$M^k$-type sharp estimates and boundedness on Morrey space for Toeplitz type operators associated to fractional integral and singular integral operator with general kernel

Abstract

In this paper, we prove the $M^k$-type sharp maximal function estimates for the Toeplitz type operators associated to the fractional integral and singular integral operator with general kernel. As an application, we obtain the weighted boundedness of the operators on the Morrey space.

1 Introduction

As the development of singular integral operators (see [6, 15]), their commutators have been well studied. In [3, 14], the authors prove that the commutators generated by the singular integral operators and ${ BMO}$ functions are bounded on $L^p(R^n)$ for $1<p<\infty$. Chanillo (see [2]) proves a similar result when singular integral operators are replaced by the fractional integral operators. In [1], some singular integral operators with general kernel are introduced, and the boundedness for the operators and their commutators generated by ${ BMO}$ and Lipschitz functions are obtained(see [1, 9]). In [7, 8], some Toeplitz type operators related to the singular integral operators and strongly singular integral operators are introduced, and the boundedness for the operators generated by ${ BMO}$ and Lipschitz functions are obtained. In this paper, we will study the Toeplitz type operators generated by the fractional integral and singular integral operators with general kernel and the ${ BMO}$ functions.

2 Preliminaries

First, let us introduce some notations. Throughout this paper, Q will denote 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

$$\begin{aligned} f^{\#}(x)=\sup _{Q\ni x}\frac{1}{|Q|}\int _Q |f(y)-f_Q|dy, \end{aligned}$$

where, and in what follows, $f_Q=|Q|^{-1}\int _Q f(x)dx$. It is well-known that (see [6, 15])

$$\begin{aligned} f^{\#}(x)\approx \sup _{Q\ni x}\inf _{c\in C}\frac{1}{|Q|}\int _Q|f(y)-c|dy. \end{aligned}$$

We say that f belongs to ${ BMO}(R^n)$ if $f^{\#}$ belongs to $L^\infty (R^n)$ and define $||f||_{{ BMO}}=||f^{\#}||_{L^\infty }$. It has been known that (see [15])

$$\begin{aligned} \left| \left| f-f_{2^kQ}\right| \right| _{{ BMO}}\le Ck||f||_{{ BMO}}. \end{aligned}$$

For $0<r<\infty$, we denote $f_r^{\#}$ by

$$\begin{aligned} f_r^{\#}(x)=\left[ \left( |f|^r\right) ^{\#}(x)\right] ^{1/r}. \end{aligned}$$

Let M be the Hardy-Littlewood maximal operator defined by

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

For $\eta >0$, let $M_{\eta }(f)=M(|f|^{\eta })^{1/\eta }.$ For $k\in N$, we denote by $M^k$ the operator M iterated k times, i.e., $M^1(f)=M(f)$ and

$$\begin{aligned} M^k(f)=M\left( M^{k-1}(f)\right) \ \text{ when }\ \ k\ge 2. \end{aligned}$$

For $0<\eta <n$ and $1\le r<\infty$, set

$$\begin{aligned} M_{\eta ,r}(f)(x)=\sup _{Q\ni x}\left( \frac{1}{|Q|^{1-r\eta /n}}\int _Q|f(y)|^rdy\right) ^{1/r}. \end{aligned}$$

Let $\Phi$ be a Young function and $\tilde{\Phi }$ be the complementary associated to $\Phi$, we denote that the $\Phi$-average by, for a function f,

$$\begin{aligned} ||f||_{\Phi ,Q}=\inf \left\{ \lambda >0: \frac{1}{|Q|}\int _Q \Phi \left( \frac{|f(y)|}{\lambda }\right) dy \le 1\right\} \end{aligned}$$

and the maximal function associated to $\Phi$ by

$$\begin{aligned} M_\Phi (f)(x)=\sup _{x\in Q}||f||_{\Phi ,Q}. \end{aligned}$$

The Young functions to be using in this paper are $\Phi (t)=t(1+logt)$ and $\tilde{\Phi }(t)=exp(t)$, the corresponding average and maximal functions denoted by $||\cdot ||_{L(logL), Q}, M_{L(logL)}$ and $||\cdot ||_{expL, Q}, M_{expL}$. Following [15], we know the generalized Hölder’s inequality and the following inequalities hold:

$$\begin{aligned} \frac{1}{|Q|}\int _Q|f(y)g(y)|dy\le & {} \left| \left| f\right| \right| _{\Phi ,Q}||g||_{\tilde{\Phi },Q},\\ ||f||_{L(logL), Q}\le & {} M_{L(logL)}(f)\le C M^2(f),\\ ||f-f_Q||_{exp L,Q}\le & {} C||f||_{{ BMO}}. \end{aligned}$$

