Some estimates for commutators of Riesz transform associated with Schrodinger type operators

Let L (1) = -Delta + V be a Schr:dinger operator and let L (2) = (-Delta)(2) + V (2) be a Schrodinger type operator on a"e (n) (n a (c) 3/4 5), where V not equal 0 is a nonnegative potential belonging to certain reverse Holder class B (s) for s a (c) 3/4 n/2. The Hardy type space is defined in terms of the maximal function with respect to the semigroup and it is identical to the Hardy space established by DziubaA"ski and Zienkiewicz. In this article, we prove the L (p) -boundedness of the commutator R (b) = bRf - R(bf) generated by the Riesz transform , where , which is larger than the space BMO(a"e (n) ). Moreover, we prove that R (b) is bounded from the Hardy space into weak .