### Hardy-type inequalities related to degenerate elliptic differential operators

We prove some Hardy-type inequalities related to quasilinear second-order degenerate elliptic differential operators ${L}_{p}u:=-{\nabla}_{L}^{*}\left({\left|{\nabla}_{L}u\right|}^{p-2}{\nabla}_{L}u\right)$. If $\phi $ is a positive weight such that $-{L}_{p}\phi \ge 0$, then the Hardy-type inequalityholds. We find an explicit value of the constant involved, which, in most cases, results optimal. As particular case we derive Hardy inequalities for subelliptic operators on Carnot Groups.