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Clifford analysis techniques for spherical PDE. (English) Zbl 1047.53023

The present paper investigates \(\alpha\)-order spherical harmonic functions on a sphere with respect to the spherical Laplace-Beltrami operator of order \(\alpha\), i.e. LB + a suitable constant term depending on \(\alpha\). The authors obtain explicit Green-type integral formulae in terms of Gegenbauer functions. They proceed and extend their treatment to higher-order spherical Dirac-type operators, giving intertwining formulae via the Cayley transformation with their Euclidean counterparts. These heavy tasks are performed using the full force of Clifford analysis, which allows factorization of operators in terms of lower-order ones; this being in general impossible within the classical analytical framework.
The paper is clearly written and paves the way for further developments in the field, already envisaged by the authors.

MSC:

53C27 Spin and Spin\({}^c\) geometry
30G35 Functions of hypercomplex variables and generalized variables
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
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