Endpoint eigenfunction bounds for the Hermite operator

We establish the optimal (formula presented), eigenfunction bound for the Hermite operator (formula presented) on R^d. Let (formula presented) denote the projection operator to the vector space spanned by the eigenfunctions of (formula presented) with eigenvalue (formula presented). The optimal L²–L^p bounds on (formula presented), have been known by the works of Karadzhov and Koch–Tataru except (formula presented), 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 (formula presented).