Projective Limits of Poletsky–Stessin Hardy Spaces

In this paper we show that on a strongly pseudoconvex domain D the projective limit of all Poletsky–Stessin Hardy spaces Hup(D), introduced by Poletsky and Stessin in 2008, is isomorphic to the space H^∞(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 Hup(D) is H^∞(D).