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 [1–9]). Let T be the Calderón-Zygmund singular integral operator and $b\in BMO(R^n)$, a classical result of Coifman et al. (see [5]) stated that the commutator $[b,T](f)=T(bf)-bT(f)$ is bounded on $L^p(R^n)$ for $1<p<\mathrm{\infty }$. 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 $[b,T]$ 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 $w(Q)={\int }_{Q}w(x)dx$, $w_Q={|Q|}^{-1}{\int }_{Q}w(x)dx$, the weighted sharp function of b is defined by

$$b^{\mathrm{\#}}(x)=\underset{Q\ni x}{sup}\frac{1}{w(Q)}{\int }_Q|b(y)-b_Q|w(y)dy.$$

We say that b belongs to $BMO(w)$ if $b^{\mathrm{\#}}$ belongs to $L^{\mathrm{\infty }}(w)$, and we define ${\parallel b\parallel }_{BMO(w)}={\parallel b^{\mathrm{\#}}\parallel }_{L^{\mathrm{\infty }}(w)}$. If $w=1$, we denote $BMO(w)=BMO(R^n)$. 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 $CMO(R^n)$, which is the space of those functions $f\in L_{\mathrm{loc}}(R^n)$ such that

$${\parallel f\parallel }_{CMO}=\underset{r>1}{sup}{|Q(0,r)|}^{-1}{\int }_Q|f(x)-f_Q|dx<\mathrm{\infty }.$$

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

$${\parallel f\parallel }_{CMO}\approx \underset{r>1}{sup}\underset{c\in C}{inf}{|Q(0,r)|}^{-1}{\int }_Q|f(x)-c|dx.$$

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

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

The $A_p$ weight is defined by (see [6])

$$\begin{array}{r}{A}_p=\{0<w\in L_{\mathrm{loc}}^1\left(R^n\right):\underset{Q}{sup}(\frac{1}{|Q|}{\int }_{Q}w(x)dx){\left(\frac{1}{|Q|}{\int }_{Q}w{(x)}^{-1/(p-1)}dx\right)}^{p-1}<\mathrm{\infty }\},\\ 1<p<\mathrm{\infty },\end{array}$$

and

$$A_1=\{0<w\in L_{\mathrm{loc}}^1\left(R^n\right):\underset{Q\ni x}{sup}\frac{1}{|Q|}{\int }_{Q}w(y)dy\le Cw(x),\text{a.e.}\}.$$

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 $L^2(R^n)$ and has a kernel K, that is, there exists a locally integrable function $K(x,y)$ on $R^n\times R^n\setminus \{(x,y)\in R^n\times R^n:x=y\}$ such that

$$T(f)(x)={\int }_{R^n}K(x,y)f(y)dy$$

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

$$\begin{array}{c}|K(x,y)|\le C{|x-y|}^{-n},\hfill \\ {\int }_{2|y-z|<|x-y|}(|K(x,y)-K(x,z)|+|K(y,x)-K(z,x)|)dx\le C,\hfill \end{array}$$

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

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

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 $1\le j\le l$,

$$R_{m_j+1}(b_j;x,y)=b_j(x)-\sum _{|\alpha |\le m_j}\frac{1}{\alpha !}{D}^{\alpha }{b}_j(y){(x-y)}^{\alpha }.$$

The multilinear operator associated to T is defined by

$$T_b(f)(x)={\int }_{R^n}\frac{{\prod }_{j=1}^l{R}_{m_j+1}(b_j;x,y)}{{|x-y|}^m}K(x,y)f(y)dy.$$

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 [1–3]). 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 [1–3]). In this paper, we will study the weighted boundedness properties of the multilinear operators $T_b$ for the extreme cases (see [12–15]).

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 $\{k^m{C}_k\}\in l^1$, $w\in A_1$ and $D^{\alpha }{b}_j\in BMO(R^n)$ for all α with $|\alpha |=m_j$ and $j=1,\dots ,l$. Then $T_b$ is bounded from $L^{\mathrm{\infty }}(w)$ to $BMO(w)$.

Theorem 2 Let T be the singular integral operator as Definition  2, and we have the sequence $\{k^m{C}_k\}\in l^1$, $1<p<\mathrm{\infty }$, $w\in A_1$, and $D^{\alpha }{b}_j\in BMO(R^n)$ for all α with $|\alpha |=m_j$ and $j=1,\dots ,l$. Then $T_b$ is bounded from $B_p(w)$ to $CMO(w)$.

3 Proofs of theorems

We begin with two preliminaries lemmas.

Lemma 1 (see [3])

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

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

where $\tilde{Q}(x,y)$ 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 $\{C_k\}\in l^1$. Then T is bounded on $L^p(R^n,w)$ for $w\in A_p$ with $1<p<\mathrm{\infty }$.

Proof of Theorem 1 It is only for us to prove that there exists a constant $C_Q$ such that

$$\frac{1}{w(Q)}{\int }_Q|T_b(f)(x)-C_Q|w(x)dx\le C{\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}$$

holds for any cube Q. Without loss of generality, we may assume $l=2$. Fix a cube $Q=Q(x_0,d)$. Let $\tilde{Q}=5\sqrt{n}Q$ and ${\tilde{b}}_j(x)=b_j(x)-{\sum }_{|\alpha |=m}\frac{1}{\alpha !}{(D^{\alpha }{b}_j)}_{\tilde{Q}}{x}^{\alpha }$, then $R_m(b_j;x,y)=R_m({\tilde{b}}_j;x,y)$ and $D^{\alpha }{\tilde{b}}_j=D^{\alpha }{b}_j-{(D^{\alpha }{b}_j)}_{\tilde{Q}}$ for $|\alpha |=m_j$. We write, for $f_1=f{\chi }_{\tilde{Q}}$ and $f_2=f{\chi }_{R^n\setminus \tilde{Q}}$,

