PreprintCC BY 4.0arXiv
Sharp weighted L1 bounds for Coulomb fields on the sphere
Abstract
Let d ≥ 2, let α1,…, αn > 0, and put A = ∑k αk. We determine, up to constants depending only on d, the smallest possible L1 norm in the unit ball 𝔹d⊂ ℝd of the Coulomb field generated by charges of strengths αk placed on the unit sphere 𝕊d−1. More precisely, we prove infx1,…,xn ∈ 𝕊d−1 ∫𝔹d | ∑k=1n αk (xk−x)/(|xk − x|d) | dx asympd A−1/(d−1) ∑k=1n αkd/(d−1). The lower bound follows from an L1 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 αk/A and diameters Od ((αk/A)1/(d−1) ). For the unit weights the sharp order is n(d−2)/(d−1), and in dimension three it is √n.
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BibTeX
@misc{doubtsov2026sharp,
title = {{Sharp weighted $L^1$ bounds for Coulomb fields on the sphere}},
author = {Evgueni Doubtsov and Ioann Vasilyev and Grigory Voinov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35123},
eprint = {2609.35123},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35123},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp weighted L^1 bounds for Coulomb fields on the sphere
AU - Evgueni Doubtsov
AU - Ioann Vasilyev
AU - Grigory Voinov
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35123
UR - https://arxiv.org/abs/2609.35123
AB - Let d ≥ 2, let α_1,…, α_n > 0, and put A = ∑_k α_k. We determine, up to constants depending only on d, the smallest possible L^1 norm in the unit ball 𝔹^d⊂ ℝ^d of the Coulomb field generated by charges of strengths α_k placed on the unit sphere 𝕊^d−1. More precisely, we prove inf_x_1,…,x_n ∈ 𝕊^d−1 ∫_𝔹^d | ∑_k=1^n α_k (x_k−x)/(|x_k − x|^d) | dx asymp_d A^−1/(d−1) ∑_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 ((α_k/A)^1/(d−1) ). For the unit weights the sharp order is n^(d−2)/(d−1), and in dimension three it is √n.
ER -