一类带有粗糙核的分数次积分广义交换子在λ-中心Morrey空间上的有界性

Boundedness of a Class of Generalized Commutators of Fractional Integrals with Rough Kernels on λ-Central Morrey Spaces

摘要: 具有粗糙核的积分算子交换子在λ-中心Morrey空间上的有界性被许多作者所研究。本文利用Hölder不等式、环分解等调和分析中的一些经典理论和方法，研究了由带有粗糙核的分数次积分算子和λ-中心BMO函数空间生成的广义交换子在λ-中心Morrey空间上的有界性。本文的研究结果推广了Fu、Lin和Lu的部分结论。

Abstract: The boundedness of integral operator commutators with rough kernels on λ-central Morrey spaces and λ-central BMO function spaces has been studied by many authors. In this paper, the generalized commutators generated by fractional integral operators with rough kernels and the λ-central BMO function space are studied to be bounded on the λ-central Morrey space by using some classical theories and methods in the harmonic analysis such as Hölder’s inequality and ring decomposition. The results in this paper extended some conclusions of Fu, Lin and Lu.

1. 引言

设$S^{n-1}$ 为$R^n$ 上的单位球面，带有通常的Lebesgue测度$d\sigma =d\sigma \left(x^{\prime }\right)$ 。则具有粗糙核的分数次积分算子定义为

$T_{\Omega ,\alpha }\left(f\right)\left(x\right):={\displaystyle {\int }_{R^n}\frac{\Omega \left(x-y\right)}{{\left|x-y\right|}^{n-\alpha }}f\left(y\right)\text{d}y}$ ，

其中$0<\alpha <n$ ，$\Omega \in L^s\left(S^{n-1}\right)$ ，$1<s<\infty$ ，并且$\Omega$ 在$R^n$ 上满足零阶齐次性。对于上述算子，文献[1] [2]研究了该分数次积分算子$T_{\Omega ,\alpha }$ 的相关性质。

设$b\in BMO$ ，由具有粗糙核的分数次积分算子$T_{\Omega ,\alpha }$ 和函数b生成的交换子定义为：

$\left[b,T_{\Omega ,\alpha }\right]f\left(x\right)=b\left(x\right)T_{\Omega ,\alpha }f\left(x\right)-T_{\Omega ,\alpha }\left(bf\right)\left(x\right)$ 。

2000年，Alvarez等人在文献[3]中提出λ-中心有界平均震荡和λ-中心Morrey空间，具体定义如下：

定义1 令$\lambda <1/n$ ，$1<q<\infty$ ，则λ-中心有界平均振荡空间$CBM{O}^{q,\lambda }$ 定义为：

$CBM{O}^{q,\lambda }\left(R^n\right)=\left\{f\in L_{loc}^q\left(R^n\right):{\Vert f\Vert }_{CBM{O}^{q,\lambda }}<\infty \right\}$ ,

其中${\Vert f\Vert }_{CBM{O}^{q,\lambda }}=\underset{R>0}{\sup }{\left(\frac{1}{{\left|B\left(0,R\right)\right|}^{1+q\lambda }}{\displaystyle {\int }_{B\left(0,R\right)}{\left|f\left(x\right)-f_{B\left(0,R\right)}\right|}^q\text{d}x}\right)}^{1/q}$ ，$f_{B\left(0,R\right)}$ 为f在$B\left(0,R\right)$ 上的积分平均。

定义2 令$\lambda \in R$ ，$1<q<\infty$ ，则λ-中心Morrey空间${\dot{B}}^{p,v}$ 定义为：

${\dot{B}}^{p,v}\left(R^n\right)=\left\{f\in L_{loc}^p\left(R^n\right):{\Vert f\Vert }_{{\dot{B}}^{p,v}}<\infty \right\}$ ，

其中${\Vert f\Vert }_{{\dot{B}}^{p,v}}=\underset{R>0}{\sup }{\left(\frac{1}{{\left|B\left(0,R\right)\right|}^{1+pv}}{\displaystyle {\int }_{B\left(0,R\right)}{\left|f\left(x\right)\right|}^p\text{d}x}\right)}^{1/p}$。

2002年，Pérez等人在文献[4]定义了奇异积分算子广义交换子，如下：

$T_{\overrightarrow{b}}^m\left(f\right)\left(x\right):=p.v.{\displaystyle {\int }_{R^n}K\left(x,y\right)\left[{\displaystyle \prod _{j=1}^m{b}_j\left(x\right)-b_j\left(y\right)}\right]f\left(y\right)\text{d}y}$。

Tao和Shi在文献[5]证明了上述积分算子广义交换子$T_{\overrightarrow{b}}^m\left(f\right)\left(x\right)$ 在λ-中心Morrey空间上的有界性。张和陶在文献[6]证明了带有粗糙核的奇异积分广义交换子$T_{\Omega ,\overrightarrow{b}}^m\left(f\right)\left(x\right)$ 的有界性。受上述启发，本文在Pérez [4]中考虑的奇异积分算子广义交换子的基础上，定义了带粗糙核的分数次积分广义交换子，并讨论该交换子在λ-中心Morrey空间上的有界性。

