  In this paper, we pursue the study of harmonic functions on the real
hyperbolic ball started by the second named author. Our focus here is on the
theory of Hardy, Hardy-Sobolev and Lipschitz spaces of these functions. We
prove here that these spaces admit Fefferman-Stein like characterizations in
terms of maximal and square functionals. We further prove that the hyperbolic
harmonic extension of Lipschitz functions on the boundary extend into Lipschitz
functions on the whole ball.
