Investigations in Harmonic Analysis

Pi和李晓春最近建立了退化Radon变换的有界性，其中Hilbert变换是在平面上选择方向来计算的。关键的一点是，方向的选择只假设是光滑的，没有对方向的选择施加几何条件。证明方法包括一组复杂的相平面方法，以及新的Kakeya极大函数。这种转变的几个方面仍然知之甚少，PI将努力解决其中的一些问题。在第二个方向上，与Sarah Ferguson和Erin Terwilleger一起的PI将Nehari定理自然地推广到乘积Hardy空间上的‘Little’Hankel算子。也就是说，这种Hankel算子有界当且仅当它们的符号在乘积BMO中。这是基本的标准，它打开了算子理论中关于多个复变量的一系列问题。这一系列问题构成了本项目将进行的第二个调查途径。这个项目要解决的一系列问题将需要在广阔的分析领域创造新的技术。研究对象自然产生于物理过程中，如电荷分布即希尔伯特变换、医学成像即Radon变换和控制论即Hankel算子。所涉及的问题涉及这些对象的行为有多好，这些问题的解决应该对这些对象的更深层次产生重要的见解。这些见解在过去导致了信号处理、成像和控制理论的重要进步。博士后助理和研究生也将参与这一项目，加强国家的科学基础设施。Pi和李晓春最近建立了退化Radon变换的有界性，其中Hilbert变换是在平面上选择方向来计算的。关键的一点是，方向的选择只假设是光滑的，没有对方向的选择施加几何条件。证明方法包括一组复杂的相平面方法，以及新的Kakeya极大函数。这种转变的几个方面仍然知之甚少，PI将努力解决其中的一些问题。在第二个方向上，与Sarah Ferguson和Erin Terwilleger一起的PI将Nehari定理自然地推广到乘积Hardy空间上的‘Little’Hankel算子。也就是说，这种Hankel算子有界当且仅当它们的符号在乘积BMO中。这是基本的标准，它打开了算子理论中关于多个复变量的一系列问题。这一系列问题构成了本项目将进行的第二个调查途径。这个项目要解决的一系列问题将需要在广阔的分析领域创造新的技术。研究对象自然产生于物理过程中，如电荷分布即希尔伯特变换、医学成像即Radon变换和控制论即Hankel算子。所涉及的问题涉及到谁表现良好这些对象，这些问题的解决应该产生对这些对象的更深层次的重要见解。这些见解在过去导致了信号处理、成像和控制理论的重要进步。博士后助理和研究生也将参与这一项目，加强国家的科学基础设施。

The PI and Xiaochun Li have recently established the boundedness of a degenerate Radon transform, in which a Hilbert transform is computed on choice of directions in the plane. The critical point is that the choice of directions is only assumed to be smooth, with no geometric condition imposed on the choice of directions. The method of proof involves an intricate set of phase plane methods, together with novel Kakeya maximal function. There are several aspects of this transform that remain poorly understood, and the PI will work to resolve some of these issues. In a second direction, the PI, with Sarah Ferguson and Erin Terwilleger have provided the natural extension of the Nehari theorem to 'little' Hankel operators on product Hardy space. Namely, such Hankel operators are bounded iff their symbol is in product BMO. This is fundamental criteria, which opens up a range of questions in operator theory in several complex variables. This range of questions forms the second avenue of investigation that will be pursued in this project. The range of problems to be pursued in this project will require the creation of new techniques in the broad area of analysis. The objects studies arise naturally in physical processes, such as charge distribution namely the Hilbert transform, medical imaging, namely Radon transforms, and control theory, namely Hankel operators. The questions addressed concern how well behaved these objects are, and the resolution of these questions should yield important insights into deeper aspects of these objects. These insights have in the past lead to important advances in signal processing, imaging, and control theory. Postdoctoral associates and graduate students will also be engaged in this project, enhancing the scientific infrastructure of the country. The PI and Xiaochun Li have recently established the boundedness of a degenerate Radon transform, in which a Hilbert transform is computed on choice of directions in the plane. The critical point is that the choice of directions is only assumed to be smooth, with no geometric condition imposed on the choice of directions. The method of proof involves an intricate set of phase plane methods, together with novel Kakeya maximal function. There are several aspects of this transform that remain poorly understood, and the PI will work to resolve some of these issues. In a second direction, the PI, with Sarah Ferguson and Erin Terwilleger have provided the natural extension of the Nehari theorem to 'little' Hankel operators on product Hardy space. Namely, such Hankel operators are bounded iff their symbol is in product BMO. This is fundamental criteria, which opens up a range of questions in operator theory in several complex variables. This range of questions forms the second avenue of investigation that will be pursued in this project. The range of problems to be pursued in this project will require the creation of new techniques in the broad area of analysis. The objects studies arise naturally in physical processes, such as charge distribution namely the Hilbert transform, medical imaging, namely Radon transforms, and control theory, namely Hankel operators. The questions addressed concern who well behaved these objects are, and the resolution of these questions should yield important insights into deeper aspects of these objects. These insights have in the past lead to important advances in signal processing, imaging, and control theory. Postdoctoral associates and graduate students will also be engaged in this project, enhancing the scientific infrastructure of the country.