A family of Dirichlet-Morrey spaces

To each weighted Dirichlet space $\mathcal{D}_p$, $0<p<1$, we associate a family of Morrey-type spaces ${\mathcal{D}}_p^{\lambda}$, $0< \lambda < 1$, constructed by imposing growth conditions on the norm of hyperbolic translates of functions. We indicate some of the properties of these spaces, mention the characterization in terms of boundary values, and study integration and multiplication operators on them.