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

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Research AbstractTime-Frequency Analysis of Multilinear Operatorsand More General Nonlinear OperatorsThis proposal focusses on the study of multilinear singularintegrals. Singular integrals have played an important rolein harmonic analysis over the past fifty years and are veryuseful 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 recentlyto control the interaction of shells around different points infrequency space, and the purpose of this project is to exploit andfurther devellope these techniques and therefore better understand multilinear singular integrals. Multilinear singular integrals appearfor example in formal Taylor series of nonlinear operators, or asexplicit correction terms in an iterative scheme. A particularemphasis of the project will be to investigate possible applicationsof this theory to partial differential equations and other fieldsin mathematics.