Problems in Harmonic Analysis Relating to Curvature

欧几里德调和分析的领域起源于使用傅里叶级数将自然信号(如声音)分解为相干线性波(如音符)的叠加。这种分解最初是用来证明和研究用于模拟物理过程的某些依赖时间的偏微分方程解的存在性和性质(特别是热方程和波动方程)。随着量子力学、医学成像和信号编码/压缩等应用的需求，该领域得到了扩展。用数学函数近似自然信号，将这些函数分解为较简单部分的叠加，通过对部分进行数学运算来模拟物理过程，最后通过求和来重构每个部分必然会导致误差。理论调和分析试图建立一个一般的数学框架，通过该框架，我们可以说，只要初始近似和模型接近实际，由分解、数学运算和重构步骤引入的后续误差很小。有一种方法从数学上证明了这种近似通过“限制”某些线性算子而导致可控的损失，并且通过确定哪种类型的数据导致最大可能的输出来理解相应的“反向不等”也是很有意义的。这个项目试图界定和研究某些数学算子的逆不等式，称为傅立叶限制和平均算子，其中一些潜在流形的曲率起着重要作用。例如，在研究多维傅里叶级数的截断方法时，以及在物理问题引发的某些偏微分方程式的研究中，就会出现这样的算子。曲率导致这些运算符的行为比简单地计算流形的维度所预测的要好，但许多悬而未决的问题仍然是，到底好到什么程度。当曲率变负或沿着某个非空集消失时，或者当底层流形缺乏预期的光滑度时，这些情况是特别有趣的。这些科学努力与研究人员帮助培养下一代数学家的努力密不可分。这一劳动力发展包括两个主要方向：在数学方面为博士生提供建议和指导，并为所有职业阶段的数学家创造在会议和其他会议上见面和互动的机会。本项目将沿着调和分析中算子的勒贝格空间界限的三条线进行调查，其中一些潜在对象的曲率在其中发挥重要作用。一是证明了傅里叶变换对曲率为负值或沿非空集为零的流形的限制的新界；二是证明了线性和多线性广义Radon变换的新的不等式；最后是对这类算子的集中紧性方法的使用和发展。这份提案的主要部分考虑了这样的问题，在病理情况下，流形的曲率为负或沿着某个非空集消失，特别关注通过使用给曲率较小的区域赋予较小权重的度量或通过改变正在考虑的勒贝格指数来进行最优估计。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.