# On Kakeya–Nikodym averages, $L_{p}$-norms and lower bounds for nodal sets of eigenfunctions in higher dimensions

### Matthew D. Blair

University of New Mexico, Albuquerque, USA### Christopher D. Sogge

The Johns Hopkins University, Baltimore, USA

## Abstract

We extend a result of the second author [27, Theorem 1.1] to dimensions $d≥3$ which relates the size of $L_{p}$-norms of eigenfunctions for $2<p<2(d+1)/d−1$ to the amount of $L_{2}$-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] $L_{2}$ oscillatory integral theorem and the Cartan–Hadamard theorem to show that, under the assumption of nonpositive curvature, the $L_{2}$-norm of eigenfunctions $e_{λ}$ over unit-length tubes of width $λ_{−1/2}$ goes to zero. Using our main estimate, we deduce that, in this case, the $L_{p}$-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.

## Cite this article

Matthew D. Blair, Christopher D. Sogge, On Kakeya–Nikodym averages, $L_{p}$-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