论文标题
爱好光谱投影操作员
Hermite spectral projection operator
论文作者
论文摘要
我们研究$ l^p $ - $ l^q $估计频谱投影操作员$π_λ$与HERMITE操作员相关的$ h = | x | X |^2-δ$ in $ \ \ \ \ \ m rathbb r^d $。在这里，$π_λ$表示由Hermite函数跨越的子空间的投影，这是$ h $的特征函数，带有特征值$λ$。此类估计以前仅适用于$ q = p'$，同等用$ p = 2 $或$ q = 2 $（按$ tt^*$参数），除了这些估计之间插值的直接后果外。如Karadzhov，Thangavelu和Koch和Tataru的作品所示，$π_λ$的本地和全球估计是不同的。特别是，$π_λ$在集合附近表现出复杂的行为，$ \sqrtλ\ mathbb s^{d-1} $。与与Laplacian相关的频谱投影操作员相比，$ l^p $ - $ l^q $估计$π_λ$现在对一般$ p，q $现在的理解还不太理解。在本文中，我们考虑$ l^p $ - $ l^q $估计在一般框架中，包括本地和全球估计，$ 1 \ le p \ le p \ le 2 \ le q \ le q \ le \ infty $，并承担表征$π_λ$的急剧界限的工作。我们在$ p，q $的扩展范围内建立了各种新的尖锐估计。首先，我们提供了$π_λ$的局部估计值的完整表征，这首先由Thangavelu考虑。其次，对于$ d \ ge5 $，我们证明了端点$ l^2 $ - $ l^{2（d+3）/（d+1）} $估计$π_λ$，这是自Koch和Tataru工作以来就已经打开的。第三，我们将操作员$π_λ$从$ l^p $统一到$ l^q $的$ p，q $范围。
We study $L^p$-$L^q$ estimate for the spectral projection operator $Π_λ$ associated to the Hermite operator $H=|x|^2-Δ$ in $\mathbb R^d$. Here $Π_λ$ denotes the projection to the subspace spanned by the Hermite functions which are the eigenfunctions of $H$ with eigenvalue $λ$. Such estimates were previously available only for $q=p'$, equivalently with $p=2$ or $q=2$ (by $TT^*$ argument) except for the estimates which are straightforward consequences of interpolation between those estimates. As shown in the works of Karadzhov, Thangavelu, and Koch and Tataru, the local and global estimates for $Π_λ$ are of different nature. Especially, $Π_λ$ exhibits complicated behaviors near the set $\sqrtλ\mathbb S^{d-1}$. Compared with the spectral projection operator associated to the Laplacian, $L^p$-$L^q$ estimate for $Π_λ$ is not so well understood up to now for general $p,q$. In this paper we consider $L^p$--$L^q$ estimate for $Π_λ$ in a general framework including the local and global estimates with $1\le p\le 2\le q\le \infty$ and undertake the work of characterizing the sharp bounds on $Π_λ$. We establish various new sharp estimates in extended ranges of $p,q$. First of all, we provide a complete characterization of the local estimate for $Π_λ$ which was first considered by Thangavelu. Secondly, for $d\ge5$, we prove the endpoint $L^2$--$L^{2(d+3)/(d+1)}$ estimate for $Π_λ$ which has been left open since the work of Koch and Tataru. Thirdly, we extend the range of $p,q$ for which the operator $Π_λ$ is uniformly bounded from $L^p$ to $L^q$.
