A note for Riesz transforms associated with Schrödinger operators on the Heisenberg Group

Abstract

Let ${\mathbb {H}^n}$ be the Heisenberg group and $Q=2n+2$ be its homogeneous dimension. The Schrödinger operator is denoted by $- {\Delta _{{\mathbb {H}^n}}} + V$, where ${\Delta _{{\mathbb {H}^n}}}$ is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class ${B_{{q_1}}}$ for ${q_1} \ge \frac{Q}{2}$. Let $H^p_L(\mathbb {H}^n)$ be the Hardy space associated with the Schrödinger operator for $\frac{Q}{Q+\delta _0}<p\le 1$, where $\delta _0=\min \{1,2-\frac{Q}{q_1}\}$. In this note we show that the operators ${T_1} = V{( - {\Delta _{{\mathbb {H}^n}}} + V)^{ - 1}}$ and ${T_2} = {V^{1/2}}{( - {\Delta _{{\mathbb {H}^n}}} + V)^{ - 1/2}}$ are bounded from $H_L^p({\mathbb {H}^n})$ into ${L^p}({\mathbb {H}^n})$. Our results are also valid on the stratified Lie group.