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New weighted norm inequalities for multilinear Calderón–Zygmund operators with kernels of Dini’s type and their commutators

Abstract

In this paper, we introduce certain classes of multilinear Calderón–Zygmund operators with kernels of Dini’s type. Applying the sharp method and \(A_{\vec{p}}^{\infty }(\varphi )\) functions, we first establish some weighted norm inequalities for multilinear Calderón–Zygmund operators with kernels of Dini’s type, including pointwise estimates, strong type, and weak endpoint estimates. Furthermore, similar weighted norm inequalities for commutators with \(\mathrm{BMO}_{\theta }(\varphi )\) functions are also obtained, but the weak endpoint estimate is of \(L({\mathrm{log}}L)\) type.

Introduction

In the 1980s, the multilinear Calderón–Zygmund theory was first studied by Coifman and Meyer [1, 2]. The multilinear Calderón–Zygmund operators with standard kernels were then further investigated by many authors, such as [38]. Meanwhile, many authors weakened the standard kernel conditions to rough associated kernel conditions; see [913]. Particularly, in 1985, Yabuta [10] introduced the Calderón–Zygmund operators of type \(\omega (t)\) (the definition given below) and obtained weighted norm inequalities of the Calderón–Zygmund operators of type \(\omega (t)\) on \(L^{p}\) spaces, here weight functions belong to Muckenhoupt’s class \(A_{p}\). In 2014, Lu and Zhang [12] obtained the weighted boundedness of multilinear Calderón–Zygmund operators of type \(\omega (t)\) and their commutators with BMO functions from weighted \(L^{p}\) spaces to weighted product of \(L^{p}\) spaces. In 2016, Zhang and Sun [13] further considered weighted norm inequalities of iterated commutators that multilinear Calderón–Zygmund operators of type \(\omega (t)\) with BMO functions.

Throughout this paper, \(\omega (t):[0,\infty )\rightarrow [0,\infty )\) is a nondecreasing function with \(0<\omega (1)<\infty \).

For \(a>0\), we say that \(\omega \in {\mathrm{Dini}}(a)\), if

$$ \vert \omega \vert _{{\mathrm{Dini}}(a)}:= \int _{0}^{1}\frac{\omega ^{a}(t)}{t}\,dt< \infty. $$

It is worth mentioning that \({\mathrm{Dini}}(a_{1})\subset {\mathrm{Dini}}(a_{2})\) when \(0< a_{1}< a_{2}\).

Definition 1.1

Let \(K(x,y_{1},\ldots,y_{m})\) be a locally integrable function, defined away from the diagonal \(x=y_{1}=\cdots =y_{m}\) in \(( \mathbb{R}^{n})^{m+1}\), it is said to belong a certain class of multilinear Calderón–Zygmund kernel of type \(\omega (t)\), if there exist constant \(A>0\), \(N>0\) such that

$$\begin{aligned} \bigl\vert K(x,y_{1},\ldots,y_{m}) \bigr\vert \leq \frac{A}{ (\sum_{j=1}^{m} \vert x-y_{j} \vert )^{mn} (1+\sum_{j=1}^{m} \vert x-y_{j} \vert )^{N}} \end{aligned}$$
(1.1)

for all \((x,y_{1},\ldots,y_{m})\in (\mathbb{R}^{n})^{m+1}\) with \(x\neq y_{j}\) for some \(j=1,2,\ldots,m \) and

$$\begin{aligned} &\bigl\vert K(x,y_{1},\ldots,y_{m})-K\bigl(x',y_{1}, \ldots,y_{m}\bigr) \bigr\vert \\ &\quad \leq \frac{A}{ (\sum_{j=1}^{m} \vert x-y_{j} \vert )^{mn} (1+\sum_{j=1}^{m} \vert x-y_{j} \vert )^{N}} \omega \biggl(\frac{ \vert x-x' \vert }{\sum_{j=1}^{m} \vert x-y_{j} \vert } \biggr) \end{aligned}$$
(1.2)

whenever \(|x-x'|\leq \frac{1}{2}\max_{1\leq j\leq m}|x-y_{j}|\), and

$$\begin{aligned} &\bigl\vert K(x,y_{1},\ldots,y_{j}\ldots,y_{m})-K \bigl(x,y_{1},\ldots,y'_{j}, \ldots,y_{m}\bigr) \bigr\vert \\ &\quad \leq \frac{A}{ (\sum_{j=1}^{m} \vert x-y_{j} \vert )^{mn} (1+\sum_{j=1}^{m} \vert x-y_{j} \vert )^{N}} \omega \biggl(\frac{ \vert y_{j}-y'_{j} \vert }{\sum_{j=1}^{m} \vert x-y_{j} \vert } \biggr) \end{aligned}$$
(1.3)

whenever \(|y_{j}-y'_{j}|\leq \frac{1}{2}\max_{1\leq j \leq m}|x-y_{j}|\).

Let \(T:\mathcal{S}(\mathbb{R}^{n})\times \cdots \times \mathcal{S}( \mathbb{R}^{n})\rightarrow \mathcal{S'}(\mathbb{R}^{n})\) (from the product of Schwarz spaces to the space of tempered distributions) be a multilinear operator with certain classes of multilinear Calderón–Zygmund kernels of type \(\omega (t)\) if there exists a \(K(x,y_{1},\ldots,y_{m})\) that satisfies (1.1)–(1.3), such that

$$\begin{aligned} T(f_{1},\ldots,f_{m}) (x)= \int _{(\mathbb{R}^{n})^{m}}K(x,y_{1}, \ldots,y_{m})f_{1}(y_{1}) \cdots f_{m}(y_{m})\,dy_{1}\cdots \,dy_{m} \end{aligned}$$
(1.4)

whenever \(x\notin \bigcap_{j=1}^{m}\) \(\operatorname{supp} f_{j}\) and each \(f_{j}\in C_{c}^{\infty }(\mathbb{R}^{n}),j=1,\ldots,m\).

If T can be extended to a bounded multilinear operator:

$$\begin{aligned} L^{q_{1}}\bigl(\mathbb{R}^{n}\bigr)\times \cdots \times L^{q_{m}}\bigl(\mathbb{R}^{n}\bigr) \rightarrow L^{q} \bigl(\mathbb{R}^{n}\bigr) \end{aligned}$$
(1.5)

for some \(1< q_{1},\ldots,q_{m}<\infty \) with \(1/q_{1}+\cdots +1/q_{m}=1/q\), or

$$\begin{aligned} L^{q_{1}}\bigl(\mathbb{R}^{n}\bigr)\times \cdots \times L^{q_{m}}\bigl(\mathbb{R}^{n}\bigr) \rightarrow L^{q,\infty } \bigl(\mathbb{R}^{n}\bigr) \end{aligned}$$
(1.6)

for some \(1\leq q_{1},\ldots,q_{m}<\infty \) with \(1/q_{1}+\cdots +1/q_{m}=1/q\), then T is said to belong to the class of multilinear Calderón–Zygmund operators of type \(\omega (t)\).

Remark

When \(N=0\) in Eqs. (1.1)–(1.3), such kernels have a standard kernel of type \(\omega (t)\) as Lu and Zhang [12] and Zhang and Sun [13] considered.

The assumption of (1.6) is reasonable, one may refer to [12, Theorem 1.2].

Let T be a multilinear operator and \(\vec{b}=(b_{1},\ldots,b_{m})\) be a locally integrable vector function in \({\mathrm{BMO}}^{m}(\mathbb{R}^{n})\), the multilinear commutators of T with b⃗ is defined by

$$ T_{\Sigma _{\vec{b}}}(f_{1},\ldots,f_{m})=\sum_{j=1}^{m}T_{ \vec{b}}^{j}(\vec{f}), $$

where

$$ T_{\vec{b}}^{j}(\vec{f})\equiv b_{j}T(f_{1}, \ldots,f_{j},\ldots,f_{m})-T(f_{1}, \ldots,b_{j}f_{j},\ldots,f_{m}). $$

In 2003, Pérez and Torres [14] first introduced multilinear commutators of multilinear Calderón–Zygmund operators and established their boundedness from \(L^{q_{1}}(\mathbb{R}^{n})\times \cdots \times L^{q_{m}}(\mathbb{R}^{n})\) to \(L^{q}(\mathbb{R}^{n})\) for \(1< q,q_{1},\ldots,q_{m}<\infty \) with \(1/q_{1}+\cdots +1/q_{m}=1/q\), also, from \(L^{q_{1}}(\mathbb{R}^{n})\times \cdots \times L^{q_{m}}(\mathbb{R}^{n})\) to \(L^{q,\infty }(\mathbb{R}^{n})\) for \(1\leq q,q_{1},\ldots,q_{m}<\infty \) with \(1/q_{1}+\cdots +1/q_{m}=1/q\). In 2009, Lerner et al. [8] obtained some weighted boundedness of multilinear commutators as follows:

$$ \bigl\Vert T_{\Sigma _{\vec{b}}}(\vec{f}) \bigr\Vert _{L^{q}(v_{\vec{w}})}\leq C\sum _{j=1}^{m} \Vert {b_{j}} \Vert _{{\mathrm{BMO}}}\prod_{j=1}^{m} \Vert f_{j} \Vert _{L^{p_{j}}(w_{j})}, $$

and for the weak end-point also it was proved that

$$ v_{\vec{w}} \bigl\{ x\in \mathbb{R}^{n}: \bigl\vert T_{\Sigma _{\vec{b}}}(\vec{f}) (x) \bigr\vert >t^{m} \bigr\} \leq C\prod _{j=1}^{m} \biggl( \int _{\mathbb{R}^{n}}\Phi \biggl( \frac{ \vert f_{j}(x) \vert }{t} \biggr)w_{j}(x) \,dx \biggr)^{\frac{1}{m}}. $$

To clarify the notation, if T is associated in the usual way with a kernel \(K(x,y_{1},\ldots,y_{m})\) satisfying (1.1)–(1.3), then at a formal level

$$\begin{aligned} &T_{\Pi _{\vec{b}}}(f_{1},f_{2},\ldots,f_{m}) (x)\\ &\quad= \int _{(\mathbb{R}^{n})^{m}} \prod_{j=1}^{m} \bigl(b_{j}(x)-b_{j}(y_{j})\bigr)K(x,y_{1}, \ldots,y_{m})f_{1}(y_{1}) \cdots f_{m}(y_{m})\,dy_{1}\cdots \,dy_{m}. \end{aligned}$$

Lerner [8] obtained weighted norm inequalities of classical multilinear Calderón–Zygmund operators and their commutators with BMO functions through new maximal functions. In 2014, end-point estimates for iterated commutators of multilinear singular integrals were shown by Pérez et al. [15]. Lu and Zhang [12] studied multilinear Calderón–Zygmund operators with type \(\omega (t)\) and multilinear commutators with BMO functions. Simultaneously, they established some weighted norm inequalities, such as strong type and weak end-point estimates. The corresponding result of iterated commutators by Zhang and Sun [13] was shown, where the weights belong to \(A_{\vec{p}}\). In 2015, Pan and Tang [16] and Bui [17], respectively, established weighted norm inequalities for certain classes of multilinear Calderón–Zygmund operators and their commutators with \(\mathrm{BMO}_{\theta }(\varphi )\). The difference is that Pan and Tang also considered weak end-point results. In 2019, Hu and Zhou [18] obtained weighted norm inequalities of Calderón–Zygmund operators of type \(\omega (t)\) and their commutators with \({\mathrm{BMO}}_{\theta }(\varphi )\) functions, here weights belong to \(A_{p}(\varphi )\) functions.

Inspired by the work above, this paper’s primary purpose is to obtain weighted norm inequalities for certain classes of multilinear operators of type \(\omega (t)\) and their commutators, including the pointwise estimate, strong type, and weak end-point estimates.

Some preliminaries and notations

In this section, we first recall some notations. For a measure set E, we define \(|E|\) as the Lebesgue measure of E and \(\chi _{E}\) as the characteristic function of E. \(Q(x,r)\) denotes the cube centered at x with the side length r and \(\lambda Q=Q(x,\lambda r)\). \(\vec{q}=(q_{1},q_{2},\ldots,q_{m})\) and θ⃗= \((\theta _{1},\theta _{2},\ldots,\theta _{m})\). For a locally integrable function f, \(f_{Q}\) denotes the average \(f_{Q}=(1/|Q|)\) \(\int _{Q} f(y)\,dy\). In this paper, let \(\varphi _{\theta }(Q)=(1+r)^{\theta }\), where r is the side length of the cube Q.

The \(A_{\vec{p}}^{\infty }(\varphi )\) weights

According to [16], we say that a weight w belongs to the class \(A_{p}^{\theta }(\varphi )\) for \(1< p<\infty \), if there exists a constant C such that, for all cubes Q,

$$ \biggl(\frac{1}{\varphi _{\theta }(Q) \vert Q \vert } \int _{Q}w(y)\,dy \biggr) \biggl( \frac{1}{\varphi _{\theta }(Q) \vert Q \vert } \int _{Q}w(y)^{\frac{-1}{p-1}}\,dy \biggr)^{p-1} \leq C. $$

In particular, when \(p=1\),

$$ \biggl(\frac{1}{\varphi _{\theta }(Q) \vert Q \vert } \int _{Q}w(y)\,dy \biggr)\leq C \inf _{x\in Q}w(x). $$

Notice that \(A^{\infty }_{p}(\varphi )=\bigcup_{\theta \geq 0}A_{p}^{\theta }( \varphi )\), \(A^{\infty }_{\infty }(\varphi )=\bigcup_{p\geq 1}A_{p}^{\infty }( \varphi )\) and \(A_{p}^{0}(\varphi )\) is equivalent to the Muckenhoupt class of weights \(A_{p}\) in [19] for all \(1\leq p<\infty \). However, in general, the class \(A_{p}^{\infty }(\varphi )\) is strictly larger than the class \(A_{p}\) for all \(1\leq p<\infty \).

Next, we give some necessary properties of \(A_{\vec{p}}^{\theta }(\varphi )\) functions.

