# Sharp weighted inequalities for iterated commutators of a class of
multilinear operators

Jiawei Tan  Jiawei Tan: School of Mathematical Sciences
Beijing Normal University
Laboratory of Mathematics and Complex Systems
Ministry of Education
Beijing 100875
People’s Republic of China<EMAIL_ADDRESS>and  Qingying Xue∗ Qingying
Xue: School of Mathematical Sciences
Beijing Normal University
Laboratory of Mathematics and Complex Systems
Ministry of Education
Beijing 100875
People’s Republic of China<EMAIL_ADDRESS>
###### Abstract.

In this paper, the sharp quantitative weighted bounds for the iterated
commutators of a class of multilinear operators were systematically studied.
This class of operators contains multilinear Calderón-Zygmund operators,
multilinear Fourier integral operators, and multilinear Littlewood-Paley
square operators as its typical examples. These were done only under two
pretty much general assumptions of pointwise sparse domination estimates. We
first established local decay estimates and quantitative weak $A_{\infty}$
decay estimates for iterated commutators of this class of operators. Then, we
considered the corresponding Coifman-Fefferman inequalities and the mixed weak
type estimates associated with Sawyer’s conjecture. Beyond that, the
Fefferman-Stein inequalities with respect to arbitrary weights and weighted
modular inequalities were also given. As applications, it was shown that all
the conclusions aforementioned can be applied to multilinear
$\omega$-Calderón-Zygmund operators, multilinear maximal singular integral
operators, multilinear pseudo-differential operators, Stein’s square
functions, and higher order Calderón commutators.

###### Key words and phrases:

multilinear operators, iterated commutators, sparse operators, modular
inequalities
2020 Mathematics Subject Classification. Primary 42B20, Secondary 42B25.

The authors were partly supported by the National Key R&D Program of China
(No. 2020YFA0712900) and NNSF of China (No. 12271041).

∗ Corresponding author, e-mail address<EMAIL_ADDRESS>
##  1\. Introduction and main results

###  1.1. Motivation

The main purpose of this paper is to develop a systematic sharp quantitative
weighted theory for iterated commutators of a class of multilinear operators,
which includes the classical Calderón-Zygmund operators as well as numerous
operators beyond the multilinear Calderón-Zygmund theory. Our motivation lies
in three aspects:

(1). During the past two decades, the theory of sparse domination has been
developing rapidly. It was well known that, the sparse domination approach can
significantly simplify the proof of the $A_{2}$ conjecture [30], which states
that if $T$ is a Calderón-Zygmund operator with a Hölder-Lipschitz kernel,
then whether it holds that

$\|Tf\|_{L^{2}(w)}\leq c_{n,T}[w]_{A_{2}}\|f\|_{L^{2}(w)},$

where the definition of $[w]_{A_{2}}$ is listed in Section 2.2. Let
$\mathcal{X}$ be a Banach functions space and define

$\mathcal{A}_{\mathcal{S}}f(x)=\sum_{Q\in\mathcal{S}}\frac{1}{|Q|}\int_{Q}|f(x)|dx\chi_{Q}(x),$

where each $Q$ is a cube with its sides parallel to the axis and $\mathcal{S}$
is a sparse family of such cubes. In [35], Lerner proved that any standard
Calderón-Zygmund operator can be controlled in norm by a family of sparse
operators in the way that

$\|Tf\|_{\mathcal{X}}\leq\sup_{\mathcal{S}}\left\|\mathcal{A}_{\mathcal{S}}f\right\|_{\mathcal{X}}.$

This estimate combined with the following inequality obtained in [18]

$\left\|\mathcal{A}_{\mathcal{S}}\right\|_{L^{2}(w)\rightarrow L^{2}(w)}\leq
c_{n}[w]_{A_{2}}$

gives an simple proof of the $A_{2}$ conjecture.

On the other hand, Lerner and Nazarov [37], Conde-Alonso and Rey [16]
independently showed that the Calderón-Zygmund operator $T$ can be dominated
pointwisely by a finite number of sparse operators,

$|Tf(x)|\leq c_{n,T}\sum_{j=1}^{3^{n}}\mathcal{A}_{\mathcal{S}_{j}}f(x).$

With this estimate in hand, Conde-Alonso and Rey [16] answered an open
question originally posed by Lerner [35].

These two examples illustrate that the method of sparse domination plays an
important role in modern analysis. In fact, this method was widely used in the
study of several important operators in Harmonic analysis, such as Bochner-
Riesz multipliers [3], singular integrals satisfying the $L^{r}$-Hörmander
condition [40], rough singular integrals [15], as well as singular non-
integral type operators [5].

Due to the importance of pointwise sparse domination and the fact that all the
operators mentioned enjoy some kind of sparse domination estimates. It is
quite natural to ask if only with the assumation of pointwise sparse
domination estimates, what kind of properties could be obtained in general for
these operators. This is the first motivation and the starting point of this
paper.

(2). Secondly, multilinear theory is an essential extension for linear theory.
Analogous to multivariable functional calculus, multivariable calculus
provides a robust approach to the study of functions of more than one
variable, and it broadens the narrow perspective of studying a single variable
by freezing other variables. Multilinear analysis focuses on the investigation
of operators linearly related to more than one function, treating all inputs
as variables rather than just dealing with certain parameters. The study of
multilinear theories, often based on the simultaneous decomposition of
multiple variables, is naturally more complex than linear analyses, but it is
more far-reaching and the results are more flexible. In Harmonic analysis,
there are numerous examples of linear operators with fixed parameters that can
be considered as multilinear operators: multiplier operators, Littlewood-Paley
operators, Calderón commutators, and Cauchy integrals along Lipschitz curves
(see [26]).

The development of multilinear Calderón-Zygmund theory has made great progress
in recent decades, see for example [38, 27]. At the same time, however, it is
noted that some important operators beyond the multilinear Calderón-Zygmund
theory, including multilinear singular integrals with non-smooth kernels [22],
multilinear pseudo-differential operators [9], Calderón commutators [21],
Stein’s square functions[13], etc., have various properties that imitate the
Calderón-Zygmund operators. For example, they enjoy the same sparse
domination, $L^{p}$ boundedness, end-point weak-type estimate, and so on.

Based on certain assumptions of sparse domination, our attention has been
drawn to the question of how to extract and analyze commonalities among
various multilinear operators and their iterated commutators. These properties
encompass the weighted local exponential decay property, the Coifman-Fefferman
inequality, mixed weighted estimation, and the weighted modular inequality.

(3). We note that local decay estimates and mixed weak type inequalities of
the commutators of a class of multilinear bounded oscillation operators have
been established in [8]. The method used to prove the local decay estimates in
[8] relies heavily on the local Coifman-Fefferman inequality as well as on the
Rubio de Francia algorithm, under which the dependence of the constants on the
weights cannot be obtained. Then one may wonder whether it is possible to give
quantitatively weighted local decay estimates for the multilinear operators
and their commutators, and obtain a multilinear version of the mixed weighted
inequality with optimized or refined constants. These are the sources of our
third motivation.

###  1.2. Two fundamental hypotheses

We need to present some definitions. Let us start with the definition of
general commutators. Let $\mathcal{T}$ be a $m$-linear operator from
$\mathscr{X}_{1}\times\cdots\times$ $\mathscr{X}_{m}$ into $\mathscr{Y}$,
where $\mathscr{X}_{1},\ldots,\mathscr{X}_{m}$ are some normed spaces and
$\mathscr{Y}$ is a quasi-normed space. In our following statements,
$\mathscr{X}_{1},\ldots,\mathscr{X}_{m}$ and $\mathscr{Y}$ will be
appropriately weighted Lebesgue spaces.

######  Definition 1.1 ($k$-th order commutators).

Given
$\vec{f}:=\left(f_{1},\ldots,f_{m}\right)\in\mathscr{X}_{1}\times\cdots\times\mathscr{X}_{m},\vec{b}=\left(b_{i_{1}},\ldots,b_{i_{l}}\right)$
of measurable functions with
$I:=\\{i_{1},\ldots,i_{l}\\}\subseteq\\{1,\ldots,m\\}$, and $k\in\mathbb{N}$,
we define, whenever it makes sense, the $k$-th order commutator of
$\mathcal{T}$ in the $i$-th entry of $\mathcal{T}$ as

$[\mathcal{T},\vec{b}]_{ke_{i}}(\vec{f})(x):=\mathcal{T}\left(f_{1},\ldots,\left(b_{i}(x)-b_{i}\right)^{k}f_{i},\ldots,f_{m}\right)(x),$

where $e_{i}$ is the basis of $\mathbb{R}^{n}$ with the $i$-th component being
1 and other components being 0.
Furthermore, if $k=1$, we write

$[\mathcal{T},\vec{b}]_{e_{i}}(\vec{f})(x)=b_{i}(x)\mathcal{T}\left(f_{1},\ldots,f_{m}\right)(x)-\mathcal{T}\left(f_{1},\ldots,b_{i}f_{i},\ldots,f_{m}\right)(x).$

Then, for a multi-index
$\vec{\alpha}=\left(\alpha_{1},\ldots,\alpha_{m}\right)\in\mathbb{N}^{m}$, we
denote

$[\mathcal{T},\vec{b}]_{\vec{\alpha}}:=\left[\cdots\left[[\mathcal{T},\vec{b}]_{\alpha_{1}e_{1}},\vec{b}\right]_{\alpha_{2}e_{2}}\cdots,\vec{b}\right]_{\alpha_{m}e_{m}}.$

Using this notation, the iterated commutator of $\mathcal{T}$ is defined as
follows.

######  Definition 1.2 (iterated commutators).

Given $l\leq
m,\vec{f}:=\left(f_{1},\ldots,f_{m}\right)\in\mathscr{X}_{1}\times\cdots\times\mathscr{X}_{m},\vec{b}=\left(b_{i_{1}},\ldots,b_{i_{l}}\right)$
of measurable functions with
$I:=\\{i_{1},\ldots,i_{l}\\}\subseteq\\{1,\ldots,m\\}$. The $m$-linear
iterated commutator of $\mathcal{T}$ is given by

$\mathcal{T}_{\vec{b}}(\vec{f})(x):=\left[\cdots\left[[\mathcal{T},\vec{b}]_{e_{i_{1}}},\vec{b}\right]_{e_{i_{2}}}\cdots,\vec{b}\right]_{e_{i_{l}}}.$

When $\vec{b}=(b,\ldots,b)$, we denote
$\mathcal{T}_{\vec{b}}:=\mathcal{T}_{b}$. In particular, if $\mathcal{T}$ is
an $m$-linear operator with a kernel representation of the form

$\mathcal{T}(\vec{f})(x)=\int_{\mathbb{R}^{nm}}K(x,\vec{y})f_{1}\left(y_{1}\right)\cdots
f_{m}\left(y_{m}\right)d\vec{y}$

where $d\vec{y}=dy_{1}\cdots dy_{m},$ then one can easily verify that
$\mathcal{T}_{\vec{b}}$ has the following expression:

$\mathcal{T}_{\vec{b}}(\vec{f})(x)=\int_{\mathbb{R}^{nm}}\prod_{s=1}^{l}\left(b_{i_{s}}(x)-b_{i_{s}}(y_{i_{s}})\right)K(x,\vec{y})\prod_{s=1}^{m}f_{s}\left(y_{s}\right)d\vec{y}.$

We should mention that the commutators given in Definition 1.1 were originally
introduced by Pérez and Torres [54] in the study of the $m$-linear Calderón-
Zygmund operators. Weighted strong as well as weak type endpoint estimates for
the iterated commutators of the Calderón-Zygmund operators in Definition 1.2
were proved in [50] by Pérez et al. Recently, using the method of
extrapolation, the weighted boundedness results for the general commutator
$[\mathcal{T},\vec{b}]_{\alpha}$ have also been proved in [4]. We refer to [9,
8, 39] for more information about commutators.

To introduce our hypotheses, we need to fix some notation. Given $r>0,$ we set

$\langle|f|^{r}\rangle_{Q}=\frac{1}{|Q|}\int_{Q}|f(y)|^{r}dy,$

particularly, $\langle f\rangle_{Q}=\frac{1}{|Q|}\int_{Q}f(y)dy.$ Let $l(Q)$
be the side length of a cube $Q$ and $rQ$ be the unique cube with sides
parallel to the axes having the same center as $Q$ and having side length
$l(rQ)=rl(Q)$.

The following two hypotheses are crucial to our forthcoming discussion of
$\mathcal{T}_{\vec{b}}.$

######  Hypothesis 1.

Let $I=\\{i_{1},\ldots,i_{l}\\}\subseteq\\{1,\ldots,m\\}$ and
$\vec{b}=\left(b_{i_{1}},\ldots,b_{i_{l}}\right)$ be locally integrable
functions defined on $\mathbb{R}^{n}$. Let $\mathcal{T}$ be an $m$-linear
operator and $\mathcal{T}_{\vec{b}}$ be its commutator given in Definition
1.2. Suppose that for all $Q_{0}$ cubes in $\mathbb{R}^{n}$ and for any
bounded functions $\vec{f}=\left(f_{1},\ldots,f_{m}\right)$ with compact
support, there exists a sparse collection
$\mathcal{F}\subseteq\mathcal{D}(Q_{0})$ (see Section 2.1) such that for a.e.
$x\in Q_{0}$,

$\displaystyle{\left|\mathcal{T}_{\vec{b}}(f_{1}\chi_{3Q_{0}},\ldots,f_{m}\chi_{3Q_{0}})(x)\right|}$
$\displaystyle\leq
C\sum_{Q\in{\mathcal{F}}}\left(\sum_{\vec{\gamma}\in{\\{1,2\\}^{l}}}\prod_{s=1}^{l}\mathcal{R}(b_{i_{s}},f_{i_{s}},Q,\gamma_{i_{s}})\right)\prod_{s\notin
I}{\langle|f_{s}|\rangle}_{3Q}\chi_{Q}(x),$

where

$\mathcal{R}(b,f,Q,\gamma)=\begin{cases}|b-{\langle
b\rangle}_{3Q}|{\langle|f|\rangle}_{3Q},&\text{ if }\gamma=1,\\\
{\langle|(b-{\langle b\rangle}_{3Q})f|\rangle}_{3Q},&\text{ if
}\gamma=2.\end{cases}$

######  Hypothesis 2.

Let $I=\\{i_{1},\ldots,i_{l}\\}\subseteq\\{1,\ldots,m\\}$ and
$\vec{b}=\left(b_{i_{1}},\ldots,b_{i_{l}}\right)$ be locally integrable
functions defined on $\mathbb{R}^{n}$. Let $\mathcal{T}$ be an $m$-linear
operator and $\mathcal{T}_{\vec{b}}$ be its commutator given in Definition
1.2. Suppose that for any bounded functions
$\vec{f}=\left(f_{1},\ldots,f_{m}\right)$ with compact support, there exist
$3^{n}$ sparse collections $\left\\{\mathcal{S}_{j}\right\\}_{j=1}^{3^{n}}$
such that

$|\mathcal{T}_{\vec{b}}(\vec{f})(x)|\leq
C\left(\sum_{j=1}^{3^{n}}\sum_{\vec{\gamma}\in\\{1,2\\}^{l}}\mathcal{A}_{\mathcal{S}_{j},b}^{\vec{\gamma}}(\vec{f})(x)\right),\quad\text{
a.e. }x\in\mathbb{R}^{n},$

where

$\begin{gathered}\mathcal{A}_{\mathcal{S}_{j},b}^{\vec{\gamma}}(\vec{f})(x):=\sum_{Q\in\mathcal{S}_{j}}\left(\prod_{s=1}^{l}\mathcal{U}\left(b_{i_{s}},f_{i_{s}},Q,\gamma_{i_{s}}\right)(x)\right)\left(\prod_{s\notin
I}\left\langle\left|f_{s}\right|\right\rangle_{Q}\right)\chi_{Q}(x),\quad\text{
with }\\\ \mathcal{U}(b,f,Q,\gamma)(x)=\begin{cases}\left|b(x)-\langle
b\rangle_{Q}\right|\langle|f|\rangle_{Q}&\text{ if }\gamma=1,\\\
\left\langle\left|\left(b-\langle
b\rangle_{Q}\right)f\right|\right\rangle_{Q}&\text{ if
}\gamma=2.\end{cases}\end{gathered}$

######  Remark 1.3.

We now make some comments on these two hypotheses. First of all, the estimate
in the form of Hypothesis 1 holds for many operators, e.g., multilinear
$w$-Calderón-Zygmund operators, multilinear pseudo-differential operators,
etc., for more details see [31, p. 166]. Secondly, if $I=\emptyset$, then
Hypothesis 2 can be rewritten as

$|\mathcal{T}(\vec{f})(x)|\leq
C\sum_{j=1}^{3^{n}}\sum_{Q\in\mathcal{S}_{j}}\prod_{s=1}^{m}\left\langle\left|f_{s}\right|\right\rangle_{Q}\chi_{Q}(x),\quad\text{
a.e. }x\in\mathbb{R}^{n},$

which is consistent with the sparse domination obtained for classical
multilinear Calderón-Zygmund operators [20, Theorem 1.4] and multilinear
pseudo-differential operators [9, Proposition 4.1], and so on.

###  1.3. Main results

The main contributions of this paper are as follows:

  1. $\bullet$

Our general framework gives a unified approach to study the quantitatively
weighted estimations of commutators for a class of multilinear operators. This
class of operators includes the multilinear Calderón-Zygmund operators,
multilinear Littlewood-Paley square operators, as well as other operators
beyond multilinear Calderón-Zygmund theory, such as Fourier integral operators
and Calderón commutators, etc., (cf. Sect. 8). We only assume that some sparse
domination estimates holds for this class of operators. It should also be
pointed out that all the results in this paper still hold for this class of
multilinear operators itself (with no commutators), but one has to modify some
places if needed.

  2. $\bullet$

The first main result, Theorem 1.4, gives a sharp weighted local sub-
exponential decay estimate for the iterated commutators of a class of
multilinear operators which essentially improves the result in [8] and it is
sharp for sub-exponential decay. These results accurately reflect the extent
that an operator is locally controlled by certain maximal operator in the
weighted case, thus improving the corresponding good-$\lambda$ inequalities. A
quick comparison with [8] reveals that the methodology and the whole proof
scheme differ in a number of key points. For instance, we take a more direct
approach and obtain a quantitatively weighted exponential decay estimate
directly via pointwise sparse domination, see Remark 1.5.

  3. $\bullet$

The second main result, Theorem 1.10, presents a weighted mixed weak type
inequality, which improves the classical endpoint weighted inequality. In
order to obtain the exact constant estimate, we used endpoint extrapolation
techniques from [41] and gave a quantitative weighted Coifman-Fefferman
inequality (Theorem 1.8).

  4. $\bullet$

Theorem 1.14 focuses on the multilinear Fefferman-Stein inequalities with
respect to arbitrary weights and Theorem 1.18 establishes two weighted modular
inequalities. The dependence between the constants and the weight functions is
given, respectively.

We are now in a position to state our main theorems. The first one is the
local decay estimate of $\mathcal{T}_{\vec{b}}$ as follows:

######  Theorem 1.4.

Let $I=\\{i_{1},\ldots,i_{l}\\}=\\{1,\ldots,l\\}\subseteq\\{1,\ldots,m\\}.$
Let $Q$ be a cube and $f_{s}\in L_{c}^{\infty}\left(\mathbb{R}^{n}\right)$
such that $\operatorname{supp}\left(f_{s}\right)\subset Q$ for $1\leq s\leq
m$. If $\vec{b}\in\mathrm{BMO}^{l}$ and $\mathcal{T}_{\vec{b}}$ satisfies the
Hypothesis 1, then there are constants $\alpha_{1},c_{1}>0$ such that

(1.1) $\displaystyle\left|\left\\{x\in
Q:\big{|}\mathcal{T}_{\vec{b}}(\vec{f})(x)\big{|}>t\min{\\{\mathcal{M}_{L(\log
L)}^{(1,l)}(\vec{f})(x),\mathcal{M}(\vec{f_{0}})(x)\\}}\right\\}\right|$
$\displaystyle\hskip 142.26378pt\qquad\leq
c_{1}e^{-\alpha_{1}(\frac{t}{\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}})^{1/(l+1)}}|Q|,\quad
t>0$

where
$\vec{f_{0}}=(\mathcal{A}_{{\mathcal{S}}^{*}}f_{1},\ldots,\mathcal{A}_{{\mathcal{S}}^{*}}f_{l},f_{l+1},\ldots,f_{m})$
with a sparse family ${\mathcal{S}}^{*}$ and

$\mathcal{M}_{L(\log L)}^{(1,l)}(\vec{f})(x)=\sup\limits_{x\in
Q}\prod\limits_{s=1}\limits^{l}\|f_{s}\|_{L(\log
L),Q}\prod\limits_{s=l+1}\limits^{m}{\langle|f_{s}|\rangle}_{Q}.$

Moreover, the local decay estimate in $(\ref{ie7})$ is sharp in the sense that
it does not hold for any $l_{0}>\frac{1}{1+l}.$

######  Remark 1.5.

We now make some comments on Theorem 1.4. Note that

$\min{\\{\mathcal{M}_{L(\log
L)}^{(1,l)}(\vec{f})(x),\mathcal{M}(\vec{f_{0}})(x)\\}}\leq\mathcal{M}_{L(\log
L)}(\vec{f})(x)\leq\mathcal{M}(Mf_{1},\ldots,Mf_{m}),$

then this shows that Theorem 1.4 improves Theorem 1.6 in [8] substantially.
Furthermore, when $m=1$ and $\mathcal{T}$ is a Calderón-Zygmund operator,
Theorem 1.4 coincides with the main conclusion in [45].

As a corollary of Theorem 1.4, we have the following weighted decay estimates
of $\mathcal{T}_{\vec{b}}$.

######  Corollary 1.6.

Let $w\in
A_{\infty}^{\text{weak}},I=\\{i_{1},\ldots,i_{l}\\}=\\{1,\ldots,l\\}\subseteq\\{1,\ldots,m\\}.$
Let $Q$ be a cube and $f_{s}\in L_{c}^{\infty}\left(\mathbb{R}^{n}\right)$
such that $\operatorname{supp}\left(f_{s}\right)\subset Q$ for $1\leq s\leq
m$. If $\vec{b}\in\mathrm{BMO}^{l}$ and $\mathcal{T}_{\vec{b}}$ satisfies the
Hypothesis 1, then there are constants $\alpha_{2},c_{2}>0$ independent of $w$
such that

(1.2) $\displaystyle w\left(\left\\{x\in
Q:\big{|}\mathcal{T}_{\vec{b}}(\vec{f})(x)\big{|}>t\mathcal{M}_{L(\log
L)}(\vec{f})(x)\right\\}\right)$ $\displaystyle\hskip 85.35826pt\qquad\leq
c_{2}e^{-{\frac{\alpha_{2}}{[w]_{A_{\infty}}^{weak}+1}}\left({\frac{t}{\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}}}\right)^{\frac{1}{l+1}}}w(2Q),\quad
t>0,$

######  Remark 1.7.

Clearly, when $m=1$ and $w\equiv 1,$ Theorem 2.8 in [31] is just a special
case of Corollary 1.6. In addition, since $A_{\infty}\subsetneq
A_{\infty}^{\text{weak}},$ Corollary 1.6 is also valid for any $w\in
A_{\infty}$, see Corollary 3.1.

