球面上Coulomb场的尖锐加权$L^1$界
Sharp weighted $L^1$ bounds for Coulomb fields on the sphere
- St. Petersburg Department of Steklov Mathematical Institute（圣彼得堡斯捷克洛夫数学研究所圣彼得堡分部）
- Department of Mathematics and Computer Science, St. Petersburg State University（圣彼得堡国立大学数学与计算机科学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文确定了单位球面上带权电荷产生的Coulomb场在单位球内的最小$L^1$范数的精确阶，给出了上下界证明，并得到单位权重时阶为$n^{(d-2)/(d-1)}$，三维时为$\sqrt{n}$。
AI中文摘要：
设$d \ge 2$，令$\alpha_1,\dots, \alpha_n > 0$，并设$A = \sum_k \alpha_k$。我们确定了放置在单位球面$\mathbb{S}^{d-1}$上、强度为$\alpha_k$的电荷所产生的Coulomb场在单位球$\mathbb{B}^d\subset \mathbb{R}^d$内的最小可能的$L^1$范数，其精确到仅依赖于$d$的常数。更精确地，我们证明\\[ \inf_{x_1,\dots,x_n \in \mathbb{S}^{d-1}} \int_{\mathbb{B}^d} \left| \sum_{k=1}^n \alpha_k \frac{x_k-x}{|x_k - x|^d} \right| dx \asymp_d A^{-1/(d-1)} \sum_{k=1}^n \alpha_k^{d/(d-1)}. \\]下界来自一个$L^1$ Lipschitz迹估计和球面上的一个显式加权帐篷函数。对于上界，我们将球面划分为质量分别为$\alpha_k/A$、直径为$O_d \left((\alpha_k/A)^{1/(d-1)} \right)$的单元。对于单位权重，尖锐阶为$n^{(d-2)/(d-1)}$，在三维情形下为$\sqrt{n}$。
英文摘要：
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 $\sqrt{n}$.
