Weighted boundedness of a multilinear operator associated to a singular integral operator with general kernels

Abstract

In this paper, we establish the weighted sharp maximal function inequalities for a multilinear operator associated to a singular integral operator with general kernels. As an application, we obtain the boundedness of the operator on weighted Lebesgue and Morrey spaces.

MSC:42B20, 42B25.

1 Introduction and preliminaries

As the development of singular integral operators (see [1–3]), their commutators and multilinear operators have been well studied. In [4–6], the authors prove that the commutators generated by singular integral operators and BMO functions are bounded on $L^p(R^n)$ for $1<p<\mathrm{\infty }$. Chanillo (see [7]) proves a similar result when singular integral operators are replaced by fractional integral operators. In [8, 9], the boundedness for the commutators generated by singular integral operators and Lipschitz functions on Triebel-Lizorkin and $L^p(R^n)$ ($1<p<\mathrm{\infty }$) spaces is obtained. In [10, 11], the boundedness for the commutators generated by singular integral operators and weighted BMO and Lipschitz functions on $L^p(R^n)$ ($1<p<\mathrm{\infty }$) spaces is obtained. In [12], some singular integral operators with general kernel are introduced, and the boundedness for the operators and their commutators generated by BMO and Lipschitz functions is obtained (see [8, 12, 13]). Motivated by these, in this paper, we study the multilinear operator generated by the singular integral operator with general kernel and the weighted Lipschitz and BMO functions.

First, let us introduce some notations. Throughout this paper, Q denotes a cube of $R^n$ with sides parallel to the axes. For any locally integrable function f, the sharp maximal function of f is defined by

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

where, and in what follows, $f_Q={|Q|}^{-1}{\int }_{Q}f(x)dx$. It is well known that (see [1, 2])

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

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 w, set

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

We write $M_{\eta ,p,w}(f)=M_{p,w}(f)$ if $\eta =0$.

The $A_p$ weight is defined by (see [1]), for $1<p<\mathrm{\infty }$,

$$A_p=\{w\in L_{\mathrm{loc}}^1\left(R^n\right):\underset{Q}{sup}(\frac{1}{|Q|}{\int }_{Q}w(x)dx){\left(\frac{1}{|Q|}{\int }_{Q}w{(x)}^{-1/(p-1)}dx\right)}^{p-1}<\mathrm{\infty }\}$$

and

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

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

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

For $0<\beta <1$ and the non-negative weight function w, the weighted Lipschitz space ${\mathit{Lip}}_{\beta }(w)$ is the space of functions b such that

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

and the weighted BMO space $\mathit{BMO}(w)$ is the space of functions b such that

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

Remark (1) It has been known that (see [11, 14]), for $b\in {\mathit{Lip}}_{\beta }(w)$, $w\in A_1$ and $x\in Q$,

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

• (2) It has been known that (see [2, 14]), for $b\in \mathit{BMO}(w)$, $w\in A_1$ and $x\in Q$,

  $$|b_Q-b_{2^{k}Q}|\le Ck{\parallel b\parallel }_{\mathit{BMO}(w)}w(x).$$

• (3) Let $b\in {\mathit{Lip}}_{\beta }(w)$ or $b\in \mathit{BMO}(w)$ and $w\in A_1$. By [14], we know that spaces ${\mathit{Lip}}_{\beta }(w)$ or $\mathit{BMO}(w)$ coincide and the norms ${\parallel b\parallel }_{{\mathit{Lip}}_{\beta }(w)}$ or ${\parallel b\parallel }_{\mathit{BMO}(w)}$ are equivalent with respect to different values $1\le p<\mathrm{\infty }$.

Definition 1 Let φ be a positive, increasing function on $R^{+}$, and there exists a constant $D>0$ such that

$$\phi (2t)\le D\phi (t)\text{for }t\ge 0.$$

Let w be a non-negative weight function on $R^n$ and f be a locally integrable function on $R^n$. Set, for $1\le p<\mathrm{\infty }$,

$${\parallel f\parallel }_{L^{p,\phi }(w)}=\underset{x\in R^n,d>0}{sup}{\left(\frac{1}{\phi (d)}{\int }_{Q(x,d)}\right|f(y){|}^{p}w(y)dy)}^{1/p},$$

where $Q(x,d)=\{y\in R^n:|x-y|<d\}$. The generalized weighted Morrey space is defined by

$$L^{p,\phi }(R^n,w)=\{f\in L_{\mathrm{loc}}^1\left(R^n\right):{\parallel f\parallel }_{L^{p,\phi }(w)}<\mathrm{\infty }\}.$$

If $\phi (d)=d^{\delta }$, $\delta >0$, then $L^{p,\phi }(R^n,w)=L^{p,\delta }(R^n,w)$, which is the classical Morrey space (see [15, 16]). If $\phi (d)=1$, then $L^{p,\phi }(R^n,w)=L^p(R^n,w)$, which is the weighted Lebesgue space (see [1]).

As the Morrey space may be considered as an extension of the Lebesgue space, it is natural and important to study the boundedness of the operator on the Morrey spaces (see [17–20]).

