Kakeya-Nikodym norms of Maass forms on $\rm{U}(2,1)$

TL;DR

This paper establishes power savings for the Kakeya-Nikodym norms of Hecke-Maass forms on complex hyperbolic surfaces, leading to improved bounds on their L^p norms, using amplification methods.

Contribution

It introduces a novel application of amplification to obtain power savings for Kakeya-Nikodym norms of Maass forms on $ m{U}(2,1)$, advancing understanding of their size and distribution.

Findings

Power savings over trivial bounds for Kakeya-Nikodym norms.

Improved bounds on $ orm{ ext{Maass form}}_p$ for 2<p<10/3.

Application of amplification method in this geometric setting.

Abstract

Let $\psi$ be a Hecke-Maass form with a large spectral parameter on a compact arithmetic complex hyperbolic surface. We apply the amplification method to obtain a power saving over the trivial bound for the Kakeya-Nikodym norm of $\psi$. As a consequence, we obtain power savings over the local bound of Sogge for $\|\psi\|_p$ when $2<p<10/3$.