Commutators of Riesz transforms of magnetic Schrödinger operators

Abstract

Let ${A=-(\nabla-i{\vec a})\cdot (\nabla-i{\vec a}) +V}$ be a magnetic Schrödinger operator acting on ${L^2({\mathbb R}^n)}$, n ≥  1, where ${{\vec a}=(a_1, \ldots, a_n)\in L^2_{\rm loc}({\mathbb R}^n, {\mathbb R}^n)}$ and ${0\leq V\in L^1_{\rm loc}({\mathbb R}^n)}$. In this paper, we show that when a function ${b\in {\rm BMO}({\mathbb R}^n)}$, the commutators [b, T _k ]f = T _k (b f) − b T _k f, k = 1, . . . , n, are bounded on ${L^p({\mathbb R}^n)}$ for all 1 < p < 2, where the operators T _k are Riesz transforms (∂/∂x _k  − i a _k )A ^−1/2 associated with A.