Lemma 2.1

([20])

The following statements hold:

  1. (i)

    \(A_{p}^{\infty }(\varphi )\subset A_{q}^{\infty }(\varphi )\) for \(1\leq p\leq q<\infty \).

  2. (ii)

    If \(w\in A_{p}^{\infty }(\varphi )\), with \(p>1\) then there exists \(\epsilon >0\) such that \(w\in A_{p-\epsilon }^{\infty }(\varphi )\). Consequently, \(A_{p}^{\infty }(\varphi )=\bigcup_{q< p}A_{q}^{\infty }(\varphi )\).

  3. (iii)

    If \(w\in A_{p}^{\infty }(\varphi )\) with \(p\geq 1\), then exist positive numbers \(\delta,l\) and C so that, for all cubes Q,

    $$ \biggl(\frac{1}{ \vert Q \vert } \int _{Q}w^{1+\delta }(x)\,dx \biggr)^{ \frac{1}{1+\delta }} \leq C \biggl(\frac{1}{ \vert Q \vert } \int _{Q}w(x)\,dx \biggr) \varphi ^{l}(Q). $$

Lemma 2.2

([21])

The following statements hold:

  1. (i)

    \(w\in A_{p}^{\theta }(\varphi )\) if and only if \(w^{-\frac{1}{p-1}}\in A_{p'}^{\theta }(\varphi )\), where \(\frac{1}{p}+\frac{1}{p'}=1\);

  2. (ii)

    if \(w_{1}\), \(w_{2}\in A_{p}^{\theta }(\varphi )\), \(p\geq 1\), then \(w_{1}^{\alpha }w_{2}^{1-\alpha }\in A_{p}^{\theta }(\varphi )\) for any \(0<\alpha <1\);

  3. (iii)

    if \(w\in A_{p}^{\theta }(\varphi )\), for \(1\leq p<\infty \), then

    $$ \frac{1}{\varphi _{\theta }(Q) \vert Q \vert } \int _{Q} \bigl\vert f(y) \bigr\vert \,dy\leq C \biggl( \frac{1}{w(5Q)} \int _{Q} \bigl\vert f(y) \bigr\vert ^{p}w(y) \,dy \biggr)^{\frac{1}{p}}. $$

    In particular, let \(f=\chi _{E}\) for any measurable set \(E\subset Q\),

    $$ \frac{ \vert E \vert }{\varphi _{\theta }(Q) \vert Q \vert }\leq C \biggl( \frac{w(E)}{w(5Q)} \biggr)^{\frac{1}{p}}. $$

Let \(\vec{p}=(p_{1},\ldots,p_{m})\) and \(1/p=1/p_{1}+\cdots +1/p_{m}\) with \(1\leq p_{1},\ldots,p_{m}<\infty \). Given \(\vec{w}=(w_{1},\ldots,w_{m})\), each \(w_{j}\) being nonnegative measurable, we set

$$ v_{\vec{w}}=\prod_{j=1}^{m}w_{j}^{p/p_{j}}. $$

For \(\theta \geq 0\), we say that w⃗ satisfies the \(A^{\theta }_{\vec{p}}(\varphi )\) condition and denote \(\vec{w}\in A^{\theta }_{\vec{p}}(\varphi )\), if

$$ \sup_{Q} \biggl(\frac{1}{\varphi _{\theta }(Q) \vert Q \vert } \int _{Q}v_{\vec{w}}(x)\,dx \biggr)^{1/p}\prod _{j=1}^{m} \biggl(\frac{1}{\varphi _{\theta }(Q) \vert Q \vert } \int _{Q}w_{j}(x)^{1-p'_{j}}\,dx \biggr)^{1/p'_{j}}< \infty, $$

where the supremum is taken over all cubes \(Q\subset \mathbb{R}^{n}\), and the term \((\frac{1}{|Q|}\int _{Q}w_{j}(x)^{1-p'_{j}} )^{1/p'_{j}}\) coincides with \((\inf_{x \in Q}w_{j} )^{-1}\) when \(p_{j}=1\) \(j=1,2,\ldots,m\).

For \(1\leq p_{1},\ldots,p_{m}<\infty \), set \(A_{\vec{p}}^{\infty }(\varphi )=\bigcup_{\theta \geq 0}A_{ \vec{p}}^{\theta }(\varphi )\). When \(\theta =0\), the class \(A_{\vec{p}}^{0}(\varphi )\) coincides with the class of multiple weights \(A_{\vec{p}}\) introduced by [15].

Lemma 2.3

([17])

Let \(1\leq p_{1},\ldots,p_{m}<\infty \) and \(\vec{w}=(w_{1},\ldots,w_{m})\). Then the following statements are equivalent:

  1. (i)

    \(\vec{w}\in A_{\vec{p}}^{\infty }(\varphi )\);

  2. (ii)

    \(w_{j}^{1-p'_{j}}\in A_{mp'_{j}}^{\infty },j=1,\ldots,m\), and \(v_{\vec{w}}\in A_{mp}^{\infty }(\varphi )\).

The class \(A_{\vec{p}}^{\infty }(\varphi )\) is not increasing, which means that, for \(\vec{p}=(p_{1},\ldots,p_{m})\) and \(\vec{q}=(q_{1},\ldots,q_{m})\) with \(p_{j}\leq q_{j},j=1,\ldots,m\), the following may not be true \(A_{\vec{p}}^{\infty }(\varphi )\subset A_{\vec{q}}^{\infty }(\varphi )\).

Lemma 2.4

([17])

Let \(1\leq p_{1},\ldots,p_{m}<\infty \) and \(\vec{w}=(w_{1},\ldots,w_{m})\in A_{\vec{p}}^{\infty }(\varphi )\). Then

  1. (i)

    for any \(r\geq 1, \vec{w}\in A_{r\vec{p}}^{\infty }(\varphi )\);

  2. (ii)

    if \(1< p_{1},\ldots,p_{m}<\infty \), then there exists \(r>1\) so that \(\vec{w}\in A_{\vec{p}/r}^{\infty }(\varphi )\).

\({\mathrm{BMO}}_{\infty }(\varphi )\) spaces

Now, recall the definition and properties of the \({\mathrm{BMO}}_{\infty }\) spaces introduced by [20].

A locally integrable function b is in \(\mathrm{BMO}_{\theta }(\varphi )(\theta \geq 0)\) if

$$ \Vert b \Vert _{\mathrm{BMO}_{\theta }(\varphi )}:=\sup_{Q} \frac{1}{\varphi _{\theta }(Q) \vert Q \vert } \int _{Q} \bigl\vert b(y)-b_{Q} \bigr\vert \,dy< \infty. $$

When \(\theta =0,{\mathrm{BMO}}_{0}(\varphi )={\mathrm{BMO}}(\mathbb{R}^{n})\). Clearly \({\mathrm{BMO}}(\mathbb{R}^{n})\subset \mathrm{BMO}_{\theta }(\varphi )\) and \(\mathrm{BMO}_{\theta _{1}}(\varphi )\subset {\mathrm{BMO}}_{\theta _{2}}(\varphi )\) for \(\theta _{1}\leq \theta _{2}\). We denote \(\mathrm{BMO}_{\infty }(\varphi )=\bigcup_{\theta \geq 0}\mathrm{BMO}_{\theta }( \varphi )\).

Lemma 2.5

([20])

Let \(\theta >0,s\geq 1\). If \(b\in {\mathrm{BMO}}_{\theta }(\varphi )\) then for all cubes \(Q=Q(x,r)\)

  1. (i)

    \((\frac{1}{|Q|}\int _{Q}|b(y)-b_{Q}|^{s}\,dy )^{ \frac{1}{s}}\leq \|b\|_{\mathrm{BMO}_{\theta }(\varphi )}\varphi _{ \theta }(Q)\);

  2. (ii)

    \((\frac{1}{|3^{k}Q|}\int _{3^{k}Q}|b(y)-b_{Q}|^{s}\,dy )^{\frac{1}{s}}\leq k\|b\|_{\mathrm{BMO}_{\theta }(\varphi )} \varphi _{\theta }(3^{k}Q)\), for all \(k\in N\).

The norm of Orlicz spaces

For \(\Phi (t)=t(1+{\mathrm{log}}^{+}t)\) and a cube Q in \(\mathbb{R}^{n}\), we will consider the average \(\|f\|_{\Phi,Q}\) of a function f given by the Luxemburg norm

$$ \Vert f \Vert _{\Phi,Q}=\inf \biggl\{ \lambda >0:\frac{1}{ \vert Q \vert } \int _{Q}\Phi \biggl(\frac{ \vert f(x) \vert }{\lambda } \biggr)\,dx\leq 1 \biggr\} . $$

The generalized Hölder inequality in Orlicz spaces together with the corresponding John–Nirenberg inequality in [18, Lemma 2.5] implies that

$$ \frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert \bigl(b(y)-b_{Q}\bigr)f(y) \bigr\vert \,dy \leq \Vert b \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \Vert f \Vert _{L(\mathrm{Log}L),Q}\varphi _{\theta }(Q). $$

Maximal functions and Sharp maximal functions

Maximal functions and sharp maximal functions play an important role in the proof of the main theorem. Next, recall the relevant definition.

For \(0<\eta <\infty \), the maximal operator \(M_{\varphi,\eta }\) is defined by

$$ M_{\varphi,\eta }f(x)=\sup_{x\in Q} \frac{1}{\varphi (Q)^{\eta } \vert Q \vert } \int _{Q} \bigl\vert f(y) \bigr\vert \,dy. $$

Definition 2.6

([21])

Let \(0<\eta <\infty \), then the dyadic maximal function \(M_{\varphi,\eta }^{d}\) is defined by

$$ M_{\varphi,\eta }^{d}f(x)=\sup_{x\in Q(dyadiccube)} \frac{1}{\varphi (Q)^{\eta } \vert Q \vert } \int _{Q} \bigl\vert f(y) \bigr\vert \,dy. $$

Let Q be a dyadic cube; f is a locally integral function, then the dyadic sharp maximal function \(M_{\varphi,\eta }^{\sharp,d}\) is defined by

$$\begin{aligned} M_{\varphi,\eta }^{\sharp,d}f(x) &=\sup_{x\in Q,r< 1} \frac{1}{ \vert Q \vert } \int _{Q(x_{0},r)} \bigl\vert f(y)-f_{Q} \bigr\vert \,dy+\sup_{x\in Q,r \geq 1}\frac{1}{\varphi (Q)^{\eta } \vert Q \vert } \int _{Q(x_{0},r)} \bigl\vert f(y) \bigr\vert \,dy \\ &\simeq \sup_{x\in Q,r< 1}\inf_{C} \frac{1}{ \vert Q \vert } \int _{Q(x_{0},r)} \bigl\vert f(y)-C \bigr\vert \,dy+ \sup _{x\in Q,r\geq 1}\frac{1}{\varphi (Q)^{\eta } \vert Q \vert } \int _{Q(x_{0},r)} \bigl\vert f(y) \bigr\vert \,dy, \end{aligned}$$

where \(f_{Q}=\frac{1}{|Q|}\int _{Q}f(y)\,dy\).

From the above definition, the variants of the dyadic maximal operator and the dyadic sharp maximal operator are as follows:

$$M_{\delta,\varphi,\eta }^{d}f(x)= \bigl[M_{\varphi,\eta }^{d}\bigl(|f|^{\delta }\bigr) \bigr]^{\frac{1}{\delta }},\qquad M_{\delta,\varphi,\eta }^{\sharp,d}f(x)= \bigl[M_{\varphi,\eta }^{ \sharp,d}\bigr(|f|^{\delta }) \bigr]^{\frac{1}{\delta }}. $$

Lemma 2.7

([21])

Let \(1< p<\infty \), \(w\in A_{p}^{\infty }\), \(0<\eta <\infty \) and \(f\in L^{p}(w)\), then

$$ \Vert f \Vert _{L^{p}(w)}\leq \bigl\Vert M_{\varphi,\eta }^{d}f \bigr\Vert _{L^{p}(w)}\leq \bigl\Vert M_{ \varphi,\eta }^{\sharp,d}f \bigr\Vert _{L^{p}(w)}. $$

Lemma 2.8

([21])

Let \(1< p<\infty \), \(\omega \in A_{\infty }^{\infty }\), \(0<\eta <\infty \) and \(\delta >0\) and let \(\psi:(0,\infty )\mapsto (0,\infty )\) be doubling, that is, \(\psi (2a)\leq \psi (a)\) for \(a>0\). Then there exists a constant C depending upon the \(A_{\infty }^{\infty }\) condition of w and the doubling condition of ψ such that

$$\begin{aligned} &\sup_{\lambda >0}\psi (\lambda )w \bigl( \bigl\{ y\in \mathbb{R}^{n}:M_{ \delta,\varphi,\eta }^{d}f(y)>\lambda \bigr\} \bigr)\leq C\sup_{ \lambda >0}\psi (\lambda )w \bigl( \bigl\{ y\in \mathbb{R}^{n}:M_{ \delta,\varphi,\eta }^{\sharp,d}f(y)>\lambda \bigr\} \bigr), \\ &M_{L(\log L),\varphi,\eta }f(x)=\sup_{x\in Q} \frac{1}{\varphi (Q)^{\eta }} \Vert f \Vert _{L(\log L),Q}. \end{aligned}$$

Let \(0<\eta <\infty \), \(\vec{f}=(f_{1},f_{2},\ldots,f_{m})\), then the multilinear maximal operators \(\mathcal{M}_{\varphi,\eta }\) and \(\mathcal{M}_{L(\mathrm{Log}L),\varphi,\eta }\) are defined by

$$\begin{aligned} &\mathcal{M}_{\varphi,\eta }\vec{f}(x)=\sup_{x\in Q}\prod _{j=1}^{m} \frac{1}{\varphi (Q)^{\eta }} \Vert f_{j} \Vert _{Q}, \\ &\mathcal{M}_{L(\mathrm{Log}L),\varphi,\eta }\vec{f}(x)=\sup_{x\in Q} \prod_{j=1}^{m}\frac{1}{\varphi (Q)^{\eta }} \Vert f_{j} \Vert _{L(\log L)}. \end{aligned}$$

Lemma 2.9

([16])

Let \(1< p_{j}<\infty \), \(j=1,2,\ldots,m\), \(\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}+\cdots +\frac{1}{p_{m}}\) and \(\vec{w}\in A_{\vec{p}}^{\infty }\), then there exists some \(\eta _{0}>0\) depending on \(p,m,p_{j}\) such that

$$ \bigl\Vert \mathcal{M}_{\varphi,\eta _{0}}(\vec{f}) \bigr\Vert _{L^{p}(v_{\vec{w}})} \leq C\prod_{j=1}^{m} \Vert f_{j} \Vert _{L^{p_{j}}(w_{j})}. $$

Estimates for multilinear operators

Theorem 3.1

Let T be a multilinear Calderón–Zygmund operator of type \(\omega (t)\) as in Definition 1.1, assume that \(0<\delta <\frac{1}{m},0<\eta \) and ω is satisfying \(\omega \in {\mathrm{Dini}}(1)\). Then there exists a constant \(C>0\) such that

$$ M_{\delta,\varphi,\eta }^{\#,d} \bigl(T(\vec{f}) \bigr) (x)\leq C \mathcal{M}_{\varphi,\eta }(\vec{f}) (x) $$

for all f⃗ in \(L^{p_{1}}(\mathbb{R}^{n})\times \cdots \times L^{p_{m}}(\mathbb{R}^{n})\) with \(1\leq p_{j}<\infty \) for \(j=1,\ldots,m\).

