On characterization of Poisson integrals of Schrodinger operators with BMO traces

Let L be a Schrodinger operator of the form L=-\Delta+V acting on L^2(Rn) where the nonnegative potential V belongs to the reverse Holder class Bq for some q>= n. Let BMO_L(Rn) denote the BMO space on Rn associated to the Schrodinger operator L. In this article we will show that a function f in BMO_L(Rn) is the trace of the solution of L'u=-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 for the classical BMO space of John and Nirenberg.