# Paley type inequality of the Fourier transform on the Heisenberg group

Atef Rahmouni Department of mathematics, King Saudi University, College of
Sciences
P. O Box 2455 Riyadh 11451, Saudi Arabia<EMAIL_ADDRESS>
###### Abstract.

A paley type inequality for the Fourier transform on $H^{p}(\mathbb{H}^{n}),$
the Hardy space on the Heisenberg group, is obtained for $0<p\leq 1.$

###### Key words and phrases:

Hardy-Littlewood inequality; Heisenberg group.

##  1\.  Introduction

The study of Hardy spaces has been originated during the 1910’s in the setting
of Fourier series and complex analysis in one variable. In 1972, Fefferman and
Stein [5] introduced Hardy spaces $H^{p}$ by mean of maximal function

$f^{\ast}(x)=\sup_{r>0}|f\ast\phi_{r}(x)|$

where $\phi$ belongs to $\mathcal{S}$, the Schwartz space of rapidly
decreasing smooth functions satisfying $\int\phi(x)dx=1$. The delation
$\phi_{r}$ is given by $\phi_{r}(x)=r^{-n}\phi(x/r).$ We say that a tempered
distributions $f\in\mathcal{S}^{\prime}$ is in $H^{p}$ if $f^{\ast}$ is in
$L^{p}$.
Using the maximal function above, Coifman [4] showed that any $f$ in $H^{p}$
can be represented as a linear combination of atoms, that is

$f=\sum_{k=1}^{\infty}\beta_{k}a_{k},\quad\beta_{k}\in\mathbb{C},$

where the $a_{k}$ are atoms and the sum converges in $H^{p}$. Moreover,

$\|f\|_{H^{p}}\thickapprox\inf\Big{\\{}\sum_{k=1}^{\infty}|\beta_{k}|^{p}:\sum_{k=1}^{\infty}\beta_{k}a_{k}\mbox{
is a decomposition of $f$ into\, }atoms\Big{\\}}.$

It has been shown that the study of some analytic problems on
$H^{p}(\mathbb{R}^{n})$ is summed up to investigate some properties of these
atoms, and therefore the problems become quite simple. In 1980, Taibleson and
Weiss [17] gave the definition of molecules belonging to $H^{p},$ and showed
that every molecule is in $H^{p}$ with continuous embedding map. By the atomic
decomposition and the molecule characterization, the proof of $H^{p}$
boundedness of the operators on Hardy space becomes easier. The theory of
$H^{p}$ have been extensively studied in [7] and [6].

In the setting of the euclidian case, Hardy’s inequality for Fourier transform
asserts that for all $f\in H^{p}(\mathbb{R}^{n})$ $0<p\leq 1.$

$\int_{\mathbb{R}^{n}}{|\widehat{f}(\xi)|^{p}\over|\xi|^{n(2-p)}}d\xi\leq\|f\|^{p}_{H^{p}{(\mathbb{R}^{n}})},\qquad
0<p\leq 1$ (1.1)

where $H^{p}(\mathbb{R}^{n})$ indicates the real Hardy space. Hardy’s type
inequality for Fourier transform has been extensively studied in [16]. Kanjin
[13] proved Hardy’s inequalities for Hermit and Laguerre expansions for
functions in $H^{1}$ and for Hankel transform [12]. In connection with
properties of regularity of the spherical means on $\mathbb{C}^{n}$,
Thangavelu [18] proved a Hardy’s inequality for special Hermit functions.
These standard inequalities for higher dimensional has been studied in [14].
Recently, an extension has been given by [1], the latter establish a Hardy’s
type inequality associated with the Hankel transform for over critical
exponent $\sigma>\sigma_{0}=2-p.$ We point out here that the result obtained
for Hardy’s inequality for the Hankel transform improves the work of Kanjin
[12] in which he proved the result for $\sigma_{0}=2-p.$ Although, in [2, 3,
15] extended this form of this inequality to Laguerre hypergroup and its dual.

In this paper we are interested in the Heisenberg group $\mathbb{H}^{n}$ is
the lie group with underlying manifold
$\mathbb{H}^{n}=\mathbb{C}^{n}\times\mathbb{R}$ and multiplication
$(z,t).(z^{\prime},t^{\prime})=(z+z^{\prime},t+t^{\prime}+2Im(z.\overline{z^{\prime}}),$
where $\linebreak z=(z_{1},z_{2},...,z_{n})\in\mathbb{C}^{n}.$ If we identify
$\mathbb{C}^{n}\times\mathbb{R}$ with $\mathbb{R}^{2n+1}$ by
$z_{j}=x_{j}+ix_{j+n},~{}j=1,...,n,$ then the group law can be rewritten as

