On fractional Hardy-type inequalities in general open sets

We show that, when $sp>N$, the sharp Hardy constant hs,p of the punctured space RN\0\ in the Sobolev-Slobodecki{} space provides an optimal lower bound for the Hardy constant hs,p(Ω) of an open Ω RN. 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 Ω. Moreover, we compute the limit of hs,p as s 1, as well as the limit when p ∞. Finally, we apply our results to establish a lower bound for the non-local eigenvalue λs,p(Ω) in terms of hs,p when $sp>N$, which, in turn, gives an improved Cheeger inequality whose constant does not vanish as p ∞.