Totally null sets and capacity in Dirichlet type spaces

TL;DR

This paper establishes that in Dirichlet type spaces, the concept of totally null sets coincides with having zero capacity, linking functional analysis and potential theory, and enhances boundary interpolation theorems.

Contribution

It proves the equivalence of null sets and capacity zero in Dirichlet spaces and strengthens existing boundary interpolation results.

Findings

Totally null sets coincide with capacity zero sets in Dirichlet spaces.

The results apply to classical Dirichlet space and logarithmic capacity.

Strengthened boundary interpolation theorems are obtained.

Abstract

In the context of Dirichlet type spaces on the unit ball of $\mathbb{C}^d$, 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 strengthenings of boundary interpolation theorems of Peller and Khrushch\"{e}v and of Cohn and Verbitsky.