# Multilinear $\theta$-type Calderón–Zygmund operators and commutators on
products of weighted Morrey spaces

Xia Han and Hua Wang 111In memory of Li Xue. E-mail address:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
School of Mathematics and System Sciences, Xinjiang University, Urumqi 830046,
P. R. China

###### Abstract

In this paper, we consider the boundedness properties of multilinear
$\theta$-type Calderón–Zygmund operators $T_{\theta}$ recently introduced in
the literature. First, we prove strong type and weak type estimates for
multilinear $\theta$-type Calderón–Zygmund operators on products of weighted
Morrey spaces with multiple weights. Then we discuss strong type estimates for
both multilinear commutators and iterated commutators of $T_{\theta}$ on
products of these spaces with multiple weights. Furthermore, the weak end-
point estimates for commutators of $T_{\theta}$ and pointwise multiplication
with functions in bounded mean oscillation are established too.
MSC(2020): 42B20; 42B25; 47B38; 47G10
Keywords: Multilinear $\theta$-type Calderón–Zygmund operators; multilinear
commutators; iterated commutators; weighted Morrey spaces; multiple weights;
Orlicz spaces

###### Contents

  1. 1 Introduction
  2. 2 Main results
  3. 3 Notations and preliminaries
    1. 3.1 Multiple weights
    2. 3.2 Orlicz spaces and Luxemburg norms
  4. 4 Proofs of Theorems 2.1 and 2.2
  5. 5 Proofs of Theorems 2.3 and 2.4
  6. 6 Appendix

##  1 Introduction

In this paper, the symbols $\mathbb{R}$ and $\mathbb{N}$ stand for the sets of
all real numbers and natural numbers, respectively. Let $\mathbb{R}^{n}$ be
the $n$-dimensional Euclidean space with the Euclidean norm $|\cdot|$ and the
Lebesgue measure $dx$. Let $m\in\mathbb{N}$ and
$(\mathbb{R}^{n})^{m}=\overbrace{\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n}}^{m}$
be the $m$-fold product space. We denote by $\mathscr{S}(\mathbb{R}^{n})$ the
space of all Schwartz functions on $\mathbb{R}^{n}$ and by
$\mathscr{S}^{\prime}(\mathbb{R}^{n})$ its dual space, the set of all tempered
distributions on $\mathbb{R}^{n}$. Calderón–Zygmund singular integral
operators and their generalizations on the Euclidean space $\mathbb{R}^{n}$
have been extensively studied (see [4, 5, 6, 23] for instance). In particular,
Yabuta [27] introduced certain $\theta$-type Calderón–Zygmund operators to
facilitate his study of certain classes of pseudo-differential operators.
Following the terminology of Yabuta [27], we introduce the so-called
$\theta$-type Calderón–Zygmund operators as follows.

######  Definition 1.1.

Let $\theta$ be a nonnegative, nondecreasing function on
$\mathbb{R}^{+}:=(0,+\infty)$ with $0<\theta(1)<+\infty$ and

$\int_{0}^{1}\frac{\theta(t)}{t}\,dt<+\infty.$

A measurable function $K(x,y)$ on
$\mathbb{R}^{n}\times\mathbb{R}^{n}\setminus\\{(x,y):x=y\\}$ is said to be a
$\theta$-type Calderón–Zygmund kernel, if there exists a constant $A>0$ such
that

  1. 1.

$\big{|}K(x,y)\big{|}\leq\frac{A}{|x-y|^{n}}$, for any $x\neq y;$

  2. 2.

$\big{|}K(x,y)-K(z,y)\big{|}+\big{|}K(y,x)-K(y,z)\big{|}\leq\frac{A}{|x-y|^{n}}\cdot\theta\Big{(}\frac{|x-z|}{|x-y|}\Big{)}$,
for $|x-z|<\frac{|x-y|}{2}$.

######  Definition 1.2.

Let ${\mathcal{T}}_{\theta}$ be a linear operator from
$\mathscr{S}(\mathbb{R}^{n})$ into its dual
$\mathscr{S}^{\prime}(\mathbb{R}^{n})$. We say that ${\mathcal{T}}_{\theta}$
is a $\theta$-type Calderón–Zygmund operator with associated kernel $K$ if

  1. 1.

${\mathcal{T}}_{\theta}$ can be extended to be a bounded linear operator on
$L^{2}(\mathbb{R}^{n});$

  2. 2.

for any $f\in C^{\infty}_{0}(\mathbb{R}^{n})$ and for all
$x\notin\mathrm{supp\,}f$, there is a $\theta$-type Calderón–Zygmund kernel
$K(x,y)$ such that

${\mathcal{T}}_{\theta}f(x):=\int_{\mathbb{R}^{n}}K(x,y)f(y)\,dy,$

where $C^{\infty}_{0}(\mathbb{R}^{n})$ is the space consisting of all
infinitely differentiable functions on $\mathbb{R}^{n}$ that have compact
support.

Note that the classical Calderón–Zygmund operator with standard kernel (see
[4, 5]) is a special case of $\theta$-type operator ${\mathcal{T}}_{\theta}$
when $\theta(t)=t^{\delta}$ with $0<\delta\leq 1$.

In 2009, Maldonado and Naibo [15] considered the bilinear $\theta$-type
Calderón–Zygmund operators which are natural generalizations of the linear
case, and established weighted norm inequalities for bilinear $\theta$-type
Calderón–Zygmund operators on products of weighted Lebesgue spaces with
Muckenhoupt weights. Moreover, they applied these operators to the study of
certain paraproducts and bilinear pseudo-differential operators with mild
regularity. Later, in 2014, Lu and Zhang [14] introduced the general
$m$-linear $\theta$-type Calderón–Zygmund operators and their commutators for
$m\geq 2$, and established boundedness properties of these multilinear
operators and multilinear commutators on products of weighted Lebesgue spaces
with multiple weights. In addition, they gave some applications to the
paraproducts and bilinear pseudo-differential operators with mild regularity
and their commutators too. Following [14], we now give the definition of the
multilinear $\theta$-type Calderón–Zygmund operators.

######  Definition 1.3.

Let $\theta$ be a nonnegative, nondecreasing function on $\mathbb{R}^{+}$ with
$0<\theta(1)<+\infty$ and

$\int_{0}^{1}\frac{\theta(t)}{t}\,dt<+\infty.$ (1.1)

A measurable function $K(x,y_{1},\ldots,y_{m})$, defined away from the
diagonal $x=y_{1}=\cdots=y_{m}$ in $(\mathbb{R}^{n})^{m+1}$, is called an
$m$-linear $\theta$-type Calderón–Zygmund kernel, if there exists a constant
$A>0$ such that

  1. 1.

$\big{|}K(x,y_{1},\ldots,y_{m})\big{|}\leq\frac{A}{(|x-y_{1}|+\cdots+|x-y_{m}|)^{mn}}$
(1.2)

for all $(x,y_{1},\dots,y_{m})\in(\mathbb{R}^{n})^{m+1}$ with $x\neq y_{k}$
for some $k\in\\{1,2,\dots,m\\}$, and

  2. 2.

$\begin{split}&\big{|}K(x,y_{1},\dots,y_{m})-K(x^{\prime},y_{1},\dots,y_{m})\big{|}\\\
\leq&\frac{A}{(|x-y_{1}|+\cdots+|x-y_{m}|)^{mn}}\cdot\theta\bigg{(}\frac{|x-x^{\prime}|}{|x-y_{1}|+\cdots+|x-y_{m}|}\bigg{)}\end{split}$
(1.3)

whenever $|x-x^{\prime}|\leq\frac{\,1\,}{2}\max_{1\leq i\leq m}|x-y_{i}|$, and

  3. 3.

for each fixed $k$ with $1\leq k\leq m$,

$\begin{split}&\big{|}K(x,y_{1},\dots,y_{k},\dots,y_{m})-K(x,y_{1},\dots,y^{\prime}_{k},\dots,y_{m})\big{|}\\\
\leq&\frac{A}{(|x-y_{1}|+\cdots+|x-y_{m}|)^{mn}}\cdot\theta\bigg{(}\frac{|y_{k}-y^{\prime}_{k}|}{|x-y_{1}|+\cdots+|x-y_{m}|}\bigg{)}\end{split}$
(1.4)

whenever $|y_{k}-y^{\prime}_{k}|\leq\frac{\,1\,}{2}\max_{1\leq i\leq
m}|x-y_{i}|$.

######  Definition 1.4.

Let $m\in\mathbb{N}$ and $T_{\theta}$ be an $m$-linear operator initially
defined on the $m$-fold product of Schwartz spaces and taking values into the
space of tempered distributions, i.e.,

$T_{\theta}:\overbrace{\mathscr{S}(\mathbb{R}^{n})\times\cdots\times\mathscr{S}(\mathbb{R}^{n})}^{m}\to\mathscr{S}^{\prime}(\mathbb{R}^{n}).$

We say that $T_{\theta}$ is an $m$-linear $\theta$-type Calderón–Zygmund
operator if

  1. 1.

$T_{\theta}$ can be extended to be a bounded multilinear operator from
$L^{q_{1}}(\mathbb{R}^{n})\times\cdots\times L^{q_{m}}(\mathbb{R}^{n})$ into
$L^{q}(\mathbb{R}^{n})$ for some $q_{1},\ldots,q_{m}\in[1,+\infty)$ and
$q\in[1/m,+\infty)$ with $1/q=\sum_{k=1}^{m}1/{q_{k}};$

  2. 2.

for any given $m$-tuples $\vec{f}=(f_{1},\ldots,f_{m})$, there is an
$m$-linear $\theta$-type Calderón–Zygmund kernel $K(x,y_{1},\dots,y_{m})$ such
that

$\begin{split}T_{\theta}(\vec{f})(x)&=T_{\theta}(f_{1},\dots,f_{m})(x)\\\
&:=\int_{(\mathbb{R}^{n})^{m}}K(x,y_{1},\dots,y_{m})f_{1}(y_{1})\cdots
f_{m}(y_{m})\,dy_{1}\cdots dy_{m}\end{split}$

whenever $x\notin\bigcap_{k=1}^{m}\mathrm{supp\,}f_{k}$ and each $f_{k}\in
C^{\infty}_{0}(\mathbb{R}^{n})$ for $k=1,2,\dots,m$.

We note that, if we simply take $\theta(t)=t^{\varepsilon}$ for some
$0<\varepsilon\leq 1$, then the multilinear $\theta$-type operator
$T_{\theta}$ is exactly the multilinear Calderón–Zygmund operator, which was
systematically studied by many authors. There is a vast literature of results
of this nature, pioneered by the work of Grafakos and Torres [8], we refer the
reader to [7, 11, 17] and the references therein for more details. In 2014,
the following weighted strong-type and weak-type estimates of multilinear
$\theta$-type Calderón–Zygmund operators on products of weighted Lebesgue
spaces were proved by Lu and Zhang in [14].

######  Theorem 1.1 ([14]).

Let $m\in\mathbb{N}$ and $T_{\theta}$ be an $m$-linear $\theta$-type
Calderón–Zygmund operator with $\theta$ satisfying the condition (1.1). If
$p_{1},\ldots,p_{m}\in(1,+\infty)$ and $p\in(1/m,+\infty)$ with
$1/p=\sum_{k=1}^{m}1/{p_{k}}$, and $\vec{w}=(w_{1},\ldots,w_{m})$ satisfies
the multilinear $A_{\vec{P}}$ condition, then there exists a constant $C>0$
independent of $\vec{f}=(f_{1},\ldots,f_{m})$ such that

$\big{\|}T_{\theta}(\vec{f})\big{\|}_{L^{p}(\nu_{\vec{w}})}\leq
C\prod_{k=1}^{m}\big{\|}f_{k}\big{\|}_{L^{p_{k}}(w_{k})},$

where $\nu_{\vec{w}}=\prod_{k=1}^{m}w_{k}^{p/{p_{k}}}$.

######  Theorem 1.2 ([14]).

Let $m\in\mathbb{N}$ and $T_{\theta}$ be an $m$-linear $\theta$-type
Calderón–Zygmund operator with $\theta$ satisfying the condition (1.1). If
$p_{1},\ldots,p_{m}\in[1,+\infty)$, $\min\\{p_{1},\ldots,p_{m}\\}=1$ and
$p\in[1/m,+\infty)$ with $1/p=\sum_{k=1}^{m}1/{p_{k}}$, and
$\vec{w}=(w_{1},\ldots,w_{m})$ satisfies the multilinear $A_{\vec{P}}$
condition, then there exists a constant $C>0$ independent of
$\vec{f}=(f_{1},\ldots,f_{m})$ such that

$\big{\|}T_{\theta}(\vec{f})\big{\|}_{WL^{p}(\nu_{\vec{w}})}\leq
C\prod_{k=1}^{m}\big{\|}f_{k}\big{\|}_{L^{p_{k}}(w_{k})},$

where $\nu_{\vec{w}}=\prod_{k=1}^{m}w_{k}^{p/{p_{k}}}$.

For any given $p\in(0,+\infty)$ and $w$(weight function), the space $L^{p}(w)$
is defined as the set of all integrable functions $f$ on $\mathbb{R}^{n}$ such
that

$\|f\|_{L^{p}(w)}:=\bigg{(}\int_{\mathbb{R}^{n}}|f(x)|^{p}w(x)\,dx\bigg{)}^{1/p}<+\infty,$

and the weak space $WL^{p}(w)$ is defined as the set of all measurable
functions $f$ on $\mathbb{R}^{n}$ such that

$\|f\|_{WL^{p}(w)}:=\sup_{\lambda>0}\lambda\cdot
w\big{(}\big{\\{}x\in\mathbb{R}^{n}:|f(x)|>\lambda\big{\\}}\big{)}^{1/p}<+\infty,$

where $w(E):=\int_{E}w(x)\,dx$ for a Lebesgue measurable set
$E\subset\mathbb{R}^{n}$. When $w\equiv 1$, we denote simply by
$L^{p}(\mathbb{R}^{n})$ and $WL^{p}(\mathbb{R}^{n})$.

######  Remark 1.1.

For the linear case $m=1$, the above weighted results were given by Quek and
Yang in [19]. For the bilinear case $m=2$, Theorems 1.1 and 1.2 were proved by
Maldonado and Naibo in [15] when some additional conditions imposed on
$\theta$. And when $\theta(t)=t^{\varepsilon}$ for some $0<\varepsilon\leq 1$,
Theorems 1.1 and 1.2 were obtained by Lerner et al. [11].

Next, we give the definition of the commutator for the multilinear
$\theta$-type Calderón–Zygmund operator. Given a collection of locally
integrable functions $\vec{b}=(b_{1},\ldots,b_{m})$, the $m$-linear commutator
of $T_{\theta}$ with $\vec{b}$ is defined by

$\begin{split}\big{[}\Sigma\vec{b},T_{\theta}\big{]}(\vec{f})(x)&=\big{[}\Sigma\vec{b},T_{\theta}\big{]}(f_{1},\ldots,f_{m})(x)\\\
&:=\sum_{k=1}^{m}\big{[}b_{k},T_{\theta}\big{]}_{k}(f_{1},\ldots,f_{m})(x),\end{split}$
(1.5)

where each term is the commutator of $b_{k}$ and $T_{\theta}$ in the $k$-th
entry of $T_{\theta}$; that is,

$\begin{split}&\big{[}b_{k},T_{\theta}\big{]}_{k}(f_{1},\ldots,f_{m})(x)\\\
&=b_{k}(x)\cdot
T_{\theta}(f_{1},\ldots,f_{k},\dots,f_{m})(x)-T_{\theta}(f_{1},\dots,b_{k}f_{k},\dots,f_{m})(x).\end{split}$

Then, at a formal level

$\begin{split}&\big{[}\Sigma\vec{b},T_{\theta}\big{]}(\vec{f})(x)=\big{[}\Sigma\vec{b},T_{\theta}\big{]}(f_{1},\ldots,f_{m})(x)\\\
&=\int_{(\mathbb{R}^{n})^{m}}\sum_{k=1}^{m}\big{[}b_{k}(x)-b_{k}(y_{k})\big{]}K(x,y_{1},\dots,y_{m})f_{1}(y_{1})\cdots
f_{m}(y_{m})\,dy_{1}\cdots dy_{m}.\end{split}$

Obviously, when $m=1$ in the above definition, this operator coincides with
the linear commutator $[b,{\mathcal{T}}_{\theta}]$(see [13, 29]), which is
defined by

$[b,{\mathcal{T}}_{\theta}](f):=b\cdot{\mathcal{T}}_{\theta}(f)-{\mathcal{T}}_{\theta}(bf).$

Let us now recall the definition of the space of
$\mathrm{BMO}(\mathbb{R}^{n})$(see [4, 9]). A locally integrable function
$b(x)$ is said to belong to $\mathrm{BMO}(\mathbb{R}^{n})$ if it satisfies

$\|b\|_{*}:=\sup_{B}\frac{1}{|B|}\int_{B}|b(x)-b_{B}|\,dx<+\infty,$

where the supremum is taken over all balls $B$ in $\mathbb{R}^{n}$, and
$b_{B}$ stands for the average of $b$ over $B$, i.e.,

$b_{B}:=\frac{1}{|B|}\int_{B}b(y)\,dy.$

In the multilinear setting, we say that
$\vec{b}=(b_{1},\ldots,b_{m})\in\mathrm{BMO}^{m}$, if each
$b_{k}\in\mathrm{BMO}(\mathbb{R}^{n})$ for $k=1,2,\dots,m$. For convenience,
we will use the following notation

$\big{\|}\vec{b}\big{\|}_{\mathrm{BMO}^{m}}:=\max_{1\leq k\leq
m}\big{\|}b_{k}\big{\|}_{\ast},\quad\mbox{for}\;\vec{b}=(b_{1},\ldots,b_{m})\in\mathrm{BMO}^{m}.$

In 2014, Lu and Zhang [14] also proved some weighted estimate and $L\log
L$-type estimate for multilinear commutators
$\big{[}\Sigma\vec{b},T_{\theta}\big{]}$ defined in (1.5) under a stronger
condition (1.6) assumed on $\theta$, if $\vec{b}\in\mathrm{BMO}^{m}$.

######  Theorem 1.3 ([14]).

Let $m\in\mathbb{N}$ and $\big{[}\Sigma\vec{b},T_{\theta}\big{]}$ be the
$m$-linear commutator generated by $\theta$-type Calderón–Zygmund operator
$T_{\theta}$ and $\vec{b}=(b_{1},\ldots,b_{m})\in\mathrm{BMO}^{m};$ let
$\theta$ satisfy

$\int_{0}^{1}\frac{\theta(t)\cdot(1+|\log t|)}{t}\,dt<+\infty.$ (1.6)

If $p_{1},\ldots,p_{m}\in(1,+\infty)$ and $p\in(1/m,+\infty)$ with
$1/p=\sum_{k=1}^{m}1/{p_{k}}$, and $\vec{w}=(w_{1},\ldots,w_{m})\in
A_{\vec{P}}$, then there exists a constant $C>0$ independent of $\vec{b}$ and
$\vec{f}=(f_{1},\ldots,f_{m})$ such that

$\big{\|}\big{[}\Sigma\vec{b},T_{\theta}\big{]}(\vec{f})\big{\|}_{L^{p}(\nu_{\vec{w}})}\leq
C\cdot\big{\|}\vec{b}\big{\|}_{\mathrm{BMO}^{m}}\prod_{k=1}^{m}\big{\|}f_{k}\big{\|}_{L^{p_{k}}(w_{k})},$

where $\nu_{\vec{w}}=\prod_{k=1}^{m}w_{k}^{p/{p_{k}}}$.

