On Characterization of Poisson Integrals of Schrödinger Operators with Morrey Traces

Let L be a Schrödinger operator of the form L = .. + V acting on L2(.n) where the nonnegative potential V belongs to the reverse Hölder class B q for some q . n. In this article we will show that a function f . L2,.(.n), 0 < . < n, is the trace of the solution of Lu = .u tt + L u = 0, u(x, 0) = f(x), where u satisfies a Carleson type condition supxB,rBr..B.rB0.B(xB,rB)t|.u(x,t)|2dxdt.C<.. Its proof heavily relies on investigate the intrinsic relationship between the classical Morrey spaces and the new Campanato spaces L 2,.L(.n) associated to the operator L, i.e. L2,.L(Rn)=L2,.(Rn). Conversely, this Carleson type condition characterizes all the L-harmonic functions whose traces belong to the space L2,.(.n) for all 0 < . < n.