AI 中文总结
该研究改进了含原点有界域上临界Hardy不等式的定量稳定性估计，将距离指数从$N^2$降至$N$，替换了函数空间框架，无需截断虚拟极值函数，还推广至带对数权的情形，证明基于带余项不等式、尺度不变Sobolev不等式与精细二进求和。
AI 中文摘要
我们建立了含原点有界域上临界Hardy不等式的定量稳定性估计。本结果改进了现有定量稳定性估计：将距离函数中的指数从$N^2$降至$N$，用临界指数Orlicz空间$\text{Exp}L^{\frac{N}{N-1}}(\u03a9)$的Luxemburg范数替代Lorentz-Zygmund框架，且无需对虚拟极值函数做任何截断修正。作为推论，我们还得到了Cianchi和Ferone所研究的带对数权的临界Hardy不等式的改进定量稳定性估计，其指数同样从$N^2$降至$N$，距离直接由虚拟极值函数度量且无需截断。证明过程完全不依赖重排，基于带余项的临界Hardy不等式、尺度不变Sobolev不等式以及精细的二进求和论证。这些要素为两种形式的临界Hardy不等式带来了更强的定量稳定性估计。
英文摘要
We establish a quantitative stability estimate for the critical Hardy inequality on bounded domains containing the origin. Our result improves the existing quantitative stability estimate by reducing the exponent in the distance function from $N^{2}$ to $N$, replacing the Lorentz-Zygmund framework with the Luxemburg norm of the critical exponential Orlicz space $\operatorname{Exp}L^{\frac{N}{N-1}}(Ω)$, and avoiding any cut-off modification of the virtual extremizers. As a consequence, we also obtain an improved quantitative stability estimate for the critical Hardy inequality with the logarithmic weight considered by Cianchi and Ferone, where the exponent is likewise reduced from $N^{2}$ to $N$ and the distance is measured directly from the virtual extremizer without truncation. The proof is completely rearrangement-free and relies on a critical Hardy inequality with a remainder term, scale-invariant Sobolev inequalities, and a refined dyadic summation argument. These ingredients yield stronger quantitative stability estimates for both forms of the critical Hardy inequality.
Comments19 pages