定义3 令$\overrightarrow{b}=\left(b_1,b_2,\cdots ,b_m\right)$ ，$b_i\in CBM{O}^{s_i,{\mu }_i}\left(R^n\right)$ ，$i=1,2,\cdots ,m$ 。则带粗糙核的分数次积分广义交换子定义为：

$\left[\overrightarrow{b},T_{\Omega ,\alpha }\right]f\left(x\right)=T_{\Omega ,\alpha }^{\overrightarrow{b},m}\left(f\right)\left(x\right):={\displaystyle {\int }_{R^n}\frac{\Omega \left(x-y\right)}{{\left|x-y\right|}^{n-\alpha }}}{\displaystyle \prod _{j=1}^m\left[b_j\left(x\right)-b_j\left(y\right)\right]}f\left(y\right)\text{d}y$。

本文的主要结果如下：

定理1 令$0<\alpha <n$ ，$\Omega \in L^s\left(S^{n-1}\right)$ ，$s>n/\left(n-\alpha \right)$ ，$1<p<n/\alpha$ 且$1/q={\displaystyle \sum _{i=1}^{m}1/s_i}+1/p-\alpha /n<1$ ， $\lambda ={\displaystyle \sum _{i=1}^m{\mu }_i}+v+\alpha /n<0$ 。如果$\overrightarrow{b}=\left(b_1,b_2,\cdots ,b_m\right)$ ，其中$b_i\in CBM{O}^{s_i,{\mu }_i}$ ，$1<s_i<\infty$ ，$0<{\mu }_i<1/n$ ，$i=1,\cdots ,m$ 。并且上述指标满足$\frac{\alpha }{n}+\frac{1}{s}\le \frac{1}{p}$ ，$v<-{\displaystyle \sum _{i=1}^m{\mu }_i-\frac{\alpha }{n}}-\frac{1}{s}$ 和${\displaystyle \sum _{i=1}^m\frac{1}{s_i}}\le \frac{1}{p^{\prime }}$ 。则对任意$f\in {\dot{B}}^{p,v}$ ，有：

${\Vert \left[\overrightarrow{b},T_{\Omega ,\alpha }\right]f\Vert }_{{\dot{B}}^{q,\lambda }}\le C{\displaystyle \prod _{i=1}^m{\Vert b_i\Vert }_{CBM{O}^{s_i,{\mu }_i}}{\Vert f\Vert }_{{\dot{B}}^{p,v}}}$ 。

2. 主要结果的证明

引理1 [2] 令$0<\alpha <n$ ，$1<p<n/\alpha$ 且$1/q=1/p-\alpha /n$ 。若$\Omega \in L^s\left(S^{n-1}\right)$ ，$s>n/\left(n-\alpha \right)$ ，则

${\Vert T_{\Omega ,\alpha }f\Vert }_{L^q}\le C{\Vert f\Vert }_{L^p}$ 。

证明 不失一般性，我们仅考虑$m=2$ 的情形，对于$m>2$ 的情形可以类似推出。

固定常数R，定义$B=B\left(0,R\right)$ 以及$2^{k+1}B=B\left(0,2^{k+1}R\right)$ 。对任意$x\in B$ ，将交换子分解如下：

$\begin{array}{c}{\left[\overrightarrow{b},T_{\Omega ,\alpha }\right]}^2\left(f\right)\left(x\right)=\left[b_1\left(x\right)-{\left(b_1\right)}_B\right]\left[b_2\left(x\right)-{\left(b_2\right)}_B\right]T_{\Omega ,\alpha }\left(f\right)\left(x\right)\\ -\left[b_1\left(x\right)-{\left(b_1\right)}_B\right]T_{\Omega ,\alpha }\left(\left(b_2\left(x\right)-{\left(b_2\right)}_B\right)f\right)\left(x\right)\\ -\left[b_2\left(x\right)-{\left(b_2\right)}_B\right]T_{\Omega ,\alpha }\left(\left(b_1\left(x\right)-{\left(b_1\right)}_B\right)f\right)\left(x\right)\\ +T_{\Omega ,\alpha }\left(\left(b_1\left(x\right)-{\left(b_1\right)}_B\right)\left(b_2\left(x\right)-{\left(b_2\right)}_B\right)f\right)\left(x\right),\end{array}$

其中${\left(b_1\right)}_B,{\left(b_2\right)}_B$ 为$b_1\left(x\right),b_2\left(x\right)$ 在B上的积分平均。

