Radon transforms: geometric combinatorics, regularity, and extensions

Radon 变换：几何组合、正则性和扩展

• 批准号: 0653755
• 负责人: Philip Gressman
• 金额: $ 10.5万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2008-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0653755&HistoricalAwards=false
• 关键词: Radon; transforms; geometric; combinatorics; regularity

The goal of this project is to address a series of interesting open problems in mathematical analysis relating to the Radon transform and its generalizations. These problems include determining the boundedness of certain multilinear functionals (nonlinear analogues of the Holder-Brascamp-Lieb inequalities) on products of Lebesgue spaces, as well as the understanding of the regularity of averaging operators (in both the standard and overdetermined cases) on Lebesgue and Lebesgue square integrable Sobolev spaces.Radon transforms and their generalizations are intimately connected to some of the greatest outstanding problems in modern analysis, including the Kakeya conjecture, the Bochner-Riesz conjecture, the Restriction conjecture, and Sogge?s local smoothing conjecture. The intellectual merit of the particular problems to be studied in this project is that their solutions require significant new theoretical insight, and they are potentially significant steps on the road to solution of some of these broader outstanding problems.A better understanding the Radon transform and its generalizations also may have broader impacts on other fields within the scientific community. Medical imaging, including CT and SPECT scans, NMR imaging, RADAR, and SONAR applications all depend on a deep theoretical and practical understanding of the Radon transform. Optical-acoustic tomography, scattering theory, and even motion-detection algorithms also depend on the Radon transform. All of these fields and more could potentially benefit from insights produced by this project.

该项目的目的是解决与ra及其概括有关的数学分析中一系列有趣的开放问题。 These problems include determining the boundedness of certain multilinear functionals (nonlinear analogues of the Holder-Brascamp-Lieb inequalities) on products of Lebesgue spaces, as well as the understanding of the regularity of averaging operators (in both the standard and overdetermined cases) on Lebesgue and Lebesgue square integrable Sobolev spaces.Radon transforms and their generalizations are intimately connected to some of the现代分析中的最杰出问题，包括Kakeya猜想，Bochner-Riesz猜想，限制猜想以及Sogge的局部平滑猜想。 在该项目中要研究的特定问题的智力优点在于，他们的解决方案需要重大的新理论见解，并且在解决这些更广泛的杰出问题的道路上可能是重要的一步。更好地了解ra及其概括也可能对科学社区内的其他领域产生更广泛的影响。医学成像，包括CT和SPECT扫描，NMR成像，雷达和声纳的应用都取决于对ra transform的深刻理论和实际理解。光学声断层扫描，散射理论甚至运动检测算法也取决于ra的变换。所有这些领域以及更多领域可能会受益于该项目产生的见解。