$$\begin{array}{rcl}{T}_b(f)(x)& =& {\int }_{R^n}\frac{{\prod }_{j=1}^2{R}_{m_j+1}({\tilde{b}}_j;x,y)}{{|x-y|}^m}K(x,y)f(y)dy\\ =& {\int }_{R^n}\frac{{\prod }_{j=1}^2{R}_{m_j}({\tilde{b}}_j;x,y)}{{|x-y|}^m}K(x,y)f_1(y)dy\\ -\sum _{|{\alpha }_1|=m_1}\frac{1}{{\alpha }_1!}{\int }_{R^n}\frac{R_{m_2}({\tilde{b}}_2;x,y){(x-y)}^{{\alpha }_1}{D}^{{\alpha }_1}{\tilde{b}}_1(y)}{{|x-y|}^m}K(x,y)f_1(y)dy\\ -\sum _{|{\alpha }_2|=m_2}\frac{1}{{\alpha }_2!}{\int }_{R^n}\frac{R_{m_1}({\tilde{b}}_1;x,y){(x-y)}^{{\alpha }_2}{D}^{{\alpha }_2}{\tilde{b}}_2(y)}{{|x-y|}^m}K(x,y)f_1(y)dy\\ +\sum _{|{\alpha }_1|=m_1,|{\alpha }_2|=m_2}\frac{1}{{\alpha }_1!{\alpha }_2!}{\int }_{R^n}\frac{{(x-y)}^{{\alpha }_1+{\alpha }_2}{D}^{{\alpha }_1}{\tilde{b}}_1(y)D^{{\alpha }_2}{\tilde{b}}_2(y)}{{|x-y|}^m}K(x,y)f_1(y)dy\\ +{\int }_{R^n}\frac{{\prod }_{j=1}^2{R}_{m_j+1}({\tilde{b}}_j;x,y)}{{|x-y|}^m}K(x,y)f_2(y)dy\\ =& T\left(\frac{{\prod }_{j=1}^2{R}_{m_j}({\tilde{b}}_j;x,\cdot )}{{|x-\cdot |}^m}{f}_1\right)\\ -T\left(\sum _{|{\alpha }_1|=m_1}\frac{1}{{\alpha }_1!}\frac{R_{m_2}({\tilde{b}}_2;x,\cdot ){(x-\cdot )}^{{\alpha }_1}{D}^{{\alpha }_1}{\tilde{b}}_1}{{|x-\cdot |}^m}{f}_1\right)\\ -T\left(\sum _{|{\alpha }_2|=m_2}\frac{1}{{\alpha }_2!}\frac{R_{m_1}({\tilde{b}}_1;x,\cdot ){(x-\cdot )}^{{\alpha }_2}{D}^{{\alpha }_2}{\tilde{b}}_2}{{|x-\cdot |}^m}{f}_1\right)\\ +T\left(\sum _{|{\alpha }_1|=m_1,|{\alpha }_2|=m_2}\frac{1}{{\alpha }_1!{\alpha }_2!}\frac{{(x-\cdot )}^{{\alpha }_1+{\alpha }_2}{D}^{{\alpha }_1}{\tilde{b}}_1{D}^{{\alpha }_2}{\tilde{b}}_2}{{|x-\cdot |}^m}{f}_1\right)\\ +T\left(\frac{{\prod }_{j=1}^2{R}_{m_j+1}({\tilde{b}}_j;x,\cdot )}{{|x-\cdot |}^m}{f}_2\right),\end{array}$$

then

$$\begin{array}{rcl}|T_b(f)(x)-T_{\tilde{b}}(f_2)(x_0)|& \le & \left|T\left(\frac{{\prod }_{j=1}^2{R}_{m_j}({\tilde{b}}_j;x,\cdot )}{{|x-\cdot |}^m}{f}_1\right)\right|\\ +\left|T\left(\sum _{|{\alpha }_1|=m_1}\frac{1}{{\alpha }_1!}\frac{R_{m_2}({\tilde{b}}_2;x,\cdot ){(x-\cdot )}^{{\alpha }_1}{D}^{{\alpha }_1}{\tilde{b}}_1}{{|x-\cdot |}^m}{f}_1\right)\right|\\ +\left|T\left(\sum _{|{\alpha }_2|=m_2}\frac{1}{{\alpha }_2!}\frac{R_{m_1}({\tilde{b}}_1;x,\cdot ){(x-\cdot )}^{{\alpha }_2}{D}^{{\alpha }_2}{\tilde{b}}_2}{{|x-\cdot |}^m}{f}_1\right)\right|\\ +\left|T\left(\sum _{|{\alpha }_1|=m_1,|{\alpha }_2|=m_2}\frac{1}{{\alpha }_1!{\alpha }_2!}\frac{{(x-\cdot )}^{{\alpha }_1+{\alpha }_2}{D}^{{\alpha }_1}{\tilde{b}}_1{D}^{{\alpha }_2}{\tilde{b}}_2}{{|x-\cdot |}^m}{f}_1\right)\right|\\ +|T_{\tilde{b}}(f_2)(x)-T_{\tilde{b}}(f_2)(x_0)|\\ =& I_1(x)+I_2(x)+I_3(x)+I_4(x)+I_5(x)\end{array}$$

and

$$\begin{array}{r}\frac{1}{w(Q)}{\int }_Q|T_b(f)(x)-T_{\tilde{b}}(f_2)(x_0)|w(x)dx\\ \le \frac{1}{w(Q)}{\int }_Q{I}_1(x)w(x)dx+\frac{1}{w(Q)}{\int }_Q{I}_2(x)w(x)dx+\frac{1}{w(Q)}{\int }_Q{I}_3(x)w(x)dx\\ +\frac{1}{w(Q)}{\int }_Q{I}_4(x)w(x)dx+\frac{1}{w(Q)}{\int }_Q{I}_5(x)w(x)dx\\ =I_1+I_2+I_3+I_4+I_5.\end{array}$$

