# Composition of rough singular integral operators on rearrangement invariant
Banach type 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.

Let $\Omega$ be a homogeneous function of degree zero and enjoy the vanishing
condition on the unit sphere $\mathbb{S}^{n-1}(n\geq 2)$. Let $T_{\Omega}$ be
the convolution singular integral operator with kernel
${\Omega(x)}{|x|^{-n}}$. In this paper, when $\Omega\in
L^{\infty}(\mathbb{S}^{n-1})$, we consider the quantitative weighted bounds of
the composite operators of $T_{\Omega}$ on rearrangement invariant Banach
function spaces. These spaces contain the classical Lorentz spaces and Orlicz
spaces as special examples. Weighted boundedness of the composite operators on
rearrangement invariant quasi-Banach spaces were also given.

###### Key words and phrases:

rough singular integral operator, composite operator, rearrangement invariant
Banach function spaces , bilinear sparse operators.
2010 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

This paper aims to establish the quantitative weighted boundedness for the
composition of rough singular integral operators in rearrangement invariant
Banach spaces and quasi-Banach spaces. It is worthy to pointing out that the
classical Lorentz spaces, Orlicz spaces and the Marcinkiewicz spaces are
special examples of these spaces. The study of these rearrangement invariant
function spaces has a long history. Indeed, in 1955, Lorentz [40] first showed
that the Hardy-Littlewood maximal operator $M$ is bounded on rearrangement
invariant Banach function space $\mathbb{X}$ if and only if
$p_{\mathbb{X}}>1$. Subsequently, Boyd [5] proved that the Hilbert transform
$H$ is also bounded on $\mathbb{X}$ if and only if $1<p_{\mathbb{X}}\leq
q_{\mathbb{X}}<\infty.$ Here $p_{\mathbb{X}}$ and $q_{\mathbb{X}}$ denote the
Boyd indices of $\mathbb{X}$ (see Section 2.1 below). These results were
originally proved for Banach spaces, but they were later generalized to the
quasi-Banach case with the same restrictions on Boyd’s index in [42]. Since
then many other contributions came to enrich the literature on this subject,
we refer the readers to [4, 21, 18] and the references therein. In particular,
by using sparse domination, Anderson and Hu [2] obtained the quantitative
weighted bounds for the maximal truncated singular integral operator on
rearrangement invariant Banach spaces.

We now give a brief review on the study of rough singular integrals. Let
$\Omega$ be a homogeneous function of degree zero, $\Omega\in
L^{1}(\mathbb{S}^{n-1})$ and satisfy the vanishing condition on the unit
sphere $\mathbb{S}^{n-1}(n\geq 2)$ as follows

(1.1) $\int_{\mathbb{S}^{n-1}}\Omega(y)d\sigma(y)=0,$

where $d\sigma(y)$ denotes the Lebesgue measure with restrictions on
$\mathbb{S}^{n-1}$. In 1956, Calderón and Zygmund [8] introduced the following
rough homogeneous singular integral operator

(1.2)
$T_{\Omega}f(x)=\text{p.v.}\int_{\mathbb{R}^{n}}\frac{\Omega\left(y/|y|\right)}{|y|^{n}}f(x-y)dy.$

Using the method of rotation, Calderón and Zygmund [8] demonstrated the
$L^{p}$ $(1<p<\infty)$ boundedness of $T_{\Omega}$ whenever $\Omega$ is odd
and $\Omega\in L^{1}(\mathbb{S}^{n-1})$, or $\Omega$ is even, $\Omega\in L\log
L(\mathbb{S}^{n-1})$ and satisfies (1.1). In 1979, Connett [16], Ricci and
Weiss [47] independently showed that $\Omega\in H^{1}({\mathbb{S}^{n-1}})$ is
sufficient to warrant the $L^{p}$ boundedness of $T_{\Omega}$. Here
$H^{1}({\mathbb{S}^{n-1}})$ denotes the Hardy space on ${\mathbb{S}^{n-1}}$
which contains $L\log L({\mathbb{S}^{n-1}})$ as a proper subspace.

Conside the weak endpoint case and the weighted case. This area has flourished
and has been enriched by several important works. Among them are the
celebrated works of Christ, Christ and Rubio de Francia, Hofmann, Seeger, and
Tao for weak type $(1,1)$ bounds of $T_{\Omega}$. In particular, in 1996,
Seeger [48] proved that $T_{\Omega}$ is bounded from $L^{1}(\mathbb{R}^{n})$
to $L^{1,\infty}(\mathbb{R}^{n})$ with a sufficient condition $\Omega\in L\log
L\left({\mathbb{S}^{n-1}}\right).$ For the weighted cases, it was
Duoandikoetxea and Rubio de Francia [19] who obtained the
$L^{p}(\mathbb{R},w(x)dx)$-boundedness of $T_{\Omega}$ for $1<p<\infty$
provided that $\Omega\in L^{\infty}(\mathbb{S}^{n-1})$ and $w$ is a
Muckenhoupt $A_{p}$ weight. This result was later improved in [20] and [51].
It is worth mentioning that, in 2017, Hytönen et al. [33] obtained the
quantitative weighted boundedness of $T_{\Omega}.$ For other works related to
$T_{\Omega}$, we refer the readers to see [12, 47, 39, 25] and the references
therein.

In general, there are two distinct approaches in the study of singular
integral operators. Consider it as a principal value operator of convolution
type or as an operator of Fourier multipliers defined by

$\widehat{T_{m}f}(\xi)=m(\xi)\hat{f}(\xi),$

where $m\in L^{\infty}\left(\mathbb{R}^{n}\right)$ and $\hat{f}$ denotes the
Fourier transform of $f$. Let $\Omega\in L\log L(\mathbb{S}^{n-1})$ be
homogeneous of degree zero and satisfy the vanishing condition (1.1). Then the
following identity transformation relationship holds between $m$ and $\Omega$

$m(\xi)=\int_{\mathbb{S}^{n-1}}\Omega\left(y^{\prime}\right)\left(\log\frac{1}{\left|\xi\cdot
y^{\prime}\right|}-\frac{i\pi}{2}\operatorname{sgn}\left(\xi\cdot
y^{\prime}\right)\right)d\sigma(y).$

However, unfortunately, this identity does not provide an exact correspondence
between various auxiliary conditions assumed on $m$ and $\Omega$.

One of the fundamental questions in operator theory is what the composition of
two operators is. This question is of great importance and attracts lots of
attention. Indeed, it was known that the composition of singular integral
operators arises typically in the study of algebra of singular integral (see
[6, 9]) and the non-coercive boundary-value problems for elliptic equations
(see [43, 45]). The answer to this question for $T_{\Omega}$ is easily
obtained by using the Fourier multiplier representation, which states that the
composition of two singular integral operators is an operator of the same form
and the multiplier of the composition is the product of the two multipliers
(see [49]). This answer is so elegant and useful that it actually forms the
basis for the calculus of pseudo-differential operators. What if it is in the
form of a principal value integral? Coifman and Meyer [14] considered the
composition of classical Calderón-Zygmund operators and pointed out that if
$T_{1},T_{2}$ are two Calderón-Zygmund operators, $T_{2}^{*}$ be the adjoint
operator of $T_{2}$ and $T_{1}(1)=T_{2}^{*}(1)=0$, then the composite operator
$T_{1}T_{2}$ is also a Calderón-Zygmund operator. It then follows that the
composition of Calderón-Zygmund operators is still strong $(p,p)$ type and
weak $(1,1)$ type. In 2018, Benea and Bernicot [3] used the sparse domination
method to reduce the above conditions to $T_{1}(1)=0$ or $T^{*}_{2}(1)=0,$ and
the weighted boundedness of the composite operator for the Calderón-Zygmund
operators can also be obtained. In addition, previous result for Hardy-
Littlewood maximal operators $M$ were obtained by Carozza and Passarelli di
Napoli [10]. They showed that the following weak type endpoint estimates hold
for the composition of $M$,

$\left|\left\\{x\in\mathbb{R}^{n}:M^{k}f(x)>\lambda\right\\}\right|\lesssim\int_{\mathbb{R}^{n}}\Psi_{k-1}\left(\frac{|f(x)|}{\lambda}\right)dx,\quad
0\leq\beta<\infty,$

where $M^{k}$ is the $k$-th iterations of $M$ and
$\Psi_{\beta}(t)=t\log^{\beta}(\mathrm{e}+t).$

By sparse domination, the results in [3] imply the conclusion that if
$T_{1},T_{2}$ be two Calderón-Zygmund operators with $T_{1}(1)=0$, then for
any $1<p<\infty,1<q<p$, and $w\in A_{p/q}\left(\mathbb{R}^{n}\right)$,

$\left\|T_{1}T_{2}f\right\|_{L^{p}\left(\mathbb{R}^{n},w\right)}\lesssim[w]_{A_{p/q}}^{\max\left\\{\frac{1}{p-q},1\right\\}}\|f\|_{L^{p}\left(\mathbb{R}^{n},w\right)},$

where the precise definitions of $A_{p}\left(\mathbb{R}^{n}\right)$ weight and
$A_{p}$ constants are listed in Section 2. It was Hu [27] who proved the
weighted bound for the composite operator $T_{1}T_{2}$ without the assumption
$T_{1}(1)=0.$ Recently, Hu [26] established the weighted weak type endpoint
estimate for $T_{1}T_{2}.$ Still more recently, using the method of sparse
domination, more accurate weighted estimate for the composition of rough
homogeneous singular integral operators were obtained by Hu, Lai and Xue [29].

######  Theorem A ([29]).

Let $\Omega_{1},\Omega_{2}$ be homogeneous of degree zero, have mean value
zero and $\Omega_{1},\Omega_{2}\in L^{\infty}\left(\mathbb{S}^{n-1}\right)$.
Then for $p\in(1,\infty)$ and $w\in A_{p}\left(\mathbb{R}^{n}\right)$,

$\displaystyle\left\|T_{\Omega_{1}}T_{\Omega_{2}}f\right\|_{L^{p}\left(\mathbb{R}^{n},w\right)}\lesssim$
$\displaystyle{[w]_{A_{p}}^{\frac{1}{p}}\left([w]_{A_{\infty}}^{\frac{1}{p^{\prime}}}+[\sigma]_{A_{\infty}}^{\frac{1}{p}}\right)\left([\sigma]_{A_{\infty}}+[w]_{A_{\infty}}\right)}$
$\displaystyle\times\min\left\\{[\sigma]_{A_{\infty}},[w]_{A_{\infty}}\right\\}\|f\|_{L^{p}\left(\mathbb{R}^{n},w\right)},$

where $p^{\prime}=p/(p-1),\sigma=w^{-1/(p-1)}.$

As we mentioned in the beginning of this paper, the main purpose of this paper
is to obtain the boundedness of the composition for rough singular integral
operators $T_{\Omega_{1}}T_{\Omega_{2}}$ on rearrangement invariant Banach
spaces (RIBFS in the sequel) and quasi-Banach spaces (RIQBFS in the sequel)
whenever both $\Omega_{1}$ and $\Omega_{2}$ belong to
$L^{\infty}\left(\mathbb{S}^{n-1}\right)$.

We summary our first main result as follows.

