Quantitative Weighted Bounds for the q-Variation of Singular Integrals with Rough Kernels

Abstract

In this paper, we study the quantitative weighted bounds for the q-variational singular integral operators with rough kernels, a stronger nonlinearity than the maximal truncations. The main result is for the truncated singular integrals itself

$$\begin{aligned} \Vert V_q\{T_{\Omega ,\varepsilon }\}_{\varepsilon >0}\Vert _{L^p(w)\rightarrow L^p(w)}\lesssim \Vert \Omega \Vert _{ L^\infty }(w)_{A_p}^{1+1/q}\{w\}_{A_p}, \end{aligned}$$

it is the best known quantitative result for this class of operators. In the course of establishing the above estimate, we obtain several quantitative weighted bounds which are of independent interest.

1 Introduction and Statement of Main Results

1.1 Background and Main Result

Let ${\mathcal {F}}=\{F_t:t\in {\mathcal {I}}\subset \mathbb {R}\}$ be a family of Lebesgue measurable functions defined on $\mathbb {R}^n$. For x in $\mathbb {R}^n$, the value of the q-variation function $V_q({\mathcal {F}})$ of the family ${\mathcal {F}}$ at x is defined by

$$\begin{aligned} V_q({\mathcal {F}})(x):= \sup \Big (\sum _{k\ge 1} |F_{t_k}(x)-F_{t_{k-1}}(x)|^q\Big )^{\frac{1}{q}} ,\quad q\ge 1, \end{aligned}$$

(1.1)

where the supremum runs over all increasing subsequences $\{t_k:k\ge 0\}\subset {\mathcal {I}}.$ Specially, when $q=\infty ,$

$$\begin{aligned} V_\infty ({\mathcal {F}})(x)=\sup _{k\ge 1} |F_{t_k}(x)-F_{t_{k-1}}(x)|, \end{aligned}$$

where the supremum runs over all increasing subsequences $\{t_k:k\ge 0\}\subset {\mathcal {I}}.$ It is trivial that $\sup _{t\in \mathbb R}|F_t(x)|\le V_\infty (\mathcal F)(x)+|F_{t_*}(x)|$ for any $t_*\in {\mathcal {I}}.$

Suppose $\mathcal {A}=\{{A}_t\}_{t\in {\mathcal {I}}}$ is a family of operators on $L^p(\mathbb R^n)\, (1\le p\le \infty )$, the associated strong q-variation operator $V_q{\mathcal {A}}$ is defined as

$$\begin{aligned} V_q{\mathcal {A}}(f)(x)=V_q(\{{A}_tf(x)\}_{t\in {\mathcal {I}}}). \end{aligned}$$

There are two elementary but important observations that motivate the development of variational inequalities in ergodic theory and harmonic analysis. The first one is that from the fact that $V_q({\mathcal {F}})(x)<\infty$ with finite q implies the convergence of $F_t(x)$ as $t\rightarrow t_0$ whenever $t_0$ is an adherent point for ${\mathcal {I}}$, it is easy to observe that $A_t(f)$ converges a.e. for $f\in L^p$ whenever $V_q{\mathcal {A}}$ for finite q is of weak type (p, p) with $p<\infty$. The second one is that q-variation function dominates pointwisely the maximal function: for any $q\ge 1$,

$$\begin{aligned} A^*(f)(x)\le A_{t_0}f(x)+ V_{q}(\mathcal {A}f)(x), \end{aligned}$$

(1.2)

where $A^*$ is the maximal operator defined by $A^*(f)(x):=\sup _{t\in {\mathcal {I}}}|{A}_t(f)(x)|$ and $t_0\in {\mathcal {I}}$ is any fixed number.

Because of the first observation, Bourgain [1] proved in ergodic theory the first variational estimate which had originated from the regularity of Brownian motion in martingale theory [28, 38] since the pointwise convergence may not hold for any non-trivial dense subclass of functions in some ergodic models and thus the maximal Banach principle does not work, see [22, 23] for further results in this direction. Because of the second observation, many maximal inequalities in harmonic analysis, such as maximal singular integrals, maximal operators of Radon type, the Carleson–Hunt theorem, Cauchy type integrals, dimension free Hardy–Littlewood maximal estimates etc, have been strengthened to variational inequalities [2, 4, 5, 24, 33,34,35,36,37, 43]. There are also many other publications on vector-valued and weighted norm estimates coming to enrich the literature on this subject (cf. e.g., [12, 16, 17, 24, 27]).

In this paper, we continue to study q-variational estimates but focus on the quantitative weighted bounds for singular integrals with rough kernels.

For $\varepsilon >0,$ suppose that $T_{\Omega ,\varepsilon }$ is the truncated homogeneous singular integral operator defined by

$$\begin{aligned} T_{\Omega , \varepsilon } f(x)=\int _{|y|>\varepsilon }\frac{\Omega (y')}{|y|^n}f(x-y)dy, \end{aligned}$$

(1.3)

where $y'={y/|y|}$, $\Omega \in L\log ^+\!\!L(\mathbb S^{n-1})$ is homogeneous of degree zero and satisfies the cancellation condition

$$\begin{aligned} \int _{\mathbb S^{n-1}}\Omega (y')d\sigma (y')=0. \end{aligned}$$

(1.4)

Denote the family $\{T_{\Omega , \varepsilon }\}_{\varepsilon >0}$ of operator by ${\mathcal {T}}_\Omega$. For $1< p<\infty$ and $f\in L^p(\mathbb R^n)$, the Calderón–Zygmund singular integral operator $T_\Omega$ with homogeneous kernel can be defined by

$$\begin{aligned} T_\Omega f(x)=\lim _{\varepsilon \rightarrow 0^+}T_{\Omega , \varepsilon } f(x), \ a.e. \ x\in \mathbb R^n. \end{aligned}$$

(1.5)

Considering the influence of (1.2) for any $f\in L^p(\mathbb R^n)$ and $x\in \mathbb R^n$ yields the inequality expression

$$\begin{aligned} T_\Omega ^*(f)(x)\le V_{q}(\mathcal {T}_\Omega f)(x)\quad \text {for}\ \ q\ge 1, \end{aligned}$$

(1.6)

where $T_\Omega ^*$ which is given by $T_\Omega ^*f(x)=\sup _{\varepsilon >0}|T_{\Omega , \varepsilon } f(x)|$ together with the maximal Banach principle.

In 2002, Campbell et al. [5] first proved the variational inequalities for $T_\Omega$ extending their previous results for the Hilbert transform [4], which gives extra information on the speed of convergence in (1.5). See also [10, 24] for more results.

Theorem A. ([5])   Let $\Omega \in L\log ^+\!\!L(\mathbb S^{n-1})$ satisfying (1.4) and ${\mathcal {T}}_\Omega$ be as defined above. Then the variational inequality

$$\begin{aligned} \Vert V_q{\mathcal {T}}_\Omega (f)\Vert _{L^p}\lesssim \Vert \Omega \Vert _{L\log ^+\!\!L(\mathbb S^{n-1})}\Vert f\Vert _{L^p} \end{aligned}$$

holds for $1<p<\infty$ and $q>2$.

The first weighted norm q-variational inequality, to the authors’ best knowledge, is due to Ma et al. [31]. Later on, their results were extended to higher dimensions [26, 32]. The weighted version of Theorem A was also considered in [7]. On the other hand, motivated by the solution of the $A_2$ conjecture, Hytönen et al. [20] established the sharp weighted inequality for the variation of the smooth truncations of Calderón–Zygmund operator with $\omega$-Dini type regularity; in [9], Franca Silva and Zorin-Kranich extended the sharp weighted estimates for smooth truncations to the case of sharp truncations. The latter result was generalized to matrix weight by Duong et al. [14].

However, the (sharp) quantitative weighted version of Theorem A, that is for operators without regularity, has not been considered up to now except the special case of $q=\infty$ due to Di Plinio et al. [11] and Lerner [30]. One main goal of the present paper is to address this question, and to provide the quantitative weighted variational inequality for ${\mathcal {T}}_\Omega$.

The arguments in [25, 39] show that strong q-variational estimates with respect to the martingale sequence in $E_{ N}f(x)=\frac{1}{|Q|}\int _Q f$ (here Q is the unique dyadic interval of sidelength $2^N$ containing x) fail whenever $q < 2$ and the strong q-variational estimates of ${\mathcal {T}}_\Omega$ depends on $\{E_{ N}\}$. Thus, throughout the paper we will only consider the case $q > 2$ for strong q-variations.

Theorem 1.1

Let $\Omega \in L^\infty ({\mathbb S}^{n-1})$ satisfying (1.4) and ${\mathcal {T}}_\Omega$ be defined as above. Then the variational inequality

$$\begin{aligned} \Vert V_q{\mathcal {T}}_\Omega (f)\Vert _{L^p(w)}\lesssim \Vert \Omega \Vert _{ L^\infty }(w)_{A_p}^{1+1/q}\{w\}_{A_p}\Vert f\Vert _{L^p(w)} \end{aligned}$$

(1.7)

holds for $q>2$, $1<p<\infty$ and $w\in A_p$, where the implicit constant $c_{p,q,n}$ is independent of f and w, and $(w)_{A_p}:=\max \{[w]_{A_\infty },\,[w^{1-p'}]_{A_\infty }\}$, $\{w\}_{A_p}:=[w]_{A_p}^{1\over p}\max \{[w]_{A_\infty }^{1\over p'},\,[w^{1-p'}]_{A_\infty }^{1\over p}\}$, and the constant $[w]_{A_p}$, $1<p\le \infty$, is given in (2.1) and (2.2).

Remark 1.2

Our quantitative estimate in (1.7) for $q=\infty$ matches the best known results in [11, 30] for $T_\Omega ^*$.

1.2 Approach and Main Methods

We now state the methods and techniques for proving our main result. Some estimates are of independent interest, and hence we will single them out as theorems or propositions. As usual, we shall prove Theorem 1.1 by verifying separately the corresponding inequalities for the long and short variations as follows (see e.g., [24, Lemma 1.3]).

Theorem 1.3

Let $q>2.$ Let $\Omega \in L^\infty ({\mathbb S}^{n-1})$ satisfying (1.4). Then for $1<p<\infty$ and $w\in A_p,$

$$\begin{aligned} \Vert V_q(\{T_{\Omega ,2^k}f\}_{k\in \mathbb Z})\Vert _{L^p(w)}\lesssim \Vert \Omega \Vert _{L^\infty }(w)_{A_p}^{1+1/q}\{w\}_{A_p}\Vert f\Vert _{L^p(w)}. \end{aligned}$$

(1.8)

Theorem 1.4

Let $q\ge 2.$ Let $\Omega \in L^\infty ({\mathbb S}^{n-1})$ satisfying (1.4). Then for $1<p<\infty$ and $w\in A_p,$

$$\begin{aligned} \Vert \mathcal {S}_q({\mathcal {T}}_\Omega f)\Vert _{L^p(w)}\lesssim \Vert \Omega \Vert _{L^\infty }(w)_{A_p}^{1/q}\{w\}_{A_p}\Vert f\Vert _{L^p(w)}, \end{aligned}$$

(1.9)

where

$$\begin{aligned} \mathcal {S}_q({\mathcal {T}}_\Omega f):=\bigg (\sum _{k\in \mathbb Z}[V_{q}(\{{ T}_{\Omega ,\varepsilon } f\}_{2^k\le \varepsilon <2^{k+1}})]^q\bigg )^{1/q}. \end{aligned}$$

Remark 1.5

It is worthy to point out the above two results are of independent interest due to the presence of different quantitative behaviors. However for $q<2$, We don’t know whether or not the $L^p$ boundedness of the short variations $\mathcal {S}_q({\mathcal {T}}_\Omega f)$ holds.

The proof of Theorem 1.3 is given in Sect. 6 and the proof of Theorem 1.4 in Sect. 7, while the important tools for the proof of these two theorems are established in Sects. 3, 4 and 5.

1.2.1 Methods for Proving Theorem 1.3

As a standard way to study variational inequalities for rough singular integrals (see e.g., [7, 10] originating from [13]), the first step is to exploit the Cotlar type decomposition of $T_{\Omega ,2^k}$. Let $\phi \in C^{\infty }_{0}(\mathbb {R}^n)$ be a non-negative radial function, supported in $\{|x|\le 1/4\}$ with $\int \phi (x)dx=1$, and denote $\phi _{k}(x):= 2^{-kn}\phi (2^{-k}x)$. Let $K_\Omega (x):={\Omega (x/|x|)}{|x|^{-n}}$, and $\delta _0$ be the Dirac measure at 0. Then for a reasonable function f, we decompose $T_{\Omega ,2^k}f$ as

$$\begin{aligned} T_{\Omega , 2^k}f&=\phi _{k}*T_\Omega f-\phi _{k}*(K_\Omega \chi _{|\cdot |\le 2^k})*f+(\delta _0-\phi _{k})*T_{\Omega ,2^k}f, \end{aligned}$$

(1.10)

which divides $\mathcal {T}_\Omega =\{T_{\Omega , 2^k}\}_{k\in \mathbb Z}$ into three families:

$$\begin{aligned} \mathcal {T}_\Omega ^1(f)&:=\{\phi _{k}*T_\Omega f\}_{k\in \mathbb Z},\,\,\, \mathcal {T}_\Omega ^2(f):=\{\phi _{k}*(K_\Omega \chi _{|\cdot |\le 2^k})*f\}_{k\in \mathbb Z},\,\,\, \mathcal {T}_\Omega ^3(f)\\&:=\{(\delta _0-\phi _{k})*T_{\Omega ,2^k}f\}_{k\in \mathbb Z}. \end{aligned}$$

Thus, it suffices to verify the weighted $L^p$ estimate for the variation of $\mathcal {T}_\Omega ^i(f)$, $i=1,2,3$.

The second usual step to deal with variational estimates for rough singular integrals is to exploit the almost orthogonality principle based on Littlewood–Paley decomposition and interpolation. However, this step is not obvious at all in the present setting for quantitative weighted estimates. Indeed, the standard Littlewood–Paley decomposition seems to be insufficient. We will exploit the one associated to a sequence of natural numbers ${\mathcal {N}}=\{N(j)\}_{j\ge 0}$ used in [21]; moreover, it is quite involved to gain the sharp kernel estimates involving the smoothing version of $T_\Omega$ which are necessary to obtain the quantitative weighted estimates. Now let us comment on the proof term by term in details.

Comments on the estimate of $\mathcal {T}_\Omega ^1(f)$. Let us first formulate below the desired estimate of this term which is of independent interest since this variational estimate strengthens the result of Lerner [30]. Denote by $\Phi (g)=\{\phi _j*g\}_{j\in \mathbb Z}.$

Theorem 1.6

Let $q>2$. Let $\phi$ be as used in (1.10) and $T_\Omega$ be given as in (1.3) with $\Omega \in L^\infty (\mathbb S^{n-1})$ satisfying (1.4). Then for $1<p<\infty$ and $w\in A_p,$

$$\begin{aligned} \Vert V_q\Phi ( T_\Omega f) \Vert _{L^p(w)}\lesssim \Vert \Omega \Vert _{ L^\infty }(w)_{A_p}^{1+1/q}\{w\}_{A_p}\Vert f\Vert _{L^p(w)}. \end{aligned}$$

(1.11)

The proof of Theorem 1.6 is given in Section 5.

To obtain (1.11), we apply the Littlewood–Paley decomposition of $T_\Omega$ in [21] to get $T_{\Omega }=\sum _{j=0}^{\infty } {T}_{j}^{{\mathcal {N}}},$ where each ${T}_{j}^{\mathcal {N}}$ is a $\omega _j$-Dini Calderón–Zygmund operator of convolution type. This yields

$$\begin{aligned} \Vert V_q\Phi (T_\Omega f)\Vert _{L^p(w)}\le \sum _{j=0}^{\infty } \Vert V_q\Phi (T_{j}^{{\mathcal {N}}}f)\Vert _{L^p(w)}. \end{aligned}$$

Then to control the summation of the right-hand side of the above inequality, it suffices to show for each term the unweighed estimate with rapid decay (with respect to j) and the weighted norm $L^p$ estimate for all $1<p<\infty$ and then to apply the Stein–Weiss interpolation—Lemma 2.2. The rapid decay estimate will follow from the $L^2$ estimate of $T^{\mathcal {N}}_j$; the involved part lies in the estimates in terms of the weighted constant and the Dini norm. The latter is not only necessary for further application but also is of independent interest, we formulate it in the following theorem.

Theorem 1.7

Let $q>2$. Let T be a $\omega$-Dini Calderón–Zygmund operator of convolution type with the kernel K satisfying the following cancellation condition: for any $0<\varepsilon ,\,R<\infty ,$

$$\begin{aligned} \int _{\varepsilon<|x|<R} K(x)\,dx=0. \end{aligned}$$

(1.12)

Then for any $1<p<\infty$ and $w\in A_p,$

$$\begin{aligned} \Vert V_q\Phi (Tf)\Vert _{L^{p}(w)}&\lesssim (\Vert {\omega }\Vert _{Dini}+\Vert \omega \Vert _{Dini}^{1/q}\Vert \omega ^{1/q'}\Vert _{Dini}\nonumber \\&\quad +\Vert \omega \Vert _{Dini}^{1/q'}\Vert \omega ^{1/2}\Vert _{Dini}^{2/q}+C_K+\Vert T\Vert _{L^2\rightarrow L^2})\{w\}_{A_p}\Vert f\Vert _{L^{p}(w)}, \end{aligned}$$

(1.13)

where $\omega$ satisfies the Dini condition (2.3) and the estimate is sharp in the sense that the inequality does not hold if we replace the exponent of the $A_{p}$ constant by a smaller one.

The proof of Theorem 1.7 is given in Section 5.

Firstly, it would be impossible to get the weighted bound in (1.13) via iteration by showing the weighted estimate of $V_q\Phi$ since the one of T is already linear (see. e.g., [21, 29]).

We provide the proof of (1.13) based on the Cotlar type decomposition

$$\begin{aligned} \phi _k*Tf&=K^k*f+\phi _{k}*K_k*f+(\phi _k-\delta _{0})*K^k*f, \end{aligned}$$

(1.14)

where $K_k=K\chi _{|\cdot |\le 2^{k}}$ and $K^k=K\chi _{|\cdot |\ge 2^{k}}.$ Now, the sharp weighted bound of $V_q(\{K^k*f\}_{j\in \mathbb Z})$ has been obtained in [9] (see also [6, 14]) with bounds $\Vert {\omega }\Vert _{Dini}+C_K+\Vert T\Vert _{L^2\rightarrow L^2}$. Regarding the second term, at a first glance, it should be easier to handle than the second term in (1.10) since the kernel K enjoys some regularity; but it turns out that they are in the same level of complexity as they will follow from Proposition 1.8 below. Finally, the third term in (1.14) is hard to deal with which will follow from Proposition 1.9.

We first control the variation norm by a stronger norm. That is,

$$\begin{aligned}{} & {} V_q(\{\phi _{k}*K_k*f\}_k)=\Vert \phi _{k}*K_k*f\Vert _{V_q}\\{} & {} \le 2\Vert \{\phi _{k}*K_k*f\}_{k}\Vert _{\ell _q}=:L_q(\{\phi _{k}*K_k*f\}_k). \end{aligned}$$

Then all the desired estimates for the left quantity involving $V_q$ will follow from that involving $L_q$.

We establish the following result.

Proposition 1.8

Let K be a kernel satisfying the mean value zero property (1.12) and the size condition (2.5). Then the following assertions hold.

(1):

Weak type (1, 1) boundedness:

$$\begin{aligned} \big \Vert \big (\sum _{k\in \mathbb Z}|{\phi }_k*K_k*f|^2\big )^{1/2}\big \Vert _{L^{1,\infty }}\lesssim C_K\Vert f\Vert _{L^1}. \end{aligned}$$

(1.15)

Weak type (1, 1) boundedness of the grand maximal truncated operator (see (2.7)):

$$\begin{aligned} \Vert {\mathcal {M}}_{(\sum _{k\in \mathbb Z}|{\phi }_k*K_k*f|^2)^{1/2}}\Vert _{L^{1,\infty }}\lesssim C_K\Vert f\Vert _{L^1}. \end{aligned}$$

(1.16)

(2):

Sharp weighted norm inequalities: for all $1<p<\infty ,$ and $w\in A_p,$

$$\begin{aligned} \bigg \Vert \bigg (\sum _{k\in \mathbb Z}|\phi _k*K_k*f|^2\bigg )^{1/2}\bigg \Vert _{L^{p}(w)}&\lesssim C_K \{w\}_{A_p}\Vert f\Vert _{L^{p}(w)}. \end{aligned}$$

(1.17)

The proof of Proposition 1.8 is given in Section 3.

Surely, by the fact that $\ell ^2\subset \ell ^q$ for $q\ge 2$, the desired estimate for the second term in the decomposition (1.14) can be deduced from the Proposition 1.8. Moreover, it is worth highlighting that in Proposition 1.8, we do not need any regularity assumption on the kernel K.

Again, we start with the comments on the estimate of the third term in the decomposition (1.14). We prove the following result.

Proposition 1.9

Let $q>2.$ Let K be a kernel satisfying the mean value zero property (1.12) and the size condition (2.5) and the Dini condition (2.6). Let $C(K,\omega ,q)= C_K+\Vert \omega \Vert _{Dini}+\Vert \omega \Vert _{Dini}^{1/q}\Vert \omega ^{1/q'}\Vert _{Dini}+\Vert \omega \Vert _{Dini}^{1/q'}\Vert \omega ^{1/2}\Vert _{Dini}^{2/q},$ where $\omega$ satisfies the Dini condition (2.3). Then the following assertions hold.

(1):

Weak type (1, 1) boundedness:

$$\begin{aligned} \big \Vert \big (\sum _{k\in \mathbb Z}|(\phi _{k}-\delta _0)*K^k*f|^q\big )^{1/q}\big \Vert _{L^{1,\infty }}&\lesssim C(K,\omega ,q)\Vert f\Vert _{L^{1}}. \end{aligned}$$

(1.18)

Weak type (1, 1) boundedness of the grand maximal truncated operator:

$$\begin{aligned} \Vert \mathcal {M}_{(\sum _{k\in \mathbb Z}|(\phi _{k}-\delta _0)*K^k*f|^q)^{1/q}}\Vert _{L^{1,\infty }}&\lesssim C(K,\omega ,q)\Vert f\Vert _{L^{1}}. \end{aligned}$$

(1.19)

(2):

Sharp weighted norm inequalities: for all $1<p<\infty ,$ and $w\in A_p,$

$$\begin{aligned} \big \Vert \big (\sum _{k\in \mathbb Z}|(\phi _k-\delta _0)*K^k*f|^q\big )^{1/q}\big \Vert _{L^p(w)}&\lesssim C(K,\omega ,q)\{w\}_{A_p}\Vert f\Vert _{L^p(w)}. \end{aligned}$$

(1.20)

This proposition will be shown in Sect. 4.

We point out that at the moment we are unable to prove the above result for $q=2$ since our argument in showing the weak type (1, 1) estimate of the localized maximal operator (1.19) depends heavily on the same estimate of q-variation operator associated to the Hardy–Littlewood averages.

In this process, together with the weak type (1, 1) estimates of variational Calderón–Zygmund operators [14], we also obtain the following result as a byproduct.

