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

  • Amna Ajaib EMAIL logo and Amjad Hussain
Published/Copyright: June 4, 2020
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Open Mathematics
From the journal Open Mathematics Volume 18 Issue 1

Abstract

In this article, we study the commutators of Hausdorff operators and establish their boundedness on the weighted Herz spaces in the setting of the Heisenberg group.

Keywords: Hausdorff operator; commutators; Herz space; CBMO function; Heisenberg group; weights
MSC 2010: 42B35; 42B30; 46E30; 22E25

1 Introduction

The matrix Hausdorff operators defined on n-dimensional Euclidean space ℝ n was first reported in [1], in the following form:

(1.1) H Φ , A f ( x ) = ∫ ℝ n Φ ( y ) f ( x A ( y ) ) d y .

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 Φ , A is bounded on Hardy spaces. Subsequently, similar boundedness of H Φ , A 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 1 ( ℝ n ) . Recently, Liflyand and Miyachi [3] extended these results on H p ( ℝ n ) spaces with 0 < p < 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 :

(1.2) H Φ , A f ( x ) = ∫ ℝ n Φ ( y ) | y | n f ( x A ( y ) ) d y .

As a subcase, when A ( y ) = diag [ 1 / | y | , 1 / | y | , … , 1 / | y | ] , they give another definition of the n-dimensional Hausdorff operator:

(1.3) H Φ f ( x ) = ∫ ℝ n Φ ( y ) | y | n f ( x | y | ) d y .

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 Φ , A to the product of Hardy-type spaces. The boundedness results regarding Hausdorff operators on H 1 ( ℝ n ) 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 Φ , A , defined by:

(1.4) H Φ , A b f ( x ) = ∫ ℝ n Φ ( y ) | y | n ( b ( x ) − b ( x A ( y ) ) ) f ( x A ( y ) ) d y ,

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 ℝ n , the matrix Hausdorff operator can be defined on the p-adic linear space ℚ p n , which is a locally compact commutative group under addition (see, for instance, [23,24]), and on the Heisenberg group ℍ n [25,26,27]. Since, we are mainly concerned with the study of the commutators of Hausdorff operators defined on the Heisenberg group ℍ n , therefore, it is mandatory to introduce this group briefly and the definition of matrix Hausdorff operators on it first.

With underlying manifold ℝ 2 n × ℝ , the Heisenberg group ℍ n is the Lie group under the law of non-commutative multiplication

x ⋅ y = ( x 1 , x 2 , … , x 2 n + 1 ) ⋅ ( y 1 , y 2 , … , y 2 n + 1 ) = ( x 1 + y 1 , … , x 2 n + y 2 n , x 2 n + 1 + y 2 n + 1 + 2 ∑ j = 1 n ( y j x n + j − x j y n + j ) ) .

The above definition suggests that for x ∈ ℍ n , we have x ⋅ 0 = x and x ⋅ − x = 0 . Therefore, the identity and inverse elements of ℍ n are same as that of ℝ 2 n + 1 Euclidean space. The basis for the corresponding Lie algebra is formed by the vector fields

X j = ∂ ∂ x j + 2 x n + j ∂ ∂ x 2 n + 1 , 1 ≤ j ≤ n , X n + j = ∂ ∂ x n + j − 2 x j ∂ ∂ x 2 n + 1 , 1 ≤ j ≤ n , X 2 n + 1 = ∂ ∂ x 2 n + 1 .

The only non-vanishing commutator relations satisfied by these vector fields are

[ X j , X n + j ] = − 4 X 2 n + 1 , 1 ≤ j ≤ n .

The dilation, on the Heisenberg group ℍ n , is defined as

δ r ( x 1 , x 2 , … , x 2 n , x 2 n + 1 ) = ( r x 1 , r x 2 , r x 2 n , r 2 x 2 n + 1 ) , r > 0 .

Also, the Haar measure on ℍ n coincides with the usual Lebesgue measure on ℝ 2 n × ℝ 1 . Thus, for any measurable set E ⊂ ℍ n , we denote its measure by | E | . Moreover, it is easy to see that

| δ r ( E ) | = r Q | E | , d ( δ r x ) = r Q d x ,

where Q = 2 n + 2 is the so-called homogeneous dimension of ℍ n .

The Heisenberg group is a homogeneous group with the norm:

| x | h = [ ( ∑ i = 1 2 n x i 2 ) 2 + x 2 n + 1 2 ] 1 / 4 ,

and the Heisenberg distance d, generated by this norm is given by

d ( p , q ) = d ( q − 1 p , 0 ) = | q − 1 p | h .

Note that d satisfies triangular inequality and is left-invariant in the sense that

d ( r ⋅ p , r ⋅ q ) = d ( p , q ) , ∀   p , q , r ∈ ℍ n .

The ball and sphere on ℍ n , for r > 0 and x ∈ ℍ n , can be defined as

B ( x , r ) = { y ∈ ℍ n : d ( x , y ) < r }

and

S ( x , r ) = { y ∈ ℍ n : d ( x , y ) = r } ,

respectively. To compute the measure of this ball on ℍ n , we proceed as follows:

| B ( x , r ) | = | B ( 0 , r ) | = Ω Q r Q ,

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 S ( 0 , 1 ) on ℍ n is w Q = Q Ω Q . 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 ℍ n . Let Φ be a locally integrable function on ℍ n . The Hausdorff operators on ℍ n are defined by:

T Φ f ( x ) = ∫ ℍ n Φ ( y ) | y | h Q f ( δ | y | h − 1 x ) d y , T Φ , A f ( x ) = ∫ ℍ n Φ ( y ) | y | h Q f ( A ( y ) x ) d y ,

where A ( y ) is a matrix-valued function, and we assume that det A ( y ) ≠ 0 almost everywhere in the support of Φ . Also, we define the commutators T Φ , A b of T Φ , A with locally integrable function b as

(1.5) T Φ , A b ( f ) = b T Φ , A ( f ) − T Φ , A ( b f ) .

In this article, we will study the boundedness of T Φ , A b on the weighted Herz spaces K ̇ q 2 α 2 , p ( ℍ n ; w ) , 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.

2 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 ℍ n , was provided in [29] and studied in [30,31]. Any non-negative, locally integrable function w on ℍ n can be given the role of a weight. The notation w(E) serves to define weighted measure of E ⊂ ℍ n , that is, w ( E ) = ∫ E w ( x ) d x . Also, if p and p′ satisfy 1 / p + 1 / p ′ = 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.

Definition 2.1

We say that w belongs to the Muckenhoupt class A p ( ℍ n ) ,  1 < p < ∞ , if there exists a C > 0 such that for every ball B ⊂ ℍ n ,

( 1 | B | ∫ B w ( x ) d x ) ( 1 | B | ∫ B w ( x ) − p ′ / p d x ) p / p ′ ≤ C .

Also, w ∈ A 1 if there exists a constant C > 0 such that for every ball B ⊂ ℍ n ,

( 1 | B | ∫ B w ( x ) d x ) ≤ C essinf x ∈ B w ( x ) .

