Sharp weak bounds for discrete Hardy operator on discrete central Morrey spaces

1.   Introduction

Let $f$ be a locally integrable function on $(0, \infty)$. The classical Hardy operator $H$ is defined by

where $x\in(0, \infty)$.

The well-known Hardy inequality yields that for $1 < p < \infty$ and any locally integrable function $f$ on $(0, \infty)$ such that

there holds

where the constant $\frac{p}{p-1}$ is optimal, see ^[1] for the details.

In 1976, the Hardy operator was extended to higher dimension by Faris ^[2]. Later, Christ and Grafakos ^[3] obtained a refined definition of the $n$-dimensional Hardy operator

where $f$ is a locally measurable function on ${\Bbb R}^n$ and $\Omega_n = \frac{\pi^{n/2}}{\Gamma(1+n/2)}$ is the volume of the unit ball in ${\Bbb R}^n$. Moreover, Christ and Grafakos proved that the operator norm of $\mathcal{H}$ on $L^p({\Bbb R}^n)$ $(1 < p < \infty)$ is also $\frac{p}{p-1}$. Since then, the sharp estimates for Hardy-type operators have been built on different function spaces. For example, Lu et al. ^[4], Wang et al. ^[5], Wei and Yan ^[6] established the sharp estimates for product Hardy-type operators on product spaces. Moreover, Fu et al. ^[7] derived the sharp constants for linear and multilinear Hardy operators on Morrey-type spaces. Other than the Euclid space, the sharp constants for Hardy-type operators were also established on $p$-adic field ^{[8,9,10,11,12,13]}, Heisenberg group ^[14,15,16], and $q$-calculus ^[17,18,19].

Note that, in ^[20], Zhao et al. obtained the sharp weak $(p, p)$ bound for $\mathcal{H}$. We give the result in details for the reader's convenience.

Theorem 1.1. For $1\leq p\leq\infty$ and $f\in L^p({\Bbb R}^n)$, we have

where the bound $1$ is best possible.

In the above, for $0 < p < \infty$, the Lebesgue space $L^p({\Bbb R}^n)$ contains all locally integrable functions $f$ on ${\Bbb R}^n$ such that

and the weak Lebesgue space $L^{p, \infty}({\Bbb R}^n)$ consists of all locally integrable functions $f$ on ${\Bbb R}^n$ such that

where $|E|$ denotes the Lebesgue measure of the measurable set $E\subseteq {\Bbb R}^n$. In addition, the space $L^{\infty}({\Bbb R}^n)$ is the collection of all measurable functions $f$ on ${\Bbb R}^n$ such that $\|f\|_{L^\infty} = {\rm ess}\sup_{x\in{\Bbb R}^n}|f(x)| < \infty$. We also denote by $L^{\infty, \infty}({\Bbb R}^n) = L^{\infty}({\Bbb R}^n)$.

In fact, when $n = 1$, Theorem 1.1 implies that the sharp constant for $H$ from $L^{p}(0, \infty)$ to $L^{p, \infty}(0, \infty)$ also equals to 1. Here the space $L^{p, \infty}(0, \infty)$ is defined by (1.1) except that we use $(0, \infty)$ instead of ${\Bbb R}^n$ there.

Recently, much attention has been paid to the sharp weak bounds for Hardy-type operators. For instance, Gao and Zhao ^[21], Gao et al. ^[22], Yu and Li ^[23] gave the sharp weak estimates for Hardy-type operators on (power-weighted) Lebesgue spaces. Moreover, Gao et al. ^[22] calculated the sharp constants for Hardy operator on central Morrey spaces. We also refer the readers to ^[24,25] for the sharp weak bounds for $p$-adic Hardy-type operators on $p$-adic Lebesgue spaces.

On the other hand, the discrete version of Hardy inequality is also of great importance. The discrete Hardy inequality (see, for instance, ^[1,26]) shows that if $p > 1$ and $\{a_k\}_1^\infty$ is a sequence of nonnegative real numbers, then

Moreover, the constant in (1.2) is the best possible one.

Similar to the Lebesgue spaces in Euclidean space, the discrete Lebesgue space $l^p(\mathbb{N})$ (here and hereafter, $\mathbb{N}$ is the collection of all positive integers) consists of all sequences $\vec{a} = \{a_k\}_1^\infty$ of real numbers such that

if $0 < p < \infty$, and for $p = \infty$, we denote by $l^{\infty}(\mathbb{N})$ the set of all sequences $\vec{a} = \{a_k\}_1^\infty$ of real numbers such that

We also define the discrete version of Hardy operator ${\bf h}$ as

for all senquences $\vec{a} = \{a_k\}_1^\infty$ of real numbers.

Using the notations in (1.3) and (1.4), the discrete Hardy inequality (1.2) can be rewritten as

for all sequences $\vec{a} = \{a_k\}_1^\infty$ such that $\vec{a}\in l^p(\mathbb{N})$. Moreover, the constant in (1.5) is optimal.

In view of the sharp weak estimates for continuous Hardy operator $H$ on Lebesgue spaces, it is natural to ask whether we can obtain the boundedness, or moreover, the sharp constants for discrete Hardy operator ${\bf h}$ from discrete Lebesgue spaces to discrete weak Lebesgue spaces. Here the discrete weak Lebesgue space $l^{p, \infty}(\mathbb{N})\; (1\leq p < \infty)$ consists of all sequences $\vec{a} = \{a_k\}_1^\infty$ of real numbers such that

where $\sharp E$ denotes the number of integers of the set $E$. In addition, for $p = \infty$, we denote by $l^{\infty, \infty}(\mathbb{N}) = l^{\infty}(\mathbb{N})$.

