# Quantitative weighted estimates for the multilinear pseudo-differential
operators in function spaces

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 weighted estimates for multilinear pseudo-differential
operators were systematically studied in rearrangement invariant Banach and
quasi-Banach spaces. These spaces contain the Lebesgue space, the classical
Lorentz space and Marcinkiewicz space as typical examples. More precisely, the
weighted boundedness and weighted modular estimates, including the weak
endpoint case, were established for multilinear pseudo-differential operators
and their commutators. As applications, we show that the above results also
hold for the multilinear Fourier multipliers, multilinear square functions,
and a class of multilinear Calderón-Zygmund operators.

###### Key words and phrases:

pseudo-differential operators, commutators, sparse operators, modular
inequalities
2020 Mathematics Subject Classification. Primary 42B20, Secondary 42B35.

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

The main purpose of the article is to establish the weighted boundedness of
the multilinear pseudo-differential operators in the rearrangement invariant
Banach and quasi-Banach spaces, and establish the associated modular
inequalities.

The study of the pseudo-differential operators can be traced back to the
celebrated works of Kohn and Nirenberg [37] and Hörmander [30]. As was well
known, the pseudo-differential operators play an important role in Harmonic
analysis as well as in other fields, such as PDEs and mathematical physics.
Particularly, the theory of the pseudo-differential operators has not only
been widely used in quantum field and exponential theory, but also played an
indispensable role in the first proof of the Atiyah-Singer index theorem [3].
Further important applications were given in the works of Hörmander [29] and
Fefferman and Kohn [26]. A generalized class of pseudo-differential operators
related to hypoelliptic equations was introduced in [29] and applied to study
the existence and regularity of the solutions. By using the tools of
microlocal analysis and the pseudo-differential operators on $\mathbb{R}^{3},$
Fefferman and Kohn [26] obtained the optimal Hölder estimates for the Bergman
projection. The corresponding estimates for the Szegö projection on 3-D CR
manifolds of finite type were also given in [26].

We now consider the boundedness of the pseudo-differential operators on
Lebesgue spaces and Hardy spaces. The minimum smoothness and decay conditions
assumed on the symbols to guarantee their boundedness in these function spaces
are what we are interested in. To make the statement of known results clearly,
we need to introduce some definitions and briefly describe the literature of
this problem.

######  Definition 1.1 (Hörmander’s class).

Let $m\in\mathbb{N},$ $r\in\mathbb{R}$ and $0\leq\rho,\delta\leq 1,$ $\sigma$
is a smooth function defined on $\mathbb{R}^{n}\times\mathbb{R}^{mn}$. We say
$\sigma\in S_{\rho,\delta}^{r}(n,m)$ if for each triple of multi-indices
$\alpha:=\left(\alpha_{1},\ldots,\alpha_{n}\right)$ and
$\beta_{1},\ldots,\beta_{m}$, there exists a constant $C_{\alpha,\beta}$ such
that

$\left|\partial_{x}^{\alpha}\partial_{\xi_{1}}^{\beta_{1}}\cdots\partial_{\xi_{m}}^{\beta_{m}}\sigma(x,\vec{\xi})\right|\leq
C_{\alpha,\beta}\left(1+\left|\xi_{1}\right|+\cdots+\left|\xi_{m}\right|\right)^{r-\rho\sum_{j=1}^{m}\left|\beta_{j}\right|+\delta|\alpha|}.$

######  Definition 1.2 ($m$-linear pseudo-differential operator).

Given a symbol $\sigma$, $m\in\mathbb{N},$ the $m$-linear pseudo-differential
operator $T_{\sigma}$ is given by

$T_{\sigma}(\vec{f})(x)=\int_{(\mathbb{R}^{n})^{m}}\sigma(x,\vec{\xi})e^{2\pi
ix\cdot(\xi_{1}+\cdots\xi_{m})}\widehat{f}_{1}(\xi_{1})\cdots\widehat{f}_{m}(\xi_{m})d\vec{\xi},$

where $\widehat{f}$ is the Fourier transform of the function $f$ defined by

$\widehat{f}(\xi)=\int_{\mathbb{R}^{n}}f(x)e^{-2\pi ix\cdot\xi}dx.$

In the linear case $m=1,$ Hörmander [31], Kumano-Go [38], Calderón and
Vaillancourt [12] proved the local and global $L^{2}$ bounds of pseudo-
differential operators in the 1970s. In 1976, Rodino [56] constructed a
symbolic function such that the pseudo-differential operator is not bounded on
$L^{2}$. Later on, using the method of the almost orthogonality principle,
Hounie [32] gave an equivalence description between the indexes of Hörmander
class and the $L^{2}$ boundedness of the pseudo-differential operators. As a
generalization of the classical Hörmander class, Coifman and Meyer [19]
studied a class of symbols with Dini type conditions and gave a sufficient and
necessary condition for the $L^{p}$ boundedness of the pseudo-differential
operators. Subsequently, Nagase [51] and Bourdaud [8] respectively improved
Coifman and Meyer’s results under two more general conditions assumed on the
symbols. In the endpoint cases, Alvarez and Hounie [1] demonstrated that if
$\sigma\in S_{\rho,\delta}^{r}$ with $0<\rho\leq 1,0\leq\delta<1$ and
$r=\frac{n}{2}(\rho-1+\min\\{0,\rho-\delta\\})$, then $T_{\sigma}$ is of weak
type $(1,1)$ and bounded from $H^{1}$ to $L^{1},$ where $H^{1}$ denotes the
classical Hardy spaces.

Great achievements have been made on the study of weighted boundedness of
$T_{\sigma}$. Let $w$ be a Muckenhoupt $A_{p}$ weight with $w\in A_{p/2}(2\leq
p<\infty)$ and $\sigma\in S_{\rho,\delta}^{-n(1-\rho)/2}(n,1)$ with
$0<\delta<\rho<1,$ the weighted boundedness of $T_{\sigma}$ on $L^{p}(w)$ has
been obtained by Chanillo and Torchinksy in [18]. Subsequently, Michalowski,
Rule and Staubach [48] considered the above condition with $\rho=\delta$.
Recently, Beltran and Cladek [4] established a sparse domination of the
pseudo-differential operators and consequently obtained a quantitative
weighted estimate.

We now turn our attention to the study of multilinear pseudo-differential
operators. The theory of bilinear pseudo-differential operators was first
originated from the famous works of Coifman and Meyer [20], in which they used
the bilinear pseudo-differential operators as a typical model for the Calderón
commutator. In [19, 21], when $\rho=1$, Coifman and Meyer showed that
$T_{\sigma}$ is bounded from $L^{p_{1}}\times\cdots\times L^{p_{m}}$ to
$L^{p}$ for symbol $\sigma\in S_{1,0}^{0}(n,m)$ with
$1<p,p_{1},\ldots,p_{m}<\infty$ and
$\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}=\frac{1}{p}.$ Grafakos and Torres [28]
considered the boundedness of multilinear pseudo-differential operators
$T_{\sigma}$ with symbol $\sigma$ belongs to $S_{1,1}^{0}(n,m).$ There is a
fundamental question in this field. It is known that the multilinear Calderón-
Zygmund operators enjoy almost all the properties of the linear Calderón-
Zygmund operators, one may ask whether the multilinear pseudo-differential
operators also have such properties? However, Bényi and Torres in [6] showed
that there exist symbols in $S_{1,1}^{0}(n,2)$ such that $T_{\sigma}$ fails to
be bounded from $L^{p_{1}}\times L^{p_{2}}$ to $L^{p}$ for $1\leq
p_{1},p_{2},p<\infty$ with $\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p}.$

The situation becomes complicated and subtle when $\rho\neq 1$. We summarize
these cases and classify the boundedness of pseudo-differential operators
according to the parameters values of the symbol. See Table 1 below. We refer
the readers to see [46, 14] and the references therein for some other weighted
boundedness properties of multilinear pseudo-differential operators and their
commutators.

Table 1. The symbol of multilinear pseudo-differential operators parameters | $L^{2}\times L^{2}\rightarrow L^{1}$ | $L^{\infty}\times L^{\infty}\rightarrow L^{\infty}$ | $L^{\infty}\times L^{\infty}\rightarrow BMO$
---|---|---|---
$\rho=\delta=r=0$ | ✗[6] |  | ✗[49]
$0\leq\delta\leq\rho\leq 1,\delta<1$ $r<-n(1-\rho)$ | ✓when $r<-\frac{n(1-\rho)}{2}$ [7] | ✓[7] | ✓
$\delta=\rho=0$ $r>-n$ | ✗[49] |  | ✗[49]
$0\leq\delta\leq\rho\leq 1$ $r=-n(1-\rho)$ | ✓when $\rho<1,\delta=\rho$ [50] |  |  ✓when $0\leq\delta\leq\rho<\frac{1}{2}$[52] or $0\leq\rho<1,\delta=\rho$ [50]

In order to state more known results, we give a brief introduction on sparse
domination. In the last decade, the problem of sharp weighted norm estimate
has led to a deep understanding of the classical Calderón-Zygmund operators in
the form of sparse domination. To be more precise, Lerner [41, 42] first
showed that any standard Calderón-Zygmund operator $T$ which satisfies a
Hölder-Lipschitz condition can be dominated pointwisely by a finite number of
sparse operators in the following way

$|Tf(x)|\leq\sup_{\mathcal{S}}\mathcal{A}_{\mathcal{S}}f(x),\text{where}~{}\mathcal{A}_{\mathcal{S}}f(x)=\sum_{Q\in\mathcal{S}}\frac{1}{|Q|}\int_{Q}|f|\chi_{Q}(x),$

where $\mathcal{S}$ is a sparse family of cubes (see Section $2.2$ below).
This estimate almost immediately leads to a simple proof of the sharp
dependence of the constant in the relevant weighted norm inequalities, the
$A_{2}$ conjecture, which has been actively investigated for more than a
decade (see [34, 42, 40]). Soon after, Lerner’s techniques were widely used
for many other operators as well as spaces (cf. e.g. [16, 22, 23]).

We need to introduce the $m$-linear commutators of pseudo-differential
operator.

######  Definition 1.3 ($m$-linear commutators of pseudo-differential
operator).

For the pseudo-differential operator $T_{\sigma}$ with symbol $\sigma$ and
locally integrable functions $\mathbf{b}=\left(b_{1},\ldots,b_{m}\right)$ with
$m\in\mathbb{N},$ the $m$-linear commutator of $T_{\sigma}$ is defined by

$T_{\sigma,\Sigma\mathbf{b}}(\vec{f})(x)=\sum_{j=1}^{m}\left[b_{j},T_{\sigma}\right]_{j}(\vec{f})(x),$

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

$\left[b_{j},T_{\sigma}\right]_{j}(\vec{f})(x)=b_{j}(x)T_{\sigma}(\vec{f})(x)-T_{\sigma}\left(f_{1},\ldots,b_{j}f_{j},\ldots,f_{m}\right)(x).$

The commutators given in Definition 1.3 were originally introduced by Pérez
and Torres [54] in the study of the $m$-linear Calderón-Zygmund operators. In
2009, Lerner et al. [43] considered the weighted strong-type as well as sharp
weak-type estimates for the $m$-linear commutators of the Calderón-Zygmund
operators. In the linear case $m=1,$ for any $r<0$ and $0\leq\delta<\rho<1$,
the $L^{p}$ boundedness of the operators $T_{\sigma,b}$ with $\sigma\in
S_{\rho,\delta}^{r}$ and $b\in\mathrm{BMO}$ has been studied by Chanillo [17].
For an overview of this area, we refer to [33, 58] and references therein. It
is worth mentioning that in [14], Cao, Xue and Yabuta obtained some sharp
results on multilinear pseudo-differential operators and their commutators,
including local exponential estimates, weighted mixed weak type inequality and
sharp weighted estimates for $T_{\sigma}$ and $T_{\sigma,\Sigma\mathbf{b}}$ by
using the method of sparse domination. More recent work can be found in [13]
in the space of generalized form. One of the main result in [14] is as
follows:

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

Assume that $\sigma\in S_{\rho,\delta}^{r}(n,m)$ with $\rho,\delta\in[0,1]$
and $r<mn(\rho-1)$. Let $\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}$
with $1<p_{i}<\infty,i=1,\ldots,m$. If $\vec{\omega}\in A_{\vec{p}}$ and
$\mathbf{b}\in\mathrm{BMO}^{m}$, then

$\left\|T_{\sigma}\right\|_{L^{p_{1}}\left(\omega_{1}\right)\times\cdots\times
L^{p_{m}}\left(\omega_{m}\right)\rightarrow
L^{p}\left(\nu_{\vec{\omega}}\right)}\leq c_{n,\vec{p}}\mathscr{N}_{\text{weak
}}[\vec{\omega}]_{A_{\vec{p}}}^{\beta(\vec{p})}$

and

$\begin{array}[]{r}\left\|T_{\sigma,\Sigma\mathbf{b}}\right\|_{L^{p_{1}}\left(\omega_{1}\right)\times\cdots\times
L^{p_{m}}\left(\omega_{m}\right)\rightarrow
L^{p}\left(\nu_{\vec{\omega}}\right)}\leq
c_{n,\vec{p}}\mathscr{N}_{weak}\|\mathbf{b}\|_{\mathrm{BMO}}[\vec{\omega}]_{A_{\vec{p}}}^{2\beta(\vec{p})},\\\
\end{array}$

where $\mathscr{N}_{\text{weak
}}=\left\|T_{\sigma}\right\|_{L^{1}\times\cdots\times L^{1}\rightarrow
L^{1/m,\infty}},\text{ and }\beta(\vec{p})=\max\limits_{1\leq i\leq
m}\left\\{1,\frac{p_{i}^{\prime}}{p}\right\\},$ for the definition of
$\vec{w}$ and $A_{\vec{p}}$ see [43, p. 1232].

Inspired by the results as above, it is natural to ask the following
questions.

######  Questions A (general Banach space).

Whether multilinear operators, such as multilinear pseudo-differential
operators $T_{\sigma}$ and their commutators $T_{\sigma,\Sigma\mathbf{b}}$,
multilinear Calderón-Zygmund singular integral operators, are bounded in more
general spaces, such as Lorentz spaces, Zygmund spaces? Furthermore, can we
find a unified way to deal with these operators and spaces?

######  Questions B (general quasi-Banach space).

Can we generalize the spaces in Theorem A to some quasi-Banach spaces ?

######  Questions C (modular inequalities).

What kinds of modular inequalities do multilinear operators $T_{\sigma}$ and
$T_{\sigma,\Sigma\mathbf{b}}$ enjoy ?

In this article, we give a firm answer to these three questions. Since
classical Lorentz spaces and Zygmund spaces are all belonging to the
rearrangement invariant Banach function spaces, this kind of function spaces
come naturally into our research scope. For the case of the quasi-spaces,
accordingly, we call the rearrangement invariant quasi-Banach function spaces.
A detailed definition of these two kinds of spaces is given in the next
section. The study of such spaces has a long history, in 1955, Lorentz [47]
first gave that a sufficient and necessary condition for Hardy-Littlewood
maximal operator $M$ to be bounded on rearrangement invariant Banach function
space $\mathbb{X}$ is $p_{\mathbb{X}}>1.$ Afterwards, Boyd [9] generalized
this result to the Hilbert transform $H$, and proved that $H$ is bounded on
$\mathbb{X}$ if and only if $1<p_{\mathbb{X}}\leqslant q_{\mathbb{X}}<\infty.$
Here $p_{\mathbb{X}}$ and $q_{\mathbb{X}}$ denote the Boyd indices of
$\mathbb{X}$ (see Section 2.3 below). For the weighted case, in 2006, Curbera
et al. [24] obtained the boundedness of Calderón-Zygmund operator in the
weighted rearrangement invariant quasi-Banach function spaces with the same
restrictions on Boyd’s index. In [2], Anderson and Hu obtained the
quantitative weighted boundedness of the maximal truncated Calderón-Zygmund
operators on $\mathbb{X}$ using sparse domination approach. For other works
related to these spaces, we refer to [5, 25] and references therein.

Now we are ready to state our main results.

######  Theorem 1.4.

Let $\mathbb{X},\mathbb{X}_{i}$ be rearrangement invariant Banach function
spaces with $1<p_{\mathbb{X}},p_{\mathbb{X}_{i}}\leq
q_{\mathbb{X}},q_{\mathbb{X}_{i}}<\infty$ for $i=1,2\cdots,m.$ Let $\sigma\in
S_{\rho,\delta}^{r}(n,m)$ with $0\leq\rho,\delta\leq 1,r<mn(\rho-1).$ Suppose
that $m$-product operator $P_{m}$ maps
$\overline{\mathbb{X}}_{1}\times\cdots\times\overline{\mathbb{X}}_{m}$ to
$\overline{\mathbb{X}},$ then for every $w\in
A_{\min\limits_{i}{\\{p_{\mathbb{X}_{i}}\\}}},$ it holds that

$\displaystyle\left\|T_{\sigma}(\vec{f})\right\|_{\mathbb{X}(w)}\lesssim[w]_{A_{\infty}}\prod\limits_{i=1}\limits^{m}[w]_{A_{p_{\mathbb{X}_{i}}}}^{\frac{1}{p_{\mathbb{X}_{i}}}}\left\|f_{i}\right\|_{\mathbb{X}_{i}(w)}.$

######  Remark 1.5.

We note that when $m=1$ and $\mathbb{X}=L^{p}$, the result of Theorem 1.4
coincides with the main conclusion in [4], with a more accurate norm constant.
When $m>1$, Theorem 1.4 is also a generalization of Theorem A, see Corollary
3.3.

