On fractional Hardy-type inequalities in general open sets

¹ Dipartimento di Matematica, Alma Mater Studiorum Università di Bologna, piazza di Porta San Donato 5, 40126 Bologna, Italy
² Dipartimento di Scienze Agrarie, Alimentari e Agro-ambientali, Università di Pisa, Via del Borghetto 80, 56124 Pisa, Italy

^* Corresponding author: francesca.prinari@unipi.it

Received: 10 July 2024
Accepted: 24 August 2024

Abstract

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 ↗ ∞.