An improved Hardy-Sobolev inequality in W1,p and its application to Schrödinger operators

In this paper we prove new Hardy-like inequalities with optimal potential singularities for functions in W1,p(Ω), where Ω is either bounded or the whole space Rn and also some extensions to arbitrary Riemannian manifolds. We also study the spectrum of perturbed Schrödinger operators for perturbations which are just below the optimality threshold for the corresponding Hardy inequality.ou