In this paper, we will give an affirmative answer to the above question. More generally, we will consider the weak type estimates for discrete Hardy operator in a more general setting. To get a better understanding of our main result in this paper, we need the definition of the weak central Morrey space and its weak version.

The classical Morrey spaces, initially introduced by Morrey ^[27], play an important role in harmonic analysis and partial differential equations. Moreover, the central version of Morrey spaces are also well used to study the mapping properties of various important integral operators in harmonic analysis. We refer the readers to ^{[28,29,30,31,32,33,34]} for the studies of Morrey-type spaces. Note that in ^[35,36,37], Gunawan et al. initially introduced the discrete version of Morrey spaces and gave some basic properties for these spaces, see also ^[38] for the mapping properties of discrete Hardy-Littlewood maximal operator and discrete fractional integral operator on discrete Morrey spaces. Inspire by ^[37], we now give the definition of discrete central Morrey spaces as follows.

Definition 1.1. Let $1\leq p < \infty$ and $-\frac{1}{p}\leq\lambda < 0$. The discrete central Morrey space ${\dot m}^{p, \lambda}(\mathbb{N})$ is defined by

where

in which $\vec{A^n} = \{A^n_k\}_1^\infty$ is the $n$-truncation sequence of the sequence $\vec{a} = \{a_k\}_1^\infty$, i.e.,

By definition, the discrete central Morrey spaces can be viewed as the central version of the discrete Morrey spaces studied in ^[37]. Similarly, one can define the discrete weak central Morrey space $w{\dot {m}}^{p, \lambda}(\mathbb{N})$:

Definition 1.2. Let $1\leq p < \infty$ and $-\frac{1}{p}\leq\lambda < 0$. The discrete weak central Morrey space $w{\dot m}^{p, \lambda}(\mathbb{N})$ is defined by

where

in which $\vec{A^n}$ is the same as in $(1.6)$.

Obviously, if we take $\lambda = -1/p$ in Definitions 1.1 and 1.2, then we recover the discrete Lebesgue space $l^p(\mathbb{N})$ and the discrete weak Lebesgue space $l^{p, \infty}(\mathbb{N})$.

In the following section, we will establish the sharp weak estimates for discrete Hardy operator on discrete central Morrey spaces. As an application, we obtain the sharp bounds for discrete Hardy operator from discrete Lebesgue spaces to discrete weak Lebesgue spaces.

2.   Main result

The main result in this paper is the following:

Theorem 2.1. Let $1\leq p < \infty$ and $-\frac{1}{p}\leq\lambda < 0$. The for $\vec{a} = \{a_k\}_1^\infty\in { {\dot m}}^{p, \lambda}(\mathbb{N})$, we have

Moreover, the constant 1 in (2.1) is the best possible one.

Proof. By using discrete Hölder's inequality, for all $n\in \mathbb{N}$, we get

where $\vec{A^n}$ is recognized as in (1.6) and $p'$ is the conjugate exponent of $p$, i.e., $1/p+1/p' = 1$.

In view of $\lambda < 0$, we have

Now we devide our discussion into two cases.

Case Ⅰ. $1\leq n\leq \left(\frac{\gamma}{\|f\|_{{\dot m}^{p, \lambda}}}\right)^{1/\lambda}$. Noting that $\lambda < 0$, we have

Csae Ⅱ. $n > \left(\frac{\gamma}{\|f\|_{{\dot m}^{p, \lambda}}}\right)^{1/\lambda}$. Since $\lambda < 0$, there holds

Combining all the estimates for Case I and Case II, we arrive at

It remains to show the constant in (2.1) is optimal. To this end, we choose a sequence $\vec{\bar a} = \{{\bar a}_k\}_1^\infty$ such that

We claim that $\vec{\bar a}\in { {\dot m}}^{p, \lambda}(\mathbb{N})$ and

In fact, for any $n\in\mathbb{N}$, we have

where $\overrightarrow{\bar{A}^n}$ is recognized as in $(1.6)$.

Since $-1/p\leq \lambda < 0$, by the definition of ${\dot m}^{p, \lambda}(\mathbb{N})$, we get

By a simple calculation, one has

for all $n\in\mathbb{N}$. Therefore,

As a consequence of (2.3) and the definition of discrete weak Morrey spaces, we obtain

Moreover, by using the boundednness of ${\bf h}$ on discrete central Morrey spaces proved above, there holds

Combining (2.4) with (2.5), we arrive at

which together with (2.2) yields that

Consequently, the constant $1$ in (2.1) is best possible. we are done.

The following corollary can be seen as the weak version of the well-known discrete Hardy inequality.

Corollary 2.1. Let $1\leq p\leq\infty$. The for $\vec{a} = \{a_k\}_1^\infty\in l^p(\mathbb{N})$, we have

Moreover, the constant 1 in (2.6) is the best possible one.

Proof. The case $1\leq p < \infty$ is a direct consequence of Theorem 2.1 by taking $\lambda = -1/p$. For $p = \infty$, it is obvious that $\|{\bf h}(\vec{a})\|_{l^{\infty}}\leq1\cdot\|\vec{a}\|_{l^{\infty}}$. On the other hand, by taking $\vec{\bar a} = \{{\bar a}_k\}_1^\infty$ such that ${\bar a}_k = 1$ for all $k\in\mathbb{N}$, there holds $\left\|{\bf h}\left({\vec{\bar a}}\right)\right\|_{l^{\infty}} = \left\|\vec{\bar a}\right\|_{l^{\infty}} = 1$. The proof is finished.

Conflict of interest

The authors declare no conflicts of interest.