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

  • L. Bernal-González
  • A. Bonilla
  • J. López-Salazar
  • J. B. Seoane-Sepúlveda
Article

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.

Keywords

Disc algebra Nowhere differentiable function \(\alpha \)-lipschitzian function Lineability Spaceability Algebrability 

Mathematics Subject Classification

Primary 30H50 Secondary 15A03 26A16 26A27 46E10 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Análisis Matemático, Facultad de Matemáticas Instituto de Matemáticas Antonio de Castro Brzezicki (IMUS)Universidad de SevillaSevillaSpain
  2. 2.Departamento de Análisis MatemáticoUniversidad de la LagunaLa Laguna (Tenerife)Spain
  3. 3.Departamento de Matemática Aplicada a las Tecnologías de la Información y las Comunicaciones Escuela Técnica Superior de Ingeniería y Sistemas de TelecomunicaciónUniversidad Politécnica de MadridMadridSpain
  4. 4.Instituto de Matemática Interdisciplinar (IMI) Departamento de Análisis y Matemática Aplicada, Facultad de Ciencias MatemáticasUniversidad Complutense de MadridMadridSpain

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