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 = (−Δ+ V ) −1V , T2 = (−Δ+V ) −1/2V 1/2 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. © 2007 Elsevier Inc. All rights reserved