Refined and Microlocal Kakeya-Nikodym Bounds of Eigenfunctions in Higher Dimensions

TL;DR

This paper establishes refined bounds on eigenfunctions in higher dimensions using microlocal analysis, leading to improved $L^q$ estimates under certain geometric conditions.

Contribution

It extends previous Kakeya-Nikodym bounds to higher dimensions and introduces a microlocal phase space decomposition invariant under geodesic flow.

Findings

Proves a Kakeya-Nikodym bound for eigenfunctions in higher dimensions.

Develops a microlocal phase space decomposition technique.

Yields improved $L^q$ bounds for eigenfunctions with nonpositive curvature.

Abstract

We prove a Kakeya-Nikodym bound on eigenfunctions and quasimodes, which sharpens a result of the authors and extends it to higher dimensions. As in the prior work, the key intermediate step is to prove a microlocal version of these estimates, which involves a phase space decomposition of these modes which is essentially invariant under the bicharacteristic/geodesic flow. In a companion paper, it will be seen that these sharpened estimates yield improved $L^q(M)$ bounds on eigenfunctions in the presence of nonpositive curvature when $2 < q < \frac{2(d+1)}{d-1}$.