Harmonic functions on the real hyperbolic ball I : Boundary values and atomic decomposition of Hardy spaces

Abstract

We study harmonic functions for the Laplace-Beltrami operator on the real hyperbolic ball. We obtain necessary and sufficient conditions for this functions and their normal derivatives to have a boundary distribution.In doing so, we put forward different behaviors of hyperbolic harmonic functions according to the parity of the dimension of the hyperbolic ball. We then study Hardy spaces of hyperbolic harmonic extensions of distributions belonging to the Hardy spaces of the sphere. In particular, we obtain an atomic decomposition of these spaces.