On fractional hardy inequalities in convex sets

Abstract

We prove a Hardy inequality on convex sets, for fractional Sobolev-Slobodeckiı spaces of order (s, p). The proof is based on the fact that in a convex set the distance from the boundary is a superharmonic function, in a suitable sense. The result holds for every 1 < p < ∞ and 0 < s < 1, with a constant which is stable as s goes to 1.