  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.

Classical Analysis and ODEs