**Heinrich Begehr**

## Iterations of Pompeiu Operators

**abstract:**

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.