Optimal Weak Type Estimates for the P-ADIC Hardy Type Operators on Higher-Dimensional Product Spaces

Abstract

In this paper, we introduce the fractional p-adic Hardy operators and its conjugate operators and obtain its optimal weak type estimates on the p-adic Lebesgue product spaces.

1. Introduction

In recent years, -adic analysis has been widely used in quantum mechanics, the probability theory, and the dynamical systems [1, 2]. Meanwhile, there is an increasing attention in pseudo-differential equations, wavelet theory, and harmonic analysis (see [3–8]).

For a prime number , let be the field of -adic numbers, a nonzero rational number is represented as , where is an integer and the integers are not divisible by . Then, the norm is defined as , and it is easy to see that the norm satisfies the following properties:(i)(ii)(iii), in the case when , we have

It is well known that is a typical model of non-Archimedean local fields. From the standard -adic analysis, any can be uniquely represented as a canonical formwhere , note that the series (1) converges with respect to the norm because one has . The space consists of elements , where , . The norm in this space is

The symbols and represent, respectively, the ball and the sphere with center at and radius , defined by

It is clear that , and we set and .

As is a locally compact commutative group with respect to addition, there exists a Harr measure on , which is unique up to a positive constant factor and is translation invariant, that is, . We normalize the measure such thatwhere denotes the Harr measure of a measure subset B of . By simple calculation, we can obtain that

The classical Hardy operator

was introduced by Hardy in [9], and a celebrated integral inequality states that

It was also shown that the constant factor is optimal, knowing its importance in analysis.

Faris in [10] and Christ and Grafakos in [11] proposed an extension of (1) and its adjoint to the n-dimensional Euclidian spaces of which the equivalent forms are

The norm of and on was evaluated and found to be equal to that of the classical Hardy operator. For more details about the boundedness of the Hardy operator and its adjoint, we included some references [12–14].

On the other hand, the fractional Hardy operator and its adjoint are obtained by merely interchanging with in (8). The weak and strong type optimal bounds for the fractional Hardy and adjoint Hardy operator have also spotlighted many researchers in the past, see [15–18].

The n-dimensional fractional p-adic Hardy and adjoint Hardy operator are defined and studied in [19], which, for and , are given as

when , the fractional p-adic Hardy and adjoint Hardy operator reduces to p-adic Hardy and adjoint Hardy operator. Some other papers showing the boundedness of p-adic Hardy-type operators are included [19–26].

In 2020, Li et al. [27] introduced the definition of the fractional Hardy operator on higher-dimensional product spaces as follows:where be a nonnegative integrable function on , , , , with . Furthermore, the corresponding operator norm on the weak Lebesgue product spaces was obtained.

Next, we will introduce the definition of the fractional Hardy operator on higher-dimensional p-adic product spaces and obtain sharp weak bounds.

Definition 1. Let , , and be a nonnegative integrable function on . Define the fractional p-adic Hardy operator on higher-dimensional product spaces bywhere with .
In 2020, Wang et al. [28] gave the definition of fractional conjugate Hardy operator on higher-dimensional product spaces as follows:where be a nonnegative integrable function on , , , , with , and they also got the corresponding operator norm on the weak Lebesgue product spaces.
Next, we will give a higher-dimensional version of the fractional p-adic conjugate Hardy operator and obtain sharp weak bounds.

Definition 2. Let , , and be a nonnegative integrable function on . Define the fractional p-adic conjugate Hardy operator on higher-dimensional product spaces bywhere with .
In this article, we will obtain sharp weak bounds for the fractional p-adic Hardy operators and its conjugate operators on the p-adic Lebesgue product spaces. Our method of proving the main results involves a frequent use of the following formula:

2. Sharp Weak Bounds for Fractional Hardy Operators

This section considers the problem of obtaining optimal weak bounds for and our results as follows.

Theorem 1. Set , let , . If , then we haveFurthermore,To obtain the desired result, we need the following lemma.

Lemma 1. Suppose that , if , then for any ,Moreover,

Proof. SinceNext, we let , then, for any , we getThus,On the other hand, we let , thenAlso,Now,Since , when , thenAlso, when , thenTherefore,Thus, as above, we getNow let us prove Theorem 1.

Proof. Without loss of generality, we consider only the situation when , and then, the case is just a repetition of the case . For , the operator can be written asWhen , we getThen by Lemma 1,Using Fubini theorem, we obtain thatObviously, , if . Then, applying the lemma again, we getCombining (31)–(33), we getfor all .
Conversely, to prove that the constant 1 is optimal, we tookAnd choosing , where , we get from the definition of thatAlso,We now that let , for , then combining both the cases, we getTherefore, we haveFor , we also divide into two cases and . As above, we getSince , by combining (37) with (38),This completes the proof.

3. Sharp Weak Bounds for Fractional Conjugate Hardy Operators

Likewise, this section contains the results having sharp weak bounds for fractional p-adic conjugate Hardy operators, and our results are as follows.

Theorem 2. Set , let . If , then we haveFurthermore,In order to prove our theorem, we need the following lemma.

Lemma 2. Suppose that , if , then for any ,Moreover,The proof of this result is almost the same as Lemma 1; here, we omit the proof details. Next, we give the proof of Theorem 2.

Proof. Without loss of generality, we only discuss the case , and then, the case is just a repetition of the case . When , the operator can be written asUsing Lemma 2 and Fubini theorem, it implies thatWe conclude thatConsequently, combining (45) and (46), we getOn the other hand, for any , we tookLet , where , thenWe haveSet and , we obtain thatNotice that when , if is small enough, tends to zero; therefore, when is small enough, we getUsing the same method for , we obtain thatLet and , it implies thatThis finishes the proof of Theorem 2.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Natural Science Foundation of Shaanxi Province (No. 2022JM-053) and the Scientific Research Project of Xi’an Traffic Engineering Institute (No. 2021KY-29).