Geometric conditions for bounded point evaluations in several complex variables

TL;DR

This paper establishes geometric criteria based on Sobolev capacity to determine when points in the closure of a domain in several complex variables serve as bounded point evaluations for analytic $L^p$ functions, extending single-variable results.

Contribution

It introduces a geometric condition involving Sobolev $q$-capacity that characterizes bounded point evaluations in several complex variables, generalizing previous single-variable findings.

Findings

Provides a geometric criterion for bounded point evaluations in several variables.

Extends single-variable results to higher dimensions.

Uses Sobolev $q$-capacity to characterize evaluation boundedness.

Abstract

Let $U$ be a bounded domain in $\mathbb C^d$ and let $L^p_a(U)$, $1 \leq p < \infty$, denote the space of functions that are analytic on $\overline{U}$ and bounded in the $L^p$ norm on $U$. A point $x \in \overline{U}$ is said to be a bounded point evaluation for $L^p_a(U)$ if the linear functional $f \to f(x)$ is bounded in $L^p_a(U)$. In this paper, we provide a purely geometric condition given in terms of the Sobolev $q$-capacity for a point to be a bounded point evaluation for $L^p_a(U)$. This extends results known only for the single variable case to several complex variables.