Some estimates for commutators of Riesz transform associated with Schrödinger type operators

Czechoslovak Mathematical Journal

Yu Liu¹  Jing Zhang²  Jie-Lai Sheng³ 

[1] (corresponding author),Jing Zhang,Jie-Lai Sheng,Li-Juan Wang, School of Mathematics and Physics, University of Science and Technology Beijing, No. 30 Xueyuan Road, Haidian, Beijing, 100083, P. R. China,

关键词: commutator;    Hardy space;    reverse Hö;    lder inequality;    Riesz transform;    Schrö;    dinger operator;    Schrö;    dinger type operator;   

DOI  :  

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

Let $\mathcal{L}_1=-\Delta+V$ be a Schrödinger operator and let $\mathcal{L}_2=(-\Delta)^2+V^2$ be a Schrödinger type operator on ${\mathbb{R}^n}$ $(n \geq5)$, where $V \neq0$ is a nonnegative potential belonging to certain reverse Hölder class $B_s$ for $s\ge{n}/2$. The Hardy type space $H^1_{\mathcal{L}_2}$ is defined in terms of the maximal function with respect to the semigroup $\{ e^{-t \mathcal{L}_2}\}$ and it is identical to the Hardy space $H^1_{\mathcal{L}_1}$ established by Dziubański and Zienkiewicz. In this article, we prove the $L^p$-boundedness of the commutator $\mathcal{R}_b=b\mathcal{R}f-\mathcal{R}(bf)$ generated by the Riesz transform $\mathcal{R}=\nabla^2\mathcal{L}_2^{-1/2}$, where $b\in BMO_\theta(\rho)$, which is larger than the space $ BMO(\mathbb{R}^n)$. Moreover, we prove that $\mathcal{R}_b$ is bounded from the Hardy space $H_{\mathcal{L}_2}^1(\mathbb{R}^n)$ into weak $L_ weak^1(\mathbb{R}^n)$.

【 授权许可】

Unknown