Boundedness of second-order Riesz transforms on weighted Hardy and $BMO$ spaces associated with Schrödinger operators

Let $d \in \lbrace 3, 4, 5, \ldots \rbrace $ and a weight $w \in A^\rho _\infty $. We consider the second-order Riesz transform $T = \nabla ^2 \, L^{-1}$ associated with the Schrödinger operator $L = -\Delta + V$, where $V \in RH_\sigma $ with $\sigma > \frac{d}{2}$. We present three main results. First $T$ is bounded on the weighted Hardy space $H^1_{w,L}(\mathbb{R}^d)$ associated with $L$ if $w$ enjoys a certain stable property. Secondly $T$ is bounded on the weighted $BMO$ space $BMO_{w,\rho }(\mathbb{R}^d)$ associated with $L$ if $w$ also belongs to an appropriate doubling class. Thirdly $BMO_{w,\rho }(\mathbb{R}^d)$ is the dual of $H^1_{w,L}(\mathbb{R}^d)$ when $w \in A^\rho _1$.