Kernel estimate and capacity in Dirichlet type spaces

TL;DR

This paper provides estimates for the reproducing kernel and capacity in Dirichlet type spaces associated with measures on the unit circle, linking kernel norms to capacity and introducing new conditions for $ extmu$-polar sets.

Contribution

It introduces new estimates for the kernel norm and $ extmu$-capacity in Dirichlet type spaces, extending classical results and providing novel conditions for $ extmu$-polar sets.

Findings

Estimated the norm of the reproducing kernel $k^ extmu$.

Connected $ extmu$-capacity of arcs to kernel norms.

Established a new condition for sets to be $ extmu$-polar.

Abstract

Let $\mu$ be a positive finite measure on the unit circle. The Dirichlet type space $\mathcal{D}(\mu)$, associated to $\mu$, consists of holomorphic functions on the unit disc whose derivatives are square integrable when weighted against the Poisson integral of $\mu$. First, we give an estimate of the norm of the reproducing kernel $k^\mu$ of $\mathcal{D}(\mu)$. Next, we study the notion of $\mu$-capacity associated to $\mathcal{D}(\mu)$, in the sense of Beurling--Deny. Namely, we give an estimate of $\mu$-capacity of arcs in terms of the norm of $k^\mu$. We also provide a new condition on closed sets to be $\mu$-polar. Note that in the particular case where $\mu$ is the Lebesgue measure, this condition coincides with Carleson's condition \cite{Ca}. 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.