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Multidimensional Hausdorff operators and commutators on Herz-type spaces

Abstract

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.

1 Introduction

Recall that for a locally integrable function Φ on (0,), the one-dimensional Hausdorff operator is defined by

h Φ f(x)= 0 Φ ( t ) t f ( x t ) dt.

The boundedness of this operator on the real Hardy space H 1 (R) was proved in [1]. Subsequently, the problem of boundedness of h Φ in H p , 0<p<1 was considered in [2, 3] and [4]. In [5], the same operator was studied on product of Hardy spaces. Due to its close relation with the summability of the classical Fourier series, it was natural to study h Φ in high-dimensional space R n . With such an objective, Chen et al. [6] considered three extensions of the one-dimensional Hausdorff operator in R n . One of them is the operator

H Φ f(x)= R n Φ ( y ) | y | n f ( x | y | ) dy.

The second multidimensional extension of the Hausdorff operator provided in [6] is the following operator:

H ˜ Φ , Ω f(x)= R n Φ ( x / | y | ) | y | n Ω ( y ) f(y)dy,

where Φ is a radial function defined on R + , and Ω( y ) is an integrable function defined on the unit sphere S n 1 . Here and in what follows, we denote H ˜ Φ , 1 = H ˜ Φ . In [6], the authors discussed the boundedness of these operators on various function spaces and found that they have better performance on Herz-type Hardy spaces H K ˙ q α , p than their performance on the Hardy spaces H p when 0<p<1.

Recently, Lin and Sun [4] defined the n-dimensional fractional Hausdorff operator initially on the Schwartz class S by

H Φ , γ = R n Φ ( | x | / | y | ) | y | n γ f(y)dy,0γ<n,

and obtained H p ( R n ) L q ( R n ) and L p ( | x | α dx) L q ( | x | α dx) boundedness for H Φ , γ . Furthermore, it is easy to show that the n-dimensional fractional Hardy operator

H γ f(x)= 1 | x | n γ | y | < | x | f(y)dy

and its adjoint operator

H γ f(x)= | y | | x | f ( y ) | y | n γ dy

are special cases of H Φ , γ if one chooses Φ(t)= Φ 1 (t)= t n + γ χ ( 1 , ) (t) and Φ(t)= Φ 2 (t)= χ ( 0 , 1 ] (t), 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 [7], 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 [8] and [9]. In [10] and [11], Fu with different co-author have considered the same problem for m-linear p-adic Hardy and classical Hardy operators, respectively.

As the development of linear as well as multilinear integral operators, their commutators have been well studied. A well-known theorem by Coifman et al. [12] states that the commutator [b,T] defined by

[b,T](f)(x)=b(x)T(f)(x)T(bf)(x),

where T is a Calderón-Zygmund singular integral operator, is bounded on L p ( R n ), 1<p<, if and only if bBMO( R n ). One can find a vast literature devoted to the study of the boundedness properties for such commutators. More recently, Gao and Jia [13] defined the commutator of the high-dimensional Hausdorff operator as

H ˜ Φ , b f(x)= R n Φ ( x / | y | ) | y | n ( b ( x ) b ( y ) ) f(y)dy

and studied it on Lebesgue and Herz-type spaces.

Motivated by the work cited above, in this paper, we obtain some sharp bounds for H Φ on Herz-type spaces. Furthermore, we give a sufficient condition for the boundedness of commutators generated by the Lipschitz functions b and the n-dimensional fractional Hausdorff operators H Φ , γ , defined by

H Φ , γ b f(x)= R n Φ ( | x | / | y | ) | y | n γ ( b ( x ) b ( y ) ) f(y)dy,

on Morrey-Herz space. Following [14], our method is direct and straightforward. In addition, the problem of boundedness of commutators of n-dimensional fractional Hardy operators [15] 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 kZ, we set B k ={x R n :|x| 2 k }, C k = B k B k 1 .

