Quantitative Weighted Estimates of the L^q-Type Rough Singular Integral Operator and Its Commutator

Abstract

Let $\Omega$ be a homogeneous function of degree zero on ${\mathbb{R}}^n$, $n\ge 2$, integrable and having mean value zero on the unit sphere ${\mathbb{S}}^{n-1}$, and let $T_{\Omega }$ be the homogeneous convolution singular integral operator with kernel ${\displaystyle \frac{\Omega \left(x\right)}{{\left|x\right|}^n}}$. By introducing reasonable refined decomposition and approximation techniques, together with sparse domination and variable measure interpolation methods, we establish the quantitative $A_1-A_{\infty }$ weighted estimates for $T_{\Omega }$ under the rough condition $\Omega \in L^q\left({\mathbb{S}}^{n-1}\right)$ for some $q\in (1,\infty )$. The results of the paper improve the previous works for the case $\Omega \in L^{\infty }\left({\mathbb{S}}^{n-1}\right)$. We also give the quantitative $A_1-A_{\infty }$ weighted estimates for the commutator $[b,T_{\Omega }]$ with $BMO$ symbol b.

1. Introduction

In harmonic analysis, singular integral operators constitute a fundamental class of linear operators. Classical theory typically requires their kernels to possess good smoothness properties to ensure the desirable behavior of the operators on function spaces. However, our research focuses on a more general class of “rough” operators, whose kernels no longer satisfy the traditional smoothness conditions. The essence of establishing quantitative weighted estimates for such operators lies in transforming the criterion for boundedness from a complex dependence on the microstructure (smoothness) of the kernel to a direct dependence on its macroscopic attributes (size) and the metric properties (non-uniformity) of the weight. This demonstrates that in weighted spaces, what truly governs the operator’s behavior is not the local details of the kernel but rather its overall strength and the geometric nature of the weight.

In this paper, we always assume that $\Omega$ is a homogeneous function of degree zero on the n-dimensional Euclidean space ${\mathbb{R}}^n$, $n\ge 2$, and $\Omega \in L^1\left({\mathbb{S}}^{n-1}\right)$, where ${\mathbb{S}}^{n-1}$ is the unit sphere in ${\mathbb{R}}^n$, and

$${\int }_{{\mathbb{S}}^{n-1}}\Omega \left(x^{\prime }\right)d{x}^{\prime }=0.$$

We will focus on studying quantitative weighted estimates of the following singular integral operator $T_{\Omega }$ defined by

$$T_{\Omega }f\left(x\right)=p.v.{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{\Omega \left(y^{\prime }\right)}{{\left|y\right|}^n}}f(x-y)dy.$$

(1)

This operator was first introduced by Calderón and Zygmund [1] in the 1950s; it has attracted considerable interest and has been studied by numerous scholars. Calderón and Zygmund [2] proved that, if $\Omega \in LlogL\left({\mathbb{S}}^{n-1}\right)$, the operator $T_{\Omega }$ is bounded on $L^p\left({\mathbb{R}}^n\right)$ for $p\in (1,\infty )$. Ricci and Weiss [3] proved the $L^p\left({\mathbb{R}}^n\right)$ boundedness of the operator $T_{\Omega }$ under the condition $\Omega \in H^1\left({\mathbb{S}}^{n-1}\right)$, which is an improvement upon the Calderón–Zygmund’s result; notably, the space $H^1\left({\mathbb{S}}^{n-1}\right)$ contains $LlogL\left({\mathbb{S}}^{n-1}\right)$. Seeger [4], in turn, proved that $\Omega \in LlogL\left({\mathbb{S}}^{n-1}\right)$ is also a sufficient condition to guarantee that $T_{\Omega }$ is bounded from $L^1\left({\mathbb{R}}^n\right)$ to $L^{1,\infty }\left({\mathbb{R}}^n\right)$. For more related works concerning the $L^p\left({\mathbb{R}}^n\right)$ boundedness of the operator $T_{\Omega }$ with rough kernels, we refer the reader to the literature [3,5,6,7,8,9,10,11] and the references therein.

In order to study the quantitative weighted estimates for the operator $T_{\Omega }$ with rough kernels, we recall the weight function class of Muckenhoupt $A_p\left({\mathbb{R}}^n\right)$. For $p\in (1,\infty )$, we say $w\in A_p\left({\mathbb{R}}^n\right)$, if w is a nonnegative and locally integrable function on ${\mathbb{R}}^n$ such that

$${\left[w\right]}_{A_p}:=\underset{Q}{sup}\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_{Q}w\left(x\right)dx\right){\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q{w}^{1-p^{\prime }}\left(x\right)dx\right)}^{p-1}<\infty ,$$

where $p^{\prime }=p/(p-1)$, the supremum is taken over all cubes $Q\subset {\mathbb{R}}^n$. We say $w\in A_1\left({\mathbb{R}}^n\right)$, if

$${\left[w\right]}_{A_1}:=\underset{x\in {\mathbb{R}}^n}{sup}{\displaystyle \frac{Mw\left(x\right)}{w\left(x\right)}}<\infty ,$$

where M is the Hardy–Littlewood maximal operator, see Equation (10) below for the definition of $Mf$. For a weight $w\in A_p\left({\mathbb{R}}^n\right)$ with $1\le p<\infty$, the constant ${\left[w\right]}_{A_p}$ defined above is called the $A_p$ constant of w, one may see ([12], Chapter 9) for the properties of $A_p\left({\mathbb{R}}^n\right)$. If $w\in {\cup }_{p\ge 1}{A}_p\left({\mathbb{R}}^n\right)$, we say $w\in A_{\infty }$, and the $A_{\infty }\left({\mathbb{R}}^n\right)$ constant of w, denoted by ${\left[w\right]}_{A_{\infty }}$, can be defined by

$${\left[w\right]}_{A_{\infty }}:=\underset{x\in {\mathbb{R}}^n}{sup}{\displaystyle \frac{1}{w\left(Q\right)}}{\int }_{Q}M\left(w{\chi }_Q\right)\left(x\right)dx,$$

see ref. [13] or ref. [14].

During the last two decades, there has been significant progress in the study of the quantitative weighted bounds with $A_p$ weights for the singular integral operators $T_{\Omega }$ with rough kernels $\Omega \in L^{\infty }\left({\mathbb{S}}^{n-1}\right)$. The quantitative weighted estimates can be traced back to Buckley [15]; he proved that for $1<p<\infty$, the Hardy–Littlewood maximal function M satisfies

$${\parallel Mf\parallel }_{L^p\left(w\right)}\le c_{n,p}{\left[w\right]}_{A_p}^{{\displaystyle \frac{1}{p-1}}}{\parallel f\parallel }_{L^p\left(w\right)}$$

for any $w\in A_p\left({\mathbb{R}}^n\right)$. Subsequently, Astala, Iwaniec and Saksman [16] proposed the famous $A_2$ conjecture for an operator T, i.e., if there exists a positive constant c independent of f and w such that ${\parallel Tf\parallel }_{L^2\left(w\right)}\le c{\left[w\right]}_{A_2}{\parallel f\parallel }_{L^2\left(w\right)}$. Later, the $A_2$ theorem had been proved for the Hilbert transform H, the Riesz transform $R_i$ and the classical Calderón–Zygmund operator T with smooth kernels. See ref. [17], among others.

Recently, the problem of establishing sharp quantitative weighted estimates for rough singular integral operators has garnered significant attention. The known study of the quantitative weighted boundedness of the operator $T_{\Omega }$ was achieved for the case where $\Omega \in L^{\infty }\left({\mathbb{S}}^{n-1}\right)$. Hytönen, Roncal and Tapiola showed the following inequality in ref. [18],

$$\begin{array}{c}\hfill {\parallel T_{\Omega }f\parallel }_{L^p\left(w\right)}\le c_{n,p}{\parallel \Omega \parallel }_{L^{\infty }\left({\mathbb{S}}^{n-1}\right)}{\left[w\right]}_{A_p}^{2max\left\{1,{\displaystyle \frac{1}{p-1}}\right\}}{\parallel f\parallel }_{L^p\left(w\right)},\end{array}$$

(2)

where $p\in (1,\infty )$ and $w\in A_p\left({\mathbb{R}}^n\right)$. This result served as an updated version of the theorem of Duoandikoetxea and Rubio de Francia [7]. Authors in ref. [18] improved the iterative bootstrap method and established precise quantitative weighted norm inequalities for the rough singular integral operators. In ref. [18], it is conjectured that the bound for $\parallel T_{\Omega }{\parallel }_{L^2\left(w\right)\to L^2\left(w\right)}$ depends linearly on the $A_2$ constant of $w\in A_2\left({\mathbb{R}}^n\right)$. It seems that proving or disproving this result is not a simple matter.

