Dirichlet spaces with superharmonic weights and de Branges-Rovnyak spaces

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 $${\cal D}_\omega(f_r)\le \frac{2r}{1+r}{\cal D}_\omega(f) \qquad(0\le r<1),$$ where ${\cal D}_\omega$ denotes the Dirichlet integral with superharmonic weight $\omega$, and $f_r(z):=f(rz)$ is the $r$-dilation of the holomorphic function $f$.