Bounds on the Hermite spectral projection operator

We study $L^p$-$L^q$ bounds on the spectral projection operator ${\mathrm{\Pi }}_{\lambda }$ associated to the Hermite operator $H=\mathrm{\mid }x{\mathrm{\mid }}^2-\mathrm{\Delta }$ in ${\mathbb{R}}^d$. We are mainly concerned with a localized operator ${\chi }_E{\mathrm{\Pi }}_{\lambda }{\chi }_E$ 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 $\sqrt{\lambda }{\mathbb{S}}^{d-1}$. Secondly, we obtain the sharp bounds as the set $E$ gets close to $\sqrt{\lambda }{\mathbb{S}}^{d-1}$. Thirdly, we extend the range of $p,q$ for which the operator ${\mathrm{\Pi }}_{\lambda }$ is uniformly bounded from $L^p({\mathbb{R}}^d)$ to $L^q({\mathbb{R}}^d)$.