Extensions of harmonic and analytic functions (Q809221)

Let \(H^{\infty}\) denote the algebra of bounded analytic functions on the open unit disk in the complex plane. The first part of this paper uses tools of analytic function theory to give simpler proofs (in the context of one complex variable) of \textit{C. Sundberg}'s results [Indiana Univ. Math. J. 33, 749-771 (1984; Zbl 0584.46019)] about extensions of BMO functions to the maximal ideal space of \(H^{\infty}.\) This next section of the paper proves that a function on the disk that has a continuous extension to the Shilov boundary of \(H^{\infty}\) must have a non-tangential limit at almost every point of the unit circle. The authors raise the question of whether the converse holds. Since this paper was written, that question has been affirmatively answered independently by \textit{Ch. Bishop} [A characterization of some algebras on the disk, preprint] and \textit{O. Ivanov} [Fatou's theorem on angular limits and problems of extendibility to the ideal boundary (Russian), Zap. Nauchn. Sem. Leningrad. Otd. Mat. Inst. Steklov 190, No.19, 101-109 (1991)]. The final section of this paper gives a new proof of \textit{J. Shapiro}'s theorem [Mich. Math. J. 34, 323-336 (1987; Zbl 0634.30033)] that for every function in VMO, the essential range equals the cluster set.