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Hardy-type inequalities related to degenerate elliptic differential. (English) Zbl 1170.35372
Summary: We prove some Hardy-type inequalities related to quasilinear second-order degenerate elliptic differential operators \(L_pu:=-\nabla^*_L(|\nabla_Lu|^{p-1} \nabla_Lu)\). If \(\varphi\) is a positive weight such that \(-L_p\varphi\geq 0\), then the Hardy-type inequality \[ c\int_\Omega\frac{|u|^p}{\varphi^p}|\nabla_L \varphi|^p\,d\xi\leq \int_\Omega|\nabla_Lu|^p\,d\xi\quad(u\in{\mathcal C}^1_0 (\Omega)) \] 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.

MSC:
35H10 Hypoelliptic equations
22E30 Analysis on real and complex Lie groups
26D10 Inequalities involving derivatives and differential and integral operators
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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