发表机构
Indian Statistical Institute（印度统计研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立局部与分数阶加权Hardy不等式的定量稳定性，证明Hardy亏量控制到虚拟极值函数族的距离，并推广至加权Sobolev嵌入，方法无需重排。
AI 中文摘要
我们在局部和分数阶情形下建立了加权Hardy不等式的定量稳定性估计，表明Hardy亏量控制归一化容许函数到虚拟极值函数族的距离。在局部情形下，对于$p>1$，$\alpha\in[0,1)$，$0<p-\alpha p<N$，以及满足$\int_{\mathbb{R}^N}|u|^p|x|^{\alpha p-p}\\,dx=1$的$u\in C_c^1(\mathbb{R}^N)$，我们证明\begin{equation*} \int_{\mathbb{R}^N}|\nabla u|^p|x|^{\alpha p}\\,dx-\Big(\frac{N-p+\alpha p}{p}\Big)^p\int_{\mathbb{R}^N}\frac{|u|^p}{|x|^{p-\alpha p}}\\,dx \ge C\\,\operatorname{dist}_{\alpha}(u,\mathcal{M}_\alpha)^{\max\{4,2p\}}, \end{equation*}其中$\mathcal{M}_\alpha$由具有尺度不变距离的虚拟极值函数生成。我们将此推广到加权分数阶Hardy不等式：对于$s\in(0,1)$，$\alpha=\alpha_1+\alpha_2\ge0$，$\alpha_1p,\alpha_2p\in(-N,sp)$，且$0<sp-\alpha p<N$，分数阶Hardy亏量满足\begin{equation*} \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}|x|^{\alpha_1p}|y|^{\alpha_2p}\\,dx\\,dy-\mathcal{C}\int_{\mathbb{R}^N}\frac{|u|^p}{|x|^{sp-\alpha p}}\\,dx \ge C\\,\operatorname{dist}_{s,\alpha}(u,\mathcal{M}_{s,\alpha})^{\max\{4,2p\}}. \end{equation*}对于$p\ge2$，距离在$u$上度量；而对于$1<p<2$，距离在$u$的幂型变换上度量。当权重消失时，两者均恢复无权重结果。我们还证明了在有界光滑域（避开原点）上的加权Sobolev和分数阶Sobolev延拓及嵌入不等式。证明无需重排，结合了这些不等式与环上的尺度不变Poincaré–Sobolev估计、伸缩振荡分解以及加权Lorentz空间嵌入。$p\ge2$与$1<p<2$的区别源于用于从亏量获得余项项的凸性估计。
英文摘要
We establish quantitative stability estimates for weighted Hardy inequalities in the local and fractional settings, showing that the Hardy deficit controls the distance of a normalized admissible function to the family of virtual extremizers. In the local case, for $p>1$, $α\in[0,1)$, $0<p-αp<N$, and $u\in C_c^1(\mathbb{R}^N)$ normalized by $\int_{\mathbb{R}^N}|u|^p|x|^{αp-p}\,dx=1$, we prove \begin{equation*} \int_{\mathbb{R}^N}|\nabla u|^p|x|^{αp}\,dx-\Big(\frac{N-p+αp}{p}\Big)^p\int_{\mathbb{R}^N}\frac{|u|^p}{|x|^{p-αp}}\,dx \ge C\,\operatorname{dist}_α(u,\mathcal{M}_α)^{\max\{4,2p\}}, \end{equation*} where $\mathcal{M}_α$ is generated by the virtual extremizer with scale-invariant distance. We extend this to the weighted fractional Hardy inequality: for $s\in(0,1)$, $α=α_1+α_2\ge0$, $α_1p,α_2p\in(-N,sp)$, and $0<sp-αp<N$, the fractional Hardy deficit satisfies \begin{equation*} \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}|x|^{α_1p}|y|^{α_2p}\,dx\,dy-\mathcal{C}\int_{\mathbb{R}^N}\frac{|u|^p}{|x|^{sp-αp}}\,dx \ge C\,\operatorname{dist}_{s,α}(u,\mathcal{M}_{s,α})^{\max\{4,2p\}}. \end{equation*} For $p\ge2$ the distance is measured on $u$, while for $1<p<2$ it is measured on a power-type transformation of $u$. Both recover the unweighted results when the weights vanish. We also prove weighted Sobolev and fractional Sobolev extension and embedding inequalities on bounded smooth domains avoiding the origin. The proof is rearrangement-free, combining these inequalities with scale-invariant Poincaré--Sobolev estimates on annuli, telescoping oscillation decompositions, and weighted Lorentz-space embeddings. The distinction between $p\ge2$ and $1<p<2$ stems from the convexity estimate used to obtain a remainder term from the deficit.
Comments46 pages, 2 figures
