Vietnam J Math DOI 10.1007/s10013-014-0065-3 The Second-Order Riesz Transforms Related to Schrödinger Operators Acting on BMO-Type Spaces Nguyen Ngoc Trong Received: 11 January 2013 / Accepted: 11 October 2013 © Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2014 Abstract Let L =−Δ + V be a Schrödinger operator on R d , where V is a nonnega- tive potential satisfying the suitable reverse Hölder inequality. We prove the boundedness of the operator (−Δ) −1 ∇ 2 on BMO β (w) and the operator (−Δ + V) −1 ∇ 2 on BMO β L (w). We also show that the operator (−Δ + V) −1 (−Δ) is bounded from BMO to BMO L and (−Δ + V) −1 ∇ 2 is bounded from BMO β to BMO β L . Keywords Riesz transform · Schrödinger operator · BMO-type space · Weights · Boundedness Mathematics Subject Classification (2000) 42B20 · 42B35 1 Introduction Let L =−Δ + V be a Schrödinger operator on R d (d ≥ 3), where V is a nonnegative potential belonging to the reverse Hölder class RH q for some q ≥ d/2, i.e., V satisfies the reverse Hölder inequality  1 |B|  B V (x) q dx  1 q ≤ C |B|  B V(x)dx for all balls B ⊂ R d . It is well known that if V ∈ RH q for some q> 1, then there exists ε> 0, which depends only on d and the constant C, such that V ∈ RH q+ε . N.N. Trong (B ) Department of Mathematics, University of Pedagogy, HoChiMinh City, Vietnam e-mail: trongtrt@gmail.com N.N. Trong (B ) e-mail: ngoctrong37@yahoo.com