Commutators of Riesz transforms associated with higher order Schrödinger type operators

Let m be a nonnegative integer, and let \(n\ge 2^{m+1}+1.\) $n\ge 2^{m+1}+1.$ In this paper, we consider the higher order Schrödinger type operator \({\mathcal {H}}_{2^m}=(-\Delta )^{2^m}+V^{2^m} \) ${\mathcal{H}}_{2^m}={(-\mathrm{\Delta })}^{2^m}+V^{2^m}$ on \({\mathbb {R}}^n,\) ${\mathbb{R}}^n,$ and establish the \(L^p({\mathbb {R}}^n)\) $L^p({\mathbb{R}}^n)$ boundedness of Riesz transforms \(\nabla ^j {\mathcal {H}}_{2^m}^{-\frac{j}{2^{m+1}}} (j=1,2,\cdot \cdot \cdot ,2^{m+1}-1)\) ${\mathrm{\nabla }}^j{\mathcal{H}}_{2^m}^{-\frac{j}{2^{m+1}}}(j=1,2,···,2^{m+1}-1)$ and their commutators. Here, V is a nonnegative potential belonging to both the reverse Hölder class \(RH_s\) $R{H}_s$ for \(s \ge \frac{n}{2}\) $s\ge \frac{n}{2}$ , and the Gaussian class associated with \((-\Delta )^{2^m}\) ${(-\mathrm{\Delta })}^{2^m}$ .