For the commutators of multilinear pseudo-differential operators, we have

######  Theorem 1.6.

Let $1<r<\infty,1\leq i\leq m,$ $\mathbb{X},\mathbb{X}_{i}$ be rearrangement
invariant Banach function spaces with $r<p_{\mathbb{X}},p_{\mathbb{X}_{i}}\leq
q_{\mathbb{X}},q_{\mathbb{X}_{i}}<\infty.$ Assume that $\sigma\in
S_{\rho,\delta}^{l}(n,m)$ with $0\leq\rho,\delta\leq 1,l<mn(\rho-1).$ Suppose
that $P_{m}$ maps
$\overline{\mathbb{X}}_{1}\times\cdots\times\overline{\mathbb{X}}_{m}$ to
$\overline{\mathbb{X}}$ and $w\in
A_{p_{0}/r},\vec{b}=\left(b_{1},\ldots,b_{m}\right)\in\mathrm{BMO}^{m},$ then
there exist $q_{i}>1,i=1,2\cdots,m,$ such that

$\displaystyle\left\|T_{\sigma,\Sigma\mathbf{b}}(\vec{f})\right\|_{\mathbb{X}(w)}\lesssim\|\vec{b}\|_{\mathrm{BMO}}\cdot[w]_{A_{\infty}}\left([w]_{A_{\infty}}\prod\limits_{i=1}\limits^{m}[w]_{A_{p_{\mathbb{X}_{i}}}}^{\frac{1}{p_{\mathbb{X}_{i}}}}+\prod\limits_{i=1}\limits^{m}[w]_{A_{p_{{\mathbb{X}}_{i}}/r}}^{\frac{1}{q_{i}r}}\right)\prod\limits_{i=1}\limits^{m}\left\|f_{i}\right\|_{\mathbb{X}_{i}(w)},$

where $p_{0}=\min\limits_{i}{\\{p_{\mathbb{X}_{i}}\\}},$
$\|\vec{b}\|_{\mathrm{BMO}}:=\sup\limits_{1\leq j\leq
m}\left\|b_{j}\right\|_{\mathrm{BMO}}.$

When $\mathbb{X}$ is a space of rearrangement invariant quasi-Banach type, we
obtain the following quantitative weighted estimates.

######  Theorem 1.7.

Let $\mathbb{X}$ be a rearrangement invariant quasi-Banach function space with
$p$-convex property for some $0<p\leq 1.$ For $i=1,2\cdots m$, let
$\mathbb{X}_{i}$ be rearrangement invariant quasi-Banach function spaces with
$1<p_{\mathbb{X}},p_{\mathbb{X}_{i}}\leq
q_{\mathbb{X}},q_{\mathbb{X}_{i}}<\infty$ and each of them is $p_{i}$-convex
for some $0<p_{i}\leq 1.$ Assume that $\sigma\in S_{\rho,\delta}^{r}(n,m)$
with $0\leq\rho,\delta\leq 1,r<mn(\rho-1).$ Suppose that $P_{m}$ maps
$\overline{\mathbb{X}}_{1}\times\cdots\times\overline{\mathbb{X}}_{m}$ to
$\overline{\mathbb{X}},$ then for every $w\in
A_{\min\limits_{i}{\\{p_{\mathbb{X}_{i}}\\}}},$

$\displaystyle\left\|T_{\sigma}(\vec{f})\right\|_{\mathbb{X}(w)}\lesssim[w]_{A_{\infty}}^{\frac{1}{p}}\prod\limits_{i=1}\limits^{m}[w]_{A_{p_{\mathbb{X}_{i}}}}^{\frac{1}{p_{\mathbb{X}_{i}}}}\left\|f_{i}\right\|_{\mathbb{X}_{i}(w)}.$

######  Theorem 1.8.

Let $1<r<\infty,$ $\mathbb{X}$ be a rearrangement invariant quasi-Banach
function space with $p$-convex property for some $0<p\leq 1.$ For $i=1,2\cdots
m$, let $\mathbb{X}_{i}$ be rearrangement invariant quasi-Banach function
spaces with $r<p_{\mathbb{X}},p_{\mathbb{X}_{i}}\leq
q_{\mathbb{X}},q_{\mathbb{X}_{i}}<\infty$ and each of them is $p_{i}$-convex
for some $0<p_{i}\leq 1.$ Assume that $\sigma\in S_{\rho,\delta}^{l}(n,m)$
with $0\leq\rho,\delta\leq 1,l<mn(\rho-1).$ Suppose that $P_{m}$ maps
$\overline{\mathbb{X}}_{1}\times\cdots\times\overline{\mathbb{X}}_{m}$ to
$\overline{\mathbb{X}}$ and $w\in
A_{p_{0}/r},\vec{b}=\left(b_{1},\ldots,b_{m}\right)\in\mathrm{BMO}^{m},$ then
there exist $q_{i}>1,i=1,2\cdots,m,$ such that

$\displaystyle\left\|T_{\sigma,\Sigma\mathbf{b}}(\vec{f})\right\|_{\mathbb{X}(w)}\lesssim\|\vec{b}\|_{\mathrm{BMO}}\cdot[w]_{A_{\infty}}^{\frac{1}{p}}\left([w]_{A_{\infty}}^{p}\prod\limits_{i=1}\limits^{m}[w]_{A_{p_{\mathbb{X}_{i}}}}^{\frac{p}{p_{\mathbb{X}_{i}}}}+\prod\limits_{i=1}\limits^{m}[w]_{A_{p_{{\mathbb{X}}_{i}}/r}}^{\frac{p}{q_{i}r}}\right)^{\frac{1}{p}}\prod\limits_{i=1}\limits^{m}\left\|f_{i}\right\|_{\mathbb{X}_{i}(w)},$

where $p_{0}=\min\limits_{i}{\\{p_{\mathbb{X}_{i}}\\}}.$

######  Remark 1.9.

Clearly, when $\sigma\in S_{\rho,\delta}^{l}(n,m)$ with $0\leq\rho,\delta\leq
1,l<mn(\rho-1),$ Theorem 4.5 in [48] is just a special case of Theorem 1.6 and
1.8.

Note that, to deal with the boundedness of operators in some Banach spaces, or
even quasi-Banach spaces, one often uses dual space method as one of the basic
tools. However, in Harmonic analysis, some estimates are not completely
related to Banach space or dual spaces. For example, Pérez [53] proved that

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

where $T$ is Calderón-Zygmund operator,
$\varphi(\lambda)=\frac{\lambda}{1+\log^{+}\lambda^{-1}}$ and $w\in
A_{\infty}.$ This inequality is very important in illustrating the endpoint
estimates of the commutators. Notice that there is a function $\varphi$ on
both sides of the above inequality which is not homogeneous and hence each
side in above inequality is not a norm or quasi-norm. Therefore, this type of
inequality not only reflects the properties of the operator itself, but also
reduces the effect of the dual space method.

Based on the above analysis, we present the following weighted modular
inequalities for multilinear pseudo-differential operators and their
commutators, which are completely new even in the unweighted case.

######  Theorem 1.10.

Assume that $\sigma\in S_{\rho,\delta}^{r}(n,m)$ with $0\leq\rho,\delta\leq
1,r<mn(\rho-1).$ Let $\phi$ be a $N$-function with sub-multiplicative
property. If $1<i_{\phi}<\infty$ then there exist two constants
$\alpha,\beta,$ such that for every $1<q<{i_{\phi}}$ and $w\in A_{q},$

$\int_{\mathbb{R}^{n}}\phi(|T_{\sigma}(\vec{f})(x)|)w(x)dx\leq
C(\phi,w)\left(\prod\limits_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi^{m}(|f_{i}(x)|)w(x)dx\right)^{\frac{1}{m}},$

and if $i_{\phi}=1,w\in A_{1},$

$\sup\limits_{\lambda}\phi(\lambda)w\left(\\{x\in\mathbb{R}^{n}:|T_{\sigma}(\vec{f})(x)|>\lambda\\}\right)^{m}\leq
C\prod\limits_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi(|f_{i}(x)|)w(x)dx,$

where the definitions of $N$-function, $i_{\phi}$ and $C_{1}$ are listed in
Section 2, and

$C(\phi,w)=\begin{cases}[w]_{A_{\infty}}^{1+\alpha
C_{1}},&\beta[w]_{A_{q}}^{\frac{1}{q}}\leq 2,\\\ [w]_{A_{\infty}}^{1+\alpha
C_{1}}[w]_{A_{q}}^{\frac{mC_{1}}{q}},&\beta[w]_{A_{q}}^{\frac{1}{q}}>2.\end{cases}$

######  Theorem 1.11.

Let $\sigma\in S_{\rho,\delta}^{l}(n,m)$ with $0\leq\rho,\delta\leq
1,l<mn(\rho-1).$ $\vec{b}=\left(b_{1},\ldots,b_{m}\right)\in\mathrm{BMO}^{m}.$
Assume that $\phi$ is a $N$-function with sub-multiplicative property. For
each $1\leq r<\infty,$ if $r<i_{\phi}<\infty,$ then there exist two constants
$\alpha,C(\phi,w,r)$ such that for every $1<q<\frac{i_{\phi}}{r}$ and $w\in
A_{q},$

$\displaystyle\int_{\mathbb{R}^{n}}\phi(|T_{\sigma,\Sigma\mathbf{b}}(\vec{f})(x)|)w(x)dx\leq$
$\displaystyle C(\phi,w,r)\|\vec{b}\|_{\mathrm{BMO}}^{1+\alpha C_{1}}$
$\displaystyle\times\left(\prod\limits_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi^{m}(|f_{i}(x)|)w(x)dx\right)^{\frac{1}{m}},$

where $C_{1}$ is the constant in (5.10).

######  Remark 1.12.

We would like to give two comments. First of all, using the fact that
$A_{p}=\bigcup_{1<q<p}A_{q}$ for all $1<p<\infty,$ we know that the results of
Theorem 1.10 and 1.11 also hold for the endpoint case $w\in A_{i_{\phi}/r}.$
Secondly, for the special case of $m=1$ and $\phi(t)=t^{p}$ with $1<p<\infty,$
the result in Theorem 1.10 covers the conclusion in [4, Theorem 1.3 and
Corollary 4.2].

The organization of this article is as follows: In Section 2, we will present
some notations and definitions, as well as introduce some important properties
which will be used later. The proofs of Theorems 1.4 and 1.6 will be given in
Section 3, which will play a important role in the modular inequalities. The
purpose of Section 4 is to prove Theorem 1.7 and Theorem 1.8. Section 5 will
be devoted to give the proofs of Theorems 1.10-1.11. Finally, some
applications will be given in Section 6.

Throughout this paper, we always use $C$ to denote a positive constant, which
is independent of the main parameters, but it may vary from line to line. If,
for any $a,b\in\mathbb{R},a\leq Cb$ ($a\geq Cb,$ respectively), we then write
$a\lesssim b$ where $C$ is independent of $a$ and $b$, $(a\gtrsim b,$
respectively).

##  2\. Preliminary

First, we present some basic facts for sparse family, Orlicz maximal
operators, RIBFS, RIQBFS and modular inequalities.

###  2.1.  Sparse family

Some basic facts on dyadic calculus will be taken from [41, 42]. We begin with
the definition of dyadic lattice.

######  Definition 2.1.

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

  1. (i).

For any $Q\in\mathcal{D}$ its sidelength $\ell_{Q}$ is of the form
$2^{k},k\in\mathbb{Z}$;

  2. (ii).

$Q\cap R\in\\{Q,R,\emptyset\\}$ for any $Q,R\in\mathcal{D}$;

  3. (iii).

The cubes of a fixed sidelength $2^{k}$ form a partition of $\mathbb{R}^{n}$.

In dyadic calculus, Three Lattice Theorem (see [41, Proposition 5.1]) plays an
important role. It asserts that there are $3^{n}$ dyadic lattices
$\mathcal{D}^{(j)}$ such that for every cube $Q\subset\mathbb{R}^{n}$, there
is a cube $R\in\mathcal{D}^{(j)}$ for some $j$, for which $3Q\subset R$ and
$|R|\leq 9^{n}|Q|$.

Based on the Definition 2.1, we have the following definition of the sparse
family.

######  Definition 2.2.

Let $\mathcal{D}$ be a dyadic lattice. $\mathcal{S}\subset\mathcal{D}$ 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 trivial calculation shows that the sets $E(Q)$ are pairwisely disjoint
and $|E(Q)|\geq\eta|Q|$.

Given a dyadic lattice $\mathcal{D}$ and a $\eta$-sparse family
$\mathcal{S}\subseteq\mathcal{D}$, we define the sparse operator as

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

where $r>0$ and $\langle|f|^{r}\rangle_{Q}=\frac{1}{|Q|}\int_{Q}|f(y)|^{r}dy$.

###  2.2.  Young function and Orlicz maximal operators

We need to recall some basic facts from the theory of Orlicz spaces. For more
information and a lively exposition about these spaces, we refer the readers
to [55].

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

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

We can also define the $\phi$-norm of $f$ over a cube $Q$ as

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

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

It is a simple but important observation that each Young function $\phi$
satisfies the following 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 $\phi$.

Then we use $M_{\phi}$ to represent the dyadic Orlicz maximal operator defined
by:

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

where the supremum is taken over all the dyadic cubes containing $x$ and
$\mathcal{D}$ is the given dyadic grid.

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

  * •

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 $M_{\phi}=M_{L\log L^{\alpha}}$. We have
that $M\leq M_{\phi}\lesssim M_{r}$ for all $1<r<\infty.$ Moreover, it can be
showed that $M_{\phi}\approx M^{l+1},$ where $\alpha=l\in\mathbb{N}$ and
$M^{l+1}$ is $M$ iterated $l+1$ times.

###  2.3.  RIBFS and RIQBFS

This subsection is devoted to giving the definitions and properties of
rearrangement invariant Banach function spaces, denoted by RIBFS, and
rearrangement invariant quasi-Banach function spaces (RIQBFS).

$\circ$ Basic definitions of RIBFS. We write $\mathcal{M}$ for the set of
measurable functions on $\left(\mathbb{R}^{n},dx\right)$ and $\mathcal{M}^{+}$
for the nonnegative ones. A rearrangement invariant Banach norm is a mapping
$\rho:\mathcal{M}^{+}\mapsto[0,\infty]$ satisfying that

  1. i).

$\rho(f)=0\Leftrightarrow f=0,$ a.e.;
$\rho(f+g)\leq\rho(f)+\rho(g);\rho(af)=a\rho(f)$ for $a\geq 0$;

  2. ii).

If $0\leq f\leq g,$ a.e., then $\rho(f)\leq\rho(g)$;

  3. iii).

If $f_{n}\uparrow f,$ a.e., then $\rho\left(f_{n}\right)\uparrow\rho(f)$;

  4. iv).

If $E$ is a measurable set with $|E|<\infty,$ then
$\rho\left(\chi_{E}\right)<\infty,$ and $\int_{E}fdx\leq$ $C_{E}\rho(f),$ for
some constant $0<C_{E}<\infty,$ depending on $E$ and $\rho,$ but independent
of $f$;

  5. v).

$\rho(f)=\rho(g),$ if $f$ and $g$ are equimeasurable, that is,
$d_{f}(\lambda)=d_{g}(\lambda),\lambda\geq 0$ where $d_{f}\left(d_{g}\right.$
respectively) denotes the distribution function of $f$ ($g$ respectively).

A rearrangement invariant Banach function space $\mathbb{X}$ is defined by:

$\mathbb{X}=\left\\{f\in\mathcal{M}:\|f\|_{\mathbb{X}}:=\rho(|f|)<\infty\right\\}.$

Let $\mathbb{X}^{\prime}$ be the associated Banach function space of
$\mathbb{X}$ in the form that

$\mathbb{X}^{\prime}=\left\\{f\in\mathcal{M}:\|f\|_{\mathbb{X}^{\prime}}:=\sup\left\\{\int_{\mathbb{R}^{n}}fgdx:g\in\mathcal{M}^{+},\rho(g)\leq
1\right\\}<\infty\right\\}.$

An important fact is that $\mathbb{X}$ is a RIBFS if and only if
$\mathbb{X}^{\prime}$ is a $\mathrm{RIBFS}.$ For any $f\in\mathbb{X}$ and
$g\in\mathbb{X}^{\prime},$ it also follows that:

$\int_{\mathbb{R}^{n}}|fg|dx\leqslant\|f\|_{\mathbb{X}}\|g\|_{\mathbb{X}^{\prime}},$

which can be regarded as the generalized Hölder’s inequality.

Another key observation in a RIBFS $\mathbb{X}$ is that the Lorentz-Luxemburg
theorem is still true, which states that

$\|f\|_{\mathbb{X}}=\sup\left\\{\left|\int_{\mathbb{R}^{n}}fgdx\right|:g\in\mathbb{X}^{\prime},\|g\|_{\mathbb{X}^{\prime}}\leq
1\right\\}.$

Recall the decreasing rearrangement of $f$ is the function $f^{*}$ defined by

$f^{*}(t)=\inf\left\\{\lambda\geq 0:d_{f}(\lambda)\leq
t\right\\},\text{for}~{}t\geq 0.$

The main property of $f^{*}$ is that it is equimeasurable with $f$. This
allows one to use Luxemburg’s representation theorem to obtain a
representation of $\mathbb{X}$. In other words, there exists a RIBFS
$\overline{\mathbb{X}}$ over $\left(\mathbb{R}^{+},dt\right)$ such that
$f\in\mathbb{X}$ if and only if $f^{*}\in\overline{\mathbb{X}}$, and in this
case $\|f\|_{\mathbb{X}}=\left\|f^{*}\right\|_{\overline{\mathbb{X}}}$.
Moreover, it can be verified that
$\overline{\mathbb{X}}^{\prime}=\overline{\mathbb{X}^{\prime}}$ and
$\|f\|_{\mathbb{X}^{\prime}}=\left\|f^{*}\right\|_{\overline{\mathbb{X}}^{\prime}}$
for the associated space.

