Commutators for multilinear singular integrals on weighted Morrey spaces

Abstract

In this paper we study the iterated commutators for multilinear singular integrals on weighted Morrey spaces. A strong type estimate and a weak endpoint estimate for the commutators are obtained. In the last section we present a problem for the multilinear Fourier multiplier with limited smooth condition.

MSC: 42B20, 42B25.

1 Introduction

As an important direction of harmonic analysis, the theory of multilinear Calderón-Zygmund singular integral operators has attracted more and more attention, which originated from the work of Coifman and Meyer [1], and it systematically was studied by Grafakos and Torres [2, 3]. The literature of the standard theory of multilinear Calderón-Zygmund singular integrals is by now quite vast, for example see [2, 4–6]. In 2009, the authors [7] introduced the new multiple weights and new maximal functions and obtained some weighted estimates for multilinear Calderón-Zygmund singular integrals. They also resolved some problems opened up in [8] and [9].

Let $\mathcal{S}({\mathbb{R}}^n)$ and ${\mathcal{S}}^{\prime }({\mathbb{R}}^n)$ be the Schwartz spaces of all rapidly decreasing functions and tempered distributions, respectively. Having fixed $m\in \mathbb{N}$, let T be a multilinear operator initially defined on the m-fold product of Schwartz spaces and taking values into the space of tempered distributions,

$$T:\mathcal{S}\left({\mathbb{R}}^n\right)\times \cdots \times \mathcal{S}\left({\mathbb{R}}^n\right)\to {\mathcal{S}}^{\prime }\left({\mathbb{R}}^n\right).$$

Following [2], the m-multilinear Calderón-Zygmund operator T satisfies the following conditions:

• (S1) there exist $q_i<\mathrm{\infty }$ ($i=1,\dots ,m$), it extends to a bounded multilinear operator from $L^{q_1}\times \cdots \times L^{q_m}$ to $L^q$ , where $\frac{1}{q}=\frac{1}{q_1}+\cdots +\frac{1}{q_m}$;

• (S2) there exists a function K, defined off the diagonal $x=y_1=\cdots =y_m$ in ${({\mathbb{R}}^n)}^{m+1}$, satisfying

  $$T(\overrightarrow{f})(x)=T(f_1,\dots ,f_m)(x)={\int }_{{({\mathbb{R}}^m)}^n}K(x,y_1,\dots ,y_m)f_1(y_1)\cdots f_m(y_m)d{y}_1\cdots d{y}_m$$

  (1)

for all $x\notin {\bigcap }_{j=1}^{m}supp{f}_j$ and $f_1,\dots ,f_m\in \mathcal{S}({\mathbb{R}}^n)$, where

$$|K(y_0,y_1,\dots ,y_m)|\le \frac{A}{{({\sum }_{l,k=0}^m|y_l-y_k|)}^{mn}}$$

(2)

and

$$|K(y_0,\dots ,y_j,\dots ,y_m)-K(y_0,\dots ,y_j^{\prime },\dots ,y_m)|\le \frac{A{|y_j-y_j^{\prime }|}^{ϵ}}{{({\sum }_{l,k=0}^m|y_l-y_k|)}^{mn+ϵ}}$$

(3)

for some $ϵ>0$ and all $0\le j\le m$, whenever $|y_j-y_j^{\prime }|\le \frac{1}{2}{max}_{0\le k\le m}|y_j-y_k|$.

We also use some notation following [10]. Given a locally integrable vector function $\mathbf{b}=(b_1,\dots ,b_m)\in {(\mathit{BMO})}^m$, the commutator of b and the m-linear Calderón-Zygmund operator T, denoted here by $T_{\mathrm{\Sigma }\mathbf{b}}$, was introduced by Pérez and Torres in [9] and is defined via

$$T_{\mathrm{\Sigma }\mathbf{b}}(\overrightarrow{f})=\sum _{j=1}^m{T}_{b_j}^j(\overrightarrow{f}),$$

where

$$T_{b_j}^j(\overrightarrow{f})=b_{j}T(\overrightarrow{f})-T(f_1,\dots ,b_j{f}_j,\dots ,f_N).$$

The iterated commutator $T_{\mathrm{\Pi }\mathbf{b}}$ is defined by

$$T_{\mathrm{\Pi }\mathbf{b}}(\overrightarrow{f})={[b_1,\dots ,{[b_{m-1},{[b_m,T]}_m]}_{m-1}\cdots ]}_1(\overrightarrow{f}).$$

To clarify the notations, if T is associated in the usual way with a Calderón-Zygmund kernel K, then at a formal level

$$T_{\mathrm{\Sigma }\mathbf{b}}(\overrightarrow{f})(x)={\int }_{{({\mathbb{R}}^n)}^m}\sum _{j=1}^m(b_j(x)-b_j(y_j))K(x,y_1,\dots ,y_m)f_1(y_1)\cdots f_m(y_m)d{y}_1\cdots d{y}_m$$

and

$$T_{\mathrm{\Pi }\mathbf{b}}(\overrightarrow{f})(x)={\int }_{{({\mathbb{R}}^n)}^m}\prod _{j=1}^m(b_j(x)-b_j(y_j))K(x,y_1,\dots ,y_m)f_1(y_1)\cdots f_m(y_m)d{y}_1\cdots d{y}_m.$$

It was shown in [2] that if $\frac{1}{q}=\frac{1}{q_1}+\cdots +\frac{1}{q_m}$, then an m-linear Calderón-Zygmund operator T maps from $L^{q_1}\times \cdots \times L^{q_m}$ to $L^q$, when $1<q_j<\mathrm{\infty }$ for all $j=1,\dots ,m$; and from $L^{q_1}\times \cdots \times L^{q_m}$ to $L^{q,\mathrm{\infty }}$, when $1\le q_j<\mathrm{\infty }$ for all $j=1,\dots ,m$, and ${min}_{1\le j\le m}{q}_j=1$. The weighted strong and weak $L^q$ boundedness of T is also true for weights in the class $A_{\overrightarrow{P}}$ which will be introduced in next section (see Corollary 3.9 [7]). It was proved in [9] that $T_{\mathrm{\Sigma }\mathbf{b}}$ is bounded from $L^{q_1}\times \cdots \times L^{q_m}$ to $L^q$ for all indices satisfying $\frac{1}{q}=\frac{1}{q_1}+\cdots +\frac{1}{q_m}$ with $q>1$ and $q_j>1$, $j=1,\dots ,m$. The result was extended in [7] to all $q>1/m$. In fact, the authors obtained the weighted $L^q$-version bounds as follows, for all $\overrightarrow{\omega }\in A_{\overrightarrow{P}}$:

$${\parallel T_{\mathrm{\Sigma }\mathbf{b}}(\overrightarrow{f})\parallel }_{L^q({\nu }_{\overrightarrow{\omega }})}\le C{\parallel \overrightarrow{b}\parallel }_{{\mathit{BMO}}^m}\prod _{j=1}^m{\parallel f_j\parallel }_{L^{q_j}({\omega }_j)}.$$

