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On the growth of eigenfunction averages: Microlocalization and geometry

Canzani, Y; Galkowski, J; (2019) On the growth of eigenfunction averages: Microlocalization and geometry. Duke Mathematical Journal , 168 (16) pp. 2991-3055. 10.1215/00127094-2019-0020. Green open access

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Abstract

Let (M,g) be a smooth, compact Riemannian manifold, and let {φ_{h]} be an L² -normalized sequence of Laplace eigenfunctions, -h²Δ_{g}φ_{h} = φ_{h}. Given a smooth submanifold H ⊂ M of codimension K ≥ 1, we find conditions on the pair ({φ_{h}, H) for which |∫_{H}φ_{h} δσ_{H}| = o (h\frac{1-k}{2}), h→0⁺. 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.

Type: Article
Title: On the growth of eigenfunction averages: Microlocalization and geometry
Open access status: An open access version is available from UCL Discovery
DOI: 10.1215/00127094-2019-0020
Publisher version: http://doi.org/10.1215/00127094-2019-0020
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10087655
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