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Frontiers of Information Technology & Electronic Engineering
ISSN 2095-9184 (print), ISSN 2095-9230 (online)
2015 Vol.16 No.8 P.654-657
Boundedness of Marcinkiewicz integral with rough kernel on Triebel-Lizorkin spaces
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.
Key words: Marcinkiewicz integral, Triebel-Lizorkin spaces
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。
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DOI:
10.1631/FITEE.1500082
CLC number:
O174.5
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On-line Access:
2015-08-04
Received:
2015-03-18
Revision Accepted:
2015-06-01
Crosschecked:
2015-07-08