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 $(0,\mathrm{\infty })$, the one-dimensional Hausdorff operator is defined by

$$h_{\mathrm{\Phi }}f(x)={\int }_0^{\mathrm{\infty }}\frac{\mathrm{\Phi }(t)}{t}f\left(\frac{x}{t}\right)dt.$$

The boundedness of this operator on the real Hardy space $H^1(R)$ was proved in [1]. Subsequently, the problem of boundedness of $h_{\mathrm{\Phi }}$ 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 $h_{\mathrm{\Phi }}$ 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

$$H_{\mathrm{\Phi }}f(x)={\int }_{R^n}\frac{\mathrm{\Phi }(y)}{{|y|}^n}f\left(\frac{x}{|y|}\right)dy.$$

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

$${\tilde{H}}_{\mathrm{\Phi },\mathrm{\Omega }}f(x)={\int }_{R^n}\frac{\mathrm{\Phi }(x/|y|)}{{|y|}^n}\mathrm{\Omega }\left(y^{\prime }\right)f(y)dy,$$

where Φ is a radial function defined on $R^{+}$, and $\mathrm{\Omega }(y^{\prime })$ is an integrable function defined on the unit sphere $S^{n-1}$. Here and in what follows, we denote ${\tilde{H}}_{\mathrm{\Phi },1}={\tilde{H}}_{\mathrm{\Phi }}$. 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 $H{\dot{K}}_q^{\alpha ,p}$ 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

$$H_{\mathrm{\Phi },\gamma }={\int }_{R^n}\frac{\mathrm{\Phi }(|x|/|y|)}{{|y|}^{n-\gamma }}f(y)dy,0\le \gamma <n,$$

and obtained $H^p(R^n)\to L^q(R^n)$ and $L^p({|x|}^{\alpha }dx)\to L^q({|x|}^{\alpha }dx)$ boundedness for $H_{\mathrm{\Phi },\gamma }$. Furthermore, it is easy to show that the n-dimensional fractional Hardy operator

$$H_{\gamma }f(x)=\frac{1}{{|x|}^{n-\gamma }}{\int }_{|y|<|x|}f(y)dy$$

and its adjoint operator

$$H_{\gamma }^{\ast }f(x)={\int }_{|y|\ge |x|}\frac{f(y)}{{|y|}^{n-\gamma }}dy$$

are special cases of $H_{\mathrm{\Phi },\gamma }$ if one chooses $\mathrm{\Phi }(t)={\mathrm{\Phi }}_1(t)=t^{-n+\gamma }{\chi }_{(1,\mathrm{\infty })}(t)$ and $\mathrm{\Phi }(t)={\mathrm{\Phi }}_2(t)={\chi }_{(0,1]}(t)$, 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 $L^p(R^n)$, $1<p<\mathrm{\infty }$, if and only if $b\in \mathit{BMO}(R^n)$. One can find a vast literature devoted to the study of the boundedness properties for such commutators. More recently, Gao and Jia [13] defined the commutator of the high-dimensional Hausdorff operator as

$${\tilde{H}}_{\mathrm{\Phi },b}f(x)={\int }_{R^n}\frac{\mathrm{\Phi }(x/|y|)}{{|y|}^n}(b(x)-b(y))f(y)dy$$

and studied it on Lebesgue and Herz-type spaces.

Motivated by the work cited above, in this paper, we obtain some sharp bounds for $H_{\mathrm{\Phi }}$ 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 $H_{\mathrm{\Phi },\gamma }$, defined by

$$H_{\mathrm{\Phi },\gamma }^{b}f(x)={\int }_{R^n}\frac{\mathrm{\Phi }(|x|/|y|)}{{|y|}^{n-\gamma }}(b(x)-b(y))f(y)dy,$$

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 $k\in Z$, we set $B_k=\{x\in R^n:|x|\le 2^k\}$, $C_k=B_k\mathrm{\setminus }{B}_{k-1}$.

Definition 1.1 ([16])

Let $\alpha \in R$, $0<p\le \mathrm{\infty }$, $0<q<\mathrm{\infty }$. The homogeneous Herz space ${\dot{K}}_q^{\alpha ,p}(R^n)$ is defined by

$${\dot{K}}_q^{\alpha ,p}\left(R^n\right)=\{f\in L_{\mathrm{loc}}^q(R^n\mathrm{\setminus }\{0\}):{\parallel f\parallel }_{{\dot{K}}_q^{\alpha ,p}(R^n)}<\mathrm{\infty }\},$$

where

$${\parallel f\parallel }_{{\dot{K}}_q^{\alpha ,p}(R^n)}={\left(\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{2}^{k\alpha p}{\parallel f{\chi }_{C_k}\parallel }_{L^q(R^n)}^p\right)}^{1/p},$$

with the usual modification made when $p=\mathrm{\infty }$.

Remark 1.2 ${\dot{K}}_q^{\alpha ,p}(R^n)$ is the generalization of $L^q(R^n,{|x|}^{\alpha })$, the Lebesgue space with power weights. Also, it is easy to see that ${\dot{K}}_q^{0,q}(R^n)=L^q(R^n)$ and ${\dot{K}}_q^{\alpha /q,q}(R^n)=L^q(R^n,{|x|}^{\alpha })$.

