On the Boundedness of Non-standard Rough Singular Integral Operators

Abstract

Let $\Omega$ be a homogeneous function of degree zero, have vanishing moment of order one on the unit sphere $\mathbb {S}^{d-1}$($d\ge 2$). In this paper, our object of investigation is the following rough non-standard singular integral operator

$$\begin{aligned} T_{\Omega ,\,A}f(x)=\mathrm{p.\,v.}\int _{{\mathbb {R}}^d}\frac{\Omega (x-y)}{|x-y|^{d+1}}\big (A(x)-A(y)-\nabla A(y)(x-y)\big )f(y)\textrm{d}y, \end{aligned}$$

where A is a function defined on ${\mathbb {R}}^d$ with derivatives of order one in ${\textrm{BMO}}({\mathbb {R}}^d)$. We show that $T_{\Omega ,\,A}$ enjoys the endpoint $L\log L$ type estimate and is $L^p$ bounded if $\Omega \in L(\log L)^{2}({\mathbb {S}}^{d-1})$. These results essentially improve the previous known results given by Hofmann (Stud Math 109:105–131, 1994) for the $L^p$ boundedness of $T_{\Omega ,\,A}$ under the condition $\Omega \in L^{q}({\mathbb {S}}^{d-1})$ $(q>1)$, Hu and Yang (Bull Lond Math Soc 35:759–769, 2003) for the endpoint weak $L\log L$ type estimates when $\Omega \in \textrm{Lip}_{\alpha }({\mathbb {S}}^{d-1})$ for some $\alpha \in (0,\,1]$.

1 Introduction

This paper will be devoted to study the boundedness of certain non-standard Calderón-Zygmund operators with rough kernels. To begin with, let $d\ge 2$, ${\mathbb {R}}^d$ be the d-dimensional Euclidean space and ${\mathbb {S}}^{d-1}$ be the unit sphere in ${\mathbb {R}}^d$. Let $\Omega$ be a function of homogeneous of degree zero, $\Omega \in L^1({{\mathbb {S}}^{d-1}})$ and satisfy the vanishing condition

$$\begin{aligned} {\int _{{\mathbb {S}}^{d-1}}\Omega (x)x_j{ d}x=0},\quad j=1,..., d. \end{aligned}$$

(1.1)

Define the non-standard rough Calderón-Zygmund operator by

$$\begin{aligned} \quad \quad T_{\Omega ,\,A}f(x)=\mathrm{p. v.}\int _{{\mathbb {R}}^d}\frac{\Omega (x-y)}{|x-y|^{d+1}}\big (A(x)-A(y)-\nabla A(y)(x-y)\big )f(y)dy,\nonumber \\ \end{aligned}$$

(1.2)

where A is a function on ${\mathbb {R}}^d$ such that $\nabla A\in \textrm{BMO}({\mathbb {R}}^d)$, that is, $\partial _n A\in \textrm{BMO}({\mathbb {R}}^d)$ for all n with $1\le n\le d$. This class of singular integrals is of interest in Harmonic analysis. It was well-known that $T_{\Omega ,\,A}$ is closely related to the study of Calderón commutators [1, 2]. Even for smooth kernel $\Omega$, since ${L^\infty ({\mathbb {R}}^d)}\varsubsetneq \textrm{BMO}({\mathbb {R}}^d)$, the kernel of the operator $T_{\Omega ,\,A}$ may fail to satisfy the classical standard kernel conditions. This is the main reason why one calls them nonstandard singular integral operators.

Recall that if $\nabla A\in L^\infty ({\mathbb {R}}^d)$, then the $L^p({\mathbb {R}}^d)$ boundedness of $T_{\Omega ,\,A}$ follows by using the methods of rotation in the nice work of Caldéron [2], Bainshansky and Coifman [1]. Since the method of rotations doesn’t work in the case of $\nabla A\in \textrm{BMO}({\mathbb {R}}^d)$, Cohen [7] and Hu [24] obtained the $L^p({\mathbb {R}}^d)$ boundedness of $T_{\Omega ,\,A}$ with smooth kernels by means of a good-$\lambda$ inequality. More precisely, if $\Omega \in \textrm{Lip}_\alpha ({\mathbb {S}}^{d-1})$ ($0<\alpha \le 1$), then Cohen [7] proved that $T_{\Omega ,A}$ is a bounded operator on $L^p({\mathbb {R}}^d)$ for $1<p<\infty$. Later on, the result of Cohen [7] was improved by Hofmann [19]. It was shown that $\Omega \in \cup _{q>1}L^q({\mathbb {S}}^{d-1})$ is a sufficient condition for the $L^p({\mathbb {R}}^d)$ boundedness of $T_{\Omega , A}$. If $\Omega \in L^{\infty }({\mathbb {S}}^{d-1})$, Hofmann [19] demonstrated that $T_{\Omega ,A}$ is bounded on $L^p({\mathbb {R}}^d,\,w)$ for all $p\in (1,\,\infty )$ and $w\in A_p({\mathbb {R}}^d)$, where and in what follows, $A_p({\mathbb {R}}^d)$ denotes the weight function class of Muckenhoupt, see [12, Chap. 9] for properties of $A_p({\mathbb {R}}^d)$.

It is quite natural to ask if one can establish weak type inequalities for $T_{\Omega ,\,A}$ or not. Hu and Yang [23] considered the operator

$$\begin{aligned} T_{a}f(x)=\mathrm{p.\,v.}\int _{{\mathbb {R}}}\frac{a(x)-a(y)-a'(y)(x-y)}{(x-y)^2}f(y)dy, \end{aligned}$$

where a is a function on ${\mathbb {R}}$ such that $a'\in \textrm{BMO}({\mathbb {R}})$. Hu and Yang showed that, $T_{a}$ may fail to be of weak type $(1,\,1)$, which differs in this aspect from the property of the classical singular integral operators, see Remark 3 in [23, p. 762]. As a replacement of weak (1, 1) boundedness, it was shown in [23] that, when $\Omega \in \textrm{Lip}_{\alpha }(S^{d-1})$ with $\alpha \in (0,\,1]$, $T_{\Omega ,A}$ still enjoys the endpoint $L\log L$ type estimates. This, tells us that, when $\Omega$ satisfies suitable regularity condition, the endpoint estimates of $T_{\Omega , A}$ parallels to that of the commutator of Calderón-Zygmund operators with symbol in $\textrm{BMO}({\mathbb {R}}^d)$. For the endpoint estimates of the commutator of Calderón-Zygmund operators, see [22, 29] and the references therein.

Now, we recall some known results of classical singular integrals and make a comparative analysis. It was first shown by Calderón and Zygmund [3] that the singular integrals $T_{\Omega }$ defined by

$$\begin{aligned} T_{\Omega }f(x)=\text {p.v.}\int _{{\mathbb {R}}^d}\frac{\Omega (y/|y|)}{|y|^d}f(x-y)dy \end{aligned}$$

is bounded on $L^p({\mathbb {R}}^d)$ $(1<p<\infty )$ either when $\Omega$ is an odd function and $\Omega \in L^1({\mathbb {S}}^{d-1})$, or $\Omega$ is an even function with $\int _{{\mathbb {S}}^{d-1}}\Omega \,d\sigma =0$ and $\Omega \in L\log L({\mathbb {S}}^{d-1})$. Later on, the condition $\Omega \in L\log L({\mathbb {S}}^{d-1})$ was improved to $\Omega \in H^1({\mathbb {S}}^{d-1})$ by Connett [8], Ricci and Weiss [30], independently. Since then, great achievements have been made in this field. Among them are the celebrated works of the weak type (1, 1) bounds given by Christ [5], Christ and Rubio de Francia [6], Hofmann [17], Seeger [31], and Tao [33]. It was shown that $\Omega \in L\log L({\mathbb {S}}^{d-1})$ is sufficient condition for the weak type (1, 1) estimate of $T_{\Omega }$. Recently, this result was generalized by Ding and Lai [9] for the operator $T_{\Omega }^*$ defined by

$$\begin{aligned} {T}^*_{\Omega }f(x)=\mathrm{p.\,v.}\int _{{\mathbb {R}}^d}\Omega (x-y)K(x,\,y)f(y)dy, \end{aligned}$$

where the kernel $\Omega \in L\log L({\mathbb {S}}^{d-1})$ and K needs to satisfy some size and regularity conditions. For other related contributions, we refer the readers to references [10, 11, 15, 22, 25,26,27,28, 32, 34, 35] and the references therein.

Consider now the $L^p({\mathbb {R}}^d)$ boundedness and endpoint estimates for the operator $T_{\Omega ,\,A}$ when $\Omega$ satisfies only size condition, things become more subtle. Hu [21] considered the $L^2({\mathbb {R}}^d)$ boundedness of $T_{\Omega , A}$ when $\Omega \in GS_{\beta }({\mathbb {S}}^{d-1})$, which means,

$$\begin{aligned} \sup _{\zeta \in S^{d-1}}\int _{{\mathbb {S}}^{d-1}}\vert \Omega (\theta )\vert \log ^{\beta }\bigg (\frac{1}{\vert \zeta \cdot \theta \vert }\bigg )d\theta <\infty . \end{aligned}$$

(1.3)

The main result in [21] can be summarized as follows:

Theorem A

Let $\Omega$ be homogeneous of degree zero which satisfies the vanishing condition (1.1), A be a function on ${\mathbb {R}}^d$ such that $\nabla A\in \textrm{BMO}({\mathbb {R}}^d)$. Suppose that $\Omega \in GS_{\beta }(S^{d-1})$ for some $\beta >3$, then $T_{\Omega ,\,A}$ is bounded on $L^2({\mathbb {R}}^d)$.

This size condition was introduced by Grafakos and Stefanov [14], to study the $L^p({\mathbb {R}}^d)$ boundedness of the homogeneous singular integral operator. As it was pointed out in [14], there exist integrable functions on ${\mathbb {S}}^{d-1}$ which are not in $H^1({\mathbb {S}}^{d-1})$ but satisfy (1.3) for all $\beta \in (1,\,\infty )$. Thus, $GS_{\beta }({\mathbb {S}}^{d-1})$ is also a minimum size condition for functions on ${\mathbb {S}}^{d-1}$. It is easy to verify that

$$\begin{aligned} \cup _{q>1}L^q({\mathbb {S}}^{d-1})\subset \cap _{\beta >1}GS_{\beta }({\mathbb {S}}^{d-1}),\,\,L(\log L)^{\beta }({\mathbb {S}}^{d-1})\subset GS_{\beta }({\mathbb {S}}^{d-1}). \end{aligned}$$

For the $L^p({\mathbb {R}}^{d})$ ($1<p<\infty$) boundedness of $T_{\Omega , A}$, the best known condition $\Omega \in \cup _{q>1}L^q({\mathbb {S}}^{d-1})$ is given in [19]. There is no any endpoint estimate for $T_{\Omega , A}$ when $\Omega$ only satisfies some size condition, even if $\Omega \in L^{\infty }({\mathbb {S}}^{d-1})$. Note that the following inclusion relationship holds

$$\begin{aligned}{} & {} \textrm{Lip}_\alpha ({\mathbb {S}}^{d-1}) (0<\alpha \le 1)\subsetneq L^q({\mathbb {S}}^{d-1})(q>1)\subsetneq L(\log L)^2({\mathbb {S}}^{d-1})\nonumber \\{} & {} \quad \subsetneq L\log L({\mathbb {S}}^{d-1})\subsetneq H^1({\mathbb {S}}^{d-1}). \end{aligned}$$

(1.4)

Therefore, it is quite natural to ask the following question:

Question: What is the minimal condition such that $T_{\Omega ,\,A}$ is bounded on $L^p({\mathbb {R}}^d)$ for all $p\in (1,\,\infty )$? Does the endpoint estimate of $L\log L$ type still holds true when $\Omega$ only satisfies size condition?

The main purpose of this paper is to show that $\Omega \in L(\log L)^2({\mathbb {S}}^{d-1})$ is a sufficient condition for the $L^p({\mathbb {R}}^d)$ boundedness and weak type $L\log L$ estimate for $T_{\Omega , A}$. Our first result can be stated as follows.

Theorem 1.1

Let $\Omega$ be homogeneous of degree zero, satisfy the vanishing moment (1.1), and A be a function on ${\mathbb {R}}^d$ such that $\nabla A\in \textrm{BMO}({\mathbb {R}}^d)$. Suppose that $\Omega \in L(\log L)^{2}({\mathbb {S}}^{d-1})$. Then $T_{\Omega ,\,A}$ is bounded on $L^2({\mathbb {R}}^d)$.

Let ${\widetilde{T}}_{\Omega ,\,A}$ be the dual operator of $T_{\Omega ,\,A}$, defined as

$$\begin{aligned} {\widetilde{T}}_{\Omega ,A}f(x)=\mathrm{p.\,v.}\int _{{\mathbb {R}}^d}\frac{\Omega (x-y)}{|x-y|^{d+1}}\big (A(x)-A(y)-\nabla A(x)(x-y)\big )f(y)dy. \end{aligned}$$

(1.5)

Theorem 1.2

Let $\Omega$ be homogeneous of degree zero, satisfy the vanishing condition (1.1), and A be a function on ${\mathbb {R}}^d$ such that $\nabla A\in \textrm{BMO}({\mathbb {R}}^d)$. Suppose that $\Omega \in L(\log L)^{2}({\mathbb {S}}^{d-1})$. Then for any $\lambda >0$ and $\mathrm{\Phi (t)=t\log ({e}+t)}$, the following inequalities hold

$$\begin{aligned}{} & {} \big |\{x\in {\mathbb {R}}^d:\, |T_{\Omega ,A}f(x)|>\lambda \}\big |\lesssim \int _{{\mathbb {R}}^d} \Phi \left( \frac{|f(x)|}{\lambda }\right) dx; \end{aligned}$$

(1.6)

$$\begin{aligned}{} & {} \big |\{x\in {\mathbb {R}}^d:\, |{\widetilde{T}}_{\Omega ,A}f(x)|> \lambda \}\big |\lesssim \lambda ^{-1}\Vert f\Vert _{L^1({\mathbb {R}}^d)}. \end{aligned}$$

(1.7)

As far as we know, there is no previous study concerning the weak type endpoint estimates for ${\widetilde{T}}_{\Omega , A}$, even if $\Omega \in \textrm{Lip}_{\alpha }({\mathbb {S}}^{d-1})$ for $\alpha \in (0,\,1]$. We consider this operator mainly to deduce the following precise $L^p({\mathbb {R}}^d)$ bounds of $T_{\Omega ,\,A}$.

Theorem 1.3

Let $\Omega$ be homogeneous of degree zero, satisfy the vanishing condition (1.1), and A be a function on ${\mathbb {R}}^d$ such that $\nabla A\in \textrm{BMO}({\mathbb {R}}^d)$. Suppose that $\Omega \in L(\log L)^{2}({\mathbb {S}}^{d-1})$. Then

$$\begin{aligned} \Vert T_{\Omega ,\,A}f\Vert _{L^p({\mathbb {R}}^d)}\lesssim \Big \{\begin{array}{ll} p'^2\Vert f\Vert _{L^p({\mathbb {R}}^d)},\,&{}p\in (1,\,2];\\ p\Vert f\Vert _{L^p({\mathbb {R}}^d)},\,&{}p\in (2,\,\infty ).\end{array} \end{aligned}$$

Remark 1.4

Theorem 1.1, along with Theorem 1.3, shows that $\Omega \in L(\log L)^{2}({\mathbb {S}}^{d-1})$ is a sufficient condition such that $T_{\Omega ,\,A}$ is bounded on $L^p({\mathbb {R}}^d)$ for all $p\in (1,\,\infty )$. This improves essentially the result obtained in [19, Theorem 1.1], in which, it was shown that if $\Omega \in \cup _{q>1}L^q({\mathbb {S}}^{d-1})$, then $T_{\Omega ,\,A}$ is bounded on $L^p({\mathbb {R}}^d)$ for all $p\in (1,\,\infty )$.

Remark 1.5

As it was pointed out, for $\beta \in [1,\,\infty )$, $L(\log L)^{\beta }({\mathbb {S}}^{d-1})\subset GS_{\beta }(S^{d-1})$. However, it is unknown whether $L(\log L)^{\beta }({\mathbb {S}}^{d-1})\subset GS_{\beta '}(S^{d-1})$ when $\beta '>\beta$. We conjecture that there is no inclusion relationship between $L(\log L)^{\beta }({\mathbb {S}}^{d-1})$ and $GS_{\beta '}(S^{d-1})$ when $\beta '>\beta$, and believe Theorem A and Theorem 1.3 do not imply each other in the case $p=2$.

We believe that the condition $\Omega \in L(\log L)^{2}({\mathbb {S}}^{d-1})$ is the weakest condition for these weak type results to hold, in the following sense.

Conjecture 1.6

$\Omega \in L(\log L)^{2}({\mathbb {S}}^{d-1})$ is the minimal condition for the weak $L\log L$ type estimate of $T_{\Omega ,\,A}$, and weak (1, 1) estimate of ${\widetilde{T}}_{\Omega ,A}$, in the sense that the power 2 can’t be replaced by any real number smaller than 2.

The article is organized as follows. Section 2 will be devoted to demonstrate the $L^2$ boundedness of $T_{\Omega ,A}.$ In Sect. 3, we will prove Theorem 1.2 and Theorem 1.3. The proof of Theorem 1.2 is not short and will be divided into several cases and steps. Smoothness trunction method will play an important role and will be used several times.

Let’s explain a little bit about the proofs of the main results. In Sect. 2, we will introduce a convolution operator $Q_s$ with the property that

$$\begin{aligned} \int ^{\infty }_0Q_s^4\frac{ds}{s}=I. \end{aligned}$$

This makes it possible to commutate with the paraproducts appeared in the proof and thus obtains more freedom in dealing with the estimates of the $L^2$ boundedness. Moreover, the method of dyadic analysis has been applied in the delicate decomposition of $L^2$ norm of $T_{\Omega ,\,A}$. At some key points, we will use some properties of Carleson measure.

The key ingredient in our proof of Theorem 1.2 is to estimate the bad part in the Calderón-Zygmund decomposition of f. In the work of [31], Seeger showed that if $\Omega \in L\log L({\mathbb {S}}^{d-1})$, then $T_{\Omega }$ is bounded from $L^1({\mathbb {R}}^d)$ to $L^{1,\,\infty }({\mathbb {R}}^d)$. Ding and Lai [9] proved that if $\Omega \in L\log L({\mathbb {S}}^{d-1})$ and for some $\delta \in (0,\,1]$, the function K satisfies

$$\begin{aligned}{} & {} |K(x,\,y)|\lesssim \frac{1}{|x-y|^d}; \end{aligned}$$

(1.8)

$$\begin{aligned}{} & {} |K(x_1,\,y)-K(x_2,\,y)|\lesssim \frac{|x_1-x_2|^{\delta }}{|x_1-y|^{d+\delta }},\,\,|x_1-y|\ge 2|x_1-x_2|, \end{aligned}$$

(1.9)

$$\begin{aligned}{} & {} |K(x,\,y_1)-K(x,\,y_2)|\lesssim \frac{|y_1-y_2|^{\delta }}{|x-y_1|^{d+\delta }},\,\,|x-y_1|\ge 2|y_1-y_2|, \end{aligned}$$

(1.10)

and ${T}_{\Omega }^*$ is bounded on $L^2({\mathbb {R}}^d)$, then ${T}_{\Omega }^*$ is bounded from $L^1({\mathbb {R}}^d)$ to $L^{1,\,\infty }({\mathbb {R}}^d)$. However, when A has derivatives of order one in $\textrm{BMO}({\mathbb {R}}^d)$, the function $[{A(x)-A(y)-\nabla A(y)(x-y)}]{|x-y|^{-d-1}}$ does not satisfy the conditions (1.8)–(1.10). Let f be a bounded function with compact support, $b=\sum _{L}b_L$ be the bad part in the Calderón-Zygmund decomposition of f. In order to overcome this essential difficulty, we write

$$\begin{aligned} T_{\Omega ,\,A}b(x)= & {} \sum _{L}\sum _{s}\int _{{\mathbb {R}}^d}\frac{\Omega (x-y)}{|x-y|^{d+1}}\phi _s(x-y)\big (A_L(x)-A_L(y)\big )b_{L}(y)dy\\ {}{} & {} +\text {error terms}, \end{aligned}$$

where $A_{L}(y)=A(y)-\sum _{n=1}^d\langle \partial _n A\rangle _Ly_n.$ $\phi _s(x)=\phi (2^{-s}x)$. Here, $\langle \partial _n A\rangle _L$ denotes the mean value of $\partial _n A$ on the cube L, $\phi$ is a smooth radial nonnegative function on ${\mathbb {R}}^d$ such that $\textrm{supp}\, \phi \subset \{x:\frac{1}{4}\le |x|\le 1\}$ and $\sum _s\phi _s(x)=1$ for all $x\in {\mathbb {R}}^d\backslash \{0\}$. Then, our key observation is that, for each $s\in {\mathbb {Z}}$ and L with side length $\ell (L)=2^{s-j}$, the kernel ${|x-y|^{-d-1}}\phi _s(x-y)\big (A_L(x)-A_L(y)\big )\chi _{L}(y)$ instead satisfies (1.9) and (1.10).

In what follows, C always denotes a positive constant which is independent of the main parameters involved but whose value may differ from line to line. We use the symbol $A\lesssim B$ to denote that there exists a positive constant C such that $A\le CB$. Specially, we use $A\lesssim _{n,p} B$ to denote that there exists a positive constant C depending only on $n,\,p$ such that $A\le CB$. Constant with subscript such as $c_1$, does not change in different occurrences. For any set $E\subset {\mathbb {R}}^d$, $\chi _E$ denotes its characteristic function. For a cube $Q\subset {\mathbb {R}}^d$, $\ell (Q)$ denotes the side length of Q, and for $\lambda \in (0,\,\infty )$, we use $\lambda Q$ to denote the cube with the same center as Q and whose side length is $\lambda$ times that of Q. For a suitable function f, ${\widehat{f}}$ denotes the Fourier transform of f. For $p\in [1,\,\infty ]$, $p'$ denotes the dual exponent of p, namely, $1/p'=1-1/p$.