As may be expected from the situation in the linear case, $T_{\mathrm{\Sigma }\mathbf{b}}$ is not bounded from $L^1\times \cdots \times L^1$ to $L^{1,\mathrm{\infty }}$. However, a sharp weak-type estimate in a very general sense was obtained in [7], that is, for all $\overrightarrow{\omega }\in A_{(1,\dots ,1)}$,

$${\nu }_{\overrightarrow{\omega }}\{x\in {\mathbb{R}}^n:|T_{\mathrm{\Sigma }\mathbf{b}}(\overrightarrow{f})(x)|>t^m\}\le C\prod _{j=1}^m{({\int }_{{\mathbb{R}}^n}\mathrm{\Phi }\left(\frac{|f_j(x)|}{t}\right){\omega }_j(x)dx)}^{1/m},$$

where $\mathrm{\Phi }(t)=t(1+{log}^{+}t)$. When $m=1$, the above endpoint estimate was obtained in [11]. The same as for $T_{\mathrm{\Sigma }\mathbf{b}}$, the strong type bound and the endpoint estimate for $T_{\mathrm{\Pi }\mathbf{b}}$ were also established by Pérez et al. in [10].

The weighted Morrey spaces $L^{p,k}(w)$ was introduced by Komori and Shirai [6]. Moreover, they showed that some classical integral operators and corresponding commutators are bounded in weighted Morrey spaces. Some other authors have been interested in this space for sublinear operators, see [12–14]. In [15], Ye proved two results similar to Pérez and Trujillo-González [11] for the multilinear commutators of the normal Calderón-Zygmund operators on weighted Morrey spaces. Wang and Yi [16] considered the multilinear Calderón-Zygmund operators on weighted Morrey spaces and obtained some results similar to weighted Lebesgue spaces.

We will prove the following strong type bound for $T_{\mathrm{\Pi }\mathbf{b}}$ on weighted Morrey spaces.

Theorem 1.1 Let T be an m-linear Calderón-Zygmund operator; $\overrightarrow{\omega }\in A_{\overrightarrow{P}}\cap {(A_{\mathrm{\infty }})}^m$ with

$$\frac{1}{p}=\frac{1}{p_1}+\cdots +\frac{1}{p_m}$$

and $1<p_j<\mathrm{\infty }$, $j=1,\dots ,m$; and $\mathbf{b}\in {\mathit{BMO}}^m$. Then, for any $0<k<1$, there exists a constant C such that

$${\parallel T_{\mathrm{\Pi }\mathbf{b}}(\overrightarrow{f})\parallel }_{L^{p,k}({\nu }_{\overrightarrow{\omega }})}\le C\prod _{j=1}^m{\parallel b_j\parallel }_{\mathit{BMO}}\prod _{j=1}^m{\parallel f_j\parallel }_{L^{p_j,k}({\omega }_j)}.$$

The following endpoint estimate will also be proved.

Theorem 1.2 Let T be an m-linear Calderón-Zygmund operator; $0<k<1$, $\overrightarrow{\omega }\in A_{(1,\dots ,1)}\cap {(A_{\mathrm{\infty }})}^m$, and $\mathbf{b}\in {\mathit{BMO}}^m$. Then, for any $\lambda >0$ and cube Q, there exists a constant C such that

$$\frac{1}{{\nu }_{\overrightarrow{\omega }}{(Q)}^k}{\nu }_{\overrightarrow{\omega }}\{x\in Q:|T_{\mathrm{\Pi }\mathbf{b}}(\overrightarrow{f})(x)|>\lambda \}\le C\prod _{j=1}^m{\left[{\parallel f_j/\lambda \parallel }_{L^{{\mathrm{\Phi }}^{(m)},k}({\omega }_j)}\right]}^{1/m},$$

where, $\mathrm{\Phi }(t)=t(1+{log}^{+}t)$ and ${\parallel f\parallel }_{L^{{\mathrm{\Phi }}^{(m)},k}(\omega )}={\parallel {\mathrm{\Phi }}^{(m)}(|f|)\parallel }_{L^{1,k}(\omega )}$.

Remark 1.1 Here we remark that the above estimate is also valid for $T_{\mathrm{\Sigma }\mathbf{b}}$.

2 Some definitions and results

In this section, we introduce some definitions and results used later.

Definition 2.1 ($A_p$ weights)

A weight ω is a nonnegative, locally integrable function on ${\mathbb{R}}^n$. Let $1<p<\mathrm{\infty }$, a weight function ω is said to belong to the class $A_p$, if there is a constant C such that for any cube Q,

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

and to the class $A_1$, if there is a constant C such that for any cube Q,

$$\frac{1}{|Q|}{\int }_Q\omega (x)dx\le C\underset{x\in Q}{inf}\omega (x).$$

We denote $A_{\mathrm{\infty }}={\bigcup }_{p>1}{A}_p$.

Definition 2.2 (Multiple weights)

For m exponents $p_1,\dots ,p_m\in [1,\mathrm{\infty })$, we often write p for the number given by $p={\sum }_{j=1}^m{p}_j$ and denote by $\overrightarrow{P}$ the vector $(p_1,\dots ,p_m)$. A multiple weight $\overrightarrow{\omega }=({\omega }_1,\dots ,{\omega }_m)$ is said to satisfy the $A_{\overrightarrow{P}}$ condition if for

$${\nu }_{\overrightarrow{\omega }}=\prod _{j=1}^m{\omega }^{p/p_j},$$

we have

$$\underset{Q}{sup}{(\frac{1}{|Q|}{\int }_Q{\nu }_{\overrightarrow{\omega }}(x)dx)}^{1/p}\prod _{j=1}^m{\left(\frac{1}{|Q|}{\int }_Q{\omega }_j{(x)}^{1-p_j^{\prime }}dx\right)}^{1/p_j^{\prime }}<\mathrm{\infty },$$

when $p_j=1$, ${(\frac{1}{|Q|}{\int }_Q{\omega }_j{(x)}^{1-p_j^{\prime }}dx)}^{1/p_j^{\prime }}$ is understood as ${({inf}_x\omega (x))}^{-1}$. As remarked in [7], ${\prod }_{j=1}^m{A}_{p_j}$ is strictly contained in $A_{\overrightarrow{P}}$, moreover, in general $\overrightarrow{\omega }\in A_{\overrightarrow{P}}$ does not imply ${\omega }_j\in L_{\mathrm{loc}}^1$ for any j, but instead

$$\overrightarrow{\omega }\in A_{\overrightarrow{P}}\iff \{\begin{array}{l}{({\nu }_{\overrightarrow{\omega }})}^p\in A_{mp},\\ {\omega }_j^{1-p_j^{\prime }}\in A_{m{p}_j^{\prime }},j=1,\dots ,m,\end{array}$$

where the condition ${\omega }_j^{1-p_j^{\prime }}\in A_{m{p}_j^{\prime }}$ in the case $p_j=1$ is understood as ${\omega }_j^{1/m}\in A_1$.