$(x_{1},x_{2},...,x_{2n},t).(y_{1},y_{2},...,y_{2n},t^{\prime})=(x_{1}+y_{2},...,x_{n}+y_{n},t+t^{\prime}-2\sum_{j=1}^{n}(x_{j}y_{j+n}-y_{j}x_{j+n})).$

The reverse element of $u=(z,t)$ is $u^{-1}=(-z,-t)$ and we write the identity
of $\mathbb{H}^{n}$ as $0=(0,0).$

Set $X_{j},X_{j+n}$ and $T$ is a basis for the left invariant vector fields on
$\mathbb{H}^{n}.$ The corresponding complex vector fields are

$Z_{j}=\frac{1}{2}(X_{j}-iX_{j+n})=\frac{\partial}{\partial
z_{j}}+i\overline{z}_{j}\frac{\partial}{\partial
t},~{}\overline{Z}_{j}=\frac{1}{2}(X_{j}+iX_{j+n})=\frac{\partial}{\partial\overline{z}_{j}}-iz_{j}\frac{\partial}{\partial
t},~{}j=1,...,n.$

The Heisenberg group is a connected, simply connected nilpotent Lie group. We
define one-parameter dilations on $\mathbb{H}^{n},$ for $R>0,$ by
$\rho_{R}(z,t)=(Rz,R^{2}t).$ These dilations are group automorphisms and the
Jacobian determinant is $R^{Q},$ where $Q=2n+2$ is the homogeneous dimension
of $\mathbb{H}^{n}.$ We will denoted by
$f_{\rho}(z,t)=\rho^{-Q}f((z,t)_{\rho})$ the dilated of the function $f$
defined on $\mathbb{H}^{n}.$

A homogeneous norm on $\mathbb{H}^{n}$ is given by

$|(z,t)|_{\mathbb{H}^{n}}=(|z|^{4}+4t^{2})^{1/4},$

With this norm, we define the Heisenberg ball centered at $u=(z,t)$ of radius
$r,$ i.e., the set

$B(u,r)=\\{v\in\mathbb{H}^{n}:~{}|uv^{-1}|_{\mathbb{H}^{n}}<R\\},$

and we denote by
$B_{R}=B(0,R)=\\{v\in\mathbb{H}^{n}:~{}|v|_{\mathbb{H}^{n}}<R\\}$ the open
ball centered at 0, the identity element of $\mathbb{H}^{n},$ with radius $R.$
The volume of the ball $B(u,R)$ is $C_{Q}R^{Q},$ where $C_{Q}$ is the volume
of the unit ball $B_{1}.$

The Haar measure $dV$ on $\mathbb{H}^{n}$ coincides with the Lebesgue measure
on $\mathbb{C}^{n}\times\mathbb{R}$ which is denoted by $dzd\overline{z}dt.$

Let
$J=(j^{1},j^{2},j^{0})\in\mathbb{Z}^{n}_{+}\times\mathbb{Z}^{n}_{+}\times\mathbb{Z}_{+},$
where $\mathbb{Z}_{+}$ the set of all nonnegative integers, we set
$h(J)=|j^{1}|+|j^{2}|+2j^{0},$ where, if $j^{1}=(j^{1}_{1},...,j^{n}_{n}),$
then $|j^{1}|=\sum_{k=1}^{n}j^{1}_{k}.$ If $P(z,t)=\sum_{J}a_{J}(z,t)^{J}$ is
a polynomial where $(z,t)^{J}=z^{j^{1}}\overline{z}^{j^{2}}t^{j^{0}},$ then we
call $\max\\{h(J):a_{J}\neq 0\\}$ the homogeneous degree of $P(z,t).$ The set
of all polynomials whose homogeneous degree $\leq s$ is denoted by
$\mathcal{P}_{s}.$ Schwartz space on $\mathbb{H}^{n}$ write as
$\mathcal{S}(\mathbb{H}^{n}).$

Fix $\lambda>0,$ let $\mathcal{H}_{\lambda}$ be the Bargmann’s space :

$\mathcal{H}_{\lambda}=\Big{\\{}F~{}\mbox{holomorphic
on}~{}\mathbb{C}^{n}:\|F\|^{2}=\big{(}\frac{2\lambda}{\pi}\big{)}^{n}\int_{\mathbb{C}^{n}}|F(\zeta)|^{2}e^{-2\lambda|\zeta|^{2}}d\zeta<\infty\Big{\\}}.$

