Skip to main content
SpringerLink
Log in

Atomic Decompositions of Hardy Spaces with Variable Exponents and its Application to Bounded Linear Operators

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

As applications of atomic decomposition results of Hardy spaces with variable exponents, we shall prove the boundedness of commutators and the fractional integral operators as well as the Hardy operators. There are many ways to prove such boundedness. For example, the boundedness of commutators can be proved by the sharp maximal inequalities. But here, we propose a different method based upon our atomic decomposition.

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. Adams D.R.: A note on Riesz potentials. Duke Math. J. 42, 765–778 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  2. Almeida A., Hästö P.: Besov spaces with variable smoothness and integrability. J. Funct. Anal. 258, 1628–1655 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cruz-Uribe D., Diening L., Hästö P.: The maximal operator on weighted variable Lebesgue spaces. Fract. Calc. Appl. Anal. 14(3), 361–374 (2011)

    MathSciNet  Google Scholar 

  4. Cruz-Uribe SFO, D., Fiorenza, A.: Variable Lebesgue Spaces. Foundations and Harmonic Analysis, Birkhäuser (2013)

  5. Cruz-Uribe, SFO, D., Fiorenza, A., Martell, J.M., Pérez, C.: The boundedness of classical operators on variable L p spaces, Ann. Acad. Sci. Fenn. Math. 31, 239–264 (2006)

  6. Diening L.: Maximal functions on generalized \({L^{p(\cdot)}}\) spaces. Math. Inequal. Appl. 7(2), 245–253 (2004)

    MathSciNet  MATH  Google Scholar 

  7. Diening, L., Harjulehto, P., Hästö P., Ruǔička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017. Springer, Berlin (2011)

  8. Diening L., Hästö P., Roudenko R.: Function spaces of variable smoothness and integrability. J. Funct. Anal. 256, 1731–1768 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Diening L., Harjulehto P., Hästö P., Mizuta Y., Shimomura T.: Maximal functions in variable exponent spaces: limiting cases of the exponent. Ann. Acad. Sci. Fenn. Math. 34(2), 503–522 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Diening, L., Samko, S.: Hardy inequality in variable exponent Lebesgue spaces. Fract. Calc. Appl. Anal. 10(1), 1–18 (2007)

    Google Scholar 

  11. Hästö P.: Local-to-global results in variable exponent spaces. Math. Res. Lett. 16(2), 263–278 (2009)

    MathSciNet  MATH  Google Scholar 

  12. Ho K.P.: Atomic decomposition of Hardy spaces and characterization of BMO via Banach function spaces. Anal. Math. 38(3), 173–185 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gunawan H., Sawano Y., Sihwaningrum I.: Fractional integral operators in nonhomogeneous spaces. Bull. Aust. Math. Soc. 80(2), 324–334 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Izuki M.: Boundedness of vector-valued sublinear operators on Herz-Morrey spaces with variable exponent. Math. Sci. Res. J. 13(10), 243–253 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Izuki M.: Boundedness of commutators on Herz spaces with variable exponent. Rend. Circ. Mat. Palermo 59(2), 199–213 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Izuki M.: Fractional integrals on Herz-Morrey spaces with variable exponent. Hiroshima Math. J. 40(3), 343–355 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Iida T., Sato E., Sawano Y., Tanaka H.: Weighted norm inequalities for multilinear fractional operators on Morrey spaces. Studia Math. 205(2), 139–170 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Karapetyants N.K., Ginzburg A.I.: Fractional integrals and singular integrals in the Hölder classes of variable order. Integral Transform. Spec. Funct. 2(2), 91–106 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Karapetyants N.K., Ginzburg A.I.: Fractional integro-differentiation in Hölder classes of variable order. Dokl. Akad. Nauk 339(4), 439–441 (1994)

    Google Scholar 

  20. Kokilashvili V., Paatashvili V.: On Hardy classes of analytic functions with a variable exponent. Proc. A. Razmadze Math. Inst 142, 134–137 (2006)

    MathSciNet  MATH  Google Scholar 

  21. Kov’ačik O., R’akosn’k J.: On spaces L p(x) and W k,p(x). Czechoslovak Math. J. 41(116), 592–618 (1991)

    MathSciNet  Google Scholar 

  22. Kokilashvili V., Samko S.: Maximal and fractional operators in weighted L p(x) spaces. Rev. Mat. Iberoamericana 20(2), 493–515 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ky, L.D.: New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators, arXiv:1103.3757

  24. Ephremidze L., Kokilashvili V., Samko S.: Fractional, maximal and singular operators in variable exponent Lorentz spaces. Fract. Calc. Appl. Anal. 11(4), 407–420 (2008)

    MathSciNet  MATH  Google Scholar 

  25. Luxenberg, W.: Banach Function Spaces. Technische Hogeschool te Delft Assen, Assen (1955)

  26. Mizuta Y., Shimomura T.: Sobolev embeddings for Riesz potentials of functions in Morrey spaces of variable exponent. J. Math. Soc. Japan 60, 583–602 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mizuta Y., Shimomura T.: Continuity properties for Riesz potentials of functions in Morrey spaces of variable exponent. Math. Inequal. Appl. 13, 99–122 (2010)

