Riesz transforms for bi-Schrödinger operators on weighted Lebesgue spaces

Let $d\in \{5,6,7,\dots \}$ and a weight $w\in A_{\infty }^{\rho }$. We consider the fourth-order Riesz transform $T={\nabla }^4{L}^{-1}$ associated with the bi-Schrödinger operator $L={\Delta }^2+V^2$, where $V\in R{H}_{\sigma }\cap G_2$ with $\sigma >\frac{2d}{3}$ and $G_2$ stands for a Gaussian class of potentials. We show that T is bounded on $L_w^p\left(R^d\right)$ for all p in a suitable range depending on σ. If more conditions are imposed on either σ or V, the range for p can be extended to $(1,\infty )$.