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

Journal of Inequalities and Applications20142014:371

https://doi.org/10.1186/1029-242X-2014-371

• Received: 9 February 2014
• Accepted: 29 August 2014
• Published:

## Abstract

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

## Keywords

• multilinear commutator
• singular integral operator
• BMO

## 1 Introduction

Let $b\in \mathit{BMO}\left({R}^{n}\right)$ and T be the Calderón-Zygmund operator, the commutator $\left[b,T\right]$ generated by b and T is defined by $\left[b,T\right]\left(f\right)\left(x\right)=b\left(x\right)T\left(f\right)\left(x\right)-T\left(bf\right)\left(x\right)$. A classical result of Coifman et al. (see ) proved that the commutator $\left[b,T\right]$ is bounded on ${L}^{p}\left({R}^{n}\right)$ ($1). 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 ). 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 ${b}_{Q}={|Q|}^{-1}{\int }_{Q}b\left(x\right)\phantom{\rule{0.2em}{0ex}}dx$ and $b\left(Q\right)={\int }_{Q}b\left(x\right)\phantom{\rule{0.2em}{0ex}}dx$, the sharp function of b is defined by
${b}^{\mathrm{#}}\left(x\right)=\underset{Q\ni x}{sup}\frac{1}{|Q|}{\int }_{Q}|b\left(y\right)-{b}_{Q}|\phantom{\rule{0.2em}{0ex}}dy.$
It is well known that (see )
${b}^{\mathrm{#}}\left(x\right)=\underset{Q\ni x}{sup}\underset{c\in C}{inf}\frac{1}{|Q|}{\int }_{Q}|b\left(y\right)-c|\phantom{\rule{0.2em}{0ex}}dy.$
Moreover, for a weight function ω (that is, a non-negative locally integrable function), b is said to belong to $\mathit{BMO}\left(\omega \right)$ if ${b}^{\mathrm{#}}\in {L}^{\mathrm{\infty }}\left(\omega \right)$ and define ${\parallel b\parallel }_{\mathit{BMO}\left(\omega \right)}={\parallel {b}^{\mathrm{#}}\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}$, if $\omega =1$, we denote $\mathit{BMO}\left(\omega \right)=\mathit{BMO}\left({R}^{n}\right)$. It is well known that (see )
${\parallel b-{b}_{{2}^{k}Q}\parallel }_{\mathit{BMO}}\le Ck{\parallel b\parallel }_{\mathit{BMO}}.$
The ${A}_{p}$ weight is defined by (see )
$\begin{array}{c}{A}_{p}=\left\{0<\omega \in {L}_{\mathrm{loc}}^{1}\left({R}^{n}\right):\underset{Q}{sup}\left(\frac{1}{|Q|}{\int }_{Q}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right){\left(\frac{1}{|Q|}{\int }_{Q}\omega {\left(x\right)}^{-1/\left(p-1\right)}\phantom{\rule{0.2em}{0ex}}dx\right)}^{p-1}<\mathrm{\infty }\right\},\hfill \\ \phantom{\rule{1em}{0ex}}1
and
${A}_{1}=\left\{0<\omega \in {L}_{\mathrm{loc}}^{1}\left({R}^{n}\right):\underset{Q\ni x}{sup}\frac{1}{|Q|}{\int }_{Q}\omega \left(y\right)\phantom{\rule{0.2em}{0ex}}dy\le c\omega \left(x\right),\text{a.e.}\right\}.$
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 ${a}_{t}\left(x,y\right)$ in the following sense:
${D}_{t}\left(f\right)\left(x\right)={\int }_{{R}^{n}}{a}_{t}\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy$
for every $f\in {L}^{p}\left({R}^{n}\right)$ with $p\ge 1$, and ${a}_{t}\left(x,y\right)$ satisfies
$|{a}_{t}\left(x,y\right)|\le {h}_{t}\left(x,y\right)=C{t}^{-n/2}s\left({|x-y|}^{2}/t\right),$
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 ${L}^{2}\left({R}^{n}\right)$ and associated with a kernel $K\left(x,y\right)$ 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 continuous function f with compact support, and for almost all x not in the support of f.
1. (1)
There exists an ‘approximation to the identity’ $\left\{{B}_{t},t>0\right\}$ such that $T{B}_{t}$ has associated kernel ${k}_{t}\left(x,y\right)$ and there exist ${c}_{1},{c}_{2}>0$ so that

2. (2)
There exists an ‘approximation to the identity’ $\left\{{A}_{t},t>0\right\}$ such that ${A}_{t}T$ has associated kernel ${K}_{t}\left(x,y\right)$ which satisfies
and

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
${T}_{b}\left(f\right)\left(x\right)={\int }_{{R}^{n}}\left[\prod _{j=1}^{m}\left({b}_{j}\left(x\right)-{b}_{j}\left(y\right)\right)\right]K\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy.$
Definition 3 Given the ‘approximations to the identity’ $\left\{{A}_{t},t>0\right\}$ and a weight function ω.
1. (1)
The weighted BMO space associated with $\left\{{A}_{t},t>0\right\}$ is defined by
${\mathit{BMO}}_{A}\left(\omega \right)=\left\{f\in {L}_{\mathrm{loc}}^{1}\left({R}^{n}\right):{\parallel f\parallel }_{{\mathit{BMO}}_{A}\left(\omega \right)}<\mathrm{\infty }\right\},$
where
${\parallel f\parallel }_{{\mathit{BMO}}_{A}\left(\omega \right)}=\underset{Q}{sup}\frac{1}{\omega \left(Q\right)}{\int }_{Q}|f\left(x\right)-{A}_{{t}_{Q}}\left(f\right)\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx,$

${t}_{Q}=l{\left(Q\right)}^{2}$ and $l\left(Q\right)$ denotes the side length of Q.

2. (2)
The weighted central BMO space associated with $\left\{{A}_{t},t>0\right\}$ is defined by
${\mathit{CMO}}_{A}\left(\omega \right)=\left\{f\in {L}_{\mathrm{loc}}^{1}\left({R}^{n}\right):{\parallel f\parallel }_{{\mathit{CMO}}_{A}\left(\omega \right)}<\mathrm{\infty }\right\},$
where
${\parallel f\parallel }_{\mathit{CMO}\left(\omega \right)}=\underset{r>1}{sup}\frac{1}{\omega \left(Q\left(0,r\right)\right)}{\int }_{Q}|f\left(x\right)-{A}_{{t}_{Q}}f\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx,$

and ${t}_{Q}={r}^{2}$.

Definition 4 Let $1 and ω be a weighted function on ${R}^{n}$. We shall call ${B}_{p}\left(\omega \right)$ the space of those functions f on ${R}^{n}$, such that
${\parallel f\parallel }_{{B}_{p}\left(\omega \right)}=\underset{r>1}{sup}{\left[\omega \left(Q\left(0,r\right)\right)\right]}^{-1/p}{\parallel f{\chi }_{Q\left(0,r\right)}\parallel }_{{L}^{p}\left(\omega \right)}<\mathrm{\infty }.$

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

## 2 Theorems and proofs

We begin with some preliminaries lemmas.

Lemma 1 ([5, 7])

Let $\omega \in {A}_{1}$, $1, and T be the singular integral operators with non-smooth kernels. Then T is boundedness on ${L}^{p}\left(w\right)$.

Lemma 2 Let $\omega \in {A}_{1}$, $\left\{{A}_{t},t>0\right\}$ be anapproximation to the identityand $b\in \mathit{BMO}\left({R}^{n}\right)$. Then
1. (a)
for every $f\in {L}^{\mathrm{\infty }}\left({R}^{n}\right)$, $1\le p<\mathrm{\infty }$, and any cube Q,
${\left(\frac{1}{|Q|}{\int }_{Q}|{A}_{{t}_{Q}}\left(\left(b-{b}_{Q}\right)f\right)\left(y\right){|}^{p}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p}\le C{\parallel b\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}};$

2. (b)
for every $f\in {B}_{p}\left(\omega \right)$, $1\le r, and any cube Q,
${\left(\frac{1}{\omega \left(Q\right)}{\int }_{Q}|{A}_{{t}_{Q}}\left(\left(b-{b}_{Q}\right)f\right)\left(y\right){|}^{r}\omega \left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/r}\le C{\parallel b\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{B}_{p}\left(\omega \right)},$

where ${t}_{Q}=l{\left(Q\right)}^{2}$ and $l\left(Q\right)$ denotes the side length of Q.