则有：

$\begin{array}{c}{\left({\displaystyle {\int }_B{\left|{\left[\overrightarrow{b},T_{\Omega ,\alpha }\right]}^2\left(f\right)\left(x\right)\right|}^q\text{d}x}\right)}^{\frac{1}{q}}\le {\left({\displaystyle {\int }_B{\left|\left[b_1\left(x\right)-{\left(b_1\right)}_B\right]\left[b_2\left(x\right)-{\left(b_2\right)}_B\right]T_{\Omega ,\alpha }\left(f\right)\left(x\right)\right|}^q\text{d}x}\right)}^{\frac{1}{q}}\\ -{\left({\displaystyle {\int }_B{\left|\left[b_1\left(x\right)-{\left(b_1\right)}_B\right]T_{\Omega ,\alpha }\left(\left(b_2\left(x\right)-{\left(b_2\right)}_B\right)f\right)\left(x\right)\right|}^q\text{d}x}\right)}^{\frac{1}{q}}\\ -{\left({\displaystyle {\int }_B{\left|\left[b_2\left(x\right)-{\left(b_2\right)}_B\right]T_{\Omega ,\alpha }\left(\left(b_1\left(x\right)-{\left(b_1\right)}_B\right)f\right)\left(x\right)\right|}^q\text{d}x}\right)}^{\frac{1}{q}}\\ +{\left({\displaystyle {\int }_B{\left|T_{\Omega ,\alpha }\left(\left(b_1\left(x\right)-{\left(b_1\right)}_B\right)\left(b_2\left(x\right)-{\left(b_2\right)}_B\right)f\right)\left(x\right)\right|}^q\text{d}x}\right)}^{\frac{1}{q}}\\ ={{\rm I}}_1\left(x\right)+{{\rm I}}_2\left(x\right)+{{\rm I}}_3\left(x\right)+{{\rm I}}_4\left(x\right).\end{array}$

记$f=f{\chi }_{2B}+f{\chi }_{{\left(2B\right)}^C}=f_1+f_2$ ，其中${\chi }_{2B}$ 为2B上的特征函数。下面分别估计${{\rm I}}_1$ 、${{\rm I}}_2$ 、${{\rm I}}_3$ 以及${{\rm I}}_4$ 。

1) ${{\rm I}}_1$ 部分

首先由f的分解得：

$\begin{array}{c}{{\rm I}}_1\left(x\right)\le {\left({\displaystyle {\int }_B{\left|\left[b_1\left(x\right)-{\left(b_1\right)}_B\right]\left[b_2\left(x\right)-{\left(b_2\right)}_B\right]T_{\Omega ,\alpha }\left(f_1\right)\left(x\right)\right|}^q\text{d}x}\right)}^{\frac{1}{q}}\\ +\left({\displaystyle {\int }_B{\left|\left[b_1\left(x\right)-{\left(b_1\right)}_B\right]\left[b_2\left(x\right)-{\left(b_2\right)}_B\right]T_{\Omega ,\alpha }\left(f_2\right)\left(x\right)\right|}^q\text{d}x}\right)\\ ={{\rm I}}_{11}+{{\rm I}}_{12}.\end{array}$

对于${{\rm I}}_{11}$ ，令$\frac{1}{t}=\frac{1}{q}-\frac{1}{s_1}-\frac{1}{s_2}=\frac{1}{p}-\frac{\alpha }{n}$ ，由Hölder不等式及$T_{\Omega ,\alpha }$ 从$L^p$ 到$L^t$ 的有界性可得：

$\begin{array}{c}{{\rm I}}_{11}\le {\left({\displaystyle {\int }_B{\left(b_1\left(x\right)-{\left(b_1\right)}_B\right)}^{s_1}\text{d}x}\right)}^{\frac{1}{s_1}}{\left({\displaystyle {\int }_B{\left(b_2\left(x\right)-{\left(b_2\right)}_B\right)}^{s_2}\text{d}x}\right)}^{\frac{1}{s_2}}{\left({\displaystyle {\int }_B{\left|T_{\Omega ,\alpha }\left(f_1\right)\left(x\right)\right|}^t\text{d}x}\right)}^{\frac{1}{t}}\\ \le C{\left|B\right|}_{}^{\frac{1}{s_1}+{\mu }_1+\frac{1}{s_2}+{\mu }_2}{\displaystyle \prod _{i=1}^2{\Vert b_i\Vert }_{CBM{O}^{s_i,{\mu }_i}}}{\left({\displaystyle {\int }_{2B}{\left|f\left(x\right)\right|}^p\text{d}x}\right)}^{\frac{1}{p}}\le C{\left|B\right|}_{}^{\lambda +\frac{1}{q}}{\displaystyle \prod _{i=1}^2{\Vert b_i\Vert }_{CBM{O}^{s_i,{\mu }_i}}}{\Vert f\Vert }_{{\dot{B}}^{p,v}}.\end{array}$

对于${{\rm I}}_{12}$ ，由$\frac{\alpha }{n}+\frac{1}{s}\le \frac{1}{p}$ ，可得$\frac{1}{q}-\frac{1}{s_1}-\frac{1}{s_2}-\frac{1}{s}>0$ ，由Minkowski不等式，Hölder不等式以及 $v<-{\displaystyle \sum _{i=1}^m{\mu }_i-\frac{\alpha }{n}}-\frac{1}{s}<-\frac{\alpha }{n}-\frac{1}{s}$ ，我们有：

