extends the result past the regime where the critical points of Green’s function form an interpolating sequence.

The basic outline of the proof is as follows. We first break the domain into a number of simply connected domains $T^n_J$ which, roughly speaking, are the cross-shaped domains $Q^n_J\setminus\bigcup_{j=1}^4Q^{n+1}_{J,j}$. The fundamental idea is that found in [@garnettjones], namely to apply Carleson’s original theorem for simply connected domains and then constructively solve a $\bar\partial$-problem to obtain corona solutions for the whole domain.

To obtain the necessary cancellations, however, we require that the corona solutions on our simply connected subdomains have a certain amount of agreement on the overlaps of those domains. To achieve this, we build these special solutions inductively, the solutions at stage $n$ obtained first by choosing solutions according to Carleson’s corona theorem for simply connected domains and then solving a $\bar\partial$-problem to alter the solutions to match the neighboring solutions already constructed. The method for solving the $\bar\partial$-problem is much like in [@garnettjones], employing an interpolating sequence in the simply connected domains $T^n_J$ to build our solutions. For our solution to have the desired special properties, however, we must choose our interpolating functions to have certain special properties, and it 