Bounds on the Hermite spectral projection operator
Eunhee Jeong, Sanghyuk Lee, Jaehyeon Ryu
备注：
The paper is a modified version of a part of the paper Hermite spectral projection operator (arXiv:2006.11762v3). The previous paper will remain unpublished
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 $L^q(\mathbb R^d)$.
Random weighted Sobolev inequalities on $\mathbb{R}^d$ and application to Hermite functions
我们扩展了随机化方法，介绍了Shiffman-Zelditch和开发的Burq-Lebeau紧流形的拉普拉斯算子，$\mathbb{R}^d$的情况下，与谐振子。我们构造措施，由于概率定律，满足浓度的测量属性，在支持下，我们证明最佳加权Sobolev估计$\mathbb{R}^d$。这种构造依赖于对非紧位形空间中的谱函数的精确估计。作为应用，我们证明了在$L^{\infty}(\mathbb{R}^d$）$, when $ d \geq 2$中存在具有良好衰减性质的Hermite函数的基.
