonal $x=y_{1}=\cdots=y_{m}$ in $(\mathbb {R}^{n})^{m+1}$ and denote $\vec{y}=(y_{1},\ldots,y_{m})$. We say that $K_{v}$ satisfies the integral condition of C-Z type I, if for some positive constants γ, A, and $B>1$, the following inequalities hold:

$$\begin{aligned}& \biggl( \int_{0}^{\infty}\bigl\vert K_{v}(x,\vec{y}) \bigr\vert ^{2} \biggr)^{1/2}\frac{dv}{v}\leq \frac{A}{(\sum_{j=1}^{m}|x-y_{j}|)^{mn}}, \end{aligned}$$

(1.1)

$$\begin{aligned}& \biggl( \int_{0}^{\infty}\bigl\vert K_{v}(z, \vec{y})-K_{v}(x,\vec{y})\bigr\vert ^{2}\frac {dv}{v} \biggr)^{1/2}\leq\frac{A|z-x|^{\gamma}}{(\sum_{j=1}^{m}|x-y_{j}|)^{mn+\gamma}}, \end{aligned}$$

(1.2)

whenever $|z-x|\leq\frac{1}{B}\max_{j=1}^{m}{|x-y_{j}|}$; and

$$ \biggl( \int_{0}^{\infty}\bigl\vert K_{v}(x, \vec{y})-K_{v}\bigl(x,y_{1},\ldots,y_{i}', \ldots ,y_{m}\bigr)\bigr\vert ^{2}\frac{dv}{v} \biggr)^{1/2}\leq\frac{A|y_{i}-y_{i}'|^{\gamma}}{(\sum_{j=1}^{m}|x-y_{j}|)^{mn+\gamma}} $$

(1.3)

for any $i\in\{1,\ldots,m\}$, whenever $|y_{i}-y_{i}'|\leq\frac{1}{B}|x-y_{i}|$.

We define the multilinear square function T by

$$ T(\vec{f}) (x)= \Biggl( \int_{0}^{\infty}\Biggl\vert \int_{(\mathbb {R}^{n})^{m}}K_{v}(x,y_{1}, \ldots,y_{m})\prod_{j=1}^{m} f_{j}(y_{j})\, d\vec{y}\Biggr\vert ^{2} \frac{dv}{v} \Biggr)^{1/2} $$

(1.4)

for any $\vec{f}=(f_{1},\ldots,f_{m})\in\mathcal{S}(\mathbb{R}^{n})\times \cdots\times\mathcal{S}(\mathbb{R}^{n})$ and for all $x\notin\bigcap_{j=1}^{m} \operatorname{supp} f_{j}$.

In order to state their results, we first give the definition of multiple weights $A_{\vec{p}}$.

Definition 2

(Multiple weights) (see [10])

Let $1\leq p_{1},\ldots, p_{m} <\infty$, and $\frac{1}{p}=\frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}$. For any $\vec{\omega} =(\omega_{1},\ldots,\omega_{m})$, denote $v_{\vec{\omega}}=\prod_{i=1}^{m} \omega_{i}^{p/p_{i}}$. If

$$ \sup_{B} \biggl(\frac{1}{|B|} \int_{B} v_{\vec{\omega}} \biggr)^{1/p}\prod _{i=1}^{m} \biggl(\frac{1}{|B|} \int_{B}\omega_{i}^{1-p_{i}'} \biggr)^{1/p_{i}'}< \infty $$

(1.5)

holds, we say that ω⃗ satisfies the $A_{\vec{p}}$ condition. Specially, when $p_{i}=1$, $(\frac{1}{|B|}\int_{B}\omega _{i}^{1-p_{i}'} )^{1/p_{i}'}$ is understood as $(\inf_{B} \omega_{i})^{-1}$.

We will need the easy fact: if each $\omega_{j}\in A_{p_{j}}$, then $\prod_{j=1}^{m}A_{p_{j}}\subset A_{\vec{p}}$.

In [10], the multilinear maximal operator $\mathcal{M}$ was defined by

$$ \mathcal{M}(\vec{f}) (x)=\sup_{x\in Q}\prod _{i=1}^{m} \frac{1}{|Q|} \int_{Q} \bigl\vert f_{i}(y_{i})\bigr\vert \, dy_{i}, $$

(1.6)

where the supremum is taken over all cubes Q containing x. The easy fact is that $\mathcal{M}(\vec{f})(x)\leq\prod_{i=1}^{m} Mf_{i}(x)$, where M is the Hardy-Littlewood maximal operator.

Theorem A

(see [1])

Let T be the multilinear square operator defined in (1.4) with the kernel satisfying the integral smooth condition of C-Z type I. Let $1< p_{1}, p_{2}, \ldots, p_{m}<\infty$ and $\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}$. If ω⃗ satisfies the $A_{\vec{p}}$ condition, there exists a constant C such that

$$ \bigl\Vert T(\vec{f})\bigr\Vert _{L^{p}(v_{\vec{\omega}})}\leq C\prod _{i=1}^{m}\|f_{i}\| _{L^{p_{i}}(\omega_{i})}. $$

(1.7)

Theorem B

(see [1])

Let T be the operator defined in (1.4) with the kernel satisfying the integral smooth condition of C-Z type I. Let $0<\delta<1/m$, the following inequality holds:

$$ M_{\delta}^{\sharp}T(\vec{f}) (x)\leq C\mathcal{M}(\vec{f}) (x) $$

(1.8)

for any bounded and compact supported functions $f_{i}$, $i=1, \ldots, m$.

In order to extend it to a more general case, we recall a class of integral operators $\{A_{t}\}_{t>0}$ defined in [11], where the operators $A_{t}$ associated with the kernels $a_{t}(x,y)$ are defined by

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

for every function $f\in L^{p}(\mathbb{R}^{n})$, $1\leq p\leq\infty$, and $a_{t}(x,y)$ satisfies the following size condition:

$$ \bigl\vert a_{t}(x,y)\bigr\vert \leq h_{t}(x,y):=t^{-n/s}h\biggl(\frac{|x-y|}{t^{1/s}}\biggr)\quad \text{for a fixed constant } s>0, $$

(1.9)

where h is a positive, bounded, decreasing function satisfying

$$ \lim_{r\rightarrow0}r^{n+\eta}h\bigl(r^{s} \bigr)=0 $$

(1.10)

for some $\eta>0$. The above conditions indicate that for some $C>0$ and all $0<\eta\leq\eta'$, the kernels $a_{t}(x,y)$ satisfy

$$ \bigl\vert a_{t}(x,y)\bigr\vert \leq Ct^{-n/s} \bigl(1+t^{-1/s}\vert x-y\vert \bigr)^{-n-\eta'}. $$

Assumption (H1)

Assume that for each $i=1, \ldots, m$, there exist operators $\{A_{t}^{(i)}\}_{t>0}$ with kernels $a_{t}^{(i)}(x,y)$ satisfying conditions (1.9) and (1.10) with constants s and η and that for every $i=1, \ldots, m$, there exist kernels $K_{t,v}^{(i)}$ such that

$$\begin{aligned}& \bigl\langle T\bigl(f_{1},\ldots, A_{t}^{(i)}f_{i}, \ldots, f_{m}\bigr),g\bigr\rangle \\& \quad = \int_{\mathbb{R}^{n}} \Biggl( \int_{0}^{\infty}\Biggl\vert \int_{(\mathbb {R}^{n})^{m}}K_{t,v}^{(i)}(x,y_{1}, \ldots,y_{m})\prod_{i=1}^{m}f_{i}(y_{i}) \, d\vec {y}\Biggr\vert ^{2}\frac{dv}{v} \Biggr)^{1/2}g(x) \, dx \end{aligned}$$

(1.11)

for all Schwartz functions $f_{1}, \ldots, f_{m}$, g with $\bigcap_{k=1}^{m} \operatorname{supp} f_{k}\cap \operatorname{supp} g= \emptyset$.

There exists a function $\phi\in C(\mathbb{R})$ with $\operatorname{supp} \phi\in [-1,1]$ and a constant $\epsilon>0$ so that, for every $i=1, \ldots, m$, we have

$$\begin{aligned}& \biggl( \int_{0}^{\infty}\bigl\vert K_{v}(x, \vec{y})-K_{t,v}^{(i)}(x,\vec{y})\bigr\vert ^{2} \frac {dv}{v} \biggr)^{1/2} \\& \quad \leq\frac{A}{(\sum_{j=1}^{m}|x-y_{j}|)^{mn}}\sum_{k=1,k\neq i}^{m} \phi\biggl(\frac {|y_{i}-y_{k}|}{t^{1/s}}\biggr)+\frac{At^{\epsilon/s}}{(\sum_{j=1}^{m}|x-y_{j}|)^{mn+\epsilon}}, \end{aligned}$$

(1.12)

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

Assumption (H2)

Assume that there exist operators $\{A_{t}\} _{t>0}$ with kernels $a_{t}(x,y)$ that satisfy conditions (1.9) and (1.10) with constants s and η, and there exist kernels $K_{t,v}^{(0)}(x,\vec{y})$ such that

$$ K_{t,v}^{(0)}(x,\vec{y})= \int_{\mathbb{R}^{n}} K_{v}(z,\vec{y})a_{t}(x,z)\, dz $$

(1.13)

makes sense for all $(x,\vec{y})\in(\mathbb{R}^{n})^{m+1}$ and $t>0$. Assume also that there exists a function $\phi\in C(\mathbb{R})$ and $\operatorname{supp} \phi\subset[-1,1]$ and a constant $\epsilon>0$ such that

$$ \biggl( \int_{0}^{\infty}\bigl\vert K_{t,v}^{(0)}(x, \vec{y})\bigr\vert ^{2}\frac{dv}{v} \biggr)^{1/2}\leq \frac{A}{(\sum_{k=1}^{m}|x-y_{k}|)^{mn}}, $$

(1.14)

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

$$\begin{aligned}& \biggl( \int_{0}^{\infty}\bigl\vert K_{v}(x, \vec{y})-K_{t,v}^{(0)}(x,\vec{y})\bigr\vert ^{2} \frac {dv}{v} \biggr)^{1/2} \\& \quad \leq\frac{A}{(\sum_{j=1}^{m}|x-y_{j}|)^{mn}}\sum_{k=1,k\neq i}^{m} \phi\biggl(\frac {|x-y_{k}|}{t^{1/s}}\biggr)+\frac{At^{\epsilon/s}}{(\sum_{j=1}^{m}|x-y_{j}|)^{mn+\epsilon}} \end{aligned}$$

(1.15)

for some $A>0$, whenever $2t^{1/s}\leq\max_{1\leq j\leq m}|x-y_{j}|$.

