Upper bound estimates of eigenvalues for Hörmander operators on non-equiregular sub-Riemannian manifolds

Abstract

We study the closed eigenvalue problem and Dirichlet eigenvalue problem of self-adjoint Hörmander operators on non-equiregular sub-Riemannian manifolds. By Rayleigh-Ritz formula and the subelliptic heat kernel estimates, we establish the upper bounds of eigenvalues which depend on the volume of the subunit ball and the measure of the manifold. Under a certain condition, we obtain the explicit upper bounds of eigenvalues which have the polynomially growth in k with the optimal order related to the non-isotropic dimension of the manifold.