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

Let H-n be the Heisenberg group and Q = 2n + 2 be its homogeneous dimension. The Schrodinger operator is denoted by - Delta(n)(H) + V, where Delta(n)(H) is the sub- Laplacian and the nonnegative potential V belongs to the reverse Holder class Bq(1) for q(1) >= Q/2 . Let H-L(P)(H-n) L (Hn) be the Hardy space associated with the Schrodinger operator for Q Q/Q+delta(0) < p <= 1, where delta(0) = min{1, 2 - Q/q(1)}. q1}. In this note we show that the operators T1 = V(- Delta(n)(H) + V)(-1) and T-2 = V-1/2(- Delta(n)(H) + 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.