Boundedness of Commutators of High-Dimensional Hausdorff Operators

whereΦ is a locally integrable function on 0,∞ . The operator hΦ 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 R was studied. Three extensions of the one-dimensional Hausdorff operator in R were recently introduced and studied in 4 . One of them is the operator


Introduction
The one-dimensional Hausdorff operator is defined as where Φ is a locally integrable function on 0, ∞ .The operator h Φ 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 R n was studied.Three extensions of the one-dimensional Hausdorff operator in R n were recently introduced and studied in 4 .One of them is the operator where Φ is a radial function defined on R .Replacing Φ with 3 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 onesided dyadic CMO functions on the Lebesgue space.
These observations motivate us to study the boundedness of commutators of n-dimensional Hausdorff operators on some function spaces.Let b x be a real measurable, locally integrable function; we define the commutators of Hausdorff operators as follows: 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 B x, r the ball centered at x with radius r.C is a constant which may vary from line to line.
Definition 1.3 see 13 .Let α ∈ R, 0 < p < ∞, and 0 < q < ∞.The homogeneous Herz space Kα,p q R n is defined by where Obviously, K0,q q R n L q R n and Kα/q,q q R n L q R n , |x| α , so the Herz space is the natural generalization of the Lebesgue spaces with power weight |x| α .Definition 1.4 see 13 .Let α ∈ R, 0 < p ≤ ∞, 0 < q < ∞ and λ ≥ 0. The homogeneous Morrey-Herz space M Kα,λ p,q R n is defined by where with the usual modification made when p ∞.
In 14 , the Morrey space M λ q R n is defined by Obviously, M Kα,0 p,q R n Kα,p q R n and M λ q R n ⊂ M K0,λ q,q R n .

Main Results
Now, we state our main results.
Theorem 2.1.Let 1 < p < ∞, 1 < r < min p, p , and b ∈ C ṀO max p,pr/ p−r R n : We can check that Φ t satisfies 2.1 ; therefore, the boundedness of commutator of Hardy operator is obtained.
We also know that Φ t 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 1 < p < ∞, 1 < r < p, 0 ≤ λ < min r /r, r /p , and b ∈ CMO max p,pr/ p−r R n .If 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 .

Proof of the Main Results
We first give several lemmas. 6

Journal of Function Spaces and Applications
Proof.

3.2
Lemma 3.2 see 9 .Let b be a C ṀO 1 R n function and i, k ∈ Z, then

3.5
For I, using Lemma 3.1 i , we have 3.6 For II, by Lemma 3.2, we have

3.7
For II 1 , by Lemma 3.1 i and H ölder's inequality, we have 3.8 For II 2 , by Lemma 3.1 i , we have 3.9 Following the estimates of I, II 1 , and II 2 , we can obtain that Obviously,

3.12
For the case p > 1, it follows from H ölder's inequality that

3.15
For JJ, by Lemma 3.2, we have

3.16
For JJ 1 , by Lemma 3.1 ii and H ölder's inequality, similarly to II 1 , we have 3.17 For JJ 2 , also due to Lemma 3.1 ii , we have

3.18
The remaining proof is similar to the proof of a , so that of b can be obtained easily.

3.20
The remaining proof Is similar to the proof of a of Theorem 2.6, so that of a can be obtained easily.b When Φ satisfies 2.2 and α > −n 1/q 2 − 1/r , by Lemmas 3.3, 3.1 ii , and 1/q 1 − 1/q 2 γ/n, we get

3.21
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 λ > 0. 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 I, II 1 , and II 2 above, we have 2 i−k n/r−n/q fχ i L q p 1/p CA Φ,r b C ṀO qr/ q−r R n sup k − i 2 i−k n/r−n/q fχ i L q p 1/p : E 1 E 2 E 3 .

3.22
For E 1 , noting that α < n 1/r − 1/q λ, we have CA Φ,r b C ṀO q R n f M Kα,λ p,q R n .

3.23
Similarly to the proof of E 1 , we have p,q R n .

3.24
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.