The $H^p(\mathbb{Z}^n)-H^q(\mathbb{Z}^n)$ boundedness of the discrete Riesz potential

TL;DR

This paper extends the boundedness results of the discrete Riesz potential operator from a limited range of p to the full range 0 < p ≤ 1 on discrete Hardy spaces, broadening its applicability.

Contribution

It generalizes previous boundedness results of the discrete Riesz potential to the entire range 0 < p ≤ 1, enhancing understanding of its operator behavior.

Findings

Boundedness holds for the full range 0 < p ≤ 1

Extended the operator's boundedness from partial to complete p-range

Provides a broader framework for discrete Riesz potential analysis

Abstract

In [J. Class. Anal., vol. 26 (1) (2025), 63-76], we proved that the discrete Riesz potential $I_{\alpha}$ is a bounded operator $H^p(\mathbb{Z}^n) \to H^q(\mathbb{Z}^n)$ for $\frac{n-1}{n} < p \leq 1$, $\frac{1}{q} = \frac{1}{p} - \frac{\alpha}{n}$ and $0 < \alpha < n$. In this note, we extend such boundedness on the full range $0 < p \leq 1$.

Equations

$$({\mathcal{R}}_{\alpha }f)(x)={\int }_{{\mathbb{R}}^n}f(y)|x-y{|}^{\alpha -n}dy.$$

$$f=\sum _j{\lambda }_j{a}_j,$$

$$\Vert a{\Vert }_{\infty }\le |Q{|}^{-1/p}\text{and}\int x^{\alpha }a(x)dx=0\text{for all}|\alpha |\le n\left(\frac{1}{p}-1\right).$$

$$(I_{\alpha }b)(j)=\sum _{i\in {\mathbb{Z}}^n\setminus \{j\}}\frac{b(i)}{|i-j{|}^{n-\alpha }},j\in {\mathbb{Z}}^n.$$

$$\Vert I_{\alpha }b{\Vert }_{H^q({\mathbb{Z}}^n)}\le C\Vert b{\Vert }_{H^p({\mathbb{Z}}^n)}$$

$$\mathcal{S}({\mathbb{R}}^n)=\left\{\varphi \in C^{\infty }({\mathbb{R}}^n):\underset{x\in {\mathbb{R}}^n}{\sup }(1+|x|{)}^N|({\partial }^{\beta }\varphi )(x)|<\infty \forall N\in {\mathbb{Z}}_{+},\beta \in {\mathbb{Z}}_{+}^n\right\}.$$

$$\Vert \varphi {\Vert }_{\mathcal{S}({\mathbb{R}}^n),N}=\sum _{[\beta ]\le N}\underset{x\in {\mathbb{R}}^n}{\sup }(1+|x|{)}^N|({\partial }^{\beta }\varphi )(x)|,(N\in {\mathbb{Z}}_{+}).$$

$$\Vert b{\Vert }_{{\ell }^p({\mathbb{Z}}^n)}:={\left(\sum _{i\in {\mathbb{Z}}^n}|b(i){|}^p\right)}^{1/p}<\infty .$$

$$\Vert b{\Vert }_{{\ell }^{\infty }({\mathbb{Z}}^n)}:=\underset{i\in {\mathbb{Z}}^n}{\sup }|b(i)|<\infty .$$

$$(b\ast c)(j)=\sum _{i\in {\mathbb{Z}}^n}b(i)c(j-i),$$

$$(M_{\alpha }b)(j)=\underset{Q\ni j}{\sup }\frac{1}{(\#Q{)}^{1-\frac{\alpha }{n}}}\sum _{i\in Q}|b(i)|,j\in {\mathbb{Z}}^n,$$

$$H^p({\mathbb{Z}}^n):=\left\{b\in {\ell }^p({\mathbb{Z}}^n):\underset{t>0}{\sup }|({\Phi }_t^d\ast b)|\in {\ell }^p({\mathbb{Z}}^n)\right\},$$

$$\Vert b{\Vert }_{H^p({\mathbb{Z}}^n)}:=\Vert b{\Vert }_{{\ell }^p({\mathbb{Z}}^n)}+\Vert \underset{t>0}{\sup }|({\Phi }_t^d\ast b)|{\Vert }_{{\ell }^p({\mathbb{Z}}^n)}.$$

$$\sum _{i\in Q_{k^0,m}\setminus \{k\}}|k-i{|}^{\alpha -n}\le C(2m+1{)}^{\alpha }$$

