Skip to main content
SpringerLink
Log in

Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces

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

Abstract

In this paper, we study the boundedness of the Hausdorff operator Hϕ on the real line ℝ. First, we start with an easy case by establishing the boundedness of the Hausdorff operator on the Lebesgue space Lp(ℝ) and the Hardy space H1(ℝ). The key idea is to reformulate Hϕ as a Calderón-Zygmund convolution operator, from which its boundedness is proved by verifying the Hörmander condition of the convolution kernel. Secondly, to prove the boundedness on the Hardy space Hp(ℝ) with 0 < p < 1; we rewrite the Hausdorff operator as a singular integral operator with the non-convolution kernel. This novel reformulation, in combination with the Taibleson-Weiss molecular characterization of Hp(ℝ) spaces, enables us to obtain the desired results. Those results significantly extend the known boundedness of the Hausdorff operator on H1(ℝ).

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. Chen J, Fan D, Li J. Hausdorff operators on function spaces. Chin Ann Math Ser B, 2012, 33: 537–556

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen J, Fan D, Lin X, et al. The fractional Hausdorff operator on Hardy spaces. Anal Math, 2016, 42: 1–17

    Article  MathSciNet  Google Scholar 

  3. Chen J, Fan D, Wang S. Hausdorff operators on Euclidean spaces. Appl Math J Chinese Univ Ser B, 2013, 28: 548–564

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen J, Fan D, Zhang C. On multilinear Hausdorff operators. Acta Math Sinica Ser B, 2012, 28: 1521–1530.

    Article  MATH  Google Scholar 

  5. Chen J, Fan D, Zhu X. The Hausdorff operator on the Hardy space H 1(ℝ1). Acta Math Hungar, 2016, 150: 142–152

    Article  MathSciNet  Google Scholar 

  6. Chen J, Zhu X. Boundedness of multidimensional Hausdorff operators on H 1(ℝn). J Math Anal Appl, 2014, 409: 428–434

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Fan X, He J, Li B, et al. Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces. Sci China Math, 2017, 60: 2093–2154

    Article  MathSciNet  MATH  Google Scholar 

  9. Fan D, Lin X. Hausdorff operator on real Hardy spaces. Analysis, 2014, 34: 319–337

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Fu X, Lin H, Yang D, et al. Hardy spaces H p over non-homogeneous metric measure spaces and their applications. Sci China Math, 2015, 58: 309–388

    Article  MathSciNet  MATH  Google Scholar 

  12. Gao G, Wu X, Guo W. Some results for Hausdorff operators. Math Inequal Appl, 2015, 18: 155–168

    MathSciNet  MATH  Google Scholar 

  13. Georgakis C. The Hausdorff mean of a Fourier-Stieltjes transform. Proc Amer Math Soc, 1992, 116: 465–471

    Article  MathSciNet  MATH  Google Scholar 

  14. Kanjin Y. The Hausdorff operators on the real Hardy spaces H 1(ℝ). Studia Math, 2001, 148: 37–45

    Article  MathSciNet  MATH  Google Scholar 

  15. Koskela P, Soto T, Wang Z. Traces of weighted function spaces: Dyadic norms and Whitney extensions. Sci China Math, 2017, 60: 1981–2010

    Article  MathSciNet  MATH  Google Scholar 

  16. Ky D. New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators. Integral Equations Operator Theory, 2014, 78: 115–150

    Article  MathSciNet  MATH  Google Scholar 

  17. Lerner K, Li yand E. Multidimensional Hausdorff operators on the real Hardy spaces. J Aust Math Soc, 2007, 83: 79–86

    Article  MathSciNet  Google Scholar 

  18. Li yand E. Boundedness of multidimensional Hausdorff operators on H 1(ℝn)). Acta Sci Math (Szeged), 2008, 74: 845–851

    MathSciNet  Google Scholar 

  19. Li yand E. Hausdorff operators on Hardy spaces. Eurasian Math J, 2013, 4: 101–141

    MathSciNet  Google Scholar 

  20. Li yand E, Miyachi A. Boundedness of the Hausdorff operators in H p spaces, 0 < p < 1. Studia Math, 2009, 194: 279–292

    Article  MathSciNet  Google Scholar 

  21. Li yand E, Móricz F. The Hausdorff operator is bounded on the real Hardy space H 1(ℝ). Proc Amer Math Soc, 2000, 128: 1391–1396

    Article  MathSciNet  Google Scholar 

  22. Li yand E, Móricz F. The multi-parameter Hausdorff operator is bounded on the product Hardy space H 1(ℝ × ℝ). Analysis, 2001, 21: 107–118

