#  A NOTE ON HARDY SPACES AND BOUNDED OPERATORS

Pablo Rocha  Universidad Nacional del Sur, INMABB (Conicet), Bahía Blanca,
8000 Buenos Aires, Argentina<EMAIL_ADDRESS>
###### Abstract.

In this note we show that if $f\in H^{p}(\mathbb{R}^{n})\cap
L^{s}(\mathbb{R}^{n})$, where $0<p\leq 1<s<\infty$, then there exists a
$(p,\infty)$-atomic decomposition which converges to $f$ in
$L^{s}(\mathbb{R}^{n})$. From this fact, we prove that a bounded operator $T$
on $L^{s}(\mathbb{R}^{n})$ can be extended to a bounded operator from
$H^{p}(\mathbb{R}^{n})$ into $L^{p}(\mathbb{R}^{n})$ if and only if $T$ is
bounded uniformly in $L^{p}$ norm on all $(p,\infty)$-atoms. A similar result
is also obtained from $H^{p}(\mathbb{R}^{n})$ into $H^{p}(\mathbb{R}^{n})$.

Key words and phrases: Hardy Spaces, Atomic decomposition.

2.010 Math. Subject Classification:

##  1\. Introduction

M. Bownik in [1] gives an example of a linear functional defined on a dense
subspace of Hardy space $H^{1}(\mathbb{R}^{n})$, which maps all atoms into
bounded scalars, but it can not be extended to a bounded functional on the
whole space $H^{1}(\mathbb{R}^{n})$. That example is in certain sense
pathological. In [6], F. Ricci and J. Verdera show that when $0<p<1$ no
example like the one in [1] can be exhibited. They also studied the extension
problem for operators defined on the space of finite linear combinations of
$(p,\infty)$-atoms, $0<p<1$, and taking values in a Banach space $B$, and
proved that if the operator is uniformly bounded on $(p,\infty)$-atoms, then
it extends to a bounded operator from $H^{p}(\mathbb{R}^{n})$ into $B$.
The extension property for operators defined on the space of finite linear
combinations of $(1,q)$-atoms with $1<q<\infty$, and taking values in Banach
spaces was studied by S. Meda, P. Sjögren and M. Vallarino in [4]. For
$0<p\leq 1$ and $(p,2)$-atoms and operators taking values in quasi-Banach
spaces, by D. Yang and Y. Zhou in [8].
Y. Han and K. Zhao in [3], using the Calderón reproducing formula, give a
$(p,2)$-atomic decomposition for members of $H^{p}(\mathbb{R}^{n})\cap
L^{2}(\mathbb{R}^{n})$, where the composition converges also in
$L^{2}(\mathbb{R}^{n})$ rather than only in the distributions sense. Then,
using this atomic decomposition, they prove that a bounded operator $T$ on
$L^{2}(\mathbb{R}^{n})$ can be extended to a bounded operator from
$H^{p}(\mathbb{R}^{n})$ into $L^{p}(\mathbb{R}^{n})$ if and only if $T$ is
bounded uniformly on all $(p,2)$-atoms.
In this note we will show, by a simple argument, that if $f\in
H^{p}(\mathbb{R}^{n})\cap L^{s}(\mathbb{R}^{n})$ with $0<p\leq 1<s<\infty$,
then the $(p,\infty)$-atomic decomposition given in [7], (Theorem 2, p. 107),
whose sum converges to $f$ in $H^{p}$ norm also converges to $f$ in $L^{s}$
norm. As a consequence of this, we prove that a bounded operator $T$ on
$L^{s}(\mathbb{R}^{n})$ can be extended to a bounded operator from
$H^{p}(\mathbb{R}^{n})$ into $L^{p}(\mathbb{R}^{n})$ if and only if $T$ is
bounded uniformly in $L^{s}$ norm on all $(p,\infty)$-atoms. A similar result
is also obtained from $H^{p}(\mathbb{R}^{n})$ into $H^{p}(\mathbb{R}^{n})$.
See Theorem 5 and Corollary 6 and 7 below.

