# Multiplication between elements in Martingale Hardy spaces and their dual
spaces

Odysseas Bakas  BCAM - Basque Center for Applied Mathematics, 48009 Bilbao,
Spain<EMAIL_ADDRESS>,  Zhendong Xu  Laboratoire de Mathématiques,
Université de Bourgogne Franche-Comté, 25030 Besançon Cedex, France
<EMAIL_ADDRESS>,  Yujia Zhai  School of Mathematical and
Statistical Sciences, Clemson University, 29634 South Carolina, USA
<EMAIL_ADDRESS>and  Hao Zhang  Laboratoire de Mathématiques, Université de
Bourgogne Franche-Comté, 25030 Besançon Cedex, France  hao.zhang@univ-
fcomte.fr

###### Abstract.

In this paper, we establish continuous bilinear decompositions that arise in
the study of products between elements in martingale Hardy spaces $H^{p}\
(0<p\leqslant 1)$ and functions in their dual spaces. Our decompositions are
based on martingale paraproducts. As a consequence of our work, we also obtain
analogous results for dyadic martingales on spaces of homogeneous type
equipped with a doubling measure.

2020 Mathematics Subject Classification. Primary: 47A07, 60G42, 60G46.
Secondary: 42B30, 46E30, 46F10

Key words: Paraproducts; Martingales; Hardy–Orlicz spaces; Musielak–Orlicz
spaces; Doubling spaces

O. Bakas is partially supported by the projects CEX2021-001142-S,
RYC2018-025477-I, PID2021-122156NB-I00/AEI/10.13039/501100011033 funded by
Agencia Estatal de Investigación and acronym “HAMIP”, Juan de la Cierva
Incorporación IJC2020-043082-I and grant BERC 2022-2025 of the Basque
Government.

Y. Zhai acknowledges partial support from ERC project FAnFArE no. 637510 and
the region Pays de la Loire.

##  1\. Introduction

The pointwise product of a function in the classical Hardy space
$H^{1}(\mathbb{R}^{n})$ and a function of bounded mean oscillation on
$\mathbb{R}^{n}$ need not be in $L^{1}(\mathbb{R}^{n})$; see e.g. §6.2 in
Chapter IV in [24]. However, using Fefferman’s duality theorem [10] and the
fact that the pointwise product of a $BMO$-function and a
$C_{0}^{\infty}$-function is in $BMO(\mathbb{R}^{n})$, Bonami, Iwaniec, Jones
and Zinsmeister defined in [5] the product $f\times g$ of a function $f\in
H^{1}(\mathbb{R}^{n})$ and a function $g\in BMO(\mathbb{R}^{n})$ as a
distribution given by

(1.1) $\langle f\times g,\phi\rangle:=\langle g\cdot\phi,f\rangle,\quad\phi\in
C_{0}^{\infty}(\mathbb{R}^{n}),$

where in the right-hand side of (1.1) the duality between $f\in
H^{1}(\mathbb{R}^{n})$ and $g\cdot\phi\in BMO(\mathbb{R}^{n})$ is employed.
Moreover, it is shown in [5] that for any fixed $f\in H^{1}(\mathbb{R}^{n})$
there exist two linear continuous operators $S_{f}$ from $BMO(\mathbb{R}^{n})$
to $L^{1}(\mathbb{R}^{n})$ and $T_{f}$ from $BMO(\mathbb{R}^{n})$ to a
weighted Hardy–Orlicz space such that

$f\times g=S_{f}(g)+T_{f}(g)$

for all $g\in BMO({}^{n})$; see [5, Theorem 1.6].

In [4], using wavelet analysis, Bonami, Grellier and Ky showed that there
exist two bilinear continuous operators $S$ from $H^{1}(\mathbb{R}^{n})\times
BMO(\mathbb{R}^{n})$ to $L^{1}(\mathbb{R}^{n})$ and $T$ from
$H^{1}(\mathbb{R}^{n})\times BMO(\mathbb{R}^{n})$ to $H^{\log}({}^{n})$ such
that

$f\times g=S(f,g)+T(f,g)$

for all $f\in H^{1}({}^{n})$ and for all $g\in BMO({}^{n})$; see [4, Theorem
1.1]. The Musielak Hardy–Orlicz space $H^{\log}({}^{n})$ is defined as the
class consisting of all distributions $h$ on $\mathbb{R}^{n}$ whose grand
maximal function ${\mathcal{M}}h$ satisfies

$\int_{{}^{n}}\frac{|{\mathcal{M}}h(x)|}{\log(e+|x|)+\log(e+|{\mathcal{M}}h(x)|)}dx<\infty$

and is smaller than the weighted Hardy–Orlicz space appearing in [5]. In fact,
as explained in [5], in view of the results of Nakai and Yabuta [22] on
pointwise multipliers of $BMO({}^{n})$ and duality, the Musielak Hardy–Orlicz
space $H^{\log}({}^{n})$ is optimal.

In addition, continuous bilinear decomposition theorems for products of
elements in $H^{p}({}^{n})$, for $0<p<1$, and their dual spaces were
established in [3].

Using the theory of wavelets on spaces of homogeneous type, which was
developed by Auscher and Hytönen in [1], the aforementioned results have been
extended to spaces of homogeneous type by Liu, Yang and Yuan [15] and Xing,
Yang and Liang [29]. More precisely, in [15] and [29], continuous bilinear
decompositions for products between elements in atomic Hardy spaces
$H^{p}_{\rm at}(\Omega)$ (in the sense of Coifman and Weiss [7]) and their
dual spaces were established in the case where $p\in(\frac{n}{n+1},1]$. Here
$n$ is defined as the dimension of the homogeneous space $\Omega$.

Recently, in [2], a dyadic variant of the aforementioned results of Bonami,
Grellier, and Ky was established; see [2, Theorem 24], which in turn was used
to deduce a periodic version of [4, Theorem 1.1]; see [2, Theorem 28].

Motivated by [2], the first part of this article is concerned with the study
of multiplication between Hardy spaces and their dual spaces for martingales
on a probability space $\Omega$. More specifically, we study multiplications
between functions in the martingale Hardy space $H^{1}(\Omega)$ and its dual
space $BMO(\Omega)$ as stated in our first result, Theorem 1.1. We also
investigate the case $0<p<1$, namely multiplication between elements in
$H^{p}(\Omega)$ and their dual spaces, the so-called martingale Lipschitz
spaces $\Lambda_{1}(\alpha_{p})$ with $\alpha_{p}:=\frac{1}{p}-1$, see Theorem
1.2. Since the dual space $(H^{p}(\Omega))^{*}$ could be $\\{0\\}$ for some
irregular martingales, we shall only consider regular martingales where every
$\sigma-$algebra ${\mathcal{F}}_{k}$ in the corresponding filtration is
generated by countably many atoms.

We would like to mention that Yong Jiao, Guangheng Xie, Dachun Yang, and
Dejian Zhou have independently obtained Theorem 1.1, and derived from it
interesting applications on the boundedness of operators involving commutators
in [14].

######  Theorem 1.1.

Let $(\Omega,{\mathcal{F}},P)$ be a probability space equipped with the
filtration $\\{{\mathcal{F}}_{k}\\}_{k\geq 1}$.

There exist continuous bilinear operators $\Pi_{1}:H^{1}(\Omega)\times
BMO(\Omega)\to L^{1}(\Omega),\ \Pi_{2}:H^{1}(\Omega)\times BMO(\Omega)\to
H^{1}(\Omega)$ and $\Pi_{3}:H^{1}(\Omega)\times BMO(\Omega)\to
H^{\Phi}(\Omega)$ such that

$f\cdot g=\Pi_{1}(f,g)+\Pi_{2}(f,g)+\Pi_{3}(f,g)$

for all $f\in H^{1}(\Omega)$ and $g\in BMO(\Omega)$, where $f\cdot g$ is in
the sense of the pointwise multiplication.

In Theorem 1.1, $H^{\Phi}(\Omega)$ is a martingale Hardy–Orlicz space defined
in terms of the growth function $\Phi(t)$; see Definition 2.15 and (2.3)
below. We shall refer to the terms $\Pi_{2}(f,g)$ and $\Pi_{3}(f,g)$ as the
martingale paraproducts.

Theorem 1.1 can be regarded as an extension of [2, Theorem 24] to the general
case of martingales.

For $0<p<1$, if $f\in H^{p}(\Omega)$, $g\in\Lambda_{1}(\alpha_{p})$ and
$f_{0}=g_{0}=0$, then their product can be regarded as a continuous linear
functional on $L^{\infty}(\Omega)\cap\Lambda_{1}(\alpha_{p})$. To be more
precise, for any $h\in L^{\infty}(\Omega)\cap\Lambda_{1}(\alpha_{p})$, define

$\langle f\times g,h\rangle:=\langle h\cdot g,f\rangle,$

where in the right-hand side the duality between $H^{p}(\Omega)$ and
$\Lambda_{1}(\alpha_{p})$ is invoked. Note that $h\cdot g$ belongs to
$\Lambda_{1}(\alpha_{p})$ since $h$ is a pointwise multiplier on
$\Lambda_{1}(\alpha_{p})$ (see [21]).

Our following theorem establishes a continuous bilinear decomposition for
products between elements in $H^{p}(\Omega)$ and functions in the dual space
$\Lambda_{1}(\alpha_{p})$ when $0<p<1$.

######  Theorem 1.2.

Let $(\Omega,{\mathcal{F}},P)$ be a probability space equipped with the
filtration $\\{{\mathcal{F}}_{k}\\}_{k\geq 1}$, where ${\mathcal{F}}_{k}$ is
generated by countably many atoms for any $k\geq 1$.

If $H^{p}(\Omega)\ (0<p<1)$ are martingale Hardy spaces, then there exist
continuous bilinear operators
$\Pi_{1}:H^{p}(\Omega)\times\Lambda_{1}(\alpha_{p})\to L^{1}(\Omega),\
\Pi_{2}:H^{p}(\Omega)\times\Lambda_{1}(\alpha_{p})\to H^{1}(\Omega)$ and
$\Pi_{3}:H^{p}(\Omega)\times\Lambda_{1}(\alpha_{p})\to H^{p}(\Omega)$ such
that

$f\times g=\Pi_{1}(f,g)+\Pi_{2}(f,g)+\Pi_{3}(f,g)$

for all $f\in H^{p}(\Omega)$ and $g\in\Lambda_{1}(\alpha_{p})$.

In the second part of this paper, we study analogues of Theorems 1.1 and 1.2
for the case of dyadic martingales on spaces of homogeneous type. Such
martingales were first constructed in [13]. We investigate the corresponding
martingale Hardy spaces and extend Mei’s results in [17] to this general
setting. Compared with the probability setting, the case of spaces of
homogeneous type is more difficult to deal with since backward martingales
arise, and the underlying measures on homogeneous spaces may be infinite.

The present paper is organized as follows. In section 2, we set down notation
and give some background on martingale Hardy–Orlicz spaces. In section 3, we
prove Theorem 1.1. In section 4, we present a characterization of martingale
Lipschitz spaces $\Lambda_{1}(\alpha_{p})$, which is of independent interest
(see Theorem 4.4 and Remark 4.5 below), and then we show Theorem 1.2. The
remaining sections are concerned with spaces of homogeneous type. For the
convenience of the reader, in section 5, we recall some definitions and facts
regarding Hardy spaces and Lipschitz spaces on spaces of homogeneous type in
the sense of Coifman and Weiss [7]. In section 6, we give some new proofs of
results in [7] based on martingale methods and the existence of dyadic
martingales on homogeneous spaces. In section 7, we establish analogues of
Theorems 1.1 and 1.2 for dyadic martingales on spaces of homogeneous type; see
Theorem 7.7 below. In the last section, we apply Theorem 7.7 to obtain a
decomposition of products of functions in Hardy spaces and their dual spaces
on spaces of homogeneous type.

##  2\. Notation and Background

In this section, we provide some notation and background that will be used in
this paper.

###  2.1. Notation

In several parts of this paper, we consider sums and intersections of quasi-
normed vector spaces. For the convenience of the reader we recall these
notions below.

######  Definition 2.1.

Let $(X_{1},\|\cdot\|_{X_{1}}),(X_{2},\|\cdot\|_{X_{2}})$ be two quasi-normed
vector spaces and let $X$ be a topological vector space $X$ such that
$X_{1},X_{2}\subset X$ continuously.

  1. (1)

$(X_{1}\cap X_{2},\|\cdot\|_{X_{1}\cap x_{2}})$ is the intersection of $X_{1}$
and $X_{2}$, where

$\|x\|_{X_{1}\cap X_{2}}:=\max\\{\|x\|_{X_{1}},\|x\|_{X_{2}}\\}$

for all $x\in X_{1}\cap X_{2}$;

  2. (2)

$(X_{1}+X_{2},\|\cdot\|_{X_{1}+X_{2}})$ is the sum of $X_{1}$ and $X_{2}$,
where

$\|x\|_{X_{1}+X_{2}}:=\inf\\{\|x_{1}\|_{X_{1}}+\|x_{2}\|_{X_{2}}:\
x=x_{1}+x_{2},\ x_{1}\in X_{1},\ x_{2}\in X_{2}\\}$

for all $x\in X_{1}+X_{2}$.

For convenience, the sum $X_{1}+X_{2}+\cdots+X_{n}$ and the intersection
$X_{1}\cap X_{2}+\cdots\cap X_{n}$ will also be denoted by
$\sum\limits_{k=1}^{n}X_{k}$ and $\bigcap_{k=1}^{n}X_{k}$, respectively.

Note that $(X_{1}\cap X_{2},\|\cdot\|_{X_{1}\cap X_{2}})$ and
$(X_{1}+X_{2},\|\cdot\|_{X_{1}+X_{2}})$ are both quasi-normed vector spaces.
Moreover, if $(X_{1},\|\cdot\|_{X_{1}})$ and $(X_{2},\|\cdot\|_{X_{2}})$ are
Banach spaces, then $(X_{1}\cap X_{2},\|\cdot\|_{X_{1}\cap X_{2}})$ and
$(X_{1}+X_{2},\|\cdot\|_{X_{1}+X_{2}})$ are both Banach spaces.

In this article we shall use the following standard notation: $A\lesssim B$
(resp. $A\lesssim_{p}B$) means that $A\leq CB$ (resp. $A\leq C_{p}B$) for some
absolute positive constant $C$ (resp. a positive constant $C_{p}$ depending
only on a parameter $p$). If $A\lesssim B$ and $B\lesssim A$ (resp.
$A\lesssim_{p}B$ and $B\lesssim_{p}A$), we write $A\approx B$ (resp.
$A\approx_{p}B$).

Throughout the paper, the terms “homogeneous spaces” and “spaces of
homogeneous type” will be used interchangeably.

###  2.2. Musielak–Orlicz-type spaces

We shall first recall some definitions and properties of Orlicz-type spaces
and Musielak–Orlicz-type spaces. In what follows, $(\Omega,{\mathcal{F}},\mu)$
denotes a $\sigma$-finite measure space.

A function $\Phi:[0,\infty)\to[0,\infty)$ is called an Orlicz function if it
is strictly positive on $(0,\infty)$, non-decreasing, unbounded and
$\Phi(0)=0$. A measurable function $\Psi:\Omega\times[0,\infty)\to[0,\infty)$
is called a Musielak–Orlicz function if for all $x\in\Omega$, $\Psi(x,\cdot)$
is an Orlicz function.

The Musielak–Orlicz-type space $L^{\Psi}(\Omega)$ is the set consisting of all
measurable functions $f$ on $\Omega$ such that

$\int_{\Omega}\Psi(x,\lambda^{-1}|f(x)|)d\mu<\infty$

for some $\lambda>0$. We equip $L^{\Psi}(\Omega)$ with the Luxembourg quasi-
norm

$\|f\|_{L^{\Psi}(\Omega)}:=\inf\left\\{\lambda>0:\int_{\Omega}\Psi(x,\lambda^{-1}|f(x)|)d\mu\leqslant
1\right\\},\quad f\in L^{\Psi}(\Omega).$

Let $p\in\mathbb{R}$. A Musielak–Orlicz function is said to be of uniformly
lower type (respectively, upper type) $p$ if there exists a positive constant
$C$ such that

$\Psi(x,st)\leqslant Cs^{p}\Psi(x,t)$

for all $x\in\Omega$, $t\geqslant 0$ and $s\in(0,1)$ (respectively,
$s\in[1,\infty)$). In particular, if $\Psi$ is of uniformly lower type $p$
with $0<p<1$ and of uniformly upper type $1$ then

(2.1) $\Psi(x,ct)\approx_{c}\Psi(x,t)\quad\text{for all }c>0.$

In the sequel, $\Psi(x,t)$ is always assumed to be of uniformly lower type $p$
with $0<p<1$ and of uniformly upper type $1$, and to be continuous in the $t$
variable. For more information on Musielak–Orlicz spaces, we refer the reader
to [4] and [30].

###  2.3. Martingales

Let $(\Omega,{\mathcal{F}},P)$ be a fixed probability space. Given a
filtration which consists of a sequence of $\sigma$-algebras

${\mathcal{F}}_{1}\subset\dots\subset{\mathcal{F}}_{k}\subset\dots\subset{\mathcal{F}}$

such that $\sigma(\cup_{k=1}^{\infty}{\mathcal{F}}_{k})={\mathcal{F}}$, for a
random variable $f\in L^{1}(\Omega,{\mathcal{F}},P)$ and $k\in\mathbb{N}_{+}$,
we set

$f_{k}=\mathbb{E}\left(f\mid{{\mathcal{F}}}_{k}\right),\quad
d_{k}f=f_{k}-f_{k-1},$

where we adopt the convention that $f_{0}=0$. We shall also denote $f_{k}$ by
${\mathbb{E}}_{k}(f)$. The sequence $\\{f_{k}\\}_{k\geq 0}$ is called the
martingale of $f$, and $\\{d_{k}f\\}_{k\geq 1}$ is called the martingale
difference of $f$. If $f$ and $\\{f_{k}\\}_{k\geq 0}$ are as above, we shall
also write $f=\\{f_{k}\\}_{k\geq 0}$. To simplify notation, we write
$L^{p}(\Omega)$ instead of $L^{p}(\Omega,{\mathcal{F}},P)$, $0<p<\infty$.

######  Definition 2.2.

If $f$, $\\{f_{k}\\}_{k\geq 0}$ and $\\{d_{k}f\\}_{k\geq 1}$ are as above, we
define:

  1. (1)

the maximal function

$f^{*}:=\sup\limits_{k\geqslant 0}|f_{k}|;$

  2. (2)

the square function

$S(f):=\left(\sum_{k=1}^{\infty}|d_{k}f|^{2}\right)^{\frac{1}{2}};$

  3. (3)

the conditional square function

$s(f):=\left(\sum_{k=1}^{\infty}{\mathbb{E}}_{k-1}|d_{k}f|^{2}\right)^{\frac{1}{2}}.$

There are several types of martingale Hardy spaces, which are defined in terms
of maximal functions, square functions and conditional square functions.

######  Definition 2.3.

For $1\leq p<\infty$, the martingale Hardy spaces
$h^{p}(\Omega),H^{p}(\Omega),H^{p}_{m}(\Omega)$ are defined as follows

$\displaystyle h^{p}(\Omega)$ $\displaystyle:=\\{f\in
L^{1}(\Omega):\|f\|_{h^{p}}:=\|s(f)\|_{p}<\infty\\},$ $\displaystyle
H^{p}(\Omega)$ $\displaystyle:=\\{f\in
L^{1}(\Omega):\|f\|_{H^{p}}:=\|S(f)\|_{p}<\infty\\},$ $\displaystyle
H^{p}_{m}(\Omega)$ $\displaystyle:=\\{f\in
L^{1}(\Omega):\|f\|_{H^{p}_{m}}:=\|f^{*}\|_{p}<\infty\\},$

respectively.