Proof (a) Write
$\begin{array}{c}{\left(\frac{1}{|Q|}{\int }_{Q}|{A}_{{t}_{Q}}\left(\left(b-{b}_{Q}\right)f\right)\left(y\right){|}^{p}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\left(\frac{1}{|Q|}{\int }_{Q}{\int }_{{R}^{n}}{h}_{{t}_{Q}}{\left(x,y\right)}^{p}|\left(b\left(y\right)-{b}_{Q}\right)f\left(y\right){|}^{p}\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\left(\frac{1}{|Q|}{\int }_{Q}{\int }_{2Q}{h}_{{t}_{Q}}{\left(x,y\right)}^{p}|\left(b\left(y\right)-{b}_{Q}\right)f\left(y\right){|}^{p}\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\hfill \\ \phantom{\rule{2em}{0ex}}+{\left(\sum _{k=1}^{\mathrm{\infty }}\frac{1}{|Q|}{\int }_{Q}{\int }_{{2}^{k+1}Q\mathrm{\setminus }{2}^{k}Q}{h}_{{t}_{Q}}{\left(x,y\right)}^{p}|\left(b\left(y\right)-{b}_{Q}\right)f\left(y\right){|}^{p}\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\hfill \\ \phantom{\rule{1em}{0ex}}={I}_{1}+{I}_{2}.\hfill \end{array}$
We have, by Hölder’s inequality,
$\begin{array}{rcl}{I}_{1}& \le & {\left(\frac{C}{|Q||2Q|}{\int }_{Q}{\int }_{2Q}|\left(b\left(y\right)-{b}_{Q}\right)f\left(y\right){|}^{p}\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\\ \le & C{\parallel f\parallel }_{{L}^{\mathrm{\infty }}}{\left(\frac{1}{|2Q|}{\int }_{2Q}|b\left(y\right)-{b}_{Q}{|}^{p}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p}\\ \le & C{\parallel b\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}}.\end{array}$
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 ${h}_{{t}_{Q}}\left(x,y\right)\le C\frac{s\left({2}^{2\left(k-1\right)}\right)}{|Q|}$. Thus
$\begin{array}{rcl}{I}_{2}& \le & C\sum _{k=1}^{\mathrm{\infty }}{2}^{\left(k-1\right)n}s\left({2}^{2\left(k-1\right)}\right){\left(\frac{1}{|{2}^{k+1}Q|}{\int }_{{2}^{k+1}Q}|\left(b\left(y\right)-{b}_{Q}\right)f\left(y\right){|}^{p}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p}\\ \le & C{\parallel f\parallel }_{{L}^{\mathrm{\infty }}}\sum _{k=1}^{\mathrm{\infty }}{2}^{\left(k-1\right)n}s\left({2}^{2\left(k-1\right)}\right){\left(\frac{1}{|{2}^{k+1}Q|}{\int }_{{2}^{k+1}Q}|b\left(y\right)-{b}_{{2}^{k+1}Q}{|}^{p}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p}\\ +C{\parallel f\parallel }_{{L}^{\mathrm{\infty }}}\sum _{k=1}^{\mathrm{\infty }}{2}^{\left(k-1\right)n}s\left({2}^{2\left(k-1\right)}\right)|{b}_{Q}-{b}_{{2}^{k+1}Q}|\\ \le & C{\parallel f\parallel }_{{L}^{\mathrm{\infty }}}\sum _{k=1}^{\mathrm{\infty }}{2}^{\left(k-1\right)n}s\left({2}^{2\left(k-1\right)}\right)\left(k+1\right){\parallel b\parallel }_{\mathit{BMO}}\\ \le & C{\parallel b\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}},\end{array}$
where the last inequality follows from
$\sum _{k=2}^{\mathrm{\infty }}{2}^{\left(k-1\right)n}s\left({2}^{2\left(k-1\right)}\right)\left(k+1\right)\le C\sum _{k=2}^{\mathrm{\infty }}k{2}^{-\left(k-1\right)ϵ}<\mathrm{\infty }$
for some $ϵ>0$.
1. (b)
Write
$\begin{array}{c}{\left(\frac{1}{\omega \left(Q\right)}{\int }_{Q}|{A}_{{t}_{Q}}\left(\left(b-{b}_{Q}\right)f\right)\left(y\right){|}^{r}\omega \left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/r}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\left(\frac{1}{\omega \left(Q\right)}{\int }_{Q}{\int }_{{R}^{n}}{h}_{{t}_{Q}}{\left(x,y\right)}^{r}|\left(b\left(y\right)-{b}_{Q}\right)f\left(y\right){|}^{r}\omega \left(y\right)\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/r}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\left(\frac{1}{\omega \left(Q\right)}{\int }_{Q}{\int }_{2Q}{h}_{{t}_{Q}}{\left(x,y\right)}^{p}|\left(b\left(y\right)-{b}_{Q}\right)f\left(y\right){|}^{r}\omega \left(y\right)\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/r}\hfill \\ \phantom{\rule{2em}{0ex}}+{\left(\sum _{k=1}^{\mathrm{\infty }}\frac{1}{\omega \left(Q\right)}{\int }_{Q}{\int }_{{2}^{k+1}Q\mathrm{\setminus }{2}^{k}Q}{h}_{{t}_{Q}}{\left(x,y\right)}^{r}|\left(b\left(y\right)-{b}_{Q}\right)f\left(y\right){|}^{r}\omega \left(y\right)\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/r}\hfill \\ \phantom{\rule{1em}{0ex}}=I+\mathit{II}.\hfill \end{array}$

