A~Moebius invariant space of \(H\)-harmonic functions on the ball

We~describe a Dirichlet-type space of \(H\)-harmonic functions, i.e. functions annihilated by the hyperbolic Laplacian on~the unit ball of the real \(n\)-space, as~the analytic continuation (in~the spirit of Rossi and Vergne) of the corresponding weighted Bergman spaces. Characterizations in terms o...

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Bibliographic Details
Published in:arXiv.org 2024-04
Main Authors: Blaschke, Petr, Engliš, Miroslav, El-Hassan, Youssfi
Format: Article
Language:English
Subjects:
Dirichlet problem
Harmonic functions
Hyperbolic functions
Invariants
Mathematical analysis
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Summary:We~describe a Dirichlet-type space of \(H\)-harmonic functions, i.e. functions annihilated by the hyperbolic Laplacian on~the unit ball of the real \(n\)-space, as~the analytic continuation (in~the spirit of Rossi and Vergne) of the corresponding weighted Bergman spaces. Characterizations in terms of derivatives are given, and the associated semi-inner product is shown to be Moebius invariant. We~also give a formula for the corresponding reproducing kernel. Our~results solve an open problem addressed by M.~Stoll in his book ``Harmonic and subharmonic function theory on the hyperbolic ball'' (Cambridge University Press, 2016).
ISSN:2331-8422