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

$$\begin{aligned} A_p\!=\! & {} \left\{ w\!\in \! L_{loc}^1(R^n):\sup _Q\left( \frac{1}{|Q|}\int _Q\! w(x)dx\right) \left( \frac{1}{|Q|}\int _Q\!w(x)^{-1/(p-1)}dx\right) ^{p-1}\!\!<\!\infty \right\} \!,\\&\quad 1<p<\infty , \end{aligned}$$

and

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

Definition 1

Let $\varphi$ be a positive, increasing function on $R^+$ and there exists a constant $D>0$ such that

$$\begin{aligned} \varphi (2t)\le D\varphi (t) \ \ \text{ for } \ \ t\ge 0. \end{aligned}$$

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

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

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

$$\begin{aligned} L^{p,\varphi }(R^n, w)=\left\{ f\in L^1_{loc}\left( R^n\right) : ||f||_{L^{p,\varphi }(w)}<\infty \right\} . \end{aligned}$$

If $\varphi (d)=d^\delta , \delta >0$, then $L^{p,\varphi }(R^n, w)=L^{p,\delta }(R^n, w)$, which is the classical weighted Morrey spaces (see [12, 13]). If $\varphi (d)=1$, then $L^{p,\varphi }(R^n, w)=L^p(R^n, w)$, which is the weighted Lebesgue spaces (see [10]).

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 [4, 5, 10, 11]).

In this paper, we will study some singular integral operators as following(see [1]).

Definition 2

Let $T: S\rightarrow S'$ be a linear operator such that T is bounded on $L^2(R^n)$ and 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

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

for every bounded and compactly supported function f, where K satisfies: there is a sequence of positive constant numbers $\{C_j\}$ such that for any $j\ge 1$,

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

and

$$\begin{aligned}&\left( \int _{2^j|z-y|\le |x-y|<2^{j+1}|z-y|}(|K(x,y)-K(x,z)|+|K(y,x)-K(z,x)|)^qdy\right) ^{1/q}\\&\quad \le C_j(2^j|z-y|)^{-n/q'}, \end{aligned}$$

where $1<q'<2$ and $1/q+1/q'=1$.

Moreover, let b be a locally integrable function on $R^n$. The Toeplitz type operators associated to T are defined by

$$\begin{aligned} T_b=\sum _{k=1}^mT^{k,1}M_bT^{k,2} \end{aligned}$$

and

$$\begin{aligned} S_b=\sum _{k=1}^m\left( T^{k,3}M_bI_\alpha T^{k,4}+T^{k,5}I_\alpha M_bT^{k,6}\right) , \end{aligned}$$

where $T^{k,1}$ and $T^{k,3}$ are T or $\pm I$(the identity operator), $T^{k,2}, T^{k,4}$ and $T^{k,6}$ are the bounded linear operators on $L^p(R^n,w)$ for $1<p<\infty$ and $w\in A_1, T^{k,5}=\pm I, k=1,...,m, M_b(f)=bf$ and $I_\alpha$ is the fractional integral operator$(0<\alpha <n)$(see [2]).

Note that the classical Calderón-Zygmund singular integral operator satisfies Definition 2 with $C_j=2^{-j\delta }$(see [6, 15]). And Note that the commutator $[b, T](f)=bT(f)-T(bf)$ is a particular operator of the Toeplitz type operators $T_b$ and $S_b$. The Toeplitz type operators are the non-trivial generalizations of the commutator. It is well known that commutators are of great interest in harmonic analysis and have been widely studied by many authors (see [15]). The main purpose of this paper is to prove the sharp maximal inequalities for the Toeplitz type operators $T_b$ and $S_b$. As the application, we obtain the boundedness on the Morrey space for the Toeplitz type operators $T_b$ and $S_b$.

3 Theorems and Lemmas

We shall prove the following theorems.

