数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (5): 1971-1980.doi: 10.1007/s10473-022-0514-0

• 论文 • 上一篇    

GLEASON’S PROBLEM ON THE SPACE Fp,q,s (B) IN $\mathbb{C}^n$

Pengcheng TANG, Xuejun ZHANG   

  1. College of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, China
  • 收稿日期:2020-11-13 修回日期:2022-05-30 发布日期:2022-11-02
  • 通讯作者: Xuejun Zhang,E-mail:xuejunttt@263.net E-mail:xuejunttt@263.net
  • 基金资助:
    The research was supported by the National Natural Science Foundation of China (11942109) and the Natural Science Foundation of Hunan Province (2022JJ30369).

GLEASON’S PROBLEM ON THE SPACE Fp,q,s (B) IN $\mathbb{C}^n$

Pengcheng TANG, Xuejun ZHANG   

  1. College of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, China
  • Received:2020-11-13 Revised:2022-05-30 Published:2022-11-02
  • Contact: Xuejun Zhang,E-mail:xuejunttt@263.net E-mail:xuejunttt@263.net
  • Supported by:
    The research was supported by the National Natural Science Foundation of China (11942109) and the Natural Science Foundation of Hunan Province (2022JJ30369).

摘要: Let $\Omega$ be a domain in $ \mathbb{C}^{n}$ and let $Y$ be a function space on $\Omega$. If $a\in \Omega$ and $g\in Y$ with $g(a)=0$, do there exist functions $f_{1},f_{2},\cdots ,f_{n}\in Y$ such that $$g(z)=\sum_{l=1}^{n}(z_{l}-a_{l})\ f_{l}(z) \ \ \mbox{ for all $z=(z_{1},z_{2},\cdots ,z_{n})\in \Omega$} \ ? $$ This is Gleason's problem. In this paper, we prove that Gleason's problem is solvable on the boundary general function space $F^{p,q,s}(B)$ in the unit ball $B$ of $ \mathbb{C}^{n}$.

关键词: boundary general function space, Gleason’s problem, solvability, unit ball

Abstract: Let $\Omega$ be a domain in $ \mathbb{C}^{n}$ and let $Y$ be a function space on $\Omega$. If $a\in \Omega$ and $g\in Y$ with $g(a)=0$, do there exist functions $f_{1},f_{2},\cdots ,f_{n}\in Y$ such that $$g(z)=\sum_{l=1}^{n}(z_{l}-a_{l})\ f_{l}(z) \ \ \mbox{ for all $z=(z_{1},z_{2},\cdots ,z_{n})\in \Omega$} \ ? $$ This is Gleason's problem. In this paper, we prove that Gleason's problem is solvable on the boundary general function space $F^{p,q,s}(B)$ in the unit ball $B$ of $ \mathbb{C}^{n}$.

Key words: boundary general function space, Gleason’s problem, solvability, unit ball

中图分类号: 

  • 32A37