Corollary 1.10

Let $q>2$ and T be as in Theorem 1.7. Then

$$\begin{aligned} \Vert V_q\Phi (Tf)\Vert _{L^{1,\infty }}&\lesssim (\Vert {\omega }\Vert _{Dini}+\Vert \omega \Vert _{Dini}^{1/q}\Vert \omega ^{1/q'}\Vert _{Dini}+\Vert \omega \Vert _{Dini}^{1/q'}\Vert \omega ^{1/2}\Vert _{Dini}^{2/q}\\ \nonumber&\quad +C_K+\Vert T\Vert _{L^2\rightarrow L^2})\Vert f\Vert _{L^1},\end{aligned}$$

where $\omega$ satisfies the Dini condition (2.3).

Comments on the estimate of $\mathcal {T}_\Omega ^2(f)$. About $\mathcal {T}_\Omega ^2(f),$ we start with the comments on the estimate of the second term in the decomposition (1.14) since the latter involving regular kernel K seems to be easier to handle. In particular, Proposition 1.8 applies well to $\mathcal {T}_\Omega ^2(f)$, and thus the almost orthogonality technique can be avoided, improving the corresponding argument in [7].

Comments on the estimate of $\mathcal {T}_\Omega ^3(f)$. For the rough kernel $K_\Omega$ in $\mathcal {T}_\Omega ^3(f)$, we have to decompose it by using the Littlewood–Paley decomposition from [21], repeat the argument of Proposition 1.9, require a slightly more involved proof and then exploit the almost orthogonality principle.

1.2.2 Methods for proving Theorem 1.4

Following from the sharp weighted boundedness of the Hardy–Littlewood maximal operator (see (7.1)),

$$\begin{aligned} \Vert \mathcal {S}_{\infty }({\mathcal {T}}_\Omega f)\Vert _{L^p(w)}&\le c_{p,n} \Vert \Omega \Vert _{L^\infty }\{w\}_{A_p}\Vert f\Vert _{L^p(w)}. \end{aligned}$$

Thus by interpolation, it suffices to prove that

$$\begin{aligned} \Vert \mathcal {S}_{2}(\mathcal {T}_\Omega f)\Vert _{L^p(w)}&\le c_{p,n} \Vert \Omega \Vert _{L^\infty }(w)_{A_p}^{1/2}\{w\}_{A_p}\Vert f\Vert _{L^p(w)}. \end{aligned}$$

(1.21)

To verify this, we write

$$\begin{aligned} \mathcal {S}_{2}({\mathcal {T}}_\Omega f)(x)&=\Big (\sum _{k\in \mathbb {Z}}|V_{2,k}( f)(x)|^2\Big )^{\frac{1}{2}}: =\Big (\sum _{k\in \mathbb {Z}}\Vert \{T_{k,t}f(x)\}_{t\in [1,2]}\Vert _{V_2}^2\Big )^{\frac{1}{2}}, \end{aligned}$$

(1.22)

where $T_{k,t}f(x)=v_{k,t}*f(x)$ for $t\in [1,2]$, and $v_{0,t}(x)={\Omega (x/|x|)}{|x|^{-n}}\chi _{_{t\le |x|\le 2}}(x)$ and $v_{k,t}(x)={2^{-kn}}\nu _{0,t}(2^{-k}x)$ for $k\in \mathbb {Z}$. Next, by using the Littlewood–Paley decomposition as in [21], we further have $T_{k,t}=\sum _{j=0}^\infty T^\mathcal {N}_{k, t, j}$, whose definition will be given in Section 7. Therefore, by the Minkowski inequality, we get

$$\begin{aligned} \mathcal {S}_2({\mathcal {T}}_\Omega f)(x)&\le \sum ^\infty _{j=0}\mathcal {S}_{2,j}^\mathcal {N}( f)(x), \end{aligned}$$

where

$$\begin{aligned} \mathcal {S}_{2,j}^\mathcal {N}( f)(x):= \bigg (\sum _{k\in \mathbb {Z}}\Vert \{T_{k,t,j}^\mathcal {N}f(x)\}_{t\in [1,2]}\Vert _{V_2}^2\bigg )^\frac{1}{2}. \end{aligned}$$

Hence, to prove (1.21), it again suffices to verify for $\mathcal {S}_{2,j}^\mathcal {N}( f)$ the unweighted norm with rapid decay in j and the weighted norm estimate and then to apply the Stein–Weiss interpolation—Lemma 2.2.

Notation. From now on, $p'=p/(p-1)$ represents the conjugate number of $p\in [1,\infty )$; $X\lesssim Y$ stands for $X\le C Y$ for a constant $C>0$ which is independent of the essential variables living on $X\ \& \ Y$; and $X\approx Y$ denotes $X\lesssim Y\lesssim X$.

2 Preliminaries

2.1 Muckenhoupt Weights

Let w be a non-negative locally integrable function defined on $\mathbb R^n$. For $1<p<\infty$, we say that $w\in A_p$ if there exists a constant $C>0$ such that

$$\begin{aligned} {[}w]_{A_p}:=\sup _Q\bigg (\frac{1}{|Q|}\int _Qw(x)dx\bigg )\bigg (\frac{1}{|Q|}\int _Qw(x)^{1-p'}dx\bigg )^{p-1}\le C. \end{aligned}$$

(2.1)

We will adopt the following definition for the $A_\infty$ constant for a weight w introduced by Fujii [15] and later by Wilson [42]:

$$\begin{aligned} {[}w]_{A_\infty }:=\sup _Q\frac{1}{w(Q)}\int _QM(w\chi _Q)(x)\,dx. \end{aligned}$$

(2.2)

Here $w(Q):=\int _Qw(x)\,dx,$ and the supremum above is taken over all cubes with edges parallel to the coordinate axes. When the supremum is finite, we will say that w belongs to the $A_\infty$ class. $A_\infty :=\bigcup _{p\ge 1} A_p$. Recall in Theorem 1.1, $(w)_{A_p}:=\max \{[w]_{A_\infty },\,[w^{1-p'}]_{A_\infty }\}.$ Using the facts that $[w^{1-p'}]_{A_p'}^{1/p'}=[w]_{A_p}^{1/p}$ and $[w]_{A_\infty }\le C[w]_{A_p}$(see [19]), one easily checks that $(w)_{A_p}\le [w]_{A_p}^{\max (1,1/(p-1))}$ (see [21]).

2.2 $\omega$-Dini Calderón–Zygmund Operators of Convolution Type

A modulus of continuity $\omega$ if it is increasing and subadditive in the sense that

$$\begin{aligned} u\le t+s\Rightarrow \omega (u)\le \omega (t)+\omega (s). \end{aligned}$$

$\omega$ satisfies the classical Dini condition

$$\begin{aligned} \Vert \omega \Vert _{Dini}:=\int _0^1\omega (t)\frac{dt}{t}<\infty . \end{aligned}$$

(2.3)

For any $c>0$ the integral can be equivalently (up to a c-dependent multiplicative constant) replaced by the sum over $2^{-j/c}$ with $j\in \mathbb N.$

Let T be a bounded linear operator on $L^2({\mathbb R}^n)$ represented as

$$\begin{aligned} Tf(x)=\int _{{\mathbb R}^n}K(x-y)f(y)\,dy,\,\forall x \notin \textrm{supp} f. \end{aligned}$$

(2.4)

We say that T is an $\omega$-Dini Calderón–Zygmund operators of convolution type if the kernel K satisfies the following size and smoothness conditions:

for any $x\in {\mathbb R}^n\backslash \{0\}$,

$$\begin{aligned} |K(x)|\le \frac{C_K}{|x|^n}; \end{aligned}$$

(2.5)

for any $x,\,y\in {\mathbb R}^n$ with $2|y|\le |x|$,

$$\begin{aligned} |K(x-y)-K(x)|\le \frac{\omega (|y|/|x|)}{|x|^n}. \end{aligned}$$

(2.6)

2.3 Criterion for Sharp Weighted Norm Estimate

The grand maximal truncated operator $\mathcal {M}_U$, associated to a sub-linear operator U, is defined by

$$\begin{aligned} \mathcal {M}_{Uf}(x)=\mathcal {M}_Uf(x)=\sup _{Q\ni x}\, \textrm{ess} \sup _{\xi \in Q} |U(f\chi _{{\mathbb R}^n\setminus {3Q}})(\xi )|,\end{aligned}$$

(2.7)

where the supremum is taken over all cubes $Q\subset {\mathbb R}^n$ containing x.

Lemma 2.1

([29]) Assume that both U and $\mathcal {M}_U$ be of weak type (1, 1), then for every compactly supported $f\in L^p({\mathbb R}^n)\,(1<p<\infty )$ and $w\in A_p,$

$$\begin{aligned} \Vert Uf\Vert _{L^p(w)}\lesssim K\{w\}_{A_p}\Vert f\Vert _{L^p(w)}, \end{aligned}$$

where $K=\Vert U\Vert _{L^1\rightarrow L^{1,\infty }}+\Vert \mathcal {M}_U\Vert _{L^1\rightarrow L^{1,\infty }}.$

2.4 The Stein–Weiss Interpolation Theorem with Change of Measures

The following interpolation result with change of measures due to Stein and Weiss plays an important role in dealing with rough singular integrals. Again, we will use frequently this tool.

Lemma 2.2

([40]) Assume that $1\le p_0,p_1\le \infty$, that $w_0$ and $w_1$ be positive weights, and T be a sublinear operator satisfying

$$\begin{aligned} T:L^{p_i}(w_i)\rightarrow L^{p_i}(w_i),~~~~i=0,1, \end{aligned}$$

with quasi-norms $M_0$ and $M_1$, respectively. Then

$$\begin{aligned} T:L^{p}(w)\rightarrow L^{p}(w), \end{aligned}$$

with quasi-norm $M\le M_0^\lambda M_1^{1-\lambda }$, where

$$\begin{aligned} \frac{1}{p}=\frac{\lambda }{p_0}+\frac{1-\lambda }{p_1},~~~w=w_0^{\frac{p\lambda }{p_0}}w_1^{\frac{p(1-\lambda )}{p_1}}. \end{aligned}$$

3 Proof of Proposition 1.8

We first give a useful Lemma which plays an important role in the proof of Proposition 1.8.

Lemma 3.1

For any fixed $k\in \mathbb Z,$ let $\mu _k$ be a function such that supp $\mu _k\subset \{x:|x| \lesssim 2^k\}.$ If there exist constants $\gamma _1,\,\gamma _2,\,\gamma _3$ such that $\mu _k$ satisfy

$$\begin{aligned} \bigg \Vert \left( \sum _{k\in \mathbb Z}|\mu _k*f|^2\right) ^{1/2}\bigg \Vert _{L^{2}}\lesssim \gamma _1\Vert f\Vert _{L^2},\end{aligned}$$

(3.1)

and

$$\begin{aligned} |\mu _k(x)|\lesssim \frac{\gamma _2}{2^{kn}},\,\,\,k\in \mathbb Z, \end{aligned}$$

(3.2)

and for $0<2|y|\le |x|,$

$$\begin{aligned} \sum _{k\in \mathbb Z}|\mu _k(x-y)-\mu _k(x)|&\lesssim \gamma _3\frac{|y|^\gamma }{|x|^{n+\gamma }}.\end{aligned}$$

(3.3)

Then for $1<p<\infty ,$ and $w\in A_p,$

$$\begin{aligned} \bigg \Vert \bigg (\sum _{k\in \mathbb Z}|\mu _k*f|^2\bigg )^{1/2}\bigg \Vert _{L^{p}(w)}&\lesssim (\gamma _1+\gamma _2+\gamma _3) \{w\}_{A_p}\Vert f\Vert _{L^{p}(w)}.\end{aligned}$$

(3.4)

Proof

To prove (3.4), by Lemma 2.1, it suffices to verify that

$$\begin{aligned} \bigg \Vert \bigg (\sum _{k\in \mathbb Z}|\mu _k*f|^2\bigg )^{1/2}\bigg \Vert _{L^{1,\infty }}&\lesssim (\gamma _1+\gamma _2+\gamma _3)\Vert f\Vert _{L^1},\end{aligned}$$

(3.5)

and

$$\begin{aligned} \Vert \mathcal {M}_{(\sum _{k\in \mathbb Z}|\mu _k*f|^{2})^{1/2}}\Vert _{L^{1,\infty }}&\lesssim (\gamma _1+\gamma _2+\gamma _3) \Vert f\Vert _{L^{1}}. \end{aligned}$$

(3.6)

By virtue of (3.1)–(3.3), we know $\{\mu _k*f\}$ is a vector Calderón–Zygmund operator. By using a Calderón–Zygmund decomposition and a trivial computation, it is easy to verify that (3.5) holds. Next, we verify (3.6). Take x and $\xi$ in any fixed cube Q. Let $B_x=B(x,2\sqrt{n}\ell (Q)),$ then $3Q\subset B_x.$ By using the triangle inequality, we get

$$\begin{aligned}&\left( \sum _{k\in \mathbb Z}|\mu _k*(f{\chi _{{\mathbb R}^n\setminus 3Q}})(\xi )|^2\right) ^{1/2}\\&\le \left| \left( \sum _{k\in \mathbb Z}|\mu _k*(f{\chi _{{\mathbb R}^n\setminus B_x}})(\xi )|^2\right) ^{1/2}-\left( \sum _{k\in \mathbb Z}|\mu _k*(f{\chi _{{\mathbb R}^n\setminus B_x}})(x)|^2\right) ^{1/2}\right| \\&\quad +\left( \sum _{k\in \mathbb Z}|\mu _k*f{\chi _{B_x\setminus 3Q}})(\xi )|^2\right) ^{1/2}\\&\quad +\left( \sum _{k\in \mathbb Z}|\mu _k*(f{\chi _{{\mathbb R}^n\setminus B_x}})(x)|^2\right) ^{1/2}=:I+I\!I+I\!I\!I. \end{aligned}$$

For the term I,  since x and $\xi$ in Q, then $|x-\xi |\le \sqrt{n}\ell (Q),$ and $|x-y|\ge 2\sqrt{n}\ell (Q)$, then we get $2|x-\xi |\le |x-y|$, by (3.3) we have

$$\begin{aligned} I&\le \int _{{\mathbb R}^n \setminus B_x} \sum _{k\in \mathbb Z}|\mu _k(\xi -y)-\mu _k(x-y)||f(y)|\,dy\\&\le \int _{{\mathbb R}^n \setminus B_x}\frac{\gamma _3|x-\xi |}{|x-y|^{n+1}}|f(y)|\,dy \lesssim \gamma _3Mf(x).\nonumber \end{aligned}$$

For the term $I\!I$, since $\xi$ in Q and $y\in B_x{\setminus } 3Q$, then $|y-\xi |\ge 3\ell (Q)$ and $|y-\xi |^{-n}\lesssim \ell (Q)^{-n}$ and $|x-y|\le 2\sqrt{n}\ell (Q)$, then by the size condition (3.2), we get

$$\begin{aligned} I\!I&\le \int _{{\mathbb R}^n}\sum _{k\in \mathbb Z}|\mu _k(\xi -y)||f|{\chi _{B_x\setminus 3Q}}(y)\,dy\\&\lesssim \gamma _2\int _{3\ell (Q)\le |x-y|\le 2\sqrt{n}\ell (Q)}|\xi -y|^{-n}|f(y)|\,dy\lesssim \gamma _2Mf(x). \end{aligned}$$

For the last term, recalling that supp $\mu _k\subset \{x:|x|\lesssim 2^{k}\}$ and that $B_{x}=\{y:|y-x|\le 2\sqrt{n} \ell (Q)\}$, we have that $\mu _k*f\chi _{\mathbb {R}^{n}{\setminus } B_{x}}=\int _{2\sqrt{n} \ell (Q)<|x-y|\lesssim 2^{k}}\mu _k(x-y)f(y)dy.$ Thus,

$$\begin{aligned} I\!I\!I&= \bigg (\sum _{2\sqrt{n} \ell (Q)\lesssim 2^{k}}|\mu _k*f\chi _{\mathbb {R}^{n}\setminus B_{x}}(x)|^{2}\bigg )^{1/2}\\&\le \bigg (\sum _{k\in \mathbb Z}|\mu _k*f(x)|^{2}\bigg )^{1/2}+\bigg (\sum _{2\sqrt{n} \ell (Q)\lesssim 2^{k}}|\mu _k*f\chi _{ B_{x}}(x)|^{2}\bigg )^{1/2}. \end{aligned}$$

By using the size estimate (3.2), we have

$$\begin{aligned}&\bigg (\sum _{2\sqrt{n} \ell (Q)\lesssim 2^{k}}|\mu _k*f\chi _{ B_{x}}(x)|^{2}\bigg )^{1/2}\\&\lesssim \sum _{2\sqrt{n} \ell (Q)\lesssim 2^{k}}\frac{\gamma _2}{2^{kn}}\int _{|x-y|\le 2\sqrt{n} \ell (Q)}|f(y)|dy\lesssim \gamma _2Mf(x), \end{aligned}$$

which gives

$$\begin{aligned} I\!I\!I \le \bigg (\sum _{k\in \mathbb Z}|\mu _k*f(x)|^{2}\bigg )^{1/2}+\gamma _2Mf(x). \end{aligned}$$

Combining the estimates of $I,\,I\!I$ and $I\!I\!I,$ we get

$$\begin{aligned} \mathcal {M}_{(\sum _{k\in \mathbb Z}|\mu _k*f(x)|^{2})^{1/2}}&\lesssim \bigg (\sum _{k\in \mathbb Z}|\mu _k*f(x)|^{2}\bigg )^{1/2}+(\gamma _2+\gamma _3)Mf(x). \end{aligned}$$

Then the weak type (1, 1) of the Hardy–Littlewood maximal function M and the estimate in (3.5) imply

$$\begin{aligned} \Vert \mathcal {M}_{(\sum _{k\in \mathbb Z}|\mu _k*f|^{2})^{1/2}}\Vert _{L^{1,\infty }}&\lesssim (\gamma _1+\gamma _2+\gamma _3) \Vert f\Vert _{L^{1}}, \end{aligned}$$

which shows that (3.6) holds. The proof of Lemma 3.1 is complete. $\square$

Now we return to the proof of Proposition 1.8. We point out that to estimate $\{\phi _{k}*K_{k}\},$ we do not need any regularity condition on K,  we only assume K satisfies the size condition (2.5) and the cancellation condition (1.12). We first verify the size estimate of $\phi _k*K_k(x)$ for any fixed $k\in \mathbb Z.$ Since supp ${\phi }_k\subset \{x: |x|\le 2^k/4\},$ we have supp $\phi _k*K_k\subset \{x:|x|\le 2^{k+1}\}.$ Then from (2.5) and (1.12) we have for some $\theta \in (0,1)$

$$\begin{aligned} |{{\phi }}_k*K_k(x)|&= \bigg |\int _{|z|\le 2^{k}} ({\phi }_k(x-z)-{\phi }_k(x))K(z)\,dz\bigg |\nonumber \\&\lesssim \frac{C_K}{2^k}\int _{|z|\le 2^{k}} |(\nabla \phi )_k(x-\theta z)|\frac{1}{|z|^{n-1}}\,dz \nonumber \\&\lesssim \frac{C_K}{2^{k(n+1)}}\int _{|z|\le 2^{k}} \frac{1}{|z|^{n-1}}\,dz\chi _{|x|\le 2^{k+1}}(x)\nonumber \\&\lesssim \frac{C_K}{2^{k{n}}}\chi _{|x|\le 2^{k+1}}(x). \end{aligned}$$

(3.7)

Now, we estimate $\nabla \phi _{k}*K_k(x)$. Recall that $\phi \in C_0^\infty ({\mathbb R}^n)$ with $\int \phi =1$ and supp $\phi \subset \{x: |x|\le 1/4\}.$ It is easy to verify that $\int \nabla \phi =0$ and supp $(\nabla \phi )\subset \{x: |x|\le 1/4\}$ with $\nabla \phi \in C_0^\infty ({\mathbb R}^n).$ Since supp $(\nabla \phi )_k\subset \{x:|x|\le 2^k/4\}$ and supp $K_k\subset \{x:|x|\le 2^k\}$ then we get supp $(\nabla \phi )_k*K_k\subset \{x:|x|\le 2^{k+1}\}.$ Then repeating the argument of (3.7), we get

$$\begin{aligned} |\nabla \phi _{k}*K_k(x)|&=\frac{1}{2^k}|(\nabla \phi )_{k}*K_k(x)|\lesssim \frac{C_K}{2^{k{(n+1)}}}\chi _{|x|\le 2^{k+1}}(x). \end{aligned}$$

(3.8)

Thus for $|x|\ge 2|y|,$ and for some $\theta \in (0,1),$

$$\begin{aligned} \sum _{k\in \mathbb Z}|{{\phi }}_{k}*K_k(x-y)-{{\phi }}_{k}*K_k(x)|&\lesssim \sum _{k\in \mathbb Z}|\nabla {\phi }_k*K_k(x-\theta y )|y|. \end{aligned}$$

Since $|x|\ge 2|y|,$ then $x/2\le |x-\theta y|\le 3x/2$, we get $|\nabla {\phi }_k*K_k(x )|\le \frac{C_K}{2^{k{(n+1)}}}\chi _{|x|\le 2^{k+1}}.$ Thus for $|x|\ge 2|y|,$ we get

$$\begin{aligned} \sum _{k\in \mathbb Z}|{{\phi }}_{k}*K_k(x-y)-{{\phi }}_{k}*K_k(x)|&\lesssim \sum _{k\in \mathbb Z}\frac{C_K}{2^{k{(n+1)}}}\chi _{|x|\le 2^{k+1}}|y|\\&\lesssim C_K\sum _{k\in \mathbb Z}{2^{-k{(n+1)}}}\chi _{2^{k}\ge |x|/2}|y|\\&\lesssim C_K|y|\sum _{2^k\ge |x|/2}{2^{-k{(n+1)}}}\\&\lesssim C_K\frac{|y|}{|x|^{n+1}}. \end{aligned}$$

On the other hand, it is easy to verify that

$$\begin{aligned}|{{\phi }}_k*K_k(x)|&\lesssim \frac{C_K}{2^{k{n}}}\chi _{|x|\le 2^{k+1}}\lesssim C_K\frac{2^k}{(2^k+|x|)^{n+1}}, \end{aligned}$$

and

$$\begin{aligned}|{{\phi }}_k*K_k(x+h)-{{\phi }}_k*K_k(x)|&\lesssim C_K\frac{|h|^\eta }{(2^k+|x|)^{n+\eta }}, \qquad |h|\le 2^k \end{aligned}$$

for some $\eta \in (0,1)$. In addition, since ${{\phi }}_k*K_k*1=0,$ by applying the T1 Theorem for square functions (see [8]) we have

$$\begin{aligned} \bigg \Vert \bigg (\sum _{k\in \mathbb Z}|\phi _k*K_k*f|^2\bigg )^{1/2}\bigg \Vert _{L^{2}}&\lesssim C_K\Vert f\Vert _{L^2}.\end{aligned}$$

(3.9)

Then applying Lemma 3.1 to $\{\mu _k\}=\{\phi _k*K_k\},$ we get

$$\begin{aligned}&\bigg \Vert \bigg (\sum _{k\in \mathbb Z}|\phi _{k}*K_k*f|^2\bigg )^{1/2}\bigg \Vert _{L^{1,\infty }}\\&\quad \lesssim C_K\Vert f\Vert _{L^{1}},\,\,\, \big \Vert \mathcal {M}_{(\sum _{k\in \mathbb Z}|\phi _{k}*K_k*f|^2)^{1/2}}\big \Vert _{L^{1,\infty }}\lesssim C_K\Vert f\Vert _{L^{1}}\end{aligned}$$

and then by Lemma 2.1, for $1<p<\infty$ and $w\in A_p,$

$$\begin{aligned} \bigg \Vert \bigg (\sum _{k\in \mathbb Z}|\phi _{k}*K_k*f|^2\bigg )^{1/2}\bigg \Vert _{L^{p}(w)}&\lesssim C_K\{w\}_{A_p}\Vert f\Vert _{L^{p}(w)}.\end{aligned}$$

This verifies (1.15)–(1.17). The proof of Proposition 1.8 is complete. $\square$

4 Proof of Proposition 1.9

Step 1. First, we would like to prove that

$$\begin{aligned}&\bigg \Vert \bigg (\sum _{k\in \mathbb Z}|({{\phi }}_k-\delta _0)*K^k*f|^q\bigg )^{1/q}\bigg \Vert _{L^{2}} \nonumber \\&\quad \lesssim (C_{K}+\Vert \omega \Vert _{Dini}+\Vert \omega \Vert _{Dini}^{1/q'}\Vert \omega ^{1/2}\Vert _{Dini}^{2/q})\Vert f\Vert _{L^2}.\end{aligned}$$

(4.1)

We first use Fourier transform and Plancherel theorem to deal with the $L^2$-norm of $(\sum _{k\in \mathbb Z}|({{\phi }}_k-\delta _0)*K^k*f|^2)^{1/2}.$ For $|2^k\xi |<1,$ write

$$\begin{aligned} \widehat{K^k}(\xi )&=\int _{2^{k}<|x|<|\xi |^{-1}}K(x)e^{-2\pi ix \cdot \xi } dx \\&\quad +\int _{|\xi |^{-1}<|x|} K(x) e^{-2\pi i x \cdot \xi } dx =: I_{1,k}(\xi )+I_{2,k}(\xi ). \end{aligned}$$

For the term $I_{1,k}$, by the size estimate (2.5) and cancellation (1.12), we have

$$\begin{aligned} |I_{1,k}(\xi )|= \bigg |\int _{2^{k}<|x|\le |\xi |^{-1}}K(x)(e^{-2\pi ix \cdot \xi }-1) dx\bigg |\lesssim 2\pi |\xi |\int _{|x| \le |\xi |^{-1}}|x||K(x)|dx\lesssim C_{K} \end{aligned}$$

uniformly in k.

