BMOL(Hn) spaces and Carleson measures for Schrödinger operators

Let L=−ΔHn+V be a Schrödinger operator on the Heisenberg group Hn, where ΔHn is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class . Here Q is the homogeneous dimension of Hn. In this article we investigate the dual space of the Hardy-type space associated with the Schrödinger operator L, which is a kind of BMO-type space BMOL(Hn) defined by means of a revised sharp function related to the potential V. We give the Fefferman–Stein type decomposition of BMOL-functions with respect to the (adjoint) Riesz transforms for L, and characterize BMOL(Hn) in terms of the Carleson measure. We also establish the BMOL-boundedness of some operators, such as the (adjoint) Riesz transforms , the Littlewood–Paley function , the Lusin area integral , the Hardy–Littlewood maximal function, and the semigroup maximal function. All results hold for stratified groups as well.