Boundedness Estimates for Commutators of Riesz Transforms Related to Schrödinger Operators

Abstract

Let L=−∆+V be a Schrödinger operator on Rn(n≥ 3), where the nonnegative potential V belongs to reverse Hölder class RHq1 for q1 > n 2 . Let H p L(R n) be the Hardy space associated with L. In this paper, we consider the commutator [b,Tα], which associated with the Riesz transform Tα =Vα(−∆+V)−α with 0< α≤ 1, and a locally integrable function b belongs to the new Campanato space Λβ(ρ). We establish the boundedness of [b,Tα] from Lp(Rn) to Lq(Rn) for 1 < p < q1/α with 1/q= 1/p−β/n. We also show that [b,Tα] is bounded from H L(R n) to Lq(Rn) when n/(n+β)< p≤ 1,1/q = 1/p−β/n. Moreover, we prove that [b,Tα] maps H n n+β L (R n) continuously into weak L1(Rn).