Weighted endpoint estimates for multilinear commutator of singular integral operators with non-smooth kernels

Abstract

In this paper, we prove the weighted endpoint estimates for multilinear commutator of singular integral operators with non-smooth kernels.

1 Introduction

Let $b\in \mathit{BMO}(R^n)$ and T be the Calderón-Zygmund operator, the commutator $[b,T]$ generated by b and T is defined by $[b,T](f)(x)=b(x)T(f)(x)-T(bf)(x)$. A classical result of Coifman et al. (see [1]) proved that the commutator $[b,T]$ is bounded on $L^p(R^n)$ ($1<p<\mathrm{\infty }$). In [2, 3], the boundedness properties of the commutators for the extreme values of p are obtained. In this paper, we will introduce the multilinear commutator of singular integral operators with non-smooth kernels and prove the weighted boundedness properties of the operator for the extreme cases.

First let us introduce some notations (see [3–12]). In this paper, Q will denote a cube of $R^n$ with sides parallel to the axes. For a cube Q and a function b, let $b_Q={|Q|}^{-1}{\int }_{Q}b(x)dx$ and $b(Q)={\int }_{Q}b(x)dx$, the sharp function of b is defined by

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

It is well known that (see [6])

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

Moreover, for a weight function ω (that is, a non-negative locally integrable function), b is said to belong to $\mathit{BMO}(\omega )$ if $b^{\mathrm{\#}}\in L^{\mathrm{\infty }}(\omega )$ and define ${\parallel b\parallel }_{\mathit{BMO}(\omega )}={\parallel b^{\mathrm{\#}}\parallel }_{L^{\mathrm{\infty }}(\omega )}$, if $\omega =1$, we denote $\mathit{BMO}(\omega )=\mathit{BMO}(R^n)$. It is well known that (see [11])

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

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

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

and

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

Definition 1 A family of operators $D_t$, $t>0$, is said to be an ‘approximation to the identity’ if, for every $t>0$, $D_t$ can be represented by the kernel $a_t(x,y)$ in the following sense:

$$D_t(f)(x)={\int }_{R^n}{a}_t(x,y)f(y)dy$$

for every $f\in L^p(R^n)$ with $p\ge 1$, and $a_t(x,y)$ satisfies

$$|a_t(x,y)|\le h_t(x,y)=C{t}^{-n/2}s({|x-y|}^2/t),$$

where s is a positive, bounded, and decreasing function satisfying

$$\underset{r\to \mathrm{\infty }}{lim}{r}^{n+ϵ}s\left(r^2\right)=0$$

for some $ϵ>0$.

Definition 2 A linear operator T is called the singular integral operators with non-smooth kernels if T is bounded on $L^2(R^n)$ and associated with a kernel $K(x,y)$ such that

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

for every continuous function f with compact support, and for almost all x not in the support of f.

• (1) There exists an ‘approximation to the identity’ $\{B_t,t>0\}$ such that $T{B}_t$ has associated kernel $k_t(x,y)$ and there exist $c_1,c_2>0$ so that

  $${\int }_{|x-y|>c_1{t}^{1/2}}|K(x,y)-k_t(x,y)|dx\le c_2\text{for all }y\in R^n.$$

• (2) There exists an ‘approximation to the identity’ $\{A_t,t>0\}$ such that $A_{t}T$ has associated kernel $K_t(x,y)$ which satisfies

  $$|K_t(x,y)|\le c_4{t}^{-n/2}\text{if }|x-y|\le c_3{t}^{1/2}$$

  and

  $$|K(x,y)-K_t(x,y)|\le c_4{t}^{\delta /2}{|x-y|}^{-n-\delta }\text{if }|x-y|\ge c_3{t}^{1/2}$$

  for some $c_3,c_4>0$, $\delta >0$.

Given some locally integrable functions $b_j$ ($j=1,\dots ,m$). The multilinear operator associated to T is defined by

$$T_b(f)(x)={\int }_{R^n}\left[\prod _{j=1}^m(b_j(x)-b_j(y))\right]K(x,y)f(y)dy.$$

Definition 3 Given the ‘approximations to the identity’ $\{A_t,t>0\}$ and a weight function ω.

• (1) The weighted BMO space associated with $\{A_t,t>0\}$ is defined by

  $${\mathit{BMO}}_A(\omega )=\{f\in L_{\mathrm{loc}}^1\left(R^n\right):{\parallel f\parallel }_{{\mathit{BMO}}_A(\omega )}<\mathrm{\infty }\},$$

  where

  $${\parallel f\parallel }_{{\mathit{BMO}}_A(\omega )}=\underset{Q}{sup}\frac{1}{\omega (Q)}{\int }_Q|f(x)-A_{t_Q}(f)(x)|\omega (x)dx,$$

  $t_Q=l{(Q)}^2$ and $l(Q)$ denotes the side length of Q.

• (2) The weighted central BMO space associated with $\{A_t,t>0\}$ is defined by

  $${\mathit{CMO}}_A(\omega )=\{f\in L_{\mathrm{loc}}^1\left(R^n\right):{\parallel f\parallel }_{{\mathit{CMO}}_A(\omega )}<\mathrm{\infty }\},$$

  where

  $${\parallel f\parallel }_{\mathit{CMO}(\omega )}=\underset{r>1}{sup}\frac{1}{\omega (Q(0,r))}{\int }_Q|f(x)-A_{t_Q}f(x)|\omega (x)dx,$$

  and $t_Q=r^2$.

Definition 4 Let $1<p<\mathrm{\infty }$ and ω be a weighted function on $R^n$. We shall call $B_p(\omega )$ the space of those functions f on $R^n$, such that

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

For $b_j\in \mathit{BMO}(R^n)$ ($j=1,\dots ,m$), set ${\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}={\prod }_{j=1}^m{\parallel b_j\parallel }_{\mathit{BMO}}$. Given a positive integer m and $1\le j\le m$, we denote by $C_j^m$ the family of all finite subsets $\sigma =\{\sigma (1),\dots ,\sigma (j)\}$ of $\{1,\dots ,m\}$ of j different elements. For $\sigma \in C_j^m$, set ${\sigma }^c=\{1,\dots ,m\}\mathrm{\setminus }\sigma$. For $\overrightarrow{b}=(b_1,\dots ,b_m)$ and $\sigma =\{\sigma (1),\dots ,\sigma (j)\}\in C_j^m$, set ${\overrightarrow{b}}_{\sigma }=(b_{\sigma (1)},\dots ,b_{\sigma (j)})$, $b_{\sigma }=b_{\sigma (1)}\cdots b_{\sigma (j)}$ and ${\parallel {\overrightarrow{b}}_{\sigma }\parallel }_{\mathit{BMO}}={\parallel b_{\sigma (1)}\parallel }_{\mathit{BMO}}\cdots {\parallel b_{\sigma (j)}\parallel }_{\mathit{BMO}}$.

2 Theorems and proofs

We begin with some preliminaries lemmas.

Lemma 1 ([5, 7])

Let $\omega \in A_1$, $1<p\le \mathrm{\infty }$, and T be the singular integral operators with non-smooth kernels. Then T is boundedness on $L^p(w)$.

