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Abstract

Poletsky–Stessin Hardy (PS–Hardy) spaces are the natural generalizations of classical Hardy spaces of the unit disc to general bounded, hyperconvex domains. On a bounded hyperconvex domain $\Omega$, the PS–Hardy space $H^{p}_{u}(\Omega)$ is generated by a continuous, negative, plurisubharmonic exhaustion function $u$ of the domain. Poletsky and Stessin considered the general properties of these spaces and mainly concentrated on the spaces $H^{p}_{u}(\Omega)$ where the Monge–Ampère measure $(dd^{c}u)^{n}$ has compact support for the associated exhaustion function $u$. In this study we consider PS–Hardy spaces in two different settings. In one variable case we examine PS–Hardy spaces that are generated by exhaustion functions with finite Monge–Ampère mass but $(dd^{c}u)^{n}$ does not necessarily have compact support. For $n \gt 1$, we focus on PS–Hardy spaces of complex ellipsoids which are generated by specific exhaustion functions. In both cases we will give results regarding the boundary value characterization and polynomial approximation.