2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/119306We consider Hardy inequalities in $I R^n$, $n \geq 3$, with best constant that involve either distance to the boundary or distance to a surface of co-dimension $k<n$, and we show that they can still be improved by adding a multiple of a whole range of critical norms that at the extreme case become precisely the critical Sobolev norm.22 pagesAnalysis of PDEsFunctional Analysis35J65, 46E35 (26D10, 58J05)text