For I, since $\omega \in {A}_{1}$, ω satisfies the reverse of Hölder’s inequality
${\left(\frac{1}{|Q|}{\int }_{Q}\omega {\left(x\right)}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}\le \frac{C}{|Q|}{\int }_{Q}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx$
for some $1, and $\frac{\omega \left({Q}_{2}\right)}{|{Q}_{2}|}\frac{|{Q}_{1}|}{\omega \left({Q}_{1}\right)}\le C$ for all cubes ${Q}_{1}$, ${Q}_{2}$ with ${Q}_{1}\subset {Q}_{2}$, $\omega \in {A}_{p/ru}$ for $1 with ${u}^{\prime }v=q$ and $p>ru$ (see ). We have, by Hölder’s inequality,
$\begin{array}{c}I\le {\left(\frac{C}{\omega \left(Q\right)|Q|}{\int }_{Q}{\int }_{2Q}|\left(b\left(y\right)-{b}_{Q}\right)f\left(y\right){|}^{r}\omega \left(y\right)\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/r}\hfill \\ \phantom{I}\le C{\left(\frac{1}{\omega \left(Q\right)}{\int }_{2Q}|\left(b\left(y\right)-{b}_{Q}\right)f\left(y\right){|}^{r}\omega \left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/r}\hfill \\ \phantom{I}\le C{\left[\frac{|2Q|}{\omega \left(Q\right)}{\left(\frac{1}{|2Q|}{\int }_{2Q}|b\left(y\right)-{b}_{Q}{|}^{r{u}^{\prime }}\omega {\left(y\right)}^{{u}^{\prime }}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/{u}^{\prime }}{\left(\frac{1}{|2Q|}{\int }_{2Q}|f\left(y\right){|}^{ru}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/u}\right]}^{1/r}\hfill \\ \phantom{I}\le C{\left(\frac{|2Q|}{\omega \left(Q\right)}\right)}^{1/r}{\left(\frac{1}{|2Q|}{\int }_{2Q}|b\left(y\right)-{b}_{Q}{|}^{r{u}^{\prime }{v}^{\prime }}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/r{u}^{\prime }{v}^{\prime }}{\left(\frac{1}{|2Q|}{\int }_{2Q}\omega {\left(y\right)}^{{u}^{\prime }v}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/r{u}^{\prime }v}\hfill \\ \phantom{I\le }×{\left(\frac{1}{|2Q|}{\int }_{2Q}|f\left(y\right){|}^{ru}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/ru}\hfill \\ \phantom{I}\le C{\parallel b\parallel }_{\mathit{BMO}}{\left(\frac{|2Q|}{\omega \left(2Q\right)}\right)}^{1/r}{\left(\frac{\omega \left(2Q\right)}{|2Q|}\right)}^{1/r}{\left(\frac{1}{|2Q|}{\int }_{2Q}|f\left(y\right){|}^{ru}\omega {\left(y\right)}^{\frac{ru}{p}}\omega {\left(y\right)}^{-\frac{ru}{p}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/ru}\hfill \\ \phantom{I}\le C{\parallel b\parallel }_{\mathit{BMO}}{\left(\frac{1}{|2Q|}{\int }_{2Q}{\left(|f\left(y\right){|}^{ru}\omega {\left(y\right)}^{\frac{ru}{p}}\right)}^{\frac{p}{ru}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p}{\left(\frac{1}{|2Q|}{\int }_{2Q}\omega {\left(y\right)}^{-\frac{ru}{p}\frac{p}{p-ru}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{\left(p-ru\right)/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}\left(\omega \right)}{\left(\frac{1}{|2Q|}{\int }_{2Q}\omega \left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{-1/p}\hfill \\ \phantom{I\le }×{\left[\left(\frac{1}{|2Q|}{\int }_{2Q}\omega \left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right){\left(\frac{1}{|2Q|}{\int }_{2Q}\omega {\left(y\right)}^{-\frac{1}{\frac{p}{ru}-1}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{\frac{p}{ru}-1}\right]}^{1/p}\hfill \\ \phantom{I}\le C{\parallel b\parallel }_{\mathit{BMO}}\omega {\left(2Q\right)}^{-1/p}{\parallel f{\chi }_{2Q}\parallel }_{{L}^{p}\left(\omega \right)}\hfill \\ \phantom{I}\le C{\parallel b\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{B}_{p}\left(\omega \right)};\hfill \\ \mathit{II}\le C{\left(\frac{|Q|}{\omega \left(Q\right)}\right)}^{1/r}\sum _{k=1}^{\mathrm{\infty }}{2}^{\left(k-1\right)n}s\left({2}^{2\left(k-1\right)}\right){\left(\frac{1}{|{2}^{k+1}Q|}{\int }_{{2}^{k+1}Q}|\left(b\left(y\right)-{b}_{Q}\right)f\left(y\right){|}^{r}\omega \left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/r}\hfill \\ \phantom{\mathit{II}}\le C{\left(\frac{|Q|}{\omega \left(Q\right)}\right)}^{1/r}\sum _{k=1}^{\mathrm{\infty }}{2}^{\left(k-1\right)n}s\left({2}^{2\left(k-1\right)}\right){\left(\frac{1}{|{2}^{k+1}Q|}{\int }_{{2}^{k+1}Q}|b\left(y\right)-{b}_{{2}^{k+1}Q}{|}^{r{u}^{\prime }}\omega {\left(y\right)}^{{u}^{\prime }}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/r{u}^{\prime }}\hfill \\ \phantom{\mathit{II}\le }×{\left(\frac{1}{|{2}^{k+1}Q|}{\int }_{{2}^{k+1}Q}|f\left(y\right){|}^{ru}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/ru}\hfill \\ \phantom{\mathit{II}}\le C{\left(\frac{|Q|}{\omega \left(Q\right)}\right)}^{1/r}\sum _{k=1}^{\mathrm{\infty }}{2}^{\left(k-1\right)n}s\left({2}^{2\left(k-1\right)}\right){\left(\frac{1}{|{2}^{k+1}Q|}{\int }_{{2}^{k+1}Q}|b\left(y\right)-{b}_{{2}^{k+1}Q}{|}^{r{u}^{\prime }{v}^{\prime }}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/r{u}^{\prime }{v}^{\prime }}\hfill \\ \phantom{\mathit{II}\le }×{\left(\frac{1}{|{2}^{k+1}Q|}{\int }_{{2}^{k+1}Q}\omega {\left(y\right)}^{{u}^{\prime }v}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/r{u}^{\prime }v}{\left(\frac{1}{|{2}^{k+1}Q|}{\int }_{{2}^{k+1}Q}f{\left(y\right)}^{ru}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/ru}\hfill \\ \phantom{\mathit{II}}\le C{\parallel b\parallel }_{\mathit{BMO}}\sum _{k=1}^{\mathrm{\infty }}{2}^{\left(k-1\right)n}s\left({2}^{2\left(k-1\right)}\right)\left(k+1\right){\left(\frac{|Q|}{\omega \left(Q\right)}\cdot \frac{\omega \left({2}^{k+1}Q\right)}{|{2}^{k+1}Q|}\right)}^{1/r}\hfill \\ \phantom{\mathit{II}\le }×{\left(\frac{1}{|{2}^{k+1}Q|}{\int }_{{2}^{k+1}Q}f{\left(y\right)}^{ru}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/ru}\hfill \\ \phantom{\mathit{II}}\le C\sum _{k=1}^{\mathrm{\infty }}{2}^{\left(k-1\right)n}s\left({2}^{2\left(k-1\right)}\right)\left(k+1\right){\parallel b\parallel }_{\mathit{BMO}}\omega {\left({2}^{k+1}Q\right)}^{-1/p}{\parallel f{\chi }_{{2}^{k+1}Q}\parallel }_{{L}^{p}\left(\omega \right)}\hfill \\ \phantom{\mathit{II}}\le C{\parallel b\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{B}_{p}\left(w\right)}.\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 $\stackrel{\to }{b}=\left({b}_{1},\dots ,{b}_{m}\right)$ with ${b}_{j}\in \mathit{BMO}\left({R}^{n}\right)$ for $1\le j\le m$. Then ${T}_{b}$ is bounded from ${L}^{\mathrm{\infty }}\left(\omega \right)$ to ${\mathit{BMO}}_{A}\left(\omega \right)$.

Proof It suffices to prove, for $f\in {C}_{0}^{\mathrm{\infty }}\left({R}^{n}\right)$, the following inequality holds:
$\frac{1}{\omega \left(Q\right)}{\int }_{Q}|{T}_{b}\left(f\right)\left(x\right)-{A}_{{t}_{Q}}{T}_{b}\left(f\right)\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\le C{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}.$

We fix a cube $Q=Q\left({x}_{0},d\right)$. We decompose f into $f={f}_{1}+{f}_{2}$ with ${f}_{1}=f{\chi }_{Q}$, ${f}_{2}=f{\chi }_{\left({R}^{n}\mathrm{\setminus }Q\right)}$.

