Radon transforms: geometric combinatorics, regularity, and extensions

这个项目的目标是解决数学分析中与Radon变换及其推广有关的一系列有趣的开放问题。这些问题包括确定勒贝格空间乘积上的某些多线性泛函(Holder-BrasCamp-Lieb不等式的非线性相似形式)的有界性，以及理解勒贝格和勒贝格平方可积Sobolv空间上平均算子(在标准和超定情况下)的正则性。Radon变换及其推广与现代分析中的一些最突出的问题密切相关，包括Kakeya猜想、Bochner-Riesz猜想、限制猜想和Sogge？S局部光滑猜想。这个项目要研究的特定问题的学术价值在于，它们的解决方案需要重大的新的理论见解，它们可能是在解决其中一些更广泛的悬而未决问题的道路上迈出的重要一步。更好地理解Radon变换及其推广也可能对科学界的其他领域产生更广泛的影响。医学成像，包括CT和SPECT扫描、核磁共振成像、雷达和声纳应用，都依赖于对Radon变换的深入理论和实践理解。光声层析成像、散射理论，甚至运动检测算法也依赖于Radon变换。所有这些领域和更多领域都可能从该项目产生的见解中受益。

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.