Weighted composition operators acting on Dirichlet type spaces

Abstract

In this paper we study the weighted composition operator $\psi C_\phi$ on the Dirichlet type spaces $\mathit{D}^p_\alpha(\mathbb{D})$. We show tht if theweighted composion operator $\psi C_\phi$ is bounded on such spaces (for 1\leq p<2$ then the related measure $\mu_{p,\alpha,\psi}$ is a Carleson measure. Also we show that if the weighted composition operator $\psi C_\phi$ is an isomety on $\mathit{D}^p_\alpha(\mathbb{D})$ then $\psi.\phi$ is a rotation map on $\mathbb{D}$.