AI 中文总结
该研究针对非紧致型对称空间上的$L^2$归一化球谐联合本征函数，改进了Sarnak的局部上确界界，得到局部一致的$o(T^{(n-r)/2})$界，在有限体积实双曲流形的特定范围及单射半径有下界时还给出了对应全局估计。
AI 中文摘要
设$X=G/K$为非紧致型对称空间，维数为$n$，秩为$r$，令$Y=\Gamma\backslash X$。对于大小为$T$、具有正则 tempered 参数的$L^2$归一化球谐联合本征函数，Sarnak的局部界为$\\|\phi\\|_\infty\ll T^{(n-r)/2}$。我们证明该界在每个商空间上局部一致地为$o(T^{(n-r)/2})$。对于有限体积实双曲流形，该界在扩展尖点范围$y\leq T^\beta$（$\beta<1/2$）内是一致的；若单射半径有下界，我们证明全局估计为$T^{(n-r)/2}(\log T)^{-r/2}$。
英文摘要
Let $X=G/K$ be a symmetric space of noncompact type, of dimension $n$ and rank $r$, and let $Y=Γ\backslash X$. Sarnak's local bound for an $L^2$-normalized spherical joint eigenfunction with regular tempered parameter of size $T$ is $\|ϕ\|_\infty\ll T^{(n-r)/2}$. We prove $o(T^{(n-r)/2})$ locally uniformly on every quotient. On finite-volume real hyperbolic manifolds this is uniform in the expanding cusp range $y\leq T^β$, $β<1/2$. If the injectivity radius is bounded below, we prove the global estimate $T^{(n-r)/2}(\log T)^{-r/2}$. The proof makes use of a novel kernel argument and a uniform bound on the spherical function.
Comments17 pages
