1 Introduction

Sparse domination is a relatively new tool to prove weighted estimates for singular integrals. The sparse method has been developed over the past five years by many researchers in harmonic analysis, with significant works by Lerner [1], Lacey [2], Conde-Alonso [3], and the references therein.

A collection \({{\mathcal{S}}}\) is an η-sparse family of cubes in \(\mathbb{R}^{d}\) if for every \(Q\in {{\mathcal{S}}}\) there exists \(E_{Q}\subset Q\) such that \(|E_{Q}|\geq \eta |Q|\) (here \(0<\eta <1\)), and \(E_{Q}\cap E_{Q'}=\varnothing \) when \(Q\neq Q'\). For an η-sparse collection of cubes \({{\mathcal{S}}}\), we use the notation

$$ \langle f \rangle _{p,Q}:= \vert Q \vert ^{-\frac{1}{p}} \Vert f \chi _{Q} \Vert _{L^{p}}. $$

This averaging form is then easily controlled, facilitating the proof of weighted \(A_{p}\)-type estimates.

Lerner [1] proved a pointwise sparse domination for ω Calderón–Zygmund operators. The key role in his proof is played by the grand maximal operator

$$ \mathcal{M}_{T}(f) (x):=\sup_{Q \ni x} \mathop{ \mathrm{esssup}}\limits _{\xi \in Q} \bigl\vert T\bigl(f\mathbb{R}^{d}\backslash {3Q}\bigr) ( \xi ) \bigr\vert . $$

Later, Li [4] established sparse domination theorem for multilinear singular integral operators with the kernel satisfying the \({L}^{r}\)-Hörmander condition. Cao and Yabuta [5] developed sparse dominations for the multilinear Littlewood–Paley operators with the same kernel condition. Wen, Wu, and Xue [6] gave a sparse domination for the iterated commutators of multilinear pseudo-differential operators. Recently Lerner and Ombrosi [7] improved the results in [1] by weakening the assumption on T and by replacing \(\mathcal{M}_{T}(f)(x)\) with a more flexible operator. Motivated by the above works, the purpose of this paper is to establish a sparse domination for the iterated commutators of multi(sub)linear operator with weaker hypotheses than [4, 5].

Motivated by Lerner and Ombrosi [7], we assume that T is an operator satisfying the following \(W_{q}\) property instead of assuming T is bounded from \(L^{q}\times \cdots \times L^{q}\rightarrow L^{q/m , \infty } \): there is a nonincreasing function \(\psi _{T,q}\) such that, for any \(f_{j}\in L^{q}(Q)\) with \(j=1,\dots ,m\) and any cube Q,

$$\begin{aligned} \Biggl|\Biggl\{ x\in Q: \bigl\vert T(\vec{f}\chi _{Q}) (x) \bigr\vert >\psi _{T,q}(\lambda )\prod_{j=1}^{m} \langle f_{j}\rangle _{q,Q} \Biggr\} \Biggr|\leq \lambda \vert Q \vert \quad (0< \lambda < 1). \end{aligned}$$
(1.1)

It is easy to see that \(L^{q}\times \cdots \times L^{q}\rightarrow L^{q/m , \infty } \) implies that T satisfies the \(W_{q}\) property with \(\psi _{T,q}=\|T\|_{L^{q}\times \cdots \times L^{q}\rightarrow L^{q/m , \infty }}\lambda ^{-m/q}\).

Let \(\alpha >0\), we define

$$ \mathcal{M}^{\sharp }_{T,\alpha }(\vec{f}) (x):=\sup _{Q \ni x} \mathop{\mathrm{esssup}}\limits _{\xi , \xi '\in Q} \bigl\vert T( \vec{f}\chi _{ \mathbb{R}^{d}\backslash {\alpha Q}}) (\xi )-T(\vec{f}\chi _{\mathbb{R}^{d} \backslash {\alpha Q}}) \bigl(\xi '\bigr) \bigr\vert . $$

Given an operator T, \(\alpha >0\), the iterated commutators of T are defined by

$$\begin{aligned} T^{\ell }_{\vec{b}}(\vec{f}) (x)= \bigl[b_{1}, \bigl[b_{2},\ldots [b_{\ell }, T]_{ \ell }\cdots \bigr]_{2}\bigr]_{1} (\vec{f}) (x), \end{aligned}$$
(1.2)

where \(\vec{b}=\vec{b}^{(\ell )}=(b_{1},\dots ,b_{\ell })\) (\(1\leq \ell \leq m\)). Throughout this paper, \(\tau _{m}= \{1,\ldots , m\}\). The symbol \(|\tau |\) denotes the number of the elements in τ. \(\tau '=\tau _{m}\backslash \tau \) is the complementary set.

Our main results of this paper are as follows.

Theorem 1.1

Assume that the multi(sub)linear integral T satisfies the \(W_{q}\) condition and \(\mathcal{M}^{\sharp }_{T,\alpha }\) is bounded from \(L^{r}\times \cdots \times L^{r}\rightarrow L^{r/m , \infty } \) for some \(\alpha \geq 3\). Let \(1\leq q,r<\infty \), and \(s=\max \{q, r\}\). Then, for any compactly supported functions \(f_{i} \in L^{s}(\mathbb{R}^{d})\), \(i=1,\dots ,m\), there exist \(3^{d}\) sparse families \(\mathcal{S}_{j}\) such that, for a.e. \(x\in \mathbb{R}^{d} \),

$$\begin{aligned} & \bigl\vert T^{\ell }_{\vec{b}}(\vec{f}) (x) \bigr\vert \\ &\quad \leq C\sum_{j=1}^{3^{d}} \sum _{\tau \subset \tau _{ \ell }}\sum_{Q\in \mathcal{S}_{j}} \prod _{ i \in \tau } \langle f_{i} \rangle _{s,Q} \bigl\vert b_{i}(x)-b_{i,Q} \bigr\vert \times \prod _{ k \in \tau _{\ell }\backslash \tau } \bigl\langle (b_{k}-b_{k,Q})f_{k} \bigr\rangle _{s,Q} \times \prod_{ t \in \tau _{m}\backslash \tau _{\ell }} \langle f_{t} \rangle _{s,Q} \chi _{Q}, \end{aligned}$$

where \(C=c_{d,s,\alpha } (\psi _{T,q}(1/12\cdot (2\alpha )^{d}) +\| \mathcal{M}^{\sharp }_{T,\alpha } \|_{L^{r}\times \cdots \times L^{r} \rightarrow L^{r/m,\infty }} )\).

