Research Article | Open Access

# Sharp Estimates of -Linear -Adic Hardy and Hardy-Littlewood-Pólya Operators

**Academic Editor:**Mark A. Petersen

#### Abstract

The sharp estimates of the -linear -adic Hardy and Hardy-Littlewood-Pólya operators on Lebesgue spaces with power weights are obtained in this paper.

#### 1. Introduction

In recent years, -adic numbers are widely used in theoretical and mathematical physics (cf. [1–8]), such as string theory, statistical mechanics, turbulence theory, quantum mechanics, and so forth.

For a prime number , let be the field of -adic numbers. It is defined as the completion of the field of rational numbers with respect to the non-Archimedean -adic norm . This norm is defined as follows: ; If any nonzero rational number is represented as , where and are integers which are not divisible by and is an integer, then . It is not difficult to show that the norm satisfies the following properties: From the standard -adic analysis [6], we see that any nonzero -adic number can be uniquely represented in the canonical series where are integers, , . The series (1.2) converges in the -adic norm because . Denote by and .

The space consists of points , where . The -adic norm on is Denote by the ball with center at and radius , and Since is a locally compact commutative group under addition, it follows from the standard analysis that there exists a Haar measure on , which is unique up to positive constant multiple and is translation invariant. We normalize the measure by the equality where denotes the Haar measure of a measurable subset of . By simple calculation, we can obtain that for any . For a more complete introduction to the -adic field, see [6] or [9].

The space consists of points , where , . The -adic norm of -tuple is

Recently, -adic analysis has received a lot of attention due to its application in mathematical physics. There are numerous papers on -adic analysis, such as [10, 11] about Riesz potentials, [12–16] about -adic pseudodifferential equations, and so forth. The harmonic analysis on -adic field has been drawing more and more concern (cf. [17–21] and references therein).

The well-known Hardy’s integral inequality [22] tells us that for , where the classical Hardy operator is defined by for nonnegative integral function on , and the constant is the best possible. Thus the norm of Hardy operator on is

Faris [23] introduced the following -dimensional Hardy operator, for nonnegative function on , where is the volume of the unit ball in . Christ and Grafakos [24] obtained that the norm of on is which is the same as that of the 1-dimension Hardy operator. In [25], Fu et al. introduced the -linear Hardy operator, which is defined by where and are nonnegative locally integrable functions on . And they obtained the precise norms of on Lebesgue spaces with power weight. The authors of [26] also got the best constants of -linear Hilbert, Hardy and Hardy-Littlewood-Pólya operators on Lebesgue spaces.

The study of multilinear averaging operators in Euclidean spaces is a byproduct of the recent interest in multilinear singular integral operator theory. This subject was established by Coifman and Meyer [27] in 1975. In this article, we consider the sharp estimates of -linear -adic Hardy and Hardy-Littlewood-Pólya operators. In contrast with [25], we use a new technique in calculations based on the feature of -adic field, and Theorem 3.1 is also new. They cannot be obtained immediately by [25]. In [28], we defined the -adic Hardy operator.

*Definition 1.1. *For a function on , we define the -adic Hardy operator as follows
where is a ball in with center at and radius .

It is obvious that , where is the Hardy-Littlewood maximal operator [17] defined by The Hardy-Littlewood maximal operator plays an important role in harmonic analysis. The boundedness of on has been solved (see, e.g., [9]). But the best estimate of on , even that of Hardy-Littlewood maximal operator on Euclidean spaces is very difficult to obtain. Instead, we have obtained the sharp estimates of (and -adic Hardy-Littlewood-Pólya operator) elsewhere.

*Definition 1.2. *Let be a positive integer and be nonnegative locally integrable functions on . The *-*linear-adic Hardy operator is defined by
where .

The Hardy-Littlewood-Pólya’s linear operator [26] is defined by In [26], the authors obtained that the norm of Hardy-Littlewood-Pólya’s operator on (see also [22, page 254]), , is

We define the -adic Hardy-Littlewood-Pólya operator as (see [28])

*Definition 1.3. *Let be a positive integer and be nonnegative locally integrable functions on . The -linear -adic Hardy-Littlewood-Pólya operator is defined by

We obtain the sharp estimates of the -linear -adic Hardy operator on Lebesgue spaces with power weights in Section 2. In Section 3, we get the best estimate of -linear -adic Hardy-Littlewood-Pólya operator on Lebesgue spaces with power weights.

In the following sequel, for , we denote and .

#### 2. Sharp Estimates of -Linear -Adic Hardy Operator

Theorem 2.1. *Let , , , , , , and . Then
**
where
**
is the best constant.*

When , we get the sharp estimates of the -linear -adic Hardy operator on Lebesgue spaces.

Corollary 2.2. *Let , , , and . Then
*

*Proof of Theorem 2.1. *Since the proof of the case when is similar to and even simpler than that of the case when , for simplicity, we will only give the proof of case when . To make the proof clearer, we will discuss it in two parts.*(I) When *

Firstly, we claim that the operator and its restriction to the functions satisfying have the same operator norm on . In fact, set
It’s clear that , , and

By Hölder's inequality, we get
Therefore,
which implies the claim. In the following, without loss of generality, we may assume that , , which satisfy that , .

By changing of variables , , we have
Then using Minkowski's integral inequality and Hölder's inequality (), we get

By calculation, we have
Therefore,

Now let us prove that our estimate is sharp. For and , we take
Then by calculation, we have

It is clear that when , . But when ,

Since , we get

By the same calculation as that in (2.10), we obtain that
Therefore,

Now take , . Then . Letting approach to , then approaches to 0 and approaches to 1. Since , , we have
Then (2.11) and (2.18) imply that
*(II) When *

The proof of the upper bound in this case is similar to that of the previous case, and we can obtain that
where

Let
It is clear that
and . Then

Now let us calculate , , respectively,

Similarly, for , we have
Therefore,

To show that is the best constant, we should prove that it is also the lower bound of the norm of from to . For and , we take
By simple calculation, we have
And when , . But when ,
Then by the similar discussion to that in previous case, we can obtain that
Theorem 2.1 is established by (2.20), (2.27), and (2.31).

#### 3. Sharp Estimate of -Linear -Adic Hardy-Littlewood-Pólya Operator

We get the following best estimate of -linear -adic Hardy-Littlewood-Pólya operator on Lebesgue spaces with power weights.

Theorem 3.1. *Let , , , , , , and . Then
**
where
**
is the best constant.*

In particular, when , we obtain the norm of the -linear -adic Hardy-Littlewood-Pólya operator on Lebesgue spaces.

Corollary 3.2. *Let , , , and . Then
*

*Proof of Theorem 3.1. *Just as the proof of Theorem 2.1, we will only give the proof of case when .*(I) Case *

By definition and the change of variables , , we have

By Minkowski’s integral inequality and Hölder's inequality , we get

By calculation, we have

By definition,
Similarly,

Substituting (3.7) and (3.8) into (3.6), we get
Then (3.5) and (3.9) imply that

On the other hand, for and , we take
Then

Since , we have