$L^q$ bounds on restrictions of spectral clusters to submanifolds for low regularity metrics

Abstract

We prove $L^q$ bounds on the restriction of spectral clusters to submanifolds in Riemannian manifolds equipped with metrics of $C^{1,\alpha }$ regularity for $0\le \alpha \le 1$. Our results allow for Lipschitz regularity when $\alpha =0$, meaning they give estimates on manifolds with boundary. When $0<\alpha \le 1$, the scalar second fundamental form for a codimension 1 submanifold can be defined, and we show improved estimates when this form is negative definite. This extends results of Burq-Gérard-Tzvetkov and Hu to manifolds with low regularity metrics.

arXiv comment22 pages, corrected significant error in previous draft

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