# Weighted boundedness of multilinear singular integral operator with general kernels for the extreme cases

## Abstract

We prove the weighted boundedness properties for the multilinear operator associated to the singular integral operator with general kernels for the extreme cases.

MSC:42B20, 42B25.

## 1 Introduction and preliminaries

As for the development of singular integral operators, their commutators and multilinear operators have been well studied (see [19]). Let T be the Calderón-Zygmund singular integral operator and $b\in BMO\left({R}^{n}\right)$, a classical result of Coifman et al. (see [5]) stated that the commutator $\left[b,T\right]\left(f\right)=T\left(bf\right)-bT\left(f\right)$ is bounded on ${L}^{p}\left({R}^{n}\right)$ for $1. In [10], the authors obtain the boundedness properties of the commutators for the extreme values of p (that is, $p=1$ and $p=\mathrm{\infty }$). Note that $\left[b,T\right]$ is not bounded for the end point boundedness. The purpose of this paper is to introduce some multilinear operator associated to the singular integral operator with general kernels (see [11]) and prove the weighted boundedness properties of the multilinear operators for the extreme cases.

First, let us introduce some preliminaries (see [6, 9]). Throughout this paper, Q will denote a cube of ${R}^{n}$ with sides parallel to the axes. For a locally integrable functions b and a weight function w (that is, a non-negative locally integrable function), let $w\left(Q\right)={\int }_{Q}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx$, ${w}_{Q}={|Q|}^{-1}{\int }_{Q}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx$, the weighted sharp function of b is defined by

${b}^{\mathrm{#}}\left(x\right)=\underset{Q\ni x}{sup}\frac{1}{w\left(Q\right)}{\int }_{Q}|b\left(y\right)-{b}_{Q}|w\left(y\right)\phantom{\rule{0.2em}{0ex}}dy.$

We say that b belongs to $BMO\left(w\right)$ if ${b}^{\mathrm{#}}$ belongs to ${L}^{\mathrm{\infty }}\left(w\right)$, and we define ${\parallel b\parallel }_{BMO\left(w\right)}={\parallel {b}^{\mathrm{#}}\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}$. If $w=1$, we denote $BMO\left(w\right)=BMO\left({R}^{n}\right)$. It has been known that (see [9])

${\parallel b-{b}_{{2}^{k}Q}\parallel }_{BMO}\le Ck{\parallel b\parallel }_{BMO}.$

We also define the central $BMO$ space by $CMO\left({R}^{n}\right)$, which is the space of those functions $f\in {L}_{\mathrm{loc}}\left({R}^{n}\right)$ such that

${\parallel f\parallel }_{CMO}=\underset{r>1}{sup}{|Q\left(0,r\right)|}^{-1}{\int }_{Q}|f\left(x\right)-{f}_{Q}|\phantom{\rule{0.2em}{0ex}}dx<\mathrm{\infty }.$

It is well known that (see [6, 9])

${\parallel f\parallel }_{CMO}\approx \underset{r>1}{sup}\underset{c\in C}{inf}{|Q\left(0,r\right)|}^{-1}{\int }_{Q}|f\left(x\right)-c|\phantom{\rule{0.2em}{0ex}}dx.$

Definition 1 Let $1 and w be a non-negative weight functions on ${R}^{n}$. We shall call ${B}_{p}\left(w\right)$ the space of those functions f on ${R}^{n}$ such that

${\parallel f\parallel }_{{B}_{p}\left(w\right)}=\underset{r>1}{sup}{\left[w\left(Q\left(0,r\right)\right)\right]}^{-1/p}{\parallel f{\chi }_{Q\left(0,r\right)}\parallel }_{{L}^{p}\left(w\right)}<\mathrm{\infty }.$

The ${A}_{p}$ weight is defined by (see [6])

$\begin{array}{r}{A}_{p}=\left\{0

and

${A}_{1}=\left\{0

## 2 Theorems

In this paper, we will study the following multilinear singular integral operator (see [11]).

Definition 2 Let $T:S\to {S}^{\prime }$ be a linear operator such that T is bounded on ${L}^{2}\left({R}^{n}\right)$ and has a kernel K, that is, there exists a locally integrable function $K\left(x,y\right)$ on ${R}^{n}×{R}^{n}\setminus \left\{\left(x,y\right)\in {R}^{n}×{R}^{n}:x=y\right\}$ such that

$T\left(f\right)\left(x\right)={\int }_{{R}^{n}}K\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy$

for every bounded and compactly supported function f, where K satisfies

$\begin{array}{c}|K\left(x,y\right)|\le C{|x-y|}^{-n},\hfill \\ {\int }_{2|y-z|<|x-y|}\left(|K\left(x,y\right)-K\left(x,z\right)|+|K\left(y,x\right)-K\left(z,x\right)|\right)\phantom{\rule{0.2em}{0ex}}dx\le C,\hfill \end{array}$

and there is a sequence of positive constant numbers $\left\{{C}_{k}\right\}$ such that for any $k\ge 1$,

$\begin{array}{r}{\left({\int }_{{2}^{k}|z-y|\le |x-y|<{2}^{k+1}|z-y|}{\left(|K\left(x,y\right)-K\left(x,z\right)|+|K\left(y,x\right)-K\left(z,x\right)|\right)}^{q}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/q}\\ \phantom{\rule{1em}{0ex}}\le {C}_{k}{\left({2}^{k}|z-y|\right)}^{-n/{q}^{\prime }},\end{array}$

where $1<{q}^{\prime }<2$ and $1/q+1/{q}^{\prime }=1$.

Let ${m}_{j}$ be the positive integers ($j=1,\dots ,l$), ${m}_{1}+\cdots +{m}_{l}=m$ and ${b}_{j}$ be the functions on ${R}^{n}$ ($j=1,\dots ,l$). Set, for $1\le j\le l$,

${R}_{{m}_{j}+1}\left({b}_{j};x,y\right)={b}_{j}\left(x\right)-\sum _{|\alpha |\le {m}_{j}}\frac{1}{\alpha !}{D}^{\alpha }{b}_{j}\left(y\right){\left(x-y\right)}^{\alpha }.$

The multilinear operator associated to T is defined by

${T}_{b}\left(f\right)\left(x\right)={\int }_{{R}^{n}}\frac{{\prod }_{j=1}^{l}{R}_{{m}_{j}+1}\left({b}_{j};x,y\right)}{{|x-y|}^{m}}K\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy.$

Note that the classical Calderón-Zygmund singular integral operator satisfies Definition 2 (see [7, 8]). Also note that when $m=0$, ${T}_{b}$ is just a multilinear commutator of T and b (see [13]). It is well known that a multilinear operator, as a non-trivial extension of a commutator, is of great interest in harmonic analysis and has been widely studied by many authors (see [13]). In this paper, we will study the weighted boundedness properties of the multilinear operators ${T}_{b}$ for the extreme cases (see [1215]).