$\begin{array}{c}{{\rm I}}_{12}\le {\left({\displaystyle {\int }_B{\left|\left[b_1\left(x\right)-{\left(b_1\right)}_B\right]\left[b_2\left(x\right)-{\left(b_2\right)}_B\right]T_{\Omega ,\alpha }\left(f_2\right)\left(x\right)\right|}^q\text{d}x}\right)}^{\frac{1}{q}}\\ \le {\left({\displaystyle {\int }_B{\left|\left[b_1\left(x\right)-{\left(b_1\right)}_B\right]\left[b_2\left(x\right)-{\left(b_2\right)}_B\right]{\displaystyle \sum _{k=1}^{\infty }{\displaystyle {\int }_{2^{k+1}B\backslash 2^{k}B}\frac{\Omega \left(x-y\right)}{{\left|x-y\right|}^{n-\alpha }}f\left(y\right)\text{d}y}}\right|}^q\text{d}x}\right)}^{\frac{1}{q}}\\ \le C{{\displaystyle \sum _{k=1}^{\infty }{\left|2^{k}B\right|}^{\frac{\alpha }{n}-1}{\displaystyle {\int }_{2^{k+1}B}\left({\displaystyle {\int }_B{\left|\Omega \left(x-y\right)\right|}^q{\left|b_1\left(x\right)-{\left(b_1\right)}_B\right|}^q{\left|b_2\left(x\right)-{\left(b_2\right)}_B\right|}^q\text{d}x}\right)}}}^{\frac{1}{q}}f\left(y\right)\text{d}y\\ \le C{\displaystyle \sum _{k=1}^{\infty }{\left|2^{k}B\right|}^{\frac{\alpha }{n}-1}}{\left({\displaystyle {\int }_B{\left|b_1\left(x\right)-{\left(b_1\right)}_B\right|}^{s_1}\text{d}x}\right)}^{\frac{1}{s_1}}{\left({\displaystyle {\int }_B{\left|b_2\left(x\right)-{\left(b_2\right)}_B\right|}^{s_2}\text{d}x}\right)}^{\frac{1}{s_2}}{\left|B\right|}_{}^{\frac{1}{q}-\frac{1}{s_1}-\frac{1}{s_2}-\frac{1}{s}}\\ {\displaystyle {\int }_{2^{k+1}B}{\left({\displaystyle {\int }_B{\left|\Omega \left(x-y\right)\right|}^s\text{d}x}\right)}^{\frac{1}{s}}}\left|f\left(y\right)\right|\text{d}y.\end{array}$

由$x\in B$ ，$y\in 2^{k+1}B$ 可得$x-y\in 2^{k+2}B$ ，又$\Omega \in L^s\left(S^{n-1}\right)$ ，从而有：

$\begin{array}{c}{\left({\displaystyle {\int }_B{\left|\Omega \left(x-y\right)\right|}^s\text{d}x}\right)}^{\frac{1}{s}}={\left({\displaystyle {\int }_{2^{k+2}B}{\left|\Omega \left(z\right)\right|}^s\text{d}z}\right)}^{\frac{1}{s}}\\ ={\left({\displaystyle {\int }_0^{2^{k+1}R}{\displaystyle {\int }_{S^{n-1}}{\left|\Omega \left(z^{\prime }\right)\right|}^s\text{d}\sigma \left(z^{\prime }\right)r^{n-1}\text{d}r}}\right)}^{1/s}\\ =C{\left|2^{k+1}B\right|}^{1/s}{\Vert \Omega \Vert }_{L^s\left(S^{n-1}\right)}\le C{\left|2^{k}B\right|}^{1/s}\end{array}$ (2.1)

由Hölder不等式可得：

$\begin{array}{c}{{\rm I}}_{12}\le C{\left|B\right|}_{}^{\frac{1}{q}+{\mu }_1+{\mu }_2-\frac{1}{s}}{\displaystyle \prod _{i=1}^2{\Vert b_i\Vert }_{CBM{O}^{s_i,{\mu }_i}}}{\displaystyle \sum _{k=1}^{\infty }{\left|2^{k}B\right|}^{\frac{\alpha }{n}-1}}{\left|2^{k}B\right|}^{\frac{1}{s}}{\left({\displaystyle {\int }_{2^{k+1}B}{\left|f\left(y\right)\right|}^p\text{d}y}\right)}^{\frac{1}{p}}{\left|2^{k+1}B\right|}^{1-\frac{1}{p}}\\ \le C{\left|B\right|}_{}^{\frac{1}{q}+{\mu }_1+{\mu }_2-\frac{1}{s}}{\displaystyle \sum _{k=1}^{\infty }{\left|2^{k}B\right|}^{\frac{\alpha }{n}+\frac{1}{s}+v}}{\displaystyle \prod _{i=1}^2{\Vert b_i\Vert }_{CBM{O}^{s_i,{\mu }_i}}}{\Vert f\Vert }_{{\dot{B}}^{p,v}}\\ \le C{\left|B\right|}_{}^{\lambda +\frac{1}{q}}{\displaystyle \prod _{i=1}^2{\Vert b_i\Vert }_{CBM{O}^{s_i,{\mu }_i}}}{\Vert f\Vert }_{{\dot{B}}^{p,v}}.\end{array}$