######  Theorem 1.1.

Let $\Omega_{1},\Omega_{2}$ be homogeneous of degree zero, have the vanishing
moment (1.1) and $\Omega_{1},\Omega_{2}\in
L^{\infty}\left(\mathbb{S}^{n-1}\right)$. Let $1<r<\infty$ and $\mathbb{X}$ be
a RIBFS with $1<p_{\mathbb{X}}\leq q_{\mathbb{X}}<\infty$, then there exist
$q,q_{0}>1$ such that

$\left\|T_{\Omega_{1}}T_{\Omega_{2}}f\right\|_{\mathbb{X}(w)}\lesssim\left\\{\begin{array}[]{ll}[w]_{A_{\infty}}^{2}[w]_{A_{p_{\mathbb{X}}/r}}^{\frac{2}{rq}}\left\|f\right\|_{\mathbb{X}(w)},&\text{
if }r<p_{\mathbb{X}}\leq q_{\mathbb{X}},w\in A_{\frac{p_{\mathbb{X}}}{r}};\\\
{[w]_{A_{\infty}}^{2}}[w]_{A_{p_{\mathbb{X}}}}^{\frac{1}{p_{\mathbb{X}}}}\left([w]_{A_{\infty}}+[w]_{A_{p_{\mathbb{X}}}}^{\frac{1}{p_{\mathbb{X}}}}\right)\left\|f\right\|_{\mathbb{X}(w)},&\text{
if }1<p_{\mathbb{X}}<q_{0},w\in A_{p_{\mathbb{X}}}.\end{array}\right.$

######  Remark 1.2.

If we set $\mathbb{X}=L^{p}$ with $1<p<\infty$, then
$p_{\mathbb{X}}=q_{\mathbb{X}}=p$ and the result in Theorem 1.1 covers the
conclusion in Theorem A as a special case. Furthermore, to the best knowledge
of the author, even the quantitative weighted estimates for a single rough
singular integral operator $T_{\Omega}$ on $\mathbb{X}$ is new.

As a consequence of Theorem 1.1, it follows that:

######  Corollary 1.3.

Let $\Omega_{1},\Omega_{2}$ be homogeneous of degree zero, have mean value
zero and $\Omega_{1},\Omega_{2}\in L^{\infty}\left(\mathbb{S}^{n-1}\right)$.
Let $1<r<\infty$ and $\mathbb{X}$ be a RIQBFS, which is $p$-convex for some
$p>0$. If $p<p_{\mathbb{X}}\leq q_{\mathbb{X}}<\infty$, then there exist
$q,q_{0}>1$ such that

$\left\||T_{\Omega_{1}}T_{\Omega_{2}}f|^{\frac{1}{p}}\right\|_{\mathbb{X}(w)}\lesssim\left\\{\begin{array}[]{ll}[w]_{A_{\infty}}^{\frac{2}{p}}[w]_{A_{\frac{p_{\mathbb{X}}}{pr}}}^{\frac{2}{prq}}\left\||f|^{\frac{1}{p}}\right\|_{\mathbb{X}(w)},&pr<p_{\mathbb{X}},w\in
A_{\frac{p_{\mathbb{X}}}{pr}};\\\
{[w]_{A_{\infty}}^{\frac{2}{p}}}[w]_{A_{\frac{p_{\mathbb{X}}}{p}}}^{\frac{1}{p_{\mathbb{X}}}}\left([w]^{\frac{1}{p}}_{A_{\infty}}+[w]_{A_{\frac{p_{\mathbb{X}}}{p}}}^{\frac{1}{p_{\mathbb{X}}}}\right)\left\||f|^{\frac{1}{p}}\right\|_{\mathbb{X}(w)},&p_{\mathbb{X}}<pq_{0},w\in
A_{\frac{p_{\mathbb{X}}}{p}}.\end{array}\right.$

It is well known that some important facts may be significantly different
between Banach spaces and quasi-Banach spaces. For example, the direction of
the Hölder’s inequality in $L^{p}(0<p<1)$ and $L^{q}(q\geq 1)$ is opposite.
When $\mathbb{X}$ is a space of rearrangement invariant quasi-Banach type, we
obtain the following quantitative weighted bounds for the composition of rough
singular integral operators.

######  Theorem 1.4.

Let $\Omega_{1},\Omega_{2}$ be homogeneous of degree zero, have mean value
zero and $\Omega_{1},\Omega_{2}\in L^{\infty}\left(\mathbb{S}^{n-1}\right)$.
Let $\mathbb{X}$ be a RIQBFS, which is p-convex with $0<p\leq 1$. If
$1<p_{\mathbb{X}}\leq q_{\mathbb{X}}<2p-\frac{1}{1+p_{\mathbb{X}}}p$, then for
every $w\in A_{{p_{\mathbb{X}}}}$,

$\left\|T_{\Omega_{1}}T_{\Omega_{2}}f\right\|_{\mathbb{X}(w)}\lesssim\left([w]_{A_{\infty}}^{1+\frac{1}{p}}+[w]_{A_{\infty}}^{2+\frac{1}{p}}\right)\left([w]_{A_{p_{\mathbb{X}}}}^{\frac{1}{p_{\mathbb{X}}}}+[w]_{A_{p_{\mathbb{X}}}}^{\frac{2}{p_{\mathbb{X}}}}\right)\left\|f\right\|_{\mathbb{X}(w)}.$

This paper is organized as follows. In Section 2, we present some lemmas and
related definitions, such as RIBFS and RIQBFS, dyadic cubes, some maximal
operators and bi-sublinear sparse operators. The proofs of Theorem 1.1 and
Corollary 1.3 will be given in Section 3. In Section 4, we demonstrate Theorem
1.4. An application of Theorem 1.1 will be given in Section 5.

In what follows, $C$ always denotes a positive constant that is independent of
the main parameters involved but whose value may differ from line to line. For
any $a,b\in\mathbb{R},a\lesssim b$ $(a\gtrsim b,$ respectively) denotes that
there exists a constant $C>0$ such that $a\leq Cb;$ and $a\simeq b$ denotes
$a\lesssim b$ and $b\lesssim a.$ $p^{\prime}$ will always denote the conjugate
of $p,$ namely, $1/p+1/p^{\prime}=1$.

##  2\. Preliminary

First, we recall some basic properties for RIBFS, RIQBFS, sparse family and
Orlicz maximal operators.

###  2.1.  RIBFS and RIQBFS

Let’s start with some simple definitions.

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

  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 such that $|E|<\infty,$ 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).

By means of $\rho,$ a rearrangement invariant Banach function space (RIBFS)
$\mathbb{X}$ can be defined:

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

Let $\mathbb{X}^{\prime}$ be the associated space of $\mathbb{X}$, which is
also a Banach function space given by

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

Note that in the present setting, $\mathbb{X}$ is a RIBFS if and only if
$\mathbb{X}^{\prime}$ is a $\mathrm{RIBFS}$ ([4, Chapter 2, Corollary 4.4]).
By definition, the following generalized Hölder’s inequality holds for any
$f\in\mathbb{X},g\in\mathbb{X}^{\prime}$ :

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

A key fact in a RIBFS $\mathbb{X}$ is that the Lorentz-Luxemburg theorem
holds:

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

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

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

An important property of $f^{*}$ is that it is equimeasurable with $f$. This
allows one to obtain a representation of $\mathbb{X}$, i.e., Luxemburg’s
representation theorem ([4, Chapter 2, Theorem 4.10]), which asserts that
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
$\|f\|_{\mathbb{X}}=\left\|f^{*}\right\|_{\overline{\mathbb{X}}}$. From this
it can be seen that the mapping $f\mapsto f^{*}$ is an isometry. In addition,
notice that $\overline{\mathbb{X}}^{\prime}=\overline{\mathbb{X}^{\prime}}$
and
$\|f\|_{\mathbb{X}^{\prime}}=\left\|f^{*}\right\|_{\overline{\mathbb{X}}^{\prime}}$
hold for the associated space.

$\circ$ Weighted versions of RIBFS $\mathbb{X}$. Before we consider weighted
versions of RIBFS $\mathbb{X}$, we need to recall the definitions of
Muckenhoupt weights. Let $w$ be a non-negative locally integrable function
defined on $\mathbb{R}^{n}.$ We say that a weight $w$ 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 $Q\subset\mathbb{R}^{n}.$ The
$A_{1}$ constant is defined by

$[w]_{A_{1}}:=\sup_{x\in\mathbb{R}^{n}}\frac{Mw(x)}{w(x)},$

where $M$ is the Hardy-Littlewood maximal operator.

Some important properties of weights are listed in the following lemmas.

######  Lemma 2.1 ([44]).

Let $1<p<\infty$ and let $w\in A_{p}.$ Then $w\in A_{p-\varepsilon}$ with

$\varepsilon:=\varepsilon(p)=\frac{p-1}{1+2^{n+1}[\sigma]_{A_{\infty}}}$

where $\sigma=w^{1-p^{\prime}}$ is the dual weight. Furthermore
$[w]_{A_{p-\varepsilon}}\leq 2^{p-1}[w]_{A_{p}}.$

######  Lemma 2.2 ([32]).

Let $w\in A_{\infty}\left(\mathbb{R}^{n}\right)$. Then for any cube $Q$ and
$\delta\in\left(1,1+\frac{1}{2^{11+n}[w]_{A_{\infty}}}\right]$,

$\left(\frac{1}{|Q|}\int_{Q}w^{\delta}(x)\mathrm{d}x\right)^{\frac{1}{\delta}}\leq\frac{2}{|Q|}\int_{Q}w(x)\mathrm{d}x.$

We now give the weighted version of the RIBFS $\mathbb{X}$. First, the
distribution function and the decreasing rearrangement with respect to $w$ are
defined by

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

In this way, the weighted version of the space $\mathbb{X}$ is given by

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

Then, the same procedure applying on the associate spaces yields
$\mathbb{X}^{\prime}(w)=\mathbb{X}(w)^{\prime}$ (see [18, p. 168]).

$\circ$ Boyd indices and $r$ exponent. Next, we recall the Boyd indices of a
RIBFS, which are closely related to some interpolation properties, see [4,
Chapter 3] for more details. We start with the dilation operator $D_{t}$ of
$\mathbb{X}$,

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

and its norm

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

The lower and upper Boyd indices are defined, respectively, by the following
form:

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

A simple calculation shows that $1\leq p_{\mathbb{X}}\leq
q_{\mathbb{X}}\leq\infty,$ which follows from the fact that
$h_{\mathbb{X}}(t)$ is submultiplicative. In order to give the above
definition a general explanation, we consider a special case. If
$\mathbb{X}=L^{p}$ with $1<p<\infty$, then $h_{\mathbb{X}}(t)=t^{\frac{1}{p}}$
and thus $p_{\mathbb{X}}=q_{\mathbb{X}}=p.$

The relationship between the Boyd indices of $\mathbb{X}$ and
$\mathbb{X}^{\prime}$ are as follows :
$p_{\mathbb{X}^{\prime}}=\left(q_{\mathbb{X}}\right)^{\prime}$ and
$q_{\mathbb{X}^{\prime}}=\left(p_{\mathbb{X}}\right)^{\prime},$ where $p$ and
$p^{\prime}$ are conjugate exponents.(see [41, Chapter 11, Corollary 11.6]).

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

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

and the norm
$\|f\|_{\mathbb{X}^{r}}=\left\||f|^{r}\right\|_{\mathbb{X}}^{\frac{1}{r}}$. By
the definition of Boyd indices it is easy to verify 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. Analogy to $L^{p}$ space, for each $r\geq 1,$
$\mathbb{X}^{r}$ is still a RIBFS when $\mathbb{X}$ is a RIBFS. However, if
$0<r<1,$ the space $\mathbb{X}^{r}$ is not necessarily a Banach space (see
[18, p. 269]). Hence, it is natural to consider the quasi-Banach case.

