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An L2 Ahlfors--Beurling inequality in every dimension
Abstract
Let Ω⊂ℝN (N≥2) be measurable with 0<|Ω|<∞, and let uΩ be its logarithmic potential (N=2) or Newtonian potential (N≥3), normalized so that ΔuΩ=χΩ. We prove the sharp inequality ∫Ω|∇ uΩ|2 d x ≤1/(2NωN−1) iintΩ×Ω(d x d y)/(|x−y|N−2), where ωN−1=|𝕊N−1|, with equality if and only if Ω is a ball up to a null set. In dimension two this settles the L2 Ahlfors--Beurling conjecture of Fleeman and Khavinson. For N≥3 it is equivalent to the scale-free bound that at most the fraction $(N-2)/(2N)$ of the total field energy lies inside Ω. We also obtain an explicit deficit identity, a sharp extension to bounded nonnegative densities, and consequences for torsional rigidity, Bergman analytic content, Brownian motion, and aggregation equations.
Related
- Sharp Higher Integrability Theory, Part I: The Lp-norm of the Beurling--Ahlfors transform
- The sharp Lp bound for the imaginary part of the Beurling-Ahlfors operator on the lattice
- Failure of dimension drop for harmonic measure on Ahlfors regular boundaries of dimension close to $1$
- Failure of dimension drop for harmonic measure on Ahlfors regular flat boundaries
- The anisotropic Michael-Simon inequality for surfaces in every codimension
BibTeX
@misc{wang2026ahlforsbeurling,
title = {{An $L^2$ Ahlfors--Beurling inequality in every dimension}},
author = {Guofang Wang and Mingwei Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37886},
eprint = {2610.37886},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37886},
}
RIS
TY - UNPB
TI - An L^2 Ahlfors--Beurling inequality in every dimension
AU - Guofang Wang
AU - Mingwei Zhang
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37886
UR - https://arxiv.org/abs/2610.37886
AB - Let Ω⊂ℝ^N (N≥2) be measurable with 0<|Ω|<∞, and let u_Ω be its logarithmic potential (N=2) or Newtonian potential (N≥3), normalized so that Δu_Ω=χ_Ω. We prove the sharp inequality ∫_Ω|∇ u_Ω|^2 d x ≤1/(2Nω_N−1) iint_Ω×Ω(d x d y)/(|x−y|^N−2), where ω_N−1=|𝕊^N−1|, with equality if and only if Ω is a ball up to a null set. In dimension two this settles the L^2 Ahlfors--Beurling conjecture of Fleeman and Khavinson. For N≥3 it is equivalent to the scale-free bound that at most the fraction $(N-2)/(2N)$ of the total field energy lies inside Ω. We also obtain an explicit deficit identity, a sharp extension to bounded nonnegative densities, and consequences for torsional rigidity, Bergman analytic content, Brownian motion, and aggregation equations.
ER -