When $m=1$, set ${\left({b}_{1}\right)}_{Q}={|Q|}^{-1}{\int }_{Q}{b}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy$, we have
$\begin{array}{rcl}{T}_{{b}_{1}}\left(f\right)\left(x\right)& =& {\int }_{{R}^{n}}\left[\left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right)-\left({b}_{1}\left(y\right)-{\left({b}_{1}\right)}_{Q}\right)\right]K\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ =& \left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right){\int }_{{R}^{n}}K\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy-{\int }_{{R}^{n}}\left({b}_{1}\left(y\right)-{\left({b}_{1}\right)}_{Q}\right)K\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\end{array}$
and
${A}_{{t}_{Q}}{T}_{{b}_{1}}\left(f\right)\left(x\right)=\left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right){\int }_{{R}^{n}}{K}_{t}\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy-{\int }_{{R}^{n}}\left({b}_{1}\left(y\right)-{\left({b}_{1}\right)}_{Q}\right){K}_{t}\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy.$
Then
$\begin{array}{c}|{T}_{{b}_{1}}\left(f\right)\left(x\right)-{A}_{{t}_{Q}}{T}_{{b}_{1}}\left(f\right)\left(x\right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le |\left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right){\int }_{{R}^{n}}K\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\hfill \\ \phantom{\rule{2em}{0ex}}+|{\int }_{{R}^{n}}\left({b}_{1}\left(y\right)-{\left({b}_{1}\right)}_{Q}\right)K\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\hfill \\ \phantom{\rule{2em}{0ex}}+|\left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right){\int }_{{R}^{n}}{K}_{t}\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\hfill \\ \phantom{\rule{2em}{0ex}}+|{\int }_{{R}^{n}}\left({b}_{1}\left(y\right)-{\left({b}_{1}\right)}_{Q}\right){K}_{t}\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\hfill \\ \phantom{\rule{2em}{0ex}}+|{\int }_{{R}^{n}}\left({b}_{1}\left(y\right)-{\left({b}_{1}\right)}_{Q}\right)\left(K\left(x,y\right)-{K}_{t}\left(x,y\right)\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\hfill \\ \phantom{\rule{1em}{0ex}}={I}_{1}\left(x\right)+{I}_{2}\left(x\right)+{I}_{3}\left(x\right)+{I}_{4}\left(x\right)+{I}_{5}\left(x\right).\hfill \end{array}$
For ${I}_{1}\left(x\right)$, let $1/p+1/{p}^{\prime }=1$, $1/q+1/{q}^{\prime }=1$, by the reverse of Hölder’s inequality with $1, Lemma 1, and Hölder’s inequality, we have
$\begin{array}{c}\frac{1}{\omega \left(Q\right)}{\int }_{Q}|{I}_{1}\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{C}{\omega \left(Q\right)}{\left({\int }_{Q}|{b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}{|}^{{p}^{\prime }}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{p}^{\prime }}{\left({\int }_{{R}^{n}}|T\left(f\right)\left(x\right){|}^{p}\omega \left(x\right){\chi }_{Q}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{C}{\omega \left(Q\right)}{\left({\int }_{Q}|{b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}{|}^{{p}^{\prime }}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{p}^{\prime }}{\left({\int }_{{R}^{n}}|f\left(x\right){|}^{p}\omega \left(x\right){\chi }_{Q}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{C}{\omega \left(Q\right)}{\left({\int }_{Q}|{b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}{|}^{{p}^{\prime }}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{p}^{\prime }}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}{\left({\int }_{Q}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{C}{\omega \left(Q\right)}{\left[{\left({\int }_{Q}|{b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}{|}^{{p}^{\prime }{q}^{\prime }}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{q}^{\prime }}{\left({\int }_{Q}\omega {\left(x\right)}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}\right]}^{1/{p}^{\prime }}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}\omega {\left(Q\right)}^{1/p}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\omega {\left(Q\right)}^{1/p-1}{|Q|}^{1/{p}^{\prime }}{\parallel {b}_{1}\parallel }_{\mathit{BMO}}{\left(\frac{1}{|Q|}{\int }_{Q}\omega {\left(x\right)}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{p}^{\prime }q}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\parallel {b}_{1}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}.\hfill \end{array}$
For ${I}_{2}\left(x\right)$, taking $p>1$, by Hölder’s inequality, we have
$\begin{array}{c}\frac{1}{\omega \left(Q\right)}{\int }_{Q}|{I}_{2}\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\le {\left(\frac{1}{\omega \left(Q\right)}{\int }_{{R}^{n}}|T\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right){f}_{1}\right)\left(x\right){|}^{p}\omega \left(x\right){\chi }_{Q}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\omega {\left(Q\right)}^{-1/p}{\left({\int }_{{R}^{n}}|\left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right){f}_{1}\left(x\right){|}^{p}\omega \left(x\right){\chi }_{Q}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\omega {\left(Q\right)}^{-1/p}{\left[{\left({\int }_{Q}|{b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}{|}^{p{q}^{\prime }}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{q}^{\prime }}{\left({\int }_{Q}|f\left(x\right){|}^{pq}\omega {\left(x\right)}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}\right]}^{1/p}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\omega {\left(Q\right)}^{-1/p}{\left({\int }_{Q}|{b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}{|}^{p{q}^{\prime }}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p{q}^{\prime }}{\left({\int }_{Q}|f\left(x\right){|}^{pq}\omega {\left(x\right)}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/pq}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\omega {\left(Q\right)}^{-1/p}{\left({\int }_{Q}|{b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}{|}^{p{q}^{\prime }}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p{q}^{\prime }}{\left({\int }_{Q}\omega {\left(x\right)}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/pq}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\omega {\left(Q\right)}^{-1/p}{|Q|}^{1/p{q}^{\prime }}{\parallel {b}_{1}\parallel }_{\mathit{BMO}}{|Q|}^{1/pq}{\left(\frac{1}{|Q|}{\int }_{Q}\omega {\left(x\right)}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/pq}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\parallel {b}_{1}\parallel }_{\mathit{BMO}}{\left(\frac{|Q|}{\omega \left(Q\right)}\right)}^{1/p}{\left(\frac{1}{|Q|}{\int }_{Q}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\parallel {b}_{1}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}.\hfill \end{array}$