综上可得${{\rm I}}_1\le C{\left|B\right|}_{}^{\lambda +\frac{1}{q}}{\displaystyle \prod _{i=1}^2{\Vert b_i\Vert }_{CBM{O}^{s_i,{\mu }_i}}}{\Vert f\Vert }_{{\dot{B}}^{p,v}}$ 。

2) ${{\rm I}}_2$ 部分

$\begin{array}{c}{{\rm I}}_2\le {\left({\displaystyle {\int }_B{\left|\left[b_1\left(x\right)-{\left(b_1\right)}_B\right]T_{\Omega ,\alpha }\left(\left(b_2\left(x\right)-{\left(b_2\right)}_B\right)f_1\right)\left(x\right)\right|}^q\text{d}x}\right)}^{\frac{1}{q}}\\ +{\left({\displaystyle {\int }_B{\left|\left[b_1\left(x\right)-{\left(b_1\right)}_B\right]T_{\Omega ,\alpha }\left(\left(b_2\left(x\right)-{\left(b_2\right)}_B\right)f_2\right)\left(x\right)\right|}^q\text{d}x}\right)}^{\frac{1}{q}}\\ ={{\rm I}}_{21}+{{\rm I}}_{22}.\end{array}$

对于${{\rm I}}_{21}$ ，令$\frac{1}{q_2}=\frac{1}{q}-\frac{1}{s_1}=\frac{1}{p_2}-\frac{\alpha }{n}$ ，由$1/q={\displaystyle \sum _{i=1}^{m}1/s_i+1/p-\alpha /n}<1$ ，可得$\frac{1}{p_2}=\frac{1}{s_2}+\frac{1}{p}$ ，则$1<p_2<\frac{n}{\alpha }$ 。通过Hölder不等式以及$T_{\Omega ,\alpha }$ 从$L^{p_2}$ 到$L^{q_2}$ 的有界性，有：

$\begin{array}{c}{{\rm I}}_{21}\le {\left({\displaystyle {\int }_B{\left(b_1\left(x\right)-{\left(b_1\right)}_B\right)}^{s_1}}\text{d}x\right)}^{\frac{1}{s_1}}{\left({\displaystyle {\int }_B{\left|T_{\Omega ,\alpha }\left(\left(b_2\left(x\right)-{\left(b_2\right)}_B\right)f_1\right)\left(x\right)\right|}^{q_2}\text{d}x}\right)}^{\frac{1}{q_2}}\\ \le C{\left|B\right|}_{}^{\frac{1}{s_1}+{\mu }_1}{\Vert b_1\Vert }_{CBM{O}^{s_1,{\mu }_1}}{\left({\displaystyle {\int }_B{\left|\left(b_2\left(x\right)-{\left(b_2\right)}_B\right)f_1\left(x\right)\right|}^{p_2}\text{d}x}\right)}^{\frac{1}{p_2}}\\ \le C{\left|B\right|}_{}^{\frac{1}{s_1}+{\mu }_1}{\Vert b_1\Vert }_{CBM{O}^{s_1,{\mu }_1}}{\left({\displaystyle {\int }_B{\left|b_2\left(x\right)-{\left(b_2\right)}_B\right|}^{s_2}\text{d}x}\right)}^{\frac{1}{s_2}}{\left({\displaystyle {\int }_{2B}{\left|f\left(x\right)\right|}^p\text{d}x}\right)}^{\frac{1}{p}}\\ \le C{\left|B\right|}_{}^{\lambda +\frac{1}{q}}{\displaystyle \prod _{i=1}^2{\Vert b_i\Vert }_{CBM{O}^{s_i,{\mu }_i}}}{\Vert f\Vert }_{{\dot{B}}^{p,v}}.\end{array}$

对于${{\rm I}}_{22}$ ，由$\frac{\alpha }{n}+\frac{1}{s}\le \frac{1}{p}$ ，${\displaystyle \sum _{i=1}^m\frac{1}{s_i}}\le \frac{1}{p^{\prime }}$ ，可得$\frac{1}{q}-\frac{1}{s}-\frac{1}{s_1}>0$ ，$1-\frac{1}{s_1}-\frac{1}{p}>0$ 。又 $v<-{\displaystyle \sum _{i=1}^m{\mu }_i-\frac{\alpha }{n}}-\frac{1}{s}<-{\mu }_1-\frac{\alpha }{n}-\frac{1}{s}$ ，通过Minkowski不等式，(2.1)和Hölder不等式可得：

