First Order Differential Operators Associated to the Space of Monogenic Functions in Parameter-Depending Clifford Algebras

Abstract

This paper is aimed at finding all linear first order partial differential operators \({\mathcal{F}}\) with parameter-depending Clifford-algebra-valued coefficients, that are associated to the generalized Cauchy-Riemann operator. In order to obtain the conditions on the coefficients of \({\mathcal{F}}\), a Leibniz rule for functions with values in a more general Clifford-type algebra is proved. Using the theory of associated spaces, we show the construction of solutions of initial value problems involving these operators.