Theorem 1

Let T be the singular integral operator as Definition 2, the sequence $\{jC_j\}\in l^1, 0<r<1, q'\le s<\infty$ and $b\in { BMO}(R^n)$. If $g\in L^\rho (R^n)(1<\rho <\infty )$ and $T_1(g)=0$, then there exists a constant $C>0$ such that, for any $f\in C_0^\infty (R^n)$ and $\tilde{x}\in R^n$,

$$\begin{aligned} (T_b(f))_r^{\#}(\tilde{x})\le C||b||_{{ BMO}}\sum _{k=1}^m\left( M^2(T^{k,2}(f)\right) (\tilde{x})+M_s\left( T^{k,2}(f)\right) (\tilde{x})). \end{aligned}$$

Theorem 2

Let T be the singular integral operator as Definition 2, the sequence $\{jC_j\}\in l^1, 0<r<1, q'\le s<\infty$ and $b\in { BMO}(R^n)$. If $g\in L^\rho (R^n)(1<\rho <\infty )$ and $S_1(g)=0$, then there exists a constant $C>0$ such that, for any $f\in C_0^\infty (R^n)$ and $\tilde{x}\in R^n$,

$$\begin{aligned} (S_b(f))^{\#}_r(\tilde{x})\le & {} C||b||_{{ BMO}}\sum _{k=1}^m\Big (M^2\left( I_\alpha T^{k,4}(f)\Big )(\tilde{x})+M_s\left( I_\alpha T^{k,4}(f)\right) (\tilde{x})\right. \\&\left. +M_{\alpha ,s}\left( T^{k,6}(f)\right) (\tilde{x})\right) . \end{aligned}$$

Theorem 3

Let T be the singular integral operator as Definition 2, the sequence $\{jC_j\}\in l^1, q'<p<\infty , 0<D<2^n, w\in A_1$ and $b\in { BMO}(R^n)$. If $g\in L^\rho (R^n)(1<\rho <\infty )$ and $T_1(g)=0$, then $T_b$ is bounded on $L^{p,\varphi }(R^n, w)$.

Theorem 4

Let T be the singular integral operator as Definition 2, the sequence $\{jC_j\}\in l^1, 0<D<2^n, q'<u<n/\alpha , 1/v=1/u-\alpha /n, w\in A_1$ and $b\in { BMO}(R^n)$. If $g\in L^\rho (R^n)(1<\rho <\infty )$ and $S_1(g)=0$, then $S_b$ is bounded from $L^{u,\varphi }(R^n, w)$ to $L^{v,\varphi }(R^n, w)$.

To prove the theorems, we need the following lemmas.

Lemma 1

([6, p. 485]) Let $0<p<q<\infty$ and for any function $f\ge 0$. We define that, for $1/r=1/p-1/q$

$$\begin{aligned} ||f||_{WL^q}= & {} \sup _{\lambda >0}\lambda \left| \left\{ x\in R^n: f(x)>\lambda \right\} \right| ^{1/q}, \\ N_{p,q}(f)= & {} \sup _E ||f\chi _E||_{L^p}/ ||\chi _E||_{L^r}, \end{aligned}$$

where the sup is taken for all measurable sets E with $0<|E|<\infty$. Then

$$\begin{aligned} ||f||_{WL^q}\le N_{p,q}(f)\le (q/(q-p))^{1/p} ||f||_{WL^q}. \end{aligned}$$

Lemma 2

(see [15]) We have

$$\begin{aligned} \frac{1}{|Q|}\int _Q |f(x)g(x)|dx\le ||f||_{expL, Q}||g||_{L(logL), Q}. \end{aligned}$$

Lemma 3

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

Lemma 4

(see [6]) Let $0<p, \eta <\infty$ and $w\in \cup _{1\le r<\infty } A_r$. Then, for any smooth function f for which the left-hand side is finite,

$$\begin{aligned} \int _{R^n}M_\eta (f)(x)^pw(x)dx\le C\int _{R^n}f^{\#}_\eta (x)^pw(x)dx. \end{aligned}$$

Lemma 5

(see [2, 6]) Let $w\in A_1, 0<\alpha <n, 1\le s<p<n/\alpha$ and $1/r=1/p-\alpha /n$. Then

$$\begin{aligned} ||M_s(f)||_{L^p(w)}\le & {} C||f||_{L^p(w)},\\ ||M_{\alpha , s}(f)||_{L^r(w)}\le & {} C||f||_{L^p(w)} \end{aligned}$$

and

$$\begin{aligned} ||I_\alpha (f)||_{L^q(w)}\le C||f||_{L^p(w)}. \end{aligned}$$

Lemma 6

(see [4, 10]) Let $1<p<\infty , w\in A_1$ and $0<D<2^n$. Then, for any smooth function f for which the left-hand side is finite,

$$\begin{aligned} ||M(f)||_{L^{p,\varphi }(w)}\le C||f||_{L^{p,\varphi }(w)}. \end{aligned}$$

