Geometric conditions for bounded point evaluations in several complex variables

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.