Heisenberg 群上加幂权 Hardy 算子的精确估计

doi:10.12066/j.issn.1007-2861.1817

上海大学学报(自然科学版) ›› 2018, Vol. 24 ›› Issue (2): 257-264.doi: 10.12066/j.issn.1007-2861.1817

Heisenberg 群上加幂权 Hardy 算子的精确估计

陈国霁¹(), 董建锋²

1. 上海大学上海市应用数学和力学研究所, 上海 200072
2. 上海大学理学院, 上海 200444

收稿日期:2016-05-09 出版日期:2018-04-30 发布日期:2018-05-07
通讯作者: 陈国霁 E-mail:mathcgj@sina.com
基金资助:
国家自然科学基金资助项目(11471207)

Sharp estimates for Hardy operator with power weight on Heisenberg group

CHEN Guoji¹(), DONG Jianfeng²

1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China
2. College of Sciences, Shanghai University, Shanghai 200444, China

Received:2016-05-09 Online:2018-04-30 Published:2018-05-07
Contact: CHEN Guoji E-mail:mathcgj@sina.com

摘要/Abstract

摘要：

研究了 Hardy 算子在 $L^{p}({H}^{n},|x|_{h}^{\alpha}{ d}x)$ 函数空间的有界性问题, 其中 Heisenberg 群记为 ${H}^{n}$. 证明了 Hardy 算子是 $(p,p)$ 型 $(1< p\leqslant \infty)$ 和弱 $(1,1)$ 型, 并得到了 $(p,p)$ 型的最佳常数和弱 $(1,1)$ 型的最佳常数的上下界.

关键词: Hardy 算子, Heisenberg 群, 幂权, 精确估计

Abstract:

In this paper, the $n$-dimensional Hardy operator with power weight on the Heisenberg group $H^n$ is studied. It is proved that the Hardy operator is a strong type of ($p, p$) ($1<p\leqslant \infty$) and a weak type of (1,1) on $L^p$($H^n$, $|x|_h^a$d$x$) and $L^1$ ($H^n$, $|x|_h^a$d$x$), respectively. Moreover, the results show that such ($p, p$) estimate is sharp, and obtain the upper and the lower bounds of the best constant of weak (1,1) type.

Key words: Hardy operator, Heisenberg group, power weight, sharp estimate

中图分类号:

陈国霁, 董建锋. Heisenberg 群上加幂权 Hardy 算子的精确估计[J]. 上海大学学报(自然科学版), 2018, 24(2): 257-264.

CHEN Guoji, DONG Jianfeng. Sharp estimates for Hardy operator with power weight on Heisenberg group[J]. Journal of Shanghai University（Natural Science Edition）, 2018, 24(2): 257-264.

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Heisenberg 群上加幂权 Hardy 算子的精确估计

Sharp estimates for Hardy operator with power weight on Heisenberg group

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