Li, Pérez, Rivera-Ríos and Roncal [19] improved the Equation (2) and proved that for $p\in (1,\infty )$ and $w\in A_p\left({\mathbb{R}}^n\right)$,

$$\begin{array}{cc}\hfill {\parallel T_{\Omega }f\parallel }_{L^p\left(w\right)}& \le c_{n,p}{\left[w\right]}_{A_p}^{{\displaystyle \frac{1}{p}}}\left({\left[w\right]}_{A_{\infty }}^{{\displaystyle \frac{1}{p^{\prime }}}}+{\left[w^{1-p^{\prime }}\right]}_{A_{\infty }}^{{\displaystyle \frac{1}{p}}}\right)min\left\{{\left[w^{1-p^{\prime }}\right]}_{A_{\infty }},{\left[w\right]}_{A_{\infty }}\right\}{\parallel f\parallel }_{L^p\left(w\right)}.\hfill \end{array}$$

On the other hand, Lerner, Ombrosi and Pérez [20] studied the sharp $A_1$ bounds for the Calderón–Zygmund operator T, which is defined by

$$Tf\left(x\right)=p.v.{\int }_{{\mathbb{R}}^n}K(x,y)f\left(y\right)dy,$$

where $K(x,y)$ is the Calderón–Zygmund kernel satisfying

$$\left|K(x,y)\right|\le {\displaystyle \frac{C}{{|x-y|}^n}},$$

$$|K(x_1,y)-K(x_2,y)|+|K(y,x_1)-K(y,x_2)|\le C{\displaystyle \frac{|x_1-x_2{|}^{ϵ}}{|x_1{-y|}^{n+ϵ}}},|x_1-y|\ge 2|x_1-x_2|.$$

Lerner, Ombrosi and Pérez [20] obtained the following quantitative $A_1$ weighted estimate for Calderón–Zygmund operator T,

$$\begin{array}{c}\hfill {\parallel Tf\parallel }_{L^{1,\infty }\left(w\right)}\le c{\left[w\right]}_{A_1}log(e+{\left[w\right]}_{A_1}){\parallel f\parallel }_{L^1\left(w\right)}.\end{array}$$

The key to obtaining this estimate is to establish the following two-weight $L^p\left({\mathbb{R}}^n\right)$ estimate of Fefferman–Stein type with good boundedness, i.e., for $1<p<\infty$ and $1<r<\infty$, it holds

$${\parallel Tf\parallel }_{L^p\left(w\right)}\le c_{T}p{p}^{\prime }{\left(r^{\prime }\right)}^{{\displaystyle \frac{1}{p^{\prime }}}}{\parallel f\parallel }_{L^p\left(M_{r}w\right)},$$

for any weight function $w\ge 0$, where $M_r$ is the maximal operator defined in (10). It is easy to see that the sharp estimate above follows that

$${\parallel Tf\parallel }_{L^p\left(w\right)}\le cp{p}^{\prime }{\left[w\right]}_{A_1}{\parallel f\parallel }_{L^p\left(w\right)}.$$

(3)

Related to the weighted estimate Equation (3) for Calderón–Zygmund operator T, Pérez, Rivera-Ríos and Roncal [21] recently established the following $A_1-A_{\infty }$ estimate for the rough singular integral operator $T_{\Omega }$ in case $\Omega \in L^{\infty }\left({\mathbb{S}}^{n-1}\right)$, $1<p<\infty$,

$${\parallel T_{\Omega }f\parallel }_{L^p\left(w\right)}\le c_n{\parallel \Omega \parallel }_{L^{\infty }\left({\mathbb{S}}^{n-1}\right)}{\left[w\right]}_{A_1}^{{\displaystyle \frac{1}{p}}}{\left[w\right]}_{A_{\infty }}^{1+{\displaystyle \frac{1}{p^{\prime }}}}{\parallel f\parallel }_{L^p\left(w\right)}.$$

(4)

Rivera-Ríos and Israel [22] improved the Equation (4) and obtained that

$${\parallel T_{\Omega }f\parallel }_{L^p\left(w\right)}\le c_n{\parallel \Omega \parallel }_{L^{\infty }\left({\mathbb{S}}^{n-1}\right)}{\left[w\right]}_{A_1}^{{\displaystyle \frac{1}{p}}}{\left[w\right]}_{A_{\infty }}^{{\displaystyle \frac{1}{p^{\prime }}}}{\parallel f\parallel }_{L^p\left(w\right)}.$$

(5)

Motivated by the works in ref. [21] and ref. [22], our goal is to obtain quantitative weighted estimates for $T_{\Omega }$ under the weaker condition that $\Omega \in L^q\left({\mathbb{S}}^{n-1}\right)$ for some $q>1$.

Our approach is to decompose $T_{\Omega }$ into a sequence of operators whose regularity satisfies the $L^q$-Hörmander condition. Via sparse domination for every piece in the operator sequence, we establish a two-weight estimate and weighted $L^p\left({\mathbb{R}}^n\right)$ estimate with decay bound for the operators. By using suitable interpolation with change in measures and taking summation for the sequence, we obtain $A_1-A_{\infty }$ estimates for $T_{\Omega }$.

In this paper, we will also consider the quantitative weighted estimates for the commutator $[b,T_{\Omega }]$. Given a linear operator T and $b\in BMO$, the commutator $[b,T]$ in the sense of Coifman–Rochberg–Weiss is defined by

$$[b,T]f\left(x\right)=b\left(x\right)Tf\left(x\right)-T\left(bf\right)\left(x\right).$$

Recall that $b\in BMO$ means that b is a locally integrable function ${\mathbb{R}}^n\to \mathbb{R}$ with

$${\parallel b\parallel }_{BMO}=\underset{Q}{sup}{\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q|b\left(y\right)-b_Q|dy<\infty ,$$

where the supremum is taken over all cubes $Q\in {\mathbb{R}}^n$ with sides parallel to the axes, and $b_Q={\displaystyle \frac{1}{\left|Q\right|}}{\int }_{Q}b\left(y\right)dy$.

If $\Omega \in Li{p}_{\alpha }\left({\mathbb{S}}^{n-1}\right)$, the class of Lipschitz continuous functions of $\alpha$-power with $\alpha \in (0,1)$, Coifman, Rochberg and Weiss [23] established that $b\in BMO$ is a sufficient and necessary condition for the $L^p\left({\mathbb{R}}^n\right)(p>1)$ boundedness of commutator $[b,T_{\Omega }]$. By combining the weighted estimates for $T_{\Omega }$ with the connection between $A_p$ weights and $BMO$ functions, Alvarez, Bagby, Kurtz and Pérez [24] established the $L^p\left({\mathbb{R}}^n\right)$ boundedness of the commutator $[b,T_{\Omega }]$ for the case $\Omega \in L^q\left({\mathbb{S}}^{n-1}\right)$ for some $q\in (1,\infty )$. Hu [25] improved these theorems and showed that if $\Omega \in L{(logL)}^2\left({\mathbb{S}}^{n-1}\right)$, the commutator $[b,T_{\Omega }]$ can also map $L^p\left({\mathbb{R}}^n\right)$ to $L^p\left({\mathbb{R}}^n\right)$ for all $p\in (1,\infty )$ with bound ${C\parallel b\parallel }_{BMO\left({\mathbb{R}}^n\right)}$. For other works about the boundedness for $[b,T_{\Omega }]$, we can also refer to the recent literature [9,11,26,27] and the references in their papers.

We will prove in Section 5 the weighted boundedness of the commutator $[b,T_{\Omega }]$, and give quantitative $A_p-A_{\infty }$ weighted estimates for the commutator $[b,T_{\Omega }]$.

2. Main Results

We will prove the following two-weight $L^p\left({\mathbb{R}}^n\right)$ estimates of Fefferman–Stein type:

Theorem 1. 

Let Ω be a homogeneous function of degree zero on ${\mathbb{R}}^n$, $n\ge 2$, integrable and have mean value zero on the unit sphere ${\mathbb{S}}^{n-1}$. Let $T_{\Omega }$ be defined as Equation (1) with $\Omega \in L^q\left({\mathbb{S}}^{n-1}\right)$ for $1<q<\infty$. Then, for $q^{\prime }<p<\infty$, any weight $w>0$, $0<\theta <1$ and $r>1$, we have

$${\parallel T_{\Omega }f\parallel }_{L^p\left(w\right)}\le {\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}{\left(c_{n,p,q}{p}^{\prime }\right)}^{{\displaystyle \frac{q^{\prime }(p-1)\theta }{p-q^{\prime }}}}{\left(r^{\prime }\right)}^{{\displaystyle \frac{\theta }{p^{\prime }}}}{\displaystyle \frac{1}{1-\theta }}{\parallel f\parallel }_{L^p\left(M_{r/\theta }w\right)},$$

(6)

and

$${\parallel T_{\Omega }f\parallel }_{L^p\left(w\right)}\le {\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}{\left(c_{n,p,q}{p}^{\prime }\right)}^{{\displaystyle \frac{q^{\prime }(p-1)}{p-q^{\prime }}}}{\left(r^{\prime }\right)}^{1+{\displaystyle \frac{1}{p^{\prime }}}}{\parallel f\parallel }_{L^p\left(M_{r}w\right)}.$$

(7)

In particular, let w be any specific weight as $w\in A_{\infty }$ and $w\in A_1$; we can use Theorem 1 to obtain the following quantitative $A_1-A_{\infty }$ weighted estimates.

Corollary 1. 

Let Ω be a homogeneous function of degree zero on ${\mathbb{R}}^n$, $n\ge 2$, integrable and have mean value zero on the unit sphere ${\mathbb{S}}^{n-1}$. Let $T_{\Omega }$ be defined as Equation (1) with $\Omega \in L^q\left({\mathbb{S}}^{n-1}\right)$ for $1<q<\infty$. Then, for $p>q^{\prime }$, any weight $w>0$ and $w\in A_{\infty }$,

$${\parallel T_{\Omega }f\parallel }_{L^p\left(w\right)}\le c_{n,p,q}{\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}{\left(p^{\prime }\right)}^{{\displaystyle \frac{q^{\prime }(p-1)}{p-q^{\prime }}}}\left({(p/q^{\prime })}^{\prime }\right)p{\left[w\right]}_{A_{\infty }}^{1+{\displaystyle \frac{1}{p^{\prime }}}}{\parallel f\parallel }_{L^p\left(M\left(w\right)\right)}.$$

(8)

moreover, if $w\in A_1$, then

$${\parallel T_{\Omega }f\parallel }_{L^p\left(w\right)}\le c_{n,p,q}{\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}{\left(p^{\prime }\right)}^{{\displaystyle \frac{q^{\prime }(p-1)}{p-q^{\prime }}}}\left({(p/q^{\prime })}^{\prime }\right)p{\left[w\right]}_{A_1}^{{\displaystyle \frac{1}{p}}}{\left[w\right]}_{A_{\infty }}^{1+{\displaystyle \frac{1}{p^{\prime }}}}{\parallel f\parallel }_{L^p\left(w\right)}.$$

(9)

Remark 1. 

