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

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


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.


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

Mathematics Subject Classification

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


  1. 1.
    Albuquerque, N., Bernal-González, L., Pellegrino, D., Seoane-Sepúlveda, J.B.: Peano curves on topological vector spaces. Linear Algebra Appl. 460, 81–96 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aron, R., Bernal-González, L., Pellegrino, D., Seoane-Sepúlveda, J.B.: Lineability: The search for linearity in Mathematics, Monographs and Research Notes in Mathematics. Chapman & Hall/CRC, Boca Raton (2016)zbMATHGoogle Scholar
  3. 3.
    Aron, R.M., García-Pacheco, F.J., Pérez-García, D., Seoane-Sepúlveda, J.B.: On dense-lineability of sets of functions on \(\mathbb{R}\). Topology 48(2–4), 149–156 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Aron, R.M., Gurariy, V.I., Seoane-Sepúlveda, J.B.: Lineability and spaceability of sets of functions on \(\mathbb{R}\). Proc. Am. Math. Soc. 133(3), 795–803 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Aron, R.M., Pérez-García, D., Seoane-Sepúlveda, J.B.: Algebrability of the set of nonconvergent Fourier series. Studia Math. 175(1), 83–90 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Balcerzak, M., Bartoszewicz, A., Filipczak, M.: Nonseparable spaceability and strong algebrability of sets of continuous singular functions. J. Math. Anal. Appl. 407(2), 263–269 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Banach, S.: Über die Bairesche Kategorie gewisser Funktionenmengen. Studia Math. 3, 174–179 (1931)CrossRefzbMATHGoogle Scholar
  8. 8.
    Baranski, K.: On some lacunary power series. Michigan Math. J. 54(1), 65–79 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bartoszewicz, A., Gła̧b, S.: Large function algebras with certain topological properties. J. Funct. Spaces (2015) (Article ID 761924, 7 pages)Google Scholar
  10. 10.
    Bastin, F., Conejero, J.A., Esser, C., Seoane-Sepúlveda, J.B.: Algebrability and nowhere Gevrey differentiability. Israel J. Math. 205(1), 127–143 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bayart, F., Quarta, L.: Algebras in sets of queer functions. Israel J. Math. 158(1), 285–296 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bernal-González, L.: Dense-lineability in spaces of continuous functions. Proc. Am. Math. Soc. 136(9), 3163–3169 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bernal-González, L.: Vector spaces of non-extendable holomorphic functions. J. Anal. Math. 134, 769–786 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bernal-González, L., Bonilla, A., López-Salazar, J., Seoane-Sepúlveda, J.B.: Nowhere hölderian functions and Pringsheim singular functions in the disc algebra. Monatsh Math. (2018)Google Scholar
  15. 15.
    Bernal-González, L., Calderón-Moreno, M.C., Prado-Bassas, J.A.: The set of space-filling curves: topological and algebraic structures. Linear Algebra Appl. 467, 57–74 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bernal-González, L., López-Salazar, J., Seoane-Sepúlveda, J.B.: On Weierstrass’ monsters in the disc algebra. Bull. Belg. Math. Soc. Simon Stevin 25 (2018)Google Scholar
  17. 17.
    Bernal-González, L., Ordóñez Cabrera, M.: Lineability criteria, with applications. J. Funct. Anal. 266(6), 3997–4025 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bernal-González, L., Pellegrino, D., Seoane-Sepúlveda, J.B.: Linear subsets of nonlinear sets in topological vector spaces. Bull. Amer. Math. Soc. (N.S.) 51(1), 71–130 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Cantón, A., Granados, A., Pommerenke, Ch.: Borel images and analytic functions. Michigan Math. J. 52(2), 279–287 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Cariello, Daniel, Seoane-Sepúlveda, Juan B.: Basic sequences and spaceability in \(\ell _p\) spaces. J. Funct. Anal. 266(6), 3797–3814 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Dong, X.H., Lau, K.S., Liu, J.C.: Cantor boundary behavior of analytic functions. Adv. Math. 232, 543–570 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Duren, P.L.: Theory of \(H^p\) spaces. Dover, Mineola, New York (2000)Google Scholar
  23. 23.
