Concerning LpLp resolvent estimates for simply connected manifolds of constant curvature

We prove families of uniform (Lr,Ls)(Lr,Ls) resolvent estimates for simply connected manifolds of constant curvature (negative or positive) that imply the earlier ones for Euclidean space of Kenig, Ruiz and the second author [7]. In the case of the sphere we take advantage of the fact that the half-wave group of the natural shifted Laplacian is periodic. In the case of hyperbolic space, the key ingredient is a natural variant of the Stein–Tomas restriction theorem.