Definition 2.3 (Weighted Morrey spaces)

Let $0<p<\mathrm{\infty }$, $0<k<1$, and ω be a weight function on ${\mathbb{R}}^n$. The weighted Morrey space is defined by

$$L^{p,k}(\omega )=\{f\in L_{\mathrm{loc}}^p:{\parallel f\parallel }_{L^{p,k}(\omega )}<\mathrm{\infty }\},$$

where

$${\parallel f\parallel }_{L^{p,k}(\omega )}=\underset{Q}{sup}{\left(\frac{1}{\omega {(Q)}^k}{\int }_Q\right|f(x){|}^p\omega (x))}^{1/p}.$$

The weighted weak Morrey space is defined by

$$W{L}^{p,k}(\omega )=\{f\text{ measurable}:{\parallel f\parallel }_{W{L}^{p,k}(\omega )}<\mathrm{\infty }\},$$

where

$${\parallel f\parallel }_{W{L}^{p,k}(\omega )}=\underset{Q}{sup}\underset{\lambda >0}{inf}\frac{\lambda }{\omega {(Q)}^{k/p}}\omega {\left(\{x\in Q:|f|(x)>\lambda \}\right)}^{1/p}.$$

Definition 2.4 (Maximal function)

For $\mathrm{\Phi }(t)=t(1+{log}^{+}t)$ and a cube Q in ${\mathbb{R}}^n$ we will consider the average ${\parallel f\parallel }_{\mathrm{\Phi },Q}$ of a function f given by the Luxemburg norm

$${\parallel f\parallel }_{\mathrm{\Phi },Q}=inf\{\lambda >0:\frac{1}{|Q|}{\int }_Q\mathrm{\Phi }\left(\frac{|f(x)|}{\lambda }\right)dx\le 1\},$$

and the corresponding maximal is naturally defined by

$$M_{\mathrm{\Phi }}f(x)=\underset{Q\ni x}{sup}{\parallel f\parallel }_{\mathrm{\Phi },Q},$$

and the multilinear maximal operator ${\mathcal{M}}_{\mathrm{\Phi },Q}$ is given by

$${\mathcal{M}}_{\mathrm{\Phi }}(\overrightarrow{f})(x)=\underset{Q\ni x}{sup}\prod _{j=1}^m{\parallel f_j\parallel }_{\mathrm{\Phi },Q}.$$

The following pointwise equivalence is very useful:

$$M_{\mathrm{\Phi }}f(x)\approx M^{2}f(x),$$

where M is the Hardy-Littlewood maximal function. We refer reader to [7, 10] and their references for details.

We say that a weight ω satisfies the doubling condition, simply denoted $\omega \in {\mathrm{\Delta }}_2$, if there is a constant $C>0$ such that $\omega (2Q)\le C\omega (Q)$ holds for any cube Q. If $\omega \in A_p$ with $1\le p<\mathrm{\infty }$, we know that $\omega (\lambda Q)\le {\lambda }^{np}{[\omega ]}_{A_p}\omega (Q)$ for all $\lambda >1$; then $\omega \in {\mathrm{\Delta }}_2$.

Lemma 2.1 ([6])

Suppose $\omega \in {\mathrm{\Delta }}_2$, then there exists a constant $D>1$ such that

$$\omega (2Q)\ge D\omega (Q)$$

for any cube.

Lemma 2.2 ([16])

If ${\omega }_j\in A_{\mathrm{\infty }}$, then for any cube Q, we have

$${\int }_Q\prod _{j=1}^m{\omega }_j^{{\theta }_j}(x)dx\ge \prod _{j=1}^m{\left(\frac{{\int }_Q{\omega }_j(x)dx}{{[{\omega }_j]}_{\mathrm{\infty }}}\right)}^{{\theta }_j},$$

where ${\sum }_{j=1}^m{\theta }_j=1$, $0\le {\theta }_j\le 1$.

Lemma 2.3 ([17])

Suppose $\omega \in A_{\mathrm{\infty }}$, then ${\parallel b\parallel }_{\mathit{BMO}(\omega )}\approx {\parallel b\parallel }_{\mathit{BMO}}$. Here

$$\mathit{BMO}(\omega )=\{b:{\parallel b\parallel }_{\mathit{BMO}(\omega )}=\underset{Q}{sup}\frac{1}{\omega (Q)}{\int }_Q|b(x)-b_{Q,\omega }|\omega (x)dx<\mathrm{\infty }\},$$

and $b_{Q,\omega }=\frac{1}{\omega (Q)}{\int }_{Q}b(x)\omega (x)dx$.

From the fact $|b_{2^{j}Q}-b_Q|\le Cj{\parallel b\parallel }_{\mathit{BMO}}$ and Lemma 2.3, we deduce that $|b_{2^{j}Q,\omega }-b_{Q,\omega }|\le Cj{\parallel b\parallel }_{\mathit{BMO}}$. The following lemma is the multilinear version of the Fefferman-Stein type inequality.

Lemma 2.4 (Theorem 3.12 [7])

Assume that ${\omega }_i$ is a weight in $A_1$ for all $i=1,\dots ,m$, and set $\nu ={({\prod }_{i=1}^m{\omega }_i)}^{1/m}$. Then

$${\parallel \prod _{j=1}^{m}M(f_j)\parallel }_{L^{p,\mathrm{\infty }}(\nu )}\le \prod _{j=1}^m{\parallel f_j\parallel }_{L^1(M{\omega }_j)}.$$

Lemma 2.5 (Proposition 3.13 [7])

Let $\frac{1}{p}=\frac{1}{p_1}+\cdots +\frac{1}{p_m}$. If $1\le p_j\le \mathrm{\infty }$, $j=1,\dots ,m$, then

$${\parallel \mathcal{M}(\overrightarrow{f})\parallel }_{L^{p,\mathrm{\infty }}({\nu }_{\overrightarrow{\omega }})}\le \prod _{j=1}^m{\parallel f_j\parallel }_{L^{p_j}(M{\omega }_j)}.$$

Lemma 2.6 (Theorem 3.2 [10])

Let $p>0$ and let ω be a weight in $A_{\mathrm{\infty }}$. Suppose that $\mathbf{b}\in {\mathit{BMO}}^m$. Then there exist $C_{\omega }$ (independent of b) and $C_{\omega ,\mathbf{b}}$ such that

$${\int }_{{\mathbb{R}}^n}|T_{\mathrm{\Pi }\mathbf{b}}(\overrightarrow{f})(x)|\omega (x)dx\le C_{\omega }\prod _{j=1}^m{\parallel b_j\parallel }_{\mathit{BMO}}{\int }_{{\mathbb{R}}^n}{\mathcal{M}}_{\mathrm{\Phi }}(\overrightarrow{f}){(x)}^p\omega (x)dx$$

and

$$\begin{array}{c}\underset{t>0}{sup}\frac{1}{{\mathrm{\Phi }}^{(m)}(\frac{1}{t})}\omega \left(\{y\in {\mathbb{R}}^n:|T_{\mathrm{\Pi }\mathbf{b}}(\overrightarrow{f})(y)|>t^m\}\right)\hfill \\ \le C_{\omega ,\mathbf{b}}\underset{t>0}{sup}\frac{1}{{\mathrm{\Phi }}^{(m)}(\frac{1}{t})}\omega \left(\{y\in {\mathbb{R}}^n:|{\mathcal{M}}_{\mathrm{\Phi }}(\overrightarrow{f})(y)|>t^m\}\right)\hfill \end{array}$$

for all $\overrightarrow{f}=(f_1,\dots ,f_m)$ bounded with compact support.

