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1 November 2019 On the growth of eigenfunction averages: Microlocalization and geometry
Yaiza Canzani, Jeffrey Galkowski
Duke Math. J. 168(16): 2991-3055 (1 November 2019). DOI: 10.1215/00127094-2019-0020

Abstract

Let (M,g) be a smooth, compact Riemannian manifold, and let {ϕh} be an L2-normalized sequence of Laplace eigenfunctions, h2Δgϕh=ϕh. Given a smooth submanifold HM of codimension k1, we find conditions on the pair ({ϕh},H) for which |HϕhdσH|=o(h1k2),h0+. One such condition is that the set of conormal directions to H that are recurrent has measure 0. In particular, we show that the upper bound holds for any H if (M,g) is a surface with Anosov geodesic flow or a manifold of constant negative curvature. The results are obtained by characterizing the behavior of the defect measures of eigenfunctions with maximal averages.

Citation

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Yaiza Canzani. Jeffrey Galkowski. "On the growth of eigenfunction averages: Microlocalization and geometry." Duke Math. J. 168 (16) 2991 - 3055, 1 November 2019. https://doi.org/10.1215/00127094-2019-0020

Information

Received: 18 October 2017; Revised: 26 September 2018; Published: 1 November 2019
First available in Project Euclid: 14 October 2019

zbMATH: 07154834
MathSciNet: MR4027827
Digital Object Identifier: 10.1215/00127094-2019-0020

Subjects:
Primary: 35P20
Secondary: 35P15

Keywords: Anosov , averages , defect measures , Eigenfunctions , quasimodes , recurrence

Rights: Copyright © 2019 Duke University Press

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Vol.168 • No. 16 • 1 November 2019
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