We now consider $I_{2,k}(\xi )$. Let $z=\frac{\xi }{2|\xi |^2}$ so that $e^{2\pi i z \cdot \xi }=-1$ and $2|z|=|\xi |^{-1}$. By changing variables $x=x'-z$, we rewrite $I_{2,k}(\xi )$ as

$$\begin{aligned} I_{2,k}(\xi )=-\int _{|\xi |^{-1}<|x'-z|}K(x'-z)e^{-2\pi ix'\cdot \xi }dx'. \end{aligned}$$

Taking the average of the above inequality and the original definition of $I_{2,k}$ gives

$$\begin{aligned} I_{2,k}(\xi )=\frac{1}{2}\int _{|\xi |^{-1}<|x|}K(x)e^{-2\pi i x \cdot \xi }dx -\frac{1}{2}\int _{|\xi |^{-1}<|x-z|}K(x-z)e^{-2\pi i x \cdot \xi } dx. \end{aligned}$$

Now we split

$$\begin{aligned} I_{2,k}(\xi )&= \frac{1}{2}\int _{|\xi |^{-1}<|x|}(K(x)-K(x-z))e^{-2\pi i x\cdot \xi } dx\\&\quad +\frac{1}{2}\int _{{\mathbb R}^n} K(x-z)(\chi _{\{|\xi |^{-1}<|x|\}}-\chi _{\{|\xi |^{-1}<|x-z|\}}) e^{-2\pi i x\cdot \xi }dx. \end{aligned}$$

For the first term, by noting that $2|z|=|\xi |^{-1}$ and using (2.6) we get

$$\begin{aligned} \left| \int _{|\xi |^{-1}<|x|}(K(x)-K(x-z))e^{-2\pi i x\cdot \xi } dx\right|&\le \int _{|\xi |^{-1}<|x|}\frac{1}{|x|^n}\omega \left( \frac{|z|}{|x|}\right) dx\\ {}&\le \int _{|\xi |^{-1}<|x|}\frac{1}{|x|^n}\omega \left( \frac{|\xi |^{-1}}{2|x|}\right) dx\\ {}&= \int _{|\xi |^{-1}<r}\omega \left( \frac{|\xi |^{-1}}{2r}\right) \frac{dr}{r}\int _{\mathbb S^{n-1}}d\sigma (\theta )\\&\lesssim \Vert \omega \Vert _{Dini}. \end{aligned}$$

For the second term, we note that $\chi _{|\xi |^{-1}<|x|}-\chi _{|\xi |^{-1}<|x-z|}$ is nonzero if and only if $|\xi |^{-1}-|z|\le |x-z|\le |\xi |^{-1}+|z|$. Thus by (2.5)

$$\begin{aligned}&\left| \int _{{\mathbb R}^n} K(x-z)(\chi _{\{|\xi |^{-1}<|x|\}}-\chi _{\{|\xi |^{-1}<|x-z|\}}) e^{-2\pi i x\cdot \xi }dx\right| \\&\quad \lesssim \int _{|\xi |^{-1}-|z|\le |x|\le |\xi |^{-1}+|z|}\frac{C_{K}}{|x|^{n}}dx \lesssim \int _{\frac{|\xi |^{-1}}{2}\le |x|\le \frac{3|\xi |^{-1}}{2}}\frac{C_{K}}{|x|^{n}}dx\lesssim C_{K}. \end{aligned}$$

Combining the above estimates, we get $|I_{2,k}(\xi )|\lesssim C_{K}+\Vert \omega \Vert _{Dini},$ which, together with the estimate of $|I_{1,k}(\xi )|$, gives

$$\begin{aligned} |\widehat{K^{k}}(\xi )|\lesssim C_{K}+\Vert \omega \Vert _{Dini}. \end{aligned}$$

(4.2)

For $|2^k\xi |>1,$ let $z=\frac{\xi }{2|\xi |^2}$ so that $e^{2\pi i z \cdot \xi }=-1$ and $2|z|=|\xi |^{-1}$. Similarly to $I_{2,k}(\xi )$, we get

$$\begin{aligned} \widehat{K^k}(\xi )&= \frac{1}{2}\int _{2^k<|x|}(K(x)-K(x-z))e^{-2\pi i x\cdot \xi } dx\\&\quad +\frac{1}{2}\int _{{\mathbb R}^n} K(x-z)(\chi _{\{2^k<|x|\}}-\chi _{\{2^k<|x-z|\}}) e^{-2\pi i x\cdot \xi }dx\\&=: J_{1,k}(\xi )+J_{2,k}(\xi ). \end{aligned}$$

For the term $J_{1,k}$, by noting that $2|z|=|\xi |^{-1}$ and using (2.6) we get

$$\begin{aligned} |J_{1,k}(\xi )|&\le \frac{1}{2}\int _{2^k<|x|}\frac{1}{|x|^n}\omega \left( \frac{|z|}{|x|}\right) dx \le \frac{1}{2}\int _{2^k<|x|}\frac{1}{|x|^n}\omega \left( \frac{|\xi |^{-1}}{2|x|}\right) dx\\&=\frac{1}{2}\int _{2^k<r}\omega \left( \frac{|\xi |^{-1}}{2r}\right) \frac{dr}{r}\int _{\mathbb S^{n-1}}d\sigma (\theta )\\&\lesssim \omega ^{1/2}(|2^k\xi |^{-1})\int _{2^k<r}\omega \left( \frac{2^k}{2r}\right) ^{1/2}\frac{dr}{r}\int _{\mathbb S^{n-1}}d\sigma (\theta )\\ {}&\lesssim \Vert \omega ^{1/2}\Vert _{Dini}\omega ^{1/2}(|2^k\xi |^{-1}). \end{aligned}$$

For the term $J_{2,k}$, we note that $\chi _{2^k<|x|}-\chi _{2^k<|x-z|}$ is nonzero if and only if $2^k-|z|\le |x-z|\le 2^k+|z|$. Thus by (2.5)

$$\begin{aligned} |J_{2,k}(\xi )|&\lesssim \int _{2^k-|z|\le |x-z|\le 2^k+|z|} |K(x-z)|dx \lesssim \int _{2^k-|z|\le |x|\le 2^k+|z|}\frac{C_{K}}{|x|^{n}}dx\\&\lesssim \frac{1}{2^k}\int _{2^k-|z|\le |x|\le 2^k+|z|}\frac{C_{K}}{|x|^{n-1}}dx\lesssim C_{K}|2^k\xi |^{-1}. \end{aligned}$$

Combining the estimates of $J_{1,k}(\xi )$ and $J_{2,k}(\xi ),$ we get

$$\begin{aligned} |\widehat{K^k}(\xi )|\lesssim C_{K}|2^k\xi |^{-1}+\Vert \omega ^{1/2}\Vert _{Dini}\omega ^{1/2}(|2^k\xi |^{-1}). \end{aligned}$$

(4.3)

Since ${\widehat{\phi }}\in \mathcal {S}({\mathbb R}^n)$ and ${\widehat{\phi }}(0)=1,$ then $|1-\widehat{\phi _k}(\xi )|\lesssim \min (|2^k\xi |, 1).$ By the Plancherel Theorem, (4.2) and (4.3), we get

$$\begin{aligned}&\bigg \Vert \bigg (\sum _{k\in \mathbb Z}|(\delta _0-\phi _k)*K^k*f|^2\bigg )^{1/2}\bigg \Vert _{ L^2}^2 \nonumber \\&=\sum _{k\in \mathbb {Z}}\int _{{\mathbb R}^n}|\widehat{(\delta _0-\phi _k)}(\xi )|^2|\widehat{K^k}(\xi )|^2|{\widehat{f}}(\xi )|^2d\xi \nonumber \\&\lesssim \int _{\mathbb R^n}\big ((C_{K}+\Vert \omega \Vert _{Dini})^2\sum _{|2^k\xi |\le 1}|2^k\xi |^{2}+C_K^2\sum _{|2^k\xi |\ge 1}|2^{k}\xi |^{-2} \nonumber \\&\quad +\Vert \omega ^{1/2}\Vert _{Dini}^2\sum _{|2^k\xi |\ge 1}\omega (|2^k\xi |^{-1})\big )|{\widehat{f}}(\xi )|^2d\xi \nonumber \\&\lesssim (C_{K}+\Vert \omega \Vert _{Dini}+\Vert \omega ^{1/2}\Vert _{Dini}\Vert \omega \Vert _{Dini}^{1/2})^2\Vert f\Vert _{L^2}^2. \end{aligned}$$

(4.4)

On the other hand, we claim that for any fixed $k\in \mathbb Z,$

$$\begin{aligned} |(\delta _0-\phi _k)*K^k(x)|&\lesssim \omega (2^k/|x|)\frac{1}{|x|^{n}}\chi _{|x|>3\cdot 2^k/4}+\frac{C_K}{|x|^n}\chi _{\frac{3}{4}\cdot 2^k\le |x|\le \frac{5}{4}\cdot 2^k}, \end{aligned}$$

(4.5)

which will be proved in Step 2. Then we can get

$$\begin{aligned} \big \Vert \sup _{k\in \mathbb Z}|(\delta _0-\phi _k)*K^k*f|\big \Vert _{ L^2}\lesssim (C_{K}+\Vert \omega \Vert _{Dini})\Vert Mf\Vert _{L^2}\lesssim (C_{K}+\Vert \omega \Vert _{Dini})\Vert f\Vert _{L^2}. \end{aligned}$$

(4.6)

Interpolating between (4.4) and (4.6), we get for $q\ge 2$

$$\begin{aligned} \bigg \Vert \bigg (\sum _{k\in \mathbb Z}|(\delta _0-\phi _k)*K^k*f|^q\bigg )^{1/q}\bigg \Vert _{ L^2}\lesssim (C_{K}+\Vert \omega \Vert _{Dini}+\Vert \omega \Vert _{Dini}^{1/q'}\Vert \omega ^{1/2}\Vert _{Dini}^{2/q})\Vert f\Vert _{L^2}. \end{aligned}$$

We get (4.1).

Step 2. We now estimate $|(\delta _0-\phi _k)*K^k(x)|$ for any fixed $k\in \mathbb Z.$

$$\begin{aligned} |(\delta _0-\phi _k)*K^k(x)|&=\bigg |\int _{{\mathbb R}^n} {\phi }_k(x-z)(K^k(z)-K^k(x))\,dz\bigg |\\&\le \bigg |\int _{{\mathbb R}^n} {\phi }_k(x-z)(K(z)-K(x))\chi _{|z|> 2^{k}}\,dz\bigg |\\&\quad +\bigg |\int _{{\mathbb R}^n} {\phi }_k(x-z)K(x))(\chi _{|z|> 2^{k}}-\chi _{|x|> 2^{k}})\,dz\bigg |\\ {}&=:A_{k,1}(x)+A_{k,2}(x). \end{aligned}$$

We first consider $A_{k,1}$. By noting that supp ${\phi }_k\subset \{x: |x|\le 2^k/4\}$ and that $|z|>2^{k},$ we have $|x|\ge \frac{3|z|}{4}> 3\cdot 2^k/4.$ Then by the regularity condition of K in (2.6),

$$\begin{aligned} A_{k,1}(x)&\lesssim \int _{|z|>2^{k}} |{\phi }_k(x-z)|\frac{\omega (| x-z|/|x|)}{|x|^{n}}\,dz\lesssim \frac{\omega (2^k/|x|)}{|x|^{n}}\chi _{|x|>3\cdot 2^k/4}(x). \end{aligned}$$

(4.7)

For the term $A_{k,2}$, we first note that $\chi _{|z|> 2^{k}}-\chi _{|x|> 2^{k}}$ is nonzero if and only if $\frac{3}{4}\cdot 2^k\le |x|\le \frac{5}{4}\cdot 2^k$ since $|x-z|\le 2^k/4$. Thus,

$$\begin{aligned}A_{k,2}(x)\lesssim |K(x)|\chi _{\frac{3}{4}\cdot 2^k\le |x|\le \frac{5}{4}\cdot 2^k}(x)\int _{{\mathbb R}^n} |{\phi }_k(x-z)|\,dz\lesssim \frac{C_K}{|x|^n}\chi _{\frac{3}{4}\cdot 2^k\le |x|\le \frac{5}{4}\cdot 2^k}(x). \end{aligned}$$

Combining the two cases, we get that for any fixed $k\in \mathbb Z,$

$$\begin{aligned} |({{\phi }}_k-\delta _0)*K^k(x)|&\lesssim \omega (2^k/|x|)\frac{1}{|x|^{n}}\chi _{|x|>3\cdot 2^k/4}+\frac{C_K}{|x|^n}\chi _{\frac{3}{4}\cdot 2^k\le |x|\le \frac{5}{4}\cdot 2^k}, \end{aligned}$$

(4.8)

which verifies (4.5) and implies that

$$\begin{aligned} \sum _{k\in \mathbb Z}|({{\phi }}_k-\delta _0)*K^k(x)|&\lesssim (\Vert \omega \Vert _{Dini}+C_K)\frac{1}{|x|^{n}}. \end{aligned}$$

(4.9)

We now estimate $\Vert \{({{\phi }}_k-\delta _0)*K^k(x-y)-({{\phi }}_k-\delta _0)*K^k(x)\}_{k}\Vert _{\ell ^q}$ for $0<|y|\le |x|/2.$ To begin with, we claim that for $|y|\le \frac{2^k}{2},$

$$\begin{aligned}&|({{\phi }}_k-\delta _0)*K^k(x-y)-({{\phi }}_k-\delta _0)*K^k(x)|\\&\lesssim \bigg (\frac{\omega (|y|/|x|)}{|x|^{n}}+ C_K\frac{|y|^\theta }{|x|^{n+\theta }}\bigg )\chi _{|x|\ge 2^{k}/2}(x)+\frac{C_K}{|x|^n}\chi _{2^{k}-|y|\le |x|\le 2^{k}+|y|}(x).\nonumber \end{aligned}$$

(4.10)

Assuming (4.10) for the moment, we write

$$\begin{aligned}&\bigg (\sum _{k\in \mathbb Z}|({{\phi }}_{k}-\delta _0)*K^k(x-y)-({{\phi }}_{k}-\delta _0)*K^k(x)|^q\bigg )^{1/q}\\&\quad \le \bigg (\big (\sum _{ {\frac{2^k}{2}}\le |y|}+\sum _{ {\frac{2^k}{2}}\ge |y|}\big )|({{\phi }}_{k}-\delta _0)*K^k(x-y)-({{\phi }}_{k}-\delta _0)*K^k(x)|^q\bigg )^{1/q}=:I\!I_1+I\!I_2. \end{aligned}$$

For the term $I\!I_1$, by noting that $|x|\ge 2|y|>0$ and by using (4.8) we get

$$\begin{aligned}I\!I_1&\le \bigg (\sum _{ \frac{{2^k}}{2}\le |y|}\big (|({{\phi }}_{k}-\delta _0)*K^k(x)|+|({{\phi }}_{k}-\delta _0)*K^k(x-y)|\big )^q\bigg )^{1/q}\\&\lesssim \sum _{ \frac{{2^k}}{2}\le |y|}\frac{C_K}{|x|^n}\chi _{\frac{3}{4}\cdot 2^k\le |x|\le \frac{5}{4}\cdot 2^k}(x)+\bigg (\sum _{ \frac{{2^k}}{2}\le |y|}\big (\frac{\omega (2^k/|x|)}{|x|^{n}}\chi _{|x|>3\cdot 2^k/4}\big )^{q}\bigg )^{1/q}\\&\lesssim \frac{C_K|y|}{|x|^{n+1}}+\frac{\omega (|y|/|x|)^{\frac{1}{q'}}}{|x|^{n}}\bigg (\sum _{|x|>3\cdot 2^k/4}\omega (2^k/|x|)\bigg )^{1/q} \\&\lesssim \frac{C_K|y|}{|x|^{n+1}}+\frac{\omega (|y|/|x|)^{\frac{1}{q'}}}{|x|^{n}}\Vert \omega \Vert _{Dini}^{1/q}. \end{aligned}$$

For the term $I\!I_2$, by using (4.10),

$$\begin{aligned} I\!I_2&\lesssim \sum _{\frac{{2^k}}{2}\ge |y|}C_K\frac{|y|^\theta }{|x|^{n+\theta }}\chi _{|x|\ge 2^{k}/4}\\&\quad +\bigg (\sum _{\frac{{2^k}}{2}\ge |y|}\bigg (\frac{\omega (|y|/|x|)}{|x|^{n}}\chi _{|x|\ge \frac{2^{k}}{4}}\bigg )^q\bigg )^{1/q}+\sum _{\frac{{2^k}}{2}\ge |y|}\frac{C_K}{|x|^n}\chi _{2^{k}-|y|\le |x|\le 2^{k}+|y|}\\&\lesssim \frac{C_K |y|^{\theta /2}}{|x|^{n+\theta /2}}+\frac{\omega (|y|/|x|)^{\frac{1}{q'}}}{|x|^{n}}\bigg (\sum _{{\frac{2^k}{2}}\ge |y|}\omega (|y|/2^k)\bigg )^{1/q}+\sum _{{\frac{2^k}{2}}\ge |y|}\frac{C_K}{|x|^n}\chi _{2^{k}-|y|\le |x|\le 2^{k}+|y|}\\&\lesssim \frac{C_K |y|^{\theta /2}}{|x|^{n+\theta /2}}+\frac{\omega (|y|/|x|)^{\frac{1}{q'}}}{|x|^{n}}\Vert \omega \Vert _{Dini}^{1/q}+\sum _{{\frac{2^k}{2}}\ge |y|}\frac{C_K}{|x|^n}\chi _{2^{k}-|y|\le |x|\le 2^{k}+|y|}. \end{aligned}$$

Combining the estimates of $I\!I_1$ and $I\!I_2$, we get

$$\begin{aligned}&\bigg (\sum _{k\in \mathbb Z}|({{\phi }}_{k}-\delta _0)*K^k(x-y)-({{\phi }}_{k}-\delta _0)*K^k(x)|^q\bigg )^{1/q}\\&\quad \lesssim \frac{C_K |y|^{\theta /2}}{|x|^{n+\theta /2}}+\frac{\omega (|y|/|x|)^{\frac{1}{q'}}}{|x|^{n}}\Vert \omega \Vert _{Dini}^{1/q}+\sum _{\frac{{2^k}}{2}\ge |y|}\frac{C_K}{|x|^n}\chi _{2^{k}-|y|\le |x|\le 2^{k}+|y|}.\nonumber \end{aligned}$$

(4.11)

Now we return to give the proof of (4.10). We write

$$\begin{aligned}&|({{\phi }}_k-\delta _0)*K^k(x-y)-({{\phi }}_k-\delta _0)*K^k(x)|\\&\le \bigg |\int _{{\mathbb R}^n}\phi _{k}(x-z)\big (K^{k}(x-y)-K^{k}(x)\big )\,dz\bigg |\\ {}&\quad +\bigg |\int _{{\mathbb R}^n}\phi _{k}(x-z)\big (K^{k}(z-y)-K^{k}(z)\big )\,dz\bigg |\\&=:I\!I\!I_1+I\!I\!I_2. \end{aligned}$$

For the term $I\!I\!I_1$, recall that $|y|\le \frac{2^k}{2}$. By noting the facts that $0<|y|\le \frac{2^k}{2}$ and $|x-y|\ge 2^{k}$ imply $|x|>\frac{2^k}{2}$ and that $\chi _{|x-y|>2^{k}}-\chi _{|x|>2^{k}}$ is nonzero if and only if $2^{k}-|y|\le |x|\le 2^{k}+|y|$, we have

$$\begin{aligned} I\!I\!I_1&\lesssim \int _{{\mathbb R}^n}|\phi _{k}(x-z)||K(x-y)-K(x)|\chi _{|x-y|>2^{k}}\,dz\\&\quad +\int _{{\mathbb R}^n}|\phi _{k}(x-z)||K(x)||\chi _{|x-y|>2^{k}}-\chi _{|x|>2^{k}}|\,dz\\&\lesssim \frac{\omega (|y|/|x|)}{|x|^{n}}\chi _{|x|>\frac{2^{k}}{2}}+\frac{C_K}{|x|^n}\chi _{2^{k}-|y|\le |x|\le 2^{k}+|y|}.\end{aligned}$$

Similarly, for the term $I\!I\!I_2$, we get

$$\begin{aligned} I\!I\!I_2&\lesssim \int _{{\mathbb R}^n}|\phi _{k}(x-z)|\frac{\omega (|y|/|z|)}{|z|^{n}}\chi _{|z|>\frac{2^{k}}{2}}\,dz+C_K\int _{{\mathbb R}^n}\frac{|\phi _{k}(x-z)|}{|z|^n}\chi _{2^{k}-|y|\le |z|\le 2^{k}+|y|}\,dz. \end{aligned}$$

Since $|x-z|\le 2^k/4$ and $|z|>\frac{2^k}{2}$, we get $3|z|/2\ge |x|\ge |z|/2>2^k/4$. Thus, there exists some $\eta \in (0,1)$,

$$\begin{aligned} I\!I\!I_2&\lesssim \frac{\omega (|y|/|x|)}{|x|^{n}}\chi _{|x|\ge 2^{k}/4}+C_K\frac{|y|^\eta }{|x|^{n+\eta }}\chi _{|x|\ge 2^{k}/4}. \end{aligned}$$

Combining the estimates of $I\!I\!I_1$ and $I\!I\!I_2$, we get (4.10).

