Analysis on geodesic balls of sub-elliptic operators (Q1906499)

We consider a geodesic ball \(B\) associated with a sub-elliptic second order differential operator and obtain sharp Sobolev inequalities and eigenvalue estimates for the Neumann problem on \(B\). Our approach bases on a previous work of D. Jerison. We recover and extend some recent results of G. Lu. The proofs are based on a Whitney covering argument where a family of coverings at different scales is used. This technique allows us to deduce a Sobolev inequality on \(B\) from Poincaré inequalities on smaller balls. By the same token, we give a two-sided estimate of the Weil counting function of the Neumann eigenvalues in \(B\). The bound we obtain for the counting function is more precise than the one resulting directly from the Sobolev inequality. These results apply to the Laplace-Beltrami operators of Riemannian manifolds with Ricci curvature bounded below and to invariant sub-elliptic operators on Lie groups.