Singular Integrals with Modulation or Rotational Symmetry

具有调制或旋转对称性的奇异积分

基本信息

批准号：
2000510
负责人：
Brett Wick
金额：
$ 13.16万
依托单位：
Washington University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-09-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2000510&HistoricalAwards=false
关键词：
Singular Integrals Modulation Rotational Symmetry

项目摘要

Harmonic Analysis is the branch of Mathematics concerned with the representation and reconstruction of signals (functions) as a superposition of basic harmonics--signals of well-specified duration, intensity and frequency--as well as the study of how suitable operations (filtering, denoising, compression, etc.) affect the reconstructed signal. Concrete versions of this decomposition/filtering/reconstruction process, sometimes referred to as the "time-frequency" method, are performed in a broad range of real-world applications, such as audio or image compression, image pattern or facial recognition, data assimilation, compressed sensing and many others. A similar procedure is employed in tomographic imaging, where the shape of a solid body is reconstructed by means of sampling the body's density along penetrating waves, which can be mathematically described as lines in three dimensional space. The first main component of this mathematics research project deals with toy mathematical models of sampling three and higher dimensional objects (for instance, solid bodies) along lower dimensional sets such as lines or planes. The second, deeply related component of this project is concerned with extending the time-frequency decomposition method to suitable vector-valued signals. Integral components of the project are the training of graduate and undergraduate students within the active research group in Harmonic Analysis and Partial Differential Equation at University of Virginia, as well as the mentoring and research start-up of undergraduates, graduate students and researchers coming from underrepresented groups in the profession.This Harmonic Analysis research project deals with singular integral operators exhibiting further invariance properties, such as modulation or rotational symmetries, in addition to those (translation and dilation invariance) characterizing Calderon-Zygmund operators: a fundamental example is the Carleson maximal operator dictating pointwise convergence of the Fourier series of square-integrable functions.The first part of this research project deals with rotation invariant singular integrals: in particular, with the Hilbert transform along Lipschitz vector fields. The PI will work on a novel characterization of those vector fields giving rise to a bounded directional Hilbert transform, in terms of boundedness of the related directional maximal function. The PI also proposes an array of model problems, of independent interest, obtained by constraining the range of the vector field. A novelty is that questions set up in higher dimensional ambient spaces are considered. The intrinsic multi-parameter nature of directional operators leads naturally to connected outstanding questions on the theory of double Fourier series: the parabolic and the polygonal summation problems. The second, related circle of problems investigated in this project concerns linear and multilinear singular integrals acting on Banach space valued functions: among other questions, the PI will investigate T(1)-type operator valued theorems in the multilinear setting, and fully noncommutative analogues of Carleson's theorem. The strength and relevance of operator-valued type theorems are that they self-improve to their multi-parameter analogues, which are of interest for applications and are often not attainable with direct techniques.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.

谐波分析是与信号（函数）的表示和重建相关的数学分支，作为基本谐波的叠加 - 指定持续时间，强度和频率的信号 - 以及研究合适的操作（过滤，demoisising，demoisising，stromission等）如何影响重建信号。该分解/过滤/重建过程的具体版本有时称为“时频”方法，在广泛的真实世界应用中进行，例如音频或图像压缩，图像模式或面部识别，数据同化，压缩感应以及许多其他。在断层成像中采用了类似的过程，其中固体体的形状通过沿着穿透波进行采样密度来重建，这可以数学上描述为三维空间中的线。该数学研究项目的第一个主要组成部分介绍了沿较低维组（例如线路或飞机）采样三个和更高维对物体（例如，固体体）的玩具数学模型。该项目的第二个密切相关的组件与将时频分解方法扩展到合适的矢量值信号。该项目的整体组成部分是培训弗吉尼亚大学的谐波分析和部分差异方程式的活跃研究小组中的毕业生和本科生，以及本科生的指导和研究成立，研究生和研究人员来自该职业中体现不足的小组的研究生和研究人员与这些既定的综合群体的统治范围或腐败的整体竞争，以旋转繁殖型群体，以旋转繁殖的旋转，以适用于旋转式的旋转式旋转式的旋转，以适用于旋转的旋转范围。那些（翻译和扩张不变性）表征了Calderon-Zygmund操作员：一个基本的例子是Carleson最大运算符，该操作员指示了一系列方形 - 综合功能的角度融合。该研究项目的第一部分涉及旋转不变的单数积分：尤其是Hilbert the Hilbert Tronsellast the Hilbert Tronsellys the Hilbert the Hilbert the Lipschitz vector vector vector vector vector vector vector vector vector vector vector vectorsssssssssss。 PI将在相关定向最大函数的界面方面对这些向量场的新颖表征进行新的表征，从而引起有界的Hilbert变换。 PI还提出了通过约束向量场的范围而获得的一系列独立感兴趣的模型问题。一个新颖的是，考虑了在较高维度的环境空间中提出的问题。定向运算符的固有多参数性质自然导致了关于双傅里叶序列理论的杰出问题：抛物线派和多边形求和问题。该项目中研究的第二个相关问题涉及作用于Banach空间有价值功能的线性和多线性奇异积分：除其他问题外，PI将调查T（1）型操作员在多线性设置中评估定理，并完全不同意Carleson Theorem的类似物。操作员可价值类型定理的强度和相关性是，它们自我激发到其多参数类似物中，这些类似物对应用程序感兴趣，并且通常无法通过直接技术来实现。该奖项反映了NSF的法定任务，并被认为是通过基金会的智力优点和广泛的影响来评估CRETERIA的评估。