Abstract
In this paper, we give necessary and sufficient conditions for the boundedness of the n-dimensional Hausdorff operators on Herz-type spaces. In addition, the sufficient condition for the boundedness of commutators generated by Lipschitz functions and the fractional Hausdorff operators on Morrey-Herz space is also provided.
MSC:26D15, 42B35, 42B99.
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1 Introduction
Recall that for a locally integrable function Φ on , the one-dimensional Hausdorff operator is defined by
The boundedness of this operator on the real Hardy space was proved in [1]. Subsequently, the problem of boundedness of in , was considered in [2, 3] and [4]. In [5], the same operator was studied on product of Hardy spaces. Due to its close relation with the summability of the classical Fourier series, it was natural to study in high-dimensional space . With such an objective, Chen et al. [6] considered three extensions of the one-dimensional Hausdorff operator in . One of them is the operator
The second multidimensional extension of the Hausdorff operator provided in [6] is the following operator:
where Φ is a radial function defined on , and is an integrable function defined on the unit sphere . Here and in what follows, we denote . In [6], the authors discussed the boundedness of these operators on various function spaces and found that they have better performance on Herz-type Hardy spaces than their performance on the Hardy spaces when .
Recently, Lin and Sun [4] defined the n-dimensional fractional Hausdorff operator initially on the Schwartz class S by
and obtained and boundedness for . Furthermore, it is easy to show that the n-dimensional fractional Hardy operator
and its adjoint operator
are special cases of if one chooses and , respectively.
In recent years, the interest in obtaining sharp bounds for integral operators has grown rapidly, mainly because of their appearance in various branches of pure and applied sciences. In [7], Xaio obtained the sharp bounds for the Hardy Littlewood averaging operator on Lebesgue and BMO spaces. Later on the problem was extended to p-adic fields in [8] and [9]. In [10] and [11], Fu with different co-author have considered the same problem for m-linear p-adic Hardy and classical Hardy operators, respectively.
As the development of linear as well as multilinear integral operators, their commutators have been well studied. A well-known theorem by Coifman et al. [12] states that the commutator defined by
where T is a Calderón-Zygmund singular integral operator, is bounded on , , if and only if . One can find a vast literature devoted to the study of the boundedness properties for such commutators. More recently, Gao and Jia [13] defined the commutator of the high-dimensional Hausdorff operator as
and studied it on Lebesgue and Herz-type spaces.
Motivated by the work cited above, in this paper, we obtain some sharp bounds for on Herz-type spaces. Furthermore, we give a sufficient condition for the boundedness of commutators generated by the Lipschitz functions b and the n-dimensional fractional Hausdorff operators , defined by
on Morrey-Herz space. Following [14], our method is direct and straightforward. In addition, the problem of boundedness of commutators of n-dimensional fractional Hardy operators [15] is also achieved as a special case of our results. Before going into the detailed proof of these results, let us first recall some definitions. For any , we set , .
Definition 1.1 ([16])
Let , , . The homogeneous Herz space is defined by
where
with the usual modification made when .
Remark 1.2 is the generalization of , the Lebesgue space with power weights. Also, it is easy to see that and .
Definition 1.3 Let , , and . The homogeneous Morrey-Herz space is defined by
where
with the usual modification made when .
In [17] the Morrey space is defined by
Obviously, and .
Definition 1.4 ([18])
Let . The Lipschitz space is defined by
In the next section we will obtain some sharp bounds for . Finally, the Lipschitz estimates for the commutators will be studied in the last section.
2 Sharp bounds for
The main result of this section is as follows:
Theorem 2.1 Let , , . If Φ is a non-negative valued function and
then is a bounded operator on .
Conversely, suppose that is a bounded operator on . If , or if , then . In addition, the operator satisfies the following operator norm:
Proof By definition and using Minkowski’s inequality
Now, it is easy to see that for [6]
Therefore, by Minkowski’s inequality, we get
Hence, we conclude that
Conversely, suppose that is bounded on . Then we consider the following two cases.
Case I: .
In this case, we choose , such that
An easy computation shows that
where denotes the volume of unit sphere . Now, by definition
On the other hand, it is easy to check that
Under the assumption that is bounded on , we get
Furthermore, combing (2.2) with (2.1), we immediately obtain
Case II: .
In this case, we have . To prove the converse relation we take the sequence of function () as follows:
Obviously for , we have . Hence, for , we obtain
Therefore,
On the other hand, we write
This implies that for . Thus for , we get
Therefore, for any , we have
Now, it is easy to show that
Consequently,
Finally, we let to obtain
In view of (2.3) with (2.1), we get
Thus, we finish the proof of Theorem 2.1. □
3 Lipschitz estimates for n-dimensional fractional Hausdorff operator
In this section, we will prove that the commutator generated by Lipschitz function b and the fractional Hausdorff operator is bounded on the Morrey-Herz space. Similar estimates for high-dimensional fractional Hardy operators are also obtained as a special case of the following theorem.
Theorem 3.1 Let , , , , . If
then is bounded from to and satisfies the following inequality:
In proving Theorem 3.1, we need the following lemmas.
Lemma 3.2 For , we have
Proof The lemma can be proved in a way similar to Theorem 3.1 in [6]. □
Lemma 3.3 ([18])
For any , if , , then . Furthermore, for any cube , , where .
Lemma 3.4 ([18])
Let , , Q and are cubes in . If , then
Proof of Theorem 3.1 Notice that
Let . Then by Hölder’s inequality, Lemma 3.2, and Lemma 3.3, we have
Now, using polar coordinates, Minkowski’s inequality and Hölder’s inequality, we approximate J as
Again by means of polar decomposition and change of the variables, we obtain
For , using Hölder’s inequality and Lemma 3.3, we have
Observe that if , then , while the reverse is true for . Hence, by Lemma 3.4, we obtain
Note that for , , we have . Therefore, by combining the estimates for I, , and , we get
Following [19], we let such that , then is contained in two adjacent annuli and . Therefore,
Hereafter, we use the notation for simplicity. Then for , we get
Here, we approximate as
Similarly,
Now, we consider the case . By Minkowski’s inequality, we write
Here, we estimate as
Similarly,
Thus, we finish the proof of Theorem 3.1. □
Now, we deduce the Lipschitz estimates for the commutators of n-dimensional fractional Hardy operators on the Morrey-Herz space as a special case of Theorem 3.1.
Corollary 3.5 If , then under the same conditions as in Theorem 3.1, the commutator of the n-dimensional fractional Hardy operator [15],
is bounded from to .
Proof In the operator , we replace
then we obtain the commutator of the n-dimensional fractional Hardy operator,
Hence, by Theorem 3.1
Thus, the corollary is proved. □
Corollary 3.6 If , then under the same conditions as in Theorem 3.1, the commutator of the adjoint fractional Hardy operator [15],
is bounded from to .
Proof In the operator , we replace
then we obtain the commutator of the n-dimensional adjoint Hardy operator
Thus, by Theorem 3.1
With this we finish the proof of Corollary 3.6. □
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Hussain, A., Gao, G. Multidimensional Hausdorff operators and commutators on Herz-type spaces. J Inequal Appl 2013, 594 (2013). https://doi.org/10.1186/1029-242X-2013-594
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DOI: https://doi.org/10.1186/1029-242X-2013-594