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Assume that (X,d,μ) is a metric measure space that satisfies a Q-doubling condition with Q&gt;1 and supports an L2-Poincaré inequality. Let  be a nonnegative operator generalized by a Dirichlet form E and V be a Muckenhoupt weight belonging to a reverse Hölder class RHq(X) for some q≥(Q+1)/2. In this paper, we consider the Dirichlet problem for the Schrödinger equation −∂t2u+u+Vu=0 on the upper half-space X×R+, which has f as its the boundary value on X. We show that a solution u of the Schrödinger 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 RQ to the metric measure space X and improves the reverse Hölder index from q≥Q to q≥(Q+1)/2.

Schrödinger Harmonic Functions with Morrey Traces on Dirichlet Metric Measure Spaces