Weighted sharp maximal function inequalities and boundedness of multilinear singular integral operator satisfying a variant of Hörmander’s condition

Abstract

In this paper, we establish the weighted sharp maximal function inequalities for the multilinear operator associated with the singular integral operator satisfying a variant of Hörmander’s condition. As an application, we obtain the boundedness of the operator on weighted Lebesgue spaces.

MSC:42B20, 42B25.

1 Introduction

As the development of singular integral operators (see [1, 2]), their commutators and multilinear operators have been well studied. In [3–5], the authors proved that the commutators generated by the singular integral operators and BMO functions are bounded on $L^p(R^n)$ for $1<p<\mathrm{\infty }$. Chanillo (see [6]) proved a similar result when singular integral operators are replaced by the fractional integral operators. In [7, 8], the boundedness for the commutators generated by the singular integral operators and Lipschitz functions on Triebel-Lizorkin and $L^p(R^n)$ ($1<p<\mathrm{\infty }$) spaces are obtained. In [9, 10], the boundedness for the commutators generated by the singular integral operators and the weighted BMO and Lipschitz functions on $L^p(R^n)$ ($1<p<\mathrm{\infty }$) spaces are obtained (also see [11]). In [12, 13], the authors studied some multilinear singular integral operators as follows (also see [14]):

$$T^b(f)(x)=\int \frac{R_{m+1}(b;x,y)}{{|x-y|}^m}K(x,y)f(y)dy,$$

and they obtained some variant sharp function estimates and boundedness of the multilinear operators if $D^{\alpha }b\in \mathit{BMO}(R^n)$ for all α with $|\alpha |=m$. In [15], some singular integral operators satisfying a variant of Hörmander’s condition are introduced, and the boundedness for the operators and their commutators is obtained (see [16, 17]). Motivated by these results, in this paper, we will study the multilinear operator generated by the singular integral operator satisfying a variant of Hörmander’s condition and the weighted Lipschitz and BMO functions, that is, $D^{\alpha }b\in \mathit{BMO}(w)$ or $D^{\alpha }b\in {Lip}_{\beta }(w)$ for all α with $|\alpha |=m$.

2 Preliminaries

First, let us introduce some notation. Throughout this paper, Q will denote a cube of $R^n$ with sides parallel to the axes. For a non-negative integrable function ω, let $\omega (Q)={\int }_Q\omega (x)dx$ and ${\omega }_Q={|Q|}^{-1}{\int }_Q\omega (x)dx$.

For any locally integrable function f, the sharp maximal function of f is defined by

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

It is well known that (see [1])

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

Let

$$M(f)(x)=\underset{Q\ni x}{sup}\frac{1}{|Q|}{\int }_Q|f(y)|dy.$$

For $\eta >0$, let $M_{\eta }^{\mathrm{\#}}(f)(x)=M^{\mathrm{\#}}{({|f|}^{\eta })}^{1/\eta }(x)$ and $M_{\eta }(f)(x)=M{({|f|}^{\eta })}^{1/\eta }(x)$.

For $0<\eta <n$, $1\le p<\mathrm{\infty }$ and the non-negative weight function ω, set

$$M_{\eta ,p,\omega }(f)(x)=\underset{Q\ni x}{sup}{\left(\frac{1}{\omega {(Q)}^{1-p\eta /n}}{\int }_Q\right|f(y){|}^p\omega (y)dy)}^{1/p}$$

and

$$M_{\omega }(f)(x)=\underset{Q\ni x}{sup}\frac{1}{\omega (Q)}{\int }_Q|f(y)|\omega (y)dy.$$

The $A_p$ weight is defined by (see [1])

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

and

$$A_1=\{\omega \in L_{\mathrm{loc}}^p\left(R^n\right):M(\omega )(x)\le Cw(x),\text{a.e.}\}.$$

Given a non-negative weight function ω. For $1\le p<\mathrm{\infty }$, the weighted Lebesgue space $L^p(R^n,\omega )$ is the space of functions f such that

$${\parallel f\parallel }_{L^p(\omega )}={\left({\int }_{R^n}\right|f(x){|}^p\omega (x)dx)}^{1/p}<\mathrm{\infty }.$$

Given the non-negative weight function ω, the weighted BMO space $\mathit{BMO}(\omega )$ is the space of functions b such that

$${\parallel b\parallel }_{\mathit{BMO}(\omega )}=\underset{Q}{sup}\frac{1}{\omega (Q)}{\int }_Q|b(y)-b_Q|dy<\mathrm{\infty }.$$

For $0<\beta <1$, the weighted Lipschitz space ${Lip}_{\beta }(\omega )$ is the space of functions b such that

$${\parallel b\parallel }_{{Lip}_{\beta }(\omega )}=\underset{Q}{sup}\frac{1}{\omega {(Q)}^{\beta /n}}{\left(\frac{1}{\omega (Q)}{\int }_Q\right|b(y)-b_Q{|}^p\omega {(x)}^{1-p}dy)}^{1/p}<\mathrm{\infty }.$$

Remark (1) It has been known that (see [18]), for $b\in {Lip}_{\beta }(\omega )$, $\omega \in A_1$ and $x\in Q$,

$$|b_Q-b_{2^{k}Q}|\le Ck{\parallel b\parallel }_{{Lip}_{\beta }(\omega )}\omega (x)\omega {\left(2^{k}Q\right)}^{\beta /n}.$$

• (2) Let $b\in {Lip}_{\beta }(\omega )$ and $\omega \in A_1$. By [18], we know that spaces ${Lip}_{\beta }(\omega )$ coincide and the norms ${\parallel b\parallel }_{{Lip}_{\beta }(\omega )}$ are equivalent with respect to different values $1\le p<\mathrm{\infty }$.

Definition 1 Let $\mathrm{\Phi }=\{{\varphi }_1,\dots ,{\varphi }_l\}$ be a finite family of bounded functions in $R^n$. For any locally integrable function f, the Φ sharp maximal function of f is defined by

$$M_{\mathrm{\Phi }}^{\mathrm{\#}}(f)(x)=\underset{Q\ni x}{sup}\underset{\{c_1,\dots ,c_l\}}{inf}\frac{1}{|Q|}{\int }_Q|f(y)-\sum _{i=1}^l{c}_i{\varphi }_i(x_Q-y)|dy,$$

where the infimum is taken over all m-tuples $\{c_1,\dots ,c_l\}$ of complex numbers and $x_Q$ is the center of Q. For $\eta >0$, let

$$M_{\mathrm{\Phi },\eta }^{\mathrm{\#}}(f)(x)=\underset{Q\ni x}{sup}\underset{\{c_1,\dots ,c_l\}}{inf}{\left(\frac{1}{|Q|}{\int }_Q\right|f(y)-\sum _{i=1}^l{c}_j{\varphi }_i(x_Q-y){|}^{\eta }dy)}^{1/\eta }.$$

Remark We note that $M_{\mathrm{\Phi }}^{\mathrm{\#}}\approx f^{\mathrm{\#}}$ if $l=1$ and ${\varphi }_1=1$.

