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Let L 1 = −∆ + V be a Schrödinger operator and let L 2 = (−∆) 2 + V 2 be a Schrödinger type operator on R n (n 5), where V = 0 is a nonnegative potential belonging to certain reverse Hölder class Bs for s n/2. The Hardy type space H 1 L2 is defined in terms of the maximal function with respect to the semigroup {e −tL2 } and it is identical to the Hardy space H 1 L1 established by Dziubański and Zienkiewicz. In this article, we prove the L p -boundedness of the commutator R b = bRf − R(bf ) generated by the Riesz transform R = ∇ 2 L −1/2 2 , where b ∈ BMO θ (̺), which is larger than the space BMO(R n ). Moreover, we prove that R b is bounded from the Hardy space H 1 L2 (R n ) into weak L 1 weak (R n ).

Let = −Δ + be a Schrödinger operator, where Δ is the laplacian on R and the nonnegative potential belongs to the reverse Hölder class 1 for some 1 ≥ ( /2). Assume that ∈ 1 (R ). Denote by 1 ( ) the weighted Hardy space related to the Schrödinger operator = −Δ + . Let R = [ , R] be the commutator generated by a function ∈ BMO (R ) and the Riesz transform R = ∇(−Δ + ) −(1/2) . Firstly, we show that the operator R is bounded from 1 ( ) into 1 weak ( ). Secondly, we obtain the endpoint estimates for the commutator [ , R]. Namely, it is bounded from the weighted Hardy space 1 ( ) into 1 weak ( ).

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