To see this, we first give the definition of the quasi-Banach function norm.
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 i),ii),iii) and v) with the triangle inequality
replaced by the quasi-triangle inequality as follows:

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

where $C$ is an absolute positive constant. Similarly, a rearrangement
invariant quasi-Banach function space is a collection that consist of all
measurable functions which satisfies $\rho^{\prime}(|f|)<\infty$. In order to
get a better study in transformation between RIBFS and RIQBFS, we need to
consider the following $p$-convex property with $p>0$ on $\mathbb{X}$ which is
a RIQBFS (see [24, p. 3]):

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

A very important conclusion states that the $p$-convex property is equivalent
to the fact that $\mathbb{X}^{\frac{1}{p}}$ is a RIBFS (see [18, p. 269]).
According to the above results and using Lorentz-Luxemburg’s theorem again, we
have

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

where $\mathbb{Y}^{\prime}$ is the associated space of the RIBFS
$\mathbb{Y}=\mathbb{X}^{\frac{1}{p}}$. Simultaneously, for each $w\in
A_{\infty}$ and $0<r<\infty$, one may define $\mathbb{X}(w)$ for a RIQBFS
$\mathbb{X}$ and it enjoys that $\mathbb{X}(w)^{r}=\mathbb{X}^{r}(w)$(see [18,
p. 269]).

######  Remark 2.3.

It is necessary for us to make a remark about the newly added $p$-convex
property. As Grafakos and Kalton ([24]) said “all practical spaces are
$p$-convex for some $p>0$”. This is because there are only very few spaces
that do not satisfy the $p$-convex property for any $p>0$ (see [34]).

###  2.2.  Young function and Orlicz maximal operators

In this subsection, we present some fundamental facts about Young functions
and Orlicz local averages which will play an important role in our analysis.
We refer the readers to [46] for more information.

Firstly, let $\Phi$ be the set of functions
$\phi:[0,\infty)\longrightarrow[0,\infty)$ which are 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. Next, we can define the average of the Luxemburg
norm, namely $\phi$-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 the sake of notation, if $\mu$ is the Lebesgue measure, we write
$\|f\|_{\phi,Q}$, and we denote $\|f\|_{\phi(w),Q},$ if $\mu=wdx$ is an
absolutely continuous measure with respect to the Lebesgue measure.

Each Young function $\phi$ enjoys the following generalized Hölder’s
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 can naturally represent the Orlicz maximal operator $M_{\phi}f$
associated to the Young function $\phi$ with the following form:

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

Finally, we present some particular examples of maximal operators related to
certain Young functions.

  * •

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

  * •

If we consider $\phi(t)=t\log^{\alpha}(e+t)$ 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 proved that $M_{\phi}\simeq M^{l+1},$ where
$\alpha=l\in\mathbb{N}$ and $M^{l+1}$ is $M$ iterated $l+1$ times.

  * •

$M_{\phi}=M_{L(\log L)^{\alpha}(\log\log L)^{\beta}}$ given by the function
$\phi(t)=t\log^{\alpha}(e+t)\log^{\beta}(e+\log(e+t))$ with $\alpha,\beta>0.$

###  2.3.  Sparse family

We also need a system of dyadic calculus from [37, 38], so in this subsection,
we introduce a part of it.

######  Definition 2.4.

By a dyadic lattice $\mathcal{D},$ we mean a collection of cubes which
satisfies the following properties:

  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 $\mathbb{R}^{n}$.

An interesting and crucial theorem in dyadic calculus is Three Lattice
Theorem, which asserts that given a dyadic lattice $\mathcal{D},$ there exist
$3^{n}$ dyadic lattices $\mathcal{D}_{1},\ldots,\mathcal{D}_{3^{n}}$ such that
for each cube $Q\in\mathcal{D}$, we can find a cube $R_{Q}$ in some
$\mathcal{D}_{j}$ such that $Q\subseteq R_{Q}$ and $3l_{Q}=l_{R_{Q}}$.

According to the method of taking dyadic lattice $\mathcal{D}$ in Definition
2.4, we can give the definition of sparse family $\mathcal{S}$ as follows.

######  Definition 2.5.

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

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

There is another equivalent definitions of sparsity for a collection of sets.
If we define

$E(Q)=Q\backslash\bigcup_{P\in\mathcal{S},P\subsetneq Q}P,$

then it is easy to deduce that the sets $E(Q)$ are pairwise disjoint and
$|E(Q)|\geq\eta|Q|$.

Let $\mathcal{D}$ be a dyadic lattice and $\mathcal{S}\subseteq\mathcal{D}$ be
a $\eta$-sparse family, the sparse operator is defined by

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

where $r>0$ and
$\langle|f|\rangle_{r,Q}^{r}=\frac{1}{|Q|}\int_{Q}|f(y)|^{r}dy$. Furthermore,
associated with the constants $\beta\in[0,\infty)$ and
$r_{1},r_{2},r\in[1,\infty)$, we can define the bilinear sparse operators
$\mathcal{A}_{\mathcal{S};L(\log L)^{\beta},L^{r}}$ and
$\mathcal{A}_{\mathcal{S};L^{r_{1}},L^{r_{2}}}$ by

$\mathcal{A}_{\mathcal{S};L(\log
L)^{\beta},L^{r}}(f,g)=\sum_{Q\in\mathcal{S}}|Q|\|f\|_{L(\log
L)^{\beta},Q}\langle|g|\rangle_{r,Q},$
$\mathcal{A}_{\mathcal{S};L^{r_{1}},L^{r_{2}}}(f,g)=\sum_{Q\in\mathcal{S}}|Q|\langle|f|\rangle_{r_{1},Q}\langle|g|\rangle_{r_{2},Q}.$

According to the above definitions, we can give the condition that the
operator $T$ satisfies the bilinear sparse domination. More specifically, for
$\beta,q\in(0,\infty)$, we say that a sublinear operator $T$ acting on
$\cup_{p\geq 1}L^{p}\left(\mathbb{R}^{n}\right)$ enjoys a $\left(L(\log
L)^{\beta},L^{q}\right)$-bilinear sparse domination with bound $A$, if for
each bounded function $f$ with compact support, there exists a sparse family
$\mathcal{S}$ of cubes, such that

$\left|\int_{\mathbb{R}^{n}}g(x)Tf(x)dx\right|\leq
A\mathcal{A}_{\mathcal{S},L(\log L)^{\beta},L^{q}}(f,g),$

holds for all bounded function $g$.

The following results of bilinear sparse domination are crucial in our
analysis ([29, Corollary 5.1]).

######  Lemma 2.6 ([29]).

Let $\Omega_{1},\Omega_{2}$ be homogeneous of degree zero, have mean value
zero and $\Omega_{1},\Omega_{2}\in L^{\infty}\left(\mathbb{S}^{n-1}\right)$.
Let $r\in(1,3/2]$. Then for each bounded function $f$ with compact support,
there exists a $\frac{1}{2}\frac{1}{9^{n}}$-sparse family of cubes
$\mathcal{S}=\\{Q\\}$, and functions $J_{1}$ and $J_{2}$, such that for each
function $g$,

(2.1)
$\displaystyle\left|\int_{\mathbb{R}^{n}}J_{1}(x)g(x)\mathrm{d}x\right|\lesssim
r^{\prime}\mathcal{A}_{\mathcal{S};L(\log L),L^{r}}(f,g),$
$\displaystyle\left|\int_{\mathbb{R}^{n}}J_{2}(x)g(x)\mathrm{d}x\right|\lesssim
r^{\prime 2}\mathcal{A}_{\mathcal{S};L^{1},L^{r}}(f,g),$

and for a. e. $x\in\mathbb{R}^{n}$,

(2.2) $T_{\Omega_{1}}T_{\Omega_{2}}f(x)=J_{1}(x)+J_{2}(x).$

Here, we give a brief introduction to sparse domination. The study of sparse
domination is a very active research area in Harmonic analysis in recent
years. It helps to simplify the proof of some well known results and even can
be used to obtain the dependence of the norm constant on the weight functions.
For example, Lerner [36] introduced a class of sparse operators and gave an
alternative and simple proof of the $A_{2}$ conjecture. Later on, Lerner [37]
obtained quantitative weighted bounds for Calderón-Zygmund operator $T$ which
satisfies a Hölder-Lipschitz condition. Since then, great attentions have been
paid to the study of the sparse bounds. We refer the readers to [11, 15, 17]
and the references therein for more informations.

##  3\.  Proofs of Theorems 1.1

This section is devoted to prove Theorem 1.1 and Corollary 1.3. To demonstrate
Theorem 1.1, we need the following lemma in [2, Lemma 3.3].

######  Lemma 3.1 ([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 1.1.

We define the weighted dyadic Hardy-Littlewood maximal operators
$M_{w}^{\mathcal{D}}$ and $M_{w,r}^{\mathcal{D}}$ by

$M_{w}^{\mathcal{D}}f(x):=\sup_{x\in
R,R\in\mathcal{D}}\frac{1}{w(R)}\int_{R}|f(y)|w(y)dy,\quad f\in
L_{\operatorname{loc}}^{1}\left(\mathbb{R}^{n}\right),$

$M_{w,r}^{\mathcal{D}}f(x):=\sup_{x\in
R,R\in\mathcal{D}}\left(\frac{1}{w(R)}\int_{R}|f(y)|^{r}w(y)dy\right)^{\frac{1}{r}},\quad
f\in L_{\operatorname{loc}}^{r}\left(\mathbb{R}^{n}\right),$

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

First, we consider the case of $1<p_{\mathbb{X}}\leq
q_{\mathbb{X}}<2-\frac{1}{1+p_{\mathbb{X}}}=:q_{0}.$ For a fixed $w\in
A_{p_{\mathbb{X}}}$, by (2.2), we obtain

$\displaystyle\left\|T_{\Omega_{1}}T_{\Omega_{2}}f\right\|_{\mathbb{X}(w)}$
$\displaystyle=\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\left|\int_{\mathbb{R}^{n}}T_{\Omega_{1}}T_{\Omega_{2}}f(x)g(x)w(x)dx\right|$
$\displaystyle\leq\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\left|\int_{\mathbb{R}^{n}}J_{1}(x)g(x)w(x)dx\right|+\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\left|\int_{\mathbb{R}^{n}}J_{2}(x)g(x)w(x)dx\right|$
$\displaystyle=:I_{1}+I_{2},$

where $I_{i}=\sup\limits_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\left|\int_{\mathbb{R}^{n}}J_{i}(x)g(x)w(x)dx\right|,i=1,2.$

Consider the contribuion of $I_{1}$. Take and fix any
$g\in\mathbb{X}^{\prime}(w)$ with $\|g\|_{\mathbb{X}^{\prime}(w)}\leq 1$, by
(2.1),

$\left|\int_{\mathbb{R}^{n}}J_{1}(x)g(x)w(x)dx\right|\lesssim
r^{\prime}\sum_{Q\in\mathcal{S}}\left\|f\right\|_{L(\log{L}),Q}{\langle|gw|\rangle}_{r,Q}|Q|.$

