Poletsky-Stessin-Hardy spaces in the plane

TL;DR

This paper characterizes Poletsky-Stessin-Hardy spaces in the complex plane through boundary value descriptions and harmonic majorants, extending classical results and providing new examples.

Contribution

It offers two complete characterizations of these spaces and extends Beurling's invariant subspace theorem to this context.

Findings

Boundary value characterization as weighted L^p spaces

Description of functions via harmonic majorants with growth conditions

Extension of Beurling's theorem to Poletsky-Stessin-Hardy spaces

Abstract

In this paper we give two complete characterizations of the Poletsky- 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. Additionally we provide non-trivial examples.