Now, let us estimate $I_1$, $I_2$, $I_3$, $I_4$, and $I_5$, respectively. First, for $x\in Q$ and $y\in \tilde{Q}$, by Lemma 1, we get

$$R_m({\tilde{b}}_j;x,y)\le C{|x-y|}^m\sum _{|{\alpha }_j|=m}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO},$$

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

$$\begin{array}{rcl}{I}_1& \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right)\frac{1}{w(Q)}{\int }_Q|T(f_1)(x)|w(x)dx\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right){(\frac{1}{w(Q)}{\int }_{R^n}{|T(f_1)(x)|}^{p}w(x)dx)}^{1/p}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right){(\frac{1}{w(Q)}{\int }_{R^n}{|f_1(x)|}^{p}w(x)dx)}^{1/p}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right){\left(\frac{w(\tilde{Q})}{w(Q)}\right)}^{1/p}{\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right){\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}.\end{array}$$

For $I_2$, since $w\in A_1$, w satisfies the reverse of Hölder’s inequality:

$${\left(\frac{1}{|Q|}{\int }_{Q}w{(x)}^{p_0}dx\right)}^{1/p_0}\le \frac{C}{|Q|}{\int }_{Q}w(x)dx$$

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

$$\begin{array}{rcl}{I}_2& \le & C\sum _{|{\alpha }_2|=m_2}{\parallel D^{{\alpha }_2}{b}_2\parallel }_{BMO}\sum _{|{\alpha }_1|=m_1}\frac{1}{w(Q)}{\int }_Q|T\left(D^{{\alpha }_1}{\tilde{b}}_1{f}_1\right)(x)|w(x)dx\\ \le & C\sum _{|{\alpha }_2|=m_2}{\parallel D^{{\alpha }_2}{b}_2\parallel }_{BMO}\sum _{|{\alpha }_1|=m_1}{(\frac{1}{w(Q)}{\int }_{R^n}{|T\left(D^{{\alpha }_1}{\tilde{b}}_1{f}_1\right)(x)|}^{p}w(x)dx)}^{1/p}\\ \le & C\sum _{|{\alpha }_2|=m_2}{\parallel D^{{\alpha }_2}{b}_2\parallel }_{BMO}\sum _{|{\alpha }_1|=m_1}{(\frac{1}{w(Q)}{\int }_{R^n}{|D^{{\alpha }_1}{\tilde{b}}_1(x)f_1(x)|}^{p}w(x)dx)}^{1/p}\\ \le & C\sum _{|{\alpha }_2|=m_2}{\parallel D^{{\alpha }_2}{b}_2\parallel }_{BMO}\sum _{|{\alpha }_1|=m_1}{\left(\frac{1}{|Q|}{\int }_{\tilde{Q}}{|D^{{\alpha }_1}{b}_1(x)-{\left(D^{{\alpha }_1}{b}_1\right)}_{\tilde{Q}}|}^{p{p}_0^{\prime }}dx\right)}^{1/p{p}_0^{\prime }}\\ \times w{(Q)}^{-1/p}{|Q|}^{1/p}{\left(\frac{1}{|\tilde{Q}|}{\int }_{\tilde{Q}}w{(x)}^{p_0}dx\right)}^{1/p{p}_0}{\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right)w{(Q)}^{-1/p}{|Q|}^{1/p}{(\frac{1}{|\tilde{Q}|}{\int }_{\tilde{Q}}w(x)dx)}^{1/p}{\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right){\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}.\end{array}$$

For $I_3$, similar to the proof of $I_2$, we get

$$I_3\le C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right){\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}.$$

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,

$$\begin{array}{rcl}{I}_4& \le & C\sum _{|{\alpha }_1|=m_1,|{\alpha }_2|=m_2}\frac{1}{w(Q)}{\int }_Q|T\left(D^{{\alpha }_1}{\tilde{b}}_1{D}^{{\alpha }_2}{\tilde{b}}_2{f}_1\right)(x)|w(x)dx\\ \le & C\sum _{|{\alpha }_1|=m_1,|{\alpha }_2|=m_2}{(\frac{1}{w(Q)}{\int }_{R^n}{|T\left(D^{{\alpha }_1}{\tilde{b}}_1{D}^{{\alpha }_2}{\tilde{b}}_2{f}_1\right)(x)|}^{p}w(x)dx)}^{1/p}\\ \le & C\sum _{|{\alpha }_1|=m_1,|{\alpha }_2|=m_2}w{(Q)}^{-1/p}{({\int }_{R^n}{|D^{{\alpha }_1}{\tilde{b}}_1(x)D^{{\alpha }_2}{\tilde{b}}_2(x)f_1(x)|}^{p}w(x)dx)}^{1/p}\\ \le & C\sum _{|{\alpha }_1|=m_1,|{\alpha }_2|=m_2}{\left(\frac{1}{|\tilde{Q}|}{\int }_{\tilde{Q}}{|D^{{\alpha }_1}{\tilde{b}}_1(x)|}^{p{r}_1}dx\right)}^{1/p{r}_1}{\left(\frac{1}{|\tilde{Q}|}{\int }_{\tilde{Q}}{|D^{{\alpha }_2}{\tilde{b}}_2(x)|}^{p{r}_2}dx\right)}^{1/p{r}_2}\\ \times w{(Q)}^{-1/p}{|Q|}^{1/p}{\left(\frac{1}{|\tilde{Q}|}{\int }_{\tilde{Q}}w{(x)}^{p_0}dx\right)}^{1/p{p}_0}{\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right){\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}.\end{array}$$

