Projective Limits of Poletsky–Stessin Hardy Spaces

Abstract

In this paper we show that on a strongly pseudoconvex domain D the projective limit of all Poletsky–Stessin Hardy spaces \(H^p_u(D)\), introduced by Poletsky and Stessin in 2008, 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 Caratheodory balls lie in approach regions, establish a sharp inequality for the Monge–Ampere 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 \(H^\infty (D)\).