Spectral projection estimates restricted to uniformly embedded submanifolds

Abstract

Let $M$ be a manifold with nonpositive sectional curvature and bounded geometry, and let $\hat{I}\pounds$ be a uniformly embedded submanifold of $M.$ We estimate the $L^2(M)\to L^q(\hat{I}\pounds )$ norm of a $\log$-scale spectral projection operator. It is a generalization of result of X. Chen to noncompact cases. We also prove sharp spectral projection estimates of spectral windows of any small size restricted to nontrapped geodesics on even asymptotically hyperbolic surfaces with bounded geometry and curvature pinched below 0.