As mentioned in Section 1, Pérez, Rivera-Ríos and Roncal [21] established the $A_1-A_{\infty }$ weighted $L^p\left({\mathbb{R}}^n\right)$ estimate Equation (4) for rough singular integral $T_{\Omega }$ in the case $\Omega \in L^{\infty }\left({\mathbb{S}}^{n-1}\right)$, and Rivera-Ríos and Israel [22] established the $A_1-A_{\infty }$ weighted $L^p\left({\mathbb{R}}^n\right)$ estimate Equation (5) for rough singular integral $T_{\Omega }$ in the case $\Omega \in L^{\infty }\left({\mathbb{S}}^{n-1}\right)$. The value and significance of our theorem and corollary lie in establishing similar conclusions but only requiring the weak condition $\Omega \in L^q\left({\mathbb{S}}^{n-1}\right)$ with $1<q<\infty$.

Remark 2. 

It is worthy to remark that, in these weighted Equations (6)–(9) of Theorem 1 and Corollary 1, the condition for Ω is weaker than that in ref. [19], where Li, Pérez, Rivera-Ríos and Roncal proved that, if $\Omega \in L^{q,1}logL\left({\mathbb{S}}^{n-1}\right)L^q\left({\mathbb{S}}^{n-1}\right)$, $1<r<\infty$, then for $p>q^{\prime }$,

$${\parallel T_{\Omega }f\parallel }_{L^p\left(w\right)}\le c_n{q\parallel \Omega \parallel }_{L^{q,1}logL\left({\mathbb{S}}^{n-1}\right)}p{\left(r^{\prime }\right)}^{{\displaystyle \frac{1}{p^{\prime }}}}{\left({\displaystyle \frac{p}{q^{\prime }}}\right)}^{\prime }{\left(c_n{p}^{\prime }\right)}^{{\displaystyle \frac{q^{\prime }(p-1)}{p-q^{\prime }}}}{\parallel f\parallel }_{L^p\left(M_{r}w\right)},$$

and

$${\parallel T_{\Omega }f\parallel }_{L^p\left(w\right)}\le c_{n,p,q}{\parallel \Omega \parallel }_{L^{q,1}logL\left({\mathbb{S}}^{n-1}\right)}{\left[w\right]}_{A_1}^{{\displaystyle \frac{1}{p}}}{\left[w\right]}_{A_{\infty }}^{{\displaystyle \frac{1}{p^{\prime }}}}{\parallel f\parallel }_{L^p\left(w\right)}.$$

We note that $L^{q,1}logL\left({\mathbb{S}}^{n-1}\right)$ is the Lorentz space defined by

$${\parallel \Omega \parallel }_{L^{q,1}logL\left({\mathbb{S}}^{n-1}\right)}:=q{\int }_0^{\infty }tlog(e+t){\left|\left\{\theta \in {\mathbb{S}}^{n-1}:|\Omega \left(\theta \right)|>t\right\}\right|}^{{\displaystyle \frac{1}{q}}}{\displaystyle \frac{dt}{t}},$$

and there is a continuous embedding relationship $L^{q+\epsilon }\left({\mathbb{S}}^{n-1}\right)\hookrightarrow L^{q,1}logL\left({\mathbb{S}}^{n-1}\right)\hookrightarrow L^q\left({\mathbb{S}}^{n-1}\right)$ for all $1<q\le \infty$ and $\epsilon >0$.

In the paper, we will also prove the following quantitative $A_p-A_{\infty }$ weighted estimates for the commutator $[b,T_{\Omega }]$:

Theorem 2. 

Let Ω be a homogeneous function of degree zero on ${\mathbb{R}}^n$, $n\ge 2$, integrable and have mean value zero on the unit sphere ${\mathbb{S}}^{n-1}$. Let $T_{\Omega }$ be defined as Equation (1) with $\Omega \in L^{\infty }\left({\mathbb{S}}^{n-1}\right)$ and $b\in BMO$. Then, for weight $w>0$, we have

$$\begin{array}{cc}\hfill \parallel [b,T_{\Omega }]{\parallel }_{L^p\left(w\right)}& \lesssim {\parallel \Omega \parallel }_{L^{\infty }}{\parallel b\parallel }_{BMO}{\left[w\right]}_{A_p}^{{\displaystyle \frac{1}{p}}}max\{{\left[w\right]}_{A_{\infty }}^{{\displaystyle \frac{1}{p^{\prime }}}},{\left[w^{1-p^{\prime }}\right]}_{A_{\infty }}^{{\displaystyle \frac{1}{p}}}\}min\{{\left[w\right]}_{A_{\infty }},{\left[w^{1-p^{\prime }}\right]}_{A_{\infty }}\}\hfill \\ \hfill & \times ({\left[w\right]}_{A_{\infty }}+{\left[w^{1-p^{\prime }}\right]}_{A_{\infty }}){\parallel f\parallel }_{L^p\left(w\right)}.\hfill \end{array}$$

The rest of the paper is organized as follows. In Section 3, we will give some basic definitions and give a suitable decomposition of $T_{\Omega }$ into a sequence of $L^q$-Hörmander type operators and related properties. We will prove the unweighted $L^p$ estimates and two-weight estimates for the pieces of the operator sequence. In Section 4, we prove the main results. In Section 5, we prove the weighted boundedness of the commutator $[b,T_{\Omega }]$. In Appendices Appendix A and Appendix B, we provide the proofs of Lemmas 3 and 4.

3. Lemmas

3.1. Some Preliminary Lemmas

Let $\Psi$ be a Young function that is a continuous, convex, increasing function that satisfies $\Psi \left(0\right)=0$ and $\Psi \left(t\right)\to \infty$ as $t\to \infty$. A Young function $\Psi$ is said to be doubling if there exists a positive constant C such that $\Psi \left(2t\right)\le C\Psi \left(t\right)$. Let f be a measurable function defined on a set E with finite measure in ${\mathbb{R}}^n$. The $\Psi -norm$ of f over E is defined by

$${\parallel f\parallel }_{\Psi \left(L\right),E}:=inf\left\{\lambda >0:{\displaystyle \frac{1}{\left|E\right|}}{\int }_E\Psi \left({\displaystyle \frac{\left|f\right(x\left)\right|}{\lambda }}\right)dx\le 1\right\}.$$

If $\Psi$ is a Young function and f is Lebesgue measurable, we define the Orlicz maximal operator $M_{\Psi \left(L\right)}$ by

$$M_{\Psi \left(L\right)}f\left(x\right)=\underset{Q\ni x}{sup}{\parallel f\parallel }_{\Psi \left(L\right),Q},$$

where Q are cubes in ${\mathbb{R}}^n$. In particular, if $\Psi \left(t\right)=t^r$ for $1\le r<\infty$, then $M_{\Psi \left(L\right)}$ coincides with the maximal operator defined by

$$M_{r}f\left(x\right)=\underset{Q\ni x}{sup}{\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q{\left|f\right|}^{r}dt\right)}^{1/r}.$$

(10)

Obviously, for $r=1$, $M_r$ is the usual Hardy–Littlewood maximal function, denoted simply by M. If $\Psi \left(t\right)=tlog(e+t)$, we denote the maximal operator $M_{\Psi \left(L\right)}$ by $M_{L(logL)}$.

For each Young function A, we can define its complementary function

$$\overline{A}\left(s\right)=\underset{t>0}{sup}\{st-A\left(t\right)\},s\ge 0.$$

$\overline{A}$ is also a Young function and enjoys the following properties:

$$st\le A\left(t\right)+\overline{A}\left(s\right),t,s\ge 0$$

and

$$t\le A^{-1}\left(t\right){\overline{A}}^{-1}\left(t\right)\le 2t,t>0.$$

Let $1<p<\infty$. A doubling Young function A is said to satisfy the $B_p$ condition, if there is a positive constant c such that

$${\beta }_p\left(A\right):={\left({\int }_c^{\infty }{\displaystyle \frac{A\left(t\right)}{t^p}}{\displaystyle \frac{dt}{t}}\right)}^{{\displaystyle \frac{1}{p}}}\approx {\left({\int }_c^{\infty }{\left({\displaystyle \frac{t^{p^{\prime }}}{\overline{A}\left(t\right)}}\right)}^{p-1}{\displaystyle \frac{dt}{t}}\right)}^{{\displaystyle \frac{1}{p}}}<\infty ,$$

in this case, we say that $A\in B_p$.

Lemma 1 

(Ref. [19], Lemma 2.4). Let A be a Young function satisfying the $B_p$ condition. Then,

$$\parallel M_A{\parallel }_{L^p}\le c_n{\beta }_p\left(A\right).$$

In particular, for $A\left(t\right)=t^{pr}$ with $1<p,r<\infty$, then $\overline{A}\left(t\right)\le t^{{\left(rp\right)}^{\prime }}$, and

$${\parallel M_{\overline{A}}\parallel }_{L^{p^{\prime }}}\le {\parallel M_{{\left(rp\right)}^{\prime }}\parallel }_{L^{p^{\prime }}}\le c_{n}p{\left(r^{\prime }\right)}^{{\displaystyle \frac{1}{p^{\prime }}}},$$

see Section 2.5 in ref. [19].

