Curvature-Related Problems in Harmonic Analysis

Betsy Stovall的这项数学研究项目将研究欧几里德调和分析中出现的与曲率有关的问题，以及它们在色散偏微分方程中的应用。这项工作将包括两个方向。其一是证明欧氏空间中曲线族的平均(线性和多线性)定义的广义Radon变换的新的与曲率无关的界。另一个方向是证明傅立叶变换对超曲面的限制的新的估计，包括线性和双线性，并将这些估计和现有的估计应用到某些色散偏微分方程中。关于平移不变广义Radon变换和傅立叶限制算子的曲率无关界的研究是近年来一个热门且卓有成效的研究方向。根据这项提议开展的工作将概括、推广和强化其中一些成果。曲线上平均的特殊情况一直是高维流形上平均的一个重要模型，Stovall证明曲率无关估计的工作很有可能促进这些方向的未来研究。Betsy Stovall的这项数学研究项目属于调和分析的一般领域，重点研究所谓的广义Radon变换和傅立叶限制算子，以及它们与曲率几何概念的关系。多年来，这两种类型的算子一直是调和分析领域研究的重要焦点，他们的研究是一个更广泛计划的一部分，目的是了解曲率对一些在工程和物理中非常自然地出现的算子的影响。例如，被提出研究的一类特定的算子，一类受限的X射线变换，与(非受限的)X射线变换有关，这是医学成像中使用的基本工具。近年来，对傅里叶限制算子的研究尤为活跃。这一活动既是由于对这些算符的内在兴趣，也是因为他们的研究在理解量子力学和光学等领域中出现的方程方面取得了重要进展。斯托瓦尔提出的调查解决了这两个动机。作为该项目的一部分，斯托瓦尔将致力于增加来自代表性不足群体的学生和初级科学家的参与。

This mathematics research project by Betsy Stovall will study curvature-related problems arising in Euclidean harmonic analysis and their applications to dispersive partial differential equations. This work will encompass two directions. One is to prove new, curvature-independent bounds for generalized Radon transforms defined by averages, linear and multilinear, along families of curves in Euclidean space. Another direction will be to prove new estimates, both linear and bilinear, for the restriction of the Fourier transform to hypersurfaces and apply these and existing estimates to certain dispersive partial differential equations. The study of curvature-independent bounds for translation-invariant generalized Radon transforms and Fourier restriction operators has been a popular and fruitful line of research during recent years. Work performed under this proposal will generalize, extend, and sharpen some of these results. The particular case of averages on curves has been an important model for averages on higher dimensional manifolds, and there is a strong potential that Stovall's work to prove curvature-independent estimates will facilitate future research in these directions. This mathematics research project by Betsy Stovall is in the general area of harmonic analysis, with a focus on the study of so-called generalized Radon transforms and Fourier restriction operators, and their relation with the geometric notion of curvature. Both types of operators have been a significant focus of research in the harmonic analysis community for many years, and their study is part of a broader program to understand the effects of curvature on some operators that arise quite naturally in engineering and physics. For example, one of the specific classes of operators whose study is proposed, a family of restricted X-ray transforms, is related to the (unrestricted) X-ray transform, which is a fundamental tool used in medical imaging. In recent years, research into Fourier restriction operators has been particularly active. This activity is due both to intrinsic interest in these operators and because their study has led to important advances in the understanding of equations arising in fields such as quantum mechanics and optics. The inquiries proposed by Stovall address both of these motivations. As part of this project, Stovall will make a dedicated effort to increase the participation of students and junior scientists from under-represented groups.