Existence and non-existence of minimizers for Poincaré–Sobolev inequalities

The Lieb–Thirring Inequality for Interacting Systems in Strong-Coupling Limit

“…This bound implies the Lieb-Thirring inequality (1) with a non-sharp constant because the gradient term is bounded by the kinetic energy, thanks to the Hoffmann-Ostenhof inequality [18]. See [21] for a related upper bound and the application in local density approximation, and see [5,6] for discussions on related interpolation inequalities.…”