For the Coifman-Fefferman inequality with the multilinear form of the iterated
commutator of $\mathcal{T}$, we have

######  Theorem 1.8.

Let $I=\\{i_{1},\ldots,i_{l}\\}=\\{1,\ldots,l\\}\subseteq\\{1,\ldots,m\\}.$ If
$\vec{b}\in\mathrm{BMO}^{l}$ and $\mathcal{T}_{\vec{b}}$ satisfies the
Hypothesis 2, then for any $0<p<\infty,w\in A_{\infty},$

(1.3)
$\int_{\mathbb{R}^{n}}\big{|}\mathcal{T}_{\vec{b}}(\vec{f})(x)\big{|}^{p}w(x)dx\lesssim\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}[w]_{A_{\infty}}^{pl}[w]_{A_{\infty}}^{\max\\{2,p\\}}\int_{\mathbb{R}^{n}}\left(\mathcal{M}_{L(\log
L)}(\vec{f})(x)\right)^{p}w(x)dx.$

######  Remark 1.9.

In general, the Coifman-Fefferman inequalities are obtained by extrapolation
(see for example [9, Proposition 5.1]), here we use sparse domination to
obtain better estimates of the weight constants. This provides convenience for
the following quantitatively weighted mixed weak type estimation.

For the endpoint case, we establish weighted mixed weak type inequalities with
a precisely weighted constant.

######  Theorem 1.10.

Let $I=\\{i_{1},\ldots,i_{l}\\}=\\{1,\ldots,l\\}\subseteq\\{1,\ldots,m\\}.$ If
$\vec{b}\in\mathrm{BMO}^{l}$ and $\mathcal{T}_{\vec{b}}$ satisfies the
Hypothesis $\ref{hyp2}.$ Let $\vec{w}=\left(w_{1},\ldots,w_{m}\right)$ and
$u=\prod_{i=1}^{m}w_{i}^{1/m}$. If $\vec{w}\in A_{\vec{1}}$ and $v\in
A_{\infty},$ then there exists $t>1$ depending only on $v$, such that

$\left\|\frac{\mathcal{T}_{\vec{b}}(\vec{f})}{v}\right\|_{L^{\frac{1}{m},\infty}(uv^{\frac{1}{m}})}\lesssim
K_{0}^{2l+6m}[v^{\frac{1}{m}}]_{A_{t}}^{2l+4m}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\left\|\frac{\mathcal{M}_{L(\log
L)}(\vec{f})}{v}\right\|_{L^{\frac{1}{m},\infty}(uv^{\frac{1}{m}})},$

where
$K_{0}=4C_{n}p_{0}p_{0}^{\prime}([u]_{A_{1}}+2^{p_{0}-1}C_{n}^{t}[v^{\frac{1}{m}}]_{A_{t}}^{2}[u]_{A_{1}}^{p_{0}-1})+1$
with $p_{0}=2^{n+3}(t-1)[u]_{A_{1}}+1.$

######  Remark 1.11.

Very recently, in the linear case, Ibáñez-Firnkorn and Rivera-Ríos [32]
established mixed-weighted endpoint estimates for the commutators of a class
of linear operators and gave some weighted constant estimates. We would like
to point out that Theorem 1.10 improves the results in [8, 41] in two aspects.
It yields a more accurate norm constant for the weights $u$ and $v,$ and
reduces the condition $v^{\frac{1}{m}}\in A_{\infty}$ to $v\in A_{\infty}.$

######  Remark 1.12.

In particular, under the above theorem conditions, if $\vec{w}\in A_{\vec{1}}$
and $v\in A_{p}(1<p<\infty),$ then we have

$\left\|\frac{\mathcal{T}_{\vec{b}}(\vec{f})}{v}\right\|_{L^{\frac{1}{m},\infty}(uv^{\frac{1}{m}})}\lesssim\tilde{K}_{0}^{2l+6m}[v^{\frac{1}{m}}]_{A_{p}}^{2l+4m}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\left\|\frac{\mathcal{M}_{L(\log
L)}(\vec{f})}{v}\right\|_{L^{\frac{1}{m},\infty}(uv^{\frac{1}{m}})},$

where
$\tilde{K}_{0}=C_{n}\tilde{p}_{0}\tilde{p}_{0}^{\prime}2^{\tilde{p}_{0}-1}([v^{\frac{1}{m}}]_{A_{p}}^{2}[u]_{A_{1}}^{\tilde{p}_{0}})$
with $\tilde{p}_{0}=2^{n+3}(p-1)[u]_{A_{1}}+1.$

As a corollary of Theorem 1.10, we can easily obtain the following weak type
estimates for iterated commutators with the type of Coifman-Fefferman
inequalities.

######  Corollary 1.13.

Let $I=\\{i_{1},\ldots,i_{l}\\}=\\{1,\ldots,l\\}\subseteq\\{1,\ldots,m\\}.$ If
$\vec{b}\in\mathrm{BMO}^{l}$ and $\mathcal{T}_{\vec{b}}$ satisfies the
Hypothesis 2. Let $\vec{w}=\left(w_{1},\ldots,w_{m}\right)$ and
$u=\prod_{i=1}^{m}w_{i}^{1/m}$. If $\vec{w}\in A_{\vec{1}},$ then

$\left\|\mathcal{T}_{\vec{b}}(\vec{f})\right\|_{L^{\frac{1}{m},\infty}(u)}\lesssim(2[u]_{A_{1}})^{2^{n+7}m[u]_{A_{1}}}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\left\|\mathcal{M}_{L(\log
L)}(\vec{f})\right\|_{L^{\frac{1}{m},\infty}(u)}.$

In order to present the Fefferman-Stein inequalities with arbitrary weights,
given a weight $w$ and $0<p<\infty,$ we define a class of weighted BMO spaces
$\mathrm{BMO}_{p}(w)$ by

$\mathrm{BMO}_{p}(w):=\left\\{f\in
L_{loc}^{1}\left(\mathbb{R}^{n}\right):\|f\|_{\mathrm{BMO}_{p}(w)}<\infty\right\\}$

where

$\|f\|_{\mathrm{BMO}_{p}(w)}:=\sup_{Q}\left(\frac{1}{w(Q)}\int_{Q}\left|f(x)-\langle
f\rangle_{Q}\right|^{p}w(x)dx\right)^{\frac{1}{p}}.$

When $p=1,$ we denote
$\|f\|_{\mathrm{BMO}(w)}:=\sup_{Q}\frac{1}{w(Q)}\int_{Q}\left|f(x)-\langle
f\rangle_{Q}\right|w(x)dx.$

######  Theorem 1.14.

Let $m\geq
2,I=\\{i_{1},\ldots,i_{l}\\}=\\{1,\ldots,l\\}\subseteq\\{1,\ldots,m\\},\vec{b}=(b_{1},\ldots,b_{l}).$
Let $1<p_{1},\ldots,p_{m}<\infty$ and
$\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}.$ Assume that for all
weights
$\vec{w}=\left(w_{1},\ldots,w_{m}\right),\nu_{\vec{w}}=\prod_{s=1}^{m}w_{s}^{p/p_{s}},$
$b_{s}\in\mathrm{BMO}_{p_{s}}(w_{s})\cap\mathrm{BMO}$ with $1\leq s\leq l,$
and $\mathcal{T}_{\vec{b}}$ satisfies the Hypothesis 2. If $0<p\leq 1,$ then

$\left\|\mathcal{T}_{\vec{b}}(\vec{f})\right\|_{L^{p}\left(\nu_{\vec{w}}\right)}\leq
C\|\vec{b}\|_{\mathrm{BMO}}^{*}\prod_{s=1}^{m}\left\|f_{s}\right\|_{L^{p_{s}}(Mw_{s})},$

where $C$ is independent of $\vec{w}$ and $\vec{b},$ and

$\|\vec{b}\|_{\mathrm{BMO}}^{*}:=\max_{\vec{\gamma}\in\\{1,2\\}^{l}}\\{\prod_{s:\gamma_{s}=1}\|b_{s}\|_{\mathrm{BMO}_{p_{s}}(w_{s})}\prod_{s:\gamma_{s}=2}\|b_{s}\|_{\mathrm{BMO}}\\}.$

######  Remark 1.15.

Since the weights $\vec{w}=\left(w_{1},\ldots,w_{m}\right)$ are arbitrary, the
conditions $b_{s}\in\mathrm{BMO}_{p_{s}}(w_{s})\cap\mathrm{BMO}(1\leq s\leq
l)$ are required. However, the following two corollaries show that when the
weights $\vec{w}$ satisfies some restrictive conditions ( e.g., $w_{s}\in
A_{\infty}^{\text{weak}}$ with $1\leq s\leq l$ ), the conditions of the
Theorem 1.14 can be reduced to $b_{s}\in\mathrm{BMO}(1\leq s\leq l).$

######  Corollary 1.16.

Let $m\geq
2,I=\\{i_{1},\ldots,i_{l}\\}=\\{1,\ldots,l\\}\subseteq\\{1,\ldots,m\\},\vec{b}=(b_{1},\ldots,b_{l}).$
Let $0<p\leq 1,$ $1<p_{1},\ldots,p_{m}<\infty$ and
$\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}.$ Assume that
$\vec{w}=\left(w_{1},\ldots,w_{m}\right)$,
$\nu_{\vec{w}}=\prod_{s=1}^{m}w_{s}^{p/p_{s}},$ $\vec{b}\in\mathrm{BMO}^{l}$
and $\mathcal{T}_{\vec{b}}$ satisfies the Hypothesis 2. Then it holds that

  1. (1)

Suppose that $w_{s}\in A_{\infty}$ for any $1\leq s\leq m.$ Then

$\left\|\mathcal{T}_{\vec{b}}(\vec{f})\right\|_{L^{p}\left(\nu_{\vec{w}}\right)}\leq
C\prod_{s=1}^{l}[w_{s}]_{A_{\infty}}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\prod_{s=1}^{m}\left\|f_{s}\right\|_{L^{p_{s}}(Mw_{s})},$

  2. (2)

Suppose that $w_{s}\in A_{\infty}^{\text{weak}}$ for any $1\leq s\leq m.$ Then

(1.4)
$\left\|\mathcal{T}_{\vec{b}}(\vec{f})\right\|_{L^{p}\left(\nu_{\vec{w}}\right)}\leq
C\sum_{\vec{\gamma}\in\\{1,2\\}^{l}}\big{(}\prod_{s:\gamma_{s}=1}[w_{s}]_{A_{\infty}}^{\text{weak}}\big{)}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\prod_{s=1}^{m}\left\|f_{s}\right\|_{L^{p_{s}}(Mw_{s})},$

where the constant $C$ is independent of $\vec{w}$ and $\vec{b}.$

######  Remark 1.17.

According to [2, Example 3.2], when $w(x)=e^{x}$,
$[w]_{A_{\infty}}^{\text{weak}}$ is less than one. Thus
$\prod_{s=1}^{k}[w_{s}]_{A_{\infty}}^{\text{weak}}$ is not monotonically
increasing with $0\leq k\leq l$. This is the reason why the term
$\sum_{\vec{\gamma}\in\\{1,2\\}^{l}}(\prod_{s:\gamma_{s}=1}[w_{s}]_{A_{\infty}}^{\text{weak}})$
appears in (1.4).

Finally, we present the weighted modular inequalities for multilinear iterated
commutators $\mathcal{T}_{\vec{b}}$, which are completely new even in the
unweighted case.

######  Theorem 1.18.

Let
$I=\\{i_{1},\ldots,i_{l}\\}=\\{1,\ldots,l\\}\subseteq\\{1,\ldots,m\\},\vec{b}=(b_{1},\ldots,b_{l})\in\mathrm{BMO}^{l},$
and $\mathcal{T}_{\vec{b}}$ satisfies the Hypothesis 2. Let $\phi$ be an
$N$-function with sub-multiplicative property. Then for any $1<r<\infty,$ we
have

  1. (1)

if $r<i_{\phi}<\infty,$ then there exists a constant $\alpha$ such that for
any $1<q<\frac{i_{\phi}}{r}$ and $w\in A_{q},$

$\displaystyle\int_{\mathbb{R}^{n}}\phi\left(\mathcal{T}_{\vec{b}}(\vec{f})(x)\right)w(x)dx\lesssim$
$\displaystyle[w]_{A_{\infty}}^{(l+1)(\alpha
C_{1}+1)}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}^{1+\alpha C_{1}}$
$\displaystyle\times\left(\prod_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi^{m}\left([w]_{A_{q}}^{\frac{1}{qr}}\left|f_{i}(x)\right|\right)w(x)dx\right)^{\frac{1}{m}};$

  2. (2)

if $1<i_{\phi}\leq r,$ then there exists a constant $\alpha$ such that for any
$1<q<i_{\phi}$ and $w\in A_{q},$

$\displaystyle\int_{\mathbb{R}^{n}}\phi\left(\mathcal{T}_{\vec{b}}(\vec{f})(x)\right)w(x)dx\lesssim$
$\displaystyle[w]_{A_{\infty}}^{(l+1)(\alpha
C_{1}+1)+1+mC_{1}}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}^{1+\alpha C_{1}}$
$\displaystyle\times\left(\prod_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi^{m}\left([w]_{A_{q}}^{\frac{2}{q}}\left|f_{i}(x)\right|\right)w(x)dx\right)^{\frac{1}{m}},$

where the definitions of the $N$-function, $i_{\phi}$ and $C_{1}$ are listed
in Section 2.4.

###  1.4. Historical background

Local decay estimates. A local decay estimate is an inequality of the form:

(1.5) $\varphi(t):=\frac{1}{|Q|}\big{|}\left\\{x\in
Q:\left|T_{1}f(x)\right|>t\left|T_{2}f(x)\right|\right\\}\big{|}\leq
c_{1}e^{-c_{2}t},\quad t>0,$

where $T_{1}$ is a linear or sublinear operator and $T_{2}$ is an appropriate
maximal function. Inequality (1.5) reflects accurately the extent that an
operator is locally controlled by cerain maximal operator and provides us with
enough information to measure the size of $T_{1}$ and $T_{2}.$ More precisely,
local decay estimation greatly improves the Coifman-Fefferman inequality as
follows:

$\left\|T_{1}f\right\|_{L^{p}(w)}\leq c\|T_{2}f\|_{L^{p}(w)}$

with $0<p<\infty$ and $w\in A_{\infty}.$ In [14], it is shown that the above
inequality holds for maximal singular integral operator $T_{1}=T^{*}$ and
Hardy-Littlewood maximal operator $T_{2}=M$. But its proof depends on the
good-$\lambda$ technique, so there is no access to the dependence of the
constant $c$ on the weight $w$ or $p.$ In 1993, Buckley [7] obtained an
improved good-$\lambda$ inequality with a local exponential decay in $t,$

(1.6) $\left|\left\\{x\in Q:T^{*}f(x)>2\lambda,Mf(x)\leq
t\lambda\right\\}\right|\leq ce^{-c/t}|Q|.$

As an application, the sharp constant dependence on weighted norm of $T^{*}$
was given in [7] by using inequality (1.6)

The estimate of type (1.5) is an improved version of inequality (1.6) due to
Karagulyan [33]. Unfortunately, it is unknown whether Karagulyan’s method
could be applied to operators besides maximal singular integral operator
$T^{*}.$ In 2013, Ortiz-Caraballo et al. [45] used a different approach to
obtain estimates in the form of (1.5), and their approach is generalized
enough to obtain local decay estimates for operators such as Calderón-Zygmund
operators, vector-valued extensions of the maximal function or Calderón-
Zygmund operators, commutators of singular integrals with BMO functions and
higher order commutators. For the most recent results on local decay
estimations, we refer to [9, 8, 64] and the references therein.

Mixed weak type estimates. Mixed weak type estimates, also known as Sawyer-
type inequalities, mean that for the weight functions $u,v$ and the operator
$T,$ the following inequality holds:

(1.7)
$\left\|\frac{T(fv)}{v}\right\|_{L^{1,\infty}\left(\mathbb{R}^{n},uv\right)}\lesssim\|f\|_{L^{1}\left(\mathbb{R}^{n},uv\right)}.$

The study of this type of inequalities has a long history. In 1977,
Muckenhoupt and Wheeden [43] first introduced a new weak type inequality which
takes into account the perturbation of the Hardy-Littlewood maximal operator
with $A_{1}$ weights,

$|\\{x\in\mathbb{R}:w(x)Mf(x)>t\\}|\leq
c_{w}\frac{1}{t}\int_{\mathbb{R}}|f|w(x)dx.$

It is worth mentioning that $w\in A_{1}$ is no longer a necessary condition to
guarantee the validity of weak endpoint results [43, Section 5].

Later on, in order to give a new proof of Muckenhoupt’s theorem, Sawyer [58]
obtained the following result:

(1.8)
$uv\left(\left\\{x\in\mathbb{R}:\frac{M(fv)(x)}{v(x)}>t\right\\}\right)\leq
c_{u,v}\frac{1}{t}\int_{\mathbb{R}}|f|u(x)v(x)dx,$

where $u,v\in A_{1}.$ Sawyer [58] conjectured that (1.8) should be true for
the Hilbert transform. In 2005, Cruz-Uribe et al. [17] extended (1.8) to
higher dimensions and actually showed that Sawyer’s conjecture also holds for
Calderón-Zygmund operators. For $u\in A_{1}$, and $v\in A_{1}$ or $uv\in
A_{\infty},$ it was demonstrated in [17] that the following estimate is valid
for either the Calderón-Zygmund operator or the Hardy-Littlewood maximal
operator,

(1.9)
$uv\left(\left\\{x\in\mathbb{R}^{n}:\frac{|T(fv)(x)|}{v(x)}>t\right\\}\right)\leq\frac{C}{t}\int_{\mathbb{R}^{n}}|f(x)|u(x)v(x)dx,\quad
t>0.$

The left side of this inequality has no difference from the standard weak-type
estimates except a weight function appearing in the level set of the operator
$T$. However, it would be extraordinarily difficult to deal with the left side
of (1.9). There are two main obstacles. The first one is that the product of
$uv$ may become more singular. For example, if one chooses
$u(x)=v(x)=|x|^{-1/2}$ in $\mathbb{R}$, then, $u,v\in A_{1}$. But the product
$uv$ is not locally integrable. The second drawback is that the structure of
such sets in the left side of (1.9) makes it impossible or very difficult to
measure them using classical tools such as Vitali covering lemma or
interpolation theorem. But once (1.9) holds for some operator $T$, one may
obtain immediately a weak-type endpoint estimate for it by taking $u\in A_{1}$
and $v\equiv 1.$

Recently, the study of mixed weak type estimates have attracted much
attention. Among them are the works for multilinear Hardy-Littlewood maximal
operators in [41], the multilinear bounded oscillation operators in [8],
Lorentz spaces extensions in [53], the multilinear maximal operators and
bilinear sparse operators in homogeneous spaces in [32].

Fefferman-Stein inequalities. For an operator $T$ and any weight $w,$ recall
that the Fefferman-Stein type inequalities are the estimates of the form:

(1.10)
$w\left(\left\\{x\in\mathbb{R}^{n}:|Tf(x)|>\lambda\right\\}\right)\leq\frac{C}{\lambda}\int_{\mathbb{R}^{n}}|f(x)|Mw(x)dx$

and

(1.11) $\int_{\mathbb{R}^{n}}|Tf(x)|^{p}w(x)dx\leq
C\int_{\mathbb{R}^{n}}|f(x)|^{p}Mw(x)dx,\qquad\hbox{for $1<p<\infty.$}$

In 1971, Fefferman and Stein [23] showed that (1.10) and (1.11) hold for
Hardy-Littlewood maximal operator $T=M.$ This type of inequalities was
extended to many operators, and (1.10) was shown to be true for square
functions [12], but false for fractional integral operators [11].

It was shown by Pérez [48] that (1.10) holds if $M$ is replaced by iterated
operator $M^{2}$ or even by the operator $M_{L(\log L)^{\varepsilon}}$ with
$\varepsilon>0,$ and (1.11) is true if $M$ is replaced by $M^{\lfloor
p\rfloor+1}$ where $\lfloor p\rfloor$ is the integer part of $p$. However, in
2012, Reguera and Thiele [57] gave an example to show that the estimate of
(1.10) does not hold for the Hilbert transform (this disproves the so-called
Muckenhoupt-Wheeden conjecture). For the recent progress of Fefferman-Stein
inequalities, see [28, 55].

Modular inequalities. The modular inequalities concern the following
estimates:

(1.12) $\int_{\mathbb{R}^{n}}\phi(|T_{1}f(x)|)w(x)dx\leq
C\int_{\mathbb{R}^{n}}\phi(|T_{2}f(x)|)w(x)dx$

and its corresponding weak version,

$\sup_{\lambda>0}\phi(\lambda)w\big{(}\\{x\in\mathbb{R}^{n}:|T_{1}f(x)|>\lambda\\}\big{)}\leq
C\sup_{\lambda>0}\phi(\lambda)w\big{(}\\{x\in\mathbb{R}^{n}:|T_{2}f(x)|>\lambda\\}\big{)},$

where $T_{1},T_{2}$ are linear or sublinear operators, $w\in A_{\infty}$ and
$\phi\geq 0$ is an increasing function satisfying some very mild condition.

The modular inequality comes from the study of endpoint estimates for
operators. It is well known that the commutators of the Calderón-Zygmund
operators $T$ with BMO function $b$ have the following Coifman-Fefferman
inequality [47]

$\int_{\mathbb{R}^{n}}|[b,T]f(x)|^{p}w(x)dx\leq
C\|b\|_{\mathrm{BMO}}^{p}\int_{\mathbb{R}^{n}}M^{2}f(x)^{p}w(x)dx$

with any $0<p<\infty$ and any $w\in A_{\infty}$. Notice that the commutators
of the Calderón-Zygmund operators are not weak (1,1) type since the maximal
operator used to control the commutator in the above estimate is $M^{2}$, but
there is no weak (1,1) property for $M^{2}$ (see [34]), which is replaced by
the following weak $L\log L$-type estimate:

$\left|\left\\{x\in\mathbb{R}^{n}:M^{2}f(x)>\lambda\right\\}\right|\leq
C\int_{\mathbb{R}^{n}}\phi\left(\frac{|f(x)|}{\lambda}\right)dx,$

where $\phi(t)=t\left(1+\log^{+}t\right)$. This result, however, is not
compatible with classical function spaces. Therefore this type of estimates is
called modular inequality (see [34]), and it gives an appropriate endpoint
result for $M^{2}$ and this type of estimates has good interpolation
properties. Based on this, one may wonder whether such estimates are also
available for commutators or not. It was Pérez [46] who considered a weak type
of Coifman-Fefferman inequality with the form

(1.13)
$\sup_{\lambda>0}\varphi(\lambda)w\left(\\{x\in\mathbb{R}^{n}:|[b,T]f(x)|>\lambda\\}\right)\leq
C\sup_{\lambda>0}\varphi(\lambda)w\left(\\{x\in\mathbb{R}^{n}:M^{2}f(x)>\lambda\\}\right),$

where $T$ is Calderón-Zygmund operator,
$\varphi(\lambda)=\frac{\lambda}{1+\log^{+}\lambda^{-1}}$ and $w\in
A_{\infty}.$ Inequality (1.13) is very important in illustrating the endpoint
estimates of the commutators since as a consequence, it gives that

$\left|\left\\{x\in\mathbb{R}^{n}:|[b,T]f(x)|>\lambda\right\\}\right|\leq
C_{\|b\|_{\mathrm{BMO}}}\int_{\mathbb{R}^{n}}\phi\left(\frac{|f(x)|}{\lambda}\right)dx.$

Notice that there is a function $\varphi$ on both sides of (1.13) which is not
homogeneous and hence each side of this inequality is not a norm or quasi-
norm. But this type of inequality reflects the properties of the operator
itself. Since then, many works have enriched the literature in this area. For
example, the extrapolation theorem for modular inequality [19], the modular
inequalities of strong type for maximal truncated Calderón-Zygmund operators
[2], and for variation operators of singular integrals and their commutators
[61].