Now we introduce a decomposition for the rough homogeneous singular integral operator $T_{\Omega }$ defined by Equation (1); one can write

$$T_{\Omega }f:=\sum _{m\in \mathbb{Z}}{\tilde{T}}_{m}f:=\sum _{m\in \mathbb{Z}}{K}^{\left(m\right)}\ast f,$$

where

$$K^{\left(m\right)}\left(x\right):={\displaystyle \frac{\Omega \left(x\right)}{{\left|x\right|}^n}}{\chi }_{2^m<\left|x\right|<2^{m+1}}m\in \mathbb{Z}.$$

(11)

We further consider the following partition of unity. Let $\phi \in C_0^{\infty }\left({\mathbb{R}}^n\right)$ be a radial nonnegative function with $supp\phi \subset \left\{x\in {\mathbb{R}}^n:\left|x\right|<{\displaystyle \frac{1}{4}}\right\}$, and $\int \phi \left(x\right)dx=1$. Let ${\phi }_d\left(x\right)=2^{-nd}\phi \left(2^{-d}x\right)$ and $S_d\left(f\right)={\phi }_d\ast f$, then $S_{d}f$ converges in f when $d\to -\infty$. For any sequence of integer numbers ${\left\{N\left(j\right)\right\}}_{j=0}^{\infty }$, with

$$1=N\left(0\right)<N\left(1\right)<N\left(2\right)<\cdots <N\left(j\right)\to \infty ,$$

we have the identity

$${\tilde{T}}_m={\tilde{T}}_m{S}_{m-1}+\sum _{j=0}^{\infty }{\tilde{T}}_m\left(S_{m-N(j+1)}-S_{m-N\left(j\right)}\right),$$

thus

$$T_{\Omega }=\sum _{m\in Z}{\tilde{T}}_m=T_0^N+\sum _{j=0}^{\infty }\left(T_{j+1}^N-T_j^N\right),$$

where

$$T_j^{N}f:=\sum _{m\in Z}{\tilde{T}}_m{S}_{m-N\left(j\right)}f:=K_j^N\ast f,j⩾0.$$

(12)

Lemma 2. 

Let $K\left(x\right)={\displaystyle \frac{\Omega \left(x\right)}{{\left|x\right|}^n}}$, $K_j^N$ be defined in Equation (12), and Ω be homogeneous of degree zero and have mean value zero on the unit sphere ${\mathbb{S}}^{n-1}$ and $\Omega \in L^q\left({\mathbb{S}}^{n-1}\right)$ with $q>1$. Then there are constants C and $0<\delta <{\displaystyle \frac{1}{q^{\prime }}}$ independent of ξ and $j\in \mathbb{N}$ such that for all $j\ge 0$,

$$\left(1\right)\left|\widehat{K}\left(\xi \right)\right|+\left|\widehat{K_j^N}\left(\xi \right)\right|\le {C\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)};\left(2\right)\left|\widehat{K}\left(\xi \right)-\widehat{K_j^N}\left(\xi \right)\right|\le C{2}^{-\delta N\left(j\right)}{\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}.$$

Proof. 

We claim that, using the properties of $\Omega$, we have the following inequality:

$$|\widehat{K^{\left(m\right)}}{\left(\xi \right)|\lesssim \parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}min\{{|2^m\xi |}^{\alpha },{|2^m\xi |}^{-\alpha }\},$$

(13)

for $0<\alpha <1/q^{\prime }$ independent of $\Omega$ and $m\in \mathbb{Z}$.

It is obvious that $\widehat{K^{\left(m\right)}}\left(\xi \right)=\widehat{K^{\left(0\right)}}\left(2^m\xi \right)$, now we only need to prove that $\left|\widehat{K^{\left(0\right)}}\left(\xi \right)\right|\lesssim min\left\{{\left|\xi \right|}^{\alpha },{\left|\xi \right|}^{-\alpha }\right\}$ for $\alpha \in (0,1/q^{\prime })$. Using the vanishing condition of $\Omega$, we have for $\left|\xi \right|\le 1$ that,

$$\begin{array}{c}\hfill \left|\widehat{K^{\left(0\right)}}\left(\xi \right)\right|=\left|{\int }_{{\mathbb{S}}^{n-1}}\Omega \left(\theta \right){\int }_1^2{\displaystyle \frac{e^{-2\pi ri\theta ·\xi }-1}{r}}drd\theta \right|\lesssim min\left\{{\left|\xi \right|}^{\alpha },{\left|\xi \right|}^{-\alpha }\right\},\end{array}$$

for any $0<\alpha <1$. On the other hand, using integration by parts, we obtain

$$\begin{array}{c}\hfill \left|{\int }_1^2{\displaystyle \frac{1}{r}}{e}^{-2\pi ri\theta ·\xi }dr\right|\lesssim min\left\{{\displaystyle \frac{1}{|\theta ·\xi |}},1\right\}\le {\displaystyle \frac{C}{{|\theta ·\xi |}^{\alpha }}},\end{array}$$

for $\alpha \in (0,1)$. Therefore, by Hölder’s inequality and the fact $0<\alpha q^{\prime }<1$, we have for $\left|\xi \right|\ge 1$ that,

$$\begin{array}{c}\hfill \left|\widehat{K^{\left(0\right)}}\left(\xi \right)\right|\lesssim min\left\{{\left|\xi \right|}^{\alpha },{\left|\xi \right|}^{-\alpha }\right\},\end{array}$$

which implies the desired Equation (13).

Fix any $\xi \ne 0$, and take an integral $m_0\in \mathbb{Z}$ satisfying $2^{m_0}\left|\xi \right|\simeq 1$; we receive from Equation (13) that

$$\left|\widehat{K}\left(\xi \right)\right|\le \sum _{m\in \mathbb{Z}}\left|\widehat{K^{\left(m\right)}}\left(\xi \right)\right|\lesssim {\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}\left(\sum _{m=-\infty }^{m_0}{\left|2^m\xi \right|}^{\alpha }+\sum _{m=m_0}^{\infty }{\left|2^m\xi \right|}^{-\alpha }\right)\lesssim {\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)},$$

and, since $\left|\widehat{{\phi }_{m-N\left(j\right)}}\left(\xi \right)\right|\lesssim 1$, we have that

$$\left|{\widehat{K}}_j^N\left(\xi \right)\right|\le \sum _{m\in \mathbb{Z}}\left|\widehat{K^{\left(m\right)}}\left(\xi \right)\widehat{{\phi }_{m-N\left(j\right)}}\left(\xi \right)\right|\le \sum _{m\in \mathbb{Z}}\left|\widehat{K^{\left(m\right)}}\left(\xi \right)\right|\lesssim {\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}.$$

Moreover, we can verify that $\left|\widehat{{\phi }_{m-N\left(j\right)}}\left(\xi \right)-1\right|\lesssim min\{1,{\left|2^{m-N\left(j\right)}\xi \right|}^{\delta }\}$ for any $\delta \in (0,1)$, and then we can deduce from Equation (13) that

$$\begin{array}{c}\hfill \left|\widehat{K}\left(\xi \right)-{\widehat{K}}_j^N\left(\xi \right)\right|\le \sum _{m\in \mathbb{Z}}\left|\widehat{K^{\left(m\right)}}\left(\xi \right)\right|\left|\widehat{{\phi }_{m-N\left(j\right)}}\left(\xi \right)-1\right|\lesssim 2^{-\delta N\left(j\right)}{\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)},\end{array}$$

as long as we take $0<\delta <\alpha$. □

It is easy to see from Lemma 2 that the sum

$$T_{\Omega }=T_0^N+\sum _{j=0}^{\infty }\left(T_{j+1}^N-T_j^N\right),$$

converges strongly in the $L^2$-operator norm. Therefore, the rough singular integral operator $T_{\Omega }$ can be approximated by the sequence of operators $T_j^N$, $j=0,1,2,\cdots$, in the sense of the $L^2$-operator norm.

Moreover, we will proved that each operator $T_j^N$ is bounded on $L^p\left({\mathbb{R}}^n\right)$ for $1<p<\infty$, and of weak $(1,1)$ boundedness. We will also proved that the grand maximal operator of $T_j^N$ is bounded from $L^{q^{\prime }}\left({\mathbb{R}}^n\right)$ to $L^{q^{\prime },\infty }\left({\mathbb{R}}^n\right)$.

Lemma 3. 

Let $\Omega \in L^q\left({\mathbb{S}}^{n-1}\right)$ with $q>1$ be a homogeneous function of degree zero and have mean value zero on the unit sphere ${\mathbb{S}}^{n-1}$, and let $T_j^N$ be defined in Equation (12). Then for any $j\in \mathbb{N}$, $T_j^N$ maps from $L^1\left({\mathbb{R}}^n\right)$ to $L^{1,\infty }\left({\mathbb{R}}^n\right)$ with bound ${CN\left(j\right)\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}$. Moreover, $T_j^N$ is bounded on $L^p\left({\mathbb{R}}^n\right)$, $1<p<\infty$, with bound ${CN\left(j\right)\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}$.

Lemma 4. 

