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Boundary-Nonregular Functions in the Disc Algebra and in Holomorphic Lipschitz Spaces

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Abstract

We prove in this paper the existence of dense linear subspaces in the classical holomorphic Lipschitz spaces in the disc all of whose non-null functions are nowhere differentiable at the boundary. Infinitely generated free algebras as well as infinite dimensional Banach spaces consisting of Lipschitz functions enjoying the mentioned property almost everywhere on the boundary are also exhibited. It is also investigated the algebraic size of the family of functions in the disc algebra that either do not preserve Borel sets on the unit circle or possess the Cantor boundary behavior on the disc.

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Authors and Affiliations

  1. Departamento de Análisis Matemático, Facultad de Matemáticas Instituto de Matemáticas Antonio de Castro Brzezicki (IMUS), Universidad de Sevilla, Avenida Reina Mercedes, Sevilla, 41080, Spain

    L. Bernal-González

  2. Departamento de Análisis Matemático, Universidad de la Laguna, La Laguna (Tenerife), 38271, Spain

    A. Bonilla

  3. Departamento de Matemática Aplicada a las Tecnologías de la Información y las Comunicaciones Escuela Técnica Superior de Ingeniería y Sistemas de Telecomunicación, Universidad Politécnica de Madrid, Carretera de Valencia, Km. 7., Madrid, 28031, Spain

    J. López-Salazar

  4. Instituto de Matemática Interdisciplinar (IMI) Departamento de Análisis y Matemática Aplicada, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, Madrid, 28040, Spain

    J. B. Seoane-Sepúlveda

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  1. L. Bernal-González
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  2. A. Bonilla
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  3. J. López-Salazar
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  4. J. B. Seoane-Sepúlveda
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Correspondence to J. López-Salazar.

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Bernal-González, L., Bonilla, A., López-Salazar, J. et al. Boundary-Nonregular Functions in the Disc Algebra and in Holomorphic Lipschitz Spaces. Mediterr. J. Math. 15, 114 (2018). https://doi.org/10.1007/s00009-018-1160-6

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  • DOI: https://doi.org/10.1007/s00009-018-1160-6

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