Hardy type estimates for commutators of fractional integrals associated with Schrodinger operators

We consider the Schrodinger operator L = −∆ + V on Rn, where n ≥ 3 and the nonnegative potential V belongs to n reverse Holder class RHq1 for some q1 n/2 . Let Iα be the fractional integral associated with L, and let b belong to a new Campanato space Λθ (ρ). In this paper, we establish the boundedness of the commutators [b, Iα] from Lp(Rn) to Lq(Rn) whenever 1/q = 1/p − (α + β)/n, 1 p n/(α + β). When n/n+β p ≤ 1, 1/q = 1/p − (α + β)/n, we show that [b, Iα] is n n bounded from Hp (Rn) to Lq(Rn). Moreover, we also prove that [b, Iα] maps H n+β (Rn) continuously into weak L n/n−α (Rn).