Lemma 7

Let $1<p<\infty , 0<\eta <\infty , w\in A_1$ and $0<D<2^n$. Then, for any smooth function f for which the left-hand side is finite,

$$\begin{aligned} ||M_\eta (f)||_{L^{p,\varphi }(w)}\le C||f^{\#}_\eta ||_{L^{p,\varphi }(w)}. \end{aligned}$$

Proof

For any cube $Q=Q(x_0, d)$ in $R^n$, we know $M(w\chi _Q)\in A_1$ for any cube $Q=Q(x,d)$ by [6]. If $x\in Q^c$, by Lemma 4, we have, for $f\in L^{p,\varphi }(R^n, w)$,

$$\begin{aligned}&\int _Q\left| M_\eta (f)(y)\right| ^pw(y)dy \\&\quad =\int _{R^n} M_\eta (f)(y)^pw(y)\chi _Q(y)dy \\&\quad \le \int _{R^n}\left| M_\eta (f)(y)\right| ^pM(w\chi _Q)(y)dy \\&\quad \le C\int _{R^n}\left| f^{\#}_\eta (y)\right| ^p M(w\chi _Q)(y)dy \\&\quad = C\left( \int _Q\left| f^{\#}_\eta (y)\right| ^p M(w\chi _Q)(y)dy+\sum _{k=0}^\infty \int _{2^{k+1}Q\setminus 2^kQ}\left| f^{\#}_\eta (y)\right| ^pM(w\chi _Q)(y)dy\right) \\&\quad \le C\left( \int _Q\left| f^{\#}_\eta (y)\right| ^pw(y)dy+\sum _{k=0}^\infty \int _{2^{k+1}Q\setminus 2^kQ}\left| f^{\#}_\eta (y)\right| ^p\frac{w(Q)}{|2^{k+1}Q|}dy\right) \\&\quad \le C\left( \int _Q\left| f^{\#}_\eta (y)\right| ^pw(y)dy+\sum _{k=0}^\infty \int _{2^{k+1}Q}\left| f^{\#}_\eta (y)\right| ^p\frac{M(w)(y)}{2^{n(k+1)}}dy\right) \\&\quad \le C\left( \int _Q\left| f^{\#}_\eta (y)\right| ^pw(y)dy+\sum _{k=0}^\infty \int _{2^{k+1}Q}\left| f^{\#}_\eta (y)\right| ^p\frac{w(y)}{2^{nk}}dy\right) \\&\quad \le C\left| \left| f^{\#}_\eta \right| \right| _{L^{p,\varphi }(w)}^p\sum _{k=0}^\infty 2^{-nk}\varphi \left( 2^{k+1}d\right) \\&\quad \le C\left| \left| f^{\#}_\eta \right| \right| _{L^{p,\varphi }(w)}^p\sum _{k=0}^\infty \left( 2^{-n}D\right) ^k \varphi (d) \\&\quad \le C\left| \left| f^{\#}_\eta \right| \right| _{L^{p,\varphi }(w)}^p \varphi (d), \end{aligned}$$

thus

$$\begin{aligned} \left( \frac{1}{\varphi (d)}\int _QM_\eta (f)(x)^pw(x)dx\right) ^{1/p}\le C\left( \frac{1}{\varphi (d)}\int _Qf^{\#}_\eta (x)^pw(x)dx\right) ^{1/p} \end{aligned}$$

and

$$\begin{aligned} \left| \left| M_\eta (f)\right| \right| _{L^{p,\varphi }(w)}\le C\left| \left| f^{\#}_\eta \right| \right| _{L^{p,\varphi }(w)}. \end{aligned}$$

This finishes the proof. $\square$

Lemma 8

Let $w\in A_1, 0<\alpha <n, 0<D<2^n, 1\le s<p<n/\alpha$ and $1/r=1/p-\alpha /n$. Then

$$\begin{aligned} \left| \left| M_s(f)\right| \right| _{L^{p,\varphi }(w)}\le & {} C\left| \left| f\right| \right| _{L^{p,\varphi }(w)},\\ ||M_{\alpha , s}(f)||_{L^{r,\varphi }(w)}\le & {} C||f||_{L^{p,\varphi }(w)} \end{aligned}$$

and

$$\begin{aligned} ||I_\alpha (f)||_{L^{r,\varphi }(w)}\le C||f||_{L^{p,\varphi }(w)}. \end{aligned}$$

Lemma 9

Let T be the bounded linear operators on $L^q(R^n,w)$ for any $1<q<\infty$ and $w\in A_1$. Then, for $1<p<\infty , w\in A_1$ and $0<D<2^n$,

$$\begin{aligned} \left| \left| T(f)\right| \right| _{L^{p,\varphi }(w)}\le C\left| \left| f\right| \right| _{L^{p,\varphi }(w)}. \end{aligned}$$

The proofs of two lemmas are similar to that of Lemma 7 by Lemma 5, we omit the details.