Remark 1.1

Our Theorem 1.1 is the commutators result of [7]. Compared with the hypotheses in [4, 5], the \(\mathcal{M}^{\sharp }_{T,\alpha }\) makes our proof concise and clear. Further, the \(W_{q} \) condition of T is weaker than the assumption \(L^{q}\times \cdots \times L^{q}\rightarrow L^{q/m , \infty } \) of T in [4, 5].

Let \(1\leq p_{1},\dots , p_{m} \leq \infty \), \(\vec{w}=(w_{1},\dots ,w_{m})\), and each \(w_{i}\) is a nonnegative function on \(\mathbb{R}^{d}\). w⃗ is said to satisfy the following \(A_{\vec{p}/r}\) condition if

$$ [\vec{w}]_{A_{\vec{p}/r}}:=\sup_{Q} \biggl(\frac{1}{ \vert Q \vert } \int _{Q} {{v_{ \vec{w}}}} \,\mathrm{d}x \biggr) \prod _{k=1}^{m} \biggl(\frac{1}{ \vert Q \vert } \int _{Q}w_{k}^{- \frac{r}{p_{i}-r}}(x) \,\mathrm{d}x \biggr)^{\frac{p(p_{i}-r)}{p_{i}r}}, $$

where \({v_{\vec{w}}}=\prod_{k=1}^{m}w_{k}^{p/p_{k}}\). When \(r=1\), \(A_{{\vec{p}}/{r}}\) is the \(A_{\vec{p}}\) weight class defined by Lerner et al. [8].

With the pointwise sparse domination result, we can obtain the following quantitative weighted estimates and endpoint estimates from Sect. 3 in [5] immediately.

Corollary 1.2

Assume that the multi(sub)linear integral T satisfies the \(W_{q}\) condition and \(\mathcal{M}^{\sharp }_{T,\alpha }\) is bounded from \(L^{r}\times \cdots \times L^{r}\rightarrow L^{r/m , \infty } \) for some \(\alpha \geq 3\). Let \(1\leq q,r<\infty \), and \(s=\max \{q, r\}\), \(\frac{1}{p}=\frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}} \) with \(1\leq s< p_{i}< \infty \). Set \(\sigma _{i}=w_{i}^{1-(p_{i}/s)'}\), \(i=1,\ldots ,m\). Let \(T^{\ell }_{\vec{b}}(\vec{f})\) be defined as (1.2).

\((1)\) For \(\vec{w}\in A_{{\vec{p}}/{s}}\), there exists a constant \(C=C_{m,\vec{p},d,s }\) such that

$$ \begin{aligned} &\bigl\Vert T^{\ell }_{\vec{b}}(\vec{f}) \bigr\Vert _{L^{p}(v_{\vec{w}})} \\ &\quad \leq C [\vec{w}]_{A_{{ \vec{p}}/{s}}}^{\max _{1\leq i\leq m}\{1,\frac{1}{p} ( \frac{p_{i}}{s} )'\}} \biggl( \sum _{\tau \subset \tau _{ \ell }} [v_{\vec{w}}]_{A_{\infty }}^{ \vert \tau \vert } \prod_{j\in \tau _{\ell }\backslash \tau } [\sigma _{j}]_{A_{\infty }} \biggr) \prod_{i=1}^{\ell } \Vert b_{i} \Vert _{BMO}\prod_{j=1}^{m} \Vert f_{j} \Vert _{L^{p_{j}}(w_{j})}. \end{aligned} $$

\((2)\) If \(\vec{w}\in A_{\vec{1}}\), then for any \(\lambda >0\) and \(\Phi _{s,\ell }=t^{s}(1+\log ^{+} t)^{s\ell }\) it holds that

$$ v_{\vec{w}} \bigl(\bigl\{ x: \bigl\vert T^{\ell }_{\vec{b}}( \vec{f}) \bigr\vert >\lambda ^{m}\bigr\} \bigr) \leq C_{\vec{b},s,\ell } \prod_{i=1}^{m} \biggl( \int _{ \mathbb{R}^{d}}{ \Phi _{s,\ell } \biggl(\frac{ \vert f_{i}(x) \vert }{\lambda } \biggr)}w_{i} \,dx \biggr)^{1/m}. $$

We also obtain the local decay estimate, the Coifman–Fefferman inequality with \(w\in A_{\infty }\) weight, and the Fefferman–Stein inequality with arbitrary weights regarding the iterated commutators. To the best knowledge of the author, these results are new for the iterated commutators of multi(sub)linear operator.

Theorem 1.3

Assume that the multi(sub)linear integral T satisfies the \(W_{q}\) condition and \(\mathcal{M}^{\sharp }_{T,\alpha }\) is bounded from \(L^{r}\times \cdots \times L^{r}\rightarrow L^{r/m , \infty } \) for some \(\alpha \geq 3\). Let \(1\leq q,r<\infty \), \(s=\max \{q, r\}\), and \(a>s\), then for any functions \(\operatorname{supp} f_{i}\subset Q\), \(i=1,\dots ,m\), there exist constants \(\alpha _{d,m}\) and \(c_{d,m}\) such that

$$\begin{aligned} \biggl\vert \biggl\{ x\in Q: \frac{ \vert T^{\ell }_{\vec{b}}(\vec{f}) \vert }{M_{a}(\vec{f})}>\lambda \biggr\} \biggr\vert \leq {\alpha }_{d,m} e^{-2c_{d,m} \frac{\lambda }{\prod _{i\in \tau _{\ell }} \Vert b_{i} \Vert _{BMO}c_{T}}} \vert Q \vert , \quad \lambda >0. \end{aligned}$$

Theorem 1.4

Assume that the multi(sub)linear integral T satisfies the \(W_{q}\) condition and \(\mathcal{M}^{\sharp }_{T,\alpha }\) is bounded from \(L^{r}\times \cdots \times L^{r}\rightarrow L^{r/m , \infty } \) for some \(\alpha \geq 3\). Let \(1\leq q,r<\infty \), \(s=\max \{q, r\}\), and \(a>s\), then for any \(1\leq p<\infty \) and any weight \(w\in A_{\infty }\),

$$ \bigl\Vert T_{b}^{\ell }(\vec{f}) \bigr\Vert _{L^{p}(w)}\leq c_{d,m} \sum_{\tau \subset \tau _{\ell }}[w]_{A_{\infty }}^{ \vert \tau \vert } { \prod_{i\in \tau _{\ell }}} \Vert b_{i} \Vert _{BMO} \bigl\Vert M_{a}( \vec{f}) \bigr\Vert _{L^{p}(w)}. $$

