Dirichlet spaces with superharmonic weights and de Branges-Rovnyak spaces

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We consider Dirichlet spaces with superharmonic weights. This class contains both the harmonic weights and the power weights. Our main result is a characterization of the Dirichlet spaces with superharmonic weights that can be identified as de Branges-Rovnyak spaces. As an application, we obtain the dilation inequality Dω(fᵣ)≤ 2r/1+r Dω(f) (0≤ r<1), where Dω denotes the Dirichlet integral with superharmonic weight ω, and fᵣ(z):=f(rz) is the r-dilation of the holomorphic function f.