Dirichlet spaces with superharmonic weights and de Branges-Rovnyak spaces

Omar El-Fallah (1) , Karim Kellay (2) , Hubert Klaja (3) , Javad Mashreghi (3) , Thomas Ransford (3)

1 Département de Mathématiques
2 IMB - Institut de Mathématiques de Bordeaux
3 Département de Mathématiques et de Statistiques

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.

Domaines: Variables complexes [math.CV]; Analyse classique [math.CA]; Analyse fonctionnelle [math.FA]

Omar El-Fallah, Karim Kellay, Hubert Klaja, Javad Mashreghi, Thomas Ransford. Dirichlet spaces with superharmonic weights and de Branges-Rovnyak spaces. Complex Analysis and Operator Theory, 2015, 10 (1), pp.97-107. ⟨hal-01195728⟩