On Second Order Elliptic Systems of Partial Differential Equations in Clifford Analysis

Abstract

The paper deals with two second order elliptic systems of partial differential equations in Clifford analysis. They are of the form \({^\phi \!\underline{\partial }}f{^\psi \!\underline{\partial }}=0\) and \(f{^\phi \!\underline{\partial }}{^\psi \!\underline{\partial }}=0\), where \({^\phi \!\underline{\partial }}\) stands for the Dirac operator related to a structural set \(\phi \). Their solutions, known as \((\phi ,\psi )\)-inframonogenic and \((\phi ,\psi )\)-harmonic functions, not every enjoy the nice properties and usual structure of the harmonic ones. We describe the precise relation between these two classes of functions and show their strong link to the Laplace operator. Finally, we apply a multi-dimensional Ahlfors-Beurling transform, to prove that some relative function spaces are indeed isomorphic.