Möbius covariance of iterated Dirac operators. (English) Zbl 1043.31500

Peetre, Jaak; Qian, Tao

J. Aust. Math. Soc., Ser. A 56, No. 3, 403-414 (1994).

In this paper the authors consider Möbius covariance of the iterated Dirac operators \(\underline D^l=(\sum_{i=1,n}e_i\partial_i)^l\). These questions are based on fundamental papers of L. Ahlfors in the period of 1984–1986. Previous results by B. Bojarski, mostly in a 1989 paper by him [in Proc. XXth Iranian Mathematics Conference (Tehran, 1989), Iranian Math. Soc., Mashhad (1989)], can be obtained by using a Fourier transform technique. The close relationship between fundamental solutions and conformal weights is also worked out. For the operator \(D=\partial_0+\underline D\) a corresponding covariance result could be obtained. The advantage of this approach seems to lie in the possiblity of generalizing this to some Lie groups.
In this connection the reviewer would like to mention some additional references: R. J. Baston, Duke Math. J. 63, 81–112, 113–138 (1991; Zbl 0724.53019 and Zbl 0724.53020); R. J. Baston and M. G. Eastwood, in Twistors in mathematics and physics, Cambridge Univ. Press, Cambridge 1990, 129–163 (1990; Zbl 0702.53003); T. P. Branson, Proc. Am. Math. Soc. 126, 1031–1042 (1998 Zbl 0890.47030); V. Wünsch, Math. Nachr. 129, 269–281 (1986; Zbl 0619.53008); J. Slovák, in Differential geometry and its applications (Opava, 1992), Silesian Univ.. Math. Publ. (Opava) 1, 335–349 (1993; Zbl 0805.53011).

Reviewer: Wolfgang Sprössig (MR1271529)

MSC:

31B99	Higher-dimensional potential theory
30G35	Functions of hypercomplex variables and generalized variables
35J99	Elliptic equations and elliptic systems

Citations:

Zbl 0724.53019; Zbl 0724.53020; Zbl 0702.53003; Zbl 0619.53008; Zbl 0890.47030; Zbl 0805.53011