$$\sum _{i\in Q_{k^0,m}\setminus \{k\}}|k-i{|}^{\alpha -n}\le \sum _{i\in Q_{k^0,m}\setminus \{k\}}|k-i{|}_{\infty }^{\alpha -n}=\sum _{i\in I_{k^0,k,m}}|k-i{|}_{\infty }^{\alpha -n}+\sum _{i\in J_{k^0,k,m}}|k-i{|}_{\infty }^{\alpha -n},$$

$$I_{k^0,k,m}=\{i\in Q_{k^0,m}:|k-i{|}_{\infty }>m\}\text{and}{J}_{k^0,k,m}=\{i\in Q_{k^0,m}:0<|k-i{|}_{\infty }\le m\}.$$

$$\sum _{i\in I_{k^0,k,m}}|k-i{|}_{\infty }^{\alpha -n}\le m^{\alpha -n}\cdot \#Q_{k^0,m}\le C(2m+1{)}^{\alpha }.$$

$$\sum _{i\in J_{k^0,k,m}}|k-i{|}_{\infty }^{\alpha -n}\le \sum _{s=0}^{s_0-1}\sum _{2^{-(s+1)}m<|k-i{|}_{\infty }\le 2^{-s}m}|k-i{|}_{\infty }^{\alpha -n}$$

$$\le m^{\alpha }\sum _{s=0}^{s_0-1}{2}^{-(s+1)\alpha }\left((2^{-(s+1)}m{)}^{-n}\sum _{i:|k-i{|}_{\infty }\le 2^{-s}m}1\right)$$

$$\le C(2m+1{)}^{\alpha }\sum _{s=0}^{\infty }{2}^{-(s+1)\alpha }.$$

$$\underset{t>0}{\sup }|({\Phi }_t^d\ast I_{\alpha }a)(j)|\le C(\#Q{)}^{\frac{\alpha }{n}-\frac{1}{p}},$$

$$({\Phi }_t^d\ast I_{\alpha }a)(j)=0,$$

$$|({\Phi }_t^d\ast I_{\alpha }a)(j)|$$

$$\le C(\#Q{)}^{\frac{\alpha }{n}-\frac{1}{p}}\left(t^{-n}\sum _{|k{|}_{\infty }\le \frac{t}{4}}|\Phi (k/t)|\right)$$

$$\le C(\#Q{)}^{\frac{\alpha }{n}-\frac{1}{p}}\left(\lfloor t{\rfloor }^{-n}\sum _{|k{|}_{\infty }\le \lfloor t\rfloor }|\Phi (k/t)|\right)$$

$$\le C\Vert \Phi {\Vert }_{\infty }(\#Q{)}^{\frac{\alpha }{n}-\frac{1}{p}},$$

$$\Vert \underset{t>0}{\sup }|({\Phi }_t^d\ast I_{\alpha }a)|{\Vert }_{{\ell }^q({\mathbb{Z}}^n)}\le C,$$

$$\sum _{j\in {\mathbb{Z}}^n}|\underset{t>0}{\sup }|({\Phi }_t^d\ast I_{\alpha }a)(j){|}^q$$

$$+\sum _{j\in {\mathbb{Z}}^n\setminus 4\lfloor \sqrt{n}\rfloor Q_{k^0}}|\underset{t>0}{\sup }|({\Phi }_t^d\ast I_{\alpha }a)(j){|}^q.$$

$$\sum _{j\in 4\lfloor \sqrt{n}\rfloor Q_{k^0}}|\underset{t>0}{\sup }|({\Phi }_t^d\ast I_{\alpha }a)(j){|}^q$$

$$({\Phi }_t^d\ast I_{\alpha }a)(j)=0$$

$$({\Phi }_t^d\ast I_{\alpha }a)(j)=\sum _{k:|j-k{|}_{\infty }\le \frac{t}{4}}{\Phi }_t^d(j-k)\sum _{i\in Q_{k^0}}a(i)|i-k{|}^{\alpha -n}.$$

Full text

††2020 Mathematics Subject Classification. 43A17, 42B30, 42B25. ††Key words and phrases: Discrete Hardy Spaces; Atomic Decomposition; Discrete Riesz Potential

1 Introduction

Let $0<\alpha<n$, the Riesz potential $\mathcal{R}_{\alpha}$ on $\mathbb{R}^{n}$ is defined by

[TABLE]

