Projective limits of Poletsky--Stessin Hardy spaces

TL;DR

This paper demonstrates that the projective limit of Poletsky--Stessin Hardy spaces on a strongly pseudoconvex domain is isomorphic to the space of bounded holomorphic functions, revealing a deep connection between these function spaces.

Contribution

It establishes the isomorphism between the projective limit of Poletsky--Stessin Hardy spaces and $H^(D)$, providing new insights into their structure and properties.

Findings

The projective limit of $H^p_u(D)$ spaces is isomorphic to $H^(D)$.

CarathE9odory balls are contained in approach regions.

The intersection of all Poletsky--Stessin Hardy spaces equals $H^(D)$.

Abstract

In this paper we show that that on a strongly pseudoconvex domain $D$ the projective limit of all Poletsky--Stessin Hardy spaces $H^p_u(D)$, introduced in \cite{PS}, is isomorphic to the space $H^\infty(D)$ of bounded holomorphic functions on $D$ endowed with a special topology. To prove this we show that Carath\'eodory balls lie in approach regions, establish a sharp inequality for the Monge--Amp\'ere mass of the envelope of plurisubharmonic exhaustion functions and use these facts to demonstrate that the intersection of all Poletsky--Stessin Hardy spaces $H^p_u(D)$ is $H^\infty(D)$.