##  2\. Preliminaries

For a given cube $Q$ in $\mathbb{R}^{n}$, we denote by $l(Q)$ its length. The
following proposition gives the well known Whitney decomposition for an open
nonempty proper subset of $\mathbb{R}^{n}$.

######  Proposition 1.

$($See p. $463$ in [2]$)$ Let $\mathcal{O}$ be an open nonempty proper subset
of $\mathbb{R}^{n}$. Then there exists a family of closed cubes
$\\{Q_{k}\\}_{k}$ such that
a) $\bigcup_{k}Q_{k}=\mathcal{O}$ and the $Q_{k}$’s have disjoint interiors.
b) $\sqrt{n}\,\,l(Q_{k})\leq dist\left(Q_{k},\mathcal{O}^{c}\right)\leq
4\sqrt{n}\,\,l(Q_{k}).$
c) If the boundaries ot two cubes $Q_{i}$ and $Q_{k}$ touch, then

$\frac{1}{4}\leq\frac{l(Q_{i})}{l(Q_{k})}\leq 4.$

d) For a given $Q_{k}$ there exist at most $12^{n}$ cubes $Q_{i}$’s that touch
it.

######  Remark 2.

Let $\mathcal{F}=\\{Q_{k}\\}$ be a Whitney decomposition of a proper open
subset $\mathcal{O}$ of $\mathbb{R}^{n}$. Fix $0<\epsilon<\frac{1}{4}$ and
denote $Q_{k}^{*}$ the cube with the same center as $Q_{k}$ but with side
length $(1+\epsilon)$ times that $Q_{k}$. Then $Q_{k}$ touch $Q_{i}$ if and
only if $Q_{k}^{*}\cap Q_{i}^{*}\neq\emptyset$. Consequently, every point in
$\mathcal{O}$ is contained in at most $12^{n}$ cubes $Q_{k}^{*}$. Moreover,
$0<\epsilon<\frac{1}{4}$ can be chosen such that
$\bigcup_{k}Q_{k}^{\ast}=\mathcal{O}$.

######  Remark 3.

We consider $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ two open nonempty proper
subset of $\mathbb{R}^{n}$ such that $\mathcal{O}_{2}\subset\mathcal{O}_{1}$,
and for $j=1,2$ let $\mathcal{F}_{j}=\\{Q^{j}_{k}\\}$ be the Whitney
decomposition of $\mathcal{O}_{j}$. Since
$\mathcal{O}_{2}\subset\mathcal{O}_{1}$, from Proposition 1-b, we obtain that
if $Q_{i}^{2}\cap Q_{k}^{1}\neq\emptyset$ then

$l(Q_{i}^{2})\leq 5\,\,l(Q_{k}^{1}).$

From this, it follows that for a given cube $Q_{i}^{2}\in\mathcal{F}_{2}$
there exist at most $7^{n}$ cubes in $\mathcal{F}_{1}$ such that
$Q_{i}^{2}\cap Q_{k_{j}}^{1}\neq\emptyset$, for $j=1,...,7^{n}$ and
$Q_{i}^{2}\subset\bigcup_{j=1}^{7^{n}}Q_{k_{j}}^{1}$; thus
$Q_{i}^{2*}\subset\bigcup_{j=1}^{7^{n}}Q_{k_{j}}^{1*}$. Finally, from
Proposition 1-d and Remark 2, we obtain that there exist at most $84^{n}$
cubes $Q_{k}^{1*}$’s that intersect to $Q_{i}^{2*}$

We conclude this preliminaries with the following

######  Lemma 4.

Let $\\{\mathcal{O}_{j}\\}_{j\in\mathbb{Z}}$ be a family of subset of
$\mathbb{R}^{n}$ such that $\mathcal{O}_{j+1}\subset\mathcal{O}_{j}$ and
$\left|\bigcap_{j=1}^{\infty}\mathcal{O}_{j}\right|=0$ . Then

