Kernel estimate and capacity in Dirichlet type spaces

2014/11/04 by Omar El-Fallah, El-Fallah, O., Y. Elmadani +3
Mathematics · #Advanced Harmonic Analysis Research #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1411.1036

openalex publication_date 2014/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let μ be a positive finite measure on the unit circle. The Dirichlet type space D(μ), associated to μ, consists of holomorphic functions on the unit disc whose derivatives are square integrable when weighted against the Poisson integral of μ. First, we give an estimate of the norm of the reproducing kernel k^μ of D(μ). Next, we study the notion of μ-capacity associated to D(μ), in the sense of Beurling--Deny. Namely, we give an estimate of μ-capacity of arcs in terms of the norm of k^μ. We also provide a new condition on closed sets to be μ-polar. Note that in the particular case where μ is the Lebesgue measure, this condition coincides with Carleson's condition \citeCa. Our method is based on sharp estimates of norms of some outer test functions which allow us to transfer these problems to an estimate of the reproducing kernel of an appropriate weighted Sobolev space.

Related

• On the theory of capacities on locally compact spaces and its interaction with the theory of balayage
• A reproducing kernel approach to Lebesgue decomposition
• On a Capacity for Modular Spaces
• The Relative Capacity
• On-diagonal Heat Kernel Lower Bound for Strongly Local Symmetric Dirichlet Forms