The Hardy inequality for Sobolev-Slobodeckiı̆ spaces
We discuss the Hardy inequality for fractional Sobolev-Slobodeckiı̆ spaces defined on open sets of the Euclidean space. We give some properties of the sharp constant for such an inequality in general open sets and then particularize the discussion to the case of open convex sets.
In this last case, we show how to determine the sharp constant, by constructing explicit supersolutions based on the distance function. We also show that this method works only for the mildly nonlocal regime and it is bound to fail for the strongly nonlocal one. We conclude by presenting some open problems.
Some of the results presented are issued from papers in collaboration with Francesca Bianchi (Ferrara & Parma), Eleonora Cinti (Bologna) and Anna Chiara Zagati (Ferrara & Parma).