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Hardy Type Inequalities Related to Degenerate Elliptic Differential Operators

Lorenzo D'Ambrosio
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Abstract

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 ϕis a positive weight such that -L_pϕ>= 0, then the Hardy type inequality c\int_Ω\frac{\abs u^p}{ϕ^p}\abs{\nabla_L ϕ}^p dξ\le \int_Ω\abs{\nabla_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.

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Lorenzo D'Ambrosio. 2006-03-08. Hardy Type Inequalities Related to Degenerate Elliptic Differential Operators. https://arxiv.org/abs/math/0603187

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