For ${I}_{3}\left(x\right)$ and ${I}_{4}\left(x\right)$, 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 \left(Q\right)}{\int }_{Q}|{I}_{3}\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{C}{\omega \left(Q\right)}{\int }_{Q}|{b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}||{A}_{{t}_{Q}}\left(f\right)\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\le C\frac{|Q|}{\omega \left(Q\right)}{\left(\frac{1}{|Q|}{\int }_{Q}|{b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}{|}^{{p}_{1}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{p}_{1}}\hfill \\ \phantom{\rule{2em}{0ex}}×{\left(\frac{1}{|Q|}{\int }_{Q}|{A}_{{t}_{Q}}\left(f\right)\left(x\right){|}^{{p}_{2}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{p}_{2}}{\left(\frac{1}{|Q|}{\int }_{Q}\omega {\left(x\right)}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\frac{|Q|}{\omega \left(Q\right)}{\parallel {b}_{1}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}\frac{\omega \left(Q\right)}{|Q|}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\parallel {b}_{1}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)},\hfill \\ \frac{1}{\omega \left(Q\right)}{\int }_{Q}|{I}_{4}\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{\omega \left(Q\right)}{\int }_{{R}^{n}}|{A}_{{t}_{Q}}\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right){f}_{1}\right)\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\le C\frac{|Q|}{\omega \left(Q\right)}{\left(\frac{1}{|Q|}{\int }_{Q}|{A}_{{t}_{Q}}\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right){f}_{1}\right)\left(x\right){|}^{{q}^{\prime }}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{q}^{\prime }}{\left(\frac{1}{|Q|}{\int }_{Q}\omega {\left(x\right)}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\frac{|Q|}{\omega \left(Q\right)}{\parallel {b}_{1}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}\frac{\omega \left(Q\right)}{|Q|}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\parallel {b}_{1}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}.\hfill \end{array}$
For ${I}_{5}\left(x\right)$, we have
$\begin{array}{rcl}{I}_{5}\left(x\right)& =& |{\int }_{{R}^{n}}\left({b}_{1}\left(y\right)-{\left({b}_{1}\right)}_{Q}\right)\left(K\left(x,y\right)-{K}_{t}\left(x,y\right)\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\\ \le & C\sum _{k=0}^{\mathrm{\infty }}{\int }_{{2}^{k+1}Q\mathrm{\setminus }{2}^{k}Q}|{b}_{1}\left(y\right)-{\left({b}_{1}\right)}_{Q}||f\left(y\right)|\frac{{d}^{\delta }}{{|{x}_{0}-y|}^{n+\delta }}\phantom{\rule{0.2em}{0ex}}dy\\ \le & C\sum _{k=1}^{\mathrm{\infty }}\frac{{d}^{\delta }}{{\left({2}^{k-1}d\right)}^{n+\delta }}|{2}^{k}Q|{\left(\frac{1}{|{2}^{k}Q|}{\int }_{{2}^{k}Q}|f\left(y\right){|}^{p}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p}\\ ×{\left(\frac{1}{|{2}^{k}Q|}{\int }_{{2}^{k}Q}|{b}_{1}\left(y\right)-{\left({b}_{1}\right)}_{Q}{|}^{{p}^{\prime }}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/{p}^{\prime }}\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{k}^{m}{2}^{-k\delta }{\parallel {b}_{1}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}\\ \le & C{\parallel {b}_{1}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)},\end{array}$
so
$\frac{1}{\omega \left(Q\right)}{\int }_{Q}|{I}_{5}\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\le C{\parallel {b}_{1}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}.$
When $m>1$, set ${\stackrel{\to }{b}}_{Q}=\left({\left({b}_{1}\right)}_{Q},\dots ,{\left({b}_{m}\right)}_{Q}\right)\in {R}^{n}$, where ${\left({b}_{j}\right)}_{Q}={|Q|}^{-1}{\int }_{Q}{b}_{j}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy$, $1\le j\le m$, we have
$\begin{array}{rcl}{T}_{b}\left(f\right)\left(x\right)& =& \prod _{j=1}^{m}\left({b}_{j}\left(x\right)-{\left({b}_{j}\right)}_{Q}\right){\int }_{{R}^{n}}K\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ +\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\left(-1\right)}^{m-j}{\left(b\left(x\right)-{\left(b\right)}_{Q}\right)}_{\sigma }{\int }_{{R}^{n}}{\left(b\left(y\right)-{\left(b\right)}_{Q}\right)}_{{\sigma }^{c}}K\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ +{\left(-1\right)}^{m}{\int }_{{R}^{n}}\prod _{j=1}^{m}\left({b}_{j}\left(y\right)-{\left({b}_{j}\right)}_{Q}\right)K\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\end{array}$
and
$\begin{array}{rcl}{A}_{{t}_{Q}}{T}_{b}\left(f\right)\left(x\right)& =& \prod _{j=1}^{m}\left({b}_{j}\left(x\right)-{\left({b}_{j}\right)}_{Q}\right){\int }_{{R}^{n}}{K}_{t}\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ +\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\left(-1\right)}^{m-j}{\left(b\left(x\right)-{\left(b\right)}_{Q}\right)}_{\sigma }{\int }_{{R}^{n}}{\left(b\left(y\right)-{\left(b\right)}_{Q}\right)}_{{\sigma }^{c}}{K}_{t}\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ +{\left(-1\right)}^{m}{\int }_{{R}^{n}}\prod _{j=1}^{m}\left({b}_{j}\left(y\right)-{\left({b}_{j}\right)}_{Q}\right){K}_{t}\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy,\end{array}$
then
$\begin{array}{c}|{T}_{b}\left(f\right)\left(x\right)-{A}_{{t}_{Q}}{T}_{b}\left(f\right)\left(x\right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le |\prod _{j=1}^{m}\left({b}_{j}\left(x\right)-{\left({b}_{j}\right)}_{Q}\right){\int }_{{R}^{n}}K\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\hfill \\ \phantom{\rule{2em}{0ex}}+|\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\left(b\left(x\right)-{\left(b\right)}_{Q}\right)}_{\sigma }{\int }_{{R}^{n}}{\left(b\left(y\right)-{\left(b\right)}_{Q}\right)}_{{\sigma }^{c}}K\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\hfill \\ \phantom{\rule{2em}{0ex}}+|{\int }_{{R}^{n}}\prod _{j=1}^{m}\left({b}_{j}\left(y\right)-{\left({b}_{j}\right)}_{Q}\right)K\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\hfill \\ \phantom{\rule{2em}{0ex}}+|\prod _{j=1}^{m}\left({b}_{j}\left(x\right)-{\left({b}_{j}\right)}_{Q}\right){\int }_{{R}^{n}}{K}_{t}\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\hfill \\ \phantom{\rule{2em}{0ex}}+|\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\left(b\left(x\right)-{\left(b\right)}_{Q}\right)}_{\sigma }{\int }_{{R}^{n}}{\left(b\left(y\right)-{\left(b\right)}_{Q}\right)}_{{\sigma }^{c}}{K}_{t}\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\hfill \\ \phantom{\rule{2em}{0ex}}+|{\int }_{{R}^{n}}\prod _{j=1}^{m}\left({b}_{j}\left(y\right)-{\left({b}_{j}\right)}_{Q}\right){K}_{t}\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\hfill \\ \phantom{\rule{2em}{0ex}}+|{\int }_{{R}^{n}}\prod _{j=1}^{m}\left({b}_{j}\left(y\right)-{\left({b}_{j}\right)}_{Q}\right)\left(K\left(x,y\right)-{K}_{t}\left(x,y\right)\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\hfill \\ \phantom{\rule{1em}{0ex}}={J}_{1}\left(x\right)+{J}_{2}\left(x\right)+{J}_{3}\left(x\right)+{J}_{4}\left(x\right)+{J}_{5}\left(x\right)+{J}_{6}\left(x\right)+{J}_{7}\left(x\right).\hfill \end{array}$
For ${J}_{1}\left(x\right)$, same as $m=1$, for some $1, 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 \left(Q\right)}{\int }_{Q}|{J}_{1}\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{C}{\omega \left(Q\right)}{\left({\int }_{Q}|\left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}\left(x\right)-{\left({b}_{m}\right)}_{Q}\right){|}^{{p}^{\prime }}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{p}^{\prime }}\hfill \\ \phantom{\rule{2em}{0ex}}×{\left({\int }_{Q}|T\left(f\right)\left(x\right){|}^{p}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{C}{\omega \left(Q\right)}{\left({\int }_{Q}|{b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}{|}^{{p}^{\prime }}\cdots |{b}_{m}\left(x\right)-{\left({b}_{m}\right)}_{Q}{|}^{{p}^{\prime }}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{p}^{\prime }}\hfill \\ \phantom{\rule{2em}{0ex}}×{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}{\left({\int }_{Q}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{C}{\omega \left(Q\right)}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}\omega {\left(Q\right)}^{1/p}\prod _{j=1}^{m}{\left[{\left({\int }_{Q}|{b}_{j}\left(x\right)-{\left({b}_{j}\right)}_{Q}{|}^{{p}^{\prime }{q}_{j}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{q}_{j}}{\left({\int }_{Q}\omega {\left(x\right)}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}\right]}^{1/{p}^{\prime }}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}\omega {\left(Q\right)}^{1/{p}^{\prime }+1/p-1}{|Q|}^{1/{p}^{\prime }\left(1/{q}_{1}+\cdots +1/{q}_{m}+1/q-1\right)}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}.\hfill \end{array}$
For ${J}_{2}\left(x\right)$, by Hölder’s inequality and the reverse of Hölder’s inequality, we have