Proof

If \(\omega \in \mathrm{Dini(1)}\), then

$$ \sum_{k=1}^{\infty }\omega \bigl(2^{-k} \bigr)\thickapprox \int _{0}^{1} \frac{\omega (t)}{t}\,dt< \infty. $$

For a fixed point \(x\in \mathbb{R}^{n}\) and let \(x\in Q=Q{(x_{0},r)}\), Q is a dyadic cube. To complete the proof, we consider the following two cases of the side length r: \(r\leq 1\) and \(r>1\).

Case 1. When \(r\leq 1\). Since \(0<\delta <\frac{1}{m}<1\), \(\eta >0\) and \(||a|^{t}-|b|^{t}|<|a-b|^{t}\) for \(0< t<1\), for any number C we can estimate

$$ \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert \bigl\vert T(\vec{f}) (z) \bigr\vert ^{\delta }- \vert C \vert ^{ \delta } \bigr\vert \,dz \biggr)^{\frac{1}{\delta }}\leq \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T(\vec{f}) (z)-C \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }}. $$

Let \(Q^{*}=8\sqrt{n}Q\), we decompose \(f_{j}=f_{j}^{0}+f_{j}^{\infty }\) for each \(f_{j}\), where \(f_{j}^{0}=f_{j}\chi _{Q^{*}}\). Then

$$ \prod_{j=1}^{m}f_{j}(y_{j})= \sum_{\alpha _{1},\ldots, \alpha _{m}\in \{0,\infty \}}f_{1}^{\alpha _{1}}(y_{1}) \cdots f_{m}^{ \alpha _{m}}(y_{m})=\prod_{j=0}^{m}f_{j}^{0}(y_{j})+ \sum_{ \alpha _{1},\ldots,\alpha _{m}\in \mathscr{L}}f_{1}^{\alpha _{1}}(y_{1}) \cdots f_{m}^{\alpha _{m}}(y_{m}), $$

where \(\mathscr{L}=\{(\alpha _{1},\ldots,\alpha _{m}): \mathrm{there~is~at~least~one~}\alpha _{j}\neq 0\}\).

Let \(C=\sum_{\alpha _{1},\ldots,\alpha _{m}\in \mathscr{L}}C_{ \alpha _{1},\ldots,\alpha _{m}}\), from this condition, we can get the following a series of estimates:

$$\begin{aligned} &\biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T(\vec{f}) (z)-C \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }}\\ &\quad\leq \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl(f_{1}^{0}, \ldots,f_{m}^{0}\bigr) (z) \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }} \\ &\qquad{} +\sum_{\alpha _{1},\ldots,\alpha _{m}\in \mathscr{L}} \biggl( \frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl(f_{1}^{\alpha _{1}}, \ldots, f_{m}^{\alpha _{m}}\bigr) (z)-C_{ \alpha _{1},\ldots,\alpha _{m}} \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }} \\ &\quad :=I+\mathit{II}. \end{aligned}$$

Since \(T:L^{1}\times \cdots \times L^{1}\rightarrow L^{\frac{1}{m},\infty }\) and using the Kolmogorov inequality with \(p=\delta \) and \(q=\frac{1}{m}\), we have

$$\begin{aligned} I&\leq C \bigl\Vert T\bigl(f_{1}^{0}, \ldots,f_{m}^{0}\bigr) \bigr\Vert _{L^{\frac{1}{m},\infty }(Q, \frac{dx}{ \vert Q \vert })} \\ &\leq C\prod_{j=1}^{m}\frac{1}{ \vert Q^{*} \vert } \int _{Q^{*}} \bigl\vert f_{j}(z) \bigr\vert \,dz \\ &\leq C\prod_{j=1}^{m}\frac{1}{\varphi (Q^{*})^{\eta } \vert Q^{*} \vert } \int _{Q^{*}} \bigl\vert f_{j}(z) \bigr\vert \,dz \\ &\leq C\mathcal{M}_{\varphi,\eta }(\vec{f}) (x). \end{aligned}$$

To estimate II, we choose \(C_{\alpha _{1},\ldots,\alpha _{m}}=T(f_{1}^{\alpha _{1}},\ldots, f_{m}^{ \alpha _{m}})(x)\), for any \(z\in Q\), the following estimate holds:

$$ \sum_{\alpha _{1},\ldots,\alpha _{m}\in \mathscr{L}} \bigl\vert T\bigl(f_{1}^{ \alpha _{1}}, \ldots, f_{m}^{\alpha _{m}}\bigr) (z)-T\bigl(f_{1}^{\alpha _{1}}, \ldots, f_{m}^{\alpha _{m}}\bigr) (x) \bigr\vert \leq C \mathcal{M}_{\varphi, \eta }(\vec{f}) (x). $$

We consider first the case when \(\alpha _{1}=\cdots =\alpha _{m}=\infty \). For any \(z\in Q\), we get

$$\begin{aligned} &\bigl\vert T\bigl(f_{1}^{\infty },\ldots, f_{m}^{\infty } \bigr) (z)-T\bigl(f_{1}^{\infty }, \ldots, f_{m}^{\infty } \bigr) (x) \bigr\vert \\ &\quad \leq \int _{(\mathbb{R}^{n}\setminus Q^{*})^{m}} \bigl\vert K(z, \vec{y})-K(x,\vec{y}) \bigr\vert \prod_{j=1}^{m} \bigl\vert f(y_{j}) \bigr\vert \,d\vec{y} \\ &\quad \leq \sum_{k=1}^{\infty } \int _{(\Omega _{k})^{m}} \bigl\vert K(z,\vec{y})-K(x, \vec{y}) \bigr\vert \prod_{j=1}^{m} \bigl\vert f(y_{j}) \bigr\vert \,d\vec{y}, \end{aligned}$$

where \(\Omega _{k}=(2^{k+3}\sqrt{n}Q)\setminus (2^{k+2}\sqrt{n}Q)\) for \(k=1,2,\ldots \) .

Note that, for \(x,z\in Q\) and any \((y_{1},\ldots,y_{m})\in (\Omega _{k})^{m}\),

$$ \vert z-y_{j} \vert \geq 2^{k}\sqrt{n}r \quad \text{and}\quad \vert z-x \vert \leq \sqrt{n}r, $$

since ω is nondecreasing, and through the kernel condition (1.2), we have

$$\begin{aligned} \bigl\vert K(z,\vec{y})-K(x,\vec{y}) \bigr\vert &\leq \frac{A}{ (\sum_{j=1}^{m} \vert z-y_{j} \vert )^{mn} (1+\sum_{j=1}^{m} \vert z-y_{j} \vert )^{N}} \omega \biggl(\frac{ \vert z-x \vert }{\sum_{j=1}^{m} \vert z-y_{j} \vert } \biggr) \\ &\leq \frac{C\omega (2^{-k})}{ \vert 2^{k}\sqrt{n}Q \vert ^{m}(1+2^{k}\sqrt{n}r)^{N}}. \end{aligned}$$

Then, taking \(N\geq m\eta \),

$$\begin{aligned} &\bigl\vert T\bigl(f_{1}^{\infty },\ldots, f_{m}^{\infty } \bigr) (z)-T\bigl(f_{1}^{\infty }, \ldots, f_{m}^{\infty } \bigr) (x) \bigr\vert \\ &\quad\leq C\sum_{k=1}^{\infty } \omega \bigl(2^{-k}\bigr) \int _{(\Omega _{k})^{m}} \frac{1}{ \vert 2^{k}\sqrt{n}Q \vert ^{m}(1+2^{k}\sqrt{n}r)^{N}}\prod _{j=1}^{m} \bigl\vert f(y_{j}) \bigr\vert \,d \vec{y} \\ &\quad\leq C\sum_{k=1}^{\infty }\omega \bigl(2^{-k}\bigr)\prod_{j=1}^{m} \frac{1}{ \vert 2^{k}\sqrt{n}Q \vert (1+2^{k}\sqrt{n}r)^{\frac{N}{m}}} \int _{2^{k+3} \sqrt{n}Q} \bigl\vert f_{j}(y_{j}) \bigr\vert \,dy_{i} \\ &\quad\leq C \vert \omega \vert _{\mathrm{Dini(1)}}\mathcal{M}_{\varphi,\eta }( \overrightarrow{f}) (x). \end{aligned}$$

We are now to consider \(\alpha _{j_{1}}=\cdots =\alpha _{j_{l}}=0\) for \(1\leq l< m\). Let \(\mathscr{J}:=\{j_{1},\ldots,j_{l}\}\) then \(\alpha _{j}=\infty \) for \(j\notin \mathscr{J}\). Thus

$$\begin{aligned} &\bigl\vert T\bigl(f_{1}^{\alpha _{1}},\ldots, f_{m}^{\alpha _{m}} \bigr) (z)-T\bigl(f_{1}^{ \alpha _{1}},\ldots, f_{m}^{\alpha _{m}} \bigr) (x) \bigr\vert \\ &\quad\leq \int _{(\mathbb{R}^{n})^{m}} \bigl\vert K(z,\vec{y})-K(x,\vec{y}) \bigr\vert \prod_{j=1}^{m} \bigl\vert f_{j}^{ \alpha _{j}}(y_{j}) \bigr\vert \,d\vec{y} \\ &\quad\leq \int _{(Q^{*})^{l}}\prod_{j\in \mathscr{J}} \bigl\vert f_{j}(y_{j}) \bigr\vert \int _{(\mathbb{R}^{n}\backslash Q^{*})^{m-l}} \bigl\vert K(z,\vec{y})-K(x, \vec{y}) \bigr\vert \prod_{j\notin \mathscr{J}} \bigl\vert f_{j}(y_{j}) \bigr\vert \,d\vec{y} \\ &\quad\leq \int _{(Q^{*})^{l}}\prod_{j\in \mathscr{J}} \bigl\vert f_{j}(y_{j}) \bigr\vert \sum_{k=1}^{\infty } \int _{(\Omega _{k})^{m-l}} \bigl\vert K(z,\vec{y})-K(x, \vec{y})\bigr\vert \prod_{j\notin \mathscr{J}} \bigl\vert f_{j}(y_{j}) \bigr\vert \,d\vec{y}. \end{aligned}$$

Similar to the above discussion, taking \(N\geq m\eta \), we have

$$\begin{aligned} &\bigl\vert T\bigl(f_{1}^{\alpha _{1}},\ldots, f_{m}^{\alpha _{m}} \bigr) (z)-T\bigl(f_{1}^{ \alpha _{1}},\ldots, f_{m}^{\alpha _{m}} \bigr) (x) \bigr\vert \\ &\quad \leq C \int _{(Q^{*})^{l}}\prod_{j\in \mathscr{J}} \bigl\vert f_{j}(y_{j}) \bigr\vert \sum_{k=1}^{\infty }\omega \bigl(2^{-k}\bigr) \int _{(\Omega _{k})^{m-l}} \frac{1}{ \vert 2^{k}\sqrt{n}Q \vert ^{m}(1+2^{k}\sqrt{n}r)^{N}}\prod_{j \notin \mathscr{J}} \bigl\vert f_{j}(y_{j}) \bigr\vert \,d\vec{y} \\ &\quad \leq C\sum_{k=1}^{\infty }\omega \bigl(2^{-k}\bigr) \frac{1}{ \vert 2^{k}\sqrt{n}Q \vert ^{m}(1+2^{k}\sqrt{n}r)^{N}} \biggl(\prod_{j\in \mathscr{J}} \int _{Q^{*}} \bigl\vert f_{j}(y_{j}) \bigr\vert \,dy_{j} \biggr) \\ &\qquad{}\times \biggl( \prod_{j\notin \mathscr{J}} \int _{2^{k+3}\sqrt{n}Q} \bigl\vert f_{j}(y_{j}) \bigr\vert \,dy_{j} \biggr) \\ &\quad \leq C\sum_{k=1}^{\infty }\omega \bigl(2^{-k}\bigr)\prod_{j=1}^{m} \frac{1}{ \vert 2^{k}\sqrt{n}Q \vert (1+2^{k}\sqrt{n}r)^{\frac{N}{m}}} \int _{2^{k+3} \sqrt{n}Q} \bigl\vert f_{j}(y_{j}) \bigr\vert \,dy_{i} \\ &\quad \leq C \vert \omega \vert _{\mathrm{Dini(1)}}\mathcal{M}_{\varphi,\eta }( \vec{y}) (x). \end{aligned}$$

