Science in China Series A: Mathematics

, Volume 42, Issue 12, pp 1233–1245

A class of singular integrals on then-complex unit sphere

  • Michael Cowling
  • Tao Qian
Article

Abstract

The operaton on the n-complex unit sphere under study have three forms: the singular integrals with holomorphic kernels, the bounded and holomorphic Fourier multipliers, and the Cauchy-Dunford bounded and holomorphic functional calculus of the radial Dirac operator\(D = \sum\nolimits_{k = 1}^n {z_k \frac{\partial }{{\partial _{z_k } }}} \). The equivalence between the three fom and the strong-type (p,p), 1 <p < ∞, and weak-type (1,1)-boundedness of the operators is proved. The results generalise the work of L. K. Hua, A. Korányli and S. Vagi, W. Rudin and S. Gong on the Cauchy-Szegö, kemel and the Cauchy singular integral operator.

Keywords

singular integral Fourier multiplier the unit sphere in Cn lunetional calculus 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Hua, L., Hannonic analysis of several complex variables in the classical domains,Amer. Math. Soc. Tranl. Math. Monograph 6, 1963.Google Scholar
  2. 2.
    Korónyi, A., Vagi, S., Singular integrals in homogeneous spaces and some problems of classical analysis,Ann. Scuola Normale Superiore Pisa, 1971, 25: 575.Google Scholar
  3. 3.
    Rudin, W.,Function Theory in the Unit Ball of C n. New York: Springer-Verlag, 1980.Google Scholar
  4. 4.
    Gong, S., Integrals of Cauchy type on the ball,Monograph in Analysis, Hong Kong: International Press, 1993, 6.Google Scholar
  5. 5.
    Stein, E. M.,Singular Integrals and Differentiability Properties of Functions, Princeton: Frinceton University Press, 1970.MATHGoogle Scholar
  6. 6.
    David, G. Journé, J. L., et Semmes, S., Opérateurs de Calderén-Zygmund fonctions persaccrétives et interpolation,Rev. Mat. Iberoamricana, 1985,1: 1.MATHGoogle Scholar
  7. 7.
    Coifman, R., Meyer, Y., Fourier analysis of multilinear convolutions, Calderón’ s theorem, and analysis on Lipschitz curves,Lecture Notes in Mathematics, New York: Springer-Verlag, 1980, 779: 104.Google Scholar
  8. 8.
    McIntosh, A., Qian, T., Convolution singular integral operators on Lipschitz curves, inProc. of the Special Year on Harmonic Analysis at Nankai Inst. of. Math., Tianjin, China, 1991, 1494: 142.MathSciNetGoogle Scholar
  9. 9.
    McIntosh, A., Qian, T.,L p Fourier multipliers on Lipschitz curves,Trans. Amer. Math. Soc., 1992, 333: 157.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Li, C., McIntosh, A., Qian, T., Clifford algebras, Fourier transforms, and singular convolution operators on Lipschitz surfaces,Revista Matemátical Iberomericana, 1994, 10(3): 665.MATHMathSciNetGoogle Scholar
  11. 11.
    Li, C., McIntosh, A., Semmes, S., Convolution singular integrals on Lipschitz surfaces,J. Amer. Math. Soc. 1992, 5: 455.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Gaudry, G., Long, R. L., Qian, T., A Martingale proof ofL 2-boundedness of Clifford-valued singular integrals,Annali di Mathematica Pura Ed Applicata, 1993, 165: 369.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Tao, T., Convolution operators on Lipschitz graphs with harmonic kernels,Adaarces in Applied Cligord Algebras, 1996, 6(2): 207.MATHGoogle Scholar
  14. 14.
    Gaudry, G., Qian, T., Wang, S. L., Boundedness of singular integral operators with holomorphic kernels on star-shaped Lipschitz curves,Colloq. Math. 1996, LXX: 133.MathSciNetGoogle Scholar
  15. 15.
    Qian, T., Ryan, J., Conformal transformations and Hardy spaces arising in CliEod analysis,Journal of Operator theory, 1996, 35: 349.MATHMathSciNetGoogle Scholar
  16. 16.
    Qian, T., Singular inkgals with holomorphic kernels andH∞-Fourier multipliers on star-shaped Liphits curves,Shudia Mathematica. 1997, 123(3): 195.MATHGoogle Scholar
  17. 17.
    Qian, T., A holomorphic extension result,Complex Variables, 1997, 32(1): 59.MathSciNetGoogle Scholar
  18. 18.
    Qian. T., Singular integrals on star-shaped Iipschitz surfaces in the quaternionic space,Mothematische Annalen, 1998, 310 (4): 601.MATHCrossRefGoogle Scholar
  19. 19.
    Qian, T., Generalization of Fueter’ s result to ℝn n+1,Rend. Mat. Acc. Lincei, 1997, 9(8): 111.Google Scholar
  20. 20.
    Qian, T., Singular integrals on the m-torus and its Lipschitz perturbations, Cltfird Algebras in Analysis and Related Topics, Studies in Advanced Mathematics (ed. Ryan, J.), Florida: CRC Press, 1995, 94–108.Google Scholar
  21. 21.
    Khavinson, D., A remark on a paper of Qian,Complex Variables, 1997, 32: 341.MATHMathSciNetGoogle Scholar
  22. 22.
    McIntosh, A., Operators which have anH∞-functional calculus,Miniconference on 0perator Theory and Partial Differential Equations, Proc Centre Math Analysis, A. N. U., Canberra, 1986, 14: 210.MathSciNetGoogle Scholar
  23. 23.
    Cowling, M., Doust, I., McIntosh, A. et al., Banach space operators with a boundedH∞ functional calculus,J. Austral. Math. Soc., Ser. A, 1996, 60: 51.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Science in China Press 1999

Authors and Affiliations

  • Michael Cowling
    • 1
  • Tao Qian
    • 2
  1. 1.Department of Pure MathematicsUniversity of New South WalseAustralia
  2. 2.School of Mathematical and Computer SciencesUniversity of New England ArmidaleAustralia

Personalised recommendations