Vol. 6, No. 6, 2013

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
$L^q$ bounds on restrictions of spectral clusters to submanifolds for low regularity metrics

Matthew D. Blair

Vol. 6 (2013), No. 6, 1263–1288
Abstract

We prove Lq bounds on the restriction of spectral clusters to submanifolds in Riemannian manifolds equipped with metrics of C1,α regularity for 0 α 1. Our results allow for Lipschitz regularity when α = 0, meaning they give estimates on manifolds with boundary. When 0 < α 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