AI 中文总结
研究二维埃尔米特算子谱投影的最优\(L^2\to L^{10/3}\)端点估计，结合之前更高维结果，完全解决各维度下该算子\(L^2\to L^q\)本征函数界问题，方法基于之前并通过多尺度分解等克服局限。
AI 中文摘要
本文建立了与\(\mathbb{R}^2\)上埃尔米特算子相关的谱投影算子的最优\(L^2(\mathbb{R}^2)\to L^{10/3}(\mathbb{R}^2)\)端点估计。这完成了对埃尔米特算子尖锐本征函数界的长期研究。在更高维\(d\geq3\)时，作者们最近建立了相应的端点估计\(L^2(\mathbb{R}^d)\to L^{\frac{2(d + 3)}{d + 1}}(\mathbb{R}^d)\)。结合早期结果，本工作完全解决了所有维度下埃尔米特谱投影的最优\(L^2\to L^q\)本征函数界问题。我们的方法基于之前的方法，但通过相对于退化集的时空多尺度分解、输入侧的非对称细化和几乎正交性克服了其局限性。
英文摘要
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.
Comments57 pages, 2 figures