Definition 2 Given a positive and locally integrable function f in $R^n$, we say that f satisfies the reverse Hölder’s condition (write this as $f\in R{H}_{\mathrm{\infty }}(R^n)$), if for any cube Q centered at the origin we have

$$0<\underset{x\in Q}{sup}f(x)\le C\frac{1}{|Q|}{\int }_{Q}f(y)dy.$$

In this paper, we will study some singular integral operators as follows (see [15]).

Definition 3 Let $K\in L^2(R^n)$ and satisfy

$$\begin{array}{c}{\parallel K\parallel }_{L^{\mathrm{\infty }}}\le C,\hfill \\ |K(x)|\le C{|x|}^{-n},\hfill \end{array}$$

there exist functions $B_1,\dots ,B_l\in L_{\mathrm{loc}}^1(R^n-\{0\})$ and $\mathrm{\Phi }=\{{\varphi }_1,\dots ,{\varphi }_l\}\subset L^{\mathrm{\infty }}(R^n)$ such that ${|det[{\varphi }_j(y_i)]|}^2\in R{H}_{\mathrm{\infty }}(R^{nl})$, and for a fixed $\delta >0$ and any $|x|>2|y|>0$,

$$|K(x-y)-\sum _{i=1}^l{B}_i(x){\varphi }_i(y)|\le C\frac{{|y|}^{\delta }}{{|x-y|}^{n+\delta }}.$$

For $f\in C_0^{\mathrm{\infty }}$, we define the singular integral operator related to the kernel K by

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

Moreover, let m be the positive integer and b be the function on $R^n$. Set

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

The multilinear operator related to the operator T is defined by

$$T^b(f)(x)={\int }_{R^n}\frac{R_{m+1}(b;x,y)}{{|x-y|}^m}K(x-y)f(y)dy.$$

Note that the commutator $[b,T](f)=bT(f)-T(bf)$ is a particular operator of the multilinear operator $T^b$ if $m=0$. The multilinear operator $T^b$ are the non-trivial generalizations of the commutator. It is well known that commutators and multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see [12–14]). The main purpose of this paper is to prove the sharp maximal inequalities for the multilinear operator $T^b$. As the application, we obtain the weighted $L^p$-boundedness for the multilinear operator $T^b$.

We give some preliminary lemmas.

Lemma 1 (see [[1], p.485])

Let $0<p<q<\mathrm{\infty }$ and for any function $f\ge 0$. We define, for $1/r=1/p-1/q$,

$${\parallel f\parallel }_{W{L}^q}=\underset{\lambda >0}{sup}\lambda |\{x\in R^n:f(x)>\lambda \}{|}^{1/q},N_{p,q}(f)=\underset{Q}{sup}{\parallel f{\chi }_Q\parallel }_{L^p}/{\parallel {\chi }_Q\parallel }_{L^r},$$

where the sup is taken for all measurable sets Q with $0<|Q|<\mathrm{\infty }$. Then

$${\parallel f\parallel }_{W{L}^q}\le N_{p,q}(f)\le {(q/(q-p))}^{1/p}{\parallel f\parallel }_{W{L}^q}.$$

Lemma 2 (see [15])

Let T be the singular integral operator as Definition  2. Then T is bounded on $L^p(R^n,\omega )$ for $\omega \in A_p$ with $1<p<\mathrm{\infty }$, and weak $(L^1,L^1)$ bounded.

Lemma 3 (see [9])

Let $b\in \mathit{BMO}(\omega )$. Then

$$|b_Q-b_{2^{j}Q}|\le Cj{\parallel b\parallel }_{\mathit{BMO}(\omega )}{\omega }_{Q_j},$$

where ${\omega }_{Q_j}={max}_{1\le i\le j}{|2^{i}Q|}^{-1}{\int }_{2^{i}Q}\omega (x)dx$.

Lemma 4 (see [9])

Let $\omega \in A_p$, $1<p<\mathrm{\infty }$. Then there exists $\epsilon >0$ such that ${\omega }^{-r/p}\in A_r$ for any $p^{\prime }\le r\le p^{\prime }+\epsilon$.

Lemma 5 (see [9])

Let $b\in \mathit{BMO}(\omega )$, $\omega ={(\mu {\nu }^{-1})}^{1/p}$, $\mu ,\nu \in A_p$ and $p>1$. Then there exists $\epsilon >0$ such that for $p^{\prime }\le r\le p^{\prime }+\epsilon$,

$${\int }_Q|b(x)-b_Q{|}^r\mu {(x)}^{-r/p}dx\le C{\parallel b\parallel }_{\mathit{BMO}(\omega )}^r{\int }_Q\nu {(x)}^{-r/p}dx.$$

Lemma 6 (see [9])

Let $\omega \in A_p$, $1<p<\mathrm{\infty }$. Then there exists $0<\delta <1$ such that ${\omega }^{1-r^{\prime }/p}\in A_{p/r^{\prime }}(d\mu )$ for any $p^{\prime }<r<p^{\prime }(1+\delta )$, where $d\mu ={\omega }^{r^{\prime }/p}dx$.

Lemma 7 (see [9])

Let $\mu ,\nu \in A_p$, $\omega ={(\mu {\nu }^{-1})}^{1/p}$, $1<p<\mathrm{\infty }$. Then there exists $1<q<p$ such that

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

Lemma 8 (see [1, 6])

Let $0\le \eta <n$, $1\le s<p<n/\eta$, $1/q=1/p-\eta /n$ and $\omega \in A_1$. Then

$${\parallel M_{\eta ,s,\omega }(f)\parallel }_{L^q(\omega )}\le C{\parallel f\parallel }_{L^p(\omega )}.$$

Lemma 9 (see [15, 17])

Let $1<p<\mathrm{\infty }$, $0<\eta <\mathrm{\infty }$, $\omega \in A_{\mathrm{\infty }}$ and $\mathrm{\Phi }=\{{\varphi }_1,\dots ,{\varphi }_l\}\subset L^{\mathrm{\infty }}(R^n)$ such that ${|det[{\varphi }_j(y_i)]|}^2\in R{H}_{\mathrm{\infty }}(R^{nl})$. Then, for any smooth function f for which the left-hand side is finite,

$${\int }_{R^n}{M}_{\eta }(f){(x)}^p\omega (x)dx\le C{\int }_{R^n}{M}_{\mathrm{\Phi },\eta }^{\mathrm{\#}}(f){(x)}^p\omega (x)dx.$$

Lemma 10 (see [13])

Let b be a function on $R^n$ and $D^{\alpha }A\in L^s(R^n)$ for all α with $|\alpha |=m$ and any $s>n$. Then

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

where $\tilde{Q}$ is the cube centered at x and having side length $5\sqrt{n}|x-y|$.

3 Theorems and proofs

We shall prove the following theorems.

