Journal of Function Spaces and ApplicationsVolume 2012 (2012), Article ID 541205, 12 pagesdoi:10.1155/2012/541205

Research Article

Boundedness of Commutators of High-Dimensional Hausdorff Operators

Guilian Gao and Houyu Jia

Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Received 15 February 2012; Accepted 16 March 2012

Academic Editor: Dashan Fan

Copyright © 2012 Guilian Gao and Houyu Jia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We prove the boundedness of commutators of high-dimensional Hausdorff operators on Lebesgue space with central BMO function or Lipschitz function. Furthermore, the boundedness on Herz spaces and Morrey-Herz spaces is also obtained.

1. Introduction

The one-dimensional Hausdorff operator is defined as where is a locally integrable function on . The operator has a deep root in the study of the one-dimensional Fourier analysis. Particularly, it is closely related to the summability of the classical Fourier series. The reader can see ([1–3]) to find details of background and recent development of the Hausdorff operator.

Recently, Hausdorff operator in the high dimensional space was studied. Three extensions of the one-dimensional Hausdorff operator in were recently introduced and studied in [4]. One of them is the operator where is a radial function defined on . Replacing with becomes the high dimensional Hardy operator and its adjoint operator respectively. It is well known that Hardy operators are important operators in Harmonic analysis and a quite number of papers have appeared in [5–9]. And in [10], the authors have obtained the boundedness of commutators of one-dimensional Hausdorff operator with one-sided dyadic CMO functions on the Lebesgue space.

These observations motivate us to study the boundedness of commutators of -dimensional Hausdorff operators on some function spaces. Let be a real measurable, locally integrable function; we define the commutators of Hausdorff operators as follows:

Definition 1.1 (see [11]). Let . A function is said to belong to the space if where . Obviously, we have for all , and , .

Definition 1.2 (see [12]). Let . The Besov-Lipschitz space is defined by

Herz-type spaces are important function spaces in harmonic analysis. It should be pointed out that Lu and Yang make tremendous contributions on this spaces. Their book (joint with Hu) [13] is the unique research book in this topic. Below, we briefly recall the definition of the Herz-type spaces. We denote by the ball centered at with radius . is a constant which may vary from line to line. For , let , , and denote the characteristic function of the set .

Definition 1.3 (see [13]). Let , and . The homogeneous Herz space is defined by where

Obviously, and , so the Herz space is the natural generalization of the Lebesgue spaces with power weight .

Definition 1.4 (see [13]). Let ,, and . The homogeneous Morrey-Herz space is defined by where with the usual modification made when .

In [14], the Morrey space is defined by Obviously, and .

2. Main Results

Now, we state our main results.

Theorem 2.1. Let , , and : (a)if one sets then is bounded on ;(b)if one sets then is bounded on ;(c)if satisfies (2.1) and (2.2), then is bounded on .

Remark 2.2. When , . We can check that satisfies (2.1); therefore, the boundedness of commutator of Hardy operator is obtained. When , . We also know that satisfies (2.2); therefore, we get the boundedness of commutator of the adjoint Hardy operator see [9].

Remark 2.3. Using the same method, we can get the generalized result of Theorem 2.1(b). Let ,, , and . If then is bounded on .

Theorem 2.4. Let ,,,, and . Then (a)if satisfies (2.1), then are bounded from to ;(b)if satisfies (2.2), then are bounded from to ;(c)if satisfies (2.1) and (2.2), then is bounded from to .

Remark 2.5. Just like in Remark 2.2, we get Lipschitz estimates for commutator of Hardy operator or the adjoint Hardy operator. See the details in [15].

Theorem 2.6. Let ,,, and . Then (a)if satisfies (2.1) and , then is bounded on ;(b)if satisfies (2.2) and , then is bounded on ;(c)if satisfies (2.1), (2.2), and , then is bounded on .

Remark 2.7. Let in Theorem 2.6, then Theorem 2.1 can be obtained.

Theorem 2.8. Let , , , , , and . Then (a)if satisfies (2.1) and , then is bounded from to ;(b)if satisfies (2.2) and , then are bounded from to ;(c)if satisfies (2.1), (2.2), and , then is bounded from to .

Remark 2.9. Let in Theorem 2.8, then Theorem 2.4 can be obtained.

Theorem 2.10. Let , , ,, and . Then (a)if satisfies (2.1) and , then is bounded on ;(b)if satisfies (2.2) and , then is bounded on ;(c)if satisfies (2.1), (2.2), and , then is bounded on .

Theorem 2.11. Let ,,,,,, and . (a)if satisfies (2.1) and , then is bounded from to ;(b)if satisfies (2.2) and , then is bounded from to ;(c)if satisfies (2.1), (2.2), and , then is bounded from to .

3. Proof of the Main Results

We first give several lemmas.

Lemma 3.1. Let and , then (i), (ii).

Proof. (i)By Hölder’s inequality and polar coordinate, we obtain (ii)By Hölder’s inequality and polar coordinate, we have

Lemma 3.2 (see [9]). Let be a function and , , then

Lemma 3.3 (see [12]). Let and . Then

Proof of Theorem 2.6. (a) When satisfies (2.1) and , we get For , using Lemma 3.1(i), we have For , by Lemma 3.2, we have For , by Lemma 3.1(i) and Hölder’s inequality, we have For , by Lemma 3.1(i), we have Following the estimates of , and , we can obtain that Obviously, When , we get For the case , it follows from Hölder’s inequality that When satisfies (2.2) and , we get For , using Lemma 3.1(ii), we have For , by Lemma 3.2, we have For , by Lemma 3.1(ii) and Hölder’s inequality, similarly to , we have For , also due to Lemma 3.1(ii), we have The remaining proof is similar to the proof of (a), so that of (b) can be obtained easily.(c) When satisfies (2.1), (2.2), and , due to therefore, by (a) and (b), we can easily get (c).

Proof of Theorem 2.8. (a) When satisfies (2.1) and , by Lemmas 3.3, 3.1(i), and , we get The remaining proof Is similar to the proof of (a) of Theorem 2.6, so that of (a) can be obtained easily.(b) When satisfies (2.2) and , by Lemmas 3.3, 3.1(ii), and , we get The remaining proof is similar to the proof of (a) of Theorem 2.6, so that of (b) can be obtained easily.(c) The proof is the same as the proof of (c) of Theorem 2.6.

Proof of Theorem 2.10. We only give the proof of (a) when . The proof of (b) is similar to that for (a), and the proof of (c) is similar to that for (c) of Theorem 2.6. By the definition of Morrey-Herz spaces and the estimates for , , and above, we have For , noting that , we have Similarly to the proof of , we have This finishes the proof of Theorem 2.10.

Proof of Theorem 2.11. The proof is similar to the proof of Theorem 2.10, so we omit the details.

Acknowledgment

The authors are supported in part by the NSF of China, no: 10931001, 10871173.

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