A geometrical approach to the sharp Hardy inequality in Sobolev-Slobodecki\u{\i} spaces

TL;DR

This paper explores the sharp Hardy inequality in Sobolev-Slobodecki spaces using a geometric approach, relating the sharp constant to the Cheeger constant and providing new bounds, especially for the case p=1.

Contribution

It introduces a geometric reformulation of the sharp constant in fractional Hardy inequalities and offers new bounds for the one-dimensional case, including non-convex sets.

Findings

Reformulation of the sharp constant as the Cheeger constant for fractional perimeter.

Partial negative answer to an open question on the sharp constant in convex sets.

New lower bounds for the sharp constant in 1D, some optimal for p=1.

Abstract

We give a partial negative answer to a question left open in a previous work by Brasco and the first and third-named authors concerning the sharp constant in the fractional Hardy inequality on convex sets. Our approach has a geometrical flavor and equivalently reformulates the sharp constant in the limit case $p=1$ as the Cheeger constant for the fractional perimeter and the Lebesgue measure with a suitable weight. As a by-product, we obtain new lower bounds on the sharp constant in the $1$-dimensional case, even for non-convex sets, some of which optimal in the case $p=1$.