A direct computation gives that

$\displaystyle{\langle|gw|\rangle}_{r,Q}$
$\displaystyle=\left(\frac{1}{|Q|}\int_{Q}|g(x)w(x)|^{r}dx\right)^{\frac{1}{r}}=\left(\frac{1}{|Q|}\int_{Q}|g(x)|^{r}|w(x)|^{\frac{1}{s}}|w(x)|^{r-\frac{1}{s}}dx\right)^{\frac{1}{r}}$
$\displaystyle\leq\left(\frac{1}{|Q|}\int_{Q}|g(x)|^{rs}w(x)dx\right)^{\frac{1}{rs}}\left(\frac{1}{|Q|}\int_{Q}|w(x)|^{(r-\frac{1}{s})s^{\prime}}dx\right)^{\frac{1}{rs^{\prime}}}.$

Take appropriate $r$ and $s$ such that
$(r-\frac{1}{s})s^{\prime}<1+\frac{1}{2^{11+n}[w]_{A_{\infty}}}.$ For example,
let us choose
$r=1+\frac{1}{2^{13+n}p_{\mathbb{X}}[w]_{A_{\infty}}},s=1+\frac{1}{2p_{\mathbb{X}}},$
then

$(r-\frac{1}{s})s^{\prime}=1+\frac{1}{2^{13+n}p_{\mathbb{X}}[w]_{A_{\infty}}}+\frac{1}{2^{12+n}[w]_{A_{\infty}}}<1+\frac{1}{2^{11+n}[w]_{A_{\infty}}}.$

Thus, combining Lemma 2.2, we deduce that

$\displaystyle\frac{1}{|Q|}\int_{Q}(w(x))^{(r-\frac{1}{s})s^{\prime}}dx$
$\displaystyle\leq\left(\frac{2}{|Q|}\int_{Q}w(x)dx\right)^{1-\frac{1}{rs}}.$

Now one can get

$\displaystyle{\langle|gw|\rangle}_{r,Q}\cdot|Q|$
$\displaystyle\leq\left(\frac{1}{|w(Q)|}\int_{Q}|g(x)|^{rs}w(x)dx\right)^{\frac{1}{rs}}w(Q)=:g_{Q,w}^{rs}\cdot
w(Q).$

Taking into account the generalized Hölder’s inequality and recalling that
$M^{2}$ is $M$ iterated 2 times, we have

$\displaystyle\sum_{Q\in\mathcal{S}}\left\|f\right\|_{L(\log
L),Q}{\langle|gw|\rangle}_{r,Q}|Q|$
$\displaystyle\leq\sum_{Q\in\mathcal{S}}\left\|f\right\|_{L(\log
L),Q}w(Q)g_{Q,w}^{rs}$
$\displaystyle\leq\sum_{B\in\mathcal{B}}\left\|f\right\|_{L(\log
L),B}g_{B,w}^{2s}\sum_{\begin{subarray}{c}R\in\mathcal{S}\\\
\pi(R)=B\end{subarray}}w(R)$ $\displaystyle\leq
C(n)[w]_{A_{\infty}}\sum_{B\in\mathcal{B}}\left\|f\right\|_{L(\log
L),B}w(B)g_{B,w}^{2s}$
$\displaystyle=C(n)[w]_{A_{\infty}}\int_{\mathbb{R}^{n}}\sum_{B\in\mathcal{B}}\left\|f\right\|_{L(\log
L),B}g_{B,w}^{2s}\chi_{B}(x)w(x)dx$ $\displaystyle\leq
C(n)[w]_{A_{\infty}}\int_{\mathbb{R}^{n}}M_{L(\log
L)}f(x)M_{w,2s}^{\mathcal{D}}g(x)w(x)dx$ $\displaystyle\leq
C(n)[w]_{A_{\infty}}\int_{\mathbb{R}^{n}}M^{2}f(x)M_{w,2s}^{\mathcal{D}}g(x)w(x)dx$
$\displaystyle\leq
C(n)[w]_{A_{\infty}}\left\|M^{2}f\right\|_{\mathbb{X}(w)}\left\|M_{w,2s}^{\mathcal{D}}g\right\|_{\mathbb{X}^{\prime}(w)},$

where $\mathcal{B}$ is the family of the principal cubes in the usual sense
and $\pi(R)$ is the minimal principal cube which contains $R$. That is

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

with $\mathcal{B}_{0}:=\\{$ maximal cubes in $\mathcal{S}\\}$ 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 $\tau(R)=\|f\|_{{L(\log L)},R}g_{R,w}^{2s}$.

Now we observe that, using Lemma 3.1,

$\displaystyle I$ $\displaystyle\leq
C(n)[w]_{A_{\infty}}^{2}[w]_{A_{p_{\mathbb{X}}}}^{\frac{2}{p_{\mathbb{X}}}}\left\|f\right\|_{\mathbb{X}(w)}\left\|\left(M_{w}^{\mathcal{D}}(|g|^{2s})\right)^{\frac{1}{2s}}\right\|_{\mathbb{X}^{\prime}(w)}$
$\displaystyle=C(n)[w]_{A_{\infty}}^{2}[w]_{A_{p_{\mathbb{X}}}}^{\frac{2}{p_{\mathbb{X}}}}\left\|f\right\|_{\mathbb{X}(w)}\left\|\left(M_{w}^{\mathcal{D}}(|g|^{2s})\right)\right\|_{\mathbb{X}^{\prime\frac{1}{2s}}(w)}^{\frac{1}{2s}}$
$\displaystyle\lesssim[w]_{A_{\infty}}^{2}[w]_{A_{p_{\mathbb{X}}}}^{\frac{2}{p_{\mathbb{X}}}}\left\|f\right\|_{\mathbb{X}(w)}\left\||g|^{2s}\right\|_{\mathbb{X}^{\prime\frac{1}{2s}}(w)}^{\frac{1}{2s}}$
$\displaystyle\leq[w]_{A_{\infty}}^{2}[w]_{A_{p_{\mathbb{X}}}}^{\frac{2}{p_{\mathbb{X}}}}\left\|f\right\|_{\mathbb{X}(w)},$

where in the second inequality of the countdown, we have used the fact that
the boundedness of $M_{w}^{\mathcal{D}}$( see [18, Theorem 3.2]). Indeed,
using the condition $q_{\mathbb{X}}<\frac{2s}{2s-1}$, it’s easy to deduce that

$p_{\mathbb{X}^{\prime\frac{1}{2s}}}=\frac{p_{{\mathbb{X}}^{\prime}}}{2s}=\frac{{(q_{\mathbb{X}})}^{{}^{\prime}}}{2s}>1.$

Now we turn our attention to the estimate of $I_{2}.$ For this case, arguing
as in the first case, we obtain

$\displaystyle\sum_{Q\in\mathcal{S}}\langle|f|\rangle_{Q}\langle|gw|\rangle_{r,Q}|Q|$
$\displaystyle\lesssim\sum_{Q\in\mathcal{S}}\langle|f|\rangle_{Q}g_{Q,w}^{rs}w(Q)$
$\displaystyle\leq\sum_{Q\in\mathcal{S}}\frac{1}{w(Q)}\left(\int_{Q}(Mf(x))^{\frac{1}{2}}(M_{w,2s}^{\mathcal{D}}g(x))^{\frac{1}{2}}w(x)dx\right)^{2}.$

That fact together with Lemma 2.6 yields

(3.1) $\left|\int_{\mathbb{R}^{n}}J_{2}(x)g(x)w(x)dx\right|\lesssim r^{\prime
2}\sum_{Q\in\mathcal{S}}\frac{1}{w(Q)}\left(\int_{Q}(Mf(x))^{\frac{1}{2}}(M_{w,2s}^{\mathcal{D}}g(x))^{\frac{1}{2}}w(x)dx\right)^{2}.$

Now we note that $\mathcal{S}$ is a $\frac{1}{2\cdot 9^{n}}$ -sparse family
(i.e., for any $Q\in\mathcal{S},$ there exists $E(Q)$ such that
$|E(Q)|\geq\frac{1}{2\cdot 9^{n}}|Q|$ ). Hence, for each dyadic cube
$Q\in\mathcal{S}$, we have

(3.2) $\displaystyle\sum_{R\subseteq Q}w(R)=\sum_{R\subseteq
Q}\frac{w(R)}{|R|}|R|$ $\displaystyle\leq 2\cdot 9^{n}\sum_{R\subseteq
Q}\frac{w(R)}{|R|}\cdot|E(R)|$ $\displaystyle\leq 2\cdot 9^{n}\sum_{R\subseteq
Q}\int_{E(R)}M(w\chi_{R})(x)dx$ $\displaystyle\leq 2\cdot
9^{n}\int_{Q}M(w\chi_{Q})(x)dx$ $\displaystyle\leq 2\cdot
9^{n}[w]_{A_{\infty}}w(Q).$

Combining (3.2) with (3.1) and the Carleson embedding theorem, together with
the generalized Hölder’s inequality, one may obtain

$\displaystyle I_{2}$
$\displaystyle\lesssim[w]_{A_{\infty}}^{2}\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\sum_{Q\in\mathcal{S}}\frac{1}{w(Q)}\left(\int_{Q}(Mf(x))^{\frac{1}{2}}(M_{w,2s}^{\mathcal{D}}g(x))^{\frac{1}{2}}w(x)dx\right)^{2}$
$\displaystyle\lesssim[w]_{A_{\infty}}^{3}\left\|Mf\right\|_{\mathbb{X}(w)}\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\left\|M_{w,2s}^{\mathcal{D}}g\right\|_{\mathbb{X}^{\prime}(w)}$
$\displaystyle\leq[w]_{A_{\infty}}^{3}[w]_{A_{p_{\mathbb{X}}}}^{\frac{1}{p_{\mathbb{X}}}}\left\|f\right\|_{\mathbb{X}(w)}\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\left\||g|^{2s}\right\|_{\mathbb{X}^{\prime\frac{1}{2s}}(w)}^{\frac{1}{2s}}$
$\displaystyle\leq[w]_{A_{\infty}}^{3}[w]_{A_{p_{\mathbb{X}}}}^{\frac{1}{p_{\mathbb{X}}}}\left\|f\right\|_{\mathbb{X}(w)}.$

This inequality, combined with the estimate of $I_{1}$ yields

$\left\|T_{\Omega_{1}}T_{\Omega_{2}}f\right\|_{\mathbb{X}(w)}\lesssim[w]_{A_{\infty}}^{2}[w]_{A_{p_{\mathbb{X}}}}^{\frac{1}{p_{\mathbb{X}}}}\left([w]_{A_{\infty}}+[w]_{A_{p_{\mathbb{X}}}}^{\frac{1}{p_{\mathbb{X}}}}\right)\left\|f\right\|_{\mathbb{X}(w)}.$

To end the proof we consider the case of $r<p_{\mathbb{X}}\leq
q_{\mathbb{X}}<\infty,$ where $1<r<\infty$ is any fixed constant.

If $w\in A_{p_{\mathbb{X}}/r},$

$\displaystyle\left\|T_{\Omega_{1}}f\right\|_{\mathbb{X}(w)}$
$\displaystyle=\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\left|\int_{\mathbb{R}^{n}}T_{\Omega_{1}}f(x)g(x)w(x)dx\right|$
$\displaystyle\lesssim\sup_{\|g\|_{\mathbb{X}^{\prime}(w)}\leq
1}\sum_{Q\in{\mathcal{S}}}\langle|f|\rangle_{r,Q}\langle|gw|\rangle_{1,Q}|Q|,$

where the last step follows from [15, Theorem A] or [35, Corollary 3.4].

