## 1 Introduction

As the development of the singular integral operators, their commutators and multilinear operators have been well studied (see [15]). In [13, 57], the authors proved that the commutators and multilinear operators generated by the singular integral operators and BMO functions are bounded on ${L}^{p}\left({R}^{n}\right)$ for $1. Chanillo (see [8]) proved a similar result when singular integral operators are replaced by the fractional integral operators. In [912], the boundedness for the commutators and multilinear operators generated by the singular integral operators and Lipschitz functions on ${L}^{p}\left({R}^{n}\right)$ ($1) and Triebel-Lizorkin spaces are obtained. In [13, 14], the weighted boundedness for the commutators generated by the singular integral operators and BMO or Lipschitz functions on ${L}^{p}\left({R}^{n}\right)$ ($1) spaces are obtained. The purpose of this paper is to study the weighted boundedness for some multilinear operators associated to the fractional singular integral operators and the weighted Lipschitz functions. As application, the weighted boundedness for the multilinear operators associated to the Calderón-Zygmund singular integral operator and the fractional integral operator is obtained.

## 2 Notations and theorems

In this paper, we are going to consider some multilinear operators as follows.

Let ${m}_{j}$ be positive integers ($j=1,\dots ,k$), ${m}_{1}+\cdots +{m}_{k}=m$, and let ${b}_{j}$ be locally integrable functions on ${R}^{n}$ ($j=1,\dots ,k$). Set

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

Definition 1 Let $T:S\to {S}^{\prime }$ be a linear operator, and there exists a locally integrable function $K\left(x,y\right)$ on ${R}^{n}×{R}^{n}$ such that

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

for every bounded and compactly supported function f, where K satisfies: for fixed $0\le \delta and $0<\epsilon \le 1$,

$|K\left(x,y\right)|\le C{|x-y|}^{-n+\delta },$
(1)

and

(2)

Given bounded and compactly supported functions f defined on ${R}^{n}$, the multilinear operator associated to T is defined by

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

Note that when $m=0$, ${T}^{b}$ is just a multilinear commutator of T and b (see [15, 16]), while when $m>0$, it is non-trivial generalization of the commutators; when $\eta =0$, ${T}^{b}$ is just a multilinear commutator of the singular integral operator, when $0\le \eta , ${T}^{b}$ is just a multilinear commutator of the fractional integral operator. It is well known that multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see [14, 6, 7, 9, 10, 15, 16]). The purpose of this paper is to study the weighted boundedness properties for the multilinear operator.

Throughout this paper, Q will denote a cube of ${R}^{n}$ with sides parallel to the axes. For a cube Q and a locally integrable function f, let $f\left(Q\right)={\int }_{Q}f\left(x\right)\phantom{\rule{0.2em}{0ex}}dx$, ${f}_{Q}={|Q|}^{-1}{\int }_{Q}f\left(x\right)\phantom{\rule{0.2em}{0ex}}dx$ and

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

It is well known that (see [17, 18])

${f}^{\mathrm{#}}\left(x\right)\approx \underset{Q\ni x}{sup}\underset{c\in C}{inf}\frac{1}{|Q|}{\int }_{Q}|f\left(y\right)-c|\phantom{\rule{0.2em}{0ex}}dy.$

For $1\le p<\mathrm{\infty }$ and $0\le \eta , let

${M}_{\eta ,p}\left(f\right)\left(x\right)=\underset{Q\ni x}{sup}{\left(\frac{1}{{|Q|}^{1-p\eta /n}}{\int }_{Q}{|f\left(y\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p},$

which is the Hardy-Littlewood maximal function when $p=1$ and $\eta =0$.

The ${A}_{p}$ weight is defined by (see [17])

$\begin{array}{r}{A}_{p}=\left\{w:\underset{Q}{sup}\left(\frac{1}{|Q|}{\int }_{Q}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right){\left(\frac{1}{|Q|}{\int }_{Q}w{\left(x\right)}^{-1/\left(p-1\right)}\phantom{\rule{0.2em}{0ex}}dx\right)}^{p-1}<\mathrm{\infty }\right\},\phantom{\rule{1em}{0ex}}10:M\left(w\right)\left(x\right)\le Cw\left(x\right),\text{a.e.}\right\},\end{array}$

and ${A}_{\mathrm{\infty }}={\bigcup }_{p\ge 1}{A}_{p}$. We know, for $w\in {A}_{1}$, w satisfies the double condition, that is, for any cube Q,

$w\left(2Q\right)\le Cw\left(Q\right).$

The $A\left(p,q\right)$ weight is defined by (see [19])

$\begin{array}{rcl}A\left(p,q\right)& =& \left\{w>0:\underset{Q}{sup}{\left(\frac{1}{|Q|}{\int }_{Q}w{\left(x\right)}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}{\left(\frac{1}{|Q|}{\int }_{Q}w{\left(x\right)}^{-p/\left(p-1\right)}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\left(p-1\right)/p}<\mathrm{\infty }\right\},\\ 1

Given a weight function w. For $1, the weighted Lebesgue space ${L}^{p}\left(w\right)$ is the space of functions f such that

${\parallel f\parallel }_{{L}^{p}\left(w\right)}={\left({\int }_{{R}^{n}}{|f\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}<\mathrm{\infty }.$

For $\beta >0$ and $p>1$, let ${\stackrel{˙}{F}}_{p}^{\beta ,\mathrm{\infty }}\left(w\right)$ be the weighted homogeneous Triebel-Lizorkin space. For $0<\beta <1$, the weighted Lipschitz space ${Lip}_{\beta }\left(w\right)$ is the space of functions f such that

${\parallel f\parallel }_{{Lip}_{\beta }\left(w\right)}=\underset{Q}{sup}\frac{1}{w{\left(Q\right)}^{1+\beta /n}}{\int }_{Q}|f\left(y\right)-{f}_{Q}|\phantom{\rule{0.2em}{0ex}}dy<\mathrm{\infty }.$

We shall prove the following theorems in Section 3.

Theorem 1 Suppose that ${T}^{b}$ is the multilinear operator as Definition 1 such that T is bounded from ${L}^{u}\left({R}^{n}\right)$ to ${L}^{v}\left({R}^{n}\right)$ for any $0\le \eta , $1 and $1/u-1/v=\eta /n$. Let $0<\beta \le 1$, $\beta +\delta , $w\in {A}_{1}$ and ${D}^{\alpha }{b}_{j}\in {Lip}_{\beta }\left(w\right)$ for all α with $|\alpha |={m}_{j}$ and $j=1,\dots ,k$. Then ${T}^{b}$ is bounded from ${L}^{p}\left(w\right)$ to ${L}^{q}\left({w}^{1-q\left(k-\left(\delta +k\right)/n\right)}\right)$ for any $1 and $1/p-1/q=\left(\delta +k+k\beta \right)/n$.