$\begin{array}{c}{{\rm I}}_{22}\le C{\left({\displaystyle {\int }_B{\left|\left[b_1\left(x\right)-{\left(b_1\right)}_B\right]{\displaystyle \sum _{k=1}^{\infty }{\displaystyle {\int }_{2^{k+1}B\backslash 2^{k}B}\frac{\Omega \left(x-y\right)}{{\left|x-y\right|}^{n-\alpha }}}}\left[b_2\left(y\right)-{\left(b_2\right)}_B\right]f\left(y\right)\text{d}y\right|}^q\text{d}x}\right)}^{\frac{1}{q}}\\ \le C{{\displaystyle \sum _{k=1}^{\infty }{\left|2^{k}B\right|}^{\frac{\alpha }{n}-1}{\displaystyle {\int }_{2^{k+1}B}\left({\displaystyle {\int }_B{\left|\Omega \left(x-y\right)\right|}^q{\left|b_1\left(x\right)-{\left(b_1\right)}_B\right|}^q\text{d}x}\right)}}}^{\frac{1}{q}}\left(b_2\left(y\right)-{\left(b_2\right)}_B\right)f\left(y\right)\text{d}y\\ \le C{\displaystyle \sum _{k=1}^{\infty }{\left|2^{k}B\right|}^{\frac{\alpha }{n}-1}}{\left({\displaystyle {\int }_B{\left|b_1\left(x\right)-{\left(b_1\right)}_B\right|}^{s_1}\text{d}x}\right)}^{\frac{1}{s_1}}{\left|B\right|}_{}^{\frac{1}{q}-\frac{1}{s_1}-\frac{1}{s}}\times {\displaystyle {\int }_{2^{k+1}B}{\left({\displaystyle {\int }_B{\left|\Omega \left(x-y\right)\right|}^s\text{d}x}\right)}^{\frac{1}{s}}}\left(b_2\left(x\right)-{\left(b_2\right)}_B\right)f\left(y\right)\text{d}y\\ \le C{\displaystyle \sum _{k=1}^{\infty }{\left|2^{k}B\right|}^{\frac{\alpha }{n}-1+\frac{1}{s}}}{\left|B\right|}_{}^{\frac{1}{q}+{\mu }_1-\frac{1}{s}}{\Vert b_1\Vert }_{CBM{O}^{s_1,{\mu }_1}}{\left({\displaystyle {\int }_B{\left|b_2\left(y\right)-{\left(b_2\right)}_B\right|}^{s_2}\text{d}x}\right)}^{\frac{1}{s_2}}\times {\left({\displaystyle {\int }_{2^{k+1}B}{\left|f\left(y\right)\right|}^p\text{d}y}\right)}^{\frac{1}{p}}{\left|2^{k+1}B\right|}_{}^{1-\frac{1}{p}-\frac{1}{s_2}}\\ \le {\displaystyle \sum _{k=1}^{\infty }{\left|2^{k}B\right|}^{\frac{\alpha }{n}+\frac{1}{s}+{\mu }_2+v}}{\left|B\right|}_{}^{\frac{1}{q}+{\mu }_1-\frac{1}{s}}{\displaystyle \prod _{i=1}^2{\Vert b_i\Vert }_{CBM{O}^{s_i,{\mu }_i}}}{\Vert f\Vert }_{{\dot{B}}^{p,v}}\\ \le C{\left|B\right|}_{}^{\lambda +\frac{1}{q}}{\displaystyle \prod _{i=1}^2{\Vert b_i\Vert }_{CBM{O}^{s_i,{\mu }_i}}}{\Vert f\Vert }_{{\dot{B}}^{p,v}}.\end{array}$

综上可得${{\rm I}}_2\le C{\left|B\right|}_{}^{\lambda +\frac{1}{q}}{\displaystyle \prod _{i=1}^2{\Vert b_i\Vert }_{CBM{O}^{s_i,{\mu }_i}}}{\Vert f\Vert }_{{\dot{B}}^{p,v}}$ 。

3) ${{\rm I}}_3$ 部分

由于${{\rm I}}_3$ 部分和${{\rm I}}_2$ 部分是对称的，因此类似于${{\rm I}}_2$ 的证明方法可得：

${{\rm I}}_3\le C{\left|B\right|}_{}^{\lambda +\frac{1}{q}}{\displaystyle \prod _{i=1}^2{\Vert b_i\Vert }_{CBM{O}^{s_i,{\mu }_i}}}{\Vert f\Vert }_{{\dot{B}}^{p,v}}$ 。

4) ${{\rm I}}_4$ 部分

$\begin{array}{c}{{\rm I}}_4\left(x\right)\le {\left({\displaystyle {\int }_B{\left|T_{\Omega ,\alpha }\left(\left[b_1\left(x\right)-{\left(b_1\right)}_B\right]\left[b_2\left(x\right)-{\left(b_2\right)}_B\right]f_1\right)\left(x\right)\right|}^q\text{d}x}\right)}^{\frac{1}{q}}\\ +\left({\displaystyle {\int }_B{\left|T_{\Omega ,\alpha }\left(\left[b_1\left(x\right)-{\left(b_1\right)}_B\right]\left[b_2\left(x\right)-{\left(b_2\right)}_B\right]f_2\right)\left(x\right)\right|}^q\text{d}x}\right)\\ ={{\rm I}}_{41}+{{\rm I}}_{42}.\end{array}$

