The Endpoint Eigenfunction Bound for the Hermite Operator in Two Dimensions

In this paper, we establish the optimal $L^2(\mathbb{R}^2)\to L^{10/3}(\mathbb{R}^2)$ 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.