#  Hardy Space Decompositions of $L^{p}(\mathbb{R}^{n})$ for $0<p<1$ with
Rational Approximation

Guan-Tie Deng, Hai-Chou Li, Tao Qian  Guan-Tie Deng , Hai-Chou Li , Tao Qian
School of Mathematical Sciences, Beijing Normal University, Beijing, 100875,
China<EMAIL_ADDRESS>This work was partially supported by NSFC
(Grant 11271045) and by SRFDP (Grant 20100003110004)Corresponding author.
College of Mathematics and informatics, South China Agricultural University,
Guangzhou, China. Email<EMAIL_ADDRESS>(or,
lihaichou@126.com).Department of Mathematics, University of Macau, Macao (Via
Hong Kong). Email<EMAIL_ADDRESS>The work was partially supported by Research
Grant of University of Macau, MYRG115(Y1-L4)-FST13-QT, Macao Government FDCT
098/2012/A3

Abstract

This paper aims to obtain decompositions of higher dimensional
$L^{p}(\mathbb{R}^{n})$ functions into sums of non-tangential boundary limits
of the corresponding Hardy space functions on tubes for the index range
$0<p<1$. In the one-dimensional case, Deng and Qian [5] recently obtained such
Hardy space decomposition result: for any function $f\in L^{p}(\mathbb{R}),\
0<p<1$, there exist functions $f_{1}$ and $f_{2}$ such that $f=f_{1}+f_{2}$,
where $f_{1}$ and $f_{2}$ are, respectively, the non-tangential boundary
limits of some Hardy space functions in the upper-half and lower-half planes.
In the present paper, we generalize the one-dimensional Hardy space
decomposition result to the higher dimensions, and discuss the uniqueness
issue of such decomposition.

Key words: Hardy spaces on tube domain, decomposition, rational approximation

##  1  Introduction

We begin with a survey on the case of one dimensional Hardy spaces
$H^{p}(\mathbb{C}^{\pm})$ for $0<p\leq\infty$. For the classical case $p=2$,
the famous Paley-Wiener Theorem states that, for an $L^{2}(\mathbb{R})$
function $f,$ it is the non-tangential boundary limit of a function in
$H^{2}(\mathbb{C}^{+})$ if and only if ${\rm supp}\hat{f}\subset[0,\infty),$
the latter being equivalent with

$\displaystyle\hat{f}=\chi_{[0,\infty)}\hat{f},$ (1)

where $\chi_{[0,\infty)}$ is the characteristic function of $[0,\infty)$.
Similarly, $f\in H^{2}(\mathbb{C}^{-})$ is equivalent with

$\displaystyle\hat{f}=\chi_{(-\infty,0]}\hat{f}.$ (2)

From the relations (1) and (2), a canonical decomposition result is obtained
as

$f=f^{+}+f^{-},$

for $f\in L^{2}(\mathbb{R}),$ where $f^{+}=(\chi_{[0,\infty)}\hat{f})^{\vee}$
and $f^{-}=(\chi_{(-\infty,0]}\hat{f})^{\vee}$ belong to, respectively, the
Hardy spaces $H^{2}(\mathbb{C}^{+})$ and $H^{2}(\mathbb{C}^{-}).$ In the
present paper we call such decomposition as Hardy space decomposition. It can
be alternatively obtained through the Hilbert transformation in view of the
Plemelj formula in relation to the Cauchy integral transform. For the relevant
knowledge we refer the reader to, for instance, [10] and [15].

The significance of the $L^{2}(\mathbb{R})$ space decomposition into the Hardy
spaces $H^{2}(\mathbb{C}^{\pm})$ lays on the fact that the $H^{2}$ functions
have much better properties than the $L^{2}$-functions. The functions
$f^{\pm}$ are non-tangential boundary limits of $H^{2}(\mathbb{C}^{\pm})$
functions, the latter being analytically defined in their respective domains.
We note that a function in $L^{p}$ is only a.e. determined, that is, if $f=g$,
a.e., then $f$ and $g$ are considered as the same function in $L^{p}.$ In this
regard one cannot assume a general $L^{p}$-function to have any smoothness. On
the other hand, any two a.e. identical $L^{p}$ functions correspond to the
same Hardy space decomposition, the latter, being unique, having all kinds of
smoothness in their domains of definition. The Hardy $H^{2}$ spaces are
reproducing kernel Hilbert spaces. In particular, Cauchy’s theory and
techniques are available to Hardy space functions. Furthermore, there is an
isometric correspondence between the Hardy space functions and their non-
tangential boundary limits. Based on what just mentioned, analysis on the
$L^{2}$-functions can be reduced to that of their Hardy space components. Such
treatment of $L^{2}$-functions has found significant applications in both the
pure and applied mathematics. In particular, some demonstrative results in
signal analysis have recently been obtained ( [3], [13], [14],).

There have been studies on Hardy space decomposition for extended index range
of $p$. Both the above mentioned Paley-Wiener Theorem related Fourier spectrum
characterization and the Plemelj formula can be extended to Hardy
$H^{p}(\mathbb{R})$ spaces with $p\neq 2.$ Systematic studies on the spectrum
properties as well as the $L^{p}$ decomposition are carried out in the works
of Qian et al. [15] among others. Their work is summarized as $f\in
H^{p}(\mathbb{C}^{+})$ if and only if $f\in L^{p}(\mathbb{R})$ and in the
distributional sense ${\rm supp}\hat{f}\subset[0,\infty)$ for all $1\leq
p\leq\infty.$ Comparatively the Plemelj formula approach is more applicable
than Fourier transformation, for on the $L^{p}(\mathbb{R}),\ p>2,$ spaces
Fourier transforms are distributions. As a consequence, the Hardy space
decomposition results also hold for $f\in L^{p}(\mathbb{R}),\ 1<p<\infty.$
There also exists an analogous theory on the unit circle for $1<p\leq\infty.$
The latter context corresponds to Fourier series other than Fourier transforms
([10], [2]).

Turning to the case of $0<p\leq 1,$ due to loss of integrability neither on
the real line, nor on the unit circle, the Plemelj formula, or the Hilbert
transformation are available. The references [2] and [5] study the Hardy space
decomposition of $L^{p}(\mathbb{R})$ for $0<p<1.$ There is no Fourier
transformation theory for functions in such $L^{p}(\mathbb{R})$ as they are
even not distributions. One, however, can still have the Hardy space
decompositions. The reference [2] uses the real analysis methods of harmonic
analysis, making use of a dense subclass of the $L^{p}(\mathbb{R})$-functions
with vanishing moment conditions and the Hilbert transforms. In contrast, [5]
uses the complex analysis methods, and, in particular, rational functions
approximation to achieve Hardy space decompositions. The methods in [5] are
direct and constructive. The following Theorem A is obtained by Deng and Qian
in [5].
Theorem A  ([5]) Suppose that $0<p<1$ and $f\in L^{p}(\mathbb{R})$. Then,
there exist a positive constant $A_{p}$ and two sequences of rational
functions $\\{P_{k}\\}$ and $\\{Q_{k}\\}$ such that $P_{k}\in
H^{p}(\mathbb{C}_{+})$, $Q_{k}\in H^{p}(\mathbb{C}_{-})$ and

$\sum_{k=1}^{\infty}\left(\|P_{k}\|^{p}_{H^{p}_{+}}+\|Q_{k}\|^{p}_{H^{p}_{-}}\right)\leq
A_{p}\|f\|_{p}^{p},$
$\lim_{n\rightarrow\infty}||f-\sum_{k=1}^{n}(P_{k}+Q_{k})||_{p}=0.$

Moreover,

$g(z)=\sum_{k=1}^{\infty}P_{k}(z)\in H^{p}(\mathbb{C}_{+}),\ \
h(z)=\sum_{k=1}^{\infty}Q_{k}(z)\in H^{p}(\mathbb{C}_{-}),$

and $g(x)$ and $h(x)$ are the non-tangential boundary limits of functions for
$g\in H^{p}(\mathbb{C}_{+})$ and $h\in H^{p}(\mathbb{C}_{-})$, respectively,
$f(x)=g(x)+h(x)$ almost everywhere, and

$\|f\|_{p}^{p}\leq\|g\|_{p}^{p}+\|h\|_{p}^{p}\leq A_{p}\|f\|_{p}^{p}.$

That is, in the sense of $L^{p}(\mathbb{R})$,

$L^{p}(\mathbb{R})=H^{p}_{+}(\mathbb{R})+H^{p}_{-}(\mathbb{R}),$

where $H_{+}(\mathbb{R})$ and $H_{-}(\mathbb{R})$ denote the set of non-
tangential boundary limits of functions in the Hardy spaces
$H^{p}(\mathbb{C}_{+})$ and $H^{p}(\mathbb{C}_{-})$, respectively.

Recently we, in [7], obtain the Hardy space decomposition of $L^{p}$ for
$0<p\leq 1$ on the unite circle $\partial\mathbb{D}$ by using polynomial
approximation other than general rational approximation. The result is stated
as follows. Denoting by $L^{p}_{I}(\partial\mathbb{D})$ and
$L^{p}_{O}(\partial\mathbb{D})$ the closed subspaces of
$L^{p}(\partial\mathbb{D}),$ consisting of respectively, the non-tangential
boundary limits of the functions of $H^{p}(\mathbb{D})$ and
$H^{p}(\mathbb{D}_{O}),$ then we have

$L^{p}(\partial\mathbb{D})=L^{p}_{I}(\partial\mathbb{D})+L^{p}_{O}(\partial\mathbb{D}),$

where the right-hand-side is not a direct sum. As a matter of fact, the
intersection $L^{p}_{I}(\partial\mathbb{D})$ and
$L^{p}_{O}(\partial\mathbb{D})$ contains non zero functions. The work on the
unit circle exposes the particular features adaptable to higher dimensions.

All the Hardy space results mentioned above for dimension one can be
generalized to dimension n in the setting of Hardy spaces on tubes with
correspondingly the right notions. In fact, [17] already contain some basic
results, mostly for $p=2$, while [8] gives a systematic treatment for general
indices $p\in[1,\infty]$, including Fourier spectrum characterization of Hardy
spaces on tubes, the Cauchy integral and Poisson integral representation of
the Hardy space functions, the Plemelj formulas in relation to Hilbert
transforms, and the Hardy space decompositions of functions in the
$L^{p}(\mathbb{R}^{n})$ spaces. The purpose of this article is to prove the
Hardy space decomposition of the $L^{p}(\mathbb{R}^{n})$ space functions for
$p\in(0,1)$. In doing so neither the Cauchy integral formula nor the Fourier
transformation can be directly used, for the functions defined on
$\mathbb{R}^{n}$ are lack of integrability. They are even not distributions.

In the present paper, inspired by the idea of [5], with the rational
approximation method, we obtain decompositions of functions in
$L^{p}(\mathbb{R})$ for $0<p<1$ into sums of boundary limits of the
corresponding Hardy space functions on tubes, $H^{p}(T_{\Gamma_{\sigma_{j}}})\
(j=1,2,...,2^{n})$, through the rational functions approximation. The idea of
using rational approximation is motivated by the studies [18] of Takenaka-
Malmquist systems.

