论文标题
Hermite操作员的端点本征界
Endpoint eigenfunction bounds for the Hermite operator
论文作者
论文摘要
我们建立了最佳$ l^p $，$ p = 2（d+3）/（d+1），$ eigenFunction in $ \ mathbb r^d $上的HERMITE操作员$ \ MATHCAL $ \ MATHCAL H =-Δ+| X |^2 $。令$π_λ$表示投影操作员用$ \ MATHCAL H $的特征函数跨越特征值$λ$。 $π_λ$，$ 2 \ le p \ le \ le \ infty $上的最佳$ l^2 $ - $ l^p $ bunds已被Karadzhov和Koch-Tataru的作品所知，但$ p = 2（d+3）/（d+1）$。对于$ d \ ge 3 $，我们证明了缺少端点情况的最佳界限。我们的结果建立在一种新现象的基础上：由于球附近的不对称定位$ \sqrtλ\ Mathbb s^{d-1} $的改进，因此界限的改进。
We establish the optimal $L^p$, $p=2(d+3)/(d+1),$ eigenfunction bound for the Hermite operator $\mathcal H=-Δ+|x|^2$ on $\mathbb R^d$. Let $Π_λ$ denote the projection operator to the vector space spanned by the eigenfunctions of $\mathcal H$ with eigenvalue $λ$. The optimal $L^2$--$L^p$ bounds on $Π_λ$, $2\le p\le \infty$, have been known by the works of Karadzhov and Koch-Tataru except $p=2(d+3)/(d+1)$. For $d\ge 3$, we prove the optimal bound for the missing endpoint case. Our result is built on a new phenomenon: improvement of the bound due to asymmetric localization near the sphere $\sqrtλ\mathbb S^{d-1}$.