######  Theorem 1.4 ([14]).

Let $m\in\mathbb{N}$ and $\big{[}\Sigma\vec{b},T_{\theta}\big{]}$ be the
$m$-linear commutator generated by $\theta$-type Calderón–Zygmund operator
$T_{\theta}$ and $\vec{b}=(b_{1},\ldots,b_{m})\in\mathrm{BMO}^{m};$ let
$\theta$ satisfy the condition (1.6). If $p_{k}=1$, $k=1,2,\dots,m$ and
$\vec{w}=(w_{1},\ldots,w_{m})\in A_{(1,\dots,1)}$, then for any given
$\lambda>0$, there exists a constant $C>0$ independent of $\vec{b}$,
$\vec{f}=(f_{1},\ldots,f_{m})$ and $\lambda$ such that

$\begin{split}&\nu_{\vec{w}}\Big{(}\Big{\\{}x\in\mathbb{R}^{n}:\big{|}\big{[}\Sigma\vec{b},T_{\theta}\big{]}(\vec{f})(x)\big{|}>\lambda^{m}\Big{\\}}\Big{)}\\\
&\leq
C\cdot\Phi\big{(}\big{\|}\vec{b}\big{\|}_{\mathrm{BMO}^{m}}\big{)}^{1/m}\prod_{k=1}^{m}\bigg{(}\int_{\mathbb{R}^{n}}\Phi\bigg{(}\frac{|f_{k}(x)|}{\lambda}\bigg{)}w_{k}(x)\,dx\bigg{)}^{1/m},\end{split}$

where $\nu_{\vec{w}}=\prod_{k=1}^{m}w_{k}^{1/{m}}$,
$\Phi(t):=t\cdot(1+\log^{+}t)$ and $\log^{+}t:=\max\\{\log t,0\\}$.

######  Remark 1.2.

As is well known, (multilinear) commutator has a greater degree of singularity
than the underlying (multilinear) $\theta$-type operator, so more regular
condition imposed on $\theta(t)$ is reasonable. Obviously, our condition (1.6)
is slightly stronger than the condition (1.1). For such type of commutators,
the condition that $\theta(t)$ satisfying (1.6) is needed in the linear case
(see [13, 29] for more details), so does in the multilinear case. Moreover, it
is straightforward to check that when $\theta(t)=t^{\varepsilon}$ for some
$\varepsilon>0$,

$\int_{0}^{1}\frac{t^{\varepsilon}\cdot(1+|\log
t|)}{t}\,dt=\int_{0}^{1}t^{\varepsilon-1}\cdot\bigg{(}1+\log\frac{\,1\,}{t}\bigg{)}dt<+\infty.$

Thus, the multilinear Calderón–Zygmund operator is also the multilinear
$\theta$-type operator $T_{\theta}$ with $\theta(t)$ satisfying (1.6).

######  Remark 1.3.

When $m=1$, the above weighted endpoint estimate for the linear commutator
$[b,\mathcal{T}_{\theta}]$ was given by Zhang and Xu in [29] (for the
unweighted case, see [13]). Since $\mathcal{T}_{\theta}$ is bounded on
$L^{p}(w)$ for $1<p<+\infty$ and $w\in A_{p}$ as mentioned earlier, then by
the well-known boundedness criterion for commutators of linear operators,
which was obtained by Alvarez et al. in [1], we know that
$[b,\mathcal{T}_{\theta}]$ is also bounded on $L^{p}(w)$ for all $1<p<+\infty$
and $w\in A_{p}$, whenever $b\in\mathrm{BMO}(\mathbb{R}^{n})$.

######  Remark 1.4.

When $m\geq 2$, $w_{1}=\cdots=w_{m}\equiv 1$ and $\theta(t)=t^{\varepsilon}$
for some $\varepsilon>0$, Pérez and Torres [17] proved that if
$\vec{b}=(b_{1},\ldots,b_{m})\in\mathrm{BMO}^{m}$, then

$\big{[}\Sigma\vec{b},T_{\theta}\big{]}:L^{p_{1}}(\mathbb{R}^{n})\times\cdots\times
L^{p_{m}}(\mathbb{R}^{n})\to L^{p}(\mathbb{R}^{n})$

for $1<p_{k}<+\infty$ and $1<p<+\infty$ with $1/p=1/{p_{1}}+\cdots+1/{p_{m}}$,
where $k=1,2,\dots,m$. And when $m\geq 2$ and $\theta(t)=t^{\varepsilon}$ for
some $\varepsilon>0$, Theorems 1.3 and 1.4 were obtained by Lerner et al. in
[11]. Namely, Lerner et al.[11] proved that if
$\vec{b}=(b_{1},\ldots,b_{m})\in\mathrm{BMO}^{m}$ and
$\vec{w}=(w_{1},\ldots,w_{m})\in A_{\vec{P}}$, then

$\big{[}\Sigma\vec{b},T_{\theta}\big{]}:L^{p_{1}}(w_{1})\times\cdots\times
L^{p_{m}}(w_{m})\to L^{p}(\nu_{\vec{w}})$

for $1<p_{k}<+\infty$ and $1/m<p<+\infty$ with
$1/p=1/{p_{1}}+\cdots+1/{p_{m}}$, where $k=1,2,\dots,m$.

######  Remark 1.5.

We will give alternative proof of Theorem 1.3, which shows that the conclusion
of Theorem 1.3 still holds provided that $\theta(t)$ only fulfills (1.1), see
the Appendix section for more details.

Motivated by [18] and [14], we will consider another type of commutators on
$\mathbb{R}^{n}$. Assume that $\vec{b}=(b_{1},\dots,b_{m})$ is a collection of
locally integrable functions, we define the iterated commutator
$\big{[}\Pi\vec{b},T_{\theta}\big{]}$ as

$\begin{split}\big{[}\Pi\vec{b},T_{\theta}\big{]}(\vec{f})(x)&=\big{[}\Pi\vec{b},T_{\theta}\big{]}(f_{1},\ldots,f_{m})(x)\\\
&:=[b_{1},[b_{2},\dots[b_{m-1},[b_{m},T_{\theta}]_{m}]_{m-1}\dots]_{2}]_{1}(f_{1},\ldots,f_{m})(x),\end{split}$

where

$\begin{split}&\big{[}b_{k},T_{\theta}\big{]}_{k}(f_{1},\ldots,f_{m})(x)\\\
&=b_{k}(x)\cdot
T_{\theta}(f_{1},\ldots,f_{k},\dots,f_{m})(x)-T_{\theta}(f_{1},\dots,b_{k}f_{k},\dots,f_{m})(x).\end{split}$

Then $\big{[}\Pi\vec{b},T_{\theta}\big{]}$ could be expressed in the following
way

$\begin{split}&\big{[}\Pi\vec{b},T_{\theta}\big{]}(\vec{f})(x)=\big{[}\Pi\vec{b},T_{\theta}\big{]}(f_{1},\ldots,f_{m})(x)\\\
&=\int_{(\mathbb{R}^{n})^{m}}\prod_{k=1}^{m}\big{[}b_{k}(x)-b_{k}(y_{k})\big{]}K(x,y_{1},\dots,y_{m})f_{1}(y_{1})\cdots
f_{m}(y_{m})\,dy_{1}\cdots dy_{m}.\end{split}$ (1.7)

Following the arguments used in [18] and [14] with some minor modifications,
we can also establish the corresponding results (strong type and weak endpoint
estimates) for iterated commutators of multilinear $\theta$-type
Calderón–Zygmund operators.

######  Theorem 1.5.

Let $m\in\mathbb{N}$ and $\big{[}\Pi\vec{b},T_{\theta}\big{]}$ be the iterated
commutator generated by $\theta$-type Calderón–Zygmund operator $T_{\theta}$
and $\vec{b}=(b_{1},\ldots,b_{m})\in\mathrm{BMO}^{m};$ let $\theta$ satisfy
the condition (1.1). If $p_{1},\ldots,p_{m}\in(1,+\infty)$ and
$p\in(1/m,+\infty)$ with $1/p=\sum_{k=1}^{m}1/{p_{k}}$, and
$\vec{w}=(w_{1},\ldots,w_{m})\in A_{\vec{P}}$, then there exists a constant
$C>0$ independent of $\vec{b}$ and $\vec{f}=(f_{1},\ldots,f_{m})$ such that

$\big{\|}\big{[}\Pi\vec{b},T_{\theta}\big{]}(\vec{f})\big{\|}_{L^{p}(\nu_{\vec{w}})}\leq
C\cdot\prod_{k=1}^{m}\big{\|}b_{k}\big{\|}_{*}\prod_{k=1}^{m}\big{\|}f_{k}\big{\|}_{L^{p_{k}}(w_{k})},$

where $\nu_{\vec{w}}=\prod_{k=1}^{m}w_{k}^{p/{p_{k}}}$.

######  Theorem 1.6.

Let $m\in\mathbb{N}$ and $\big{[}\Pi\vec{b},T_{\theta}\big{]}$ be the iterated
commutator generated by $\theta$-type Calderón–Zygmund operator $T_{\theta}$
and $\vec{b}=(b_{1},\ldots,b_{m})\in\mathrm{BMO}^{m};$ let $\theta$ satisfy

$\int_{0}^{1}\frac{\theta(t)\cdot(1+|\log t|^{m})}{t}\,dt<+\infty.$ (1.8)

If $p_{k}=1$, $k=1,2,\dots,m$ and $\vec{w}=(w_{1},\ldots,w_{m})\in
A_{(1,\dots,1)}$, then for any given $\lambda>0$, there exists a constant
$C>0$ independent of $\vec{f}=(f_{1},\ldots,f_{m})$ and $\lambda$ such that

$\begin{split}&\nu_{\vec{w}}\Big{(}\Big{\\{}x\in\mathbb{R}^{n}:\big{|}\big{[}\Pi\vec{b},T_{\theta}\big{]}(\vec{f})(x)\big{|}>\lambda^{m}\Big{\\}}\Big{)}\\\
&\leq
C\cdot\prod_{k=1}^{m}\bigg{(}\int_{\mathbb{R}^{n}}\Phi^{(m)}\bigg{(}\frac{|f_{k}(x)|}{\lambda}\bigg{)}w_{k}(x)\,dx\bigg{)}^{1/m},\end{split}$

where $\nu_{\vec{w}}=\prod_{k=1}^{m}w_{k}^{1/{m}}$,
$\Phi(t)=t\cdot(1+\log^{+}t)$ and
$\Phi^{(m)}:=\overbrace{\Phi\circ\cdots\circ\Phi}^{m}$.

######  Remark 1.6.

It was proved in [18] that when $\theta(t)=t^{\varepsilon}$ for some
$\varepsilon>0$, the estimate in Theorem 1.6 is sharp in the sense that
$\Phi^{(m)}$ cannot be replaced by $\Phi^{(k)}$ for any $k<m$.

On the other hand, the classical Morrey spaces $L^{p,\kappa}(\mathbb{R}^{n})$
were originally introduced by Morrey in [16] to study the local regularity of
solutions to second order elliptic partial differential equations. Nowadays
these spaces have been studied intensively in the literature, and found a wide
range of applications in harmonic analysis, potential theory and nonlinear
dispersive equations. In 2009, Komori and Shirai [10] defined and investigated
the weighted Morrey spaces $L^{p,\kappa}(w)$ for $1\leq p<+\infty$, which
could be viewed as an extension of weighted Lebesgue spaces, and obtained the
boundedness of some classical integral operators on these weighted spaces. In
order to deal with the multilinear case $m\geq 2$, we consider the weighted
Morrey spaces $L^{p,\kappa}(w)$ here for all $0<p<+\infty$. We will extend the
results obtained in [14] for $m$-linear $\theta$-type Calderón–Zygmund
operators to the product of weighted Morrey spaces with multiple weights.
Moreover, the corresponding weighted estimates for both multilinear
commutators and iterated commutators are also considered. Let us first recall
the definition of the spaces $L^{p,\kappa}(w)$ and $WL^{p,\kappa}(w)$.

######  Definition 1.5 ([10]).

Let $0<p<+\infty$, $0\leq\kappa<1$ and let $w$ be a weight on
$\mathbb{R}^{n}$. The weighted Morrey space $L^{p,\kappa}(w)$ is defined to be
the set of all locally integrable functions $f$ on $\mathbb{R}^{n}$ satisfying

$\|f\|_{L^{p,\kappa}(w)}:=\sup_{B}\bigg{(}\frac{1}{w(B)^{\kappa}}\int_{B}|f(x)|^{p}w(x)\,dx\bigg{)}^{1/p}<+\infty,$

where the supremum is taken over all balls $B$ in $\mathbb{R}^{n}$.

######  Definition 1.6 ([10]).

Let $0<p<+\infty$, $0\leq\kappa<1$ and let $w$ be a weight on
$\mathbb{R}^{n}$. The weighted weak Morrey space $WL^{p,\kappa}(w)$ is defined
to be the set of all measurable functions $f$ on $\mathbb{R}^{n}$ satisfying

$\begin{split}\|f\|_{WL^{p,\kappa}(w)}:=&\sup_{B}\frac{1}{m(B)^{\kappa/p}}\sup_{\lambda>0}\lambda\cdot
w\big{(}\big{\\{}x\in
B:|f(x)|>\lambda\big{\\}}\big{)}^{1/p}<+\infty,\end{split}$

where the supremum is taken over all balls $B$ in $\mathbb{R}^{n}$ and all
$\lambda>0$.

Note that when $w\in\Delta_{2}$, then $L^{p,0}(w)=L^{p}(w)$,
$WL^{p,0}(w)=WL^{p}(w)$ and $L^{p,1}(w)=L^{\infty}(w)$ by the Lebesgue
differentiation theorem with respect to $w$.

In order to deal with the end-point case of the commutators, we have to
consider the following $L\log L$-type space, which was introduced by the
second author in [25, 26] (for the unweighted case, see also [12] and [21]).

######  Definition 1.7.

Let $p=1$, $0\leq\kappa<1$ and let $w$ be a weight on $\mathbb{R}^{n}$. We
denote by $(L\log L)^{1,\kappa}(w)$ the weighted Morrey space of $L\log L$
type, the space of all locally integrable functions $f$ defined on
$\mathbb{R}^{n}$ with finite norm $\big{\|}f\big{\|}_{(L\log
L)^{1,\kappa}(w)}$.

$(L\log L)^{1,\kappa}(w):=\Big{\\{}f:\big{\|}f\big{\|}_{(L\log
L)^{1,\kappa}(w)}<\infty\Big{\\}},$

where

$\begin{split}\big{\|}f\big{\|}_{(L\log
L)^{1,\kappa}(w)}:=&\sup_{B}w(B)^{1-\kappa}\big{\|}f\big{\|}_{L\log
L(w),B}.\end{split}$

Here $\|\cdot\|_{L\log L(w),B}$ denotes the weighted Luxemburg norm, whose
definition will be given in Section 3 below. Note that $t\leq
t\cdot(1+\log^{+}t)$ for any $t>0$. By definition, for any ball $B$ in
$\mathbb{R}^{n}$ and $w\in A_{\infty}$, then we have

$\big{\|}f\big{\|}_{L(w),B}\leq\big{\|}f\big{\|}_{L\log L(w),B},$

which means that the following inequality (it can be viewed as a generalized
Jensen’s inequality)

$\big{\|}f\big{\|}_{L(w),B}=\frac{1}{w(B)}\int_{B}|f(x)|w(x)\,dx\leq\big{\|}f\big{\|}_{L\log
L(w),B}$ (1.9)

holds for any ball $B\subset\mathbb{R}^{n}$. Hence, for all $0<\kappa<1$ and
$w\in A_{\infty}$, we can further obtain the following inclusion from (1.9):

$(L\log L)^{1,\kappa}(w)\hookrightarrow L^{1,\kappa}(w).$

It is known that $L^{p,\kappa}$ is an extension of $L^{p}$ in the sense that
$L^{p,0}=L^{p}$. Motivated by the works in [10, 14, 15], the main purpose of
this paper is to establish boundedness properties of multilinear $\theta$-type
Calderón–Zygmund operators and their commutators on products of weighted
Morrey spaces with multiple weights.

In what follows, the letter $C$ always stands for a positive constant
independent of the main parameters and not necessarily the same at each
occurrence. The symbol $\mathbf{X}\lesssim\mathbf{Y}$ means that there is a
constant $C>0$ such that $\mathbf{X}\leq C\mathbf{Y}$. The symbol
$\mathbf{X}\approx\mathbf{Y}$ means that there is a constant $C>0$ such that
$C^{-1}\mathbf{Y}\leq\mathbf{X}\leq C\mathbf{Y}$.

##  2 Main results

Our first two results on the boundedness properties of multilinear
$\theta$-type Calderón–Zygmund operators can be formulated as follows.

######  Theorem 2.1.

Let $m\geq 2$ and $T_{\theta}$ be an $m$-linear $\theta$-type Calderón–Zygmund
operator with $\theta$ satisfying the condition (1.1). If
$1<p_{1},\dots,p_{m}<+\infty$ and $1/m<p<+\infty$ with
$1/p=\sum_{i=1}^{m}1/{p_{i}}$, and $\vec{w}=(w_{1},\ldots,w_{m})\in
A_{\vec{P}}$ with $w_{1},\ldots,w_{m}\in A_{\infty}$, then for any
$0<\kappa<1$, the multilinear operator $T_{\theta}$ is bounded from
$L^{p_{1},\kappa}(w_{1})\times L^{p_{2},\kappa}(w_{2})\times\cdots\times
L^{p_{m},\kappa}(w_{m})$ into $L^{p,\kappa}(\nu_{\vec{w}})$ with
$\nu_{\vec{w}}=\prod_{i=1}^{m}w_{i}^{p/{p_{i}}}$.

######  Theorem 2.2.

Let $m\geq 2$ and $T_{\theta}$ be an $m$-linear $\theta$-type Calderón–Zygmund
operator with $\theta$ satisfying the condition (1.1). If $1\leq
p_{1},\dots,p_{m}<+\infty$, $\min\\{p_{1},\ldots,p_{m}\\}=1$ and $1/m\leq
p<+\infty$ with $1/p=\sum_{i=1}^{m}1/{p_{i}}$, and
$\vec{w}=(w_{1},\ldots,w_{m})\in A_{\vec{P}}$ with $w_{1},\ldots,w_{m}\in
A_{\infty}$, then for any $0<\kappa<1$, the multilinear operator $T_{\theta}$
is bounded from $L^{p_{1},\kappa}(w_{1})\times
L^{p_{2},\kappa}(w_{2})\times\cdots\times L^{p_{m},\kappa}(w_{m})$ into
$WL^{p,\kappa}(\nu_{\vec{w}})$ with
$\nu_{\vec{w}}=\prod_{i=1}^{m}w_{i}^{p/{p_{i}}}$.

Our next theorem concerns norm inequalities for the multilinear commutator
$\big{[}\Sigma\vec{b},T_{\theta}\big{]}$ with $\vec{b}\in\mathrm{BMO}^{m}$.

######  Theorem 2.3.

Let $m\geq 2$ and $\big{[}\Sigma\vec{b},T_{\theta}\big{]}$ be the $m$-linear
commutator of $\theta$-type Calderón–Zygmund operator $T_{\theta}$ with
$\theta$ satisfying the condition (1.1) and $\vec{b}\in\mathrm{BMO}^{m}$. If
$1<p_{1},\dots,p_{m}<+\infty$ and $1/m<p<+\infty$ with
$1/p=\sum_{i=1}^{m}1/{p_{i}}$, and $\vec{w}=(w_{1},\ldots,w_{m})\in
A_{\vec{P}}$ with $w_{1},\ldots,w_{m}\in A_{\infty}$, then for any
$0<\kappa<1$, the multilinear commutator
$\big{[}\Sigma\vec{b},T_{\theta}\big{]}$ is bounded from
$L^{p_{1},\kappa}(w_{1})\times L^{p_{2},\kappa}(w_{2})\times\cdots\times
L^{p_{m},\kappa}(w_{m})$ into $L^{p,\kappa}(\nu_{\vec{w}})$ with
$\nu_{\vec{w}}=\prod_{i=1}^{m}w_{i}^{p/{p_{i}}}$.

For the endpoint case $p_{1}=p_{2}=\cdots=p_{m}=1$, we will also prove the
following weak-type $L\log L$ estimate for the multilinear commutator
$\big{[}\Sigma\vec{b},T_{\theta}\big{]}$ in the weighted Morrey spaces with
multiple weights.