Lemma 2 Let $\omega \in A_1$, $\{A_t,t>0\}$ be an ‘approximation to the identity’ and $b\in \mathit{BMO}(R^n)$. Then

• (a) for every $f\in L^{\mathrm{\infty }}(R^n)$, $1\le p<\mathrm{\infty }$, and any cube Q,

  $${\left(\frac{1}{|Q|}{\int }_Q\right|A_{t_Q}((b-b_Q)f)(y){|}^{p}dy)}^{1/p}\le C{\parallel b\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}};$$

• (b) for every $f\in B_p(\omega )$, $1\le r<p<\mathrm{\infty }$, and any cube Q,

  $${\left(\frac{1}{\omega (Q)}{\int }_Q\right|A_{t_Q}((b-b_Q)f)(y){|}^r\omega (y)dy)}^{1/r}\le C{\parallel b\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{B_p(\omega )},$$

where $t_Q=l{(Q)}^2$ and $l(Q)$ denotes the side length of Q.

Proof (a) Write

$$\begin{array}{c}{\left(\frac{1}{|Q|}{\int }_Q\right|A_{t_Q}((b-b_Q)f)(y){|}^{p}dy)}^{1/p}\hfill \\ \le {\left(\frac{1}{|Q|}{\int }_Q{\int }_{R^n}{h}_{t_Q}{(x,y)}^p\right|(b(y)-b_Q)f(y){|}^{p}dydx)}^{1/p}\hfill \\ \le {\left(\frac{1}{|Q|}{\int }_Q{\int }_{2Q}{h}_{t_Q}{(x,y)}^p\right|(b(y)-b_Q)f(y){|}^{p}dydx)}^{1/p}\hfill \\ +{\left(\sum _{k=1}^{\mathrm{\infty }}\frac{1}{|Q|}{\int }_Q{\int }_{2^{k+1}Q\mathrm{\setminus }{2}^{k}Q}{h}_{t_Q}{(x,y)}^p\right|(b(y)-b_Q)f(y){|}^{p}dydx)}^{1/p}\hfill \\ =I_1+I_2.\hfill \end{array}$$

We have, by Hölder’s inequality,

$$\begin{array}{rcl}{I}_1& \le & {\left(\frac{C}{|Q||2Q|}{\int }_Q{\int }_{2Q}\right|(b(y)-b_Q)f(y){|}^{p}dydx)}^{1/p}\\ \le & C{\parallel f\parallel }_{L^{\mathrm{\infty }}}{\left(\frac{1}{|2Q|}{\int }_{2Q}\right|b(y)-b_Q{|}^{p}dy)}^{1/p}\\ \le & C{\parallel b\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}}.\end{array}$$

For $I_2$, for $x\in Q$ and $y\in 2^{k+1}Q\mathrm{\setminus }{2}^{k}Q$, we have $|x-y|\ge 2^{k-1}{t}_Q$ and $h_{t_Q}(x,y)\le C\frac{s(2^{2(k-1)})}{|Q|}$. Thus

$$\begin{array}{rcl}{I}_2& \le & C\sum _{k=1}^{\mathrm{\infty }}{2}^{(k-1)n}s\left(2^{2(k-1)}\right){\left(\frac{1}{|2^{k+1}Q|}{\int }_{2^{k+1}Q}\right|(b(y)-b_Q)f(y){|}^{p}dy)}^{1/p}\\ \le & C{\parallel f\parallel }_{L^{\mathrm{\infty }}}\sum _{k=1}^{\mathrm{\infty }}{2}^{(k-1)n}s\left(2^{2(k-1)}\right){\left(\frac{1}{|2^{k+1}Q|}{\int }_{2^{k+1}Q}\right|b(y)-b_{2^{k+1}Q}{|}^{p}dy)}^{1/p}\\ +C{\parallel f\parallel }_{L^{\mathrm{\infty }}}\sum _{k=1}^{\mathrm{\infty }}{2}^{(k-1)n}s\left(2^{2(k-1)}\right)|b_Q-b_{2^{k+1}Q}|\\ \le & C{\parallel f\parallel }_{L^{\mathrm{\infty }}}\sum _{k=1}^{\mathrm{\infty }}{2}^{(k-1)n}s\left(2^{2(k-1)}\right)(k+1){\parallel b\parallel }_{\mathit{BMO}}\\ \le & C{\parallel b\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}},\end{array}$$

where the last inequality follows from

$$\sum _{k=2}^{\mathrm{\infty }}{2}^{(k-1)n}s\left(2^{2(k-1)}\right)(k+1)\le C\sum _{k=2}^{\mathrm{\infty }}k{2}^{-(k-1)ϵ}<\mathrm{\infty }$$

for some $ϵ>0$.

• (b) Write

  $$\begin{array}{c}{\left(\frac{1}{\omega (Q)}{\int }_Q\right|A_{t_Q}((b-b_Q)f)(y){|}^r\omega (y)dy)}^{1/r}\hfill \\ \le {\left(\frac{1}{\omega (Q)}{\int }_Q{\int }_{R^n}{h}_{t_Q}{(x,y)}^r\right|(b(y)-b_Q)f(y){|}^r\omega (y)dydx)}^{1/r}\hfill \\ \le {\left(\frac{1}{\omega (Q)}{\int }_Q{\int }_{2Q}{h}_{t_Q}{(x,y)}^p\right|(b(y)-b_Q)f(y){|}^r\omega (y)dydx)}^{1/r}\hfill \\ +{\left(\sum _{k=1}^{\mathrm{\infty }}\frac{1}{\omega (Q)}{\int }_Q{\int }_{2^{k+1}Q\mathrm{\setminus }{2}^{k}Q}{h}_{t_Q}{(x,y)}^r\right|(b(y)-b_Q)f(y){|}^r\omega (y)dydx)}^{1/r}\hfill \\ =I+\mathit{II}.\hfill \end{array}$$

For I, since $\omega \in A_1$, ω satisfies the reverse of Hölder’s inequality

$${\left(\frac{1}{|Q|}{\int }_Q\omega {(x)}^{q}dx\right)}^{1/q}\le \frac{C}{|Q|}{\int }_Q\omega (x)dx$$

for some $1<q<\mathrm{\infty }$, and $\frac{\omega (Q_2)}{|Q_2|}\frac{|Q_1|}{\omega (Q_1)}\le C$ for all cubes $Q_1$, $Q_2$ with $Q_1\subset Q_2$, $\omega \in A_{p/ru}$ for $1<u,v<\mathrm{\infty }$ with $u^{\prime }v=q$ and $p>ru$ (see [6]). We have, by Hölder’s inequality,

