On fractional Hardy-type inequalities in general open sets

We show that, when sp>N, the sharp Hardy constant hₒ, ₏ of the punctured space RN\0\ in the Sobolev-Slobodecki space provides an optimal lower bound for the Hardy constant hₒ, ₏ () of an open 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. Moreover, we compute the limit of hₒ, ₏ 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 ₒ, ₏ () in terms of hₒ, ₏ when sp>N, which, in turn, gives an improved Cheeger inequality whose constant does not vanish as p.