A class of infinitely connected domains and the corona

Abstract:

Let D be a bounded domain in the complex plane. Let ${H^\infty }(D)$ be the Banach algebra of bounded analytic functions on D. The corona problem asks whether D is $\text {weak}^\ast$ dense in the space $\mathfrak {M}(D)$ of maximal ideals of ${H^\infty }(D)$. Carleson [3] proved that the open unit disc ${\Delta _0}$ is dense in $\mathfrak {M}({\Delta _0})$. Stout [9] extended Carleson’s result to finitely connected domains. Behrens [2] found a class of infinitely connected domains for which the corona problem has an affirmative answer. In this paper we will use Behrens’ idea to extend the results to more general domains. See [11] for further extensions and applications of these techniques.