Definition 1.3 Let $\alpha \in R$, $0<p\le \mathrm{\infty }$, $0<q<\mathrm{\infty }$ and $\lambda \ge 0$. The homogeneous Morrey-Herz space $M{\dot{K}}_{p,q}^{\alpha ,\lambda }(R^n)$ is defined by

$$M{\dot{K}}_{p,q}^{\alpha ,\lambda }\left(R^n\right)=\{f\in L_{\mathrm{loc}}^q(R^n\mathrm{\setminus }\{0\}):{\parallel f\parallel }_{M{\dot{K}}_{p,q}^{\alpha ,\lambda }(R^n)}<\mathrm{\infty }\},$$

where

$${\parallel f\parallel }_{M{\dot{K}}_{p,q}^{\alpha ,\lambda }(R^n)}=\underset{k_0\in Z}{sup}{2}^{-k_0\lambda }{\left(\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\parallel f{\chi }_{C_k}\parallel }_{L^q(R^n)}^p\right)}^{1/p},$$

with the usual modification made when $p=\mathrm{\infty }$.

In [17] the Morrey space $M_q^{\lambda }(R^n)$ is defined by

$$M_q^{\lambda }\left(R^n\right)=\{f\in L_{\mathrm{loc}}^q\left(R^n\right):\underset{\lambda >0,x\in R^n}{sup}\frac{1}{r^{\lambda }}{\int }_{|x-y|<r}{|f(y)|}^{q}dy<\mathrm{\infty }\}.$$

Obviously, $M{\dot{K}}_{p,q}^{\alpha ,0}(R^n)={\dot{K}}_q^{\alpha ,p}(R^n)$ and $M_q^{\lambda }(R^n)\subset M{\dot{K}}_{q,q}^{0,\lambda }(R^n)$.

Definition 1.4 ([18])

Let $0<\beta <1$. The Lipschitz space ${\dot{\mathrm{\Lambda }}}_{\beta }(R^n)$ is defined by

$${\parallel f\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}=\underset{x,h\in R^n}{sup}\frac{|f(x+h)-f(x)|}{{|h|}^{\beta }}<\mathrm{\infty }.$$

In the next section we will obtain some sharp bounds for $H_{\mathrm{\Phi }}$. Finally, the Lipschitz estimates for the commutators $H_{\mathrm{\Phi },\gamma }^b$ will be studied in the last section.

2 Sharp bounds for $H_{\mathrm{\Phi }}$

The main result of this section is as follows:

Theorem 2.1 Let $\alpha \in R$, $\lambda \ge 0$, $1<p,q<\mathrm{\infty }$. If Φ is a non-negative valued function and

$$A_1={\int }_{R^n}\frac{\mathrm{\Phi }(y)}{{|y|}^n}{|y|}^{\alpha +\frac{n}{q}-\lambda }dy<\mathrm{\infty },$$

then $H_{\mathrm{\Phi }}$ is a bounded operator on $M{\dot{K}}_{p,q}^{\alpha ,\lambda }(R^n)$.

Conversely, suppose that $H_{\mathrm{\Phi }}$ is a bounded operator on $M{\dot{K}}_{p,q}^{\alpha ,\lambda }(R^n)$. If $\lambda =0$, or if $\lambda >max\{0,\alpha \}$, then $A_1<\mathrm{\infty }$. In addition, the operator $H_{\mathrm{\Phi }}$ satisfies the following operator norm:

$${\parallel H_{\mathrm{\Phi }}\parallel }_{M{\dot{K}}_{p,q}^{\alpha ,\lambda }(R^n)\to M{\dot{K}}_{p,q}^{\alpha ,\lambda }(R^n)}=A_1.$$

Proof By definition and using Minkowski’s inequality

$$\begin{array}{rl}{\parallel H_{\mathrm{\Phi }}f\parallel }_{M{\dot{K}}_{p,q}^{\alpha ,\lambda }(R^n)}& =\underset{k_0\in Z}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\parallel (H_{\mathrm{\Phi }}f){\chi }_{C_k}\parallel }_{L^q(R^n)}^p\right\}}^{\frac{1}{p}}\\ =\underset{k_0\in Z}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\left({\int }_{C_k}\right|{\int }_{R^n}\frac{\mathrm{\Phi }(y)}{{|y|}^n}f\left(\frac{x}{|y|}\right)dy{|}^{q}dx)}^{\frac{p}{q}}\right\}}^{\frac{1}{p}}\\ \le \underset{k_0\in Z}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\left(\sum _{j=-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{C_j}\frac{\mathrm{\Phi }(y)}{{|y|}^n}{\parallel f\left(\frac{\cdot }{|y|}\right)\parallel }_{L^q(C_k)}dy\right)}^p\right\}}^{\frac{1}{p}}.\end{array}$$

Now, it is easy to see that for $y\in C_j$ [6]

$${\parallel f\left(\frac{\cdot }{|y|}\right)\parallel }_{L^q(C_k)}={|y|}^{\frac{n}{q}}{\parallel f{\chi }_{C_{k-j}}\parallel }_{L^q(R^n)}.$$

Therefore, by Minkowski’s inequality, we get

$$\begin{array}{rl}{\parallel H_{\mathrm{\Phi }}f\parallel }_{M{\dot{K}}_{p,q}^{\alpha ,\lambda }(R^n)}& \le \sum _{j=-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{C_j}\frac{\mathrm{\Phi }(y)}{{|y|}^n}{|y|}^{\frac{n}{q}}\underset{k_0\in Z}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\parallel f{\chi }_{C_{k-j}}\parallel }_{L^q(R^n)}^p\right\}}^{\frac{1}{p}}dy\\ \le {\parallel f\parallel }_{M{\dot{K}}_{p,q}^{\alpha ,\lambda }(R^n)}\sum _{j=-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{C_j}\frac{\mathrm{\Phi }(y)}{{|y|}^n}{|y|}^{\frac{n}{q}}{2}^{-j(\lambda -\alpha )}dy\\ \le {\parallel f\parallel }_{M{\dot{K}}_{p,q}^{\alpha ,\lambda }(R^n)}{\int }_{R^n}\frac{\mathrm{\Phi }(y)}{{|y|}^n}{|y|}^{\alpha +\frac{n}{q}-\lambda }dy.\end{array}$$