Assumption (H3)

Assume that there exist operators $\{A_{t}\} _{t>0}$ with kernels $a_{t}(x,y)$ that satisfy condition (1.9) and (1.10) with constant s and η. Also assume that there exist kernels $K_{t,v}^{(0)}$ satisfying (1.13) and positive constants A and ϵ such that

$$ \biggl( \int_{0}^{\infty}\bigl\vert K_{t,v}^{(0)}(x, \vec{y})-K_{t,v}^{(0)}\bigl(x',\vec {y}\bigr) \bigr\vert ^{2}\frac{dv}{v} \biggr)^{1/2} \leq \frac{At^{\epsilon/s}}{(\sum_{j=1}^{m}|x-y_{j}|)^{mn+\epsilon}}, $$

(1.16)

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

We say that the kernels $K_{v}$ generalized the square function kernels if they satisfy (1.1), (1.11), and (1.12) with parameters m, A, s, η, ϵ, and we denote their collection by $m-\operatorname{GSFK}(A,s,\eta,\epsilon)$. We say that T is of class $m-\operatorname{GSFO}(A,s,\eta,\epsilon)$ if T has an associated kernel $K_{v}$ in $m-\operatorname{GSFK}(A,s,\eta,\epsilon)$.

Theorem C

(see [2])

Let T be a multilinear operator in $m-\operatorname{GSFO}(A,s,\eta,\epsilon)$ with a kernel satisfying Assumptions (H2) and (H3). For $1< p_{1}, \ldots, p_{m}<\infty$, $p\geq1$ with $\frac {1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}$, $\omega\in A_{p}$, the following inequality holds:

$$ \bigl\Vert T(\vec{f})\bigr\Vert _{L^{p}(\omega)}\leq C\prod _{i=1}^{m}\|f_{i}\|_{L^{p_{i}}(\omega)}. $$

(1.17)

Theorem D

(see [2])

Let $0<\delta<1/m$ and T be a multilinear operator in $m-\operatorname{GSFO}(A,s, \eta,\epsilon)$ with a kernel satisfying Assumptions (H2) and (H3). Then there exists a constant C such that

$$ M_{\delta}^{\sharp }T(\vec{f}) (x)\leq C \prod _{j=1}^{m} Mf_{j}(x) $$

(1.18)

holds for any bounded and compact supported function $f_{i}$, $i=1, 2, \ldots, m$.

Moreover, the corresponding multilinear maximal square function $T^{*}$ is defined by

$$ T^{*}(\vec{f}) (x)=\sup_{\delta>0} \Biggl( \int_{0}^{\infty} \Biggl| \int_{\sum _{i=1}^{m}|x-y_{i}|^{2}>\delta^{2}}K_{v}(x,\vec{y})\prod _{k=1}^{m}f_{j}(y_{j})\, d\vec {y} \Biggr|^{2}\frac{dv}{v} \Biggr)^{1/2}. $$

(1.19)

Theorem E

(see [2])

Let T be a multilinear operator in $m-\operatorname{GSFO}(A,s,\eta,\epsilon)$ with a kernel satisfying Assumptions (H2) and (H3). For $1< p_{1}, \ldots, p_{m}<\infty$, $p\geq1$ with $\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}$, $\omega\in A_{p}$, the following inequality holds:

$$ \bigl\Vert T^{*}(\vec{f})\bigr\Vert _{L^{p}(\omega)}\leq C \prod _{i=1}^{m}\|f_{i}\| _{L^{p_{i}}(\omega)}. $$

(1.20)

Theorem F

(see [2])

Let T be a multilinear operator in $m-\operatorname{GSFO}(A,s,\eta,\epsilon)$ with a kernel satisfying Assumptions (H2) and (H3). For any $\eta>0$, there is a constant $C<\infty$ depending on η such that the following inequality holds:

$$ T^{*}(\vec{f}) (x)\leq C \Biggl(M_{\eta}T({\vec{f}}) (x)+\prod _{j=1}^{m}Mf_{j}(x) \Biggr),\quad \forall x \in\mathbb{R}^{n} $$

(1.21)

for all f⃗ in any product of $L^{q_{j}}(\mathbb{R}^{n})$ spaces, with $1\leq q_{j}<\infty$.

2 Main results

In this section, we first list some results about vector-valued multilinear operator $T_{q}$ and the corresponding vector-valued maximal multilinear operator $T_{q}^{*}$ which are defined, respectively, by

$$\begin{aligned}& T_{q}(\vec{f}) (x)=\bigl\Vert T(\vec{f}) (x)\bigr\Vert _{{\ell}^{q}}= \Biggl(\sum_{k=1}^{\infty } \bigl\vert T(f_{1k},\ldots,f_{mk}) (x)\bigr\vert ^{q} \Biggr)^{1/q}, \end{aligned}$$

(2.1)

$$\begin{aligned}& T_{q}^{*}(\vec{f}) (x)=\bigl\Vert T^{*}(\vec{f}) (x)\bigr\Vert _{{\ell}^{q}}= \Biggl(\sum_{k=1}^{\infty} \bigl\vert T^{*}(f_{1k},\ldots,f_{mk}) (x)\bigr\vert ^{q} \Biggr)^{1/q}, \end{aligned}$$

(2.2)

where $\vec{f}=(f_{1},\ldots,f_{m})$ with $f_{i}=\{f_{ik}\}_{k=1}^{\infty}$.

Theorem 3

Assume that T is a multilinear square operator defined in (1.4) with the kernel satisfying the integral condition of C-Z type I. Let $1< p_{1}, p_{2}, \ldots, p_{m}<\infty$, $1< q_{1}, q_{2}, \ldots, q_{m}<\infty$, and $1/m< p,q<\infty$ with $\frac{1}{p}=\frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}$, $\frac{1}{q}=\frac{1}{q_{1}}+\cdots+\frac{1}{q_{m}}$. If $(\omega_{1}^{p_{1}},\ldots,\omega_{m}^{p_{m}})\in(A_{p_{1}}, \ldots, A_{p_{m}})$, the following inequality holds:

$$ \bigl\Vert T_{q}(\vec{f})\bigr\Vert _{L^{p}(\prod_{j=1}^{m}\omega_{j}^{p})}\leq C\prod _{j=1}^{m} \bigl\Vert \Vert f_{j}\Vert _{\ell^{q_{j}}}\bigr\Vert _{L^{p_{j}}(\omega_{j}^{p_{j}})}. $$

(2.3)

Theorem 4

Assume that T is a multilinear square operator defined in (1.4) with the kernel satisfying the integral condition of C-Z type I. Let $1\leq p_{1}, p_{2}, \ldots, p_{m}<\infty$, $1< q_{1}, q_{2}, \ldots, q_{m}<\infty$, and $0< p,q<\infty$ with $\frac{1}{p}=\frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}$, $\frac{1}{q}=\frac{1}{q_{1}}+\cdots+\frac{1}{q_{m}}$.

• (i) If $1\leq p_{1}, p_{2}, \ldots, p_{m}<\infty$ and $\omega\in A_{p_{1}}\cap\cdots\cap A_{p_{m}}$, there exists a constant $C>0$ such that

  $$ \bigl\Vert T_{q}(\vec{f})\bigr\Vert _{L^{p}(\omega)}\leq C\prod _{j=1}^{m}\bigl\Vert \Vert f_{j}\Vert _{\ell ^{q_{j}}}\bigr\Vert _{L^{p_{j}}(\omega)}. $$

  (2.4)

• (ii) If at least one $p_{j}=1$ and $\omega\in A_{1}$, there exists a constant $C>0$ such that

  $$ \bigl\Vert T_{q}(\vec{f})\bigr\Vert _{L^{p,\infty}(\omega)}\leq C\prod _{j=1}^{m}\bigl\Vert \Vert f_{j}\Vert _{\ell^{q_{j}}}\bigr\Vert _{L^{p_{j}}(\omega)}. $$

  (2.5)

Next, we show the results for the multilinear square operator T with non-smooth kernels and its corresponding maximal operator $T^{*}$. Meanwhile, we also establish multiple weighted inequalities for their corresponding iterated commutator generated by the vector-valued multilinear operator and BMO function. We will state our results as follows.

Theorem 5

Let T be a multilinear operator in $m-\operatorname{GSFO}(A,s,\eta,\epsilon)$ with a kernel satisfying Assumptions (H2) and (H3). Let $1< p_{1}, p_{2}, \ldots, p_{m}<\infty$, $1< q_{1}, q_{2}, \ldots, q_{m}<\infty$ and $1/m< p,q<\infty$ with $\frac {1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}$, $\frac{1}{q}=\frac {1}{q_{1}}+\cdots+\frac{1}{q_{m}}$. If $(\omega_{1}^{p_{1}},\ldots,\omega _{m}^{p_{m}})\in(A_{p_{1}}, \ldots, A_{p_{m}})$, there exists a constant $C>0$ such that

$$ \bigl\Vert T_{q}(\vec{f})\bigr\Vert _{L^{p}(\prod_{j=1}^{m}\omega_{j}^{p})}\leq C\prod _{j=1}^{m} \bigl\Vert \Vert f_{j}\Vert _{\ell^{q_{j}}}\bigr\Vert _{L^{p_{j}}(\omega_{j}^{p_{j}})}. $$

(2.6)

A similar estimate also holds true for the corresponding maximal operator $T^{*}$.

