Dirichlet spaces with superharmonic weights and de Branges-Rovnyak spaces
Résumé
Wec onsider 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 $D_\omega(f_r)≤ \frac{2r}{1+r} D_\omega(f) $ $(0\leq r<1)$, where $D_\omega$ denotes the Dirichlet integral with superharmonic weight $\omega$, and $f_r(z) := f(rz)$
is the r-dilation of the holomorphic function f.