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Mathematical Notes

, Volume 104, Issue 5–6, pp 628–635 | Cite as

Exact Calculation of Sums of the Lorentz Spaces Λα and Applications

  • E. I. BerezhnoiEmail author
Article
  • 11 Downloads

Abstract

The norm on the sum of Lorentz spaces endowed with norms equal to the products of the classical norm by some numbers is exactly calculated. The obtained result makes it possible to prove an extrapolation theorem for collections of Lorentz, Lebesgue, and Marcinkiewicz spaces with a sharp constant.

Keywords

sum of Lorentz spaces extrapolation theorems 

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References

  1. 1.
    T. Iwaniec and C. Sbordone, “On the integrability of the Jacobian under minimal hypothesis,” Arch. Ration. Mech. Anal. 119, 129–143 (1992).CrossRefzbMATHGoogle Scholar
  2. 2.
    A. Fiorenza, “Duality and reflexivity in grand Lebesgue spaces,” Collect.Math. 51, 131–148 (2000).MathSciNetzbMATHGoogle Scholar
  3. 3.
    A. Fiorenza and G. E. Karadzhov, “Grand and small Lebesgue spaces and their analogs,” Z. Anal. Anwendungen 23 (4), 657–681 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    C. Sbordone, “Grand Sobolev spaces and their applications to variational problems,” Matematiche (Catania) 51 (2), 335–347 (1996).MathSciNetzbMATHGoogle Scholar
  5. 5.
    T. Iwaniec and C. Sbordone, “Weak minima of variational integrals,” J. Reine Angew. Math. 454, 143–161 (1994).MathSciNetzbMATHGoogle Scholar
  6. 6.
    A. Fiorenza, B. Gupta, and P. Jain, “The maximal theorem for weighted Grand Lebesgue spaces,” Studia Math. 188 (2), 123–133 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    B. Jawerth and M. Milman, “Extrapolation theory with applications,” in Mem. Amer. Math. Soc. (Amer. Math. Soc., Providence, RI, 1991), Vol. 89, No. 440, pp. 1–82.MathSciNetzbMATHGoogle Scholar
  8. 8.
    M. Milman, Extrapolation and OptimalDecomposition with Applications to Analysis, in Lecture Notes in Math. (Springer-Verlag, Berlin, 1994), Vol. 1580.Google Scholar
  9. 9.
    G. E. Karadzhov and M. Milman, “Extrapolation theory: new results and applications,” J. Approx. Theory 133, 38–99 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators (Nauka, Moscow, 1978) [in Russian].Google Scholar
  11. 11.
    C. Bennett and R. Sharpley, Interpolation Operators (Academic Press, Boston, 1988).zbMATHGoogle Scholar
  12. 12.
    E. I. Berezhnoi, “A sharp extrapolation theorem for Lorentz spaces,” Sibirsk. Mat. Zh. 54 (3), 520–535 (2013) [Sib.Math. J. 54 (3), 406–418 (2013)].MathSciNetzbMATHGoogle Scholar
  13. 13.
    E. I. Berezhnoi and A. A. Perfil’ev, “A sharp extrapolation theorem for operators,” Funktsional. Anal. Prilozhen. 34 (3), 66–68 (2000) [Functional Anal. Appl. 34 (3), 211–213 (2000)].MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    C. Capone, A. Fiorenza, and M. Krbec, “On extrapolation blowups in the L p scale,” J. Inequal. Appl. 1, 1–18 (2006).CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Demidov Yaroslavl State UniversityYaroslavlRussia

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