Decoupling Theory, Time-Frequency Analysis and Related Oscillatory Integrals

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This project involves work in the fields of harmonic analysis and analytic number theory. Analytic number theory is a branch of number theory which uses tools from mathematical analysis to solve problems about the integers. Harmonic analysis has its roots in the work of Fourier, who studied the way general functions may be represented by sums of simpler trigonometric functions. It has applications in various areas including mathematical physics, signal processing, and digital image processing. The project involves a series of problems in harmonic analysis related to Radon transforms and X-ray transforms. A deeper understanding of these transforms will produce a number of applications in areas such as computed tomography (CT) scan, sonar techniques, and radar techniques. On the other hand, the proposed project will stimulate interactions between harmonic analysis and analytic number theory. The principal investigator proposes to study a few famous open problems in analytic number theory, via tools recently developed in harmonic analysis. The project involves work in three directions, focusing separately on decoupling theory, time-frequency analysis, and certain oscillatory integrals that connect the previous two topics. One problem that is proposed in decoupling theory concerns Parsell-Vinogradov systems in all dimensions. The goal is to establish certain sharp decoupling inequalities that would imply sharp upper bounds on the number of integer solutions of these systems. Progress on this problem will allow a representation of every polynomial of a given degree by fewer linear forms. In the direction of time-frequency analysis, the principal investigator proposes to study an important special case of the famous Zygmund conjecture. This conjecture states that the maximal operator associated with a planar Lipschitz vector field is weakly bounded on the space of square-integrable functions. The principal investigator proposes to study the conjecture by further assuming the vector field to be constant along each vertical line on the plane. One problem that is proposed in the area of oscillatory integrals aims at proving sharp Sobolev regularity estimates for an averaging operator along the moment curve in every dimension. Sharp decoupling inequalities established by the principal investigator and collaborator will provide necessary tools from decoupling theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.