Lower bounds of Dirichlet eigenvalues for a class of finitely degenerate Grushin type elliptic operators

Let Ω be a bounded open domain in Rn with smooth boundary ∂Ω, X=(X1,X2,⋯,Xm) be a system of real smooth vector fields defined on Ω and the boundary ∂Ω is non-characteristic for X. If X satisfies the Hörmander's condition, then the vector field is finitely degenerate and the sum of square operator ΔX=∑j=1mXj2 is a finitely degenerate elliptic operator. In this paper, we shall study the sharp estimate of the Dirichlet eigenvalue for a class of general Grushin type degenerate elliptic operators ΔX on Ω.