Potential theory and boundary behavior in the Drury-Arveson space

TL;DR

This paper introduces a capacity concept for the Drury-Arveson space, establishing boundary limit properties, sharpness of capacity zero sets, and applications to cyclicity, with potential extensions to other function spaces.

Contribution

It develops a capacity theory for $H^2_d$, characterizes boundary behavior, and links capacity zero sets to total nullity, advancing understanding of boundary limits in this space.

Findings

Functions in $H^2_d$ have Korányi limits outside capacity zero sets

Capacity zero sets are equivalent to totally null sets in $H^2_d$

Results extend to applications in cyclicity and other function spaces

Abstract

We develop a notion of capacity for the Drury-Arveson space $H^2_d$ of holomorphic functions on the Euclidean unit ball. We show that every function in $H^2_d$ has a non-tangential limit (in fact Kor\'anyi limit) at every point in the sphere outside of a set of capacity zero. Moreover, we prove that the capacity zero condition is sharp, and that it is equivalent to being totally null for $H^2_d$. We also provide applications to cyclicity. Finally, we discuss generalizations of these results to other function spaces on the ball.