Advertisement

Positivity

, Volume 22, Issue 1, pp 63–81 | Cite as

Local rearrangement invariant spaces and distribution of Rademacher series

  • Javier Carrillo-Alanís
Article

Abstract

We prove that a local version of Khintchine inequality holds for arbitrary rearrangement invariant (r.i.) spaces on an non-empty open set \(E\subset [0,1]\). For this, we give a definition of local r.i. space which is compatible with the notion of systems equivalent in distribution and prove that the Rademacher system \((r_{k+N})_{k=1}^\infty \) on an non-empty open set E is equivalent in distribution to \((r_k)_{k=1}^\infty \) on [0, 1], with N depending on E. The result can be generalized to a wider class of sets.

Keywords

Function spaces Rademacher series Rearrangement invariant spaces Distribution function 

Mathematics Subject Classification

46E30 60E99 

Notes

Acknowledgements

This work is part of the Ph.D. thesis of the author which is being prepared at University of Sevilla under the supervision of Prof. G. P. Curbera.

References

  1. 1.
    Kolmogorov, A., Khintchine, A.: Über konvergenz von reihen, deren glieder durch den zufall bestimmt werden. Mat. Sb., 32, 668–677Google Scholar
  2. 2.
    Astashkin, S.: Systems of random variables equivalent in distribution to the Rademacher systems, and the \({K}\)-closed representability of Banach pairs. Mat. Sb. 191(6), 3–30 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Astashkin, S.: About interpolation of subspaces of rearrangement invariant spaces generated by Rademacher system. Int. J. Math. Math. Sci. 25(7), 451–465 (2001)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Astashkin, S.: Rademacher functions in symmetric spaces. Sovrem. Mat. Fundam. Napravl. 32, 3–161 (2009)MathSciNetGoogle Scholar
  5. 5.
    Astashkin, S., Curbera, G.P.: Local Khintchine inequality in rearrangement invariant spaces. Ann. Mat. Pura Appl. (4) 194(3), 619–643 (2015)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Astashkin, S.V.: Khintchine inequality for sets of small measure. Funct. Anal. Appl. 48(4), 235–241 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Pure and Applied Mathematics, vol. 129. Academic Press Inc, Boston (1988)MATHGoogle Scholar
  8. 8.
    Carrillo-Alanís, J.: On local Khintchine inequalities for spaces of exponential integrability. Proc. Am. Math. Soc. 139(8), 2753–2757 (2011)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hitczenko, P.: Domination inequality for martingale transforms of a Rademacher sequence. Isr. J. Math. 84(1–2), 161–178 (1993)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces. Lecture Notes in Mathematics, vol. 338. Springer, Berlin (1973)MATHGoogle Scholar
  11. 11.
    Rademacher, H.: Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen. Math. Ann. 87(1–2), 112–138 (1922)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Semenov, E.M., Rodin, V.A.: Rademacher series in symmetric spaces. Anal. Math. 1(3), 207–222 (1975)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Zhou, K.C., Sagher, Y.: A local version of a theorem of Khinchin. In: Sadosky C (ed.) Analysis and Partial Differential Equations, volume 122 of Lecture Notes in Pure and Applied Mathematics, pp. 327–330. Dekker, New York (1990)Google Scholar
  14. 14.
    Zhou, K.C., Sagher, Y.: Exponential integrability of Rademacher series. In: Bergelson V, March P, Rosenblatt J (eds.) Convergence in Ergodic Theory and Probability (Columbus, OH, 1993), volume 5 of Ohio State University Mathematics Research Institute Publications, pp. 389–395. de Gruyter, Berlin (1996)Google Scholar
  15. 15.
    Zygmund, A.: Trigonometric Series, vol. I, II. Cambridge University Press, Cambridge (1977)MATHGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Universidad de Sevilla SevilleSevilleSpain

Personalised recommendations