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

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

doi:10.1631/FITEE.1500082

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

计算数学 | Updated：2022-06-14

- 带粗糙核的Marcinkiewicz积分在Triebel-Lizorkin空间的有界性
- Boundedness of Marcinkiewicz integral with rough kernel on Triebel-Lizorkin spaces
- 信息与电子工程前沿（英文） 2015年16卷第8期页码：654-657
- 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：
  
  Project supported by the National Natural Science Foundation of China (Nos. 11201103 and 11471288)
- DOI：10.1631/FITEE.1500082
  中图分类号： O174.5
- 收稿：2015-03-18，
  
  修回：2015-07-08，
  
  纸质出版：2015-08
- Accepted：
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带粗糙核的Marcinkiewicz积分在Triebel-Lizorkin空间的有界性[J]. 信息与电子工程前沿（英文）, 2015,16(8):654-657.

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.
带粗糙核的Marcinkiewicz积分在Triebel-Lizorkin空间的有界性[J]. 信息与电子工程前沿（英文）, 2015,16(8):654-657. DOI： 10.1631/FITEE.1500082.

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

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