Rectifiability of the straight visible boundary and the limits of segment-based Hardy criteria
Abstract
Weighted Hardy inequalities on a domain Ω⊂Rn can be deduced from lower Hausdorff-content bounds on suitable visible parts of the boundary, and it is natural to look for visible sets described by straight segments, which are easier to verify than the curve-based visual boundary of Koskela and Lehrbäck. We show that this approach is subject to an intrinsic obstruction. If a boundary point ξ is seen from x∈Ω along a segment satisfying a uniform thickness condition, then Ω contains a truncated cone with vertex ξ; consequently the quantitatively straight visible boundary ∂strx,αΩ∩ B(x,C0dΩ(x)) is a finite union of Lipschitz graphs, is (n−1)-rectifiable, and carries no λ-Hausdorff content for any λ>n−1. Since content conditions with λ≤ n−1 already imply the (p,β)-Hardy inequality for β<p−n+λ without any visibility assumption, segment-based criteria are either unsatisfiable or redundant: they never reach the regime β≥ p−1 in which visibility is genuinely needed. We show that the exponent n−1 is attained, so that the rectifiability theorem is sharp; we exhibit a boundary point lying in the visual boundary of Koskela and Lehrbäck but in no ∂strx,αΩ; and we observe that raw straight visibility, without the thickness condition, escapes the obstruction, is sufficient for λ≤ n−1, and remains open for λ>n−1.
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BibTeX
@misc{abbas2026rectifiability,
title = {{Rectifiability of the straight visible boundary and the limits of segment-based Hardy criteria}},
author = {H. Abbas and A. Azzouz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22407},
eprint = {2609.22407},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22407},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rectifiability of the straight visible boundary and the limits of segment-based Hardy criteria
AU - H. Abbas
AU - A. Azzouz
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22407
UR - https://arxiv.org/abs/2609.22407
AB - Weighted Hardy inequalities on a domain Ω⊂R^n can be deduced from lower Hausdorff-content bounds on suitable visible parts of the boundary, and it is natural to look for visible sets described by straight segments, which are easier to verify than the curve-based visual boundary of Koskela and Lehrbäck. We show that this approach is subject to an intrinsic obstruction. If a boundary point ξ is seen from x∈Ω along a segment satisfying a uniform thickness condition, then Ω contains a truncated cone with vertex ξ; consequently the quantitatively straight visible boundary ∂^str_x,αΩ∩ B(x,C_0d_Ω(x)) is a finite union of Lipschitz graphs, is (n−1)-rectifiable, and carries no λ-Hausdorff content for any λ>n−1. Since content conditions with λ≤ n−1 already imply the (p,β)-Hardy inequality for β<p−n+λ without any visibility assumption, segment-based criteria are either unsatisfiable or redundant: they never reach the regime β≥ p−1 in which visibility is genuinely needed. We show that the exponent n−1 is attained, so that the rectifiability theorem is sharp; we exhibit a boundary point lying in the visual boundary of Koskela and Lehrbäck but in no ∂^str_x,αΩ; and we observe that raw straight visibility, without the thickness condition, escapes the obstruction, is sufficient for λ≤ n−1, and remains open for λ>n−1.
ER -