Characterizations of Dirichlet-type Spaces

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.