Harmonic approximation by finite sums of moduli

Let h(Bd) denote the space of real-valued harmonic functions on the unit ball Bd of Rd, d≥2. Given a radial weight w on Bd, consider the following problem: construct a finite family {f1,f2,…,fJ} in h(Bd) such that the sum |f1|+|f2|+⋯+|fJ| 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.