Sufficient conditions for interpolation and sampling hypersurfaces in the Bergman ball

We give sufficient conditions for a closed smooth hypersurface W in the ra-dimensional Bergman ball to be interpolating or sampling. As in the recent work [5] of Ortega-Cerdà, Schuster and the second author on the Bargmann-Fock space, our sufficient conditions are expressed in terms of a geometric density of the hypersurface that, though less natural, is shown to be equivalent to Bergman ball analogs of the Beurling-type densities used in [S]. In the interpolation theorem we interpolate L² data from W to the ball using the method of Ohsawa-Takegoshi, extended to the present setting, rather than the Cousin I approach used in [5], In the sampling theorem, our proof is completely different from [5]. We adapt the more natural method of Berndtsson and Ortega-Cerdà [1] to higher dimensions. This adaptation motivated the notion of density that we introduced. The approaches of [5] and the present paper both work in either the case of the Bergman ball or of the Bargmann-Fock space.