Theorem 1.5

Assume that the multi(sub)linear integral T satisfies the \(W_{q}\) condition and \(\mathcal{M}^{\sharp }_{T,\alpha }\) is bounded from \(L^{r}\times \cdots \times L^{r}\rightarrow L^{r/m , \infty } \) for some \(\alpha \geq 3\). Let \(1\leq q,r<\infty \), the exponents \(\frac{1}{p}=\frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}\) with \(p>s\) and \(1< p_{1},\ldots , p_{m}<\infty \), then for all weights \(\vec{w}=(w_{1}, \ldots , w_{m})\), \(\nu _{\vec{w}}=\prod_{i=1}^{m} w_{i}^{p/p_{i}}\),

$$\begin{aligned} \bigl\Vert T_{b}^{\ell }(\vec{f}) \bigr\Vert _{L^{p}(\nu _{\vec{w}})} \lesssim \prod_{i=1}^{m} \Vert f_{i} \Vert _{L^{p_{i}}(Mw_{i})}. \end{aligned}$$
(1.3)

The article is organized as follows. Section 2 contains some definitions and main lemmas. The proof of theorems is given in Sect. 3. We present some variations of Theorem 1.1 in Sect. 4.

2 Definitions and main lemmas

We begin by introducing some definitions and notations.

Lemma 2.1

(Generalized Hölder inequality, [9])

Let \({\Phi }_{0},{\Phi }_{1},,\ldots ,{\Phi }_{k} \) be Young functions. If

$$\begin{aligned} {\Phi }^{-1}_{1}(t){\Phi }^{-1}_{2} (t) \cdots {\Phi }^{-1}_{k}(t) \leq \kappa {\Phi }^{-1}_{0}(t) , \end{aligned}$$

then the following inequality holds:

$$\begin{aligned} \Vert f_{1} f_{2} \cdots f_{k} \Vert _{\Phi _{0},Q}\leq \kappa \Vert f_{1} \Vert _{ \Phi _{1},Q} \Vert f_{2} \Vert _{\Phi _{2},Q}\cdots \Vert f_{k} \Vert _{\Phi _{k},Q} \end{aligned}$$

for all functions \(f_{1},\ldots ,f_{m}\) and all cubes Q.

In particular, if \(\sum_{i=1}^{k}\frac{1}{s_{i}}=\frac{1}{s}\) with each \(s_{i}\ge 1\), then it holds that

$$\begin{aligned} \frac{1}{Q} \int _{Q} \vert f_{1} f_{2} \cdots f_{k} g \vert \leq C_{s} \Vert f_{1} \Vert _{ \exp L^{s_{1}},Q} \Vert f_{2} \Vert _{\exp L^{s_{2}},Q}\cdots \Vert f_{k} \Vert _{\exp L^{s_{k}},Q} \Vert g \Vert _{L(\log L)^{\frac{1}{s}},Q}. \end{aligned}$$

We introduce the weighted maximal operator and the multi(sub)linear maximal operator which will be used in the proof of our theorem. Let w be a weight and \(a\geq 1\),

$$ M_{w}f(x) = \sup_{Q\ni x} \frac{1}{w(Q)} \int _{Q} \vert f \vert w \,dy, \qquad M_{a}( \vec{f})=\sup_{Q\ni x} \prod_{i=1}^{m} \biggl( \frac{1}{ \vert Q \vert } \int _{Q} \vert {f_{i}} \vert ^{a} d{y_{i}} \biggr)^{\frac{1}{a}}. $$

Lemma 2.2

([5])

Let \(s>1\), \(t>0\), and \(w\in A_{\infty }\), then there holds that

$$ \Vert fw \Vert _{L(\log L)^{t},Q}\lesssim [w]_{A_{\infty }}^{t} \inf_{x \in Q}M_{w}\bigl( \vert f \vert ^{s}\bigr)^{1/s}\langle w\rangle _{Q}. $$

Definition 2.1

The sidelength of Q is denoted by \(\ell (Q)\). Given a cube \(Q_{0} \subset \mathbb{R}^{d}\), let \(\mathcal{D}(Q_{0})\) denote the generation of \(Q_{0}\), that is, the cubes obtained by repeated subdivision of \(Q_{0}\). Given dyadic grids \(\mathcal{D}\), for any \(j\in \mathbb{Z}\), the set \(\mathcal{D}_{j}=\{Q\in \mathcal{D},\ell {(Q)}=2^{j}\}\) forms a partition of \(\mathbb{R}^{d}\).

3 Proof of theorems

Now we devote to proving Theorem 1.1 with the case \(\ell =m\). The other cases are similar so we omit their proof here. The basic idea of our proof is borrowed from Sect. 3 in [5], but the definition of \(W_{q} \) and \(\mathcal{M}^{\sharp }_{T,\alpha }\) makes our proof more convenient than [5].

Proof of Theorem 1.1

Fix a cube \(Q_{0}\in \mathbb{R}^{d}\) and let \(Q_{0}^{*}=\alpha Q_{0}\). Set

$$ \widetilde{M}_{T}(\vec{f})=\max \bigl( \bigl\vert {T}(\vec{f}) \bigr\vert , \mathcal{M}^{ \sharp }_{T,\alpha }(\vec{f}) \bigr). $$

We define the set E as follows:

$$ E= {\Biggl\{ } {x\in Q_{0}: \max \biggl( \frac{ M_{s} (\vec{f}\chi _{Q_{0}^{*}} )(x) }{c}, \frac{ \widetilde{M}_{T}(\vec{f})}{A} \biggr)}>\prod_{j=1}^{m} \langle f_{j} \rangle _{s,Q^{*}_{0}} {\Biggr\} }. $$

By the weak endpoint estimates of \(M_{s}(\vec{f})\) and (1.1), we can choose \(c=c_{d,s,\alpha }\) and \(A=2\psi _{T,q}(1/12\cdot (2\alpha )^{d}) +\| \mathcal{M}^{\sharp }_{T, \alpha } \|_{L^{r}\rightarrow L^{r,\infty }}\) such that \(|E|\leq \frac{1}{2^{d+2}}|Q_{0}|\). Then, applying the local Calderón–Zygmund decomposition to \(\chi _{E}\) on \(Q_{0}\) at \(\lambda =\frac{1}{2^{d+1}}\), we can get a family of pairwisely disjoint cubes \(\{P_{l}\}\subset \mathcal{D}{(Q_{0})}\) such that

$$ \frac{1}{2^{d+1} \vert P_{l} \vert } < \vert E\cap P_{l} \vert \leq \frac{1}{2} \vert P_{l} \vert , \qquad \biggl\vert E \setminus \bigcup_{l} P_{l} \biggr\vert =0. $$

