Multilinear inequalities: Combinatorial and geometric aspects, and extremization

数学研究的目的是发现适用于任何可以通过数学精确描述或建模的可能现象的普遍数学规律。基本的物理定律通常通过系统最小化或最大化（即极值）某些量的原则来表达。例如，光线通过具有不同折射率的介质时，所遵循的路径可以使传播时间最短。系统使各种作用或能量最小化。其他物理定律都用偏微分方程来表示。数学分析提供了工具，可以用来理解这类微分方程和许多类型的函数的极值。本项目关注的是对与经典（非相对论）物理空间的基本代数和几何结构密切相关的不等式极值或近似极值的函数和集合的分析技术的发展。近极值器的表征提供了一个更强大的理解精确极值器，它考虑到小的缺陷和扰动。在要研究的不等式中，有一些最基本和最广泛使用的数学分析不等式，包括支配傅里叶变换的不等式，卷积和欧几里德空间中的集合和。这些不等式中的每一个都是多线性的，涉及乘积或其他成对相互作用，或者具有潜在的多线性方面。多线性泛函处于线性和全非线性现象之间的边界。其他多线性不等式也将作为这个项目的一部分进行研究，包括傅里叶变换的变体。研究勒贝格空间中的基本多元线性不等式及其相关范数。原型包括Riesz-Sobolev不等式（关于集合对称上的积分），Brunn-Minkowski不等式（关于集合的和），Hausdorff-Young不等式（关于傅里叶变换）和Young不等式（关于卷积）。在这个项目的主要部分，注意力将集中在函数和集合的性质和数量性质上，这些函数和集合几乎是极值的，但不是完全极值的。这将建立更强大的版本的现有的精确极值器的特征。建立紧致极值序列的方法，基于加法组合的考虑，而不是浓度，将发展。这些工具包括具有小集合的集合的特征，具有中等大小但大小可控的集合的集合的特征，以及具有大可加能量的集合的特征。精细化的不等式将被表述和建立，其中包含测量结构而不是大小的第二项。等差数列、区间、椭球、凸集和高斯函数将为这种测量提供上下文。改进对仿射不变多线性不等式中近似群结构和近极值之间相互作用的理解是一个主要目标。首席研究员还将离散多元线性不等式应用于计算机科学。将证明内存层次元素之间通信的严格下界，并设计计算这些边界的方法。其他需要研究的问题包括与正复线束的高次幂相关的Bergman核的非对角行为有关的逆问题，与Hausdorff-Young不等式相关的多线性振荡积分算子的不等式，以及先前由Holder， Rogers, Riesz， Sobolev和brascampp - lieb - luttinger分析的一类多线性不等式中等式的可能表征。

Mathematical research aims to discover universal mathematical laws that apply to any possible phenomenon that can be accurately described or modeled through mathematics. Fundamental physical laws are often expressed through the principle that a system minimizes or maximizes (that is, extremizes) some quantity. Light rays, for instance, follow paths that minimize travel time through media with varying indices of refraction. Systems minimize various actions or energies. Other physical laws are encoded in partial differential equations. Mathematical analysis provides tools that can be used to understand both such differential equations, and the extremization of many types of functionals. This project is concerned with development of techniques for the analysis of functions and sets that extremize, or nearly extremize, inequalities that are closely tied to the underlying algebraic and geometric structures of classical (nonrelativistic) physical space. Characterization of near-extremizers provides a more robust understanding of exact extremizers, which takes into account small imperfections and perturbations. Among the inequalities to be studied are some of the most fundamental and widely used inequalities of mathematical analysis, including inequalities governing the Fourier transform, convolutions, and sums of sets in Euclidean space. Each of these inequalities is multilinear, involving products or other pairwise interactions, or has underlying multilinear aspects. Multilinear functionals lie on the frontier between linear and fully nonlinear phenomena. Other multilinear inequalities will also be investigated as part of this project, including variants of the Fourier transform. Fundamental multilinear inequalities in Lebesgue space and related norms will be investigated. Prototypes include inequalities of Riesz-Sobolev (concerning integrals over symmetrizations of sets), Brunn-Minkowski (concerning sums of sets), Hausdorff-Young (concerning the Fourier transform) and Young (concerning convolutions). In a major part of the project, attention will focus on the nature and quantitative properties of functions and sets that nearly, but not exactly, extremize such inequalities. This will establish more robust versions of existing characterizations of exact extremizers. Methodology for establishing compactness for extremizing sequences, based on additive combinatorial considerations rather than on concentration, will be developed. These tools include characterizations of sets with small sumsets, of sets with sumsets of moderately large but controlled size, and of sets with large additive energies. Refined inequalities will be formulated and established, incorporating second terms measuring structure, rather than size. Arithmetic progressions, intervals, ellipsoids, convex sets, and Gaussian functions will provide the context for such measurement. An improved understanding of the interplay between approximate group structure and near extremality in affine-invariant multilinear inequalities is a primary goal. The principal investigator will also apply discrete multilinear inequalities to computer science. Rigorous lower bounds for communication between elements of memory hierarchies will be proved, and methods for computing these bounds will be devised. Other research problems to be investigated include an inverse problem concerning the off-diagonal behavior of Bergman kernels associated to high powers of positive complex line bundles,inequalities for multilinear oscillatory integral operators related to the Hausdorff-Young inequality,and a possible characterization of equality in a class of multilinear inequalities previously analyzed by Holder, Rogers, Riesz, Sobolev, and Brascamp-Lieb-Luttinger.