Kernel estimate and capacity in the Dirichlet spaces

Omar El-Fallah (1) , Youssef Elmadani (1) , Karim Kellay (2)

1 Département de Mathématiques
2 IMB - Institut de Mathématiques de Bordeaux

Let µ be a positive finite measure on the unit circle. The associated Dirichlet space D(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 with 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. 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.

Domaines: Variables complexes [math.CV]; Analyse classique [math.CA]; Analyse fonctionnelle [math.FA]

Omar El-Fallah, Youssef Elmadani, Karim Kellay. Kernel estimate and capacity in the Dirichlet spaces. 2014. ⟨hal-01083234v1⟩