Hence, we conclude that

$${\parallel H_{\mathrm{\Phi }}\parallel }_{M{\dot{K}}_{p,q}^{\alpha ,\lambda }(R^n)\to M{\dot{K}}_{p,q}^{\alpha ,\lambda }(R^n)}\le A_1.$$

(2.1)

Conversely, suppose that $H_{\mathrm{\Phi }}$ is bounded on $M{\dot{K}}_{p,q}^{\alpha ,\lambda }(R^n)$. Then we consider the following two cases.

Case I: $\lambda >0$.

In this case, we choose $f_0\in L_{\mathrm{loc}}^q(R^n\mathrm{\setminus }\{0\})$, such that

$$f_0(x)={|x|}^{-\alpha -\frac{n}{q}+\lambda }.$$

An easy computation shows that

$${\parallel f_0{\chi }_{C_k}\parallel }_{L^q(R^n)}=2^{k(\lambda -\alpha )}{\left[\frac{(1-2^{q(\alpha -\lambda )})|S^{n-1}|}{\lambda -\alpha }\right]}^{\frac{1}{q}},$$

where $|S^{n-1}|$ denotes the volume of unit sphere $S^{n-1}$. Now, by definition

$$\begin{array}{rl}{\parallel f_0\parallel }_{M{\dot{K}}_{p,q}^{\alpha ,\lambda }(R^n)}& =\underset{k_0\in Z}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\parallel f_0{\chi }_{C_k}\parallel }_{L^q(R^n)}^p\right\}}^{\frac{1}{p}}\\ ={\left[\frac{(1-2^{q(\alpha -\lambda )})|S^{n-1}|}{\lambda -\alpha }\right]}^{\frac{1}{q}}\underset{k_0\in Z}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\lambda p}\right\}}^{\frac{1}{p}}\\ ={\left[\frac{(1-2^{q(\alpha -\lambda )})|S^{n-1}|}{\lambda -\alpha }\right]}^{\frac{1}{q}}\frac{2^{\lambda }}{{(2^{\lambda p}-1)}^{\frac{1}{p}}}<\mathrm{\infty }.\end{array}$$

On the other hand, it is easy to check that

$$H_{\mathrm{\Phi }}{f}_0(x)=f_0(x){\int }_{R^n}\frac{\mathrm{\Phi }(y)}{{|y|}^n}{|y|}^{\alpha +\frac{n}{q}-\lambda }dy.$$

Under the assumption that $H_{\mathrm{\Phi }}$ is bounded on $M{\dot{K}}_{p,q}^{\alpha ,\lambda }(R^n)$, we get

$${\int }_{R^n}\frac{\mathrm{\Phi }(y)}{{|y|}^n}{|y|}^{\alpha +\frac{n}{q}-\lambda }dy\le {\parallel H_{\mathrm{\Phi }}\parallel }_{M{\dot{K}}_{p,q}^{\alpha ,\lambda }(R^n)\to M{\dot{K}}_{p,q}^{\alpha ,\lambda }(R^n)}<\mathrm{\infty }.$$

(2.2)

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

$${\parallel H_{\mathrm{\Phi }}\parallel }_{M{\dot{K}}_{p,q}^{\alpha ,\lambda }(R^n)\to M{\dot{K}}_{p,q}^{\alpha ,\lambda }(R^n)}={\int }_{R^n}\frac{\mathrm{\Phi }(y)}{{|y|}^n}{|y|}^{\alpha +\frac{n}{q}-\lambda }dy.$$

Case II: $\lambda =0$.

In this case, we have $M{\dot{K}}_{p,q}^{\alpha ,\lambda }(R^n)={\dot{K}}_q^{\alpha ,p}(R^n)$. To prove the converse relation we take the sequence of function $\{f_m\}$ ($m\ge 0$) as follows:

$$f_m(x)=\{\begin{array}{cc}0\hfill & \text{if }|x|<1,\hfill \\ {|x|}^{-\alpha -\frac{n}{q}-2^{-m}}\hfill & \text{if }|x|\ge 1.\hfill \end{array}$$

Obviously for $k<0$, we have $f_m{\chi }_{C_k}=0$. Hence, for $k\ge 0$, we obtain

$$\begin{array}{rl}{\parallel f_m{\chi }_{C_k}\parallel }_{L^q(R^n)}^q& ={\int }_{C_k}{|x|}^{-\alpha -\frac{n}{q}-2^{-m}}dx\\ =\frac{(2^{q(\alpha +2^{-m})}-1)}{q(\alpha +2^{-m})}{|S|}^{n-1}{2}^{-kq(\alpha +2^{-m})}.\end{array}$$

Therefore,

$$\begin{array}{rl}{\parallel f_m\parallel }_{{\dot{K}}_q^{\alpha ,p}(R^n)}& ={\left\{\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{2}^{k\alpha p}{\parallel f_m{\chi }_{C_k}\parallel }_{L^q(R^n)}^p\right\}}^{\frac{1}{p}}\\ ={\left[\frac{(2^{q(\alpha +2^{-m})}-1)}{q(\alpha +2^{-m})}{|S|}^{n-1}\right]}^{\frac{1}{q}}{\left\{\sum _{k=0}^{\mathrm{\infty }}{2}^{k\alpha p}{2}^{-kp(\alpha +2^{-m})}\right\}}^{\frac{1}{p}}\\ ={\left[\frac{(2^{q(\alpha +2^{-m})}-1)}{q(\alpha +2^{-m})}{|S|}^{n-1}\right]}^{\frac{1}{q}}\frac{2^{\frac{1}{2^m}}}{{(2^{\frac{p}{2^m}}-1)}^{\frac{1}{p}}}<\mathrm{\infty }.\end{array}$$

