On characterization of Poisson integrals of Schrödinger operators with BMO traces

Let LL be a Schrödinger operator of the form L=−Δ+VL=−Δ+V acting on L2(Rn)L2(Rn) where the nonnegative potential V   belongs to the reverse Hölder class BqBq for some q⩾nq⩾n. Let BMOL(Rn)BMOL(Rn) denote the BMO space on RnRn associated to the Schrödinger operator LL. In this article we will show that a function f∈BMOL(Rn)f∈BMOL(Rn) is the trace of the solution of Lu=−utt+Lu=0Lu=−utt+Lu=0, u(x,0)=f(x)u(x,0)=f(x), where u satisfies a Carleson conditionsupxB,rBrB−n∫0rB∫B(xB,rB)t|∇u(x,t)|2dxdt⩽C<∞. Conversely, this Carleson condition characterizes all the LL-harmonic functions whose traces belong to the space BMOL(Rn)BMOL(Rn). This result extends the analogous characterization founded by Fabes, Johnson and Neri in [13] for the classical BMO space of John and Nirenberg.