When p = ∞ , we define A ∞ = ∪ 1 ≤ p < ∞ A p .

According to Proposition 2.2 in [25], we have A p ( ℍ n ) ⊂ A q ( ℍ n ) , for 1 ≤ p < q < ∞ , and if w ∈ A p ( ℍ n ) , 1 < p < ∞ , then there is an ε > 0 such that p − ε > 1 and w ∈ A p − ε ( ℍ n ) . Therefore, we may use q w ≔ inf { q > 1 : w ∈ A q } to denote the critical index of w.

Definition 2.2

We say that w belongs to the reverse Hölder class R H r ( ℍ n ) , if there exists a fixed constant C > 0 and r > 1 , such that for every ball B ⊂ ℍ n ,

( 1 | B | ∫ B w r ( x ) d x ) 1 / r ≤ C | B | ∫ B w ( x ) d x .

In [31], it was proved that w ∈ A ∞ ( ℍ n ) if and only if there exist some r > 1 such that w ∈ R H r ( ℍ n ) . In addition, if w ∈ R H r ( ℍ n ) , r > 1 , then for some ε > 0 we have w ∈ R H r + ε ( ℍ n ) . We therefore use r w ≔ sup { r > 1 : w ∈ R H r ( ℍ n ) } to denote the critical index of w for the reverse Hölder condition.

A particular case of Muckenhoupt A p ( ℍ n ) weights is the power weight function | x | h α . From Proposition 2.3 in [25], for x ∈ ℍ n , we have | x | h α ∈ A 1 ( ℍ n ) if and only if − Q < α ≤ 0 . Also, for 1 < p < ∞ , | x | h ∈ A p ( ℍ n ) , if and only if − Q < α < Q ( p − 1 ) . In view of these observations, it is easy to see that for 0 < α < ∞ ,

| x | h α ∈ ⋂ Q + α Q < p < ∞ A p ( ℍ n ) ,

where ( Q + α ) / Q is known as the critical index of | x | h α .

The following two Propositions, proved in [25], concerning A p ( ℍ n ) weights will be useful in establishing weighted estimates for T Φ, A b on Herz-type spaces on ℍ n .

Proposition 2.3

Let w ∈ A p ∩ R H r ( ℍ n ) ,   p ≥ 1 and r > 1 . Then, there exist constants C 1 , C 2 > 0 such that

C 1 ( | E | | B | ) p ≤ w ( E ) w ( B ) ≤ C 2 ( | E | | B | ) ( r − 1 ) / r ,

for any measurable subset E of a ball B. In general, for any λ > 1 ,

w ( B ( x 0 , λ R ) ≤ λ Q p w ( B ( x 0 , R ) ) .

Proposition 2.4

If w ∈ A p ( ℍ n ) , 1 ≤ p < ∞ , then for any f ∈ L loc 1 ( ℍ n ) and any ball B ⊂ ℍ n

1 | B | ∫ B | f ( x ) | d x ≤ C ( 1 w ( B ) ∫ B | f ( x ) | p w ( x ) d x ) 1 / p .

For any measurable set E ⊂ ℍ n , the weighted Lebesgue space L p ( E ; w ) is the space of all functions f satisfying the norm condition

∥ f ∥ L p ( E ; w ) = ( ∫ E | f ( x ) | p w ( x ) d x ) 1 / p < ∞ ,

where 1 ≤ p < ∞ and w is a weight function on ℍ n . When p = ∞ , we have L ∞ ( ℍ n ; w ) = L ∞ ( ℍ n ) and ∥ f ∥ L ∞ ( ℍ n ; w ) = ∥ f ∥ L ∞ ( ℍ n ) .

Let B k ≔ { x ∈ ℍ n : | x | h < 2 k } , E k = B k / B k − 1 for k ∈ Z . Then, the homogeneous weighted Herz space in the setting of the Heisenberg group can be defined as follows.

Definition 2.5

[25] Let α ∈ ℝ , 0 < p , q < ∞ , and w is a weight function on ℍ n . The homogeneous weighted Herz space K ̇ q α , p ( ℍ n ) is defined by

K ̇ q α , p ( ℍ n ; w ) ≔ { f ∈ L loc q ( ℍ n / { 0 } ; w ) : ∥ f ∥ K ̇ q α , p ( ℍ n ; w ) < ∞ } ,

where

∥ f ∥ K ̇ q α , p ( ℍ n ; w ) = { ∑ k = − ∞ ∞ w ( B k ) α p / Q ∥ f ∥ L p ( E k ; w ) p } 1 / p .

When w = 1 , we obtain K ̇ q α , p ( ℍ n ) introduced in [34]. It is easy to verify that K ̇ p α , p ( ℍ n ) = L p ( ℍ n , | ⋅ | h α p ) . 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].

Definition 2.6

[44] Let 1 < q < ∞ and w be a weight function on ℍ n . Then, we say a function f ∈ L loc q ( ℍ n ; w ) belongs to the weighted central bounded mean oscillation (CBMO) space C M ̇ O q ( ℍ n ; w ) if

∥ f ∥ C M ̇ O q ( ℍ n ; w ) = sup R > 0 ( 1 w ( B ( 0 , R ) ) ∫ B ( 0 , R ) | f ( x ) − f B | q w ( x ) ) 1 / q < ∞ ,

where

(2.1) f B = 1 | B ( 0 , r ) | ∫ B ( 0 , r ) f ( x ) d x .

For detailed study of CBMO space on ℝ n , we refer the reader to [45,46].

Recently, weighted boundedness of matrix Hausdorff operators and their commutators defined on different underlying spaces are reported in [44,47,48,49,50,51,52,53].

Lemma 2.7

[25] Suppose that the ( 2 n + 1 ) × ( 2 n + 1 ) matrix M is invertible. Then,

(2.2) ∥ M ∥ − Q ≤ | det M − 1 | ≤ ∥ M − 1 ∥ Q ,

where

(2.3) ∥ M ∥ = sup x ∈ ℍ n , x ≠ 0 | M x | h | x | h .

Also, when A p weights are reduced to the power function, we shall use the notation v ( ⋅ ) instead of w ( ⋅ ) , that is, v ( ⋅ ) = | ⋅ | h β . In that case, an easy computation results in:

(2.4) v ( B k ) = ∫ | x | h ≤ 2 k | x | h β d x = ω Q 2 k ( Q + β ) / ( β + Q ) .

Moreover, in the case of boundedness of T Φ , A b on the power-weighted Herz space, we shall frequently use the piecewise defined function G:

G ( M , δ β ) = { ∥ M ∥ δ β if β > 0 , ∥ M − 1 ∥ − δ β if β ≤ 0 ,

where M is any invertible matrix, α ∈ ℝ and δ is a positive real number. Then, it is easy to see that G satisfies:

(2.5) G ( M , β ( 1 / q + 1 / p ) ) = G ( M , β / q ) G ( M , β / p ) ,

where p , q ∈ Z + .

