On the Representation of Generalized Analytic Vectors

Abstract

This paper is concerned with the representation of generalized analytic vectors which are defined as solutions of certain elliptic systems of linear first-order partial differential equations in two complex variables. For the explicite representation of solutions we give integral operators of Vekua type as well as differential operators of Bauer type which map holomorphic vectors onto the set of solutions. For the existence of the differential operators we give a necessary and sufficient condition. We discuss properties of these operators and show how to construct the kernels of the integral operators using the differential operators. Finally we present examples for systems of differential equations for which all the solutions can be represented explicitely.