$M^2$-Type sharp estimates and boundedness on a Morrey space for Toeplitz-type operators associated to singular integral operators satisfying a variant of Hörmander’s condition

Abstract

In this paper, we prove the $M^2$-type sharp maximal function estimates for the Toeplitz-type operators associated to certain singular integral operators satisfying a variant of Hörmander’s condition. As an application, we obtain the weighted boundedness of the operators on the Lebesgue and Morrey spaces.

MSC:42B20, 42B25.

1 Introduction

As the development of singular integral operators (see [1, 2]), their commutators have been well studied. In [3, 4], the authors prove that the commutators generated by the singular integral operators and BMO functions are bounded on $L^p(R^n)$ for $1<p<\mathrm{\infty }$. Chanillo (see [5]) proves a similar result when singular integral operators are replaced by the fractional integral operators. In [6, 7], 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 is obtained. In [6], some singular integral operators satisfying a variant of Hörmander’s condition are introduced, and the boundedness for the operators is obtained (see [6], [20]). In this paper, we prove the sharp maximal function inequalities for the Toeplitz-type operator related to some singular integral operators satisfying a variant of Hörmander’s condition. As an application, we obtain the weighted boundedness of the Toeplitz-type operator on Lebesgue and Morrey spaces.

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

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

where, and in what follows, $f_Q={|Q|}^{-1}{\int }_{Q}f(x)dx$. We say that f belongs to $\mathit{BMO}(R^n)$ if $f^{\mathrm{\#}}$ belongs to $L^{\mathrm{\infty }}(R^n)$ and define ${\parallel f\parallel }_{\mathit{BMO}}={\parallel f^{\mathrm{\#}}\parallel }_{L^{\mathrm{\infty }}}$. It has been known that (see [2])

$${\parallel f-f_{2^{k}Q}\parallel }_{\mathit{BMO}}\le Ck{\parallel f\parallel }_{\mathit{BMO}}.$$

Let M be the Hardy-Littlewood maximal operator defined by

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

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

$$M^k(f)=M(M^{k-1}(f))\text{when }k\ge 2.$$

Let Φ be a Young function and $\tilde{\mathrm{\Phi }}$ be the complementary associated to Φ. We denote the Φ-average by, for a function f,

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

and the maximal function associated to Φ by

$$M_{\mathrm{\Phi }}(f)(x)=\underset{x\in Q}{sup}{\parallel f\parallel }_{\mathrm{\Phi },Q}.$$

The Young functions to be used in this paper are $\mathrm{\Phi }(t)=t(1+logt)$ and $\tilde{\mathrm{\Phi }}(t)=exp(t)$, the corresponding average and maximal functions denoted by ${\parallel \cdot \parallel }_{L(logL),Q}$, $M_{L(logL)}$ and ${\parallel \cdot \parallel }_{expL,Q}$, $M_{expL}$. Following [2], we know that the generalized Hölder inequality and the following inequalities hold:

$$\begin{array}{c}\frac{1}{|Q|}{\int }_Q|f(y)g(y)|dy\le {\parallel f\parallel }_{\mathrm{\Phi },Q}{\parallel g\parallel }_{\tilde{\mathrm{\Phi }},Q},\hfill \\ {\parallel f\parallel }_{L(logL),Q}\le M_{L(logL)}(f)\le C{M}^2(f),\hfill \\ {\parallel f-f_Q\parallel }_{expL,Q}\le C{\parallel f\parallel }_{\mathit{BMO}}\hfill \end{array}$$

and

$${\parallel f-f_Q\parallel }_{expL,2^{k}Q}\le Ck{\parallel f\parallel }_{\mathit{BMO}}.$$

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

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

and

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

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

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

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

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

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 condition (write this as $f\in R{H}_{\mathrm{\infty }}(R^n)$) if for any cube Q centered at the origin, we have

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

Definition 3 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 weight function and f be a locally integrable function on $R^n$. Set, for $1\le p<\mathrm{\infty }$,

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

where $Q(x,d)=\{y\in R^n:|x-y|<d\}$. The generalized 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 $\phi (d)=d^{\eta }$, $\eta >0$, then $L^{p,\phi }(R^n,w)=L^{p,\eta }(R^n,w)$, which is the classical weighted Morrey spaces (see [8, 9]). If $\phi (d)=1$, then $L^{p,\phi }(R^n,w)=L^p(R^n,w)$, which is the weighted Lebesgue spaces (see [6]).