$\begin{array}{c}\frac{1}{\omega \left(Q\right)}{\int }_{Q}|{J}_{2}\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\le \sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}\frac{C}{\omega \left(Q\right)}{\left({\int }_{Q}|{\left(b\left(x\right)-{b}_{Q}\right)}_{\sigma }{|}^{{p}^{\prime }}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{p}^{\prime }}\hfill \\ \phantom{\rule{2em}{0ex}}×{\left({\int }_{Q}|T\left({\left(b-{b}_{Q}\right)}_{{\sigma }^{c}}f\right)\left(x\right){|}^{p}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\left(\frac{1}{\omega \left(Q\right)}{\int }_{Q}|{\left(b\left(x\right)-{b}_{Q}\right)}_{\sigma }{|}^{{p}^{\prime }}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{p}^{\prime }}\hfill \\ \phantom{\rule{2em}{0ex}}×{\left(\frac{1}{\omega \left(Q\right)}{\int }_{Q}|T\left({\left(b-{b}_{Q}\right)}_{{\sigma }^{c}}f\right)\left(x\right){|}^{p}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}\omega {\left(Q\right)}^{-1/{p}^{\prime }}{\left[{\left({\int }_{Q}|{\left(b\left(x\right)-{b}_{Q}\right)}_{\sigma }{|}^{{p}^{\prime }{q}^{\prime }}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{q}^{\prime }}{\left({\int }_{Q}\omega {\left(x\right)}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}\right]}^{1/{p}^{\prime }}\hfill \\ \phantom{\rule{2em}{0ex}}×\omega {\left(Q\right)}^{-1/p}{\left({\int }_{{R}^{n}}|{\left(b\left(x\right)-{b}_{Q}\right)}_{{\sigma }^{c}}f\left(x\right){|}^{p}\omega \left(x\right){\chi }_{Q}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}\omega {\left(Q\right)}^{-1/{p}^{\prime }}{|Q|}^{1/{p}^{\prime }{q}^{\prime }+1/{p}^{\prime }q-1/{p}^{\prime }}\omega {\left(Q\right)}^{1/{p}^{\prime }}{\parallel {\stackrel{\to }{b}}_{\sigma }\parallel }_{\mathit{BMO}}\hfill \\ \phantom{\rule{2em}{0ex}}×\omega {\left(Q\right)}^{-1/p}{\left({\int }_{Q}|{\left(b\left(x\right)-{b}_{Q}\right)}_{{\sigma }^{c}}{|}^{p{q}^{\prime }}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p{q}^{\prime }}{\left({\int }_{Q}|f\left(x\right){|}^{pq}{\omega }^{q}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/pq}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\parallel {\stackrel{\to }{b}}_{\sigma }\parallel }_{\mathit{BMO}}{\parallel {\stackrel{\to }{b}}_{{\sigma }^{c}}\parallel }_{\mathit{BMO}}{\left(\frac{|Q|}{\omega \left(Q\right)}\right)}^{1/p}\hfill \\ \phantom{\rule{2em}{0ex}}×{\left(\frac{1}{|Q|}{\int }_{Q}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}.\hfill \end{array}$
For ${J}_{3}\left(x\right)$, taking $p>1$, by the ${L}^{p}\left(\omega \right)$-boundedness of T, we have
$\begin{array}{c}\frac{1}{\omega \left(Q\right)}{\int }_{Q}|{J}_{3}\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\le {\left(\frac{1}{\omega \left(Q\right)}{\int }_{{R}^{n}}|T\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}-{\left({b}_{m}\right)}_{Q}\right){f}_{1}\right)\left(x\right){|}^{p}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\omega {\left(Q\right)}^{-1/p}{\left({\int }_{{R}^{n}}|\left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}\left(x\right)-{\left({b}_{m}\right)}_{Q}\right){f}_{1}\left(x\right){|}^{p}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\omega {\left(Q\right)}^{-1/p}{|Q|}^{1/p{q}^{\prime }}{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{|Q|}^{1/pq}{\left(\frac{1}{|Q|}{\int }_{Q}{\omega }^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/pq}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\left(\frac{|Q|}{\omega \left(Q\right)}\right)}^{1/p}{\left(\frac{\omega \left(Q\right)}{|Q|}\right)}^{1/p}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}.\hfill \end{array}$
For ${J}_{4}\left(x\right)$, ${J}_{5}\left(x\right)$, and ${J}_{6}\left(x\right)$, choose $1, $j=1,\dots ,m$, such that $1/p+1/{q}_{1}+\cdots +1/{q}_{m}+1/q$, by Lemma 2 and similar to the proofs of ${J}_{1}\left(x\right)$, ${J}_{2}\left(x\right)$, and ${J}_{3}\left(x\right)$, we get
$\begin{array}{c}\frac{1}{\omega \left(Q\right)}{\int }_{Q}|{J}_{4}\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\le C\frac{|Q|}{\omega \left(Q\right)}\prod _{j=1}^{m}{\left(\frac{1}{|Q|}{\int }_{Q}|\left({b}_{j}\left(x\right)-{\left({b}_{j}\right)}_{Q}\right){|}^{{q}_{j}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{q}_{j}}\hfill \\ \phantom{\rule{2em}{0ex}}×{\left(\frac{1}{|Q|}{\int }_{Q}|{A}_{{t}_{Q}}\left(f\right)\left(x\right){|}^{p}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}{\left(\frac{1}{|Q|}{\int }_{Q}\omega {\left(x\right)}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)},\hfill \\ \frac{1}{\omega \left(Q\right)}{\int }_{Q}|{J}_{5}\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\le C\frac{|Q|}{\omega \left(Q\right)}\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\left(\frac{1}{|Q|}{\int }_{Q}{|{\left(b\left(x\right)-{b}_{Q}\right)}_{\sigma }|}^{{q}^{\prime }}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{q}^{\prime }}\hfill \\ \phantom{\rule{2em}{0ex}}×{\left(\frac{1}{|Q|}{\int }_{Q}|{A}_{{t}_{Q}}\left({\left(b-{b}_{Q}\right)}_{{\sigma }^{c}}f\right)\left(x\right){|}^{p}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}{\left(\frac{1}{|Q|}{\int }_{Q}\omega {\left(x\right)}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\frac{|Q|}{\omega \left(Q\right)}\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\parallel {\stackrel{\to }{b}}_{\sigma }\parallel }_{\mathit{BMO}}{\parallel {\stackrel{\to }{b}}_{{\sigma }^{c}}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}\frac{\omega \left(Q\right)}{|Q|}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)},\hfill \\ \frac{1}{\omega \left(Q\right)}{\int }_{Q}|{J}_{6}\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\le C\frac{|Q|}{\omega \left(Q\right)}{\left(\frac{1}{|Q|}{\int }_{Q}|{A}_{{t}_{Q}}\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}-{\left({b}_{m}\right)}_{Q}\right){f}_{1}\right)\left(x\right){|}^{{q}^{\prime }}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{q}^{\prime }}\hfill \\ \phantom{\rule{2em}{0ex}}×{\left(\frac{1}{|Q|}{\int }_{Q}\omega {\left(x\right)}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}.\hfill \end{array}$
For ${J}_{7}\left(x\right)$, 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
$\begin{array}{rcl}{J}_{7}\left(x\right)& \le & C\sum _{k=0}^{\mathrm{\infty }}{\int }_{{2}^{k+1}Q\mathrm{\setminus }{2}^{k}Q}\prod _{j=1}^{m}|\left({b}_{j}\left(y\right)-{\left({b}_{j}\right)}_{Q}\right)||f\left(y\right)|\frac{{d}^{\delta }}{{|{x}_{0}-y|}^{n+\delta }}\phantom{\rule{0.2em}{0ex}}dy\\ \le & C\sum _{k=1}^{\mathrm{\infty }}\frac{{d}^{\delta }}{{\left({2}^{k-1}d\right)}^{n+\delta }}|{2}^{k}Q|{\left(\frac{1}{|{2}^{k}Q|}{\int }_{{2}^{k}Q}|f\left(y\right){|}^{r}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/r}\\ ×\prod _{j=1}^{m}{\left(\frac{1}{|{2}^{k}Q|}{\int }_{{2}^{k}Q}|{b}_{j}\left(y\right)-{\left({b}_{j}\right)}_{Q}{|}^{{q}_{j}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/{q}_{j}}\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{2}^{-k\delta }{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}\prod _{j=1}^{m}{\left(\frac{1}{|{2}^{k}Q|}{\int }_{{2}^{k}Q}|{b}_{j}\left(y\right)-{\left({b}_{j}\right)}_{Q}{|}^{{q}_{j}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/{q}_{j}}\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{k}^{m}{2}^{-k\delta }\prod _{j=1}^{m}{\parallel {b}_{j}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}\\ \le & C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)},\end{array}$
so
$\frac{1}{\omega \left(Q\right)}{\int }_{Q}|{J}_{7}\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{\mathrm{\infty }}\left(\omega \right)}.$