######  Theorem 2.4.

Let $m\geq 2$ and $\big{[}\Sigma\vec{b},T_{\theta}\big{]}$ be the $m$-linear
commutator of $\theta$-type Calderón–Zygmund operator $T_{\theta}$ with
$\theta$ satisfying the condition (1.6) and $\vec{b}\in\mathrm{BMO}^{m}$.
Assume that $\vec{w}=(w_{1},\ldots,w_{m})\in A_{(1,\dots,1)}$ with
$w_{1},\ldots,w_{m}\in A_{\infty}$. If $p_{i}=1$, $i=1,2,\ldots,m$ and
$p=1/m$, then for any given $\lambda>0$ and any ball $B\subset\mathbb{R}^{n}$,
there exists a constant $C>0$ such that

$\begin{split}&\frac{1}{\nu_{\vec{w}}(B)^{m\kappa}}\cdot\Big{[}\nu_{\vec{w}}\Big{(}\Big{\\{}x\in
B:\big{|}\big{[}\Sigma\vec{b},T_{\theta}\big{]}(\vec{f})(x)\big{|}>\lambda^{m}\Big{\\}}\Big{)}\Big{]}^{m}\\\
&\leq
C\cdot\Phi\big{(}\big{\|}\vec{b}\big{\|}_{\mathrm{BMO}^{m}}\big{)}\prod_{i=1}^{m}\bigg{\|}\Phi\bigg{(}\frac{|f_{i}|}{\lambda}\bigg{)}\bigg{\|}_{(L\log
L)^{1,\kappa}(w_{i})},\end{split}$

where $\nu_{\vec{w}}=\prod_{i=1}^{m}w_{i}^{1/{m}}$ and
$\Phi(t)=t\cdot(1+\log^{+}t)$.

######  Remark 2.1.

From the above definitions and Theorem 2.4, we can roughly say that the
multilinear commutator $\big{[}\Sigma\vec{b},T_{\theta}\big{]}$ is bounded
from $(L\log L)^{1,\kappa}(w_{1})\times(L\log
L)^{1,\kappa}(w_{2})\times\cdots\times(L\log L)^{1,\kappa}(w_{m})$ into
$WL^{1/m,\kappa}(\nu_{\vec{w}})$ with
$\nu_{\vec{w}}=\prod_{i=1}^{m}w_{i}^{1/{m}}$.

##  3 Notations and preliminaries

###  3.1 Multiple weights

For any $r>0$ and $x\in\mathbb{R}^{n}$, let
$B(x,r)=\big{\\{}y\in\mathbb{R}^{n}:|x-y|<r\big{\\}}$ denote the open ball
centered at $x$ with radius $r$,
$B(x,r)^{\complement}=\mathbb{R}^{n}\backslash B(x,r)$ denote its complement
and $|B(x,r)|$ be the Lebesgue measure of the ball $B(x,r)$. We also use the
notation $\chi_{B(x,r)}$ to denote the characteristic function of $B(x,r)$.
For some $t>0$, the notation $tB$ stands for the ball with the same center as
$B$ whose radius is $t$ times that of $B$.

A weight $w$ is said to belong to the Muckenhoupt class $A_{p}$ for
$1<p<+\infty$, if there exists a constant $C>0$ such that

$\bigg{(}\frac{1}{|B|}\int_{B}w(x)\,dx\bigg{)}^{1/p}\bigg{(}\frac{1}{|B|}\int_{B}w(x)^{-p^{\prime}/p}\,dx\bigg{)}^{1/{p^{\prime}}}\leq
C$

for every ball $B$ in $\mathbb{R}^{n}$, where $p^{\prime}$ is the conjugate
exponent of $p$ such that $1/p+1/{p^{\prime}}=1$. The class $A_{1}$ is defined
replacing the above inequality by

$\frac{1}{|B|}\int_{B}w(x)\,dx\leq C\cdot\underset{x\in
B}{\mbox{ess\,inf}}\;w(x)$

for every ball $B$ in $\mathbb{R}^{n}$. Since the $A_{p}$ classes are
increasing with respect to $p$, the $A_{\infty}$ class of weights is defined
in a natural way by

$A_{\infty}:=\bigcup_{1\leq p<+\infty}A_{p}.$

Moreover, the following characterization will often be used in the sequel.
There are positive constants $C$ and $\delta$ such that for any ball $B$ and
any measurable set $E$ contained in $B$,

$\frac{w(E)}{w(B)}\leq C\bigg{(}\frac{|E|}{|B|}\bigg{)}^{\delta}.$ (3.1)

Given a Lebesgue measurable set $E$, we denote the characteristic function of
$E$ by $\chi_{E}$. We say that a weight $w$ satisfies the doubling condition,
simply denoted by $w\in\Delta_{2}$, if there is an absolute constant $C>0$
such that

$w(2B)\leq C\,w(B)$ (3.2)

holds for any ball $B$ in $\mathbb{R}^{n}$. If $w\in A_{p}$ with $1\leq
p<+\infty$ (or $w\in A_{\infty}$), then we have that $w\in\Delta_{2}$.

Recently, the theory of multiple weights adapted to multilinear
Calderón–Zygmund operators was developed by Lerner et al. in [11]. New more
refined multilinear maximal function was defined and used in [11] to
characterize the class of multiple $A_{\vec{P}}$ weights, and to obtain some
weighted estimates for multilinear Calderón–Zygmund operators. Now let us
recall the definition of multiple weights. For $m$ exponents
$p_{1},\ldots,p_{m}\in[1,+\infty)$, we will often write $\vec{P}$ for the
vector $\vec{P}=(p_{1},\ldots,p_{m})$, and $p$ for the number given by
$1/p=\sum_{k=1}^{m}1/{p_{k}}$ with $p\in[1/m,+\infty)$. Given
$\vec{w}=(w_{1},\ldots,w_{m})$, let us set

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

We say that $\vec{w}$ satisfies the multilinear $A_{\vec{P}}$ condition if it
satisfies

$\sup_{B}\bigg{(}\frac{1}{|B|}\int_{B}\nu_{\vec{w}}(x)\,dx\bigg{)}^{1/p}\prod_{k=1}^{m}\bigg{(}\frac{1}{|B|}\int_{B}w_{k}(x)^{-p^{\prime}_{k}/{p_{k}}}\,dx\bigg{)}^{1/{p^{\prime}_{k}}}<+\infty.$
(3.3)

When $p_{k}=1$ for some $k\in\\{1,2,\dots,m\\}$, the condition
$\big{(}\frac{1}{|B|}\int_{B}w_{k}(x)^{-p^{\prime}_{k}/{p_{k}}}\,dx\big{)}^{1/{p^{\prime}_{k}}}$
is understood as $\big{(}\inf_{x\in B}w_{k}(x)\big{)}^{-1}$. In particular,
when each $p_{k}=1$, $k=1,2,\dots,m$, we denote $A_{\vec{1}}=A_{(1,\dots,1)}$.
One can easily check that $A_{(1,\dots,1)}$ is contained in $A_{\vec{P}}$ for
each $\vec{P}$, however, the classes $A_{\vec{P}}$ are NOT increasing with the
natural partial order (see [11, Remark 7.3]). It was shown in [11] that these
are the largest classes of weights for which all multilinear Calderón–Zygmund
operators are bounded on weighted Lebesgue spaces. Moreover, in general, the
condition $\vec{w}\in A_{\vec{P}}$ does not imply $w_{k}\in
L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$ for any $1\leq k\leq m$ (see [11, Remark
7.2]), but instead

######  Lemma 3.1 ([11]).

Let $p_{1},\ldots,p_{m}\in[1,+\infty)$ and $1/p=\sum_{k=1}^{m}1/{p_{k}}$. Then
$\vec{w}=(w_{1},\ldots,w_{m})\in A_{\vec{P}}$ if and only if

$\left\\{\begin{aligned} &\nu_{\vec{w}}\in A_{mp},\\\
&w_{k}^{1-p^{\prime}_{k}}\in A_{mp^{\prime}_{k}},\quad
k=1,\ldots,m,\end{aligned}\right.$ (3.4)

where $\nu_{\vec{w}}=\prod_{k=1}^{m}w_{k}^{p/{p_{k}}}$ and the condition
$w_{k}^{1-p^{\prime}_{k}}\in A_{mp^{\prime}_{k}}$ in the case $p_{k}=1$ is
understood as $w_{k}^{1/m}\in A_{1}$.

Observe that in the linear case $m=1$ both conditions included in (3.4)
represent the same $A_{p}$ condition. However, in the multilinear case $m\geq
2$ neither of the conditions in (3.4) implies the other. We refer the reader
to [11] for further details.

###  3.2 Orlicz spaces and Luxemburg norms

Next we recall some basic definitions and facts from the theory of Orlicz
spaces. For more information about these spaces the reader may consult the
book [20]. Let $\mathcal{A}:[0,+\infty)\rightarrow[0,+\infty)$ be a Young
function. That is, a continuous, convex and strictly increasing function with
$\mathcal{A}(0)=0$ and such that $\mathcal{A}(t)\to+\infty$ as $t\to+\infty$.
Given a Young function $\mathcal{A}$ and a ball $B$ in $\mathbb{R}^{n}$, we
consider the $\mathcal{A}$-average of a function $f$ over a ball $B$, which is
given by the following Luxemburg norm:

$\big{\|}f\big{\|}_{\mathcal{A},B}:=\inf\bigg{\\{}\lambda>0:\frac{1}{|B|}\int_{B}\mathcal{A}\bigg{(}\frac{|f(x)|}{\lambda}\bigg{)}dx\leq
1\bigg{\\}}.$

When $\mathcal{A}(t)=t^{p}$ with $1\leq p<+\infty$, it is easy to see that

$\big{\|}f\big{\|}_{\mathcal{A},B}=\bigg{(}\frac{1}{|B|}\int_{B}\big{|}f(x)\big{|}^{p}\,dx\bigg{)}^{1/p};$

that is, the Luxemburg norm coincides with the normalized $L^{p}$ norm.
Associated to each Young function $\mathcal{A}$, one can define its
complementary function $\bar{\mathcal{A}}$ by

$\bar{\mathcal{A}}(s):=\sup_{0\leq
t<+\infty}\big{[}st-\mathcal{A}(t)\big{]},\quad 0\leq s<+\infty.$

It is not difficult to check that such $\bar{\mathcal{A}}$ is also a Young
function. A standard computation shows that for all $t>0$,

$t\leq\mathcal{A}^{-1}(t)\bar{\mathcal{A}}^{-1}(t)\leq 2t.$

From this, it follows that the following generalized Hölder’s inequality in
Orlicz spaces holds for any given ball $B$ in $\mathbb{R}^{n}$.

$\frac{1}{|B|}\int_{B}\big{|}f(x)\cdot g(x)\big{|}\,dx\leq
2\big{\|}f\big{\|}_{\mathcal{A},B}\big{\|}g\big{\|}_{\bar{\mathcal{A}},B}.$

A particular case of interest, and especially in this paper, is the Young
function $\Phi(t)=t\cdot(1+\log^{+}t)$, and we know that its complementary
Young function is given by $\bar{\Phi}(t)\approx\exp(t)-1$. The corresponding
averages will be denoted by

$\big{\|}f\big{\|}_{\Phi,B}=\big{\|}f\big{\|}_{L\log
L,B}\qquad\mbox{and}\qquad\big{\|}g\big{\|}_{\bar{\Phi},B}=\big{\|}g\big{\|}_{\exp
L,B}.$

Consequently, from the above generalized Hölder’s inequality in Orlicz spaces,
we also get

$\frac{1}{|B|}\int_{B}\big{|}f(x)\cdot g(x)\big{|}\,dx\leq
2\big{\|}f\big{\|}_{L\log L,B}\big{\|}g\big{\|}_{\exp L,B}.$ (3.5)

To obtain endpoint weak-type estimates for the multilinear and iterated
commutators on the product of weighted Morrey spaces, we need to define the
$\mathcal{A}$-average of a function $f$ over a ball $B$ by means of the
weighted Luxemburg norm; that is, given a Young function $\mathcal{A}$ and
$w\in A_{\infty}$, we define (see [20, 28])

$\big{\|}f\big{\|}_{\mathcal{A}(w),B}:=\inf\bigg{\\{}\sigma>0:\frac{1}{w(B)}\int_{B}\mathcal{A}\bigg{(}\frac{|f(x)|}{\sigma}\bigg{)}\cdot
w(x)\,dx\leq 1\bigg{\\}}.$

When $\mathcal{A}(t)=t$, this norm is denoted by $\|\cdot\|_{L(w),B}$, when
$\Phi(t)=t\cdot(1+\log^{+}t)$, this norm is also denoted by $\|\cdot\|_{L\log
L(w),B}$. The complementary Young function of $\Phi(t)$ is
$\bar{\Phi}(t)\approx\exp(t)-1$ with the corresponding Luxemburg norm denoted
by $\|\cdot\|_{\exp L(w),B}$. For $w\in A_{\infty}$ and for every ball $B$ in
$\mathbb{R}^{n}$, we can also show the weighted version of (3.5). Namely, the
following generalized Hölder’s inequality in the weighted context is true for
$f,g$ (see [28] for instance).

$\frac{1}{w(B)}\int_{B}\big{|}f(x)\cdot g(x)\big{|}w(x)\,dx\leq
C\big{\|}f\big{\|}_{L\log L(w),B}\big{\|}g\big{\|}_{\exp L(w),B}.$ (3.6)

This estimate will play an important role in the proof of Theorem 2.4.

##  4 Proofs of Theorems 2.1 and 2.2

This section is concerned with the proofs of Theorems 2.1 and 2.2. Before
proving the main theorems of this section, we first state the following
important results without proof (see [4] and [6]).

######  Lemma 4.1 ([6]).

Let $\big{\\{}f_{k}\big{\\}}_{k=1}^{N}$ be a sequence of $L^{p}(\nu)$
functions with $0<p<+\infty$ and $\nu\in A_{\infty}$. Then we have

$\Big{\|}\sum_{k=1}^{N}f_{k}\Big{\|}_{L^{p}(\nu)}\leq\mathcal{C}(p,N)\sum_{k=1}^{N}\big{\|}f_{k}\big{\|}_{L^{p}(\nu)},$

where $\mathcal{C}(p,N)=\max\big{\\{}1,N^{\frac{1-p}{p}}\big{\\}}$. More
specifically, $\mathcal{C}(p,N)=1$ for $1\leq p<+\infty$, and
$\mathcal{C}(p,N)=N^{\frac{1-p}{p}}$ for $0<p<1$.

######  Lemma 4.2 ([6]).

Let $\big{\\{}f_{k}\big{\\}}_{k=1}^{N}$ be a sequence of $WL^{p}(\nu)$
functions with $0<p<+\infty$ and $\nu\in A_{\infty}$. Then we have

$\Big{\|}\sum_{k=1}^{N}f_{k}\Big{\|}_{WL^{p}(\nu)}\leq\mathcal{C}^{\prime}(p,N)\sum_{k=1}^{N}\big{\|}f_{k}\big{\|}_{WL^{p}(\nu)},$

where $\mathcal{C}^{\prime}(p,N)=\max\big{\\{}N,N^{\frac{\,1\,}{p}}\big{\\}}$.
More specifically, $\mathcal{C}^{\prime}(p,N)=N$ for $1\leq p<+\infty$, and
$\mathcal{C}^{\prime}(p,N)=N^{\frac{\,1\,}{p}}$ for $0<p<1$.

######  Lemma 4.3 ([4]).

Let $w\in A_{\infty}$. Then for any ball $B$ in $\mathbb{R}^{n}$, the
following reverse Jensen’s inequality holds.

$\int_{B}w(x)\,dx\leq C|B|\cdot\exp\bigg{(}\frac{1}{|B|}\int_{B}\log
w(x)\,dx\bigg{)}.$

We are now in a position to prove Theorems 2.1 and 2.2.

###### Proof of Theorem 2.1.

Let $1<p_{1},\dots,p_{m}<+\infty$ and $\vec{f}=(f_{1},\dots,f_{m})$ be in
$L^{p_{1},\kappa}(w_{1})\times\cdots\times L^{p_{m},\kappa}(w_{m})$ with
$\vec{w}=(w_{1},\dots,w_{m})\in A_{\vec{P}}$ and $0<\kappa<1$. For any given
ball $B$ in $\mathbb{R}^{n}$(denote by $x_{0}$ the center of $B$, and $r>0$
the radius of $B$), it is enough for us to show that

$\frac{1}{\nu_{\vec{w}}(B)^{\kappa/p}}\bigg{(}\int_{B}\big{|}T_{\theta}(f_{1},\dots,f_{m})(x)\big{|}^{p}\nu_{\vec{w}}(x)\,dx\bigg{)}^{1/p}\lesssim\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}.$
(4.1)

To this end, for any $1\leq i\leq m$, we represent $f_{i}$ as

$f_{i}=f_{i}\cdot\chi_{2B}+f_{i}\cdot\chi_{(2B)^{\complement}}:=f^{0}_{i}+f^{\infty}_{i};$

and $2B=B(x_{0},2r)$. Then we write

$\begin{split}\prod_{i=1}^{m}f_{i}(y_{i})&=\prod_{i=1}^{m}\Big{(}f^{0}_{i}(y_{i})+f^{\infty}_{i}(y_{i})\Big{)}\\\
&=\sum_{\beta_{1},\ldots,\beta_{m}\in\\{0,\infty\\}}f^{\beta_{1}}_{1}(y_{1})\cdots
f^{\beta_{m}}_{m}(y_{m})\\\
&=\prod_{i=1}^{m}f^{0}_{i}(y_{i})+\sum_{(\beta_{1},\dots,\beta_{m})\in\mathfrak{L}}f^{\beta_{1}}_{1}(y_{1})\cdots
f^{\beta_{m}}_{m}(y_{m}),\end{split}$

where

$\mathfrak{L}:=\big{\\{}(\beta_{1},\dots,\beta_{m}):\beta_{k}\in\\{0,\infty\\},\mbox{there
is at least one $\beta_{k}\neq 0$},1\leq k\leq m\big{\\}};$

that is, each term of $\sum$ contains at least one $\beta_{k}\neq 0$. Since
$T_{\theta}$ is an $m$-linear operator, then by Lemma 4.1($N=2^{m}$), we have

$\displaystyle\frac{1}{\nu_{\vec{w}}(B)^{\kappa/p}}\bigg{(}\int_{B}\big{|}T_{\theta}(f_{1},\dots,f_{m})(x)\big{|}^{p}\nu_{\vec{w}}(x)\,dx\bigg{)}^{1/p}$
$\displaystyle\leq\frac{C}{\nu_{\vec{w}}(B)^{\kappa/p}}\bigg{(}\int_{B}\big{|}T_{\theta}(f^{0}_{1},\dots,f^{0}_{m})(x)\big{|}^{p}\nu_{\vec{w}}(x)\,dx\bigg{)}^{1/p}$
$\displaystyle+\sum_{(\beta_{1},\dots,\beta_{m})\in\mathfrak{L}}\frac{C}{\nu_{\vec{w}}(B)^{\kappa/p}}\bigg{(}\int_{B}\big{|}T_{\theta}(f^{\beta_{1}}_{1},\ldots,f^{\beta_{m}}_{m})(x)\big{|}^{p}\nu_{\vec{w}}(x)\,dx\bigg{)}^{1/p}$
$\displaystyle:=I^{0,\dots,0}+\sum_{(\beta_{1},\dots,\beta_{m})\in\mathfrak{L}}I^{\beta_{1},\dots,\beta_{m}}.$
(4.2)

By the weighted strong-type estimate of $T_{\theta}$ (see Theorem 1.1), we
have

$\displaystyle I^{0,\dots,0}$ $\displaystyle\leq
C\cdot\frac{1}{\nu_{\vec{w}}(B)^{\kappa/p}}\prod_{i=1}^{m}\bigg{(}\int_{2B}|f_{i}(x)|^{p_{i}}w_{i}(x)\,dx\bigg{)}^{1/{p_{i}}}$
$\displaystyle\leq
C\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}\cdot\frac{1}{\nu_{\vec{w}}(B)^{\kappa/p}}\prod_{i=1}^{m}w_{i}(2B)^{\kappa/{p_{i}}}.$
(4.3)

Let $p_{1},\ldots,p_{m}\in[1,+\infty)$ and $p\in[1/m,+\infty)$ with
$1/p=\sum_{i=1}^{m}1/{p_{i}}$. We first claim that under the assumptions of
Theorem 2.1 (or Theorem 2.2), the following result holds for any ball
$\mathcal{B}$ in $\mathbb{R}^{n}$:

$\prod_{i=1}^{m}\bigg{(}\int_{\mathcal{B}}w_{i}(x)\,dx\bigg{)}^{p/{p_{i}}}\lesssim\int_{\mathcal{B}}\nu_{\vec{w}}(x)\,dx,$
(4.4)

provided that $w_{1},\ldots,w_{m}\in A_{\infty}$ and
$\nu_{\vec{w}}=\prod_{i=1}^{m}w_{i}^{p/{p_{i}}}$. In fact, since
$w_{1},\ldots,w_{m}\in A_{\infty}$, by using Lemma 4.3, then we have