Theorem 2 Suppose that ${T}^{b}$ is the multilinear operator as Definition 1 such that T is bounded from ${L}^{u}\left({R}^{n}\right)$ to ${L}^{v}\left({R}^{n}\right)$ for any $0\le \eta , $1 and $1/u-1/v=\eta /n$. Let $0<\beta , $\beta +\delta , $w\in {A}_{1}$ and ${D}^{\alpha }{b}_{j}\in {Lip}_{\beta }\left(w\right)$ for all α with $|\alpha |={m}_{j}$ and $j=1,\dots ,k$. Then ${T}^{b}$ is bounded from ${L}^{p}\left(w\right)$ to ${\stackrel{˙}{F}}_{q}^{k\beta ,\mathrm{\infty }}\left({w}^{1-q\left(k-\left(\delta +k-k\beta \right)/n\right)}\right)$ for any $1 and $1/p-1/q=\left(\delta +k\right)/n$.

## 3 Proofs of theorems

We begin with some preliminary lemmas.

Lemma 1 (see [1])

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

$|{R}_{m}\left(b;x,y\right)|\le C{|x-y|}^{m}\sum _{|\alpha |=m}{\left(\frac{1}{|\stackrel{˜}{Q}\left(x,y\right)|}{\int }_{\stackrel{˜}{Q}\left(x,y\right)}{|{D}^{\alpha }b\left(z\right)|}^{q}\phantom{\rule{0.2em}{0ex}}dz\right)}^{1/q},$

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

Lemma 2 (see [11, 12])

For $0<\beta <1$, $1 and $w\in {A}_{\mathrm{\infty }}$, we have

$\begin{array}{rcl}{\parallel f\parallel }_{{\stackrel{˙}{F}}_{p}^{\beta ,\mathrm{\infty }}\left(w\right)}& \approx & {\parallel \underset{Q\ni \cdot }{sup}\frac{1}{{|Q|}^{1+\beta /n}}{\int }_{Q}|f\left(x\right)-{f}_{Q}|\phantom{\rule{0.2em}{0ex}}dx\parallel }_{{L}^{p}\left(w\right)}\\ \approx & {\parallel \underset{Q\ni \cdot }{sup}\underset{c}{inf}\frac{1}{{|Q|}^{1+\beta /n}}{\int }_{Q}|f\left(x\right)-c|\phantom{\rule{0.2em}{0ex}}dx\parallel }_{{L}^{p}\left(w\right)}.\end{array}$

Lemma 3 (see [8])

Suppose that $1\le s, $1/q=1/p-\eta /n$ and $w\in A\left(p,q\right)$. Then

${\parallel {M}_{\eta ,s}\left(f\right)\parallel }_{{L}^{q}\left({w}^{q}\right)}\le C{\parallel f\parallel }_{{L}^{p}\left({w}^{p}\right)}.$

Lemma 4 (see [17, 20])

For $0<\beta <1$, $1\le p\le \mathrm{\infty }$ and $w\in {A}_{1}$, we have

${\parallel b\parallel }_{{Lip}_{\beta }\left(w\right)}\approx \underset{Q}{sup}\frac{1}{w{\left(Q\right)}^{\beta /n}}{\left(\frac{1}{w\left(Q\right)}{\int }_{Q}{|b\left(x\right)-{b}_{Q}|}^{p}w{\left(y\right)}^{1-p}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}.$

Lemma 5 (see [20])

For any cube Q, $b\in {Lip}_{\beta }\left(w\right)$, $0<\beta <1$ and $w\in {A}_{1}$, we have

$\underset{x\in Q}{sup}|b\left(x\right)-{b}_{Q}|\le C{\parallel b\parallel }_{{Lip}_{\beta }\left(w\right)}w{\left(Q\right)}^{1+\beta /n}{|Q|}^{-1}.$

To prove the theorems, we need the following lemmas.

Key Lemma 1 Suppose that ${T}^{b}$ is the multilinear operator as Definition 1 such that T is bounded from ${L}^{u}\left({R}^{n}\right)$ to ${L}^{v}\left({R}^{n}\right)$ for any $0\le \eta , $1 and $1/u-1/v=\eta /n$. Let $0<\beta \le 1$, $\beta +\delta , $w\in {A}_{1}$ and ${D}^{\alpha }{b}_{j}\in {Lip}_{\beta }\left(w\right)$ for all α with $|\alpha |={m}_{j}$ and $j=1,\dots ,k$. Then there exists a constant ${C}_{0}$ such that for every $f\in {C}_{0}^{\mathrm{\infty }}\left({R}^{n}\right)$, $1 and $\stackrel{˜}{x}\in {R}^{n}$,

$\begin{array}{r}\underset{Q\ni \stackrel{˜}{x}}{sup}\frac{1}{|Q|}{\int }_{Q}|{T}^{b}\left(f\right)\left(x\right)-{C}_{0}|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C\prod _{j=1}^{k}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)w{\left(\stackrel{˜}{x}\right)}^{k+k\beta /n}{M}_{\delta +k+k\beta ,s}\left(f\right)\left(\stackrel{˜}{x}\right).\end{array}$

Key Lemma 2 Suppose that ${T}^{b}$ is the multilinear operator as Definition 1 such that T is bounded from ${L}^{u}\left({R}^{n}\right)$ to ${L}^{v}\left({R}^{n}\right)$ for any $0\le \eta , $1 and $1/u-1/v=\eta /n$. Let $0<\beta , $\beta +\delta , $w\in {A}_{1}$ and ${D}^{\alpha }{b}_{j}\in {Lip}_{\beta }\left(w\right)$ for all α with $|\alpha |={m}_{j}$ and $j=1,\dots ,k$. Then there exists a constant ${C}_{0}$ such that for every $f\in {C}_{0}^{\mathrm{\infty }}\left({R}^{n}\right)$, $1 and $\stackrel{˜}{x}\in {R}^{n}$,

$\begin{array}{r}\underset{Q\ni \stackrel{˜}{x}}{sup}\frac{1}{{|Q|}^{1+k\beta /n}}{\int }_{Q}|{T}^{b}\left(f\right)\left(x\right)-{C}_{0}|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C\prod _{j=1}^{k}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)w{\left(\stackrel{˜}{x}\right)}^{k+k\beta /n}{M}_{\delta +k,s}\left(f\right)\left(\stackrel{˜}{x}\right).\end{array}$

