Orlicz spaces with a oO type structure

  • Francesca AngrisaniEmail author
  • Giacomo Ascione
  • Gianluigi Manzo


We study non reflexive Orlicz spaces \(L^\varPsi \) and their Morse subspace \(M^\varPsi \), i.e. the closure of \(L^\infty \) in \(M^\varPsi \) to determine when \((M^\varPsi ,L^\varPsi )\) can be described as having an oO type structure with respect to an equivalent norm on \(L^\varPsi \). Examples of classes of Young functions for which the answer is affirmative are provided, but also examples are given to show that this is not possible for all non-reflexive Orlicz spaces. An equivalent expression of the distance in \(L^\varPsi \) to \(M^\varPsi \), induced by the new norm, is also provided.


Orlicz space M-ideal Rearrangement Distance 

Mathematics Subject Classification




  1. 1.
    Bennett, C., Sharpley, R.C.: Interpolation of Operators, vol. 129. Academic Press, Cambridge (1988)zbMATHGoogle Scholar
  2. 2.
    Bourgain, J., Brezis, H., Mironescu, P.: A new function space and applications. J. Eur. Math. Soc. (JEMS) 6 17(9), 2083–2101 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Capone, C., Formica, M.: The distance to \({L}^\infty \) from the morrey space \({L}^{p,\lambda }\). Rendiconto della Academia delle scienze fisiche e matematiche 62, 291–299 (1995)zbMATHGoogle Scholar
  4. 4.
    Capone, C., Formica, M.R.: A decomposition of the dual space of some Banach function spaces. J. Funct. Spaces Appl. 2012, 737534-1–737534-10 (2012)Google Scholar
  5. 5.
    Carozza, M., Sbordone, C.: The distance to \({L}^\infty \) in some function spaces and applications. Differ. Integral Equ. 10(4), 599–607 (1997)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Caruso, A.: Two properties of norms in Orlicz spaces. Le Matematiche 56(1), 183–194 (2001)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chen, S.: Geometry of Orlicz spaces. Instytut Matematyczny Polskiej Akademi Nauk, Warszawa (1996)Google Scholar
  8. 8.
    Chen, S., Hudzik, H.: \({E}^\phi \) and \(h^\phi \) fail to be M-ideals in \({L}^\phi \) and \(l^\phi \) in the case of the Orlicz norm. Rendiconti del Circolo Matematico di Palermo 44(2), 283–292 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    D’Onofrio, L., Sbordone, C., Schiattarella, R.: Duality and distance formulas in Banach function spaces. J. Elliptic Parabol. Equ. 1–23 (2018).
  10. 10.
    Farroni, F., Giova, R.: Quasiconformal mappings and sharp estimates for the distance to \({L}^\infty \) in some function spaces. J. Math. Anal. Appl. 395(2), 694–704 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fusco, N., Moscariello, G., Sbordone, C.: \({BMO}\)-type seminorms and Sobolev functions. ESAIM: Control Optim. Calc. Var. 24(2), 835–847 (2018)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Harmand, P., Werner, D., Werner, W.: M-Ideals in Banach Spaces and Banach Algebras. Springer, Berlin (2006)zbMATHGoogle Scholar
  13. 13.
    Kaminska, A., Lee, H.: M-ideal properties in Marcinkiewicz spaces. Comment. Math. Prace Mat. Tomus specialis in Honorem Juliani Musielak. 123–144 (2004)Google Scholar
  14. 14.
    Kamińska, A., Lee, H.J., Tag, H.J.: \({M}\)-ideal properties in Orlicz–Lorentz spaces. arXiv preprint arXiv:1705.10451 (2017)
  15. 15.
    Kinnunen, J., Shukla, P.: The distance of \({L}^\infty \) from \({BMO}\) on metric measure spaces. Adv. Pure Appl. Math. 5(2), 117–129 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lefevre, P., Li, D., Queffélec, H., Rodríguez-Piazza, L.: A criterion of weak compactness for operators on subspaces of Orlicz spaces. J. Funct. Spaces 6(3), 277–292 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Milman, M.: Marcinkiewicz spaces, Garsia-Rodemich spaces and the scale of John–Nirenberg self improving inequalities. arXiv preprint arXiv:1508.05057 (2015)
  18. 18.
    Perfekt, K.M.: Duality and distance formulas in spaces defined by means of oscillation. Arkiv för matematik 51(2), 345–361 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Perfekt, K.M.: On M-ideals and \(o\)\({O}\) type spaces. arXiv preprint arXiv:1412.5486 (2014)
  20. 20.
    Perfekt, K.M.: Weak compactness of operators acting on o–O type spaces. Bull. Lond. Math. Soc. 47(4), 677–685 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Pick, L., Kufner, A., John, O., Fucík, S.: Function Spaces. De Gruyter Series in Nonlinear Analysis and Applications 14, vol. 1. Walter de Gruyter & Co., Berlin (2013)Google Scholar
  22. 22.
    Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. M. Dekker, New York (1991)zbMATHGoogle Scholar
  23. 23.
    Werner, D.: New classes of Banach spaces which are \({M}\)-ideals in their biduals. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 111, pp. 337–354. Cambridge University Press, Cambridge (1992)Google Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”Università degli Studi di Napoli Federico IINaplesItaly

Personalised recommendations