$\begin{split}\prod_{i=1}^{m}\bigg{(}\int_{\mathcal{B}}w_{i}(x)\,dx\bigg{)}^{p/{p_{i}}}&\leq
C\prod_{i=1}^{m}\bigg{[}|\mathcal{B}|\cdot\exp\bigg{(}\frac{1}{|\mathcal{B}|}\int_{\mathcal{B}}\log
w_{i}(x)\,dx\bigg{)}\bigg{]}^{p/{p_{i}}}\\\
&=C\prod_{i=1}^{m}\bigg{[}|\mathcal{B}|^{p/{p_{i}}}\cdot\exp\bigg{(}\frac{1}{|\mathcal{B}|}\int_{\mathcal{B}}\log
w_{i}(x)^{p/{p_{i}}}\,dx\bigg{)}\bigg{]}\\\
&=C\cdot\big{(}|\mathcal{B}|\big{)}^{\sum_{i=1}^{m}p/{p_{i}}}\cdot\exp\bigg{(}\sum_{i=1}^{m}\frac{1}{|\mathcal{B}|}\int_{\mathcal{B}}\log
w_{i}(x)^{p/{p_{i}}}\,dx\bigg{)}.\end{split}$

Note that

$\sum_{i=1}^{m}p/{p_{i}}=1\quad\mathrm{and}\quad\nu_{\vec{w}}(x)=\prod_{i=1}^{m}w_{i}(x)^{p/{p_{i}}}.$

Thus, by Jensen’s inequality, we obtain

$\begin{split}\prod_{i=1}^{m}\bigg{(}\int_{\mathcal{B}}w_{i}(x)\,dx\bigg{)}^{p/{p_{i}}}&\leq
C\cdot|\mathcal{B}|\cdot\exp\bigg{(}\frac{1}{|\mathcal{B}|}\int_{\mathcal{B}}\log\nu_{\vec{w}}(x)\,dx\bigg{)}\\\
&\leq C\int_{\mathcal{B}}\nu_{\vec{w}}(x)\,dx.\end{split}$

This gives (4.4). Moreover, in view of Lemma 3.1, we have that
$\nu_{\vec{w}}\in A_{mp}$ with $1/m<p<+\infty$. This fact, together with (4.4)
and (3.2), implies that

$\begin{split}I^{0,\dots,0}&\leq
C\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}\cdot\frac{\nu_{\vec{w}}(2B)^{\kappa/p}}{\nu_{\vec{w}}(B)^{\kappa/p}}\\\
&\leq
C\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}.\end{split}$
(4.5)

To estimate the remaining terms in (4), let us first consider the case when
$\beta_{1}=\cdots=\beta_{m}=\infty$. By a simple geometric observation, we
know that

$\overbrace{\big{(}\mathbb{R}^{n}\backslash
2B\big{)}\times\cdots\times\big{(}\mathbb{R}^{n}\backslash
2B\big{)}}^{m}\subset(\mathbb{R}^{n})^{m}\backslash(2B)^{m},$

and

$(\mathbb{R}^{n})^{m}\backslash(2B)^{m}=\bigcup_{j=1}^{\infty}(2^{j+1}B)^{m}\backslash(2^{j}B)^{m},$

where we have used the notation $E^{m}=\overbrace{E\times\cdots\times E}^{m}$
for a measurable set $E$ and a positive integer $m$. By the size condition
(1.2) of the $\theta$-type Calderón–Zygmund kernel $K$, for any $x\in B$, we
obtain

$\displaystyle\big{|}T_{\theta}(f^{\infty}_{1},\ldots,f^{\infty}_{m})(x)\big{|}$
$\displaystyle\lesssim\int_{(\mathbb{R}^{n})^{m}\backslash(2B)^{m}}\frac{|f_{1}(y_{1})\cdots
f_{m}(y_{m})|}{(|x-y_{1}|+\cdots+|x-y_{m}|)^{mn}}\,dy_{1}\cdots dy_{m}$
$\displaystyle=\sum_{j=1}^{\infty}\int_{(2^{j+1}B)^{m}\backslash(2^{j}B)^{m}}\frac{|f_{1}(y_{1})\cdots
f_{m}(y_{m})|}{(|x-y_{1}|+\cdots+|x-y_{m}|)^{mn}}\,dy_{1}\cdots dy_{m}$
$\displaystyle\lesssim\sum_{j=1}^{\infty}\bigg{(}\frac{1}{|2^{j+1}B|^{m}}\int_{(2^{j+1}B)^{m}\backslash(2^{j}B)^{m}}\big{|}f_{1}(y_{1})\cdots
f_{m}(y_{m})\big{|}\,dy_{1}\cdots dy_{m}\bigg{)}$
$\displaystyle\leq\sum_{j=1}^{\infty}\bigg{(}\frac{1}{|2^{j+1}B|^{m}}\prod_{i=1}^{m}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}\bigg{)}$
$\displaystyle=\sum_{j=1}^{\infty}\bigg{(}\prod_{i=1}^{m}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}\bigg{)},$
(4.6)

where we have used the fact that $|x-y_{1}|+\cdots+|x-y_{m}|\approx
2^{j+1}r\approx|2^{j+1}B|^{1/n}$ when $x\in B$ and
$(y_{1},\dots,y_{m})\in(2^{j+1}B)^{m}\backslash(2^{j}B)^{m}$. Furthermore, by
using Hölder’s inequality, the multiple $A_{\vec{P}}$ condition on $\vec{w}$,
we can deduce that

$\begin{split}&\big{|}T_{\theta}(f^{\infty}_{1},\ldots,f^{\infty}_{m})(x)\big{|}\\\
\lesssim&\sum_{j=1}^{\infty}\bigg{\\{}\prod_{i=1}^{m}\frac{1}{|2^{j+1}B|}\bigg{(}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}^{p_{i}}w_{i}(y_{i})\,dy_{i}\bigg{)}^{1/{p_{i}}}\bigg{(}\int_{2^{j+1}B}w_{i}(y_{i})^{-p^{\prime}_{i}/{p_{i}}}\,dy_{i}\bigg{)}^{1/{p^{\prime}_{i}}}\bigg{\\}}\\\
\leq&\sum_{j=1}^{\infty}\bigg{\\{}\frac{1}{|2^{j+1}B|^{m}}\cdot\frac{|2^{j+1}B|^{1/p+\sum_{i=1}^{m}(1-1/{p_{i}})}}{\nu_{\vec{w}}(2^{j+1}B)^{1/p}}\prod_{i=1}^{m}\bigg{(}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}w_{i}(2^{j+1}B)^{\kappa/{p_{i}}}\bigg{)}\bigg{\\}}\\\
=&\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}\sum_{j=1}^{\infty}\bigg{\\{}\frac{1}{\nu_{\vec{w}}(2^{j+1}B)^{1/p}}\cdot\prod_{i=1}^{m}w_{i}(2^{j+1}B)^{\kappa/{p_{i}}}\bigg{\\}},\end{split}$

where in the last step we have used the fact that
$1/p+\sum_{i=1}^{m}(1-1/{p_{i}})=m$. Hence, from the above pointwise estimate
and (4.4), we obtain

$\begin{split}I^{\infty,\dots,\infty}&\lesssim\frac{\nu_{\vec{w}}(B)^{1/p}}{\nu_{\vec{w}}(B)^{\kappa/p}}\cdot\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}\sum_{j=1}^{\infty}\frac{\nu_{\vec{w}}(2^{j+1}B)^{\kappa/p}}{\nu_{\vec{w}}(2^{j+1}B)^{1/p}}\\\
&=\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}\sum_{j=1}^{\infty}\frac{\nu_{\vec{w}}(B)^{{(1-\kappa)}/p}}{\nu_{\vec{w}}(2^{j+1}B)^{{(1-\kappa)}/p}}.\end{split}$

Since $\nu_{\vec{w}}\in A_{mp}\subset A_{\infty}$ by Lemma 3.1, then it
follows directly from the inequality (3.1) with exponent $\delta>0$ that

$\frac{\nu_{\vec{w}}(B)}{\nu_{\vec{w}}(2^{j+1}B)}\lesssim\bigg{(}\frac{|B|}{|2^{j+1}B|}\bigg{)}^{\delta},$
(4.7)

which further implies

$\begin{split}I^{\infty,\dots,\infty}&\lesssim\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}\sum_{j=1}^{\infty}\bigg{(}\frac{|B|}{|2^{j+1}B|}\bigg{)}^{{\delta(1-\kappa)}/p}\\\
&\lesssim\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})},\end{split}$
(4.8)

where in the last estimate we have used the fact that $0<\kappa<1$ and
$\delta>0$. We now consider the case where exactly $\ell$ of the $\beta_{i}$
are $\infty$ for some $1\leq\ell<m$. We only give the arguments for one of
these cases. The rest are similar and can be easily obtained from the
arguments below by permuting the indices. In this case, by the same reason as
above, we also have

$\overbrace{\big{(}\mathbb{R}^{n}\backslash
2B\big{)}\times\cdots\times\big{(}\mathbb{R}^{n}\backslash
2B\big{)}}^{\ell}\subset(\mathbb{R}^{n})^{\ell}\backslash(2B)^{\ell},$

and

$(\mathbb{R}^{n})^{\ell}\backslash(2B)^{\ell}=\bigcup_{j=1}^{\infty}(2^{j+1}B)^{\ell}\backslash(2^{j}B)^{\ell},\quad
1\leq\ell<m.$

Using the size condition (1.2) again, we deduce that for any $x\in B$,

$\displaystyle\big{|}T_{\theta}(f^{\infty}_{1},\ldots,f^{\infty}_{\ell},f^{0}_{\ell+1},\ldots,f^{0}_{m})(x)\big{|}$
$\displaystyle\lesssim\int_{(\mathbb{R}^{n})^{\ell}\backslash(2B)^{\ell}}\int_{(2B)^{m-\ell}}\frac{|f_{1}(y_{1})\cdots
f_{m}(y_{m})|}{(|x-y_{1}|+\cdots+|x-y_{m}|)^{mn}}dy_{1}\cdots dy_{m}$
$\displaystyle\lesssim\prod_{i=\ell+1}^{m}\int_{2B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}\times\sum_{j=1}^{\infty}\frac{1}{|2^{j+1}B|^{m}}\int_{(2^{j+1}B)^{\ell}\backslash(2^{j}B)^{\ell}}\big{|}f_{1}(y_{1})\cdots
f_{\ell}(y_{\ell})\big{|}\,dy_{1}\cdots dy_{\ell}$
$\displaystyle\leq\prod_{i=\ell+1}^{m}\int_{2B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}\times\sum_{j=1}^{\infty}\frac{1}{|2^{j+1}B|^{m}}\prod_{i=1}^{\ell}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}$
$\displaystyle\leq\sum_{j=1}^{\infty}\bigg{(}\prod_{i=1}^{m}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}\bigg{)},$
(4.9)

where in the last inequality we have used the inclusion relation $2B\subseteq
2^{j+1}B$ with $j\in\mathbb{N}$, and hence we arrive at the same expression
considered in the previous case. Hence, we can now argue exactly as we did in
the estimation of $I^{\infty,\dots,\infty}$ to obtain that for all $m$-tuples
$(\beta_{1},\dots,\beta_{m})\in\mathfrak{L}$,

$\displaystyle I^{\beta_{1},\dots,\beta_{m}}$
$\displaystyle\lesssim\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}\sum_{j=1}^{\infty}\frac{\nu_{\vec{w}}(B)^{{(1-\kappa)}/p}}{\nu_{\vec{w}}(2^{j+1}B)^{{(1-\kappa)}/p}}$
$\displaystyle\lesssim\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}\sum_{j=1}^{\infty}\bigg{(}\frac{|B|}{|2^{j+1}B|}\bigg{)}^{{\delta(1-\kappa)}/p}$
$\displaystyle\lesssim\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}.$
(4.10)

Combining these estimates (4.5), (4.8) and (4), then (4.1) holds and concludes
the proof of the theorem. ∎

###### Proof of Theorem 2.2.

Let $1\leq p_{1},\dots,p_{m}<+\infty$, $\min\\{p_{1},\ldots,p_{m}\\}=1$ and
$\vec{f}=(f_{1},\dots,f_{m})$ be in $L^{p_{1},\kappa}(w_{1})\times\cdots\times
L^{p_{m},\kappa}(w_{m})$ with $\vec{w}=(w_{1},\dots,w_{m})\in A_{\vec{P}}$ and
$0<\kappa<1$. For an arbitrary ball $B=B(x_{0},r)\subset\mathbb{R}^{n}$ with
$x_{0}\in\mathbb{R}^{n}$ and $r>0$, we need to show that the following
estimate holds.

$\frac{1}{\nu_{\vec{w}}(B)^{\kappa/p}}\lambda\cdot\nu_{\vec{w}}\big{(}\big{\\{}x\in
B:\big{|}T_{\theta}(f_{1},\dots,f_{m})\big{|}>\lambda\big{\\}}\big{)}^{1/p}\lesssim\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}.$
(4.11)

To this end, we represent $f_{i}$ as

$f_{i}=f_{i}\cdot\chi_{2B}+f_{i}\cdot\chi_{(2B)^{\complement}}:=f^{0}_{i}+f^{\infty}_{i},\quad\mbox{for}~{}~{}i=1,2,\dots,m.$

By using Lemma 4.2($N=2^{m}$), one can write

$\displaystyle\frac{1}{\nu_{\vec{w}}(B)^{\kappa/p}}\lambda\cdot\nu_{\vec{w}}\big{(}\big{\\{}x\in
B:\big{|}T_{\theta}(f_{1},\dots,f_{m})\big{|}>\lambda\big{\\}}\big{)}^{1/p}$
$\displaystyle\leq\frac{C}{\nu_{\vec{w}}(B)^{\kappa/p}}\lambda\cdot\nu_{\vec{w}}\big{(}\big{\\{}x\in
B:\big{|}T_{\theta}(f^{0}_{1},\dots,f^{0}_{m})\big{|}>\lambda/{2^{m}}\big{\\}}\big{)}^{1/p}$
$\displaystyle+\sum_{(\beta_{1},\dots,\beta_{m})\in\mathfrak{L}}\frac{C}{\nu_{\vec{w}}(B)^{\kappa/p}}\lambda\cdot\nu_{\vec{w}}\big{(}\big{\\{}x\in
B:\big{|}T_{\theta}(f^{\beta_{1}}_{1},\dots,f^{\beta_{m}}_{m})\big{|}>\lambda/{2^{m}}\big{\\}}\big{)}^{1/p}$
$\displaystyle:=I^{0,\dots,0}_{\ast}+\sum_{(\beta_{1},\dots,\beta_{m})\in\mathfrak{L}}I^{\beta_{1},\dots,\beta_{m}}_{\ast},$
(4.12)

where

$\mathfrak{L}=\big{\\{}(\beta_{1},\dots,\beta_{m}):\beta_{k}\in\\{0,\infty\\},\mbox{there
is at least one $\beta_{k}\neq 0$},1\leq k\leq m\big{\\}}.$

By the weighted weak-type estimate of $T_{\theta}$ (see Theorem 1.2), we can
estimate the first term on the right hand side of (4) as follows.

$\displaystyle I^{0,\dots,0}_{\ast}$ $\displaystyle\leq
C\cdot\frac{1}{\nu_{\vec{w}}(B)^{\kappa/p}}\prod_{i=1}^{m}\bigg{(}\int_{2B}|f_{i}(x)|^{p_{i}}w_{i}(x)\,dx\bigg{)}^{1/{p_{i}}}$
$\displaystyle\leq
C\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}\frac{1}{\nu_{\vec{w}}(B)^{\kappa/p}}\cdot\prod_{i=1}^{m}w_{i}(2B)^{\kappa/{p_{i}}}.$
(4.13)

Moreover, in view of Lemma 3.1 again, we also have $\nu_{\vec{w}}\in A_{mp}$
with $1/m\leq p<+\infty$. Then we apply the inequalities (3.2) and (4.4) to
obtain that

$\begin{split}I^{0,\dots,0}_{\ast}&\leq
C\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}\frac{\nu_{\vec{w}}(2B)^{\kappa/p}}{\nu_{\vec{w}}(B)^{\kappa/p}}\\\
&\leq
C\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}.\end{split}$
(4.14)

In the proof of Theorem 2.1, we have already showed the following pointwise
estimate for all $m$-tuples $(\beta_{1},\dots,\beta_{m})\in\mathfrak{L}$ (see
(4) and (4)).

$\begin{split}&\big{|}T_{\theta}(f^{\beta_{1}}_{1},\ldots,f^{\beta_{m}}_{m})(x)\big{|}\lesssim\sum_{j=1}^{\infty}\bigg{(}\prod_{i=1}^{m}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}\bigg{)}.\end{split}$
(4.15)

Without loss of generality, we may assume that

$p_{1}=\cdots=p_{\ell}=\min\\{p_{1},\ldots,p_{m}\\}=1\quad\mathrm{and}\quad
p_{\ell+1},\ldots,p_{m}>1$

with $1\leq\ell<m$. The case that $p_{1}=\cdots=p_{m}=1$ can be dealt with
quite similarly and more easily. Using Hölder’s inequality, the multiple
$A_{\vec{P}}$ condition on $\vec{w}$, we obtain that for any $x\in B$,

$\begin{split}&\big{|}T_{\theta}(f^{\beta_{1}}_{1},\ldots,f^{\beta_{m}}_{m})(x)\big{|}\\\
&\lesssim\sum_{j=1}^{\infty}\bigg{(}\prod_{i=1}^{\ell}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}\bigg{)}\times\bigg{(}\prod_{i=\ell+1}^{m}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}\bigg{)}\\\
&\lesssim\sum_{j=1}^{\infty}\prod_{i=1}^{\ell}\frac{1}{|2^{j+1}B|}\bigg{(}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}w_{i}(y_{i})\,dy_{i}\bigg{)}\bigg{(}\inf_{y_{i}\in
2^{j+1}B}w_{i}(y_{i})\bigg{)}^{-1}\\\
&\times\prod_{i=\ell+1}^{m}\frac{1}{|2^{j+1}B|}\bigg{(}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}^{p_{i}}w_{i}(y_{i})\,dy_{i}\bigg{)}^{1/{p_{i}}}\bigg{(}\int_{2^{j+1}B}w_{i}(y_{i})^{-p^{\prime}_{i}/{p_{i}}}\,dy_{i}\bigg{)}^{1/{p^{\prime}_{i}}}\\\
&\lesssim\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}\sum_{j=1}^{\infty}\bigg{\\{}\frac{1}{\nu_{\vec{w}}(2^{j+1}B)^{1/p}}\cdot\prod_{i=1}^{m}w_{i}(2^{j+1}B)^{\kappa/{p_{i}}}\bigg{\\}}\\\
&\lesssim\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}\sum_{j=1}^{\infty}\frac{1}{\nu_{\vec{w}}(2^{j+1}B)^{{(1-\kappa)}/p}},\end{split}$

where in the last inequality we have invoked (4.4). Observe that
$\nu_{\vec{w}}\in A_{mp}$ with $1\leq mp<\infty$. Thus, it follows directly
from Chebyshev’s inequality and the above pointwise estimate that

$\displaystyle I^{\beta_{1},\dots,\beta_{m}}_{\ast}$
$\displaystyle\leq\cdot\frac{C}{\nu_{\vec{w}}(B)^{\kappa/p}}\bigg{(}\int_{B}\big{|}T_{\theta}(f^{\beta_{1}}_{1},\ldots,f^{\beta_{m}}_{m})(x)\big{|}^{p}\nu_{\vec{w}}(x)\,dx\bigg{)}^{1/p}$
$\displaystyle\leq
C\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}\sum_{j=1}^{\infty}\frac{\nu_{\vec{w}}(B)^{{(1-\kappa)}/p}}{\nu_{\vec{w}}(2^{j+1}B)^{{(1-\kappa)}/p}}.$

Moreover, in view of (4.7), we obtain that for all $m$-tuples
$(\beta_{1},\dots,\beta_{m})\in\mathfrak{L}$,

$\begin{split}I^{\beta_{1},\dots,\beta_{m}}_{\ast}&\lesssim\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}\sum_{j=1}^{\infty}\bigg{(}\frac{|B|}{|2^{j+1}B|}\bigg{)}^{{\delta(1-\kappa)}/p}\\\
&\lesssim\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})},\end{split}$
(4.16)

where in the last step we have used the fact $\delta>0$ and $0<\kappa<1$.
Putting the estimates (4.14) and (4.16) together produces the required
inequality (4.11). Thus, by taking the supremum over all $\lambda>0$, we
finish the proof of Theorem 2.2. ∎

Let $1\leq p_{1},\dots,p_{m}\leq+\infty$. We say that
$\vec{w}=(w_{1},\ldots,w_{m})\in\prod_{i=1}^{m}A_{p_{i}}$, if each $w_{i}$ is
in $A_{p_{i}}$, $i=1,2,\dots,m$. By using Hölder’s inequality, it is not
difficult to check that

$\prod_{i=1}^{m}A_{p_{i}}\subset A_{\vec{P}}.$

Moreover, it was shown in [11, Remark 7.2] that this inclusion is strict. It
is clear that $\prod_{i=1}^{m}A_{p_{i}}\subset\prod_{i=1}^{m}A_{\infty}$. So
we have

$\prod_{i=1}^{m}A_{p_{i}}\subset A_{\vec{P}}\bigcap\prod_{i=1}^{m}A_{\infty}.$
(4.17)

A natural question appearing here is whether the above inclusion relation is
also strict. Thus, as a direct consequence of Theorems 2.1 and 2.2, we
immediately obtain the following results.