Then, $\mathcal{H}_{\lambda}$ is a Hilbert space and the monomials

$F_{\alpha,\lambda}(\zeta)=\sqrt{\frac{(2\lambda)^{|\alpha|}}{\alpha!}}\zeta^{\alpha},~{}~{}~{}\alpha=(\alpha_{1},\alpha_{2},...,\alpha_{n})\in\mathbb{Z}^{n}_{+}$

form an orthonormal basis for $\mathcal{H}_{\lambda},$ where
$\alpha!=\alpha_{1}!\alpha_{2}!...\alpha_{n}!,~{}|\alpha|=(\alpha_{1},\alpha_{2},...,\alpha_{n})$
and
$\linebreak\zeta^{\alpha}=\zeta^{\alpha_{1}}_{1}\zeta^{\alpha_{2}}_{2}...\zeta^{\alpha_{n}}_{n}.$
Suppose $W_{k,\lambda}$ and $W^{+}_{k,\lambda}$ are the closed operators on
$\mathcal{H}_{\lambda}$ such that

$\displaystyle W_{k,\lambda}F_{\alpha,\lambda}$ $\displaystyle=$
$\displaystyle(2(\alpha_{k}+1)\lambda)^{1/2}F_{\alpha+e_{k},\lambda},$
$\displaystyle W_{k,\lambda}^{+}F_{\alpha,\lambda}$ $\displaystyle=$
$\displaystyle(2\alpha_{k}\lambda)^{1/2}F_{\alpha-
e_{k},\lambda},\qquad\mbox{for}~{}~{}\lambda>0,$

and

$\displaystyle W_{k,\lambda}$ $\displaystyle=$ $\displaystyle
W_{k,-\lambda}^{+},$ $\displaystyle W_{k,\lambda}^{+}$ $\displaystyle=$
$\displaystyle W_{k,-\lambda},\qquad\mbox{for}~{}~{}\lambda<0,$

where $e_{k}=(0,...,1,...,0)\in\mathbb{Z}^{n}$ with the 1 in the $k$-th
position. Then

$\prod_{\lambda}(z,t)=exp^{i\lambda
t}exp^{(-z.W_{\lambda}+\overline{z}.W_{\lambda}^{+})}$

is an irreducible unitary representation of $\mathbb{H}^{n}$ on
$\mathcal{H}_{\lambda},$ where
$z.W_{\lambda}=\sum_{k=1}^{n}z_{k}.W_{k,\lambda}.$

The group Fourier transform of $f\in L^{1}(\mathbb{H}^{n})\cap
L^{2}(\mathbb{H}^{n})$ is an operator-valued function defined by

$\mathcal{F}(f)(\lambda)=\int_{\mathbb{H}^{n}}f(z,t)\prod_{\lambda}(z,t)dV.$
(1.2)

Obviously, $\|\mathcal{F}(f)(\lambda)\|\leq\|f\|_{L^{1}}.$ Here, $\|-\|$
denotes the operator norm. Similar as in $\mathbb{R}^{n},$ for $f\in
L^{1}(\mathbb{H}^{n})\cap L^{2}(\mathbb{H}^{n}),$ we have the following
Plancherel and inversion formulas :

$\|f\|_{2}^{2}=\frac{2^{n-1}}{\pi^{n+1}}\int_{\mathbb{R}}\|\mathcal{F}(f)(\lambda)\|^{2}_{HS}|\lambda|^{n}d\lambda,\qquad
f\in L^{1}(\mathbb{H}^{n})\cap L^{2}(\mathbb{H}^{n}),$ (1.3)
$\int_{\mathbb{R}}tr\Big{(}\prod^{*}_{\lambda}(z,t)\mathcal{F}(f)(\lambda)\Big{)}|\lambda|^{n}d\lambda=\frac{(2\pi)^{n+1}}{4^{n}}f(u)$
(1.4)

where $tr$ is the canonical semifinite trace and $\|-\|_{HS}$ denotes the
Hilbert-Schmidt norm.
For
$(\lambda,m,\alpha)\in\mathbb{R}^{*}\times\mathbb{Z}^{n}\times\mathbb{Z}^{n}_{+},$
where $\mathbb{R}^{*}=\mathbb{R}\backslash\\{0\\},$ we use the notations

| $m_{i}^{+}=\max\\{m_{i},0\\},\qquad$ | $m_{i}^{-}=-\min\\{m_{i},0\\},$
---|---|---
| $m^{+}=(m_{1}^{+},m_{2}^{+},...,m_{n}^{+})\qquad$ | $m^{-}=(m_{1}^{-},m_{2}^{-},...,m_{n}^{-}).$

The partial isometry operator $W^{m}_{\alpha}(\lambda)$ on
$\mathcal{H}_{|\lambda|}$ by

| $W_{k,\alpha}(\lambda)F_{\beta,\lambda}=(-1)^{|m^{+}|}\delta_{\alpha+m^{+},\beta}F_{\alpha+m^{-},\lambda},$ | $\qquad\mbox{for}~{}~{}\lambda>0;$
---|---|---
| $W_{\alpha}^{m}(\lambda)=[W_{\alpha}^{m}(-\lambda)]^{*},$ | $\qquad\mbox{for}~{}~{}\lambda<0.$

