Skip to main content
SpringerLink
Log in

Singular and fractional integral operators on preduals of Campanato spaces with variable growth condition

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

We investigate the boundedness of singular and fractional integral operators on generalized Hardy spaces defined on spaces of homogeneous type, which are preduals of Campanato spaces with variable growth condition. To do this we introduce molecules with variable growth condition. Our results are new even for ℝn case.

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. Alvarez J. Continuity of Calderón-Zygmund type operators on the predual of a Morrey space. In: Clifford Algebra in Analysis and Related Topics. Studies in Advanced Mathematics. Boca Raton: CRC Press, 1996, 309–319

    Google Scholar 

  2. Bownik M. Boundedness of operators on Hardy spaces via atomic decompositions. Proc Amer Math Soc, 2005, 133: 3535–3542

    Article  MATH  MathSciNet  Google Scholar 

  3. Coifman R R, Weiss G. Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes. Berlin-New York: Springer-Verlag, 1971

    Book  MATH  Google Scholar 

  4. Coifman R R, Weiss G. Extensions of Hardy spaces and their use in analysis. Bull Amer Math Soc, 1977, 83: 569–645

    Article  MATH  MathSciNet  Google Scholar 

  5. Fefferman C, Stein E M. H p spaces of several variables. Acta Math, 1972, 129: 137–193

    Article  MATH  MathSciNet  Google Scholar 

  6. Gatto A E, Segovia C, Vági S. On fractional differentiation and integration on spaces of homogeneous type. Rev Mat Iberoamericana, 1996, 12: 111–145

    Article  MATH  MathSciNet  Google Scholar 

  7. Gatto A E, Vági S. Fractional integrals on spaces of homogeneous type. In: Analysis and Partial Differential Equations. New York: Dekker, 1990, 171–216

    Google Scholar 

  8. Grafakos L, Liu L, Yang D. Maximal function characterizations of Hardy spaces on RD-spaces and their applications. Sci China Ser A, 2008, 51: 2253–2284

    Article  MATH  MathSciNet  Google Scholar 

  9. Han Y, Müller D, Yang D. Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type. Math Nachr, 2006, 279: 1505–1537

    Article  MATH  MathSciNet  Google Scholar 

  10. Harboure E, Viviani B. Boundedness of singular integral operators on H ω. Rev Un Mat Argentina, 1993, 38: 219–245

    MATH  MathSciNet  Google Scholar 

  11. Hu G, Yang D, Zhou Y. Boundedness of singular integrals in Hardy spaces on spaces of homogeneous type. Taiwanese J Math, 2009, 13: 91–135

    Article  MATH  MathSciNet  Google Scholar 

  12. Komori Y. Calderón-Zygmund operators on the predual of a Morrey space. Acta Math Sin (Engl Ser), 2003, 19: 297–302

    Article  MATH  MathSciNet  Google Scholar 

  13. Lemarié P G. Algèbres d’operateurs et semi-groupes de Poisson sur un espace de nature homogène. Orsay: Publ Math Orsay, 1984

    MATH  Google Scholar 

  14. Liu L, Yang D, Zhou Y. Boundedness of generalized Riesz potentials on spaces of homogeneous type. Math Inequal Appl, 2010, 13: 867–885

    MATH  MathSciNet  Google Scholar 

  15. Macías R A, Segovia C. Lipschitz functions on spaces of homogeneous type. Adv Math, 1979, 33: 257–270

    Article  MATH  MathSciNet  Google Scholar 

  16. Macías R A, Segovia C. Singular integrals on generalized Lipschitz and Hardy spaces. Studia Math, 1979, 65: 55–75

    Article  MATH  MathSciNet  Google Scholar 

  17. Meda S, Sjögren P, Vallarino M. On the H 1-L 1 boundedness of operators. Proc Amer Math Soc, 2008, 136: 2921–2931

    Article  MATH  MathSciNet  Google Scholar 

  18. Nakai E. Pointwise multipliers for functions of weighted bounded mean oscillation. Studia Math, 1993, 105: 105–119

    Article  MATH  MathSciNet  Google Scholar 

  19. Nakai E. Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces. Math Nachr, 1994, 166: 95–103

    Article  MATH  MathSciNet  Google Scholar 

  20. Nakai E. Pointwise multipliers on weighted BMO spaces. Studia Math, 1997, 125: 35–56

    Article  MATH  MathSciNet  Google Scholar 

  21. Nakai E. The Campanato, Morrey and Hölder spaces on spaces of homogeneous type. Studia Math, 2006, 176: 1–19

    Article  MATH  MathSciNet  Google Scholar 

  22. Nakai E. A generalization of Hardy spaces H p by using atoms. Acta Math Sin (Engl Ser), 2008, 24: 1243–1268

    Article  MATH  MathSciNet  Google Scholar 

  23. Nakai E. Singular and fractional integral operators on Campanato spaces with variable growth conditions. Rev Mat Complut, 2010, 23: 355–381

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  25. Nakai E, Sawano Y. Orlicz-Hardy spaces and their duals. Sci China Math, 2014, 57: 903–962

    Article  MATH  MathSciNet  Google Scholar 

  26. Nakai E, Yabuta K. Pointwise multipliers for functions of bounded mean oscillation. J Math Soc Japan, 1985, 37: 207–218

    Article  MATH  MathSciNet  Google Scholar 

  27. Nakai E, Yabuta K. Pointwise multipliers for functions of weighted bounded mean oscillation on spaces of homogeneous type. Math Japon, 1997, 46: 15–28

    MATH  MathSciNet  Google Scholar 

  28. Nakai E, Yoneda T. Riesz transforms on generalized Hardy spaces and a uniqueness theorem for the Navier-Stokes equations. Hokkaido Math J, 2011, 40: 67–88

    Article  MATH  MathSciNet  Google Scholar 

  29. Nakamura S. Generalized weighted Morrey spaces and classical operators. Math Nachr, 2016, 289: 2235–2262

    Article  MATH  MathSciNet  Google Scholar 

  30. Nakamura S, Noi T, Sawano Y. Generalized Morrey spaces and trace operator. Sci China Math, 2016, 59: 281–336

    Article  MATH  MathSciNet  Google Scholar 

  31. Yabuta K. A remark on the (H 1;L 1) boundedness. Bull Fac Sci Ibaraki Univ Ser A, 1993, 25: 19–21

    Article  MATH  MathSciNet  Google Scholar 

  32. Yang D, Zhou Y. A boundedness criterion via atoms for linear operators in Hardy spaces. Constr Approx, 2009, 29: 207–218

    Article  MATH  MathSciNet  Google Scholar 

  33. Zorko C T. Morrey space. Proc Amer Math Soc, 1986, 98: 586–592

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by Grant-in-Aid for Scientific Research (B) (Grant No. 15H03621), Japan Society for the Promotion of Science. The author thanks the referees for their careful reading and useful comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, Ibaraki University, Mito, Ibaraki, 310-8512, Japan

    Eiichi Nakai

Authors
  1. Eiichi Nakai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eiichi Nakai.

Additional information

Dedicated to the memory of Professor CHENG MinDe on the occasion of the centenary of his birth

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nakai, E. Singular and fractional integral operators on preduals of Campanato spaces with variable growth condition. Sci. China Math. 60, 2219–2240 (2017). https://doi.org/10.1007/s11425-017-9154-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-017-9154-y

Keywords

MSC(2010)

Search

Navigation