$\circ$ Weighted versions of RIBFS $\mathbb{X}$. This paragraph mainly
introduces the weighted versions of RIBFS $\mathbb{X}$, before that, we recall
the definitions of Muckenhoupt weights. We say that a weight $w$ which is a
nonnegative and locally integrable function on $\mathbb{R}^{n}$ belongs to the
Muckenhoupt class $A_{p}$ with $1<p<\infty,$ if

$[w]_{A_{p}}=\sup_{Q}\left(\frac{1}{|Q|}\int_{Q}w(x)dx\right)\left(\frac{1}{|Q|}\int_{Q}w(x)^{1-p^{\prime}}dx\right)^{p-1}<\infty,$

and for $w\in A_{\infty},$

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

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

A weight $w$ belongs to the class $A_{1}$ if there is a constant $C$ such that

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

and the infimum of these constants $C$ is called the $A_{1}$ constant of $w.$
Based on this, we can give the distribution function and the decreasing
rearrangement with respect to $w$ as follows

$w_{f}(\lambda)=w\left(\\{x\in\mathbb{R}^{n}:|f(x)|>\lambda\\}\right);\quad
f_{w}^{*}(t)=\inf\left\\{\lambda\geqslant 0:w_{f}(\lambda)\leqslant
t\right\\}.$

Then, we define the weighted version of the space $\mathbb{X}$,

$\mathbb{X}(w)=\left\\{f\in\mathcal{M}:\|f\|_{\mathbb{X}(w)}:=\left\|f_{w}^{*}\right\|_{\overline{\mathbb{X}}}<\infty\right\\}.$

It is well known that $\mathbb{X}^{\prime}(w)=\mathbb{X}(w)^{\prime}$ (see
[24, p. 168]).

$\circ$ Boyd indices and $r$ exponent. The Boyd indices play a key role in
RIBFS and are closely related to many properties. Let’s start with the
dilation operator

$D_{t}f(s)=f\left(\frac{s}{t}\right),0<t<\infty,f\in\overline{\mathbb{X}},$

and its norm

$h_{\mathbb{X}}(t)=\left\|D_{t}\right\|_{\overline{\mathbb{X}}\mapsto\overline{\mathbb{X}}},0<t<\infty.$

We can define the lower and upper Boyd indices $p_{\mathbb{X}}$ and
$q_{\mathbb{X}}$, by

$p_{\mathbb{X}}=\lim_{t\rightarrow\infty}\frac{\log t}{\log
h_{\mathbb{X}}(t)}=\sup_{1<t<\infty}\frac{\log t}{\log
h_{\mathbb{X}}(t)},\quad q_{\mathbb{X}}=\lim_{t\rightarrow 0^{+}}\frac{\log
t}{\log h_{\mathbb{X}}(t)}=\inf_{0<t<1}\frac{\log t}{\log h_{\mathbb{X}}(t)}.$

By a trivial calculation, we can see that $1\leq p_{\mathbb{X}}\leq
q_{\mathbb{X}}\leq\infty,$ which follows from the fact that
$h_{\mathbb{X}}(t)$ is submultiplicative. Here, we take a special case to
illustrate these two Boyd indices. If $\mathbb{X}=L^{p}$, then
$h_{\mathbb{X}}(t)=t^{\frac{1}{p}}$ and thus
$p_{\mathbb{X}}=q_{\mathbb{X}}=p.$

The Boyd indices of $\mathbb{X}$ and $\mathbb{X}^{\prime}$ have the following
relationships: $p_{\mathbb{X}^{\prime}}=\left(q_{\mathbb{X}}\right)^{\prime}$
and $q_{\mathbb{X}^{\prime}}=\left(p_{\mathbb{X}}\right)^{\prime},$ where $p$
and $p^{\prime}$ are conjugate exponents.

Now, we consider the following $r$ exponent, for each $0<r<\infty$, of RIBFS
$\mathbb{X}$:

$\mathbb{X}^{r}=\left\\{f\in\mathcal{M}:|f|^{r}\in\mathbb{X}\right\\},$

with the norm
$\|f\|_{\mathbb{X}^{r}}=\left\||f|^{r}\right\|_{\mathbb{X}}^{\frac{1}{r}}$. It
is easy to see that $p_{\mathbb{X}^{r}}=p_{\mathbb{X}}\cdot r$ and
$q_{\mathbb{X}^{r}}=q_{\mathbb{X}}\cdot r$.

$\circ$ The case of RIQBFS. Samely as $L^{p}$ space, if $\mathbb{X}$ is a
RIBFS, then $\mathbb{X}^{r}$ is still a RIBFS for $r\geq 1.$ However, in
general, the space $\mathbb{X}^{r}$ is not necessarily a Banach space with
$0<r<1,$ (see [24, p. 269]). Therefore, it is natural to consider the case of
quasi-Banach space.

We say a mapping $\rho^{\prime}:\mathcal{M}^{+}\mapsto[0,\infty)$ is a
rearrangement invariant quasi-Banach function norm if $\rho^{\prime}$
satisfies the basic condition ii),iii), v) and i) is replaced by the following
inequality:

$\rho^{\prime}(f+g)\leq C\left(\rho^{\prime}(f)+\rho^{\prime}(g)\right),$

where $C$ is an absolute positive constant. Similarly, a rearrangement
invariant quasi-Banach function space (RIQBFS) is a collection of all
measurable functions which satisfies $\rho^{\prime}(|f|)<\infty$. In order to
build the connection between RIBFS and RIQBFS, we impose the following
$p$-convex condition with $0<p<1$ on $\mathbb{X}$ which is a RIQBFS (see [27,
p. 3]):

$\left\|\left(\sum_{j=1}^{N}\left|f_{j}\right|^{p}\right)^{\frac{1}{p}}\right\|_{\mathbb{X}}\lesssim\left(\sum_{j=1}^{N}\left\|f_{j}\right\|_{\mathbb{X}}^{p}\right)^{\frac{1}{p}}.$

In this case, a basic observation is that the $p$-convex property is
equivalent to the fact that $\mathbb{X}^{\frac{1}{p}}$ is a RIBFS. Thereby,
applying the Lorentz-Luxemburg’s theorem, we can give a norm inscription of
the RIQBFS, that is

$\|f\|_{\mathbb{X}}\simeq\sup\left\\{\left(\int_{\mathbb{R}^{n}}|f(x)|^{p}g(x)dx\right)^{\frac{1}{p}}:g\in\mathcal{M}^{+},\|g\|_{\mathbb{Y}^{\prime}}\leq
1\right\\},$

where $\mathbb{Y}^{\prime}$ is the associated space of the RIBFS
$\mathbb{Y}=\mathbb{X}^{\frac{1}{p}}$. Following the previous definition, we
note that if $\mathbb{X}$ is a RIQBFS, then
$p_{\mathbb{X}^{r}}=p_{\mathbb{X}}\cdot r$ and equivalently for
$q_{\mathbb{X}}$. In a similar fashion, we can define $\mathbb{X}(w)$ for a
RIQBFS $\mathbb{X}$ and it enjoys that $\mathbb{X}(w)^{r}=\mathbb{X}^{r}(w)$
with $w\in A_{\infty}$ and $0<r<\infty$, (see [24, p. 269]).

###  2.4.  Modular inequality

In this subsection, we review some concepts related to Young functions and
modular inequalities. Let’s start with the definition of quasi-convex. A
function $\phi\in\Phi$ is said to be quasi-convex means that there exist a
convex function $\widetilde{\phi}$ and $a_{1}\geqslant 1$ 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 $\phi,$ similar to the Boyd indices, we
can also 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 $h_{\phi}(t)=\sup_{s>0}\frac{\phi(st)}{\phi(s)},t>0.$ As mentioned
earlier, the dilation indices have a relationship with Boyd indices. For
instance, if $\mathbb{X}$ is the Orlicz space induced by $\phi$, then
$p_{\mathbb{X}}=i_{\phi}$ and $q_{\mathbb{X}}=I_{\phi}$ (see [24, p. 274]).

Now we turn to the $\Delta_{2}$ condition. A function $\phi\in\Phi$ satisfies
the $\Delta_{2}$ condition if $\phi$ is doubling (we write
$\phi\in\Delta_{2}$). A crucial fact is that if $\phi$ is quasi-convex, then
$i_{\phi}\geqslant 1$ 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 the second subsection.

Let $w\in A_{\infty}$ and $\phi\in\Phi$, we define the modular of $f$ as

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

The collection of functions

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

is called as a modular space. A multilinear 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 1.6

This section is devoted to prove Theorem 1.4 and Theorem 1.6. To this aim, let
us give some notations and lemmas. Given a sparse family $\mathcal{S}$, we can
define multilinear sparse operators $\mathcal{A}_{\mathcal{S}},$
$\mathcal{A}_{\mathcal{S},b_{j}}$ and $\mathcal{A}_{\mathcal{S},b_{j}}^{*}$
with $1\leq j\leq m$ as follows,

$\displaystyle\mathcal{A}_{\mathcal{S}}(\vec{f})(x)$
$\displaystyle:=\sum_{Q\in\mathcal{S}}\prod_{i=1}^{m}\left\langle\left|f_{i}\right|\right\rangle_{Q}\chi_{Q}(x);$
$\displaystyle\mathcal{A}_{\mathcal{S},b_{j}}(\vec{f})(x)$
$\displaystyle:=\sum_{Q\in\mathcal{S}}\left|b_{j}(x)-b_{j,Q}\right|\left\langle\left|f_{j}\right|\right\rangle_{Q}\prod_{i\neq
j}\left\langle\left|f_{i}\right|\right\rangle_{Q}\chi_{Q}(x)$

and

$\mathcal{A}_{\mathcal{S},b_{j}}^{*}(\vec{f})(x):=\sum_{Q\in\mathcal{S}}\left\langle\left|\left(b_{j}-b_{j,Q}\right)f_{j}\right|\right\rangle_{Q}\prod_{i\neq
j}\left\langle\left|f_{i}\right|\right\rangle_{Q}\chi_{Q}(x).$

The following results of sparse domination (see [14, Proposition 4.1]) play a
crucial role in our analysis and will be used several times later.

######  Lemma 3.1 ([14]).

Let $\sigma\in S_{\rho,\delta}^{r}(n,m)$ with $0\leq\rho,\delta\leq 1$ and
$r<mn(\rho-1)$. Then, for every compactly supported functions
$f_{i},i=1,\ldots,m$, there exist $3^{n}+1$ sparse collections $\mathcal{S}$
and $\left\\{\mathcal{S}_{i}\right\\}_{i=1}^{3^{n}}$ such that

$\left|T_{\sigma}(\vec{f})(x)\right|\lesssim\mathcal{A}_{\mathcal{S}}(\vec{f})(x),\quad\text{
a.e. }x\in\mathbb{R}^{n},$

and

$\left|T_{\sigma,\Sigma\mathbf{b}}(\vec{f})(x)\right|\lesssim\sum_{i=1}^{3^{n}}\sum_{j=1}^{m}\left(\mathcal{A}_{\mathcal{S}_{i},b_{j}}(\vec{f})(x)+\mathcal{A}_{\mathcal{S}_{i},b_{j}}^{*}(\vec{f})(x)\right),\quad\text{
a.e. }x\in\mathbb{R}^{n}.$

We also need the following Lemma in [2, Lemma 3.3].

######  Lemma 3.2 ([2]).

Let $\mathbb{X}$ be an RIQBFS which is $p$-convex for some $0<p\leq 1$. If
$1<p_{\mathbb{X}}<\infty$, then for all $w\in A_{p_{\mathbb{X}}},$ we have

$\|M\|_{\mathbb{X}(w)\mapsto\mathbb{X}(w)}\leq
C[w]_{A_{p_{\mathbb{X}}}}^{1/p_{\mathbb{X}}},$

where $C$ is an absolute constant only depending on $p_{\mathbb{X}}$ and $n$.

###### Proof of Theorem 1.4.

We define the $m$-product operator $P_{m}$ by

$P_{m}(\vec{f})(x)=P_{m}\left(f_{1},f_{2},\ldots,f_{m}\right)(x)=\prod_{j=1}^{m}f_{j}(x).$

Recall that for any $1<p<\infty,A_{p}=\bigcup\limits_{q\in(1,p)}A_{q}.$
Therefore, each $w\in A_{\underset{i}{\min}{\\{p_{\mathbb{X}_{i}}\\}}}$
implies that $w\in A_{p_{\mathbb{X}_{i}}}$ with $1\leq i\leq m,$ and
$[w]_{A_{p_{\mathbb{X}_{i}}}}\leq[w]_{A_{\underset{i}{\min}{\\{p_{\mathbb{X}_{i}}\\}}}}<\infty.$
By the definition of $\mathbb{X}$ and Lemma 3.1, we have

(3.1) $\displaystyle{\left\|T_{\sigma}(\vec{f})\right\|}_{\mathbb{X}(w)}$
$\displaystyle=\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\left|\int_{\mathbb{R}^{n}}T_{\sigma}(\vec{f})(x)g(x)w(x)dx\right|$
$\displaystyle\lesssim\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\sum_{Q\in\mathcal{S}}\prod_{i=1}^{m}{\langle|f_{i}|\rangle}_{Q}\int_{Q}|g(x)|w(x)dx$
$\displaystyle\leq\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\sum_{Q\in\mathcal{S}}\left(\frac{1}{w(Q)}\int_{Q}\left(\mathcal{M}(\vec{f})(x)\right)^{\frac{1}{2}}\left(M_{w}^{\mathcal{D}}g(x)\right)^{\frac{1}{2}}w(x)dx\right)^{2}w(Q).$

Assume that $\mathcal{S}$ is a $\eta$-sparse family (i.e., $|E(Q)|\geq\eta|Q|$
holds for any $Q\in\mathcal{S}$) where $\eta$ is an absolute constant
depending only on $n.$ Therefore, for each dyadic cube $R\in\mathcal{S},$ it
holds that

(3.2) $\displaystyle\sum_{Q\subseteq R}w(Q)$ $\displaystyle=\sum_{Q\subseteq
R}\frac{w(Q)}{|Q|}|Q|\leq\frac{1}{\eta}\sum_{Q\subseteq
R}\frac{w(Q)}{|Q|}|E(Q)|$ $\displaystyle\leq\frac{1}{\eta}\sum_{Q\subseteq
R}\int_{E(Q)}M(w\chi_{Q})(x)dx$
$\displaystyle\leq\frac{1}{\eta}\int_{R}M(w\chi_{R})(x)dx$
$\displaystyle\leq\frac{1}{\eta}w(R)[w]_{A_{\infty}}.$

Combining inequality (3.2) with the Carleson embedding theorem ([35, Theorem
4.5]) and using the generalized Hölder’s inequality together with (3.1), it
may lead to

$\displaystyle{\left\|T_{\sigma}(\vec{f})\right\|}_{\mathbb{X}(w)}$
$\displaystyle\lesssim\frac{4}{\eta}[w]_{A_{\infty}}\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\int_{\mathbb{R}^{n}}\mathcal{M}(\vec{f})(x)M_{w}^{\mathcal{D}}g(x)w(x)dx$
$\displaystyle\lesssim[w]_{A_{\infty}}\left\|\mathcal{M}(\vec{f})\right\|_{\mathbb{X}(w)}\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}{\left\|M_{w}^{\mathcal{D}}g\right\|}_{\mathbb{X}^{\prime}(w)}$
$\displaystyle\leq[w]_{A_{\infty}}{\left\|P_{m}((Mf_{1},\ldots,Mf_{n}))\right\|}_{\mathbb{X}(w)},$

where the last inequality follows from the boundedness of
$M_{w}^{\mathcal{D}}$ in [2] or [24, Theorem 2.3].

Finally, Lemma 3.2, together with the boundedness of $P_{m}$ and [24,
Corollary 6.11], implies that

(3.3) $\displaystyle{\left\|T_{\sigma}(\vec{f})\right\|}_{\mathbb{X}(w)}$
$\displaystyle\lesssim[w]_{A_{\infty}}\prod_{i=1}^{m}{\left\|Mf_{i}\right\|}_{\mathbb{X}_{i}(w)}\lesssim[w]_{A_{\infty}}\prod_{i=1}^{m}[w]^{\frac{1}{p_{\mathbb{X}_{i}}}}_{A_{p_{\mathbb{X}_{i}}}}{\left\|f_{i}\right\|}_{\mathbb{X}_{i}(w)}.$

Then we finish the proof of Theorem 1.4. ∎

As a corollary of Theorem 1.4, we obtain the following result.

