  We study $L^p$-$L^q$ bounds on the spectral projection operator $\Pi_\lambda$
associated to the Hermite operator $H=|x|^2-\Delta$ in $\mathbb R^d$. We are
mainly concerned with a localized operator $\chi_E\Pi_\lambda\chi_E$ 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 $\Pi_\lambda$ is uniformly
bounded from $L^p(\mathbb R^d)$ to $L^q(\mathbb R^d)$.