For $0<p<1$, $h^{p}(\Omega)$ is defined as the completion of the space
$\\{f\in L^{1}(\Omega):\|f\|_{h^{p}}:=\|s(f)\|_{p}<\infty\\}$ with respect to
the norm $\|\cdot\|_{h^{p}}$. Similarly, $H^{p}(\Omega)$ is defined as the
completion of the space $\\{f\in
L^{1}(\Omega):\|f\|_{H^{p}}:=\|S(f)\|_{p}<\infty\\}$ with respect to the norm
$\|\cdot\|_{H^{p}}$, and $H^{p}_{m}(\Omega)$ is defined as the completion of
the space $\\{f\in L^{1}(\Omega):\|f\|_{H_{m}^{p}}:=\|f^{*}\|_{p}<\infty\\}$
with respect to the norm $\|\cdot\|_{H_{m}^{p}}$.

In general, the above three martingale Hardy spaces are different. However,
for $1\leq p<\infty$, $H^{p}(\Omega)=H^{p}_{m}(\Omega)$ (see [6], [9], [26]).

######  Definition 2.4.

(Regular filtration) A filtration is regular if there exists a constant $C>0$
such that for all $k\geq 2,F_{k}\in{\mathcal{F}}_{k}$, there exists a
$G_{k}\in{\mathcal{F}}_{k-1}$ satisfying

$F_{k}\subset G_{k},\quad P(G_{k})\leq C\cdot P(F_{k}).$

In addition, a martingale $f=\\{f_{k}\\}_{k\geq 0}$ with respect to such a
regular filtration is called a regular martingale.

######  Remark 2.5.

Suppose that for a positive random variable $f\in L^{1}(\Omega)$ the
corresponding martingale $\\{f_{k}\\}_{k\geqslant 0}$ is regular. Then for any
$k\geq 2$

$f_{k}\leqslant A\cdot f_{k-1},$

where $A>0$ is a constant that depends only on the constant $C$ of Definition
2.4.

See [16] for more information about regular filtrations and martingales.

######  Remark 2.6.

For regular martingale filtrations,
$H^{p}(\Omega)=h^{p}(\Omega)=H^{p}_{m}(\Omega)$ when $0<p<\infty$. See [27],
[26] and [16] for more information.

An important aspect of martingale Hardy spaces is that they admit atomic
decompositions. The definition of atoms in the martingale setting is given
below.

######  Definition 2.7.

A random variable $a:\Omega\to{\mathbb{C}}$ is called a martingale simple
$(p,q)$-atom ($0<p\leq 1,1\leq q\leqslant\infty$) if there exist
$k\in{\mathbb{N}}$ and $A\in{\mathcal{F}}_{k}$ such that

  1. (1)

${\mathbb{E}}_{k}(a)=0$;

  2. (2)

$\text{supp}(a)\subset A$;

  3. (3)

$\|a\|_{q}\leqslant P(A)^{\frac{1}{q}-\frac{1}{p}}$,

where $\frac{1}{q}:=0$ when $q=\infty$ as convention.

######  Definition 2.8.

We define the martingale atomic Hardy spaces $H_{\rm at}^{p,q}(\Omega)$ for
$0<p<1\leqslant q\leqslant\infty$ or $p=1,\ 1<q\leq\infty$ as follows

$H_{\rm at}^{p,q}(\Omega):=\left\\{f=\sum_{j=0}^{\infty}\lambda_{j}a^{j}\text{
where $a^{j}$ is a simple $(p,q)$-atom and
}\sum_{j=0}^{\infty}|\lambda_{j}|^{p}<\infty\right\\},$

where for $f\in H_{\rm at}^{p,q}(\Omega)$

$\|f\|_{H^{p,q}_{\rm
at}(\Omega)}:=\inf\left\\{\big{(}\sum_{j=0}^{\infty}|\lambda_{j}|^{p}\big{)}^{\frac{1}{p}}:f=\sum_{j=0}^{\infty}\lambda_{j}a^{j},\text{
where $a^{j}$ is a simple $(p,q)$-atom}\right\\}.$

It is well-known that $h^{p}(\Omega)=H_{\rm at}^{p,2}(\Omega)$ when $0<p\leq
1$ (see [27]). In particular, if the martingale filtration is regular, then
$h^{p}(\Omega)=H_{\rm at}^{p,q}(\Omega)$ when $0<p\leq 1$ and $1<q\leq\infty$.
The following result is the atomic decomposition of $H^{1}(\Omega)$, which
follows from the noncommutative result in [23]. In particular, it reveals the
relationship between $H^{1}(\Omega)$ and $h^{1}(\Omega)$ and shows that
$H^{1}(\Omega)\neq h^{1}(\Omega)$ for general martingales.

######  Theorem 2.9.

We have $H^{1}(\Omega)=h^{1}(\Omega)+h^{1}_{d}(\Omega)$, where $h^{1}_{d}$
denotes the diagonal Hardy space of martingale differences

$h^{1}_{d}(\Omega):=\left\\{h\in
L^{1}(\Omega):\|h\|_{h^{1}_{d}(\Omega)}:=\sum_{k=1}^{\infty}\|d_{k}h\|_{1}<\infty\right\\}.$

We shall now introduce the martingale $BMO$ and $bmo$ spaces, which are the
duals of $H^{1}(\Omega)$ and $h^{1}(\Omega)$, respectively (see Theorem 2.13
below).

######  Definition 2.10.

Assume $f,g\in L^{2}(\Omega)$. We say that $f$ is a martingale $BMO$ function
if

$\|f\|_{BMO(\Omega)}:=\sup_{n\geqslant
1}\|{\mathbb{E}}_{n}|f-f_{n-1}|^{2}\|_{\infty}^{\frac{1}{2}}<\infty.$

We say that $g$ is a martingale $bmo$ function if

$\|g\|_{bmo(\Omega)}:=\sup_{n\geqslant
0}\|{\mathbb{E}}_{n}|g-g_{n}|^{2}\|_{\infty}^{\frac{1}{2}}<\infty.$

Denote by $BMO(\Omega)$ and $bmo(\Omega)$ the spaces consisting of all
martingale $BMO$ and $bmo$ functions, respectively.

For regular martingales, $BMO(\Omega)=bmo(\Omega)$. The following result is
the so-called martingale John–Nirenberg inequality and can be found in [12].

######  Theorem 2.11.

There exists a sufficiently small constant $\kappa>0$ such that for any $f\in
BMO(\Omega)$ with $\|f\|_{BMO(\Omega)}\leq\kappa$, we have

${\mathbb{E}}\left(e^{|f|}\right)\leqslant 1.$

######  Remark 2.12.

From the martingale John–Nirenberg inequality, we have for any $1\leq
p<\infty$,

$\|f\|_{BMO(\Omega)}\approx_{p}\sup_{n\geqslant
1}\|{\mathbb{E}}_{n}|f-f_{n-1}|^{p}\|_{\infty}^{\frac{1}{p}}.$

However, the above John–Nirenberg inequality fails for $bmo(\Omega)$ in the
general martingale setting.

For the following duality theorem, see [12], [16], [26].

######  Theorem 2.13.

$(H^{1}(\Omega))^{*}=BMO(\Omega)$ and $\ (h^{1}(\Omega))^{*}=bmo(\Omega)$.

The following proposition, which can be found in [8] and [12], is a
consequence of Theorems 2.9 and 2.13 and it gives a description of the
relationship between $BMO(\Omega)$ and $bmo(\Omega)$. In particular, it
implies that $BMO(\Omega)\subsetneqq bmo(\Omega)$ for general martingales.

######  Proposition 2.14.

Assume $f$ is a martingale $BMO$ function. Then

(2.2) $\|f\|_{BMO(\Omega)}\approx\|f\|_{bmo(\Omega)}+\sup\limits_{k\geq
1}\|d_{k}f\|_{\infty}.$

We end this section with the definition of martingale Musielak–Orlicz Hardy
spaces and the generalized Hölder inequality.

######  Definition 2.15.

The martingale Musielak–Orlicz Hardy space $H^{\Psi}(\Omega)$ (where $\Psi$ is
described in Section 2.2) is the space consisting of all martingales
$f=\\{f_{k}\\}_{k\geq 0}$ such that the square function $S(f)\in
L^{\Psi}(\Omega)$. Moreover, we define the quasi-norm

$\|f\|_{H^{\Psi}(\Omega)}:=\|S(f)\|_{L^{\Psi}(\Omega)}.$

If $\Psi$ is replaced by an Orlicz function $\Phi$, the corresponding
Hardy–Orlicz space $H^{\Phi}(\Omega)$ is defined in an analogous way.

To obtain the generalized Hölder inequality, we shall introduce a particular
Orlicz space $L^{\Phi}(\Omega)$, where

(2.3) $\Phi(t):=\frac{t}{\log(e+t)},\quad t\geq 0.$

Note that $\Phi$ is an Orlicz function of uniformly lower type $p\ (0<p<1)$
and upper type $1$, which guarantees that the vector space $L^{\Phi}(\Omega)$
is a quasi-normed space. Note that $L^{1}(\Omega)\subset L^{\Phi}(\Omega)$.

######  Remark 2.16.

It follows from [19] that if $f=\\{f_{k}\\}_{k\geq 0}$ is a regular
martingale, then the martingale Hardy–Orlicz space $H^{\Phi}(\Omega)$ can also
be characterized by martingale maximal functions and conditional square
functions. For any $f\in H^{\Phi}(\Omega)$ one has

$\|f\|_{H^{\Phi}(\Omega)}=\|S(f)\|_{L^{\Phi}(\Omega)}\approx\|f^{*}\|_{L^{\Phi}(\Omega)}\approx\|s(f)\|_{L^{\Phi}(\Omega)}.$

The following lemma is a variant of [5, Proposition 2.1] in the martingale
setting.

######  Lemma 2.17.

Assume $(\Omega,{\mathcal{F}},P)$ is a probability space, $f\in L^{1}(\Omega)$
and $g\in BMO(\Omega)$. Then $f\cdot g\in L^{\Phi}(\Omega)$ and

(2.4) $\|f\cdot g\|_{L^{\Phi}(\Omega)}\lesssim\|f\|_{1}\|g\|_{BMO(\Omega)}.$

###### Proof.

The proof is similar to the proof of the corresponding Euclidean result and we
shall only outline it here for the convenience of the reader. By [5, Lemma
2.1], one has

(2.5) $\dfrac{st}{M+\log(e+st)}\leq e^{t-M}+s.$

for all $M\geq 0,s\geq 0,t\geq 0$.

When $\|f\|_{1}=0$ or $\|g\|_{BMO(\Omega)}=0$, $(\ref{2.4})$ trivially holds.
Assume $g\in BMO(\Omega)$ with $\|g\|_{BMO(\Omega)}>0$ and $f\in
L^{1}(\Omega)$ with $\|f\|_{1}>0$. Let $\kappa$ be the constant in Theorem
2.11, $M=0$, $t=\frac{\kappa|g(x)|}{\|g\|_{BMO(\Omega)}}$ and
$s=\frac{|f(x)|}{\|f\|_{1}}$. Then by Theorem 2.11 and (2.5), we have

(2.6) $\displaystyle\int_{\Omega}\Phi\left(\frac{|f(x)\cdot
g(x)|}{\kappa^{-1}\|f\|_{1}\|g\|_{BMO(\Omega)}}\right)dP$
$\displaystyle\leq\int_{\Omega}e^{\frac{\kappa|g(x)|}{\|g\|_{BMO(\Omega)}}}dP+\left\|\frac{f}{\|f\|_{1}}\right\|_{1}\leq
2.$

Hence, from (2.1) we conclude

$\|f\cdot
g\|_{L^{\Phi}(\Omega)}\lesssim\kappa^{-1}\|f\|_{1}\|g\|_{BMO(\Omega)},$

which completes the proof of the lemma. ∎

We shall refer to (2.4) as the generalized Hölder inequality.

##  3\. Bilinear decompositions on $H^{1}(\Omega)\times BMO(\Omega)$

In this section we prove Theorem 1.1. Let $(\Omega,{\mathcal{F}},P)$ be a
fixed probability space and let $f\in H^{1}(\Omega),\ g\in BMO(\Omega)$. If we
assume that $f$ and $g$ have finite martingale expansions, then we may write
their pointwise product $f\cdot g$ as follows

(3.1) $f\cdot g=\Pi_{1}(f,g)+\Pi_{2}(f,g)+\Pi_{3}(f,g),$

where

$\Pi_{1}(f,g):=\sum_{k=1}^{\infty}d_{k}fd_{k}g,\quad\Pi_{2}(f,g):=\sum_{k=1}^{\infty}f_{k-1}d_{k}g\quad\text{and}\quad\Pi_{3}(f,g):=\sum_{k=1}^{\infty}g_{k-1}d_{k}f.$

We shall estimate $\Pi_{1}(f,g),\Pi_{2}(f,g)$, $\Pi_{3}(f,g)$ separately. To
do so, we shall make use of the atomic decomposition of $H^{1}(\Omega)$. It
follows from our arguments below that the operators $\Pi_{1}$, $\Pi_{2}$ and
$\Pi_{3}$ are well-defined (in a pointwise sense) on the product space
$H^{1}(\Omega)\times BMO(\Omega)$. Hence, the proof of Theorem will follow
from the boundedness properties of $\Pi_{1}$, $\Pi_{2}$ and $\Pi_{3}$, (3.1)
and a limiting argument.

In §3.4, we present a direct way to deal with $\Pi_{3}(f,g)$, which avoids the
use of the atomic decomposition.

###### Proof of Theorem 1.1.

By Theorem 2.9, there always exist two functions $f^{h}$ and $f^{d}$ such that
$f=f^{h}+f^{d}$, where $f^{h}\in h^{1}(\Omega)$ and $f^{d}\in
h^{1}_{d}(\Omega)$. For any such decomposition of $f$, since $f^{h}\in
h^{1}(\Omega)$, there exist $\left\\{\lambda_{j}\right\\}_{j\geqslant
1}\subset\real$ and simple $(1,2)$-atoms $\left\\{a^{j}\right\\}_{j\geqslant
1}$ such that

(3.2)
$f^{h}=\sum_{j=1}^{\infty}\lambda_{j}a^{j},\quad\|f^{h}\|_{h^{1}(\Omega)}\approx\sum_{j=1}^{\infty}|\lambda_{j}|,$

where we assume $\text{supp}(a^{j})\subset A_{n_{j}}$ and
$A_{n_{j}}\in{\mathcal{F}}_{n_{j}}$ with $P(A_{n_{j}})>0$ for $j\geq 1$. Then

(3.3)
$\Pi_{i}(f,g)=\sum_{j=1}^{\infty}\lambda_{j}\Pi_{i}(a^{j},g)+\Pi_{i}(f^{d},g),\quad
i=1,2,3.$

###  3.1. Estimates for $\Pi_{1}(f^{h},g)$ and $\Pi_{1}(f^{d},g)$

We are going to show that $\Pi_{1}$ is a bounded bilinear operator from
$H^{1}(\Omega)\times BMO(\Omega)$ to $L^{1}(\Omega)$. In fact, the boundedness
of $\Pi_{1}$ follows naturally from the duality between $H^{1}(\Omega)$ and
$BMO(\Omega)$, i.e. Theorem 2.13 (see [12]). For the reader’s convenience, we
give a proof based on atomic decompositions.

We first focus on $\Pi_{1}(f^{h},g)$, which can further be decomposed into
atoms as described in (4.6). It thus suffices to consider

$\Pi_{1}(a^{j},g)=\sum_{k=1}^{\infty}d_{k}a^{j}d_{k}g,$

which can further be localized as $d_{k}a^{j}=1_{A_{n_{j}}}d_{k}a^{j}$ when
$k\geq n_{j}+1$ since $A_{n_{j}}\in{\mathcal{F}}_{n_{j}}$, namely

$\Pi_{1}(a^{j},g)=\sum_{k=n_{j}+1}^{\infty}1_{A_{n_{j}}}d_{k}a^{j}d_{k}g.$

Now, by applying the Cauchy-Schwarz inequality, we derive the estimate

$\displaystyle\|\Pi_{1}(a^{j},g)\|_{1}$
$\displaystyle={\mathbb{E}}\left(\left|\sum_{k=n_{j}+1}^{\infty}1_{A_{n_{j}}}d_{k}a^{j}d_{k}g\right|\right)$
$\displaystyle\leqslant\left[{\mathbb{E}}\left(\sum_{k=n_{j}+1}^{\infty}|d_{k}a^{j}|^{2}\right)\right]^{\frac{1}{2}}\left[{\mathbb{E}}\left(\sum_{k=n_{j}+1}^{\infty}1_{A_{n_{j}}}|d_{k}g|^{2}\right)\right]^{\frac{1}{2}}$
$\displaystyle\leqslant\|a^{j}\|_{2}\left[{\mathbb{E}}{\mathbb{E}}_{n_{j}}\left(\sum_{k=n_{j}+1}^{\infty}1_{A_{n_{j}}}|d_{k}g|^{2}\right)\right]^{\frac{1}{2}}$
$\displaystyle\leqslant
P(A_{n_{j}})^{-\frac{1}{2}}\left[{\mathbb{E}}\left(1_{A_{n_{j}}}{\mathbb{E}}_{n_{j}}\left(\sum_{k=n_{j}+1}^{\infty}|d_{k}g|^{2}\right)\right)\right]^{\frac{1}{2}}$
$\displaystyle\leqslant
P(A_{n_{j}})^{-\frac{1}{2}}\|g\|_{bmo(\Omega)}P(A_{n_{j}})^{\frac{1}{2}}$

where the fourth inequalitiy follows from the definition of the atom. Hence,
we deduce from the definition of the $bmo-$norm that

(3.4) $\|\Pi_{1}(a^{j},g)\|_{1}\leq\|g\|_{bmo(\Omega)}.$

By using (3.4) and (3.3), we have by Theorem 2.14

$\displaystyle\|\Pi_{1}(f,g)\|_{1}$
$\displaystyle\leqslant\sum_{j=1}^{\infty}|\lambda_{j}|\|g\|_{bmo(\Omega)}+\left\|\sum_{k=1}^{\infty}d_{k}f^{d}d_{k}g\right\|_{1}$
$\displaystyle\lesssim\|f^{h}\|_{h^{1}}\|g\|_{bmo(\Omega)}+\left(\sup\limits_{k\geq
1}\|d_{k}g\|_{\infty}\right)\left(\sum_{k=1}^{\infty}\|d_{k}f^{d}\|_{1}\right)$
$\displaystyle\lesssim\left(\|f^{h}\|_{h^{1}(\Omega)}+\|f^{d}\|_{h_{1}^{d}(\Omega)}\right)\|g\|_{BMO(\Omega)}.$

Since the decomposition of $f=f^{h}+f^{d}$ is arbitrary, by Theorem 2.9 we
conclude

(3.5) $\|\Pi_{1}(f,g)\|_{1}\lesssim\|f\|_{H^{1}(\Omega)}\|g\|_{BMO(\Omega)}.$

###  3.2. Estimates for $\Pi_{2}(f^{h},g)$ and $\Pi_{2}(f^{d},g)$

We are going to show that $\Pi_{2}$ is a bounded bilinear operator from
$H^{1}(\Omega)\times BMO(\Omega)$ to $H^{1}(\Omega)$. Arguing as in section
3.1, we perform the localization on each term