Case 2. When \(r>1\), since \(0<\delta <\frac{1}{m}<1\), and \(\eta >0\), it follows that

$$\begin{aligned} & \biggl(\frac{1}{\varphi (Q)^{\eta } \vert Q \vert } \int _{Q} \bigl\vert T(\vec{f}) (x) \bigr\vert ^{ \delta }\,dx \biggr)^{\frac{1}{\delta }} \\ &\quad\leq \frac{1}{\varphi (Q)^{\eta /\delta }} \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl(f_{1}^{0}, \ldots,f_{m}^{0}\bigr) (x) \bigr\vert ^{\delta }\,dx \biggr)^{\frac{1}{\delta }} \\ &\qquad{} +\sum_{\alpha _{1},\ldots,\alpha _{m}\in \mathscr{L}} \frac{1}{\varphi (Q)^{\eta /\delta }} \biggl( \frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl(f_{1}^{ \alpha _{1}}, \ldots, f_{m}^{\alpha _{m}}\bigr) (z) \bigr\vert ^{\delta }\,dz \biggr)^{ \frac{1}{\delta }} \\ &\quad:=I+\mathit{II}. \end{aligned}$$

For I, by the Kolmogorov inequality and \(T:L^{1}\times \cdots \times L^{1}\rightarrow L^{\frac{1}{m},\infty }\), we have

$$\begin{aligned} I&\leq C\frac{1}{\varphi (Q)^{m\eta }} \bigl\Vert T\bigl(f_{1}^{0}, \ldots,f_{m}^{0}\bigr) \bigr\Vert _{L^{\frac{1}{m},\infty }(Q,\frac{dx}{ \vert Q \vert })} \\ &\leq C\frac{1}{\varphi (Q)^{m\eta }}\prod_{j=1}^{m} \frac{1}{ \vert Q^{*} \vert } \int _{Q^{*}} \bigl\vert f_{j}(z) \bigr\vert \,dz \\ &\leq C\mathcal{M}_{\varphi,\eta }(\vec{f}) (x). \end{aligned}$$

To estimate II, note that, for \(z\in Q\) and any \((y_{1},\ldots,y_{m})\in (\Omega _{k})^{m},|z-y_{j}|\geq 2^{k} \sqrt{n}r\). Consider now \(\alpha _{1}=\cdots =\alpha _{m}=\infty \), taking \(N\geq m\eta +1\), the following estimate holds:

$$\begin{aligned} &\frac{1}{\varphi (Q)^{\eta /\delta }} \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl(f_{1}^{ \infty }, \ldots, f_{m}^{\infty }\bigr) (z) \bigr\vert ^{\delta }\,dz \biggr)^{ \frac{1}{\delta }} \\ &\quad\leq \frac{C}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl(f_{1}^{\infty }, \ldots, f_{m}^{\infty }\bigr) (z) \bigr\vert \,dz \\ &\quad\leq \frac{C}{ \vert Q \vert } \int _{Q}\sum_{k=1}^{\infty } \int _{(\Omega _{k})^{m}} \bigl\vert K(z, \vec{y}) \bigr\vert \prod _{j=1}^{m} \bigl\vert f(y_{j}) \bigr\vert \,d\vec{y}\,dz \\ &\quad\leq \frac{C}{ \vert Q \vert } \int _{Q}\sum_{k=1}^{\infty } \int _{(\Omega _{k})^{m}} \frac{\prod_{j=1}^{m} \vert f(y_{j}) \vert }{ (\sum_{j=1}^{m} \vert z-y_{j} \vert )^{mn} (1+\sum_{j=1}^{m} \vert z-y_{j} \vert )^{N}}\,d \vec{y}\,dz \\ &\quad\leq \sum_{k=1}^{m} \frac{C}{ \vert 2^{k+3}\sqrt{n}Q \vert ^{m} (1+2^{k+3}\sqrt{n}r )^{N}} \int _{(2^{k+3}\sqrt{n}Q)^{m}}\prod_{j=1}^{m} \bigl\vert f(y_{j}) \bigr\vert \,dy_{j} \\ &\quad\leq \sum_{k=1}^{m} \frac{C}{ (1+2^{k+3}\sqrt{n}r )^{N}}\prod_{j=1}^{m} \frac{1}{ \vert 2^{k+3}\sqrt{n}Q \vert } \int _{2^{k+3}\sqrt{n}Q} \bigl\vert f(y_{j}) \bigr\vert \,dy_{j} \\ &\quad\leq \sum_{k=1}^{m} \frac{C}{ (1+2^{k+3}\sqrt{n}r )}\mathcal{M}_{\varphi, \eta }(\vec{f}) (x) \\ &\quad\leq C\mathcal{M}_{\varphi,\eta }(\vec{f}) (x). \end{aligned}$$

When \(\alpha _{j_{1}}=\cdots =\alpha _{j_{l}}=0\) for \(1\leq l< m\). Let \(\mathscr{J}:=\{j_{1},\ldots,j_{l}\}\) then \(\alpha _{j}=\infty \) for \(j\notin \mathscr{J}\), taking \(N\geq m\eta +1\), then

$$\begin{aligned} &\frac{1}{\varphi (Q)^{\eta /\delta }} \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl(f_{1}^{ \alpha _{1}}, \ldots, f_{m}^{\alpha _{m}}\bigr) (z) \bigr\vert ^{\delta }\,dz \biggr)^{ \frac{1}{\delta }} \\ &\quad\leq C \int _{(Q^{*})^{l}}\prod_{j\in \mathscr{J}} \bigl\vert f_{j}(y_{j}) \bigr\vert \int _{(R^{n}\backslash Q^{*})^{m-l}} \frac{\prod_{j\notin \mathscr{J}} \vert f_{j}(y_{j}) \vert }{ (\sum_{j=1}^{m} \vert z-y_{j} \vert )^{mn} (1+\sum_{j=1}^{m} \vert z-y_{j} \vert )^{N}}\,d \vec{y} \\ &\quad\leq C\prod_{j\in \mathscr{J}} \int _{Q^{*}} \bigl\vert f_{j}(y_{j}) \bigr\vert \sum_{k=1}^{\infty } \int _{(\Omega _{k})^{m-l}} \frac{\prod_{j\notin \mathscr{J}} \vert f_{j}(y_{j}) \vert }{ \vert 2^{k+3}\sqrt{n}Q \vert ^{m}(1+2^{k+3}\sqrt{n}r)^{N}}\,d \vec{y} \\ &\quad\leq C\sum_{k=1}^{\infty } \frac{1}{ \vert 2^{k+3}\sqrt{n}Q \vert ^{m}(1+2^{k+3}\sqrt{n}r)^{N}} \biggl(\prod_{j\in \mathscr{J}} \int _{Q^{*}} \bigl\vert f_{j}(y_{j}) \bigr\vert \,dy_{j} \biggr) \biggl( \prod _{j\notin \mathscr{J}} \int _{2^{k+3}\sqrt{n}Q} \bigl\vert f_{j}(y_{j}) \bigr\vert \,dy_{j} \biggr) \\ &\quad\leq C\mathcal{M}_{\varphi,\eta }(\vec{f}) (x). \end{aligned}$$

Pan and Tang in [16, Lemma 2.7] proved the result in our framework, which is similar to the classical Fefferman–Stein inequalities. Next, using Lemma 2.7 of our paper, we obtain the result as follows. □

Corollary 3.2

Let T be a multilinear operator satisfying (1.1)(1.5), and suppose that ω is satisfying \(\omega \in {\mathrm{Dini}}(1)\), \(w\in A_{\infty }^{\infty }\), \(\eta >0\) and \(p>0\). Then there exist constants \(C>0\), such that

$$ \bigl\Vert T(\vec{f}) \bigr\Vert _{L^{p}(w)}\leq C \bigl\Vert \mathcal{M}_{\varphi,\eta }(\vec{f}) \bigr\Vert _{L^{p}(w)} $$

and

$$ \bigl\Vert T(\vec{f}) \bigr\Vert _{L^{p,\infty }(w)}\leq C \bigl\Vert \mathcal{M}_{\varphi,\eta }( \vec{f}) \bigr\Vert _{L^{p,\infty }(w)}. $$

Proof

From Lemma 2.7 and Theorem 3.1, we get

$$\begin{aligned} \bigl\Vert T(\vec{f}) \bigr\Vert _{L^{p}(w)}&\leq \bigl\Vert M_{\varphi,\eta }^{d}\bigl(T(\vec{f})\bigr) \bigr\Vert _{L^{p}(w)} \\ &\leq C \bigl\Vert M_{\varphi,\eta }^{\sharp,d}\bigl(T(\vec{f})\bigr) \bigr\Vert _{L^{p}(w)} \\ &\leq C \bigl\Vert \mathcal{M}_{\varphi,\eta }(\vec{f}) \bigr\Vert _{L^{p}(w)}. \end{aligned}$$

 □

Similarly, with the help of Lemma 2.8, the weak-type estimate is obtained.

Theorem 3.3

Let T be a multilinear operator satisfying (1.1)(1.5), \(\vec{w}\in A_{\vec{p}}^{\infty }(\varphi )\) and \(1/p=1/p_{1}+\cdots +1/p_{m}\). If ω is satisfying \(\omega \in {\mathrm{Dini}}(1)\), then there exists a constant \(C>0\), such that:

  1. (i)

    If \(1< p_{j}<\infty \), \(j=1,\ldots,m\), then

    $$ \bigl\Vert T(\vec{f}) \bigr\Vert _{L^{p}(v_{\vec{w}})}\leq C\prod_{j=1}^{m} \Vert f_{j} \Vert _{L^{p_{j}}(w_{j})}. $$
  2. (ii)

    If \(1\leq p_{j}<\infty \), \(j=1,\ldots,m\), and at least one of \(p_{j}=1\), then

    $$ \bigl\Vert T(\vec{f}) \bigr\Vert _{L^{p,\infty }(v_{\vec{w}})}\leq C\prod_{j=1}^{m} \Vert f_{j} \Vert _{L^{p_{j}}(w_{j})}. $$

Proof

The desired result directly is obtained from Theorem 3.1, Corollary 3.2, Lemma 2.4 and Lemma 2.9. The proof is completed. □

Estimates for multilinear commutators

To ensure the fluency of the demonstration in this section, we need first to explain the meaning of some notations. We write

$$ C_{j}^{m}=\bigl\{ \sigma:\sigma =\bigl\{ \sigma (1),\sigma (2),\ldots,\sigma (j) \bigr\} , 1\leq j\leq m\bigr\} , $$

We always take \(\sigma (i)\leq \sigma (j)\) if \(i\leq j\).

For any \(\sigma '\in C_{j}^{m}\), we have \(\sigma '=\{\sigma (1),\sigma (2),\ldots,\sigma (m)\}\setminus \sigma \) and \(\sigma '\in C_{m-j}^{m}\).

Let b⃗ be m-tuple functions and \(\sigma \in C_{j}^{m}\), we have the j-tuple function \(\vec{b}=(b_{\sigma (1)},b_{\sigma (2)}, \ldots, b_{\sigma (j)})\). For all \(b_{\sigma (j)}\in {\mathrm{BMO}}_{\theta }(\varphi )\), \(1\leq j\leq m\), we have \(\vec{b}=(b_{\sigma (1)},b_{\sigma (2)},\ldots,b_{\sigma (m)})\in { \mathrm{BMO}}_{\vec{\theta }}^{m}(\varphi )\). See [15, 16].

Corresponding to the classical form, can define the following form of the iterated commutators:

$$ T_{\Pi _{\vec{b_{\sigma }}}}(\vec{f}) (x)= \int _{(\mathbb{R}^{n})^{m}} \prod_{i=1}^{m} \bigl(b_{\sigma (i)}(x)-b_{\sigma (i)}(y_{\sigma (i)}) \bigr)K(x,y_{1}, \ldots,y_{m})f_{1}(y_{1}) \cdots f_{m}(y_{m})\,dy_{1}\cdots dy_{m}. $$

Theorem 4.1

Let T be a multilinear Calderón–Zygmund operator of type \(\omega (t)\) as in Definition 1.1, \(T_{\Pi _{\vec{b}}}\) be a multilinear commutator with \(\vec{b}\in {\mathrm{BMO}}^{m}_{\vec{\theta }}(\varphi )\). We have \(0<\delta <\epsilon <1/m\) and \(\eta >(\theta _{1},\ldots,\theta _{m})/ (1/\delta -1/\epsilon )\), assume that ω is satisfying

$$\begin{aligned} \int _{0}^{1}\frac{\omega (t)}{t} \biggl(1+\log \frac{1}{t} \biggr)^{m}\,dt< \infty. \end{aligned}$$
(4.1)

Then there exists a constant \(C>0\) such that

$$\begin{aligned} M^{\#,d}_{\delta,\varphi,\eta } \bigl(T_{\Pi _{\vec{b}}}(\vec{f}) \bigr) (x) \leq{}& C \prod_{j=1}^{m} \Vert b_{j} \Vert _{{\mathrm{BMO}}_{\theta _{j}}(\varphi )} \bigl(\mathcal{M}_{L(\log L),\varphi,\eta }( \vec{f}) (x)+M_{\epsilon, \varphi,\eta }^{d} \bigl(T(\vec{f}) \bigr) (x) \bigr) \\ &{} +C\sum_{j=1}^{m-1}\sum_{\sigma \in C_{j}^{m}} \prod_{i=1}^{j} \Vert b_{\sigma (i)} \Vert _{{\mathrm{BMO}}_{\sigma (i)}}M_{ \epsilon,\varphi,\eta } \bigl(T_{\Pi _{\vec{b}_{\sigma '}}}(\vec{f}) \bigr) (x), \end{aligned}$$

for all m-tuples \(\vec{f}=(f_{1},\ldots,f_{m})\) of bounded measurable functions with compact support.

Proof

For simplicity, we only prove the case \(m=2\) and \(\theta _{1}=\theta _{2}=\theta \).