Definition 1.1 ([16])

Let αR, 0<p, 0<q<. The homogeneous Herz space K ˙ q α , p ( R n ) is defined by

K ˙ q α , p ( R n ) = { f L loc q ( R n { 0 } ) : f K ˙ q α , p ( R n ) < } ,

where

f K ˙ q α , p ( R n ) = ( k = 2 k α p f χ C k L q ( R n ) p ) 1 / p ,

with the usual modification made when p=.

Remark 1.2 K ˙ q α , p ( R n ) is the generalization of L q ( R n , | x | α ), the Lebesgue space with power weights. Also, it is easy to see that K ˙ q 0 , q ( R n )= L q ( R n ) and K ˙ q α / q , q ( R n )= L q ( R n , | x | α ).

Definition 1.3 Let αR, 0<p, 0<q< and λ0. The homogeneous Morrey-Herz space M K ˙ p , q α , λ ( R n ) is defined by

M K ˙ p , q α , λ ( R n ) = { f L loc q ( R n { 0 } ) : f M K ˙ p , q α , λ ( R n ) < } ,

where

f M K ˙ p , q α , λ ( R n ) = sup k 0 Z 2 k 0 λ ( k = k 0 2 k α p f χ C k L q ( R n ) p ) 1 / p ,

with the usual modification made when p=.

In [17] the Morrey space M q λ ( R n ) is defined by

M q λ ( R n ) = { f L loc q ( R n ) : sup λ > 0 , x R n 1 r λ | x y | < r | f ( y ) | q d y < } .

Obviously, M K ˙ p , q α , 0 ( R n )= K ˙ q α , p ( R n ) and M q λ ( R n )M K ˙ q , q 0 , λ ( R n ).

Definition 1.4 ([18])

Let 0<β<1. The Lipschitz space Λ ˙ β ( R n ) is defined by

f Λ ˙ β ( R n ) = sup x , h R n | f ( x + h ) f ( x ) | | h | β <.

In the next section we will obtain some sharp bounds for H Φ . Finally, the Lipschitz estimates for the commutators H Φ , γ b will be studied in the last section.

2 Sharp bounds for H Φ

The main result of this section is as follows:

Theorem 2.1 Let αR, λ0, 1<p,q<. If Φ is a non-negative valued function and

A 1 = R n Φ ( y ) | y | n | y | α + n q λ dy<,

then H Φ is a bounded operator on M K ˙ p , q α , λ ( R n ).

Conversely, suppose that H Φ is a bounded operator on M K ˙ p , q α , λ ( R n ). If λ=0, or if λ>max{0,α}, then A 1 <. In addition, the operator H Φ satisfies the following operator norm:

H Φ M K ˙ p , q α , λ ( R n ) M K ˙ p , q α , λ ( R n ) = A 1 .

Proof By definition and using Minkowski’s inequality

H Φ f M K ˙ p , q α , λ ( R n ) = sup k 0 Z 2 k 0 λ { k = k 0 2 k α p ( H Φ f ) χ C k L q ( R n ) p } 1 p = sup k 0 Z 2 k 0 λ { k = k 0 2 k α p ( C k | R n Φ ( y ) | y | n f ( x | y | ) d y | q d x ) p q } 1 p sup k 0 Z 2 k 0 λ { k = k 0 2 k α p ( j = C j Φ ( y ) | y | n f ( | y | ) L q ( C k ) d y ) p } 1 p .

Now, it is easy to see that for y C j [6]

f ( | y | ) L q ( C k ) = | y | n q f χ C k j L q ( R n ) .

Therefore, by Minkowski’s inequality, we get

H Φ f M K ˙ p , q α , λ ( R n ) j = C j Φ ( y ) | y | n | y | n q sup k 0 Z 2 k 0 λ { k = k 0 2 k α p f χ C k j L q ( R n ) p } 1 p d y f M K ˙ p , q α , λ ( R n ) j = C j Φ ( y ) | y | n | y | n q 2 j ( λ α ) d y f M K ˙ p , q α , λ ( R n ) R n Φ ( y ) | y | n | y | α + n q λ d y .