######  Corollary 3.3.

Assume that $\sigma\in S_{\rho,\delta}^{r}(n,m)$ with $0\leq\rho,\delta\leq 1$
and $r<mn(\rho-1)$. Let $\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}$,
with $1<p_{1},\cdots,p_{m}<\infty.$ If $w\in
A_{\underset{i}{\min}{\\{p_{i}\\}}},$ then $T_{\sigma}$ is bounded from
$L^{p_{1}}(w)\times\cdots\times L^{p_{m}}(w)$ to $L^{p}(w).$

###### Proof.

The proof of Corollary 3.3 follows from the same scheme as in the proof of
[24, Corollary 6.11], and hence we outline it briefly.
According to Theorem 1.4, when $\mathbb{X}=L^{p},\mathbb{X}_{i}=L^{p_{i}}$
with $i=1,\ldots,m,$ it suffices to show that for
$\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}$, $P_{m}$ is bounded from
$\overline{\mathbb{X}}_{1}\times\cdots\times\overline{\mathbb{X}}_{m}$ to
$\overline{\mathbb{X}}.$

For $i=1,\ldots,m,$ we take and fix non-negative functions
$f_{i}\in\overline{\mathbb{X}}_{i}.$ Then it is clear that

$\displaystyle\left\|P_{m}(\vec{f})\right\|_{\overline{\mathbb{X}}}$
$\displaystyle=\sup_{\|g\|_{{\overline{\mathbb{X}}}^{{}^{\prime}}}\leq
1}\left|\int_{0}^{\infty}g(t)\prod_{i=1}^{m}f_{i}(t)dt\right|$
$\displaystyle\leqslant\sup_{\|g\|_{{\overline{\mathbb{X}}}^{{}^{\prime}}}\leq
1}\int_{0}^{\infty}\prod_{i=1}^{m}(f_{i}(t)|g(t)|^{\frac{p}{p_{i}}})dt$
$\displaystyle\leqslant\sup_{\|g\|_{{\overline{\mathbb{X}}}^{{}^{\prime}}}\leq
1}\prod_{i=1}^{m}\left(\int_{0}^{\infty}f_{i}^{\frac{p_{i}}{p}}(t)|g(t)|dt\right)^{\frac{p}{p_{i}}}$
$\displaystyle\leqslant\sup_{\|g\|_{{\overline{\mathbb{X}}}^{{}^{\prime}}}\leq
1}\prod_{i=1}^{m}\|f_{i}^{\frac{p_{i}}{p}}\|^{\frac{p}{p_{i}}}_{{\overline{\mathbb{X}}}}\|g\|^{\frac{p}{p_{i}}}_{{\overline{\mathbb{X}}}^{{}^{\prime}}}=\prod_{i=1}^{m}\left\|f_{i}\right\|_{\overline{\mathbb{X}}_{i}}.$

For the last inequality, it is enough to show that for any $0<r<\infty,$
$\|f\|_{\overline{{\mathbb{X}}^{r}}}=\|f\|_{\overline{{\mathbb{X}}}^{r}}.$ In
fact, let $g^{*}=f,$ by the fact that
$\|g\|_{\mathbb{X}}=\|g^{*}\|_{\overline{{\mathbb{X}}}},$ we deduce that

$\|f\|_{\overline{{\mathbb{X}}^{r}}}=\|g^{*}\|_{\overline{{\mathbb{X}}^{r}}}=\|g\|_{\mathbb{X}^{r}}=\||g|^{r}\|_{\mathbb{X}}^{\frac{1}{r}}=\|(|g|^{r})^{*}\|_{\overline{\mathbb{X}}}^{\frac{1}{r}},$

which, together with the property in [27, P.50], further implies that

$\|f\|_{\overline{{\mathbb{X}}^{r}}}=\|(|g|^{r})^{*}\|_{\overline{\mathbb{X}}}^{\frac{1}{r}}=\|(g^{*})^{r}\|_{\overline{\mathbb{X}}}^{\frac{1}{r}}=\|g^{*}\|_{\overline{{\mathbb{X}}}^{r}}=\|f\|_{\overline{{\mathbb{X}}}^{r}}.$

∎

######  Remark 3.4.

Note that Corollary 3.3 is a single-weight version of the main results in
[14]. The condition that $P_{m}$ is bounded in the Theorem 1.4 is natural,
since it holds automatically when we return back to the classical spaces of
$L^{p_{1}}\times\cdots\times L^{p_{m}}$, and the method is still valid for the
classical Lorentz spaces. However, when dealing with the more general
rearrangement invariant Banach function spaces, we do need this condition to
deal with the multilinear maximal function.

As another consequence of Theorem 1.4, we have the following corollary.

######  Corollary 3.5.

Assume that $\sigma\in S_{\rho,\delta}^{r}(n,m)$ with $0\leq\rho,\delta\leq
1,r<mn(\rho-1).$ Let $\mathbb{X},\mathbb{X}_{i}$ be rearrangement quasi-
invariant Banach function spaces with $p$-convex for some $p>0,$ and satisfy
$1<p_{\mathbb{X}},p_{\mathbb{X}_{i}}\leq
q_{\mathbb{X}},q_{\mathbb{X}_{i}}<\infty$ for $i=1,2\cdots,m.$ Assume that
$m$-product operator $P_{m}$ maps
$\overline{\mathbb{X}}_{1}\times\cdots\times\overline{\mathbb{X}}_{m}$ to
$\overline{\mathbb{X}},$ then for every $w\in
A_{\min\limits_{i}{\\{p_{\mathbb{X}_{i}}\\}}/p},$

$\displaystyle\left\||T_{\sigma}(\vec{f})|^{\frac{1}{p}}\right\|_{\mathbb{X}(w)}\lesssim[w]_{A_{\infty}}^{\frac{1}{p}}\prod\limits_{i=1}\limits^{m}[w]_{A_{p_{\mathbb{X}_{i}}/p}}^{\frac{1}{pp_{\mathbb{X}_{i}}}}\left\||f_{i}|^{\frac{1}{p}}\right\|_{\mathbb{X}_{i}(w)}.$

###### Proof.

This corollary follows easily from the fact that $\mathbb{X}^{\frac{1}{p}}$ is
a RIBFS and Boyd indices satisfy
$p_{\mathbb{X}^{\frac{1}{p}}}=\frac{p_{\mathbb{X}}}{p}.$ ∎

In order to prove Theorem 1.6, we need the following lemma which can be found
in [15, Lemma 3.1].

######  Lemma 3.6 ([15]).

Let $s>1,t>0$, and $\omega\in A_{\infty}$. Then there holds

$\displaystyle\left\|\omega^{1/s}\right\|_{L^{s}(\log
L)^{st},Q}\lesssim[\omega]_{A_{\infty}}^{t}\langle\omega\rangle_{Q}^{1/s},$
$\displaystyle\|f\omega\|_{L(\log
L)^{t},Q}\lesssim[\omega]_{A_{\infty}}^{t}\inf_{x\in
Q}M_{\omega}\left(|f|^{s}\right)(x)^{1/s}\langle\omega\rangle_{Q}.$

###### Proof of Theorem 1.6.

By Lemma 3.1 and the definition of $\mathbb{X}(w),$ we can control the
boundedness of $T_{\sigma,\Sigma\mathbf{b}}$ by

(3.4)
$\displaystyle{||T_{\sigma,\Sigma\mathbf{b}}(\vec{f})||}_{\mathbb{X}{(w)}}$
$\displaystyle=\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\left|\int_{\mathbb{R}^{n}}T_{\sigma,\Sigma\mathbf{b}}(\vec{f})(x)g(x)w(x)dx\right|$
$\displaystyle\leq\sum_{i=1}^{3^{n}}\sum_{j=1}^{m}\left(\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\int_{\mathbb{R}^{n}}\mathcal{A}_{\mathcal{S}_{i},b_{j}}(\vec{f})(x)|g(x)|w(x)dx\right.$
$\displaystyle\left.\qquad+\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\int_{\mathbb{R}^{n}}\mathcal{A}_{\mathcal{S}_{i},b_{j}}^{*}(\vec{f})(x)|g(x)|w(x)dx\right)$
$\displaystyle=:\sum_{i=1}^{3^{n}}\sum_{j=1}^{m}\left(\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\mathcal{I}_{i,j}^{1}(g)+\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\mathcal{I}_{i,j}^{2}(g)\right),$

where

$\displaystyle\mathcal{I}_{i,j}^{1}(g)=\int_{\mathbb{R}^{n}}\mathcal{A}_{\mathcal{S}_{i},b_{j}}(\vec{f})(x)|g(x)|w(x)dx,$
$\displaystyle\mathcal{I}_{i,j}^{2}(g)=\int_{\mathbb{R}^{n}}\mathcal{A}_{\mathcal{S}_{i},b_{j}}^{*}(\vec{f})(x)|g(x)|w(x)dx,$

with $1\leq j\leq m,1\leq i\leq 3^{n}.$

First, we consider the contribuion of $\mathcal{I}_{i,j}^{1}(g).$ Via the
definition of $\mathcal{A}_{\mathcal{S}_{i},b_{j}},$ the fact that
$||b-b_{Q}||_{\exp L,Q}\lesssim||b||_{\mathrm{BMO}}$ (see [36, Lemma 4.7]) and
Lemma 3.6, the generalized Hölder’s inequality tells us that for any $s>1,$

(3.5) $\displaystyle\mathcal{I}_{i,j}^{1}(g)$
$\displaystyle=\sum_{Q\in\mathcal{S}_{i}}\int_{Q}\left|b_{j}(x)-b_{j,Q}\right||g(x)|w(x)\prod_{i=1}^{m}{\langle\left|f_{i}\right|\rangle}_{Q}dx$
$\displaystyle\leq 2\sum_{Q\in\mathcal{S}_{i}}|Q|{||b_{j}-b_{j,Q}||}_{\exp
L,Q}||gw||_{L(\log L),Q}\prod_{i=1}^{m}\langle|f_{i}|\rangle_{Q}$
$\displaystyle\lesssim[w]_{A_{\infty}}\left\|b_{j}\right\|_{\mathrm{BMO}}\sum_{Q\in\mathcal{S}_{i}}|Q|\underset{x\in
Q}{\inf}\left(M_{w}^{\mathcal{D}}(|g|^{s})(x)\right)^{\frac{1}{s}}{\langle
w\rangle}_{Q}\prod_{i=1}^{m}\langle|f_{i}|\rangle_{Q}.$

Therefore, an elementary calculation, together with Theorem 1.4, yields that

(3.6) $\displaystyle\mathcal{I}_{i,j}^{1}(g)$
$\displaystyle\lesssim[w]_{A_{\infty}}\left\|b_{j}\right\|_{\mathrm{BMO}}\int_{\mathbb{R}^{n}}\left(M_{w}^{\mathcal{D}}(|g|^{s})\right)^{\frac{1}{s}}(x)\mathcal{A}_{\mathcal{S}_{i}}(\vec{f})(x)w(x)dx$
$\displaystyle\leq[w]_{A_{\infty}}\left\|b_{j}\right\|_{\mathrm{BMO}}\|\mathcal{A}_{\mathcal{S}_{i}}(\vec{f})\|_{\mathbb{X}(w)}\|\left(M_{w}^{\mathcal{D}}(|g|^{s})\right)^{\frac{1}{s}}\|_{\mathbb{X}^{\prime}(w)}$
$\displaystyle\lesssim[w]_{A_{\infty}}^{2}\left\|b_{j}\right\|_{\mathrm{BMO}}\||g|^{s}\|^{\frac{1}{s}}_{{\mathbb{X}^{\prime}}^{\frac{1}{s}}(w)}\prod_{i=1}^{m}[w]^{\frac{1}{p_{\mathbb{X}_{i}}}}_{A_{p_{\mathbb{X}_{i}}}}{\left\|f_{i}\right\|}_{\mathbb{X}_{i}(w)},$

where in the last inequality, we have used $1<p_{\mathbb{X}}\leq
q_{\mathbb{X}}<\infty$ and pick
$1<s<p_{\mathbb{X}^{\prime}}=\frac{q_{\mathbb{X}}}{q_{\mathbb{X}}-1}$ such
that $p_{(\mathbb{X}^{\prime})^{1/s}}>1.$

Now we turn to the proof of $\mathcal{I}_{i,j}^{2}(g).$ For any $1<r<\infty,$
we apply Hölder’s inequality, together with the fact that
$\langle\left|b-b_{Q}\right|^{s}\rangle_{Q}^{1/s}\leq
2^{n+1}e^{3}s\|b\|_{\mathrm{BMO}}$ (see [59, p. 19]), to obtain

$\displaystyle\mathcal{I}_{i,j}^{2}(g)$
$\displaystyle=\int_{\mathbb{R}^{n}}\mathcal{A}_{\mathcal{S}_{i},b_{j}}^{*}(\vec{f})(x)|g(x)|w(x)dx$
$\displaystyle\leq\sum_{Q\in\mathcal{S}_{i}}\left\langle\left|f_{j}\left(b_{j}-b_{j,Q}\right)\right|\right\rangle_{Q}\int_{Q}|g(x)|w(x)dx\prod_{i\neq
j}\left\langle\left|f_{i}\right|\right\rangle_{Q}$
$\displaystyle\leq\sum_{Q\in\mathcal{S}_{i}}\langle\left|b_{j}-b_{j,Q}\right|^{r^{\prime}}\rangle_{Q}^{1/r^{\prime}}\int_{Q}|g(x)|w(x)dx\langle\left|f_{j}\right|^{r}\rangle_{Q}^{1/r}\prod_{i\neq
j}\left\langle\left|f_{i}\right|^{r}\right\rangle_{Q}^{1/r}$
$\displaystyle\leq
2^{n+1}e^{3}r^{\prime}\|b_{j}\|_{\mathrm{BMO}}\sum_{Q\in\mathcal{S}_{i}}\frac{1}{w(Q)}\int_{Q}|g(x)|w(x)dxw(Q)\prod_{i=1}^{m}\left\langle\left|f_{i}\right|^{r}\right\rangle_{Q}^{\frac{1}{r}}.$

Let $\mathcal{M}_{r}$ be the multilinear maximal operator with power $r$
defined by

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

We note that each $\mathcal{S}_{i}$ is a $\frac{1}{2\cdot 12^{n}}$-sparse
family (see [15, p. 1227]) and for each $w\in A_{p_{0}/r}$, $M_{r}$ is bounded
on $\mathbb{X}(w)$ where $M_{r}(f)=M\left(|f|^{r}\right)^{1/r}$ and $M$ is the
Hardy-Littlewood maximal operator (see [57, Lemma 3.1]). Therefore, by the
Carleson embedding theorem, there exist constants $q_{i}>1(i=1,\ldots,m)$ such
that

(3.7) $\displaystyle\mathcal{I}_{i,j}^{2}(g)$
$\displaystyle\lesssim[w]_{A_{\infty}}\|b_{j}\|_{\mathrm{BMO}}\int_{\mathbb{R}^{n}}\mathcal{M}_{r}(\vec{f})(x)M_{w}^{\mathcal{D}}g(x)w(x)dx$
$\displaystyle\lesssim[w]_{A_{\infty}}\|b_{j}\|_{\mathrm{BMO}}\left\|\mathcal{M}_{r}(\vec{f})\right\|_{\mathbb{X}(w)}\left\|M_{w}^{\mathcal{D}}g\right\|_{\mathbb{X}^{\prime}(w)}$
$\displaystyle\lesssim[w]_{A_{\infty}}\|b_{j}\|_{\mathrm{BMO}}\left\|P_{m}\left(M_{r}(f_{1}),\ldots,M_{r}(f_{m})\right)\right\|_{\mathbb{X}(w)}\left\|g\right\|_{\mathbb{X}^{\prime}(w)}$
$\displaystyle\lesssim[w]_{A_{\infty}}\|b_{j}\|_{\mathrm{BMO}}\left\|g\right\|_{\mathbb{X}^{\prime}(w)}\prod_{i=1}^{m}\left\|M_{r}(f_{i})\right\|_{\mathbb{X}_{i}(w)}$
$\displaystyle\lesssim[w]_{A_{\infty}}\|b_{j}\|_{\mathrm{BMO}}\left\|g\right\|_{\mathbb{X}^{\prime}(w)}\prod_{i=1}^{m}[w]^{\frac{1}{q_{i}r}}_{A_{p_{\mathbb{X}_{i}}/{r}}}\|f_{i}\|_{{\mathbb{X}_{i}}(w)}.$

Finally, taking the supremum over all $g\in\mathbb{X}^{\prime}(w)$ with
$\|g\|_{\mathbb{X}^{\prime}(w)}\leq 1$ in (3.6) and (3.7), together with the
estimate of (3.4), it yields that

(3.8)
$\displaystyle{||{T}_{\sigma,\Sigma\mathbf{b}}(\vec{f})||}_{\mathbb{X}{(w)}}$
$\displaystyle\lesssim\sum_{i=1}^{3^{n}}\sum_{j=1}^{m}\left(\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\mathcal{I}_{i,j}^{1}(g)+\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\mathcal{I}_{i,j}^{2}(g)\right)$
$\displaystyle\lesssim[w]_{A_{\infty}}\sum_{i}\sum_{j}\|b_{j}\|_{\mathrm{BMO}}\left(\|\mathcal{A}_{\mathcal{S}_{i}}(\vec{f})\|_{\mathbb{X}(w)}+\prod_{i=1}^{m}\left\|M_{r}(f_{i})\right\|_{\mathbb{X}_{i}(w)}\right)$
$\displaystyle\lesssim\|\vec{b}\|_{\mathrm{BMO}}[w]_{A_{\infty}}\left([w]_{A_{\infty}}\prod\limits_{i=1}\limits^{m}[w]_{A_{p_{\mathbb{X}_{i}}}}^{\frac{1}{p_{\mathbb{X}_{i}}}}+\prod\limits_{i=1}\limits^{m}[w]_{A_{p_{{\mathbb{X}}_{i}}/r}}^{\frac{1}{q_{i}r}}\right)\prod\limits_{i=1}\limits^{m}\left\|f_{i}\right\|_{\mathbb{X}_{i}(w)}.$

This completes the proof of Theorem 1.6. ∎

##  4\.  Proofs of Theorems 1.7 and 1.8

###### Proof of Theorem 1.7.

Note that $\mathbb{Y}:=\mathbb{X}^{\frac{1}{p}}$ is a RIBFS since $\mathbb{X}$
is $p$-convex and $\mathbb{X}_{i}$ is $p_{i}$-convex with
$0<p,p_{i}<1,i=1,\ldots,m$, respectively. We may assume that $\mathcal{S}$ is
a $\eta$-sparse family. Thus, by Lemma 3.1, it holds that

$\displaystyle\left\|{T}_{\sigma}(\vec{f})\right\|_{\mathbb{X}(w)}^{p}$
$\displaystyle\simeq\sup_{{\|g\|}_{\mathbb{Y}^{\prime}(w)}\leq
1}\int_{\mathbb{R}^{n}}\left|{T}_{\sigma}(\vec{f})(x)\right|^{p}g(x)w(x)dx$
$\displaystyle\lesssim\sup_{{\|g\|}_{\mathbb{Y}^{\prime}(w)}\leq
1}\int_{\mathbb{R}^{n}}\Big{(}\sum_{Q\in\mathcal{S}}\prod_{i=1}^{m}\langle|f_{i}|\rangle_{Q}\chi_{Q}(x)\Big{)}^{p}g(x)w(x)dx$
$\displaystyle\lesssim\sup_{{\|g\|}_{\mathbb{Y}^{\prime}(w)}\leq
1}\sum_{Q\in\mathcal{S}}\prod_{i=1}^{m}\langle|f_{i}|\rangle_{Q}^{p}\int_{Q}g(x)w(x)dx$
$\displaystyle\leq\sup_{{\|g\|}_{\mathbb{Y}^{\prime}(w)}\leq
1}\sum_{Q\in\mathcal{S}}\left(\frac{1}{w(Q)}\int_{Q}\Big{(}\mathcal{M}(\vec{f})(x)\Big{)}^{\frac{p}{2}}(M_{w}^{\mathcal{D}}g(x))^{\frac{1}{2}}w(x)dx\right)^{2}w(Q).$

Combining the Carleson embedding theorem with the above estimates gives that

$\displaystyle\left\|{T}_{\sigma}(\vec{f})\right\|_{\mathbb{X}(w)}$
$\displaystyle\lesssim[w]_{A_{\infty}}^{\frac{1}{p}}\sup_{{\|g\|}_{\mathbb{Y}^{\prime}(w)}\leq
1}\left(\int_{\mathbb{R}^{n}}(\mathcal{M}(\vec{f})(x))^{p}M_{w}^{\mathcal{D}}g(x)\textperiodcentered
w(x)dx\right)^{\frac{1}{p}}$
$\displaystyle\leq[w]_{A_{\infty}}^{\frac{1}{p}}\left\|(\mathcal{M}(\vec{f}))^{p}\right\|_{\mathbb{Y}(w)}^{\frac{1}{p}}\sup_{{\|g\|}_{\mathbb{Y}^{\prime}(w)}\leq
1}\|M_{w}^{\mathcal{D}}g\|_{\mathbb{Y}^{\prime}(w)}^{\frac{1}{p}}.$

Observe that for $0<p<1<p_{\mathbb{X}}\leq q_{\mathbb{X}}<\infty,$

$p_{\mathbb{Y}^{\prime}}=\left(q_{\mathbb{Y}}\right)^{\prime}=\frac{q_{\mathbb{Y}}}{q_{\mathbb{Y}}-1}=\frac{q_{\mathbb{X}}}{q_{\mathbb{X}}-p}>1,$

which indicates that
$\left\|M_{w}^{\mathcal{D}}g\right\|_{\mathbb{Y}^{\prime}(w)}\lesssim\|g\|_{\mathbb{Y}^{\prime}(w)}$.

