Discrete Hardy’s inequalities with 



0
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p
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1

Kwok-Pun Ho

doi:10.1016/j.jksus.2017.03.007

Discrete Hardy’s inequalities with $0 < p ⩽ 1$

Kwok-Pun Ho

Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, Hong Kong, China

Received: 2017-2-23, Accepted: 2017-3-19, Published: 2017-10-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

We generalize the famous discrete Hardy inequality to $0 < p ⩽ 1$ . We obtain this generalization by using the atomic decompositions of the discrete Hardy spaces.

1 Introduction

The main theme of this paper is the following discrete Hardy inequalities.

Let $0 < p ⩽ 1$ . There exists a constant $C > 0$ such that for any sequence $a = {\{a (n)\}}_{n \in Z}$ with $a (- n) = 0, n \in N \cup {0}$ , we have

(1)

{(\sum_{m \in N} {(\frac{1}{m} \sum_{j = 1}^{m} a (j))}^{p})}^{\frac{1}{p}} ⩽ C {(\sum_{m \in N} | a (m) |^{p} + \sum_{\min Z} {|\sum_{j \neq m} \frac{a (j)}{m - j}|}^{p})}^{\frac{1}{p}} .

The family of Hardy inequalities, which consists of the discrete form and the integral form, is one of the most important inequalities in analysis. For the history of the Hardy inequalities, the reader is referred to Kufner et al. (2006, 2007). For the applications and further developments of this famous inequality, the reader may consult (Kufner et al., 2007; Kufner and Persson, 2003; Opic and Kufner, 1990).

Recently, the integral form Hardy inequality had been extended to Hardy spaces on $R$ . In Ho (2016), the Hardy inequalities in Hardy spaces are established by using the atomic decompositions of Hardy spaces. This method is also used in Ho (2016, 2017a,b) to study the Hardy inequalities on Hardy-Morrey spaces with variable exponents and weak Hardy-Morrey spaces.

In this paper, we use the idea from Ho (2016) to obtain the above generalization (1) of Hardy’s inequality to $0 < p ⩽ 1$ .

To apply the method in Ho (2016), we need to consider the discrete analogue of the classical Hardy spaces on $R$ . The discrete Hardy spaces had been introduced in Boza and Carro (1998, 2002). Moreover, the atomic decompositions of the discrete Hardy spaces were also obtained in Boza and Carro (1998, 2002). These atomic decomposition are precisely what we need to establish the discrete Hardy inequality with $0 < p ⩽ 1$ .

Thus, on one hand, we extend the classical discrete Hardy inequalities to $0 < p ⩽ 1$ , on the other hand, the main result of this paper gives an application of the atomic decompositions established in Boza and Carro (1998, 2002).

This paper is organized as follows. In Section 2, we recall the definition of the discrete Hardy spaces. The atomic decompositions for the discrete Hardy spaces are also presented in this section. The discrete Hardy inequalities with $0 < p ⩽ 1$ are established in Section 3.

2 Discrete Hardy spaces

In this section, we first recall the definition of the discrete Hardy spaces by discrete Hilbert transform on $Z$ .

For any sequence $a = {\{a (n)\}}_{n \in Z}$ , the discrete Hilbert transform of a is defined by $(H^{d} a) (m) = \sum_{n \neq m} \frac{a (n)}{m - n} .$

For any $B \subset Z$ , let $| B |$ denote the cardinality of B.

We use the definition of discrete Hardy spaces from (Boza and Carro, 1998, Definition 3.1).

Definition 2.1

Let $0 < p ⩽ 1$ . The discrete Hardy spaces $H^{p} (Z)$ consists of those sequence $a = {\{a (n)\}}_{n \in Z}$ satisfying $‖ a ‖_{H^{p} (Z)} = ‖ a ‖_{l^{p} (Z)} + ‖ H^{d} a ‖_{l^{p} (Z)} < \infty .$

In view of the above definition, we see the reason why the second summation on the left hand side of (1) is taking over $Z$ .

We now present the atomic characterization of $H^{p} (Z)$ , we begin with the definition of $H^{p} (Z)$ -atom (Boza and Carro, 1998, Definition 3.9).

Definition 2.2

Let $0 < p ⩽ 1$ . A sequence $a = {\{a (k)\}}_{k \in Z}$ is an $H^{p} (Z)$ -atom if it satisfies

supp a is contained in a ball in $Z$ of cardinality $2 n + 1, n ⩾ 1$ .
$‖ a ‖_{l^{\infty} (Z)} ⩽ {(2 n + 1)}^{- 1 / p}$ .
$\sum_{n \in Z} n^{α} a (n) = 0$ for every $α \in N$ with $α ⩽ \frac{1}{p} - 1$ .

