Geometric conditions for bounded point evaluations in several complex variables
Stephen Deterding
TL;DR
The paper addresses geometric conditions for the existence of bounded point evaluations on $L^p_a(U)$, the $L^p$-closure of analytic functions on a bounded domain $U\subset\mathbb{C}^d$, extending the one-variable Sobolev-capacity criterion to higher dimensions. It builds on the Bochner–Martinelli integral formula to represent evaluations as integrals and derives capacity-based bounds using annuli $A_n(x)$ and the Sobolev capacity $\Gamma_q$, with $q=p/(p-1)$ and $p>\frac{2d}{2d-1}$. The main result is a sufficient condition: if $\sum_{n=1}^\infty 2^{n(2d-1)q} \Gamma_q(A_n(x)\setminus U) < \infty$, then $x$ is a bounded point evaluation for $L^p_a(U)$; the bound is explicit in terms of this capacity sum. This work generalizes classical geometric criteria to several complex variables and ties bounded evaluation functionals to Sobolev capacity geometry, informing $L^p$-approximation and related analytic structures in higher dimensions.
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.