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

TL;DR

This paper extends bounds relating eigenfunction norms and geodesic tubes to higher dimensions, using advanced oscillatory integral estimates, and improves lower bounds for nodal sets under nonpositive curvature assumptions.

Contribution

It generalizes a key relation between eigenfunction norms and geodesic tubes to dimensions three and higher, employing new bilinear oscillatory integral estimates.

Findings

Eigenfunction $L^p$-norms are small in nonpositive curvature settings.

Improved lower bounds for nodal sets in higher dimensions.

Application of bilinear oscillatory integral estimates to spectral geometry.

Abstract

We extend a result of the second author \cite[Theorem 1.1]{soggekaknik} to dimensions $d \geq 3$ which relates the size of $L^p$-norms of eigenfunctions for $2<p<\frac{2(d+1)}{d-1}$ to the amount of $L^2$-mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee \cite{leebilinear} and a variable coefficient variant of an "$\veps$ removal lemma" of Tao and Vargas \cite{tv1}. We also use H\"ormander's \cite{HorOsc} $L^2$ oscillatory integral theorem and the Cartan-Hadamard theorem to show that, under the assumption of nonpositive curvature, the $L^2$-norm of eigenfunctions $e_\la$ over unit-length tubes of width $\la^{-\frac12}$ 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 $d\ge3$ of Colding and Minicozzi \cite{CM} in the special case of (variable) nonpositive curvature.