If ω is satisfying (4.1), then \(\omega \in \mathrm{Dini(1)}\) and

$$ \sum_{k=1}^{\infty }k^{m}\cdot \omega \bigl(2^{-k}\bigr)\thickapprox \int _{0}^{1} \frac{\omega (t)}{t} \biggl(1+\log \frac{1}{t} \biggr)^{m}< \infty. $$

For \(b_{1},b_{2}\in \mathrm{BMO}_{\theta }(\varphi )\), it suffices to prove that

$$\begin{aligned} M^{\#,d}_{\delta,\varphi,\eta } \bigl(T_{\Pi _{\vec{b}}}(f_{1},f_{2}) \bigr) (x)\leq{}& C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \Vert b_{2} \Vert _{\mathrm{BMO}_{ \theta }(\varphi )}\bigl(\mathcal{M}_{L(\log L),\varphi,\eta }(f_{1},f_{2}) (x) \\ & {}+CM_{\epsilon,\varphi,\eta }^{d}\bigl(T(f_{1},f_{2}) \bigr) (x)\bigr) \\ &{} +C\bigl( \Vert b_{2} \Vert _{\mathrm{BMO}_{\theta }(\varphi )}M_{\epsilon,\varphi, \eta } \bigl(T_{b_{1}}^{1}\bigr) (f_{1},f_{2}) (x) \\ & {}+C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )}M_{\epsilon,\varphi, \eta } \bigl(T_{b_{2}}^{2}\bigr) (f_{1},f_{2}) (x)\bigr). \end{aligned}$$

For any constants \(\lambda _{1},\lambda _{2}\), it follows that

$$\begin{aligned} T_{\Pi _{\vec{b}}}(f_{1},f_{2}) (x)={}&\bigl(b_{1}(x)- \lambda _{1}\bigr) \bigl(b_{2}(x)- \lambda _{2} \bigr)T(f_{1},f_{2}) (x)-\bigl(b_{1}(x)-\lambda _{1}\bigr)T\bigl(f_{1},(b_{2}- \lambda _{2})f_{2}\bigr) (x) \\ &{} -\bigl(b_{2}(x)-\lambda _{2}\bigr)T\bigl((b_{1}- \lambda _{1})f_{1},f_{2}\bigr) (x)+T \bigl((b_{1}- \lambda _{1})f_{1},(b_{2}- \lambda _{2})f_{2}\bigr) (x) \\ ={}&{-}\bigl(b_{1}(x)-\lambda _{1}\bigr) \bigl(b_{2}(x)- \lambda _{2}\bigr)T(f_{1},f_{2}) (x)+ \bigl(b_{1}(x)- \lambda _{1}\bigr)T^{2}_{b_{2}-\lambda _{2}}(f_{1},f_{2}) (x) \\ &{} +\bigl(b_{2}(x)-\lambda _{2}\bigr)T^{1}_{b_{1}-\lambda _{1}}(f_{1},f_{2}) (x)+T\bigl((b_{1}- \lambda _{1})f_{1},(b_{2}- \lambda _{2})f_{2}\bigr) (x), \end{aligned}$$

where

$$\begin{aligned} T^{1}_{b_{1}-\lambda _{1}}(f_{1},f_{2}) (x)= \bigl(b_{1}(x)-\lambda _{1}\bigr)T(f_{1},f_{2}) (x)-T\bigl((b_{1}- \lambda _{1})f_{1},f_{2} \bigr) (x) \end{aligned}$$
(4.2)

and

$$\begin{aligned} T^{2}_{b_{2}-\lambda _{2}}(f_{1},f_{2}) (x)= \bigl(b_{2}(x)-\lambda _{2}\bigr)T(f_{1},f_{2}) (x)-T\bigl(f_{1},(b_{2}- \lambda _{2})f_{2} \bigr) (x). \end{aligned}$$
(4.3)

Now, we fix \(x\in \mathbb{R}^{n}\), a dyadic cube \(Q\ni x\) and a constant c, then, since \(0<\delta <\frac{1}{2}\), we only need to consider the two cases \(r\leq 1\) and \(r>1\).

Case 1: When \(r\leq 1\), the following estimate holds:

$$\begin{aligned} &\biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert \bigl\vert T_{\Pi _{\vec{b}}}(f_{1},f_{2}) (z) \bigr\vert ^{ \delta }- \vert c \vert ^{\delta } \bigr\vert \,dz \biggr)^{\frac{1}{\delta }} \\ &\quad\leq \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T_{\Pi _{\vec{b}}}(f_{1},f_{2}) (z)-c \bigr\vert ^{ \delta }\,dz \biggr)^{\frac{1}{\delta }} \\ &\quad\leq \biggl( \frac{C}{ \vert Q \vert } \int _{Q} \bigl\vert \bigl(b_{1}(z)-\lambda _{1}\bigr) \bigl(b_{2}(z)- \lambda _{2} \bigr)T(f_{1},f_{2}) (z) \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }} \\ &\qquad{} + \biggl( \frac{C}{ \vert Q \vert } \int _{Q} \bigl\vert \bigl(b_{1}(z)-\lambda _{1}\bigr)T^{2}_{b_{2}- \lambda _{2}}(f_{1},f_{2}) (z) \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }} \\ & \qquad{}+ \biggl( \frac{C}{ \vert Q \vert } \int _{Q} \bigl\vert \bigl(b_{2}(z)-\lambda _{2}\bigr)T^{1}_{b_{1}- \lambda _{1}}(f_{1},f_{2}) (z) \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }} \\ &\qquad{} + \biggl( \frac{C}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b_{1}-\lambda _{1})f_{1},(b_{2}- \lambda _{2})f_{2} \bigr) (z)-c \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }} \\ &\quad:=I+\mathit{II}+\mathit{III}+\mathit{IV}. \end{aligned}$$

Let \(Q^{*}=8\sqrt{n}Q\) and let \(\lambda _{j}=(b_{j})_{Q^{*}}\) be the average of \(b_{j}\) on \(Q^{*},j=1,2\). For any \(1< r_{1},r_{2},r_{3}<\infty \) with \(1/r_{1}+1/r_{2}+1/r_{3}=1\), choosing a δ to make \(\delta r_{i}<1,i=1,2\) and \(r_{3}<\epsilon /\delta \).

By Hölder’s inequality, we have

$$\begin{aligned} I\leq{}& C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert b_{1}(z)-(b_{1})_{Q^{*}} \bigr\vert ^{r_{1} \delta }\,dz \biggr)^{\frac{1}{r_{1}\delta }} \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert b_{2}(z)-(b_{2})_{Q^{*}} \bigr\vert ^{r_{2} \delta }\,dz \biggr)^{\frac{1}{r_{2}\delta }} \\ &{} \times \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T(f_{1},f_{2}) \bigr\vert ^{r_{3} \delta }\,dz \biggr)^{\frac{1}{r_{3}\delta }} \\ \leq{}& C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \Vert b_{2} \Vert _{\mathrm{BMO}_{ \theta }(\varphi )}M_{\epsilon,\varphi,\eta }^{d} \bigl(T(f_{1},f_{2})\bigr) (x)). \end{aligned}$$

For II, let \(1< t_{1},t_{2}<\infty \) with \(1=1/t_{1}+1/t_{2}\) and \(t_{2}<\epsilon /\delta \). By Hölder’s inequality,

$$\begin{aligned} \mathit{II}&\leq C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert b_{1}(z)-(b_{1})_{Q^{*}} \bigr\vert ^{t_{1} \delta }\,dz \biggr)^{\frac{1}{t_{1}\delta }} \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T^{2}_{b_{2}- \lambda _{2}}(f_{1},f_{2}) (z) \bigr\vert ^{t_{2}\delta }\,dz \biggr)^{ \frac{1}{t_{2}\delta }} \\ &\leq C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )}M_{t_{2}\delta,\varphi, \eta } \bigl(T^{2}_{b_{2}-\lambda _{2}}(f_{1},f_{2})\bigr) (x) \\ &\leq C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )}M_{\epsilon,\varphi, \eta } \bigl(T^{2}_{b_{2}-\lambda _{2}}(f_{1},f_{2})\bigr) (x) \\ &=C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )}M_{\epsilon,\varphi,\eta } \bigl(T^{2}_{b_{2}}(f_{1},f_{2})\bigr) (x). \end{aligned}$$

Similarly, we obtain

$$ \mathit{III}\leq C \Vert b_{2} \Vert _{\mathrm{BMO}_{\theta }(\varphi )}M_{\epsilon, \varphi,\eta } \bigl(T^{1}_{b_{1}}(f_{1},f_{2})\bigr) (x). $$

Now for the last term IV. We split each \(f_{j}\) as \(f_{j}=f_{j}^{0}+f_{j}^{\infty }\) where \(f_{i}^{0}=f_{j}\chi _{Q^{*}}\) and \(f^{\infty }_{j}=f_{j}-f_{j}^{0}\).

Let \(c=c_{1}+c_{2}+c_{3}\), where

$$\begin{aligned} &c_{1}=T\bigl((b_{1}-\lambda _{1})f_{1}^{0},(b_{2}- \lambda _{2})f_{2}^{ \infty }\bigr) (x), \\ &c_{2}=T\bigl((b_{1}-\lambda _{1})f_{1}^{\infty },(b_{2}- \lambda _{2})f_{2}^{0}\bigr) (x), \\ &c_{3}=T\bigl((b_{1}-\lambda _{1})f_{1}^{\infty },(b_{2}- \lambda _{2})f_{2}^{ \infty }\bigr) (x). \end{aligned}$$

Then

$$\begin{aligned} \mathit{IV}\leq{}& C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b_{1}-\lambda _{1})f_{1}^{0},(b_{2}- \lambda _{2})f_{2}^{0}\bigr) (z) \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }} \\ &{}+C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b_{1}-\lambda _{1})f_{1}^{0},(b_{2}- \lambda _{2})f_{2}^{\infty }\bigr) (z)-c_{1} \bigr\vert ^{\delta }\,dz \biggr)^{ \frac{1}{\delta }} \\ &{}+C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b_{1}-\lambda _{1})f_{1}^{\infty },(b_{2}- \lambda _{2})f_{2}^{0}\bigr) (z)-c_{2} \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }} \\ &{}+C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b_{1}-\lambda _{1})f_{1}^{\infty },(b_{2}- \lambda _{2})f_{2}^{\infty }\bigr) (z)-c_{3} \bigr\vert ^{\delta }\,dz \biggr)^{ \frac{1}{\delta }} \\ :={}&\mathit{IV}_{1}+\mathit{IV}_{2}+\mathit{IV}_{3}+\mathit{IV}_{4}. \end{aligned}$$

For \(\mathit{IV}_{1}\), choosing \(1< p<\frac{1}{2\delta }\) and applying Kolmogorov’s inequality with \(p=\delta <\frac{1}{2}\), \(q=\frac{1}{2}\),

$$\begin{aligned} \mathit{IV}_{1}&=C \bigl\Vert T\bigl((b_{1}-\lambda _{1})f_{1}^{0},(b_{2}-\lambda )f_{2}^{0}\bigr) \bigr\Vert _{L^{\delta }(Q,\frac{dx}{ \vert Q \vert })} \\ &\leq C \bigl\Vert T\bigl((b_{1}-\lambda _{1})f_{1}^{0},(b_{2}- \lambda )f_{2}^{0}\bigr) \bigr\Vert _{L^{ \frac{1}{2},\infty }(Q,\frac{dx}{ \vert Q \vert })} \\ &\leq C\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert \bigl(b_{1}(z)-\lambda _{1}\bigr)f_{1}^{0}(z) \bigr\vert \,dz \frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert \bigl(b_{2}(z)-\lambda _{2}\bigr)f_{2}^{0}(z) \bigr\vert \,dz \\ &\leq C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \Vert b_{2} \Vert _{\mathrm{BMO}_{ \theta }(\varphi )}\mathcal{M}_{L(\log L),\varphi,\eta }(f_{1},f_{2}) (x). \end{aligned}$$

Next to estimate \(\mathit{IV}_{2}\). For any \(z\in Q\), let \(c_{1}=T((b_{1}-\lambda _{1})f_{1}^{0},(b_{2}-\lambda _{2})f_{2}^{ \infty })(x)\), we have

$$\begin{aligned} \mathit{IV}_{2}= {}&\biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b_{1}-\lambda _{1})f_{1}^{0},(b_{2}- \lambda _{2})f_{2}^{\infty }\bigr) (z)-T\bigl((b_{1}- \lambda _{1})f_{1}^{0},(b_{2}- \lambda _{2})f_{2}^{\infty }\bigr) (x) \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }} \\ \leq{}& \int _{(\mathbb{R}^{n})^{2}} \bigl\vert K(z,y_{1},y_{2})-K(x,y_{1},y_{2}) \bigr\vert \bigl\vert \bigl(b_{1}(y_{1})- \lambda _{1}\bigr)f_{1}^{0}(y_{1}) \bigr\vert \bigl\vert \bigl(b_{2}(y_{2})-\lambda _{2} \bigr)f_{2}^{ \infty }(y_{2}) \bigr\vert \,dy_{1}\,dy_{2} \\ \leq{}& \int _{Q^{*}} \bigl\vert \bigl(b_{1}(y_{1})- \lambda _{1}\bigr)f_{1}(y_{1}) \bigr\vert \\ &{}\times \biggl( \int _{\mathbb{R}^{n}\backslash Q^{*}} \bigl\vert K(z,y_{1},y_{2})-K(x,y_{1},y_{2}) \bigr\vert \bigl\vert \bigl(b_{2}(y_{2})- \lambda _{2}\bigr)f_{2}(y_{2}) \bigr\vert \,dy_{2} \biggr)\,dy_{1}. \end{aligned}$$