We will discuss the non-uniqueness of the Hardy space decomposition via
rational approximation method. We conclude that the sum is not a direct sum.
In fact, there is a non-trivial intersection of all those summed spaces. The
intersection of those spaces $H_{\sigma_{j}}^{p}(\mathbb{R}^{n})\
(j=1,2,...,2^{n})$ is identical with the $L^{p}$ closure of the set of
functions $f\in L^{p}(\mathbb{R}^{n})$ of the following form

$f(x_{1},x_{2},...,x_{n})=\frac{P(x_{1},...,x_{n})}{\prod\limits_{k=1}^{n}\prod\limits_{j=1}^{m}(x_{k}-a_{kj})},$

where
$P(x_{1},...,x_{n})=\sum\limits_{s_{1}=0}^{l_{1}}\cdot\cdot\cdot\sum\limits_{s_{n}=0}^{l_{n}}a_{s_{1}...s_{n}}x_{1}^{s_{1}}\cdot\cdot\cdot
x_{n}^{s_{n}},\ l=\max\\{l_{1},...,l_{n}\\},$ $(m-l)p>1,$ and $a_{kj}\neq
a_{km}\in\mathbb{R}$ as $j\neq m$ for $k=1,2,...,n.$

The writing plan of this paper is as follows: In § 2, some basic definitions
and notations are given. In § 3 we devote to establishing the higher
dimensional Hardy space decomposition of $L^{p}(\mathbb{R}^{n}),\ 0<p<1.$ The
decomposition is a sum of boundary limit functions of Hardy spaces on tubes,
$H^{p}(T_{\Gamma_{\sigma_{k}}})$, for all $k=1,2,...,2^{n}.$ In § 4, we
discuss the uniqueness of such Hardy space decomposition.

##  2 Preliminary Knowledge

In this section, we introduce some useful basic definitions and notions. For
more information, see e.g. [10] and [17]. The classical Hardy spaces
$H^{p}(\mathbb{C}_{k}),\ 0<p<+\infty,\ k=\pm 1$, consists of the functions $f$
analytic in the half plane $\mathbb{C}_{k}=\\{z=x+iy:ky>0\\}.$ They are Banach
spaces for $1\leq p<\infty$ under the norms

$\|f\|_{H^{p}_{k}}=\sup_{ky>0}\left(\int_{-\infty}^{\infty}|f(x+iy)|^{p}dx\right)^{\frac{1}{p}};$

and complete metric spaces for $0<p<1$ under the metric functions

$d(f,g)=\sup_{ky>0}\int_{-\infty}^{\infty}|f(x+iy)|^{p}dx.$

Let $B$ be an open subset of $\mathbb{R}^{n}$. Then the tube $T_{B}$ with base
$B\subset\mathbb{R}^{n}$ is the set

$T_{B}=\\{z=x+iy\in\mathbb{C}^{n}:\ x\in\mathbb{R}^{n},\ y\in B\\}.$

For example, when $n=1$, the classical upper-half complex plane
$\mathbb{C}^{+}$ and lower-half complex plane $\mathbb{C}^{-}$ are the tubes
in $\mathbb{C}$ with the base $B_{+}=\\{y\in\mathbb{R}:\ y>0\\}$ and the base
$B_{-}=\\{y\in\mathbb{R}:\ y<0\\}$, respectively. That is,
$\mathbb{C}^{+}=T_{B_{+}}=\\{z=x+iy:\ x\in\mathbb{R},\ y>0\\}$ and
$\mathbb{C}^{-}=T_{B_{-}}=\\{z=x+iy:\ x\in\mathbb{R},\ y<0\\}$. Obviously, the
tube $T_{B}$ are generalizations of $\mathbb{C}^{+}$ and $\mathbb{C}^{-}$.

It is known that $n$-dimensional real Euclidean space $\mathbb{R}^{n}$ has
$2^{n}$ octants. To denote the octants we adopt the following notations. Let
$\sigma_{k}=(\sigma_{k}(1),\sigma_{k}(2),...,\sigma_{k}(n))$, where
$\sigma_{k}(j)=+1\ {\rm or}-1$ for $k=1,2,...,2^{n}$. Then $2^{n}$ octants of
$\mathbb{R}^{n}$ are denoted by $\Gamma_{\sigma_{k}},\ \ k=1,2,...,2^{n}$,
where

$\Gamma_{\sigma_{k}}=\\{y=(y_{1},y_{2},...,y_{n})\in\mathbb{R}^{n}:\
\sigma_{k}(j)y_{j}>0,\ j=1,2,...,n\\}.$

In this paper, we denote $\Gamma_{\sigma_{1}}$ the first octant of
$\mathbb{R}^{n}$, that is

$\Gamma_{\sigma_{1}}=\\{y=(y_{1},y_{2},...,y_{n})\in\mathbb{R}^{n}:\ y_{j}>0,\
j=1,2,...,n\\}.$

Correspondingly, $\mathbb{C}^{n}$ can be decomposed into $2^{n}$ tubes,
denoted by $T_{\Gamma_{\sigma_{k}}},\ k=1,2,...,2^{n}$. That is

$T_{\Gamma_{\sigma_{k}}}=\\{z=x+iy\in\mathbb{C}^{n}:\ x\in\mathbb{R}^{n},\
y\in\Gamma_{\sigma_{k}}\\}.$

A function $F(z)$ is said to belong to the space $H^{p}(T_{B}),\ 0<p<\infty,$
if it is holomorphic in the tube $T_{B}$, and satisfies

$\|F\|_{H^{p}}=\sup\left\\{\left(\int_{\mathbb{R}^{n}}|F(x+iy)|^{p}dx\right)^{\frac{1}{p}}:\
y\in B\right\\}<\infty.$

Hence,

$H^{p}(T_{B})=\\{F:F\mbox{ holomorphic on }\ T_{B}\mbox{ and }\
\|F\|_{H^{p}}<\infty\\}.$

The spaces $H^{p}(T_{\Gamma_{\sigma_{k}}})$ are defined through replacing $B$
by $\Gamma_{\sigma_{k}},\ k=1,...,2^{n}.$

A function, $f$, defined in tube $T_{\Gamma}$, is said to have non-tangential
boundary limit (NTBL) $l$ in each component of the variable at
$x_{0}\in\mathbb{R}^{n}$ if $f(z)=f(x+iy)=f(x_{1}+iy_{1},...,x_{n}+iy_{n})$
tends to $l$ as the point $z=(x_{1},y_{1};x_{2},y_{2};...;x_{n},y_{n})$ tends
to $x_{0}=(x_{0}^{(1)},0;x_{0}^{(2)},0;...;x_{0}^{(n)},0)$ within the
Cartesian product

$\gamma_{\alpha}(x_{0})=\Gamma_{\alpha_{1}}(x_{0}^{(1)})\times\Gamma_{\alpha_{2}}(x_{0}^{(2)})\times\cdot\cdot\cdot\times\Gamma_{\alpha_{n}}(x_{0}^{(n)})\subset
T_{\Gamma},$

for all $n$-tuples $\alpha=(\alpha_{1},\alpha_{2},...,\alpha_{n})$ of positive
real numbers, where

$\Gamma_{\alpha_{j}}(x_{0}^{(j)})=\left\\{(x_{j},y_{j})\in\mathbb{C}^{+}:\
|x_{j}-x_{0}^{(j)}|<\alpha_{j}y_{j}\right\\},\ j=1,2,...,n.$

As an important property of the Hardy spaces, it is shown that if $f$ is a
function in a Hardy space $H^{p},0<p<\infty,$ then for almost all $x_{0},\ f$
has NTBL ([17]).

Since the mapping that maps the functions in the Hardy spaces to their NTBLs
is an isometric isomorphism, we denote by $H_{\sigma_{k}}^{p}(\mathbb{R}^{n})$
for $0<p<1$ the NTBLs of the functions in $H^{p}(T_{\Gamma_{\sigma_{k}}})$,
that is

$H_{\sigma_{k}}^{p}(\mathbb{R}^{n})=\\{f:\ f\mbox{ is the NTBL of a function
in }H^{p}(T_{\Gamma_{\sigma_{k}}})\\}$

for all $k=1,2,...,2^{n}$. The non-tangential boundary limit of $F(z)\in
H^{p}(T_{\Gamma_{\sigma_{k}}})$ as $y\rightarrow 0$ in the tube are denoted by

$F_{\sigma_{k}}(x)=\lim\limits_{y\in\Gamma_{\sigma_{k}},y\rightarrow
0}F(x+iy)=\lim\limits_{\sigma_{k}(1)y_{1}\rightarrow
0^{+},...,\sigma_{k}(n)y_{n}\rightarrow
0^{+}}F(x_{1}+iy_{1},...,x_{n}+iy_{n}).$ (3)

##  3  Hardy Space Decomposition of $L^{p}(\mathbb{R}^{n})$

As previously mentioned in the introduction, for one dimension, in [5], the
authors use the rational approximation method to obtain the Hardy space
decomposition for the range $0<p<1$. As is well known, rational approximation
has a long history, and is naturally related to complex approximation [18]. As
obtained in [5] through a rational approximation method, for a given real-
valued function $f\in L^{p}(\mathbb{R}),\ 0<p<1$, there exists the relation
$f=f_{1}+f_{2}$, where $f_{+}$ and $f_{-}$ are the non-tangential boundary
limit functions of some analytic $H^{p}$ functions in, respectively, the
upper-half and the lower-half complex planes [5]. Precisely, the analytic
Hardy space function $f_{+}\in H^{p}_{+}$, and thus its boundary limit
function as well, are defined through a sequence of rational functions whose
poles are in the lower-half plane, and $f_{-}\in H^{p}_{-}$, through a
sequence of rational functions whose poles are in the upper-half plane. We
note that the Hardy spaces decompositions for functions in
$L^{p}(\mathbb{R}),\ 0<p<1,$ are not unique. This amounts to saying that the
intersection $H^{p}_{+}\bigcap H^{p}_{-}$ a non-empty set.

In this section by using a higher dimensional rational approximation method,
we will generalize the above type of Hardy space decomposition of
$L^{p}(\mathbb{R})$ to higher dimensional $L^{p}(\mathbb{R}^{n}),\ 0<p<1,$ and
obtain Theorem 3.1. Specifically, for any real-valued function $f\in
L^{p}(\mathbb{R}^{n}),\ 0<p<1$, it is proved to have the Hardy space
decomposition $f(x)=\sum\limits_{j=1}^{2^{n}}f_{\sigma_{j}}(x),$ where for
each $j$, $f_{\sigma_{j}}(x)\ (j=1,2,...,2^{n})$ is the non-tangential
boundary limit of some $H^{p}(T_{\Gamma_{\sigma_{j}}})$-function. In fact,
each analytic Hardy space function $f_{\sigma_{j}}(x)\in H^{p}_{\sigma_{j}}$,
and thus its boundary limit function as well, may be approximated by a
sequence of rational $L^{p}(\mathbb{R}^{n})$-functions whose poles are not in
the octant $T_{\Gamma_{\sigma_{j}}}$. We will call such rational functions
rational atoms.

###### Theorem 3.1

Suppose that $f\in L^{p}(\mathbb{R}^{n}),\ 0<p<1$. Then there exist $2^{n}$
sequences of rational functions $\\{R_{{k\sigma}_{j}}(z)\\}\in
H^{p}(T_{\Gamma_{\sigma_{j}}}),$ and $f_{\sigma_{j}}(z)\in
H^{p}(T_{\Gamma_{\sigma_{j}}}),\ j=1,2,...,2^{n},$ such that the following
properties hold
(i)

$\sum\limits_{k=1}^{\infty}\sum\limits_{j=1}^{2^{n}}||R_{{k\sigma}_{j}}||^{p}_{H^{p}_{\sigma_{j}}}\leq
A_{np}\|f\|_{p}^{p},$ (4)

where $A_{np}$ is a constant only depending on $(n,p)$;
(ii)

$\lim_{m\rightarrow\infty}||f-\sum_{k=1}^{m}\sum\limits_{j=1}^{2^{n}}R_{{k\sigma}_{j}}||_{p}=0;$
(5)

(iii)

$\lim_{m\rightarrow\infty}||f_{\sigma_{j}}-\sum_{k=1}^{m}R_{{k\sigma}_{j}}||_{H^{p}}=0,$
(6)

for all $j=1,2,...,2^{n};$
(iv) $f_{\sigma_{j}}(x)$ are the non-tangential boundary limits of functions
$f_{\sigma_{j}}(z)$, and

$f(x)=\sum\limits_{j=1}^{2^{n}}f_{\sigma_{j}}(x),\ \ \ \ \ a.e.\ \
x\in\mathbb{R}^{n};$

(v)

$\|f\|^{p}_{p}\leq\sum\limits_{j=1}^{2^{n}}\|f_{\sigma_{j}}\|^{p}_{p}\leq
A_{np}\|f\|_{p}^{p}.$

(vi) In summary,

$L^{p}(\mathbb{R}^{n})=\sum\limits_{j=1}^{2^{n}}H_{\sigma_{j}}^{p}(\mathbb{R}^{n}).$

###### Remark 3.1

For the non-uniqueness issue of the Hardy space decomposition,we will show
that $\bigcap\limits_{j=1}^{2^{n}}H_{\sigma_{j}}^{p}(\mathbb{R}^{n})$ is a
non-empty set. We will prove it in § 4.

