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 := −∇ L (|∇ L u| p−2 ∇ L u). If φ is a positive weight such that −L p φ ≥ 0, then the Hardy-type inequality c |u| p |∇ φ| p dξ ≤ L φp |∇ u| p dξ L 1 u ∈ C 0( ) 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.