Function Spaces Related to the Dirichlet Space

TL;DR

This paper explores function spaces connected to the Dirichlet space, drawing analogies with Hardy space theory, and investigates their properties and roles in complex analysis.

Contribution

It introduces and analyzes new function spaces related to the Dirichlet space, highlighting their similarities to $H^{1}$ and $BMO$ in Hardy space theory.

Findings

Identification of spaces analogous to $H^{1}$ and $BMO$ within the Dirichlet space framework

Establishment of properties and roles of these spaces in holomorphic function theory

Insights into the structure and relationships of Dirichlet-related function spaces

Abstract

We present results about spaces of holomorphic functions associated to the classical Dirichlet space. The spaces we consider have roles similar to the roles of $H^{1}$ and $BMO$ in the Hardy space theory and we emphasize those analogies.