A Moebius invariant space of H-harmonic functions on the ball

Abstract

It has been an open problem — at least since M. Stoll's book “Harmonic and subharmonic function theory on the hyperbolic ball” (Cambridge University Press, 2016) — whether there exists a Moebius invariant Hilbert space of hyperbolic-harmonic functions on the unit ball of the real n-space, i.e. of functions annihilated by the hyperbolic Laplacian on the ball. We give an answer by describing a Dirichlet-type space of hyperbolic-harmonic functions, 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.