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A note on constructing families of sharp examples for $ L^{p}$ growth of eigenfunctions and quasimodes

Author: Melissa Tacy
Journal: Proc. Amer. Math. Soc. 146 (2018), 2909-2924
MSC (2010): Primary 35P99
Published electronically: April 4, 2018
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Abstract: In this note we analyse $ L^{p}$ estimates for Laplacian eigenfunctions and quasimodes and their associated sharp examples. In particular, we use previously determined estimates to produce a new set of estimates for restriction to thickened neighbourhoods of submanifolds. In addition, we produce a family of flat model quasimode examples that can be used to determine sharpness of estimates on Laplacian eigenfunctions restricted to subsets. For each quasimode in the family we show that there is a corresponding spherical harmonic that displays the same growth properties. Therefore it is enough to check $ L^{p}$ growth estimates against the simple flat model examples. Finally, we present a heuristic that for any subset determines which quasimode in the family is expected to produce sharp examples.

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Melissa Tacy
Affiliation: School of Mathematics and Statistics, University of Otago, Otago, 9016 New Zealand

Received by editor(s): May 24, 2016
Received by editor(s) in revised form: August 17, 2017
Published electronically: April 4, 2018
Communicated by: Michael Hitrik
Article copyright: © Copyright 2018 American Mathematical Society

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