In order to prove Theorem 3.1, we need the following lemmas. We note that the
proof of Theorem 3.1 is at the end of this section.

###### Lemma 3.1

Let $\Gamma_{\sigma_{1}}$ be the first octant of $\mathbb{R}^{n}$. Suppose
that $0<p<1$ and $R$ is a rational function with the form

$R(z)=\frac{P(z)}{Q(z)},$

where $P(z)$ is a polynomial of $z$, and
$Q(z)=Q_{1}(z_{1})Q_{2}(z_{2})\cdot\cdot\cdot\cdot Q_{n}(z_{n})$, $Q(z_{k})$
is a polynomial of $z_{k},\ k=1,2,...,n$. If $R\in L^{p}(\mathbb{R}^{n})$ and
$R(z)$ is holomorphic in $T_{\Gamma_{\sigma_{1}}}$, then $R(z)\in
H^{p}(T_{\Gamma_{\sigma_{1}}}).$

Deng-Qian [5] proved the special case $n=1$ of Lemma 3.1, and obtained the
following Lemma 3.1-1, which is needed for the proof of Lemma 3.1.

Lemma 3.1-1 ([5]) Suppose that $0<p<1$ and $R$ is a rational function with
$R\in L^{p}(\mathbb{R}).$ For $k=\pm 1$, if $R(z)$ is analytic in the half
plane $\mathbb{C}_{k}$, then $R\in H^{p}(\mathbb{C}_{k})$.

In order to prove Lemma 3.1, we need the following Lemma 3.1-2 which obtained
in our recently work [9]:

Lemma 3.1-2 ([9]) If $f(z)\in H^{p}(T_{\Gamma_{\sigma_{j}}}),\ 0<p<\infty,\
j=1,2,...,2^{n}$, and $f(x)$ is the boundary limit of $f(z)$. Then
$\varphi(y)$ is continuous convex and bounded in
$\overline{\Gamma}_{\sigma_{j}}$, moreover,

$\|f\|^{p}_{H^{p}_{\sigma_{j}}}=\sup\limits_{y\in\Gamma_{\sigma_{j}}}\varphi(y)=\varphi(0,...,0)=\|f\|^{p}_{p},$

where $\varphi(y)=\int_{\mathbb{R}^{n}}|f(x+iy)|^{p}dx,\
y\in\overline{\Gamma}_{\sigma_{j}},\ j=1,2,...,2^{n}.$
Proof of Lemma 3.1  We are to prove Lemma 3.1 by mathematical induction.

When $n=1$, Lemma 3.1 is just Lemma 3.1-1.

Next, when $n>1,$ we assume that Lemma 3.1 holds for $n-1$. Take $n_{1}=n-1,$
and fix $t_{n}\in\mathbb{R}\backslash Z_{0}(Q_{n})$, where
$Z_{0}(Q_{n})=\\{z_{n}:\ Q_{n}(z_{n})=0\\}$. We consider the function $r$ of
$n-1$ real variables defined by

$r(x_{1},x_{2}...,x_{n-1})=R(x_{1},...,x_{n-1},t_{n}),$

where $(x_{1},x_{2}...,x_{n-1})\in\mathbb{R}^{n-1}.$ The Fubini Theorem
ensures that $r(x_{1},x_{2}...,x_{n-1})$ belongs to $L^{p}(\mathbb{R}^{n-1})$
for almost all $t_{n}\in\mathbb{R}\backslash Z_{0}(Q_{n}).$ Moreover, it is
easy to see that the rational function $r(z_{1},...,z_{n-1})$ satisfies the
assumptions of Lemma 3.1.

Therefore, by the induction hypothesis, we obtain that
$r(z_{1},...,z_{n-1})\in H^{p}(T_{\Gamma^{n-1}_{\sigma_{1}}}),$ where
$T_{\Gamma^{n-1}_{\sigma_{1}}}$ denotes the tube with the first octant of
$\mathbb{R}^{n-1}$ as base.

By Lemma 3.1-2,

$\int_{\mathbb{R}^{n-1}}|r(x_{1}+iy_{1},...,x_{n-1}+iy_{n-1})|^{p}\
dx_{1}\cdot\cdot\cdot
dx_{n-1}\leq\int_{\mathbb{R}^{n-1}}|r(x_{1},...,x_{n-1})|^{p}\
dx_{1}\cdot\cdot\cdot dx_{n-1}.$

That is

$\begin{split}&\int_{\mathbb{R}^{n-1}}|R(x_{1}+iy_{1},...,x_{n-1}+iy_{n-1},x_{n})|^{p}\
dx_{1}\cdot\cdot\cdot dx_{n-1}\\\
&\leq\int_{\mathbb{R}^{n-1}}|R(x_{1},...,x_{n-1},x_{n})|^{p}\
dx_{1}\cdot\cdot\cdot dx_{n-1}.\end{split}$ (7)

Integrating both sides of the last inequality with respect to $x_{n}$, we have

$\int_{\mathbb{R}^{n}}|R(x_{1}+iy_{1},...,x_{n-1}+iy_{n-1},x_{n})|^{p}\
dx_{1}\cdot\cdot\cdot dx_{n}\leq\int_{\mathbb{R}^{n}}|R(x_{1},...,x_{n})|^{p}\
dx_{1}\cdot\cdot\cdot dx_{n}.$ (8)

By Fubini Theorem,

$\int_{\mathbb{R}^{n-1}}\left(\int_{\mathbb{R}}|R(x_{1}+iy_{1},...,x_{n-1}+iy_{n-1},x_{n})|^{p}\
dx_{n}\right)dx_{1}\cdot\cdot\cdot
dx_{n-1}\leq\int_{\mathbb{R}^{n}}|R(x)|^{p}\ dx.$ (9)

So, fix $(y_{1},...,y_{n-1})$, for almost all
$(x_{1},...,x_{n-1})\in\mathbb{R}^{n-1}$, we have

$\int_{\mathbb{R}}|R(x_{1}+iy_{1},...,x_{n-1}+iy_{n-1},x_{n})|^{p}\
dx_{n}<\infty.$

That is, rational function $R(x_{1}+iy_{1},...,x_{n-1}+iy_{n-1},x_{n})$ as a
function of $x_{n}$ belongs to $L^{p}(\mathbb{R})$. Moreover, since
$R(z_{1},...,z_{n})$ is holomorphic in $T_{\Gamma_{\sigma_{1}}}$,
$R(z_{1},...,z_{n})$ as a function of $z_{n}$ is also holomorphic in upper-
half plane $\mathbb{C}^{+}$. Due to the result for $n=1$,
$R(z_{1},...,z_{n-1},z_{n})$ as a function of $z_{n}$ is a member of
$H^{p}(\mathbb{C}^{+})$. By Lemma 3.1-2 again,

$\int_{\mathbb{R}}R(z_{1},...,z_{n-1},x_{n}+iy_{n})|^{p}\
dx_{n}\leq\int_{\mathbb{R}}|R(z_{1},...,z_{n-1},x_{n})|^{p}\ dx_{n}.$

Integrating both sides of the last inequality with respect to
$x_{1},...,x_{n-1}$, we have

$\begin{split}&\int_{\mathbb{R}^{n-1}}\left(\int_{\mathbb{R}}|R(x_{1}+iy_{1},...,x_{n-1}+iy_{n-1},x_{n}+iy_{n})|^{p}\
dx_{n}\right)dx_{1}\cdot\cdot\cdot dx_{n-1}\\\
&=\int_{\mathbb{R}^{n}}|R(x+iy)|^{p}\ dx\\\
&\leq\int_{\mathbb{R}^{n-1}}\left(\int_{\mathbb{R}}|R(x_{1}+iy_{1},...,x_{n-1}+iy_{n-1},x_{n})|^{p}\
dx_{n}\right)dx_{1}\cdot\cdot\cdot dx_{n-1}.\end{split}$ (10)

Together with (9) and (10), we obtain

$\int_{\mathbb{R}^{n}}|R(x+iy)|^{p}\ dx\leq\int_{\mathbb{R}^{n}}|R(x)|^{p}\
dx.$

Due to the analyticity of the rational function $R(z)$ in
$T_{\Gamma_{\sigma_{1}}}$, we get $R(z)\in H^{p}(T_{\Gamma_{\sigma_{1}}}).$
The proofs for the other octants are similar. So the proof of Lemma 3.1 is
complete.

Similarly to prove Lemma 3.1, we also can get the analogous results about the
other octants as follows.

###### Corollary 3.1

Let $\Gamma_{\sigma_{j}}\ (j=1,2,...,2^{n})$ be all the octants of
$\mathbb{R}^{n}$. Suppose that $0<p<1$ and $R$ is a rational function with the
form

$R(z)=\frac{P(z)}{Q(z)},$

where $P(z)$ is a polynomial of $z$, and
$Q(z)=Q_{1}(z_{1})Q_{2}(z_{2})\cdot\cdot\cdot\cdot Q_{n}(z_{n})$, where for
each $k$, $Q_{k}(z_{k})$ is a polynomial of $z_{k},\ k=1,2,...,n$. If $R\in
L^{p}(\mathbb{R}^{n})$ and $R(z)$ is holomorphic in $T_{\Gamma_{\sigma_{j}}}$,
then $R(z)\in H^{p}(T_{\Gamma_{\sigma_{j}}}),$ $j=1,2,...,2^{n}.$

###### Lemma 3.2

If $0<p<1$, $f\in L^{p}(\mathbb{R}^{n})$, then, for $\varepsilon>0$, there
exist a sequence of rational functions $\\{R_{k}(x)\\}$,
$R_{k}\in\mathcal{A}$, such that

$\sum_{k=1}^{\infty}||R_{k}||^{p}_{p}\leq(1+\varepsilon)\|f\|_{p}^{p},$ (11)

and

$\lim_{n\rightarrow\infty}||f-\sum_{k=1}^{n}R_{k}||_{p}=0,$ (12)

where

$\mathcal{A}=\left\\{R(x)=\frac{P(x)}{(1+x_{1}^{2})^{l_{1}}\cdot\cdot\cdot(1+x_{n}^{2})^{l_{n}}}:\
2l_{j}>{\rm deg}_{j}P,\ j=1,2,...,n\right\\},$

$P(x)=\sum_{(k)}\alpha_{k}x_{1}^{k_{1}}\cdot\cdot\cdot x_{n}^{k_{n}}$ is a
polynomial of $x=(x_{1},x_{2},...,x_{n})\in\mathbb{R}^{n}$, $\alpha_{k}$ are
constants, $k=(k_{1},k_{2},...,k_{n})$, and the notation $\sum_{(k)}$ means
that
$\sum_{(k)}=\sum\limits_{k_{n}=0}^{s_{n}}\cdot\cdot\cdot\sum\limits_{k_{1}=0}^{s_{1}},$
${\rm deg}_{j}P=s_{j},\ j=1,2,...,n.$

Proof  We assume that $\|f\|_{p}>0,$ and let

$C_{0}(\mathbb{R}^{n})=\left\\{f:\ f\ {\rm is\ continuous\ in}\
\mathbb{R}^{n},\ \lim\limits_{|x|\rightarrow\infty}f(x)=0\right\\}.$

It is obviously that $\mathcal{A}$ is a subalgebra of $C_{0}(\mathbb{R}^{n})$
and $\mathcal{A}$ separates points. Since $\mathbb{R}^{n}$ is a local compact
Hausdorff space, $C_{0}(\mathbb{R}^{n})$ is a Banach algebra with the supremum
norm $\|f\|=\sup\\{f(x):x\in\mathbb{R}^{n}\\}$. The Stone-Weierstrass theorem
assures that $\mathcal{A}$ is dense in $C_{0}(\mathbb{R}^{n})$.

