Investigations in Harmonic Analysis

PI和李晓春最近建立了简并Radon变换的有界性，其中希尔伯特变换是在平面上的方向选择上计算的。关键的一点是，方向的选择只假设是光滑的，没有几何条件强加于方向的选择。证明方法涉及一套复杂的相平面方法，以及一种新颖的Kakeya极大函数。这种转变的几个方面仍然没有得到很好的理解，PI将努力解决其中的一些问题。在第二个方向上，PI与Sarah Ferguson和Erin Terwilleger一起将Nehari定理自然地推广到乘积Hardy空间上的“小”Hankel算子。也就是说，如果汉高算子的符号在产品BMO中，那么它们是有界的。这是一个基本准则，它开辟了一系列复杂变量算子理论的问题。这一系列问题构成了本项目将进行的第二个调查途径。在这个项目中要解决的一系列问题将需要在广泛的分析领域中创造新的技术。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，医学成像即氡变换，控制理论即汉克尔算子。这些问题涉及到这些物体的表现如何，这些问题的解决将产生对这些物体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展。博士后和研究生也将参与该项目，加强国家的科学基础设施。PI和李晓春最近建立了简并Radon变换的有界性，其中希尔伯特变换是在平面上的方向选择上计算的。关键的一点是，方向的选择只假设是光滑的，没有几何条件强加于方向的选择。证明方法涉及一套复杂的相平面方法，以及一种新颖的Kakeya极大函数。这种转变的几个方面仍然没有得到很好的理解，PI将努力解决其中的一些问题。在第二个方向上，PI与Sarah Ferguson和Erin Terwilleger一起将Nehari定理自然地推广到乘积Hardy空间上的“小”Hankel算子。也就是说，如果汉高算子的符号在产品BMO中，那么它们是有界的。这是一个基本准则，它开辟了一系列复杂变量算子理论的问题。这一系列问题构成了本项目将进行的第二个调查途径。在这个项目中要解决的一系列问题将需要在广泛的分析领域中创造新的技术。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，医学成像即氡变换，控制理论即汉克尔算子。这些问题涉及到这些物体表现良好的人，这些问题的解决将产生对这些物体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展。博士后和研究生也将参与该项目，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.