$\sum_{j=-\infty}^{\infty}2^{j}\chi_{\mathcal{O}_{j}}(x)\leq
2\sum_{j=-\infty}^{\infty}2^{j}\chi_{\mathcal{O}_{j}\setminus\mathcal{O}_{j+1}}(x),\,\,\,\,\,p.p.x\in\mathbb{R}^{n}.$

###### Proof.

We consider $N,M\in\mathbb{N}$. Now a computation gives

$\sum_{j=-N}^{M}2^{j}\chi_{\mathcal{O}_{j}}=\frac{1}{2}\sum_{j=-N+1}^{M}2^{j}\chi_{\mathcal{O}_{j}}+\sum_{j=-N}^{M-1}2^{j}\chi_{\mathcal{O}_{j}\setminus\mathcal{O}_{j+1}}+2^{M}\chi_{\mathcal{O}_{M}}$
$\leq\frac{1}{2}\sum_{j=-N}^{M}2^{j}\chi_{\mathcal{O}_{j}}+\sum_{j=-N}^{M-1}2^{j}\chi_{\mathcal{O}_{j}\setminus\mathcal{O}_{j+1}}+2^{M}\chi_{\mathcal{O}_{M}}.$

Since
$\lim_{M\rightarrow\infty}\chi_{\mathcal{O}_{M}}=\chi_{\bigcap_{j=1}^{\infty}\mathcal{O}_{j}}$
and $\left|\bigcap_{j=1}^{\infty}\mathcal{O}_{j}\right|=0$ the lemma follows.
∎

##  3\. Main Result

We recall the definition of Hardy space and $(p,\infty)$-atoms.
Define $\mathcal{F}_{N}=\left\\{\varphi\in
S(\mathbb{R}^{n}):\sum\limits_{\left|\mathbf{\beta}\right|\leq
N}\sup\limits_{x\in\mathbb{R}^{n}}\left(1+\left|x\right|\right)^{N}\left|\partial^{\mathbf{\beta}}\varphi(x)\right|\leq
1\right\\}$. Denote by $\mathcal{M}$ the grand maximal operator given by

$\mathcal{M}f(x)=\sup\limits_{t>0}\sup\limits_{\varphi\in\mathcal{F}_{N}}\left|\left(t^{-n}\varphi(t^{-1}.)\ast
f\right)\left(x\right)\right|,$

where $f\in S^{\prime}(\mathbb{R}^{n})$ and $N$ is a large and fix integer.
For $0<p<\infty$, the Hardy space $H^{p}\left(\mathbb{R}^{n}\right)$ is the
set of all $f\in S^{\prime}(\mathbb{R}^{n})$ for which
$\left\|\mathcal{M}f\right\|_{p}<\infty$. In this case we define
$\left\|f\right\|_{H^{p}}=\left\|\mathcal{M}f\right\|_{p}$.
For $1<p<\infty$, it is well known that $H^{p}(\mathbb{R}^{n})\cong
L^{p}(\mathbb{R}^{n})$, $H^{1}(\mathbb{R}^{n})\subset L^{1}(\mathbb{R}^{n})$
strictly, and for $0<p<1$ the spaces $H^{p}(\mathbb{R}^{n})$ and
$L^{p}(\mathbb{R}^{n})$ are not comparable. One of the principal interest of
$H^{p}(\mathbb{R}^{n})$ theory is that it gives a natural extension of the
results for singular integrals, originally developed for $L^{p}$ ($p>1$), to
the range $0<p\leq 1$. This is achieved to decompose elements in $H^{p}$ as
sums of $(p,\infty)$-atoms.

For $0<p\leq 1$, a $(p,\infty)$-atom is a measurable function $a$ supported on
a ball $B$ of $\mathbb{R}^{n}$ satisfying
$(i)$ $\|a\|_{\infty}\leq|B|^{-\frac{1}{p}}$,
$(ii)$ $\int x^{\alpha}a(x)dx=0$, for all multiindex $\alpha$ with
$|\alpha|\leq n(p^{-1}-1)$.