4 Proofs of Theorems

Proof of Theorem 1

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

$$\begin{aligned} \left( \frac{1}{|Q|}\int _Q\big |T_b(f)(x)-C_0\big |^rdx\right) ^{1/r}\le & {} C||b||_{{ BMO}}\sum _{k=1}^m\left( M^2(T^{k,2}(f)\right) (\tilde{x})\\&+M_s\left( T^{k,2}(f)\right) (\tilde{x})). \end{aligned}$$

Without loss of generality, we may assume $T^{k,1}$ are $T(k=1,\ldots ,m)$. Fix a cube $Q=Q(x_0,d)$ and $\tilde{x}\in Q$. By $T_1(g)=0$, we have

$$\begin{aligned} T_b(f)(x)= & {} T_{b-b_{2Q}}(f)(x)=T_{(b-b_{2Q})\chi _{2Q}}(f)(x)+T_{(b-b_{2Q})\chi _{(2Q)^c}}(f)(x)\\= & {} f_1(x)+f_2(x) \end{aligned}$$

and

$$\begin{aligned}&\left( \frac{1}{|Q|}\int _Q\big |T_b(f)(x)-f_2(x_0)\big |^rdx\right) ^{1/r} \\&\quad \le C\left( \frac{1}{|Q|}\int _Q\big |f_1(x)\big |^rdx\right) ^{1/r}+C\left( \frac{1}{|Q|}\int _Q\big |f_2(x)-f_2(x_0)\big |^rdx\right) ^{1/r} \\&\quad = I+II. \end{aligned}$$

For I, by Lemmas 1, 2 and 3, we obtain

$$\begin{aligned}&\left( \frac{1}{|Q|}\int _Q|T^{k,1}M_{(b-b_{2Q})\chi _{2Q}}T^{k,2}(f)(x)|^rdx\right) ^{1/r}\\&\quad \le |Q|^{-1}\frac{||T^{k,1}M_{(b-b_{2Q})\chi _{2Q}}T^{k,2}(f)\chi _Q||_{L^r}}{|Q|^{1/r-1}}\\&\quad \le C|Q|^{-1}||T^{k,1}M_{(b-b_{2Q})\chi _{2Q}}T^{k,2}(f)||_{WL^1} \\&\quad \le C|Q|^{-1}||M_{(b-b_{2Q})\chi _{2Q}}T^{k,2}(f)||_{L^1} \\&\quad \le C|Q|^{-1}\int _{2Q}|b(x)-b_{2Q}||T^{k,2}(f)(x)|dx \\&\quad \le C||b-b_{2Q}||_{expL, 2Q}||T^{k,2}(f)||_{L(logL),2Q} \\&\quad \le C||b||_{{ BMO}}M^2(T^{k,2}(f))(\tilde{x}), \end{aligned}$$

thus

$$\begin{aligned} I\le & {} \sum _{k=1}^m\left( \frac{1}{|Q|}\int _Q|T^{k,1}M_{(b-b_{2Q})\chi _{2Q}}T^{k,2}(f)(x)|^rdx\right) ^{1/r} \\\le & {} C||b||_{{ BMO}}\sum _{k=1}^mM^2(T^{k,2}(f))(\tilde{x}). \end{aligned}$$

For II, recalling that $s>q'$, taking $1<p<\infty , 1<t<s$ with $1/p+1/q+1/t=1$, by the Hölder’s inequality, we get, for $x\in Q$,

