On fractional Hardy-type inequalities in general open sets

We show that, when sp > N, the sharp Hardy constant h(s,p) of the punctured space R-N \ {0} in the Sobolev-Slobodeckii space provides an optimal lower bound for the Hardy constant h(s,p)(Omega) of an open set Omega subset of 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 Omega. Moreover, we compute the limit of h(s,p) as s NE arrow 1, as well as the limit when p NE arrow infinity. Finally, we apply our results to establish a lower bound for the non-local eigenvalue lambda(s,p)(Omega) in terms of h(s,p) when sp > N, which, in turn, gives an improved Cheeger inequality whose constant does not vanish as p NE arrow infinity.