Bulletin of the
Korean Mathematical Society BKMS

We consider the Schr\"odinger type operator \(\mathcal{L}=(-\Delta_{\mathbb{H}^n})^2+V^2 \) on the Heisenberg group $\mathbb{H}^n,$ where $\Delta_{\mathbb{H}^n}$ is the sub-Laplacian and the non-negative potential \(V\) belongs to the reverse H\"older class \(RH_s$ for $ s\geq Q/2$ and $Q\geq 6.\) We shall establish the $(L^p,L^q)$ estimates for the Riesz transforms $ T_{\alpha,\beta,j} =V^{2\alpha}\nabla_{\mathbb{H}^n}^j \mathcal{L}^{-\beta},~j=0,1,2,3,$ where $\nabla_{\mathbb{H}^n}$ is the gradient operator on $\mathbb{H}^n,~ 0<\alpha\leq 1-j/4,~ j/4<\beta\leq1,$ and $\beta-\alpha\geq j/4.$

Keywords: Schr\"odinger type operator, reverse H\"older class, Riesz transform, Heisenberg group

MSC numbers: Primary 42B25, 22E30, 35J10