To present the atomic decomposition of $H^{p} (Z)$ , we recall the atomic version of $H^{p} (Z)$ from (Boza and Carro, 1998, p.43).

Definition 2.3

Let $0 < p ⩽ 1$ . The atomic discrete Hardy space $H_{at}^{p} (Z)$ consists of those sequence $a = {\{a (n)\}}_{n \in Z}$ such that $a = \sum_{j \in N} λ_{j} a_{j}$ where $a_{j}$ are $H^{p} (Z)$ -atoms and $‖ a ‖_{H_{at}^{p} (Z)} = \inf \{{(\sum_{j \in N} | λ_{j} |^{p})}^{\frac{1}{p}}\}$ where the infimum is taken over all possible representations of a in terms of $H^{p} (Z)$ -atoms.

The following result gives the atomic decomposition of $H^{p} (Z)$ .

Theorem 1

Let $0 < p ⩽ 1$ . Then, there exist constants $B, C > 0$ such that for any sequence $a = {\{a (n)\}}_{n \in Z}$ , we have $B ‖ a ‖_{H^{p} (Z)} ⩽ ‖ a ‖_{H_{at}^{p} (Z)} ⩽ C ‖ a ‖_{H^{p} (Z)} .$

The reader is referred to (Boza and Carro, 1998, Theorems 3.10 and 3.14) for the proof of the above theorem.

We can also characterize discrete Hardy spaces by using Poisson integral and area functions, see (Boza and Carro, 1998, Theorems 3.4 and 3.8). The reader is also referred to Boza (2012) and Kanjin and Satake (2000), Komori (2002) for the factorization theorem and the molecular characterizations of discrete Hardy spaces, respectively.

The reader is also referred to Herz (1973), Ho (2009, 2012), Jiao et al. (2017), Weisz (1994) for some other applications of the atomic decompositions such as the characterizations of BMO and martingale BMO.

3 Hardy’s inequalities

We establish the main result of this paper in this section. We first introduce the Hardy operators in order to simplify our presentation. For any $α \in N \cup {0}$ and $μ > 0$ , define $(T_{α, μ} a) (m) = \frac{1}{m^{α - μ + 1}} \sum_{j = 1}^{m} j^{α} a (j), m \in N .$

Notice that when $α = μ = 0$ , we have

(2)

(T_{0, 0} a) (m) = \frac{1}{m} \sum_{j = 1}^{m} a (j) .

It is precisely the Hardy-Littlewood average for the sequence $a = {\{a (k)\}}_{k = 1}^{\infty}$ .

We are now ready to present the main result of this paper, the mapping properties of $T_{α, μ}$ on discrete Hardy spaces $H^{p} (Z)$ .

Theorem 2

Let $0 < p ⩽ 1$ and $0 ⩽ μ < 1$ . Suppose that $α \in N \cup {0}$ satisfies $α ⩽ \frac{1}{p} - 1$ . If

(3)

\frac{1}{p} = \frac{1}{r} + μ,

then there exists a constant

C > 0

such that for any sequence

a \in H^{p} (Z)

with support contained in

N

, we have

‖ T_{α, μ} a ‖_{l^{r} (N)} ⩽ C ‖ a ‖_{H^{p} (Z)} .

When $0 < p ⩽ 1$ and $α = μ = 0$ , we have $p = r$ and the above theorem yields $‖ T_{0, 0} a ‖_{l^{p} (N)} ⩽ C ‖ a ‖_{H^{p} (Z)} .$

In view of (2), it establishes the discrete Hardy inequality (1).

We need several supporting results to obtain the proof of Theorem 2. We start with the mapping property of $T_{α, μ}$ on $H^{p} (Z)$ -atoms.

Lemma 3

Let $0 < p ⩽ 1$ , $0 ⩽ μ < 1$ and $α \in N \cup {0}$ with $α ⩽ \frac{1}{p} - 1$ . Suppose that $\frac{1}{q} - \frac{1}{r} < μ ⩽ \frac{1}{q}$ for some $q > 1$ . If $a = {\{a (n)\}}_{n \in Z}$ satisfies

$supp a$ is contained in a ball B in $N ⧹ {0}$ of cardinality $2 n + 1, n ⩾ 1$ ,
$‖ a ‖_{l^{\infty} (N)} ⩽ {(2 n + 1)}^{- 1 / p}$ ,
$\sum_{n = 0}^{\infty} n^{α} a (n) = 0$ ,

then, we have

‖ T_{α, μ} a ‖_{l^{r} (N)} ⩽ C | B |^{μ + \frac{1}{r} - \frac{1}{p}} .

