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Complex Analysis and Operator Theory

, Volume 6, Issue 5, pp 1011–1023 | Cite as

Factorization of Laplace Operators on Higher Spin Representations

  • David Eelbode
  • Dalibor ŠmídEmail author
Article

Abstract

This paper deals with the problem of factorizing integer powers of the Laplace operator acting on functions taking values in higher spin representations. This is a far-reaching generalization of the well-known fact that the square of the Dirac operator is equal to the Laplace operator. Using algebraic properties of projections of Stein–Weiss gradients, i.e. generalized Rarita–Schwinger and twistor operators, we give a sharp upper bound on the order of polyharmonicity for functions with values in a given representation with half-integral highest weight.

Keywords

Clifford algebras Laplace operator Dirac and Rarita–Schwinger operators 

Mathematics Subject Classification (2000)

Primary 30G35 Secondary 20G05 31A30 

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References

  1. 1.
    Brackx F., Delanghe R., Souček V.: Clifford Analysis. Research Notes in Mathematics, vol. 76. Pitman, London (1982)Google Scholar
  2. 2.
    Brackx F., Eelbode D., Van de Voorde L.: The polynomial null solutions of a higher spin Dirac operator in two vector variables. Adv. Appl. Clifford Algebras 21(3), 455–476 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bureš J.: The Rarita-Schwinger operator and spherical monogenic forms. Complex Variables Theory Appl. 43(1), 77–108 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bureš J., Sommen F., Souček V., Van Lancker P.: Rarita-Schwinger type operators in Clifford analysis. J. Funct. Anal. 185, 425–456 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bureš J., Sommen F., Souček V., Van Lancker P.: Symmetric analogues of Rarita-Schwinger equations. Ann. Global Anal. Geom. 21(3), 215–240 (2001)Google Scholar
  6. 6.
    Constales D., Sommen F., Van Lancker P.: Models for irreducible representations of Spin(m). Adv. Appl. Clifford Algebras 11(S1), 271–289 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    De Schepper H., Eelbode D., Raeymaekers T.: On a special type of solutions of arbitrary higher spin Dirac operators. J. Phys. A 43(32), 1–13 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Delanghe R., Sommen F., Souček V.: Clifford Analysis and Spinor Valued Functions. Kluwer, Dordrecht (1992)CrossRefGoogle Scholar
  9. 9.
    Eelbode D., Šmíd D.: Algebra of invariants for the Rarita-Schwinger operators. Ann. Acad. Sci. Fenn. Math. 34, 637–649 (2009)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Eelbode, D., Šmíd, D.: Polynomial invariants for the Rarita-Schwinger operator. In: Hypercomplex Analysis, pp. 125–135. Birkhäuser Basel, Basel (2009)Google Scholar
  11. 11.
    Rarita W., Schwinger J.: On a theory of particles with half-integral spin. Phys. Rev. Lett. 60(1), 61 (1941)zbMATHGoogle Scholar
  12. 12.
    Stein E., Weiss G.: Generalization of the Cauchy-Riemann equations and representations of the rotation group. Am. J. Math. 90, 163–196 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Van Lancker P.: Rarita-Schwinger fields in the half space. Complex Variables Elliptic Equ. 51(5–6), 563–579 (2006)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of AntwerpAntwerpBelgium
  2. 2.Mathematical InstituteCharles University PraguePraha 8Czech Republic

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