Let $T_j^N$ be defined as in Equation (12) and let ${\mathcal{M}}_{T_j^N}$ be the grand maximal operator of $T_j^N$ defined by

$${\mathcal{M}}_{T_j^N}f\left(x\right)=\underset{Q\ni x}{sup}\underset{\xi \in Q}{ess\; sup}\left|T_j^N\left(f{\chi }_{{\mathbb{R}}^n\setminus 3Q}\right)\left(\xi \right)\right|,$$

then

$${\mathcal{M}}_{T_j^N}\left(f\right)\left(x\right)\le c_n{\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}\left(N\left(j\right)M_{q^{\prime }}f\left(x\right)+M{T}_j^{N}f\left(x\right)+N\left(j\right)Mf\left(x\right)\right),$$

moreover, ${\mathcal{M}}_{T_j^N}$ is bounded from $L^{q^{\prime }}\left({\mathbb{R}}^n\right)$ to $L^{q^{\prime },\infty }\left({\mathbb{R}}^n\right)$ with bound ${CN\left(j\right)\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}$.

Lemmas 3 and 4 were in fact given by the authors in ([28], Lemmas 3.4 and 3.5). For the sake of completeness, we will provide different methods in the Appendices Appendix A and Appendix B to give lines of the proof of Lemmas 3 and 4.

Now we introduce the sparse collection and sparse operator. The collection $\mathcal{S}$ of cubes is $\eta$-sparse for $0<\eta <1$ if for each fixed $Q\in \mathcal{S}$, there exists a measurable set $E_Q\subset Q$ such that $\left|E_Q\right|\ge \eta \left|Q\right|$ and the sets ${\left\{E_Q\right\}}_{Q\in \mathcal{S}}$ are pairwise disjoint. Usually $\eta$ will depend only on the dimension, and when this parameter is unessential we will skip it.

Given a sparse family $\mathcal{S}$ and $r\in (0,\infty )$, we define the sparse operator ${\mathcal{A}}_{r,\mathcal{S}}$ by

$${\mathcal{A}}_{r,\mathcal{S}}f\left(x\right)=\sum _{Q\in \mathcal{S}}{\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q{\left|f\left(y\right)\right|}^{r}dy\right)}^{{\displaystyle \frac{1}{r}}}{\chi }_Q\left(x\right),$$

(14)

A general dyadic grid $\mathcal{D}$ is the collection of cubes satisfying the following properties: (i) for any cube $Q\in \mathcal{D}$, its side length $\ell \left(Q\right)$ is of the form $2^k$ for some $k\in \mathbb{Z}$; (ii) for any cubes $Q_1,Q_2\in \mathcal{D}$, $Q_1\cap Q_2\in \{Q_1,Q_2,\varnothing \}$; (iii) for each $k\in \mathbb{Z}$, the cubes of side length $2^k$ form a partition of ${\mathbb{R}}^n$.

Lemma 5 

(Ref. [29]). Assume that T is a bounded linear operator from $L^q\left({\mathbb{R}}^n\right)$ to $L^{q,\infty }\left({\mathbb{R}}^n\right)$ and ${\mathcal{M}}_T$ is bounded from $L^r\left({\mathbb{R}}^n\right)$ to $L^{r,\infty }\left({\mathbb{R}}^n\right)$, with $1\le q\le r<\infty$. Then, for every $f\in L^r\left({\mathbb{R}}^n\right)$ with compact support, there exists a sparse family $\mathcal{S}$ such that for a.e. $x\in {\mathbb{R}}^n$,

$$\left|Tf\left(x\right)\right|\le C{\mathcal{A}}_{r,\mathcal{S}}\left(\left|f\right|\right)\left(x\right),$$

where $C=C_{n,q,r}\left({∥T∥}_{L^q({\mathbb{R}}^n,w)\to L^{q,\infty }({\mathbb{R}}^n,w)}+{∥{\mathcal{M}}_T∥}_{L^r({\mathbb{R}}^n,w)\to L^{r,\infty }({\mathbb{R}}^n,w)}\right)$.

We need results concerning the sharp reverse Hölder’s inequality (RHI).

Lemma 6 

(Ref. [21], Theorem 2.7). If $w\in A_{\infty }$, there exists constant ${\tau }_n>1$, when $1\le r<r_w=1+{\displaystyle \frac{1}{{\tau }_n{\left[w\right]}_{A_{\infty }}}}$,

$${\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q{w}^{r_w}dx\right)}^{{\displaystyle \frac{1}{r_w}}}\le {\displaystyle \frac{2}{\left|Q\right|}}{\int }_{Q}w,$$

note $r_w^{\prime }=1+{\tau }_n{\left[w\right]}_{A_{\infty }}\approx {\left[w\right]}_{A_{\infty }}$.

We will also need following interpolation theorems:

Lemma 7 

(Riesz–Thorin interpolation theorem). Let $1\le p_0,p_1,q_0,q_1\le \infty$, and $T:L^{p_0}+L^{p_1}\mapsto L^{q_0}+L^{q_1}$ be a linear operator satisfying

$${\parallel Tf\parallel }_{L^{q_0}\left({\mathbb{R}}^n\right)}\le M_0{\parallel f\parallel }_{L^{p_0}\left({\mathbb{R}}^n\right)},\forall f\in L^{p_0}\left({\mathbb{R}}^n\right),$$

and

$${\parallel Tf\parallel }_{L^{q_1}\left({\mathbb{R}}^n\right)}\le M_1{\parallel f\parallel }_{L^{p_1}\left({\mathbb{R}}^n\right)},\forall f\in L^{p_1}\left({\mathbb{R}}^n\right),$$

then

$${\parallel Tf\parallel }_{L^{q_t}\left({\mathbb{R}}^n\right)}\le {M_0}^{1-t}{M_1}^t{\parallel f\parallel }_{L^{p_t}\left({\mathbb{R}}^n\right)},\forall f\in L^{p_t}\left({\mathbb{R}}^n\right),$$

where ${\displaystyle \frac{1}{p_t}}={\displaystyle \frac{1-t}{p_0}}+{\displaystyle \frac{t}{p_1}}$, ${\displaystyle \frac{1}{q_t}}={\displaystyle \frac{1-t}{q_0}}+{\displaystyle \frac{t}{q_1}}$, $0\le t\le 1$.

Lemma 8 

(Stein–Weiss interpolation theorem). Assume that $1\le p_0,p_1\le \infty$, that $w_0$ and $w_1$ are positive weights, and that T is a sublinear operator satisfying

$${∥Tf∥}_{L^{p_0}\left(w_0\right)}\le M_0{∥f∥}_{L^{p_0}\left(w_0\right)},\forall f\in L^{p_0}\left(w_0\right),$$

$${∥Tf∥}_{L^{p_1}\left(w_1\right)}\le M_1{∥f∥}_{L^{p_1}\left(w_1\right)},\forall f\in L^{p_1}\left(w_1\right),$$

then,

$${∥Tf∥}_{L^p\left(w\right)}\le M{∥f∥}_{L^p\left(w\right)},\forall f\in L^p\left(w\right),$$

where

$$M\le M_0^{\lambda }{M}_1^{1-\lambda },{\displaystyle \frac{1}{p}}={\displaystyle \frac{\lambda }{p_0}}+{\displaystyle \frac{(1-\lambda )}{p_1}},w=w_0^{p\lambda /p_0}{w}_1^{p(1-\lambda )/p_1},0\le \lambda \le 1.$$

3.2. The Decay Estimates for $T_j^N$ on $L^p$ Space

We first study the unweighted $L^p$ estimates with good decay for $T_{j+1}^N-T_j^N$. By Lemma 2, the Plancherel Theorem and the triangle inequality, we have

$${∥\left(T_{j+1}^N-T_j^N\right)f∥}_{L^2\left({\mathbb{R}}^n\right)}\le C{2}^{-\delta N\left(j\right)}{\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}{\parallel f\parallel }_{L^2\left({\mathbb{R}}^n\right)},0<\delta <{\displaystyle \frac{1}{q^{\prime }}}.$$

(15)

Lemma 3 tells us, for $1<p_1<\infty$,

$${∥\left(T_j^N-T_{j+1}^N\right)f∥}_{L^{p_1}\left({\mathbb{R}}^n\right)}\le {CN(j+1)\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}{\parallel f\parallel }_{L^{p_1}\left({\mathbb{R}}^n\right)},$$

(16)

by the interpolation method between the Equations (15) and (16), we can show the following decay estimates for $T_j^N$ on $L^2$ space.

Lemma 9. 

Let $\Omega \in L^q\left({\mathbb{S}}^{n-1}\right)$ with $q>1$ be a homogeneous function of degree zero and have mean value zero on the unit sphere ${\mathbb{S}}^{n-1}$, and let $T_j^N$ be defined in Equation (12), then for $1<p<\infty$,

$${∥\left(T_j^N-T_{j+1}^N\right)f∥}_{L^p\left({\mathbb{R}}^n\right)}\le c_{n,p}{2}^{{-\delta }_{p,q}N\left(j\right)}{N(j+1)\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}{\parallel f\parallel }_{L^p\left({\mathbb{R}}^n\right)},$$

for some constant ${\delta }_{p,q}>0$ independent of $T_{\Omega }$, j and the function $N(·)$.

Proof. 

