Problems in Harmonic Analysis Relating to Curvature

欧几里德谐波分析领域的发展源于使用傅立叶级数将自然信号（如声音）分解为相干线性波（如音符）的叠加。 这种分解最初是为了证明某些依赖于时间的偏微分方程（特别是热和波动方程）的解的存在性和研究其性质而开发的，这些方程用于模拟物理过程。 随着量子力学、医学成像和信号编码/压缩等应用的需求，该领域已经扩大。 用数学函数来近似自然信号，将这些函数分解为简单部分的叠加，通过对部分的数学运算来模拟物理过程，最后通过求和来重构部分，每一个都必然会导致错误。 理论谐波分析试图建立一个通用的数学框架，通过这个框架，我们可以说，如果初始近似和模型接近现实，那么由分解、数学运算和重构步骤引入的后续误差很小。 一个数学证明，这样的近似导致可控的损失“绑定”某些线性算子，它也是有趣的，以了解相应的“反向不等式”，确定什么样的数据导致最大可能的输出。该项目旨在约束和研究某些数学算子的反向不等式，称为傅立叶限制和平均算子，其中一些基本流形的曲率起着重要作用。 例如，在研究多维傅立叶级数的截断方法时，以及在物理学问题中激发的某些偏微分方程时，出现了这样的算子。 曲率使这些算子的行为比通过简单地计算流形的维数所预测的要好，然而，许多悬而未决的问题仍然是精确地好多少。 特别令人感兴趣的是，当曲率沿着某个非空集变负或沿着消失时，或者当底层流形缺乏预期的光滑度时。 这些科学的努力是密不可分的调查人员的努力，以帮助培养下一代的数学家。这种劳动力发展包括两个主要方向：咨询和指导博士。学生在数学和创造机会的数学家在各个职业阶段，以满足和互动的会议和其他meeting.This项目将遵循三条线的调查有关勒贝格空间边界的运营商在调和分析，其中曲率的一些基本对象起着重要的作用。一个是证明新的限制的傅立叶变换的流形的曲率要么去负或消失沿着一些非空集的界限;另一个是证明新的不等式的线性和多线性广义Radon变换;最后，是使用和发展的集中紧性方法，这样的运营商。该建议的主要部分考虑了病态情况下的此类问题，其中流形的曲率为负或沿沿着某些非空集消失，特别关注通过使用对曲率小的区域给予小权重的措施或通过改变所考虑的勒贝格指数来进行最佳估计。该奖项反映了NSF的法定使命，并被认为值得通过评估予以支持使用基金会的知识价值和更广泛的影响审查标准。

The field of Euclidean harmonic analysis grew out of the use of Fourier series to decompose natural signals (such as sounds) as superpositions of coherent linear waves (such as notes). This decomposition was originally developed to prove the existence and study the properties of solutions to certain time-dependent partial differential equations (particularly the heat and wave equations) that are used to model physical processes. With the demands of such applications as quantum mechanics, medical imaging, and signal encoding/compression, the field has expanded. Approximating natural signals by mathematical functions, decomposing these functions as superpositions of simpler parts, modeling physical processes by mathematical operations on the parts, and, finally, reconstituting the parts by summation each necessarily leads to errors. Theoretical harmonic analysis seeks to establish a general mathematical framework by which we can say that, provided the initial approximation and model are close to reality, subsequent errors introduced by the decomposition, mathematical operation, and reconstitution steps are small. One mathematically proves that such approximations lead to manageable losses by “bounding” certain linear operators, and it is also of interest to understand the corresponding “reverse inequalities” by determining what kinds of data lead to the largest possible output. This project seeks to bound and study reverse inequalities for certain mathematical operators, called Fourier restriction and averaging operators in which the curvature of some underlying manifold plays an important role. Such operators arise, for instance, in studying truncation methods for multidimensional Fourier series, as well as certain partial differential equations motivated by questions in physics. Curvature causes these operators to behave better than would be predicted by simply counting the dimension of the manifold, and yet many open questions remain about precisely how much better. The cases when the curvature goes negative or vanishes along some nonempty set or when the underlying manifold lacks the expected degree of smoothness are of particular interest. These scientific endeavors are inextricably linked with the investigator's efforts to help train the next generation of mathematicians. This workforce development encompasses two main directions: advising and mentoring Ph.D. students in mathematics and creating opportunities for mathematicians at all career stages to meet and interact at conferences and other meetings.This project will follow three lines of inquiry regarding Lebesgue space bounds for operators in harmonic analysis in which the curvature of some underlying object plays an important role. One is to prove new bounds for the restriction of the Fourier transform to manifolds whose curvature either goes negative or vanishes along some nonempty set; another is to prove new inequalities for linear and multilinear generalized Radon transforms; finally, is the use and development of concentration–compactness methods for such operators. The main part of this proposal considers such problems in pathological situations where the curvature of the manifold goes negative or vanishes along some nonempty set, with a particular focus on optimal estimates by using a measure that gives small weight to regions where the curvature is small or by changing the Lebesgue exponents under consideration.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.