Lemma 2.7 (Theorem 4.1 [10])

Let $\omega \in A_{(1,\dots ,1)}$. Then there exists a constant C such that

$${\nu }_{\overrightarrow{\omega }}\left(\{x\in {\mathbb{R}}^n:|{\mathcal{M}}_{LlogL}(\overrightarrow{f})(x)|>t^m\}\right)\le C\prod _{j=1}^m{({\int }_{{\mathbb{R}}^n}{\mathrm{\Phi }}^{(m)}\left(\frac{|f_j(x)|}{t}\right){\omega }_j(x)dx)}^{1/m}.$$

By the above two inequalities, Pérez and Trujillo-González obtained the following results.

Lemma 2.8 (Theorem 1.1 [10])

Let T be an m-linear Calderón-Zygmund operator; $\overrightarrow{\omega }\in A_{\overrightarrow{P}}$ with

$$\frac{1}{p}=\frac{1}{p_1}+\cdots +\frac{1}{p_m}$$

and $1<p_j<\mathrm{\infty }$, $j=1,\dots ,m$; and $\mathbf{b}\in {\mathit{BMO}}^m$. Then there exists a constant C such that

$${\parallel T_{\mathrm{\Pi }\mathbf{b}}(\overrightarrow{f})\parallel }_{L^p({\nu }_{\overrightarrow{\omega }})}\le C\prod _{j=1}^m{\parallel b_j\parallel }_{\mathit{BMO}}\prod _{j=1}^m{\parallel f_j\parallel }_{L^{p_j}({\omega }_j)}.$$

Lemma 2.9 (Theorem 1.2 [10])

Let T be an m-linear Calderón-Zygmund operator; $\overrightarrow{\omega }\in A_{(1,\dots ,1)}$, and $\mathbf{b}\in {\mathit{BMO}}^m$. Then, for any $\lambda >0$ and cube Q, there exists a constant C such that

$${\nu }_{\overrightarrow{\omega }}\{x\in {\mathbb{R}}^n:|T_{\mathrm{\Pi }\mathbf{b}}(\overrightarrow{f})(x)|>\lambda \}\le C\prod _{j=1}^m{({\int }_{{\mathbb{R}}^n}{\mathrm{\Phi }}^{(m)}\left(\frac{|f_j(x)|}{t}\right){\omega }_j(x)dx)}^{1/m},$$

where $\mathrm{\Phi }(t)=t(1+{log}^{+}t)$ and.

3 Proofs of theorems

We only present the case $m=2$ for simplicity, but, as the reader will immediately notice, a complicated notation and a similar procedure can be followed to obtain the general case. Our arguments will be standard.

Proof of Theorem 1.1 For any cube Q, we split $f_j$ into $f_j^0+f_j^{\mathrm{\infty }}$, where $f_j^0=f_j{\chi }_{2Q}$ and $f_j^{\mathrm{\infty }}=f_j-f_j^0$, $j=1,2$. Then we only need to verify the following inequalities:

$$\begin{array}{c}I={\left(\frac{1}{{\nu }_{\overrightarrow{\omega }}{(Q)}^k}{\int }_Q\right|T_{\mathrm{\Pi }\mathbf{b}}(f_1^0,f_2^0)(x){|}^p{\nu }_{\overrightarrow{\omega }}(x)dx)}^{1/p}\le C\prod _{j=1}^2{\parallel b_j\parallel }_{\mathit{BMO}}\prod _{j=1}^2{\parallel f_j\parallel }_{L^{p_j,k}({\omega }_j)},\hfill \\ \mathit{II}={\left(\frac{1}{{\nu }_{\overrightarrow{\omega }}{(Q)}^k}{\int }_Q\right|T_{\mathrm{\Pi }\mathbf{b}}(f_1^0,f_2^{\mathrm{\infty }})(x){|}^p{\nu }_{\overrightarrow{\omega }}(x)dx)}^{1/p}\le C\prod _{j=1}^2{\parallel b_j\parallel }_{\mathit{BMO}}\prod _{j=1}^2{\parallel f_j\parallel }_{L^{p_j,k}({\omega }_j)},\hfill \\ \mathit{III}={\left(\frac{1}{{\nu }_{\overrightarrow{\omega }}{(Q)}^k}{\int }_Q\right|T_{\mathrm{\Pi }\mathbf{b}}(f_1^{\mathrm{\infty }},f_2^0)(x){|}^p{\nu }_{\overrightarrow{\omega }}(x)dx)}^{1/p}\le C\prod _{j=1}^2{\parallel b_j\parallel }_{\mathit{BMO}}\prod _{j=1}^2{\parallel f_j\parallel }_{L^{p_j,k}({\omega }_j)},\hfill \\ \mathit{IV}={\left(\frac{1}{{\nu }_{\overrightarrow{\omega }}{(Q)}^k}{\int }_Q\right|T_{\mathrm{\Pi }\mathbf{b}}(f_1^{\mathrm{\infty }},f_2^{\mathrm{\infty }})(x){|}^p{\nu }_{\overrightarrow{\omega }}(x)dx)}^{1/p}\le C\prod _{j=1}^2{\parallel b_j\parallel }_{\mathit{BMO}}\prod _{j=1}^2{\parallel f_j\parallel }_{L^{p_j,k}({\omega }_j)}.\hfill \end{array}$$

From Lemma 2.8 and Lemma 2.2, we get

$$\begin{array}{rl}I& \le C\frac{1}{{\nu }_{\overrightarrow{\omega }}{(Q)}^{k/p}}\prod _{j=1}^2{\parallel b_j\parallel }_{\mathit{BMO}}{\left({\int }_{{\mathbb{R}}^n}\right|f_j^0(x){|}^{p_j}{\omega }_j(x)dx)}^{1/p_j}\\ \le C\frac{1}{{\nu }_{\overrightarrow{\omega }}{(Q)}^{k/p}}\prod _{j=1}^2\left[{\parallel b_j\parallel }_{\mathit{BMO}}{\omega }_j{(2Q)}^{k/p_j}{\parallel f_j\parallel }_{L^{p_j,k}({\omega }_j)}\right]\\ \le C\prod _{j=1}^2\left[{\parallel b_j\parallel }_{\mathit{BMO}}{\parallel f_j\parallel }_{L^{p_j,k}({\omega }_j)}\right].\end{array}$$

