Dominating Sets for Analytic and Harmonic Functions and Completeness of Weighted Bergman Spaces

Abstract

A set E c Q is holomorphically dominating for Q if sUP,cE If(z)I = SUpzcn If(z)l for all holomorphic functions f on Q. As follows from a result of Stray, this property is equivalent to the inaccessibility of the Aleksandrov compactification point * (of Q) from Q x E. Moreover, it is equivalent to a large number of other statements (old and new) of holomorphic, harmonic and topological nature, including that a certain weighted Bergman space with p = oo is a Banach space. We extend this to the cases of harmonic functions in R' and holomorphic functions in cV. We also present some results on when weighted Bergman spaces are (quasi)-Banach spaces, the case p = oo being characterised by the result mentioned above.