Proposition 2.8

Suppose that the ( 2 n + 1 ) × ( 2 n + 1 ) matrix M is invertible. Let β > − n , v ( x ) = | x | h β and x ∈ ℍ n , then

v ( M x ) ≤ { ∥ M ∥ β v ( x ) if β > 0 , ∥ M − 1 ∥ − β v ( x ) if β ≤ 0 , = G ( M , β ) v ( x ) .

From this point forward, the notations A ≼ B will imply that A ≤ C B , for some C > 0 . Similarly, for some positive constants C 1 and C 2 , if A ≤ C 1 B and B ≤ C 2 A , then we will write A ≃ B . Also, we set λ B ( 0 , R ) = B ( 0 , λ R ) , for λ > 0 .

3 Main results and their proofs

This section contains the main results of this study and the relevant proofs. Our first result is as follows.

Theorem 3.1

Let 1 ≤ p , q , q 1 , q 2 ≤ ∞ and α 1 , α 2 ∈ ℝ with α 1 < 0 . Suppose that 1 / s = 1 / q 1 + 1 / q and α 1 / Q + 1 / q 1 = α 2 / Q + 1 / q 2 . In addition, let w ∈ A 1 with the critical index r w for the reverse Hölder condition and s > q 2 r w / ( r w − 1 ) .

  1. If 1 / q 1 + α 1 / Q ≥ 0 , then for any 1 < δ < r w ,

    ∥ T Φ , A b f ∥ K ̇ q 2 α 2 , p ( ℍ n ; w ) ≤ K 1 ∥ b ∥ C M ̇ O q ( ℍ n ; w ) ∥ f ∥ K ̇ q 1 α 1 , p ( ℍ n ; w ) ,

    where

    K 1 = ∫ ∥ A ( y ) ∥ < 1 | Φ ( y ) | | y | h Q ( 1 + | det A − 1 ( y ) | 1 / q ∥ A ( y ) ∥ Q / q ) | det A − 1 ( y ) | 1 / q 1 ∥ A ( y ) ∥ − α 1 log 2 ∥ A ( y ) ∥ d y + ∫ ∥ A ( y ) ∥ ≥ 1 | Φ ( y ) | | y | h Q ( 1 + | det A − 1 ( y ) | 1 / q ∥ A ( y ) ∥ Q / q ) | det A − 1 ( y ) | 1 / q 1 ∥ A ( y ) ∥ Q / q 1 − ( α 1 + Q / q 1 ) ( δ − 1 ) / δ log 2 ∥ A ( y ) ∥ d y ;

  2. If α 1 / Q + 1 / q 1 < 0 , then for any 1 < δ < r w

∥ T Φ , A b ∥ K ̇ q 2 α 2 , p ( ℍ n ; w ) ≤ K 2 ∥ b ∥ C M ̇ O q ( ℍ n ; w ) ∥ f ∥ K ̇ q 1 α 1 , p ( ℍ n ; w ) ,

where

K 2 = ∫ ∥ A ( y ) ∥ ≥ 1 | Φ ( y ) | | y | h Q ( 1 + | det A − 1 ( y ) | 1 / q ∥ A ( y ) ∥ Q / q ) | det A − 1 ( y ) | 1 / q 1 ∥ A ( y ) ∥ − α 1 log 2 ∥ A ( y ) ∥ d y + ∫ ∥ A ( y ) ∥ < 1 | Φ ( y ) | | y | h Q ( 1 + | det A − 1 ( y ) | 1 / q ∥ A ( y ) ∥ Q / q ) | det A − 1 ( y ) | 1 / q 1 ∥ A ( y ) ∥ Q / q 1 − ( α 1 + Q / q 1 ) ( δ − 1 ) / δ log 2 ∥ A ( y ) ∥ d y .

When general weights are reduced to power weights, then the next theorem is as follows.

Theorem 3.2

Let 1 ≤ p < ∞ , 1 < q , q 1 , q 2 < ∞ and β > − n . If 1 / q 2 = 1 / q + 1 / q 1 and 1 / q + α 2 / Q = α 1 / Q , then we have

∥ T Φ , A b ∥ K ̇ q 2 α 2 , p ( ℍ n ; v ) ≼ K 3 ∥ b ∥ C M ̇ O q ( ℍ n ; v ) ∥ f ∥ K ̇ q 1 α 1 , p ( ℍ n ; v ) ,

where K 3 is

K 3 = { ∫ ℍ n Θ ( y ) ( 1 + log 2 ( ∥ A − 1 ( y ) ∥ ∥ A ( y ) ∥ ) ) d y , if α 1 = 0 , ∫ ℍ n Θ ( y ) G ( A − 1 ( y ) , α 1 ( Q + β ) / Q ) d y , if α 1 ≠ 0 ,

and

Θ ( y ) = | Φ ( y ) | | y | h Q | det A − 1 ( y ) | 1 / q 1 ( log 2 ∥ A ( y ) ∥ χ { ∥ A ( y ) ∥ < 1 } + log 2 ∥ A ( y ) ∥ χ { ∥ A ( y ) ∥ ≥ 1 } ) × G ( A − 1 ( y ) , β / q 1 ) ( 1 + | det A − 1 ( y ) | 1 / q G ( A − 1 ( y ) , β / q ) ∥ A ( y ) ∥ ( Q + β ) / q ) .

3.1 Proof of Theorem 3.1

Here, we have to show that

{ ∑ k = − ∞ ∞ w ( B k ) α 2 p / Q ∥ T Φ , A b f ∥ L q 2 ( E k , w ) p } 1 / p ≼ ∥ f ∥ K ̇ q 2 α 2 , p ( ℍ n ; w ) .

By the Minkowski inequality and necessary splitting, an upper bound for the inner norm ∥ T Φ , A b f ∥ L q 2 ( E k , w ) p can be obtained as:

(3.1) ∥ ( T Φ , A b f ) ∥ L q 2 ( E k ; w ) = ∥ ( ∫ ℍ n Φ ( y ) | y | h Q ( b ( x ) − b ( A ( y ) x ) ) f ( A ( y ) x ) d y ) ∥ L q 2 ( E k ; w ) ≤ ∫ ℍ n Φ ( y ) | y | h Q ∥ ( b ( x ) − b ( A ( y ) x ) ) f ( A ( y ) x ) ∥ L q 2 ( E k ; w ) d y ≤ ∫ ℍ n Φ ( y ) | y | h Q ∥ ( b ( x ) − b B k ) f ( A ( y ) x ) ∥ L q 2 ( E k ; w ) d y + ∫ ℍ n Φ ( y ) | y | h Q ∥ ( b ( A ( y ) x ) − b ∥ A ( y ) ∥ B k ) f ( A ( y ) x ) ∥ L q 2 ( E k ; w ) d y + ∫ ℍ n Φ ( y ) | y | h Q ∥ ( b B k − b ∥ A ( y ) ∥ B k ) f ( A ( y ) x ) ∥ L q 2 ( E k ; w ) d y = I 1 + I 2 + I 3 .

