On Kakeya–Nikodym averages, -norms and lower bounds for nodal sets of eigenfunctions in higher dimensions
Matthew D. Blair
University of New Mexico, Albuquerque, USAChristopher D. Sogge
The Johns Hopkins University, Baltimore, USA
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Abstract
We extend a result of the second author [27, Theorem 1.1] to dimensions which relates the size of -norms of eigenfunctions for to the amount of -mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee [22] and a variable coefficient variant of an " removal lemma" of Tao and Vargas [35]. We also use Hörmander's [20] oscillatory integral theorem and the Cartan–Hadamard theorem to show that, under the assumption of nonpositive curvature, the -norm of eigenfunctions over unit-length tubes of width goes to zero. Using our main estimate, we deduce that, in this case, the -norms of eigenfunctions for the above range of exponents is relatively small. As a result, we can slightly improve the known lower bounds for nodal sets in dimensions of Colding and Minicozzi [10] in the special case of (variable) nonpositive curvature.
Cite this article
Matthew D. Blair, Christopher D. Sogge, On Kakeya–Nikodym averages, -norms and lower bounds for nodal sets of eigenfunctions in higher dimensions. J. Eur. Math. Soc. 17 (2015), no. 10, pp. 2513–2543
DOI 10.4171/JEMS/564