The Cauchy–Kovalevskaya extension theorem in Hermitian Clifford analysis

Abstract

Hermitian Clifford analysis is a higher dimensional function theory centered around the simultaneous null solutions, called Hermitian monogenic functions, of two Hermitian conjugate complex Dirac operators. As an essential step towards the construction of an orthogonal basis of Hermitian monogenic polynomials, in this paper a Cauchy–Kovalevskaya extension theorem is established for such polynomials. The minimal number of initial polynomials needed to obtain a unique Hermitian monogenic extension is determined, along with the compatibility conditions they have to satisfy. The Cauchy–Kovalevskaya extension principle then allows for a dimensional analysis of the spaces of spherical Hermitian monogenics, i.e. homogeneous Hermitian monogenic polynomials. A version of this extension theorem for specific real-analytic functions is also obtained.