二维Hermite算子的尖锐端点本征函数估计
Sharp Endpoint Eigenfunction Estimates for the Two-Dimensional Hermite Operator
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中文总结 AI 辅助
针对二维Hermite算子的谱投影，通过多种分析方法结合，证明了其从$L^2$到$L^{10/3}$的无对数尖锐端点估计，拓展了算子谱分析的相关结果。
中文摘要 AI 辅助
设$\boldsymbol{\textit{H}}=-\boldsymbol{\textit{\u0394}}+|\boldsymbol{\textit{x}}|^2$为$\boldsymbol{\textit{R}}^2$上的Hermite算子，$\boldsymbol{\textit{\u03a0}}_\boldsymbol{\textit{\u03bb}}$对应$\boldsymbol{\textit{\u03bb}}=2\boldsymbol{\textit{N}}+2$的谱投影。我们证明了无对数的尖锐端点估计$\boldsymbol{\textit{||\u03a0}}_\boldsymbol{\textit{\u03bb}||_{L^2(\boldsymbol{\textit{R}}^2)\to L^{10/3}(\boldsymbol{\textit{R}}^2)}\boldsymbol{\textit{\u2272}}\boldsymbol{\textit{\u03bb}}^{-1/10}$，证明采用极坐标下的谱分解，结合Koch-Tataru局域谱投影界、Liouville-Green表示、指数和的van der Corput估计及径向尺度间的加权$\boldsymbol{\textit{TT}}^*$论证。
英文摘要
Let $\mathcal H=-Δ+|x|^2$ be the Hermite operator on $\mathbb R^2$, and let $Π_λ$ denote the spectral projection corresponding to $λ=2N+2$. We prove the sharp log-free endpoint estimate $||Π_λ||_{L^2(\mathbb R^2)\to L^{10/3}(\mathbb R^2)}\lesssimλ^{-1/10}$. The proof uses a spectral decomposition in polar coordinates and combines Koch-Tataru localized spectral projection bounds with a Liouville-Green representation, van der Corput estimates for exponential sums, and a weighted $TT^*$ argument across radial scales.