The well known Hardy-Littlewood-Sobolev inequality establishes that if $1<p<\frac{n}{\alpha}$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$, then the operator $\mathcal{R}_{\alpha}$ is bounded from $L^{p}(\mathbb{R}^{n})$ into $L^{q}(\mathbb{R}^{n})$. For $p=1$ and $q=n/(n-\alpha)$, $\mathcal{R}_{\alpha}$ is of weak-type $(1,q)$ (see [18]). For the endpoint $p=\frac{n}{\alpha}$ is known that $\mathcal{R}_{\alpha}$ is not a bounded operator $L^{\frac{n}{\alpha}}(\mathbb{R}^{n})\to L^{\infty}(\mathbb{R}^{n})$, however it can see that $\mathcal{R}_{\alpha}:L^{\frac{n}{\alpha}}(\mathbb{R}^{n})\to BMO(\mathbb{R}^{n})$ is bounded. This last estimate can also be obtained by duality. Indeed, $\mathcal{R}_{\alpha}:H^{1}(\mathbb{R}^{n})\to L^{\frac{n}{n-\alpha}}(\mathbb{R}^{n})$ is bounded (see [21]), where $H^{1}(\mathbb{R}^{n})$ is the $1$-Hardy space on $\mathbb{R}^{n}$. In [5], C. Fefferman and E. Stein proved that $(H^{1})^{*}=BMO$, this remarkable result joint with the fact that $(L^{\frac{n}{n-\alpha}})^{*}=L^{\frac{n}{\alpha}}$ allow to obtain, by duality, the $L^{\frac{n}{\alpha}}(\mathbb{R}^{n})\to BMO(\mathbb{R}^{n})$ boundedness of $\mathcal{R}_{\alpha}$.

The real Hardy spaces $H^{p}(\mathbb{R}^{n})$, $\frac{n-1}{n}<p\leq\infty$, were first introduced by E. Stein and G. Weiss in [21]. Therein, the authors describe the $H^{p}$ theory in terms of systems of conjugate harmonic functions, for it the theory was only developed on the range $\frac{n-1}{n}<p\leq 1$ (see [19, §5.16]). They also showed that $\mathcal{R}_{\alpha}$ is a bounded operator from $H^{p}(\mathbb{R}^{n})$ into $H^{q}(\mathbb{R}^{n})$, when $\frac{n-1}{n}<p<\frac{n}{\alpha}$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Afterward, C. Fefferman and E. Stein in [5] introduced real variable methods to characterize the Hardy spaces by means of maximal functions on the full range $0<p\leq\infty$. This second approach brought greater flexibility to the theory. For $1<p\leq\infty$, one has 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.

The spaces $H^{p}(\mathbb{R}^{n})$ can also be characterized by atomic decompositions. That is, every distribution $f\in H^{p}$ can be expressed by

[TABLE]

where the $a_{j}$’s are $p$ - atoms, $\{\lambda_{j}\}\in\ell^{p}$ and $\|f\|^{p}_{H^{p}}\approx\sum_{j}|\lambda_{j}|^{p}$. For $0<p\leq 1$, an $p$ - atom is a function $a(\cdot)$ supported on a cube $Q$ such that

[TABLE]

Such decompositions were obtained by R. Coifman in [4] for the case $n=1$ and by R. Latter in [11] for the case $n\geq 1$. These decompositions, in principle, allow to study the behavior of certain operators on $H^{p}(\mathbb{R}^{n})$ by focusing one’s attention on individual atoms (see [1], [13], [14]). For instance, using such atomic decomposition, it obtains that the Riesz potential $\mathcal{R}_{\alpha}$ is bounded from $H^{p}(\mathbb{R}^{n})$ into $L^{q}(\mathbb{R}^{n})$, for $0<p\leq 1$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. In [22], M. H. Taibleson and G. Weiss, by means of a molecular decomposition for members in $H^{p}(\mathbb{R}^{n})$, proved the $H^{p}(\mathbb{R}^{n})\to H^{q}(\mathbb{R}^{n})$ boundedness for $\mathcal{R}_{\alpha}$ on the full range $0<p\leq 1$ (see also [10]). This extends the result obtained by E. Stein and G. Weiss above mentioned. A study about the behavior of Riesz potential $\mathcal{R}_{\alpha}$ on others function spaces was made by E. Nakai in [12].

For $0<\alpha<n$, the discrete counterpart of (1) is the discrete Riesz potential $I_{\alpha}$ on $\mathbb{Z}^{n}$, which is defined by

[TABLE]

As one would expect, (2) has a similar behavior to that of (1). Indeed, in [8, p. 288], for $n=1$, $0<\alpha<1$, $1<p<1/\alpha$ and $\frac{1}{q}=\frac{1}{p}-\alpha$, G. H. Hardy et al established the $\ell^{p}(\mathbb{Z})-\ell^{q}(\mathbb{Z})$ boundedness of $I_{\alpha}$ (for the case $n\geq 1$ see [20, Proposition $(a)$], other proof was given in [16, Theorem 3.1]).