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 [5, 8–11]).

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

Definition 4 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 b be a locally integrable function on $R^n$. The Toeplitz-type operator related to T is defined by

$$T^b=\sum _{j=1}^m{T}^{j,1}{M}_b{T}^{j,2},$$

where $T^{j,1}$ are T or ±I (the identity operator), $T^{j,2}$ are the bounded linear operators on $L^p(w)$ for $1<p<\mathrm{\infty }$ and $w\in A_1$, $j=1,\dots ,m$, $M_b(f)=bf$.

Remark Note that the classical Calderón-Zygmund singular integral operator satisfies Definition 4 (see [13], [19]). Also note that the commutator $[b,T](f)=bT(f)-T(bf)$ is a particular operator of the Toeplitz-type operators $T^b$. The Toeplitz-type operators $T^b$ 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 [12, 14]). The main purpose of this paper is to prove the sharp maximal inequalities for the Toeplitz-type operator $T^b$. As the application, we obtain the weighted $L^p$-norm inequality and Morrey space boundedness for the Toeplitz-type operators $T^b$.

3 Theorems and lemmas

We shall prove the following theorems.

Theorem 1 Let T be the singular integral operator as Definition 4, $0<r<1$ and $b\in \mathit{BMO}(R^n)$. If $T^1(g)=0$ for any $g\in L^u(R^n)$ ($1<u<\mathrm{\infty }$), then there exists a constant $C>0$ such that, for any $f\in C_0^{\mathrm{\infty }}(R^n)$ and $\tilde{x}\in R^n$,

$$M_{\mathrm{\Phi },r}^{\mathrm{\#}}(T^b(f))(\tilde{x})\le C{\parallel b\parallel }_{\mathit{BMO}}\sum _{j=1}^m{M}^2(T^{j,2}(f))(\tilde{x}).$$

Theorem 2 Let T be the singular integral operator as Definition 4, $1<p<\mathrm{\infty }$, $w\in A_1$ and $b\in \mathit{BMO}(R^n)$. If $T^1(g)=0$ for any $g\in L^u(R^n)$ ($1<u<\mathrm{\infty }$), then $T^b$ is bounded on $L^p(w)$.

Theorem 3 Let T be the singular integral operator as Definition 4, $0<D<2^n$, $1<p<\mathrm{\infty }$, $w\in A_1$ and $b\in \mathit{BMO}(R^n)$. If $T^1(g)=0$ for any $g\in L^u(R^n)$ ($1<u<\mathrm{\infty }$), then $T^b$ is bounded on $L^{p,\phi }(R^n,w)$.

To prove the theorems, we need the following lemmas.

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

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

$$\begin{array}{c}{\parallel f\parallel }_{W{L}^q}=\underset{\lambda >0}{sup}\lambda {\left|\{x\in R^n:f(x)>\lambda \}\right|}^{1/q},\hfill \\ N_{p,q}(f)=\underset{E}{sup}{\parallel f{\chi }_E\parallel }_{L^p}/{\parallel {\chi }_E\parallel }_{L^r},\hfill \end{array}$$

where the sup is taken for all measurable sets E with $0<|E|<\mathrm{\infty }$. 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 [2])

We have

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

Lemma 3 (see [15])

Let T be the singular integral operator as Definition 4. Then T is bounded on $L^p(w)$ for $1<p<\mathrm{\infty }$, $w\in A_1$ and weak $(L^1,L^1)$ bounded.

