Advertisement

Distribution Function and Nonincreasing Rearrangement

  • René Erlín Castillo
  • Humberto Rafeiro
Chapter
Part of the CMS Books in Mathematics book series (CMSBM)

Abstract

In this chapter we study the distribution function which is a tool that provides information about the size of a function but not about its pointwise behavior or locality; for example, a function f and its translation are the same in terms of their distributions. Based on the distribution function we study the nonincreasing rearrangement and establish its basic properties. We obtain sub-additive and sub-multiplicative type inequalities for the decreasing rearrangement. The maximal function associated with the decreasing rearrangement is introduced and some important relations are obtained, e.g., Hardy’s inequality. In the last section of this chapter we deal with the rearrangement of the Fourier transform.

Keywords

Distribution Function Measure Space Maximal Function Lorentz Space Part Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [1]
    C. Bennett and R. Sharpley. Interpolation of operators, volume 129 of Pure and Applied Mathematics. Academic Press, Inc., Boston, MA, 1988.Google Scholar
  2. [4]
    A.-P. Calderón. Intermediate spaces and interpolation, the complex method. Studia Math., 24:113–190, 1964.MathSciNetzbMATHGoogle Scholar
  3. [7]
    K. M. Chong and N. M. Rice. Equimeasurable rearrangements of functions. Queen’s University, Kingston, Ont., 1971. Queen’s Papers in Pure and Applied Mathematics, No. 28.Google Scholar
  4. [15]
    P. W. Day. Rearrangements of measurable functions. ProQuest LLC, Ann Arbor, MI, 1970. Thesis (Ph.D.)–California Institute of Technology.Google Scholar
  5. [24]
    A. Grothendieck. Réarrangements de fonctions et inégalités de convexité dans les algèbres de von Neumann munies d’une trace. In Séminaire Bourbaki, Vol. 3, pages Exp. No. 113, 127–139. Soc. Math. France, Paris, 1995.Google Scholar
  6. [28]
    G. H. Hardy and J. E. Littlewood. A maximal theorem with function-theoretic applications. Acta Math., 54(1):81–116, 1930.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [30]
    G. H. Hardy, J. E. Littlewood, and G. Pólya. Inequalities. Cambridge, at the University Press, 1952. 2d ed.Google Scholar
  8. [43]
    G. G. Lorentz. Some new functional spaces. Ann. Math. (2), 51:37–55, 1950.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [63]
    J. V. Ryff. Measure preserving transformations and rearrangements. J. Math. Anal. Appl., 31:449–458, 1970.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [70]
    G. Sinnamon. The Fourier transform in weighted Lorentz spaces. Publ. Mat., 47(1):3–29, 2003.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • René Erlín Castillo
    • 1
  • Humberto Rafeiro
    • 2
  1. 1.Universidad Nacional de ColombiaBogotáColombia
  2. 2.Pontificia Universidad JaverianaBogotáColombia

Personalised recommendations