The theory for Hardy spaces on $\mathbb{Z}^{n}$ was developed by S. Boza and M. Carro in [3] (see also [2]). There, the authors gave a variety of distinct approaches to characterize the discrete Hardy spaces $H^{p}(\mathbb{Z}^{n})$ analogous to the ones given for the Hardy spaces $H^{p}(\mathbb{R}^{n})$. They also described an atomic decomposition for elements in $H^{p}(\mathbb{Z}^{n})$.

Y. Kanjin and M. Satake in [9], based on some results of [2], constructed a molecular decomposition for $H^{p}(\mathbb{Z})$ analogous to the ones given by M. Taibleson and G. Weiss in [22] for the Hardy spaces $H^{p}(\mathbb{R}^{n})$. With this framework, they obtain the Marcinkiewicz multiplier theorem on $H^{p}(\mathbb{Z})$ and proved the $H^{p}(\mathbb{Z})\to H^{q}(\mathbb{Z})$ boundedness of discrete Riesz potential $I_{\alpha}$, for $n=1$, $0<p<\alpha^{-1}$ and $\frac{1}{q}=\frac{1}{p}-\alpha$. The $H^{p}(\mathbb{Z})\to\ell^{q}(\mathbb{Z})$ boundedness of (2), with $n=1$, was observed by the author in [15, Remark 11].

In [16], by means of the atomic characterization of $H^{p}(\mathbb{Z}^{n})$ developed in [3] and the boundedness of discrete fractional maximal operator, we proved that $I_{\alpha}$ is a bounded operator $H^{p}(\mathbb{Z}^{n})\to\ell^{q}(\mathbb{Z}^{n})$ for $n\geq 1$, $0<p<\frac{n}{\alpha}$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$.

The $H^{p}(\mathbb{Z}^{n})\to H^{q}(\mathbb{Z}^{n})$ boundedness of $I_{\alpha}$ for $\frac{n-1}{n}<p\leq q\leq 1$, was proved by the author in [17]. To achieve this result we used the characterization of $H^{p}(\mathbb{Z}^{n})$ by the discrete Riesz transforms and furnished a molecular decomposition, as in [9], for the elements of $H^{p}(\mathbb{Z}^{n})$ on the range $\frac{n-1}{n}<p\leq 1$ with $n\geq 1$. We point out that the lower bound $\frac{n-1}{n}$ in this decomposition is a consequence of Theorem 2.6 in [3].

The purpose of this note is to extend the $H^{p}(\mathbb{Z}^{n})\to H^{q}(\mathbb{Z}^{n})$ boundedness of $I_{\alpha}$ obtained in [17] to the full range $0<p\leq 1$.

The main result of this note is contained in the following theorem, which will be proved in Section 3 by means of the characterization maximal for $H^{p}(\mathbb{Z}^{n})$ established in [3, Theorem 2.7] and the atomic decomposition for $H^{p}(\mathbb{Z}^{n})$ also given in [3], joint with some auxiliary results of Section 2.

Theorem 3.2. For $0<\alpha<n$, let $I_{\alpha}$ be the discrete Riesz potential given by (2). If $0<p\leq 1$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$, then

[TABLE]

for all $b\in H^{p}(\mathbb{Z}^{n})$, where $C$ does not depend on $b$.

Notation. Throughout this paper, $C$ will denote a positive real constant not necessarily the same at each occurrence. We set $\mathbb{Z}_{+}=\mathbb{N}\cup\{0\}$. For every $A\subset\mathbb{Z}^{n}$, we denote by $\#A$ and $\chi_{A}$ the cardinality of the set $A$ and the characteristic sequence of $A$ on $\mathbb{Z}^{n}$ respectively. Given a real number $s\geq 0$, we write $\lfloor s\rfloor$ for the integer part of $s$. For $j=(j_{1},...,j_{n})\in\mathbb{Z}^{n}$ we consider the two following norms $|j|=(j_{1}^{2}+\cdot\cdot\cdot+j_{n}^{2})^{1/2}$ and $|j|_{\infty}=\max\{|j_{k}|:k=1,...,n\}$. Given $k^{0}\in\mathbb{Z}^{n}$ and $m\in\mathbb{Z}_{+}$ fix, we put $Q_{k^{0},m}=\{i\in\mathbb{Z}^{n}:|i-k^{0}|_{\infty}\leq m\}$, that is the discrete cube centered at $k^{0}$ and side length $2m+1$. If $\beta$ is the multiindex $\beta=(\beta_{1},...,\beta_{n})\in\mathbb{Z}_{+}^{n}$, then $[\beta]:=\beta_{1}+...+\beta_{n}$ and $j^{\beta}:=j_{1}^{\beta_{1}}\cdot\cdot\cdot j_{n}^{\beta_{n}}$ for every $j=(j_{1},...,j_{n})\in\mathbb{Z}^{n}$. By $B_{r}({\bf 0})$ we denote the Euclidean ball in $\mathbb{R}^{n}$ with center at ${\bf 0}$ and radius $r>0$. Given a function $F$ defined on $\mathbb{R}^{n}$, we shall use the notation $F^{d}$ to indicate its restriction on $\mathbb{Z}^{n}$.