Theorem 1 Let T be the singular integral operator as Definition  3, $1<p<\mathrm{\infty }$, $\mu ,\nu \in A_p$, $\omega ={(\mu {\nu }^{-1})}^{1/p}$, $0<\eta <1$ and $D^{\alpha }b\in \mathit{BMO}(\omega )$ for all α with $|\alpha |=m$. Then there exist a constant $C>0$, $\epsilon >0$, $0<\delta <1$, $1<q<p$ and $p^{\prime }<r<min(p^{\prime }+\epsilon ,p^{\prime }(1+\delta ))$ such that, for any $f\in C_0^{\mathrm{\infty }}(R^n)$ and $\tilde{x}\in R^n$,

$$\begin{array}{rcl}{M}_{\mathrm{\Phi },\eta }^{\mathrm{\#}}(T^b(f))(\tilde{x})& \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}({[M_{\nu }\left(\right|\omega T(f){|}^q)(\tilde{x})]}^{1/q}\\ +{[M_{{\nu }^{r^{\prime }/p}}\left({|\omega f|}^{r^{\prime }}\right)(\tilde{x})]}^{1/r^{\prime }}+{[M_{\nu }\left({|\omega f|}^q\right)(\tilde{x})]}^{1/q}).\end{array}$$

Theorem 2 Let T be the singular integral operator as Definition  3, $\omega \in A_1$, $0<\eta <1$, $1<r<\mathrm{\infty }$, $0<\beta <1$ and $D^{\alpha }b\in {Lip}_{\beta }(\omega )$ for all α with $|\alpha |=m$. Then there exists a constant $C>0$ such that, for any $f\in C_0^{\mathrm{\infty }}(R^n)$ and $\tilde{x}\in R^n$,

$$M_{\mathrm{\Phi },\eta }^{\mathrm{\#}}(T^b(f))(\tilde{x})\le C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}\omega (\tilde{x})M_{\beta ,r,\omega }(f)(\tilde{x}).$$

Theorem 3 Let T be the singular integral operator as Definition  3, $1<p<\mathrm{\infty }$, $\mu ,\nu \in A_p$, $\omega ={(\mu {\nu }^{-1})}^{1/p}$ and $D^{\alpha }b\in \mathit{BMO}(\omega )$ for all α with $|\alpha |=m$. Then $T^b$ is bounded from $L^p(R^n,\mu )$ to $L^p(R^n,\nu )$.

Theorem 4 Let T be the singular integral operator as Definition  3, $\omega \in A_1$, $0<\beta <1$, $1<p<n/\beta$, $1/q=1/p-\beta /n$ and $D^{\alpha }b\in {Lip}_{\beta }(\omega )$ for all α with $|\alpha |=m$. Then $T^b$ is bounded from $L^p(R^n,\omega )$ to $L^q(R^n,{\omega }^{1-q})$.

Corollary Let $[b,T](f)=bT(f)-T(bf)$ be the commutator generated by the singular integral operator T as Definition  2 and b. Then Theorems 1-4 hold for $[b,T]$.

Proof of Theorem 1 It suffices to prove for $f\in C_0^{\mathrm{\infty }}(R^n)$ and some constant $C_0$, the following inequality holds:

$$\begin{array}{rcl}{\left(\frac{1}{|Q|}{\int }_Q\right|T^b(f)(x)-C_0{|}^{\eta }dx)}^{1/\eta }& \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}({[M_{\nu }\left(\right|\omega T(f){|}^q)(\tilde{x})]}^{1/q}\\ +{[M_{{\nu }^{r^{\prime }/p}}(|\omega f{|}^{r^{\prime }})(\tilde{x})]}^{1/r^{\prime }}+{[M_{\nu }(|\omega f{|}^q)(\tilde{x})]}^{1/q}),\end{array}$$

where Q is any a cube centered at $x_0$, $C_0={\sum }_{j=1}^l{c}_j{\varphi }_j(x_0-x)$ and $c_j={\int }_{R^n}\frac{K(x_0,y)}{{|x_0-y|}^m}{B}_j(x_0-y)f_2(y)dy$. Fix a cube $Q=Q(x_0,d)$ and $\tilde{x}\in Q$. Let $\tilde{Q}=5\sqrt{n}Q$ and $\tilde{b}(x)=b(x)-{\sum }_{|\alpha |=m}\frac{1}{\alpha !}{(D^{\alpha }b)}_{\tilde{Q}}{x}^{\alpha }$, then $R_m(b;x,y)=R_m(\tilde{b};x,y)$ and $D^{\alpha }\tilde{b}=D^{\alpha }b-{(D^{\alpha }b)}_{\tilde{Q}}$ for $|\alpha |=m$. We write, for $f_1=f{\chi }_{\tilde{Q}}$ and $f_2=f{\chi }_{R^n\setminus \tilde{Q}}$,

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

then

$$\begin{array}{c}{\left(\frac{1}{|Q|}{\int }_Q\right|T^b(f)(x)-C_0{|}^{\eta }dx)}^{1/\eta }\hfill \\ \le C{\left(\frac{1}{|Q|}{\int }_Q\right|T\left(\frac{R_m(\tilde{b};x,\cdot )}{{|x-\cdot |}^m}{f}_1\right){|}^{\eta }dx)}^{1/\eta }\hfill \\ +C{\left(\frac{1}{|Q|}{\int }_Q\right|T\left(\sum _{|\alpha |=m}\frac{{(x-\cdot )}^{\alpha }{D}^{\alpha }\tilde{b}}{{|x-\cdot |}^m}{f}_1\right){|}^{\eta }dx)}^{1/\eta }\hfill \\ +C{\left(\frac{1}{|Q|}{\int }_Q\right|T^{\tilde{b}}(f_2)(x)-C_0{|}^{\eta }dx)}^{1/\eta }\hfill \\ =I_1+I_2+I_3.\hfill \end{array}$$

For $I_1$, noting that $\omega \in A_1$, w satisfies the reverse of Hölder’s inequality:

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

for all cubes Q and some $1<p_0<\mathrm{\infty }$ (see [1]). We take $s=r{p}_0/(r+p_0-1)$ in Lemma 10 and have $1<s<r$ and $p_0=s(r-1)/(r-s)$, then by Lemma 10 and Hölder’s inequality, we obtain

