Dirichlet spaces with superharmonic weights and de Branges-Rovnyak spaces
Omar El-Fallah, Karim Kellay, Hubert Klaja, Javad Mashreghi, Thomas, Ransford

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 denotes the Dirichlet integral with superharmonic weight , and is the -dilation of the holomorphic function .
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