$\Pi_{2}(a^{j},g)=\sum_{k=1}^{\infty}a^{j}_{k-1}d_{k}g=\sum_{k=n_{j}+2}^{\infty}1_{A_{n_{j}}}a^{j}_{k-1}d_{k}g.$

It is easy to verify that

$d_{k}(\Pi_{2}(a^{j},g))=a^{j}_{k-1}d_{k}g,\quad k\geq n_{j}+2\ \text{ and }\
d_{k}(\Pi_{2}(a^{j},g))=0,\quad 1\leqslant k\leqslant n_{j}+1.$

We consider the corresponding square function

$\displaystyle S\left(\Pi_{2}(a^{j},g)\right)$
$\displaystyle=\left(\sum_{k=n_{j}+2}^{\infty}\left(|a^{j}_{k-1}|^{2}1_{A_{n_{j}}}|d_{k}g|^{2}\right)\right)^{\frac{1}{2}}$
$\displaystyle\leqslant|(a^{j})^{*}|\left(\sum_{k=n_{j}+2}^{\infty}1_{A_{n_{j}}}\left(|d_{k}g|^{2}\right)\right)^{\frac{1}{2}}.$

Then by invoking the Cauchy-Schwarz inequality, we have that

$\displaystyle\|\Pi_{2}(a^{j},g)\|_{H^{1}(\Omega)}$
$\displaystyle={\mathbb{E}}\left[S\left(\Pi_{2}(a^{j},g)\right)\right]$
$\displaystyle\leqslant\|(a^{j})^{*}\|_{2}\left[{\mathbb{E}}\left(\sum_{k=n_{j}+2}^{\infty}1_{A_{n_{j}}}(|d_{k}g|^{2})\right)\right]^{\frac{1}{2}}$
$\displaystyle\leqslant\|a^{j}\|_{2}\left[{\mathbb{E}}\left(1_{A_{n_{j}}}{\mathbb{E}}_{n_{j}}\left(\sum_{k=n_{j}+2}^{\infty}|d_{k}g|^{2}\right)\right)\right]^{\frac{1}{2}}$
$\displaystyle\leqslant
P(A_{n_{j}})^{-\frac{1}{2}}\|g\|_{BMO(\Omega)}P(A_{n_{j}})^{\frac{1}{2}}$

and hence,

(3.6) $\|\Pi_{2}(a^{j},g)\|_{H^{1}(\Omega)}\leqslant\|g\|_{BMO(\Omega)}.$

Similarly, by Theorem 2.14

$\displaystyle\left\|\Pi_{2}(f^{d},g)\right\|_{H^{1}(\Omega)}$
$\displaystyle={\mathbb{E}}\left[S\left(\Pi_{2}(\sum_{m=1}^{\infty}d_{m}f^{d},g)\right)\right]$
$\displaystyle\leqslant\sum_{m=1}^{\infty}{\mathbb{E}}\left[S\left(\Pi_{2}(d_{m}f^{d},g)\right)\right]$
$\displaystyle=\sum_{m=1}^{\infty}{\mathbb{E}}\left[{\mathbb{E}}_{m}\left(\sum_{k=m+1}^{\infty}|d_{m}f^{d}|^{2}|d_{k}g|^{2}\right)^{\frac{1}{2}}\right]$
$\displaystyle=\sum_{m=1}^{\infty}{\mathbb{E}}\left[|d_{m}f^{d}|{\mathbb{E}}_{m}\left(\sum_{k=m+1}^{\infty}|d_{k}g|^{2}\right)^{\frac{1}{2}}\right]$
$\displaystyle\leqslant\sum_{m=1}^{\infty}\left(\|d_{m}f^{d}\|_{1}\left\|{\mathbb{E}}_{m}\left(\sum_{k=m+1}^{\infty}|d_{k}g|^{2}\right)\right\|_{\infty}^{\frac{1}{2}}\right)$

and hence,

(3.7)
$\left\|\Pi_{2}(f^{d},g)\right\|_{H^{1}(\Omega)}\leqslant\|f^{d}\|_{h^{1}_{d}(\Omega)}\|g\|_{BMO(\Omega)}.$

By using (3.7), (3.6) and (3.3), we have by Theorem 2.9

$\displaystyle\|\Pi_{2}(f,g)\|_{H^{1}(\Omega)}$
$\displaystyle\leq\left\|\Pi_{2}(f^{h},g)\right\|_{H^{1}(\Omega)}+\left\|\Pi_{2}(f^{d},g)\right\|_{H^{1}(\Omega)}$
$\displaystyle\leqslant\sum_{j=1}^{\infty}|\lambda_{j}|\|g\|_{BMO(\Omega)}+\|f^{d}\|_{h_{1}^{d}(\Omega)}\|g\|_{BMO(\Omega)}$
$\displaystyle\lesssim\left(\|f^{h}\|_{h^{1}(\Omega)}+\|f^{d}\|_{h^{1}_{d}(\Omega)}\right)\|g\|_{BMO(\Omega)}.$

Since the decomposition of $f=f^{h}+f^{d}$ is arbitrary, by Theorem 2.9 we
conclude

(3.8)
$\|\Pi_{2}(f,g)\|_{H^{1}(\Omega)}\lesssim\|f\|_{H^{1}(\Omega)}\|g\|_{BMO(\Omega)}.$

###  3.3. Estimates for $\Pi_{3}(f^{h},g)$ and $\Pi_{3}(f^{d},g)$

We are going to show that $\Pi_{3}$ is a bounded bilinear operator from
$H^{1}(\Omega)\times BMO(\Omega)$ to $H^{\Phi}(\Omega)$. To this end, we first
deal with $\Pi_{3}(f^{h},g)$. Note that

$\displaystyle S(\Pi_{3}(f^{h},g))$
$\displaystyle=S\left(\sum_{k=1}^{\infty}\sum_{j=1}^{\infty}\lambda_{j}g_{k-1}d_{k}a^{j}\right)\leq\sum_{j=1}^{\infty}\lambda_{j}S\left(\sum_{k=1}^{\infty}g_{k-1}d_{k}a^{j}\right)$
$\displaystyle=\sum_{j=1}^{\infty}\lambda_{j}\left(\sum_{k=n_{j}+1}^{\infty}|g_{k-1}|^{2}|d_{k}a^{j}|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq\sum_{j=1}^{\infty}\lambda_{j}\left(\sum_{k=n_{j}+1}^{\infty}|g_{k-1}-g_{n_{j}}|^{2}|d_{k}a^{j}|^{2}\right)^{\frac{1}{2}}+\sum_{j=1}^{\infty}\lambda_{j}|g_{n_{j}}|S(a^{j})$
$\displaystyle=:I_{1}+I_{2}.$

It thus suffices to handle $I_{1}$ and $I_{2}$. For $I_{1}$, we have

$\displaystyle{\mathbb{E}}(I_{1})$
$\displaystyle\leqslant\sum_{j=1}^{\infty}|\lambda_{j}|{\mathbb{E}}\left(\sum_{k=n_{j}+1}^{\infty}1_{A_{n_{j}}}|g_{k-1}-g_{n_{j}}|^{2}|d_{k}a^{j}|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leqslant\sum_{j=1}^{\infty}|\lambda_{j}|\left[{\mathbb{E}}\left(\sum_{k=n_{j}+1}^{\infty}1_{A_{n_{j}}}|g_{k-1}-g|^{2}|d_{k}a^{j}|^{2}\right)^{\frac{1}{2}}+{\mathbb{E}}\left(\sum_{k=n_{j}+1}^{\infty}1_{A_{n_{j}}}|g-g_{n_{j}}|^{2}|d_{k}a^{j}|^{2}\right)^{\frac{1}{2}}\right]$
$\displaystyle\leqslant\sum_{j=1}^{\infty}|\lambda_{j}|\left\\{P(A_{n_{j}})^{\frac{1}{2}}\left[{\mathbb{E}}\left(\sum_{k=n_{j}+1}^{\infty}|g_{k-1}-g|^{2}|d_{k}a^{j}|^{2}\right)\right]^{\frac{1}{2}}+{\mathbb{E}}\left(1_{A_{n_{j}}}|g-g_{n_{j}}|S(a^{j})\right)\right\\}$
$\displaystyle\leqslant\sum_{j=1}^{\infty}|\lambda_{j}|\left\\{P(A_{n_{j}})^{\frac{1}{2}}\left[{\mathbb{E}}\left(\sum_{k=n_{j}+1}^{\infty}|d_{k}a^{j}|^{2}{\mathbb{E}}_{k}(|g_{k-1}-g|^{2})\right)\right]^{\frac{1}{2}}+\|a^{j}\|_{2}P(A_{n_{j}})^{\frac{1}{2}}\|g\|_{BMO(\Omega)}\right\\}$
$\displaystyle\leqslant
2\sum_{j=1}^{\infty}|\lambda_{j}|P(A_{n_{j}})^{\frac{1}{2}}\|g\|_{BMO(\Omega)}\|a^{j}\|_{2}$

and so,

(3.9)
${\mathbb{E}}(I_{1})\lesssim\|f^{h}\|_{h^{1}(\Omega)}\|g\|_{BMO(\Omega)}.$

Next, we obtain an estimate for $I_{2}$. To this end, notice that

$\displaystyle I_{2}$
$\displaystyle\leq\left(\sum_{j=1}^{\infty}1_{A_{n_{j}}}|\lambda_{j}|S(a^{j})\right)\cdot|g|+\sum_{j=1}^{\infty}|\lambda_{j}|1_{A_{n_{j}}}|g_{n_{j}}-g|S(a^{j})$
$\displaystyle=:I_{3}+I_{4}.$

Since $a^{j}$ is a simple $(1,2)$-atom, we have
$\|1_{A_{n_{j}}}S(a^{j})\|_{1}\leq 1$ and

$\left\|\sum\limits_{j=1}^{\infty}1_{A_{n_{j}}}|\lambda_{j}|S(a^{j})\right\|_{1}\leq\sum\limits_{j=1}^{\infty}|\lambda_{j}|\lesssim\|f^{h}\|_{h^{1}(\Omega)}.$

By Lemma 2.17, we have

(3.10)
$\|I_{3}\|_{L^{\Phi}(\Omega)}\lesssim\left\|\sum_{j=1}^{\infty}1_{A_{n_{j}}}|\lambda_{j}|S(a^{j})\right\|_{1}\|g\|_{BMO(\Omega)}\lesssim\|f^{h}\|_{h^{1}(\Omega)}\|g\|_{BMO(\Omega)}.$

The following estimate is implicit in the proof of (3.9):

(3.11)
${\mathbb{E}}(I_{4})\leqslant\sum_{j=1}^{\infty}|\lambda_{j}|P(A_{n_{j}})^{\frac{1}{2}}\|g\|_{BMO(\Omega)}\|a^{j}\|_{2}\lesssim\|f^{h}\|_{h^{1}(\Omega)}\|g\|_{BMO(\Omega)}.$

By combining (3.10) and (3.11), we deduce that

(3.12)
$\|I_{2}\|_{L^{\Phi}(\Omega)}\lesssim\|f^{h}\|_{h^{1}(\Omega)}\|g\|_{BMO(\Omega)}.$

In conclusion, by (3.9) and (3.12) we get

(3.13)
$\|\Pi_{3}(f^{h},g)\|_{H^{\Phi}(\Omega)}\lesssim\|f^{h}\|_{h^{1}(\Omega)}\|g\|_{BMO(\Omega)}.$

It remains to deal with $\Pi_{3}(f^{d},g)$. We have

$\displaystyle S(\Pi_{3}(f^{d},g))$
$\displaystyle=\left(\sum_{k=1}^{\infty}|g_{k-1}|^{2}|d_{k}f^{d}|^{2}\right)^{\frac{1}{2}}\leqslant\sum_{k=1}^{\infty}|g_{k-1}||d_{k}f^{d}|$
$\displaystyle\leq\sum_{k=1}^{\infty}|g_{k-1}-g||d_{k}f^{d}|+|g|\left(\sum_{k=1}^{\infty}|d_{k}f^{d}|\right).$

By Lemma 2.17,

(3.14)
$\left\|g\left(\sum_{k=1}^{\infty}|d_{k}f^{d}|\right)\right\|_{L^{\Phi}(\Omega)}\lesssim\left(\sum_{k=1}^{\infty}\|d_{k}f^{d}\|_{1}\right)\|g\|_{BMO(\Omega)}=\|f^{d}\|_{h^{1}_{d}(\Omega)}\|g\|_{BMO(\Omega)}.$

For the remaining term, we have

$\displaystyle{\mathbb{E}}\left(\sum_{k=1}^{\infty}|g_{k-1}-g||d_{k}f^{d}|\right)$
$\displaystyle={\mathbb{E}}\left(\sum_{k=1}^{\infty}|d_{k}f^{d}|{\mathbb{E}}_{k}|g_{k-1}-g|\right)$
$\displaystyle\leqslant\|g\|_{BMO(\Omega)}\left(\sum_{k=1}^{\infty}\|d_{k}f^{d}\|_{1}\right)$

and so

(3.15)
${\mathbb{E}}\left(\sum_{k=1}^{\infty}|g_{k-1}-g||d_{k}f^{d}|\right)\leqslant\|f^{d}\|_{h^{1}_{d}(\Omega)}\|g\|_{BMO(\Omega)}.$

Hence, by (3.14) and (3.15), we get

(3.16)
$\|\Pi_{3}(f^{d},g)\|_{H^{\Phi}(\Omega)}\lesssim\|f^{d}\|_{h^{1}_{d}(\Omega)}\|g\|_{BMO(\Omega)}.$

By (3.13) and (3.16), we obtain

$\|\Pi_{3}(f,g)\|_{H^{\Phi}(\Omega)}\lesssim\left(\|f^{h}\|_{h^{1}(\Omega)}+\|f^{d}\|_{h^{1}_{d}(\Omega)}\right)\|g\|_{BMO(\Omega)}.$

Thus we conclude

(3.17)
$\|\Pi_{3}(f,g)\|_{H^{\Phi}(\Omega)}\lesssim\|f\|_{H^{1}(\Omega)}\|g\|_{BMO(\Omega)}.$

This completes the proof of Theorem 1.1 ∎

###  3.4. Another method for handling $\Pi_{3}(f,g)$

In this section we present a different method for dealing with $\Pi_{3}(f,g)$,
which is much neater and simpler than the one presented above, and it relies
on the following theorem which has been shown in [12].

######  Theorem 3.1.

If $g\in BMO(\Omega)$ and $g_{0}=0$, then
$(g^{*})_{0}\lesssim\|g\|_{BMO(\Omega)}$ and $g^{*}\in BMO(\Omega)$. Moreover,
$\|g^{*}\|_{BMO(\Omega)}\lesssim\|g\|_{BMO(\Omega)}.$

We begin with a pointwise estimate for $S(\Pi_{3}(f,g))$. Towards this aim,
note that $d_{k}(\Pi_{3}(f,g))=g_{k-1}d_{k}f$, which implies that

$S(\Pi_{3}(f,g))=\left(\sum_{k=1}^{\infty}|g_{k-1}|^{2}|d_{k}f|^{2}\right)^{\frac{1}{2}}\leqslant|g^{*}|S(f)\leqslant
J_{1}+J_{2},$

where

$J_{1}:=|g^{*}-(g^{*})_{0}|S(f)\quad\text{and}\quad
J_{2}:=S(f)\|g\|_{BMO(\Omega)}.$

Clearly,

(3.18) $\|J_{2}\|_{1}\lesssim\|f\|_{H^{1}(\Omega)}\|g\|_{BMO(\Omega)}.$

By Theorem 3.1, we get $g^{*}\in BMO(\Omega)$, and hence by Lemma 2.17

(3.19)
$\|J_{1}\|_{L^{\Phi}(\Omega)}\lesssim\|g^{*}\|_{BMO(\Omega)}\|S(f)\|_{1}\lesssim\|f\|_{H^{1}(\Omega)}\|g\|_{BMO(\Omega)}.$

As $S(\Pi_{3}(f,g))\leq J_{1}+J_{2}$, by combining (3.18) with (3.19), and by
the fact $L^{1}(\Omega)\subset L^{\Phi}(\Omega)$, we conclude

$\|\Pi_{3}(f,g)\|_{H^{\Phi}(\Omega)}=\|S(\Pi_{3}(f,g))\|_{L^{\Phi}(\Omega)}\lesssim\|f\|_{H^{1}(\Omega)}\|g\|_{BMO(\Omega)},$

as desired.

We would like to end this section with the comparison between our proof and
the one provided in [14]. Though both arguments heavily rely on the atomic
decomposition of $H^{1}(\Omega)$, they further use weak atom decomposition for
the diagonal Hardy space while our proof proceeds more directly. Moreover, the
treatment of the most technical term $\Pi_{3}$ is significantly simplified in
this section thanks to Theorem 3.1.

##  4\. Bilinear decompositions on
$H^{p}(\Omega)\times\Lambda_{1}(\alpha_{p})$ for $0<p<1$

In this section, we give a proof of Theorem 1.2. Arguing as in the proof of
Theorem 1.1, it suffices to establish appropriate estimates for the bilinear
operators $\Pi_{1}$, $\Pi_{2}$ and $\Pi_{3}$.

Let $(\Omega,{\mathcal{F}},P)$ be a fixed probability space. If we consider
the filtration ${\mathcal{F}}_{0}=\\{\emptyset,\Omega\\}$ and
${\mathcal{F}}_{k}={\mathcal{F}}$ for all $k\geqslant 1$, then
$H^{p}(\Omega)=L^{p}(\Omega)$ for $0<p<\infty$. It is well-known that
$(L^{p}(\Omega))^{*}\neq\\{0\\}$ if and only if the probability space
$(\Omega,{\mathcal{F}},P)$ contains at least one atom with non-zero measure
when $0<p<1$. This means that $(H^{p}(\Omega))^{*}=\\{0\\}$ may occur.
Therefore, we are only concerned with regular martingales where every
${\mathcal{F}}_{k}$ is generated by countably many atoms.

To prove Theorem 1.2, we start with the following lemma, which holds for
general martingales that are not necessarily regular. This shall be familiar
to the experts in the area, but we will enclose the proof here for the sake of
completeness.

######  Lemma 4.1.

For any $0<p<1$, we have $L^{1}(\Omega)\subset H^{p}_{m}(\Omega)$.

###### Proof.

By Doob’s maximal inequality, for any $f\in L^{1}(\Omega)$ and for any
$\lambda>0$ we have

(4.1)
$P(f^{*}>\lambda)\leqslant\dfrac{1}{\lambda}\int_{\left\\{f^{*}>\lambda\right\\}}|f|dP.$

Without loss of generality, we may assume $\|f\|_{1}\leqslant 1$. Then

$\displaystyle\|f^{*}\|_{p}^{p}$
$\displaystyle=\int_{\Omega}|f^{*}|^{p}dP=p\int_{0}^{\infty}P(|f^{*}|>\lambda)\lambda^{p-1}d\lambda$
$\displaystyle=p\int_{0}^{1}P(f^{*}>\lambda)\lambda^{p-1}d\lambda+p\int_{1}^{\infty}P(f^{*}>\lambda)\lambda^{p-1}d\lambda$
$\displaystyle\leqslant
p\int_{0}^{1}\lambda^{p-1}d\lambda+p\int_{1}^{\infty}\dfrac{1}{\lambda}\left(\int_{\left\\{f^{*}>\lambda\right\\}}|f|dP\right)\lambda^{p-1}d\lambda$
$\displaystyle=1+p\int_{\\{f^{*}>1\\}}|f|\left(\int_{1}^{f^{*}}\lambda^{p-2}d\lambda\right)dP$
$\displaystyle=1+\dfrac{p}{1-p}\int_{\\{f^{*}>1\\}}|f|\left(1-|f^{*}|^{p-1}\right)dP$
$\displaystyle\leqslant
1+\dfrac{p}{1-p}\int_{\\{f^{*}>1\\}}|f|dP\leqslant\dfrac{1}{1-p}.$

This implies that for any $f\in L^{1}(\Omega)$

$\|f\|_{H^{p}_{m}(\Omega)}\leqslant\big{(}\dfrac{1}{1-p}\big{)}^{\frac{1}{p}}\|f\|_{1},$

which yields the desired result. ∎

For regular martingales, we have $L^{1}(\Omega)\subset
H^{p}_{m}(\Omega)=H^{p}(\Omega)=h^{p}(\Omega)$. In what follows, the
martingales are always assumed to be regular and every ${\mathcal{F}}_{k}$ is
generated by countable atoms.