This completes the proof of Theorem 1. □

Theorem 2 Let $1, $\omega \in {A}_{1}$ and $\stackrel{\to }{b}=\left({b}_{1},\dots ,{b}_{m}\right)$ with ${b}_{j}\in \mathit{BMO}\left({R}^{n}\right)$ for $1\le j\le m$. Then ${T}_{b}$ is bounded from ${B}_{p}\left(\omega \right)$ to ${\mathit{CMO}}_{A}\left(\omega \right)$.

Proof It suffices to prove for $f\in {C}_{0}^{\mathrm{\infty }}\left({R}^{n}\right)$, the following inequality holds:
$\frac{1}{\omega \left(Q\right)}{\int }_{Q}|{T}_{b}\left(f\right)\left(x\right)-{A}_{{t}_{Q}}{T}_{b}\left(f\right)\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\le C{\parallel f\parallel }_{{B}_{p}\left(\omega \right)}$
for any cube $Q=Q\left(0,d\right)$ with $d>1$. Fix a cube $Q=Q\left(0,d\right)$ with $d>1$. Set ${f}_{1}=f{\chi }_{Q}$, ${f}_{2}=f{\chi }_{\left({R}^{n}\mathrm{\setminus }Q\right)}$ and ${\stackrel{\to }{b}}_{Q}=\left({\left({b}_{1}\right)}_{Q},\dots ,{\left({b}_{m}\right)}_{Q}\right)\in {R}^{n}$, where ${\left({b}_{j}\right)}_{Q}={|Q|}^{-1}{\int }_{Q}|{b}_{j}\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy$, $1\le j\le m$, we have
$\begin{array}{c}|{T}_{b}\left(f\right)\left(x\right)-{A}_{{t}_{Q}}{T}_{b}\left(f\right)\left(x\right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le |\prod _{j=1}^{m}\left({b}_{j}\left(x\right)-{\left({b}_{j}\right)}_{Q}\right){\int }_{{R}^{n}}K\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\hfill \\ \phantom{\rule{2em}{0ex}}+|\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\left(b\left(x\right)-{\left(b\right)}_{Q}\right)}_{\sigma }{\int }_{{R}^{n}}{\left(b\left(y\right)-{\left(b\right)}_{Q}\right)}_{{\sigma }^{c}}K\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\hfill \\ \phantom{\rule{2em}{0ex}}+|{\int }_{{R}^{n}}\prod _{j=1}^{m}\left({b}_{j}\left(y\right)-{\left({b}_{j}\right)}_{Q}\right)K\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\hfill \\ \phantom{\rule{2em}{0ex}}+|\prod _{j=1}^{m}\left({b}_{j}\left(x\right)-{\left({b}_{j}\right)}_{Q}\right){\int }_{{R}^{n}}{K}_{t}\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\hfill \\ \phantom{\rule{2em}{0ex}}+|\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\left(b\left(x\right)-{\left(b\right)}_{Q}\right)}_{\sigma }{\int }_{{R}^{n}}{\left(b\left(y\right)-{\left(b\right)}_{Q}\right)}_{{\sigma }^{c}}{K}_{t}\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\hfill \\ \phantom{\rule{2em}{0ex}}+|{\int }_{{R}^{n}}\prod _{j=1}^{m}\left({b}_{j}\left(y\right)-{\left({b}_{j}\right)}_{Q}\right){K}_{t}\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\hfill \\ \phantom{\rule{2em}{0ex}}+|{\int }_{{R}^{n}}\prod _{j=1}^{m}\left({b}_{j}\left(y\right)-{\left({b}_{j}\right)}_{Q}\right)\left(K\left(x,y\right)-{K}_{t}\left(x,y\right)\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\hfill \\ \phantom{\rule{1em}{0ex}}={L}_{1}\left(x\right)+{L}_{2}\left(x\right)+{L}_{3}\left(x\right)+{L}_{4}\left(x\right)+{L}_{5}\left(x\right)+{L}_{6}\left(x\right)+{L}_{7}\left(x\right).\hfill \end{array}$
For ${L}_{1}\left(x\right)$, we have
$\begin{array}{c}\frac{1}{\omega \left(Q\right)}{\int }_{Q}|{L}_{1}\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{C}{\omega \left(Q\right)}{\left({\int }_{Q}|\left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}\left(x\right)-{\left({b}_{m}\right)}_{Q}\right){|}^{{p}^{\prime }}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{p}^{\prime }}\hfill \\ \phantom{\rule{2em}{0ex}}×{\left({\int }_{Q}|T\left(f\right)\left(x\right){|}^{p}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{C}{\omega \left(Q\right)}{\left[{\left({\int }_{Q}|\left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}\left(x\right)-{\left({b}_{m}\right)}_{Q}\right){|}^{{p}^{\prime }{q}^{\prime }}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{q}^{\prime }}{\left({\int }_{Q}\omega {\left(x\right)}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}\right]}^{1/{p}^{\prime }}\hfill \\ \phantom{\rule{2em}{0ex}}×{\left({\int }_{Q}|f\left(x\right){|}^{p}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{C}{\omega \left(Q\right)}{|Q|}^{1/{p}^{\prime }{q}^{\prime }}{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{|Q|}^{1/{p}^{\prime }q}{\left(\frac{\omega \left(Q\right)}{|Q|}\right)}^{1/{p}^{\prime }}{\parallel f{\chi }_{Q}\parallel }_{{L}^{p}\left(\omega \right)}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}\omega {\left(Q\right)}^{-1/p}{\parallel f{\chi }_{Q}\parallel }_{{L}^{p}\left(\omega \right)}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{B}_{p}\left(\omega \right)}.\hfill \end{array}$
For ${L}_{2}\left(x\right)$, taking $1, and $1/s+1/{s}^{\prime }=1$, we have
$\begin{array}{c}\frac{1}{\omega \left(Q\right)}{\int }_{Q}|{L}_{2}\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\le C\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\left(\frac{1}{\omega \left(Q\right)}{\int }_{Q}|{\left(b\left(x\right)-{b}_{Q}\right)}_{\sigma }{|}^{{s}^{\prime }}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{s}^{\prime }}\hfill \\ \phantom{\rule{2em}{0ex}}×{\left(\frac{1}{\omega \left(Q\right)}{\int }_{Q}|T\left({\left(b-{b}_{Q}\right)}_{{\sigma }^{c}}f\right)\left(x\right){|}^{s}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/s}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}\omega {\left(Q\right)}^{-1/{s}^{\prime }}{\left[{\left({\int }_{Q}|{\left(b\left(x\right)-{b}_{Q}\right)}_{\sigma }{|}^{{s}^{\prime }{q}^{\prime }}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{q}^{\prime }}{\left({\int }_{Q}{\omega }^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}\right]}^{1/{s}^{\prime }}\hfill \\ \phantom{\rule{2em}{0ex}}×\omega {\left(Q\right)}^{-1/s}{\left({\int }_{Q}|{\left(b\left(x\right)-{b}_{Q}\right)}_{{\sigma }^{c}}f\left(x\right){|}^{s}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/s}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}\omega {\left(Q\right)}^{-1/{s}^{\prime }}{|Q|}^{1/{s}^{\prime }{q}^{\prime }+1/{s}^{\prime }q-1/{s}^{\prime }}\omega {\left(Q\right)}^{1/{s}^{\prime }}{\parallel {\stackrel{\to }{b}}_{\sigma }\parallel }_{\mathit{BMO}}\hfill \\ \phantom{\rule{2em}{0ex}}×\omega {\left(Q\right)}^{-1/s}{|Q|}^{1/rs}{\parallel {\stackrel{\to }{b}}_{{\sigma }^{c}}\parallel }_{\mathit{BMO}}{\left({\int }_{Q}|f\left(x\right){|}^{p}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}{\left({\int }_{Q}\omega {\left(x\right)}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\left(p-s\right)/pqs}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\parallel {\stackrel{\to }{b}}_{\sigma }\parallel }_{\mathit{BMO}}{\parallel {\stackrel{\to }{b}}_{{\sigma }^{c}}\parallel }_{\mathit{BMO}}\omega {\left(Q\right)}^{-1/p}{\parallel f{\chi }_{Q}\parallel }_{{L}^{p}\left(\omega \right)}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{B}_{p}\left(\omega \right)}.\hfill \end{array}$
For ${L}_{3}\left(x\right)$, we have
