Behaviour of Schrödinger Riesz transforms over smoothness spaces

As it was shown by Shen, the Riesz transforms associated to Schrödinger operator L=−Δ+V are not bounded on Lp(Rd)-spaces for all p,1<p<∞, under only assumption that potential satisfies a reverse Hölder condition of order d/2, d≥3. Furthermore, they p in some finite interval type (1,p0), so can be expected preserve regularity spaces. In this work we search kind minimal additional conditions obtain boundedness appropriate weighted BMO spaces first and second transforms, namely operators ∇L−1/2, V1/2L−1/2, ∇2L−1, VL−1 V1/2∇L−1. We also explore what extent such extra necessary.