Hence, we conclude that

H Φ M K ˙ p , q α , λ ( R n ) M K ˙ p , q α , λ ( R n ) A 1 .
(2.1)

Conversely, suppose that H Φ is bounded on M K ˙ p , q α , λ ( R n ). Then we consider the following two cases.

Case I: λ>0.

In this case, we choose f 0 L loc q ( R n {0}), such that

f 0 (x)= | x | α n q + λ .

An easy computation shows that

f 0 χ C k L q ( R n ) = 2 k ( λ α ) [ ( 1 2 q ( α λ ) ) | S n 1 | λ α ] 1 q ,

where | S n 1 | denotes the volume of unit sphere S n 1 . Now, by definition

f 0 M K ˙ p , q α , λ ( R n ) = sup k 0 Z 2 k 0 λ { k = k 0 2 k α p f 0 χ C k L q ( R n ) p } 1 p = [ ( 1 2 q ( α λ ) ) | S n 1 | λ α ] 1 q sup k 0 Z 2 k 0 λ { k = k 0 2 k λ p } 1 p = [ ( 1 2 q ( α λ ) ) | S n 1 | λ α ] 1 q 2 λ ( 2 λ p 1 ) 1 p < .

On the other hand, it is easy to check that

H Φ f 0 (x)= f 0 (x) R n Φ ( y ) | y | n | y | α + n q λ dy.

Under the assumption that H Φ is bounded on M K ˙ p , q α , λ ( R n ), we get

R n Φ ( y ) | y | n | y | α + n q λ dy H Φ M K ˙ p , q α , λ ( R n ) M K ˙ p , q α , λ ( R n ) <.
(2.2)

Furthermore, combing (2.2) with (2.1), we immediately obtain

H Φ M K ˙ p , q α , λ ( R n ) M K ˙ p , q α , λ ( R n ) = R n Φ ( y ) | y | n | y | α + n q λ dy.

Case II: λ=0.

In this case, we have M K ˙ p , q α , λ ( R n )= K ˙ q α , p ( R n ). To prove the converse relation we take the sequence of function { f m } (m0) as follows:

f m (x)={ 0 if  | x | < 1 , | x | α n q 2 m if  | x | 1 .

Obviously for k<0, we have f m χ C k =0. Hence, for k0, we obtain

f m χ C k L q ( R n ) q = C k | x | α n q 2 m d x = ( 2 q ( α + 2 m ) 1 ) q ( α + 2 m ) | S | n 1 2 k q ( α + 2 m ) .

Therefore,

f m K ˙ q α , p ( R n ) = { k = 2 k α p f m χ C k L q ( R n ) p } 1 p = [ ( 2 q ( α + 2 m ) 1 ) q ( α + 2 m ) | S | n 1 ] 1 q { k = 0 2 k α p 2 k p ( α + 2 m ) } 1 p = [ ( 2 q ( α + 2 m ) 1 ) q ( α + 2 m ) | S | n 1 ] 1 q 2 1 2 m ( 2 p 2 m 1 ) 1 p < .

On the other hand, we write

H Φ f m (x)={ 0 if  | x | < 1 , | x | α n q 2 m | y | | x | Φ ( y ) | y | n | y | α + n q + 2 m d y if  | x | 1 .

This implies that ( H Φ f m ) χ C k =0 for k<0. Thus for k0, we get

( H Φ f m ) χ C k L q ( R n ) q = C k ( | x | α n q 2 m | y | | x | Φ ( y ) | y | n | y | α + n q + 2 m d y ) q dx.

Therefore, for any mk, we have

( H Φ f m ) χ C k L q ( R n ) ( C k | x | α q n 2 m q d x ) 1 q | y | 2 m 1 Φ ( y ) | y | n | y | α + n q + 2 m d y = 2 k ( α + 2 m ) [ ( 2 q ( α + 2 m ) 1 ) q ( α + 2 m ) | S | n 1 ] 1 q × | y | 2 m 1 Φ ( y ) | y | n | y | α + n q + 2 m d y .