Proof

Let $B = {i \in N : M ⩽ i ⩽ N}, M, N \in N$ . We have $| B | = N - M + 1$ . For any $i < M$ , we have $(T_{α, μ} a) (i) = \frac{1}{i^{α - μ + 1}} \sum_{j = 1}^{i} j^{α} a (j) = 0 .$

Similarly, for any $i > N$ , Items (1) and (3) assure that $(T_{α, μ} a) (i) = \frac{1}{i^{α - μ + 1}} \sum_{j = 1}^{i} j^{α} a (j) = \frac{1}{i^{α - μ + 1}} \sum_{j \in N} j^{α} a (j) = 0 .$

Therefore, we find that $supp (T_{α, μ} a) \subseteq B$ .

The Hölder inequality yields $| \sum_{j = 1}^{m} j^{α} a (j) | ⩽ {(\sum_{j = 1}^{m} | a (j) |^{q})}^{1 / q} {(\sum_{j = 1}^{m} j^{α q'})}^{1 / q'} ⩽ C ‖ a ‖_{l^{q}} m^{α + \frac{1}{q'}}$ for some $C > 0$ . Therefore, $| (T_{α, μ} a) (m) | ⩽ C \frac{1}{m^{α - μ + 1}} ‖ a ‖_{l^{q}} m^{α + \frac{1}{q'}} = {Cm}^{μ - \frac{1}{q}} ‖ a ‖_{l^{q}} .$

Furthermore, as $supp (T_{α, μ} a) \subseteq B$ and $μ ⩽ \frac{1}{q}$ , we obtain $\begin{matrix} ‖ T_{α, μ} a ‖_{l^{r}}^{r} ⩽ & C ‖ a ‖_{l^{q}}^{r} \sum_{m = M}^{N} m^{r μ - \frac{r}{q}} ⩽ C \int_{M}^{N + 1} x^{r μ - \frac{r}{q}} dx \\ ⩽ & C ‖ a ‖_{l^{q}}^{r} (N^{r μ - \frac{r}{q} + 1} - M^{r μ - \frac{r}{q} + 1}) . \end{matrix}$

Since $\frac{1}{q} - \frac{1}{r} < μ ⩽ \frac{1}{q}$ , we find that $0 < r μ - \frac{r}{q} + 1 ⩽ 1$ . Consequently, $(N^{r μ - \frac{r}{q} + 1} - M^{r μ - \frac{r}{q} + 1}) ⩽ {(N - M)}^{r μ - \frac{r}{q} + 1} ⩽ | B |^{r μ - \frac{r}{q} + 1} .$

Hence, $‖ T_{α, μ} a ‖_{l^{r}} ⩽ C ‖ a ‖_{l^{q}} | B |^{μ - \frac{r}{q} + \frac{1}{r}} ⩽ C ‖ a ‖_{l^{\infty}} | B |^{\frac{1}{q}} | B |^{μ - \frac{1}{q} + \frac{1}{r}} ⩽ C | B |^{μ + \frac{1}{r} - \frac{1}{p}}$ for some $C > 0$ . □

Lemma 3 applies to those $H^{p} (Z)$ -atom $a = {\{a (n)\}}_{n \in Z}$ with support in $N ⧹ {0}$ . On the other hand, for any $a \in H^{p} (Z)$ with support in $N ⧹ {0}, a$ does not necessarily possess an atomic decomposition with all the $H^{p} (Z)$ -atoms supported in $N ⧹ {0}$ . To tackle this difficulty, we consider the odd and the even extensions of $a \in H^{p} (Z)$ .

For any sequence $a = {\{a (n)\}}_{n \in Z}$ , the even part of a and the odd part of a, denoted by $a_{e}$ and $a_{o}$ , are defined by $a_{e} = {\{\frac{a (n) + a (- n)}{2}\}}_{n \in Z} and a_{o} = {\{\frac{a (n) - a (- n)}{2}\}}_{n \in Z},$ respectively.

