Analytic continuation of weighted $H$-harmonic Bergman spaces

We provide a partial answer to Problems 1 and 2 raised in the recent article by Blaschke et al., concerning the analytic continuation of weighted $H$-harmonic Bergman spaces. These are spaces of functions annihilated by the Möbius-invariant Laplacian on the unit ball. More precisely, we identify some of the discrete Wallach sets and show, among others, that structure depends on the parity of the dimension.