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

$\begin{array}{rcl}{T}^{b}\left(f\right)\left(x\right)& =& {\int }_{{R}^{n}}\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,y\right)}{{|x-y|}^{m-2}}K\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ -\sum _{|{\alpha }_{1}|={m}_{1}}\frac{1}{{\alpha }_{1}!}{\int }_{{R}^{n}}\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};x,y\right){\left(x-y\right)}^{{\alpha }_{1}}}{{|x-y|}^{m-2}}{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right)K\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ -\sum _{|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{2}!}{\int }_{{R}^{n}}\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};x,y\right){\left(x-y\right)}^{{\alpha }_{2}}}{{|x-y|}^{m-2}}{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right)K\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ +\sum _{|{\alpha }_{1}|={m}_{1},\phantom{\rule{0.25em}{0ex}}|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{1}!{\alpha }_{2}!}{\int }_{{R}^{n}}\frac{{\left(x-y\right)}^{{\alpha }_{1}+{\alpha }_{2}}{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right){D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right)}{{|x-y|}^{m-2}}K\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ +{\int }_{{R}^{n}}\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}+1}\left({\stackrel{˜}{b}}_{j};x,y\right)}{{|x-y|}^{m-2}}K\left(x,y\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy,\end{array}$

then

$\begin{array}{r}\frac{1}{|Q|}{\int }_{Q}|{T}^{b}\left(f\right)\left(x\right)-{T}^{\stackrel{˜}{b}}\left({f}_{2}\right)\left({x}_{0}\right)|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{|Q|}{\int }_{Q}|{\int }_{{R}^{n}}\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,y\right)}{{|x-y|}^{m-2}}K\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+\frac{C}{|Q|}{\int }_{Q}|\sum _{|{\alpha }_{1}|={m}_{1}}{\int }_{{R}^{n}}\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};x,y\right){\left(x-y\right)}^{{\alpha }_{1}}}{{|x-y|}^{m-2}}{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right)K\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+\frac{C}{|Q|}{\int }_{Q}|\sum _{|{\alpha }_{2}|={m}_{2}}{\int }_{{R}^{n}}\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};x,y\right){\left(x-y\right)}^{{\alpha }_{2}}}{{|x-y|}^{m-2}}{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right)K\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+\frac{C}{|Q|}{\int }_{Q}|\sum _{|{\alpha }_{1}|={m}_{1},\phantom{\rule{0.25em}{0ex}}|{\alpha }_{2}|={m}_{2}}{\int }_{{R}^{n}}\frac{{\left(x-y\right)}^{{\alpha }_{1}+{\alpha }_{2}}{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right){D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right)}{{|x-y|}^{m-2}}K\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+\frac{1}{|Q|}{\int }_{Q}|{T}^{\stackrel{˜}{b}}\left({f}_{2}\right)\left(x\right)-{T}^{\stackrel{˜}{b}}\left({f}_{2}\right)\left({x}_{0}\right)|\phantom{\rule{0.2em}{0ex}}dx:={I}_{1}+{I}_{2}+{I}_{3}+{I}_{4}+{I}_{5}.\end{array}$

Now, let us estimate ${I}_{1}$, ${I}_{2}$, ${I}_{3}$, ${I}_{4}$ and ${I}_{5}$, respectively. First, by Lemmas 1 and 5, we get

$\begin{array}{rl}|{R}_{m}\left({\stackrel{˜}{b}}_{j};x,y\right)|& \le C{|x-y|}^{m}\sum _{|\alpha |=m}\underset{x\in \stackrel{˜}{Q}}{sup}|{D}^{\alpha }{b}_{j}\left(x\right)-{\left({D}^{\alpha }{b}_{j}\right)}_{\stackrel{˜}{Q}}|\\ \le C{|x-y|}^{m}\frac{w{\left(\stackrel{˜}{Q}\right)}^{1+\beta /n}}{|Q|}\sum _{|\alpha |=m}{\parallel {D}^{\alpha }{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}.\end{array}$

Thus, by the $\left({L}^{s},{L}^{t}\right)$-boundedness of T with $1 and $1/t=1/s-\delta /n$, we obtain

$\begin{array}{rcl}{I}_{1}& \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)\left(\frac{{|Q|}^{2/n}w{\left(\stackrel{˜}{Q}\right)}^{2+2\beta /n}}{{|Q|}^{2}}\right)\frac{1}{|Q|}{\int }_{Q}|T\left({f}_{1}\right)\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)\left(\frac{{|Q|}^{2/n}w{\left(\stackrel{˜}{Q}\right)}^{2+2\beta /n}}{{|Q|}^{2}}\right){\left(\frac{1}{|Q|}{\int }_{{R}^{n}}{|T\left({f}_{1}\right)\left(x\right)|}^{t}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/t}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)\left(\frac{{|Q|}^{2/n}w{\left(\stackrel{˜}{Q}\right)}^{2+2\beta /n}}{{|Q|}^{2}}\right){|Q|}^{-1/t}{\left({\int }_{{R}^{n}}{|{f}_{1}\left(x\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/s}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right){\left(\frac{w\left(\stackrel{˜}{Q}\right)}{|\stackrel{˜}{Q}|}\right)}^{2+2\beta /n}{\left(\frac{1}{{|\stackrel{˜}{Q}|}^{1-s\left(\delta +2+2\beta \right)/n}}{\int }_{\stackrel{˜}{Q}}{|f\left(y\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/s}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)w{\left(\stackrel{˜}{x}\right)}^{2+2\beta /n}{M}_{\delta +2+2\beta ,s}\left(f\right)\left(\stackrel{˜}{x}\right).\end{array}$

For ${I}_{2}$, by the $\left({L}^{s},{L}^{t}\right)$-boundedness of T with $1, $1/t=1/s-\delta /n$ and Lemma 5, we get

$\begin{array}{rcl}{I}_{2}& \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{b}_{2}\parallel }_{{Lip}_{\beta }\left(w\right)}\frac{{|Q|}^{2/n}w{\left(\stackrel{˜}{Q}\right)}^{1+\beta /n}}{|Q|}\sum _{|{\alpha }_{1}|={m}_{1}}\frac{1}{|Q|}{\int }_{Q}|T\left({D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}{f}_{1}\right)\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx\\ \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{b}_{2}\parallel }_{{Lip}_{\beta }\left(w\right)}\frac{{|Q|}^{2/n}w{\left(\stackrel{˜}{Q}\right)}^{1+\beta /n}}{|Q|}\\ ×\sum _{|{\alpha }_{1}|={m}_{1}}{\left(\frac{1}{|Q|}{\int }_{{R}^{n}}{|T\left({D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}{f}_{1}\right)\left(x\right)|}^{t}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/t}\\ \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{b}_{2}\parallel }_{{Lip}_{\beta }\left(w\right)}\frac{{|Q|}^{2/n}w{\left(\stackrel{˜}{Q}\right)}^{1+\beta /n}}{|Q|}{|Q|}^{-1/t}\\ ×\sum _{|{\alpha }_{1}|={m}_{1}}{\left({\int }_{{R}^{n}}{|{D}^{{\alpha }_{1}}{b}_{1}\left(x\right)-{\left({D}^{\alpha }{b}_{1}\right)}_{\stackrel{˜}{Q}}|}^{s}{|{f}_{1}\left(x\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/s}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)\frac{{|Q|}^{2/n}w{\left(\stackrel{˜}{Q}\right)}^{2+2\beta /n}}{{|Q|}^{2}}{|Q|}^{-1/t}{\left(\frac{1}{|\stackrel{˜}{Q}|}{\int }_{\stackrel{˜}{Q}}{|f\left(y\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/s}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right){\left(\frac{w\left(\stackrel{˜}{Q}\right)}{|\stackrel{˜}{Q}|}\right)}^{2+2\beta /n}{\left(\frac{1}{{|\stackrel{˜}{Q}|}^{1-s\left(\delta +2+2\beta \right)/n}}{\int }_{\stackrel{˜}{Q}}{|f\left(y\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/s}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)w{\left(\stackrel{˜}{x}\right)}^{2+2\beta /n}{M}_{\delta +2+2\beta ,s}\left(f\right)\left(\stackrel{˜}{x}\right).\end{array}$

