  We prove uniform $L^p$ estimates for resolvents of higher order elliptic
self-adjoint differential operators on compact manifolds without boundary,
generalizing a corresponding resul of [3] in the case of Laplace-- Beltrami
operators on Riemannian manifolds. In doing so, we follow the methods,
developed in [1] very closely. We also show that spectral regions in our $L^p$
resolvent estimates are optimal.