On the other hand, we write

$$H_{\mathrm{\Phi }}{f}_m(x)=\{\begin{array}{cc}0\hfill & \text{if }|x|<1,\hfill \\ {|x|}^{-\alpha -\frac{n}{q}-2^{-m}}{\int }_{|y|\le |x|}\frac{\mathrm{\Phi }(y)}{{|y|}^n}{|y|}^{\alpha +\frac{n}{q}+2^{-m}}dy\hfill & \text{if }|x|\ge 1.\hfill \end{array}$$

This implies that $(H_{\mathrm{\Phi }}{f}_m){\chi }_{C_k}=0$ for $k<0$. Thus for $k\ge 0$, we get

$${\parallel (H_{\mathrm{\Phi }}{f}_m){\chi }_{C_k}\parallel }_{L^q(R^n)}^q={\int }_{C_k}{\left({|x|}^{-\alpha -\frac{n}{q}-2^{-m}}{\int }_{|y|\le |x|}\frac{\mathrm{\Phi }(y)}{{|y|}^n}{|y|}^{\alpha +\frac{n}{q}+2^{-m}}dy\right)}^{q}dx.$$

Therefore, for any $m\le k$, we have

$$\begin{array}{rl}{\parallel (H_{\mathrm{\Phi }}{f}_m){\chi }_{C_k}\parallel }_{L^q(R^n)}\ge & {\left({\int }_{C_k}{|x|}^{-\alpha q-n-2^{-m}q}dx\right)}^{\frac{1}{q}}{\int }_{|y|\le 2^{m-1}}\frac{\mathrm{\Phi }(y)}{{|y|}^n}{|y|}^{\alpha +\frac{n}{q}+2^{-m}}dy\\ =& 2^{-k(\alpha +2^{-m})}{\left[\frac{(2^{q(\alpha +2^{-m})}-1)}{q(\alpha +2^{-m})}{|S|}^{n-1}\right]}^{\frac{1}{q}}\\ \times {\int }_{|y|\le 2^{m-1}}\frac{\mathrm{\Phi }(y)}{{|y|}^n}{|y|}^{\alpha +\frac{n}{q}+2^{-m}}dy.\end{array}$$

Now, it is easy to show that

$$\begin{array}{rl}{\parallel H_{\mathrm{\Phi }}{f}_m\parallel }_{{\dot{K}}_q^{\alpha ,p}(R^n)}\ge & {\left[\frac{(2^{q(\alpha +2^{-m})}-1)}{q(\alpha +2^{-m})}{|S|}^{n-1}\right]}^{\frac{1}{q}}{\left\{\sum _{k=m}^{\mathrm{\infty }}{2}^{-\frac{kp}{2^m}}\right\}}^{\frac{1}{p}}\\ \times {\int }_{|y|\le 2^{m-1}}\frac{\mathrm{\Phi }(y)}{{|y|}^n}{|y|}^{\alpha +\frac{n}{q}+2^{-m}}dy\\ =& {\left[\frac{(2^{q(\alpha +2^{-m})}-1)}{q(\alpha +2^{-m})}{|S|}^{n-1}\right]}^{\frac{1}{q}}{\left\{\sum _{k=0}^{\mathrm{\infty }}{2}^{-\frac{kp}{2^m}}\right\}}^{\frac{1}{p}}\\ \times 2^{-\frac{m}{2^m}}{\int }_{|y|\le 2^{m-1}}\frac{\mathrm{\Phi }(y)}{{|y|}^n}{|y|}^{\alpha +\frac{n}{q}+2^{-m}}dy\\ =& {\parallel f_m\parallel }_{{\dot{K}}_q^{\alpha ,p}(R^n)}{2}^{-\frac{m}{2^m}}{\int }_{|y|\le 2^{m-1}}\frac{\mathrm{\Phi }(y)}{{|y|}^n}{|y|}^{\alpha +\frac{n}{q}+2^{-m}}dy.\end{array}$$

Consequently,

$${\parallel H_{\mathrm{\Phi }}\parallel }_{{\dot{K}}_q^{\alpha ,p}(R^n)\to {\dot{K}}_q^{\alpha ,p}(R^n)}\ge 2^{-\frac{m}{2^m}}{\int }_{|y|\le 2^{m-1}}\frac{\mathrm{\Phi }(y)}{{|y|}^n}{|y|}^{\alpha +\frac{n}{q}+2^{-m}}dy.$$

Finally, we let $m\to +\mathrm{\infty }$ to obtain

$${\parallel H_{\mathrm{\Phi }}\parallel }_{{\dot{K}}_q^{\alpha ,p}(R^n)\to {\dot{K}}_q^{\alpha ,p}(R^n)}\ge {\int }_{R^n}\frac{\mathrm{\Phi }(y)}{{|y|}^n}{|y|}^{\alpha +\frac{n}{q}}dy.$$

(2.3)

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

$${\parallel H_{\mathrm{\Phi }}\parallel }_{{\dot{K}}_q^{\alpha ,p}(R^n)\to {\dot{K}}_q^{\alpha ,p}(R^n)}={\int }_{R^n}\frac{\mathrm{\Phi }(y)}{{|y|}^n}{|y|}^{\alpha +\frac{n}{q}}dy.$$