$$\begin{aligned}&|T^{k,1}M_{(b-b_Q)\chi _{(2Q)^c}}(f)(x)-T^{k,1}M_{(b-b_Q)\chi _{(2Q)^c}}T^{k,2}(f)(x_0)| \\&\quad \le \sum _{j=1}^\infty \int _{2^{j}d\le |y-x_0|<2^{j+1}d}|K(x,y)-K(x_0,y)||b(y)-b_{2Q}||T^{k,2}(f)(y)|dy\\&\quad \le \sum _{j=1}^\infty \left( \int _{2^{j}d\le |y-x_0|<2^{j+1}d}|K(x,y)-K(x_0,y)|^qdy\right) ^{1/q} \\&\qquad \times \left( \int _{2^{j+1}Q}\left| b(y)-b_Q\right| ^pdy\right) ^{1/p}\left( \int _{2^{j+1}Q}\left| T^{k,2}(f)(y)\right| ^tdy\right) ^{1/t} \\&\quad \le C\left| \left| b\right| \right| _{{ BMO}}\sum _{j=1}^\infty C_j(2^jd)^{-n/q'}j(2^jd)^{n/p}(2^jd)^{n/s}\\&\qquad \times \left( \frac{1}{|2^{j+1}Q|}\int _{2^{j+1}Q}\left| T^{k,2}(f)(y)\right| ^sdy\right) ^{1/s} \\&\quad \le C||b||_{{ BMO}}M_s\left( T^{k,2}(f)\right) (\tilde{x})\sum _{j=1}^\infty jC_j \\&\quad \le C||b||_{{ BMO}}M_s\left( T^{k,2}(f)\right) (\tilde{x}), \end{aligned}$$

thus

$$\begin{aligned} II\le & {} \frac{1}{|Q|}\int _Q\sum _{k=1}^m|T^{k,1}M_{(b-b_Q)\chi _{(2Q)^c}}T^{k,2}(f)(x)\!-\!T^{k,1}M_{(b-b_Q)\chi _{(2Q)^c}}T^{k,2}(f)(x_0)|dx\\\le & {} C||b||_{{ BMO}}\sum _{k=1}^lM_s\left( T^{k,2}(f)\right) (\tilde{x}). \end{aligned}$$

This completes the proof of Theorem 1. $\square$

Proof of Theorem 2

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

$$\begin{aligned} \left( \frac{1}{|Q|}\int _Q\big |S_b(f)(x)-C_0\big |^rdx\right) ^{1/r}\le & {} C||b||_{{ BMO}}\sum _{k=1}^m(M^2\left( I_\alpha T^{k,4}(f)\right) (\tilde{x})\\&+M_s\left( I_\alpha T^{k,4}(f)\right) (\tilde{x})\!+\!M_{\alpha ,s}\left( T^{k,6}(f)\right) (\tilde{x})). \end{aligned}$$

Without loss of generality, we may assume $T^{k,3}$ are $T(k=1,\ldots ,m)$. Fix a cube $Q=Q(x_0,d)$ and $\tilde{x}\in Q$. Write

$$\begin{aligned} S_b(f)(x)= & {} \sum _{k=1}^mT^{k,3}M_bI_\alpha T^{k,4}(f)(x)+\sum _{k=1}^mT^{k,5}I_\alpha M_bT^{k,6}(f)(x)\\= & {} A_b(x)+B_b(x)=A_{b-b_Q}(x)+B_{b-b_Q}(x), \end{aligned}$$

where

$$\begin{aligned} A_{b-b_Q}(x)= & {} \sum _{k=1}^mT^{k,3}M_{(b-b_Q)\chi _{2Q}}I_\alpha T^{k,4}(f)(x)\\&+\sum _{k=1}^mT^{k,3} M_{(b-b_Q)\chi _{(2Q)^c}}I_\alpha T^{k,4}(f)(x)=A_1(x)+A_2(x) \end{aligned}$$

and

$$\begin{aligned} B_{b-b_Q}(x)= & {} \sum _{k=1}^mT^{k,5}I_\alpha M_{(b-b_Q)\chi _{2Q}}T^{k,6}(f)(x)\\&+\sum _{k=1}^mT^{k,5}I_\alpha M_{(b-b_Q)\chi _{(2Q)^c}}T^{k,6}(f)(x)=B_1(x)+B_2(x). \end{aligned}$$

Then

$$\begin{aligned}&\left( \frac{1}{|Q|}\int _Q\left| A_{b-b_Q}(f)(x)-A_2(x_0)\right| ^rdx\right) ^{1/r} \\&\quad \le C\left( \frac{1}{|Q|}\int _Q|A_1(x)|^rdx\right) ^{1/r}+C\left( \frac{1}{|Q|}\int _Q|A_2(x)-A_2(x_0)|^rdx\right) ^{1/r} \\&\quad =I_1+I_2 \end{aligned}$$

and

$$\begin{aligned}&\left( \frac{1}{|Q|}\int _Q\left| B_{b-b_Q}(f)(x)-B_2(x_0)\right| ^rdx\right) ^{1/r} \\&\quad \le C\left( \frac{1}{|Q|}\int _Q|B_1(x)|^rdx\right) ^{1/r}+C\left( \frac{1}{|Q|}\int _Q|B_2(x)-B_2(x_0)|^rdx\right) ^{1/r} \\&\quad =I_3+I_4. \end{aligned}$$

