Analysis & PDE

Rademacher functions in Nakano spaces

Sergey Astashkin and Mieczysław Mastyło

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Abstract

The closed span of Rademacher functions is investigated in Nakano spaces Lp() on [0,1] equipped with the Lebesgue measure. The main result of this paper states that under some conditions on distribution of the exponent function p the Rademacher functions form in Lp() a basic sequence equivalent to the unit vector basis in 2.

Article information

Source
Anal. PDE, Volume 9, Number 1 (2016), 1-14.

Dates
Received: 27 March 2015
Revised: 24 July 2015
Accepted: 7 September 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843201

Digital Object Identifier
doi:10.2140/apde.2016.9.1

Mathematical Reviews number (MathSciNet)
MR3461299

Zentralblatt MATH identifier
1347.46009

Subjects
Primary: 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Secondary: 46B20: Geometry and structure of normed linear spaces 46B42: Banach lattices [See also 46A40, 46B40]

Keywords
Rademacher functions Nakano spaces symmetric spaces

Citation

Astashkin, Sergey; Mastyło, Mieczysław. Rademacher functions in Nakano spaces. Anal. PDE 9 (2016), no. 1, 1--14. doi:10.2140/apde.2016.9.1. https://projecteuclid.org/euclid.apde/1510843201


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References

  • S. V. Astashkin, “Rademacher functions in symmetric spaces”, Sovrem. Mat. Fundam. Napravl. 32 (2009), 3–161. In Russian; translated in J. Math. Sci. $($N.Y.$)$ 169:6 (2010), 725–886.
  • S. V. Astashkin and L. Maligranda, “Rademacher functions in Cesàro type spaces”, Studia Math. 198:3 (2010), 235–247.
  • S. V. Astashkin and E. M. Semënov, “\cyr Prostranstva, opredelyaemye funktsieĭ Pe1li”, Mat. Sb. 204:7 (2013), 3–24. Translated as “Spaces defined by the Paley function” in Sbornik Math. 204:7 (2013), 937–957.
  • S. V. Astashkin, M. V. Leibov, and L. Maligranda, “Rademacher functions in BMO”, Studia Math. 205:1 (2011), 83–100.
  • C. Bennett and R. Sharpley, Interpolation of operators, Pure and Applied Mathematics 129, Academic Press, Boston, 1988.
  • R. Blei, Analysis in integer and fractional dimensions, Cambridge Studies in Advanced Mathematics 71, Cambridge University Press, 2001.
  • D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue spaces: foundations and harmonic analysis, Birkhäuser, Heidelberg, 2013.
  • D. V. Cruz-Uribe, A. Fiorenza, M. V. Ruzhansky, and J. Wirth, Variable Lebesgue spaces and hyperbolic systems (Barcelona, 2011), edited by S. Tikhonov, Birkhäuser, Basel, 2014.
  • L. Diening, P. Harjulehto, P. Häst ö, and M. R\ružička, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics 2017, Springer, Heidelberg, 2011.
  • J. Diestel, H. Jarchow, and A. Tonge, Absolutely summing operators, Cambridge Studies in Advanced Mathematics 43, Cambridge University Press, 1995.
  • A. Fiorenza and J. M. Rakotoson, “Relative rearrangement and Lebesgue spaces $L\sp {p(\cdot)}$ with variable exponent”, J. Math. Pures Appl. $(9)$ 88:6 (2007), 506–521.
  • W. B. Johnson, B. Maurey, G. Schechtman, and L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 19:217, American Mathematical Society, Providence, RI, 1979.
  • S. G. Kreĭn, J. \=I. Petunīn, and E. M. Semënov, Interpolation of linear operators, Translations of Mathematical Monographs 54, American Mathematical Society, Providence, RI, 1982.
  • J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, II: Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete 97, Springer, Berlin, 1979.
  • G. G. Lorentz, “On the theory of spaces $\Lambda$”, Pacific J. Math. 1 (1951), 411–429.
  • B. Maurey and G. Pisier, “Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach”, Studia Math. 58:1 (1976), 45–90.
  • J. Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics 1034, Springer, Berlin, 1983.
  • G. Pisier, Factorization of linear operators and geometry of Banach spaces, CBMS Regional Conference Series in Mathematics 60, American Mathematical Society, Providence, RI, 1986.
  • V. A. Rodin and E. M. Semënov, “Rademacher series in symmetric spaces”, Anal. Math. 1:3 (1975), 207–222.
  • V. A. Rodin and E. M. Semënov, “\cyr O dopolnyaemosti podprostranstva, porozhdennogo sistemoĭ Rademakhera, v simmetrichnom prostranstve”, Funktsional. Anal. i Prilozhen. 13:2 (1979), 91–92. Translated as “Complementability of the subspace generated by the Rademacher system in a symmetric space” in Funct. Anal. Appl. 13:2 (1979), 150–151.
  • Y. Sagher and K. C. Zhou, “A local version of a theorem of Khinchin”, pp. 327–330 in Analysis and partial differential equations, edited by C. Sadosky, Lecture Notes in Pure and Applied Mathematics 122, Dekker, New York, 1990.
  • I. I. Sharapudinov, “\cyr O bazisnosti sistemy KHaara v prostranstve ${\mathscr L}\sp {p(t)}([0,1])$ \cyr i printsipe lokalizatsii v srednem”, Mat. Sb. $($N.S.$)$ 130(172):2 (1986), 275–283. Translated as “On the basis property of the Haar system in the space ${\mathscr L}\sp {p(t)}([0,1])$ and the principle of localization in the mean” in Math. USSR Sbornik 58:1 (1987), 279–287.
  • S. J. Szarek, “On the best constants in the Khinchin inequality”, Studia Math. 58:2 (1976), 197–208.