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Regular Poles and β-Numbers in the Theory of Holomorphic Semigroups

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Abstract

We introduce the notion of regular (boundary) poles for infinitesimal generators of semigroups of holomorphic self-maps of the unit disc. We characterize such regular poles in terms of β-points (i.e., pre-images of values with positive Carleson–Makarov β-numbers) of the associated semigroup and of the associated Königs intertwining function. We also define a natural duality operation in the cone of infinitesimal generators and show that the regular poles of an infinitesimal generator correspond to the regular null poles of the dual generator. Finally we apply such a construction to study radial multi-slits and give an example of a nonisolated radial slit whose tip does not have not a positive Carleson–Makarov β-number.

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Acknowledgements

This research was partially supported by the Ministerio de Ciencia e Innovación and the European Union (FEDER), project MTM2009-14694-C02-02, by the ESF Networking Programme “Harmonic and Complex Analysis and its Applications”, by La Consejería de Ecónomía, Innovación y Ciencia de la Junta de Andalucía (research group FQM-133) and by the ERC grant “HEVO—Holomorphic Evolution Equations” No. 277691.

This work started while the first and third named authors were visiting the Mittag-Leffler Institute, during the program “Complex Analysis and Integrable Systems” in Fall 2011, and it was completed while the first named author was visiting the Departamento de Matemática Aplicada II in Seville. The authors thank both the Mittag-Leffler Institute and the University of Seville for the kind hospitality and the atmosphere experienced there. The authors want to thank the referee for his/her comments and remarks which improved the original manuscript.

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Authors and Affiliations

  1. Dipartimento Di Matematica, Università di Roma “Tor Vergata”, Via Della Ricerca Scientifica 1, 00133, Roma, Italy

    Filippo Bracci

  2. Camino de los Descubrimientos, s/n, Departamento de Matemática Aplicada II, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, 41092, Sevilla, Spain

    Manuel D. Contreras & Santiago Díaz-Madrigal

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  1. Filippo Bracci
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  2. Manuel D. Contreras
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  3. Santiago Díaz-Madrigal
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Correspondence to Filippo Bracci.

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Communicated by Stephan Ruscheweyh.

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Bracci, F., Contreras, M.D. & Díaz-Madrigal, S. Regular Poles and β-Numbers in the Theory of Holomorphic Semigroups. Constr Approx 37, 357–381 (2013). https://doi.org/10.1007/s00365-013-9180-8

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  • DOI: https://doi.org/10.1007/s00365-013-9180-8

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