    Enflo, P.H., Gurariy, V.I., Seoane-Sepúlveda, J.B.: Some results and open questions on spaceability in function spaces. Trans. Am. Math. Soc. 366(2), 611–625 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Eskenazis, A.: Topological genericity of nowhere differentiable functions in the disc algebra. Arch. Math. (Basel) 103(1), 85–92 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Eskenazis, A., Makridis, K.: Topological genericity of nowhere differentiable functions in the disc and polydisc algebras. J. Math. Anal. Appl. 420(1), 435–446 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Fonf, V.P., Gurariy, V.I., Kadets, M.I.: An infinite dimensional subspace of \(C[0,1]\) consisting of nowhere differentiable functions. C. R. Acad. Bulgare Sci. 52(11–12), 13–16 (1999)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Gámez, J.L., Muñoz-Fernández, G.A., Seoane-Sepúlveda, J.B.: Lineability and additivity in \(\mathbb{R}^{\mathbb{R}}\). J. Math. Anal. Appl. 369(1), 265–272 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Gámez-Merino, J.L., Seoane-Sepúlveda, J.B.: An undecidable case of lineability in \(\mathbb{R}^{\mathbb{R}}\). J. Math. Anal. Appl. 401(2), 959–962 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    García, D., Grecu, B.C., Maestre, M., Seoane-Sepúlveda, J.B.: Infinite dimensional Banach spaces of functions with nonlinear properties. Math. Nachr. 283(5), 712–720 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Girela, D., González, C.: Some results on mean Lipschitz spaces of analytic functions. Rocky Mt. J. Math. 30(3), 901–922 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Gurariĭ, V.I.: Linear spaces composed of everywhere nondifferentiable functions. C. R. Acad. Bulgare Sci. 44(5), 13–16 (1991). (in Russian)MathSciNetGoogle Scholar
  32. 32.
    Hencl, S.: Isometrical embeddings of separable Banach spaces into the set of nowhere approximatively differentiable and nowhere Hölder functions. Proc. Am. Math. Soc. 128(12), 3505–3511 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Horváth, J.: Topological Vector Spaces and Distributions, vol. I. Addison-Wesley, Reading (1966)zbMATHGoogle Scholar
  34. 34.
    Jiménez-Rodríguez, P., Muñoz-Fernández, G.A., Seoane-Sepúlveda, J.B.: On Weierstrass’ Monsters and lineability. Bull. Belg. Math. Soc. Simon Stevin 20(4), 577–586 (2013)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Kavvadias, K., Makridis, K.: Nowhere differentiable functions with respect to the position, arXiv:1701.04875v1 [math.CV], Preprint (2017)
  36. 36.
    Kitson, D., Timoney, R.M.: Operator ranges and spaceability. J. Math. Anal. Appl. 378(2), 680–686 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Liu, J.C., Dong, X.H., Pen, S.M.: A note on Cantor boundary behavior. J. Math. Anal. Appl. 408, 795–801 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Lusky, W.: On weighted spaces of harmonic and holomorphic functions. J. London Math. Soc. 51, 309–320 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Mazurkiewicz, S.: Sur les fonctions non dérivables. Studia Math. 3, 92–94 (1931)CrossRefzbMATHGoogle Scholar
  40. 40.
    O’Farrell, Anthony G.: Hausdorff content and rational approximation in fractional Lipschitz norms. Trans. Am. Math. Soc. 228, 187–206 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Oxtoby, J.C.: Measure and Category, 2nd edn. Springer, New York (1980)CrossRefzbMATHGoogle Scholar
  42. 42.
    Pavlovic, Miroslav: Function Classes on the Unit Disc. An Introduction. Walter de Gruyter, Berlin (2014)zbMATHGoogle Scholar
  43. 43.
    Rodríguez-Piazza, L.: Every separable Banach space is isometric to a space of continuous nowhere differentiable functions. Proc. Am. Math. Soc. 123(12), 3649–3654 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Rudin, W.: Real and complex analysis, 3rd edn. McGraw-Hill Book Co., New York (1987)zbMATHGoogle Scholar
  45. 45.
    Sagan, H.: Space-filling Curves, Universitext. Springer, New York (1994)CrossRefzbMATHGoogle Scholar
  46. 46.
    Seoane-Sepúlveda, J.B.: Chaos and lineability of pathological phenomena in analysis, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)–Kent State UniversityGoogle Scholar
  47. 47.
    Thim, J.: Continuous nowhere differentiable functions, LuleåUniversity of Technology, 2003. Master ThesisGoogle Scholar
  48. 48.
    Van Rooij, A.C.M., Schikhof, W.H.: A Second Course on Real Functions. Cambridge University Press, Cambridge (1982)CrossRefzbMATHGoogle Scholar

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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