论文标题
部分可观测时空混沌系统的无模型预测
Bounds on the Hermite spectral projection operator
论文作者
论文摘要
我们研究$ l^p $ - $ l^q $在频谱投影运算符上$π_λ$与HERMITE操作员相关的$ h = | x | x |^2-Δ$ in $ \ mathbb r^d $。我们主要关注局部运算符$χ_eπ_λχ_e$对于子集$ e \ subset \ mathbb r^d $，并承担表征尖锐的$ l^p $ - $ l^q $ bungs的任务。我们在$ p，q $的扩展范围内获得急剧的界限。首先，当$ e $远离$ \sqrtλ\ mathbb s^{d-1} $时，我们提供了尖锐的$ l^p $ - $ l^q $界的完整表征。其次，我们获得尖锐的界限，因为集合$ e $接近$ \sqrtλ\ mathbb s^{d-1} $。第三，我们将操作员$π_λ$从$ l^p（\ mathbb r^d）$均匀界定的$ p，q $范围扩展到$ l^q（\ mathbb r^d）$。
We study $L^p$-$L^q$ bounds on the spectral projection operator $Π_λ$ associated to the Hermite operator $H=|x|^2-Δ$ in $\mathbb R^d$. We are mainly concerned with a localized operator $χ_EΠ_λχ_E$ for a subset $E\subset\mathbb R^d$ and undertake the task of characterizing the sharp $L^p$--$L^q$ bounds. We obtain sharp bounds in extended ranges of $p,q$. First, we provide a complete characterization of the sharp $L^p$--$L^q$ bounds when $E$ is away from $\sqrtλ\mathbb S^{d-1}$. Secondly, we obtain the sharp bounds as the set $E$ gets close to $\sqrtλ\mathbb S^{d-1}$. Thirdly, we extend the range of $p,q$ for which the operator $Π_λ$ is uniformly bounded from $L^p(\mathbb R^d)$ to $L^q(\mathbb R^d)$.
