Conformally invariant powers of the Dirac operator in Clifford analysis. (English) Zbl 1201.30065

Conformal invariance of operators is important, because it is a characteristic of massless field case in physical applications. This paper deals with conformally invariant higher-order operators acting on spinor-valued functions, such that their symbols are given by powers of the Dirac operator. A general classification result proves that these operators are unique, up to a multiplicative constant. A general construction for such invariant operators on manifolds with a given conformal spin structure was described in another paper, generalizing the case of powers of the Laplace operator. Although there is no hope to obtain explicit formulae for higher powers of the Laplace or Dirac operator on a general manifold, it is possible to write down an explicit formula on Einstein manifolds in case of the Laplace operator.
In this paper the question how to explicitly write down the conformally invariant version of powers of the Dirac operator on the sphere is considered. The explicit form of such operators is computed and it is shown that they coincide with operators studied previously by H. Liu and J. Ryan. The methods used in this paper are coming from representation theory combined with traditional Clifford analysis techniques.


30G35 Functions of hypercomplex variables and generalized variables
35J30 Higher-order elliptic equations
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