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.
