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Complex Analysis and Operator Theory

, Volume 5, Issue 1, pp 113–130 | Cite as

A Holomorphic Extension Theorem using Clifford Analysis

  • Ricardo Abreu Blaya
  • Juan Bory Reyes
  • Dixan Peña PeñaEmail author
  • Frank Sommen
Article

Abstract

In this paper a new holomorphic extension theorem is presented using Clifford analysis.

Keywords

Clifford analysis Isotonic functions Holomorphic extension 

Mathematics Subject Classification (2000)

30G35 32A40 

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References

  1. 1.
    Abreu Blaya R., Bory Reyes J., Peña Peña D., Sommen F.: The isotonic Cauchy transform. Adv. Appl. Clifford Algebr. 17(2), 145–152 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Abreu Blaya R., Bory Reyes J., Shapiro M.: On the notion of the Bochner–Martinelli integral for domains with rectifiable boundary. Complex Anal. Oper. Theory 1(2), 143–168 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Aronov A.M., Kytmanov A.M.: The holomorphy of functions that are representable by the Martinelli–Bochner integral. Funkcional. Anal. i Priložen 9(3), 83–84 (1975)MathSciNetGoogle Scholar
  4. 4.
    Bory Reyes J., Peña Peña D., Sommen F.: A Davydov theorem for the isotonic Cauchy transform. J. Anal. Appl. 5(2), 109–121 (2007)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis, Research Notes in Mathematics, vol. 76. Pitman (Advanced Publishing Program), Boston (1982)Google Scholar
  6. 6.
    Brackx F., De Schepper H., Sommen F.: The Hermitian Clifford analysis toolbox. Adv. Appl. Clifford Algebr. 18(3–4), 451–487 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Chen S.J.: The boundary properties of Cauchy type integral in several complex variables. J. Math. Res. Expo. 14(3), 391–398 (1994)zbMATHGoogle Scholar
  8. 8.
    David G., Semmes S.: Analysis of and on Uniformly Rectifiable Sets. Mathematical Surveys and Monographs, vol. 38. American Mathematical Society, Providence (1993)Google Scholar
  9. 9.
    Delanghe R., Sommen F., Souček V.: Clifford Algebra and Spinor-Valued Functions, Mathematics and its Applications, vol. 53. Kluwer, Dordrecht (1992)Google Scholar
  10. 10.
    Falconer K.J.: The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, vol. 85. Cambridge University Press, Cambridge (1986)Google Scholar
  11. 11.
    Federer H.: Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York (1969)Google Scholar
  12. 12.
    Gaziev A.: Limit values of a Martinelli–Bochner integral. Izv. Vyssh. Uchebn. Zaved. Mat. 9(196), 25–30 (1978)MathSciNetGoogle Scholar
  13. 13.
    Gaziev A.: Necessary and sufficient conditions for continuity of the Martinelli–Bochner integral. Izv. Vyssh. Uchebn. Zaved. Mat. 9, 13–17 (1983)MathSciNetGoogle Scholar
  14. 14.
    Gaziev A.: Some properties of an integral of Martinelli–Bochner type with continuous density. Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 1, 16–22 (1985) 93MathSciNetGoogle Scholar
  15. 15.
    Gilbert J., Murray M.: Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge Studies in Advanced Mathematics, vol. 26. Cambridge University Press, Cambridge (1991)CrossRefGoogle Scholar
  16. 16.
    Gunning, R.C.: Introduction to Holomorphic Functions of Several Variables, 3 vols. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey (1990)Google Scholar
  17. 17.
    Gürlebeck K., Sprössig W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, New York (1997)zbMATHGoogle Scholar
  18. 18.
    Guseĭnov, A.I., Muhtarov, H.Š.: Introduction to the theory of nonlinear singular integral equations, “Nauka”, Moscow (1980)Google Scholar
  19. 19.
    Hörmander L.: An Introduction to Complex Analysis in Several Variables, 3rd edn. North-Holland Mathematical Library, vol. 7. North-Holland, Amsterdam (1990)Google Scholar
  20. 20.
    Krantz, S.G.: Function Theory of Several Complex Variables, 2nd edn. The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove (1992)Google Scholar
  21. 21.
    Kytmanov A.M.: The Bochner–Martinelli Integral and its Applications. Birkhäuser, Basel (1995)zbMATHGoogle Scholar
  22. 22.
    Kytmanov A.M., Aĭzenberg L.A.: The holomorphy of continuous functions that are representable by the Martinelli–Bochner integral. Izv. Akad. Nauk Armjan. SSR Ser. Mat. 13(2), 158–169 (1978) 173zbMATHMathSciNetGoogle Scholar
  23. 23.
    Ma Z.T., Zhang Q.Q.: Boundary properties of Bochner–Martinelli type integrals. Pure Appl. Math. (Xi’an) 18(4), 313–316 (2002)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Martinelli E.: Sulle estensioni della formula integrale di Cauchy alle funzioni analitiche di più variabili complesse. Ann. Mat. Pura Appl. 34(4), 277–347 (1953)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Mattila P.: Geometry of Sets and Measures in Euclidean Spaces, Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  26. 26.
    Mitelman I.M., Shapiro M.V.: Differentiation of the Martinelli–Bochner integrals and the notion of hyperderivability. Math. Nachr. 172, 211–238 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Range R.M.: Holomorphic Functions and Integral Representations in Several Complex Variables, Graduate Texts in Mathematics, vol. 108. Springer, New York (1986)Google Scholar
  28. 28.
    Range R.M.: Complex analysis: a brief tour into higher dimensions. Am. Math. Mon. 110(2), 89–108 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Rocha Chávez R., Shapiro M., Sommen F.: On the singular Bochner–Martinelli integral. Integral Equ. Oper. Theory 32(3), 354–365 (1998)zbMATHCrossRefGoogle Scholar
  30. 30.
    Rocha Chávez R., Shapiro M., Sommen F.: Integral Theorems for Functions and Differential Forms in \({\mathbb C\sp m}\) , Research Notes in Mathematics, vol. 428. Chapman & Hall/CRC, Boca Raton (2002)Google Scholar
  31. 31.
    Sabadini I., Sommen F.: Hermitian Clifford analysis and resolutions. Math. Methods Appl. Sci. 25(16–18), 1395–1413 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Schneider B.: On the Bochner–Martinelli operator. Appl. Comput. Math. 4(2), 200–209 (2005)MathSciNetGoogle Scholar
  33. 33.
    Sommen F., Peña Peña D.: Martinelli–Bochner formula using Clifford analysis. Archiv. Math. 88(4), 358–363 (2007)zbMATHCrossRefGoogle Scholar
  34. 34.
    Tarkhanov D.: Operator algebras related to the Bochner–Martinelli integral. Complex Var. Elliptic Equ. 51(3), 197–208 (2006)zbMATHMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Ricardo Abreu Blaya
    • 1
  • Juan Bory Reyes
    • 2
  • Dixan Peña Peña
    • 3
    Email author
  • Frank Sommen
    • 4
  1. 1.Facultad de Informática y MatemáticaUniversidad de HolguínHolguínCuba
  2. 2.Departamento de MatemáticaUniversidad de OrienteSantiago de CubaCuba
  3. 3.Department of MathematicsAveiro UniversityAveiroPortugal
  4. 4.Department of Mathematical AnalysisGhent UniversityGhentBelgium

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