###  1.5. Structure of the paper

The organization of the rest of this article is as follows: In Section 2 we
present some preliminaries, including the properties of Muckenhoupt weights,
weak $A_{\infty}$ weights, Young functions and Orlicz maximal operators.
Section 3 contains the proof of local exponential decay estimates (1.1) of
iterated commutators, both in quantitatively weighted and unweighted versions.
In Section 4, the proof of Coifman-Fefferman inequality (Theorem 1.8) will be
given, which will be used later. Section 5 is devoted to establishing mixed
weak type estimates and the purpose of Section 5 is to prove Theorem 1.14. The
modular inequalities are proved in Section 7 based on the sparse domination.
Finally, some applications, including multilinear $\omega$-Calderón-Zygmund
operators, multilinear Fourier integral operators and Stein’s square
functions, will be given in Section 8.

Throughout this paper, we always use $C$ to denote a positive constant, which
is independent of the main parameters, but it may change at each occurrence.
Let $L_{\text{loc }}^{1}(\mathbb{R}^{n})$ denote the set of all locally
integrable functions on $\mathbb{R}^{n},$ and $\mathbb{N}=\\{0,1,2,\ldots\\}.$
For any set $E$, we use $\chi_{E}$ to denote its characteristic function. Let
$\mathscr{S}\left(\mathbb{R}^{n}\right)$ denote the collection of all Schwartz
functions on $\mathbb{R}^{n}$, equipped with the classical well-known topology
determined by a countable family of norms, and
$\mathscr{S}^{\prime}\left(\mathbb{R}^{n}\right)$ its topological dual. If,
for any $a,b\in\mathbb{R},a\leq Cb$ ($a\geq Cb,$ respectively), we then denote
$a\lesssim b$ where $C$ is independent of $a$ and $b$, $(a\gtrsim b,$
respectively). If $a\lesssim b\lesssim a$, we then denote $a\backsimeq b$.

##  2\. Preliminary

We begin by presenting some basic facts for sparse family, modular
inequalities, Muckenhoupt weights and Orlicz maximal operators.

###  2.1.  Sparse family

In this subsection, we will introduce a quite useful tool, dyadic calculus,
which is taken from [37, 36].

In the following, let $\mathcal{D}\left(Q\right)$ be the set of all dyadic
cubes with respect to cube $Q$. These cubes obtained by repeated subdivision
$Q$ and its descendants in $2^{n}$ cubes with the same side length.

######  Definition 2.1.

A collection, $\mathcal{D}$ of cubes is said to be a dyadic lattice if it
satisfies the following properties:

  1. (i).

If $Q\in\mathcal{D}$, then each descendant of $Q$ is in $\mathcal{D}$ as well;

  2. (ii).

For every cubes $Q_{1},Q_{2}\in\mathcal{D}$, we can find a common ancestor
$Q\in\mathcal{D}$ such that $Q_{1},Q_{2}\in D(Q)$;

  3. (iii).

For any compact set $K\subset\mathbb{R}^{n}$, there exists a cube
$Q\in\mathcal{D}$ containing $K$.

In dyadic calculus, the following Three Lattice Theorem (see [37, Theorem
3.1]) plays an important role, which provides a clearly understanding of the
structure for dyadic lattics.

######  Lemma 2.2 ([37]).

Given a dyadic lattice $\mathcal{D}$, there exist $3^{n}$ dyadic lattices
$\\{\mathcal{D}_{j}\\}_{j=1}^{3^{n}}$ such that

$\\{3Q:Q\in\mathcal{D}\\}=\bigcup_{j=1}^{3^{n}}\mathcal{D}_{j}$

and for each cube $Q\in\mathcal{D}$, there is a cube $R_{Q}\in\mathcal{D}_{j}$
for some $j$ such that $Q\subseteq R_{Q}$ and $3l_{Q}=l_{R_{Q}}$.

We need to introduce one more definition.

######  Definition 2.3.

Given a dyadic lattice $\mathcal{D},$ a subset $\mathcal{S}$ of $\mathcal{D}$
is said to be a $\eta$-sparse family with $\eta\in(0,1)$ if for every cube
$Q\in\mathcal{S},$

$|\bigcup_{P\in\mathcal{S},P\subsetneq Q}P|\leq(1-\eta)\left|Q\right|.$

Equivalently, if we define
$E(Q)=Q\backslash\bigcup_{P\in\mathcal{S},P\subsetneq Q}P,$ then a simple
calculation shows that the sets $E(Q)$ are pairwisely disjoint and
$|E(Q)|\geq\eta|Q|$.

Let $\mathcal{D}$ be a dyadic lattice and $\mathcal{S}\subseteq\mathcal{D}$ be
a $\eta$-sparse family, we define the sparse operator
$\mathcal{A}_{r,\mathcal{S}}$ with $r>0$ as

$\mathcal{A}_{r,\mathcal{S}}f(x)=\sum_{Q\in\mathcal{S}}\langle|f|^{r}\rangle_{Q}^{1/r}\chi_{Q}(x)=\sum_{Q\in\mathcal{S}}\left(\frac{1}{|Q|}\int_{Q}|f(y)|^{r}dy\right)^{\frac{1}{r}}\chi_{Q}(x).$

The following lemma in [39, Lemma 5.1] for the oscillation estimation of
sparse families will play a crucial role in proving the local estimation of
the commutators.

######  Lemma 2.4 ([39]).

Let $\mathscr{D}$ be a dyadic lattice and let $\mathcal{S}\subset\mathscr{D}$
be a $\gamma$-sparse family. Assume that $b\in L_{\text{loc
}}^{1}(\mathbb{R}^{n})$. Then there exists a
$\frac{\gamma}{2(1+\gamma)}$-sparse family
$\widetilde{\mathcal{S}}\subset\mathscr{D}$ such that
$\mathcal{S}\subset\widetilde{\mathcal{S}}$ and for every cube
$Q\in\widetilde{\mathcal{S}}$,

$\left|b(x)-{\langle b\rangle}_{Q}\right|\leq
2^{n+2}\sum_{R\in\widetilde{\mathcal{S}},R\subseteq Q}\langle\left|b-{\langle
b\rangle}_{R}\right|\rangle_{R}\chi_{R}(x)$

for a.e. $x\in Q$.

###  2.2.  The classical weights

In this subsection, we will present the relevant definitions of classical
Muckenhoupt weights, multiple weights, and the weak $A_{\infty}$ class, as
well as their main properties. We will start with the multilinear maximal
function $\mathcal{M}$ defined by

$\mathcal{M}(\vec{f})(x):=\sup_{Q\ni
x}\prod_{i=1}^{m}\frac{1}{|Q|}\int_{Q}\left|f_{i}(y)\right|dy,$

where the supremum is taken over all cubes $Q$ containing $x$.
The following multiple weights associated with $\mathcal{M}$ were introduced
in [38].

######  Definition 2.5 ([38]).

Let $\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}$ with $1\leq
p_{1},\ldots,p_{m}<\infty$, $\vec{w}=\left(w_{1},\ldots,w_{m}\right)$, where
each $w_{i}$ is a nonnegative and locally integrable function on
$\mathbb{R}^{n}$, we say that $\vec{w}$ belongs to multiple weights
$A_{\vec{p}}$ if

$[\vec{w}]_{A_{\vec{p}}}:=\sup_{Q}\left(\frac{1}{|Q|}\int_{Q}\nu_{\vec{w}}(x)dx\right)\prod_{j=1}^{m}\left(\frac{1}{|Q|}\int_{Q}w_{i}^{1-p_{i}^{\prime}}(x)dx\right)^{p/p_{i}^{\prime}}<\infty,$

where the supremum is taken over all cubes $Q\subset\mathbb{R}^{n}$ and
$\nu_{\vec{w}}=\prod_{j=1}^{m}w_{j}^{p/p_{j}}$. When $p_{i}=1$ for some $i,$
$\left(\frac{1}{|Q|}\int_{Q}w_{i}{}^{1-p_{i}^{\prime}}\right)^{1/p_{i}^{\prime}}$
is understood as $\left(\inf_{Q}w_{i}\right)^{-1}$.

If $m=1$, the multiple $A_{\vec{p}}$ weights coincide with the classical
Muckenhoupt $A_{p}$ weights. In the linear case, we say a weight $w$ belongs
to the class $A_{1}$ if there is a constant $C$ such that

$\frac{1}{|Q|}\int_{Q}w(y)dy\leq C\inf_{Q}w,$

and the infimum of these constants $C$ is called the $A_{1}$ constant of $w.$

The following characterization of multiple weights $A_{\vec{p}}$ is given in
[38, Theorem 3.6].

######  Lemma 2.6 ([38]).

Let $\vec{w}=\left(w_{1},\ldots,w_{m}\right)$ and $1\leq
p_{1},\ldots,p_{m}<\infty$. Then $\vec{w}\in A_{\vec{p}}$ if and only if

$\left\\{\begin{array}[]{l}w_{j}^{1-p_{j}^{\prime}}\in
A_{mp_{j}^{\prime}},\quad j=1,\ldots,m,\\\ \nu_{\vec{w}}\in
A_{mp},\end{array}\right.$

where the condition $w_{j}^{1-p_{j}^{\prime}}\in A_{mp_{j}^{\prime}}$ in the
case $p_{j}=1$ is understood as $w_{j}^{1/m}\in A_{1}$.

We now introduce definitions of several other classes of weights which will be
used later. For more information about them, see [2, 44, 49]. Since the
$A_{p}$ classes are increasing with respect to $p$, the $A_{\infty}$ class of
weights is defined in a natural way by

$A_{\infty}=\bigcup_{p>1}A_{p}.$

A weight $w\in A_{\infty}$ if and only if

$[w]_{A_{\infty}}:=\sup_{Q}\frac{1}{w(Q)}\int_{Q}M\left(w\chi_{Q}\right)(x)dx<\infty.$

This form of $[w]_{A_{\infty}}$ is known as the Fujii-Wilson $A_{\infty}$
constant and was first introduced in [24]. Next we give a class of weights
that is more general than $A_{\infty}$. A weight $w$ belonging to weak
$A_{\infty}$ class means that there exist $0<c,\delta<\infty$ such that for
all cubes $Q$ and all measurable subsets $E$ of $Q$:

$w(E)\leq c\left(\frac{|E|}{|Q|}\right)^{\delta}w(2Q).$

This class of weights is derived from [59], but is very interesting in its own
way as it appears in many contexts, such as quasiregular mappings theory or
the regularity for solutions of elliptic PDE’s (see [6]).

Similar to the Fujii-Wilson $A_{\infty}$ constants, the weak $A_{\infty}$
constants can be expressed as follows

$[w]_{A_{\infty}}^{\text{weak
}}:=\sup_{Q}\frac{1}{w(2Q)}\int_{Q}M\left(w\chi_{Q}\right)(x)dx.$

It is shown in [2] that the constant $2$ in the average could be replaced by
any parameter $\kappa>1,$ and the following lemma holds.

######  Lemma 2.7 ([2]).

Let $w\in A_{\infty}^{\text{weak}},$ then for all $Q$ cubes in
$\mathbb{R}^{n}$ with sides parallel to the axes,

$\left(\frac{1}{|Q|}\int_{Q}w^{r}(x)dx\right)^{\frac{1}{r}}\leq\frac{2}{|2Q|}\int_{2Q}w(x)dx,$

with

$1<r\leq 1+\frac{1}{\tau_{n}[w]_{A_{\infty}}^{weak}},$

where $\tau_{n}$ is a dimensional constant with the property $\tau_{n}\simeq
2^{n}.$

###  2.3.  Young function and Orlicz maximal operators

We need to recall some fundamental facts about Young functions and Orlicz
spaces. For more information and a lively exposition about these spaces, we
refer the readers to [56].

Let $\Phi$ be the set of functions $\phi:[0,\infty)\longrightarrow[0,\infty)$
which are non-negative, increasing, $\lim_{t\rightarrow\infty}\phi(t)=\infty$
and $\lim_{t\rightarrow 0}\phi(t)=0.$ $\phi$ is said to be a Young function If
$\phi\in\Phi$ is convex. Given a Young function $\phi,$ the Orlicz space
$L_{\phi}(\mu)$ with respect to the measure $\mu$ is defined to be the set of
measurable functions $f$, such that for some $\lambda>0$,

$\int_{\mathbb{R}^{n}}\phi\left(\frac{|f(x)|}{\lambda}\right)d\mu<\infty.$

The Luxemburg norm of $f$ over a cube $Q$ is defined by

$\|f\|_{\phi(\mu),Q}:=\inf\left\\{\lambda>0:\frac{1}{\mu(Q)}\int_{Q}\phi\left(\frac{|f(x)|}{\lambda}\right)d\mu\leq
1\right\\}.$

For the sake of convenience, we denote $\|f\|_{\phi(\mu),Q}=\|f\|_{\phi,Q}$ if
$\mu$ is the Lebesgue measure and $\|f\|_{\phi(\mu),Q}=\|f\|_{\phi(w),Q}$ if
$\mu=wdx$ is an absolutely continuous measure with respect to the Lebesgue
measure.

A simple yet important observation in Orlicz space $L_{\phi}(\mu)$ is that
each Young function $\phi$ satisfies the generalized Hölder inequality:

$\frac{1}{\mu(Q)}\int_{Q}|fg|d\mu\leq
2\|f\|_{\phi(\mu),Q}\|g\|_{\bar{\phi}(\mu),Q},$

where $\bar{\phi}(t)=\sup_{s>0}\\{st-\phi(s)\\}$ is the complementary function
of $\phi$.

Let $\mathcal{D}$ be a dyadic grid and $M_{\phi}^{\mathcal{D}}$ be the dyadic
Orlicz maximal operator defined by

$M_{\phi}^{\mathcal{D}}f(x):=\sup_{Q\ni x,Q\in\mathcal{D}}\|f\|_{\phi,Q},$

where the supremum is taken over all the dyadic cubes containing $x$.
Similarly, we denote the classical Orlicz maximal operator by $M_{\phi}$.

We will employ the following particular examples of maximal operators several
times.

  * •

If $\phi(t)=t^{r}$ with $r>1,$ then $M_{\phi}=M_{r}$.

  * •

If $\phi(t)=t\log(e+t)^{\alpha}$ with $\alpha>0,$ then $\bar{\phi}(t)\simeq
e^{t^{1/\alpha}}-1$ and we denote $M_{\phi}=M_{L\log L^{\alpha}}$. Then $M\leq
M_{\phi}\lesssim M_{r}$ for all $1<r<\infty.$ Moreover, $M_{\phi}\backsimeq
M^{l+1},$ where $\alpha=l\in\mathbb{N}$ and $M^{l+1}$ is $M$ iterated $l+1$
times.

We end this subsection by the definition of the multilinear $L(\log
L)$-maximal operators

$\mathcal{M}_{L(\log L)}(\vec{f})(x):=\sup_{Q\ni
x}\prod_{i=1}^{m}\left\|f_{i}\right\|_{L(\log L),Q}.$

###  2.4.  Modular inequality

In this subsection, we will collect some concepts related to Young functions
and modular inequalities from [19].

A function $\phi\in\Phi$ is said to be quasi-convex if there exist a convex
function $\widetilde{\phi}$ and $a_{1}\geq 1$ such that

$\widetilde{\phi}(t)\leq\phi(t)\leq
a_{1}\widetilde{\phi}\left(a_{1}t\right),\quad t\geq 0.$

Given a positive increasing function $\phi,$ we define the lower and upper
dilation indices of $\phi,$ respectively, as follows:

$i_{\phi}=\lim_{t\rightarrow 0^{+}}\frac{\log h_{\phi}(t)}{\log
t}=\sup_{0<t<1}\frac{\log h_{\phi}(t)}{\log t},\quad
I_{\phi}=\lim_{t\rightarrow\infty}\frac{\log h_{\phi}(t)}{\log
t}=\inf_{1<t<\infty}\frac{\log h_{\phi}(t)}{\log t},$

where $h_{\phi}(t)=\sup_{s>0}\frac{\phi(st)}{\phi(s)},t>0.$

Now we turn to the $\Delta_{2}$ condition. A function $\phi\in\Phi$ satisfies
the $\Delta_{2}$ condition (we denote $\phi\in\Delta_{2}$) if $\phi$ is
doubling, that is, $\phi(2t)\leq C\phi(t)$. A key fact is that if $\phi$ is
quasi-convex, then $i_{\phi}\geq 1$ and that $\phi\in\Delta_{2}$ if and only
if $I_{\phi}<\infty.$ Moreover, $\bar{\phi}\in\Delta_{2}$ if and only if
$i_{\phi}>1,$ where $\bar{\phi}$ is the complementary function of $\phi$
defined in Section 2.3.

Given a weight $w\in A_{\infty}$ and $\phi\in\Phi$, the modular of $f$ is
defined by

$\rho_{w}^{\phi}(f)=\int_{\mathbb{R}^{n}}\phi(|f(x)|)w(x)dx.$

The collection

$Z_{w}^{\phi}=\left\\{f:\rho_{w}^{\phi}(f)<\infty\right\\}$

is called as a modular space. A multi(sub)linear operator $T$ is said to
satisfy a modular inequality on $Z_{w}^{\phi}$ if there exist constants
$c_{i}^{(1)},c_{i}^{(2)}>0$ with $i=1,\ldots,m,$ such that

$\rho_{w}^{\phi}(T\vec{f})\leq\prod_{i=1}^{m}c_{i}^{(1)}\rho_{w}^{\phi}\left(c_{i}^{(2)}f_{i}\right).$

##  3\.  Proofs of Theorems 1.4 and Corollary 1.6

In this section, we will prove Theorem 1.4 in the unweighted setting and
Corollary 1.6 in the weighted case. We begin with the proof of Theorem 1.4.

###### Proof of Theorem 1.4.

Let $I=\\{1,\ldots,l\\}\subseteq\\{1,\ldots,m\\}$ and fix a cube $Q_{0}$ such
that $\operatorname{supp}\left(f_{s}\right)\subset Q_{0}$ for $1\leq s\leq m.$
By the Hypothesis 1, we can see that there exists a $\eta$-sparse family
$\mathcal{F}\subset\mathcal{D}\left(Q_{0}\right)$ such that for a.e. $x\in
Q_{0}$,

$\displaystyle{\left|\mathcal{T}_{\vec{b}}\vec{f}(x)\right|}$
$\displaystyle=\left|\mathcal{T}_{\vec{b}}(\vec{f}\chi_{3Q_{0}})(x)\right|$
$\displaystyle\leq
C\sum_{Q\in{\mathcal{F}}}\sum_{\vec{\gamma}\in{\\{1,2\\}^{l}}}\left(\prod_{s=1}^{l}\mathcal{R}(b_{s},f_{s},Q,\gamma_{s})\right)\prod_{s=l+1}^{m}{\langle|f_{s}|\rangle}_{3Q}\chi_{Q}(x),$

where

$\mathcal{R}(b,f,Q,\gamma)=\begin{cases}|b-{\langle
b\rangle}_{3Q}|{\langle|f|\rangle}_{3Q},&\text{if}\quad\gamma=1,\\\
{\langle|(b-{\langle
b\rangle}_{3Q})f|\rangle}_{3Q},&\text{if}\quad\gamma=2.\end{cases}$

Consider the pointwisely estimate of

$\displaystyle\mathcal{T}_{\mathcal{F}}^{\vec{\gamma}}(\vec{f},\vec{b})(x):=\sum_{Q\in{\mathcal{F}}}\prod_{s=1}^{l}\mathcal{R}(b_{s},f_{s},Q,\gamma_{s})\prod_{s=l+1}^{m}{\langle|f_{s}|\rangle}_{3Q}\chi_{Q}(x).$

Without loss of generality, we may assume that
$\vec{\gamma}=(\overbrace{1,\ldots,1}^{l_{1}},\overbrace{2,\ldots,2}^{l-l_{1}})$
and write

$\displaystyle\mathcal{T}_{\mathcal{F}}^{\vec{\gamma}}(\vec{f},\vec{b})(x)=$
$\displaystyle\sum_{Q\in{\mathcal{F}}}\prod_{s=1}^{l_{1}}\left|b_{s}(x)-{\langle
b_{s}\rangle}_{3Q}\right|{\langle|f_{s}|\rangle}_{3Q}\prod_{s=l_{1}+1}^{l}{\langle|(b_{s}-{\langle
b_{s}\rangle}_{3Q})f_{s}|\rangle}_{3Q}$
$\displaystyle\times\prod_{s=l+1}^{m}{\langle|f_{s}|\rangle}_{3Q}\chi_{Q}(x).$

First, we observe that

(3.1) $|b_{s}-{\langle b_{s}\rangle}_{3Q}|\leq|b_{s}-{\langle
b_{s}\rangle}_{Q}|+C_{n}\|b_{s}\|_{\mathrm{BMO}},\quad 1\leq s\leq l_{1}.\\\ $

By applying the Lemma 2.4 to $b_{1},$ we know that there exists a
$\frac{\gamma}{2(1+\gamma)}$-sparse family
$\tilde{{{\mathcal{S}}_{1}}}\subset\mathscr{D}(Q_{0})$ such that
$\mathcal{F}\subset\tilde{{{\mathcal{S}}_{1}}}$ and

$\left|b_{1}(x)-{\langle b_{1}\rangle}_{Q}\right|\leq
2^{n+2}\sum_{R\in\tilde{{{\mathcal{S}}_{1}}},R\subseteq
Q}{\langle\left|b_{1}-{\langle
b_{1}\rangle}_{R}\right|\rangle}_{R}\chi_{R}(x)$

for a.e. $x\in Q$.

