Fractional Hardy inequalities on C^(1,1) open sets

Let $\Omega$ be a bounded open set of class $C^{1,1}$ in $\mathbb{R}^N$ and $s\in(\frac{1}{2}, 1)$. We study a family of fractional Hardy-type inequalities \begin{equation} \frac{c_{N,s}}{2}\displaystyle\iint_{\Omega\times\Omega}\frac{(u(x)-u(y))^2}{|x-y|^{N+2s}}\ dxdy-\displaystyle\lambda\int_{\Omega}u^2\ dx\geq C\displaystyle\int_{\Omega}\frac{u^2}{\delta^{2s}}\ dx,~~~\quad\forall\lambda\in\mathbb{R},~~~~~~~(0.1) \end{equation} with $u\in C_c^\infty(\Omega)$ and $C=C(\Omega,s,N,\lambda)>0$. We show that the best constant in $(0.1)$ is achieved if and only if $\lambda>\lambda^*(s,\Omega)$, for some $\lambda^*(s,\Omega)\in\mathbb{R}$. As a by-product, we derive in particular that the best constant in Hardy inequality $\mu_{N,s}(\Omega)$ is achieved if and only if $\mu_{N,s}(\Omega)<\mathfrak{h}_{N,s}$, with $\mathfrak{h}_{N,s}$ being the best constant for the fractional Hardy inequality in the half space. Moreover, if $\Omega$ is a convex open set, we obtain a lower bound for $\lambda^*(s,\Omega)$ in terms of the volume of $\Omega$. Specifically, we prove that $\lambda^*(s,\Omega)\geq a(N,s)|\Omega|^{-\frac{2s}{N}}$ with an explicit constant $a(N,s)>0$. Finally, for bounded $C^{1,1}$ domains, we prove that, for $s$ sufficiently close to $\frac{1}{2}$, the optimal Hardy constant is independent of both the geometry and the topology of $\Omega$. More precisely, we establish that $\mu_{N,s}(\Omega)=\mathfrak{h}_{N,s}$. This behavior is in sharp contrast with the local case, where the topology/geometry of the domain strongly influences the value of the optimal constant, and reveals a new rigidity phenomenon in the nonlocal setting.