$$\begin{array}{c}I\le {\left(\frac{C}{\omega (Q)|Q|}{\int }_Q{\int }_{2Q}\right|(b(y)-b_Q)f(y){|}^r\omega (y)dydx)}^{1/r}\hfill \\ \phantom{I}\le C{\left(\frac{1}{\omega (Q)}{\int }_{2Q}\right|(b(y)-b_Q)f(y){|}^r\omega (y)dy)}^{1/r}\hfill \\ \phantom{I}\le C{\left[\frac{|2Q|}{\omega (Q)}{\left(\frac{1}{|2Q|}{\int }_{2Q}\right|b(y)-b_Q{|}^{r{u}^{\prime }}\omega {(y)}^{u^{\prime }}dy)}^{1/u^{\prime }}{\left(\frac{1}{|2Q|}{\int }_{2Q}\right|f(y){|}^{ru}dy)}^{1/u}\right]}^{1/r}\hfill \\ \phantom{I}\le C{\left(\frac{|2Q|}{\omega (Q)}\right)}^{1/r}{\left(\frac{1}{|2Q|}{\int }_{2Q}\right|b(y)-b_Q{|}^{r{u}^{\prime }{v}^{\prime }}dy)}^{1/r{u}^{\prime }{v}^{\prime }}{\left(\frac{1}{|2Q|}{\int }_{2Q}\omega {(y)}^{u^{\prime }v}dy\right)}^{1/r{u}^{\prime }v}\hfill \\ \phantom{I\le }\times {\left(\frac{1}{|2Q|}{\int }_{2Q}\right|f(y){|}^{ru}dy)}^{1/ru}\hfill \\ \phantom{I}\le C{\parallel b\parallel }_{\mathit{BMO}}{\left(\frac{|2Q|}{\omega (2Q)}\right)}^{1/r}{\left(\frac{\omega (2Q)}{|2Q|}\right)}^{1/r}{\left(\frac{1}{|2Q|}{\int }_{2Q}\right|f(y){|}^{ru}\omega {(y)}^{\frac{ru}{p}}\omega {(y)}^{-\frac{ru}{p}}dy)}^{1/ru}\hfill \\ \phantom{I}\le C{\parallel b\parallel }_{\mathit{BMO}}{\left(\frac{1}{|2Q|}{\int }_{2Q}{\left(\right|f(y){|}^{ru}\omega {(y)}^{\frac{ru}{p}})}^{\frac{p}{ru}}dy\right)}^{1/p}{\left(\frac{1}{|2Q|}{\int }_{2Q}\omega {(y)}^{-\frac{ru}{p}\frac{p}{p-ru}}dy\right)}^{(p-ru)/pru}\hfill \\ \phantom{I}\le C{\parallel b\parallel }_{\mathit{BMO}}{\left(\frac{1}{|2Q|}\right)}^{1/p}{\parallel f{\chi }_{2Q}\parallel }_{L^p(\omega )}{(\frac{1}{|2Q|}{\int }_{2Q}\omega (y)dy)}^{-1/p}\hfill \\ \phantom{I\le }\times {\left[(\frac{1}{|2Q|}{\int }_{2Q}\omega (y)dy){\left(\frac{1}{|2Q|}{\int }_{2Q}\omega {(y)}^{-\frac{1}{\frac{p}{ru}-1}}dy\right)}^{\frac{p}{ru}-1}\right]}^{1/p}\hfill \\ \phantom{I}\le C{\parallel b\parallel }_{\mathit{BMO}}\omega {(2Q)}^{-1/p}{\parallel f{\chi }_{2Q}\parallel }_{L^p(\omega )}\hfill \\ \phantom{I}\le C{\parallel b\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{B_p(\omega )};\hfill \\ \mathit{II}\le C{\left(\frac{|Q|}{\omega (Q)}\right)}^{1/r}\sum _{k=1}^{\mathrm{\infty }}{2}^{(k-1)n}s\left(2^{2(k-1)}\right){\left(\frac{1}{|2^{k+1}Q|}{\int }_{2^{k+1}Q}\right|(b(y)-b_Q)f(y){|}^r\omega (y)dy)}^{1/r}\hfill \\ \phantom{\mathit{II}}\le C{\left(\frac{|Q|}{\omega (Q)}\right)}^{1/r}\sum _{k=1}^{\mathrm{\infty }}{2}^{(k-1)n}s\left(2^{2(k-1)}\right){\left(\frac{1}{|2^{k+1}Q|}{\int }_{2^{k+1}Q}\right|b(y)-b_{2^{k+1}Q}{|}^{r{u}^{\prime }}\omega {(y)}^{u^{\prime }}dy)}^{1/r{u}^{\prime }}\hfill \\ \phantom{\mathit{II}\le }\times {\left(\frac{1}{|2^{k+1}Q|}{\int }_{2^{k+1}Q}\right|f(y){|}^{ru}dy)}^{1/ru}\hfill \\ \phantom{\mathit{II}}\le C{\left(\frac{|Q|}{\omega (Q)}\right)}^{1/r}\sum _{k=1}^{\mathrm{\infty }}{2}^{(k-1)n}s\left(2^{2(k-1)}\right){\left(\frac{1}{|2^{k+1}Q|}{\int }_{2^{k+1}Q}\right|b(y)-b_{2^{k+1}Q}{|}^{r{u}^{\prime }{v}^{\prime }}dy)}^{1/r{u}^{\prime }{v}^{\prime }}\hfill \\ \phantom{\mathit{II}\le }\times {\left(\frac{1}{|2^{k+1}Q|}{\int }_{2^{k+1}Q}\omega {(y)}^{u^{\prime }v}dy\right)}^{1/r{u}^{\prime }v}{\left(\frac{1}{|2^{k+1}Q|}{\int }_{2^{k+1}Q}f{(y)}^{ru}dy\right)}^{1/ru}\hfill \\ \phantom{\mathit{II}}\le C{\parallel b\parallel }_{\mathit{BMO}}\sum _{k=1}^{\mathrm{\infty }}{2}^{(k-1)n}s\left(2^{2(k-1)}\right)(k+1){(\frac{|Q|}{\omega (Q)}\cdot \frac{\omega (2^{k+1}Q)}{|2^{k+1}Q|})}^{1/r}\hfill \\ \phantom{\mathit{II}\le }\times {\left(\frac{1}{|2^{k+1}Q|}{\int }_{2^{k+1}Q}f{(y)}^{ru}dy\right)}^{1/ru}\hfill \\ \phantom{\mathit{II}}\le C\sum _{k=1}^{\mathrm{\infty }}{2}^{(k-1)n}s\left(2^{2(k-1)}\right)(k+1){\parallel b\parallel }_{\mathit{BMO}}\omega {\left(2^{k+1}Q\right)}^{-1/p}{\parallel f{\chi }_{2^{k+1}Q}\parallel }_{L^p(\omega )}\hfill \\ \phantom{\mathit{II}}\le C{\parallel b\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{B_p(w)}.\hfill \end{array}$$

This completes the proof. □

Theorem 1 Let T be the singular integral operators with non-smooth kernels, $\omega \in A_1$ and $\overrightarrow{b}=(b_1,\dots ,b_m)$ with $b_j\in \mathit{BMO}(R^n)$ for $1\le j\le m$. Then $T_b$ is bounded from $L^{\mathrm{\infty }}(\omega )$ to ${\mathit{BMO}}_A(\omega )$.

Proof It suffices to prove, for $f\in C_0^{\mathrm{\infty }}(R^n)$, the following inequality holds:

$$\frac{1}{\omega (Q)}{\int }_Q|T_b(f)(x)-A_{t_Q}{T}_b(f)(x)|\omega (x)dx\le C{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}.$$

We fix a cube $Q=Q(x_0,d)$. We decompose f into $f=f_1+f_2$ with $f_1=f{\chi }_Q$, $f_2=f{\chi }_{(R^n\mathrm{\setminus }Q)}$.

