Lp boundedness of commutators of Riesz transforms associated to Schrödinger operator

In this paper we consider Lp boundedness of some commutators of Riesz transforms associated to Schrödinger operator P=−Δ+V(x) on Rn, n⩾3. We assume that V(x) is non-zero, non-negative, and belongs to Bq for some q⩾n/2. Let T1=−1(−Δ+V)V, T2=(−Δ+V)−1/2V1/2 and T3=(−Δ+V)−1/2∇. We obtain that [b,Tj] (j=1,2,3) are bounded operators on Lp(Rn) when p ranges in a interval, where b∈BMO(Rn). Note that the kernel of Tj (j=1,2,3) has no smoothness.