For $I_5$, we write

$$\begin{array}{r}{T}_{\tilde{b}}(f_2)(x)-T_{\tilde{b}}(f_2)(x_0)\\ ={\int }_{R^n}(\frac{K(x,y)}{{|x-y|}^m}-\frac{K(x_0,y)}{{|x_0-y|}^m})\prod _{j=1}^2{R}_{m_j}({\tilde{b}}_j;x,y)f_2(y)dy\\ +{\int }_{R^n}(R_{m_1}({\tilde{b}}_1;x,y)-R_{m_1}({\tilde{b}}_1;x_0,y))\frac{R_{m_2}({\tilde{b}}_2;x,y)}{{|x_0-y|}^m}K(x_0,y)f_2(y)dy\\ +{\int }_{R^n}(R_{m_2}({\tilde{b}}_2;x,y)-R_{m_2}({\tilde{b}}_2;x_0,y))\frac{R_{m_1}({\tilde{b}}_1;x_0,y)}{{|x_0-y|}^m}K(x_0,y)f_2(y)dy\\ -\sum _{|{\alpha }_1|=m_1}\frac{1}{{\alpha }_1!}{\int }_{R^n}[\frac{R_{m_2}({\tilde{b}}_2;x,y){(x-y)}^{{\alpha }_1}}{{|x-y|}^m}K(x,y)-\frac{R_{m_2}({\tilde{b}}_2;x_0,y){(x_0-y)}^{{\alpha }_1}}{{|x_0-y|}^m}K(x_0,y)]\\ \times D^{{\alpha }_1}{\tilde{b}}_1(y)f_2(y)dy\\ -\sum _{|{\alpha }_2|=m_2}\frac{1}{{\alpha }_2!}{\int }_{R^n}[\frac{R_{m_1}({\tilde{b}}_1;x,y){(x-y)}^{{\alpha }_2}}{{|x-y|}^m}K(x,y)-\frac{R_{m_1}({\tilde{b}}_1;x_0,y){(x_0-y)}^{{\alpha }_2}}{{|x_0-y|}^m}K(x_0,y)]\\ \times D^{{\alpha }_2}{\tilde{b}}_2(y)f_2(y)dy\\ +\sum _{|{\alpha }_1|=m_1,|{\alpha }_2|=m_2}\frac{1}{{\alpha }_1!{\alpha }_2!}{\int }_{R^n}[\frac{{(x-y)}^{{\alpha }_1+{\alpha }_2}}{{|x-y|}^m}K(x,y)-\frac{{(x_0-y)}^{{\alpha }_1+{\alpha }_2}}{{|x_0-y|}^m}K(x_0,y)]\\ \times D^{{\alpha }_1}{\tilde{b}}_1(y)D^{{\alpha }_2}{\tilde{b}}_2(y)f_2(y)dy\\ =I_5^{(1)}(x)+I_5^{(2)}(x)+I_5^{(3)}(x)+I_5^{(4)}(x)+I_5^{(5)}(x)+I_5^{(6)}(x).\end{array}$$

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

$$|b_{Q_1}-b_{Q_2}|\le Clog(|Q_2|/|Q_1|){\parallel b\parallel }_{BMO}\text{for }{Q}_1\subset Q_2,$$

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

$$\begin{array}{rcl}|R_{m_j}({\tilde{b}}_j;x,y)|& \le & C{|x-y|}^{m_j}\sum _{|\alpha |=m_j}({\parallel D^{\alpha }{b}_j\parallel }_{BMO}+|{\left(D^{\alpha }{b}_j\right)}_{\tilde{Q}(x,y)}-{\left(D^{\alpha }{b}_j\right)}_{\tilde{Q}}|)\\ \le & Ck{|x-y|}^{m_j}\sum _{|\alpha |=m_j}{\parallel D^{\alpha }{b}_j\parallel }_{BMO}.\end{array}$$

Note that $|x-y|\sim |x_0-y|$ for $x\in Q$ and $y\in R^n\setminus \tilde{Q}$, we obtain, by the conditions on K,

$$\begin{array}{rcl}|I_5^{(1)}(x)|& \le & {\int }_{R^n}|\frac{1}{{|x-y|}^m}-\frac{1}{{|x_0-y|}^m}||K(x,y)|\prod _{j=1}^2|R_{m_j}({\tilde{b}}_j;x,y)||f_2(y)|dy\\ +{\int }_{R^n}|K(x,y)-K(x_0,y)|{|x_0-y|}^{-m}\prod _{j=1}^2|R_{m_j}({\tilde{b}}_j;x,y)||f_2(y)|dy\\ \le & \sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}|\frac{1}{{|x-y|}^m}-\frac{1}{{|x_0-y|}^m}||K(x,y)|\prod _{j=1}^2|R_{m_j}({\tilde{b}}_j;x,y)||f(y)|dy\\ +\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}|K(x,y)-K(x_0,y)|{|x_0-y|}^{-m}\prod _{j=1}^2|R_{m_j}({\tilde{b}}_j;x,y)||f(y)|dy\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right)\sum _{k=0}^{\mathrm{\infty }}{k}^2{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}\frac{|x-x_0|}{{|x_0-y|}^{n+1}}|f(y)|dy\\ +C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right)\sum _{k=0}^{\mathrm{\infty }}{k}^2{\left({\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}{|f(y)|}^{q^{\prime }}dy\right)}^{1/q^{\prime }}\\ \times {\left({\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}{|K(x,y)-K(x_0,y)|}^{q}dy\right)}^{1/q}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right)\sum _{k=1}^{\mathrm{\infty }}{k}^2(2^{-k}+C_k){\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right){\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}.\end{array}$$

For $I_5^{(2)}(x)$, by the formula (see [3]):

$$R_{m_j}({\tilde{b}}_j;x,y)-R_{m_j}({\tilde{b}}_j;x_0,y)=\sum _{|\beta |<m}\frac{1}{\beta !}{R}_{m-|\beta |}(D^{\beta }{\tilde{b}}_j;x,x_0){(x-y)}^{\beta }$$

and Lemma 1, we have

$$|R_{m_j}({\tilde{b}}_j;x,y)-R_{m_j}({\tilde{b}}_j;x_0,y)|\le C\sum _{|\beta |<m_j}\sum _{|\alpha |=m_j}{|x-x_0|}^{m_j-|\beta |}{|x-y|}^{|\beta |}{\parallel D^{\alpha }{b}_j\parallel }_{BMO},$$

thus

$$\begin{array}{rcl}|I_5^{(2)}(x)|& \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right)\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}k\frac{|x-x_0|}{{|x_0-y|}^{n+1}}|f(y)|dy\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right)\sum _{k=1}^{\mathrm{\infty }}k{2}^{-k}{\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right){\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}.\end{array}$$