Assume that $p>2$ and take $2<p<p_1$ and ${\displaystyle \frac{1}{p}}={\displaystyle \frac{1-\theta }{2}}+{\displaystyle \frac{\theta }{p_1}}$, we have $0<1-\theta :={\displaystyle \frac{\frac{1}{p_1}-\frac{1}{p}}{\frac{1}{p_1}-\frac{1}{2}}}<{\displaystyle \frac{2}{p}}<1.$ By Lemma 7, combining with Equations (15) and (16), we have that

$$\begin{array}{cc}\hfill {∥(T_j^N-T_{j+1}^N)f∥}_{L^p\left({\mathbb{R}}^n\right)}& \le {∥T_j^N-T_{j+1}^N∥}_{L^2\left({\mathbb{R}}^n\right)\to L^2\left({\mathbb{R}}^n\right)}^{1-\theta }{∥T_j^N-T_{j+1}^N∥}_{L^{p_1}\left({\mathbb{R}}^n\right)\to L^{p_1}\left({\mathbb{R}}^n\right)}^{\theta }{\parallel f\parallel }_{L^p\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le {\left(c_n{\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}{2}^{-\delta N\left(j\right)}\right)}^{1-\theta }{\left(c_{n,p_1}{\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}\left(N(j+1)\right)\right)}^{\theta }{\parallel f\parallel }_{L^p\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le c_{n,p}{\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}{2}^{-{\delta }_{p,q}N\left(j\right)}N(j+1){\parallel f\parallel }_{L^p\left({\mathbb{R}}^n\right)},\hfill \end{array}$$

where ${\delta }_{p,q}=\delta (1-\theta )\le 2\delta /p<{\displaystyle \frac{2}{q^{\prime }p}}$.

On the other hand, if $p<2$, let us take $p_1<p<2$, in this case, $0<1-\theta :={\displaystyle \frac{\frac{1}{p_1}-\frac{1}{p}}{\frac{1}{p_1}-\frac{1}{2}}}<{\displaystyle \frac{2}{p^{\prime }}}<1$. Then, by interpolating between $L^{p_1}\left({\mathbb{R}}^n\right)$ and $L^2\left({\mathbb{R}}^n\right)$, we obtain the estimate on $L^p\left({\mathbb{R}}^n\right)$. □

Next we consider the two-weight estimate for $T_{j+1}^N-T_j^N$.

Lemma 10 

(Ref. [19], Theorem 1.12). Let ${\mathcal{A}}_{q^{\prime },\mathcal{S}}$ be defined in (14), w be a weight and $\mathcal{S}$ be a sparse family. Let A be a Young function such that $\overline{A}\in B_{p^{\prime }}$. For $f\ge 0$, $p>q^{\prime }$, there holds

$${\parallel {\mathcal{A}}_{q^{\prime },\mathcal{S}}f\parallel }_{L^p\left(w\right)}\le {\left(c_n{p}^{\prime }\right)}^{{\displaystyle \frac{q^{\prime }(p-1)}{p-q^{\prime }}}}{\left({\displaystyle \frac{p}{q^{\prime }}}\right)}^{\prime }\parallel M_{\overline{A}}{\parallel }_{L^{p^{\prime }}}{\parallel f\parallel }_{L^p\left(M_{A_p}w\right)},$$

where $A_p\left(t\right)=A\left(t^{1/p}\right)$.

Lemma 11 

(Two-weight estimate for $T_j^N$). Let $\Omega \in L^q\left({\mathbb{S}}^{n-1}\right)$ with $q>1$ be a homogeneous function of degree zero and have mean value zero on the unit sphere ${\mathbb{S}}^{n-1}$, and let $T_j^N$ be defined in Equation (12). Then for any weight w, $f\ge 0$, $p>q^{\prime }$, $1<r<\infty$,

$${∥T_j^{N}f∥}_{L^p\left(w\right)}\le {CN\left(j\right)\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}{\left(c_n{p}^{\prime }\right)}^{{\displaystyle \frac{q^{\prime }(p-1)}{p-q^{\prime }}}}{\left({\displaystyle \frac{p}{q^{\prime }}}\right)}^{\prime }p{\left(r^{\prime }\right)}^{{\displaystyle \frac{1}{p^{\prime }}}}{\parallel f\parallel }_{L^p\left(M_{r}w\right)}.$$

Proof. 

Using Lemmas 10 and 1 by taking $A\left(t\right)=t^{pr}$ with $1<r<\infty$, we have that

$${\parallel {\mathcal{A}}_{q^{\prime },\mathcal{S}}f\parallel }_{L^p\left(w\right)}\le {\left(c_n{p}^{\prime }\right)}^{{\displaystyle \frac{q^{\prime }(p-1)}{p-q^{\prime }}}}{\left({\displaystyle \frac{p}{q^{\prime }}}\right)}^{\prime }{c}_{n}p{\left(r^{\prime }\right)}^{{\displaystyle \frac{1}{p^{\prime }}}}{\parallel f\parallel }_{L^p\left(M_{r}w\right)},$$

(17)

on the other hand, Lemmas 3–5 imply that

$$\begin{array}{c}\hfill {\parallel T_j^{N}f\parallel }_{L^p\left(w\right)}\le {CN\left(j\right)\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}{\parallel {\mathcal{A}}_{q^{\prime },\mathcal{S}}f\parallel }_{L^p\left(w\right)},\end{array}$$

(18)

therefore, combining Equations (17) and (18), we obtain the two-weight estimate for $T_j^N$. □

4. Proof of Theorem 1 and Corollary 1

Proof of Theorem 1. 

Let us establish the two-weight inequality of Fefferman–Stein type for the operators $T_{j+1}^N-T_j^N$. Combining with Lemmas 9 and 11 and via the method of interpolation with change in measures in ref. [30], we obtain that, for any $0<\theta <1$, any $1<r<\infty$ and any weight function $w>0$,

$$\begin{array}{cc}\hfill {∥\left(T_{j+1}^N-T_j^N\right)f∥}_{L^p\left(w^{\theta }\right)}\le c_{n,p}{\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}& N(j+1)2^{-{\delta }_{p,q}N\left(j\right)(1-\theta )}\hfill \\ \hfill & ·{\left(c_n{p}^{\prime }\right)}^{{\displaystyle \frac{q^{\prime }(p-1)\theta }{p-q^{\prime }}}}{\left({(p/q^{\prime })}^{\prime }\right)}^{\theta }{p}^{\theta }{\left(r^{\prime }\right)}^{{\displaystyle \frac{\theta }{p^{\prime }}}}{\parallel f\parallel }_{L^p\left(M_{r/\theta }\left(w^{\theta }\right)\right)},\hfill \end{array}$$

we can replace $w^{\theta }$ by w and $r/\theta$ by r, then

$$\begin{array}{cc}\hfill {∥\left(T_{j+1}^N-T_j^N\right)f∥}_{L^p\left(w\right)}\le c_{n,p}{\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}& N(j+1)2^{-{\delta }_{p,q}N\left(j\right)(1-\theta )}\hfill \\ \hfill & ·{\left(c_n{p}^{\prime }\right)}^{{\displaystyle \frac{q^{\prime }(p-1)\theta }{p-q^{\prime }}}}{\left({(p/q^{\prime })}^{\prime }\right)}^{\theta }{p}^{\theta }{\left({\left(r\theta \right)}^{\prime }\right)}^{{\displaystyle \frac{\theta }{p^{\prime }}}}{\parallel f\parallel }_{L^p\left(M_r\left(w\right)\right)},\hfill \end{array}$$

(19)

Thus, by Lemma 11 and the Equation (19),

$$\begin{array}{cc}\hfill {\parallel T_{\Omega }f\parallel }_{L^p\left(w\right)}& \le {∥T_0^{N}f∥}_{L^p\left(w\right)}+\sum _{j=0}^{\infty }{∥\left(T_{j+1}^N-T_j^N\right)f∥}_{L^p\left(w\right)}\hfill \\ \hfill & \le c_{n,p}{\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}{\left(c_n{p}^{\prime }\right)}^{{\displaystyle \frac{q^{\prime }(p-1)}{p-q^{\prime }}}}{\left({\displaystyle \frac{p}{q^{\prime }}}\right)}^{\prime }p{\left({\left(r\theta \right)}^{\prime }\right)}^{{\displaystyle \frac{1}{p^{\prime }}}}{\parallel f\parallel }_{L^p\left(M_{r}w\right)}·\left(1+\sum _{j=0}^{\infty }N(j+1)2^{-{\delta }_{p,q}(1-\theta )N\left(j\right)}\right).\hfill \end{array}$$

Now we take $N\left(j\right)=2^j$, and note $2^x>e^{x/2}>x^2/8$ for any $x>0$, we have

$$\begin{array}{cc}\hfill 1+& \sum _{j=0}^{\infty }N(j+1)2^{-{\delta }_{p,q}(1-\theta )N\left(j\right)}=1+\sum _{j=1}^{\infty }{2}^j{2}^{-{\delta }_{p,q}(1-\theta )2^{j-1}}\hfill \\ \hfill & \le 1+\sum _{2\le 2^j<{\displaystyle \frac{1}{1-\theta }}}{2}^j+8\sum _{2^j\ge {\displaystyle \frac{1}{1-\theta }}}{2}^j{\left({\delta }_{p,q}(1-\theta )2^{j-1}\right)}^{-2}\le {\displaystyle \frac{c_{p,q}}{1-\theta }},\hfill \end{array}$$

this implies

$${\parallel T_{\Omega }f\parallel }_{L^p\left(w\right)}\le c_{n,p,q}{\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}{\left(p^{\prime }\right)}^{{\displaystyle \frac{q^{\prime }(p-1)}{p-q^{\prime }}}}{\left({\displaystyle \frac{p}{q^{\prime }}}\right)}^{\prime }p{\left({\left(r\theta \right)}^{\prime }\right)}^{{\displaystyle \frac{1}{p^{\prime }}}}{\displaystyle \frac{1}{1-\theta }}{\parallel f\parallel }_{L^p\left(M_r\left(w\right)\right)}.$$

(20)

Particularly, we choose $\theta ={\displaystyle \frac{2}{r+1}}={\displaystyle \frac{{\left(2r^{\prime }\right)}^{\prime }}{r}}$. By a direct compute, we observe that $\theta \in ({\displaystyle \frac{1}{r}},1)$ and ${\displaystyle \frac{1}{1-\theta }}={\displaystyle \frac{r+1}{r-1}}=2r^{\prime }-1$. Then the Equation (20) yields

$${\parallel T_{\Omega }f\parallel }_{L^p\left(w\right)}\le c_{n,p,q}{\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}{\left(p^{\prime }\right)}^{{\displaystyle \frac{q^{\prime }(p-1)}{p-q^{\prime }}}}{\left({\displaystyle \frac{p}{q^{\prime }}}\right)}^{\prime }p{\left(r^{\prime }\right)}^{1+{\displaystyle \frac{1}{p^{\prime }}}}{\parallel f\parallel }_{L^p\left(M_r\left(w\right)\right)},$$

(21)

which completes the proof of Theorem 1. □

Proof of Corollary 1. 