This fact, together with Lemma 3.2, further implies that

$\displaystyle\left\|{T}_{\sigma}(\vec{f})\right\|_{\mathbb{X}(w)}$
$\displaystyle\lesssim[w]_{A_{\infty}}^{\frac{1}{p}}\left\|\mathcal{M}(\vec{f})\right\|_{\mathbb{X}(w)}$
$\displaystyle\leq[w]_{A_{\infty}}^{\frac{1}{p}}\left\|P_{m}\left(Mf_{1},\ldots,Mf_{m}\right)\right\|_{\mathbb{X}(w)}$
$\displaystyle\lesssim[w]_{A_{\infty}}^{\frac{1}{p}}\prod_{i=1}^{m}[w]_{A_{p_{\mathbb{X}_{i}}}}^{\frac{1}{p_{\mathbb{X}_{i}}}}\left\|f_{i}\right\|_{\mathbb{X}_{i}(w)},$

which finishes the proof of Theorem 1.7. ∎

######  Remark 4.1.

In [24, Theorem 6.10], in order to ensure the extrapolation theorem holds for
multilinear Calderón-Zygmund operators, one need to assume that RIQBFS
$\mathbb{X}$ and $\mathbb{X}_{i}$ with $i=1,\ldots,m$ enjoy the same
$p$-convex property. However, in Theorem 1.7, we showed some weaker conditions
for the extrapolation theorem such that each RIQBFS $\mathbb{X}_{i}$ may
enjoys different $p_{i}$-convex property.

###### Proof of Theorem 1.8.

We first note that $\mathbb{Y}=\mathbb{X}^{\frac{1}{p}}$ is a RIBFS since
$\mathbb{X}$ is $p$-convex. By Lemma 3.1, it holds that

(4.1)
$\displaystyle{\left\|{T}_{\sigma,\Sigma\mathbf{b}}(\vec{f})\right\|}_{\mathbb{X}{(w)}}$
$\displaystyle\simeq\sup_{{\|h\|}_{\mathbb{Y}^{\prime}(w)}\leq
1}\left(\int_{\mathbb{R}^{n}}\left|{T}_{\sigma,\Sigma\mathbf{b}}(\vec{f})(x)\right|^{p}h(x)w(x)dx\right)^{\frac{1}{p}}$
$\displaystyle\lesssim\sup_{{\|h\|}_{\mathbb{Y}^{\prime}(w)}\leq
1}\left(\int_{\mathbb{R}^{n}}\sum_{i=1}^{3^{n}}\sum_{j=1}^{m}\Big{(}\mathcal{A}_{\mathcal{S}_{i,b_{j}}}(\vec{f})(x)+\mathcal{A}_{\mathcal{S}_{i,b_{j}}}^{*}(\vec{f})(x)\Big{)}^{p}h(x)w(x)dx\right)^{\frac{1}{p}}$
$\displaystyle\lesssim\sup_{{\|h\|}_{\mathbb{Y}^{\prime}(w)}\leq
1}\sum_{i=1}^{3^{n}}\sum_{j=1}^{m}\left(\mathcal{H}_{i,j}^{1}+\mathcal{H}_{i,j}^{2}\right)^{\frac{1}{p}},$

where
$\mathcal{H}_{i,j}^{1}:=\int_{\mathbb{R}^{n}}(\mathcal{A}_{\mathcal{S}_{i,b_{j}}}(\vec{f})(x))^{p}h(x)w(x)dx,$
and
$\mathcal{H}_{i,j}^{2}:=\int_{\mathbb{R}^{n}}(\mathcal{A}_{\mathcal{S}_{i,b_{j}}}^{*}(\vec{f})(x))^{p}h(x)w(x)dx.$

Take and fix $0\leq h\in\mathbb{Y}^{\prime}(w)$ with
$\|h\|_{\mathbb{Y}^{\prime}(w)}\leq 1.$ For each $1\leq i\leq 3^{n},1\leq
j\leq m$ and $0<p\leq 1,$ a trivial calculation shows that

(4.2) $\displaystyle\mathcal{H}_{i,j}^{1}$
$\displaystyle\leq\sum_{Q\in\mathcal{S}_{i}}\frac{1}{w(Q)}\int_{Q}\left|b_{j}(x)-b_{j,Q}\right|^{p}|h(x)|w(x)dxw(Q)\prod_{i=1}^{m}\langle|f_{i}|\rangle_{Q}^{p}$
$\displaystyle\leq
2\sum_{Q\in\mathcal{S}_{i}}\left\|\left|b_{j}-b_{j,Q}\right|^{p}\right\|_{\exp
L^{\frac{1}{p}}(w),Q}\|h\|_{L(\log
L)^{p}(w),Q}w(Q)\prod_{i=1}^{m}\langle|f_{i}|\rangle_{Q}^{p}$
$\displaystyle\lesssim[w]_{A_{\infty}}^{p}\|b_{j}\|_{\mathrm{BMO}}^{p}\sum_{Q\in\mathcal{S}_{i}}\|h\|_{L(\log
L)^{p}(w),Q}w(Q)\prod_{i=1}^{m}\langle|f_{i}|\rangle_{Q}^{p},$

where in the above inequality we have used the fact
$\left\|\left|b-b_{Q}\right|^{p}\right\|_{\exp
L^{\frac{1}{p}}(w),Q}\lesssim[w]_{A_{\infty}}^{p}\|b\|_{\mathrm{BMO}}^{p}$
([36, Lemma 4.7]) where

$\|f\|_{\exp
L^{\frac{1}{p}}(w),Q}=\inf\left\\{\lambda>0:\frac{1}{w(Q)}\int_{Q}\exp\left(\frac{|f(x)|}{\lambda}\right)^{\frac{1}{p}}-1dw<1\right\\}.$

For any $1<\alpha<\infty$, we define $M_{\alpha(w)}^{\mathcal{D}}f$ by

$M_{\alpha(w)}^{\mathcal{D}}f(x):=\sup_{x\in
R,R\in\mathcal{D}}\left(\frac{1}{w(R)}\int_{R}|f(y)|^{\alpha}w(y)dy\right)^{\frac{1}{\alpha}}.$

Then inequality (4.2) implies that

$\displaystyle\mathcal{H}_{i,j}^{1}$
$\displaystyle\lesssim[w]_{A_{\infty}}^{p}\|b_{j}\|_{\mathrm{BMO}}^{p}\sum_{B\in\mathcal{B}}\|h\|_{L(\log
L)^{p}(w),B}\prod_{i=1}^{m}\langle|f_{i}|\rangle_{B}^{p}\sum_{Q\in
S_{i},\pi(Q)=B}w(Q)$
$\displaystyle\lesssim[w]_{A_{\infty}}^{p+1}\|b_{j}\|_{\mathrm{BMO}}^{p}\sum_{B\in\mathcal{B}}\|h\|_{L(\log
L)^{p}(w),B}w(B)\prod_{i=1}^{m}\langle|f_{i}|\rangle_{B}^{p}$
$\displaystyle\lesssim[w]_{A_{\infty}}^{p+1}\|b_{j}\|_{\mathrm{BMO}}^{p}\int_{\mathbb{R}^{n}}\left(\mathcal{M}(\vec{f})(x)\right)^{p}M_{L(\log
L)^{p}(w)}^{\mathcal{D}}h(x)w(x)dx$
$\displaystyle\lesssim[w]_{A_{\infty}}^{p+1}\|b_{j}\|_{\mathrm{BMO}}^{p}\int_{\mathbb{R}^{n}}\left(\mathcal{M}(\vec{f})(x)\right)^{p}M_{\alpha(w)}^{\mathcal{D}}h(x)w(x)dx,$

where $\pi(R)$ is the minimal principal cube which contains $R$ and

$\mathcal{B}=\cup_{k=0}^{\infty}\mathcal{B}_{k}$

is the family of the principal cubes ([45, P.2542]) with
$\mathcal{B}_{0}:=\\{$ maximal cubes in $\mathcal{S}_{j}\\}$ and

$\mathcal{B}_{k+1}:=\underset{B\in\mathcal{B}_{k}}{\cup}\operatorname{ch}_{\mathcal{B}}(B),\quad\operatorname{ch}_{\mathcal{B}}(B)=\\{R\subsetneq
B\text{ maximal s.t. }\tau(R)>2\tau(B)\\},$

where $\tau(R)=\|h\|_{L(\log
L)^{p}(w),R}\prod_{i=1}^{m}\langle|f_{i}|\rangle_{R}^{p}$.

Combining this with the generalized Hölder’s inequality, we deduce that

$\displaystyle\mathcal{H}_{i,j}^{1}$
$\displaystyle\lesssim[w]_{A_{\infty}}^{p+1}\|b_{j}\|_{\mathrm{BMO}}^{p}\left\|\mathcal{M}(\vec{f})\right\|_{\mathbb{X}(w)}^{p}\left\|M_{w}^{\mathcal{D}}h^{\alpha}\right\|_{(\mathbb{Y}^{\prime})^{1/\alpha}(w)}^{\frac{1}{\alpha}}.$

In order to use [2, Lemma 3.4], we may take an $\alpha$ with
$1<\alpha<(\frac{q_{\mathbb{X}}}{p})^{\prime}$, in this case
$p_{(\mathbb{Y}^{\prime})^{1/\alpha}}=\frac{p_{\mathbb{Y}^{\prime}}}{\alpha}>1$.

Then Lemma 3.2 yields that

(4.3) $\displaystyle\mathcal{H}_{i,j}^{1}$
$\displaystyle\lesssim[w]_{A_{\infty}}^{p+1}\|b_{j}\|_{\mathrm{BMO}}^{p}\left\|h\right\|_{\mathbb{Y}^{\prime}(w)}\prod_{i=1}^{m}[w]_{A_{p_{\mathbb{X}_{i}}}}^{\frac{p}{p_{\mathbb{X}_{i}}}}\|f_{i}\|_{\mathbb{X}_{i}(w)}^{p}$
$\displaystyle\leq[w]_{A_{\infty}}^{p+1}\|b_{j}\|_{\mathrm{BMO}}^{p}\prod_{i=1}^{m}[w]_{A_{p_{\mathbb{X}_{i}}}}^{\frac{p}{p_{\mathbb{X}_{i}}}}\|f_{i}\|_{\mathbb{X}_{i}(w)}^{p}.$

Now we consider $\mathcal{H}_{i,j}^{2}.$ Note that for $1<r<\infty,$ by the
definition of $\mathcal{H}_{i,j}^{2},$ we have

$\displaystyle\mathcal{H}_{i,j}^{2}$
$\displaystyle\leq\sum_{Q\in\mathcal{S}_{i}}\langle\left|f_{j}\left(b_{j}-b_{j,Q}\right)\right|\rangle_{Q}^{p}\left(w(Q)\prod_{i\neq
j}\langle|f_{i}|\rangle_{Q}^{p}\right)\frac{1}{w(Q)}\int_{Q}h(x)w(x)dx$
$\displaystyle\leq\sum_{Q\in\mathcal{S}_{i}}w(Q)\langle\left|b_{j}-b_{j,Q}\right|^{r^{\prime}}\rangle_{Q}^{p/r^{\prime}}\frac{1}{w(Q)}\int_{Q}h(x)w(x)dx\langle\left|f_{j}\right|^{r}\rangle_{Q}^{p/r}\prod_{i\neq
j}\left\langle\left|f_{i}\right|^{r}\right\rangle_{Q}^{p/r}$
$\displaystyle\lesssim\|b_{j}\|_{\mathrm{BMO}}^{p}\sum_{Q\in\mathcal{S}_{i}}\frac{1}{w(Q)}\int_{Q}h(x)w(x)dxw(Q)\prod_{i=1}^{m}\left\langle\left|f_{i}\right|^{r}\right\rangle_{Q}^{\frac{p}{r}}.$

Consequently, applying the Carleson embedding theorem and the generalized
Hölder’s inequality, it gives that

$\displaystyle\mathcal{H}_{i,j}^{2}$
$\displaystyle\lesssim[w]_{A_{\infty}}\|b_{j}\|_{\mathrm{BMO}}^{p}\int_{\mathbb{R}^{n}}\left(\mathcal{M}_{r}(\vec{f})(x)\right)^{p}M_{w}^{\mathcal{D}}h(x)w(x)dx$
$\displaystyle\leq[w]_{A_{\infty}}\|b_{j}\|_{\mathrm{BMO}}^{p}\left\|(\mathcal{M}_{r}(\vec{f}))^{p}\right\|_{\mathbb{Y}(w)}\left\|M_{w}^{\mathcal{D}}h\right\|_{\mathbb{Y}^{\prime}(w)}$
$\displaystyle\lesssim[w]_{A_{\infty}}\|b_{j}\|_{\mathrm{BMO}}^{p}\left\|h\right\|_{\mathbb{Y}^{\prime}(w)}\prod_{i=1}^{m}[w]^{\frac{p}{q_{i}r}}_{A_{p_{\mathbb{X}_{i}}/{r}}}\|f_{i}\|_{{\mathbb{X}_{i}}(w)}^{p}.$

This inequality, together with (4.3) and (4.1), indicates that

$\left\|T_{\sigma,\Sigma\mathbf{b}}(\vec{f})\right\|_{\mathbb{X}(w)}\lesssim\|\vec{b}\|_{\mathrm{BMO}}[w]_{A_{\infty}}^{\frac{1}{p}}\left([w]_{A_{\infty}}^{p}\prod\limits_{i=1}\limits^{m}[w]_{A_{p_{\mathbb{X}_{i}}}}^{\frac{p}{p_{\mathbb{X}_{i}}}}+\prod\limits_{i=1}\limits^{m}[w]_{A_{p_{{\mathbb{X}}_{i}}/r}}^{\frac{p}{q_{i}r}}\right)^{\frac{1}{p}}\prod\limits_{i=1}\limits^{m}\left\|f_{i}\right\|_{\mathbb{X}_{i}(w)},$

which completes the proof of Theorem 1.8. ∎

##  5\.  Proofs of Theorems 1.10 and 1.11

Before giving the proof, we need to introduce the definition of $N$-function.
A Young function $\phi$ is called $N$-function if it satisfies the following

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

Here, we list some main properties 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
satisfies the following inequalities:

(5.1) $t\leq\phi^{-1}(t)\bar{\phi}^{-1}(t)\leq 2t,t\geq 0;$ (5.2)
$\bar{\phi}\left(\frac{\phi(t)}{t}\right)\leq\phi(t),t>0.$

  * •

If $\phi$ is 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 $C_{1}$ such
that for any $\lambda\geq 2$,

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

In addition, in order to prove Theorem 1.10-1.11, we need the multilinear
weighted dyadic Hardy-Littlewood maximal operator
$\mathcal{M}_{w}^{\mathcal{D}},$ given by

$\mathcal{M}_{w}^{\mathcal{D}}(\vec{f})(x):=\sup_{x\in
R,R\in\mathcal{D}}\prod_{i=1}^{m}\frac{1}{w(R)}\int_{R}|f_{i}(y)|w(y)dy,$

where $w\in A_{\infty}$ and $\mathcal{D}$ is the given dyadic grid.

The following estimate of modular inequality for multilinear weighted dyadic
maximal operator will play a crucial role in our proof, which has some
interests of its own.

