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Boundedness for commutators of fractional p-adic Hardy operators
Journal of Inequalities and Applications volume 2012, Article number: 293 (2012)
Abstract
In this paper we prove that the commutators generated by the fractional p-adic Hardy operators and the central BMO function are bounded on weighted homogeneous Herz spaces.
MSC:11E95, 11K70, 42B99.
1 Introduction
In recent years, p-adic fields have been introduced into some aspects of mathematical physics. There are a lot of articles where different applications of the p-adic analysis in the string theory, quantum mechanics, stochastics, the theory of dynamical systems, cognitive sciences, and psychology are studied [1–9] (see also the references therein). As a consequence, new mathematical problems have emerged, among them, the study of harmonic analysis on a p-adic field has been drawing more and more concern (cf. [10–14] and the references therein).
For a prime number p, let be the field of p-adic numbers. It is defined as the completion of the field of rational numbers ℚ with respect to the non-Archimedean p-adic norm . This norm is defined as follows: . If any non-zero rational number x is represented as , where m and n are integers which are not divisible by p, and γ is an integer, then . It is not difficult to show that the norm satisfies the following properties:
It follows from the second property that when , then . From the standard p-adic analysis [7], we see that any non-zero p-adic number can be uniquely represented in the canonical series
where are integers, , . The series (1.1) converges in the p-adic norm because . Set .
The space consists of points , where , . The p-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 dx on , which is unique up to a positive constant multiple and is translation invariant. We normalize the measure dx by the equality
where denotes the Haar measure of a measurable subset E of . By simple calculation, we can obtain that
for any . For a more complete introduction to the p-adic field, see [15] or [7].
The classical Hardy operators are defined by
for a non-negative integrable function f on . Obviously, â„‹ and satisfy
The well-known Hardy integral inequality [16] tells us that for ,
where the constant is the best possible. The generalized result [17] is that
and
where .
The Hardy integral inequalities have received considerable attention due to their usefulness in analysis and their applications. There are numerous papers dealing with their various generalizations, variants and applications (cf. [18–20] and the references cited therein). We have obtained the Hardy integral inequalities for p-adic Hardy operators and their commutators [21]. The boundedness of commutators is an active topic in harmonic analysis because of its important applications; for example, it can be applied to characterizing some function spaces. There are a lot of works about the boundedness of commutators of various Hardy-type operators on Euclidean spaces (cf. [22, 23], etc.). In this paper, we will establish the Hardy integral inequalities for commutators generated by fractional p-adic Hardy operators and CMO functions.
Definition 1.1 For a function f on , we define the p-adic Hardy operators as follows:
where is a ball in with center at and radius .
Definition 1.2 Let , . The fractional p-adic Hardy operators are defined by
where is the ball as in Definition 1.1.
It is clear that when , then becomes .
Definition 1.3 Let , . The commutators of fractional p-adic Hardy operators are defined by
In [24–26], the CMO spaces (central BMO spaces) on have been introduced and studied. CMO spaces bear a simple relationship with BMO: precisely when g and all of its translates belong to BMO spaces uniformly a.e. Many precise analogies exist between CMO spaces and BMO spaces from the point of view of real Hardy spaces. Similarly, we define the spaces on .
Definition 1.4 Let , a function is said to be in if
where
Remark 1.1 It is obvious that .
Let , and be the characteristic function of the set .
Definition 1.5 [27]
Suppose that , and . The homogeneous p-adic Herz space is defined by
where
with the usual modifications made when or .
Remark 1.2 is the generalization of , and , for all and .
Motivated by [22], we get the following operator boundedness results. Throughout this paper, we use C to denote different positive constants which are independent of the essential variables.
Theorem 1.1 Suppose that , , , , , . Then
-
(1)
If , then
(1.6) -
(2)
If , then
(1.7)
When , , , we can get the following result.
Corollary 1.1 Suppose that , , , , , . Then
and
When , we can get the boundedness of a p-adic Hardy operator in [21].
Corollary 1.2 Let , , . Then
-
(1)
If , then
(1.10) -
(2)
If , then
(1.11)
By the similar proof of Theorem 1.1, we can obtain the following result.
Corollary 1.3 Suppose that , , , , . Then
-
(1)
If , then
(1.12) -
(2)
If , then
(1.13)
2 Boundedness of commutators of fractional p-adic Hardy operator
In order to prove Theorem 1.1, we firstly give the following lemmas.
Lemma 2.1 Suppose that b is a CMO function and , then and .
Proof For any , by Hölder’s inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2012-293/MediaObjects/13660_2012_Article_458_Equo_HTML.gif)
Therefore, and . This completes the proof. □
Lemma 2.2 Suppose that b is a CMO function, , then
Proof For , recall that , we have
For , without loss of generality, we can assume that , by (2.2), we get
The lemma is proved. □
Proof of Theorem 1.1 Denote .
-
(1)
By definition,
Now let us estimate I and II, respectively. For I, by Hölder’s inequality , we have
For II, by Lemma 2.2, we get
For and , by Hölder’s inequality, we obtain
and
Then the above inequalities together with Lemma 2.1 imply that
For the case , since , we have
For the case , by Hölder’s inequality, we have
Then (1.6) follows from (2.4)-(2.6).
-
(2)
By definition,
By Hölder’s inequality, we get
By Lemma 2.2, we have
For and , by Hölder’s inequality, we obtain
and
Then (2.7)-(2.8) together with Lemma 2.1 imply that
For the case , since , we have
For the case , by Hölder’s inequality, we have
Then the above inequalities imply (1.7). Theorem 1.1 is proved. □
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Acknowledgements
The author sincerely thanks Professor Zunwei Fu for his useful discussions. This work was supported by NSF of China (Grant No. 11126203), NSF of Shandong Province (Grant Nos. ZR2010AL006).
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Wu, Q.Y. Boundedness for commutators of fractional p-adic Hardy operators. J Inequal Appl 2012, 293 (2012). https://doi.org/10.1186/1029-242X-2012-293
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DOI: https://doi.org/10.1186/1029-242X-2012-293