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

We study the capacity in the sense of Beurling-Deny associated with the Dirichlet space $\mathcal{D}(\mu)$ where $\mu$ is a finite positive Borel measure on the unit circle. First, we obtain a sharp asymptotic estimate of the norm of the reproducing kernel of $\mathcal{D}(\mu)$. It allows us to give an estimates of the capacity of points and arcs of the unit circle. We also provide a new conditions on closed sets to be polar. Our method is based on sharp estimates of norms of some outer 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. Journal of Functional Analysis, 2019, 276 (3), pp.867-895. ⟨hal-01083234v2⟩