On fractional Hardy-type inequalities in general open sets

We show that, when $sp>N$, the sharp Hardy constant ${\mathfrak{h}}_{s,p}$ of the punctured space ${\mathbb{R}}^N\setminus \{0\}$ in the Sobolev-Slobodeckiĭ space provides an optimal lower bound for the Hardy constant ${\mathfrak{h}}_{s,p}(\mathrm{\Omega })$ of an open $\mathrm{\Omega }⊊{\mathbb{R}}^N$. The proof exploits the characterization of Hardy’s inequality in the fractional setting in terms of positive local weak supersolutions of the relevant Euler-Lagrange equation and relies on the construction of suitable supersolutions by means of the distance function from the boundary of $\mathrm{\Omega }$. Moreover, we compute the limit of ${\mathfrak{h}}_{s,p}$ as $s\nearrow 1$, as well as the limit when $p\nearrow \mathrm{\infty }$. Finally, we apply our results to establish a lower bound for the non-local eigenvalue ${\lambda }_{s,p}(\mathrm{\Omega })$ in terms of ${\mathfrak{h}}_{s,p}$ when $sp>N$, which, in turn, gives an improved Cheeger inequality whose constant does not vanish as $p\nearrow \mathrm{\infty }$.