2 Proof of Theorem 1.1

This section will be devoted to prove Theorem 1.1, the $L^2({\mathbb {R}}^{d})$ boundedness of $T_{\Omega ,\,A}$ when $\Omega \in L(\log L)^{2}({\mathbb {S}}^{d-1})$. We will employ some ideas from [19], together with many more refined estimates. We begin with some notions and lemmas. Let $\psi \in C^{\infty }_0({\mathbb {R}}^d)$ be a radial function with integral zero, $\textrm{supp}\,\psi \subset B(0,\,1)$, $\psi _s(x)=s^{-d}\psi (s^{-1}x)$ and assume that

$$\begin{aligned} \int ^{\infty }_0[{\widehat{\psi }}(s)]^4\frac{ds}{s}=1. \end{aligned}$$

Consider the convolution operator $Q_sf(x)=\psi _s*f(x).$ It enjoys the property that

$$\begin{aligned} \int ^{\infty }_0Q_s^4\frac{ds}{s}=I. \end{aligned}$$

(2.1)

Moreover, by the classical Littlewood-Paley theory, it follows that

$$\begin{aligned} \Big \Vert \Big (\int ^{\infty }_0|Q_sf|^2\frac{ds}{s}\Big )^{1/2}\Big \Vert _{L^2({\mathbb {R}}^d)}\lesssim \Vert f\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

(2.2)

Let $\phi$ be a smooth radial nonnegative function on ${\mathbb {R}}^d$ with $\textrm{supp}\, \phi \subset \{x:\frac{1}{4}\le |x|\le 1\}$, $\sum _s\phi _s(x)=1$ with $\phi _j(x)=2^{-jd}\phi (2^{-j}x)$ for all $x\in {\mathbb {R}}^d\backslash \{0\}$. For each fixed $j\in {\mathbb {Z}}$, define

$$\begin{aligned} T_{\Omega ,\,A;\,j}f(x)=\int _{{\mathbb {R}}^d}K_{A,\,j}(x,\,y)f(y)dy, \end{aligned}$$

(2.3)

where

$$\begin{aligned} K_{A,\,j}(x,\,y)=\frac{\Omega (x-y)}{|x-y|^{d+1}}(A(x)-A(y)-\nabla A(y)(x-y)\big ) {\phi _j(x-y).} \end{aligned}$$

The following lemmas are needed in our analysis.

Lemma 2.1

([19]) Let $\Omega$ be homogeneous of degree zero, satisfies the vanishing condition (1.1) and $\Omega \in L^1 ({\mathbb {S}}^{d-1})$. Let A be a function on ${\mathbb {R}}^d$ such that $\nabla A\in \textrm{BMO}({\mathbb {R}}^d)$. Then for any $k_1,\,k_2\in {\mathbb {Z}}$ with $k_1<k_2$, the following inequality holds

$$\begin{aligned} \Big |\sum _{k_1\le j\le k_2}\int _{{\mathbb {R}}^d}K_{A,\,j}(x,\,y)dy\Big |\lesssim \Vert \Omega \Vert _{L^1({\mathbb {S}}^{d-1})}. \end{aligned}$$

Lemma 2.2

([19]) Let $\Omega$ be homogeneous of degree zero, integrable on ${\mathbb {S}}^{d-1}$ and satisfy the vanishing moment (1.1). Let A be a function on ${\mathbb {R}}^d$ such that $\nabla A\in \textrm{BMO}({\mathbb {R}}^d)$. Then there exists a constant $\epsilon \in (0,\,1)$, such that for $s\in (0,\,\infty )$ and $j\in {\mathbb {Z}}$ with $s2^{-j}\le 1$,

$$\begin{aligned} \Vert Q_sT_{\Omega ,\,A;\,j}1\Vert _{L^{\infty }({\mathbb {R}}^d)}\lesssim \Vert \Omega \Vert _{L^1({\mathbb {S}}^{d-1})}(2^{-j}s)^{\epsilon }. \end{aligned}$$

Lemma 2.3

([19]) Let $\Omega$ be homogeneous of degree zero and $\Omega \in L^{\infty }({\mathbb {S}}^{d-1})$. Let A be a function on ${\mathbb {R}}^d$ such that $\nabla A\in \textrm{BMO}({\mathbb {R}}^d)$. Then there exists a constant $\varepsilon \in (0,\,1)$, such that for $s\in (0,\,\infty )$ and $j\in {\mathbb {Z}}$ with $2^{-j}s\le 1$,

$$\begin{aligned} \Vert Q_sT_{\Omega ,\,A;\,j}f\Vert _{L^2({\mathbb {R}}^d)}\lesssim \Vert \Omega \Vert _{L^{\infty }({\mathbb {S}}^{d-1})}(2^{-j}s)^{\varepsilon }\Vert f\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

Lemma 2.4

([20]) Let $\Omega$ be homogeneous of degree zero, have mean value zero on ${\mathbb {S}}^{d-1}$ and $\Omega \in L(\log L)^2({S}^{d-1})$. Then for $b\in \textrm{BMO}({\mathbb {R}}^d)$, $[b,\,T_{\Omega }]$, the commutator of $T_{\Omega }$ with symbol b, defined by

$$\begin{aligned}{}[b,\,T_{\Omega }]f(x)=b(x)T_{\Omega }f(x)-T_{\Omega }(bf)(x),\,\,\,f\in C^{\infty }_0({\mathbb {R}}^d), \end{aligned}$$

is bounded on $L^p({\mathbb {R}}^d)$ for all $p\in (1,\,\infty )$.

Lemma 2.5

([19]) Let $\Omega$ be homogeneous of degree zero, and integrable on ${\mathbb {S}}^{d-1}$ and satisfy the vanishing moment (1.1), A be a function in ${\mathbb {R}}^d$ with derivatives of order one in $\textrm{BMO}({\mathbb {R}}^d)$. Then for any $r\in (0,\,\infty )$, functions ${\widetilde{\eta }}_1,\,{\widetilde{\eta }}_2\in C^{\infty }_0({\mathbb {R}}^d)$ whose supported on balls of radius r,

$$\begin{aligned} \Big |\int _{{\mathbb {R}}^d}{\widetilde{\eta }}_2(x)T_{\Omega ,\,A}{\widetilde{\eta }}_1(x)dx\Big |\lesssim \Vert \Omega \Vert _{L^1({\mathbb {S}}^{d-1})} r^{-d}\prod _{j=1}^2\big (\Vert {\widetilde{\eta }}_j\Vert _{L^{\infty }({\mathbb {R}}^d)}+r\Vert \nabla {\widetilde{\eta }}_j\Vert _{L^{\infty }({\mathbb {R}}^d)}). \end{aligned}$$

The following lemma plays an important role in our analysis.

Lemma 2.6

([4]) Let A be a function on ${\mathbb {R}}^d$ with derivatives of order one in $L^q({\mathbb {R}}^d)$ for some $q\in (d,\,\infty ]$. Then

$$\begin{aligned} |A(x)-A(y)| \lesssim |x-y|\Big (\frac{1}{|I_{(x,|x-y|)}|}\int _{I_{(x,|x-y|)}}|\nabla A(z)|^q\textrm{d}z\Big )^{\frac{1}{q}}, \end{aligned}$$

where $I_{(x,|x-y|)}$ is a cube which is centered at x with length $2|x-y|.$

We need a lemma from the book of Grafakos.

Lemma 2.7

([12, p. 140]) Let $\Phi$ be a function on ${\mathbb {R}}^d$ satisfying for some $0<C,\delta <\infty ,$ $|\Phi (x)|\le C {(1+|x)|)^{-d-\delta }}$. For $t>0$, set $\Phi _t(x)=t^{-d}\Phi (t^{-1}x)$. Then a measure $\mu$ on ${\mathbb {R}}_+^{d+1}$ is a Carleson if and only if for every p with $1<p<\infty$ there is a constant $C_{p,d,\mu }$ such that for all $f\in L^p({\mathbb {R}}^d)$ we have

$$\begin{aligned} \int _{{\mathbb {R}}_+^{d+1}} |\Phi _t*f(x)|^p d\mu (x,t)\le C_{p,d,\mu }\int _{{\mathbb {R}}^d}|f(x)|^pdx. \end{aligned}$$

Proof of Theorem 1.1

Invoking (2.1), to prove that $T_{\Omega ,\,A}$ is bounded on $L^2({\mathbb {R}}^d)$, it suffices to show the following inequalities hold for $f,\,g\in C^{\infty }_0({\mathbb {R}}^d)$,

$$\begin{aligned}{} & {} \Big |\int ^{\infty }_0\int _{0}^t\int _{{\mathbb {R}}^d}Q_s^4T_{\Omega ,\,A}Q_t^4f(x)g(x)dx\frac{ds}{s}\frac{dt}{t}\Big |\lesssim \Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}; \end{aligned}$$

(2.4)

$$\begin{aligned}{} & {} \Big |\int ^{\infty }_0\int _{t}^{\infty }\int _{{\mathbb {R}}^d}Q_s^4T_{\Omega ,\,A}Q_t^4f(x)g(x)dx\frac{ds}{s}\frac{dt}{t}\Big |\lesssim \Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

(2.5)

First, we will prove (2.4). To this aim, the kernel $\Omega$ will be decomposed into disjoint forms. Let

$$\begin{aligned} E_0=\{\theta \in {\mathbb {S}}^{d-1}:\, |\Omega (\theta )|\le 1\} \ \hbox {and } E_i=\{\theta \in {\mathbb {S}}^{d-1}:\,2^{i-1}<|\Omega (\theta )|\le 2^i\},\quad i\in {\mathbb {N}}. \end{aligned}$$

Set

$$\begin{aligned} \Omega _0(\theta )=\Omega (\theta )\chi _{E_0}(\theta ),\quad \,\Omega _i(\theta )=\Omega (\theta )\chi _{E_i}(\theta )\,\,(i\in {\mathbb {N}}). \end{aligned}$$

For $i\in {\mathbb {N}}\cup \{0\}$, let $T_{\Omega ,\,A;\,j}^i$ be the same as in (2.3) for $T_{\Omega ,\,A;\,j}$ with $\Omega$ replaced by $\Omega _i$. Then

$$\begin{aligned}{} & {} \int ^{\infty }_0\int _{0}^t\int _{{\mathbb {R}}^d}Q_s^4T_{\Omega ,\,A}Q_t^4f(x)g(x)dx\frac{ds}{s}\frac{dt}{t}\nonumber \\{} & {} \quad =\sum _i\sum _j\int ^{\infty }_0\int _{0}^t\int _{{\mathbb {R}}^d}Q_s^4T_{\Omega ,\,A;\,j}^iQ_t^4f(x)g(x)dx\frac{ds}{s}\frac{dt}{t}. \end{aligned}$$

(2.6)

Let $\alpha \in \big (\frac{d+1}{d+2},\,1\big )$ be a constant. Fix $j\in {\mathbb {Z}}$, we decompose the set $\{(s,t):\, 0<t<\infty ,\,0<s\le t\}$ into three regions:

$$\begin{aligned}&E_1 (j, s,t)=\{(s,\,t):\,0\le t\le 2^j,\,0< s\le t\};\\&E_2 (j, s,t)=\big \{(s,\,t):\,2^j\le t<(2^js^{-\alpha })^{\frac{1}{1-\alpha }},\,\, 0<s\le t\big \};\\&E_3 (j, s,t)=\big \{(s,\,t):\,\max \{2^j,\, (2^js^{-\alpha })^{\frac{1}{1-\alpha }} \}\le t<\infty ,\,0< s\le t\big \}. \end{aligned}$$

In the following three subsections, we will discuss the contribution of each $E_{j,s,t}$ on the right ride of (2.6) to inequality (2.4). $\square$

2.1 Contribution of ${E_1(j,s,t)}$

Let $\varepsilon$ be the same constant appeared in Lemma 2.3 and denote $N=2(\lfloor \varepsilon ^{-1}\rfloor +1)$. For each fixed $i\in {\mathbb {N}}$, we introduce the notion $E_{1,1}^i$ and $E_{1,2}^i$ as follows

$$\begin{aligned}&E_{1,1}^i(j,s,\,t)=\{(j,s,\,t):\,0\le t\le 2^j,\,0\le s\le t,\, 2^j\le s2^{iN}\};\\&E_{1,2}^i(j,s,\,t)=\{(j,s,\,t):\,0\le t\le 2^j,\,0\le s\le t,\,2^j>s2^{iN}\}. \end{aligned}$$

Then, one gets obviously that ${E_1(j,s,t)}=E_{1,1}^i(j,s,\,t)\cup E_{1,2}^i(j,s,\,t):=E_{1,1}^i\cup E_{1,2}^i.$ Therefore

$$\begin{aligned}{} & {} \Big |\sum _{i=0}^{\infty }\sum _j\int ^{\infty }_0\int _{0}^t\int _{{\mathbb {R}}^d}{\chi _{E_1(j,s,t)}}Q_s^4T_{\Omega ,\,A;\,j}^iQ_t^4f(x)g(x)dx\frac{ds}{s}\frac{dt}{t}\Big | \\{} & {} \quad \le \sum _{i=1}^{\infty }\sum _j\int ^{\infty }_0\int _{0}^{\infty }\chi _{E^i_{1,1}} \Big |\int _{{\mathbb {R}}^d}Q^4_s{T}^i_{\Omega ,\,A;\,j}Q_t^4f(x)g(x)dx\Big |\frac{ds}{s}\frac{dt}{t}\\{} & {} \qquad + \sum _{i=1}^{\infty }\sum _j\int ^{\infty }_0\int _{0}^{\infty }\chi _{E^i_{1,2}}\Big |\int _{{\mathbb {R}}^d} Q_s^4 {T}^i_{\Omega ,\,A;\,j}Q_t^4f(x)g(x){dx}\Big |\frac{ds}{s}\frac{dt}{t}\\{} & {} \qquad + \sum _j\int ^{\infty }_0\int _{0}^{\infty }{\chi _{E_1(j,s,t)}}\Big |\int _{{\mathbb {R}}^d} Q_s^4 {T}^0_{\Omega ,\,A;\,j}Q_t^4f(x)g(x)dx\Big |\frac{ds}{s}\frac{dt}{t}=:\textrm{I}+\textrm{II}+\textrm{III}. \end{aligned}$$

We first consider term I. Let $\{I_l\}_{l}$ be a sequence of cubes having disjoint interiors and side lengths $2^j$, such that

$$\begin{aligned} {\mathbb {R}}^d=\mathop {\cup }\limits _{l}I_l. \end{aligned}$$

(2.7)

For each fixed l, let $\zeta _l\in C^{\infty }_0({\mathbb {R}}^d)$ such that $\textrm{supp}\,\zeta _l\subset 48dI_l$, $0\le \zeta _l\le 1$ and $\zeta _l(x)\equiv 1$ when $x\in 32dI_l$. Let $x_l$ be a point on the boundary of $50dI_l$ and

$$\begin{aligned} {\widetilde{A}}_{I_l}(y)=A(y)-{\sum _{m=1}^d}\langle \partial _m A\rangle _{I_l}y_m,\,\,A_{I_l}(y)=A_{I_l}^*(y)\zeta _l(y),\,\,\,y\in {\mathbb {R}}^d, \end{aligned}$$

with $A_{I_l}^*(y)={\widetilde{A}}_{I_l}(y)-{\widetilde{A}}_{I_l}(x_l).$ Note that for $x\in 30 dI_l$ and $y\in {\mathbb {R}}^d$ with ${|x-y|\le 2^{j}}$, we have

$$\begin{aligned} A(x)-A(y)-\nabla A(y)(x-y)=A_{I_l}(x)-A_{I_l}(y)-\nabla A_{I_l}(y)(x-y). \end{aligned}$$

An application of Lemma 2.6 then implies that $\Vert A_{I_l}\Vert _{L^{\infty }({\mathbb {R}}^d)}\lesssim 2^j.$

For each fixed $j\in {\mathbb {Z}}$, consider the operators $W^i_{\Omega ,\,j}$ and $U^i_{\Omega ,\,m;j}$ defined by

$$\begin{aligned} W_{\Omega ,j}^ih(x)=\int _{{\mathbb {R}}^d}\frac{\Omega _i(x-y)}{|x-y|^{d+1}}\phi _j(x-y)h(y)dy \end{aligned}$$

and

$$\begin{aligned} U_{\Omega ,m;j}^ih(x)=\int _{{\mathbb {R}}^d}\frac{\Omega _i(x-y)(x_m-y_m)}{|x-y|^{d+1}}\phi _j(x-y)h(y)dy. \end{aligned}$$

The method of rotation of Caldeón-Zygmund states that for $p\in (1,\,\infty )$, they enjoy the following properties:

$$\begin{aligned} \Vert W_{\Omega ,j}^ih\Vert _{L^p({\mathbb {R}}^d)}\lesssim & {} 2^{-j}\Vert \Omega _i\Vert _{L^1(S^{d-1})}\Vert h\Vert _{L^p({\mathbb {R}}^d)};\\ \Vert U_{\Omega ,\,m,j}^ih\Vert _{L^p({\mathbb {R}}^d)}\lesssim & {} \Vert \Omega _i\Vert _{L^1(S^{d-1})} \Vert h\Vert _{L^p({\mathbb {R}}^d)}, \end{aligned}$$

see [12, pp. 272–274]. For each fixed l, let $h_{s,l}(x)=Q_sg(x)\chi _{I_l}(x)$ and $I_l^*=60dI_l$. For $x\in \textrm{supp}h_{s,l}$, we have

$$\begin{aligned} {T}_{\Omega ,A,j}^iQ_t^4f(x)= & {} A_{I_l}(x)W^i_{\Omega ,j}Q_t^4f(x)-W^i_{\Omega ,j}(A_{I_l}Q_t^4f)(x)\\{} & {} -\sum _{m=1}^dU_{\Omega ,m,j}^i(\partial _mA_{I_l}Q_t^4f)(x). \end{aligned}$$

Hence, to show the estimate for $\textrm{I}$, we need to consider the following three terms.

$$\begin{aligned} \textrm{R}_{i}^1= & {} \sum _{j}\int ^{2^j}_{2^{j-Ni}}\int ^{2^j}_{2^{j-Ni}} \Big |\sum _l\int _{{\mathbb {R}}^d}A_{I_l}(x)Q_s^3h_{s,l}(x)W^i_{\Omega ,j}Q_t^4f(x)dx\Big |\frac{dt}{t}\frac{ds}{s};\\ \textrm{R}_{i}^2= & {} \sum _{j}\int ^{2^j}_{2^{j-Ni}}\int ^{2^j}_{2^{j-Ni}} \Big |\sum _l\int _{{\mathbb {R}}^d}Q_s^3h_{s,l}(x)W^i_{\Omega ,j}(A_{I_l}Q_t^4f)(x)dx\Big |\frac{dt}{t}\frac{ds}{s}; \end{aligned}$$

and

$$\begin{aligned} \textrm{R}_{i}^3&=\sum _{m=1}^d\sum _{j}\int ^{2^j}_{2^{j-Ni}}\int ^{2^j}_{2^{j-Ni}} \Big |\sum _l\int _{{\mathbb {R}}^d} Q_s^3h_{s,l}(x)U^i_{\Omega ,m,j}(\partial _mA_{I_l}Q^4_tf\big )(x)dx\Big |\frac{ds}{s}\frac{dt}{t}\\&=:\sum _{m=1}^d\textrm{R}_{i,m}^3. \end{aligned}$$

For $\textrm{R}^1_{i},$ note that

$$\begin{aligned} \sum _j\sum _l\int ^{2^j}_{2^{j-iN}}\Vert Q_s^3h_{s,l}\Vert _{L^2({\mathbb {R}}^d)}^2\frac{ds}{s}\lesssim & {} iN\sum _j\int ^{2^j}_{2^{j-1}}\sum _l\Vert h_{s,l}\Vert _{L^2({\mathbb {R}}^d)}^2\frac{ds}{s}\\\lesssim & {} i\int ^{\infty }_0 \Vert Q_sg\Vert _{L^2({\mathbb {R}}^d)}^2\frac{ds}{s}. \end{aligned}$$

Then, the well-known Littlewood-Paley theory for g-function leads to that

$$\begin{aligned} \sum _j\sum _l\int ^{2^j}_{2^{j-iN}}\Vert Q_s^3h_{s,l}\Vert _{L^2({\mathbb {R}}^d)}^2\frac{ds}{s} \lesssim i {{\bigg \Vert \bigg (\int ^{\infty }_0 |Q_sg( \cdot )|^2\frac{ds}{s}\bigg )^{1/2} \bigg \Vert ^{2}_{L^2({\mathbb {R}}^d)}}} \lesssim i\Vert g\Vert _{L^2({\mathbb {R}}^d)}^2. \end{aligned}$$

For $x\in 48dI_l,$ since $\sup \{\phi _j\}\subset [2^{j-2},2^j]$ and note that $\phi _j(x-y)Q_t^4f(y)=\chi _{I_l^*}(y)\phi _j(x-y){Q_t^4f(y),}$ then, $W_{\Omega ,j}^i(Q_t^4f)=W_{\Omega ,j}^i(\chi _{I_l^*}Q_t^4f)$. It then follows from Hölder’s inequality, Cauchy-Schwarz inequality and the boundedness of $W_{\Omega ,j}^i$ that

$$\begin{aligned} |\textrm{R}^1_{i}|\le & {} \Big (\sum _j\sum _l\int ^{2^j}_{2^{j-iN}}\int ^{2^j}_{2^{j-iN}} \Vert Q_s^3h_{s,l}\Vert _{L^2({\mathbb {R}}^d)}^2 \frac{ds}{s}\frac{dt}{t}\Big )^{1/2}\\{} & {} \times \Big (\sum _j\sum _l\int ^{2^j}_{2^{j-iN}}\int ^{2^j}_{2^{j-iN}} \Vert A_{I_l}W_{\Omega ,j}^i(\chi _{I_l^*}Q_t^4f)\Vert _{L^2({\mathbb {R}}^d)}^2 \frac{dt}{t}\frac{ds}{s}\Big )^{1/2}\\\lesssim & {} \Vert \Omega _i\Vert _{L^1(S^{d-1})}\Big (\sum _j\sum _l\int ^{2^j}_{2^{j-iN}} \int ^{2^j}_{2^{j-iN}} \Vert Q_s^3h_{s,l}\Vert _{L^2({\mathbb {R}}^d)}^2\frac{dt}{t}\frac{ds}{s}\Big )^{1/2}\\{} & {} \times \Big (\sum _j\sum _l\int ^{2^j}_{2^{j-iN}}\int ^{2^j}_{2^{j-iN}}\Vert \chi _{I_l^*}Q_t^4f\Vert _{L^2({\mathbb {R}}^d)}^2\frac{dt}{t}\frac{ds}{s}\Big )^{1/2}\\\lesssim & {} i^{2}\Vert \Omega _i\Vert _{L^1(S^{d-1})}\Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}, \end{aligned}$$

where in the last inequality we have used the fact that the cubes $\{60 dI_l\}_l$ have bounded overlaps.