It is easy to have that \(\sum_{l} |P_{l}|\leq \frac{1}{2}|Q_{0}|\) and \(P_{l}\bigcap E^{c} \neq \emptyset \).

$$\begin{aligned} & \bigl\vert T_{\vec{b}}(\vec{f}\chi _{Q_{0}^{*}} ) (x) \bigr\vert \chi _{Q_{0}} (x) \\ &\quad \leq \bigl\vert T_{\vec{b}}(\vec{f}\chi _{Q_{0}^{*}} ) (x) \bigr\vert \chi _{Q_{0} \backslash \cup _{l} P_{l}}+ \sum_{l} \bigl\vert T_{\vec{b}}(\vec{f}\chi _{Q_{0}^{*} \backslash P_{l}^{*}} ) (x) \bigr\vert \chi _{P_{l}} + \sum_{l} \bigl\vert T_{\vec{b}}( \vec{f}\chi _{P_{l}^{*}} ) (x) \bigr\vert \chi _{P_{l}} \\ & \quad :=I+II+III. \end{aligned}$$
(3.1)

From [10, Remark 5.1], there exist \(3^{d}\) dyadic lattices \(\mathcal{D}_{j}\) such that for every cube \(Q \subset \mathbb{R}^{d}\) we can find a cube \(R_{Q}\in \mathcal{D}_{j}\) satisfying \(3Q\subset R_{Q}\) and \(|R_{Q}|\leq 9^{n}|Q|\). Note that \(\prod_{i=1}^{m} (b_{i}(x)-b_{i}(y_{i}) )=\sum_{\tau \subset \tau _{m}}\prod_{i\in \tau } (b_{i}(x)-b_{i, R_{Q_{0}}} ) \prod_{j\in \tau '} (b_{j, R_{Q_{0}}}-b_{j}(y_{j}) ) \). Then we can write

$$ T_{\vec{b}}(\vec{f}) (x)= \sum_{\tau \subset \tau _{m}}\prod _{i\in \tau } \bigl(b_{i}(x)-b_{i, R_{Q_{0}}} \bigr) T \biggl( \prod_{i\in \tau } f_{i}(y_{i}) \prod_{j\in \tau '} \bigl(b_{j, R_{Q_{0}}}-b_{j}(y_{j}) \bigr) f_{j}(y_{j}) \biggr) (x). $$

Since \(|E\setminus \bigcup_{l} P_{l}|=0\), it follows that

$$\begin{aligned} I\lesssim A \sum_{\tau \subset \tau _{m}}\prod _{i \in \tau } \langle f_{i} \rangle _{s,Q_{0}^{*}} \bigl\vert b_{i}(x)- b_{i,R_{Q_{0}}} \bigr\vert \prod _{j\in \tau '} \bigl\langle (b_{j}- b_{j,R_{Q_{0}}})f_{j} \bigr\rangle _{s,Q_{0}^{*}} \chi _{Q_{0}} . \end{aligned}$$
(3.2)

Now we calculate II in (3.1). For \(x\in P_{l} \) and \(x'\in P_{l}\backslash E\),

$$\begin{aligned} &\sum_{l} \bigl\vert T_{\vec{b}}(\vec{f} \chi _{Q_{0}^{*}\backslash P_{l}^{*}} ) (x) \bigr\vert \chi _{P_{l}} \\ &\quad \leq \sum_{l} \sum_{\tau \subset \tau _{m}} \prod_{i \in \tau } \bigl\vert b_{i}(x)-b_{i, R_{Q_{0}}} \bigr\vert \biggl\vert T \biggl( \prod_{i\in \tau } f_{i}\chi _{Q_{0}^{*}\backslash P_{l}^{*}}\prod_{j\in \tau '} (b_{j, R_{Q_{0}}}-b_{j} ) f_{j}\chi _{Q_{0}^{*} \backslash P_{l}^{*}} \biggr) (x) \biggr\vert . \end{aligned}$$

Denoting \(\vec{g}\chi _{Q_{0}^{*}\backslash P_{l}^{*}}(x)=\prod_{i \in \tau } f_{i}\chi _{Q_{0}^{*}\backslash P_{l}^{*}}\prod_{j \in \tau '} (b_{j, R_{Q_{0}}}-b_{j} ) f_{j}\chi _{Q_{0}^{*} \backslash P_{l}^{*}}(x)\), we can write

$$\begin{aligned} \bigl\vert T\vec{g}\chi _{Q_{0}^{*}\backslash P_{l}^{*}}(x) \bigr\vert & \leq \inf _{P_{l}} \mathcal{M}^{\sharp }_{T,\alpha }(\vec{g} \chi _{Q_{0}^{*}\backslash P_{l}^{*}}) (x) + \bigl\vert T(\vec{g}\chi _{Q_{0}^{*} \backslash P_{l}^{*}}) \bigl(x'\bigr) \bigr\vert \\ & \lesssim A \prod_{i\in \tau } \langle f_{i} \rangle _{s,Q_{0}^{*}} \prod_{j\in \tau '} \bigl\langle (b_{j}- b_{j,R_{Q_{0}}})f_{j} \bigr\rangle _{s,Q_{0}^{*}} + \bigl\vert T(\vec{g}\chi _{Q^{*}_{0}}) \bigl(x'\bigr) \bigr\vert + \bigl\vert T(\vec{g}\chi _{P_{l}^{*}}) \bigl(x'\bigr) \bigr\vert \\ & \lesssim 2A\prod_{i\in \tau } \langle f_{i} \rangle _{s,Q_{0}^{*}} \prod_{j\in \tau '} \bigl\langle (b_{j}- b_{j,R_{Q_{0}}})f_{j} \bigr\rangle _{s,Q_{0}^{*}} + \bigl\vert T(\vec{g}\chi _{P_{l}^{*}}) \bigl(x'\bigr) \bigr\vert . \end{aligned}$$
(3.3)

By \(|P_{l}\setminus E|\geq \frac{|P_{l}|}{2}\) and \(| {\{} x\in P_{l}: |T(\vec{f}\chi _{P_{l}^{*}})(x') |>A \prod_{j=1}^{m}\langle f_{j} \rangle _{s,P^{*}_{l}} {\}} |\leq \frac{|P_{l}|}{2^{d+2}}\), we can get that

$$ \inf_{P_{l}\setminus E} \bigl\vert T(\vec{g}\chi _{P_{l}^{*}}) \bigr\vert \leq A\prod_{j=1}^{m}\langle g_{j} \rangle _{s,P_{l}^{*}} \leq cA\prod _{j=1}^{m} \langle g_{j} \rangle _{s,Q_{0}^{*}} . $$

