Schrodinger Harmonic Functions with Morrey Traces on Dirichlet Metric Measure Spaces

Assume that (X,d,mu) is a metric measure space that satisfies a Q-doubling condition with Q > 1 and supports an L-2-Poincare inequality. Let L be a nonnegative operator generalized by a Dirichlet form E and V be a Muckenhoupt weight belonging to a reverse Holder class RHq(X) for some q >= (Q + 1)/2. In this paper, we consider the Dirichlet problem for the Schrodinger equation -partial derivative(2)(t)u + Lu + Vu = 0 on the upper half-space X x R+, which has f as its the boundary value on X. We show that a solution u of the Schrodinger equation satisfies the Carleson type condition if and only if there exists a square Morrey function f such that u can be expressed by the Poisson integral of f. This extends the results of Song-Tian-Yan [Acta Math. Sin. (Engl. Ser.) 34 (2018), 787-800] from the Euclidean space R-Q to the metric measure space X and improves the reverse Holder index from q >= Q to q >= (Q + 1)/2.