2 Preliminaries

The Schwartz space $\mathcal{S}(\mathbb{R}^{n})$ is defined by

[TABLE]

We topologize the space $\mathcal{S}(\mathbb{R}^{n})$ with the following family of seminorms

[TABLE]

For $0<p<\infty$ and a sequence $b=\{b(i)\}_{i\in\mathbb{Z}^{n}}$ we say that $b$ belongs to $\ell^{p}(\mathbb{Z}^{n})$ if

[TABLE]

For $p=\infty$, we say that $b$ belongs to $\ell^{\infty}(\mathbb{Z}^{n})$ if

[TABLE]

Given two sequences $b=\{b(i)\}_{i\in\mathbb{Z}^{n}}$ and $c=\{c(i)\}_{i\in\mathbb{Z}^{n}}$, their convolution $b\ast c$ is defined by

[TABLE]

when the series converges (e.g. [6, Section 1.2.3, p. 21-25]). Moreover, $b\ast c=c\ast b$.

We recall that a discrete cube $Q$ centered at $j=(j_{1},...,j_{n})\in\mathbb{Z}^{n}$ can be written of the form $Q=Q_{j,m}=\prod_{1\leq l\leq n}[[j_{l}-m,j_{l}+m]]$, where for each $l=1,...,n$, $[[j_{l}-m,j_{l}+m]]:=[j_{l}-m,j_{l}+m]\cap\mathbb{Z}=\{j_{l}-m,...,j_{l},...,j_{l}+m\}$ with $m\in\mathbb{Z}_{+}$. It is clear that $\#Q=(2m+1)^{n}$.

Let $0\leq\alpha<n$, given a sequence $b=\{b(i)\}_{i\in\mathbb{Z}^{n}}$ we define the centered fractional maximal sequence $M_{\alpha}b$ by

[TABLE]

where the supremum is taken over all discrete cubes $Q$ centered at $j$. We observe that if $\alpha=0$, then $M_{0}=M$ where $M$ is the centered discrete maximal operator of Hardy-Littlewood. For $0\leq\alpha<n$, by Theorem 2.3 and Proposition 2.4 in [16], we have that the operator $M_{\alpha}$ is bounded from $\ell^{p}(\mathbb{Z}^{n})$ into $\ell^{q}(\mathbb{Z}^{n})$ for $1<p<\frac{n}{\alpha}$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$.

We adopt the following definition of discrete Hardy space on $\mathbb{Z}^{n}$ (see [3, Theorem 2.7]). Let $\Phi\in\mathcal{S}(\mathbb{R}^{n})$ with $\int_{\mathbb{R}^{n}}\Phi=1$. Then, for $t>0$, we put $\Phi_{t}^{d}(j)=t^{-n}\Phi(j/t)$ if $j\in\mathbb{Z}^{n}\setminus\{{\bf 0}\}$ and $\Phi_{t}^{d}({\bf 0})=0$. Now, for $0<p\leq\infty$, we define

[TABLE]

equipped with the quasi-norm given by

[TABLE]

In [3], Boza and Carro also gave an atomic characterization of $H^{p}(\mathbb{Z}^{n})$ for $0<p\leq 1$. Before establishing this result we recall the definition of $(p,\infty,N_{p})$-atom in $H^{p}(\mathbb{Z}^{n})$.

Definition 2.1**.**

Let $0<p\leq 1$ and $N_{p}:=\lfloor n(p^{-1}-1)\rfloor$. We say that a sequence $a=\{a(j)\}_{j\in\mathbb{Z}^{n}}$ is an $(p,\infty,N_{p})$-atom centered at a discrete cube $Q\subset\mathbb{Z}^{n}$ if the following three conditions hold:

(a1) $\mathop{\rm supp}a\subset Q$,

(a2.$p$) $\|a\|_{\ell^{\infty}(\mathbb{Z}^{n})}\leq(\#Q)^{-1/p}$,

(a3.$p$) $\displaystyle{\sum_{j\in Q}}j^{\beta}a(j)=0$ for every multi-index $\beta=(\beta_{1},...,\beta_{n})\in\mathbb{Z}_{+}^{n}$ with $[\beta]\leq N_{p}$.

