Multilinear inequalities: Combinatorial and geometric aspects, and extremization

数学研究的目的是发现适用于任何可能的现象的普遍数学规律，这些现象可以通过数学准确地描述或建模。基本的物理定律通常通过系统最小化或最大化(即极值)某个量的原理来表达。例如，光线沿着通过具有不同折射率的介质的传播时间最小化的路径。系统最大限度地减少各种动作或能量。其他物理定律被编码在偏微分方程式中。数学分析提供了工具，既可以用来理解这样的微分方程，也可以用来理解许多类型泛函的极值。这个项目致力于开发分析函数和集合的技术，这些函数和集合使与经典(非相对论)物理空间的基本代数和几何结构密切相关的不平等变得极端或接近极端。对近极值的描述提供了对精确极值的更可靠的理解，这考虑到了小的缺陷和扰动。在将要研究的不等式中，有一些数学分析中最基本和最广泛使用的不等式，包括控制欧几里德空间中的傅立叶变换、卷积和集合和的不等式。这些不平等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。多线性泛函介于线性现象和完全非线性现象之间。作为该项目的一部分，还将研究其他多线性不等式，包括傅里叶变换的变体。我们将研究勒贝格空间中的基本多线性不等式及其相关范数。原型包括Riesz-Sobolv(关于集合对称化上的积分)、Brunn-Minkowski(关于集合的和)、Hausdorff-Young(关于傅立叶变换)和Young(关于卷积)的不等式。在项目的主要部分，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(但不完全是)使这种不平等达到极值。这将为现有的精确极值者的刻画建立更可靠的版本。将开发基于加性组合考虑而不是集中的方法来建立极值序列的紧致性。这些工具包括具有小和集的集合的刻画，具有中等大但可控大小的和集的集合的刻画，以及具有大附加能量的集合的刻画。将制定和建立精细的不平等，纳入衡量结构的第二项，而不是大小。算术级数、区间、椭球体、凸集和高斯函数将为这种测量提供背景。提高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。首席研究员还将把离散的多线性不等式应用于计算机科学。将证明内存层次结构元素之间通信的严格下限，并将设计计算这些下限的方法。其他要研究的问题包括与正复线丛的高次方有关的Bergman核的非对角线行为的反问题，与Hausdorff-Young不等式有关的多线性振荡积分算子的不等式，以及之前由Holder，Rogers，Riesz，Sobolv和BrasCamp-Lieb-Luttinger分析的一类多线性不等式中等式的可能特征。

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.