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 $H_{\mathrm{\Phi },\gamma }$ 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 $b\in {\dot{\mathrm{\Lambda }}}_{\beta }(R^n)$, $0<\beta <1<q_2<q_1<\mathrm{\infty }$, $0<p<\mathrm{\infty }$, $\lambda >0$, $\mu =\alpha +\beta +\gamma +\frac{n}{q_2}-\frac{n}{q_1}$. If

$$A_2={\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t}{t}^{\alpha +\frac{n}{q_2}-\lambda }max\{1,t^{\beta }\}dt<\mathrm{\infty },$$

then $H_{\mathrm{\Phi },\gamma }^b$ is bounded from $M{\dot{K}}_{p,q_1}^{\mu ,\lambda }(R^n)$ to $M{\dot{K}}_{p,q_2}^{\alpha ,\lambda }(R^n)$ and satisfies the following inequality:

$${\parallel H_{\mathrm{\Phi },\gamma }^{b}f\parallel }_{M{\dot{K}}_{p,q_2}^{\alpha ,\lambda }(R^n)}\le C{A}_2{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\parallel f\parallel }_{M{\dot{K}}_{p,q_1}^{\mu ,\lambda }(R^n)}.$$

In proving Theorem 3.1, we need the following lemmas.

Lemma 3.2 For $1<p<\mathrm{\infty }$, we have

$${\parallel (H_{\mathrm{\Phi },\gamma }f){\chi }_{C_k}\parallel }_{L^p(R^n)}\le 2^{k\gamma }\left|S^{n-1}\right|{\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{t}^{\frac{n}{p}}{\parallel f{\chi }_{t^{-1}{C}_k}\parallel }_{L^p(R^n)}dt.$$

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

Lemma 3.3 ([18])

For any $x,y\in R^n$, if $f\in {\dot{\mathrm{\Lambda }}}_{\beta }(R^n)$, $0<\beta <1$, then $|f(x)-f(y)|\le {|x-y|}^{\beta }{\parallel f\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}$. Furthermore, for any cube $Q\subset R^n$, ${sup}_{x\in Q}|f(x)-f_Q|\le C{|Q|}^{\frac{\beta }{n}}{\parallel f\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}$, where $f_Q=\frac{1}{|Q|}{\int }_{Q}f$.

Lemma 3.4 ([18])

Let $f\in {\dot{\mathrm{\Lambda }}}_{\beta }(R^n)$, $0<\beta <1$, Q and $Q^{\ast }$ are cubes in $R^n$. If $Q^{\ast }\subset Q$, then

$$|f_{Q^{\ast }}-f_Q|\le C{|Q|}^{\frac{\beta }{n}}{\parallel f\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}.$$

Proof of Theorem 3.1 Notice that

$$\begin{array}{rl}{\parallel \left(H_{\mathrm{\Phi },\gamma }^{b}f\right){\chi }_{C_k}\parallel }_{L^{q_2}(R^n)}=& {\parallel ({\int }_{R^n}\frac{\mathrm{\Phi }(|x|/|y|)}{{|y|}^{n-\gamma }}(b(x)-b(y))f(y)dy){\chi }_{C_k}\parallel }_{L^{q_2}(R^n)}\\ \le & {\parallel ({\int }_{R^n}\frac{\mathrm{\Phi }(|x|/|y|)}{{|y|}^{n-\gamma }}(b(x)-b_{B_k})f(y)dy){\chi }_{C_k}\parallel }_{L^{q_2}(R^n)}\\ +{\parallel ({\int }_{R^n}\frac{\mathrm{\Phi }(|x|/|y|)}{{|y|}^{n-\gamma }}(b(y)-b_{B_k})f(y)dy){\chi }_{C_k}\parallel }_{L^{q_2}(R^n)}\\ =& I+J.\end{array}$$

Let $\frac{1}{r}=\frac{1}{q_2}-\frac{1}{q_1}$. Then by Hölder’s inequality, Lemma 3.2, and Lemma 3.3, we have

$$\begin{array}{rl}I& \le {\left({\int }_{C_k}{|b(x)-b_{B_k}|}^{r}dx\right)}^{\frac{1}{r}}{\left({\int }_{C_k}\right|{\int }_{R^n}\frac{\mathrm{\Phi }(|x|/|y|)}{{|y|}^{n-\gamma }}f(y)dy{|}^{q_1}dx)}^{\frac{1}{q_1}}\\ \le C{|B_k|}^{\frac{\beta }{n}+\frac{1}{r}}{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\parallel (H_{\mathrm{\Phi },\gamma }f){\chi }_{C_k}\parallel }_{L^{q_1}(R^n)}\\ \le C{2}^{k(\beta +\gamma +\frac{n}{r})}{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{t}^{\frac{n}{q_1}}{\parallel f{\chi }_{t^{-1}{C}_k}\parallel }_{L^{q_1}(R^n)}dt.\end{array}$$