Thus
$\\{W^{m}_{\alpha}(\lambda):m\in\mathbb{Z}^{n},\alpha\in\mathbb{Z}^{n}\\}$ is
an orthonormal basis for the Hilbert-Schmidt operators on
$\mathcal{H}_{|\lambda|}.$ Given a function $f\in L^{2}(\mathbb{H}^{n})$ such
that

$f(z,t)=\sum_{m,\alpha}f_{m}(r_{1},...,r_{n},t)e^{i(m_{1}\theta_{1}+...+m_{n}\theta_{n})},\qquad\mbox{where}\qquad
z_{j}=r_{j}e^{i\theta_{j}},$

then

$\mathcal{F}(f)(\lambda)=\sum_{m,\alpha}R_{f}(\lambda,m,\alpha)W^{m}_{\alpha}(\lambda),$

where

$R_{f}(\lambda,m,\alpha)=\int_{\mathbb{H}^{n}}f_{m}(r_{1},...,r_{n},t)e^{i\lambda
t}\ell_{\alpha_{1}}^{|m_{1}|}(2|\lambda|r_{1}^{2})...\ell_{\alpha_{n}}^{|m_{n}|}(2|\lambda|r_{n}^{2})dV,$

and $\ell_{\alpha}^{|m|}$ is the Larguerre function of type $|m|$ and degree
$|\alpha|.$

Let $P$ be a polynomial in $z_{j},\overline{z}_{j},t$ on $\mathbb{H}^{n},$ and
we define the difference-differential operator $\Delta_{P}$ acting on the
Fourier transform of $f\in L^{1}\cap L^{2}(\mathbb{H}^{n})$ by

$\Delta_{P}\Big{(}\sum_{m,\alpha}R_{f}(\lambda,m,\alpha)W_{\alpha}^{m}(\lambda)\Big{)}=\sum_{m,\alpha}R_{Pf}(\lambda,m,\alpha)W_{\alpha}^{m}(\lambda),$

namely, $\Delta_{P}\mathcal{F}(f)(\lambda)=\widehat{P(.)f(.)}(\lambda).$ In
[9] and [10], the authors gave the explicit expressions for
$\Delta_{t},\Delta_{z_{j}}$ and $\Delta_{\overline{z}_{j}}.$ For convenience,
we shall write $\Delta^{J}_{(z,t)}=\Delta^{J}.$

The paper is organized as follows. In the Second section we give an
appropriate definition of atoms and investigate the atoms characterization of
Hardy spaces $H^{p}(\mathbb{H}^{n})$ for $0<p\leq 1.$ In the last section we
state and prove our main result:

######  Theorem 1.1.

Let $0<p\leq 1,$ and $s\geq J=[Q({1/p}-1)],$ the greatest integer not
exceeding $\linebreak Q({1/p}-1).$ Then for any $f\in H^{p}(\mathbb{H}^{n})$
the Fourier transform of $f$ satisfies the following Hardy’s type inequality

$\int_{\mathbb{R}}\frac{\|\mathcal{F}(f)(\lambda)\|_{HS}^{p}}{\big{(}(2|\alpha|+n)|\lambda|\big{)}^{\sigma}}|\lambda|^{n}d\lambda\leq
C(p,n)\|f\|^{p}_{H^{p}(\mathbb{H}^{n})},$ (1.5)

provided that

$\frac{Q}{2}(2-p)\leq\sigma<\frac{Q}{2}+p(\frac{J+1}{2})$ (1.6)

where $C(p,n)$ depend only on $p$ and $n.$

Finally, we mention that $C$ will be always used to denote a suitable positive
constant that is not necessarily the same in each occurrence.

##  2\. Atomic decomposition for $H^{p}(\mathbb{H}^{n})$

Now we state the definition of atomic Hardy spaces in the setting of the
Heisenberg group $H^{p}(\mathbb{H}^{n})$, $0<p\leq 1$. To this end, we
introduce the following kind of atoms, which is closely related to the Haar
measure $dV.$

######  Definition 2.1.

Let $0<p\leq 1\leq q\leq\infty,p\neq q,s\in\mathbb{Z}$ and $s\geq
J=[Q(1/p-1)].$ (Such an ordered triple $(p,q,s)$ is called admissible). A
$(p,q,s)$-atom centered at $x_{0}\in\mathbb{H}^{n}$ is a function $a\in
L^{q}(\mathbb{H}^{n}),$ supported on a ball $B(x_{0},R)\subset\mathbb{H}^{n}$
with centre $x_{0}=(z_{0},t_{0})$ and satisfying the following

  * (i)

$\|a\|_{L^{q}(\mathbb{H}^{n})}\leq|B(0,r)|^{\frac{1}{q}-{1\over p}}$, a.e,

  * (ii)

$\displaystyle\int_{\mathbb{H}^{n}}a(x)P(x)dV(x)=0,$ for every
$P\in\mathcal{P}_{s}.$

Here, $(i)$ means that the size condition of atoms, and $(ii)$ is called the
cancelation moment condition.

