Skip to main content
SpringerLink
Log in

Hausdorff Operators on Real Hardy Spaces H1 Over Homogeneous Spaces with Local Doubling Property

  • Published:
Analysis Mathematica Aims and scope Submit manuscript

Abstract

We give conditions for boundedness of Hausdorff operators on real Hardy spaces H1 over homogeneous spaces of locally compact groups with local doubling property. Special cases of the hyperbolic plane and 2-sphere are considered. In this cases Hausdorff operators look as orbital like integrals.

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. N. Bourbaki, Éléments de mathématique. Topologie générale, Chapitres 5 à 10, Springer-Verlag (Berlin-Heidelberg, 2007).

    Book  Google Scholar 

  2. N. Bourbaki, Elements of Mathematics. Lie Groups and Lie Algebras, Chapters 1-3, Springer-Verlag (Berlin-Heidelberg, 1989).

    MATH  Google Scholar 

  3. N. Bourbaki, Éléments de mathématique. Intégration, Chapters 1-9, 2nd ed., Hermann (Paris, 1965–1989).

  4. N. Bourbaki, Elements de mathematique. Livre VI. Integration. Ch. 1-9. 2nd ed., Ch. 1-9. Hermann, Paris (1965–1969)

  5. A. Carbonaro, G. Mauceri, and S. Meda, H1 and BMO for certain locally doubling metric measure spaces, Ann. Sci. Norm. Super. Pisa Cl. Sci., (5)8 (2009), 543–582.

    MATH  Google Scholar 

  6. J. Chen, J. Dai, D. Fan, and X. Zhu, Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces, Science China Math., 61 (2018), 1647–1664.

    Article  MathSciNet  Google Scholar 

  7. J. Chen, D. Fan, and S. Wang, Hausdorff operators on Euclidean space (a survey article), Appl. Math. J. Chinese Univ. Ser. B., 28 (2013), 548–564.

    Article  MathSciNet  Google Scholar 

  8. J. Chen, X. Zhu, Boundedness of multidimensional Hausdorff operators on H1(ℝn), J. Math. Anal. Appl., 409 (2014), 428–434.

    Article  MathSciNet  Google Scholar 

  9. J. Y. Chu, Z. W. Fu, and Q. Y. Wu, Lp and BMO bounds for weighted Hardy operators on the Heisenberg group, J. Inequal. Appl., 282 (2016), 1–12.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. J. Delsarte, Hypergroupes et opérateurs de permutation et de transmutation, in: La théorie des équations aux dérivées partielles (Nancy, 9-15 avril 1956), Colloq. Internat. Centre Nat. Rech. Sci., LXXI, Centre National de la Recherche Scientifique (Paris, 1956), pp. 29–45.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Richard R. Goldberg, Certain operators and Fourier transforms on L2, Proc. Amer. Math. Soc., 10 (1959), 385–390.

    MathSciNet  MATH  Google Scholar 

  14. G. H. Hardy, Divergent Series, Clarendon Press (Oxford, 1949).

    MATH  Google Scholar 

  15. P. Jain, S. Jain, and V. D. Stepanov, LCT based integral transforms and Hausdorff operators, Eurasian Math. J., 11 (2020), 57–71.

    Article  MathSciNet  Google Scholar 

  16. T. Kawazoe, Atomic Hardy spaces on semisimple Lie groups, Japan. J. Math., 11 (1985), 293–343.

    Article  MathSciNet  Google Scholar 

  17. T. Kawazoe and J. Liu, On a weak L1 property of maximal operators on non-compact semisimple Lie groups, Tokyo J. Math., 25 (2002), 165–180.

    Article  MathSciNet  Google Scholar 

  18. P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd Ed., Academic Press, Inc. (Orlando, FL, 1985).

    MATH  Google Scholar 

  19. S. Lang, SL2(ℝ), Addison-Wesley (Reading, Mass.-London-Amsterdam, 1975).

    Google Scholar 

  20. A. Lerner and E. Liflyand, Multidimensional Hausdorff operators on the real Hardy space, J. Austr. Math. Soc., 83 (2007), 79–86.

    Article  MathSciNet  Google Scholar 

  21. B.M. Levitan, The Theory of Generalized Displacement Operators, Nauka (Moscow) (In Russian): English translation: Generalized Translation Operators and Some of their Applications, Don Goelman Israel Program for Scientific Translations, Daniel Davey & Co., Inc. (Jerusalem, 1964).

  22. E. Liflyand, Hausdorff operators on Hardy spaces, Eurasian Math. J., 4 (2013), 101–141.

    MathSciNet  MATH  Google Scholar 

  23. E. Liflyand and F. Móricz, The Hausdorff operator is bounded on the real Hardy space H1(R), Proc. Amer. Math. Soc., 128 (2000) 1391–1396.

    Article  MathSciNet  Google Scholar 

  24. A. R. Mirotin, On the boundedness of Hausdorff operators on real Hardy spaces H1 over homogeneous spaces of groups with local doubling property, arXiv:2007.10836.

  25. A. R. Mirotin, On the general form of linear functionals on the Hardy spaces H1 over compact Abelian groups and some of its applications, Indag. Math., 28 (2017), 451–462.

    Article  MathSciNet  Google Scholar 

  26. A. R. Mirotin, Boundedness of Hausdorff operators on Hardy spaces H1 over locally compact groups, J. Math. Anal. Appl., 473 (2019), 519–533.

    Article  MathSciNet  Google Scholar 

  27. A. R. Mirotin, Hausdorff operators on homogeneous spaces of locally compact groups, J. Beloruss. Gos. Univ. Math. Inform., 2 (2020), 28–35.

    MathSciNet  Google Scholar 

  28. J. Ruan, D. Fan, and Q. Wu, Weighted Morrey estimates for Hausdorff operator and its commutator on the Heisenberg group, Math. Inequal. Appl., 22 (2019), 307–329.

    MathSciNet  MATH  Google Scholar 

  29. S. Stahl, Poincare Half-plane. Gateway to Modern Geometry, Jones and Bartlett Publishers (Boston-London, 1993).

    MATH  Google Scholar 

  30. G. Stylogiannis, Hausdorff operators on Bergman spaces of the upper half plane, Concr. Oper., 7 (2020), 69–80.

    Article  MathSciNet  Google Scholar 

  31. S. S. Volosivets, Hausdorff operators on p-adic linear spaces and their properties in Hardy, BMO, and Hölder spaces, Math. Notes, 93 (2013), 382–391.

    Article  Google Scholar 

  32. Q. Wu, and D. Fan, Hardy space estimates of Hausdorff operators on the Heisenberg group, Nonlinear Anal., 164 (2017), 135–154.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgement

The author thanks the referees for very helpful suggestions and comments that improve the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics and Programming Technologies, Francisk Skorina Gomel State University, Sovietskaya, 104, Gomel, 246019, Belarus

    A. R. Mirotin

Authors
  1. A. R. Mirotin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. R. Mirotin.

Additional information

This work was supported by the State Program of Scientific Research of the Republic of Belarus.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mirotin, A.R. Hausdorff Operators on Real Hardy Spaces H1 Over Homogeneous Spaces with Local Doubling Property. Anal Math 47, 385–403 (2021). https://doi.org/10.1007/s10476-021-0087-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10476-021-0087-5

Key words and phrases

Mathematics Subject Classification

Search

Navigation