Acta Mathematica Hungarica

, Volume 149, Issue 1, pp 177–189 | Cite as

Atomic decompositions of martingale hardy–morrey spaces



We establish the atomic decompositions of martingale Hardy– Morrey spaces. The martingale Hardy–Morrey spaces are generalizations of martingale Hardy spaces. Therefore, the atomic decompositions presented in this paper are extensions of the atomic decompositions of martingale Hardy spaces.

Mathematics Subject Classification

60G42 60G46 46E30 42B20 

Key words and phrases

martingale atomic decomposition Hardy space Morrey space 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Burkholder D., Gundy R.: Extrapolation and interpolation of quasi-linear operators on martingales. Acta Math., 124, 249–304 (1970)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Chiarenza F., Frasca M.: Morrey Spaces and Hardy–Littlewood Maximal Function. Rend. Mat. Appl., 7, 273–279 (1987)MathSciNetMATHGoogle Scholar
  3. 3.
    Coifman R., Weiss G.: Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc., 83, 569–645 (1977)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ho K.-P.: Littlewood-Paley spaces. Math. Scand., 108, 77–102 (2011)MathSciNetMATHGoogle Scholar
  5. 5.
    Ho K.-P.: Atomic decompositions of weighted Hardy–Morrey spaces. Hokkaido Math. J., 42, 131–157 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ho K.-P.: Vector-valued singular integral operators on Morrey type spaces and variable Triebel-Lizorkin-Morrey spaces. Ann. Acad. Sci. Fenn. Math., 37, 375–406 (2012)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ho K.-P.: Atomic decompositions, dual spaces and interpolations of martingale Hardy– Lorenta–Karamata spaces. Q.J. Math., 65, 985–1009 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    K.-P. Ho, Vector-valued operators with singular kernel and Triebel–Lizorkin-block spaces with variable exponents (accepted in Kyoto J. Math.).Google Scholar
  9. 9.
    Jia H., Wang H.: Decomposition of Hardy–Morrey spaces. J. Math. Anal. Appl., 354, 99–110 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Jiao Y., Xie G., Zhou D.: Dual spaces and John–Nirenberg inequalities on martingale Hardy–Lorenta–Karamata spaces. Q.J. Math 66, 605–623 (2015)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Jiao Y., Peng L., Liu P.: Atomic decompositions of Lorentz martingale spaces and applications. J. Funct. Space Appl., 7, 153–166 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Long R.: Sur I’espace Hp de martingale régulières (\({0 < p \leqq 1}\)). Ann. Inst. H. Poincare (B), 17, 123–142 (1981)Google Scholar
  13. 13.
    Miyamoto T., Nakai E., Sadasue G.: Martingale Orlicz–Hardy spaces. Math. Nachr., 285, 670–686 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Morrey C.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc., 43, 126–166 (1938)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Nakai E.: Hardy–Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces. Math. Nachr., 166, 95–104 (1994)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Nakai E.: Orlicz–Morrey spaces and the Hardy–Littlewood maximal function. Studia Math., 188, 193–221 (2008)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    E. Nakai and G. Sadasue, Martingale Morrey–Campanato spaces and fractional integrals, J. Funct. Spaces Appl. (2012), Article ID 673929, 29 pp.Google Scholar
  18. 18.
    E. Nakai, G. Sadasue and Y. Sawano, Martingale Morrey–Hardy and Canpanato– Hardy spaces, J. Funct. Spaces Appl. (2013), Article ID 690258, 14 pp.Google Scholar
  19. 19.
    Sawano Y., Tanaka H.: Decompositions of Besov–Morrey spaces and Triebel- Lizorkin-Morrey spaces. Math. Z., 257, 871–905 (2007)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    F. Schipp, W. Wade, P. Simon and J. Pál, Walsh Series: An Introduction to Dyadic Harmonic Analysis, Adam Hilger (Bristol–New York, 1990).Google Scholar
  21. 21.
    Weisz F.: Martingale Hardy spaces for \({0 < p \leqq 1}\). Probab. Th. Rel. Fields, 84, 361–376 (1990)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    F.Weisz, Martingale Hardy Spaces and their Applications in Fourier Analysis, Lecture Notes in Math., Vol. 1568, Springer-Verlag (1994).Google Scholar
  23. 23.
    Weisz F.: Bounded linear operator on weak Hardy spaces and applications. Acta Math. Hungar., 80, 249–264 (1998)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    W. Yuan, W. Sickel and D. Yang, Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Math., Vol. 2005, Springer-Verlag (Berlin, 2010).Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Information TechnologyThe Hong Kong Institute of EducationTai PoChina

Personalised recommendations