Projective limits of Poletsky--Stessin Hardy spaces

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 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 $H^p_u(D)$ is $H^\infty(D)$.

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