In this paper, we study some singular integral operators as follows (see [12]).

Definition 2 Let $T:S\to S^{\prime }$ be a linear operator such that T is bounded on $L^2(R^n)$ and has a kernel K, that is, there exists a locally integrable function $K(x,y)$ on $R^n\times R^n\setminus \{(x,y)\in R^n\times R^n:x=y\}$ such that

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

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

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

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

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

where $1<q^{\prime }<2$ and $1/q+1/q^{\prime }=1$. Moreover, let m be a positive integer and b be a 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 classical Calderón-Zygmund singular integral operator satisfies Definition 1 (see [1–3, 5, 6]) and 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 $T^b$ is a non-trivial generalization 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 [21–23]). The main purpose of this paper is to prove the sharp maximal inequalities for the multilinear operator $T^b$. As application, we obtain the weighted $L^p$-norm inequality and Morrey space boundedness for the multilinear operator $T^b$.

2 Theorems and lemmas

We shall prove the following theorems.

Theorem 1 Let T be a singular integral operator as in Definition  2, the sequence $\{k{C}_k\}\in l^1$, $w\in A_1$, $0<\eta <1$, $q^{\prime }<r<\mathrm{\infty }$ and $D^{\alpha }b\in \mathit{BMO}(w)$ 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 $\tilde{x}\in R^n$,

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

Theorem 2 Let T be a singular integral operator as in Definition  2, the sequence $\{k{C}_k\}\in l^1$, $w\in A_1$, $0<\eta <1$, $q^{\prime }<r<\mathrm{\infty }$, $0<\beta <1$ and $D^{\alpha }b\in {\mathit{Lip}}_{\beta }(w)$ 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 $\tilde{x}\in R^n$,

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

Theorem 3 Let T be a singular integral operator as in Definition  2, the sequence $\{k{C}_k\}\in l^1$, $w\in A_1$, $q^{\prime }<u<\mathrm{\infty }$ and $D^{\alpha }b\in \mathit{BMO}(w)$ for all α with $|\alpha |=m$. Then $T^b$ is bounded from $L^u(R^n,w)$ to $L^u(R^n,w^{1-u})$.

Theorem 4 Let T be a singular integral operator as in Definition  2, the sequence $\{k{C}_k\}\in l^1$, $w\in A_1$, $q^{\prime }<u<\mathrm{\infty }$, $0<D<2^n$ and $D^{\alpha }b\in \mathit{BMO}(w)$ for all α with $|\alpha |=m$. Then $T^b$ is bounded from $L^{u,\phi }(R^n,w)$ to $L^{u,\phi }(R^n,w^{1-u})$.

Theorem 5 Let T be a singular integral operator as in Definition  2, the sequence $\{k{C}_k\}\in l^1$, $w\in A_1$, $0<\beta <1$, $q^{\prime }<u<n/\beta$, $1/v=1/u-\beta /n$ and $D^{\alpha }b\in {\mathit{Lip}}_{\beta }(w)$ for all α with $|\alpha |=m$. Then $T^b$ is bounded from $L^u(R^n,w)$ to $L^v(R^n,w^{1-v})$.

Theorem 6 Let T be a singular integral operator as in Definition  2, the sequence $\{k{C}_k\}\in l^1$, $w\in A_1$, $0<\beta <1$, $0<D<2^n$, $q^{\prime }<u<n/\beta$, $1/v=1/u-\beta /n$ and $D^{\alpha }b\in {\mathit{Lip}}_{\beta }(w)$ for all α with $|\alpha |=m$. Then $T^b$ is bounded from $L^{u,\phi }(R^n,w)$ to $L^{v,\phi }(R^n,w^{1-v})$.

To prove the theorems, we need the following lemmas.

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

Let $0<u<v<\mathrm{\infty }$ and for any function $f\ge 0$, we define that, for $1/r=1/u-1/v$,

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

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

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

Lemma 2 (see [12])

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

Lemma 3 (see [1, 7])

Let $0\le \eta <n$, $1\le s<u<n/\eta$, $1/v=1/u-\eta /n$ and $w\in A_1$. Then

$${\parallel M_{\eta ,s,w}(f)\parallel }_{L^v(w)}\le C{\parallel f\parallel }_{L^u(w)}.$$

Lemma 4 (see [1])

Let $0<p,\eta <\mathrm{\infty }$ and $w\in {\bigcup }_{1\le r<\mathrm{\infty }}{A}_r$. Then, for any smooth function f for which the left-hand side is finite,

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

Lemma 5 (see [17, 20])

Let $0<p<\mathrm{\infty }$, $0<\eta <\mathrm{\infty }$, $0<D<2^n$ and $w\in A_1$. Then, for any smooth function f for which the left-hand side is finite,

$${\parallel M_{\eta }(f)\parallel }_{L^{p,\phi }(w)}\le C{\parallel M_{\eta }^{\mathrm{\#}}(f)\parallel }_{L^{p,\phi }(w)}.$$

Lemma 6 (see [17, 20])

