Singular Integrals with Modulation or Rotational Symmetry

调和分析是数学的一个分支，涉及将信号(函数)表示和重构为基本谐波的叠加--具有明确规定的持续时间、强度和频率的信号--以及研究如何进行适当的运算(滤波、去噪、压缩等)。影响重建的信号。这种分解/过滤/重建过程的具体版本，有时被称为“时频”方法，在诸如音频或图像压缩、图像模式或面部识别、数据同化、压缩传感等许多实际应用中执行。在断层成像中使用了类似的过程，通过沿穿透波采样实体密度来重建实体的形状，在数学上可以将其描述为三维空间中的线。这个数学研究项目的第一个主要部分涉及沿低维集合(如线或平面)采样三维和更高维对象(例如，实体)的玩具数学模型。该项目的第二个深层次相关部分是将时频分解方法扩展到合适的矢量值信号。该项目的组成部分是对弗吉尼亚大学活跃的研究小组中的研究生和本科生进行调和分析和偏微分方程的培训，以及对来自该专业中代表性不足群体的本科生、研究生和研究人员的指导和研究启动。这个调和分析研究项目涉及除了那些具有Calderon-Zygmund算子特征的(平移和伸缩不变)外，还表现出更不变性质的奇异积分算子：一个基本的例子是Carleson极大算子，它指示了傅立叶平方可积函数级数的逐点收敛。本研究项目的第一部分涉及旋转不变奇异积分：特别是利用沿着Lipschitz矢量场的希尔伯特变换。PI将根据相关方向极大函数的有界性，对那些产生有界方向希尔伯特变换的矢量场进行新的表征。PI还提出了一系列独立感兴趣的模型问题，这些问题是通过限制矢量场的范围获得的。一个新奇之处在于，问题设置在更高维度的环境空间中被考虑。方向算子固有的多参数性质自然导致了双重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。第二个相关的问题是作用在Banach空间值函数上的线性和多线性奇异积分：PI将研究多线性环境下的T(1)型算子值定理，以及Carleson定理的完全非对易类似。算子值类型定理的优势和相关性在于，它们可以自我改进到其多参数模拟，这对于应用来说是有意义的，而且通常无法通过直接技术实现。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.