Zero sets of analytic function spaces on the unit disk

Abstract

We survey some known results on the zero sets of two families of analytic function spaces and another single space de ned on the unit disk in the complex plane. We investigate mostly the basic properties of the zero sets of these spaces that are comparable to those of the Hardy spaces and to each other. The spaces we consider are standard weighted Bergman spaces, the Dirichlet space, and certain Dirichlet-type spaces that are very close to both Hardy spaces and the Bergman spaces. The completely known zero sets of Hardy spaces are easy to describe, characterized by the Blaschke condition and the same for all the spaces in the family. The zero sets of the other spaces considered have started to be investigated relatively recently and are far from a complete description. Yet it is possible to nd conditions similar to the Blaschke condition for the zero sets of Bergman spaces and Dirichlet-type spaces. For the zero sets of the true Dirichlet space, the known results are sporadic and do not form a general theory yet.