Harmonic approximation by finite sums of moduli

Abstract

Let $h(B_d)$ denote the space of real-valued harmonic functions on the unit ball $B_d$ of $\mathbb{R}^d$, $d\ge 2$. Given a radial weight $w$ on $B_d$, consider the following problem: construct a finite family $\{f_1, f_2, \dots, f_J\}$ in $h(B_d)$ such that the sum $|f_1| + |f_2|+\dots + |f_J|$ is equivalent to $w$. We solve the problem for weights $w$ with a doubling property. Moreover, if $d$ is even, then we characterize those $w$ for which the problem has a solution.