2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/160382We obtain optimal generalized versions of Hardy inequalities, which as special cases contain Hardy's inequality and Hardy's inequality involving the distance function to the boundary of $ Ω$. In addition we obtain neccesary and sufficient conditions to add improvements in the form of non negative potentials.Analysis of PDEs26D10text