Slow Growth for Universal Harmonic Functions

Abstract

Given any continuous increasing function ϕ:[0,+∞[→]0,+∞[ such that lim⁡t→∞log⁡ϕ(t)/log⁡t=+∞, we show that there are harmonic functions H on ℝN satisfying the inequality |H(x)|≤ϕ(∥x∥) for every x∈ℝN, which are universal with respect to translations. This answers positively a problem of D. H. Armitage (2005). The proof combines techniques of Dynamical Systems and Operator Theory, and it does not need any result from Harmonic Analysis.