Embedding model and de Branges-Rovnyak spaces in Dirichlet spaces

In this paper we study embeddings between de Branges-Rovnyak spaces $ H(b) $ H(b) and harmonically weighted Dirichlet spaces $ \mathcal {D}(\mu ) $ D(mu) in terms of the boundary spectrum of b and the support of the measure mu, by using elementary reproducing kernel estimates. We completely characterize the embedding between the model spaces $ K_u $ Ku and the local Dirichlet spaces $ \mathcal {D}_\zeta $ D zeta, and we discuss some applications.