Geometric conditions for bounded point evaluations in several complex variables

Let $U$ be a bounded domain in ${\mathbb{C}}^d$ and let $L_a^p(U)$, $1\le p<\mathrm{\infty }$, denote the space of functions that are analytic on $\bar{U}$ and bounded in the $L^p$ norm on $U$. A point $x\in \bar{U}$ is said to be a bounded point evaluation for $L_a^p(U)$ if the linear functional $f\to f(x)$ is bounded in $L_a^p(U)$. 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 $L_a^p(U)$. This extends results known only for the single variable case to several complex variables.