Geometric conditions for bounded point evaluations in spaces of several complex variables

Abstract

Let U be a bounded domain in \(\mathbb C^d\) and let \(L^p_a(U)\), \(1 \le p < \infty \), 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 \(f \rightarrow f(x)\) 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.