######  Corollary 4.2.

For $0<p<1$ and $1\leq q\leq\infty$, $H^{p}(\Omega)=H^{p,q}_{\rm at}(\Omega)$.

###### Proof.

By considering the aforementioned atomic decomposition of $H^{p}(\Omega)$ and
Definition 2.8, we have $H^{p}(\Omega)=H^{p,\infty}_{\rm at}(\Omega)$. It is
easy to see $H^{p,\infty}_{\rm at}(\Omega)\subset H^{p,q}_{\rm
at}(\Omega)\subset H^{p,1}_{\rm at}(\Omega)$. It thus suffices to show that
$H^{p,1}_{\rm at}(\Omega)\subset H^{p}(\Omega)$. By Lemma 4.1, if $a$ is a
simple $(p,1)$-atom, then

$\|a\|_{H^{p}(\Omega)}\lesssim_{p}\|a\|_{1},$

which implies that $a\in H^{p}(\Omega)$. Hence, $H^{p,1}_{\rm
at}(\Omega)\subset H^{p}(\Omega)$ and so, $H^{p}(\Omega)=H^{p,q}_{\rm
at}(\Omega)$. ∎

###  4.1. Characterization of martingale Lipschitz spaces

In this subsection, we give a characterization of martingale Lipschitz spaces
that appears to be new and useful in our argument. We shall first recall the
definition of martingale Lipschitz spaces. For $0<p<1$ define

(4.2) $\Lambda_{q}(\alpha_{p}):=\left\\{f\in
L^{2}(\Omega):\|f\|_{\Lambda_{q}(\alpha_{p})}=\sup_{n\geq
0}\sup_{A\in{\mathcal{F}}_{n}}P(A)^{-\frac{1}{q}-\alpha_{p}}\left(\int_{A}|f-f_{n}|^{q}dP\right)^{\frac{1}{q}}<\infty\right\\},$

where $q=1$ or $q=2$, $\alpha_{p}:=\frac{1}{p}-1>0$.

In [27], Weisz showed that $(H^{p}(\Omega))^{*}=\Lambda_{1}(\alpha_{p})$ and
$\Lambda_{1}(\alpha_{p})=\Lambda_{2}(\alpha_{p})$.

######  Corollary 4.3.

For any $g\in\Lambda_{1}(\alpha_{p})$, we have
$\|g-g_{0}\|_{\infty}\lesssim_{p}\|g\|_{\Lambda_{1}(\alpha_{p})}$.

###### Proof.

By duality and Lemma 4.1, for any $f\in L^{2}(\Omega)$,

$|{\mathbb{E}}\big{(}f(g-g_{0})\big{)}|=\big{|}{\mathbb{E}}\left(g(f-f_{0})\right)\big{|}\lesssim_{p}\|f\|_{H^{p}}\|g\|_{\Lambda_{1}(\alpha_{p})}\lesssim_{p}\|f\|_{1}\|g\|_{\Lambda_{1}(\alpha_{p})}.$

The above estimate together with the fact
$\big{(}L^{1}\big{(}\Omega))^{*}=L^{\infty}(\Omega)$ yields

$\|g-g_{0}\|_{\infty}\lesssim_{p}\|g\|_{\Lambda_{1}(\alpha_{p})},$

which finishes the proof. ∎

By virtue of Corollary 4.3, we have the following property of martingale
Lipschitz spaces.

######  Theorem 4.4.

If $g\in\Lambda_{1}(\alpha_{p})$, we have
$\|1_{A}\cdot|g-g_{n}|\|_{\infty}\lesssim_{p}P(A)^{\alpha_{p}}\|g\|_{\Lambda_{1}(\alpha_{p})}$,
for any $n\in{\mathbb{N}}$ and any $A\in{\mathcal{F}}_{n}$.

###### Proof.

Note that when $P(A)=0$, the desired result holds trivially. Fix
$n\in{\mathbb{N}}$ and $A\in{\mathcal{F}}_{n}$ with $P(A)\neq 0$. For $k\geq
0$, let ${\mathcal{F}}_{k}^{A}:=\left\\{B\in{\mathcal{F}}_{k+n}:B\subseteq
A\right\\}$. Note that the union ${\mathcal{F}}^{A}$ of all
${\mathcal{F}}_{k}^{A}$ is exactly $\\{B\in{\mathcal{F}}|B\subset A\\}$.
Hence, if we define

$P_{A}(B):=\dfrac{P(B)}{P(A)}\quad(B\in{\mathcal{F}}^{A})$

then $(A,{\mathcal{F}}^{A},P_{A})$ is a probability space. Note that for any
$g\in L^{1}(A,{\mathcal{F}}^{A},P_{A})$ one has

$\mathbb{E}(g|{\mathcal{F}}^{A}_{k})=1_{A}\cdot{\mathbb{E}}(g|{\mathcal{F}}_{k+n}).$

Denote $\mathbb{E}(\cdot|{\mathcal{F}}^{A}_{k})$ by ${\mathbb{E}}_{k}^{A}$. It
is easy to verify $\\{{\mathbb{E}}_{k}^{A}(g)\\}_{k\geq 0}$ is also a regular
martingale on $(A,{\mathcal{F}}^{A},P_{A})$. If $g\in\Lambda_{1}(\alpha_{p})$,
then for $B\in{\mathcal{F}}_{k}^{A}$ with $P(B)\neq 0$,

$\displaystyle
P_{A}(B)^{-1-\alpha_{p}}\left(\int_{B}|g-{\mathbb{E}}_{k}^{A}(g)|dP_{A}\right)$
$\displaystyle=P(A)^{\alpha_{p}}\left(P(B)^{-1-\alpha_{p}}\left(\int_{B}|g-g_{k+n}|dP\right)\right)$
$\displaystyle\leq P(A)^{\alpha_{p}}\|g\|_{\Lambda_{1}(\alpha_{p})}$

which implies that by Corollary 4.3,

$\|1_{A}\cdot|g-g_{n}|\|_{\infty}=\|1_{A}\cdot|g-{\mathbb{E}}_{0}^{A}(g)|\|_{\infty}\lesssim_{p}P(A)^{\alpha_{p}}\|g\|_{\Lambda_{1}(\alpha_{p})}.$

This completes the proof of the theorem. ∎

######  Remark 4.5.

By Theorem 4.4 and (4.2), we conclude that for $g\in\Lambda_{1}(\alpha_{p})$
we have the characterization

(4.3) $\|g\|_{\Lambda_{1}(\alpha_{p})}\approx_{p}\sup_{n\geq
0}\sup_{A\in{\mathcal{F}}_{n}}P(A)^{-\alpha_{p}}\|1_{A}\cdot|g-g_{n}|\|_{\infty}.$

Note that the results in [18] can be deduced from (4.3).

###  4.2. Proof of Theorem 1.2

As in the proof of Theorem 1.1, we divide the proof into three parts. Without
loss of generality, we may assume that $f_{0}=g_{0}=0$.

####  4.2.1. Estimates for $\Pi_{1}(f,g)$ and $\Pi_{3}(f,g)$

The boundedness of $\Pi_{1}$ from $H^{p}(\Omega)\times\Lambda_{1}(\alpha_{p})$
to $L^{1}(\Omega)$ follows directly from the duality between $H^{p}(\Omega)$
and $\Lambda_{1}(\alpha_{p})$, we omit the details.

We shall also prove that $\Pi_{3}$ is a bounded bilinear operator from
$H^{p}(\Omega)\times\Lambda_{1}(\alpha_{p})$ to $H^{p}(\Omega)$. Note that

(4.4)
$S(\Pi_{3}(f,g))^{2}=\sum_{k=1}^{\infty}|g_{k-1}|^{2}|d_{k}f|^{2}\leqslant(g^{*})^{2}S(f)^{2}.$

Hence we conclude from Corollary 4.3 and the $L^{\infty}$ boundedness of the
maximal function that

(4.5)
$\|\Pi_{3}(f,g)\|^{p}_{H^{p}(\Omega)}\lesssim\|g^{*}\|_{\infty}^{p}{\mathbb{E}}(S(f)^{p})\leq\|g\|_{\infty}^{p}\|f\|_{H^{p}(\Omega)}^{p}\lesssim_{p}\|f\|_{H^{p}(\Omega)}^{p}\|g\|_{\Lambda_{1}(\alpha_{p})}^{p},$

as desired.

####  4.2.2. Estimates for $\Pi_{2}(f,g)$.

We will show that $\Pi_{2}$ is a bounded bilinear operator from
$H^{p}(\Omega)\times\Lambda_{1}(\alpha_{p})$ to $H^{1}(\Omega)$. Note that
$H^{p}(\Omega)=h^{p}(\Omega)$, and $h^{p}(\Omega)$ admits an atomic
decomposition.

Then there exist $\left\\{\lambda_{j}\right\\}_{j\geqslant 1}\subset\real$ and
simple $(p,\infty)$-atoms $\left\\{a^{j}\right\\}_{j\geqslant 1}$ such that

(4.6)
$f=\sum_{j=1}^{\infty}\lambda_{j}a^{j},\quad\|f\|_{H^{p}(\Omega)}\approx_{p}\left(\sum_{j=1}^{\infty}|\lambda_{j}|^{p}\right)^{\frac{1}{p}},$

where we assume $\text{supp}(a^{j})\subset A_{n_{j}}$ and
$A_{n_{j}}\in{\mathcal{F}}_{n_{j}}$ with $P(A_{n_{j}})>0$ for $j\geq 1$. By
arguing as in the corresponding case in the proof of Theorem 1.1,

(4.7)
$S(\Pi_{2}(a^{j},g))=\left(\sum_{k=n_{j}+1}^{\infty}1_{A_{n_{j}}}|a^{j}_{k-1}|^{2}|d_{k}g|^{2}\right)^{\frac{1}{2}}\leqslant|(a^{j})^{*}|\left(\sum_{k=n_{j}+1}^{\infty}1_{A_{n_{j}}}|d_{k}g|^{2}\right)^{\frac{1}{2}}.$

Hence,

$\displaystyle{\mathbb{E}}\left[S(\Pi_{2}(a^{j},g))\right]$
$\displaystyle\leqslant\|(a^{j})^{*}\|_{\infty}\left[{\mathbb{E}}\left(1_{A_{n_{j}}}\sum_{k=n_{j}+1}^{\infty}|d_{k}g|^{2}\right)^{\frac{1}{2}}\right]$
$\displaystyle\leqslant\|a^{j}\|_{\infty}\left(P(A_{n_{j}})\left({\mathbb{E}}\sum_{k=n_{j}+1}^{\infty}|d_{k}g|^{2}\right)\right)^{\frac{1}{2}}$
$\displaystyle\leqslant
P(A_{n_{j}})^{-\frac{1}{p}}\left(P(A_{n_{j}})\|g\|_{\Lambda_{2}(\alpha_{p})}^{2}P(A_{n_{j}})^{1+2\alpha_{p}}\right)^{\frac{1}{2}}$
$\displaystyle=\|g\|_{\Lambda_{2}(\alpha_{p})}P(A_{n_{j}})^{-\frac{1}{p}}P(A_{n_{j}})^{1+\alpha_{p}}$
$\displaystyle\leqslant\|g\|_{\Lambda_{2}(\alpha_{p})}\lesssim_{p}\|g\|_{\Lambda_{1}(\alpha_{p})},$

where the last inequality follows from the condition that
$\alpha_{p}=\frac{1}{p}-1$. As a consequence of the above estimates, we have
that

(4.8)
$\|\Pi_{2}(f,g)\|_{H^{1}(\Omega)}^{p}\leqslant\sum_{j=1}^{\infty}|\lambda_{j}|^{p}\left[{\mathbb{E}}S(\Pi_{2}(a^{j},g))\right]^{p}\lesssim_{p}\|f\|_{H^{p}(\Omega)}^{p}\|g\|_{\Lambda_{1}(\alpha_{p})}^{p}.$

This completes the proof of the theorem.

##  5\. Homogeneous spaces

In this section, we introduce some fundamental concepts and important theorems
for homogeneous spaces, which can be found in [7]. We begin with the
definition of homogeneous spaces. Recall that $d$ is a quasi-metric on
$\Omega$ if

  1. (1)

$d(x,x)\geq 0$, $\forall x\in\Omega$;

  2. (2)

$d(x,y)=d(y,x)$, $\forall x,y\in\Omega$;

  3. (3)

there exists a constant $A_{0}\geq 1$ such that

(5.1) $d(x,y)\leqslant A_{0}(d(x,z)+d(z,y)),\quad\forall x,y,z\in\Omega.$

Denote by $B(x,r):=\\{y\in\Omega:d(y,x)<r\\}$ the open ball centered at $x$
with radius $r$. In this paper, all quasi-metric spaces are assumed to have
the doubling property: there exists a positive integer $A_{1}\in{\mathbb{N}}$
such that for every $x\in\Omega$ and for every $r>0$, the ball $B(x,r)$ can be
covered by at most $A_{1}$ balls $B(x_{i},\frac{r}{2})$ for some
$x_{i}\in\Omega$.

######  Definition 5.1.

A $\sigma$-finite measure space $(\Omega,{\mathcal{F}},\mu)$ equipped with a
quasi-metric $d$ is called a homogeneous space if $\mu$ is a Borel measure of
homogeneous type:

(5.2) $0<\mu\left(B(x,2r)\right)\leqslant
2^{C_{\mu}}\mu\left(B(x,r)\right)<\infty,\quad\forall x\in\Omega,r>0,$

where the constant $C_{\mu}$ is independent of $x$ and $r$.

In [7], Coifman and Weiss defined Hardy spaces on homogeneous spaces by
regarding their elements as linear functionals acting on some appropriate
quasi-normed spaces. In order to state the definition of Coifman and Weiss, we
need to introduce the notions of atoms, $BMO$ and Lipschitz spaces on
homogeneous spaces.

######  Definition 5.2.

If $0<p\leqslant 1\leqslant q\leq\infty$ and $p<q$, we say that a function $a$
is a $(p,q)$-atom if

  1. (1)

$\text{supp}(a)\subset B$ where $B$ is a ball;

  2. (2)

$\|a\|_{q}\leqslant\left(\mu(B)\right)^{\frac{1}{q}-\frac{1}{p}}$;

  3. (3)

$\int_{\Omega}ad\mu=\int_{B}ad\mu=0.$

######  Definition 5.3.

A locally integrable function $f$ is called a $BMO$ function if

$\|f\|_{BMO}:=\sup_{B}\dfrac{1}{\mu(B)}\int_{B}|f-f_{B}|d\mu<\infty,$

where $f_{B}:=\dfrac{1}{\mu(B)}\int_{B}fd\mu$, and the supremum runs over all
balls $B$. Denote by $BMO(\mu)$ the $BMO$ space consisting of all $BMO$
functions.

######  Definition 5.4.

For $\alpha>0$, a locally integrable function $l$ is called a Lipschitz
function if

(5.3) $|l(x)-l(y)|\leqslant C_{\alpha}\left(\mu(B)\right)^{\alpha}\text{ for
any $x,y\in\Omega$ and any ball $B$ containing $x,y$. }$

Moreover,

(5.4) $\|l\|_{{\mathcal{L}}_{\alpha}}:=\inf\\{C_{\alpha}:|l(x)-l(y)|\leqslant
C_{\alpha}\left(\mu(B)\right)^{\alpha},\ \forall x,y\in B\\},$

where the infimum runs over all balls $B$. Denote by
${\mathcal{L}}_{\alpha}(\mu)$ the space consisting of all Lipschitz functions.

It is well-known that each $BMO$ function can be regarded as a continuous
linear functional on the vector space generated by finite linear combinations
of $(1,q)$-atoms for $1<q\leq\infty$ (cf. [7]). Hence we can define the atomic
Hardy space $H^{1,q}_{\rm at}(\mu)$ $(1<q\leq\infty)$ as follows:

(5.5) $H^{1,q}_{\rm at}(\mu):=\\\ \left\\{f\in\left(BMO(\mu)\right)^{*}:\
f=\sum_{j=0}^{\infty}\lambda_{j}a^{j},\text{ where $a^{j}$ is a $(1,q)$-atom
and }\sum_{j=0}^{\infty}|\lambda_{j}|<\infty\right\\}$

endowed with the norm

$\|f\|_{H^{1,q}_{\rm
at}(\mu)}:=\inf\left\\{\sum_{j=0}^{\infty}|\lambda_{j}|:f=\sum_{j=0}^{\infty}\lambda_{j}a^{j},\text{
where $a^{j}$ is a $(1,q)$-atom}\right\\}.$

Similarly, each Lipschitz function $l\in{\mathcal{L}}_{\alpha_{p}}(\mu)$ can
be also regarded as a continuous linear functional of the vector space
generated by finite linear combinations of $(p,q)$-atoms where $0<p<1\leq
q\leq\infty$ and $\alpha_{p}=\frac{1}{p}-1$ (cf. [7]). We define the atomic
Hardy spaces $H^{p,q}_{\rm at}(\mu)$ as follows:

(5.6) $H^{p,q}_{\rm at}(\mu)\\\
:=\left\\{f\in\left({\mathcal{L}}_{\alpha_{p}}(\mu)\right)^{*}:\
f=\sum_{j=0}^{\infty}\lambda_{j}a^{j},\text{ where $a^{j}$ is a $(p,q)$-atom
and }\sum_{j=0}^{\infty}|\lambda_{j}|^{p}<\infty\right\\}$

endowed with the quasi-norm

$\|f\|_{H^{p,q}_{\rm
at}(\mu)}:=\inf\left\\{\left(\sum_{j=0}^{\infty}|\lambda_{j}|^{p}\right)^{\frac{1}{p}}:f=\sum_{j=0}^{\infty}\lambda_{j}a^{j},\text{
where $a^{j}$ is a $(p,q)$-atom}\right\\}.$

Although the Hardy spaces vary with $p$ and $q$ according to the above
definitions, the following theorem, which can be found in [7], shows that the
Hardy spaces actually depend only on $p$. Consequently, this enables us to
define the Hardy spaces $H^{p}_{\rm at}(\mu)$ for $0<p\leqslant 1$ to be any
one of the spaces $H^{p,q}_{\rm at}(\mu)$ for $0<p<q\leqslant\infty,1\leqslant
q\leq\infty$.

######  Theorem 5.5.

$H^{p,q}_{\rm at}(\mu)=H_{\rm at}^{p,\infty}(\mu)$ whenever $0<p\leq
1\leqslant q\leqslant\infty$ and $p<q$.