Now, it is easy to show that

H Φ f m K ˙ q α , p ( R n ) [ ( 2 q ( α + 2 m ) 1 ) q ( α + 2 m ) | S | n 1 ] 1 q { k = m 2 k p 2 m } 1 p × | y | 2 m 1 Φ ( y ) | y | n | y | α + n q + 2 m d y = [ ( 2 q ( α + 2 m ) 1 ) q ( α + 2 m ) | S | n 1 ] 1 q { k = 0 2 k p 2 m } 1 p × 2 m 2 m | y | 2 m 1 Φ ( y ) | y | n | y | α + n q + 2 m d y = f m K ˙ q α , p ( R n ) 2 m 2 m | y | 2 m 1 Φ ( y ) | y | n | y | α + n q + 2 m d y .

Consequently,

H Φ K ˙ q α , p ( R n ) K ˙ q α , p ( R n ) 2 m 2 m | y | 2 m 1 Φ ( y ) | y | n | y | α + n q + 2 m dy.

Finally, we let m+ to obtain

H Φ K ˙ q α , p ( R n ) K ˙ q α , p ( R n ) R n Φ ( y ) | y | n | y | α + n q dy.
(2.3)

In view of (2.3) with (2.1), we get

H Φ K ˙ q α , p ( R n ) K ˙ q α , p ( R n ) = R n Φ ( y ) | y | n | y | α + n q dy.

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 H Φ , γ 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.

Theorem 3.1 Let b Λ ˙ β ( R n ), 0<β<1< q 2 < q 1 <, 0<p<, λ>0, μ=α+β+γ+ n q 2 n q 1 . If

A 2 = 0 | Φ ( t ) | t t α + n q 2 λ max { 1 , t β } dt<,

then H Φ , γ b is bounded from M K ˙ p , q 1 μ , λ ( R n ) to M K ˙ p , q 2 α , λ ( R n ) and satisfies the following inequality:

H Φ , γ b f M K ˙ p , q 2 α , λ ( R n ) C A 2 b Λ ˙ β ( R n ) f M K ˙ p , q 1 μ , λ ( R n ) .

In proving Theorem 3.1, we need the following lemmas.

Lemma 3.2 For 1<p<, we have

( H Φ , γ f ) χ C k L p ( R n ) 2 k γ | S n 1 | 0 | Φ ( t ) | t 1 + γ t n p f χ t 1 C k L p ( R n ) dt.

Proof The lemma can be proved in a way similar to Theorem 3.1 in [6]. □

Lemma 3.3 ([18])

For any x,y R n , if f Λ ˙ β ( R n ), 0<β<1, then |f(x)f(y)| | x y | β f Λ ˙ β ( R n ) . Furthermore, for any cube Q R n , sup x Q |f(x) f Q |C | Q | β n f Λ ˙ β ( R n ) , where f Q = 1 | Q | Q f.

Lemma 3.4 ([18])

Let f Λ ˙ β ( R n ), 0<β<1, Q and Q are cubes in R n . If Q Q, then

| f Q f Q |C | Q | β n f Λ ˙ β ( R n ) .

Proof of Theorem 3.1 Notice that

( H Φ , γ b f ) χ C k L q 2 ( R n ) = ( R n Φ ( | x | / | y | ) | y | n γ ( b ( x ) b ( y ) ) f ( y ) d y ) χ C k L q 2 ( R n ) ( R n Φ ( | x | / | y | ) | y | n γ ( b ( x ) b B k ) f ( y ) d y ) χ C k L q 2 ( R n ) + ( R n Φ ( | x | / | y | ) | y | n γ ( b ( y ) b B k ) f ( y ) d y ) χ C k L q 2 ( R n ) = I + J .