We shall prove the following theorems in Section 3.

Theorem 1 Let T be the singular integral operator as Definition  2, and we have the sequence $\left\{{k}^{m}{C}_{k}\right\}\in {l}^{1}$, $w\in {A}_{1}$ and ${D}^{\alpha }{b}_{j}\in BMO\left({R}^{n}\right)$ for all α with $|\alpha |={m}_{j}$ and $j=1,\dots ,l$. Then ${T}_{b}$ is bounded from ${L}^{\mathrm{\infty }}\left(w\right)$ to $BMO\left(w\right)$.

Theorem 2 Let T be the singular integral operator as Definition  2, and we have the sequence $\left\{{k}^{m}{C}_{k}\right\}\in {l}^{1}$, $1, $w\in {A}_{1}$, and ${D}^{\alpha }{b}_{j}\in BMO\left({R}^{n}\right)$ for all α with $|\alpha |={m}_{j}$ and $j=1,\dots ,l$. Then ${T}_{b}$ is bounded from ${B}_{p}\left(w\right)$ to $CMO\left(w\right)$.

## 3 Proofs of theorems

We begin with two preliminaries lemmas.

Lemma 1 (see [3])

Let b be a function on ${R}^{n}$ and ${D}^{\alpha }b\in {L}^{q}\left({R}^{n}\right)$ for $|\alpha |=m$ and some $q>n$. Then

$|{R}_{m}\left(b;x,y\right)|\le C{|x-y|}^{m}\sum _{|\alpha |=m}{\left(\frac{1}{|\stackrel{˜}{Q}\left(x,y\right)|}{\int }_{\stackrel{˜}{Q}\left(x,y\right)}{|{D}^{\alpha }b\left(z\right)|}^{q}dz\right)}^{1/q},$

where $\stackrel{˜}{Q}\left(x,y\right)$ is the cube centered at x and having side length $5\sqrt{n}|x-y|$.

Lemma 2 (see [11])

Let T be the singular integral operator as Definition  2, and we have the sequence $\left\{{C}_{k}\right\}\in {l}^{1}$. Then T is bounded on ${L}^{p}\left({R}^{n},w\right)$ for $w\in {A}_{p}$ with $1.

Proof of Theorem 1 It is only for us to prove that there exists a constant ${C}_{Q}$ such that

$\frac{1}{w\left(Q\right)}{\int }_{Q}|{T}_{b}\left(f\right)\left(x\right)-{C}_{Q}|w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\le C{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}$

holds for any cube Q. Without loss of generality, we may assume $l=2$. Fix a cube $Q=Q\left({x}_{0},d\right)$. Let $\stackrel{˜}{Q}=5\sqrt{n}Q$ and ${\stackrel{˜}{b}}_{j}\left(x\right)={b}_{j}\left(x\right)-{\sum }_{|\alpha |=m}\frac{1}{\alpha !}{\left({D}^{\alpha }{b}_{j}\right)}_{\stackrel{˜}{Q}}{x}^{\alpha }$, then ${R}_{m}\left({b}_{j};x,y\right)={R}_{m}\left({\stackrel{˜}{b}}_{j};x,y\right)$ and ${D}^{\alpha }{\stackrel{˜}{b}}_{j}={D}^{\alpha }{b}_{j}-{\left({D}^{\alpha }{b}_{j}\right)}_{\stackrel{˜}{Q}}$ for $|\alpha |={m}_{j}$. We write, for ${f}_{1}=f{\chi }_{\stackrel{˜}{Q}}$ and ${f}_{2}=f{\chi }_{{R}^{n}\setminus \stackrel{˜}{Q}}$,

$\begin{array}{rcl}{T}_{b}\left(f\right)\left(x\right)& =& {\int }_{{R}^{n}}\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}+1}\left({\stackrel{˜}{b}}_{j};x,y\right)}{{|x-y|}^{m}}K\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ =& {\int }_{{R}^{n}}\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,y\right)}{{|x-y|}^{m}}K\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ -\sum _{|{\alpha }_{1}|={m}_{1}}\frac{1}{{\alpha }_{1}!}{\int }_{{R}^{n}}\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};x,y\right){\left(x-y\right)}^{{\alpha }_{1}}{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right)}{{|x-y|}^{m}}K\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ -\sum _{|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{2}!}{\int }_{{R}^{n}}\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};x,y\right){\left(x-y\right)}^{{\alpha }_{2}}{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right)}{{|x-y|}^{m}}K\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ +\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{1}!{\alpha }_{2}!}{\int }_{{R}^{n}}\frac{{\left(x-y\right)}^{{\alpha }_{1}+{\alpha }_{2}}{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right){D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right)}{{|x-y|}^{m}}K\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ +{\int }_{{R}^{n}}\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}+1}\left({\stackrel{˜}{b}}_{j};x,y\right)}{{|x-y|}^{m}}K\left(x,y\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ =& T\left(\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,\cdot \right)}{{|x-\cdot |}^{m}}{f}_{1}\right)\\ -T\left(\sum _{|{\alpha }_{1}|={m}_{1}}\frac{1}{{\alpha }_{1}!}\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};x,\cdot \right){\left(x-\cdot \right)}^{{\alpha }_{1}}{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}}{{|x-\cdot |}^{m}}{f}_{1}\right)\\ -T\left(\sum _{|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{2}!}\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};x,\cdot \right){\left(x-\cdot \right)}^{{\alpha }_{2}}{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}}{{|x-\cdot |}^{m}}{f}_{1}\right)\\ +T\left(\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{1}!{\alpha }_{2}!}\frac{{\left(x-\cdot \right)}^{{\alpha }_{1}+{\alpha }_{2}}{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}}{{|x-\cdot |}^{m}}{f}_{1}\right)\\ +T\left(\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}+1}\left({\stackrel{˜}{b}}_{j};x,\cdot \right)}{{|x-\cdot |}^{m}}{f}_{2}\right),\end{array}$

then

$\begin{array}{rcl}|{T}_{b}\left(f\right)\left(x\right)-{T}_{\stackrel{˜}{b}}\left({f}_{2}\right)\left({x}_{0}\right)|& \le & |T\left(\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,\cdot \right)}{{|x-\cdot |}^{m}}{f}_{1}\right)|\\ +|T\left(\sum _{|{\alpha }_{1}|={m}_{1}}\frac{1}{{\alpha }_{1}!}\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};x,\cdot \right){\left(x-\cdot \right)}^{{\alpha }_{1}}{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}}{{|x-\cdot |}^{m}}{f}_{1}\right)|\\ +|T\left(\sum _{|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{2}!}\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};x,\cdot \right){\left(x-\cdot \right)}^{{\alpha }_{2}}{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}}{{|x-\cdot |}^{m}}{f}_{1}\right)|\\ +|T\left(\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{1}!{\alpha }_{2}!}\frac{{\left(x-\cdot \right)}^{{\alpha }_{1}+{\alpha }_{2}}{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}}{{|x-\cdot |}^{m}}{f}_{1}\right)|\\ +|{T}_{\stackrel{˜}{b}}\left({f}_{2}\right)\left(x\right)-{T}_{\stackrel{˜}{b}}\left({f}_{2}\right)\left({x}_{0}\right)|\\ =& {I}_{1}\left(x\right)+{I}_{2}\left(x\right)+{I}_{3}\left(x\right)+{I}_{4}\left(x\right)+{I}_{5}\left(x\right)\end{array}$