对于${{\rm I}}_{41}$ ，令$\frac{1}{q_1}=\frac{1}{q}+\frac{\alpha }{n}=\frac{1}{s_1}+\frac{1}{s_2}+\frac{1}{p}$ ，有$1<q_1<\frac{n}{\alpha }$ 。由$T_{\Omega ,\alpha }$ 从$L^{q_1}$ 到$L^q$ 的有界及Hölder不等式，得：

$\begin{array}{c}{{\rm I}}_{41}\le {\left({\displaystyle {\int }_B{\left|\left(b_1\left(x\right)-{\left(b_1\right)}_B\right)\left(\left(b_2\left(x\right)-{\left(b_2\right)}_B\right)f_1\right)\left(x\right)\right|}^{q_1}\text{d}x}\right)}^{\frac{1}{q_1}}\\ \le C{\left({\displaystyle {\int }_B{\left|b_1\left(x\right)-{\left(b_1\right)}_B\right|}^{s_1}\text{d}x}\right)}^{\frac{1}{s_1}}{\left({\displaystyle {\int }_B{\left|b_2\left(x\right)-{\left(b_2\right)}_B\right|}^{s_2}\text{d}x}\right)}^{\frac{1}{s_2}}{\left({\displaystyle {\int }_{2B}{\left|f\left(x\right)\right|}^p\text{d}x}\right)}^{\frac{1}{p}}\\ \le C{\left|B\right|}_{}^{\lambda +\frac{1}{q}}{\displaystyle \prod _{i=1}^2{\Vert b_i\Vert }_{CBM{O}^{s_i,{\mu }_i}}}{\Vert f\Vert }_{{\dot{B}}^{p,v}}.\end{array}$

对于${{\rm I}}_{42}$ ，由$\frac{1}{q}-\frac{1}{s}>0$ ，$1-\frac{1}{s_1}-\frac{1}{s_2}-\frac{1}{p}>0$ 又$v<-{\displaystyle \sum _{i=1}^2{\mu }_i-\frac{\alpha }{n}}-\frac{1}{s}$ ，通过Minkowski不等式，(2.1)和Hölder不等式可得：

$\begin{array}{c}{{\rm I}}_{42}\le {\left({\displaystyle {\int }_B{\left|T_{\Omega ,\alpha }\left(b_1\left(x\right)-{\left(b_1\right)}_B\right)\left(b_2\left(x\right)-{\left(b_2\right)}_B\right)\left(f_2\right)\left(x\right)\right|}^q\text{d}x}\right)}^{\frac{1}{q}}\\ \le {\left({\displaystyle {\int }_B{\left|{\displaystyle \sum _{k=1}^{\infty }{\displaystyle {\int }_{2^{k+1}B\backslash 2^{k}B}\frac{\Omega \left(x-y\right)}{{\left|x-y\right|}^{n-\alpha }}}}\left(b_1\left(y\right)-{\left(b_1\right)}_B\right)\left(b_2\left(y\right)-{\left(b_2\right)}_B\right)f\left(y\right)\text{d}y\right|}^q\text{d}x}\right)}^{\frac{1}{q}}\\ \le C{{\displaystyle \sum _{k=1}^{\infty }{\left|2^{k}B\right|}_{}^{\frac{\alpha }{n}-1}{\displaystyle {\int }_{2^{k+1}B}\left({\displaystyle {\int }_B{\left|\Omega \left(x-y\right)\right|}^q\text{d}x}\right)}}}^{\frac{1}{q}}\left(b_1\left(y\right)-{\left(b_1\right)}_B\right)\left(b_2\left(y\right)-{\left(b_2\right)}_B\right)f\left(y\right)\text{d}y\\ \le C{\displaystyle \sum _{k=1}^{\infty }{\left|2^{k}B\right|}_{}^{\frac{\alpha }{n}-1}}{\displaystyle {\int }_{2^{k+1}B}{\left({\displaystyle {\int }_B{\left|\Omega \left(x-y\right)\right|}^s\text{d}x}\right)}^{\frac{1}{s}}{\left|B\right|}_{}^{\frac{1}{q}-\frac{1}{s}}}\left(b_1\left(y\right)-{\left(b_1\right)}_B\right)\left(b_2\left(y\right)-{\left(b_2\right)}_B\right)f\left(y\right)\text{d}y\\ \le C{\displaystyle \sum _{k=1}^{\infty }{\left|2^{k}B\right|}_{}^{\frac{\alpha }{n}-1+\frac{1}{s}}}{\left|B\right|}_{}^{\frac{1}{q}-\frac{1}{s}}{\left({\displaystyle {\int }_{2^{k+1}B}{\left|b_1\left(y\right)-{\left(b_1\right)}_B\right|}^{s_1}\text{d}y}\right)}^{\frac{1}{s_1}}{\left({\displaystyle {\int }_{2^{k+1}B}{\left|b_2\left(y\right)-{\left(b_2\right)}_B\right|}^{s_2}\text{d}y}\right)}^{\frac{1}{s_2}}\\ \times {\left({\displaystyle {\int }_{2^{k+1}B}{\left|f\left(y\right)\right|}^p\text{d}y}\right)}^{\frac{1}{p}}{\left|2^{k+1}B\right|}_{}^{1-\frac{1}{s_1}-\frac{1}{s_2}-\frac{1}{p}},\end{array}$