Theorem 6

Let T be a multilinear operator in $m-\operatorname{GSFO}(A,s,\eta,\epsilon)$ with a kernel satisfying Assumptions (H2) and (H3). Let $1\leq p_{1}, p_{2}, \ldots, p_{m}<\infty$, $1< q_{1}, q_{2}, \ldots, q_{m}<\infty$ and $0< p,q<\infty$ with $\frac {1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}$, $\frac{1}{q}=\frac {1}{q_{1}}+\cdots+\frac{1}{q_{m}}$.

• (i) If $1\leq p_{1}, p_{2}, \ldots, p_{m}<\infty$ and $\omega\in A_{p_{1}}\cap\cdots\cap A_{p_{m}}$, the following inequality holds:

  $$ \bigl\Vert T_{q}(\vec{f})\bigr\Vert _{L^{p}(\omega)}\leq C\prod _{j=1}^{m}\bigl\Vert \Vert f_{j}\Vert _{\ell ^{q_{j}}}\bigr\Vert _{L^{p_{j}}(\omega)}. $$

  (2.7)

• (ii) If at least one $p_{j}=1$ and $\omega\in A_{1}$, the following inequality holds:

  $$ \bigl\Vert T_{q}(\vec{f})\bigr\Vert _{L^{p,\infty}(\omega)}\leq C\prod _{j=1}^{m}\bigl\Vert \Vert f_{j}\Vert _{\ell^{q_{j}}}\bigr\Vert _{L^{p_{j}}(\omega)}. $$

  (2.8)

Similar estimates also hold true for the corresponding maximal operators $T^{*}$.

The commutator associated with T is given by

$$\begin{aligned} T_{\Pi\vec{b}}(\vec{f}) (x)&=\bigl[b_{1},\bigl[b_{2}, \ldots\bigl[b_{\ell-1},[b_{\ell },T]_{\ell} \bigr]_{\ell-1}\cdots\bigr]_{2}\bigr]_{1}(\vec{f}) (x) \\ &= \int_{(\mathbb{R}^{n})^{m}}\prod_{j=1}^{\ell} \bigl(b_{j}(x)-b_{j}(y)\bigr)K(x,y_{1},\ldots ,y_{m})\prod_{i=1}^{m}f_{i}(y_{i}) \, d\vec{y}, \end{aligned}$$

where $1\leq\ell\leq m$.

For simplicity of notation, for the sequence $\{\vec{f}_{k}\} _{k=1}^{\infty}=\{f_{1k},\ldots,f_{mk}\}_{k=1}^{\infty}$ of vector functions, the commutator associated with a vector-valued $T_{q}$ can be defined by

$$ T_{\Pi\vec{b},q}(\vec{f}) (x)=\bigl\Vert T_{\Pi\vec{b}}(\vec{f}) (x)\bigr\Vert _{\ell ^{q}}= \Biggl(\sum_{k=1}^{\infty} \bigl\vert T_{\Pi\vec{b}}(\vec{f}_{k}) (x)\bigr\vert ^{q} \Biggr)^{1/q}. $$

(2.9)

Theorem 7

Assume that T is a multilinear operator in $m-\operatorname{GSFO}(A, s, \eta,\epsilon)$ with kernel satisfying Assumptions (H2) and (H3). Let $1/m< p<\infty$, $\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}$ with $1< p_{1},\ldots ,p_{m}<\infty$, $1/m< q<\infty$, and $\frac{1}{q}=\frac{1}{q_{1}}+\cdots +\frac{1}{q_{m}}$ with $1< q_{1},\ldots,q_{m}<\infty$. Suppose that $\vec {\omega}\in A_{\vec{p}}$ and $\vec{b}\in(\mathit{BMO})^{\ell}$.

• (i) There exists a constant $C>0$ such that

  $$ \bigl\Vert T_{\Pi\vec{b},q}(\vec{f})\bigr\Vert _{L^{p}(v_{\omega})}\leq\prod _{j=1}^{\ell}\| b_{j} \|_{\mathit{BMO}}\prod_{j=1}^{m}\bigl\Vert \Vert f\Vert _{q_{j}}\bigr\Vert _{L^{p_{j}}(M\omega_{j})}. $$

  (2.10)

• (ii) If $\omega_{j}\in A_{p_{j}}$, there exists a constant $C>0$ such that

  $$ \bigl\Vert T_{\Pi\vec{b},q}(\vec{f})\bigr\Vert _{L^{p}(v_{\omega})}\leq\prod _{j=1}^{\ell}\| b_{j} \|_{\mathit{BMO}}\prod_{j=1}^{m}\bigl\Vert \Vert f\Vert _{q_{j}}\bigr\Vert _{L^{p_{j}}(\omega_{j})}. $$

  (2.11)

3 The proof of Theorem 3

Since $\omega_{j}\in A_{p_{j}}$, by the previous statement, $\prod_{j=1}^{m} \omega_{j}^{p_{j}}\in A_{\vec{p}}$. Writing $v_{\vec{\omega}}=\prod_{j=1}^{m}(\omega_{j}^{p_{j}})^{p/p_{j}}=\prod_{j=1}^{m}\omega_{j}^{p}$, Theorem A implies that

$$ \bigl\Vert T(f)\bigr\Vert _{L^{p}(\prod_{j=1}^{m}\omega_{j}^{p})}\leq C\prod _{j=1}^{m}\|f_{j}\| _{L^{p_{j}}}\bigl( \omega_{j}^{p_{j}}\bigr). $$

(3.1)

We will apply the following lemma to get the desirable result.

Lemma 8

(see [12])

Let $\mathcal{T}$ be an m-linear operator, and let $1< s_{1},\ldots ,s_{m}<\infty$ and $1/m< s<\infty$ be fixed indices such that $\frac {1}{s}=\frac{1}{s_{1}}+\cdots+\frac{1}{s_{m}}$. For $(\omega_{1}^{s_{1}},\ldots ,\omega_{m}^{s_{m}})\in(A_{s_{1}}, \ldots, A_{s_{m}})$, the following estimate holds:

$$ \bigl\Vert \mathcal{T}(\vec{f})\bigr\Vert _{L^{s}(\prod_{j=1}^{m}\omega_{j}^{s})}\prod _{j=1}^{m}\| f_{j}\|_{L^{s_{j}}}\bigl( \omega_{j}^{s_{j}}\bigr). $$

(3.2)

Then, for all indices, $1< p_{1},\ldots,p_{m}<\infty$ and $1/m< p<\infty$ satisfy $\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}$, $1< q_{1},\ldots ,q_{m}<\infty$, and $1/m< q<\infty$ such that $\frac{1}{q}=\frac {1}{q_{1}}+\cdots+\frac{1}{q_{m}}$, and all $(\omega_{1}^{p_{1}},\ldots,\omega _{m}^{p_{m}})\in(A_{p_{1}}, \ldots, A_{p_{m}})$. Then the following inequality holds:

$$ \bigl\Vert \mathcal{T}_{q}(\vec{f})\bigr\Vert _{L^{p}(\prod_{j=1}^{m}\omega_{j}^{p})}\leq C\prod_{j=1}^{m}\bigl\Vert \Vert f_{j}\Vert _{\ell^{q_{j}}}\bigr\Vert _{L^{p_{j}}(\omega_{j}^{p_{j}})}. $$

(3.3)

4 The proof of Theorem 5

We first state the following Fefferman-Stein inequality.

Lemma 9

(see [13])

Let $0< p,\delta<\infty$ and ω be any Mockenhaupt $A_{\infty}$ weight. Then there exists a constant C independent of f such that the inequality

$$ \int_{\mathbb{R}^{n}}\bigl(M_{\delta}f(x)\bigr)^{p} \omega(x)\, dx\leq C \int_{\mathbb {R}^{n}}\bigl(M_{\delta}^{\sharp}f(x) \bigr)^{p}\omega(x)\, dx, $$

(4.1)

holds for any function f for which the left-hand side is finite.

Lemma 10

(see [12])

For $(\omega_{1},\ldots,\omega_{m})\in(A_{p_{1}},\ldots,A_{p_{m}})$ with $1\leq p_{1}, \ldots, p_{m}<\infty$ and for $0<\theta_{1}, \ldots, \theta _{m}<1$ such that $\theta_{1}+\cdots+\theta_{m}=1$, we have $\omega_{1}^{\theta _{1}}\cdots\omega_{1}^{\theta_{1}}\in A_{\max\{p_{1},\ldots,p_{m}\}}$.

Note that $(\omega_{1}^{p_{1}},\ldots,\omega_{m}^{p_{m}})\in(A_{p_{1}},\ldots ,A_{p_{m}})$, and Lemma 10 indicates that $\prod_{j=1}^{m}\omega _{j}^{p}= \prod_{j=1}^{m}(\omega_{j}^{p_{j}})^{p/p_{j}}\in A_{\max\{p_{1},\ldots,p_{m}\} }\subset A_{\infty}$.

Exploiting Lemma 3.3 in [2] and the standard argument, we obtain $\|M_{\delta}T(\vec{f})\|_{L^{p}(\prod_{j=1}^{m}\omega_{j}^{p})}<\infty$. Together with Theorem D, we have

$$\begin{aligned} \bigl\Vert T(\vec{f})\bigr\Vert _{L^{p}(\prod_{j=1}^{m}\omega_{j}^{p})} &\leq\bigl\Vert M_{\delta}T(\vec{f})\bigr\Vert _{L^{p}(\prod_{j=1}^{m}\omega_{j}^{p})} \\ &\leq C\bigl\Vert M_{\delta}^{\sharp}T(\vec{f})\bigr\Vert _{L^{p}(\prod_{j=1}^{m}\omega _{j}^{p})} \\ &\leq C\Biggl\Vert \prod_{j=1}^{m}Mf_{j} \Biggr\Vert _{L^{p}(\prod_{j=1}^{m}\omega_{j}^{p})} \\ &\leq C\prod_{j=1}^{m}\Vert Mf_{j}\Vert _{L^{p_{j}}(\omega_{j}^{p_{j}})} \\ &\leq C\prod_{j=1}^{m}\Vert f_{j}\Vert _{L^{p_{j}}(\omega_{j}^{p_{j}})}. \end{aligned}$$

By Lemma 8, we finish the proof of Theorem 5.

