CAREER: Time-Frequency Analysis of Multilinear Operators and More General Nonlinear Operators

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

Research Abstract Time-Frequency Analysis of Multilinear Operators and More General Nonlinear Operators This proposal focusses on the study of multilinear singular integrals. Singular integrals have played an important role in harmonic analysis over the past fifty years and are very useful in other mathematical disciplines such as for example partial differential equations. The main tool in the subject is Littlewood Paley theory, whose main message is that functions on Euclidean space should be decomposed into pieces whose frequency content lie in concentric shells around the origin. Passing to multilinear singular integrals one encounters the new phenomenon that these multilinear singular integrals have translation symmetries in frequency space. Therefore it doesnt suffice to study concentric shells around the origin, but one has to study shells around any point in frequency space. Techniques have been develloped recently to control the interaction of shells around different points in frequency space, and the purpose of this project is to exploit and further devellope these techniques and therefore better understand multilinear singular integrals. Multilinear singular integrals appear for example in formal Taylor series of nonlinear operators, or as explicit correction terms in an iterative scheme. A particular emphasis of the project will be to investigate possible applications of this theory to partial differential equations and other fields in mathematics.