多线性分数次积分与Lipschitz函数生成的交换子的有界性

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多线性分数次积分与Lipschitz函数生成的交换子的有界性

Commutators Generated by Multilinear Fractional Integrals and Lipschitz Functions

多线性分数次积分与Lipschitz函数生成的交换子的有界性

Commutators Generated by Multilinear Fractional Integrals and Lipschitz Functions

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数学物理学报 ›› 2011, Vol. 31 ›› Issue (5): 1447-1458.

• 论文 • 上一篇

默会霞¹, 张志英²

1.北京邮电大学理学院北京 100876;2.浙江理工大学理学院杭州 310018

收稿日期:2009-12-17 修回日期:2010-12-30 出版日期:2011-10-25 发布日期:2011-10-25
基金资助:
国家自然科学基金(10871024)和中央高校基本科研业务费专项资金(BUPT 2009RC0703)资助

MO Hui-Xia¹, ZHANG Zhi-Ying²

1.School of Science, Beijing University of Post and Telecommunications, Beijing 100876;
2.School of Science, Zhejiang Sci-Tech University, Hangzhou 310018

Received:2009-12-17 Revised:2010-12-30 Online:2011-10-25 Published:2011-10-25
Supported by:
国家自然科学基金(10871024)和中央高校基本科研业务费专项资金(BUPT 2009RC0703)资助

摘要：

设 $m\in{\Bbb N}$ , $\vec{b}=(b_{1},\cdots ,b_{m})$ 是一个局部可积函数族, 且 $\vec{f}=(f_{1},\cdots ,f_{m}),$ 其中 $f_{1},\cdots ,f_{m}\in L_{c}^{\infty}({\mathbf{R}}^{n}).$ 设 $x\notin\bigcap\limits_{i=1}^{m}\mbox{supp}f_{i},$ 则由多线性分数次积分与函数族 $\vec{b}=(b_{1},\cdots ,b_{m})$ 生成的交换子定义为

I_{α, m}^{\vec{b}} (\vec{f}) (x) = \dint_{(R^{n})^{m}} K (x, y_{1}, \dots, y_{m}) \prod_{i = 1}^{m} (b_{i} (x) - b_{i} (y_{i})) f_{i} (y_{i}) d y_{1} \dots d y_{m} .

$I_{\alpha,m}^{\vec{b}}(\vec{f})(x) =\dint_{({\mathbf{R}}^{n})^{m}}K(x,y_{1},\cdots ,y_{m})\prod\limits_{i=1}^{m}(b_{i}(x)-b_{i}(y_{i}))f_{i}(y_i){\rm d}y_1\cdots {\rm d}y_m.$

当 $b_{j}\in\dot{\Lambda}_{\beta_{j}}({\mathbf{R}}^{n})~(1\leq j\leq m)$ 时, 作者考虑 $I_{\alpha,m}^{\vec{b}}$ 在乘积 Lebeasgue 空间,Triebel-Lizorkin 空间和Lipschitz 函数空间的有界性

关键词: 多线性分数次积分, 交换子, Triebel-Lizorkin 空间, Lipschitz 函数空间

Abstract:

Let $m\in{\Bbb N}$ , $\vec{b}=(b_{1},\cdots ,b_{m})$ whose components are of locally integrable functions, and $\vec{f}=(f_{1},\cdots ,f_{m})$ where $f_{1},\cdots ,f_{m}\in L_{c}^{\infty}({\mathbf{R}}^{n}).$ Let $x\notin\bigcap\limits_{i=1}^{m}\mbox{supp}f_{i},$ then the commutator generated by the multilinear fractional integral is given by

I_{α, m}^{\vec{b}} (\vec{f}) (x) = \dint_{(R^{n})^{m}} K (x, y_{1}, \dots, y_{m}) \prod_{i = 1}^{m} (b_{i} (x) - b_{i} (y_{i})) f_{i} (y_{i}) d y_{1} \dots d y_{m} .

$I_{\alpha,m}^{\vec{b}}(\vec{f})(x) =\dint_{({\mathbf{R}}^{n})^{m}}K(x,y_{1},\cdots ,y_{m})\prod\limits_{i=1}^{m}(b_{i}(x)-b_{i}(y_{i}))f_{i}(y_i){\rm d}y_1\cdots {\rm d}y_m.$

When $b_{j}\in\dot{\Lambda}_{\beta_{j}}({{\mathbf{R}}}^{n})~(1\leq j\leq m)$ , the authors establish the boundedness of $I_{\alpha,m}^{\vec{b}}$ on product Lebeasgue spaces, Triebel-Lizorkin spaces and Lipschitz spaces.

Key words: Multilinear fractional integral, Commutator, Triebel-Lizorkin space, Lipschitz function space

中图分类号:

42B25

默会霞, 张志英. 多线性分数次积分与Lipschitz函数生成的交换子的有界性[J]. 数学物理学报, 2011, 31(5): 1447-1458.

MO Hui-Xia, ZHANG Zhi-Ying. Commutators Generated by Multilinear Fractional Integrals and Lipschitz Functions[J]. Acta mathematica scientia,Series A, 2011, 31(5): 1447-1458.

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