Composition operators on Dirichlet-type spaces

The Dirichlet-type space D-p (1 less than or equal to p less than or equal to 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 o Phi for f is an element of D-p. The operator C-Phi : D-p --> 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 less than or equal to q less than or equal to p) and on the weighted Dirichlet space D2-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.