2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/173998We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in the rearrangement inequality in fractional Sobolev spaces.AMSLaTeX, 22 pages; extension of the result to the case N<psAnalysis of PDEsMathematical PhysicsSpectral Theorytext