By using the same argument as in the proof of Theorem 1, we get, for $1<p<\infty , 1<t_1<s$ with $1/p+1/q+1/t_1=1, 1/t_2=1/p-\alpha /n$ with $1<p<s$,

$$\begin{aligned} I_1\le & {} C\sum _{k=1}^m\left( \frac{1}{|Q|}\int _{R^n}|T^{k,3}M_{(b-b_Q)\chi _{2Q}}I_\alpha T^{k,4}(f)(x)|^rdx\right) ^{1/r}\\\le & {} \sum _{k=1}^m|Q|^{-1}\frac{\left| \left| T^{k,3}M_{(b-b_Q)\chi _{2Q}}I_\alpha T^{k,4}(f)\chi _Q\right| \right| _{L^r}}{|Q|^{1/r-1}}\\\le & {} C\sum _{k=1}^m|Q|^{-1}\left| \left| T^{k,3}M_{(b-b_Q)\chi _{2Q}}I_\alpha T^{k,4}(f)\right| \right| _{WL^1} \\\le & {} C\sum _{k=1}^m|Q|^{-1}\left| \left| M_{(b-b_Q)\chi _{2Q}}I_\alpha T^{k,4}(f)\right| \right| _{L^1} \\\le & {} C\sum _{k=1}^m|Q|^{-1}\int _{2Q}\big |b(x)-b_{2Q}\big |\left| I_\alpha T^{k,4}(f)(x)\right| dx \\\le & {} C\sum _{k=1}^m||b-b_{2Q}||_{expL, 2Q}||I_\alpha T^{k,4}(f)||_{L(logL),2Q} \\\le & {} C||b||_{{ BMO}}\sum _{k=1}^mM^2\left( I_\alpha T^{k,4}(f)\right) (\tilde{x}), \end{aligned}$$

$$\begin{aligned} I_2\le & {} C\sum _{k=1}^m\frac{1}{|Q|}\int _Q\sum _{j=1}^\infty \int _{2^{j}d\le |y-x_0|<2^{j+1}d}|b(y)-b_{2Q}||K(x,y)\\&-K(x_0,y)||I_\alpha T^{k,4}(f)(y)|dydx\\\le & {} C\sum _{k=1}^m\frac{C}{|Q|}\int _Q\sum _{j=1}^\infty \left( \int _{2^{j}d\le |y-x_0|<2^{j+1}d}|K(x,y)-K(x_0,y)|^qdy\right) ^{1/q} \\&\times \left( \int _{2^{j+1}Q}|b(y)-b_Q|^pdy\right) ^{1/p}\left( \int _{2^{j+1}Q}|I_\alpha T^{k,4}(f)(y)|^{r_2}dy\right) ^{1/{t_1}}dx \\\le & {} C||b||_{{ BMO}}\sum _{k=1}^m\sum _{j=1}^{\infty }C_j(2^jd)^{-n/q'}j(2^jd)^{n/p}(2^jd)^{n/s}\\&\quad \times \left( \frac{1}{|2^{j+1}Q|}\int _{2^{j+1}Q}|I_\alpha T^{k,4}(f)(y)|^sdy\right) ^{1/s} \\\le & {} C||b||_{{ BMO}}\sum _{k=1}^mM_s\left( I_\alpha T^{k,4}(f)\right) (\tilde{x})\sum _{j=1}^{\infty }jC_j \\\le & {} C||b||_{{ BMO}}\sum _{k=1}^mM_s\left( I_\alpha T^{k,4}(f)\right) (\tilde{x}), \end{aligned}$$

$$\begin{aligned} I_3\le & {} C\sum _{k=1}^m\left( \frac{1}{|Q|}\int _{R^n}|I_\alpha M_{(b-b_Q)\chi _{2Q}}T^{k,6}(f)(x)|^{t_2}dx\right) ^{1/t_2}\\\le & {} C\sum _{k=1}^m|Q|^{-1/{t_2}}\left( \int _{2Q}(|b(x)-b_Q||T^{k,6}(f)(x)|)^pdx\right) ^{1/p}\\\le & {} C\sum _{k=1}^m\left( \frac{1}{|Q|}\int _{2Q}|b(x)-b_Q|^{ps/(s-p)}dx\right) ^{(s-p)/ps}\\&\times \left( \frac{1}{|Q|^{1-s\alpha /n}}\int _{2Q}|T^{k,6}(f)(x)|^sdx\right) ^{1/s}\\\le & {} C||b||_{{ BMO}}\sum _{k=1}^mM_{\alpha ,s}\left( T^{k,6}(f)\right) (\tilde{x}), \end{aligned}$$

