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Growth of high $L^p$ norms for eigenfunctions: an application of geodesic beams

Yaiza Canzani and Jeffrey Galkowski

Vol. 16 (2023), No. 10, 2267–2325
Abstract

This work concerns Lp norms of high energy Laplace eigenfunctions: (Δg λ2)ϕλ = 0, ϕλL2 = 1. Sogge (1988) gave optimal estimates on the growth of ϕλLp for a general compact Riemannian manifold. Here we give general dynamical conditions guaranteeing quantitative improvements in Lp estimates for p > pc, where pc 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 Lp 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 M. Moreover, we give a structure theorem for eigenfunctions which saturate the quantitatively improved Lp bound. Modulo an error, the theorem describes these eigenfunctions as finite sums of quasimodes which, roughly, approximate zonal harmonics on the sphere scaled by 1log λ.

Keywords
Eigenfunctions, high energy, $L^p$ norms, microlocal, second microlocal
Mathematical Subject Classification
Primary: 35P05, 58J50
Milestones
Received: 18 March 2021
Revised: 13 January 2022
Accepted: 25 March 2022
Published: 11 December 2023
Authors
Yaiza Canzani
Department of Mathematics
University of North Carolina at Chapel Hill
Chapel Hill, NC
United States
Jeffrey Galkowski
Department of Mathematics
University College London
London
United Kingdom

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