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^*(\abs{\nabla_L u}^{p-2}\nabla_L u). If \phi is a positive weight such that -L_p\phi>= 0, then the Hardy type inequality c\int_\Omega \frac{\abs u^p}{\phi ^p}\abs{\nabla_L \phi}^p d\xi \le \int_\Omega\abs{\nabla_L u}^p d\xi holds. 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.