A characterization of $H^{p}(\mathbb{H}^{n})$ is included in the following
statements.

######  Proposition 2.1.

Let $0<p\leq 1.$ If $\\{a_{k}\\}_{k=0}^{\infty}$ is a sequence of $p$-atoms,
and $\\{\lambda_{k}\\}_{k=0}^{\infty}$ is a sequence of complex numbers with

$\Big{(}\sum_{k=0}^{\infty}|\lambda_{k}|^{p}\Big{)}^{1/p}<\infty,$

then $\sum_{k=0}^{\infty}\lambda_{k}a_{k}$ converges in
$H^{p}(\mathbb{H}^{n})$ and

$\Big{\|}\sum_{k}\lambda_{k}a_{k}\Big{\|}_{H^{p}(\mathbb{H}^{n})}\leq
C(p,n)\Big{(}\sum_{k}|\lambda_{k}|^{p}\Big{)}^{1/p}.$

Conversely, if $f\in H^{p}(\mathbb{H}^{n})$ there exists a sequence
$\\{a_{k}\\}_{k=0}^{\infty}$ of $p$-atoms, and a sequence
$\\{\lambda_{k}\\}_{k=0}^{\infty}$ of complex numbers such that

$f=\sum_{k}\lambda_{k}a_{k}\qquad\mbox{and}\qquad\Big{(}\sum_{k}|\lambda_{k}|^{p}\Big{)}^{1/p}\leq
C(p,n)\|f\|_{H^{p}(\mathbb{H}^{n})},$

where $C(p,n)$ depends on $p$ and $n.$

##  3\. Proof of the main result

Now we are in a position to give the proof of the main result. First we stat
the following proposition which has its own interest.

######  Proposition 3.1.

For all $(z,t)\in\mathbb{H}^{n}$ the function $\prod_{\lambda}(z,t)$ satisfies

$\prod_{\lambda}(z,t)=\sum_{2k+\ell\leq
J}\omega_{k,\ell}(\lambda,n)~{}z^{k}t^{\ell}+R_{\theta}(z,t),\,\,0<\theta<1,$
(3.1)

where

$R_{\theta}(z,t)=\sum_{2k+\ell=J+1}\frac{(i\lambda
t)^{k}}{k!}.\frac{(z.W_{\lambda}-\overline{z}.W^{+}_{\lambda})^{\ell}}{\ell!}.$
(3.2)

Here $\omega_{k,\ell}(\lambda,n)$ are functions expressed by mean of
$\lambda,n.$

Set $\mathcal{H}_{|\lambda|}^{N}$ be the subspace of $\mathcal{H}_{|\lambda|}$
spanned by $\\{W^{0}_{\alpha}(\lambda):|\alpha|\leq N\\}.$ Remark that (see
[9, 11]) $z.W_{\lambda}-\overline{z}.W^{+}_{\lambda}$ is bounded from
$\mathcal{H}_{|\lambda|}^{N}$ to $\mathcal{H}_{|\lambda|}^{N+1}$ and whose
bound $<((2|\alpha|+n)|\lambda|)^{1/2}|z|.$ Then

$R_{\theta}(z,t)\leq
C\sum_{2k+\ell=J+1}\omega_{k,\ell}~{}\big{(}(2|\alpha|+n)|\lambda|\big{)}^{k+\frac{\ell}{2}}~{}z^{\ell}~{}t^{k}.$

Proof of Theorem 1.1. Let $f=\sum_{k=0}^{\infty}\beta_{k}a_{k}\in
H^{p}(\mathbb{H}^{n}),$ being element of $H^{p}(\mathbb{H}^{n})$ where $a_{k}$
are atoms. Since $0<p\leq 1$ it follows

$\int_{\mathbb{R}}\frac{\|\mathcal{F}(f)(\lambda)\|_{HS}^{p}}{\big{(}(2|\alpha|+n)|\lambda|\big{)}^{\sigma}}|\lambda|^{n}d\lambda\leq
C\sum_{k=0}^{\infty}|\beta_{k}|^{p}\int_{\mathbb{R}}\frac{\|\mathcal{F}(a_{k})(\lambda)\|_{HS}^{p}}{\big{(}(2|\alpha|+n)|\lambda|\big{)}^{\sigma}}|\lambda|^{n}d\lambda.$

In order to prove Theorem 1.1, it is enough to prove,

$\int_{\mathbb{R}}\frac{\|\mathcal{F}(a_{k})(\lambda)\|_{HS}^{p}}{\big{(}(2|\alpha|+n)|\lambda|\big{)}^{\sigma}}|\lambda|^{n}d\lambda\leq
C.$ (3.3)

