Weighted Caffarelli-Kohn-Nirenberg type inequalities related to Grushin type operators

Abstract

We consider the Grushin type operator on ${\mathbb{R}}_x^d\times {\mathbb{R}}_y^k$ with the form \begin{equation*} G_μ=\overset{d}{\underset{i=1}{\sum}}\partial_{x_i}^2+\left(\overset{d}{\underset{i=1}{\sum}}x_i^2\right)^{2μ}\overset{k}{\underset{j=1}{\sum}}\partial_{y_j}^2. \end{equation*} and derive weighted Hardy-Sobolev type inequalities and weighted Caffarelli-Kohn-Nirenberg type inequalities related to $G_{\hat{I}}¼$.