Let $0\le \eta <n$, $0<D<2^n$, $1\le s<u<n/\eta$, $1/v=1/u-\eta /n$ and $w\in A_1$. Then

$${\parallel M_{\eta ,s,w}(f)\parallel }_{L^{v,\phi }(w)}\le C{\parallel f\parallel }_{L^{u,\phi }(w)}.$$

Lemma 7 (see [22])

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

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

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

3 Proofs of theorems

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

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

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 $f_2=f{\chi }_{R^n\setminus \tilde{Q}}$,

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

then

$$\begin{array}{c}{\left(\frac{1}{|Q|}{\int }_Q\right|T^b(f)(x)-T^{\tilde{b}}(f_2)(x_0){|}^{\eta }dx)}^{1/\eta }\hfill \\ \le C{\left(\frac{1}{|Q|}{\int }_Q\right|T\left(\frac{R_m(\tilde{b};x,\cdot )}{{|x-\cdot |}^m}{f}_1\right){|}^{\eta }dx)}^{1/\eta }\hfill \\ +C{\left(\frac{1}{|Q|}{\int }_Q\right|T\left(\sum _{|\alpha |=m}\frac{{(x-\cdot )}^{\alpha }{D}^{\alpha }\tilde{b}}{{|x-\cdot |}^m}{f}_1\right){|}^{\eta }dx)}^{1/\eta }\hfill \\ +C{\left(\frac{1}{|Q|}{\int }_Q\right|T^{\tilde{b}}(f_2)(x)-T^{\tilde{b}}(f_2)(x_0){|}^{\eta }dx)}^{1/\eta }\hfill \\ =I_1+I_2+I_3.\hfill \end{array}$$

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

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

for all cube Q and some $1<p_0<\mathrm{\infty }$ (see [1]). We take $u=r{p}_0/(r+p_0-1)$ in Lemma 7 and have $1<u<r$ and $p_0=u(r-1)/(r-u)$. Then by Lemma 7 and Hölder’s inequality, we get

$$\begin{array}{c}|R_m(\tilde{b};x,y)|\hfill \\ \le C{|x-y|}^m\sum _{|\alpha |=m}{\left(\frac{1}{|\tilde{Q}(x,y)|}{\int }_{\tilde{Q}(x,y)}\right|D^{\alpha }\tilde{b}(z){|}^{u}dz)}^{1/u}\hfill \\ \le C{|x-y|}^m\sum _{|\alpha |=m}{|\tilde{Q}|}^{-1/u}{\left({\int }_{\tilde{Q}(x,y)}\right|D^{\alpha }\tilde{b}(z){|}^{u}w{(z)}^{u(1-r)/r}w{(z)}^{u(r-1)/r}dz)}^{1/u}\hfill \\ \le C{|x-y|}^m\sum _{|\alpha |=m}{|\tilde{Q}|}^{-1/u}{\left({\int }_{\tilde{Q}(x,y)}\right|D^{\alpha }\tilde{b}(z){|}^{r}w{(z)}^{1-r}dz)}^{1/r}\hfill \\ \times {\left({\int }_{\tilde{Q}(x,y)}w{(z)}^{u(r-1)/(r-u)}dz\right)}^{(r-u)/ru}\hfill \\ \le C{|x-y|}^m\sum _{|\alpha |=m}{|\tilde{Q}|}^{-1/u}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}w{(\tilde{Q})}^{1/r}{|\tilde{Q}|}^{(r-u)/ru}\hfill \\ \times {\left(\frac{1}{|\tilde{Q}(x,y)|}{\int }_{\tilde{Q}(x,y)}w{(z)}^{p_0}dz\right)}^{(r-u)/ru}\hfill \\ \le C{|x-y|}^m\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}{|\tilde{Q}|}^{-1/u}w{(\tilde{Q})}^{1/r}{|\tilde{Q}|}^{1/u-1/r}{(\frac{1}{|\tilde{Q}(x,y)|}{\int }_{\tilde{Q}(x,y)}w(z)dz)}^{(r-1)/r}\hfill \\ \le C{|x-y|}^m\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}{|\tilde{Q}|}^{-1/u}w{(\tilde{Q})}^{1/r}{|\tilde{Q}|}^{1/u-1/r}w{(\tilde{Q})}^{1-1/r}{|\tilde{Q}|}^{1/r-1}\hfill \\ \le C{|x-y|}^m\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}\frac{w(\tilde{Q})}{|\tilde{Q}|}\hfill \\ \le C{|x-y|}^m\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}w(\tilde{x}).\hfill \end{array}$$