$$\begin{array}{rcl}|R_m(b;x,y)|& \le & C{|x-y|}^m\sum _{|\alpha |=m}{\left(\frac{1}{|\tilde{Q}(x,y)|}{\int }_{\tilde{Q}(x,y)}\right|D^{\alpha }b(z){|}^{s}dz)}^{1/s}\\ \le & C{|x-y|}^m\sum _{|\alpha |=m}{|\tilde{Q}|}^{-1/s}{\left({\int }_{\tilde{Q}(x,y)}\right|D^{\alpha }b(z){|}^s\omega {(z)}^{s(1-r)/r}\omega {(z)}^{s(r-1)/r}dz)}^{1/s}\\ \le & C{|x-y|}^m\sum _{|\alpha |=m}{|\tilde{Q}|}^{-1/s}{\left({\int }_{\tilde{Q}(x,y)}\right|D^{\alpha }b(z){|}^r\omega {(z)}^{1-r}dz)}^{1/r}\\ \times {\left({\int }_{\tilde{Q}(x,y)}\omega {(z)}^{s(r-1)/(r-s)}dz\right)}^{(r-s)/rs}\\ \le & C{|x-y|}^m\sum _{|\alpha |=m}{|\tilde{Q}|}^{-1/s}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}\omega {(\tilde{Q})}^{1/r}{|\tilde{Q}|}^{(r-s)/rs}\\ \times {\left(\frac{1}{|\tilde{Q}(x,y)|}{\int }_{\tilde{Q}(x,y)}\omega {(z)}^{p_0}dz\right)}^{(r-s)/rs}\\ \le & C{|x-y|}^m\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}{|\tilde{Q}|}^{-1/q}\omega {(\tilde{Q})}^{1/r}{|\tilde{Q}|}^{1/s-1/r}\\ \times {(\frac{1}{|\tilde{Q}(x,y)|}{\int }_{\tilde{Q}(x,y)}\omega (z)dz)}^{(r-1)/r}\\ \le & C{|x-y|}^m\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}{|\tilde{Q}|}^{-1/q}\omega {(\tilde{Q})}^{1/r}{|\tilde{Q}|}^{1/s-1/r}\omega {(\tilde{Q})}^{1-1/r}{|\tilde{Q}|}^{1/r-1}\\ \le & C{|x-y|}^m\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}\frac{\omega (\tilde{Q})}{|\tilde{Q}|},\end{array}$$

thus, by Lemma 7, we obtain

$$\begin{array}{rcl}{I}_1& \le & \frac{C}{|Q|}{\int }_Q\left|T\left(\frac{R_m(\tilde{b};x,\cdot )}{{|x-\cdot |}^m}{f}_1\right)\right|dx\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}\frac{\omega (\tilde{Q})}{|\tilde{Q}|}\frac{1}{|Q|}{\int }_Q|T(f)(y)|\omega (y)\nu {(y)}^{1/q}\omega {(y)}^{-1}\nu {(y)}^{-1/q}dy\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}{\omega }_{\tilde{Q}}{\left(\frac{1}{|Q|}{\int }_Q\right|\omega (y)T(f)(y){|}^q\nu (y)dy)}^{1/q}\\ \times {\left(\frac{1}{|Q|}{\int }_Q\omega {(y)}^{-q^{\prime }}\nu {(y)}^{-q^{\prime }/q}dy\right)}^{1/q^{\prime }}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}{\omega }_Q{({\nu }_Q)}^{1/q}{\left(\frac{1}{\nu (Q)}{\int }_Q\right|\omega (y)T(f)(y){|}^q\nu (y)dy)}^{1/q}\\ \times {\left(\frac{1}{|Q|}{\int }_Q\omega {(y)}^{-q^{\prime }}\nu {(y)}^{-q^{\prime }/q}dy\right)}^{1/q^{\prime }}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}{[M_{\nu }\left(\right|\omega T(f){|}^q)(\tilde{x})]}^{1/q}\\ \times {\omega }_Q{({\nu }_Q)}^{1/q}{\left(\frac{1}{|Q|}{\int }_Q\omega {(y)}^{-q^{\prime }}\nu {(y)}^{-q^{\prime }/q}dy\right)}^{1/q^{\prime }}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}{[M_{\nu }\left(\right|\omega T(f){|}^q)(\tilde{x})]}^{1/q}.\end{array}$$

For $I_2$, we know ${\nu }^{-r/p}\in A_r$ by Lemma 4, thus

$${\left(\frac{1}{|Q|}{\int }_Q\nu {(x)}^{-r/p}dx\right)}^{1/r}\le C{\left(\frac{1}{|Q|}{\int }_Q\nu {(x)}^{r^{\prime }/p}dx\right)}^{-1/r^{\prime }},$$

then, by the weak $(L^1,L^1)$ boundedness of T (see Lemma 2) and Kolmogorov’s inequality (see Lemma 1), we obtain, by Lemma 5,

$$\begin{array}{rcl}{I}_2& \le & C\sum _{|\alpha |=m}{\left(\frac{1}{|Q|}{\int }_Q\right|T\left(D^{\alpha }\tilde{b}{f}_1\right)(x){|}^{\eta }dx)}^{1/\eta }\\ \le & C\sum _{|\alpha |=m}\frac{{|Q|}^{1/\eta -1}}{{|Q|}^{1/\eta }}\frac{{\parallel T(D^{\alpha }\tilde{b}{f}_1){\chi }_Q\parallel }_{L^{\eta }}}{{\parallel {\chi }_Q\parallel }_{L^{\eta /(1-\eta )}}}\\ \le & C\sum _{|\alpha |=m}\frac{1}{|Q|}{\parallel T\left(D^{\alpha }\tilde{b}{f}_1\right)\parallel }_{W{L}^1}\\ \le & C\sum _{|\alpha |=m}\frac{1}{|Q|}{\int }_{R^n}|D^{\alpha }\tilde{b}(x)f_1(x)|dx\\ =& C\sum _{|\alpha |=m}\frac{1}{|Q|}{\int }_{\tilde{Q}}|D^{\alpha }b(x)-{\left(D^{\alpha }b\right)}_{\tilde{Q}}|\mu {(x)}^{-1/p}|f(x)|\omega (x)\nu {(x)}^{1/p}dx\\ \le & C\sum _{|\alpha |=m}{\left(\frac{1}{|\tilde{Q}|}{\int }_{\tilde{Q}}\right|(D^{\alpha }b(x)-{\left(D^{\alpha }b\right)}_{\tilde{Q}}){|}^r\mu {(x)}^{-r/p}dx)}^{1/r}\\ \times {\left(\frac{1}{|\tilde{Q}|}{\int }_{\tilde{Q}}\right|f(x){|}^{r^{\prime }}\omega {(x)}^{r^{\prime }}\nu {(x)}^{r^{\prime }/p}dx)}^{1/r^{\prime }}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}{\left(\frac{1}{|\tilde{Q}|}{\int }_{\tilde{Q}}\nu {(x)}^{-r/p}dx\right)}^{1/r}{\left(\frac{1}{|\tilde{Q}|}{\int }_{\tilde{Q}}\right|f(x)\omega (x){|}^{r^{\prime }}\nu {(x)}^{r^{\prime }/p}dx)}^{1/r^{\prime }}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}{\left(\frac{1}{|\tilde{Q}|}{\int }_{\tilde{Q}}\nu {(x)}^{r^{\prime }/p}dx\right)}^{-1/r^{\prime }}\\ \times {\left(\frac{1}{|\tilde{Q}|}{\int }_{\tilde{Q}}\right|f(x)\omega (x){|}^{r^{\prime }}\nu {(x)}^{r^{\prime }/p}dx)}^{1/r^{\prime }}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}{\left(\frac{1}{\nu {(\tilde{Q})}^{r^{\prime }/p}}{\int }_{\tilde{Q}}\right|f(x)\omega (x){|}^{r^{\prime }}\nu {(x)}^{r^{\prime }/p}dx)}^{1/r^{\prime }}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}{[M_{{\nu }^{r^{\prime }/p}}(|\omega f{|}^{r^{\prime }})(\tilde{x})]}^{1/r^{\prime }}.\end{array}$$

