Abstract
In this article, we prove the ellipticity of higher order conformally invariant differential operators in the higher spin spaces, where functions take values in certain irreducible representations of the spin group in the Euclidean space. We introduce the product formulas for these operators obtained in our previous work to greatly simplify our proofs. In both the bosonic cases and fermionic cases, our product formula enables us to use similar methods as for the higher spin Laplace operator, exploiting the structure of the conformal Lie algebra and branching rules.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brackx, F., Eelbode, D., Van de Voorde, L.: Higher spin Dirac operators between spaces of simplicial monogenics in two vector variables. Math. Phys. Anal. Geom. 14(1), 1–20 (2011)
Bureš, J., Sommen, F., Souček, V., Van Lancker, P.: Rarita–Schwinger type operators in Clifford analysis. J. Funct. Anal. 185(2), 425–455 (2001)
De Bie, H., Eelbode, D., Roels, M.: The higher spin Laplace operator. Potent. Anal. 47(2), 123–149 (2017)
Ding, C., Walter, R., Ryan, J.: Construction of arbitrary order conformally invariant operators in higher spin spaces. J. Geom. Anal. 27, 2418–2452 (2017)
Ding, C., Walter, R., Ryan, J.: Third-order fermionic and fourth-order bosonic operators. In: Breaz, D., Rassias, M. (eds.) Advancements in Complex Analysis. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-40120-7_4
Ding, C., Walter, R., Ryan, J.: Higher order fermionic and bosonic operators. In: Bernstein, S. (eds.) Topics in Clifford Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-23854-4_17
Dunkl, C.F., Li, J., Ryan, J.: Some Rarita–Schwinger type operators. Comput. Methods Funct. Theory 13(3), 397–424 (2013)
Eelbode, D., Roels, M.: Generalised Maxwell equations in higher dimensions. Complex Anal. Oper. Theory 10(2), 267–293 (2016)
Li, J., Ryan, J.: Some operators associated to Rarita–Schwinger type operators. Complex Var. Ellipt. Equ. 57(7–8), 885–902 (2012)
Miller, W.: Symmetry and Separation of Variables. Addison-Wesley Publishing Company, Massachusetts (1977)
Slovák, J.: Natural operators on conformal manifolds, Habilitation thesis. Masaryk University, Brno (1993)
Souček, V.: Higher spins and conformal invariance in Clifford analysis. In: Proc. Symp. Analytical and Numerical methods in Clifford Analysis, Seiffen, pp. 175–186 (1966)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
R.W. acknowledges this material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-0957325 and the University of Arkansas Graduate School Distinguished Doctoral Fellowship in Mathematics and Physics.
This article is part of the Topical Collection on Proceedings ICCA 12, Hefei, 2020, edited by Guangbin Ren, Uwe Kähler, Rafał Abłamowicz, Fabrizio Colombo, Pierre Dechant, Jacques Helmstetter, G. Stacey Staples, Wei Wang.
Rights and permissions
About this article
Cite this article
Ding, C., Walter, R. & Ryan, J. Ellipticity of Some Higher Order Conformally Invariant Differential Operators. Adv. Appl. Clifford Algebras 32, 15 (2022). https://doi.org/10.1007/s00006-022-01198-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00006-022-01198-z