The atomic decomposition mentioned for $H^{p}(\mathbb{Z}^{n})$ is established in the following theorem.

Theorem 2.2**.**

([3, Theorem 3.7]) Let $0<p\leq 1$, $N_{p}=\lfloor n(p^{-1}-1)\rfloor$ and $b\in H^{p}(\mathbb{Z}^{n})$. Then there exist a sequence of $(p,\infty,N_{p})$-atoms $\{a_{k}\}_{k=0}^{+\infty}$, a sequence of scalars $\{\lambda_{k}\}_{k=0}^{+\infty}$ and a positive constant $C$, which depends only on $p$ and $n$, with $\sum_{k=0}^{+\infty}|\lambda_{k}|^{p}\leq C\|b\|_{H^{p}(\mathbb{Z}^{n})}^{p}$ such that $b=\sum_{k=0}^{+\infty}\lambda_{k}a_{k}$, where the series converges in $H^{p}(\mathbb{Z}^{n})$.

Remark 2.3**.**

From [15, Corollary 4], it follows for every $0<p\leq 1$ that $H^{p}(\mathbb{Z}^{n})\subset\ell^{p}(\mathbb{Z}^{n})$ strictly. If $\Phi\in\mathcal{S}(\mathbb{R}^{n})$ is such that $\mathop{\rm supp}(\Phi)\subset B_{1/4}({\bf 0})$, then $\sup_{t>0}|(\Phi_{t}^{d}\ast b)(j)|\leq C(Mb)(j)$ for all $j\in\mathbb{Z}^{n}$, where $M$ is the discrete maximal operator of Hardy-Littlewood. Thus, by [16, Theorem 2.3], $H^{p}(\mathbb{Z}^{n})=\ell^{p}(\mathbb{Z}^{n})$ for $1<p\leq\infty$, with equivalent norms.

We conclude these preliminaries with the following two auxiliary results.

Lemma 2.4**.**

For $0<\alpha<n$, $m\in\mathbb{N}$ and $k^{0},k\in\mathbb{Z}^{n}$ fix, the estimate

[TABLE]

holds with $C$ independent of $m\in\mathbb{N}$ and $k^{0},k\in\mathbb{Z}^{n}$.

Proof..

Since $|j|_{\infty}\leq|j|$ for all $j\in\mathbb{Z}^{n}$, we have

[TABLE]

where

[TABLE]

Now, one has

[TABLE]

On the other hand, there exists a unique $s_{0}\in\mathbb{N}$ such that $2^{s_{0}-1}\leq m<2^{s_{0}}$. Then,

[TABLE]

[TABLE]

[TABLE]

Thus, (5) follows. ∎

Lemma 2.5**.**

Let $0<\alpha<n$ and $\Phi\in\mathcal{S}(\mathbb{R}^{n})$ such that $\mathop{\rm supp}(\Phi)\subset B_{1/4}({\bf 0})$. If $I_{\alpha}$ is the discrete Riesz potential given by (2), then there exists a positive constant $C=C_{\Phi,n,\alpha}$ such that

[TABLE]

holds for all $j\in\mathbb{Z}^{n}$ and all $p$-discrete atoms $a(\cdot)$ supported on $Q$.

Proof..

Let $a(\cdot)$ be a $p$-atom supported by the cube $Q_{k^{0},m}=\{i:|i-k^{0}|_{\infty}\leq m\}$ centered at $k^{0}$ and side length $2m+1$. Since by hypothesis $\mathop{\rm supp}(\Phi)\subset B_{1/4}({\bf 0})$ and $\Phi^{d}_{t}({\bf 0})=0$, we have

[TABLE]

for all $0<t<1$ and $j\in\mathbb{Z}^{n}$.

For $t\geq 1$ we have $1\leq\lfloor t\rfloor\leq t$, then the condition (a2.$p$) of the atom $a(\cdot)$ and the estimate (5) give

[TABLE]

for all $1\leq t<\infty$ and $j\in\mathbb{Z}^{n}$. Thus, (6) follows. ∎

3 Main result

In this section we establish the $H^{p}(\mathbb{Z}^{n})-H^{q}(\mathbb{Z}^{n})$ boundedness of the discrete Riesz potential $I_{\alpha}$ on the full range $0<p\leq 1$, where $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. For them, we first show that the operator $I_{\alpha}$ is bounded uniformly in the $H^{q}(\mathbb{Z}^{n})$-norm on all $(p,\infty,N_{p})$-atoms.