Since II and III are symmetric we only estimate II. Taking ${\lambda }_j={(b_j)}_{Q,{\omega }_j}$, the operator $T_{\mathrm{\Pi }\mathbf{b}}$ can be divided into four parts:

$$\begin{array}{r}{T}_{\mathrm{\Pi }\mathbf{b}}(f_1^0,f_2^{\mathrm{\infty }})(x)\\ =(b_1(x)-{\lambda }_1)(b_2(x)-{\lambda }_2)T(f_1^0,f_2^{\mathrm{\infty }})(x)-(b_1(x)-{\lambda }_1)T(f_1^0,(b_2-{\lambda }_2)f_2^{\mathrm{\infty }})(x)\\ -(b_2(x)-{\lambda }_2)T((b_1-{\lambda }_1)f_1^0,f_2^{\mathrm{\infty }})(x)+T((b_1-{\lambda }_1)f_1^0,(b_2-{\lambda }_2)f_2^{\mathrm{\infty }})(x)\\ ={\mathit{II}}_1+{\mathit{II}}_2+{\mathit{II}}_3+{\mathit{II}}_4.\end{array}$$

Using the size condition (2) of K, Definition 2.2, and Lemma 2.2, we deduce that for any $x\in Q$,

$$\begin{array}{c}|T(f_1^0,f_2^{\mathrm{\infty }})(x)|\hfill \\ \le C{\int }_{2Q}{\int }_{{\mathbb{R}}^n\mathrm{\setminus }2Q}\frac{|f_1(y_1)f_2(y_2)|}{{(|x-y_1|+|x-y_2|)}^{2n}}d{y}_{2}d{y}_1\hfill \\ \le C{\int }_{2Q}|f_1(y_1)|d{y}_1\sum _{l=1}^{\mathrm{\infty }}\frac{1}{{|2^{l}Q|}^2}{\int }_{2^{l+1}Q\mathrm{\setminus }{2}^{l}Q}|f_2(y_2)|d{y}_2\hfill \\ \le C\sum _{l=1}^{\mathrm{\infty }}\prod _{j=1}^2\frac{1}{|2^{l+1}Q|}{\int }_{2^{l+1}Q}|f_j(y_j)|d{y}_j\hfill \\ \le C\sum _{l=1}^{\mathrm{\infty }}\prod _{j=1}^2\frac{1}{|2^{l+1}Q|}{\left({\int }_{2^{l+1}Q}\right|f_j(y_j){|}^{p_j}{\omega }_j(y_j)d{y}_j)}^{1/p_j}\hfill \\ \times {\left({\int }_{2^{l+1}Q}{\omega }_j{(y_j)}^{1-p_j^{\prime }}d{y}_j\right)}^{1/p_j^{\prime }}\hfill \\ \le C\sum _{l=1}^{\mathrm{\infty }}\frac{1}{{|2^{l+1}Q|}^2}\frac{{|2^{l+1}Q|}^{\frac{1}{p}+\frac{1}{p_1^{\prime }}+\frac{1}{p_2^{\prime }}}}{{\nu }_{\overrightarrow{\omega }(2^{l+1}Q)}}\prod _{j=1}^2{\parallel f_j\parallel }_{L^{p_j,k}({\omega }_j)}{\omega }_j{\left(2^{l+1}Q\right)}^{k/p_j}\hfill \\ \le C\prod _{j=1}^2{\parallel f_j\parallel }_{L^{p_j,k}({\omega }_j)}\sum _{l=1}^{\mathrm{\infty }}{\nu }_{\overrightarrow{\omega }}{\left(2^{l+1}Q\right)}^{(k-1)/p}.\hfill \end{array}$$

Taking the above estimate together with Hölder’s inequality and Lemma 2.3, we have

$$\begin{array}{c}{(\frac{1}{{\nu }_{\overrightarrow{\omega }}{(Q)}^k}{\int }_Q{|{\mathit{II}}_1|}^p{\nu }_{\overrightarrow{\omega }}(x)dx)}^{1/p}\hfill \\ \le \frac{1}{{\nu }_{\overrightarrow{\omega }}{(Q)}^{k/p}}{\left({\int }_Q\right|(b_1(x)-{\lambda }_1)(b_2(x)-{\lambda }_2){|}^p{\nu }_{\overrightarrow{\omega }}(x)dx)}^{1/p}\hfill \\ \times \prod _{j=1}^2{\parallel f_j\parallel }_{L^{p_j,k}}\sum _{l=1}^{\mathrm{\infty }}{\nu }_{\overrightarrow{\omega }}{\left(2^{l+1}Q\right)}^{(k-1)/p}\hfill \\ \le \frac{{\nu }_{\overrightarrow{\omega }}{(Q)}^{1/p}}{{\nu }_{\overrightarrow{\omega }}{(Q)}^{k/p}}\prod _{j=1}^2{\left(\frac{1}{{\nu }_{\overrightarrow{\omega }}(Q)}{\int }_Q\right|(b_j(x)-{\lambda }_1){|}^{2p}{\nu }_{\overrightarrow{\omega }}(x)dx)}^{1/2p}\hfill \\ \times \prod _{j=1}^2{\parallel f_j\parallel }_{L^{p_j,k}}\sum _{l=1}^{\mathrm{\infty }}{\nu }_{\overrightarrow{\omega }}{\left(2^{l+1}Q\right)}^{(k-1)/p}\hfill \\ \le \prod _{j=1}^2{\parallel b_j\parallel }_{\mathit{BMO}}{\parallel f_j\parallel }_{L^{p_j,k}({\omega }_j)},\hfill \end{array}$$

where the last inequality is obtained by the property of $A_{\mathrm{\infty }}$: there is a constant $\delta >0$ such that

$$\frac{{\nu }_{\overrightarrow{\omega }}(Q)}{{\nu }_{\overrightarrow{\omega }}(2^{l+1}Q)}\le C{\left(\frac{|Q|}{|2^{l+1}Q|}\right)}^{\delta }.$$

