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

Abstract

Let $d\ge 2$, let ${\alpha }_1,\dots ,{\alpha }_n>0$, and put $A={\sum }_k{\alpha }_k$. 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 ${\alpha }_k$ 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 ${\alpha }_k/A$ and diameters $O_d\left(({\alpha }_k/A{)}^{1/(d-1)}\right)$. For the unit weights the sharp order is $n^{(d-2)/(d-1)}$, and in dimension three it is $\sqrt{n}$.

12 pages, no figures. Comments are welcome!