For $I_3$, note that $|x-y|\approx |x_0-y|$ for $x\in Q$ and $y\in R^n\setminus Q$, we write

$$\begin{array}{rcl}|T^{\tilde{b}}(f_2)(x)-C_0|& \le & {\int }_{R^n}|\frac{R_m(\tilde{b};x,y)}{{|x-y|}^m}-\frac{R_m(\tilde{b};x_0,y)}{{|x_0-y|}^m}||K(x-y)||f_2(y)|dy\\ +{\int }_{R^n}\frac{|R_{m+1}(\tilde{b};x_0,y)|}{{|x_0-y|}^m}|K(x-y)-\sum _{j=1}^l{B}_j(x_0-y){\varphi }_j(x_0-x)||f_2(y)|dy\\ +C\sum _{|\alpha |=m}{\int }_{R^n}|\frac{{(x-y)}^{\alpha }}{{|x-y|}^m}-\frac{{(x_0-y)}^{\alpha }}{{|x_0-y|}^m}||K(x-y)||D^{\alpha }\tilde{b}(y)||f_2(y)|dy\\ =& I_3^{(1)}(x)+I_3^{(2)}(x)+I_3^{(3)}(x).\end{array}$$

For $I_3^{(1)}$, by the formula (see [13]):

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

and Lemma 10, we have, similar to the proof of $I_1$ and for $k\ge 0$,

$$|R_m(\tilde{b};x,y)-R_m(\tilde{b};x_0,y)|\le C\sum _{|\gamma |<m}\sum _{|\alpha |=m}{|x-x_0|}^{m-|\gamma |}{|x-y|}^{|\gamma |}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}\frac{\omega (2^k\tilde{Q})}{|2^k\tilde{Q}|}$$

and

$$|R_m(\tilde{b};x,y)|\le C{|x-y|}^m\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}\frac{\omega (2^k\tilde{Q})}{|2^k\tilde{Q}|},$$

thus

$$\begin{array}{rcl}{I}_3^{(1)}(x)& \le & \sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}|R_m(\tilde{b};x,y)-R_m(\tilde{b};x_0,y)|\frac{|K(x-y)|}{{|x-y|}^m}|f(y)|dy\\ +\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}|\frac{1}{{|x-y|}^m}-\frac{1}{{|x_0-y|}^m}||R_m(\tilde{b};x_0,y)||f(y)|dy\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}\sum _{k=0}^{\mathrm{\infty }}\frac{\omega (2^{k+1}\tilde{Q})}{|2^{k+1}\tilde{Q}|}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}\frac{|x-x_0|}{{|x_0-y|}^{n+1}}|f(y)|dy\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}\sum _{k=1}^{\mathrm{\infty }}{\omega }_{2^k\tilde{Q}}\frac{d}{{(2^{k}d)}^{n+1}}{\int }_{2^k\tilde{Q}}|f(y)|\omega (y)\nu {(y)}^{1/q}\omega {(y)}^{-1}\nu {(y)}^{-1/q}dy\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}\sum _{k=1}^{\mathrm{\infty }}{2}^{-k}{\omega }_{2^k\tilde{Q}}{\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}\right|\omega (y)f(y){|}^q\nu (y)dy)}^{1/q}\\ \times {\left(\frac{1}{|2^{k}Q|}{\int }_{2^k\tilde{Q}}\omega {(y)}^{-q^{\prime }}\nu {(y)}^{-q^{\prime }/q}dy\right)}^{1/q^{\prime }}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}\sum _{k=1}^{\mathrm{\infty }}{2}^{-k}{\omega }_{2^k\tilde{Q}}{({\nu }_{2^k\tilde{Q}})}^{1/q}\\ \times {\left(\frac{1}{\nu (2^k\tilde{Q})}{\int }_{2^k\tilde{Q}}\right|\omega (y)f(y){|}^q\nu (y)dy)}^{1/q}\\ \times {\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}\omega {(y)}^{-q^{\prime }}\nu {(y)}^{-q^{\prime }/q}dy\right)}^{1/q^{\prime }}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}{[M_{\nu }\left({|\omega f|}^q\right)(\tilde{x})]}^{1/q}\sum _{k=1}^{\mathrm{\infty }}k{2}^{-k}\\ \times {\omega }_{2^{k}Q}{({\nu }_{2^{k}Q})}^{1/q}{\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}\omega {(y)}^{-q^{\prime }}\nu {(y)}^{-q^{\prime }/q}dy\right)}^{1/q^{\prime }}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}{[M_{\nu }\left({|\omega f|}^q\right)(\tilde{x})]}^{1/q}\sum _{k=1}^{\mathrm{\infty }}k{2}^{-k}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}{[M_{\nu }\left({|\omega f|}^q\right)(\tilde{x})]}^{1/q}.\end{array}$$

For $I_3^{(2)}$, we get

$$\begin{array}{rcl}{I}_3^{(2)}(x)& \le & C\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}\frac{|R_m(\tilde{b};x_0,y)|}{{|x_0-y|}^m}|K(x-y)-\sum _{j=1}^l{B}_j(x_0-y){\varphi }_j(x_0-x)||f(y)|dy\\ +C\sum _{|\alpha |=m}\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}\frac{|D^{\alpha }\tilde{b}(y)||{(x_0-y)}^{\alpha }|}{{|x_0-y|}^m}\\ \times |K(x-y)-\sum _{j=1}^l{B}_j(x_0-y){\varphi }_j(x_0-x)||f(y)|dy\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}\frac{\omega (2^{k+1}\tilde{Q})}{|2^{k+1}\tilde{Q}|}\frac{{|x-x_0|}^{\delta }}{{|x_0-y|}^{n+\delta }}|f(y)|dy\\ +C\sum _{|\alpha |=m}\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}|D^{\alpha }b(y)-{\left(D^{\alpha }b\right)}_{2^{k+1}\tilde{Q}}|\frac{{|x-x_0|}^{\delta }}{{|x_0-y|}^{n+\delta }}|f(y)|dy\\ +C\sum _{|\alpha |=m}\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}|{\left(D^{\alpha }b\right)}_{2^{k+1}\tilde{Q}}-{\left(D^{\alpha }b\right)}_{\tilde{Q}}|\frac{{|x-x_0|}^{\delta }}{{|x_0-y|}^{n+\delta }}|f(y)|dy\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}{[M_{\nu }\left({|\omega f|}^q\right)(\tilde{x})]}^{1/q}\sum _{k=1}^{\mathrm{\infty }}k{2}^{-k\delta }\\ \times {\omega }_{2^k\tilde{Q}}{({\nu }_{2^k\tilde{Q}})}^{1/q}{\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}\omega {(y)}^{-q^{\prime }}\nu {(y)}^{-q^{\prime }/q}dy\right)}^{1/q^{\prime }}\\ +C\sum _{k=1}^{\mathrm{\infty }}{2}^{-k\delta }{\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}\right|D^{\alpha }b(y)-{\left(D^{\alpha }b\right)}_{2^k\tilde{Q}}{|}^r\mu {(y)}^{-r/p}dy)}^{1/r}\\ \times {\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}\right|f(y){|}^{r^{\prime }}\omega {(y)}^{r^{\prime }}\nu {(y)}^{r^{\prime }/p}dy)}^{1/r^{\prime }}\\ +C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}\sum _{k=1}^{\mathrm{\infty }}k{2}^{-k\delta }{\omega }_{2^k\tilde{Q}}{\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}\right|\omega (y)f(y){|}^q\nu (y)dy)}^{1/q}\\ \times {\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}\omega {(y)}^{-q^{\prime }}\nu {(y)}^{-q^{\prime }/q}dy\right)}^{1/q^{\prime }}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}\sum _{k=1}^{\mathrm{\infty }}{2}^{-k\delta }{\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}\nu {(y)}^{-r/p}dy\right)}^{1/r}\\ \times {\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}\right|f(y)\omega (y){|}^{r^{\prime }}\nu {(y)}^{r^{\prime }/p}dy)}^{1/r^{\prime }}\\ +C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}{[M_{\nu }\left({|\omega f|}^q\right)(\tilde{x})]}^{1/q}\sum _{k=1}^{\mathrm{\infty }}k{2}^{-k\delta }\\ \times {\omega }_{2^k\tilde{Q}}{({\nu }_{2^k\tilde{Q}})}^{1/q}{\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}\omega {(y)}^{-q^{\prime }}\nu {(y)}^{-q^{\prime }/q}dy\right)}^{1/q^{\prime }}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}({[M_{{\nu }^{r^{\prime }/p}}\left({|\omega f|}^{r^{\prime }}\right)(\tilde{x})]}^{1/r^{\prime }}+{[M_{\nu }\left({|\omega f|}^q\right)(\tilde{x})]}^{1/q}).\end{array}$$

