Characterization of Temperatures Associated to Schrödinger Operators with Initial Data in Morrey Spaces

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 L-p,L-lambda(R-n) 0 <= lambda < n denote the Morrey space on R-n. In this paper, we will show that a function f is an element of L-2,L-lambda(R-n) is the trace of the solution of Lu := u(t) + L-u = 0, u(x, 0) = f(x), where u satisfies a Carleson-type condition sup(xB, rB) r(B)(-lambda )integral(r2B)(0) integral(B(xB, rB) )vertical bar del u(x,t)vertical bar(2 )dxdt <= C < infinity. Conversely, this Carleson-type condition characterizes all the L-carolic functions whose traces belong to the Morrey space L-2,L-lambda(R-n) for all 0 <= lambda < n. This result extends the analogous characterization found by Fabes and Neri in [8] for the classical BMO space of John and Nirenberg.