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Complex interpolation of grand Lebesgue spaces

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The aim of this paper is to obtain the description of the first and second complex interpolations of grand Lebesgue spaces. We also investigate the complex interpolation of closed subspaces satisfying the lattice property.

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References

  1. Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Springer, Berlin (1976)

  2. Calderón, A.P.: Intermediate spaces and interpolation, the complex method. Stud. Math. 24, 113–190 (1964)

    MathSciNet  MATH  Google Scholar 

  3. Fiorenza, A.: Duality and reflexivity in grand Lebesgue spaces. Collect. Math. 51, 131–148 (2000)

    MathSciNet  MATH  Google Scholar 

  4. Greco, L., Iwaniec, T., Sbordone, C.: Inverting the \(p\)-harmonic operator. Manuscr. Math. 92, 249–258 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fiorenza, A., Gupta, B., Jain, P.: The maximal theorem in weighted grand Lebesgue spaces. Stud. Math. 188(2), 123–133 (2008)

    Article  MATH  Google Scholar 

  6. Hakim, D.I., Sawano, Y.: Interpolation of generalized Morrey spaces. Rev. Math. Complut. 29(2), 295–340 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hakim, D.I., Sawano, Y.: Calderón’s First and Second Complex Interpolations of Closed Subspaces of Morrey Spaces. J. Fourier Anal. Appl. (2016). doi:10.1007/s00041-016-9503-9

  8. Iwaniec, T., Sbordone, C.: On the integrability of the Jacobian under minimal hypotheses. Arch. Rational Mech. Anal. 119(2), 129–143 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kokilashvili, V., Meskhi, A., Rafeiro, H., Samko, S.: Integral Operators in Non-standard Function Spaces: Variable Exponent Hölder, Morrey–Campanato and Grand Spaces, vol. 2. Springer, Heidelberg (2016)

    MATH  Google Scholar 

  10. Kozono, H., Yamazaki, M.: Semilinear heat equations and the Navier–Stokes equation with distributions in new function spaces as initial data. Commun. PDE 19, 959–1014 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lemarié-Rieusset, P.G.: Multipliers and Morrey spaces. Potential Anal. 38, 741–752 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lemarié-Rieusset, P.G.: Erratum to: Multipliers and Morrey spaces. Potential Anal. 41, 1359–1362 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lu, Y., Yang, D., Yuan, W.: Interpolation of Morrey spaces on metric measure spaces. Canad. Math. Bull. 57, 598–608 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mazzucato, A.L.: Decomposition of Besov–Morrey spaces. Harmonic analysis at Mount Holyoke (South Hadley, MA, 2001), 279–294, Contemp. Math. 320, Am. Math. Soc., Providence, RI (2003)

  15. Mazzucato, A.L.: Besov–Morrey spaces: function space theory and applications to non-linear PDE. Trans. Am. Math. Soc. 355(4), 1297–1364 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Najafov, A.M.: Some properties of functions from the intersection of Besov–Morrey type spaces with dominant mixed derivatives. Proc. A. Razmadze Math. Inst. 139, 71–82 (2005)

    MathSciNet  MATH  Google Scholar 

  17. Sawano, Y., Tanaka, H.: Decompositions of Besov–Morrey spaces and Triebel–Lizorkin–Morrey spaces. Math. Z. 257(4), 871–905 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sawano, Y., Tanaka, H.: Besov–Morrey spaces and Triebel–Lizorkin–Morrey spaces for non-doubling measures. Math. Nachr. 282(12), 1788–1810 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sawano, Y., Tanaka, H.: The Fatou property of block spaces. J. Math. Sci. Univ. Tokyo 22, 663–683 (2015)

    MathSciNet  MATH  Google Scholar 

  20. Sestakov, V.A.: On complex interpolation of Banach space of measurable functions. Vestn. Leningr. 19(4), 64–68 (1974)

    MathSciNet  Google Scholar 

  21. Tang, L., Xu, J.: Some properties of Morrey type Besov–Triebel spaces. Math. Nachr. 278, 904–917 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Triebel, H.: Hybrid Function Spaces, Heat and Navier–Stokes Equations. Tracts in Mathematics 24, European Mathematical Society (2015)

  23. Yang, D., Yuan, W.: A new class of function spaces connecting Triebel–Lizorkin spaces and \(Q\) spaces. J. Funct. Anal. 255, 2760–2809 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yang, D., Yuan, W.: New Besov-type spaces and Triebel–Lizorkin-type spaces including \(Q\) spaces. Math. Z. 265, 451–480 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yang, D., Yuan, W.: Dual properties of Triebel–Lizorkin-type spaces and their applications. Z. Anal. Anwend. 30, 29–58 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer, Berlin (2010)

  27. Yuan, W., Sickel, W., Yang, D.: Interpolation of Morrey–Campanato and related smoothness spaces. Sci. China Math. 58(9), 1835–1908 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji, Tokyo, 192-0397, Japan

    Denny Ivanal Hakim & Yoshihiro Sawano

  2. Department of Mathematics Education, Faculty of Education, Okayama University, 3-1-1, Tsushimanaka, Okayama, 700-8530, Japan

    Mitsuo Izuki

Authors
  1. Denny Ivanal Hakim
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  2. Mitsuo Izuki
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  3. Yoshihiro Sawano
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Corresponding author

Correspondence to Denny Ivanal Hakim.

Additional information

Communicated by A. Constantin.

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Hakim, D.I., Izuki, M. & Sawano, Y. Complex interpolation of grand Lebesgue spaces. Monatsh Math 184, 245–272 (2017). https://doi.org/10.1007/s00605-017-1022-5

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