Hardy空间上维林肯型系统下的极大算子

数学物理学报 ›› 2022, Vol. 42 ›› Issue (5): 1294-1305.

Hardy空间上维林肯型系统下的极大算子

张传洲(),王超越(),张学英*()

^{武汉科技大学理学院武汉 430065
^{武汉科技大学冶金工业过程系统科学湖北省重点实验室武汉 430081}}

收稿日期:2021-11-15 出版日期:2022-10-26 发布日期:2022-09-30
通讯作者: 张学英 E-mail:zczwust@163.com;974882481@qq.com;zhangxueying@wust.edu.cn
作者简介:张传洲, E-mail: zczwust@163.com|王超越, 974882481@qq.com
基金资助:
国家自然科学基金(11871195)

The Maximal Operator of Vilenkin-like System on Hardy Spaces

Chuanzhou Zhang(),Chaoyue Wang(),Xueying Zhang*()

^{College of Science, Wuhan University of Science and Technology, Wuhan 430065
^{Hubei Province Key Laboratory of Systems Science in Metallurgical Process, Wuhan University of Science and Technology, Wuhan 430081}}

Received:2021-11-15 Online:2022-10-26 Published:2022-09-30
Contact: Xueying Zhang E-mail:zczwust@163.com;974882481@qq.com;zhangxueying@wust.edu.cn
Supported by:
the NSFC(11871195)

摘要/Abstract

摘要：

维林肯型系统(或 $\psi\alpha$ 系统)是维林肯系统的推广, 该文研究有界维林肯型系统下的极大算子的有界性. 该文证明当 $0 < p <1/2$ 时, 极大算子 $\tilde{\sigma}_p^*f=\sup\limits_{n\in {\Bbb N}}\frac{|\sigma_nf|}{(n+1)^{1/p-2}}$ 是从鞅 Hardy 空间 $H_p$ 到 $L_p$ 有界的, 其中 $\sigma_nf$ 是关于有界维林肯型系统的 Fej\'er 均值. 并通过构造反例, 证明当 $0 < p <1/2$ 且 $\mathop{\overline{\lim}}\limits_{n\rightarrow \infty}\frac{(n+1)^{1/p-2}}{\varphi(n)}=+\infty$ 时, 极大算子 $\sup\limits_{n\in {\Bbb N}}\frac{|\sigma_nf|}{\varphi(n)}$ 不是从鞅 Hardy 空间 $H_{p}$ 到 $L_{p,\infty}$ 有界的.

关键词: 维林肯型系统, 极大算子, Fejér均值, Hardy空间

Abstract:

In this paper, we discuss the boundedness of maximal operator with respect to bounded Vilenkin-like system (or $\psi\alpha$ system) which is generalization of bounded Vilenkin system. We prove that when $0 < p <1/2$ the maximal operator $\tilde{\sigma}_p^*f=\sup\limits_{n\in {\Bbb N}}\frac{|\sigma_nf|}{(n+1)^{1/p-2}}$ is bounded from the martingale Hardy space $H_p$ to the space $L_p$, where $\sigma_nf$ is $n$-th Fej\'er mean with respect to bounded Vilenkin-like system. By a counterexample, we also prove that the maximal operator $\sup\limits_{n\in {\Bbb N}}\frac{|\sigma_nf|}{\varphi(n)}$ is not bounded from the martingale Hardy space $H_{p}$ to the space $L_{p,\infty}$ when $0 < p <1/2$ and $\mathop{\overline{\lim}}\limits_{n\rightarrow \infty}\frac{(n+1)^{1/p-2}}{\varphi(n)}=+\infty$.

Key words: Vilenkin-like system, Maximal operator, Fejér mean, Hardy spaces

中图分类号:

O174.2

张传洲,王超越,张学英. Hardy空间上维林肯型系统下的极大算子[J]. 数学物理学报, 2022, 42(5): 1294-1305.

Chuanzhou Zhang,Chaoyue Wang,Xueying Zhang. The Maximal Operator of Vilenkin-like System on Hardy Spaces[J]. Acta mathematica scientia,Series A, 2022, 42(5): 1294-1305.

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