Similarly,

$$|I_5^{(3)}(x)|\le C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right){\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}.$$

For $I_5^{(4)}(x)$, similar to the proof of $I_5^{(1)}(x)$ and $I_5^{(2)}(x)$, we get

$$\begin{array}{rcl}|I_5^{(4)}(x)|& \le & C\sum _{|{\alpha }_1|=m_1}{\int }_{R^n}|R_{m_2}({\tilde{b}}_2;x,y)-R_{m_2}({\tilde{b}}_2;x_0,y)|\frac{|{(x_0-y)}^{{\alpha }_1}K(x_0,y)|}{{|x_0-y|}^m}\\ \times |D^{{\alpha }_1}{\tilde{b}}_1(y)||f_2(y)|dy\\ +C\sum _{|{\alpha }_1|=m_1}{\int }_{R^n}|\frac{{(x-y)}^{{\alpha }_1}}{{|x-y|}^m}-\frac{{(x_0-y)}^{{\alpha }_1}}{{|x_0-y|}^m}||K(x,y)||R_{m_2}({\tilde{b}}_2;x,y)|\\ \times |D^{{\alpha }_1}{\tilde{b}}_1(y)||f_2(y)|dy\\ +C\sum _{|{\alpha }_1|=m_1}{\int }_{R^n}|K(x,y)-K(x_0,y)|\left|\frac{{(x_0-y)}^{{\alpha }_1}}{{|x_0-y|}^m}\right||R_{m_2}({\tilde{b}}_2;x,y)|\\ \times |D^{{\alpha }_1}{\tilde{b}}_1(y)||f_2(y)|dy\\ \le & C\sum _{|{\alpha }_2|=m_2}{\parallel D^{{\alpha }_2}{b}_2\parallel }_{BMO}\sum _{k=1}^{\mathrm{\infty }}k{2}^{-k}\sum _{|{\alpha }_1|=m_1}\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}|D^{{\alpha }_1}{\tilde{b}}_1(y)|dy\right){\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}\\ +C\sum _{|{\alpha }_2|=m_2}{\parallel D^{{\alpha }_2}{b}_2\parallel }_{BMO}\sum _{|{\alpha }_1|=m_1}\sum _{k=0}^{\mathrm{\infty }}k{\left({\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}{|K(x,y)-K(x_0,y)|}^{q}dy\right)}^{1/q}\\ \times {\left({\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}{|D^{{\alpha }_1}{\tilde{b}}_1(y)|}^{q^{\prime }}dy\right)}^{1/q^{\prime }}{\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right)\sum _{k=1}^{\mathrm{\infty }}{k}^2(2^{-k}+C_k){\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right){\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}.\end{array}$$

Similarly,

$$|I_5^{(5)}(x)|\le C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right){\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}.$$

For $I_5^{(6)}(x)$, taking $1<r_1,r_2<\mathrm{\infty }$ such that $1/r_1+1/r_2=1$, then

$$\begin{array}{rcl}|I_5^{(6)}(x)|& \le & C\sum _{|{\alpha }_1|=m_1,|{\alpha }_2|=m_2}{\int }_{R^n}|\frac{{(x-y)}^{{\alpha }_1+{\alpha }_2}K(x,y)}{{|x-y|}^m}-\frac{{(x_0-y)}^{{\alpha }_1+{\alpha }_2}K(x_0,y)}{{|x_0-y|}^m}|\\ \times |D^{{\alpha }_1}{\tilde{b}}_1(y)||D^{{\alpha }_2}{\tilde{b}}_2(y)||f_2(y)|dy\\ \le & C\sum _{|{\alpha }_1|=m_1,|{\alpha }_2|=m_2}\sum _{k=1}^{\mathrm{\infty }}(2^{-k}+C_k){\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}\\ \times {\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}{|D^{{\alpha }_1}{\tilde{b}}_1(y)|}^{r_1}dy\right)}^{1/r_1}{\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}{|D^{{\alpha }_2}{\tilde{b}}_2(y)|}^{r_2}dy\right)}^{1/r_2}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right)\sum _{k=1}^{\mathrm{\infty }}{k}^2(2^{-k}+C_k){\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right){\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}.\end{array}$$

Thus

$$I_5\le C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right){\parallel f\parallel }_{L^{\mathrm{\infty }}(w)}.$$

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

$$\frac{1}{w(Q)}{\int }_Q|T_b(f)(x)-C_Q|w(x)dx\le C{\parallel f\parallel }_{B_p(w)}$$

holds for any cube $Q=Q(0,d)$ with $d>1$. Without loss of generality, we may assume $l=2$. Fix a cube $Q=Q(0,d)$ with $d>1$. Let $\tilde{Q}=5\sqrt{n}Q$ and ${\tilde{b}}_j(x)=b_j(x)-{\sum }_{|\alpha |=m_j}\frac{1}{\alpha !}{(D^{\alpha }{b}_j)}_{\tilde{Q}}{x}^{\alpha }$, then $R_{m_j}(b_j;x,y)=R_{m_j}({\tilde{b}}_j;x,y)$ and $D^{\alpha }{\tilde{b}}_j=D^{\alpha }{b}_j-{(D^{\alpha }{b}_j)}_{\tilde{Q}}$ for $|\alpha |=m_j$. Similar to the proof of Theorem 1, we write, for $f_1=f{\chi }_{\tilde{Q}}$ and $f_2=f{\chi }_{R^n\setminus \tilde{Q}}$,

