Sobolev improvements on sharp Rellich inequalities

There are two Rellich inequalities for the bilaplacian, that is, for integral(Delta u)(2)dx, the one involving vertical bar del u vertical bar and the other involving vertical bar u vertical bar 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 vertical bar del u vertical bar 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 vertical bar u vertical bar is obtained.