Equivalent and attained version of Hardy's inequality in $\mathbb{R}^n$

We investigate connections between Hardy's inequality in the whole space $\mathbb{R}^n$ and embedding inequalities for Sobolev-Lorentz spaces. In particular, we complete previous results due to [A. Alvino, Sulla diseguaglianza di Sobolev in spazi di Lorentz, (1977)] and [G. Talenti, An inequality between $u^*$ and $|{\rm{grad}} u^*|$, (1992)] by establishing optimal embedding inequalities for the Sobolev-Lorentz quasinorm $\|\nabla\,\cdot\,\|_{p,q}$ also in the range $p < q<\infty$, which remained essentially open since the work of Alvino. Attainability of the best embedding constants is also studied, as well as the limiting case when $q=\infty$. 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 [Brezis-V\'azquez, Blow-up solutions of some nonlinear elliptic problems, (1977)], 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 by Brezis and V\'azquez.