Widom 集上具有奇异连续分量的无反射测度
Reflectionless measures with singular continuous components on Widom sets
- Texas A&M University（德克萨斯农工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文详细实现了质量分裂构造，用于在 Widom 集上构建同时具有绝对连续和奇异连续分量的无反射概率测度。
AI中文摘要：
我们详细实现了 Nazarov、Volberg 和 Yuditskii 关于具有奇异连续分量的无反射测度的质量分裂构造。有限逼近允许具有孤立原子。一个精确的局部操作将每个活跃原子替换为一个短的永久区间和两个子原子，从而消除了未来分裂尺度对已规定带宽的依赖。对已完成世代和固定柱体的可求和估计在零 Cantor 集上保留正质量，并在极限中排除原子。永久区间附近的均匀解析估计确立了无反射性并排除了端点奇异性。一个针对薄区间对的独立 Green 函数估计给出了 Widom 条件和极限紧集的正则性。所得概率测度同时具有非零绝对连续分量和非零奇异连续分量。
英文摘要：
We give a detailed implementation of the mass-splitting construction of Nazarov, Volberg and Yuditskii for reflectionless measures with singular continuous components. The finite approximants are allowed to have isolated atoms. An exact local operation replaces each active atom by a short permanent interval and two child atoms, thereby eliminating the dependence of a future splitting scale on an already prescribed band width. Summable estimates for completed generations and fixed cylinders retain positive mass on a null Cantor set and exclude atoms in the limit. Uniform analytic estimates near the permanent intervals establish reflectionlessness and exclude endpoint singularities. A separate Green function estimate for thin pairs of intervals gives the Widom condition and regularity of the limiting compact set. The resulting probability measure has both a nonzero absolutely continuous component and a nonzero singular continuous component.