The same reasoning applies to $\textrm{R}^2_{i}$ with small and straightforward modifications yields that

$$\begin{aligned} |\textrm{R}^2_{i}|\lesssim & {} i\Vert \Omega _i\Vert _{L^1(S^{d-1})}\Big (\sum _j \int ^{2^j}_{2^{j-iN}} \Vert Q_sg\Vert _{L^2({\mathbb {R}}^d)}^2\frac{ds}{s}\Big )^{1/2}\\{} & {} \times \Big (\sum _j\sum _l\int ^{2^j}_{2^{j-iN}}\Vert \zeta _lQ_t^4f\Vert _{L^2({\mathbb {R}}^d)}^2\frac{dt}{t}\Big )^{1/2}\\\lesssim & {} i^2\Vert \Omega _i\Vert _{L^1(S^{d-1})}\Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

Now we are in a position to consider each term $\textrm{R}_{i,m}^3$. For $x\in 32dI_l$, it is easy to check

$$\begin{aligned} \partial _mA_{I_l}(x)Q_t^4f(x)= & {} \zeta _l(x)[\partial _m{A},\,Q_t] Q_t^3f(x)+\zeta _l(x)Q_t{([\partial _m{A},\, Q_t]Q_t^2f)}(x)\\{} & {} +\zeta _l(x)Q_t^2{(\partial _m{\widetilde{A}}_{I_l}Q^2_tf)}(x). \end{aligned}$$

Therefore $\textrm{R}_{i,m}^3$ can be controlled by the sum of the following terms:

$$\begin{aligned} {\textrm{R}_{i,m}^{3,1}}= & {} \sum _{j}\int ^{2^j}_{2^{j-Ni}}\int ^{2^j}_{2^{j-iN}} \Big |\sum _l\int _{{\mathbb {R}}^d} Q_s^3h_{s,l}(x)U^i_{\Omega ,m,j}\big ([\partial _m A,\, Q_t]Q^3_tf\big )(x)dx\Big |\frac{dt}{t}\frac{ds}{s};\\ \textrm{R}_{i,m}^{3,2}= & {} \sum _{j}\int ^{2^j}_{2^{j-Ni}}\int ^{2^j}_{2^{j-iN}}\Big |\sum _l\int _{{\mathbb {R}}^d} Q_s^3h_{s,l}(x)U^i_{\Omega ,m,j}Q_t\big ([\partial _m A,\,Q_t]Q_t^2f\big )(x)dx\Big |\frac{dt}{t}\frac{ds}{s};\\ {\textrm{R}_{i,m}^{3,3}}= & {} \sum _{j}\int ^{2^j}_{2^{j-Ni}}\int ^{2^j}_{2^{j-iN}}\Big |\sum _l\int _{{\mathbb {R}}^d} Q_s^3h_{s,l}(x)U^i_{\Omega ,m,j}Q_t^2(\partial _m{\widetilde{A}}_{I_l}Q_t^2)f(x)dx\Big |\frac{dt}{t}\frac{ds}{s}. \end{aligned}$$

Observe that $|[\partial _mA,\,Q_t]h(x)|\lesssim M_{\partial _mA}h(x),$ where $M_{\partial _m A}$ is the commutator of the Hardy-Littlewood maximal operator defined by

$$\begin{aligned} M_{\partial _m A}h(x)=\sup _{r>0}r^{-d}\int _{|x-y|<r}|\partial _m A(x)-\partial _m A(y)|{|h(y)|}dy. \end{aligned}$$

Hölder’s inequality, along with the $L^2({\mathbb {R}}^d)$ boundedness of $M_{\partial _m A}$ and $U_{\Omega ,m,j}^i$, it yields that

$$\begin{aligned} |\textrm{R}_{i,m}^{3,1}|\le & {} \Big (\sum _j\int ^{2^j}_{2^{j-Ni}}\int ^{2^j}_{2^{j-iN}}\Big \Vert Q_s^3\big (\sum _{l}h_{s,l}\big )\Big \Vert _{L^2({\mathbb {R}}^d)}^2\frac{dt}{t}\frac{ds}{s}\Big )^{1/2}\\{} & {} \times \Big (\sum _j\int ^{2^j}_{2^{j-Ni}}\int ^{2^j}_{2^{j-iN}}\Vert U_{\Omega ,m,j}^i([\partial _mA,\, Q_t]Q_t^3f\Vert _{L^2({\mathbb {R}}^d)}^2 \frac{dt}{t}\frac{ds}{s}\Big )^{1/2}\\\lesssim & {} i^{2}\Vert \Omega _i\Vert _{L^1(S^{d-1})}\Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

Exactly the same reasoning applies to $\textrm{R}_{i,m}^{3,2}$, we obtain

$$\begin{aligned} |\textrm{R}_{i,m}^{3,2}|\lesssim i^{2}\Vert \Omega _i\Vert _{L^1(S^{d-1})}\Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

As for $\textrm{R}_{i,m}^{3,3},$ observing that for fixed $l\in {\mathbb {Z}}$, $s,\,t\le 2^j$, one gets

$$\begin{aligned} Q_t(\partial _m{\widetilde{A}}_{I_l}Q_t^2f)(x)= & {} Q_t(\partial _m{\widetilde{A}}_{I_l}\chi _{I_l^*}Q_t^2f)(x),\\ U^i_{\Omega ,\,m,j}Q_s= & {} Q_sU^i_{\Omega ,m,j}\ \hbox {and }\ Q_sQ_t=Q_tQ_s. \end{aligned}$$

Henceforth we have

$$\begin{aligned} \textrm{R}_{i,m}^{3,3}= & {} \sum _j\int ^{2^j}_{2^{j-Ni}}\int ^{2^j}_{2^{j-iN}} \Big |\sum _l\int _{{\mathbb {R}}^d}Q_tQ_s^2h_{s,l}(x)Q_sU^i_{\Omega ,m,j}Q_t(\partial _m{\widetilde{A}}_{I_l}\chi _{I_l^*}Q_t^2f)(x)dx \frac{dt}{t}\frac{ds}{s}\Big |\\\le & {} \Big (\sum _j\sum _{l}\int ^{2^j}_{2^{j-Ni}}\int ^{2^j}_{2^{j-iN}} \Vert Q_tQ^2_sh_{s,l}\Vert ^2_{L^2({\mathbb {R}}^d)}\frac{dt}{t}\frac{ds}{s}\Big )^{1/2}\\{} & {} \times \Big (\sum _j\sum _l\int ^{2^j}_{2^{j-Ni}}\int ^{2^j}_{2^{j-iN}} \Big \Vert Q_s\big (U^i_{\Omega ,m,j}Q_t(\partial _m{\widetilde{A}}_{I_l}\chi _{I_l^*}Q_t^2f)\big )\Big \Vert ^2_{L^2({\mathbb {R}}^d)} \frac{dt}{t}\frac{ds}{s}\Big )^{1/2}. \end{aligned}$$

Let $x\in 48dI_l$, $q\in (1,\,2)$ and $s\in (2^{j-1}, 2^j)$. A straightforward computation involving Hölder’s inequality and the John-Nirenberg inequality gives us that

$$\begin{aligned} |{Q_s(\partial _m{\tilde{A}}_{I_l}h)(x)}|\le & {} \int _{{\mathbb {R}}^d}|\psi _s(x-y)||\partial _mA(y)-\langle \partial _mA\rangle _{I(x,\,s)}||h(y)|dy\nonumber \\{} & {} +|\langle \partial _m A\rangle _{I_l}-\langle \partial _mA\rangle _{I(x,\,s)}|\int _{{\mathbb {R}}^d}|\psi _s(x-y)||h(y)|dy \nonumber \\\lesssim & {} M_qh(x)+\log (1+2^j/s)Mh(x)\nonumber \\\lesssim & {} M_qh(x), \end{aligned}$$

(2.8)

where $I(x,\,s)$ is the cube center at x and having side length s.

This inequality, together with the boundedness of $U^i_{\Omega ,m,j}$ and maximal function $M_qh$, implies that

$$\begin{aligned}{} & {} \Big (\sum _j\sum _l\int ^{2^j}_{2^{j-Ni}}\int ^{2^j}_{2^{j-iN}} \big \Vert Q_s\big (U^i_{\Omega ,m,j}Q_t({\partial _m{\tilde{A}}_{I_l}}\chi _{I_l^*}Q_t^2f)\big )\big \Vert ^2_{L^2({\mathbb {R}}^d)} \frac{dt}{t}\frac{ds}{s}\Big )^{1/2}\\{} & {} \quad \lesssim {\Big (i}\sum _j\sum _l\int _{2^{j-1}}^{2^j} \big \Vert U^i_{\Omega ,m,j}Q_t({\partial _m{\tilde{A}}_{I_l}}\chi _{I_l^*}Q_t^2f)\big \Vert ^2_{L^2({\mathbb {R}}^d)}\frac{dt}{t}\Big )^{1/2}\\{} & {} \quad \lesssim i\Vert \Omega _i\Vert _{L^1(S^{d-1})} \Vert f\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

On the other hand, by the $L^2$ boundedness of convolution operators and the Littlewood-Paley theory for g-function again, we have that

$$\begin{aligned}{} & {} \sum _j\sum _{l}\int ^{2^j}_{2^{j-Ni}}\int ^{2^j}_{2^{j-iN}} \Vert Q_tQ^2_sh_{s,l}\Vert ^2_{L^2({\mathbb {R}}^d)}\frac{dt}{t}\frac{ds}{s}\\{} & {} \quad {~\lesssim ~} i^2 \int ^{\infty }_{0}\Vert Q_sg\Vert ^2_{L^2({\mathbb {R}}^d)}\frac{ds}{s}\lesssim i^2\Vert g\Vert _{L^2({\mathbb {R}}^d)}^2. \end{aligned}$$

Therefore

$$\begin{aligned} \textrm{R}_{i,m}^{3,3}\lesssim i^2\Vert \Omega _i\Vert _{L^1(S^{d-1})}\Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

Combining the estimates for $\textrm{R}_{i}^1$, $\textrm{R}_i^2$ and $\textrm{R}_{i,m}^{3,n}$ (with $1\le m\le d, \,n=1,\,2,\,3$) in all yields that

$$\begin{aligned} \textrm{I}\lesssim & {} \sum _{i=1}^{\infty }i^2\Vert \Omega _i\Vert _{L^1(S^{d-1})}\Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}\lesssim \Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}, \end{aligned}$$

(2.9)

since

$$\begin{aligned} \sum _{i=1}^{\infty }i^2\Vert \Omega _i\Vert _{L^1(S^{d-1})}\lesssim \Vert \Omega \Vert _{L(\log L)^2({\mathbb {S}}^{d-1})}. \end{aligned}$$

It remains to discuss the contribution of terms II and III. For $i\in {\mathbb {N}}\cup \{0\}$, by Lemma 2.3, one gets

$$\begin{aligned}{} & {} \sum _j\int ^{\infty }_0\int _{0}^{\infty }\chi _{E^i_{1,2}}\Vert Q_sT^i_{\Omega ,\,A;\,j}Q_t^4f\Vert _{L^2({\mathbb {R}}^d)} \Vert Q_s^3g\Vert _{L^2({\mathbb {R}}^d)}\frac{ds}{s}\frac{dt}{t}\nonumber \\{} & {} \quad \lesssim \Vert \Omega _i\Vert _{L^{\infty }({\mathbb {S}}^{d-1})} \Big (\int ^{\infty }_0\int _{0}^{\infty } \sum _{j} \chi _{E^i_{1,2}}(2^{-j}s)^{\varepsilon }\Vert Q_s^3g\Vert ^2_{L^2({\mathbb {R}}^d)} \frac{ds}{s}\frac{dt}{t}\Big )^{\frac{1}{2}}\nonumber \\{} & {} \qquad \times \Big (\int ^{\infty }_0\int _{0}^{\infty } \sum _{j} \chi _{E^i_{1,2}}(2^{-j}s)^{\varepsilon }\Vert Q_t^4f\Vert _{L^2({\mathbb {R}}^d)}^2 \frac{ds}{s}\frac{dt}{t}\Big )^{\frac{1}{2}}. \end{aligned}$$

(2.10)

Note that

$$\begin{aligned} E_{1,2}^i(j,s,\,t)\subset \big \{(j,s,\,t):\,0\le t\le 2^j,\,0\le s\le t,\,2^j\ge \max \{t,\,s2^{iN}\}\big \}, \end{aligned}$$

Thus

$$\begin{aligned}\sum _{j} \chi _{E_{1,2}^i}(2^{-j}s)^{\varepsilon }\le 2^{-iN\varepsilon /2}\big (\frac{s}{t}\big )^{\varepsilon /2}\chi _{\{(s,t):\,s\le t\}}(s,\,t), \end{aligned}$$

which further implies that

$$\begin{aligned}{} & {} \Big (\int ^{\infty }_0\int _{0}^{\infty } \sum _{j} \chi _{E_{1,2}^i}(2^{-j}s)^{\varepsilon }\Vert Q_s^3g\Vert ^2_{L^2({\mathbb {R}}^d)} \frac{ds}{s}\frac{dt}{t}\Big )^{\frac{1}{2}}\\{} & {} \quad \lesssim 2^{-Ni\varepsilon /4}\Big (\int ^{\infty }_0\int ^{\infty }_s\big (\frac{s}{t}\big )^{\varepsilon /2} \frac{dt}{t}\Vert Q_sg\Vert _{L^2({\mathbb {R}}^d)}^2\frac{ds}{s}\Big )^{1/2} \lesssim 2^{-Ni\varepsilon /4}\Vert g\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

Similarly, we have that

$$\begin{aligned} \Big (\int ^{\infty }_0\int _{0}^{\infty } \sum _{j} \chi _{E_{1,2}^i}(2^{-j}s)^{\varepsilon }\Vert Q_t^4f\Vert ^2_{L^2({\mathbb {R}}^d)} \frac{ds}{s}\frac{dt}{t}\Big )^{\frac{1}{2}}\lesssim 2^{-Ni\varepsilon /4}\Vert f\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

Therefore, these inequalities, together with the fact that $E_{1,1}^0=\emptyset$ may lead to

$$\begin{aligned} \mathrm{II+III}\lesssim \sum _{i=0}^{\infty }2^{i}2^{-Ni\varepsilon /2} \Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}\lesssim \Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

(2.11)

Inequality (2.11), together with the estimate (2.9) for $\textrm{I}$, gives that

$$\begin{aligned}{} & {} \Big |\sum _i\sum _j\int ^{\infty }_0\int _{0}^t\int _{{\mathbb {R}}^d} \chi _{E_1(j,s,t)}Q_s^4T_{\Omega ,\,A;\,j}^iQ_t^4f(x)g(x)dx\frac{ds}{s}\frac{dt}{t}\Big |\nonumber \\{} & {} \quad \lesssim \Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

(2.12)

2.2 Contribution of ${E_2(j,s,t)}$

Let $\alpha \in (\frac{d+1}{d+2},\,1)$, $i\in {\mathbb {N}}\cup \{0\}$, and write

$$\begin{aligned}{} & {} \sum _i\sum _{j\in {\mathbb {Z}}}\int ^{\infty }_{2^j}\int ^{(2^jt^{\alpha -1})^{\frac{1}{\alpha }}}_0 \Big |\int _{{\mathbb {R}}^d}Q_s^4{T}_{\Omega , A,\,j}^iQ_t^4f(x)g(x)dx\Big |\frac{ds}{s}\frac{dt}{t}\nonumber \\{} & {} \quad \le \sum _{i}\sum _{j\in {\mathbb {Z}}}\int ^{2^j}_{2^{j-Ni}}\int ^{(2^js^{-\alpha })^{\frac{1}{1-\alpha }}}_{2^j} \Big |\int _{{\mathbb {R}}^d}Q_s^4 {T}_{\Omega , A,\,j}^iQ_t^4f(x)g(x)dx\Big |\frac{dt}{t}\frac{ds}{s}\nonumber \\{} & {} \qquad +\sum _{i}\sum _{j\in {\mathbb {Z}}}\int ^{2^{j-Ni}}_0\int ^{(2^js^{-\alpha })^{\frac{1}{1-\alpha }}}_{2^j} \Big |\int _{{\mathbb {R}}^d}Q_s^4 {T}_{\Omega , A,\,j}^iQ_t^4f(x)g(x)dx\Big |\frac{dt}{t}\frac{ds}{s}\nonumber \\{} & {} \quad =:\mathrm{IV+V}. \end{aligned}$$

(2.13)

Firstly, we consider the term $\textrm{IV}$. When $i=0$, the integral $\int ^{2^j}_{2^{j-Ni}}\int ^{(2^js^{-\alpha })^{1/{(1-\alpha )}}}_{2^j}\frac{dt}{t}\frac{ds}{s}$ vanishes, we only need to consider the case $i\in {\mathbb {N}}$. Since $s>2^{j-Ni}$, then $(2^j s^{-\alpha })^{\frac{1}{1-\alpha }}\le 2^j 2^{iN\frac{\alpha }{1-\alpha }}$. Therefore

$$\begin{aligned} \textrm{IV}= & {} \sum _{i}\sum _{j\in {\mathbb {Z}}}\int ^{2^j}_{2^{j-Ni}}{\int ^{(2^js^{-\alpha })^{\frac{1}{1-\alpha }}}_{2^j}} \Big |\int _{{\mathbb {R}}^d}Q_s^4{T}^i_{\Omega , A,\,j}Q_t^4f(x)g(x)dx\Big |\frac{dt}{t}\frac{ds}{s}\\\le & {} \sum _{i}\sum _j\int ^{2^j}_{2^{j-Ni}}\int ^{2^j2^{iN\frac{\alpha }{1-\alpha }}}_{2^j}\Big |\int _{{\mathbb {R}}^d} T_{\Omega ,A,j}^iQ_t^4f(x)Q_s^4g(x)dx\Big |\frac{dt}{t}\frac{ds}{s}\\\le & {} \sum _{i}\sum _j\int ^{2^j}_{2^{j-Ni}}\int ^{2^j2^{iN\frac{\alpha }{1-\alpha }}}_{2^j}\Big |\sum _l\int _{{\mathbb {R}}^d} T_{\Omega ,A,j}^iQ_t^4f(x)Q_s^3h_{s,l}(x)dx\Big |\frac{dt}{t}\frac{ds}{s}, \end{aligned}$$

where $h_{s,l}(x)=Q_sg(x)\chi _{I_l}(x)$, and $\{I_l\}_{l}$ be the cubes in (2.7).

