2005/04/12 by Tamás Forgács, Tamas Forgacs, Dror Varolin +2
Mathematics · #32A36 #Advanced Harmonic Analysis Research #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #math.CV #msc:32A36
paper · pdf · doi:10.48550/arxiv.math/0504252
Final Version. To appear in International Journal of Mathematics
openalex publication_date 2005/04/12 · arxiv created 2006/07/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Cn by Ortega-Cerda, Schuster and the second author. For the case of sampling, our proofs are different than those in the Cn 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 Cn case, except that we use an improved dbar theorem, due to Ohsawa, to solve our Cousin problem with L2 bounds. As in the case of Cn, it is expected that our sufficient conditions are also necessary.