Radon-like Transforms: Possible Applications to Partial Differential Equations and Inverse Problems

摘要：在我的论文和我在过去一年半的扩展中，我研究了Radon变换的变体，即奇异Radon变换，推广了Christ， Nagel， Stein和Wainger的早期工作。在我提出的研究中，我打算扩展我的方法，以包括更广泛的算子类，其中将包括分数和更传统的光滑Radon变换，证明Sobolev和L^pspace上的有界性。我还打算研究偏微分方程的可能应用；我使用的概念和efferman和Phong在亚椭圆PDE理论中的结果有相似之处，这可能会在那个方向上产生有趣的结果。此外，我想把我的工作推广到更多的应用方向。例如，断层扫描是医学成像和相关领域的重要工具，需要对氡变换进行反演。由于PDE与逆问题之间的密切关系，我希望能够将我的方法应用到断层成像等逆问题的解决中。在最近的工作中，我研究了被称为奇异拉东变换的数学对象。它们是标准Radon变换的变体，在数学和科学中都有重要的应用。例如，在医学成像中，人们经常使用x射线断层扫描来生成内部器官的图像，这种图像很难通过其他成像技术进行分析。x射线断层扫描涉及反转拉东变换算子，换句话说，从该图像的拉东变换确定图像。在我提出的研究中，除了将我之前的研究成果扩展到严格的数学问题之外，我希望能够将我的研究扩展到帮助解决更多的应用问题，比如这些问题。可能需要来自不同数学和科学领域的想法。

ABSTRACT:In my thesis and in extensions I have made over the past year and a half, I have studied variants of Radon transforms known as singular Radontransforms, generalizing earlier work of Christ, Nagel, Stein and Wainger. In my proposed research, I intend to extend my methods to include a broaderclass of operators, which would include fractional and more traditional smooth Radon transforms, proving boundedness properties on Sobolev and L^pspaces. I also intend to research possible applications to partial differentialequations; there is a similarity between concepts that I use and results ofFefferman and Phong in subelliptic PDE theory that might lead to interestingresults in that direction. Further, I would like to generalize my work in moreapplied directions. For example, tomography, an important tool in medical imaging and related areas, requires the inversion of Radon transforms. Due to the close relation between PDE's and inverse problems, I hope to be able to apply my methods to help solve inverse problems such as those in tomography.In recent work, I have studied mathematical objects known as singular Radontransforms. They are variants of standard Radon transforms, which haveimportant applications in both mathematics as well as the sciences. Forexample, in medical imaging one frequently uses X-ray tomography to generatean image of an internal organ that one can not easily analyze through otherimaging techniques. X-ray tomography involves inverting a Radon transformoperator, in other words, the determination of an image from the Radontransform of this image. In my proposed research, in addition to extending myearlier results to strictly mathematical questions, I hope to be able to extend my research to help tackle more applied problems such as these. Ideas from disparate areas of mathematics and science might be needed.