As the even part is a “reflection” about $k = 0$ , the term $a (0)$ will be counted twice in the even part. Thus, for the case when $α$ equals to zero, we need some further modifications. The details of these modifications and the uses of the even and odd parts are given in the subsequent results.

Lemma 4

Let $0 < p ⩽ 1$ and $α \in N \cup {0}$ with $α ⩽ \frac{1}{p} - 1$ . If $a \in H^{p} (Z)$ with $supp a \subset N ⧹ {0}$ , then there exist a sequence of scalars ${\{λ_{j}\}}_{j \in N}$ and sequences ${\{a_{j}\}}_{j \in N}$ satisfying

supp $a_{j}$ is contained in a ball in $N ⧹ {0}$ of cardinality $2 n + 1, n ⩾ 1$ ,
$‖ a_{j} ‖_{l^{\infty} (Z)} ⩽ 2 {(2 n + 1)}^{- 1 / p}$ ,
$\sum_{n = 0}^{\infty} n^{α} a_{j} (n) = 0$ ,

such that

a = \sum_{j \in N} λ_{j} a_{j}

and

{(\sum_{j \in N} | λ_{j} |^{p})}^{1 / p} ⩽ C ‖ a ‖_{H^{p} (Z)} .

Proof

In view of Theorem 1, we have a sequence of scalars ${\{λ_{j}\}}_{j \in N}$ and $H^{p} (Z)$ -atoms ${\{a_{j}\}}_{j \in N}$ such that $a = \sum_{j \in N} λ_{j} a_{j}$ and ${\sum_{j \in N} | λ_{j} |^{p})}^{1 / p} ⩽ C ‖ a ‖_{H^{p} (Z)}$ .

We split the proof into three cases, $α$ is a positive even integer, $α$ is a positive odd integer and $α$ equals zero.

$α$ is a positive even integer. We consider the even part of a, $a_{e} (k) = \sum_{j \in N} λ_{j} (\frac{a_{j} (k) + a_{j} (- k)}{2}) .$

Since $supp a \subset N ⧹ {0}$ , we have $a = 2 a_{e} χ_{N ⧹ {0}}$ . That is, $a (k) = \sum_{j \in N} λ_{j} (a_{j} (k) + a_{j} (- k)) χ_{N ⧹ {0}} (k) .$

Write $A_{j} = {\{A_{j} (k)\}}_{k \in Z}$ where $A_{j} (k) = (a_{j} (k) + a_{j} (- k)) χ_{N ⧹ {0}} (k)$ . We are going to show that $A_{j}$ fulfills Item (1)–(3).

As $α$ is even, we find that $\sum_{k \in Z} k^{α} a_{j} (k) = \sum_{k \in Z} k^{α} a_{j} (- k) = 0 .$

If $supp a_{j} \subset N ⧹ {0}, A_{j} = a_{j}$ satisfies Item (1)–(3). If $supp a_{j} \subset Z ⧹ (N \cup {0})$ , then $A_{j} (\cdot) = a_{j} (- \cdot)$ also satisfies Item (1)–(3).

If $0 \in supp a_{j}$ , we find that $\sum_{n = 0}^{\infty} n^{α} A_{j} (n) = \sum_{n = 0}^{\infty} n^{α} (a_{j} (n) + a_{j} (- n)) = \sum_{n \in Z} n^{α} a_{j} (n) = 0$ because $α$ is a positive even integer.

We have $‖ A_{j} ‖_{l^{\infty} (N)} ⩽ 2 ‖ a_{j} ‖_{l^{\infty} (N)}$ . Moreover, $supp A_{j} \subseteq (supp a_{j} \cup supp a_{j} (- \cdot)) \cap (N ⧹ {0})$ . Hence, $| supp A_{j} | ⩽ | supp a_{j} |$ .

Consequently, $A_{j}$ fulfills Item (1)–(3).
$α$ is a positive odd integer. We consider the odd part of a, $a_{o} (k) = \sum_{j \in N} λ_{j} (\frac{a_{j} (k) - a_{j} (- k)}{2}) .$

Since $supp a \subset N ⧹ {0}$ , we have $a = 2 a_{o} χ_{N ⧹ {0}}$ . That is, $a (k) = \sum_{j \in N} λ_{j} (a_{j} (k) - a_{j} (- k)) χ_{N ⧹ {0}} (k) .$

Write $A_{j} = {\{A_{j} (k)\}}_{k \in Z}$ where $A_{j} (k) = (a_{j} (k) - a_{j} (- k)) χ_{N ⧹ {0}} (k)$ .