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

${I}_{3}\le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)w{\left(\stackrel{˜}{x}\right)}^{2+2\beta /n}{M}_{\delta +2+2\beta ,s}\left(f\right)\left(\stackrel{˜}{x}\right).$

Similarly, for ${I}_{4}$, we obtain, for $1 and $1/t=1/s-\delta /n$,

$\begin{array}{rcl}{I}_{4}& \le & C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\frac{{|Q|}^{2/n}}{|Q|}{\int }_{Q}|T\left({D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}{f}_{1}\right)\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx\\ \le & C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}{|Q|}^{2/n}{\left(\frac{1}{|Q|}{\int }_{{R}^{n}}{|T\left({D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}{f}_{1}\right)\left(x\right)|}^{t}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/t}\\ \le & C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}{|Q|}^{2/n}{|Q|}^{-1/t}{\left({\int }_{{R}^{n}}{|{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(x\right){D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(x\right)|}^{s}{|{f}_{1}\left(x\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/s}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)\frac{{|Q|}^{2/n}w{\left(\stackrel{˜}{Q}\right)}^{2+2\beta /n}}{{|Q|}^{2}}{|Q|}^{-1/t}{\left(\frac{1}{|\stackrel{˜}{Q}|}{\int }_{\stackrel{˜}{Q}}{|f\left(y\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/s}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right){\left(\frac{w\left(\stackrel{˜}{Q}\right)}{|\stackrel{˜}{Q}|}\right)}^{2+2\beta /n}{\left(\frac{1}{{|\stackrel{˜}{Q}|}^{1-s\left(\delta +2+2\beta \right)/n}}{\int }_{\stackrel{˜}{Q}}{|f\left(y\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/s}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)w{\left(\stackrel{˜}{x}\right)}^{2+2\beta /n}{M}_{\delta +2+2\beta ,s}\left(f\right)\left(\stackrel{˜}{x}\right).\end{array}$

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

$\begin{array}{r}{T}^{\stackrel{˜}{b}}\left({f}_{2}\right)\left(x\right)-{T}^{\stackrel{˜}{b}}\left({f}_{2}\right)\left({x}_{0}\right)\\ \phantom{\rule{1em}{0ex}}={\int }_{{R}^{n}}\left(\frac{K\left(x,y\right)}{{|x-y|}^{m}}-\frac{K\left({x}_{0},y\right)}{{|{x}_{0}-y|}^{m}}\right)\prod _{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,y\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}+{\int }_{{R}^{n}}\left({R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};x,y\right)-{R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};{x}_{0},y\right)\right)\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};x,y\right)}{{|{x}_{0}-y|}^{m}}K\left({x}_{0},y\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}+{\int }_{{R}^{n}}\left({R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};x,y\right)-{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};{x}_{0},y\right)\right)\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};{x}_{0},y\right)}{{|{x}_{0}-y|}^{m}}K\left({x}_{0},y\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}-\sum _{|{\alpha }_{1}|={m}_{1}}\frac{1}{{\alpha }_{1}!}{\int }_{{R}^{n}}{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right){f}_{2}\left(y\right)\\ \phantom{\rule{2em}{0ex}}×\left[\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};x,y\right){\left(x-y\right)}^{{\alpha }_{1}}}{{|x-y|}^{m}}K\left(x,y\right)-\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};{x}_{0},y\right){\left({x}_{0}-y\right)}^{{\alpha }_{1}}}{{|{x}_{0}-y|}^{m}}K\left({x}_{0},y\right)\right]\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}-\sum _{|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{2}!}{\int }_{{R}^{n}}{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right){f}_{2}\left(y\right)\\ \phantom{\rule{2em}{0ex}}×\left[\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};x,y\right){\left(x-y\right)}^{{\alpha }_{2}}}{{|x-y|}^{m}}K\left(x,y\right)-\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};{x}_{0},y\right){\left({x}_{0}-y\right)}^{{\alpha }_{2}}}{{|{x}_{0}-y|}^{m}}K\left({x}_{0},y\right)\right]\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}+\sum _{|{\alpha }_{1}|={m}_{1},\phantom{\rule{0.25em}{0ex}}|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{1}!{\alpha }_{2}!}{\int }_{{R}^{n}}\left[\frac{{\left(x-y\right)}^{{\alpha }_{1}+{\alpha }_{2}}}{{|x-y|}^{m}}K\left(x,y\right)-\frac{{\left({x}_{0}-y\right)}^{{\alpha }_{1}+{\alpha }_{2}}}{{|{x}_{0}-y|}^{m}}K\left({x}_{0},y\right)\right]\\ \phantom{\rule{2em}{0ex}}×{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right){D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{1em}{0ex}}={I}_{5}^{\left(1\right)}+{I}_{5}^{\left(2\right)}+{I}_{5}^{\left(3\right)}+{I}_{5}^{\left(4\right)}+{I}_{5}^{\left(5\right)}+{I}_{5}^{\left(6\right)}.\end{array}$

Note that $|x-y|\sim |{x}_{0}-y|$ for $x\in Q$ and $y\in {R}^{n}\setminus \stackrel{˜}{Q}={\bigcup }_{l=0}^{\mathrm{\infty }}\left({2}^{l+1}\stackrel{˜}{Q}\setminus {2}^{l}\stackrel{˜}{Q}\right)$. By Lemmas 1 and 5, we get

$|{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,y\right)|\le C\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\left(\frac{w{\left({2}^{l+1}\stackrel{˜}{Q}\right)}^{1+\beta /n}}{|{2}^{l+1}Q|}\right){|x-y|}^{{m}_{j}}.$

Then, by the conditions on K, we obtain

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

For ${I}_{5}^{\left(2\right)}$, by the formula (see [1])

${R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,y\right)-{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};{x}_{0},y\right)=\sum _{|\eta |<{m}_{j}}\frac{1}{\eta !}{R}_{{m}_{j}-|\eta |}\left({D}^{\eta }{\stackrel{˜}{b}}_{j};x,{x}_{0}\right){\left(x-y\right)}^{\eta }$

and Lemma 5, we get

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

Similarly,

$|{I}_{5}^{\left(3\right)}|\le C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)w{\left(\stackrel{˜}{x}\right)}^{2+2\beta /n}{M}_{\delta +2+2\beta ,s}\left(f\right)\left(\stackrel{˜}{x}\right).$