Step 3. By Lemma 2.1, to prove (1.20), we need to verify that

$$\begin{aligned}&\left\| \left( \sum _{k\in \mathbb Z}|(\phi _{k}-\delta _0)*K^k*f|^{q}\right) ^{1/q}\right\| _{L^{1,\infty }}\nonumber \\&\quad \lesssim (C_K+\Vert \omega \Vert _{Dini}+\Vert \omega \Vert _{Dini}^{1/q}\Vert \omega ^{1/q'}\Vert _{Dini} +\Vert \omega \Vert _{Dini}^{1/q'}\Vert \omega ^{1/2}\Vert _{Dini}^{2/q})\Vert f\Vert _{L^{1}}\end{aligned}$$

(4.12)

and

$$\begin{aligned}&\Vert \mathcal {M}_{(\sum _{k\in \mathbb Z}|(\phi _{k}-\delta _0)*K^k*f|^q)^{1/q}}\Vert _{L^{1,\infty }}\lesssim (C_K+\Vert \omega \Vert _{Dini}\nonumber \\&+\Vert \omega \Vert _{Dini}^{1/q}\Vert \omega ^{1/q'}\Vert _{Dini}+\Vert \omega \Vert _{Dini}^{1/q'}\Vert \omega ^{1/2}\Vert _{Dini}^{2/q})\Vert f\Vert _{L^{1}}. \end{aligned}$$

(4.13)

The two above inequalities just are (1.18) and (1.19).

To verify (4.12), we apply the Calderón-Zygmund decomposition to f at height $\alpha$ to obtain that there is a disjoint family of dyadic cubes $\{Q\}$ with total measure $\sum _Q|Q| \lesssim {\alpha }^{-1}\Vert f\Vert _{L^1},$ and that $f=g+h,$ with $\Vert g\Vert _{L^\infty }\lesssim \alpha ,$ and $\Vert g\Vert _{L^1}\lesssim \Vert f\Vert _{L^1}$, $h=\sum _{Q}h_Q,$ where supp$(h_Q)\subseteq Q,$ $\int _{\mathbb R^n} h_Q(x)\,dx=0$ and $\sum \Vert h_Q\Vert _{L^1}\lesssim \Vert f\Vert _{L^1}.$

By Chebychev’s inequality and (4.1), we get

$$\begin{aligned}&\big | \big \{x: \Big (\sum _{k\in \mathbb Z}|({{\phi }}_k-\delta _0)*K^k*g(x)|^q\Big )^{1/q}> \alpha \big \}\big |\\&\quad \lesssim ( C_K+\Vert \omega \Vert _{Dini}+\Vert \omega \Vert _{Dini}^{1/q'}\Vert \omega ^{1/2}\Vert _{Dini}^{2/q}) \alpha ^{-2}\Vert g\Vert _{L^2}^2\\&\quad \lesssim ( C_K+\Vert \omega \Vert _{Dini}+\Vert \omega \Vert _{Dini}^{1/q'}\Vert \omega ^{1/2}\Vert _{Dini}^{2/q})\frac{\Vert f\Vert _{L^1}}{\alpha }. \end{aligned}$$

Then it suffices to estimate $(\sum _{k\in \mathbb Z}|({{\phi }}_k-\delta _0)*K^k*h(x)|^q)^{1/q}$ away from $\bigcup {\widetilde{Q}}$ with ${\widetilde{Q}}=2Q.$ Let $y_Q$ denote the center of Q. Since $\int _{\mathbb R^n} h_Q(x)\,dx=0$, then we can write

$$\begin{aligned}&({{\phi }}_{k}-\delta _0)*K^k*h(x)=\sum _Q({{\phi }}_{k}-\delta _0)*K^k*h_Q(x)\\&\quad =\sum _Q\int _Q(\phi _k-\delta _0)*K^k(x-y)h_Q(y)\,dy\\ {}&=\sum _Q\int _Q[(\phi _k-\delta _0)*K^k(x-y)-(\phi _k-\delta _0)*K^k(x-y_Q)]h_Q(y)\,dy. \end{aligned}$$

Now by Chebychev’s inequality and (4.11),

$$\begin{aligned}&\alpha \big |\{x\notin \cup {\widetilde{Q}}:\Big (\sum _{k\in \mathbb Z}|({{\phi }}_{k}-\delta _0)*K^k*h(x)|^q\Big )^{1/q}>\alpha \}\big |\\&\le \displaystyle \sum _{Q}\displaystyle \int _{x\notin {\widetilde{Q}}}\displaystyle \int _Q|h_Q(y)|\Big (\sum _{k\in \mathbb Z}\big |(\phi _k-\delta _0)*K^k(x-y)-(\phi _k-\delta _0)*K^k(x-y_Q)\big |^q\Big )^{1/q}dy\,dx\\&\lesssim \displaystyle \sum _{Q}\displaystyle \int _Q|h_Q(y)|\displaystyle \int _{|x-y_Q|\ge 2\ell (Q)}\bigg (\frac{C_K |y-y_Q|^{\eta /2}}{|x-y_Q|^{n+\eta /2}}+\frac{\omega (|y-y_Q|/|x-y_Q|)^{\frac{1}{q'}}}{|x-y_Q|^{n}}\Vert \omega \Vert _{Dini}^{1/q}\\ {}&\quad +\sum _{\frac{{2^k}}{2}\ge |y-y_Q|}\frac{C_K}{|x-y_Q|^n}\chi _{2^{k}-|y-y_Q|\le |x-y_Q|\le 2^{k}+|y-y_Q|}\bigg )\,dxdy\\ {}&\lesssim \displaystyle \sum _{Q}\displaystyle \int _Q|h_Q(y)|\sum _{j=1}^\infty \displaystyle \int _{2^{j}\ell (Q)<|x-y_Q|\le 2^{j+1}\ell (Q)}\frac{C_K |y-y_Q|^{\eta /2}}{|x-y_Q|^{n+\eta /2}} \\&\quad +\frac{\omega (|y-y_Q|/|x-y_Q|)^{\frac{1}{q'}}}{|x-y_Q|^{n}}\Vert \omega \Vert _{Dini}^{1/q}\,dx\,dy\\&\quad +\displaystyle \sum _{Q}\displaystyle \int _Q|h_Q(y)|\sum _{\frac{{2^k}}{2}\ge |y-y_Q|}\int _{2^{k}-|y-y_Q|\le |x-y_Q|\le 2^{k}+|y-y_Q|}\frac{C_K}{|x-y_Q|^n}\,dxdy. \end{aligned}$$

If $|y-y_Q|\le 2^k/2,$ then $|x-y_Q|\ge 2^k-|y-y_Q|\ge 2^k/2,$ thereby $|x-y_Q|^{-1}\le 2\cdot 2^{-k},$

$$\begin{aligned}&\alpha \big |\{x\notin \cup {\widetilde{Q}}:\Big (\sum _{k\in \mathbb Z}|({{\phi }}_{k}-\delta _0)*K^k*h(x)|^q\Big )^{1/q}>\alpha \}\big |\\&\lesssim \displaystyle \sum _{Q}\displaystyle \int _Q|h_Q(y)|\sum _{j=1}^\infty \displaystyle \int _{2^{j}\ell (Q)<|x-y_Q|\le 2^{j+1}\ell (Q)}\\&\quad \frac{C_K \ell (Q)^{\eta /2}}{2^{j(n+\eta /2)}\ell (Q)^{(n+\eta /2)}}+\frac{\omega (2^{-j})^{\frac{1}{q'}}}{(2^j\ell (Q))^{n}}\Vert \omega \Vert _{Dini}^{1/q}\,dx\,dy\\&\quad + C_K\displaystyle \sum _{Q}\displaystyle \int _Q|h_Q(y)|\sum _{\frac{{2^k}}{2}\ge |y-y_Q|}2^{-k}\int _{2^{k}-|y-y_Q|\le r\le 2^{k}+|y-y_Q|}\,dr\int _{\mathbb S^{n-1}}\,d\sigma (\theta )dy\\&\lesssim C_K\displaystyle \sum _{Q}\displaystyle \int _Q|h_Q(y)|\,dy\sum _{j=1}^\infty (2^{-\eta j/2} +\omega (2^{-j})^{\frac{1}{q'}}\Vert \omega \Vert _{Dini}^{1/q})\\&\quad + C_K\displaystyle \sum _{Q}\displaystyle \int _Q|h_Q(y)|\sum _{\frac{2^{k}}{2}\ge |y-y_Q|}\frac{|y-y_Q|}{2^{k}}dy\\&\lesssim (C_K+\Vert {\omega }^{1/q'}\Vert _{Dini}\Vert \omega \Vert _{Dini}^{1/q})\displaystyle \sum _{Q}\Vert h_Q\Vert _{L^1}\\&\quad + C_K\displaystyle \sum _{Q}\displaystyle \int _Q|h_Q(y)||y-y_Q|\sum _{\frac{2^{k}}{2}\ge |y-y_Q|}{2^{-k}}dy\\&\lesssim (C_K+\Vert {\omega }^{1/q'}\Vert _{Dini}\Vert \omega \Vert _{Dini}^{1/q})\displaystyle \sum _{Q}\Vert h_Q\Vert _{L^1}+ C_K\displaystyle \sum _{Q}\displaystyle \int _Q|h_Q(y)|dy\\&\lesssim (C_K+\Vert {\omega }^{1/q'}\Vert _{Dini}\Vert \omega \Vert _{Dini}^{1/q})\Vert f\Vert _{L^1} \end{aligned}$$

This finishes the proof of (4.12).

Now, we verify (4.13). Let Q be a cube, and take $x,\,\xi \in Q.$ Let $B(x)=B(x,2\sqrt{n}\ell (Q)),$ then $3Q\subset B_x.$ By the triangle inequality, we get

$$\begin{aligned}&\bigg (\sum _{k\in \mathbb Z}|(\phi _{k}-\delta _0)*K^k*(f{\chi _{{\mathbb R}^n\setminus 3Q}})(\xi )|^q\bigg )^{1/q}\\ {}&\le \big |\bigg (\sum _{k\in \mathbb Z}|(\phi _{k}-\delta _0)*K^k*(f{\chi _{{\mathbb R}^n\setminus B_x}})(\xi )|^q\bigg )^{1/q}-\bigg (\sum _{k\in \mathbb Z}|(\phi _{k}-\delta _0)*K^k*(f{\chi _{{\mathbb R}^n\setminus B_x}})(x)|^q\bigg )^{1/q}\big |\\ {}&\quad +\bigg (\sum _{k\in \mathbb Z}|(\phi _{k}-\delta _0)*K^k*f{\chi _{B_x\setminus 3Q}})(\xi )|^q\bigg )^{1/q}+\bigg (\sum _{k\in \mathbb Z}|(\phi _{k}-\delta _0)*K^k*(f{\chi _{{\mathbb R}^n\setminus B_x}})(x)|^q\bigg )^{1/q}\\ {}&=:I+II+III. \end{aligned}$$

For the term I,  by using the triangular inequality,

$$\begin{aligned} I&\lesssim \bigg ( \sum _{|x-\xi |\ge \frac{2^k}{2}}\left( \int _{{\mathbb R}^n \setminus B_x}|(\phi _{k}-\delta _{0})*K^{k}(\xi -y)-(\phi _{k}-\delta _{0})*K^{k}(x-y)||f(y)|\,dy\right) ^q\bigg )^{1/q}\\ {}&\quad +\bigg ( \sum _{|x-\xi |\le \frac{2^k}{2}}\left( \int _{{\mathbb R}^n \setminus B_x}|(\phi _{k}-\delta _{0})*K^{k}(\xi -y)-(\phi _{k}-\delta _{0})*K^{k}(x-y)||f(y)|\,dy\right) ^q\bigg )^{1/q} \\&=:I_1+I_2. \end{aligned}$$

By using the size estimate (4.8) and the fact that $\frac{1}{2}|x-y|\le |\xi -y|\le \frac{3}{2}|x-y|$ (since $2|x-\xi |\le |x-y|$), we get

$$\begin{aligned} I_1&\lesssim \bigg ( \sum _{|x-\xi |\ge \frac{2^k}{2}}\bigg (\int _{{\mathbb R}^n \setminus B_x}\frac{\omega (2^k/|x-y|)}{|x-y|^{n}}\chi _{|x-y|> \frac{3}{4}\cdot 2^{k}}|f(y)|\,dy\bigg )^q\bigg )^{1/q} \\&\quad +C_K\int _{{\mathbb R}^n \setminus B_x}\sum _{|x-\xi |\ge \frac{2^k}{2}}\frac{|f(y)|}{|x-y|^n}\chi _{\frac{3}{4}\cdot 2^{k}\le |x-y|\le \frac{5}{4}\cdot 2^{k}}\,dy\\ {}&\lesssim \bigg ( \sum _{|x-\xi |\ge \frac{2^k}{2}}\bigg (\sum _{j\ge 1}\int _{2^{j}\sqrt{n}\ell (Q)\le |x-y|\le 2^{j+1}\sqrt{n}\ell (Q)}\omega (\frac{2^k}{2^{j}\ell (Q)})\frac{|f(y)|}{(2^{j}\ell (Q))^{n}}\,dy\bigg )^q\bigg )^{1/q}\\&\quad + C_K\int _{{\mathbb R}^n \setminus B_x}\frac{|x-\xi |}{|x-y|^{n+1}}|f(y)|\,dy\\ {}&\lesssim \Bigg ( \sum _{|x-\xi |\ge \frac{2^k}{2}}\omega \bigg (\frac{2^k}{\ell (Q)}\bigg )\Bigg )^{1/q}\sum _{j\ge 1}\omega ^{1/q'}(2^{-j})Mf(x)+C_KMf(x)\\&\lesssim (\Vert \omega \Vert _{Dini}^{1/q}\Vert \omega ^{1/q'}\Vert _{Dini}+C_K)Mf(x). \end{aligned}$$

For the term $I_2$, by (4.10), we get

$$\begin{aligned} I_2&\lesssim \left( \sum _{|x-\xi |\le \frac{2^k}{2}}\left( \int _{{\mathbb R}^n \setminus B_x}\left( \frac{\omega (|x-\xi |/|x-y|)}{|x-y|^{n}}+ C_K\frac{|x-\xi |^\eta }{|x-y|^{n+\eta }}\right) \chi _{|x-y|\ge 2^{k}/4}|f(y)|\,dy\right) ^q\right) ^{1/q}\\ {}&\quad +C_K\bigg ( \sum _{|x-\xi |\le \frac{2^k}{2}}\Big (\int _{{\mathbb R}^n \setminus B_x}\frac{|f(y)|}{|x-y|^n}\chi _{2^{k}-|x-\xi |\le |x-y|\le 2^{k}+|x-\xi |}\,dy\Big )^q\bigg )^{1/q}\\ {}&\lesssim (\Vert \omega \Vert _{Dini}^{1/q}\Vert \omega ^{1/q'}\Vert _{Dini}+C_K)Mf(x)+{\widetilde{I}}_2. \end{aligned}$$

To estimate ${\widetilde{I}}_2$, it suffices to consider the form

$$\begin{aligned} C_K\bigg (\sum _{|x-\xi |\le \frac{2^k}{2}}\bigg (\int _{s_{k}\le |x-y|\le s_{k+1}}\frac{1}{|x-y|^n}| f(y)|\,dy\bigg )^{q}\bigg )^{1/q}, \end{aligned}$$

where $|s_{k+1}-s_{k}| \le 2|x-\xi |,$ and $\frac{2^{k}}{2}\le s_k\le \frac{3}{2}\cdot 2^{k}.$ Using the hypothesis that $|x-\xi |<\ell (Q)$ and the kernel estimate we can bound the above by a dimensional constant times

$$\begin{aligned} C_{K}\bigg (\sum _{|x-\xi |\le \frac{2^k}{2}}\bigg (s_{k}^{-n} \int _{s_{k}\le |x-y|\le s_{k+1}}|f(y)|\,dy\bigg )^{q}\bigg )^{1/q}. \end{aligned}$$

The above $\ell ^{q}$ norm can be written as

$$\begin{aligned}&~\bigg (\sum _{|x-\xi |\le \frac{2^k}{2}}(s_{k}^{-n}\int _{|x-y|\le s_{k+1}}|f(y)|\,dy-s_{k}^{-n}\int _{|x-y|\le s_{k}}|f(y)|\,dy)^{q}\bigg )^{1 / q} \\&\le \bigg (\sum _{k\in \mathbb Z}\bigg (s_{k+1}^{-n} \int _{|x-y|\le s_{k+1}}|f(y)|\,dy-s_{k}^{-n} \int _{|x-y|\le s_{k}}|f(y)|\,dy\bigg )^{q}\bigg )^{1/q}\\&\quad +\bigg (\sum _{|x-\xi |\le \frac{2^k}{2}}((s_{k}^{-n}-s_{k+1}^{-n}) \int _{|x-y|\le s_{k+1}}|f(y)|\,dy)^{q}\bigg )^{1/q} \\&\lesssim {V}_{q}\mathcal {A}(|f|)(x)+Mf(x)\bigg (\sum _{|x-\xi |\le \frac{2^k}{2}}((s_{k}^{-n}-s_{k+1}^{-n}) / s_{k+1}^{-n})^{q}\bigg )^{1/q}, \end{aligned}$$

where $V_{q}\mathcal {A} (|f|)(x)=V_q\{A_{t}(|f|)(x)\}_{t>0},$ and

$$\begin{aligned} A_{t}(f)(x)=\frac{1}{|B_{t}|} \int _{B_{t}} f(x+y) dy, \quad x \in \mathbb {R}^{n}, t>0. \end{aligned}$$

Here $B_{t}$ denotes the open ball in $\mathbb {R}^{n}$ of center at the origin and radius t. Also note that

$$\begin{aligned} \bigg (\sum _{|x-\xi |\le \frac{2^k}{2}}((s_{k}^{-n}-s_{k+1}^{-n}) / s_{k+1}^{-n})^{q}\bigg )^{1/q}&\lesssim \sum _{|x-\xi |\le \frac{2^k}{2}} \frac{s_{k+1}-s_{k}}{2^k} \lesssim \sum _{|x-\xi |\le \frac{2^k}{2}} \frac{|x-\xi |}{2^k} \le c_n. \end{aligned}$$

Thus we get

$$\begin{aligned} {\widetilde{I}}_2&\lesssim C_K({V}_{q}\mathcal {A}(|f|)(x)+Mf(x)). \end{aligned}$$

(4.14)

Thus combining the estimates of $I_1$ and $I_2,$ we get

$$\begin{aligned} I&\le (\Vert \omega \Vert _{Dini}^{1/q}\Vert \omega ^{1/q'}\Vert _{Dini}+C_K)Mf(x)+C_K {V}_{q}\mathcal {A}(|f|)(x). \end{aligned}$$

For the term II, by using (4.9) and the fact that $|x-y|\simeq |\xi -y|$ (since $2|x-\xi |\le |x-y|$), we get

$$\begin{aligned} II&\le \int _{{\mathbb R}^n} \sum _{k\in \mathbb Z}|(\phi _{k}-\delta _{0})*K^{k}(\xi -y)||f|{\chi _{B_x\setminus 3Q}}(y)\,dy\\ {}&\lesssim (\Vert \omega \Vert _{Dini}+C_K)\int _{3\ell (Q)\le |x-y|\le 2\sqrt{n}\ell (Q)}|\xi -y|^{-n}|f(y)|\,dy\lesssim (\Vert \omega \Vert _{Dini}+C_K)Mf(x).\nonumber \end{aligned}$$

(4.15)

For the term III, since supp $(\phi _{k}-\delta _{0})*K^{k}\subset \{x:|x|\ge \frac{3}{4}\cdot 2^k\},$ we obtain that

$$\begin{aligned} III&\le \bigg (\sum _{\frac{3}{4}\cdot 2^k\ge 2\sqrt{n} l(Q)}|(\phi _{k}-\delta _{0})*K^{k}*f\chi _{{\mathbb R}^n\setminus B_x}(x)|^{q}\bigg )^{1/q}\\ {}&\quad + \bigg (\sum _{\frac{3}{4}\cdot 2^k\le 2\sqrt{n} l(Q)}|(\phi _{k}-\delta _{0})*K^{k}*f\chi _{{\mathbb R}^n\setminus B_x}(x)|^{q}\bigg )^{1/q}\\ {}&\le \bigg (\sum _{k\in \mathbb Z}|(\phi _{k}-\delta _{0})*K^{k}*f(x)|^{q}\Bigg )^{1/q}\\ {}&\quad + \bigg (\sum _{\frac{3}{4}\cdot 2^k\le 2\sqrt{n} l(Q)}|(\phi _{k}-\delta _{0})*K^{k}*f\chi _{{\mathbb R}^n\setminus B_x}(x)|^{q}\bigg )^{1/q}. \end{aligned}$$

By using the size estimate (4.8) we get

$$\begin{aligned}&|(\phi _{k}-\delta _{0})*K^{k}*f\chi _{\mathbb {R}^{n}\backslash B_{x}}(x)|\\&\quad \lesssim \int _{|x-y|>2\sqrt{n} l(Q)}\Big (\omega (2^k/|x-y|)+\frac{C_K2^k}{|x-y|}\chi _{\frac{3}{4}\cdot 2^k\le |x-y|\le \frac{5}{4}\cdot 2^k}\Big )\frac{|f(y)|}{|x-y|^n}dy\\&\quad \lesssim \sum _{j\ge 1}\int _{2^{j}\sqrt{n} l(Q)\le |x-y|\le 2^{j+1}\sqrt{n} l(Q)}\bigg (\omega (\frac{2^k}{2^j\ell (Q)})+C_K\frac{2^k}{2^j\ell (Q)}\bigg )\frac{|f(y)|}{(2^j\ell (Q))^n}dy\\&\quad \lesssim \Big (\sum _{j\ge 1}\omega (2^{-j})^{1/q'}\omega (\frac{2^k}{\ell (Q)})^{1/q}+C_K\frac{2^k}{\ell (Q)}\Big )Mf(x)\\&\quad \lesssim \bigg (\Vert \omega ^{1/q'}\Vert _{Dini}\omega (\frac{2^k}{\ell (Q)})^{1/q}+C_K\frac{2^k}{\ell (Q)}\bigg )Mf(x), \end{aligned}$$

which implies

$$\begin{aligned}&\bigg (\sum _{\frac{3}{4}\cdot 2^{k}\le 2\sqrt{n} l(Q)}|(\phi _{k}-\delta _{0})*K^{k}*f\chi _{\mathbb {R}^{n}\backslash B_{x}}(x)|^{q}\bigg )^{1/q} \\ {}&\quad \lesssim \big (\Vert \omega \Vert _{Dini}^{1/q}\Vert \omega ^{1/q'}\Vert _{Dini}+C_K\big )Mf(x). \end{aligned}$$