Suppose that $0<p<\infty,$ and $f\in L^{p}(\mathbb{R}^{n}).$ Let

$f_{N}(x)=f(x)\mathcal{X}_{\\{x:|x|\leq N;|f(x)|\leq N\\}}(x).$

It is clear to see that, $|f_{N}(x)|\leq N$, and

${\rm supp}f_{N}\subset\overline{B(0,N)}=\\{x:\ |x|\leq N\\}.$

Moreover, $|f_{N}-f|\leq 2|f|\in L^{p}(\mathbb{R}^{n})$, and,
$\lim\limits_{N\rightarrow\infty}|f_{N}-f|^{p}=0.$ Thus, by the Lebesgue
dominated convergence theorem, we have

$\lim\limits_{N\rightarrow\infty}\int_{\mathbb{R}^{n}}|f_{N}-f|^{p}dx=\int_{\mathbb{R}^{n}}\lim\limits_{N\rightarrow\infty}|f_{N}-f|^{p}dx=0.$

Therefore, for $\varepsilon_{0}>0$, there exists an integer $N>1$, such that

$||f_{N}-f||_{p}^{p}<\varepsilon_{0}/3.$ (13)

Because $f_{N}(x)$ is a measurable function, according Lusin Theorem, there
exists a function $g_{0}\in C_{0}(\mathbb{R}^{n}),$ such that

${\rm supp}g_{0}\subset B(0,N),\ \ \ |g_{0}(x)|\leq N,\ \ \
m(E_{N})<\varepsilon_{0}/3(2N)^{p},$

where $E_{N}=\\{x:\ g_{0}(x)\neq f_{N}(x)\\}$. Therefore,

$\begin{split}\int_{\mathbb{R}^{n}}|g_{0}(x)-f_{N}|^{p}dx&=\int_{B(0,N)}|g_{0}(x)-f_{N}|^{p}dx\\\
&\leq(2N)^{p}\int_{E_{N}}dx<\varepsilon_{0}/3.\end{split}$ (14)

Thus we obtain

$\|g_{0}-f\|_{p}^{p}=\|g_{0}-f_{N}+f_{N}-f\|_{p}^{p}\leq\|g_{0}-f_{N}\|_{p}^{p}+\|f_{N}-f\|_{p}^{p}<\varepsilon_{0}/3+\varepsilon_{0}/3=2\varepsilon_{0}/3.$
(15)

Taking integers $\widetilde{l}_{k}$ such that $p\widetilde{l}_{k}>1,\
k=1,2,...,n,$ the fact ${\rm supp}g_{0}\subset B(0,N)$ implies

$g_{0}(x)\prod\limits_{k=1}^{n}(1+x_{k}^{2})^{\widetilde{l}_{k}}\in
C_{0}(\mathbb{R}^{n}).$

Since $\mathcal{A}$ is dense in $C_{0}(\mathbb{R}^{n})$, there exists rational
functions $r(x)\in\mathcal{A}$, such that

$\left|r(x)-g_{0}(x)\prod\limits_{k=1}^{n}(1+x_{k}^{2})^{\widetilde{l}_{k}}\right|<(\varepsilon_{0}/3)^{1/p}.$
(16)

In fact, $r(x)$ can be written as

$r(x)=\frac{P(x)}{\prod\limits_{k=1}^{n}(1+x_{k}^{2})^{l_{k}}},$

where $P(x)=\sum_{(k)}\alpha_{k}x_{1}^{k_{1}}\cdot\cdot\cdot x_{n}^{k_{n}},\
$deg${}_{j}P<2l_{j},\ j=1,2,...,n$. Then, by (16),

$\left|P(x)/\prod\limits_{k=1}^{n}(1+x_{k}^{2})^{l_{k}}-g_{0}(x)\prod\limits_{k=1}^{n}(1+x_{k}^{2})^{\widetilde{l}_{k}}\right|<\varepsilon_{0}/3\pi^{n},$

or

$\left|P(x)/\prod\limits_{k=1}^{n}(1+x_{k}^{2})^{l_{k}+\widetilde{l}_{k}}-g_{0}(x)\right|<\frac{(\varepsilon_{0}/3)^{1/p}}{\pi^{n}\prod\limits_{k=1}^{n}(1+x_{k}^{2})^{\widetilde{l}_{k}}}.$

Obviously,
$Q(x)=P(x)/\prod\limits_{k=1}^{n}(1+x_{k}^{2})^{l_{k}+\widetilde{l}_{k}}\in\mathcal{A}$.
Thus,

$\int_{\mathbb{R}^{n}}|Q(x)-g_{0}|^{p}dx<\frac{1}{\pi^{n}}\int_{\mathbb{R}^{n}}\frac{(\varepsilon_{0}/3)}{\prod\limits_{k=1}^{n}(1+x_{k}^{2})^{p\widetilde{l}_{k}}}\
dx<\varepsilon_{0}/3.$

Therefore, for any $\varepsilon_{0}>0$, there exists $Q(x)\in\mathcal{A}$ such
that

$\|Q-f\|_{p}^{p}=\|Q-g_{0}+g_{0}-f\|_{p}^{p}\leq\|Q-g_{0}\|_{p}^{p}+\|g_{0}-f\|_{p}^{p}<\varepsilon_{0}.$

Thus, for any $\varepsilon>0,$ taking
$\varepsilon_{k}=\frac{\|f\|_{p}^{p}\varepsilon_{0}}{4^{k+3}},\ k=1,2,...,$
there exist $Q_{k}(x)\in\mathcal{A}$, such that

$\|Q_{k}-f\|_{p}^{p}<\varepsilon_{k}.$

Moreover, the function $Q_{k}(z)$ is a rational function satisfying

$\|Q_{k}\|_{p}^{p}=\|Q_{k}-f+f\|_{p}^{p}\leq\|Q_{k}-f\|_{p}^{p}+\|f\|_{p}^{p}<\varepsilon_{k}+\|f\|_{p}^{p}=(1+\frac{\varepsilon}{4^{k+3}})\|f\|_{p}^{p}.$

Thus the sequence of rational functions $Q_{k}(z)$ can be chosen such that

$\|Q_{k}-Q_{k-1}\|_{p}^{p}=\|Q_{k}-f+f-Q_{k-1}\|_{p}^{p}\leq\|Q_{k}-f\|_{p}^{p}+\|f-Q_{k-1}\|_{p}^{p}<2\varepsilon_{k},\
\ \ (k=2,3,\cdot\cdot\cdot).$

Let

$R_{1}(z)=Q_{1}(z),\ R_{k}(z)=Q_{k}(z)-Q_{k-1}(z),\ \ \
(k=2,3,\cdot\cdot\cdot).$

Then $\\{R_{k}(z)\\}$ is a sequence of rational functions satisfying (11) and
(12). This completes the proof of Lemma 3.2.

###### Lemma 3.3

Suppose that $0<p<1$ and $R\in L^{p}(\mathbb{R}^{n})\bigcap\mathcal{A}$ is a
rational function, where $\mathcal{A}$ is the same as the $\mathcal{A}$ in
Lemma 3.2. Then, there exist $2^{n}$ rational functions $R_{\sigma_{j}}(z)\in
H^{p}(T_{\Gamma_{\sigma_{j}}}),\ j=1,2,...,2^{n}$, such that

$R(z)=\sum\limits_{j=1}^{2^{n}}R_{\sigma_{j}}(z),$

and

$\sum\limits_{j=1}^{2^{n}}\|R_{\sigma_{j}}\|^{p}_{p}\leq C_{np}\|R\|^{p}_{p},$
(17)

where $C_{np}=2^{n}\left(\frac{2^{1-p}\pi}{1-p}\right)^{n}.$

Proof Let $R(x)\in L^{p}(\mathbb{R}^{n})\bigcap\mathcal{A}$ be a rational
function. $R(z)$ can be written as

$R(z)=\frac{P(z)}{\prod\limits_{j=1}^{n}(1+z_{j}^{2})^{l_{j}}}=\frac{\sum_{(k)}\alpha_{k}z_{1}^{k_{1}}\cdot\cdot\cdot
z_{n}^{k_{n}}}{\prod\limits_{j=1}^{n}(1+z_{j}^{2})^{l_{j}}},$

where $p(2l_{j}-{\rm deg}_{j}P)>1$ for all $j=1,2,...,n.$

Let

$\beta(\xi)=\frac{i-\xi}{i+\xi},\ \ \ \xi\in\mathbb{C}.$

It is easy to know that $\beta(x)=e^{i\theta(x)},$ where
$\theta(x)=\arg(i-x)-\arg(i+x)\in(-\pi,\pi)$ for $x\in\mathbb{R}$.

For each $\varphi=(\varphi_{1},\varphi_{2},...,\varphi_{n})\in\mathbb{R}^{n}$,
$z\in T_{\Gamma_{\sigma_{j}}}$. Rational functions $R_{\sigma_{j}}(z,\varphi)\
(j=1,2,...,2^{n})$ are defined as follows.

$R_{\sigma_{j}}(z,\varphi)=\frac{\prod\limits_{k=1}^{n}(\beta(z_{k}))^{m_{k}\sigma_{j}(k)}}{\prod\limits_{k=1}^{n}((\beta(z_{k}))^{m_{k}\sigma_{j}(k)}-e^{i\varphi_{k}\sigma_{j}(k)})}R(z),$

where $m_{k}>l_{k}+n\ (k=1,2,...,n)$ are positive integers .