Thus, by the $L^s$-boundedness of T (see Lemma 2) for $1<s<r$ and $w\in A_1\subseteq A_{r/s}$, we obtain

$$\begin{array}{rcl}{I}_1& \le & \frac{C}{|Q|}{\int }_Q\left|T\left(\frac{R_m(\tilde{b};x,\cdot )}{{|x-\cdot |}^m}{f}_1\right)\right|dx\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}w(\tilde{x}){\left(\frac{1}{|Q|}{\int }_{R^n}\right|T(f_1)(x){|}^{s}dx)}^{1/s}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}w(\tilde{x}){|Q|}^{-1/s}{\left({\int }_{R^n}\right|f_1(x){|}^{s}dx)}^{1/s}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}w(\tilde{x}){|Q|}^{-1/s}{\left({\int }_{\tilde{Q}}\right|f(x){|}^{s}w{(x)}^{s/r}w{(x)}^{-s/r}dx)}^{1/s}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}w(\tilde{x}){|Q|}^{-1/s}{\left({\int }_{\tilde{Q}}\right|f(x){|}^{r}w(x)dx)}^{1/r}{\left({\int }_{\tilde{Q}}w{(x)}^{-s/(r-s)}dx\right)}^{(r-s)/rs}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}w(\tilde{x}){|Q|}^{-1/s}w{(\tilde{Q})}^{1/r}{\left(\frac{1}{w(\tilde{Q})}{\int }_{\tilde{Q}}\right|f(x){|}^{r}w(x)dx)}^{1/r}\\ \times {\left(\frac{1}{|\tilde{Q}|}{\int }_{\tilde{Q}}w{(x)}^{-s/(r-s)}dx\right)}^{(r-s)/rs}{(\frac{1}{|\tilde{Q}|}{\int }_{\tilde{Q}}w(x)dx)}^{1/r}{|\tilde{Q}|}^{1/s}w{(\tilde{Q})}^{-1/r}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}w(\tilde{x})M_{r,w}(f)(\tilde{x}).\end{array}$$

For $I_2$, by the weak $(L^1,L^1)$ boundedness of T (see Lemma 2) and Kolmogoro’s inequality (see Lemma 1), we obtain

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

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

$$\begin{array}{c}|T^{\tilde{b}}(f_2)(x)-T^{\tilde{b}}(f_2)(x_0)|\hfill \\ \le {\int }_{R^n}|R_m(\tilde{b};x,y)-R_m(\tilde{b};x_0,y)|\frac{|K(x,y)|}{{|x-y|}^m}|f_2(y)|dy\hfill \\ +{\int }_{R^n}|\frac{K(x,y)}{{|x-y|}^m}-\frac{K(x_0,y)}{{|x_0-y|}^m}||R_m(\tilde{b};x_0,y)||f_2(y)|dy\hfill \\ +\sum _{|\alpha |=m}\frac{1}{\alpha !}{\int }_{R^n}|\frac{K(x,y){(x-y)}^{\alpha }}{{|x-y|}^m}-\frac{K(x_0,y){(x_0-y)}^{\alpha }}{{|x_0-y|}^m}||D^{\alpha }\tilde{b}(y)||f_2(y)|dy\hfill \\ =I_3^{(1)}(x)+I_3^{(2)}(x)+I_3^{(3)}(x).\hfill \end{array}$$

For $I_3^{(1)}(x)$, by the formula (see [22])

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

and Lemma 7, we have, similar to the proof of $I_1$,

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

and

$$|R_m(\tilde{b};x_0,y)|\le C\sum _{|\alpha |=m}{|x-x_0|}^m{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}w(\tilde{x}).$$

Thus, by $w\in A_1\subseteq A_r$, we get

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

For $I_3^{(2)}(x)$, we take $1<p<\mathrm{\infty }$ such that $1/p+1/q+1/r=1$. Recalling $r>q^{\prime }$ and $w\in A_1\subseteq A_{r/p+1}$, we get

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

Similarly, we have, for $r<p_1^{\prime }<\mathrm{\infty }$, $1<s_1,s_2<\mathrm{\infty }$ with $1/p_1+1/q+1/r+1/s_1=1$ and $1/q+1/r+1/s_2=1$,