Observe that when $x\in 4dI_l$, $T_{\Omega ,A,j}^i(Q_t^4f)(x)Q_s^3h_{s,l}(x)= T_{\Omega ,A,j}^i(\zeta _lQ_t^4f)(x)Q_s^3h_{s,l}(x),$ we rewrite

$$\begin{aligned}{} & {} {T}_{\Omega ,A,j}^i(\zeta _lQ_t^4f)(x)\\{} & {} \quad =\Big (A_{I_l}(x)W^i_{\Omega ,j}Q_t^4f(x)-W^i_{\Omega ,j}(A_{I_l}Q_t^4f)(x)-\sum _{m=1}^dU_{\Omega ,m,j}^i(\zeta _{l}[\partial _m A,Q_t]Q_t^3f)(x)\\{} & {} \qquad -\sum _{m=1}^dU_{\Omega ,m,j}^i(\zeta _lQ_t[\partial _{m}A,\,Q_t]Q_t^2f)(x)\\{} & {} \qquad -\sum _{m=1}^dU_{\Omega ,\,m,j}^i(\zeta _lQ_tQ_t(\partial _m\widetilde{A_{I_l}}Q_t^2f)(x)\Big )\chi _{4dI_l}(x). \end{aligned}$$

Similar to the estimate for $\textrm{R}_i^1$ and $\textrm{R}_i^2$, we know that

$$\begin{aligned}{} & {} \sum _j\int ^{2^j}_{2^{j-Ni}}\int ^{2^j2^{iN\frac{\alpha }{1-\alpha }}}_{2^j}\Big |\sum _l\int _{{\mathbb {R}}^d} W_{\Omega ,j}^i(A_{I_l}Q_t^4f)(x)Q_s^3h_{s,l}(x)dx\Big |\frac{dt}{t}\frac{ds}{s}\nonumber \\{} & {} \quad \lesssim i^2\Vert \Omega _i\Vert _{L^1({\mathbb {R}}^d)}\Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}, \end{aligned}$$

(2.14)

and

$$\begin{aligned}{} & {} \sum _j\int ^{2^j}_{2^{j-Ni}}\int ^{2^j2^{iN\frac{\alpha }{1-\alpha }}}_{2^j}\sum _l\Big |\int _{{\mathbb {R}}^d} A_{I_l}(x)W_{\Omega ,j}^iQ_t^4f(x)Q_s^3h_{s,l}(x)dx\Big |\frac{dt}{t}\frac{ds}{s}\nonumber \\{} & {} \quad \lesssim i^2\Vert \Omega _i\Vert _{L^1({\mathbb {R}}^d)}\Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

(2.15)

On the other hand, for each fixed $1\le m\le d$, the same reasoning as what we have done for ${\textrm{R}^{3,1}_{i,m}}$ and ${\textrm{R}^{3,2}_{i,m}}$ yields that

$$\begin{aligned}{} & {} \sum _j\sum _l\int ^{2^j}_{2^{j-Ni}}\int ^{2^j2^{iN\frac{\alpha }{1-\alpha }}}_{2^j}\Big |\int _{{\mathbb {R}}^d} U_{\Omega ,m,j}^i(\zeta _l[\partial _mA,Q_t]Q_t^3f)(x)Q_s^3h_{s,l}(x)dx\Big |\frac{dt}{t}\frac{ds}{s}\nonumber \\{} & {} \quad \lesssim i^2\Vert \Omega _i\Vert _{L^1({\mathbb {R}}^d)}\Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}, \end{aligned}$$

(2.16)

and

$$\begin{aligned}{} & {} \sum _j\sum _l\int ^{2^j}_{2^{j-Ni}}\int ^{2^j2^{iN\frac{\alpha }{1-\alpha }}}_{2^j}\Big |\int _{{\mathbb {R}}^d} U_{\Omega ,m,j}^i(\zeta _lQ_t[\partial _mA,Q_t]Q_t^2f)(x)Q_s^3h_{s,l}(x)dx\Big |\frac{dt}{t}\frac{ds}{s}\nonumber \\{} & {} \quad \lesssim i^2\Vert \Omega _i\Vert _{L^1({\mathbb {R}}^d)}\Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}, \end{aligned}$$

(2.17)

Note that if $x\in 4dI_l(x)$, then $U_{\Omega ,\,m,j}^i(Q_tQ_t(\partial _m\widetilde{A_{I_l}}Q_t^2f))(x)=U_{\Omega ,\,m,j}^i(\zeta _lQ_tQ_t(\partial _m\widetilde{A_{I_l}}Q_t^2f))(x).$ Since the kernel of $Q_t$ is radial and it enjoys the property that

$$\begin{aligned}<U_{\Omega ,\,m,j}^i(\zeta _lQ_tf),g>= <U_{\Omega ,\,m,j}^i(\zeta _lf), Q_tg>. \end{aligned}$$

Hence, we have

$$\begin{aligned} \begin{aligned}&\int _{{\mathbb {R}}^d}U_{\Omega ,\,m,j}^i(\zeta _lQ_tQ_t(\partial _m\widetilde{A_{I_l}}Q_t^2f))(x){Q_s^3h_{s,l}(x)}dx\\&\quad =\int _{{\mathbb {R}}^d}U_{\Omega ,\,m,j}^iQ_s(\partial _m\widetilde{A_{I_l}}Q_sQ_t^2f)(x) Q_t^2Q_sh_{s,l}(x)dx\\&\qquad -\int _{{\mathbb {R}}^d}U_{\Omega ,\,m,j}^iQ_s[\partial _m A,\,Q_s]Q_t^2f(x)Q_t^2Q_sh_{s,l}(x)dx. \end{aligned} \end{aligned}$$

A trivial argument then yields that

$$\begin{aligned}{} & {} \Big |\sum _{j}\sum _l\int ^{2^j}_{2^{j-iN}}\int ^{2^j2^{iN\frac{\alpha }{1-\alpha }}}_{2^j}\int _{{\mathbb {R}}^d}U_{\Omega ,\,m,j}^iQ_s[\partial _m A,\,Q_s]Q_t^2f(x)Q_t^2{Q_sh_{s,l}(x)}dx\frac{dt}{t}\frac{ds}{s}\Big |\nonumber \\{} & {} \quad \lesssim i^2\Vert \Omega _i\Vert _{L^1(S^{d-1})}\Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

(2.18)

Now we write

$$\begin{aligned} \begin{aligned}{}&\int _{{\mathbb {R}}^d}U_{\Omega ,\,m,j}^iQ_s(\partial _m\widetilde{A_{I_l}}Q_sQ_t^2f)(x) Q_t^2Q_sh_{s,l}(x)dx\\&=\int _{{\mathbb {R}}^d}Q_sQ_t^2f(x) [\partial _mA,\,Q_s]U^i_{\Omega ,m,j}Q_t^2Q_sh_{s,l}(x)dx\\&\quad +\int _{{\mathbb {R}}^d}Q_sQ_t^2f(x) Q_s[\partial _mA,\,U^i_{\Omega ,m,j}]Q_t^2Q_sh_{s,l}(x)dx\\ {}&\quad +\int _{{\mathbb {R}}^d}Q_sQ_t^2f(x) Q_sU^i_{\Omega ,m,j}[\partial _mA,Q_t^2]Q_sh_{s,l}(x)dx\\&\quad +\int _{{\mathbb {R}}^d}Q_sQ_t^2f(x)Q_sU^i_{\Omega ,m,j}Q_t^2[\partial _mA,Q_s]h_{s,l}(x)dx\\&\quad +\int _{{\mathbb {R}}^d}Q_sQ_t^2f(x)Q_sU_{\Omega ,m,j}^iQ_t^2Q_s(\partial _m{\widetilde{A}}_{I_l}h_{s,l})(x)dx:= \sum _{k=1}^5\textrm{S}_{i,m,l}^k. \end{aligned} \end{aligned}$$

A standard argument involving Hölder’s inequality leads to that

$$\begin{aligned}{} & {} \sum _j\int ^{2^j}_{2^{j-Ni}}\int ^{2^j2^{iN\frac{\alpha }{1-\alpha }}}_{2^j}\big |\sum _l\textrm{S}_{i,m,l}^1\big | \frac{dt}{t}\frac{ds}{s}\nonumber \\{} & {} \quad \lesssim \sum _j\int ^{2^j}_{2^{j-Ni}}\int ^{2^j2^{iN\frac{\alpha }{1-\alpha }}}_{2^j} \Vert Q_sQ_t^2f\Vert _{L^2({\mathbb {R}}^d)} \Big \Vert [\partial _mA,\,Q_s]U^i_{\Omega ,m,j}Q_t^2Q_s^2g\Big \Vert _{L^2({\mathbb {R}}^d)} \frac{dt}{t}\frac{ds}{s}\nonumber \\{} & {} \quad \lesssim \Vert \Omega _i\Vert _{L^1(S^{d-1})}\Big (\sum _j\int ^{2^j}_{2^{j-Ni}}\int ^{2^j2^{iN\frac{\alpha }{1-\alpha }}}_{2^j} \Vert Q_sQ_t^2f\Vert _{L^2({\mathbb {R}}^d)}^2\frac{dt}{t}\frac{ds}{s}\Big )^{1/2}\nonumber \\{} & {} \qquad \times \Big (\sum _j\int ^{2^j}_{2^{j-Ni}}\int ^{2^j2^{iN\frac{\alpha }{1-\alpha }}}_{2^j} \Big \Vert Q_t^2Q_s^2g\Big \Vert _{L^2({\mathbb {R}}^d)}^2\frac{dt}{t}\frac{ds}{s}\Big )^{1/2}\nonumber \\{} & {} \quad \lesssim i^2|\Omega _i\Vert _{L^1(S^{d-1})}\Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

(2.19)

Similarly, one can verify that

$$\begin{aligned} \sum _j\int ^{2^j}_{2^{j-Ni}}\int ^{2^j2^{iN\frac{\alpha }{1-\alpha }}}_{2^j}\big |\sum _l\textrm{S}_{i,m,l}^3\big | \frac{dt}{t}\frac{ds}{s}\lesssim i^2\Vert \Omega _i\Vert _{L^1(S^{d-1})}\Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}.\nonumber \\ \end{aligned}$$

(2.20)

and

$$\begin{aligned} \sum _j\int ^{2^j}_{2^{j-Ni}}\int ^{2^j2^{iN\frac{\alpha }{1-\alpha }}}_{2^j}\big |\sum _l\textrm{S}_{i,m,l}^4\big | \frac{dt}{t}\frac{ds}{s}\lesssim i^2\Vert \Omega _i\Vert _{L^1(S^{d-1})}\Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}.\nonumber \\ \end{aligned}$$

(2.21)

On the other hand, the fact (see [20, Lemma 4 and Lemma 3])

$$\begin{aligned} \Vert [\partial _mA,\,U^i_{\Omega ,m,j}]h\Vert _{L^2({\mathbb {R}}^d)}\lesssim \big (2^{-i}+i\Vert \Omega _i\Vert _{L^1(S^{d-1})}\big )\Vert h\Vert _{L^2({\mathbb {R}}^d)}, \end{aligned}$$

implies that

$$\begin{aligned}{} & {} \sum _j\int ^{2^j}_{2^{j-Ni}}\int ^{2^j2^{iN\frac{\alpha }{1-\alpha }}}_{2^j}\big |{\sum _l\textrm{S} _{i,m,l}^2}\big | \frac{dt}{t}\frac{ds}{s}\nonumber \\{} & {} \quad \lesssim \big (i2^{-i}+i^2\Vert \Omega _i\Vert _{L^1(S^{d-1})}\big )\Vert f\Vert _{L^2({\mathbb {R}}^d)} \Vert g\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

(2.22)

Applying Hölder’s inequality and inequality (2.8) in the case $s\in (2^{j-1}, 2^j)$, we obtain

$$\begin{aligned}{} & {} \sum _j\int ^{2^j}_{2^{j-Ni}}\int ^{2^j2^{iN\frac{\alpha }{1-\alpha }}}_{2^j}\big |{\sum _l\textrm{S} _{i,m,l}^5}\big | \frac{dt}{t}\frac{ds}{s}\nonumber \\{} & {} \quad \lesssim \Vert \Omega _i\Vert _{L^1(S^{d-1})}\Big (\sum _j\int ^{2^j}_{2^{j-Ni}}\int ^{2^j2^{iN\frac{\alpha }{1-\alpha }}}_{2^j} \Vert Q_sQ_t^2f\Vert _{L^2({\mathbb {R}}^d)}^2\frac{dt}{t}\frac{ds}{s}\Big )^{1/2}\nonumber \\{} & {} \qquad \times \Big (\sum _j\int ^{2^j}_{2^{j-Ni}}\int ^{2^j2^{iN\frac{\alpha }{1-\alpha }}}_{2^j} \Big \Vert Q_t^2Q_s\big (\sum _l\partial _m{\widetilde{A}}_{I_l}h_{s,l}\big )\Big \Vert _{L^2({\mathbb {R}}^d)}^2 \frac{dt}{t}\frac{ds}{s}\Big )^{1/2} \nonumber \\{} & {} \quad \lesssim i^{\frac{3}{2}}\Vert \Omega _i\Vert _{L^1(S^{d-1})}\Vert f\Vert _{L^2({\mathbb {R}}^d)} \Big (\sum _j\int ^{2^j}_{2^{j-Ni}} \Big \Vert \sum _lQ_s\big (\partial _m{\widetilde{A}}_{I_l}h_{s,l}\big )\Big \Vert _{L^2({\mathbb {R}}^d)}^2 \frac{ds}{s}\Big )^{1/2}\nonumber \\{} & {} \quad \lesssim i^{2}\Vert \Omega _i\Vert _{L^1(S^{d-1})}\Vert f\Vert _{L^2({\mathbb {R}}^d)} \Big (\sum _j\int ^{2^j}_{2^{j-1}} \Vert M_qh\Vert _{L^2({\mathbb {R}}^d)}^2\frac{ds}{s}\Big )^{1/2}\nonumber \\{} & {} \quad \lesssim i^{2}\Vert \Omega _i\Vert _{L^1(S^{d-1})}\Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

(2.23)

Collecting the estimates from (2.14) to (2.23) in all, we deduce that

$$\begin{aligned} \textrm{IV}{} & {} =\sum _i\sum _j\int ^{2^j}_{2^{j-Ni}}{\int ^{(2^js^{-\alpha })^{\frac{1}{1-\alpha }}}_{2^j}}\Big |\int _{{\mathbb {R}}^d} Q_t^4f(x)T_{\Omega ,A,j}^iQ_s^4g(x)dx\Big |\frac{dt}{t}\frac{ds}{s}\nonumber \\{} & {} \lesssim \Big (\sum _{i}i2^{-i}+\sum _ii^2\Vert \Omega _i\Vert _{L^1(S^{d-1})}\Big )\Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}\nonumber \\{} & {} \lesssim \Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

(2.24)

To show the estimate for $\mathrm V$, note that for each fixed j, it holds that

$$\begin{aligned}{} & {} \big \{(s,\,t):\,0\le s\le 2^{j-Ni},\,2^j\le t<(2^js^{-\alpha })^{\frac{1}{1-\alpha }}\big \}\\{} & {} \quad \subset \big \{(s,\,t):\,2^j\le t<\infty , 0<s\le \min \{2^{j-Ni},\,{(2^jt^{\alpha -1})^{\frac{1}{\alpha }}}\}\big \}. \end{aligned}$$

It then follows that

$$\begin{aligned}{} & {} \sum _j\int ^{2^{j-Ni}}_0\int ^{(2^js^{-\alpha })^{\frac{1}{1-\alpha }}}_{2^j} \Vert Q_t^4f\Vert ^2_{L^2({\mathbb {R}}^d)} \frac{dt}{t}(2^{-j}s)^{\varepsilon }\frac{ds}{s}\\{} & {} \quad \le 2^{-Ni\varepsilon /2}\int ^{\infty }_0\sum _{j:\,2^j\le t}\int _0^{(2^jt^{\alpha -1})^{\frac{1}{\alpha }}}(2^{-j}s)^{\frac{\varepsilon }{2}} \frac{ds}{s}\Vert Q_t^4f\Vert ^2_{L^2({\mathbb {R}}^d)} \frac{dt}{t}\\{} & {} \quad \lesssim 2^{-Ni\varepsilon /2}\Vert f\Vert _{L^2({\mathbb {R}}^d)}^2, \end{aligned}$$

and

$$\begin{aligned}{} & {} \sum _j\int ^{2^{j-Ni}}_0\int ^{(2^js^{-\alpha })^{\frac{1}{1-\alpha }}}_{2^j} \frac{dt}{t}\Vert Q_s^3g\Vert ^2_{L^2({\mathbb {R}}^d)} (2^{-j}s)^{\varepsilon }\frac{ds}{s}\\{} & {} \quad \le 2^{-Ni\varepsilon /2}\int ^{\infty }_0\Big (\sum _{j:\,2^j\ge s2^{Ni}}\int _{2^j}^{(2^js^{-\alpha })^{\frac{1}{1-\alpha }}} \frac{dt}{t}(2^{-j}s)^{\frac{\varepsilon }{2}}\Big )\Vert Q_s^3g\Vert ^2_{L^2({\mathbb {R}}^d)} \frac{ds}{s}\\{} & {} \quad \lesssim 2^{-Ni\varepsilon /2}\Vert g\Vert _{L^2({\mathbb {R}}^d)}^2, \end{aligned}$$

Thus, by Lemma 2.3, we obtain

$$\begin{aligned} \textrm{V}\le & {} \sum _i\sum _j\int ^{2^{j-Ni}}_0\int ^{(2^js^{-\alpha })^{\frac{1}{1-\alpha }}}_{2^j} \Vert Q_sT^i_{\Omega ,\,A;\,j}Q_t^4f\Vert _{L^2({\mathbb {R}}^d)}\Vert Q_s^3g\Vert _{L^2({\mathbb {R}}^d)}{\frac{dt}{t}\frac{ds}{s}} \nonumber \\\le & {} \sum _i2^i\Big (\sum _j\int ^{2^{j-Ni}}_0\int ^{(2^js^{-\alpha })^{\frac{1}{1-\alpha }}}_{2^j} \Vert Q_t^4f\Vert ^2_{L^2({\mathbb {R}}^d)} \frac{dt}{t}(2^{-j}s)^{\varepsilon }\frac{ds}{s}\Big )^{\frac{1}{2}}\nonumber \\{} & {} \times \Big (\sum _j\int ^{2^{j-Ni}}_0\int ^{(2^js^{-\alpha })^{\frac{1}{1-\alpha }}}_{2^j} \Vert Q_s^3g\Vert ^2_{L^2({\mathbb {R}}^d)} (2^{-j}s)^{\varepsilon }{\frac{dt}{t}\frac{ds}{s}}\Big )^{1/2}\nonumber \\\lesssim & {} \sum _i2^i2^{-Ni\varepsilon /2} \Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

(2.25)

Combining estimates (2.24)–(2.25) yields

$$\begin{aligned}{} & {} \Big |\sum _i\sum _j\int ^{\infty }_{2^j}\int _{0}^{(2^jt^{\alpha -1})^{1/\alpha }}\int _{{\mathbb {R}}^d}Q_s^4 T_{\Omega ,A,j}^iQ_t^4f(x)g(x)dx \frac{ds}{s}\frac{dt}{t}\Big |\nonumber \\{} & {} \quad \lesssim \Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

(2.26)

Therefore, by (2.13), (2.24) and (2.26), it holds that

$$\begin{aligned}{} & {} \Big |\sum _i\sum _j\int ^{\infty }_0\int _{0}^t\int _{{\mathbb {R}}^d}\chi _{E_2(j,,s,t)}Q_s^4T_{\Omega ,\,A;\,j}^iQ_t^4f(x)g(x)dx\frac{ds}{s}\frac{dt}{t}\Big |\\{} & {} \quad \lesssim IV+V \lesssim \Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}, \end{aligned}$$

which gives the contribution of $E_2(j,s,t)$.

To finish the proof of (2.4), it remains to show the contribution of the term $E_3^i(j,s,t)$.

2.3 Contribution of ${E_3(j,s,t)}$

Our aim is to prove

$$\begin{aligned}{} & {} \Big |\sum _{i=0}^{\infty }\sum _j\int ^{\infty }_0\int _{0}^t\int _{{\mathbb {R}}^d}{\chi _{E_{3}}(j,s,t)}Q_s^4T_{\Omega ,\,A;\,j}^iQ_t^4f(x)g(x)dx\frac{ds}{s}\frac{dt}{t}\Big |\nonumber \\{} & {} \quad \lesssim \Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}, \end{aligned}$$

(2.27)

where

$$\begin{aligned} T_{\Omega _i,\,A;\,j}f(x)= & {} \int _{{\mathbb {R}}^d}\frac{\Omega _i(x-y)}{|x-y|^{d+1}}\nonumber \\ {}{} & {} \times (A(x)-A(y)-\nabla A(y)(x-y)\big ) \phi _j(|x-y|)f(y)dy, \end{aligned}$$

(2.28)

Since the sum of i and the sum of j are independent and the sum of j depends only on the functions $\phi _j$ and ${\chi _{E_{3}}(j,s,t)},$ one may put $\phi _j \cdot {\chi _{E_{3}}(j,s,t)}$ together in the place of $\phi _j$ in (2.28), and temporary moves the summation over j before $\phi _j \cdot {\chi _{E_{3}}(j,s,t)}$, which indicates that it is possible to move the summation over i inside the integral again before $\Omega _i$ to obtain $\Omega$. After that, one may move the sum of j outside the integral. Therefore, to prove (2.27), it suffices to show that

$$\begin{aligned}{} & {} \sum _j\int ^{\infty }_{2^j}\int ^t_{(2^jt^{\alpha -1})^{1/\alpha }}\Big |\int _{{\mathbb {R}}^d}Q_s^4 {T}_{\Omega ,A,j}Q_t^4f(x)g(x)dx\Big | \frac{ds}{s}\frac{dt}{t}\nonumber \\{} & {} \quad \lesssim \Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

(2.29)

To this purpose, we set

$$\begin{aligned} h^{(1)}(x,\,y)=\int \int {{\psi _s}(x-z)}\sum _{j:2^j\le s^{\alpha }t^{1-\alpha }}K_{A,\,j}(z,\,u){\big [{\psi _t}(u-y)-{\psi _t}(x-y)]}dudz. \end{aligned}$$

Let $H^{(1)}$ be the integral operator corresponding to kernel $h^{(1)}$. It then follows that

$$\begin{aligned}{} & {} \Big |\sum _j\int ^{\infty }_{2^j}\int ^t_{(2^jt^{\alpha -1})^{1/\alpha }} \int _{{\mathbb {R}}^d}Q_s^4T_{\Omega ,A,j}Q_t^4f(x)g(x)dx \frac{ds}{s}\frac{dt}{t}\Big |\nonumber \\{} & {} \quad \le \int ^{\infty }_0\int ^t_0\Vert H^{(1)}Q_t^3f\Vert _{L^2({\mathbb {R}}^d)} \Vert Q_s^3g\Vert _{L^2({\mathbb {R}}^d)}\frac{ds}{s}\frac{dt}{t}\nonumber \\{} & {} \qquad +\sum _j\int ^{\infty }_{2^j}\int ^t_{(2^jt^{\alpha -1})^{1/\alpha }} \int _{{\mathbb {R}}^d}(Q_sT_{\Omega ,A;\,j}1)(x)Q_t^4f(x)Q_s^3g(x)dx\frac{ds}{s}\frac{dt}{t}. \end{aligned}$$

(2.30)

Applying Lemma 2.5 and reasoning as the same argument as in [18, p. 1282] give us that

$$\begin{aligned} {|h^{(1)}(x,\,y)|}\lesssim \big (\frac{s}{t}\big )^{\gamma }t^{-d}\chi _{\{(x,\,y):\,|x-y|\le Ct\}}(x,\,y), \end{aligned}$$

where $\gamma =(d+2)\alpha -d-1$. This in turn indicates that $|H^{(1)}Q_tf(x)|\lesssim \big (\frac{s}{t}\big )^{\gamma }M(Q_tf)(x).$ Therefore

$$\begin{aligned}{} & {} \int ^{\infty }_0\int ^t_0\Vert H^{(1)}Q_t^3f\Vert _{L^2({\mathbb {R}}^d)}\Vert Q_s^3g\Vert _{L^2({\mathbb {R}}^d)} \frac{ds}{s}\frac{dt}{t}\nonumber \\{} & {} \quad \lesssim \Big ( \int ^{\infty }_0\int ^t_0\big (\frac{s}{t}\big )^{\gamma }\Vert M(Q_t^3f)\Vert _{L^2({\mathbb {R}}^d)}^2 \frac{ds}{s}\frac{dt}{t}\Big )^{\frac{1}{2}}\nonumber \\{} & {} \qquad \times \Big (\int ^{\infty }_0\Vert Q_s^3g\Vert _{L^2({\mathbb {R}}^d)}^2\int _{s}^{\infty }\big (\frac{s}{t}\big )^{\gamma } \frac{dt}{t}\frac{ds}{s}\Big )^{\frac{1}{2}}\nonumber \\{} & {} \quad \lesssim \Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

(2.31)

It remains to show the corresponding estimate for the second term on the rightside of (2.30).

