Endpoint eigenfunction bounds for the Hermite operator

Abstract

We establish the optimal $L^p$, $p=2(d+3)/(d+1)$, eigenfunction bound for the Hermite operator $\mathcal{H}=-\Delta +|x{|}^2$ on ${\mathbb{R}}^d$. Let ${\Pi }_{\lambda }$ denote the projection operator to the vector space spanned by the eigenfunctions of $\mathcal{H}$ with eigenvalue $\lambda$. The optimal $L^2$–$L^p$ bounds on ${\Pi }_{\lambda }$, $2\le p\le \infty$, have been known by the works of Karadzhov and Koch–Tataru except $p=2(d+3)/(d+1)$. For $d\ge 3$, we prove the optimal bound for the missing endpoint case. Our result is built on a new phenomenon: improvement of the bound due to asymmetric localization near the sphere $\sqrt{\lambda }{\mathbb{S}}^{d-1}$.