Boundedness and compactness of commutators related with Schrödinger operators on Heisenberg groups

Let L=-Δ _ℍ^n+V be a Schrödinger operator on Heisenberg groups ℍ^n , where Δ _ℍ^n denotes the sub-Laplacian on ℍ^n and the nonnegative potential V belongs to the reverse Hölder class B_q , q> (2n+2)/2 . We establish the L^p(ℍ^n) estimates for operators T including Calderón-Zygmund operators of Schrödinger type, maximal operators for Riesz transforms, the Littlewood-Paley function, the area function and their commutators [ b , T ], where b belongs to a new class of BMO type spaces related with L . Furthermore, we investigate the corresponding compactness of commutators [ b , T ] on L^p(ℍ^n) .