While targeting I 1, we first compute ∥ ( b ( x ) − b ( A ( y ) x ) ) f ( A ( y ) x ) ∥ L q 2 ( E k ; w ) . The condition s > q 2 r w / ( r w − 1 ) implies that there exist 1 < r < r w such that s = q 2 r ′ . Therefore, by the Hölder inequality and the reverse Hölder condition, we have

(3.2) ∥ ( b ( ⋅ ) − b B k ) f ( A ( y ) ⋅ ) ∥ L q 2 ( E k ; w ) = ( ∫ E k | ( b ( x ) − b B k ) f ( A ( y ) x ) | s d x ) 1 / s ( ∫ E k w ( x ) r d x ) 1 / r q 2 ≼ | B k | − 1 / s w ( B k ) 1 / q 2 ∥ ( b ( ⋅ ) − b B k ) f ( A ( y ) ⋅ ) ∥ L s ( E k ) .

Next, using the condition 1 / s = 1 / q + 1 / q 1 , we can have

(3.3) ∥ ( b ( ⋅ ) − b B k ) f ( A ( y ) ⋅ ) ∥ L s ( E k ) ≤ ∥ b ( ⋅ ) − b B k ∥ L q ( B k ) ∥ f ( A ( y ) ⋅ ) ∥ L q 1 ( B k ) .

In second factor, on the right side of inequality (3.3), a change of variables along with Proposition 2.4 yields

(3.4) ∥ f ( A ( y ) ⋅ ) ∥ L q 1 ( B k ) = | det A − 1 ( y ) | 1 / q 1 ( ∫ A ( y ) B k | f ( x ) | q 1 d x ) 1 / q 1 ≼ | det A − 1 ( y ) | 1 / q 1 | B ( 0 , 2 k ∥ A ( y ) ∥ ) | 1 / q 1 × ( 1 w ( B ( 0 , 2 k ∥ A ( y ) ∥ ) ) ∫ B ( 0 , 2 k ∥ A ( y ) ∥ ) | f ( x ) | q 1 w ( x ) d x ) 1 / q 1 ≼ ( | det A − 1 ( y ) | ∥ A ( y ) ∥ Q | B k | ) 1 / q 1 w ( ∥ A ( y ) ∥ B k ) − 1 / q 1 ∥ f ∥ L q 1 ( ∥ A ( y ) ∥ B k ; w ) .

Similarly, the other factor on the right hand of inequality (3.3), in view of Proposition 2.4, gives

(3.5) ∥ b ( ⋅ ) − b B k ∥ L q ( B k ) ≼ | B k | 1 / q ∥ b ∥ C M ̇ O q ( ℍ n ; w ) .

Inequalities (3.2)–(3.5) together yield

∥ ( b ( ⋅ ) − b B k ) f ( A ( y ) ⋅ ) ∥ L q 2 ( E k ; w ) ≼ ∥ b ∥ C M ̇ O q ( ℍ n ; w ) ∥ f ∥ L q 1 ( ∥ A ( y ) ∥ B k ; w ) ( | det A − 1 ( y ) | ∥ A ( y ) ∥ Q ) 1 / q 1 w ( B k ) 1 / q 2 w ( ∥ A ( y ) ∥ B k ) 1 / q 1 .

Hence, we obtain the following estimate for I 1 :

I 1 ≼ ∥ b ∥ C M ̇ O q ( ℍ n ; w ) ∫ ℍ n | Φ ( y ) | | y | h Q ( | det A − 1 ( y ) | ∥ A ( y ) ∥ Q ) 1 / q 1 w ( B k ) 1 / q 2 w ( ∥ A ( y ) ∥ B k ) 1 / q 1 ∥ f ∥ L q 1 ( ∥ A ( y ) ∥ B k ; w ) d y .

Next, we fix to estimate I 2 , which is given by

I 2 = ∫ ℍ n | Φ ( y ) | | y | h Q ∥ ( b ( A ( y ) ⋅ ) − b ∥ A ( y ) ∥ B k ) f ( A ( y ) ⋅ ) ∥ L q 2 ( E k ; w ) d y .

Since s = q 2 r ′ , therefore, we infer from (3.2) that

(3.6) ∥ ( b ( A ( y ) ⋅ ) − b ∥ A ( y ) ∥ B k ) f ( A ( y ) ⋅ ) ∥ L q 2 ( E k ; w ) ≼ | B k | − 1 / s w ( B k ) 1 / q 2 ∥ ( b ( A ( y ) ⋅ ) − b ∥ A ( y ) ∥ B k ) f ( A ( y ) ⋅ ) ∥ L s ( E k ) .

Applying the change of variables formula, Proposition 2.4 and Hölder’s inequality, we have

(3.7) ∥ ( b ( A ( y ) . ) − b ∥ A ( y ) ∥ B k ) f ( A ( y ) . ) ∥ L s ( E k ) = | det A − 1 ( y ) | 1 / s ( ∫ A ( y ) B k | ( b ( x ) − b ∥ A ( y ) ∥ B k ) f ( x ) | s d x ) 1 / s ≼ | det A − 1 ( y ) | 1 / s | ∥ A ( y ) ∥ B k | 1 / s ( 1 w ( ∥ A ( y ) ∥ B k ) ∫ ∥ A ( y ) ∥ B k | ( b ( x ) − b ∥ A ( y ) ∥ B k ) f ( x ) | s w ( x ) d x ) 1 / s ≼ | det A − 1 ( y ) | 1 / s | B k | 1 / s ∥ A ( y ) ∥ Q 1 / s w ( ∥ A ( y ) ∥ B k ) − 1 / s × ( ∫ ∥ A ( y ) ∥ B k | b ( x ) − b ∥ A ( y ) ∥ B k | q w ( x ) d x ) 1 / q ( ∫ ∥ A ( y ) ∥ B k | f ( x ) | q 1 w ( x ) d x ) 1 / q 1 ≼ | det A − 1 ( y ) | 1 / s | B k | 1 / s ∥ A ( y ) ∥ Q 1 / s w ( ∥ A ( y ) ∥ B k ) − 1 / q 1 ∥ f ∥ L q 1 ( ∥ A ( y ) ∥ B k ; w ) ∥ b ∥ C M ̇ O q ( ℍ n ; w ) .

By virtue of (3.6) and (3.7), the expression for I 2 assumes the following form:

I 2 ≼ ∥ b ∥ C M ̇ O q ( ℍ n , w ) ∫ ℍ n | Φ ( y ) | | y | h Q ( | det A − 1 ( y ) | ∥ A ( y ) ∥ Q ) 1 / s w ( B k ) 1 / q 2 w ( ∥ A ( y ) ∥ B k ) 1 / q 1 ∥ f ∥ L q 1 ( ∥ A ( y ) ∥ B k ; w ) d y .