$$\begin{array}{r}\frac{1}{w(Q)}{\int }_Q|T_b(f)(x)-T_{\tilde{b}}(f_2)(0)|w(x)dx\\ \le \frac{1}{w(Q)}{\int }_Q\left|T\left(\frac{{\prod }_{j=1}^2{R}_{m_j}({\tilde{b}}_j;x,\cdot )}{{|x-\cdot |}^m}{f}_1\right)\right|w(x)dx\\ +\frac{1}{w(Q)}{\int }_Q\left|T\left(\sum _{|{\alpha }_1|=m_1}\frac{1}{{\alpha }_1!}\frac{R_{m_2}({\tilde{b}}_2;x,\cdot ){(x-\cdot )}^{{\alpha }_1}{D}^{{\alpha }_1}{\tilde{b}}_1}{{|x-\cdot |}^m}{f}_1\right)\right|w(x)dx\\ +\frac{1}{w(Q)}{\int }_Q\left|T\left(\sum _{|{\alpha }_2|=m_2}\frac{1}{{\alpha }_2!}\frac{R_{m_1}({\tilde{b}}_1;x,\cdot ){(x-\cdot )}^{{\alpha }_2}{D}^{{\alpha }_2}{\tilde{b}}_2}{{|x-\cdot |}^m}{f}_1\right)\right|w(x)dx\\ +\frac{1}{w(Q)}{\int }_Q\left|T\left(\sum _{|{\alpha }_1|=m_1,|{\alpha }_2|=m_2}\frac{1}{{\alpha }_1!{\alpha }_2!}\frac{{(x-\cdot )}^{{\alpha }_1+{\alpha }_2}{D}^{{\alpha }_1}{\tilde{b}}_1{D}^{{\alpha }_2}{\tilde{b}}_2}{{|x-\cdot |}^m}{f}_1\right)\right|w(x)dx\\ +\frac{1}{w(Q)}{\int }_Q|T_{\tilde{b}}(f_2)(x)-T_{\tilde{b}}(f_2)(0)|w(x)dx\\ =L_1+L_2+L_3+L_4+L_5.\end{array}$$

Similar to the proof of Theorem 1, we get

$$\begin{array}{rcl}{L}_1& \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right){(\frac{1}{w(Q)}{\int }_{R^n}{|T(f_1)(x)|}^{p}w(x)dx)}^{1/p}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right)w{(\tilde{Q})}^{-1/p}{\parallel f{\chi }_{\tilde{Q}}\parallel }_{L^p(w)}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right){\parallel f\parallel }_{B_p(w)}.\end{array}$$

For $L_2$, taking $r,s,t>1$ such that $r<p$, $t=p{p}_0/(p-r)$, and $1/s+1/(p/r)+1/t=1$, then, by the reverse of Hölder’s inequality,

$$\begin{array}{rcl}{L}_2& \le & C\sum _{|{\alpha }_2|=m_2}{\parallel D^{{\alpha }_2}{b}_2\parallel }_{BMO}\sum _{|{\alpha }_1|=m_1}{(\frac{1}{w(Q)}{\int }_{R^n}{|T\left(D^{{\alpha }_1}{\tilde{b}}_1{f}_1\right)(x)|}^{r}w(x)dx)}^{1/r}\\ \le & C\sum _{|{\alpha }_2|=m_2}{\parallel D^{{\alpha }_2}{b}_2\parallel }_{BMO}w{(Q)}^{-1/r}\sum _{|{\alpha }_1|=m_1}{({\int }_{R^n}{|D^{{\alpha }_1}{\tilde{b}}_1(x)f_1(x)|}^{r}w(x)dx)}^{1/r}\\ \le & C\sum _{|{\alpha }_2|=m_2}{\parallel D^{{\alpha }_2}{b}_2\parallel }_{BMO}w{(Q)}^{-1/r}\sum _{|{\alpha }_1|=m_1}{\left({\int }_{\tilde{Q}}{|D^{\alpha }{\tilde{b}}_1(x)|}^{rs}dx\right)}^{1/rs}\\ \times {({\int }_{\tilde{Q}}{|f(x)|}^{p}w(x)dx)}^{1/p}{\left({\int }_{\tilde{Q}}w{(x)}^{(1-r/p)t}dx\right)}^{1/rt}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right)w{(Q)}^{-1/r}{|Q|}^{1/rs}{\parallel f{\chi }_{\tilde{Q}}\parallel }_{L^p(w)}{|Q|}^{1/rt}\\ \times {\left(\frac{1}{|\tilde{Q}|}{\int }_{\tilde{Q}}w{(x)}^{p_0}dx\right)}^{1/rt}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right)w{(Q)}^{-1/r}{|Q|}^{1/rs}{\parallel f{\chi }_{\tilde{Q}}\parallel }_{L^p(w)}{(\frac{1}{|\tilde{Q}|}{\int }_{\tilde{Q}}w(x)dx)}^{p_0/rt}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right)w{(\tilde{Q})}^{-1/p}{\parallel f{\chi }_{\tilde{Q}}\parallel }_{L^p(w)}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right){\parallel f\parallel }_{B_p(w)},\\ L_3& \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right){\parallel f\parallel }_{B_p(w)}.\end{array}$$