Then $R_{\sigma_{j}}(z,\varphi)$ can be written as,

$\begin{split}R_{\sigma_{j}}(z,\varphi)&=\prod\limits_{\sigma_{j}(k)=+1}\frac{(\beta(z_{k}))^{m_{k}}}{(\beta(z_{k}))^{m_{k}}-e^{i\varphi_{k}}}\prod\limits_{\sigma_{j}(k)=-1}\frac{(\beta(z_{k}))^{-m_{k}}}{(\beta(z_{k}))^{-m_{k}}-e^{-i\varphi_{k}}}R(z)\\\
&=\prod\limits_{\sigma_{j}(k)=+1}\frac{(\beta(z_{k}))^{m_{k}}}{(\beta(z_{k}))^{m_{k}}-e^{i\varphi_{k}}}\prod\limits_{\sigma_{j}(k)=-1}\frac{e^{i\varphi_{k}}(\beta(z_{k}))^{m_{k}}}{(e^{i\varphi_{k}}-(\beta(z_{k}))^{m_{k}})(\beta(z_{k}))^{m_{k}}}R(z)\\\
&=\prod\limits_{\sigma_{j}(k)=+}\frac{(\beta(z_{k}))^{m_{k}}}{(\beta(z_{k}))^{m_{k}}-e^{i\varphi_{k}}}\prod\limits_{\sigma_{j}(k)=-1}\frac{(\beta(z_{k}))^{m_{k}}}{(\beta(z_{k}))^{m_{k}}-e^{i\varphi_{k}}}\prod\limits_{\sigma_{j}(k)=-1}\frac{-e^{i\varphi_{k}}}{(\beta(z_{k}))^{m_{k}}}R(z)\\\
&=\prod\limits_{k=1}^{n}\frac{(\beta(z_{k}))^{m_{k}}}{(\beta(z_{k}))^{m_{k}}-e^{i\varphi_{k}}}\prod\limits_{\sigma_{j}(k)=-1}\frac{-e^{i\varphi_{k}}}{(\beta(z_{k}))^{m_{k}}}R(z)\\\
&=\frac{\prod\limits_{\sigma_{j}(k)=-1}\frac{-e^{i\varphi_{k}}}{(\beta(z_{k}))^{m_{k}}}}{\prod\limits_{k=1}^{n}\left(1-\frac{e^{i\varphi_{k}}}{(\beta(z_{k}))^{m_{k}}}\right)}R(z).\\\
\end{split}$

Therefore,

$\sum\limits_{j=1}^{2^{n}}R_{\sigma_{j}}(z,\varphi)=\frac{\sum\limits_{j=1}^{2^{n}}\prod\limits_{\sigma_{j}(k)=-1}\frac{-e^{i\varphi_{k}}}{(\beta(z_{k}))^{m_{k}}}}{\prod\limits_{k=1}^{n}\left(1-\frac{e^{i\varphi_{k}}}{(\beta(z_{k}))^{m_{k}}}\right)}R(z)=R(z).$

Next we are to prove that $R_{\sigma_{j}}(z,\varphi)\in
H^{p}(T_{\Gamma_{\sigma_{j}}}),$ for all $j=1,2,...,2^{n}.$

Now we only consider the case that the base is the first octant of
$\mathbb{R}^{n}$, because proofs of the other octants are similar. For any
$z\in T_{\Gamma_{\sigma_{1}}}$,

$\begin{split}R_{\sigma_{1}}(z,\varphi)&=\frac{\prod\limits_{k=1}^{n}(\beta(z_{k}))^{m_{k}}}{\prod\limits_{k=1}^{n}((\beta(z_{k}))^{m_{k}}-e^{i\varphi_{k}})}R(z)\\\
&=\frac{\prod\limits_{k=1}^{n}(\frac{i-z_{k}}{i+z_{k}})^{m_{k}}}{\prod\limits_{k=1}^{n}((\beta(z_{k}))^{m_{k}}-e^{i\varphi_{k}})}\frac{P(z)}{\prod\limits_{k=1}^{n}(z_{k}-i)^{l_{k}}(z_{k}+i)^{l_{k}}}\\\
&=P(z)\prod\limits_{j=1}^{n}\frac{(-1)^{m_{j}}(z_{j}-i)^{m_{j}-l_{j}}}{(z_{j}+i)^{l_{j}+m_{j}}((\beta(z_{j}))^{m_{j}}-e^{i\varphi_{j}})}.\end{split}$
(18)

Since $m_{k}>l_{k},$ and $|\beta(z_{k})|<1,\ |e^{i\varphi_{k}}|=1,$ for all
$k=1,2,...,n$, the function $R_{\sigma_{1}}(z)$ is a rational function which
is holomorphic in the tube $T_{\Gamma_{\sigma_{1}}}$.

Moreover, set

$I_{\sigma_{1}}=\int_{\mathbb{R}^{n}}\int_{(-\pi,\pi)^{n}}|R_{\sigma_{1}}(x,\varphi)|^{p}\
d\varphi dx.$

Then,

$\begin{split}I_{\sigma_{1}}&=\int_{\mathbb{R}^{n}}\int_{(-\pi,\pi)^{n}}\left|\frac{\prod\limits_{k=1}^{n}(\beta(x_{k}))^{m_{k}\sigma_{1}(k)}}{\prod\limits_{k=1}^{n}((\beta(x_{k}))^{m_{k}\sigma_{1}(k)}-e^{i\varphi_{k}\sigma_{1}(k)})}R(x)\right|^{p}\
d\varphi_{1}\cdot\cdot\cdot d\varphi_{n}dx\\\
&=\int_{\mathbb{R}^{n}}\int_{-\pi}^{\pi}\cdot\cdot\cdot\int_{-\pi}^{\pi}\prod\limits_{k=1}^{n}\frac{|R(x)|^{p}}{|1-e^{i\varphi_{k}\sigma_{1}(k)-im_{k}\theta(x_{k})}|^{p}}\
d\varphi_{1}\cdot\cdot\cdot d\varphi_{n}dx.\\\ \end{split}$ (19)

Observe that

$\begin{split}&\int_{-\pi}^{\pi}\frac{1}{|1-e^{i\varphi_{k}\sigma_{j}(k)-im_{k}\theta(w_{k})}|^{p}}\
d\varphi_{k}\\\ &=\int_{-\pi}^{\pi}\frac{1}{|1-e^{i\theta\sigma_{j}(k)}|^{p}}\
d\theta\\\
&=\int_{-\pi}^{\pi}\frac{d\theta}{2^{p}|\sin^{p}(\frac{\theta}{2})|}\leq\frac{4}{2^{p}}\int_{0}^{\frac{\pi}{2}}\frac{d\theta}{(\frac{2\theta}{\pi})^{p}}\leq\frac{2^{1-p}\pi}{1-p}.\end{split}$
(20)

By (19) and (20),

$I_{\sigma_{1}}\leq\left(\frac{2^{1-p}\pi}{1-p}\right)^{n}\int_{\mathbb{R}^{n}}|R(x)|^{p}\
dx=\left(\frac{2^{1-p}\pi}{1-p}\right)^{n}\|R\|_{p}^{p}.$

Similarly,

$I_{\sigma_{k}}\leq\left(\frac{2^{1-p}\pi}{1-p}\right)^{n}\|R\|_{p}^{p}$

for $k=2,3,...,2^{n}.$

Therefore,

$\sum\limits_{k=1}^{2^{n}}I_{\sigma_{k}}=\int_{(-\pi,\pi)^{n}}\sum\limits_{k=1}^{2^{n}}\int_{\mathbb{R}^{n}}|R_{\sigma_{k}}(x,\varphi)|^{p}\
dxd\varphi\leq 2^{n}\left(\frac{2^{1-p}\pi}{1-p}\right)^{n}\|R\|^{p}_{p}.$

Thus, there exists a $\varphi=(\varphi_{1},...,\varphi_{n})\in(-\pi,\pi)^{n}$,
such that

$\sum\limits_{k=1}^{2^{n}}\|R_{\sigma_{k}}\|^{p}_{p}\leq
2^{n}\left(\frac{2^{1-p}\pi}{1-p}\right)^{n}\|R\|^{p}_{p},$

where $R_{\sigma_{k}}(x)=R_{\sigma_{k}}(x,\varphi),\ k=1,2,...,2^{n}$. This
shows that the inequality (17) holds. It is easy to know that

$\|R_{\sigma_{k}}\|^{p}_{p}\leq
2^{n}\left(\frac{2^{1-p}\pi}{1-p}\right)^{n}\|R\|^{p}_{p},$

for all $k=1,2,...,2^{n}$. So rational functions $R_{\sigma_{k}}(x)\in
L^{p}(\mathbb{R}^{n})$. By Lemma 3.1, and that $R_{\sigma_{k}}(z)$ is
holomorphic in $T_{\Gamma_{\sigma_{k}}}$, we have

$R_{\sigma_{k}}(z)\in H^{p}(T_{\Gamma_{\sigma_{k}}})$

for all $k=1,2,...,2^{n}$.

Thus, the proof of Lemma 3.3 is complete.

We still need the following lemmas.

###### Lemma 3.4

([17]) Let $B$ be an open cone in $\mathbb{R}^{n}$. Suppose $F\in
H^{p}(T_{B}),\ p>0,$ and $B_{0}\subset B$ satisfies
$d(B_{0},B^{c})=\inf\\{|y_{1}-y_{2}|;\ y_{1}\in B_{0},\ y_{2}\in
B^{c}\\}\geq\varepsilon>0$, then there exists a constant $C_{p}(\varepsilon),$
depending on $\varepsilon$ and $p$ but not on $F$, such that

$\sup\limits_{z\in T_{B_{0}}}|F(z)|\leq C_{p}(\varepsilon)\|F\|_{H^{p}}.$

Given below offers a more precise estimation than that obtained in above Lemma
3.4.

###### Lemma 3.5

Suppose that $f\in H^{p}(T_{\Gamma}),\ p>0,$ and $T_{\Gamma}$ is the tube with
its base $\Gamma$ as the first octant of $\mathbb{R}^{n}$. If let
$f_{\delta}(z)=f(z+i\delta)$, for any $z=x+iy\in T_{\Gamma}$ and
$\delta=(\delta_{1},\delta_{2},...,\delta_{n})\in\Gamma$, then there holds

$\sup\limits_{z\in T_{\Gamma}}|f_{\delta}(z)|\leq
C_{p}\|f\|_{H^{p}}(\delta_{1}\cdot\cdot\cdot\delta_{n})^{-\frac{1}{p}},$

where $C_{p}=(\frac{2}{\pi})^{\frac{n}{p}}.$

We note that the proof of Lemma 3.5 is obtained in our recently work [9].

###  3.1  Proof of Theorem 3.1

Based on the above lemmas, we are now to prove Theorem 3.1.

Proof of Theorem 3.1 When $0<p<1$, by Lemma 3.2, for any $f(x)\in
L^{p}(\mathbb{R}^{n})$, and $\varepsilon>0,$ there exists a sequence of
rational functions $\\{R_{k}(x)\\}$, such that

$\sum_{k=1}^{\infty}||R_{k}||^{p}_{p}\leq(1+\varepsilon)\|f\|_{p}^{p},$ (21)

and

$\lim_{m\rightarrow\infty}||f-\sum_{k=1}^{m}R_{k}||_{p}=0.$ (22)

For each $k=1,2,...$, by Lemma 3.3, there exist $2^{n}$ rational functions
$R_{k\sigma_{j}}(z)\in H^{p}(T_{\Gamma_{\sigma_{j}}})\ (j=1,2,...,2^{n}),$
such that

$R_{k}(z)=\sum\limits_{j=1}^{2^{n}}R_{k\sigma_{j}}(z),$ (23)

and

$\sum\limits_{j=1}^{2^{n}}\|R_{k\sigma_{j}}\|^{p}_{p}\leq
C_{np}\|R_{k}\|^{p}_{p}.$ (24)

Therefore,

$\sum_{k=1}^{\infty}\sum\limits_{j=1}^{2^{n}}||R_{k\sigma_{j}}||^{p}_{p}\leq\sum_{k=1}^{\infty}C_{np}\|R_{k}\|^{p}_{p}\leq(1+\varepsilon)C_{np}\|f\|_{p}^{p},$
$\lim_{m\rightarrow\infty}||f-\sum_{k=1}^{m}\sum\limits_{j=1}^{2^{n}}R_{k\sigma_{j}}||_{p}=0.$

By Lemma 3.1-2, there are

$\|R_{k\sigma_{j}}\|^{p}_{H^{p}_{\sigma_{j}}}=\|R_{k\sigma_{j}}\|^{p}_{p},\ \
j=1,2,...,2^{n},$

which imply that the properties (4) and (5) hold.