Thus, we obtain that

$$\begin{aligned} III&\lesssim \bigg (\sum _{k\in \mathbb {Z}}|(\phi _{k}-\delta _{0})*K^{k}*f(x)|^{q}\bigg )^{1/q}+ (\Vert \omega \Vert _{Dini}^{1/q}\Vert \omega ^{1/q'}\Vert _{Dini}+C_K)Mf(x). \end{aligned}$$

Combining the estimates of $I,\,II$ and III,  we get

$$\begin{aligned}&\mathcal { M}_{(\sum _{j\in \mathbb Z}|(\phi _{k}-\delta _0)*K^k*f(x)|^{q})^{1/q}}\\&\quad \lesssim \bigg (\sum _{k\in \mathbb Z}|(\phi _{k}-\delta _0)*K^k*f(x)|^{q}\bigg )^{1/q}+(C_K+ \Vert \omega \Vert _{Dini}^{1/q}\Vert \omega ^{1/q'}\Vert _{Dini})Mf(x)\\&\qquad +C_K {V}_{q}\mathcal {A}(|f|)(x). \end{aligned}$$

Then by the weak type (1, 1) of M (see [41]) and ${V}_{q}\mathcal {A}(|f|)$(see [23]), and (4.12), we get

$$\begin{aligned}&\Vert \mathcal {M}_{(\sum _{k\in \mathbb Z}|(\phi _{k}-\delta _0)*K^k*f|^{q})^{1/q}}\Vert _{L^{1,\infty }}\\ {}&\quad \lesssim (C_K+ \Vert \omega \Vert _{Dini}^{1/q}\Vert \omega ^{1/q'}\Vert _{Dini}+\Vert \omega \Vert _{Dini}^{1/q'}\Vert \omega ^{1/2}\Vert _{Dini}^{2/q}+\Vert \omega \Vert _{Dini})\Vert f\Vert _{L^{1}}, \end{aligned}$$

which gives the proof of (4.13). The proof of Proposition 1.9 is complete.$\square$

5 Proofs of Theorems 1.6 and 1.7

Proof of Theorem 1.7

Recalling that for any fixed $k\in \mathbb Z,$ we denote by

$$\begin{aligned}K_{k}(x)=K(x)\chi _{|x|\le 2^{k}},\,\, K^{k}(x)=K(x)\chi _{|x|>2^{k}}.\end{aligned}$$

Then for any fixed $k\in \mathbb Z,$ write

$$\begin{aligned} \phi _{k}*K(x)=\phi _{k}*K_{k}(x)+(\phi _{k}-\delta _{0})*K^{k}(x)+K^{k}(x), \end{aligned}$$

where $\delta _0$ is the Dirac measure at 0. Thus by the triangle inequality, we get

$$\begin{aligned} V_{q}\{\phi _{k}*K*f\}_{k\in \mathbb {Z}}&\le V_{q}\{\phi _{k}*K_{k}*f\}_{k\in \mathbb {Z}}+V_{q}\{(\phi _{k}-\delta _{0})*K^{k}*f\}_{k\in \mathbb {Z}}+V_{q}\{K^{k}*f\}_{k\in \mathbb {Z}}\\&\le \bigg (\sum _{k\in \mathbb {Z}}|\phi _{k}*K_{k}*f|^{q}\bigg )^{1/q}+\bigg (\sum _{k\in \mathbb {Z}}|(\phi _{k}-\delta _{0})*K^{k}*f|^{q}\bigg )^{1/q}\\&\quad + V_{q}\{K^{k}*f\}_{k\in \mathbb {Z}}. \end{aligned}$$

Since the following results have been established in [9, Theorem 1.5], [14, Prosition 2.1] and [6, Theorem 1.3],

$$\begin{aligned} \Vert V_{q}\{K^{k}*f\}_{k\in \mathbb {Z}}\Vert _{L^{1,\infty }}&\lesssim (\Vert \omega \Vert _{Dini}+C_K+\Vert T\Vert _{L^2\rightarrow L^2})\Vert f\Vert _{L^{1}}. \end{aligned}$$

(5.1)

and for $1<p<\infty ,$ $w\in A_p,$

$$\begin{aligned} \Vert V_{q}\{K^{k}*f\}_{k\in \mathbb {Z}}\Vert _{L^{p}(w)}&\lesssim (\Vert \omega \Vert _{Dini}+C_K+\Vert T\Vert _{L^2\rightarrow L^2})\{w\}_{A_p}\Vert f\Vert _{L^{p}(w)}. \end{aligned}$$

(5.2)

Then combining Propositions 1.8 and 1.9, we finish the proof of Theorem 1.7. $\square$

Proof of Theorem 1.6

Recall the definition of the operator $T_{\Omega }$ given in the introduction. It can be written as

$$\begin{aligned} T_{\Omega }f=\sum _{k \in \mathbb {Z}} T_{k} f=\sum _{k \in \mathbb {Z}} v_{k}*f, \quad v_{k}=\frac{\Omega (x^{\prime })}{|x|^{n}} \chi _{\{2^{k}<|x|\le 2^{k+1}\}}. \end{aligned}$$

(5.3)

We consider the following partition of unity. Let $\varphi \in C_{c}^{\infty }(\mathbb {R}^{n})$ be such that supp $\varphi \subset \{x:|x| \le \frac{1}{100}\}$ and $\int \varphi d x=1,$ and so that ${\widehat{\varphi }} \in \mathcal {S}(\mathbb {R}^{n}).$ Let us also define $\psi$ by ${\widehat{\psi }}(\xi )={\widehat{\varphi }}(\xi )-{\widehat{\varphi }}(2 \xi ).$ Then, with this choice of $\psi ,$ it follows that $\int \psi d x=0$. We write $\varphi _{j}(x)=\frac{1}{2^{j n}} \varphi (\frac{x}{2^{j}}),$ and $\psi _{j}(x)=\frac{1}{2^{j n}} \psi (\frac{x}{2^{j}})$. We now define the partial sum operators $S_{j}$ by $S_{j}(f)=f*\varphi _{j}$. Their differences are given by

$$\begin{aligned} S_{j}(f)-S_{j+1}(f)=f*\psi _{j}. \end{aligned}$$

(5.4)

Since $S_{j}f \rightarrow f$ as $j \rightarrow -\infty$, for any sequence of integers $\mathcal {N}=\{N(j)\}_{j=0}^{\infty }$, with $0 = N(0)< N(1)<\cdot \cdot \cdot < N(j)\rightarrow \infty$, we have the identity

$$\begin{aligned} T_{k}=T_{k} S_{k}+\sum _{j=1}^{\infty } T_{k}(S_{k-N(j)}-S_{k-N(j-1)}). \end{aligned}$$

(5.5)

In this way,

$$\begin{aligned} T_{\Omega }=\sum _{j=0}^{\infty } {\widetilde{T}}_{j}=\sum _{j=0}^{\infty } {T}_{j}^{\mathcal {N}},\end{aligned}$$

(5.6)

where

$$\begin{aligned} {\widetilde{T}}_{0}:={T}_{0}^{\mathcal {N}}:=\sum _{k \in \mathbb {Z}} T_{k} S_{k} \end{aligned}$$

(5.7)

and, for $j\ge 1$,

$$\begin{aligned} {\widetilde{T}}_{j}&:=\sum _{k \in \mathbb {Z}} T_{k}(S_{k-j}-S_{k-(j-1)}),\nonumber \\ {T}_{j}^{\mathcal {N}}&:=\sum _{k \in \mathbb {Z}} T_{k}(S_{k-N(j)}-S_{k-N(j-1)})=\sum _{i=N(j-1)+1}^{N(j)} {{\widetilde{T}}}_{i}. \end{aligned}$$

(5.8)

Therefore,

$$\begin{aligned}\Vert V_q\{\phi _l*T_\Omega f\}_{l\in \mathbb Z}\Vert _{L^p(w)}\le \sum _{j=0}^{\infty } \Vert V_q\{\phi _l*{T}_{j}^{\mathcal {N}}f\}_{l\in \mathbb Z}\Vert _{L^p(w)}. \end{aligned}$$

To prove Theorem 1.6, we claim that the following inequalities hold for $1<p<\infty$ and $w\in A_p$:

$$\begin{aligned} \Vert V_q\{\phi _{l}*{T}_{j}^{\mathcal {N}}f\}_{l\in \mathbb Z}\Vert _{L^p}&\lesssim \Vert \Omega \Vert _{L^{\infty }}2^{-\theta N(j-1)}(1+N(j))^{1+1/q}\Vert f\Vert _{L^p} \end{aligned}$$

(5.9)

and

$$\begin{aligned} \Vert V_q\{\phi _{l}*{T}_{j}^{\mathcal {N}}f\}_{l\in \mathbb Z}\Vert _{L^p(w)}&\lesssim \Vert \Omega \Vert _{L^{\infty }}(1+N(j))^{1+1/q}\{w\}_{A_p}\Vert f\Vert _{L^p(w)}. \end{aligned}$$

(5.10)

Assume the above two claims at the moment. Set $\varepsilon :=\frac{1}{2}c_n/(w)_{A_p},$ $c_n$ is small enough (see [21, Corollary 3.18]). By (5.10), we have, for this choice of $\varepsilon$,

$$\begin{aligned} \Vert V_q(\{\phi _{l}*{T}_{j}^{\mathcal {N}}f\}_{l\in \mathbb Z})\Vert _{L^p(w^{1+\varepsilon })}&\lesssim \Vert \Omega \Vert _{L^\infty }(1+N(j))^{1+1/q}\{w^{1+\varepsilon }\}_{A_p}\Vert f\Vert _{L^p(w^{1+\varepsilon })}\\&\lesssim \Vert \Omega \Vert _{L^\infty }(1+N(j))^{1+1/q}\{w\}_{A_p}^{1+\varepsilon }\Vert f\Vert _{L^p(w^{1+\varepsilon })}\nonumber . \end{aligned}$$

(5.11)

Now we are in position to apply the interpolation theorem with change of measures by Stein and Weiss—Lemma 2.2. We apply it to $T=V_q\{\phi _{l}*{T}_{j}^{\mathcal {N}}\}_{l\in \mathbb Z}$ with $p_0=p_1=p,~~w_0=w^0=1$ and $w_1=w^{1+\varepsilon }$, so that by $\lambda =\varepsilon /(1+\varepsilon )$, (5.9) and (5.11), one has for some $\theta ,\,\gamma >0$ such that

$$\begin{aligned}&\Vert V_q\{\phi _{l}*{T}_{j}^{\mathcal {N}}\}_{l\in \mathbb Z}\Vert _{L^p(w)\rightarrow L^p(w)}\\ {}&\quad \lesssim \Vert V_q\{\phi _{l}*{T}_{j}^{\mathcal {N}}\}_{l\in \mathbb Z}\Vert _{L^p\rightarrow L^p}^{\varepsilon /(1+\varepsilon )}\Vert V_q\{\phi _{l}*{T}_{j}^{\mathcal {N}}\}_{l\in \mathbb Z}\Vert _{L^p(w^{1+\varepsilon })\rightarrow L^p(w^{1+\varepsilon })}^{1/(1+\varepsilon )}\\&\quad \lesssim \Vert \Omega \Vert _{L^\infty }(1+N(j))^{1+1/q}2^{-\theta N(j-1)\varepsilon /(1+\varepsilon )}\{w\}_{A_p}\\&\quad \lesssim \Vert \Omega \Vert _{L^\infty }(1+N(j))^{1+1/q}2^{-\gamma N(j-1)/{(w)}_{A_p}}\{w\}_{A_p}. \end{aligned}$$

Thus

$$\begin{aligned}&\Vert V_q\{\phi _{l}*T_\Omega \}_{l\in \mathbb Z}\Vert _{L^p(w)\rightarrow L^p(w)}\\ {}&\quad \lesssim \Vert \Omega \Vert _{L^\infty }\{w\}_{A_p}\sum ^\infty _{j=0}(1+N(j))^{1+1/q}2^{-\gamma N(j-1)/{(w)}_{A_p}}. \end{aligned}$$

We now choose $N(j)=2^j$ for $j\ge 1$. Then, using $e^{-x}\le 2x^{-2}$, we have

$$\begin{aligned} \sum ^\infty _{j=0}&(1+N(j))^{1+1/q}2^{-\gamma N(j-1)/{(w)}_{A_p}}\\&\quad \lesssim \sum _{j:2^{j}\le (w)_{A_p}}2^{j(1+1/q)}+\sum _{j:2^j\ge (w)_{A_p}}2^{j(1+1/q)}\big (\frac{(w)_{A_p}}{2^j}\big )^2 \lesssim (w)_{A_p}^{1+1/q}, \end{aligned}$$

by summing the two geometric series in the last step. This implies

$$\begin{aligned} \Vert V_q\{\phi _{l}*T_\Omega f\}_{l\in \mathbb Z}\Vert _{L^p(w)}\lesssim \Vert \Omega \Vert _{L^\infty }\{w\}_{A_p}(w)_{A_p}^{1+1/q}\Vert f\Vert _{L^p(w)}, \end{aligned}$$

and hence the proof of Theorem 1.6 is complete under the assumptions that (5.9) and (5.10) hold.

Now we turn to proving the claims (5.9) and (5.10). The following inequality can be found in [21, Lemma 3.7],

$$\begin{aligned} \Vert {T}_{j}^{\mathcal {N}}f\Vert _{L^{2}}&\lesssim \Vert \Omega \Vert _{L^{\infty }} 2^{-\alpha N(j-1)}\Vert f\Vert _{L^{2}}, \end{aligned}$$

(5.12)

for some $0< \alpha < 1$ independent of $T_{\Omega }$ and j. Then by (see [10, 24])

$$\begin{aligned} \Vert V_q\{\phi _l*f\}_{l\in \mathbb Z}\Vert _{L^2}\lesssim \Vert f\Vert _{L^2}, \end{aligned}$$

(5.13)

we get

$$\begin{aligned} \Vert V_q\{\phi _l*{T}_{j}^{\mathcal {N}}f\}_{l\in \mathbb Z}\Vert _{L^2} \lesssim \Vert {T}_{j}^{\mathcal {N}}f\Vert _{L^{2}}\lesssim \Vert \Omega \Vert _{L^{\infty }} 2^{-\alpha N(j-1)}\Vert f\Vert _{L^{2}}. \end{aligned}$$

(5.14)

The operator ${T}_{j}^{\mathcal {N}}$ is a $\omega$-Calderón–Zygmund operator with the kernel $K_j^\mathcal {N}$ (see [21, Lemma 3.10]) satisfying

$$\begin{aligned} |K_j^\mathcal {N}(x)| \lesssim \frac{\Vert \Omega \Vert _{L^{\infty }}}{|x|^n} \end{aligned}$$

(5.15)

and for $2|y|\le |x|,$

$$\begin{aligned} |K_j^\mathcal {N}(x-y)-K_j^\mathcal {N}(x)| \lesssim \frac{\omega _j(|y|/|x|)}{|x|^n}, \end{aligned}$$

(5.16)

where $\omega _{j}(t)\le \Vert \Omega \Vert _{L^ \infty } \min (1,2^{N(j)}t),$ and $\Vert \omega _j\Vert _{Dini}\lesssim \Vert \Omega \Vert _{L^{\infty }}(1+N(j)).$ For $r>1,$ the Dini norm of $\omega _{j}^{1/r}$ is estimated as

$$\begin{aligned} \Vert \omega _j^{1/r}\Vert _{Dini}&\lesssim \Vert \Omega \Vert _{L^{\infty }}^{1/r} \int _0^{2^{-N(j)}}2^{N(j)/r}t^{1/r}\frac{dt}{t}+\Vert \Omega \Vert _{L^{\infty }}^{1/r}\int _{2^{-N(j)}}^{1}\frac{dt}{t}\nonumber \\ {}&\lesssim \Vert \Omega \Vert _{L^{\infty }}^{1/r}(1+N(j)). \end{aligned}$$

(5.17)

Applying Theorem 1.7 to $T={T}_{j}^{\mathcal {N}}$, we get for $1<p<\infty$ and $w\in A_p$

$$\begin{aligned} \Vert V_q\{\phi _{l}*{T}_{j}^{\mathcal {N}}f\}_{l\in \mathbb Z}\Vert _{L^p(w)}&\lesssim \Vert \Omega \Vert _{L^{\infty }}(1+N(j))^{1+1/q}\{w\}_{A_p}\Vert f\Vert _{L^p(w)}, \end{aligned}$$

which gives the proof of (5.10). Taking $w=1,$ we get for $1<p<\infty ,$

$$\begin{aligned} \Vert V_q\{\phi _{l}*{T}_{j}^{\mathcal {N}}f\}_{l\in \mathbb Z}\Vert _{L^p}&\lesssim \Vert \Omega \Vert _{L^{\infty }}(1+N(j))^{1+1/q}\Vert f\Vert _{L^p}. \end{aligned}$$

(5.18)

Interpolating between (5.14) and (5.18), we get

$$\begin{aligned} \Vert V_q\{\phi _{l}*{T}_{j}^{\mathcal {N}}f\}_{l\in \mathbb Z}\Vert _{L^p}&\lesssim \Vert \Omega \Vert _{L^{\infty }}2^{-\theta N(j-1)}(1+N(j))^{1+1/q}\Vert f\Vert _{L^p} \end{aligned}$$

which establishes the proof of (5.9). The proof of Theorem 1.6 is complete. $\square$

6 Proof of Theorem 1.3

Recalling that for any fixed $k\in \mathbb Z,$

$$\begin{aligned} T_{\Omega ,2^k}f(x)=\int _{|x-y|>2^k}\frac{\Omega (x-y)}{|x-y|^n}f(y)\,dy, \end{aligned}$$

and $K_\Omega (x)=\frac{\Omega (x')}{|x|^n}$. Following [13], for any fixed $k\in \mathbb Z,$ we write

$$\begin{aligned} T_{\Omega , 2^k}f(x)&=\phi _{k}*T_\Omega f-\phi _{k}*K_\Omega \chi _{|\cdot |\le 2^k}*f+(\delta _0-\phi _{k})*T_{\Omega ,2^k} f, \end{aligned}$$

which splits $\mathcal {T}_\Omega =\{T_{\Omega , 2^k}\}_{k\in \mathbb Z}$ into three families:

$$\begin{aligned} \mathcal {T}_\Omega ^1(f)&:=\{\phi _{k}*T_\Omega f\}_{k\in \mathbb Z},\quad \mathcal {T}_\Omega ^2(f):=\{\phi _{k}*(K_\Omega \chi _{|\cdot |\le 2^k})*f\}_{k\in \mathbb Z},\\ \mathcal {T}_\Omega ^3(f)&:=\{(\delta _0-\phi _{k})*T_{\Omega ,2^k}f\}_{k\in \mathbb Z}. \end{aligned}$$

Thus, it suffices to estimate the weighted $L^p$ norm of $\mathcal {T}_\Omega ^i(f)$, $i=1,2,3$.

Part 1 Let us first consider $\mathcal {T}_\Omega ^1(f)$. By Theorem 1.6, we get that for $1<p<\infty$ and $w\in A_p,$

$$\begin{aligned} \Vert V_q(\mathcal {T}_\Omega ^1(f))\Vert _{L^p(w)}\lesssim \Vert \Omega \Vert _{L^{\infty }} (w)_{A_p}^{1+1/q} \{w\}_{A_p}\Vert f\Vert _{L^p(w)}. \end{aligned}$$

Part 2 For the term $\mathcal {T}_\Omega ^2(f)$, by the Minkowski inequality, we get for $1<p<\infty$ and $w\in A_p,$

$$\begin{aligned} \Vert V_q(\mathcal {T}_\Omega ^2(f))\Vert _{L^p(w)} \lesssim \bigg \Vert \bigg (\sum _{k\in \mathbb Z}|\phi _k*K_\Omega \chi _{|\cdot |\le 2^k}*f|^q\bigg )^{1/q}\bigg \Vert _{L^p(w)}. \end{aligned}$$

It is easy to verify that $K_\Omega$ satisfies the cancellation condition (1.12) and the size estimate $|K_\Omega (x)|\le \frac{\Vert \Omega \Vert _{L^\infty }}{|x|^n}.$ Then applying Proposition 1.8 to $K=K_\Omega$, we get

$$\begin{aligned}\bigg \Vert \bigg (\sum _{k\in \mathbb Z}|\phi _k*K_\Omega \chi _{|\cdot |\le 2^k}*f|^q\bigg )^{1/q}\bigg \Vert _{L^p(w)}\lesssim \Vert \Omega \Vert _{L^\infty }\{w\}_{A_p}\Vert f\Vert _{L^p(w)},\end{aligned}$$

by $\ell ^2\subset \ell ^q$ for $q>2.$

Part 3 We now estimate $\mathcal {T}_\Omega ^3(f).$ We claim that for $1<p<\infty$ and $w\in A_p,$

$$\begin{aligned} \Vert V_q(\mathcal {T}_\Omega ^3(f))\Vert _{L^p(w)}\lesssim \Vert \Omega \Vert _{L^\infty }(w)_{A_p}^{1+1/q}\{w\}_{A_p}\Vert f\Vert _{L^p(w)}, \end{aligned}$$

(6.1)

which, together with the estimates of $\mathcal {T}_\Omega ^1(f)$, $\mathcal {T}_\Omega ^2(f)$ and $\mathcal {T}_\Omega ^3(f),$ gives

$$\begin{aligned}&\Vert V_q\mathcal {T}_\Omega (f) \Vert _{L^p(w)}\lesssim \Vert \Omega \Vert _{L^\infty }(w)_{A_p}^{1+1/q}\{w\}_{A_p}\Vert f\Vert _{L^p(w)}. \end{aligned}$$

This finishes the proof of Theorem 1.3.

