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Fourier Multipliers and Singular Integrals on \(\mathbb {C}^{n}\)

Chapter
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Abstract

In this chapter, we introduce a class of singular integral operators on the n-complex unit sphere. This class of singular integral operators corresponds to bounded Fourier multipliers. Similar to the results of Chaps.  6 and  7, we also develop the fractional Fourier multiplier theory on the unit complex sphere.

References

  1. 1.
    Hua L. Harmonic analysis of several complex in the classical domains. Am Math Soc Transl Math Monogr. 1963;6.Google Scholar
  2. 2.
    Korányi A, Vagi S. Singular integrals in homogeneous spaces and some problems of classical analysis. Ann Sc Norm Super Pisa. 1971;25:575.zbMATHGoogle Scholar
  3. 3.
    Rudin W. Function theory in the unit ball of \({\mathbb{C}}{n}\). New York: Springer; 1980.CrossRefGoogle Scholar
  4. 4.
    Gong S. Integrals of Cauchy type on the ball. Monographs in analysis. Hong Kong: International Press; 1993.Google Scholar
  5. 5.
    David G, Journé J-L, Semmes S. Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation. Rev Mat Iberoam. 1985;1:1–56.CrossRefGoogle Scholar
  6. 6.
    Gaudry G, Qian T, Wang S. Boundedness of singular integral operators with holomorphic kernels on star-shaped Lipschitz curves. Colloq Math. 1996;70:133–50.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gaudry G, Long R, Qian T. A martingale proof of \(L^{2}\)-boundedness of Clifford-valued singular integrals. Ann Math Pura Appl. 1993;165:369–94.MathSciNetCrossRefGoogle Scholar
  8. 8.
    McIntosh A, Qian T. Fourier theory on Lipschitz curves. In: Minicoference on Harmonic Analysis, Proceedings of the Center for Mathematical Analysis, ANU, Canberra, vol. 15; 1987. p. 157–66.Google Scholar
  9. 9.
    McIntosh A, Qian T. \(L^{p}\) Fourier multipliers on Lipschitz curves. Center for mathematical analysis research report, R36-88, ANU, Canberra; 1988.Google Scholar
  10. 10.
    McIntosh A, Qian T. Convolution singular integral operators on Lipschitz curves. Lecture notes in mathematics, vol. 1494, Berlin: Springer;1991. p. 142–62.Google Scholar
  11. 11.
    Qian T. Singular integrals with holomorphic kernels and \(H^{\infty }\)–Fourier multipliers on star-shaped Lipschitz curves. Stud Math. 1997;123:195–216.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Qian T. A holomorphic extension result. Complex Var. 1996;32:58–77.Google Scholar
  13. 13.
    Qian T. Singular integrals with monogenic kernels on the m-torus and their Lipschitz perturbations. In: Ryan J, editor. Clifford algebras in analysis and related topics studies. Advanced Mathematics Series, Boca Raton, CRC Press; 1996. p. 94–108.Google Scholar
  14. 14.
    Qian T. Transference between infinite Lipschitz graphs and periodic Lipschitz graphs. In: Proceeding of the center for mathematics and its applications, ANU, vol. 33; 1994. p. 189–194.Google Scholar
  15. 15.
    Qian T. Singular integrals on star-shaped Lipschitz surfaces in the quaternionic spaces. Math Ann. 1998;310:601–30.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Qian T. Generalization of Fueter’s result to \(R^{n+1}\). Rend Mat Acc Lincei. 1997;8:111–7.MathSciNetCrossRefGoogle Scholar
  17. 17.
    McIntosh A. Operators which have an \(H_{\infty }\)–functional calculus. In: Miniconference on operator theory and partial differential equations, proceedings of the center for mathematical analysis, ANU: Canberra, vol. 14; 1986.Google Scholar
  18. 18.
    Cowling M, Doust I, McIntosh A, Yagi A. Bacach space operators with \(H_{\infty }\) functional calculus. J Aust Math Soc Ser A. 1996;60:51–89.CrossRefGoogle Scholar
  19. 19.
    Li P, Lv J, Qian T. A class of unbounded Fourier multipliers on the unit complex ball. Abstr Appl Anal. 2014; Article ID 602121, p. 8.Google Scholar
  20. 20.
    Li C, McIntosh A, Semmes S. Convolution singular integrals on Lipschitz surfaces. J Am Math Soc. 1992;5:455–81.MathSciNetCrossRefGoogle Scholar
  21. 21.
    Qian T. Fourier analysis on starlike Lipschitz surfaces. J Funct Anal. 2001;183:370–412.MathSciNetCrossRefGoogle Scholar

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© Springer Nature Singapore Pte Ltd. and Science Press 2019

Authors and Affiliations

  1. 1.Macau Institute of Systems EngineeringMacau University of Science and TechnologyMacaoChina
  2. 2.School of Mathematics and StatisticsQingdao UniversityQingdaoChina

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