The estimate for $T^{*}$ will follow from Lemma 8, Theorem F, and the following argument:

$$\begin{aligned} \bigl\Vert T^{*}(\vec{f})\bigr\Vert _{L^{p}(\prod_{j=1}^{m}\omega_{j}^{p})} &\leq C \Biggl(\bigl\Vert M_{\eta}T(\vec{f})\bigr\Vert _{L^{p}(\prod_{j=1}^{m}\omega_{j}^{p})}+\Biggl\Vert \prod_{j=1}^{m}Mf_{j} \Biggr\Vert _{L^{p}(\prod_{j=1}^{m}\omega_{j}^{p})} \Biggr) \\ &\leq C \Biggl(\bigl\Vert M\bigl\vert T(\vec{f})\bigr\vert ^{\eta} \bigr\Vert ^{\frac{1}{\eta}}_{L^{\frac {p}{\eta}}(\prod_{j=1}^{m}\omega_{j}^{p})}+\Biggl\Vert \prod _{j=1}^{m}Mf_{j}\Biggr\Vert _{L^{p}(\prod _{j=1}^{m}\omega_{j}^{p})} \Biggr) \\ &\leq C \Biggl(\bigl\Vert \bigl\vert T(\vec{f})\bigr\vert ^{\eta} \bigr\Vert ^{\frac{1}{\eta}}_{L^{\frac{p}{\eta }}(\prod_{j=1}^{m}\omega_{j}^{p})}+\Biggl\Vert \prod _{j=1}^{m}Mf_{j}\Biggr\Vert _{L^{p}(\prod _{j=1}^{m}\omega_{j}^{p})} \Biggr) \\ &\leq C \Biggl(\bigl\Vert T(\vec{f})\bigr\Vert _{L^{p}(\prod_{j=1}^{m}\omega_{j}^{p})}+\Biggl\Vert \prod_{j=1}^{m}Mf_{j} \Biggr\Vert _{L^{p}(\prod_{j=1}^{m}\omega_{j}^{p})} \Biggr) \\ &\leq C\Biggl\Vert \prod_{j=1}^{m}Mf_{j} \Biggr\Vert _{L^{p}(\prod_{j=1}^{m}\omega_{j}^{p})}. \end{aligned}$$

(4.2)

5 The proofs of Theorem 4 and Theorem 6

In order to prove these theorems, first we introduce the following lemmas.

Let $\mathcal{F}$ denote a family of ordered pairs of non-negative, measurable functions $(f, g)$, if we say that for some p, $0< p<\infty$, and $\omega\in A_{\infty}$,

$$ \int_{\mathbb{R}^{n}}f(x)^{p}\omega(x)\, dx\leq C \int_{\mathbb {R}^{n}}g(x)^{p}\omega(x)\, dx, $$

(5.1)

and we denote it by $(f, g)\in\mathcal{F}$.

Lemma 11

(see [14])

Given a family $\mathcal{F}$, suppose that for some $p_{0}$, $0< p_{0}<\infty$, and for every weight $\omega\in A_{\infty}$, $(f, g)\in\mathcal{F}$. Then we have, for all $0< p,q<\infty$ and $\omega\in A_{\infty}$,

$$ \biggl\Vert \biggl(\sum_{k}(f_{k})^{q} \biggr)^{1/q}\biggr\Vert _{L^{p}(\omega)}\leq C\biggl\Vert \biggl( \sum_{k}(g_{k})^{q} \biggr)^{1/q}\biggr\Vert _{L^{p}(\omega)},\quad \bigl\{ (f_{k},g_{k}) \bigr\} _{k}\subset\mathcal{F}. $$

(5.2)

For all $0< p,q<\infty$, $0< s\leq\infty$, and $\omega\in A_{\infty}$,

$$ \biggl\Vert \biggl(\sum_{k}(f_{k})^{q} \biggr)^{1/q}\biggr\Vert _{L^{p,s}(\omega)}\leq C\biggl\Vert \biggl( \sum_{k}(g_{k})^{q} \biggr)^{1/q}\biggr\Vert _{L^{p,s}(\omega)}, \quad \bigl\{ (f_{k},g_{k}) \bigr\} _{k}\subset\mathcal{F}. $$

(5.3)

Lemma 12

(see [15])

• (i) Let $1< q<\infty$ and $1\leq p<\infty$, there is a constant $C_{r,p}$ such that

  $$ \biggl\Vert \biggl(\sum_{k}\bigl\vert Mf_{k}(x)\bigr\vert ^{q}\biggr)^{1/q}\biggr\Vert _{L^{p,\infty }(\omega)}\leq C_{q,p}\biggl\Vert \biggl(\sum _{k}\bigl\vert f_{k}(x)\bigr\vert ^{q}\biggr)^{1/q} \biggr\Vert _{L^{p}(\omega)} $$

  (5.4)

  if and only if $\omega\in A_{p}$.

• (ii) Let $1< q<\infty$ and $1< p<\infty$, there is a constant $C_{q,p}$ such that

  $$ \biggl\Vert \biggl(\sum_{k}\bigl\vert Mf_{k}(x)\bigr\vert ^{q}\biggr)^{1/q}\biggr\Vert _{L^{p}(\omega)}\leq C_{q,p}\biggl\Vert \biggl(\sum _{k}\bigl\vert f_{k}(x)\bigr\vert ^{q}\biggr)^{1/q}\biggr\Vert _{L^{p}(\omega)} $$

  (5.5)

  if and only if $\omega\in A_{p}$.

By Theorem B, Theorem D, and Theorem F, together with the argument from Section 3 and (4.2), we have

$$ \bigl\Vert \mathcal{T}(\vec{f})\bigr\Vert _{L^{p}(\omega)}\leq C\Biggl\Vert \prod_{j=1}^{m}Mf_{j} \Biggr\Vert _{L^{p}(\omega)}. $$

(5.6)

Here $\mathcal{T}$ can be replaced by T and $T^{*}$ which are from Theorem 4 and Theorem 6.

We apply Lemma 11 to $(T(\vec{f}), \prod_{j=1}^{m}Mf_{j})\in \mathcal{F}$, and by Lemma 12 we get the desirable results.

6 The proof of Theorem 7

In order to prove Theorem 7, first we will list some notations and lemmas:

$$\begin{aligned}& \mathcal{M}\bigl(\Vert \vec{f}\Vert _{\ell^{q}}\bigr) (x):=\sup _{x\in Q}\prod_{j=1}^{m} \frac {1}{|Q|} \int_{Q}\bigl\Vert f_{j}(y_{j})\bigr\Vert _{\ell^{q_{j}}}\, dy_{j}, \end{aligned}$$

(6.1)

$$\begin{aligned}& \mathcal{M}_{L(\log L)}\bigl(\Vert \vec{f}\Vert _{\ell^{q}}\bigr) (x):=\sup_{x\in Q}\prod_{j=1}^{m} \bigl\Vert \Vert f_{j}\Vert _{\ell^{q_{j}}}\bigr\Vert _{L(\log L),Q}, \end{aligned}$$

(6.2)

$$\begin{aligned}& \mathcal{M_{\rho}}\bigl(\Vert \vec{f}\Vert _{\ell^{q}}\bigr) (x) \\& \quad :=\sup_{x\in Q}\sum_{\nu =0}^{\infty}2^{-\nu n \ell} \prod_{j\in\rho}\frac{1}{|Q|} \int_{Q}\bigl\Vert f_{j}(y_{j})\bigr\Vert _{\ell^{q_{j}}}dy_{j}\prod_{j\in\rho'} \frac{1}{|2^{\nu}Q|} \int _{2^{\nu}Q}\bigl\Vert f_{j}(y_{j})\bigr\Vert _{\ell^{q_{j}}}\, dy_{j}, \end{aligned}$$

(6.3)

where $\rho=\{j_{1},\ldots,j_{\ell}\}\subset\{1,\ldots,m\}$, $1\leq\ell < m$ and $\rho'=\{1,\ldots,m\}\backslash\rho$.

Lemma 13

(see [10])

Let $1< p_{1},\ldots,p_{m}<\infty$, $\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac {1}{p_{m}}$, $\vec{P}=(p_{1},\ldots,p_{m})$, $\vec{\omega}\in A_{\vec{P}}$, and $\rho=\{j_{1},\ldots,j_{\ell}\}\subset\{1,\ldots,m\}$, $1\leq\ell < m$. Then $\mathcal{M}$, $\mathcal{M}_{L(\log L)}$, $\mathcal{M}_{\rho }$ are bounded from $L^{p_{1}}(\omega_{1})\times\cdots\times L^{p_{m}}(\omega_{m})$ to $L^{p}(v_{\omega})$.

Lemma 14

Let T be a multilinear operator in $m-\operatorname{GSFO}(A, s, \eta,\epsilon)$ with kernel satisfying Assumptions (H2) and (H3). Assume that $1\leq\ell < m$, $\rho=\{j_{1},\ldots,j_{\ell}\}$, and $1/m< q<\infty$, $1\leq q_{1},\ldots,q_{m}<\infty$ with $\frac{1}{q}=\frac {1}{q_{1}}+\cdots+\frac{1}{q_{m}}$. Then there exists a constant $C>0$ such that

$$ M_{\delta}^{\sharp}T_{q}(\vec{f}) (x)\leq C \bigl(\mathcal{M}\bigl(\Vert \vec{f}\Vert _{\ell^{q}}\bigr) (x)+ \mathcal{M}_{\rho}\bigl(\Vert \vec{f}\Vert _{\ell^{q}}\bigr) (x) \bigr). $$

(6.4)

Proof

For a point x and a cube $Q\ni x$, to obtain (6.4), it suffices to prove for $0<\delta<1/m$,

$$ \biggl(\frac{1}{|Q|} \int_{Q}\bigl\Vert T(\vec{f}) (z)-c\bigr\Vert _{\ell^{q}}^{\delta}\, dz \biggr)^{1/\delta}\leq C \bigl( \mathcal{M}\bigl(\Vert \vec{f}\Vert _{\ell^{q}}\bigr) (x)+\mathcal {M}_{\rho}\bigl(\Vert \vec{f}\Vert _{\ell^{q}}\bigr) (x) \bigr) $$

(6.5)

for some constant c to be determined later.

