数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (1): 39-66.doi: 10.1007/s10473-021-0103-7

• 论文 • 上一篇    下一篇

ISOMORPHISMS OF VARIABLE HARDY SPACES ASSOCIATED WITH SCHRÖDINGER OPERATORS

张俊强1, 杨大春2   

  1. 1. School of Science, China University of Mining and Technology-Beijing, Beijing 100083, China;
    2. Laboratory of Mathematics and Complex Systems(Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
  • 收稿日期:2020-01-22 修回日期:2020-03-30 出版日期:2021-02-25 发布日期:2021-04-06
  • 通讯作者: Dachun YANG E-mail:dcyang@bnu.edu.cn
  • 作者简介:Junqiang ZHANG,E-mail:jqzhang@cumtb.edu.cn
  • 基金资助:
    Junqiang Zhang was supported by the National Natural Science Foundation of China (11801555 and 11971058) and the Fundamental Research Funds for the Central Universities (2020YQLX02). Dachun Yang was supported by the National Natural Science Foundation of China (11971058, 11761131002 and 11671185).

ISOMORPHISMS OF VARIABLE HARDY SPACES ASSOCIATED WITH SCHRÖDINGER OPERATORS

Junqiang ZHANG1, Dachun YANG2   

  1. 1. School of Science, China University of Mining and Technology-Beijing, Beijing 100083, China;
    2. Laboratory of Mathematics and Complex Systems(Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
  • Received:2020-01-22 Revised:2020-03-30 Online:2021-02-25 Published:2021-04-06
  • Contact: Dachun YANG E-mail:dcyang@bnu.edu.cn
  • About author:Junqiang ZHANG,E-mail:jqzhang@cumtb.edu.cn
  • Supported by:
    Junqiang Zhang was supported by the National Natural Science Foundation of China (11801555 and 11971058) and the Fundamental Research Funds for the Central Universities (2020YQLX02). Dachun Yang was supported by the National Natural Science Foundation of China (11971058, 11761131002 and 11671185).

摘要: Let $L:=-\Delta+V$ be the Schrödinger operator on $\mathbb{R}^n$ with $n\geq3$, where $V$ is a non-negative potential satisfying $\Delta^{-1}(V)\in L^\infty(\mathbb{R}^n)$. Let $w$ be an $L$-harmonic function, determined by $V$, satisfying that there exists a positive constant $\delta$ such that, for any $x\in\mathbb{R}^n$, $0<\delta\leq w(x)\leq 1$. Assume that $p(\cdot):\ \mathbb{R}^n\to (0,\,1]$ is a variable exponent satisfying the globally $\log$-Hölder continuous condition. In this article, the authors show that the mappings $H_L^{p(\cdot)}(\mathbb{R}^n)\ni f\mapsto wf\in H^{p(\cdot)}(\mathbb{R}^n)$ and $H_L^{p(\cdot)}(\mathbb{R}^n)\ni f\mapsto (-\Delta)^{1/2}L^{-1/2}(f)\in H^{p(\cdot)}(\mathbb{R}^n)$ are isomorphisms between the variable Hardy spaces $H_L^{p(\cdot)}(\mathbb{R}^n)$, associated with $L$, and the variable Hardy spaces $H^{p(\cdot)}(\mathbb{R}^n)$.

关键词: variable Hardy space, Schrödinger operator, L-harmonic function, isomorphism, atom

Abstract: Let $L:=-\Delta+V$ be the Schrödinger operator on $\mathbb{R}^n$ with $n\geq3$, where $V$ is a non-negative potential satisfying $\Delta^{-1}(V)\in L^\infty(\mathbb{R}^n)$. Let $w$ be an $L$-harmonic function, determined by $V$, satisfying that there exists a positive constant $\delta$ such that, for any $x\in\mathbb{R}^n$, $0<\delta\leq w(x)\leq 1$. Assume that $p(\cdot):\ \mathbb{R}^n\to (0,\,1]$ is a variable exponent satisfying the globally $\log$-Hölder continuous condition. In this article, the authors show that the mappings $H_L^{p(\cdot)}(\mathbb{R}^n)\ni f\mapsto wf\in H^{p(\cdot)}(\mathbb{R}^n)$ and $H_L^{p(\cdot)}(\mathbb{R}^n)\ni f\mapsto (-\Delta)^{1/2}L^{-1/2}(f)\in H^{p(\cdot)}(\mathbb{R}^n)$ are isomorphisms between the variable Hardy spaces $H_L^{p(\cdot)}(\mathbb{R}^n)$, associated with $L$, and the variable Hardy spaces $H^{p(\cdot)}(\mathbb{R}^n)$.

Key words: variable Hardy space, Schrödinger operator, L-harmonic function, isomorphism, atom

中图分类号: 

  • 42B30