For ${I}_{5}^{\left(4\right)}$, similar to the estimates of ${I}_{5}^{\left(1\right)}$ and ${I}_{5}^{\left(2\right)}$, and noticing that for $b\in {Lip}_{\beta }\left(w\right)$, $w\in {A}_{1}$ and $x\in Q$, by Lemma 4, we have

$|{b}_{Q}-{b}_{{2}^{l}Q}|\le Clw\left(x\right)w{\left({2}^{l}Q\right)}^{\beta /n}{\parallel b\parallel }_{{Lip}_{\beta }\left(w\right)}.$

Thus, we obtain

$\begin{array}{rcl}|{I}_{5}^{\left(4\right)}|& \le & C\sum _{|{\alpha }_{1}|={m}_{1}}{\int }_{{R}^{n}\setminus \stackrel{˜}{Q}}|\frac{{\left(x-y\right)}^{{\alpha }_{1}}K\left(x,y\right)}{{|x-y|}^{m-2}}-\frac{{\left({x}_{0}-y\right)}^{{\alpha }_{1}}K\left({x}_{0},y\right)}{{|{x}_{0}-y|}^{m-2}}|\\ ×|{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};x,y\right)||{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right)||f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ +C\sum _{|{\alpha }_{1}|={m}_{1}}{\int }_{{R}^{n}\setminus \stackrel{˜}{Q}}|{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};x,y\right)-{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};{x}_{0},y\right)|\\ ×\frac{|{\left({x}_{0}-y\right)}^{{\alpha }_{1}}K\left({x}_{0},y\right)|}{{|{x}_{0}-y|}^{m-2}}|{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right)f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{b}_{2}\parallel }_{{Lip}_{\beta }\left(w\right)}\sum _{|\alpha |={m}_{1}}\sum _{l=0}^{\mathrm{\infty }}{\int }_{{2}^{l+1}\stackrel{˜}{Q}\setminus {2}^{l}\stackrel{˜}{Q}}\left(\frac{|x-{x}_{0}|}{{|{x}_{0}-y|}^{n-1-\delta }}+\frac{{|x-{x}_{0}|}^{\epsilon }}{{|{x}_{0}-y|}^{n+\epsilon -2-\delta }}\right)\\ ×\frac{w{\left({2}^{l+1}\stackrel{˜}{Q}\right)}^{1+\beta /n}}{|{2}^{l+1}\stackrel{˜}{Q}|}\left(|{D}^{{\alpha }_{1}}{b}_{1}\left(y\right)-{\left({D}^{{\alpha }_{1}}{b}_{1}\right)}_{{2}^{l+1}\stackrel{˜}{Q}}|\\ +|{\left({D}^{{\alpha }_{1}}{b}_{1}\right)}_{{2}^{l+1}\stackrel{˜}{Q}}-{\left({D}^{{\alpha }_{1}}{b}_{1}\right)}_{\stackrel{˜}{Q}}|\right)|f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \le & C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)\sum _{l=0}^{\mathrm{\infty }}{\int }_{{2}^{l+1}\stackrel{˜}{Q}\setminus {2}^{l}\stackrel{˜}{Q}}\left(\frac{d}{{\left({2}^{l}d\right)}^{n-1-\delta }}+\frac{{d}^{\epsilon }}{{\left({2}^{l}d\right)}^{n+\epsilon -2-\delta }}\right)\\ ×\frac{w{\left({2}^{l+1}\stackrel{˜}{Q}\right)}^{1+\beta /n}}{|{2}^{l+1}\stackrel{˜}{Q}|}\left[\frac{w{\left({2}^{l+1}\stackrel{˜}{Q}\right)}^{1+\beta /n}}{|{2}^{l+1}\stackrel{˜}{Q}|}+lw\left(\stackrel{˜}{x}\right)w{\left({2}^{l+1}\stackrel{˜}{Q}\right)}^{\beta /n}\right]|f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \le & C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)\sum _{l=0}^{\mathrm{\infty }}w\left(\stackrel{˜}{x}\right){\left(\frac{w\left({2}^{l+1}\stackrel{˜}{Q}\right)}{|{2}^{l+1}\stackrel{˜}{Q}|}\right)}^{1+2\beta /n}\\ ×{\left(\frac{1}{{|{2}^{l+1}\stackrel{˜}{Q}|}^{1-s\left(\delta +2+2\beta \right)/n}}{\int }_{{2}^{l+1}\stackrel{˜}{Q}}{|f\left(y\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/s}l\left({2}^{-l}+{2}^{-l\epsilon }\right)\\ \le & C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)w{\left(\stackrel{˜}{x}\right)}^{2+2\beta /n}{M}_{\delta +2+2\beta ,s}\left(f\right)\left(\stackrel{˜}{x}\right).\end{array}$

Similarly,

$|{I}_{5}^{\left(5\right)}|\le C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)w{\left(\stackrel{˜}{x}\right)}^{2+2\beta /n}{M}_{\delta +2+2\beta ,s}\left(f\right)\left(\stackrel{˜}{x}\right).$