For ${\mathit{II}}_2$, from the size condition (2) of K, the $A_{\overrightarrow{P}}$ condition, Lemma 2.2, and Lemma 2.3, it follows that

$$\begin{array}{c}|T(f_1^0,(b_2-{\lambda }_2)f_2^{\mathrm{\infty }})(x)|\hfill \\ \le C{\int }_{2Q}|f_1(y_1)|d{y}_1\sum _{l=1}^{\mathrm{\infty }}\frac{1}{{|2^{l}Q|}^2}{\int }_{2^{l+1}Q\mathrm{\setminus }{2}^{l}Q}|(b_2(y_2)-{\lambda }_2)f_2(y_2)|d{y}_2\hfill \\ \le C\sum _{l=1}^{\mathrm{\infty }}\frac{1}{{|2^{l+1}Q|}^2}{\left({\int }_{2^{l+1}Q}\right|f_1(y_1){|}^{p_1}{\omega }_j(y_1)d{y}_1)}^{1/p_1}{\left({\int }_{2^{l+1}Q}{\omega }_1{(y_1)}^{1-p_1^{\prime }}d{y}_j\right)}^{1/p_1^{\prime }}\hfill \\ \times {\left({\int }_{2^{l+1}Q}\right|f_2(y_2){|}^{p_2}{\omega }_2(y_2)d{y}_2)}^{1/p_2}\hfill \\ \times {\left({\int }_{2^{l+1}Q}\right|b_2(y_2)-{\lambda }_2{|}^{p_2^{\prime }}{\omega }_2{(y_2)}^{-p_2^{\prime }/p_2}d{y}_2)}^{1/p_2^{\prime }}\hfill \\ \le C\sum _{l=1}^{\mathrm{\infty }}l\prod _{j=1}^2\frac{1}{|2^{l+1}Q|}{\left({\int }_{2^{l+1}Q}\right|f_j(y_j){|}^{p_j}{\omega }_j(y_j)d{y}_j)}^{1/p_j}{\left({\int }_{2^{l+1}Q}{\omega }_j{(y_j)}^{1-p_j^{\prime }}d{y}_j\right)}^{1/p_j^{\prime }}\hfill \\ \le C\prod _{j=1}^2{\parallel f_j\parallel }_{L^{p_j,k}({\omega }_j)}\sum _{l=1}^{\mathrm{\infty }}l{\nu }_{\overrightarrow{\omega }}{\left(2^{l+1}Q\right)}^{(k-1)/p}.\hfill \end{array}$$

The third inequality can be deduced by the fact that

$${\left(\frac{1}{\omega (2^{j+1}Q)}{\int }_{2^{l+1}Q}\right|b(y)-b_{Q,\omega }{|}^p\omega (y)dy)}^{1/p}\le Cl{\parallel b\parallel }_{\mathit{BMO}(\omega )}.$$

Hölder’s inequality and Lemma 2.3 tell us

$$\begin{array}{c}{(\frac{1}{{\nu }_{\overrightarrow{\omega }}{(Q)}^k}{\int }_Q{|{\mathit{II}}_2|}^p{\nu }_{\overrightarrow{\omega }}(x)dx)}^{1/p}\hfill \\ \le C\frac{1}{{\nu }_{\overrightarrow{\omega }}{(Q)}^{k/p}}{\left({\int }_Q\right|(b_1(x)-{\lambda }_1){|}^p{\nu }_{\overrightarrow{\omega }}(x)dx)}^{1/p}\prod _{j=1}^2{\parallel f_j\parallel }_{L^{p_j,k}}\sum _{l=1}^{\mathrm{\infty }}l{\nu }_{\overrightarrow{\omega }}{\left(2^{l+1}Q\right)}^{(k-1)/p}\hfill \\ \le C\frac{{\nu }_{\overrightarrow{\omega }}{(Q)}^{1/p}}{{\nu }_{\overrightarrow{\omega }}{(Q)}^{k/p}}\prod _{j=1}^2{\parallel f_j\parallel }_{L^{p_j,k}}\sum _{l=1}^{\mathrm{\infty }}l{\nu }_{\overrightarrow{\omega }}{\left(2^{l+1}Q\right)}^{(k-1)/p}\hfill \\ \le C\prod _{j=1}^2{\parallel b_j\parallel }_{\mathit{BMO}}{\parallel f_j\parallel }_{L^{p_j,k}({\omega }_j)}.\hfill \end{array}$$

Similarly, we get

$$\begin{array}{c}|T(f_1^0,(b_2-{\lambda }_2)f_2^{\mathrm{\infty }})(x)|\hfill \\ \le C\sum _{l=1}^{\mathrm{\infty }}\frac{1}{{|2^{l+1}Q|}^2}{\left({\int }_{2^{l+1}Q}\right|f_1(y_1){|}^{p_1}{\omega }_j(y_1)d{y}_1)}^{1/p_1}\hfill \\ \times {\left({\int }_{2^{l+1}Q}\right|b_1(y_1)-{\lambda }_1{|}^{p_1^{\prime }}{\omega }_1{(y_1)}^{1-p_1^{\prime }}d{y}_j)}^{1/p_1^{\prime }}\hfill \\ \times {\left({\int }_{2^{l+1}Q}\right|f_2(y_2){|}^{p_2}{\omega }_2(y_2)d{y}_2)}^{1/p_2}{\left({\int }_{2^{l+1}Q}{\omega }_2{(y_2)}^{-p_2^{\prime }/p_2}d{y}_2\right)}^{1/p_2^{\prime }}\hfill \\ \le C\prod _{j=1}^2{\parallel f_j\parallel }_{L^{p_j,k}({\omega }_j)}\sum _{l=1}^{\mathrm{\infty }}l{\nu }_{\overrightarrow{\omega }}{\left(2^{l+1}Q\right)}^{(k-1)/p},\hfill \end{array}$$

and so

$${(\frac{1}{{\nu }_{\overrightarrow{\omega }}{(Q)}^k}{\int }_Q{|{\mathit{II}}_3|}^p{\nu }_{\overrightarrow{\omega }}(x)dx)}^{1/p}\le C\prod _{j=1}^2{\parallel b_j\parallel }_{\mathit{BMO}}{\parallel f_j\parallel }_{L^{p_j,k}({\omega }_j)}.$$

The term ${\mathit{II}}_4$ is estimated in a similar way and we deduce

$$\begin{array}{c}|T((b_1-{\lambda }_1)f_1^0,(b_2-{\lambda }_2)f_2^{\mathrm{\infty }})(x)|\hfill \\ \le C\sum _{l=1}^{\mathrm{\infty }}\frac{1}{{|2^{l+1}Q|}^2}\prod _{j=1}^2{\left({\int }_{2^{l+1}Q}\right|f_j(y_j){|}^{p_j}{\omega }_j(y_j)d{y}_j)}^{1/p_j}\hfill \\ \times {\left({\int }_{2^{l+1}Q}\right|b_j(y_j)-{\lambda }_j{|}^{p_j^{\prime }}{\omega }_j{(y_j)}^{-p_j^{\prime }/p_j}d{y}_j)}^{1/p_j^{\prime }}\hfill \\ \le C\prod _{j=1}^2{\parallel f_j\parallel }_{L^{p_j,k}({\omega }_j)}\sum _{l=1}^{\mathrm{\infty }}{l}^2{\nu }_{\overrightarrow{\omega }}{\left(2^{l+1}Q\right)}^{(k-1)/p}.\hfill \end{array}$$

So,

$${(\frac{1}{{\nu }_{\overrightarrow{\omega }}{(Q)}^k}{\int }_Q{|{\mathit{II}}_4|}^p{\nu }_{\overrightarrow{\omega }}(x)dx)}^{1/p}\le C\prod _{j=1}^2{\parallel b_j\parallel }_{\mathit{BMO}}{\parallel f_j\parallel }_{L^{p_j,k}({\omega }_j)}.$$

