Interpolation and Sampling Hypersurfaces for weighted Bergman spaces on the unit ball

Abstract

We present sufficient conditions on a smooth uniformly flat hypersurface W in the unit ball to be an interpolation hypersurface or a sampling hypersurface for generalized Bergman spaces associated to the unit ball with its Bergman metric. The conditions are phrased in terms of a geometric Beurling-type densities, similar to the densities defined for hypersurfaces in C^n by Ortega-Cerda, Schuster and the second author. For the case of sampling, our proofs are different than those in the C^n case; we use an approach closer in spirit to the work of Berndtsson and Ortega-Cerda in the 1-dimensional case. For the case of interpolation we use a modified version of the Ohsawa-Takegoshi method to carry out the interpolation in the situation where the density of W is a little less than optimal. For the general case, we use the same approach as in the C^n case, except that we use an improved dbar theorem, due to Ohsawa, to solve our Cousin problem with L^2 bounds. As in the case of C^n, it is expected that our sufficient conditions are also necessary.