We end this section with the following duality theorem obtained in [7].

######  Theorem 5.6.

$\left(H^{1}_{\rm at}(\mu)\right)^{*}=BMO(\mu)$, and $\left(H^{p}_{\rm
at}(\mu)\right)^{*}={\mathcal{L}}_{\alpha_{p}}(\mu)$ for $0<p<1$.

The proofs of Theorem 5.5 and Theorem 5.6 that appeared in [7] are very
technical. In the following sections, by employing martingale methods, we give
much simpler proofs of these facts. Our approach is based on the fact that
$H^{p}_{\rm at}(\mu)$ for $0<p\leq 1$ is the finite sum of several dyadic
martingale Hardy spaces.

##  6\. Dyadic systems on homogeneous spaces

In this section, we start with introducing dyadic systems on homogeneous
spaces, which first appeared in the work of Hytönen and Kairema [13]. With the
help of these dyadic structures, we then show that $H^{p}_{\rm at}(\mu)$ is
exactly the finite sum of martingale Hardy spaces associated with some
adjacent dyadic martingales, which extends Mei’s result [17] to homogeneous
spaces.

The following theorem concerning the existence of dyadic structures is due to
Hytönen and Kairema [13].

######  Theorem 6.1.

Let $\Omega$ denote a homogeneous space. Suppose that the constants
$0<c_{0}\leqslant C_{0}<\infty$ and $\delta\in(0,1)$ satisfy

$12A_{0}^{3}C_{0}\delta\leqslant c_{0},$

where $A_{0}$ is specified in the definition of quasi-metric, see (5.1).

Given a set of reference points $\\{z^{k}_{\alpha}\\}_{\alpha},\
\alpha\in{\mathcal{A}}_{k}$ (an index set), for every $k\in{\mathbb{Z}}$, with
the properties that

$d(z^{k}_{\alpha},z^{k}_{\beta})\geqslant c_{0}\delta^{k},\
(\alpha\neq\beta)\quad\min_{\alpha}d(x,z^{k}_{\alpha})<C_{0}\delta^{k},\text{
for all }x\in\Omega,$

one can construct families of sets $\tilde{Q}^{k}_{\alpha}\subseteq
Q^{k}_{\alpha}\subseteq\bar{Q}^{k}_{\alpha}$, called open, half-open and
closed dyadic cubes respectively, such that:

(6.1) $\displaystyle\tilde{Q}^{k}_{\alpha}\text{ and
}\bar{Q}^{k}_{\alpha}\text{are the interior and closure of }Q^{k}_{\alpha};$
(6.2) $\displaystyle\text{if }k\leqslant l,\text{ then either
}Q^{l}_{\beta}\subseteq Q^{k}_{\alpha}\text{ or }Q^{l}_{\beta}\cap
Q^{k}_{\alpha}=\emptyset;$ (6.3) $\displaystyle
X=\bigcup\limits_{\alpha}Q^{k}_{\alpha}\text{ (disjoint union) for all
}k\in{\mathbb{Z}};$ (6.4) $\displaystyle
B(z^{k}_{\alpha},c_{1}\delta^{k})\subseteq Q^{k}_{\alpha}\subseteq
B(z^{k}_{\alpha},C_{1}\delta^{k})=:B(Q^{k}_{\alpha})\text{ where
}c_{1}=(3A_{0}^{2})^{-1}c_{0}\text{ and }C_{1}=2A_{0}C_{0};$ (6.5)
$\displaystyle\text{if }k\leqslant l\text{ and }Q^{l}_{\beta}\subseteq
Q^{k}_{\alpha}\text{ then }B(Q^{l}_{\beta})\subseteq B(Q^{k}_{\alpha}).$

The open and closed cubes $\tilde{Q}^{k}_{\alpha}$ and $\bar{Q}^{k}_{\alpha}$
depend only on the points $z^{l}_{\beta}$ for $l\geqslant k$. The half-open
cubes $Q^{k}_{\alpha}$ depend on $z^{l}_{\beta}$ for
$l\geqslant\min(k,k_{0})$, where $k_{0}\in{\mathbb{Z}}$ is a preassigned
number entering the construction.

It is obvious that the construction of the above dyadic systems is not unique,
and it depends on the set of the reference points
$\\{z^{k}_{\alpha}\\}_{\alpha}$. We denote this dyadic system by
${\mathscr{D}}=\\{Q^{k}_{\alpha}\\}_{k,\alpha}$. Let
${\mathcal{F}}_{k}=\sigma(\\{Q^{k}_{\alpha}\\}_{\alpha})$ be the
$\sigma$-algebra generated by $\\{Q^{k}_{\alpha}\\}_{\alpha}$. Then it is
clear that

$\cdots\subset{\mathcal{F}}_{k-1}\subset{\mathcal{F}}_{k}\subset\cdots,$

which implies that $\\{{\mathcal{F}}_{k}\\}_{k\in{\mathbb{Z}}}$ is a
filtration generated by atoms. Let
${\mathcal{F}}=\sigma{(\cup_{k\in{\mathbb{Z}}}{\mathcal{F}}_{k})}$. Note that
each $Q^{k}_{\alpha}\text{ is an atom of }{\mathcal{F}}_{k}$.

######  Remark 6.2.

The standard dyadic grid on the real line is a dyadic system given by

${\mathcal{F}}_{k}=\\{[2^{-k}m,2^{-k}(m+1)):m\in{\mathbb{Z}}\\}\quad\text{ for
all }k\in{\mathbb{Z}}.$

Similarly, an example of a dyadic system on n is given by the family of
standard dyadic cubes in n.

Recall that, for $f\in L^{1}_{\text{loc}}(\Omega,{\mathcal{F}},\mu)$, the
martingale maximal function, the square function and the conditional square
function of $f$ associated with $({\mathcal{F}}_{k})_{k\in{\mathbb{Z}}}$ are
given by

$f^{*}:=\max\limits_{k\in{\mathbb{Z}}}|f_{k}|,\quad
S(f):=\left(\sum\limits_{k\in{\mathbb{Z}}}|d_{k}f|^{2}\right)^{\frac{1}{2}}\quad\text{and}\quad
s(f):=\left(\sum\limits_{k\in{\mathbb{Z}}}{\mathbb{E}}_{k-1}|d_{k}f|^{2}\right)^{\frac{1}{2}},$

respectively.

Let $0<p\leq 1$. The martingale Hardy space $H^{p}_{m,{\mathscr{D}}}(\mu)$ is
defined as the completion of the space consisting of all $f\in
L^{1}_{\text{loc}}(\Omega)$ such that $f^{\ast}\in L^{p}(\Omega)$ with respect
to the quasi-norm $\|f\|_{H^{p}_{m,{\mathscr{D}}}(\mu)}:=\|f^{*}\|_{p}$.

We define $H^{p}_{{\mathscr{D}}}(\mu)$ and $h^{p}_{{\mathscr{D}}}(\mu)$ by the
square functions and the conditional square functions respectively, with the
additional assumption that

(6.6) $\lim\limits_{n\rightarrow-\infty}\int_{\Omega}\sup_{k\leq
n}|f_{k}|^{p}d\mu=0.$

From (6.6), we have

$\lim\limits_{n\rightarrow-\infty}\sup_{k\leq n}|f_{k}|=0.$

Analogously, define the martingale atomic Hardy spaces $H^{p,q}_{\rm
at,{\mathscr{D}}}(\mu)$ $(0<p<1\leq q\leq\infty\ \text{or}\
p=1,1<q\leq\infty)$ like Definition 2.8.

In order to show Theorem 5.6, we introduce the dual spaces of these atomic
martingale Hardy spaces. For $0<p<1$, $q=1\ \text{or}\ 2$ and
$\alpha_{p}=\frac{1}{p}-1$, define

$\displaystyle BMO^{{\mathscr{D}}}(\mu)$ $\displaystyle:=\left\\{f\in
L^{1}_{\text{loc}}(\Omega,\mu)\ :\
\|f\|_{BMO^{{\mathscr{D}}}(\mu)}:=\sup_{Q\in{\mathscr{D}}}\dfrac{1}{\mu(Q)}\int_{Q}|f-f_{Q}|d\mu<\infty\right\\},$
$\displaystyle\Lambda_{q}^{{\mathscr{D}}}(\alpha_{p})$
$\displaystyle:=\left\\{f\in L^{1}_{\text{loc}}(\Omega,\mu)\ :\
\|f\|_{\Lambda_{q}^{{\mathscr{D}}}(\alpha_{p})}:=\sup_{Q\in{\mathscr{D}}}{\mu(Q)}^{-\frac{1}{q}-\alpha_{p}}\left(\int_{Q}|f-f_{Q}|^{q}d\mu\right)^{\frac{1}{q}}<\infty\right\\}.$

The spaces $\Lambda_{q}^{{\mathscr{D}}}(\alpha_{p})$ are called the martingale
Lipschitz spaces with respect to ${\mathscr{D}}$. Note that
$\Lambda_{1}^{{\mathscr{D}}}(\alpha_{p})=\Lambda_{2}^{{\mathscr{D}}}(\alpha_{p})$.

Arguing as in [27], one can show that

$\left(H^{1}_{\rm at,{\mathscr{D}}}(\mu)\right)^{*}=BMO^{{\mathscr{D}}}(\mu),$

and for $0<p<1$,

$\left(H^{p}_{\rm
at,{\mathscr{D}}}(\mu)\right)^{*}=\Lambda_{q}^{{\mathscr{D}}}(\alpha_{p}).$

######  Remark 6.3.

Since every simple $(p,q)$-atom is locally supported, by Corollary 4.2, we
conclude that for $0<p<1\leq q\leq\infty\ \text{or}\ p=1,1<q\leq\infty$

$H^{p,q}_{\rm at,{\mathscr{D}}}(\mu)=H^{p,\infty}_{\rm
at,{\mathscr{D}}}(\mu).$

Thus we are only concerned with $H^{p}_{\rm
at,{\mathscr{D}}}(\mu):=H^{p,\infty}_{\rm at,{\mathscr{D}}}(\mu)$.

######  Proposition 6.4.

For $0<p\leqslant 1$, the martingale Hardy spaces defined above are mutually
equivalent. Namely,
$H^{p}_{{\mathscr{D}}}(\mu)=H^{p}_{m,{\mathscr{D}}}(\mu)=h^{p}_{{\mathscr{D}}}(\mu)=H^{p}_{\rm
at,{\mathscr{D}}}(\mu)$.

###### Proof.

Let $p\in(0,1]$ be fixed. First, we show
$H^{p}_{{\mathscr{D}}}(\mu)=H^{p}_{m,{\mathscr{D}}}(\mu)$. Suppose that $f\in
H^{p}_{m,{\mathscr{D}}}(\mu)$. Then for any $n>0$, by a well-known inequality
of Burkholder–Davis–Gundy,

$\int_{\Omega}\left(|f_{-n}|^{2}+\sum_{k=-n+1}^{n}|d_{k}f|^{2}\right)^{\frac{p}{2}}d\mu\lesssim\int_{\Omega}\sup_{-n\leq
k\leq n}|f_{k}|^{p}d\mu\lesssim\int_{\Omega}(f^{*})^{p}d\mu$

which yields by letting $n\rightarrow\infty$ and by Fatou’s lemma

$\|S(f)\|_{p}\lesssim\|f^{*}\|_{p}.$

Thus $H^{p}_{m,{\mathscr{D}}}(\mu)\subset H^{p}_{{\mathscr{D}}}(\mu)$.

Conversely, if $f\in H^{p}_{{\mathscr{D}}}(\mu)$, then for $n>0$,

$\int_{\Omega}\sup_{-n\leq k\leq
n}|f_{k}|^{p}d\mu\lesssim\int_{\Omega}\left(|f_{-n}|^{2}+\sum_{k=-n+1}^{n}|d_{k}f|^{2}\right)^{\frac{p}{2}}d\mu,$

and hence

(6.7) $\int_{\Omega}\sup_{-n\leq k\leq
n}|f_{k}|^{p}d\mu\lesssim\int_{\Omega}\sup_{k\leq-n}|f_{k}|^{p}d\mu+\int_{\Omega}|S(f)|^{p}d\mu<\infty.$

Then by letting $n\rightarrow\infty$ and applying Fatou’s lemma, we obtain
$\|f^{*}\|_{p}<\infty$ and

$\|f^{*}\|_{p}\lesssim\|S(f)\|_{p}.$

Therefore, $H^{p}_{{\mathscr{D}}}(\mu)\subset H^{p}_{m,{\mathscr{D}}}(\mu)$
and $H^{p}_{m,{\mathscr{D}}}(\mu)=H^{p}_{{\mathscr{D}}}(\mu)$.

One shows $H^{p}_{m,{\mathscr{D}}}(\mu)=h^{p}_{{\mathscr{D}}}(\mu)$ in a
completely analogous way. To show $h^{p}_{{\mathscr{D}}}(\mu)=H^{p}_{\rm
at,{\mathscr{D}}}(\mu)$, one can argue by mimicking the corresponding proof in
[27] and [26]. We omit the details. ∎

The following theorem can be found in [13] and ensures that there exist enough
dyadic cubes to cover all balls on homogeneous spaces.

######  Theorem 6.5.

Given a set of reference points
$\\{z^{k}_{\alpha}\\},k\in{\mathbb{Z}},\alpha\in{\mathcal{A}}_{k}$, suppose
that there exists constant $\delta\in(0,1)$ that satisfies
$96A^{6}_{0}\delta\leqslant 1$. Then there exists a finite collection of
families ${\mathscr{D}}^{t},t=1,2,\cdots,K=K(A_{0},A_{1},\delta)<\infty$,
where each ${\mathscr{D}}^{t}$ is a collection of dyadic cubes, associated to
dyadic points
$\\{z^{k}_{\alpha}\\},k\in{\mathbb{Z}},\alpha\in{\mathcal{A}}_{k}$, with the
properties (6.1)-(6.5) in Theorem 6.1.

In addition, the following property is satisfied:

(6.8) $\text{for every }B(x,r)\subseteq\Omega,\text{ there exist $t$ and
$Q\in{\mathscr{D}}^{t}$ with $B(x,r)\subseteq Q$ and $diam(Q)\leqslant Cr$.}$

The constant $C<\infty$ in (6.8) only depends on the quasi-metric constant
$A_{0}$ and the parameter $\delta$.

By virtue of Proposition 6.4 and Theorem 6.5, we have the following theorem,
which extends Mei’s result in [17].

######  Theorem 6.6.

For $0<p\leqslant 1$, we have

(6.9) $H^{p}_{\rm at}(\mu)=\sum_{t=1}^{K}H^{p}_{\rm
at,{\mathscr{D}}^{t}}(\mu)=\sum_{t=1}^{K}H^{p}_{{\mathscr{D}}^{t}}(\mu)=\sum_{t=1}^{K}H^{p}_{m,{\mathscr{D}}^{t}}(\mu)=\sum_{t=1}^{K}h^{p}_{{\mathscr{D}}^{t}}(\mu).$

###### Proof.

Let $p\in(0,1]$ be fixed. In view of Proposition 6.4, it suffices to show
$H^{p}_{\rm at}(\mu)=\sum\limits_{t=1}^{K}H^{p}_{\rm
at,{\mathscr{D}}^{t}}(\mu)$. We prove it via comparing the atoms. Let $a$ be a
$(p,\infty)$-atom in $H^{p}_{\rm at}(\mu)$. Then there exists a ball $B$ such
that

$\text{supp}(a)\subset B,\
\|a\|_{\infty}\leqslant\left(\mu(B)\right)^{-\frac{1}{p}},\
\int_{B}a(x)d\mu=0.$

By Theorem 6.5, there exist $t$ and a cube $Q\in{\mathscr{D}}^{t}$ such that
$B\subset Q$, and $\mu(Q)\lesssim\mu(B)$. Then

$\text{supp}(a)\subset B\subset Q,\
\|a\|_{\infty}\leqslant\left(\mu(B)\right)^{-\frac{1}{p}}\lesssim\left(\mu(Q)\right)^{-\frac{1}{p}},\
\int_{Q}ad\mu=0,$

which implies that $a$ is a constant multiple of a simple $(p,\infty)$-atom in
$H^{p}_{\rm at,{\mathscr{D}}^{t}}(\mu)$. Thus

(6.10) $H^{p}_{\rm at}(\mu)\subset\sum_{t=1}^{K}H^{p}_{\rm
at,{\mathscr{D}}^{t}}(\mu).$

For any $t=1,2,\cdots,K$ and for any given simple $(p,\infty)$-atom $b$ in
$H^{p}_{\rm at,{\mathscr{D}}^{t}}(\mu)$, there exists $Q\in{\mathscr{D}}^{t}$
such that

$\text{supp}(b)\subset Q,\
\|b\|_{\infty}\leqslant\left(\mu(Q)\right)^{-\frac{1}{p}},\ \int_{Q}bd\mu=0.$

By Theorem 6.1, there exists a ball $B$ such that $Q\subset B$ and
$\mu(Q)\gtrsim\mu(B)$. Hence

$\text{supp}(b)\subset Q\subset B,\
\|b\|_{\infty}\leqslant\left(\mu(Q)\right)^{-\frac{1}{p}}\lesssim\left(\mu(B)\right)^{-\frac{1}{p}},\
\int_{B}bd\mu=0,$

which implies that a multiple of $b$ is also a $(p,\infty)$-atom in
$H^{p}_{\rm at}(\mu)$, thus

(6.11) $\sum_{t=1}^{K}H^{p}_{\rm at,{\mathscr{D}}^{t}}(\mu)\subset H^{p}_{\rm
at}(\mu).$

To complete the proof of the theorem, combine (6.10) and (6.11). ∎

######  Remark 6.7.

Theorem 5.5 follows immediately from Corollary 4.2, Proposition 6.4 and
Theorem 6.6, which simplifies the original proof by Coifman and Weiss in [7].

By duality and Theorem 6.6, we recover the following result of [13], which is
an extension of a result due to Mei [17]:

(6.12) $BMO(\mu)=\bigcap_{t=1}^{K}BMO^{{\mathscr{D}}^{t}}(\mu).$

We will now establish an analogous result for
${\mathcal{L}}_{\alpha_{p}}(\mu)$ $(0<p<1)$.

