A note on constructing families of sharp examples for $L^{p}$ growth of eigenfunctions and quasimodes
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- by Melissa Tacy PDF
- Proc. Amer. Math. Soc. 146 (2018), 2909-2924 Request permission
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.References
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Additional Information
- Melissa Tacy
- Affiliation: School of Mathematics and Statistics, University of Otago, Otago, 9016 New Zealand
- MR Author ID: 922506
- Email: mtacy@maths.otago.ac.nz
- 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
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 2909-2924
- MSC (2010): Primary 35P99
- DOI: https://doi.org/10.1090/proc/14028
- MathSciNet review: 3787353