Hardy Type Inequalities Related to Degenerate Elliptic Differential Operators

TL;DR

This paper establishes Hardy type inequalities for degenerate elliptic operators, providing explicit constants and applications to subelliptic operators on Carnot groups, advancing understanding of weighted inequalities in degenerate PDEs.

Contribution

It introduces new Hardy inequalities with explicit, often optimal constants for degenerate elliptic operators, including subelliptic cases on Carnot groups.

Findings

Derived Hardy inequalities with explicit constants

Established conditions for inequalities to hold

Applied results to subelliptic operators on Carnot groups

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 \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.