We study weighted Bergman and Dirichlet spaces of hyperbolic harmonic functions and, more generally, eigenfunctions of the hyperbolic Laplacian on the unit ball of \(\mathbb {R}^n\) . For radial weights belonging to the classes \(\widehat{\mathcal {D}}\) and , we establish norm inequalities relating the hyperbolic gradient to the function itself. These results significantly extend Stoll’s foundational work on the standard weights to a much broader class of radial weights. In particular, we obtain hyperbolic analogues of the classical Littlewood–Paley inequalities. Moreover, we resolve an open question of Stoll on the hyperbolic subharmonicity of powers of the hyperbolic gradient.