By a direct computation and the generalized Hölder’s inequality, it follows
that

$\displaystyle\sum_{Q\in{\mathcal{S}}}\langle|f|\rangle_{r,Q}\langle|gw|\rangle_{1,Q}|Q|$
$\displaystyle=\sum_{Q\in{\mathcal{S}}}\langle|f|\rangle_{r,Q}\frac{1}{w(Q)}\int_{Q}|g(x)|w(x)dxw(Q)$
$\displaystyle\lesssim[w]_{A_{\infty}}\int_{\mathbb{R}^{n}}M_{r}f(x)M_{w}^{\mathcal{D}}g(x)w(x)dx$
$\displaystyle\lesssim[w]_{A_{\infty}}\left\|M_{r}f\right\|_{\mathbb{X}(w)}\left\|M_{w}^{\mathcal{D}}g\right\|_{\mathbb{X}^{\prime}(w)}$
$\displaystyle\lesssim[w]_{A_{\infty}}\left\|M_{r}f\right\|_{\mathbb{X}(w)}\left\|g\right\|_{\mathbb{X}^{\prime}(w)},$

where in the first inequality we apply the Carleson embedding theorem.
Hence, by the fact that
$\left\|M_{r}f\right\|_{\mathbb{X}(w)}\lesssim[w]_{A_{p_{\mathbb{X}}/r}}^{\frac{1}{rq}}\left\|f\right\|_{\mathbb{X}(w)}$
(see [50, Lemma 3.1]), where
$q=p_{\mathbb{X}}/{r}-\varepsilon(p_{\mathbb{X}}/{r})$ and $\varepsilon(p)$ is
defined in Lemma 2.1, we obtain

$\left\|T_{\Omega_{1}}f\right\|_{\mathbb{X}(w)}\leq
C[w]_{A_{\infty}}[w]_{A_{p_{\mathbb{X}}/{r}}}^{\frac{1}{rq}}\left\|f\right\|_{\mathbb{X}(w)}.$

This inequality, combining with the definition of
$T_{\Omega_{1}}T_{\Omega_{2}}$ gives

$\left\|T_{\Omega_{1}}T_{\Omega_{2}}f\right\|_{\mathbb{X}(w)}\leq
C[w]_{A_{\infty}}^{2}[w]_{A_{p_{\mathbb{X}}/{r}}}^{\frac{2}{rq}}\left\|f\right\|_{\mathbb{X}(w)},$

This finish the proof of the second case.

Therefore, we obtain

$\left\|T_{\Omega_{1}}T_{\Omega_{2}}f\right\|_{\mathbb{X}(w)}\lesssim\left\\{\begin{array}[]{ll}[w]_{A_{\infty}}^{2}[w]_{A_{p_{\mathbb{X}}/{r}}}^{\frac{2}{rq}}\left\|f\right\|_{\mathbb{X}(w)},&\text{
if }r<p_{\mathbb{X}}\leq q_{\mathbb{X}},w\in A_{\frac{p_{\mathbb{X}}}{r}};\\\
{[w]_{A_{\infty}}^{2}}[w]_{A_{p_{\mathbb{X}}}}^{\frac{1}{p_{\mathbb{X}}}}\left([w]_{A_{\infty}}+[w]_{A_{p_{\mathbb{X}}}}^{\frac{1}{p_{\mathbb{X}}}}\right)\left\|f\right\|_{\mathbb{X}(w)},&\text{
if }1<p_{\mathbb{X}}<q_{0},w\in A_{p_{\mathbb{X}}},\end{array}\right.$

where $q_{0}=2-\frac{1}{1+p_{\mathbb{X}}}$. ∎

Using Theorem 1.1, we can easily get Corollary 1.3.

###### Proof of Corollary 1.3.

Recall that if $\mathbb{X}$ be a RIQBFS with $p$-convex then
$\mathbb{X}^{\frac{1}{p}}$ is a RIBFS. This together with Theorem 1.1 lead to
the desired result. As a matter of fact, it suffices for us to prove that for
any $r>0$,
$\left\||f|^{r}\right\|^{1/r}_{\mathbb{X}(w)}=\left\|f\right\|_{\mathbb{X}^{r}(w)}$
and $p_{\mathbb{X}^{r}}=r\cdot p_{\mathbb{X}},q_{\mathbb{X}^{r}}=r\cdot
q_{\mathbb{X}}$ hold for RIQBFS $\mathbb{X}$. To see this, using the fact that
$\mathbb{X}^{1/p}$ is a RIBFS, we have

$p_{\mathbb{X}^{r}}=p_{\mathbb{X}^{\frac{1}{p}pr}}=pr\cdot
p_{\mathbb{X}^{\frac{1}{p}}}.$

On the other hand,

$p_{\mathbb{X}}=p_{\mathbb{X}^{\frac{1}{p}p}}=p\cdot
p_{\mathbb{X}^{\frac{1}{p}}}.$

Therefore $p_{\mathbb{X}^{r}}=r\cdot p_{\mathbb{X}}$ and it also works for
$q_{\mathbb{X}}$.
From the definitions of the norm of $\mathbb{X}^{r},$ it is easy to check that

$\left\|f\right\|_{\mathbb{X}^{r}(w)}=\left\||f|^{pr}\right\|^{\frac{1}{pr}}_{\mathbb{X}^{\frac{1}{p}}(w)}$

and

$\left\||f|^{r}\right\|_{\mathbb{X}(w)}=\left\||f|^{pr}\right\|^{\frac{1}{p}}_{\mathbb{X}^{\frac{1}{p}}(w)}.$

Then we have
$\left\||f|^{r}\right\|^{1/r}_{\mathbb{X}(w)}=\left\|f\right\|_{\mathbb{X}^{r}(w)}.$

∎

##  4\.  Proofs of Theorem 1.4

The section will be devoted to prove Theorem 1.4. Using sparse domination
method, we will show that the bilinear sparse domination of the composition of
rough singular integral operators holds, which implies Theorem 1.4.

We start with some definitions. For a linear operator $T$ and $1\leq
r<\infty$, we define the corresponding grand maximal truncated operator
$\mathscr{M}_{T,r}$ by

$\mathscr{M}_{T,r}f(x):=\sup_{Q\ni
x}|Q|^{-\frac{1}{r}}\left\|T\left(f\chi_{\mathbb{R}^{n}\backslash
3Q}\right)\chi_{Q}\right\|_{L^{r}\left(\mathbb{R}^{n}\right)},$

where the supremum is taken over all cubes $Q\subset\mathbb{R}^{n}$ containing
$x$. It is well known that the operator $\mathscr{M}_{T,r}$ was introduced by
Lerner [35] and it plays a key role in establishing bilinear sparse domination
of rough operator. Let $T_{1},T_{2}$ be two linear operators. We define the
grand maximal operator $\mathscr{M}_{T_{1}T_{2},r}^{*}$ by

$\mathscr{M}_{T_{1}T_{2},r}^{*}f(x):=\sup_{Q\ni
x}\left(\frac{1}{|Q|}\int_{Q}\left|T_{1}\left(\chi_{\mathbb{R}^{n}\backslash
3Q}T_{2}\left(f\chi_{\mathbb{R}^{n}\backslash
9Q}\right)\right)(\xi)\right|^{r}\mathrm{~{}d}\xi\right)^{\frac{1}{r}}.$

Now, we can state bilinear sparse domination of the composition operator
associated with RIQBFS $\mathbb{X}.$

######  Lemma 4.1.

Let $1<r<\frac{3}{2}$, $0\leq\beta_{1},\beta_{2}<\infty.$ Let $T_{1},T_{2}$ be
two linear operators satisfying additionally the following conditions

  1. (i).

the operator $T_{1}$ is bounded on $L^{r^{\prime}}\left(\mathbb{R}^{n}\right)$
with bound $A$;

  2. (ii).

there exists $A_{0}>0$ such that for each $\lambda>0$,

$\left|\left\\{x\in\mathbb{R}^{n}:\left|T_{1}T_{2}f(x)\right|>\lambda\right\\}\right|\lesssim\int_{\mathbb{R}^{n}}\frac{A_{0}|f(x)|}{\lambda}\log^{\beta_{1}}\left(\mathrm{e}+\frac{A_{0}|f(x)|}{\lambda}\right)\mathrm{d}x;$

  3. (iii).

there exists $A_{1},A_{2}>0$ such that for each $\lambda>0$,

$\left|\left\\{x\in\mathbb{R}^{n}:\mathscr{M}_{T_{1},r^{\prime}}T_{2}f(x)>\lambda\right\\}\right|\lesssim\int_{\mathbb{R}^{n}}\frac{A_{1}|f(x)|}{\lambda}\log^{\beta_{1}}\left(\mathrm{e}+\frac{A_{1}|f(x)|}{\lambda}\right)\mathrm{d}x$

and

$\left|\left\\{x\in\mathbb{R}^{n}:\mathscr{M}_{T_{2},r^{\prime}}f(x)>\lambda\right\\}\right|\lesssim\int_{\mathbb{R}^{n}}\frac{A_{2}|f(x)|}{\lambda}\log^{\beta_{2}}\left(\mathrm{e}+\frac{A_{2}|f(x)|}{\lambda}\right)\mathrm{d}x.$

Then for each $0<p\leq 1$ and a bounded function $f$ with compact support,
there exists a $\frac{1}{2\cdot 9^{n}}$-sparse family of cubes
$\mathcal{S}=\\{Q\\}$, and functions $H_{1}$ and $H_{2}$, such that for each
function $g$,

$\int_{\mathbb{R}^{n}}|H_{1}(x)|^{p}|g(x)|dx\lesssim\left(A_{0}^{p}+A_{1}^{p}\right)\mathcal{A}^{(p,1)}_{\mathcal{S};L(\log
L)^{\beta_{1}},L^{(r^{{}^{\prime}}/p)^{{}^{\prime}}}}(f,g),$
$\int_{\mathbb{R}^{n}}|H_{2}(x)|^{p}|g(x)|dx\lesssim
A^{p}A_{2}^{p}\mathcal{A}^{(p,1)}_{\mathcal{S};L(\log
L)^{\beta_{2}},L^{(r^{{}^{\prime}}/p)^{{}^{\prime}}}}(f,g),$

and for a.e. $x\in\mathbb{R}^{n}$,

$T_{1}T_{2}f(x)=H_{1}(x)+H_{2}(x),$

where

$\mathcal{A}^{(a,b)}_{\mathcal{S};L(\log
L)^{\beta},L^{t}}(f,g):=\sum_{Q\in\mathcal{S}}|Q|\|f\|^{a}_{L(\log
L)^{\beta},Q}\langle|g|\rangle^{b}_{t,Q}$

with $0\leq a,b,\beta<\infty,1\leq t<\infty.$

###### Proof of Lemma 4.1.

For any $0<p\leq 1$, in order to establish bilinear sparse domination over
operator $(T_{1}T_{2}f)^{p}$, we are going to follow the scheme of the proof
of [35, Theorem 3.1], together with some ideas in [26]. For a fixed cube
$Q_{0}$, we should also consider a local version of operators
$\mathscr{M}_{T_{2},r^{\prime}}$ and $\mathscr{M}_{T_{1}T_{2},r^{\prime}}^{*}$
by

