基于微局Kakeya-Nikodym估计的算术特征函数$L^q$-norm界
$L^q$-norm bounds for arithmetic eigenfunctions via microlocal Kakeya-Nikodym estimate
AI总结：
本文通过微局Kakeya-Nikodym估计改进了算术特征函数的$L^6$-norm界，利用微局分解和算术放大技术获得更优的全局界。
摘要：
设$X$为紧算术全息黎曼面，令$ψ$为在$X$上$L^2$正交的Hecke-Maass形式，具有足够大的谱参数$λ$。本文给出一种新证明，获得全局$L^6$-norm$\|ψ\|_{L^6(X)}\lesssim_\varepsilonλ^{\frac{5}{36}+\varepsilon}$，优于Sogge的局部界$\|ψ\|_{L^6(X)}\lesssimλ^{\frac{1}{6}}$。方法利用$ψ$的微局分解，将$L^6$-norm问题转化为微局Kakeya-Nikodym估计，通过Iwaniec和Sarnak发展的算术放大技术建立改进的微局Kakeya-Nikodym估计。
英文摘要：
Let $X$ be a compact arithmetic congruence hyperbolic surface, and let $ψ$ be an $L^2$-normalized Hecke-Maass form on $X$ with sufficiently large spectral parameter $λ$. We give a new proof to obtain a power saving for the global $L^6$-norm $\|ψ\|_{L^6(X)}\lesssim_\varepsilonλ^{\frac{5}{36}+\varepsilon}$ over the local bound $\|ψ\|_{L^6(X)}\lesssimλ^{\frac{1}{6}}$ of Sogge. Our method uses a microlocal decomposition for $ψ$ and reduces the $L^6$-norm problem to microlocal Kakeya-Nikodym estimates for $ψ$, and we establish improved microlocal Kakeya-Nikodym estimates via arithmetic amplification developed by Iwaniec and Sarnak.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
