We note that these regions remain simply connected, so that Carleson’s corona theorem for the unit disc $\mathbb D$ provides corona solutions $\{g^{(n,J)}_\mu\}_{\mu=1}^M$ corresponding to $\{f_\mu\}_1^M$ in $\hat T^n_J$. We note that $\sum\chi_{\hat T^n_J}\le2$. The most important result for the present construction is embodied in the following proposition.

\[basicconstruction\] Given corona data $\{f_\mu\}_1^M$ in $\Omega$ as above, if the angle $\alpha$ defining the lozenges $L^n_{J,j}$ is sufficiently small then there are corona solutions $\{g^{(n,J)}_\mu\}_1^M$ in each $\hat T^n_J$ such that $\Vert g^{(n,J)}_\mu\Vert_\infty\le C(M,\delta,K)$ and if $(n,J)$ and $(m,I)$ are indices with $\hat T^n_J\cap \hat T^m_I=L^n_{J,j}$ then for $1\le\mu\le M$, $$|g^{(n,J)}_\mu(z)-g_\mu^{(m,I)}(z)|\lesssim\frac1{n^3}\frac{d(z,K)}{\sigma_n\delta_n},\qquad z\in L^n_{J,j}.$$\

Assuming this proposition, let us prove the theorem. Let $\{\phi_{(n,J)}\}$ to be a partition of unity on $\Omega$ subordinate to $\hat T^n_J$ satisfying $|\nabla\phi_{(n,J)}(z)|\lesssim d(z,K)^{-1}$ and define $$\label{dbar1}
    \tilde g_\mu=\sum_{(n,J)}\phi_{(n,J)}g^{(n,J)}_\mu.$$ We note that the functions $\tilde g_\mu$, while not analytic, have the desired property that $\sum f_\mu\tilde g_\mu\equiv1$ on $\Omega$. An observation due to Hörmander [@Hormander], now reduces us to solving the $\bar\partial$-problem $$\bar\partial a_{\mu\nu}=\tilde g_\mu\bar\partial\tilde g_\nu,\qquad a_{\mu\nu}\in L^\infty(\Omega).$$ Indeed, given such functions $a_{\mu\nu}$, if we define $$G_\mu=\tilde g_\mu+\sum_{\nu=1}^M(a_{\mu\nu}-a_{\nu\mu})f_\nu$$ the antisymmetry of the matrix $A=[a_{\mu\nu}]$ gives us $\sum f_\mu G_\mu\equiv1$ on $\Omega$, while (\[dbar1\]) provides $\bar\partial G_\mu\equiv0$ on 