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

Abstract

In this article we study harmonic functions for the Laplace-Beltrami operator on the real hyperbolic space Bn. 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 Bn. We then study Hardy spaces H p (Bn), 0 < p < 1, whose elements appear as the hyperbolic harmonic extensions of distributions belonging to the Hardy spaces of the sphere H p (S n 1 ). In particular, we obtain an atomic decomposition of this spaces.