Spectral projection estimates restricted to uniformly embedded submanifolds

TL;DR

The paper advances spectral projection estimates restricted to submanifolds on manifolds with nonpositive curvature and bounded geometry, deriving log-scale bounds from $L^2(M)$ to $L^q(\Sigma)$ for uniformly embedded $\Sigma$. It develops unit-band estimates and a localization framework, then deduces log-scale results via TT^* with careful time-parameter balancing, extending Chen’s compact-manifold results to noncompact settings. It further analyzes even asymptotically hyperbolic surfaces, constructing a background model and proving sharp lossless estimates for arbitrarily small spectral windows along nontrapped geodesics, and establishes critical log-scale bounds on compact geodesics. Finally, the paper demonstrates sharpness through explicit hyperbolic-plane examples, underscoring the geometric dependence and the necessity of bounded geometry for the obtained rates.

Abstract

Let $M$ be a manifold with nonpositive sectional curvature and bounded geometry, and let $Σ$ be a uniformly embedded submanifold of $M.$ We estimate the $L^2(M)\to L^q(Σ)$ 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.

Theorems & Definitions (24)

• Theorem 1
• Theorem 2
• Definition 2.1: Manifold with bounded geometry
• Definition 2.2: Uniformly embedded submanifold
• Remark
• Lemma 2.1: Local equivalence of distance
• Lemma 2.2: Uniformly locally finite cover of $M$
• Lemma 2.3: Submanifold of bounded geometry
• Proposition 2.1
• proof