Dirichlet spaces with superharmonic weights and de Branges-Rovnyak spaces

2015/10/27 by El-Fallah, Omar, Kellay, Karim, Klaja, Hubert +2 · 3 citations
#Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1510.08130

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_ω(f_r)≤ (2r)/(1+r)\cal D_ω(f) (0≤ rlt;1), where \cal D_ω denotes the Dirichlet integral with superharmonic weight ω, and f_r(z):=f(rz) is the r-dilation of the holomorphic function f.

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• Douglas Weighted Dirichlet Spaces: Analytic and Probabilistic Aspects
• Higher order weighted Dirichlet type spaces with poly-superharmonic weights and Dirichlet type operators of finite order
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