Note that, if ${\mathcal{S}}_{1}$ is a $\eta_{1}$-sparse family and
${\mathcal{S}}_{2}$ is a $\eta_{2}$-sparse family, then
${\mathcal{S}}_{1}\bigcup{\mathcal{S}}_{2}$ is a
$\frac{\eta_{1}\eta_{2}}{\eta_{1}+\eta_{2}}$-sparse family. In fact, by the
fact that ${\mathcal{S}}_{i}$ is $\frac{1}{\eta_{i}}$-Carleson ($i=1,2$), (see
[37, p. 22]), for any $Q\in\mathscr{D}(Q_{0}),$ it holds that

(3.2) $\sum_{P\in\mathcal{S}_{1}\bigcup\mathcal{S}_{2},P\subseteq
Q}|P|\leq\sum_{P\in\mathcal{S}_{1},P\subseteq
Q}|P|+\sum_{P\in\mathcal{S}_{2},P\subseteq
Q}|P|\leq\frac{\eta_{1}+\eta_{2}}{\eta_{1}\eta_{2}}|Q|,$

which implies that ${\mathcal{S}}_{1}\bigcup{\mathcal{S}}_{2}$ is a
$\frac{\eta_{1}\eta_{2}}{\eta_{1}+\eta_{2}}$-sparse family.

For $b_{2}$, there exists a $\frac{\gamma}{2(1+\gamma)}$-sparse family
$\tilde{{{\mathcal{S}}_{2}}}\subset\mathscr{D}(Q_{0})$ such that

$\left|b_{2}(x)-{\langle b_{2}\rangle}_{Q}\right|\leq
2^{n+2}\sum_{R\in\tilde{{\mathcal{S}_{1}}}\bigcup\tilde{\mathcal{S}_{2}},R\subseteq
Q}{\langle\left|b_{2}-{\langle
b_{2}\rangle}_{R}\right|\rangle}_{R}\chi_{R}(x).$

This estimate is also valid for $b_{1}$ and
$\tilde{{\mathcal{S}_{1}}}\bigcup\tilde{\mathcal{S}_{2}}$ is a
$\frac{\eta}{4(1+\eta)}$-sparse family. It is also convenient to denote
$\tilde{{\mathcal{S}_{1}}}\bigcup\tilde{\mathcal{S}_{2}}$ by
$\tilde{\mathcal{S}_{2}}.$ We iterate this procedure $l$ times to obtain a
$\frac{\eta}{2^{l}(1+\eta)}$-sparse family
$\tilde{{\mathcal{S}_{l}}}=:\tilde{{\mathcal{S}}}$ satisfying the following
property

(3.3) $\left|b_{t}(x)-{\langle b_{t}\rangle}_{Q}\right|\leq
2^{n+2}\sum_{R\in\tilde{{\mathcal{S}}},R\subseteq
Q}{\langle\left|b_{t}-{\langle
b_{t}\rangle}_{R}\right|\rangle}_{R}\chi_{R}(x),\quad 1\leq t\leq l.$

This indicates that there exists a sparse family
$\tilde{{{\mathcal{S}}}}\subset\mathscr{D}(Q_{0})$ such that for any $1\leq
s\leq l_{1}$ and $Q\in\mathcal{F}\subseteq\tilde{{{\mathcal{S}}}},$

$|b_{s}(x)-{\langle b_{s}\rangle}_{Q}|\leq
2^{n+2}\|b_{s}\|_{\mathrm{BMO}}\sum_{R\in\tilde{{{\mathcal{S}}}},R\subseteq
Q}\chi_{R}(x),$

which, together with (3.1), gives that

(3.4) $|b_{s}-{\langle b_{s}\rangle}_{3Q}|\leq
C_{n}\|b_{s}\|_{\mathrm{BMO}}(1+\sum_{R\in\tilde{{{\mathcal{S}}}},R\subseteq
Q}\chi_{R}(x)),\quad 1\leq s\leq l_{1}.$

We now turn to ${\langle|(b_{s}-{\langle
b_{s}\rangle}_{3Q})f_{s}|\rangle}_{3Q}$ with $l_{1}+1\leq s\leq l$, there are
two different ways to deal with it.
Method one: By the generalized Hölder’s inequality, we have

$\displaystyle{\langle|(b_{s}-{\langle b_{s}\rangle}_{3Q})f_{s}|\rangle}_{3Q}$
$\displaystyle\leq 2\left\|b_{s}-{\langle b_{s}\rangle}_{3Q}\right\|_{\exp
L,3Q}\|f_{s}\|_{L(\log L),3Q}$
$\displaystyle\lesssim\|b_{s}\|_{\mathrm{BMO}}\|f_{s}\|_{L(\log L),3Q}.$

This fact together with (3.4) easily yields

$\displaystyle\mathcal{T}_{\mathcal{F}}^{\vec{\gamma}}(\vec{b},\vec{f})(x)$
$\displaystyle\lesssim\sum_{Q\in{\mathcal{F}}}\prod_{s=1}^{l_{1}}\left(\|b_{s}\|_{\mathrm{BMO}}\left(1+\sum_{R\in\tilde{{{\mathcal{S}}}},R\subseteq
Q}\chi_{R}(x)\right){\langle|f_{s}|\rangle}_{3Q}\right)$
$\displaystyle\quad\quad\times\prod_{s=l_{1}+1}^{l}\|b_{s}\|_{\mathrm{BMO}}\|f_{s}\|_{L(\log
L),3Q}\prod_{s=l+1}^{m}{\langle|f_{s}|\rangle}_{3Q}\chi_{Q}(x)$
$\displaystyle\leq\left(1+\sum_{R\in\tilde{{{\mathcal{S}}}},R\subseteq
Q_{0}}\chi_{R}(x)\right)^{l_{1}}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\sum_{Q\in{\mathcal{F}}}\prod_{s=1}^{l}\|f_{s}\|_{L(\log
L),3Q}$
$\displaystyle\quad\times\prod_{s=l+1}^{m}{\langle|f_{s}|\rangle}_{3Q}\chi_{Q}(x)$
$\displaystyle\lesssim\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\sum_{Q\in{\mathcal{F}},Q\subseteq
Q_{0}}\prod_{s=1}^{l}\|f_{s}\|_{L(\log
L),3Q}\prod_{s=l+1}^{m}{\langle|f_{s}|\rangle}_{3Q}\chi_{Q}(x)$
$\displaystyle\quad+\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\left(\sum_{Q\in{\mathcal{F}},Q\subseteq
Q_{0}}\prod_{s=1}^{l}\|f_{s}\|_{L(\log
L),3Q}\prod_{s=l+1}^{m}{\langle|f_{s}|\rangle}_{3Q}\chi_{Q}(x)\right)$
$\displaystyle\quad\times\left(\sum_{R\in\tilde{{\mathcal{S}}},R\subseteq
Q_{0}}\chi_{R}(x)\right)^{l_{1}}.$

By the definition of $M_{L(\log L)}^{(1,l)},$ we have

(3.5)
$\displaystyle\mathcal{T}_{\mathcal{F}}^{\vec{\gamma}}(\vec{b},\vec{f})(x)$
$\displaystyle\lesssim\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\mathcal{M}_{L(\log
L)}^{(1,l)}(\vec{f})(x)\sum_{Q\in{\mathcal{F}},Q\subseteq Q_{0}}\chi_{Q}(x)$
$\displaystyle\quad+\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\mathcal{M}_{L(\log
L)}^{(1,l)}(\vec{f})(x)\left(\sum_{Q\in{\tilde{{\mathcal{S}}}},Q\subseteq
Q_{0}}\chi_{Q}(x)\right)^{l_{1}+1}$
$\displaystyle\lesssim\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\mathcal{M}_{L(\log
L)}^{(1,l)}(\vec{f})(x)\left(\sum_{Q\in{\tilde{{\mathcal{S}}}},Q\subseteq
Q_{0}}\chi_{Q}(x)\right)^{l_{1}+1}.$

Method two: For each $l_{1}+1\leq s\leq l$, using (3.3) again, we obtain

$\displaystyle{\langle|(b_{s}-{\langle b_{s}\rangle}_{3Q})f_{s}|\rangle}_{3Q}$
$\displaystyle\leq\frac{1}{|3Q|}\int_{3Q}\left|b_{s}(x)-{\langle
b_{s}\rangle}_{Q}\right||f_{s}(x)|dx$ $\displaystyle\quad+|{\langle
b_{s}\rangle}_{Q}-{\langle
b_{s}\rangle}_{3Q}|\frac{1}{|3Q|}\int_{3Q}|f_{s}(x)|dx$
$\displaystyle\lesssim\|b_{s}\|_{\mathrm{BMO}}\frac{1}{|3Q|}\int_{3Q}\sum_{R\in{\mathcal{\tilde{S}}},R\subseteq
Q}\chi_{R}(x)|f_{s}(x)|dx+\|b_{s}\|_{\mathrm{BMO}}{\langle|f_{s}|\rangle}_{3Q}$
$\displaystyle=\|b_{s}\|_{\mathrm{BMO}}\frac{1}{|3Q|}\Big{(}\sum_{R\in{\mathcal{\tilde{S}}},R\subseteq
Q}\int_{R}|f_{s}(x)|dx+\int_{3Q}|f_{s}(x)|dx\Big{)}.$

This estimate yields that

$\displaystyle{\langle|(b_{s}-{\langle b_{s}\rangle}_{3Q})f_{s}|\rangle}_{3Q}$
$\displaystyle\lesssim\frac{2\|b_{s}\|_{\mathrm{BMO}}}{|3Q|}\sum_{R\in{\mathcal{\tilde{S}}},R\in
Q}\int_{3R}|f_{s}(x)|dx$
$\displaystyle=\frac{2\|b_{s}\|_{\mathrm{BMO}}}{|3Q|}\sum_{R\in{\mathcal{\tilde{S}}},R\subseteq
Q}\frac{3^{n}}{|3R|}\int_{3R}|f_{s}(x)|dx\int_{Q}\chi_{R}(x)dx$
$\displaystyle=\frac{2\|b_{s}\|_{\mathrm{BMO}}}{|Q|}\int_{Q}\sum_{R\in{\mathcal{\tilde{S}}},R\subseteq
Q}{\langle|f_{s}|\rangle}_{3R}\chi_{R}(x)dx.$

Let $\mathcal{S}^{*}=\\{3R:R\in\mathcal{\tilde{S}}\\},$ then $\mathcal{S}^{*}$
is also a sparse family and

$\sum_{R\in{\mathcal{\tilde{S}}},R\subseteq
Q}{\langle|f_{s}|\rangle}_{3R}\chi_{R}(x)\leq\sum_{Q\in{\mathcal{S}^{*}}}{\langle|f_{s}|\rangle}_{Q}\chi_{Q}(x)=\mathcal{A}_{{S}^{*}}(f_{s})(x).$

Therefore

${\langle|(b_{s}-{\langle
b_{s}\rangle}_{3Q})f_{s}|\rangle}_{3Q}\lesssim\|b_{s}\|_{\mathrm{BMO}}\frac{1}{|Q|}\int_{Q}\mathcal{A}_{{\mathcal{S}}^{*}}(f_{s})(x)dx,\\\
$

which together with (3.4) implies that

(3.6)
$\displaystyle\mathcal{T}_{\mathcal{F}}^{\vec{\gamma}}(\vec{f},\vec{b})(x)$
$\displaystyle\lesssim\left(1+\sum_{R\in\tilde{{{\mathcal{S}}}},R\subseteq
Q_{0}}\chi_{R}(x)\right)^{l_{1}}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}$
$\displaystyle\quad\times\sum_{Q\in{\mathcal{F}}}\prod_{s=1}^{l_{1}}{\langle|f_{s}|\rangle}_{3Q}\prod_{s=l+1}^{m}{\langle|f_{s}|\rangle}_{3Q}\prod_{s=l_{1}+1}^{m}{\langle\mathcal{A}_{{{\mathcal{S}}}^{*}}f_{s}\rangle}_{3Q}\chi_{Q}(x)$
$\displaystyle\leq\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\left(1+\sum_{R\in\tilde{{{\mathcal{S}}}},R\subseteq
Q_{0}}\chi_{R}(x)\right)^{l_{1}}\mathcal{M}(\vec{f_{0}})(x)\sum_{Q\in{\mathcal{F}}}\chi_{Q}(x),$

where
$\vec{f_{0}}:=(\mathcal{A}_{{\mathcal{S}}^{*}}f_{1},\cdots,\mathcal{A}_{{{\mathcal{S}}}^{*}}(f_{l}),f_{l+1},\cdots,f_{m})$

Combining all the estimates obtained in (3.5) and (3.6), it yields that

(3.7)
$\displaystyle\mathcal{T}_{\mathcal{F}}^{\vec{\gamma}}(\vec{f},\vec{b})(x)$
$\displaystyle\leq
C\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\min\left\\{\mathcal{M}_{L(\log
L)}^{(1,l)}(\vec{f}),\mathcal{M}(\vec{f_{0}})\right\\}\left(\sum_{Q\in\tilde{{{\mathcal{S}}}},Q\subseteq
Q_{0}}\chi_{Q}(x)\right)^{l_{1}+1}.$

Recall that in [45, Theorem 2], it was shown that

(3.8) $\left|\left\\{x\in Q:\sum_{Q^{\prime}\in\mathcal{S},Q^{\prime}\subseteq
Q}\chi_{Q^{\prime}}(x)>t\right\\}\right|\leq ce^{-\alpha t}|Q|,\quad\forall Q\
\text{and}\ t>0.$

Keeping this significant observation in mind, then we have

$\displaystyle\left|\left\\{x\in
Q_{0}:\mathcal{T}_{\mathcal{F}}^{\vec{\gamma}}(\vec{f},\vec{b})(x)>t\min\left\\{\mathcal{M}_{L(\log
L)}^{(1,l)}(\vec{f})(x),\mathcal{M}(\vec{f_{0}})(x)\right\\}\right\\}\right|$
$\displaystyle\hskip 113.81102pt\leq\left|\left\\{x\in
Q_{0}:\sum_{Q\in{\tilde{{\mathcal{S}}}},Q\subseteq
Q_{0}}\chi_{Q}(x)>(\frac{t}{C\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}})^{\frac{1}{l+1}}\right\\}\right|$
$\displaystyle\hskip 113.81102pt\leq
Ce^{-\alpha\left(\frac{t}{\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}}\right)^{\frac{1}{l+1}}}|Q_{0}|.$

It then follows that

$\displaystyle\left|\left\\{x\in
Q_{0}:\left|\mathcal{T}_{\vec{b}}(\vec{f})(x)\right|>t\min\left\\{\mathcal{M}_{L(\log
L)}^{(1,l)}(\vec{f})(x),\mathcal{M}(\vec{f_{0}})(x)\right\\}\right\\}\right|$
$\displaystyle\quad\leq\sum_{\vec{\gamma}\in(1,2)^{l}}\left|\left\\{x\in
Q_{0}:\mathcal{T}_{\mathcal{F}}^{\vec{\gamma}}(\vec{f},\vec{b})(x)>\frac{t}{C2^{l}}\min\left\\{\mathcal{M}_{L(\log
L)}^{(1,l)}(\vec{f})(x),\mathcal{M}(\vec{f_{0}})(x)\right\\}\right\\}\right|$
$\displaystyle\quad\leq
C_{1}e^{\frac{-\alpha_{1}t^{1/(l+1)}}{\left(\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\right)^{1/(l+1)}}}|Q_{0}|.$

This finishes the proof of (1.1).

Finally, we need to prove that the exponent of the result in Theorem 1.4 is
sharp. To see this, let $m=1,I=\\{1\\}$ and $T$ be an $\omega$-Calderón-
Zygmund operator with $\omega$ satisfying the Dini condition
$[\omega]_{\text{Dini }}=\int_{0}^{1}\omega(t)\frac{dt}{t}<\infty$. Note that
for any $b\in\mathrm{BMO},$ $T_{b}$ satisfies Hypothesis 1 (see [39, Theorem
1.1]) and $M_{L\log L}f\simeq M^{2}f$ where $M^{2}$ denotes the composition of
Hardy-Littlewood maximal operator $M.$
Applying Theorem 1.4 to $T_{b},$ we have

(3.9) $\displaystyle\left|\left\\{x\in
Q_{0}:\left|T_{b}(f)(x)\right|>tM^{2}f(x)\right\\}\right|\leq
C_{1}e^{-\sqrt{\frac{\alpha_{1}t}{\|b\|_{\mathrm{BMO}}}}}|Q_{0}|.$

In particular, if $n=1,Q_{0}=(0,1)$, and $f(x)=\chi_{(0,1)}(x),$ then for any
$x\in Q_{0},$ $Mf(x)=1$ which implies $M^{2}f(x)=1.$ Let
$b=\log|x|\in\mathrm{BMO}$ and $T=H$ (the Hilbert transform). Then the
following estimate was proved in [52, p. 6]:

$\left|\left\\{x\in
Q_{0}:\left|T_{b}\left(\chi_{Q_{0}}\right)(x)\right|>t\right\\}\right|\geq
e^{-\sqrt{c_{0}t}},$

which holds for some absolute constant $c_{0}.$

Comparing this result with (3.9), we know that the exponent in local decay
estimate is sharp and this completes the proof of Theorem 1.4. ∎

We now turn our attention to the proof of Corollary 1.6.

###### Proof of Corollary 1.6.

Set

$E=\left\\{x\in Q_{0}:|\mathcal{T}_{\vec{b}}(\vec{f})(x)|>t\mathcal{M}_{L(\log
L)}(\vec{f})(x)\right\\}\subseteq Q_{0}.$

Then Theorem 1.4 gives that

$|E|\leq
C_{1}e^{-\alpha_{1}\left(\frac{t}{\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}}\right)^{\frac{1}{l+1}}}|Q_{0}|.$

It suffices to show that for each $w\in A_{\infty}^{\text{weak}},$ there exist
constants $C_{0},\varepsilon_{0}$ which depend on $w$ such that for every
$A\subseteq Q,$

(3.10) $\frac{w(A)}{w(2Q)}\leq
C_{0}\left(\frac{|A|}{|Q|}\right)^{\varepsilon_{0}}.$

In view of this, $w(E)$ automatically satisfies

$w(E)\leq
C_{1}^{\varepsilon_{0}}C_{0}e^{-\alpha_{1}\varepsilon_{0}\left(\frac{t}{\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}}\right)^{\frac{1}{l+1}}}w(2Q_{0}).$

In order to obtain the quantitative weighted estimate (1.2), it remains to
prove (3.10) and determine the dependence of $C_{0},\varepsilon_{0}$ and $w.$
Let $r_{w}=1+\frac{1}{\tau_{n}[w]_{A_{\infty}}^{weak}},$ then
$r_{w}^{\prime}=\frac{r_{w}}{r_{w}-1}=\tau_{n}[w]_{A_{\infty}}^{weak}+1$ and
Lemma 2.7 imply that

$\left(\frac{1}{|Q|}\int_{Q}w^{r_{w}}(x)dx\right)^{\frac{1}{r_{w}}}\leq\frac{2}{|2Q|}\int_{2Q}w(x)dx.$

Hölder’s inequality with exponents $r_{w}$ further gives

$\displaystyle w(E)$
$\displaystyle\leq\left(\int_{E}w^{r_{w}}(x)dx\right)^{\frac{1}{r_{w}}}|E|^{\frac{1}{r_{w}^{\prime}}}\leq\frac{2}{|2Q_{0}|}\int_{2Q_{0}}w(x)dx|Q_{0}|^{\frac{1}{r_{w}}}|E|^{\frac{1}{r_{w}^{\prime}}},$

where we have used the fact that $E\subseteq Q_{0}.$
Therefore,

$\displaystyle\frac{w(E)}{w(2Q_{0})}\leq
2^{1-n}\left(\frac{|E|}{|Q_{0}|}\right)^{\frac{1}{r_{w}^{\prime}}}\leq
2^{1-n}\left(\frac{|E|}{|Q_{0}|}\right)^{\frac{1}{\tau_{n}[w]_{A_{\infty}}^{weak}+1}}.$

Picking $C_{0}=2^{1-n}$ and
$\varepsilon_{0}=\frac{1}{\tau_{n}[w]_{A_{\infty}}^{weak}+1},$ we have now
proved that

$\displaystyle w\left(\left\\{x\in
Q_{0}:|\mathcal{T}_{\vec{b}}(\vec{f})(x)|>t\mathcal{M}_{L(\log
L)}\vec{f}(x)\right\\}\right)\leq
C_{2}e^{-{\frac{\alpha_{2}c_{*}}{[w]_{A_{\infty}}^{weak}+1}}t^{\frac{1}{l+1}}}w(2Q_{0}),$

where $c_{*}^{-l-1}=\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}.$ This finishes
the proof of the Corollary $\ref{cor1.1}.$ ∎

Using Theorem 1.4 and Corollary 1.6, we can directly obtain the following
result.

######  Corollary 3.1.

Let $\mathcal{T}_{\vec{b}}$ be defined as in Theorem 1.4. If $w\in
A_{\infty}$, then

$\displaystyle w\left(\left\\{x\in
Q_{0}:\left|\mathcal{T}_{\vec{b}}(\vec{f})(x)\right|>t\mathcal{M}_{L(\log
L)}(\vec{f})(x)\right\\}\right)$ $\displaystyle\hskip 142.26378pt\qquad\leq
c_{2}e^{-{\frac{\alpha_{2}}{[w]_{A_{\infty}}}}\left({\frac{t}{\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}}}\right)^{\frac{1}{l+1}}}w(Q_{0}),\quad
t>0.$

###### Proof of Corollary 3.1.

Following the definitions in the proof of Corollary 1.6, we have

(3.11) $|E|\leq C_{1}e^{-\alpha_{1}c_{*}t^{\frac{1}{l+1}}}|Q_{0}|$

with $c_{*}^{-l-1}=\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}.$ Using the
estimate

$\frac{w(E)}{w(Q_{0})}\leq
2\left(\frac{|E|}{|Q_{0}|}\right)^{\frac{1}{c_{n}[w]_{A_{\infty}}}}$

proved in Lemma 4.6 in [31] for every $w\in A_{\infty}$ and (3.11), it is easy
to verfy that Corollary 3.1 is valid. ∎

######  Remark 3.2.

Without using Lemma 4.6 in [31], Corollary 3.1 may also be proved via the
doubling property of $w\in A_{\infty}$. But the constant $c_{2}$ would depend
on the doubling constant of the measure $wdx$ and thus on $[w]_{A_{\infty}}$,
which yields the fact that $c_{2}$ has a worse dependence on
$[w]_{A_{\infty}}$.

##  4\.  Proofs of Theorems 1.8

In this section, we aim to establish the Coifman-Fefferman inequality for
$\mathcal{T}_{\vec{b}}.$ Before doing it, we present a generalized Hölder’s
inequality of multilinear version, which is a generalization of [51, Lemma 1]
under the general measure.

