Anisotropic $L^2$-weighted Hardy and $L^2$-Caffarelli-Kohn-Nirenberg inequalities

We establish sharp remainder terms of the $L^2$-Caffarelli-Kohn-Niren-berg inequalities on homogeneous groups, yielding the inequalities with best constants. Our methods also give new sharp Caffarelli-Kohn-Nirenberg type inequalities in ${\mathbb{R}}^n$ with arbitrary quasi-norms. We also present explicit examples to illustrate our results for different weights and in abelian cases.