$\begin{array}{c}\frac{1}{\omega \left(Q\right)}{\int }_{Q}|{L}_{3}\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\left(\frac{1}{\omega \left(Q\right)}{\int }_{{R}^{n}}|T\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}-{\left({b}_{m}\right)}_{Q}\right){f}_{1}\right)\left(x\right){|}^{s}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/s}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\omega {\left(Q\right)}^{-1/s}{\left({\int }_{Q}|\left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}\left(x\right)-{\left({b}_{m}\right)}_{Q}\right)f\left(x\right){|}^{s}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/s}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\omega {\left(Q\right)}^{-1/p}{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel f{\chi }_{Q}\parallel }_{{L}^{p}\left(\omega \right)}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{B}_{p}\left(\omega \right)}.\hfill \end{array}$
For ${L}_{4}\left(x\right)$, ${L}_{5}\left(x\right)$, and ${L}_{6}\left(x\right)$, by Lemma 2, we have
$\begin{array}{c}\frac{1}{\omega \left(Q\right)}{\int }_{Q}|{L}_{4}\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\left(\frac{1}{\omega \left(Q\right)}{\int }_{Q}|\left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}\left(x\right)-{\left({b}_{m}\right)}_{Q}\right){|}^{{s}^{\prime }}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{s}^{\prime }}\hfill \\ \phantom{\rule{2em}{0ex}}×{\left(\frac{1}{\omega \left(Q\right)}{\int }_{Q}|{A}_{{t}_{Q}}\left(f\right)\left(x\right){|}^{s}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/s}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\left(\frac{1}{\omega \left(Q\right)}\right)}^{1/{s}^{\prime }}\left[{\left({\int }_{Q}|\left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}\left(x\right)-{\left({b}_{m}\right)}_{Q}\right){|}^{{s}^{\prime }{q}^{\prime }}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{q}^{\prime }}\hfill \\ \phantom{\rule{2em}{0ex}}{×{\left({\int }_{Q}\omega {\left(x\right)}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}\right]}^{1/{s}^{\prime }}{\parallel f\parallel }_{{B}_{p}\left(w\right)}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\left(\frac{1}{\omega \left(Q\right)}\right)}^{1/{s}^{\prime }}{|Q|}^{1/{s}^{\prime }{q}^{\prime }}{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{|Q|}^{1/{s}^{\prime }q}{\left(\frac{\omega \left(Q\right)}{|Q|}\right)}^{1/{s}^{\prime }}{\parallel f\parallel }_{{B}_{p}\left(w\right)}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{B}_{p}\left(\omega \right)};\hfill \\ \frac{1}{\omega \left(Q\right)}{\int }_{Q}|{L}_{5}\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\le C\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\left(\frac{1}{\omega \left(Q\right)}{\int }_{Q}|{\left(b\left(x\right)-{b}_{Q}\right)}_{\sigma }{|}^{{s}^{\prime }}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{s}^{\prime }}\hfill \\ \phantom{\rule{2em}{0ex}}×{\left(\frac{1}{\omega \left(Q\right)}{\int }_{Q}|{A}_{{t}_{Q}}\left({\left(b-{b}_{Q}\right)}_{{\sigma }^{c}}f\right)\left(x\right){|}^{s}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/s}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}\omega {\left(Q\right)}^{-1/{s}^{\prime }}{\left[{\left({\int }_{Q}|{\left(b\left(x\right)-{b}_{Q}\right)}_{\sigma }{|}^{{s}^{\prime }{q}^{\prime }}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{q}^{\prime }}{\left({\int }_{Q}{\omega }^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}\right]}^{1/{s}^{\prime }}\hfill \\ \phantom{\rule{2em}{0ex}}×{\parallel {\stackrel{\to }{b}}_{{\sigma }^{c}}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{B}_{p}\left(\omega \right)}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}\omega {\left(Q\right)}^{-1/{s}^{\prime }}{|Q|}^{1/{s}^{\prime }{q}^{\prime }+1/{s}^{\prime }q-1/{s}^{\prime }}\omega {\left(Q\right)}^{1/{s}^{\prime }}{\parallel {\stackrel{\to }{b}}_{\sigma }\parallel }_{\mathit{BMO}}{\parallel {\stackrel{\to }{b}}_{{\sigma }^{c}}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{B}_{p}\left(\omega \right)}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\parallel {\stackrel{\to }{b}}_{\sigma }\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{B}_{p}\left(\omega \right)};\hfill \\ \frac{1}{\omega \left(Q\right)}{\int }_{Q}|{L}_{6}\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\le {\left(\frac{1}{\omega \left(Q\right)}{\int }_{{R}^{n}}|{A}_{{t}_{Q}}\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}-{\left({b}_{m}\right)}_{Q}\right){f}_{1}\right)\left(x\right){|}^{s}\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/s}\hfill \\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{B}_{p}\left(\omega \right)}.\hfill \end{array}$
For ${L}_{7}\left(x\right)$, note that $|x-y|\ge d={t}^{1/2}$, taking $1, then
$\begin{array}{rcl}{L}_{7}\left(x\right)& \le & C{\int }_{{Q}^{c}}\prod _{j=1}^{m}|{b}_{j}\left(y\right)-{\left({b}_{j}\right)}_{Q}||f\left(y\right)|\frac{{d}^{\delta }}{{|{x}_{0}-y|}^{n+\delta }}\phantom{\rule{0.2em}{0ex}}dy\\ \le & C\sum _{k=0}^{\mathrm{\infty }}{\int }_{{2}^{k+1}Q\mathrm{\setminus }{2}^{k}Q}\prod _{j=1}^{m}|{b}_{j}\left(y\right)-{\left({b}_{j}\right)}_{Q}||f\left(y\right)|\frac{{d}^{\delta }}{{|{x}_{0}-y|}^{n+\delta }}\phantom{\rule{0.2em}{0ex}}dy\\ \le & C\sum _{k=1}^{\mathrm{\infty }}\frac{{d}^{\delta }}{{\left({2}^{k-1}d\right)}^{n+\delta }}|{2}^{k}Q|{\left(\frac{1}{|{2}^{k}Q|}{\int }_{{2}^{k}Q}|f\left(y\right){|}^{u}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/u}\\ ×{\left(\frac{1}{|{2}^{k}Q|}{\int }_{{2}^{k}Q}\prod _{j=1}^{m}|{b}_{j}\left(y\right)-{\left({b}_{j}\right)}_{Q}{|}^{{u}^{\prime }}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/{u}^{\prime }}\\ \le & C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}\sum _{k=1}^{\mathrm{\infty }}{k}^{m}{2}^{-k\delta }{\left(\frac{1}{|{2}^{k}Q|}\right)}^{1/u}\\ ×{\left[{\left({\int }_{{2}^{k}Q}|f\left(y\right){|}^{p}\omega \left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{\frac{u}{p}}{\left({\int }_{{2}^{k}Q}\omega {\left(y\right)}^{-\frac{u}{p-u}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{\frac{p-u}{p}}\right]}^{1/u}\\ \le & C{\parallel \stackrel{\to }{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}\left(\omega \right)}{\left(\frac{\omega \left({2}^{k}Q\right)}{|{2}^{k}Q|}\right)}^{-1/p}|{2}^{k}Q{|}^{\left(\frac{p}{u}-1\right)\frac{1}{p}}\\ ×{\left[\left(\frac{1}{|{2}^{k}Q|}{\int }_{{2}^{k}Q}\omega \left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right){\left(\frac{1}{|{2}^{k}Q|}{\int }_{{2}^{k}Q}\omega {\left(y\right)}^{-\frac{1}{\frac{p}{u}-1}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{\frac{p}{u}-1}\right]}^{1/p}\\ \le & C{\parallel \stackrel{\to }{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}\left(\omega \right)}\\ \le & C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{B}_{p}\left(\omega \right)},\end{array}$
so
$\frac{1}{\omega \left(Q\right)}{\int }_{Q}|{L}_{7}\left(x\right)|\omega \left(x\right)\phantom{\rule{0.2em}{0ex}}dx\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{B}_{p}\left(\omega \right)}.$

This completes the proof of Theorem 2. □

## Declarations

### Acknowledgements

Project was supported by the National Natural Science Foundation of China (No. 11061003).

## Authors’ Affiliations

(1)
College of Science, Guangxi University of Science and Technology, Liuzhou, 545006, P.R. China

## References 