######  Lemma 4.1.

Let $\Phi_{0},\Phi_{1},\Phi_{2},\ldots,\Phi_{m}$ be Young functions. If

$\Phi_{1}^{-1}(t)\Phi_{2}^{-1}(t)\cdots\Phi_{m}^{-1}(t)\leq
D\Phi_{0}^{-1}(t),$

then for all functions $f_{1},\ldots,f_{m}$ and all cubes $Q$ we have that

(4.1) $\left\|f_{1}f_{2}\cdots f_{m}\right\|_{\Phi_{0}(\mu),Q}\leq
mD\left\|f_{1}\right\|_{\Phi_{1}(\mu),Q}\left\|f_{2}\right\|_{\Phi_{2}(\mu),Q}\cdots\left\|f_{m}\right\|_{\Phi_{m}(\mu),Q}.$

In particular, for any weight $w$ and $s_{1},\ldots,s_{m}\geq 1.$ Let
$\frac{1}{s}=\sum_{i=1}^{m}\frac{1}{s_{i}}$. Then we have

(4.2) $\frac{1}{w(Q)}\int_{Q}\left|f_{1}(x)\cdots
f_{m}(x)g(x)\right|w(x)dx\leq
2^{\frac{1}{s}}(1+\frac{1}{s})^{\frac{1}{s}}\prod_{i=1}^{m}\left\|f_{i}\right\|_{\exp
L^{s_{i}}(w),Q}\|g\|_{L(\log L)^{\frac{1}{s}}(w),Q}.$

###### Proof.

We first prove that if $\Phi_{0},\ldots,\Phi_{k}$ are continuous, nonnegative,
strictly increasing functions on $[0,\infty)$ with $\Phi_{i}(0)=0$ and
$\lim_{t\rightarrow\infty}\Phi_{i}(t)=\infty$ $(0\leq i\leq m)$ such that

$\Phi_{1}^{-1}(t)\Phi_{2}^{-1}(t)\cdots\Phi_{k}^{-1}(t)\leq\Phi_{0}^{-1}(t),\quad
t\geq 0,$

then for all $0\leq x_{1},x_{2},\ldots,x_{k}<\infty$

$\Phi_{0}\left(x_{1}x_{2}\cdots
x_{k}\right)\leq\Phi_{1}\left(x_{1}\right)+\Phi_{2}\left(x_{2}\right)+\cdots+\Phi_{k}\left(x_{k}\right).$

To see this, fix any
$\vec{x}=\left(x_{1},\ldots,x_{m}\right)\in(\mathbb{R}^{+})^{m}$ and let
$t_{0}=\Phi_{1}\left(x_{1}\right)+\Phi_{2}\left(x_{2}\right)+\cdots+\Phi_{m}\left(x_{m}\right)$.
Then the condition in Lemma 4.1 gives

$\Phi_{0}\left(\frac{\Phi_{1}^{-1}\left(t_{0}\right)\Phi_{2}^{-1}\left(t_{0}\right)\cdots\Phi_{m}^{-1}\left(t_{0}\right)}{D}\right)\leq\Phi_{0}\left(\frac{D\Phi_{0}^{-1}(t_{0})}{D}\right)=t_{0}.$

By the nonnegativity of $\Phi_{i}$, for any $i\in\\{1,\ldots,m\\},$ we have
$t_{0}\geq\Phi_{i}(x_{i}).$ Then

$\Phi_{i}^{-1}\left(t_{0}\right)\geq\Phi_{i}^{-1}\left(\Phi_{i}\left(x_{i}\right)\right)=x_{i},$

which indicates that

(4.3) $\Phi_{0}\left(\frac{x_{1}x_{2}\cdots x_{m}}{D}\right)\leq
t_{0}=\Phi_{1}\left(x_{1}\right)+\Phi_{2}\left(x_{2}\right)+\cdots+\Phi_{m}\left(x_{m}\right).$

Now consider the proof of (4.1). By using the convexity of $\Phi_{0}$, for any
$\lambda\in(0,1),$ it holds that

$\Phi_{0}(\lambda
t_{1}+(1-\lambda)t_{2})\leq\lambda\Phi_{0}(t_{1})+(1-\lambda)\Phi_{0}(t_{2}).$

Let $t_{2}=0$, then $\Phi_{0}(\lambda t)\leq\lambda\Phi_{0}(t)(t>0)$ since
$\Phi_{0}(0)=0$. Recalling the definition of $\|f\|_{A(\mu),Q}$, we get

$\|f\|_{A(\mu),Q}\leq
1\Leftrightarrow\frac{1}{\mu(Q)}\int_{Q}A(|f(x)|)d\mu(x)\leq 1.$

Using this fact and by (4.3), for any
$t_{i}>\left\|f_{i}\right\|_{\Phi_{i},Q}$ with $1\leq i\leq m,$ we obtain

$\displaystyle\frac{1}{\mu(Q)}\int_{Q}\Phi_{0}\left(\frac{\left|f_{1}\cdots
f_{m}\right|}{mDt_{1}\cdots t_{m}}\right)d\mu$
$\displaystyle\leq\frac{1}{m\mu(Q)}\int_{Q}\Phi_{0}\left(\frac{\left|f_{1}\cdots
f_{m}\right|}{Dt_{1}\cdots t_{m}}\right)d\mu$
$\displaystyle\leq\frac{1}{m}\sum_{i=1}^{m}\frac{1}{\mu(Q)}\int_{Q}\Phi_{i}\left(\frac{\left|f_{i}(x)\right|}{t_{i}}\right)d\mu(x)$
$\displaystyle\leq 1.$

This inequality implies

$\left\|f_{1}\cdots f_{m}\right\|_{\Phi_{0}(\mu),Q}\leq mDt_{1}\cdots t_{m}.$

and it is enough to take the infimum on each $t_{i}$ to finish the proof of
the (4.1).

Finally, we give the proof of (4.2). For $x\geq 0,t>0,$ we denote
$\varphi_{t}(x)=e^{x^{t}}-1,$ and $\Phi_{t}(x)=x(\log(e+x))^{t}.$ It is easy
to see that $\varphi_{t}^{-1}(x)=(\log(x+1))^{\frac{1}{t}}.$ In order to prove
(4.2), by (4.1) with $d\mu=wdx$, it suffices to show that

(4.4)
$\varphi_{s_{1}}^{-1}(x)\varphi_{s_{2}}^{-1}(x)\cdots\varphi_{s_{m}}^{-1}(x)\Phi_{1/s}^{-1}(x)\leq
Dx.$

First, we claim that

$\Phi_{t}^{-1}(x)\simeq\frac{x}{(\log(e+x))^{t}}.$

In fact, we only need to show
$x\simeq\frac{\Phi_{t}(x)}{(\log(e+\Phi_{t}(x)))^{t}}.$ Note that
$\Phi_{t}(x)\geq x,$ then

$\frac{\Phi_{t}(x)}{(\log(e+\Phi_{t}(x)))^{t}}\leq\frac{\Phi_{t}(x)}{(\log(e+x))^{t}}=x.$

On the other hand, since $\Phi_{t}(x)\leq(x+1)^{t+1}$, we have

$\displaystyle x(\log(e+\Phi_{t}(x)))^{t}$ $\displaystyle\leq
x(\log(e+(x+1)^{t+1}))^{t}\leq x(\log(e+x+1)^{t+1})^{t}$ $\displaystyle\leq
2^{t}(t+1)^{t}x(\log(e+x))^{t}=2^{t}(t+1)^{t}\Phi_{t}(x).$

We now continue with the proof of (4.4). By the fact that
$\varphi_{t}^{-1}(x)=(\log(x+1))^{\frac{1}{t}}\leq(\log(e+x))^{\frac{1}{t}},$
it may lead to

$\displaystyle\varphi_{s_{1}}^{-1}(x)\varphi_{s_{2}}^{-1}(x)\cdots\varphi_{s_{m}}^{-1}(x)\Phi_{1/s}^{-1}(x)$
$\displaystyle\leq
2^{\frac{1}{s}}(1+\frac{1}{s})^{\frac{1}{s}}(\log(e+x))^{\frac{1}{s_{1}}+\cdots+\frac{1}{s_{m}}}\frac{x}{(\log(e+x))^{\frac{1}{s}}}$
$\displaystyle=2^{\frac{1}{s}}(1+\frac{1}{s})^{\frac{1}{s}}x,$

which together with (4.1) completes the proof of Lemma 4.1. ∎

The following weighted John-Nirenberg inequality for BMO functions provides a
foundation for our analysis.

######  Lemma 4.2 ([31]).

Let $b\in\mathrm{BMO}$ and $w\in A_{\infty}$. Then we have

$\left\|b-b_{Q}\right\|_{\exp L(w),Q}\leq
c_{n}[w]_{A_{\infty}}\|b\|_{\mathrm{BMO}}.$

Furthermore, if $j>0$ then

$\left\|\left|b-b_{Q}\right|^{j}\right\|_{\exp L^{\frac{1}{j}}(w),Q}\leq
c_{n,j}[w]_{A_{\infty}}^{j}\|b\|_{\mathrm{BMO}}^{j}.$

We are now ready to prove Theorem 1.8.

###### Proof of Theorem 1.8.

According to the Hypothesis 2, we have

$\displaystyle\big{|}\mathcal{T}_{\vec{b}}(\vec{f})(x)\big{|}\leq
C\cdot\sum_{j=1}^{3^{n}}\sum_{\vec{\gamma}\in\\{1,2\\}^{l}}\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)\quad
a.e.\ x\in\mathbb{R}^{n}.$

By symmetry, we may assume that
$\vec{\gamma}=(\overbrace{1,\ldots,1}^{l_{1}},\overbrace{2,\ldots,2}^{l-l_{1}}).$
In order to show inequality (1.3), by using the triangle inequality, it
suffices to show that

(4.5)
$\|\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})\|_{L^{p}(w)}\lesssim\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}^{1/p}[w]_{A_{\infty}}^{l}[w]_{A_{\infty}}^{\max\\{2/p,1\\}}\|\mathcal{M}_{L(\log
L)}(\vec{f})\|_{L^{p}(w)},$

for every $1\leq j\leq 3^{n}.$

To prove (4.5), we first consider the case $p>1$. For any $w\in A_{\infty},$
by duality, we obtain

(4.6)
$\displaystyle\left\|\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})\right\|_{L^{p}(w)}=\sup_{\|g\|_{L^{p^{\prime}}(w)}\leq
1}\left|\int_{\mathbb{R}^{n}}\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)g(x)w(x)dx\right|.$

For any fixed nonnegative function $g\in L^{p^{\prime}}(w)$ with
$\|g\|_{L^{p^{\prime}}(w)}\leq 1,$ Lemmas 4.1 and 4.2 give that

$\displaystyle\left|\int_{\mathbb{R}^{n}}\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)g(x)w(x)dx\right|$
$\displaystyle\leq\sum_{Q\in\mathcal{S}_{j}}\prod_{s=1}^{l_{1}}{\langle|f_{s}|\rangle}_{Q}\int_{Q}\prod_{s=1}^{l_{1}}\left|b_{s}(x)-{\langle
b_{s}\rangle}_{Q}\right|g(x)w(x)dx$
$\displaystyle\quad\quad\times\prod_{s=l_{1}+1}^{m}{\langle|(b_{s}-{\langle
b_{s}\rangle}_{Q})f_{s}|\rangle}_{Q}\prod_{s=l+1}^{m}{\langle|f_{s}|\rangle}_{Q}$
$\displaystyle\lesssim\sum_{Q\in\mathcal{S}_{j}}w(Q)\prod_{s=1}^{l_{1}}{\langle|f_{s}|\rangle}_{Q}\prod_{s=1}^{l_{1}}\left\|b_{s}-{\langle
b_{s}\rangle}_{Q}\right\|_{\exp L(w),Q}$
$\displaystyle\quad\quad\times\|g\|_{L(\log
L)^{l_{1}}(w),Q}\prod_{s=l_{1}+1}^{l}\left\|b_{s}-{\langle
b_{s}\rangle}_{Q}\right\|_{\exp L,Q}\|f_{s}\|_{L(\log L),Q}$
$\displaystyle\quad\quad\times\prod_{s=l+1}^{m}{\langle|f_{s}|\rangle}_{Q}$
$\displaystyle\lesssim\sum_{Q\in\mathcal{S}_{j}}\|g\|_{L(\log
L)^{l_{1}}(w),Q}\left(\prod_{s=1}^{m}\|f_{s}\|_{L(\log L),Q}\right)w(Q)$
$\displaystyle\quad\times[w]_{A_{\infty}}^{l_{1}}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}.$

Then the Carleson embedding theorem combining with Hölder’s inequality yields
that

$\displaystyle\left|\int_{\mathbb{R}^{n}}\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}f(x)g(x)w(x)dx\right|$
$\displaystyle\lesssim[w]_{A_{\infty}}^{l_{1}}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\sum_{Q\in\mathcal{S}_{j}}w(Q)$
$\displaystyle\quad\times\left(\frac{1}{w(Q)}\int_{Q}\left(\mathcal{M}_{L(\log
L)}(\vec{f})(x)M_{L(\log
L)^{l_{1}}(w)}^{\mathcal{D}_{j}}g(x)\right)^{\frac{1}{2}}w(x)dx\right)^{2}$
$\displaystyle\lesssim[w]_{A_{\infty}}^{l_{1}+1}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}$
$\displaystyle\quad\times\int_{\mathbb{R}^{n}}\mathcal{M}_{L(\log
L)}(\vec{f})(x)M_{L(\log L)^{l_{1}}(w)}^{\mathcal{D}_{j}}g(x)w(x)dx$
$\displaystyle\lesssim[w]_{A_{\infty}}^{l_{1}+1}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\left\|\mathcal{M}_{L(\log
L)}(\vec{f})\right\|_{L^{p}(w)}\left\|(M_{w}^{\mathcal{D}_{j}})^{l_{1}+1}g\right\|_{L^{p^{\prime}}(w)},$

here we used that $M_{L(\log
L)^{k}(w)}^{\mathcal{D}}f\simeq(M_{w}^{\mathcal{D}})^{k+1}f$
$(k\in\mathbb{N}^{*})$ ([46, p. 179]) in the last step.
For any weight $w$, since $\left\|M_{w}^{\mathcal{D}}f\right\|_{L^{p}(w)}\leq
C\left\|f\right\|_{L^{p}(w)}$ ([37, Theorem 15.1]), then

(4.7)
$\displaystyle\left\|\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})\right\|_{L^{p}(w)}\lesssim[w]_{A_{\infty}}^{l_{1}+1}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\left\|\mathcal{M}_{L(\log
L)}(\vec{f})\right\|_{L^{p}(w)}.$

We now turn our attention to the case $0<p\leq 1.$ By duality, it follows that

(4.8)
$\displaystyle\|\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})\|_{L^{p}(w)}=\sup_{\|g\|_{L^{2}(w)}\leq
1}\Big{|}\int_{\mathbb{R}^{n}}\left(\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)\right)^{\frac{p}{2}}g(x)w(x)dx\Big{|}^{\frac{2}{p}}.$

Therefore,

$\displaystyle\Big{|}\int_{\mathbb{R}^{n}}\left(\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)\right)^{\frac{p}{2}}g(x)w(x)dx\Big{|}\leq$
$\displaystyle\sum_{Q\in\mathcal{S}_{j}}\prod_{s=1}^{l_{1}}{\langle|f_{s}|\rangle}_{Q}^{\frac{p}{2}}\int_{Q}\prod_{s=1}^{l_{1}}\left|b_{s}(x)-{\langle
b_{s}\rangle}_{Q}\right|^{\frac{p}{2}}|g(x)|w(x)dx$
$\displaystyle\quad\quad\times\prod_{s=l_{1}+1}^{m}{\langle|(b_{s}-{\langle
b_{s}\rangle}_{Q})f_{s}|\rangle}_{Q}^{\frac{p}{2}}\prod_{s=l+1}^{m}{\langle|f_{s}|\rangle}_{Q}^{\frac{p}{2}}.$

Using Lemma 4.1 and the same argument as in the case of $p>1$, we deduce that

$\displaystyle\int_{\mathbb{R}^{n}}\left(\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)\right)^{\frac{p}{2}}|g(x)|w(x)dx$
$\displaystyle\leq\sum_{Q\in\mathcal{S}_{j}}(l_{1}+1)(1+\frac{pl_{1}}{2})^{\frac{pl_{1}}{2}}2^{\frac{pl_{1}}{2}}w(Q)\left(\prod_{s=1}^{l_{1}}{\langle|f_{s}|\rangle}_{Q}^{\frac{p}{2}}\right)$
$\displaystyle\quad\times\prod_{s=1}^{l_{1}}\left\||b_{s}-{\langle
b_{s}\rangle}_{Q}|^{\frac{p}{2}}\right\|_{\exp
L^{\frac{p}{2}}(w),Q}\|g\|_{L(\log L)^{{}^{\frac{pl_{1}}{2}}}(w),Q}$
$\displaystyle\quad\quad\times
C_{n}^{\frac{pl}{2}}\prod_{s=l_{1}+1}^{l}\|b_{s}\|_{\mathrm{BMO}}^{\frac{p}{2}}\|f_{s}\|_{L(\log
L),Q}^{\frac{p}{2}}\prod_{s=l+1}^{m}{\langle|f_{s}|\rangle}_{Q}$
$\displaystyle\leq(l_{1}+1)(1+\frac{pl_{1}}{2})^{\frac{pl_{1}}{2}}2^{\frac{pl_{1}}{2}}C_{n}^{\frac{pl}{2}}[w]_{A_{\infty}}^{\frac{pl_{1}}{2}}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}^{\frac{p}{2}}$
$\displaystyle\quad\times\sum_{Q\in\mathcal{S}_{j}}\|g\|_{L(\log
L)^{\frac{pl_{1}}{2}}(w),Q}\left(\prod_{s=1}^{m}\|f_{s}\|_{L(\log
L),Q}^{\frac{p}{2}}\right)w(Q).$

Then the Carleson embedding theorem yields the following inequality

(4.9)
$\displaystyle\int_{\mathbb{R}^{n}}\left(\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)\right)^{\frac{p}{2}}|g(x)|w(x)dx$
$\displaystyle\leq(l_{1}+1)(1+\frac{pl_{1}}{2})^{\frac{pl_{1}}{2}}2^{\frac{pl_{1}}{2}}C_{n}^{\frac{pl}{2}}[w]_{A_{\infty}}^{1+\frac{pl_{1}}{2}}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}^{\frac{p}{2}}$
$\displaystyle\quad\times\int_{\mathbb{R}^{n}}(\mathcal{M}_{L(\log
L)}(\vec{f})(x))^{\frac{p}{2}}M_{L(\log
L)^{(pl_{1})/2}(w)}^{\mathcal{D}_{j}}g(x)w(x)dx$
$\displaystyle\leq(l_{1}+1)(1+\frac{pl_{1}}{2})^{\frac{pl_{1}}{2}}2^{\frac{pl_{1}}{2}}C_{n}^{\frac{pl}{2}}[w]_{A_{\infty}}^{1+\frac{pl_{1}}{2}}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}^{\frac{p}{2}}$
$\displaystyle\quad\times\left\|(\mathcal{M}_{L(\log
L)}(\vec{f}))^{\frac{p}{2}}\right\|_{L^{2}(w)}\left\|M_{L(\log
L)^{(pl_{1})/2}(w)}^{\mathcal{D}_{j}}g\right\|_{L^{2}(w)}.$

Note that $\|M_{L(\log
L)^{(pl_{1})/2}(w)}^{\mathcal{D}_{j}}g\|_{L^{2}(w)}\leq\|M_{L(\log
L)^{l_{1}}(w)}^{\mathcal{D}_{j}}g\|_{L^{2}(w)}\leq C_{n,l}\|g\|_{L^{2}(w)},$
which together with (4.8) and (4.9) gives

(4.10)
$\displaystyle\|\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})\|_{L^{p}(w)}$
$\displaystyle\leq
C_{n,l}C_{n}^{\frac{4}{p}}(l+1)^{\frac{2}{p}}(1+\frac{pl}{2})^{l}[w]_{A_{\infty}}^{l+\frac{2}{p}}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\left\|\mathcal{M}_{L(\log
L)}(\vec{f})\right\|_{L^{p}(w)}.$

Combining (4.7) with (4.10), we conclude that for any $0<p<\infty$ and $w\in
A_{\infty},$

$\int_{\mathbb{R}^{n}}\left|\mathcal{T}_{\vec{b}}(\vec{f})(x)\right|^{p}w(x)dx\lesssim\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}[w]_{A_{\infty}}^{pl}[w]_{A_{\infty}}^{\max\\{2,p\\}}\int_{\mathbb{R}^{n}}\left(\mathcal{M}_{L(\log
L)}(\vec{f})(x)\right)^{p}w(x)dx,$

which yields the required estimate (1.3). ∎

######  Remark 4.3.

For the Coifman-Fefferman inequality, we focus only on the precise
$A_{\infty}$ weight constant. But in proving the mixed weighted weak type
estimates, such as Theorem 1.10, we need the dependence of the constants in
(4.10) with respect to $p$ .

##  5\.  Proofs of Theorems 1.10 and Corollary 1.13

To prove Theorem 1.10, we need the following lemma with a more precise
constant estimate, a previous version of which can be found in [17, Lemma
2.3].

######  Lemma 5.1.

If $u\in A_{1},v\in A_{p},1\leq p<\infty,$ then $uv^{\varepsilon}\in A_{p}$
for all $0<\varepsilon<\frac{1}{2^{n+2}[u]_{A_{1}}}.$

###### Proof.

Since $u\in A_{1},$ by [49, Lemma 3.26], for each cube $Q$ it follows that

$\left(\frac{1}{|Q|}\int_{Q}u^{r_{0}}(x)dx\right)^{1/r_{0}}\leq\frac{2}{|Q|}\int_{Q}u(x)dx,$

where $r_{0}=1+\frac{1}{2^{n+1}[u]_{A_{1}}}.$ For any
$0<\varepsilon<\frac{1}{2^{n+2}[w]_{A_{1}}},$ let
$t=(\frac{1}{\varepsilon})^{\prime}.$ Then the Hölder’s inequality yields that

$\left(\frac{1}{|Q|}\int_{Q}u^{t}(x)dx\right)^{1/t}\leq\frac{2}{|Q|}\int_{Q}u(x)dx.$

Consider first the case $p=1$. Since $u,v\in A_{1}$, for any cube $Q$ and
almost every $x\in Q$,

$\displaystyle\frac{1}{|Q|}\int_{Q}u(x)v^{\varepsilon}(x)dx$
$\displaystyle\leq\left(\frac{1}{|Q|}\int_{Q}u^{t}(x)dx\right)^{1/t}\left(\frac{1}{|Q|}\int_{Q}v^{\varepsilon
t^{\prime}}(x)dx\right)^{1/t^{\prime}}\leq
2[u]_{A_{1}}[v]_{A_{1}}^{\varepsilon}u(x)v^{\varepsilon}(x),$

which implies that $uv^{\varepsilon}\in A_{1}$ and
$\left[uv^{\varepsilon}\right]_{A_{1}}\leq
2[u]_{A_{1}}[v]_{A_{1}}^{\varepsilon}$.

