such that $$h_j(z_j)=1,$$ $$\Vert h_j\Vert_{L^\infty(\mathbb D)}\le A^2,$$ $$\sum_j|h_j(z)|\le\frac{\log8A^2}{\log\beta^{-1}}A^2,$$ and such that $$\sum_j|h_j(z)|\le\frac{\log8A^2}{\log\beta^{-1}}\frac{A^2}{n^6}\left(\frac{d(z,\{\zeta_1,\zeta_2\})}{{{\rm{diam}}}(L_j)}\right)^2,\qquad z\in L_j,$$ where $L_j$ and $\zeta_1(j),\zeta_2(j)$ are defined as in the Lemma \[outerfunction\].

[**Proof:** ]{} This is a mild refinement of an argument of Varopoulos [@Var]. We note that the sequence $S$ can be split into $\frac{\log8A^2}{\log\beta^{-1}}$ disjoint sequences $S_m$ such that (\[sepbeta\]) holds. Restricting our attention to a subsequence $S_m$, it suffices to consider the case that $S_m$ is finite, $S_m=\{z_1,\ldots,z_{n_0}\}$, as one may then employ normal families to the construction below. Set $\omega=e^\frac{2\pi i}{n_0}$. Employing the remarks above, by Lemma \[outerfunction\] we may choose $f_j\in H^\infty(\mathbb D)$ such that $f_j(z_k)=\omega^{jk}$, $\Vert f_j\Vert_{L^\infty(\mathbb D)}\le A$, and $|f_j(z)|\le \frac{A}{n^3}\frac{d(z,\{\zeta_1,\zeta_2\})}{{{\rm{diam}}}(L_\ell)}$ when $z\in L_\ell$ for each $1\le\ell\le4$. If we now define $$h_j(z)=\left(\frac1{n_0}\sum_{k=1}^{n_0}\omega^{-jk}f_k(z)\right)^2,$$ then $$h_j(z_i)=\left(\frac1{n_0}\sum^{n_0}_{k=1}\omega^{(i-j)k}\right)^2
        =\left\{\begin{array}{ll}1\qquad&\mbox{if }i=j,\\0& \mbox{else,}\end{array}\right.$$ $$\Vert h_j\Vert_{L^\infty(\mathbb D)}\le A^2,$$ and $$|h_j(z)|\le \frac{A^2}{n^6}\left(\frac{d(z,\{\zeta_1,\zeta_2\})}{{{\rm{diam}}}(L_\ell)}\right)^2$$ for $z\in L_\ell$. Also, $$\begin{aligned}
    \sum^{n_0}_{j=1}|h_j(z)| &=& \frac1{n_0^2}\sum^{n_0}_{k=1}\sum_{j,\ell}\omega^{-jk}\omega^{j\ell}f_j(z)\bar f_\ell(z)    \\
        &=& \frac1{n_0^2}\sum^{n_0}_{j=1}n_0|f_j(z)|^2,
    \end{aligned}$$ so that $$\sum_{j=1}^{n_0}|h_j(z)|\le A^2$$ throughout the unit disc, and $$\sum^{n_0}_{j=1}|h_j(z)|\le \frac{A^2}{n^6}\left(\frac{d(z,\{\zeta_1,\zeta_2\})}{{{\rm{diam}}}(L_\ell)}\right)^2,$$ for $z\in L_\ell$ as desired.\
With these lemmas in hand we turn now to constructing our special corona solutions. By Carleson’s original corona theorem [@car-disc], there are corona solutions $\{g_\mu\}$ in $\mathbb D$ with $\Vert g_\mu\Vert\le c(M,\eta)$. To generate corona solutions close 