Behaviour of Schrodinger Riesz transforms over smoothness spaces

As it was shown by Shen, the Riesz transforms associated to the Schrodinger operator L = -triangle + V are not bounded on L-p(R-d)-spaces for all p, 1 < p < infinity, under the only assumption that the potential satisfies a reverse Holder condition of order d/2, d >= 3. Furthermore, they are bounded only for p in some finite interval of the type (1, p0), so it can not be expected to preserve regularity spaces. In this work we search for some kind of minimal additional conditions on the potential in order to obtain boundedness on appropriate weighted BMO type regularity spaces for all first and second order Riesz transforms, namely for the operators del L-1/2, (VL-1/2)-L-1/2, del L-2(-1), V L-1 and V-1/2 del L-1. We also explore to what extent such extra conditions are also necessary. (C) 2022 Elsevier Inc. All rights reserved.

Automated summary

In this work, we investigate the boundedness of Riesz transforms associated with the Schrodinger operator and the additional conditions required on the potential. We explore the boundedness of first and second order Riesz transforms and their effects on regularity spaces.