######  Lemma 5.1.

Let $\phi\in\Phi$ with sub-multiplicative property. If there exists
$0<\alpha<1$ for which $\phi^{\alpha}$ is a quasi-convex function. Then, there
exists some constant $a_{2}^{\prime}>1,$ such that for each $w\in A_{\infty},$

(5.4)
$\int_{\mathbb{R}^{n}}\phi\left(\mathcal{M}_{w}^{\mathcal{D}}(\vec{f})(x)\right)w(x)dx\leq
a_{2}^{\prime}\left(\prod_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi^{m}\left(a_{2}^{\prime}|f_{i}(x)|\right)w(x)dx\right)^{\frac{1}{m}},$

where $\mathcal{D}$ is the given dyadic grid.

###### Proof.

Since $\phi^{\alpha}$ is a quasi-convex function, then there exists convex
function $\psi$ such that

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

Thus for each dyadic cube $Q\in\mathcal{D},$ using Jensen’s inequality,
convexity of $\psi$ and for any $\lambda\geq 1,t\geq
0,\lambda\psi(t)\leq\psi(\lambda t),$ we have

(5.5)
$\displaystyle\phi^{\alpha}\left(\prod_{i=1}^{m}\frac{1}{w(Q)}\int_{Q}|f_{i}(y)|w(y)dy\right)$
$\displaystyle\leq
a_{1}\psi\left(a_{1}\prod_{i=1}^{m}\frac{1}{w(Q)}\int_{Q}|f_{i}(y)|w(y)dy\right)$
$\displaystyle\leq
a_{1}\left(\frac{1}{(w(Q))^{m}}\int_{Q^{m}}\psi(a_{1}\prod_{i=1}^{m}|f_{i}(y_{i})|)\prod_{i=1}^{m}w(y_{i})dy_{i}\right)$
$\displaystyle\leq\prod_{i=1}^{m}\frac{1}{w(Q)}\int_{Q}\phi^{\alpha}(a_{1}^{\frac{2}{m}}|f_{i}(y)|)w(y)dy.$

Denote the vector of $\phi$ by
$\vec{f}_{\phi}=\left(\phi^{\alpha}(a_{1}^{\frac{2}{m}}|f_{1}|),\phi^{\alpha}(a_{1}^{\frac{2}{m}}|f_{2}|),\ldots,\phi^{\alpha}(a_{1}^{\frac{2}{m}}|f_{m}|)\right).$
Using this notation and (5.5), one may obtain

$\phi\left(\mathcal{M}_{w}^{\mathcal{D}}(\vec{f})(x)\right)=\left(\phi^{\alpha}(\mathcal{M}_{w}^{\mathcal{D}}(\vec{f})(x))\right)^{\frac{1}{\alpha}}\leq\left(\mathcal{M}_{w}^{\mathcal{D}}(\vec{f}_{\phi})(x)\right)^{\frac{1}{\alpha}}.$

Therefore,

(5.6)
$\displaystyle\int_{\mathbb{R}^{n}}\phi\left(\mathcal{M}_{w}^{\mathcal{D}}(\vec{f})(x)\right)w(x)dx$
$\displaystyle\leq\int_{\mathbb{R}^{n}}\left(\mathcal{M}_{w}^{\mathcal{D}}(\vec{f}_{\phi})(x)\right)^{\frac{1}{\alpha}}w(x)dx$
$\displaystyle\leq
C\prod_{i=1}^{m}\left(\int_{\mathbb{R}^{n}}\phi^{\alpha}(a_{1}^{\frac{2}{m}}|f_{i}(x)|)^{\frac{m}{\alpha}}w(x)dx\right)^{\frac{1}{m}}$
$\displaystyle\leq
a_{2}^{\prime}\prod_{i=1}^{m}\left(\int_{\mathbb{R}^{n}}\phi^{\alpha}(a_{2}^{\prime}|f_{i}(x)|)^{\frac{m}{\alpha}}w(x)dx\right)^{\frac{1}{m}},$

where we have used the fact that $\mathcal{M}_{w}^{\mathcal{D}}(\vec{f})$
satisfy $L^{\frac{m}{\alpha}}(w)\times\cdots\times
L^{\frac{m}{\alpha}}(w)\rightarrow L^{\frac{1}{\alpha}}(w)$ boundedly with
$0<\alpha<1\leq m,$ which derived directly from inequality
$\mathcal{M}_{w}^{\mathcal{D}}(\vec{f})(x)\leq\prod_{i=1}^{m}M_{w}^{\mathcal{D}}(f_{i})(x),$
the boundedness of $M_{w}^{\mathcal{D}}$ and the Hölder’s inequality. ∎

######  Remark 5.2.

From the proof, it is clear that Lemma 5.1 holds with $m=1$ without assume the
sub-multiplication condition of $\phi$.

We also need the following version of the modular inequalities for the
multilinear maximal operator.

######  Lemma 5.3.

Let $\phi\in\Phi$ be a quasi-convex function with sub-multiplicative property.
For each $1\leq r<\infty,$ if $r<i_{\phi}<\infty$ then there exists constant
$a_{3}$ such that for every $1<q<\frac{i_{\phi}}{r}$ and $w\in A_{q},$

(5.7)
$\int_{\mathbb{R}^{n}}\phi\left(\mathcal{M}_{r}(\vec{f})(x)\right)w(x)dx\leq
a_{3}\left(\prod_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi^{m}\left(a_{3}[w]_{A_{q}}^{\frac{1}{qr}}\left|f_{i}(x)\right|\right)w(x)dx\right)^{\frac{1}{m}}.$

###### Proof.

Observe that for any dyadic cube $Q\in\mathcal{D}$, for each $x\in Q$, it
holds that

$\displaystyle\prod_{i=1}^{m}\frac{1}{|Q|}\int_{Q}|f_{i}|^{r}(y)dy$
$\displaystyle\leq\prod_{i=1}^{m}\left(\frac{1}{|Q|}\int_{Q}|f_{i}|^{qr}(y)w(y)dy\right)^{\frac{1}{q}}\left(\frac{1}{|Q|}\int_{Q}w^{-\frac{q^{\prime}}{q}}(y)dy\right)^{\frac{1}{q^{\prime}}}$
$\displaystyle\leq\prod_{i=1}^{m}\left[\left(\frac{1}{|Q|}\int_{Q}w(y)dy\right)\left(\frac{1}{|Q|}\int_{Q}w^{-\frac{1}{q-1}}(y)dy\right)^{q-1}\right]^{\frac{1}{q}}$
$\displaystyle\quad\times\left(\frac{1}{w(Q)}\int_{Q}|f_{i}|^{qr}(y)w(y)dy\right)^{\frac{1}{q}}$
$\displaystyle\leq[w]^{\frac{m}{q}}_{A_{q}}\left(\mathcal{M}_{w}^{\mathcal{D}}((\vec{f})^{qr})(x)\right)^{\frac{1}{q}},$

where $(\vec{f})^{s}$ denotes $(f_{1}^{s},\ldots,f_{m}^{s}).$

Therefore

$\mathcal{M}_{r}(\vec{f})(x)\leq[w]^{\frac{m}{rq}}_{A_{q}}\left(M_{w}^{\mathcal{D}}((\vec{f})^{qr})(x)\right)^{\frac{1}{rq}}.$

For $s,t>0$, set $\phi_{t}(s)=\phi(s^{\frac{1}{t}}).$ By the properties of
$\phi$, we have

$\displaystyle\phi(\mathcal{M}_{r}(\vec{f})(x))$
$\displaystyle\leq\phi\left([w]^{\frac{m}{rq}}_{A_{q}}\left(\mathcal{M}_{w}^{\mathcal{D}}((\vec{f})^{qr})(x)\right)^{\frac{1}{rq}}\right)$
$\displaystyle=\phi_{qr}\left(\mathcal{M}_{w}^{\mathcal{D}}\left([w]_{A_{q}}(\vec{f})^{qr}\right)(x)\right).$

Since $1<rq<i_{\phi}$, then by [24, Lemma 5.2], there exists a constant
$0<\alpha<1$, such that
$\phi_{qr}(s)^{\alpha}=\phi\left(s^{\frac{1}{qr}}\right)^{\alpha}$ is a quasi-
convex function. This fact, together with Lemma 5.1 further implies that

$\displaystyle\int_{\mathbb{R}^{n}}\phi(\mathcal{M}_{r}(\vec{f})(x))w(x)dx$
$\displaystyle\leq\int_{\mathbb{R}^{n}}\phi_{qr}\left(M_{w}^{\mathcal{D}}\left([w]_{A_{q}}(\vec{f})^{qr}\right)(x)\right)w(x)dx$
$\displaystyle\leq
a_{2}^{\prime}\left(\prod_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi_{qr}^{m}(a_{2}^{\prime}[w]_{A_{q}}|f_{i}(x)|^{qr})w(x)dx\right)^{\frac{1}{m}}$
$\displaystyle\leq
a_{3}\left(\prod_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi^{m}\left(a_{3}[w]_{A_{q}}^{\frac{1}{qr}}\left|f_{i}(x)\right|\right)w(x)dx\right)^{\frac{1}{m}}.$

This finishes the proof of Lemma 5.3. ∎

For the sake of convenience, we will make a summary of the constants in the
inequality which we need to use later.

(5.8) $constant~{}a_{1}:\quad\psi(t)\leq\phi(t)\leq a_{1}\psi(a_{1}t),t>0.$
(5.9)
$constant~{}a_{2}^{\prime}:\quad\int_{\mathbb{R}^{n}}\phi\left(M_{w}^{\mathcal{D}}(\vec{f})(x)\right)w(x)dx\leq
a_{2}^{\prime}\left(\prod_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi^{m}\left(a_{2}^{\prime}|f_{i}(x)|\right)w(x)dx\right)^{\frac{1}{m}}$
(5.10) $constant~{}C_{1}:\quad\phi(\lambda t)\leq
2^{C_{1}}\lambda^{C_{1}}\phi(t),\quad\lambda\geq 2,t>0.$

The following lemma gives an estimate of the modular inequality for the sparse
operator.

######  Lemma 5.4.

Let $\phi$ be a $N$-function with sub-multiplicative property. Then there
exists $0<\alpha<1,$ such that for any dyadic lattice $\mathcal{D},$ sparse
family $\mathcal{S}\subseteq\mathcal{D}$ and $w\in A_{\infty},$

$\int_{\mathbb{R}^{n}}\phi\left(\mathcal{A}_{\mathcal{S}}(\vec{f})(x)\right)w(x)dx\lesssim[w]_{A_{\infty}}^{1+\alpha
C_{1}}\int_{\mathbb{R}^{n}}\phi\left(\mathcal{M}(\vec{f})(x)\right)w(x)dx,$

where $C_{1}$ is the same constant defined in $(\ref{ie5.3})$.

###### Proof.

Since $\phi$ satisfies sub-multiplicative condition, that is, for each
$t_{1},t_{2}\geq 0,$ $\phi(t_{1}t_{2})\leq\phi(t_{1})\phi(t_{2}).$ Fixed
$t_{1}=2,$ for $t\geq 0$ we have $\phi(2t)\leq\phi(2)\phi(t)$ which implies
that $\phi\in\Delta_{2}.$ Therefore, there exists some $0<\alpha<1,$ such that
$\bar{\phi}^{\alpha}$ is quasi-convex, i.e., there exists some convex function
$\psi$ and $a_{1}>1$ such that

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

Observe that, we can always assume $a_{1}\geq 2$ by iteration. We also assume
that $\mathcal{S}$ is $\eta$-sparse family where $\eta$ is an absolute
constant only depending on $n.$

Note that $\phi(\mathcal{A}_{\mathcal{S}}(\vec{f})(x))=0$ when
$\mathcal{A}_{\mathcal{S}}(\vec{f})(x)=0$ since $\phi$ is a $N$-function. Now,
we define the function $h$ having the following form.

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

Let $E_{1}=\\{x\in\mathbb{R}^{n}:\mathcal{A}_{\mathcal{S}}(\vec{f})(x)=0\\}.$
By the definition of $\mathcal{A}_{\mathcal{S}}(\vec{f})$ and the embedding
theorem, it shows that

$\displaystyle\int_{\mathbb{R}^{n}}\phi(\mathcal{A}_{\mathcal{S}}(\vec{f})(x))w(x)dx$
$\displaystyle=\int_{\mathbb{R}^{n}\setminus
E_{1}}\phi(\mathcal{A}_{\mathcal{S}}(\vec{f})(x))w(x)dx$
$\displaystyle\leq\frac{4}{\eta}[w]_{A_{\infty}}\int_{\mathbb{R}^{n}}\mathcal{M}(\vec{f})(x)(M_{w}^{\mathcal{D}}h(x))w(x)dx.$

We now take an $\varepsilon$ with the property that

$0<\varepsilon=\min\left\\{\frac{1}{2},~{}\frac{1}{a_{1}a_{2}^{\prime}},~{}(\frac{\eta}{8[w]_{A_{\infty}}a_{2}^{\prime}})^{\alpha}\cdot\frac{1}{a_{2}^{\prime}a_{1}^{2}}\right\\}.$

Using the Young’s inequality of $\phi,$ it is easy to verify that

$\displaystyle\int_{\mathbb{R}^{n}}\phi(\mathcal{A}_{\mathcal{S}}(\vec{f})(x))w(x)dx$
$\displaystyle\leq\frac{4}{\eta}[w]_{A_{\infty}}\int_{\mathbb{R}^{n}}\frac{\mathcal{M}(\vec{f})(x)}{\varepsilon}(M_{w}^{\mathcal{D}}(\varepsilon
h(x))w(x)dx$
$\displaystyle\leq\frac{4}{\eta}[w]_{A_{\infty}}\int_{\mathbb{R}^{n}}\left[\phi(\frac{\mathcal{M}(\vec{f})(x)}{\varepsilon})+\bar{\phi}\left(M_{w}^{\mathcal{D}}(\varepsilon
h(x))\right)\right]w(x)dx$
$\displaystyle\leq\frac{4}{\eta}[w]_{A_{\infty}}2^{C_{1}}\varepsilon^{-C_{1}}\int_{\mathbb{R}^{n}}\phi(\mathcal{M}(\vec{f})(x))w(x)dx$
$\displaystyle\quad+\frac{4}{\eta}[w]_{A_{\infty}}a_{2}^{\prime}\int_{\mathbb{R}^{n}}\bar{\phi}(a_{2}^{\prime}\varepsilon
h(x))w(x)dx,$

where the last inequality follows from the fact that $\phi\in\Delta_{2}$ with
$\varepsilon\leq\frac{1}{2}$ and Remark 5.2.

Now we turn our attention to
$\int_{\mathbb{R}^{n}}\bar{\phi}(a_{2}^{\prime}\varepsilon h(x))w(x)dx.$
Observe that $\psi$ is a convex function, then for any $0\leq\lambda\leq
1,t\geq 0,$ $\psi(\lambda t)\leq\lambda\psi(t).$

Therefore, by $(\ref{ie5.4})$, we get

$\bar{\phi}^{\alpha}(a_{2}^{\prime}\varepsilon h(x))\leq
a_{1}^{2}a_{2}^{\prime}\varepsilon\psi(h(x))\leq
a_{1}^{2}a_{2}^{\prime}\varepsilon\bar{\phi}^{\alpha}(h(x)),$

where in the above inequality we have used $a_{1}a_{2}^{\prime}\varepsilon\leq
1.$

This means that

(5.12) $\bar{\phi}(a_{2}^{\prime}\varepsilon
h(x))\leq(a_{1}^{2}a_{2}^{\prime}\varepsilon)^{\frac{1}{\alpha}}\bar{\phi}(h(x)).$

By the definition of $h$ and
$\bar{\phi}\left(\frac{\phi(t)}{t}\right)\leq\phi(t),t>0,$ we see that if
$\mathcal{A}_{\mathcal{S}}(\vec{f})(x)\neq 0,$ then

$\bar{\phi}(h(x))=\bar{\phi}\left(\frac{\phi(\mathcal{A}_{\mathcal{S}}(\vec{f})(x))}{\mathcal{A}_{\mathcal{S}}(\vec{f})(x)}\right)\leq\phi(\mathcal{A}_{\mathcal{S}}(\vec{f})(x)).$

If $\mathcal{A}_{\mathcal{S}}(\vec{f})(x)=0,$ by the definitions of
$\bar{\phi},$ it is easy to check that

$\bar{\phi}(h(x))=\bar{\phi}\left(0\right)=0\leq\phi(\mathcal{A}_{\mathcal{S}}(\vec{f})(x)).$

Combining the above two cases and integrating (5.12), we deduce that

$\displaystyle\int_{\mathbb{R}^{n}}\phi(\mathcal{A}_{\mathcal{S}}(\vec{f})(x))w(x)dx$
$\displaystyle\leq\frac{4}{\eta}[w]_{A_{\infty}}2^{C_{1}}\varepsilon^{-C_{1}}\int_{\mathbb{R}^{n}}\phi(\mathcal{M}(\vec{f})(x))w(x)dx$
$\displaystyle\quad+\frac{4}{\eta}[w]_{A_{\infty}}a_{2}^{\prime}(a_{1}^{2}a_{2}^{\prime}\varepsilon)^{\frac{1}{\alpha}}\int_{\mathbb{R}^{n}}\bar{\phi}(h(x))w(x)dx$
$\displaystyle\leq\frac{4}{\eta}[w]_{A_{\infty}}2^{C_{1}}\varepsilon^{-C_{1}}\int_{\mathbb{R}^{n}}\phi(\mathcal{M}(\vec{f})(x))w(x)dx$
$\displaystyle\quad+\frac{1}{2}\int_{\mathbb{R}^{n}}\phi(\mathcal{A}_{\mathcal{S}}(\vec{f})(x))w(x)dx.$

Moreover, a direct calculation shows that
$[w]_{A_{\infty}}\varepsilon^{-C_{1}}\lesssim[w]_{A_{\infty}}^{1+\alpha
C_{1}},$ which enable us to conclude that

$\int_{\mathbb{R}^{n}}\phi\left(\mathcal{A}_{\mathcal{S}}(\vec{f})(x)\right)w(x)dx\lesssim[w]_{A_{\infty}}^{1+\alpha
C_{1}}\int_{\mathbb{R}^{n}}\phi\left(\mathcal{M}(\vec{f})(x)\right)w(x)dx.$

This completes the proof of Lemma 5.4. ∎

Now we begin the proof of the main theorems.