Write $\vec{f}_{k}=\vec{f}_{k}^{0}+\vec{f}_{k}^{\infty}$, where $\{\vec {f}_{k}^{0}\}_{k=1}^{\infty}=\{\vec{f}_{k}^{0}\chi_{Q^{*}}\}_{k=1}^{\infty}=\{ f_{1k}\chi_{Q^{*}},\ldots,f_{mk}\chi_{Q^{*}}\}$ and $Q^{*}=(8\sqrt{n}+4)Q$. Let $c=\sum_{\alpha_{1},\ldots,\alpha_{m}}T(\vec{f}^{\alpha})(x)$ and in the sum each $\alpha_{j}=0$ or ∞ and in each term there is at least one $\alpha_{j}=\infty$. Then

$$\begin{aligned}& \biggl(\frac{1}{|Q|} \int_{Q}\bigl\Vert T(\vec{f}) (z)-c\bigr\Vert _{\ell^{q}}^{\delta}\, dz \biggr)^{1/\delta} \\& \quad \leq C \biggl(\frac{1}{|Q|} \int_{Q}\bigl\vert T_{q}\bigl( \vec{f}^{0}\bigr) (z)\bigr\vert ^{\delta}\, dz \biggr)^{1/\delta} \\& \qquad {}+C\sum_{\alpha_{1},\ldots,\alpha_{m}} \biggl( \frac{1}{|Q|} \int_{Q}\bigl\Vert T\bigl(\vec{f}^{\alpha}\bigr) (z)-T \bigl(\vec{f}^{\alpha }\bigr) (x)\bigr\Vert _{\ell^{q}}^{\delta} \, dz \biggr)^{1/\delta} \\& \quad :=I+\sum_{\alpha_{1},\ldots,\alpha_{m}}\mathit{II}_{\alpha_{1},\ldots,\alpha_{m}}, \end{aligned}$$

where in each term of the last sum there is at least one $\alpha _{j}=\infty$.

Kolmogorov’s inequality and Theorem 6 implies that

$$\begin{aligned} I \leq& C\bigl\Vert T_{q}\bigl(\vec{f}^{0}\bigr)\bigr\Vert _{L^{1/m,\infty}(Q,\frac{dz}{|Q|})} \\ \leq& C\prod_{j=1}^{m} \frac{1}{|Q|} \int_{Q}\bigl\Vert f_{j}(z)\bigr\Vert _{\ell^{q_{j}}}\,dz \\ \leq& C\mathcal{M}\bigl(\Vert \vec{f}\Vert _{\ell^{q}}\bigr). \end{aligned}$$

(6.6)

We proceed to the estimate for $\mathit{II}_{\alpha_{1},\ldots,\alpha_{m}}$. Here we choose $t=[2\sqrt{n}\ell(Q)]^{s}$. If $\alpha_{1}=\cdots\alpha_{m}=\infty$, we have

$$\begin{aligned} \mathit{II}_{\infty,\ldots,\infty}&\leq\frac{C}{|Q|} \int_{Q}\bigl\Vert T\bigl(\vec{f}^{\alpha }\bigr) (z)-T \bigl(\vec{f}^{\alpha}\bigr) (x)\bigr\Vert _{\ell^{q}}\, dz \\ &\leq\frac{C}{|Q|} \int_{Q} \Biggl(\sum_{k=1}^{\infty} \bigl\vert T\bigl(\vec{f}_{k}^{\infty }\bigr) (z)-T\bigl( \vec{f}_{k}^{\infty}\bigr) (x)\bigr\vert ^{q} \Biggr)^{1/q}\, dz \\ &\leq\frac{C}{|Q|} \int_{Q} \Biggl(\sum_{k=1}^{\infty} \Biggl\vert \int_{0}^{\infty }\Biggl\vert \int_{(\mathbb{R}^{n}\backslash Q^{*})^{m}}\bigl(K_{v}(z,\vec{y})-K_{v}(x, \vec {y})\bigr)\prod_{j=1}^{m}f_{jk}(y_{j}) \, d\vec{y}\Biggr\vert ^{2}\frac{dv}{v}\Biggr\vert ^{q/2} \Biggr)^{1/q}\, dz, \end{aligned}$$

applying Minkowski’s inequality, we get

$$\begin{aligned}& \frac{C}{\vert Q\vert } \int_{Q}\Biggl(\sum_{k=1}^{\infty} \Biggl\vert \int_{0}^{\infty }\Biggr\vert \int_{(\mathbb{R}^{n}\backslash Q^{*})^{m}}\bigl(K_{v}(z,\vec{y})-K_{v}(x, \vec {y})\bigr)\prod_{j=1}^{m}f_{jk}(y_{j}) \, d\vec{y}\Biggl\vert ^{2}\frac{dv}{v} \Biggr\vert ^{q/2}\Biggr)^{1/q}\,dz \\& \quad \leq\frac{C}{\vert Q\vert } \int_{Q}\Biggl(\sum_{k=1}^{\infty} \Biggl\vert \int_{(\mathbb {R}^{n}\backslash Q^{*})^{m}}\biggl( \int_{0}^{\infty}\bigl\vert K_{v}(z,\vec {y})-K_{v}(x,\vec{y})\bigr\vert ^{2}\frac{dv}{v} \biggr)^{1/2}\prod_{j=1}^{m}f_{jk}(y_{j}) \, d\vec{y}\Biggr\vert ^{q }\Biggr)^{1/q}\,dz \\& \quad \leq\frac{C}{\vert Q\vert } \int_{Q}\Biggl(\sum_{k=1}^{\infty} \Biggl\vert \int_{(\mathbb {R}^{n}\backslash Q^{*})^{m}}\biggl( \int_{0}^{\infty}\bigl\vert K_{v}(z,\vec {y})-K_{t,v}^{(0)}(z,\vec{y})\bigr\vert ^{2} \frac{dv}{v}\biggr)^{1/2}\prod_{j=1}^{m}f_{jk}(y_{j}) \, d\vec{y}\Biggr\vert ^{q }\Biggr)^{1/q}\,dz \\& \qquad {} +\frac{C}{\vert Q\vert } \int_{Q}\Biggl(\sum_{k=1}^{\infty} \Biggl\vert \int _{(\mathbb{R}^{n}\backslash Q^{*})^{m}}\biggl( \int_{0}^{\infty} \bigl\vert K_{t,v}^{(0)}(z, \vec{y})-K_{t,v}^{(0)}(x,\vec{y})\bigr\vert ^{2} \frac {dv}{v}\biggr)^{1/2} \\& \qquad {}\times\prod_{j=1}^{m}f_{jk}(y_{j}) \, d\vec{y}\Biggr\vert ^{q }\Biggr)^{1/q}\,dz \\& \qquad {} +\frac{C}{\vert Q\vert } \int_{Q}\Biggl(\sum_{k=1}^{\infty} \Biggl\vert \int _{(\mathbb{R}^{n}\backslash Q^{*})^{m}}\biggl( \int_{0}^{\infty} \bigl\vert K_{t,v}^{(0)}(x, \vec{y})-K_{v}(x,\vec{y})\bigr\vert ^{2}\frac{dv}{v} \biggr)^{1/2}\prod_{j=1}^{m}f_{jk}(y_{j}) \, d\vec{y}\Biggr\vert ^{q }\Biggr)^{1/q}\,dz \\& \quad :=\mathit{II}_{\infty,\ldots,\infty}^{1}+\mathit{II}_{\infty,\ldots,\infty}^{2}+ \mathit{II}_{\infty ,\ldots,\infty}^{3}. \end{aligned}$$

Because of $z\in Q$ and $y_{j}\in\mathbb{R}^{n}\backslash(8\sqrt{n}+4)Q$, we obtain $|y_{j}-z|>(4\sqrt{n}+1)\ell(Q)>2t^{1/s}$ for all $j=1,\ldots ,m$. Assumption (H2) gives

$$\begin{aligned} \mathit{II}_{\infty,\ldots,\infty}^{1} &\leq\frac{C}{|Q|} \int_{Q} \int_{(\mathbb{R}^{n}\backslash Q^{*})^{m}}\frac {At^{\epsilon/s}}{(\sum_{j=1}^{m}|z-y_{j}|)^{mn+\epsilon}}\prod_{j=1}^{m} \bigl\Vert f_{j}(y_{j})\bigr\Vert _{\ell^{q_{j}}}\, d \vec{y}\,dz \\ &\leq\sum_{k=1}^{\infty}\frac{1}{2^{k\epsilon}} \prod_{j=1}^{m}\frac {1}{2^{(k+1)n}|Q^{*}|} \int_{2^{k+1}Q^{*}}\bigl\Vert f_{j}(y_{j})\bigr\Vert _{\ell^{q_{j}}}\, dy_{j} \\ &\leq C\mathcal{M}\bigl(\Vert \vec{f}\Vert _{\ell^{q}}\bigr) (x). \end{aligned}$$

Since $x, z\in Q$, $|z-x|\leq\sqrt{n}\ell(Q)\leq1/2t^{1/s}$. Noting that $|y_{j}-z|>(4\sqrt{n}+1)\ell(Q)>2t^{1/s}$ for all $j=1,\ldots,m$, applying Assumption (H3) and a similar argument to $\mathit{II}^{1}$, we have $\mathit{II}_{\infty,\ldots,\infty}^{2}\leq C\mathcal{M}(\|\vec{f}\|_{\ell ^{q}})(x)$. Similarly, we also get $\mathit{II}_{\infty,\ldots,\infty}^{3}\leq C\mathcal{M}(\|\vec{f}\|_{\ell^{q}})(x)$.

