Extensions of harmonic and analytic functions

Abstract

Introduction. This introduction will present a quick survey of our results; the complete definitions necessary to state these results precisely are given in later sections. Let H°° denote the algebra of bounded analytic functions on the open unit disk in the complex plane. The maximal ideal space of H°° is denoted by M. We can think of the open unit disk as a dense subset of M. Carl Sundberg [11] proved that every function in BMO extends to a continuous function from M into the Riemann sphere; he also described several properties of these extensions. Sundberg was working in the context of functions of several real variables. In the next section of this paper we take advantage of the tools offered by analytic function theory to give considerably simpler proofs (in the context of one complex variable) of Sundberg's results about extensions of BMO functions. In the section of this paper on nontangential limits, we prove that a function on the disk that has a continuous extension to a small subset of M must have a nontangential limit at almost every point of the unit circle. We then use this result to produce a class of functions in the little Bloch space that cannot be extended to. be continuous functions from this small subset of M to the Riemann sphere. The section of this paper dealing with cluster sets and essential ranges shows how those sets can be computed from the appropriate continuous extensions. We use these results to give a new proof of Joel Shapiro's theorem [10] that for every function in VMO, the essential range equals the cluster set. The final section of the paper discusses some open questions.