Radon transforms: geometric combinatorics, regularity, and extensions

这个项目的目标是解决一系列有趣的开放问题，在数学分析有关的拉东变换及其推广。 这些问题包括确定某些多线性泛函的有界性（Holder-Brascamp-Lieb不等式的非线性类似物）关于Lebesgue空间的乘积，以及对平均算子规律性的理解（在标准和超定情况下）Radon变换及其推广与Lebesgue和Lebesgue平方可积Sobolev空间中的一些最伟大的悬而未决的问题密切相关，现代分析，包括挂谷猜想，Bochner-Riesz猜想，限制猜想，和Sogge？的局部光滑化猜想。 在这个项目中要研究的特定问题的智力价值是，他们的解决方案需要重要的新的理论见解，他们是潜在的重要步骤，在解决这些更广泛的突出问题的道路上，更好地理解Radon变换及其推广也可能有更广泛的影响，在科学界的其他领域。医学成像，包括CT和SPECT扫描，NMR成像，雷达和声纳应用都依赖于对Radon变换的深刻理论和实践理解。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受益于该项目产生的见解。

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.