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arXiv · 1510.07724

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

Matthew D. Blair·Christopher D. Sogge
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Abstract

We prove a Kakeya-Nikodym bound on eigenfunctions and quasimodes, which sharpens a result of the authors 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 which is essentially invariant under the bicharacteristic/geodesic flow. In a companion paper, 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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Matthew D. Blair, Christopher D. Sogge. 2017-05-25. Refined and Microlocal Kakeya-Nikodym Bounds of Eigenfunctions in Higher Dimensions. https://arxiv.org/abs/1510.07724

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