$\mathscr{M}_{T_{2};r^{\prime};Q_{0}}f(x)=\sup_{Q\ni x,Q\subset
Q_{0}}|Q|^{-\frac{1}{r^{\prime}}}\left\|\chi_{Q}T_{2}\left(f\chi_{3Q_{0}\backslash
3Q}\right)\right\|_{L^{r^{\prime}}\left(\mathbb{R}^{n}\right)}$

and

$\mathscr{M}_{T_{1}T_{2},r^{\prime};Q_{0}}^{*}f(x)=\sup_{Q\ni x,Q\subset
Q_{0}}\left(\frac{1}{|Q|}\int_{Q}\left|T_{1}\left(\chi_{\mathbb{R}^{n}\backslash
3Q}T_{2}\left(f\chi_{9Q_{0}\backslash
9Q}\right)\right)(\xi)\right|^{r^{\prime}}\mathrm{d}\xi\right)^{\frac{1}{r^{\prime}}},$

respectively. Then we define three sets $E_{i}$ with $i=1,2,3$ by

$E_{1}=\left\\{x\in
Q_{0}:\left|T_{1}T_{2}\left(f\chi_{9Q_{0}}\right)(x)\right|>DA_{0}\|f\|_{L(\log
L)^{\beta_{1}},9Q_{0}}\right\\};$ $E_{2}=\left\\{x\in
Q_{0}:\mathscr{M}_{T_{2},r^{\prime};Q_{0}}f(x)>DA_{2}\|f\|_{L(\log
L)^{\beta_{2},},9Q_{0}}\right\\};$ $E_{3}=\left\\{x\in
Q_{0}:\mathscr{M}_{T_{1}T_{2},r^{\prime};Q_{0}}^{*}f(x)>DA_{1}\|f\|_{L(\log
L)^{\beta_{1}},9Q_{0}}\right\\},$

with $D$ a positive constant. Let $E=\cup_{i=1}^{3}E_{i}.$ Hence, by our
hypothesis $(ii),$ $(iii)$ and [29, Lemma 4.3], taking $D$ large enough, we
deduce that

$|E|\leq\frac{1}{2^{n+2}}\left|Q_{0}\right|.$

Now, by using the Calderón-Zygmund decomposition to the function $\chi_{E}$ on
$Q_{0}$ at height $\lambda=$ $\frac{1}{2^{n+1}}$, we obtain pairwise disjoint
cubes $\left\\{P_{l}\right\\}_{l}\subset\mathcal{D}\left(Q_{0}\right)$ such
that

$\chi_{E}(x)\leq\frac{1}{2^{n+1}}$

for a.e. $x\notin\bigcup_{l}P_{l}$. Together with this we immediately obtain
$\left|E\backslash\bigcup_{l}P_{l}\right|=0.$ At the same time, for each
$l\geq 1$, we also have

$\frac{1}{2^{n+1}}\left|P_{l}\right|\leq\left|P_{l}\cap
E\right|\leq\frac{1}{2}\left|P_{l}\right|.$

Observe that $\sum_{l}\left|P_{l}\right|\leq
2^{n+1}|E|\leq\frac{1}{2}\left|Q_{0}\right|,$ $P_{l}\cap E^{c}\neq\emptyset$.
Let

$\displaystyle G_{1}(x):=$ $\displaystyle
T_{1}T_{2}\left(f\chi_{9Q_{0}}\right)(x)\chi_{Q_{0}\backslash\cup_{l}P_{l}}(x)$
$\displaystyle+\sum_{l}T_{1}\left(\chi_{\mathbb{R}^{n}\backslash
3P_{l}}T_{2}\left(f\chi_{9Q_{0}\backslash
9P_{l}}\right)\right)(x)\chi_{P_{l}}(x).$

Then we have the following claim:

$\int_{\mathbb{R}^{n}}|G_{1}(x)|^{p}|g(x)|dx\lesssim\left(A_{0}^{p}+A_{1}^{p}\right)\|f\|^{p}_{L(\log
L)^{\beta_{1}},9Q_{0}}\langle|g|\rangle_{(\frac{r^{\prime}}{p})^{\prime},Q_{0}}\left|Q_{0}\right|.$

Indeed, noting that $0<p\leq 1$ and the definition of $G_{1}$ we have

$\displaystyle|G_{1}(x)|^{p}\leq$
$\displaystyle\left|T_{1}T_{2}\left(f\chi_{9Q_{0}}\right)(x)\right|^{p}\chi_{Q_{0}\backslash\cup_{l}P_{l}}(x)$
$\displaystyle+\sum_{l}|T_{1}\left(\chi_{\mathbb{R}^{n}\backslash
3P_{l}}T_{2}\left(f\chi_{9Q_{0}\backslash
9P_{l}}\right)\right)(x)|^{p}\chi_{P_{l}}(x)$
$\displaystyle=:L_{1}(x)+\sum_{l}L^{p}_{2,l}(x),$

where $L_{2,l}(x)=|T_{1}\left(\chi_{\mathbb{R}^{n}\backslash
3P_{l}}T_{2}\left(f\chi_{9Q_{0}\backslash
9P_{l}}\right)\right)(x)|\chi_{P_{l}}(x)$ with $l=1,2,\ldots$.
Therefore,

$\displaystyle\int_{\mathbb{R}^{n}}|G_{1}(x)|^{p}|g(x)|dx$
$\displaystyle\leq\int_{Q_{0}\backslash\cup_{l}P_{l}}L_{1}(x)|g(x)|dx+\sum_{l}\int_{P_{l}}L_{2,l}^{p}(x)|g(x)|dx$
$\displaystyle\leq\left(\int_{Q_{0}\backslash
E}+\int_{E\backslash\cup_{l}P_{l}}\right)L_{1}(x)|g(x)|dx+\sum_{l}\int_{P_{l}}L_{2,l}^{p}(x)|g(x)|dx$
$\displaystyle=\int_{Q_{0}\backslash
E}L_{1}(x)|g(x)|dx+\sum_{l}\int_{P_{l}}L_{2,l}^{p}(x)|g(x)|dx,$

where the last inequality is due to the fact that
$\left|E\backslash\bigcup_{l}P_{l}\right|=0.$

Note that for any $x\in Q_{0}\backslash E$ yields that $x\notin E_{1}$ which
implies that

$\left|T_{1}T_{2}\left(f\chi_{9Q_{0}}\right)(x)\right|\leq DA_{0}\|f\|_{L(\log
L)^{\beta_{1}},9Q_{0}}.$

This estimate combined with Hölder’s inequality for any $t\geq 1$ yields

(4.1) $\displaystyle\int_{Q_{0}\backslash E}L_{1}(x)|g(x)|dx$
$\displaystyle\leq D^{p}A_{0}^{p}\|f\|^{p}_{L(\log
L)^{\beta_{1}},9Q_{0}}{\langle|g|\rangle}_{1,Q_{0}}|Q_{0}|$ $\displaystyle\leq
D^{p}A_{0}^{p}\|f\|^{p}_{L(\log
L)^{\beta_{1}},9Q_{0}}{\langle|g|\rangle}_{t,Q_{0}}|Q_{0}|.$

For each $l$, using the fact that $P_{l}\cap E^{c}\neq\emptyset,$ we observe
that there exists some $x_{l}\in P_{l}\cap E^{c}$ such that

$\inf_{\xi\in
P_{l}}\mathscr{M}_{T_{1}T_{2},r^{\prime};Q_{0}}^{*}f(\xi)\leq\mathscr{M}_{T_{1}T_{2},r^{\prime};Q_{0}}^{*}f(x_{l}).$

Combining this inequality and by the Hölder’s inequality with
$\frac{r^{\prime}}{p}\geq r^{\prime}>1$, it gives

(4.2) $\displaystyle\sum_{l}\int_{P_{l}}L_{2,l}^{p}(x)|g(x)|dx$
$\displaystyle\lesssim\sum_{l}\left(\int_{P_{l}}L_{2,l}^{r^{\prime}}(x)dx\right)^{\frac{p}{r^{\prime}}}\left(\int_{P_{l}}|g(x)|^{(\frac{r^{\prime}}{p})^{\prime}}dx\right)^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle=\sum_{l}\left(\frac{1}{|P_{l}|}\int_{P_{l}}L_{2,l}^{r^{\prime}}(x)dx\right)^{\frac{p}{r^{\prime}}}|P_{l}|^{\frac{p}{r^{\prime}}}\left(\int_{P_{l}}|g(x)|^{(\frac{r^{\prime}}{p})^{\prime}}dx\right)^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle\leq\sum_{l}\left(\inf_{\xi\in
P_{l}}\mathscr{M}_{T_{1}T_{2},r^{\prime};Q_{0}}^{*}f(\xi)\right)^{p}|P_{l}|^{\frac{p}{r^{\prime}}}\left(\int_{P_{l}}|g(x)|^{(\frac{r^{\prime}}{p})^{\prime}}dx\right)^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle\leq D^{p}A_{1}^{p}\|f\|^{p}_{L(\log
L)^{\beta_{1}},9Q_{0}}\sum_{l}|P_{l}|^{\frac{p}{r^{\prime}}}\left(\int_{P_{l}}|g(x)|^{(\frac{r^{\prime}}{p})^{\prime}}dx\right)^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle\leq D^{p}A_{1}^{p}\|f\|^{p}_{L(\log
L)^{\beta_{1}},9Q_{0}}\left(\sum_{l}|P_{l}|\right)^{\frac{p}{r^{\prime}}}\left(\sum_{l}\int_{P_{l}}|g(x)|^{(\frac{r^{\prime}}{p})^{\prime}}dx\right)^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle\leq A_{1}^{p}\|f\|^{p}_{L(\log
L)^{\beta_{1}},9Q_{0}}\left|Q_{0}\right|^{1-\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}\left(\int_{Q_{0}}|g(x)|^{(\frac{r^{\prime}}{p})^{\prime}}dx\right)^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle=A_{1}^{p}\|f\|^{p}_{L(\log
L)^{\beta_{1}},9Q_{0}}\langle|g|\rangle_{(\frac{r^{\prime}}{p})^{\prime},Q_{0}}\left|Q_{0}\right|.$

Combining this bounds with (4.1), we see that our claim holds with
$t=(\frac{r^{\prime}}{p})^{\prime}$.