    Google Scholar 

  23. Li yand E, Móricz F. Commutating relations for Hausdorff operators and Hilbert transforms on real Hardy space. Acta Math. Hungar, 2002, 97: 133–143

    Article  MathSciNet  Google Scholar 

  24. Lin H, Yang D, Yang D. Hardy spaces H p over non-homogeneous metric measure spaces and their applications. Sci China Math, 2015, 58: 309–388

    Article  MathSciNet  MATH  Google Scholar 

  25. Liu J, Yang D, Yuan W. Anisotropic Hardy-Lorentz spaces and their applications. Sci China Math, 2016, 59: 1669–1720

    Article  MathSciNet  MATH  Google Scholar 

  26. Lu S. Four Lectures on Real Hardy Spaces. Beijing: World Scientic, 1995

    Book  MATH  Google Scholar 

  27. Miyachi A. Boundedness of the Cesàro operator in Hardy space. J Fourier Anal Appl, 2004, 10: 83–92

    Article  MathSciNet  MATH  Google Scholar 

  28. Móricz F. Multivariate Hausdorff operators on the spaces H 1(ℝn) and BMO((ℝn). Anal Math, 2005, 31: 31–41

    Article  MathSciNet  Google Scholar 

  29. Ruan J, Fan D. Hausdorff operators on the power weighted Hardy spaces. J Math Anal Appl, 2016, 433: 31–48

    Article  MathSciNet  MATH  Google Scholar 

  30. Ruan J, Fan D. Hausdorff operators on the weighted Herz-type Hardy spaces. Math Inequal Appl, 2016, 19: 565–587

    MathSciNet  MATH  Google Scholar 

  31. Ruan J, Fan D. Hausdorff type operators on the power weighted Hardy spaces \(H_{| \cdot {|^\alpha }}^p\)(Rn). Math Nachr, 2017, https://doi.org/10.1002/mana.201600257

    Google Scholar 

  32. Ruan J, Fan D, Wu Q. Weighted Herz space estimates for Hausdorff operators on the Heisenberg group. Banach J Math Anal, 2017, 11: 513–535

    Article  MathSciNet  MATH  Google Scholar 

  33. Stein M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, vol. 43. Monographs in Harmonic Analysis, III. Princeton: Princeton University Press, 1993

  34. Stein M, Weiss G. On the theory of harmonic functions of several variables. I. The theory of H p-spaces. Acta Math, 1960, 103: 25–62

    Article  MathSciNet  MATH  Google Scholar 

  35. Taibleson M, Weiss G. The molecular characterization of certain Hardy spaces. Astérisque, 1980, 77: 67–151

    MathSciNet  MATH  Google Scholar 

  36. Triebel H. Theory of Function Spaces. Monographs in Mathematics, vol. 78. Basel: Birkhäuser Verlag, 1983

  37. Volosivets S S. Hausdorff operator of special kind in Morrey and Herz p-adic spaces. p-Adic Num Ultrametr Anal Appl, 2012, 4: 222–230

    Article  MathSciNet  MATH  Google Scholar 

  38. Weisz F. The boundedness of the Hausdorff operator on multi-dimensional Hardy spaces. Analysis, 2004, 24: 183–195

    Article  MathSciNet  MATH  Google Scholar 

  39. Wu X, Chen J. Best constants for Hausdorff operators on n-dimensional product spaces. Sci China Math, 2014, 57: 569–578

    Article  MathSciNet  MATH  Google Scholar 

  40. Yang D, Liang Y, Ky D. Real-Variable Theory of Musielak-Orlicz Hardy Spaces. Lecture Notes in Mathematics, vol. 2182. Cham: Springer, 2017

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

    Article  MathSciNet  MATH  Google Scholar 

  42. Yuan W, Sickel W, Yang D. Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Berlin: Springer-Verlag, 2010

  43. Zhao F, Lu S. Endpoint estimates for n-dimensional Hardy operators and their commutators. Sci China Math, 2012, 55: 1977–1990

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11671363, 11471288 and 11601456). Also, the authors are indebted to the referees who gave them many helpful comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

    Jiecheng Chen, Jiawei Dai & Xiangrong Zhu

  2. Department of Mathematics, University of Wisconsin-Milwaukee, Milwaukee, WI, 53201, USA

    Dashan Fan

Authors
  1. Jiecheng Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiawei Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dashan Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xiangrong Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiangrong Zhu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, J., Dai, J., Fan, D. et al. Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces. Sci. China Math. 61, 1647–1664 (2018). https://doi.org/10.1007/s11425-017-9246-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-017-9246-7

Keywords

MSC(2010)

Search

Navigation