@Article{ATA-34-45, author = {Wanqing Ma and Yu Liu}, 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 $1/p−1/q =
α/n$.