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Sharp Estimates of the Solutions to Bézout’s Polynomial Equation and a Corona Theorem

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Abstract

The main result of this paper is a corona theorem for the multipliers of a class of de Branges–Rovnyak spaces. A key to this involves estimates for the solutions to the classical Bézout equation that are analogous to Carleson’s solution to the corona theorem.

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Acknowledgements

We would like to thank Hervé Queffélec and Benoît Merlet for their useful comments. In particular, B. Merlet indicated a significant simplification of the proof of Theorem 1.5.

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Authors and Affiliations

  1. Laboratoire Paul Painlevé, Université de Lille, 59 655, Villeneuve d’Ascq Cédex, France

    Emmanuel Fricain

  2. Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, 33400, Talence, France

    Andreas Hartmann

  3. Department of Mathematics and Statistics, University of Richmond, Richmond, VA, 23173, USA

    William T. Ross

  4. Simion Stoilow Institute of Mathematics of the Romanian Academy, PO Box 1-764, 014700, Bucharest, Romania

    Dan Timotin

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  1. Emmanuel Fricain
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  2. Andreas Hartmann
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  4. Dan Timotin
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Correspondence to William T. Ross.

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Emmanuel Fricain was supported by Labex CEMPI (ANR-11-LABX-0007-01) and the Project FRONT (ANR-17-CE40 - 0021). Andreas Hartmann was supported by the Project REPKA (ANR-18-CE40-0035). Dan Timotin’s work was partially supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS/CCCDI – UEFISCDI, Project Number PN-III-P4-ID-PCE-2020-0458, within PNCDI III.

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Fricain, E., Hartmann, A., Ross, W.T. et al. Sharp Estimates of the Solutions to Bézout’s Polynomial Equation and a Corona Theorem. J Geom Anal 33, 256 (2023). https://doi.org/10.1007/s12220-023-01315-9

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