Now, the estimation of I 3 , given by

I 3 = ∫ ℍ n | Φ ( y ) | | y | h Q ∥ f ( A ( y ) ⋅ ) ∥ L q 2 ( E k ) | b B k − b ∥ A ( y ) ∥ B k | d y ,

requires the bounds for ∥ f ( A ( y ) ⋅ ) ∥ L q 2 ( E k ) and | b B k − b ∥ A ( y ) ∥ B k | . First, we consider ∥ f ( A ( y ) ⋅ ) ∥ L q 2 ( E k , w ) . In view of the condition s = q 2 r ′ , we use the Hölder inequality and the reverse Hölder condition to obtain

(3.8) ∥ f ( A ( y ) ⋅ ) ∥ L q 2 ( E k , w ) ≤ ( ∫ B k | f ( A ( y ) x ) | q 2 w ( x ) d x ) 1 / q 2 ≤ ( ∫ B k | f ( A ( y ) x ) | s d x ) 1 / s ( ∫ B k w ( x ) r d x ) 1 / r q 2 ≼ | B k | − 1 / s w ( B k ) 1 / q 2 ∥ f ( A ( y ) ⋅ ) ∥ L s ( B k ) .

Furthermore, the condition 1 / s = 1 / q + 1 / q 1 and inequality (3.4) help us to write

(3.9) ∥ f ( A ( y ) ⋅ ) ∥ L s ( B k ) = | B k | 1 / q ∥ f ( A ( y ) ⋅ ) ∥ L q 1 ( B k ) ≼ | B k | 1 / s ( | det A − 1 ( y ) | ∥ A ( y ) ∥ Q ) 1 / q 1 w ( ∥ A ( y ) ∥ B k ) − 1 / q 1 ∥ f ∥ L q 1 ( ∥ A ( y ) ∥ B k ; w ) .

We combine inequalities (3.8) and (3.9) to substitute the result in the expression for I 3 , which now becomes

I 3 ≼ ∫ ℍ n | Φ ( y ) | | y | h Q ( | det A − 1 ( y ) | ∥ A ( y ) ∥ Q ) 1 / q 1 w ( B ( 0 , 2 k ) ) 1 / q 2 w ( ∥ A ( y ) ∥ ( B k ) ) 1 / q 1 ∥ f ∥ L q 1 ( w ( ∥ A ( y ) ∥ B k ) , w ) | b B k − b ∥ A ( y ) ∥ B k | d y .

Now, it turns to bound | b B k − b ∥ A ( y ) ∥ B k | . For this purpose, we split the integral as follows:

I 3 ≼ ∫ ∥ A ( y ) ∥ < 1 | b B k − b ∥ A ( y ) ∥ B k | Ψ ( y ) d y + ∫ ∥ A ( y ) ∥ ≥ 1 | b B k − b ∥ A ( y ) ∥ B k | Ψ ( y ) d y = I 31 + I 32 ,

where, for the convenience’s sake, we used the following notation:

Ψ ( y ) = | Φ ( y ) | | y | h Q ( | det A − 1 ( y ) | ∥ A ( y ) ∥ Q ) 1 / q 1 w ( B ( 0 , 2 k ) ) 1 / q 2 w ( ∥ A ( y ) ∥ ( B k ) ) 1 / q 1 ∥ f ∥ L q 1 ( ∥ A ( y ) ∥ B k ; w ) .

Further decomposition of integral for I 31 results in:

I 31 = ∑ j = 0 ∞ ∫ 2 − j − 1 ≤ ∥ A ( y ) ∥ < 2 − j Ψ ( y ) { ∑ i = 1 j | b 2 − 1 B k − b 2 − i + 1 B k | + | b 2 − j B k − b ∥ A ( y ) ∥ B k | } d y .

The first term inside the curly brackets can be approximated using Proposition 2.4, that is,

| b 2 − i B k − b 2 − i + 1 B k | ≤ 1 | 2 − i B k | ∫ 2 − i B k | b ( y ) − b 2 − i + 1 B k | d y ≤ 1 w ( 2 − i B k ) ∫ 2 − i B k | b ( y ) − b 2 − i + 1 B k | w ( y ) d y ≤ 1 w ( 2 − i B k ) ( ∫ 2 − i + 1 B k | b ( y ) − b 2 − i + 1 B k | q w ( y ) d y ) 1 q ( ∫ 2 − i + 1 B k w ( y ) d y ) 1 / q ′ ≤ w ( 2 − i + 1 B k ) w ( 2 − i B k ) ( 1 w ( 2 − i + 1 B k ) ∫ 2 − i + 1 B k | b ( y ) − b 2 − i + 1 B k | q w ( y ) d y ) 1 q ≼ ∥ b ∥ C M ̇ O q ( ℍ n ; w ) .

Similarly, for the second term inside the curly brackets in the expression of I 31 , we have

| b 2 − j B k − b ∥ A ( y ) ∥ B k | ≼ ∥ b ∥ C M ̇ O q ( ℍ n ; w ) .

Therefore, we finish the estimation of I 31 by writing

I 31 ≼ ∥ b ∥ C M ̇ O q ( ℍ n ; w ) ∑ j = 0 ∞ ∫ 2 − j − 1 ≤ ∥ A ( y ) ∥ < 2 − j Ψ ( y ) ( j + 1 ) d y ≼ ∥ b ∥ C M ̇ O q ( ℍ n ; w ) ∫ ∥ A ( y ) ∥ < 1 Ψ ( y ) log 2 ∥ A ( y ) ∥ d y .

In a similar fashion, the integral I 32 gives us

I 32 = ∫ ∥ A ( y ) ∥ ≥ 1 Ψ ( y ) | b B k − b ∥ A ( y ) ∥ B k | d y = ∑ j = 0 ∞ ∫ 2 j ≤ ∥ A ( y ) ∥ < 2 j + 1 Ψ ( y ) { ∑ i = 1 j | b 2 i B k − b 2 i + 1 B k | + | b 2 j + 1 B k − b ∥ A ( y ) ∥ B k | } d y ≼ ∥ b ∥ C M ̇ O q ( ℍ n ; w ) ∫ ∥ A ( y ) ∥ ≥ 1 Ψ ( y ) log 2 ∥ A ( y ) ∥ d y .

A combination of expressions for I 1, I 2, I 31 and I 32 gives

∥ T Φ , A b f ∥ L q 2 ( E k ; w ) ≼ ∥ b ∥ C M ̇ O q ( ℍ n ; w ) ∫ ℍ n | Φ ( y ) | | y | h Q ( | det A − 1 ( y ) | ∥ A ( y ) ∥ Q ) 1 / q 1 × ( 1 + | det A − 1 ( y ) | 1 / q ∥ A ( y ) ∥ Q / q ) w ( B ( 0 , 2 k ) ) 1 / q 2 w ( ∥ A ( y ) ∥ B k ) ) 1 / q 1 ∥ f ∥ L q 1 ( ∥ A ( y ) ∥ B k ; w ) × max { log 2 ∥ A ( y ) ∥ , log ( 2 ∥ A ( y ) ∥ ) } d y .