When $m=1$, set ${(b_1)}_Q={|Q|}^{-1}{\int }_Q{b}_1(y)dy$, we have

$$\begin{array}{rcl}{T}_{b_1}(f)(x)& =& {\int }_{R^n}[(b_1(x)-{(b_1)}_Q)-(b_1(y)-{(b_1)}_Q)]K(x,y)f(y)dy\\ =& (b_1(x)-{(b_1)}_Q){\int }_{R^n}K(x,y)f(y)dy-{\int }_{R^n}(b_1(y)-{(b_1)}_Q)K(x,y)f(y)dy\end{array}$$

and

$$A_{t_Q}{T}_{b_1}(f)(x)=(b_1(x)-{(b_1)}_Q){\int }_{R^n}{K}_t(x,y)f(y)dy-{\int }_{R^n}(b_1(y)-{(b_1)}_Q)K_t(x,y)f(y)dy.$$

Then

$$\begin{array}{c}|T_{b_1}(f)(x)-A_{t_Q}{T}_{b_1}(f)(x)|\hfill \\ \le |(b_1(x)-{(b_1)}_Q){\int }_{R^n}K(x,y)f(y)dy|\hfill \\ +|{\int }_{R^n}(b_1(y)-{(b_1)}_Q)K(x,y)f_1(y)dy|\hfill \\ +|(b_1(x)-{(b_1)}_Q){\int }_{R^n}{K}_t(x,y)f(y)dy|\hfill \\ +|{\int }_{R^n}(b_1(y)-{(b_1)}_Q)K_t(x,y)f_1(y)dy|\hfill \\ +|{\int }_{R^n}(b_1(y)-{(b_1)}_Q)(K(x,y)-K_t(x,y))f_2(y)dy|\hfill \\ =I_1(x)+I_2(x)+I_3(x)+I_4(x)+I_5(x).\hfill \end{array}$$

For $I_1(x)$, let $1/p+1/p^{\prime }=1$, $1/q+1/q^{\prime }=1$, by the reverse of Hölder’s inequality with $1<q<\mathrm{\infty }$, Lemma 1, and Hölder’s inequality, we have

$$\begin{array}{c}\frac{1}{\omega (Q)}{\int }_Q|I_1(x)|\omega (x)dx\hfill \\ \le \frac{C}{\omega (Q)}{\left({\int }_Q\right|b_1(x)-{(b_1)}_Q{|}^{p^{\prime }}\omega (x)dx)}^{1/p^{\prime }}{\left({\int }_{R^n}\right|T(f)(x){|}^p\omega (x){\chi }_Q(x)dx)}^{1/p}\hfill \\ \le \frac{C}{\omega (Q)}{\left({\int }_Q\right|b_1(x)-{(b_1)}_Q{|}^{p^{\prime }}\omega (x)dx)}^{1/p^{\prime }}{\left({\int }_{R^n}\right|f(x){|}^p\omega (x){\chi }_Q(x)dx)}^{1/p}\hfill \\ \le \frac{C}{\omega (Q)}{\left({\int }_Q\right|b_1(x)-{(b_1)}_Q{|}^{p^{\prime }}\omega (x)dx)}^{1/p^{\prime }}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}{({\int }_Q\omega (x)dx)}^{1/p}\hfill \\ \le \frac{C}{\omega (Q)}{\left[{\left({\int }_Q\right|b_1(x)-{(b_1)}_Q{|}^{p^{\prime }{q}^{\prime }}dx)}^{1/q^{\prime }}{\left({\int }_Q\omega {(x)}^{q}dx\right)}^{1/q}\right]}^{1/p^{\prime }}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}\omega {(Q)}^{1/p}\hfill \\ \le C\omega {(Q)}^{1/p-1}{|Q|}^{1/p^{\prime }}{\parallel b_1\parallel }_{\mathit{BMO}}{\left(\frac{1}{|Q|}{\int }_Q\omega {(x)}^{q}dx\right)}^{1/p^{\prime }q}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}\hfill \\ \le C{\parallel b_1\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}.\hfill \end{array}$$

For $I_2(x)$, taking $p>1$, by Hölder’s inequality, we have

$$\begin{array}{c}\frac{1}{\omega (Q)}{\int }_Q|I_2(x)|\omega (x)dx\hfill \\ \le {\left(\frac{1}{\omega (Q)}{\int }_{R^n}\right|T\left((b_1-{(b_1)}_Q)f_1\right)(x){|}^p\omega (x){\chi }_Q(x)dx)}^{1/p}\hfill \\ \le C\omega {(Q)}^{-1/p}{\left({\int }_{R^n}\right|(b_1(x)-{(b_1)}_Q)f_1(x){|}^p\omega (x){\chi }_Q(x)dx)}^{1/p}\hfill \\ \le C\omega {(Q)}^{-1/p}{\left[{\left({\int }_Q\right|b_1(x)-{(b_1)}_Q{|}^{p{q}^{\prime }}dx)}^{1/q^{\prime }}{\left({\int }_Q\right|f(x){|}^{pq}\omega {(x)}^{q}dx)}^{1/q}\right]}^{1/p}\hfill \\ \le C\omega {(Q)}^{-1/p}{\left({\int }_Q\right|b_1(x)-{(b_1)}_Q{|}^{p{q}^{\prime }}dx)}^{1/p{q}^{\prime }}{\left({\int }_Q\right|f(x){|}^{pq}\omega {(x)}^{q}dx)}^{1/pq}\hfill \\ \le C\omega {(Q)}^{-1/p}{\left({\int }_Q\right|b_1(x)-{(b_1)}_Q{|}^{p{q}^{\prime }}dx)}^{1/p{q}^{\prime }}{\left({\int }_Q\omega {(x)}^{q}dx\right)}^{1/pq}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}\hfill \\ \le C\omega {(Q)}^{-1/p}{|Q|}^{1/p{q}^{\prime }}{\parallel b_1\parallel }_{\mathit{BMO}}{|Q|}^{1/pq}{\left(\frac{1}{|Q|}{\int }_Q\omega {(x)}^{q}dx\right)}^{1/pq}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}\hfill \\ \le C{\parallel b_1\parallel }_{\mathit{BMO}}{\left(\frac{|Q|}{\omega (Q)}\right)}^{1/p}{(\frac{1}{|Q|}{\int }_Q\omega (x)dx)}^{1/p}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}\hfill \\ \le C{\parallel b_1\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}.\hfill \end{array}$$

For $I_3(x)$ and $I_4(x)$, we get, for $1<p_1,p_2<\mathrm{\infty }$ with $1/p_1+1/p_2+1/q=1$,

$$\begin{array}{c}\frac{1}{\omega (Q)}{\int }_Q|I_3(x)|\omega (x)dx\hfill \\ \le \frac{C}{\omega (Q)}{\int }_Q|b_1(x)-{(b_1)}_Q||A_{t_Q}(f)(x)|\omega (x)dx\hfill \\ \le C\frac{|Q|}{\omega (Q)}{\left(\frac{1}{|Q|}{\int }_Q\right|b_1(x)-{(b_1)}_Q{|}^{p_1}dx)}^{1/p_1}\hfill \\ \times {\left(\frac{1}{|Q|}{\int }_Q\right|A_{t_Q}(f)(x){|}^{p_2}dx)}^{1/p_2}{\left(\frac{1}{|Q|}{\int }_Q\omega {(x)}^{q}dx\right)}^{1/q}\hfill \\ \le C\frac{|Q|}{\omega (Q)}{\parallel b_1\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}\frac{\omega (Q)}{|Q|}\hfill \\ \le C{\parallel b_1\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )},\hfill \\ \frac{1}{\omega (Q)}{\int }_Q|I_4(x)|\omega (x)dx\hfill \\ \le \frac{1}{\omega (Q)}{\int }_{R^n}|A_{t_Q}\left((b_1-{(b_1)}_Q)f_1\right)(x)|\omega (x)dx\hfill \\ \le C\frac{|Q|}{\omega (Q)}{\left(\frac{1}{|Q|}{\int }_Q\right|A_{t_Q}\left((b_1-{(b_1)}_Q)f_1\right)(x){|}^{q^{\prime }}dx)}^{1/q^{\prime }}{\left(\frac{1}{|Q|}{\int }_Q\omega {(x)}^{q}dx\right)}^{1/q}\hfill \\ \le C\frac{|Q|}{\omega (Q)}{\parallel b_1\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}\frac{\omega (Q)}{|Q|}\hfill \\ \le C{\parallel b_1\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}.\hfill \end{array}$$

