Projective limits of Dirichlet spaces and embeddings of Bloch-type spaces

In this paper, we prove that the projective limit of a certain class of Dirichlet-type spaces is isomorphic to the Bloch-type spaces B-alpha. We provide a complete characterization of bounded or compact embeddings of B-alpha into a family of Lebesgue spaces by the derivative operator. As an application, we prove precise necessary and sufficient criteria for boundedness/compactness of the generalized composition operators. The Dirichlet-type intermediate spaces constructed in this paper serve very useful.