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Generalization of the notion of grand Lebesgue space

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Abstract

We introduce families of weighted grand Lebesgue spaces which generalize weighted grand Lebesgue spaces (known also as Iwaniec-Sbordone spaces). The generalization admits a possibility of expanding usual (weighted) Lebesgue spaces to grand spaces by various ways by means of additional functional parameter. For such generalized grand spaces we prove a theorem on the boundedness of linear operators under the information of their boundedness in ordinary weighted Lebesgue spaces. By means of this theorem we prove boundedness of the Hardy-Littlewood maximal operator and the Calderon-Zygmund singular operators in the weighted grand spaces.

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References

  1. T. Iwaniec and C. Sbordone, “On the Integrability of the Jacobian Under Minimal Hypotheses,” Arch. Rational Mech. Anal. 119, 129–143 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  2. C. Capone and A. Fiorenza, “On Small Lebesgue Spaces,” J. Function Spaces and Appl. 3, 73–89 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  3. G. Di Fratta and A. Fiorenza, “A Direct Approach to the Duality of Grand and Small Lebesgue Spaces,” Nonlinear Anal.: Theory, Methods and Appl. 70(7), 2582–2592 (2009).

    Article  MATH  Google Scholar 

  4. A. Fiorenza, “Duality and Reflexivity in Grand Lebesgue Spaces,” Collect.Math. 51(2), 131–148 (2000).

    MATH  MathSciNet  Google Scholar 

  5. A. Fiorenza, B. Gupta, and P. Jain, “The Maximal Theorem in Weighted Grand Lebesgue Spaces,” Studia Math. 188(2), 123–133 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  6. A. Fiorenza and G. E. Karadzhov, “Grand and Small Lebesgue Spaces and Their Analogs,” J. Anal. and Appl. 23(4), 657–681 (2004).

    MATH  MathSciNet  Google Scholar 

  7. A. Fiorenza and J. M. Rakotoson, “Petits Espaces de Lebesgue et Quelques Applications,” C. R. Math., Acad. Sci. Paris 334(1), 23–26 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  8. L. Greco, T. Iwaniec, and C. Sbordone, “Inverting the p-Harmonic Operator,” ManuscriptaMath. 92, 249–258 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  9. V. Kokilashvili, “Boundedness Criterion for the Cauchy Singular Integral Operator in Weighted Grand Lebesgue Spaces and Application to the Riemann Problem,” Proc. A. Razmadze Math. Inst. 151, 129–133 (2009).

    MATH  MathSciNet  Google Scholar 

  10. V. Kokilashvili, “The Riemann Boundary Value Problem for Analytic Functions in the Frame of Grand L p) Spaces,” Bull. Georgian Nat. Acad. Sci. 4(1), 5–7 (2010).

    MATH  MathSciNet  Google Scholar 

  11. V. Kokilashvili and A. Meskhi, “A Note on the Boundedness of the Hilbert Transform in Weighted Grand Lebesgue Spaces,” Georgian Math. J. 16(3), 547–551 (2009).

    MATH  MathSciNet  Google Scholar 

  12. V. Kokilashvili and S. Samko, “Boundedness ofWeighted Singular Integral Operators on a Carleson Curves in Grand Lebesgue Spaces,” in ICNAAM 2010: Intern. Conf. Numer. Anal. Appl. Math. (AIP Confer. Proc., 2010), Vol. 1281, pp. 490–493.

    Google Scholar 

  13. A. Meskhi, “Maximal Functions and Singular Integrals in Morrey Spaces Associated with Grand Lebesgue Spaces,” Proc. A. RazmadzeMath. Inst. 151, 139–143 (2009).

    MATH  MathSciNet  Google Scholar 

  14. S. G. Samko and S. M. Umarkhadzhiev, “On Iwaniec-Sbordone Spaces on Sets Which May Have Infinite Measure,” Azerb. J. Math. 1(1), 67–84 (2011).

    MATH  MathSciNet  Google Scholar 

  15. S. G. Samko and S. M. Umarkhadzhiev, “On Iwaniec-Sbordone Spaces on Sets Which May Have Infinite Measure: Addendum,” Azerb. J. Math. 1(2), 143–144 (2011).

    MATH  MathSciNet  Google Scholar 

  16. V. Kokilashvili, “Boundedness Criteria for Singular Integrals inWeighted Grand Lebesgue Spaces,” J.Math. Sci. 170(1), 20–33 (2010).

    Article  MathSciNet  Google Scholar 

  17. V. Kokilashvili and S. Samko, “Boundedness of Weighted Singular Integral Operators in Grand Lebesgue Spaces,” 18 (2), 259–269 (2011).

    Google Scholar 

  18. J. Bergh and J. Löfström, Interpolation Spaces. An Introduction (Springer Verlag, Berlin-Heidelberg-New York, 1976;Mir, Moscow, 1980).

    Book  MATH  Google Scholar 

  19. E. M. Stein and G. Weiss, “Interpolation of Operators with Change of Measures,” Trans. Amer.Math. Soc. 87, 159–172 (1958).

    Article  MATH  MathSciNet  Google Scholar 

  20. R. R. Coifman and Y. Meyer, Au dela des Opérateurs Pseudo-Differentiels (Asterisque, 1978).

    MATH  Google Scholar 

  21. B. Muckenhoupt, “Weighted Norm Inequalities for the Hardy Maximal Function,” Trans. Amer. Math. Soc. 165, 207–226 (1972).

    Article  MATH  MathSciNet  Google Scholar 

  22. J. Duoandikoetxea, Fourier Analysis (Amer. Math. Soc., Graduate Studies, 2001), Vol. 29.

    MATH  Google Scholar 

  23. A. de la Torre, “On the Adjoint of theMaximal Function,” in Proceedings of Conference ‘Function Spaces, Differential Operators and Nonlinear Analysis,’ Ed. by Rákosnik and Jiři, Paseky nad Jizerou, Czech Republic, 1995 (Prometheus, Prague, 1996), pp. 189–194.

    Google Scholar 

  24. M. Lacey, E. Sawyer, and I. Uriarte-Tuero, “Two Weight Inequalities for Discrete Positive Operators,” http://arxiv.org/abs/0911.3437, 2010.

    Google Scholar 

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Authors and Affiliations

  1. Chechen State University, ul. Kievskaya 33, Grozny, 364037, Russia

    S. M. Umarkhadzhiev

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  1. S. M. Umarkhadzhiev
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Correspondence to S. M. Umarkhadzhiev.

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Original Russian Text © S.M. Umarkhadzhiev, 2014, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 4, pp. 18–24.

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Umarkhadzhiev, S.M. Generalization of the notion of grand Lebesgue space. Russ Math. 58, 35–43 (2014). https://doi.org/10.3103/S1066369X14040057

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