On pointwise bounded approximation (Q839749)

Let \(G\) be an open subset in the extended complex plane \(\overline{\mathbb C}\) and let \(A(G)\) denote the Banach algebra of all functions analytic on \(G\) and continuous on \(\overline G\). Let \(H^\infty(G)\) denote the Banach algebra of all bounded analytic functions on \(G\) with the supremum norm. A connected open subset is called a circular domain if its boundary consists of a finite number of disjoint circles. It is a classical result that every finitely connected domain whose boundary contains no single point components is conformally equivalent to a circular domain. Let \(G\) be a domain conformally equivalent to a circular domain \(W\) and let \(\alpha\) be a conformal map from \(W\) onto \(G\). Then \(\alpha\) has nontangential limits at almost every point of \(\partial W\) with respect to the arclength measure. Thus, up to a zero-arclength measurable subset, \(\alpha\) is well-defined on \(\partial W\). We call \(G\) multi-nicely connected if \(\alpha\) is univalent on the subset of \(\partial W\) on which it is defined. If a multi-nicely connected domain is also simply connected, then it is called a nicely connected domain. In this work, the author investigates the following problem: When \(A(G)\) is pointwise boundedly dense in \(H^\infty(G)\)? Davie showed if the complement of \(G\) is connected, then \(A(G)\) is pointwise boundedly dense in \(H^\infty(G)\) if and only if the harmonic measures of the components of \(G\) are mutually singular and each component is nicely connected. A special case when the complement of \(G\) is an arc was first investigated by Browder and Wermer. Davie extended the Browder-Wermer result to the general case. In this paper, the author extends Davie's characterization to those open subsets whose components are finitely connected. The class of the open subsets, that the following Theorem covers, is significantly larger since it is not required that the complement of \(G\) has finitely many components, which is equivalent to requiring that all, except for finitely many components of \(G\) must be simply connected. {Theorem.} Let \(G\) be a bounded open subset such that each of its components is finitely connected and the components of \(G\) have no single point boundary components. Then \(A(G)\) is pointwise boundedly dense in \(H^\infty(G)\) if and only if the harmonic measures of the components of \(G\) are mutually singular and every component of \(G\) is multi-nicely connected. An example is given which shows that there is a domain \(G\) such that \(G\) is multi-nicely connected, but \(A(G)\) is not a hypo-Dirichlet algebra.