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

Abstract

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 $\mathbb{S}^{d-1}$. More precisely, we prove \[ \inf_{x_1,\dots,x_n \in \mathbb{S}^{d-1}} \int_{\mathbb{B}^d} \left| \sum_{k=1}^n α_k \frac{x_k-x}{|x_k - x|^d} \right| dx \asymp_d A^{-1/(d-1)} \sum_{k=1}^n α_k^{d/(d-1)}. \] The lower bound follows from an $L^1$ Lipschitz trace estimate and an explicit weighted tent function on the sphere. For the upper bound we use a partition of the sphere into cells of masses $α_k/A$ and diameters $O_d \left((α_k/A)^{1/(d-1)} \right)$. For the unit weights the sharp order is $n^{(d-2)/(d-1)}$, and in dimension three it is $\sqrt{n}$.