$$\begin{array}{rcl}{I}_3^{(3)}(x)& \le & \sum _{|\alpha |=m}\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}|K(x,y)-K(x_0,y)|\frac{|{(x-y)}^{\alpha }|}{{|x-y|}^m}\\ \times |D^{\alpha }b(y)-{\left(D^{\alpha }b\right)}_{2^{k+1}\tilde{Q}}|w{(y)}^{(1-p_1)/p_1}|f(y)|w{(y)}^{1/r}w{(y)}^{(p_1-1)/p_1-1/r}dy\\ +\sum _{|\alpha |=m}\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}|K(x,y)-K(x_0,y)|\frac{|{(x-y)}^{\alpha }|}{{|x-y|}^m}\\ \times |{\left(D^{\alpha }b\right)}_{2^{k+1}\tilde{Q}}-{\left(D^{\alpha }b\right)}_{\tilde{Q}}||f(y)|w{(y)}^{1/r}w{(y)}^{-1/r}dy\\ +\sum _{|\alpha |=m}{\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_0,y)||f(y)||D^{\alpha }\tilde{b}(y)|dy\\ \le & C\sum _{|\alpha |=m}\sum _{k=0}^{\mathrm{\infty }}{\left({\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}\right|K(x,y)-K(x_0,y){|}^{q}dy)}^{1/q}{\left({\int }_{2^{k+1}\tilde{Q}}\right|f(y){|}^{r}w(y)dy)}^{1/r}\\ \times {\left({\int }_{2^{k+1}\tilde{Q}}\right|D^{\alpha }b(y)-{\left(D^{\alpha }b\right)}_{2^{k+1}\tilde{Q}}{|}^{p_1}w{(y)}^{1-p_1}dy)}^{1/p_1}\\ \times {\left({\int }_{2^{k+1}\tilde{Q}}w{(y)}^{-(1/r-1/p_1^{\prime })s_1}dy\right)}^{1/s_1}\\ +C\sum _{|\alpha |=m}\sum _{k=0}^{\mathrm{\infty }}{\left({\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}\right|K(x,y)-K(x_0,y){|}^{q}dy)}^{1/q}\\ \times {\left({\int }_{2^{k+1}\tilde{Q}}\right|f(y){|}^{r}w(y)dy)}^{1/r}\\ \times |{\left(D^{\alpha }b\right)}_{2^{k+1}\tilde{Q}}-{\left(D^{\alpha }b\right)}_{\tilde{Q}}|{\left({\int }_{2^{k+1}\tilde{Q}}w{(y)}^{-s_2/r}dy\right)}^{1/s_2}\\ +C\sum _{|\alpha |=m}\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}\frac{|x-x_0|}{{|x_0-y|}^{n+1}}\\ \times |f(y)||D^{\alpha }b(y)-{\left(D^{\alpha }b\right)}_{2^{k+1}\tilde{Q}}|w{(y)}^{1/r}w{(y)}^{-1/r}dy\\ +C\sum _{|\alpha |=m}\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}\frac{|x-x_0|}{{|x_0-y|}^{n+1}}\\ \times |f(y)||{\left(D^{\alpha }b\right)}_{2^{k+1}\tilde{Q}}-{\left(D^{\alpha }b\right)}_{\tilde{Q}}|w{(y)}^{1/r}w{(y)}^{-1/r}dy\\ \le & C\sum _{|\alpha |=m}\sum _{k=1}^{\mathrm{\infty }}{C}_k|2^{k}Q{|}^{-1/q^{\prime }}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}w{\left(2^k\tilde{Q}\right)}^{1/p_1}w{\left(2^k\tilde{Q}\right)}^{1/r}\\ \times {\left(\frac{1}{w(2^k\tilde{Q})}{\int }_{2^k\tilde{Q}}\right|f(y){|}^{r}w(y)dy)}^{1/r}\\ \times {(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}w(y)dy)}^{1/r-1/p_1^{\prime }}{\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}w{(y)}^{-(1/r-1/p_1^{\prime })s_1}dy\right)}^{1/s_1}\\ \times |2^k\tilde{Q}{|}^{1/s_1+1/r-1/p_1^{\prime }}w{\left(2^k\tilde{Q}\right)}^{-1/r+1/p_1^{\prime }}\\ +C\sum _{|\alpha |=m}\sum _{k=1}^{\mathrm{\infty }}{C}_k|2^{k}Q{|}^{-1/q^{\prime }}k{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}w(\tilde{x})w{\left(2^k\tilde{Q}\right)}^{1/r}\\ \times {\left(\frac{1}{w(2^k\tilde{Q})}{\int }_{2^k\tilde{Q}}\right|f(y){|}^{r}w(y)dy)}^{1/r}{(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}w(y)dy)}^{1/r}\\ \times {\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}w{(y)}^{-s_2/r}dy\right)}^{1/s_2}|2^k\tilde{Q}{|}^{1/s_2+1/r}w{\left(2^k\tilde{Q}\right)}^{-1/r}\\ +C\sum _{|\alpha |=m}\sum _{k=1}^{\mathrm{\infty }}\frac{d}{{(2^{k}d)}^{n+1}}{\left({\int }_{2^k\tilde{Q}}\right|D^{\alpha }b(y)-{\left(D^{\alpha }b\right)}_{2^k\tilde{Q}}{|}^{r^{\prime }}w{(y)}^{1-r^{\prime }}dy)}^{1/r^{\prime }}\\ \times {\left({\int }_{2^{k+1}\tilde{Q}}\right|f(y){|}^{r}w(y)dy)}^{1/r}\\ +C\sum _{|\alpha |=m}\sum _{k=1}^{\mathrm{\infty }}\frac{d}{{(2^{k}d)}^{n+1}}k{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}w(\tilde{x}){\left({\int }_{2^k\tilde{Q}}w{(y)}^{-r^{\prime }/r}dy\right)}^{1/r^{\prime }}\\ \times {\left({\int }_{2^{k+1}\tilde{Q}}\right|f(y){|}^{r}w(y)dy)}^{1/r}\\ \le & \sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}\sum _{k=1}^{\mathrm{\infty }}{C}_k\frac{w(2^k\tilde{Q})}{|2^k\tilde{Q}|}{\left(\frac{1}{w(2^k\tilde{Q})}{\int }_{2^k\tilde{Q}}\right|f(y){|}^{r}w(y)dy)}^{1/r}\\ +\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}w(\tilde{x})\sum _{k=1}^{\mathrm{\infty }}k{C}_k{\left(\frac{1}{w(2^k\tilde{Q})}{\int }_{2^k\tilde{Q}}\right|f(y){|}^{r}w(y)dy)}^{1/r}\\ +C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}\sum _{k=1}^{\mathrm{\infty }}{2}^{-k}\frac{w(2^k\tilde{Q})}{|2^k\tilde{Q}|}{\left(\frac{1}{w(2^k\tilde{Q})}{\int }_{2^k\tilde{Q}}\right|f(y){|}^{r}w(y)dx)}^{1/r}\\ +C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}w(\tilde{x})\sum _{k=1}^{\mathrm{\infty }}k{2}^{-k}{\left(\frac{1}{w(2^k\tilde{Q})}{\int }_{2^k\tilde{Q}}\right|f(y){|}^{r}w(y)dx)}^{1/r}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}w(\tilde{x})M_{r,w}(f)(\tilde{x}).\end{array}$$

