Painlevé null sets, dimension and compact embedding of weighted holomorphic spaces
Volume 213 / 2012
Studia Mathematica 213 (2012), 169-187
MSC: Primary 32K05; Secondary 32C18, 32C22, 46B20.
DOI: 10.4064/sm213-2-4
Abstract
We obtain, in terms of associated weights, natural criteria for compact embedding of weighted Banach spaces of holomorphic functions on a wide class of domains in the complex plane. Our study is based on a complete characterization of finite-dimensional weighted spaces and canonical weights for them. In particular, we show that for a domain whose complement is not a Painlevé null set each nontrivial space of holomorphic functions with $O$-growth condition is infinite-dimensional.