and

$\begin{array}{r}\frac{1}{w\left(Q\right)}{\int }_{Q}|{T}_{b}\left(f\right)\left(x\right)-{T}_{\stackrel{˜}{b}}\left({f}_{2}\right)\left({x}_{0}\right)|w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{w\left(Q\right)}{\int }_{Q}{I}_{1}\left(x\right)w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx+\frac{1}{w\left(Q\right)}{\int }_{Q}{I}_{2}\left(x\right)w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx+\frac{1}{w\left(Q\right)}{\int }_{Q}{I}_{3}\left(x\right)w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+\frac{1}{w\left(Q\right)}{\int }_{Q}{I}_{4}\left(x\right)w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx+\frac{1}{w\left(Q\right)}{\int }_{Q}{I}_{5}\left(x\right)w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}={I}_{1}+{I}_{2}+{I}_{3}+{I}_{4}+{I}_{5}.\end{array}$

Now, let us estimate ${I}_{1}$, ${I}_{2}$, ${I}_{3}$, ${I}_{4}$, and ${I}_{5}$, respectively. First, for $x\in Q$ and $y\in \stackrel{˜}{Q}$, by Lemma 1, we get

${R}_{m}\left({\stackrel{˜}{b}}_{j};x,y\right)\le C{|x-y|}^{m}\sum _{|{\alpha }_{j}|=m}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO},$

thus, by the ${L}^{p}\left(w\right)$-boundedness of T for $1 (Lemma 2) and Hölder’s inequality, we obtain

$\begin{array}{rcl}{I}_{1}& \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right)\frac{1}{w\left(Q\right)}{\int }_{Q}|T\left({f}_{1}\right)\left(x\right)|w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\left(\frac{1}{w\left(Q\right)}{\int }_{{R}^{n}}{|T\left({f}_{1}\right)\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\left(\frac{1}{w\left(Q\right)}{\int }_{{R}^{n}}{|{f}_{1}\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\left(\frac{w\left(\stackrel{˜}{Q}\right)}{w\left(Q\right)}\right)}^{1/p}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}.\end{array}$

For ${I}_{2}$, since $w\in {A}_{1}$, w satisfies the reverse of Hölder’s inequality:

${\left(\frac{1}{|Q|}{\int }_{Q}w{\left(x\right)}^{{p}_{0}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{p}_{0}}\le \frac{C}{|Q|}{\int }_{Q}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx$

for all cube Q and some $1<{p}_{0}<\mathrm{\infty }$ (see [13]), thus, by the ${L}^{p}$-boundedness of T for $p>1$, we get

$\begin{array}{rcl}{I}_{2}& \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{b}_{2}\parallel }_{BMO}\sum _{|{\alpha }_{1}|={m}_{1}}\frac{1}{w\left(Q\right)}{\int }_{Q}|T\left({D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}{f}_{1}\right)\left(x\right)|w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{b}_{2}\parallel }_{BMO}\sum _{|{\alpha }_{1}|={m}_{1}}{\left(\frac{1}{w\left(Q\right)}{\int }_{{R}^{n}}{|T\left({D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}{f}_{1}\right)\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\\ \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{b}_{2}\parallel }_{BMO}\sum _{|{\alpha }_{1}|={m}_{1}}{\left(\frac{1}{w\left(Q\right)}{\int }_{{R}^{n}}{|{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(x\right){f}_{1}\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\\ \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{b}_{2}\parallel }_{BMO}\sum _{|{\alpha }_{1}|={m}_{1}}{\left(\frac{1}{|Q|}{\int }_{\stackrel{˜}{Q}}{|{D}^{{\alpha }_{1}}{b}_{1}\left(x\right)-{\left({D}^{{\alpha }_{1}}{b}_{1}\right)}_{\stackrel{˜}{Q}}|}^{p{p}_{0}^{\prime }}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p{p}_{0}^{\prime }}\\ ×w{\left(Q\right)}^{-1/p}{|Q|}^{1/p}{\left(\frac{1}{|\stackrel{˜}{Q}|}{\int }_{\stackrel{˜}{Q}}w{\left(x\right)}^{{p}_{0}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p{p}_{0}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right)w{\left(Q\right)}^{-1/p}{|Q|}^{1/p}{\left(\frac{1}{|\stackrel{˜}{Q}|}{\int }_{\stackrel{˜}{Q}}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}.\end{array}$

For ${I}_{3}$, similar to the proof of ${I}_{2}$, we get

${I}_{3}\le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}.$

Similarly, for ${I}_{4}$, choose $1<{r}_{1},{r}_{2}<\mathrm{\infty }$ such that $1/{r}_{1}+1/{r}_{2}+1/{p}_{0}=1$, we obtain, by Hölder’s inequality and the reverse of Hölder’s inequality,

$\begin{array}{rcl}{I}_{4}& \le & C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\frac{1}{w\left(Q\right)}{\int }_{Q}|T\left({D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}{f}_{1}\right)\left(x\right)|w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \le & C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}{\left(\frac{1}{w\left(Q\right)}{\int }_{{R}^{n}}{|T\left({D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}{f}_{1}\right)\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\\ \le & C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}w{\left(Q\right)}^{-1/p}{\left({\int }_{{R}^{n}}{|{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(x\right){D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(x\right){f}_{1}\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\\ \le & C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}{\left(\frac{1}{|\stackrel{˜}{Q}|}{\int }_{\stackrel{˜}{Q}}{|{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(x\right)|}^{p{r}_{1}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p{r}_{1}}{\left(\frac{1}{|\stackrel{˜}{Q}|}{\int }_{\stackrel{˜}{Q}}{|{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(x\right)|}^{p{r}_{2}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p{r}_{2}}\\ ×w{\left(Q\right)}^{-1/p}{|Q|}^{1/p}{\left(\frac{1}{|\stackrel{˜}{Q}|}{\int }_{\stackrel{˜}{Q}}w{\left(x\right)}^{{p}_{0}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p{p}_{0}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}.\end{array}$