Let $F_x^j(s,t)=(Q_sT_{\Omega ,A;\,j}1)(x)Q_t^4f(x)Q_s^3g(x)$. Then

$$\begin{aligned} {}&\int ^{\infty }_{2^j}\int ^t_{(2^jt^{\alpha -1})^{1/\alpha }}F_x^j(s,t)\frac{dsdt}{st}\nonumber \\&\quad =\int ^{\infty }_0\int _0^tF_x^j(s,t)\frac{dsdt}{st}-\int ^{2^j}_0\int _0^t~F_x^j(s,t)\frac{dsdt}{st}\nonumber \\&\qquad -\int ^{\infty }_{2^j}\int _0^{(2^jt^{\alpha -1})^{1/\alpha }}~F_x^j(s,t)\frac{dsdt}{st} \end{aligned}$$

(2.32)

Therefore, it is sufficient to consider the contributions of each terms in Eq. (2.32) to the second term in (2.30).

Consider the first term in (2.32). Let $P_s=\int ^{\infty }_sQ_t^4\frac{dt}{t}.$ Han and Sawyer [16] observed that the kernel $\Phi$ of the convolution operator $P_s$ is a radial bounded function with bound $cs^{-d}$, supported on a ball of radius Cs and has integral zero. Therefore, it is easy to see that $\Phi$ is a Schwartz function. Since ${P_sg=\Phi _s * g}$, it then follows from the Littlewood-Paley theory that

$$\begin{aligned} \int ^{\infty }_0\Vert P_sg\Vert _{L^2({\mathbb {R}}^d)}^2\frac{ds}{s}\lesssim \Vert g\Vert _{L^2({\mathbb {R}}^d)}^2. \end{aligned}$$

On the other hand, whenever $\Omega \in L^1({\mathbb {R}}^d)$, it was shown in [19, p.121, Lemma 4.1] that $T_{\Omega ,A}1\equiv b\in \textrm{BMO}({\mathbb {R}}^d)$. Therefore, by [19, p.114, (3.1)], $\int ^{\infty }_0Q_s^3(Q_sbP_s)\frac{ds}{s}$ defines an operator which is bounded on $L^2({\mathbb {R}}^d)$. However, we can’t use this boundedness directly in our case, since once using Hölder’s inequality, we have to put the absolute value inside the integral and the $L^2({\mathbb {R}}^d)$ boundedness may fail in this case. To overcome this obstacle, we apply the property of Carleson measure.

Note that $|Q_s T_{\Omega ,\,A}1(x)|^2\frac{dxds}{s}$ is a Carleson measure since $T_{\Omega ,\,A}1\in \textrm{BMO}({\mathbb {R}}^d)$. By Hölder’s inequality, Lemma 2.7, it yields that

$$\begin{aligned}{} & {} \Big |\int ^{\infty }_0\int _0^t\int _{{\mathbb {R}}^d}\sum _jQ_s T_{\Omega ,\,A;j}1(x)Q_t^4f(x)Q^3_sg(x)dx\frac{ds}{s}\frac{dt}{t}\Big |\nonumber \\{} & {} \quad \lesssim {\bigg (\int ^{\infty }_0\int _{{\mathbb {R}}^d}|Q_s^3g(x)|^2dx\frac{ds}{s}\bigg )^{\frac{1}{2}}\bigg (\int _{\mathbb {R}_+^{n+1}}|P_sf(x)|^2|Q_s T_{\Omega ,\,A}1(x)|^2\frac{dxds}{s}\bigg )^{\frac{1}{2}}}\nonumber \\{} & {} \quad \lesssim \Vert f\Vert _{L^2({\mathbb {R}}^d)} \Vert g\Vert _{L^2({\mathbb {R}}^d)} \end{aligned}$$

(2.33)

On the other hand, by Lemma 2.2, one gets

$$\begin{aligned} \Vert Q_sT_{\Omega ,\,A;\,j}1\Vert _{L^{\infty }({\mathbb {R}}^d)}\lesssim \Vert \Omega \Vert _{L^1({\mathbb {S}}^{d-1})}(2^{-j}s)^{\epsilon }. \end{aligned}$$

Denote by $D^1_{j,s,t}=\{(j,s,t): s\le t\le 2^j\}$, $D^2_{j,s,t}=\{(j,s,\,t):\,s\le t,s^{\alpha }t^{1-\alpha }\le 2^j\le t\}$. It then follows that

$$\begin{aligned}{} & {} \sum _j\int ^{2^j}_0\int _0^t\int _{{\mathbb {R}}^d} \Big |Q_sT_{\Omega ,\,A;j}1(x)Q_t^4f(x)Q_s^3g(x)\Big |dx\frac{ds}{s}\frac{dt}{t}\nonumber \\{} & {} \qquad +\sum _j\int ^{\infty }_{2^j}\int _0^{(2^jt^{\alpha -1})^{1/\alpha }}\int _{{\mathbb {R}}^d} \Big |Q_sT_{\Omega ,\,A;j}1(x)Q_t^4f(x)Q^3_sg(x)\Big |dx\frac{ds}{s}\frac{dt}{t}\nonumber \\{} & {} \quad \lesssim \sum _{i=1}^2\bigg \{\Big (\int _0^\infty \int _0^\infty \sum _j(2^{-j}s)^{\epsilon }\chi _{D_{j,s,t}^i}(j,s,t)\Vert Q_tf\Vert _{L^2({\mathbb {R}}^d)}^2\frac{ds}{s}\frac{dt}{t}\Big )^{1/2}\nonumber \\{} & {} \qquad \times \Big (\int _0^\infty \int _0^\infty \sum _j(2^{-j}s)^{\epsilon }\chi _{D_{j,s,t}^i}(j,s,t)\Vert {Q_sg\Vert _{L^2({\mathbb {R}}^d)}^2}\frac{ds}{s}\frac{dt}{t}\Big )^{1/2}\bigg \}\nonumber \\{} & {} \quad \lesssim \Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}, \end{aligned}$$

(2.34)

where in the last inequality, we used the property (2.2).

Combining (2.32)–(2.34), we have

$$\begin{aligned}{} & {} \Big |\sum _j\int ^{\infty }_{2^j}\int ^t_{(2^jt^{\alpha -1})^{1/\alpha }}\int _{{\mathbb {R}}^d}(Q_sT_{\Omega ,A;\,j}1)(x)Q_t^4f(x)Q^3_sg(x)dx\frac{ds}{s}\frac{dt}{t} \Big |\\{} & {} \quad \lesssim \Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}, \end{aligned}$$

which, together with (2.31), leads to (2.27). This finishes the proof of $E_3(j,s,t)$, and also completes the proof of inequality (2.4).

2.4 Proof of (2.5)

To finish the proof of Theorem 1.1, it remains to show the estimate (2.5). Observe that

$$\begin{aligned}{} & {} \int ^{\infty }_0\int _{t}^{\infty }\int _{{\mathbb {R}}^d}Q_s^4{T}_{\Omega ,\,A}Q_t^4f(x)g(x)dx \frac{ds}{s}\frac{dt}{t}\\{} & {} \quad =-\int ^{\infty }_0\int _{0}^{s}\int _{{\mathbb {R}}^d}Q_t^4{\widetilde{T}}_{{\widetilde{\Omega }},\,A}Q_s^4g(x)f(x)dx \frac{dt}{t} \frac{ds}{s}, \end{aligned}$$

where ${\widetilde{\Omega }}(x)=\Omega (-x)$ and ${\widetilde{T}}_{{\widetilde{\Omega }},\,A}$ is the operator defined by (1.5), with $\Omega$ replaced by ${\widetilde{\Omega }}$. Let $T_{{\widetilde{\Omega }},\,m}$ be the operator defined by

$$\begin{aligned} T_{{\widetilde{\Omega }},\,m}h(x)=\mathrm{p.\,v.}\int _{{\mathbb {R}}^d}\frac{{\widetilde{\Omega }}(x-y)(x_m-y_m)}{|x-y|^{d+1}}h(y)dy. \end{aligned}$$

It then follows that

$$\begin{aligned} {\widetilde{T}}_{{\widetilde{\Omega }},\,A}h(x)={T_{{\widetilde{\Omega }},\,A}h(x)-\sum _{m=1}^d[\partial _mA,\, T_{{\widetilde{\Omega }},\,m}]h(x).} \end{aligned}$$

Inequality (2.4) tells us that

$$\begin{aligned} \Big |\int ^{\infty }_0\int _{0}^t\int _{{\mathbb {R}}^d}Q_s^4 {T}_{{\widetilde{\Omega }},\,A}Q_t^4g(x)f(x)dx{\frac{ds}{s}\frac{dt}{t}}\Big |\lesssim \Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

(2.35)

For each fixed m with $1\le m\le d$, by duality, involving Lemma 2.4 and Hölder’s inequality may lead to

$$\begin{aligned}{} & {} \Big |\int ^{\infty }_0\int _{0}^t\int _{{\mathbb {R}}^d}Q_s^4[\partial _mA,\,T_{{\widetilde{\Omega }},\,m}]Q_t^4g(x)f(x)dx \frac{ds}{s}\frac{dt}{t}\Big |\\{} & {} \quad \lesssim \int ^{\infty }_0\Vert [\partial _mA,\,T_{{\widetilde{\Omega }},\,m}]Q_s^4f\Vert _{L^2({\mathbb {R}}^d)} \Big \Vert \int _{s}^{\infty }Q_t^4 g\frac{dt}{t}\Big \Vert _{L^2({\mathbb {R}}^d)} \frac{ds}{s}\\{} & {} \quad \lesssim \Big (\int ^{\infty }_0\Vert Q_s^4f\Vert ^2_{L^2({\mathbb {R}}^d)}\frac{ds}{s}\Big )^{1/2} \Big (\int ^{\infty }_0\Vert P_sg\Vert ^2_{L^2({\mathbb {R}}^d)} \frac{ds}{s}\Big )^{1/2}\\{} & {} \quad \lesssim \Vert f\Vert _{L^2({\mathbb {R}}^d)}\Vert g\Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

This estimate, together with (2.35), leads to (2.5) and then completes the proof of Theorem 1.1.

3 Proof of Theorems 1.2 and 1.3

This section is devoted to prove Theorem 1.2, the weak type endpoint estimates for $T_{\Omega ,\,A}$ and ${\widetilde{T}}_{\Omega ,A}$. To this end, we first introduce the definition of standard dyadic grid. Recall that the standard dyadic grid in ${\mathbb {R}}^d$, denoted by ${\mathcal {D}}$, consists of all cubes of the form

$$\begin{aligned} 2^{-k}([0,\,1)^d+j),\,k\in {\mathbb {Z}},\,\,j\in {\mathbb {Z}}^d. \end{aligned}$$

For each fixed $j\in {\mathbb {Z}}$, set ${\mathcal {D}}_j=\{Q\in {\mathcal {D}}:\, \ell (Q)=2^j\}$.

3.1 Proof of (1.6) in Theorem 1.2

The key ingredient of our proof lies in the step of dealing with the bad part of the Calderón-Zygmund decomposition of f. By homogeneity, it suffices to prove (1.6) for the case $\lambda =1$. Applying the Calderón-Zygmund decomposition to $|f|\log (\textrm{e}+|f|)$ at level 1, we can obtain a collection of non-overlapping closed dyadic cubes ${\mathcal {S}}=\{{\mathbb {L}}\}$, such that

• (i) $\Vert f\Vert _{L^{\infty }({\mathbb {R}}^d\backslash \cup _{{\mathbb {L}}\in {\mathcal {S}}}{\mathbb {L}})}\lesssim 1;$

• (ii) $\int _{{\mathbb {L}}}|f(x)|\log (\textrm{e}+|f(x)|)dx \lesssim |{\mathbb {L}}|;$

• (iii) $\sum _{{\mathbb {L}}\in {\mathcal {S}}}|{\mathbb {L}}|\lesssim \int _{{\mathbb {R}}^d}|f(x)|\log (\textrm{e}+|f(x)|)dx$.

Let g be the good part and b be the bad part of the decomposition of f, which are defined by

$$\begin{aligned} g(x)= & {} f(x)\chi _{{\mathbb {R}}^d\backslash \cup _{{\mathbb {L}}\in {\mathcal {S}}}{\mathbb {L}}}(x)+\sum _{{\mathbb {L}}\in {\mathcal {S}}}\langle f\rangle _{{\mathbb {L}}}\chi _{{\mathbb {L}}}(x) \quad \hbox { and }\\ b(x)= & {} \sum _{{\mathbb {L}}\in {\mathcal {S}}}(f-\langle f\rangle _{{\mathbb {L}}})\chi _{{\mathbb {L}}}(x)=\sum _{{\mathbb {L}}\in {\mathcal {S}}}b_{{\mathbb {L}}}(x). \end{aligned}$$

It is easy to see that $\Vert g\Vert _{L^{\infty }({\mathbb {R}}^d)}\lesssim 1$, and for $E=\cup _{{\mathbb {L}}\in {\mathcal {S}}}100d{\mathbb {L}}$, it holds that

$$\begin{aligned} |E|\lesssim \int _{{\mathbb {R}}^d}|f(x)|\log (\textrm{e}+|f(x)|)dx. \end{aligned}$$

The $L^2({\mathbb {R}}^d)$ boundedness of $T_{\Omega ,\,A}$ then yields that

$$\begin{aligned} \big |\{x\in {\mathbb {R}}^d:\, |T_{\Omega ,\,A}g(x)|\ge & {} 1/2\}\big |\lesssim \Vert T_{\Omega ,\,A}g\Vert _{L^2({\mathbb {R}}^d)}^2\nonumber \\ {}\lesssim & {} \Vert g\Vert ^2_{L^2({\mathbb {R}}^d)}\lesssim \Vert f\Vert _{L^1({\mathbb {R}}^d)}. \end{aligned}$$

(3.1)

Therefore, it is sufficient to show that

$$\begin{aligned} \big |\{x\in {\mathbb {R}}^d:\, |T_{\Omega ,\,A}b(x)|\ge 1/2\}\big | \lesssim \int _{{\mathbb {R}}^d}|f(x)|\log (\textrm{e}+|f(x)|)dx. \end{aligned}$$

(3.2)

To prove (3.2), let $\phi$ be a smooth radial nonnegative function on ${\mathbb {R}}^d$ with $\textrm{supp}\, \phi \subset \{x:\frac{1}{4}\le |x|\le 1\}$ and $\sum _s\phi _s(x)=1$ with $\phi _s(x)=\phi (2^{-s}x)$ for all $x\in {\mathbb {R}}^d\backslash \{0\}$. Set ${\mathcal {S}}_{j}=\{{\mathbb {L}}\in {\mathcal {S}}:\, \ell ({\mathbb {L}})=2^j\}$. Then, we have

$$\begin{aligned}{} & {} \int _{{\mathbb {R}}^d}\frac{\Omega (x-y)}{|x-y|^{d+1}}(A(x)-A(y))b(y)dy\\{} & {} \quad =\int _{{\mathbb {R}}^d}\frac{\Omega (x-y)}{|x-y|^{d+1}}(A(x)-A(y))\sum _s\phi _{s}(x-y)\sum _j\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}}b_{\mathbb {L}}(y)dy\\{} & {} \quad = \int _{{\mathbb {R}}^d}\frac{\Omega (x-y)}{|x-y|^{d+1}}(A(x)-A(y))\sum _j\sum _s\phi _{s}(x-y)\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}}b_{\mathbb {L}}(y)dy\\{} & {} \quad = \sum _j\sum _s\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}} T_{\Omega ,\,A;\,s,j}b_{\mathbb {L}}(x), \end{aligned}$$

where

$$\begin{aligned} T_{\Omega ,\,A;\,s,j}b_{\mathbb {L}}(x)=\int _{{\mathbb {R}}^d} \phi _{s}(x-y)\frac{\Omega (x-y)}{|x-y|^{d+1}}(A(x)-A(y))b_{\mathbb {L}}(y)dy. \end{aligned}$$

(3.3)

Let $A_{{\mathbb {L}}}(y)=A(y)-\sum _{n=1}^d\langle \partial _n A\rangle _{\mathbb {L}}y_n.$ A trivial computation leads to the fact that

Now write $T_{\Omega ,A}b$ as

$$\begin{aligned} T_{\Omega ,A}b(x)=\sum _{j}\sum _{s}\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}}T_{\Omega ,\,A_{\mathbb {L}};s,j}b_{{\mathbb {L}}}(x)-\sum _{n=1}^dT_{\Omega }^n\Big (\sum _{{\mathbb {L}}\in {\mathcal {S}}}b_{\mathbb {L}}\partial _nA_{{\mathbb {L}}}\Big )(x), \end{aligned}$$

where

$$\begin{aligned} T_{\Omega }^nh(x)=\mathrm{p.\, v.}\int _{{\mathbb {R}}^d}\frac{\Omega (x-y)}{|x-y|^{d+1}}(x_n-y_n)h(y)dy, \quad \hbox { for } 1\le n\le d. \end{aligned}$$

Fixed $1\le n\le d$, since the kernel ${\Omega (x)x_n}{|x|^{-1}}$ is still in $L\log L {({\mathbb {S}}^{d-1})}$, homogenous of degree zero and satisfies the vanishing condition on the unit sphere, by the weak endpoint estimate of the operators $T_{\Omega }^n$ (see [31] or [9]), it follows that

$$\begin{aligned}{} & {} \Big |\Big \{x\in {\mathbb {R}}^d\backslash E:\Big | T_{\Omega }^n\Big (\sum _{{\mathbb {L}}\in {\mathcal {S}}}b_{\mathbb {L}}\partial _nA_{\mathbb {L}}\Big )(x)\Big |>\frac{1}{4d}\Big \}\Big |\lesssim \Big \Vert \sum _{{\mathbb {L}}\in {\mathcal {S}}}b_{\mathbb {L}}\partial _nA_{\mathbb {L}}\Big \Vert _{L^1({\mathbb {R}}^d)}\nonumber \\{} & {} \lesssim \sum _{{\mathbb {L}}\in {\mathcal {S}}}|{\mathbb {L}}|\Vert b_{\mathbb {L}}||_{L\log L,\,{\mathbb {L}}}\nonumber \\{} & {} \lesssim \int _{{\mathbb {R}}^d}|f(x)|\log (\textrm{e}+|f(x)|)dx, \end{aligned}$$

(3.4)

where in the last inequality, we have used the fact that $\Vert b_{\mathbb {L}}\Vert _{L\log L,\,{\mathbb {L}}}\lesssim 1$ for each cube ${\mathbb {L}}\in {\mathcal {S}}$.

Therefore, to prove inequality (1.6), by (3.1), (3.2) and (3.4), it is sufficient to show that

$$\begin{aligned}{} & {} \Big |\Big \{x\in {\mathbb {R}}^d\backslash E:\, \Big |\sum _{j}\sum _{s}\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}}T_{\Omega ,A_{\mathbb {L}};s,j}b_{\mathbb {L}}(x)\Big |>1/4\Big \}\Big |\lesssim {\Vert f\Vert }_{L^1({\mathbb {R}}^d)}. \end{aligned}$$

(3.5)

In order to prove inequality (3.5), we first give some estimate for $\sum _s\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}}T_{\Omega ,\,A_{\mathbb {L}};s,j}b_{{\mathbb {L}}}$. For this purpose, we need to introduce some notations.

For ${\mathbb {L}}\in {\mathcal {S}}_{s-j}$, s, $j\,\in {\mathbb {Z}}$ with $j\ge \log _2(100d/2)=:j_0$. Let $L_{j,1}=2^{j+2}d{\mathbb {L}}$, $L_{j,2}=2^{j+4}d{\mathbb {L}}$, $L_{j,3}=2^{j+6}d{\mathbb {L}}$, and $y^j_{{\mathbb {L}}}$ be a point on the boundary of $L_{j,3}$. Set

$$\begin{aligned} A_{\varphi _{{\mathbb {L}}}}(y)=\varphi _{{\mathbb {L}}}(y)\big (A_{{\mathbb {L}}}(y)-A_{{\mathbb {L}}}(y^j_{{\mathbb {L}}})), \end{aligned}$$

where $\varphi _{{\mathbb {L}}}\in C^{\infty }_c({\mathbb {R}}^d)$, $\textrm{supp}\,\varphi _{{\mathbb {L}}}\subset L_{j,1}$, $\varphi _{{\mathbb {L}}}\equiv 1$ on $3\cdot 2^{j}d{\mathbb {L}}$, and $\Vert \nabla \varphi _{{{\mathbb {L}}}}\Vert _{L^{\infty }({\mathbb {R}}^d)}\lesssim 2^{-s}.$ Let $y_0$ be the center point of ${\mathbb {L}}$. Observe that for $x\in {\mathbb {R}}^d\backslash E$, $j\le j_0$, $y\in {\mathbb {L}}$, we have $|x-y|\ge |x-y_0|-|y-y_0|>2^s$. The support condition of $\phi$ then implies that $T_{\Omega ,A_{{\mathbb {L}}};s,j}b_{\mathbb {L}}(x)=0$ if $j\le j_0$. For $y\in {\mathbb {L}}\in {\mathcal {S}}_{s-j}$, $s,\,j\in {\mathbb {Z}}$ with $j> j_0$, we have $\varphi _{{\mathbb {L}}}(y)= 1$. By the support condition of $\phi$, it follows that $|x-y_0|\le |x-y|+|y-y_0|\le 1.5d2^s$. Hence $x\in 3\cdot 2^{j}d{\mathbb {L}}$ and $\varphi _{{\mathbb {L}}}(x)= 1$. Collecting these facts in all, it follows that

The kernel $\Omega$ will be decomposed into disjoint forms as in Section 2 as follows:

$$\begin{aligned} \Omega _0(\theta )=\Omega (\theta )\chi _{E_0}(\theta ),\quad \,\Omega _k(\theta )=\Omega (\theta )\chi _{E_k}(\theta )\,\,(k\in {\mathbb {N}}), \end{aligned}$$

where $E_0=\{\theta \in {\mathbb {S}}^{d-1}:\, |\Omega (\theta )|\le 1\} \ \hbox {and } E_k=\{\theta \in {\mathbb {S}}^{d-1}:\,2^{k-1}<|\Omega (\theta )|\le 2^k\}$ for $k\in {\mathbb {N}}.$

Let the operator $T^i_{\Omega ,A_{{\mathbb {L}}};\,s,j}b_{\mathbb {L}}$ be defined in the same form as $T_{\Omega ,A_{{\mathbb {L}}};\,s,j}b_{\mathbb {L}}$, with $\Omega$ replaced by $\Omega _i$. Then we can divide the summation of $T_{\Omega ,A_{{\mathbb {L}}};\,s}b_{\mathbb {L}}$ into two terms as follows

$$\begin{aligned} \sum _{ j> j_0}\sum _{s}\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}} T_{\Omega ,A_{{\mathbb {L}}};\,s,j}b_{\mathbb {L}}(x)&=\sum \limits _{i=0}^{\infty } \sum _{ j> j_0}\sum _{s}\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}} T^i_{\Omega ,A_{{\mathbb {L}}};\,s,j}b_{\mathbb {L}}(x) \\&=\sum \limits _{i=0}^{\infty } \sum _{j_0< j\le Ni} \sum _{s}\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}} T^i_{\Omega ,A_{{\mathbb {L}}};\,s,j}b_{\mathbb {L}}(x)\\&\quad +\sum \limits _{i=0}^{\infty } \sum _{ j> Ni}\sum _{s}\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}} T^i_{\Omega ,A_{{\mathbb {L}}};\,s,j}b_{\mathbb {L}}(x) \\&:=\textrm{D}_1(x)+\textrm{D}_2(x), \end{aligned}$$

where N is some constant which will be chosen later. If we can verify that

$$\begin{aligned} \Vert \textrm{D}_{1}\Vert _{L^1({\mathbb {R}}^d)}\lesssim \Vert \Omega \Vert _{L(\log L)^2({\mathbb {S}}^{d-1})}\Vert f\Vert _{L^1({\mathbb {R}}^d)}. \end{aligned}$$

(3.6)

and

$$\begin{aligned} |[x\in {\mathbb {R}}^d:\, |D_2(x)|>1/8\}|\lesssim \Vert f\Vert _{L^1({\mathbb {R}}^d)}, \end{aligned}$$

(3.7)

the inequality (1.6) then follows directly. The proofs of these two estimate will be given in the next two subsections respectively.