    MathSciNet  MATH  Google Scholar 

  28. Mizuta Y., Shimomura T.: Continuity properties of Riesz potentials of Orlicz functions. Tohoku Math. J. 61, 225–240 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mizuta Y., Shimomura T.: Continuity properties for Riesz potentials of functions in Morrey spaces of variable exponent. Math. Inequal. Appl. 13(1), 99–122 (2010)

    MathSciNet  MATH  Google Scholar 

  30. Mizuta Y., Shimomura T., Sobukawa T.: Sobolev’s inequality for Riesz potentials of functions in non-doubling Morrey spaces. Osaka J. Math. 46, 255–271 (2009)

    MathSciNet  MATH  Google Scholar 

  31. Nakai E., Sawano Y.: Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262, 3665–3748 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Nakano, H.: Modulared Semi-Ordered Linear Spaces. pp. i+288, MR0038565, Maruzen Co., Ltd., Tokyo (1950)

  33. Nakano, H.: Topology of Linear Topological Spaces. pp. viii+281, MR0046560, Maruzen Co., Ltd., Tokyo, (1951)

  34. Nekvinda A.: Hardy–Littlewood maximal operator on \({L^{p(x)}({{\mathbb{R}}})}\) . Math. Inequal. Appl. 7(2), 255–265 (2004)

    MathSciNet  MATH  Google Scholar 

  35. Olsen, A.: Fractional integration, Morrey spaces and a Schröndinger equation, Commun. Partial Diff. Equ. 20 (11 and 12), 2005–2055 (1995)

  36. Orlicz, W.: Über Räume (L M). Bull. Acad. Pol. Sc. Lett. Ser. A 93–107 (1936)

  37. Rafeiro H., Samko S.G.: Approximative method for the inversion of the Riesz potential operator in variable Lebesgue spaces. Fract. Calc. Appl. Anal. 11(3), 269–280 (2008)

    MathSciNet  MATH  Google Scholar 

  38. Rafeiro, H., Samko, S.G.: Variable exponent Campanato spaces. J. Math. Sci. (N. Y.), 172 (1), 143–164 (2011) (Problems in mathematical analysis. No. 51)

  39. Samko S.G.: Some classical operators of variable order in variable exponent analysis. Oper. Theory Adv. Appl. 193, 281–301 (2009)

    MathSciNet  Google Scholar 

  40. Samko, S.: Fractional integration and differentiation of variable order: an overview. Nonlinear Dyn. 71(4), 653–662 (2013)

    Google Scholar 

  41. Sawano Y., Sugano S., Tanaka H.: Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces. Trans. Am. Math. Soc. 363(12), 6481–6503 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  42. Sawano Y., Sugano S., Tanaka H.: Orlicz-Morrey spaces and fractional operators. Potential Anal. 36, 517–556 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  43. Samko S.: Convolution and potential type operators in the space L p(x). Integral Transform Special Funct. 7(3–4), 261–284 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  44. Samko N., Vakulov B.: Spherical fractional and hypersingular integrals in generalized Hölder spaces with variable characteristic. Math. Nachrichten 284, 355–369 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  45. Samko N., Samko S.G., Vakulov B.: Fractional integrals and hypersingular integrals in variable order Hölder spaces on homogeneous spaces. Armen. J. Math. 2(2), 38–64 (2009)

    MathSciNet  Google Scholar 

  46. Sugano S.: Some inequalities for generalized fractional integral operators on generalized Morrey spaces. Math Inequal. Appl. 14(4), 849–865 (2011)

    MathSciNet  MATH  Google Scholar 

  47. Stein E.M.: Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  48. Strömberg, J.O., Torchinsky, A.: Weighted Hardy spaces. Lecture Notes in Mathematics, vol. 1381. Springer, Berlin (1989)

  49. Vakulov, B.G.: Spherical potentials in weighted Hölder spaces of variable order. Dokl. Akad. Nauk, 400(1) (2005), 7–10. Translated in Doklady Mathematics 71(1), 1–4 (2005)

    Google Scholar 

  50. Vakulov B.G.: Spherical potentials of complex order in the variable order Hölder spaces. Integral Transforms Spec. Funct. 16(5-6), 489–497 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. 1-1 Minami-Ohsawa, Hachioji, Tokyo, 192-0397, Japan

    Yoshihiro Sawano

Authors
  1. Yoshihiro Sawano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yoshihiro Sawano.

Additional information

The author was supported by Grant-in-Aid for Young Scientists (B), No. 24740085, Japan Society for the Promotion of Science.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sawano, Y. Atomic Decompositions of Hardy Spaces with Variable Exponents and its Application to Bounded Linear Operators. Integr. Equ. Oper. Theory 77, 123–148 (2013). https://doi.org/10.1007/s00020-013-2073-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00020-013-2073-1

Mathematics Subject Classification (2010)

Keywords

Search

Navigation