Boundary Values Properties of Functions in Weighted Hardy Spaces

TL;DR

This paper investigates the boundary value properties of harmonic and holomorphic functions within weighted Hardy spaces on the unit disk, extending classical Hardy space theory to these generalized spaces.

Contribution

It demonstrates that the boundary value theory for weighted Hardy spaces on the unit disk parallels the classical Hardy space boundary theory.

Findings

Boundary values exist similarly to classical Hardy spaces

Weighted Hardy spaces generalize classical Hardy spaces

Boundary behavior is analogous to classical results

Abstract

In this paper we study the boundary values of harmonic and holo- morphic functions in the weighted Hardy spaces on the unit disk $\mathbb{D}$. These spaces were introduced by Poletsky and Stessin in [6] for plurisubharmonic functions on hyperconvex domains $D \subset \mathbb{C}^n$ as generalizations of classical Hardy spaces. We show that in the case when $D$ is the unit disk $\mathbb{D}$ the theory of boundary values for functions in these spaces is analogous to the classical one.