Monge–Ampère measures and Poletsky–Stessin Hardy spaces on bounded hyperconvex domains

Abstract

In this thesis study, we consider Poletsky-Stessin Hardy (PS-Hardy) spaces that are generated by continuous, plurisubharmonic exhaustion functions on hyperconvex domains. In the first part of this study we examine these spaces on domains in the complex plane that are bounded by an analytic Jordan curve. In this setting we focus on PS-Hardy spaces generated by exhaustion functions that have finite Monge-Ampre mass but are not necessarily maximal outside of a compact set. This choice gives us new Banach spaces strictly contained in classical Hardy spaces. We characterize PS-Hardy spaces through their boundary values and we show factorization results analogous to unit disc case. Using functional analysis techniques we prove that the algebra of holomorphic functions which are continuous on the boundary are dense in PS-Hardy spaces. Moreover, we consider the composition operators with holomorphic symbols acting on PS-Hardy spaces and show that contrary to classical case, not all composition operators are bounded on PS-Hardy spaces. In the second part, we study PS-Hardy spaces on polydisc, complex ellipsoid and on strongly convex domains. On complex ellipsoid case, we prove the existence of radial boundary values and then by applying a classical method given by Stein we show the existence of boundary values along admissible approach regions. As an application of this method , we also obtain that polynomials are dense in PS-Hardy spaces on complex ellipsoids. Lastly, we examine the boundedness of composition operators on PS-Hardy spaces on hyperconvex domains in several variables.