Refined and Microlocal Kakeya–Nikodym Bounds of Eigenfunctions in Higher Dimensions
Refined and Microlocal Kakeya–Nikodym Bounds of Eigenfunctions in Higher Dimensions
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DOI：
10.1007/s00220-017-2977-8
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发表时间：
2017-12
影响因子：
2.4
通讯作者：
Matthew D. Blair;C. Sogge
中科院分区：
文献类型：
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作者：
Matthew D. Blair;C. Sogge
我们证明了本征函数和准模态上的 Kakeya-Nikodym 界，这使作者的结果更加清晰（Blair 和 Sogge in Anal PDE 8:747–764, 2015）并将其扩展到更高的维度。与之前的工作一样，关键的中间步骤是证明这些估计的微局部版本，其中涉及这些模式的相空间分解，该分解在双特征/测地线流下本质上是不变的。在一篇配套论文（Blair 和 Sogge in J Differ Geom，2015）中，可以看出，在存在非正曲率的情况下，这些锐化估计会产生改进的特征函数 Lq(M) 界限。
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 improvedLq(M) bounds on eigenfunctions in the presence of nonpositive curvature when.