Two new proofs of Chui's Conjecture in Weighted Bergman Spaces
Abstract
Chui's conjecture asks whether the average electrostatic field generated by N unit point charges on the unit circle is minimized when the charges are equally spaced. Abakumov, Borichev, and Fedorovskiy proved an analogue of this conjecture in weighted Bergman spaces, showing that the corresponding norm is minimized uniquely by the Nth roots of unity. In this paper, we give two alternative approaches to this weighted minimization problem. The first is a direct calculus-based proof. We reduce the norm to a pairwise interaction energy, identify the equally spaced configuration as a critical point, and analyze the resulting gradient system and Hessian. The second approach connects the problem with discrete energy minimization on spheres by Cohn and Kumar. By rewriting the Bergman interaction as a completely monotonic function of squared Euclidean distance, we recover the minimizing property of the regular N-gon. This second argument also extends the minimization result beyond the concave-weight setting of Abakumov, Borichev, and Fedorovskiy, including the power weights gα(t)=tα for every α>0.
Related
- Norm of the generalized Hilbert operator on weighted Bergman spaces
- Compact linear combination of composition operators on weighted Bergman-Orlicz spaces induced by doubling weights
- The Critical Value for the Beurling-Type Theorem in Weighted Bergman Spaces
- Localization and weighted theory on the Bergman spaces
- Characterizations for quasiconformal weighted Bergman spaces
BibTeX
@misc{corbett2026proofs,
title = {{Two new proofs of Chui's Conjecture in Weighted Bergman Spaces}},
author = {Georgia Corbett},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21121},
eprint = {2609.21121},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21121},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Two new proofs of Chui's Conjecture in Weighted Bergman Spaces
AU - Georgia Corbett
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.21121
UR - https://arxiv.org/abs/2609.21121
AB - Chui's conjecture asks whether the average electrostatic field generated by N unit point charges on the unit circle is minimized when the charges are equally spaced. Abakumov, Borichev, and Fedorovskiy proved an analogue of this conjecture in weighted Bergman spaces, showing that the corresponding norm is minimized uniquely by the Nth roots of unity. In this paper, we give two alternative approaches to this weighted minimization problem. The first is a direct calculus-based proof. We reduce the norm to a pairwise interaction energy, identify the equally spaced configuration as a critical point, and analyze the resulting gradient system and Hessian. The second approach connects the problem with discrete energy minimization on spheres by Cohn and Kumar. By rewriting the Bergman interaction as a completely monotonic function of squared Euclidean distance, we recover the minimizing property of the regular N-gon. This second argument also extends the minimization result beyond the concave-weight setting of Abakumov, Borichev, and Fedorovskiy, including the power weights g_α(t)=t^α for every α>0.
ER -