A generalization of the corona theorem in the unit disc. (English) Zbl 0702.30038

Cegrell, Urban

Math. Z. 203, No. 2, 255-261 (1990).

Let \(H^{\infty}\) be the standard Hardy space in the unit disk \({\mathbb{D}}\) and functions \(g,f_ 1,f_ 2\in H^{\infty}\) satisfy \[ | g(z)| \leq (| f_ 1(z)|^ 2+| f_ 2(z)|^ 2)^{1+\epsilon},\quad \forall z\in {\mathbb{D}} \] for some \(\epsilon >0\). The author proves existence of functions \(g_ 1,g_ 2\in H^{\infty}\) such that \(g=g_ 1f_ 1+g_ 2f_ 2\). The proof is based on \({\bar \partial}\)-estimates and uses neither the notion of Carleson measure, nor factorization theorems for analytic functions.

Reviewer: J.Ljubarsky

MSC:

30D55

\(H^p\)-classes (MSC2000)

Keywords:

corona problem; \({\bar \partial }\)-equation

References:

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