We now provide the proof of (6.1). Recall that $\mathcal {T}_\Omega ^3(f):=\{(\delta _0-\phi _{k})*T_{\Omega ,2^k}f\}_{k\in \mathbb Z}$. For $k\in \mathbb Z$, we define $v_{k}$ as $v_{k}(x)=\frac{\Omega (x')}{|x|^n}\chi _{2^k<|x|\le 2^{k+1}}(x).$ Then we can write $T_{\Omega , 2^k}f(x)=\sum _{s\ge 0}v_{k+s}*f(x).$ Recall that $S_{j}(f)=f*\varphi _{j}$ and their differences are given by $S_{j}(f)-S_{j+1}(f)=f*\psi _{j},$ where $\varphi _{j}$ and $\psi _{j}$ are defined as in Section 5. Denote by $T_{k}f(x)=v_k*f(x).$ Similarly, for any sequence of integers $\mathcal {N}=\{N(j)\}_{j=0}^{\infty }$, with $0 = N(0)< N(1)<\cdot \cdot \cdot < N(j)\rightarrow \infty$, we also have the identity.

$$\begin{aligned} T_{k+s}&=T_{k+s}S_{k+s}+\sum _{j=1}^\infty T_{k+s}(S_{k+s-N(j)}-S_{k+s-N(j-1)}) =:\sum _{j=0}^\infty T^\mathcal {N}_{k+s,j}, \end{aligned}$$

(6.2)

where $T_{k+s,0}^\mathcal {N}=T_{k+s}S_{k+s}$ and for $j\ge 1$,

$$\begin{aligned} T^\mathcal {N}_{k+s,j}&=\sum _{i=N(j-1)+1}^{N(j)}T_{k+s}(S_{k+s-i}-S_{k+s-(i-1)})=:\sum _{i=N(j-1)+1}^{N(j)}T_{k+s,i}. \end{aligned}$$

(6.3)

Then

$$\begin{aligned} (\delta _0-\phi _k)*v_{k+s}*f(x) =\sum ^\infty _{j=0}(\delta _0-\phi _k)*T^\mathcal {N}_{k+s,j}f(x). \end{aligned}$$

(6.4)

So

$$\begin{aligned} \Vert V_q(\mathcal {T}_\Omega ^3(f))\Vert _{L^p(w)}&\le \sum ^\infty _{j=0}\bigg \Vert \bigg (\sum _{k\in \mathbb Z}|(\delta _0-\phi _k)*\sum _{s\ge 0} T^\mathcal {N}_{k+s,j}f|^q\bigg )^{1/q}\bigg \Vert _{L^p(w)}. \end{aligned}$$

We first estimate the $L^2-$norm of $(\sum _{k\in \mathbb Z}|(\delta _0-\phi _k)*\sum _{s\ge 0} T^\mathcal {N}_{k+s,j}f|^q)^{1/q}$.

Lemma 6.1

We have for $j\ge 0$, there is a positive $0<\tau <1$ such that

$$\begin{aligned} \bigg \Vert \bigg (\sum _{k\in \mathbb Z}|(\delta _0-\phi _k)*\sum _{s\ge 0} T^\mathcal {N}_{k+s,j}f|^q\bigg )^{1/q}\bigg \Vert _{ L^2}\lesssim \Vert \Omega \Vert _{L^\infty }2^{-\tau N(j-1)} \Vert f\Vert _{L^2}.\end{aligned}$$

(6.5)

Proof

By the Plancherel Theorem and (6.3), we get for $j\ge 1,$

$$\begin{aligned}&\bigg \Vert \bigg (\sum _{k\in \mathbb Z}|(\delta _0-\phi _k)*\sum _{s\ge 0}T^\mathcal {N}_{k+s,j}f|^2\bigg )^{1/2}\bigg \Vert _{ L^2}\\&\le \sum _{s\ge 0}\sum _{i=N(j-1)+1}^{N(j)}\bigg (\sum _{k\in \mathbb {Z}}\int _{{\mathbb R}^n}|\widehat{(\delta _0-\phi _k)}(\xi )|^2|\widehat{T_{k+s,i}f}(\xi )|^2d\xi \bigg )^{1/2}. \end{aligned}$$

Since ${\widehat{\phi }}\in \mathcal {S}({\mathbb R}^n)$ and ${\widehat{\phi }}(0)=1,$ $|1-{\widehat{\phi }}(\xi )|\lesssim \min (1, |\xi |).$ Also by ${\widehat{\psi }}\in \mathcal {S}({\mathbb R^n})$ and ${\widehat{\psi }}(0)=0,$ then $|{\widehat{\psi }}(\xi )|\lesssim \min (1, |\xi |).$ Therefore, $|1-\widehat{\phi _k}(\xi )|\lesssim \min (1,|2^k\xi |^\alpha )$, $|{\widehat{\psi }}(2^{k+s-i}\xi )|\lesssim \min (|2^{k+s-i}\xi |^\gamma ,1),\,|{\widehat{v}}_{k+s}(\xi )|\lesssim \Vert \Omega \Vert _{L^\infty }|2^{k+s}\xi |^{-\beta }$ (see [13]) for some $0<\beta ,\,\gamma ,\,\alpha <1$, $\alpha +\gamma -\beta >0,$ $\gamma <\beta$ and $\alpha <\beta ,$

$$\begin{aligned}&\sum _{k\in \mathbb {Z}}|\widehat{(\delta _0-\phi _k)}(\xi )|^2|\widehat{T_{k+s,i}f}(\xi )|^2\\&\quad =\sum _{k\in \mathbb {Z}}|1-\widehat{\phi _k}(\xi )|^2|{\widehat{v}}_{k+s}(\xi )|^2|{\widehat{\psi }}(2^{k+s-i}\xi )|^2|{\widehat{f}}(\xi )|^2\\&\quad \lesssim \Vert \Omega \Vert _{L^\infty }^2\bigg (\sum _{|2^k\xi |\le 2^i}|2^k\xi |^{2\alpha }|2^{k+s-i}\xi |^{2\gamma }|2^{k+s}\xi |^{-2\beta }+\sum _{|2^k\xi |\ge 2^i}|2^{k+s}\xi |^{-2\beta }\bigg )|{\widehat{f}}(\xi )|^2\\&\quad \lesssim \Vert \Omega \Vert _{L^\infty }^22^{-2(\beta -\alpha )i}2^{-2(\beta -\gamma )s}|{\widehat{f}}(\xi )|^2. \end{aligned}$$

Therefore, by summing the geometric series and the Plancherel theorem, we get for some $\tau \in (0,1),$

$$\begin{aligned}&\bigg \Vert \bigg (\sum _{k\in \mathbb Z}|(\delta _0-\phi _k)*\sum _{s\ge 0}T^\mathcal {N}_{k+s,j}f|^2\bigg )^{1/2}\bigg \Vert _{L^2}\lesssim \Vert \Omega \Vert _{L^\infty }2^{-\tau N(j-1)} \Vert f\Vert _{L^2}. \end{aligned}$$

For $j=0$, $|1-{\widehat{\phi }}(2^{k}\xi )|\lesssim \min (|2^{k}\xi |,1),\,|{\widehat{\varphi }}(2^{k}\xi )|\lesssim 1,\,|{\widehat{v}}_{k+s}(\xi )|\lesssim \Vert \Omega \Vert _{L^\infty }|2^{k+s}\xi |^{-\beta }$ for some $0<\beta <1$ (see [13]). Therefore, by the Plancherel theorem, we get

$$\begin{aligned}&\bigg \Vert \bigg (\sum _{k\in \mathbb Z}|(\delta _0-\phi _k)*\sum _{s\ge 0} T^\mathcal {N}_{k+s,0}f|^2\bigg )^{1/2}\bigg \Vert _{L^2}\\&\quad \le \sum _{s\ge 0}\bigg (\int _{{\mathbb R}^n}\sum _{k\in \mathbb {Z}}|1-\widehat{\phi _k}(\xi )|^2|{\widehat{v}}_{k+s}(\xi )|^2|{\widehat{\varphi }}(2^{k}\xi )|^2|{\widehat{f}}(\xi )|^2d\xi \bigg )^{1/2}\\&\quad \lesssim \sum _{s\ge 0}\bigg (\int _{{\mathbb R}^n}\Vert \Omega \Vert _{L^\infty }^2\bigg (\sum _{|2^k\xi |\le 1}|2^{k}\xi |^2|2^{k+s}\xi |^{-2\beta }+\sum _{|2^k\xi |\ge 1}|2^{k+s}\xi |^{-2\beta }\bigg )|{\widehat{f}}(\xi )|^2d\xi \bigg )^{1/2}\\&\quad \lesssim \Vert \Omega \Vert _{L^\infty } \Vert f\Vert _{L^2}. \end{aligned}$$

The proof of Lemma 6.1 is complete by $\ell ^2\subset \ell ^q$ for $q>2$. $\square$

Denote $T_{k,j}^\mathcal {N}:=\sum _{s\ge 0} T^\mathcal {N}_{k+s,j}$, and let $K_{k,j}^\mathcal {N}$ be the kernel of $T_{k,j}^\mathcal {N}$ given by

$$\begin{aligned} K_{k,j}^\mathcal {N}:=\sum _{s\ge 0} v_{k+s}*(\varphi _{{k+s-N(j)}}-\varphi _{{k+s-N(j-1)}})\end{aligned}$$

(6.6)

for $j\ge 1$ and for $j=0,$

$$\begin{aligned} K_{k,0}^\mathcal {N}:=\sum _{s\ge 0} v_{k+s}*\varphi _{{k+s}}.\end{aligned}$$

(6.7)

In the following, we will verify that $K_{k,j}^\mathcal {N}$ is a $\omega$-Dini Calderón–Zygmund kernel satisfying (2.5) and (2.6).

Lemma 6.2

For $j\ge 0$ and $k\in \mathbb Z$. Then we have the size estimate

$$\begin{aligned} |K_{k,j}^\mathcal {N}(x)|&\lesssim \frac{\Vert \Omega \Vert _{L^\infty }}{|x|^n}\chi _{ |x|\ge \frac{3}{4}\cdot 2^{k}}(x) \end{aligned}$$

(6.8)

and the regularity estimate

$$\begin{aligned} |K_{k,j}^\mathcal {N}(x-h)-K_{k,j}^\mathcal {N}(x)|&\lesssim \frac{\omega _j(\frac{|h|}{|x|})}{|x|^n}\chi _{ |x|\ge \frac{3}{4}\cdot 2^{k}}(x),\quad 0<|h|<\frac{|x|}{2}, \end{aligned}$$

(6.9)

where $\omega _j(t)\le \Vert \Omega _j\Vert _{L^{\infty }}\min (1, 2^{N(j)}t).$

Proof

In order to get the required estimates for the kernel $K_{k,j}^\mathcal {N}$, we first study the kernel of each $H_{k, s, N(j)}$ which is defined by

$$\begin{aligned} H_{k, s, N(j)}:= v_{k+s}*\varphi _{{k+s-N(j)}}. \end{aligned}$$

First, we estimate $|H_{k, s, N(j)}(x)|.$ Since supp $\varphi \subset \{x: |x|\le \frac{1}{100}\}$, a simple computation gives that

$$\begin{aligned} |H_{k,s,N(j)}(x)|&\le \int _{2^{k+s}\le |y|\le 2^{k+s+1}}\frac{|\Omega (y)|}{|y|^{n}}|\varphi _{{k+s-N(j)}}(x-y)|dy\nonumber \\&\lesssim \frac{\Vert \Omega \Vert _{L^{\infty }}}{|x|^{n}}\chi _{\frac{3}{4}\cdot 2^{k+s}\le |x|\le \frac{5}{4}\cdot 2^{k+s}}. \end{aligned}$$

(6.10)

From the triangular inequality, and $N(j-1)<N(j)$, we obtain that the kernel $K_{k,j}^\mathcal {N}:=\sum _{s\ge 0}(H_{k,s,N(j)}-H_{k,s,N(j-1)})$ satisfies

$$\begin{aligned} |K_{k,j}^\mathcal {N}(x)|&\lesssim \sum _{s\ge 0}\frac{\Vert \Omega \Vert _{L^{\infty }}}{|x|^{n}}\chi _{\frac{3}{4}\cdot 2^{k+s}\le |x|\le \frac{5}{4}\cdot 2^{k+s}}\lesssim \frac{\Vert \Omega \Vert _{L^{\infty }}}{|x|^{n}}\chi _{ |x|\ge \frac{3}{4}\cdot 2^{k}}. \end{aligned}$$

(6.11)

On the other hand, we compute the gradient. Again by the support of $\varphi$, we have

$$\begin{aligned} \nabla H_{k,s,N(j)}(x)&=v_{k+s}*\nabla \varphi _{{k+s-N(j)}}(x)=\frac{1}{2^{k+s-N(j)}}\int _{2^{k+s}< |y|\le 2^{k+s+1}}\\ {}&\quad \times \frac{\Omega (y)}{|y|^{n}}(\nabla \varphi )_{{k+s-N(j)}}(x-y)dy. \end{aligned}$$

Since $|x-y|\le \frac{2^{k+s}}{100}, 2^{k+s}< |y|\le 2^{k+s+1}$, then $|x|\simeq |y|$ and $\frac{3}{4}\cdot 2^{k+s}\le |x|\le \frac{5}{4}\cdot 2^{k+s}.$ Thus, we get

$$\begin{aligned} |\nabla H_{k,s,N(j)}(x)|&\lesssim \Vert \Omega \Vert _{L^{\infty }}\frac{2^{N(j)}}{|x|^{n+1}}\chi _{\frac{3}{4}\cdot 2^{k+s}\le |x|\le \frac{5}{4}\cdot 2^{k+s}}. \end{aligned}$$

(6.12)

Therefore, if $|h|<\frac{|x|}{2},$ we get $|x-\theta h|\simeq |x|$ and

$$\begin{aligned} |K_{k,j}^\mathcal {N}(x-h)-K_{k,j}^\mathcal {N}(x)|&\le |\nabla K_{k,j}^\mathcal {N}(x-\theta h)||h|\nonumber \\ {}&\lesssim \sum _{s\ge 0}\Vert \Omega \Vert _{L^{\infty }}\frac{2^{N(j)}}{|x|^{n+1}}\chi _{\frac{3}{4}\cdot 2^{k+s}\le |x|\le \frac{5}{4}\cdot 2^{k+s}}(x)\cdot |h|\nonumber \\ {}&\lesssim \Vert \Omega \Vert _{L^{\infty }}\frac{2^{N(j)}|h|}{|x|^{n+1} } \chi _{ |x|\ge \frac{3}{4}\cdot 2^k}(x). \end{aligned}$$

(6.13)

If $|h|<\frac{|x|}{2},$ combining (6.11) and (6.13), we get for $j\ge 0$ and $k\in \mathbb Z,$

$$\begin{aligned} |K_{k,j}^\mathcal {N}(x-h)-K_{k,j}^\mathcal {N}(x)|&\lesssim \frac{\omega _j(\frac{|h|}{|x|})}{|x|^n}\chi _{ |x|\ge \frac{3}{4}\cdot 2^k}(x), \end{aligned}$$

where $\omega _j(t)\le \Vert \Omega \Vert _{L^{\infty }}\min (1, 2^{N(j)}t).$ The proof of Lemma 6.2 is complete. $\square$

We now continue the proof of (6.1). By noting that supp $K_{k,j}^\mathcal {N}\subset \{x: |x|\ge \frac{3}{4}\cdot 2^k\}$ and that $K_{k,j}^\mathcal {N}$ satisfies mean value zero, (6.8) and (6.9), repeating the argument of Proposition 1.9 only with $K^k$ replaced by $K_{k,j}^\mathcal {N},$ we can also get for $1<p<\infty$ and $w\in A_p$

$$\begin{aligned} \bigg \Vert \bigg (\sum _{k\in \mathbb Z}|(\delta _0-\phi _k)*T_{k,j}^\mathcal {N} f|^q\bigg )^{1/q}\bigg \Vert _{L^p(w)}&\lesssim \Vert \Omega \Vert _{L^{\infty }}(1+N(j))^{1+1/q}\Vert f\Vert _{L^p(w)}. \end{aligned}$$

(6.14)

Taking $w=1$ in (6.14) we get for $1<p<\infty ,$

$$\begin{aligned} \bigg \Vert \bigg (\sum _{k\in \mathbb Z}|(\delta _0-\phi _k)*T_{k,j}^\mathcal {N} f|^q\bigg )^{1/q}\bigg \Vert _{L^p}&\lesssim \Vert \Omega \Vert _{L^{\infty }}(1+N(j))^{1+1/q}\Vert f\Vert _{L^p}. \end{aligned}$$

(6.15)

Interpolating between (6.5) and (6.15), we get

$$\begin{aligned} \bigg \Vert \bigg (\sum _{k\in \mathbb Z}|(\delta _0-\phi _k)*T_{k,j}^\mathcal {N} f|^q\bigg )^{1/q}\bigg \Vert _{L^p}&\lesssim \Vert \Omega \Vert _{L^{\infty }}2^{-\tau N(j-1)}(1+N(j))^{1+1/q}\Vert f\Vert _{L^p}. \end{aligned}$$

(6.16)

Similar to the proof of Theorem 1.6, based on (6.14) and (6.16), applying the interpolation theorem with change of measures (Lemma 2.2), we get

$$\begin{aligned} \bigg \Vert \bigg (\sum _{k\in \mathbb Z}|(\delta _0-\phi _k)*T_{k,j}^\mathcal {N} f|^q\bigg )^{1/q}\bigg \Vert _{L^p(w)}\lesssim \Vert \Omega \Vert _{L^{\infty }}\{w\}_{A_p}(w)_{A_p}^{1+1/q}\Vert f\Vert _{L^p(w)}, \end{aligned}$$

which gives (6.1). The proof of Theorem 1.3 is complete.$\square$

7 Proof of Theorem 1.4

Recall that we can write

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {S}_q( \mathcal {{T}}_\Omega f)(x)=\left( \sum \limits _{k\in \mathbb Z}[V_{q,k}( f)(x)]^q\right) ^{1/q};\\ V_{q,k}( f)(x)=\left( \sup \limits _ {2^k\le t_0<\cdots<t_\lambda <2^{k+1}}{\sum \limits _{l=0}^{\lambda -1}}| T_{k,t_{l+1}}f(x)- T_{k,t_l}f(x)|^q\right) ^{1/q}, \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned}T_{k,t_{l}}f(x)=\int _{t_l\le |x-y|\le 2^{k+1}}\frac{\Omega (x-y)}{|x-y|^n}f(y)\,dy.\end{aligned}$$

Observe that

$$\begin{aligned} \mathcal {S}_{\infty }(\mathcal {{T}}_\Omega f)(x)\le \Vert \Omega \Vert _{L^\infty }Mf(x). \end{aligned}$$

Then by the sharp weighted boundedness of the Hardy–Littlewood maximal operator M (see Hytönen–Pérez [19, Corollary 1.10], the original version was due to Buckley [3]),

$$\begin{aligned} \Vert Mf\Vert _{L^p(w)}\le c_n \cdot p' \cdot [w]_{A_p}^{1\over p} [w^{1-p'}]_{A_\infty }^{1\over p}\Vert f\Vert _{L^p(w)},\quad 1<p<\infty , \end{aligned}$$

we get

$$\begin{aligned} \Vert \mathcal {S}_{\infty }({\mathcal {T}}_\Omega f)\Vert _{L^p(w)}&\le c_n\cdot p'\cdot \Vert \Omega \Vert _{L^\infty }\{w\}_{A_p}\Vert f\Vert _{L^p(w)}. \end{aligned}$$

(7.1)

Now we claim that

$$\begin{aligned} \Vert \mathcal {S}_{2}(\mathcal {T}_\Omega f)\Vert _{L^p(w)}&\lesssim \Vert \Omega \Vert _{L^\infty }(w)_{A_p}^{1/2}\{w\}_{A_p}\Vert f\Vert _{L^p(w)}. \end{aligned}$$

(7.2)

In fact, interpolating between (7.1) and (7.2), we get for $q\ge 2,$

$$\begin{aligned} \Vert \mathcal {S}_{q}({\mathcal {T}}_\Omega f)\Vert _{L^p(w)}&\lesssim \Vert \Omega \Vert _{L^\infty }(w)_{A_p}^{1/q}\{w\}_{A_p}\Vert f\Vert _{L^p(w)}. \end{aligned}$$

(7.3)

Now we turn to verify (7.2). Recall that $S_{j}(f)=f*\varphi _{j}$ and their differences are given by

$$\begin{aligned} S_{j}(f)-S_{j+1}(f)=f*\psi _{j}, \end{aligned}$$

(7.4)

where $\varphi _{j}$ and $\psi _{j}$ are defined as in Section 5. Similarly, for any sequence of integers $\mathcal {N}=\{N(j)\}_{j=0}^{\infty }$, with $0 = N(0)< N(1)<\cdot \cdot \cdot < N(j)\rightarrow \infty$, we also have the identity

$$\begin{aligned} T_{k,t}&=T_{k,t}S_k+\sum _{j=1}^\infty T_{k,t}(S_{k-N(j)}-S_{k-N(j-1)})=:\sum _{j=0}^\infty T^\mathcal {N}_{k, t, j}, \end{aligned}$$

(7.5)

where $T_{k,t,0}:=T_{k,t,0}^\mathcal {N}:=T_{k,t}S_k$ and for $j\ge 1$,

$$\begin{aligned} T^\mathcal {N}_{k,t,j}&=\sum _{i=N(j-1)+1}^{N(j)}T_{k,t}(S_{k-i}-S_{k-(i-1)})=:\sum _{i=N(j-1)+1}^{N(j)}T_{k,t,i}. \end{aligned}$$

(7.6)

Therefore, by the Minkowski inequality, we get

$$\begin{aligned} \mathcal {S}_2({\mathcal {T}}_\Omega f)(x)&=\bigg (\sum _{k\in \mathbb {Z}}\bigg \Vert \bigg \{\sum ^\infty _{j=0}T_{k,t,j}^\mathcal {N}f(x)\bigg \}_{t\in [1,2]}\bigg \Vert _{V_2}^2\bigg )^\frac{1}{2}\\&\le \sum ^\infty _{j=0}\bigg (\sum _{k\in \mathbb {Z}}\Vert \{T_{k,t,j}^\mathcal {N}f(x)\}_{t\in [1,2]}\Vert _{V_2}^2\bigg )^\frac{1}{2}=:\sum ^\infty _{j=0}\mathcal {S}_{2,j}^\mathcal {N}( f)(x)\nonumber . \end{aligned}$$

(7.7)

Then for $1<p<\infty$ and $w\in A_p,$

$$\begin{aligned} \Vert \mathcal {S}_2({\mathcal {T}}_\Omega f)\Vert _{L^p(w)}\le \sum ^\infty _{j=0}\Vert \mathcal {S}_{2,j}^\mathcal {N}( f)\Vert _{L^p(w)}. \end{aligned}$$

Part 1. We first give the $L^2$-norm of $\mathcal {S}_{2,j}^\mathcal {N}( f)(x)$.

