Investigations in Harmonic Analysis

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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.