Let 1 r = 1 q 2 1 q 1 . Then by Hölder’s inequality, Lemma 3.2, and Lemma 3.3, we have

I ( C k | b ( x ) b B k | r d x ) 1 r ( C k | R n Φ ( | x | / | y | ) | y | n γ f ( y ) d y | q 1 d x ) 1 q 1 C | B k | β n + 1 r b Λ ˙ β ( R n ) ( H Φ , γ f ) χ C k L q 1 ( R n ) C 2 k ( β + γ + n r ) b Λ ˙ β ( R n ) 0 | Φ ( t ) | t 1 + γ t n q 1 f χ t 1 C k L q 1 ( R n ) d t .

Now, using polar coordinates, Minkowski’s inequality and Hölder’s inequality, we approximate J as

J = ( 0 S n 1 Φ ( | x | / r ) r 1 γ ( b ( r y ) b B k ) f ( r y ) d σ ( y ) d r ) χ C k L q 2 ( R n ) = ( 0 S n 1 Φ ( t ) t ( | x | t 1 ) γ ( b ( | x | t 1 y ) b B k ) f ( | x | t 1 y ) d σ ( y ) d t ) χ C k L q 2 ( R n ) 2 k γ 0 | Φ ( t ) | t 1 + γ ( S n 1 ( b ( | x | t 1 y ) b B k ) f ( | x | t 1 y ) d σ ( y ) ) χ C k L q 2 ( R n ) d t 2 k γ | S n 1 | 1 q 2 0 | Φ ( t ) | t 1 + γ ( C k S n 1 | ( b ( | x | t 1 y ) b B k ) f ( | x | t 1 y ) | q 2 d σ ( y ) d x ) 1 q 2 d t .

Again by means of polar decomposition and change of the variables, we obtain

J 2 k γ | S n 1 | 0 | Φ ( t ) | t 1 + γ ( 2 k 1 2 k s n 1 S n 1 | ( b ( s t 1 y ) b B k ) f ( s t 1 y ) | q 2 d σ ( y ) d s ) 1 q 2 d t = C 2 k γ 0 | Φ ( t ) | t 1 + γ t n q 2 ( t 1 C k | ( b ( y ) b B k ) f ( y ) | q 2 d y ) 1 q 2 d t C 2 k γ 0 | Φ ( t ) | t 1 + γ t n q 2 ( t 1 C k | ( b ( y ) b t 1 B k ) f ( y ) | q 2 d y ) 1 q 2 d t + C 2 k γ 0 | Φ ( t ) | t 1 + γ t n q 2 ( t 1 C k | ( b B k b t 1 B k ) f ( y ) | q 2 d y ) 1 q 2 d t = J 1 + J 2 .

For J 1 , using Hölder’s inequality and Lemma 3.3, we have

J 1 C 2 k γ 0 | Φ ( t ) | t 1 + γ t n q 2 ( t 1 C k | b ( x ) b t 1 B k | r d x ) 1 r ( t 1 C k | f ( y ) | q 1 d y ) 1 q 1 d t C 2 k ( β + γ + n r ) b Λ ˙ β ( R n ) 0 | Φ ( t ) | t 1 + γ t n q 1 f χ t 1 C k L q 1 ( R n ) t β d t .

Observe that if t<1, then B k t 1 B k , while the reverse is true for t>1. Hence, by Lemma 3.4, we obtain

J 2 = C 2 k γ 0 | Φ ( t ) | t 1 + γ t n q 2 ( t 1 C k | f ( y ) | q 2 d y ) 1 q 2 | b B k b t 1 B k | d t C 2 k γ | B k | 1 r 0 | Φ ( t ) | t 1 + γ t n q 1 ( t 1 C k | f ( y ) | q 1 d y ) 1 q 1 | b B k b t 1 B k | d t C 2 k ( β + γ + n r ) b Λ ˙ β ( R n ) 0 1 | Φ ( t ) | t 1 + γ t n q 1 f χ t 1 C k L q 1 ( R n ) t β d t + C 2 k ( β + γ + n r ) b Λ ˙ β ( R n ) 1 | Φ ( t ) | t 1 + γ t n q 1 f χ t 1 C k L q 1 ( R n ) d t C 2 k ( β + γ + n r ) b Λ ˙ β ( R n ) 0 | Φ ( t ) | t 1 + γ t n q 1 f χ t 1 C k L q 1 ( R n ) max { 1 , t β } d t .

