JournalsjemsVol. 17, No. 10pp. 2513–2543

On Kakeya–Nikodym averages, LpL^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
On Kakeya–Nikodym averages, $L^p$-norms and lower bounds for nodal sets of eigenfunctions in higher dimensions cover
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Abstract

We extend a result of the second author [27, Theorem 1.1] to dimensions d3d \geq 3 which relates the size of LpL^p-norms of eigenfunctions for 2<p<2(d+1)/d12 < p < 2(d+1) / d-1 to the amount of L2L^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\epsilon removal lemma" of Tao and Vargas [35]. We also use Hörmander's [20] L2L^2 oscillatory integral theorem and the Cartan–Hadamard theorem to show that, under the assumption of nonpositive curvature, the L2L^2-norm of eigenfunctions eλe_{\lambda} over unit-length tubes of width λ1/2\lambda^{-1/2} goes to zero. Using our main estimate, we deduce that, in this case, the LpL^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 d3d \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, LpL^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