Thus

$$I_3\le C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}w(\tilde{x})M_{r,w}(f)(\tilde{x}).$$

These 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:

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

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 $f_2=f{\chi }_{R^n\setminus \tilde{Q}}$,

$$\begin{array}{c}{\left(\frac{1}{|Q|}{\int }_Q\right|T^b(f)(x)-T^{\tilde{b}}(f_2)(x_0){|}^{\eta }dx)}^{1/\eta }\hfill \\ \le C{\left(\frac{1}{|Q|}{\int }_Q\right|T\left(\frac{R_m(\tilde{b};x,\cdot )}{{|x-\cdot |}^m}{f}_1\right){|}^{\eta }dx)}^{1/\eta }\hfill \\ +C{\left(\frac{1}{|Q|}{\int }_Q\right|T\left(\sum _{|\alpha |=m}\frac{{(x-\cdot )}^{\alpha }{D}^{\alpha }\tilde{b}}{{|x-\cdot |}^m}{f}_1\right){|}^{\eta }dx)}^{1/\eta }\hfill \\ +C{\left(\frac{1}{|Q|}{\int }_Q\right|T^{\tilde{b}}(f_2)(x)-T^{\tilde{b}}(f_2)(x_0){|}^{\eta }dx)}^{1/\eta }\hfill \\ =J_1+J_2+J_3.\hfill \end{array}$$

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

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

Thus

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

For $J_3$, we have

$$\begin{array}{c}|R_m(\tilde{b};x,y)-R_m(\tilde{b};x_0,y)|\hfill \\ \le C\sum _{|\gamma |<m}\sum _{|\alpha |=m}{|x-x_0|}^{m-|\gamma |}{|x-y|}^{|\gamma |}{\parallel D^{\alpha }b\parallel }_{{\mathit{Lip}}_{\beta }(w)}w(\tilde{x})w\left(2^k\tilde{Q}\right)\hfill \end{array}$$

and

$$|R_m(\tilde{b};x,y)|\le C\sum _{|\alpha |=m}{|x-x_0|}^m{\parallel D^{\alpha }b\parallel }_{{\mathit{Lip}}_{\beta }(w)}w(\tilde{x})w\left(2^k\tilde{Q}\right).$$

Thus, for $1<p<\mathrm{\infty }$ with $1/p+1/q+1/r=1$ and $r<p_1^{\prime }<\mathrm{\infty }$, $1<s_1,s_2<\mathrm{\infty }$ with $1/p_1+1/q+1/r+1/s_1=1$ and $1/q+1/r+1/s_2=1$, we obtain

