Semiclassical L^p estimates of quasimodes on submanifolds

Let P = P(h) be a semiclassical pseudodifferential operator on a Riemannian manifold M. Suppose that u(h) is a localized, L² normalized family of functions such that P(h)u(h) is O(h) in L², as h → 0. Then, for any submanifold Y ⊂ M, we obtain estimates on the L^p norm of u(h) restricted to Y, with exponents that are sharp for h → 0. These results generalize those of Burq et al. [4] on L^p norms for restriction of Laplacian eigenfunctions. As part of the technical development we prove some extensions of the abstract Strichartz estimates of Keel and Tao [8].