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(R-n) where the nonnegative potential. V belongs to the reverse Holder class B-q for some q >= n. Let BMOL (R-n) denote the BMO space on R-n associated to the Schrodinger operator L. In this article we will show that a function f is an element of BMOL (R-n) is the trace of the solution of L-u = -u(tt) + Lu = 0, u(x, 0) = f (x), where u satisfies a Carleson condition sup(xB,rB) r(B)(-n) integral(rB)(0) integral(B(xB,rB)) t vertical bar del u(x,t)vertical bar(2)dxdt <= C < infinity. Conversely, this Carleson condition characterizes all the L-harmonic functions whose traces belong to the space BMOL (R-n). This result extends the analogous characterization founded by Fabes, Johnson and Neri in [13] for the classical BMO space of John and Nirenberg. (C) 2013 Elsevier Inc. All rights reserved.