Calculus of Variations and Geometric Measure Theory

F. Bianchi - L. Brasco - A. C. Zagati

On the sharp Hardy inequality in Sobolev-Slobodeckii spaces

created by zagati on 07 Sep 2022
modified by brasco on 30 Dec 2023


Accepted Paper

Inserted: 7 sep 2022
Last Updated: 30 dec 2023

Journal: Math. Ann.
Pages: 47
Year: 2023

ArXiv: 2209.03012 PDF


We study the sharp constant in the Hardy inequality for fractional Sobolev spaces defined on open subsets of the Euclidean space. We first list some properties of such a constant, as well as of the associated variational problem. We then restrict the discussion to open convex sets and compute such a sharp constant, by constructing suitable supersolutions by means of the distance function. Such a method of proof works only for $s\,p\ge 1$ or for $\Omega$ being a half-space. We exhibit a simple example suggesting that this method can not work for $s\,p<1$ and $\Omega$ different from a half-space. The case $s\,p<1$ for a generic convex set is left as an interesting open problem, except in the Hilbertian setting (i.e. for $p=2$): in this case we can compute the sharp constant in the whole range $0<s<1$. This completes a result which was left open in the literature.

Keywords: fractional Sobolev spaces, Nonlocal operators, Hardy inequality