Lemma 7.1

There is a positive $0<\tau <1$ such that

$$\begin{aligned} \Vert \mathcal {S}_{2,j}^\mathcal {N}(f)\Vert _{ L^2}\lesssim \Vert \Omega \Vert _{L^\infty }2^{-\tau N(j-1)}(N(j)+1))^{1/2} \Vert f\Vert _{L^2},\quad j\ge 0. \end{aligned}$$

(7.8)

Proof

For $t\in [1,2]$, we define $v_{k,t}$ as $v_{k,t}(x)=\frac{\Omega (x')}{|x|^n}\chi _{_{2^kt<|x|\le 2^{k+1}}}(x).$ Then $T_{k,t}$ can be expressed as $T_{k,t}f(x)=v_{k,t}*f(x)$, $t\in [1,2]$. Recall that

$$\begin{aligned} \mathcal {S}_{2,j}^\mathcal {N}( f)(x)=\bigg (\sum _{k\in \mathbb Z}\Vert \{T_{k,t,j}^\mathcal {N}f(x)\}_{t\in [1,2]}\Vert _{V_2}^2\bigg )^{1/2}. \end{aligned}$$

Note that

$$\begin{aligned} \Vert a\Vert _{V_{2}}\le C\Vert a\Vert ^{1/2}_{L^2}\Vert a'\Vert ^{1/2}_{L^2},\end{aligned}$$

(7.9)

where $a'=\{\frac{d}{dt}a_{t}:t\in \mathbb {R}\}$ (see [24]). We have applied (7.9) to $a=\{T_{k,t,j}^\mathcal {N}f(x)\}_{t\in [1,2]},$ then we have

$$\begin{aligned} \Vert \{T_{k,t,j}^\mathcal {N}f(x)\}_{t\in [1,2]}\Vert _{V_2}^2&\le C\Vert T_{k,t,j}^\mathcal {N}f(x)\Vert _{L_t^2([1,2])}\Vert \frac{d}{dt}T_{k,t,j}^\mathcal {N}f(x)\Vert _{L_t^2([1,2])} \\ {}&=C\bigg (\int _1^2|T_{k,t,j}^\mathcal {N}f(x)|^2\frac{dt}{t}\bigg )^\frac{1}{2} \bigg (\int _1^2|\frac{d}{dt}T_{k,t,j}^\mathcal {N}f(x)|^2\frac{dt}{t}\bigg )^\frac{1}{2}. \end{aligned}$$

Then by the above inequalities, we get

$$\begin{aligned} \mathcal {S}_{2,j}^\mathcal {N}( f)(x)^2&=\sum _{k\in \mathbb Z}\Vert \{T_{k,t,j}^\mathcal {N}f(x)\}_{t\in [1,2]}\Vert _{V_2}^2&\le C\sum _{k\in \mathbb Z}\bigg (\int _1^2|T_{k,t,j}^\mathcal {N}f(x)|^2\frac{dt}{t}\bigg )^\frac{1}{2}\\&\quad \bigg (\int _1^2|\frac{d}{dt}T_{k,t,j}^\mathcal {N}f(x)|^2\frac{dt}{t}\bigg )^\frac{1}{2}. \end{aligned}$$

This, along with the Cauchy–Schwartz inequality, yields

$$\begin{aligned} \Vert \mathcal {S}_{2,j}^\mathcal {N}( f)\Vert _{L^2}^2&\le \bigg \Vert \bigg (\int _1^2\sum _{k\in \mathbb {Z}}|T_{k,t,j}^\mathcal {N}f|^2\frac{dt}{t}\bigg )^\frac{1}{2}\bigg \Vert _{L^2} \bigg \Vert \bigg (\int _1^2\sum _{k\in \mathbb {Z}}|\frac{d}{dt}T_{k,t,j}^\mathcal {N}f|^2\frac{dt}{t}\bigg )^\frac{1}{2}\bigg \Vert _{L^2}. \end{aligned}$$

(7.10)

By the Plancherel theorem and (7.6), we get for $j\ge 1,$

$$\begin{aligned} \bigg \Vert \bigg (\int _1^2\sum _{k\in \mathbb {Z}}|T_{k,t,j}^\mathcal {N}f|^2\frac{dt}{t}\bigg )^\frac{1}{2}\bigg \Vert _{L^2}&=\bigg (\int _1^2\sum _{k\in \mathbb {Z}}|\widehat{T_{k,t,j}^\mathcal {N}f}(\xi )|^2\frac{dt}{t}d\xi \bigg )^{1/2}\nonumber \\&\le \sum _{i=N(j-1)+1}^{N(j)}\bigg (\int _1^2\sum _{k\in \mathbb {Z}}|\widehat{T_{k,t,i}f}(\xi )|^2\frac{dt}{t}d\xi \bigg )^{1/2}. \end{aligned}$$

(7.11)

Since $|{\widehat{\psi }}(2^{k-i}\xi )|\lesssim \min (|2^{k-i}\xi |,1),\,|{\widehat{v}}_{k,t}(\xi )|\lesssim \Vert \Omega \Vert _{L^\infty }|2^{k}\xi |^{-\beta }$ for some $0<\beta <1$ (see [13]),

$$\begin{aligned} \sum _{k\in \mathbb {Z}}|\widehat{T_{k,t,i}f}(\xi )|^2&=\sum _{k\in \mathbb {Z}}|{\widehat{v}}_{k,t}(\xi )|^2|{\widehat{\psi }}(2^{k-i}\xi )|^2|{\widehat{f}}(\xi )|^2\\&\lesssim \Vert \Omega \Vert _{L^\infty }^2\bigg (\sum _{|2^k\xi |\le 2^i}|2^{k-i}\xi |^{2}|2^{k}\xi |^{-2\beta }+\sum _{|2^k\xi |\ge 2^i}|2^{k}\xi |^{-2\beta }\bigg )|{\widehat{f}}(\xi )|^2\\&\lesssim \Vert \Omega \Vert _{L^\infty }^22^{-2\beta i}|{\widehat{f}}(\xi )|^2. \end{aligned}$$

Then by summing the geometric series and the Plancherel theorem, we get

$$\begin{aligned} \bigg \Vert \bigg (\int _1^2\sum _{k\in \mathbb {Z}}|T_{k,t,j}^\mathcal {N}f|^2\frac{dt}{t}\bigg )^\frac{1}{2}\bigg \Vert _{L^2}&\lesssim \Vert \Omega \Vert _{L^{\infty }}2^{-\beta N(j-1)}\Vert f\Vert _{L^2}. \end{aligned}$$

(7.12)

For $j=0$, $|{\widehat{v}}_{k,t}(\xi )|\lesssim \Vert \Omega \Vert _{L^\infty }\min (|2^k\xi |,\,|2^{k}\xi |^{-\beta })$ for some $0<\beta <1$ and $|{\widehat{\varphi }}(2^{k}\xi )|\lesssim 1$,

$$\begin{aligned} \sum _{k\in \mathbb {Z}}|\widehat{T_{k,t,0}f}(\xi )|^2&=\sum _{k\in \mathbb {Z}}|{\widehat{v}}_{k,t}(\xi )|^2|{\widehat{\varphi }}(2^{k}\xi )|^2|{\widehat{f}}(\xi )|^2\\&\lesssim \Vert \Omega \Vert _{L^\infty }^2\bigg (\sum _{|2^k\xi |\le 1}|2^{k}\xi |^2+\sum _{|2^k\xi |\ge 1}|2^{k}\xi |^{-2\beta }\bigg )|{\widehat{f}}(\xi )|^2\lesssim \Vert \Omega \Vert _{L^\infty }^2|{\widehat{f}}(\xi )|^2. \end{aligned}$$

Then by the Plancherel theorem, we get

$$\begin{aligned} \bigg \Vert \bigg (\int _1^2\sum _{k\in \mathbb {Z}}|T_{k,t,0}^\mathcal {N}f|^2\frac{dt}{t}\bigg )^\frac{1}{2}\bigg \Vert _{L^2}&\lesssim \Vert \Omega \Vert _{L^{\infty }}\Vert f\Vert _{L^2}. \end{aligned}$$

Combining the above estimates for $j>0$ and $j=0,$ we get

$$\begin{aligned} \bigg \Vert \bigg (\int _1^2\sum _{k\in \mathbb {Z}}|T_{k,t,j}^\mathcal {N}f|^2\frac{dt}{t}\bigg )^\frac{1}{2}\bigg \Vert _{L^2}&\lesssim 2^{-\beta N(j-1)}\Vert \Omega \Vert _{L^{\infty }}\Vert f\Vert _{L^2}. \end{aligned}$$

(7.13)

On the other hand, we recall the elementary fact: for any Schwartz function h,

$$\begin{aligned} \frac{d}{dt}[v_{k,t}*h(x)]&=\frac{d}{dt}\bigg [\int _{{\mathbb S}^{n-1}}\Omega (y')\int _{2^kt}^{2^{k+1}}\frac{1}{r}h(x-ry')drd\sigma (y')\bigg ]\nonumber \\&=\frac{1}{t}\int _{{\mathbb S}^{n-1}}\Omega (y')h(x-2^kty')d\sigma (y'). \end{aligned}$$

Then by $\int _{\mathbb S^{n-1}}\Omega (y')d\sigma (y')=0$ and $t\in [1,2]$, we get

$$\begin{aligned} \big |\frac{d}{dt}[v_{k,t}*h]^{\bigwedge }(\xi )\big |&=|{\widehat{h}}(\xi )| \frac{1}{t} \Big |\int _{{\mathbb S}^{n-1}}\Omega (y')(e^{2\pi i2^kty'\cdot \xi }-1)d\sigma (y')\Big |\nonumber \\&\lesssim \Vert \Omega \Vert _{L^\infty }|{\widehat{h}}(\xi )|\min (|2^k\xi |,1). \end{aligned}$$

(7.14)

Therefore, by the Plancherel theorem, we get for $j\ge 1$

$$\begin{aligned} \bigg \Vert \bigg (\int _1^2\sum _{k\in \mathbb {Z}}|\frac{d}{dt}T_{k,t,j}^\mathcal {N}f|^2\frac{dt}{t}\bigg )^\frac{1}{2}\bigg \Vert _{L^2}&\lesssim \sum _{i=N(j-1)+1}^{N(j)}\bigg (\int _1^2\sum _{k\in \mathbb {Z}}|\widehat{\frac{d}{dt}T_{k,t,i}f}(\xi )|^2\frac{dt}{t}d\xi \bigg )^{1/2}. \end{aligned}$$

By (7.14) and $|{\widehat{\psi }}(2^{k-i}\xi )|\lesssim \min (|2^{k-i}\xi |, \,|2^{k-i}\xi |^{-\beta })$ for some $0<\beta <1,$

$$\begin{aligned}&\sum _{k\in \mathbb {Z}}\big |\widehat{\frac{d}{dt}T_{k,t,i}f}(\xi )\big |^2\\&\quad =\sum _{k\in \mathbb {Z}}\big |(\frac{d}{dt}v_{j,t}*\psi _{k-i}*f)^{\bigwedge }(\xi )\big |^2\\&\quad \lesssim \Vert \Omega \Vert _{L^\infty }^2\sum _{k\in \mathbb {Z}}|{\widehat{\psi }}(2^{k-i}\xi )|^2|{\widehat{f}}(\xi )|^2\\&\quad \lesssim \bigg (\sum _{|2^k\xi |\le 2^i}|2^{k-i}\xi |^2+\sum _{|2^k\xi |> 2^i}|2^{k-i}\xi |^{-2\beta }\bigg )\ \Vert \Omega \Vert _{L^\infty }^2|{\widehat{f}}(\xi )|^2\\ {}&\quad \lesssim \Vert \Omega \Vert _{L^\infty }^2|{\widehat{f}}(\xi )|^2. \end{aligned}$$

Then by the Plancherel theorem, we get for $j\ge 1$

$$\begin{aligned}&\bigg \Vert \bigg (\int _1^2\sum _{k\in \mathbb {Z}}|\frac{d}{dt}T_{k,t,j}^\mathcal {N}f|^2\frac{dt}{t}\bigg )^\frac{1}{2}\bigg \Vert _{L^2}\lesssim N(j)\Vert \Omega \Vert _{L^\infty }\Vert f\Vert _{L^2}. \end{aligned}$$

(7.15)

For $j=0,$

$$\begin{aligned} \sum _{k\in \mathbb {Z}}\big |\widehat{\frac{d}{dt}T_{k,t,0}^\mathcal {N}f}(\xi )\big |^2|{\widehat{f}}(\xi )|^2&=\sum _{k\in \mathbb {Z}}\big |\Big (\frac{d}{dt}v_{k,t}*\varphi _{k}*f\Big )^{\bigwedge }(\xi )\big |^2. \end{aligned}$$

Then by (7.14), we have

$$\begin{aligned} \big |\Big (\frac{d}{dt}v_{k,t}*\varphi _{k}*f\Big )^{\bigwedge }(\xi )\big |^2&\le \Vert \Omega \Vert _{L^\infty }^2\min (|2^k\xi |,1)^2|\widehat{\varphi _{k}*f}(\xi )|^2\\ {}&\le \Vert \Omega \Vert _{L^\infty }^2\min (|2^k\xi |,1)^2|{\widehat{\varphi }}(2^{k}\xi )|^2|{\widehat{f}}(\xi )|^2. \end{aligned}$$

By $|{\widehat{\varphi }}(2^{k}\xi )|\lesssim \min (1, \,|2^{k}\xi |^{-\beta }),$ then

$$\begin{aligned} \big |\Big (\frac{d}{dt}v_{k,t}*\varphi _{k}*f\Big )^{\bigwedge }(\xi )\big |^2&\lesssim \Vert \Omega \Vert _{L^\infty }^2\min (|2^k\xi |^2,1)\min (1, \,|2^{k}\xi |^{-2\beta })|{\widehat{f}}(\xi )|^2. \end{aligned}$$

Then by the above inequalities, we get

$$\begin{aligned}&\sum _{k\in \mathbb {Z}}\big |\widehat{\frac{d}{dt}T_{k,t,0}^\mathcal {N}f}(\xi )\big |^2|{\widehat{f}}(\xi )|^2\\&\quad \lesssim \sum _{k\in \mathbb {Z}}\big |\Big (\frac{d}{dt}v_{k,t}*\varphi _{k}*f\Big )^{\bigwedge }(\xi )\big |^2\\&\quad \lesssim \Vert \Omega \Vert _{L^\infty }^2\min (|2^k\xi |^2,1)\min (1, \,|2^{k}\xi |^{-2\beta })|{\widehat{f}}(\xi )|^2\\&\quad \lesssim \Vert \Omega \Vert _{L^\infty }^2\Big (\sum _{|2^k\xi |\le 1}|2^{k}\xi |^2+\sum _{|2^k\xi |\ge 1}|2^{k}\xi |^{-2\beta }\Big )\ |{\widehat{f}}(\xi )|^2 \lesssim \Vert \Omega \Vert _{L^\infty }^2 |{\widehat{f}}(\xi )|^2. \end{aligned}$$

Then by the Plancherel theorem and the above inequality, we get

$$\begin{aligned}&\bigg \Vert \bigg (\int _1^2\sum _{k\in \mathbb {Z}}|\frac{d}{dt}T_{k,t,0}^\mathcal {N}f|^2\frac{dt}{t}\bigg )^\frac{1}{2}\bigg \Vert _{L^2}\lesssim \Vert \Omega \Vert _{L^\infty }\Vert f\Vert _{L^2}. \end{aligned}$$

(7.16)

Combining the case of $j>0$ and $j=0,$ we get

$$\begin{aligned}&\bigg \Vert \bigg (\int _1^2\sum _{k\in \mathbb {Z}}|\frac{d}{dt}T_{k,t,j}^\mathcal {N}f|^2\frac{dt}{t}\bigg )^\frac{1}{2}\bigg \Vert _{L^2}\lesssim (1+N(j)) \Vert \Omega \Vert _{L^\infty }\Vert f\Vert _{L^2}. \end{aligned}$$

(7.17)

This along with (7.10) and (7.13), we get for $0<\beta <1,$

$$\begin{aligned}&\big \Vert \mathcal {S}_{2,j}^\mathcal {N}( f)\big \Vert _{L^2}\lesssim 2^{-\frac{\beta }{2} N(j-1)}(N(j)+1)^{1/2}\Vert f\Vert _{L^2}, \end{aligned}$$

which proves Lemma 7.1. $\square$

Part 2. Next, we give the $L^p(w)$-norm of $\mathcal {S}_{2,j}^N( f)(x).$ For $j\ge 1,$ we denote by $S_{k,j}:=S_{k-N(j-1)}-S_{k-N(j)}$. For $j=0,$ we denote by $S_{k,0}:=S_{k}.$ We have the following observation:

$$\begin{aligned} \mathcal {S}_{2,j}^\mathcal {N}( f)(x)&=\Big (\displaystyle \sum _{k\in \mathbb {Z}}\displaystyle \sup _{\begin{array}{c} t_1<\cdots<t_\lambda \\ {[}t_l,t_{l+1}]\subset [1,2] \end{array}}\displaystyle \sum _{l=1}^{\lambda -1}|T_{k,t_l} S_{k,j}f(x)-T_{k,t_{l+1}} S_{k,j}f(x)|^2\Big )^{\frac{1}{2}}\\&=: \Big (\displaystyle \sum _{k\in \mathbb {Z}}\displaystyle \sup _{\begin{array}{c} t_1<\cdots <t_\lambda \\ {[}t_l,t_{l+1}]\subset [1,2] \end{array}}\displaystyle \sum _{l=1}^{\lambda -1}|{T}_{k,t_l,t_{l+1}}S_{k,j} f(x)|^2\Big )^{\frac{1}{2}}, \end{aligned}$$

where the operator ${{T}}_{k,t_l,t_{l+1}}$ is given by

$$\begin{aligned} {{T}}_{k,t_l,t_{l+1}}f(x):= & {} \displaystyle \int _{2^kt_l<|x-y|\le 2^{k}t_{l+1}}\frac{\Omega (x-y)}{|x-y|^n}f(y)dy\\ {}= & {} v_{k,t_l,t_{l+1}}*f(x),\,\,[t_l,t_{l+1}]\subset [1,2]. \end{aligned}$$

Denote by

$$\begin{aligned} K_{k,l, j}:=v_{k, t_l, t_{l+1}}*(\varphi _{{k-N(j)}}-\varphi _{{k-N(j-1)}})\end{aligned}$$

(7.18)

the kernel of ${T}_{k,t_l,t_{l+1}}S_{k,j }$ for $j\ge 1$ and by

$$\begin{aligned} K_{k,l, 0}:=v_{k, t_l, t_{l+1}}*\varphi _{{k}}\end{aligned}$$

(7.19)

the kernel of ${T}_{k,t_l,t_{l+1}}S_{k,0 }$ for $j=0.$ Then

$$\begin{aligned} \mathcal {S}_{2,j}^\mathcal {N}( f)(x)&= \bigg (\displaystyle \sum _{k\in \mathbb {Z}}\displaystyle \sup _{\begin{array}{c} t_1<\cdots <t_\lambda \\ {[}t_l,t_{l+1}]\subset [1,2] \end{array}}\displaystyle \sum _{l=1}^{\lambda -1}|K_{k,l, j}*f(x)|^2\bigg )^{\frac{1}{2}}, \end{aligned}$$

(7.20)

In the following, we give the kernel estimates.

Lemma 7.2

For every $x \in \mathbb {R}^{n}\backslash \{0\}$, $j\ge 0$ and $k\in \mathbb Z,$

$$\begin{aligned} \displaystyle \sup _{\begin{array}{c} t_1<\cdots <t_\lambda \\ {[}t_l,t_{l+1}]\subset [1,2] \end{array}}\displaystyle \sum _{l=1}^{\lambda -1}|K_{k,l, j}(x)|&\lesssim \Vert \Omega \Vert _{L^{\infty }}\frac{2^k}{|x|^{n+1}}\chi _{2^{k-1}\le |x|\le 2^{k+2}}.\end{aligned}$$

(7.21)

If $0<|y|\le \frac{|x|}{2},$

$$\begin{aligned} \displaystyle \sup _{\begin{array}{c} t_1<\cdots <t_\lambda \\ {[}t_l,t_{l+1}]\subset [1,2] \end{array}}\displaystyle \sum _{l=1}^{\lambda -1}|K_{k,l, j}(x-y)-K_{k,l, j}(x)|&\lesssim \frac{\omega _j\left( \frac{|y|}{|x|}\right) }{|x|^n}\chi _{2^{k-1}\le |x|\le 2^{k+2}}, \end{aligned}$$

(7.22)

where $\omega _j(t)\le \Vert \Omega \Vert _{L^\infty }\min (1, 2^{N(j)}t).$

Proof

Let $x \in \mathbb {R}^{n}\backslash \{0\}$. Since supp $\varphi \subset \{x:|x| \le \frac{1}{100}\},$ we get that

$$\begin{aligned}&\displaystyle \sup _{\begin{array}{c} t_1<\cdots<t_\lambda \\ {[}t_l,t_{l+1}]\subset [1,2] \end{array}}\displaystyle \sum _{l=1}^{\lambda -1}|v_{k,t_l,t_{l+1}} *\varphi _{k-N(j)}(x)|\nonumber \\&\quad =\displaystyle \sup _{\begin{array}{c} t_1<\cdots<t_\lambda \\ {[}t_l,t_{l+1}]\subset [1,2] \end{array}}\displaystyle \sum _{l=1}^{\lambda -1}\bigg |\int _{{\mathbb R}^n}\frac{\Omega (y^{\prime })}{|y|^{n}} \chi _{2^{k}t_l<|y|<2^{k}t_{l+1}} \varphi _{k-N(j)}(x-y)d y\bigg | \\&\quad \lesssim \Vert \Omega \Vert _{L^{\infty }} \frac{1}{|x|^{n}} \chi _{2^{k-1}\le |x|\le 2^{k+2}}.\nonumber \end{aligned}$$

(7.23)

Then we get

$$\begin{aligned} \displaystyle \sup _{\begin{array}{c} t_1<\cdots <t_\lambda \\ {[}t_l,t_{l+1}]\subset [1,2] \end{array}}\displaystyle \sum _{l=1}^{\lambda -1}|v_{k,t_l,t_{l+1}} *\varphi _{k-N(j)}(x)|&\lesssim \Vert \Omega \Vert _{L^{\infty }} \frac{2^k}{|x|^{n+1}} \chi _{2^{k-1}\le |x|\le 2^{k+2}}. \end{aligned}$$

(7.24)

On the other hand, we compute the gradient. Again by taking into account the support of $\varphi$, we obtain that

$$\begin{aligned}&\displaystyle \sup _{\begin{array}{c} t_1<\cdots<t_\lambda \\ {[}t_l,t_{l+1}]\subset [1,2] \end{array}}\displaystyle \sum _{l=1}^{\lambda -1}|\nabla (v_{k,t_l,t_{l+1}}*\varphi _{k-N(j)})(x)|\nonumber \\&\quad =\displaystyle \sup _{\begin{array}{c} t_1<\cdots<t_\lambda \\ {[}t_l,t_{l+1}]\subset [1,2] \end{array}}\displaystyle \sum _{l=1}^{\lambda -1}\left| \int _{{\mathbb R}^n} \frac{\Omega (y^{\prime })}{|y|^{n}} \chi _{2^{k}t_l<|y|<2^{k}t_{l+1}} 2^{-(k-N(j))(n+1)} \nabla \varphi \left( \frac{x-y}{2^{k-N(j)}}\right) d y\right| \\&\quad \lesssim \int _{{\mathbb R}^n} \frac{|\Omega (y^{\prime })|}{|y|^{n}}\chi _{2^{k}\le |y|\le 2^{k+1} } 2^{-(k-N(j))(n+1)} \left| \nabla \varphi \left( \frac{x-y}{2^{k-N(j)}}\right) \right| d y\nonumber \\&\quad \lesssim \Vert \Omega \Vert _{L^{\infty }} \frac{2^{N(j)}}{|x|^{n+1}} \chi _{2^{k-1}\le |x|\le 2^{k+2}}.\nonumber \end{aligned}$$

(7.25)

Therefore, by the gradient estimate, for $|y| \le \frac{1}{2}|x|$ we have

$$\begin{aligned}&\displaystyle \sup _{\begin{array}{c} t_1<\cdots <t_\lambda \\ {[}t_l,t_{l+1}]\subset [1,2] \end{array}}\displaystyle \sum _{l=1}^{\lambda -1}|v_{k,t_l,t_{l+1}}*\varphi _{k-N(j)}(x-y)-v_{k,t_l,t_{l+1}}*\varphi _{k-N(j)}(x)|\nonumber \\&\quad \lesssim \frac{\Vert \Omega \Vert _{L^{\infty }}}{|x|^{n}} 2^{N(j)} \frac{|y|}{|x|}\chi _{2^{k-1}\le |x|\le 2^{k+2}}.\end{aligned}$$