For $L_4$, taking $r,s_1,s_2,t>1$ such that $r<p$, $t=p{p}_0/(p-r)$, and $1/s_1+1/s_2+1/(p/r)+1/t=1$, then, by the reverse of Hölder’s inequality,

$$\begin{array}{rcl}{L}_4& \le & C\sum _{|{\alpha }_1|=m_1,|{\alpha }_2|=m_2}{(\frac{1}{w(Q)}{\int }_{R^n}{|T\left(D^{{\alpha }_1}{\tilde{b}}_1{D}^{{\alpha }_2}{\tilde{b}}_2{f}_1\right)(x)|}^{r}w(x)dx)}^{1/r}\\ \le & Cw{(Q)}^{-1/r}\sum _{|{\alpha }_1|=m_1,|{\alpha }_2|=m_2}{({\int }_{R^n}{|D^{{\alpha }_1}{\tilde{b}}_1(x)D^{{\alpha }_2}{\tilde{b}}_2(x)f_1(x)|}^{r}w(x)dx)}^{1/r}\\ \le & Cw{(Q)}^{-1/r}\sum _{|{\alpha }_1|=m_1,|{\alpha }_2|=m_2}{\left({\int }_{\tilde{Q}}{|D^{\alpha }{\tilde{b}}_1(x)|}^{r{s}_1}dx\right)}^{1/r{s}_1}{\left({\int }_{\tilde{Q}}{|D^{\alpha }{\tilde{b}}_2(x)|}^{r{s}_2}dx\right)}^{1/r{s}_2}\\ \times {({\int }_{\tilde{Q}}{|f(x)|}^{p}w(x)dx)}^{1/p}{\left({\int }_{\tilde{Q}}w{(x)}^{(1-r/p)t}dx\right)}^{1/rt}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right)w{(Q)}^{-1/r}{|Q|}^{1/r{s}_1+1/r{s}_2+1/rt}{\parallel f{\chi }_{\tilde{Q}}\parallel }_{L^p(w)}\\ \times {(\frac{1}{|\tilde{Q}|}{\int }_{\tilde{Q}}w(x)dx)}^{p_0/rt}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right)w{(\tilde{Q})}^{-1/p}{\parallel f{\chi }_{\tilde{Q}}\parallel }_{L^p(w)}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right){\parallel f\parallel }_{B_p(w)}.\end{array}$$

For $L_5$, similar to the proof of the proof of $I_5$ in Theorem 1, we have

$$\begin{array}{r}{T}_{\tilde{b}}(f_2)(x)-T_{\tilde{b}}(f_2)(0)\\ ={\int }_{R^n}(\frac{K(x,y)}{{|x-y|}^m}-\frac{K(0,y)}{{|y|}^m})\prod _{j=1}^2{R}_{m_j}({\tilde{b}}_j;x,y)f_2(y)dy\\ +{\int }_{R^n}(R_{m_1}({\tilde{b}}_1;x,y)-R_{m_1}({\tilde{b}}_1;0,y))\frac{R_{m_2}({\tilde{b}}_2;x,y)}{{|y|}^m}K(0,y)f_2(y)dy\\ +{\int }_{R^n}(R_{m_2}({\tilde{b}}_2;x,y)-R_{m_2}({\tilde{b}}_2;0,y))\frac{R_{m_1}({\tilde{b}}_1;x_0,y)}{{|y|}^m}K(0,y)f_2(y)dy\\ -\sum _{|{\alpha }_1|=m_1}\frac{1}{{\alpha }_1!}{\int }_{R^n}[\frac{R_{m_2}({\tilde{b}}_2;x,y){(x-y)}^{{\alpha }_1}}{{|x-y|}^m}K(x,y)-\frac{R_{m_2}({\tilde{b}}_2;0,y){(-y)}^{{\alpha }_1}}{{|y|}^m}K(0,y)]\\ \times D^{{\alpha }_1}{\tilde{b}}_1(y)f_2(y)dy\\ -\sum _{|{\alpha }_2|=m_2}\frac{1}{{\alpha }_2!}{\int }_{R^n}[\frac{R_{m_1}({\tilde{b}}_1;x,y){(x-y)}^{{\alpha }_2}}{{|x-y|}^m}K(x,y)-\frac{R_{m_1}({\tilde{b}}_1;0,y){(-y)}^{{\alpha }_2}}{{|y|}^m}K(0,y)]\\ \times D^{{\alpha }_2}{\tilde{b}}_2(y)f_2(y)dy\\ +\sum _{|{\alpha }_1|=m_1,|{\alpha }_2|=m_2}\frac{1}{{\alpha }_1!{\alpha }_2!}{\int }_{R^n}[\frac{{(x-y)}^{{\alpha }_1+{\alpha }_2}}{{|x-y|}^m}K(x,y)-\frac{{(-y)}^{{\alpha }_1+{\alpha }_2}}{{|y|}^m}K(0,y)]\\ \times D^{{\alpha }_1}{\tilde{b}}_1(y)D^{{\alpha }_2}{\tilde{b}}_2(y)f_2(y)dy\\ =L_5^{(1)}(x)+L_5^{(2)}(x)+L_5^{(3)}(x)+L_5^{(4)}(x)+L_5^{(5)}(x)+L_5^{(6)}(x).\end{array}$$

For $L_5^{(1)}(x)$, taking $1<r<\mathrm{\infty }$ such that $1/p+1/q+1/r=1$, by $w\in A_1\subset A_{p/r+1}$, we get

$$\begin{array}{rcl}|L_5^{(1)}(x)|& \le & C\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}|\frac{1}{{|x-y|}^m}-\frac{1}{{|y|}^m}||K(x,y)|\prod _{j=1}^2|R_{m_j}({\tilde{b}}_j;x,y)||f(y)|dy\\ +\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}|K(x,y)-K(0,y)|{|y|}^{-m}\\ \times \prod _{j=1}^2|R_{m_j}({\tilde{b}}_j;x,y)||f(y)|w{(y)}^{1/p}w{(y)}^{-1/p}dy\\ \le & C\prod _{j=1}^2\left(\sum _{|\alpha |=m_j}{\parallel D^{\alpha }{b}_j\parallel }_{BMO}\right)\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}{k}^2\frac{d}{{(2^{k}d)}^{n+1}}|f(y)|dy\\ +C\prod _{j=1}^2\left(\sum _{|\alpha |=m_j}{\parallel D^{\alpha }{b}_j\parallel }_{BMO}\right)\sum _{k=0}^{\mathrm{\infty }}{k}^2{\left({\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}{|K(x,y)-K(0,y)|}^{q}dy\right)}^{1/q}\\ \times {({\int }_{2^{k+1}\tilde{Q}}{|f(y)|}^{p}w(y)dy)}^{1/p}{\left({\int }_{2^{k+1}\tilde{Q}}w{(y)}^{-r/p}dy\right)}^{1/r}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right)\sum _{k=1}^{\mathrm{\infty }}{k}^2{2}^{-k}w{\left(2^k\tilde{Q}\right)}^{-1/p}{({\int }_{2^k\tilde{Q}}{|f(y)|}^{p}w(y)dy)}^{1/p}\\ \times {(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}w(y)dy)}^{1/p}{\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}w{(y)}^{-1/(p-1)}dy\right)}^{(p-1)/p}\\ +C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right)\sum _{k=1}^{\mathrm{\infty }}{k}^2{C}_{k}w{\left(2^k\tilde{Q}\right)}^{-1/p}{({\int }_{2^k\tilde{Q}}{|f(y)|}^{p}w(y)dy)}^{1/p}\\ \times {(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}w(y)dy)}^{1/p}{\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}w{(y)}^{-r/p}dy\right)}^{1/r}\\ \le & C\prod _{j=1}^2\left(\sum _{|{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right){\parallel f\parallel }_{B_p(w)}.\end{array}$$

