Endpoint estimates for commutators of Riesz transforms related to Schrödinger operators

We consider the Schrödinger operator L = - Δ + V on R n , where n ≥ 3 and the nonnegative potential V belongs to reverse Hölder class R H s for s > n 2 . In this paper, we discuss the boundedness of Riesz transform T α , β = V α L - β and its commutator at the endpoint. We show that T α , β is bounded from L 1 ( R n ) into L p 0 ( R n ) , and prove that [ b , T α , β ] is bounded from H L 1 ( R n ) (Hardy space related to L ) into L p 0 ( R n ) , where p 0 = n n - 2 ( β - α ) and b belongs to the BMO type space introduced by Bongioanni, Harboure and Salinas.