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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 . Subsequently, the problem of boundedness of $h Φ$ in $H p$, $0 was considered in [2, 3] and . In , 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.  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  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 , 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.

Recently, Lin and Sun  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,0≤γ

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

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

$H γ ∗ f(x)= ∫ | y | ≥ | x | f ( y ) | y | n − γ dy$

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 , 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  and . In  and , 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.  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, if and only if $b∈BMO( R n )$. One can find a vast literature devoted to the study of the boundedness properties for such commutators. More recently, Gao and Jia  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 , our method is direct and straightforward. In addition, the problem of boundedness of commutators of n-dimensional fractional Hardy operators  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 $k∈Z$, we set $B k ={x∈ R n :|x|≤ 2 k }$, $C k = B k ∖ B k − 1$.

Definition 1.1 ()

Let $α∈R$, $0, $0. 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 $α∈R$, $0, $0 and $λ≥0$. 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  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 ()

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 $α∈R$, $λ≥0$, $1. 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 $y∈ C j$ 

$∥ f ( ⋅ | y | ) ∥ L q ( C k ) = | y | n q ∥ f χ C k − j ∥ L q ( R n ) .$

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 ${ f m }$ ($m≥0$) as follows:

Obviously for $k<0$, we have $f m χ C k =0$. Hence, for $k≥0$, 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

This implies that $( H Φ f m ) χ C k =0$ for $k<0$. Thus for $k≥0$, 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 $m≤k$, 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, $λ>0$, $μ=α+β+γ+ n q 2 − n q 1$. 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, 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 . □

Lemma 3.3 ()

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 ()

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 , we let $m∈Z$ such that $m−1<− log 2 t≤m$, 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, 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. 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 ,

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

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 ,

$H γ , b ∗ f(x)= ∫ | y | ≥ | x | 1 | y | n − γ ( b ( x ) − b ( y ) ) f(y)dy,$

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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