Quantitative stability for fractional Hardy inequalities: Rearrangement-free techniques and Emden-Fowler analysis
A classical result due to Frank and Seiringer asserts that for $1\leq p<\frac Ns$, there exists a sharp constant $\mathcal{C}_{N,s,p}>0$ such that $$ \delta_{s,p}(u):=\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}\,dx\,dy-\mathcal{C}_{N,s,p}\int_{\mathbb{R}^N}\frac{|u(x)|^p}{|x|^{sp}}\,dx\ge0, $$ for all $u\in W^{s,p}(\mathbb{R}^N)$. The optimal constant is explicitly known. We investigate quantitative refinements of this inequality. Our first result shows that, under the normalization $ \int_{\mathbb{R}^N}\frac{|u(x)|^p}{|x|^{sp}}\,dx=1,$ the inequality \[ \delta_{s,p}(u)\gtrsim\bigl(\mathrm{dist}_{s,p}(u,\mathcal{Z})\bigr)^\alpha, \] holds, where $\alpha=\max\{4,2p\}$, $\mathcal{Z}$ denotes the family of ``virtual'' extremals, and the distance is measured in Marcinkiewicz (weak-$L^{p_s^*}$) space. The stability exponent remains constant for $p\le2$, while it depends on $p$ for $p>2$. Our approach is based on a localized Poincar\'e-Sobolev inequality combined with suitable rescaling and Lorentz embeddings. We exploit a decomposition of the nonlocal energy together with Lorentz estimates, which enables us to control the deficit $\delta_{s,p}(u)$ in terms of the distance to $\mathcal{Z}$. The method also applies to the local case $s=1$, the argument is rearrangement-free and the exponent in the stability estimate improves the existing literature. For $p=2$, via an Emden-Fowler correspondence and pseudo-differential operators, we show that the nonlocal Hardy deficit coincides with the local one and obtain quantitative stability on $\mathbb{R}\times\mathbb{S}^{N-1}$ using the diagonalization of the fractional Hardy quadratic form due to Frank, Lieb, and Seiringer. As an application, we establish a Hardy-Heisenberg-type uncertainty principle in the nonlocal setting, which appears to be new in the literature.