Endpoint eigenfunction bounds for the Hermite operator

TL;DR

This paper proves the optimal eigenfunction bounds for the Hermite operator at a critical endpoint, revealing a new localization phenomenon that improves existing bounds.

Contribution

It establishes the missing endpoint eigenfunction bound for the Hermite operator in higher dimensions, introducing a novel localization effect near the sphere.

Findings

Proved the optimal $L^p$ eigenfunction bounds at the critical endpoint.

Identified a new localization phenomenon improving bounds near the sphere.

Extended known bounds to include the previously unresolved endpoint case.

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 $\sqrt\lambda \mathbb S^{d-1}$.