A generalization of the corona theorem in the unit disc

Abstract

The purpose of this note is to prove a variant of the corona theorem in the unit disc without using the notation of Carleson measure or any particular duality or factorization theorem for analytic functions. The corona theorem was first proved by Carleson [1]. For more background and further references, see Gamelin 1-2] and Garnett I-3, 4]. Notations. D is the unit disc in ~ ; T its boundary; H(D) the analytic functions on D; He(D)={ f sH(D) , sup ~ l f f d a < + ~ } where dcr is the normalized O < r < l Lebesgue measure on T. H ~ (D) = H(O) c~ L ~ (O).