Hermite spectral projection operator

We study $L^p$-$L^q$ estimate for the spectral projection operator $\Pi_\lambda$ associated to the Hermite operator $H=|x|^2-\Delta$ in $\mathbb R^d$. Here $\Pi_\lambda$ denotes the projection to the subspace spanned by the Hermite functions which are the eigenfunctions of $H$ with eigenvalue $\lambda$. 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 $\Pi_\lambda$ are of different nature. Especially, $\Pi_\lambda$ exhibits complicated behaviors near the set $\sqrt\lambda\mathbb S^{d-1}$. Compared with the spectral projection operator associated to the Laplacian, $L^p$-$L^q$ estimate for $\Pi_\lambda$ is not so well understood up to now for general $p,q$. In this paper we consider $L^p$--$L^q$ estimate for $\Pi_\lambda$ 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 $\Pi_\lambda$. 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 $\Pi_\lambda$ which was first considered by Thangavelu. Secondly, for $d\ge5$, we prove the endpoint $L^2$--$L^{2(d+3)/(d+1)}$ estimate for $\Pi_\lambda$ which has been left open since the work of Koch and Tataru. Thirdly, we extend the range of $p,q$ for which the operator $\Pi_\lambda$ is uniformly bounded from $L^p$ to $L^q$.