Finally, we still split $T_{\mathrm{\Pi }\mathbf{b}}(f_1^{\mathrm{\infty }},f_2^{\mathrm{\infty }})(x)$ into four terms:

$$\begin{array}{c}{T}_{\mathrm{\Pi }\mathbf{b}}(f_1^{\mathrm{\infty }},f_2^{\mathrm{\infty }})(x)\hfill \\ =(b_1(x)-{\lambda }_1)(b_2(x)-{\lambda }_2)T(f_1^{\mathrm{\infty }},f_2^{\mathrm{\infty }})(x)-(b_1(x)-{\lambda }_1)T(f_1^{\mathrm{\infty }},(b_2-{\lambda }_2)f_2^{\mathrm{\infty }})(x)\hfill \\ -(b_2(x)-{\lambda }_2)T((b_1-{\lambda }_1)f_1^{\mathrm{\infty }},f_2^{\mathrm{\infty }}+T((b_1-{\lambda }_1)f_1^{\mathrm{\infty }},(b_2-{\lambda }_2)f_2^{\mathrm{\infty }})(x))(x)\hfill \\ ={\mathit{IV}}_1+{\mathit{IV}}_2+{\mathit{IV}}_3+{\mathit{IV}}_4.\hfill \end{array}$$

Because each term of ${\mathit{IV}}_j$ is completely analogous to ${\mathit{II}}_j$, $j=1,2,3,4$ with a small difference, we only estimate ${\mathit{IV}}_1$:

$$\begin{array}{rl}|T(f_1^{\mathrm{\infty }},f_2^{\mathrm{\infty }})(x)|& \le C{\int }_{{({\mathbb{R}}^n)}^2\mathrm{\setminus }{(2Q)}^2}\frac{|f_1(y_1)f_2(y_2)|}{{(|x-y_1|+|x-y_2|)}^{2n}}d{y}_{2}d{y}_1\\ \le C\sum _{l=1}^{\mathrm{\infty }}{\int }_{{(2^{l+1}Q)}^2\mathrm{\setminus }{(2^{l}Q)}^2}\frac{|f_1(y_1)f_2(y_2)|}{{(|x-y_1|+|x-y_2|)}^{2n}}d{y}_{2}d{y}_1\\ \le C\sum _{l=1}^{\mathrm{\infty }}\frac{1}{{|2^{l+1}Q|}^2}{\int }_{{(2^{l+1}Q)}^2}\prod _{j=1}^2|f_j(y_j)|d{y}_j\\ \le C\prod _{j=1}^2{\parallel f_j\parallel }_{L^{p_j,k}({\omega }_j)}\sum _{l=1}^{\mathrm{\infty }}{\nu }_{\overrightarrow{\omega }}{\left(2^{l+1}Q\right)}^{(k-1)/p}.\end{array}$$

Hence,

$${(\frac{1}{{\nu }_{\overrightarrow{\omega }}{(Q)}^k}{\int }_Q{|{\mathit{IV}}_1|}^p{\nu }_{\overrightarrow{\omega }}(x)dx)}^{1/p}\le C\prod _{j=1}^2{\parallel b_j\parallel }_{\mathit{BMO}}{\parallel f_j\parallel }_{L^{p_j,k}({\omega }_j)}.$$

Combining all estimates, we complete the proof of Theorem 1.1. □

We now turn to the proof of Theorem 1.2.

Proof of Theorem 1.2 By homogeneity, we may assume that $\lambda ={\parallel b_1\parallel }_{\mathit{BMO}}={\parallel b_2\parallel }_{\mathit{BMO}}=1$ and we only need to prove that

$${\nu }_{\overrightarrow{\omega }}\{x\in Q:|T_{\mathrm{\Pi }\mathbf{b}}(f_1,f_2)(x)|>1\}\le C{\nu }_{\overrightarrow{\omega }}{(Q)}^k\prod _{j=1}^2{\left({\parallel f_j\parallel }_{L^{{\mathrm{\Phi }}^{(2)},k}({\omega }_j)}\right)}^{1/2}.$$

To prove the above inequality, we can write

$$\begin{array}{c}{\nu }_{\overrightarrow{\omega }}\{x\in Q:|T_{\mathrm{\Pi }\mathbf{b}}(f_1,f_2)(x)|>1\}\hfill \\ \le {\nu }_{\overrightarrow{\omega }}\{x\in Q:|T_{\mathrm{\Pi }\mathbf{b}}(f_1^0,f_2^0)(x)|>1/4\}+{\nu }_{\overrightarrow{\omega }}\{x\in Q:|T_{\mathrm{\Pi }\mathbf{b}}(f_1^0,f_2^{\mathrm{\infty }})(x)|>1/4\}\hfill \\ +{\nu }_{\overrightarrow{\omega }}\{x\in Q:|T_{\mathrm{\Pi }\mathbf{b}}(f_1^{\mathrm{\infty }},f_2^0)(x)|>1/4\}+{\nu }_{\overrightarrow{\omega }}\{x\in Q:|T_{\mathrm{\Pi }\mathbf{b}}(f_1^{\mathrm{\infty }},f_2^{\mathrm{\infty }})(x)|>1/4\}\hfill \\ =V+\mathit{VI}+\mathit{VII}+\mathit{VIII}\hfill \end{array}$$

for any cube Q. Employing Lemma 2.9 and Lemma 2.2, we have

$$\begin{array}{rl}V& \le C\prod _{j=1}^2{({\int }_{{\mathbb{R}}^n}{\mathrm{\Phi }}^{(m)}\left(\right|f_j(x)\left|\right){\omega }_j(x)dx)}^{1/2}\\ \le C\prod _{j=1}^2{\left[{\omega }_j{(Q)}^k{\parallel f_j\parallel }_{L^{{\mathrm{\Phi }}^{(m)},k}({\omega }_j)}\right]}^{1/2}\\ \le C{\nu }_{\overrightarrow{\omega }}{(Q)}^k\prod _{j=1}^2{\left[{\parallel f_j\parallel }_{L^{{\mathrm{\Phi }}^{(m)},k}({\omega }_j)}\right]}^{1/2}.\end{array}$$