For any \(z\in Q,y_{1}\in Q^{*}\) and \(y_{2}\in \Omega _{k}\),

$$\begin{aligned} &\bigl\vert K(z,y_{1},y_{2})-K(x,y_{1},y_{2}) \bigr\vert \\ &\quad\leq \frac{A}{( \vert z-y_{1} \vert + \vert z-y_{2} \vert )^{2n}(1+ \vert z-y_{1} \vert + \vert z-y_{2} \vert )^{N}} \omega \biggl(\frac{ \vert z-x \vert }{ \vert z-y_{1} \vert + \vert z-y_{2} \vert } \biggr) \\ &\quad\leq C \frac{\omega (2^{-k})}{ \vert 2^{k+3}\sqrt{n}Q \vert ^{2}(1+2^{k+3}\sqrt{n}r)^{N}}, \end{aligned}$$

then we have

$$\begin{aligned} \mathit{IV}_{2}\leq{}& C \int _{Q^{*}} \bigl\vert \bigl(b_{1}(y_{1})- \lambda _{1}\bigr)f_{1}(y_{1}) \bigr\vert \\ &{}\times\Biggl( \sum_{k=1}^{\infty } \int _{\Omega _{k}} \frac{\omega (2^{-k})}{ \vert 2^{k+3}\sqrt{n}Q \vert ^{2}(1+2^{k+3}\sqrt{n}r)^{N}} \bigl\vert \bigl(b_{2}(y_{2})- \lambda _{2} \bigr)f_{2}(y_{2}) \bigr\vert \,dy_{2} \Biggr) \,dy_{1} \\ \leq{}& C \int _{Q^{*}} \bigl\vert \bigl(b_{1}(y_{1})- \lambda _{1}\bigr)f_{1}(y_{1}) \bigr\vert \\ &{}\times\Biggl( \sum_{k=1}^{\infty }\omega {\bigl(2^{-k} \bigr)} \int _{2^{k+3}\sqrt{n}Q} \frac{ \vert (b_{2}(y_{2})-\lambda _{2})f_{2}(y_{2}) \vert }{ \vert 2^{k+3}\sqrt{n}Q \vert ^{2}(1+2^{k+3}\sqrt{n}r)^{N}}\,dy_{2} \Biggr) \,dy_{1} \\ \leq{}& C\sum_{k=1}^{\infty } \frac{\omega {(2^{-k})}}{ \vert 2^{k+3}\sqrt{n}Q \vert ^{2}(1+2^{k+3}\sqrt{n}r)^{N}}\\ &{}\times \int _{(2^{k+3}\sqrt{n}Q)^{2}} \bigl\vert b_{1}(y_{1})- \lambda _{1} \bigr\vert \bigl\vert f_{1}(y_{1}) \bigr\vert \bigl\vert b_{2}(y_{2})- \lambda _{2} \bigr\vert \bigl\vert f_{2}(y_{2}) \bigr\vert \,dy_{1}\,dy_{2}. \end{aligned}$$

Note that, for the constant \(\lambda _{j}=(b_{j})_{Q^{*}}\), the following holds:

$$ \int _{2^{k+3}\sqrt{n}Q} \bigl\vert b(y)-b_{Q^{*}} \bigr\vert \bigl\vert f(y) \bigr\vert \,dy\leq Ck \bigl\vert 2^{k+3} \sqrt{n}Q \bigr\vert \varphi \bigl(2^{k+3}\sqrt{n}Q\bigr) \Vert b \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \Vert f \Vert _{L\log L,2^{k+3}\sqrt{n}Q}. $$

Taking \(N\geq 2\eta \), then

$$\begin{aligned} \mathit{IV}_{2}&\leq C\sum_{k=1}^{\infty }k^{2} \omega \bigl(2^{-k}\bigr) \Vert b_{1} \Vert _{\mathrm{BMO}_{ \theta }(\varphi )} \Vert b_{2} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \Vert f_{1} \Vert _{L \log L,2^{k+3}\sqrt{n}Q} \Vert f_{2} \Vert _{L\log L,2^{k+3}\sqrt{n}Q} \\ &\leq C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \Vert b_{2} \Vert _{\mathrm{BMO}_{ \theta }(\varphi )}\mathcal{M}_{L(\log L),\varphi,\eta }(f_{1},f_{2}) (x). \end{aligned}$$

Similarly to \(\mathit{IV}_{2}\), we can estimate

$$ \mathit{IV}_{3}\leq C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \Vert b_{2} \Vert _{ \mathrm{BMO}_{\theta }(\varphi )}\mathcal{M}_{L(\log L),\varphi,\eta }(f_{1},f_{2}) (x). $$

Now for the term \(\mathit{IV}_{4}\). For any \(z\in Q\) and \(y_{1},y_{2}\in \Omega _{k}\),

$$\begin{aligned} &\bigl\vert K(z,y_{1},y_{2})-K(x,y_{1},y_{2}) \bigr\vert \\ &\quad\leq \frac{A}{( \vert z-y_{1} \vert + \vert z-y_{2} \vert )^{2n}(1+ \vert z-y_{1} \vert + \vert z-y_{2} \vert )^{N}} \omega \biggl(\frac{ \vert z-x \vert }{ \vert z-y_{1} \vert + \vert z-y_{2} \vert } \biggr) \\ &\quad\leq C \frac{\omega (2^{-k})}{ \vert 2^{k+3}\sqrt{n}Q \vert ^{2}(1+2^{k+3}\sqrt{n}r)^{N}}. \end{aligned}$$

Note \(c_{3}=T((b_{1}-\lambda _{1})f_{1}^{\infty },(b_{2}-\lambda _{2})f_{2}^{ \infty })(x)\). Then

$$\begin{aligned} \mathit{IV}_{4}\leq{}& C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b_{1}-\lambda _{1})f_{1}^{ \infty },(b_{2}-\lambda _{2})f_{2}^{\infty }\bigr) (z)\\ &{}-T\bigl((b_{1}- \lambda _{1})f_{1}^{ \infty },(b_{2}-\lambda _{2})f_{2}^{\infty }\bigr) (x) \bigr\vert ^{\delta }\,dz \biggr)^{ \frac{1}{\delta }} \\ \leq{}& C \int _{(\mathbb{R}^{n}\backslash Q^{*})^{2}} \bigl\vert K(z,y_{1},y_{2})-K(x,y_{1},y_{2}) \bigr\vert \Biggl(\prod_{j=1}^{2} \bigl\vert \bigl(b_{j}(y_{j})-\lambda _{j} \bigr)f_{j}(y_{j}) \bigr\vert \Biggr)\,dy_{1} \,dy_{2} \\ \leq{}& C\sum_{k=1}^{\infty } \int _{(\Omega _{k})^{2}} \bigl\vert K(z,y_{1},y_{2})-K(x,y_{1},y_{2}) \bigr\vert \Biggl(\prod_{j=1}^{2} \bigl\vert \bigl(b_{j}(y_{j})-\lambda _{j} \bigr)f_{j}(y_{j}) \bigr\vert \Biggr)\,dy_{1} \,dy_{2} \\ \leq{}& C\sum_{k=1}^{\infty } \int _{(2^{k+3}\sqrt{n}Q)^{2}} \frac{\omega (2^{-k})}{ \vert 2^{k+3}\sqrt{n}Q \vert ^{2}(1+2^{k+3}\sqrt{n}r)^{N}} \Biggl(\prod _{j=1}^{2} \bigl\vert b_{j}(y_{j})- \lambda _{j} \bigr\vert \bigl\vert f_{j}(y_{j}) \bigr\vert \Biggr)\,dy_{1}\,dy_{2} \\ \leq{}& C\sum_{k=1}^{\infty }k^{2} \omega \bigl(2^{-k}\bigr) \Vert b_{1} \Vert _{\mathrm{BMO}_{ \theta }(\varphi )} \Vert b_{2} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \Vert f_{1} \Vert _{L \log L,2^{k+3}\sqrt{n}Q} \Vert f_{2} \Vert _{L\log L,2^{k+3}\sqrt{n}Q} \\ \leq{}& C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \Vert b_{2} \Vert _{\mathrm{BMO}_{ \theta }(\varphi )}\mathcal{M}_{L(\log L),\varphi,\eta }(f_{1},f_{2}) (x). \end{aligned}$$

Case 2: When \(r>1\). Let \(0<\delta <\epsilon <1\), the following holds:

$$\begin{aligned} &\biggl(\frac{1}{\varphi (Q)^{\eta } \vert Q \vert } \int _{Q} \bigl\vert T_{\Pi _{\vec{b}}}( \vec{f}) (z) \bigr\vert ^{\delta }\,dy \biggr)^{\frac{1}{\delta }} \\ &\quad\leq C\frac{1}{\varphi (Q)^{\eta /\delta }} \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert (b_{1}- \lambda _{1}) (b_{2}-\lambda _{2})T(f_{1},f_{2}) \bigr\vert ^{\delta }\,dz \biggr)^{ \frac{1}{\delta }} \\ &\qquad{} + C\frac{1}{\varphi (Q)^{\eta /\delta }} \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert (b_{1}-\lambda _{1})T\bigl(f_{1},(b_{2}-\lambda _{2})f_{2}\bigr) \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }} \\ &\qquad{} + C\frac{1}{\varphi (Q)^{\eta /\delta }} \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert (b_{2}-\lambda _{2})T\bigl((b_{1}-\lambda _{1})f_{1},f_{2} \bigr) \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }} \\ &\qquad{} + C\frac{1}{\varphi (Q)^{\eta /\delta }} \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b_{1}-\lambda _{1})f_{1},(b_{2}-\lambda _{2})f_{2} \bigr) \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }} \\ &\quad:=I+\mathit{II}+\mathit{III}+\mathit{IV}. \end{aligned}$$

Let \(Q^{*}=8\sqrt{n}Q\) and let \(\lambda _{j}=(b_{j})_{Q^{*}}\) be the average of \(b_{j}\) on \(Q^{*},j=1,2\). For any \(1< r_{1},r_{2},r_{3}<\infty \) with \(1/r_{1}+1/r_{2}+1/r_{3}=1\), we choose a δ small enough to make \(\delta r_{i}<1,i=1,2\) and \(r_{3}<\epsilon /\delta \).

Using Hölder’s inequality, choosing η so that \(\eta (\frac{1}{\delta }-\frac{1}{\epsilon })>2\theta \), then

$$\begin{aligned} I\leq{}& C\frac{1}{\varphi (Q)^{\frac{\eta }{\delta }}} \biggl( \frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert b_{1}(z)-\lambda _{1} \bigr\vert ^{r_{1}\delta }\,dz \biggr)^{ \frac{1}{r_{1}\delta }} \\ &{} \times \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert b_{2}(z)-\lambda _{2} \bigr\vert ^{r_{2} \delta }\,dz \biggr)^{\frac{1}{r_{2}\delta }} \biggl( \frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T(f_{1},f_{2}) (z) \bigr\vert ^{r_{3} \delta }\,dz \biggr)^{\frac{1}{r_{3}\delta }} \\ \leq{}& C \frac{\varphi (Q)^{\eta /\epsilon }}{\varphi (Q)^{\eta /\delta }} \varphi _{\theta }^{2} \bigl(Q^{*}\bigr) \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \Vert b_{2} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \biggl(\frac{1}{\varphi (Q)^{\eta } \vert Q \vert } \int _{Q} \bigl\vert T(f_{1},f_{2}) (z) \bigr\vert ^{\epsilon }\,dz \biggr)^{\frac{1}{\epsilon }} \\ \leq{}& C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \Vert b_{2} \Vert _{\mathrm{BMO}_{\theta }(\varphi )}M_{\epsilon,\varphi,\eta }^{d} \bigl(T(f_{1},f_{2})\bigr) (x). \end{aligned}$$

By the Hölder inequality, and Lemma 2.5, we have

$$\begin{aligned} \mathit{II} \leq{}& C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \frac{1}{\varphi (Q)^{\eta /\delta -\theta }} \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl(f_{1}^{0},(b_{2}- \lambda _{2})f_{2}^{0}\bigr) (z) \bigr\vert ^{p\delta } \,dz \biggr)^{\frac{1}{p\delta }} \\ &{} + C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \frac{1}{\varphi (Q)^{\eta /\delta -\theta }} \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl(f_{1}^{0},(b_{2}- \lambda _{2})f_{2}^{\infty }\bigr) (z) \bigr\vert ^{p\delta } \,dz \biggr)^{ \frac{1}{p\delta }} \\ &{} + C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \frac{1}{\varphi (Q)^{\eta /\delta -\theta }} \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl(f_{1}^{\infty },(b_{2}- \lambda _{2})f_{2}^{0}\bigr) (z) \bigr\vert ^{p\delta } \,dz \biggr)^{ \frac{1}{p\delta }} \\ &{} + C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \frac{1}{\varphi (Q)^{\eta /\delta -\theta }} \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl(f_{1}^{\infty },(b_{2}- \lambda _{2})f_{2}^{\infty }\bigr) (z) \bigr\vert ^{p\delta } \,dz \biggr)^{ \frac{1}{p\delta }} \\ :={}&\mathit{II}_{1}+\mathit{II}_{2}+\mathit{II}_{3}+\mathit{II}_{4}. \end{aligned}$$

Now to estimate \(\mathit{II}_{1}\), using the Kolmogorov inequality and the boundedness of operators, we have

$$\begin{aligned} \mathit{II}_{1}&\leq C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \frac{1}{\varphi (Q)^{\eta /\delta -\theta }} \bigl\Vert T\bigl(f_{1}^{0},(b_{2}- \lambda _{2})f_{2}^{0}\bigr) \bigr\Vert _{L^{\frac{1}{2},\infty }(Q,\frac{dx}{ \vert Q \vert })} \\ &\leq C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \frac{1}{\varphi (Q)^{\eta /\delta -\theta }} \frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert f_{1}^{0}(z) \bigr\vert \,dz \int _{Q} \bigl\vert (b_{2}-\lambda _{2})f_{2}^{0}(z) \bigr\vert \,dz \\ &\leq C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \frac{1}{\varphi (Q)^{\eta /\delta -\theta }} \Vert b_{2} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \Vert f_{2} \Vert _{L(\mathrm{Log}L),Q}\varphi _{\theta }(Q) \vert f_{1} \vert _{Q} \\ &\leq C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \frac{1}{\varphi (Q)^{\eta (1/\delta -2)-2\theta }} \Vert b_{2} \Vert _{\mathrm{BMO}_{\theta }(\varphi )}\mathcal{M}_{L(\mathrm{Log}L),\varphi,\eta }(f_{1},f_{2}) (x) \\ &\leq C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \Vert b_{2} \Vert _{\mathrm{BMO}_{\theta }(\varphi )}\mathcal{M}_{L(\log L),\varphi,\eta }(f_{1},f_{2}) (x). \end{aligned}$$