Similarly, we have

$$\begin{array}{rcl}{I}_3^{(3)}(x)& \le & C\sum _{|\alpha |=m}\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}|D^{\alpha }b(y)-{\left(D^{\alpha }b\right)}_{2^{k+1}\tilde{Q}}|\frac{|x-x_0|}{{|x_0-y|}^{n+1}}|f(y)|dy\\ +C\sum _{|\alpha |=m}\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}|{\left(D^{\alpha }b\right)}_{2^{k+1}\tilde{Q}}-{\left(D^{\alpha }b\right)}_{\tilde{Q}}|\frac{|x-x_0|}{{|x_0-y|}^{n+1}}|f(y)|dy\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}\sum _{k=1}^{\mathrm{\infty }}{2}^{-k}{\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}\nu {(y)}^{-r/p}dy\right)}^{1/r}\\ \times {\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}\right|f(y)\omega (y){|}^{r^{\prime }}\nu {(y)}^{r^{\prime }/p}dy)}^{1/r^{\prime }}\\ +C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}{[M_{\nu }\left({|\omega f|}^q\right)(\tilde{x})]}^{1/q}\sum _{k=1}^{\mathrm{\infty }}k{2}^{-k}\\ \times {\omega }_{2^k\tilde{Q}}{({\nu }_{2^k\tilde{Q}})}^{1/q}{\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}\omega {(y)}^{-q^{\prime }}\nu {(y)}^{-q^{\prime }/q}dy\right)}^{1/q^{\prime }}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}({[M_{{\nu }^{r^{\prime }/p}}\left({|\omega f|}^{r^{\prime }}\right)(\tilde{x})]}^{1/r^{\prime }}+{[M_{\nu }\left({|\omega f|}^q\right)(\tilde{x})]}^{1/q}).\end{array}$$

Thus

$$I_3\le C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}({[M_{{\nu }^{r^{\prime }/p}}\left({|\omega f|}^{r^{\prime }}\right)(\tilde{x})]}^{1/r^{\prime }}+{[M_{\nu }\left({|\omega f|}^q\right)(\tilde{x})]}^{1/q}).$$

These results complete the proof of Theorem 1. □

Proof of Theorem 2 It suffices to prove for $f\in C_0^{\mathrm{\infty }}(R^n)$ and some constant $C_0$ that the following inequality holds:

$${\left(\frac{1}{|Q|}{\int }_Q\right|T^b(f)(x)-C_0{|}^{\eta }dx)}^{1/\eta }\le C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}w(\tilde{x})M_{\beta ,r,\omega }(f)(\tilde{x}),$$

where Q is any cube centered at $x_0$, $C_0={\sum }_{j=1}^m{c}_j{\varphi }_j(x_0-x)$ and $c_j={\int }_{R^n}\frac{K(x_0,y)}{{|x_0-y|}^m}{B}_j(x_0-y)f_2(y)dy$. Fix a cube $Q=Q(x_0,d)$ and $\tilde{x}\in Q$. Similar to the proof of Theorem 1, we have, for $f_1=f{\chi }_{\tilde{Q}}$ and $f_2=f{\chi }_{R^n\setminus \tilde{Q}}$,

$$\begin{array}{c}{\left(\frac{1}{|Q|}{\int }_Q\right|T^b(f)(x)-C_0{|}^{\eta }dx)}^{1/\eta }\hfill \\ \le C{\left(\frac{1}{|Q|}{\int }_Q\right|T\left(\frac{R_m(\tilde{b};x,\cdot )}{{|x-\cdot |}^m}{f}_1\right){|}^{\eta }dx)}^{1/\eta }\hfill \\ +C{\left(\frac{1}{|Q|}{\int }_Q\right|T\left(\sum _{|\alpha |=m}\frac{{(x-\cdot )}^{\alpha }{D}^{\alpha }\tilde{b}}{{|x-\cdot |}^m}{f}_1\right){|}^{\eta }dx)}^{1/\eta }\hfill \\ +C{\left(\frac{1}{|Q|}{\int }_Q\right|T^{\tilde{b}}(f_2)(x)-C_0{|}^{\eta }dx)}^{1/\eta }\hfill \\ =J_1+J_2+J_3.\hfill \end{array}$$