This allows us to continue (3.3) with

$$ \bigl\vert T\vec{g}\chi _{Q_{0}^{*}\backslash P_{l}^{*}}(x) \bigr\vert \lesssim (2+c)A \prod _{i\in \tau } \langle f_{i} \rangle _{s,Q_{0}^{*}} \prod_{j\in \tau '} \bigl\langle (b_{j}- b_{j,R_{Q_{0}}})f_{j} \bigr\rangle _{s,Q_{0}^{*}} . $$

Therefore

$$\begin{aligned} II\lesssim (2+c)A \sum_{\tau \subset \tau _{m}}\prod _{i\in \tau } \langle f_{i} \rangle _{s,Q_{0}^{*}} \bigl\vert b_{i}(x)- b_{i,R_{Q_{0}}} \bigr\vert \prod _{j\in \tau '} \bigl\langle (b_{j}- b_{j,R_{Q_{0}}})f_{j} \bigr\rangle _{s,Q_{0}^{*}} \chi _{Q_{0}}. \end{aligned}$$
(3.4)

\(III\) in (3.1) is the term we need. Combining (3.2) with (3.4), it follows that

$$\begin{aligned} \bigl\vert T_{\vec{b}}(\vec{f}\chi _{Q_{0}^{*}} ) \bigr\vert \chi _{Q_{0}} (x)&\lesssim (3+c)A \sum_{\tau \subset \tau _{m}}\prod _{i\in \tau } \langle f_{i} \rangle _{s,Q_{0}^{*}} \bigl\vert b_{i}(x)- b_{i,R_{Q_{0}}} \bigr\vert \prod_{j\in \tau '} \bigl\langle (b_{j}- b_{j,R_{Q_{0}}})f_{j} \bigr\rangle _{s,Q_{0}^{*}} \chi _{Q_{0}} \\ &\quad{}+ \sum_{l} \bigl\vert T_{\vec{b}}( \vec{f}\chi _{P_{l}^{*}} ) (x) \bigr\vert \chi _{P_{l}} . \end{aligned}$$

Integrating the above estimates, we can get a \(\frac{1}{2}\)-sparse family \(\mathcal{F}\subset \mathcal{D}(Q_{0})\) such that for a.e. \(x\in Q_{0}\)

$$\begin{aligned} \bigl\vert T_{\vec{b}}(\vec{f}\chi _{Q_{0}^{*}} ) \bigr\vert \lesssim &(3+c)A\sum_{Q\in \mathcal{F}} \sum _{\tau \subset \tau _{m}} \prod_{i\in \tau } \langle f_{i} \rangle _{s,Q^{*}} \bigl\vert b_{i}(x)- b_{i,R_{Q}} \bigr\vert \prod_{j\in \tau '} \bigl\langle (b_{j}- b_{j,R_{Q}})f_{j} \bigr\rangle _{s,Q^{*}} \chi _{Q}. \end{aligned}$$
(3.5)

The remaining procedure which transfers local setting \(Q_{0}\) to global setting \(\mathbb{R}^{d}\) can be referred to Sect. 4.2 in [5]. We omit the details to avoid redundancy. The proof of Theorem 1.1 is finished now. □

Proof of Theorem 1.3

Let \(\ell =m\). The proof of other cases is similar. Assume \(\operatorname{supp} f_{i}\subset Q_{0}\), \(i=1,\dots ,m\), and denote

$$\mathcal{A}_{\mathcal{F},\tau }(\vec{b},\vec{f}):=\sum_{Q \in \mathcal{F}} \prod_{i\in \tau } \langle f_{i} \rangle _{s,Q^{*}} \bigl|b_{i}(x)- b_{i,R_{Q}}\bigr| \prod_{j\in \tau '} \bigl\langle (b_{j}- b_{j,R_{Q}})f_{j} \bigr\rangle _{s,Q^{*}} \chi _{Q}.$$

It is straightforward to see that we can replace \(b_{R_{Q}}\) with \(b_{Q^{*}}\) in \(\mathcal{A}_{\mathcal{F},\tau }(\vec{b},\vec{f})\). Then, by (3.5), we can get a \(\frac{1}{2}\)-sparse family \(\mathcal{F}\subset \mathcal{D}(Q_{0})\) such that, for almost every \(x\in Q_{0}\),

$$ \bigl\vert T_{\vec{b}}(\vec{f} ) \bigr\vert = \bigl\vert T_{\vec{b}}(\vec{f}\chi _{Q_{0}^{*}} ) \bigr\vert \lesssim \sum _{\tau \subset \tau _{m}}\mathcal{A}_{ \mathcal{F},\tau }(\vec{b},\vec{f}). $$

First we observe that \(|b_{i}(x)-b_{i,Q^{*}}|\lesssim \|b_{i}\|_{BMO}+|b_{i}(x)-b_{i,Q}| \). From [11, Lemma 5.1], we can construct a sparse family \(\tilde{\mathcal{F}}\subset \mathcal{D}(Q_{0})\) such that, for every \(Q\in \mathcal{F}\subset \tilde{\mathcal{F}} \),

$$ \bigl\vert b_{i}(x)-b_{i,Q} \bigr\vert \lesssim c_{d}\sum_{P\in \tilde{\mathcal{F}},P\subseteq Q} \biggl(\frac{1}{ \vert P \vert } \int _{P} \vert b_{i}-b_{i,P} \vert \,dx \biggr)\chi _{P}(x) \lesssim c_{d} \Vert b_{i} \Vert _{BMO}\sum_{P \in \tilde{\mathcal{F}},P\subseteq Q_{0}} \chi _{P}(x). $$

For any \(a>s\), Lemma 2.1 gives that

$$ \bigl\langle (b_{j}- b_{j,{Q^{*}}})f_{j} \bigr\rangle _{s,Q^{*}} \lesssim \Vert b_{j} \Vert _{BMO} \langle f_{j}\rangle _{a,Q^{*}}. $$

