Poletsky-Stessin Hardy Spaces on Domains Bounded by An Analytic Jordan Curve in $\mathbb{C}$

TL;DR

This paper explores Poletsky-Stessin Hardy spaces on domains bounded by an analytic Jordan curve, highlighting their factorization properties and the boundedness of composition operators, which differ from classical Hardy spaces.

Contribution

It extends the theory of Poletsky-Stessin Hardy spaces by considering non-harmonic exhaustion functions with finite Monge-Ampère mass and analyzes their algebraic and operator-theoretic properties.

Findings

Functions have a factorization similar to classical Hardy spaces.

The algebra A(Ω) is dense in these spaces.

Composition operators may not always be bounded on these spaces.

Abstract

We study Poletsky-Stessin Hardy spaces that are generated by continuous, subharmonic exhaustion functions on a domain $\Omega\subset\mathbb{C}$, that is bounded by an analytic Jordan curve. Different from Poletsky & Stessin's work these exhaustion functions are not necessarily harmonic outside of a compact set but have finite Monge-Amp\'ere mass. We have showed that functions belonging to Poletsky-Stessin Hardy spaces have a factorization analogous to classical Hardy spaces and the algebra $A(\Omega)$ is dense in these spaces as in the classical case ; however, contrary to the classical Hardy spaces, composition operators with analytic symbols on these Poletsky-Stessin Hardy spaces need not always be bounded.