Sharp Estimates of -Linear -Adic Hardy and Hardy-Littlewood-Pólya Operators
The sharp estimates of the -linear -adic Hardy and Hardy-Littlewood-Pólya operators on Lebesgue spaces with power weights are obtained in this paper.
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 , 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  or .
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  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  introduced the following -dimensional Hardy operator, for nonnegative function on , where is the volume of the unit ball in . Christ and Grafakos  obtained that the norm of on is which is the same as that of the 1-dimension Hardy operator. In , 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  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  in 1975. In this article, we consider the sharp estimates of -linear -adic Hardy and Hardy-Littlewood-Pólya operators. In contrast with , 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 . In , 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  defined by The Hardy-Littlewood maximal operator plays an important role in harmonic analysis. The boundedness of on has been solved (see, e.g., ). 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 .
We define the -adic Hardy-Littlewood-Pólya operator as (see )
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 Therefore,