Note that for t>1, 0<β<1, we have 0< t β <1. Therefore, by combining the estimates for I, J 1 , and J 2 , we get

( H Φ , γ b f ) χ C k L q 2 ( R n ) C 2 k ( β + γ + n r ) b Λ ˙ β ( R n ) × 0 | Φ ( t ) | t 1 + γ t n q 1 f χ t 1 C k L q 1 ( R n ) max { 1 , t β } d t .

Following [19], we let mZ such that m1< log 2 tm, then t 1 C k is contained in two adjacent annuli C k + m and C k + m 1 . Therefore,

( H Φ , γ b f ) χ C k L q 2 ( R n ) C 2 k ( β + γ + n r ) b Λ ˙ β ( R n ) × 0 | Φ ( t ) | t 1 + γ t n q 1 i = 0 1 f χ C k + m i L q 1 ( R n ) max { 1 , t β } d t .

Hereafter, we use the notation Φ ˜ (t)= | Φ ( t ) | t 1 + γ t n q 1 max{1, t β } for simplicity. Then for 0<p<1, we get

H Φ , γ b f M K ˙ p , q 2 α , λ ( R n ) C b Λ ˙ β ( R n ) sup k 0 Z 2 k 0 λ { k = k 0 2 k μ p ( 0 Φ ˜ ( t ) f χ C k + m L q 1 ( R n ) d t ) p } 1 p + C b Λ ˙ β ( R n ) sup k 0 Z 2 k 0 λ { k = k 0 2 k μ p ( 0 Φ ˜ ( t ) f χ C k + m 1 L q 1 ( R n ) d t ) p } 1 p = K 1 + K 2 .

Here, we approximate K 1 as

K 1 sup k 0 Z 2 k 0 λ { k = k 0 2 k μ p × ( 0 Φ ˜ ( t ) 2 ( k + m ) λ ( i = k + m 2 i μ p f χ C i L q 1 ( R n ) p ) 1 p 2 ( k + m ) ( λ μ ) d t ) p } 1 p C b Λ ˙ β ( R n ) C b Λ ˙ β ( R n ) f M K ˙ p , q 1 μ , λ ( R n ) sup k 0 Z 2 k 0 λ { k = k 0 2 k λ p ( 0 Φ ˜ ( t ) 2 m ( λ μ ) d t ) p } 1 p C b Λ ˙ β ( R n ) f M K ˙ p , q 1 μ , λ ( R n ) sup k 0 Z 2 k 0 λ { k = k 0 2 k λ p } 1 p × 0 | Φ ( t ) | t 1 + γ t n q 1 max { 1 , t β } t μ λ d t C b Λ ˙ β ( R n ) f M K ˙ p , q 1 μ , λ ( R n ) 0 | Φ ( t ) | t t α + n q 2 λ max { 1 , t β } d t .

Similarly,

K 2 C b Λ ˙ β ( R n ) f M K ˙ p , q 1 μ , λ ( R n ) sup k 0 Z 2 k 0 λ { k = k 0 2 k λ p ( 0 Φ ˜ ( t ) 2 ( m 1 ) ( λ μ ) d t ) p } 1 p C b Λ ˙ β ( R n ) f M K ˙ p , q 1 μ , λ ( R n ) 0 | Φ ( t ) | t 1 + γ t n q 1 max { 1 , t β } t μ λ d t = C b Λ ˙ β ( R n ) f M K ˙ p , q 1 μ , λ ( R n ) 0 | Φ ( t ) | t t α + n q 2 λ max { 1 , t β } d t .