Moreover, for any
$\delta=(\delta_{1},\delta_{2},...,\delta_{n})\in\Gamma_{\sigma_{j}}$, and
$j=1,2,...,2^{n}$, by Lemma 3.4 and 3.5, there exists a positive constant
$M_{\delta}<\infty$ such that

$\left|R_{k\sigma_{j}}(z+i\delta)\right|\leq
M_{\delta}\|R_{k\sigma_{j}}\|_{p}.$

Hence,

$\left|\sum_{k=1}^{m}R_{k\sigma_{j}}(z+i\delta)\right|^{p}\leq\sum_{k=1}^{m}|R_{k\sigma_{j}}(z+i\delta)|^{p}\leq
M_{\delta}\sum_{k=1}^{m}\|R_{\sigma_{j}}\|^{p}_{p},$

for $z\in T_{\Gamma_{\sigma_{j}}}.$ This implies that the series
$\sum\limits_{k=1}^{\infty}R_{k\sigma_{j}}(z)$ uniformly converges to a
function $f_{\sigma_{j}}(z)$ in the tube domain
$T_{\Gamma_{\sigma_{j}}^{\delta}}=\\{z=x+iy\in\mathbb{C}^{n}:\
x\in\mathbb{R}^{n},\ \sigma_{j}(l)y_{l}>\sigma_{j}(l)\delta_{l},\
l=1,2,...,n\\}$ for any
$\delta=(\delta_{1},\delta_{2},...,\delta_{n})\in\Gamma_{\sigma_{j}}$.
As a consequence, the function $f_{\sigma_{j}}(z)$ is holomorphic in
$T_{\Gamma_{\sigma_{j}}}$. Property (4) implies that property (6) holds. By
properties of Hardy spaces on tubes, the non-tangential boundary limit
$f_{\sigma_{j}}(x)$ of function $f_{\sigma_{j}}(z)\in
H^{p}(T_{\Gamma_{\sigma_{j}}})$ exists for every $j=1,2,...,2^{n}$.
Therefore, property (5) implies that

$f(x)=\sum\limits_{j=1}^{2^{n}}f_{\sigma_{j}}(x)$

holds almost everywhere, and moreover,

$\|f\|^{p}_{p}\leq\sum\limits_{j=1}^{2^{n}}\|f_{\sigma_{j}}\|^{p}_{p}\leq
A_{p}\|f\|_{p}^{p}.$

Thus, the proof of Theorem 3.1 is complete.

##  4  Non-uniqueness of Hardy Space Decomposition

In this section, we are to answer the questions asked in the Remark 3.1 of
Theorem 3.1. For the one dimension, A.B. Aleksandrov ([1], [2]) obtained the
following theorem.

Theorem B  ([1] and [2]) Let $0<p<1$ and $X^{p}$ denote the $L^{p}$ closure of
the set of $f\in L^{p}(\mathbb{R})$ which can be written in the form

$f(x)=\sum\limits_{j=1}^{N}\frac{c_{j}}{x-a_{j}},\ \ a_{j}\in\mathbb{R},\
c_{j}\in\mathbb{C}.$

Then

$X^{p}=H_{+}^{p}(\mathbb{R})\cap H_{-}^{p}(\mathbb{R}).$

We note that, A.B. Aleksandrov’s proof of Theorem M ([1], [2]) is rather long
involving vanishing moments and the Hilbert transformation. Deng and Qian [5]
present a more straightforward proof for Theorem M. In this section, our aim
is to extend Theorem M to higher dimensions. In order to do this, we need
first to extend the following Theorem C obtained by J.B. Garnett ([10]) and
Theorem D obtained by Deng-Qian ([5]) to higher dimensions.

Theorem C  ([10]) Let $N$ be a positive integer. For $0<p<\infty,\ Np>1,$ the
class $\mathfrak{w}_{N}$ is dense in $H^{p}(\mathbb{C}^{+})$, where
$\mathfrak{w}_{N}$ is the family of $H^{p}(\mathbb{C}^{+})$ functions
satisfying

(i) $f(z)$ is infinitely differentiable in $\overline{\mathbb{C}^{+}}$,

(ii) $|z|^{N}f(z)\rightarrow 0$ as $z\rightarrow\infty$.

Theorem D ([5]) Let $N$ be a positive integer. For $0<p<\infty,\ Np>1$, the
class $\mathfrak{R}_{N}(i)$ is dense in $H^{p}(\mathbb{C}^{+})$, and the class
$\mathfrak{R}_{N}(-i)$ is dense in $H^{p}(\mathbb{C}^{-})$. Where
$\alpha\in\mathbb{C}$ and $\mathfrak{R}_{N}(\alpha)$ is the family of rational
functions $f(z)=(z+\alpha)^{-N-1}P(\frac{1}{z+\alpha}),$ $P(w)$ are
polynomials.

We obtain the following three theorems for higher dimensions.

###### Theorem 4.1

Let $N$ be a positive integer. For $0<p<\infty,\ pN>1$, the class
$\mathfrak{w}_{N}$ is dense in $H^{p}(T_{\Gamma}),$ where $\mathfrak{w}_{N}$
is the family of $H^{p}(T_{\Gamma})$ functions satisfying

(i) $f(z)$ is infinitely differentiable in $\overline{T}_{\Gamma}$,

(ii) $|z|^{N}f(z)\rightarrow 0$ as $|z|\rightarrow\infty$, where
$|z|\rightarrow\infty$ means that $z_{j}\rightarrow\infty,\ 1\leq j\leq n,\
z=(z_{1},z_{2},...,z_{n})\in\overline{T}_{\Gamma}.$

Proof We can approximate $f(z)\in H^{p}(T_{\Gamma})$ by the smooth function
$f(z+i/m)=f(z_{1}+i/m,...,z_{n}+i/m)$. In fact, the property that the
existence of the boundary limits functions of Hardy space functions assures
that

$\|f_{m}-f\|_{H^{p}}\rightarrow 0,\ \ m\rightarrow\infty.$

We will construct the special functions $g_{k}(z)$ such that

(a) $g_{k}(z)\in\mathfrak{w}_{N}$,

(b) $|g_{k}(z)|\leq 1,\ \ z\in T_{\Gamma}$,

(c) $|g_{k}(z)|\rightarrow 1,\ \ z\in T_{\Gamma},$ as $k\rightarrow\infty$.
Before we construct the above functions $g_{k}(z)$, we note that the functions

$f_{m}(z)=g_{m}(z)f(z+i/m)$

in $\mathfrak{w}_{N}$ and then obtain the desired approximation.

That is to say, if there exist the special functions $g_{k}(z)$, we can
complete the proof of Theorem 4.1.

As the heart of the proof, we are to construct the functions $g_{k}(z)$ in the
following. Let $(w_{1},w_{2},...,w_{n})\in T_{\Gamma},\
(\alpha_{k},\alpha_{k},...,\alpha_{k})\in\mathbb{R}^{n},\ 0<\alpha_{k}<1,$ and
$\alpha_{k}\rightarrow 1$ as $k\rightarrow\infty.$
Consider the function

$h_{k}(w)=\prod\limits_{j=1}^{n}\varphi_{j}(w_{j}),$

where
$\varphi_{j}(w_{j})=\left(\frac{w_{j}+\alpha_{k}}{1+\alpha_{k}w_{j}}\right)^{N+1},\
j=1,2,...,n$ has infinite $(N+1)-$fold zero at $-\alpha_{k}$.
Then,

$|h_{k}(w)|=\left|\prod\limits_{j=1}^{n}\varphi_{j}(w_{j})\right|=\prod\limits_{j=1}^{n}\left|\frac{w_{j}+\alpha_{k}}{1+\alpha_{k}w_{j}}\right|^{N+1}<1.$
(25)

Fixing $N$, $h_{k}(w)$ converges to $1$ uniformly on the compact set
$\overline{D}\setminus\bigcup\limits_{k=1}^{n}E_{k}$, where

$D=D_{1}\times D_{2}\times\cdot\cdot\cdot\times D_{n},\ \ D_{k}=|z|<1,\
k=1,2,...,n,$

and

$E_{k}=\\{(w_{1},...,w_{j-1},-\alpha_{k},w_{j},...,w_{n}):\ w_{j}\in D_{k},\
k\neq j\\}.$

Then, for
$w=(w_{1},w_{2},...,w_{n})=\left(\frac{i-z_{1}}{i+z_{1}},...,\frac{i-z_{n}}{i+z_{n}}\right)$
and $\alpha=(\alpha_{k},\alpha_{k},...,\alpha_{k}),$ we define the functions

$g_{k}(z)=h_{k}(\alpha_{k}w).$

Below we verify that the function $g_{k}(z)$ satisfies the three conditions
(a) (b) and (c).
In fact, for condition (a), there is

$\begin{split}g_{k}(z)&=h_{k}(\alpha_{k}w)=\prod\limits_{j=1}^{n}\left(\frac{\alpha_{k}w_{j}+\alpha_{k}}{1+\alpha^{2}_{k}w_{j}}\right)^{N+1}\\\
&=\prod\limits_{j=1}^{n}\alpha_{k}^{N+1}\left(\frac{w_{j}+1}{1+\alpha^{2}_{k}w_{j}}\right)^{N+1}=\prod\limits_{j=1}^{n}\alpha_{k}^{N+1}\left(\frac{\frac{i-z_{j}}{i+z_{j}}+1}{1+\alpha^{2}_{k}\frac{i-z_{j}}{i+z_{j}}}\right)^{N+1}\\\
&=\prod\limits_{j=1}^{n}\alpha_{k}^{N+1}\left(\frac{2i}{(1+\alpha^{2}_{k})i+(1-\alpha^{2}_{k})z_{j}}\right)^{N+1}.\end{split}$

It is clearly that $g_{k}(z)$ satisfies the first condition $(i)$ of the class
$\mathfrak{w}_{N}$. Moreover, there holds $|z|^{N}g_{k}(z)\rightarrow 0$ as
$|z|\rightarrow\infty$. This implies that $g_{k}(z)$ satisfies the second
condition $(ii)$ of the class $\mathfrak{w}_{N}$. Therefore,
$g_{k}(z)\in\mathfrak{w}_{N}$, which shows that $g_{k}(z)$ satisfies the
condition $(a)$.

For condition (b), from (25), we can get that

$|g_{k}(w)|=|h_{k}(\alpha_{k}w)|<1.$

For (c), it is clear that

$g_{k}(w)=h_{k}(\alpha_{k}w)\rightarrow 1,\ \ k\rightarrow\infty,\
z\in\overline{T}_{\Gamma}.$

Thus, the proof is complete.

We shall notice that the condition $Np>1$ implies that the above class
$\mathfrak{w}_{N}$ is contained in $H^{p}(T_{\Gamma})$. Let
$\alpha=(\alpha_{1},\alpha_{2},...,\alpha_{n})\in\mathbb{C}^{n}$ and let
$\mathfrak{R}_{N}(\alpha)$ be the family of the rational functions

$f(z)=(z_{1}+\alpha_{1})^{-N-1}(z_{2}+\alpha_{2})^{-N-1}\cdot\cdot\cdot(z_{n}+\alpha_{n})^{-N-1}P(\frac{1}{z_{1}+\alpha_{1}},...,\frac{1}{z_{n}+\alpha_{n}}),$

where $z=(z_{1},z_{2},...,z_{n})\in\mathbb{C}^{n}$ and $P(w)$ are polynomials.
We notice that the class $\mathfrak{R}_{N}(\alpha)$ is contained in the class
$\mathfrak{w}_{N}$ for Im$\alpha_{j}>0,\ j=1,2,...,n.$ Thus, we obtain the
following results.

