Equivalent and attained version of Hardy's inequality in Rn

We investigate connections between Hardy's inequality in the whole space Rnand embedding inequalities for Sobolev–Lorentz spaces. In particular, we complete previous results due to [1, Alvino] and [28, Talenti] by establishing optimal embedding inequalities for the Sobolev–Lorentz quasinorm ‖∇⋅‖p,qalso in the range p<q, which remained essentially open since [1]. Attainability of the best embedding constants is also studied, as well as the limiting case when q=∞. Here, we surprisingly discover that the Hardy inequality is equivalent to the corresponding Sobolev–Marcinkiewicz embedding inequality. Moreover, the latter turns out to be attained by the so-called “ghost” extremal functions of [6, Brezis–Vazquez], in striking contrast with the Hardy inequality, which is never attained. In this sense, our functional approach seems to be more natural than the classical Sobolev setting, answering a question raised in [6].