Now let us consider the typical case of $\mathit{II}_{\alpha_{1},\ldots,\alpha _{m}}$, that is, $\alpha_{1}=\cdots=\alpha_{h}=\infty$ and $\alpha_{h+1}=\cdots =\alpha_{m}=0$, $1\leq h< m$,

$$\begin{aligned} \mathit{II}_{\infty,\ldots,0} \leq&\frac{C}{|Q|} \int_{Q}\Biggl(\sum_{k=1}^{\infty} \Biggl\vert \int_{(\mathbb {R}^{n})^{m}}\biggl( \int_{0}^{\infty}\bigl\vert K_{v}(z, \vec{y})- K_{t,v}^{(0)}(z,\vec{y})\bigr\vert ^{2} \frac{dv}{v}\biggr)^{1/2} \\ &{}\times\prod_{j=1}^{h}f_{jk}^{\infty} \prod_{j=h+1}^{m}f_{jk}^{0} \, d\vec{y}\Biggr\vert ^{q }\Biggr)^{1/q}\,dz \\ &{}+\frac{C}{|Q|} \int_{Q}\Biggl(\sum_{k=1}^{\infty} \Biggl\vert \int_{(\mathbb {R}^{n})^{m}}\biggl( \int_{0}^{\infty}\bigl\vert K_{t,v}^{(0)}(z, \vec{y})- K_{t,v}^{(0)}(x,\vec{y})\bigr\vert ^{2} \frac{dv}{v}\biggr)^{1/2} \\ &{}\times\prod_{j=1}^{h}f_{jk}^{\infty} \prod_{j=h+1}^{m}f_{jk}^{0} \, d\vec{y}\Biggr\vert ^{q }\Biggr)^{1/q}\,dz \\ &{}+\frac{C}{|Q|} \int_{Q}\Biggl(\sum_{k=1}^{\infty} \Biggl\vert \int_{(\mathbb {R}^{n})^{m}}\biggl( \int_{0}^{\infty} \bigl\vert K_{t,v}^{(0)}(x, \vec{y})-K_{v}(x,\vec{y})\bigr\vert ^{2}\frac {dv}{v} \biggr)^{1/2} \\ &{}\times\prod_{j=1}^{h}f_{jk}^{\infty} \prod_{j=h+1}^{m}f_{jk}^{0} \, d\vec{y}\Biggr\vert ^{q }\Biggr)^{1/q}\,dz \\ :=&\mathit{II}_{\infty,\ldots,0}^{1}+\mathit{II}_{\infty,\ldots,0}^{2}+ \mathit{II}_{\infty,\ldots,0}^{3}. \end{aligned}$$

For $\mathit{II}_{\infty,\ldots,0}^{1}$, by Assumption (H2), we have

$$\begin{aligned} \mathit{II}_{\infty,\ldots,0}^{1} \leq&\frac{C}{|Q|} \int_{Q} \biggl( \int_{(\mathbb {R}^{n}\backslash Q^{*})^{h}}\frac{At^{\epsilon/s}\prod_{j=1}^{h}\|f_{j}\|_{\ell ^{q_{j}}}\, dy_{j}}{(\sum_{j\in\{1,\ldots,h\}}^{m}|z-y_{j}|)^{mn+\epsilon}} \\ &{}+ \int_{(\mathbb{R}^{n}\backslash Q^{*})^{h}}\frac{A\prod_{j=1}^{h}\|f_{j}\|_{\ell ^{q_{j}}}\, dy_{j}}{(\sum_{j\in\{1,\ldots,h\}}^{m}|z-y_{j}|)^{mn}} \biggr) \prod _{j=h+1}^{m} \int_{Q^{*}}\|f_{j}\|_{\ell^{q_{j}}}\, dy_{j}\, dz \\ \leq& \Biggl(\sum_{k=1}^{\infty} \frac{A|Q^{*}|^{\epsilon /n}}{(2^{k}|Q^{*}|^{1/n})^{mn+\epsilon}} \int_{(2^{k}Q^{*})^{h}}\prod_{j=1}^{h} \| f_{j}\|_{\ell^{q_{j}}}\, dy_{j} \\ &{}+\sum_{k=1}^{\infty}\frac{A}{(2^{k}|Q^{*}|^{1/n})^{mn}} \int _{(2^{k}Q^{*})^{h}}\prod_{j=1}^{h} \|f_{j}\|_{\ell^{q_{j}}}\, dy_{j} \Biggr) \prod _{j=h+1}^{m} \int_{Q^{*}}\|f_{j}\|_{\ell^{q_{j}}}\, dy_{j} \\ \leq& C\sum_{k=0}^{\infty}\prod _{j=1}^{m}\frac{1}{2^{k\epsilon}}\frac {1}{(2^{k+1}|Q^{*}|^{1/n})^{n}} \int_{2^{k+1}Q^{*}}\|f_{j}\|_{\ell^{q_{j}}}\, dy_{j} \\ &{}+ C\sum_{k=0}^{\infty}\frac{1}{2^{kn(m-h)}} \prod_{j=h+1}^{m}\frac {1}{|Q^{*}|} \int_{Q^{*}}\|f_{j}\|_{\ell^{q_{j}}}\, dy_{j} \\ &{}\times\prod_{j=1}^{h} \frac{1}{(2^{k+1}|Q^{*}|^{1/n})^{n}} \int_{2^{k+1}Q^{*}}\|f_{j}\| _{\ell^{q_{j}}}\, dy_{j} \\ \leq& C\mathcal{M}\bigl(\Vert \vec{f}\Vert _{\ell^{q}}\bigr) (x)+ \mathcal{M}_{\rho}\bigl(\Vert \vec {f}\Vert _{\ell^{q}}\bigr) (x). \end{aligned}$$

By a similar argument, we deduce that $\mathit{II}_{\infty,\ldots,0}^{3}\leq C\mathcal{M}(\|\vec{f}\|_{\ell^{q}})(x)+\mathcal{M}_{\rho}(\|\vec{f}\| _{\ell^{q}})(x)$ and $\mathit{II}_{\infty,\ldots,0}^{2}\leq C\mathcal{M}(\|\vec{f}\|_{\ell^{q}})(x)$. □

Given any positive integer m, $\forall1\leq j\leq m$, let $C_{j}^{m}$ denote the family of all finite subset $\sigma=\{\sigma(1),\ldots,\sigma (j)\}$ of j different elements. For any $\sigma\in C_{j}^{m}$ we associate the complementary sequence $\sigma'$ given by $\sigma'=\{ 1,2,\ldots,m\}\backslash\sigma$.

Lemma 15

Let $0<\delta<\epsilon<1/m$, $1/m< q<\infty$, and $\frac{1}{q}=\frac {1}{q_{1}}+\cdots+\frac{1}{q_{m}}$ with $1< q_{1},\ldots,q_{m}<\infty$. Suppose that $\vec{b}\in(\mathit{BMO})^{\ell}$. Then there exists a constant $C>0$ depending only on δ and ϵ such that

$$\begin{aligned} M_{\delta}^{\sharp}(T_{\Pi\vec{b},q}\vec{f}) (x) \leq& C\prod _{j=1}^{\ell }\|b_{j} \|_{\mathit{BMO}} \bigl(\mathcal{M}_{L(\log L)}\|\vec{f}\|_{\ell^{q}}(x) +M_{\epsilon}\bigl(T_{q}(\vec{f})\bigr) (x) \bigr) \\ &{} +C\sum_{i=1}^{\ell-1}\sum _{\sigma\in C_{i}^{\ell}}C_{i,\ell}\prod_{j\in\sigma} \|b_{j}\|_{\mathit{BMO}}M_{\epsilon}(T_{\Pi\vec{b}_{\sigma'},q}\vec{f}) (x) \end{aligned}$$

(6.7)

for any smooth vector function $\{f_{k}\}_{k=1}^{\infty}$ for any $x\in \mathbb{R}^{n}$.

Proof

For simplicity of notation, we replace $\prod_{j=1}^{m}f_{j}(y_{j})$ by $F(\vec{y})$ and let $\lambda_{j}=\frac{1}{2|Q|}\int_{2Q}b_{j}(z)\,dz$, for $j=1,\ldots,\ell$. Let $x\in\mathbb{R}^{n}$ and Q be a cube centered at x. We have

$$\begin{aligned} T_{\Pi\vec{b}}(\vec{f}) (x) =&\Biggl( \int_{0}^{\infty}\Biggl\vert \int_{(\mathbb {R}^{n})^{m}}\prod_{j=1}^{\ell} \bigl(b_{j}(x)-b_{j}(y_{j})\bigr)K_{v}(x, \vec{y})F(\vec{y}) \, d\vec{y}\Biggr\vert ^{2}\frac{dv}{v} \Biggr)^{1/2} \\ =&\Biggl( \int_{0}^{\infty}\Biggl\vert \int_{(\mathbb{R}^{n})^{m}}\prod_{j=1}^{\ell} \bigl(\bigl(b_{j}(x)-\lambda_{j}\bigr)-\bigl(b_{j}(y_{j})- \lambda_{j}\bigr)\bigr) K_{v}(x,\vec{y})F(\vec{y})\, d \vec{y}\Biggr\vert ^{2}\frac{dv}{v}\Biggr)^{1/2} \\ \leq&\sum_{i=0}^{\ell}\sum _{\sigma\in C_{i}^{\ell}}\prod_{j\in\sigma }\bigl\vert b_{j}(x)-\lambda_{j}\bigr\vert \\ &{} \times \biggl( \int_{0}^{\infty}\biggl\vert \int_{(\mathbb{R}^{n})^{m}}\prod_{j\in\sigma'} \bigl(b_{j}(y_{j})-\lambda_{j} \bigr)K_{v}(x,\vec{y})F(\vec{y})\, d\vec{y}\biggr\vert ^{2} \frac {dv}{v}\biggr)^{1/2} \\ =&\prod_{j=1}^{\ell}\bigl\vert b_{j}(x)-\lambda_{j}\bigr\vert T(\vec{f}) (x)+T\Biggl( \prod_{j=1}^{\ell }\bigl(b_{j}( \cdot_{j})-\lambda_{j}\bigr)\vec{f}\Biggr) (x) \\ &{} +\sum_{i=1}^{\ell-1}\sum _{\sigma\in C_{i}^{\ell}}\prod_{j\in \sigma}\bigl\vert b_{j}(x)-\lambda_{j}\bigr\vert \\ &{} \times \biggl( \int_{0}^{\infty}\biggl\vert \int_{(\mathbb{R}^{n})^{m}}\prod_{j\in\sigma'} \bigl(b_{j}(y_{j})-\lambda_{j} \bigr)K_{v}(x,\vec{y})F(\vec{y})\, d\vec{y}\biggr\vert ^{2} \frac {dv}{v}\biggr)^{1/2}. \end{aligned}$$

