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Multidimensional Hausdorff operators and commutators on Herz-type spaces
Journal of Inequalities and Applicationsvolume 2013, Article number: 594 (2013)
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.
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 . Subsequently, the problem of boundedness of in , was considered in [2, 3] and . In , 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.  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  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 , 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  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 , 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  and . In  and , 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.  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  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 , our method is direct and straightforward. In addition, the problem of boundedness of commutators of n-dimensional fractional Hardy operators  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 ()
Let , , . The homogeneous Herz space is defined by
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
with the usual modification made when .
In  the Morrey space is defined by
Obviously, and .
Definition 1.4 ()
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 
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
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
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 . □
Lemma 3.3 ()
For any , if , , then . Furthermore, for any cube , , where .
Lemma 3.4 ()
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 , 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
Now, we consider the case . By Minkowski’s inequality, we write
Here, we estimate as
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 ,
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 ,
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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The authors are grateful to the referees for their valuable suggestions and comments, which improved the earlier version of the manuscript.
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.