$L^q$ 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, and Tzvetkov and Hu to manifolds with low regularity metrics.