A geometrical approach to the sharp Hardy inequality in Sobolev-Slobodecki\uı spaces

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 $p=1$.

arXiv comment29 pages. Updates: several improvements in the geometrical approach, minor typos amended

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• A bound for the perimeter of inner parallel bodies
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