Schrödinger Harmonic Functions with Morrey Traces on Dirichlet Metric Measure Spaces

Assume that ( X , d , μ ) is a metric measure space that satisfies a Q -doubling condition with Q > 1 and supports an L 2 -Poincaré inequality. Let 𝓛 be a nonnegative operator generalized by a Dirichlet form E and V be a Muckenhoupt weight belonging to a reverse Hölder class R H q ( X ) for some q ≥ ( Q + 1 ) / 2 . In this paper, we consider the Dirichlet problem for the Schrödinger equation − ∂ t 2 u + 𝓛 u + V u = 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 R Q to the metric measure space X and improves the reverse Hölder index from q ≥ Q to q ≥ ( Q + 1 ) / 2 .