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

ZHANG Chun-jie; REN Fang-fang; ZHANG Yu-huai; GAO Gui-lian

doi:10.1631/FITEE.1500082

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

Computational Mathematics | Updated：2022-06-14

- Boundedness of Marcinkiewicz integral with rough kernel on Triebel-Lizorkin spaces
- 带粗糙核的Marcinkiewicz积分在Triebel-Lizorkin空间的有界性
- Frontiers of Information Technology & Electronic Engineering Vol. 16, Issue 8, Pages: 654-657(2015)
- Affiliations：
  
  Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310016, China
- Author bio：
  
  [ "Chun-jie ZHANG, E-mail: purezhang@hdu.edu.cn" ]
  Gui-lian GAO, E-mail: gaoguilian305@163.com
- Funds：
- DOI：10.1631/FITEE.1500082
  CLC： O174.5
- Received：18 March 2015，
  
  Revised：2015-07-08，
  
  Published：2015-08
- Accepted：
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ZHANG Chun-jie, REN Fang-fang, ZHANG Yu-huai, et al. Boundedness of Marcinkiewicz integral with rough kernel on Triebel-Lizorkin spaces[J]. Frontiers of Information Technology & Electronic Engineering, 2015, 16(8): 654-657.
DOI：

ZHANG Chun-jie, REN Fang-fang, ZHANG Yu-huai, et al. Boundedness of Marcinkiewicz integral with rough kernel on Triebel-Lizorkin spaces[J]. Frontiers of Information Technology & Electronic Engineering, 2015, 16(8): 654-657. DOI： 10.1631/FITEE.1500082.

摘要

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。

Abstract

This paper is a continuation of our previous work (

Zhang and Chen

2010b

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.

关键词

Keywords

references

H.M. Al-Qassem , L.C. Cheng , Y. Pan . Boundedness of rough integral operators on Triebel-Lizorkin spaces . Publ. Mat. , 2012 . 56 ( 2 ): 261 - 277 . DOI: 10.5565/PUBLMAT\_56212\_01 http://doi.org/10.5565/PUBLMAT\_56212\_01 .

D.N. Chen , J.C. Chen , D.S. Fan . Rough singular integral operators on Hardy-Sobolev spaces . Appl. Math. J. Chin. Univ. , 2005 . 20 ( 1 ): 1 - 9 . DOI: 10.1007/s11766-005-0030-8 http://doi.org/10.1007/s11766-005-0030-8 .

J.C. Chen , C.J. Zhang . Boundedness of rough singular integral operators on the Triebel-Lizorkin spaces . J. Math. Anal. Appl. , 2008 . 337 ( 2 ): 1048 - 1052 . DOI: 10.1016/j.jmaa.2007.04.026 http://doi.org/10.1016/j.jmaa.2007.04.026 .

J.C. Chen , D.S. Fan , Y.M. Ying . Singular integral operators on function spaces . J. Math. Anal. Appl. , 2002 . 276 ( 2 ): 691 - 708 . DOI: 10.1016/S0022-247X(02)00419-5 http://doi.org/10.1016/S0022-247X(02)00419-5 .

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

J.C. Chen , H.Y. Jia , L.Y. Jiang . Boundedness of rough oscillatory singular integral on Triebel-Lizorkin spaces . J. Math. Anal. Appl. , 2005 . 306 ( 2 ): 385 - 397 . DOI: 10.1016/j.jmaa.2005.01.015 http://doi.org/10.1016/j.jmaa.2005.01.015 .

Q.L. Chen , Z.F. Zhang . Boundedness of a class of super singular integral operators and the associated commutators . Sci. China Ser. A , 2004 . 47 ( 6 ): 842 - 853 . DOI: 10.1360/03ys0099 http://doi.org/10.1360/03ys0099 .

Y.P. Chen , Y. Ding . Rough singular integrals on Triebel-Lizorkin space and Besov space . J. Math. Anal. Appl. , 2008 . 347 ( 2 ): 493 - 501 . DOI: 10.1016/j.jmaa.2008.06.039 http://doi.org/10.1016/j.jmaa.2008.06.039 .

Y.P. Chen , K. Zhu . L p bounds for the commutators of oscillatory singular integrals with rough kernels . Abs. Appl. Anal. , 2014 . 2014 393147.1 - 393147.8 . DOI: 10.1155/2014/393147 http://doi.org/10.1155/2014/393147 .

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

C.J. Zhang , J.C. Chen . Boundedness of singular integrals and maximal singular integrals on Triebel-Lizorkin spaces . Int. J. Math. , 2010a . 21 ( 2 ): 157 - 168 . DOI: 10.1142/S0129167X10005982 http://doi.org/10.1142/S0129167X10005982 .

C.J. Zhang , J.C. Chen . Boundedness of Marcinkiewicz integral on Triebel-Lizorkin spaces . Appl. Math. J. Chin. Univ. , 2010b . 25 ( 1 ): 48 - 54 . DOI: 10.1007/s11766-010-2086-3 http://doi.org/10.1007/s11766-010-2086-3 .

C.J. Zhang , Y.D. Zhang . Boundedness of oscillatory singular integral with rough kernels on Triebel-Lizorkin spaces . Appl. Math. J. Chin. Univ. , 2013 . 28 ( 1 ): 90 - 100 . DOI: 10.1007/s11766-013-3033-x http://doi.org/10.1007/s11766-013-3033-x .

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