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Refined and Microlocal Kakeya–Nikodym Bounds of Eigenfunctions in Higher Dimensions

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Abstract

We prove a Kakeya–Nikodym bound on eigenfunctions and quasimodes, which sharpens a result of the authors (Blair and Sogge in Anal PDE 8:747–764, 2015) and extends it to higher dimensions. As in the prior work, the key intermediate step is to prove a microlocal version of these estimates, which involves a phase space decomposition of these modes that is essentially invariant under the bicharacteristic/geodesic flow. In a companion paper (Blair and Sogge in J Differ Geom, 2015), it will be seen that these sharpened estimates yield improved L q(M) bounds on eigenfunctions in the presence of nonpositive curvature when \({2 < q < \frac{2(d+1)}{d-1}}\) .

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Correspondence to Matthew D. Blair.

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Communicated by J. Marklof

The first author was supported in part by the National Science Foundation Grant DMS-1301717, and the second by the National Science Foundation Grant DMS-1361476.

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Blair, M.D., Sogge, C.D. Refined and Microlocal Kakeya–Nikodym Bounds of Eigenfunctions in Higher Dimensions. Commun. Math. Phys. 356, 501–533 (2017). https://doi.org/10.1007/s00220-017-2977-8

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  • DOI: https://doi.org/10.1007/s00220-017-2977-8

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