Reproducing kernels in hermitian Clifford analysis

Presenter: Michael Wutzig, Ghent University, Belgium

Joint work with Hendrik De Bie, Frank Sommen

The study of homogeneous polynomials plays an important role in various settings of mathematical analysis. Harmonic homogenous polynomials (which are called spherical harmonics) are a powerful tool in harmonic analysis and provide a connection to Fourier analysis. In Clifford analysis one studies homogeneous polynomials that are null-solutions of the Dirac operator and are therefore named spherical monogenics. In all settings one essential property of homogeneous polynomials is the existence of a unique reproducing kernel. We will give an overview of these kernels, show a way to determine the (yet unknown) reproducing kernel in the hermitian setting of Clifford analysis and show its connection to the case of (complex) harmonic analysis.