The Endpoint Eigenfunction Bound for the Hermite Operator in Two Dimensions

In this paper, we establish the optimal $L^2({\mathbb{R}}^2)\to L^{10\mathrm{/}3}({\mathbb{R}}^2)$ endpoint estimate for the spectral projection operator associated with the Hermite operator on ${\mathbb{R}}^2$. This completes a long-standing line of inquiry into sharp eigenfunction bounds for the Hermite operator, developed through the works of Thangavelu, Karadzhov, Koch–Tataru, and others. In higher dimensions $d\ge 3$, the corresponding endpoint estimates $$L^2({\mathbb{R}}^d)\to L^{\frac{2(d+3)}{d+1}}({\mathbb{R}}^d)$$ were recently established by the present authors. Together with these earlier results, the present work fully resolves the problem of optimal $L^2\to L^q$ eigenfunction bounds for Hermite spectral projections in all dimensions. Although our approach builds on our previous method, we overcome its limitations through a multiscale decomposition in space and time relative to the degeneracy set, combined with an asymmetric refinement on the input side and almost orthogonality.