Now we can write

$$\begin{aligned} & \mathcal{A}_{\mathcal{F},\tau }(\vec{b},\vec{f}) \\ &\quad \lesssim \sum_{Q\in \mathcal{F}} \prod _{i=1}^{m} \langle f_{i} \rangle _{a,Q^{*}} \Vert b_{i} \Vert _{BMO}\chi _{Q}+c_{d,m} \sum_{Q\in \mathcal{F}} \prod _{i=1}^{m} \langle f_{i} \rangle _{a,Q^{*}} \Vert b_{i} \Vert _{BMO}\sum _{P\in \tilde{\mathcal{F}},P \subseteq Q_{0}}\chi _{P}(x)\chi _{Q}. \end{aligned}$$

Hence, it is straightforward to have that

$$\begin{aligned} \frac{\mathcal{A}_{\mathcal{F},\tau }(\vec{b},\vec{f})(x)}{M_{a}\vec{f}(x)}& \leq c_{d,m} \sum_{Q\in \mathcal{F}} \prod_{i=1}^{m} \Vert b_{i} \Vert _{BMO}\chi _{Q}+c_{d,m} \sum _{Q\in \mathcal{F}} \prod_{i=1}^{m} \Vert b_{i} \Vert _{BMO}\sum _{P\in \tilde{\mathcal{F}},P\subseteq Q_{0}}\chi _{P}(x)\chi _{Q} \\ & \leq c_{d,m} \prod_{i=1}^{m} \Vert b_{i} \Vert _{BMO}\sum _{P \in \tilde{\mathcal{F}},P\subseteq Q_{0}}\chi _{P}(x). \end{aligned}$$

[12, Lemma 2.1] gives that

$$ \begin{aligned} \biggl\vert \biggl\{ x\in Q_{0} : \frac{\mathcal{A}_{\mathcal{F},\tau }(\vec{b},\vec{f})}{M_{a}(\vec{f})}> \lambda \biggr\} \biggr\vert & \leq \biggl\vert \biggl\{ x \in Q_{0} : \frac{\sum_{P\in \tilde{\mathcal{F}},P\subseteq Q_{0}}\chi _{P}(x)}{M_{a}\vec{f}}> \frac{\lambda }{c_{d,m}\prod_{i=1}^{m} \Vert b_{i} \Vert _{BMO}c_{T}} \biggr\} \biggr\vert \\ &\leq ce^{-\alpha _{d,m} \frac{\lambda }{\prod _{i=1}^{m} \Vert b_{i} \Vert _{BMO} c_{T}}} \vert Q_{0} \vert . \end{aligned} $$

This finishes the proof of Theorem 1.3. □

Proof of Theorem 1.4

Let \(\ell =m\). From Theorem 1.1, it only needs to control \(\mathcal{A}_{\mathcal{S},\tau }(\vec{b},\vec{f})\). By duality, there exists a nonnegative function \(g\in L^{p'}(w)\) satisfying \(\|g\|_{ L^{p'}(w)}=1\). Then we can write

$$\begin{aligned} & \int _{\mathbb{R}^{n}}\mathcal{A}_{\mathcal{S},\tau }(\vec{b}, \vec{f})gw\,dx \\ &\quad =\sum_{Q\in \mathcal{S}} \prod_{i\in \tau } \langle f_{i} \rangle _{s,Q} \int _{Q}\prod_{i\in \tau } \bigl\vert b_{i}(x)- b_{i,{Q}} \bigr\vert g(x)w(x)\,dx \prod _{j\in \tau '} \bigl\langle (b_{j}- b_{j,{Q}})f_{j} \bigr\rangle _{s,Q} \\ &\quad \lesssim \sum_{Q\in \mathcal{S}} \prod _{i\in \tau } \langle f_{i} \rangle _{s,Q} \vert Q \vert \bigl\Vert (b_{i}-b_{i,Q}) \bigr\Vert _{\exp L,Q} \Vert gw \Vert _{L( \log L)^{ \vert \tau \vert },Q}\prod _{j\in \tau '} \Vert b_{j} \Vert _{BMO} \langle f_{j}\rangle _{a,Q} \\ &\quad \lesssim \prod_{i=1}^{m} \Vert b_{i} \Vert _{\text{BMO}}\langle f_{i} \rangle _{a,Q} \sum_{Q\in \mathcal{S}}[w]^{ \vert \tau \vert }_{A_{\infty }} \inf_{x\in Q}M_{w}\bigl( \vert g \vert ^{q}\bigr) (x)^{1/q} w(Q), \end{aligned}$$

where \(s< a\). The second inequality follows from Lemma 2.1, and we have used Lemma 2.2 in the last inequality. By [13], we know \(M_{w}(|g|^{q})(x)^{1/q} \) is \(L^{p'}(w_{j})\) bounded when \(1< q< p'\). We can continue writing above display as

$$\begin{aligned} & \int _{\mathbb{R}^{n}}\mathcal{A}_{\mathcal{S},\tau }(\vec{b}, \vec{f})gw\,dx \\ &\quad \lesssim [w]^{ \vert \tau \vert }_{A_{\infty }} \prod_{i=1}^{m} \Vert b_{i} \Vert _{\text{BMO}} \sum _{Q\in \mathcal{S}} \int _{Q} M_{a}(\vec{f}) (x)M_{w}\bigl( \vert g \vert ^{q}\bigr) (x)^{1/q} w \,dx \\ &\quad \lesssim [w]^{ \vert \tau \vert }_{A_{\infty }} \prod_{i=1}^{m} \Vert b_{i} \Vert _{\text{BMO}} \int _{\mathbb{R}^{d}} M_{a}(\vec{f}) (x)M_{w} \bigl( \vert g \vert ^{q}\bigr) (x)^{1/q} w \,dx \\ &\quad \lesssim [w]^{ \vert \tau \vert }_{A_{\infty }} \prod_{i=1}^{m} \Vert b_{i} \Vert _{\text{BMO}} \bigl\Vert M_{a}( \vec{f}) (x) \bigr\Vert _{L^{p}(w)} \bigl\Vert M_{w}\bigl( \vert g \vert ^{q}\bigr) (x)^{1/q} \bigr\Vert _{L^{p'}(w)} \\ &\quad \lesssim [w]^{ \vert \tau \vert }_{A_{\infty }} \prod_{i=1}^{m} \Vert b_{i} \Vert _{\text{BMO}} \bigl\Vert M_{a}( \vec{f}) (x) \bigr\Vert _{L^{p}(w)} \Vert g \Vert _{L^{p'}(w)}, \end{aligned}$$

where \(1< q< p'\). This finishes the proof of Theorem 1.4. □

Proof of Theorem 1.5.

