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

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.