Projective limits of Poletsky--Stessin Hardy spaces

Abstract

In this paper we show that that on a strongly pseudoconvex domain $D$ the projective limit of all Poletsky--Stessin Hardy spaces $H_u^p(D)$, 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 Carathéodory balls lie in approach regions, establish a sharp inequality for the Monge--Ampére mass of the envelope of plurisubharmonic exhaustion functions and use these facts to demonstrate that the intersection of all Poletsky--Stessin Hardy spaces $H_u^p(D)$ is $H^{\infty }(D)$.