This was later extended by C. Moore [@moore] to the case of domains with boundary contained in the graph of a $C^{1+\epsilon}$ function.

Due to the results in [@jones-marshall], the corona problem for a domain can be solved if the critical points of Green’s function (for a fixed base point) form an interpolating sequence for $H^\infty(\Omega)$. However, it was shown by M. Gonzalez [@gonzalez] that for a large class of domains, conditions necessary for the critical points to be an interpolating sequence might fail. Peter Jones (unpublished) has given an explicit example of a class of domains where the critical point of greens function fail to form an interpolating sequence, namely the complements of certain square Cantor sets (which we describe more explicitly below).

Throughout the remainder of this work, $\Omega$ will denote the complement of a square Cantor set $K$. To fix notation, we define $K$ explicitly as $K=\bigcap_{n}K^n$, where the $K^n$ are defined inductively as follows. Fix $\{\lambda_n\}_\mathbb N\in(\frac14,\frac12)$, which we assume satisfies $\lambda_n\le\lambda_{n+1}$ for simplicity. (These conditions also ensure that $H^\infty(\Omega)$ is nontrivial.) Put $K^0=[0,1]^2$. At stage $n$, we set $K^n=\bigcup_{|J|=n}Q^n_J$ where $Q^n_J$ are squares with sides parallel to the axes and side length $\sigma_n=\prod^n_{k=1}\lambda_k$, and $J$ 