$L^q$ 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
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$L^q$ 限制谱簇对低规律性度量的子流形的限制
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发表时间：
2012
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通讯作者：
Matthew D. Blair
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作者：
Matthew D. Blair
我们证明$L^q$限制的频谱集群的子流形在黎曼流形配备度量的$C^{1，alpha}$正则性为0 leq α leq 1$。我们的结果允许Lipschitz正则性时，$阿尔法=0$，这意味着他们给估计流形的边界。当$0< alpha leq 1$时，余维1子流形的标量第二基本形式可以被定义，并且当这种形式是负定的时，我们给出了改进的估计。这将Burq-G 'erard-Tzvetkov和Hu的结果扩展到具有低正则性度量的流形。
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.