$\Omega$ for each $1\le\mu\le M$, so that $G_\mu\in H^\infty(\Omega)$. Thus the functions $\{G_\mu\}_1^M$ provide the corona solutions sought by the theorem.

Let us therefore turn our attention to solving (\[dbar1\]). The immediate thought is to consider $$a_{\mu\nu}(z)=\frac1{2\pi i}\iint_\Omega\frac{\tilde g_\mu(\zeta)\bar\partial\tilde g_\nu(\zeta)}{\zeta-z}\,d\zeta\,d\bar\zeta,$$ but it is not clear that the integral is convergent à priori, so instead we begin by viewing this formally as $$\frac1{2\pi i}\iint_\Omega\frac{\tilde g_\mu(\zeta)\bar\partial\tilde g_\nu(\zeta)}{\zeta-z}\,d\zeta\,d\bar\zeta
        =\sum\frac1{2\pi i}\iint_{L^n_{J,j}}\frac{\tilde g_\mu(\zeta)\bar\partial\tilde g_\nu(\zeta)}{\zeta-z}\,d\zeta\,d\bar\zeta,$$ and attempt to introduce additional cancellations in each term. We begin by estimating individual terms. For ease of notation, we define $$I^n_{J,j}(z) = \frac1{2\pi i}\iint_{L^n_{J,j}}\frac{\tilde g_\mu(\zeta)\bar\partial\tilde g_\nu(\zeta)}{\zeta-z}\,d\zeta\,d\bar\zeta.$$ Let us suppose that $(m,I)$ is such that $L^n_{J,j}=\hat T^n_J\cap\hat T^m_I$. We first note that, for $z\in\overline{L^n_{J,j}}\setminus\{x_1,x_2\}$, $$\begin{aligned}
    |I^n_{J,j}(z)|
        &\lesssim& \iint_{L^n_{J,j}}\frac{|g_\nu^{(n,J)}\bar\partial\phi_{(n,J)}+g_\nu^{(m,I)}\bar\partial\phi_{(m,I)}|}{|\zeta-z|}\,dx\,dy   \\
        &\lesssim& \iint_{L^n_{J,j}}\frac{|g_\nu^{(n,J)}-g_\mu^{(m,I)}||\bar\partial\phi_{(n,J)}|}{|\zeta-z|}\,dx\,dy  \\
        &\lesssim& \frac1{\sigma_n\delta_nn^3}\iint_{L^n_{J,j}}\frac{dx\,dy}{|\zeta-z|}  \\
    |I^n_{J,j}(z)| &\lesssim& \frac1{n^3},
    \end{aligned}$$ where we have exploited the fact that $\sum\chi_{\hat T^n_J}\le2$ in the second line and the proposition in the third. Proceeding in a similar manner, we can achieve estimates $$\left|\iint_{L^n_{J,j}}\tilde g_\mu\bar\partial g_\nu\,d\zeta\,d\bar\zeta\right|\lesssim\frac{\sigma_n\delta_n}{n^3}.$$

To generate further cancellations, we employ the “derivative matching trick” from rational approximation theory (see [@vitushkin] or Chapter XII of [@gamelin-ua]). To this end, we introduce functions $k^n_{J,j}$ defined as follows. Let $\tilde K^n_{J,j}$ be a “square” $K^{m'}_{I'}=K\cap Q^{m'}_{I'}$ with $\tilde 