Geodesic
Fischer decompositions in Euclidean and Hermitean Clifford analysis
Archivum mathematicum, Tome 46 (2010) no. 5, pp. 301-321

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Euclidean Clifford analysis is a higher dimensional function theory studying so–called monogenic functions, i.e. null solutions of the rotation invariant, vector valued, first order Dirac operator $\underline{\partial }$. In the more recent branch Hermitean Clifford analysis, this rotational invariance has been broken by introducing a complex structure $J$ on Euclidean space and a corresponding second Dirac operator $\underline{\partial }_J$, leading to the system of equations $\underline{\partial } f = 0 = \underline{\partial }_J f$ expressing so-called Hermitean monogenicity. The invariance of this system is reduced to the unitary group U($n$). In this paper we decompose the spaces of homogeneous monogenic polynomials into U($n$)-irrucibles involving homogeneous Hermitean monogenic polynomials and we carry out a dimensional analysis of those spaces. Meanwhile an overview is given of so-called Fischer decompositions in Euclidean and Hermitean Clifford analysis.
Classification : 30G35
Keywords: Fischer decomposition; Clifford analysis 
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     title = {Fischer decompositions in {Euclidean} and {Hermitean} {Clifford} analysis},
     journal = {Archivum mathematicum},
     pages = {301--321},
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Brackx, Fred; de Schepper, Hennie; Souček, Vladimír. Fischer decompositions in Euclidean and Hermitean Clifford analysis. Archivum mathematicum, Tome 46 (2010) no. 5, pp. 301-321. http://geodesic.mathdoc.fr/item/ARM_2010__46_5_a1/