Let $G_{2}(x)$ be the function defined by

$G_{2}(x):=\sum_{l}T_{1}\left(\chi_{3P_{l}}T_{2}\left(f\chi_{9Q_{0}\backslash
9P_{l}}\right)\right)(x)\chi_{P_{l}}(x).$

For each function $g$, applying the Hölder’s inequality and the
$L^{r^{\prime}}$ boundedness of $T_{1}$ yield

$\displaystyle\int_{\mathbb{R}^{n}}|G_{2}(x)|^{p}|g(x)|dx$
$\displaystyle\leq\sum_{l}\left(\int_{P_{l}}\left|T_{1}\left(\chi_{3P_{l}}T_{2}\left(f\chi_{9Q_{0}\backslash
9P_{l}}\right)\right)(x)\right|^{r^{\prime}}(x)dx\right)^{\frac{p}{r^{\prime}}}\langle|g|\rangle_{(\frac{r^{\prime}}{p})^{\prime},P_{l}}\left|P_{l}\right|^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle\leq
A^{p}\sum_{l}\left(\int_{3P_{l}}\left|T_{2}\left(f\chi_{9Q_{0}\backslash
9P_{l}}\right)(x)\right|^{r^{\prime}}dx\right)^{\frac{p}{r^{\prime}}}\langle|g|\rangle_{(\frac{r^{\prime}}{p})^{\prime},P_{l}}\left|P_{l}\right|^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle\leq
3^{n}A^{p}\sum_{l}\left(\frac{1}{|3P_{l}|}\int_{3P_{l}}\left|T_{2}\left(f\chi_{9Q_{0}\backslash
9P_{l}}\right)(x)\right|^{r^{\prime}}dx\right)^{\frac{p}{r^{\prime}}}\langle|g|\rangle_{(\frac{r^{\prime}}{p})^{\prime},P_{l}}\left|P_{l}\right|.$

The same reasoning as what we have done for $G_{1}$ then gives

(4.3) $\displaystyle\int_{\mathbb{R}^{n}}|G_{2}(x)|^{p}|g(x)|dx$
$\displaystyle\lesssim A^{p}\sum_{l}\left(\inf_{\xi\in
P_{l}}\mathscr{M}_{T_{2},r^{\prime};Q_{0}}^{*}f(\xi)\right)^{p}|P_{l}|^{\frac{p}{r^{\prime}}}\left(\int_{P_{l}}|g(x)|^{(\frac{r^{\prime}}{p})^{\prime}}dx\right)^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle\lesssim A^{p}A_{2}^{p}\|f\|^{p}_{L(\log
L)^{\beta_{2}},9Q_{0}}\sum_{l}|P_{l}|^{\frac{p}{r^{\prime}}}\left(\int_{P_{l}}|g(x)|^{(\frac{r^{\prime}}{p})^{\prime}}dx\right)^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle\leq A^{p}A_{2}^{p}\|f\|^{p}_{L(\log
L)^{\beta_{2}},9Q_{0}}\left(\sum_{l}|P_{l}|\right)^{\frac{p}{r^{\prime}}}\left(\sum_{l}\int_{P_{l}}|g(x)|^{(\frac{r^{\prime}}{p})^{\prime}}dx\right)^{\frac{1}{(\frac{r^{\prime}}{p})^{\prime}}}$
$\displaystyle\leq A^{p}A_{2}^{p}\|f\|^{p}_{L(\log
L)^{\beta_{2}},9Q_{0}}\langle|g|\rangle_{(\frac{r^{\prime}}{p})^{\prime},Q_{0}}\left|Q_{0}\right|.$

We note that at each point $x\in\mathbb{R}^{n},$

$T_{1}T_{2}\left(f\chi_{9Q_{0}}\right)(x)\chi_{Q_{0}}(x)=G_{1}(x)+G_{2}(x)+\sum_{l}T_{1}T_{2}\left(f\chi_{9P_{l}}\right)(x)\chi_{P_{l}}(x).$

Observe that the last term on the right-hand side is consistent with the form
on the left-hand side, so we can iterate with
$T_{1}T_{2}\left(f\chi_{9P_{l}}\right)(x)\chi_{P_{l}}(x)$ instead of
$T_{1}T_{2}\left(f\chi_{9Q_{0}}\right)(x)\chi_{Q_{0}}(x),$ and so on. For
fixed $j_{1},\ldots,j_{m-1}\in\mathbb{Z}^{+}$, let $\\{Q_{0}^{j_{1}\ldots
j_{m-1}j_{m}}\\}_{j_{m}}$ be the cubes obtained at the $m$-th stage of the
decomposition process to the cube $Q_{0}^{j_{1}\ldots j_{m-1}}$, where
$\\{Q_{0}^{j_{1}}\\}=\\{P_{j}\\}$ . For each fixed $j_{1}\ldots,j_{m}$, define
the functions $G_{Q_{0},1}^{j_{1}\ldots j_{m}}f$ and $G_{Q_{0},2}^{j_{1}\ldots
j_{m}}f$ by

$G_{Q_{0},1}^{j_{1}\ldots j_{m}}f(x)=T_{1}\left(\chi_{\mathbb{R}^{n}\backslash
3Q_{0}^{j_{1}\ldots j_{m}}}T_{2}\left(f\chi_{9Q_{0}^{j_{1}\ldots
j_{m-1}}\backslash 9Q_{0}^{j_{1}\ldots
j_{m}}}\right)\right)(x)\chi_{Q_{0}^{j_{1}\ldots j_{m}}}(x)$

and

$G_{Q_{0},2}^{j_{1}\ldots j_{m}}f(x)=T_{1}\left(\chi_{3Q_{0}^{j_{1}\ldots
j_{m}}}T_{2}\left(f\chi_{9Q_{0}^{j_{1}\ldots j_{m-1}}\backslash
9Q_{0}^{j_{1}\ldots j_{m}}}\right)\right)(x)\chi_{Q_{0}^{j_{1}\ldots
j_{m}}}(x),$

respectively.
Let
$\mathcal{F}=\left\\{Q_{0}\right\\}\cup_{m=1}^{\infty}\cup_{j_{1},\ldots,j_{m}}\left\\{Q_{0}^{j_{1}\ldots
j_{m}}\right\\}$. It is easy to check that
$\mathcal{F}\subset\mathcal{D}\left(Q_{0}\right)$ is a $\frac{1}{2}$-sparse
family with $\sum_{l}\left|P_{l}\right|\leq\frac{1}{2}\left|Q_{0}\right|$.
Then

(4.4) $\displaystyle G_{Q_{0},1}(x)=T_{1}$ $\displaystyle
T_{2}\left(f\chi_{9Q_{0}}\right)\chi_{Q_{0}\backslash\cup_{j_{1}}Q_{0}^{j_{1}}}(x)$
$\displaystyle+\sum_{m=1}^{\infty}\sum_{j_{1},\ldots,j_{m}}T_{1}T_{2}\left(f\chi_{9Q_{0}^{j_{1}\ldots
j_{m}}}\right)\chi_{Q_{0}^{j_{1}\ldots
j_{m}}\backslash\cup_{j_{m+1}}Q_{0}^{j_{1}\ldots j_{m+1}}}(x)$
$\displaystyle+\sum_{m=1}^{\infty}\sum_{j_{1},\ldots,j_{m}}G_{Q_{0},1}^{j_{1}\ldots
j_{m}}f(x)\chi_{Q_{0}^{j_{1}\ldots j_{m}}}(x).$

Similar as what we have done for $G_{2}$, we may define function $G_{Q_{0},2}$
by

$G_{Q_{0},2}(x)=\sum_{m=1}^{\infty}\sum_{j_{1}\ldots
j_{m}}G_{Q_{0},2}^{j_{1}\ldots j_{m}}f(x)\chi_{Q_{0}^{j_{1}\ldots j_{m}}}(x).$

Then for a. e. $x\in\mathbb{R}^{n}$,

$T_{1}T_{2}\left(f\chi_{9Q_{0}}\right)(x)\chi_{Q_{0}}(x)=G_{Q_{0},1}(x)+G_{Q_{0},2}(x).$

We are now ready to combine all our ingredients to finish the proof. In fact,
by applying (4.2), (4.3) and (4.1) with $t=(\frac{r^{\prime}}{p})^{\prime},$
we obtain

(4.5)
$\int_{\mathbb{R}^{n}}|G_{Q_{0},1}(x)|^{p}|g(x)|dx\lesssim\left(A_{0}^{p}+A_{1}^{p}\right)\sum_{Q\in\mathcal{F}}\|f\|^{p}_{L(\log
L)^{\beta_{1}},9Q}\langle|g|\rangle_{(\frac{r^{\prime}}{p})^{\prime},Q}\left|Q\right|,$
(4.6)
$\int_{\mathbb{R}^{n}}|G_{Q_{0},2}(x)|^{p}|g(x)|dx\lesssim\left(A^{p}A_{2}^{p}\right)\sum_{Q\in\mathcal{F}}\|f\|^{p}_{L(\log
L)^{\beta_{2}},9Q}\langle|g|\rangle_{(\frac{r^{\prime}}{p})^{\prime},Q}\left|Q\right|.$

Observe that $\bigcup_{l}Q_{l}=\mathbb{R}^{n}$, where the cubes $Q_{l}$’s have
disjoint interiors and $\operatorname{supp}f\subset 9Q_{j}$ for each $l$. To
see this, we begin by taking a cube $Q_{0}$ such that
$\operatorname{supp}f\subset Q_{0}$. And cover $9Q_{0}\backslash Q_{0}$ by
$9^{n}-1$ congruent cubes $Q_{l}$. For every $l,Q_{0}\subset 9Q_{l}$. We
continue to do the same way for $27Q_{0}\backslash 9Q_{0}$ and so on. It’s
easy to check that the union of the cubes $Q_{l}$ of this process, including
$Q_{0}$, satisfies our requirement. Applying to each $Q_{l}$, we have that the
above estimates (4.5) and (4.6) hold for $\mathcal{F}_{l}.$ Moreover, for a.e.
$x\in\mathbb{R}^{n}$,

(4.7)
$T_{1}T_{2}f(x)=\sum_{l}G_{Q_{l},1}f(x)+\sum_{l}G_{Q_{l},2}f(x)=:H_{1}f(x)+H_{2}f(x).$

Let $\mathcal{S}$ denote $\left\\{9Q:Q\in\bigcup_{l}\mathcal{F}_{l}\right\\}.$
From the definitions of $\mathcal{F}_{l}$, it is easy to check that
$\mathcal{S}$ is a $\frac{1}{2\cdot 9^{n}}$-sparse family. This fact, together
with (4.7) gives us the desired results. ∎

Applying Lemma 4.1 to the rough singular integral operators $T_{\Omega_{1}}$
and $T_{\Omega_{2}}$, we can complete the proof of Theorem 1.4.

###### Proof of Theorem 1.4.

Let $T_{1},T_{2}$ be two linear operators satisfying the conditions in Lemma
4.1. Note that $\mathbb{X}$ is a RIQBFS with $p$-convex, which implies that
$\mathbb{Y}=\mathbb{X}^{\frac{1}{p}}$ is a RIBFS. Then Lemma 4.1 tells us that
for a.e.$x\in\mathbb{R}^{n},$ $T_{1}T_{2}f(x)=H_{1}(x)+H_{2}(x).$ Therefore,

(4.8) $\displaystyle\left\|T_{1}T_{2}f\right\|_{\mathbb{X}(w)}$
$\displaystyle\lesssim\left\|H_{1}\right\|_{\mathbb{X}(w)}+\left\|H_{2}\right\|_{\mathbb{X}(w)}$
$\displaystyle\backsimeq\sum_{i=1}^{2}\sup_{\left\|g\right\|_{\mathbb{Y}^{\prime}(w)}\leq
1}\left(\int_{\mathbb{R}^{n}}|H_{i}(x)|^{p}g(x)w(x)dx\right)^{\frac{1}{p}}$
$\displaystyle=:\sup_{\left\|g\right\|_{\mathbb{Y}^{\prime}(w)}\leq
1}\left(\mathcal{L}_{1}(g)\right)^{\frac{1}{p}}+\sup_{\left\|g\right\|_{\mathbb{Y}^{\prime}(w)}\leq
1}\left(\mathcal{L}_{2}(g)\right)^{\frac{1}{p}},$

where
$\mathcal{L}_{i}(g)=\int_{\mathbb{R}^{n}}|H_{i}(x)|^{p}g(x)w(x)dx,i=1,2.$

Consider first the estimate of $\mathcal{L}_{1}(g).$ By Lemma 4.1, we have

$\displaystyle\mathcal{L}_{1}(g)$
$\displaystyle\lesssim(A_{0}^{p}+A_{1}^{p})\mathcal{A}^{(p,1)}_{\mathcal{S};L(\log
L)^{\beta_{1}},L^{(r^{{}^{\prime}}/p)^{{}^{\prime}}}}(f,gw)$
$\displaystyle=(A_{0}^{p}+A_{1}^{p})\sum_{Q\in\mathcal{S}}\left\|f\right\|_{L(\log
L)^{\beta_{1}},Q}^{p}\langle|gw|\rangle_{(\frac{r^{\prime}}{p})^{\prime},Q}|Q|.$