######  Corollary 4.1.

Let $m\geq 2$ and $T_{\theta}$ be an $m$-linear $\theta$-type Calderón–Zygmund
operator with $\theta$ satisfying the condition (1.1). If
$1<p_{1},\dots,p_{m}<+\infty$ and $1/m<p<+\infty$ with
$1/p=\sum_{i=1}^{m}1/{p_{i}}$, and
$\vec{w}=(w_{1},\ldots,w_{m})\in\prod_{i=1}^{m}A_{p_{i}}$, then for any
$0<\kappa<1$, the multilinear operator $T_{\theta}$ is bounded from
$L^{p_{1},\kappa}(w_{1})\times L^{p_{2},\kappa}(w_{2})\times\cdots\times
L^{p_{m},\kappa}(w_{m})$ into $L^{p,\kappa}(\nu_{\vec{w}})$ with
$\nu_{\vec{w}}=\prod_{i=1}^{m}w_{i}^{p/{p_{i}}}$.

######  Corollary 4.2.

Let $m\geq 2$ and $T_{\theta}$ be an $m$-linear $\theta$-type Calderón–Zygmund
operator with $\theta$ satisfying the condition (1.1). If $1\leq
p_{1},\dots,p_{m}<+\infty$, $\min\\{p_{1},\ldots,p_{m}\\}=1$ and $1/m\leq
p<+\infty$ with $1/p=\sum_{i=1}^{m}1/{p_{i}}$, and
$\vec{w}=(w_{1},\ldots,w_{m})\in\prod_{i=1}^{m}A_{p_{i}}$, then for any
$0<\kappa<1$, the multilinear operator $T_{\theta}$ is bounded from
$L^{p_{1},\kappa}(w_{1})\times L^{p_{2},\kappa}(w_{2})\times\cdots\times
L^{p_{m},\kappa}(w_{m})$ into $WL^{p,\kappa}(\nu_{\vec{w}})$ with
$\nu_{\vec{w}}=\prod_{i=1}^{m}w_{i}^{p/{p_{i}}}$.

##  5 Proofs of Theorems 2.3 and 2.4

To prove our main theorems for multilinear commutators in this section, we
need the following lemmas about $\mathrm{BMO}$ functions.

######  Lemma 5.1.

Let $b$ be a function in $\mathrm{BMO}(\mathbb{R}^{n})$. Then

  1. 1.

For every ball $B$ in $\mathbb{R}^{n}$ and for all $j\in\mathbb{N}$,

$\big{|}b_{2^{j+1}B}-b_{B}\big{|}\leq C\cdot(j+1)\|b\|_{*}.$

  2. 2.

Let $1\leq p<+\infty$. For every ball $B$ in $\mathbb{R}^{n}$ and for all
$\omega\in A_{\infty}$,

$\bigg{(}\int_{B}\big{|}b(x)-b_{B}\big{|}^{p}\omega(x)\,dx\bigg{)}^{1/p}\leq
C\|b\|_{*}\cdot\omega(B)^{1/p}.$

###### Proof.

For the proofs of the above results, we refer the reader to [24]. ∎

Based on Lemma 5.1, we now assert that for any $j\in\mathbb{N}$ and $\omega\in
A_{\infty}$, the estimate

$\bigg{(}\int_{2^{j+1}B}\big{|}b(x)-b_{B}\big{|}^{p}\omega(x)\,dx\bigg{)}^{1/p}\leq
C(j+1)\|b\|_{*}\cdot\omega(2^{j+1}B)^{1/p}$ (5.1)

holds whenever $b\in\mathrm{BMO}(\mathbb{R}^{n})$ and $1\leq p<+\infty$.
Indeed, by using Lemma 5.1 (1) and (2), we could easily obtain

$\begin{split}&\bigg{(}\int_{2^{j+1}B}\big{|}b(x)-b_{B}\big{|}^{p}\omega(x)\,dx\bigg{)}^{1/p}\\\
&\leq\bigg{(}\int_{2^{j+1}B}\big{|}b(x)-b_{2^{j+1}B}\big{|}^{p}\omega(x)\,dx\bigg{)}^{1/p}+\bigg{(}\int_{2^{j+1}B}\big{|}b_{2^{j+1}B}-b_{B}\big{|}^{p}\omega(x)\,dx\bigg{)}^{1/p}\\\
&\leq
C\|b\|_{*}\cdot\omega(2^{j+1}B)^{1/p}+C(j+1)\|b\|_{*}\cdot\omega(2^{j+1}B)^{1/p}\\\
&\leq C(j+1)\|b\|_{*}\cdot\omega(2^{j+1}B)^{1/p},\end{split}$

as desired. Next, let us set up the following result.

######  Lemma 5.2.

Let $b$ be a function in $\mathrm{BMO}(\mathbb{R}^{n})$. Then for any ball $B$
in $\mathbb{R}^{n}$ and any $\omega\in A_{\infty}$, we have

$\big{\|}b-b_{B}\big{\|}_{\exp L(\omega),B}\leq C\|b\|_{*}.$ (5.2)

###### Proof.

By the well-known John–Nirenberg’s inequality (see [9]), we know that there
exist two positive constants $C_{1}$ and $C_{2}$, depending only on the
dimension $n$, such that for any $\lambda>0$,

$\big{|}\big{\\{}x\in B:|b(x)-b_{B}|>\lambda\big{\\}}\big{|}\leq
C_{1}|B|\exp\bigg{\\{}-\frac{C_{2}\lambda}{\|b\|_{*}}\bigg{\\}}.$

This result shows that in some sense logarithmic growth is the maximum
possible for BMO functions (more precisely, we can take $C_{1}=\sqrt{2}$,
$C_{2}=\log 2/{2^{n+2}}$, see [4, p.123–125]). Applying the comparison
property (3.1) of $A_{\infty}$ weights, there is a positive number $\delta>0$
such that

$\omega\big{(}\big{\\{}x\in B:|b(x)-b_{B}|>\lambda\big{\\}}\big{)}\leq
C_{1}\omega(B)\exp\bigg{\\{}-\frac{C_{2}\delta\lambda}{\|b\|_{*}}\bigg{\\}}.$

From this, it follows that ($c_{0}$ and $C$ are two constants)

$\frac{1}{\omega(B)}\int_{B}\exp\bigg{(}\frac{|b(y)-b_{B}|}{c_{0}\|b\|_{*}}\bigg{)}\omega(y)\,dy\leq
C,$

which yields (5.2). ∎

Furthermore, by (5.2) and Lemma 5.1(1), it is easy to check that for each
$\omega$ in $A_{\infty}$ and for any ball $B$ in $\mathbb{R}^{n}$,

$\big{\|}b-b_{B}\big{\|}_{\exp L(\omega),2^{j+1}B}\leq
C(j+1)\|b\|_{\ast},\quad j\in\mathbb{N}.$ (5.3)

We are now in a position to give the proofs of Theorems 2.3 and 2.4.

###### Proof of Theorem 2.3.

Let $1<p_{1},\dots,p_{m}<+\infty$ and $\vec{f}=(f_{1},\dots,f_{m})$ be in
$L^{p_{1},\kappa}(w_{1})\times\cdots\times L^{p_{m},\kappa}(w_{m})$ with
$\vec{w}=(w_{1},\dots,w_{m})\in A_{\vec{P}}$ and $0<\kappa<1$. As was pointed
out in [11], by linearity it is enough to consider the multilinear commutator
$[\Sigma b,T_{\theta}]$ with only one symbol. Without loss of generality, we
fix $b\in\mathrm{BMO}(\mathbb{R}^{n})$, and then consider the operator

$\big{[}b,T_{\theta}\big{]}_{1}(\vec{f})(x)=b(x)\cdot
T_{\theta}(f_{1},f_{2},\dots,f_{m})(x)-T_{\theta}(bf_{1},f_{2},\dots,f_{m})(x).$

For each fixed ball $B=B(x_{0},r)\subset\mathbb{R}^{n}$, it is enough to prove
that

$\frac{1}{\nu_{\vec{w}}(B)^{\kappa/p}}\bigg{(}\int_{B}\big{|}\big{[}b,T_{\theta}\big{]}_{1}(f_{1},\dots,f_{m})(x)\big{|}^{p}\nu_{\vec{w}}(x)\,dx\bigg{)}^{1/p}\lesssim\|b\|_{\ast}\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}.$
(5.4)

As before, we decompose $f_{i}$ as $f_{i}=f^{0}_{i}+f^{\infty}_{0}$, where
$f^{i}_{0}=f_{i}\cdot\chi_{2B}$ and
$f^{\infty}_{i}=f_{i}\cdot\chi_{(2B)^{\complement}}$, $i=1,2,\dots,m$. We set
$tB=B(x_{0},tr)$ for any $t>0$. Let $\mathfrak{L}$ be the same as before. By
using Lemma 4.1($N=2^{m}$), we can write

$\displaystyle\frac{1}{\nu_{\vec{w}}(B)^{\kappa/p}}\bigg{(}\int_{B}\big{|}\big{[}b,T_{\theta}\big{]}_{1}(f_{1},\dots,f_{m})(x)\big{|}^{p}\nu_{\vec{w}}(x)\,dx\bigg{)}^{1/p}$
$\displaystyle\leq
C\cdot\frac{1}{\nu_{\vec{w}}(B)^{\kappa/p}}\bigg{(}\int_{B}\big{|}\big{[}b,T_{\theta}\big{]}_{1}(f^{0}_{1},\dots,f^{0}_{m})(x)\big{|}^{p}\nu_{\vec{w}}(x)\,dx\bigg{)}^{1/p}$
$\displaystyle+C\sum_{(\beta_{1},\dots,\beta_{m})\in\mathfrak{L}}\frac{1}{\nu_{\vec{w}}(B)^{\kappa/p}}\bigg{(}\int_{B}\big{|}\big{[}b,T_{\theta}\big{]}_{1}(f^{\beta_{1}}_{1},\ldots,f^{\beta_{m}}_{m})(x)\big{|}^{p}\nu_{\vec{w}}(x)\,dx\bigg{)}^{1/p}$
$\displaystyle:=J^{0,\dots,0}+\sum_{(\beta_{1},\dots,\beta_{m})\in\mathfrak{L}}J^{\beta_{1},\dots,\beta_{m}}.$
(5.5)

To estimate the first summand of (5), applying Theorem 1.3 along with (3.2)
and (4.4), we get

$\displaystyle J^{0,\dots,0}$ $\displaystyle\leq
C\cdot\frac{1}{\nu_{\vec{w}}(B)^{\kappa/p}}\prod_{i=1}^{m}\bigg{(}\int_{2B}|f_{i}(x)|^{p_{i}}w_{i}(x)\,dx\bigg{)}^{1/{p_{i}}}$
$\displaystyle\leq
C\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}\cdot\frac{1}{\nu_{\vec{w}}(B)^{\kappa/p}}\prod_{i=1}^{m}w_{i}(2B)^{\kappa/{p_{i}}}$
$\displaystyle\leq
C\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}\cdot\frac{\nu_{\vec{w}}(2B)^{\kappa/p}}{\nu_{\vec{w}}(B)^{\kappa/p}}$
$\displaystyle\leq
C\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}.$ (5.6)

To estimate the remaining terms in (5), let us first consider the case when
$\beta_{1}=\cdots=\beta_{m}=\infty$. It is easy to see that for any $x\in B$,

$\big{[}b,T_{\theta}\big{]}_{1}(\vec{f})(x)=[b(x)-b_{B}]\cdot
T_{\theta}(f_{1},f_{2},\dots,f_{m})(x)-T_{\theta}((b-b_{B})f_{1},f_{2},\dots,f_{m})(x).$

Hence, we divide the term $J^{\infty,\dots,\infty}$ into two parts below.

$\begin{split}J^{\infty,\dots,\infty}&\leq
C\cdot\frac{1}{\nu_{\vec{w}}(B)^{\kappa/p}}\bigg{(}\int_{B}\big{|}[b(x)-b_{B}]\cdot
T_{\theta}(f^{\infty}_{1},f^{\infty}_{2},\dots,f^{\infty}_{m})(x)\big{|}^{p}\nu_{\vec{w}}(x)\,dx\bigg{)}^{1/p}\\\
&+C\cdot\frac{1}{\nu_{\vec{w}}(B)^{\kappa/p}}\bigg{(}\int_{B}\big{|}T_{\theta}((b-b_{B})f^{\infty}_{1},f^{\infty}_{2},\dots,f^{\infty}_{m})(x)\big{|}^{p}\nu_{\vec{w}}(x)\,dx\bigg{)}^{1/p}\\\
&:=J^{\infty,\dots,\infty}_{\star}+J^{\infty,\dots,\infty}_{\star\star}.\end{split}$

Next, we estimate each term separately. In the proof of Theorem 2.1, we have
already shown that (see (4))

$\big{|}T_{\theta}(f^{\infty}_{1},f^{\infty}_{2},\ldots,f^{\infty}_{m})(x)\big{|}\lesssim\sum_{j=1}^{\infty}\bigg{(}\prod_{i=1}^{m}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}\bigg{)}.$

Note that $\nu_{\vec{w}}\in A_{mp}\subset A_{\infty}$. From Lemma 5.1(2), it
follows that

$\begin{split}J^{\infty,\dots,\infty}_{\star}&\lesssim\frac{1}{\nu_{\vec{w}}(B)^{\kappa/p}}\sum_{j=1}^{\infty}\bigg{(}\prod_{i=1}^{m}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}\bigg{)}\\\
&\times\bigg{(}\int_{B}\big{|}b(x)-b_{B}\big{|}^{p}\nu_{\vec{w}}(x)\,dx\bigg{)}^{1/p}\\\
&\lesssim\|b\|_{\ast}\cdot\nu_{\vec{w}}(B)^{1/{p}-\kappa/p}\sum_{j=1}^{\infty}\bigg{(}\prod_{i=1}^{m}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}\bigg{)}.\end{split}$

We then follow the same arguments as in the proof of Theorem 2.1 to get

$\displaystyle J^{\infty,\dots,\infty}_{\star}$
$\displaystyle\lesssim\|b\|_{\ast}\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}\sum_{j=1}^{\infty}\frac{\nu_{\vec{w}}(B)^{{(1-\kappa)}/p}}{\nu_{\vec{w}}(2^{j+1}B)^{{(1-\kappa)}/p}}$
$\displaystyle\lesssim\|b\|_{\ast}\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}.$
(5.7)

Using the same methods as in Theorem 2.1, we can also deduce that

$\begin{split}&\big{|}T_{\theta}((b-b_{B})f^{\infty}_{1},f^{\infty}_{2},\dots,f^{\infty}_{m})(x)\big{|}\\\
&\lesssim\int_{(\mathbb{R}^{n})^{m}\backslash(2B)^{m}}\frac{|(b(y_{1})-b_{B})f_{1}(y_{1})|\cdot|f_{2}(y_{2})\cdots
f_{m}(y_{m})|}{(|x-y_{1}|+\cdots+|x-y_{m}|)^{mn}}\,dy_{1}\cdots dy_{m}\\\
&=\sum_{j=1}^{\infty}\int_{(2^{j+1}B)^{m}\backslash(2^{j}B)^{m}}\frac{|(b(y_{1})-b_{B})f_{1}(y_{1})|\cdot|f_{2}(y_{2})\cdots
f_{m}(y_{m})|}{(|x-y_{1}|+\cdots+|x-y_{m}|)^{mn}}\,dy_{1}\cdots dy_{m}\\\
&\lesssim\sum_{j=1}^{\infty}\bigg{(}\frac{1}{|2^{j+1}B|^{m}}\int_{(2^{j+1}B)^{m}\backslash(2^{j}B)^{m}}|(b(y_{1})-b_{B})f_{1}(y_{1})|\cdot\big{|}f_{2}(y_{2})\cdots
f_{m}(y_{m})\big{|}\,dy_{1}\cdots dy_{m}\bigg{)}\\\
&\leq\sum_{j=1}^{\infty}\bigg{(}\frac{1}{|2^{j+1}B|^{m}}\int_{2^{j+1}B}|(b(y_{1})-b_{B})f_{1}(y_{1})|\,dy_{1}\prod_{i=2}^{m}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}\bigg{)}\\\
&=\sum_{j=1}^{\infty}\bigg{(}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}|(b(y_{1})-b_{B})f_{1}(y_{1})|\,dy_{1}\bigg{)}\bigg{(}\prod_{i=2}^{m}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}\bigg{)}.\end{split}$

Then we have

$\begin{split}J^{\infty,\dots,\infty}_{\star\star}&\lesssim\nu_{\vec{w}}(B)^{{(1-\kappa)}/p}\\\
&\times\sum_{j=1}^{\infty}\bigg{(}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}|(b(y_{1})-b_{B})f_{1}(y_{1})|\,dy_{1}\bigg{)}\bigg{(}\prod_{i=2}^{m}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}\bigg{)}.\end{split}$
(5.8)

For each $2\leq i\leq m$, by using Hölder’s inequality with exponent $p_{i}$,
we obtain that

$\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}\leq\bigg{(}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}^{p_{i}}w_{i}(y_{i})\,dy_{i}\bigg{)}^{1/{p_{i}}}\bigg{(}\int_{2^{j+1}B}w_{i}(y_{i})^{-p^{\prime}_{i}/{p_{i}}}\,dy_{i}\bigg{)}^{1/{p^{\prime}_{i}}}.$

According to Lemma 3.1, we have
$w_{i}^{1-p^{\prime}_{i}}=w_{i}^{-p^{\prime}_{i}/{p_{i}}}\in
A_{mp^{\prime}_{i}}\subset A_{\infty}$, $i=1,2,\dots,m$. By using Hölder’s
inequality again with exponent $p_{1}$ and (5.1), we deduce that

$\begin{split}&\int_{2^{j+1}B}|(b(y_{1})-b_{B})f_{1}(y_{1})|\,dy_{1}\\\
&\leq\bigg{(}\int_{2^{j+1}B}\big{|}f_{1}(y_{1})\big{|}^{p_{1}}w_{1}(y_{1})\,dy_{1}\bigg{)}^{1/{p_{1}}}\bigg{(}\int_{2^{j+1}B}|b(y_{1})-b_{B}|^{p^{\prime}_{1}}w_{1}(y_{1})^{-p^{\prime}_{1}/{p_{1}}}\,dy_{1}\bigg{)}^{1/{p^{\prime}_{1}}}\\\
&\lesssim\bigg{(}\int_{2^{j+1}B}\big{|}f_{1}(y_{1})\big{|}^{p_{1}}w_{1}(y_{1})\,dy_{1}\bigg{)}^{1/{p_{1}}}(j+1)\|b\|_{*}\cdot\bigg{(}\int_{2^{j+1}B}w_{1}(y_{1})^{-p^{\prime}_{1}/{p_{1}}}\,dy_{1}\bigg{)}^{1/{p^{\prime}_{1}}},\end{split}$

where the last inequality is valid by the fact that
$w_{1}^{-p^{\prime}_{1}/{p_{1}}}\in A_{\infty}$. Substituting the above two
estimates into the formula (5.8), we have

$\begin{split}J^{\infty,\dots,\infty}_{\star\star}&\lesssim\|b\|_{*}\cdot\nu_{\vec{w}}(B)^{{(1-\kappa)}/p}\\\
&\sum_{j=1}^{\infty}(j+1)\bigg{\\{}\prod_{i=1}^{m}\frac{1}{|2^{j+1}B|}\bigg{(}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}^{p_{i}}w_{i}(y_{i})\,dy_{i}\bigg{)}^{1/{p_{i}}}\bigg{(}\int_{2^{j+1}B}w_{i}(y_{i})^{-p^{\prime}_{i}/{p_{i}}}\,dy_{i}\bigg{)}^{1/{p^{\prime}_{i}}}\bigg{\\}}\\\
&\lesssim\|b\|_{*}\cdot\nu_{\vec{w}}(B)^{{(1-\kappa)}/p}\sum_{j=1}^{\infty}(j+1)\bigg{\\{}\frac{1}{\nu_{\vec{w}}(2^{j+1}B)^{1/p}}\prod_{i=1}^{m}\bigg{(}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}w_{i}(2^{j+1}B)^{\kappa/{p_{i}}}\bigg{)}\bigg{\\}}\\\
&\lesssim\|b\|_{\ast}\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}\sum_{j=1}^{\infty}(j+1)\cdot\frac{\nu_{\vec{w}}(B)^{{(1-\kappa)}/p}}{\nu_{\vec{w}}(2^{j+1}B)^{{(1-\kappa)}/p}},\end{split}$

where in the last two inequalities we have used the $A_{\vec{P}}$ condition
and (4.4). Moreover, in view of (4.7)(since $\nu_{\vec{w}}\in A_{mp}$ with
$1<mp<+\infty$), the last expression is bounded by

$\displaystyle\|b\|_{\ast}\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}\sum_{j=1}^{\infty}(j+1)\cdot\left(\frac{|B|}{|2^{j+1}B|}\right)^{\delta{(1-\kappa)}/p}$
$\displaystyle\lesssim$
$\displaystyle\|b\|_{\ast}\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})},$
(5.9)

where the last series is convergent since the exponent $\delta{(1-\kappa)}/p$
is positive. Consequently, combining the inequality (5) with (5), we get

$J^{\infty,\dots,\infty}\lesssim\|b\|_{\ast}\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}.$

We now consider the case where exactly $\ell$ of the $\beta_{i}$ are $\infty$
for some $1\leq\ell<m$. We only give the arguments for one of these cases. The
rest are similar and can be easily obtained from the arguments below by
permuting the indices. Meanwhile, we consider only the case $\beta_{1}=\infty$
here since the other case can be proved in the same way. We now estimate the
term
$\big{|}\big{[}b,T_{\theta}\big{]}_{1}(f^{\beta_{1}}_{1},\ldots,f^{\beta_{m}}_{m})(x)\big{|}$
when

$\beta_{1}=\cdots=\beta_{\ell}=\infty\quad\&\quad\beta_{\ell+1}=\cdots=\beta_{m}=0.$

In our present situation, we first divide the term
$J^{\beta_{1},\dots,\beta_{m}}$ into two parts as follows.

