论文标题
高$ l^p $ quarms for eigenfunctions的增长：地球梁的应用
Growth of high $L^p$ norms for eigenfunctions: an application of geodesic beams
论文作者
论文摘要
这项工作涉及$ l^p $高能拉普拉斯特征函数的规范，$（Δ_g-λ^2）ϕ_λ = 0 $，$ \ | ϕ_λ \ | _ {l^2} = 1 $。 1988年，Sogge对一般紧凑的Riemannian歧管给出了$ \ | ϕ_λ \ | _ {l^p} $的最佳估计。本文的目的是提供一般的动态条件，以确保$ p> p_c $的$ l^p $估计值的定量改进，其中$ p_c $是关键指数。我们还采用了作者的先前结果，以获得包括所有产品歧管在内的具体几何设置中的定量改进。这些是对仅需要动态假设的本征函数增长的$ l^p $增长的最初结果。与以前的改进相反，我们的假设是本地的，因为它们仅取决于经过$ m $中给定集合的缩小社区的地球化学。此外，该文章为本征函数提供了一个结构定理，该定理饱和了定量改进的$ l^p $绑定。 Modulo A错误，定理将这些本征函数描述为拟象症的有限总和，在球体上大致近似于$ 1/\ sqrt {\logλ} $的球体上的Zonal谐波。
This work concerns $L^p$ norms of high energy Laplace eigenfunctions, $(-Δ_g-λ^2)ϕ_λ=0$, $\|ϕ_λ\|_{L^2}=1$. In 1988, Sogge gave optimal estimates on the growth of $\|ϕ_λ\|_{L^p}$ for a general compact Riemannian manifold. The goal of this article is to give general dynamical conditions guaranteeing quantitative improvements in $L^p$ estimates for $p>p_c$, where $p_c$ is the critical exponent. We also apply previous results of the authors to obtain quantitative improvements in concrete geometric settings including all product manifolds. These are the first results improving estimates for the $L^p$ growth of eigenfunctions that only require dynamical assumptions. In contrast with previous improvements, our assumptions are local in the sense that they depend only on the geodesics passing through a shrinking neighborhood of a given set in $M$. Moreover, the article gives a structure theorem for eigenfunctions which saturate the quantitatively improved $L^p$ bound. Modulo an error, the theorem describes these eigenfunctions as finite sums of quasimodes which, roughly, approximate zonal harmonics on the sphere scaled by $1/\sqrt{\log λ}$.
