Dirichlet Spaces with Superharmonic Weights and de Branges–Rovnyak Spaces

Abstract

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