$\begin{split}J^{\beta_{1},\dots,\beta_{m}}&\leq
C\cdot\frac{1}{\nu_{\vec{w}}(B)^{\kappa/p}}\bigg{(}\int_{B}\big{|}[b(x)-b_{B}]\cdot
T_{\theta}(f^{\infty}_{1},\ldots,f^{\infty}_{\ell},f^{0}_{\ell+1},\ldots,f^{0}_{m})(x)\big{|}^{p}\nu_{\vec{w}}(x)\,dx\bigg{)}^{1/p}\\\
&+C\cdot\frac{1}{\nu_{\vec{w}}(B)^{\kappa/p}}\bigg{(}\int_{B}\big{|}T_{\theta}((b-b_{B})f^{\infty}_{1},\ldots,f^{\infty}_{\ell},f^{0}_{\ell+1},\ldots,f^{0}_{m})(x)\big{|}^{p}\nu_{\vec{w}}(x)\,dx\bigg{)}^{1/p}\\\
&:=J^{\beta_{1},\dots,\beta_{m}}_{\star}+J^{\beta_{1},\dots,\beta_{m}}_{\star\star}.\end{split}$

Next, we estimate each term respectively. Recall that the following result has
been proved in Theorem 2.1(see (4)).

$\big{|}T_{\theta}(f^{\infty}_{1},\ldots,f^{\infty}_{\ell},f^{0}_{\ell+1},\ldots,f^{0}_{m})(x)\big{|}\lesssim\sum_{j=1}^{\infty}\bigg{(}\prod_{i=1}^{m}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}\bigg{)}.$

From Lemma 5.1(2), it then follows that

$\begin{split}J^{\beta_{1},\dots,\beta_{m}}_{\star}&\lesssim\frac{1}{\nu_{\vec{w}}(B)^{\kappa/p}}\sum_{j=1}^{\infty}\bigg{(}\prod_{i=1}^{m}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}\bigg{)}\\\
&\times\bigg{(}\int_{B}\big{|}b(x)-b_{B}\big{|}^{p}\nu_{\vec{w}}(x)\,dx\bigg{)}^{1/p}\\\
&\lesssim\|b\|_{\ast}\cdot\nu_{\vec{w}}(B)^{1/{p}-\kappa/p}\sum_{j=1}^{\infty}\bigg{(}\prod_{i=1}^{m}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}\bigg{)}.\end{split}$

We now proceed exactly as we did in the proof of Theorem 2.1 to obtain that

$\displaystyle J^{\beta_{1},\dots,\beta_{m}}_{\star}$
$\displaystyle\lesssim\|b\|_{\ast}\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}\sum_{j=1}^{\infty}\frac{\nu_{\vec{w}}(B)^{{(1-\kappa)}/p}}{\nu_{\vec{w}}(2^{j+1}B)^{{(1-\kappa)}/p}}$
$\displaystyle\lesssim\|b\|_{\ast}\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}.$
(5.10)

On the other hand, by adopting the same method given in Theorem 2.1, we can
see that

$\displaystyle\big{|}T_{\theta}((b-b_{B})f^{\infty}_{1},\ldots,f^{\infty}_{\ell},f^{0}_{\ell+1},\ldots,f^{0}_{m})(x)\big{|}$
(5.11)
$\displaystyle\lesssim\int_{(\mathbb{R}^{n})^{\ell}\backslash(2B)^{\ell}}\int_{(2B)^{m-\ell}}\frac{|(b(y_{1})-b_{B})f_{1}(y_{1})|\cdot|f_{2}(y_{2})\cdots
f_{m}(y_{m})|}{(|x-y_{1}|+\cdots+|x-y_{m}|)^{mn}}dy_{1}\cdots dy_{m}$
$\displaystyle\lesssim\prod_{i=\ell+1}^{m}\int_{2B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}$
$\displaystyle\times\sum_{j=1}^{\infty}\frac{1}{|2^{j+1}B|^{m}}\int_{(2^{j+1}B)^{\ell}\backslash(2^{j}B)^{\ell}}|(b(y_{1})-b_{B})f_{1}(y_{1})|\cdot\big{|}f_{2}(y_{2})\cdots
f_{\ell}(y_{\ell})\big{|}\,dy_{1}\cdots dy_{\ell}$
$\displaystyle\leq\prod_{i=\ell+1}^{m}\int_{2B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}$
$\displaystyle\times\sum_{j=1}^{\infty}\frac{1}{|2^{j+1}B|^{m}}\int_{2^{j+1}B}|(b(y_{1})-b_{B})f_{1}(y_{1})|\,dy_{1}\prod_{i=2}^{\ell}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}$
$\displaystyle\leq\sum_{j=1}^{\infty}\bigg{(}\frac{1}{|2^{j+1}B|^{m}}\int_{2^{j+1}B}|(b(y_{1})-b_{B})f_{1}(y_{1})|\,dy_{1}\prod_{i=2}^{m}\int_{2^{j+1}B}\big{|}f_{i}(y_{i})\big{|}\,dy_{i}\bigg{)},$

where in the last inequality we have used the inclusion relation $2B\subseteq
2^{j+1}B$ with $j\in\mathbb{N}$. For the same reason as above, we get the
desired estimate.

$\displaystyle J^{\beta_{1},\dots,\beta_{m}}_{\star\star}$
$\displaystyle\lesssim\|b\|_{\ast}\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}\sum_{j=1}^{\infty}(j+1)\cdot\frac{\nu_{\vec{w}}(B)^{{(1-\kappa)}/p}}{\nu_{\vec{w}}(2^{j+1}B)^{{(1-\kappa)}/p}}$
$\displaystyle\lesssim\|b\|_{\ast}\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}.$
(5.12)

Combining (5) and (5), we conclude that

$J^{\beta_{1},\dots,\beta_{m}}\lesssim\|b\|_{\ast}\prod_{i=1}^{m}\big{\|}f_{i}\big{\|}_{L^{p_{i},\kappa}(w_{i})}.$

Summarizing the estimates derived above, then (5.4) holds and hence the proof
of Theorem 2.3 is complete. ∎

###### Proof of Theorem 2.4.

Given $\vec{f}=(f_{1},f_{2},\dots,f_{m})$, for any fixed ball $B=B(x_{0},r)$
in $\mathbb{R}^{n}$, as before, we decompose each $f_{i}$ as

$f_{i}=f^{0}_{i}+f^{\infty}_{i},~{}~{}i=1,2,\dots,m,$

where
$f^{0}_{i}=f_{i}\cdot\chi_{2B}$,$f^{\infty}_{i}=f_{i}\cdot\chi_{(2B)^{\complement}}$
and $2B=B(x,2r)\subset\mathbb{R}^{n}$. Again, we only consider here the
multilinear commutator with only one symbol by linearity; that is, fix
$b\in\mathrm{BMO}(\mathbb{R}^{n})$ and consider the operator

$\big{[}b,T_{\theta}\big{]}_{1}(\vec{f})(x)=b(x)\cdot
T_{\theta}(f_{1},f_{2},\dots,f_{m})(x)-T_{\theta}(bf_{1},f_{2},\dots,f_{m})(x).$

Let $\mathfrak{L}$ be the same as before. Then for any given $\lambda>0$, by
using Lemma 4.2($N=2^{m}$), one can write

$\displaystyle\frac{1}{\nu_{\vec{w}}(B)^{m\kappa}}\cdot\Big{[}\nu_{\vec{w}}\big{(}\big{\\{}x\in
B:\big{|}\big{[}{b},T_{\theta}\big{]}_{1}(\vec{f})(x)\big{|}>\lambda^{m}\big{\\}}\big{)}\Big{]}^{m}$
$\displaystyle\leq\frac{C}{\nu_{\vec{w}}(B)^{m\kappa}}\cdot\Big{[}\nu_{\vec{w}}\big{(}\big{\\{}x\in
B:\big{|}\big{[}{b},T_{\theta}\big{]}_{1}(f^{0}_{1},\dots,f^{0}_{m})(x)\big{|}>\lambda^{m}/{2^{m}}\big{\\}}\big{)}\Big{]}^{m}$
$\displaystyle+\sum_{(\beta_{1},\dots,\beta_{m})\in\mathfrak{L}}\frac{C}{\nu_{\vec{w}}(B)^{m\kappa}}\cdot\Big{[}\nu_{\vec{w}}\big{(}\big{\\{}x\in
B:\big{|}\big{[}{b},T_{\theta}\big{]}_{1}(f^{\beta_{1}}_{1},\ldots,f^{\beta_{m}}_{m})(x)\big{|}>\lambda^{m}/{2^{m}}\big{\\}}\big{)}\Big{]}^{m}$
$\displaystyle:=J^{0,\dots,0}_{\ast}+\sum_{(\beta_{1},\dots,\beta_{m})\in\mathfrak{L}}J^{\beta_{1},\dots,\beta_{m}}_{\ast}.$

Observe that the Young function $\Phi(t)=t\cdot(1+\log^{+}t)$ satisfies the
doubling condition, that is, there is a constant $C_{\Phi}>0$ such that for
every $t>0$,

$\Phi(2t)\leq C_{\Phi}\,\Phi(t).$

This fact together with Theorem 1.4 yields

$\begin{split}J^{0,\dots,0}_{\ast}&\leq\frac{C}{\nu_{\vec{w}}(B)^{m\kappa}}\prod_{i=1}^{m}\bigg{(}\int_{\mathbb{R}^{n}}\Phi\bigg{(}\frac{2|f^{0}_{i}(x)|}{\lambda}\bigg{)}\cdot
w_{i}(x)\,dx\bigg{)}\\\
&\leq\frac{C}{\nu_{\vec{w}}(B)^{m\kappa}}\prod_{i=1}^{m}\bigg{(}\int_{2B}\Phi\bigg{(}\frac{|f_{i}(x)|}{\lambda}\bigg{)}\cdot
w_{i}(x)\,dx\bigg{)}\\\
&=\frac{C}{\nu_{\vec{w}}(B)^{m\kappa}}\prod_{i=1}^{m}w_{i}(2B)\bigg{(}\frac{1}{w_{i}(2B)}\int_{2B}\Phi\bigg{(}\frac{|f_{i}(x)|}{\lambda}\bigg{)}\cdot
w_{i}(x)\,dx\bigg{)}\\\
&\leq\frac{C}{\nu_{\vec{w}}(B)^{m\kappa}}\prod_{i=1}^{m}w_{i}(2B)\cdot\bigg{\|}\Phi\bigg{(}\frac{|f_{i}|}{\lambda}\bigg{)}\bigg{\|}_{L\log
L(w_{i}),2B},\end{split}$

where in the last inequality we have used the estimate (1.9). Since
$\vec{w}=(w_{1},\ldots,w_{m})\in A_{(1,\dots,1)}$, by definition, we know that

$\bigg{(}\frac{1}{|\mathcal{B}|}\int_{\mathcal{B}}\nu_{\vec{w}}(x)\,dx\bigg{)}^{m}\leq
C\prod_{i=1}^{m}\inf_{x\in\mathcal{B}}w_{i}(x)$ (5.13)

holds for any ball $\mathcal{B}$ in $\mathbb{R}^{n}$, where
$\nu_{\vec{w}}=\prod_{i=1}^{m}w_{i}^{1/{m}}$. We can rewrite this inequality
as

$\begin{split}\bigg{(}\frac{1}{|\mathcal{B}|}\int_{\mathcal{B}}\nu_{\vec{w}}(x)\,dx\bigg{)}&\leq
C\bigg{(}\prod_{i=1}^{m}\inf_{x\in\mathcal{B}}w_{i}(x)\bigg{)}^{1/m}=C\bigg{(}\prod_{i=1}^{m}\inf_{x\in\mathcal{B}}w_{i}(x)^{1/m}\bigg{)}\\\
&\leq
C\bigg{(}\inf_{x\in\mathcal{B}}\prod_{i=1}^{m}w_{i}(x)^{1/m}\bigg{)}=C\cdot\inf_{x\in\mathcal{B}}\nu_{\vec{w}}(x),\end{split}$

which means that $\nu_{\vec{w}}\in A_{1}$. Moreover, for each $w_{i}$,
$i=1,2,\dots,m$, it is easy to see that

$\begin{split}\bigg{(}\prod_{j\neq
i}\inf_{x\in\mathcal{B}}w_{j}(x)^{1/{m}}\bigg{)}^{m}\bigg{(}\frac{1}{|\mathcal{B}|}\int_{\mathcal{B}}w_{i}(x)^{1/{m}}\,dx\bigg{)}^{m}&\leq\bigg{(}\frac{1}{|\mathcal{B}|}\int_{\mathcal{B}}w_{i}(x)^{1/{m}}\cdot\prod_{j\neq
i}w_{j}(x)^{1/{m}}\,dx\bigg{)}^{m}\\\ &\leq
C\prod_{j=1}^{m}\inf_{x\in\mathcal{B}}w_{j}(x).\end{split}$

Also observe that

$\bigg{(}\prod_{j\neq
i}\inf_{x\in\mathcal{B}}w_{j}(x)^{1/{m}}\bigg{)}^{m}=\prod_{j\neq
i}\inf_{x\in\mathcal{B}}w_{j}(x).$

From this, it follows that

$\bigg{(}\frac{1}{|\mathcal{B}|}\int_{\mathcal{B}}w_{i}(x)^{1/{m}}\,dx\bigg{)}^{m}\leq
C\cdot\inf_{x\in\mathcal{B}}w_{i}(x),$

which implies that $w_{i}^{1/{m}}\in A_{1}$ ($i=1,2,\dots,m$). Thus, by the
inequality (3.2) and (4.4)(taking $p_{1}=\cdots=p_{m}=1$ and $p=1/m$), we have

$\begin{split}J^{0,\dots,0}_{\ast}&\lesssim\prod_{i=1}^{m}\bigg{\|}\Phi\bigg{(}\frac{|f_{i}|}{\lambda}\bigg{)}\bigg{\|}_{(L\log
L)^{1,\kappa}(w_{i})}\frac{1}{\nu_{\vec{w}}(B)^{m\kappa}}\cdot\prod_{i=1}^{m}w_{i}(2B)^{\kappa}\\\
&\lesssim\prod_{i=1}^{m}\bigg{\|}\Phi\bigg{(}\frac{|f_{i}|}{\lambda}\bigg{)}\bigg{\|}_{(L\log
L)^{1,\kappa}(w_{i})}\cdot\frac{\nu_{\vec{w}}(2B)^{m\kappa}}{\nu_{\vec{w}}(B)^{m\kappa}}\\\
&\lesssim\prod_{i=1}^{m}\bigg{\|}\Phi\bigg{(}\frac{|f_{i}|}{\lambda}\bigg{)}\bigg{\|}_{(L\log
L)^{1,\kappa}(w_{i})}.\end{split}$

It remains to estimate the term $J^{\beta_{1},\dots,\beta_{m}}_{\ast}$ for
$(\beta_{1},\dots,\beta_{m})\in\mathfrak{L}$. Recall that for any $x\in B$,

$\big{[}b,T_{\theta}\big{]}_{1}(\vec{f})(x)=[b(x)-b_{B}]\cdot
T_{\theta}(f_{1},f_{2},\dots,f_{m})(x)-T_{\theta}((b-b_{B})f_{1},f_{2},\dots,f_{m})(x).$

So we can further decompose $J^{\beta_{1},\dots,\beta_{m}}_{\ast}$ as

$\begin{split}J^{\beta_{1},\dots,\beta_{m}}_{\ast}\leq&\frac{C}{\nu_{\vec{w}}(B)^{m\kappa}}\Big{[}\nu_{\vec{w}}\Big{(}\Big{\\{}x\in
B:\big{|}[b(x)-b_{B}]\cdot
T_{\theta}(f^{\beta_{1}}_{1},f^{\beta_{2}}_{2},\ldots,f^{\beta_{m}}_{m})(x)\big{|}>\lambda^{m}/{2^{m+1}}\Big{\\}}\Big{)}\Big{]}^{m}\\\
&+\frac{C}{\nu_{\vec{w}}(B)^{m\kappa}}\Big{[}\nu_{\vec{w}}\Big{(}\Big{\\{}x\in
B:\big{|}T_{\theta}((b-b_{B})f^{\beta_{1}}_{1},f^{\beta_{2}}_{2},\dots,f^{\beta_{m}}_{m})(x)\big{|}>\lambda^{m}/{2^{m+1}}\Big{\\}}\Big{)}\Big{]}^{m}\\\
:=&\widetilde{J}^{\beta_{1},\dots,\beta_{m}}_{\star}+\widetilde{J}^{\beta_{1},\dots,\beta_{m}}_{\star\star}.\end{split}$

By using the previous pointwise estimates (4) and (4) together with
Chebyshev’s inequality, we can deduce that

$\begin{split}\widetilde{J}^{\beta_{1},\dots,\beta_{m}}_{\star}&\leq\frac{C}{\nu_{\vec{w}}(B)^{m\kappa}}\times\frac{2^{m+1}}{\lambda^{m}}\bigg{(}\int_{B}\big{|}[b(x)-b_{B}]\cdot
T_{\theta}(f^{\beta_{1}}_{1},f^{\beta_{2}}_{2},\ldots,f^{\beta_{m}}_{m})(x)\big{|}^{\frac{\,1\,}{m}}\nu_{\vec{w}}(x)\,dx\bigg{)}^{m}\\\
&\leq\frac{C}{\nu_{\vec{w}}(B)^{m\kappa}}\sum_{j=1}^{\infty}\bigg{(}\prod_{i=1}^{m}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}\frac{|f_{i}(y_{i})|}{\lambda}\,dy_{i}\bigg{)}\\\
&\times\bigg{(}\int_{B}\big{|}b(x)-b_{B}\big{|}^{\frac{\,1\,}{m}}\nu_{\vec{w}}(x)\,dx\bigg{)}^{m}.\end{split}$

