$L^q$-norm bounds for arithmetic eigenfunctions via microlocal Kakeya-Nikodym estimate

TL;DR

This paper proves a power-saving bound for the $L^6$-norm of Hecke-Maass forms on hyperbolic surfaces using microlocal analysis and arithmetic amplification, improving upon local bounds.

Contribution

It introduces a microlocal decomposition approach combined with arithmetic amplification to achieve better $L^6$-norm bounds for eigenfunctions on arithmetic hyperbolic surfaces.

Findings

Established a power-saving $L^6$-norm bound of $oxed{oxed{oxed{oxed{oxed{oxed{oxed{oxed{oxed{oxed{rac{5}{36}+ ext{epsilon}}}}}}}}}}$ over the local bound.

Reduced the $L^6$-norm problem to microlocal Kakeya-Nikodym estimates for eigenfunctions.

Developed improved microlocal Kakeya-Nikodym estimates via arithmetic amplification.

Abstract

Let $X$ be a compact arithmetic congruence hyperbolic surface, and let $\psi$ be an $L^2$-normalized Hecke-Maass form on $X$ with sufficiently large spectral parameter $\lambda$. We give a new proof to obtain a power saving for the global $L^6$-norm $\|\psi\|_{L^6(X)}\lesssim_\varepsilon\lambda^{\frac{5}{36}+\varepsilon}$ over the local bound $\|\psi\|_{L^6(X)}\lesssim\lambda^{\frac{1}{6}}$ of Sogge. Our method uses a microlocal decomposition for $\psi$ and reduces the $L^6$-norm problem to microlocal Kakeya-Nikodym estimates for $\psi$, and we establish improved microlocal Kakeya-Nikodym estimates via arithmetic amplification developed by Iwaniec and Sarnak.