If $1<p<\infty$ and $v\in A_{p}.$ Then for any cube $Q$, the Hölder’s
inequality implies that

$\displaystyle\left(\frac{1}{|Q|}\int_{Q}u(x)v^{\varepsilon}(x)dx\right)\left(\frac{1}{|Q|}\int_{Q}\left(u(x)v^{\varepsilon}(x)\right)^{1-p^{\prime}}dx\right)^{p-1}$
$\displaystyle\hskip
85.35826pt\leq\left(\frac{1}{|Q|}\int_{Q}u^{t}(x)dx\right)^{1/t}\left(\frac{1}{|Q|}\int_{\mathrm{Q}}v^{\varepsilon
t^{\prime}}(x)dx\right)^{1/t^{\prime}}$ $\displaystyle\hskip
113.81102pt\times\left(\frac{1}{|Q|}\int_{Q}u^{t(1-p^{\prime})}(x)dx\right)^{\frac{p-1}{t}}\left(\frac{1}{|Q|}\int_{Q}v^{\varepsilon
t^{\prime}(1-p^{\prime})}(x)dx\right)^{\frac{p-1}{t^{\prime}}}$
$\displaystyle\hskip
85.35826pt\leq\frac{2}{|Q|}\int_{Q}u(x)dx\|u^{-1}\|_{L^{\infty}(Q)}\left[\frac{1}{|Q|}\int_{Q}v(x)dx\left(\frac{1}{|Q|}\int_{Q}v^{1-p^{\prime}}(x)dx\right)^{p-1}\right]^{\varepsilon}$
$\displaystyle\hskip 85.35826pt\leq 2[u]_{A_{1}}[v]_{A_{p}}^{\varepsilon}.$

Therefore, $uv^{\varepsilon}\in A_{p}$ with
$\left[uv^{\varepsilon}\right]_{A_{p}}\leq
2[u]_{A_{1}}[v]_{A_{p}}^{\varepsilon}$. This finishes the proof of Lemma 5.1.
∎

Now we need to show that Lemma 5.1 implies Theorem 1.10.

###### Proof of Theorem 1.10.

Some basic ideas will be taken from [41, Theorem 1.9], and these ideas have
also been used in [17, Theorem 1.7]. Note that $u=w_{1}^{\frac{1}{m}}\cdots
w_{m}^{\frac{1}{m}}\in A_{1}$ and $v\in A_{\infty}.$ Let $u\in A_{1},$ $S_{u}$
be the operator defined by

$S_{u}f(x)=\frac{M(fu)(x)}{u(x)}$

if $u(x)\neq 0,S_{u}f(x)=0$ otherwise.

For any $h\in L^{r^{\prime},1}(uv^{\frac{1}{m}})$ with $h\geq 0,$ applying the
Rubio de Francia algorithm with

$\mathcal{R}h(x)=\sum_{j=0}^{\infty}\frac{S_{u}^{j}h(x)}{(2K_{0})^{j}},$

where $K_{0}>0$ is an absolute constant which will be chosen later. A simple
calculation shows that

$\displaystyle 0\leq h(x)\leq\mathcal{R}h(x);\quad S_{u}(\mathcal{R}h)(x)\leq
2K_{0}\mathcal{R}h(x).$

Then, it follows from the second estimate that $\mathcal{R}h\cdot u\in A_{1}$
and $[\mathcal{R}h\cdot u]_{A_{1}}\leq 2K_{0}.$ Furthermore, we claim that
there exists some $r>1$ such that $\mathcal{R}h\cdot
uv^{\frac{1}{mr^{\prime}}}\in A_{\infty}$ and

$\|\mathcal{R}h\|_{L^{r^{\prime},1}(uv^{\frac{1}{m}})}\leq
2\|h\|_{L^{r^{\prime},1}(uv^{\frac{1}{m}})}.$

We postpone the proof of this claim to the end of this section.

Observe that

(5.1)
$\displaystyle\bigg{\|}\frac{\mathcal{T}_{\vec{b}}(\vec{f})}{v}\bigg{\|}_{L^{\frac{1}{m},\infty}(uv^{\frac{1}{m}})}^{\frac{1}{mr}}$
$\displaystyle=\bigg{\|}\Big{|}\frac{\mathcal{T}_{\vec{b}}(\vec{f})}{v}\Big{|}^{\frac{1}{mr}}\bigg{\|}_{L^{r,\infty}\left(uv^{\frac{1}{m}}\right)},$

then using the duality property of Lorentz spaces for $1<p<\infty$ (Exercise
1.4.12 in [25]), it follows that

$\|f\|_{L^{p,\infty}(\mu)}\backsimeq\sup_{\|g\|_{L^{p^{\prime},1}(\mu)}\leq
1}\left|\int_{\mathbb{R}^{n}}f(x)g(x)d\mu(x)\right|.$

This means that there exist $c_{1},c_{2}>0,$

(5.2) $\displaystyle c_{1}\sup_{\|g\|_{L^{p^{\prime},1}(\mu)}\leq
1}\Big{|}\int_{\mathbb{R}^{n}}f(x)g(x)d\mu(x)\Big{|}\leq\|f\|_{L^{p,\infty}(\mu)}\leq
c_{2}\sup_{\|g\|_{L^{p^{\prime},1}(\mu)}\leq
1}\Big{|}\int_{\mathbb{R}^{n}}f(x)g(x)d\mu(x)\Big{|}.$

In the following, we will calculate the exact value of $c_{1}$ and $c_{2}$.

Suppose that $\mathcal{X}$ is a quasi-Banach space and let $\mathcal{X}^{*}$
be its dual space. Then for all $T\in\mathcal{X}^{*}$, we have

$\|T\|_{\mathcal{X}^{*}}=\sup_{\begin{subarray}{c}x\in\mathcal{X}\\\
\|x\|_{\mathcal{X}}\leq 1\end{subarray}}|T(x)|.$

Let $\mathcal{X}=L^{p^{\prime},1}(\mu)$ and
$\mathcal{X}^{*}=L^{p,\infty}(\mu).$ For a fixed $f\in L^{p,\infty}(\mu),$ we
define $T_{f}(g)$ by

$T_{f}(g):=\int_{\mathbb{R}^{n}}f(x)g(x)d\mu(x),\quad g\in
L^{p^{\prime},1}(\mu).$

Therefore

(5.3) $\|T_{f}\|_{\mathcal{X}^{*}}=\sup_{\begin{subarray}{c}x\in\mathcal{X}\\\
\|x\|_{\mathcal{X}}\leq
1\end{subarray}}|T_{f}(x)|=\sup_{\|g\|_{L^{p^{\prime},1}(\mu)}\leq
1}\Big{|}\int_{\mathbb{R}^{n}}f(x)g(x)d\mu(x)\Big{|}.$

In addition, the discussion in [25, p. 59] gives

(5.4) $\|T_{f}\|_{\mathcal{X}^{*}}\leq\|f\|_{L^{p,\infty}(\mu)}\leq
p^{\prime}\|T_{f}\|_{\mathcal{X}^{*}}.$

Inserting (5.3) in (5.4) and picking $c_{1}=1,c_{2}=p^{\prime}$ yields what we
want to prove.

Let us continue to estimate (5.1). Applying (5.2) with $c_{2}=r^{\prime},$ we
obtain

(5.5)
$\displaystyle\bigg{\|}\frac{\mathcal{T}_{\vec{b}}(\vec{f})}{v}\bigg{\|}_{L^{\frac{1}{m},\infty}(uv^{\frac{1}{m}})}^{\frac{1}{mr}}$
$\displaystyle\leq r^{\prime}\sup_{\begin{subarray}{c}h\in
L^{r^{\prime},1}(uv^{\frac{1}{m}})\\\
\|h\|_{L^{r^{\prime},1}(uv^{\frac{1}{m}})}\leq
1\end{subarray}}\int_{\mathbb{R}^{n}}\Big{|}\mathcal{T}_{\vec{b}}(\vec{f})(x)\Big{|}^{\frac{1}{mr}}\mathcal{R}h(x)u(x)v^{\frac{1}{mr^{\prime}}}(x)dx.$

Note that ${mr}>1$, where $r$ depends on the weights $u$ and $v.$ Then, using
the weighted constant estimates in Theorem 1.8 with $0<p\leq 1$ gives

$\displaystyle\int_{\mathbb{R}^{n}}\Big{|}\mathcal{T}_{\vec{b}}(\vec{f})(x)\Big{|}^{\frac{1}{mr}}\mathcal{R}h(x)u(x)v^{\frac{1}{mr^{\prime}}}(x)dx$
$\displaystyle\leq
C_{n,l}^{\frac{l}{mr}}C_{n}(l+1)^{2}(1+\frac{1}{2mr})^{\frac{l}{mr}}[\mathcal{R}h\cdot
uv^{\frac{1}{mr^{\prime}}}]_{A_{\infty}}^{2+\frac{l}{mr}}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}^{\frac{1}{mr}}$
$\displaystyle\quad\times\int_{\mathbb{R}^{n}}\Big{|}\mathcal{M}_{L(\log
L)}(\vec{f})(x)\Big{|}^{\frac{1}{mr}}\mathcal{R}h(x)u(x)v^{\frac{1}{mr^{\prime}}}(x)dx$
$\displaystyle\leq C_{n,m,l}[\mathcal{R}h\cdot
uv^{\frac{1}{mr^{\prime}}}]_{A_{\infty}}^{2+\frac{l}{m}}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}^{\frac{1}{mr}}$
$\displaystyle\quad\times\int_{\mathbb{R}^{n}}\Big{|}\frac{\mathcal{M}_{L(\log
L)}(\vec{f})(x)}{v(x)}\Big{|}^{\frac{1}{mr}}\mathcal{R}h(x)u(x)v^{\frac{1}{m}}(x)dx.$

This inequality, together with the Hölder’s inequality (5.2) in Lorentz
spaces, implies that

$\displaystyle\int_{\mathbb{R}^{n}}\Big{|}\mathcal{T}_{\vec{b}}(\vec{f})(x)\Big{|}^{\frac{1}{mr}}\mathcal{R}h(x)u(x)v^{\frac{1}{mr^{\prime}}}(x)dx$
$\displaystyle\leq C_{n,m,l}[\mathcal{R}h\cdot
uv^{\frac{1}{mr^{\prime}}}]_{A_{\infty}}^{2+\frac{l}{m}}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}^{\frac{1}{mr}}$
$\displaystyle\quad\times\bigg{\|}\frac{\mathcal{M}_{L(\log
L)}(\vec{f})}{v}\bigg{\|}_{L^{\frac{1}{m},\infty}(uv^{\frac{1}{m}})}^{\frac{1}{mr}}\|\mathcal{R}h\|_{L^{r^{\prime},1}(uv^{\frac{1}{m}})}^{\frac{1}{mr}}$

Recall that $\|\mathcal{R}h\|_{L^{r^{\prime},1}(uv^{\frac{1}{m}})}\leq
2\|h\|_{L^{r^{\prime},1}(uv^{\frac{1}{m}})}.$Then for
$h\in{L^{r^{\prime},1}(uv^{\frac{1}{m}})}$ in (5.5), we have

(5.6)
$\displaystyle\bigg{\|}\frac{\mathcal{T}_{\vec{b}}(\vec{f})}{v}\bigg{\|}_{L^{\frac{1}{m},\infty}(uv^{\frac{1}{m}})}^{\frac{1}{r}}$
$\displaystyle\leq C_{n,m,l}(r^{\prime})^{m}[\mathcal{R}h\cdot
uv^{\frac{1}{mr^{\prime}}}]_{A_{\infty}}^{2m+l}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}^{\frac{1}{r}}\bigg{\|}\frac{\mathcal{M}_{L(\log
L)}(\vec{f})}{v}\bigg{\|}_{L^{\frac{1}{m},\infty}(uv^{\frac{1}{m}})}^{\frac{1}{r}}.$

To finish the proof it remains to show our foregoing claim and get the value
of $K_{0}$. The proof follows the same scheme of that in [17], but we have a
more precise estimate of the constants. Since
$\vec{w}=\left(w_{1},\ldots,w_{m}\right)\in A_{\vec{1}}$ and $v\in
A_{\infty},$ it follows that $u=w_{1}^{\frac{1}{m}}\ldots
w_{m}^{\frac{1}{m}}\in A_{1}$ and $v^{\frac{1}{m}}\in A_{\infty}.$ The former
indicates that $S_{u}$ is bounded on $L^{\infty}(uv^{\frac{1}{m}})$ with
constant $C_{0}=[u]_{A_{1}},$ that is,

(5.7)
$\|S_{u}f\|_{L^{\infty}(uv^{\frac{1}{m}})}\leq[u]_{A_{1}}\|f\|_{L^{\infty}(uv^{\frac{1}{m}})}.$

In order to apply the Marcinkiewicz interpolation theorem, we also need to
show that $S_{u}$ is bounded on $L^{p_{0}}(uv^{\frac{1}{m}})$ for some
$1<p_{0}<\infty$. Note that for $v^{1/m}\in A_{\infty},$ there exists $t>1$
such that $v^{\frac{1}{m}}\in A_{t}.$ Then $A_{p}$ factorization theorem ([25,
Theorem 7.5.1]) tells us that there exist $v_{1},v_{2}\in A_{1}$ such that
$v^{\frac{1}{m}}=v_{1}v_{2}^{1-t}.$ Using these facts, we have

$u^{1-p_{0}}v^{\frac{1}{m}}=v_{1}\left(uv_{2}^{\frac{t-1}{p_{0}-1}}\right)^{1-p_{0}}.$

Applying Lemma 5.1, for any $0<\varepsilon<\frac{1}{2^{n+2}[u]_{A_{1}}}$ and
$v_{2}\in A_{1},$ yields that $uv_{2}^{\varepsilon}\in A_{1}.$ Then we pick
$p_{0}=2^{n+3}(t-1)[u]_{A_{1}}+1$ such that $uv_{2}^{\frac{t-1}{p_{0}-1}}\in
A_{1},$ which further implies

$u^{1-p_{0}}v^{\frac{1}{m}}\in A_{p_{0}}.$

Observe that

$\int_{\mathbb{R}^{n}}|S_{u}f(x)|^{p_{0}}u(x)v^{\frac{1}{m}}(x)dx=\int_{\mathbb{R}^{n}}(M(fu)(x))^{p_{0}}u^{1-p_{0}}(x)v^{\frac{1}{m}}(x)dx.$

By this observation and Buckley’s theorem [49, Theorem 3.11], it follows that

(5.8)
$\|S_{u}f\|_{L^{p_{0}}(uv^{\frac{1}{m}})}=\|M(fu)\|_{L^{p_{0}}(u^{1-p_{0}}v^{\frac{1}{m}})}\leq
c_{n}p_{0}^{\prime}[u^{1-p_{0}}v^{\frac{1}{m}}]_{A_{p_{0}}}^{\frac{1}{p_{0}-1}}\|f\|_{L^{p_{0}}(uv^{\frac{1}{m}})},$

and thus $S_{u}:L^{p_{0}}(uv^{\frac{1}{m}})\rightarrow
L^{p_{0}}(uv^{\frac{1}{m}})$ holds with constant
$C_{1}=c_{n}p_{0}^{\prime}[u^{1-p_{0}}v^{\frac{1}{m}}]_{A_{p_{0}}}^{\frac{1}{p_{0}-1}}.$

The Marcinkiewicz interpolation theorem in [17, Proposition A.1], together
with (5.7) and (5.8), yields that $S_{u}$ is bounded on
$L^{q,1}(uv^{\frac{1}{m}})$ for all $p_{0}<q<\infty$ and enjoys the property

$\|S_{u}f\|_{L^{q,1}(uv^{\frac{1}{m}})}\leq
2^{\frac{1}{q}}\Big{(}C_{1}(\frac{1}{p_{0}}-\frac{1}{q})^{-1}+C_{0}\Big{)}\|f\|_{L^{q,1}(uv^{\frac{1}{m}})}.$

Notice that, if $q\geq 2p_{0},$ then

$2^{\frac{1}{q}}\Big{(}C_{1}(\frac{1}{p_{0}}-\frac{1}{q})^{-1}+C_{0}\Big{)}\leq
4p_{0}(C_{0}+C_{1})\leq
c_{n}p_{0}p_{0}^{\prime}([u^{1-p_{0}}v^{\frac{1}{m}}]_{A_{p_{0}}}^{\frac{1}{p_{0}-1}}+[u]_{A_{1}}).$

Using Lemma 5.1 and $A_{p}$ factorization theorem again, we obtain

$\displaystyle\ [u^{1-p_{0}}v^{\frac{1}{m}}]_{A_{p_{0}}}^{\frac{1}{p_{0}-1}}$
$\displaystyle\leq[v_{1}]_{A_{1}}\left[uv_{2}^{\frac{t-1}{p_{0}-1}}\right]_{A_{1}}^{p_{0}-1}\leq
2^{p_{0}-1}c_{n}^{t}[v^{\frac{1}{m}}]_{A_{t}}^{2}[u]_{A_{1}}^{p_{0}-1}.$

Let
$K_{0}=4c_{n}p_{0}p_{0}^{\prime}([u]_{A_{1}}+2^{p_{0}-1}c_{n}^{t}[v^{\frac{1}{m}}]_{A_{t}}^{2}[u]_{A_{1}}^{p_{0}-1})+1,$
then

$\|S_{u}f\|_{L^{q,1}(uv^{\frac{1}{m}})}\leq
K_{0}\|f\|_{L^{q,1}(uv^{\frac{1}{m}})},\quad\hbox{\ for all }q\geq 2p_{0}.$

Take
$\varepsilon=\frac{1}{2}{\min\\{\frac{1}{2^{n+3}K_{0}},\frac{1}{2p_{0}}\\}}$
and $r=(\frac{1}{\varepsilon})^{\prime},$ which implies that
$r^{\prime}>2p_{0}.$ This ensures that $S_{u}$ is bounded on
$L^{r^{\prime},1}(uv^{\frac{1}{m}})$ with constant bounded by $K_{0}$.

For any $0<\varepsilon<\frac{1}{2^{n+3}K_{0}}$ and $W_{1}\in A_{1}$ with
$[W_{1}]_{A_{1}}\leq 2K_{0}.$ Lemma 5.1 and a simple calculation yield that
$W_{1}W_{2}^{\varepsilon}\in A_{t},\hbox{for all }W_{2}\in A_{t}.$ Based on
the facts that $\mathcal{R}h\cdot u\in A_{1}$ with $[\mathcal{R}h\cdot
u]_{A_{1}}\leq 2K_{0}$ and $v^{\frac{1}{m}}\in A_{t}$, we have

$\mathcal{R}h\cdot uv^{\frac{1}{mr^{\prime}}}=\mathcal{R}h\cdot
uv^{\frac{\varepsilon}{m}}\in A_{t}\subseteq A_{\infty},$

where $\frac{1}{r^{\prime}}=\varepsilon<\frac{1}{2^{n+3}K_{0}}.$ In addition,
it is easy to see that

(5.9) $\displaystyle\ [\mathcal{R}h\cdot
uv^{\frac{1}{mr^{\prime}}}]_{A_{\infty}}$ $\displaystyle\leq[\mathcal{R}h\cdot
uv^{\frac{1}{mr^{\prime}}}]_{A_{t}}\leq 2[\mathcal{R}h\cdot
u]_{A_{1}}[v^{\frac{1}{m}}]_{A_{t}}^{\frac{1}{r^{\prime}}}\leq
4K_{0}[v^{\frac{1}{m}}]_{A_{t}}.$

This completes the proof of the claim. Note that the following two facts hold:
$r\leq 2$, which follows from $K_{0}\geq 1$ and $p_{0}>1.$ $r^{\prime}\leq
2^{n+4}K_{0}$ since $K_{0}\geq p_{0}.$

Combining these two facts and using (5.6), (5.9), we deduce

$\left\|\frac{\mathcal{T}_{\vec{b}}(\vec{f})}{v}\right\|_{L^{\frac{1}{m},\infty}(uv^{\frac{1}{m}})}\leq
C_{n,m,l}K_{0}^{2l+6m}[v^{\frac{1}{m}}]_{A_{t}}^{2l+4m}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\left\|\frac{\mathcal{M}_{L(\log
L)}(\vec{f})}{v}\right\|_{L^{\frac{1}{m},\infty}(uv^{\frac{1}{m}})},$

which completes the proof of Theorem 1.10. ∎

###### Proof of Corollary 1.13.

Let $v\equiv 1,$ applying Theorem 1.14 and Remark 1.12 with $v\in A_{2}$,
Corollary 1.13 follows easily by a simple calculation. ∎

##  6\.  Proofs of Theorems 1.14 and Corollary 1.16

###### Proof of Theorem 1.14.

Let
$\vec{\gamma}=(\overbrace{1,\ldots,1}^{l_{1}},\overbrace{2,\ldots,2}^{l-l_{1}}),$
under the Hypothesis 2 of $\mathcal{T}_{\vec{b}},$ we consider only the
contribution of
$\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})$ for
$j=1,\ldots,3^{n}.$

Fix exponents $\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}$ with
$1<p_{1},\ldots,p_{m}<\infty$ and weights
$\vec{w}=\left(w_{1},\ldots,w_{m}\right)$. It follows from the definition of
$\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})$ and $0<p\leq
1$ that

(6.1)
$\displaystyle\|\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})\|_{L^{p}(\nu_{\vec{w}})}^{p}\leq\sum_{Q\in\mathcal{S}_{j}}$
$\displaystyle\prod_{s=1}^{l_{1}}{\langle|f_{s}|\rangle}_{Q}^{p}\int_{Q}\prod_{s=1}^{l_{1}}\left|b_{s}(x)-{\langle
b_{s}\rangle}_{Q}\right|^{p}\nu_{\vec{w}}(x)dx$
$\displaystyle\times\prod_{s=l_{1}+1}^{l}{\langle|(b_{s}-{\langle
b_{s}\rangle}_{Q})f_{s}|\rangle}_{Q}^{p}\prod_{s=l+1}^{m}{\langle|f_{s}|\rangle}_{Q}^{p}.$

Let $v_{i}(x)=Mw_{i}(x)$ with $i=1,\ldots,m,$ then $v_{i}(x)\geq\langle
w_{i}\rangle_{Q}$ for a.e. $x\in Q$ where $Q$ is any dyadic cube contained in
$\mathcal{S}_{j}.$ Using the Hölder’s inequality, we obtain

(6.2) $\displaystyle\int_{Q}\prod_{s=1}^{l_{1}}\left|b_{s}(x)-{\langle
b_{s}\rangle}_{Q}\right|^{p}\nu_{\vec{w}}(x)dx\lesssim\prod_{s=1}^{l_{1}}\|b_{s}\|_{\mathrm{BMO}_{p_{s}}(w_{s})}^{p}\prod_{s=1}^{m}\left(w_{s}(Q)\right)^{\frac{p}{p_{s}}}.$

Pick $r,t\in\mathbb{R}$ such that $1<r<t<\min_{i}\\{p_{i}\\}.$ Applying the
Hölder’s inequality twice gives

(6.3) $\displaystyle{\langle|(b_{s}-{\langle
b_{s}\rangle}_{Q})f_{s}|\rangle}_{Q}$
$\displaystyle\lesssim\|b_{s}\|_{\mathrm{BMO}}{\langle|f_{s}|^{r}\rangle}_{Q}^{\frac{1}{r}}\leq\left\|b_{s}\right\|_{\mathrm{BMO}}{\langle|f_{s}|^{t}v_{s}^{\frac{t}{p_{s}}}\rangle}_{Q}^{\frac{1}{t}}{\langle
w_{s}\rangle}_{Q}^{-{\frac{1}{p_{s}}}}.$

