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

TL;DR

This paper establishes $L^q$ bounds for spectral cluster restrictions to submanifolds in Riemannian manifolds with low regularity metrics, extending previous results to metrics with only $C^{1,eta}$ regularity.

Contribution

It extends $L^q$ restriction estimates for spectral clusters to manifolds with $C^{1,eta}$ metrics, including Lipschitz and low regularity cases, and introduces improvements based on the second fundamental form.

Findings

Established $L^q$ bounds for spectral clusters on low regularity manifolds.

Extended previous results to metrics with $C^{1,eta}$ regularity.

Provided improved estimates when the second fundamental form is negative definite.

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 $0 \leq \alpha \leq 1$. 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.