Slow growth for universal harmonic functions

Abstract

Given any continuous increasing function : φ [ 0, + λ [ → ] 0, + λ [ such that lim t → log φ (t)/log t = + λ, we show that there are harmonic functions H on ℝN satisfying the inequality | H (x) | ≤ φ (x) for every x ε FN, 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. Copyright © 2010 M. Carmen Gómez-Collado et al.