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Growth of high
$L^p$ norms for eigenfunctions: an application of geodesic
beams
Yaiza Canzani and Jeffrey Galkowski |
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Vol. 16 (2023), No. 10, 2267–2325
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Abstract |
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This work concerns norms of high energy Laplace eigenfunctions: , . Sogge (1988) gave optimal estimates on the growth of for a general compact Riemannian manifold. Here we give general dynamical conditions guaranteeing quantitative improvements in estimates for , where is the critical exponent. We also apply results of an earlier paper (Canzani and Galkowski 2018) to obtain quantitative improvements in concrete geometric settings including all product manifolds. These are the first results giving quantitative improvements for estimates on the growth of eigenfunctions that only require dynamical assumptions. In contrast with previous improvements, our assumptions are local in the sense that they depend only on the geodesics passing through a shrinking neighborhood of a given set in . Moreover, we give a structure theorem for eigenfunctions which saturate the quantitatively improved bound. Modulo an error, the theorem describes these eigenfunctions as finite sums of quasimodes which, roughly, approximate zonal harmonics on the sphere scaled by . |
Keywords
Eigenfunctions, high energy, $L^p$ norms, microlocal,
second microlocal
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Mathematical Subject Classification
Primary: 35P05, 58J50
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Milestones
Received: 18 March 2021
Revised: 13 January 2022
Accepted: 25 March 2022
Published: 11 December 2023
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Authors |
| Yaiza Canzani | |
| Department of Mathematics University of North Carolina at Chapel Hill Chapel Hill, NC United States |
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| Jeffrey Galkowski | |
| Department of Mathematics University College London London United Kingdom |
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| © 2023 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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