ON DOMINATING PHENOMENON FOR BOUNDED REAL HARMONIC FUNCTIONS IN SEVERAL VARIABLES

Abstract

In this article, motivated by the work of Brown, Shields, and Zeller[2], we first consider the representation problem of zero by exponential sums in several complex variables. Then we give a complete characterization of dominating sets for bounded real harmonic functions, denoted by $h^\infty(\Re{B_n})$, on the open unit ball $\Re{B_n}$ in $\mathbb{R}^n$, $n \geq 3$, and for bounded real $\mathcal{M}$-harmonic functions, denoted by $\mathcal{M} h^\infty(B_n)$, on the open unit ball $B_n$ in $\mathbb{C}^n$, $n \ge 2$.