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 := -del(L)* (vertical bar del(L)u vertical bar(p-2)del(L)u). If phi is a positive weight such that -L-p phi >= 0, then the Hardy-type inequality c integral(Omega) vertical bar u vertical bar(p)/phi(p) vertical bar del(L)phi vertical bar(p) d xi <= integral(Omega) vertical bar del(L)u vertical bar(p) d xi (i is an element of C-0(1)(Omega))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.