Vekua systems in Hyperbolic Harmonic Analysis

Heikki Orelma, Tampere University of Technology, Finland

We consider the solutions of the equation $\mathcal{M}_\kappa f=0$, where $\mathcal{M}_\kappa$ is the so called modifier Dirac operator acting on functions $f$ defined in the upper half space and taking values in the Clifford algebra. We look for solutions $f(\underline{x},x_n)$ where the first variable is invariant under rotations. A special type of solution is generated by the so called spherical monogenic functions. These solutions may be characterize by a vekua-type system and this system may be solved using Bessel functions. We will see that the solution of the equation $\mathcal{M}_\kappa f=0$ in this case will be a product of Bessel functions.