###### Theorem 4.2

Let $N$ be a positive integer. For $0<p<\infty$ and $Np>1$, the class
$\mathfrak{R}_{N}(i,i,...,i)$ is dense in $H^{p}(T_{\Gamma}).$

###### Corollary 4.1

Let $N$ be a positive integer. For $0<p<\infty$ and $Np>1$, the class
$\mathfrak{R}_{N}(\sigma_{j}(1)i,...,\sigma_{j}(n)i)$ is dense in
$H^{p}(T_{\Gamma_{\sigma_{j}}}),\ j=1,2,...,2^{n}.$

The proof of Corollary 4.1 is similar to the proof of Theorem 4.2, so we only
prove Theorem 4.2.
Proof of Theorem 4.2 If $f(z)\in H^{p}(T_{\Gamma}),\ Np>1$, then, for any
$\varepsilon>0,$ by Theorem 4.1, there exists function $f_{N}$ in
$H^{p}(T_{\Gamma})\bigcap C^{\infty}(\overline{T}_{\Gamma})$ such that

$\lim\limits_{|z|\rightarrow\infty,z\in T_{\Gamma}}|z|^{N+1}f(z)=0,$

and

$\|f_{N}-f\|_{H^{p}}<\varepsilon.$

The fractional linear mapping

$z_{j}=\alpha(w_{j})=i\frac{1-w_{j}}{1+w_{j}},\ \ j=1,2,...,n,$

is a conformal mapping from the $n$-tuple unit disc

$\mathbb{D}=\mathbb{D}_{1}\times\mathbb{D}_{2}\times\cdot\cdot\cdot\mathbb{D}_{n}=\left\\{(w_{1},w_{2},...,w_{n})\in\mathbb{C}^{n}:|w_{j}|<1,\
j=1,2,...,n\right\\}$

to the first octant of $\mathbb{C}^{n}$,
$T_{\Gamma}=\\{(z_{1},z_{2},...,z_{n})\in\mathbb{C}^{n}:{\rm Im}z_{j}>0,\
j=1,2,...,n\\}$.
Its inverse mapping is

$w_{j}=\beta(z_{j})=\frac{i-z_{j}}{i+z_{j}},\ \ j=1,2,...,n.$

Let

$h_{N}(w)=f_{N}(\alpha(w_{1}),...,\alpha(w_{n}))$

and $h_{N}(w_{1},...,w_{j},-1,w_{j+1},...,w_{n})=0,\ \ j=1,2,...,n.$
Then $h_{N}(w)$ is continues in the closed disc $\overline{\mathbb{D}}$, and

$h_{N}(w)\left|i\frac{1-w_{1}}{1+w_{1}}\right|^{N+1}\cdot\cdot\cdot\left|i\frac{1-w_{n}}{1+w_{n}}\right|^{N+1}\rightarrow
0,$

as $w_{j}\rightarrow w_{0}\in F,\ |w-w_{0}|\rightarrow 0,$ for $\ j=1,2,...,n$
and $w\in\overline{\mathbb{D}}\backslash F$, where
$F=\bigcup\limits_{j=1}^{n}F_{j},\ \
F_{j}=\\{(w_{1},...,w_{j},-1,w_{j+1},...,w_{n}):|w_{j}|\leq 1,\
j=1,2,...,n\\}.$

Therefore, for $w_{0}\in F$,

$\frac{h_{N}(w)}{\prod\limits_{j=1}^{n}(1+w_{j})^{N+1}}\rightarrow 0,$

as $\ w_{j}\rightarrow w_{0},\ |w_{j}|\leq 1,\ j=1,2,...,n.$
Let

$\displaystyle\widetilde{h}_{N}(w)=\begin{cases}\frac{h_{N}(w)}{\prod\limits_{j=1}^{n}(1+w_{j})^{N+1}}&w\in\overline{\mathbb{D}}\backslash
F\\\ 0&w\in F\end{cases}$ (26)

Then $\widetilde{h}_{N}(w)$ is holomorphic in the $n$-tuple $\mathbb{D}$ and
continues in the closed set $\overline{\mathbb{D}}$.
By the Stone-Weierstrass Theorem, there exists a polynomial
$P_{N}(w)=P_{N}(w_{1},...,w_{n})$ such that

$\left|\widetilde{h}_{N}(w)-P_{N}(w+1)\right|<\varepsilon,\ |w_{j}|\leq 1,\
w_{j}\neq-1,\ j=1,2,...,n,$

that is

$\left|h_{N}(w)/\prod\limits_{j=1}^{n}(1+w_{j})^{N+1}-P_{N}(w+1)\right|<\varepsilon,\
|w_{j}|\leq 1,\ w_{j}\neq-1,\ j=1,2,...,n.$

Thus,

$\left|f_{N}(\alpha(w_{1}),...,\alpha(w_{n}))-\prod\limits_{j=1}^{n}(1+w_{j})^{N+1}P_{N}(w+1)\right|\leq\varepsilon\prod\limits_{j=1}^{n}|1+w_{j}|^{N+1},$

where $|w_{j}|\leq 1,\ w_{j}\neq 1,\ j=1,2,...,n.$
Since $z_{j}=\alpha(w_{j})=i\frac{1-w_{j}}{1+w_{j}}$ and
$w_{j}=\beta(z_{j})=\frac{i-z_{j}}{i+z_{j}}$ for all $j=1,2,...,n$, the last
inequality becomes

$\left|f_{N}(z)-\prod\limits_{j=1}^{n}\left(\frac{2i}{i+z_{j}}\right)^{N+1}P_{N}\left(\frac{2i}{i+z_{n}},...,\frac{2i}{i+z_{n}}\right)\right|\
\leq\varepsilon\prod\limits_{j=1}^{n}\left|\frac{2i}{i+z_{j}}\right|^{N+1},$

for $z\in T_{\Gamma},\ {\rm Im}z_{j}>0,\ j=1,2,...,n.$ Therefore, we can
obtain that

$\int_{\mathbb{R}^{n}}|f_{N}(x+iy)-R(x+iy)|^{p}\
dx<\varepsilon^{p}2^{(N+1)p}\prod\limits_{j=1}^{n}\int_{-\infty}^{+\infty}\left|(1+x^{2}_{j})\right|^{-(N+1)p}dx_{j}<\infty,$

where $z=(z_{1},z_{2},...,z_{n})\in T_{\Gamma}$ and

$R(x+iy)=\prod\limits_{j=1}^{n}\left(\frac{2i}{i+z_{j}}\right)^{N+1}P_{N}\left(\frac{2i}{i+z_{n}},...,\frac{2i}{i+z_{n}}\right)\in\mathfrak{R}_{N}(i,i,...,i).$

This concludes that the class $\mathfrak{R}_{N}(i,i,...,i)$ is dense in
$H^{p}(T_{\Gamma}).$ Therefore, the proof of Theorem 4.2 is complete.

The following result shows that the Hardy space decomposition of
$L^{p}(\mathbb{R}^{n})$ for $0<p<1$ is not unique and the intersection space
$\bigcap\limits_{j=1}^{2^{n}}H_{\sigma_{j}}^{p}(\mathbb{R}^{n})$ is a non-
empty set.

###### Theorem 4.3

Let $0<p<1$ and let $X^{p}$ denote the $L^{p}$ closure of the set of
$L^{p}(\mathbb{R}^{n})$-functions of the form

$f(x_{1},x_{2},...,x_{n})=\frac{P(x_{1},...,x_{n})}{\prod\limits_{k=1}^{n}\prod\limits_{j=1}^{m}(x_{k}-a_{kj})},$

where
$P(x_{1},...,x_{n})=\sum\limits_{s_{1}=0}^{l_{1}}\cdot\cdot\cdot\sum\limits_{s_{n}=0}^{l_{n}}a_{s_{1}...s_{n}}x_{1}^{s_{1}}\cdot\cdot\cdot
x_{n}^{s_{n}},\ l=\max\\{l_{1},...,l_{n}\\},$ $(m-l)p>1,$ and $a_{kj}\neq
a_{km}\in\mathbb{R}$ for $j\neq m$ and $k=1,2,...,n.$
Then

$X^{p}=\bigcap\limits_{j=1}^{2^{n}}H_{\sigma_{j}}^{p}(\mathbb{R}^{n})\neq\emptyset.$

Proof Firstly, we will show that
$X^{p}\subseteq\bigcap\limits_{j=1}^{2^{n}}H_{\sigma_{j}}^{p}(\mathbb{R}^{n}).$
Let

$f(z_{1},z_{2},...,z_{n})=\frac{\sum\limits_{s_{1}=0}^{l_{1}}\cdot\cdot\cdot\sum\limits_{s_{n}=0}^{l_{n}}a_{s_{1}...s_{n}}z_{1}^{s_{1}}\cdot\cdot\cdot
z_{n}^{s_{n}}}{\prod\limits_{k=1}^{n}\prod\limits_{j=1}^{m}(z_{k}-a_{kj})},\ \
$

$a_{k}=(a_{k1},...,a_{km})\in\mathbb{R}^{n},\ k=1,2,...,n.$
Then $f(z)$ is a rational function with singular part being contained in
$\bigcup\limits_{l=1}^{n}G_{l}$, where $G_{l}=\\{(z_{1},...,z_{n}):\ {\rm
Im}z_{l}=0,\ z_{k}\in\mathbb{C},\ k\neq l\\}$. It is clear that the set
$G_{j}$ is not in the octant $T_{\Gamma_{\sigma_{j}}}$ and
$\bigcup\limits_{l=1}^{n}G_{l}\subseteq\bigcap\limits_{j=1}^{2^{n}}\partial
T_{\Gamma_{\sigma_{j}}}$. So $f(z)$ is holomorphic in
$T_{\Gamma_{\sigma_{j}}}$, for all $j=1,2,...,2^{n}$.

Moreover,

$\begin{split}\int_{\mathbb{R}^{n}}|f(x+iy)|^{p}\
dx&=\int_{\mathbb{R}^{n}}\frac{\left|\sum\limits_{s_{1}=0}^{l_{1}}\cdot\cdot\cdot\sum\limits_{s_{n}=0}^{l_{n}}a_{s_{1}...s_{n}}z_{1}^{s_{1}}\cdot\cdot\cdot
z_{n}^{s_{n}}\right|^{p}}{\left|\prod\limits_{k=1}^{n}\prod\limits_{j=1}^{m}(z_{k}-a_{kj})\right|^{p}}\
dx\\\
&\leq\sum\limits_{s}|a_{s_{1}...s_{n}}|^{p}\prod\limits_{k=1}^{n}\int_{-\infty}^{+\infty}\frac{\left|z_{k}^{s_{k}}\right|^{p}}{\left|\prod\limits_{j=1}^{m}(z_{k}-a_{kj})\right|^{p}}\
dx_{k}.\end{split}$ (27)

Observe that, the function

$R_{k}(z_{k})=\frac{z_{k}^{s_{k}}}{\prod\limits_{j=1}^{m}(z_{k}-a_{kj})},$

is holomorphic in the upper-half and the lower-half complex planes. Moreover,
we can prove that $R_{k}(x_{k})\in L^{p}(\mathbb{R})$. In fact,

$\begin{split}\int_{-\infty}^{+\infty}|R_{k}(x_{k})|^{p}\
dx_{k}&=\int_{-\infty}^{+\infty}\frac{\left|x_{k}^{s_{k}}\right|^{p}}{\left|\prod\limits_{j=1}^{m}(x_{k}-a_{kj})\right|^{p}}\
dx_{k}\\\ &=\int_{|x_{k}|\leq
M}\frac{\left|x_{k}^{s_{k}}\right|^{p}}{\left|\prod\limits_{j=1}^{m}(x_{k}-a_{kj})\right|^{p}}\
dx_{k}+\int_{|x_{k}|>M}\frac{\left|x_{k}^{s_{k}}\right|^{p}}{\left|\prod\limits_{j=1}^{m}(x_{k}-a_{kj})\right|^{p}}\
dx_{k}.\end{split}$

where $M$ is sufficiently large so that the interval $(-M,\ M)$ contains all
the poles of $R_{k}(x_{k})$. Then, for the first integral in the right hand of
the last inequality,

