Quantitative stability for weighted Hardy inequalities: Local and nonlocal cases

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 $p>1$, $\alpha \in [0,1)$, $0<p-\alpha p<N$, and $u\in C_c^1({\mathbb{R}}^N)$ normalized by ${\int }_{{\mathbb{R}}^N}\mathrm{\mid }u{\mathrm{\mid }}^p\mathrm{\mid }x{\mathrm{\mid }}^{\alpha p-p}dx=1$, 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 ${\mathcal{M}}_{\alpha }$ is generated by the virtual extremizer with scale-invariant distance. We extend this to the weighted fractional Hardy inequality: for $s\in (0,1)$, $\alpha ={\alpha }_1+{\alpha }_2\ge 0$, ${\alpha }_{1}p,{\alpha }_{2}p\in (-N,sp)$, and $0<sp-\alpha p<N$, 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 $p\ge 2$ 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 $p\ge 2$ and $1<p<2$ stems from the convexity estimate used to obtain a remainder term from the deficit.