Asymptotic behaviour of Dirichlet eigenvalues for homogeneous Hörmander operators and algebraic geometry approach

Abstract

We study the Dirichlet eigenvalue problem of homogeneous Hörmander operators △ X = m j=1 X 2 j on a bounded connected open domain containing the origin, where X 1 , X 2 , . . . , X m are linearly independent smooth vector fields in R n satisfying Hörmander's condition and a suitable homogeneity property with respect to a family of non-isotropic dilations. Suppose that Ω is a bounded connected open domain in R n containing the origin, and its boundary ∂Ω is smooth and non-characteristic for X. Combining the subelliptic heat kernel estimates, the resolution of singularities in algebraic geometry and some refined analysis involving convex geometry, we establish the explicit asymptotic behaviourwhere λ k denotes the k-th Dirichlet eigenvalue of △ X on Ω, Q 0 is a positive rational number, and d 0 is a non-negative integer. Furthermore, we also give the optimal bounds of index Q 0 , which depends on the homogeneous dimension associated with vector fields X 1 , X 2 , . . . , X m .