We consider ordinary and fractional Schrödinger operators with Hardy potentials. These operators generate natural scales of homogeneous Sobolev spaces, which we compare with the ordinary homogeneous Sobolev spaces. For local operators, such a comparison was used to study the global well-posedness and scattering for non-linear Schrödinger equations with inverse-square potential. In this talk, we apply the equivalence of Sobolev norms to analyze the ground state density of large relativistic atoms close to the nucleus. The talk is based on joint works with Rupert Frank, Heinz Siedentop, and Barry Simon.