###### Proof of Theorem 1.10.

We divide the proof into two parts.
Case 1: We start with the case $i_{\phi}>1.$ By Lemma 3.1 and the sub-
multiplicative property of $\phi,$ we have

(5.13)
$\displaystyle\int_{\mathbb{R}^{n}}\phi(|T_{\sigma}(\vec{f})(x)|)w(x)dx$
$\displaystyle\lesssim\int_{\mathbb{R}^{n}}\phi(\mathcal{A}_{\mathcal{S}}(\vec{f})(x))w(x)dx$
$\displaystyle\lesssim[w]_{A_{\infty}}^{1+\alpha
C_{1}}\int_{\mathbb{R}^{n}}\phi(\mathcal{M}(\vec{f})(x))w(x)dx$
$\displaystyle\lesssim[w]_{A_{\infty}}^{1+\alpha
C_{1}}\left(\prod_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi^{m}(a_{3}[w]_{A_{q}}^{\frac{1}{q}}|f_{i}(x)|)w(x)dx\right)^{\frac{1}{m}},$

where we have applied Lemma 5.4 and Lemma 5.3 with $r=1.$

For the sake of accuracy, we discuss two scenarios.

If $a_{3}[w]_{A_{q}}^{\frac{1}{q}}\leq 2,$ by the sub-multiplicative property
and the monotonicity of $\phi$,

$\int_{\mathbb{R}^{n}}\phi(|T_{\sigma}(\vec{f})(x)|)w(x)dx\lesssim[w]_{A_{\infty}}^{1+\alpha
C_{1}}\left(\prod_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi^{m}(|f_{i}(x)|)w(x)dx\right)^{\frac{1}{m}}.$

For the case $a_{3}[w]_{A_{q}}^{\frac{1}{q}}>2,$ by (5.10), we obtain that

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

Case 2: Now, we consider the endpoint case $i_{\phi}=1.$ In this case, by
(5.13), applying the extrapolation theorem [24, Theorem 3.1] to
$(\phi(|T_{\sigma}(\vec{f})(x)|),\phi(\mathcal{M}(\vec{f})(x))$, we have

${||\phi(|T_{\sigma}(\vec{f})(x)|)||}_{\mathbb{X}(w)}\lesssim{||\phi(\mathcal{M}(\vec{f}))||}_{\mathbb{X}(w)},$

holds for RIQBFS $\mathbb{X}$ with $q_{\mathbb{X}}<\infty.$

Let $\mathbb{X}=L^{\frac{1}{m},\infty}(\mathbb{R}^{n}),$ we observe that

$\sup_{\lambda>0}\phi(\lambda)w\left(\\{x\in\mathbb{R}^{n}:|T_{b}(\vec{f})(x)|>\lambda\\}\right)^{m}\lesssim\sup_{\lambda>0}\phi(\lambda)w\left(\\{x\in\mathbb{R}^{n}:\mathcal{M}(\vec{f})(x)>\lambda\\}\right)^{m}.$

For any dyadic cube $Q,$ and $x\in Q,$ it is easy to see

(5.14) $\displaystyle\prod_{i=1}^{m}\frac{1}{|Q|}\int_{Q}|f_{i}(y)|dy$
$\displaystyle\leq[w]_{A_{1}}^{m}\prod_{i=1}^{m}\frac{1}{w(Q)}\int_{Q}|f_{i}(y)|w(y)dy$
$\displaystyle\leq[w]_{A_{1}}^{m}\mathcal{M}_{w}^{\mathcal{D}}(\vec{f})(x),$

which further implies that
$\phi(\mathcal{M}(\vec{f})(x))\leq\phi([w]_{A_{1}}^{m}\mathcal{M}_{w}^{\mathcal{D}}(\vec{f})(x)).$

This fact together with Lemma 5.1 gives that

$\phi(\mathcal{M}(\vec{f})(x))\leq\mathcal{M}_{w}^{\mathcal{D}}(\phi([w]_{A_{1}}f_{1}),\cdots,\phi([w]_{A_{1}}f_{m}))(x)=:\mathcal{M}_{w}^{\mathcal{D}}(\vec{f_{\phi}})(x).$

Using the method of Theorem 1.5 in [44], we can obtain that
$\mathcal{M}_{w}^{\mathcal{D}}(\vec{f})$ is bounded from
$L^{1}(w)\times\cdots\times L^{1}(w)$ into $L^{\frac{1}{m},\infty}(w).$

Then, for any $\lambda>0,$ it is easy to verify that

$\displaystyle w(\\{x\in\mathbb{R}^{n}:\mathcal{M}(\vec{f})(x)>\lambda\\})$
$\displaystyle=w(\\{x\in\mathbb{R}^{n}:\phi(\mathcal{M}(\vec{f})(x))>\phi(\lambda)\\})$
$\displaystyle\lesssim
w(\\{x\in\mathbb{R}^{n}:\mathcal{M}_{w}^{\mathcal{D}}(\vec{f}_{\phi})(x)>\phi(\lambda)\\})$
$\displaystyle\lesssim\left(\frac{1}{\phi(\lambda)}\prod_{i=1}^{m}{||f_{i,\phi}||}_{L^{1}(w)}\right)^{\frac{1}{m}}.$

Therefore

$\sup_{\lambda>0}\phi(\lambda)w(\\{x\in\mathbb{R}^{n}:\mathcal{M}(\vec{f})(x)>\lambda\\})^{m}\lesssim\prod_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi([w]_{A_{1}}f_{i}(x))w(x)dx,$

which finish the proof of Theorem 1.10 with the case $i_{\phi}=1,$ and also
completes the proof of Theorem 1.10. ∎

######  Remark 5.5.

The above proof shows that if $\phi$ satisfies
$\phi(\prod_{i=1}^{m}t_{i})\leq\prod_{i=1}^{m}\phi_{i}(t_{i})$ with
$N$-functions $\phi_{1},\ldots,\phi_{m},$ then Theorem 1.10 has the more
general form as follows.

$\int_{\mathbb{R}^{n}}\phi(|T_{\sigma}(\vec{f})(x)|)w(x)dx\leq
C(\phi,w)\left(\prod\limits_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi_{i}^{m}(|f_{i}(x)|)w(x)dx\right)^{\frac{1}{m}}.$

###### Proof of Theorem 1.11.

By using the sparse domination results of Lemma 3.1, we see that

$\displaystyle\left|T_{\sigma,\Sigma\mathbf{b}}(\vec{f})(x)\right|$
$\displaystyle\leq\max\\{2,3^{n}\cdot
Cm\\}\sum_{i=1}^{3^{n}}\sum_{j=1}^{m}\left(\frac{\mathcal{A}_{\mathcal{S}_{i},b_{j}}(\vec{f})(x)+\mathcal{A}_{\mathcal{S}_{i},b_{j}}^{*}(\vec{f})(x)}{3^{n}m}\right).$

Using condition $\phi\in\Delta_{2}$ and the convexity of $\phi,$ it holds that

$\displaystyle\phi\left(|T_{\sigma,\Sigma\mathbf{b}}(\vec{f})(x)|\right)$
$\displaystyle\leq\frac{C}{3^{n}2m}\sum_{i=1}^{3^{n}}\sum_{j=1}^{m}\left[\phi\left(\mathcal{A}_{\mathcal{S}_{i},b_{j}}(\vec{f})(x)\right)+\phi\left(\mathcal{A}_{\mathcal{S}_{i},b_{j}}^{*}(\vec{f})(x)\right)\right].$

From this inequality, it follows that

(5.15)
$\displaystyle\int_{\mathbb{R}^{n}}\phi\left(|T_{\sigma,\Sigma\mathbf{b}}(\vec{f})(x)|\right)w(x)dx\lesssim\sum_{i=1}^{3^{n}}\sum_{j=1}^{m}$
$\displaystyle\left(\int_{\mathbb{R}^{n}}\phi(\mathcal{A}_{\mathcal{S}_{i},b_{j}}(\vec{f})(x))w(x)dx\right.$
$\displaystyle+\left.\int_{\mathbb{R}^{n}}\phi\left(\mathcal{A}_{\mathcal{S}_{i},b_{j}}^{*}(\vec{f})(x)\right)w(x)dx\right).$

Consider now the contribution of
$\int_{\mathbb{R}^{n}}\phi(\mathcal{A}_{\mathcal{S}_{i},b_{j}}(\vec{f})(x))w(x)dx$
for $1\leq i\leq 3^{n},1\leq j\leq m.$ Using the Carleson embedding theorem,
we apply (3.5) in Theorem 1.6 to obtain

(5.16)
$\displaystyle\int_{\mathbb{R}^{n}}\phi(\mathcal{A}_{\mathcal{S}_{i},b_{j}}(\vec{f})(x))w(x)dx$
$\displaystyle=\sum_{Q\in\mathcal{S}_{i}}\int_{Q}\left|b_{j}(x)-b_{j,Q}\right|\frac{\phi(\mathcal{A}_{\mathcal{S}_{i},b_{j}}(\vec{f})(x))}{\mathcal{A}_{\mathcal{S}_{i},b_{j}}(\vec{f})(x)}w(x)dx\left(\prod_{i=1}^{m}\langle|f_{i}|\rangle_{Q}\right)$
$\displaystyle\leq
2\sum_{Q\in\mathcal{S}_{i}}\left\|b_{j}-b_{j,Q}\right\|_{\exp
L(w),Q}\|\frac{\phi(\mathcal{A}_{\mathcal{S}_{i},b_{j}}(\vec{f})(x))}{\mathcal{A}_{\mathcal{S}_{i},b_{j}}(\vec{f})(x)}\|_{L(\log
L)(w),Q}$ $\displaystyle\quad\times
w(Q)\left(\prod_{i=1}^{m}\langle|f_{i}|\rangle_{Q}\right)$ $\displaystyle\leq
C^{\prime}[w]_{A_{\infty}}^{2}\|b_{j}\|_{\mathrm{BMO}}$
$\displaystyle\quad\times\int_{\mathbb{R}^{n}}\mathcal{M}(\vec{f})(x)(M_{w}^{\mathcal{D}})^{2}\left(\frac{\phi(\mathcal{A}_{\mathcal{S}_{i},b_{j}}(\vec{f})(x))}{\mathcal{A}_{\mathcal{S}_{i},b_{j}}(\vec{f})(x)}\right)(x)w(x)dx,$

where $C^{\prime}>0$ is an absolute constant which is independent of $w$.

For any $1\leq j\leq m,$ we take some $\varepsilon_{j}$ in the following form:

$0<\varepsilon_{j}=\min\left\\{\frac{1}{2},~{}\frac{1}{a_{1}a_{2}^{\prime}},~{}\left(\frac{1}{2C^{\prime}[w]_{A_{\infty}}^{2}(a_{2}^{\prime})^{2}\|b_{j}\|_{\mathrm{BMO}}}\right)^{\alpha}\cdot\frac{1}{(a_{2}^{\prime})^{2}a_{1}^{2}}\right\\}.$

Then the Young’s inequality gives that

(5.17)
$\displaystyle\int_{\mathbb{R}^{n}}\mathcal{M}(\vec{f})(x)(M_{w}^{\mathcal{D}})^{2}$
$\displaystyle\left(\frac{\phi(\mathcal{A}_{\mathcal{S}_{i},b_{j}}(\vec{f})(x))}{\mathcal{A}_{\mathcal{S}_{i},b_{j}}(\vec{f})(x)}\right)(x)w(x)dx$
$\displaystyle\leq
2^{C_{1}}\varepsilon_{j}^{-C_{1}}\int_{\mathbb{R}^{n}}\phi(\mathcal{M}(\vec{f})(x))w(x)dx$
$\displaystyle\quad+a_{2}^{\prime}(a_{1}^{2}(a_{2}^{\prime})^{2}\varepsilon_{j})^{\frac{1}{\alpha}}\int_{\mathbb{R}^{n}}\bar{\phi}\left(\frac{\phi(\mathcal{A}_{\mathcal{S}_{i},b_{j}}(\vec{f})(x))}{\mathcal{A}_{\mathcal{S}_{i},b_{j}}(\vec{f})(x)}\right)w(x)dx$
$\displaystyle\leq
2^{C_{1}}\varepsilon_{j}^{-C_{1}}\int_{\mathbb{R}^{n}}\phi(\mathcal{M}(\vec{f})(x))w(x)dx$
$\displaystyle\quad+\frac{1}{2C^{\prime}[w]_{A_{\infty}}^{2}\|b_{j}\|_{\mathrm{BMO}}}\int_{\mathbb{R}^{n}}\phi(\mathcal{A}_{\mathcal{S}_{i},b_{j}}(\vec{f})(x))w(x)dx,$

where in the first inequality we have used Lemma 5.1 twice and obtained

$\int_{\mathbb{R}^{n}}\bar{\phi}((M_{w}^{\mathcal{D}})^{2}f(x))w(x)dx\leq(a_{2}^{\prime})^{2}\int_{\mathbb{R}^{n}}\bar{\phi}\left((a_{2}^{\prime})^{2}|f(x)|\right)w(x)dx.$

Applying the estimates (5.16) and (5.17), we may obtain

(5.18)
$\int_{\mathbb{R}^{n}}\phi(\mathcal{A}_{\mathcal{S}_{i},b_{j}}(\vec{f})(x))w(x)dx\lesssim[w]_{A_{\infty}}^{2+2\alpha
C_{1}}\|b_{j}\|_{\mathrm{BMO}}^{1+\alpha
C_{1}}\int_{\mathbb{R}^{n}}\phi\left(\mathcal{M}(\vec{f})(x)\right)w(x)dx.$

Now we turn to the proof of
$\int_{\mathbb{R}^{n}}\phi(\mathcal{A}_{\mathcal{S}_{i},b_{j}}^{*}(\vec{f})(x))w(x)dx$
for $1\leq i\leq 3^{n},1\leq j\leq m.$ The same reasoning as what we have
dealt with $\mathcal{I}_{i,j}^{2},$ together with the Hölder inequality,
yields that

$\displaystyle\int_{\mathbb{R}^{n}}\phi\left(\mathcal{A}_{\mathcal{S}_{i},b_{j}}^{*}(\vec{f})(x)\right)w(x)dx$
$\displaystyle\leq\sum_{Q\in\mathcal{S}_{i}}\langle\left|b_{j}-b_{j,Q}\right|^{r^{\prime}}\rangle_{Q}^{1/r^{\prime}}\left(\prod_{i=1}^{m}\left\langle\left|f_{i}\right|^{r}\right\rangle_{Q}^{1/r}\right)$
$\displaystyle\qquad\times\int_{Q}\frac{\phi(\mathcal{A}_{\mathcal{S}_{i},b_{j}}^{*}(\vec{f})(x))}{\mathcal{A}_{\mathcal{S}_{i},b_{j}}^{*}(\vec{f})(x)}w(x)dx$
$\displaystyle\lesssim\|b_{j}\|_{\mathrm{BMO}}\sum_{Q\in\mathcal{S}_{i}}\int_{Q}\frac{\phi(\mathcal{A}_{\mathcal{S}_{i},b_{j}}^{*}(\vec{f})(x))}{\mathcal{A}_{\mathcal{S}_{i},b_{j}}^{*}(\vec{f})(x)}w(x)dx\prod_{i=1}^{m}\left\langle\left|f_{i}\right|^{r}\right\rangle_{Q}^{1/r}$
$\displaystyle\lesssim\|b_{j}\|_{\mathrm{BMO}}[w]_{A_{\infty}}$
$\displaystyle\quad\times\int_{\mathbb{R}^{n}}\mathcal{M}_{r}(\vec{f})(x)M_{w}^{\mathcal{D}}\left(\frac{\phi(\mathcal{A}_{\mathcal{S}_{i},b_{j}}^{*}(\vec{f}))}{\mathcal{A}_{\mathcal{S}_{i},b_{j}}^{*}(\vec{f})}\right)(x)w(x)dx.$

Then, some elementary calculations give that

(5.19)
$\int_{\mathbb{R}^{n}}\phi\left(\mathcal{A}_{\mathcal{S}_{i},b_{j}}^{*}(\vec{f})(x)\right)w(x)dx\lesssim[w]_{A_{\infty}}^{1+2\alpha
C_{1}}\|b_{j}\|_{\mathrm{BMO}}^{1+\alpha
C_{1}}\int_{\mathbb{R}^{n}}\mathcal{M}_{r}(\vec{f})(x)w(x)dx.$

We are now in the position to finish our proof. First, by the estimates
(5.15), (5.18) and (5.19), we get

$\displaystyle\int_{\mathbb{R}^{n}}\phi\left(|T_{\sigma,\Sigma\mathbf{b}}(\vec{f})(x)|\right)w(x)dx\lesssim[w]_{A_{\infty}}^{1+2\alpha
C_{1}}\|\vec{b}\|_{\mathrm{BMO}}^{1+\alpha C_{1}}$
$\displaystyle\left([w]_{A_{\infty}}\int_{\mathbb{R}^{n}}\phi\left(\mathcal{M}(\vec{f})(x)\right)w(x)dx\right.$
$\displaystyle+\left.\int_{\mathbb{R}^{n}}\mathcal{M}_{r}(\vec{f})(x)w(x)dx\right).$

Then, for any $r>1$ and $1<q<\frac{i_{\phi}}{r},$ there exist $a_{3}$ and
$a_{3}^{\prime}$ satisfying Lemma 5.3, such that

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

To finish the proof, we only need to consider four cases for modular
inequalities.