For ${I}_{5}$, we write

$\begin{array}{r}{T}_{\stackrel{˜}{b}}\left({f}_{2}\right)\left(x\right)-{T}_{\stackrel{˜}{b}}\left({f}_{2}\right)\left({x}_{0}\right)\\ \phantom{\rule{1em}{0ex}}={\int }_{{R}^{n}}\left(\frac{K\left(x,y\right)}{{|x-y|}^{m}}-\frac{K\left({x}_{0},y\right)}{{|{x}_{0}-y|}^{m}}\right)\prod _{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,y\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}+{\int }_{{R}^{n}}\left({R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};x,y\right)-{R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};{x}_{0},y\right)\right)\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};x,y\right)}{{|{x}_{0}-y|}^{m}}K\left({x}_{0},y\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}+{\int }_{{R}^{n}}\left({R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};x,y\right)-{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};{x}_{0},y\right)\right)\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};{x}_{0},y\right)}{{|{x}_{0}-y|}^{m}}K\left({x}_{0},y\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}-\sum _{|{\alpha }_{1}|={m}_{1}}\frac{1}{{\alpha }_{1}!}{\int }_{{R}^{n}}\left[\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};x,y\right){\left(x-y\right)}^{{\alpha }_{1}}}{{|x-y|}^{m}}K\left(x,y\right)-\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};{x}_{0},y\right){\left({x}_{0}-y\right)}^{{\alpha }_{1}}}{{|{x}_{0}-y|}^{m}}K\left({x}_{0},y\right)\right]\\ \phantom{\rule{2em}{0ex}}×{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}-\sum _{|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{2}!}{\int }_{{R}^{n}}\left[\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};x,y\right){\left(x-y\right)}^{{\alpha }_{2}}}{{|x-y|}^{m}}K\left(x,y\right)-\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};{x}_{0},y\right){\left({x}_{0}-y\right)}^{{\alpha }_{2}}}{{|{x}_{0}-y|}^{m}}K\left({x}_{0},y\right)\right]\\ \phantom{\rule{2em}{0ex}}×{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}+\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{1}!{\alpha }_{2}!}{\int }_{{R}^{n}}\left[\frac{{\left(x-y\right)}^{{\alpha }_{1}+{\alpha }_{2}}}{{|x-y|}^{m}}K\left(x,y\right)-\frac{{\left({x}_{0}-y\right)}^{{\alpha }_{1}+{\alpha }_{2}}}{{|{x}_{0}-y|}^{m}}K\left({x}_{0},y\right)\right]\\ \phantom{\rule{2em}{0ex}}×{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right){D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{1em}{0ex}}={I}_{5}^{\left(1\right)}\left(x\right)+{I}_{5}^{\left(2\right)}\left(x\right)+{I}_{5}^{\left(3\right)}\left(x\right)+{I}_{5}^{\left(4\right)}\left(x\right)+{I}_{5}^{\left(5\right)}\left(x\right)+{I}_{5}^{\left(6\right)}\left(x\right).\end{array}$

By Lemma 1 and the inequality (see [9])

we know that, for $x\in Q$ and $y\in {2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}$,

$\begin{array}{rcl}|{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,y\right)|& \le & C{|x-y|}^{{m}_{j}}\sum _{|\alpha |={m}_{j}}\left({\parallel {D}^{\alpha }{b}_{j}\parallel }_{BMO}+|{\left({D}^{\alpha }{b}_{j}\right)}_{\stackrel{˜}{Q}\left(x,y\right)}-{\left({D}^{\alpha }{b}_{j}\right)}_{\stackrel{˜}{Q}}|\right)\\ \le & Ck{|x-y|}^{{m}_{j}}\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{b}_{j}\parallel }_{BMO}.\end{array}$

Note that $|x-y|\sim |{x}_{0}-y|$ for $x\in Q$ and $y\in {R}^{n}\setminus \stackrel{˜}{Q}$, we obtain, by the conditions on K,

$\begin{array}{rcl}|{I}_{5}^{\left(1\right)}\left(x\right)|& \le & {\int }_{{R}^{n}}|\frac{1}{{|x-y|}^{m}}-\frac{1}{{|{x}_{0}-y|}^{m}}||K\left(x,y\right)|\prod _{j=1}^{2}|{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,y\right)||{f}_{2}\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ +{\int }_{{R}^{n}}|K\left(x,y\right)-K\left({x}_{0},y\right)|{|{x}_{0}-y|}^{-m}\prod _{j=1}^{2}|{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,y\right)||{f}_{2}\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \le & \sum _{k=0}^{\mathrm{\infty }}{\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}|\frac{1}{{|x-y|}^{m}}-\frac{1}{{|{x}_{0}-y|}^{m}}||K\left(x,y\right)|\prod _{j=1}^{2}|{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,y\right)||f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ +\sum _{k=0}^{\mathrm{\infty }}{\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}|K\left(x,y\right)-K\left({x}_{0},y\right)|{|{x}_{0}-y|}^{-m}\prod _{j=1}^{2}|{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,y\right)||f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right)\sum _{k=0}^{\mathrm{\infty }}{k}^{2}{\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}\frac{|x-{x}_{0}|}{{|{x}_{0}-y|}^{n+1}}|f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ +C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right)\sum _{k=0}^{\mathrm{\infty }}{k}^{2}{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}{|f\left(y\right)|}^{{q}^{\prime }}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/{q}^{\prime }}\\ ×{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}{|K\left(x,y\right)-K\left({x}_{0},y\right)|}^{q}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/q}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right)\sum _{k=1}^{\mathrm{\infty }}{k}^{2}\left({2}^{-k}+{C}_{k}\right){\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}.\end{array}$

For ${I}_{5}^{\left(2\right)}\left(x\right)$, by the formula (see [3]):

${R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,y\right)-{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};{x}_{0},y\right)=\sum _{|\beta |

and Lemma 1, we have

$|{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,y\right)-{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};{x}_{0},y\right)|\le C\sum _{|\beta |<{m}_{j}}\sum _{|\alpha |={m}_{j}}{|x-{x}_{0}|}^{{m}_{j}-|\beta |}{|x-y|}^{|\beta |}{\parallel {D}^{\alpha }{b}_{j}\parallel }_{BMO},$

thus

$\begin{array}{rcl}|{I}_{5}^{\left(2\right)}\left(x\right)|& \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right)\sum _{k=0}^{\mathrm{\infty }}{\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}k\frac{|x-{x}_{0}|}{{|{x}_{0}-y|}^{n+1}}|f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right)\sum _{k=1}^{\mathrm{\infty }}k{2}^{-k}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}.\end{array}$

Similarly,

$|{I}_{5}^{\left(3\right)}\left(x\right)|\le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}.$

For ${I}_{5}^{\left(4\right)}\left(x\right)$, similar to the proof of ${I}_{5}^{\left(1\right)}\left(x\right)$ and ${I}_{5}^{\left(2\right)}\left(x\right)$, we get

