Endpoint estimates for commutators of singular integrals related to Schrodinger operators

Let L = -Delta + v Schrodinger operator on R-d, d >= 3, where V is a nonnegative potential, V not equal 0, and belongs to the reverse Holder class RHd/2. In this paper, we study the commutators [b, T] for T in a class K-L of sublinear operators containing the fundamental operators in harmonic analysis related to L. More precisely, when T is an element of K-L, we prove that there exists a bounded subbilinear operator R = R-T H-L(1)(R-d) x BMO(R-d) -> L-1 (Rd) such that (*) vertical bar T(S(f, b))vertical bar - R(f, b) <= vertical bar[b,T](f)I vertical bar <= R(f,b)+vertical bar T(S(f,b))vertical bar, where S is a bounded bilinear operator from H-L(1)(R-d) x BMO(R-d) into L-1 (R-d) which does not depend on T. The subbilinear decomposition (*) allows us to explain why commutators with the fundamental operators are of weak type (H-L(1), L-1), and when a commutator [b, T] is of strong type (H-L(1), L-1). Also, we discuss the H-L(1)-estimates for commutators of the Riesz transforms associated with the Schrodinger operator L.