The corona problem with restrictions on the relative position of sublevel sets (Q896041)

Let \(\Omega\) be a bounded strongly pseudoconvex domain in \(\mathbb{C}^n\), and let \(f_1,\dots,f_k\) be bounded holomorphic functions on \(\Omega\) satisfying the corona condition \[ \sum_{j=1}^k |f_j(z)|^2\geq \delta^2>0,\qquad\text{for all }\quad z\in\Omega. \] For \(1\leq j\leq k\) and \(\alpha>0\), let \(V_j(\alpha)=\{z\in \Omega: |f_j(z)|<\alpha\}\) and let \(\overline{V_j(\alpha)}\) be its closure in \(\mathbb{C}^n\). The author proves the following corona type result: If there exist \(\alpha>0\) and \(i, j \in\{1,\dots,k\}\) such that \[ \overline{V_i(\alpha)}\cap \overline{V_j(\alpha)}\cap \partial \Omega=\emptyset, \] then there exist bounded holomorphic functions \(g_1,\dots,g_k\) on \(\Omega\) such that \(f_1g_1+\dots+ f_kg_k=1\).