# 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 $d \geq 3$ 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 "$\epsilon$ 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 $e_{\lambda}$ over unit-length tubes of width $\lambda^{-1/2}$ 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 $d \ge 3$ 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