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 Schrodinger 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 Holder 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 Schrodinger 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.