Endpoint eigenfunction bounds for the Hermite operator

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} .