If $w\in A_{\infty }$, by Lemma 6, there exists a dimensional constant ${\tau }_n$, and we can choose $s_w=1+{\displaystyle \frac{1}{{\tau }_n{\left[w\right]}_{A_{\infty }}}}$ such that

$${\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q{w}^{s_w}\right)}^{{\displaystyle \frac{1}{s_w}}}\le {\displaystyle \frac{2}{\left|Q\right|}}{\int }_{Q}w.$$

Then, by taking $r=s_w$, we have $r^{\prime }=1+{\tau }_n{\left[w\right]}_{A_{\infty }}$ and $M_r\left(w\right)\le 2M\left(w\right)$, and so by the estimate Equation (21), we have that

$${\parallel T_{\Omega }f\parallel }_{L^p\left(w\right)}\le c_{n,p,q}{\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}{\left[w\right]}_{A_{\infty }}^{1+{\displaystyle \frac{1}{p^{\prime }}}}{\parallel f\parallel }_{L^p\left(M\left(w\right)\right)}.$$

Moreover, if $w\in A_1$, then $M\left(w\right)\le {\left[w\right]}_{A_1}w$, and thus

$${\parallel T_{\Omega }f\parallel }_{L^p\left(w\right)}\le c_{n,p,q}{\parallel \Omega \parallel }_{L^q\left({\mathbb{S}}^{n-1}\right)}{\left[w\right]}_{A_1}^{{\displaystyle \frac{1}{p}}}{\left[w\right]}_{A_{\infty }}^{1+{\displaystyle \frac{1}{p^{\prime }}}}{\parallel f\parallel }_{L^p\left(w\right)}.$$

The proof of Corollary 1 is complete. □

5. Proof of Theorem 2

We begin with some necessary preliminaries and lemmas.

The significance of $BMO$ stems from its exponential self-improving property, which is established in the celebrated John–Nirenberg Theorem [31]. We will use a very precise version as follows.

Lemma 12 

(Sharp John–Nirenberg ([32], Theorem 2.1)). There are dimensional constants ${\alpha }_n={\displaystyle \frac{1}{2^{n+2}}}$ and ${\beta }_n>1$ such that

$$\begin{array}{c}\hfill \underset{Q}{sup}{\displaystyle \frac{1}{\left|Q\right|}}{\int }_{Q}exp\left({\displaystyle \frac{{\alpha }_n}{{\parallel b\parallel }_{BMO}}}|b\left(y\right)-b_Q|\right)dy\le {\beta }_n.\end{array}$$

(22)

From Lemma 12, we can derive the following Lemma 13.

Lemma 13. 

Let $b\in BMO$ and let ${\alpha }_n<1<{\beta }_n$ be the dimensional constants from Equation (22). Then for all $1<p<\infty$,

$$s\in \mathbb{R},\left|s\right|\le {\displaystyle \frac{{\alpha }_n}{{\parallel b\parallel }_{BMO}}}min\left\{1,p-1\right\}\Rightarrow e^{sb}\in A_p,$$

and moreover, we have that ${\left[e^{sb}\right]}_{A_p}\le {\beta }_n^2$.

Proof. 

The lemma can be deduced directly from Lemma 12, the following equality and the Hölder’s inequality, we have that

$$\begin{array}{cc}\hfill & \left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_{Q}exp\left(sb\right)dx\right){\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_{Q}exp\left(-{\displaystyle \frac{sb}{p-1}}\right)dx\right)}^{p-1}\hfill \\ \hfill & =\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_{Q}exp\left(s(b-b_Q)\right)dx\right){\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_{Q}exp\left(-{\displaystyle \frac{s(b-b_Q)}{p-1}}\right)dx\right)}^{p-1}\le {\beta }_n^2.\hfill \end{array}$$

□

It is well-known that, for all $1<p<\infty$, $A_p\subset A_{\infty }$ and ${\left[w\right]}_{A_{\infty }}\lesssim {\left[w\right]}_{A_p}$.

Corollary 2. 

Let $b\in BMO$ and let ${\alpha }_n<1<{\beta }_n$ be the dimensional constants from Equation (22). Then

$$s\in \mathbb{R},\left|s\right|\le {\displaystyle \frac{{\alpha }_n}{{\parallel b\parallel }_{BMO}}}\Rightarrow e^{sb}\in A_{\infty },$$

and moreover, we have that ${\left[e^{sb}\right]}_{A_{\infty }}\le {\beta }_n^2$.

Let ${\left(w\right)}_{A_p}:=max\{{[w]}_{A_{\infty }},{[w^{-{\displaystyle \frac{1}{p-1}}}]}_{A_{\infty }}\}\approx {\left[w\right]}_{A_{\infty }}+{[w^{-{\displaystyle \frac{1}{p-1}}}]}_{A_{\infty }}.$

Lemma 14. 

Let $b\in BMO$, $w\in A_p$, ${\alpha }_n<1<{\beta }_n$, if $1<p<\infty$, $p\left|s\right|r^{\prime }\le {\displaystyle \frac{{\alpha }_n}{{\parallel b\parallel }_{BMO}}}min\left\{1,p-1\right\}$ with $r=1+{\displaystyle \frac{1}{{\tau }_n{\left(w\right)}_{A_{\infty }}}}$, then

$$w{e}^{psb}\in A_p,\mathit{and}{\left[w{e}^{psb}\right]}_{A_p}\lesssim {\beta }_n^2{\left[w\right]}_{A_p}.$$

Proof. 

By using Lemmas 13 and 6 and Hölder’s inequality, one can see that

$$\begin{array}{cc}\hfill & \left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_{Q}w{e}^{sbp}\right){\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q{w}^{-{\displaystyle \frac{1}{p-1}}}{e}^{-{\displaystyle \frac{spb}{p-1}}}\right)}^{p-1}\hfill \\ \hfill & \le {\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q{w}^r\right)}^{{\displaystyle \frac{1}{r}}}{\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q{w}^{-{\displaystyle \frac{r}{p-1}}}\right)}^{{\displaystyle \frac{p-1}{r}}}{\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q{e}^{r^{\prime }sbp}\right)}^{{\displaystyle \frac{1}{r^{\prime }}}}{\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q{e}^{-{\displaystyle \frac{r^{\prime }spb}{p-1}}}\right)}^{{\displaystyle \frac{p-1}{r^{\prime }}}}\hfill \\ \hfill & \lesssim {\left[w\right]}_{A_p}{\beta }_n^{{\displaystyle \frac{2}{r^{\prime }}}}\lesssim {\left[w\right]}_{A_p}{\beta }_n^2.\hfill \end{array}$$

□

Lemma 15. 

Let $b\in BMO$ and $w\in A_{\infty }$, ${\alpha }_n<1<{\beta }_n$. $\left|s\right|\lesssim {\displaystyle \frac{1}{2r^{\prime }}}{\displaystyle \frac{{\alpha }_n}{{\parallel b\parallel }_{BMO}}}$ with $r=1+{\displaystyle \frac{1}{{\tau }_n{\left[w\right]}_{A_{\infty }}}}$, then

$${\left[w{e}^{sb}\right]}_{A_{\infty }}\le C{\beta }_n^2{\left[w\right]}_{A_{\infty }}.$$

Proof. 

