Sharp weighted $L^1$ bounds for Coulomb fields on the sphere

Let $d \ge 2$, let $α_1,\dots, α_n > 0$, and put $A = \sum_k α_k$. We determine, up to constants depending only on $d$, the smallest possible $L^1$ norm in the unit ball $\mathbb{B}^d\subset \mathbb{R}^d$ of the Coulomb field generated by charges of strengths $α_k$ placed on the unit sphere $\mathbb