Lq bounds on restrictions of spectral clusters to submanifolds for low regularity metrics

Blair, Matthew

doi:10.2140/apde.2013.6.1263

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

Matthew D. Blair

Vol. 6 (2013), No. 6, 1263–1288

DOI: 10.2140/apde.2013.6.1263

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, and Tzvetkov and Hu to manifolds with low regularity metrics.

Keywords

eigenfunctions, $L^p$ estimates, spectral cluster estimates, quasimodes, wave packets, folding singularities

Mathematical Subject Classification 2010

Primary: 35P99, 35R05, 42B37

Secondary: 35L15, 42C15, 42B20

Milestones

Received: 1 March 2012

Revised: 6 August 2012

Accepted: 20 December 2012

Published: 18 November 2013

Authors

Matthew D. Blair
Department of Mathematics and Statistics University of New Mexico Albuquerque, NM 87131 USA

Vol. 6, No. 6, 2013