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As it was shown by Shen, the Riesz transforms associated to the Schr\"odingeroperator $L=-\Delta + V$ are not bounded on $L^p(\mathbb{R}^d)$-spaces for all$p, 1<p<\infty$, under the only assumption that the potential satisfies areverse H\"older condition of order $d/2$, $d\geq3$. Furthermore, they arebounded only for $p$ in some finite interval of the type $(1,p_0)$, so it cannot be expected to preserve regularity spaces. In this work we search for somekind of minimal additional conditions on the potential in order to obtainboundedness on appropriate weighted $BMO$ type regularity spaces for all firstand second order Riesz transforms, namely for the operators $\nabla L^{-1/2}$,$V^{1/2}L^{-1/2}$, $\nabla^2 L^{-1}$, $VL^{-1}$ and $V^{1/2}\nabla L^{-1}$. Wealso explore to what extent such extra conditions are also necessary.

Behaviour of Schr\"odinger Riesz transforms over smoothness spaces