Journal of Geometric Analysis

, Volume 20, Issue 3, pp 670–689

Discrete Calderón’s Identity, Atomic Decomposition and Boundedness Criterion of Operators on Multiparameter Hardy Spaces

Article

Abstract

In this paper we establish a discrete Calderón’s identity which converges in both Lq(ℝn+m) (1<q<∞) and Hardy space Hp(ℝn×ℝm) (0<p≤1). Based on this identity, we derive a new atomic decomposition into (p,q)-atoms (1<q<∞) on Hp(ℝn×ℝm) for 0<p≤1. As an application, we prove that an operator T, which is bounded on Lq(ℝn+m) for some 1<q<∞, is bounded from Hp(ℝn×ℝm) to Lp(ℝn+m) if and only if T is bounded uniformly on all (p,q)-product atoms in Lp(ℝn+m). The similar result from Hp(ℝn×ℝm) to Hp(ℝn×ℝm) is also obtained.

Keywords

Boundedness Calderón-Zygmund operator Calderón’s identity Multiparameter Hardy spaces Atomic decomposition Boundedness criterion of operators 

Mathematics Subject Classification (2000)

42B30 42B20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bownik, M.: Boundedness of operators on Hardy spaces via atomic decompositions. Proc. Am. Math. Soc. 133(12), 3535–3542 (2005) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Carleson, L.: A counterexample for measures bounded on H p for the bidisc. Mittag-Leffler Report No. 7, 1974 Google Scholar
  3. 3.
    Chang, S.Y.A., Fefferman, R.: A continuous version of duality of H 1 with BMO on the bidisc. Ann. Math. 112, 179–201 (1980) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Chang, S.Y.A., Fefferman, R.: The Calderón-Zygmund decomposition on product domains. Am. J. Math. 104(3), 445–468 (1982) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Chang, S.Y.A., Fefferman, R.: Some recent developments in Fourier analysis and H p theory on product domains. Bull. Am. Math. Soc. 12, 1–43 (1985) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Coifman, R.R.: A real variable characterization of H p. Stud. Math. 51, 269–274 (1974) MATHMathSciNetGoogle Scholar
  7. 7.
    Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Deng, D., Han, Y.: Harmonic analysis on spaces of homogeneous type. Lecture Notes in Mathematics, vol. 1966. Springer, Berlin (2009) (with a preface by Yves Meyer) MATHGoogle Scholar
  9. 9.
    Fefferman, R.: Harmonic analysis on product spaces. Ann. Math. 126, 109–130 (1987) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Han, Y., Lu, G.: Endpoint estimates for singular integral operators and Hardy spaces associated with Zygmund dilations. Preprint (2008) Google Scholar
  11. 11.
    Han, Y., Lu, G.: Discrete Littlewood-Paley-Stein theory and multi-parameter Hardy spaces associated with the flag singular integrals. http://arxiv.org/abs/0801.1701
  12. 12.
    Han, Y., Sawyer, E.: Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces. Mem. Am. Math. Soc. 110(530), 1–126 (1994) MathSciNetGoogle Scholar
  13. 13.
    Han, Y., Zhao, K.: Boundedness of operators on Hardy spaces. Taiwan. J. Math. (to appear) Google Scholar
  14. 14.
    Han, Y., Lu, G., Ruan, Z.: Boundedness criterion of Journé’s class of singular integrals on multiparameter Hardy spaces. Preprint (2009) Google Scholar
  15. 15.
    Han, Y., Lee, M., Lin, C., Lin, Y.: Calderón-Zygmund operator on product spaces: H p theory. J. Funct. Anal. (to appear) Google Scholar
  16. 16.
    Han, Y., Li, J., Lu, G.: Duality of multiparameter Hardy space H p on product spaces of homogeneous type. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (to appear) Google Scholar
  17. 17.
    Han, Y., Li, J., Lu, G., Wang, P.: H pH p boundedness implies H pL p boundedness. Forum Math. (to appear) Google Scholar
  18. 18.
    Journé, J.-L.: Calderón-Zygmund operators on product space. Rev. Mat. Iberoam. 1(3), 55–91 (1985) MATHGoogle Scholar
  19. 19.
    Latter, R.H.: A decomposition of H p(ℝn) in terms of atoms. Stud. Math 62, 92–101 (1978) MathSciNetGoogle Scholar
  20. 20.
    Meda, S., Sjögren, P., Vallarino, M.: On the H 1-L 1 boundedness of operators. Proc. Am. Math. Soc. 136, 2921–2931 (2008) MATHCrossRefGoogle Scholar
  21. 21.
    Meyer, Y.: Wavelets and Operators. Cambridge University Press, Cambridge (1992) MATHGoogle Scholar
  22. 22.
    Meyer, Y., Coifman, R.R.: Wavelets, Calderón-Zygmund and multilinear operators. Cambridge Univ. Press, Cambridge (1997) MATHGoogle Scholar
  23. 23.
    Ricci, F., Verdera, J.: Duality in spaces of finite linear combinations of atoms. arXiv:0809.1719v3
  24. 24.
    Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Univ. Press, Princeton (1993) MATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2010

Authors and Affiliations

  1. 1.Department of MathematicsAuburn UniversityAuburnUSA
  2. 2.Department of MathematicsWayne State UniversityDetroitUSA
  3. 3.College of MathematicsQingdao UniversityQingdaoChina

Personalised recommendations