We claim that for $2\leq m\in\mathbb{N}$ and $\nu_{\vec{w}}\in A_{1}$,

$\bigg{(}\int_{B}\big{|}b(x)-b_{B}\big{|}^{\frac{\,1\,}{m}}\nu_{\vec{w}}(x)\,dx\bigg{)}^{m}\lesssim\|b\|_{*}\cdot\nu_{\vec{w}}(B)^{m}.$
(5.14)

Assuming the claim (5.14) holds for the moment, then we have

$\widetilde{J}^{\beta_{1},\dots,\beta_{m}}_{\star}\lesssim\|b\|_{*}\cdot\nu_{\vec{w}}(B)^{m(1-\kappa)}\sum_{j=1}^{\infty}\bigg{(}\prod_{i=1}^{m}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}\frac{|f_{i}(y_{i})|}{\lambda}\,dy_{i}\bigg{)}.$

Furthermore, note that $t\leq\Phi(t)=t\cdot(1+\log^{+}t)$ for any $t>0$. This
fact along with the multiple $A_{(1,\dots,1)}$ condition (5.13) implies that

$\begin{split}\widetilde{J}^{\beta_{1},\dots,\beta_{m}}_{\star}&\lesssim\|b\|_{*}\cdot\nu_{\vec{w}}(B)^{m(1-\kappa)}\\\
&\times\sum_{j=1}^{\infty}\prod_{i=1}^{m}\bigg{(}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}\frac{|f_{i}(y_{i})|}{\lambda}\cdot
w_{i}(y_{i})\,dy_{i}\bigg{)}\left(\inf_{y_{i}\in
2^{j+1}B}w_{i}(y_{i})\right)^{-1}\\\
&\lesssim\|b\|_{*}\cdot\nu_{\vec{w}}(B)^{m(1-\kappa)}\times\sum_{j=1}^{\infty}\frac{1}{\nu_{\vec{w}}(2^{j+1}B)^{m}}\prod_{i=1}^{m}\int_{2^{j+1}B}\Phi\bigg{(}\frac{|f_{i}(y_{i})|}{\lambda}\bigg{)}\cdot
w_{i}(y_{i})\,dy_{i}\\\
&\lesssim\|b\|_{*}\cdot\nu_{\vec{w}}(B)^{m(1-\kappa)}\times\sum_{j=1}^{\infty}\frac{1}{\nu_{\vec{w}}(2^{j+1}B)^{m}}\prod_{i=1}^{m}w_{i}\big{(}2^{j+1}B\big{)}\bigg{\|}\Phi\bigg{(}\frac{|f_{i}|}{\lambda}\bigg{)}\bigg{\|}_{L\log
L(w_{i}),2^{j+1}B},\end{split}$

where the last inequality follows from the previous estimate (1.9). In view of
(4.4) and (4.7), the last expression is bounded by

$\begin{split}&\|b\|_{*}\cdot\nu_{\vec{w}}(B)^{m(1-\kappa)}\times\sum_{j=1}^{\infty}\frac{1}{\nu_{\vec{w}}(2^{j+1}B)^{m}}\prod_{i=1}^{m}\bigg{\|}\Phi\bigg{(}\frac{|f_{i}|}{\lambda}\bigg{)}\bigg{\|}_{(L\log
L)^{1,\kappa}(w_{i})}\prod_{i=1}^{m}w_{i}(2^{j+1}B)^{\kappa}\\\
&\lesssim\|b\|_{*}\prod_{i=1}^{m}\bigg{\|}\Phi\bigg{(}\frac{|f_{i}|}{\lambda}\bigg{)}\bigg{\|}_{(L\log
L)^{1,\kappa}(w_{i})}\times\sum_{j=1}^{\infty}\frac{\nu_{\vec{w}}(B)^{m(1-\kappa)}}{\nu_{\vec{w}}(2^{j+1}B)^{m(1-\kappa)}}\\\
&\lesssim\|b\|_{*}\prod_{i=1}^{m}\bigg{\|}\Phi\bigg{(}\frac{|f_{i}|}{\lambda}\bigg{)}\bigg{\|}_{(L\log
L)^{1,\kappa}(w_{i})}.\end{split}$

Let us return to the proof of (5.14). Since $\nu_{\vec{w}}\in A_{1}$, we know
that $\nu_{\vec{w}}$ belongs to the reverse Hölder class $RH_{s}$ for some
$1<s<+\infty$(see [4] and [7]). Here the reverse Hölder class is defined in
the following way: $\omega\in RH_{s}$, if there is a constant $C>0$ such that

$\bigg{(}\frac{1}{|B|}\int_{B}\omega(x)^{s}\,dx\bigg{)}^{1/s}\leq
C\bigg{(}\frac{1}{|B|}\int_{B}\omega(x)\,dx\bigg{)}.$

A further application of Hölder’s inequality leads to that

$\begin{split}\int_{B}\big{|}b(x)-b_{B}\big{|}^{\frac{\,1\,}{m}}\nu_{\vec{w}}(x)\,dx&\leq|B|\bigg{(}\frac{1}{|B|}\int_{B}\big{|}b(x)-b_{B}\big{|}^{s^{\prime}/m}\,dx\bigg{)}^{1/s^{\prime}}\bigg{(}\frac{1}{|B|}\int_{B}\nu_{\vec{w}}(x)^{s}\,dx\bigg{)}^{1/s}\\\
&\leq
C\nu_{\vec{w}}(B)\bigg{(}\frac{1}{|B|}\int_{B}\big{|}b(x)-b_{B}\big{|}^{s^{\prime}/m}\,dx\bigg{)}^{1/s^{\prime}}.\end{split}$

Thus, there are two cases to be considered. If $s^{\prime}/m<1$, then (5.14)
holds by using Hölder’s inequality again. If $s^{\prime}/m\geq 1$, then (5.14)
holds by using Lemma 5.1(2). On the other hand, applying the pointwise
estimates (5),(5.11) and Chebyshev’s inequality, we have

$\begin{split}\widetilde{J}^{\beta_{1},\dots,\beta_{m}}_{\star\star}&\leq\frac{C}{\nu_{\vec{w}}(B)^{m\kappa}}\times\frac{2^{m+1}}{\lambda^{m}}\bigg{(}\int_{B}\big{|}T_{\theta}((b-b_{B})f^{\beta_{1}}_{1},f^{\beta_{2}}_{2},\ldots,f^{\beta_{m}}_{m})(x)\big{|}^{\frac{\,1\,}{m}}\nu_{\vec{w}}(x)\,dx\bigg{)}^{m}\\\
&\leq
C\cdot\nu_{\vec{w}}(B)^{m(1-\kappa)}\sum_{j=1}^{\infty}\bigg{(}\prod_{i=2}^{m}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}\frac{|f_{i}(y_{i})|}{\lambda}\,dy_{i}\bigg{)}\\\
&\times\bigg{(}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}\big{|}b(y_{1})-b_{B}\big{|}\cdot\frac{|f_{1}(y_{1})|}{\lambda}\,dy_{1}\bigg{)}\\\
&\leq
C\cdot\nu_{\vec{w}}(B)^{m(1-\kappa)}\sum_{j=1}^{\infty}\bigg{(}\prod_{i=2}^{m}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}\frac{|f_{i}(y_{i})|}{\lambda}w_{i}(y_{i})\,dy_{i}\bigg{)}\\\
&\times\bigg{(}\frac{1}{|2^{j+1}B|}\int_{2^{j+1}B}\big{|}b(y_{1})-b_{B}\big{|}\cdot\frac{|f_{1}(y_{1})|}{\lambda}w_{1}(y_{1})\,dy_{1}\bigg{)}\\\
&\times\prod_{i=1}^{m}\left(\inf_{y_{i}\in
2^{j+1}B}w_{i}(y_{i})\right)^{-1}\\\ &\leq
C\cdot\nu_{\vec{w}}(B)^{m(1-\kappa)}\times\sum_{j=1}^{\infty}\frac{1}{\nu_{\vec{w}}(2^{j+1}B)^{m}}\bigg{(}\prod_{i=2}^{m}\int_{2^{j+1}B}\frac{|f_{i}(y_{i})|}{\lambda}w_{i}(y_{i})\,dy_{i}\bigg{)}\\\
&\times\bigg{(}\int_{2^{j+1}B}\big{|}b(y_{1})-b_{B}\big{|}\cdot\frac{|f_{1}(y_{1})|}{\lambda}w_{1}(y_{1})\,dy_{1}\bigg{)},\end{split}$

where in the last inequality we have used the $A_{(1,\dots,1)}$ condition
(5.13). In addition, using the fact that $t\leq\Phi(t)$ and (1.9), we get

$\begin{split}&\int_{2^{j+1}B}\frac{|f_{i}(y_{i})|}{\lambda}w_{i}(y_{i})\,dy_{i}\\\
&\leq\int_{2^{j+1}B}\Phi\bigg{(}\frac{|f_{i}(y_{i})|}{\lambda}\bigg{)}\cdot
w_{i}(y_{i})\,dy_{i}\\\ &\leq
w_{i}\big{(}2^{j+1}B\big{)}\bigg{\|}\Phi\bigg{(}\frac{|f_{i}|}{\lambda}\bigg{)}\bigg{\|}_{L\log
L(w_{i}),2^{j+1}B}.\end{split}$

Using the fact that $t\leq\Phi(t)$ and the previous estimate (3.6), we thus
obtain

$\begin{split}&\int_{2^{j+1}B}\big{|}b(y_{1})-b_{B}\big{|}\cdot\frac{|f_{1}(y_{1})|}{\lambda}w_{1}(y_{1})\,dy_{1}\\\
&\leq\int_{2^{j+1}B}\big{|}b(y_{1})-b_{B}\big{|}\cdot\Phi\bigg{(}\frac{|f_{1}(y_{1})|}{\lambda}\bigg{)}w_{1}(y_{1})\,dy_{1}\\\
&\leq C\cdot w_{1}\big{(}2^{j+1}B\big{)}\big{\|}b-b_{B}\big{\|}_{\exp
L(w_{1}),2^{j+1}B}\bigg{\|}\Phi\bigg{(}\frac{|f_{1}|}{\lambda}\bigg{)}\bigg{\|}_{L\log
L(w_{1}),2^{j+1}B}.\end{split}$

Furthermore, by the inequality (5.3),

$\begin{split}&\int_{2^{j+1}B}\big{|}b(y_{1})-b_{B}\big{|}\cdot\frac{|f_{1}(y_{1})|}{\lambda}w_{1}(y_{1})\,dy_{1}\\\
&\leq C(j+1)\|b\|_{\ast}\cdot
w_{1}\big{(}2^{j+1}B\big{)}\bigg{\|}\Phi\bigg{(}\frac{|f_{1}|}{\lambda}\bigg{)}\bigg{\|}_{L\log
L(w_{1}),2^{j+1}B}.\end{split}$

Consequently, from the above two estimates, it follows that

$\displaystyle\widetilde{J}^{\beta_{1},\dots,\beta_{m}}_{\star\star}$
$\displaystyle\lesssim\|b\|_{*}\cdot\nu_{\vec{w}}(B)^{m(1-\kappa)}$
$\displaystyle\times\sum_{j=1}^{\infty}(j+1)\frac{1}{\nu_{\vec{w}}(2^{j+1}B)^{m}}\prod_{i=1}^{m}w_{i}\big{(}2^{j+1}B\big{)}\bigg{\|}\Phi\bigg{(}\frac{|f_{i}|}{\lambda}\bigg{)}\bigg{\|}_{L\log
L(w_{i}),2^{j+1}B}$
$\displaystyle\lesssim\|b\|_{*}\cdot\nu_{\vec{w}}(B)^{m(1-\kappa)}$
$\displaystyle\times\sum_{j=1}^{\infty}(j+1)\frac{1}{\nu_{\vec{w}}(2^{j+1}B)^{m}}\prod_{i=1}^{m}\bigg{\|}\Phi\bigg{(}\frac{|f_{i}|}{\lambda}\bigg{)}\bigg{\|}_{(L\log
L)^{1,\kappa}(w_{i})}\prod_{i=1}^{m}w_{i}(2^{j+1}B)^{\kappa}$
$\displaystyle\lesssim\|b\|_{*}\prod_{i=1}^{m}\bigg{\|}\Phi\bigg{(}\frac{|f_{i}|}{\lambda}\bigg{)}\bigg{\|}_{(L\log
L)^{1,\kappa}(w_{i})}\times\sum_{j=1}^{\infty}(j+1)\frac{\nu_{\vec{w}}(B)^{m(1-\kappa)}}{\nu_{\vec{w}}(2^{j+1}B)^{m(1-\kappa)}}$
$\displaystyle\lesssim\|b\|_{*}\prod_{i=1}^{m}\bigg{\|}\Phi\bigg{(}\frac{|f_{i}|}{\lambda}\bigg{)}\bigg{\|}_{(L\log
L)^{1,\kappa}(w_{i})}.$ (5.15)

where the last two inequalities follow from (4.4) and (3.1). This completes
the proof of Theorem 2.4. ∎

For the iterated commutator $\big{[}\Pi\vec{b},T_{\theta}\big{]}$, we can also
establish the following results in the same manner as in Theorems 2.3 and 2.4.
The proof then needs appropriate but minor modifications and we leave the
details to the reader.

######  Theorem 5.1.

Let $m\geq 2$ and $\big{[}\Pi\vec{b},T_{\theta}\big{]}$ be the iterated
commutator of $\theta$-type Calderón–Zygmund operator $T_{\theta}$ with
$\theta$ satisfying the condition (1.1) and $\vec{b}\in\mathrm{BMO}^{m}$. If
$1<p_{1},\dots,p_{m}<+\infty$ and $1/m<p<+\infty$ with
$1/p=\sum_{i=1}^{m}1/{p_{i}}$, and $\vec{w}=(w_{1},\ldots,w_{m})\in
A_{\vec{P}}$ with $w_{1},\ldots,w_{m}\in A_{\infty}$, then for any
$0<\kappa<1$, the iterated commutator $\big{[}\Pi\vec{b},T_{\theta}\big{]}$ is
bounded from $L^{p_{1},\kappa}(w_{1})\times
L^{p_{2},\kappa}(w_{2})\times\cdots\times L^{p_{m},\kappa}(w_{m})$ into
$L^{p,\kappa}(\nu_{\vec{w}})$ with
$\nu_{\vec{w}}=\prod_{i=1}^{m}w_{i}^{p/{p_{i}}}$.

######  Theorem 5.2.

Let $m\geq 2$ and $\big{[}\Pi\vec{b},T_{\theta}\big{]}$ be the iterated
commutator of $\theta$-type Calderón–Zygmund operator $T_{\theta}$ with
$\theta$ satisfying the condition (1.8) and $\vec{b}\in\mathrm{BMO}^{m}$.
Assume that $\vec{w}=(w_{1},\ldots,w_{m})\in A_{(1,\dots,1)}$ with
$w_{1},\ldots,w_{m}\in A_{\infty}$. If $p_{i}=1$, $i=1,2,\ldots,m$ and
$p=1/m$, then for any given $\lambda>0$ and any ball $B\subset\mathbb{R}^{n}$,
there exists a constant $C>0$ such that

$\begin{split}&\frac{1}{\nu_{\vec{w}}(B)^{m\kappa}}\cdot\Big{[}\nu_{\vec{w}}\Big{(}\Big{\\{}x\in
B:\big{|}\big{[}\Pi\vec{b},T_{\theta}\big{]}(\vec{f})(x)\big{|}>\lambda^{m}\Big{\\}}\Big{)}\Big{]}^{m}\\\
&\leq
C\cdot\prod_{i=1}^{m}\bigg{\|}\Phi^{(m)}\bigg{(}\frac{|f_{i}|}{\lambda}\bigg{)}\bigg{\|}_{(L\log
L)^{1,\kappa}(w_{i})},\end{split}$

where $\nu_{\vec{w}}=\prod_{i=1}^{m}w_{i}^{1/{m}}$,
$\Phi(t)=t\cdot(1+\log^{+}t)$ and
$\Phi^{(m)}=\overbrace{\Phi\circ\cdots\circ\Phi}^{m}$.

Finally, in view of the relation (4.17), we have the following results.

######  Corollary 5.1.

Let $m\geq 2$ and $\vec{b}\in\mathrm{BMO}^{m}$. If
$1<p_{1},\dots,p_{m}<+\infty$ and $1/m<p<+\infty$ with
$1/p=\sum_{i=1}^{m}1/{p_{i}}$, and
$\vec{w}=(w_{1},\ldots,w_{m})\in\prod_{i=1}^{m}A_{p_{i}}$, then for any
$0<\kappa<1$, both the multilinear commutator
$\big{[}\Sigma\vec{b},T_{\theta}\big{]}$ and the iterated commutator
$\big{[}\Pi\vec{b},T_{\theta}\big{]}$ are bounded from
$L^{p_{1},\kappa}(w_{1})\times L^{p_{2},\kappa}(w_{2})\times\cdots\times
L^{p_{m},\kappa}(w_{m})$ into $L^{p,\kappa}(\nu_{\vec{w}})$ with
$\nu_{\vec{w}}=\prod_{i=1}^{m}w_{i}^{p/{p_{i}}}$, provided that $\theta$
satisfies the condition (1.1).

######  Corollary 5.2.

Let $m\geq 2$ and $\vec{b}\in\mathrm{BMO}^{m}$. Assume that
$\vec{w}=(w_{1},\ldots,w_{m})\in\prod_{i=1}^{m}A_{1}$. If $p_{i}=1$,
$i=1,2,\ldots,m$ and $p=1/m$, then for any given $\lambda>0$ and any ball
$B\subset\mathbb{R}^{n}$, there exists a constant $C>0$ such that
$(\nu_{\vec{w}}=\prod_{i=1}^{m}w_{i}^{1/{m}})$

$\begin{split}&\frac{1}{\nu_{\vec{w}}(B)^{m\kappa}}\cdot\Big{[}\nu_{\vec{w}}\Big{(}\Big{\\{}x\in
B:\big{|}\big{[}\Sigma\vec{b},T_{\theta}\big{]}(\vec{f})(x)\big{|}>\lambda^{m}\Big{\\}}\Big{)}\Big{]}^{m}\\\
&\leq
C\cdot\prod_{i=1}^{m}\bigg{\|}\Phi\bigg{(}\frac{|f_{i}|}{\lambda}\bigg{)}\bigg{\|}_{(L\log
L)^{1,\kappa}(w_{i})},\end{split}$

provided that $\theta$ satisfies the condition (1.6), and

$\begin{split}&\frac{1}{\nu_{\vec{w}}(B)^{m\kappa}}\cdot\Big{[}\nu_{\vec{w}}\Big{(}\Big{\\{}x\in
B:\big{|}\big{[}\Pi\vec{b},T_{\theta}\big{]}(\vec{f})(x)\big{|}>\lambda^{m}\Big{\\}}\Big{)}\Big{]}^{m}\\\
&\leq
C\cdot\prod_{i=1}^{m}\bigg{\|}\Phi^{(m)}\bigg{(}\frac{|f_{i}|}{\lambda}\bigg{)}\bigg{\|}_{(L\log
L)^{1,\kappa}(w_{i})},\end{split}$

provided that $\theta$ satisfies the condition (1.8).