$\begin{array}{rcl}|{I}_{5}^{\left(4\right)}\left(x\right)|& \le & C\sum _{|{\alpha }_{1}|={m}_{1}}{\int }_{{R}^{n}}|{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};x,y\right)-{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};{x}_{0},y\right)|\frac{|{\left({x}_{0}-y\right)}^{{\alpha }_{1}}K\left({x}_{0},y\right)|}{{|{x}_{0}-y|}^{m}}\\ ×|{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right)||{f}_{2}\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ +C\sum _{|{\alpha }_{1}|={m}_{1}}{\int }_{{R}^{n}}|\frac{{\left(x-y\right)}^{{\alpha }_{1}}}{{|x-y|}^{m}}-\frac{{\left({x}_{0}-y\right)}^{{\alpha }_{1}}}{{|{x}_{0}-y|}^{m}}||K\left(x,y\right)||{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};x,y\right)|\\ ×|{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right)||{f}_{2}\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ +C\sum _{|{\alpha }_{1}|={m}_{1}}{\int }_{{R}^{n}}|K\left(x,y\right)-K\left({x}_{0},y\right)||\frac{{\left({x}_{0}-y\right)}^{{\alpha }_{1}}}{{|{x}_{0}-y|}^{m}}||{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};x,y\right)|\\ ×|{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right)||{f}_{2}\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{b}_{2}\parallel }_{BMO}\sum _{k=1}^{\mathrm{\infty }}k{2}^{-k}\sum _{|{\alpha }_{1}|={m}_{1}}\left(\frac{1}{|{2}^{k}\stackrel{˜}{Q}|}{\int }_{{2}^{k}\stackrel{˜}{Q}}|{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\right){\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}\\ +C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{b}_{2}\parallel }_{BMO}\sum _{|{\alpha }_{1}|={m}_{1}}\sum _{k=0}^{\mathrm{\infty }}k{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}{|K\left(x,y\right)-K\left({x}_{0},y\right)|}^{q}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/q}\\ ×{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}{|{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right)|}^{{q}^{\prime }}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/{q}^{\prime }}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right)\sum _{k=1}^{\mathrm{\infty }}{k}^{2}\left({2}^{-k}+{C}_{k}\right){\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}.\end{array}$

Similarly,

$|{I}_{5}^{\left(5\right)}\left(x\right)|\le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}.$

For ${I}_{5}^{\left(6\right)}\left(x\right)$, taking $1<{r}_{1},{r}_{2}<\mathrm{\infty }$ such that $1/{r}_{1}+1/{r}_{2}=1$, then

$\begin{array}{rcl}|{I}_{5}^{\left(6\right)}\left(x\right)|& \le & C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}{\int }_{{R}^{n}}|\frac{{\left(x-y\right)}^{{\alpha }_{1}+{\alpha }_{2}}K\left(x,y\right)}{{|x-y|}^{m}}-\frac{{\left({x}_{0}-y\right)}^{{\alpha }_{1}+{\alpha }_{2}}K\left({x}_{0},y\right)}{{|{x}_{0}-y|}^{m}}|\\ ×|{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right)||{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right)||{f}_{2}\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \le & C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\sum _{k=1}^{\mathrm{\infty }}\left({2}^{-k}+{C}_{k}\right){\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}\\ ×{\left(\frac{1}{|{2}^{k}\stackrel{˜}{Q}|}{\int }_{{2}^{k}\stackrel{˜}{Q}}{|{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right)|}^{{r}_{1}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/{r}_{1}}{\left(\frac{1}{|{2}^{k}\stackrel{˜}{Q}|}{\int }_{{2}^{k}\stackrel{˜}{Q}}{|{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right)|}^{{r}_{2}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/{r}_{2}}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right)\sum _{k=1}^{\mathrm{\infty }}{k}^{2}\left({2}^{-k}+{C}_{k}\right){\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}.\end{array}$

Thus

${I}_{5}\le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(w\right)}.$

This completes the proof of Theorem 1. □

Proof of Theorem 2 It is only for us to prove that there exists a constant ${C}_{Q}$ such that

$\frac{1}{w\left(Q\right)}{\int }_{Q}|{T}_{b}\left(f\right)\left(x\right)-{C}_{Q}|w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\le C{\parallel f\parallel }_{{B}_{p}\left(w\right)}$

holds for any cube $Q=Q\left(0,d\right)$ with $d>1$. Without loss of generality, we may assume $l=2$. Fix a cube $Q=Q\left(0,d\right)$ with $d>1$. Let $\stackrel{˜}{Q}=5\sqrt{n}Q$ and ${\stackrel{˜}{b}}_{j}\left(x\right)={b}_{j}\left(x\right)-{\sum }_{|\alpha |={m}_{j}}\frac{1}{\alpha !}{\left({D}^{\alpha }{b}_{j}\right)}_{\stackrel{˜}{Q}}{x}^{\alpha }$, then ${R}_{{m}_{j}}\left({b}_{j};x,y\right)={R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,y\right)$ and ${D}^{\alpha }{\stackrel{˜}{b}}_{j}={D}^{\alpha }{b}_{j}-{\left({D}^{\alpha }{b}_{j}\right)}_{\stackrel{˜}{Q}}$ for $|\alpha |={m}_{j}$. Similar to the proof of Theorem 1, we write, for ${f}_{1}=f{\chi }_{\stackrel{˜}{Q}}$ and ${f}_{2}=f{\chi }_{{R}^{n}\setminus \stackrel{˜}{Q}}$,

$\begin{array}{r}\frac{1}{w\left(Q\right)}{\int }_{Q}|{T}_{b}\left(f\right)\left(x\right)-{T}_{\stackrel{˜}{b}}\left({f}_{2}\right)\left(0\right)|w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{w\left(Q\right)}{\int }_{Q}|T\left(\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,\cdot \right)}{{|x-\cdot |}^{m}}{f}_{1}\right)|w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+\frac{1}{w\left(Q\right)}{\int }_{Q}|T\left(\sum _{|{\alpha }_{1}|={m}_{1}}\frac{1}{{\alpha }_{1}!}\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};x,\cdot \right){\left(x-\cdot \right)}^{{\alpha }_{1}}{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}}{{|x-\cdot |}^{m}}{f}_{1}\right)|w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+\frac{1}{w\left(Q\right)}{\int }_{Q}|T\left(\sum _{|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{2}!}\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};x,\cdot \right){\left(x-\cdot \right)}^{{\alpha }_{2}}{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}}{{|x-\cdot |}^{m}}{f}_{1}\right)|w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+\frac{1}{w\left(Q\right)}{\int }_{Q}|T\left(\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{1}!{\alpha }_{2}!}\frac{{\left(x-\cdot \right)}^{{\alpha }_{1}+{\alpha }_{2}}{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}}{{|x-\cdot |}^{m}}{f}_{1}\right)|w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+\frac{1}{w\left(Q\right)}{\int }_{Q}|{T}_{\stackrel{˜}{b}}\left({f}_{2}\right)\left(x\right)-{T}_{\stackrel{˜}{b}}\left({f}_{2}\right)\left(0\right)|w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}={L}_{1}+{L}_{2}+{L}_{3}+{L}_{4}+{L}_{5}.\end{array}$

