On characterization of Poisson integrals of Schrödinger operators with BMO traces

Abstract

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 Bq for some q⩾n. Let BMOL(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=−utt+Lu=0, 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 L-harmonic functions whose traces belong to the space 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.