Lemma 4 (see [12])

Let $1<p<\mathrm{\infty }$, $0<\eta <\mathrm{\infty }$, $w\in A_{\mathrm{\infty }}$ 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,

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

Lemma 5 (see [5, 11])

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

$${\parallel M(f)\parallel }_{L^{p,\phi }(w)}\le C{\parallel f\parallel }_{L^{p,\phi }(w)}.$$

Lemma 6 Let $1<p<\mathrm{\infty }$, $0<\eta <\mathrm{\infty }$, $w\in A_1$, $0<D<2^n$ 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,

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

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 [3]. By Lemma 4, we have, for $f\in L^{p,\phi }(R^n,w)$,

$$\begin{array}{r}{\int }_Q{|M_{\eta }(f)(y)|}^{p}w(y)dy\\ ={\int }_{R^n}{|M_{\eta }(f)(y)|}^{p}w(y){\chi }_Q(y)dy\\ \le {\int }_{R^n}{|M_{\eta }(f)(y)|}^{p}M(w{\chi }_Q)(y)dy\\ \le C{\int }_{R^n}{|M_{\mathrm{\Phi },\eta }^{\mathrm{\#}}(f)(y)|}^{p}M(w{\chi }_Q)(y)dy\\ =C({\int }_Q{|M_{\mathrm{\Phi },\eta }^{\mathrm{\#}}(f)(y)|}^{p}M(w{\chi }_Q)(y)dy+\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}Q\setminus 2^{k}Q}{|M_{\mathrm{\Phi },\eta }^{\mathrm{\#}}(f)(y)|}^{p}M(w{\chi }_Q)(y)dy)\\ \le C({\int }_Q{|M_{\mathrm{\Phi },\eta }^{\mathrm{\#}}(f)(y)|}^{p}w(y)dy+\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}Q\setminus 2^{k}Q}{|M_{\mathrm{\Phi },\eta }^{\mathrm{\#}}(f)(y)|}^p\frac{w(Q)}{|2^{k+1}Q|}dy)\\ \le C({\int }_Q{|M_{\mathrm{\Phi },\eta }^{\mathrm{\#}}(f)(y)|}^{p}w(y)dy+\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}Q}{|M_{\mathrm{\Phi },\eta }^{\mathrm{\#}}(f)(y)|}^p\frac{M(w)(y)}{2^{n(k+1)}}dy)\\ \le C({\int }_Q{|M_{\mathrm{\Phi },\eta }^{\mathrm{\#}}(f)(y)|}^{p}w(y)dy+\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}Q}{|M_{\mathrm{\Phi },\eta }^{\mathrm{\#}}(f)(y)|}^p\frac{w(y)}{2^{nk}}dy)\\ \le C{\parallel M_{\mathrm{\Phi },\eta }^{\mathrm{\#}}(f)\parallel }_{L^{p,\phi }(w)}^p\sum _{k=0}^{\mathrm{\infty }}{2}^{-nk}\phi \left(2^{k+1}d\right)\\ \le C{\parallel M_{\mathrm{\Phi },\eta }^{\mathrm{\#}}(f)\parallel }_{L^{p,\phi }(w)}^p\sum _{k=0}^{\mathrm{\infty }}{\left(2^{-n}D\right)}^k\phi (d)\\ \le C{\parallel M_{\mathrm{\Phi },\eta }^{\mathrm{\#}}(f)\parallel }_{L^{p,\phi }(w)}^p\phi (d),\end{array}$$

thus

$${(\frac{1}{\phi (d)}{\int }_Q{M}_{\eta }(f){(x)}^{p}w(x)dx)}^{1/p}\le C{(\frac{1}{\phi (d)}{\int }_Q{M}_{\mathrm{\Phi },\eta }^{\mathrm{\#}}(f){(x)}^{p}w(x)dx)}^{1/p}$$

and

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

This finishes the proof. □

Lemma 7 Let T be the singular integral operator as Definition 3 or the bounded linear operator on $L^r(w)$ for any $1<r<\mathrm{\infty }$ and $w\in A_1$, $1<p<\mathrm{\infty }$, $w\in A_1$ and $0<D<2^n$. Then

$${\parallel T(f)\parallel }_{L^{p,\phi }(w)}\le C{\parallel f\parallel }_{L^{p,\phi }(w)}.$$

The proof of the lemma is similar to that of Lemma 6 by Lemma 3, we omit the details.