Similar to the proof of Theorem 1, we get

$\begin{array}{rcl}{L}_{1}& \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\left(\frac{1}{w\left(Q\right)}{\int }_{{R}^{n}}{|T\left({f}_{1}\right)\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right)w{\left(\stackrel{˜}{Q}\right)}^{-1/p}{\parallel f{\chi }_{\stackrel{˜}{Q}}\parallel }_{{L}^{p}\left(w\right)}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{B}_{p}\left(w\right)}.\end{array}$

For ${L}_{2}$, taking $r,s,t>1$ such that $r, $t=p{p}_{0}/\left(p-r\right)$, and $1/s+1/\left(p/r\right)+1/t=1$, then, by the reverse of Hölder’s inequality,

$\begin{array}{rcl}{L}_{2}& \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{b}_{2}\parallel }_{BMO}\sum _{|{\alpha }_{1}|={m}_{1}}{\left(\frac{1}{w\left(Q\right)}{\int }_{{R}^{n}}{|T\left({D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}{f}_{1}\right)\left(x\right)|}^{r}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/r}\\ \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{b}_{2}\parallel }_{BMO}w{\left(Q\right)}^{-1/r}\sum _{|{\alpha }_{1}|={m}_{1}}{\left({\int }_{{R}^{n}}{|{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(x\right){f}_{1}\left(x\right)|}^{r}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/r}\\ \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{b}_{2}\parallel }_{BMO}w{\left(Q\right)}^{-1/r}\sum _{|{\alpha }_{1}|={m}_{1}}{\left({\int }_{\stackrel{˜}{Q}}{|{D}^{\alpha }{\stackrel{˜}{b}}_{1}\left(x\right)|}^{rs}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/rs}\\ ×{\left({\int }_{\stackrel{˜}{Q}}{|f\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}{\left({\int }_{\stackrel{˜}{Q}}w{\left(x\right)}^{\left(1-r/p\right)t}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/rt}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right)w{\left(Q\right)}^{-1/r}{|Q|}^{1/rs}{\parallel f{\chi }_{\stackrel{˜}{Q}}\parallel }_{{L}^{p}\left(w\right)}{|Q|}^{1/rt}\\ ×{\left(\frac{1}{|\stackrel{˜}{Q}|}{\int }_{\stackrel{˜}{Q}}w{\left(x\right)}^{{p}_{0}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/rt}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right)w{\left(Q\right)}^{-1/r}{|Q|}^{1/rs}{\parallel f{\chi }_{\stackrel{˜}{Q}}\parallel }_{{L}^{p}\left(w\right)}{\left(\frac{1}{|\stackrel{˜}{Q}|}{\int }_{\stackrel{˜}{Q}}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{{p}_{0}/rt}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right)w{\left(\stackrel{˜}{Q}\right)}^{-1/p}{\parallel f{\chi }_{\stackrel{˜}{Q}}\parallel }_{{L}^{p}\left(w\right)}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{B}_{p}\left(w\right)},\\ {L}_{3}& \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{B}_{p}\left(w\right)}.\end{array}$

For ${L}_{4}$, taking $r,{s}_{1},{s}_{2},t>1$ such that $r, $t=p{p}_{0}/\left(p-r\right)$, and $1/{s}_{1}+1/{s}_{2}+1/\left(p/r\right)+1/t=1$, then, by the reverse of Hölder’s inequality,

$\begin{array}{rcl}{L}_{4}& \le & C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}{\left(\frac{1}{w\left(Q\right)}{\int }_{{R}^{n}}{|T\left({D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}{f}_{1}\right)\left(x\right)|}^{r}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/r}\\ \le & Cw{\left(Q\right)}^{-1/r}\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}{\left({\int }_{{R}^{n}}{|{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(x\right){D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(x\right){f}_{1}\left(x\right)|}^{r}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/r}\\ \le & Cw{\left(Q\right)}^{-1/r}\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}{\left({\int }_{\stackrel{˜}{Q}}{|{D}^{\alpha }{\stackrel{˜}{b}}_{1}\left(x\right)|}^{r{s}_{1}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/r{s}_{1}}{\left({\int }_{\stackrel{˜}{Q}}{|{D}^{\alpha }{\stackrel{˜}{b}}_{2}\left(x\right)|}^{r{s}_{2}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/r{s}_{2}}\\ ×{\left({\int }_{\stackrel{˜}{Q}}{|f\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}{\left({\int }_{\stackrel{˜}{Q}}w{\left(x\right)}^{\left(1-r/p\right)t}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/rt}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right)w{\left(Q\right)}^{-1/r}{|Q|}^{1/r{s}_{1}+1/r{s}_{2}+1/rt}{\parallel f{\chi }_{\stackrel{˜}{Q}}\parallel }_{{L}^{p}\left(w\right)}\\ ×{\left(\frac{1}{|\stackrel{˜}{Q}|}{\int }_{\stackrel{˜}{Q}}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{{p}_{0}/rt}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right)w{\left(\stackrel{˜}{Q}\right)}^{-1/p}{\parallel f{\chi }_{\stackrel{˜}{Q}}\parallel }_{{L}^{p}\left(w\right)}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{B}_{p}\left(w\right)}.\end{array}$

For ${L}_{5}$, similar to the proof of the proof of ${I}_{5}$ in Theorem 1, we have

