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CLC number: O174.5

On-line Access: 2015-08-04

Received: 2015-03-18

Revision Accepted: 2015-06-01

Crosschecked: 2015-07-08

Cited: 0

Clicked: 4232

Citations:  Bibtex RefMan EndNote GB/T7714

 ORCID:

Chun-jie Zhang

http://orcid.org/0000-0002-5712-8764

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Frontiers of Information Technology & Electronic Engineering  2015 Vol.16 No.8 P.654-657

http://doi.org/10.1631/FITEE.1500082


Boundedness of Marcinkiewicz integral with rough kernel on Triebel-Lizorkin spaces


Author(s):  Chun-jie Zhang, Fang-fang Ren, Yu-huai Zhang, Gui-lian Gao

Affiliation(s):  Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310016, China

Corresponding email(s):   purezhang@hdu.edu.cn, gaoguilian305@163.com

Key Words:  Marcinkiewicz integral, Triebel-Lizorkin spaces


Chun-jie Zhang, Fang-fang Ren, Yu-huai Zhang, Gui-lian Gao. Boundedness of Marcinkiewicz integral with rough kernel on Triebel-Lizorkin spaces[J]. Frontiers of Information Technology & Electronic Engineering, 2015, 16(8): 654-657.

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Abstract: 
This paper is a continuation of our previous work (Zhang and Chen, 2010b). Following the same general steps of the proof there, we make essential improvement on our previous theorem by recalculating a key inequality. Our result shows that the marcinkiewicz integral, with a bounded radial function in its kernel, is still bounded on the Triebel-Lizorkin space.

The paper studies the boundedness of Marcinkiewicz integral on Triebel-Lizorkin spaces, it is new and interesting in harmonic analysis.

带粗糙核的Marcinkiewicz积分在Triebel-Lizorkin空间的有界性

目的:研究带有径向粗糙项的Marcinkiewicz积分,证明这类积分算子也有Triebel-Lizorkin有界性。
创新点:沿用向量值奇异积分将粗糙核算子光滑化的思路,证明转后的算子具有更好的光滑性条件。
方法:首先利用本文作者之前文章的方法,把带径向粗糙项的Marckinkiewicz积分转化成研究一些具有一定光滑性的算子(需反复利用向量值奇异积分定理)。然后,利用微分指标较低时,Triebel-Lizorkin空间的一个等刻画,把Triebel-Lizorkin有界性转化成向量值的Lebesgue空间有界性。于是我们只需要研究这些有光滑性算子的向量值Lebesgue空间有界性,这整套方法是作者之前系列文章的一个整体思路。本文也利用这套思路,在该框架下,研究转化后算子的核,得到关于这个核的更精细估计,从而推广了原有结果。
结论:对于带有径向粗糙项的算子,同样可以得到一般的Marcinkiewicz积分在Triebel-Lizorkin空间的有界性。

关键词:Marcinkiewicz积分;Triebel-Lizorkin空间

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Reference

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[2]Chen, D.N., Chen, J.C., Fan, D.S., 2005. Rough singular integral operators on Hardy-Sobolev spaces. Appl. Math. J. Chin. Univ., 20(1):1-9.

[3]Chen, J.C., Zhang, C.J., 2008. Boundedness of rough singular integral operators on the Triebel-Lizorkin spaces. J. Math. Anal. Appl., 337(2):1048-1052.

[4]Chen, J.C., Fan, D.S., Ying, Y.M., 2002. Singular integral operators on function spaces. J. Math. Anal. Appl., 276(2):691-708.

[5]Chen, J.C., Fan, D.S., Ying, Y.M., 2003. Certain operators with rough singular kernels. Can. J. Math., 55(3): 504-532.

[6]Chen, J.C., Jia, H.Y., Jiang, L.Y., 2005. Boundedness of rough oscillatory singular integral on Triebel-Lizorkin spaces. J. Math. Anal. Appl., 306(2):385-397.

[7]Chen, Q.L., Zhang, Z.F., 2004. Boundedness of a class of super singular integral operators and the associated commutators. Sci. China Ser. A, 47(6):842-853.

[8]Chen, Y.P., Ding, Y., 2008. Rough singular integrals on Triebel-Lizorkin space and Besov space. J. Math. Anal. Appl., 347(2):493-501.

[9]Chen, Y.P., Zhu, K., 2014. Lp bounds for the commutators of oscillatory singular integrals with rough kernels. Abs. Appl. Anal., 2014:393147.1-393147.8.

[10]Zhang, C.J., Chen, J.C., 2009. Boundedness of g-functions on Triebel-Lizorkin spaces. Taiwan. J. Math., 13(3):973-981.

[11]Zhang, C.J., Chen, J.C., 2010a. Boundedness of singular integrals and maximal singular integrals on Triebel-Lizorkin spaces. Int. J. Math., 21(2):157-168.

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[13]Zhang, C.J., Zhang, Y.D., 2013. Boundedness of oscillatory singular integral with rough kernels on Triebel-Lizorkin spaces. Appl. Math. J. Chin. Univ., 28(1):90-100.

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