For $I_5(x)$, we have

$$\begin{array}{rcl}{I}_5(x)& =& |{\int }_{R^n}(b_1(y)-{(b_1)}_Q)(K(x,y)-K_t(x,y))f_2(y)dy|\\ \le & C\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}Q\mathrm{\setminus }{2}^{k}Q}|b_1(y)-{(b_1)}_Q||f(y)|\frac{d^{\delta }}{{|x_0-y|}^{n+\delta }}dy\\ \le & C\sum _{k=1}^{\mathrm{\infty }}\frac{d^{\delta }}{{(2^{k-1}d)}^{n+\delta }}\left|2^{k}Q\right|{\left(\frac{1}{|2^{k}Q|}{\int }_{2^{k}Q}\right|f(y){|}^{p}dy)}^{1/p}\\ \times {\left(\frac{1}{|2^{k}Q|}{\int }_{2^{k}Q}\right|b_1(y)-{(b_1)}_Q{|}^{p^{\prime }}dy)}^{1/p^{\prime }}\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{k}^m{2}^{-k\delta }{\parallel b_1\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}\\ \le & C{\parallel b_1\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )},\end{array}$$

so

$$\frac{1}{\omega (Q)}{\int }_Q|I_5(x)|\omega (x)dx\le C{\parallel b_1\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}.$$

When $m>1$, set ${\overrightarrow{b}}_Q=({(b_1)}_Q,\dots ,{(b_m)}_Q)\in R^n$, where ${(b_j)}_Q={|Q|}^{-1}{\int }_Q{b}_j(y)dy$, $1\le j\le m$, we have

$$\begin{array}{rcl}{T}_b(f)(x)& =& \prod _{j=1}^m(b_j(x)-{(b_j)}_Q){\int }_{R^n}K(x,y)f(y)dy\\ +\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{(-1)}^{m-j}{(b(x)-{(b)}_Q)}_{\sigma }{\int }_{R^n}{(b(y)-{(b)}_Q)}_{{\sigma }^c}K(x,y)f(y)dy\\ +{(-1)}^m{\int }_{R^n}\prod _{j=1}^m(b_j(y)-{(b_j)}_Q)K(x,y)f(y)dy\end{array}$$

and

$$\begin{array}{rcl}{A}_{t_Q}{T}_b(f)(x)& =& \prod _{j=1}^m(b_j(x)-{(b_j)}_Q){\int }_{R^n}{K}_t(x,y)f(y)dy\\ +\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{(-1)}^{m-j}{(b(x)-{(b)}_Q)}_{\sigma }{\int }_{R^n}{(b(y)-{(b)}_Q)}_{{\sigma }^c}{K}_t(x,y)f(y)dy\\ +{(-1)}^m{\int }_{R^n}\prod _{j=1}^m(b_j(y)-{(b_j)}_Q)K_t(x,y)f(y)dy,\end{array}$$

then

$$\begin{array}{c}|T_b(f)(x)-A_{t_Q}{T}_b(f)(x)|\hfill \\ \le |\prod _{j=1}^m(b_j(x)-{(b_j)}_Q){\int }_{R^n}K(x,y)f(y)dy|\hfill \\ +|\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{(b(x)-{(b)}_Q)}_{\sigma }{\int }_{R^n}{(b(y)-{(b)}_Q)}_{{\sigma }^c}K(x,y)f(y)dy|\hfill \\ +|{\int }_{R^n}\prod _{j=1}^m(b_j(y)-{(b_j)}_Q)K(x,y)f_1(y)dy|\hfill \\ +|\prod _{j=1}^m(b_j(x)-{(b_j)}_Q){\int }_{R^n}{K}_t(x,y)f(y)dy|\hfill \\ +|\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{(b(x)-{(b)}_Q)}_{\sigma }{\int }_{R^n}{(b(y)-{(b)}_Q)}_{{\sigma }^c}{K}_t(x,y)f(y)dy|\hfill \\ +|{\int }_{R^n}\prod _{j=1}^m(b_j(y)-{(b_j)}_Q)K_t(x,y)f_1(y)dy|\hfill \\ +|{\int }_{R^n}\prod _{j=1}^m(b_j(y)-{(b_j)}_Q)(K(x,y)-K_t(x,y))f_2(y)dy|\hfill \\ =J_1(x)+J_2(x)+J_3(x)+J_4(x)+J_5(x)+J_6(x)+J_7(x).\hfill \end{array}$$

For $J_1(x)$, same as $m=1$, for some $1<q<\mathrm{\infty }$, let $1/q_1+1/q_2+\cdots +1/q_m+1/q=1$, $1/p+1/p^{\prime }=1$, by Hölder’s inequality, and the reverse of Hölder’s inequality, we get

$$\begin{array}{c}\frac{1}{\omega (Q)}{\int }_Q|J_1(x)|\omega (x)dx\hfill \\ \le \frac{C}{\omega (Q)}{\left({\int }_Q\right|(b_1(x)-{(b_1)}_Q)\cdots (b_m(x)-{(b_m)}_Q){|}^{p^{\prime }}\omega (x)dx)}^{1/p^{\prime }}\hfill \\ \times {\left({\int }_Q\right|T(f)(x){|}^p\omega (x)dx)}^{1/p}\hfill \\ \le \frac{C}{\omega (Q)}{\left({\int }_Q\right|b_1(x)-{(b_1)}_Q{|}^{p^{\prime }}\cdots |b_m(x)-{(b_m)}_Q{|}^{p^{\prime }}\omega (x)dx)}^{1/p^{\prime }}\hfill \\ \times {\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}{({\int }_Q\omega (x)dx)}^{1/p}\hfill \\ \le \frac{C}{\omega (Q)}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}\omega {(Q)}^{1/p}\prod _{j=1}^m{\left[{\left({\int }_Q\right|b_j(x)-{(b_j)}_Q{|}^{p^{\prime }{q}_j}dx)}^{1/q_j}{\left({\int }_Q\omega {(x)}^{q}dx\right)}^{1/q}\right]}^{1/p^{\prime }}\hfill \\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}\omega {(Q)}^{1/p^{\prime }+1/p-1}{|Q|}^{1/p^{\prime }(1/q_1+\cdots +1/q_m+1/q-1)}\hfill \\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}.\hfill \end{array}$$

