Acta Mathematica Sinica

, Volume 17, Issue 2, pp 339–360 | Cite as

Remarks on Herz-Type Hardy Spaces

  • Akihiko MiyachiEmail author


Basic properties of the Herz-type Hardy spaces \( H\ifmmode\expandafter\dot\else\expandafter\.\fi{K}^{{\alpha ,p}}_{q} \), such as the boundedness of singular integral operators and the fractional integration operators, atomic decomposition, dense subspaces, etc., are established in the full range 0 < q < ∞.


Herz-type Hardy space Singular integral Fractional integration Multilinear operator 

MR (1991) Subject Classification

42B30 46F05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Beurling, Construction and analysis of some convolution algebra, Ann. Inst. Fourier (Grenoble), 1964, 14:1–32zbMATHMathSciNetGoogle Scholar
  2. 2.
    C. Herz, Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms, J. Math. Mech., 1968, 18:283–324zbMATHMathSciNetGoogle Scholar
  3. 3.
    Y. Z. Chen, K. S. Lau, On some new classes of Hardy spaces, J. Funct. Anal., 1989, 84:255–278zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    J. García-Cuerva, Hardy spaces and Beurling algebras, J. London Math. Soc. (2), 1989, 39:499–513zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    S. Lu, D. Yang, Some Hardy spaces associated with the Herz spaces and their wavelet characterizations (in Chinese), Beijing Shifan Daxue Xuebao (= J. Beijing Normal Univ. (Natur. Sci.)), 1993, 29:10–19zbMATHMathSciNetGoogle Scholar
  6. 6.
    S. Lu, D. Yang, The local versions of H p(ℝn) spaces at the origin, Studia Math., 1995, 116:103–131zbMATHMathSciNetGoogle Scholar
  7. 7.
    J. García-Cuerva, M. J. L. Herrero, A theory of Hardy spaces associated to Herz spaces, Proc. London Math. Soc. (3), 1994, 69:605–628zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    X. Li, S. Lu, D. Yang, Certain bilinear operators on Herz-type Hardy spaces, Beijing Math., 1996, 2:72–95Google Scholar
  9. 9.
    S. Lu, D. Yang, Herz-type Sobolevan d Bessel potential spaces and their applications, Science in China (Ser. A), 1997, 40:113–129zbMATHGoogle Scholar
  10. 10.
    L. Grafakos, X. Li, D. Yang, Bilinear operators on Herz-type Hardy spaces, Trans. Amer. Math. Soc., 1998, 305:1249–1275CrossRefMathSciNetGoogle Scholar
  11. 11.
    L. Tang, D. Yang, Boundedness of multilinear operators in Herz-type Hardy spaces, Acta Math. Sinica (English Ser.), 2000, 16(2):295–306zbMATHCrossRefGoogle Scholar
  12. 12.
    L. Tang, D. Yang, Boundedness of vector-valued operators on weighted Herz spaces, to appear in Approx. Theory and its Appl Google Scholar
  13. 13.
    X. Li, D. Yang, Boundedness of some sublinear operators on Herz spaces, Illinois J. Math., 1996, 40:484–501zbMATHMathSciNetGoogle Scholar
  14. 14.
    J.-O. Strömberg, A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math., 1381, Springer-Verlag, 1989Google Scholar
  15. 15.
    A. Uchiyama, On the radial maximal function of distributions, Pacific J. Math., 1986, 121:467–483zbMATHMathSciNetGoogle Scholar
  16. 16.
    S. Lu, D. Yang, Some characterizations of weighted Herz-type Hardy spaces and their applications, Acta Math. Sinica (New Ser.), 1997, 13:45–58zbMATHMathSciNetGoogle Scholar
  17. 17.
    A. Miyachi, H p spaces over open subsets of ℝn, Studia Math., 1990, 95:205–228MathSciNetGoogle Scholar
