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We consider the Grushin type operator onℝxd×ℝyk{\mathbb{R}^{d}_{x}\times\mathbb{R}^{k}_{y}}of the formGμ=∑i=1d∂xi2+(∑i=1dxi2)2⁢μ⁢∑j=1k∂yj2G_{\mu}=\sum_{i=1}^{d}\partial_{x_{i}}^{2}+\Bigl{(}\sum_{i=1}^{d}x_{i}^{2}% \Bigr{)}^{2\mu}\sum_{j=1}^{k}\partial_{y_{j}}^{2}and derive weighted Hardy–Sobolev type inequalities and weighted Caffarelli–Kohn–Nirenberg type inequalities related toGμ{G_{\mu}}.

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.