The way of estimate \(\mathit{II}_{2}\) is the same as \(\mathit{II}_{3}\), we only prove \(\mathit{II}_{2}\):

$$\begin{aligned} \mathit{II}_{2}\leq {}&C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \frac{1}{\varphi (Q)^{\eta /\delta -\theta }}\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl(f_{1}^{0},(b_{2}- \lambda _{2})f_{2}^{\infty }\bigr) (z) \bigr\vert \,dz \\ \leq{}& C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \frac{1}{\varphi (Q)^{\eta /\delta -\theta }} \frac{1}{ \vert Q \vert } \int _{Q} \biggl( \int _{Q^{*}} \bigl\vert f_{1}(y_{1}) \bigr\vert \,dy_{1} \biggr) \\ &{} \times \biggl( \int _{\mathbb{R}^{n}\backslash {Q^{*}}} \frac{ \vert (b_{2}(y_{2})-\lambda _{2})f_{2}(y_{2})\,dy_{2} \vert }{( \vert z-y_{1} \vert + \vert z-y_{2} \vert )^{2n}(1+ \vert z-y_{1} \vert + \vert z-y_{2} \vert )^{N}} \biggr)\,dz \\ \leq{}& C \frac{1}{\varphi (Q)^{\eta /\delta -\theta }} \sum_{k=1}^{\infty } \frac{k(1+2^{k+3}\sqrt{n}r)^{\theta +2\eta }}{(1+2^{k+3}\sqrt{n} \vert Q \vert ^{\frac{1}{n}})^{N}} \Vert b_{\mathrm{BMO}_{\theta }(\varphi )} \Vert b_{2} \|_{\mathrm{BMO}_{\theta }(\varphi )} \mathcal{M}_{L(\log L),\varphi,\eta }(f_{1},f_{2}) (x). \end{aligned}$$

Choosing η such that \(\eta /\delta -1>0\), \(N\geq \theta +2\eta +1\), we get

$$ \mathit{II}_{2}\leq C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \Vert b_{2} \Vert _{\mathrm{BMO}_{\theta }}\mathcal{M}_{L(\log L),\varphi,\eta }(f_{1},f_{2}) (x). $$

Let \(N\geq \theta +2\eta +1\) and \(\eta /\delta -1>0\), then

$$\begin{aligned} \mathit{II}_{4}\leq{}& C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \frac{1}{\varphi (Q)^{\eta /\delta -\theta }}\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl(f_{1}^{\infty },(b_{2}- \lambda _{2})f_{2}^{\infty }\bigr) (z) \bigr\vert \,dz \\ \leq{}& C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \frac{1}{\varphi (Q)^{\eta /\delta -\theta }} \frac{1}{ \vert Q \vert } \\ &{} \times \biggl(\sum_{k=1}^{\infty } \int _{(\Omega _{k})^{2}} \frac{ \vert f_{1}(y_{1}) \vert \vert (b_{2}(y_{2})-\lambda _{2})f_{2}(y_{2}) \vert \,dy_{1}\,dy_{2}}{( \vert z-y_{1} \vert + \vert z-y_{2} \vert )^{2n}(1+ \vert z-y_{1} \vert + \vert z-y_{2} \vert )^{N}} \biggr)\,dz \\ \leq{}& C \frac{1}{\varphi (Q)^{\eta /\delta -\theta }} \sum_{k=1}^{\infty } \frac{k(1+2^{k+3}\sqrt{n}r)^{\theta +2\eta }}{(1+2^{k+3}\sqrt{n} \vert Q \vert ^{\frac{1}{n}})^{N}} \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \Vert b_{2} \Vert _{\mathrm{BMO}_{\theta }( \varphi )}\mathcal{M}_{L(\log L),\varphi,\eta }(f_{1},f_{2}) (x) \\ \leq{}& C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \Vert b_{2} \Vert _{\mathrm{BMO}_{\theta }(\varphi )}\mathcal{M}_{L(\log L),\varphi,\eta }(f_{1},f_{2}) (x). \end{aligned}$$

Now estimate IV. We first split any function

$$\begin{aligned} \mathit{IV}\leq{}& C\frac{1}{\varphi (Q)^{\eta /\delta }} \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b_{1}- \lambda _{1})f_{1}^{0},(b_{2}-\lambda _{2})f_{2}^{0}\bigr) (z) \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }} \\ &{} + C\frac{1}{\varphi (Q)^{\eta /\delta }} \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b_{1}-\lambda _{1})f_{1}^{0},(b_{2}-\lambda _{2})f_{2}^{\infty }\bigr) (z) \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }} \\ &{} + C\frac{1}{\varphi (Q)^{\eta /\delta }} \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b_{1}-\lambda _{1})f_{1}^{\infty },(b_{2}-\lambda _{2})f_{2}^{0}\bigr) (z) \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }} \\ &{} + C\frac{1}{\varphi (Q)^{\eta /\delta }} \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b_{1}-\lambda _{1})f_{1}^{\infty },(b_{2}-\lambda _{2})f_{2}^{\infty }\bigr) (z) \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }} \\ :={}&\mathit{IV}_{1}+\mathit{IV}_{2}+\mathit{IV}_{3}+\mathit{IV}_{4}. \end{aligned}$$

Similar to the estimate of \(\mathit{II}_{1}\), taking \(\eta (\frac{1}{\delta }-2)>2\theta \), then

$$ \mathit{IV}_{1}\leq C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \Vert b_{2} \Vert _{\mathrm{BMO}_{\theta }(\varphi )}\mathcal{M}_{L(\log L),\varphi,\eta }(f_{1},f_{2}) (x). $$

\(\mathit{IV}_{2}\) and \(\mathit{IV}_{3}\) are symmetric, here, only to estimate of \(\mathit{IV}_{2}\), similar to \(\mathit{II}_{2}\), taking \(N\geq 2\theta +2\eta +1\), we get

$$ \mathit{IV}_{2}\leq C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \Vert b_{2} \Vert _{\mathrm{BMO}_{\theta }(\varphi )}\mathcal{M}_{L(\log L),\varphi,\eta }(f_{1},f_{2}) (x). $$

Finally, similarly we estimate \(\mathit{IV}_{4}\),

$$ \mathit{IV}_{4}\leq C \Vert b_{1} \Vert _{\mathrm{BMO}_{\theta }(\varphi )} \Vert b_{2} \Vert _{\mathrm{BMO}_{\theta }(\varphi )}\mathcal{M}_{L(\log L),\varphi,\eta }(f_{1},f_{2}) (x). $$

Thus, we completed the proof of Theorem 4.1. □

Theorem 4.2

Let T be a multilinear Calderón–Zygmund operator of type \(\omega (t)\) as in Definition 1.1, \(T_{\Pi _{\vec{b}}}\) be a multilinear commutator with \(\vec{b}\in {\mathrm{BMO}}^{m}_{\vec{\theta }}(\varphi )\) and \(\vec{w}\in A_{\vec{p}}^{\infty }(\varphi )\) with \(1/p=1/p_{1}+\cdots +1/p_{m}\) and \(1< p_{j}<\infty,j=1,\ldots,m\). If ω satisfies

$$ \int _{0}^{1}\frac{\omega (t)}{t} \biggl(1+\log \frac{1}{t} \biggr)^{m}\,dt< \infty, $$

then there exists a constant \(C>0\) such that

$$ \bigl\Vert T_{\Pi _{\vec{b}}}(\vec{f}) \bigr\Vert _{L^{p}(v_{\vec{w}})}\leq C \prod_{j=1}^{m} \Vert b_{j} \Vert _{{\mathrm{BMO}}_{\theta _{j}}(\varphi )}\prod_{j=1}^{m} \Vert f_{j} \Vert _{L^{p_{j}}(w_{j})}. $$

Proof

It is sufficient to prove that, \(p>0\), \(w\in A_{\infty }^{\infty }\),

$$ \int _{\mathbb{R}^{n}} \bigl\vert T_{\Pi _{\vec{b}}}(\vec{f}) (x) \bigr\vert ^{p}w(x)\leq C \prod_{i=1}^{m} \Vert b_{j} \Vert _{{\mathrm{BMO}}_{\theta _{j}}(\varphi )} \int _{ \mathbb{R}^{n}}\mathcal{M}_{L(\log L),\varphi,\eta }(\vec{f}) (x)^{p}w(x)\,dx. $$

By the related Fefferman–Stein inequality (Lemma 2.7) and Theorem 4.1, we get

$$\begin{aligned} \bigl\Vert T_{\Pi _{\vec{b}}}(\vec{f}) \bigr\Vert _{L^{p}(w)}\leq{}& \bigl\Vert \mathcal{M}_{ \delta,\varphi,\eta }^{d}\bigl(T_{\Pi _{\vec{b}}}(\vec{f}) \bigr) \bigr\Vert _{L^{p}(w)} \\ \leq{}& \bigl\Vert \mathcal{M}_{\delta,\varphi,\eta }^{\sharp,d} \bigl(T_{\Pi _{ \vec{b}}}(\vec{f})\bigr) \bigr\Vert _{L^{p}(w)} \\ \leq{}& C\prod_{j=1}^{m} \Vert b_{j} \Vert _{{\mathrm{BMO}}_{\theta _{j}(\varphi )}} \bigl\Vert \mathcal{M}_{L(\log L),\varphi,\eta }( \vec{f}) \bigr\Vert _{L^{p}(w)}+ \bigl\Vert M_{ \epsilon,\varphi,\eta }^{d} \bigl(T(\vec{f})\bigr) \bigr\Vert _{L^{p}(w)} \\ &{} +C\sum_{j=1}^{m-1}\sum _{\sigma \in C_{j}^{m}}\prod_{j=1}^{m} \Vert b_{\sigma (i)} \Vert _{{\mathrm{BMO}}_{\sigma (i)}} \bigl\Vert M_{\epsilon,\varphi, \eta } \bigl(T_{\Pi \vec{b_{\sigma '}}}(\vec{f})\bigr) \bigr\Vert _{L^{p}(w)} \\ \leq{}& C\prod_{j=1}^{m} \Vert b_{j} \Vert _{{\mathrm{BMO}}_{\theta _{j}(\varphi )}} \bigl\Vert \mathcal{M}_{L(\log L),\varphi,\eta }( \vec{f}) \bigr\Vert _{L^{p}(w)}+ \bigl\Vert M_{ \epsilon,\varphi,\eta }^{\sharp,d} \bigl(T(\vec{f})\bigr) \bigr\Vert _{L^{p}(w)} \\ &{} +C\sum_{j=1}^{m-1}\sum _{\sigma \in C_{j}^{m}}\prod_{i=1}^{m} \Vert b_{\sigma (i)} \Vert _{{\mathrm{BMO}}_{\sigma (i)}} \bigl\Vert M_{\epsilon,\varphi, \eta }^{\sharp }\bigl(T_{\Pi _{\vec{b_{\sigma '}}}}(\vec{f})\bigr) \bigr\Vert _{L^{p}(w)}. \end{aligned}$$

Here \(\epsilon,\eta \) are the same as in Theorem 4.1.

By Theorem 3.1, then

$$\begin{aligned} \bigl\Vert M_{\epsilon,\varphi,\eta }^{\sharp,d}\bigl(T(\vec{f})\bigr) \bigr\Vert _{L^{p}(w)}&\leq C \bigl\Vert \mathcal{M}_{\varphi,\eta }(\vec{f}) \bigr\Vert _{L^{p}(w)} \\ &\leq C \bigl\Vert \mathcal{M}_{L(\log L),\varphi,\eta }(\vec{f}) \bigr\Vert _{L^{p}(w)}. \end{aligned}$$

For simplicity, we only prove the case \(m=2\) in the following:

$$\begin{aligned} \bigl\Vert M_{\epsilon,\varphi,\eta }^{\sharp }\bigl(T_{\Pi _{\vec{b_{\sigma '}}}}( \vec{f}) \bigr) \bigr\Vert _{L^{p}(w)}\leq{}& C \bigl\Vert M_{\epsilon,\varphi,\eta }^{\sharp } \bigl(T_{b_{1}}^{1}(f_{1},f_{2})\bigr) \bigr\Vert _{L^{p}(w)} \\ & {}+ C \bigl\Vert M_{\epsilon,\varphi,\eta }^{\sharp }\bigl(T_{b_{2}}^{1}(f_{1},f_{2}) \bigr) \bigr\Vert _{L^{p}(w)}. \end{aligned}$$

Similar to the estimate of \(\|M_{\epsilon,\varphi,\eta }^{\sharp,d}(T(\vec{f}))\|_{L^{p}(w)}\), and Eqs. (4.2) and (4.3),

$$\begin{aligned} \bigl\Vert M_{\epsilon,\varphi,\eta }^{\sharp }\bigl(T_{b_{j}}^{j}(f_{1},f_{2}) \bigr) \bigr\Vert _{L^{p}(w)} \leq C \Vert b_{j} \Vert _{{\mathrm{BMO}}_{\theta _{j}}} \bigl\Vert \mathcal{M}_{L(\log L), \varphi,\eta }(\vec{f}) \bigr\Vert _{L^{p}(w)}. \end{aligned}$$

