On the sharp Hardy inequality in Sobolev–Slobodeckiĭ spaces

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 $sp\ge 1$ or for $\Omega$ being a half-space. We exhibit a simple example suggesting that this method can not work for $sp<1$ and $\Omega$ different from a half-space. The case $sp<1$ 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 $0<1$ . This completes a result which was left open in the literature.