  As it was shown by Shen, the Riesz transforms associated to the Schr\"odinger
operator $L=-\Delta + V$ are not bounded on $L^p(\mathbb{R}^d)$-spaces for all
$p, 1<p<\infty$, under the only assumption that the potential satisfies a
reverse H\"older condition of order $d/2$, $d\geq3$. Furthermore, they are
bounded only for $p$ in some finite interval of the type $(1,p_0)$, 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 $\nabla L^{-1/2}$,
$V^{1/2}L^{-1/2}$, $\nabla^2 L^{-1}$, $VL^{-1}$ and $V^{1/2}\nabla L^{-1}$. We
also explore to what extent such extra conditions are also necessary.
