, Volume 22, Issue 4, pp 1143–1163 | Cite as

Extrapolation results in grand Lebesgue spaces defined on product sets

  • Vakhtang Kokilashvili
  • Alexander MeskhiEmail author


Extrapolation results in weighted grand Lebesgue spaces defined with respect to product measure \(\mu \times \nu \) on \(X\times Y\), where \((X, d, \mu )\) and \((Y, \rho , \nu )\) are spaces of homogeneous type, are obtained. As applications of the derived results we prove new one-weight estimates for multiple integral operators such as strong maximal, Calderón–Zygmund and fractional integral operators with product kernels in these spaces.


Weighted extrapolation Grand Lebesgue spaces Strong maximal operators Multiple integral operators Calderón–Zygmund operators with product kernels Fractional integrals with product kernels 

Mathematics Subject Classification

46E30 42B20 42B25 


  1. 1.
    Capone, C., Fiorenza, A.: On small Lebesgue spaces. J. Funct. Spaces Appl. 3, 73–89 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Coifman, R.R., Weiss, G.: Analyse harmonique non-commutative sur certains espaces homogénes. Lecture Notes in Math, vol. 242. Springer, Berlin (1971)CrossRefGoogle Scholar
  3. 3.
    Dragičević, O., Grafakos, L., Pereyra, M.C., Petermichl, S.: Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces. Publ. Mat. 49(1), 73–91 (2005)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Duoandikoetxea, J.: Extrapolation of weights revisited: new proofs and sharp bounds. J. Funct. Anal. 260, 1886–1901 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Edmunds, D.E., Kokilashvili, V., Meskhi, A.: Bounded and Compact Integral Operators, Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht (2002)CrossRefzbMATHGoogle Scholar
  6. 6.
    Fefferman, R., Stein, E.: Singular integrals on product spaces. Adv. Math. 45, 117–143 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fiorenza, A.: Duality and reflexivity in grand Lebesgue spaces. Collect. Math. 51(2), 131–148 (2000)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Fiorenza, A., Gupta, B., Jain, P.: The maximal theorem in weighted grand Lebesgue spaces. Stud. Math. 188(2), 123–133 (2008)CrossRefzbMATHGoogle Scholar
  9. 9.
    Garcia-Cuerva, J., Rubio de Francia, R.L.: Weighted Norm Inequalities and Related Topics, vol. 116. North Holland, Amsterdam (1985)CrossRefzbMATHGoogle Scholar
  10. 10.
    Greco, L., Iwaniec, T., Sbordone, C.: Inverting the \(p\)-harmonic operator. Manuscr. Math. 9(2), 249–258 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Harboure, E., Macías, R., Segovia, C.: Extrapolation results for classes of weights. Am. J. Math. 110, 383–397 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hoffman, S.: Weighted norm inequalities and vector-valued inequalities for certain rough operators. Indiana Univ. Math. J. 42, 1–14 (1993)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hytönen, T.P., Pérez, C., Rela, E.: Sharp reverse Hölder property for \(A_{\infty }\) weights on spaces of homogeneous type. J. Funct. Anal. 263(12), 3883–3899 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Iwaniec, T., Sbordone, C.: On the integrability of the Jacobian under minimal hypotheses. Arch. Ration. Mech. Anal. 119, 129–143 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kairema, A.: Sharp weighted bounds for fractional integral operators in a space of homogeneous type. Math. Scand. 114, 226–253 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kokilashvili, V.: Weighted Lizorkin-Triebel spaces. Singular integrals, multipliers, imbedding theorems. Studies in the Theory of Differentiable Functions of Several Variables and Its Applications, IX. Trudy Mat. Inst. Steklov. 161, 125–149 (1983) (in Russian); English translation: Proc. Steklov Inst. Mat. 161, 135–162 (1984)Google Scholar
  17. 17.
    Kokilashvili, V.: 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)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kokilashvili, V.: Singular integrals and strong fractional maximal functions in weighted grand Lebesgue spaces. In: Rákosník, J.J. (ed.) Proceedings of Non-linear Analysis, Function Spaces and Application, Třešt, , vol. 9, 11–17 September 2010, pp. 261–269. Institute of Mathematics, Academy of Sciences of the Czech Republic, Praha (2011)Google Scholar
  19. 19.
    Kokilashvili, V.: Boundedness criteria for singular integrals in weighted grand Lebesgue spaces. J. Math. Sci. (N.Y.) 170(1), 20–33 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kokilashvili, V., Meskhi, A.: A note on the boundedness of the Hilbert tranform in weighted grand Lebesgue spaces. Georgian Math. J. 16(3), 547–551 (2009)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Kokilashvili, V., Meskhi, A.: Potentials with product kernels in grand Lebesgue spaces: one-weight criteria. Lithuanian Math. J. 53(1), 27–39 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kokilashvili, V., Meskhi, A.: Weighted extrapolation in Iwaniec–Sbordone spaces. Applications to integral operators and theory of approximation. In: Proceedings of the Steklov Institute of Mathematics, vol. 293, pp. 161–185 (2016). Original Russian Text published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, vol. 293, pp. 167–192 (2016)Google Scholar
  23. 23.
    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. Birkhäuser/Springer, Heidelberg (2016)zbMATHGoogle Scholar
  24. 24.
    Kokilashvili, V., Meskhi, A., Zaighum, M.A.: Sharp weighted bounds for multiple integral operators. Trans. A. Razmadze Math. Inst. 170, 75–90 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lacey, M.T., Moen, K., Perez, C., Torres, R.H.: Sharp weighted bounds for fractional integral operators. J. Funct. Anal. 259, 1073–1097 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lique, T., Pérez, C., Rela, E.: Reverse Hölder property for strong weights and general measures. J. Geom. Anal. 27(1), 162–182 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Macías, R.A., Segovia, C.: Lipschitz functions on spaces of homogeneous type. Adv. Math. 3(3), 257–270 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Meskhi, A.: Criteria for the boundedness of potential operators in grand Lebesgue spaces. Proc. A. Razmadze Math. Inst. 169, 119–132 (2015)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Rubio de Francia, J.L.: Factorization and extrapolation of weights. Bull. Am. Math. Soc. (N.S.) 7, 393–395 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Samko, S., Umarkhadziev, S.M.: On Iwaniec-Sbordone spaces on sets which may have infinite measure. Azerb. J. Math. 1(1), 67–84 (2010)MathSciNetGoogle Scholar
  31. 31.
    Strömberg, J.O., Torchinsky, A.: Weighted Hardy Spaces. Lecture Notes in Math, vol. 1381. Springer, Berlin (1989)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical Analysis, A. Razmadze Mathematical InstituteI. Javakhishvili Tbilisi State UniversityTbilisiGeorgia
  2. 2.Department of Mathematics, Faculty of Informatics and Control SystemsGeorgian Technical UniversityTbilisiGeorgia

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