For $J_1$ and $J_2$, by using the same argument as in the proof of Theorem 1, we get

$$\begin{array}{rcl}|R_m(\tilde{b};x,y)|& \le & C{|x-y|}^m\sum _{|\alpha |=m}{|\tilde{Q}|}^{-1/q}{\left({\int }_{\tilde{Q}(x,y)}\right|D^{\alpha }\tilde{b}(z){|}^q\omega {(z)}^{q(1-r)/r}\omega {(z)}^{q(r-1)/r}dz)}^{1/q}\\ \le & C{|x-y|}^m\sum _{|\alpha |=m}{|\tilde{Q}|}^{-1/q}{\left({\int }_{\tilde{Q}(x,y)}\right|D^{\alpha }\tilde{b}(z){|}^r\omega {(z)}^{1-r}dz)}^{1/r}\\ \times {\left({\int }_{\tilde{Q}(x,y)}\omega {(z)}^{q(r-1)/(r-q)}dz\right)}^{(r-q)/rq}\\ \le & C{|x-y|}^m\sum _{|\alpha |=m}{|\tilde{Q}|}^{-1/q}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}\omega {(\tilde{Q})}^{\beta /n+1/r}{|\tilde{Q}|}^{(r-q)/rq}\\ \times {\left(\frac{1}{|\tilde{Q}(x,y)|}{\int }_{\tilde{Q}(x,y)}\omega {(z)}^{p_0}dz\right)}^{(r-q)/rq}\\ \le & C{|x-y|}^m\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}{|\tilde{Q}|}^{-1/q}\omega {(\tilde{Q})}^{\beta /n+1/r}{|\tilde{Q}|}^{1/q-1/r}\\ \times {(\frac{1}{|\tilde{Q}(x,y)|}{\int }_{\tilde{Q}(x,y)}\omega (z)dz)}^{(r-1)/r}\\ \le & C{|x-y|}^m\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}{|\tilde{Q}|}^{-1/q}\omega {(\tilde{Q})}^{\beta /n+1/r}{|\tilde{Q}|}^{1/q-1/r}\omega {(\tilde{Q})}^{1-1/r}{|\tilde{Q}|}^{1/r-1}\\ \le & C{|x-y|}^m\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}\omega {(\tilde{Q})}^{\beta /n}\omega (\tilde{x}),\end{array}$$

thus

$$\begin{array}{c}{J}_1\le C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}\omega {(\tilde{Q})}^{\beta /n}\omega (\tilde{x}){|Q|}^{-1/s}{\left({\int }_{R^n}\right|f_1(x){|}^{s}dx)}^{1/s}\hfill \\ \phantom{J_1}\le C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}\omega {(\tilde{Q})}^{\beta /n}\omega (\tilde{x}){|Q|}^{-1/s}{\left({\int }_{\tilde{Q}}\right|f(x){|}^r\omega (x)dx)}^{1/r}\hfill \\ \phantom{J_1\le }\times {\left({\int }_{\tilde{Q}}\omega {(x)}^{-s/(r-s)}dx\right)}^{(r-s)/rs}\hfill \\ \phantom{J_1}\le C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}\omega (\tilde{x}){|\tilde{Q}|}^{-1/s}\omega {(\tilde{Q})}^{1/r}{\left(\frac{1}{\omega {(\tilde{Q})}^{1-r\beta /n}}{\int }_{\tilde{Q}}\right|f(x){|}^r\omega (x)dx)}^{1/r}\hfill \\ \phantom{J_1\le }\times {\left(\frac{1}{|\tilde{Q}|}{\int }_{\tilde{Q}}\omega {(x)}^{-s/(r-s)}dx\right)}^{(r-s)/rs}{(\frac{1}{|\tilde{Q}|}{\int }_{\tilde{Q}}\omega (x)dx)}^{1/r}{|\tilde{Q}|}^{1/s}\omega {(\tilde{Q})}^{-1/r}\hfill \\ \phantom{J_1}\le C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}\omega (\tilde{x})M_{\beta ,r,\omega }(f)(\tilde{x}),\hfill \\ J_2\le C\sum _{|\alpha |=m}\frac{1}{|Q|}{\int }_{\tilde{Q}}|D^{\alpha }b(x)-{\left(D^{\alpha }b\right)}_{\tilde{Q}}|\omega {(x)}^{-1/r}|f(x)|\omega {(x)}^{1/r}dx\hfill \\ \phantom{J_2}\le C\sum _{|\alpha |=m}\frac{1}{|Q|}{\left({\int }_{\tilde{Q}}\right|D^{\alpha }b(x)-{\left(D^{\alpha }b\right)}_{\tilde{Q}}{|}^{r^{\prime }}\omega {(x)}^{1-r^{\prime }}dx)}^{1/r^{\prime }}{\left({\int }_{\tilde{Q}}\right|f(x){|}^r\omega (x)dx)}^{1/r}\hfill \\ \phantom{J_2}\le C\sum _{|\alpha |=m}\frac{1}{|Q|}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}\omega {(\tilde{Q})}^{\beta /n+1/r^{\prime }}\omega {(\tilde{Q})}^{1/r-\beta /n}\hfill \\ \phantom{J_2\le }\times {\left(\frac{1}{\omega {(\tilde{Q})}^{1-r\beta /n}}{\int }_{\tilde{Q}}\right|f(x){|}^r\omega (x)dx)}^{1/r}\hfill \\ \phantom{J_2}\le C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}\frac{\omega (\tilde{Q})}{|\tilde{Q}|}{M}_{\beta ,r,\omega }(f)(\tilde{x})\hfill \\ \phantom{J_2}\le C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}\omega (\tilde{x})M_{\beta ,r,\omega }(f)(\tilde{x}).\hfill \end{array}$$

For $J_3$, we have

$$\begin{array}{rcl}|R_m(\tilde{b};x,y)-R_m(\tilde{b};x_0,y)|& \le & C\sum _{|\gamma |<m}\sum _{|\alpha |=m}{|x-x_0|}^{m-|\gamma |}{|x-y|}^{|\gamma |}\\ \times {\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}\omega (\tilde{x})\omega {\left(2^k\tilde{Q}\right)}^{\beta /n},\end{array}$$

