domain then it fails for a domain of the form $\mathbb D\setminus\bigcup B_j$, where $\{B_j\}$ is a sequence of discs clustering only at the origin. In this direction there is also a result of Gamelin [@gamelin] that the corona problem is local in that it depends only on the behavior of the domain locally about each boundary point.

The next significant progress for infinitely connected domains was again achieved by Carleson [@car-fatsets], who solved the corona problem for domains having boundary $E\subset\mathbb R$ satisfying, for some $\epsilon>0$, $$\Lambda_1(B(x,r)\cap E)\ge\epsilon r$$ for every $x\in E$ and $r>0$, where $\Lambda_1$ denotes one-dimensional Hausdorff measure. This proof followed an idea introduced in [@forelli], constructing an explicit projection from $H^\infty(\mathbb D)$ onto $H^\infty(\Omega)$. Following in the same vein, P. Jones and D. Marshall [@jones-marshall] used such projections to show that if the corona problem can be solved at the critical points of Green’s function for the domain, then it can be solved for the domain, and they provided a number of sufficient conditions for this criterion.

Later J. Garnett and P. Jones [@garnettjones] extended the result of Carleson [@car-fatsets], showing that the corona theorem holds for any domain having boundary contained in $\mathbb R$. 