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

Abstract

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