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–Slobodeckiĭ space provides an optimal lower bound for the Hardy constant h s,p (Ω) of an open set Ω ⊂ 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 Ω. Moreover, we compute the limit of h s,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 λ s,p (Ω) in terms of h s,p when sp > N , which, in turn, gives an improved Cheeger inequality whose constant does not vanish as p ↗ ∞.