Singular Integrals with Modulation or Rotational Symmetry

具有调制或旋转对称性的奇异积分

• 批准号: 1800628
• 负责人: Francesco DiPlinio
• 金额: $ 18万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2019-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800628&HistoricalAwards=false
• 关键词: Singular; Integrals; Modulation; Rotational; Symmetry

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.

谐波分析是数学的一个分支，涉及将信号（函数）表示和重建为基本谐波（具有明确的持续时间、强度和频率的信号）的叠加，以及研究适当的操作（滤波、去噪、压缩等）如何影响重建的信号。这种分解/过滤/重建过程的具体版本，有时称为“时频”方法，在广泛的现实应用中执行，例如音频或图像压缩、图像模式或面部识别、数据同化、压缩感知等等。断层扫描成像中也采用了类似的过程，其中通过沿穿透波对物体的密度进行采样来重建固体的形状，在数学上可以将其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涉及沿着较低维度集（例如直线或平面）对三维及更高维度物体（例如固体）进行采样的玩具数学模型。该项目的第二个密切相关的组成部分涉及将时频分解方法扩展到合适的矢量值信号。该项目的重要组成部分是对弗吉尼亚大学调和分析和偏微分方程活跃研究小组中的研究生和本科生进行培训，以及对来自该专业中代表性不足群体的本科生、研究生和研究人员进行指导和研究启动。该调和分析研究项目涉及奇异积分算子，这些算子表现出进一步的不变性属性，例如调制或旋转对称性，此外 这些（平移和扩张不变性）表征了 Calderon-Zygmund 算子：一个基本的例子是卡尔森最大算子，它规定了平方可积函数的傅里叶级数的点向收敛。本研究项目的第一部分涉及旋转不变奇异积分：特别是沿 Lipschitz 向量场的希尔伯特变换。 PI 将根据相关方向最大函数的有界性，对这些向量场进行新颖的表征，从而产生有界方向希尔伯特变换。 PI 还提出了一系列具有独立兴趣的模型问题，通过约束矢量场的范围获得。新颖之处在于考虑了在更高维度的环境空间中设置的问题。方向算子固有的多参数性质自然会导致双傅里叶级数理论中相关的突出问题：抛物线和多边形求和问题。本项目研究的第二个相关问题圈涉及作用于 Banach 空间值函数的线性和多线性奇异积分：除其他问题外，PI 将研究多线性环境中的 T(1) 型算子值定理，以及卡尔森定理的完全非交换类似物。算子值类型定理的优势和相关性在于，它们可以自我改进为多参数类似物，这些类似物对应用很有意义，但通常无法通过直接技术实现。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。