Our main result is the following

######  Theorem 5.

Let $f\in H^{p}(\mathbb{R}^{n})\cap L^{s}(\mathbb{R}^{n})$, with $0<p\leq
1<s<\infty$. Then there is a sequence of $(p,\infty)$-atoms $\\{a_{j}\\}$ and
a sequence of scalars $\\{\lambda_{j}\\}$ with $\sum_{j}|\lambda_{j}|^{p}\leq
c\|f\|_{H^{p}}^{p}$ such that $f=\sum_{j}\lambda_{j}a_{j}$, where the series
converges to $f$ in $L^{s}(\mathbb{R}^{n})$.

###### Proof.

Given $f\in H^{p}(\mathbb{R}^{n})\cap L^{s}(\mathbb{R}^{n})$, let
$\mathcal{O}_{j}=\\{x:\mathcal{M}f(x)>2^{j}\\}$ and let
$\mathcal{F}_{j}=\\{Q_{k}^{j}\\}_{k}$ be the Whitney decomposition associated
to $\mathcal{O}_{j}$ such that $\bigcup_{k}Q_{k}^{j\ast}=\mathcal{O}_{j}$.
Following the proof 2.3 in [7] (p. 107-109), we have a sequence of functions
$A_{k}^{j}$ such that

(i) $supp(A_{k}^{j})\subset
Q_{k}^{j\ast}\cup\bigcup_{i\,:\,Q_{i}^{j+1\ast}\cap
Q_{k}^{j\ast}\neq\emptyset}Q_{i}^{j+1\ast}$ and $|A_{k}^{j}(x)|\leq c2^{j}$
for all $k,j\in\mathbb{Z}$.

(ii) $\int x^{\alpha}A_{k}^{j}(x)dx=0$ for all $\alpha$ with $|\alpha|\leq
n(p^{-1}-1)$ and all $k,j\in\mathbb{Z}$.

(iii) The sum $\sum_{j,k}A_{k}^{j}$ converges to $f$ in the sense of
distributions.
From (i) we obtain

$\sum_{k}|A_{k}^{j}|\leq
c2^{j}\left(\sum_{k}\chi_{Q_{k}^{j\ast}}+\sum_{k}\chi_{\bigcup_{i\,:\,Q_{i}^{j+1\ast}\cap
Q_{k}^{j\ast}\neq\emptyset}Q_{i}^{j+1\ast}}\right)$

the remark 2 gives

$\leq
c2^{j}\left(\chi_{\mathcal{O}_{j}}+\sum_{k}\sum_{i\,:\,Q_{i}^{j+1\ast}\cap
Q_{k}^{j\ast}\neq\emptyset}\chi_{Q_{i}^{j+1\ast}}\right)$
$=c2^{j}\left(\chi_{\mathcal{O}_{j}}+\sum_{i}\sum_{k\,:\,Q_{i}^{j+1\ast}\cap
Q_{k}^{j\ast}\neq\emptyset}\chi_{Q_{i}^{j+1\ast}}\right)$

from remark 3 we have

$\leq
c2^{j}\left(\chi_{\mathcal{O}_{j}}+84^{n}\sum_{i}\chi_{Q_{i}^{j+1\ast}}\right)$

once again, from remark 2, we obtain

$\leq c2^{j}\left(\chi_{\mathcal{O}_{j}}+\chi_{\mathcal{O}_{j+1}}\right)\leq
c2^{j}\chi_{\mathcal{O}_{j}},$

by Lemma 4 we can conclude that

$\sum_{j,k}|A_{k}^{j}(x)|\leq
c\sum_{j}2^{j}\chi_{\mathcal{O}_{j}\setminus\mathcal{O}_{j+1}}(x),\,\,\,\,p.p.x\in\mathbb{R}^{n}.$

