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Potential theory and boundary behavior in the Drury-Arveson space

Nikolaos Chalmoukis, Michael Hartz

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 Korányi 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.

Potential theory and boundary behavior in the Drury-Arveson space

Abstract

We develop a notion of capacity for the Drury-Arveson space of holomorphic functions on the Euclidean unit ball. We show that every function in has a non-tangential limit (in fact Korányi 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 . We also provide applications to cyclicity. Finally, we discuss generalizations of these results to other function spaces on the ball.

Paper Structure

This paper contains 9 sections, 36 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Energy and capacity in RKHS
  3. Unboudnedness sets
  4. Weak type estimate of maximal function
  5. Radial limits in more general spaces
  6. Applications to cyclicity
  7. Dual formulation and Choquet capacity
  8. Totally null sets and Henkin measures
  9. Abstract capacities

Key Result

Theorem 1.1

For each $f \in H^2_d$, there exists a Borel set $E \subset \partial \mathbb{B}_d$ of outer $H^2_d$-capacity zero such that $f$ has a Korányi limit at every point in $\partial \mathbb{B}_d \setminus E$.

Theorems & Definitions (84)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Example 2.4
  • ...and 74 more