Heinrich Begehr

Iterations of Pompeiu Operators

The Pompeiu operator $T$ was extensively used by I. N. Vekua in his treatment of generalized Cauchy-Riemann systems. In the case of several complex variables when polydomains are considered, proper combinations of different $T$-operators for different components of the variable lead to a particular solution of the inhomogeneous Cauchy-Riemann system. This is applied to solve explicitely the Dirichlet problem in the unit polydisc for the inhomogeneous pluriharmonic system in the case of two complex variables.

Mathematics Subject Classification: 47G10, 35J05, 35C15.

Key words and phrases: Pompeiu operator, Cauchy Pompeiu representation, higher order equations, one and several complex variables, boundary value problems, polydomains.