$\int_{|x_{k}|\leq
M}\frac{\left|x_{k}^{s_{k}}\right|^{p}}{\left|\prod\limits_{j=1}^{m}(x_{k}-a_{kj})\right|^{p}}\
dx_{k}\leq M^{s_{k}p}\int_{|x_{k}|\leq
M}\frac{dx_{k}}{\prod\limits_{j=1}^{m}|x_{k}-a_{kj}|^{p}},$

which is finite since $0<p<1.$
For the other integral, we have

$\begin{split}\int_{|x_{k}|>M}\frac{\left|x_{k}^{s_{k}}\right|^{p}}{\left|\prod\limits_{j=1}^{m}(x_{k}-a_{kj})\right|^{p}}\
dx_{k}&=\int_{|x_{k}|>M}\frac{\left|x_{k}^{s_{k}}\right|^{p}}{\left|x^{m}_{k}\prod\limits_{j=1}^{m}(1-a_{kj}/x_{k})\right|^{p}}\
dx_{k}\\\
&\leq\int_{|x_{k}|>M}\frac{dx_{k}}{\prod\limits_{j=1}^{m}|1-a_{kj}/x_{k}|^{p}|x|^{(m-s_{k})p}_{k}}\\\
&\leq\prod\limits_{j=1}^{m}\frac{\left|M-a_{kj}\right|^{p}}{\left|M\right|^{p}}\int_{|x_{k}|>M}\frac{dx_{k}}{|x|^{(m-s_{k})p}_{k}},\end{split}$

which is also finite since $(m-s_{k})p>1.$ Therefore, $R_{k}(x_{k})\in
L^{p}(\mathbb{R}).$
By Lemma 3.1, together with $R_{k}(x_{k})\in L^{p}(\mathbb{R})$ and the
analyticity of $R_{k}(x_{k})$, we get that $R_{k}(z_{k})\in
H^{p}(\mathbb{C}^{\pm}).$ Thus, $\|R\|_{H^{p}}\leq\|R\|_{L^{p}};$ and the
relation (27) becomes

$\int_{\mathbb{R}^{n}}|f(x+iy)|^{p}\
dx\leq\sum\limits_{s}|a_{s_{1}...s_{n}}|^{p}\|R\|^{np}_{L^{p}}<\infty.$

Hence, we have

$f(z)\in\bigcap\limits_{j=1}^{2^{n}}H^{p}(T_{\Gamma_{\sigma_{j}}}),\ \
j=1,2,...,2^{n}.$

This shows that

$f(x)\in\bigcap\limits_{j=1}^{2^{n}}H_{\sigma_{j}}^{p}(\mathbb{R}^{n}).$

Hence
$X^{p}\subseteq\bigcap\limits_{j=1}^{2^{n}}H_{\sigma_{j}}^{p}(\mathbb{R}^{n})$
as desired.

Next, we will show
$X^{p}\supseteq\bigcap\limits_{j=1}^{2^{n}}H_{\sigma_{j}}^{p}(\mathbb{R}^{n}).$

Let there exist $f_{\sigma_{j}}(z)\in H^{p}(T_{\Gamma_{\sigma_{j}}}),\
j=1,2,...,2^{n},$ such that

$f(x)=f_{\sigma_{j}}(x)=f_{\sigma_{l}}(x),$

for all $1\leq j,\ l\leq 2^{n},$ a.e. $x\in\mathbb{R}^{n}.$
By Theorem 4.2 and Corollary 4.1, for any $\varepsilon>1$, there exist

$R_{\sigma_{j}}\in\mathfrak{R}_{N}(\sigma_{j}i)$

such that

$\|f_{\sigma_{j}}-R_{\sigma_{j}}\|^{p}_{H^{p}_{\sigma_{j}}}=\|f-R_{\sigma_{j}}\|^{p}_{p}<\frac{\varepsilon}{4},$

for $j=1,2,...,2^{n}.$
The fact $R_{\sigma_{j}}\in\mathfrak{R}_{N}(\sigma_{j}i)$ implies that there
exist polynomials $P_{\sigma_{j}},\ j=1,2,...,2^{n},$ such that for
$\beta(z_{k})=\frac{i-z_{k}}{i+z_{k}},\ \ k=1,2,...,n,$

$R_{\sigma_{j}}(z)=P_{\sigma_{j}}((\beta(z_{1}))^{\sigma_{j}(1)}+1,...,(\beta(z_{n}))^{\sigma_{j}(n)}+1)\prod\limits_{k=1}^{n}\left(\beta^{\sigma_{j}(k)}(z_{k})+1\right)^{N+1},$

where $(N+1)p>1,\ N+1>\max\\{{\rm deg}P_{\sigma_{j}}:\ j=1,2,...,2^{n}\\}.$

It is easy to see that the singular part of $R_{\sigma_{j}}(z)$ are contained
in $\bigcup\limits_{k=1}^{n}H_{k}$ which is not in $T_{\Gamma_{\sigma_{j}}}$,
where $H_{k}=\\{(z_{1},...,z_{n}):\ z_{k}=-\sigma_{j}(k)i,\
z_{l}\in\mathbb{C},\ k\neq l\\}$. So $R_{\sigma_{j}}(z)$ is a holomorphic
rational function in $T_{\Gamma_{\sigma_{j}}},\ j=1,2,...,2^{.}$
Let

$R(z,\varphi)=R_{\sigma_{1}}(z)+\frac{\sum\limits_{j=1}^{2^{n}}\prod\limits_{k=1}^{n}(-e^{i\varphi_{k}})^{\frac{1+\sigma_{j}(k)}{2}}\prod\limits_{k=1}^{n}(\beta(z_{k}))^{\frac{1-\sigma_{j}(k)}{2}m}(R_{\sigma_{j}}(z)-R_{\sigma_{1}}(z))}{\prod\limits_{k=1}^{n}((\beta(z_{k}))^{m}-e^{i\varphi_{k}})},$

where $(m-\max\\{{\rm deg}P_{\sigma_{j}}:\ j=1,2,...,2^{n}\\}-N-1)p>1$, and
$\varphi=(\varphi_{1},...,\varphi_{n})\in\mathbb{R}^{n}$. We thus are aware
that the singular part of the rational function $R(z,\varphi)$ is contained in
$\bigcap\limits_{j=1}^{2^{n}}\partial T_{\Gamma_{\sigma_{j}}}=\mathbb{R}^{n}$.
So $R(z,\varphi)$ is holomorphic in
$\bigcup\limits_{j=1}^{2^{n}}T_{\Gamma_{\sigma_{j}}}$.
Note that $\beta(x_{k})=e^{i\theta(x_{k})}$, where
$\theta(x_{k})=\arg(i_{k}-x_{k})-\arg(i_{k}+x_{k})\in[-\pi,\pi]$ for
$x_{k}\in\mathbb{R}$. Moreover, set

$J=\int_{\mathbb{R}^{n}}\int_{[-\pi,\pi]^{n}}|R(x,\varphi)-R_{\sigma_{1}}(x)|^{p}\
d\varphi dx.$

Then

$\begin{split}J&=\int_{\mathbb{R}^{n}}\int_{[-\pi,\pi]^{n}}\frac{\left|\sum\limits_{j=1}^{2^{n}}\prod\limits_{k=1}^{n}(-e^{i\varphi_{k}})^{\frac{1+\sigma_{j}(k)}{2}}\prod\limits_{k=1}^{n}(\beta(z_{k}))^{\frac{1-\sigma_{j}(k)}{2}m}(R_{\sigma_{j}}(x)-R_{\sigma_{1}}(x))\right|^{p}}{\left|\prod\limits_{k=1}^{n}((\beta(z_{k}))^{m}-e^{i\varphi_{k}})\right|^{p}}\
d\varphi dx\\\
&\leq\sum\limits_{j=1}^{2^{n}}\int_{\mathbb{R}^{n}}\int_{[-\pi,\pi]^{n}}|R_{\sigma_{j}}(x)-R_{\sigma_{1}}(x)|^{p}\prod\limits_{k=1}^{n}\left|\frac{(\beta(z_{k}))^{\frac{1-\sigma_{j}(k)}{2}m}}{(\beta(z_{k}))^{m}-e^{i\varphi_{k}}}\right|^{p}\
d\varphi dx\\\
&=\sum\limits_{j=1}^{2^{n}}\int_{\mathbb{R}^{n}}\int_{[-\pi,\pi]^{n}}\frac{|R_{\sigma_{j}}(x)-R_{\sigma_{1}}(x)|^{p}d\varphi
dx}{\prod\limits_{\sigma_{j}(k)=1}|e^{im\theta(x_{k})}-e^{i\varphi_{k}}|^{p}\prod\limits_{\sigma_{j}(k)=-1}|1-e^{i\varphi_{k}-im\theta(x_{k})}|^{p}}.\end{split}$

Observe that

$\begin{split}&\int_{-\pi}^{\pi}\frac{1}{|1-e^{i\varphi_{k}-im\theta(w_{k})}|^{p}}\
d\varphi_{k}\\\ &=\int_{-\pi}^{\pi}\frac{1}{|1-e^{i\theta}|^{p}}\ d\theta\\\
&=\int_{-\pi}^{\pi}\frac{d\theta}{2^{p}|\sin^{p}(\frac{\theta}{2})|}\leq\frac{4}{2^{p}}\int_{0}^{\frac{\pi}{2}}\frac{d\theta}{(\frac{2\theta}{\pi})^{p}}\leq\frac{2^{1-p}\pi}{1-p},\end{split}$
(28)

and similarly,

$\begin{split}&\int_{-\pi}^{\pi}\frac{1}{|e^{im\theta(x_{k})}-e^{i\varphi_{k}}|^{p}}\
d\varphi_{k}\\\
&=\int_{-\pi}^{\pi}\frac{d\theta}{2^{p}|\sin^{p}(\frac{\theta}{2})|}\leq\frac{4}{2^{p}}\int_{0}^{\frac{\pi}{2}}\frac{d\theta}{(\frac{2\theta}{\pi})^{p}}\leq\frac{2^{1-p}\pi}{1-p}.\end{split}$
(29)

From the above relations, we obtain

$J\leq\left(\frac{2^{1-p}\pi}{1-p}\right)^{n}\int_{\mathbb{R}^{n}}|R_{\sigma_{j}}(x)-R_{\sigma_{1}}(x)|^{p}\
dx.$

Therefore, there is a real vector $\varphi\in\mathbb{R}^{n}$ such that

$J\leq\left(\frac{2^{1-p}\pi}{1-p}\right)^{n}(\|R_{\sigma_{j}}-f\|^{p}_{p}+\|f-R_{\sigma_{1}}\|^{p}_{p})\leq\left(\frac{2\pi}{1-p}\right)^{n}\varepsilon.$

Thus, we have

$\begin{split}&\int_{\mathbb{R}^{n}}|R(x,\varphi)-f(x)|^{p}\ dx\\\
&\leq\int_{\mathbb{R}^{n}}|R(x,\varphi)-R_{\sigma_{j}}(x)|^{p}\
dx+\int_{\mathbb{R}^{n}}|R_{\sigma_{j}}(x)-f(x)|^{p}\ dx\\\
&\leq\left(\frac{2\pi}{1-p}\right)^{n}\varepsilon+\frac{\varepsilon}{4}.\end{split}$
(30)

Hence, $R(x,\varphi)\in L^{p}(\mathbb{R}^{n})$. This, together with the
definition of $R(x,\varphi)$, implies that $R(x,\varphi)\in X^{p}.$ Therefore,
$f(x)\in X^{p}$. The proof is complete.

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