Nihat Gogus (PhD '06) and Muhammed Alan
Abstract: In this paper we give two complete characterizations of the Poletky-Stessin-Hardy spaces in the complex plane: First in terms of their boundary values as a weighted subclass of the usual L^p class with respect to the arclength measure on the boundary. Second we completely describe functions in these spaces by having a harmonic majorant with a certain growth condition and we prove some basic results about these spaces. In particular, we prove approximation results in such spaces and extend the classical result of Beurling which describes the invariant subspaces of the shift operator. We introduce new spaces that are extensions of the Hardy spaces and prove a removable singularity result for holomorphic functions within these new spaces. Additionally we provide non-trivial examples.
ArXiv: 1209.2609