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:

Ψ ˜ ( y ) = | Φ ( y ) | | y | h Q ( | det A − 1 ( y ) | ∥ A ( y ) ∥ Q ) 1 / q 1 ( 1 + | det A − 1 ( y ) | 1 / q ∥ A ( y ) ∥ Q / q ) max { log 2 ∥ A ( y ) ∥ , log 2 ∥ A ( y ) ∥ } .

Then,

∥ T Φ , A b f ∥ L q 2 ( E k ; w ) ≤ ∥ b ∥ C M ̇ O q ( ℍ n ; w ) ∫ ℍ n Ψ ˜ ( y ) w ( B k ) 1 / q 2 w ( ∥ A ( y ) ∥ B k ) 1 / q 1 ∥ f ∥ L q 1 ( ∥ A ( y ) ∥ B k ; w ) d y .

Finally, we take into consideration the definition of Herz space and employ the Minkowski inequality to have

(3.10) ∥ T Φ , A b f ∥ K ̇ q 2 α 2 , p ( ℍ n ; w ) = { ∑ k = − ∞ ∞ w ( B k ) α 2 p Q ∥ T Φ , A b f ∥ L q 2 ( E k ; w ) p } 1 / p ≼ ∥ b ∥ C M ̇ O q ( ℍ n ; w ) ∫ ℍ n Ψ ˜ ( y ) { ( ∑ k = − ∞ ∞ w ( B k ) α 2 / Q + 1 / q 2 w ( ∥ A ( y ) ∥ B k ) 1 / q 1 ∥ f ∥ L q 1 ( ∥ A ( y ) ∥ B k ; w ) ) p } 1 / p d y .

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 ∥ b ∥ C M ̇ O q ( ℍ n ; w ) outside the integral. Therefore, inequality (3.10) can be written as:

(3.11) ∥ T Φ , A b f ∥ K ̇ q 2 α 2 , p ( ℍ n ; w ) ≼ ∥ b ∥ C M ̇ O q ( ℍ n ; w ) ∑ j = − ∞ ∞ ∫ 2 j − 1 < ∥ A ( y ) ∥ ≤ 2 j Ψ ˜ ( y ) { ∑ k = − ∞ ∞ [ ( w ( B k ) w ( B k + j ) ) α 1 / Q + 1 / q 1 × ∑ l = − ∞ j ( w ( B k + j ) w ( B k + l ) ) α 1 / Q w ( B k + l ) α 1 / Q ∥ f ∥ L q 1 ( E k + l ; w ) ] p } 1 / p d y ,

where the condition α 1 / Q + 1 / q 1 = α 2 / Q + 1 / q 2 is utilized in obtaining the last inequality.

Under the stated condition that α 1 < 0 and l ≤ j , we use Proposition 2.3 to have

(3.12) ( w ( B k + j ) w ( B k + l ) ) α 1 / Q ≼ ( | B k + j | | B k + l | ) α 1 ( δ − 1 ) / ( Q δ ) = 2 ( j − l ) α 1 ( δ − 1 ) / δ ,

for any 1 < δ < r w .

In view of Proposition 2.3, if α 1 / Q + 1 / q 1 ≥ 0 , then

(3.13) ( w ( B k ) w ( B k + j ) ) α 1 / Q + 1 / q 1 ≼ { 2 − j Q ( α 1 / Q + 1 / q 1 ) , if j ≤ 0 , 2 − j Q ( α 1 / Q + 1 / q 1 ) ( δ − 1 ) / δ , if j > 0 ,

and if α 1 / Q + 1 / q 1 < 0 , then

(3.14) ( w ( B k ) w ( B k + j ) ) α 1 / Q + 1/ q 1 ≼ { 2 j Q ( α 1 / Q + 1/ q 1 ) ( δ − 1)/ δ , if j ≤ 0, 2 j Q ( α 1 / Q + 1/ q 1 ) , if j > 0,

for any 1 < δ < r w .

Thus, for α 1 / Q + 1 / q 1 ≥ 0 , from inequalities (3.11)–(3.13), for any 1 < δ < r w , we have

∥ T Φ , A b f ∥ K ̇ q 2 α 2 , p ( ℍ n ; w ) ≼ ∥ b ∥ C M ̇ O q ( ℍ n ; w ) ∑ j = − ∞ 0 ∫ 2 j − 1 < ∥ A ( y ) ∥ ≤ 2 j Ψ ˜ ( y ) ∥ A ( y ) ∥ − α 1 − Q / q 1 × ∑ l = − ∞ j 2 α 1 ( j − l ) ( δ − 1 ) / δ { ∑ k = − ∞ ∞ w ( B k + l ) α 1 p / Q ∥ f ∥ L q 1 ( E k + l ; w ) p } 1 / p d y + ∥ b ∥ C M ̇ O q ( ℍ n ; w ) ∑ j = 1 ∞ ∫ 2 j − 1 < ∥ A ( y ) ∥ ≤ 2 j Ψ ˜ ( y ) ∥ A ( y ) ∥ ( α 1 + Q / q 1 ) ( δ − 1 ) / δ × ∑ l = − ∞ j 2 α 1 ( j − l ) ( δ − 1 ) / δ { ∑ k = − ∞ ∞ w ( B k + l ) α 1 p / Q ∥ f ∥ L q 1 ( E k + l ; w ) p } 1 / p d y .

Replacing Ψ ˜ ( y ) with its value in the above inequality, we get

∥ T Φ , A b f ∥ K ̇ q 2 α 2 , p ( ℍ n ; w ) ≼ ∥ b ∥ C M ̇ O q ( ℍ n ; w ) ∥ f ∥ K ̇ q 1 α 1 , p ( ℍ n ; w ) × { ∫ ∥ A ( y ) ∥ < 1 | Φ ( y ) | | y | h Q ( 1 + | det A − 1 ( y ) | 1 / q ∥ A ( y ) ∥ Q / q ) | det A − 1 ( y ) | 1 / q 1 ∥ A ( y ) ∥ − α 1 log 2 ∥ A ( y ) ∥ d y + ∫ ∥ A ( y ) ∥ ≥ 1 | Φ ( y ) | | y | h Q ( 1 + | det A − 1 ( y ) | 1 / q ∥ A ( y ) ∥ Q / q ) × | det A − 1 ( y ) | 1 / q 1 ∥ A ( y ) ∥ Q / q 1 − ( α 1 + Q / q 1 ) ( δ − 1 ) / δ log ( 2 ∥ A ( y ) ∥ ) d y } .

This completes the proof of Theorem 3.1(i).

Similarly, when α 1 / Q + 1 / q 1 < 0 , 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.

