Schrödinger Harmonic Functions with Morrey Traces on Dirichlet Metric Measure Spaces

• Main Authors: Li, B. (Author), Shen, T. (Author)
• Format: Article
• Language: English
• Published: MDPI 2022
• Subjects: Dirichlet problem; metric measure space; Morrey space; Schrödinger equation
• Online Access: View Fulltext in Publisher

 Assume that (X, d, µ) is a metric measure space that satisfies a Q-doubling condition with Q > 1 and supports an L2-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 RHq (X) for some q ≥ (Q + 1)/2. In this paper, we consider the Dirichlet problem for the Schrödinger equation −∂2tu+Lu+Vu= 0 on the upper half-spaceX ×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. © 2022 by the authors. Licensee MDPI, Basel, Switzerland.