Weighted CBMO estimates for the commutators of matrix Hausdorff operator on Heisenberg group

In this article, we study the commutators of Hausdor? operator and establish their boundedness on weighted Herz space in the setting of Heisenberg group.


Introduction
The matrix Hausdorff operators defined on n-dimensional Euclidean space n was first reported in [1], in the following form: Taking into consideration the duality of the Hardy space H 1 and bounded mean oscillation (BMO) space, Lerner and Liflyand in [1] have shown that H A Φ, is bounded on Hardy spaces. Subsequently, similar boundedness of H A Φ, was reconsidered in [2] using atomic decomposition of Hardy spaces. The above cited publications are important as their results are the first attempts to study the high-dimensional Hausdorff operators on ( ) H n 1 .
Recently, Liflyand and Miyachi [3] extended these results on ( ) H p n spaces with < < p 0 1. In 2012, Chen et al. [4] modified the form of (1.1) by replacing the kernel function ( ) Φ y with ( )/ Φ y y n : As a subcase, when ( ) = [ /| | /| | … /| |] A y y y y diag 1 ,1 , ,1 , they give another definition of the n-dimensional Hausdorff operator: Their results include the boundedness of Hausdorff operators on Hardy spaces, local Hardy spaces, Herz and Herz-type Hardy spaces with a conclusion that these operators have better performance on Herz-type Hardy spaces than their performance on Hardy spaces. In the same year, with different co-authors, Chen et al. [5] extended the problem on the boundedness of H A Φ, to the product of Hardy-type spaces. The boundedness results regarding Hausdorff operators on ( ) H n 1 were improved in [6]. The continuity of (1.2) on Morrey spaces, Hardy-Morrey spaces, Block spaces and rectangularly defined spaces has also been discussed in [7], [8], [9] and [10], respectively. Similarly, some results regarding the boundedness of H Φ can be found in [11][12][13].
In the same way, the study of commutators to integral operators is important as it has many applications in the theory of partial differential equations and in characterizing function spaces (see, for instance, [14][15][16]). An attempt has been made in [17] to discuss the boundedness of commutators of H A Φ, , defined by: on function spaces when the symbol function b is either from the Lipschitz space or central BMO space. However, when the matrix ( ) A y is diagonal, we get the commutators of H Φ which were studied in [18][19][20]. For detailed history and other developments regarding Hausdorff operators, we refer the interested readers to the review articles [21,22].
Besides the Euclidean space n , the matrix Hausdorff operator can be defined on the p-adic linear space p n , which is a locally compact commutative group under addition (see, for instance, [23,24]), and on the Heisenberg group n [25][26][27]. Since, we are mainly concerned with the study of the commutators of Hausdorff operators defined on the Heisenberg group n , therefore, it is mandatory to introduce this group briefly and the definition of matrix Hausdorff operators on it first. With underlying manifold × n 2 , the Heisenberg group n is the Lie group under the law of noncommutative multiplication The above definition suggests that for ∈ x n , we have ⋅ = x x 0 and ⋅− = x x 0. Therefore, the identity and inverse elements of n are same as that of + n 2 1 Euclidean space. The basis for the corresponding Lie algebra is formed by the vector fields The only non-vanishing commutator relations satisfied by these vector fields are The dilation, on the Heisenberg group n , is defined as 2 is the so-called homogeneous dimension of n . The Heisenberg group is a homogeneous group with the norm: and the Heisenberg distance d, generated by this norm is given by Note that d satisfies triangular inequality and is left-invariant in the sense that r p r q  d p q  p q r  , , , , , .
n The ball and sphere on n , for > r 0 and ∈ x n , can be defined as : , : , , n respectively. To compute the measure of this ball on n , we proceed as follows: where Ω Q , being a function of n only, is the volume of the unit ball ( ) B 0, 1 . Also, the area of unit sphere For further readings on the Heisenberg group, we refer the interested reader to the book by Folland and Stein [28] and previous studies [29][30][31].
Now, we are in position to define the Hausdorff operator and its commutators on the Heisenberg group n . Let Φ be a locally integrable function on n . The Hausdorff operators on n are defined by: where ( ) A y is a matrix-valued function, and we assume that

5)
In this article, we will study the boundedness of T A b Φ, on the weighted Herz spaces ( ) K ẇ ; q α p n , 2 2 , defined in Section 2, with the Heisenberg group as underlying space. Section 2 contains some basic definitions and notations likewise some necessary propositions which will be used in the succeeding sections. Finally, Section 3 is reserved for the main results of this study along with their proofs.