3.2 Proof of Inequality (3.6)

We first claim that if ${\mathbb {L}}\in {\mathcal {S}}_{s-j}$, then

$$\begin{aligned} \big |T^i_{\Omega ,A_{{\mathbb {L}}};\,s,j}b_{\mathbb {L}}(x)\big |\lesssim j \int _{\{2^{s-2}\le |y|\le 2^{s+2}\}}\frac{|\Omega _i(y')|}{|y|^{d}}|b_{\mathbb {L}}(x-y)|dy. \end{aligned}$$

This claim is a consequence of the following lemma, which will also be used several times later.

Lemma 3.1

Let A be a function in ${\mathbb {R}}^d$ with derivatives of order one in $\textrm{BMO}({\mathbb {R}}^d)$. Let $s,\,j\in {\mathbb {Z}}$ and ${\mathbb {L}}\in {\mathcal {S}}_{s-j}$ with $j>j_0$ and let $R_{s,{\mathbb {L}};j}(x,y)$ be the function on ${\mathbb {R}}^d\times {\mathbb {R}}^d$ defined by

$$\begin{aligned} R_{s,{\mathbb {L}};j}(x,y)=\phi _{s}(x-y)\frac{A_{\varphi _{{\mathbb {L}}}}(x)-A_{\varphi _{{\mathbb {L}}}}(y)}{|x-y|^{d+1}}. \end{aligned}$$

Then, $R_{s,{\mathbb {L}};j}$ enjoys the properties that

• (i) For any $x,\, y\in {\mathbb {R}}^d$,

  $$\begin{aligned} \big |R_{s,{\mathbb {L}};j}(x,y)\big |\lesssim \frac{j}{|x-y|^{d}}\chi _{\{2^{s-2}\le |x-y|\le 2^{s+2}\}}(x,\,y); \end{aligned}$$

• (ii) For any $x,\,x'\in {\mathbb {R}}^d$ and $y\in {\mathbb {L}}$ with $|x-y|>2|x-x'|$,

  $$\begin{aligned} \big |R_{s,{\mathbb {L}};j}(x,y)-R_{s,{\mathbb {L}};j}(x',y)\big |\lesssim \frac{|x-x'|}{|x-y|^{d+1}}\Big (j+\Big |\log \big ({2^{s-j}}{|x-x'|^{-1}}\big )\Big |\Big ); \end{aligned}$$

• (iii) For any $x,\,y'\in {\mathbb {R}}^d$ and $y\in {\mathbb {L}}$ with $|x-y|>2|y-y'|$,

  $$\begin{aligned} \big |R_{s,{\mathbb {L}};j}(x,y)-R_{s,{\mathbb {L}};j}(x,\,y')\big |\lesssim \frac{|y-y'|}{|x-y|^{d+1}}\Big (j+\Big |\log \big ({2^{s-j}}{|y-y'|^{-1}}\big )\Big |\Big ). \end{aligned}$$

Proof

We first prove (i). It is obvious that $\textrm{supp}\,R_{s,{\mathbb {L}};j} \subset {L_{j,2}\times L_{j,2}}$. Fixed $x\in L_{j,1}$, we know that $2^{s-j}<|x-y_{\mathbb {L}}^j|$ and

$$\begin{aligned} \big |\langle \nabla A\rangle _{{\mathbb {L}}}-\langle \nabla A\rangle _{I_{(x,|x-y_{\mathbb {L}}^j|)}}\big |&\le \big |\langle \nabla A\rangle _{{\mathbb {L}}}-\langle \nabla A\rangle _{I_{(x, 2^{s-j})}}\big |+{\big |\langle \nabla A\rangle _{I_{(x, 2^{s-j})}}-\langle \nabla A\rangle _{I_{(x,|x-y_{\mathbb {L}}^j|)}}\big |}. \end{aligned}$$

Note that if $x\in 4{\mathbb {L}}$, then ${I_{(x, 2^{s-j})}}\subset 8{\mathbb {L}}$ and it holds that

$$\begin{aligned} \big |\langle \nabla A\rangle _{{\mathbb {L}}}-\langle \nabla A\rangle _{I_{(x, 2^{s-j})}}\big |&\le \big |\langle \nabla A\rangle _{{\mathbb {L}}}-\langle \nabla A\rangle _{8{\mathbb {L}}}\big |+\big |\langle \nabla A\rangle _{8{\mathbb {L}}}-\langle \nabla A\rangle _{I_{(x, 2^{s-j})}}\big | \lesssim 1. \end{aligned}$$

If $x\in L_{j,1}\backslash 4{\mathbb {L}}$, then the center of ${\mathbb {L}}$ and the center of ${I_{(x, 2^{s-j})}}$ are at a distance of $a2^{s-j}$ with $a>1$. Hence, the results in [13, Proposition 3.1.5, p. 158 and 3.1.5–3.1.6, p. 166.] gives that

$$\begin{aligned} \big |\langle \nabla A\rangle _{{\mathbb {L}}}-\langle \nabla A\rangle _{I_{(x,2^{s-j})}}\big | \lesssim j\quad \hbox {and} \quad \big |\langle \nabla A\rangle _{I_{(x, 2^{s-j})}}-\langle \nabla A\rangle _{I_{(x,|x-y_{\mathbb {L}}^j|)}}\big | \lesssim j, \end{aligned}$$

since $2^s<|x-y_{\mathbb {L}}^j|<2^{s+5+d^2}$.

Therefore, for $x\in {\mathbb {L}}_{j,1}$, it holds that

$$\begin{aligned} \big |\langle \nabla A\rangle _{{\mathbb {L}}}-\langle \nabla A\rangle _{I_{(x,|x-y_{\mathbb {L}}^j|)}}\big |\lesssim j. \end{aligned}$$

(3.8)

Lemma 2.6, together with John-Nirenberg inequality then gives that

$$\begin{aligned} |A_{\varphi _{\mathbb {L}}}(x)|&\lesssim |x-y_{{\mathbb {L}}}^j|\Big (\frac{1}{|I_{ (x,|x-y_{\mathbb {L}}^j|)}|}\int _{I_{ (x,|x-y_{\mathbb {L}}^j|)}}|\nabla A(z)-\langle \nabla A\rangle _{{\mathbb {L}}}|^qdz\Big )^{1/q}\lesssim j2^s, \end{aligned}$$

(3.9)

which finishes the proof of (i).

Now we give the proof of (ii). For any $x,\,x'\in {\mathbb {R}}^d$ and $y\in {\mathbb {L}}$ with $|x-y|>2|x-x'|$, it is easy to see that

• (1) if $x\notin L_{j,1}$ and $x'\notin L_{j,1}$, then $R_{s,{\mathbb {L}};j}(x,y)=R_{s,{\mathbb {L}};j}(x',y) =0$;

• (2) if $x\notin L_{j,1}$, then $x'\notin 3\cdot 2^{j}d{\mathbb {L}}$, hence $R_{s,{\mathbb {L}};j}(x,y)=R_{s,{\mathbb {L}};j}(x',y) =0$;

• (3) if $x'\notin L_{j,1}$, then $x\notin 3\cdot 2^{j}d{\mathbb {L}}$, hence $R_{s,{\mathbb {L}};j}(x,y)=R_{s,{\mathbb {L}};j}(x',y) =0$.

If $z\in I_{(x,|x-x'|)}$, another application of Lemma 2.6 and John-Nirenberg inequality indicates

$$\begin{aligned} |\nabla A_{\varphi _{\mathbb {L}}}(z)|\lesssim & {} 2^{-s}|A_{{\mathbb {L}}}(z)-A_{{\mathbb {L}}}(y^j_{L})|+|\nabla A(z)-\langle \nabla A\rangle _{{\mathbb {L}}}|\nonumber \\\lesssim & {} j+|\nabla A(z)-\langle \nabla A\rangle _{{\mathbb {L}}}|, \end{aligned}$$

(3.10)

and the similar method as what was used in the proof of (3.8) further implies that

$$\begin{aligned} |\langle \nabla A\rangle _{{\mathbb {L}}}-\langle \nabla A\rangle _{I_{(x,|x-x'|)}}|\le & {} |\langle \nabla A\rangle _{{\mathbb {L}}}-\langle \nabla A\rangle _{I_{(x,2^{s-j})}}|+|\langle \nabla A\rangle _{I_{(x,|x-x'|)}}\langle \nabla A\rangle _{I_{(x,2^{s-j})}}|\nonumber \\\lesssim & {} \log 2^j+\Big |\log \big ({2^{s-j}}{|x-x'|^{-1}}\big )\Big |. \end{aligned}$$

(3.11)

By Lemma 2.6, (3.10) and (3.11), we have

$$\begin{aligned} |A_{\varphi _{{\mathbb {L}}}}(x)-A_{\varphi _{{\mathbb {L}}}}(x')|\lesssim & {} |x-x'|\Big (\frac{1}{|I_{(x,|x-x'|)}|}\int _{I_{(x,|x-x'|)}}|\nabla A_{\varphi _{{\mathbb {L}}}}(z)|^qdz\Big )^{\frac{1}{q}}\nonumber \\\lesssim & {} |x-x'|{\Big (j+\frac{1}{|I_{(x,|x-x'|)}|}\int _{I_{(x,|x-x'|)}}|\nabla A(z)-\langle \nabla A_L\rangle |^q dz\Big )^{\frac{1}{q}}}\nonumber \\ \end{aligned}$$

(3.12)

Similarly, we obtain

$$\begin{aligned} |A_{\varphi _{{\mathbb {L}}}}(x)-A_{\varphi _{{\mathbb {L}}}}(x')|\lesssim & {} |x-x'|\big (j+|\langle \nabla A\rangle _{{\mathbb {L}}}-\langle \nabla A\rangle _{I_{(x,|x-x'|)}}|\big )\nonumber \\\lesssim & {} |x-x'|\Big [j+\Big |\log \big ({2^{s-j}}{|x-x'|^{-1}}\big )\Big |\Big ]. \end{aligned}$$

(3.13)

In a similar way, we have

$$\begin{aligned} |A_{\varphi _{{\mathbb {L}}}}(x)-A_{\varphi _{{\mathbb {L}}}}(y)|\lesssim & {} |x-y|\Big [j+\Big |\log \big ({2^{s-j}}{|x-y|^{-1}}\big )\Big |\Big ];\\ |A_{\varphi _{{\mathbb {L}}}}(x')-A_{\varphi _{{\mathbb {L}}}}(y)|\lesssim & {} |x'-y|\Big [j+\Big |\log \big ({2^{s-j}}{|x'-y|^{-1}}\big )\Big |\Big ]. \end{aligned}$$

Therefore,

$$\begin{aligned} \begin{aligned}&|R_{s,{\mathbb {L}};j}(x,\,y)-R_{s,{\mathbb {L}};j}(x',\,y)| \\ {}&\quad \le |\phi _s(x-y)|\Big |\frac{A_{\varphi _{{\mathbb {L}}}}(x)-A_{\varphi _{{\mathbb {L}}}}(y)}{|x-y|^{d+1}}- \frac{A_{\varphi _{{\mathbb {L}}}}(x')-A_{\varphi _{{\mathbb {L}}}}(y)}{|x'-y|^{d+1}}\Big |\\&\quad +\frac{|A_{\varphi _{{\mathbb {L}}}}(x')-A_{\varphi _{\mathbb {L}}}(y)|}{|x'-y|^{d+1}}\big |\phi _{s}(x-y)-\phi _{s}(x'-y)\big |\\&\lesssim \frac{|x-x'|}{|x-y|^{d+1}}\Big (j+\Big |\log \big ({2^{s-j}}{|x-x'|^{-1}}\big )\Big |\Big ). \end{aligned} \end{aligned}$$

This completes the proof of (ii) in Lemma 3.1. (iii) can be proved in the same way as (ii). $\square$

Let us turn back to the contribution of $D_1.$ It follows from the method of rotation of Calderón-Zygmund that

$$\begin{aligned} \Vert \textrm{D}_{1}\Vert _{L^1({\mathbb {R}}^d)}= & {} \sum \limits _{i=0}^{\infty } \sum _{j_0< j\le Ni} \sum _{s}\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}} \Vert T^i_{\Omega ,A_{{\mathbb {L}}};\,s,j}b_{\mathbb {L}}(x)\Vert _{L^1({\mathbb {R}}^d)}\\\lesssim & {} \sum \limits _{i=0}^{\infty } \sum _{j_0< j\le Ni} \sum _{s}\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}} j \\ {}{} & {} \int _{{\mathbb {R}}^{d} }\int _{2^{s-2}}^{ 2^{s+2}} \int _{{\mathbb {S}}^{d-1}}\frac{|\Omega _i(y')|}{|r|}|b_{\mathbb {L}}(x-ry')|\,dy' \,dr\,dx\\\lesssim & {} \sum _{i=0}^{\infty }\Vert \Omega _i\Vert _{L^1({\mathbb {S}}^{d-1})}\sum _{j_0< j\le Ni}j\sum _{s}\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}}\Vert b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)}\\\lesssim & {} \Vert \Omega \Vert _{L(\log L)^2({\mathbb {S}}^{d-1})}\Vert f\Vert _{L^1({\mathbb {R}}^d)}. \end{aligned}$$

This verifies (3.6).

3.3 Proof of the Inequality (3.7)

The estimate of $\mathrm{D_2}$ is long and complicated. We split the proof into three steps.

Step 1. A reduction for the estimate of $D_2$.

Let $l_{\tau }(j)=\tau j+3$, where $0<\tau <1$ will be chosen later. Let $\omega$ be a nonnegative, radial $C^{\infty }_c({\mathbb {R}}^d)$ function which is supported in $\{x\in {\mathbb {R}}^d:|x|\le 1\}$ and has integral 1. Set $\omega _{t}(x)=2^{-td}\omega (2^{-t}x)$. For $s\in {\mathbb {N}}$ and a cube ${\mathbb {L}}$, we define $R^j_{s,{\mathbb {L}}}$ as

$$\begin{aligned} R^j_{s,{\mathbb {L}}}(x,y)=\int _{{\mathbb {R}}^d}\omega _{s-l_{\tau }( j)}(x-z)\frac{1}{|z-y|^{d+1}}\phi _{s}(z-y) \big (A_{\varphi _{\mathbb {L}}}(z)-A_{\varphi _{\mathbb {L}}}(y)\big )dz. \end{aligned}$$

(3.14)

It is obvious that $\textrm{supp}R_{s,{\mathbb {L}}}^{j}(x,y)\subset \{(x,\,y):\, 2^{s-3}\le |x-y|\le 2^{s+3}\}$. Moreover, if $y\in {\mathbb {L}}$ with ${{\mathbb {L}}\in {\mathcal {S}}_{s-j}}$, then (i) of Lemma 3.1 implies that

$$\begin{aligned} |R_{s,{\mathbb {L}}}^{j}(x,y)|{~\lesssim ~} j2^{-sd}\chi _{\{2^{s-3}\le |x-y|\le 2^{s+3}\}}(x,\,y). \end{aligned}$$

(3.15)

We define the operator $T^{i,j}_{\Omega ,{\mathbb {L}};\,s}$ by

$$\begin{aligned} T^{i,j}_{\Omega ,{\mathbb {L}};\,s}h(x)= \int _{{\mathbb {R}}^d}\Omega _i(x-y)R_{s,{\mathbb {L}}}^{j}(x,y)h(y)dy, \end{aligned}$$

and let $\textrm{D}_{2}^*$ be the operator as follows

$$\begin{aligned} \textrm{D}_{2}^*(x)=\sum \limits _{i=0}^{\infty } \sum _{ j> Ni}\sum _{s}\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}} T^{i,j}_{\Omega ,{\mathbb {L}};\,s}b_{\mathbb {L}}(x). \end{aligned}$$

The following lemma indicates the intrinsically close relationship in each subtract terms between $\textrm{D}_{2}$ and $\textrm{D}_{2}^*$. Thus, the corresponding proof is transferred to verify it for each term of $\textrm{D}_{2}^*$.

Lemma 3.2

Let $\Omega$ be homogeneous of degree zero, A be a function on ${\mathbb {R}}^d$ with derivatives of order one in $\textrm{BMO}({\mathbb {R}}^d)$. For $j>j_0$ and $i\ge 0$, it holds that

$$\begin{aligned} \Vert T^i_{\Omega ,A_{{\mathbb {L}}};\,s,j}b_{\mathbb {L}}- T^{i,j}_{\Omega ,{\mathbb {L}};\,s}b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)}\lesssim j 2^{-\tau j} \Vert \Omega _i\Vert _{L^{1}({\mathbb {S}}^{d-1})}\Vert b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)}. \end{aligned}$$

Proof

For each $y\in {\mathbb {L}}$ and $z\in \textrm{supp}\,\omega _{s-l_{\tau }( j)}$, notice that $R_{s,{\mathbb {L}};j}(x,\,y)-R_{s,{\mathbb {L}};j{\tiny }}(x-z,\,y)=0$ if $x\in L_{j,1}\backslash {3\cdot 2^{j}d{\mathbb {L}}}$. In fact, since $|z|\le 2^{s-\tau j-3}$, then we have $2^{s+1}<|x-y|<3\cdot 2^s$ and $2^s<|x-y-z|<2^{s+2}$.

By Lemma 3.1, we have

$$\begin{aligned} {\big |R_{s,{\mathbb {L}};j}(x,\,y)-R_{s,{\mathbb {L}};j}(x-z,\,y)\big |\lesssim \frac{|z|}{2^{s(d+1)}}\bigg [j+\log \bigg (\frac{2^{s-j}}{|z|}\bigg )\bigg ]\chi _{\{2^{s-2}\le |x-y|\le 2^{s+2}\}}(x,y).} \end{aligned}$$

Observing that the function $\Theta (t)=t\log (\textrm{e}+\frac{1}{t})$ is bounded at $t\in (0,\,1]$, and then for $0<t\le r$,

$$\begin{aligned} t\log \left( \textrm{e}+\frac{r}{t}\right) \lesssim r, \end{aligned}$$

we deduce that

$$\begin{aligned}{} & {} \Big |\int _{{\mathbb {R}}^d}\omega _{s-l_{\tau }( j)}(z)\Big (R_{s,{\mathbb {L}};j}(x,\,y)-R_{s,{\mathbb {L}};j}(x-z,\,y)\Big )dz\Big |\\{} & {} \quad \lesssim 2^{(-s+\tau j)d}\int _{\{|z|\le 2^{s-\tau j}\}}\frac{|z|}{2^{s(d+1)}}\bigg [j+\log \bigg (\frac{2^{s-\tau j}}{|z|}\bigg )\bigg ]dz\lesssim j2^{-sd-\tau j}. \end{aligned}$$

Therefore

$$\begin{aligned}&\Vert T^i_{\Omega ,A_{{\mathbb {L}}};\,s,j}b_{\mathbb {L}}-T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s}b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)}\\&\quad \le \int _{{\mathbb {R}}^{d}}\int _{{\mathbb {R}}^{d}}|\Omega _i(x-y)|\Big |\int _{{\mathbb {R}}^d}\omega _{s-l_{\tau }( j)}(z)\Big (R_{s,{\mathbb {L}};j}(x,\,y)\\&\qquad -R_{s,{\mathbb {L}};j}(x-z,\,y)\Big )dz\Big ||b_{\mathbb {L}}(y)|dydx\\&\quad \lesssim j2^{-sd-\tau j}\int _{{\mathbb {R}}^{d}}\int _{{\mathbb {R}}^{d}}|\Omega _i(y)|\chi _{\{2^{s-2}\le |y|\le 2^{s+2}\}}(y)|b_{\mathbb {L}}(x-y)|dydx\\&\quad {\lesssim {j2^{-\tau j}} \Vert \Omega _i\Vert _{L^{1}({\mathbb {S}}^{d-1})}\Vert b_{{\mathbb {L}}}\Vert _{L^1({\mathbb {R}}^d)}}. \end{aligned}$$

This leads to the desired conclusion of Lemma 3.2. $\square$

With Lemma 3.2 in hand, we only need to estimate $\textrm{D}_{2}^*$. This is the content of the second step.

