Volume 34, Issue 1
Commutators of Singular Integral Operators Related to Magnetic Schrödinger Operators

Anal. Theory Appl., 34 (2018), pp. 45-76.

Published online: 2018-07

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• Abstract

Let $A:=−(\nabla−i\vec{a})·(\nabla−i\vec{a})+V$ be a magnetic Schrödinger operator on $L^2(\mathbb{R}^n)$, $n\geq 2$, where $\vec{a} := (a_1 ,···,a_n) \in L^2_{loc}(\mathbb{R^n}, \mathbb{R^n})$ and $0\leq V \in L^1_{loc}(\mathbb{R^n})$. In this paper, we show that for a function $b$ in Lipschitz space Lip$_{\alpha}$ $(\mathbb{R^n})$ with $\alpha\in (0,1)$, the commutator $[b, V^{1/2}A^{-1/2}]$ is bounded from $L^p(\mathbb{R^n})$ to $L^q(\mathbb{R^n})$, where $p$, $q\in (1,2]$ and $1/p−1/q = α/n$.

• Keywords

Commutator, Lipschitz space, the sharp maxical function, magnetic Schrödinger operator, Hölder inequality.

42B20, 42B35

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@Article{ATA-34-45, author = {}, title = {Commutators of Singular Integral Operators Related to Magnetic Schrödinger Operators}, journal = {Analysis in Theory and Applications}, year = {2018}, volume = {34}, number = {1}, pages = {45--76}, abstract = {

Let $A:=−(\nabla−i\vec{a})·(\nabla−i\vec{a})+V$ be a magnetic Schrödinger operator on $L^2(\mathbb{R}^n)$, $n\geq 2$, where $\vec{a} := (a_1 ,···,a_n) \in L^2_{loc}(\mathbb{R^n}, \mathbb{R^n})$ and $0\leq V \in L^1_{loc}(\mathbb{R^n})$. In this paper, we show that for a function $b$ in Lipschitz space Lip$_{\alpha}$ $(\mathbb{R^n})$ with $\alpha\in (0,1)$, the commutator $[b, V^{1/2}A^{-1/2}]$ is bounded from $L^p(\mathbb{R^n})$ to $L^q(\mathbb{R^n})$, where $p$, $q\in (1,2]$ and $1/p−1/q = α/n$.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2018.v34.n1.4}, url = {http://global-sci.org/intro/article_detail/ata/12544.html} }
TY - JOUR T1 - Commutators of Singular Integral Operators Related to Magnetic Schrödinger Operators JO - Analysis in Theory and Applications VL - 1 SP - 45 EP - 76 PY - 2018 DA - 2018/07 SN - 34 DO - http://doi.org/10.4208/ata.2018.v34.n1.4 UR - https://global-sci.org/intro/article_detail/ata/12544.html KW - Commutator, Lipschitz space, the sharp maxical function, magnetic Schrödinger operator, Hölder inequality. AB -

Let $A:=−(\nabla−i\vec{a})·(\nabla−i\vec{a})+V$ be a magnetic Schrödinger operator on $L^2(\mathbb{R}^n)$, $n\geq 2$, where $\vec{a} := (a_1 ,···,a_n) \in L^2_{loc}(\mathbb{R^n}, \mathbb{R^n})$ and $0\leq V \in L^1_{loc}(\mathbb{R^n})$. In this paper, we show that for a function $b$ in Lipschitz space Lip$_{\alpha}$ $(\mathbb{R^n})$ with $\alpha\in (0,1)$, the commutator $[b, V^{1/2}A^{-1/2}]$ is bounded from $L^p(\mathbb{R^n})$ to $L^q(\mathbb{R^n})$, where $p$, $q\in (1,2]$ and $1/p−1/q = α/n$.

Wanqing Ma & Yu Liu. (1970). Commutators of Singular Integral Operators Related to Magnetic Schrödinger Operators. Analysis in Theory and Applications. 34 (1). 45-76. doi:10.4208/ata.2018.v34.n1.4
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