$\begin{array}{r}{T}_{\stackrel{˜}{b}}\left({f}_{2}\right)\left(x\right)-{T}_{\stackrel{˜}{b}}\left({f}_{2}\right)\left(0\right)\\ \phantom{\rule{1em}{0ex}}={\int }_{{R}^{n}}\left(\frac{K\left(x,y\right)}{{|x-y|}^{m}}-\frac{K\left(0,y\right)}{{|y|}^{m}}\right)\prod _{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,y\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}+{\int }_{{R}^{n}}\left({R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};x,y\right)-{R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};0,y\right)\right)\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};x,y\right)}{{|y|}^{m}}K\left(0,y\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}+{\int }_{{R}^{n}}\left({R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};x,y\right)-{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};0,y\right)\right)\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};{x}_{0},y\right)}{{|y|}^{m}}K\left(0,y\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}-\sum _{|{\alpha }_{1}|={m}_{1}}\frac{1}{{\alpha }_{1}!}{\int }_{{R}^{n}}\left[\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};x,y\right){\left(x-y\right)}^{{\alpha }_{1}}}{{|x-y|}^{m}}K\left(x,y\right)-\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};0,y\right){\left(-y\right)}^{{\alpha }_{1}}}{{|y|}^{m}}K\left(0,y\right)\right]\\ \phantom{\rule{2em}{0ex}}×{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}-\sum _{|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{2}!}{\int }_{{R}^{n}}\left[\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};x,y\right){\left(x-y\right)}^{{\alpha }_{2}}}{{|x-y|}^{m}}K\left(x,y\right)-\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};0,y\right){\left(-y\right)}^{{\alpha }_{2}}}{{|y|}^{m}}K\left(0,y\right)\right]\\ \phantom{\rule{2em}{0ex}}×{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}+\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{1}!{\alpha }_{2}!}{\int }_{{R}^{n}}\left[\frac{{\left(x-y\right)}^{{\alpha }_{1}+{\alpha }_{2}}}{{|x-y|}^{m}}K\left(x,y\right)-\frac{{\left(-y\right)}^{{\alpha }_{1}+{\alpha }_{2}}}{{|y|}^{m}}K\left(0,y\right)\right]\\ \phantom{\rule{2em}{0ex}}×{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right){D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{1em}{0ex}}={L}_{5}^{\left(1\right)}\left(x\right)+{L}_{5}^{\left(2\right)}\left(x\right)+{L}_{5}^{\left(3\right)}\left(x\right)+{L}_{5}^{\left(4\right)}\left(x\right)+{L}_{5}^{\left(5\right)}\left(x\right)+{L}_{5}^{\left(6\right)}\left(x\right).\end{array}$

For ${L}_{5}^{\left(1\right)}\left(x\right)$, taking $1 such that $1/p+1/q+1/r=1$, by $w\in {A}_{1}\subset {A}_{p/r+1}$, we get

$\begin{array}{rcl}|{L}_{5}^{\left(1\right)}\left(x\right)|& \le & C\sum _{k=0}^{\mathrm{\infty }}{\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}|\frac{1}{{|x-y|}^{m}}-\frac{1}{{|y|}^{m}}||K\left(x,y\right)|\prod _{j=1}^{2}|{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,y\right)||f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ +\sum _{k=0}^{\mathrm{\infty }}{\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}|K\left(x,y\right)-K\left(0,y\right)|{|y|}^{-m}\\ ×\prod _{j=1}^{2}|{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,y\right)||f\left(y\right)|w{\left(y\right)}^{1/p}w{\left(y\right)}^{-1/p}\phantom{\rule{0.2em}{0ex}}dy\\ \le & C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{b}_{j}\parallel }_{BMO}\right)\sum _{k=0}^{\mathrm{\infty }}{\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}{k}^{2}\frac{d}{{\left({2}^{k}d\right)}^{n+1}}|f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ +C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{b}_{j}\parallel }_{BMO}\right)\sum _{k=0}^{\mathrm{\infty }}{k}^{2}{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}{|K\left(x,y\right)-K\left(0,y\right)|}^{q}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/q}\\ ×{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}}{|f\left(y\right)|}^{p}w\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p}{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}}w{\left(y\right)}^{-r/p}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/r}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right)\sum _{k=1}^{\mathrm{\infty }}{k}^{2}{2}^{-k}w{\left({2}^{k}\stackrel{˜}{Q}\right)}^{-1/p}{\left({\int }_{{2}^{k}\stackrel{˜}{Q}}{|f\left(y\right)|}^{p}w\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p}\\ ×{\left(\frac{1}{|{2}^{k}\stackrel{˜}{Q}|}{\int }_{{2}^{k}\stackrel{˜}{Q}}w\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p}{\left(\frac{1}{|{2}^{k}\stackrel{˜}{Q}|}{\int }_{{2}^{k}\stackrel{˜}{Q}}w{\left(y\right)}^{-1/\left(p-1\right)}\phantom{\rule{0.2em}{0ex}}dy\right)}^{\left(p-1\right)/p}\\ +C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right)\sum _{k=1}^{\mathrm{\infty }}{k}^{2}{C}_{k}w{\left({2}^{k}\stackrel{˜}{Q}\right)}^{-1/p}{\left({\int }_{{2}^{k}\stackrel{˜}{Q}}{|f\left(y\right)|}^{p}w\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p}\\ ×{\left(\frac{1}{|{2}^{k}\stackrel{˜}{Q}|}{\int }_{{2}^{k}\stackrel{˜}{Q}}w\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p}{\left(\frac{1}{|{2}^{k}\stackrel{˜}{Q}|}{\int }_{{2}^{k}\stackrel{˜}{Q}}w{\left(y\right)}^{-r/p}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/r}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{B}_{p}\left(w\right)}.\end{array}$

Similarly, we get, for $1<{r}_{1},{r}_{2},{r}_{3},{r}_{4},s<\mathrm{\infty }$ with $1/p+1/{r}_{1}+1/s=1$, $1/p+1/q+1/{r}_{2}+1/s=1$, and $1/p+1/q+1/{r}_{3}+1/{r}_{4}+1/s=1$,