Step 2. Estimate for each term of $\textrm{D}_{2}^*$.

Define $P_tf(x)=\omega _t*f(x)$. Now we split

$$\begin{aligned} T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s}=P_{s-j\kappa }T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s}+(I-P_{s-j\kappa })T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s}, \end{aligned}$$

where $\kappa \in (0,1)$ will be chosen later. In the following, we will estimate this two terms one by one. We have the following norm inequality for $P_{s-j\kappa }T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s}$.

Lemma 3.3

Let $\Omega$ be homogeneous of degree zero, A be a function in ${\mathbb {R}}^d$ with derivatives of order one in ${\textrm{BMO}}({\mathbb {R}}^d)$, $b_{\mathbb {L}}$ satisfies the vanishing moment with $\ell ({\mathbb {L}})=2^{s-j}$. For each $j\in {\mathbb {N}}$ with $j>j_0$, we have

$$\begin{aligned}{} & {} \big \Vert P_{s-j\kappa } T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s}b_{\mathbb {L}}\big \Vert _{L^1({\mathbb {R}}^d)}\\{} & {} \quad \lesssim j\big (2^{-(1-\kappa )j} +2^{-(1-\tau )j} \big ) \Vert \Omega _i\Vert _{L^{\infty }({\mathbb {S}}^{d-1})}\Vert b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)}. \end{aligned}$$

Before proving Lemma 3.3, we need the following lemma for $R_{s,{\mathbb {L}}}^{j}$.

Lemma 3.4

Let $R^j_{s,{\mathbb {L}}}$ be defined as (3.14), $\theta \in {\mathbb {S}}^{d-1}$, $y,\,y'\in {\mathbb {L}}$ with $\ell ({\mathbb {L}})=2^{s-j}$. Then

$$\begin{aligned}{} & {} \int _{{\mathbb {L}}}\int _{{\mathbb {L}}}|R_{s,{\mathbb {L}}}^{j}(y+r\theta ,\,y)-R_{s,{\mathbb {L}}}^{j}(y'+r\theta ,\,y')||b_{{\mathbb {L}}}(y)|dydy'\\{} & {} \quad \lesssim j 2^{-sd}2^{\tau j}2^{-j}|{\mathbb {L}}|\int _{{\mathbb {L}}}|b_{\mathbb {L}}(y)|dy. \end{aligned}$$

Proof

By the triangle inequality, the mean value theorem and the support condition of $\phi$, we get

$$\begin{aligned}{} & {} |R^j_{s,{\mathbb {L}}}(y'+r\theta ,\,y)-R^j_{s,{\mathbb {L}}}(y'+r\theta ,\,y')|\\{} & {} \quad \lesssim \int _{{\mathbb {R}}^d}|\omega _{s-l_{\tau }( j)}(y'+r\theta -z)||\phi _s(z-y')| \frac{|A_{\varphi _{{\mathbb {L}}}}(y)-A_{\varphi _{{\mathbb {L}}}}(y')|}{|z-y'|^{d+1}}dz\\{} & {} \qquad +\int _{{\mathbb {R}}^d}|\omega _{s-l_{\tau }( j)}(y'+r\theta -z)| \frac{|A_{\varphi _{{\mathbb {L}}}}(z)-A_{\varphi _{{\mathbb {L}}}}(y)|}{|z-y|^{d+1}}|\phi _{s}(z-y)-\phi _{s}(z-y')|dz\\{} & {} \qquad +\int _{{\mathbb {R}}^d}|\omega _{s-l_{\tau }( j)}(y'+r\theta -z)||\phi _s(z-y')| \frac{|A_{\varphi _{\mathbb {L}}}(z)-A_{\varphi _{{\mathbb {L}}}}(y)||y-y'|}{|z-y|^{d+2}}dz\\{} & {} \quad =:\mathrm{I+II+III}. \end{aligned}$$

If $r\notin [2^{s-4}, 2^{s+4}]$, by the support of $R^j_{s,{\mathbb {L}}}$, it gives that $|R^j_{s,{\mathbb {L}}}(y'+r\theta ,\,y)-R^j_{s,{\mathbb {L}}}(y'+r\theta ,\,y')|=0$.

For $y,\,y'\in {\mathbb {L}}$, (3.13) gives us that

$$\begin{aligned} |A_{\varphi _{{\mathbb {L}}}}(y')-A_{\varphi _{{\mathbb {L}}}}(y)|\lesssim |y-y'|\Bigg [j+\Bigg |\log \bigg (\frac{2^{s-j}}{|y-y'|}\bigg )\Bigg |\Bigg ]. \end{aligned}$$

For $\textrm{I}$, since $|z-y'|\ge 2^{s-2}$, $y,\,y'\in {\mathbb {L}}$, then (3.13) gives us that

$$\begin{aligned} \textrm{I}\lesssim |y-y'|\Bigg [j+\Bigg |\log \bigg (\frac{2^{s-j}}{|y-y'|}\bigg )\Bigg |\Bigg ] 2^{-s(d+1)}. \end{aligned}$$

Consider now the other two terms. If $y,\,y'\in {\mathbb {L}}$ and $|z-y'|\le 2^{s}$, (i) of Lemma 3.1 gives us that

$$\begin{aligned} |A_{\varphi _{{\mathbb {L}}}}(y)|\lesssim j2^s,\,\,\,|A_{\varphi _{{\mathbb {L}}}}(z)|\lesssim j2^s. \end{aligned}$$

On the other hand, for $j>j_0$, when $y,\,y'\in {\mathbb {L}}$ and $|z-y'|\ge 2^{s-2}$, it holds that

$$\begin{aligned} |z-y|\ge |z-y'|-|y-y'|\ge 2^{s-2}-\sqrt{d}2^{s-j}>2^{s-2}-\sqrt{d}2^{s-\log _2(100d/2)} >2^{s-3}. \end{aligned}$$

Therefore,

$$\begin{aligned} {\textrm{II}}\lesssim & {} \frac{j2^s}{(2^s)^{d+1}}\int _{{\mathbb {R}}^d}|\omega _{s-l_{\tau }( j)}(y'+r\theta -z)| |\phi _{s}(z-y)-\phi _{s}(z-y')|dz\\\lesssim & {} j2^{-s(d+1)}|y-y'|\int _{{\mathbb {R}}^d}|\omega _{s-l_{\tau }( j)}(y'+r\theta -z)| dz\lesssim j2^{-s(d+1)}|y-y'|, \end{aligned}$$

where the second inequality follows from the fact that

$$\begin{aligned} |\phi _{s}(z-y)-\phi _{s}(z-y')|\lesssim \frac{|y-y'|}{2^s}\Vert \nabla \phi \Vert _{L^{\infty }({\mathbb {R}}^d)}\lesssim \frac{|y-y'|}{2^s}. \end{aligned}$$

Similarly, we have

$$\begin{aligned} \textrm{III}\lesssim j2^{-s(d+1)}|y-y'|. \end{aligned}$$

Estimates for $\textrm{I}$, $\textrm{II}$ and $\textrm{III}$ above lead to that

$$\begin{aligned} |R^j_{s,{\mathbb {L}}}(y'+r\theta ,\,y)-R^j_{s,{\mathbb {L}}}(y'+r\theta ,\,y')|\lesssim \frac{ j|y-y'|}{2^{s(d+1)}}\Bigg [1+\Bigg |\log \bigg (\frac{2^{s-j}}{|y-y'|}\bigg )\Bigg |\Bigg ]. \end{aligned}$$

(3.16)

Similar to (3.16), we also have

$$\begin{aligned}&|R^j_{s,{\mathbb {L}}}(y+r\theta ,\,y)-R^j_{s,{\mathbb {L}}}(y'+r\theta ,\,y)|\nonumber \\&\quad \le \int _{{\mathbb {R}}^d}|\omega _{s-l_{\tau }( j)}(y+r\theta -z)-\omega _{s-l_{\tau }( j)}(y'+r\theta -z)||\phi _{s}(z-y)|\nonumber \\&\qquad \frac{|A_{\varphi _{{\mathbb {L}}}}(z)- A_{\varphi _{{\mathbb {L}}}}(y)|}{|z-y|^{d+1}}dz\nonumber \\&\quad \le j2^{-sd} 2^{-s+l_{\tau }( j)}|y-y'|\int _{{\mathbb {R}}^d}|\nabla \omega _{s-l_{\tau }( j)}(z)|dz \nonumber \\&\quad \lesssim j |y-y'|2^{-s+l_{\tau }( j)}2^{-sd}. \end{aligned}$$

(3.17)

Notice that

$$\begin{aligned}&\int _{{\mathbb {L}}}\int _{{\mathbb {L}}} |y-y' |\Big [1+\Big |\log \Big (\frac{2^{s-j}}{|y-y'|}\Big )\Big |\Big ]dy' |b_{{\mathbb {L}}}(y) |dy\le 2^{s-j}|{\mathbb {L}}|\int _{{\mathbb {L}}}|b_{\mathbb {L}}(y)|dy. \end{aligned}$$

Combining (3.16) with (3.17), it gives that

$$\begin{aligned}{} & {} \int _{{\mathbb {L}}}\int _{{\mathbb {L}}}|R_{s,{\mathbb {L}}}^{j}(y+r\theta ,\,y)-R_{s,{\mathbb {L}}}^j(y'+r\theta ,\,y') | |b_{{\mathbb {L}}}(y) |dydy'\\{} & {} \quad \lesssim j 2^{-s(d+1)}2^{l_{\tau }( j)}\int _{{\mathbb {L}}}\int _{{\mathbb {L}}}|y-y'|dy'|b_{{\mathbb {L}}}(y) |dy\\{} & {} \qquad +j 2^{-s(d+1)}\int _{{\mathbb {L}}}\int _{{\mathbb {L}}} |y-y' |\Big [1+\Big |\log \Big (\frac{2^{s-j}}{|y-y'|}\Big )\Big |\Big ]dy' |b_{{\mathbb {L}}}(y) |dy\\{} & {} \quad \lesssim j 2^{-sd}2^{l_{\tau }( j)}2^{-j}|{\mathbb {L}}|\int _{{\mathbb {L}}}|b_{\mathbb {L}}(y)|dy. \end{aligned}$$

This finishes the proof of Lemma 3.4. $\square$

With Lemma 3.4, we are ready to prove Lemma 3.3 now.

Proof of Lemma 3.3

Write

$$\begin{aligned} P_{s-j\kappa } T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s}b_{\mathbb {L}} (x)= \int _{{\mathbb {R}}^d}\Big (\int _{{\mathbb {R}}^d} \omega _{s-j\kappa }(x-z)\Omega _i(z-y) R_{s,{\mathbb {L}}}^{j}(z,y)dz \Big )b_{\mathbb {L}}(y)dy. \end{aligned}$$

Let $z-y=r\theta$. By Fubini’s theorem, $P_{s-j\kappa } T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s}b_{\mathbb {L}} (x)$ can be written as

$$\begin{aligned} \int _{{\mathbb {S}}^{d-1}} \int _{{\mathbb {R}}^d}\int _0^{\infty }\Omega _i(\theta )\omega _{s-j\kappa }(x-y-r\theta ) R_{s,{\mathbb {L}}}^{j}(y+r\theta ,y)r^{d-1} b_{\mathbb {L}}(y)drdyd\sigma _{\theta }. \end{aligned}$$

Let $y' \in {\mathbb {L}}$. By the vanishing moment of $b_{\mathbb {L}}$, we have

$$\begin{aligned}&|P_{s-j\kappa } T^{i,j}_{\Omega _i,{{\mathbb {L}}};\,s}b_L (x)|\\&\quad \le \inf \limits _{y'\in {\mathbb {L}}} \int _{{\mathbb {S}}^{d-1}}| \Omega _i(\theta )|\Big |\int _{{\mathbb {R}}^d}\int _0^{\infty } \Big (\omega _{s-j\kappa }(x-y-r\theta ) R_{s,{\mathbb {L}}}^{j}(y+r\theta ,y)\\&\qquad -\omega _{s-j\kappa }(x-y'-r\theta ) R_{s,{\mathbb {L}}}^{j}(y'+r\theta ,y')\Big ) r^{d-1}dr b_{\mathbb {L}}(y)dy\Big |d\sigma _{\theta }\\&\quad \le \int _{{\mathbb {S}}^{d-1}}| \Omega _i(\theta )|\frac{1}{|{\mathbb {L}}|}\int _{{\mathbb {L}}}\Big |\int _{{\mathbb {R}}^d}\int _0^{\infty } \Big (\omega _{s-j\kappa }(x-y-r\theta ) R_{s,{\mathbb {L}}}^{j}(y+r\theta ,y)\\&\qquad -\omega _{s-j\kappa }(x-y'-r\theta ) R_{s,{\mathbb {L}}}^{j}(y'+r\theta ,y')\Big ) r^{d-1}dr b_{\mathbb {L}}(y)dy\Big |d{y'}d\sigma _{\theta } \\&\quad \lesssim \mathrm{I+II}, \end{aligned}$$

where

$$\begin{aligned} \textrm{I}=:&\frac{1}{|{\mathbb {L}}|}\int _{{\mathbb {S}}^{d-1}} |\Omega _i(\theta )|\int _{{\mathbb {L}}} \Big |\int _{{\mathbb {R}}^d}\int _0^{\infty }\Big (\omega _{s-j\kappa }(x-y-r\theta ) -\omega _{s-j\kappa }(x-y'-r\theta ) \Big )\\&\times R_{s,{\mathbb {L}}}^{j}(y+r\theta ,y)r^{d-1}dr b_{\mathbb {L}}(y)dy \Big | d{y'}d\sigma _{\theta }, \end{aligned}$$

and

$$\begin{aligned} \textrm{II}=:&\frac{1}{|{\mathbb {L}}|}\int _{{\mathbb {S}}^{d-1}} |\Omega _i(\theta )|\int _{{\mathbb {L}}} \Big | \int _{{\mathbb {R}}^d}\int _0^{\infty }\omega _{s-j\kappa }(x-y'-r\theta ) \big (R_{s,{\mathbb {L}}}^{j}(y+r\theta ,y) \\&-R_{s,{\mathbb {L}}}^{j}(y'+r\theta ,y')\big )r^{d-1}dr b_{\mathbb {L}}(y)dy\Big |d{y'}\,d\sigma _{\theta }. \end{aligned}$$

Note that $|y-y'|\lesssim 2^{s-j}$, when y, $y'\in {\mathbb {L}}$. By (3.15) and the mean value formula, it follows that

$$\begin{aligned} \Vert I\Vert _{L^1({\mathbb {R}}^d)}&\lesssim j\int _{{\mathbb {S}}^{d-1}} |\Omega _i(\theta )|\\ {}&\quad \int _{{\mathbb {R}}^d}\int _{2^{s-3}}^{2^{s+3}} 2^{-s+j\kappa } \Vert \triangledown \omega \Vert _{L^1({\mathbb {R}}^d)} 2^{s-j} 2^{-sd}r^{d-1}dr |b_{\mathbb {L}}(y)|dyd\sigma ({\theta })\\&\lesssim j 2^{-(1-\kappa )j} \Vert \Omega _i\Vert _{L^{\infty }({\mathbb {S}}^{d-1})} \Vert b_{{\mathbb {L}}}\Vert _{L^1({\mathbb {R}}^d)}. \end{aligned}$$

By Lemma 3.4 and the Fubini’s theorem one can get

$$\begin{aligned} \Vert II\Vert _{L^1({\mathbb {R}}^d)}&\lesssim \int _{{\mathbb {S}}^{d-1}} \int _{2^{s-3}}^{2^{s+3}} |\Omega _i(\theta )|\frac{1}{|{\mathbb {L}}|}\int _{{\mathbb {L}}} \int _{{\mathbb {L}}}\Vert \omega _{s-j\kappa }(\cdot -y'-r\theta )\Vert _{ L^1({\mathbb {R}}^d) } \\&\quad \times |\big (R_{s,{\mathbb {L}}}^{j}(y+r\theta ,y) \\&\quad -R_{s,{\mathbb {L}}}^{j}(y'+r\theta ,y')\big )| |b_{\mathbb {L}}(y)|dyd{y'}\,r^{d-1}dr d\sigma _{\theta }\\&\lesssim j2^{-{(1-\tau )}j} \Vert \Omega _i\Vert _{L^{\infty }({\mathbb {S}}^{d-1})} \Vert b_{{\mathbb {L}}}\Vert _{L^1({\mathbb {R}}^d)}. \end{aligned}$$

This finishes the proof of Lemma 3.3. $\square$

To estimate the term $(I-P_{s-j\kappa })T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s}$, we introduce a partition of unity on the unit surface ${\mathbb {S}}^{d-1}$. For $j>j_0$, let ${\mathfrak {E}}^j=\{e_{\nu }^j\}$ be a collection of unit vectors on ${\mathbb {S}}^{d-1}$ such that

• (a) for $\nu \not =\nu '$, $|e_{\nu }^j-e_{\nu '}^j|>2^{-j\gamma -4}$;

• (b) for each $\theta \in {\mathbb {S}}^{d-1}$, there exists an $e^j_{\nu }$ satisfying that $|e^j_{\nu }-\theta |\le 2^{-j\gamma -4},$ where $\gamma \in (0,\,1)$ is a constant.

The set ${\mathfrak {E}}^j$ can be constructed as in [31]. Observe that $\textrm{card}({\mathfrak {E}}^j)\lesssim 2^{j\gamma (d-1)}$.

Below, we will construct an associated partition of unity on the unit surface ${\mathbb {S}}^{d-1}$. Let $\zeta$ be a smooth, nonnegative, radial function with $\zeta (u)\equiv 1$ when $|u|\le 1/2$ and $\textrm{supp}\, \zeta \subset \{|x|\le 1\}$. Set

$$\begin{aligned} {\widetilde{\Gamma }}_\nu ^j(\xi )=\zeta \Bigg (2^{j\gamma }\Bigg (\frac{\xi }{|\xi |}-e^j_{\nu }\Bigg )\Bigg ), \ \text{ and } \Gamma _{\nu }^j(\xi )={\widetilde{\Gamma }}^j_{\nu }(\xi )\Bigg (\sum _{e^j_{\nu }\in {\mathfrak {E}}^j}{\widetilde{\Gamma }}^j_{\nu }(\xi )\Bigg )^{-1}. \end{aligned}$$

It is easy to verify that $\Gamma ^j_{\nu }$ is homogeneous of degree zero, and for all j and $\xi \in {\mathbb {S}}^{d-1}$, $\sum _{\nu }\Gamma ^j_{\nu }(\xi )=1.$ Let ${\widetilde{\psi }}\in C^{\infty }_c({\mathbb {R}})$ such that $0\le {\widetilde{\psi }}\le 1$, $\textrm{supp}\, {\widetilde{\psi }}\subset [-4,\,4]$ and ${\widetilde{\psi }}(t)\equiv 1$ when $t\in [-2,\,2]$. Define the multiplier operator $G_{\nu }^j$ by

$$\begin{aligned} \widehat{G_{\nu }^jf}(\xi )={\widetilde{\psi }}\big (2^{j\gamma }\langle \xi /|\xi |, e_{\nu }^j\rangle \big ){\widehat{f}}(\xi ). \end{aligned}$$

Denote the operator $T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }$ by

$$\begin{aligned} T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }h(x)= \int _{{\mathbb {R}}^d}\Omega _i(x-y)\Gamma ^j_{\nu }(x-y)R_{s,{\mathbb {L}}}^{j}(x,y)h(y)dy. \end{aligned}$$

(3.18)

It is obvious that $T^{i,j}_{\Omega ,{\mathbb {L}};\,s}h(x)=\sum _{\nu }T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }h(x).$ For each fixed $i,\,s,\,j,\,{\mathbb {L}}$ and $\nu$, $(I-P_{s-j\kappa })T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }$ can be decomposed in the following way

$$\begin{aligned} (I-P_{s-j\kappa })T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }=G_{\nu }^j(I-P_{s-j\kappa })T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }+(1-G_{\nu }^j)(I-P_{s-j\kappa })T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }. \end{aligned}$$

Estimate for the term $G_{\nu }^j(I-P_{s-j\kappa })T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }$.

For the term $G_{\nu }^j(I-P_{s-j\kappa })T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }$, we have the following lemma.

