Skip to main content
SpringerLink
Log in

Recent Trends in Grand Lebesgue Spaces

  • Conference paper
  • First Online:
Function Spaces and Inequalities

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 206))

Abstract

The aim of this paper is two fold. Since their inception in 1992, we collect various generalizations of the grand Lebesgue spaces touching upon several of their aspects such as properties, duality, equivalent norms etc. Also, we prove certain new extrapolation results of the type of Rubio De Francia in the framework of fully measurable grand Lebesgue spaces.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Anatriello, G.: Iterated grand and small Lebesgue spaces. Collect Math. 65, 273–284 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  2. Anatriello, G., Fiorenza, A.: Fully measurable grand Lebesgue spaces. J. Math. Anal. Appl. 422, 783–797 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press (1988)

    Google Scholar 

  4. Bocardo, I.: Quelques problems de Dirichlet avec donneées dans de grands espaces de Sobolev. C.R. Acad.Sci. Paris Ser. I Math. 325, 1269–1272 (1997)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Capone, C., Fiorenza, A., Karadzhov, G.E.: Grand Orlicz spaces and global integrability of the Jacobian. Math. Scand. 102, 131–148 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Capone, C., Formica, M.R., Giova, R.: Grand Lebesgue spaces with respect to measurable functions. Nonlinear Anal. 85, 125–131 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Carozza, M., Sbordone, C.: The distance to \(L^\infty \) in some function spaces and applications. Diff. Int. Eq. 10, 599–607 (1997)

    MATH  MathSciNet  Google Scholar 

  9. Carro, M.J., Lorente, M.: Rubio de Francia’s extrapolation theorem for \(B_p\) weights. Proc. Amer. Math. Soc. 138, 629–640 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Castillo, R.E., Rafeiro, H.: Inequalities with conjugate exponents in grand Lebesgue spaces. Hacet. J. Math. Stat. 44, 33–39 (2015)

    MATH  MathSciNet  Google Scholar 

  11. De Francia, Rubio: Factorization theory and \(A_p\) weights. Amer. J. Math. 106, 533–547 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fiorenza, A., Rakotoson, J.M., Sbordone, C.: Variable exponents and grand Lebesgue spaces: some optimal results. Commun. Contemp. Math. 17, 1550023 (2015). doi:10.1142/S0219199715500236

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

    MATH  MathSciNet  Google Scholar 

  14. Fiorenza, A., Karadzhov, G.E.: Grand and small Lebesgue spaces and their analogs. J. Anal. Appl. 23, 657–681 (2004)

    MATH  MathSciNet  Google Scholar 

  15. Fiorenza, A., Krbec, M.: On the domain and range of the maximal operator. Nagoya Math. J. 158, 43–61 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Fiorenza, A., Rakotoson, J.M.: New properties of small Lebesgue spaces and their applications. Math. Ann. 326, 543–561 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Fiorenza, A., Rakotoson, J.M.: Compactness, interpolation inequalities for small Lebesgue-Sobolev spaces and applications. Calc. Var. Partial. Differ. Equ. 25, 187–203 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Fiorenza, A., Sbordone, C.: Existence and uniqueness results for solutions of nonlinear equations with right hand side in \(L^1\). Stud. Math. 127, 223–231 (1998)

    MATH  MathSciNet  Google Scholar 

  19. Fiorenza, A., Mercaldo, A., Rakotoson, J.M.: Regularity and comparison results in grand Sobolev spaces for parabolic equations with measure data. Appl. Math. Lett. 14, 979–981 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  20. Fiorenza, A., Mercaldo, A., Rakotoson, J.M.: Regularity and uniqueness results in grand Sobolev spaces for parabolic equations with measure data. Discrete Contin. Dyn. Syst. 8, 893–906 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. Fiorenza, A., Gupta, B., Jain, P.: The maximal theorem for weighted grand Lebesgue spaces. Studia Math. 188, 123–133 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Fratta, G.Di, Fiorenza, A.: A direct approach to the duality of grand and small Lebesgue spaces. Nonlinear Anal. 70, 2582–2592 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Greco, L.: A remark on the equality of detDf = DetDf. Diff. Int. Eq. 6, 1089–1100 (1993)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. Iwaniec, T., Sbordone, C.: Weak minima of variational integrals. J. Reine Angew. Math. 454, 143–161 (1994)

    MATH  MathSciNet  Google Scholar 

  27. Iwaniec, T., Sbordone, C.: Riesz transform and elliptic PDE’s with VMO coefficients. J. Analyse Math. 74, 183–212 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  28. Jain, P., Kumari, S.: On grand Lorentz spaces and the maximal operator. Georg. Math. J. 19, 235–246 (2012)

    MATH  MathSciNet  Google Scholar 

  29. Jain, P., Singh, M., Singh, A.P.: Hardy type operators on grand Lebesgue spaces for non-increasing functions. Trans. A. Razmad. Math. Inst. 170, 34–46 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  30. Jain, P., Singh, M., Singh, A.P.: Integral operators on fully measurable weighted grand Lebesgue spaces. Indag. Math. 28, 516–526 (2017)

    Google Scholar 

  31. Kokilashvili, V., Meskhi, A., Rafeiro, H.: Estimates for nondivergence elliptic equations with VMO coefficients in generalized grand Morrey spaces. Complex Var. Elliptic Equ. Int. J. 59, 1169–1184 (2013)

    Google Scholar 

  32. Kokilashvili, V., Samko, S.: Boundedness of weighted singular integral operators on a Carleson curve in grand Lebesgue spaces. Intern. Conf. Numer. Anal. Appl. Math.(ICNAAM), AIP Conf. Proc. 1281, 490-493 (2010)