$$\begin{array}{c}|T^{\tilde{b}}(f_2)(x)-T^{\tilde{b}}(f_2)(x_0)|\hfill \\ \le {\int }_{R^n}|R_m(\tilde{b};x,y)-R_m(\tilde{b};x_0,y)|\frac{|K(x,y)|}{{|x-y|}^m}|f_2(y)|dy\hfill \\ +{\int }_{R^n}|\frac{K(x,y)}{{|x-y|}^m}-\frac{K(x_0,y)}{{|x_0-y|}^m}||R_m(\tilde{b};x_0,y)||f_2(y)|dy\hfill \\ +\sum _{|\alpha |=m}\frac{1}{\alpha !}{\int }_{R^n}|\frac{K(x,y){(x-y)}^{\alpha }}{{|x-y|}^m}-\frac{K(x_0,y){(x_0-y)}^{\alpha }}{{|x_0-y|}^m}||D^{\alpha }\tilde{b}(y)||f_2(y)|dy\hfill \\ \le C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{\mathit{Lip}}_{\beta }(w)}w(\tilde{x})\sum _{k=0}^{\mathrm{\infty }}w{\left(2^{k+1}\tilde{Q}\right)}^{\beta /n}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}\frac{|x-x_0|}{{|x_0-y|}^{n+1}}|f(y)|dy\hfill \\ +C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{\mathit{Lip}}_{\beta }(w)}w(\tilde{x})\sum _{k=0}^{\mathrm{\infty }}w{\left(2^{k+1}\tilde{Q}\right)}^{\beta /n}{\left({\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}\right|K(x,y)-K(x_0,y){|}^{q}dy)}^{1/q}\hfill \\ \times {\left({\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}\right|f(y){|}^{r}w(y)dy)}^{1/r}{\left({\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}w{(y)}^{-p/r}dy\right)}^{1/p}\hfill \\ +C\sum _{|\alpha |=m}\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}\frac{|x-x_0|}{{|x_0-y|}^{n+1}}|D^{\alpha }b(y)-{\left(D^{\alpha }b\right)}_{2^{k+1}\tilde{Q}}||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-x_0|}{{|x_0-y|}^{n+1}}|{\left(D^{\alpha }b\right)}_{2^{k+1}\tilde{Q}}-{\left(D^{\alpha }b\right)}_{\tilde{Q}}||f(y)|dy\hfill \\ +C\sum _{|\alpha |=m}\sum _{k=0}^{\mathrm{\infty }}{\left({\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}\right|K(x,y)-K(x_0,y){|}^{q}dy)}^{1/q}{\left({\int }_{2^{k+1}\tilde{Q}}\right|f(y){|}^{r}w(y)dy)}^{1/r}\hfill \\ \times {\left({\int }_{2^{k+1}\tilde{Q}}\right|D^{\alpha }b(y)-{\left(D^{\alpha }b\right)}_{2^{k+1}\tilde{Q}}{|}^{p_1}w{(y)}^{1-p_1}dy)}^{1/p_1}{\left({\int }_{2^{k+1}\tilde{Q}}w{(y)}^{-(1/r-1/p_1^{\prime })s_1}dy\right)}^{1/s_1}\hfill \\ +C\sum _{|\alpha |=m}\sum _{k=0}^{\mathrm{\infty }}{\left({\int }_{2^{k+1}\tilde{Q}\setminus 2^k\tilde{Q}}\right|K(x,y)-K(x_0,y){|}^{q}dy)}^{1/q}{\left({\int }_{2^{k+1}\tilde{Q}}\right|f(y){|}^{r}w(y)dy)}^{1/r}\hfill \\ \times |{\left(D^{\alpha }b\right)}_{2^{k+1}\tilde{Q}}-{\left(D^{\alpha }b\right)}_{\tilde{Q}}|{\left({\int }_{2^{k+1}\tilde{Q}}w{(y)}^{-s_2/r}dy\right)}^{1/s_2}\hfill \\ \le C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{\mathit{Lip}}_{\beta }(w)}w(\tilde{x})\sum _{k=1}^{\mathrm{\infty }}k\frac{d}{{(2^{k}d)}^{n+1}}w{\left(2^k\tilde{Q}\right)}^{\beta /n}{\left({\int }_{2^k\tilde{Q}}\right|f(y){|}^{r}w(y)dx)}^{1/r}\hfill \\ \times {\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}w{(y)}^{-1/(r-1)}dy\right)}^{(r-1)/r}{(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}w(y)dy)}^{1/r}\left|2^k\tilde{Q}\right|w{\left(2^k\tilde{Q}\right)}^{-1/r}\hfill \\ +C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{\mathit{Lip}}_{\beta }(w)}w(\tilde{x})\sum _{k=1}^{\mathrm{\infty }}{C}_k|2^{k}Q{|}^{-1/q^{\prime }}w{\left(2^k\tilde{Q}\right)}^{\beta /n}{\left({\int }_{2^k\tilde{Q}}\right|f(y){|}^{r}w(y)dx)}^{1/r}\hfill \\ \times {\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}w{(y)}^{-p/r}dy\right)}^{1/p}{(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}w(y)dy)}^{1/r}|2^k\tilde{Q}{|}^{1/p+1/r}w{\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}}{\left({\int }_{2^k\tilde{Q}}\right|D^{\alpha }b(y)-{\left(D^{\alpha }b\right)}_{2^k\tilde{Q}}{|}^{r^{\prime }}w{(y)}^{1-r^{\prime }}dy)}^{1/r^{\prime }}\hfill \\ \times {\left({\int }_{2^k\tilde{Q}}\right|f(y){|}^{r}w(y)dy)}^{1/r}\hfill \\ +C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{\mathit{Lip}}_{\beta }(w)}w(\tilde{x})\sum _{k=1}^{\mathrm{\infty }}{C}_k|2^{k}Q{|}^{-1/q^{\prime }}w{\left(2^k\tilde{Q}\right)}^{\beta /n}{\left({\int }_{2^k\tilde{Q}}\right|f(y){|}^{r}w(y)dx)}^{1/r}\hfill \\ \times {\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}w{(y)}^{-(1/r-1/p_1^{\prime })s_1}dy\right)}^{1/s_1}{(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}w(y)dy)}^{1/r-1/p_1^{\prime }}\hfill \\ \times |2^k\tilde{Q}{|}^{1/s_1+1/r-1/p_1^{\prime }}w{\left(2^k\tilde{Q}\right)}^{-1/r+1/p_1^{\prime }}\hfill \\ +C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{\mathit{Lip}}_{\beta }(w)}w(\tilde{x})\sum _{k=1}^{\mathrm{\infty }}k{C}_k|2^{k}Q{|}^{-1/q^{\prime }}w{\left(2^k\tilde{Q}\right)}^{\beta /n}{\left({\int }_{2^k\tilde{Q}}\right|f(y){|}^{r}w(y)dy)}^{1/r}\hfill \\ \times {(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}w(y)dy)}^{1/r}{\left(\frac{1}{|2^k\tilde{Q}|}{\int }_{2^k\tilde{Q}}w{(y)}^{-s_2/r}dy\right)}^{1/s_2}|2^k\tilde{Q}{|}^{1/s_2+1/r}w{\left(2^k\tilde{Q}\right)}^{-1/r}\hfill \\ \le C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{\mathit{Lip}}_{\beta }(w)}w(\tilde{x})\sum _{k=1}^{\mathrm{\infty }}k(C_k+2^{-k})\hfill \\ \times {\left(\frac{1}{w{(2^k\tilde{Q})}^{1-r\beta /n}}{\int }_{2^k\tilde{Q}}\right|f(y){|}^{r}w(y)dx)}^{1/r}\hfill \\ +C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{\mathit{Lip}}_{\beta }(w)}\sum _{k=1}^{\mathrm{\infty }}k(C_k+2^{-k})\frac{w(2^k\tilde{Q})}{|2^k\tilde{Q}|}\hfill \\ \times {\left(\frac{1}{w{(2^k\tilde{Q})}^{1-r\beta /n}}{\int }_{2^k\tilde{Q}}\right|f(y){|}^{r}w(y)dx)}^{1/r}\hfill \\ \le C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{\mathit{Lip}}_{\beta }(w)}w(\tilde{x})M_{\beta ,r,w}(f)(\tilde{x}).\hfill \end{array}$$