Lemma 3.5

Let $\Omega$ be homogeneous of degree zero, A be a function in ${\mathbb {R}}^d$ with derivatives of order one in $\textrm{BMO}({\mathbb {R}}^d)$. For each $j\in {\mathbb {N}}$ with $j> j_0$, we have that,

$$\begin{aligned}&\Big \Vert \sum _{\nu } \sum _s\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}}G_{\nu }^j(I-P_{s-j\kappa })T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }b_{\mathbb {L}}\Big \Vert _{L^2({\mathbb {R}}^d)}^2\\&\quad \lesssim j^22^{-j\gamma }\Vert \Omega _i\Vert _{L^{\infty }({\mathbb {S}}^{d-1})}^2{\sum _s\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}}\Vert b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)}.} \end{aligned}$$

Proof

The proof is similar to the proof of Lemma 2.3 in [9]. For the sake of self-contained, we present the proof here. Observe that

$$\begin{aligned} \sup _{\xi \not =0}\sum _{\nu }|{\widetilde{\psi }}(2^{j\gamma }\langle e^j_{\nu },\xi /|\xi |\rangle )|^2\lesssim 2^{j\gamma (d-2)}. \end{aligned}$$

This, together with Plancherel’s theorem and Cauchy-Schwartz inequality, leads to that

$$\begin{aligned}{} & {} \Big \Vert \sum _{\nu } \sum _s\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}} G_{\nu }^j(I-P_{s-j\kappa })T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }b_{\mathbb {L}}\Big \Vert _{L^2({\mathbb {R}}^d)}^2\\{} & {} \quad =\Big \Vert \sum _{\nu }{\widetilde{\psi }}\Big (2^{j\gamma }\langle e^j_{\nu },\,\xi /|\xi |\rangle \Big ){\mathcal {F}}\Big (\sum _s\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}}(I-P_{s-j\kappa })T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }b_{\mathbb {L}}\Big )(\xi )\Big \Vert ^2_{L^2({\mathbb {R}}^d)}\\{} & {} \quad \lesssim 2^{j\gamma (d-2)}\sum _{\nu }\Big \Vert \sum _s\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}}(I-P_{s-j\kappa })T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }b_{\mathbb {L}}\Big \Vert _{L^2({\mathbb {R}}^d)}^2. \end{aligned}$$

Applying (3.15), we see that for each fixed s, ${\mathbb {L}}$, and $x\in {\mathbb {R}}^d$,

$$\begin{aligned}&\big | (I-P_{s-j\kappa })T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }b_{\mathbb {L}}(x)\big | \nonumber \\&\quad \lesssim \int _{{\mathbb {R}}^d}|\Omega _i(x-y)||\Gamma ^j_{\nu }(x-y)||R_{s,{\mathbb {L}}}^{j}(x,y)||b_{\mathbb {L}}(y)|dy\nonumber \\&\qquad + \int _{{\mathbb {R}}^{d}}\int _{{\mathbb {R}}^{d}} |\Omega _i(z-y)||\omega _{s-j\kappa }(x-z)| |\Gamma ^j_{\nu }(z-y)||R_{s,{\mathbb {L}}}^{j}(z,y)|dz|b_{\mathbb {L}}(y)|dy\nonumber \\&\quad \lesssim j \Vert \Omega _i\Vert _{L^{\infty }({\mathbb {S}}^{d-1})}H_{s,\nu }^j*|b_{\mathbb {L}}|(x), \end{aligned}$$

(3.19)

where $H_{s,\nu }^j(x)=2^{-sd}\chi _{{\mathcal {R}}_{s\nu }^j}(x)$, and ${\mathcal {R}}_{s\nu }^j=\{x\in {\mathbb {R}}^d:\,|\langle x,e_{\nu }^j \rangle |\le 2^{s+3},\,|x-\langle x,e_{\nu }^j\rangle e_{\nu }^j|\le 2^{s+3-j\gamma }\}.$ This means that ${\mathcal {R}}_{s\nu }^j$ is contained in a box having one long side of length $\lesssim 2^s$ and $(d-1)$ short sides of length $\lesssim 2^{s-j\gamma }$. Therefore, we have

$$\begin{aligned}{} & {} \Big \Vert \sum _s\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}}(I-P_{s-j\kappa })T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }b_{\mathbb {L}}\Big \Vert _{L^2({\mathbb {R}}^d)}^2\\{} & {} \quad \lesssim j^2\Vert \Omega _i\Vert ^2_{L^{\infty }({\mathbb {S}}^{d-1})}\sum _s\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}}\sum _{I\in {\mathcal {S}}_{s-j}} \int _{{\mathbb {R}}^d}\big (H_{s,\nu }^j*H_{s,\nu }^j*|b_I|\big )(x)|b_{\mathbb {L}}(x)|dx\\{} & {} \qquad +2j^2\Vert \Omega _i\Vert ^2_{L^{\infty }({\mathbb {S}}^{d-1})}\sum _s\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}}\sum _{i<s}\sum _{I\in {\mathcal {S}}_{i-j}}\int _{{\mathbb {R}}^d}\big (H_{s,\nu }^j*H_{i,\nu }^j* |b_I|\big )(x)|b_{\mathbb {L}}(x)|dx. \end{aligned}$$

Let $\widetilde{{\mathcal {R}}}_{s\nu }^j={{\mathcal {R}}}_{s\nu }^j+{{\mathcal {R}}}_{s\nu }^j$. As in the proof of Lemma 2.3 in [9], for each fixed ${\mathbb {L}}\in {\mathcal {S}}_{s-j}$, $x\in {\mathbb {L}}$, $\nu$ and s, we obtain

$$\begin{aligned} \sum _{i\le s}\sum _{I\in {\mathcal {S}}_{i-j}}H_{s,\nu }^j*H_{i,\nu }^j*|b_I|(x)\lesssim & {} { 2^{-j\gamma (d-1)}2^{-sd}\sum _{i\le s}\sum _{I\in {\mathcal {S}}_{i-j}}{\int _{x+\widetilde{{\mathcal {R}}}_{s\nu }^j}}|b_I(y)|dy} \\\lesssim & {} 2^{-2j\gamma (d-1)}, \end{aligned}$$

where we have used the fact that $\int _{{\mathbb {R}}^d} |b_I(y)|dy\lesssim |I|$ and the cubes $I\in {\mathcal {S}}$ are pairwise disjoint.

This, in turn, implies further that

$$\begin{aligned} \Big \Vert \sum _s\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}}(I-P_{s-j\kappa })T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }b_{\mathbb {L}}\Big \Vert _{L^2({\mathbb {R}}^d)}^2\lesssim & {} j^2\Vert \Omega _i\Vert _{L^{\infty }({\mathbb {S}}^{d-1})}^22^{-2j\gamma (d-1)}\\{} & {} \quad \times {\sum _s\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}}\Vert b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)}.} \end{aligned}$$

which finishes the proof of Lemma 3.5. $\square$

Estimate for the term $(I-G_{\nu }^j)$$(I-P_{s-j\kappa })T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }.$

We need to present a lemma for $(I-G_{\nu }^j)$$(I-P_{s-j\kappa })T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }$.

Lemma 3.6

Let $\Omega$ be homogeneous of degree zero, A be a function in ${\mathbb {R}}^d$ with derivatives of order one in $\textrm{BMO}({\mathbb {R}}^d)$. For each $j\in {\mathbb {N}}$ with $j> j_0$, $\ell ({\mathbb {L}})=2^{s-j}$, some $s_0>0$, we have that

$$\begin{aligned} \sum _{\nu } \Vert (I-{G_{\nu }^j)}(I-P_{s-j\kappa } )T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)}\lesssim j 2^{-s_0 j} \Vert \Omega _i\Vert _{L^{\infty }({\mathbb {S}}^{d-1})} \Vert b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)}. \end{aligned}$$

Next we give the estimate of $\textrm{D}_{2}^*$ and postpone the proof of Lemma 3.6 later.

Let $\varepsilon =\min \{ (1-\kappa ), (1-\tau ), s_0,\gamma \}$. With Lemma 3.3, Lemma 3.5 and Lemma 3.6, we have

$$\begin{aligned} \Big |\Big \{x\in {\mathbb {R}}^d:\, |\textrm{D}_{2}^*|>1/16\Big \}\Big |\lesssim & {} \sum \limits _{i=0}^{\infty } \sum _{ j> Ni} j^22^{-j\varepsilon }\Vert \Omega _i\Vert ^2_{L^{\infty }({\mathbb {S}}^{d-1})}\Big \Vert \sum _s\sum _{{\mathbb {L}}\in {\mathcal {S}}_{s-j}}b_{\mathbb {L}}\Big \Vert _{L^1({\mathbb {R}}^d)}\nonumber \\\lesssim & {} \Vert f\Vert _{L^1({\mathbb {R}}^d)}. \end{aligned}$$

(3.20)

The proof of Lemma 3.6 is similar to the proof of Lemma 2.4 in [9]. For the completeness of this paper, we give the proof for the remaining term $(1-G_{\nu }^j)(I-P_{s-j\kappa })T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }$ here. Let’s introduce the Littlewood-Paley decomposition first. Let $\alpha$ be a radial $C^{\infty }$ function such that $\alpha (\xi )=1$ for $|\xi | \le 1$, $\alpha (\xi )=0$ for $|\xi | \ge 2$ and $0\le \alpha (\xi )\le 1$ for all $\xi \in {\mathbb {R}}^d$. Define $\beta _k(\xi )=\alpha (2^k\xi )-\alpha (2^{k+1}\xi )$. Choose ${\tilde{\beta }}$ be a radial $C^{\infty }$ function such that ${\tilde{\beta }}(\xi )=1$ for $1/2\le |\xi | \le 2$, ${\textrm{supp}} \,{\tilde{\beta }} \in [1/4,4]$ and $0\le {\tilde{\beta }}\le 1$ for all $\xi \in {\mathbb {R}}^d$. Set ${\tilde{\beta }}_k(\xi )={\tilde{\beta }}(2^k\xi )$, then it is easy to see $\beta _k={\tilde{\beta }}_k{\beta }_k$. Define the convolution operators $\Lambda _k$ and ${\tilde{\Lambda }} _k$ with Fourier multipliers $\beta _k$ and $\tilde{\beta _k}$, respectively.

$$\begin{aligned} \widehat{\Lambda _k f}(\xi )= \beta _k(\xi ){\widehat{f}}(\xi ),\qquad \widehat{ {\tilde{\Lambda }} _k f}(\xi )={\tilde{\beta }} _k(\xi ){\widehat{f}}(\xi ). \end{aligned}$$

It is easy to have $\Lambda _k={\tilde{\Lambda }} _k \Lambda _k$.

Proof of Lemma 3.6

We first write $(I-{G_{\nu }^j)}T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }= \sum \limits _{k}(I-{G_{\nu }^j)} \Lambda _k T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }$. Then

$$\begin{aligned}&\Vert (I-{G_{\nu }^j)}(I-P_{s-j\kappa } )\Lambda _kT^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)}\\&\quad \le \Vert (I-P_{s-j\kappa } ){\tilde{\Lambda }} _k (I-{G_{\nu }^j)}\Lambda _kT^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)}\\&\quad \le \Vert (I-P_{s-j\kappa } ){\tilde{\Lambda }} _k\Vert _{L^1({\mathbb {R}}^d)\rightarrow L^1({\mathbb {R}}^d)} \Vert (I-{G_{\nu }^j)}\Lambda _kT^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)}. \end{aligned}$$

We can write

$$\begin{aligned} (I-{G_{\nu }^j)}\Lambda _kT^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }b_{\mathbb {L}}(x)&=\int _{\mathbb {L}} (I-{G_{\nu }^j)}\Lambda _k \Omega _i(x-y)\Gamma ^j_{\nu }(x-y)R_{s,{\mathbb {L}}}^{j}(x,y) b_{\mathbb {L}}(y)dy\\&:= \int _{\mathbb {L}} M_k(x,y)b_{\mathbb {L}}(y)dy, \end{aligned}$$

where $M_k$ is the kernel of the operator $(I-{G_{\nu }^j)}\Lambda _kT^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }$. Then

$$\begin{aligned} \Vert (I-{G_{\nu }^j)}\Lambda _kT^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)}\le \int _{\mathbb {L}} \Vert M_k(\cdot ,y)\Vert _{L^1({\mathbb {R}}^d)} {|b_{\mathbb {L}}(y)|}dy. \end{aligned}$$

Applying the method of Lemma 4.2 in [9], there exists $M>0$ such that

$$\begin{aligned} \Vert M_k(\cdot ,y)\Vert _{L^1({\mathbb {R}}^d)}\lesssim j 2^{\tau j-j\gamma (d-1)-s+k+j\gamma (1+2M)}{\Vert \Omega _{i}\Vert _{L^{\infty }({\mathbb {S}}^{d-1})}}. \end{aligned}$$

Hence, note that $\Vert (I-P_{s-j\kappa } ){\tilde{\Lambda }} _k\Vert _{L^1({\mathbb {R}}^d)\rightarrow L^1({\mathbb {R}}^d)} \le \Vert {\mathcal {F}}^{-1}( {\tilde{\beta }}_k)-\omega _{s-j\kappa }*{\mathcal {F}}^{-1}( {\tilde{\beta }}_k) \Vert _{L^1({\mathbb {R}}^d)}\lesssim 1$, we have

$$\begin{aligned}&\Vert (I-{G_{\nu }^j)}(I-P_{s-j\kappa } )\Lambda _kT^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)}\nonumber \\&\quad \lesssim j2^{\tau j-j\gamma (d-1)-s+k+j\gamma (1+2M)}{\Vert \Omega _{i}\Vert _{L^{\infty }({\mathbb {S}}^{d-1})}} \Vert b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)}. \end{aligned}$$

(3.21)

On the other hand, we can write

$$\begin{aligned}&\Vert (I-{G_{\nu }^j)}(I-P_{s-j\kappa } )\Lambda _kT^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)}\\&\quad \le \Vert (I-P_{s-j\kappa } ){\tilde{\Lambda }} _k\Vert _{L^1({\mathbb {R}}^d)\rightarrow L^1({\mathbb {R}}^d)} \Vert (I-{G_{\nu }^j)}\Lambda _k\Vert _{L^1({\mathbb {R}}^d)\rightarrow L^1({\mathbb {R}}^d)}\Vert T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)}. \end{aligned}$$

By (3.18), it is easy to show that

$$\begin{aligned} \Vert T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)}\lesssim j2^{-j\gamma (d-1)}\Vert \Omega _i\Vert _{L^{\infty }({\mathbb {S}}^{d-1})}\Vert b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)}. \end{aligned}$$

Let $W_{k,s,\,\kappa }^j$ be the kernel of $(I-P_{s-j\kappa } ){\tilde{\Lambda }} _k$, then by the mean value formula, we obtain

$$\begin{aligned} \int _{{\mathbb {R}}^d}|W_{k,s,\kappa }^j(y)|dy\le & {} \int _{{\mathbb {R}}^{d}} \int _{{\mathbb {R}}^{d}} \big |{\mathcal {F}}^{-1}{{\tilde{\beta }}}_k(y) -{\mathcal {F}}^{-1}{{\tilde{\beta }}}_k(y-z) \big |\omega _{s-j\kappa }(z)dzdy\nonumber \\ {}{} & {} \lesssim 2^{s-j\kappa -k}. \end{aligned}$$

(3.22)

By the proof of [26, Lemma 3.2], it holds that $\Vert (I-{G_{\nu }^j)}\Lambda _k\Vert {_{L^1({\mathbb {R}}^d)\rightarrow L^1({\mathbb {R}}^d)}}\lesssim 1$. Hence

$$\begin{aligned}&\Vert (I-{G_{\nu }^j)}(I-P_{s-j\kappa } )\Lambda _kT^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }b_{\mathbb {L}} \Vert _{L^1({\mathbb {R}}^d)} \nonumber \\&\quad \lesssim j 2^{-j\gamma (d-1)+s-k-j\kappa } {\Vert \Omega _{i}\Vert _{L^{\infty }({\mathbb {S}}^{d-1})}}\Vert b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)}. \end{aligned}$$

(3.23)

Let $m=s-[j\varepsilon _0]$, with $0<\varepsilon _0<1$. Since $\textrm{card}({\mathfrak {E}}^j)\lesssim 2^{j\gamma (d-1)}$. Then (3.21) and (3.23) lead to

$$\begin{aligned}&\sum _{\nu } \Vert (I-{G_{\nu }^j)}(I-P_{s-j\kappa } )T^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }(b_{\mathbb {L}})\Vert _{L^1({\mathbb {R}}^d)}\\&\quad \le \Big (\sum _{\nu } \sum _{k<m} + \sum _{\nu } \sum _{k\ge m} \Big ) \Vert (I-P_{s-j\kappa } ) (I-{G_{\nu }^j)}\Lambda _kT^{i,j}_{\Omega ,{{\mathbb {L}}};\,s,\nu }b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)} \\&\quad \lesssim (2^{{s_1}j}+2^{{s_2}j})j\Vert \Omega _i\Vert _{L^{\infty }({\mathbb {S}}^{d-1})} \Vert b_{\mathbb {L}}\Vert _{L^1({\mathbb {R}}^d)}, \end{aligned}$$

where $s_1=\big (\tau -\varepsilon _0+\gamma (1+2M)\big )$ and $s_2=-\kappa +\varepsilon _0$.

We can now choose $0\ll \tau \ll \gamma \ll \varepsilon _0< \kappa < 1$ such that $\max \{s_1,s_2\}<0$. Let $s_0=-\max \{s_1,s_2\}$, then the proof of Lemma 3.6 is finished now. $\square$

With Lemma 3.2 and (3.20) in hand, we can deduce (3.7) by

$$\begin{aligned} \big |\big \{x\in {\mathbb {R}}^d\backslash E:\, |\textrm{D}_2>1/8\big \}\big |\le & {} 16\Vert \textrm{D}_{2}-\textrm{D}_{2}^*\Vert _{L^1({\mathbb {R}}^d)}+\big |\big \{x\in {\mathbb {R}}^d:\, |\textrm{D}_{2}^*|>1/16\big \}\big |\\\lesssim & {} \Vert f\Vert _{L^1({\mathbb {R}}^d)}. \end{aligned}$$

3.4 Proof of (1.7) in Theorem 1.2

It suffices to prove (1.7) for $\lambda =1$. For a bounded function f with compact support, we employ the Calderón-Zygmund decomposition to |f| at level 1 then obtain a collection of non-overlapping dyadic cubes ${\mathcal {S}}=\{Q\}$, such that

$$\begin{aligned} \Vert f\Vert _{L^{\infty }({\mathbb {R}}^d\backslash \cup _{Q\in {\mathcal {S}}}Q)}\lesssim 1,\quad \int _{Q}|f(x)|dx \lesssim |Q|,\qquad \hbox {and } \sum _{Q\in {\mathcal {S}}}|Q|\lesssim \int _{{\mathbb {R}}^d}|f(x)|dx. \end{aligned}$$

Let $E=\cup _{Q\in {\mathcal {S}}}100dQ$. With the same notations as in the proof of (1.6), for $x\in {\mathbb {R}}^d\backslash E$, we write

$$\begin{aligned} {\widetilde{T}}_{\Omega ,A}b(x)=\sum _j\sum _{Q\in {\mathcal {S}}}T_{\Omega ,\,A_Q,j}b_{Q}(x)-\sum _{Q\in {\mathcal {S}}} \sum _{n=1}^d\partial _nA_{Q}(x)T_{\Omega }^nb_Q(x). \end{aligned}$$

By estimate (3.5), the proof of (1.7) can be reduced to show that for each n with $1\le n\le d$,

$$\begin{aligned} \Big |\Big \{x\in {\mathbb {R}}^d\backslash E:\, \Big |\sum _{Q\in {\mathcal {S}}}\partial _nA_{Q}(x){T_{\Omega }^nb_Q(x)}\Big |>{1/4d}\Big \}\Big |\lesssim \int _{{\mathbb {R}}^d}|f(x)|dx. \end{aligned}$$

But this inequality has already been proved in [22, inequality (3.3)]. Then the proof of (1.7) is finished. $\square$

3.5 Proof of Theorem 1.3

The proof of Theorem 1.3 is now standard. We present the proof here mainly to make the constant of the norm inequality clearly. Consider the case $p\in (1,2]$. Let

$$\begin{aligned} f_{\lambda }(x)= \Big \{\begin{array}{ll} f(x),\,&{}|f(x)|>\lambda \\ 0,\,&{}|f(x)|\le \lambda ;\end{array} \end{aligned}$$

and

$$\begin{aligned} f^{\lambda }(x)= \Big \{\begin{array}{ll} 0,\,&{}|f(x)|>\lambda \\ f(x),\,&{}|f(x)|\le \lambda \end{array} \end{aligned}$$

By (1.6), we have

$$\begin{aligned}&p\int _{0}^{\infty }\lambda ^{p-1}{|\{x\in {\mathbb {R}}^d:|{T}_{\Omega ,A}f_{\lambda }(x)|>\lambda /2\}|d\lambda }\\&\quad {\lesssim } p\int _{0}^{\infty }\lambda ^{p-1}\int _{{\mathbb {R}}^d}\frac{|f_{\lambda }(x)|}{\lambda }\log \left( e+\frac{|f_{\lambda }(x)|}{\lambda }\right) dx\,d\lambda \\&\quad \le \left( \frac{p}{p-1}\right) ^2\Vert f\Vert _{L^p({\mathbb {R}}^d)}^p. \end{aligned}$$

$L^2({\mathbb {R}}^d)$ boundedness of $T_{\Omega ,\,A}$ implies that

$$\begin{aligned}&p\int _{0}^{\infty }\lambda ^{p-1}{|\{x\in {\mathbb {R}}^d:|{T}_{\Omega ,A}f^{\lambda }(x)|>\lambda /2\}|d\lambda }\\&\quad {~\lesssim ~} p\int _{|f(x)|}^{\infty }\lambda ^{p-1}\lambda ^{-2}\Vert f^{\lambda }\Vert {^2}_{L^2({\mathbb {R}}^d)}\,d\lambda \le \frac{p}{2-p}\Vert f\Vert _{L^p({\mathbb {R}}^d)}^p. \end{aligned}$$

Since ${p\in (1,2)}$, we have

$$\begin{aligned}{} & {} \Vert T_{\Omega ,\,A}f\Vert _{L^p({\mathbb {R}}^d)}=\bigg (p\int _{0}^{\infty }\lambda ^{p-1}\big | \{x\in {\mathbb {R}}^d:|{T}_{\Omega ,A}f(x)|>\lambda \}\big |\,d\lambda \bigg )^{1/p}\\ {}{} & {} \le (p')^2\Vert f\Vert _{L^p({\mathbb {R}}^d)}. \end{aligned}$$

When $p\in (2,\infty )$, by (1.7), we know ${\widetilde{T}}_{\Omega ,A}f(x)$ is of weak type (1, 1). Combining the $L^2({\mathbb {R}}^d)$ boundedness of ${\widetilde{T}}_{\Omega ,A}f(x)$ and the Marcinkiewicz interpolation theorem, we have

$$\begin{aligned} \Vert {\widetilde{T}}_{\Omega ,\,A}f\Vert _{L^{p'}({\mathbb {R}}^d)}\le p'\Vert f\Vert _{L^{p'}({\mathbb {R}}^d)}. \end{aligned}$$

By duality, it holds that

$$\begin{aligned} \Vert {T}_{\Omega ,\,A}f\Vert _{L^{p}({\mathbb {R}}^d)}\le p\Vert f\Vert _{L^{p}({\mathbb {R}}^d)}. \end{aligned}$$

This completes the proof of Theorem 1.3. $\square$