To sum up, we obtain

$$\begin{aligned} \int _{\mathbb{R}^{n}} \bigl\vert T_{\Pi _{\vec{b}}}(\vec{f}) (x) \bigr\vert ^{p}w(x)\leq C \prod_{i=1}^{m} \Vert b_{j} \Vert _{{\mathrm{BMO}}_{\theta _{j}}(\varphi )} \int _{ \mathbb{R}^{n}}\mathcal{M}_{L(\log L),\varphi,\eta }(\vec{f}) (x)^{p}w(x)\,dx. \end{aligned}$$
(4.4)

By Lemma 2.3, we get

$$ \int _{\mathbb{R}^{n}} \bigl\vert T_{\Pi _{\vec{b}}}(\vec{f}) (x) \bigr\vert ^{p}v_{\vec{w}} \leq C\prod_{i=1}^{m} \Vert b_{i} \Vert _{{\mathrm{BMO}}_{\theta _{i}}} \int _{ \mathbb{R}^{n}}\mathcal{M}_{L(\log L),\varphi,\eta }(\vec{f}) (x)^{p}v_{ \vec{w}}\,dx. $$

If \(\mu >1\), and since \(\Phi (t)=t(1+{\mathrm{log}}^{+}(t))\leq t^{\mu } (t>1)\), we easily get

$$ \mathcal{M}_{L(\log L),\varphi,\eta }(\vec{f}) (x)\leq \mathcal{M}_{\mu, \varphi,\eta }( \vec{f}) (x). $$

By Lemma 2.9, then

$$ \bigl\Vert \mathcal{M}_{\mu,\varphi,\eta }(\vec{f}) \bigr\Vert _{L^{p}{(v_{\vec{w}})}} \leq C \prod_{j=1}^{m} \Vert f_{j} \Vert _{L^{p}_{(w_{j})}}. $$

The inequality \(\|\mathcal{M}_{\mu,\varphi,\eta }(\vec{f})\|_{L^{p}_{(v(\vec{w}))}} \leq C \prod_{j=1}^{m}\|f_{j}\|_{L^{p}_{(w_{j})}}\) is equivalent to \(\|\mathcal{M}_{\varphi,\eta }(\vec{f})\|_{L^{p/u}(v_{\vec{w}})}\leq C \prod_{j=1}^{m}\|f_{j}\|_{L^{p}_{(w_{j})}}\), which was proved in [16]. For some \(\mu >1\), using Lemma 2.4, we get

$$ \bigl\Vert T_{\Pi _{\vec{b}}}(\vec{f}) \bigr\Vert _{L^{p}(v_{\vec{w}})}\leq C \prod_{j=1}^{m} \Vert b_{j} \Vert _{{\mathrm{BMO}}_{\theta _{j}}(\varphi )}\prod_{j=1}^{m} \Vert f_{j} \Vert _{L^{p_{j}}(w_{j})}. $$

Thus, this proof is finished. By the proof of Theorem 4.2, the following results are obtained. □

Theorem 4.3

Let \(p>0\) and w be a weight in \(A_{\infty }^{\infty }(\varphi )\), and suppose that \(T_{\Pi _{\vec{b}}}\) is a multilinear iterated commutator with \(\vec{b}\in {\mathrm{BMO}}_{\vec{\theta }}^{m}(\varphi )\). Let \(\eta >0\) and ω is satisfying (4.1). Then there exist constants \(C>0\) depending on the \(A_{\infty }^{\infty }(\varphi )\) constant of w such that

$$\begin{aligned} \int _{\mathbb{R}^{n}} \bigl\vert T_{\Pi _{\vec{b}}}(\vec{f}) (x) \bigr\vert ^{p}w(x)\leq C \prod_{i=1}^{m} \Vert b_{j} \Vert _{{\mathrm{BMO}}_{\theta _{j}}(\varphi )} \int _{ \mathbb{R}^{n}}\mathcal{M}_{L(\log L),\varphi,\eta }(\vec{f}) (x)^{p}w(x)\,dx \end{aligned}$$

and

$$\begin{aligned} &\sup_{t>0}\frac{1}{\Phi ^{(m)}(\frac{1}{t})} w \bigl( \bigl\{ y\in \mathbb{R}^{n}: \bigl\vert T_{\Pi _{\vec{b}}}(\vec{f}) (y) \bigr\vert >t^{m} \bigr\} \bigr) \\ &\quad\leq C\sup_{t>0}\frac{1}{\Phi ^{(m)}(\frac{1}{t})}w \bigl( \bigl\{ y \in \mathbb{R}^{n}:\mathcal{M}_{L(\log L),\varphi,\eta }(\vec{f}) (y)>t^{m} \bigr\} \bigr). \end{aligned}$$

Proof

The first result is proved in Theorem 4.2, the proof of second result is similar to the first, also refer to the [8, Theorem 3.19], we need to use Lemma 2.8, Theorem 3.1 and Theorem 4.1. We omit the details here. □

Lemma 4.4

([21])

Let \(w\in A_{\vec{1}}^{\theta }(\varphi )\) and \(\eta >2\theta \). Then there exists a constant \(C>0\) such that

$$ v_{\vec{w}} \bigl\{ x\in \mathbb{R}^{n}:\mathcal{M}_{L(\log L),\varphi, \eta }( \vec{f}) (x)>t^{m} \bigr\} \leq C\prod_{j=1}^{m} \biggl( \int _{ \mathbb{R}^{n}}\Phi ^{(m)} \biggl(\frac{ \vert f_{j}(x) \vert }{t} \biggr)w_{j}(x)\,dx \biggr)^{\frac{1}{m}}, $$

where \(\Phi (t)=t(1+{\mathrm{log}}^{+}t)\) and \(\Phi ^{(m)}=\underbrace{\Phi \circ \cdots \circ \Phi }\).

Theorem 4.5

Let T be a multilinear Calderón–Zygmund operator of type \(\omega (t)\) as in Definition 1.1, \(T_{\Pi _{\vec{b}}}\) be a multilinear commutator with \(\vec{b}\in {\mathrm{BMO}}^{m}_{\vec{\theta }}(\varphi )\) and \(\vec{w}\in A_{\vec{1}}^{\infty }(\varphi )\), assume that ω is satisfying (4.1). Then there exists a constant \(C>0\) such that

$$ v_{\vec{w}} \bigl\{ x\in \mathbb{R}^{:} \bigl\vert T_{\Pi _{\vec{b}}}(\vec{f}) (x) \bigr\vert >t^{m} \bigr\} \leq C\prod _{j=1}^{m} \biggl( \int _{\mathbb{R}^{n}}\Phi ^{(m)} \biggl(\frac{ \vert f_{j}(x) \vert }{t} \biggr)w_{j}(x)\,dx \biggr)^{\frac{1}{m}}. $$

Proof

Now by Theorem 4.1, Theorem 4.3 and Lemma 4.4, we can get the above result. Since this argument is the same as the proof of [16, Theorem 4.2], here, we omit the proof. □

Corollary 4.6

Let T be a multilinear Calderón–Zygmund operator of type \(\omega (t)\) as in Definition 1.1, \(T_{\Sigma _{\vec{b}}}\) be a multilinear commutator with \(\vec{b}\in {\mathrm{BMO}}^{m}_{\vec{\theta }}(\varphi )\), assume that ω is satisfying (4.1), \(0<\delta <\epsilon <1/m\) and \(\eta \geq 2(\theta _{1},\ldots,\theta _{m})/(1/\delta -1/\epsilon )\). Then there exists a constant \(C>0\) such that

$$ M_{\delta,\varphi,\eta,}^{\sharp,d}\bigl(T_{\Sigma _{\vec{b}}}(\vec{f})\bigr) (x) \leq C\sum_{j=1}^{m} \Vert b_{j} \Vert _{\mathrm{BMO}_{\theta _{j}}(\varphi )} \Biggl(\sum_{j=1}^{m} \mathcal{M}_{L(\log L),\varphi,\eta }^{i}(\vec{f}) (x)+M_{ \epsilon,\varphi,\eta }^{d} \bigl(T(\vec{f})\bigr) (x) \Biggr). $$

For all m-tuples \(\vec{f}=(f_{1},\ldots,f_{m})\) of bounded measurable functions with compact support.

Proof

In fact, the multilinear commutator is a special case of iterate commutator, so we can directly obtain this result through Theorem 4.1. □

Corollary 4.7

Let \(T_{\Sigma _{\vec{b}}}\) be a multilinear commutator with \(\vec{b}\in {\mathrm{BMO}}^{m}_{\vec{\theta }}(\varphi )\), T be a multilinear Calderón–Zygmund operator of type \(\omega (t)\) as in Definition 1.1and \(\vec{w}\in A_{\vec{p}}^{\infty }(\varphi )\) with \(1/p=1/p_{1}+\cdots +1/p_{m}\) and \(1< p_{j}<\infty \), \(j=1,\ldots,m\). If ω is satisfying

$$ \int _{0}^{1}\frac{\omega (t)}{t} \biggl(1+\log \frac{1}{t} \biggr)^{m}\,dt< \infty, $$

then there exists a constant \(C>0\) such that

$$ \bigl\Vert T_{\Sigma \vec{b}}(\vec{f}) \bigr\Vert _{L^{p}(v_{\vec{w}})}\leq C \sum_{j=1}^{m} \Vert b_{j} \Vert _{\mathrm{BMO}_{\theta _{j}}(\varphi )}\prod_{j=1}^{m} \Vert f_{j} \Vert _{L^{p_{j}}(w_{j})}. $$

Proof

Obviously, the multilinear commutator is a special case of the iterate commutator, then, through Theorem 4.2 we can directly obtain this result. □

Corollary 4.8

Let \(p>0\) and w be a weight in \(A_{\infty }^{\infty }(\varphi )\), and suppose that \(T_{\Sigma _{\vec{b}}}\) be a multilinear commutator with \(\vec{b}\in {\mathrm{BMO}}^{m}_{\vec{\theta }}(\varphi )\), T be a multilinear Calderón–Zygmund operator of type \(\omega (t)\) as in Definition 1.1. Let \(\eta >0\) and ω is satisfying (4.1). Then there exist constants \(C>0\) depending on the \(A_{\infty }^{\infty }(\varphi )\) constant of w such that

$$ \int _{\mathbb{R}^{n}} \bigl\vert T_{\Sigma _{\vec{b}}}(\vec{f}) (x) \bigr\vert ^{p}w(x) \leq C\sum_{i=1}^{m} \Vert b_{j} \Vert _{\mathrm{BMO}_{\theta _{j}}(\varphi )} \int _{ \mathbb{R}^{n}}\sum_{i=1}^{m} \mathcal{M}_{L(\log L),\varphi,\eta }^{i}( \vec{f}) (x)^{p}w(x)\,dx $$

and

$$\begin{aligned} &\sup_{t>0}\frac{1}{\Phi (\frac{1}{t})}w \bigl( \bigl\{ y\in \mathbb{R}^{n}: \bigl\vert T_{ \Sigma _{\vec{b}}}(\vec{f}) (y) \bigr\vert >t^{m} \bigr\} \bigr) \\ &\quad\leq C\sup_{t>0}\frac{1}{\Phi (\frac{1}{t})}w \Biggl( \Biggl\{ y\in \mathbb{R}^{n}:\sum_{i=1}^{m} \mathcal{M}_{L(\log L),\varphi,\eta }^{i}( \vec{f}) (y)>t^{m} \Biggr\} \Biggr). \end{aligned}$$

Proof

Similar to the proof of [8, Theorem 3.19], we need to use Lemma 2.7, Lemma 2.8, Theorem 3.1, and Corollary 4.6. We omit the details here. □

Lemma 4.9

([16])

Let \(w\in A_{\vec{1}}^{\theta }(\varphi )\) and \(\eta >2\theta \). Then there exists a constant C such that

$$ v_{\vec{w}} \bigl\{ x\in \mathbb{R}^{n}:\mathcal{M}_{L(\log L),\varphi, \eta }^{i}( \vec{f}) (x)>t^{m} \bigr\} \leq C\prod_{j=1}^{m} \biggl( \int _{ \mathbb{R}^{n}}\Phi \biggl(\frac{ \vert f_{j}(x) \vert }{t} \biggr)w_{j}(x) \,dx \biggr)^{ \frac{1}{m}}, $$

where \(\Phi (t)=t(1+{\mathrm{log}}^{+}t)\).

Corollary 4.10

Let \(T_{\Sigma _{\vec{b}}}\) be a multilinear commutator with \(\vec{b}\in {\mathrm{BMO}}^{m}_{\vec{\theta }}(\varphi )\), T be a multilinear Calderón–Zygmund operator of type \(\omega (t)\) as in Definition 1.1and \(\vec{w}\in A_{\vec{1}}^{\infty }(\varphi )\), assume that ω is satisfying (4.1). Then there exists a constant C such that

$$ v_{\vec{w}} \bigl\{ x\in \mathbb{R}^{n}: \bigl\vert T_{\Sigma _{\vec{b}}}(\vec{f}) (x) \bigr\vert >t^{m} \bigr\} \leq C\prod _{j=1}^{m} \biggl( \int _{\mathbb{R}^{n}}\Phi \biggl( \frac{ \vert f_{j}(x) \vert }{t} \biggr)w_{j}(x) \,dx \biggr)^{\frac{1}{m}}. $$

Proof

Now by Corollary 4.8 and Lemma 4.9. We can get Corollary 4.10. Since this argument is the same as the proof of [16, Theorem 4.2], here, we omit the proof. □

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Acknowledgements

The authors would like to express their thanks to the referees for valuable advice regarding a previous version of this paper.

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The research was supported by Natural Science Foundation of China (Grant Nos. 12061069).

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Zhao, Y., Zhou, J. New weighted norm inequalities for multilinear Calderón–Zygmund operators with kernels of Dini’s type and their commutators. J Inequal Appl 2021, 29 (2021). https://doi.org/10.1186/s13660-021-02560-8

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  • 42B20

Keywords

  • \(A_{\vec{p}}^{\infty }(\varphi )\) weight
  • \(\mathrm{BMO}_{\theta }(\varphi )\) function
  • Dini kernel
  • Commutator