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 ϕis a positive weight such that -L_pϕ>= 0, then the Hardy type inequality c∫_Ω u^p/ϕ^p{∇_L ϕ}^p dξ≤ ∫_Ω{∇_L u}^p dξholds. 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.