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Möbius covariance of iterated Dirac operators

Part of: Generalized function theory

Published online by Cambridge University Press:  09 April 2009

Jaak Peetre  and
Tao Qian
Jaak Peetre
Affiliation:
Institut Mittag-Leffler, Auravägen 17, S-182 62 Djursholm, Sweden, and Department of Mathmatics, University of StockholmBox 6701, S-113 85 Stockholm, Sweden
Tao Qian
Affiliation:
Department of Mathmatics, University of New England, Armidale 2351, Australia.
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Abstract

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Using Fourier transforms, we give a new proof of certain identites for the fundamental solutions of the iterated Dirac operators and Ḏl = (Σ/Σx0 + Ḏ)l. Based on the close relationship between the fundamental solutions and the conformal weights we then give a simple proof of B. Bojarski's results on the conformal covariance of Ḏl. We also prove a new conformal covatiance result of D.

MSC classification

Secondary: 30G35: Functions of hypercomplex variables and generalized variables
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 56 , Issue 3 , June 1994 , pp. 403 - 414
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Ahlfors, L., ‘Old and new in Möbius groups’, Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 93–105.CrossRefGoogle Scholar
[2]Ahlfors, L., ‘Möbius transforms and Clifford Numbers’, in: Differential geometry and complex analysis (eds. Chavel, Isaac and Farkas, Hershel M.) (Springer-Verlag, Berlin, 1985) pp. 65–73.CrossRefGoogle Scholar
[3]Ahlfors, L., ‘Clifford numbers and Möbius transformations in Rn’, Clifford algebras and their role in mathematics and physics (Canterbury, 1985) (eds. Chisholm, J. R. and Common, A. K.) (Reidel, Dordrecht, 1986) pp. 167–175.Google Scholar
[4]Ahlfors, L., ‘Möbius transformations in R n expressed through 2 × 2 Clifford matrices’, Complex Variables Theory Appl. 5 (1986), 215–224.Google Scholar
[5]Bojarski, B., ‘Remarks on polyharmonic operators and conformal maps in spaces’, in: Trudyi Vsesoyuznogo Simpoziuma v Tbilisi 21–23 aprelya 1982 (Tbilisi. Gos. Univ., Tbilis, 1986) pp. 49–56 (Russian).Google Scholar
[6]Bojarski, B., ‘Conformally covariant differential operators’, in: Proc. of the XX Iranian Math. Congress (Teheran, 1989)Google Scholar
[7]Delanghe, R. and Brackx, F., ‘Hypercomplex function theory and Hilbert modules with reproducting kernel’, Proc. London Math. Soc. (3) 37 (1982), 545–576.Google Scholar
[8]Gustafasson, B. and Peetre, J., ‘Notes on projective structures on complex manifolds’, Nagoya Math. J. 116 (1989), 63–88.CrossRefGoogle Scholar
[9]Jakobsen, Hans, ‘Intertwining differential operators for Mp(n, R) and SU(n, n)’, Trans. Amer. Math. Soc. 246 (1978), 311–337.Google Scholar
[10]Jakobsen, Hans and Vergne, Michelle, ‘Wave and Dirac operators, and representations of the conformal group’, J. Funct. Anal. 24 (1977), 52–106.CrossRefGoogle Scholar
[11]Peetre, J., ‘Moebius invariant function space –the case of hyperbolic space’, Mittag-Leffler Report 5 (1990/1991).Google Scholar
[12]Ryan, John, ‘Dirac Operators, Schrödinger Type Operators in Cn and Huygens Principles’, J. Funct. Anal. 87 (1989), 321–347.CrossRefGoogle Scholar
[13]Ryan, John, ‘Iterated Dirac Operators in C n’, Z. Anal. Anwendungen 9 (1990), 385–401.CrossRefGoogle Scholar
[14]Stein, E. M., Singular integrals and differentiability properties of functions (Princeton University Press, Princeton, 1970).Google Scholar