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

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

2012-02-28

arxiv.org/abs/1202.6385v1

Mathematics

Analysis of PDEs

22 pages

Scientific paper

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 $0 \leq \alpha \leq 1$. Our results allow for Lipschitz regularity when $\alpha =0$, meaning they give estimates on manifolds with boundary. When $0< \alpha \leq 1$, 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-G\'erard-Tzvetkov and Hu to manifolds with low regularity metrics.

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