A new estimate for the vector valued corona problem
In this paper the author gives an improved estimate in the vector valued corona theorem for the unit disk. His proof splits into two parts: first, he resurrects the operator corona theorem from the 1970s, and then he uses a Hilbert space argument, based on a version of Wolff's ideas. Since he uses Hilbert space methods, the Riesz representation theorem replaces the usual \(\bar\partial\)-techniques. Then the author considers the operator version of the corona problem on the unit disk \(D\) and gives an improvement of Vasyunin's estimate. He gets estimates for analytic solutions, \(G(z)\), of \(F(z)G(z)=I\) for all \(z\in D\) of the form \[ \sup\limits_{z\in D}||G(z)||_{B(\mathbb{C}^m,\mathbb{C}^n)}\leq 2[2\sqrt{e}+ 2\sqrt{2}e]\frac 1{\varepsilon^{n+1}}\ln\left(\frac 1{\varepsilon^{2n}}\right) . \] Since estimates independent of \(m\), the case \(m=\infty\) follows from a routine argument.
scientific article; zbMATH DE number 1747022