Proposition 3.1**.**

Let $0<\alpha<n$ and $\Phi\in\mathcal{S}(\mathbb{R}^{n})$ such that $\mathop{\rm supp}(\Phi)\subset B_{1/4}({\bf 0})$. If $I_{\alpha}$ is the discrete Riesz potential given by (2), then, for $0<p\leq 1$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$, there exists a positive constant $C$ such that

[TABLE]

for all discrete $(p,\infty,N_{p})$-atom $a=\{a(i)\}$.

Proof..

To prove (7), let $a(\cdot)$ be a discrete $(p,\infty,N_{p})$-atom centered at the cube $Q_{k^{0}}=\displaystyle{\prod_{1\leq l\leq n}}\left[\left[k^{0}_{l}-m,k^{0}_{l}+m\right]\right]$. We put $4\lfloor\sqrt{n}\rfloor Q_{k^{0}}=\displaystyle{\prod_{1\leq l\leq n}}\left[\left[k^{0}_{l}-4\lfloor\sqrt{n}\rfloor m,k^{0}_{l}+4\lfloor\sqrt{n}\rfloor m\right]\right]$. Then, we split

[TABLE]

To estimate the first sum in (8), we apply Lemma 2.5 above, which leads to

[TABLE]

with $C$ independent of $k^{0}$ and $m$.

To estimate the second sum in (8), we analize the cases $0<t<1$ and $t\geq 1$ separately. For $0<t<1$, by taking into account that $\mathop{\rm supp}(\Phi)\subset B_{1/4}({\bf 0})$ and $\Phi^{d}_{t}({\bf 0})=0$, we have

[TABLE]

for all $j\in\mathbb{Z}^{n}$. For $t\geq 1$ we consider two sub-cases,

Case I: $|j-k^{0}|_{\infty}>4\lfloor\sqrt{n}\rfloor m$ and $1\leq t\leq 2|j-k^{0}|_{\infty}$. We have

[TABLE]

Now, $|j-k|_{\infty}\leq t/4\leq|j-k^{0}|_{\infty}/2$ and $|j-k^{0}|_{\infty}>4\lfloor\sqrt{n}\rfloor m$ imply that $|k-k^{0}|_{\infty}\geq|j-k^{0}|_{\infty}/2\geq 2|i-k^{0}|_{\infty}$. We put $N-1=\lfloor n(p^{-1}-1)\rfloor$. In view of the moment condition (a3.$p$) of $a(\cdot)$ we have, for $j\in\mathbb{Z}^{n}\setminus 4\lfloor\sqrt{n}\rfloor Q_{k^{0}}$, that

[TABLE]

where $q_{N}(\,\cdot\,,k)$ is the degree $N-1$ Taylor polynomial of the function $x\rightarrow|k-x|^{\alpha-n}$ expanded around $k^{0}$. By the standard estimate of the remainder term in the Taylor expansion there exists $\xi$ between $i$ and $k^{0}$ such that

[TABLE]

Since $|k-k^{0}|_{\infty}\geq 2|i-k^{0}|_{\infty}$ for any $i\in Q_{k^{0}}$, we get $|k-\xi|\geq|k-\xi|_{\infty}\geq|k-k^{0}|_{\infty}/2$. So,

[TABLE]

As also $|k-k^{0}|_{\infty}\geq|j-k^{0}|_{\infty}/2$ for any $j\notin 4\lfloor\sqrt{n}\rfloor Q_{k^{0}}$, it follows

[TABLE]

with $C$ independent of $i$, $j$, $k$ and $k^{0}$. This inequality, the condition (a2.$p$) of the atom $a(\cdot)$ and the fact that $\{k:|k|_{\infty}\leq t/4\}\subset\{k:|k|_{\infty}\leq\lfloor t\rfloor\}$, where $1\leq\lfloor t\rfloor\leq t$, allow us to conclude that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $Q_{j,2|j-k^{0}|_{\infty}}$ is the discrete cube centered at $j$ and side length $4|j-k^{0}|_{\infty}+1$. Thus

[TABLE]

for all $j\notin 4\lfloor\sqrt{n}\rfloor Q_{k^{0}}$ and $1\leq t\leq 2|j-k^{0}|_{\infty}$.

