The problem of ideals of \(H^{\infty }\): beyond the exponent 3/2. (English) Zbl 1154.46028
Let \(\psi:[0,\infty[\to [0,\infty[\) be a bounded, decreasing function with \(\int_0^\infty {\psi(x)\,dx<\infty}\) and let \(\varphi(s)=s^2\psi(\log s^{-2})\). Suppose that \(f, f_1,\dots, f_n\) are bounded holomorphic functions in the unit disk \(\mathbb D\) satisfying \(| f| \leq \varphi(\sum_{j=1}^n | f_j| )\). Then the author shows that \(f\) belongs to the ideal generated by the \(f_j\).
This generalizes a recent result of J.Pau [Proc.Am.Math.Soc.133, No.1, 167–174 (2005; Zbl 1048.30026)] and earlier results by U.Cegrell [Math.Z.203, No.2, 255–261 (1990; Zbl 0702.30038), Proc.R.Ir.Acad., Sect.A 94, No.1, 25–30 (1994; Zbl 0818.30023)]. Admissible functions are, e.g., \[ \varphi(s)= s^2\biggl/ \biggl( (\log s^{-2})(\log\log s^{-2})\dots (\underbrace{\log\log\dots\log s^{-2}}_{m\text{ times}}) (\underbrace{\log\log\dots\log s^{-2}}_{m+1\text{ times}})^{1+\varepsilon}\biggr),\;\varepsilon>0. \] Actually, the author proves such a result for countably many data.
This generalizes a recent result of J.Pau [Proc.Am.Math.Soc.133, No.1, 167–174 (2005; Zbl 1048.30026)] and earlier results by U.Cegrell [Math.Z.203, No.2, 255–261 (1990; Zbl 0702.30038), Proc.R.Ir.Acad., Sect.A 94, No.1, 25–30 (1994; Zbl 0818.30023)]. Admissible functions are, e.g., \[ \varphi(s)= s^2\biggl/ \biggl( (\log s^{-2})(\log\log s^{-2})\dots (\underbrace{\log\log\dots\log s^{-2}}_{m\text{ times}}) (\underbrace{\log\log\dots\log s^{-2}}_{m+1\text{ times}})^{1+\varepsilon}\biggr),\;\varepsilon>0. \] Actually, the author proves such a result for countably many data.
Reviewer: Raymond Mortini (Metz)
MSC:
| 46J15 | Banach algebras of differentiable or analytic functions, \(H^p\)-spaces |
| 30H05 | Spaces of bounded analytic functions of one complex variable |
| 46J20 | Ideals, maximal ideals, boundaries |
References:
| [1] | Andersson, M., The corona theorem for matrices, Math. Z., 201, 121-130 (1989) · Zbl 0648.30039 |
| [2] | Berndtsson, Bo, \( \overline{\partial}_b\) and Carleson type inequalities, (Complex Analysis, II. Complex Analysis, II, College Park, MD, 1985-1986. Complex Analysis, II. Complex Analysis, II, College Park, MD, 1985-1986, Lecture Notes in Math., vol. 1276 (1987), Springer-Verlag: Springer-Verlag Berlin), 42-54 · Zbl 0627.32014 |
| [3] | Carleson, L., Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2), 76, 547-559 (1962) · Zbl 0112.29702 |
| [4] | Cegrell, U., A generalization of the corona theorem in the unit disc, Math. Z., 203, 2, 255-261 (1990) · Zbl 0702.30038 |
| [5] | Cegrell, U., Generalisations of the corona theorem in the unit disc, Proc. Roy. Irish Acad. Sect. A, 94, 1, 25-30 (1994) · Zbl 0818.30023 |
| [6] | Garnett, J. B., Bounded Analytic Functions, Pure Appl. Math., vol. 96 (1981), Academic Press Inc. [Harcourt Brace Jovanovich Publishers]: Academic Press Inc. [Harcourt Brace Jovanovich Publishers] New York · Zbl 0469.30024 |
| [7] | Helson, H., Lectures on Invariant Subspaces (1964), Academic Press: Academic Press New York · Zbl 0119.11303 |
| [8] | Lin, Kai-Ching, On the constants in the corona theorem and the ideals of \(H^\infty \), Houston J. Math., 19, 1, 97-106 (1993) · Zbl 0780.30036 |
| [9] | Sz.-Nagy, B.; Foias˛, C., Harmonic Analysis of Operators on Hilbert Space (1970), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam, (translated from the French and revised) · Zbl 0201.45003 |
| [10] | Nikolski, N. K., Operators, Functions, and Systems: An Easy Reading, vol. 1, Hardy, Hankel, and Toeplitz, Math. Surveys Monogr., vol. 92 (2002), American Mathematical Society: American Mathematical Society Providence, RI, (translated from the French by Andreas Hartmann) · Zbl 1007.47001 |
| [11] | Nikolski, N. K., Treatise on the Shift Operator, Grundlehren Math. Wiss., vol. 273 (1986), Springer-Verlag: Springer-Verlag Berlin, (Spectral function theory, with an appendix by S.V. Hruščev [S.V. Khrushchëv] and V.V. Peller, translated from the Russian by Jaak Peetre) · Zbl 0587.47036 |
| [12] | Pau, J., On a generalized corona problem on the unit disc, Proc. Amer. Math. Soc., 133, 1, 167-174 (2005), (electronic) · Zbl 1048.30026 |
| [13] | Rosenblum, M., A corona theorem for countably many functions, Integral Equations Operator Theory, 3, 1, 125-137 (1980) · Zbl 0452.46032 |
| [14] | Rajeswara Rao, K. V., On a generalized corona problem, J. Anal. Math., 18, 277-278 (1967) · Zbl 0176.44103 |
| [15] | Tolokonnikov, V. A., Estimates in the Carleson corona theorem, ideals of the algebra \(H^\infty \), a problem of Sz.-Nagy, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 113, 178-198 (1981), 267 (Investigations on linear operators and the theory of functions, XI) · Zbl 0472.46024 |
| [16] | Treil, S., Estimates in the corona theorem and ideals of \(H^\infty \): A problem of T. Wolff, J. Anal. Math., 87, 481-495 (2002), (dedicated to the memory of Thomas H. Wolff) · Zbl 1044.46043 |
| [17] | Treil, S.; Volberg, A., A fixed point approach to Nehari’s problem and its applications, (Oper. Theory Adv. Appl., vol. 71 (1994)), 165-186 · Zbl 0818.47026 |
| [18] | Treil, S.; Wick, B., Analytic projections, corona problem and geometry of holomorphic vector bundles, preprint, arXiv: · Zbl 1206.30076 |
| [19] | Trent, T., A new estimate for the vector valued corona problem, J. Funct. Anal., 189, 1, 267-282 (2002) · Zbl 1004.30026 |
| [20] | Trent, T., An estimate for ideals in \(H^\infty(D)\), Integral Equations Operator Theory, 53, 4, 573-587 (2005) · Zbl 1083.30040 |
| [21] | A. Uchiyama, Corona theorems for countably many functions and estimates for their solution, preprint, UCLA, 1981; A. Uchiyama, Corona theorems for countably many functions and estimates for their solution, preprint, UCLA, 1981 |
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