4 Proofs of theorems

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

$${\left(\frac{1}{|Q|}{\int }_Q\right|T^b(f)(x)-C_0{|}^{r}dx)}^{1/r}\le C{\parallel b\parallel }_{\mathit{BMO}}\sum _{j=1}^m{M}^2(T^{j,2}(f))(\tilde{x}),$$

where Q is any cube centered at $x_0$, $C_0={\sum }_{j=1}^m{\sum }_{i=1}^l{g}_j^i{\varphi }_i(x_0-x)$ and $g_j^i={\int }_{R^n}{B}_i(x_0-y)M_{(b-b_Q){\chi }_{{(2Q)}^c}}{T}^{j,2}(f)(y)dy$. Without loss of generality, we may assume $T^{j,1}$ are T ($j=1,\dots ,m$). Let $\tilde{x}\in Q$. Fix a cube $Q=Q(x_0,d)$ and $\tilde{x}\in Q$. Write

$$T^b(f)(x)=T^{b-b_{2Q}}(f)(x)=T^{(b-b_{2Q}){\chi }_{2Q}}(f)(x)+T^{(b-b_{2Q}){\chi }_{{(2Q)}^c}}(f)(x)=f_1(x)+f_2(x).$$

Then

$$\begin{array}{rl}{\left(\frac{1}{|Q|}{\int }_Q\right|T^b(f)(x)-C_0{|}^{r}dx)}^{1/r}\le & C{\left(\frac{1}{|Q|}{\int }_Q{|f_1(x)|}^{r}dx\right)}^{1/r}\\ +C{\left(\frac{1}{|Q|}{\int }_Q{|f_2(x)-C_0|}^{r}dx\right)}^{1/r}=I+\mathit{II}.\end{array}$$

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

$$\begin{array}{r}{\left(\frac{1}{|Q|}{\int }_Q{|T^{j,1}{M}_{(b-b_{2Q}){\chi }_{2Q}}{T}^{j,2}(f)(x)|}^{r}dx\right)}^{1/r}\\ \le {|Q|}^{-1}\frac{{\parallel T^{j,1}{M}_{(b-b_{2Q}){\chi }_{2Q}}{T}^{j,2}(f){\chi }_Q\parallel }_{L^r}}{{|Q|}^{1/r-1}}\\ \le C{|Q|}^{-1}{\parallel T^{j,1}{M}_{(b-b_{2Q}){\chi }_{2Q}}{T}^{j,2}(f)\parallel }_{W{L}^1}\\ \le C{|Q|}^{-1}{\parallel M_{(b-b_{2Q}){\chi }_{2Q}}{T}^{j,2}(f)\parallel }_{L^1}\\ \le C{|Q|}^{-1}{\int }_{2Q}|b(x)-b_{2Q}||T^{j,2}(f)(x)|dx\\ \le C{\parallel b-b_{2Q}\parallel }_{expL,2Q}{\parallel T^{j,2}(f)\parallel }_{L(logL),2Q}\\ \le C{\parallel b\parallel }_{\mathit{BMO}}{M}^2(T^{j,2}(f))(\tilde{x}),\end{array}$$

thus

$$I\le C\sum _{j=1}^m{\left(\frac{1}{|Q|}{\int }_Q{|T^{j,1}{M}_{(b-b_{2Q}){\chi }_{2Q}}{T}^{j,2}(f)(x)|}^{r}dx\right)}^{1/r}\le C{\parallel b\parallel }_{\mathit{BMO}}\sum _{j=1}^m{M}^2(T^{j,2}(f))(\tilde{x}).$$

