connected domains to the domain $\Omega_{n_0}=\mathbb C^\ast\setminus[0,1]^2\cup\bigcup_{n\le n_0}\bigcup_{|J|=n}\tilde T^n_{J,j}$. Given $n\ge n_0$, let us suppose we have already obtained the desired corona solutions $\{g^{(m,I)}_\mu\}$ in the regions $\tilde T^m_I$ for which $m<n$. As the domains $\{\tilde T^n_J\}_{|J|=n}$ are disjoint, we may construct our corona solutions in each of those domains separately.

Fix $\tilde T^n_J$. We note that $\tilde T^n_J\cap\left(\bigcup_{m<n}\bigcup_{|I|=m}\tilde T^m_I\right)=\bigcup_1^4D^n_{J,j}$, and so in each region $D^n_{J,j}$ there are corona solutions constructed from previous generations, which we shall denote $\{g^j_\mu\}_\mu$. (Since $\sum_{n,J}\chi_{\tilde T^n_J}\le2$, these solutions are uniquely determined among those previously constructed.) We now push the situation forward to the unit disc according to the map $\phi^n_J$. Let $D_j$ denote the push-forward of the domain $D^n_{J,j}$, $F_\mu$ the push-forward of a corona datum $f_\mu$, and $G_\mu^j$ the push-forward of $g^j_\mu$. We note that $D_j$ is a lens-shaped domain by construction.

In order to construct our corona solutions, we will, as above, reduce the problem to an appropriate $\bar\partial$-problem. Obtaining the desired bounds in this manner will require the use of interpolating sequences, so before continuing with the main line of argument we make a detour to obtain the special interpolating functions that we require.

\[outerfunction\] There is a function $G\in H^\infty(\mathbb 