Geodesic
On Kakeya–Nikodym averages, $L^p$-norms and lower bounds for nodal sets of eigenfunctions in higher dimensions
Journal of the European Mathematical Society, Tome 17 (2015) no. 10, pp. 2513-2543.

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We extend a result of the second author [27, Theorem 1.1] to dimensions d≥3 which relates the size of Lp-norms of eigenfunctions for 22(d+1)/d−1 to the amount of L2-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 "ε removal lemma" of Tao and Vargas [35]. We also use Hörmander's [20] L2 oscillatory integral theorem and the Cartan–Hadamard theorem to show that, under the assumption of nonpositive curvature, the L2-norm of eigenfunctions eλ​ over unit-length tubes of width λ−1/2 goes to zero. Using our main estimate, we deduce that, in this case, the Lp-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 d≥3 of Colding and Minicozzi [10] in the special case of (variable) nonpositive curvature.
DOI : 10.4171/jems/564
Classification : 58-XX, 35-XX, 42-XX
Keywords: Eigenfunctions, Lp-norms, Kakeya averages 
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Matthew D. Blair; Christopher D. Sogge. On Kakeya–Nikodym averages, $L^p$-norms and lower bounds for nodal sets of eigenfunctions in higher dimensions. Journal of the European Mathematical Society, Tome 17 (2015) no. 10, pp. 2513-2543. doi : 10.4171/jems/564. http://geodesic.mathdoc.fr/articles/10.4171/jems/564/

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