Result: Reproducing kernels of spaces of vector valued monogenics

Let \(Cl( n) \) be a Clifford algebra generated by the vectors \(e_i\) satisfying the relations \(e_ie_j+e_je_i=-2\delta _{ij}\) for \(i,j=1,...,n\). A function \(f:Cl( n) \to Cl( n) \) is called (left) monogenic if \(\partial _xf=\sum_{j=1}^ne_j\partial _xf=0\). The author considers vector valued monogenic functions. For these functions a rich function theory exists, see for example \textit{R. Delanghe, F. Sommen} and \textit{V. Souček} [Clifford algebra and spinor valued functions, Kluwer, Dordrecht (1992; Zbl 0747.53001)]. Without the notation of Clifford algebra they have been studied under the name conjugate harmonic functions. The author studies the question of reproducing kernels for Hilbert spaces of vector valued monogenics. He proves minimum and symmetry properties for these kernels similar to the classical ones. He considers also explicitly the Bergman and the Szegő kernels for the unit ball.