Hardy Type Inequalities for ${\Delta }_{\lambda }$-Laplacians

Abstract

We derive Hardy type inequalities for a large class of sub-elliptic operators that belong to the class of ${\Delta }_{\lambda }$-Laplacians and find explicit values for the constants involved. Our results generalize previous inequalities obtained for Grushin type operators $${\Delta }_x+|x{|}^{2\alpha }{\Delta }_y,(x,y)\in {\mathbb{R}}^{N_1}\times {\mathbb{R}}^{N_2},\alpha \ge 0,$$ which were proved to be sharp.