Weighted Endpoint Estimates for Commutators of Riesz Transforms Associated with Schrodinger Operators

Let L = Delta + V be a Schrodinger operator, where Delta is the laplacian on R-n and the nonnegative potential V belongs to the reverse Holder class B-s1 for some s(1) >= (n/2). Assume that omega is an element of A(1)(R-n). Denote by H-L(1)(omega) the weighted Hardy space related to the Schrodinger operator L = -Delta + V. Let R-b = [b, R] be the commutator generated by a function b is an element of BMO theta(R-n) and the Riesz transform R = del(-Delta + V)(-(1/2)). Firstly, we show that the operator R is bounded from L-1(omega) into L-weak(1) (omega). Secondly, we obtain the endpoint estimates for the commutator [b, R]. Namely, it is bounded from the weighted Hardy space H-L(1)(omega) into L-weak(1)(omega).