For ${I}_{5}^{\left(6\right)}$, we get

$\begin{array}{rcl}|{I}_{5}^{\left(6\right)}|& \le & C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}{\int }_{{R}^{n}\setminus \stackrel{˜}{Q}}|\frac{{\left(x-y\right)}^{{\alpha }_{1}+{\alpha }_{2}}K\left(x,y\right)}{{|x-y|}^{m}}-\frac{{\left({x}_{0}-y\right)}^{{\alpha }_{1}+{\alpha }_{2}}K\left({x}_{0},y\right)}{{|{x}_{0}-y|}^{m}}|\\ ×\left(|{D}^{{\alpha }_{1}}{b}_{1}\left(y\right)-{\left({D}^{{\alpha }_{1}}{b}_{1}\right)}_{{2}^{l+1}\stackrel{˜}{Q}}|+|{\left({D}^{{\alpha }_{1}}{b}_{1}\right)}_{{2}^{l+1}\stackrel{˜}{Q}}-{\left({D}^{{\alpha }_{1}}{b}_{1}\right)}_{\stackrel{˜}{Q}}|\right)\\ ×\left(|{D}^{{\alpha }_{2}}{b}_{2}\left(y\right)-{\left({D}^{{\alpha }_{2}}{b}_{2}\right)}_{{2}^{l+1}\stackrel{˜}{Q}}|+|{\left({D}^{{\alpha }_{2}}{b}_{2}\right)}_{{2}^{l+1}\stackrel{˜}{Q}}-{\left({D}^{{\alpha }_{2}}{b}_{2}\right)}_{\stackrel{˜}{Q}}|\right)|f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)\sum _{l=0}^{\mathrm{\infty }}{\int }_{{2}^{l+1}\stackrel{˜}{Q}\setminus {2}^{l}\stackrel{˜}{Q}}\left(\frac{|x-{x}_{0}|}{{|{x}_{0}-y|}^{n-1-\delta }}+\frac{{|x-{x}_{0}|}^{\epsilon }}{{|{x}_{0}-y|}^{n+\epsilon -2-\delta }}\right)\\ ×{\left[\frac{w{\left({2}^{l+1}\stackrel{˜}{Q}\right)}^{1+\beta /n}}{|{2}^{l+1}\stackrel{˜}{Q}|}+lw\left(\stackrel{˜}{x}\right)w{\left({2}^{l+1}\stackrel{˜}{Q}\right)}^{\beta /n}\right]}^{2}|f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)\sum _{l=0}^{\mathrm{\infty }}{\left[w\left(\stackrel{˜}{x}\right){\left(\frac{w\left({2}^{l+1}\stackrel{˜}{Q}\right)}{|{2}^{l+1}\stackrel{˜}{Q}|}\right)}^{\beta /n}\right]}^{2}\\ ×{\left(\frac{1}{{|{2}^{l+1}\stackrel{˜}{Q}|}^{1-s\left(\delta +2+2\beta \right)/n}}{\int }_{{2}^{l+1}\stackrel{˜}{Q}}{|f\left(y\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/s}{l}^{2}\left({2}^{-l}+{2}^{-l\epsilon }\right)\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)w{\left(\stackrel{˜}{x}\right)}^{2+2\beta /n}{M}_{\delta +2+2\beta ,s}\left(f\right)\left(\stackrel{˜}{x}\right).\end{array}$

Thus

$|{T}^{\stackrel{˜}{b}}\left({f}_{2}\right)\left(x\right)-{T}^{\stackrel{˜}{b}}\left({f}_{2}\right)\left({x}_{0}\right)|\le C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)w{\left(\stackrel{˜}{x}\right)}^{2+2\beta /n}{M}_{\delta +2+2\beta ,s}\left(f\right)\left(\stackrel{˜}{x}\right)$

and

${I}_{5}\le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)w{\left(\stackrel{˜}{x}\right)}^{2+2\beta /n}{M}_{\delta +2+2\beta ,s}\left(f\right)\left(\stackrel{˜}{x}\right).$

This completes the proof of the lemma. □

Proof of Key Lemma 2 Without loss of generality, we may assume $k=2$. By using the same argument as in the proof of Key Lemma 1, we have

$\begin{array}{r}\frac{1}{{|Q|}^{1+2\beta /n}}{\int }_{Q}|{T}^{b}\left(f\right)\left(x\right)-{T}^{\stackrel{˜}{b}}\left({f}_{2}\right)\left({x}_{0}\right)|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{{|Q|}^{1+2\beta /n}}{\int }_{Q}|{\int }_{{R}^{n}}\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{b}}_{j};x,y\right)}{{|x-y|}^{m-2}}K\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+\frac{C}{{|Q|}^{1+2\beta /n}}{\int }_{Q}|\sum _{|{\alpha }_{1}|={m}_{1}}{\int }_{{R}^{n}}\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{b}}_{2};x,y\right){\left(x-y\right)}^{{\alpha }_{1}}}{{|x-y|}^{m-2}}{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right)K\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+\frac{C}{{|Q|}^{1+2\beta /n}}{\int }_{Q}|\sum _{|{\alpha }_{2}|={m}_{2}}{\int }_{{R}^{n}}\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{b}}_{1};x,y\right){\left(x-y\right)}^{{\alpha }_{2}}}{{|x-y|}^{m-2}}{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right)K\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+\frac{C}{{|Q|}^{1+2\beta /n}}{\int }_{Q}|\sum _{|{\alpha }_{1}|={m}_{1},\phantom{\rule{0.25em}{0ex}}|{\alpha }_{2}|={m}_{2}}{\int }_{{R}^{n}}\frac{{\left(x-y\right)}^{{\alpha }_{1}+{\alpha }_{2}}{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(y\right){D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(y\right)}{{|x-y|}^{m-2}}K\left(x,y\right){f}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+\frac{1}{{|Q|}^{1+2\beta /n}}{\int }_{Q}|{T}^{\stackrel{˜}{b}}\left({f}_{2}\right)\left(x\right)-{T}^{\stackrel{˜}{b}}\left({f}_{2}\right)\left({x}_{0}\right)|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}:={J}_{1}+{J}_{2}+{J}_{3}+{J}_{4}+{J}_{5}.\end{array}$

We obtain

$\begin{array}{rcl}{J}_{1}& \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right){|Q|}^{2/n}{\left(\frac{w\left(\stackrel{˜}{Q}\right)}{|Q|}\right)}^{2+2\beta /n}{\left(\frac{1}{|Q|}{\int }_{{R}^{n}}{|T\left({f}_{1}\right)\left(x\right)|}^{t}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/t}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right){|Q|}^{2/n}{\left(\frac{w\left(\stackrel{˜}{Q}\right)}{|Q|}\right)}^{2+2\beta /n}{|Q|}^{-1/t}{\left({\int }_{{R}^{n}}{|{f}_{1}\left(x\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/s}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right){\left(\frac{w\left(\stackrel{˜}{Q}\right)}{|\stackrel{˜}{Q}|}\right)}^{2+2\beta /n}{\left(\frac{1}{{|\stackrel{˜}{Q}|}^{1-s\left(\delta +2\right)/n}}{\int }_{\stackrel{˜}{Q}}{|f\left(y\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/s}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)w{\left(\stackrel{˜}{x}\right)}^{2+2\beta /n}{M}_{\delta +2,s}\left(f\right)\left(\stackrel{˜}{x}\right),\\ {J}_{2}& \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{b}_{2}\parallel }_{{Lip}_{\beta }\left(w\right)}\frac{{|Q|}^{2/n}w{\left(\stackrel{˜}{Q}\right)}^{1+\beta /n}}{{|Q|}^{1+2\beta /n}}\sum _{|{\alpha }_{1}|={m}_{1}}{\left(\frac{1}{|Q|}{\int }_{{R}^{n}}{|T\left({D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}{f}_{1}\right)\left(x\right)|}^{t}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/t}\\ \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{b}_{2}\parallel }_{{Lip}_{\beta }\left(w\right)}\frac{{|Q|}^{2/n}w{\left(\stackrel{˜}{Q}\right)}^{1+\beta /n}}{{|Q|}^{1+2\beta /n}}{|Q|}^{-1/t}\sum _{|{\alpha }_{1}|={m}_{1}}{\left({\int }_{{R}^{n}}{|{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(x\right)|}^{s}{|{f}_{1}\left(x\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/s}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right){\left(\frac{w\left(\stackrel{˜}{Q}\right)}{|\stackrel{˜}{Q}|}\right)}^{2+2\beta /n}{\left(\frac{1}{{|\stackrel{˜}{Q}|}^{1-s\left(\delta +2\right)/n}}{\int }_{\stackrel{˜}{Q}}{|f\left(y\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/s}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)w{\left(\stackrel{˜}{x}\right)}^{2+2\beta /n}{M}_{\delta +2,s}\left(f\right)\left(\stackrel{˜}{x}\right),\\ {J}_{3}& \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)w{\left(\stackrel{˜}{x}\right)}^{2+2\beta /n}{M}_{\delta +2,s}\left(f\right)\left(\stackrel{˜}{x}\right),\\ {J}_{4}& \le & C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\frac{{|Q|}^{2/n}}{{|Q|}^{2\beta /n}}{\left(\frac{1}{|Q|}{\int }_{{R}^{n}}{|T\left({D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}{D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}{f}_{1}\right)\left(x\right)|}^{t}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/t}\\ \le & C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\frac{{|Q|}^{2/n}}{{|Q|}^{2\beta /n}}{|Q|}^{-1/t}{\left({\int }_{{R}^{n}}{|{D}^{{\alpha }_{1}}{\stackrel{˜}{b}}_{1}\left(x\right){D}^{{\alpha }_{2}}{\stackrel{˜}{b}}_{2}\left(x\right)|}^{s}{|{f}_{1}\left(x\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/s}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right){\left(\frac{w\left(\stackrel{˜}{Q}\right)}{|\stackrel{˜}{Q}|}\right)}^{2+2\beta /n}{\left(\frac{1}{{|\stackrel{˜}{Q}|}^{1-s\left(\delta +2\right)/n}}{\int }_{\stackrel{˜}{Q}}{|f\left(y\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/s}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)w{\left(\stackrel{˜}{x}\right)}^{2+2\beta /n}{M}_{\delta +2,s}\left(f\right)\left(\stackrel{˜}{x}\right).\end{array}$

