Decomposition of Analytic Hyperbolically Harmonic Functions

Abstract

There are two standard models for hyperbolic spaces, the first based on the Poincaré metric $d{s^2} = {{{{\left| {dx} \right|}^2}} \over {{{({x_n})}^2}}}$, which is invariant under the group of sense preserving conformai mappings of the hyperplane x _n = 0, therefore called the planar model, and the hyperbolic space provided with the metric $d{s^2} = {{{{\left| {dx} \right|}^2}} \over {{{(1 - {{\left| x \right|}^2})}^2}}}$, invariant under the group of sense preserving conformal mappings of the unit sphere S ⁿ⁻¹, called the spherical model. We study here the real valued analytic solutions of the Laplace-Beltrami operator for the spherical model of hyperbolic spaces, presenting some results regarding the decomposition into solid harmonics of such functions.