thus

$$\begin{array}{c}|T^{\tilde{b}}(f_2)(x)-C_0|\hfill \\ \le \sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}|R_m(\tilde{b};x,y)-R_m(\tilde{b};x_0,y)|\frac{|K(x-y)|}{{|x-y|}^m}|f(y)|dy\hfill \\ +\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}|\frac{1}{{|x-y|}^m}-\frac{1}{{|x_0-y|}^m}||R_m(\tilde{b};x_0,y)||K(x-y)||f(y)|dy\hfill \\ +C\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}\frac{|R_m(\tilde{b};x_0,y)|}{{|x_0-y|}^m}|K(x-y)-\sum _{j=1}^l{B}_j(x_0-y){\varphi }_j(x_0-x)||f(y)|dy\hfill \\ +C\sum _{|\alpha |=m}\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}\frac{|D^{\alpha }\tilde{b}(y)||{(x_0-y)}^{\alpha }|}{{|x_0-y|}^m}\hfill \\ \times |K(x-y)-\sum _{j=1}^l{B}_j(x_0-y){\varphi }_j(x_0-x)||f(y)|dy\hfill \\ +C\sum _{|\alpha |=m}\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}|\frac{{(x-y)}^{\alpha }}{{|x-y|}^m}-\frac{{(x_0-y)}^{\alpha }}{{|x_0-y|}^m}||K(x-y)||D^{\alpha }\tilde{b}(y)||f(y)|dy\hfill \\ \le C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}\omega (\tilde{x})\sum _{k=0}^{\mathrm{\infty }}\omega {\left(2^{k+1}\tilde{Q}\right)}^{\beta /n}\hfill \\ \times {\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}(\frac{|x-x_0|}{{|x_0-y|}^{n+1}}+\frac{{|x-x_0|}^{\delta }}{{|x_0-y|}^{n+\delta }})|f(y)|dy\hfill \\ +C\sum _{|\alpha |=m}\sum _{k=1}^{\mathrm{\infty }}(\frac{|x-x_0|}{{|x_0-y|}^{n+1}}+\frac{{|x-x_0|}^{\delta }}{{|x_0-y|}^{n+\delta }})\hfill \\ \times {\int }_{2^k\tilde{Q}}|D^{\alpha }b(y)-{\left(D^{\alpha }b\right)}_{2^k\tilde{Q}}|w{(y)}^{-1/r}|f(y)|\omega {(y)}^{1/r}dy\hfill \\ +C\sum _{|\alpha |=m}\sum _{k=1}^{\mathrm{\infty }}(\frac{|x-x_0|}{{|x_0-y|}^{n+1}}+\frac{{|x-x_0|}^{\delta }}{{|x_0-y|}^{n+\delta }})\hfill \\ \times {\int }_{2^k\tilde{Q}}|{\left(D^{\alpha }b\right)}_{2^k\tilde{Q}}-{\left(D^{\alpha }b\right)}_{\tilde{Q}}||f(y)|\omega {(y)}^{1/r}\omega {(y)}^{-1/r}dy\hfill \\ \le C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}\omega (\tilde{x})\sum _{k=1}^{\mathrm{\infty }}(\frac{d}{{(2^{k}d)}^{n+1}}+\frac{d^{\delta }}{{(2^{k}d)}^{n+\delta }})\hfill \\ \times \omega {\left(2^k\tilde{Q}\right)}^{\beta /n}{\left({\int }_{2^k\tilde{Q}}\right|f(y){|}^r\omega (y)dx)}^{1/r}\hfill \\ \times {\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}\omega {(y)}^{-1/(r-1)}dy\right)}^{(r-1)/r}{(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}\omega (y)dy)}^{1/r}\left|2^k\tilde{Q}\right|\omega {\left(2^k\tilde{Q}\right)}^{-1/r}\hfill \\ +C\sum _{|\alpha |=m}\sum _{k=1}^{\mathrm{\infty }}(\frac{d}{{(2^{k}d)}^{n+1}}+\frac{d^{\delta }}{{(2^{k}d)}^{n+\delta }})\hfill \\ \times {\left({\int }_{2^k\tilde{Q}}\right|(D^{\alpha }b(y)-{\left(D^{\alpha }b\right)}_{2^k\tilde{Q}}){|}^{r^{\prime }}\omega {(y)}^{1-r^{\prime }}dy)}^{1/r^{\prime }}{\left({\int }_{2^k\tilde{Q}}\right|f(y){|}^r\omega (y)dy)}^{1/r}\hfill \\ +C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}\omega (\tilde{x})\sum _{k=1}^{\mathrm{\infty }}k\omega {\left(2^k\tilde{Q}\right)}^{\beta /n}(\frac{d}{{(2^{k}d)}^{n+1}}+\frac{d^{\delta }}{{(2^{k}d)}^{n+\delta }})\hfill \\ \times {\left({\int }_{2^k\tilde{Q}}\right|f(y){|}^r\omega (y)dx)}^{1/r}\hfill \\ \times {\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}\omega {(y)}^{-1/(r-1)}dy\right)}^{(r-1)/r}{(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}\omega (y)dy)}^{1/r}\left|2^k\tilde{Q}\right|\omega {\left(2^k\tilde{Q}\right)}^{-1/r}\hfill \\ \le C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}\omega (\tilde{x})\sum _{k=1}^{\mathrm{\infty }}k(2^{-k}+2^{-k\delta })\hfill \\ \times {\left(\frac{1}{\omega {(2^k\tilde{Q})}^{1-r\beta /n}}{\int }_{2^k\tilde{Q}}\right|f(y){|}^r\omega (y)dx)}^{1/r}\hfill \\ +C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}\sum _{k=1}^{\mathrm{\infty }}(2^{-k}+2^{-k\delta })\hfill \\ \times \frac{\omega (2^k\tilde{Q})}{|2^k\tilde{Q}|}{\left(\frac{1}{\omega {(2^k\tilde{Q})}^{1-r\beta /n}}{\int }_{2^k\tilde{Q}}\right|f(y){|}^r\omega (y)dx)}^{1/r}\hfill \\ \le C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}\omega (\tilde{x})M_{\beta ,r,\omega }(f)(\tilde{x}).\hfill \end{array}$$

This completes the proof of Theorem 2. □

Proof of Theorem 3 Notice that ${\nu }^{r^{\prime }/p}\in A_{r^{\prime }+1-r^{\prime }/p}\subset A_p$ and $\nu (x)dx\in A_{p/r^{\prime }}(\nu {(x)}^{r^{\prime }/p}dx)$ by Lemma 6, thus, by Theorem 1, Lemmas 2 and 9,

$$\begin{array}{c}{\int }_{R^n}|T^b(f)(x){|}^p\nu (x)dx\hfill \\ \le {\int }_{R^n}|M_{\eta }(T^b(f))(x){|}^p\nu (x)dx\le C{\int }_{R^n}|M_{\mathrm{\Phi },\eta }^{\mathrm{\#}}(T^b(f))(x){|}^p\nu (x)dx\hfill \\ \le C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}{\int }_{R^n}({[M_{\nu }\left(\right|\omega T(f){|}^q)(x)]}^{p/q}\hfill \\ +{[M_{{\nu }^{r^{\prime }/p}}\left({|\omega f|}^{r^{\prime }}\right)(x)]}^{p/r^{\prime }}+{[M_{\nu }\left({|\omega f|}^q\right)(x)]}^{p/q})\nu (x)dx\hfill \\ \le C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}\left({\int }_{R^n}\right|\omega (x)f(x){|}^p\nu (x)dx+{\int }_{R^n}|\omega (x)T(f)(x){|}^p\nu (x)dx)\hfill \\ =C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}\left({\int }_{R^n}\right|f(x){|}^p\mu (x)dx+{\int }_{R^n}|T(f)(x){|}^p\mu (x)dx)\hfill \\ \le C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(\omega )}{\int }_{R^n}|f(x){|}^p\mu (x)dx.\hfill \end{array}$$

This completes the proof of Theorem 3. □

Proof of Theorem 4 Choose $1<r<p$ in Theorem 2 and notice ${\omega }^{1-q}\in A_1$, then we have, by Lemmas 8 and 9,

$$\begin{array}{rcl}{\parallel T^b(f)\parallel }_{L^q({\omega }^{1-q})}& \le & {\parallel M_{\eta }(T^b(f))\parallel }_{L^q({\omega }^{1-q})}\le C{\parallel M_{\mathrm{\Phi },\eta }^{\mathrm{\#}}(T^b(f))\parallel }_{L^q({\omega }^{1-q})}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}{\parallel \omega M_{\beta ,r,\omega }(f)\parallel }_{L^q({\omega }^{1-q})}\\ =& C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}{\parallel M_{\beta ,r,\omega }(f)\parallel }_{L^q(\omega )}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{Lip}_{\beta }(\omega )}{\parallel f\parallel }_{L^p(\omega )}.\end{array}$$

This completes the proof of Theorem 4. □