The weak Behrens' property and the Corona

Abstract

In this paper we study a class of infinitely connected domains larger than the one considered by Behrens [1] and prove that the corona problem has an affirmative answer.Introduction.Let D be a bounded domain in the complex plane and H^{\infty}(D) be the algebra of bounded analytic functions on D. The corona problem asks whether D is weak* dense in the space \mathscr{M} (D) of maximal ideals of H^{\infty}(D) .Carleson [3] proved that the open unit disc \Delta is dense in \mathscr{M}(\Delta) .In [7] Stout extended Carleson's result to finitely connected domains.In [1] Behrens 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 [4] and [5] for other

Locations

• Hokkaido Mathematical Journal - View - PDF