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

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


30D55 \(H^p\)-classes (MSC2000)
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[1] Carleson, L.: Interpolation by bounded analytic functions and the coronaproblem. Ann. Math.76, 547–559 (1962) · Zbl 0112.29702
[2] Gamelin, T.W.: Wolffs proof of the corona theorem. Isr. J. Math.37, 113–119 (1980) · Zbl 0466.46050
[3] Garnett, J.B.: Bounded analytic functions. New York London: Academic Press 1981 · Zbl 0469.30024
[4] Garnett, J.B.: Corona problems, interpolation problems and inhomogeneous Cauchy-Riemann equations. Proceedings of the ICM, Berkeley, Calif, pp. 917–923 1986
[5] Hörmander, L.: Generators for some rings of analytic functions. Bull. Am. Math. Soc.73, 943–949 (1967) · Zbl 0172.41701
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