On characterization of Poisson integrals of Schrödinger operators with BMO traces

Let L be a Schrödinger operator of the form L=-δ + V acting on L²(Rⁿ) where the nonnegative potential V belongs to the reverse Hölder class B_q for some q≥n. Let BMO_L(Rⁿ) denote the BMO space on Rn associated to the Schrödinger operator L. In this article we will show that a function f∈BMOL(Rn) is the trace of the solution of Lu=-u_tt+Lu=0, u(x, 0)=f(x), where u satisfies a Carleson condition. Conversely, this Carleson condition characterizes all the L-harmonic functions whose traces belong to the space BMO_L(Rⁿ). This result extends the analogous characterization founded by Fabes, Johnson and Neri in [13] for the classical BMO space of John and Nirenberg.