Putting the estimates (6.1-(6.3) together we obtain

$\displaystyle\left\|\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})\right\|_{L^{p}(\nu_{\vec{w}})}^{p}$
$\displaystyle\leq\sum_{Q\in\mathcal{S}_{j}}\prod_{s=1}^{l_{1}}\|b_{s}\|_{\mathrm{BMO}_{p_{s}}(w_{s})}^{p}\prod_{s=l_{1}+1}^{l}\|b_{s}\|_{\mathrm{BMO}}^{p}\prod_{s=1}^{m}{\langle|f_{s}|^{t}v_{s}^{\frac{t}{p_{s}}}\rangle}_{Q}^{\frac{p}{t}}$
$\displaystyle\quad\quad\times\prod_{s=1}^{m}{\langle
w_{s}\rangle}_{Q}^{-{\frac{p}{p_{s}}}}\prod_{s=1}^{m}w_{s}(Q)^{\frac{p}{p_{s}}}$
$\displaystyle\lesssim\sum_{Q\in\mathcal{S}_{j}}\left(\prod_{s=1}^{m}\inf_{x\in
Q}\left(M_{t}(f_{s}v_{s}^{\frac{1}{p_{s}}})\right)^{p}(x)\right)|Q|$
$\displaystyle\quad\times\prod_{s=1}^{l_{1}}\|b_{s}\|_{\mathrm{BMO}_{p_{s}}(w_{s})}^{p}\prod_{s=l_{1}+1}^{l}\|b_{s}\|_{\mathrm{BMO}}^{p}$
$\displaystyle\lesssim\sum_{Q\in\mathcal{S}_{j}}\left(\prod_{s=1}^{m}\inf_{x\in
E_{Q}}\left(M_{t}(f_{s}v_{s}^{\frac{1}{p_{s}}})\right)^{p}(x)\right)|E_{Q}|$
$\displaystyle\quad\times\prod_{s=1}^{l_{1}}\|b_{s}\|_{\mathrm{BMO}_{p_{s}}(w_{s})}^{p}\prod_{s=l_{1}+1}^{l}\|b_{s}\|_{\mathrm{BMO}}^{p},$

where in the last inequality, we have used the sparseness property of the
collection $\mathcal{S}_{j}$.
Note that $1<t<\min_{i}\\{p_{i}\\}$ and $\\{E_{Q}\\}$ are pairwise disjoint,
it follows that

$\displaystyle\left\|\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})\right\|_{L^{p}(\nu_{\vec{w}})}^{p}$
$\displaystyle\lesssim\prod_{s=1}^{l_{1}}\|b_{s}\|_{\mathrm{BMO}_{p_{s}}(w_{s})}^{p}\prod_{s=l_{1}+1}^{l}\|b_{s}\|_{\mathrm{BMO}}^{p}\int_{\mathbb{R}^{n}}\prod_{s=1}^{m}\left(M_{t}(f_{s}v_{s}^{\frac{1}{p_{s}}})\right)^{p}(x)dx$
$\displaystyle\leq\left(\|\vec{b}\|_{\mathrm{BMO}}^{*}\right)^{p}\prod_{s=1}^{m}\|M_{t}(f_{s}v_{s}^{\frac{1}{p_{s}}})\|_{L^{p_{s}}(\mathbb{R}^{n})}^{p}$
$\displaystyle\lesssim\left(\|\vec{b}\|_{\mathrm{BMO}}^{*}\right)^{p}\prod_{s=1}^{m}\|f_{s}\|_{L^{p_{s}}(Mw_{s})}^{p}.$

The proof of Theorem 1.14 is finished. ∎

###### Proof of Corollary 1.16.

It was shown in [44, Corollary 2.2] that, for any $w\in A_{\infty}$ and
$f\in\mathrm{BMO},$ there exists a dimensional constant $c_{n}$ independent of
$f$ and $w$ such that for each cube $Q,$

$\displaystyle\left(\frac{1}{w(Q)}\int_{Q}\left|f(x)-{\langle
f\rangle}_{Q}\right|^{p}w(x)dx\right)^{\frac{1}{p}}\leq
c_{n}p[w]_{A_{\infty}}\|f\|_{\mathrm{BMO}}.$

Therefore, if $w_{s}\in A_{\infty}$ and $b_{s}\in\mathrm{BMO}$ with $1\leq
s\leq l,$ then $b_{s}\in{\mathrm{BMO}_{p_{s}}(w_{s})}$ and

$\|b_{s}\|_{\mathrm{BMO}_{p_{s}}(w_{s})}\leq
c_{n}p_{s}[w_{s}]_{A_{\infty}}\|b_{s}\|_{\mathrm{BMO}},$

which implies that

$\|\vec{b}\|_{\mathrm{BMO}}^{*}\leq
C_{n,p}\prod_{s=1}^{l}[w_{s}]_{A_{\infty}}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}.$

Plugging this estimate into Theorem 1.14 we achieve the desired conclusion (1)
in Corollary 1.16.

It remains to show the estimate (2) holds in Corollary 1.16. We will use the
same notation as in the proof of Theorem 1.14. Note that for every $x\in
Q\subseteq 2Q,$ if $w_{s}\in A_{\infty}^{\text{weak}},$ then
$v_{s}(x):=Mw_{s}(x)\geq{\langle w_{s}\rangle}_{2Q}$. For a fixed
$j\in\\{1,\ldots,3^{n}\\},$ we need to estimate

(6.4)
$\sum_{Q\in\mathcal{S}_{j}}\prod_{s=1}^{l_{1}}{\langle|f_{s}|\rangle}_{Q}^{p}\int_{Q}\prod_{s=1}^{l_{1}}\left|b_{s}(x)-{\langle
b_{s}\rangle}_{Q}\right|^{p}\nu_{\vec{w}}(x)dx\prod_{s=l_{1}+1}^{l}{\langle|(b_{s}-{\langle
b_{s}\rangle}_{Q})f_{s}|\rangle}_{Q}^{p}\prod_{s=l+1}^{m}{\langle|f_{s}|\rangle}_{Q}^{p}.$

A simple calculation yields that

(6.5) $\displaystyle\int_{Q}\prod_{s=1}^{l_{1}}\left|b_{s}(x)-{\langle
b_{s}\rangle}_{Q}\right|^{p}\nu_{\vec{w}}(x)dx$
$\displaystyle\leq\prod_{s=1}^{l_{1}}\left(\frac{1}{w(2Q)}\int_{Q}\left|b_{s}(x)-{\langle
b_{s}\rangle}_{Q}\right|^{p_{s}}w_{s}(x)dx\right)^{\frac{p}{p_{s}}}\prod_{s=1}^{m}(w_{s}(2Q))^{\frac{p}{p_{s}}}$
$\displaystyle\lesssim\prod_{s=1}^{l_{1}}\left([w_{s}]_{A_{\infty}}^{weak}\right)^{p}\prod_{s=1}^{l_{1}}\|b_{s}\|_{\mathrm{BMO}}^{p}\prod_{s=1}^{m}(w_{s}(2Q))^{\frac{p}{p_{s}}},$

where the last inequality follows from [44, Corollary 2.4].

Therefore, using the same ideas as what have been used in (6.3), we obtain

(6.6) $\displaystyle{\langle|(b_{s}-{\langle
b_{s}\rangle}_{Q})f_{s}|\rangle}_{Q}$
$\displaystyle\lesssim\|b_{s}\|_{\mathrm{BMO}}{\langle|f_{s}|^{t}v_{s}^{\frac{t}{p_{s}}}\rangle}_{Q}^{\frac{1}{t}}{\langle
w_{s}\rangle}_{2Q}^{-{\frac{1}{p_{s}}}}.$

This inequality, together with inqualities (6.4)-(6.6), gives that

$\displaystyle\left\|\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})\right\|_{L^{p}(\nu_{\vec{w}})}^{p}$
$\displaystyle\lesssim\prod_{s=1}^{l_{1}}\left([w_{s}]_{A_{\infty}}^{weak}\right)^{p}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}^{p}\prod_{s=1}^{m}{\langle|f_{s}|^{t}v_{s}^{\frac{t}{p_{s}}}\rangle}_{Q}^{\frac{p}{t}}\prod_{s=1}^{m}{\langle
w_{s}\rangle}_{2Q}^{-{\frac{p}{p_{s}}}}\prod_{s=1}^{m}(w_{s}(2Q))^{\frac{p}{p_{s}}}$
$\displaystyle\lesssim\prod_{s=1}^{l_{1}}\left([w_{s}]_{A_{\infty}}^{weak}\right)^{p}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}^{p}\prod_{s=1}^{m}\|f_{s}\|_{L^{p_{s}}(Mw_{s})}^{p}.$

Keeping this estimate in mind and applying the same reasoning to the other
forms of $\vec{\gamma},$ we obtain

$\displaystyle\left\|\mathcal{T}_{\vec{b}}(\vec{f})\right\|_{L^{p}(\nu_{\vec{w}})}$
$\displaystyle\lesssim\sum_{\vec{\gamma}\in\\{1,2\\}^{l}}(\prod_{s:\gamma_{s}=1}[w_{s}]_{A_{\infty}}^{\text{weak}})\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\prod_{s=1}^{m}\|f_{s}\|_{L^{p_{s}}(Mw_{s})},$

which finishes the proof of corollary 1.16. ∎

##  7\.  Proofs of Theorem 1.18

This section will be devoted to demonstrate Theorem 1.18. For this purpose, we
first present the definition of $N$-function. We say a Young function $\phi$
is an $N$-function if it satisfies

$\lim_{t\rightarrow 0^{+}}\frac{\phi(t)}{t}=0\quad\text{ and
}\quad\lim_{t\rightarrow\infty}\frac{\phi(t)}{t}=\infty.$

Moreover, an $N$-function is said to satisfy the sub-multiplicative property
if $\phi(st)\leq\phi(s)\phi(t)$ for any $s,t\geq 0,$ For convenience, we need
to state some properties as well as the lemmata of $\phi\in\Phi$ and it’s
complementary function $\bar{\phi}.$

  * •

(Young’s inequality) $st\leq\phi(s)+\bar{\phi}(t),s,t\geq 0.$

  * •

When $\phi$ is an $N$-function, then $\bar{\phi}$ is also an $N$-function, and
the following inequalities hold:

(7.1) $t\leq\phi^{-1}(t)\bar{\phi}^{-1}(t)\leq 2t,t\geq 0;$ (7.2)
$\bar{\phi}\left(\frac{\phi(t)}{t}\right)\leq\phi(t),t>0.$

  * •

Let $\phi$ be an $N$-function, then there exists $0<\alpha<1$ such that
$\phi^{\alpha}$ is quasi-convex if and only if $\bar{\phi}\in\Delta_{2},$
where $\phi^{\alpha}(t)=\phi(t)^{\alpha}.$

  * •

$\phi\in\Delta_{2}$ if and only if there exists some constant $C_{1}>0$ such
that for any $\lambda\geq 2$,

(7.3) $\phi(\lambda t)\leq 2^{C_{1}}\lambda^{C_{1}}\phi(t),t>0.$

The next two technical lemmata allow us to prove Theorem 1.18. The first one
is a modular inequality with respect to the Hardy-Littlewood maximal operator
$M.$

######  Lemma 7.1 ([1]).

Let $\phi\in\Phi$ and be quasi-convex. If $1<i_{\phi}<\infty$ and $w\in
A_{i_{\phi}}$, then

$\int_{\mathbb{R}^{n}}\phi(Mf(x))w(x)dx\leq
C\int_{\mathbb{R}^{n}}\phi\left(C[w]_{A_{i_{\phi}}}^{1/i_{\phi}}|f(x)|\right)w(x)dx,$

where $C$ is an absolute constant which only depends on $\phi$ and $\alpha$.

The next lemma concerns on a modular inequality for sparse operators (see [1,
Lemma 3.13]), which plays a foundamental role in our analysis.

######  Lemma 7.2 ([1]).

Let $\phi$ be an $N$-function with $\phi\in\Delta_{2},$ and $w\in
A_{i_{\phi}}$. If $i_{\phi}>1$, for any dyadic grid $\mathcal{D}$ and
$\mathcal{S}\subseteq\mathcal{D}$ a sparse family, we have

$\int_{\mathbb{R}^{n}}\phi\left(\mathcal{A}_{\mathcal{S}}(f)(x)\right)w(x)dx\leq
C[w]_{A_{\infty}}^{1+\widetilde{C}}\int_{\mathbb{R}^{n}}\phi(Mf(x))w(x)dx,$

where $C$ is an absolute constant only depending on $\phi$ and $\widetilde{C}$
which satisfies $\phi(\lambda t)\leq
2^{\widetilde{C}}\lambda^{\widetilde{C}}\phi(t)$ for $\lambda\geq 2$ and
$t>0.$

###### Proof of Theorem 1.18.

For $1\leq j\leq 3^{n}$ and
$\vec{\gamma}=(\overbrace{1,\ldots,1}^{l_{1}},\overbrace{2,\ldots,2}^{l-l_{1}}),$
we first consider the contribution of
$\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}$ and try to show that
there exists a constant $K$ such that

(7.4)
$\int_{\mathbb{R}^{n}}\phi\left(\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)\right)w(x)dx\lesssim\left(\prod_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi^{m}\left(K\left|f_{i}(x)\right|\right)w(x)dx\right)^{\frac{1}{m}}.$

Note that if
$\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)=0$, then
$\phi(\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x))=0$
since $\phi$ is an $N$-function. Define a function $h$ on $\mathbb{R}^{n}$ by

$h(x)=\begin{cases}0,&\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)=0,\\\
\frac{\phi(\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x))}{\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)},&otherwise.\end{cases}$

Let $H_{l_{1}}(x)=\prod_{s=1}^{l_{1}}\big{|}b_{s}(x)-{\langle
b_{s}\rangle}_{Q}\big{|}{\langle|f_{s}|\rangle}_{Q}$ and denote

$\displaystyle\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)=:\sum_{Q\in\mathcal{S}_{j}}H_{l_{1}}(x)\prod_{s=l_{1}+1}^{l}{\langle|(b_{s}-{\langle
b_{s}\rangle}_{Q})f_{s}|\rangle}_{Q}\prod_{s=l+1}^{m}{\langle|f_{s}|\rangle}_{Q}\chi_{Q}(x),$

Then the left-hand side of (7.4) can be rewritten as

(7.5)
$\displaystyle\int_{\mathbb{R}^{n}}\phi\left(\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)\right)w(x)dx=$
$\displaystyle\sum_{Q\in\mathcal{S}_{j}}\prod_{s=l_{1}+1}^{l}{\langle\big{|}(b_{s}-{\langle
b_{s}\rangle}_{Q})f_{s}\big{|}\rangle}_{Q}\prod_{s=l+1}^{m}{\langle|f_{s}|\rangle}_{Q}$
$\displaystyle\quad\times\int_{Q}H_{l_{1}}(x)h(x)w(x)dx.$

Hence for any $r>1,$ Lemma 4.1 with
$s_{1}=s_{2}=\cdots=s_{l_{1}}=1,s=\frac{1}{l_{1}}$ yields

(7.6)
$\displaystyle\int_{\mathbb{R}^{n}}\phi\left(\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)\right)w(x)dx$
$\displaystyle\lesssim\sum_{Q\in\mathcal{S}_{j}}\prod_{s=1}^{l_{1}}{\langle|f_{s}|\rangle}_{Q}\prod_{s=l+1}^{m}{\langle|f_{s}|\rangle}_{Q}\prod_{s=l_{1}+1}^{l}{\langle\big{|}(b_{s}-{\langle
b_{s}\rangle}_{Q})f_{s}\big{|}\rangle}_{Q}$
$\displaystyle\quad\quad\times\prod_{s=1}^{l_{1}}\left\|b_{s}-{\langle
b_{s}\rangle}_{Q}\right\|_{\exp L(w),Q}\|h\|_{L(\log L)^{l_{1}}(w),Q}w(Q)$
$\displaystyle\lesssim[w]_{A_{\infty}}^{l_{1}}\sum_{Q\in\mathcal{S}_{j}}\prod_{s=1}^{l_{1}}\|b_{s}\|_{\mathrm{BMO}}\prod_{s=l_{1}+1}^{l}{\langle\big{|}b_{s}-{\langle
b_{s}\rangle}_{Q}\big{|}^{r^{\prime}}\rangle}_{Q}^{\frac{1}{r^{\prime}}}$
$\displaystyle\quad\times\prod_{s=1}^{m}{\langle|f_{s}|^{r}\rangle}_{Q}^{\frac{1}{r}}\|h\|_{L(\log
L)^{l_{1}}(w),Q}w(Q)$
$\displaystyle\lesssim[w]_{A_{\infty}}^{l}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\sum_{Q\in\mathcal{S}_{j}}\prod_{s=1}^{m}{\langle|f_{s}|^{r}\rangle}_{Q}^{\frac{1}{r}}\|h\|_{L(\log
L)^{l_{1}}(w),Q}w(Q),$

where we have used $\langle\left|b-b_{Q}\right|^{t}\rangle_{Q}^{1/t}\leq
2^{n+1}e^{3}t\|b\|_{\mathrm{BMO}}$ (see [64, p. 19]) in the last inequality.
Let $\mathcal{M}_{r}$ be the multilinear maximal operator with power $r>1$
defined by

$\mathcal{M}_{r}(\vec{f})(x):=\sup\limits_{Q\ni
x}\prod\limits_{i=1}^{m}\left(\frac{1}{|Q|}\int_{Q}\left|f_{i}(y)\right|^{r}dy\right)^{\frac{1}{r}}.$

Using Carleson embedding theorem again, one may obtain

(7.7)
$\displaystyle\int_{\mathbb{R}^{n}}\phi(\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x))w(x)dx$
$\displaystyle\lesssim[w]_{A_{\infty}}^{l+1}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\int_{\mathbb{R}^{n}}\mathcal{M}_{r}\vec{f}(x)M_{L(\log
L)^{l_{1}}(w)}^{\mathcal{D}_{j}}(h)(x)w(x)dx$ $\displaystyle\leq
C^{\prime}[w]_{A_{\infty}}^{l+1}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\int_{\mathbb{R}^{n}}\mathcal{M}_{r}\vec{f}(x)(M_{w}^{\mathcal{D}_{j}})^{l+1}(h)(x)w(x)dx.$

For convenience, we give some notation for the constants $a_{1},a_{2}$ and
$\alpha.$ First, since $\phi$ has the sub-multiplicative property, it follows
that $\phi(2t)\leq\phi(2)\phi(t)$ which implies that $\phi\in\Delta_{2}.$
Hence, there exists some $0<\alpha<1,$ such that $\bar{\phi}^{\alpha}$ is
quasi-convex, which means that, there exists some convex function $\psi$ and
$a_{1}>1$ such that

$\psi(t)\leq\bar{\phi}^{\alpha}(t)\leq a_{1}\psi(a_{1}t),\ t>0.$

Afterwards, for $w\in A_{\infty}$ and $\phi\in\Phi$ which satisfies that there
exists $0<\alpha<1$ such that $\phi^{\alpha}$ be a quasi-convex function, we
recall the following modular inequality [1, Lemma 3.12] for the weighted
maximal operator

(7.8)
$\int_{\mathbb{R}^{n}}\phi\left(M_{w}^{\mathcal{D}}f(x)\right)w(x)\mathrm{d}x\leq
a_{2}\int_{\mathbb{R}^{n}}\phi\left(a_{2}|f(x)|\right)w(x)\mathrm{d}x,$

where the constant $a_{2}>1$ only depends on $\phi$ and $\alpha$, and is
independent of $w$.

Using the previous notation, we take some $\varepsilon$ such that

$0<\varepsilon\leq\min\Big{\\{}\frac{1}{2},\frac{1}{a_{1}a_{2}^{l+1}},\Big{(}\frac{1}{2C^{\prime}a_{2}^{l+1}[w]_{A_{\infty}}^{l+1}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}}\Big{)}^{\alpha}\cdot\frac{1}{a_{1}^{2}a_{2}^{l+1}}\Big{\\}},$

where $C^{\prime}$ is determined by (7.7).
Combining (7.7) with the Young’s inequality and applying (7.8) $l+1$ times,
one obtains

(7.9)
$\displaystyle\int_{\mathbb{R}^{n}}\phi\left(\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)\right)w(x)dx$
$\displaystyle\leq
C^{\prime}[w]_{A_{\infty}}^{l+1}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\int_{\mathbb{R}^{n}}\frac{\mathcal{M}_{r}\vec{f}(x)}{\varepsilon}(M_{w}^{\mathcal{D}_{j}})^{l+1}(\varepsilon
h)(x)w(x)dx$ $\displaystyle\leq
C^{\prime}[w]_{A_{\infty}}^{l+1}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\bigg{[}\int_{\mathbb{R}^{n}}\phi\Big{(}\frac{\mathcal{M}_{r}\vec{f}(x)}{\varepsilon}\Big{)}w(x)dx$
$\displaystyle\hskip
14.22636pt+\int_{\mathbb{R}^{n}}\bar{\phi}\left((M_{w}^{\mathcal{D}_{j}})^{l+1}(\varepsilon
h)(x)\right)w(x)dx\bigg{]}$ $\displaystyle\leq
C^{\prime}[w]_{A_{\infty}}^{l+1}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\cdot
2^{C_{1}}\varepsilon^{-C_{1}}\int_{\mathbb{R}^{n}}\phi\left(\mathcal{M}_{r}\vec{f}(x)\right)w(x)dx$
$\displaystyle\hskip
14.22636pt+C^{\prime}[w]_{A_{\infty}}^{l+1}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\cdot
a_{2}^{l+1}\int_{\mathbb{R}^{n}}\bar{\phi}\left(a_{2}^{l+1}\varepsilon
h(x)\right)w(x)dx.$

Consider to estimate $\int_{\mathbb{R}^{n}}\bar{\phi}(a_{2}^{l+1}\varepsilon
h(x))w(x)dx.$ Noting that $\bar{\phi}^{\alpha}$ is a quasi-convex function, it
follows that

$\displaystyle\bar{\phi}^{\alpha}\left(a_{2}^{l+1}\varepsilon h(x)\right)\leq
a_{1}\psi\left(a_{1}a_{2}^{l+1}\varepsilon h(x)\right)\leq
a_{1}^{2}a_{2}^{l+1}\varepsilon\psi\left(h(x)\right)\leq
a_{1}^{2}a_{2}^{l+1}\varepsilon\bar{\phi}^{\alpha}\left(h(x)\right),$

where in the above inequality we have used $a_{1}a_{2}^{l+1}\varepsilon\leq 1$
and $\psi(\lambda t)\leq\lambda\psi(t)(0\leq\lambda\leq 1).$ Thus, the
definition of $h$ together with the fact that
$\bar{\phi}\left(\frac{\phi(t)}{t}\right)\leq\phi(t)$ for $t>0,$ gives that

