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  and a variable coefficient variant of an " removal lemma" of Tao and Vargas . We also use Hörmander's  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  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. 2513–2543DOI 10.4171/JEMS/564