发表机构
Faculty of Economics, University of Saida; Faculty of Science, University of Naama（赛达大学经济学院; 纳马大学理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明基于直线段的可见边界判据受内在障碍限制：满足厚度条件的直线可见边界是(n-1)-可求长的，导致此类判据要么不可满足要么多余，无法达到需要可见性的Hardy不等式范围。
AI 中文摘要
在区域 $\Omega\subset\R^n$ 上的加权Hardy不等式可以从边界适当可见部分的低Hausdorff内容下界推导出来，自然希望寻找由直线段描述的可见集合，这比Koskela和Lehrbäck基于曲线的可视边界更容易验证。我们证明这种方法受到内在障碍的限制。如果边界点 $\xi$ 从 $x\in\Omega$ 沿满足均匀厚度条件的线段可见，则 $\Omega$ 包含一个顶点为 $\xi$ 的截锥；因此定量直线可见边界 $\partial^{\mathrm{str}}_{x,\alpha}\Omega\cap B(x,C_0d_\Omega(x))$ 是有限个Lipschitz图的并集，是$(n-1)$-可求长的，并且对任何 $\lambda>n-1$ 不携带 $\lambda$-Hausdorff内容。由于 $\lambda\le n-1$ 的内容条件已经在没有任何可见性假设的情况下蕴含 $\beta<p-n+\lambda$ 的$(p,\beta)$-Hardy不等式，基于线段的判据要么不可满足，要么是多余的：它们永远达不到 $\beta\ge p-1$ 这一真正需要可见性的范围。我们证明指数 $n-1$ 是可以达到的，因此可求长定理是精确的；我们展示一个位于Koskela和Lehrbäck可视边界中但不属于任何 $\partial^{\mathrm{str}}_{x,\alpha}\Omega$ 的边界点；并且我们观察到，没有厚度条件的原始直线可见性可以避开障碍，对 $\lambda\le n-1$ 是充分的，而对 $\lambda>n-1$ 仍然是开放的。
英文摘要
Weighted Hardy inequalities on a domain $Ω\subset\R^n$ 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 $x\inΩ$ along a segment satisfying a uniform thickness condition, then $Ω$ contains a truncated cone with vertex $ξ$; consequently the quantitatively straight visible boundary $\partial^{\mathrm{str}}_{x,α}Ω\cap B(x,C_0d_Ω(x))$ 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 $λ\le n-1$ 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 $β\ge p-1$ 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 $\partial^{\mathrm{str}}_{x,α}Ω$; and we observe that raw straight visibility, without the thickness condition, escapes the obstruction, is sufficient for $λ\le n-1$, and remains open for $λ>n-1$.
