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–Slobodeckiĭ space provides an optimal lower bound for the Hardy constant hs,p(Ω) of an open set 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 ↗ ∞.