Characterizations of Dirichlet-type Spaces

TL;DR

This paper provides three new characterizations of Dirichlet-type spaces, including integral, mean oscillation, and higher order derivative descriptions, along with a decomposition theorem, enhancing understanding of these function spaces.

Contribution

It introduces novel characterizations of Dirichlet-type spaces without derivatives, and establishes a decomposition theorem, advancing the theoretical framework of these spaces.

Findings

Characterization via double integral

Mean oscillation in Bergman metric characterization

Decomposition theorem for $D(mu)$

Abstract

We give three characterizations of the Dirichlet-type spaces $D(\mu)$. First we characterize $D(\mu)$ in terms of a double integral and in terms of the mean oscillation in the Bergman metric, none of them involve the use of derivatives. Next, we obtain another characterization for $D(\mu)$ in terms of higher order derivatives. Also, a decomposition theorem for $D(\mu)$ is established.