(7.26)

Then using (7.23) and (7.26), we get

$$\begin{aligned}&\displaystyle \sup _{\begin{array}{c} t_1<\cdots <t_\lambda \\ {[}t_l,t_{l+1}]\subset [1,2] \end{array}}\displaystyle \sum _{l=1}^{\lambda -1}|v_{k,t_l,t_{l+1}}*\varphi _{k-N(j)}(x-y)-v_{k,t_l,t_{l+1}}*\varphi _{k-N(j)}(x)|\nonumber \\&\quad \lesssim \frac{1}{|x|^{n}} \omega _j\Big (\frac{|y|}{|x|}\Big )\chi _{2^{k-1}\le |x|\le 2^{k+2}}, \end{aligned}$$

(7.27)

where $\omega _j(t)\le \Vert \Omega \Vert _{L^\infty }\min (1,2^{N(j)}t)$ (for $j=0,$ the subtraction is not even needed). From the triangle inequality, and $N(j-1)<N(j)$ it follows that the kernel $v_{k,t_l,t_{l+1}}*(\varphi _{k-N(j)}-\varphi _{k-N(j-1)})$ satisfies the same estimates (7.24) and (7.27) which proves Lemma 7.2. $\square$

Lemma 7.3

Let $\mathcal {S}_{2,j}^\mathcal {N}$ be defined as in (7.7). Then we get

$$\begin{aligned} \Vert \mathcal {S}_{2,j}^\mathcal {N}( f)\Vert _{ L^{1,\infty }}\lesssim \Vert \Omega \Vert _{L^\infty }(1+N(j))^{1/2}\Vert f\Vert _{L^1}. \end{aligned}$$

(7.28)

Proof. We perform the Calderón-Zygmund decomposition of f at height $\alpha$, thereby producing a disjoint family of dyadic cubes $\{Q\}$ with total measure $\sum _Q|Q| \lesssim {\alpha }^{-1}\Vert f\Vert _{L^1}$ and allowing us to write $f=g+h$ as in the proof of Theorem 1.7. From Lemma 7.1,

$$\begin{aligned} \Vert \mathcal {S}_{2,j}^\mathcal {N}( g)\Vert _{L^{1,\infty }}\lesssim \Vert \Omega \Vert _{L^\infty }(1+N(j))^{1/2}\Vert f\Vert _{L^1}. \end{aligned}$$

It suffices to estimate $\mathcal {S}_{2,j}^\mathcal {N}( h)$ away from $\bigcup {\widetilde{Q}}$ where ${\widetilde{Q}}$ is a large fixed dilate of Q. Denote

$$\begin{aligned} \mathcal {S}_{2,j,k}^\mathcal {N} h(x)=\Big (\displaystyle \sup _{\begin{array}{c} t_1<\cdots <t_\lambda \\ {[}t_l,t_{l+1}]\subset [1,2] \end{array}}\displaystyle \sum _{l=1}^{\lambda -1}|K_{k,l, j}*h(x)|^2\Big )^{\frac{1}{2}}. \end{aligned}$$

(7.29)

Then from the definition of $\mathcal {S}_{2,j}^\mathcal {N}$ in (7.20) we get $\mathcal {S}_{2,j}^\mathcal {N}(h)(x)\le \Big (\sum _{k\in \mathbb Z}|\mathcal {S}_{2,j,k}^\mathcal {N} h(x)|^2\Big )^{1/2}.$ Since

$$\begin{aligned} \sum _{k\in \mathbb Z}|\mathcal {S}_{2,j,k}^\mathcal {N} h(x)|^2\le \sum _{k\in \mathbb Z}|\mathcal {S}_{2,j,k}^\mathcal {N} h(x)| \sup _{k\in \mathbb Z}|\mathcal {S}_{2,j,k}^\mathcal {N} h(x)| \end{aligned}$$

and

$$\begin{aligned} \sum _{k\in \mathbb Z}|\mathcal {S}_{2,j,k}^\mathcal {N} h(x)|^2>\alpha ^2=(1+N(j))^{1/2}\alpha (1+N(j))^{-1/2}\alpha , \end{aligned}$$

then we get

$$\begin{aligned}&\big |\big \{x\notin \cup {\widetilde{Q}}:\Big (\displaystyle \sum _{k\in \mathbb Z}|\mathcal {S}_{2,j,k}^\mathcal {N} h(x)|^2\Big )^{1/2}>\alpha \big \}\big |\\ {}&=\{x\notin \cup {\widetilde{Q}}:\sum _{k\in \mathbb Z}|\mathcal {S}_{2,j,k}^\mathcal {N} h(x)|^2>\alpha ^2\}\\ {}&\subset \{x\notin \cup {\widetilde{Q}}:\sum _{k\in \mathbb Z}|\mathcal {S}_{2,j,k}^\mathcal {N} h(x)|>\alpha (1+N(j))^{1/2}\}\\&\quad \cup \{x\notin \cup {\widetilde{Q}}: \sup _{k\in \mathbb Z}|\mathcal {S}_{2,j,k}^\mathcal {N} h(x)|>(1+N(j))^{-1/2}\alpha \}. \end{aligned}$$

By the above inequalities we get

$$\begin{aligned}&\alpha \big |\big \{x\notin \cup {\widetilde{Q}}:\mathcal {S}_{2,j}^\mathcal {N} h(x)>\alpha \big \}\big |\\ {}&\le \alpha \big |\big \{x\notin \cup {\widetilde{Q}}:\Big (\displaystyle \sum _{k\in \mathbb Z}|\mathcal {S}_{2,j,k}^\mathcal {N} h(x)|^2\Big )^{1/2}>\alpha \big \}\big |\\ {}&\le \alpha \big |\big \{x\notin \cup {\widetilde{Q}}:\displaystyle \sum _{k\in \mathbb Z}|\mathcal {S}_{2,j,k}^\mathcal {N} h(x)|>(1+N(j))^{1/2}\alpha \big \}\big |\\&\quad +\alpha \big |\big \{x\notin \cup {\widetilde{Q}}:\sup _{k\in \mathbb Z}|\mathcal {S}_{2,j,k}^\mathcal {N} h(x)|>(1+N(j))^{-1/2}\alpha \big \}\big | \\&=: I+II. \end{aligned}$$

For the term I, by Chebyshev’s inequality we have

$$\begin{aligned}&\alpha \big |\big \{x\notin \cup {\widetilde{Q}}:\displaystyle \sum _{k\in \mathbb Z}|\mathcal {S}_{2,j,k}^\mathcal {N} h(x)|>(1+N(j))^{1/2}\alpha \big \}\big |\\&\quad \le (1+N(j))^{-1/2}\displaystyle \sum _Q\int _{({\widetilde{Q}})^c}\displaystyle \sum _{k\in \mathbb Z}\mathcal {S}_{2,j,k}^\mathcal {N} h_Q(x)\,dx. \end{aligned}$$

Denote by $y_Q$ the center of Q. For $x\notin {\widetilde{Q}}$ and $y\in Q,$ we get $2|y-y_Q|\le |x-y_Q|,$ then by $|y-y_Q|\le \ell (Q)$ and using the vanishing mean value of $h_Q$ and (7.22),

$$\begin{aligned} \displaystyle \sum _{k\in \mathbb Z}\mathcal {S}_{2,j,k}^\mathcal {N} h_Q(x)&\lesssim \displaystyle \sum _{k\in \mathbb Z}\displaystyle \int _{{\mathbb R}^n}|h_Q(y)|\frac{\omega _j\big (\frac{|y-y_Q|}{|x-y|}\big )}{|x-y|^n}\chi _{2^{k-1}\le |x-y|<2^{k+2}}dy\\ {}&\lesssim \displaystyle \int _{{\mathbb R}^n}|h_Q(y)|\frac{\omega _j\big (\frac{\ell (Q)}{|x-y|}\big )}{|x-y|^n}dy. \end{aligned}$$

Hence

$$\begin{aligned}&\alpha \big |\big \{x\notin \cup {\widetilde{Q}}:\displaystyle \sum _{k\in \mathbb Z}|\mathcal {S}_{2,j,k}^\mathcal {N} h(x)|>(1+N(j))^{1/2}\alpha \big \}\big |\\&\lesssim (1+N(j))^{-1/2}\displaystyle \sum _Q\displaystyle \int _{{\mathbb R}^n}|h_Q(y)|\int _{|x-y|\ge 2\ell (Q)}\frac{\omega _j\big (\frac{\ell (Q)}{|x-y|}\big )}{|x-y|^n}\,dx\ dy\nonumber \\&\lesssim (1+N(j))^{-1/2}\Vert \omega _j\Vert _{Dini}\displaystyle \sum _Q\Vert h_Q\Vert _{L^1}\lesssim \Vert \Omega \Vert _{L^\infty }(1+N(j))^{1/2}\Vert f\Vert _{L^1},\nonumber \end{aligned}$$

(7.30)

where the last inequality follows from $\Vert \omega _j\Vert _{Dini}\le \Vert \Omega \Vert _{L^\infty }(1+N(j)).$ On the other hand, for the term II, by (7.21), we have

$$\begin{aligned} \sup _{k\in \mathbb Z}|\mathcal {S}_{2,j,k}^\mathcal {N} h(x)|\le \Vert \Omega \Vert _{L^\infty }Mh(x). \end{aligned}$$

Then by the weak type (1, 1) of M (see [41]),

$$\begin{aligned}&\alpha \big |\big \{x\notin \cup {\widetilde{Q}}:\sup _{k\in \mathbb Z}|\mathcal {S}_{2,j,k}^\mathcal {N} h(x)|>(1+N(j))^{-1/2}\alpha \big \}\big |\lesssim \Vert \Omega \Vert _{L^\infty }(1+N(j))^{1/2}\Vert f\Vert _{L^1}. \end{aligned}$$

(7.31)

Combining the estimates of (7.30) and (7.31), we get

$$\begin{aligned}&\alpha \big |\big \{x\notin \cup {\widetilde{Q}}:\mathcal {S}_{2,j}^\mathcal {N} h(x)>\alpha \big \}\big | \lesssim \Vert \Omega \Vert _{L^\infty }(1+N(j))^{1/2}\Vert f\Vert _{L^1}. \end{aligned}$$

The proof of Lemma 7.3 is complete.$\square$

Lemma 7.4

For $j\ge 0,$ let $S_{2,j}^\mathcal {N}$ be defined as in (7.7). Then we get

$$\begin{aligned} \mathcal {M}_{\mathcal {S}_{2,j}^\mathcal {N}}f(x) \lesssim \mathcal {S}_{2,j}^N( f)(x)+\Vert \omega _j^{2}\Vert _{Dini}^{1/2}Mf(x), \end{aligned}$$

(7.32)

where $\omega _j(t)\le \Vert \Omega \Vert _{L^\infty }\min (1, 2^{N(j)}t).$

Proof

Let Q be a cube, and take $x,\,\xi \in Q.$ Let $B(x)=B(x,2\sqrt{n}\ell (Q)),$ then $3Q\subset B_x.$ By the triangular inequality, we get

$$\begin{aligned} |\mathcal {S}_{2,j}^\mathcal {N}( f\chi _{{{\mathbb R}^n\setminus 3Q}})(\xi )|&\le |\mathcal {S}_{2,j}^\mathcal {N}( f\chi _{{{\mathbb R}^n\setminus B_x}})(\xi )-\mathcal {S}_{2,j}^\mathcal {N}(f{\chi _{{\mathbb R}^n\setminus B_x}})(x)|\\&\quad +\mathcal {S}_{2,j}^\mathcal {N}( f{\chi _{B_x\setminus 3Q}})(\xi )+\mathcal {S}_{2,j}^\mathcal {N}( f{\chi _{{\mathbb R}^n\setminus B_x}})(x)\\&=:I+II+III. \end{aligned}$$

We begin by estimating term I.

$$\begin{aligned} I&\le \bigg (\bigg (\sum _{2^k\le \ell (Q)}+\sum _{2^k> \ell (Q)}\bigg )\bigg (\displaystyle \sup _{\begin{array}{c} t_1<\cdots <t_\lambda \\ {[}t_l,t_{l+1}]\subset [1,2] \end{array}}\displaystyle \sum _{l=1}^{\lambda -1}\int _{{\mathbb R}^n \setminus B_x}|K_{k,l, j}(\xi -y)-K_{k,l, j}(x-y)||f(y)|\,dy\bigg )^2\bigg )^{1/2}. \end{aligned}$$

For $2^k\le \ell (Q),$ since $|x-\xi |\le 2|x-y|,$ we can get $|x-y|\simeq |\xi -y|.$ Therefore by (7.21)

$$\begin{aligned}&\sum _{2^k\le \ell (Q)}\displaystyle \sup _{\begin{array}{c} t_1<\cdots <t_\lambda \\ {[}t_l,t_{l+1}]\subset [1,2] \end{array}}\displaystyle \sum _{l=1}^{\lambda -1}\int _{{\mathbb R}^n \setminus B_x}| K_{k,l, j}(\xi -y)-K_{k,l, j}(x-y)||f(y)|\,dy\\&\lesssim \Vert \Omega \Vert _{L^\infty }\displaystyle \sum _{2^k\le \ell (Q)}\displaystyle \int _{|x-y|\ge 2\sqrt{n}\ell (Q)}\frac{2^k}{|x-y|^{n+1}}|f(y)|\,dy \lesssim \Vert \Omega \Vert _{L^\infty } Mf(x). \end{aligned}$$

For $2^k>\ell (Q)$, by (7.22) we get

$$\begin{aligned}&\bigg (\sum _{2^k> \ell (Q)}\bigg (\displaystyle \sup _{\begin{array}{c} t_1<\cdots <t_\lambda \\ {[}t_l,t_{l+1}]\subset [1,2] \end{array}}\displaystyle \sum _{l=1}^{\lambda -1}\int _{{\mathbb R}^n \setminus B_x}|K_{k,l, j}(\xi -y)-K_{k,l, j}(x-y)||f(y)|\,dy\bigg )^2\bigg )^{1/2}\\&\quad \lesssim \bigg (\sum _{2^k> \ell (Q)}\bigg (\int _{|x-y|\ge 2\sqrt{n}\ell (Q)}\frac{\omega _j(|x-\xi |/|x-y|)}{|x-y|^{n}}\chi _{2^{k-1}\le |x-y|\le 2^{k+2}}|f(y)|\,dy\bigg )^2\bigg )^{1/2}\\&\quad \lesssim \bigg (\sum _{2^k> \ell (Q)}\omega _j^2(\ell (Q)/2^k)\bigg )^{1/2}Mf(x) \lesssim \Vert \omega _j^{2}\Vert _{Dini}^{1/2}Mf(x). \end{aligned}$$

Combining the estimates of $\sum _{2^k\le \ell (Q)}$ and $\sum _{2^k> \ell (Q)}$, we get

$$\begin{aligned} I\lesssim (\Vert \omega _j^{2}\Vert _{Dini}^{1/2}+\Vert \Omega \Vert _{L^\infty })Mf(x). \end{aligned}$$

For the term II,  by using (7.21) and noting that $|x-y|\simeq |\xi -y|$ (since $3|x-\xi |\le |x-y|$), we have

$$\begin{aligned} II&\lesssim \sum _{k\in \mathbb Z}\displaystyle \sup _{\begin{array}{c} t_1<\cdots <t_\lambda \\ {[}t_l,t_{l+1}]\subset [1,2] \end{array}}\displaystyle \sum _{l=1}^{\lambda -1}\int _{{\mathbb R}^n}|K_{k,l, j}(\xi -y)||f{\chi _{B_x\setminus 3Q}}(y)|\,dy\\ {}&\lesssim \Vert \Omega \Vert _{L^\infty }\int _{3\ell (Q)\le |x-y|\le 2\sqrt{n}\ell (Q)}\frac{1}{|x-y|^n}\sum _{k\in \mathbb Z}\chi _{2^{k-1}\le |x-y|\le 2^{k+2}}|f(y)|\,dy\\ {}&\lesssim \Vert \Omega \Vert _{L^\infty }Mf(x). \end{aligned}$$

We now turn to the term III. Note that

$$\begin{aligned} \mathcal {S}_{2,j}^\mathcal {N}( f{\chi _{{\mathbb R}^n\setminus B_x}})(x)&= \Big (\displaystyle \sum _{k\in \mathbb {Z}}\displaystyle \sup _{\begin{array}{c} t_1<\cdots <t_\lambda \\ {[}t_l,t_{l+1}]\subset [1,2] \end{array}}\displaystyle \sum _{l=1}^{\lambda -1}|K_{k,l, j}*f{\chi _{{\mathbb R}^n\setminus B_x}}(x)|^2\Big )^{\frac{1}{2}}, \end{aligned}$$

where for $j\ge 1,$

$$\begin{aligned} K_{k,l, j}*f(x)=v_{k,t_l,t_{l+1}}*(\varphi _{k-N(j)}-\varphi _{k-N(j-1)})*f(x) \end{aligned}$$

and for $j=0,$

$$\begin{aligned} K_{k,l, j}*f(x)=v_{k,t_l,t_{l+1}}*\varphi _{k}*f(x). \end{aligned}$$

Since supp $v_{k,t_l,t_{l+1}}*\varphi _{k}\subset \{x: |x|\le 2^{k+1}\}$ and supp $v_{k,t_l,t_{l+1}}*\varphi _{k-N(j)}\subset \{x: |x|\le 2^{k+1}\},$ from (7.21) we get

$$\begin{aligned} III&= \Big (\displaystyle \sum _{k\in \mathbb {Z}}\displaystyle \sup _{\begin{array}{c} t_1<\cdots<t_\lambda \\ {[}t_l,t_{l+1}]\subset [1,2] \end{array}}\displaystyle \sum _{l=1}^{\lambda -1}\bigg |\int _{|x-y|>2\sqrt{n}\ell (Q)}K_{k,l, j}(x-y)f(y)\,dy\bigg |^2\bigg )^{\frac{1}{2}}\\ {}&= \Big (\displaystyle \sum _{2\sqrt{n}\ell (Q)<2^{k+1}}\displaystyle \sup _{\begin{array}{c} t_1<\cdots<t_\lambda \\ {[}t_l,t_{l+1}]\subset [1,2] \end{array}}\displaystyle \sum _{l=1}^{\lambda -1}\bigg |\int _{|x-y|>2\sqrt{n}\ell (Q)}K_{k,l, j}(x-y)f(y)\,dy\bigg |^2\bigg )^{\frac{1}{2}} \\ {}&\lesssim \mathcal {S}_{2,j}^\mathcal {N}( f)(x)+\bigg (\displaystyle \sum _{2\sqrt{n}\ell (Q)<2^{k+1}}\bigg (\int _{|x-y|\le 2\sqrt{n}\ell (Q)} \displaystyle \sup _{\begin{array}{c} t_1<\cdots <t_\lambda \\ {[}t_l,t_{l+1}]\subset [1,2] \end{array}}\displaystyle \sum _{l=1}^{\lambda -1}|K_{k,l, j}(x-y)||f(y)|\,dy\bigg )^2\bigg )^{1/2}\\ {}&\lesssim \mathcal {S}_{2,j}^\mathcal {N}( f)(x)+ \Vert \Omega \Vert _{L^\infty }\sum _{2\sqrt{n}\ell (Q)\le 2^{k+1} }2^{-kn}\int _{|x-y|\le 2\sqrt{n}\ell (Q) } |f(y)|\,dy\\ {}&\lesssim \mathcal {S}_{2,j}^\mathcal {N}( f)(x)+ \Vert \Omega \Vert _{L^\infty }Mf(x). \end{aligned}$$

Combining the estimates of $I,\,II$ and III, we get

$$\begin{aligned} \mathcal {S}_{2,j}^\mathcal {N}( f{\chi _{{\mathbb R}^n\setminus 3Q}})(\xi )\lesssim \mathcal {S}_{2,j}^\mathcal {N}( f)(x)+(\Vert \Omega \Vert _{L^\infty }+\Vert \omega _j^2\Vert _{Dini}^{1/2})Mf(x). \end{aligned}$$

which leads to

$$\begin{aligned} \mathcal {M}_{\mathcal {S}_{2,j}^\mathcal {N}}f(x)\lesssim \mathcal {S}_{2,j}^\mathcal {N}f(x)+(\Vert \Omega \Vert _{L^\infty }+\Vert \omega _j^2\Vert _{Dini}^{1/2})Mf(x). \end{aligned}$$

The proof of Lemma 7.4 is complete. $\square$

Then by Lemma 7.3, Lemma 7.4 and the weak type (1, 1) of M (see [41]) we get

$$\begin{aligned} \Vert \mathcal {M}_{\mathcal {S}_{2,j}^\mathcal {N}}f\Vert _{ L^{1,\infty }}&\lesssim \Vert \Omega \Vert _{L^{\infty }}(1+N(j))^{1/2}\Vert f\Vert _{L^1} \end{aligned}$$

(7.33)

since $\Vert \omega _j^{2}\Vert _{Dini}^{1/2}\lesssim \Vert \Omega \Vert _{L^{\infty }}(1+N(j))^{1/2}.$

Since $\mathcal {S}_{2,j}^\mathcal {N}$ satisfies (7.28) and (7.33), therefore by applying Lemma 2.1 to $U=\mathcal {S}_{2,j}^\mathcal {N}$, we get that for $1<p<\infty$ and $w\in A_p$,

$$\begin{aligned} \Vert \mathcal {S}_{2,j}^\mathcal {N}( f)\Vert _{L^p(w)}&\lesssim \Vert \Omega \Vert _{L^{\infty }}(1+N(j))^{1/2}\{w\}_{A_p}\Vert f\Vert _{L^p(w)}. \end{aligned}$$

(7.34)

Taking $w=1$ in the above inequality gives

$$\begin{aligned} \Vert \mathcal {S}_{2,j}^\mathcal {N}( f)\Vert _{L^p}&\lesssim \Vert \Omega \Vert _{L^{\infty }}(1+N(j))^{1/2}\Vert f\Vert _{L^p}. \end{aligned}$$

(7.35)

Interpolating between (7.8) and (7.35) shows that there exists some $\theta \in (0,1)$

$$\begin{aligned} \Vert \mathcal {S}_{2,j}^\mathcal {N}( f)\Vert _{L^p}&\lesssim \Vert \Omega \Vert _{L^{\infty }}2^{-\theta N(j-1)}(1+N(j))^{1/2}\Vert f\Vert _{L^p}. \end{aligned}$$

(7.36)

Combining (7.34) and (7.36), and by using the interpolation theorem with change of measures-Lemma 2.2, we obtain that

$$\begin{aligned} \Vert \mathcal {S}_{2}(\mathcal {T}_\Omega f)\Vert _{L^p(w)}\lesssim \Vert \Omega \Vert _{L^\infty }\{w\}_{A_p}(w)_{A_p}^{1/2}\Vert f\Vert _{L^p(w)}, \end{aligned}$$

which gives the proof of (7.2). The proof of Theorem 1.4 is complete. $\square$