Case I: $a_{3}[w]_{A_{q}}^{\frac{1}{q}}<2,$
$a_{3}^{\prime}[w]_{A_{q}}^{\frac{1}{qr}}<2$.

In this case, we have

$\displaystyle\int_{\mathbb{R}^{n}}\phi\left(|T_{\sigma,\Sigma\mathbf{b}}(\vec{f})(x)|\right)w(x)dx\lesssim$
$\displaystyle[w]_{A_{\infty}}^{1+2\alpha
C_{1}}\|\vec{b}\|_{\mathrm{BMO}}^{1+\alpha C_{1}}([w]_{A_{\infty}}+1)$
$\displaystyle\times\left(\prod\limits_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi^{m}(2|f_{i}(x)|)w(x)dx\right)^{\frac{1}{m}}.$

Case II: $a_{3}[w]_{A_{q}}^{\frac{1}{q}}\geq 2,$
$a_{3}^{\prime}[w]_{A_{q}}^{\frac{1}{qr}}<2$.

For this case, by $\Delta_{2}$ condition, we obtain

$\displaystyle\int_{\mathbb{R}^{n}}\phi\left(|T_{\sigma,\Sigma\mathbf{b}}(\vec{f})(x)|\right)w(x)dx\lesssim$
$\displaystyle[w]_{A_{\infty}}^{1+2\alpha
C_{1}}\|\vec{b}\|_{\mathrm{BMO}}^{1+\alpha
C_{1}}([w]_{A_{\infty}}[w]_{A_{q}}^{\frac{C_{1}m}{q}}+1)$
$\displaystyle\times\left(\prod\limits_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi^{m}(2|f_{i}(x)|)w(x)dx\right)^{\frac{1}{m}}.$

Case III: $a_{3}[w]_{A_{q}}^{\frac{1}{q}}<2,$
$a_{3}^{\prime}[w]_{A_{q}}^{\frac{1}{qr}}\geq 2$.

To see this case, using $\Delta_{2}$ condition again, we get

$\displaystyle\int_{\mathbb{R}^{n}}\phi\left(|T_{\sigma,\Sigma\mathbf{b}}(\vec{f})(x)|\right)w(x)dx\lesssim$
$\displaystyle[w]_{A_{\infty}}^{1+2\alpha
C_{1}}\|\vec{b}\|_{\mathrm{BMO}}^{1+\alpha
C_{1}}([w]_{A_{\infty}}+[w]_{A_{q}}^{\frac{C_{1}m}{qr}})$
$\displaystyle\times\left(\prod\limits_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi^{m}(2|f_{i}(x)|)w(x)dx\right)^{\frac{1}{m}}.$

Case IV: $a_{3}[w]_{A_{q}}^{\frac{1}{q}}\geq 2,$
$a_{3}^{\prime}[w]_{A_{q}}^{\frac{1}{qr}}\geq 2$.

For this case, the result have the following representation by using
$\Delta_{2}$ condition,

$\displaystyle\int_{\mathbb{R}^{n}}\phi\left(|T_{\sigma,\Sigma\mathbf{b}}(\vec{f})(x)|\right)w(x)dx\lesssim$
$\displaystyle[w]_{A_{\infty}}^{1+2\alpha
C_{1}}[w]_{A_{q}}^{\frac{C_{1}m}{q}}\|\vec{b}\|_{\mathrm{BMO}}^{1+\alpha
C_{1}}([w]_{A_{\infty}}+1)$
$\displaystyle\times\left(\prod\limits_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi^{m}(|f_{i}(x)|)w(x)dx\right)^{\frac{1}{m}}.$

Combining the four cases, each of them has an exact expression for the
constant of $w$, we have thus completed the proof of the theorem. ∎

For $\frac{1}{p}=\frac{1}{p_{1}}+\ldots+\frac{1}{p_{m}}$ with
$1<p_{1},\ldots,p_{m}<\infty,$ we note that Theorem 1.11 cannot imply the
classical $L^{p_{1}}\times\ldots\times L^{p_{m}}$ to $L^{p}$ boundedness when
$\phi=t^{p}.$ In fact, by Theorem 1.11 we have the following corollary
consistent with the classical case.

######  Corollary 5.6.

Let $\sigma\in S_{\rho,\delta}^{l}(n,m)$ with $0\leq\rho,\delta\leq
1,l<mn(\rho-1),$ $\vec{b}=\left(b_{1},\ldots,b_{m}\right)\in\mathrm{BMO}^{m}.$
Assume that $\phi$ is a $N-$function with sub-multiplicative property. Let
$\frac{1}{p}=\frac{1}{p_{1}}+\ldots+\frac{1}{p_{m}}$ with
$1<p_{1},\ldots,p_{m}<\infty,$ then for each $1<r<\infty,$
$r<i_{\phi}<\infty,$ there exist constants $\alpha,C(\phi,w,r)$ such that for
every $1<q<\frac{i_{\phi}}{r}$ and $w\in A_{q},$

$\displaystyle\int_{\mathbb{R}^{n}}\phi(|T_{\sigma,\Sigma\mathbf{b}}(\vec{f})(x)|)w(x)dx\leq$
$\displaystyle C(\phi,w,r)\|\vec{b}\|_{\mathrm{BMO}}^{1+\alpha C_{1}}$
$\displaystyle\times\left(\prod\limits_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi^{\frac{p_{i}}{p}}(|f_{i}(x)|)w(x)dx\right)^{\frac{p}{p_{i}}}.$

In particular, if $\phi(t)=t^{p}$ then

$\|T_{\sigma,\Sigma\mathbf{b}}(\vec{f})\|_{L^{p}(w)}\lesssim
C(\phi,w,r)\|\vec{b}\|_{\mathrm{BMO}}^{1+\alpha
C_{1}}\prod\limits_{i=1}^{m}\|f_{i}\|_{L^{p_{i}}(w)}.$

###### Proof.

Notice that $\frac{1}{p}=\frac{1}{p_{1}}+\ldots+\frac{1}{p_{m}},$ by Hölder’s
inequality and Lemma 5.1, we see that

$\displaystyle\|\mathcal{M}_{w}^{\mathcal{D}}(\vec{f}_{\phi})\|_{L^{\frac{1}{\alpha}}(w)}^{\frac{1}{\alpha}}$
$\displaystyle\leq\int_{\mathbb{R}^{n}}|\prod_{i=1}^{m}M_{w}^{\mathcal{D}}(f_{\phi,i})(x)|^{\frac{1}{\alpha}}w(x)dx$
$\displaystyle\leq\prod_{i=1}^{m}\left(\int_{\mathbb{R}^{n}}|M_{w}^{\mathcal{D}}(f_{\phi,i})(x)|^{\frac{p_{i}}{p\alpha}}w(x)dx\right)^{\frac{p}{p_{i}}}$
$\displaystyle\leq\prod_{i=1}^{m}{(\frac{p_{i}}{p\alpha})^{\prime}}^{\frac{1}{\alpha}}\left(\int_{\mathbb{R}^{n}}|f_{\phi,i}(x)|^{\frac{p_{i}}{p\alpha}}w(x)dx\right)^{\frac{p}{p_{i}}}.$

This inequality, together with the estimate of (5.6), indicates that

$\int_{\mathbb{R}^{n}}\phi\left(\mathcal{M}_{w}^{\mathcal{D}}(\vec{f})(x)\right)w(x)dx\leq\prod_{i=1}^{m}{(\frac{p_{i}}{p\alpha})^{\prime}}^{\frac{1}{\alpha}}\left(\int_{\mathbb{R}^{n}}\phi^{\frac{p_{i}}{p}}(|f_{i}(x)|)w(x)dx\right)^{\frac{p}{p_{i}}}.$

Using the method in Lemma 5.3, it further gives that

(5.20)
$\int_{\mathbb{R}^{n}}\phi\left(\mathcal{M}_{r}(\vec{f})(x)\right)w(x)dx\leq
a_{3}\prod_{i=1}^{m}\left(\int_{\mathbb{R}^{n}}\phi^{\frac{p_{i}}{p}}(a_{3}[w]_{A_{q}}^{\frac{1}{qr}}|f_{i}(x)|)w(x)dx\right)^{\frac{p}{p_{i}}}.$

Then the proof process of Theorem 1.11 and (5.20) together imply the desired
result. ∎

##  6\.  applications

This section is devoted to giving some applications of our theorems for
multilinear Fourier multiplier and multilinear square function. As a corollary
of Theorem 1.4, we first give the following results of a class of the
multilinear Calderón-Zygmund singular integral operators.

######  Corollary 6.1.

Let $T_{K}$ is a multilinear Calderón-Zygmund singular integral operator with
kernel function $K$ satisfies

$K(x,\vec{y})=\int_{\mathbb{R}^{mn}}\sigma(x,\vec{\xi})e^{2\pi
i\vec{y}\cdot\vec{\xi}}d\vec{\xi},$

where $\vec{x}=(x,\ldots,x)\in\mathbb{R}^{mn},$ $\sigma\in
S_{\rho,\delta}^{r}(n,m)$ with $\rho,\delta\in[0,1]$ and $r+\delta<nm(\rho-1)$
for some $0\leq\delta\leq 1.$ Let $\mathbb{X},\mathbb{X}_{i}$ be rearrangement
invariant Banach function spaces with $1<p_{\mathbb{X}},p_{\mathbb{X}_{i}}\leq
q_{\mathbb{X}},q_{\mathbb{X}_{i}}<\infty$ for $i=1,2\cdots,m.$ Assume that
$m$-product operator $P_{m}$ maps
$\overline{\mathbb{X}}_{1}\times\cdots\times\overline{\mathbb{X}}_{m}$ to
$\overline{\mathbb{X}},$ then for every $w\in
A_{\min\limits_{i}{\\{p_{\mathbb{X}_{i}}\\}}},$ $T_{K}$ is bounded from
$\mathbb{X}_{1}(w)\times\cdots\times\mathbb{X}_{m}(w)$ to $\mathbb{X}(w).$

###### Proof.

This corollary can be directly deduced from Theorem 1.4 and [14, Proposition
3.7]. ∎

###  6.1.  Multilinear Fourier multiplier

In this subsection, we apply our results to derive the boundedness of
multilinear Fourier multipliers in rearrangement invariant Banach function
spaces which contain the $L^{p}$ space.

The motivation for multilinear Fourier multipliers comes in large part from
the fact that people began to consider a class of kernel conditions which are
weaker than the Hölder continuity kernels, and consider the boundedness of
some operators on the products of Lebesgue spaces (see [39]). For the sake of
simplicity, we only consider the bilinear case. Now we give the definition of
bilinear Fourier multiplier.

For integer $s$, we assume that $m\in
C^{s}\left(\mathbb{R}^{2n}\backslash\\{0\\}\right)$ satisfying the following
condition:

(6.1)
$\left|\partial_{\xi_{1}}^{\alpha}\partial_{\xi_{2}}^{\beta}m({\xi_{1}},{\xi_{2}})\right|\leq
C_{\alpha,\beta}(|{\xi_{1}}|+|{\xi_{2}}|)^{-(|\alpha|+|\beta|)}$

for all $|\alpha|+|\beta|\leq s$ and
$({\xi_{1}},{\xi_{2}})\in\mathbb{R}^{2n}\backslash\\{0\\}$. The bilinear
Fourier multiplier operator $T_{m}$ is defined by

$T_{m}(f,g)(x)=\frac{1}{(2\pi)^{2n}}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{ix\cdot({\xi_{1}}+{\xi_{2}})}m({\xi_{1}},{\xi_{2}})\hat{f}({\xi_{1}})\hat{g}({\xi_{2}})d{\xi_{1}}d{\xi_{2}}.$

It is clear that if $\sigma$ is independent of $x$ in multilinear pseudo-
differential operator $T_{\sigma}$, that is
$\sigma(x,\vec{\xi})=m(\vec{\xi}),$ then multilinear pseudo-differential
operators are Fourier multipliers.

As in [10, Theorem 5.10], by Theorems 1.4 and 1.7, we obtain the following
Theorem 6.2. We omit the details.

######  Theorem 6.2.

Assume that (6.1) holds for some $n+1\leq s\leq 2n$. Let $2n/s<p_{0}$ and
$\mathbb{X}$ be a rearrangement invariant quasi-Banach function space with
$p$-convex for some $0<p\leq 1.$ For $i=1,2$, let $\mathbb{X}_{i}$ be
rearrangement invariant quasi-Banach function spaces with
$p_{0}<p_{\mathbb{X}},p_{\mathbb{X}_{i}}\leq
q_{\mathbb{X}},q_{\mathbb{X}_{i}}<\infty$ and each of them is $p_{i}$-convex
for some $0<p_{i}\leq 1.$ If $P_{m}$ maps
$\overline{\mathbb{X}}_{1}\times\overline{\mathbb{X}}_{2}$ to
$\overline{\mathbb{X}},$ then for every $w\in
A_{\min\limits_{i}{\\{p_{\mathbb{X}_{i}}\\}}/{p_{0}}},$ there exist
$q_{1},q_{2}>1,$ such that

$\displaystyle\left\|T_{m}(\vec{f})\right\|_{\mathbb{X}(w)}\lesssim[w]_{A_{\infty}}^{\frac{1}{p}}\prod\limits_{i=1}\limits^{2}[w]_{A_{p_{\mathbb{X}_{i}}/{p_{0}}}}^{\frac{1}{p_{0}q_{i}}}\left\|f_{i}\right\|_{\mathbb{X}_{i}(w)}.$

###  6.2.  Multilinear square functions

This subsection is devoted to establishing modular inequality estimates for
multilinear square functions, and we start with some simple definitions.

Let $\psi(x,\vec{y})$ be a locally integrable function defined away from the
diagonal $x=y_{1}=$ $\ldots=y_{m}$ in $\mathbb{R}^{n\times(m+1)}$. We assume
that $\psi(x,\vec{y})$ satisfies the following two size and smoothness
conditions.

Size condition: There exist $\delta,A>0$ such that

$|\psi(x,\vec{y})|\leq\frac{A}{\left(1+\left|x-y_{1}\right|+\cdots+\left|x-y_{m}\right|\right)^{mn+\delta}}$

Smoothness conditions: There exist $\delta,A,\gamma>0$ so that

$|\psi(x,\vec{y})-\psi(x+h,\vec{y})|\leq\frac{A|h|^{\gamma}}{\left(1+\left|x-y_{1}\right|+\cdots+\left|x-y_{m}\right|\right)^{mn+\delta+\gamma}},$

with $|h|<\frac{1}{2}\max\limits_{j}\left|x-y_{j}\right|$, and

$\left|\psi\left(x,y_{1},\ldots,y_{i},\ldots,y_{m}\right)-\psi\left(x,y_{1},\ldots,y_{i}+h,\ldots,y_{m}\right)\right|\leq\frac{A|h|^{\gamma}}{\left(1+\left|x-y_{1}\right|+\cdots+\left|x-y_{m}\right|\right)^{mn+\delta+\gamma}},$

with $|h|<\frac{1}{2}\left|x-y_{i}\right|$ for $i=1,\ldots,m$.

Let
$\vec{f}=\left(f_{1},\ldots,f_{m}\right)\in\mathcal{S}\left(\mathbb{R}^{n}\right)\times\cdots\times\mathcal{S}\left(\mathbb{R}^{n}\right)$
and $x\notin\bigcap_{j=1}^{m}\operatorname{supp}f_{j},$ we define

$\psi_{t}(\vec{f})(x)=\frac{1}{t^{mn}}\int_{\left(\mathbb{R}^{n}\right)^{m}}\psi\left(\frac{x}{t},\frac{y_{1}}{t},\ldots,\frac{y_{m}}{t}\right)\prod_{j=1}^{m}f_{j}\left(y_{j}\right)dy_{j}$

According to $\psi(x,\vec{y}),$ we may define the multilinear square function
$g_{\lambda,\psi}^{*}$ of the following form with $\lambda>2m,\alpha>0.$

$g_{\lambda,\psi}^{*}(\vec{f})(x)=\left(\int_{\mathbb{R}_{+}^{n+1}}\left(\frac{t}{t+|x-y|}\right)^{n\lambda}\left|\psi_{t}(\vec{f})(y)\right|^{2}\frac{dydt}{t^{n+1}}\right)^{1/2}.$

Using [11, Theorem 1.2], we obtain the modular inequality of the following
form for $g_{\lambda,\psi}^{*}.$

######  Theorem 6.3.

Assume that $\lambda>2m.$ Let $\phi$ be a $N-$function with sub-multiplicative
property. If $1<i_{\phi}<\infty$ then there exist constants $\alpha,a_{3},$
such that for every $1<q<{i_{\phi}}$ and $w\in A_{q},$

$\int_{\mathbb{R}^{n}}\phi(|g_{\lambda,\psi}^{*}(\vec{f})(x)|)w(x)dx\leq
C(\phi,w)\left(\prod\limits_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi^{m}(|f_{i}(x)|)w(x)dx\right)^{\frac{1}{m}},$

and if $i_{\phi}=1,w\in A_{1},$

$\sup\limits_{\lambda}\phi(\lambda)w\left(\\{x\in\mathbb{R}^{n}:|g_{\lambda,\psi}^{*}(\vec{f})(x)|>\lambda\\}\right)^{m}\leq
C\prod\limits_{i=1}^{m}\int_{\mathbb{R}^{n}}\phi(|f_{i}(x)|)w(x)dx,$

where

$C(\phi,w)=\begin{cases}[w]_{A_{\infty}}^{1+\alpha
C_{1}},&a_{3}[w]_{A_{q}}^{\frac{1}{q}}\leq 2,\\\ [w]_{A_{\infty}}^{1+\alpha
C_{1}}[w]_{A_{q}}^{\frac{mC_{1}}{q}},&a_{3}[w]_{A_{q}}^{\frac{1}{q}}>2.\end{cases}$

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