Skip to main content
Log in

A Holomorphic Extension Theorem using Clifford Analysis

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

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

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  5. Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis, Research Notes in Mathematics, vol. 76. Pitman (Advanced Publishing Program), Boston (1982)

  6. Brackx F., De Schepper H., Sommen F.: The Hermitian Clifford analysis toolbox. Adv. Appl. Clifford Algebr. 18(3–4), 451–487 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chen S.J.: The boundary properties of Cauchy type integral in several complex variables. J. Math. Res. Expo. 14(3), 391–398 (1994)

    MATH  Google Scholar 

  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. Delanghe R., Sommen F., Souček V.: Clifford Algebra and Spinor-Valued Functions, Mathematics and its Applications, vol. 53. Kluwer, Dordrecht (1992)

    Google Scholar 

  10. Falconer K.J.: The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, vol. 85. Cambridge University Press, Cambridge (1986)

    Google Scholar 

  11. Federer H.: Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York (1969)

    Google Scholar 

  12. Gaziev A.: Limit values of a Martinelli–Bochner integral. Izv. Vyssh. Uchebn. Zaved. Mat. 9(196), 25–30 (1978)

    MathSciNet  Google Scholar 

  13. Gaziev A.: Necessary and sufficient conditions for continuity of the Martinelli–Bochner integral. Izv. Vyssh. Uchebn. Zaved. Mat. 9, 13–17 (1983)

    MathSciNet  Google Scholar 

  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) 93

    MathSciNet  Google Scholar 

  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)

    Book  Google Scholar 

  16. Gunning, R.C.: Introduction to Holomorphic Functions of Several Variables, 3 vols. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey (1990)

  17. Gürlebeck K., Sprössig W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, New York (1997)

    MATH  Google Scholar 

  18. Guseĭnov, A.I., Muhtarov, H.Š.: Introduction to the theory of nonlinear singular integral equations, “Nauka”, Moscow (1980)

  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. 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)

  21. Kytmanov A.M.: The Bochner–Martinelli Integral and its Applications. Birkhäuser, Basel (1995)

    MATH  Google Scholar 

  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) 173

    MATH  MathSciNet  Google Scholar 

  23. Ma Z.T., Zhang Q.Q.: Boundary properties of Bochner–Martinelli type integrals. Pure Appl. Math. (Xi’an) 18(4), 313–316 (2002)

    MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    Book  Google Scholar 

  26. Mitelman I.M., Shapiro M.V.: Differentiation of the Martinelli–Bochner integrals and the notion of hyperderivability. Math. Nachr. 172, 211–238 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  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. Range R.M.: Complex analysis: a brief tour into higher dimensions. Am. Math. Mon. 110(2), 89–108 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  29. Rocha Chávez R., Shapiro M., Sommen F.: On the singular Bochner–Martinelli integral. Integral Equ. Oper. Theory 32(3), 354–365 (1998)

    Article  MATH  Google Scholar 

  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. Sabadini I., Sommen F.: Hermitian Clifford analysis and resolutions. Math. Methods Appl. Sci. 25(16–18), 1395–1413 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  32. Schneider B.: On the Bochner–Martinelli operator. Appl. Comput. Math. 4(2), 200–209 (2005)

    MathSciNet  Google Scholar 

  33. Sommen F., Peña Peña D.: Martinelli–Bochner formula using Clifford analysis. Archiv. Math. 88(4), 358–363 (2007)

    Article  MATH  Google Scholar 

  34. Tarkhanov D.: Operator algebras related to the Bochner–Martinelli integral. Complex Var. Elliptic Equ. 51(3), 197–208 (2006)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Dixan Peña Peña.

Additional information

Communicated by Daniel Aron Alpay, Ph.D..

Rights and permissions

Reprints and permissions

About this article

Cite this article

Abreu Blaya, R., Bory Reyes, J., Peña Peña, D. et al. A Holomorphic Extension Theorem using Clifford Analysis. Complex Anal. Oper. Theory 5, 113–130 (2011). https://doi.org/10.1007/s11785-009-0037-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11785-009-0037-x

Keywords

Mathematics Subject Classification (2000)

Navigation