Berndtsson and T. J. Ransford [@Berndtsson-Ransford] and Z. Slodkowski [@slod1]. Besides the intrinsic interest of this result in classical function theory, the proof of this result introduced a number of tools and ideas which have proven to be of great importance in analysis.

Following the appearance of this result, it was swiftly generalized to the case of finitely connected domains, and by now a number of proofs exist for this case, e.g. [@alling1], [@alling2], [@earle-marden], [@forelli], [@slod2], [@stout1], [@stout2], [@stout3]. Although any planar domain can be exhausted by a sequence of finitely connected domains, to solve the problem on the larger domain one must control the norms of the corona solutions, $\Vert g_j\Vert_\infty$, in the approximating domains in order to take normal limits, control which is unfortunately not provided by any of the proofs just cited.

The corona problem was first solved for a class of infinitely connected domains by M. Behrens [@behrens1], [@behrens2]. These are “roadrunner” domains $\mathbb D\setminus\bigcup B_j$, where $B_j=B(c_j,r_j)$ is a disc centered at $c_j$ and radius $r_j$ such that $$\sum_{j=1}^\infty\frac{r_j}{|c_j|}<\infty\qquad\text{and}\qquad\left|\frac{c_{j+1}}{c_j}\right|<\lambda<1\text{ for all j}.$$ This summability restriction was improved somewhat in [@deeb], [@deeb-wilken]. Behrens [@behrens2] also proved that if the corona theorem fails for a plane 