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The generalized corona theorem. (English) Zbl 0834.46034
This is another beautiful paper on the ideal theory of \(H^\infty\) by one of the leading experts (R.M.) and colleagues. \(H^\infty\) is the ring of bounded holomorphic functions in the open unit disk \(\mathbb{D}\). For \(f_1,\dots, f_N\in H^\infty\), \(J:= \{f\in H^\infty: |f|\leq C_f \sum^N_{j= 1} |f_j|\), for some finite constant \(C_f\}\) is an ideal that contains the ideal \(I\) generated by \(f_1,\dots, f_N\), and simple examples show that proper containment can occur. L. Carleson’s famous Corona Theorem asserts that equality occurs if \(J= H^\infty\). It is proved that if \(N= 2\) and \(f_1\), \(f_2\) have no common non-invertible \(H^\infty\)-factor, then \(I= J\) if and only if either \(I\) or \(J\) contains an interpolating Blaschke product. [Earlier work of V. Tolokonnikov is relevant, especially J. Soc. Math. 27, 2549-2553 (1984; Zbl 0546.46046).] But for \(N= 3\) an (easy) example is constructed of \(I= J\) without this ideal containing an interpolating Blaschke product. Denote by \(M\) the maximal ideal space of \(H^\infty\), and by \(\widehat f: M\to \mathbb{C}\) the Gelfand transform of \(f\in H^\infty\). The order of \(m\in M\) as a zero of \(\widehat f\) is defined in a natural way and the minimum of these over \(f\in I\), denoted \(\text{ord}(I, m)\), plays a significant role. Use is made of an earlier result of the second author’s [Analysis 14, No. 1, 67-73 (1984; Zbl 0810.46055)]that an ideal \(I\subsetneqq H^\infty\) is generated by interpolating Blaschke products if and only if \(\text{ord}(I, m)\in \{0, 1\}\) for every \(m\in M\). Thomas Wolff showed some time ago that \(f\in J\) implies \(f^3\in I\) always holds, and he posed the problem [cf. Lect. Notes Math. 1043 (1984; Zbl 0545.30038)] whether the stronger conclusion \(f^2\in I\) always holds as well.
In the second part of this paper, the authors provide an affirmative answer under the additional assumption that at every \(m\in M\), \(\text{ord}(\widehat f_j, m)\) is finite for at least one \(j\in \{1,\dots, N\}\); this hypothesis is (nontrivially) equivalent to the \(\widehat f_j\) not all vanishing at any one-point Gleason part in \(M\). The methods used are the standard ones developed by Wolff for proving the Corona theorem and involve Carleson measures, CN (= Carleson-Newman)- sequences and CN-Blaschke products.

MSC:
46J20 Ideals, maximal ideals, boundaries
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
30D55 \(H^p\)-classes (MSC2000)
30D50 Blaschke products, etc. (MSC2000)
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