Commutators of Singular Integral Operators Related to Magnetic Schrödinger Operators

Abstract

Let A:=−(∇−i~a)·(∇−i~a)+V be a magnetic Schrödinger operator on L2(Rn), n ≥ 2, where ~a := (a1,··· ,an) ∈ Lloc(R,R) and 0 ≤ V ∈ Lloc(R). In this paper, we show that for a function b in Lipschitz space Lipα(R n) with α∈ (0,1), the commutator [b,V1/2 A−1/2] is bounded from Lp(Rn) to Lq(Rn), where p,q∈ (1,2] and 1/p−1/q= α/n.