Singular Integrals with Modulation or Rotational Symmetry

谐波分析是数学的分支，涉及将信号（函数）表示和重建为基本谐波的叠加-具有指定持续时间，强度和频率的信号-以及研究如何进行适当的操作（滤波，去噪，压缩等）。影响重建的信号。这种分解/滤波/重建过程的具体版本，有时被称为“时间-频率”方法，在广泛的现实世界应用中执行，例如音频或图像压缩，图像模式或面部识别，数据同化，压缩传感等。在断层摄影成像中采用了类似的过程，其中通过沿着穿透波对身体的密度进行采样来重建固体的形状，穿透波可以在数学上被描述为三维空间中的线。 这个数学研究项目的第一个主要组成部分涉及玩具数学模型的采样三维和更高的维对象（例如，固体）沿着较低的维集，如线或平面。第二，这个项目的深入相关的组成部分是关于扩展的时间-频率分解方法，以适当的向量值信号。该项目的组成部分是在弗吉尼亚大学的调和分析和偏微分方程的活跃研究小组内对研究生和本科生进行培训，以及指导和研究来自专业中代表性不足的群体的本科生，研究生和研究人员的研究启动。这个调和分析研究项目涉及表现出进一步不变性的奇异积分算子，如调制或旋转对称性，除了那些（平移和伸缩不变性）的特点Calderon-Zygmund运营商：一个基本的例子是Carleson极大运营商指示的平方可积函数的傅立叶级数的逐点收敛。本研究项目的第一部分涉及旋转不变奇异积分：特别是，与希尔伯特变换沿着Lipschitz向量场。PI将致力于一个新的表征这些向量场产生一个有界的方向希尔伯特变换，在相关的方向极大函数的有界性。PI还提出了一系列的模型问题，独立的利益，通过约束范围的向量场。一个新奇的是，在高维环境空间中设置的问题被认为是。方向算子的内在多参数性质自然地导致了二重傅立叶级数理论中的一些悬而未决的问题：抛物线和多边形求和问题。第二，在这个项目中调查的问题的相关循环关注线性和多线性奇异积分作用于Banach空间值函数：在其他问题中，PI将调查T（1）型算子值定理在多线性设置，和完全非交换类似物的Carleson定理。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，这对应用程序很有意义，并且通常无法通过直接技术实现。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

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.