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

$$\begin{array}{rl}J& ={\parallel \left({\int }_0^{\mathrm{\infty }}{\int }_{S^{n-1}}\frac{\mathrm{\Phi }(|x|/r)}{r^{1-\gamma }}(b\left(r{y}^{\prime }\right)-b_{B_k})f\left(r{y}^{\prime }\right)d\sigma \left(y^{\prime }\right)dr\right){\chi }_{C_k}\parallel }_{L^{q_2}(R^n)}\\ ={\parallel \left({\int }_0^{\mathrm{\infty }}{\int }_{S^{n-1}}\frac{\mathrm{\Phi }(t)}{t}{(|x|t^{-1})}^{\gamma }(b(|x|t^{-1}{y}^{\prime })-b_{B_k})f(|x|t^{-1}{y}^{\prime })d\sigma \left(y^{\prime }\right)dt\right){\chi }_{C_k}\parallel }_{L^{q_2}(R^n)}\\ \le 2^{k\gamma }{\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{\parallel \left({\int }_{S^{n-1}}(b(|x|t^{-1}{y}^{\prime })-b_{B_k})f(|x|t^{-1}{y}^{\prime })d\sigma \left(y^{\prime }\right)\right){\chi }_{C_k}\parallel }_{L^{q_2}(R^n)}dt\\ \le 2^{k\gamma }{\left|S^{n-1}\right|}^{\frac{1}{q_2^{\prime }}}{\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{\left({\int }_{C_k}{\int }_{S^{n-1}}{\left|(b(|x|t^{-1}{y}^{\prime })-b_{B_k})f(|x|t^{-1}{y}^{\prime })\right|}^{q_2}d\sigma \left(y^{\prime }\right)dx\right)}^{\frac{1}{q_2}}dt.\end{array}$$

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

$$\begin{array}{rl}J\le & 2^{k\gamma }\left|S^{n-1}\right|{\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{\left({\int }_{2^{k-1}}^{2^k}{s}^{n-1}{\int }_{S^{n-1}}{\left|(b\left(s{t}^{-1}{y}^{\prime }\right)-b_{B_k})f\left(s{t}^{-1}{y}^{\prime }\right)\right|}^{q_2}d\sigma \left(y^{\prime }\right)ds\right)}^{\frac{1}{q_2}}dt\\ =& C{2}^{k\gamma }{\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{t}^{\frac{n}{q_2}}{\left({\int }_{t^{-1}{C}_k}{|(b(y)-b_{B_k})f(y)|}^{q_2}dy\right)}^{\frac{1}{q_2}}dt\\ \le & C{2}^{k\gamma }{\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{t}^{\frac{n}{q_2}}{\left({\int }_{t^{-1}{C}_k}{|(b(y)-b_{t^{-1}{B}_k})f(y)|}^{q_2}dy\right)}^{\frac{1}{q_2}}dt\\ +C{2}^{k\gamma }{\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{t}^{\frac{n}{q_2}}{\left({\int }_{t^{-1}{C}_k}{|(b_{B_k}-b_{t^{-1}{B}_k})f(y)|}^{q_2}dy\right)}^{\frac{1}{q_2}}dt\\ =& J_1+J_2.\end{array}$$

For $J_1$, using Hölder’s inequality and Lemma 3.3, we have

$$\begin{array}{rl}{J}_1& \le C{2}^{k\gamma }{\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{t}^{\frac{n}{q_2}}{\left({\int }_{t^{-1}{C}_k}{|b(x)-b_{t^{-1}{B}_k}|}^{r}dx\right)}^{\frac{1}{r}}{\left({\int }_{t^{-1}{C}_k}\right|f(y){|}^{q_1}dy)}^{\frac{1}{q_1}}dt\\ \le C{2}^{k(\beta +\gamma +\frac{n}{r})}{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{t}^{\frac{n}{q_1}}{\parallel f{\chi }_{t^{-1}{C}_k}\parallel }_{L^{q_1}(R^n)}{t}^{-\beta }dt.\end{array}$$

Observe that if $t<1$, then $B_k\subset t^{-1}{B}_k$, while the reverse is true for $t>1$. Hence, by Lemma 3.4, we obtain

$$\begin{array}{rl}{J}_2=& C{2}^{k\gamma }{\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{t}^{\frac{n}{q_2}}{\left({\int }_{t^{-1}{C}_k}{|f(y)|}^{q_2}dy\right)}^{\frac{1}{q_2}}|b_{B_k}-b_{t^{-1}{B}_k}|dt\\ \le & C{2}^{k\gamma }{|B_k|}^{\frac{1}{r}}{\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{t}^{\frac{n}{q_1}}{\left({\int }_{t^{-1}{C}_k}{|f(y)|}^{q_1}dy\right)}^{\frac{1}{q_1}}|b_{B_k}-b_{t^{-1}{B}_k}|dt\\ \le & C{2}^{k(\beta +\gamma +\frac{n}{r})}{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\int }_0^1\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{t}^{\frac{n}{q_1}}{\parallel f{\chi }_{t^{-1}{C}_k}\parallel }_{L^{q_1}(R^n)}{t}^{-\beta }dt\\ +C{2}^{k(\beta +\gamma +\frac{n}{r})}{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\int }_1^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{t}^{\frac{n}{q_1}}{\parallel f{\chi }_{t^{-1}{C}_k}\parallel }_{L^{q_1}(R^n)}dt\\ \le & C{2}^{k(\beta +\gamma +\frac{n}{r})}{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{t}^{\frac{n}{q_1}}{\parallel f{\chi }_{t^{-1}{C}_k}\parallel }_{L^{q_1}(R^n)}max\{1,t^{-\beta }\}dt.\end{array}$$

Note that for $t>1$, $0<\beta <1$, we have $0<t^{-\beta }<1$. Therefore, by combining the estimates for I, $J_1$, and $J_2$, we get

$$\begin{array}{rl}{\parallel \left(H_{\mathrm{\Phi },\gamma }^{b}f\right){\chi }_{C_k}\parallel }_{L^{q_2}(R^n)}\le & C{2}^{k(\beta +\gamma +\frac{n}{r})}{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}\\ \times {\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{t}^{\frac{n}{q_1}}{\parallel f{\chi }_{t^{-1}{C}_k}\parallel }_{L^{q_1}(R^n)}max\{1,t^{-\beta }\}dt.\end{array}$$