3.2 Proof of Theorem 3.2

Following the proof of Theorem 3.1, we write:

∥ T Φ , A b ∥ L q 2 ( E k ; v ) ≤ J 1 + J 2 + J 3 ,

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

J 1 ≤ ∫ ℍ n | Φ ( y ) | | y | h Q ( ∫ E k | b ( x ) − b B k | q v ( x ) d x ) 1 / q ( ∫ E k | f ( A ( y ) x ) | q 1 v ( x ) d x ) 1 / q 1 d y ≤ v ( B k ) 1 / q ∥ b ∥ C M ̇ O q ( ℍ n ; v ) ∫ ℍ n | Φ ( y ) | | y | h Q | det A − 1 ( y ) | 1 / q 1 ( ∫ A ( y ) E k | f ( z ) | q 1 v ( A − 1 ( y ) z ) d z ) 1 / q 1 d y .

Using Proposition 2.8, we get

J 1 ≤ v ( B k ) 1 / q ∥ b ∥ C M ̇ O q ( ℍ n ; v ) ∫ ℍ n | Φ ( y ) | | y | h Q | det A − 1 ( y ) | 1 / q 1 ( ∫ A ( y ) E k | f ( x ) | q 1 G ( A − 1 ( y ) , β / q 1 ) v ( x ) d x ) 1 / q 1 d y ≤ v ( B k ) 1 / q ∥ b ∥ C M ̇ O q ( ℍ n ; v ) ∫ ℍ n | Φ ( y ) | | y | h Q | det A − 1 ( y ) | 1 / q 1 G ( A − 1 ( y ) , β / q 1 ) ∥ f ∥ L q 1 ( ∥ A ( y ) ∥ E k ; v ) d y .

Next, the expression for J 2 can be written as:

(3.15) J 2 = ∫ ℍ n | Φ ( y ) | | y | h Q ∥ ( b ( A ( y ) ⋅ ) − b ∥ A ( y ) ∥ B k ) f ( A ( y ) ⋅ ) ∥ L q 2 ( E k ; v ) d y .

Changing variables and using the condition q 2 / q + q 2 / q 1 = 1 , we get

(3.16) ∥ ( b ( A ( y ) x ) − b ∥ A ( y ) ∥ B k ) ( f ( A ( y ) . ) ) ∥ L q 2 ( E k ; v ) = ( ∫ E k | ( b ( A ( y ) x ) − b ∥ A ( y ) ∥ B k ) f ( A ( y ) x ) | q 2 v ( x ) d x ) 1 / q 2 = | det A − 1 ( y ) | 1 / q 2 G ( A − 1 ( y ) , β / q 2 ) ( ∫ A ( y ) E k | ( b ( x ) − b ∥ A ( y ) ∥ B k ) f ( x ) | q 2 v ( x ) d x ) 1 / q 2 ≤ | det A − 1 ( y ) | 1 / q 2 G ( A − 1 ( y ) , β / q 2 ) ( ∫ ∥ A ( y ) ∥ B k | b ( x ) − b ∥ A ( y ) ∥ B k | q v ( x ) d x ) 1 / q ( ∫ A ( y ) E k | f ( x ) | q 1 v ( x ) d x ) 1 / q 1 = | det A − 1 ( y ) | 1 / q 2 G ( A − 1 ( y ) , β / q 2 ) v ( ∥ A ( y ) ∥ B k ) 1 / q ∥ b ∥ C M ̇ O q ( ℍ n ; v ) ∥ f ∥ L q 1 ( A ( y ) E k ; v ) .

It is easy to see that v ( ∥ A ( y ) ∥ B k ) = ∥ A ( y ) ∥ Q + β v ( B k ) . Using properties (2.5) and (3.16), inequality (3.15) becomes:

J 2 = v ( B k ) 1 / q ∥ b ∥ C M ̇ O q ( ℍ n ; v ) ∫ ℍ n | Φ ( y ) | | y | h Q | det A − 1 ( y ) | 1 / q 2 G ( A − 1 ( y ) , β / q ) G ( A − 1 ( y ) , β / q 1 ) ∥ A ( y ) ∥ ( Q + β ) / q ∥ f ∥ L q 1 ( A ( y ) E k ; v ) d y .

It remains to estimate J 3 . A change of variables following the Hölder inequality and Proposition 2.8 gives us

J 3 = ∫ ℍ n | Φ ( y ) | | y | h Q ∥ ( b B k − b ∥ A ( y ) ∥ B k ) f ( A ( y ) ⋅ ) ∥ L q 2 ( E k ; v ) d y = ∫ ℍ n | Φ ( y ) | | y | h Q ∥ f ( A ( y ) x ) ∥ L q 2 ( E k ; v ) | b B k − b ∥ A ( y ) ∥ B k | d y ≤ ∫ ℍ n | Φ ( y ) | | y | h Q | det A − 1 ( y ) | 1 / q 2 G ( A − 1 ( y ) , β / q 2 ) v ( ∥ A ( y ) ∥ B k ) 1 / q ∥ f ∥ L q 1 ( A ( y ) E k ; v ) | b B k − b ∥ A ( y ) ∥ B k | d y .

Next, if ∥ A ( y ) ∥ < 1 , then there exists an integer j ≥ 0 , such that

2 − j − 1 ≤ ∥ A ( y ) ∥ < 2 − j .

Therefore,

| b B k − b ∥ A ( y ) ∥ B k | ≤ ∑ i = 1 j | b 2 − i B k − b 2 − i + 1 B k | + | b 2 − j B k − b A ( y ) B k | ≤ ∥ b ∥ C M ̇ O q ( ℍ n ; v ) log 2 ∥ A ( y ) ∥ .

Similarly, for ∥ A ( y ) ∥ ≥ 1 , we have

| b B k − b ∥ A ( y ) ∥ B k | ≤ ∥ b ∥ C M ̇ O q ( ℍ n ; v ) log 2 ∥ A ( y ) ∥ .

Hence,

J 3 ≼ v ( B k ) 1 / q ∥ b ∥ C M ̇ O q ( ℍ n ; v ) ∫ ℍ n | Φ ( y ) | | y | h Q | det A − 1 ( y ) | 1 / q 1 G ( A − 1 ( y ) , β / q 2 ) × G ( A − 1 ( y ) , β / q ) ∥ A ( y ) ∥ ( Q + β ) / q ( log 2 ∥ A ( y ) ∥ χ { ∥ A ( y ) ∥ < 1 } + log 2 ∥ A ( y ) ∥ χ { ∥ A ( y ) ∥ ≥ 1 } ) ∥ f ∥ L q 1 ( A ( y ) E k ; v ) d y .

Thus, combining J 1, J 2 and J 3, we get

(3.17) ∥ T Φ , A b ∥ L q 2 ( E k ; v ) ≼ v ( B k ) 1 / q ∥ b ∥ C M ̇ O q ( ℍ n ; v ) ∫ ℍ n | Φ ( y ) | | y | h Q | det A − 1 ( y ) | 1 / q 1 × G ( A − 1 ( y ) , β / q 1 ) ( 1 + | det A − 1 ( y ) | 1 / q G ( A − 1 ( y ) , β / q ) ∥ A ( y ) ∥ ( Q + β ) / q ) × ( log 2 ∥ A ( y ) ∥ χ { ∥ A ( y ) ∥ < 1 } + log 2 ∥ A ( y ) ∥ χ { ∥ A ( y ) ∥ ≥ 1 } ) ∥ f ∥ L q 1 ( A ( y ) E k ; v ) d y .