From Lemma 2.6 and Lemma 2.4, we deduce that

$$\begin{array}{r}{\nu }_{\overrightarrow{\omega }}\{x\in Q:|T_{\mathrm{\Pi }\mathbf{b}}(f_1^0,f_2^{\mathrm{\infty }})(x)|>1/4\}\\ \le \underset{t>0}{sup}\frac{1}{{\mathrm{\Phi }}^{(m)}(\frac{1}{t})}{\nu }_{\overrightarrow{\omega }}\{x\in Q:|T_{\mathrm{\Pi }\mathbf{b}}(f_1^0,f_2^{\mathrm{\infty }})(x)|>t^2\}\\ \le C_{{\nu }_{\overrightarrow{\omega }},\mathbf{b}}\underset{t>0}{sup}\frac{1}{{\mathrm{\Phi }}^{(m)}(\frac{1}{t})}{\nu }_{\overrightarrow{\omega }}\left(\{y\in Q:|{\mathcal{M}}_{\mathrm{\Phi }}(f_1^0,f_2^{\mathrm{\infty }})(y)|>t^2\}\right)\\ \le C_{{\nu }_{\overrightarrow{\omega }},\mathbf{b}}\underset{t>0}{sup}\frac{1}{{\mathrm{\Phi }}^{(m)}(\frac{1}{t})}{\nu }_{\overrightarrow{\omega }}\left(\{y\in Q:|M_{\mathrm{\Phi }}\left(f_1^0\right)(y)M_{\mathrm{\Phi }}\left(f_2^{\mathrm{\infty }}\right)(y)|>t^2\}\right)\\ \le \frac{C_{{\nu }_{\overrightarrow{\omega }},\mathbf{b}}}{t}{({\int }_{{\mathbb{R}}^n}\mathrm{\Phi }\left(\right|f_1^0\left|\right)(y)M({\chi }_Q{\omega }_1)(y)dy{\int }_{{\mathbb{R}}^n}\mathrm{\Phi }\left(\right|f_2^{\mathrm{\infty }}\left|\right)(y)M({\chi }_Q{\omega }_2)(y)dy)}^{1/2}\\ \le \frac{C_{{\nu }_{\overrightarrow{\omega }},\mathbf{b}}}{t}{\left[{\omega }_j{(Q)}^k{\parallel f_j\parallel }_{L^{\mathrm{\Phi },k}({\omega }_j)}\right]}^{1/2},\end{array}$$

where the last inequality holds by the (3.10) and (3.11) in [15]. Then from Lemma 2.2 and the fact that $t\mathrm{\Phi }(\frac{1}{t})>1$, we have

$$\mathit{VI}\le C{\nu }_{\overrightarrow{\omega }}(Q){\left[{\omega }_j{(Q)}^k{\parallel f_j\parallel }_{L^{\mathrm{\Phi },k}({\omega }_j)}\right]}^{1/2}.$$

A similar statement follows:

$$\begin{array}{c}\mathit{VII}\le C{\nu }_{\overrightarrow{\omega }}(Q){\left[{\omega }_j{(Q)}^k{\parallel f_j\parallel }_{L^{\mathrm{\Phi },k}({\omega }_j)}\right]}^{1/2};\hfill \\ \mathit{VIII}\le C{\nu }_{\overrightarrow{\omega }}(Q){\left[{\omega }_j{(Q)}^k{\parallel f_j\parallel }_{L^{\mathrm{\Phi },k}({\omega }_j)}\right]}^{1/2}.\hfill \end{array}$$

Thus we complete the proof of Theorem 1.2. □

4 A problem

Fix $N\in \mathbb{N}$. Let $m\in C^L({\mathbb{R}}^{Nn}\mathrm{\setminus }\{0\})$, for some positive integer L, satisfying the following condition:

$$|{\partial }_{{\xi }_1}^{{\alpha }_1}\cdots {\partial }_{{\xi }_N}^{{\alpha }_N}m({\xi }_1,\dots ,{\xi }_N)|\le C_{{\alpha }_1,\dots ,{\alpha }_N}{(|{\xi }_1|+\cdots +|{\xi }_N|)}^{|\alpha |}$$

(4)

for all $|\alpha |\le s$ and $\xi \in {\mathbb{R}}^{Nn}\mathrm{\setminus }\{0\}$, where $\alpha =({\alpha }_1,\dots ,{\alpha }_N)$ and $\xi =({\xi }_1,\dots ,{\xi }_N)$. The multilinear Fourier multiplier operator $T_N$ is defined by

$$T_m(\overrightarrow{f})(x)=\frac{1}{{(2\pi )}^{Nn}}{\int }_{({\mathbb{R}}^{Nn})}{e}^{ix({\xi }_1+\cdots +{\xi }_N)}m({\xi }_1,\dots ,{\xi }_N){\hat{f}}_1({\xi }_1)\cdots {\hat{f}}_N({\xi }_N)d{\xi }_1\cdots d{\xi }_N$$

(5)

for all $f_1,\dots ,f_N\in \mathcal{S}({\mathbb{R}}^n)$, where $\overrightarrow{f}=(f_1,\dots ,f_N)$. If ${\mathcal{F}}^{-1}m$ is an integrable function, then this can also be written as

$$T_m(\overrightarrow{f})(x)={\int }_{({\mathbb{R}}^{Nn})}{\mathcal{F}}^{-1}m(x-y_1,\dots ,x-y_N)f(y_1)\cdots f(y_N)d{y}_1\cdots d{y}_N.$$

In [18], Fujita and Tomita obtained the following theorem.

Theorem 4.1 Let $1<p_1,\dots ,p_N<\mathrm{\infty }$, $\frac{1}{p_1}+\cdots +\frac{1}{p_N}=\frac{1}{p}$ and $\frac{n}{2}<s_j\le n$ for $1\le j\le N$. Assume $pj>n/sj$ and $w_j\in A_{p_j{s}_j/n}$ for $1\le j\le N$. If $m\in L^{\mathrm{\infty }}({\mathbb{R}}^{Nn})$ satisfies

$${\parallel m_k\parallel }_{W^{(s_1,\dots ,s_N)}}={({\int }_{{\mathbb{R}}^{Nn}}{(1+{|{\xi }_1|}^2)}^{1/2}\cdots |{(1+{|{\xi }_N|}^2)}^{1/2}\hat{m}(\xi ){|}^{2}d\xi )}^{1/2}<\mathrm{\infty },$$

then $T_N$ is bounded from $L^{p_1}({\omega }_1)\times \cdots \times L^{p_N}({\omega }_N)$ to $L^p({\nu }_{\overrightarrow{\omega }})$, where

$$m_j(\xi )=m(2^j{\xi }_1,\dots ,2^j{\xi }_N)\mathrm{\Psi }({\xi }_1,\dots {\xi }_N),$$

where Ψ is the Schwarz function and satisfies

$$supp\mathrm{\Psi }\subset \{\xi \in {\mathbb{R}}^{Nn}:1/2\le |\xi |\le 2\},\sum _{k\in \mathbb{Z}}\mathrm{\Psi }(\xi /2^k)=1\mathit{\text{for all }}\xi \in {\mathbb{R}}^{Nn}\mathrm{\setminus }\{0\}.$$

A natural problem is whether the Lebesgue spaces $L^{p_j}({\omega }_j)$ and $L^p({\nu }_{\overrightarrow{\omega }})$ can be replaced by $L^{p_j,k}(\omega )$ and $L^{p,k}({\nu }_{\overrightarrow{\omega }})$. It should be pointed out that the method in this paper may not be suitable to address this problem.