Caffarelli-Kohn-Nirenberg and Sobolev type inequalities on stratified Lie groups

In this short paper, we establish a range of Caffarelli-Kohn-Nirenberg and weighted $L^p$-Sobolev type inequalities on stratified Lie groups. All inequalities are obtained with sharp constants. Moreover, the equivalence of the Sobolev type inequality and Hardy inequality is shown in the $L^2$-case.