Skip to main content
SpringerLink
Log in

Extrapolation results in grand Lebesgue spaces defined on product sets

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Capone, C., Fiorenza, A.: On small Lebesgue spaces. J. Funct. Spaces Appl. 3, 73–89 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Coifman, R.R., Weiss, G.: Analyse harmonique non-commutative sur certains espaces homogénes. Lecture Notes in Math, vol. 242. Springer, Berlin (1971)

    Book  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  4. Duoandikoetxea, J.: Extrapolation of weights revisited: new proofs and sharp bounds. J. Funct. Anal. 260, 1886–1901 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Edmunds, D.E., Kokilashvili, V., Meskhi, A.: Bounded and Compact Integral Operators, Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht (2002)

    Book  MATH  Google Scholar 

  6. Fefferman, R., Stein, E.: Singular integrals on product spaces. Adv. Math. 45, 117–143 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fiorenza, A.: Duality and reflexivity in grand Lebesgue spaces. Collect. Math. 51(2), 131–148 (2000)

    MathSciNet  MATH  Google Scholar 

  8. 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 

  9. Garcia-Cuerva, J., Rubio de Francia, R.L.: Weighted Norm Inequalities and Related Topics, vol. 116. North Holland, Amsterdam (1985)

    Book  MATH  Google Scholar 

  10. Greco, L., Iwaniec, T., Sbordone, C.: Inverting the \(p\)-harmonic operator. Manuscr. Math. 9(2), 249–258 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Harboure, E., Macías, R., Segovia, C.: Extrapolation results for classes of weights. Am. J. Math. 110, 383–397 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hoffman, S.: Weighted norm inequalities and vector-valued inequalities for certain rough operators. Indiana Univ. Math. J. 42, 1–14 (1993)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Kairema, A.: Sharp weighted bounds for fractional integral operators in a space of homogeneous type. Math. Scand. 114, 226–253 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    MathSciNet  MATH  Google Scholar 

  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)

  19. Kokilashvili, V.: Boundedness criteria for singular integrals in weighted grand Lebesgue spaces. J. Math. Sci. (N.Y.) 170(1), 20–33 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  21. Kokilashvili, V., Meskhi, A.: Potentials with product kernels in grand Lebesgue spaces: one-weight criteria. Lithuanian Math. J. 53(1), 27–39 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    MATH  Google Scholar 

  24. Kokilashvili, V., Meskhi, A., Zaighum, M.A.: Sharp weighted bounds for multiple integral operators. Trans. A. Razmadze Math. Inst. 170, 75–90 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lique, T., Pérez, C., Rela, E.: Reverse Hölder property for strong weights and general measures. J. Geom. Anal. 27(1), 162–182 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Macías, R.A., Segovia, C.: Lipschitz functions on spaces of homogeneous type. Adv. Math. 3(3), 257–270 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  28. Meskhi, A.: Criteria for the boundedness of potential operators in grand Lebesgue spaces. Proc. A. Razmadze Math. Inst. 169, 119–132 (2015)

    MathSciNet  MATH  Google Scholar 

  29. Rubio de Francia, J.L.: Factorization and extrapolation of weights. Bull. Am. Math. Soc. (N.S.) 7, 393–395 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  31. Strömberg, J.O., Torchinsky, A.: Weighted Hardy Spaces. Lecture Notes in Math, vol. 1381. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Analysis, A. Razmadze Mathematical Institute, I. Javakhishvili Tbilisi State University, 6, Tamarashvili Str., 0177, Tbilisi, Georgia

    Vakhtang Kokilashvili & Alexander Meskhi

  2. Department of Mathematics, Faculty of Informatics and Control Systems, Georgian Technical University, 77, Kostava St., Tbilisi, Georgia

    Alexander Meskhi

Authors
  1. Vakhtang Kokilashvili
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Meskhi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Meskhi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kokilashvili, V., Meskhi, A. Extrapolation results in grand Lebesgue spaces defined on product sets. Positivity 22, 1143–1163 (2018). https://doi.org/10.1007/s11117-018-0564-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11117-018-0564-7

Keywords

Mathematics Subject Classification

Search

Navigation