BMOL(H{double-struck}ⁿ) spaces and Carleson measures for Schrödinger operators

Let L=-Δ_{H{double-struck}n}+V be a Schrödinger operator on the Heisenberg group H{double-struck}ⁿ, where ΔH{double-struck}ⁿ is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class BQ2. Here Q is the homogeneous dimension of H{double-struck}ⁿ. In this article we investigate the dual space of the Hardy-type space HL1(H{double-struck}ⁿ) associated with the Schrödinger operator L, which is a kind of BMO-type space BMOL(H{double-struck}ⁿ) 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 R̃jL for L, and characterize BMOL(H{double-struck}ⁿ) in terms of the Carleson measure. We also establish the BMOL-boundedness of some operators, such as the (adjoint) Riesz transforms R̃jL, the Littlewood-Paley function sQL, the Lusin area integral SQL, the Hardy-Littlewood maximal function, and the semigroup maximal function. All results hold for stratified groups as well.