又因为

$\begin{array}{c}\left|b_{2^{k+1}B}-b_B\right|\le {\displaystyle \sum _{j=0}^k\left|b_{2^{k+1}B}-b_{2^{j}B}\right|}\le C{{\displaystyle \sum _{j=0}^k\left(\frac{1}{\left|2^{j+1}B\right|}{\displaystyle {\int }_{2^j+1B}{\left|b\left(y\right)-b_{2^{j+1}B}\right|}^{s_1}\text{d}y}\right)}}^{1/s_1}\\ \le C{\displaystyle \sum _{j=0}^k{2}^{\left(j+1\right)n{\mu }_1}}{\left|B\right|}^{{\mu }_1}{\Vert b\Vert }_{CBM{O}^{s_1,{\mu }_1}}\le C{\left|2^{k+1}B\right|}^{{\mu }_1}{\Vert b\Vert }_{CBM{O}^{s_1,{\mu }_1}},\end{array}$

有：

$\begin{array}{l}{\left({\displaystyle {\int }_{2^{k+1}B}{\left|b_1\left(y\right)-{\left(b_1\right)}_B\right|}^{s_1}\text{d}y}\right)}^{\frac{1}{s_1}}\\ \le {\left({\displaystyle {\int }_{2^{k+1}B}{\left|b_1\left(y\right)-{\left(b_1\right)}_{2^{k+1}B}\right|}^{s_1}\text{d}y}\right)}^{\frac{1}{s_1}}+\left|{\left(b_1\right)}_{2^{k+1}B}-{\left(b_1\right)}_B\right|{\left|2^{k+1}B\right|}_{}^{\frac{1}{s_1}}\\ \le C{\left|2^{k+1}B\right|}_{}^{\frac{1}{s_1}+{\mu }_1}{\Vert b_1\Vert }_{CBM{O}^{s_1,{\mu }_1}}.\end{array}$

所以

$\begin{array}{c}{{\rm I}}_{42}\le C{\left|B\right|}_{}^{\frac{1}{q}-\frac{1}{s}}{\displaystyle \prod _{i=1}^2{\Vert b_i\Vert }_{CBM{O}^{s_i,{\mu }_i}}}{\displaystyle \sum _{k=1}^{\infty }{\left|2^{k}B\right|}_{}^{\frac{\alpha }{n}+\frac{1}{s}+{\mu }_1+{\mu }_2}}{\Vert f\Vert }_{{\dot{B}}^{p,v}}\\ \le C{\left|B\right|}_{}^{\lambda +\frac{1}{q}}{\displaystyle \prod _{i=1}^2{\Vert b_i\Vert }_{CBM{O}^{s_i,{\mu }_i}}}{\Vert f\Vert }_{{\dot{B}}^{p,v}}.\end{array}$

综合可得${{\rm I}}_4\le C{\left|B\right|}_{}^{\lambda +\frac{1}{q}}{\displaystyle \prod _{i=1}^2{\Vert b_i\Vert }_{CBM{O}^{s_i,{\mu }_i}}}{\Vert f\Vert }_{{\dot{B}}^{p,v}}$ 。

综合${{\rm I}}_1,{{\rm I}}_2,{{\rm I}}_3,{{\rm I}}_4$ ，再对所有的球B取上确界可得：

${\Vert {\left[\overrightarrow{b},T_{\Omega ,\alpha }\right]}^{2}f\Vert }_{{\dot{B}}^{q,\lambda }}\le C{\displaystyle \prod _{i=1}^2{\Vert b_i\Vert }_{CBM{O}^{s_i,{\mu }_i}}{\Vert f\Vert }_{{\dot{B}}^{p,v}}}$ ，

证毕。

3. 结论

λ-中心Morrey空间和λ-中心BMO空间引起了数学工作者的极大兴趣，并且也是调和分析领域的重要空间。关于此类空间的研究及其积分算子交换子在这些空间中的有界性，可以参见[3] [5]-[7]。与此同时，积分算子广义交换子是经典交换子的推广，文献[4]对此类广义交换子做了深入研究并且得到了一些有意义的结果。本文通过对由带有粗糙核的分数次积分算子和λ-中心BMO函数空间生成的广义交换子进行环分解，利用Hölder不等式，以及带有粗糙核的分数次积分算子$T_{\Omega ,\alpha }$ 的有界性和Minkowski不等式，可得出带有粗糙核的分数次积分广义交换子在λ-中心Morrey空间上的λ-中心BMO估计。其中，本文中的主要结果是在Fu、Lu和Lin [7]，得出具有粗糙核的分数次积分算子在中心Morrey空间上的有界性以及交换子$\left[b,T_{\Omega ,\alpha }\right]$ 在λ-中心Morrey空间上的有界性的基础上进行推广，从而得出带有粗糙核的分数次积分广义交换子在λ-中心Morrey空间上的有界性。本文对研究λ-中心Morrey空间和λ-中心BMO函数空间具有重要意义。