Thus

$\int\left(\sum_{j,k}|A_{k}^{j}(x)|\right)^{s}dx\leq
c\sum_{j}\int_{\mathcal{O}_{j}\setminus\mathcal{O}_{j+1}}2^{js}dx\leq
c\sum_{j}\int_{\mathcal{O}_{j}\setminus\mathcal{O}_{j+1}}(\mathcal{M}f(x))^{s}dx$
$\leq c\int_{\mathbb{R}^{n}}(\mathcal{M}f(x))^{s}<\infty$

since $f\in L^{s}(\mathbb{R}^{n})$. Now from (iii) we obtain that the sum
$\sum_{j,k}A_{k}^{j}$ converges to $f$ in $L^{s}(\mathbb{R}^{n})$.
Finally, we set $a_{k}^{j}=\lambda^{-1}_{j,k}A_{k}^{j}$ with
$\lambda_{j,k}=c2^{j}|B_{k}^{j}|^{1/p}$, where $B_{k}^{j}$ is the smallest
ball containing $Q_{k}^{j\ast}$ as well as all the $Q_{i}^{j+1\ast}$ that
intersect $Q_{k}^{j\ast}$. Then we have a sequence $\\{a_{j,k}\\}$ of
$(p,\infty$)-atoms and a sequence of scalars $\\{\lambda_{j,k}\\}$ such that
the sum $\sum_{j,k}\lambda_{j,k}a_{j,k}$ converges to $f$ in
$L^{s}(\mathbb{R}^{n})$ with $\sum_{j,k}|\lambda_{j,k}|^{p}\leq
c\|f\|_{H^{p}}^{p}$ (see (38), in [7] p. 109). The proof of the theorem is
therefore concluded. ∎

######  Corollary 6.

Let $T$ be a bounded operator on $L^{s}(\mathbb{R}^{n})$ for some
$1<s<\infty$. Then $T$ can be extended to a bounded operator from
$H^{p}(\mathbb{R}^{n})$ into $L^{p}(\mathbb{R}^{n})$ $(0<p\leq 1)$ if and only
if $T$ is bounded uniformly in the $L^{p}$ norm on all $(p,\infty)$-atoms.

###### Proof.

Since $T$ is a bounded operator on $L^{s}(\mathbb{R}^{n})$, $T$ is well
defined on $H^{p}(\mathbb{R}^{n})\cap L^{s}(\mathbb{R}^{n})$ for $0<p\leq 1$.
If $T:H^{p}(\mathbb{R}^{n})\cap L^{s}(\mathbb{R}^{n})\rightarrow
L^{p}(\mathbb{R}^{n})$ can be extended to a bounded operator from
$H^{p}(\mathbb{R}^{n})$ into $L^{p}(\mathbb{R}^{n})$, then $\|Ta\|_{p}\leq
c_{p}\|a\|_{H^{p}}$ for all $(p,\infty)$-atom $a$. Since there exists a
universal constant $C$ such that $\|a\|_{H^{p}}\leq C<\infty$ for all
$(p,\infty)$-atom $a$; it follows that $\|Ta\|_{p}\leq C_{p}$ for all
$(p,\infty)$-atom $a$.
Reciprocally, let $f\in H^{p}(\mathbb{R}^{n})\cap L^{s}(\mathbb{R}^{n})$ by
Theorem 5 there is a $(p,\infty)$-atom decomposition such that
$\sum_{j}\lambda_{j}a_{j}=f$ in $L^{s}(\mathbb{R}^{n})$. Since $T$ is bounded
on $L^{s}(\mathbb{R}^{n})$ we have that the sum $\sum_{j}\lambda_{j}Ta_{j}$
converges a $Tf$ in $L^{s}(\mathbb{R}^{n})$, thus there exists a subsequence
of natural numbers $\\{j_{N}\\}_{N\in\mathbb{N}}$ such that
$\lim_{N\rightarrow\infty}\sum_{j=-j_{N}}^{j_{N}}\lambda_{j}Ta_{j}(x)=Tf(x)$
p.p.$x\in\mathbb{R}^{n}$, this implies

$|Tf(x)|\leq\sum_{j}|\lambda_{j}Ta_{j}(x)|,\,\,\,\,p.p.x\in\mathbb{R}^{n}.$

If $\|Ta\|_{p}\leq C_{p}<\infty$ for all $(p,\infty)$-atom $a$, and since
$0<p\leq 1$ we get