Some definitions and notations
In 1972, Muckenhoupt [32] studied the Hardy-Littlewood maximal function on weighted L p spaces and introduced the theory of A p weights as a result. The theory was well studied in the later work by García-Cuerva and Rubio de Francia [33]. An extension of this theory, in the settings of the Heisenberg group n , was provided in [29] and studied in [30,31]. Any non-negative, locally integrable function w on n can be given the role of a weight. The notation w(E) serves to define weighted measure of ⊂ E n , that is, Also, if p and p′ satisfy / + / ′ = p p 1 1 1, then they will be called mutually conjugate indices. Next, let us recall some basic definitions and properties of A p weights on the Heisenberg group which will be used in the sequel.
and only if − < < ( − ) Q α Q p 1 . In view of these observations, it is easy to see that for < < ∞ α 0 , The following two Propositions, proved in [25], concerning ( ) A p n weights will be useful in establishing weighted estimates for T A b Φ, on Herz-type spaces on n .
for any measurable subset E of a ball B. In general, for any > λ 1, , .
and w is a weight function on n . When = ∞ p , we have Then, the homogeneous weighted Herz space in the setting of the Heisenberg group can be defined as follows.
, and w is a weight function on n . The homogeneous Hence, Herz space can be considered as an extension of power weighted Lebesgue space. Some relevant papers on Herz-type spaces and Hardy spaces associated with them along with their application include [35][36][37][38][39][40][41][42][43].
and w be a weight function on n . Then, we say a function ∈ For detailed study of CBMO space on n , we refer the reader to [45,46].
Lemma 2.7. [25] Suppose that the ( + ) × ( + ) n n 2 1 2 1 matrix M is invertible. Then, Also, when A p weights are reduced to the power function, we shall use the notation In that case, an easy computation results in: , on the power-weighted Herz space, we shall frequently use the piecewise defined function G: where M is any invertible matrix, ∈ α and δ is a positive real number. Then, it is easy to see that G satisfies: From this point forward, the notations ≼ A B will imply that ≤ A CB, for some > C 0. Similarly, for some positive constants

Main results and their proofs
This section contains the main results of this study and the relevant proofs. Our first result is as follows.
In addition, let ∈ w A 1 with the critical index r w for the reverse Hölder condition and > /( − ) s q r r 1 , , 1 2 and > − β n. If / = / + / q q q 1 1 1 2 1 and / + / = / q α Q α Q 1 2 1 , then we have 1̇;  implies that there exist < < r r 1 w such that = ′ s q r 2 . Therefore, by the Hölder inequality and the reverse Hölder condition, we have In second factor, on the right side of inequality (3.3), a change of variables along with Proposition 2.4 yields Hence, we obtain the following estimate for I 1 : Next, we fix to estimate I 2 , which is given by Applying the change of variables formula, Proposition 2.4 and Hölder's inequality, we have

(3.7)
Weighted CBMO estimates for commutators of matrix Hausdorff operator  503 By virtue of (3.6) and (3.7), the expression for I 2 assumes the following form: Now, the estimation of I 3 , given by . In view of the condition = ′ s q r 2 , we use the Hölder inequality and the reverse Hölder condition to obtain .
We combine inequalities (3.8) and (3.9) to substitute the result in the expression for I 3 , which now becomes where, for the convenience's sake, we used the following notation: Further decomposition of integral for I 31 results in: The first term inside the curly brackets can be approximated using Proposition 2.4, that is, Similarly, for the second term inside the curly brackets in the expression of I 31 , we have Therefore, we finish the estimation of I 31 by writing In a similar fashion, the integral I 32 gives us A combination of expressions for I 1 , I 2 , I 31 and I 32 gives Keeping in view the definition of the Herz space, factors containing the index k in the expression of ( ) Ψ y are important. Therefore, to proceed further and to avoid repetition of unimportant factors relative to the Herz space, we have to modify and rename the expression for Ψ . Hence, in the remaining of this paper we shall use the following notation: Finally, we take into consideration the definition of Herz space and employ the Minkowski inequality to have Comparing inequality (3.10) with inequality (3.9) in [25], we found that the term inside the curly brackets is same in both these inequalities, the only difference lies in the integrands outside the curly brackets along with a constant multiple ∥ ∥ q n outside the integral. Therefore, inequality (3.10) can be written as: (3.14) for any < < δ r 1 w . Replacing ( ) Ψ ỹ with its value in the above inequality, we get , by using inequalities (3.11), (3.12) and (3.14), the second part of Theorem 3.1 can be proved easily. Hence, we complete the proof of Theorem 3.1.

Proof of Theorem 3.2
Following the proof of Theorem 3.1, we write: where J 1 , J 2 and J 3 are similar to I 1 , I 2 and I 3 in the previous theorem with (⋅) w replaced by (⋅) = |⋅| v h α . Then, by using the Hölder inequality and change of variables, we obtain  Next, the expression for J 2 can be written as: Changing variables and using the condition / + / = q q q q 1   Next, if ‖ ( )‖ < A y 1, then there exists an integer ≥ j 0, such that Similarly, for ‖ ( )‖ ≥ A y 1, we have    For the approximation of ∥ (⋅)∥ ( ( ) ) f L A y C q k , we consider the method used in [25]. Hence, the definition of E k and (2.2) imply that