$$\begin{aligned} I_4\le & {} |Q|^{-1}\sum _{k=1}^m\int _Q\int _{(2Q)^c}|b(y)-b_{2Q}|\left| \frac{1}{|x-y|^{n-\alpha }}-\frac{1}{|x_0-y|^{n-\alpha }}\right| \\&\times |T^{k,6}(f)(y)|dydx\\\le & {} C\sum _{k=1}^m\sum _{j=1}^\infty \int _{2^{j}d\le |y-x_0|<2^{j+1}d}|b(y)-b_{2Q}|\frac{d}{|x_0-y|^{n-\alpha +1}}|T^{k,6}(f)(y)|dy \\\le & {} C\sum _{k=1}^m\sum _{j=1}^{\infty }d(2^jd)^{-n+\alpha -1}(2^jd)^{n(1-1/s)}(2^jd)^{n/s-\alpha }\\&\times \left( \frac{1}{|2^{j+1}Q|}\int _{2^{j+1}Q}|b(y)-b_Q|^{s'}dy\right) ^{1/s'}\\&\times \left( \frac{1}{|2^{j+1}Q|^{1-s\alpha /n}}\int _{2^{j+1}Q}|T^{k,6}(f)(y)|^sdy\right) ^{1/s} \\\le & {} C||b||_{{ BMO}}\sum _{k=1}^mM_{\alpha ,s}\left( T^{k,6}(f)\right) (\tilde{x})\sum _{j=1}^{\infty }j2^{-j} \\\le & {} C||b||_{{ BMO}}\sum _{k=1}^mM_{\alpha ,s}\left( T^{k,6}(f)\right) (\tilde{x}). \end{aligned}$$

This completes the proof of Theorem 2. $\square$

Proof of Theorem 3

Choose $q'<s<p$ in Theorem 1, we get, by Lemmas 6, 7, 8 and 9,

$$\begin{aligned}&||T_b(f)||_{L^{p,\varphi }(w)}\le \Vert M_r(T_b(f))\Vert _{L^{p,\varphi }(w)}\\&\quad \le C\Vert (T_b(f))^{\#}_r\Vert _{L^{p,\varphi }(w)} \\&\quad \le C||b||_{{ BMO}}\sum _{k=1}^m(\Vert M^2(T^{k,2}(f))\Vert _{L^{p,\varphi }(w)}+\Vert M_s(T^{k,2}(f))\Vert _{L^{p,\varphi }(w)}) \\&\quad \le C||b||_{{ BMO}}\sum _{k=1}^m\Vert T^{k,2}(f)\Vert _{L^{p,\varphi }(w)} \\&\quad \le C||b||_{{ BMO}}\Vert f\Vert _{L^{p,\varphi }(w)}. \end{aligned}$$

This completes the proof. $\square$

Proof of Theorem 4

Choose $q'<s<p$ in Theorem 2, we get, by Lemmas 6, 7, 8 and 9,

$$\begin{aligned}&||S_b(f)||_{L^{v,\varphi }(w)}\le \Vert M_r(S_b(f))\Vert _{L^{v,\varphi }(w)}\le C\Vert (S_b(f))^{\#}_r\Vert _{L^{v,\varphi }(w)} \\&\quad \le C||b||_{{ BMO}}\sum _{k=1}^m(\Vert M^2\left( I_\alpha T^{k,4}(f)\right) \Vert _{L^{v,\varphi }(w)}+\Vert M_s\left( I_\alpha T^{k,4}(f)\right) \Vert _{L^{v,\varphi }(w)}\\&\quad \quad +\Vert M_{\alpha ,s}\left( T^{k,6}(f)\right) \Vert _{L^{v,\varphi }(w)}) \\&\quad \le C||b||_{{ BMO}}\sum _{k=1}^m\left( \Vert I_\alpha T^{k,4}(f)\Vert _{L^{v,\varphi }(w)}+\Vert T^{k,4}(f)\Vert _{L^{u,\varphi }(w)}\right) \\&\quad \le C||b||_{{ BMO}}\sum _{k=1}^m\left( \Vert T^{k,4}(f)\Vert _{L^{u,\varphi }(w)}+\Vert f\Vert _{L^{u,\varphi }(w)}\right) \\&\quad \le C||b||_{{ BMO}}\Vert f\Vert _{L^{u,\varphi }(w)}. \end{aligned}$$

This completes the proof. $\square$