Case II: $|j-k^{0}|_{\infty}>4\lfloor\sqrt{n}\rfloor m$ and $t>2|j-k^{0}|_{\infty}$. We have that

[TABLE]

[TABLE]

[TABLE]

where $k=l-i$, in this case $|j-k-i|_{\infty}\leq t/4$ implies that the inner sum is vanishing on $|k|_{\infty}>2t$, so

[TABLE]

As before, we put $N-1=\lfloor n(p^{-1}-1)\rfloor$. Then, for $j\in\mathbb{Z}^{n}\setminus 4\lfloor\sqrt{n}\rfloor Q_{k^{0}}$, the moment condition (a3.$p$) of $a(\cdot)$ gives

[TABLE]

where $\widetilde{q}_{N}(\,\cdot\,,j,k,t)$ is the degree $N-1$ Taylor polynomial of the function $x\rightarrow\Phi_{t}(j-k-x)$ expanded around $k^{0}$. By the standard estimate of the remainder term in the Taylor expansion we have that

[TABLE]

where $C$ does not depend on $i$, $j$, $k$ and $t$. This inequality and the condition (a2.$p$) of the atom $a(\cdot)$ lead to

[TABLE]

Now, the estimate (5) and $t>2|j-k^{0}|_{\infty}$ imply

[TABLE]

[TABLE]

for all $j\notin 4\lfloor\sqrt{n}\rfloor Q_{k^{0}}$ and $t>2|j-k^{0}|_{\infty}$.

Thus, (10) and the estimates obtained in the cases $I$ and $II$ above give

[TABLE]

Since $N-1=\lfloor n(\frac{1}{p}-1)\rfloor$, we have $q\frac{n+N}{n}>1$. We write $\widetilde{q}=q\frac{n+N}{n}$ and let $\frac{1}{\widetilde{p}}=\frac{1}{\widetilde{q}}+\frac{\alpha}{n+N}$, so $\frac{\widetilde{p}}{\widetilde{q}}=\frac{p}{q}$. From Proposition 2.4 in [16], we obtain

[TABLE]

Now, (11) and (12) give

[TABLE]

Finally, (9) and (13) lead to (7). Thus the proof is finished. ∎

Theorem 3.2**.**

For $0<\alpha<n$, let $I_{\alpha}$ be the discrete Riesz potential given by (2). If $0<p\leq 1$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$, then

[TABLE]

for all $b\in H^{p}(\mathbb{Z}^{n})$, where $C$ does not depend on $b$.

Proof..

We fix $p_{0}$ such that $1<p_{0}<\frac{n}{\alpha}$ and $\Phi\in\mathcal{S}(\mathbb{R}^{n})$ with $\mathop{\rm supp}(\Phi)\subset B_{1/4}({\bf 0})$ and $\int\Phi=1$. By Theorem 2.2, given $b\in H^{p}(\mathbb{Z}^{n})$, with $0<p\leq 1$, we can write $b=\sum_{k}\lambda_{k}a_{k}$ where the $a_{k}$’s are $(p,\infty,N_{p})$ atoms, the scalars $\lambda_{k}$ satisfies $\sum_{k}|\lambda_{k}|^{p}\leq C\|b\|_{H^{p}(\mathbb{Z}^{n})}^{p}$ and the series converges in $H^{p}(\mathbb{Z}^{n})$ and so in $\ell^{p_{0}}(\mathbb{Z}^{n})$ since $H^{p}(\mathbb{Z}^{n})\subset\ell^{p}(\mathbb{Z}^{n})\subset\ell^{p_{0}}(\mathbb{Z}^{n})$ embed continuously. Now for $\frac{1}{q_{0}}=\frac{1}{p_{0}}-\frac{\alpha}{n}$, by [16, Theorem 3.1], the discrete Riesz potential $I_{\alpha}$ is a bounded operator $\ell^{p_{0}}(\mathbb{Z}^{n})\to\ell^{q_{0}}(\mathbb{Z}^{n})$ and since $H^{q_{0}}(\mathbb{Z}^{n})=\ell^{q_{0}}(\mathbb{Z}^{n})$ with equivalent norms (see Remark 2.3), it follows that

[TABLE]

Then for $1\leq q\leq\frac{n}{n-\alpha}$, by (14) and Minkowski’s integral inequality on $\sigma$-finite measure spaces, we have

[TABLE]

Now for $0<q<1$, from (14), it is easy to check that

[TABLE]

Finally, since $0<p\leq 1$ and $0<p<q\leq\frac{n}{n-\alpha}$, the estimate (15) or (16) according to the case, Proposition 3.1, and the fact that $\sum_{k}|\lambda_{k}|^{p}\leq C\|b\|_{H^{p}(\mathbb{Z}^{n})}^{p}$ allow us to conclude that

[TABLE]

Since $b$ is an arbitrary element of $H^{p}(\mathbb{Z}^{n})$, the theorem follows. ∎

Remark 3.3**.**

The argument used to prove Theorem 3.2 is similar to the one given in the proof of [7, Theorem 2.4.5, see p. 140-141].

Acknowledgements. I am very grateful to the referee for the careful reading of the paper and for the useful comments and suggestions which helped me to improve the original manuscript.