Schrödinger Harmonic Functions with Morrey Traces on Dirichlet Metric Measure Spaces

Tianjun Shen and Bo Li
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Tianjun Shen: Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
Bo Li: College of Data Science, Jiaxing University, Jiaxing 314001, China

Mathematics, 2022, vol. 10, issue 7, 1-22

Abstract: 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 .

Keywords: Schrödinger equation; Morrey space; Dirichlet problem; metric measure space (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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