######  Theorem 6.8.

For $0<p<1$,

${\mathcal{L}}_{\alpha_{p}}(\mu)=\bigcap_{t=1}^{K}\Lambda_{2}^{{\mathscr{D}}^{t}}(\alpha_{p}).$

###### Proof.

By Theorem 6.1, for any $Q\in{\mathscr{D}}^{t}$ (and $t=1,2,\cdots,K$), there
exists a ball $B$ such that $Q\subset B$ and $\mu(B)\lesssim\mu(Q)$. If
$f\in{\mathcal{L}}_{\alpha_{p}}(\mu)$, then for any $x,y\in Q$, we have

$|f(x)-f(y)|\leq\|f\|_{{\mathcal{L}}_{\alpha_{p}}(\mu)}\mu(B)^{\alpha_{p}}\lesssim\|f\|_{{\mathcal{L}}_{\alpha_{p}}(\mu)}\mu(Q)^{\alpha_{p}}.$

We thus have

$\displaystyle\|f\|_{\Lambda_{2}^{{\mathscr{D}}^{t}}(\alpha_{p})}$
$\displaystyle\leq\sup_{Q\in{\mathscr{D}}^{t}}\left(\mu(Q)\right)^{-\frac{1}{2}-\alpha_{p}}\left(\mu(Q)^{-2}\int_{Q}\left(\int_{Q}|f(x)-f(y)|d\mu(y)\right)^{2}d\mu(x)\right)^{\frac{1}{2}}$
$\displaystyle\leqslant\sup_{Q\in{\mathscr{D}}^{t}}\left(\mu(Q)\right)^{-\frac{1}{2}-\alpha_{p}}\left(\int_{Q}\|f\|^{2}_{{\mathcal{L}}_{\alpha_{p}}(\mu)}\mu\left(Q\right)^{2\alpha_{p}}d\mu\right)^{\frac{1}{2}}$
$\displaystyle\lesssim\|f\|_{{\mathcal{L}}_{\alpha_{p}}(\mu)},$

which yields

(6.13)
${\mathcal{L}}_{\alpha_{p}}(\mu)\subset\bigcap_{t=1}^{K}\Lambda_{2}^{{\mathscr{D}}^{t}}(\alpha_{p}).$

Conversely, let
$f\in\bigcap\limits_{t=1}^{K}\Lambda_{2}^{{\mathscr{D}}^{t}}(\alpha_{p}).$ For
$Q\in{\mathscr{D}}^{t}$, by Theorem 4.4,

$|f(x)-f_{Q}|\lesssim\mu(Q)^{\alpha_{p}}\|f\|_{\Lambda_{2}^{{\mathscr{D}}^{t}}(\alpha_{p})}\
\ \ \forall x\in Q,$

which implies that for any $x,y\in Q$,

(6.14)
$|f(x)-f(y)|\lesssim\mu(Q)^{\alpha_{p}}\|f\|_{\Lambda_{2}^{{\mathscr{D}}^{t}}(\alpha_{p})}.$

For any ball $B\subset\Omega$, by Theorem 6.5, there exist $t$ and
$Q\in{\mathscr{D}}^{t}$ such that $B\subset Q$ and $\mu(Q)\lesssim\mu(B).$
Then for any $x,y\in B$, by (6.14)

$|f(x)-f(y)|\lesssim\mu(B)^{\alpha_{p}}\|f\|_{\Lambda_{2}^{{\mathscr{D}}^{t}}(\alpha_{p})}.$

Thus

$\|f\|_{{\mathcal{L}}_{\alpha_{p}}}\lesssim\sum_{t=1}^{K}\|f\|_{\Lambda_{2}^{{\mathscr{D}}^{t}}(\alpha_{p})},$

which implies

(6.15)
$\bigcap_{t=1}^{K}\Lambda_{2}^{{\mathscr{D}}^{t}}(\alpha_{p})\subset{\mathcal{L}}_{\alpha_{p}}(\mu).$

The theorem follows from (6.13) and (6.15). ∎

######  Remark 6.9.

Theorem 6.6 and Theorem 6.8 give a simple proof of Theorem 5.6 originally
established by Coifman and Weiss [7]:

$(H^{p}_{\rm at}(\mu))^{*}=\left(\sum_{t=1}^{K}H^{p}_{\rm
at,{\mathscr{D}}^{t}}(\mu)\right)^{*}=\bigcap_{t=1}^{K}(H^{p}_{\rm
at,{\mathscr{D}}^{t}}(\mu))^{*}=\bigcap_{t=1}^{K}\Lambda_{2}^{{\mathscr{D}}^{t}}(\alpha_{p})={\mathcal{L}}_{\alpha_{p}}(\mu).$

##  7\. Bilinear decompositions for dyadic martingales on homogeneous spaces

In this section, we focus on bilinear decompositions arising in the study of
products between elements in spaces of dyadic martingales on homogeneous
spaces introduced in the previous section. In the setting of homogeneous
spaces, due to their quasi-metrics and measures, the dyadic martingales behave
worse than martingales in probability spaces and the underlying analysis is
more intricate.

In §7.1 we prove appropriate generalized Hölder-type inequalities on
homogeneous spaces (see Lemmas 7.2 and 7.4 below). We then introduce a class
of pointwise multipliers of $\Lambda_{1,+}^{\mathscr{D}}(\alpha_{p})$ and
$BMO^{\mathscr{D}}(\mu)$; see Theorem 7.5 below. Using Theorem 7.5, we define
products between dyadic martingale Hardy spaces on homogeneous spaces and
their duals and then, in §7.2 we establish analogues of the results of
Sections 3 and 4 in the setting of homogeneous spaces.

###  7.1. A generalized Hölder-type inequality

Let $0<p\leq 1$ and ${\mathscr{D}}$ be a dyadic system, constructed as in
Theorem 6.1. The martingale Musielak–Orlicz Hardy spaces
$H^{\Psi_{p}}_{\mathscr{D}}(\mu)$ consist of all measurable functions $f$ on
$(\Omega,{\mathcal{F}},\mu)$ such that $s(f)\in L^{\Psi_{p}}(\Omega)$ where
$O\in\Omega$ is a fixed point, and

$\displaystyle\Psi_{1}(x,t)$
$\displaystyle:=\dfrac{t}{\log\left(e+d(x,O)\right)+\log(e+t)},$
$\displaystyle\Psi_{p}(x,t)$
$\displaystyle:=\frac{t}{1+\left\\{t[1+\mu(B(O,d(x,O)))]\right\\}^{1-p}}\quad(0<p<1).$

Note that $L^{\Psi_{p}}(\Omega)$ is a quasi-normed space.

Let $M:=(C_{\mu}+1)\log\left(e+d(x,O)\right)$. By (2.5) we obtain

(7.1) $\Psi_{1}(x,st)\lesssim(e+d(x,O))^{-(C_{\mu}+1)}e^{t}+s\lesssim
w(x)e^{t}+s,\quad\text{for all}\ x\in\Omega,s,t>0,$

where $w:\Omega\to{}_{+}$ is a weight function with

(7.2) $w(x)\lesssim\min\left\\{1,d(x,O)^{-(C_{\mu}+1)}\right\\}.$

Let $Q^{0}\in{\mathcal{F}}_{0}$ be the dyadic cube such that $O\in Q^{0}$. For
$g\in BMO^{\mathscr{D}}(\mu)$, define

$\|g\|_{BMO^{\mathscr{D}}_{+}(\mu)}:=\sup\limits_{\alpha\in{\mathcal{A}}_{0}}\frac{|g_{Q_{\alpha}^{0}}-g_{Q^{0}}|}{\log\left(e+d(z_{\alpha}^{0},O)\right)}+|g_{Q^{0}}|+\|g\|_{BMO^{\mathscr{D}}(\mu)},$

where $Q_{\alpha}^{0}\in{\mathcal{F}}_{0}$ is a dyadic cube with its center
$z_{\alpha}^{0}$ and ${\mathcal{A}}_{0}$ is the index set in Theorem 6.1.
Denote by $BMO^{\mathscr{D}}_{+}(\mu)$ the space consisting of all $g\in
BMO^{\mathscr{D}}(\mu)$ such that $\|g\|_{BMO^{\mathscr{D}}_{+}(\mu)}<\infty$.
It is not difficult to verify that $\|\cdot\|_{BMO^{\mathscr{D}}_{+}(\mu)}$ is
a norm on $BMO^{\mathscr{D}}_{+}(\mu)$.

######  Remark 7.1.

If we consider the dyadic martingales on n, by taking appropriate cubes
$Q^{0}$ one shows that if $g\in BMO^{\mathscr{D}}(\mu)$, then $g\in
BMO^{\mathscr{D}}_{+}(\mu)$. Note that if $g\in BMO(\mu)$, then $g\in
BMO^{\mathscr{D}}_{+}(\mu)$. Moreover,

$\|g\|_{BMO^{\mathscr{D}}_{+}(\mu)}\lesssim\|g\|_{BMO(\mu)}+|g_{Q^{0}}|.$

We now introduce the following generalized Hölder inequality for
$L^{1}(\Omega,{\mathcal{F}},\mu)$ and $BMO^{\mathscr{D}}_{+}(\mu)$.

######  Lemma 7.2.

If $f\in L^{1}(\Omega,{\mathcal{F}},\mu)$ and $g\in
BMO^{\mathscr{D}}_{+}(\mu)$, then $f\cdot g\in L^{\Psi_{1}}(\Omega)$.
Moreover,

(7.3)
$\|fg\|_{L^{\Psi_{1}}(\Omega)}\lesssim\|f\|_{1}\|g\|_{BMO^{\mathscr{D}}_{+}(\mu)}.$

###### Proof.

Without loss of generality, assume $\|f\|_{1}\leq 1$,
$\|g\|_{BMO^{\mathscr{D}}_{+}(\mu)}\leq 1$ and $g_{Q^{0}}=0$. It suffices to
show that

$\int_{\Omega}\Psi_{1}(x,|f(x)g(x)|)d\mu\lesssim 1.$

Let $S_{k}:=B(O,C_{0}\delta^{k})\setminus B(O,C_{0}\delta^{k+1})$ for $k<0$
and $S_{0}:=B(O,C_{0})$, where $\delta\in(0,1)$ is the constant in Theorem
6.1. Then for each $k\leq 0$, there exists a finite index subset
${\mathcal{B}}_{k}\subset{\mathcal{A}}_{0}$ such that
$B(O,C_{0}\delta^{k})\subset\bigcup\limits_{\alpha\in{\mathcal{B}}_{k}}Q_{\alpha}^{0}$
(where $Q^{0}_{\alpha}\in{\mathcal{F}}_{0}$) and

$\sum\limits_{\alpha\in{\mathcal{B}}_{k}}\mu\left(Q_{\alpha}^{0}\right)=\mu\left(\bigcup\limits_{\alpha\in{\mathcal{B}}_{k}}Q_{\alpha}^{0}\right)\leq\mu\left(B(O,2A_{0}C_{0}\delta^{k})\right)\lesssim\delta^{C_{\mu}k}.$

Take $s=\nu^{-1}|f(x)|,t=\nu|g(x)|$ in (7.1), one has

$\displaystyle\int_{\Omega}\Psi_{1}(x,|f(x)g(x)|)d\mu$
$\displaystyle=\sum_{k=-\infty}^{0}\sum\limits_{\alpha\in{\mathcal{B}}_{k}}\int_{S_{k}\cap
Q^{0}_{\alpha}}\Psi_{1}(x,|f(x)g(x)|)d\mu$
$\displaystyle\lesssim\sum_{k=-\infty}^{0}\sum\limits_{\alpha\in{\mathcal{B}}_{k}}\int_{S_{k}\cap
Q^{0}_{\alpha}}w(x)e^{\nu|g(x)|}d\mu+\nu^{-1}\|f\|_{1}.$

Therefore,

(7.4) $\int_{\Omega}\Psi_{1}(x,|f(x)g(x)|)d\mu\lesssim
T_{1}+\nu^{-1}\|f\|_{1},$

where

$T_{1}:=\sum_{k=-\infty}^{0}\sum\limits_{\alpha\in{\mathcal{B}}_{k}}\int_{S_{k}\cap
Q^{0}_{\alpha}}w(x)e^{\nu\left|g(x)-g_{Q^{0}_{\alpha}}\right|}e^{\nu\left|g_{Q^{0}_{\alpha}}\right|}d\mu.$

Let $\nu:=\frac{\min\\{\kappa,1\\}}{2}>0$ (where $\kappa$ is defined in
Theorem 2.11), by (7.2) and Theorem 4.1, one has

$\displaystyle T_{1}$
$\displaystyle\lesssim\sum_{k=-\infty}^{0}\sum\limits_{\alpha\in{\mathcal{B}}_{k}}\dfrac{\mu(Q^{0}_{\alpha})\left(e+d(z^{0}_{\alpha},O)\right)^{\frac{1}{2}}}{\delta^{(k+1)(C_{\mu}+1)}}$
$\displaystyle\lesssim\sum_{k=-\infty}^{0}\sum\limits_{\alpha\in{\mathcal{B}}_{k}}\dfrac{\mu(Q^{0}_{\alpha})\delta^{\frac{k}{2}}}{\delta^{(k+1)(C_{\mu}+1)}}\lesssim\sum_{k=-\infty}^{0}\dfrac{\delta^{C_{\mu}k}\delta^{\frac{k}{2}}}{\delta^{C_{\mu}k+k}}$
$\displaystyle\lesssim\sum_{k=-\infty}^{0}\delta^{-\frac{1}{2}k},$

and hence

(7.5) $T_{1}\lesssim 1.$

Combine (7.4), (7.5) and the fact that $\nu^{-1}\|f\|_{1}\lesssim 1$, and the
proof is complete. ∎

We consider the case $0<p<1$. Define

$\|g\|_{\Lambda_{1,+}^{\mathscr{D}}(\alpha_{p})}:=\sup\limits_{\alpha\in{\mathcal{A}}_{0}}\frac{|g_{Q_{\alpha}^{0}}-g_{Q^{0}}|}{1+\mu\left\\{B\left(O,d(z^{0}_{\alpha},O)\right)\right\\}^{\alpha_{p}}}+|g_{Q^{0}}|+\|g\|_{\Lambda_{1}^{\mathscr{D}}(\alpha_{p})},$

Denote by $\Lambda_{1,+}^{\mathscr{D}}(\alpha_{p})$ the space consisting of
all $g\in\Lambda_{1}^{\mathscr{D}}(\alpha_{p})$ such that
$\|g\|_{\Lambda_{1,+}^{\mathscr{D}}(\alpha_{p})}<\infty$. It is easy to verify
that $\|\cdot\|_{\Lambda_{1,+}^{\mathscr{D}}(\alpha_{p})}$ is a norm on
$\Lambda_{1,+}^{\mathscr{D}}(\alpha_{p})$.

######  Remark 7.3.

If we consider the dyadic martingales on n, by taking appropriate cubes
$Q^{0}$ one can show that if $g\in\Lambda_{1}^{\mathscr{D}}(\alpha_{p})$, then
$g\in\Lambda_{1,+}^{\mathscr{D}}(\alpha_{p})$. Note that if
$g\in{\mathcal{L}}_{\alpha_{p}}(\mu)$, then
$g\in\Lambda_{1,+}^{\mathscr{D}}(\alpha_{p})$. Moreover,

$\|g\|_{\Lambda_{1,+}^{\mathscr{D}}(\alpha_{p})}\lesssim\|g\|_{{\mathcal{L}}_{\alpha_{p}}(\mu)}+|g_{Q^{0}}|.$

Next we present a generalized Hölder inequality for
$L^{p}(\Omega,{\mathcal{F}},\mu)$ and
$\Lambda_{1,+}^{\mathscr{D}}(\alpha_{p})$ for $0<p<1$.

######  Lemma 7.4.

If $f\in L^{p}(\Omega,{\mathcal{F}},\mu)$ and
$g\in\Lambda_{1,+}^{{\mathscr{D}}}(\alpha_{p})$ for $0<p<1$, then $f\cdot g\in
L^{\Psi_{p}}(\mu)$. Moreover,

(7.6)
$\|fg\|_{L^{\Psi_{p}}(\Omega)}\lesssim\|f\|_{p}\|g\|_{\Lambda_{1,+}^{{\mathscr{D}}}(\alpha_{p})}.$

###### Proof.

Without loss of generality, assume $\|f\|_{p}\leq 1$,
$\|g\|_{\Lambda^{\mathscr{D}}_{1,+}(\alpha_{p})}\leq 1$ and $g_{Q^{0}}=0$. It
suffices to show that

$\int_{\Omega}\Psi_{p}(x,|f(x)g(x)|)d\mu\lesssim 1.$

Take the same family of sets $\\{S_{k}\\}_{k\leq 0}$ as above. From Theorem
4.4, we know that for $x\in Q^{0}_{\alpha}$,

$\displaystyle|g(x)|$
$\displaystyle=|g(x)-g_{Q^{0}}|\leqslant\left|g(x)-g_{Q^{0}_{\alpha}}\right|+\left|g_{Q_{\alpha}^{0}}-g_{Q^{0}}\right|$
$\displaystyle\leq\left(\mu(Q^{0}_{\alpha})\right)^{\alpha_{p}}+\mu\left\\{B\left(O,d(z^{0}_{\alpha},O)\right)\right\\}^{\alpha_{p}}+1$
$\displaystyle\lesssim\mu\left(B(O,2A_{0}C_{0}\delta^{k})\right)^{\alpha_{p}}+1.$

Therefore

$\displaystyle\int_{\Omega}\Psi_{p}(x,|f(x)g(x)|)d\mu$
$\displaystyle=\sum_{k=-\infty}^{0}\sum\limits_{\alpha\in{\mathcal{B}}_{k}}\int_{S_{k}\cap
Q^{0}_{\alpha}}\frac{|g(x)||f(x)|}{1+\left\\{|g(x)||f(x)|\left[1+\mu\left(B(O,d(x,O)\right)\right]\right\\}^{1-p}}d\mu$
$\displaystyle\lesssim\sum_{k=-\infty}^{0}\sum\limits_{\alpha\in{\mathcal{B}}_{k}}\int_{S_{k}\cap
Q^{0}_{\alpha}}\frac{|g(x)|^{p}|f(x)|^{p}}{1+\mu\left(B(O,d(x,O)\right)^{1-p}}d\mu$
$\displaystyle\lesssim\sum_{k=-\infty}^{0}\sum\limits_{\alpha\in{\mathcal{B}}_{k}}\int_{S_{k}\cap
Q^{0}_{\alpha}}\frac{\mu\left(B(O,2A_{0}C_{0}\delta^{k})\right)^{\alpha_{p}p}+1}{\left\\{1+\mu\left(B(O,C_{0}\delta^{k+1}\right)\right\\}^{1-p}}|f(x)|^{p}d\mu$
$\displaystyle\lesssim 1,$

which finishes the proof. ∎

We are now about to present the analogues of the results in Sections 3 and 4
concerning bilinear decompositions for dyadic martingales on homogeneous
spaces. To this end, we need to define the product between martingale Hardy
spaces and their dual spaces first. As in the probability setting, we regard
the product in the sense of distribution as follows: for $0<p<1$,

$\langle f\times g,h\rangle:=\langle h\cdot g,f\rangle,\quad f\in
H^{p}_{\rm{at},{\mathscr{D}}}(\mu),\
g\in\Lambda_{1,+}^{\mathscr{D}}(\alpha_{p}),$

where $h$ is a test function such that $h\cdot g$ is in
$\Lambda_{1,+}^{\mathscr{D}}(\alpha_{p})$. For $p=1$, we may define the
product between $H^{1}_{\rm{at},{\mathscr{D}}}(\mu)$ and
$BMO^{\mathscr{D}}(\mu)$ analogously. To this end, we need to introduce some
pointwise multipliers of $\Lambda_{1,+}^{\mathscr{D}}(\alpha_{p})$ and
$BMO^{\mathscr{D}}(\mu)$.

Denote the space of test functions by ${\mathcal{H}}(\alpha_{p})$ $(0<p\leq
1)$, and a measurable function $h$ is a test function if it satisfies the
following properties:

(7.7)
$|h(x)|\lesssim\frac{1}{\left(1+\mu(B(O,d(x,O)))^{\alpha_{p}}\right)\log(e+d(x,O))},\quad\forall
x\in\Omega,$

and

(7.8)
$|h(y)-h(z)|\lesssim\frac{\mu(B)^{\alpha_{p}}}{\left(1+\mu[B(O,1+r+d(c_{B},O))]^{\alpha_{p}}\right)\log(e+r+d(c_{B},O))}$

whenever $y,z$ are both contained in a ball $B$ with center $c_{B}$ and radius
$r\leq\frac{d(c_{B},O)}{2A_{0}}+1$.