$\|Tf\|_{p}^{p}\leq\sum_{j}|\lambda_{j}|^{p}\|Ta_{j}\|_{p}^{p}\leq
C_{p}^{p}\sum_{j}|\lambda_{j}|^{p}\leq C_{p}^{p}\|f\|_{H^{p}}^{p}$

for all $f\in H^{p}(\mathbb{R}^{n})\cap L^{s}(\mathbb{R}^{n})$. Since
$H^{p}(\mathbb{R}^{n})\cap L^{s}(\mathbb{R}^{n})$ is a dense subspace of
$H^{p}(\mathbb{R}^{n})$, the corollary follows. ∎

Since many operators that appear in the practice are bounded on $L^{s}$
$(1<s<\infty)$, in view of Corollary 6 is sufficient to prove that our
operator is bounded uniformly in the $L^{p}$ norm on all $(p,\infty)$-atoms to
assure the boundedness $H^{p}-L^{p}$.

######  Corollary 7.

Let $T$ be a bounded operator on $L^{s}(\mathbb{R}^{n})$ for some
$1<s<\infty$. Then $T$ can be extended to a bounded operator on
$H^{p}(\mathbb{R}^{n})$ $(0<p\leq 1)$ if and only if $T$ is bounded uniformly
in the $H^{p}$ norm on all $(p,\infty)$-atoms.

###### Proof.

The only if is similar to the only if of the corollary 6. By the other hand,
once again by Theorem 5, for $f\in H^{p}(\mathbb{R}^{n})\cap
L^{s}(\mathbb{R}^{n})$, we have a $(p,\infty)$-atomic decomposition such that
$\sum_{j}\lambda_{j}a_{j}=f$ in $L^{s}$ norm. Since $T$ is bounded on $L^{s}$
we obtain

$|\mathcal{M}(Tf)(x)|\leq\sum_{j}|\lambda_{j}\mathcal{M}(Ta_{j})(x)|,\,\,\,\,p.p.x\in\mathbb{R}^{n}.$

Now, if $\|Ta\|_{H^{p}}\leq C_{p}<\infty$ for all $(p,\infty)$-atom $a$ with
$0<p\leq 1$, we get

$\|Tf\|_{H^{p}}^{p}=\|\mathcal{M}(Tf)\|_{p}^{p}\leq\sum_{j}|\lambda_{j}|^{p}\|\mathcal{M}(Ta_{j})\|_{p}^{p}\leq
C_{p}^{p}\sum_{j}|\lambda_{j}|^{p}\leq C_{p}^{p}\|f\|_{H^{p}}^{p},$

for all $f\in H^{p}(\mathbb{R}^{n})\cap L^{s}(\mathbb{R}^{n})$. Since
$H^{p}(\mathbb{R}^{n})\cap L^{s}(\mathbb{R}^{n})$ is a dense subspace of
$H^{p}(\mathbb{R}^{n})$, the corollary follows.

∎

######  Remark 8.

The statement of Theorem 5 appears in [5], as well as (of implicity way) Lemma
4 (see p. 3692).

## References

  * [1] M. Bownik, Boundedness of operators on Hardy spaces via atomic decompositions, Proc. Amer. Math. Soc. 133, 3535 3542, (2005).
  * [2] L. Grafakos, Classical Fourier Analysis, Second Edition, Springer Science+Business Media, LLC, (2008).
  * [3] Y. Han and K. Zhao, Boundedness of operators on Hardy spaces, Taiwanese Journal of Mathematics. Vol. 14, No. 2, 319-327, (2010).
  * [4] S. Meda, P. Sjögren and M. Vallarino, On the $H^{1}-L^{1}$ boundedness of operators, Proc. Amer. Math. Soc. 136, 2921-2931, (2008).
  * [5] E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, Journal of Functional Analysis, 262, 3665-3748, (2012).
  * [6] F. Ricci and J. Verdera, Duality in spaces of finite linear combinations of atoms, http://arxiv.org/abs/0809.1719.
  * [7] E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, (1993).
  * [8] D. Yang, Y. Zhou, A boundedness criterion via atoms for linear operators in Hardy spaces, Constr. Approx. 29, 207-218, (2009).