Commutators of Singular Integral Operators Related to Magnetic Schrödinger Operators (Analysis in Theory and Applications, 2018)
Let $A:=−( abla−i\vec{a})·( abla−i\vec{a})+V$ be a magnetic Schrödinger operator on $L^2(\mathbb{R}^n)$, $n≥ 2$, where $\vec{a} := (a_1,···,a_n) \in L^2_{loc}(\mathbb{R^n}, \mathbb{R^n})$ and $0≤ V \in L^1_{loc}(\mathbb{R^n})$. In this paper, weshow that for a function $b$ in Lipschitz space Lip$_{α}$ $(\mathbb{R^n})$ with $α\in (0,1)$, 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 $1/p−1/q =α/n$.