For the approximation of ∥ f ( ⋅ ) ∥ L q ( A ( y ) C k ) , we consider the method used in [25]. Hence, the definition of E k and (2.2) imply that

A ( y ) E k ⊂ { x : ∥ A − 1 ( y ) ∥ − 1 2 k − 1 < | x | h < ∥ A ( y ) ∥ 2 k } .

Now, there exists an integer l such that for any y ∈ supp ( Φ ) , we have

(3.18) 2 l < ∥ A − 1 ( y ) ∥ − 1 < 2 l + 1 .

Finally, the inequality ∥ A − 1 ( y ) ∥ − 1 ≤ ∥ A ( y ) ∥ implies that there exists a non-negative integer m satisfying:

(3.19) 2 l + m < ∥ A ( y ) ∥ < 2 l + m + 1 .

We infer from (3.18) and (3.19) that:

log 2 ( ∥ A ( y ) ∥ ∥ A − 1 ( y ) ∥ / 2 ) < m < log 2 ( 2 ∥ A ( y ) ∥ ∥ A − 1 ( y ) ∥ ) .

Therefore,

A ( y ) E k ⊂ { x : 2 l + k − 1 < | x | h < 2 k + l + m + 1 } .

Hence,

(3.20) ∥ f ∥ L q 1 ( A ( y ) E k ; v ) ≤ ∑ j = l l + m + 1 ∥ f ∥ L q 1 ( E k + j ; v ) .

Incorporating inequality (3.20) into inequality (3.17), we obtain

(3.21) ∥ T Φ , A b ∥ L q 2 ( E k ; v ) ≤ v ( B k ) 1 / q ∥ b ∥ C M ̇ O q ( ℍ n ; v ) ∫ ℍ n Θ ( y ) ∑ j = l l + m + 1 ∥ f ∥ L q 1 ( E k + j ; v ) d y ,

where

Θ ( y ) = | Φ ( y ) | | y | h Q | det A − 1 ( y ) | 1 / q 1 ( log 2 ∥ A ( y ) ∥ χ { ∥ A ( y ) ∥ < 1 } + log 2 ∥ A ( y ) ∥ χ { ∥ A ( y ) ∥ ≥ 1 } ) × G ( A − 1 ( y ) , β / q 1 ) ( 1 + | det A − 1 ( y ) | 1 / q G ( A − 1 ( y ) , β / q ) ∥ A ( y ) ∥ ( Q + β ) / q ) .

Using the Minkowski inequality and the condition 1 / q + α 2 / Q = α 1 / Q yields

∥ T Φ , A b ∥ K ̇ q 2 α 2 , p ( ℍ n ; v ) ≼ ∥ b ∥ C M ̇ O q ( ℍ n ; v ) { ∑ k = − ∞ ∞ ( v ( B k ) 1 / q + α 2 / Q ∫ ℍ n Θ ( y ) ∑ j = l l + m + 1 ∥ f ∥ L q 1 ( E k + j ; v ) d y ) p } 1 / p ≼ ∥ b ∥ C M ̇ O q ( ℍ n ; v ) ∫ ℍ n Θ ( y ) ∑ j = l l + m + 1 v ( B − j ) α 1 / Q { ∑ k = − ∞ ∞ ( v ( B k + j ) α 1 / Q ∥ f ∥ L q 1 ( E k + j ; v ) ) p } 1 / p d y ≼ ∥ b ∥ C M ̇ O q ( ℍ n ; v ) ∥ f ∥ K ̇ q 1 α 1 , p ( ℍ n ; v ) ∫ ℍ n Θ ( y ) ∑ j = l l + m + 1 v ( B − j ) α 1 / Q d y .

It is easy to see that

∑ j = l l + m + 1 v ( B − j ) α 1 / Q ≃ ∑ j = l l + m + 1 2 − j α 1 ( Q + β ) / Q .

Next, for α 1 = 0 ,

∑ j = l l + m + 1 2 − j α 1 ( Q + β ) / Q = m + 2 ≼ 1 + log 2 ( ∥ A − 1 ( y ) ∥ ∥ A ( y ) ∥ ) ,

and for α 1 ≠ 0 ,

∑ j = l l + m + 1 2 − j α 1 ( Q + β ) / Q ≃ 2 − l α 1 ( Q + β ) / Q ≼ { ∥ A − 1 ( y ) ∥ α 1 ( Q + β ) / Q , if α 1 > 0 , ∥ A ( y ) ∥ − α 1 ( Q + β ) / Q , if α 1 < 0 , = G ( A − 1 ( y ) , α 1 ( Q + β ) / Q ) .

Therefore,

∥ T Φ , A b ∥ K ̇ q 2 α 2 , p ( ℍ n ; v ) ≼ ∥ b ∥ C M ̇ O q ( ℍ n ; v ) ∥ f ∥ K ̇ q 1 α 1 , p ( ℍ n ; v ) { ∫ ℍ n Θ ( y ) ( 1 + log 2 ( ∥ A − 1 ( y ) ∥ ∥ A ( y ) ∥ ) ) d y , if α 1 = 0 , ∫ ℍ n Θ ( y ) G ( A − 1 ( y ) , α 1 ( Q + β ) / Q ) d y , if α 1 ≠ 0 , = K 3 ∥ b ∥ C M ̇ O q ( ℍ n ; v ) ∥ f ∥ K ̇ q 1 α 1 , p ( ℍ n ; v ) .

Thus, we complete the proof of Theorem 3.2.

4 Conclusion

In this article, the authors obtained the boundedness of the commutator of the matrix Hausdorff operator on the homogeneous weighted Herz space in the settings of the Heisenberg group. As an application, the authors also investigated the particular case of Muckenhoupt A p weights, namely, the power weights. Some potential directions for the future works include the boundedness of the same operator on homogeneous weighted Herz-Morrey spaces and weighted Herz-type Hardy spaces defined on homogeneous groups.

Acknowledgments

The authors would like to thank the referees for careful reading of the paper and valuable suggestions. Amjad Hussain is supported by Higher Education Commission (HEC) of Pakistan through the National Research Program for Universities (NRPU) Project No: 7098/Federal/NRPU/R&D/HEC/2017 and the Quaid-I-Azam University Research Fund (URF) Project.

  1. Conflicts of interest: The authors declare that they have no competing interest.

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Received: 2019-08-07
Revised: 2020-04-09
Accepted: 2020-04-28
Published Online: 2020-06-04

© 2020 Amna Ajaib and Amjad Hussain, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2020-0175
Keywords for this article
Hausdorff operator; commutators; Herz space; CBMO function; Heisenberg group; weights
Creative Commons
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Articles in the same Issue

  1. Regular Articles
  2. Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians
  3. Strong and weak convergence of Ishikawa iterations for best proximity pairs
  4. Curve and surface construction based on the generalized toric-Bernstein basis functions
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  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
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  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
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  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
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  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
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