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

Let H n be the Heisenberg group and Q = 2 n + 2 be its homogeneous dimension. The Schrödinger operator is denoted by - Δ H n + V , where Δ H n is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class B q 1 for q 1 ≥ Q 2 . Let H L p ( H n ) be the Hardy space associated with the Schrödinger operator for Q Q + δ 0 < p ≤ 1 , where δ 0 = min { 1 , 2 - Q q 1 } . In this note we show that the operators T 1 = V ( - Δ H n + V ) - 1 and T 2 = V 1 / 2 ( - Δ H n + V ) - 1 / 2 are bounded from H L p ( H n ) into L p ( H n ) . Our results are also valid on the stratified Lie group.