Fix cube $Q\in {\mathbb{R}}^n$ and $x\in Q$, we have, by Hölder’s inequality,

$$M\left(w{e}^{sb}{\chi }_Q\right)\left(x\right)\le M_r\left(w{\chi }_Q\right)\left(x\right)M_{r^{\prime }}\left(e^{sb}{\chi }_Q\right)\left(x\right),$$

note $M_r\left(w{\chi }_Q\right)\in A_1$ and ${\left[M_r\left(w{\chi }_Q\right)\right]}_{A_1}\approx r^{\prime }$. We take $\delta =1+{\displaystyle \frac{1}{c_n{\left[M_r\left(w\right)\right]}_{A_1}}}$, by choosing $c_n>0$ large enough, $\delta \approx 1+{\displaystyle \frac{1}{c_n{r}^{\prime }}}\approx 1+{\displaystyle \frac{1}{{\tau }_n{\left[w\right]}_{A_{\infty }}}}$, one can see that ${\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q{M}_r{\left(w{\chi }_Q\right)}^{\delta }dy\right)}^{1/\delta }\le {\displaystyle \frac{2}{\left|Q\right|}}{\int }_Q{M}_r\left(w{\chi }_Q\right)dy$ by Lemma 6. Thus by Hölder’s inequality, and ${\delta }^{\prime }>r^{\prime }$,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\left|Q\right|}}{\int }_{Q}M\left(w{e}^{sb}{\chi }_Q\right)\left(x\right)dx\le {\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q{M}_r{\left(w{\chi }_Q\right)}^{\delta }dx\right)}^{{\displaystyle \frac{1}{\delta }}}{\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q{M}_{r^{\prime }}{\left(e^{sb}{\chi }_Q\right)}^{{\delta }^{\prime }}dx\right)}^{{\displaystyle \frac{1}{{\delta }^{\prime }}}}\hfill \\ \hfill & \lesssim C\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q{M}_r\left(w{\chi }_Q\right)dx\right){\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q{e}^{sb{\delta }^{\prime }}dx\right)}^{{\displaystyle \frac{1}{{\delta }^{\prime }}}},\hfill \end{array}$$

then we have

$$\begin{array}{cc}\hfill {\int }_{Q}M\left(w{e}^{sb}{\chi }_Q\right)dx& \le C{\int }_Q{M}_r\left(w{\chi }_Q\right)dx{\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q{e}^{s(b-b_Q){\delta }^{\prime }}dx\right)}^{{\displaystyle \frac{1}{{\delta }^{\prime }}}}{e}^{s{b}_Q}\hfill \\ \hfill & \lesssim C{\beta }_n^{{\displaystyle \frac{1}{{\delta }^{\prime }}}}{e}^{s{b}_Q}{\int }_{Q}M\left(w{\chi }_Q\right)dx,\hfill \end{array}$$

(23)

on the other hand,

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\left|Q\right|}}{\int }_{Q}wdx& ={\displaystyle \frac{1}{\left|Q\right|}}{\int }_{Q}w{e}^{sb}{e}^{-sb}dx\hfill \\ \hfill & \le {\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q{\left(w{e}^{sb}\right)}^{r}dx\right)}^{{\displaystyle \frac{1}{r}}}{\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q{e}^{-sb{r}^{\prime }}dx\right)}^{{\displaystyle \frac{1}{r^{\prime }}}}\hfill \\ \hfill & \le \left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_{Q}w{e}^{sb}dx\right){\left({\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q{e}^{-s(b-b_Q)r^{\prime }}dx\right)}^{{\displaystyle \frac{1}{r^{\prime }}}}{e}^{-s{b}_Q},\hfill \end{array}$$

therefore

$$\begin{array}{c}\hfill {\left({\int }_{Q}w{e}^{sb}dx\right)}^{-1}\le C{\beta }_n^{{\displaystyle \frac{1}{r^{\prime }}}}{e}^{-s{b}_Q}{\left({\int }_{Q}wdx\right)}^{-1},\end{array}$$

(24)

combining with Equations (23) and (24),

$${\left[w{e}^{sb}\right]}_{A_{\infty }}\le C{\beta }_n^2\underset{Q}{sup}{\displaystyle \frac{1}{w\left(Q\right)}}{\int }_{Q}M\left(w{\chi }_Q\right)dx\le C{\beta }_n^2{\left[w\right]}_{A_{\infty }}.$$

□

Proof of Theorem 2. 

Let $T_{z}f=e^{zb}{T}_{\Omega }\left(e^{zb}f\right)$ with complex variable z, then we can write that

$$[b,T_{\Omega }]f={\displaystyle \frac{d}{dz}}{T}_z{f|}_{z=0}={\displaystyle \frac{1}{2\pi i}}{\int }_{\left|z\right|=\epsilon }{\displaystyle \frac{T_z\left(f\right)}{z^2}}dz.$$

Using Minkowski’s inequality, we get

$$\begin{array}{c}\hfill {\parallel [b,T_{\Omega }]f\parallel }_{L^p\left(w\right)}\le {\displaystyle \frac{1}{2\pi {\epsilon }^2}}{\int }_{\left|z\right|=\epsilon }\parallel T_z{f\parallel }_{L^p\left(w\right)}\left|dz\right|.\end{array}$$

(25)

Now by the quantitative weighted estimates for $T_{\Omega }$ with $\Omega \in L^{\infty }\left({\mathbb{S}}^{n-1}\right)$, see ref. [19], we can get

$$\begin{array}{cc}\hfill {\parallel T_{z}f\parallel }_{L^p\left(w\right)}& =\parallel T_{\Omega }\left(e^{-zb}f\right){\parallel }_{L^p\left(w{e}^{pRezb}\right)}\hfill \\ \hfill & \lesssim {\parallel \Omega \parallel }_{L^{\infty }}\phi \left(w{e}^{pRezb}\right){\parallel e^{-zb}f\parallel }_{L^p\left(w{e}^{pRezb}\right)}\hfill \\ \hfill & ={\parallel \Omega \parallel }_{L^{\infty }}\phi \left(w{e}^{pRezb}\right){\parallel f\parallel }_{L^p\left(w\right)},\hfill \end{array}$$

(26)

where we have denoted by $\phi \left(w\right)={\left[w\right]}_{A_p}^{{\displaystyle \frac{1}{p}}}max\{{\left[w\right]}_{A_{\infty }}^{{\displaystyle \frac{1}{p^{\prime }}}},{\left[w^{1-p^{\prime }}\right]}_{A_{\infty }}^{{\displaystyle \frac{1}{p}}}\}min\{{\left[w\right]}_{A_{\infty }},{\left[w^{1-p^{\prime }}\right]}_{A_{\infty }}\}$. Recall Lemma 14, we let $p\left|Rez\right|\le {\displaystyle \frac{{\alpha }_n}{{\parallel b\parallel }_{BMO}}}min\{1,p-1\}{\displaystyle \frac{1}{r^{\prime }}}$, which means that

$$\left|Rez\right|\le {\displaystyle \frac{{\alpha }_n}{{\parallel b\parallel }_{BMO}}}min\left\{{\displaystyle \frac{1}{p}},{\displaystyle \frac{1}{p^{\prime }}}\right\}{\displaystyle \frac{1}{{\tau }_n{\left(w\right)}_{A_p}}},$$

then we have

$${\left[e^{p{r}^{\prime }Rezb}\right]}_{A_p}\le {\beta }_n^2,$$

and

$${\left[w{e}^{pRezb}\right]}_{A_p}\le {\left[w\right]}_{A_p}{\beta }_n.$$

According to the Lemma 15, when $\left|Rez\right|\le {\displaystyle \frac{{\alpha }_n}{{\parallel b\parallel }_{BMO}}}{\displaystyle \frac{1}{p}}{\displaystyle \frac{1}{{\tau }_n{\left[w\right]}_{A_{\infty }}}}$, we have

$${\left[w{e}^{pRezb}\right]}_{A_{\infty }}\le {\left[w\right]}_{A_{\infty }}{\beta }_n,$$

on the other hand, when $\left|Rez\right|\le {\displaystyle \frac{{\alpha }_n}{{\parallel b\parallel }_{BMO}}}{\displaystyle \frac{1}{p^{\prime }}}{\displaystyle \frac{1}{{\tau }_n{\left[w^{1-p^{\prime }}\right]}_{A_{\infty }}}}$, we have

$${\left[w^{1-p^{\prime }}{e}^{-p^{\prime }Rezb}\right]}_{A_{\infty }}\le {\left[w^{1-p^{\prime }}\right]}_{A_{\infty }}{\beta }_n.$$

Therefore,

$$\begin{array}{cc}\hfill \phi \left(w{e}^{pRezb}\right)& \lesssim {\left[w\right]}_{A_p}^{{\displaystyle \frac{1}{p}}}{\beta }_n^{{\displaystyle \frac{1}{p}}}max\{{\left[w\right]}_{A_{\infty }}^{{\displaystyle \frac{1}{p^{\prime }}}}{\beta }_n^{{\displaystyle \frac{1}{p^{\prime }}}},{\left[w^{1-p^{\prime }}\right]}_{A_{\infty }}^{{\displaystyle \frac{1}{p}}}{\beta }_n^{{\displaystyle \frac{1}{p}}}\}min\{{\left[w\right]}_{A_{\infty }}{\beta }_n,{\left[w^{1-p^{\prime }}\right]}_{A_{\infty }}{\beta }_n\}\hfill \\ \hfill & \lesssim {\beta }_n^3\phi \left(w\right).\hfill \end{array}$$

Combining the Equations (25) and (26) together with the estimate above yields that

$${\parallel [b,T_{\Omega }]f\parallel }_{L^p\left(w\right)}\lesssim {\parallel \Omega \parallel }_{L^{\infty }}{\displaystyle \frac{1}{2\pi \epsilon }}{\beta }_n^3\phi \left(w\right){\parallel f\parallel }_{L^p\left(w\right)},$$

now we take $\epsilon ={\displaystyle \frac{{\alpha }_n}{{\parallel b\parallel }_{BMO}}}min\left\{{\displaystyle \frac{1}{p}},{\displaystyle \frac{1}{p^{\prime }}}\right\}{\displaystyle \frac{1}{{\tau }_n{\left(w\right)}_{A_{\infty }}}}$, then we obtain that

$${\parallel [b,T_{\Omega }]f\parallel }_{L^p\left(w\right)}\lesssim {\parallel \Omega \parallel }_{L^{\infty }}{\parallel b\parallel }_{BMO}max\{p,p^{\prime }\}\phi \left(w\right){\left(w\right)}_{A_{\infty }}{\parallel f\parallel }_{L^p\left(w\right)},$$

which implies Theorem 2. □