The basic idea of our proof is borrowed from Sect. 4.3 in [14] or [15]. From Theorem 1.1, it is enough to control \(\mathcal{A}_{\mathcal{S},\tau }(\vec{b},\vec{f})\). Let \(\ell =m\).

$$\begin{aligned} \bigl\Vert \mathcal{A}_{\mathcal{S},\tau }(\vec{b},\vec{f}) \bigr\Vert _{L^{p}(\nu _{ \vec{w}})}^{p} \leq \sum_{Q\in \mathcal{S}} \prod _{i \in \tau } \langle f_{i} \rangle ^{p} _{s,Q} \int _{Q}\prod_{i \in \tau } \bigl\vert b_{i}(x)- b_{i,{Q}} \bigr\vert ^{p} \nu _{w}(x)\,dx \prod_{j \in \tau '} \bigl\langle (b_{j}- b_{j,{Q}})f_{j} \bigr\rangle ^{p}_{s,Q}. \end{aligned}$$

We denote \(v_{i}(x):=Mw_{i}(x)\), then it is easy to see that \(\langle w_{i} \rangle _{Q}\leq v_{i}(x) \) for any cube Q containing x. We can choose constants a, b with \(s< a< p< b\). [11, Lemma 5.1] gives us a sparse family \(\tilde{\mathcal{S}}\) such that, for every \(Q\in \mathcal{S}\subset \tilde{\mathcal{S}} \), it holds that

$$ \bigl\vert b_{i}(x)-b_{i,Q} \bigr\vert \lesssim \Vert b_{i} \Vert _{BMO}\sum_{P\in \tilde{\mathcal{S}},P\subseteq Q} \chi _{P}(x). $$

By Lemma 2.1, it holds that

$$\begin{aligned} & \bigl\Vert \mathcal{A}_{\mathcal{S}}(\vec{b},\vec{f}) \bigr\Vert _{L^{p}(\nu _{\vec{w}})}^{p} \\ &\quad \lesssim \sum_{Q\in \mathcal{S}} \prod _{i\in \tau } \langle f_{i} \rangle ^{p}_{s,Q} \Vert b_{i} \Vert ^{p}_{BMO}\sum _{P \in \tilde{\mathcal{S}},P\subseteq Q}\chi _{P}(x)\prod _{j\in \tau '} \bigl\langle (b_{j}- b_{j,{Q}})f_{j} \bigr\rangle ^{p}_{s,Q} \nu _{ \vec{w} }(P) \\ &\quad \lesssim \sum_{Q\in \mathcal{S}} \prod _{i\in \tau } \langle f_{i} \rangle ^{p}_{s,Q} \Vert b_{i} \Vert ^{p}_{BMO}\sum _{P \in \tilde{\mathcal{S}},P\subseteq Q}\prod_{j\in \tau '} \Vert b_{j} \Vert ^{p}_{\text{BMO}} \langle f_{j} \rangle ^{p}_{a,Q} \langle \nu _{ \vec{w}}\rangle _{P} \vert P \vert \\ &\quad \lesssim \sum_{Q\in \mathcal{S}} \prod _{i=1}^{m} \bigl\langle f_{i} v_{i}^{\frac{1}{p_{i}}}\bigr\rangle ^{p}_{p,Q} \bigl\langle v_{i}^{- \frac{1}{p_{i}}}\bigr\rangle ^{p}_{b,Q} \Vert b_{i} \Vert ^{p}_{BMO}\sum _{P \in \tilde{\mathcal{S}},P\subseteq Q} \langle \nu _{ \vec{w}}\rangle _{P} \vert P \vert \\ &\quad \lesssim \sum_{Q\in \mathcal{S}} \prod _{i=1}^{m} \bigl\langle f_{i} v_{i}^{\frac{1}{p_{i}}}\bigr\rangle ^{p}_{p,Q} \Vert b_{i} \Vert ^{p}_{BMO} \sum _{P\in \tilde{\mathcal{S}},P\subseteq Q} \langle w_{i}\rangle ^{-\frac{p}{p_{i}}}_{P} \langle \nu _{ \vec{w}} \rangle _{P} \vert P \vert . \end{aligned}$$

The sparseness property of collection \({{\mathcal{S}}}\), \({\tilde{\mathcal{S}}}\) allows us to continue with

$$\begin{aligned} \bigl\Vert \mathcal{A}_{\mathcal{S}}(\vec{b},\vec{f}) \bigr\Vert _{L^{p}(\nu _{\vec{w}})}^{p} & \lesssim \sum_{Q\in \mathcal{S}} \prod_{i=1}^{m} \bigl\langle f_{i} v_{i}^{\frac{1}{p_{i}}}\bigr\rangle ^{p}_{p,Q} \Vert b_{i} \Vert ^{p}_{BMO} \sum _{P\in \tilde{\mathcal{S}},P\subseteq Q} \vert E_{P} \vert \\ & \lesssim \sum_{Q\in \mathcal{S}} \prod _{i=1}^{m} \Bigl( \inf_{Q} M_{p}\bigl(f_{i} v_{i}^{\frac{1}{p_{i}}}\bigr) \Bigr)^{p} \Vert b_{i} \Vert ^{p}_{BMO} \vert Q \vert \\ & \lesssim \int _{\mathbb{R}^{d}}\prod_{i=1}^{m} \bigl( M_{p}\bigl(f_{i} v_{i}^{ \frac{1}{p_{i}}} \bigr) \bigr)^{p} \,dx \prod_{i=1}^{m} \Vert b_{i} \Vert ^{p}_{BMO} \\ &\lesssim \prod_{i=1}^{m} \Vert b_{i} \Vert ^{p}_{BMO} \Vert f_{i} \Vert _{L^{p_{i}}(Mw_{i})}^{p}. \end{aligned}$$

This finishes the proof of Theorem (1.3). □

4 Variations

Let us introduce some notions first. T is a Calderón–Zygmund operator defined as

$$ T\vec{f}(x)= \int _{(\mathbb{R}^{d})^{m} } K(x,\vec{y})\prod_{j=1}^{m}{f_{j}}(y_{j}) \,d\vec{y}, \quad x\notin \bigcap _{j=1}^{m}\operatorname{supp} f_{j}. $$
(4.1)

Definition 4.1

([4])

