双重希尔伯特变换的不变集
Invariant sets of the double Hilbert transform
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中文总结 AI 辅助
本文研究双重希尔伯特变换及其二进模型的不变集，构造了特征函数为近似特征向量及指示函数被固定的例子，并完整分类了边界在可数多条直线上的不变集，通过射影对偶与Sylvester-Gallai构型及行波叠加相联系。
中文摘要 AI 辅助
我们给出一个例子，表明双重希尔伯特变换 $\H$ 可以将 $\mathbb{R} \times \mathbb{R}$ 中有限测度开集的特征函数作为特征值 $1$ 的近似特征向量。对于 $\H$ 的二进模型，即二进双重移位，同样成立。我们进一步构造了非对角集合，其指示函数被 $\H$ 固定——这些集合是沿 Boole 曲线 $x=y-1/y$ 弯曲的带状区域，更一般地，沿具有纯奇异测度的实 Herglotz 函数的水平曲线弯曲的带状区域——并研究了它们所生成的不变集类。在正密度下，我们构造了一个基于三族平行线的例子，该例子与进位余循环 $\{u\}+\{v\}-\{u+v\}$ 相关，以及扇形——即奇数条共点线的交替扇形——它们是不变锥的最简单成员，我们完整描述了这一类。然后我们研究边界位于可数多条直线上的不变集的分类。其组合部分已完全解决：一个局部有限的直线排列，其中没有点恰好位于两条直线上，则它要么是一族平行线，要么是一个共点线束，要么是三角格子的仿射像。该分类进一步通过射影对偶与 Sylvester--Gallai 型构型相联系，并通过变量替换与行波的两值叠加相联系。
英文摘要
We present an example that shows that the double Hilbert transform $\H$ can have the characteristic function of an open set of finite measure in $\mathbb{R} \times \mathbb{R}$ as an approximate eigenvector for the eigenvalue $1$. The same holds for the dyadic model of $\H$, the double dyadic shift. We further construct non-diagonal sets whose indicators are fixed by $\H$ -- strips bent along the Boole curve $x=y-1/y$ and, more generally, along level curves of real Herglotz functions with purely singular measures -- and study the class of invariant sets which they generate. At positive density we construct an example built on three families of parallel lines, related to the carry cocycle $\{u\}+\{v\}-\{u+v\}$, and the fans -- alternate sectors of an odd number of concurrent lines -- which are the simplest members of the invariant cones, a class that we describe completely. We then study the classification of the invariant sets whose boundary lies on countably many lines. Its combinatorial half is settled completely: a locally finite arrangement of lines in which no point lies on exactly two lines is a family of parallel lines, a pencil of concurrent lines, or an affine image of the triangular lattice. The classification is further connected, via projective duality, with configurations of Sylvester--Gallai type, and, via a change of variables, with two-valued superpositions of traveling waves.
