Jean Dolbeault, Maria J. Esteban, Michael Loss, Luis Vega An analytical proof of Hardy-like inequalities related to the Dirac operator (361K, PDF) ABSTRACT. We prove some sharp Hardy type inequalities related to the Dirac operator by elementary, direct methods. Some of these inequalities have been obtained previously using spectral information about the Dirac-Coulomb operator. Our results are stated under optimal conditions on the asymptotics of the potentials near zero and near infinity.