It is obvious that ${\mathcal{H}}(\alpha_{p})\subset L^{\infty}(\Omega)$. The
following theorem shows that if $h\in{\mathcal{H}}(\alpha_{p})$, then $h$ is a
pointwise multiplier of $\Lambda_{1,+}^{\mathscr{D}}(\alpha_{p})$.

######  Theorem 7.5.

For $0<p<1$ and any dyadic system ${\mathscr{D}}$, ${\mathcal{H}}(\alpha_{p})$
is a space of pointwise multipliers of
$\Lambda_{1,+}^{\mathscr{D}}(\alpha_{p})$. For $p=1$, ${\mathcal{H}}(0)$ is a
space of pointwise multipliers of $BMO^{\mathscr{D}}_{+}(\mu)$. More
precisely, for any $g\in\Lambda_{1,+}^{{\mathscr{D}}}(\alpha_{p})$ and
$h\in{\mathcal{H}}(\alpha_{p})$, we have

$\|g\cdot
h\|_{\Lambda_{1,+}^{{\mathscr{D}}}(\alpha_{p})}\lesssim\|g\|_{\Lambda_{1,+}^{{\mathscr{D}}}(\alpha_{p})}\left(\|h\|_{L^{\infty}(\Omega)}+1\right),$

and for any $g\in BMO^{\mathscr{D}}_{+}(\mu)$ and $h\in{\mathcal{H}}(0)$, we
have

$\|g\cdot
h\|_{BMO^{\mathscr{D}}_{+}(\mu)}\lesssim\|g\|_{BMO^{\mathscr{D}}_{+}(\mu)}\left(\|h\|_{L^{\infty}(\Omega)}+1\right).$

###### Proof.

First, we consider the case $0<p<1$. Assume that
$g\in\Lambda_{1,+}^{{\mathscr{D}}}(\alpha_{p})$ and
$h\in{\mathcal{H}}(\alpha_{p})$. According to [20], it suffices to show that

(7.9)
$\sup_{Q}\frac{|g_{Q}|}{\mu(Q)^{\alpha_{p}+1}}\left(\int_{Q}|h(x)-h_{Q}|dx\right)<\infty,$

where $Q$ runs over all dyadic cubes in ${\mathscr{D}}$.

If $Q\subset Q^{0}_{\beta}$ for some $\beta\in{\mathcal{A}}_{0}$, there exists
a collection of cubes $Q=Q_{0}\subset Q_{1}\subset\cdots\subset
Q_{N}=Q^{0}_{\beta}$ such that there exists a universal constant
$0<\delta^{{}^{\prime}}<1$ with
$\mu(Q_{k-1})\leq\delta^{{}^{\prime}}\mu(Q_{k})$. Hence

$\displaystyle|g_{Q}-g_{Q^{0}_{\beta}}|$
$\displaystyle\leq\sum_{k=1}^{N}|g_{Q_{k}}-g_{Q_{k-1}}|\lesssim\sum_{k=1}^{N}\mu(Q_{k})^{\alpha_{p}}\|g\|_{\Lambda_{1,+}^{{\mathscr{D}}}(\alpha_{p})}$
$\displaystyle\lesssim\|g\|_{\Lambda_{1,+}^{{\mathscr{D}}}(\alpha_{p})}\sum_{k=1}^{N}\int^{\mu(Q_{k})}_{\mu(Q_{k-1})}t^{\alpha_{p}-1}dt$
$\displaystyle\lesssim\mu(Q^{0}_{\beta})^{\alpha_{p}}\|g\|_{\Lambda_{1,+}^{{\mathscr{D}}}(\alpha_{p})}.$

Similarly, if $Q_{\beta}^{0}\subset Q$, we have

$|g_{Q}-g_{Q^{0}_{\beta}}|\lesssim\mu(Q)^{\alpha_{p}}\|g\|_{\Lambda_{1,+}^{{\mathscr{D}}}(\alpha_{p})}.$

By Theorem 6.1, there exists a ball $B$, with center $c_{B}$ and radius $r$,
such that $Q\subset B$ and $\mu(B)\lesssim\mu(Q)$.

If $Q^{0}_{\beta}\subset Q$ and $r>\frac{d(O,c_{B})}{2A_{0}}+1$, for any $x\in
B(O,r)$, we have $d(c_{B},x)\leq
A_{0}(d(c_{B},O)+d(O,x))<(2A_{0}^{2}+A_{0})r$. Then
$\mu(Q)\gtrsim\mu(B)\gtrsim
C_{\mu}^{-(2A_{0}^{2}+A_{0})}\mu\left(B(O,r)\right)\gtrsim 1$. Similarly, we
also have $d(z_{\beta}^{0},O)<(2A_{0}^{2}+A_{0})r$ and
$\mu\left\\{B\left(O,d(z^{0}_{\beta},O)\right)\right\\}\lesssim\mu(B)\lesssim\mu(Q)$.
Thus

$\displaystyle\frac{|g_{Q}|}{\mu(Q)^{\alpha_{p}+1}}\left(\int_{Q}|h(x)-h_{Q}|dx\right)$
$\displaystyle\lesssim\frac{|g_{Q}-g_{Q_{\beta}^{0}}|+|g_{Q_{\beta}^{0}}-g_{Q^{0}}|+|g_{Q^{0}}|}{\mu(Q)^{\alpha_{p}}}\cdot\|h\|_{L^{\infty}(\Omega)}$
$\displaystyle\lesssim\frac{\mu(Q)^{\alpha_{p}}+\mu\left\\{B\left(O,d(z^{0}_{\beta},O)\right)\right\\}^{\alpha_{p}}+1}{\mu(Q)^{\alpha_{p}}}\cdot\|g\|_{\Lambda_{1,+}^{{\mathscr{D}}}(\alpha_{p})}\|h\|_{L^{\infty}(\Omega)}$
$\displaystyle\lesssim\|g\|_{\Lambda_{1,+}^{{\mathscr{D}}}(\alpha_{p})}\|h\|_{L^{\infty}(\Omega)}.$

If $Q^{0}_{\beta}\subset Q$ and $r\leq\frac{d(O,c_{B})}{2A_{0}}+1$, for any
$x\in B$, we have $d(x,O)\leq A_{0}(d(O,c_{B})+r)$, then
$\mu(Q)\lesssim\mu\left(B(O,A_{0}(d(O,c_{B})+r)\right)$. Thus

$\displaystyle\frac{|g_{Q}|}{\mu(Q)^{\alpha_{p}+1}}\left(\int_{Q}|h(x)-h_{Q}|dx\right)$
$\displaystyle\lesssim\frac{|g_{Q}-g_{Q_{\beta}^{0}}|+|g_{Q_{\beta}^{0}}-g_{Q^{0}}|+|g_{Q^{0}}|}{\mu(Q)^{\alpha_{p}+1}}\frac{\mu(B)^{\alpha_{p}+1}}{\left(1+\mu[B(O,1+r+d(O,c_{B}))]\right)^{\alpha_{p}}}$
$\displaystyle\lesssim\frac{\left(\mu(Q)^{\alpha_{p}}+\mu\left\\{B\left(O,d(z^{0}_{\beta},O)\right)\right\\}^{\alpha_{p}}+1\right)\|g\|_{\Lambda_{1,+}^{{\mathscr{D}}}(\alpha_{p})}}{\left(1+\mu[B(O,1+r+d(O,c_{B}))]\right)^{\alpha_{p}}}$
$\displaystyle\lesssim\|g\|_{\Lambda_{1,+}^{{\mathscr{D}}}(\alpha_{p})}.$

If $Q\subset Q^{0}_{\beta}$, from Theorem 6.1, we can choose $C_{0}$
sufficiently small such that $C_{1}=2A_{0}C_{0}\leq 1$, then $r\leq
C_{1}\leq\frac{d(c_{B},O)}{2A_{0}}+1$. For any $x\in Q^{0}_{\beta}$, we have
$d(O,x)\leq A_{0}(d(O,z^{0}_{\beta})+C_{1})$. Then

$\mu(Q^{0}_{\beta})\lesssim\mu\left\\{B\left(O,A_{0}(d(O,z_{\beta}^{0})+C_{1})\right)\right\\}.$

By a calculation similar to the one presented above, we get the desired
result.

Combining the above estimates, we finish our proof for $0<p<1$. The case for
$p=1$ is similar. ∎

######  Remark 7.6.

Note that in Theorem 7.5, the dyadic system ${\mathscr{D}}$ is arbitrary. Then
from Theorem 6.8 and (6.12), we conclude that ${\mathcal{H}}(\alpha_{p})$ is a
space of pointwise multipliers of ${\mathcal{L}}_{\alpha_{p}}(\mu)$ and
${\mathcal{H}}(0)$ is a space of pointwise multipliers of $BMO(\mu)$.

###  7.2. Bilinear decompositions

Assume $f\in H^{1}_{{\mathscr{D}}}(\mu),g\in BMO^{{\mathscr{D}}}_{+}(\mu)$ or
$f\in H^{p}_{{\mathscr{D}}}(\mu),g\in\Lambda^{\mathscr{D}}_{1,+}(\alpha_{p})$,
$0<p<1$.

Denote by $\mathcal{H}^{p}_{{\mathscr{D}},\text{fin}}(\mu)\ (0<p\leq 1)$ the
linear space consisting of all functions which can be written as a finite sum
of simple $(p,\infty)$-atoms. Thus if
$f\in\mathcal{H}^{p}_{{\mathscr{D}},\text{fin}}(\mu)$, $f$ is locally
supported, $f\in L^{1}(\Omega)\cap L^{\infty}(\Omega)$ and
$\int_{\Omega}fd\mu=0$. Note that
$\mathcal{H}^{p}_{{\mathscr{D}},\text{fin}}(\mu)$ is dense in
$H^{p}_{{\mathscr{D}}}(\mu)$ with respect to the norm
$\|\cdot\|_{H^{p}_{\mathscr{D}}(\mu)}$.

In the following, we shall only consider the case where
$f\in\mathcal{H}^{p}_{{\mathscr{D}},\text{fin}}(\mu)$. Then $f\cdot g\in
L^{1}(\Omega)$, and we can write

(7.10) $f\cdot g=\Pi_{1}(f,g)+\Pi_{2}(f,g)+\Pi_{3}(f,g),$

where

$\Pi_{1}(f,g):=\sum_{k=1}^{\infty}d_{k}fd_{k}g,\quad\Pi_{2}(f,g):=\sum_{k=1}^{\infty}f_{k-1}d_{k}g\quad\text{and}\quad\Pi_{3}(f,g):=\sum_{k=1}^{\infty}g_{k-1}d_{k}f.$

######  Theorem 7.7.

We have the following:

  1. (1)

$\Pi_{1}$ is a bilinear bounded operator from
$H^{1}_{{\mathscr{D}}}(\mu)\times BMO^{{\mathscr{D}}}_{+}(\mu)$ to
$L^{1}(\Omega)$, and
when $0<p<1$, $\Pi_{1}$ is a bilinear bounded operator from
$H^{p}_{{\mathscr{D}}}(\mu)\times\Lambda_{1,+}^{{\mathscr{D}}}(\alpha_{p})$ to
$L^{1}(\Omega)$.

  2. (2)

$\Pi_{2}$ is a bilinear bounded operator from
$H^{1}_{{\mathscr{D}}}(\mu)\times BMO^{{\mathscr{D}}}_{+}(\mu)$ to
$H^{1}_{\mathscr{D}}(\mu)$, and
when $0<p<1$, $\Pi_{2}$ is a bilinear bounded operator from
$H^{p}_{{\mathscr{D}}}(\mu)\times\Lambda_{1,+}^{{\mathscr{D}}}(\alpha_{p})$ to
$H^{1}_{\mathscr{D}}(\mu)$.

  3. (3)

$\Pi_{3}$ is a bilinear bounded operator from
$H^{1}_{{\mathscr{D}}}(\mu)\times BMO^{{\mathscr{D}}}_{+}(\mu)$ to
$H^{\Psi_{1}}_{\mathscr{D}}(\mu)$, and
when $0<p<1$, $\Pi_{3}$ is a bilinear bounded operator from
$H^{p}_{{\mathscr{D}}}(\mu)\times\Lambda_{1,+}^{{\mathscr{D}}}(\alpha_{p})$ to
$H^{\Psi_{p}}_{\mathscr{D}}(\mu)$.

###### Proof.

For $\Pi_{1}$ and $\Pi_{2}$, we can argue as in the corresponding part of the
proof of Theorem 1.1. As for $\Pi_{3}$, we can also argue as in the
corresponding part of the proof Theorem 1.1, where in the homogeneous setting
one needs to apply Lemma 7.2 and Lemma 7.4. We omit the details. ∎

######  Remark 7.8.

For $\Pi_{1}$ and $\Pi_{2}$, the condition $H^{1}_{{\mathscr{D}}}(\mu)\times
BMO^{{\mathscr{D}}}_{+}(\mu)$ and
$H^{p}_{{\mathscr{D}}}(\mu)\times\Lambda_{1,+}^{{\mathscr{D}}}(\alpha_{p})$
can be in fact replaced by $H^{1}_{{\mathscr{D}}}(\mu)\times
BMO^{{\mathscr{D}}}(\mu)$ and
$H^{p}_{{\mathscr{D}}}(\mu)\times\Lambda_{1}^{{\mathscr{D}}}(\alpha_{p})$,
respectively.

##  8\. Applications to Homogeneous spaces

In the first part of this section we show that
$H^{\Psi_{p}}_{\mathscr{D}}(\mu)$ admits an atomic decomposition for $0<p<1$,
which allows us to integrate several adjacent dyadic systems on homogeneous
spaces.

For a given dyadic system ${\mathscr{D}}$ on $\Omega$, we define the dyadic
$H^{\Psi_{p}}_{\rm at,{\mathscr{D}}}$-atom as follows.

######  Definition 8.1.

A measurable function $a$ is said to be an $H^{\Psi_{p}}_{\rm
at,{\mathscr{D}}}$-atom if

  1. (i)

$\text{supp}(a)\subset Q$ where $Q\in{\mathscr{D}}$ is a cube;

  2. (ii)

$\int_{\Omega}ad\mu=0$;

  3. (iii)

$\|a\|_{\infty}\leqslant\|1_{Q}\|_{L^{\Psi_{p}}(\Omega)}^{-1}$.

The atomic dyadic martingale Musielak–Orlicz Hardy spaces $H^{\Psi_{p}}_{\rm
at,{\mathscr{D}}}(\mu)\ (0<p<1)$ are defined in a way analogous to (5.5) and
(5.6). We first introduce the space $BMO_{\Psi_{p}}^{\mathscr{D}}(\mu)$, which
is a subspace of continuous linear functionals on finite sums of atoms.

######  Definition 8.2.

A locally integrable function $g$ is said to be a dyadic
$BMO_{\Psi_{p}}^{\mathscr{D}}(\mu)$ function associated with a dyadic system
${\mathscr{D}}$ if

$\|g\|_{BMO_{\Psi_{p}}^{\mathscr{D}}(\mu)}:=\sup_{k\in{\mathbb{Z}}}\sup_{Q\in{\mathcal{F}}_{k}}\frac{1}{\|1_{Q}\|_{L^{\Psi_{p}}(\Omega)}}\int_{Q}|g(x)-g_{k}(x)|dx<\infty.$

Then we define the atomic Musielak–Orlicz martingale Hardy spaces
$H^{\Psi_{p}}_{\rm at,{\mathscr{D}}}(\mu)$ as follows:

$H^{\Psi_{p}}_{\rm at,{\mathscr{D}}}(\mu):=\\\
\left\\{f\in\left(BMO_{\Psi_{p}}^{\mathscr{D}}(\mu)\right)^{*}:\
f=\sum_{i=0}^{\infty}\lambda_{i}a_{i},\text{ where $a_{i}$ is an
$H^{\Psi_{p}}_{\rm at,{\mathscr{D}}}(\mu)$-atom supported on a cube
$Q_{i}.$}\right\\},$

where

$\sum_{i=0}^{\infty}\int_{Q_{i}}\Psi_{p}(x,|\lambda_{i}|\|a_{i}\|_{\infty})d\mu<\infty.$

Moreover,

$\|f\|_{H^{\Psi_{p}}_{\rm
at,{\mathscr{D}}}(\mu)}:=\inf\left\\{\rho>0:\sum_{i=0}^{\infty}\int_{Q_{i}}\Psi_{p}(x,\rho^{-1}|\lambda_{i}|\|a_{i}\|_{\infty})d\mu\leqslant
1\right\\}$

Arguing as in [28], one can show that for $0<p<1$

(8.1) $H^{\Psi_{p}}_{{\mathscr{D}}}(\mu)=H^{\Psi_{p}}_{\rm
at,{\mathscr{D}}}(\mu).$

We shall now introduce the atomic Musielak–Orlicz Hardy spaces
$H^{\Psi_{p}}_{\rm at}(\mu)\ (0<p\leq 1)$ on the homogeneous space $\Omega$.
First, we present the definition of atoms for $H^{\Psi_{p}}_{\rm at}(\mu)$.

######  Definition 8.3.

A measurable function $a(x)$ is said to be an $H^{\Psi_{p}}_{\rm
at}(\mu)$-atom if

  1. (i)

$\text{supp}(a)\subset B$ where $B\subset\Omega$ is a ball;

  2. (ii)

$\int_{\Omega}ad\mu=0$;

  3. (iii)

$\|a\|_{\infty}\leqslant\|1_{B}\|_{L^{\Psi_{p}}(\Omega)}^{-1}$.

######  Definition 8.4.

A locally integrable function $g$ is said to be a $BMO_{\Psi_{p}}(\mu)$
function if

$\|g\|_{BMO_{\Psi_{p}}(\mu)}:=\sup_{B}\frac{1}{\|1_{B}\|_{L^{\Psi_{p}}(\Omega)}}\int_{B}|g(x)-g_{B}|dx<\infty,$

where $B$ runs over all balls in $\Omega$.

######  Definition 8.5.

The atomic Musielak–Orlicz Hardy spaces $H^{\Psi_{p}}_{\rm at}(\mu)\ (0<p\leq
1)$ are defined as follows:

$H^{\Psi_{p}}_{\rm at}(\mu):=\\\
\left\\{f\in\left(BMO_{\Psi_{p}}(\mu)\right)^{*}:f=\sum_{i=0}^{\infty}\lambda_{i}a_{i},\text{
where $a_{i}$ is an $H^{\Psi_{p}}_{\rm at}(\mu)$-atom supported on a ball
$B_{i}$}\right\\},$

where

$\sum_{i=0}^{\infty}\int_{B_{i}}\Psi_{p}(x,|\lambda_{i}|\|a_{i}\|_{\infty})d\mu<\infty.$

Moreover,

$\|f\|_{H^{\Psi_{p}}_{\rm
at}(\mu)}:=\inf\left\\{\rho>0:\sum_{i=0}^{\infty}\int_{B_{i}}\Psi_{p}(x,\rho^{-1}|\lambda_{i}|\|a_{i}\|_{\infty})d\mu\leqslant
1\right\\}.$

Let ${\mathscr{D}}^{t}\ (1\leq t\leq K)$ be the adjacent systems of Theorem
6.5. By arguing as in the proof of Theorem 6.6, we have the following:

######  Lemma 8.6.

For $0<p<1$, $H^{\Psi_{p}}_{\rm at}(\mu)=H^{\Psi_{p}}_{\rm
at,{\mathscr{D}}^{1}}(\mu)+H^{\Psi_{p}}_{\rm
at,{\mathscr{D}}^{2}}(\mu)+\cdots+H^{\Psi_{p}}_{\rm
at,{\mathscr{D}}^{K}}(\mu)$.

