Critical points of Green's function, harmonic measure, and the corona problem (Q1073957)

It is still an open problem if the corona theorem is true for arbitrary planar domains G, i.e. for the space \(H^{\infty}(G)\) of all bounded holomorphic functions on G. Carleson gave in 1980 an affirmative answer for a special class of planar domains of the form \(G={\mathbb{C}}\setminus E\), where E is a homogeneous subset of \({\mathbb{R}}\). The heart of the paper under review consist of a discussion and a presentation of a new proof of Carleson's result. Author's remark: In the meantime a paper of \textit{J. Garnett} and the first author appeared [Acta Math. 155, 27-40 (1985; Zbl 0578.30043)] in which the corona theorem is proven for all Denjoy domains. The techniques are quite different from those used in the paper under review.