Observe that $(\frac{r^{\prime}}{p})^{\prime}\leq r$ for $0<p\leq 1,$ and then
a direct calculation shows that

$|Q|\langle|gw|\rangle_{(\frac{r^{\prime}}{p})^{\prime},Q}\leq\langle|gw|\rangle_{r,Q}|Q|\leq
w(Q)\cdot g_{Q,w}^{rs},$

where the definitions of $s$ and $g_{Q,w}^{rs}$ are as the same as in the
proof of Theorem 1.1.
Hence, by the Carleson embedding theorem,

$\displaystyle\mathcal{L}_{1}(g)$
$\displaystyle\lesssim(A_{0}^{p}+A_{1}^{p})\sum_{Q\in\mathcal{S}}\left\|f\right\|_{L(\log
L)^{\beta_{1}},Q}^{p}w(Q)g_{Q,w}^{rs}$
$\displaystyle\lesssim(A_{0}^{p}+A_{1}^{p})[w]_{A_{\infty}}\int_{\mathbb{R}^{n}}\left(M_{L(\log
L)^{\beta_{1}}}f(x)\right)^{p}M_{w,2s}^{\mathcal{D}}g(x)w(x)dx$
$\displaystyle\lesssim(A_{0}^{p}+A_{1}^{p})[w]_{A_{\infty}}\int_{\mathbb{R}^{n}}\left(M^{\beta_{1}+1}f(x)\right)^{p}M_{w,2s}^{\mathcal{D}}g(x)w(x)dx,$

which, together with the generalized Hölder’s inequality, implies that

$\displaystyle\mathcal{L}_{1}(g)$
$\displaystyle\lesssim(A_{0}^{p}+A_{1}^{p})[w]_{A_{\infty}}\left\|(M^{\beta_{1}+1}f)^{p}\right\|_{\mathbb{Y}(w)}\left\|M_{w,2s}^{\mathcal{D}}g\right\|_{\mathbb{Y}^{\prime}(w)}$
$\displaystyle=(A_{0}^{p}+A_{1}^{p})[w]_{A_{\infty}}\left\|M^{\beta_{1}+1}f\right\|_{\mathbb{X}(w)}^{p}\left\|M_{w}^{\mathcal{D}}(g^{2s})\right\|_{\mathbb{Y}^{\prime\frac{1}{2s}}(w)}^{\frac{1}{2s}}$
$\displaystyle\lesssim(A_{0}^{p}+A_{1}^{p})[w]_{A_{\infty}}[w]_{A_{p_{\mathbb{X}}}}^{\frac{\beta_{1}+1}{p_{\mathbb{X}}}p}\left\|f\right\|^{p}_{\mathbb{X}(w)},$

where the last inequality follows from Lemma 3.1 and the fact
$p_{\mathbb{Y}^{\prime\frac{1}{2s}}}=\frac{p_{\mathbb{Y}^{\prime}}}{2s}=\frac{1}{2s}\frac{q_{\mathbb{X}}}{q_{\mathbb{X}}-p}>1.$
Therefore

(4.9)
$\left\|H_{1}\right\|_{\mathbb{X}(w)}\lesssim(A_{0}^{p}+A_{1}^{p})^{\frac{1}{p}}[w]_{A_{\infty}}^{\frac{1}{p}}[w]_{A_{p_{\mathbb{X}}}}^{\frac{\beta_{1}+1}{p_{\mathbb{X}}}}\left\|f\right\|_{\mathbb{X}(w)}.$

Now we turn to the proof of $\left\|H_{2}\right\|_{\mathbb{X}(w)}.$ The same
reasoning as what we have done for $\mathcal{L}_{1}(g)$ yields that

$\mathcal{L}_{2}(g)=\int_{\mathbb{R}^{n}}|H_{2}(x)|^{p}|g(x)|w(x)dx\lesssim
A^{p}A_{2}^{p}[w]_{A_{\infty}}[w]_{A_{p_{\mathbb{X}}}}^{\frac{\beta_{2}+1}{p_{\mathbb{X}}}p}\left\|f\right\|_{\mathbb{X}(w)}^{p}\left\|g\right\|_{\mathbb{Y}^{\prime}(w)}.$

By taking the supermum over $\|g\|_{\mathbb{Y}^{\prime}(w)}\leq 1,$ we obtain

(4.10) $\left\|H_{2}\right\|_{\mathbb{X}(w)}\lesssim
AA_{2}[w]_{A_{\infty}}^{\frac{1}{p}}[w]_{A_{p_{\mathbb{X}}}}^{\frac{\beta_{2}+1}{p_{\mathbb{X}}}}\left\|f\right\|_{\mathbb{X}(w)}.$

Combining the above estimates (4.8), (4.9) and (4.10), we conclude that

$\left\|T_{1}T_{2}f\right\|_{\mathbb{X}(w)}\lesssim\left[(A_{0}^{p}+A_{1}^{p})^{\frac{1}{p}}+AA_{2}\right][w]_{A_{\infty}}^{\frac{1}{p}}\left([w]_{A_{p_{\mathbb{X}}}}^{\frac{\beta_{1}+1}{p_{\mathbb{X}}}}+[w]_{A_{p_{\mathbb{X}}}}^{\frac{\beta_{2}+1}{p_{\mathbb{X}}}}\right)\left\|f\right\|_{\mathbb{X}(w)}.$

Finally, we set
$T_{1}=T_{\Omega_{1}},T_{2}=T_{\Omega_{2}},A_{0}=1,A_{1}=A=A_{2}=r^{\prime},\beta_{1}=1,\beta_{2}=0.$
This together with the proof of Corollary 5.1 in [29] yields

$\left\|T_{\Omega_{1}}T_{\Omega_{2}}f\right\|_{\mathbb{X}(w)}\lesssim\left([w]_{A_{\infty}}^{1+\frac{1}{p}}+[w]_{A_{\infty}}^{2+\frac{1}{p}}\right)\left([w]_{A_{p_{\mathbb{X}}}}^{\frac{1}{p_{\mathbb{X}}}}+[w]_{A_{p_{\mathbb{X}}}}^{\frac{2}{p_{\mathbb{X}}}}\right)\left\|f\right\|_{\mathbb{X}(w)}.$

This finishes the proof of Theorem 1.4. ∎

##  5\.  Applications

This section will be devoted to give an application of Theorem 1.1. We
consider the boundedness of certain non-standard Calderón-Zygmund operators
with rough kernels in RIBFS $\mathbb{X}$. For fixed $n\geq 2$, let $\Omega$ be
a function with homogeneous of degree zero, integrable on the unit sphere
$\mathbb{S}^{n-1}$ and satisfy the vanishing moment condition that for all
$1\leq j\leq n$,

(5.1) $\int_{\mathbb{S}^{n-1}}\Omega\left(x\right)x_{j}d\sigma(x)=0,$

where $x_{j}(1\leq j\leq n)$ denote the $j$-th variable of
$x\in\mathbb{R}^{n}$. Note that the vanishing condition here is different from
(1.1).

Let $A$ be a function on $\mathbb{R}^{n}$ whose derivatives of order one in
$\mathrm{BMO}\left(\mathbb{R}^{n}\right),$ namely, $\nabla A\in\mathrm{BMO}$.
Then we can define the non-standard rough Calderón-Zygmund operator
$T_{\Omega,A}$ by

(5.2)
$T_{\Omega,A}f(x)=\text{p.v.}\int_{\mathbb{R}^{n}}\frac{\Omega(x-y)}{|x-y|^{n+1}}\left(A(x)-A(y)-\nabla
A(y)\cdot(x-y)\right)f(y)dy.$

The dual operator of $T_{\Omega,A}$ has the following form

$\widetilde{T}_{\Omega,A}f(x)=\text{p.v.}\int_{\mathbb{R}^{n}}\frac{\Omega(x-y)}{|x-y|^{n+1}}(A(x)-A(y)-\nabla
A(x)\cdot(x-y))f(y)dy.$

The operator ${T}_{\Omega,A}$ is closely related to the Calderón commutator,
of interest in PDE, and was first studied by Cohen [13]. An interesting aspect
of this operator is that it may not satisfy the classical standard kernel
condition, even if the kernel $\Omega$ is a smooth kernel. This is also the
main reason why people call it the non-standard singular integral operator. We
refer the reader to [13, 31, 28] and their references for more details on this
topic. It is worth mentioning that Hu et al. [30] recently got the endpoint
$L\log L$ type estimate and the $L^{p}$ boundedness of ${T}_{\Omega,A}$ with
$\Omega\in L(\log L)^{2}(\mathbb{S}^{n-1}).$ In addition, they also obtained
the following results:

######  Theorem C ([30]).

Let $\Omega\in L^{\infty}\left(\mathbb{S}^{n-1}\right)$ be homogeneous of
degree zero, satisfy the vanishing condition (5.1), and $A$ be a function on
$\mathbb{R}^{n}$ with derivatives of order one in
$\mathrm{BMO}\left(\mathbb{R}^{n}\right)$. Then for $p\in(1,\infty)$ and $w\in
A_{p}\left(\mathbb{R}^{n}\right)$, the following weighted norm inequality
holds

$\left\|T_{\Omega,A}f\right\|_{L^{p}\left(\mathbb{R}^{n},w\right)}\lesssim[w]_{A_{p}}^{\frac{1}{p}}\left([w]_{A_{\infty}}^{\frac{1}{p}}+[\sigma]_{A_{\infty}}^{\frac{1}{p}}\right)[\sigma]_{A_{\infty}}\min\left\\{[\sigma]_{A_{\infty}},[w]_{A_{\infty}}\right\\}\|f\|_{L^{p}\left(\mathbb{R}^{n},w\right)}.$

As in [30, Theorem 4.11] and [30, Theorem 5.6], by Theorem 1.1, we obtain

######  Theorem 5.1.

Let $\Omega$ be homogeneous of degree zero, have the vanishing moment (5.1)
and $\Omega\in L^{\infty}\left(\mathbb{S}^{n-1}\right)$, and $A$ be a function
on $\mathbb{R}^{n}$ with derivatives of order one in
$\mathrm{BMO}\left(\mathbb{R}^{n}\right)$. Let $1<r<\infty$ and $\mathbb{X}$
be a RIBFS with $1<p_{\mathbb{X}}\leq q_{\mathbb{X}}<\infty$, then there exist
$q>1$ such that

$\left\|T_{\Omega,A}f\right\|_{\mathbb{X}(w)}\lesssim\left\\{\begin{array}[]{ll}[w]_{A_{\infty}}[w]_{A_{p_{\mathbb{X}}/r}}^{\frac{1}{rq}}\left\|f\right\|_{\mathbb{X}(w)},&\text{
if }r<p_{\mathbb{X}}\leq q_{\mathbb{X}}<\infty,w\in
A_{\frac{p_{\mathbb{X}}}{r}};\\\
{[w]_{A_{\infty}}^{2}}[w]_{A_{p_{\mathbb{X}}}}^{\frac{2}{p_{\mathbb{X}}}}\left\|f\right\|_{\mathbb{X}(w)},&\text{
if }1<p_{\mathbb{X}}\leq q_{\mathbb{X}}<2-\frac{1}{1+p_{\mathbb{X}}},w\in
A_{p_{\mathbb{X}}}.\end{array}\right.$

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