特征函数的Kakeya-Nikodym平均和$L^p$ -范数
Kakeya-Nikodym averages and $L^p$-norms of eigenfunctions
摘要：
本文给出了二维紧致无边界黎曼流形$M$上的Laplacian的平方根的本征函数的$L^p$ -范数$2<p<6$小于本征值$λ$的自然幂的充要条件.确保这一点的条件是，当$λ$较大时，它们在任意单位测地线的$O(λ^{-1/2})$邻域上的$L^2$范数较小（例如，对于$S^2$上的最高权重球谐函数，情况并非如此）。证明利用高斯引理和事实，即双线性振荡积分在Hörmander的Carleson-Sjölin定理的证明中变得越来越好，远离对角线。我们的研究结果是有关最近的工作Bourgain谁表明，$L^2$平均测地线的本征函数是小的自然电源的本征值$λ$提供的$L^4(M)$规范同样小。我们的结果意味着QUE不能保持在一个紧凑的二维无边界黎曼流形$(M,g)$，如果$L^p$ -范数是饱和的，对于一个给定的$2<p<6$。我们还表明，本征函数不能有一个最大的$L^2$ -质量集中沿单位部分的测地线是不顺利关闭的速率。
英文摘要：
We provide a necessary and sufficient condition that $L^p$-norms, $2<p<6$, of eigenfunctions of the square root of minus the Laplacian on 2-dimensional compact boundaryless Riemannian manifolds $M$ are small compared to a natural power of the eigenvalue $λ$. The condition that ensures this is that their $L^2$ norms over $O(λ^{-1/2})$ neighborhoods of arbitrary unit geodesics are small when $λ$ is large (which is not the case for the highest weight spherical harmonics on $S^2$ for instance). The proof exploits Gauss' lemma and the fact that the bilinear oscillatory integrals in Hörmander's proof of the Carleson-Sjölin theorem become better and better behaved away from the diagonal. Our results are related to a recent work of Bourgain who showed that $L^2$ averages over geodesics of eigenfunctions are small compared to a natural power of the eigenvalue $λ$ provided that the $L^4(M)$ norms are similarly small. Our results imply that QUE cannot hold on a compact boundaryless Riemannian manifold $(M,g)$ of dimension two if $L^p$-norms are saturated for a given $2<p<6$. We also show that eigenfunctions cannot have a maximal rate of $L^2$-mass concentrating along unit portions of geodesics that are not smoothly closed.
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