Composition operators on Dirichlet-type spaces

The Dirichlet-type space $D^{p} (1 \leq p \leq 2$) is the Banach space of functions analytic in the unit disc with derivatives belonging to the Bergman space $A^{p}$. Let $\Phi$ be an analytic self-map of the disc and define $C_{\Phi}(f) = f \circ \Phi$ for $f \in D^{p}$. The operator $C_{\Phi}: D^{p} \rightarrow D^{p}$ is bounded (respectively, compact) if and only if a related measure $\mu_{p}$ is Carleson (respectively, compact Carleson). If $C_{\Phi}$ is bounded (or compact) on $D^{p}$, then the same behavior holds on $D^{q} (1 \leq q < p$) and on the weighted Dirichlet space $D_{2-p}$. Compactness on $D^{p}$ implies that $C_{\Phi}$ is compact on the Hardy spaces and the angular derivative exists nowhere on the unit circle. Conditions are given which, together with the angular derivative condition, imply compactness on the space $D^{p}$. Inner functions which induce bounded composition operators on $D^{p}$ are discussed briefly.