  18. 18.
    S. Lu, F. Soria, On the Herz spaces with power weights, Fourier Analysis and PDE’s (Garc´ıa-Cuerva et al. eds), Proc. of Miraflores Conference 1992, Studies in Advanced Math., CRC Press, 1995, 227–236Google Scholar
  19. 19.
    S. Lu, D. Yang, The weighted Herz-type Hardy spaces and its applications, Science in China (Ser. A), 1995, 38:662–673zbMATHGoogle Scholar
  20. 20.
    E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1970Google Scholar
  21. 21.
    S. Lu, D. Yang, The decomposition of the weighted Herz spaces and its applications, Science in China (Ser. A), 1995, 38:147–158zbMATHGoogle Scholar
  22. 22.
    E. Hernández, D. Yang, Interpolation of Herz spaces and applications, Math. Nachr., 1999, 205:69–87zbMATHMathSciNetGoogle Scholar
  23. 23.
    S. Lu, D. Yang, Hardy-Littlewood-Sobolevt heorems of fractional integration on Herz-type spaces and its applications, Canad. J. Math., 1996, 48:363–380zbMATHMathSciNetGoogle Scholar
  24. 24.
    A. P. Calderón, A. Torchinsky, Parabolic maximal functions associated with a distribution, II, Adv. in Math., 1977, 24:101–171zbMATHCrossRefGoogle Scholar
  25. 25.
    A. Miyachi, Hardy space estimate for the product of singular integrals, Canadian J. Math., 2000, 52(2):381–411zbMATHMathSciNetGoogle Scholar
  26. 26.
    S. Lu, D. Yang, The Continuity of commutators on Herz-type spaces, Michigan Math. J., 1997, 44:255–281zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    W. Chen, D. Yang, Z. Zhou, Singular integrals and commutators in parabolic Herz spaces and their applications, Northeast. Math. J., 1998, 14(4):440–454zbMATHMathSciNetGoogle Scholar
  28. 28.
    E. Hernández, D. Yang, Interpolation of Herz-type Hardy spaces, Illinois J. Math., 1998, 42:564–581zbMATHMathSciNetGoogle Scholar
  29. 29.
    G. Hu, S. Lu, D. Yang, The weak Herz spaces, J. Beijing Normal Univ. (Natur. Sci.), 1997, 33:27–34zbMATHGoogle Scholar
  30. 30.
    G. Hu, S. Lu, D. Yang, The applications of weak Herz spaces, Adv. in Math. (China), 1997, 26:417–428zbMATHGoogle Scholar
  31. 31.
    G. Hu, S. Lu, D. Yang, Boundedness of rough singular integral operators on homogeneous Herz spaces, J. Austral. Math. Soc. (Series A), 1999, 66:201–223zbMATHCrossRefGoogle Scholar
  32. 32.
    S. Lu, F. Soria, Sublinear operators on the Beurling algebras with power weights, J. Beijing Normal Univ., 1994, 30:170–175zbMATHGoogle Scholar
  33. 33.
    S. Lu, L. Tang, D. Yang, Boundedness of commutators on homogeneous Herz spaces, Science in China (Ser. A), 1998, 41:1023–1033zbMATHCrossRefGoogle Scholar
  34. 34.
    S. Lu, D. Yang, The Littlewood-Paley function and φ-transform characterizations of a new Hardy space HK 2 associated with the Herz space, Studia Math., 1992, 101:285–298zbMATHMathSciNetGoogle Scholar
  35. 35.
    S. Lu, D. Yang, Multiplier theorems for Herz type Hardy spaces, Proc. Amer. Math. Soc., 1998, 126: 3337–3346zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    D. Yang, Real-variable characterizations of the Hardy spaces HK p(ℝn) (in Chinese), Adv. in Math. (China), 1995, 24:63–73zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  1. 1.Department of MathematicsTokyo Woman's Christian UniversityTokyo 167-8585Japan

Personalised recommendations