Endpoint Estimates for Commutators of Riesz Transforms Associated with Schrödinger Operators

In this paper, we discuss the H1 L -boundedness of commutators of Riesz transforms associated with the Schrödinger operator L =−4+ V , where H1 L (R n) is the Hardy space associated with L . We assume that V (x) is a nonzero, nonnegative potential which belongs to Bq for some q > n/2. Let T1 = V (x)(−4+ V )−1, T2 = V 1/2(−4+ V )−1/2 and T3 =∇(−4+ V )−1/2. We prove that, for b ∈ BMO(Rn), the commutator [b, T3] is not bounded from H1 L (R n) to L1(Rn) as T3 itself. As an alternative, we obtain that [b, Ti ], ( i = 1, 2, 3) are of (H1 L , L 1 weak)-boundedness. 2000 Mathematics subject classification: primary 47B32, 47A75; secondary 42C40, 94A40.