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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm213-2-4,
author = {Alexander V. Abanin and Pham Trong Tien},
title = {Painlev\'e null sets, dimension and compact embedding of weighted holomorphic spaces},
journal = {Studia Mathematica},
volume = {209},
year = {2012},
pages = {169-187},
zbl = {1275.30032},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm213-2-4}
}
Alexander V. Abanin; Pham Trong Tien. Painlevé null sets, dimension and compact embedding of weighted holomorphic spaces. Studia Mathematica, Tome 209 (2012) pp. 169-187. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm213-2-4/