$\begin{array}{r}|{L}_{5}^{\left(2\right)}\left(x\right)+{L}_{5}^{\left(3\right)}\left(x\right)+{L}_{5}^{\left(4\right)}\left(x\right)+{L}_{5}^{\left(5\right)}\left(x\right)+{L}_{5}^{\left(6\right)}\left(x\right)|\\ \phantom{\rule{1em}{0ex}}\le C\left(\sum _{|\alpha |={m}_{2}}{\parallel {D}^{\alpha }{b}_{2}\parallel }_{BMO}\right)\sum _{|\alpha |={m}_{1}}\sum _{k=0}^{\mathrm{\infty }}{\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}k\frac{d}{{\left({2}^{k}d\right)}^{n+1}}|{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right)||f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}+C\left(\sum _{|\alpha |={m}_{1}}{\parallel {D}^{\alpha }{b}_{1}\parallel }_{BMO}\right)\sum _{|\alpha |={m}_{2}}\sum _{k=0}^{\mathrm{\infty }}{\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}k\frac{d}{{\left({2}^{k}d\right)}^{n+1}}|{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right)||f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}+C\left(\sum _{|\alpha |={m}_{2}}{\parallel {D}^{\alpha }{b}_{2}\parallel }_{BMO}\right)\sum _{|\alpha |={m}_{1}}\sum _{k=0}^{\mathrm{\infty }}k{\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}|K\left(x,y\right)-K\left(0,y\right)|\\ \phantom{\rule{2em}{0ex}}×|{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right)||f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}+C\left(\sum _{|\alpha |={m}_{1}}{\parallel {D}^{\alpha }{b}_{1}\parallel }_{BMO}\right)\sum _{|\alpha |={m}_{2}}\sum _{k=0}^{\mathrm{\infty }}k{\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}|K\left(x,y\right)-K\left(0,y\right)|\\ \phantom{\rule{2em}{0ex}}×|{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right)||f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}+C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\sum _{k=0}^{\mathrm{\infty }}{\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}|K\left(x,y\right)-K\left(0,y\right)||{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right)||{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right)||f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{1em}{0ex}}\le C\left(\sum _{|\alpha |={m}_{2}}{\parallel {D}^{\alpha }{b}_{2}\parallel }_{BMO}\right)\sum _{|\alpha |={m}_{1}}\sum _{k=0}^{\mathrm{\infty }}k{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}}{|{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right)|}^{{r}_{1}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/{r}_{1}}\\ \phantom{\rule{2em}{0ex}}×{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}}{|f\left(y\right)|}^{p}w\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p}{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}}w{\left(y\right)}^{-s/p}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/s}\\ \phantom{\rule{2em}{0ex}}+C\left(\sum _{|\alpha |={m}_{1}}{\parallel {D}^{\alpha }{b}_{1}\parallel }_{BMO}\right)\sum _{|\alpha |={m}_{2}}\sum _{k=0}^{\mathrm{\infty }}k{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}}{|{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right)|}^{{r}_{1}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/{r}_{1}}\\ \phantom{\rule{2em}{0ex}}×{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}}{|f\left(y\right)|}^{p}w\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p}{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}}w{\left(y\right)}^{-s/p}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/s}\\ \phantom{\rule{2em}{0ex}}+C\left(\sum _{|\alpha |={m}_{2}}{\parallel {D}^{\alpha }{b}_{2}\parallel }_{BMO}\right)\sum _{|\alpha |={m}_{1}}\sum _{k=0}^{\mathrm{\infty }}k{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}{|K\left(x,y\right)-K\left(0,y\right)|}^{q}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/q}\\ \phantom{\rule{2em}{0ex}}×{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}}{|{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right)|}^{{r}_{2}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/{r}_{2}}{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}}{|f\left(y\right)|}^{p}w\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p}{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}}w{\left(y\right)}^{-s/p}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/s}\\ \phantom{\rule{2em}{0ex}}+C\left(\sum _{|\alpha |={m}_{1}}{\parallel {D}^{\alpha }{b}_{1}\parallel }_{BMO}\right)\sum _{|\alpha |={m}_{2}}\sum _{k=0}^{\mathrm{\infty }}k{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}{|K\left(x,y\right)-K\left(0,y\right)|}^{q}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/q}\\ \phantom{\rule{2em}{0ex}}×{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}}{|{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right)|}^{{r}_{2}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/{r}_{2}}{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}}{|f\left(y\right)|}^{p}w\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p}{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}}w{\left(y\right)}^{-s/p}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/s}\\ \phantom{\rule{2em}{0ex}}+C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\sum _{k=0}^{\mathrm{\infty }}{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}{|K\left(x,y\right)-K\left(0,y\right)|}^{q}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/q}{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}}w{\left(y\right)}^{-s/p}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/s}\\ \phantom{\rule{2em}{0ex}}×{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}}{|{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right)|}^{{r}_{3}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/{r}_{3}}{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}}{|{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right)|}^{{r}_{4}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/{r}_{4}}\\ \phantom{\rule{2em}{0ex}}×{\left({\int }_{{2}^{k+1}\stackrel{˜}{Q}}{|f\left(y\right)|}^{p}w\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p}\\ \phantom{\rule{1em}{0ex}}\le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right)\sum _{k=1}^{\mathrm{\infty }}{k}^{2}\left({2}^{-k}+{C}_{k}\right)w{\left({2}^{k}\stackrel{˜}{Q}\right)}^{-1/p}{\left({\int }_{{2}^{k}\stackrel{˜}{Q}}{|f\left(y\right)|}^{p}w\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p}\\ \phantom{\rule{1em}{0ex}}\le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{B}_{p}\left(w\right)}.\end{array}$

Thus

${L}_{5}\le C\prod _{j=1}^{2}\left(\sum _{{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{B}_{p}\left(w\right)}.$

This finishes the proof of Theorem 2. □

## References

1. Cohen J:A sharp estimate for a multilinear singular integral on ${R}^{n}$. Indiana Univ. Math. J. 1981, 30: 693-702. 10.1512/iumj.1981.30.30053

2. Cohen J, Gosselin J:On multilinear singular integral operators on ${R}^{n}$. Stud. Math. 1982, 72: 199-223.

3. Cohen J, Gosselin J: A BMO estimate for multilinear singular integral operators. Ill. J. Math. 1986, 30: 445-465.

4. Coifman R, Meyer Y Cambridge Studies in Advanced Math. 48. In Wavelets, Calderón-Zygmund and Multilinear Operators. Cambridge University Press, Cambridge; 1997.

5. Coifman RR, Rochberg R, Weiss G: Factorization theorems for Hardy spaces in several variables. Ann. Math. 1976, 103: 611-635. 10.2307/1970954

6. Garcia-Cuerva J, Rubio de Francia JL North-Holland Mathematics Studies 116. In Weighted Norm Inequalities and Related Topics. North-Holland, Amsterdam; 1985.

7. Pérez C, Pradolini G: Sharp weighted endpoint estimates for commutators of singular integral operators. Mich. Math. J. 2001, 49: 23-37. 10.1307/mmj/1008719033

8. Pérez C, Trujillo-Gonzalez R: Sharp weighted estimates for multilinear commutators. J. Lond. Math. Soc. 2002, 65: 672-692. 10.1112/S0024610702003174

9. Stein EM: Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton; 1993.

10. Harboure E, Segovia C, Torrea JL: Boundedness of commutators of fractional and singular integrals for the extreme values of p . Ill. J. Math. 1997, 41: 676-700.

11. Chang DC, Li JF, Xiao J: Weighted scale estimates for Calderón-Zygmund type operators. Contemp. Math. 2007, 446: 61-70.

12. Lin Y: Sharp maximal function estimates for Calderón-Zygmund type operators and commutators. Acta Math. Sci. Ser. A Chin. Ed. 2011, 31: 206-215.

13. Liu LZ: Weighted boundedness of multilinear operators for the extreme cases. Taiwan. J. Math. 2006, 10: 669-690.

14. Liu LZ: Weighted Herz type spaces estimates of multilinear singular integral operators for the extreme cases. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 2007, 101: 87-98.

15. Liu LZ: Endpoint estimates for multilinear fractional singular integral operators on some Hardy spaces. Math. Notes 2010, 88: 735-752.

## Acknowledgements

The work is supported by NNSFC (No. 11301097), Master Foundation of Guangxi University of Science and Technology (No. 0816208), Guangxi Education Institution Scientific Research Item (No. 2013YB170), GXNSF Grant (No. 2013GXNSFAA019001).

## Author information

Authors

### Corresponding author

Correspondence to Yanfeng Guo.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The authors completed the paper. MZ carried out the ideas and methods of studies, YG participated in the design of the study and performed the statistical analysis, the sequence alignment and drafted the manuscript. The authors read and approved the final manuscript.