Noting that $b_{j}(y_{j})-\lambda_{j}=(b_{j}(y_{j})-b_{j}(x))+(b_{j}(x)-\lambda_{j})$, we get

$$\begin{aligned} T_{\Pi\vec{b},q}(\vec{f}) (z) =&\prod_{j=1}^{\ell} \bigl\vert b_{j}(z)-\lambda_{j}\bigr\vert T_{q}(\vec{f}) (z)+T_{q}\Biggl(\prod _{j=1}^{\ell}\bigl(b_{j}( \cdot_{j})-\lambda_{j}\bigr)\vec{f}\Biggr) (z) \\ &{} +\sum_{i=1}^{\ell-1}\sum _{\sigma\in C_{i}^{\ell}}C_{i,\ell}\prod_{j\in\sigma} \bigl\vert b_{j}(z)-\lambda_{j}\bigr\vert T_{\Pi\vec{b}_{\sigma '},q}\vec{f}(z). \end{aligned}$$

Here $C_{i,\ell}$ depends only on i and ℓ.

Let $c_{0}=\|c\|_{\ell^{q}}=(\sum_{k=1}^{\infty}|c_{k}|^{q})^{1/q}$. Since $0<\delta<1/m<1$, we have

$$\begin{aligned}& \biggl(\frac{1}{|Q|} \int_{Q}\bigl\vert \bigl\vert T_{\Pi\vec{b},q}\vec{f}(z) \bigr\vert ^{\delta }-|c_{0}|^{\delta}\bigr\vert \,dz \biggr)^{1/\delta} \\& \quad \leq C \biggl(\frac{1}{|Q|} \int_{Q}\bigl\Vert T_{\Pi\vec{b}}(\vec{f}) (z)-c\bigr\Vert _{\ell ^{q}}^{\delta}\,dz \biggr)^{1/\delta} \\& \quad \leq C \Biggl(\frac{1}{|Q|} \int_{Q}\Biggl\Vert \prod_{j=1}^{\ell} \bigl\vert b_{j}(x)-\lambda _{j}\bigr\vert T(\vec{f}) (z)\Biggr\Vert _{\ell^{q}}^{\delta}\,dz \Biggr)^{1/\delta} \\& \qquad {}+C\sum_{i=1}^{\ell-1}\sum _{\sigma\in C_{i}^{\ell}}C_{i,\ell} \biggl(\frac{1}{|Q|} \int_{Q}\prod_{j\in\sigma} \bigl(\bigl\vert b_{j}(z)-\lambda_{j}\bigr\vert T_{\Pi \vec{b}_{\sigma'},q} \vec{f}(z) \bigr)^{\delta}\,dz \biggr)^{1/\delta} \\& \qquad {}+C \Biggl(\frac{1}{|Q|} \int_{Q}\Biggl\Vert T\Biggl(\prod _{j=1}^{\ell}\bigl(b_{j}(\cdot _{j})-\lambda_{j}\bigr)\vec{f}\Biggr) (z)-c\Biggr\Vert _{\ell^{q}}^{\delta}\,dz \Biggr)^{1/\delta} \\& \quad :=I+\mathit{II}+\mathit{III}. \end{aligned}$$

Now exploiting the standard Hölder inequality for finitely many functions with $1< p<\epsilon/\delta$, it follows that

$$\begin{aligned}& I\leq C\prod_{j=1}^{\ell}\|b_{j} \|_{\mathit{BMO}}M_{\epsilon}(T_{q}\vec{f}) (x), \\& \mathit{II}\leq C\sum_{i=1}^{\ell-1}\sum _{\sigma\in C_{i}^{\ell}}C_{j,\ell}\prod _{j\in\sigma}\|b_{j}\|_{\mathit{BMO}}M_{\epsilon}(T_{\Pi\vec{b}_{\sigma '}} \vec{f}) (x). \end{aligned}$$

Next let us address part III. Set $\vec{f}_{j}=\vec{f}_{j}^{0}+\vec {f}_{j}^{\infty}$, where $\vec{f}_{j}^{0}=\vec{f}_{j}\chi_{Q^{*}}$. Let $\vec {f}^{\alpha}=f_{1}^{\alpha_{1}}\cdots f_{m}^{\alpha_{m}}$ and $Q^{*}=(8\sqrt{n}+4)Q$. Taking $c_{0}=\sum_{\alpha_{1},\ldots,\alpha_{m}}\| T((b_{1}(\cdot_{1})-\lambda_{1})\cdots(b_{\ell}(\cdot_{\ell})-\lambda_{\ell }))f_{1}^{\alpha_{1}}\cdots f_{m}^{\alpha_{m}}(x)\|_{\ell^{q}}$, we have

$$\begin{aligned}& \Biggl\Vert T\Biggl(\prod_{j=1}^{\ell} \bigl(b_{j}(\cdot_{j})-\lambda_{j}\bigr)\vec{f} \Biggr) (z)-c\Biggr\Vert _{\ell ^{q}} \\& \quad \leq\Biggl\vert T_{q}\Biggl(\prod _{j=1}^{\ell}\bigl(b_{j}( \cdot_{j})-\lambda_{j}\bigr)\vec{f}^{0}\Biggr) (z) \Biggr\vert \\& \qquad {} + C\sum_{\alpha_{1},\ldots,\alpha_{m}}\Biggl\Vert T\Biggl(\prod _{j=1}^{\ell }\bigl(b_{j}( \cdot_{j})-\lambda_{j}\bigr)\vec{f}^{\alpha}\Biggr) (z)-T\Biggl(\prod_{j=1}^{\ell } \bigl(b_{j}(\cdot_{j})-\lambda_{j}\bigr) \vec{f}^{\alpha}\Biggr) (x)\Biggr\Vert _{\ell^{q}}, \end{aligned}$$

where in the last sum each $\alpha_{j}=0\mbox{ or }\infty$ and in each term there is at least one $\alpha_{j}=\infty$.

If $\alpha_{1}=\cdots=\alpha_{m}=0$, Kolmogorov’s inequality and Theorem 6 imply

$$\begin{aligned}& \Biggl(\frac{1}{|Q|}\Biggl\vert T_{q}\Biggl(\prod _{j=1}^{\ell}\bigl(b_{j}( \cdot_{j})-\lambda_{j}\bigr)\vec {f}^{0}\Biggr) (z)\Biggr\vert ^{\delta} \Biggr)^{1/\delta} \\& \quad \leq C\Biggl\Vert T_{q}\Biggl(\prod _{j=1}^{\ell}\bigl(b_{j}( \cdot_{j})-\lambda_{j}\bigr)\vec{f}^{0}\Biggr) \Biggr\Vert _{L^{1/m,\infty}(Q,\frac{dz}{|Q|})} \\& \quad \leq C \prod_{j=1}^{\ell} \|b_{j}\|_{\mathit{BMO}}\bigl\Vert \Vert f_{j}\Vert _{\ell ^{q_{j}}}\bigr\Vert _{L(\log L),Q}\prod_{j=\ell+1}^{m} \frac{1}{|Q|} \int_{Q}\| f_{j}\|_{\ell^{q_{j}}}\,dz \\& \quad \leq C\prod_{j=1}^{\ell} \|b_{j}\|_{\mathit{BMO}}\mathcal{M}_{L(\log L)}\bigl(\Vert f_{j}\Vert _{\ell^{q_{j}}}\bigr) (x). \end{aligned}$$

