Advertisement

Ukrainian Mathematical Journal

, Volume 57, Issue 2, pp 186–199 | Cite as

Estimate for a Rearrangement of a Function Satisfying the “Reverse Jensen Inequality”

  • A. A. Korenovskii
Article
  • 60 Downloads

Abstract

We show that an equimeasurable rearrangement of any function satisfying the “reverse Jensen inequality” with respect to various multidimensional segments also satisfies the “reverse Jensen inequality” with the same constant.

Keywords

Jensen Inequality Equimeasurable Rearrangement Reverse Jensen Inequality 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities [Russian translation], Inostrannaya Literatura, Moscow (1948).Google Scholar
  2. 2.
    F. W. Gehring, “The L p-integrability of the partial derivatives of a quasiconformal mapping,” Acta Math., 130, 265–277 (1973).Google Scholar
  3. 3.
    B. Muckenhoupt, “Weighted norm inequalities for the Hardy maximal function,” Trans. Amer. Math. Soc., 165, 207–226 (1972).Google Scholar
  4. 4.
    M. Franciosi and G. Moscariello, “Higher integrability results,” Manuscr. Math., 52, No.1–3, 151–170 (1985).CrossRefGoogle Scholar
  5. 5.
    I. Wik, “On Muckenhoupt classes of weight functions,” Dep. Math. Univ. Umea [Publ.], No. 3, 1–13 (1987).Google Scholar
  6. 6.
    I. Wik, “On Muckenhoupt classes of weight functions,” Stud. Math., 94, No.3, 245–255 (1989).Google Scholar
  7. 7.
    C. Sbordone, “Rearrangement of functions and reverse Jensen inequalities,” Proc. Symp. Pure Math., 45, No.2, 325–329 (1986).Google Scholar
  8. 8.
    A. A. Korenovskii, “On exact extension of the reverse Holder inequality and Muckenhoupt conditions,” Mat. Zametki, 52, Issue 6, 32–44 (1992).Google Scholar
  9. 9.
    A. A. Korenovskii, “Reverse Holder inequality, Muckenhoupt condition, and equimeasurable rearrangements of functions,” Dokl. Akad. Nauk SSSR, 323, No.2, 229–232 (1992).Google Scholar
  10. 10.
    N. A. Malaksiano, “On exact inclusions of Gehring classes into Muckenhoupt classes,” Mat. Zametki, 70, Issue 5, 742–750 (2001).Google Scholar
  11. 11.
    N. A. Malaksiano, “The precise embeddings of the one-dimensional Muckenhoupt classes in Gehring classes,” Acta Sci. Math. (Szeged.), 68, 237–248 (2002).Google Scholar
  12. 12.
    C. Sbordone, “Rearrangement of functions and reverse Holder inequalities,” Res. Notes Math., 125, 139–148 (1983).Google Scholar
  13. 13.
    L. d’Apuzzo and C. Sbordone, “Reverse Holder inequalities. A sharp result,” Rend. Mat., 10, Ser. VII, 357–366 (1990).Google Scholar
  14. 14.
    J. Kinnunen, “Sharp result on reverse Holder inequalities,” Ann. Acad. Sci. Fenn., Ser. A I. Math. Diss., 95, 1–34 (1994).Google Scholar
  15. 15.
    A. P. Calderon and A. Zygmund, “On the existence of certain singular integrals,” Acta Math., 88, 85–139 (1952).Google Scholar
  16. 16.
    F. Riesz, “Sur un theoreme de maximum de MM. Hardy et Littlewood,” J. London Math. Soc., 7, 10–13 (1932).Google Scholar
  17. 17.
    I. Klemes, “A mean oscillation inequality,” Proc. Amer. Math. Soc., 93, No.3, 497–500 (1985).Google Scholar
  18. 18.
    A. A. Korenovskii, “Riesz ‘rising-sun’ lemma for many variables and John-Nirenberg inequality,” Mat. Zametki, 77, No.1, 53–66 (2005).Google Scholar
  19. 19.
    A. Popoli, “Weighted reverse Holder inequalities,” Rend. Accad. Sci. Fis. Mat., 62, 187–212 (1995).Google Scholar
  20. 20.
    A. A. Korenovskyy, A. K. Lerner, and A. M. Stokolos, On Multidimensional F. Riesz’s “Rising Sun” Lemma, Preprint, Arxiv: Math. CA No. 0308211 (2003).Google Scholar
  21. 21.
    A. A. Korenovskyy, A. K. Lerner, and A. M. Stokolos, “On a multidimensional form of F. Riesz’s ‘rising sun’ lemma,” Proc. Amer. Math. Soc., 133, No.5, 1437–1440 (2005).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • A. A. Korenovskii
    • 1
  1. 1.Institute of Mathematics, Economics, and MechanicsOdessa National UniversityOdessaUkraine

Personalised recommendations