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

We study Poletsky-Stessin Hardy spaces that are generated by continuous, subharmonic exhaustion functions on a domain $\mathrm{\Omega }\subset \mathbb{C}$, 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 $A(\mathrm{\Omega })$ 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.