AI 中文总结
研究时间频率局部化算子的反问题，核心方法是基于特征函数确定局部化域，主要贡献包括给出局部化域的一般反构造、恢复圆盘相关特征、证明指数最优性及扩展集中不等式并给出定理新证明。
AI 中文摘要
我们为时间频率局部化算子发展了一种反问题理论，其核心思想是自由边界问题：局部化域未知，需从规定的谱数据中恢复其边界。该方法基于特征函数可视为确定局部化域的几何数据这一原理，规定特征函数对相关变分问题有重大影响。此主要框架得出四个主要结果。首先，若\(f_0\)是足够接近高斯函数的多项式且\(\lambda\in(0,1)\)，我们构造一个实解析域\(U_\lambda\)，使\(f_0\)是与\(U_\lambda\)相关且特征值为\(\lambda\)的局部化算子的特征函数，给出局部化域的一般反构造，这是文献中的首次。其次，在单连通情形下，我们恢复了圆盘作为埃尔米特多项式唯一局部化域的零集不变阿布雷乌 - 多弗勒特征。第三，我们证明了戈麦斯 - 格拉 - 拉莫斯 - 蒂利定量稳定性不等式中指数\(1/2\)的最优性，回答了这些作者提出的问题。最后，我们表明高斯法伯 - 克拉恩问题的局部极大值是圆盘，这也将尼古拉 - 蒂利集中不等式扩展到局部情形，并且实际上，作为福克空间集中 - 紧致性轮廓分解的结果，我们能够用此给出尼古拉 - 蒂利定理的一个新的、不同的证明。
英文摘要
We develop an inverse theory for time--frequency localization operators, formulated as a free-boundary problem in which the localization domain is unknown and its boundary is recovered from prescribed spectral data. The guiding principle is that an eigenfunction can itself be regarded as geometric data determining a localization domain, with strong consequences for the associated variational problem. We obtain five main results. First, for every polynomial $f_0$ sufficiently close to the Gaussian and every $λ\in(0,1)$, we construct a real-analytic domain $U_λ$ such that $f_0$ is an eigenfunction of the corresponding localization operator with eigenvalue $λ$, providing a general inverse construction of localization domains. Second, we recover the null-set invariant Abreu--D"orfler characterization of disks as the only simply connected localization domains having a Hermite polynomial as eigenfunction. Third, we show that simple connectivity is essential: a weighted Hele--Shaw flow yields an infinite-dimensional family of non-radial, doubly connected analytic domains for which the Gaussian is an eigenfunction. Fourth, we prove the optimality of the exponent $1/2$ in the quantitative stability inequality of G'omez--Guerra--Ramos--Tilli. Finally, we prove that every local maximizer of the Gaussian Faber--Krahn problem is a disk and, using a Fock-space concentration--compactness profile decomposition, obtain a new proof of the Nicola--Tilli concentration theorem.
CommentsVersion 2: 76 pages, one result added, typos corrected