$\displaystyle\int_{\mathbb{R}^{n}}\bar{\phi}\left(a_{2}^{l+1}\varepsilon
h(x)\right)w(x)dx$
$\displaystyle\leq\left(a_{1}^{2}a_{2}^{l+1}\varepsilon\right)^{\frac{1}{\alpha}}\int_{\mathbb{R}^{n}}\bar{\phi}\bigg{(}\frac{\phi(\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x))}{\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)}\bigg{)}w(x)dx$
$\displaystyle\leq\left(a_{1}^{2}a_{2}^{l+1}\varepsilon\right)^{\frac{1}{\alpha}}\int_{\mathbb{R}^{n}}\phi\left(\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)\right)w(x)dx.$

Plugging the above estimate into (7.9), we obtain

$\displaystyle\int_{\mathbb{R}^{n}}\phi\left(\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)\right)w(x)dx$
$\displaystyle\leq
C^{\prime}[w]_{A_{\infty}}^{l+1}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\cdot
2^{C_{1}}\varepsilon^{-C_{1}}\int_{\mathbb{R}^{n}}\phi\left(\mathcal{M}_{r}(\vec{f})(x)\right)w(x)dx$
$\displaystyle\quad+C^{\prime}[w]_{A_{\infty}}^{l+1}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\cdot
a_{2}^{l+1}$
$\displaystyle\quad\quad\times\left(a_{1}^{2}a_{2}^{l+1}\varepsilon\right)^{\frac{1}{\alpha}}\int_{\mathbb{R}^{n}}\phi\left(\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)\right)w(x)dx,$

which further implies that for $1\leq j\leq 3^{n},$

(7.10)
$\displaystyle\int_{\mathbb{R}^{n}}\phi\left(\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)\right)w(x)dx\lesssim\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\cdot[w]_{A_{\infty}}^{(l+1)(\alpha{C_{1}}+1)}\int_{\mathbb{R}^{n}}\phi\left(\mathcal{M}_{r}(\vec{f})(x)\right)w(x)dx.$

We will give another proof of the modular inequality of
$\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}},$ which has the
advantage to give the best possible range of $i_{\phi}.$

Similarly as the argument for (3.3), one can verfy that there exists a sparse
family $\tilde{{\mathcal{S}}}$ such that for any $l_{1}+1\leq t\leq l,$

$\displaystyle\big{|}b_{t}(x)-{\langle b_{t}\rangle}_{Q}\big{|}\leq
2^{n+2}\sum_{R\in\tilde{{\mathcal{S}}},R\subseteq
Q}{\langle\left|b_{t}-{\langle
b_{t}\rangle}_{R}\right|\rangle}_{R}\chi_{R}(x).$

Therefore

$\displaystyle\prod_{s=l_{1}+1}^{l}{\langle\big{|}(b_{s}-{\langle
b_{s}\rangle}_{Q})f_{s}\big{|}\rangle}_{Q}$
$\displaystyle\lesssim\prod_{s=l_{1}+1}^{l}\frac{1}{|Q|}\int_{Q}\sum_{R\in\tilde{{\mathcal{S}}},R\subseteq
Q}{\langle\left|b_{s}-{\langle
b_{s}\rangle}_{R}\right|\rangle}_{R}\chi_{R}(x)|f_{s}(x)|dx$
$\displaystyle\lesssim\prod_{s=l_{1}+1}^{l}\|b_{s}\|_{\mathrm{BMO}}\sum_{R\in\tilde{{\mathcal{S}}},R\subseteq
Q}\frac{1}{|Q|}\int_{R}|f_{s}(x)|dx$
$\displaystyle\lesssim\prod_{s=l_{1}+1}^{l}\|b_{s}\|_{\mathrm{BMO}}\frac{1}{|Q|}\int_{Q}\mathcal{A}_{\tilde{\mathcal{S}}}(f_{s})(x)dx.$

It then follows from the above estimate and (7.6) that

$\displaystyle\int_{\mathbb{R}^{n}}\phi\left(\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)\right)w(x)dx$
$\displaystyle\lesssim\sum_{Q\in\tilde{\mathcal{S}}_{j}}\prod_{s=1}^{l_{1}}{\langle|f_{s}|\rangle}_{Q}\prod_{s=l+1}^{m}{\langle|f_{s}|\rangle}_{Q}\prod_{s=l_{1}+1}^{l}{\langle\big{|}(b_{s}-{\langle
b_{s}\rangle}_{Q})f_{s}\big{|}\rangle}_{Q}$
$\displaystyle\quad\quad\times\prod_{s=1}^{l_{1}}\left\|b_{s}-{\langle
b_{s}\rangle}_{Q}\right\|_{\exp L(w),Q}\|h\|_{L(\log L)^{l_{1}}(w),Q}w(Q)$
$\displaystyle\lesssim[w]_{A_{\infty}}^{l_{1}}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\prod_{s=1}^{l_{l}}{\langle|f_{s}|\rangle}_{Q}\prod_{s=l+1}^{m}{\langle|f_{s}|\rangle}_{Q}\prod_{s=l_{1}+1}^{l}{\langle\mathcal{A}_{\tilde{\mathcal{S}}}f_{s}\rangle}_{Q}$
$\displaystyle\quad\times\|h\|_{L(\log L)^{l_{1}}(w),Q}w(Q)$
$\displaystyle\lesssim[w]_{A_{\infty}}^{l+1}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\int_{\mathbb{R}^{n}}\mathcal{M}(\vec{f^{*}})(x)M_{L(\log
L)^{l_{1}}(w)}^{\mathcal{D}_{j}}(h)(x)w(x)dx.$

where
$\vec{f^{*}}=\left(f_{1},\ldots,f_{l_{1}},\mathcal{A}_{\tilde{\mathcal{S}}}(f_{l_{1}+1}),\ldots,\mathcal{A}_{\tilde{\mathcal{S}}}(f_{l}),f_{l+1},\ldots,f_{m}\right).$

The same reasoning as what we have done with the case for $\mathcal{M}_{r}$
then gives

(7.11)
$\displaystyle\int_{\mathbb{R}^{n}}\phi\left(\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)\right)w(x)dx$
$\displaystyle\leq
C^{\prime}[w]_{A_{\infty}}^{(1+l)(\alpha{C_{1}}+1)}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\int_{\mathbb{R}^{n}}\phi\left(\mathcal{M}(\vec{f^{*}})(x)\right)w(x)dx.$

Hence, by (7.10) and (7.11), we can dominate the left side of 7.11 by a
constant times

(7.12)
$\displaystyle[w]_{A_{\infty}}^{(1+l)(\alpha{C_{1}}+1)}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\min\left\\{\int_{\mathbb{R}^{n}}\phi\left(\mathcal{M}_{r}(\vec{f})(x)\right)w(x)dx,\int_{\mathbb{R}^{n}}\phi\left(\mathcal{M}(\vec{f^{*}})(x)\right)w(x)dx\right\\}.$

Having obtained the above estimate, we are in a position to finish our proof.
First, for any $r\geq 1,$ $w\in A_{q}$ with $1<q<\frac{i_{\phi}}{r}$, in [62,
Lemma 5.3], it was proved that there exists a constant $a_{3}>1$ such that

(7.13)
$\displaystyle\int_{\mathbb{R}^{n}}\phi\left(\mathcal{M}_{r}(\vec{f})(x)\right)w(x)dx\leq
a_{3}\left(\prod_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi^{m}\big{(}a_{3}[w]_{A_{q}}^{\frac{1}{q_{r}}}|f_{i}(x)|\big{)}w(x)dx\right)^{\frac{1}{m}}.$

In particular, when $r=1$ then it holds for every $w\in A_{q}$ with
$1<q<i_{\phi}$ that

$\displaystyle\int_{\mathbb{R}^{n}}\phi\left(\mathcal{M}(\vec{f^{*}})(x)\right)w(x)dx$
$\displaystyle\leq a_{3}\Bigg{(}\prod_{i\in\\{1,\cdots,m\\}\atop
i\notin\\{l_{1}+1,\cdots,l\\}}\int_{\mathbb{R}^{n}}\phi^{m}\Big{(}a_{3}[w]_{A_{q}}^{\frac{1}{q}}|f_{i}(x)|\Big{)}w(x)dx\Bigg{)}^{\frac{1}{m}}$
$\displaystyle\quad\times\Bigg{(}\prod_{i=l_{1}+1}^{l}\int_{\mathbb{R}^{n}}\phi^{m}\left(a_{3}[w]_{A_{q}}^{\frac{1}{q}}\mathcal{A}_{\tilde{\mathcal{S}}}(f_{i})(x)\right)w(x)dx\Bigg{)}^{\frac{1}{m}}.$

In order to apply
$\int_{\mathbb{R}^{n}}\phi^{m}\big{(}a_{3}[w]_{A_{q}}^{\frac{1}{q}}\mathcal{A}_{\tilde{\mathcal{S}}}(f_{i})(x)\big{)}w(x)dx$
$(i=l_{1}+1,\cdots,l)$ to Lemma 7.2, we need the following observation: for
any $m\in\mathbb{N}^{*},$ $\phi^{m}$ is an $N$-function and
$I_{\phi^{m}}<\infty.$ First we show that $\phi^{m}$ is a convex function. To
see this, it suffices to prove that if $f,g$ are $N$-functions then $fg$ is a
convex function, this means for every $x\geq y>0,\lambda\in(0,1),$

(7.14) $f(\lambda x+(1-\lambda)y)g(\lambda x+(1-\lambda)y)\leq\lambda
f(x)g(x)+(1-\lambda)f(y)g(y).$

In deed, we can dominate $f(\lambda x+(1-\lambda)y)g(\lambda x+(1-\lambda)y)$
by

$\displaystyle\lambda^{2}f(x)g(x)+(1-\lambda)^{2}f(y)g(y)+\lambda(1-\lambda)(f(x)g(y)+f(y)g(x)).$

Then, a simple calculation gives

$\displaystyle f(\lambda x+(1-\lambda)y)g(\lambda x+(1-\lambda)y)\leq\lambda
f(x)g(x)+(1-\lambda)f(y)g(y),$

where in the last inequality we have used the fact that $f,g$ are
$N$-functions. This proves (7.14).
On the one hand, it is easy to verify that

$\lim_{t\rightarrow 0^{+}}\frac{\phi^{m}(t)}{t}=0.$

On the other hand, using $\lim_{t\rightarrow\infty}\phi(t)=\infty,$ we obtain
that there exists $M>0$ such that $\phi(t)>1$ holds for any $t>M.$ Therefore,

$\frac{\phi^{m}(t)}{t}\geq\frac{\phi(t)}{t}\rightarrow\infty(t\rightarrow\infty),$

which implies that $\phi^{m}$ is an $N$-function.
It remains to prove that $I_{\phi^{m}}<\infty$ is valid. Recall that
$h_{\phi}(t)=\sup_{s>0}\frac{\phi(st)}{\phi(s)},t>0,$ and

$I_{\phi}=\lim_{t\rightarrow\infty}\frac{\log h_{\phi}(t)}{\log
t}=\inf_{1<t<\infty}\frac{\log h_{\phi}(t)}{\log t}.$

Thus, we have $h_{{\phi}^{m}}(t)=h_{\phi}^{m}(t),$ which indicates that

$I_{{\phi}^{m}}=\inf_{1<t<\infty}\frac{\log h_{\phi}^{m}(t)}{\log
t}=mI_{\phi}<\infty.$

In virtue of the preceding observation, applying Lemma 7.2 with
$\tilde{C}=mC_{1}$, we have

$\displaystyle\Bigg{(}\prod_{i=l_{1}+1}^{l}\int_{\mathbb{R}^{n}}\phi^{m}\left(a_{3}[w]_{A_{q}}^{\frac{1}{q}}\mathcal{A}_{\tilde{\mathcal{S}}}(f_{i})(x)\right)w(x)dx\Bigg{)}^{\frac{1}{m}}$
$\displaystyle\quad\quad\quad\quad\lesssim[w]_{A_{\infty}}^{1+mC_{1}}\left(\prod_{i=l_{1}+1}^{l}\int_{\mathbb{R}^{n}}\phi^{m}\Big{(}a_{3}^{2}[w]_{A_{q}}^{\frac{2}{q}}|f_{i}(x)|\Big{)}w(x)dx\right)^{\frac{1}{m}}.$

Therefore

$\int_{\mathbb{R}^{n}}\phi\left(\mathcal{M}(\vec{f^{*}})(x)\right)w(x)dx\lesssim[w]_{A_{\infty}}^{1+mC_{1}}\left(\prod_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi^{m}\left([w]_{A_{q}}^{\frac{2}{q}}|f_{i}(x)|\right)w(x)dx\right)^{\frac{1}{m}}.$

This inequality, together with (7.12) and (7.13), yields that

  1. (i)

if $r<i_{\phi}<\infty,$ then for every $1<q<\frac{i_{\phi}}{r}$ and $w\in
A_{q},$

(7.15)
$\displaystyle\int_{\mathbb{R}^{n}}\phi\left(\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)\right)w(x)dx\lesssim$
$\displaystyle[w]_{A_{\infty}}^{(1+l)(\alpha
C_{1}+1)}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}^{1+\alpha C_{1}}$
$\displaystyle\times\left(\prod_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi^{m}\left([w]_{A_{q}}^{\frac{1}{qr}}\left|f_{i}(x)\right|\right)w(x)dx\right)^{\frac{1}{m}};$

  2. (ii)

if $1<i_{\phi}\leq r<\infty,$ then for every $1<q<i_{\phi}$ and $w\in A_{q},$

(7.16)
$\displaystyle\int_{\mathbb{R}^{n}}\phi\left(\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})(x)\right)w(x)dx\lesssim$
$\displaystyle[w]_{A_{\infty}}^{(l+1)(\alpha
C_{1}+1)+1+mC_{1}}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}^{1+\alpha C_{1}}$
$\displaystyle\times\left(\prod_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi^{m}\left([w]_{A_{q}}^{\frac{2}{q}}\left|f_{i}(x)\right|\right)w(x)dx\right)^{\frac{1}{m}}.$

Finally, applying the convexity of $\phi$ and replacing
$\mathcal{A}_{\mathcal{S}_{j},\vec{b}}^{\vec{\gamma}}(\vec{f})$ with
$\mathcal{T}_{\vec{b}}(\vec{f})$, inequalities (7.15) and (7.16) still hold,
which completes the proof of Theorem 1.18.

∎

##  8\.  applications

In this section, we present some applications of the results obtained in
Section 1.3. We will see that the Hypothesis 1 or Hypothesis 2 hold for
multilinear $\omega$-Calderón-Zygmund operators, multilinear pseudo-
differential operators, higher order Calderón commutators, and Stein’s square
functions. Furthermore, we will establish weighted modular estimates for them.

###  8.1.  Multilinear $\omega$-Calderón-Zygmund operators

We recall the definition of multilinear Calderón-Zygmund operator of type
$\omega$.

######  Definition 8.1 (Multilinear $\omega$-Calderón-Zygmund operator).

Let $\omega(t):[0,\infty)\rightarrow[0,\infty)$ be a nondecreasing function. A
locally integrable function $K\left(x,y_{1},\ldots,y_{m}\right)$, defined away
from the diagonal $x=y_{1}=$ $\cdots=y_{m}$ in
$\left(\mathbb{R}^{n}\right)^{m+1}$, is called an $m$-linear Calderón-Zygmund
kernel of type $\omega$ if, for some constants $0<\tau<1$, there exists a
constant $A>0$ such that

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

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

$\displaystyle\left|K\left(x,y_{1},\ldots,y_{m}\right)-K\left(x^{\prime},y_{1},\ldots,y_{m}\right)\right|$
$\displaystyle\qquad\leq\frac{A}{\left(\left|x-y_{1}\right|+\cdots+\left|x-y_{m}\right|\right)^{mn}}\omega\left(\frac{\left|x-x^{\prime}\right|}{\left|x-y_{1}\right|+\cdots+\left|x-y_{m}\right|}\right)$

whenever $\left|x-x^{\prime}\right|\leq\tau\max_{1\leq j\leq
m}\left|x-y_{j}\right|$, and for all $1\leq i\leq m$

$\displaystyle\left|K\left(x,y_{1},\ldots,y_{i},\ldots,y_{m}\right)-K\left(x,y_{1},\ldots,y_{i}^{\prime},\ldots,y_{m}\right)\right|$
$\displaystyle\qquad\leq\frac{A}{\left(\left|x-y_{1}\right|+\cdots+\left|x-y_{m}\right|\right)^{mn}}\omega\left(\frac{|y_{i}-y_{i}^{\prime}|}{\left|x-y_{1}\right|+\cdots+\left|x-y_{m}\right|}\right)$

whenever $\left|y_{i}-y_{i}^{\prime}\right|\leq\tau\max_{1\leq j\leq
m}\left|x-y_{j}\right|$. Particularly, when $\omega(t)=t^{\delta}$ with
$\delta\in(0,1],K$ is called an $m$-linear standard Calderón-Zygmund kernel.
We say $T$:
$\mathscr{S}\left(\mathbb{R}^{n}\right)\times\cdots\times\mathscr{S}\left(\mathbb{R}^{n}\right)\rightarrow\mathscr{S}^{\prime}\left(\mathbb{R}^{n}\right)$
is an $m$-linear operator with an $m$-linear Calderón-Zygmund kernel of type
$\omega,K\left(x,y_{1},\ldots,y_{m}\right)$, if

$T\left(f_{1},\ldots,f_{m}\right)(x)=\int_{(\mathbb{R}^{n})^{m}}K\left(x,y_{1},\ldots,y_{m}\right)f_{1}\left(y_{1}\right)\cdots
f_{m}\left(y_{m}\right)dy_{1}\cdots dy_{m}$

whenever $x\notin\bigcap_{j=1}^{m}\operatorname{supp}f_{j}$ and each $f_{j}$
with $j=1,\ldots,m$ is a bounded function with compact support. If $T$ can be
extended to a bounded multilinear operator from
$L^{q_{1}}\left(\mathbb{R}^{n}\right)\times\cdots\times$
$L^{q_{m}}\left(\mathbb{R}^{n}\right)$ to
$L^{q,\infty}\left(\mathbb{R}^{n}\right)$ for some
$1/q=1/q_{1}+\cdots+1/q_{m}$ with $1\leq q_{1},\ldots,q_{m}<\infty$, then $T$
is called an $m$-linear Calderón-Zygmund operator of type $\omega$.

######  Definition 8.2 ($\log$-Dini condition ).

Let $\omega(t):[0,\infty)\rightarrow[0,\infty)$ be a nondecreasing function.
For $a>0,m\in\mathbb{N}$, we say that $\omega$ satisfies the
$\log$-Dini$(a,m)$ condition, denote $\omega\in$ $\log$-Dini$(a,m)$, if

$\|\omega\|_{\log\mathrm{-Dini}(a,m)}:=\int_{0}^{1}\frac{\omega^{a}(t)}{t}\left(1+\log\frac{1}{t}\right)^{m}dt<\infty.$

It is worth mentioning that, in 2014, Lu and Zhang [42] obtained the weighted
inequalities for commutators of multilinear Calderón-Zygmund operators $T$ of
type $\omega$ with $\omega\in$ $\log$-Dini$(1,m).$ Very recently, Cao et al.
[8] proved the local exponential decay, mixed weak type estimate for $T$ with
$\omega\in$ $\log$-Dini$(1,0),$ and obtained the weighted compactness for
commutators.

As was shown in [8, Theorem 2.5], both Hypothesis 1 and Hypothesis 2 hold. By
Theorems 1.4, 1.10 and 1.14, we obtain

######  Theorem 8.3.

Let $I=\\{i_{1},\ldots,i_{l}\\}=\\{1,\ldots,l\\}\subseteq\\{1,\ldots,m\\},$
$T$ be an $m$-linear $\omega$-Calderón-Zygmund operator with $\omega\in$
$\log$-Dini$(1,m).$ If $\vec{b}\in\mathrm{BMO}^{l},$ then we have

  1. (a).

Let $w\in A_{\infty}^{\text{weak}},$ $Q_{0}$ be a cube and $f_{s}\in
L_{c}^{\infty}\left(\mathbb{R}^{n}\right)$ such that
$\operatorname{supp}\left(f_{s}\right)\subset Q_{0}$ for $1\leq s\leq m.$ Then
there are constants $\alpha,c>0$ independent of $w$ such that

$\displaystyle w\left(\left\\{x\in
Q_{0}:\left|T_{\vec{b}}(\vec{f})(x)\right|>t\mathcal{M}_{L(\log
L)}(\vec{f})(x)\right\\}\right)$ $\displaystyle\hskip 142.26378pt\qquad\leq
ce^{-{\frac{\alpha}{[w]_{A_{\infty}}^{weak}+1}}\left({\frac{t}{\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}}}\right)^{\frac{1}{l+1}}}w(2Q_{0}),\quad
t>0.$

  2. (b).

Let $\vec{w}=\left(w_{1},\ldots,w_{m}\right)$ and
$u=\prod_{i=1}^{m}w_{i}^{1/m}$. If $\vec{w}\in A_{\vec{1}}$ and $v\in
A_{\infty},$ then there exists $t>1$ depending only on $v$, such that

$\left\|\frac{T_{\vec{b}}(\vec{f})}{v}\right\|_{L^{\frac{1}{m},\infty}(uv^{\frac{1}{m}})}\lesssim
K_{0}^{2l+6m}[v^{\frac{1}{m}}]_{A_{t}}^{2l+4m}\prod_{s=1}^{l}\|b_{s}\|_{\mathrm{BMO}}\left\|\frac{\mathcal{M}_{L(\log
L)}(\vec{f})}{v}\right\|_{L^{\frac{1}{m},\infty}(uv^{\frac{1}{m}})},$

where
$K_{0}=4C_{n}p_{0}p_{0}^{\prime}([u]_{A_{1}}+2^{p_{0}-1}C_{n}^{t}[v^{\frac{1}{m}}]_{A_{t}}^{2}[u]_{A_{1}}^{p_{0}-1})+1$
with $p_{0}=2^{n+3}(t-1)[u]_{A_{1}}+1.$

  3. (c).

Let $1<p_{1},\ldots,p_{m}<\infty$ and
$\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}.$ Assume that $m\geq 2$
and for all weights
$\vec{w}=\left(w_{1},\ldots,w_{m}\right),\nu_{\vec{w}}=\prod_{s=1}^{m}w_{s}^{p/p_{s}},$
$b_{s}\in\mathrm{BMO}_{p_{s}}(w_{s})\cap\mathrm{BMO}$ with $1\leq s\leq l.$ If
$0<p\leq 1,$ then

$\left\|T_{\vec{b}}(\vec{f})\right\|_{L^{p}\left(\nu_{\vec{w}}\right)}\leq
C\|\vec{b}\|_{\mathrm{BMO}}^{*}\prod_{s=1}^{m}\left\|f_{s}\right\|_{L^{p_{s}}(Mw_{s})},$

where $C$ is independent of $\vec{w}$ and $\vec{b},$ and

$\|\vec{b}\|_{\mathrm{BMO}}^{*}=\max_{\vec{\gamma}\in\\{1,2\\}^{l}}\\{\prod_{s:\gamma_{s}=1}\|b_{s}\|_{\mathrm{BMO}_{p_{s}}(w_{s})}\prod_{s:\gamma_{s}=2}\|b_{s}\|_{\mathrm{BMO}}\\}.$