Now, we consider the case 1<p<. By Minkowski’s inequality, we write

H Φ , γ b f M K ˙ p , q 2 α , λ ( R n ) C b Λ ˙ β ( R n ) sup k 0 Z 2 k 0 λ 0 Φ ˜ ( t ) { k = k 0 2 k μ p f χ C k + m 1 L q 1 ( R n ) p } 1 p d t + C b Λ ˙ β ( R n ) sup k 0 Z 2 k 0 λ 0 Φ ˜ ( t ) { k = k 0 2 k μ p f χ C k + m L q 1 ( R n ) p } 1 p d t = L 1 + L 2 .

Here, we estimate L 1 as

L 1 C b Λ ˙ β ( R n ) 0 Φ ˜ ( t ) sup k 0 Z 2 ( k 0 + m 1 ) λ { k = k 0 + m 1 2 k μ p f χ C k L q 1 ( R n ) p } 1 p 2 ( m 1 ) ( λ μ ) d t C b Λ ˙ β ( R n ) f M K ˙ p , q 1 μ , λ ( R n ) 0 | Φ ( t ) | t 1 + γ t n q 1 max { 1 , t β } t μ λ d t = C b Λ ˙ β ( R n ) f M K ˙ p , q 1 μ , λ ( R n ) 0 | Φ ( t ) | t t α + n q 2 λ max { 1 , t β } d t .

Similarly,

L 2 C b Λ ˙ β ( R n ) f M K ˙ p , q 1 μ , λ ( R n ) 0 | Φ ( t ) | t 1 + γ t n q 1 max { 1 , t β } t μ λ d t = C b Λ ˙ β ( R n ) f M K ˙ p , q 1 μ , λ ( R n ) 0 | Φ ( t ) | t t α + n q 2 λ max { 1 , t β } d t .

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.

Corollary 3.5 If α+β+γ< n q 2 +λ, then under the same conditions as in Theorem 3.1, the commutator of the n-dimensional fractional Hardy operator [15],

H γ , b f(x)= 1 | x | n + γ | y | < | x | ( b ( x ) b ( y ) ) f(y)dy,

is bounded from M K ˙ p , q 1 μ , λ ( R n ) to M K ˙ p , q 2 α , λ ( R n ).

Proof In the operator H Φ , γ b f(x), we replace

Φ(t)= Φ 1 (t)= t n + γ χ ( 1 , ) (t),

then we obtain the commutator of the n-dimensional fractional Hardy operator,

H Φ 1 , γ b f(x)= H γ , b f(x).

Hence, by Theorem 3.1

H γ , b f M K ˙ p , q 2 α , λ ( R n ) C b Λ ˙ β ( R n ) f M K ˙ p , q 1 μ , λ ( R n ) 1 t α + β + γ n q 2 λ 1 d t C b Λ ˙ β ( R n ) f M K ˙ p , q 1 μ , λ ( R n ) .

Thus, the corollary is proved. □

Corollary 3.6 If α+ n q 2 >λ, then under the same conditions as in Theorem 3.1, the commutator of the adjoint fractional Hardy operator [15],

H γ , b f(x)= | y | | x | 1 | y | n γ ( b ( x ) b ( y ) ) f(y)dy,

is bounded from M K ˙ p , q 1 μ , λ ( R n ) to M K ˙ p , q 2 α , λ ( R n ).

Proof In the operator H Φ , γ b f(x), we replace

Φ(t)= Φ 2 (t)= χ ( 0 , 1 ] (t),

then we obtain the commutator of the n-dimensional adjoint Hardy operator

H Φ 2 , γ b f(x)= H γ , b f(x).

Thus, by Theorem 3.1

H γ , b f M K ˙ p , q 2 α , λ ( R n ) C b Λ ˙ β ( R n ) f M K ˙ p , q 1 μ , λ ( R n ) 0 1 t α + n q 2 λ 1 d t . C b Λ ˙ β ( R n ) f M K ˙ p , q 1 μ , λ ( R n ) .

With this we finish the proof of Corollary 3.6. □

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Acknowledgements

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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Correspondence to Amjad Hussain.

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Keywords

  • Hausdorff operators
  • Herz-type spaces
  • Lipschitz functions
  • commutators