Radial symmetry and its breaking in the Caffarelli-Kohn-Nirenberg type inequalities for $p=1$

The main purpose of this article is to study the Caffarelli-Kohn-Nirenberg type inequalities (1.2) with $p=1$. We show that symmetry breaking of the best constants occurs provided that a parameter $|\gamma|$ is large enough. In the argument we effectively employ equivalence between the Caffarelli-Kohn-Nirenberg type inequalities with $p=1$ and the isoperimetric inequalities with weights.