    Google Scholar 

  33. Kokilashvili, V.: Boundedness criterion for the Cauchy singular integral operator in weighted grand Lebesgue spaces and application to the Riemann problem. Proc. A. Razmad. Math. Inst. 151, 129–133 (2009)

    MATH  MathSciNet  Google Scholar 

  34. Kokilashvili, V.: The Riemann boundary value problem for analytic functions in the frame of grand \(L^{p)}\) spaces. Bull. Georg. Nat. Acad. Sci. 4, 5–7 (2010)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  36. Kokilashvili, V., Meskhi, A.: A note on the boundedness of the Hilbert transform in weighted grand Lebesgue spaces. Georgian Math. J. 16, 547–551 (2009)

    MATH  MathSciNet  Google Scholar 

  37. Kokilashvili, V., Meskhi, A.: Trace inequalities for fractional integrals in grand Lebesgue spaces. Studia Math. 210, 159–176 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  38. Kokilashvili, V., Meskhi, A., Rafeiro, H.: Boundedness of commutators of singular and potential operators in generalized grand Morrey spaces and some applications. Studia Math. 217, 159–178 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  39. Kokilashvili, V., Meskhi, A., Rafeiro, H.: Riesz type potential operators in generalized grand Morrey spaces. Georgian Math. J. 20, 43–64 (2013)

    MATH  MathSciNet  Google Scholar 

  40. Kokilashvili, V., Meskhi, A., Rafeiro, H.: Grand Bochner Lebesgue space and its associate space. J. Func. Anal. 266, 2125–2136 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  41. Kolmogorov, A.: Über kompaktheit der funktionen mengen bei der konvergenz im mittel, Nachr. Ges. Wiss. Gottingen Math. Phys. Ki. 60–63 (1931)

    Google Scholar 

  42. Kufner, A., Maligranda, L., Persson, L.E.: The Hardy Inequality: About its History and Some Related Results. Vydavatelsky Sewis, Pilsen (2007)

    MATH  Google Scholar 

  43. Liflyand, E., Ostrovsky, E., Sirota, L.: Structural properties of bilateral grand Lebesgue spaces. Turk. J. Math. 34, 207–219 (2010)

    MATH  Google Scholar 

  44. Meskhi, A.: Maximal functions and singular integrals in Morrey spaces associated with grand Lebesgue spaces. Proc. A. Razmadze Math. Inst. 151, 139–143 (2009)

    MATH  MathSciNet  Google Scholar 

  45. Meskhi, A.: Weighted criteria for Hardy transform under the \(B_p\) condition in grand Lebesgue spaces and some applications. J. Math. Sci. 178, 622–636 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  46. Meskhi, A.: Maximal functions, potentials and singular integrals in grand Morrey spaces. Complex Var. Elliptic Eqn. 56, 1003–1019 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  48. Neugebauer, C.J.: Weighted norm inequalities for averaging operators of monotone functions. Pub. Mat. 35, 429–447 (1991)

    Google Scholar 

  49. Ostrovsky, E., Rogover, E., Sirota, L.: Integral operators in bilateral grand Lebesgue spaces (2011). arXiv: 0912.2538v1 [math.FA]

  50. Ostrovsky, E., Sirota, L.: Composed grand Lebesgue spaces. (2009). arXiv:1110.4880v1 [math.FA]

  51. Rafeiro, H., Vargas, A.: On the compactness in grand spaces. Georg. Math J. 22, 141–152 (2015)

    MATH  MathSciNet  Google Scholar 

  52. Samko, S.G., Umarkhadzhiev, S.M.: On Iwaniec Sbordone spaces on sets which may have infinite measure. Azerb. J. Math. 1, 67–84 (2010)

    MATH  MathSciNet  Google Scholar 

  53. Sbordone, C.: Nonlinear elliptic equations with right hand side in nonstandard spaces, pp. 361–368. Rend. Sem. Mat. Fis. Modena, Supplemento al XLVI (1998)

    Google Scholar 

  54. Sbordone, C.: Grand Sobolev spaces and their applications to variational problems. Le Matematiche 51, 335–347 (1996)

    MATH  MathSciNet  Google Scholar 

  55. Zaneen, A.C.: Integration. North-Holland (1967)

    Google Scholar 

Download references

Acknowledgements

The first author acknowledges with thanks CSIR, India for the research support (Ref. No. 25(0242)/15/EMR-II).

Author information

Authors and Affiliations

  1. Department of Mathematics, South Asian University, Akbar Bhawan, Chanakya Puri, New Delhi, 110021, India

    Pankaj Jain

  2. Department of Mathematics, Lady Shri Ram College For Women (University of Delhi), Lajpat Nagar, New Delhi, 110024, India

    Monika Singh

  3. Department of Mathematics, Dyal Singh College (University of Delhi), Lodhi Road, New Delhi, 110003, India

    Arun Pal Singh

Authors
  1. Pankaj Jain
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Monika Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Arun Pal Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pankaj Jain .

Editor information

Editors and Affiliations

  1. Department of Mathematics, South Asian University, New Delhi, Delhi, India

    Pankaj Jain

  2. Faculty of Mathematics & Computer Science, Friedrich Schiller University Jena, Jena, Thüringen, Germany

    Hans-Jürgen Schmeisser

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer Nature Singapore Pte Ltd

About this paper

Cite this paper

Jain, P., Singh, M., Singh, A.P. (2017). Recent Trends in Grand Lebesgue Spaces. In: Jain, P., Schmeisser, HJ. (eds) Function Spaces and Inequalities. Springer Proceedings in Mathematics & Statistics, vol 206. Springer, Singapore. https://doi.org/10.1007/978-981-10-6119-6_6

Download citation

Publish with us

Policies and ethics

Search

Navigation