##  6 Appendix

As pointed out in Remark 1.6, the conclusion of Theorem 1.3 also holds with
(1.6) replaced by the weaker condition (1.1). In the last section, we shall
give the proof of Theorem 1.5 since the proof of Theorem 1.3 is quite similar
and easier. Let $\vec{b}\in\mathrm{BMO}^{m}$ and
$\big{[}\Pi\vec{b},T_{\theta}\big{]}$ be the iterated commutator of
$\theta$-type Calderón–Zygmund operator $T_{\theta}$ with $\theta$ satisfying
the condition (1.1). We prove that if $\vec{w}=(w_{1},\ldots,w_{m})\in
A_{\vec{P}}$, then there exists a constant $C>0$ independent of $\vec{f}$ such
that

$\big{\|}\big{[}\Pi\vec{b},T_{\theta}\big{]}(\vec{f})\big{\|}_{L^{p}(\nu_{\vec{w}})}\leq
C\prod_{k=1}^{m}\big{\|}f_{k}\big{\|}_{L^{p_{k}}(w_{k})},$

where $\nu_{\vec{w}}=\prod_{k=1}^{m}w_{k}^{p/{p_{k}}}$. The method used here
is different from the one in [14]. The basic idea of the proof is taken from
[1, 3] and [17, Proposition 3.1]. For $b_{k}\in\mathrm{BMO}(\mathbb{R}^{n})$
with $1\leq k\leq m$, we denote $F_{k}(\xi)=e^{\xi[b_{k}(x)-b_{k}(y)]}$,
$\xi\in\mathbb{C}$. Then by the analyticity of $F_{k}(\xi)$ on $\mathbb{C}$
and the Cauchy integral formula, we get

$\begin{split}b_{k}(x)-b_{k}(y)&=F^{\prime}_{k}(0)=\frac{1}{2\pi
i}\int_{|\xi|=1}\frac{F_{k}(\xi)}{\xi^{2}}\,d\xi\\\
&=\frac{1}{2\pi}\int_{0}^{2\pi}e^{e^{i\varphi_{k}}[b_{k}(x)-b_{k}(y)]}\cdot
e^{-i\varphi_{k}}d\varphi_{k}.\end{split}$ (6.1)

Hence, by (1.7) and (6.1), we can see that

$\begin{split}&\big{[}\Pi\vec{b},T_{\theta}\big{]}(f_{1},\ldots,f_{m})(x)\\\
&=\int_{(\mathbb{R}^{n})^{m}}\prod_{k=1}^{m}\big{[}b_{k}(x)-b_{k}(y_{k})\big{]}K(x,y_{1},\dots,y_{m})f_{1}(y_{1})\cdots
f_{m}(y_{m})\,dy_{1}\cdots dy_{m}\\\
&=\int_{(\mathbb{R}^{n})^{m}}\prod_{k=1}^{m}\bigg{(}\frac{1}{2\pi}\int_{0}^{2\pi}e^{e^{i\varphi_{k}}[b_{k}(x)-b_{k}(y_{k})]}\cdot
e^{-i\varphi_{k}}d\varphi_{k}\bigg{)}\\\ &\times
K(x,y_{1},\dots,y_{m})f_{1}(y_{1})\cdots f_{m}(y_{m})\,dy_{1}\cdots dy_{m}\\\
&=\int_{(\mathbb{R}^{n})^{m}}\bigg{[}\frac{1}{(2\pi)^{m}}\int_{[0,2\pi]^{m}}\bigg{(}\prod_{k=1}^{m}e^{e^{i\varphi_{k}}b_{k}(x)}\cdot
e^{-i\varphi_{k}}\bigg{)}d\varphi_{1}\cdots d\varphi_{m}\bigg{]}\\\ &\times
K(x,y_{1},\dots,y_{m})\prod_{k=1}^{m}e^{-e^{i\varphi_{k}}b_{k}(y_{k})}\cdot
f_{k}(y_{k})\,dy_{1}\cdots dy_{m}\\\
&=\frac{1}{(2\pi)^{m}}\int_{[0,2\pi]^{m}}T_{\theta}\Big{(}e^{-e^{i\varphi_{1}}b_{1}}\cdot
f_{1},\dots,e^{-e^{i\varphi_{m}}b_{m}}\cdot
f_{m}\Big{)}(x)\bigg{(}\prod_{k=1}^{m}e^{e^{i\varphi_{k}}b_{k}(x)}\cdot
e^{-i\varphi_{k}}\bigg{)}d\varphi_{1}\cdots d\varphi_{m}.\end{split}$

From this, it follows that

$\begin{split}&\big{|}\big{[}\Pi\vec{b},T_{\theta}\big{]}(f_{1},\ldots,f_{m})(x)\big{|}\\\
\leq&\frac{1}{(2\pi)^{m}}\int_{[0,2\pi]^{m}}\Big{|}T_{\theta}\Big{(}e^{-e^{i\varphi_{1}}b_{1}}\cdot
f_{1},\dots,e^{-e^{i\varphi_{m}}b_{m}}\cdot
f_{m}\Big{)}(x)\Big{|}\bigg{(}\prod_{k=1}^{m}e^{\cos\varphi_{k}b_{k}(x)}\bigg{)}d\varphi_{1}\cdots
d\varphi_{m}.\end{split}$

For any $(\varphi_{1},\dots,\varphi_{m})\in[0,2\pi]^{m}$, define $m$-tuples

$\vec{g}_{\varphi}=\big{(}g^{1}_{\varphi_{1}},\dots,g^{m}_{\varphi_{m}}\big{)},\quad\mbox{where}\;\;g^{k}_{\varphi_{k}}=e^{-e^{i\varphi_{k}}b_{k}}\cdot
f_{k},\;\;k=1,2,\dots,m,$

and define

$\vec{w}_{\varphi}=\big{(}w^{1}_{\varphi_{1}},\dots,w^{m}_{\varphi_{m}}\big{)},\quad\mbox{where}\;\;w^{k}_{\varphi_{k}}=w_{k}\cdot
e^{p_{k}\cos\varphi_{k}b_{k}},\;\;k=1,2,\dots,m.$

Set

$\nu^{\ast}_{\vec{w}}=\prod_{k=1}^{m}\big{(}w^{k}_{\varphi_{k}}\big{)}^{p/{p_{k}}}.$

Then we have

$\nu^{\ast}_{\vec{w}}=\prod_{k=1}^{m}\big{(}w_{k}\cdot
e^{p_{k}\cos\varphi_{k}b_{k}}\big{)}^{p/{p_{k}}}=\nu_{\vec{w}}\cdot\prod_{k=1}^{m}e^{p\cos\varphi_{k}b_{k}}.$

Using Minkowski’s inequality, we thus obtain

$\begin{split}\Big{\|}\big{[}\Pi\vec{b},T_{\theta}\big{]}(\vec{f})\Big{\|}_{L^{p}(\nu_{\vec{w}})}&\leq\frac{1}{(2\pi)^{m}}\int_{[0,2\pi]^{m}}\bigg{\|}T_{\theta}(\vec{g}_{\varphi})\prod_{k=1}^{m}e^{\cos\varphi_{k}b_{k}}\bigg{\|}_{L^{p}(\nu_{\vec{w}})}d\varphi_{1}\cdots
d\varphi_{m}\\\
&=\frac{1}{(2\pi)^{m}}\int_{[0,2\pi]^{m}}\big{\|}T_{\theta}(\vec{g}_{\varphi})\big{\|}_{L^{p}(\nu^{\ast}_{\vec{w}})}d\varphi_{1}\cdots
d\varphi_{m}.\end{split}$

Since $\vec{w}=(w_{1},\ldots,w_{m})\in A_{\vec{P}}$, we have $\nu_{\vec{w}}\in
A_{mp}$ and $w_{k}^{1-p^{\prime}_{k}}\in
A_{mp^{\prime}_{k}}$,$k=1,2,\ldots,m$, by using Lemma 3.1. Hence, by the self-
improvement property of $A_{p}$ weights (see [4, 5]), there exist some
positive numbers
$\varepsilon^{\prime},\varepsilon_{1},\dots,\varepsilon_{m}>0$ such that

$\nu_{\vec{w}}^{1+\varepsilon^{\prime}}\in
A_{mp}\quad\&\quad\big{(}w_{k}^{1-p^{\prime}_{k}}\big{)}^{1+\varepsilon_{k}}\in
A_{mp^{\prime}_{k}},~{}k=1,2,\dots,m.$

Now choose

$\varepsilon:=\min\big{\\{}\varepsilon^{\prime},\varepsilon_{1},\dots,\varepsilon_{m}\big{\\}}.$

Then we have

$\nu_{\vec{w}}^{1+\varepsilon}\in
A_{mp}\quad\&\quad\big{(}w_{k}^{1-p^{\prime}_{k}}\big{)}^{1+\varepsilon}=\big{(}w_{k}^{1+\varepsilon}\big{)}^{1-p^{\prime}_{k}}\in
A_{mp^{\prime}_{k}},~{}k=1,2,\dots,m,$

which implies
$(\vec{w})^{1+\varepsilon}:=(w_{1}^{1+\varepsilon},\ldots,w_{m}^{1+\varepsilon})\in
A_{\vec{P}}$ by using Lemma 3.1 again. Note that

$\prod_{k=1}^{m}\big{(}w_{k}^{1+\varepsilon}\big{)}^{p/{p_{k}}}=\bigg{(}\prod_{k=1}^{m}w_{k}^{p/{p_{k}}}\bigg{)}^{1+\varepsilon}=(\nu_{\vec{w}})^{1+\varepsilon}.$

Thus by Theorem 1.1,

$T_{\theta}:L^{p_{1}}(w_{1}^{1+\varepsilon})\times\cdots\times
L^{p_{m}}(w_{m}^{1+\varepsilon})\longrightarrow
L^{p}((\nu_{\vec{w}})^{1+\varepsilon}).$ (6.2)

On the other hand, for any fixed $\eta>0$, it is known that when
$b\in\mathrm{BMO}(\mathbb{R}^{n})$ with
$\|b\|_{*}<\min\\{C_{2}/{\eta},C_{2}(p-1)/{\eta}\\}$, where $C_{2}$ is the
constant in the John–Nirenberg inequality mentioned above, we have $e^{\eta
b(x)}\in A_{p}$ for $1<p<+\infty$ (see [3, Lemma 1]). For
$b_{k}\in\mathrm{BMO}(\mathbb{R}^{n})$($1\leq k\leq m$), we now choose

$\eta_{k}:=\frac{p_{k}(1+\varepsilon)}{\varepsilon}.$

For such $\eta_{k}>0$, we may assume that
$\|b_{k}\|_{*}<\min\\{C_{2}/{\eta_{k}},C_{2}(p_{k}-1)/{\eta_{k}}\\}$. The
general case can be proved using the linearity of $T_{\theta}$ as well. Then
for any $\varphi_{k}\in[0,2\pi]$, we have $\cos\varphi_{k}\cdot
b_{k}(x)\in\mathrm{BMO}(\mathbb{R}^{n})$, and

$\|\cos\varphi_{k}\cdot
b_{k}\|_{*}\leq\|b_{k}\|_{*}<\min\big{\\{}C_{2}/{\eta_{k}},C_{2}(p_{k}-1)/{\eta_{k}}\big{\\}},$

which implies that each $\nu_{k}(x):=e^{\eta_{k}\cos\varphi_{k}b_{k}(x)}\in
A_{p_{k}}$ for $1<p_{k}<+\infty$, $k=1,2,\dots,m$. Notice that

$\prod_{k=1}^{m}\big{(}e^{\frac{1+\varepsilon}{\varepsilon}p_{k}\cos\varphi_{k}b_{k}}\big{)}^{p/{p_{k}}}=\prod_{k=1}^{m}\big{(}e^{\frac{1+\varepsilon}{\varepsilon}p\cos\varphi_{k}b_{k}}\big{)}=\bigg{(}\prod_{k=1}^{m}e^{p\cos\varphi_{k}b_{k}}\bigg{)}^{\frac{1+\varepsilon}{\varepsilon}}.$

This fact along with (4.17) and Theorem 1.1 gives us that

$T_{\theta}:L^{p_{1}}\big{(}e^{\frac{1+\varepsilon}{\varepsilon}p_{1}\cos\varphi_{1}b_{1}}\big{)}\times\cdots\times
L^{p_{m}}\big{(}e^{\frac{1+\varepsilon}{\varepsilon}p_{m}\cos\varphi_{m}b_{m}}\big{)}\longrightarrow
L^{p}\Big{(}\big{(}\prod_{k=1}^{m}e^{p\cos\varphi_{k}b_{k}}\big{)}^{\frac{1+\varepsilon}{\varepsilon}}\Big{)}.$
(6.3)

Interpolating between (6.2) and (6.3)(see [2, 22]) we obtain that

$T_{\theta}:L^{p_{1}}\big{(}w_{1}e^{p_{1}\cos\varphi_{1}b_{1}}\big{)}\times\cdots\times
L^{p_{m}}\big{(}w_{m}e^{p_{m}\cos\varphi_{m}b_{m}}\big{)}\longrightarrow
L^{p}\Big{(}\nu_{\vec{w}}\prod_{k=1}^{m}e^{p\cos\varphi_{k}b_{k}}\Big{)},$

that is

$T_{\theta}:L^{p_{1}}\big{(}w^{1}_{\varphi_{1}}\big{)}\times\cdots\times
L^{p_{m}}\big{(}w^{m}_{\varphi_{m}}\big{)}\longrightarrow
L^{p}\big{(}\nu^{\ast}_{\vec{w}}\big{)}.$ (6.4)

By (6.4) we have

$\big{\|}T_{\theta}(\vec{g}_{\varphi})\big{\|}_{L^{p}(\nu^{\ast}_{\vec{w}})}\leq
C\prod_{k=1}^{m}\big{\|}g^{k}_{\varphi_{k}}\big{\|}_{L^{p_{k}}(w^{k}_{\varphi_{k}})}.$
(6.5)

Since $f_{k}\in L^{p_{k}}(w_{k})$, it is easy to check that for any
$\varphi_{k}\in[0,2\pi]$,

$\begin{split}\big{\|}g^{k}_{\varphi_{k}}\big{\|}_{L^{p_{k}}(w^{k}_{\varphi_{k}})}&=\bigg{(}\int_{\mathbb{R}^{n}}\big{|}g^{k}_{\varphi_{k}}(x)\big{|}^{p_{k}}w_{k}(x)\cdot
e^{p_{k}\cos\varphi_{k}b_{k}(x)}dx\bigg{)}^{1/{p_{k}}}\\\
&=\bigg{(}\int_{\mathbb{R}^{n}}\big{|}f_{k}(x)\big{|}^{p_{k}}e^{-p_{k}\cos\varphi_{k}b_{k}(x)}\cdot
w_{k}(x)\cdot e^{p_{k}\cos\varphi_{k}b_{k}(x)}dx\bigg{)}^{1/{p_{k}}}\\\
&=\bigg{(}\int_{\mathbb{R}^{n}}\big{|}f_{k}(x)\big{|}^{p_{k}}w_{k}(x)dx\bigg{)}^{1/{p_{k}}}=\big{\|}f_{k}\big{\|}_{L^{p_{k}}(w_{k})}.\end{split}$

Therefore

$\begin{split}\Big{\|}\big{[}\Pi\vec{b},T_{\theta}\big{]}(\vec{f})\Big{\|}_{L^{p}(\nu_{\vec{w}})}&\leq
C\frac{1}{(2\pi)^{m}}\int_{[0,2\pi]^{m}}\prod_{k=1}^{m}\big{\|}g^{k}_{\varphi_{k}}\big{\|}_{L^{p_{k}}(w^{k}_{\varphi_{k}})}d\varphi_{1}\cdots
d\varphi_{m}\\\
&=C\frac{1}{(2\pi)^{m}}\int_{[0,2\pi]^{m}}\prod_{k=1}^{m}\big{\|}f_{k}\big{\|}_{L^{p_{k}}(w_{k})}d\varphi_{1}\cdots
d\varphi_{m}\\\ &\leq
C\prod_{k=1}^{m}\big{\|}f_{k}\big{\|}_{L^{p_{k}}(w_{k})},\end{split}$

which is our desired estimate.

### Acknowledgment

This work was supported by the Natural Science Foundation of China (Grant No.
XJEDU2020Y002 and 2022D01C407).

## References

  * [1] J. Alvarez, R. J. Bagby, D. S. Kurtz and C. Pérez, _Weighted estimates for commutators of linear operators_ , Studia Math, 104(1993), 195–209.
  * [2] J. Bergh and J. Löfström, _Interpolation Spaces_. An Introduction, Springer–Verlag, 1976.
  * [3] Y. Ding, S. Z. Lu and K. Yabuta, _On commutators of Marcinkiewicz integrals with rough kernel_ , J. Math. Anal. Appl, 275(2002), 60–68.
  * [4] J. Duoandikoetxea, _Fourier Analysis_ , American Mathematical Society, Providence, Rhode Island, 2000.
  * [5] J. Garcia-Cuerva and J. L. Rubio de Francia, _Weighted Norm Inequalities and Related Topics_ , North-Holland, Amsterdam, 1985.
  * [6] L. Grafakos, _Classical Fourier Analysis_ , Third Edition, Springer-Verlag, 2014
  * [7] L. Grafakos, _Modern Fourier Analysis_ , Third Edition, Springer-Verlag, 2014.
  * [8] L. Grafakos and R. H. Torres, _Multilinear Calderón–Zygmund theory_ , Adv. Math., 165(2002), 124–164.
  * [9] F. John and L. Nirenberg, _On functions of bounded mean oscillation_ , Comm. Pure Appl. Math, 14(1961), 415–426.
  * [10] Y. Komori and S. Shirai, _Weighted Morrey spaces and a singular integral operator_ , Math. Nachr, 282(2009), 219–231.
  * [11] A. K. Lerner, S. Ombrosi, C. Pérez, R. H. Torres and R. Trujillo-González, _New maximal functions and multiple weights for the multilinear Calderón–Zygmund theory_ , Adv. Math., 220(2009), 1222–1264.
  * [12] T. Iida, E. Sato, Y. Sawano and H. Tanaka, _Multilinear fractional integrals on Morrey spaces_ , Acta Math. Sin. (Engl. Ser.), 28(2012), 1375–1384.
  * [13] Z. G. Liu and S. Z. Lu, _Endpoint estimates for commutators of Calderón–Zygmund type operators_ , Kodai Math. J., 25(2002), 79–88.
  * [14] G. Z. Lu and P. Zhang, _Multilinear Calderón–Zygmund operators with kernels of Dini’s type and applications_ , Nonlinear Anal., 107(2014), 92–117.
  * [15] D. Maldonado and V. Naibo, _Weighted norm inequalities for paraproducts and bilinear pseudodifferential operators with mild regularity_ , J. Fourier Anal. Appl., 15(2009), 218–261.
  * [16] C. B. Morrey, _On the solutions of quasi-linear elliptic partial differential equations_ , Trans. Amer. Math. Soc, 43(1938), 126–166.
  * [17] C. Pérez and R. H. Torres, _Sharp maximal function estimates for multilinear singular integrals_ , Contemp. Math., 320(2003), 323–331.
  * [18] C. Pérez, G. Pradolini, R. H. Torres and R. Trujillo-González, _End-point estimates for iterated commutators of multilinear singular integrals_ , Bull. Lond. Math. Soc., 46(2014), 26–42.
  * [19] T. S. Quek and D. C. Yang, _Calderón–Zygmund-type operators on weighted weak Hardy spaces over $\mathbb{R}^{n}$_, Acta Math. Sinica (Engl. Ser), 16(2000), 141–160.
  * [20] M. M. Rao and Z. D. Ren, _Theory of Orlicz Spaces_ , Marcel Dekker, New York, 1991.
  * [21] Y. Sawano, S. Sugano and H. Tanaka, _Orlicz–Morrey spaces and fractional operators_ , Potential Anal., 36(2012), 517–556.
  * [22] E. M. Stein and G. Weiss, _Interpolation of operators with change of measures_ , Trans. Amer. Math. Soc, 87(1958), 159–172.
  * [23] E. M. Stein, _Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals_ , Princeton Univ. Press, Princeton, New Jersey, 1993.
  * [24] H. Wang, _Intrinsic square functions on the weighted Morrey spaces_ , J. Math. Anal. Appl, 396(2012), 302–314.
  * [25] H. Wang, _Weighted inequalities for fractional integral operators and linear commutators in the Morrey-type spaces_ , J. Inequal. Appl, 2017, Paper No. 6, 33 pp.
  * [26] H. Wang, _Boundedness of $\theta$-type Calderón–Zygmund operators and commutators in the generalized weighted Morrey spaces_, J. Funct. Spaces, 2016, Art. ID 1309348, 18 pp.
  * [27] K. Yabuta, _Generalizations of Calderón–Zygmund operators_ , Studia Math, 82(1985), 17–31.
  * [28] P. Zhang, _Weighted endpoint estimates for commutators of Marcinkiewicz integrals_ , Acta Math. Sinica (Engl. Ser), 26(2010), 1709–1722.
  * [29] P.Zhang and H.Xu, _Sharp weighted estimates for commutators of Calderón–Zygmund type operators_ , Acta Math. Sinica(Chin. Ser), 48(2005), 625–636.