Symmetry Parameter Analysis of Singular Integrals

谐波分析是数学的一个分支，它涉及信号（函数）及其处理（算子）的严格描述。信号的例子有声音、图像、时间序列和天气数据。这些信号通过叠加（叠加）具有指定持续时间、强度和频率的基本谐波进行分析。这些基本谐波是称为小波的函数。图像或音频的去噪、压缩或模式识别都是通过滤波处理来完成的，滤波处理是指在滤波器对其作用后，将每个小波进行适当的叠加。这也被称为时频方法。重建过程的特定具体示例被用在断层摄影成像中，其中固体的形状是从身体的样本沿着穿透波的沿着一维或二维射线重建的，所述穿透波的一维或二维射线可以在数学上被描述为三维空间中的线或平面。这个数学研究项目的一个组成部分集中在一类奇异积分算子的小波描述的一个新的家庭的方法，例如在高度振荡信号的时频分析中产生。这个研究项目的另一个组成部分是关于沿着沿着或平面采样固体对象的数学特性。综合的更广泛影响活动侧重于加强社会经济弱势群体、少数民族学生（代表性不足的群体）的数学研究生学位储备，并提高保留率。还将开展与分析和专题知识传播方面的研究生培训和指导有关的活动。 第一圈问题的广泛目标是产生一类奇异积分的表示公式，即所谓的模型算子保持相同的不变性结构。这种范式适用于Zygmund型算子和类似于双线性希尔伯特变换的调制不变算子，这两者都是二进概率方法无法达到的。具体应用来自椭圆和色散偏微分方程，算子理论和拟共形映射。第二个相关的家庭的问题是出于点态收敛的双线性遍历平均值的巴拿赫值函数，一个著名的定理布尔甘在标量的情况下。该方法是基于截断双线性希尔伯特变换的Banach值变分估计。中心项目在进一步的一组问题有关的方向奇异积分是一个版本的Kakeya最大的估计，其中管状平均值被替换为平均奇异线段，更一般地说，n维子空间。动机的一个来源是与更高余维中的傅立叶限制的联系。该奖项反映了NSF的法定使命，并已被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Harmonic Analysis is the branch of mathematics concerned with the rigorous description of signals (functions) and of their processing (operators). Examples of signals are sound, images, time series and weather data. Such signals are analyzed via the overlaying (superposition) of basic harmonics of well-specified duration, intensity and frequency. These basic harmonics are functions called wavelets. Image or audio denoising, compression, or pattern recognition are accomplished by filter processing, which refers to a suitable superposition of each wavelet after the action of the filter on it. This is also known as the time-frequency method. A particular concrete example of a re-construction process is used in tomographic imaging, where the shape of a solid body is re-composed from samples of the body along one or two-dimensional rays of penetrating waves, which can be mathematically described as lines or planes in three dimensional space. One component of this mathematics research project focuses on a new family of methods for the wavelet description of the class of singular integral operators, arising for instance in the time-frequency analysis of highly oscillatory signals. Another component of this research project is concerned with the mathematical properties of sampling solid objects along lines or planes. The integrated broader impact activities focus on strengthening the pool of socioeconomically disadvantaged, ethnical minority students (underrepresented groups) in graduate degrees in mathematics and improving retention. Activities connected to training and mentoring of graduate students in Analysis and topical dissemination of knowledge will also be carried out. The broad aim of the first circle of questions is to produce representation formulas for classes of singular integrals in terms of so-called model operators conserving the same invariance structure. This paradigm applies to Zygmund-type operators and modulation invariant operators akin to the bilinear Hilbert transform, both of which are out of reach for dyadic-probabilistic methods. Concrete applications come from elliptic and dispersive PDE, operator theory and quasi-conformal mappings. The second related family of questions is motivated by pointwise convergence of bilinear ergodic averages for Banach-valued functions, a celebrated theorem by Bourgain in the scalar case. The approach is based on Banach-valued variational estimates for the truncated bilinear Hilbert transform. The central item in a further set of questions concerning directional singular integrals is a version of the Kakeya maximal estimate where tubular averages are replaced with averages over singular line segments, and more generally, n-dimensional subspaces. One source of motivation is the connection with Fourier restriction in higher codimensions. The methods involve algebra-geometric techniques such as polynomial partitioning on manifolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.