For $J_2(x)$, by Hölder’s inequality and the reverse of Hölder’s inequality, we have

$$\begin{array}{c}\frac{1}{\omega (Q)}{\int }_Q|J_2(x)|\omega (x)dx\hfill \\ \le \sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}\frac{C}{\omega (Q)}{\left({\int }_Q\right|{(b(x)-b_Q)}_{\sigma }{|}^{p^{\prime }}\omega (x)dx)}^{1/p^{\prime }}\hfill \\ \times {\left({\int }_Q\right|T\left({(b-b_Q)}_{{\sigma }^c}f\right)(x){|}^p\omega (x)dx)}^{1/p}\hfill \\ \le C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{\left(\frac{1}{\omega (Q)}{\int }_Q\right|{(b(x)-b_Q)}_{\sigma }{|}^{p^{\prime }}\omega (x)dx)}^{1/p^{\prime }}\hfill \\ \times {\left(\frac{1}{\omega (Q)}{\int }_Q\right|T\left({(b-b_Q)}_{{\sigma }^c}f\right)(x){|}^p\omega (x)dx)}^{1/p}\hfill \\ \le C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}\omega {(Q)}^{-1/p^{\prime }}{\left[{\left({\int }_Q\right|{(b(x)-b_Q)}_{\sigma }{|}^{p^{\prime }{q}^{\prime }}dx)}^{1/q^{\prime }}{\left({\int }_Q\omega {(x)}^{q}dx\right)}^{1/q}\right]}^{1/p^{\prime }}\hfill \\ \times \omega {(Q)}^{-1/p}{\left({\int }_{R^n}\right|{(b(x)-b_Q)}_{{\sigma }^c}f(x){|}^p\omega (x){\chi }_Q(x)dx)}^{1/p}\hfill \\ \le C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}\omega {(Q)}^{-1/p^{\prime }}{|Q|}^{1/p^{\prime }{q}^{\prime }+1/p^{\prime }q-1/p^{\prime }}\omega {(Q)}^{1/p^{\prime }}{\parallel {\overrightarrow{b}}_{\sigma }\parallel }_{\mathit{BMO}}\hfill \\ \times \omega {(Q)}^{-1/p}{\left({\int }_Q\right|{(b(x)-b_Q)}_{{\sigma }^c}{|}^{p{q}^{\prime }}dx)}^{1/p{q}^{\prime }}{\left({\int }_Q\right|f(x){|}^{pq}{\omega }^q(x)dx)}^{1/pq}\hfill \\ \le C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{\parallel {\overrightarrow{b}}_{\sigma }\parallel }_{\mathit{BMO}}{\parallel {\overrightarrow{b}}_{{\sigma }^c}\parallel }_{\mathit{BMO}}{\left(\frac{|Q|}{\omega (Q)}\right)}^{1/p}\hfill \\ \times {(\frac{1}{|Q|}{\int }_Q\omega (x)dx)}^{1/p}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}\hfill \\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}.\hfill \end{array}$$

For $J_3(x)$, taking $p>1$, by the $L^p(\omega )$-boundedness of T, we have

$$\begin{array}{c}\frac{1}{\omega (Q)}{\int }_Q|J_3(x)|\omega (x)dx\hfill \\ \le {\left(\frac{1}{\omega (Q)}{\int }_{R^n}\right|T((b_1-{(b_1)}_Q)\cdots (b_m-{(b_m)}_Q)f_1)(x){|}^p\omega (x)dx)}^{1/p}\hfill \\ \le C\omega {(Q)}^{-1/p}{\left({\int }_{R^n}\right|(b_1(x)-{(b_1)}_Q)\cdots (b_m(x)-{(b_m)}_Q)f_1(x){|}^p\omega (x)dx)}^{1/p}\hfill \\ \le C\omega {(Q)}^{-1/p}{|Q|}^{1/p{q}^{\prime }}{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{|Q|}^{1/pq}{\left(\frac{1}{|Q|}{\int }_Q{\omega }^{q}dx\right)}^{1/pq}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}\hfill \\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\left(\frac{|Q|}{\omega (Q)}\right)}^{1/p}{\left(\frac{\omega (Q)}{|Q|}\right)}^{1/p}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}\hfill \\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}.\hfill \end{array}$$

For $J_4(x)$, $J_5(x)$, and $J_6(x)$, choose $1<p,q_j<\mathrm{\infty }$, $j=1,\dots ,m$, such that $1/p+1/q_1+\cdots +1/q_m+1/q$, by Lemma 2 and similar to the proofs of $J_1(x)$, $J_2(x)$, and $J_3(x)$, we get

$$\begin{array}{c}\frac{1}{\omega (Q)}{\int }_Q|J_4(x)|\omega (x)dx\hfill \\ \le C\frac{|Q|}{\omega (Q)}\prod _{j=1}^m{\left(\frac{1}{|Q|}{\int }_Q\right|(b_j(x)-{(b_j)}_Q){|}^{q_j}dx)}^{1/q_j}\hfill \\ \times {\left(\frac{1}{|Q|}{\int }_Q\right|A_{t_Q}(f)(x){|}^{p}dx)}^{1/p}{\left(\frac{1}{|Q|}{\int }_Q\omega {(x)}^{q}dx\right)}^{1/q}\hfill \\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )},\hfill \\ \frac{1}{\omega (Q)}{\int }_Q|J_5(x)|\omega (x)dx\hfill \\ \le C\frac{|Q|}{\omega (Q)}\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{\left(\frac{1}{|Q|}{\int }_Q{\left|{(b(x)-b_Q)}_{\sigma }\right|}^{q^{\prime }}dx\right)}^{1/q^{\prime }}\hfill \\ \times {\left(\frac{1}{|Q|}{\int }_Q\right|A_{t_Q}\left({(b-b_Q)}_{{\sigma }^c}f\right)(x){|}^{p}dx)}^{1/p}{\left(\frac{1}{|Q|}{\int }_Q\omega {(x)}^{q}dx\right)}^{1/q}\hfill \\ \le C\frac{|Q|}{\omega (Q)}\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{\parallel {\overrightarrow{b}}_{\sigma }\parallel }_{\mathit{BMO}}{\parallel {\overrightarrow{b}}_{{\sigma }^c}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}\frac{\omega (Q)}{|Q|}\hfill \\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )},\hfill \\ \frac{1}{\omega (Q)}{\int }_Q|J_6(x)|\omega (x)dx\hfill \\ \le C\frac{|Q|}{\omega (Q)}{\left(\frac{1}{|Q|}{\int }_Q\right|A_{t_Q}((b_1-{(b_1)}_Q)\cdots (b_m-{(b_m)}_Q)f_1)(x){|}^{q^{\prime }}dx)}^{1/q^{\prime }}\hfill \\ \times {\left(\frac{1}{|Q|}{\int }_Q\omega {(x)}^{q}dx\right)}^{1/q}\hfill \\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}.\hfill \end{array}$$

For $J_7(x)$, note that $|x-y|\ge d=t^{1/2}$, taking $1<q_j<\mathrm{\infty }$, $j=1,\dots ,m$ such that $1/q_1+\cdots +1/q_m+1/r=1$, then

$$\begin{array}{rcl}{J}_7(x)& \le & C\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}Q\mathrm{\setminus }{2}^{k}Q}\prod _{j=1}^m\left|(b_j(y)-{(b_j)}_Q)\right||f(y)|\frac{d^{\delta }}{{|x_0-y|}^{n+\delta }}dy\\ \le & C\sum _{k=1}^{\mathrm{\infty }}\frac{d^{\delta }}{{(2^{k-1}d)}^{n+\delta }}\left|2^{k}Q\right|{\left(\frac{1}{|2^{k}Q|}{\int }_{2^{k}Q}\right|f(y){|}^{r}dy)}^{1/r}\\ \times \prod _{j=1}^m{\left(\frac{1}{|2^{k}Q|}{\int }_{2^{k}Q}\right|b_j(y)-{(b_j)}_Q{|}^{q_j}dy)}^{1/q_j}\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{2}^{-k\delta }{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}\prod _{j=1}^m{\left(\frac{1}{|2^{k}Q|}{\int }_{2^{k}Q}\right|b_j(y)-{(b_j)}_Q{|}^{q_j}dy)}^{1/q_j}\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{k}^m{2}^{-k\delta }\prod _{j=1}^m{\parallel b_j\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}\\ \le & C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )},\end{array}$$

so

$$\frac{1}{\omega (Q)}{\int }_Q|J_7(x)|\omega (x)dx\le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{\mathrm{\infty }}(\omega )}.$$

This completes the proof of Theorem 1. □

Theorem 2 Let $1<p<\mathrm{\infty }$, $\omega \in A_1$ and $\overrightarrow{b}=(b_1,\dots ,b_m)$ with $b_j\in \mathit{BMO}(R^n)$ for $1\le j\le m$. Then $T_b$ is bounded from $B_p(\omega )$ to ${\mathit{CMO}}_A(\omega )$.