For II, we get, for $x\in Q$,

$$\begin{array}{r}|T^{j,1}{M}_{(b-b_Q){\chi }_{{(2Q)}^c}}{T}^{j,2}(f)(x)-\sum _{i=1}^l{g}_j^i{\varphi }_i(x_0-x)|\\ \le |{\int }_{R^n}(K(x-y)-\sum _{i=1}^l{B}_i(x_0-y){\varphi }_i(x_0-x))(b(y)-b_{2Q}){\chi }_{{(2Q)}^c}(y)T^{j,2}(f)(y)dy|\\ \le \sum _{k=1}^{\mathrm{\infty }}{\int }_{2^{k}d\le |y-x_0|<2^{k+1}d}|K(x-y)-\sum _{i=1}^l{B}_i(x_0-y){\varphi }_i(x_0-x)||b(y)-b_{2Q}||T^{j,2}(f)(y)|dy\\ \le C\sum _{k=1}^{\mathrm{\infty }}{\int }_{2^{k}d\le |y-x_0|<2^{k+1}d}\frac{{|x-x_0|}^{\delta }}{{|y-x_0|}^{n+\delta }}|b(y)-b_{2Q}||T^{j,2}(f)(y)|dy\\ \le C\sum _{k=1}^{\mathrm{\infty }}\frac{d^{\delta }}{{(2^{k}d)}^{n+\delta }}{\left(2^{k}d\right)}^n{\parallel b-b_{2Q}\parallel }_{expL,2^{k+1}Q}{\parallel T^{j,2}(f)\parallel }_{L(logL),2^{k+1}Q}\\ \le C{\parallel b\parallel }_{\mathit{BMO}}{M}^2(T^{j,2}(f))(\tilde{x})\sum _{k=1}^{\mathrm{\infty }}k{2}^{-k\delta }\\ \le C{\parallel b\parallel }_{\mathit{BMO}}{M}^2(T^{j,2}(f))(\tilde{x}),\end{array}$$

thus

$$\mathit{II}\le \frac{1}{|Q|}{\int }_Q\sum _{j=1}^m|T^{j,1}{M}_{(b-b_Q){\chi }_{{(2Q)}^c}}{T}^{j,2}(f)(x)-C_0|dx\le C{\parallel b\parallel }_{\mathit{BMO}}\sum _{j=1}^m{M}^2(T^{j,2}(f))(\tilde{x}).$$

This completes the proof of Theorem 1. □

Proof of Theorem 2 By Theorem 1 and Lemmas 3-4, we have

$$\begin{array}{rl}{\parallel T^b(f)\parallel }_{L^p(w)}& \le {\parallel M_r((T^b(f))\parallel }_{L^p(w)}\le C{\parallel M_{\mathrm{\Phi },r}^{\mathrm{\#}}(T^b(f))\parallel }_{L^p(w)}\\ \le C{\parallel b\parallel }_{\mathit{BMO}}\sum _{j=1}^m{\parallel M^2(T^{j,2}(f))\parallel }_{L^p(w)}\le C{\parallel b\parallel }_{\mathit{BMO}}\sum _{j=1}^m{\parallel T^{j,2}(f)\parallel }_{L^p(w)}\\ \le C{\parallel b\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^p(w)}.\end{array}$$

This completes the proof. □

Proof of Theorem 3 By Theorem 1 and Lemmas 5-7, we have

$$\begin{array}{rl}{\parallel T^b(f)\parallel }_{L^{p,\phi }(w)}& \le {\parallel M_r(T^b(f))\parallel }_{L^{p,\phi }(w)}\le C{\parallel M_{\mathrm{\Phi },r}^{\mathrm{\#}}(T^b(f))\parallel }_{L^{p,\phi }(w)}\\ \le C{\parallel b\parallel }_{\mathit{BMO}}\sum _{j=1}^m{\parallel M^2(T^{j,2}(f))\parallel }_{L^{p,\phi }(w)}\le C{\parallel b\parallel }_{\mathit{BMO}}\sum _{j=1}^m{\parallel T^{j,2}(f)\parallel }_{L^{p,\phi }(w)}\\ \le C{\parallel b\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{p,\phi }(w)}.\end{array}$$

This completes the proof. □