This follows as $f=\sum_{k=0}^{\infty}\beta_{k}a_{k}$ implies
$\mathcal{F}(a_{k})(\lambda)^{p}\leq\Big{|}\sum_{k}\beta_{k}\mathcal{F}(a_{k})(\lambda)\Big{|}^{p}\leq\sum_{k=0}^{\infty}|\beta_{k}|^{p}|\mathcal{F}(a_{k})(\lambda)|^{p}$
and hence

$\displaystyle\int_{\mathbb{R}}\frac{\|\mathcal{F}(f)(\lambda)\|_{HS}^{p}}{\big{(}(2|\alpha|+n)|\lambda|\big{)}^{\sigma}}|\lambda|^{n}d\lambda$
$\displaystyle\leq$ $\displaystyle
C\sum_{k=0}^{\infty}|\beta_{k}|^{p}\int_{\mathbb{R}}\frac{\|\mathcal{F}(a_{k})(\lambda)\|_{HS}^{p}}{\big{(}(2|\alpha|+n)|\lambda|\big{)}^{\sigma}}|\lambda|^{n}d\lambda$
$\displaystyle\leq$ $\displaystyle
C\Big{\\{}\sum_{k=0}^{\infty}|\beta_{k}|^{p}\Big{\\}}^{1/p}$
$\displaystyle\leq$ $\displaystyle C\|f\|_{H^{p}(\mathbb{H}^{n})}.$

Let us now take $\gamma$ an arbitrary nonnegative real number, and decomposing
the left hand side of (3.3) as

$\displaystyle\int_{\mathbb{R}}\frac{\|\mathcal{F}(a_{k})(\lambda)\|_{HS}^{p}}{\big{(}(2|\alpha|+n)|\lambda|\big{)}^{\sigma}}|\lambda|^{n}d\lambda\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\int_{0<|\lambda|\leq\gamma}\frac{\|\mathcal{F}(a_{k})(\lambda)\|_{HS}^{p}}{\big{(}(2|\alpha|+n)|\lambda|\big{)}^{\sigma}}|\lambda|^{n}d\lambda$
$\displaystyle+$
$\displaystyle\int_{|\lambda|>\gamma}\frac{\|\mathcal{F}(a_{k})(\lambda)\|_{HS}^{p}}{\big{(}(2|\alpha|+n)|\lambda|\big{)}^{\sigma}}|\lambda|^{n}d\lambda$
$\displaystyle\,\,\,\,:=$ $\displaystyle S_{1}+S_{2}.$

To estimate $S_{1}$ we may use Proposition 3.1, and cancelation property of
atoms. Hence, by the cancelation property of atom,

$\mathcal{F}(a_{k})(\lambda)=\int_{\mathbb{H}^{n}}\Big{[}\sum_{2k+\ell\leq
J}\omega_{k,\ell}(\lambda,n)~{}z^{k}t^{\ell}+R_{\theta}(z,t)\Big{]}a(z,t)~{}dV(z,t).$

Now with the help of properties $(i),(ii)$ for $a(p,\infty,s)$-atoms of
$H^{p}(\mathbb{H}^{n})$ together with Proposition 3.1, we get

$\displaystyle\mathcal{F}(a_{k})(\lambda)$ $\displaystyle\leq$ $\displaystyle
C\sum_{2k+\ell=J+1}\omega_{k,\ell}~{}\big{(}(2|\alpha|+n)|\lambda|\big{)}^{k+\frac{\ell}{2}}\int_{B(o,R)}~{}z^{\ell}~{}t^{k}|B(0,R)|^{-\frac{1}{p}}~{}dV(z,t)$
$\displaystyle\leq$ $\displaystyle
C~{}\sum_{2k+\ell=J+1}\omega_{k,\ell}~{}R^{Q(1-\frac{1}{p})+2k+\frac{\ell}{2}}~{}\big{(}(2|\alpha|+n)|\lambda|\big{)}^{k+\frac{\ell}{2}}.$

Integrating with respect to the measure
$d\gamma_{n}(\lambda)=|\lambda|^{n}d\lambda$ over the domain
$0\leq|\lambda|\leq\gamma,$ we obtain

$\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle\int_{0<|\lambda|\leq\gamma}\frac{\|\mathcal{F}(a_{k})(\lambda)\|^{p}}{\big{(}(2|\alpha|+n)|\lambda|\big{)}^{\sigma}}|\lambda|^{n}d\lambda$
$\displaystyle\leq$ $\displaystyle
C~{}\sum_{2k+\ell=J+1}\omega_{k,\ell}~{}R^{Q(p-1)+p(2k+\frac{\ell}{2})}\int_{0<|\lambda|\leq\gamma}\big{(}(2|\alpha|+n)|\lambda|\big{)}^{p(k+\frac{\ell}{2})-{\sigma}}~{}|\lambda|^{n}d\lambda$
$\displaystyle\leq$ $\displaystyle
2C~{}\sum_{\ell=0}^{J+1}~{}\omega_{\ell}~{}R^{Q(p-1)+p(J+1-\frac{\ell}{2})}\int_{0}^{\gamma}\big{(}(2|\alpha|+n)|\lambda|\big{)}^{p(\frac{J+1}{2})-{\sigma}}~{}|\lambda|^{n}d\lambda.$

