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We consider the Grushin type operator on ℝxd×ℝyk{\mathbb{R}^{d}_{x}\times\mathbb{R}^{k}_{y}} of the form

We consider the Grushin type operator on ℝdx × ℝky of the form Gμ = ∑i=1d∂xi2 + (∑i=1dxi2)2μ∑j=1k∂yj2 and derive weighted Hardy–Sobolev type inequalities and weighted Caffarelli–Kohn–Nirenberg type inequalities related to Gμ. [ABSTRACT FROM AUTHOR]

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 ∣∣γ 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 [ABSTRACT FROM AUTHOR]