Multidimensional Hausdorff operators and commutators on Herz-type spaces
© Hussain and Gao; licensee Springer. 2013
Received: 1 August 2013
Accepted: 12 December 2013
Published: 30 December 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.
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 .
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.
and studied it on Lebesgue and Herz-type spaces.
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 ()
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 .
with the usual modification made when .
Obviously, and .
Definition 1.4 ()
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:
then is a bounded operator on .
Conversely, suppose that is bounded on . Then we consider the following two cases.
Case I: .
Case II: .
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.
In proving Theorem 3.1, we need the following lemmas.
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 ()
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.
is bounded from to .
Thus, the corollary is proved. □
is bounded from to .
With this we finish the proof of Corollary 3.6. □
The authors are grateful to the referees for their valuable suggestions and comments, which improved the earlier version of the manuscript.
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