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Sharp Extrapolation Theorems for Local Morrey Spaces

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Abstract

We show that the calculation of the sum of local Morrey spaces can be reduced to the calculation of the sum of sequence spaces that appear as parameters in the definition of local Morrey spaces. The presence of such a reduction allows us to obtain new extrapolation theorems for local Morrey spaces with sharp constants.

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References

  1. E. I. Berezhnoi, “A sharp extrapolation theorem for Lorentz spaces,” Sib. Math. J. 54 (3), 406–418 (2013) [transl. from Sib. Mat. Zh. 54 (3), 520–535 (2013)].

    Article  MathSciNet  Google Scholar 

  2. E. I. Berezhnoi, “Can Yano’s extrapolation theorem be strengthened?,” Math. Notes 49 (2), 145–147 (2015) [transl. from Mat. Zametki 49 (2), 82–85 (2015)].

    MathSciNet  MATH  Google Scholar 

  3. E. I. Berezhnoi, “A discrete version of local Morrey spaces,” Izv. Math. 81 (1), 1–28 (2017) [transl. from Izv. Ross. Akad. Nauk, Ser. Mat. 81 (1), 3–30 (2017)].

    Article  MathSciNet  Google Scholar 

  4. E. I. Berezhnoi, “Exact calculation of sums of the Lorentz spaces \(\Lambda ^\alpha \) and applications,” Math. Notes 104 (5), 628–635 (2018) [transl. from Mat. Zametki 104 (5), 649–658 (2018)].

    Article  MathSciNet  Google Scholar 

  5. E. I. Berezhnoi, “Exact calculation of sums of cones in Lorentz spaces,” Funct. Anal. Appl. 52 (2), 134–138 (2018) [transl. from Funkts. Anal. Prilozh. 52 (2), 66–71 (2018)].

    Article  MathSciNet  Google Scholar 

  6. E. I. Berezhnoi, “Extremal extrapolation spaces,” Funct. Anal. Appl. 54 (1), 1–6 (2020) [transl. from Funkts. Anal. Prilozh. 54 (1), 3–10 (2020)].

    Article  MathSciNet  Google Scholar 

  7. E. I. Berezhnoi and A. A. Perfil’ev, “A sharp extrapolation theorem for operators,” Funct. Anal. Appl. 34 (3), 211–213 (2000) [transl. from Funkts. Anal. Prilozh. 34 (3), 66–68 (2000)].

    Article  MathSciNet  Google Scholar 

  8. V. I. Burenkov, “Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces. I,” Eurasian Math. J. 3 (3), 11–32 (2012).

    MathSciNet  MATH  Google Scholar 

  9. V. I. Burenkov, “Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces. II,” Eurasian Math. J. 4 (1), 21–45 (2013).

    MathSciNet  MATH  Google Scholar 

  10. V. I. Burenkov, E. D. Nursultanov, and D. K. Chigambayeva, “Description of the interpolation spaces for a pair of local Morrey-type spaces and their generalizations,” Proc. Steklov Inst. Math. 284, 97–128 (2014) [transl. from Tr. Mat. Inst. Steklova 284, 105–137 (2014)].

    Article  MathSciNet  Google Scholar 

  11. M. J. Carro, “On the range space of Yano’s extrapolation theorem and new extrapolation estimates at infinity,” Publ. Mat., Barc., Extra vol., 27–37 (2002).

    Article  MathSciNet  Google Scholar 

  12. A. Fiorenza, B. Gupta, and P. Jain, “The maximal theorem for weighted Grand Lebesgue spaces,” Stud. Math. 188 (2), 123–133 (2008).

    Article  MathSciNet  Google Scholar 

  13. A. Fiorenza and G. E. Karadzhov, “Grand and small Lebesgue spaces and their analogs,” Z. Anal. Anwend. 23 (4), 657–681 (2004).

    Article  MathSciNet  Google Scholar 

  14. T. Iwaniec and C. Sbordone, “On the integrability of the Jacobian under minimal hypotheses,” Arch. Ration. Mech. Anal. 119 (2), 129–143 (1992).

    Article  MathSciNet  Google Scholar 

  15. B. Jawerth and M. Milman, Extrapolation Theory with Applications (Am. Math. Soc., Providence, RI, 1991), Mem. Am. Math. Soc. 89 (440).

    MATH  Google Scholar 

  16. G. E. Karadzhov and M. Milman, “Extrapolation theory: New results and applications,” J. Approx. Theory 133 (1), 38–99 (2005).

    Article  MathSciNet  Google Scholar 

  17. S. G. Kreĭn, Yu. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators (Nauka, Moscow, 1977; Am. Math. Soc, Providence, RI, 1982).

    Google Scholar 

  18. A. Kufner, L. Maligranda, and L.-E. Persson, The Hardy Inequality: About Its History and Some Related Results (Vydavatelský Servis, Pilsen, 2007).

    MATH  Google Scholar 

  19. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. I: Sequence Spaces; II: Function Spaces (Springer, Berlin, 1977, 1979), Ergebn. Math. Grenzgeb. 92, 97.

    MATH  Google Scholar 

  20. M. Milman, Extrapolation and Optimal Decompositions: With Applications to Analysis (Springer, Berlin, 1994), Lect. Nothes Math. 1580.

    Book  Google Scholar 

  21. C. B. Morrey Jr., “On the solutions of quasi-linear elliptic partial differential equations,” Trans. Am. Math. Soc. 43 (1), 126–166 (1938).

    Article  MathSciNet  Google Scholar 

  22. C. Sbordone, “Grand Sobolev spaces and their application to variational problems,” Matematiche 51 (2), 335–347 (1996).

    MathSciNet  MATH  Google Scholar 

  23. S. Yano, “Notes on Fourier analysis. XXIX: An extrapolation theorem,” J. Math. Soc. Japan 3 (2), 296–305 (1951).

    Article  MathSciNet  Google Scholar 

  24. A. Zygmund, Trigonometric Series (Univ. Press, Cambridge, 1959), Vol. 2.

    MATH  Google Scholar 

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Funding

This work was supported by the Russian Foundation for Basic Research, project no. 18-51-06005.

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Authors and Affiliations

  1. Mathematics Faculty, P. G. Demidov Yaroslavl State University, ul. Soyuznaya 144, Yaroslavl, 150008, Russia

    E. I. Berezhnoi

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  1. E. I. Berezhnoi
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Correspondence to E. I. Berezhnoi.

Additional information

Translated from Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 312, pp. 82–97 https://doi.org/10.4213/tm4126.

Translated by I. Nikitin

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Berezhnoi, E.I. Sharp Extrapolation Theorems for Local Morrey Spaces. Proc. Steklov Inst. Math. 312, 76–90 (2021). https://doi.org/10.1134/S0081543821010041

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