Following [19], we let $m\in Z$ such that $m-1<-{log}_{2}t\le m$, then $t^{-1}{C}_k$ is contained in two adjacent annuli $C_{k+m}$ and $C_{k+m-1}$. Therefore,

$$\begin{array}{rl}{\parallel \left(H_{\mathrm{\Phi },\gamma }^{b}f\right){\chi }_{C_k}\parallel }_{L^{q_2}(R^n)}\le & C{2}^{k(\beta +\gamma +\frac{n}{r})}{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}\\ \times {\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{t}^{\frac{n}{q_1}}\sum _{i=0}^1{\parallel f{\chi }_{C_{k+m-i}}\parallel }_{L^{q_1}(R^n)}max\{1,t^{-\beta }\}dt.\end{array}$$

Hereafter, we use the notation $\tilde{\mathrm{\Phi }}(t)=\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{t}^{\frac{n}{q_1}}max\{1,t^{-\beta }\}$ for simplicity. Then for $0<p<1$, we get

$$\begin{array}{r}{\parallel H_{\mathrm{\Phi },\gamma }^{b}f\parallel }_{M{\dot{K}}_{p,q_2}^{\alpha ,\lambda }(R^n)}\\ \le C{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}\underset{k_0\in Z}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\mu p}{({\int }_0^{\mathrm{\infty }}\tilde{\mathrm{\Phi }}(t){\parallel f{\chi }_{C_{k+m}}\parallel }_{L^{q_1}(R^n)}dt)}^p\right\}}^{\frac{1}{p}}\\ +C{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}\underset{k_0\in Z}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\mu p}{({\int }_0^{\mathrm{\infty }}\tilde{\mathrm{\Phi }}(t){\parallel f{\chi }_{C_{k+m-1}}\parallel }_{L^{q_1}(R^n)}dt)}^p\right\}}^{\frac{1}{p}}\\ =K_1+K_2.\end{array}$$

Here, we approximate $K_1$ as

$$\begin{array}{rl}{K}_1\le & \underset{k_0\in Z}{sup}{2}^{-k_0\lambda }\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\mu p}\\ {\times {({\int }_0^{\mathrm{\infty }}\tilde{\mathrm{\Phi }}(t)2^{-(k+m)\lambda }{\left(\sum _{i=-\mathrm{\infty }}^{k+m}{2}^{i\mu p}{\parallel f{\chi }_{C_i}\parallel }_{L^{q_1}(R^n)}^p\right)}^{\frac{1}{p}}{2}^{(k+m)(\lambda -\mu )}dt)}^p\}}^{\frac{1}{p}}C{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}\\ \le & C{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\parallel f\parallel }_{M{\dot{K}}_{p,q_1}^{\mu ,\lambda }(R^n)}\underset{k_0\in Z}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\lambda p}{({\int }_0^{\mathrm{\infty }}\tilde{\mathrm{\Phi }}(t)2^{m(\lambda -\mu )}dt)}^p\right\}}^{\frac{1}{p}}\\ \le & C{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\parallel f\parallel }_{M{\dot{K}}_{p,q_1}^{\mu ,\lambda }(R^n)}\underset{k_0\in Z}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\lambda p}\right\}}^{\frac{1}{p}}\\ \times {\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{t}^{\frac{n}{q_1}}max\{1,t^{-\beta }\}{t}^{\mu -\lambda }dt\\ \le & C{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\parallel f\parallel }_{M{\dot{K}}_{p,q_1}^{\mu ,\lambda }(R^n)}{\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t}{t}^{\alpha +\frac{n}{q_2}-\lambda }max\{1,t^{\beta }\}dt.\end{array}$$

Similarly,

$$\begin{array}{rl}{K}_2& \le C{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\parallel f\parallel }_{M{\dot{K}}_{p,q_1}^{\mu ,\lambda }(R^n)}\underset{k_0\in Z}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\lambda p}{({\int }_0^{\mathrm{\infty }}\tilde{\mathrm{\Phi }}(t)2^{(m-1)(\lambda -\mu )}dt)}^p\right\}}^{\frac{1}{p}}\\ \le C{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\parallel f\parallel }_{M{\dot{K}}_{p,q_1}^{\mu ,\lambda }(R^n)}{\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{t}^{\frac{n}{q_1}}max\{1,t^{-\beta }\}{t}^{\mu -\lambda }dt\\ =C{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\parallel f\parallel }_{M{\dot{K}}_{p,q_1}^{\mu ,\lambda }(R^n)}{\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t}{t}^{\alpha +\frac{n}{q_2}-\lambda }max\{1,t^{\beta }\}dt.\end{array}$$

Now, we consider the case $1<p<\mathrm{\infty }$. By Minkowski’s inequality, we write

$$\begin{array}{rl}{\parallel H_{\mathrm{\Phi },\gamma }^{b}f\parallel }_{M{\dot{K}}_{p,q_2}^{\alpha ,\lambda }(R^n)}\le & C{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}\underset{k_0\in Z}{sup}{2}^{-k_0\lambda }{\int }_0^{\mathrm{\infty }}\tilde{\mathrm{\Phi }}(t){\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\mu p}{\parallel f{\chi }_{C_{k+m-1}}\parallel }_{L^{q_1}(R^n)}^p\right\}}^{\frac{1}{p}}dt\\ +C{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}\underset{k_0\in Z}{sup}{2}^{-k_0\lambda }{\int }_0^{\mathrm{\infty }}\tilde{\mathrm{\Phi }}(t){\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\mu p}{\parallel f{\chi }_{C_{k+m}}\parallel }_{L^{q_1}(R^n)}^p\right\}}^{\frac{1}{p}}dt\\ =& L_1+L_2.\end{array}$$