Proof It suffices to prove for $f\in C_0^{\mathrm{\infty }}(R^n)$, the following inequality holds:

$$\frac{1}{\omega (Q)}{\int }_Q|T_b(f)(x)-A_{t_Q}{T}_b(f)(x)|\omega (x)dx\le C{\parallel f\parallel }_{B_p(\omega )}$$

for any cube $Q=Q(0,d)$ with $d>1$. Fix a cube $Q=Q(0,d)$ with $d>1$. Set $f_1=f{\chi }_Q$, $f_2=f{\chi }_{(R^n\mathrm{\setminus }Q)}$ and ${\overrightarrow{b}}_Q=({(b_1)}_Q,\dots ,{(b_m)}_Q)\in R^n$, where ${(b_j)}_Q={|Q|}^{-1}{\int }_Q|b_j(y)|dy$, $1\le j\le m$, we have

$$\begin{array}{c}|T_b(f)(x)-A_{t_Q}{T}_b(f)(x)|\hfill \\ \le |\prod _{j=1}^m(b_j(x)-{(b_j)}_Q){\int }_{R^n}K(x,y)f(y)dy|\hfill \\ +|\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{(b(x)-{(b)}_Q)}_{\sigma }{\int }_{R^n}{(b(y)-{(b)}_Q)}_{{\sigma }^c}K(x,y)f(y)dy|\hfill \\ +|{\int }_{R^n}\prod _{j=1}^m(b_j(y)-{(b_j)}_Q)K(x,y)f_1(y)dy|\hfill \\ +|\prod _{j=1}^m(b_j(x)-{(b_j)}_Q){\int }_{R^n}{K}_t(x,y)f(y)dy|\hfill \\ +|\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{(b(x)-{(b)}_Q)}_{\sigma }{\int }_{R^n}{(b(y)-{(b)}_Q)}_{{\sigma }^c}{K}_t(x,y)f(y)dy|\hfill \\ +|{\int }_{R^n}\prod _{j=1}^m(b_j(y)-{(b_j)}_Q)K_t(x,y)f_1(y)dy|\hfill \\ +|{\int }_{R^n}\prod _{j=1}^m(b_j(y)-{(b_j)}_Q)(K(x,y)-K_t(x,y))f_2(y)dy|\hfill \\ =L_1(x)+L_2(x)+L_3(x)+L_4(x)+L_5(x)+L_6(x)+L_7(x).\hfill \end{array}$$

For $L_1(x)$, we have

$$\begin{array}{c}\frac{1}{\omega (Q)}{\int }_Q|L_1(x)|\omega (x)dx\hfill \\ \le \frac{C}{\omega (Q)}{\left({\int }_Q\right|(b_1(x)-{(b_1)}_Q)\cdots (b_m(x)-{(b_m)}_Q){|}^{p^{\prime }}\omega (x)dx)}^{1/p^{\prime }}\hfill \\ \times {\left({\int }_Q\right|T(f)(x){|}^p\omega (x)dx)}^{1/p}\hfill \\ \le \frac{C}{\omega (Q)}{\left[{\left({\int }_Q\right|(b_1(x)-{(b_1)}_Q)\cdots (b_m(x)-{(b_m)}_Q){|}^{p^{\prime }{q}^{\prime }}dx)}^{1/q^{\prime }}{\left({\int }_Q\omega {(x)}^{q}dx\right)}^{1/q}\right]}^{1/p^{\prime }}\hfill \\ \times {\left({\int }_Q\right|f(x){|}^p\omega (x)dx)}^{1/p}\hfill \\ \le \frac{C}{\omega (Q)}{|Q|}^{1/p^{\prime }{q}^{\prime }}{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{|Q|}^{1/p^{\prime }q}{\left(\frac{\omega (Q)}{|Q|}\right)}^{1/p^{\prime }}{\parallel f{\chi }_Q\parallel }_{L^p(\omega )}\hfill \\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}\omega {(Q)}^{-1/p}{\parallel f{\chi }_Q\parallel }_{L^p(\omega )}\hfill \\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{B_p(\omega )}.\hfill \end{array}$$

For $L_2(x)$, taking $1<s,s^{\prime }<\mathrm{\infty }$, and $1/s+1/s^{\prime }=1$, we have

$$\begin{array}{c}\frac{1}{\omega (Q)}{\int }_Q|L_2(x)|\omega (x)dx\hfill \\ \le C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{\left(\frac{1}{\omega (Q)}{\int }_Q\right|{(b(x)-b_Q)}_{\sigma }{|}^{s^{\prime }}\omega (x)dx)}^{1/s^{\prime }}\hfill \\ \times {\left(\frac{1}{\omega (Q)}{\int }_Q\right|T\left({(b-b_Q)}_{{\sigma }^c}f\right)(x){|}^s\omega (x)dx)}^{1/s}\hfill \\ \le C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}\omega {(Q)}^{-1/s^{\prime }}{\left[{\left({\int }_Q\right|{(b(x)-b_Q)}_{\sigma }{|}^{s^{\prime }{q}^{\prime }}dx)}^{1/q^{\prime }}{\left({\int }_Q{\omega }^{q}dx\right)}^{1/q}\right]}^{1/s^{\prime }}\hfill \\ \times \omega {(Q)}^{-1/s}{\left({\int }_Q\right|{(b(x)-b_Q)}_{{\sigma }^c}f(x){|}^s\omega (x)dx)}^{1/s}\hfill \\ \le C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}\omega {(Q)}^{-1/s^{\prime }}{|Q|}^{1/s^{\prime }{q}^{\prime }+1/s^{\prime }q-1/s^{\prime }}\omega {(Q)}^{1/s^{\prime }}{\parallel {\overrightarrow{b}}_{\sigma }\parallel }_{\mathit{BMO}}\hfill \\ \times \omega {(Q)}^{-1/s}{|Q|}^{1/rs}{\parallel {\overrightarrow{b}}_{{\sigma }^c}\parallel }_{\mathit{BMO}}{\left({\int }_Q\right|f(x){|}^p\omega (x)dx)}^{1/p}{\left({\int }_Q\omega {(x)}^{q}dx\right)}^{(p-s)/pqs}\hfill \\ \le C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{\parallel {\overrightarrow{b}}_{\sigma }\parallel }_{\mathit{BMO}}{\parallel {\overrightarrow{b}}_{{\sigma }^c}\parallel }_{\mathit{BMO}}\omega {(Q)}^{-1/p}{\parallel f{\chi }_Q\parallel }_{L^p(\omega )}\hfill \\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{B_p(\omega )}.\hfill \end{array}$$