This completes the proof of Theorem 2. □

Proof of Theorem 3 Choose $1<r<u$ in Theorem 1 and notice $w^{1-u}\in A_1$, then we have, by Lemmas 3 and 4,

$$\begin{array}{rcl}{\parallel T^b(f)\parallel }_{L^u(w^{1-u})}& \le & {\parallel M_{\eta }(T^b(f))\parallel }_{L^u(w^{1-u})}\le C{\parallel M_{\eta }^{\mathrm{\#}}(T^b(f))\parallel }_{L^u(w^{1-u})}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}{\parallel w{M}_{r,w}(f)\parallel }_{L^u(w^{1-u})}\\ =& C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}{\parallel M_{r,w}(f)\parallel }_{L^u(w)}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}{\parallel f\parallel }_{L^u(w)}.\end{array}$$

This completes the proof of Theorem 3. □

Proof of Theorem 4 Choose $1<r<u$ in Theorem 1 and notice $w^{1-u}\in A_1$, then we have, by Lemmas 5 and 6,

$$\begin{array}{rcl}{\parallel T^b(f)\parallel }_{L^{u,\phi }(w^{1-u})}& \le & {\parallel M_{\eta }(T^b(f))\parallel }_{L^{u,\phi }(w^{1-u})}\le C{\parallel M_{\eta }^{\mathrm{\#}}(T^b(f))\parallel }_{L^{u,\phi }(w^{1-u})}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}{\parallel w{M}_{r,w}(f)\parallel }_{L^{u,\phi }(w^{1-u})}\\ =& C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}{\parallel M_{r,w}(f)\parallel }_{L^{u,\phi }(w)}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{\mathit{BMO}(w)}{\parallel f\parallel }_{L^{u,\phi }(w)}.\end{array}$$

This completes the proof of Theorem 4. □

Proof of Theorem 5 Choose $1<r<u$ in Theorem 2 and notice $w^{1-v}\in A_1$, then we have, by Lemmas 3 and 4,

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

This completes the proof of Theorem 5. □

Proof of Theorem 6 Choose $1<r<u$ in Theorem 2 and notice $w^{1-v}\in A_1$, then we have, by Lemmas 5 and 6,

$$\begin{array}{rcl}{\parallel T^b(f)\parallel }_{L^{v,\phi }(w^{1-v})}& \le & {\parallel M_{\eta }(T^b(f))\parallel }_{L^{v,\phi }(w^{1-v})}\le C{\parallel M_{\eta }^{\mathrm{\#}}(T^b(f))\parallel }_{L^{v,\phi }(w^{1-v})}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{\mathit{Lip}}_{\beta }(w)}{\parallel w{M}_{\beta ,r,w}(f)\parallel }_{L^{v,\phi }(w^{1-v})}\\ =& C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{\mathit{Lip}}_{\beta }(w)}{\parallel M_{\beta ,r,w}(f)\parallel }_{L^{v,\phi }(w)}\\ \le & C\sum _{|\alpha |=m}{\parallel D^{\alpha }b\parallel }_{{\mathit{Lip}}_{\beta }(w)}{\parallel f\parallel }_{L^{u,\phi }(w)}.\end{array}$$

This completes the proof of Theorem 6. □