On radial symmetry and its breaking in the Caffarelli-Kohn-Nirenberg type inequalities for p=1 On radial symmetry and its breaking in the Caffarelli-Kohn-Nirenberg type inequalities for <i>p </i>= 1

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