Functions simultaneously harmonic and M-harmonic in the unit polydisc

Abstract

We give a complete characterization of functions which are at the same time harmonic and M-harmonic in the unit polydisc. These are precisely functions which are harmonic in each of the variables, or equivalently those that can be written as a linear combination of functions which are in each of the variables either holomorphic or conjugate holomorphic. As a consequence we obtain characterization of functions u such that both u and u^s (s ≥ 2) have this property. If we additionally assume that u is real valued, then u is constant. Our results stand in contrast to results known for such functions in the unit ball in C^n.