Some problems in time-frequency analysis

本项目将探讨调制不变时频分析领域的几个相互关联的问题。第一组问题涉及保测度系统的返回时间现象。这里，我们感兴趣的是动力系统中沿流动的加权平均的收敛性，特别是当权重由第二个动力系统中的流动产生时。两个特别有趣的问题是确定收敛所需的Lp正则性的数量，特别是当平均值被奇异积分取代时，以及根据变范数估计确定收敛发生的速度。这些问题应该通过证明Carleson-Hunt定理的适当扩展来解决。第二组问题涉及双线性希尔伯特变换。我们感兴趣的一个问题是找到新的估计值这些估计值在决定变换奇异集的参数中是一致的。第二个问题是确定截断变换的收敛速度和相关的最大算子。虽然，乍一看，这两组问题似乎没有关联，但它们的解决方案预计将在很大程度上由一组通用工具指导。拟议的研究属于谐波分析的广泛范畴，谐波分析是一门研究将许多类型的数据分解成不同频率的基波组合的学科。这些分解已广泛应用于物理、化学、生物和金融等领域。这里特别相关的一个应用程序是数据压缩，它寻求在使用最小存储空间的情况下准确地表示信号，例如电影或音轨。收敛速率估计，如前一段所述，原则上可以用来描述需要多少空间来给出一个信号的“良好”表示。该项目还与谐波分析以外的数学领域进行了重要的接触，包括遍历理论和加性组合，并有望激发与路易斯安那州立大学数学系的教师和学生以及全国各地的合作者的许多积极讨论。

This project will explore several interconnected questions in the area of modulation-invariant time-frequency analysis. The first group of questions is concerned with the return times phenomenon for measure-preserving systems. Here, one is interested in the convergence of weighted averages along flows in dynamical systems, specifically when the weights are generated by a flow in a second dynamical system. Two problems of particular interest are to determine the amount of Lp regularity required for convergence, especially when the averages are replaced by singular integrals, and to determine, in terms of variation-norm estimates, how quickly this convergence occurs. These problems should be addressed by proving suitable extensions of the Carleson-Hunt theorem. The second group of questions concerns the bilinear Hilbert transform. One problem of interest here is to find new estimates which are uniform in the parameter which determines the singularity set for the transform. A second problem is to determine the rate of convergence for truncated transforms and the associated maximal operator. Although, at first glance, these two groups of questions may not seem to be related, their resolution is expected to be largely guided by a common set of tools. The proposed research falls under the broad umbrella of harmonic analysis, a discipline which studies the decomposition of many types of data into combinations of basic waves of varying frequencies. These decompositions have been broadly applied in areas such as physics, chemistry, biology, and finance. An application which is especially relevant here is data compression, where one seeks to represent a signal, such as a movie or audio track, accurately while using a minimum amount of storage space. Convergence rate estimates, as in the previous paragraph, can in principle be used to characterize how much space is needed to give a "good" representation of a signal. The project also makes significant contact with areas of mathematics outside of harmonic analysis, including ergodic theory and additive combinatorics, and can be expected to stimulate many active discussions with faculty and students of the LSU math department, and with collaborators across the country.