An upper bound for the distance to finitely generated ideals in Douglas algebras (Q2773389)

In [Ann. Inst. Fourier 35, No. 4, 163-174 (1985; Zbl 0564.46044)], \textit{J. Bourgain} proved that if \(\alpha\) is a real-valued function satisfying \(\alpha(t)/t\to 0\) at \(t\to 0\) and if \(f, f_1,\dots,f_n\in H^{\infty}(\mathbb T)\) are such that \(|f|\leq\alpha(|f_1|+\dots+|f_n|)\) on the unit disk \({\mathbb D}\), then \(f\) is in the closed ideal \(I\) generated by \(f_1,\dots,f_n\). NEWLINENEWLINENEWLINEIn this paper the authors examine Bourgain's proof and extend it from Banach algebra \(H^{\infty}(\mathbb T)\) to arbitrary Douglas algebras. The essential difference in this case is that some estimates involving \({\overline{\partial}}\)-equations are no longer valid on \(D\) but rather on regions of \(D\) that are asymptotically close to the maximal ideal space of the Douglas algebra. In particular, an estimate for distance \(\text{dist}_A (f, I)\) from a function \(f\) in the Douglas algebra \(A\) to a finitely generated ideal \(I\) in \(A\) is obtained.