ANALYTIC MULTIFUNCTIONS, THE 3-EQUATΪON, AND A PROOF OF THE CORONA THEOREM

The purpose of this article is to give some applications of a recent theorem by Alexander-Wermer and Slodkowski on the structure of certain polynomial hulls. We want to show that this theorem gives a useful method of constructing analytic functions with prescribed proper-ties in the disc. In particular it yields a rather easy proof of the Corona Theorem for two generators, and also implies Wolff's Theorem on the d-equation. 1. Background on hulls and analytic multifunctions. We begin by formulating the theorem on polynomial hulls proved by Alexander and Wermer ([1]), and independently by Slodkowski [16]. In what follows, Δ denotes the open unit disc { λ e C: | λ | < l} , and T denotes the unit circle THEOREM 1.1. Let X be a compact subset of T X C, and assume that for each f e T the fibre X ξ = { z E C: ( ί, z) e I} is convex. Suppose moreover that X, the polynomial hull of X, contains the point (0, z 0) for some z0 e C. Then there exists h e i/°°(Δ) such that for almost all ζ G T where h * denotes the radial boundary values ofh. Sketch proof (see [1] for further details). Let μ be a probability measure on X which represents (0, z 0), in the sense that for any poly-nomial p, (1) p(09z0)=fp(ζ9z)dμ(ζ9z)