Totally null sets and capacity in Dirichlet type spaces

Abstract

In the context of Dirichlet type spaces on the unit ball of ℂ𝑑, also known as Hardy–Sobolev or Besov–Sobolev spaces, we compare two notions of smallness for compact subsets of the unit sphere. We show that the functional analytic notion of being totally null agrees with the potential theoretic notion of having capacity zero. In particular, this applies to the classical Dirichlet space on the unit disc and logarithmic capacity. In combination with a peak interpolation result of Davidson and the second named author, we obtain strengthening of boundary interpolation theorems of Peller and Khrushchëv and of Cohn and Verbitsky.