Commutators associated with Schrödinger operators on the nilpotent Lie group

Assume that G is a nilpotent Lie group. Denote by [Formula: see text] the Schrödinger operator on G, where Δ is the sub-Laplacian, the nonnegative potential W belongs to the reverse Hölder class [Formula: see text] for some [Formula: see text] and D is the dimension at infinity of G. Let [Formula: see text] be the Riesz transform associated with L. In this paper we obtain some estimates for the commutator [Formula: see text] for [Formula: see text] , where [Formula: see text] is a function space which is larger than the classical Lipschitz space.