For ${J}_{5}$, similar to the proof of ${I}_{5}$ in Key Lemma 1, we obtain

$\begin{array}{r}|{T}^{\stackrel{˜}{b}}\left({f}_{2}\right)\left(x\right)-{T}^{\stackrel{˜}{b}}\left({f}_{2}\right)\left({x}_{0}\right)|\\ \phantom{\rule{1em}{0ex}}\le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)\sum _{l=0}^{\mathrm{\infty }}\frac{w{\left({2}^{l+1}\stackrel{˜}{Q}\right)}^{2+2\beta /n}}{{|{2}^{l+1}Q|}^{2}}\\ \phantom{\rule{2em}{0ex}}×{\int }_{{2}^{l+1}\stackrel{˜}{Q}\setminus {2}^{l}\stackrel{˜}{Q}}\left(\frac{d}{{\left({2}^{l}d\right)}^{n-1-\delta }}+\frac{{d}^{\epsilon }}{{\left({2}^{l}d\right)}^{n+\epsilon -2-\delta }}\right)|f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}+C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)\sum _{l=0}^{\mathrm{\infty }}\frac{w{\left({2}^{l+1}Q\right)}^{2+2\beta /n}}{{|{2}^{l+1}Q|}^{2}}{\int }_{{2}^{l+1}\stackrel{˜}{Q}\setminus {2}^{l}\stackrel{˜}{Q}}\frac{d}{{\left({2}^{l}d\right)}^{n-1-\delta }}|f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}+C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)\sum _{l=0}^{\mathrm{\infty }}{\int }_{{2}^{l+1}\stackrel{˜}{Q}\setminus {2}^{l}\stackrel{˜}{Q}}\left(\frac{d}{{\left({2}^{l}d\right)}^{n-1-\delta }}+\frac{{d}^{\epsilon }}{{\left({2}^{l}d\right)}^{n+\epsilon -2-\delta }}\right)\\ \phantom{\rule{2em}{0ex}}×\frac{w{\left({2}^{l+1}\stackrel{˜}{Q}\right)}^{1+\beta /n}}{|{2}^{l+1}\stackrel{˜}{Q}|}\left[\frac{w{\left({2}^{l+1}\stackrel{˜}{Q}\right)}^{1+\beta /n}}{|{2}^{l+1}\stackrel{˜}{Q}|}+lw\left(\stackrel{˜}{x}\right)w{\left({2}^{l+1}\stackrel{˜}{Q}\right)}^{\beta /n}\right]|f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}+C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)\sum _{l=0}^{\mathrm{\infty }}{\int }_{{2}^{l+1}\stackrel{˜}{Q}\setminus {2}^{l}\stackrel{˜}{Q}}\left(\frac{d}{{\left({2}^{l}d\right)}^{n-1-\delta }}+\frac{{d}^{\epsilon }}{{\left({2}^{l}d\right)}^{n+\epsilon -2-\delta }}\right)\\ \phantom{\rule{2em}{0ex}}×{\left[\frac{w{\left({2}^{l+1}\stackrel{˜}{Q}\right)}^{1+\beta /n}}{|{2}^{l+1}\stackrel{˜}{Q}|}+lw\left(\stackrel{˜}{x}\right)w{\left({2}^{l+1}\stackrel{˜}{Q}\right)}^{\beta /n}\right]}^{2}|f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{1em}{0ex}}\le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right){|Q|}^{2\beta /n}\sum _{l=0}^{\mathrm{\infty }}{\left(\frac{w\left({2}^{l+1}\stackrel{˜}{Q}\right)}{|{2}^{l+1}Q|}\right)}^{2+2\beta /n}\left({2}^{l\left(2\beta -1\right)}+{2}^{l\left(2\beta -\epsilon \right)}\right)\\ \phantom{\rule{2em}{0ex}}×{\left(\frac{1}{{|{2}^{l+1}\stackrel{˜}{Q}|}^{1-s\left(\delta +2\right)/n}}{\int }_{{2}^{l+1}\stackrel{˜}{Q}}{|f\left(y\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/s}\\ \phantom{\rule{2em}{0ex}}+C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right){|Q|}^{2\beta /n}\sum _{l=0}^{\mathrm{\infty }}w\left(\stackrel{˜}{x}\right){\left(\frac{w\left({2}^{l+1}\stackrel{˜}{Q}\right)}{|{2}^{l+1}\stackrel{˜}{Q}|}\right)}^{1+2\beta /n}\\ \phantom{\rule{2em}{0ex}}×{\left(\frac{1}{{|{2}^{l+1}\stackrel{˜}{Q}|}^{1-s\left(\delta +2\right)/n}}{\int }_{{2}^{l+1}\stackrel{˜}{Q}}{|f\left(y\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/s}l\left({2}^{l\left(2\beta -1\right)}+{2}^{l\left(2\beta -\epsilon \right)}\right)\\ \phantom{\rule{2em}{0ex}}+C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right){|Q|}^{2\beta /n}\sum _{l=0}^{\mathrm{\infty }}{\left[w\left(\stackrel{˜}{x}\right){\left(\frac{w\left({2}^{l+1}\stackrel{˜}{Q}\right)}{|{2}^{l+1}\stackrel{˜}{Q}|}\right)}^{\beta /n}\right]}^{2}\\ \phantom{\rule{2em}{0ex}}×{\left(\frac{1}{{|{2}^{l+1}\stackrel{˜}{Q}|}^{1-s\left(\delta +2\right)/n}}{\int }_{{2}^{l+1}\stackrel{˜}{Q}}{|f\left(y\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/s}{l}^{2}\left({2}^{l\left(2\beta -1\right)}+{2}^{l\left(2\beta -\epsilon \right)}\right)\\ \phantom{\rule{1em}{0ex}}\le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right){|Q|}^{2\beta /n}w{\left(\stackrel{˜}{x}\right)}^{2+2\beta /n}{M}_{\delta +2,s}\left(f\right)\left(\stackrel{˜}{x}\right).\end{array}$

