Specification of the Munsell book of color

Abstract

We study the capacity in the sense of Beurling–Deny associated with the Dirichlet space D(μ) where μ 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 D(μ). It allows us to give an estimate of the capacity of points and arcs of the unit circle. We also provide 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.