That is

$S_{1}\leq
C~{}R^{Q(p-1)+p(J+1-\frac{\ell}{2})}\gamma^{p(\frac{J+1}{2})+{\frac{Q}{2}}-\sigma},~{}~{}\forall\ell=0,1,...,J+1,$
(3.4)

provided that $p(\frac{J+1}{2})+\frac{Q}{2}-\sigma>0,$ which follows from the
inequality (1.6).
Now to estimate $S_{2},$ we may apply Hölder’s inequality for $q=\frac{2}{p}$
and Plancherel formula. Thus, we immediately obtain

$\displaystyle S_{2}$ $\displaystyle\leq$
$\displaystyle\Bigg{(}\int_{\mathbb{R}}(\|\mathcal{F}(a_{k})(\lambda)\|^{p})^{\frac{2}{p}}|\lambda|^{n}d\lambda\Bigg{)}^{\frac{p}{2}}\Bigg{(}\int_{|\lambda|>\gamma}\big{(}(2|\alpha|+n)|\lambda|\big{)}^{\frac{2\sigma}{p-2}}|\lambda|^{n}d\lambda\Bigg{)}^{\frac{2-p}{2}}$
$\displaystyle\leq$ $\displaystyle
C\|\mathcal{F}(a_{k})\|^{p}_{\mathcal{L}^{2}}\Bigg{(}\int_{|\lambda|>\gamma}\big{(}(2|\alpha|+n)|\lambda|\big{)}^{\frac{2\sigma}{p-2}}|\lambda|^{n}d\lambda\Bigg{)}^{\frac{2-p}{2}}$
$\displaystyle\leq$ $\displaystyle
2C\|\mathcal{F}(a_{k})\|^{p}_{\mathcal{L}^{2}}\Bigg{(}\int_{\gamma}^{\infty}\big{(}(2|\alpha|+n)|\lambda|\big{)}^{\frac{2\sigma}{p-2}}|\lambda|^{n}d\lambda\Bigg{)}^{\frac{2-p}{2}}$
$\displaystyle\leq$ $\displaystyle
C\|\mathcal{F}(a_{k})\|^{p}_{\mathcal{L}^{2}}\gamma^{{Q\over 4}(2-p)-\sigma}$

provided that ${Q\over 4}(2-p)-\sigma<0,$ which is a consequence of the left
hand side of (1.6). Thanks to Plancherel’s formula for Laguerre Fourier
transform it follows

$\displaystyle\|\mathcal{F}(a_{k})\|^{2}_{\mathcal{L}^{2}}=\|a_{k}\|_{L^{2}(\mathbb{H}^{n})}^{2}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{H}^{n}}|a_{k}(z,t)|^{2}~{}dV(z,t)$
$\displaystyle\leq$ $\displaystyle|B(0,R)|^{1-\frac{2}{p}}$
$\displaystyle\leq$ $\displaystyle C~{}R^{-Q(\frac{2-p}{p})}.$

That is

$\|\mathcal{F}(a_{k})\|^{p}_{\mathcal{L}^{2}}\leq C~{}R^{-\frac{Q}{2}(2-p)},$

and hence,

$S_{2}\leq C~{}R^{-\frac{Q}{2}(2-p)}\gamma^{{{Q\over 4}(2-p)}-\sigma}.$ (3.5)

However, to prove that $S_{1}+S_{2}\leq C,$ we shall discuss the cases $0<R<1$
and $R\geq 1.$ Hence, in order to deal with the case $0<R<1,$ we need more
precise estimates, so we consider the set $\Gamma_{\gamma};$ the collection of
all numbers $\gamma$ satisfying

$\Gamma_{\gamma}=\Big{\\{}\gamma>0,~{}\frac{\frac{Q}{2}(2-p)}{{\frac{Q}{4}(2-p)}-\sigma}\log(R)\leq\log(\gamma)\leq\frac{Q(1-p)-p(J+1)}{p(\frac{J+1}{2})+{\frac{Q}{2}}-\sigma}\log(R)\Big{\\}}.$

We mention that the collection $\Gamma_{\gamma}$ above is an nonempty set if
and only if

$\frac{\frac{Q}{2}(2-p)}{\frac{Q}{4}(2-p)-\sigma}\times\frac{p(\frac{J+1}{2})+{\frac{Q}{2}}-\sigma}{Q(1-p)-p(J+1)}\leq
1$

which is a different formulation of the hand side of (1.6), that is
$\frac{Q}{2}(2-p)\leq\sigma.$

