Dirichlet spaces with superharmonic weights and de Branges-Rovnyak spaces

TL;DR

This paper characterizes Dirichlet spaces with superharmonic weights that are equivalent to de Branges-Rovnyak spaces and derives a dilation inequality for these weighted Dirichlet integrals.

Contribution

It provides a characterization of superharmonic-weighted Dirichlet spaces as de Branges-Rovnyak spaces and establishes a related dilation inequality.

Findings

Characterization of superharmonic-weighted Dirichlet spaces as de Branges-Rovnyak spaces

Derivation of a dilation inequality for these spaces

Inclusion of harmonic and power weights within this class

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 \[ {\cal D}_\omega(f_r)\le \frac{2r}{1+r}{\cal D}_\omega(f) \qquad(0\le r<1), \] 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 $f$.