Hardy Type Estimates for Riesz Transforms Associated with Schrdinger Operators on the Heisenberg Group

Let Hnbe the Heisenberg group and Q=2n +2 be its homogeneous dimension. In this paper, we consider the Schrdinger operator-?Hn+V, where ?Hn is the sub-Laplacian and V is the nonnegative potential belonging to the reverse Ho ¨lder class Bq1 for q1≥ Q/2. We show that the operators T1= V(-?Hn +V)-1 and T2=V1/2(-?Hn +V)-1/2 are both bounded from H1L(Hn) into L1(Hn). Our results are also valid on the stratified Lie group.