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arXiv · 1202.6385

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

Matthew D. Blair
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Abstract

We prove $L^q$ bounds on the restriction of spectral clusters to submanifolds in Riemannian manifolds equipped with metrics of $C^{1,α}$ regularity for $0 \leq α\leq 1$. Our results allow for Lipschitz regularity when $α=0$, meaning they give estimates on manifolds with boundary. When $0< α\leq 1$, 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-Gérard-Tzvetkov and Hu to manifolds with low regularity metrics.

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Matthew D. Blair. 2012-08-06. $L^q$ bounds on restrictions of spectral clusters to submanifolds for low regularity metrics. https://doi.org/10.2140/apde.2013.6.1263

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