If $\alpha_{1}=\cdots=\alpha_{m}=\infty$, applying Hölder’s inequality and Minkowski’s inequality, then we get

$$\begin{aligned}& \Biggl(\frac{1}{|Q|} \int_{Q}\Biggl\Vert T\Biggl(\prod _{j=1}^{\ell}\bigl(b_{j}( \cdot_{j})-\lambda _{j}\bigr)\vec{f}^{\alpha}\Biggr) (z)-T\Biggl(\prod_{j=1}^{\ell} \bigl(b_{j}(y_{j})-\lambda_{j}\bigr) \vec{f}^{\alpha}\Biggr) (x)\Biggr\Vert _{\ell^{q}}^{\delta} \Biggr)^{1/\delta} \\ & \quad \leq\frac{C}{|Q|} \int_{Q}\Biggl(\sum_{k=1}^{\infty} \Biggl( \int_{0}^{\infty }\Biggl\vert \int_{(\mathbb{R}^{n}\backslash Q^{*})^{m}}\bigl\vert K_{v}(z,\vec{y})-K_{v}(x, \vec {y})\bigr\vert \\ & \qquad {} \times\prod_{j=1}^{\ell} \bigl(b_{j}(y_{j})-\lambda_{j} \bigr)f_{1k}(y_{1})\cdots f_{mk}(y_{m}) \, d\vec{y}\Biggr\vert ^{2}\frac{dv}{v}\Biggr)^{q/2} \Biggr)^{1/q}\,dz \\ & \quad \leq\frac{C}{|Q|} \int_{Q}\Biggl(\sum_{k=1}^{\infty} \Biggl\vert \int_{(\mathbb {R}^{n}\backslash Q^{*})^{m}}\biggl( \int_{0}^{\infty}\bigl\vert K_{v}(z, \vec{y})-K_{v}(x,\vec {y})\bigr\vert ^{2}\frac{dv}{v} \biggr)^{1/2} \\ & \qquad {} \times\prod_{j=1}^{\ell} \bigl(b_{j}(y_{j})-\lambda_{j} \bigr)f_{1k}(y_{1})\cdots f_{mk}(y_{m}) \, d\vec{y}\Biggr\vert ^{q}\Biggr)^{1/q}\,dz \\ & \quad \leq\frac{C}{|Q|} \int_{Q}\Biggl(\sum_{k=1}^{\infty} \Biggl\vert \int_{(\mathbb {R}^{n}\backslash Q^{*})^{m}}\biggl( \int_{0}^{\infty}\bigl\vert K_{v}(z,\vec {y})-K_{v,t}^{0}(z,\vec{y})\bigr\vert ^{2} \frac{dv}{v}\biggr)^{1/2} \\ & \qquad {} \times\prod_{j=1}^{\ell} \bigl(b_{j}(y_{j})-\lambda_{j} \bigr)f_{1k}(y_{1})\cdots f_{mk}(y_{m}) \, d\vec{y}\Biggr\vert ^{q}\Biggr)^{1/q}\,dz \\ & \qquad {} + \frac{C}{|Q|} \int_{Q}\Biggl(\sum_{k=1}^{\infty} \Biggl\vert \int _{(\mathbb{R}^{n}\backslash Q^{*})^{m}}\biggl( \int_{0}^{\infty}\bigl\vert K_{v,t}^{0}(z, \vec {y})-K_{v,t}^{0}(x,\vec{y})\bigr\vert ^{2} \frac{dv}{v}\biggr)^{1/2} \\ & \qquad {} \times\prod_{j=1}^{\ell} \bigl(b_{j}(y_{j})-\lambda_{j} \bigr)f_{1k}(y_{1})\cdots f_{mk}(y_{m}) \, d\vec{y}\Biggr\vert ^{q}\Biggr)^{1/q}\,dz \\ & \qquad {} + \frac{C}{|Q|} \int_{Q}\Biggl(\sum_{k=1}^{\infty} \Biggl\vert \int _{(\mathbb{R}^{n}\backslash Q^{*})^{m}}\biggl( \int_{0}^{\infty}\bigl\vert K_{v,t}^{0}(x, \vec {y})-K_{v}(x,\vec{y})\bigr\vert ^{2}\frac{dv}{v} \biggr)^{1/2} \\ & \qquad {} \times\prod_{j=1}^{\ell} \bigl(b_{j}(y_{j})-\lambda_{j} \bigr)f_{1k}(y_{1})\cdots f_{mk}(y_{m}) \, d\vec{y}\Biggr\vert ^{q}\Biggr)^{1/q}\,dz \\ & \quad :=\mathit{III}_{1}+\mathit{III}_{2}+ \mathit{III}_{3}. \end{aligned}$$

First we consider $\mathit{III}_{1}$. Taking $t=[2\sqrt{n}\ell(Q)]^{s}$, by Assumption (H2) we have

$$\begin{aligned} \mathit{III}_{1} \leq& \frac{C}{|Q|} \int_{Q} \int_{(\mathbb{R}^{n}\backslash Q^{*})^{m}}\frac {At^{\epsilon/s}}{(\sum_{j=1}^{m}|z-y_{j}|)^{mn+\epsilon}} \prod _{j=1}^{\ell}\bigl\vert b_{j}(y_{j})- \lambda_{j}\bigr\vert \|f_{1}\|_{\ell^{q_{1}}}\cdots\| f_{m}\|_{\ell^{q_{m}}}\, d\vec{y}\,dz \\ \leq& C \sum_{k=1}^{\infty}\frac{1}{2^{k\epsilon}} \prod_{j=1}^{\ell }\frac{1}{2^{(k+1)n}|Q^{*}|} \int_{2^{k+1}Q^{*}}\bigl\vert b_{j}(y_{j})- \lambda_{j}\bigr\vert \|f_{j}\| _{\ell^{q_{j}}}\, dy_{j} \\ &{} \times\prod_{j=\ell+1}^{m} \frac{1}{2^{(k+1)n}|Q^{*}|} \int _{2^{k+1}Q^{*}}\|f_{j}\|_{\ell^{q_{j}}}\, dy_{j} \\ \leq& C \sum_{k=1}^{\infty}\frac{1}{2^{k\epsilon}} \prod_{j=1}^{\ell}\| b_{j} \|_{\mathit{BMO}}\bigl\Vert \Vert f_{j}\Vert _{\ell^{q_{j}}} \bigr\Vert _{L(\log L),2^{k+1}Q^{*}} \\ &{} \times\prod_{j=\ell+1}^{m} \frac{1}{2^{(k+1)n}|Q^{*}|} \int _{2^{k+1}Q^{*}}\|f_{j}\|_{\ell^{q_{j}}}\, dy_{j} \\ \leq& C\mathcal{M}_{L(\log L)}\bigl(\Vert \vec{f}\Vert _{\ell^{q}} \bigr) (x). \end{aligned}$$

Similarly, we have $\mathit{III}_{2}\leq C\mathcal{M}_{L(\log L)}(\|\vec{f}\|_{\ell ^{q}})(x)$ and $\mathit{III}_{3}\leq C\mathcal{M}_{L(\log L)}(\|\vec{f}\|_{\ell ^{q}})(x)$. Now it remains to consider the typical case of III,

$$\begin{aligned}& \Biggl(\frac{1}{|Q|} \int_{Q}\Biggl\Vert T\Biggl(\prod _{j=1}^{\ell}\bigl(b_{j}( \cdot_{j})-\lambda _{j}\bigr)f_{1}^{\infty}, \ldots,f_{\ell}^{\infty},f_{\ell+1}^{0}, \ldots,f_{m}^{0}\Biggr) (z) \\& \qquad {} -T\Biggl(\prod_{j=1}^{\ell} \bigl(b_{j}(y_{j})-\lambda_{j} \bigr)f_{1}^{\infty},\ldots,f_{\ell}^{\infty},f_{\ell +1}^{0}, \ldots,f_{m}^{0}\Biggr) (x)\Biggr\Vert _{\ell^{q}}^{\delta} \Biggr)^{1/\delta} \\& \quad \leq\frac{C}{|Q|} \int_{Q} \Biggl(\sum_{k=1}^{\infty} \Biggl( \int_{0}^{\infty } \Biggl\vert \int_{(\mathbb{R}^{n}\backslash Q^{*})^{m}}\bigl\vert K_{v}(z,\vec{y})-K_{v}(x, \vec {y})\bigr\vert \prod_{j=1}^{\ell} \bigl(b_{j}(y_{j})-\lambda_{j}\bigr) \\& \qquad {}\times f_{1k}^{\infty}(y_{1})\cdots f_{\ell k}^{\infty}(y_{\ell}) f_{\ell+1,k}^{0}(y_{\ell+1}) \cdots f_{mk}^{0}(y_{m})\, d\vec{y} \Biggr\vert ^{2}\frac {dv}{v} \Biggr)^{q/2} \Biggr)^{1/q}\,dz \\& \quad \leq \biggl( \int_{(\mathbb{R}^{n}\backslash Q^{*})^{\ell}}\frac{t^{\epsilon /s}\prod_{j=1}^{\ell}(b_{j}(y_{j})-\lambda_{j})\|f_{j}(y_{j})\|_{\ell ^{q_{j}}}\, dy_{j}}{(\sum_{j=1}^{\ell}|z-y_{j}|)^{mn+\epsilon}} \\& \qquad {}+ \int_{(\mathbb{R}^{n}\backslash Q^{*})^{\ell}}\frac{\prod_{j=1}^{\ell}(b_{j}(y_{j})-\lambda_{j})\|f_{j}(y_{j})\|_{\ell^{q_{j}}}dy_{j}}{(\sum_{j=1}^{\ell}|z-y_{j}|)^{mn}} \biggr)\prod _{j=\ell+1}^{m} \int_{Q^{*}}\bigl\Vert f(y_{j})\bigr\Vert _{\ell^{q_{j}}}\, dy_{j} \\& \quad \leq C\prod_{j=1}^{\ell} \|b_{j}\|_{\mathit{BMO}}\mathcal{M}_{L(\log L)}\bigl(\Vert \vec {f}\Vert _{\ell^{q}}\bigr) (x). \end{aligned}$$

 □

Lemma 16

Let $0< p<\infty$, $1/m< q<\infty$, and $\frac{1}{q}=\frac{1}{q_{1}}+\cdots +\frac{1}{q_{m}}$ with $1< q_{1},\ldots,q_{m}<\infty$ and let $\omega\in A_{\infty}$. Suppose that $\vec{b}\in(\mathit{BMO})^{\ell}$. Then there exists a constant $C>0$ such that

$$ \int_{\mathbb{R}^{n}}|T_{\Pi\vec{b},q}\vec{f}|^{p}\omega(x)\, dx \leq C\prod_{j=1}^{\ell}\|b_{j} \|_{\mathit{BMO}}^{p} \int_{\mathbb{R}^{n}} \bigl(\mathcal {M}_{L(\log L)}\bigl(\Vert \vec{f}\Vert _{\ell^{q}}\bigr) (x) \bigr)^{p}\omega(x)\, dx. $$

(6.8)

The proof is similar to [16], so we omit it here.

Based on the above lemmas, the proof of Theorem 1.3 in [16] provides the main ideas for the proof of Theorem 7.