As $α$ is odd, we find that $\sum_{k \in Z} k^{α} a_{j} (k) = - \sum_{k \in Z} k^{α} a_{j} (- k) = 0 .$

If $supp a_{j} \subset N ⧹ {0}, A_{j} = a_{j}$ satisfies Item (1)–(3). If $supp a_{j} \subset Z ⧹ (N \cup {0})$ , then $A_{j} (\cdot) = a_{j} (- \cdot)$ also satisfies Item (1)–(3).

If $0 \in supp a_{j}$ , we find that $\sum_{n = 0}^{\infty} n^{α} A_{j} (n) = \sum_{n = 0}^{\infty} n^{α} (a_{j} (n) - a_{j} (- n)) = \sum_{n \in Z} n^{α} a_{j} (n) = 0$ because $α$ is a positive odd integer.

Therefore, $A_{j}$ fulfills Item (1)–(3).
$α$ equals to zero. In this case, we also consider the even part of a, $a_{e} (k) = \sum_{j \in N} λ_{j} (\frac{a_{j} (k) + a_{j} (- k)}{2}) .$ Write $b_{j} (k) = \{\begin{matrix} a_{j} (k), & k \neq 0, \\ \frac{a_{j} (0)}{2}, & k = 0 . \end{matrix}$ When $k \neq 0$ , we obviously have $a_{e} (k) = \sum_{j \in N} λ_{j} (\frac{a_{j} (k) + a_{j} (- k)}{2}) = \sum_{j \in N} λ_{j} (\frac{b_{j} (k) + b_{j} (- k)}{2}) .$ Since $supp a \subset N ⧹ {0}$ , we have $a (0) = \sum_{j \in N} λ_{j} a_{j} (0) = 0 .$ Therefore, for $k = 0$ , we have $a_{e} (0) = a (0) = 0 = \sum_{j \in N} λ_{j} a_{j} (0) = \sum_{j \in N} λ_{j} (\frac{b_{j} (0) + b_{j} (0)}{2}) .$ As $supp a \subset N ⧹ {0}$ , we have $a = 2 a_{e} χ_{N ⧹ {0}}$ . That is, $a (k) = \sum_{j \in N} λ_{j} (b_{j} (k) + b_{j} (- k)) χ_{N ⧹ {0}} (k) .$

Write $B_{j} = {\{B_{j} (k)\}}_{k \in Z}$ where $B_{j} (k) = (b_{j} (k) + b_{j} (- k)) χ_{N ⧹ {0}} (k)$ . It remains to show that $B_{j}$ satisfies Item (1)–(3).

When $supp a_{j} \subset N ⧹ {0}, B_{j} = a_{j}$ satisfies Item (1)–(3). When $supp a_{j} \subset Z ⧹ (N \cup {0})$ , then $B_{j} (\cdot) = a_{j} (- \cdot)$ also satisfies Item (1)–(3).

If $0 \in supp a_{j}$ , we find that $\begin{matrix} \sum_{n = 0}^{\infty} B_{j} (n) = & \sum_{n = 0}^{\infty} (b_{j} (n) + b_{j} (- n)) = (\sum_{n = 1}^{\infty} a_{j} (n) + \sum_{n = - \infty}^{- 1} a_{j} (n)) + 2 b_{j} (0) \\ = & \sum_{n \in Z} a_{j} (n) = 0 . \end{matrix}$

Thus,

B_{j}

fulfills Item (1)–(3). □

Proof of Theorem 2.

In view of Lemma 4, for any $a \in H^{p} (Z)$ with $supp a \subset N ⧹ {0}$ , there exist a sequence of scalars ${\{λ_{j}\}}_{j \in N}$ and ${\{a_{j}\}}_{j \in N}$ satisfying.

supp $a_{j}$ is contained in a ball in $N ⧹ {0}$ of cardinality $2 n + 1, n ⩾ 1$ ,
$‖ a_{j} ‖_{l^{\infty} (Z)} ⩽ 2 {(2 n + 1)}^{- 1 / p}$ ,
$\sum_{n = 0}^{\infty} n^{α} a_{j} (n) = 0$ ,

such that

a = \sum_{j \in N} λ_{j} a_{j}

and

(4)

{(\sum_{j \in N} | λ_{j} |^{p})}^{1 / p} ⩽ C ‖ a ‖_{H^{p} (Z)} .

