Sobolev improvements on sharp Rellich inequalities

There are two Rellich inequalities for the bilaplacian, that is, for ∫ (Δ u)²dx , the one involving |∇ u| and the other involving |u| at the RHS. In this article, we consider these inequalities with sharp constants and obtain sharp Sobolev-type improvements. More precisely, in our first result, we improve the Rellich inequality with |∇ u| obtained by Beckner in dimensions n=3,4 by a sharp Sobolev term, thus complementing existing results for the case n≥ 5 . In the second theorem, the sharp constant of the Sobolev improvement for the Rellich inequality with |u| is obtained.