Now let us choose $\gamma\in\Gamma_{\gamma}$ and using the fact that
$\frac{Q}{2}+p\frac{(J+1)}{2}-\sigma>0$ together with the right hand side of
(1.6) it follows that

$S_{1}\leq
C~{}R^{Q(p-1)+p(J+1)}\gamma^{p(\frac{J+1}{2})+{\frac{Q}{2}}-\sigma}.$ (3.6)

Also, with the same choose of $\gamma\in\Gamma_{\gamma}$ and under the
condition $\frac{Q}{2}(2-p)<\sigma,$ together with the help of the left hand
side of (1.6) we obtain

$S_{2}\leq C.$ (3.7)

Combining (3.6) and (3.7) we obtain

$S_{1}+S_{2}\leq C\qquad\mbox{for}\qquad 0<R<1.$ (3.8)

Now, to deal with the case $R\geq 1,$ we may take

$\gamma=R^{\frac{Q(1-p)-p(J+1)}{p(\frac{J+1}{2})+{\frac{Q}{2}}-\sigma}}$ (3.9)

so, using the fact that $R\geq 1,$ we obtain

$\gamma\leq R^{\frac{\frac{Q}{2}(2-p)}{\frac{Q}{4}(2-p)-\sigma}}.$ (3.10)

which leads to

$S_{1}+S_{2}\leq C\qquad\mbox{for}\qquad R\geq 1.$ (3.11)

Hence, to prove (3.3), it is enough to combine (3.8) and (3.11). The proof of
the main theorem is completed.

Acknowledgements. This project was supported by King Saud University, Deanship
of Scientific Research, College of Science Research Center.

## References

  * [1] M. Assal, Hardy’s type inequality associated with the Hankel transform for overcritical exponent, Integr. Transf. Spec. F., (2010), 1-6.
  * [2] M. Assal and A. Rahmouni, Hardy’s type inequality associated with the Laguerre Fourier transform, Integr. Transf. Spec. F., 24, (2013), 156–163.
  * [3] M. Assal and A. Rahmouni, An improved Hardy’s inequality associated with the Laguerre Fourier transform, Collect. Math., (2013), 1–11.
  * [4] R. R. Coifman, A real-variable characterization of $H^{p},$ Studia Math. 51, (1974), 269-274.
  * [5] C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129, (1972), 137-193.
  * [6] G. B. Folland and E. M. Stein, _Hardy Spaces on Homogeneous Groups,_ Princeton University Press, Princeton, NJ, 1982.
  * [7] J. Garcia-Cuerva and J. Rubio de Francia, _Weighted Norm Inequalities and Related Topics,_ North Holland, 1985.
  * [8] S. Giulini, _Bernstein and Jackson theorems for th Heisenberg group,_ J. Austral. Math. Soc. (Series A) 38 (1985), 241-254.
  * [9] H. P. Liu, The group Fourier transforms and mulitipliers of the Hardy spaces on the Heisenberg group, Approx. Theory $\&$ Its Appl., 7 (1991), 106-117.
  * [10] C. C. Lin, $L^{p}$ multipliers and their $H^{1}-L^{1}$ estimates on the Heisenberg group, Revista Math. Ibero., 11 (1995), 269-308.
  * [11] C. C. Lin, H$\ddot{o}$rmander’s $H^{p}$ Multiplier theorem for the Heisenberg group, J. London Math. Soc. (3) 67 (2003) 686-700.
  * [12] Y. Kanjin, On Hardy-Type Inequalities and Hankel Transforms, Monatshefte $f\ddot{u}$r Mathematik, 127, (1999), 311-319.
  * [13] Y. Kanjin, Hardy’s inequalities for Hermite and Laguerre expansions, Bull. London Math. Soc., 29, (1997), 331-337.
  * [14] R. Radha and S. Thangavelu, Hardy’s inequalities for Hermite and Laguerre expansions, Proc. Amer. Math. Soc., 132, (12), (2004), 3525-3536.
  * [15] A. Rahmouni and M. Assal, Hardy’s type inequality for the critical exponent associated with the inverse Laguerre Fourier transform, Integr. Transf. Spec. F., (2013) 1–9.
  * [16] E. M. Stein, _Harmonic Analysis, real variable Methods, orthogonality and oscillatory integrals,_ Princeton Univ. Press, Princeton, NJ, 1993.
  * [17] M. H. Taibleson and G. Weiss, The molecular characterization of certain Hardy spaces, Astérisque 77, (1980), Société Math. de France, Paris, 67-149.
  * [18] S. Thangavelu, On regularity of twisted spherical means and special Hermite expansion, Proc. Ind. Acad. Sci., 103, (1993), 303-320.