On the capacity approach to non-attainability of Hardy's inequality in $\mathbbR^N$

In this note we exploit nonlinear capacity estimates in the spirit of Mitidieri-Pohozaev [15] in the context of Lorentz spaces. This from one side yields a simple proof, though non-optimal, of non-attainability of Hardy's inequality in RN, on the other side gives a partial positive answer to a conjecture raised in [15].