Let \(1\leq r <\infty \), \(r'=\frac{r}{r-1}\). The kernel \(K(x, \vec{y})\) is called to satisfy the multilinear \(L^{r}\)-Hörmander condition if

$$ K_{r} :=\sup_{Q}\sup_{x,z\in \frac{1}{2}Q} \sum_{k=1}^{\infty } \bigl\vert 2^{k}Q \bigr\vert ^{\frac{m}{r}} \biggl( \int _{(2^{k}Q)^{m} \backslash (2^{k-1}Q)^{m}} \bigl\vert K(x,\vec{y})-K(z,\vec{y}) \bigr\vert ^{r'} \,d\vec{y} \biggr)^{\frac{1}{r'}} < \infty . $$

When \(r=1\), it should be understood as

$$ K_{1} :=\sup_{Q}\sup_{x,z\in \frac{1}{2}Q} \sum_{k=1}^{\infty } \bigl\vert 2^{k}Q \bigr\vert ^{\frac{m}{r}} \mathop{ \mathrm{esssup}}\limits _{\vec{y}\in (2^{k}Q)^{m}\backslash (2^{k-1}Q)^{m}} \bigl\vert K(x,\vec{y})-K(z,\vec{y}) \bigr\vert < \infty . $$

We first show that the multilinear \(L^{r}\)-Hörmander condition implies that \(\mathcal{M}^{\sharp }_{T,\alpha }(\vec{f})\) is bounded from \(L^{r}\times \cdots \times L^{r}\rightarrow L^{r/m,\infty }\) with \(1\leq r<\infty \).

Lemma 4.1

Suppose \(1\leq r<\infty \), \(\alpha \geq 3\). T is an operator whose kernel satisfies the bilinear \(L^{r}\)-Hörmander condition. Then \(\mathcal{M}^{\sharp }_{T,\alpha }(\vec{f})\) is bounded from \(L^{r}\times \cdots \times L^{r}\rightarrow L^{r/m,\infty }\). More specifically, for any \(x\in \mathbb{R}^{d}\), it holds that

$$\begin{aligned} \mathcal{M}^{\sharp }_{T,\alpha }(\vec{f}) (x) \leq K_{r} \mathcal{M}_{r}( \vec{f}) (x), \end{aligned}$$
(4.2)

where \(\mathcal{M}_{r}(\vec{f})(x)=\mathrm{{M}}(|\vec{f}|^{r})^{\frac{1}{r}}\) and M is the Hardy–Littlewood maximal function.

Proof

Let \(\xi , \xi '\in Q\).

$$\begin{aligned} & \bigl\vert T(\vec{f}\chi _{\mathbb{R}^{d}\backslash {\alpha Q}}) (\xi )-T( \vec{f}\chi _{\mathbb{R}^{d}\backslash {\alpha Q}}) \bigl(\xi '\bigr) \bigr\vert \\ &\quad \leq \Biggl| \int _{(\mathbb{R}^{d})^{m} {\backslash } (\alpha Q)^{m}} \bigl( K(\xi ,\vec{y})-K\bigl(\xi ', \vec{y}\bigr) \bigr) \prod_{j=1}^{m}{f_{j}}(y_{j}) \,d\vec{y} \Biggr| \\ &\quad \leq \sum_{k=1}^{\infty } \bigl\vert 2^{k}Q \bigr\vert ^{\frac{m}{r}} \biggl( \int _{(2^{k}Q)^{m} \backslash (2^{k-1}Q)^{m}} \bigl\vert K(\xi ,\vec{y})-K\bigl(\xi ',\vec{y}\bigr) \bigr\vert ^{r'} \,d \vec{y} \biggr)^{\frac{1}{r'}} \biggl( \frac{1}{ \vert 2^{k}Q \vert ^{m}} \int _{ (2^{k}Q)^{m}} \vert \vec{f} \vert ^{r} \,d\vec{y} \biggr)^{\frac{1}{r}} \\ &\quad \leq K_{r}\mathcal{M}_{r}(\vec{f}) (x). \end{aligned}$$

Since \(\mathcal{M}_{r}(\vec{f})(x)\) is bounded from \(L^{r}\times \cdots \times L^{r}\rightarrow L^{r/m , \infty } \), we can get the weak type estimates of \(\mathcal{M}^{\sharp }_{T,\alpha }(\vec{f})\). □

Theorem 4.2

Let the multi(sub)linear integral T satisfy the \(W_{q}\) condition and the kernel satisfy the multilinear \(L^{r}\)-Hörmander condition with \(1\leq q,r<\infty \), then all the estimates in Sect. 1hold for this operator T.

We give another variation of Theorem 1.1. In this case, we do not need to assume that T is a multi(sub)linear operator. Given an operator \(T(\vec{f})\) and \(\alpha >0\), we define

$$ \mathcal{M}^{\sharp }_{T,\alpha }(\vec{f}) (x):=\sup _{Q \ni x} \mathop{\mathrm{esssup}}\limits _{\xi , \xi '\in Q} \bigl\vert \bigl(T \vec{f}- T(\vec{f}\chi _{{\alpha Q}}) \bigr) (\xi )- \bigl(T\vec{f}- T( \vec{f}\chi _{{\alpha Q}}) \bigr) \bigl(\xi '\bigr) \bigr\vert . $$

Theorem 4.3

Let \(1\leq q,r<\infty \), \(\alpha \geq 3\). There are nonincreasing functions ψ and ϕ such that, for any cube Q,

$$\begin{aligned} &\Biggl|\Biggl\{ x\in Q: \bigl\vert T(\vec{f}\chi _{Q}) (x) \bigr\vert >\psi ( \lambda )\prod_{j=1}^{m} \langle f_{j}\rangle _{q,Q}\Biggr\} \Biggr|\leq \lambda \vert Q \vert \quad (0< \lambda < 1); \\ & \Biggl|\Biggl\{ x\in Q:\bigl|\mathcal{M}^{\sharp }_{T,\alpha }\vec{f}\chi _{Q}) (x)\bigr|> \phi (\lambda )\prod_{j=1}^{m} \langle f_{j}\rangle _{r,Q} \Biggr\} \Biggr|\leq \lambda \vert Q \vert \quad (0< \lambda < 1). \end{aligned}$$

Then all the estimates in Sect. 1hold for this operator T.

Remark 4.2

The assumption of \(\mathcal{M}^{\sharp }_{T,\alpha }\vec{f}\) satisfying the \(W_{r}\) condition allows us not to prove Lemma 4.1. In addition, we should use \(T_{\vec{b}}(\vec{f}\chi _{Q_{0}} )(x)-T_{\vec{b}}(\vec{f}\chi _{ P_{l}^{*}} )(x)\) instead of \(T_{\vec{b}}(\vec{f}\chi _{Q_{0}^{*}\backslash P_{l}^{*}} )(x)\) to avoid using the multi(sub)linear property.