Here, we estimate $L_1$ as

$$\begin{array}{rl}{L}_1& \le C{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\int }_0^{\mathrm{\infty }}\tilde{\mathrm{\Phi }}(t)\underset{k_0\in Z}{sup}{2}^{-(k_0+m-1)\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0+m-1}{2}^{k\mu p}{\parallel f{\chi }_{C_k}\parallel }_{L^{q_1}(R^n)}^p\right\}}^{\frac{1}{p}}{2}^{(m-1)(\lambda -\mu )}dt\\ \le C{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\parallel f\parallel }_{M{\dot{K}}_{p,q_1}^{\mu ,\lambda }(R^n)}{\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{t}^{\frac{n}{q_1}}max\{1,t^{-\beta }\}{t}^{\mu -\lambda }dt\\ =C{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\parallel f\parallel }_{M{\dot{K}}_{p,q_1}^{\mu ,\lambda }(R^n)}{\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t}{t}^{\alpha +\frac{n}{q_2}-\lambda }max\{1,t^{\beta }\}dt.\end{array}$$

Similarly,

$$\begin{array}{rl}{L}_2& \le C{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\parallel f\parallel }_{M{\dot{K}}_{p,q_1}^{\mu ,\lambda }(R^n)}{\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t^{1+\gamma }}{t}^{\frac{n}{q_1}}max\{1,t^{-\beta }\}{t}^{\mu -\lambda }dt\\ =C{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\parallel f\parallel }_{M{\dot{K}}_{p,q_1}^{\mu ,\lambda }(R^n)}{\int }_0^{\mathrm{\infty }}\frac{|\mathrm{\Phi }(t)|}{t}{t}^{\alpha +\frac{n}{q_2}-\lambda }max\{1,t^{\beta }\}dt.\end{array}$$

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 $\alpha +\beta +\gamma <\frac{n}{q_2^{\prime }}+\lambda$, then under the same conditions as in Theorem 3.1, the commutator of the n-dimensional fractional Hardy operator [15],

$$H_{\gamma ,b}f(x)=\frac{1}{{|x|}^{n+\gamma }}{\int }_{|y|<|x|}(b(x)-b(y))f(y)dy,$$

is bounded from $M{\dot{K}}_{p,q_1}^{\mu ,\lambda }(R^n)$ to $M{\dot{K}}_{p,q_2}^{\alpha ,\lambda }(R^n)$.

Proof In the operator $H_{\mathrm{\Phi },\gamma }^{b}f(x)$, we replace

$$\mathrm{\Phi }(t)={\mathrm{\Phi }}_1(t)=t^{-n+\gamma }{\chi }_{(1,\mathrm{\infty })}(t),$$

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

$$H_{{\mathrm{\Phi }}_1,\gamma }^{b}f(x)=H_{\gamma ,b}f(x).$$

Hence, by Theorem 3.1

$$\begin{array}{rcl}{\parallel H_{\gamma ,b}f\parallel }_{M{\dot{K}}_{p,q_2}^{\alpha ,\lambda }(R^n)}& \le & C{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\parallel f\parallel }_{M{\dot{K}}_{p,q_1}^{\mu ,\lambda }(R^n)}{\int }_1^{\mathrm{\infty }}{t}^{\alpha +\beta +\gamma -\frac{n}{q_2^{\prime }}-\lambda -1}dt\\ \le & C{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\parallel f\parallel }_{M{\dot{K}}_{p,q_1}^{\mu ,\lambda }(R^n)}.\end{array}$$

Thus, the corollary is proved. □

Corollary 3.6 If $\alpha +\frac{n}{q_2}>\lambda$, then under the same conditions as in Theorem 3.1, the commutator of the adjoint fractional Hardy operator [15],

$$H_{\gamma ,b}^{\ast }f(x)={\int }_{|y|\ge |x|}\frac{1}{{|y|}^{n-\gamma }}(b(x)-b(y))f(y)dy,$$

is bounded from $M{\dot{K}}_{p,q_1}^{\mu ,\lambda }(R^n)$ to $M{\dot{K}}_{p,q_2}^{\alpha ,\lambda }(R^n)$.

Proof In the operator $H_{\mathrm{\Phi },\gamma }^{b}f(x)$, we replace

$$\mathrm{\Phi }(t)={\mathrm{\Phi }}_2(t)={\chi }_{(0,1]}(t),$$

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

$$H_{{\mathrm{\Phi }}_2,\gamma }^{b}f(x)=H_{\gamma ,b}^{\ast }f(x).$$

Thus, by Theorem 3.1

$$\begin{array}{rcl}{\parallel H_{\gamma ,b}^{\ast }f\parallel }_{M{\dot{K}}_{p,q_2}^{\alpha ,\lambda }(R^n)}& \le & C{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\parallel f\parallel }_{M{\dot{K}}_{p,q_1}^{\mu ,\lambda }(R^n)}{\int }_0^1{t}^{\alpha +\frac{n}{q_2}-\lambda -1}dt.\\ \le & C{\parallel b\parallel }_{{\dot{\mathrm{\Lambda }}}_{\beta }(R^n)}{\parallel f\parallel }_{M{\dot{K}}_{p,q_1}^{\mu ,\lambda }(R^n)}.\end{array}$$

With this we finish the proof of Corollary 3.6. □