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Integral Formulas for Higher Order Conformally Invariant Fermionic Operators

  • Chao DingEmail author
Article
  • 19 Downloads

Abstract

In this paper, we establish higher order Borel–Pompeiu formulas for conformally invariant fermionic operators in higher spin theory, which is the theory of functions on m-dimensional Euclidean space taking values in arbitrary irreducible representations of the Spin group. As applications, we provide higher order Cauchy’s integral formulas for those fermionic operators. This paper continues the work of building up basic integral formulas for conformally invariant differential operators in higher spin theory.

Keywords

Fermionic operators Stokes’ theorem Borel–Pompeiu formula Cauchy’s integral formula 

Mathematics Subject Classification

Primary 30Gxx Secondary 42Bxx 46F12 58Jxx 

Notes

Acknowledgements

The author is grateful to the anonymous referees for detailed comments.

References

  1. 1.
    Begehr, H.: Iterated integral operators in Clifford analysis. Z. Anal. Anwendungen 18(2), 361–377 (1999)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Begehr, H., Du, J.Y., Zhang, Z.X.: On higher order Cauchy–Pompeiu formula in Clifford analysis and its applications. Gen. Math. 11, 5–26 (2003)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Begehr, H., Zhang, Z.X., Vu, T.N.H.: Generalized integral representations in Clifford analysis. Complex Var. Elliptic Equ. 51(8–11), 745–762 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Research Notes in Mathematics, vol. 76. Pitman (Advanced Publishing Program), Boston (1982)Google Scholar
  5. 5.
    Bureš, J., Sommen, F., Souček, V., Van Lancker, P.: Rarita–Schwinger type operators in clifford analysis. J. Funct. Anal. 185(2), 425–455 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    De Bie, H., Eelbode, D., Roels, M.: The higher spin Laplace operator. Potential Anal. 47(2), 123–149 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    De Schepper, H., Eelbode, D., Raeymaekers, T.: Twisted higher spin Dirac operators. Complex Anal. Oper. Theory 8(2), 429–447 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ding, C., Ryan, J.: On Some Conformally Invariant Operators in Euclidean Space, Clifford Analysis and Related Topics: In Honor of Paul A. M. Dirac, CART (2014)Google Scholar
  9. 9.
    Ding, C., Ryan, J.: Some properties for the higher spin Laplace operator. Trans. Am. Math. Soc.  https://doi.org/10.1090/tran/7404 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ding, C., Walter, R., Ryan, J.: Construction of arbitrary order conformally invariant operators in higher spin spaces. J. Geometr. Anal. 27(3), 2418–2452 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dunkl, C.F., Li, J., Ryan, J., Van Lancker, P.: Some Rarita–Schwinger type operators. Computat. Methods Funct. Theory 13(3), 397–424 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Eelbode, D., Raeymaekers, T.: Construction of conformally invariant higher spin operators using transvector algebras. J. Math. Phys. (2014).  https://doi.org/10.1063/1.4898772 ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Eelbode, D., Roels, M.: Generalised Maxwell equations in higher dimensions. Complex Anal. Oper. Theory 10(2), 267–293 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Li, J., Ryan, J.: Some operators associated with Rarita–Schwinger type operators. Complex Var. Elliptic Equ. 57(7–8), 885–902 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Porteous, I.: Clifford Algebra and the Classical Groups. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  16. 16.
    Reyes, J.B., De Schepper, H., Adán, A.G., Sommen, F.: Higher order Borel–Pompeiu representations in Clifford analysis. Math. Methods Appl. Sci. 39(16), 4787–4796 (2016)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Ryan, J.: Iterated Dirac operators and conformal transformations in \(R^{n}\). In Proceedings of the XV International Conference on Differential Geometric Methods in Theoretical Physics, World Scientific, pp. 390–399 (1987)Google Scholar
  18. 18.
    Shapiro, M.V.: On some boundary value problems for functions with values in Clifford algebras. Matem. Vesnik. Beograd 40, 321–326 (1988)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Zhang, Z.X.: A revised higher order Cauchy–Pompeiu formulas in Clifford analysis and its application. J. Appl. Funct. Anal. 2(3), 269–278 (2007)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Mathematics/PhysicsBauhaus-Universität WeimarWeimarGermany

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