For $L_3(x)$, we have

$$\begin{array}{c}\frac{1}{\omega (Q)}{\int }_Q|L_3(x)|\omega (x)dx\hfill \\ \le C{\left(\frac{1}{\omega (Q)}{\int }_{R^n}\right|T((b_1-{(b_1)}_Q)\cdots (b_m-{(b_m)}_Q)f_1)(x){|}^s\omega (x)dx)}^{1/s}\hfill \\ \le C\omega {(Q)}^{-1/s}{\left({\int }_Q\right|(b_1(x)-{(b_1)}_Q)\cdots (b_m(x)-{(b_m)}_Q)f(x){|}^s\omega (x)dx)}^{1/s}\hfill \\ \le C\omega {(Q)}^{-1/p}{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel f{\chi }_Q\parallel }_{L^p(\omega )}\hfill \\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{B_p(\omega )}.\hfill \end{array}$$

For $L_4(x)$, $L_5(x)$, and $L_6(x)$, by Lemma 2, we have

$$\begin{array}{c}\frac{1}{\omega (Q)}{\int }_Q|L_4(x)|\omega (x)dx\hfill \\ \le C{\left(\frac{1}{\omega (Q)}{\int }_Q\right|(b_1(x)-{(b_1)}_Q)\cdots (b_m(x)-{(b_m)}_Q){|}^{s^{\prime }}\omega (x)dx)}^{1/s^{\prime }}\hfill \\ \times {\left(\frac{1}{\omega (Q)}{\int }_Q\right|A_{t_Q}(f)(x){|}^s\omega (x)dx)}^{1/s}\hfill \\ \le C{\left(\frac{1}{\omega (Q)}\right)}^{1/s^{\prime }}[{\left({\int }_Q\right|(b_1(x)-{(b_1)}_Q)\cdots (b_m(x)-{(b_m)}_Q){|}^{s^{\prime }{q}^{\prime }}dx)}^{1/q^{\prime }}\hfill \\ {\times {\left({\int }_Q\omega {(x)}^{q}dx\right)}^{1/q}]}^{1/s^{\prime }}{\parallel f\parallel }_{B_p(w)}\hfill \\ \le C{\left(\frac{1}{\omega (Q)}\right)}^{1/s^{\prime }}{|Q|}^{1/s^{\prime }{q}^{\prime }}{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{|Q|}^{1/s^{\prime }q}{\left(\frac{\omega (Q)}{|Q|}\right)}^{1/s^{\prime }}{\parallel f\parallel }_{B_p(w)}\hfill \\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{B_p(\omega )};\hfill \\ \frac{1}{\omega (Q)}{\int }_Q|L_5(x)|\omega (x)dx\hfill \\ \le C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{\left(\frac{1}{\omega (Q)}{\int }_Q\right|{(b(x)-b_Q)}_{\sigma }{|}^{s^{\prime }}\omega (x)dx)}^{1/s^{\prime }}\hfill \\ \times {\left(\frac{1}{\omega (Q)}{\int }_Q\right|A_{t_Q}\left({(b-b_Q)}_{{\sigma }^c}f\right)(x){|}^s\omega (x)dx)}^{1/s}\hfill \\ \le C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}\omega {(Q)}^{-1/s^{\prime }}{\left[{\left({\int }_Q\right|{(b(x)-b_Q)}_{\sigma }{|}^{s^{\prime }{q}^{\prime }}dx)}^{1/q^{\prime }}{\left({\int }_Q{\omega }^{q}dx\right)}^{1/q}\right]}^{1/s^{\prime }}\hfill \\ \times {\parallel {\overrightarrow{b}}_{{\sigma }^c}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{B_p(\omega )}\hfill \\ \le C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}\omega {(Q)}^{-1/s^{\prime }}{|Q|}^{1/s^{\prime }{q}^{\prime }+1/s^{\prime }q-1/s^{\prime }}\omega {(Q)}^{1/s^{\prime }}{\parallel {\overrightarrow{b}}_{\sigma }\parallel }_{\mathit{BMO}}{\parallel {\overrightarrow{b}}_{{\sigma }^c}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{B_p(\omega )}\hfill \\ \le C{\parallel {\overrightarrow{b}}_{\sigma }\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{B_p(\omega )};\hfill \\ \frac{1}{\omega (Q)}{\int }_Q|L_6(x)|\omega (x)dx\hfill \\ \le {\left(\frac{1}{\omega (Q)}{\int }_{R^n}\right|A_{t_Q}((b_1-{(b_1)}_Q)\cdots (b_m-{(b_m)}_Q)f_1)(x){|}^s\omega (x)dx)}^{1/s}\hfill \\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{B_p(\omega )}.\hfill \end{array}$$

For $L_7(x)$, note that $|x-y|\ge d=t^{1/2}$, taking $1<u<p$, then

$$\begin{array}{rcl}{L}_7(x)& \le & C{\int }_{Q^c}\prod _{j=1}^m|b_j(y)-{(b_j)}_Q||f(y)|\frac{d^{\delta }}{{|x_0-y|}^{n+\delta }}dy\\ \le & C\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}Q\mathrm{\setminus }{2}^{k}Q}\prod _{j=1}^m|b_j(y)-{(b_j)}_Q||f(y)|\frac{d^{\delta }}{{|x_0-y|}^{n+\delta }}dy\\ \le & C\sum _{k=1}^{\mathrm{\infty }}\frac{d^{\delta }}{{(2^{k-1}d)}^{n+\delta }}\left|2^{k}Q\right|{\left(\frac{1}{|2^{k}Q|}{\int }_{2^{k}Q}\right|f(y){|}^{u}dy)}^{1/u}\\ \times {\left(\frac{1}{|2^{k}Q|}{\int }_{2^{k}Q}\prod _{j=1}^m\right|b_j(y)-{(b_j)}_Q{|}^{u^{\prime }}dy)}^{1/u^{\prime }}\\ \le & C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}\sum _{k=1}^{\mathrm{\infty }}{k}^m{2}^{-k\delta }{\left(\frac{1}{|2^{k}Q|}\right)}^{1/u}\\ \times {\left[{\left({\int }_{2^{k}Q}\right|f(y){|}^p\omega (y)dy)}^{\frac{u}{p}}{\left({\int }_{2^{k}Q}\omega {(y)}^{-\frac{u}{p-u}}dy\right)}^{\frac{p-u}{p}}\right]}^{1/u}\\ \le & C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}\sum _{k=1}^{\mathrm{\infty }}{k}^m{2}^{-k\delta }{\left(\frac{1}{|2^{k}Q|}\right)}^{1/u}{\parallel f{\chi }_{2^{k}Q}\parallel }_{L^p(\omega )}{\left(\frac{\omega (2^{k}Q)}{|2^{k}Q|}\right)}^{-1/p}|2^{k}Q{|}^{(\frac{p}{u}-1)\frac{1}{p}}\\ \times {\left[(\frac{1}{|2^{k}Q|}{\int }_{2^{k}Q}\omega (y)dy){\left(\frac{1}{|2^{k}Q|}{\int }_{2^{k}Q}\omega {(y)}^{-\frac{1}{\frac{p}{u}-1}}dy\right)}^{\frac{p}{u}-1}\right]}^{1/p}\\ \le & C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}\sum _{k=1}^{\mathrm{\infty }}{k}^m{2}^{-k\delta }\omega {\left(2^{k}Q\right)}^{-1/p}{\parallel f{\chi }_{2^{k}Q}\parallel }_{L^p(\omega )}\\ \le & C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{B_p(\omega )},\end{array}$$

so

$$\frac{1}{\omega (Q)}{\int }_Q|L_7(x)|\omega (x)dx\le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{B_p(\omega )}.$$

This completes the proof of Theorem 2. □