When $0 < r ⩽ 1$ , Lemma 3 and (3) assure that $‖ T_{α, μ} a ‖_{l^{r} (N)}^{r} ⩽ \sum_{j \in N} | λ_{j} |^{r} ‖ T_{α, μ} a_{j} ‖_{l^{r} (N)}^{r} ⩽ C \sum_{j \in N} | λ_{j} |^{r}$ for some $C > 0$ . Furthermore, (3) also guarantees that $r > p$ . Thus, (4) yields $‖ T_{α, μ} a ‖_{l^{r} (N)} ⩽ C {(\sum_{j \in N} | λ_{j} |^{r})}^{1 / r} ⩽ C {(\sum_{j \in N} | λ_{j} |^{p})}^{1 / p} = C ‖ a ‖_{H^{p} (Z)}$ for some $C > 0$ .

When $r > 1$ , we have $\begin{matrix} ‖ T_{α, μ} a ‖_{l^{r} (N)} ⩽ & \sum_{j \in N} | λ_{j} | ‖ T_{α, μ} a_{j} ‖_{l^{r} (N)} ⩽ C \sum_{j \in N} | λ_{j} | \\ ⩽ & C {(\sum_{j \in N} | λ_{j} |^{p})}^{1 / p} = C ‖ a ‖_{H^{p} (Z)} \end{matrix}$ because $0 < p ⩽ 1$ . □

Acknowledgment

The author would like to thank the referees for their helpful suggestions.

References

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Introduction

Discrete Hardy spaces

Hardy’s inequalities

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[1] Boza S., Carro M., . Discrete Hardy spaces. Stud. Math.. 1998;129:31-50.
[Google Scholar]

[2] Boza S., Carro M., . Hardy spaces on $Z^{n}$ . Proc. R. Soc. Edinburgh. 2002;132A:25-43.
[Google Scholar]

[3] Boza S., . Factorization of sequences in discrete Hardy spaces. Stud. Math.. 2012;209:53-69.
[Google Scholar]

[4] Herz, C., 1973/74. $H_{p}$ -spaces of martingales, $0 < p ⩽ 1$ . Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 28, 189–205.

[5] Ho K.-P., . Characterization of BMO in terms of Rearrangement-invariant Banach function spaces. Expo. Math.. 2009;27:363-372.
[Google Scholar]

[6] Ho K.-P., . Atomic decomposition of Hardy spaces and characterization of BMO via Banach function spaces. Anal. Math.. 2012;38:173-185.
[Google Scholar]

[7] Ho K.-P., . Hardy’s inequality on Hardy spaces. Proc. Jpn. Acad. Ser. A. 2016;92:125-130.
[Google Scholar]

[8] Ho, K.-P., 2016. Hardy’s inequality on Hardy-Morrey spaces. Georgian Math. J. (accepted).

[9] Ho K.-P., . Hardy’s inequality on Hardy-Morrey spaces with variable exponents. Mediter. J. Math. 2017
[CrossRef] [Google Scholar]

[10] Ho K.-P., . Atomic decompositions and Hardy’s inequality on weak Hardy-Morrey spaces. Sci. China Math.. 2017;60:449-468.
[Google Scholar]

[11] Jiao J., . The predual and John-Nirenberg inequalities on generalized BMO martingale spaces. Trans. Am. Math. Soc.. 2017;369:537-553.
[Google Scholar]

[12] Kanjin Y., Satake M., . Inequalities for discrete Hardy spaces. Acta Math. Hungar.. 2000;89:301-313.
[Google Scholar]

[13] Kufner A., . The Prehistory of the Hardy Inequality. Am. Math. Monthly. 2006;113:715-732.
[Google Scholar]

[14] Kufner A., . The Hardy Inequality, About its History and Some Related Results. Pilsen: Vydavatelsky Publishing House; 2007.

[15] Kufner A., Persson L.-E., . Weighted inequalities of Hardy type. River Edge, NJ: World Scientific Publishing Co., Inc.; 2003.

[16] Komori Y., . The atomic decomposition of molecule on discrete Hardy spaces. Acta Math. Hungar.. 2002;95:21-27.
[Google Scholar]

[17] Opic, B., Kufner, A., 1990. Hardy-type inequalities, Pitman Reserach Notes in Math. Series 219, Longman Sci. and Tech, Harlow.

[18] Weisz F., . Martingale Hardy spaces and their applications in Fourier analysis.Lecture Notes in Mathematics. Vol vol. 1568. Berlin: Springer-Verlag; 1994.

Discrete Hardy’s inequalities with 0<p⩽1