Abstract: In this paper we obtain the fundamental solution for a class of weighted Baouendi-Grushin type operator L_p,γ,αu = ▽_γ·(|▽_γu|^p-2ρ^α▽_γu) on R^m+n with singularity at the origin, where ▽_γ is the gradient operator defined by ▽_γ =(▽_x,|x|^γ▽_y) and ρ is the distance function. As an application, we get some Hardy type inequalities associated with ▽_γ.

Key words: fundamental solution, weighted Baouendi-Grushin type operator, Hardy type inequality