The Corona Construction

Abstract

This chapter is an extensive discussion of Carleson’s corona theorem. Several proofs of the theorem will be presented, because the ideas behind each proof have proved useful for other problems. We first give T.Wolff’s recent, very elegant proof, which is based on Littlewood–Paley integrals and which employs analyticity in a decisiveway. Then we take up Carleson’s original proof. It consists of a geometric construction that has led to many of the deeper results in this theory and that applies to harmonic functions and to more general situations.