###### Proof.

It suffices to show that any dyadic $H^{\Psi_{p}}_{{\mathscr{D}}^{t}}$-atom
$a$ is a constant multiple of an $H^{\Psi_{p}}(\mu)$-atom, and any
$H^{\Psi_{p}}(\mu)$-atom $b$ is a constant multiple of a dyadic
$H^{\Psi_{p}}_{{\mathscr{D}}^{t}}$-atom.

If $B:=B(x_{0},r)$, then denote the ball $B(x_{0},Dr)$ by $DB$ for $D\geq 1$.
Denote $d:=d(x_{0},O)$. In what follows, $C(D,p,A_{0},C_{\mu})$ denotes a
constant that depends on $D,p,A_{0},C_{\mu}$ and may differ from line to line.
We first show that if

$\int_{B}\frac{1}{1+[1+\mu(B(O,d(x,O)))]^{1-p}}d\mu(x)=1,$

then

(8.2) $\int_{DB}\frac{1}{1+[1+\mu(B(O,d(x,O)))]^{1-p}}d\mu(x)\leqslant
C(D,p,A_{0},C_{\mu}).$

Notice that

$\displaystyle 1$
$\displaystyle=\int_{B}\frac{1}{1+[1+\mu(B(O,d(x,O)))]^{1-p}}d\mu(x)$
$\displaystyle\geqslant\frac{\mu(B)}{\sup\limits_{x\in
B}\left\\{1+[1+\mu(B(O,d(x,O)))]^{1-p}\right\\}}$
$\displaystyle\geqslant\frac{\mu(B)}{1+[1+\mu(B(O,A_{0}(d+r)))]^{1-p}},$

which implies

$\mu(B)\leqslant 1+[1+\mu(B(O,A_{0}(d+r)))]^{1-p}.$

If $d\leqslant 2A_{0}Dr$, we have

$\displaystyle\mu(B)$ $\displaystyle\leqslant
1+[1+\mu(B(O,A_{0}(2A_{0}D+1)r))]^{1-p}$ $\displaystyle\leqslant
1+\left\\{1+\mu[B(x_{0},A_{0}(A_{0}+1)(2A_{0}D+1)r)]\right\\}^{1-p}$
$\displaystyle\leqslant
1+\left\\{1+\left[A_{0}(A_{0}+1)(2A_{0}D+1)\right]^{C_{\mu}}\mu(B)\right\\}^{1-p},$

and thus $\mu(B)\leqslant C(D,p,A_{0},C_{\mu}).$

Then

(8.3)
$\int_{DB}\frac{1}{1+[1+\mu(B(O,d(x,O)))]^{1-p}}d\mu(x)\leqslant\mu(DB)\leqslant
D^{C_{\mu}}\mu(B)\leqslant C(D,p,A_{0},C_{\mu}).$

If $d>2A_{0}Dr$, then

$\displaystyle\int_{DB}\frac{1}{1+[1+\mu(B(O,d(x,O)))]^{1-p}}d\mu(x)$
$\displaystyle\leqslant$ $\displaystyle\frac{\mu(DB)}{\inf\limits_{x\in
DB}\left\\{1+[1+\mu(B(O,d(x,O)))]^{1-p}\right\\}}$ $\displaystyle\leqslant$
$\displaystyle\frac{D^{C_{\mu}}\mu(B)}{1+[1+\mu(B(O,d/A_{0}-Dr))]^{1-p}},$
$\displaystyle\leqslant$
$\displaystyle\frac{D^{C_{\mu}}\left\\{1+\mu[B\left(O,[A_{0}+1/(2D)]d\right)]\right\\}^{1-p}+D^{C_{\mu}}}{1+\left\\{1+\mu[B(O,d/(2A_{0}))]\right\\}^{1-p}}$
$\displaystyle\leqslant$
$\displaystyle\frac{D^{C_{\mu}}\left\\{1+\left[(2A_{0}+1/D)A_{0}\right]^{C_{\mu}}\mu[B(O,d/(2A_{0}))]\right\\}^{1-p}}{1+\left\\{1+\mu[B(O,d/(2A_{0}))]\right\\}^{1-p}}+D^{C_{\mu}}.$

Hence

(8.4) $\int_{DB}\frac{1}{1+[1+\mu(B(O,d(x,O)))]^{1-p}}d\mu(x)\leqslant
C(D,p,A_{0},C_{\mu})$

Combining (8.3) with (8.4), we get (8.2).

Assume $a$ is an $H^{\Psi_{p}}(\mu)$-atom supported on $B$. By Theorem 6.5,
there exist $t$ and a cube $Q\in{\mathscr{D}}^{t}$ such that $B\subset Q$ and
$\mathrm{diam}(Q)\leqslant Cr$, hence $B\subset Q\subset CB$.

Note that $\text{supp}(a)\subset Q,\int_{Q}a(x)d\mu(x)=0$ and

$\|1_{Q}\|_{L^{\Psi_{p}}(\mu)}\leqslant\|1_{CB}\|_{L^{\Psi_{p}}(\mu)}\leqslant
C(C,p,A_{0},C_{\mu})\|1_{B}\|_{L^{\Psi_{p}}(\mu)},$

which follows from (8.2). Thus

$\|a\|_{\infty}\leqslant\|1_{B}\|_{L^{\Psi_{p}}(\mu)}^{-1}\lesssim\|1_{Q}\|_{L^{\Psi_{p}}(\mu)}^{-1},$

which implies $a$ is a multiple of dyadic
$H^{\Psi_{p}}_{{\mathscr{D}}^{t}}$-atom supported on $Q$.

For any $t=1,2\cdots,K$, assume $b$ is a dyadic
$H^{\Psi_{p}}_{{\mathscr{D}}^{t}}$-atom supported on ${Q^{k}_{\beta}}$. By
Theorem 6.1, there exists two balls such that
$B(z_{\beta}^{k},c_{1}\delta^{k})\subset Q^{k}_{\beta}\subset
B(z_{\beta}^{k},C_{1}\delta^{k})$.

Thus $\text{supp}(b)\subset
B(z_{\beta}^{k},C_{1}\delta^{k}),\int_{B(z_{\beta}^{k},C_{1}\delta^{k})}b(x)d\mu(x)=0$
and

$\|1_{B(z_{\beta}^{k},C_{1}\delta^{k})}\|_{L^{\Psi_{p}}(\mu)}\leqslant
C\left(\frac{C_{1}}{c_{1}},p,A_{0},C_{\mu}\right)\|1_{B(z_{\beta}^{k},c_{1}\delta^{k})}\|_{L^{\Psi_{p}}(\mu)}\lesssim\|1_{Q^{k}_{\beta}}\|_{L^{\Psi_{p}}(\mu)},$

which follows from (8.2). Therefore,

$\|b\|_{\infty}\leqslant\|1_{Q^{k}_{\beta}}\|_{L^{\Psi_{p}}(\mu)}^{-1}\lesssim\|1_{B(z_{\beta}^{k},C_{1}\delta^{k})}\|_{L^{\Psi_{p}}(\mu)}^{-1},$

which implies $b$ is a multiple of dyadic $H^{\Psi_{p}}$-atom supported on
$B(z_{\beta}^{k},C_{1}\delta^{k})$. ∎

######  Remark 8.7.

In [11], Fu, Ma and Yang defined another kind of Musielak–Orlicz Hardy spaces
by grand maximal function and they also proved that these Musielak–Orlicz
Hardy spaces are equivalent to $H^{\Psi_{p}}_{\rm at}(\mu)$ with respect to
the corresponding norms when $p\in(\frac{C_{\mu}}{C_{\mu}+1},1]$.

Let $B_{1}:=B(O,1)$. Define

$\|g\|_{BMO_{+}(\mu)}:=|g_{B_{1}}|+\|g\|_{BMO(\mu)},\quad\text{for }g\in
BMO(\mu),$

and

$\|g\|_{{\mathcal{L}}_{+,\alpha}(\mu)}:=|g_{B_{1}}|+\|g\|_{{\mathcal{L}}_{\alpha_{p}}(\mu)},\quad\text{for
}g\in{\mathcal{L}}_{\alpha_{p}}(\mu).$

Thus $\|\cdot\|_{BMO_{+}(\mu)}$ and
$\|\cdot\|_{{\mathcal{L}}_{+,\alpha_{p}}(\mu)}$ are quasi-norms on $BMO(\mu)$
and ${\mathcal{L}}_{\alpha_{p}}(\mu)$, respectively.

######  Theorem 8.8.

Let $0<p<1$ and $f\in H^{p}_{\rm at}(\mu)$. There exist three linear
continuous operators $\Pi^{f}_{1}:{\mathcal{L}}_{\alpha_{p}}(\mu)\rightarrow
L^{1}(\Omega)$, $\Pi^{f}_{2}:{\mathcal{L}}_{\alpha_{p}}(\mu)\rightarrow
H^{1}_{\rm at}(\mu)$ and
$\Pi^{f}_{3}:{\mathcal{L}}_{\alpha_{p}}(\mu)\rightarrow H^{\Psi_{p}}_{\rm
at}(\mu)$ such that

$f\cdot g=\Pi^{f}_{1}(g)+\Pi^{f}_{2}(g)+\Pi^{f}_{3}(g)\quad\text{for all
}g\in{\mathcal{L}}_{\alpha_{p}}(\mu),$

where ${\mathcal{L}}_{\alpha_{p}}(\mu)$ is endowed with the quasi-norm
$\|\cdot\|_{{\mathcal{L}}_{+,\alpha_{p}}(\mu)}$.

###### Proof.

Let $f\in H^{p}_{\rm at}(\mu)$. By Theorem 6.6 there exist $f^{t}\in
H^{p}_{{\mathscr{D}}^{t}}(\mu)\ (t=1,2,\cdots,K)$ such that
$f=f^{1}+f^{2}+\cdots+f^{K}$, and

$\sum_{t=1}^{K}\|f^{t}\|_{H^{p}_{{\mathscr{D}}^{t}}(\mu)}\approx\|f\|_{H^{p}_{\rm
at}(\mu)}.$

Define $\Pi_{i}^{f}(g):=\sum\limits_{t=1}^{K}\Pi_{i}(f^{t},g)$ for $i=1,2,3$
and $g\in{\mathcal{L}}_{\alpha_{p}}(\mu)$ ($\Pi_{i}$ defined as in Theorem
7.7). Then

$f\cdot g=\Pi_{1}^{f}(g)+\Pi_{2}^{f}(g)+\Pi_{3}^{f}(g).$

By Theorem 7.7, Theorem 6.6 and Lemma 8.6, we have

$\displaystyle\|\Pi_{1}^{f}(g)\|_{1}\lesssim\sum_{t=1}^{K}\|\Pi_{1}(f^{t},g)\|_{1}\lesssim\sum_{t=1}^{K}\|f^{t}\|_{H^{p}_{{\mathscr{D}}^{t}}(\mu)}\|g\|_{\Lambda_{1,+}^{{\mathscr{D}}^{t}}(\alpha_{p})}\lesssim\|f\|_{H^{p}_{\rm
at}(\mu)}\|g\|_{{\mathcal{L}}_{+,\alpha_{p}}(\mu)},$
$\displaystyle\|\Pi_{2}^{f}(g)\|_{H^{1}_{\rm
at}(\mu)}\lesssim\sum_{t=1}^{K}\|\Pi_{2}(f^{t},g)\|_{H^{1}_{{\mathscr{D}}^{t}}(\mu)}\lesssim\sum_{t=1}^{K}\|f^{t}\|_{H^{p}_{{\mathscr{D}}^{t}}(\mu)}\|g\|_{\Lambda_{1,+}^{{\mathscr{D}}^{t}}(\alpha_{p})}\lesssim\|f\|_{H^{p}_{\rm
at}(\mu)}\|g\|_{{\mathcal{L}}_{+,\alpha_{p}}(\mu)},$
$\displaystyle\|\Pi_{3}^{f}(g)\|_{H^{\Psi_{p}}_{\rm
at}(\mu)}\lesssim\sum_{t=1}^{K}\|\Pi_{3}(f^{t},g)\|_{H^{\Psi_{p}}_{{\mathscr{D}}^{t}}(\mu)}\lesssim\sum_{t=1}^{K}\|f^{t}\|_{H^{p}_{{\mathscr{D}}^{t}}(\mu)}\|g\|_{\Lambda_{1,+}^{{\mathscr{D}}^{t}}(\alpha_{p})}\lesssim\|f\|_{H^{p}_{\rm
at}(\mu)}\|g\|_{{\mathcal{L}}_{+,\alpha_{p}}(\mu)}.$

which finishes the proof. ∎

######  Remark 8.9.

If the homogeneous space $(\Omega,\mu)$ satisfies the reverse doubling
condition, then Lemma 8.6 holds for $p=1$. Then we conclude the following.

Let $f\in H^{1}_{\rm at}(\mu)$. There exist three linear continuous operators
$\Pi^{f}_{1}:BMO(\mu)\rightarrow L^{1}(\Omega)$,
$\Pi^{f}_{2}:BMO(\mu)\rightarrow H^{1}_{\rm at}(\mu)$ and
$\Pi^{f}_{3}:BMO(\mu)\rightarrow H^{\Psi_{1}}_{\rm at}(\mu)$ such that

$f\cdot g=\Pi^{f}_{1}(g)+\Pi^{f}_{2}(g)+\Pi^{f}_{3}(g)\quad\text{for all }g\in
BMO(\mu),$

where $BMO(\mu)$ is endowed with the norm $\|\cdot\|_{BMO_{+}(\mu)}$.

### Acknowledgments

We thank Professor Quanhua Xu for helpful discussions and suggestions. O.B.
and Y.Z. would like to express their gratitude to Professor Xu for his kind
invitation and hospitality during their visit to Besançon in March 2022.

We would also like to thank Yong Jiao, Guangheng Xie, Dachun Yang, Dejian Zhou
for personal communications on their work with us.

## References

  * [1] P. Auscher and T. Hytönen. Orthonormal bases of regular wavelets in spaces of homogeneous type. Appl. Comput. Harmon. Anal. 34 (2013), no. 2, 266–296; Addendum to Orthonormal bases of regular wavelets in spaces of homogeneous type. Appl. Comput. Harmon. Anal. 39 (2015), no. 3, 568–569.
  * [2] O. Bakas, S. Pott, S. Rodríguez-López and A. Sola. Notes on $H^{\log}$: structural properties, dyadic variants, and bilinear $H^{1}$-$BMO$ mappings. Ark. Mat. 60 (2022), no. 2, 231–275.
  * [3] A. Bonami, J. Cao, L. D. Ky, L. Liu, D. Yang and W. Yuan. Multiplication between Hardy spaces and their dual spaces. J. Math. Pures Appl. (9) 131 (2019), 130–170.
  * [4] A. Bonami, S. Grellier and L. D. Ky. Paraproducts and products of functions in $BMO({}^{n})$ and $H^{1}({}^{n})$ through wavelets. J. Math. Pures Appl. (9) 97 (2012), no. 3, 230–241.
  * [5] A. Bonami, T. Iwaniec, P. Jones and M. Zinsmeister. On the product of functions in $BMO$ and $H^{1}$. Ann. Inst. Fourier (Grenoble) 57 (2007), no. 5, 1405–1439.
  * [6] D. L. Burkholder. Martingale transforms. Ann. Math. Statist. 37 (1966), no. 6, 1494–1504.
  * [7] R. R. Coifman and G. Weiss. Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc, 83 (1977), no. 4, 569–645
  * [8] J. M. Conde-Alonso and J. Parcet. Atomic Blocks for Noncommutative Martingales. Indiana Univ. Math. J. 65 (2016), no. 4, 1425–1443.
  * [9] B. Davis. On the integrability of the martingale square function. Israel J. Math. 8 (1970), 187–190.
  * [10] C. Fefferman. Characterizations of bounded mean oscillation. Bull. Amer. Math. Soc. 77 (1971), 587–588.
  * [11] X. Fu, T. Ma, and D. Yang. Real-variable characterizations of Musielak-Orlicz Hardy spaces on spaces of homogeneous type. Ann. Acad. Sci. Fenn. Math. 45 (2020), no. 1, 343–410.
  * [12] A. M. Garsia. Martingale inequalities: Seminar notes on recent progress. Mathematics Lecture Note Series. W. A. Benjamin, Inc., Reading, Mass.-London-Amsterdam, 1973. viii+184 pp.
  * [13] T. Hytönen and A. Kairema. Systems of dyadic cubes in a doubling metric space. Colloq. Math.126 (2012), no. 1, 1–33.
  * [14] Y. Jiao, G. Xie, D. Yang, D. Zhou. Bilinear decomposition for product of martingale Hardy space $H^{1}(\Omega)$ and martingale $BMO(\Omega)$. Manuscript.
  * [15] L. Liu, D. Yang and W. Yuan. Bilinear decompositions for products of Hardy and Lipschitz spaces on spaces of homogeneous type. Dissertationes Math. 533 (2018), 93 pp.
  * [16] R. Long. Martingale spaces and inequalities. Peking University Press, Beijing; Friedr. Vieweg & Sohn, Braunschweig, 1993. iv+346 pp.
  * [17] T. Mei. BMO is the intersection of two translates of dyadic BMO. C. R. Math. Acad. Sci. Paris, 336 (2003), no. 12, 1003–1006.
  * [18] N. G. Meyers. Mean oscillation over cubes and Hölder continuity. Proc. Amer. Math. Soc., 15 (1964), 717–721.
  * [19] T. Miyamoto, E. Nakai and G. Sadasue. Martingale Orlicz-Hardy spaces. Math. Nachr. 285 (2012), no. 5-6, 670–686.
  * [20] E. Nakai. Pointwise multipliers on weighted $BMO$ spaces. Studia Math. 125 (1997), no. 1, 35–56.
  * [21] E. Nakai and G. Sadasue. Pointwise multipliers on martingale Campanato spaces. Studia Math. 220 (2014), no. 1, 87–100.
  * [22] E. Nakai and K. Yabuta. Pointwise multipliers for functions of bounded mean oscillation. J. Math. Soc. Japan 37 (1985), no. 2, 207–218.
  * [23] M. Perrin. A noncommutative Davis’ decomposition for martingales. J. Lond. Math. Soc. (2) 80 (2009), no. 3, 627–648.
  * [24] E. M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S. Murphy. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993. xiv+695 pp.
  * [25] B. E. Viviani. An atomic decomposition of the predual of $BMO(\rho)$. Rev. Mat. Iberoamericana 3 (1987), no. 3-4, 401–425
  * [26] F. Weisz. Martingale Hardy spaces and their applications in Fourier analysis. Lecture Notes in Mathematics, 1568. Springer-Verlag, Berlin, 1994. viii+218 pp.
  * [27] F. Weisz. Martingale Hardy spaces for $0<p\leq 1$. Probab. Theory Related Fields 84 (1990), no. 3, 361–376.
  * [28] G. Xie, Y. Jiao and D. Yang. Martingale Musielak-Orlicz Hardy spaces. Sci. China Math. 62 (2019), no. 8, 1567–1584.
  * [29] F. Xing, D. Yang and Y. Liang. Products of Functions in $BMO(\mathcal{X})$ and $H^{1}_{\rm at}(\mathcal{X})$ via wavelets over spaces of homogeneous type. J. Fourier Anal. Appl. 23 (2017), no. 4, 919–990.
  * [30] D. Yang, Y. Liang and L. D. Ky. Real-variable theory of Musielak-Orlicz Hardy spaces. Lecture Notes in Mathematics, 2182. Springer, Cham, 2017. xiii+466 pp.