(untitled) (Analysis in Theory and Applications, 2022)
Let Ω be a bounded open domain in R ^n with smooth boundary Ω. Let X=(X_1,X_2,…,X_m) be a system of general Grushin type vector fields defined on Ω and the boundary Ω is non-characteristic for X. For Δ _X=_j=1^mX_j^2, we denote λ_k as the k-th eigenvalue for the bi-subelliptic operator Δ _X^2 on Ω. In this paper, by using the sharp sub-elliptic estimates and maximally hypoelliptic estimates, we give the optimal lower bound estimates of λ_k for the operator Δ _X^2.