Lower bounds of eigenvalues for a class of bi-subelliptic operators

Let Ω be a bounded open domain in Rn with smooth boundary and X=(X1,X2,⋯,Xm) be a system of real smooth vector fields defined on Ω with the boundary ∂Ω which is non-characteristic for X. If X satisfies the Hörmander's condition, then the vector fields are finitely degenerate and the sum of square operators △X=∑i=1mXi2 is a subelliptic operator. Let λk be the k-th eigenvalue for the bi-subelliptic operator △X2 on Ω. In this paper, we introduce the generalized Métivier's condition and study the lower bounds of Dirichlet eigenvalues for the operator △X2 on some finitely degenerate systems of vector fields X which satisfy the Hörmander's condition or the generalized Métivier's condition. By using the subelliptic estimates, we shall give a explicit lower bound estimates of λk which is polynomial increasing in k with the order relating to the Hörmander index or the generalized Métivier index.