Uniform Resolvent Estimates on Manifolds of Bounded Curvature

Abstract

We establish $$L^{q*}\rightarrow L^q$$ bounds for the resolvent of the Laplacian on compact Riemannian manifolds assuming only that the sectional curvatures of the manifold are uniformly bounded. When the resolvent parameter lies outside a parabolic neighborhood of $$[0,\infty )$$, the operator norm of the resolvent is shown to depend only on upper bounds for the sectional curvature and diameter and lower bounds for the volume. The resolvent bounds are derived from square-function estimates for the wave equation, an approach that admits the use of paradifferential approximations in the parametrix construction.