A generalization of Hardy spaces Hp by using atoms

Article

Abstract

Let X = (X, d, µ) be a space of homogeneous type in the sense of Coifman and Weiss. The purpose of this paper is to generalize the definition of Hardy space Hp(X) and prove that the generalized Hardy spaces have the same property as Hp(X). Our definition includes a kind of Hardy-Orlicz spaces and a kind of Hardy spaces with variable exponent. The results are new even for the ℝn case. Let (X, δ, µ) be the normalized space of (X, d, µ) in the sense of Macías and Segovia. We also study the relations of our function spaces for (X, d, µ) and (X, δ, µ).

Keywords

Hardy space Hardy-Orlicz space variable exponent Campanato space space of homogeneous type 

MR(2000) Subject Classification

42B30 46E30 42B35 46E15 

References

  1. [1]
    Coifman, R. R., Weiss, G.: Analyse harmonique non-commutative sur certains espaces homogenes, Lecture Notes in Math., vol. 242, Springer-Verlag, Berlin and New York, 1971MATHGoogle Scholar
  2. [2]
    Coifman, R. R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc., 83, 569–645 (1977)MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Nakai, E., Yabuta, K.: Pointwise multipliers for functions of bounded mean oscillation. J. Math. Soc. Japan, 37, 207–218 (1985)MATHMathSciNetGoogle Scholar
  4. [4]
    Nakai, E.: Pointwise multipliers for functions of weighted bounded mean oscillation. Studia Math., 105, 105–119 (1993)MATHMathSciNetGoogle Scholar
  5. [5]
    Nakai, E.: Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces. Math. Nachr., 166, 95–103 (1994)MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Nakai, E.: Pointwise multipliers on weighted BMO spaces. Studia Math., 125, 35–56 (1997)MATHMathSciNetGoogle Scholar
  7. [7]
    Nakai, E.: Pointwise multipliers on the Morrey Spaces. Mem. Osaka Kyoiku Univ. III Natur. Sci. Appl. Sci., 46, 1–11 (1997)MathSciNetGoogle Scholar
  8. [8]
    Nakai, E., Yabuta, K.: Pointwise multipliers for functions of weighted bounded mean oscillation on spaces of homogeneous type. Math. Japon., 46, 15–28 (1997)MATHMathSciNetGoogle Scholar
  9. [9]
    Nakai, E.: The Campanato, Morrey and Hölder spaces on spaces of homogeneous type. Studia Math., 176, 1–19 (2006)MATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    Zorko, C. T.: Morrey space. Proc. Amer. Math. Soc., 98, 586–592 (1986)MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Taibleson, M. H., Weiss, G.: Certain function spaces connected with almost everywhere convergence of Fourier series, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), 95–113, Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983Google Scholar
  12. [12]
    Lu, S., Taibleson, M. H., Weiss, G.: On the almost everywhere convergence of Bochner-Riesz means of multiple Fourier series, Harmonic analysis (Minneapolis, Minn., 1981), 311–318, Lecture Notes in Math., 908, Springer, Berlin-New York, 1982Google Scholar
  13. [13]
    Long, R.: The spaces generated by blocks. Sci. Sinica Ser. A, 27, 16–26 (1984)MATHMathSciNetGoogle Scholar
  14. [14]
    Macías, R. A., Segovia, C.: Lipschitz functions on spaces of homogeneous type. Adv. Math., 33, 257–270 (1979)MATHCrossRefGoogle Scholar
  15. [15]
    Hu, G., Yang, D., Zhou, Y.: Boundedness of singular integrals in Hardy spaces on spaces of homogeneous type, to appear in Taiwanese J. Math.Google Scholar
  16. [16]
    Han, Y., Müller, D., Yang, D.: Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type. Math. Nachr., 279, 1505–1537 (2006)MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Duong, X. T., Yan, L.: Hardy spaces of spaces of homogeneous type. Proc. Amer. Math. Soc., 131, 3181–3189 (2003)MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    Grafakos, L., Liu, L., Yang, D.: Maximal function characterizations of Hardy spaces on RD-spaces and their applications, preprintGoogle Scholar
  19. [19]
    Han, Y.: Triebel-Lizorkin spaces on spaces of homogeneous type. Studia Math., 108, 247–273 (1994)MATHMathSciNetGoogle Scholar
  20. [20]
    Li, W. M.: A maximal function characterization of Hardy spaces on spaces of homogeneous type. Approx. Theory Appl. (N. S.), 14(2), 12–27 (1998)MathSciNetGoogle Scholar
  21. [21]
    Macías, R. A., Segovia, C.: A decomposition into atomes of distributions on spaces of homogeneous type. Adv. Math., 33, 270–309 (1979)Google Scholar
  22. [22]
    Lemarié, P. G.: Algèbres d’operateurs et semi-groupes de Poisson sur un espace de nature homogène, Publ. Math. Orsay, 84–83, 1984Google Scholar
  23. [23]
    Alvarez, J.: Continuity of Calderón-Zygmund type operators on the predual of a Morrey space, Clifford Algebra in Analysis and Related Topics, Studies in Advanced Mathematics, CRC Press, 309–319, 1996Google Scholar
  24. [24]
    Komori, Y.: Calderón-Zygmund operators on the predual of a Morrey space. Acta Mathematica Sinica, English Series, 19(2), 297–302 (2003)MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    Meyer, Y., Taibleson, M. H., Weiss, G.: Some functional analytic properties of the spaces B q generated by blocks. Indiana Univ. Math. J., 34(3), 493–515 (1985)MATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    Soria, F.: Characterizations of classes of functions generated by blocks and associated Hardy spaces. Indiana Univ. Math. J., 34(3), 463–492 (1985)MATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    Lu, S., Taibleson, M. H., Weiss, G.: Spaces generated by blocks, Publishing House of Beijing Normal University, 1989Google Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH 2008

Authors and Affiliations

  1. 1.Department of MathematicsOsaka Kyoiku UniversityKashiwara, OsakaJapan

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