Thus

${J}_{5}\le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)w{\left(\stackrel{˜}{x}\right)}^{2+2\beta /n}{M}_{\delta +2,s}\left(f\right)\left(\stackrel{˜}{x}\right).$

This completes the proof of the lemma. □

Proof of Theorem 1 By Key Lemma 1, we get the sharp function estimate of ${T}^{b}$ as follows:

${\left({T}^{b}\left(f\right)\right)}^{\mathrm{#}}\left(\stackrel{˜}{x}\right)\le C\prod _{j=1}^{k}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right)w{\left(\stackrel{˜}{x}\right)}^{k+k\beta /n}{M}_{\delta +k+k\beta ,s}\left(f\right)\left(\stackrel{˜}{x}\right).$

Now, choose $1 in Key Lemma 1, by using Lemma 3 and notice that ${w}^{1-q\left(k-\left(\delta +k\right)/n\right)}\in {A}_{\mathrm{\infty }}$ and ${w}^{1/p}\in A\left(p,q\right)$. We get

$\begin{array}{r}{\parallel {T}^{b}\left(f\right)\parallel }_{{L}^{q}\left({w}^{1-q\left(k-\left(\delta +k\right)/n\right)}\right)}\\ \phantom{\rule{1em}{0ex}}\le {\parallel M\left({T}^{b}\left(f\right)\right)\parallel }_{{L}^{q}\left({w}^{1-q\left(k-\left(\delta +k\right)/n\right)}\right)}\\ \phantom{\rule{1em}{0ex}}\le C{\parallel {\left({T}^{b}\left(f\right)\right)}^{\mathrm{#}}\parallel }_{{L}^{q}\left({w}^{1-q\left(k-\left(\delta +k\right)/n\right)}\right)}\\ \phantom{\rule{1em}{0ex}}\le C\prod _{j=1}^{k}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right){\parallel {M}_{\delta +k+k\beta ,s}\left(f\right){w}^{k+k\beta /n}\parallel }_{{L}^{q}\left({w}^{1-q\left(k-\left(\delta +k\right)/n\right)}\right)}\\ \phantom{\rule{1em}{0ex}}=C\prod _{j=1}^{k}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right){\parallel {M}_{\delta +k+k\beta ,s}\left(f\right)\parallel }_{{L}^{q}\left({w}^{q/p}\right)}\\ \phantom{\rule{1em}{0ex}}\le C\prod _{j=1}^{k}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right){\parallel f\parallel }_{{L}^{p}\left(w\right)}.\end{array}$

This completes the proof of the theorem. □

Proof of Theorem 2 Choose $1 in Key Lemma 2, notice that ${w}^{1-q\left(k-\left(\delta +k-k\beta \right)/n\right)}\in {A}_{\mathrm{\infty }}$ and ${w}^{1/p}\in A\left(p,q\right)$. By using Lemmas 2 and 3, we obtain

$\begin{array}{r}{\parallel {T}^{b}\left(f\right)\parallel }_{{\stackrel{˙}{F}}_{q}^{k\beta ,\mathrm{\infty }}\left({w}^{1-q\left(k-\left(\delta +k-k\beta \right)/n\right)}\right)}\\ \phantom{\rule{1em}{0ex}}\le C\prod _{j=1}^{k}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right){\parallel {M}_{\delta +k,s}\left(f\right){w}^{k+k\beta /n}\parallel }_{{L}^{q}\left({w}^{1-q\left(k-\left(\delta +k-k\beta \right)/n\right)}\right)}\\ \phantom{\rule{1em}{0ex}}=C\prod _{j=1}^{k}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right){\parallel {M}_{\delta +k,s}\left(f\right)\parallel }_{{L}^{q}\left({w}^{q/p}\right)}\\ \phantom{\rule{1em}{0ex}}\le C\prod _{j=1}^{k}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{b}_{j}\parallel }_{{Lip}_{\beta }\left(w\right)}\right){\parallel f\parallel }_{{L}^{p}\left(w\right)}.\end{array}$

This completes the proof of the theorem. □

## 4 Applications

In this section we shall apply Theorems 1 and 2 of the paper to some particular operators such as the Calderón-Zygmund singular integral operator and the fractional integral operator.

Application 1 Calderón-Zygmund singular integral operator.

Let T be the Calderón-Zygmund operator (see [17, 18]). The multilinear operator related to T is defined by

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

Then Theorem 1 holds for ${T}^{b}$ with $\delta =0$.

Application 2 Fractional integral operator with rough kernel.

For $0<\delta , let ${T}_{\delta }$ be the fractional integral operator with rough kernel defined by (see [8, 17, 18])

${T}_{\delta }f\left(x\right)={\int }_{{R}^{n}}\frac{\mathrm{\Omega }\left(x-y\right)}{{|x-y|}^{n-\delta }}f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy,$

which is the fractional integral operator when $\mathrm{\Omega }=1$ (see [8]). The multilinear operator related to ${T}_{\delta }$ is defined by

${T}_{\delta }^{b}f\left(x\right)={\int }_{{R}^{n}}\frac{{R}_{m+1}\left(b;x,y\right)}{{|x-y|}^{m+n-k-\delta }}\mathrm{\Omega }\left(x-y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy,$

where Ω is homogeneous of degree zero on ${R}^{n}$, ${\int }_{{S}^{n-1}}\mathrm{\Omega }\left({x}^{\prime }\right)\phantom{\rule{0.2em}{0ex}}d\sigma \left({x}^{\prime }\right)=0$ and $\mathrm{\Omega }\in {Lip}_{\epsilon }\left({S}^{n-1}\right)$ for some $0<\epsilon \le 1$, that is, there exists a constant $M>0$ such that for any $x,y\in {S}^{n-1}$, $|\mathrm{\Omega }\left(x\right)-\mathrm{\Omega }\left(y\right)|\le M{|x-y|}^{\epsilon }$. Then Theorem 2 holds for ${T}_{\delta }^{b}$.