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^{(2)}(x)+L_5^{(3)}(x)+L_5^{(4)}(x)+L_5^{(5)}(x)+L_5^{(6)}(x)|\\ \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}\tilde{Q}\setminus 2^k\tilde{Q}}k\frac{d}{{(2^{k}d)}^{n+1}}|D^{{\alpha }_1}{\tilde{b}}_1(y)||f(y)|dy\\ +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}\tilde{Q}\setminus 2^k\tilde{Q}}k\frac{d}{{(2^{k}d)}^{n+1}}|D^{{\alpha }_2}{\tilde{b}}_2(y)||f(y)|dy\\ +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}\tilde{Q}\setminus 2^k\tilde{Q}}|K(x,y)-K(0,y)|\\ \times |D^{{\alpha }_1}{\tilde{b}}_1(y)||f(y)|dy\\ +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}\tilde{Q}\setminus 2^k\tilde{Q}}|K(x,y)-K(0,y)|\\ \times |D^{{\alpha }_2}{\tilde{b}}_2(y)||f(y)|dy\\ +C\sum _{|{\alpha }_1|=m_1,|{\alpha }_2|=m_2}\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}|K(x,y)-K(0,y)||D^{{\alpha }_1}{\tilde{b}}_1(y)||D^{{\alpha }_2}{\tilde{b}}_2(y)||f(y)|dy\\ \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}\tilde{Q}}{|D^{{\alpha }_1}{\tilde{b}}_1(y)|}^{r_1}dy\right)}^{1/r_1}\\ \times {({\int }_{2^{k+1}\tilde{Q}}{|f(y)|}^{p}w(y)dy)}^{1/p}{\left({\int }_{2^{k+1}\tilde{Q}}w{(y)}^{-s/p}dy\right)}^{1/s}\\ +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}\tilde{Q}}{|D^{{\alpha }_2}{\tilde{b}}_2(y)|}^{r_1}dy\right)}^{1/r_1}\\ \times {({\int }_{2^{k+1}\tilde{Q}}{|f(y)|}^{p}w(y)dy)}^{1/p}{\left({\int }_{2^{k+1}\tilde{Q}}w{(y)}^{-s/p}dy\right)}^{1/s}\\ +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}\tilde{Q}\setminus 2^k\tilde{Q}}{|K(x,y)-K(0,y)|}^{q}dy\right)}^{1/q}\\ \times {\left({\int }_{2^{k+1}\tilde{Q}}{|D^{{\alpha }_1}{\tilde{b}}_1(y)|}^{r_2}dy\right)}^{1/r_2}{({\int }_{2^{k+1}\tilde{Q}}{|f(y)|}^{p}w(y)dy)}^{1/p}{\left({\int }_{2^{k+1}\tilde{Q}}w{(y)}^{-s/p}dy\right)}^{1/s}\\ +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}\tilde{Q}\setminus 2^k\tilde{Q}}{|K(x,y)-K(0,y)|}^{q}dy\right)}^{1/q}\\ \times {\left({\int }_{2^{k+1}\tilde{Q}}{|D^{{\alpha }_2}{\tilde{b}}_2(y)|}^{r_2}dy\right)}^{1/r_2}{({\int }_{2^{k+1}\tilde{Q}}{|f(y)|}^{p}w(y)dy)}^{1/p}{\left({\int }_{2^{k+1}\tilde{Q}}w{(y)}^{-s/p}dy\right)}^{1/s}\\ +C\sum _{|{\alpha }_1|=m_1,|{\alpha }_2|=m_2}\sum _{k=0}^{\mathrm{\infty }}{\left({\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}{|K(x,y)-K(0,y)|}^{q}dy\right)}^{1/q}{\left({\int }_{2^{k+1}\tilde{Q}}w{(y)}^{-s/p}dy\right)}^{1/s}\\ \times {\left({\int }_{2^{k+1}\tilde{Q}}{|D^{{\alpha }_1}{\tilde{b}}_1(y)|}^{r_3}dy\right)}^{1/r_3}{\left({\int }_{2^{k+1}\tilde{Q}}{|D^{{\alpha }_2}{\tilde{b}}_2(y)|}^{r_4}dy\right)}^{1/r_4}\\ \times {({\int }_{2^{k+1}\tilde{Q}}{|f(y)|}^{p}w(y)dy)}^{1/p}\\ \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}+C_k)w{\left(2^k\tilde{Q}\right)}^{-1/p}{({\int }_{2^k\tilde{Q}}{|f(y)|}^{p}w(y)dy)}^{1/p}\\ \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(w)}.\end{array}$$

Thus

$$L_5\le C\prod _{j=1}^2\left(\sum _{{\alpha }_j|=m_j}{\parallel D^{{\alpha }_j}{b}_j\parallel }_{BMO}\right){\parallel f\parallel }_{B_p(w)}.$$

This finishes the proof of Theorem 2. □